JP2015505611A - System and method for measuring perturbations using low speed fiber optic Bragg grating sensors - Google Patents

Abstract

System and method for measuring perturbations using low speed fiber optic Bragg grating sensors. An optical device, a method of configuring the optical device, and a method of using a fiber Bragg grating are provided. The optical device includes a fiber Bragg grating, a narrow band light source, and at least one photodetector. A fiber Bragg grating has a power transmission spectrum with one or more resonant peaks as a function of wavelength, each of which has a local maximum and two regions with non-zero slopes, There is a local maximum between the two regions. The light generated by the narrowband light source has a wavelength in a region where the slope of the resonance peak selected so that one or more of the next quantities evaluated at the resonance peak is at a maximum value is not zero. That is, (a) the product of the group delay spectrum and the power transmission spectrum, and (b) the product of the group delay spectrum and 1 minus the power reflection spectrum.

Description

[Priority claim]
This application claims the benefit of priority from US Provisional Application No. 61 / 589,248, filed Jan. 20, 2012, the entire disclosure of which is incorporated herein by reference. This application is a continuation-in-part of US Patent Application No. 13 / 224,985, filed September 2, 2011, the entire disclosure of which is incorporated herein by reference.
[Related applications]

  This application is related to US patent application Ser. No. 12 / 792,631, filed Jun. 2, 2010, the entire disclosure of which is incorporated herein by reference.

The present application relates generally to optical devices utilizing fiber Bragg gratings and slow light, and more specifically to optical sensors utilizing fiber Bragg gratings and slow light.
[Description of prior art]

  Fiber Bragg gratings (FBGs) are widely used in research and industry, especially for numerous photonics applications such as communication systems, fiber lasers, and fiber sensors. They are used as filters, high reflectors or partial reflectors, dispersion compensators, frequency standards, frequency stabilizers, spectrum analyzers and the like. In the field of fiber sensors, a key area closely related to some of the embodiments described herein, changes to many perturbations that are applied individually or simultaneously to the FBG, primarily strain and temperature. FBG is used to sense. Optical strain sensors based on FBG have been applied in many areas, including structural monitoring, robotics, and the aerospace industry.

  For example, if a temperature change is applied to the FBG, three of the FBG parameters, namely its length (due to thermal expansion) and hence the grating period, the effective refractive index of the mode propagating in the core (thermal As well as the size of the fiber core (again due to thermal expansion). Of these three effects, the one that contributes the most to the performance of the FBG is usually the thermo-optic effect. These three changes, when combined, result in a change in Bragg wavelength, which can be measured to extract the temperature change applied to the grating. A similar principle is commonly used to measure longitudinal distortion applied to the FBG. As the fiber is distorted, the above three parameters also change, thereby causing a shift in the Bragg wavelength. The FBG is undoubtedly the most widely used light sensitive component in the field of fiber sensors. The reason for this is mainly due to its small size and economics, its ease of manufacture, and its relative stability, after considering that it is a very sensitive multi-wave interferometer. Is.

  In some embodiments, an optical device is provided that includes a fiber Bragg grating, a narrowband light source, and at least one photodetector. The fiber Bragg grating has a refractive index modulation that is substantially periodic along the length of the fiber Bragg grating. The fiber Bragg grating has a power reflection spectrum as a function of wavelength, a power transmission spectrum as a function of wavelength, and a group delay spectrum as a function of wavelength. The narrowband light source is in optical communication with the fiber Bragg grating and is configured to transmit light to the fiber Bragg grating. As a result, the light-transmitted portion is transmitted along the length of the fiber Bragg grating, and the light-reflected portion is reflected from the fiber Bragg grating. The power transmission spectrum as a function of wavelength has one or more resonance peaks, each of which has a local maximum and two regions with non-zero slopes between the two regions. Has a local maximum. At least one photodetector is adapted to detect the optical power of the light transmitted portion, the light reflected portion, or both the light transmitted portion and the light reflected portion. Composed. The light has a wavelength in a region where the inclination of one resonance peak among one or a plurality of resonance peaks is not zero. The resonance peak is selected such that one or more of the following quantities evaluated at the resonance peak is at a maximum value. That is, (a) the product of the group delay spectrum and the power transmission spectrum, and (b) the product of the group delay spectrum and 1 minus the power reflection spectrum.

  In some embodiments, a method of using a fiber Bragg grating is provided. The method includes providing a fiber Bragg grating, generating light from a narrow band light source, transmitting light, reflecting light, or transmitting light and light. Detecting with both at least one photodetector the optical power of both of the reflected portions of The fiber Bragg grating has a refractive index modulation that is substantially periodic along the length of the fiber Bragg grating. The fiber Bragg grating has a power reflection spectrum as a function of wavelength, a power transmission spectrum as a function of wavelength, and a group delay spectrum as a function of wavelength. The power transmission spectrum as a function of wavelength has one or more resonance peaks, each of which has a local maximum and two regions with non-zero slopes between the two regions. Has a local maximum. The narrow band light source is in optical communication with the fiber Bragg grating so that the transmitted portion of light is transmitted along the length of the fiber Bragg grating and the reflected portion of light is reflected from the fiber Bragg grating. . The light has a wavelength in a region where the inclination of one resonance peak among one or a plurality of resonance peaks is not zero. The resonance peak is selected such that one or more of the following quantities evaluated at the local maximum value of the resonance peak is at the maximum value. That is, (a) the product of the group delay spectrum and the power transmission spectrum, and (b) the product of the group delay spectrum and 1 minus the power reflection spectrum.

  In some embodiments, a method for configuring an optical device to be used as an optical sensor is provided. The optical device comprises a fiber Bragg grating and a narrowband light source in optical communication with the fiber Bragg grating. Narrowband light sources transmit light to the fiber Bragg grating so that the transmitted portion of light is transmitted along the length of the fiber Bragg grating and the reflected portion of light is reflected from the fiber Bragg grating. Configured to let The fiber Bragg grating has a refractive index modulation that is substantially periodic along the length of the fiber Bragg grating. The method comprises determining at least one of a power reflection spectrum of light as a function of wavelength for a fiber Bragg grating and a power transmission spectrum of light as a function of wavelength for a fiber Bragg grating. The power transmission spectrum as a function of wavelength has one or more resonance peaks, each of which has a local maximum and two regions with non-zero slopes between the two regions. Has a local maximum. The method further comprises determining a group delay spectrum of light as a function of wavelength for the fiber Bragg grating. The method further comprises selecting one resonance peak of the one or more resonance peaks. The resonance peak is selected such that one or more of the following quantities evaluated at the local maximum value of the selected resonance peak is at the maximum value. That is, (a) the product of the group delay spectrum and the power transmission spectrum, and (b) the product of the group delay spectrum and 1 minus the power reflection spectrum. Configuring the fiber Bragg grating and the narrow band light source so that the light from the narrow band light source has a wavelength in one of the two regions where the slope of the selected resonance peak is not zero. The method further comprises:

1 shows a general overview of the device used to measure the shift of the Bragg wavelength.

FIG. 2 shows a comprehensive implementation diagram of signal processing utilizing a Mach-Zehnder interferometer with a fiber Bragg grating sensor.

The coherence length of the light reflected by the FBG is shown as a function of the refractive index contrast of the FBG. Calculations were made assuming lossless gratings for a wavelength of 1.55 microns for three different FBG lengths.

FIG. 3 shows the calculated maximum sensitivity versus temperature for the Bragg reflection mode FBG sensor of FIG. 2 as a function of refractive index contrast. Calculations were made for various FBG lengths assuming a lossless grating for a wavelength of 1.55 microns.

FIG. 3 shows the calculated maximum sensitivity as a function of FBG length for the temperature of the FBG sensor in the Bragg reflection mode of FIG. The various refractive index contrasts were calculated assuming a lossless grating for a wavelength of 1.55 microns.

1 is a schematic of an optical sensor according to some embodiments described herein.

1 is a schematic of an optical sensor according to some embodiments described herein.

FIG. 6 shows a diagram of an embodiment of an apparatus that utilizes FBGs used in low light transmission mode, in accordance with some embodiments described herein.

FIG. 6 shows a diagram of an embodiment of an apparatus utilizing FBG used in slow light reflection mode, according to some embodiments described herein.

FIG. 4 shows the calculated power transmission spectrum for an example FBG used in transmission at a wavelength of 1.064 microns, in accordance with some embodiments described herein. Calculation was performed assuming a lossless grating. FIG. 4 shows the calculated power transmission spectrum for an example FBG used in transmission at a wavelength of 1.55 microns, in accordance with some embodiments described herein. Calculation was performed assuming a lossless grating. (The “unheated” curve shows no thermal perturbation and the “heated” curve shows thermal perturbation.)

FIG. 6 shows the phase of a transmitted signal calculated as a function of wavelength for the example of an FBG used in transmission at a wavelength of 1.064 microns, according to some embodiments described herein. Calculation was performed assuming a lossless grating. FIG. 5 shows the phase of the transmitted signal calculated as a function of wavelength for the example of an FBG used in a 1.55 micron transmission according to some embodiments described herein. Calculation was performed assuming a lossless grating.

FIG. 9 shows the group index calculated as a function of wavelength for the example of an FBG used in transmission at a wavelength of 1.064 microns, according to some examples described herein. Calculation was performed assuming a lossless grating. FIG. 6 shows the group index calculated as a function of wavelength for the example of FBG used in transmission at a wavelength of 1.55 microns, according to some examples described herein. Calculation was performed assuming a lossless grating.

The calculated temperature sensitivity is shown for the FBG example used in transmission at a wavelength of 1.064 microns, according to some examples described herein. Calculation was performed assuming a lossless grating. The calculated temperature sensitivity is shown for the FBG example used in the 1.55 micron transmission according to some examples described herein. Calculation was performed assuming a lossless grating.

Phase versus temperature as a function of power transmission for the FBG example used in the transmission mode of λ _B = 1.064 μm in the vicinity of the slow light wavelength according to some embodiments described herein. The relationship between the sensitivities is shown. Calculation was performed assuming a lossless grating. Phase versus temperature as a function of power transmission for the FBG example used in the transmission mode of λ _B = 1.55 μm in the vicinity of the slow light wavelength according to some embodiments described herein. The relationship between the sensitivities is shown. Calculation was performed assuming a lossless grating.

A slow light transmission mode with λ _B = 1.064 μm (dashed line) according to some embodiments described herein, a slow light reflection mode according to some embodiments described herein (dotted line), and FIG. 5 shows the relationship between the sensitivity of power to temperature as a function of refractive index contrast for a fixed length (eg 2 cm) FBG used in the Bragg reflection mode (solid line). Calculation was performed assuming a lossless grating. Slow light transmission mode with λ _B = 1.55 μm (dashed line) according to some embodiments described herein, Slow light reflection mode (dotted line) according to some embodiments described herein, and FIG. 5 shows the relationship between the sensitivity of power to temperature as a function of refractive index contrast for a fixed length (eg 2 cm) FBG used in the Bragg reflection mode (solid line). Calculation was performed assuming a lossless grating.

A fixed length (2 cm) FBG used in slow light reflection mode (solid line) and slow light transmission mode (dashed line) for λ _B = 1.064 μm, according to some embodiments described herein. In contrast, the group refractive index calculated as a function of the refractive index contrast Δn is shown. Calculation was performed assuming a lossless grating. Fixed length (2 cm) FBG used in slow light reflection mode (solid line) and slow light transmission mode (dashed line) for λ _B = 1.55 μm, according to some embodiments described herein. In contrast, the group refractive index calculated as a function of the refractive index contrast Δn is shown. Calculation was performed assuming a lossless grating.

Fixed refraction at λ _B = 1.064 μm for slow light reflection mode (dashed line) and slow light transmission mode (dashed line) and Bragg reflection mode (solid line) according to some embodiments described herein. FIG. 6 shows the relationship between the sensitivity of power to temperature as a function of length for a rate contrast (1.5 × 10 ⁻⁴ ) FBG. Calculation was performed assuming a lossless grating. Fixed refraction for slow light reflection mode (dotted line) and slow light transmission mode (dashed line) and Bragg reflection mode (solid line) according to some embodiments described herein at λ _B = 1.55 μm FIG. 6 shows the relationship between the sensitivity of power to temperature as a function of length for a rate contrast (1.5 × 10 ⁻⁴ ) FBG. Calculation was performed assuming a lossless grating.

For λ _B = 1.064 μm, the smallest detectable temperature change calculated for a 1 cm long FBG used in the Bragg reflection mode is shown as a function of the refractive index contrast Δn. Calculation was performed assuming a lossless grating. For λ _B = 1.55 μm, the smallest detectable temperature change calculated for a 2 cm long FBG used in the Bragg reflection mode is shown as a function of the refractive index contrast Δn. Calculation was performed assuming a lossless grating.

In accordance with some examples described herein, for λ _B = 1.064 μm, the minimum detectable temperature change calculated for a 1 cm long FBG used in the slow light transmission mode is the refractive index. It is shown as a function of contrast Δn. Calculation was performed assuming a lossless grating. In accordance with some examples described herein, for λ _B = 1.55 μm, the minimum detectable temperature change calculated for a 2 cm long FBG used in the slow light transmission mode is the refractive index. It is shown as a function of contrast Δn. Calculation was performed assuming a lossless grating.

Laser line widths in slow light reflection mode (solid line) and slow light transmission mode (dashed line) for a 2 cm long and Δn = 1.5 × 10 ⁻⁴ FBG according to some embodiments described herein. The dependence of the power sensitivity on is shown. Calculation was performed assuming a lossless grating.

The group index and power transmission are shown as a function of length for various losses in the example of a strong uniform FBG (Δn = 1.0 × 10 ⁻³ ).

In the hydrogen loaded FBG example, the group index and power transmission for a loss of 2m ⁻¹ are shown as a function of length.

Figure 5 shows the refractive index profile for an FBG with a uniform profile, a type A apodization profile, and a type B apodization profile.

The group refractive index spectrum of a type A apodized grating is shown.

Figure 6 shows a plot of a power asymmetric spectrum for the FBG example.

FIG. 20 shows a plot of the asymmetric spectrum of group index for the example FBG used in FIG. 19C.

The group index and power transmission as a function of length are shown for various losses in the example of type B strong apodized FBG.

FIG. 6 shows group index and power transmission as a function of FWHM of Gaussian apodization in an example of a type B strong apodized FBG.

FIG. 5 shows group index and power transmission as a function of length for an example of a hydrogen loaded FBG.

FIG. 23 shows group index and power transmission as a function of FWHM of Gaussian apodization in the hydrogen loaded FBG example of FIG.

Figure 2 shows an example of the experimental setup used to measure the group delay of FBG.

Figure 2 shows the measured and theoretical transmission spectra of the FBG example.

FIG. 26 shows measured and theoretical group index spectra for the same FBG example used in FIG. 25A.

The entire transmission spectrum measured for the FBG example is shown.

FIG. 26B shows the short wavelength portion of the measured and theoretical transmission spectrum shown in FIG. 26A.

FIG. 26B shows measured and theoretical group index spectra for the example FBG used in FIG. 26A.

2 is a flowchart of an example method of light sensing, according to some embodiments described herein. 2 is a flowchart of an example method of light sensing, according to some embodiments described herein.

The transmission spectrum of a π-shifted grating is shown.

FIG. 30B shows the group index spectrum of the π-shifted grating used in FIG. 29A.

An example of a transmission spectrum calculated for an example of an apodized grating and a calculated group index spectrum are shown.

FIG. 3 shows a diagram of an embodiment of an apparatus utilizing FBG used in low light transmission mode according to some embodiments described herein.

An example of a figure of merit, which is a product of the square root of the transmission spectrum and the group refractive index spectrum of FIG. 30, is shown.

2 is a flowchart of an example of a light sensing method according to some embodiments described herein. 2 is a flowchart of an example of a light sensing method according to some embodiments described herein. 2 is a flowchart of an example of a light sensing method according to some embodiments described herein.

Figure 2 shows the measured transmission spectrum for the FBG example.

FIG. 36B shows a group index spectrum for the example FBG used in FIG. 36A.

An example of setting up an experiment using FBG in a Mach-Zehnder interferometer to test its performance as a slow light sensor is shown.

FIG. 36 shows the sensitivity measured at the four slow light peaks measured in the experimental setup example of FIG.

Figure 5 shows the FWHM bandwidth as a function of refractive index contrast in an example of a uniform grating with length L = 2 cm and no loss.

No loss (upper solid line) with length L = 2 cm and utilized in the slow light transmission mode according to some embodiments described herein and some implementations described herein Refractive index for lossy (various dotted and dashed lines) utilized in slow light transmission mode according to example, and conventional reflection mode (lower solid line) FBG with Mach-Zehnder (MZ) interferometer processing Fig. 4 shows the relationship between sensitivity to distortion as a function of contrast.

FIG. 6 shows the power transmission and group index spectra of an example uniform FBG according to some embodiments described herein. FIG.

Fig. 3 schematically illustrates an example configuration of an FBG sensor in a transmission scheme according to some embodiments described herein.

Fig. 4 schematically illustrates an example configuration of an FBG sensor in a reflection scheme according to some embodiments described herein.

2 is a flowchart of an example method for using FBGs in accordance with some embodiments described herein.

2 is a flowchart of an example method for configuring an optical device for use as an optical sensor according to some embodiments described herein.

FIG. 4 shows (a) a measured power transmission spectrum and (b) a measured group delay spectrum for an example FBG according to some embodiments described herein.

Figure 6 schematically illustrates an example configuration for measuring strain sensitivity according to some embodiments described herein.

FIG. 4 shows measured and predicted sensitivity spectra for an example FBG according to some examples described herein.

FIG. 6 schematically illustrates an example configuration for measuring distortion sensitivity in an MZ-based scheme with two feedback loops in accordance with some embodiments described herein.

Fiber Bragg gratings (FBGs) can take many forms that differ in their details, but FBGs are typically one-dimensional photonic bandgap structures and are typically guided in optical waveguides (eg, optical fibers). It includes a grating with a periodic refractive index having a period Λ manufactured with the wave region. The presence of a periodic structure in the waveguide region of the FBG induces a photonic band gap. This is a finite bandwidth band in the optical frequency space where light is not allowed to propagate forward through the grating. The center wavelength of this band gap is known as the Bragg wavelength λ _B. When light having a wavelength in the vicinity of λ _B is incident on the core of the FBG, the light is substantially reflected from the FBG, whereas light having a wavelength sufficiently far from λ _B is along the length of the FBG. Substantially transparent. The physical explanation for this reflection is that each ripple in the refractive index of the core region reflects a small and small portion of incident light into the back-propagating fundamental mode of the fiber. This reflection is physically due to Fresnel reflection occurring at the interface between two dielectric media of different refractive indices. Thus, a small fraction of the light reflected at each ripple (from the electric field point of view) is proportional to a very small number Δn. However, since FBGs typically contain tens of thousands of periods, all these reflections can be added to make a fairly large total reflection. At the Bragg wavelength λ _B (λ _B = 2n _e Λ, where n _e is the effective refractive index and Λ is the grating period), substantially all individual reflections are in phase with each other. Then, all reflections are added so as to strengthen each other, so that a back propagation mode is obtained. Eventually, a large portion of incident light power can be transmitted. In an FBG having a sufficiently long length and a strong refractive index modulation Δn, essentially 100% of the incident light can be reflected. When a perturbation (eg distortion) is applied to the FBG, both the refractive index and FBG length perturbations are shifted in λ _B proportional to the perturbation (eg distortion) and equal in the transmission and reflection spectra Cause shift

In the field of fiber sensors, most conventional FBGs have been used for what is referred to herein as the Bragg reflection mode. An outline of this mode of operation is shown in FIG. Light (eg, from a broadband light source) is emitted into the FBG via a fiber coupler. The portion of the optical spectrum centered at λ _B that is reflected from the FBG is split by the same coupler and directed to a wavelength monitor device, eg, an optical spectrum analyzer (OSA) that measures λ _B. Alternatively, a portion of the light spectrum transmitted by the FBG can also be measured by a second wavelength monitoring instrument that provides a measurement of λ _B , such as OSA. When a temperature change is applied to the FBG, λ _B changes, and this change in λ _B (or the changed value of λ _B ) is measured by one or both wavelength monitoring instruments. Then, from the change of the measured lambda _B (or changed values of lambda _B), it is possible to calculate the absolute value of the temperature change. The same principle is used to measure the absolute (or relative) magnitude of any other perturbation applied to the FBG that changes λ _B , such as strain or acceleration. Many examples of this mode of operation of the FBG as a sensor are described in the literature. All of them have this point in common. That is, in order to extract the measurement amount, these depend on the measurement of the Bragg wavelength λ _B (or the changed value of λ _B ).

The sensitivity (eg, minimum detectable distortion) of such a sensing scheme is limited by the resolution of OSA and is usually quite low, about 0.1-0.5 nm. In order to improve the sensitivity of FBGs used in the Bragg reflection mode, it is essential to improve the ability to measure very small changes in wavelength, for example, less than ^10-13 meters. This can be done by using OSA with high resolution. Commercially available OSAs with sufficiently high wavelength resolution are available. For example, Yokogawa Electric in Tokyo, Japan, sells OSA with a resolution of 0.05 nm, and Anritsu, Atsugi, Japan, provides an OSA with a resolution of 0.07 nm.

Another solution that provides much higher wavelength resolution than conventional OSA, such as 10 ⁻¹² m resolution, is to use an unbalanced MZ interferometer to monitor the wavelength. See, for example, AD Kersey, TA Berkoff, and WW Morey, "High resolution fiber-grating based strain sensor with interferometric wavelength-shift detection," Electronic Letters, Vol. 28, No. 3. A diagram of a comprehensive implementation of this concept is shown in FIG. The signal having the wavelength λ _B to be measured and reflected from the FBG passes through the two arms of the MZ interferometer. Since the two arms have different lengths L ₁ and L ₂ with appropriate phase bias, eg L ₁ is 50 cm and L ₂ is 51 cm, exiting from one output port of the MZ interferometer The incoming signal is proportional to sin (Δφ / 2). Here is Δφ = 2πnΔL / λ _B, n is the mode effective index in the MZ fiber, and ΔL ₌ L 1 -L _2. As a result of perturbations applied to the FBG, if the Bragg wavelength of the FBG varies by δλ _B , the phase difference between the two arms of the MZ interferometer will vary by the following amount:

The interferometer converts the phase difference into a change in output power, which is a quantity measured at the output of the interferometer. With the appropriate phase bias of the MZ interferometer, the power detected in the presence of perturbations is proportional to sin (Δφ / 2) and therefore varies with sin (πnΔLδλ _B / λ _B ² ). By measuring this variation in power, δλ _B can be extracted. For small perturbations, δλ _B is small and therefore δΔφ is also small. Therefore, the change in power is proportional to ΔLδλ _B / λ _B ² . Thus, in principle, this technique can give a very high resolution of δλ _B by increasing ΔL to a very high value. This is simple because optical fibers usually have very low loss (so long lengths can be used without the penalty of increased signal loss, thus reducing the signal-to-noise ratio) and are inexpensive Can be executed. In one embodiment of the present principle having a ΔL of 100 m, AD Kersey et al. Measured a minimum detectable strain of 0.6 nε / √Hz.

同様に、グレーティングから反射される光の線幅は、およそ次のように与えられる。

ここでＮ＝Ｌ／Λはグレーティング中の周期の数であり、ＬはＦＢＧ長である。例えば、Y. J. Raoの"In-fibre Bragg grating sensors"Meas. Sci. Technol. Vol. 8, 355-375 (1997)を参照のこと。従って、例えば、ＦＢＧの長さを増大させることのように、ＦＢＧの屈折率変調を減少させること、および／または、周期の数を増大させることによって、反射される光のコヒーレンス長を増大させるので、第２の条件（狭反射線幅）を満たすことができる。しかしながら、屈折率変調は実際のところは限られた量だけしか低減されず、また、ＦＢＧの長さの増大は、その熱的不安定性を増大させる。ＭＺ干渉計の経路長差を増大させることも、熱変動に対してＭＺ干渉計を安定化させることをさらに困難にする。より最近になって、２つのＦＢＧにより形成されるファイバファブリペロー（ＦＦＰ）歪みセンサによって、Ｐｏｕｎｄ−Ｄｒｅｖｅｒ−Ｈａｌｌ周波数ロッキング技法を利用してレーザ周波数を安定化させることにより、１３０ｆε／√Ｈｚの検出可能な最小の歪みが実証された（G. Gagliardi, M. A. Salza, P. S. Ferraro, P. De Natale, "Probing the ultimate limit of fiber-optic strain sensing", Science, Vol.330, Iss. 6007, 1081-1084 (2010))。この装置は有望であるが、ここに記載される低速光を用いたいくつかの実施例は、はるかに簡単な実施形態によって、名目上、同じ検出可能な最小の歪みを実現することができ、また、究極的には同じ感度を実現することができる。 The approach of FIG. 2 has two main limitations. The first is that the unbalanced ΔL cannot be increased without limit. The basic operation of the MZ interferometer is that they are recombined (eg, in the second combiner of the MZ interferometer) with a high degree of temporal coherence so that the two signals can interfere. I need. This means that the optical length mismatch ΔL is selected such that it does not substantially exceed the coherence length L _{c of the} signal traveling through the MZ interferometer. This coherence length is related to the frequency line width Δν (or wavelength line width Δλ) of the signal reflected by the FBG as follows.

Similarly, the line width of the light reflected from the grating is given as follows.

Here, N = L / Λ is the number of periods in the grating, and L is the FBG length. See, for example, YJ Rao's “In-fibre Bragg grating sensors” Meas. Sci. Technol. Vol. 8, 355-375 (1997). Thus, for example, increasing the coherence length of the reflected light by decreasing the refractive index modulation of the FBG and / or increasing the number of periods, such as increasing the length of the FBG. The second condition (narrow reflection line width) can be satisfied. However, refractive index modulation is actually reduced only by a limited amount, and increasing the length of the FBG increases its thermal instability. Increasing the path length difference of the MZ interferometer also makes it more difficult to stabilize the MZ interferometer against thermal fluctuations. More recently, the detection of 130 fε / √Hz is achieved by stabilizing the laser frequency using the Pound-Drever-Hall frequency locking technique by a fiber Fabry-Perot (FFP) strain sensor formed by two FBGs. The smallest possible strain has been demonstrated (G. Gagliardi, MA Salza, PS Ferraro, P. De Natale, "Probing the ultimate limit of fiber-optic strain sensing", Science, Vol. 330, Iss. 6007, 1081- 1084 (2010)). Although this device is promising, some examples using slow light described herein can achieve the same nominal minimum detectable distortion, with much simpler embodiments, Ultimately, the same sensitivity can be achieved.

When the line width of the reflected signal is narrow, the signal coherence length is long, and a large length imbalance can be used in the MZ interferometer, so that the sensitivity can be increased. However, the line width of the reflected signal cannot be arbitrarily reduced. The line width is constrained by the grating, ie by the number of periods N and the relative refractive index contrast Δn / n, through equation (3). Using very weak gratings (very small relative refractive index contrast (or modulation) Δn / n, and very long gratings) to be able to use large path mismatch ΔL it can. For example, to use a 1 m path mismatch at a wavelength of 1.55 μm, a 1 m coherence length is used. Or, according to equation (3), for example, a relative refractive index contrast of 10 ⁻⁵ and a grating length greater than 16 cm are used.

ここでは、ＦＢＧのブラッグ波長の式、λ_Ｂ＝２ｎΛが使用されている。シリカファイバーにおいては、ｎ≒１．４５であり、故に式（４）中の２ｎ／πは０．９２に等しい。それにより、図３において予測されるように、Ｌ_ｃはＬに近くなる。与えられる屈折率コントラストに対しては、長さが増大するに連れてコヒーレンス長もまた平坦域に達する。ここを越えてさらにＦＢＧ長を増大させても、いかなる顕著な様式でもコヒーレンス長を増大させない。従って、長いコヒーレンス長を得るためには、低コントラストで長さの長いグレーティングを使用することができる。しかしながら、典型的なコントラスト値（１０^−４から１０^−６、１０^−５が大体最も典型的である）に対しては、典型的で妥当な値（数ｃｍ）を越えるＦＢＧ長に対してさえも、コヒーレンス長はたかだか１０ｃｍ程度か、あるいはそれよりも小さい。これは、Kersey等のリポートと矛盾しない。そこでは線幅Δλ＝０．２ｎｍ（式（２）に従うと、コヒーレンス長〜３．８ｍｍに相当する）を有するＦＢＧからの反射信号を処理するＭＺ干渉計において、１０ｍｍの長さ不整合が使用されている。 FIG. 3 shows the coherence length of the light reflected from the FBG as a function of its refractive index contrast Δn for three different FBG lengths at a wavelength of 1.064 microns. Calculated assuming a lossless grating. For a given FBG length, the coherence length increases to a certain maximum as the refractive index contrast decreases. This maximum value increases as the grating length increases. This maximum coherence length is approximately equal to the grating length (see FIG. 3). This result is expected from equations (2) and (3). In the limit where Δn / n is negligible, Δλ approaches λ _B / N, and therefore L _c approaches the following equation:

Here, the formula of Bragg wavelength of FBG, λ _B = 2nΛ is used. In silica fiber, n≈1.45, so 2n / π in equation (4) is equal to 0.92. Thereby, L _c is close to L, as predicted in FIG. For a given index contrast, the coherence length also reaches a plateau as the length increases. Increasing the FBG length beyond this does not increase the coherence length in any significant manner. Therefore, in order to obtain a long coherence length, a long grating with a low contrast can be used. However, for typical contrast values (10 ⁻⁴ to 10 ⁻⁶ , 10 ⁻⁵ is most typical), even for FBG lengths exceeding typical and reasonable values (several centimeters) However, the coherence length is at most about 10 cm or smaller. This is consistent with Kersey et al. There, a 10 mm length mismatch is used in an MZ interferometer that processes reflected signals from an FBG having a line width Δλ = 0.2 nm (corresponding to a coherence length of ~ 3.8 mm according to equation (2)). Has been.

Based on the above, the sensitivity of the Bragg reflector configuration of FIG. 2 is limited by the length mismatch. The length mismatch itself is limited by the coherence length of the reflected signal, which is itself imposed by Δn and L according to FIG. For the length mismatch ΔL, the change in the phase difference Δφ between the two arms of the MZ interferometer caused by the change in the wavelength δλ _B reflected by the FBG is given by equation (1). First, assuming that this change in wavelength is primarily due to changes in fiber refractive index with temperature (eg, ignoring the effect of changes in FBG length and fiber cross-sectional dimensions), the sensitivity of the sensor of FIG. Can be written explicitly as:

Sensitivity is a simple linear function of ΔL. For silica fibers, dn / dT≈1.1 × 10 ⁻⁵ ° C.− ¹ . For the exemplary maximum length mismatch of 10 cm used in FIG. 3 and for a Bragg wavelength of about 1.064 μm, the phase sensitivity to temperature is about 6.5 rad / ° C. Equation (5) states: If the minimum phase change detectable at the output of the MZ interferometer is 1 μrad (standard and good value), the minimum detectable temperature change is 10 ⁻⁶ /6.5≈1.54×10 ^−7. ° C.

The above description assumes a specific arm length mismatch of 10 cm. (This is applicable, for example, for a grating length of about 10 cm and a contrast of less than 10 ^−5, see FIG. 3.) In fact, the maximum possible imbalance is according to FIG. Determined by refractive index contrast and length, less than 10 cm. FIG. 4 shows temperature sensitivity plotted as a function of refractive index contrast for different grating lengths for a wavelength of 1.064 microns. Calculated assuming a lossless grating. For a given grating length, the sensitivity increases to an asymptotic maximum as the refractive index contrast decreases. This asymptotic maximum value increases as the grating length increases. FIG. 4 shows that much higher sensitivity results if high contrast is used even though the length of the device is increased.

FIG. 5 shows temperature sensitivity plotted as a function of grating length for different refractive index contrasts for a wavelength of 1.064 microns. Calculated assuming a lossless grating. As the grating length increases, the phase sensitivity increases to an asymptotic maximum. In order to achieve high sensitivity in the Bragg reflection mode, the refractive index contrast of the FBG can be selected to be low and its length can be selected to be long. Furthermore, the maximum practical sensitivity is about 10 rad / ° C. (see FIG. 5), or the minimum detectable temperature is about 10 ⁻⁷ ° C. Further, by reducing the refractive index contrast and increasing the length, a smaller value is obtained, but at the cost of a longer device.

A second limitation of the approach of FIG. 2 is that an unbalanced MZ interferometer is very sensitive to temperature fluctuations and becomes more sensitive as the unbalance increases. As the signal propagates through each arm, the signal undergoes a phase shift proportional to the length of this arm (eg, φ ₁ = 2πnL ₁ / λ _B for arm 1). If the entire MZ interferometer is subjected to a temperature change ΔT for some reason, the phase of the signals in arm 1 and arm 2 will vary by different amounts. As a result, the phase bias of the MZ interferometer will change. It is desirable that this phase bias does not fluctuate too much. Otherwise, the sensitivity of the MZ interferometer will vary over time between the optimal value (for optimal bias) and zero. Therefore, it is desirable to stabilize the temperature of the MZ interferometer. For larger length mismatches ΔL, it is desirable that this temperature control be more stringent, which is difficult to implement in practice. For example, consider an example of a fiber made of silica having a signal wavelength of 1.064 μm and an arm length mismatch of 10 cm. In order to keep the phase difference between the two arms less than ± 0.02 rad (reasonable bias stability requirement), the temperature is desirably controlled at about ± 0.003 ° C. This can be a substantial engineering task, increasing complexity, power consumption, and cost of the final sensor system.

To increase the dependence of wavelength shift on perturbations applied to FBG, KP Koo and AD Kersey, "Bragg grating-based laser sensors systems with interferometric interrogation and wavelength division multiplexing," J. Lightwave Technol., Vol. 13, This same approach has been used in other methods, for example by placing an FBG in the laser cavity, as described in Issue 7 (July 1995). However, the difficulties arising from the need to stabilize the temperature of an unbalanced MZ interferometer remain the same. In summary, increasing the length mismatch can trigger a larger difference in λ _B variations, but this comes at the price of greater instability in the MZ interferometer.

  Electromagnetically induced transmission (EIT) (LV Hau, SE Harris, Z. Dutton, and CH Behroozi, "Light speed reduction to 17 meters per second in an ultracold atomic gas"), and photonic crystal waveguide (Nature 397 (6720), 594-598 (1999); TF Krauss, "Slow light in photonic crystal waveguides," J. Phys. D: Appl. Phys. 40, 2666-2670 (2007)) has been experimentally demonstrated. It was. Slow light can be generated in the FBG even at narrow resonance peaks present at the edge of the band gap or at the edges. When a slow optical medium is subjected to external perturbations such as strain or temperature changes, the phase shift induced in the device by the perturbation is enhanced for light exhibiting a low group velocity (or high group index). The sensitivity of this phase increases proportionally with group delay (M. Soljacic, SG Johnson, S. Fan, M. Ibanescu, E. Ippen, and JD Joannopoulos, "Photonic-crystal slow-light enhancement of nonlinear phase sensitivity, "JOSA B, Vol. 19, No. 9, 2052-2059 (2002)).

  Some embodiments described herein advantageously utilize a new mode of operation of the FBG sensor. These new modes offer several substantial benefits over previous uses of FBG as sensors in the Bragg reflection mode. The largest of these is a large increase in sensitivity to a measured quantity (eg, distortion) at a given FBG length and / or a large reduction in FBG length at a given sensitivity. In some embodiments, the increase in sensitivity and / or the decrease in length is in the range of a factor of 1 to several orders of magnitude.

  Two examples of an optical device 10 according to some embodiments described herein are schematically illustrated in FIGS. 6A and 6B. In each of FIGS. 6A and 6B, the optical device 10 comprises an FBG 20 having a refractive index modulation that is substantially periodic along the length of the FBG 20. The FBG 20 has a power transmission spectrum having a plurality of local transmittance minimum values. Each pair of adjacent local transmission minimums has a local transmission maximum between them. The local transmission maximum has the maximum power at the transmission peak wavelength. The optical device 10 includes a narrow band light source 30 that is in optical communication with the first optical path 31 and the second optical path 32. Narrowband light source 30 is configured to generate light having a wavelength between two adjacent local transmission minimums. The wavelength is at or near the local transmission maximum, or the local transmission maximum and two local transmission minimums adjacent to the local transmission maximum The power transmission spectrum is at or near the wavelength having the maximum slope.

  As used herein, the term “being there or in the vicinity” for a particular wavelength has its broadest reasonable interpretation and is not limited to a particular Or at a wavelength sufficiently close to a particular wavelength such that the performance of the optical device 10 is substantially equivalent to the performance of the optical device 10 at the particular wavelength. For example, a wavelength that is “at or near a particular wavelength” can mean that the wavelength is within an amount Δ from a particular target wavelength. Here, Δ is a part of the FWHM line width of the transmission peak. This portion may be, for example, 1%, or 5%, or 10%, or 20%, depending on the application requirements. For example, for Δ = 10%, if the FWHM line width is 2 pm, a wavelength within 0.2 pm of a specific target wavelength is considered to be in the vicinity of this target wavelength and is 2 pm away from this target wavelength The wavelength is not considered to be in the vicinity of this target wavelength.

  In some embodiments, the optical device 10 is an optical sensor and further includes at least one photodetector 40 that is in optical communication with the FBG 20. The light generated by the narrow band light source 30 is branched into the first portion 33a and the second portion 33b. The first portion 33 a is transmitted along the first optical path 31 that extends along the entire length of the FBG 20. In some embodiments, the at least one photodetector 40 is configured to receive the first portion 33a, the second portion 33b, or both the first and second portions 33a, 33b.

  In some embodiments, the wavelength of the light generated by the narrowband light source 30 is such that the transmission of the local transmittance maximum is such that a substantial portion of the incident light from the narrowband light source 30 is transmitted by the FBG 20. At or near peak wavelength. In some such embodiments, as schematically illustrated in FIG. 6A, the first portion 33a includes light incident on the FBG 20 from the narrowband light source 30 and transmitted along the FBG 20, and second The portion 33 b includes light that does not substantially interact with the FBG 20. Accordingly, the first portion 33a is substantially affected by perturbations applied to the FBG 20 while the first portion 33a is transmitted along the FBG 20. On the other hand, the second portion 33b is not substantially affected by the perturbation applied to the FBG 20.

  In some other embodiments, the wavelength of the light generated by the narrowband light source 30 is a local transmission maximum and two adjacent locals on either side of the local transmission maximum. Between one of the typical minimum transmission values. As a result, the FBG 20 transmits a substantial part of the incident light from the narrow band light source 30 and reflects a substantial part of the incident light from the narrow band light source 30. In some such embodiments, as schematically illustrated in FIG. 6B, the first portion 33a includes light incident on the FBG 20 from the narrowband light source 30 and transmitted along the FBG 20, The portion 33 b includes light reflected from the FBG 20. Thus, in some embodiments, the first portion 33a is substantially affected by perturbations applied to the FBG 20 while the first portion 33a is transmitted along the FBG 20. The second portion 33b is substantially affected by perturbations applied to the FBG 20 while the second portion 33b is reflected from the FBG 20.

As described more fully below, the light transmitted along the FBG 20 is at a wavelength such that it has a slower group velocity than light at most other wavelengths propagating through the FBG 20 In addition, the light generated by the narrow band light source 30 is selected. For example, in some embodiments, the wavelength of light generated by the narrowband light source 30 is the speed of light in vacuum relative to the group speed of light transmitted through the FBG 20 (approximately 3 × 10 ⁵ km / s). ) Ratio greater than 5, or greater than 10, or greater than 30, or greater than 50, or greater than 100, or greater than 300, or greater than 500, or greater than 1,000. Greater than, or greater than 3,000, or greater than 5,000, or greater than 10,000, or greater than 30,000, or greater than 50,000, or greater than 100,000. Or greater than 300,000, or greater than 500,000, or greater than 1,000,000 Can be selected. In some other embodiments, the wavelength of the light generated by the narrowband light source 30 is the speed of light in vacuum relative to the group speed of light transmitted through the FBG 20 (approximately 3 × 10 ⁵ km / s). ) Ratio between 5 and 10, or 5 and 30, or 10 and 50, or 30 and 100, or 50 and 300, or 100 and 500 Or between 300 and 1,000, or between 500 and 3,000, or between 1,000 and 5,000, or between 3,000 and 10,000, or 5,000 Between 30,000, or between 10,000 and 50,000, or between 30,000 and 100,000, or between 50,000 and 300,000, or 100,000 and 500, Between 000 or 300 Can be selected to be between 1,000,000,000, or 500,000 and 3,000,000, or 1,000,000 and 5,000,000 .

  In some embodiments, the substantially periodic refractive index modulation in the FBG 20 has a constant period along the length of the FBG 20. In some other embodiments, the substantially periodic refractive index modulation has a period that varies along the length of the FBG 20, such as a chirped grating. In some embodiments, the amplitude of the refractive index modulation can vary along the length, such as an apodized grating.

  The FBG 20 can be manufactured by exposing the core of the optical fiber to a spatially modulated UV beam or by many other means. The refractive index modulation can be a sine function. Alternatively, any number of other spatial distributions can be taken. In some embodiments, the optical fiber is a conventional single mode fiber such as SMF-28® optical fiber commercially available from Corning, Inc. of Corning, NY. However, the fiber in other embodiments is a multimode fiber. In some other embodiments, the fiber is doped with a specific element to be substantially photosensitive so that exposure to spatially varying light induces the desired modulation in the refractive index. (Eg, substantially responsive to UV light). The fiber can be composed of silica, hydrogen loaded silica, phosphate glass, chalcogenide glass, or other materials.

The perturbation or modulation of the refractive index of the grating in the FBG 20 can be weak (eg, Δn≈10 ⁻⁵ ) or very high (eg, Δn≈0.015). The refractive index grating of the FBG 20 may in some cases extend into the cladding that directly surrounds the core, but is usually limited to the core. Although the FBG 20 having a length exceeding 1 meter or as short as 1 mm has been created, the length of the FBG 20 is usually several millimeters to several centimeters.

In some embodiments, narrowband light source 30 comprises a semiconductor laser that is an Er-Yb doped fiber laser having a wavelength range between 1530 nm and 1565 nm, for example, by NP Photonics, Tucson, Arizona. In other embodiments, the narrowband light source 30 comprises an Nd: YAG laser having a wavelength of 1064.2 nm. In some embodiments, narrowband light source 30 has a line width less than or equal to 10 ⁻¹³ meters. Other wavelengths (eg, 1.3 microns) and other line widths also correspond to some embodiments described herein.

  In some embodiments, the light generated by the narrowband light source 30 is split into a first portion 33a and a second portion 33b. The first portion 33 a is transmitted along the first optical path 31 that extends along the length of the FBG 20. The second portion 33 b is transmitted along the second optical path 32 that does not extend along the length of the FBG 20. In some embodiments, the first optical path 31 is different from the second optical path 32, as shown in FIG. 6A. For example, as shown in FIG. 6A, the first optical path 31 does not overlap with the second optical path 32. For some other embodiments, the first optical path 31 and the second optical path 32 may overlap each other. For example, as shown in FIG. 6B, both the first optical path 31 and the second optical path 32 include a common portion between the narrowband light source 30 and the FBG 20. In some embodiments, the first optical path 31 and / or the second optical path 32 may traverse the empty space or may traverse various optical elements. For example, one or both of the first optical path 31 and the second optical path 32 can traverse an optical element that is, for example, a fiber coupler described more fully below. In some embodiments, the first optical path 31 and / or the second optical path 32 may traverse regions having different refractive indices. For example, in FIG. 6A, the first optical path 31 traverses the FBG 20 having a substantially periodic refractive index modulation along the length of the FBG 20.

  In some embodiments, the optical device 10 includes at least one photodetector 40 that is in optical communication with the FBG 20. The at least one photodetector 40 is configured to receive a first portion of light 33a, a second portion of light 33b, or both first and second portions of light 33a, 33b. In some embodiments, the photodetector 40 is a New Focus general purpose photodetector, a model 1811 low noise photodetector. However, the photodetector 40 may be one of a variety of low noise photodetectors well known in the art, but detectors that have not yet been devised may be used as well.

In some embodiments, a mode of operation, referred to herein as a slow light transmission mode, may be used (eg, by the structure schematically shown in FIGS. 6A and 7). In these embodiments, light having a narrow band spectrum is emitted to the FBG 20. The light has a wavelength that is at a wavelength (eg, at or near that wavelength) such that the FBG 20 transmits more light than reflects light. For example, the wavelength of the light is selected to be at a transmission peak wavelength (eg, at or near that wavelength) that corresponds to the local transmittance maximum of the power transmission spectrum of the FBG 20. The possible locations of these two wavelengths, referred to herein as λ ₁ and λ ′ ₁ , are discussed in more detail below, particularly with reference to FIGS. 9A-9H. At these wavelengths, light undergoes considerable group delay. That is, light travels at a much slower group velocity than the group velocity of light at a wavelength far away from the band gap of the FBG 20. For example, the slow light group velocity can be as low as 300 km / s. In contrast, the non-slow light group velocity is typically about 207,000 km / s for light traveling in silica. Light having such a slower group velocity in the vicinity of the edge of the band gap of the FBG 20 is referred to as low-speed light here. Slow light has been previously investigated in other situations. Specifically, for example, to evaluate the availability of FBG for dispersion compensation in an optical communication system. For example, F. Ouellette, PA Krug, T. Stephens, G. Dhosi, and B. Eggleton, "Broadband and WDM dispersion compensation using chirped sample fiber Bragg gratings," Electronic Letters, Vol. 31, No. 11 (May 1995) checking ...

In some embodiments, the advantage of the slow light transmission mode of operation is that the power transmission is a local maximum (eg, it is close to 1 or near the slow light wavelength, eg, λ ₁ or λ ′ _1). Equal). As a result, the loss experienced by the signal is small as it propagates along or through the FBG 20. In some embodiments, another advantage is that perturbations applied to the FBG 20 for light traveling through the FBG 20 at or near _one of the slow light wavelengths λ ₁ or λ ′ ₁ (eg, Distortion) changes the phase of the light traveling through the FBG 20, but not the amplitude. More precisely, in some embodiments the perturbation changes the light phase primarily and secondarily the light amplitude. This is in contrast to the FBG's Bragg reflection mode, where perturbations to the FBG change the frequency of light that is maximally reflected. Thus, in some embodiments (eg, FIGS. 6A and 7) that use a slow light transmission mode, the FBG 20 can act as a phase sensor. For example, this can be done in one of any number of interferometers to convert the phase modulation induced by the perturbation (measurement quantity) into a change in power (this is the quantity that the user measures). Can be placed directly.

  FIG. 7 schematically shows an example of a configuration comprising a nominally balanced MZ interferometer. The configuration of FIG. 7 is an example of a general configuration schematically shown in FIG. 6A. In FIG. 7, the optical device or optical sensor 10 includes a first fiber that is in optical communication with a narrowband light source 30, a first optical path 31, and a second optical path 32 that does not extend the entire length along the FBG 20. A coupler 51 is provided. The light generated by the narrow band light source 30 is branched into a first portion 33a and a second portion 33b by a first fiber coupler 51 having a power division ratio of 3 dB, for example. In this embodiment, the first portion 33 a is transmitted along the first optical path 31, and the second portion 33 b is transmitted along the second optical path 32. The first portion 33a propagates along the FBG 20, while the second portion 33b does not substantially interact with the FBG 20. In this embodiment, the first portion 33a includes information regarding the perturbation of the FBG 20. In contrast, the second portion 33b remains unaffected by such perturbations.

  In FIG. 7, the optical sensor 10 further includes a second fiber coupler 52 having a power division ratio of, for example, 3 dB and optically communicating with the first optical path 31 and the second optical path 32. The first portion 33 a and the second portion 33 b are recombined by the second fiber coupler 52 and transmitted to the at least one photodetector 40. By this recombination, the first portion 33a and the second portion 33b can interfere with each other, and a combined signal including information on the phase difference between the first portion 33a and the second portion 33b is generated. In some embodiments, the at least one photodetector 40 comprises a single photodetector at one of the plurality of output ports of the second fiber coupler 52. In some other embodiments, as shown schematically in FIG. 7, the at least one photodetector 40 includes a first photodetector 40a at one output port of the second fiber coupler 52, In addition, the second optical detector 40 b is provided at the other output port of the second fiber coupler 52. The plurality of signals detected by these two photodetectors 40a, 40b vary in opposite directions with respect to each other. (For example, if the power detected at one output port of the second fiber coupler 52 increases, the power detected at the other output port of the second fiber coupler 52 decreases.) The difference between the outputs from the detectors 40a, 40b can be used as a sensor signal. Such a detection scheme can provide various advantages, including common mode rejection and higher signals. In some embodiments, the phase difference indicates the amount of distortion applied to the FBG 20. In some other embodiments, the phase difference indicates the temperature of the FBG 20.

  In some embodiments, a balanced MZ interferometer configuration is used with slow light so that between the first portion 33a and the second portion 33b as shown schematically in FIG. By detecting and measuring the phase difference, it is possible to accurately detect and measure perturbations applied to the FBG. In contrast, in the Bragg reflection mode, a change in wavelength (or frequency) is detected (eg, as shown in FIG. 1) or converted into a change in power. This can be done with high accuracy by stabilizing an unbalanced interferometer (eg as shown in FIG. 2). Also, in some embodiments, the use of a balanced MZ interferometer in the slow light transmission mode advantageously avoids the high sensitivity of the unbalanced MZ interferometer to temperature. This provides a significant improvement in the temperature stability of the MZ interferometer and hence its phase bias stability. This also greatly simplifies engineering by reducing the amount of temperature control to be used.

When light travels through the medium and the group velocity is low, the matter-field interaction increases. Since light takes longer to travel through the medium, the compression of local energy density produces enhanced physical effects, including phase shift. The dependence of the induced phase on the dk shift is significantly enhanced when the group velocity ν _g = dω / dk is small. M. Soljacic, SG Johnson, S. Fan, M. Ibanescu, E. Ippen, and JD Joannopoulos, "Photonic-crystal slow-light enhancement of nonlinear phase sensitivity," JOSA B, Vol. 19, Issue 9 (September 2002) This effect can be quantified as follows by relating the phase shift to the group velocity.

This relationship states that the phase shift is inversely proportional to the group velocity ν _g = dω / dk or proportional to the group refractive index _ng = c / ν _g . Here, c is the speed of light in a vacuum. According to some embodiments described herein, the main advantage of operating in this slow light transmission mode is that the light is high while everything else is the same, as stated before without demonstration That is, a given perturbation will induce a much larger phase perturbation in a device where the light has a lower group velocity than in a device having a group velocity. As demonstrated below using numerical simulation, an optical sensor 10 having an FBG 20 that is operated in a slow light transmission mode in accordance with some embodiments described herein, therefore, determines the phase of the signal traveling through the grating. Much greater sensitivity can be shown for any measured quantity that changes.

  The MZ interferometer in the configuration of FIG. 7 is just one of many interferometers that can be used to convert a phase shift induced by perturbations to light traveling along the FBG 20 into a change in intensity. Any interferometer that converts phase modulation to amplitude modulation can be used in place of the MZ interferometer. For example, the interferometer may be a Michelson interferometer, a Fabry-Perot interferometer, or a Sagnac interferometer (if the perturbation is time dependent). For example, in some embodiments, the optical sensor 10 includes a fiber loop (eg, a Sagnac interferometer loop) that includes asymmetrically disposed fiber Bragg gratings. The first optical path extends in a first direction along the fiber loop, the second optical path extends in a second direction along the fiber loop, and the second direction is opposite to the first direction. In some such embodiments, the optical sensor 10 further comprises at least one fiber coupler optically coupled to the narrowband light source and the fiber loop. Here, the light generated by the narrowband light source is transmitted by at least one fiber coupler such that the first portion propagates along the first optical path and the second portion propagates along the second optical path. Branches into a first part and a second part. The first portion and the second portion are recombined by at least one fiber coupler after propagating along the entire length of the fiber Bragg grating. The at least one photodetector comprises an optical phase detector configured to receive the recombined first and second portions and detect a phase difference between the first portion and the second portion. Such a Sagnac configuration can be used to detect time-varying perturbations to the FBG 20. In some embodiments, the at least one fiber coupler is configured to allow power detection at each other output port.

  6B and 8 schematically illustrate a second mode of operation disclosed herein and referred to as the slow light reflection mode. The configuration of FIG. 8 is one example of the general configuration shown schematically in FIG. 6B. The optical device 10 includes an FBG 20. The FBG 20 has a substantially periodic refractive index modulation along the length of the FBG 20. The FBG 20 has a power transmission spectrum having a plurality of local transmittance minimum values. Each pair of adjacent local transmission minimums has a local transmission maximum between them. The local transmission maximum has the maximum power at the transmission peak wavelength. The optical sensor 10 includes a narrow-band light source 30 that is in optical communication with the first optical path 31 and the second optical path 32. The narrow-band light source 30 is located between two adjacent local transmission minimum values (for example, between a local transmission maximum value and a local transmission minimum value adjacent to the local transmission maximum value). ) To generate light having a wavelength. In some embodiments, the optical device 10 is an optical sensor and further comprises at least one photodetector 40a, 40b in optical communication with the FBG 20. The light generated by the narrow-band light source 30 is branched by the FBG 20 into the first portion 33a and the second portion 33b. The first portion 33 a is transmitted along the first optical path 31 that extends along the length of the FBG 20. The second portion 33 b is transmitted along the second optical path 32 and reflected from the FBG 20. Therefore, it does not extend along the length of the FBG 20, as schematically shown in FIG. In some embodiments, the at least one photodetector 40a, 40b is configured to receive the first portion 33a, the second portion 33b, or both the first and second portions 33a, 33b.

  In the embodiment shown in FIG. 8, the light generated by the narrowband light source 30 traverses the fiber coupler 51. However, unlike the embodiment shown in FIG. 7, the first optical path 31 and the second optical path 32 overlap each other. The first optical path 31 in FIG. 8 extends from the narrowband light source 30 through the fiber coupler 51 and along the length of the FBG 20 over the entire length. The second optical path 32 in FIG. 8 extends from the narrowband light source 30 through the fiber coupler 51 to the FBG 20. The light is reflected by the FBG 20 and returned to the fiber coupler 51. The transmitted first portion 33a is formed by light incident on the FBG 20 in the first direction. When interacting with the FBG 20, some of this incident light interferes forward (in the first direction) so as to intensify. The reflected second portion 33b is formed by light incident on the FBG 20 in the first direction. When interacting with the FBG 20, some of this incident light interferes backwards (in a second direction opposite to the first direction) so as to intensify. In this embodiment, the second portion 33b is reflected from the FBG 20 and therefore does not propagate along the FBG 20. The photodetector 40b is configured to receive the first portion 33a. In contrast, the photodetector 40 a is configured to receive the second portion 33 b after the second portion 33 b is returned to traverse through the fiber coupler 51.

In some embodiments, the FBG 20 is probed by the narrowband laser 30, the first portion 33 a is transmitted along the FBG 20 and the second portion 33 b is reflected from the FBG 20. The wavelength of the light that examines the FBG 20 is the local transmittance maximum of the power transmission spectrum (eg, λ ₁ , λ ₂ , λ ₃ , λ ′ ₁ , λ ′ ₂ , λ ′ ₃ , or λ _i , or λ ′ _i , where i ≧ 1, and will be described more fully with reference to FIGS. 9A and 9E below), and the local transmission minimum adjacent to the local transmission maximum To be between. In some such embodiments, light incident on the FBG 20 is reflected by the FBG 20 reflection peaks (eg, λ _a , λ _b , λ _c , λ _d , λ _e , λ _f , λ ′ _a , λ ′ _b , λ ′ _c , λ ′ _d , λ ′ _e , λ ′ _f , or other such wavelengths between wavelengths λ ′ _i where i ≧ 1, and with reference to FIGS. 9A and 9E, With wavelengths λ _A and λ ′ _A at or near the steepest part).

For example, in some embodiments, the FBG 20 includes a Bragg wavelength and a first edge wavelength (eg, the first local transmission maximum transmission peak wavelength λ ₁ , described more fully below). To light in the wavelength range up to the second end wavelength (eg, the second local transmission maximum transmission peak wavelength λ ′ ₁ , described more fully below). The reflected light has a maximum intensity at a reflection peak wavelength (eg, Bragg wavelength) within the band gap (eg, between the first and second end wavelengths). The region between the two transmission peak wavelengths λ ₁ and λ ′ ₁ can be regarded as a local minimum transmission value of the power transmission spectrum of the FBG 20. In some such embodiments, the resonance edge or power transmission is the maximum of power transmission at the transmission peak wavelengths λ ₁ and λ ′ ₁ of the first or second local transmission maximum. The wavelength can be selected to be a slow light peak that is a selected portion of the value (eg, within about half or between 1/5 and 4/5).

When an external perturbation is applied to the FBG 20, the reflection peak shifts in wavelength. This shift of λ _B causes a change in the first portion 33a transmitted by the FBG 20 and the second portion 33b reflected by the FBG 20. For example, it is a change in the power of light reflected at the wavelength of light incident on the FBG 20. In some embodiments, the at least one photodetector 40 includes a photodiode 40a configured to receive and detect the optical power of the second portion 33b. As shown in FIG. 8, the second portion 33b of the laser signal reflected by the FBG 20 is separated from the FBG input port by a fiber coupler 51 (eg, a fiber coupler having a power split ratio of approximately 3 dB). . Then, the optical power of the second portion 33b is measured by the photodetector 40a. In some embodiments, the at least one photodetector 40 includes a photodiode 40b configured to receive and detect the optical power of the first portion 33a. In FIG. 8, a change in power can be detected at the output of the FBG 20 by a photodetector 40b that is in optical communication with the output of the FBG 20.

  In some embodiments, the detected optical power indicates the amount of distortion applied to the FBG 20. In some other embodiments, the detected optical power is indicative of the temperature of the FBG 20.

  In some embodiments operating in the slow light reflection mode, the signal experiences a slow group velocity as it travels through the FBG 20. However, it is not as slow as some embodiments in the slow light transmission mode of FIG. Thus, some such embodiments also advantageously provide a large increase in sensitivity to FBGs operated in Bragg reflection mode. Furthermore, since the slow light reflection mode does not involve measuring wavelength shifts, unlike the Bragg reflection mode (eg, FIG. 2), it does not utilize an unbalanced MZ interferometer. This eliminates the thermal stability problem of the MZ interferometer and greatly simplifies the design. In the slow light reflection mode of operation, according to some embodiments described herein, the power transmission of the FBG 20 is not as high as some embodiments in the slow light transmission mode. However, the transmittance is still high. (~ 70%, excluding loss. Depends on design details.) Thus, the loss experienced by the signal as it propagates through the FBG 20 is still small. Thus, in some embodiments, the optical power transmitted by the FBG 20 can be detected and measured (eg, by the photodiode 40b) to measure perturbations applied to the FBG 20.

  The sensitivity in some embodiments of the optical sensor 10 operated in one of the new reflection and transmission modes described herein is directly dependent on how much the group velocity of light can be reduced in the FBG 20. doing. Numerous computer simulations described below illustrate this principle and quantify the magnitude of the sensitivity improvement provided by some embodiments of these new modes of operation. For comparison, these simulations also modeled the sensitivity of the FBG to a specific measurement, ie temperature, in the Bragg reflection mode outlined above. The result would be substantially the same as a simulation modeling the effect of another measurand such as distortion. These are well known equations for the phase of a signal traveling through a grating with known parameters: sinusoidal index modulation of period Λ and amplitude Δn, grating length L, and uniform small temperature change ΔT. (See, for example, A. Yariv and P. Yeh, Optical waves in crystals: propagation and control of laser radiation, pp. 155-214 (New York: Wiley 1984)).

9A-9D show the calculated characteristics for a temperature sensor using a sinusoidal FBG with the following parameter values. Calculations were made assuming Δn = 1.5 × 10 ⁻⁴ , L = 1 cm, and Λ = 0.37 μm (which gives a Bragg wavelength λ _B of about 1.064 μm) and a lossless grating. 9E-9H uses Δn = 2.0 × 10 ⁻⁴ , L = 2 cm, loss = 1 m ⁻¹ , and Λ = 0.53 μm (which gives a Bragg wavelength λ _B of about 1.55 μm). The calculated characteristic of the temperature sensor using the sine wave FBG calculated on the assumption of a lossless grating is shown. 9A and 9E are calculated at two temperatures in the vicinity of λ _B , namely room temperature (300K) (shown as not heated in FIGS. 9A-9C and 9E-9G). The power transmission of this grating is also shown. For the 1.064 micron Bragg wavelength shown in FIGS. 9A-9C, the heated curve corresponds to room temperature plus ΔT = 0.01 ° C. For the Bragg wavelength of 1.55 microns shown in FIGS. 9E-9G, the heated curve corresponds to room temperature plus ΔT = 0.01 ° C. Since the temperature change is very small, the heated and unheated curves of FIGS. 9A-9C and 9E-9G will be very close to each other and indistinguishable on the graph. Figures 9A-9C and 9E-9G show the wavelength dependence of FBG power transmission, phase, and group index for Bragg wavelengths of 1.064 microns and 1.55 microns, respectively. In the vicinity of λ _B , the FBG acts as a reflector and its transmission is close to zero as expected, as shown in FIGS. 9A and 9E. Such a low transmission region has a high reflectivity and constitutes an FBG band gap. The full width at half maximum (FWHM) of this grating for λ _B = 1.064 μm is about 126 pm as shown by FIG. 9A, and the FWHM of the band gap for λ _B = 1.55 μm is shown by FIG. 9E. Is about 202 pm. Outside of the band gap region, the transmittance reaches a first resonance peak, then the vibration so that the amplitude decreases leaves away from lambda _B. At sufficiently far from λ _B (outside the range shown in these figures), the transmission approaches asymptotically to 1.

As described above, the first wavelength at which the transmittance reaches the resonance peak is defined as λ ₁ (in the case of the short wavelength side of λ _B ) and λ ′ ₁ (in the case of the long wavelength side of λ _B ). Call. The higher-order wavelengths at which the transmittance reaches the resonance peak are called λ _i (for the shorter wavelength side of λ _B ) and λ ′ _i (for the longer wavelength side of λ _B ). Here, i ≧ 2. In some embodiments, the local transmittance maximum (eg, also referred to herein as the resonant peak or slow light peak, λ ₁ , λ ₂ , λ ₃ , λ ₄ etc., and λ ′ ₁ , Narrowband light source generates light having a wavelength at or near one of λ ′ ₂ , λ ′ ₃ , λ ′ _4, etc.). In some embodiments, the local transmittance maximum (eg, also referred to herein as the resonant peak or slow light peak, λ ₁ , λ ₂ , λ ₃ , λ ₄ etc., and λ ′ ₁ , between one of λ ′ ₂ , λ ′ ₃ , λ ′ _4, etc.) and the adjacent local transmittance minimum (λ _a , λ _b , λ _c , λ _d, etc.), And _a narrowband light source generates light having λ ′ _a , λ ′ _b , λ ′ _c , λ ′ _d, etc.).

For example, the power transmission spectrum has a first local transmission between a first local transmission minimum including the Bragg wavelength and a second local transmission minimum on the short wavelength side of the Bragg wavelength. It has a rate maximum value lambda _1, and a second local transmission between the third local transmission minimum of the short wavelength side of the second local transmission minimum and the Bragg wavelength In some embodiments, such as having a rate maximum λ ₂ , the wavelength of the light generated by the narrowband light source is a first local transmission minimum and a second local transmission minimum Between the first local transmittance maximum value λ ₁ and the first local transmittance minimum value or the second local transmittance minimum value. Between the second local transmission minimum and the third local transmission minimum between either one of the values. To a maximum value or between the second local transmittance maximum and either the second local transmittance minimum or the third local transmittance minimum You can choose. Similarly, either the third local transmittance maximum value, the short wavelength side of the Bragg wavelength in the fourth local transmittance maximum value, or the third or fourth local transmittance maximum value A wavelength can be selected between either one and the adjacent local transmission minimum.

As another example, if the power transmission spectrum has a first local transmission minimum including the Bragg wavelength and a second local transmission minimum on the long wavelength side of the Bragg wavelength, the first A second local transmission maximum between λ ′ ₁ and a second local transmission minimum between a second local transmission minimum and a third local transmission minimum on the longer wavelength side of the Bragg wavelength; In some embodiments, such as having a local maximum transmittance λ ′ ₂ , the wavelength of the light generated by the narrowband light source may be the first local minimum transmittance and the second local Between the first local transmittance maximum value and the first local transmittance maximum value and the first local transmittance minimum value or the second local transmittance value. Between the second local transmission minimum value and the third local transmission minimum value, between the second local transmission minimum value and the second local transmission minimum value. A maximum transmission value or between the second local transmission maximum value and either the second local transmission minimum value or the third local transmission minimum value Can be selected. Similarly, either the third local transmittance maximum value, the longer wavelength side of the Bragg wavelength in the fourth local transmittance maximum value, or the third or fourth local transmittance maximum value A wavelength can be selected between either one and the adjacent local transmission minimum.

9B and 9F show the calculated phase as a function of wavelength after the narrowband signal has traveled through this grating. This calculation was performed again for the temperatures used for FIGS. 9A and 9E. At certain wavelengths λ ₁ and λ ′ ₁ , and around λ ₂ and λ ′ ₂ , the phase is more abrupt with wavelength than around the center of the curve (around λ _B where the FBG reflects strongly) fluctuate. This increase in phase dependence on wavelength at or near the edge wavelength is a result of the larger group delay of the signal as it travels through the grating. In other words, in the vicinity of these two wavelengths, for example ± 5 pm, the grating sustains slow light.

FIGS. 9C and 9G show the group refraction of light traveling through this FBG, calculated as a function of wavelength, by applying equation (6) to the phase dependence on the wavelength of FIGS. 9B and 9F. The rate is plotted. 9C and 9G show that the group index _ng increases significantly in the vicinity of λ ₁ and λ ′ ₁ . The same applies in the vicinity of other transmission resonances such as λ ₂ and λ ′ ₂ shown in the figure, but in the vicinity of other resonances λ _i and λ ′ _i outside the wavelength range shown in the figure. Is also true. Specifically, for an exemplary grating with λ _B = 1.064 μm, _ng reaches a value of about 4.2 at or near the edge wavelength. In the fiber and in the FBG for wavelengths far from λ _B , the group index of light is approximately c / n, or about 207,000 km / s (this value is weakly dependent on the wavelength of the light). is there. In contrast, around the two edge wavelengths, the group index is only slightly less than the speed of light in free space by a factor of about 4.2, or about 71,400 km / s. In another example of a grating having a lambda _B = 1.55 .mu.m, the end wavelength, or in the vicinity thereof, _{n g} reaches the group velocity of a value of about 8.7, or about 34,500km / s.

9A-9C shows the general behavior outlined in FIG. 9C, and the power transmission spectrum, transmitted phase, and group index of FBGs with sinusoidal index perturbations have been previously reported in different situations. I came. See M. Lee et al, "Improved slow-light delay performance of a broadband stimulated Brillouin scattering system using fiber Bragg gratings," Applied Optics Vol. 47, No. 34, pp. 6404-6415, December 1, 2008. . In this reference, the authors have shown by numerical simulation that the refractive index contrast Δn with parameters similar to those used in FIGS. 9A-9C, ie, length L = 2.67 cm, 10 ⁻⁴ , and FBGs with sinusoidal refractive index modulation with a grating period Λ = 533 nm (1550 nm Bragg wavelength) were modeled. Their conclusion was that light traveling through the FBG would exhibit a higher group delay than usual when the center of the frequency of the light was near the first transmission peak on the Bragg wavelength reflection peak side. This property is used in this reference to increase group delay in SBS-based optical delay lines without increasing power consumption by adding one or two FBGs to the SBS delay line. It was. However, this reference does not recognize or teach that the group delay increases nonlinearly with length. In fact, this reference discloses that group delay is doubled when using two gratings instead of one, as opposed to some example aspects described herein. Yes. In particular, as will be described more fully below, the group delay increases with higher powers of the grating length. Further, but not limited to, (1) the desirability of increasing the index contrast to increase group delay, (2) the phase that accumulates faster at this wavelength, (3) other The presence of resonant wavelengths (grating transmission shows local maximum) and the possibility of operation at these wavelengths, and (4) the use of FBG as a sensor in slow light mode and its advantages, In addition, this reference remains silent on the various aspects described herein, including means for optimizing its performance characteristics.

In some embodiments described herein, the FBG is designed or configured to produce a very large group delay, or equivalently, a very large group index. As a result, very high sensitivity results when this FBG is used as a sensor in one of the operations in the slow light mode described herein. In comparison, previous work on FBG has produced a relatively small group index. For example, in the previously cited M. Lee et al., The maximum value of the group refractive index calculated from FIG. 2A in the reference is about 3.3. As another example, in “Dispersionless slow light using gap solitons,” Nature Physics, Vol. 21, pp. 775-780, November 2006 by Joe T. Mok, the refractive index contrast Δn = 1.53 × 10 ⁻⁴ is set. In a 10 cm long apodized FBG having a group index of about 5 has been reported. In contrast, some embodiments disclosed herein have a group index in the range of a few hundred, if not much higher, as described more fully below. Utilize innovative concepts that allow the creation of FBGs with.

  Some embodiments described herein advantageously provide FBGs with fairly large group indices, ranging from 10 s to 100 s or more. Such a grating, as described below, has a significantly increased sensitivity and a fiber sensor that has improved tens to hundreds, or more, for most measurements compared to existing FBG-based sensors. Can be used to manufacture. They can also be used in any application that utilizes or benefits from a large group index or large group delay. Applications include, but are not limited to, solitary waves, group delay lines, dispersion compensation, and optical filters.

Based on equation (6), the group index value of about 4.2 achieved by the FBG of FIGS. 9A-9C or the group index of about 8.7 achieved by the FBG of FIGS. 9E-9G. For light of value, the sensitivity to temperature of the slow light FBG sensor according to some embodiments described herein is significantly greater than if this same FBG was used in Bragg transmission mode. 9A-9C show two curves for λ _B = 1.064 μm, respectively, calculated at two temperatures separated by ΔT = 0.01 ° C., respectively. 9E-9G show two curves for λ _B = 1.55 μm, respectively, calculated at two temperatures separated by ΔT = 0.01 ° C., respectively. By taking the difference between the two phase curves in FIG. 9B and dividing by ΔT because ΔT is small, the phase derivative dφ / dT with respect to temperature for λ _B = 1.064 μm, as shown in FIG. 9D. A close approximation is obtained. Similarly, by dividing the difference between the two phase curves of FIG. 9F by ΔT, for λ _B = 1.55 μm, a close approximation of the phase differential coefficient dφ / dT with respect to temperature is obtained, as shown in FIG. 9H. Can be acquired. The maximum value of sensitivity occurs in the vicinity of λ ₁ and λ ′ ₁ . At either of these wavelengths, the group index is 4.2 for λ _B = 1.064 μm and the sensitivity dφ / dT is 2.9 rad / ° C. and the group index for λ _B = 1.55 μm is 8.7, and the sensitivity dφ / dT is 8.1 rad / ° C. Therefore, at these slow light wavelengths, the sensitivity to temperature is very high.

FIGS. 10A and 10B show in more detail the relationship between phase sensitivity and transmission, calculated assuming a lossless grating for λ _B = 1.064 μm and λ _B = 1.55 μm, respectively. Show. It is shown that the sensitivity does not have a maximum value exactly at λ ₁ and λ ′ ₁ (where power transmission is equal to 1), but at a maximum in the vicinity thereof. By definition, the transmittance is maximum at these two wavelengths (and is equal to 1), so FIGS. 10A and 10B (at least in these examples) maximize the transmittance (the signal has a grating). It shows that there is no single wavelength that achieves both the maximum loss of the group index (desired to minimize sensitivity) and the maximum group index (desired to maximize sensitivity). However, since the wavelengths at which the transmission and group index are maximized are in the vicinity of each other, the compromise to be made is relatively small. For example, for λ _B = 1.064 μm, the transmittance is 1 at the first transmission resonance peak, the group refractive index is 4.0, and the sensitivity dφ / dT is 2.6 rad / ° C. At the wavelength where the group index is maximum (and equal to 4.2), the sensitivity is 2.85 rad / ° C. and the transmission is equal to 94%. As another example, for λ _B = 1.55 μm, the group refractive index is 8.38 at the first resonance peak (transmittance≈89%), and the sensitivity dφ / dT is 7.84 rad / ° C. At the wavelength where the group index is maximum (and equal to 8.7), the sensitivity is 8.1 rad / ° C. and the transmission is equal to 85%. Considering the loss, the resonance peak does not reach 100% power transmission. Since these values only differ by less than 10% from their corresponding maximum values, in some embodiments, either transmittance or group index, depending on the criteria imposed by the particular intended application. Wavelengths can be selected to maximize this. Regardless of the exact operating point, both transmission and group index (and hence sensitivity) are close to their corresponding maximum values over a fairly broad wavelength range. This is a useful feature for both in some embodiments. This qualitative conclusion is valid for a very wide range of FBG parameter values, values that produce a much higher group index (eg, 10 ⁵ ) than is used in this particular numerical example. Even effective against.

言い換えると、これは、感度に対する測定基準として上記で使用された位相の感度の半分に等しい。 The figure described above shows the refractive index contrast given (Δn = 1.5 × 10 ⁻⁴ for λ _B = 1.064 μm, Δn = 2.0 × 10 ⁻⁴ for λ _B = 1.55 μm). Generated by modeling FBG with. As the refractive index contrast increases, the group delay further increases, and according to equation (6), the sensitivity to the measured quantity also increases. For example, when FBG is produced in hydrogen-loaded fiber (eg Δn is 0.015, eg PJ Lemaire, RM Atkins, V. Mizrahi, and WA Reed, “High pressure H ₂ loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO ₂ doped optical fibers, "Electronic Letters, Vol. 29, No. 13 (June 1993)), the FBG Δn is much higher than this modeled value. Since it can be high, the increased Δn results in a group delay and a substantial increase in sensitivity. To quantify this improvement, sensitivity was calculated as a function of refractive index contrast for a grating used in a low light transmission configuration according to some examples described herein. To show the dependence of group index and sensitivity on index modulation and length only, the grating is assumed to have zero loss at both wavelengths. FIG. 11A (in the case of λ _B = 1.064 μm) and FIG. 11B (in the case of λ _B = 1.55 μm) show (i) with this sensitivity of the configuration of slow light transmission at λ ₁ or equivalent λ ′ ₁ ( ii) Sensitivity to gratings used in low light reflection configurations at λ _a or equivalent λ ′ _a according to some embodiments described herein, and (iii) Sensitivity to gratings used in Bragg reflection configurations Is calculated assuming a lossless grating. All three curves in each of FIGS. 11A and 11B were calculated for an exemplary grating length of L = 2 cm. To perform this comparison, phase sensitivity was converted to power sensitivity in a slow light transmission scheme as follows. When the MZ interferometer is biased in phase with respect to the maximum sensitivity, the output power at one of the interferometer output powers is given by P = P ₀ sin (Δφ / 2) It is done. Where P ₀ is the total output power (including both ports) and Δφ is the phase difference between the two arms. When a small perturbation δΔφ is applied to the FBG placed on one of the two arms of the MZ interferometer, the output power varies as δP≈P ₀ δΔφ / 2. Therefore, the sensitivity of the sensor power is defined as follows.

In other words, this is equal to half of the phase sensitivity used above as a metric for sensitivity.

In the low light transmission configuration, the sensitivity is constant when the grating length is lower than a specific refractive index contrast. If the index contrast is sufficiently large (typically above about 10 ⁻⁴ ), the sensitivity increases as a higher power of Δn. For example, for a 2 cm grating length operating at 1.064 μm, the sensitivity of power to temperature is estimated to be Δn ^1.95 . As another example, for a 2 cm grating length operating at 1.55 μm, the sensitivity of power to temperature is estimated to be Δn ^1.99 . In comparison, in the low-speed light reflection configuration, the sensitivity increases monotonically as the refractive index contrast increases (see FIGS. 11A and 11B). In this example, for a refractive index contrast greater than about ^10-4 , the sensitivity of the two slow light schemes is very close to each other.

In contrast, FIG. 11A (1.064 μm) and FIG. 11B (1.55 μm) are also more specific in Bragg reflection mode than a specific index contrast (about 10 ⁻⁵ for this specific grating length). At low points, the sensitivity is constant, just as in the slow light transmission scheme. Beyond this refractive index contrast, the sensitivity decreases. The reason for this decrease has been described above. In these simulations, the sensitivity of the Bragg reflector configuration is achieved by maximizing the length imbalance ΔL of the MZ interferometer (eg, length imbalance equals the coherence length of the light reflected by the FBG). Maximized) Since this coherence length depends on the refractive index contrast of the FBG (see equations (2) and (3)), this value was adjusted for each value of Δn used in the simulation. As the refractive index contrast increases, the coherence length of the reflected light decreases, the length imbalance ΔL decreases, and therefore the sensitivity decreases (see equation (5)).

FIG. 11A shows a slow light reflection scheme and a slow light reflection scheme for λ _B = 1.064 μm for Δn of 1.5 × 10 ⁻² that can also be achieved in practice (see eg Lemaire et al.). The sensitivity of power at is as high as ˜6.5 × 10 ⁴ ° C. ⁻¹ (the upper end of the curve), indicating that it corresponds to a phase sensitivity of 1.3 × 10 ⁵ rad / ° C. This is nearly 110,000 times higher than the best expected value (1.18 rad / ° C.) for a grating operated in Bragg reflection mode (L = 2 cm, Δn = 10 ⁻⁵ ). FIG. 11B shows that the sensitivity of power in the low-speed light transmission scheme and the low-speed light reflection scheme for λ _B = 1.55 μm is about 1.9 × 10 ⁴ ° C. ⁻¹ for Δn of 1.5 × 10 ⁻² . It shows that it corresponds to a phase sensitivity of 3.8 × 10 ⁴ rad / ° C. This is ˜46,000 times higher than the best predicted value (0.82 rad / ° C.) for a grating operated in Bragg reflection mode (L = 2 cm, Δn = 10 ⁻⁵ ).

The reason why the two slow light configurations show almost the same sensitivity for large Δn (see FIGS. 11A and 11B) is that the wavelength λ ₁ (or λ ′ ₁ ) in the slow light transmission configuration. This is because the group refractive index of light used at the same time and the group refractive index of light used at λ ₂ (or λ ′ ₂ ) in the low-speed light reflection configuration are almost the same. This can be seen in FIG. 12A (1.064 μm) and FIG. 12B (1.55 μm). These plot the group index calculated as a function of Δn against an FBG of length L = 2 cm. Calculated assuming a lossless grating. These plots were again calculated at λ ₁ (or equivalent λ ′ ₁ ) for the transmission mode and at λ _a (or equivalent λ ′ _a ) for the reflection mode. The slow light reflection configuration and the slow light transmission configuration produce almost the same group index, the latter being slightly less than the reflection configuration. In both schemes, the group index increases by about Δn ^1.95 relative to Δn. In the slow light reflection scheme, the signal wavelength deviates more strongly from the wavelength producing the slowest light than in the slow light transmission scheme. FIG. 12A also demonstrates that it is possible to achieve extreme low speed light in an optical fiber grating with λ _B = 1.064 μm. In this example, the actual maximum value of _ng for an FBG having a Δn of 0.015 (hydrogen loaded FBG) is about 10 ⁵ . This corresponds to a group velocity of only 3,000 m / s. This is about 20,000 times slower than previously proven in FBG either experimentally or by simulation. By increasing the FBG length from 2 cm (the value used in this simulation) to 10 cm, the numbers of the group index increases from about ^{5 1.99} ≒ 25 group velocity of 2.5 million -120m / s, until To do. In the example for λ _B = 1.55 μm shown in FIG. 12B, the group index increases again with Δn of about Δn ^1.99 for both schemes. The actual maximum value of _ng that occurs for an FBG having a Δn of 0.015 (hydrogen loaded FBG) is about 4.2 × 10 ⁴ . This corresponds to a group velocity of only 7,100 m / s. This is about 8,000 times slower than previously proven in FBG either experimentally or by simulation. By increasing the FBG length from 2 cm to 10 cm, this number increases to a group velocity of about 1 million-300 m / s. This property has tremendous implications for many applications, including all of the above applications (eg, including but not limited to optical data storage, optical buffers, optical data or pulse delays). . Thus, in some embodiments, optical device 10 is an optical data storage device, an optical buffer, or an optical delay device.

To determine the effect of FBG length on sensitivity, FIG. 13A (1.064 μm) and FIG. 13B (1.55 μm) are 1.5 × 10 ⁻⁴ according to some examples described herein. Was generated to show power sensitivity versus grating length for a fixed Δn. Calculation was performed assuming a lossless grating. For slow light transmission schemes (also evaluated here at λ ₁ or equivalent λ ′ ₁ ), the phase sensitivity to operation at 1.064 μm is estimated to be approximately L ^2.75 . This dependency is not exactly universal, but close to it. For example, running the same simulation (eg, using the Yariv and Yeh calculation schemes) with Δn = 7.5 × 10 ⁻⁴ resulted in a sensitivity varying at L ^2.98 . As Δn is further increased to 1.5 × 10 ⁻³ , the sensitivity increases at L ^2.97 . These numbers also depend on the exact spatial profile of the refractive index modulation (sine function, square, etc.). Nevertheless, the conclusion is that sensitivity is abruptly dependent on length. For operation of the slow light transmission scheme at 1.55 μm, the phase sensitivity is estimated to be approximately L ^2.89 . For slow light reflection schemes at 1.064 μm (also evaluated here at λ _a or equivalent λ ′ _a ), the sensitivity also increases at L ^2.91 . This is similar to the relationship seen in the slow light transmission scheme. For a slow light reflection scheme at 1.55 μm, the sensitivity also increases at L ^2.85 .

In the above example of an FBG having λ _B = 1.064 μm, Δn of 1.5 × 10 ⁻⁴ , and a length of 2 cm, the power sensitivity in the slow light transmission mode was ˜8 ° C. ⁻¹ . FIG. 13A shows the sensitivity of this power when the length of this grating is increased from 2 cm to 10 cm (an exemplary high value that is considered reasonable for use in a reflective grating, see FIG. 4). Indicates an increase to 877 ° C ^-1 . These figures are dramatic of the sensitivity that can be obtained by increasing the length and / or refractive index contrast of an FBG operated in slow light transmission mode according to some embodiments described herein. Showing improvement. For the other example at 1.550 μm for FBG shown in FIG. 13B and having a Δn of 1.5 × 10 ⁻⁴ and a length of 2 cm, the sensitivity of power in slow light transmission mode is ˜2.7. ° C ^-1 . FIG. 13B shows that increasing the length of the grating from 2 cm to 10 cm increases the sensitivity of this power to 243 ° C. ⁻¹ .

In some embodiments, the length L and the refractive index contrast Δn are greater than 10, greater than 20, greater than 30, greater than 40, greater than 50, greater than 100, greater than 500. Can be selected to provide a group index of refraction _ng that is greater than 1,000, greater than 5,000, or greater than 10,000.

  In one embodiment, as shown in FIG. 7, the FBG is placed in one arm of an MZ interferometer made of, for example, an optical fiber. Some example MZ interferometers are substantially balanced except for biasing the two arms (eg, π / 2) to maximize sensitivity to small phase changes. When a small temperature change is applied to the FBG, the phase of the signal traveling through the FBG changes while the phase of the signal traveling through the reference arm does not change. When these two signals are recombined in the second combiner of the MZ interferometer, they interfere in a manner that depends on their relative phase shift being π / 2 + δφ. Where δφ = (dφ / dT) ΔT, where dφ / dT is the sensitivity previously described and calculated (eg, in FIGS. 10A and 10B). As a result of this relative phase shift, the output power of the signal at either port of the MZ interferometer changes by an amount proportional to δφ.

Fiber MZ interferometers typically have a detectable minimum phase value (MDP) on the order of 0.1 to 1 μrad. As an example, for an MZ interferometer operating at 1.55 microns with an MDP of 1 μrad, a refractive index contrast of 0.015, and a grating length of 10 cm, the phase sensitivity is 4.8 × 10 ⁶ rad / ° C. Since the MDP is 1 μrad, this MZ low-speed optical sensor configuration can detect a temperature change as small as 2.1 × 10 ⁻¹³ ° C. Again, this is nearly 5 million times larger than that of the reflective FBG optimized for the same length.

A further example of this principle is shown in FIGS. 14A and 14B. Here, for each of λ _B = 1.064 μm and λ _B = 1.55 μm, the calculated minimum detectable temperature as a function of FBG refractive index contrast for the FBG used in the Bragg reflection mode. It is plotted and calculated assuming a lossless grating. The grating length in this simulation was 1 cm for λ _B = 1.064 μm, 2 cm for λ _B = 1.55 μm, and the MZ interferometer had an MDP of 1 μrad. As predicted in previous simulations, particularly from FIGS. 11A and 11B, the sensitivity decreases as the refractive index contrast increases, and thus the minimum detectable temperature increases. FIGS. 15A and 15B illustrate the same dependency on the same length (and MDP) FBGs operated in slow light transmission mode in accordance with some embodiments described herein. Calculated assuming a lossless grating. In marked contrast to the Bragg reflection mode, as the refractive index contrast increases, the minimum detectable temperature decreases monotonically and eventually reaches a very small value.

  This example clearly demonstrates the advantages over conventional Bragg reflection mode of operation provided by some embodiments described herein. First, the sensitivity is quite large for both slow light configurations according to some embodiments described herein. Second, for a low light transmission configuration according to some embodiments described herein, the MZ interferometer need not be unbalanced. Therefore, both of its arms can have a very short length and can therefore be very stable against temperature changes. Third, for both slow light configurations according to some embodiments described herein, the sensor can utilize a commercially available laser as the light source. This requires a broadband light source in one case (see, for example, Kersey et al.), And in the second case a unique laser (see, for example, Koo and Kersey). Different from mode configuration. In some embodiments, a commercially available laser can be selected that has a very narrow linewidth and low noise limited by shot noise. In contrast, in the first case in a Bragg reflection configuration (eg, FIG. 1), the broadband light source is much more noisy and will add phase and intensity noise to the detected output signal. It will then further increase MDP (and hence the minimum detectable temperature). In the second case of the Bragg reflection configuration (eg FIG. 2), the light source is essentially a special laser containing FBG. This will require precise wavelength stabilization to reduce laser linewidth broadening and keep noise low. While this can be done, it still requires a significant amount of engineering. It is manufactured and sold in large quantities and is more costly than a commercially available narrow linewidth laser that benefits from scale merit.

This ability to detect very small temperatures is excessive for most applications. However, in practical applications, this high sensitivity and exchange can provide a shorter length. In the example of slow light transmission mode quoted above for λ _B = 1.064 μm, the sensor has a sensitivity of 2.2 × 10 ⁷ rad / ° C. for a length of 10 cm. The FBG length to 800 [mu] m, ^or by reducing the coefficient to 125 according ^{L 2.88} dependent and will be less sensitive by a factor of ~1.77 × 10 ⁶ to 12.4rad / ℃. For the second slow light transmission mode example operating at 1.55 μm, the sensor has a phase sensitivity of 4.8 × 10 ⁶ rad / ° C. for a length of 10 cm. By reducing this FBG length to 800 μm, the sensitivity will drop to 4.8 rad / ° C. with a factor of about 1 × 10 ⁶ . These sensors still have about the same sensitivity as the optimized FBG used in the Bragg reflection mode (see FIG. 4). However, the length is only 800 μm, not 10 cm, and is therefore quite small.

  The above analysis was performed for the case where the temperature was a measured quantity. The same conclusion applies when the measured quantity is another quantity, such as longitudinal distortion applied directly to the FBG.

  By using slow light, both strain sensitivity and temperature sensitivity are increased. Thus, one advantage of the slow light sensor according to some embodiments described herein is that it is more sensitive to temperature fluctuations while being a more sensitive strain sensor. In some embodiments, the sensor can be stabilized against temperature fluctuations, but such stabilization may not be desired. However, sensitivity and length can always be exchanged for each other. Thus, in some example low speed light sensors described herein, the sensitivity of strain and the sensitivity of temperature are enhanced at approximately the same ratio. Therefore, the physical length L of the grating can be reduced to bring the strain sensitivity and the temperature sensitivity to the same level as the best case of the Bragg reflection FBG. The difference and advantage of the slow light configuration is that for the same sensitivity, the slow light FBG is much shorter. This can be important for many applications where small size and economics are critical. Any compromise between length and sensitivity is also possible, which will design the slow light sensor so that it must be more sensitive while being somewhat shorter than a normal reflection grating. In addition, many engineering solutions that have been applied to distinguish between changes in strain applied to the grating and changes in temperature are also applicable in this configuration of slow light sensors. Specifically, for example, two gratings can be arranged in parallel in a region where strain and temperature change. One of the gratings is distorted while the other is not distorted, while both are subject to (same) temperature changes. Comparison between the readings of the two sensors can provide both a common temperature change and a strain change applied to one grating.

The simulation also shows that in some embodiments described herein, the line width of the light source used to examine an FBG operated in either slow light mode is quite reasonable. FIG. 16 shows a plot of power sensitivity dependence on laser linewidth for a grating having length L = 2 cm and Δn = 1.5 × 10 ⁻⁴ and operating at λ _B = 1.064 microns. This point is shown. Calculated assuming a lossless grating. As the laser linewidth increases, the sensitivity remains constant up to a linewidth of about 10 ⁻¹³ m. Beyond this value, the sensitivity begins to decline because the laser signal is exploring a wider spectral region that extends beyond the peak in the group index spectrum. In other words, some photons are considered high group index (photons at the peak of the group index spectrum and frequencies around it), others are considered lower group index (photons at frequencies off this peak). ). This curve shows that in order to obtain the maximum value of sensitivity, it is advantageous that the laser line width is selected to be no greater than about 10 ⁻¹³ m or a frequency line width of 26 MHz. This is a linewidth readily available from a number of commercially available semiconductor lasers, such as, for example, an Er-Yb doped fiber laser from NP Photonics, Tucson, Arizona. A similar simulation performed on an FBG of the same length but with a large refractive index contrast (Δn = 1.5 × 10 ⁻³ ) for a laser linewidth of about 2 × 10 ⁻¹⁵ m (530 kHz) Give the maximum value. Such laser line widths are also readily available commercially. Therefore, the laser line width can be reduced as the refractive index contrast of the grating increases or as its length increases.

All simulations were performed on FBGs with a Bragg wavelength of either 1064 nm (the main wavelength of the Nd: YAG laser) or 1.55 μm. These wavelengths were selected because they are commonly used. However, wavelength has nothing to do with the general trend outlined in some of the embodiments described herein. The characteristics of similar FBGs centered at another wavelength, such as around 1.3 μm, are not substantially different from those presented here, and are the same as those presented and cited here. Can be used to model. The relative advantages that slow light schemes according to some embodiments described herein have over the Bragg reflection described herein remain substantially unchanged.
Optimization process

The properties of uniform grating transmission and group index spectrum are uniquely determined by three parameters: refractive index modulation, length, and loss. In the lossless grating as described above, the group refractive index can be increased by infinitely increasing the refractive index contrast and length. In practice, when light travels through the grating, it encounters losses due to scattering that induce coupling into the radiation mode. In the presence of loss, as the length of the grating increases, the light travels a longer distance in the grating and faces higher losses accordingly. This effect is intensified when the group refractive index of the FBG is large. This is because the light faces more losses as the light travels back and forth through the grating. Therefore, for a given loss, as the grating length increases, the group index first increases as described above. As the group index increases further, as in the Fabry-Perot interferometer, losses begin to limit the maximum number of round trips, and the group index begins to decrease with some further increase in length. . As a result, there is a grating length that maximizes the group index at resonance for a given loss factor. Similarly, as the length increases, the transmittance of the grating at these resonances is also limited by the loss. When designing an FBG for slow light applications, use the above model, for example, to determine the optimum length of the grating given its profile type, refractive index modulation, and loss It is advisable to carry out a survey of conversion. A number of standard techniques known to those skilled in the art can be used to measure the loss factor of the FBG. The measured power loss factor of FBG is 1 m ⁻¹ (Y. Liu, L. Wei, and J. Lit, “Transmission loss of phase-shifted fiber Bragg gratings in lossy materials: a theoretical and From experimental investigation, "Applied Optics, 2007", over 2m ^-1 in hydrogen loaded gratings (D. Johlen, F. Knappe, H. Renner, and E. Brinkmeyer, "UV-induced absorption, scattering and transition losses in UV side-written fibers, "in Optical Fiber Communication Conference and the International Conference on Integrated Optics and Optical Fiber Communication, 1999 OSA Technical Digest Series (Optical Society of America, Washington, DC, 1999), paper ThD1, pp. 50 -52).

This behavior is shown in FIG. 17 for the example of a uniform grating operating at 1.55 μm. For loss factors α that vary between ¹ m ⁻¹ and 2 m ⁻¹ , in a strongly Ge-doped grating, for example a grating with a large refractive index contrast (in this example Δn = 1.0 × 10 ⁻³ ) The figure shows the dependence of the group index and transmittance at the first resonance (λ ₁ ) on the length. For a given loss factor, when the grating length is short, the grating does not have a sufficient period and the group index is low. If the length is long, as the light travels through the grating, the light encounters greater loss and the group index decreases. Somewhere between these two limits, the group index is at its maximum. As shown in the example of FIG. 17, when the loss of the grating is 1 m ⁻¹ , the highest group refractive index is 84 and the transmittance is about 10% at the grating length of 2.25 cm. As the loss increases to 2 m ⁻¹ , the highest group index decreases to 53 at the shorter optimal length of 1.8 cm, and the transmission at this length is roughly the same (11%). Also, as explained above, the transmittance decreases steadily as the length increases. In applications where the highest group index is desirable and transmission is not a concern, operation at or near the optimum length is preferred. In applications where transmission is a greater concern, a compromise can be made to achieve the highest possible group index without excessively reducing the transmission. In addition to the first resonance wavelength modeled in this example, the designer of the slow light FBG can select other resonance wavelengths.

FIG. 18 shows the same dependency calculated for the example of a stronger FBG produced in a hydrogen loaded fiber. The value of Δn used in this simulation is 0.01, the value reported for a grating written in a hydrogen loaded fiber, and the loss factor is 2m ⁻¹ . See PJ Lemaire, RM Atkins, V. Mizrahi, and WA Reed, "High-pressure H ₂ -loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO ₂ -doped optical fibers," Electron. Lett., 1993. That. For this calculation, the FBG was assumed to be uniform. The highest group index for this example occurs for a length of 0.37 and is equal to 243. The transmittance of the grating at this group refractive index is 12%.

Apodization also affects the relationship between group refractive index, transmittance, and length. Here, there are two types of types, called raised-Gaussian-apodized with zero dc refractive index change as type A and Gaussian-apodized as type B. An example of apodization is shown in FIG. 19A. See T. Erdogan, "Fiber grating spectra," J. of Lightwave Technology, Vol. 15, pp. 1277-1294, 1997. As shown in FIG. 19A, in Type A, the refractive index profile is modulated above and below some average refractive index value. In type B, the refractive index profile is modulated completely beyond some average value. In either type, the refractive index modulation envelope can have any profile, for example a cosine function or Gaussian. In the following simulations, for both Type A and Type B FBGs, the envelope is assumed to be Gaussian with a full width at half maximum (FWHM) denoted W. FIG. 19B shows the group index spectra calculated for a type A apodized FBG with Δn = 1.0 × 10 ⁻³ , L = 2 cm, loss factor 1.3 m ⁻¹ , and W = 2L. The highest group index peak occurs at the second resonance, but its value is smaller than that of a uniform grating with the same index modulation, length, and loss. There are four relevant parameters controlling the slow light behavior of such apodized gratings: refractive index modulation maximum, length, loss factor, and full width at half maximum (FWHM) of the envelope of the refractive index profile. W. As shown in FIGS. 19C and 19D for Δn = 1.0 × 10 ⁻³ , L = 2 cm, loss = 1.3 m ⁻¹ , and W = 2L, type B gratings have asymmetric transmittance and group A refractive index spectrum is generated. The highest group refractive index occurs at the first resonance peak on the short wavelength side. With proper apodization, the value can be made larger than that of a uniform grating with the same refractive index modulation and length. This can be achieved in some embodiments by the optimization process described below.

Two parameters that can be optimized to maximize the group index for a given index contrast Δn maximum and a given loss factor are length and FWHM W. In the most common approach well known in the optimization process, a simple two-dimensional parameter search can be performed. As an example, FIG. 20 shows the group refractive index, transmittance for a type A apodized FBG with FWHM equal to twice the length, Δn of 1.0 × 10 ⁻³ , and a loss factor of 1 m ⁻¹ , And shows the relationship between lengths. The optimum length for the maximum value of the group index is 1.43 cm. At this length, the group index is so high that it reaches 178. At this time, the transmittance is 1.4%. Increasing the loss to 1.5 m ⁻¹ decreases the group index and reduces the optimum length as expected. However, the transmission increases slightly to about 1.7%. Compared to FIG. 17, which models a uniform FBG with the same refractive index contrast, for a given length and refractive index modulation, a Type B apodized grating with the same length and refractive index modulation as a uniform grating is It is shown to produce a higher group index and lower transmission than a uniform grating. Thus, adjusting the refractive index profile of the FBG can have a significant impact on the design of some embodiments described herein of FBGs used as low speed light devices.

The width of the apodization envelope can also play an important role in group index and transmission. If the FWHM is small, the effective length of the grating is small, resulting in a lower group index. When the FWHM is large, the grating profile is similar to a uniform grating. Therefore, in this limit, the dependence on the group refractive index and the transmission length converges to the respective dependence in the corresponding uniform grating. In FIG. 21, in the example in the case of a loss factor of 1 m ⁻¹ , the maximum value of refractive index contrast Δn = 1.0 × 10 ⁻³ , and a length of 1.43 cm, the group refractive index and transmission of the envelope with respect to FWHM W The rate is plotted. The optimum FWHM for the maximum value of the group index is 1.4 cm. In this FWHM, the group refractive index is 200. This is even higher than in FIG. 20, but the power transmission is very low. Again, depending on the application requirements, a compromise can be made to determine which optimal length to choose. However, curves such as FIGS. 20 and 21 clearly provide information that allows this selection to be made.

As shown in FIG. 22, the same optimization process can be applied to a Type B hydrogen loaded apodized FBG. If the FWHM of the Gaussian envelope is equal to twice the length, the optimal length for a Δn of 1.0 × 10 ⁻² and a loss factor of 2m ⁻¹ is 0.17 cm. The group index for this example reaches 744, and the power transmission at this group index is 5%.

  FIG. 23 shows that the optimal FWHM is 0.17 cm, assuming that the length of the hydrogen loaded FBG modeled in FIG. 22 is chosen to be 0.17 cm. In this FWHM, the group index reaches 868, but the power transmission at this point is very low. The FWHM that produces the highest group index is in this case approximately equal to the length of the grating. The advantage of this particular FWHM is a very high group index and the disadvantage is a low power transmission. Here, as with all other examples cited previously, the trade-off between power transmission and group index is unique for each application.

In addition to uniform and apodized gratings, π-shifted gratings are another type of general grating profile that can produce slow light. The π-shifted grating has a phase shift π located at the center of the grating profile. This type of grating broadens the narrow transmission resonance at the Bragg wavelength and broadens the transmission spectrum. The lowest group velocity for this type of grating is not located on the longer band gap edge, located in the center of the band gap lambda _[pi rather. This shows the transmittance and group index spectra calculated for a uniform π-shifted grating with Δn = 2.0 × 10 ⁻⁴ , L = 2 cm, and zero loss. 29B.

These predictions have been experimentally verified by measuring the group delay of light traveling through various FBGs having a Bragg wavelength near 1550 nm. Light traveling at a wavelength where a large group index occurs produces a large group delay proportional to the group index. Group delay was determined by measuring the time difference between the arrival times of two signals of different wavelengths, both provided by the same tunable laser. The first wavelength was located far (˜2 nm) from the band gap edge of the FBG so that light traveled through the FBG at normal group velocity. At this first wavelength, the group index is very close to the phase index, and the phase index itself is very close to the refractive index n _{0 of the} material, eg about 1.45. The second wavelength was adjusted to be near the band gap edge (within 200 pm). Here the group refractive index and hence the group delay is larger. The signals at the first and second wavelengths were both amplitude modulated at the same frequency before entering the FBG. The difference between the group delays measured at the two wavelengths provided a basis for the group index increase induced by FBG.

第２の波長における群屈折率は、下記の式を用いて微分群遅延から計算することができる。

The experimental setup used for this measurement is shown in FIG. A beam from a tunable laser (Hewlett-Packard HP 81689A) was sent through an optical isolator and polarization controller, then an amplitude modulator. The polarization controller was used to adjust the state of polarization (SOP) of the light entering the modulator and thus maximize the power transmitted by the modulator (whose operation happened to be polarization dependent). Having a frequency f _m of between 25MHz to 100 MHz, sinusoidal signal from the function generator was fed into the modulator for modulating the power of the laser signal. A sinusoidally modulated laser beam was sent through the test FBG. The signal leaving the FBG was split in two using a 50/50 fiber coupler. One of the output signals was sent to a power meter to measure its power, and hence (if the laser wavelength was varied) to measure the transmission spectrum of the FBG. The other beam was sent to a photodetector that followed a lock-in amplifier that measured its phase. The first measurement was made at a wavelength of 1548.000 nm, which is not subject to slow light and is far enough (2 nm) from the bandgap edge, and was therefore used as a reference signal. Next, the laser was adjusted to a low-speed light wavelength close to the band gap edge, and the phase measurement was repeated. From the phase change Δφ measured between these two wavelengths, the difference in group delay between the first wavelength and the second wavelength was calculated using the following equation.

The group refractive index at the second wavelength can be calculated from the differential group delay using the following equation:

Table 1 lists the commercial fiber Bragg gratings tested. These were all manufactured by Canadian OE-Land. The table lists the length provided by the manufacturer, whether it was a non-thermal grating, and whether the refractive index profile of the grating was uniform. Also listed are the refractive index contrasts Δn (peak values for non-uniform FBGs) for each grating.

FIG. 25A shows, by way of example, the measured transmission spectrum of grating # 1 having a nominally uniform refractive index contrast of about 1 × 10 ⁻⁴ in length 3 cm and a value specified by the vendor. The transmission spectrum shows the shape expected for a uniform grating. That is, the center of the narrow reflection peak is located at the Bragg wavelength (in this case, λ _B ≈1549.948 nm), and the both sides are surrounded by vibration whose amplitude decreases with increasing distance from this peak. The solid curve in FIG. 25A is a transmission spectrum calculated from theory for a uniform FBG. The only parameter adjusted to match the theoretical curve with the experiment is the refractive index contrast. This fitting was used because the index contrast value specified by the manufacturer was not accurate enough. The fitted value used to generate FIG. 25A is Δn = 1.10 × 10 ⁻⁴ , which is close to the vendor value. These simulations assumed zero loss. These curves show that there are transmission peaks very close to 100% on either side of the Bragg reflection peak.

FIG. 25B shows the group index spectra measured for the same grating (# 1), along with the theoretical spectra calculated for the same Δn and length as for the solid curve in FIG. 25A. The light is slowest at wavelengths λ ₁ ≈15499.881 nm and λ ′ ₁ ≈1550.012 nm. They are located in contrast to λ _B and also coincide with the first high transmission peak on either side of the band gap of the FBG. The highest measured group refractive index at these two wavelengths is ˜3.7, which is in good agreement with the expected value. As in FIG. 25A, we see good agreement between theory and experiment.

When FBGs with higher refractive index contrast were tested, they gave higher maximum group refractive index values, as expected. As an example, FIGS. 26A-26C show corresponding curves for grating # 3 having a length of 2 cm and a Δn of ˜1.0 × 10 ⁻³ . FIG. 26A shows that the entire measured transmission spectrum is not symmetric around the Bragg wavelength, indicating that it is an apodized grating, as explained above. FIG. 26B shows an enlarged view of the short wavelength portion of the same measured transmission spectrum for convenience. Superimposed on this measured spectrum is four parameters: refractive index contrast (Δn = 1.402 × 10 ^{−3) in} order to obtain the nominally best fit to the measured spectrum. ), A fitted transmission spectrum predicted by a model fitted with length (L = 20 mm), Gaussian apodization FWHM (W = 42 mm), and loss factor (γ = 1.6 m ⁻¹ ). is there. The agreement between the measurement results and the theory is still very good. The fitted group index Δn is close to the value specified by the manufacturer and the loss factor is within the value reported for the FBG. FIG. 26C shows the group index spectra measured for this grating, as well as the predicted spectra calculated for the same fitted parameter values used in FIG. 26B. The maximum value of the measured group refractive index occurs at the second slow light peak (wavelength λ ₂ ) and is 69. This is the highest value ever reported in a fiber Bragg grating (previous record is ~ 5, which is JT Mok, C. Martijn de Sterke, ICM Littler and BJ Eggleton, "Dispersionless slow-light. using gap solitons, "reported in Nature Physics 2, 775-780 (2006)), and also the highest value reported in optical fibers (previous record is -10, which is CJ Misas, P. Petropoulos, and DJ Richardson, "Slowing of Pulses to c / 10 With Subwatt Power Levels and Low Latency Using Brillouin Amplification in a Bismuth-Oxide Optical Fiber", J. of Lightwave Technology, Vol. 25, No. 1, January Reported in 2007). This corresponds to a group velocity of ~ 4,350 km / s, which is a much lower value than ever reported for optical fibers. The agreement between the experimental spectrum and the theoretical spectrum is very good. This value of n _g = 69 was observed at the second slow light peak (wavelength λ ₂ ) from the Bragg wavelength. The first slow light peak could not be measured. This is because, as shown in FIG. 26B, the first peak (around the wavelength λ ₁ = 1549.692 nm) is too weak to be measured. The transmittance of FBG at the second peak was ˜0.5%. The group index measured at the third peak (wavelength λ ₃ ) was only slightly lower (˜68), but the transmission of FBG was significantly higher, about 8%. The corresponding value for the _fourth peak (wavelength λ ₄ ) was n _g ≈43 and the transmittance was 32%. FIG. 26C also shows that the bandwidth of the slow light peak increases as the order i of the slow light peak increases. This also shows that a given FBG can provide a wide range of group index / group index bandwidth / transmittance combinations, based on the desired performance for the intended application. The user can select this.

The last column in Table 1 summarizes the maximum _ng value measured in the five gratings tested. In all cases except for grating # 4, the agreement between the predicted and measured values was very good. In the case of Grating # 4, the length was so long that the calculation did not converge and a reliable value could not be provided.

As the group index increases, for example, as the refractive index contrast or length of the grating increases, the line width of the slow light peak tends to decrease. In order to obtain the maximum benefit from the slow light FBG sensor or from the slow light FBG used for other purposes, a laser having a line width smaller than the line width of the slow light peak being used can be selected. If the line width of the laser is greater than the line width of the slow light peak, the laser photons at the peak maximum will receive the maximum sensitivity, but photons that deviate from the peak will receive lower sensitivity. Therefore, the average sensitivity will be reduced. This can be indicated by the laser line width used in the measurement. For grating # 1 having a moderate group refractive index maximum, the line width of the group refractive index of this slow light peak (λ ₁ ) is relatively wide, and its transmittance and group refractive index spectrum (FIGS. 25A and 25B). ) Could be probed using a laser with a line width equal to ˜1 pm. For grating # 3, which has a much higher group index maximum, the line width of the group index of the slow light peaks (λ ₂ , λ ₃ , and λ ₄ ) is much narrower, its transmittance and group index. The spectra (FIGS. 26B and 26C) were probed using a laser with a line width equal to ˜0.8 fm (100 kHz in frequency). The line width of the slow light peak can be easily calculated using the above theoretical model, as shown by FIGS. 25A-B and FIGS. 26A-C. From this line width prediction, it is easy to calculate the sensitivity of the sensor as a function of the laser line width. Alternatively, the sensor can be operated using a wider line width. The disadvantage is lower sensitivity (as in FIG. 16), but greater stability against temperature changes is an advantage.

Temperature affects the slow light spectrum. Due to the combination of thermal expansion and / or temperature dependence of the refractive index dependence of the fiber material, its period Λ, effective mode refractive index, and length all vary as the temperature of the FBG changes. To do. These effects are well known and can be easily predicted using well-established mathematical models. As an example, applying these basic effects to an FBG having L = 2 cm, Δn≈1.5 × 10 ⁻⁴ , and λ _B = 1.55 μm, the first transmission peak wavelength (λ ₁ and The relative temperature sensitivity of λ ′ ₁ ) is expected to be approximately Δλ ₁ / λ ₁ = 10 pm per 1 ° C. For example, if an FBG is used as a strain sensor, the sensitivity to strain will generally vary as the grating temperature changes. This is because the transmission peak wavelength varies with temperature. In practice, this can be avoided by controlling the temperature of the FBG to the extent that it depends on the group index of refraction at the operating wavelength (generally, the higher the group index, the more severe the temperature control). Alternatively, a non-thermal FBG, a commercially available device, that has been partially compensated for the inherent temperature dependence of the FBG spectrum by properly packaging the grating can be used. Such devices are commercially available from, for example, OE Land or Teraxion, Canada.

  Fiber Bragg gratings are affected by phase or amplitude disorder, ie, random variations along the longitudinal axis z of the grating, either in the period of the grating or in the refractive index contrast of the grating. It is well known that the presence of such disorder changes the properties of FBG. In particular, such disorder generally results in a broadening of the reflection peak and a reduction in its power reflection coefficient. Similarly, phase or amplitude disorder will result in a change in the slow light spectrum of the FBG, particularly in the direction that generally reduces the slow light peak transmission and group index. If these effects are considered detrimental for the application under consideration, steps may be taken to minimize phase or amplitude disorder during the manufacture of slow light FBGs.

  FIG. 27 is a flowchart of an exemplary method 1000 for the use of a fiber Bragg grating, according to some embodiments described herein. Method 1000 comprises providing FBG 20 having a refractive index perturbation that is substantially periodic along the length of FBG 20, as shown in operation block 1010. The FBG 20 has a power transmission spectrum having a plurality of local transmittance minimum values. Each pair of adjacent local transmission minimums has a local transmission maximum between them. The local maximum transmittance has the maximum power at the transmission peak wavelength. The method 1000 also includes generating light having a wavelength between two adjacent local transmission minima from a narrowband light source, as shown in operation block 1020 of FIG. In some embodiments, the generated light has a line width that is narrower than the line width of the transmission peak. The method 1000 transmits a first portion 33a of light along a first optical path 31 that extends the entire length along the length of the FBG 20 at an action block 1030, and a second optical path 32 at an action block 1040. And transmitting the second portion 33b of light along the line. In some embodiments in which FBGs are used in the optical sensor, the method 1000 includes, at operation block 1050, the first portion 33a, the second portion 33b, or both the first and second portions 33a, 33b, The method further includes detecting using the photodetector 40.

  In some embodiments of method 1000, the substantially periodic refractive index perturbation has a constant period along the length of FBG 20. In some other embodiments, the substantially periodic refractive index perturbation has a period that varies along the length of the FBG 20 such that the FBG 20 is a chirped grating. In some embodiments, the substantially periodic refractive index perturbation has an amplitude that varies along the length of the FBG 20 such that the FBG 20 is an apodized grating.

  In some embodiments of the method 1000, the method 1000 further comprises recombining the first and second portions 33a, 33b and transmitting them to the photodetector 40. For example, in some embodiments, the method 1000 provides a first fiber coupler 51 in optical communication with the narrowband light source 30, the first optical path 31, and the second optical path 32; Providing a second fiber coupler 52 in optical communication with the optical path 31 and the second optical path 32. In these embodiments, the method 1000 includes splitting the light generated by the narrowband light source 30 into a first portion 33a and a second portion 33b by the first fiber coupler 51. Thus, in these embodiments, the recombination and transmission steps are performed by the second fiber coupler 52. Also, the detecting step 1050 in these embodiments comprises detecting a phase difference between the first portion 33a and the second portion 33b. In some embodiments, the first optical path 31 and the second optical path 32 form a nominally balanced Mach-Zehnder interferometer.

  In some embodiments, the phase difference indicates the amount of distortion applied to the FBG 20. In some embodiments, the phase difference indicates the temperature of the FBG 20.

  In some embodiments of the method 1000, the step 1040 of transmitting the second portion 33b of light along the second optical path 32 comprises the step of reflecting the second portion 33b from the FBG 20. The detecting step 1050 in these embodiments may comprise detecting the optical power of the first portion 33a, the second portion 33b, or both the first and second portions 33a, 33b. In some embodiments, the detected optical power indicates the amount of distortion applied to the FBG 20. In some embodiments, the detected optical power is indicative of the temperature of the FBG 20. In some embodiments of the method 1000, the first portion 33a that is transmitted along the FBG 20 has a first group velocity. The first group velocity is smaller than the second group velocity of light having a wavelength outside the reflected range of wavelengths transmitted along the FBG 20. In some of these embodiments, the ratio of the first group speed to the second group speed is equal to or less than 1/3. In some embodiments, the ratio of the first group speed to the second group speed is equal to or less than 1/10.

FIG. 28 is a flowchart of another embodiment of a method 2000 using a fiber Bragg grating, according to some embodiments described herein. The method 2000 includes providing the FBG 20 with a refractive index perturbation that is substantially periodic along the length of the FBG 20, as shown in operation block 2010. In some embodiments of the method 2000, the method 2000 includes generating light having a certain wavelength from the narrowband light source 30, as shown in operation block 2020. In some embodiments, the wavelength is near the slow light peak of FBG 20. The method 2000 includes a first optical path 31 that extends the entire length along the length of the FBG 20 having a group velocity such that the ratio of the velocity of light in the vacuum to the group velocity is greater than 5 at operation block 2030. The method further includes transmitting the first portion of light 33 a and transmitting the second portion of light 33 b along the second optical path 32 in the operation block 2040. In some embodiments in which the FBG 20 is used for light sensing, the method 2000 may perform at operation block 2050 the first portion 33a, the second portion 33b, or both the first and second portions 33a, 33b. The method further includes detecting using the detector 40.
Optimization using figure of merit

Operation in some embodiments of a novel type of fiber sensor utilizing a fiber Bragg grating (FBG) as a sensing element is described herein and is also described in US patent application Ser. 792,631, the entire disclosure of which is incorporated herein by reference. The difference from other FBG sensors that have been reported in the literature so far is that some embodiments described herein sense using slow light. Slow light can be excited in the FBG by allowing light of a specific wavelength to enter the FBG. In some embodiments implementing this concept, this wavelength can be selected in the vicinity of one of a plurality of high transmission peaks present at the edge of the band gap of the FBG for a particular FBG. On the short wavelength side of the band gap shown in FIGS. 9A and 9E, these wavelengths are denoted as λ _j . Here, j is an integer equal to or greater than 1, and the peak at j = 1 is the peak closest to the band gap. On the long wavelength side of the band gap, these wavelengths are denoted as λ ′ _j . Here, j is an integer equal to or greater than 1, and the peak at j = 1 is the peak closest to the band gap. At these wavelengths, FBGs can sustain slow light characterized by group velocities that can be substantially lower than the group velocities of light that normally travel through optical fibers. This low group velocity ν _g is characterized by a high group index n _g = c / ν _g . Here, c is the speed of light in vacuum.

  When a perturbation to be sensed (eg, distortion) is applied to a phase sensor according to some embodiments, the resulting phase perturbation of the light traveling through the sensor is proportional to the inverse of the group velocity To do. Thus, in these embodiments, operating the FBG in the vicinity of one of these multiple transmission peaks can result in increased sensitivity to the measurand. In particular, this is not limited to the following, but is true for temperature, strain, displacement, and relative rotation. Thus, in some embodiments of FBG sensors that use slow light, the sensitivity to the measured quantity can be increased, such as the group index. Thus, in some embodiments, if all else is the same, the higher the group index or the slower the group velocity, the higher the sensitivity.

In some of the embodiments described herein, and as disclosed in US patent application Ser. No. 12 / 792,631, light is generally the lowest group velocity at the first (j = 1) peak. Can have. For example, some embodiments of uniform FBGs can have the lowest group velocities on either side of the bandgap, ie, both λ ₁ and λ ′ ₁ . As another example, some examples of apodized FBGs described herein and described in US patent application Ser. No. 12 / 792,631 are lowest on the short wavelength side of the bandgap, ie, λ ₁ Can have group velocities. Thanks to the asymmetry in the spectral response of these particular apodized gratings, some embodiments may show very little to no noticeable high transmission peaks on the long wavelength side of the bandgap.

FIG. 30 shows an example of a transmission spectrum calculated for an example of an apodized grating. An example of a transmission spectrum is Gaussian refraction with length L = 1.2 cm, loss 1 m ⁻¹ , and peak refractive index modulation Δn = 1.04 × 10 ⁻³ and full width at half maximum (FWHM) W = 0.98 cm. Calculated for an apodized grating with a rate profile envelope. The peak refractive index modulation is 0.06 cm away from the center position of the grating at 0.6 cm. If there is apodization shift, _{n g} at the peak number 1 in FIG. 30 is a 218.6 a 126.8 in the peak number 2. When the shift is not, _{n g} at the peak number 1 is 225.2, the peak 2 is 148.4. Shifting the peak refractive index modulation from the center position of the grating will reduce _ng , but in this case the change in group index is not significant. This is because the shift is small compared to the length of the grating. For clarity, only the short wavelength side of the spectrum is shown. As described above for some embodiments, the transmission spectrum shows a sharp peak at the edge of the bandgap. In FIG. 30, λ _B indicates the position of the Bragg wavelength of the FBG. In this example, the transmission peak has a peak transmission value that increases with peak number j. In this example, the transmission peak also broadens as the peak number increases.

FIG. 30 also shows the calculated group index spectrum for the apodized grating example. Again, for clarity, only the short wavelength side is shown. As explained here and for some of the examples described in US patent application Ser. No. 12 / 792,631, this spectrum can also be used for the transmission peak wavelengths λ _j and λ ′ _j A sharp resonance centered at a wavelength approaching the can be shown. In some embodiments, the first peak (j = 1) indicates the highest n _g , eg, the slowest group velocity. In these embodiments, the maximum group index of subsequent slow light peaks decreases as the peak number (j) increases.

  Based on these concepts, two general types of sensors according to some embodiments are disclosed here and in US patent application Ser. No. 12 / 792,631. In some embodiments of this first type, referred to as transmission mode, an FBG is placed on one of the nominally balanced arms of the MZ interferometer and the interferometer at wavelengths near the slow light peak of the FBG. Can be explored. An exemplary diagram of this approach is described above and is shown in FIG. In some embodiments of the second type of sensor, the wavelength is adjusted to be on either side of the slow light peak at or near the wavelength where the slope of the group index spectrum is at its maximum. You can choose. Thus, these sensors can be used in the reflective mode as described above and shown in FIG.

Based on the fact that the sensitivity is proportional only to the group index, it may be tempting to conclude to probe the sensor at wavelengths near the maximum of the first slow light peak to achieve the highest possible sensitivity. Absent. However, for some embodiments, this is not necessarily so, as described below.
a. Transparent configuration

  FIG. 31 shows a diagram of an embodiment of an apparatus utilizing FBG used in slow light transmission mode in accordance with some embodiments described herein. As shown in FIG. 31, the optical device 110 may comprise an FBG 120 having a refractive index modulation that is substantially periodic along the length of the FBG 120. The FBG 120 may have a power transmission spectrum having a plurality of local transmittance maximum values. Each of the local transmission maxima has the maximum power at the transmission peak wavelength. The FBG 120 has a group index spectrum as a function of wavelength. The optical device 110 can include a narrowband light source 130 that is in optical communication with the first optical path 131 and the second optical path 132. The narrow band light source 130 can be configured to generate light that can be configured to be branched into a first portion 133a and a second portion 133b. The first portion 133a may be transmitted at a group velocity along a first optical path 131 that extends the entire length along the length of the FBG 120. The light is at or near a wavelength where the product of the group index spectrum and the square root of the power transmission spectrum is at a maximum (eg, higher than for any other wavelength near the edge of the band gap of the FBG 120). Can have a wavelength.

  In some embodiments, the FBG 120 may be similar to the FBG 20 described herein. For example, the FBG 120 can be manufactured by exposing the core of the optical fiber to a spatially modulated UV beam. The refractive index modulation can take any number of spatial distributions. The optical fiber may be a conventional single mode fiber or a multimode fiber. The optical fiber may be doped with a specific element so that the desired modulation in refractive index can be induced by exposure to spatially varying light. The spatially periodic refractive index modulation in the FBG 120 can have a constant period along the length of the FBG 120 as a uniform grating, and a period that varies along the length of the FBG 120 as a chirp grating. Alternatively, the apodized grating can have a refractive index modulation amplitude variation along the length of the FBG 120. As shown in FIGS. 9A and 9E, the FBG 120 can have a power transmission spectrum with multiple local transmission maxima. Each of the local transmission maxima has the maximum power at the transmission peak wavelength.

In some embodiments, the narrowband light source 130 may be similar to the narrowband light source 30 described herein. For example, the narrowband light source 130 can comprise a semiconductor laser or a fiber laser, such as an Er-doped fiber laser having a wavelength range between about 1530 nm and 1565 nm. As another example, narrowband light source 130 may comprise an Nd: YAG laser having a wavelength of 1064.2 nm. In some embodiments, the narrowband light source 130 can have a narrow linewidth, for example, less than or equal to 10 ⁻¹³ .

  The narrow band light source 130 can optically communicate with the first optical path 131 and the second optical path 132. The light generated by the narrow band light source 130 can be branched into the first portion 133a and the second portion 133b. In some embodiments, the first portion 133a can be transmitted at a group velocity along a first optical path 131 that extends the entire length along the length of the FBG 120. In some embodiments, the second portion 133 b can be transmitted along the second optical path 132 that does not extend along the length of the FBG 120. In some embodiments, the first optical path 131 may be different from the second optical path 132, as shown in FIG. In other embodiments, the first optical path 131 and the second optical path 132 may at least partially overlap each other. The first optical path 131 and the second optical path 132 may cross an empty space or may cross a plurality of optical elements. For example, the first optical path 131 and / or the second optical path 132 may traverse an optical element that is, for example, a fiber coupler as described below. In some embodiments, the light generated by narrowband light source 130 can have a wavelength at or near the wavelength where the product of the group index and the square root of power transmission is highest. This is also fully described below.

  In some embodiments, the optical device 110 can include at least one photodetector 140. Photodetector 140 can be configured to optically communicate with FBG 120. The photodetector 140 can be configured to receive and detect the optical power of the first portion 133a of light, the second portion 133b of light, or both the first portion 133a and second portion 133b of light. . In FIG. 31, the photodetector 140 can receive and detect both the first portion 133a and the second portion 133b of light. In some embodiments, the photodetector 140 may be similar to the photodetector 40 described herein. For example, the photodetector 140 may be a general-purpose low noise photodetector.

  In some embodiments, the optical device 110 can include a first fiber coupler 151 that is in optical communication with the narrowband light source 130, the first optical path 131, and the second optical path 132. As shown in FIG. 31, the light generated by the narrow-band light source 130 can be branched into a first portion 133a and a second portion 133b by a first fiber coupler 151, which is a 3 dB fiber coupler, for example. it can. The first portion 133a may be transmitted along the first optical path 131. The second portion 133b may be transmitted along the second optical path 132. While the first portion 133a can propagate along the FBG 120, the second portion 133b may not substantially interact with the FBG 120. The first portion 133a in this embodiment may include information regarding the perturbation of the FBG 120. In contrast, the second portion 133b in this embodiment may remain unaffected by perturbations.

  The optical sensor 110 may further include a second fiber coupler 152 that is, for example, a 3 dB fiber coupler and is in optical communication with the first optical path 131 and the second optical path 132. The first portion 133 a and the second portion 133 b can be recombined by the second fiber coupler 152 and transmitted to the at least one photodetector 140. As described herein, this recombination allows the first portion 133a and the second portion 133b to interfere with each other and includes information regarding the phase difference between the first portion 133a and the second portion 133b. A combined signal that can be generated is generated. As shown in FIG. 31, the photodetector 140 can comprise a single photodetector at one of the plurality of output ports of the second fiber coupler 152. In some other embodiments, as schematically illustrated in FIG. 7, the photodetector 140 includes a first photodetector 40a and a second detector at one output port of the second fiber coupler 52. A second photodetector 40b at the other output port of the fiber coupler 52 can be provided. In some embodiments, the phase difference can indicate the amount of distortion applied to the FBG 120. In some other embodiments, the phase difference can indicate the temperature of the FBG 120.

ここでＥ_０は狭帯域光源１３０と第１ファイバ結合器１５１への入射によって生成される場であり、√ηは第１光結合器１５１の場結合係数であるか、または、同等にηが第１光結合器１５１のパワー結合係数であり、φ_１およびφ_２はそれぞれ第１光学経路１３１および第２光学経路１３２を通って伝搬する光によって蓄積される位相であり、ｔ_１およびｔ_２はそれぞれ第１光学経路１３１および第２光学経路１３２の場の透過率である。位相の項ｅｘｐ（ｉπ／２）は、光が結合器を通って結合される場合に得る、よく知られたπ／２の位相シフトを構成する。光デバイス１１０の上方出力ポート１５２ａにおいては、Ｅ_１とＥ_２とのコヒーレント和として場が与えられ、出力パワーＰ_ｏｕｔはこの全場の絶対値の平方に比例する。従って、次の通り。

ここでＰ_０は光デバイス１１０の第１ファイバ結合器１５１に入射されるパワーであり、Δφ＝φ_２−φ_１は２つの光学経路１３１、１３２中の２つの信号が受ける位相間の差異である。式（１１）の最後の等式における平方を展開すると次式を与える。

As described above, the light generated by the narrowband light source 130 can have a wavelength at or near the wavelength where the product of the group refractive index and the square root of power transmission is highest. For example, as shown in FIG. 31, the optical device 110 may be an optical sensor operating in a transmissive mode, such as the MZ interferometer described herein. For example, the signal at the output of the optical device, which is an MZ interferometer, is a field E ₁ that is transmitted to the port 152 a by the first optical path 131 and a field E ₂ that is transmitted to the port 152 b by the second optical path 132. It may be a coherent sum. These fields can be written as:

Here, E ₀ is a field generated by incidence on the narrow-band light source 130 and the first fiber coupler 151, and √η is the field coupling coefficient of the first optical coupler 151, or η is equivalently It is a power coupling coefficient of the first optical coupler 151, and φ ₁ and φ ₂ are phases accumulated by light propagating through the first optical path 131 and the second optical path 132, respectively, and t ₁ and t ₂ Are the field transmissivities of the first optical path 131 and the second optical path 132, respectively. The phase term exp (iπ / 2) constitutes the well known π / 2 phase shift obtained when light is coupled through the coupler. Above the output port 152a of the optical device 110, the field is given as a coherent sum of _{E 1} and _{E 2, the} output power _{P out} is proportional to the square of the absolute value of the total field. Therefore:

Here, P ₀ is the power incident on the first fiber coupler 151 of the optical device 110, and Δφ = φ ₂ −φ ₁ is the difference between the phases received by the two signals in the two optical paths 131 and 132. is there. Expanding the square in the last equation of equation (11) gives:

  The first term in the right side of equation (12) is a DC term that is independent of the phase and perturbation of the phase applied to the FBG 120. The second term contains the interference term between the two waves and can therefore contain important phase information.

According to some embodiments, when a perturbation δψ is added to the FBG 120 in the optical device 110 of FIG. 31, this perturbation can induce a change in the phase of the signal propagating through the FBG 120. This phase perturbation can be proportional to the perturbation (for a sufficiently small perturbation) and to the group index of light in the FBG 120, as seen above. So we can write:

The phase difference Δφ in the equation (12) is the sum between a constant term that is an intrinsic phase difference between the two optical paths 131 and 132 and a perturbation δφ of this phase. When this inherent phase difference is selected to be π / 2 (π is modulo), the output power P _out depends to the maximum on a small perturbation δψ. Therefore, the part depending on the phase of the output power (the second term on the right side of Equation (12)) can be written as follows.

式の最も右側のδφを置換するために式（１３）を用いている。 In some embodiments, if the phase perturbation is small (as in an attempt to measure a very small perturbation applied to the FBG 120), sin (δφ) ≈δφ and Equation (14) is It becomes like this.

Equation (13) is used to replace the rightmost δφ in the equation.

Thus, in some embodiments, the output power, which is a signal provided by the optical device 110, eg, an MZ interferometer, is η (1-η) t ₁ t _{2 ng} as a result of perturbations applied to the FBG 120. Proportional. In order to maximize this signal and hence maximize the sensitivity of the optical device 110 according to some embodiments described herein, firstly the product η (1-η) Can be maximized. This can be achieved when η = 0.5. When the first fiber coupler 151 and the second fiber coupler 152 have a power coupling coefficient of 50%, for example, the sensitivity of the optical device 110 that is an MZ interferometer becomes the maximum value. The second item that can be maximized to maximize output power is the product t ₁ t _{2 ng} . This can be achieved first of all by maximizing the transmittance t ₂ of the second optical path 132, for example the reference arm of the MZ interferometer. The second step is to maximize the product t _{1 ng} . The field transmittance t ₁ of the first optical path 131 can be expressed more simply as √T ₁ . Here, T ₁ is the power transmittance of the first optical path 131. Thus, the sensitivity of the optical device of FIG. 31 according to some embodiments can be maximized not only when _ng is the highest, but also when the product _ng √T ₁ is the highest. Therefore, the associated figure of merit that is maximized to maximize the sensitivity of this optical device 110 is F _t = n _g √T ₁ .

FIG. 32 shows an example of a figure of merit that is the product of the square root of the transmission spectrum of FIG. 30 and the group refractive index spectrum of FIG. As explained previously, both T ₁ (plotted in FIG. 30) and _ng (plotted in FIG. 30) are strongly wavelength dependent. Therefore, the figure of merit F _t is also strongly dependent on the wavelength, as shown in FIG. FIG. 32 shows that in this particular apodized FBG example, the figure of merit (or sensitivity) is not maximum at the wavelength of the first slow light peak. As shown in FIG. 30, _ng is the highest at the first slow light peak, but T ₁ is very weak in this example. On the other hand, at the second peak, _ng is only slightly smaller, but T ₁ is significantly higher. Therefore, in this example, _Ft is higher at the second peak than at the first peak. In this example, for the subsequent slow light peak, _ng decreases more than T ₁ increases. Therefore, the figure of merit at these other peaks is lower than that at the second peak. The net result is that operating at the second peak (wavelength λ ₂ ) gives the highest figure of merit in this particular example. Therefore, the most sensitive optical device 110 is obtained. Plotting diagrams such as FIGS. 30 and 32 can be useful in finding the optimal wavelength that operates at the maximum sensitivity for a particular FBG 120.

Plotting diagrams such as FIGS. 30 and 32 may not be necessary in some embodiments. For example, in some embodiments using a uniform grating, the peaks in the transmission spectrum all have a maximum value of 1. The wavelength of maximum sensitivity is determined by the group index spectrum, which roughly corresponds to the wavelengths λ _j and λ ′ _j . Therefore, the wavelength that gives the highest sensitivity in these examples is the wavelength that maximizes the group index, which is λ ₁ and λ ′ ₁ . However, to determine the bandwidth in some embodiments of the slow light sensor, or to determine the maximum perturbation equivalent that can be detected, the spectrum of the figure of merit F can be plotted. Can be advantageous because it contains the spectrum of the detected signal and hence bandwidth information.
b. Reflection configuration

  Some embodiments described herein utilize FBGs used in slow light reflection mode, for example as shown in FIG. The optical device 10 can include a narrowband light source 30 that is in optical communication with the first optical path 31 and the second optical path 32. The narrowband light source 30 can be configured to generate light configured to be branched into a first portion 33a and a second portion 33b by the FBG 20. When functioning as a wavelength, the slope of the product of the group index spectrum and 1 minus the power transmission spectrum is the maximum (eg, for any other wavelength near the edge of the band gap of the FBG 20) The light can have a wavelength at or near such a wavelength. In some embodiments, the light has a maximum slope of the product of the group index spectrum and the power transmission spectrum as a function of wavelength (eg, any other wavelength near the edge of the band gap of the FBG 20). The wavelength can be at or near such a wavelength.

  As further described above, some embodiments of the optical device 10 shown in FIG. 8 may further comprise at least one photodetector 40a and / or 40b. The optical device 10 of FIG. 8 can also include at least one fiber coupler 51.

  In some embodiments, the light generated by the narrowband light source 30 can be split by the FBG 20 into a first portion 33a and a second portion 33b. In some embodiments, at least one photodetector 40a and / or 40b is configured to receive a first portion 33a, a second portion 33b, or both first and second portions 33a, 33b of light. can do.

  In some embodiments, the FBG 20 is examined by the narrowband light source 30, and the light generated by the narrowband light source 30 may be split by the FBG 20 into a first portion 33a and a second portion 33b. The wavelength of the light for investigating the FBG 20 can be set to a wavelength at which the slope of the product of the group refractive index when 1 is a function of the wavelength and 1 minus power transmission is at or near the maximum . A more complete description is given below. In some embodiments, the wavelength of the light that examines the FBG 20 is at or near a wavelength that maximizes the slope of the product of the group index and power transmission as a function of wavelength. Can do. A more complete description is given below.

As explained above, the reflection peak can shift the wavelength when an external perturbation is applied to the FBG 20. This shift in λ _B is a change in the first portion 33a of the light transmitted by the FBG 20 and the second portion 33b of the light reflected by the FBG 20, for example, the power of the light reflected at the wavelength of the light incident on the FBG 20 Can bring about changes.

For reflection mode operation (e.g., FIG. 8), the output power of the optical device 10 in accordance to some embodiments, for example, as the transmission mode of FIG. 31, proportional to n _g to that power reflection coefficient times FBG20 To do. When the output power is the power reflected by the FBG 20, the power reflection coefficient is R ₁ = 1−T ₁ . Thus, the relevant figure of merit is F _r = (1−T ₁ ) _ng . If the output power is the power transmitted by the FBG 20, the associated figure of merit is F ′ _r = T _{1 ng} . Depending on which output is measured as the output of the optical device 10, a plot similar to FIG. 32 can be plotted to obtain either F _r or F ′ _r spectra. Therefore, the operating wavelength that maximizes the sensitivity is given by the wavelength at which the slope of the figure of merit becomes the maximum value.
c. Additional examples

  FIG. 33A is a flowchart of an exemplary method 3000 using a fiber Bragg grating, according to some embodiments described herein. The method 3000 comprises providing the FBG 20 with a refractive index perturbation that is substantially periodic along the length of the FBG 20, as shown in operation block 3010. The FBG 20 can have a power transmission spectrum having a plurality of local transmission maximums. Each local transmission maximum has a maximum power at the transmission peak wavelength. The method 3000 also includes generating light, as shown in operation block 3020 of FIG. 33A. Light can be generated by a narrow band light source 30. In some embodiments, narrowband light source 30 is in optical communication with first optical path 31 and second optical path 32. The light is also split into a first portion 33a of light and a second portion 33b of light. The method 3000 may further comprise transmitting the first portion 33a of light at a group velocity along the first optical path 31 that extends the entire length along the length of the FBG 20 at operation block 3030. In some embodiments, the light has a maximum product of the group index and the square root of power transmission (eg, higher than for any other wavelength near the edge of the band gap of the FBG 20). The wavelength can be at or near the wavelength.

  Some embodiments of method 3000 include receiving light first portion 33a, second portion 33b, or both first portion 33a and second portion 33b using photodetector 40; Detecting the optical power of the first portion 33a, the second portion 33b, or both the first portion 33a and the second portion 33b. The method 3000 may further comprise transmitting the second portion 33b of light along the second optical path 32. The second optical path 32 may not extend along the entire length along the FBG 20. In various embodiments, the method 3000 can further include providing a first fiber coupler 51 in optical communication with the narrowband light source 30, the first optical path 31, and the second optical path 32. The method 3000 can further include providing a second fiber coupler 52 in optical communication with the photodetector 40, the first optical path 31, and the second optical path 32.

  In some embodiments of the method 3000, the method 3000 may further comprise recombining the first and second portions of light 33a, 33b and transmitting them to the photodetector 40. In these embodiments, the method 3000 may include splitting the light generated by the narrowband light source 30 into a first portion 33a and a second portion 33b by the first fiber coupler 51. Thus, in these embodiments, the recombination and transmission steps can be accomplished by the second fiber coupler 52. Further, the detecting step in these embodiments may include a step of detecting a phase difference between the first portion 33a and the second portion 33b. In some embodiments, the first optical path 31 and the second optical path 32 may form a nominally balanced Mach-Zehnder interferometer. In some embodiments, the phase difference can indicate the amount of distortion applied to the FBG 20. In other embodiments, the phase difference can indicate the temperature of the FBG 20.

  In some embodiments of the method 3000, the substantially periodic refractive index perturbation can have a constant period along the length of the FBG 20. In some other embodiments, the substantially periodic refractive index perturbation can have a period that varies along the length of the FBG 20, such that the FBG 20 is a chirped grating. In some embodiments, the substantially periodic refractive index perturbation can have an amplitude that varies along the length of the FBG 20 such that the FBG 20 is an apodized grating.

  FIG. 33B is a flowchart of another embodiment of a method 4000 using a fiber Bragg grating in accordance with certain embodiments disclosed herein. The method 4000 may comprise providing the FBG 20 with a refractive index perturbation that is substantially periodic along the length of the FBG 20, as shown in operation block 4010. In some embodiments of the method 4000, the method 4000 may comprise generating light having a certain wavelength, as shown in operation block 4020. Light can be generated from a narrow band light source 30. In some embodiments, the narrowband light source 30 can be in optical communication with the first optical path 31 and the second optical path 32. The light can be branched into a first portion 33a of light and a second portion 33b of light. The method 4000 may further include transmitting the first portion of light 33a at a group velocity along the first optical path 31 that extends the entire length along the length of the FBG 20 at operation block 4030. In some embodiments, the light has a maximum slope of the product of the group index and 1 minus the power transmission as a function of wavelength (eg, near the edge of the band gap of the FBG 20). It has a wavelength at or near such a wavelength (which is the maximum compared to this amount of value for other wavelengths).

  Some embodiments of the method 4000 include receiving the first portion 33a, the second portion 33b, or both the first portion 33a and the second portion 33b using the photodetector 40, and the light Detecting the optical power of the first portion 33a, the second portion 33b, or both the first portion 33a and the second portion 33b. The method 4000 may further comprise reflecting the second portion 33b of light along the second optical path 32.

  In some embodiments of the method 4000, the substantially periodic refractive index perturbation has a constant period along the length of the FBG 20. In some other embodiments, the substantially periodic refractive index perturbation has a period that varies along the length of the FBG 20 such that the FBG 20 is a chirped grating. In some embodiments, the substantially periodic refractive index perturbation has an amplitude that varies along the length of the FBG 20 such that the FBG 20 is an apodized grating.

  FIG. 33C is a flowchart of another embodiment of a method 5000 using a fiber Bragg grating in accordance with certain embodiments described herein. The method 5000 may comprise providing the FBG 20 with a refractive index perturbation that is substantially periodic along the length of the FBG 20, as shown in operation block 5010. In some embodiments of the method 5000, the method 5000 may comprise generating light having a wavelength, as shown in operation block 5020. Light can be generated from a narrow band light source 30. In some embodiments, the narrowband light source 30 can be in optical communication with the first optical path 31 and the second optical path 32. The light can be branched into a first portion 33a of light and a second portion 33b of light. The method 5000 may further include, at operation block 5030, transmitting the first portion 33a of light at a group velocity along a first optical path 31 that extends the entire length along the length of the FBG 20. In some embodiments, the light has a maximum slope of the product of the group index and power transmission as a function of wavelength (eg, this amount for any other wavelength near the edge of the band gap of the FBG 20). The wavelength is at or near the wavelength that is the maximum value compared to the value of.

  Some embodiments of the method 5000 include receiving the first portion 33a, the second portion 33b, or both the first portion 33a and the second portion 33b using the photodetector 40, and the light Detecting the optical power of the first portion 33a, the second portion 33b, or both the first portion 33a and the second portion 33b. The method 5000 may further comprise reflecting the second portion 33b of light along the second optical path 32.

In some embodiments of the method 5000, the substantially periodic refractive index perturbation has a constant period along the length of the FBG 20. In some other embodiments, the substantially periodic refractive index perturbation has a period that varies along the length of the FBG 20 such that the FBG 20 is a chirped grating. In some embodiments, the substantially periodic refractive index perturbation has an amplitude that varies along the length of the FBG 20 such that the FBG 20 is an apodized grating.
Example

FIG. 24 shows an exemplary experimental setup used to measure FBG transmission and group index spectra. This optimization using a figure of merit results in a nominal Δn of 1.035 × 10 ⁻³ , a length L = 1.2 cm, an estimated loss factor of 1.16 m ⁻¹ , and FWHM W = 0.9 cm This was performed by testing FBGs having a Gaussian apodized refractive index profile with Group delay and group refractive index were measured as described above in connection with FIG. 24 and equations (8) and (9).

FIG. 34A shows the measured transmission spectrum, and FIG. 34B shows the measured group index spectrum of the signal transmitted by the FBG. The corresponding solid curve is the theoretical prediction calculated using the model after adjusting Δn, W, and α so that they are best fitted to the experimental spectrum. The fitted values are Δn = 1.35 × 10 ⁻³ , W = 0.9 cm, and α = 1.16 m ⁻¹ . The former agrees well with the manufacturer's estimate. Both measured spectra show multiple ripples that are in good agreement with theory. The maximum value of the measured group refractive index occurs in the vicinity of the second transmission peak (λ≈15499.6976 nm) and is equal to 127. The transmittance at this wavelength is 0.8%. This is one of the slowest group velocities (2,362 km / s) reported so far in either FBG or optical fiber. The peak closest to the band edge (λ≈1.54974 μm) should have an even higher group index (˜217), but in this example the transmitted power at this peak is too small to be detected. It was not possible to measure.

  The FBG was mounted on a piezoelectric (PZT) ring to apply controlled and calibrated strain to the FBG. An AC voltage was applied to the ring, which applied a sinusoidal stretch to the FBG. Then, as shown in FIG. 35, an FBG was placed in the MZ interferometer to test its performance as a slow light sensor. To minimize the laser phase noise being converted by the MZ interferometer into intensity noise that would increase the minimum detectable distortion, the two arms of the MZ interferometer are of the same length. Care was taken to ensure that it has According to estimation from independent interferometry, the difference in arm length was 1-3 mm. A voltage of known amplitude was applied to the PZT at a frequency ω = 25 kHz by a function generator. The output of the MZ interferometer was measured with an output detector. This output includes a slowly varying component that is induced by a slowly varying phase difference between the two arms due to temperature variations. This fluctuating signal was sent to a proportional integral derivative (PID) controller. In order to offset this slow drift, the latter applied just the right voltage to the second PZT located in the lower arm. The purpose of this closed loop is to stabilize the interferometer and, as previously explained, its phase bias (if there is no modulation applied to the FBG / sensing arm to maximize sensitivity) The phase difference between the two arms of) was kept equal to π / 2. Another portion of the detector output was sent to a lock-in amplifier that extracted the output component modulated by ω from this portion. This output component was a sensor signal. By varying the wavelength of the tunable laser, the sensor's response to the same perturbation (distortion) amplitude and frequency applied to the FBG as a function of wavelength can be measured, as in this example.

  The PZT to which the FBG is attached is pre-calibrated using conventional techniques. A known length of fiber was wrapped around it. This fiber was placed inside the MZ interferometer. The interferometer was then used to measure the amount of phase shift that occurred in the fiber as a function of the voltage applied to the PZT.

ここでｄＰ_ｏｕｔは、歪みｄεの変化の結果生じるＭＺ干渉計の出力におけるパワーの変化である。Ｓは歪みの逆数を単位としている。式（１５）を用いると、感度は次の式に比例することがわかった。

ここでＴ_１およびＴ_２はそれぞれＦＢＧアームおよび参照アームのパワー透過率である。式（１７）に示されていない比例係数は、材料の屈折率が歪みに対してどのように依存するかを表す材料パラメータに依存する（例えば、H. Wen, M. Terrel, S. Fan, and M. Digonnet, "Sensing with slow light in fiber Bragg gratings," Sensor Journal, IEEE Vol. 12, Issue 1, 156-163 (2012)に記載されており、その内容全体をここに参照により組み込む）。式（１７）は、ＭＺベースの低速光ＦＢＧセンサについて、与えられたＦＢＧセンサに対する感度の最大値に最適な波長を選択する（例えば、

の最大の値を有するピークを選択する）ための性能指数として使用することができる。 Based on the above, the sensitivity of the exemplary sensor of FIG. 35 has a spectrum proportional to F _t (λ) = n _g (λ) √T ₁ (λ). Here, _ng (λ) is the measured group refractive index spectrum (for example, FIG. 34B), and T ₁ (λ) is the transmission spectrum of the FBG (for example, FIG. 34A). The spectrum F _t (λ) calculated from these two measured spectra can be plotted, for example, as shown in FIG. As previously demonstrated in connection with FIG. 32, the sensitivity is expected to reach a maximum near peak j = 3 (the first peak j = 1 is too weak as pointed out previously. Not shown because it was not measured). Sensor sensitivity to strain is defined as:

Here, dP _out is the change in power at the output of the MZ interferometer resulting from the change in strain dε. S is in units of reciprocal distortion. Using equation (15), the sensitivity was found to be proportional to:

Here, T ₁ and T ₂ are the power transmittances of the FBG arm and the reference arm, respectively. Proportional coefficients not shown in equation (17) depend on material parameters that describe how the refractive index of the material depends on strain (eg, H. Wen, M. Terrel, S. Fan, and M. Digonnet, “Sensing with slow light in fiber Bragg gratings,” Sensor Journal, IEEE Vol. 12, Issue 1, 156-163 (2012), the entire contents of which are incorporated herein by reference). Equation (17) selects the optimum wavelength for the maximum sensitivity for a given FBG sensor for an MZ-based slow light FBG sensor (eg,

Can be used as a figure of merit to select the peak with the largest value of

(For example, H. Wen, M. Terrel, S. Fan, and M. Digonnet, "Sensing with slow light in fiber Bragg gratings," Sensor Journal, IEEE Vol. 12, Issue 1, 156-163 (2012). Inserting the complete equation for δφ into equation (15) yields the following equation for sensitivity to silica fiber:

の最大の値を有するＦＢＧセンサを選択する場合、あるいは、

の大きな値を有するピークを持つようなＦＢＧセンサを設計する場合）に、積

を、性能指数として使用することができる。 Equation (18) can be used to provide a figure of merit that characterizes the absolute and relative performance of various MZ-based FBG sensors. According to equation (18), the sensitivity is maximum when η = 0.5. For some embodiments utilizing this MZ scheme operating in a given wavelength range (eg around 1.5 μm), when comparing multiple MZ-based slow light FBG sensors, or as an FBG sensor When designing (for example,

Select the FBG sensor with the largest value of, or

When designing an FBG sensor having a peak with a large value of

Can be used as a figure of merit.

である。ここでＴ_１はやはりＦＢＧを含むアームのパワー透過であり、従って、全ての意図および目的に対し、動作波長におけるＦＢＧの透過率である。 In fact, we generally strive to maintain the transmittance of the reference (eg, lower) arm at a maximum value, and this maximum value is practically independent of the characteristics of the FBG contained in the other (upper) arm. The practical figure of merit for an MZ-based scheme operated in a specific wavelength range is

It is. Where T ₁ is also the power transmission of the arm containing the FBG, and is therefore the transmission of the FBG at the operating wavelength for all intentions and purposes.

の最大の値を有するＦＢＧを選択する場合、あるいは、

の大きな値を有するピークを持つようにＦＢＧを設計する場合）に、動作の波長において評価される積

を性能指数として使用することができる。同等に、τ_ｇ＝ｎ_ｇＬ／ｃなので、等価な性能指数は

である。式（１８）は、小さな歪みに対するＦＢＧの感度を最大化するために、動作の波長で評価される積

を最大化できるということを述べている。例えば式（１８）は、ＭＺベースの構成にて使用される特定のＦＢＧにおいて、どのピークおよびどの動作波長が感度の最大値（あるいは特定の目標値を有する感度）を与えるかを選択するために使用することができる。与えられるＦＢＧに対して長さＬは固定されているので、与えられるＦＢＧに対して積

を性能指数として使用することができる。 For some embodiments utilizing this MZ scheme, for FBGs to be used for a given sensor in a given wavelength range, when comparing multiple FBGs to be used in an MZ-based sensor Or when designing an FBG to be used to have a maximum sensitivity (eg,

Select the FBG with the largest value of, or

The product evaluated at the wavelength of operation when the FBG is designed to have a peak with a large value of

Can be used as a figure of merit. Equivalently, since τ _g = n _g L / c, the equivalent figure of merit is

It is. Equation (18) is a product evaluated at the wavelength of operation to maximize the sensitivity of the FBG to small distortions.

It can be maximized. For example, equation (18) can be used to select which peak and which operating wavelength gives the maximum sensitivity (or sensitivity with a specific target value) for a particular FBG used in an MZ-based configuration. Can be used. Since the length L is fixed for a given FBG, the product for the given FBG

Can be used as a figure of merit.

Two parameters τ _g and T ₁ (described for example by H. Wen et al., The entire disclosure of which is hereby incorporated by reference) are calculated or measured, and these values are then expressed in equation (18) Equation (18) may also be used to calculate or predict the maximum sensitivity for small distortions of a specific FBG with a specific set of parameters by substituting and predicting the maximum sensitivity. it can.

The results of the strain sensing experiment using the settings of FIG. 35 are also plotted in FIG. 36 in the form of sensitivity measured at the four observed slow light peaks. The measured sensitivity agrees reasonably well with the value shown by the spectrum. Again, this was predicted from the measured group index and transmission spectrum of the grating. Thus, in some embodiments, the sensitivity of the slow light sensor according to some embodiments in transmissive mode, as predicted by theory, is a figure of merit F _t (λ) = n _g (λ) √ Estimated by T ₁ (λ). Furthermore, slow light can play a significant role in increasing the sensitivity of the sensor to strain in some embodiments. The maximum value of the measurement shown in FIG. 36 is about 3.14 × 10 ⁵ strain ⁻¹ , probably the largest reported so far for a strain sensor. Subsequent work (H. Wen et al.) Showed that this MZ-based approach provides a strain sensor with a minimum detectable strain of 880 fε√Hz. In MZ-based schemes, the MZ interferometer is kept square to maximize sensitivity. This means stabilizing the MZ interferometer. The probe laser wavelength is also locked to the slow light peak of the FBG to stabilize the strain sensitivity over time. In contrast, in some embodiments, the transmission and reflection techniques described herein do not utilize additional interferometers. They are therefore more stable in temperature. However, they can also achieve increased sensitivity.

The minimum distortion that can be measured by the sensor of FIG. 35 according to some embodiments can be calculated from the sensitivity. For the power used in our measurements (the average detected power is P ₀ = 36 μW), the total noise at the sensor output (measured in the absence of added distortion) is about 1 at 3 kHz. From 0 μV / √Hz to about 0.45 μV / √Hz at 30 kHz, or after calibration, noise power P _noise ≈25 pW to 11 pW. At this output power level, this noise consisted of laser relative intensity noise (RIN) and photodetector noise at higher frequencies, and dominant lock-in amplifier noise at lower frequencies. As mentioned earlier, since the MZ interferometer is almost balanced, the phase noise of the laser is a negligible component and as a result, the phase noise has not been converted to significant intensity noise. Minimum distortion detectable is a distortion as just generating a variation equal to the noise in the output power P _out. From the sensitivity definition (Equation (16)), the minimum detectable distortion (MDS) can be written as:

Maximum sensitivity measured at the measured noise power (25 pW), input power (36 μW), and most sensitive peak (j = 3, see FIG. 36) 3.14 × 10 ⁵ strain ⁻¹ Thus, the MDS is ˜2.2 picostrain (pε) at 3 kHz and ˜1 picodistortion at 30 kHz. As a comparison, 5 pε MDS at frequencies greater than 100 kHz has been reported in π-shifted FBGs operated by reflection (D. Gatti, G. Galzerano, D. Janner, S. Longhi, and P. Laporta, "Fiber strain sensor based on a π-phase-shifted Bragg grating and the Pound-Drever-Hall technique," Optics Express, Vol. 16, No. 3, 1945-50 (Feb. 4, 2008)). Both of these devices are passive. These two references would be the smallest detectable distortion reported prior to this application for passive sensors based on FBG. References such as Gatti et al. Do not mention "slow light" at all, they just speculate whether they used slow light. In KP Koo and AD Kersey's "Bragg grating-based laser sensors systems with interferometric interrogation and wavelength division multiplexing," J. Lightwave Technol., Vol. 13, No. 7, 1243-49 (July 1995), 56 femto distortion (Femtostrain) (or 0.056 pε) MDS has been reported in fiber lasers using FBG as a reflector. The sensor is an active device that requires pump power (~ 70mW) and a strong unbalanced MZ interferometer (~ 100m difference in arm length), which makes the MZ interferometer used for readout external temperature fluctuations It was very difficult to stabilize. In contrast, devices according to some embodiments described herein use much lower power (only 36 μW) and their MZ interferometer is nominally balanced. This implies that it is much more thermally stable.

FIG. 37 shows the full width at half maximum of the bandwidth of the first slow light peak predicted from the model as a function of refractive index contrast in a uniform grating with length L = 2 cm and substantially no loss. This figure shows that the bandwidth in this example ranges from about 20 pm for a weak grating (refractive index contrast 10 ⁻⁴ ) to about 1 fm for a strong grating (refractive index contrast 1.5 × 10 ⁻² ). It shows that. For some embodiments, at the FBG example's Bragg wavelength of 1.55 μm, the line width of the laser used to probe the sensor should be less than this bandwidth value; Depending on the refractive index contrast of the grating, it should be on the order of 1 fm (or lower) to 10 pm (or higher). Such lasers are readily available from a number of manufacturers, such as Redfern Integrated Optics Inc. (RIO) and Agilent Technologies, both headquartered in Santa Clara, California. This bandwidth depends on a number of parameters, and is advantageously calculated or experimentally measured to determine the line width of the laser to be used in some embodiments of the sensor. I can do it. FIG. 37 shows a description of a process that could be used to determine this line width.

FIG. 38 shows the sensitivity to strain as a function of refractive index contrast for length L = 2 cm and lossless FBG used in slow light transmission mode (upper side), according to some embodiments described herein. ) And the conventional reflection mode (lower solid line) by the MZ process. As shown in FIG. 38, for the conventional FBG sensor, as the refractive index contrast increases, the reflection peak spreads and the resolution becomes worse. On the other hand, in either low-speed light sensor, as the refractive index contrast (and group refractive index) increases, the sensitivity can increase to, for example, 30,000 times higher sensitivity. This improvement can be orders of magnitude, even for a moderate L, for example a few centimeters, and a moderate refractive index contrast (eg 10 ⁻³ ). FIG. 38 shows a numerical example different from FIG. 11B, and models the sensitivity of strain opposite to the temperature sensitivity of FIG. 11B.

FIG. 38 is a function of refractive index contrast for a length L = 2 cm and lossy FBG used in slow light transmission mode (dotted and dashed), according to some embodiments described herein. The relationship between the sensitivity to distortion is also shown. As shown in FIG. 38, if, for example, even has a loss such as loss of range of 2m ^-1 from 0.02 m ^-1, some embodiments of the sensor described herein, An improvement in sensitivity of 14-1000 can be expected.

Some examples have demonstrated that very slow light is sustained in the FBG. The group index can be increased dramatically by the length L of the FBG, for example L ^2.9 , and by the index contrast, for example Δn ^1.8 . Certain apodized FBGs can provide slower light. In addition, values of 10,000 and higher have been predicted for low loss FBGs. For example, compared to about 5 in silica fiber and about 10 in Bragg fiber, the maximum group index of 127 reported in an optical fiber, eg, 1.2 cm FBG has been shown as described herein. It was. This value of group index corresponds to a group velocity as low as 2,360 km / s. Thus, some embodiments of the FBG sensor described herein that utilize slow light have increased sensitivity, such that the minimum detectable distortion is, for example, about 1 pε in a passive FBG sensor.
Further discussion of transmission and reflection schemes

ここでＱは、共振ピークの線幅に対する周波数の比で定義され、共振の鋭さの基準である。線幅が減少するに連れて、群遅延は増大する。透過スキームおよび反射スキームにおいて、ＦＢＧを探査するレーザの周波数は、これらのピークの１つにおける２つの急峻な縁部の一方に、またはその近傍に調整される。例えば、波長に対する透過率の微分係数が局所的な最大値となるような波長である。図３９は、λ_１、λ_２、λ_３等、およびλ'_１、λ'_２、λ'_３、等のようないくつかの波長を示しており、これらにおいては、透過ピークまたは反射ピークが局所的に最も急峻であり、歪みまたはその他の外部摂動に対する感度が局所的に最大である。ＦＢＧが歪まされた場合、あるいはその温度が変えられた場合、そのスペクトルは変形しないが、歪みまたは温度変化に比例する量だけ波長空間においてシフトする。探査波長での透過スペクトルにおける傾きの急峻さのため、透過スペクトルのこのシフトは、図３９から容易にわかるように、透過されるパワーに大きな変化を生成する。 FIG. 39 shows the calculated transmission and group index spectra of a typical uniform FBG having a length of 5 mm, a refractive index contrast of 5 × 10 ⁻³ , and a propagation loss of 0.02 m ⁻¹ at 1.55 μm. Indicates. This indicates that there is a resonance peak in both the spectrum of group delay τ _g and the transmittance (or reflection) spectrum at the edge of the FBG photonic band gap (centered at Bragg wavelength λ _B ). . As shown in FIG. 39, the power transmission spectrum has a plurality of resonance peaks, each peak having a local maximum value and a slope of 0 on each side of the local maximum value. (Ie, the local maximum is between the two regions where the slope is not zero). At least a portion of one of the non-zero slope regions has a steep edge with a large positive slope as a function of wavelength (or frequency). At least a portion of the other region has a sharp edge with a large negative slope as a function of wavelength (or frequency). The peak of the power transmission spectrum and the peak of the group refractive index spectrum occur at almost the same frequency (or wavelength). Therefore, the resonant wavelength, the peak transmission factor T _o and a peak group delay tau _g characterized _(or equivalently group velocity, or equivalently group index) by. Both peaks (transmission or reflection and group delay) have essentially the same frequency line width Δω. In general, τ _g is related to the quality factor Q = ω _res / Δω of the interferometer (T. Sunner, M. Gellner, A. Loffler, M. Kamp, and A. Forchel, “Group delay measurements on photonic crystal resonators, "Appl. Phys. Lett. 90, 151117 (2007)).

Here, Q is defined by the ratio of the frequency to the line width of the resonance peak, and is a reference for the sharpness of resonance. As the line width decreases, the group delay increases. In transmission and reflection schemes, the frequency of the laser probing the FBG is tuned to or near one of the two sharp edges at one of these peaks. For example, the wavelength is such that the differential coefficient of transmittance with respect to the wavelength becomes a local maximum value. FIG. 39 shows several wavelengths, such as λ ₁ , λ ₂ , λ ₃ , and λ ′ ₁ , λ ′ ₂ , λ ′ ₃ , etc., in which the transmission or reflection peak is Locally steepest and locally sensitive to strain or other external perturbations. If the FBG is distorted or its temperature is changed, its spectrum will not be deformed, but will shift in the wavelength space by an amount proportional to the strain or temperature change. Due to the steepness of the slope in the transmission spectrum at the exploration wavelength, this shift in the transmission spectrum produces a large change in the transmitted power, as can easily be seen from FIG.

A shift in either the power transmission or power reflection spectrum resulting from the applied perturbation at the operating or exploration wavelength can be measured as a change in power at the output. 40 and 41 schematically illustrate example configurations in a transmission scheme and a reflection scheme, respectively, as described above in connection with FIGS. 6A-6B and FIG. In transmission schemes (eg, FIGS. 6A-6B and 40), light at the operating wavelength is emitted into the FBG and the transmitted power is measured. For example, when a perturbation that is a distortion is applied to the FBG, the transmitted power changes. The magnitude of the perturbation can be taken from the measured quantity. In reflection schemes (eg, FIGS. 8 and 41), light is emitted to the FBG through a fiber coupler or fiber circulator. The reflected signal returning from the FBG is measured at the second input port of the coupler or the third port of the isolator. As in the transmission scheme, when a perturbation is applied to the FBG, the reflected power changes.
Sensitivity modeling

  The optical device can be designed to have the maximum achievable sensitivity with an appropriate figure of merit for the FBG in a particular configuration of the optical device. As described in more detail below and with reference to FIGS. 40 and 41, in some embodiments, the optical device 210 comprises an FBG 220 and a narrowband light source 230 (eg, a tunable narrowband light source). The FBG 220 has a refractive index modulation that is substantially periodic along the length of the FBG 220. The FBG 220 has a power reflection spectrum as a function of wavelength, a power transmission spectrum as a function of wavelength, and a group delay spectrum as a function of wavelength. The power transmission spectrum as a function of wavelength has one or more resonance peaks, each of which has a local maximum and two regions with non-zero slopes between the two regions. Has a local maximum. Narrowband light source 230 is in optical communication with FBG 220 and is configured to transmit light to FBG 220 such that a portion of the light is transmitted along the length of FBG 220 and a portion of the light is reflected from FBG 220. Is done. The optical device has at least one light detection configured to detect light power of the light transmissive portion, the light reflected portion, or both the light transmissive portion and the light reflected portion. A vessel is further provided. The light from the narrow-band light source 230 has a wavelength in a region where the inclination at one resonance peak of one or a plurality of resonance peaks is not zero. The resonance peak is selected such that one or more of the next quantities evaluated at the resonance peak is at a maximum value. That is, (a) the product of the group delay spectrum and the power transmission spectrum, and (b) the product of the group delay spectrum and 1 minus the power reflection spectrum. For example, the wavelength is such that the derivative of the power transmission spectrum with respect to the wavelength is a local maximum for the selected resonance peak.

  In the transmission scheme schematically illustrated in FIG. 40, a portion of the light transmitted along the length of the FBG 220 is detected by at least one photodetector 240 (eg, one or more photodetectors). . In the reflection scheme schematically illustrated in FIG. 41, a portion of the light reflected from the FBG 220 is detected by at least one photodetector 240 (eg, one or more photodetectors).

ここでＴ（λ）は、探査する波長（例えばλ_１）におけるＦＢＧの波長依存性パワー透過であり、εは加えられる歪みである。微分係数ｄＴ／ｄεは次のように表される。

ここでｄλ／ｄεは、加えられる小さな歪みｄεによるＦＢＧの透過スペクトル中のシフトである。例えば、シリカファイバーにおいてはｄλ／ｄε＝０．７９λ_Ｂである（A.D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, "Fiber Grating Sensors," J. of Lightwave Technol. Vol. 15, No. 8, 1442-1463 (1997)を参照のこと）。式（２１）と式（２２）とを結合すると、シリカＦＢＧに対する歪みの感度は次のようになる。

Distortion sensitivity (input, for example, a configuration having at least one photodetector 240 configured to detect the optical power of a portion of the light that is schematically illustrated by FIG. 40 and transmitted through the FBG 220) Normalized to power) is defined as:

Here, T (λ) is the wavelength-dependent power transmission of the FBG at the wavelength to be searched (for example, λ ₁ ), and ε is the applied strain. The differential coefficient dT / dε is expressed as follows.

Here, dλ / dε is a shift in the transmission spectrum of the FBG due to the small strain dε applied. For example, in silica fiber, dλ / dε = 0.79λ _B (AD Kersey, MA Davis, HJ Patrick, M. LeBlanc, KP Koo, CG Askins, MA Putnam, and EJ Friebele, “Fiber Grating Sensors,” J of Lightwave Technol. Vol. 15, No. 8, 1442-1463 (1997)). Combining Equation (21) and Equation (22) gives the following strain sensitivity to silica FBG.

  The first conclusion of this analysis is that the sensitivity spectrum of a given FBG can be easily calculated by simply taking the derivative with respect to the wavelength of the measured or calculated transmittance T (λ). is there.

In order to relate the transmission derivative to the group index, and to clearly show the dependence of sensitivity on group delay (or group index), the spectrum of the transmission resonance peak is expressed as the resonance wavelength of λ _o and Δλ. Can be modeled as Lorentzian linear with FWHM.

である場合に最大化されるべきことが容易に示される。これは、透過ピークの傾きが最大値となる、すなわち、センサの感度が名目上最大値となるような波長である。この波長における微分係数は

である。 The differential coefficient dT / dλ is

Is easily shown to be maximized. This is the wavelength at which the slope of the transmission peak has a maximum value, that is, the sensor sensitivity has a nominal maximum value. The derivative at this wavelength is

It is.

By substituting Δλ expressed as a function of τ _g from equation (20) into this last equation and substituting the resulting differential coefficient dT / dλ into equation (23), the maximum distortion sensitivity You can get a simple expression for.

Performance index of the FBG for maximum sensitivity attainable in transmission scheme, therefore, at a particular wavelength range of operation, the product of the power transmission spectrum _{T o} of the group delay spectrum tau _g and FBG of FBG (τ _g T _o) is there. Equivalently, since τ _g = _ng L / c, the equivalent figure of merit is the product of the group index spectrum, power transmission spectrum, and length ( _ng LT _o ). Here, _ng is the group index of light in the FBG at the operating wavelength, and L is the length of the FBG. To maximize the sensitivity of the FBG to small distortions in transmission scheme, that a product tau _g T _o to be evaluated at the peak wavelength of the low-speed optical resonance wavelength of operation is located can be maximized formula ( 25) states. Equivalently, the product n _g LT _o evaluated at the peak wavelength of the slow optical resonance where the operating wavelength is located can be maximized.

ここでｄＰ_ｒｅｆは、加えられる歪みεによって生じる反射パワー中の変化であり、Ｒ（λ）はパワー反射スペクトルである。損失の無いＦＢＧにおいては、透過されるパワーと反射されるパワーとの合計は１であり、Ｒ（λ）＝１−Ｔ（λ）である。従って、低損失極限においては、どちらのスキームにおいても感度は（負の符号であることを除いて）同一である。

伝搬損失が存在する場合感度は異なるものの、通常の場合がそうであるように、損失が小さい場合はほんのわずかに異なるのみである。反射スペクトルの形状は、共振の中心部に大きさＲ_ｏの窪みを有する反転ローレンツ線形である。従って、反射スキームにおける感度の最大値は、シリカファイバーについては次の通り。

Distortion sensitivity in a reflection scheme (eg, a configuration having at least one photodetector 240 configured to detect the optical power of the reflected portion of the light reflected by the FBG 220 and schematically illustrated in FIG. 41). The corresponding derivative for is similar to that in the transmission scheme. Normalized sensitivity is defined as:

Where dP _ref is the change in reflected power caused by the applied strain ε, and R (λ) is the power reflection spectrum. In an FBG without loss, the sum of transmitted power and reflected power is 1, and R (λ) = 1−T (λ). Thus, in the low loss limit, the sensitivity is the same (except for the negative sign) in both schemes.

The sensitivity is different when there is a propagation loss, but only slightly different when the loss is small, as is the normal case. The shape of the reflection spectrum is inverted Lorentz lineshape having a recess size R _o in the center of the resonance. Therefore, the maximum sensitivity in the reflection scheme is as follows for silica fibers.

  If there is a loss, the sensitivity spectrum can be simply calculated instead by taking the derivative dR (λ) / dλ of the measured reflection spectrum, as shown in equation (26).

Therefore, the figure of merit for the maximum sensitivity achievable in the reflection scheme subtracted the FBG power reflection spectrum from the FBG group delay spectrum τ _g and 1 evaluated at the peak of the slow optical resonance considered (1 -R _o ) (τ _g (1-R _o )). Equivalently, since τ _g = n _g L / c, the equivalent figure of merit is the group refractive index spectrum that is evaluated both at the peak of the low-speed optical resonance considered, and 1 minus the power reflection spectrum , As well as the product of the FBG length ( _ng L (1-R ₀ )). In order to maximize the sensitivity of the FBG to small distortions in the reflection scheme, it is possible to maximize the product τ _g (1-R _o ) evaluated at the peak of the slow optical resonance considered (Equation 28) Says. Equivalently, the product n _g L (1-R _o ) evaluated at the peak of the slow optical resonance considered can be maximized. In the low loss limit, the power transmission spectrum is equal to 1 minus the power reflection spectrum, and the reflection scheme has the same figure of merit as the transmission scheme described above has.

Equation (25) or Equation (28) gives which peak and which operating wavelength in a particular FBG gives the maximum sensitivity (or sensitivity with a particular target value) for a transmission or reflection scheme, respectively. Can be used to select. It designed a peak with a maximum value for the transmission scheme tau _g T _o or _n g LT _o to have the FBG, or in reflection scheme τ _{g (1-R o)} or _n g L (1- Equation (25) or equation (28) can also be used to design an FBG with a maximum sensitivity value by designing the FBG to have a peak with a maximum value for R _o ). Furthermore, equation (25) or equation (28) selects which peak in a particular FBG gives the maximum sensitivity (or sensitivity with a particular target value) for a transmission scheme or a reflection scheme, respectively. Can be used for. In some transmission scheme applications where the FBG length L is fixed, the figure of merit may be τ _g T _o or _ng T _o , either of which is the maximum achievable at any resonance peak Provides a quick but reliable metric for sensitivity. In some reflection scheme applications where the FBG length L is fixed, the figure of merit may be _ng (1-R _o ) or τ _g (1-R _o ), either of which is arbitrary Provides a quick but reliable metric for the maximum achievable sensitivity at the resonance peak.

For example, by calculating or measuring the two parameters (τ _g and T _o ) of equation (25) for the transmission scheme, or for the reflection scheme, two parameters (τ _{g of} equation (28). And R _o ) to calculate or predict the maximum value of the sensitivity of a particular FBG with a particular set of parameters, for small distortions, by calculating or measuring (25) or ( 28) can also be used. For both schemes, the maximum value of sensitivity corresponds to the peak where the product of the two corresponding parameters is the maximum value. These parameters can be calculated or measured (eg, as described by H. Wen et al., The entire disclosure of which is incorporated herein by reference), and then the maximum sensitivity is predicted if necessary In order to do so, it can be substituted into equation (25) or equation (28).

  As noted above, applying Equation (25) and Equation (28) as a figure of merit for a slow light FBG sensor operating in transmissive or reflective mode, respectively, is a performance for an MZ-based slow light FBG scheme. Similar to applying equation (18) in giving an index. For some embodiments, equation (25) or equation (28) provides a figure of merit to characterize the relative performance of multiple peaks for different FBG sensors, FBGs, or a particular FBG. Can be used for For example, when comparing multiple FBG sensors to a sensor that utilizes either a transmission scheme or a reflection scheme operated in a given wavelength range, or when designing an FBG sensor (eg, the highest figure of merit) When selecting an FBG sensor with a value, when designing an FBG sensor to have a peak with a large figure of merit value, or for a maximum sensitivity (or sensitivity with a specific target value) at a specific FBG This figure of merit can be used to select which peak should be used for).

The above differentiation is performed assuming that the slow light peak has a Lorentzian shape. This is a convenient and fairly accurate approximation, resulting in a simple closed form equation for the maximum sensitivity. This assumption only affects the value of the constant coefficient (3.22) in equation (25). In practice, depending on the refractive index profile of the FBG, the linearity will not be exactly Lorentzian. In that case, this assumption may be modified and a line that more faithfully represents the slow light peak line can be used. However, in general, a large difference in the final result, ie this constant factor, is not expected. As an example, for a numerical calculation of the maximum value of the sensitivity of a specific FBG, a differential coefficient with respect to the wavelength of the measured transmission spectrum is calculated numerically, and this differential coefficient spectrum is calculated as a basic definition of sensitivity (formula (21 )) (The differential coefficient dλ / dε included in this equation is a simple scale factor equal to 0.79λ _B as described above and is easily evaluated). The maximum sensitivity value S _max was then obtained by looking for the wavelength with the highest sensitivity in the spectrum S (λ). The value of S _max found in this way, ie by using the actual linearity of the transmission spectrum instead of assuming it has a Lorentzian line, is It was very close. For example, if the FBG described below, more accurate numerical simulations value _{S max} to τ _g T _o / a ratio obtained by normalizing relative to _{λ S max / (τ g T} o / λ) value Gave a constant coefficient of 3.09 compared to 3.22 predicted by equation (25). When the group delay is high, the slow light peak is similar to the Lorentzian shape, so the coefficient value approaches the theoretical value of 3.22.

As explained above, the slow light peak closest to the band gap often has the highest group index. However, if the propagation loss of the FBG is large enough, this first peak (first in the sense of being closest to the band gap) will have a low transmission and a relatively low _ng LT ₀ product. Thus, it is quite possible that the second or third peak, or higher order peak, will have a higher n _g LT ₀ product and therefore have a higher sensitivity maximum. An equivalent method associated with an MZ-based slow light scheme is described in more detail in US patent application Ser. No. 13 / 224,985 filed on Sep. 2, 2011, the entire disclosure of which is hereby incorporated herein by reference. Incorporate into.

  FIG. 42 is a flowchart of an example method 300 for using FBGs in accordance with some embodiments described herein. Although method 300 is described in connection with optical device 210 and FBG 220 of FIGS. 40 and 41, other configurations of optical device and FBG also correspond to some embodiments of method 300. In operation block 310, the method 300 includes providing the FBG 220 with a refractive index modulation that is substantially periodic along the length of the FBG 220. The FBG 220 has a power reflection spectrum as a function of wavelength, a power transmission spectrum as a function of wavelength, and a group delay spectrum as a function of wavelength. The power transmission spectrum as a function of wavelength has one or more resonance peaks, each of which has a local maximum and two regions with non-zero slopes between the two regions. Has a local maximum. In operation block 320, the method 300 is in optical communication with the FBG 220 such that the transmitted portion of light is transmitted along the length of the FBG 220 and the reflected portion of light is reflected from the FBG 220. The method further includes generating light from the narrow band light source 230. The light has a wavelength in a region where the inclination at one resonance peak of one or a plurality of resonance peaks is not zero. Here, the resonance peak is selected such that one or more of the following quantities evaluated at the local maximum value of the resonance peak is at the maximum value. That is, (a) the product of the group delay spectrum and the power transmission spectrum, and (b) the product of the group delay spectrum and 1 minus the power reflection spectrum. In some embodiments, the method 300 uses at least one light power of a light transmissive portion, a light reflected portion, or both a light transmissive portion and a light reflected portion. The method further comprises detecting using a photodetector (eg, one or more photodetectors).

  The wavelength in some embodiments is such that the derivative of the power transmission spectrum with respect to wavelength is a local maximum for the selected resonance peak. In some embodiments, the step of generating light includes generating a light having a wavelength in a region where one of the two regions where the slope of the resonance peak is not zero is not zero. The step of adjusting can be provided.

  FIG. 43 is a flowchart of an example method 400 for configuring an optical device to be used as an optical sensor according to some embodiments described herein. Although the method 400 is described in connection with the optical device 210 and the FBG 220 of FIGS. 40 and 41, other configurations of the optical device and FBG are also compatible with some embodiments of the method 400. In operation block 410, the method 400 includes determining at least one of a power transmission spectrum of light as a function of wavelength for the FBG 220 and a power reflection spectrum of light as a function of wavelength for the FBG 220. In some embodiments, determining at least one of the power transmission spectrum and the power reflection spectrum may comprise performing a spectral measurement. In contrast, in some other embodiments, determining at least one of the power transmission spectrum and the power reflection spectrum includes obtaining or utilizing such data from a separate source ( For example, using a data sheet from the manufacturer of the FBG 220). The power transmission spectrum as a function of wavelength has one or more resonance peaks, each of which has a local maximum and two regions with non-zero slopes between the two regions. Has a local maximum.

  In operation block 420, the method 400 further comprises determining for the FBG 220 a group delay spectrum of light as a function of wavelength. In some embodiments, determining the group delay spectrum may comprise performing a spectrum measurement. In contrast, determining the group delay spectrum in some other embodiments involves obtaining or utilizing such data from a separate source (eg, a data sheet from the manufacturer of FBG 220). Use stage).

  In operation block 430, the method 400 further comprises selecting one of the one or more resonance peaks. The resonance peak is selected such that one or more of the next quantities evaluated at the local maximum value of the selected resonance peak is at the maximum value. That is, (a) the product of the group delay spectrum and the power transmission spectrum, and (b) the product of the group delay spectrum and 1 minus the power reflection spectrum. In operation block 440, the method 400 may cause the FBG 220 and the narrowband light such that the light from the narrowband light source 230 has a wavelength in a region where one of the two regions where the slope of the selected resonance peak is not zero is non-zero. The method further includes configuring the band light source 230. For example, the wavelength is such a wavelength that the differential coefficient of the power transmission spectrum with respect to the wavelength becomes a local maximum with respect to the selected resonance peak. In some embodiments, the narrowband light source 230 is adjusted to generate light having a wavelength in a region where one of the two regions where the slope of the selected resonance peak is not zero is non-zero. Can do. Alternatively, an FBG 220 that satisfies the condition that light has a wavelength in a region where one of the two regions where the slope of the resonance peak is not 0 is not 0 can be selected from a plurality of FBGs.

In some embodiments, method 300 and method 400 can be performed by replacing the group delay spectrum with a group index spectrum without loss of generality, as described above.
Transmittance and group delay measurements

In order to improve the sensitivity of the low-speed optical strain sensor, the propagation loss of the FBG can be minimized. This increases both the group index and the transmittance. It is also possible to increase the refractive index modulation of the grating (see eg H. Wen et al.). This also increases the group index. Any FBG having various losses, refractive index modulations, apodization profiles, and lengths can be used to benefit from some of the embodiments described herein. The FBG used in the configuration described here was purchased from O / E Land, Quebec, Canada. The FBG has a refractive index contrast Δn of 1.0 × 10 ⁻³ and a length of 1. × 10 ⁻³ as estimated by fitting the measured transmittance and group index spectra to a numerical model. It was 2 cm. FIG. 44 is (a) the power transmission spectrum and (b) the group delay spectrum of the signal transmitted through the FBG, both measured using the method described by H. Wen et al.

Both the transmittance and group delay spectrum of FIG. 44 show several slow light peaks on the short wavelength side of the band gap (the band gap starts from the right side of the rightmost peak). The maximum value of the measured group delay occurs at peak # 1 (λ≈1549.7112 nm) and is equal to 4.20 ns, corresponding to the group index n _g = cτ _g / L = 105. The transmittance measured at this wavelength was 1.3%. The maximum sensitivity predicted for this peak is obtained by substituting these values into equation (25), and is 3.4 × 10 ⁴ distortion− ¹ . Peak # 2 (λ≈1549.6435 nm) has a lower group delay of 1.48 ns, but a much higher transmission of 12.55%. The sensitivity calculated at the steepest slope of this second peak is 1.16 × 10 ⁵ strain− ¹ . Thus, this peak gives the highest sensitivity among all the peaks in FIG.
Measuring strain sensitivity

  The measurement of the strain sensitivity spectrum of the FBG was performed using the configuration shown in FIG. Light from the tunable laser source was sent through a polarization controller (PC) and then coupled into the FBG. The FBG was attached to a piezoelectric (PZT) ring that stretched the FBG sinusoidally and added sinusoidal strain to the FBG. The optical signal exiting the FBG is split into two output ports by a 3 dB fiber coupler, the light coming from one output port is detected by a power meter that measures its power, and the light coming from the other port is locked The distortion signal was extracted after being sent to a detector that followed the in-amplifier. An AC voltage of 10 mV at 25 kHz was applied to the PZT ring. This induced a strain of 10 nε in the FBG (this value was obtained by a well-known calibration procedure). The sensitivity spectrum was obtained by dividing the measured spectrum by the known distortion applied and the input optical power. The parts used for these measurements are commercially available. For example, a tunable laser source from Agilent Technologies, Santa Clara, California (model 81682A), a polarization controller from ProtoDel, Surrey, UK, a fiber coupler (model NoTail coupler) from General Photonics, Chino, California, and Santa, CA Clara's New Focus photodetector (model 1811) and Irport, California Newport Corporation photodetector (model 818IR).

  The configuration schematically shown in FIG. 45 uses a stabilization scheme. Here, it is believed that the transmitted optical power exiting the FBG has two components: a first slowly varying component and a second faster varying component. The first component is caused by environmental effects on the FBG, such as variations in its temperature or time-dependent external strain applied to the FBG. As the temperature of the FBG shifts, the spectrum shifts. If the laser wavelength is constant, the transmittance varies, and the transmitted power also varies. This first component usually has a time constant that appears as a power variation at low frequencies (typically in the range from about dc to several hundred Hz or several kHz). The second component is caused by dynamic distortion applied to the FBG, which is typical (but not necessarily) at higher frequencies, eg, over a few hundred Hz.

The light output can be split into two parts as shown in FIG. A portion is sent to a detector that leads to a lock-in amplifier. This acts as a filter and detects fast components at the frequency f ₀ of dynamic distortion in a narrow frequency bandwidth (eg 1 Hz). If the dynamic distortion has multiple frequency components, or has a complex and generally unknown spectrum, the electronics can be designed to scan the frequency of the lock-in amplifier. Thereby it is possible to look for where the signals are and measure the amplitude and frequency of these signals. In simpler cases where there is a distortion at a nominally single frequency f ₀ , the lock-in amplifier measures the amplitude of the detector signal at f ₀ . The detector in this branch is not fast enough, but sometimes has a higher frequency, but is usually fast enough to detect f ₀ of tens of kHz or less can do. The other part of the optical signal may be sent to a power meter that responds only to slow changes, for example slower than a few hundred Hz.

When the FBG undergoes a temperature change, its transmission spectrum shifts. The average value of the voltage from the power meter (average in the vicinity of dc) will change. Since the power meter responds only to changes slowly, the independent This changes the ac signal at f ₀ power meter can be detected. A comparator that is part of the feedback loop can then generate an error voltage equal to the difference between this measured voltage and a stable reference voltage. The feedback loop can then apply a signal proportional to this error voltage to the laser frequency. The error voltage changes the laser frequency by just the right amount to return to the new (temperature shifted) operating frequency.

  This stabilization scheme for FIG. 45 works best when distortion is applied at higher frequencies as the environment provides. Different feedback loops can be used when detecting lower frequency distortions.

40 and 41 schematically show a scheme using a single detector. In this case, the electrical signal coming from the detector is filtered by a low pass filter in the feedback control branch. The feedback control system 250 of FIGS. 40 and 41 can comprise such a low pass filter to remove the f ₀ component in a manner known to those skilled in the art of feedback circuits.

The result of the strain sensing measurement is plotted as a solid curve in FIG. FIG. 46 also shows the theoretical calculated from equation (23) (eg, by taking the derivative of the measured transmission spectrum of FIG. 44 and multiplying this spectrum by 0.79λ _B according to equation (23)). The sensitivity spectrum is shown by the dashed curve. As predicted from the above discussion regarding the predicted maximum sensitivity in relation to FIG. 44, the measured maximum sensitivity is the steepest peak # 2 shown in the power transmission spectrum of FIG. Inclined. At this wavelength of maximum sensitivity (λ≈1549.6435 nm), the measured sensitivity is 1.2 × 10 ⁵ strain ⁻¹ . This value agrees very well with the predicted value of 1.16 × 10 ⁵ strain ⁻¹ derived above. Since the laser is adjusted in the vicinity of this low-speed light peak, dT / dλ increases, so the sensitivity also increases. Beyond the peak apex, T (λ) reaches its maximum value, and therefore its differential coefficient dT / dλ drops, so the sensitivity begins to decline. Finally, on the other side of the transmission peak, the other steep slope region of the transmission peak is reached, so that the sensitivity increases again. Thus, each peak has two wavelengths with the highest sensitivity. In this specific FBG, the side closer to the band edge has slightly higher sensitivity. There is a very good agreement between the predicted sensitivity spectrum and the measured sensitivity spectrum.

To compare the performance of slow light sensors with other FBG-based sensors, the most frequently used metric is the minimum detectable distortion, the definition of which is given by equation (19). At the maximum sensitivity, the noise measured in the detected signal was 57 μV / √Hz. The noise was dominated by optical and electrical shot noise of the detector. Minimum detectable distortion by using a noise level of 57 nV / √Hz, normalized to an equivalent optical input voltage of 0.41 V and then divided by a sensitivity of 1.2 × 10 ⁵ distortion- ¹ Is found. Therefore, the minimum detectable strain of this slow light strain sensor is 1.2 pε / √Hz at the highest sensitivity wavelength.
Thermal stability

ここでα＝５×１０^−７Ｋ^−１は熱膨張係数であり、δｎ／δＴ＝１．１×１０^−５Ｋ^−１はシリカファイバの熱光学係数である。１．５５μｍにおいては、ｄλ／ｄＴは１２．５ｐｍ／℃である。歪みの感度と同様に、透過スキームにおける正規化された熱的感度は次の通りである。

動作波長において、低速光共振に対して再びローレンツ線形を仮定すると、ｄＴ（λ）／ｄλは、

である。従って、この式およびｄλ／ｄＴ＝１２．５ｐｍ／℃を式（３０）に代入することにより、透過スキームにおける熱的感度の最大値は次の通りである。

同様に反射スキームにおいては次の通りである。

When the FBG is subjected to a temperature perturbation ΔT, both the effective refractive index of the mode and the length of the FBG change. These two changes induce a shift in the transmission and reflection spectra given by:

Here, α = 5 × 10 ⁻⁷ K ⁻¹ is a thermal expansion coefficient, and δn / δT = 1.1 × 10 ⁻⁵ K ⁻¹ is a thermo-optic coefficient of the silica fiber. At 1.55 μm, dλ / dT is 12.5 pm / ° C. As with strain sensitivity, the normalized thermal sensitivity in the transmission scheme is:

Assuming a Lorentzian shape again for the slow optical resonance at the operating wavelength, dT (λ) / dλ is

It is. Therefore, by substituting this equation and dλ / dT = 12.5 pm / ° C. into equation (30), the maximum value of thermal sensitivity in the transmission scheme is as follows.

Similarly, the reflection scheme is as follows.

  Thermal sensitivity in both transmission and reflection schemes increases as the group delay increases. When operating in the slow light mode, it may be advantageous to use feedback control to stabilize the sensor against thermal fluctuations. For example, as shown in FIGS. 40 and 41, a feedback control system 250 (eg, having a servo loop or proportional integral derivative controller) can be used in a transmission or reflection scheme. The DC output voltage measured at the operating wavelength is used as the set point for the proportional integral derivative (PID) controller. If the sensor's DC output deviates from the set point as a result of the FBG temperature change, an error voltage that adjusts the laser wavelength to return to a new operating wavelength (eg, the FBG operating wavelength at the new temperature), Generated by the PID controller. In order to operate at the highest sensitivity, it is sufficient to have one feedback loop for either the transmission scheme or the reflection scheme.

In contrast to transmission and reflection schemes, MZ-based schemes utilize two feedback loops to stabilize against temperature changes. As shown in FIG. 47, the first feedback loop ensures that the laser remains tuned to a temperature dependent operating wavelength. In contrast, the second feedback loop ensures that the phase difference between the two arms of the interferometer is kept square (± π / 2, ± 3π / 2, etc.). Thus, in some embodiments, the implementation of the transmission and reflection schemes is nominally simpler than the implementation of MZ-based schemes because it utilizes a single feedback loop rather than two feedback loops. Can do.
Sensing other parameters

  The slow light sensor can measure many perturbations in addition to distortion. Other perturbations such as temperature, magnetic field and electric field can also shift the transmission and reflection spectra of the FBG. In a temperature sensor, when thermal perturbation is applied to the FBG, the refractive index of the material changes due to the thermo-optic effect, and the FBG is stretched due to the thermal expansion effect. The combination of both effects induces a shift in the above spectrum. This can be measured by using slow light sensors in transmission, reflection, or MZ based schemes. In a magnetic field sensor, FBG can be coupled to a ferromagnetic material such as magneto-optic glass. When a DC or AC magnetic field is applied to the magneto-optic glass, changes that occur during the magnetization of the material induce magnetostrictive strain, which changes the length of the material, and hence the length of the FBG, and thus in the spectrum Generate a shift. The same concept can be applied to measuring the electric field by FBG on the electrostrictive material. Under the influence of a DC or AC electric field, the dimensions of the material will change, changing the length of the FBG and also creating a shift in the spectrum. Other parameters other than those listed here can also be measured using this and similar techniques.

  Various embodiments of the invention have been described above. Although the invention has been described with reference to these specific embodiments, the description is intended to be illustrative of the invention and is not intended to be limiting. Various changes and uses will occur to those skilled in the art without departing from the true spirit and scope of the invention as defined herein.

Claims (20)

An optical device,
A fiber Bragg grating, having a substantially periodic refractive index modulation along the length of the fiber Bragg grating, a power reflection spectrum as a function of wavelength, a power transmission spectrum as a function of wavelength, and A fiber Bragg grating having a group delay spectrum as a function of wavelength;
Optically communicating with the fiber Bragg grating, such that a portion through which light is transmitted is transmitted along the length of the fiber Bragg grating, and a portion through which light is reflected is reflected from the fiber Bragg grating. A narrowband light source that transmits the light into the fiber Bragg grating, wherein the power transmission spectrum as a function of wavelength has one or more resonance peaks, each of which has a local maximum A narrowband light source having a value and two regions whose slope is non-zero, wherein the local maximum is between the two regions;
At least one photodetector for detecting the optical power of the transmitted portion of the light, the reflected portion of the light, or both the transmitted portion of the light and the reflected portion of the light When,
With
The light has a wavelength in a region where the slope of one resonance peak of the one or more resonance peaks is not 0,
The one resonance peak is an amount evaluated at the one resonance peak (a) a product of the group delay spectrum and the power transmission spectrum, and (b) the power reflection from the group delay spectrum and 1 An optical device that is selected such that one or more of its products with the subtracted spectrum is at its maximum value.   The product of the group delay spectrum and the power transmission spectrum evaluated at the one resonance peak is at a maximum value, and the at least one photodetector detects the optical power of the transmitted portion of the light. The optical device according to claim 1 for detection.   The product of the group delay spectrum evaluated at the one resonance peak and 1 minus the power reflection spectrum is at a maximum value, and the at least one photodetector is the reflected of the light The optical device according to claim 1, wherein the optical power of a portion is detected.   The optical device according to claim 1, wherein the detected optical power indicates an amount of distortion applied to the fiber Bragg grating.   The optical device according to any one of claims 1 to 4, wherein the detected optical power indicates a temperature of the fiber Bragg grating.   The optical device according to any one of claims 1 to 5, wherein the detected optical power indicates a magnitude of a magnetic field or an electric field applied to the fiber Bragg grating.   The optical device according to claim 1, further comprising a feedback control system that stabilizes the optical device against thermal fluctuations. A method using a fiber Bragg grating,
Power reflection spectrum as a function of wavelength, power transmission spectrum as a function of wavelength, and group delay as a function of wavelength, with substantially periodic refractive index modulation along the length of the fiber Bragg grating The power transmission spectrum as a function of wavelength has one or more resonance peaks, each of the resonance peaks having a local maximum and two regions with non-zero slopes; Providing a fiber Bragg grating in which the local maximum is between the two regions;
Generating light from a narrowband light source, wherein the transmitted portion of the light is transmitted along the length of the fiber Bragg grating and the reflected portion of the light is reflected from the fiber Bragg grating. The narrowband light source is in optical communication with the fiber Bragg grating;
The optical power of the transmitted portion of the light, the reflected portion of the light, or both the transmitted portion of the light and the reflected portion of the light is by at least one photodetector. Detecting, and
With
The light has a wavelength in a region where the slope of one resonance peak of the one or more resonance peaks is not 0,
The one resonance peak is an amount evaluated at the local maximum value of the one resonance peak (a) a product of the group delay spectrum and the power transmission spectrum, and (b) the group delay. A method selected such that one or more of the product of the spectrum and 1 minus the power reflection spectrum is at a maximum.   The product of the group delay spectrum and the power transmission spectrum evaluated at the local maximum value of the one resonance peak is at a maximum value, and detecting the optical power includes transmitting the light. 9. The method of claim 8, comprising detecting the optical power of a portion.   The product of the group delay spectrum evaluated at the local maximum value of the one resonance peak and 1 minus the power reflection spectrum is at a maximum value, and detecting the optical power comprises: 9. The method of claim 8, comprising detecting the optical power of the reflected portion of the light.   The method according to any one of claims 8 to 10, wherein the detected optical power indicates an amount of strain applied to the fiber Bragg grating.   The method according to claim 8, wherein the detected optical power indicates a temperature of the fiber Bragg grating.   The method according to claim 8, wherein the detected optical power indicates a magnitude of a magnetic field or an electric field applied to the fiber Bragg grating.   14. The any of claims 8 to 13, wherein the substantially periodic refractive index modulation has an amplitude that varies along the length of the fiber Bragg grating such that the fiber Bragg grating is an apodized grating. The method according to claim 1. A method of configuring an optical device used as an optical sensor, wherein the optical device comprises a fiber Bragg grating, and a narrowband light source in optical communication with the fiber Bragg grating, the narrowband light source comprising: Transmitting the light to the fiber Bragg grating such that a transmitted portion of light is transmitted along the length of the fiber Bragg grating, and a reflected portion of the light is reflected from the fiber Bragg grating; The fiber Bragg grating is a method having a substantially periodic refractive index modulation along the length of the fiber Bragg grating,
Determining at least one of a power reflection spectrum of the light as a function of wavelength for the fiber Bragg grating and a power transmission spectrum of the light as a function of wavelength for the fiber Bragg grating, wherein The power transmission spectrum has one or a plurality of resonance peaks, each of the resonance peaks having a local maximum value and two regions whose slopes are not 0, and the local maximum value is 2 A stage between two territories,
Determining a group delay spectrum of the light for the fiber Bragg grating as a function of wavelength;
Selecting one resonance peak of the one or more resonance peaks, wherein the one resonance peak is an amount evaluated at the local maximum of the selected one resonance peak. One or more of (a) a product of the group delay spectrum and the power transmission spectrum, and (b) a product of the group delay spectrum and 1 minus the power reflection spectrum is at a maximum value. The stage to choose
The fiber Bragg grating and the narrow band light source so that the light from the narrow-band light source has a wavelength in one region of which the slope of the selected resonance peak is not zero. Configuring a band light source; and
A method comprising:   The optical device detects optical power of the transmitted portion of the light, the reflected portion of the light, or both the transmitted portion of the light and the reflected portion of the light The method of claim 15, further comprising at least one photodetector.   The product of the group delay spectrum and the power transmission spectrum evaluated at the local maximum value of the selected one resonance peak is at a maximum value, and the at least one photodetector is The method according to claim 15 or 16, wherein the optical power of the transmitted part is detected.   The product of the group delay spectrum evaluated at the local maximum value of the selected one resonance peak and 1 minus the power reflection spectrum is at a maximum value, and the at least one light detection 17. A method according to claim 15 or claim 16, wherein the device detects the optical power of the reflected portion of the light.   The step of configuring the fiber Bragg grating and the narrow-band light source includes light having the wavelength in one region where the slope is not 0 out of two regions where the slope of the selected resonance peak is not 0. 19. A method according to any one of claims 15 to 18 comprising adjusting the narrowband light source to generate.   The step of configuring the fiber Bragg grating and the narrow-band light source is such that the wavelength is in one region where the slope is not 0 of two regions where the slope of the selected resonance peak is not 0. 20. A method according to any one of claims 15 to 19, comprising selecting the fiber Bragg grating.

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