JP2024029318A - Pulse light source and system using the same - Google Patents

Abstract

To provide a pulse light source capable of generating convex or concave pulses (positive bright, positive dark, negative bright, and negative dark) with a positive or negative amplitude in any shape directly from a mode-locked laser and a system using the same.SOLUTION: The pulse light source has a frequency-modulated mode-locked laser including an optical phase modulator and an optical filter in a laser resonator. Amplitude and phase characteristics are tunable by setting the amplitude and phase characteristics of the optical filter according to the shape of a desired optical pulse and the amount of continuous wave offset. The pulse light source is configured to generate dark and bright pulses having the desired optical pulse shape having the continuous wave offset amount.SELECTED DRAWING: Figure 2

Description

The present invention relates to a pulsed light source that can realize bright pulses with convex amplitudes and dark pulses with concave amplitudes in both positive and negative phases and in various shapes, and a system using the same.

Mode-locked lasers have a wide range of uses as optical pulse light sources used in optical communications, optical measurement, optical signal processing, and the like. It is known that the pulse shape generated from a mode-locked laser is generally a Gaussian pulse (for example, see Non-Patent Documents 1 and 2). Furthermore, a mode-locked laser can generate a sech pulse by using a nonlinear optical effect called a soliton effect within a resonator (see, for example, Non-Patent Document 3).

On the other hand, as a method for directly generating pulses other than Gaussian and sech pulses from a mode-locked laser, the present inventors used an optical filter with appropriately designed amplitude and phase characteristics according to the shape of the optical pulse to be generated. have proposed an optical function generator that can generate a pulse train with a desired time waveform even if it is not Gaussian or Sec by inserting it into a resonator (for example, see Non-Patent Documents 4, 5, 6, and 7). ). This technique allows Nyquist pulses with single-exponential, double-exponential, triangular, parabolic, rectangular, and even arbitrary roll-off rates to be modulated using amplitude modulation (AM) or frequency modulation (FM) mode locking. It has been revealed that it can be generated using a laser.

The waveforms of optical pulses that have been achieved so far with these mode-locked lasers are all shapes in which the amplitude is convex at positive electric field values and the bases at both ends are zero, as shown in Figure 1(a). It was a positive pulse. On the other hand, in addition to this optical pulse, there is also a dark pulse having a concave shape. A dark pulse is a pulse in which a part of continuous wave (CW) light is shaved off in a certain shape, as shown in FIG. 1(c). A known example is the dark soliton. A dark soliton is an optical pulse with an amplitude of tanh(t/T) and an intensity of tanh ² (t/T) = 1 - sech ² (t/T). A dark soliton is a nonlinear pulse that propagates stably by balancing the normal dispersion and Kerr effect in an optical fiber (see, for example, Non-Patent Documents 8 and 9), and is a non-linear pulse that propagates stably (see, for example, Non-Patent Documents 8 and 9), and is a pulse for optical transmission (for example, see Non-Patent Document 10). ), but also as an optical frequency comb (for example, see Non-Patent Document 11).

D. Kuizenga and A. Siegman, “FM and AM mode locking of the homogeneous laser - Part I: Theory”, IEEE J. Quantum Electron., November 1970, vol. 6, no. 11, pp. 694-708 H. A. Haus, “A theory of forced mode locking”, IEEE J. Quantum Electron., July 1975, vol. QE-11, no. 7, pp. 323-330 M. Nakazawa and E. Yoshida, "A 40-GHz 850-fs regeneratively FM mode-locked polarization-maintaining erbium fiber ring laser", IEEE Photonics Technology Letters, Dec. 2000, vol. 12, no. 12, pp. 1613 -1615 M. Nakazawa and T. Hirooka, “Theory of AM Mode-Locking of a Laser as an Arbitrary Optical Function Generator”, IEEE J. Quantum Electron., December 2021, vol. 57, no. 6, 1300320 M. Nakazawa and T. Hirooka, “Theory of FM Mode-Locking of a Laser as an Arbitrary Optical Function Generator”, IEEE J. Quantum Electron., April 2022, vol. 58, no. 2, 1300125 M. Nakazawa, M. Yoshida, and T. Hirooka, “Experiments on an AM Mode-Locked Laser as an Arbitrary Optical Function Generator”, IEEE J. Quantum Electron., June 2022, vol. 58, no. 3, 1300218 M. Nakazawa, M. Yoshida, and T. Hirooka, “Experiments on an FM Mode-Locked Laser as an Arbitrary Optical Function Generator”, IEEE J. Quantum Electron., June 2022, vol. 58, no. 3, 1300316 A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. 2. Normal dispersion”, Appl. Phys. Lett., August 1973, vol. 23, no. 4, pp. 142-144 A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, “Experimental observation of the fundamental dark soliton in optical fibers”, Phys. Rev. Lett., November 1988, vol. 61, no. .21, pp.2445-2448 M. Nakazawa and K. Sizuki, "10Gbit/s pseudorandom dark soliton data transmission over 1200 km", Electron. Lett. June 1995, vol. 31, No. 13, pp. 1076 - 1077 X. Xue, Y. Xuan, Y. Liu, P. H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators”, Nature Photon., 2015, vol. 9, 594-600

However, although dark solitons as shown in Fig. 1(c) can be obtained by balancing the normal dispersion and the Kerr effect in an optical fiber, the conditions are limited, and above all, it is difficult to obtain a pulsed soliton with a mode-locked laser. However, there was a problem in that this hindered practicality. On the other hand, in contrast to the dark pulse, a bright pulse can be considered, in which a CW offset is superimposed on a convex optical pulse, as shown in FIG. 1(b).

Conventional optical pulses are positive pulses whose amplitude is defined by a positive value (see Figure 1 (a)), but negative pulses whose amplitude is defined by the value of a negative electric field (see Figure 1 (d)). ) also exists. Since negative pulses can be generated in this way, it is possible to turn on and off light with a high extinction ratio through interference with positive pulses, and there are growing expectations for its wide range of applications in optical signal processing.

Therefore, if the aforementioned bright pulses and dark pulses can be easily generated, there is a possibility that expectations for a wide range of applications in optical signal processing will be greatly expanded, similar to the relationship between positive pulses and negative pulses.

As is clear from the above, when positive/negative CW and positive/negative pulses are combined, (1) positive bright pulse (see Figure 1(b)), (2) positive dark pulse (see Figure 1 (c)), (3) negative bright pulse (see FIG. 1(e)), and (4) negative dark pulse (see FIG. 1(f)). (1) is a positive pulse + positive CW offset, (2) is a negative pulse + positive CW offset, (3) is a positive pulse + negative CW offset, and (4) is a combination of negative pulse + negative CW offset. defined.

The present invention is intended to solve such problems, and is capable of generating a mode-locked laser with a convex or concave pulse (positive bright, positive dark, negative bright, negative dark) having a positive or negative amplitude in an arbitrary shape. The purpose of the present invention is to provide a pulsed light source that can be directly generated from a pulsed light source and a system using the same.

In order to achieve such an object, the pulsed light source according to the present invention has a frequency modulation mode-locked laser that includes an optical phase modulator (frequency modulator) and an optical filter in a laser resonator, and generates a desired optical pulse. By setting the amplitude and phase characteristics of the optical filter according to the shape and continuous wave offset amount, the amplitude and the phase characteristics are variable, and the shape of the desired optical pulse having the continuous wave offset amount can be set. The invention is characterized in that it is configured to generate dark pulses and bright pulses with

Further, in the pulsed light source according to the present invention, the optical phase modulator is driven by a sine wave with a repetition angular frequency Ω _m , and the amplitude and the phase characteristics of the optical filter are the modulation degree of the optical phase modulator and the desired output. The pulse spectrum A(ω), the sinc function S(ω) = sin(ωτ/2)/ω (τ= 2π/Ω _m ) giving the continuous wave offset amount, and A(ω) and S(ω) Functions A(ω-nΩ _m ), A(ω+nΩ _m ), S(ω-nΩ _{m )} , S(ω+nΩ _m ), ( n: an integer). Furthermore, in the optical filter, a band limit may be provided to the amplitude and phase characteristics of the optical pulse depending on the spectral width of the optical pulse to be output.

Further, in the pulsed light source according to the present invention, the optical filter is configured to adjust the sign of the spectrum A(ω) and the sinc function S(ω) giving the continuous wave offset amount to be positive or negative depending on the combination thereof. It may be configured to generate four types of pulses: a positive bright pulse, a positive dark pulse, a negative bright pulse, and a negative dark pulse.

A system using a pulsed light source according to the present invention is characterized by using the pulsed light source according to the present invention.

According to the present invention, a pulsed light source capable of directly generating convex or concave pulses (positive bright, positive dark, negative bright, negative dark) having positive or negative amplitude in any shape from a mode-locked laser, and a pulsed light source using the same, can be used. We can provide a system that The pulsed light source according to the present invention generates positive/negative, bright/dark pulse trains of various waveforms with a high optical signal-to-noise ratio (OSNR; Optical Signal to Noise Ratio). Therefore, arbitrary positive/negative, bright/dark pulses can be generated with a simple configuration using a mode-locked laser, making it possible to provide a highly functional and high-quality pulsed light source for optical measurement and optical signal processing.

To explain the definitions of (a) positive pulse, (b) positive bright pulse, (c) positive dark pulse, (d) negative pulse, (e) negative bright pulse, and (f) negative dark pulse. FIG. 1 is a schematic diagram showing the configuration of a pulsed light source according to an embodiment of the present invention. Optical filter for generating (a) positive bright Gaussian pulse, (b) positive dark Gaussian pulse, (c) negative bright Gaussian pulse, and (d) negative dark Gaussian pulse in the first embodiment of the present invention It is a graph showing the shape (black: absolute value, gray: phase). (a) Positive bright Gaussian pulse, (b) positive dark Gaussian pulse, (c) negative bright Gaussian pulse, (d) negative dark Gaussian pulse obtained by computer analysis using the optical filter shown in Figure 3. It is a graph showing the waveform of a steady pulse (black: electric field amplitude, gray: phase). Optical filter for generation of (a) positive bright sech pulses, (b) positive dark sech pulses, (c) negative bright sech pulses, and (d) negative dark sech pulses in a second embodiment of the invention It is a graph showing the shape (black: absolute value, gray: phase). (a) Positive bright sech pulse, (b) positive dark sech pulse, (c) negative bright sech pulse, (d) negative dark sech pulse obtained by computer analysis using the optical filter shown in Figure 5. It is a graph showing the waveform of a steady pulse (black: electric field amplitude, gray: phase). In a third embodiment of the invention, (a) positive bright biexponential pulse, (b) positive dark biexponential pulse, (c) negative bright biexponential pulse, (d) negative dark biexponential pulse. It is a graph showing the shape of an optical filter for generating a function pulse (black: absolute value, gray: phase). (a) Positive bright biexponential pulse, (b) Positive dark biexponential pulse, (c) Negative bright biexponential pulse, (d) obtained by computer analysis using the optical filter shown in FIG. ) is a graph showing a steady pulse waveform (black: electric field amplitude, gray: phase) of a negative dark biexponential pulse. (a) positive bright triangular pulse, (b) positive dark triangular pulse, (c) negative bright triangular pulse, and (d) negative dark triangular pulse defined by electric field amplitude in the fourth embodiment of the present invention. It is a graph showing the shape of the optical filter for generation (black: absolute value, gray: phase). Using the optical filter shown in Figure 9, (a) positive bright triangular pulse, (b) positive dark triangular pulse, (c) negative bright triangular pulse, (d) defined by electric field amplitude determined by computer analysis. It is a graph showing a steady pulse waveform (black: electric field amplitude, gray: phase) of a negative dark triangular pulse. (a) positive bright triangular pulse, (b) positive dark triangular pulse, (c) negative bright triangular pulse, and (d) negative dark triangular pulse defined by electric field strength in the fifth embodiment of the present invention. It is a graph showing the shape of the optical filter for generation (black: absolute value, gray: phase). (a) Positive bright triangular pulse, (b) Positive dark triangular pulse, (c) Negative bright triangular pulse, (d) Defined by electric field strength determined by computer analysis using the optical filter shown in Figure 11. It is a graph showing a steady pulse waveform (black: electric field amplitude, gray: phase) of a negative dark triangular pulse. (a) positive bright parabolic pulse, (b) positive dark parabolic pulse, (c) negative bright parabolic pulse, and (d) negative dark parabolic pulse defined by electric field amplitude in the sixth embodiment of the present invention. It is a graph showing the shape of the optical filter for generation (black: absolute value, gray: phase). (a) Positive bright parabolic pulse, (b) Positive dark parabolic pulse, (c) Negative bright parabolic pulse, (d) Defined by electric field amplitude determined by computer analysis using the optical filter shown in Fig. 13 It is a graph showing a steady pulse waveform (black: electric field amplitude, gray: phase) of a negative dark parabolic pulse. (a) positive bright parabolic pulse, (b) positive dark parabolic pulse, (c) negative bright parabolic pulse, and (d) negative dark parabolic pulse defined by electric field strength in the seventh embodiment of the present invention. It is a graph showing the shape of the optical filter for generation (black: absolute value, gray: phase). (a) Positive bright parabolic pulse, (b) Positive dark parabolic pulse, (c) Negative bright parabolic pulse, (d) Defined by electric field strength determined by computer analysis using the optical filter shown in Fig. 15. It is a graph showing a steady pulse waveform (black: electric field amplitude, gray: phase) of a negative dark parabolic pulse. Optical filter for generation of (a) positive bright rectangular pulse, (b) positive dark rectangular pulse, (c) negative bright rectangular pulse, and (d) negative dark rectangular pulse in the eighth embodiment of the present invention It is a graph showing the shape (black: absolute value, gray: phase). (a) Positive bright rectangular pulse, (b) Positive dark rectangular pulse, (c) Negative bright rectangular pulse, (d) Negative dark rectangular pulse obtained by computer analysis using the optical filter shown in FIG. It is a graph showing the waveform of a steady pulse (black: electric field amplitude, gray: phase). Optical filter for generating (a) positive bright Nyquist pulse, (b) positive dark Nyquist pulse, (c) negative bright Nyquist pulse, and (d) negative dark Nyquist pulse in the ninth embodiment of the present invention It is a graph showing the shape (black: absolute value, gray: phase). (a) Positive bright Nyquist pulse, (b) positive dark Nyquist pulse, (c) negative bright Nyquist pulse, (d) negative dark Nyquist pulse obtained by computer analysis using the optical filter shown in FIG. It is a graph showing the waveform of a steady pulse (black: electric field amplitude, gray: phase). It is a graph which shows the shape ((a) absolute value, (b) phase) of the optical filter for dark soliton generation in the 10th Embodiment of this invention. 22 is a graph showing the waveform ((a) electric field strength, (b) phase) of a steady pulse of a dark soliton obtained by computer analysis using the optical filter shown in FIG. 21. 1 is a schematic diagram showing the configuration of an optical fiber ring resonator used in an experiment of a pulsed light source according to an embodiment of the present invention. 3 shows (a) transmittance characteristics and (b) phase characteristics of an optical filter F _Fpd (ω) used for generating a dark Gaussian pulse implemented by an LCoS element in the first embodiment of the present invention. These are the waveforms of the dark Gaussian pulse obtained using the optical filter of FIG. 24 ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting it into amplitude), and (c) optical spectrum. 3 shows (a) transmittance characteristics and (b) phase characteristics of an optical filter F _Fpb (ω) used for generating bright Gaussian pulses implemented with an LCoS element according to the first embodiment of the present invention. These are the waveforms of the bright Gaussian pulse obtained using the optical filter of FIG. 26 ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting it into amplitude), and (c) optical spectrum. 3 shows (a) transmittance characteristics and (b) phase characteristics of an optical filter F _Fpd (ω) used for generating a dark sech pulse implemented with an LCoS element according to the second embodiment of the present invention. These are the waveforms of the dark sech pulse obtained using the optical filter of FIG. 28 ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting it into amplitude), and (c) optical spectrum. 3 shows (a) transmittance characteristics and (b) phase characteristics of an optical filter F _Fpb (ω) used to generate a bright sech pulse implemented by an LCoS element according to the second embodiment of the present invention. These are the waveforms of the bright sech pulse obtained using the optical filter of FIG. 30 ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting it into amplitude), and (c) optical spectrum. These are (a) transmittance characteristics and (b) phase characteristics of an optical filter F _Fpd (ω) used to generate a dark biexponential pulse implemented with an LCoS element according to the third embodiment of the present invention. These are the waveforms ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting into amplitude), and (c) optical spectrum of the dark biexponential pulse obtained using the optical filter of FIG. 32. 3 shows (a) transmittance characteristics and (b) phase characteristics of an optical filter F _Fpb (ω) used to generate a bright double-exponential pulse implemented by an LCoS element according to the third embodiment of the present invention. These are the waveforms ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting into amplitude), and (c) optical spectrum of the bright biexponential pulse obtained using the optical filter of FIG. 34. (a) Transmittance characteristics and (b) phase characteristics of the optical filter F _Fpd (ω) used to generate a dark triangular pulse defined by electric field amplitude and implemented with an LCoS element according to the fourth embodiment of the present invention. be. Waveforms of dark triangular pulses defined by electric field amplitude obtained using the optical filter in Figure 36 ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting to amplitude), (c) light It is a spectrum. (a) Transmittance characteristics and (b) phase characteristics of the optical filter F _Fpb (ω) used to generate a bright triangular pulse defined by electric field amplitude and implemented with an LCoS element in the fourth embodiment of the present invention. be. Waveforms of bright triangular pulses defined by electric field amplitude obtained using the optical filter in Figure 38 ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting to amplitude), (c) light It is a spectrum. In the fifth embodiment of the present invention, (a) transmittance characteristics and (b) phase characteristics of the optical filter F _Fpd (ω) used to generate a dark triangular pulse defined by electric field strength and implemented with an LCoS element. be. Waveforms of dark triangular pulses defined by electric field intensity obtained using the optical filter in Figure 40 ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting to amplitude), (c) light It is a spectrum. (a) Transmittance characteristics and (b) phase characteristics of the optical filter F _Fpb (ω) used to generate a bright triangular pulse defined by electric field strength, implemented with an LCoS element in the fifth embodiment of the present invention. be. Waveforms of bright triangular pulses defined by electric field intensity obtained using the optical filter in Figure 42 ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting to amplitude), (c) light It is a spectrum. In the sixth embodiment of the present invention, (a) transmittance characteristics and (b) phase characteristics of the optical filter F _Fpd (ω) used to generate a dark parabolic pulse defined by electric field amplitude and implemented with an LCoS element. be. Waveforms of dark parabolic pulses defined by electric field amplitude obtained using the optical filter in Figure 44 ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting to amplitude), (c) light It is a spectrum. (a) Transmittance characteristics and (b) phase characteristics of the optical filter F _Fpb (ω) used for generating bright parabolic pulses defined by electric field amplitude and implemented with an LCoS element according to the sixth embodiment of the present invention. be. Waveforms of bright parabolic pulses defined by electric field amplitude obtained using the optical filter in Figure 46 ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting to amplitude), (c) light It is a spectrum. (a) Transmittance characteristics and (b) phase characteristics of the optical filter F _Fpd (ω) used to generate a dark parabolic pulse defined by electric field intensity, implemented with an LCoS element according to the seventh embodiment of the present invention. be. Waveforms of dark parabolic pulses defined by electric field intensity obtained using the optical filter in Figure 48 ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting to amplitude), (c) light It is a spectrum. In the seventh embodiment of the present invention, (a) transmittance characteristics and (b) phase characteristics of the optical filter F _Fpb (ω) used to generate a bright parabolic pulse defined by electric field intensity and implemented with an LCoS element. be. Waveforms of bright parabolic pulses defined by electric field intensity obtained using the optical filter in Figure 50 ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting to amplitude), (c) light It is a spectrum. These are (a) transmittance characteristics and (b) phase characteristics of an optical filter F _Fpd (ω) used to generate a dark rectangular pulse implemented by an LCoS element in the eighth embodiment of the present invention. These are the waveforms of the dark rectangular pulse obtained using the optical filter of FIG. 52 ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting it into amplitude), and (c) optical spectrum. These are (a) transmittance characteristics and (b) phase characteristics of an optical filter F _Fpb (ω) used to generate a bright rectangular pulse implemented by an LCoS element in the eighth embodiment of the present invention. These are the waveforms of the bright rectangular pulse obtained using the optical filter in FIG. 54 ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting it into amplitude), and (c) optical spectrum. These are (a) transmittance characteristics and (b) phase characteristics of an optical filter F _Fpd (ω) used to generate a dark Nyquist pulse implemented by an LCoS element in the ninth embodiment of the present invention. These are the waveforms of the dark Nyquist pulse obtained using the optical filter of FIG. 56 ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting it into amplitude), and (c) optical spectrum. These are (a) transmittance characteristics and (b) phase characteristics of an optical filter F _Fpb (ω) used to generate a bright Nyquist pulse implemented by an LCoS element in the ninth embodiment of the present invention. These are the waveforms of the bright Nyquist pulse obtained using the optical filter of FIG. 58 ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting it into amplitude), and (c) optical spectrum. These are (a) transmittance characteristics and (b) phase characteristics of an optical filter F _tanh (ω) used for dark soliton generation implemented with an LCoS element in the tenth embodiment of the present invention. These are the waveforms of the dark soliton obtained using the optical filter of FIG. 60 ((a) intensity waveform, (b) waveform obtained by taking √ in (a) and converting it into amplitude), and (c) optical spectrum.

Embodiments of the present invention will be described below based on the drawings.
FIG. 2 shows a schematic diagram of the configuration of a pulsed light source in an embodiment of the present invention. An optical fiber 1 constitutes a laser ring resonator, and the resonator includes an optical amplifier 2 used as a gain medium, an optical phase modulator 3 used as a mode locker, and an optical filter 4 whose amplitude and phase characteristics can be set arbitrarily. Equipped with. As the optical amplifier 2, a general optical amplifier is used. Preferably, an erbium-doped fiber amplifier (EDFA), a semiconductor optical amplifier (SOA), a solid-state laser element, or the like can be used. The optical phase modulator 3 uses a modulator that modulates the phase of output light by changing the refractive index of an electro-optic crystal using an electric signal. Preferably, a phase modulator using LN (LiNbO ₃ ) crystal can be used. The optical phase modulator 3 is externally driven with a sine wave M _PM cos(Ω _m t) having a frequency f _m (angular frequency Ω _m = 2πf _m ) and a modulation degree M _PM . As the optical filter 4, a programmable optical filter composed of LCoS or the like can be used. As will be described later, it is desirable that the optical filter 4 be able to control not only the amplitude (transmission) characteristics but also the phase characteristics, assuming that a pulse waveform with an asymmetric shape is generated.

Using this pulsed light source, in order to generate an optical pulse whose amplitude is given by a _p (t) (the positive pulse in Fig. 1(a)), the transfer function F _Fp (ω) of the optical filter 4 is expressed as (1) may be designed (for example, see Non-Patent Document 5). Here, A _p (ω) is the spectrum of a _p (t), n is an integer from −∞ to ∞, and J _n (x) is an n-th order Bessel function of the first kind. The transfer function in equation (1) is obtained from equation (2), which is the master equation for the optical pulse a _p (t) given by the phase modulation exp(iM _PM cosΩ _m t) and the optical filter F _Fp (ω). F stands for Fourier transform). Here, G is the gain, L is the loss of the resonator, and GL = 1.

In equation (2), consider the master equation of an optical pulse (negative pulse) whose amplitude is given by a _n (t) = -a _p (t) and whose spectrum is given by A _n (ω) = - A _p (ω). Letting the transfer function of the optical filter 4 for generating a _n (t) be F _Fn (ω), the master equation of a _n (t) is given by equation (3). Since equation (3) can also be written as equation (4), from equation (2) and equation (4), F _Fn (ω) is given by F _Fn (ω)=−F _Fp (ω).

Next, the generation of the positive bright pulse a _pb (t) shown in FIG. 1(b) will be described. a _pb (t) is defined as a pulse with a positive CW offset superimposed on a _p (t). Considering that this CW offset is superimposed on a _p (t) every cycle (time width τ =2π/Ω _m ), the waveform of a _pb (t) is a rectangular wave R _τ (t) with a pulse width of τ. is given by equation (5). Here, γ represents the amplitude of the rectangular wave and represents the height of the offset. The spectrum A _pb (ω) of a _pb (t) is given by Equation (6) from the Fourier transform of Equation (5). Here, the first item on the right side, A _p (ω), is the Fourier transform of a _p (t), and the second item on the right side, the sinc function 2sin(ωτ/2)/ω, is the Fourier transform of the rectangular wave R _τ (t). Therefore, the transfer function F _Fpb (ω) of the optical filter 4 for generating a _pb (t) is given by equation (7) by substituting equation (6) into equation (1).

Similarly, for the positive dark pulse shown in FIG. 1(c), its amplitude a _pd (t) and spectrum A _pd (ω) are given by equations (8) and (9), respectively. Therefore, the transfer function F _Fpb (ω) of the optical filter 4 for generating a _pd (t) is given by equation (10) by substituting equation (9) into equation (1).

Similarly, for the negative bright pulse and negative dark pulse shown in FIGS. 1(e) and 1(f), respectively, their amplitudes a _nb (t), a _nd (t) and spectra A _nb (ω) , A _nd (ω) are given by equations (11), (12), (13), and (14), respectively. Therefore, the transfer function F _Fpb (ω) of the optical filter 4 for generating a _nb (t) and a _nd (t) can be calculated by substituting equations (13) and (14) into equation (1), and formula ( 15) and equation (16).

In other words, the spectrum A(ω) of the pulse to be output is the transfer function F _Fpb (ω) of the optical filter 4, and the sinc function S(ω) = sin(ωτ/2)/ω(τ= 2π/ Ω _m ), and the functions A(ω-nΩ _{m )} , A(ω+nΩ _m ), S(ω -nΩ _m ), S(ω+nΩ _m ) (n: integer).

[First embodiment]
In a first embodiment of the invention, positive or negative bright or dark Gaussian pulses can be generated. The waveforms a _p (t), a _n (t) of the positive and negative Gaussian pulses are given by equation (17). From the Fourier transform of equation (17), the spectra A _p (ω) and A _n (ω) of positive and negative Gaussian pulses are given by equation (18). Substituting equation (18) into equations (7), (10), (15), and (16), we obtain the positive bright Gaussian pulse, positive dark Gaussian pulse, negative bright Gaussian pulse, and negative dark Gaussian pulse, respectively. The shape of the optical filter 4 to be generated can be determined.

For example, the transfer function F _Fpb (ω) of the optical filter 4 for generating a positive bright Gaussian pulse is given by equation (19). Further, the transfer function F _Fpd (ω) of the optical filter 4 for generating a positive dark Gaussian pulse is given by equation (20). By inserting these optical filters into the resonator, positive bright or dark Gaussian pulses can be generated. Optical filters F _Fnb (ω) and F _Fnd (ω) for generating negative bright and dark pulses are calculated using F Fpb (ω) _{and F Fpd (} ω) of equations (19) and (20), It is given by F _Fnb (ω) = - F _Fpd (ω), F _Fnd (ω) = - F _Fpb (ω).

In this embodiment, an example of the shape of the transfer function F _Fpb (ω) of the optical filter 4 used to generate a positive bright Gaussian pulse is shown in FIG. 3(a). Here, pulse width T = 6 ps, modulation angular frequency Ω _m = 2π×10 GHz, offset amplitude γ = 1, and modulation index M _PM = 2. The black dots in the figure represent frequencies where longitudinal modes (10 GHz intervals) of the optical spectrum exist. F _Fpb (ω) is given by a complex function, and the black line in FIG. 3(a) shows its absolute value |F _Fpb (ω)|, and the gray line shows the phase arg F _Fpb (ω). F _Fpb (ω) may be band-limited to the extent that the waveform is not impaired. In FIG. 3(a), a band limit of 200 GHz (±100 GHz) is given. FIG. 4(a) shows the results obtained by computer analysis of the steady state solution of the laser using this filter. The black line in FIG. 4(a) shows the waveform a _pb (t) of the steady pulse, and the gray line shows its phase arg a _pb (t). It can be seen that the phase of the steady pulse is almost constant, and a bright Gaussian pulse without chirp can be output.

Next, in this embodiment, an example of the shape of the transfer function F _Fpd (ω) of the optical filter 4 used to generate a positive dark Gaussian pulse is shown in FIG. 3(b). Here, pulse width T = 6 ps, modulation angular frequency Ω _m = 2π×10 GHz, offset amplitude γ = 1, and modulation index M _PM = 1. Here, F _Fpd (ω) is given a band limit of 200 GHz (±100 GHz). The steady state solution of the laser was obtained by computer analysis using this filter, and the results are shown in FIG. 4(b). The black line in FIG. 4(b) shows the waveform a _pd (t) of the steady pulse, and the gray line shows its phase arg a _pd (t). It can be seen that the phase of the steady pulse is almost constant, and a dark Gaussian pulse without chirp can be output.

Furthermore, in this embodiment, examples of the shapes of the transfer functions F _Fnb (Ω) and F _Fnd (Ω) of the optical filter 4 used to generate negative bright Gaussian pulses and dark Gaussian pulses are shown in FIGS. d). The results obtained by computer analysis of the steady state solution of the laser using this filter are shown in FIGS. 4(c) and 4(d). The steady pulse has a phase difference of π compared to a _pd (t) in Fig. 4(b) and a _pb (t) in Fig. 4(a), and negative bright/dark Gaussian pulses can be output. I understand.

[Second embodiment]
In a second embodiment of the invention, positive or negative bright or dark sech pulses can be generated. The waveforms a _p (t), a _n (t) of the positive and negative sech pulses are given by equation (21). From the Fourier transform of equation (21), the spectra A _p (ω) and A _n (ω) of positive and negative sech pulses are given by equation (22). Substituting equation (22) into equations (7), (10), (15), and (16), we obtain the positive bright sech pulse, positive dark sech pulse, negative bright sech pulse, and negative dark sech pulse, respectively. The shape of the optical filter 4 to be generated can be determined.

For example, the transfer function F _Fpb (ω) of the optical filter 4 for generating a positive bright sech pulse is given by equation (23). Further, the transfer function F _Fpd (ω) of the optical filter 4 for generating a positive dark sech pulse is given by equation (24). By inserting these optical filters into the resonator, positive bright or dark sech pulses can be generated. Optical filters F _Fnb (ω) and F _Fnd (ω) for generating negative bright and dark pulses are calculated using F Fpb (ω) _{and F Fpd (} ω) of equations (23) and (24), It is given by F _Fnb (ω) = - F _Fpd (ω), F _Fnd (ω) = - F _Fpb (ω).

In this embodiment, an example of the shape of the transfer function F _Fpb (ω) of the optical filter 4 used to generate the positive bright sech pulse is shown in FIG. 5(a). Here, pulse width T = 6 ps, modulation angular frequency Ω _m = 2π×10 GHz, offset amplitude γ = 1, and modulation index M _PM = 2. Here, F _Fpb (ω) is given a band limit of 200 GHz (±100 GHz). The black dots in the figure represent the frequencies where the longitudinal modes (10 GHz intervals) of the optical spectrum exist. F _Fpb (ω) is given by a complex function, and the black line in FIG. 5(a) shows its absolute value |F _Fpb (ω)|, and the gray line shows the phase arg F _Fpb (ω). FIG. 6(a) shows the results obtained by computer analysis of the steady state solution of the laser using this filter. The black line in FIG. 6(a) shows the waveform a _pb (t) of the steady pulse, and the gray line shows its phase arg a _pb (t). It can be seen that the phase of the steady pulse is almost constant, and a bright sech pulse without chirp can be output.

Next, in this embodiment, an example of the shape of the transfer function F _Fpd (ω) of the optical filter 4 used to generate the positive dark sech pulse is shown in FIG. 5(b). Here, pulse width T = 6 ps, modulation angular frequency Ω _m = 2π×10 GHz, offset amplitude γ = 1, and modulation index M _PM = 1. Here, F _Fpd (ω) is given a band limit of 200 GHz (±100 GHz). FIG. 6(b) shows the results obtained by computer analysis of the steady state solution of the laser using this filter. The black line in FIG. 6(b) shows the waveform a _pd (t) of the steady pulse, and the gray line shows its phase arg a _pd (t). It can be seen that the phase of the steady pulse is almost constant, and a chirp-free dark sech pulse can be output.

Furthermore, in this embodiment, examples of the shapes of the transfer functions F _Fnb (ω) and F _Fnd (ω) of the optical filter 4 used to generate negative bright sech pulses and dark sech pulses are shown in FIGS. d). Using this filter, the steady state solution of the laser was obtained by computer analysis, and the results are shown in FIGS. 6(c) and 6(d). The steady pulse has a phase difference of π compared to a _pd (t) in Fig. 6(b) and a _pb (t) in Fig. 6(a), and negative bright/dark sech pulses can be output. I understand.

[Third embodiment]
In a third embodiment of the invention, positive or negative bright or dark biexponential pulses can be generated. The waveforms a _p (t), a _n (t) of both positive and negative exponential pulses are given by equation (25). From the Fourier transform of equation (25), the spectra A _p (ω) and A _n (ω) of both positive and negative exponential pulses are given by equation (26) (Lorentz function, which is the Fourier transform of both exponential functions). It will be done. Substituting equation (26) into equations (7), (10), (15), and (16) yields a positive bright biexponential pulse, a positive dark biexponential pulse, a negative bright biexponential pulse, and a negative bright biexponential pulse. It is possible to determine the shape of the optical filter 4 for generating the bright biexponential pulses.

For example, the transfer function F _Fpb (ω) of the optical filter 4 for generating a positive bright biexponential pulse is given by equation (27). Further, the transfer function F _Fpd (ω) of the optical filter 4 for generating a positive dark biexponential pulse is given by equation (28). By inserting these optical filters into the resonator, positive bright or dark biexponential pulses can be generated. Optical filters F _Fnb (ω) and F _Fnd (ω) for generating negative bright _and dark pulses can _be calculated using It is given by _Fnb (ω) = - F _Fpd (ω), F _Fnd (ω) = - F _Fpb (ω).

In this embodiment, an example of the shape of the transfer function F _Fpb (ω) of the optical filter 4 used to generate the positive bright biexponential pulse is shown in FIG. 7(a). Here, pulse width T = 6.25 ps, modulation angular frequency Ω _m = 2π×10 GHz, offset amplitude γ = 1, and modulation index M _PM = 2. Here, F _Fpb (ω) is given a band limit of 440 GHz (±220 GHz). The black dots in the figure represent frequencies where longitudinal modes (10 GHz intervals) of the optical spectrum exist. F _Fpb (ω) is given by a complex function, and the black line in FIG. 7(a) shows its absolute value |F _Fpb (ω)|, and the gray line shows the phase arg F _Fpb (ω). FIG. 8(a) shows the results obtained by computer analysis of the steady state solution of the laser using this filter. The black line in FIG. 8(a) shows the waveform a _pb (t) of the steady pulse, and the gray line shows its phase arg a _pb (t). It can be seen that the phase of the steady pulse is almost constant, and a bright biexponential pulse without chirp can be output.

Next, in this embodiment, an example of the shape of the transfer function F _Fpd (ω) of the optical filter 4 used to generate the positive dark biexponential pulse is shown in FIG. 7(b). Here, pulse width T = 6.25 ps, modulation angular frequency Ω _m = 2π×10 GHz, offset amplitude γ = 1, and modulation index M _PM = 1. Here, a band limit of 440 GHz (±220 GHz) is given to F _Fpd (ω). FIG. 8(b) shows the results obtained by computer analysis of the steady state solution of the laser using this filter. The black line in FIG. 8(b) shows the waveform a _pd (t) of the steady pulse, and the gray line shows its phase arg a _pd (t). It can be seen that the phase of the steady pulse is almost constant, and a chirp-free dark biexponential pulse can be output.

Furthermore, in this embodiment, an example of the shapes of the transfer functions F _Fnb (ω) and F _Fnd (ω) of the optical filter 4 used to generate negative bright biexponential pulses and dark biexponential pulses is shown in FIG. They are shown in c) and (d), respectively. Using this filter, the steady state solution of the laser was obtained by computer analysis, and the results are shown in FIGS. 8(c) and 8(d). The steady pulse has a phase difference of π compared to a _pd (t) in Figure 8(b) and a _pb (t) in Figure 8(a), and the negative bright and dark exponential pulses are It turns out that it can be output.

[Fourth embodiment]
In a fourth embodiment of the invention, positive or negative bright or dark pulses can be generated whose electric field amplitude is defined by a triangle. The waveforms a _p (t) and a _n (t) of the positive and negative triangular pulses are given by equation (29). From the Fourier transform of equation (29), the spectra A _p (ω) and A _n (ω) of the positive and negative triangular pulses are given by equation (30) (square of the sinc function). Substituting equation (30) into equations (7), (10), (15), and (16), we obtain the positive bright triangular pulse, positive dark triangular pulse, negative bright triangular pulse, and negative bright triangular pulse, respectively. The shape of the optical filter 4 to be generated can be found.

For example, the transfer function F _Fpb (ω) of the optical filter 4 for generating a positive bright triangular pulse is given by equation (31). Further, the transfer function F _Fpd (ω) of the optical filter 4 for generating a positive dark triangular pulse is given by equation (32). By inserting these optical filters into the resonator, positive bright or dark triangular pulses can be generated. Optical filters F _Fnb (ω) and F _Fnd (ω) for generating negative bright _and dark pulses can _be calculated using It is given by _Fnb (ω) = - F _Fpd (ω), F _Fnd (ω) = - F _Fpb (ω).

In this embodiment, an example of the shape of the transfer function F _Fpb (ω) of the optical filter 4 used to generate the positive bright triangular pulse is shown in FIG. 9(a). Here, pulse width T = 25 ps, modulation angular frequency Ω _m = 2π×10 GHz, offset amplitude γ = 1, and modulation index M _PM = 2.4. Here, F _Fpd (ω) is given a band limit of 240 GHz (±120 GHz). The black dots in the figure represent frequencies where longitudinal modes (10 GHz intervals) of the optical spectrum exist. F _Fpb (ω) is given by a complex function, and the black line in FIG. 9(a) shows its absolute value |F _Fpb (ω)|, and the gray line shows the phase arg F _Fpb (ω). FIG. 10(a) shows the results obtained by computer analysis of the steady state solution of the laser using this filter. The black line in FIG. 10(a) shows the waveform a _pb (t) of the steady pulse, and the gray line shows its phase arg a _pb (t). It can be seen that the phase of the steady pulse is almost constant, and a bright triangular pulse without chirp can be output.

Next, in this embodiment, an example of the shape of the transfer function F _Fpd (ω) of the optical filter 4 used to generate the positive dark triangular pulse is shown in FIG. 9(b). Here, pulse width T = 25 ps, modulation angular frequency Ω _m = 2π×10 GHz, offset amplitude γ = 1, and modulation index M _PM = 2. Here, F _Fpd (ω) is given a band limit of 240 GHz (±120 GHz). FIG. 10(b) shows the results obtained by computer analysis of the steady state solution of the laser using this filter. The black line in FIG. 10(b) shows the waveform a _pd (t) of the steady pulse, and the gray line shows its phase arg a _pd (t). It can be seen that the phase of the steady pulse is almost constant, and a dark triangular pulse without chirp can be output.

Furthermore, in this embodiment, examples of the shapes of the transfer functions F _Fnb (ω) and F _Fnd (ω) of the optical filter 4 used to generate negative bright triangular pulses and dark triangular pulses are shown in FIGS. d). The results obtained by computer analysis of the steady state solution of the laser using this filter are shown in FIGS. 10(c) and 10(d). The steady pulse has a phase difference of π compared to a _pd (t) in Fig. 10(b) and a _pb (t) in Fig. 10(a), and it is possible to output negative bright/dark triangular pulses. I understand.

[Fifth embodiment]
In the fifth embodiment of the present invention, instead of the triangular pulse defined by the electric field amplitude in the fourth embodiment, a positive or negative bright or dark pulse whose intensity (square of the electric field) is given by a triangle is generated. I can do it. The waveform (electric field amplitude) a _pb (t) of the positive bright pulse is given by equation (33). At this time, |a _pb (t)| ² represents a bright triangular pulse. From the Fourier transform of equation (33), the spectrum A _pb (ω) of this pulse is given by equation (34) (series of Bessel function of the first kind J _2k (ω) (k: integer)). Here, φ = cos ^-1 (1/(1+γ)). In particular, when γ = 1, φ = π/3, and equation (34) is expressed by equation (35). By substituting equation (34) or (35) into equation (7), the shape F _Fpb (ω) of the optical filter 4 for generating a positive bright pulse whose electric field strength is given by a triangle can be determined.

Next, the waveform (electric field amplitude) a _pd (t) of the positive dark pulse is given by equation (36). At this time, |a _pd (t)| ² represents the dark triangular pulse. From the Fourier transform of equation (36), the spectrum A _pd (ω) of this pulse is given by equation (37). Here, φ = cos ^-1 (1/(1+γ)). In particular, when γ = 1, φ = π/2, and equation (37) is expressed by equation (38). By substituting equation (37) or (38) into equation (10), the shape F _Fpd (ω) of the optical filter 4 for generating a positive dark pulse whose electric field strength is given by a triangle can be determined.

The transfer function F _Fnb (ω) of the optical filter 4 for generating a negative bright pulse is obtained by inverting the sign of the last ω ^-1 term from + to - in equation (37) or equation (38). It can be found by substituting into equation (10). The transfer function F _Fnd (ω) of the optical filter 4 for generating a negative dark pulse is obtained by inverting the sign of the last ω ^-1 term from + to - in equation (34) or equation (35). It can be found by substituting into equation (7).

In this embodiment, an example of the shape of the transfer function F _Fpb (ω) of the optical filter 4 used to generate a positive bright pulse whose electric field strength is given by a triangle is shown in FIG. 11(a). Here, pulse width T = 20 ps, modulation angular frequency Ω _m = 2π×10 GHz, offset amplitude γ = 1, and modulation index M _PM = 2. Here, F _Fpd (ω) is given a band limit of 240 GHz (±120 GHz). The black dots in the figure represent frequencies where longitudinal modes (10 GHz intervals) of the optical spectrum exist. F _Fpb (ω) is given by a complex function, and the black line in FIG. 11(a) indicates its absolute value |F _Fpb (ω)|, and the gray line indicates the phase arg F _Fpb (ω). FIG. 12(a) shows the results obtained by computer analysis of the steady state solution of the laser using this filter. The black line in FIG. 12(a) shows the waveform a _pb (t) of the steady pulse, and the gray line shows its intensity |a _pb (t)| ² . As shown by the gray line and the right vertical axis in FIG. 12(a), it can be seen that a positive bright pulse with a triangular electric field strength is obtained.

Next, in this embodiment, an example of the shape of the transfer function F _Fpd (ω) of the optical filter 4 used to generate a positive dark pulse whose electric field strength is given by a triangle is shown in FIG. 11(b). Here, the pulse width T = 20 ps, the modulation angular frequency Ω _m = 2π × 10 GHz, the offset amplitude γ = 1, and the modulation index M _PM = 1. Here, F _Fpd (ω) is given a band limit of 240 GHz (±120 GHz). The black dots in the figure represent frequencies where longitudinal modes (10 GHz intervals) of the optical spectrum exist. F _Fpd (ω) is given by a complex function, and the black line in FIG. 11(b) shows its absolute value |F _Fpd (ω)|, and the gray line shows the phase arg F _Fpd (ω). FIG. 12(b) shows the results obtained by computer analysis of the steady state solution of the laser using this filter. The black line in FIG. 12(b) shows the waveform a _pd (t) of the steady pulse, and the gray line shows its intensity |a _pd (t)| ² . As shown by the gray line and the right vertical axis in FIG. 12(b), it can be seen that a positive dark pulse with a triangular electric field strength is obtained.

Furthermore, in this embodiment, an example of the shape of the transfer functions F _Fnb (ω) and F _Fnd (ω) of the optical filter 4 used to generate negative bright pulses and dark pulses whose electric field strength is given by a triangle is shown in FIG. They are shown in (c) and (d), respectively. The results of a steady-state laser solution obtained by computer analysis using this filter are shown in FIGS. 12(c) and 12(d). The steady pulse has a waveform that is the phase inversion of a _pd (t) in Fig. 12(b) and a _pb (t) in Fig. 12(a), and the gray lines and As shown by the vertical axis on the right, it can be seen that a negative bright/dark pulse with a triangular electric field strength can be output.

[Sixth embodiment]
In a sixth embodiment of the invention, it is possible to generate positive or negative bright or dark pulses in which the electric field amplitude is given by a parabola. The waveforms a _p (t), a _n (t) of the positive and negative parabolic pulses are given by equation (39). From the Fourier transform of equation (39), the spectra A _p (ω) and A _n (ω) of positive and negative parabolic pulses can be expressed as equation (40) (the sum of the functions sin ω/ω ³ and cos ω/ω ² or (difference). Substituting equation (40) into equations (7), (10), (15), and (16), we obtain positive bright parabolic pulse, positive dark parabolic pulse, negative bright parabolic pulse, and negative bright parabolic pulse, respectively. The shape of the optical filter 4 to be generated can be found.

For example, the transfer function F _Fpb (ω) of the optical filter 4 for generating a positive bright parabolic pulse is given by equation (41). Further, the transfer function F _Fpd (ω) of the optical filter 4 for generating a positive dark parabolic pulse is given by equation (42). By inserting these optical filters into the resonator, positive bright or dark parabolic pulses can be generated. Optical filters F _Fnb (ω) and F _Fnd (ω) for generating negative bright and dark pulses can be calculated using F Fpb (ω) _{and F Fpd (} ω) in equations (41) and (42). It is given by _Fnb (ω) = - F _Fpd (ω), F _Fnd (ω) = - F _Fpb (ω).

In this embodiment, an example of the shape of the transfer function F _Fpb (ω) of the optical filter 4 used to generate a positive bright parabolic pulse is shown in FIG. 13(a). Here, pulse width T = 25 ps, modulation angular frequency Ω _m = 2π×10 GHz, offset amplitude γ = 1, and modulation index M _PM = 2. Here, F _Fpb (ω) is given a band limit of 640 GHz (±320 GHz). The black dots in the figure represent frequencies where longitudinal modes (10 GHz intervals) of the optical spectrum exist. F _Fpb (ω) is given by a complex function, and the black line in FIG. 13(a) indicates its absolute value |F _Fpb (ω)|, and the gray line indicates the phase arg F _Fpb (ω). FIG. 14(a) shows the results obtained by computer analysis of the steady state solution of the laser using this filter. The black line in FIG. 14(a) shows the waveform a _pb (t) of the steady pulse, and the gray line shows its phase arg a _pb (t). It can be seen that the phase of the steady pulse is almost constant, and a bright parabolic pulse without chirp can be output.

Next, in this embodiment, an example of the shape of the transfer function F _Fpd (ω) of the optical filter 4 used to generate a positive dark parabolic pulse is shown in FIG. 13(b). Here, pulse width T = 25 ps, modulation angular frequency Ω _m = 2π × 10 GHz, offset amplitude γ = 1, and modulation index M _PM = 1. Here, F _Fpd (ω) is given a band limit of 640 GHz (±320 GHz). FIG. 14(b) shows the results obtained by computer analysis of the steady state solution of the laser using this filter. The black line in FIG. 14(b) shows the waveform a _pd (t) of the steady pulse, and the gray line shows its phase arg a _pd (t). It can be seen that the phase of the steady pulse is almost constant, and a dark parabolic pulse without chirp can be output.

Furthermore, in this embodiment, examples of the shapes of the transfer functions F _Fnb (ω) and F _Fnd (ω) of the optical filter 4 used to generate negative bright parabolic pulses and dark parabolic pulses are shown in FIGS. d). The results obtained by computer analysis of the steady state solution of the laser using this filter are shown in FIGS. 14(c) and 14(d). The steady pulse has a phase difference of π compared to a _pd (t) in Fig. 14(b) and a _pb (t) in Fig. 14(a), and it is possible to output negative bright/dark parabolic pulses. I understand.

[Seventh embodiment]
In the seventh embodiment of the present invention, instead of the parabolic pulse defined by the electric field amplitude in the sixth embodiment, a positive or negative bright or dark pulse whose intensity (square of the electric field) is given by a parabola is generated. I can do it. The waveform (electric field amplitude) a _pb (t) of the positive bright pulse is given by equation (43). At this time, |a _pb (t)| ² represents a bright parabolic pulse. From the Fourier transform of Equation (43), the spectrum A _pb (ω) of this pulse is expressed as Equation (44) (function J ₁ (ω)/ω (J ₁ (ω): first-order Bessel function of the first kind)) is given by Here, φ = sin ^-1 (1/(1+γ) ^1/2 ). In particular, when γ = 1, φ = π/4, and equation (44) is expressed by equation (45). By substituting equation (44) or equation (45) into equation (7), the shape F _Fpb (ω) of the optical filter 4 for generating a positive bright pulse whose electric field strength is given by a parabola can be determined.

Next, the waveform (electric field amplitude) a _pd (t) of the positive dark pulse is given by equation (46). At this time, |a _pd (t)| ² represents the dark parabolic pulse. From the Fourier transform of equation (46), the spectrum A _pd (ω) of this pulse is given by equation (47). Here, φ = sin ^-1 (1/(1+γ) ^1/2 ). In particular, when γ = 1, φ = π/2, and equation (47) is expressed by equation (48). By substituting equation (47) or equation (48) into equation (10), the shape F _Fpd (ω) of the optical filter 4 for generating a positive dark pulse whose electric field strength is given by a parabola can be determined.

The transfer function F _Fnb (ω) of the optical filter 4 for generating a negative bright pulse is obtained by inverting the sign of the last ω ⁻¹ term from + to − in equation (47) or equation (48). It can be found by substituting into equation (10). The transfer function F _Fnd (ω) of the optical filter 4 for generating a negative dark pulse is obtained by inverting the sign of the last ω ^-1 term from + to - in equation (44) or equation (45). It can be found by substituting into equation (7).

In this embodiment, an example of the shape of the transfer function F _Fpb (ω) of the optical filter 4 used to generate a positive bright pulse whose electric field strength is given by a parabola is shown in FIG. 15(a). Here, pulse width T = 25 ps, modulation angular frequency Ω _m = 2π×10 GHz, offset amplitude γ = 1, and modulation index M _PM = 2. Here, F _Fpb (ω) is given a band limit of 240 GHz (±120 GHz). The black dots in the figure represent frequencies where longitudinal modes (10 GHz intervals) of the optical spectrum exist. F _Fpb (ω) is given by a complex function, and the black line in FIG. 15(a) shows its absolute value |F _Fpb (ω)|, and the gray line shows the phase arg F _Fpb (ω). FIG. 16(a) shows the results obtained by computer analysis of the steady state solution of the laser using this filter. The black line in FIG. 16(a) shows the waveform a _pb (t) of the steady pulse, and the gray line shows its intensity |a _pb (t)| ² . As shown by the gray line and the right vertical axis in FIG. 16(a), it can be seen that a positive bright pulse with a parabolic electric field strength is obtained.

Next, in this embodiment, an example of the shape of the transfer function F _Fpd (ω) of the optical filter 4 used to generate a positive dark pulse whose electric field strength is given by a parabola is shown in FIG. 15(b). Here, pulse width T = 25 ps, modulation angular frequency Ω _m = 2π×10 GHz, offset amplitude γ = 1, and modulation index M _PM = 2. Here, F _Fpd (ω) is given a band limit of 240 GHz (±120 GHz). The black dots in the figure represent frequencies where longitudinal modes (10 GHz intervals) of the optical spectrum exist. F _Fpd (ω) is given by a complex function, and the black line in FIG. 15(b) shows its absolute value |F _Fpd (ω)|, and the gray line shows the phase arg F _Fpd (ω). The steady state solution of the laser was obtained by computer analysis using this filter, and the results are shown in FIG. 16(b). The black line in FIG. 16(b) shows the waveform a _pd (t) of the steady pulse, and the gray line shows its intensity |a _pd (t)| ² . As shown by the gray line and the right vertical axis in FIG. 16(b), it can be seen that a positive dark pulse with a parabolic electric field strength is obtained.

Furthermore, in this embodiment, an example of the shape of the transfer functions F _Fnb (ω) and F _Fnd (ω) of the optical filter 4 used to generate negative bright pulses and dark pulses whose electric field strength is given by parabola is shown in FIG. They are shown in (c) and (d), respectively. The results obtained by computer analysis of the steady state solution of the laser using this filter are shown in FIGS. 16(c) and 16(d). The steady pulse has a waveform that is the phase inversion of a _pd (t) in Fig. 16(b) and a _pb (t) in Fig. 16(a), and the gray line in Fig. 16(c) and (d) As shown by the vertical axis on the right, it can be seen that negative bright/dark pulses with parabolic electric field strength can be output.

[Eighth embodiment]
In an eighth embodiment of the invention, positive or negative bright or dark rectangular pulses can be generated. The waveforms a _p (t) and a _n (t) of the positive and negative rectangular pulses are given by equation (49). From the Fourier transform of Equation (49), the spectra A _p (ω) and A _n (ω) of positive and negative rectangular pulses are expressed as Equation (50) (sin(ωT)/(ωT) given by pulse width T) is given by Substituting equation (50) into equations (7), (10), (15), and (16), we obtain the positive bright rectangular pulse, positive dark rectangular pulse, negative bright rectangular pulse, and negative bright rectangular pulse, respectively. The shape of the optical filter 4 to be generated can be determined.

For example, the transfer function F _Fpb (ω) of the optical filter 4 for generating a positive bright rectangular pulse is given by equation (51). Further, the transfer function F _Fpd (ω) of the optical filter 4 for generating a positive dark rectangular pulse is given by equation (52). By inserting these optical filters into the resonator, positive bright or dark rectangular pulses can be generated. Optical filters F _Fnb (ω) and F _Fnd (ω) for generating negative bright and dark pulses can be calculated using F Fpb (ω) _{and F Fpd (} ω) in equations (51) and (52). It is given by _Fnb (ω) = - F _Fpd (ω), F _Fnd (ω) = - F _Fpb (ω).

In this embodiment, an example of the shape of the transfer function F _Fpb (ω) of the optical filter 4 used to generate the positive bright rectangular pulse is shown in FIG. 17(a). Here, pulse width T = 25 ps, modulation angular frequency Ω _m = 2π × 10 GHz, offset amplitude γ = 1, and modulation index M _PM = 1. Here, F _Fpb (ω) is given a band limit of 640 GHz (±320 GHz). The black dots in the figure represent frequencies where longitudinal modes (10 GHz intervals) of the optical spectrum exist. F _Fpb (ω) is given by a complex function, and the black line in FIG. 17(a) indicates its absolute value |F _Fpb (ω)|, and the gray line indicates the phase arg F _Fpb (ω). FIG. 18(a) shows the results obtained by computer analysis of the steady state solution of the laser using this filter. The black line in FIG. 18(a) shows the waveform a _pb (t) of the steady pulse, and the gray line shows its phase arg a _pb (t). It can be seen that the phase of the steady pulse is almost constant, and a bright rectangular pulse without chirp can be output.

Next, in this embodiment, an example of the shape of the transfer function F _Fpd (ω) of the optical filter 4 used to generate the positive dark rectangular pulse is shown in FIG. 17(b). Here, pulse width T = 25 ps, modulation angular frequency Ω _m = 2π×10 GHz, offset amplitude γ = 1, and modulation index M _PM = 1. Here, F _Fpd (ω) is given a band limit of 640 GHz (±320 GHz). FIG. 18(b) shows the results obtained by computer analysis of the steady state solution of the laser using this filter. The black line in FIG. 18(b) shows the waveform a _pd (t) of the steady pulse, and the gray line shows its phase arg a _pd (t). It can be seen that the phase of the steady pulse is almost constant, and a dark rectangular pulse without chirp can be output.

Furthermore, in this embodiment, examples of the shapes of the transfer functions F _Fnb (ω) and F _Fnd (ω) of the optical filter 4 used to generate negative bright rectangular pulses and dark rectangular pulses are shown in FIGS. d). The results obtained by computer analysis of the steady state solution of the laser using this filter are shown in FIGS. 18(c) and 18(d). The steady pulse has a phase difference of π compared to a _pd (t) in Fig. 18(b) and a _pb (t) in Fig. 18(a), and a negative bright/dark rectangular pulse can be output. I understand.

[Ninth embodiment]
In a ninth embodiment of the invention, positive or negative bright or dark Nyquist pulses can be generated. The waveforms a _p (t) and a _n (t) of the positive and negative Nyquist pulses are given by equation (53). Here, ω _N = π/T and α (0≦α≦1) are parameters called roll-off rate. From the Fourier transform of equation (53), the spectra A _p (ω) and A _n (ω) of positive and negative Nyquist pulses are given by equation (54). Substituting equation (54) into equations (7), (10), (15), and (16), we obtain the positive bright Nyquist pulse, positive dark Nyquist pulse, negative bright Nyquist pulse, and negative bright Nyquist pulse, respectively. The shape of the optical filter 4 to be generated can be found.

For example, the transfer function F _Fpb (ω) of the optical filter 4 for generating a positive bright Nyquist pulse is given by equation (55). Further, the transfer function F _Fpd (ω) of the optical filter 4 for generating a positive dark Nyquist pulse is given by equation (56). By inserting these optical filters into the resonator, positive bright or dark Nyquist pulses can be generated. Optical filters F _Fnb (ω) and F _Fnd (ω) for generating negative bright and dark pulses can _{be calculated using F Fpb (ω) and F Fpd (} ω) in equations (55) and (56). It is given by _Fnb (ω) = - F _Fpd (ω), F _Fnd (ω) = - F _Fpb (ω).

In this embodiment, an example of the shape of the transfer function F _Fpb (ω) of the optical filter 4 used to generate a positive bright Nyquist pulse is shown in FIG. 19(a). Here, T = 12.5 ps, modulation angular frequency Ω _m = 2π×10 GHz, offset amplitude γ = 1, and modulation index M _PM = 2. Here, F _Fpb (ω) is given a band limit by a Nyquist filter with α=0.08. The black dots in the figure represent frequencies where longitudinal modes (10 GHz intervals) of the optical spectrum exist. F _Fpb (ω) is given by a complex function, and the black line in FIG. 19(a) shows its absolute value |F _Fpb (ω)|, and the gray line shows the phase arg F _Fpb (ω). FIG. 20(a) shows the results obtained by computer analysis of the steady state solution of the laser using this filter. The black line in FIG. 20(a) shows the waveform a _pb (t) of the steady pulse, and the gray line shows its phase arg a _pb (t). It can be seen that the phase of the steady pulse is almost constant, and a bright Nyquist pulse without chirp can be output.

Next, in this embodiment, an example of the shape of the transfer function F _Fpd (ω) of the optical filter 4 used to generate the positive dark Nyquist pulse is shown in FIG. 19(b). Here, T = 12.5 ps, modulation angular frequency Ω _m = 2π × 10 GHz, offset amplitude γ = 1, and modulation index M _PM = 1. Here, F _Fpd (ω) is given a band limit by a Nyquist filter with α=0.08. FIG. 20(b) shows the results obtained by computer analysis of the steady state solution of the laser using this filter. The black line in FIG. 20(b) shows the waveform a _pd (t) of the steady pulse, and the gray line shows its phase arg a _pd (t). It can be seen that the phase of the steady pulse is almost constant, and a dark Nyquist pulse without chirp can be output.

Furthermore, in this embodiment, examples of the shapes of the transfer functions F _Fnb (ω) and F _Fnd (ω) of the optical filter 4 used to generate negative bright Nyquist pulses and dark Nyquist pulses are shown in FIGS. d). FIGS. 20(c) and 20(d) show the results obtained by computer analysis of the steady state solution of the laser using this filter. The steady pulse has a phase difference of π compared to a _pd (t) in Fig. 20(b) and a _pb (t) in Fig. 20(a), indicating that negative bright/dark Nyquist pulses can be output. I understand.

[Tenth embodiment]
In the tenth embodiment of the present invention, it is possible to generate dark solitons with an electric field amplitude of tanh and an electric field strength of 1 - sech ² . The dark soliton waveform d(t) defined by one cycle of pulse repetition (-τ/2 < t < τ/2) is given by equation (57). From the Fourier transform of equation (57), the spectrum D(ω) of the dark soliton is given by equation (58). From equation (58), the transfer function F _tanh (ω) of the optical filter 4 for generating dark solitons is given by equation (59). By inserting this optical filter into a resonator, dark solitons can be generated.

FIG. 21 shows an example of the shape of the transfer function F _tanh (ω) of the optical filter 4 used to generate dark solitons in this embodiment. Here, the pulse width T = 2.5 ps, the modulation angular frequency Ω _m = 2π × 10 GHz (repetition frequency 20 GHz), and the modulation index M _PM = 2.4. Here, F _tanh (ω) is given a band limit of 200 GHz (±100 GHz). The black dots in the figure represent frequencies where longitudinal modes (10 GHz intervals) of the optical spectrum exist. F _tanh (ω) is given by a complex function, and FIG. 21(a) shows its absolute value |F _tanh (ω)|, and FIG. 21(b) shows the phase arg F _tanh (ω). FIG. 22 shows the results obtained by computer analysis of the steady state solution of the laser using this filter. FIG. 22(a) shows the intensity |d(t)| ² of the steady pulse, and FIG. 22(b) shows its phase arg d(t). It can be seen that the phase of the steady pulse is almost constant, and dark solitons without chirp can be output. It can also be seen that the phase is reversed by π at t = ± 50 ps.

A specific experimental example will be shown below. In the experiment, an optical fiber ring resonator (resonator length 15 m) shown in Fig. 23 was used. The optical fiber ring resonator uses a high-power 1.48 μm semiconductor laser (LD; Laser Diode) 6 (a harmonic FM mode-locked erbium fiber laser with a repetition rate of 20 GHz that oscillates at a wavelength of 1.56 μm) and excitation from the semiconductor laser 6. A wavelength division multiplexing (WDM) coupler 7 that couples light to an optical fiber resonator, and a polarization-maintaining ervidium fiber 5 (fiber length is 5 m; Figure 2) that amplifies the pump light coupled by the WDM coupler 7. ), a PZT (lead zirconate titanate) element 10 that makes the resonator length variable, a coupler 11 that makes 20% of the laser output and the remaining 80% to the optical fiber resonator, and a coupler an isolator 12 that rectifies the light from 11; a phase modulator 8 (corresponding to the optical phase modulator 3 in FIG. 2) connected to the isolator 12, frequency synthesizer 16 (20 GHz), phase shifter 15, and amplifier 14, respectively; It consists of an LCoS element 13 (corresponding to the optical filter 4 in FIG. 2) that receives light from the phase modulator 8 via a wavelength filter (etalon) 9. Light from the LCoS element 13 is coupled by a WDM coupler 7.

The amplitude and phase characteristics of the filter functions F _Fpb (ω), F _Fpd (ω), F _Fnb (ω), and F _Fnd (ω) shown in the first to ninth embodiments are determined using software. It is mounted on the optical filter 4). The frequency resolution of the LCoS element is 1 GHz.

Below, the optical filter shapes and generated pulses of each embodiment of the present invention will be shown. In addition, the above-mentioned positive dark pulse and positive bright pulse will be explained below, but the negative bright pulse and negative dark pulse are different from those whose intensity waveforms are a positive dark pulse and a positive bright pulse, respectively. Since the intensity waveforms are the same, they will be omitted below.

FIG. 24 shows the shape of the optical filter used to generate the dark Gaussian pulse in the first embodiment of the present invention. Here, T = 3 ps and M _PM = 1. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that band-limits F _Fpd (ω) (see Figure 3(b)) in equation (20) at 400 GHz, and the black line is a filter that approximates this with a step function every 20 GHz and implements it in the LCoS element. It has a filter shape. Although the LCoS element used in the experiment originally has a frequency resolution of 1 GHz, F _Fpd (ω) was approximated in steps at every 20 GHz, which is the longitudinal mode interval, in consideration of fluctuations in the longitudinal mode frequency of the laser. , which implements this.

FIG. 25 shows the waveform and optical spectrum of a dark Gaussian pulse generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that the dark Gaussian pulse can be generated as designed using the optical filter of FIG.

Next, FIG. 26 shows the shape of the optical filter used to generate bright Gaussian pulses in the first embodiment of the present invention. Here, T = 3 ps and M _PM = 2. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that band-limits F _Fpb (ω) (see Figure 3 (a)) in equation (19) at 400 GHz, and the black line is a filter that approximates this with a step function every 20 GHz and implements it in the LCoS element. It has a filter shape.

FIG. 27 shows the waveform and optical spectrum of a bright Gaussian pulse generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that bright Gaussian pulses can be generated as designed using the optical filter of FIG.

FIG. 28 shows the shape of the optical filter used to generate the dark sech pulse in the second embodiment of the present invention. Here, T = 3 ps and M _PM = 1. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that band-limits F _Fpd (ω) (see Figure 5(b)) in equation (24) at 400 GHz, and the black line is a filter that approximates this with a step function every 20 GHz and implements it in the LCoS element. It has a filter shape.

FIG. 29 shows the waveform and optical spectrum of a dark sech pulse generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking its √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that the dark sech pulse can be generated as designed using the optical filter of FIG.

Next, FIG. 30 shows the shape of the optical filter used to generate the bright sech pulse in the second embodiment of the present invention. Here, T = 3 ps and M _PM = 2. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that band-limits F _Fpb (ω) (see Figure 5(a)) in Equation (23) at 400 GHz, and the black line is a filter that approximates this with a step function every 20 GHz and implements it in the LCoS element. It has a filter shape.

FIG. 31 shows the waveform and optical spectrum of a bright sech pulse generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that bright sech pulses can be generated as designed using the optical filter of FIG. 30.

FIG. 32 shows the shape of the optical filter used to generate the dark biexponential pulse in the third embodiment of the present invention. Here, T = 6.25 ps and M _PM = 1. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that limits the band of F _Fpd (ω) in equation (28) (see Figure 7 (b)) to 440 GHz, and the black line is a filter that approximates this with a step function every 20 GHz and implements it in the LCoS element. It has a filter shape.

FIG. 33 shows the waveform and optical spectrum of a dark biexponential pulse generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that the dark biexponential pulse can be generated as designed using the optical filter of FIG. 32.

Next, FIG. 34 shows the shape of the optical filter used to generate bright biexponential pulses in the third embodiment of the present invention. Here, T = 6.25 ps and M _PM = 2. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that band-limits F _Fpb (ω) (see Figure 7 (a)) in equation (27) at 440 GHz, and the black line is a filter that approximates this with a step function every 20 GHz and implements it in the LCoS element. It has a filter shape.

FIG. 35 shows the waveform and optical spectrum of a bright biexponential pulse generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that bright biexponential pulses can be generated as designed using the optical filter of FIG.

FIG. 36 shows the shape of the optical filter used to generate the dark triangular pulse defined by the electric field amplitude in the fourth embodiment of the present invention. Here, T = 12.5 ps and M _PM = 2. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that band-limits F _Fpd (ω) (see Figure 9(b)) in equation (32) at 320 GHz, and the black line is a filter that approximates this with a step function every 20 GHz and implements it in the LCoS element. It has a filter shape.

FIG. 37 shows the waveform and optical spectrum of a dark triangular pulse defined by the electric field amplitude generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that a dark triangular pulse defined by the electric field amplitude can be generated as designed using the optical filter of FIG. 36.

Next, FIG. 38 shows the shape of the optical filter used to generate the bright triangular pulse defined by the electric field amplitude in the fourth embodiment of the present invention. Here, T = 12.5 ps and M _PM = 2.4. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that band-limits F _Fpb (ω) (see Figure 9(a)) in Equation (31) to 320 GHz, and the black line is a filter that approximates this with a step function every 20 GHz and implements it in the LCoS element. It has a filter shape.

FIG. 39 shows the waveform and optical spectrum of a bright triangular pulse defined by the electric field amplitude generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that a bright triangular pulse defined by the electric field amplitude can be generated as designed using the optical filter of FIG.

FIG. 40 shows the shape of an optical filter used to generate a dark triangular pulse defined by intensity (square of electric field) in the fifth embodiment of the present invention. Here, T = 10 ps and M _PM = 1. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that band-limits F _Fpd (ω) (see Figure 11(b)) at 600 GHz, and the black line is the filter shape that is approximated by a step function every 20 GHz and implemented in the LCoS element. .

FIG. 41 shows the waveform and optical spectrum of a dark triangular pulse defined by intensity (square of electric field) generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that the dark triangular pulse defined by the intensity as designed can be generated using the optical filter of FIG.

Next, FIG. 42 shows the shape of an optical filter used to generate a bright triangular pulse defined by intensity (square of electric field) in the fifth embodiment of the present invention. Here, T = 10 ps and M _PM = 2. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that band-limits F _Fpb (ω) (see Figure 11(a)) to 540 GHz, and the black line is the filter shape that is approximated by a step function every 20 GHz and implemented in the LCoS element. .

FIG. 43 shows the waveform and optical spectrum of an intensity (square of electric field) bright triangular pulse generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that the bright triangular pulse defined by the intensity can be generated as designed using the optical filter of FIG.

FIG. 44 shows the shape of an optical filter used to generate a dark parabolic pulse defined by electric field amplitude in the sixth embodiment of the present invention. Here, T = 12.5 ps and M _PM = 1. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that band-limits F _Fpd (ω) (see Figure 13(b)) in equation (42) at 600 GHz, and the black line is a filter that approximates this with a step function every 20 GHz and implements it in the LCoS element. It has a filter shape.

FIG. 45 shows the waveform and optical spectrum of a dark parabolic pulse defined by the electric field amplitude generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that the dark parabolic pulse defined by the electric field amplitude can be generated as designed using the optical filter of FIG.

Next, FIG. 46 shows the shape of an optical filter used to generate a bright parabolic pulse defined by electric field amplitude in the sixth embodiment of the present invention. Here, T = 12.5 ps and M _PM = 2. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that band-limits F _Fpb (ω) (see Figure 13(a)) in equation (41) at 600 GHz, and the black line is a filter that approximates this with a step function every 20 GHz and implements it in the LCoS element. It has a filter shape.

FIG. 47 shows the waveform and optical spectrum of a bright parabolic pulse defined by the electric field amplitude generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that a bright parabolic pulse defined by the electric field amplitude can be generated as designed using the optical filter shown in FIG.

FIG. 48 shows the shape of an optical filter used to generate a dark parabolic pulse defined by intensity (square of electric field) in the seventh embodiment of the present invention. Here, T = 12.5 ps and M _PM = 2. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that band-limits F _Fpd (ω) (see Figure 15(b)) at 300 GHz, and the black line is the filter shape that is approximated by a step function every 20 GHz and implemented in the LCoS element. .

FIG. 49 shows the waveform and optical spectrum of a dark parabolic pulse defined by intensity (square of electric field) generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that a dark parabolic pulse defined by the intensity as designed can be generated using the optical filter of FIG. 48.

Next, FIG. 50 shows the shape of an optical filter used to generate a bright parabolic pulse defined by intensity (square of electric field) in the seventh embodiment of the present invention. Here, T = 10 ps and M _PM = 2. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that band-limits F _Fpb (ω) (see Figure 15(a)) at 600 GHz, and the black line is the filter shape that is approximated by a step function every 20 GHz and implemented in the LCoS element. .

FIG. 51 shows the waveform and optical spectrum of an intensity (square of electric field) bright parabolic pulse generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking its √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that bright parabolic pulses defined by the intensity as designed can be generated using the optical filter of FIG.

FIG. 52 shows the shape of the optical filter used to generate dark rectangular pulses in the eighth embodiment of the present invention. Here, T = 12.5 ps and M _PM = 1. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that band-limits F _Fpd (ω) (see Figure 17(b)) in equation (52) to 640 GHz, and the dashed line is a filter that is approximated by a step function every 20 GHz and implemented in the LCoS element. It has a filter shape.

FIG. 53 shows the waveform and optical spectrum of a dark rectangular pulse generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that dark rectangular pulses can be generated as designed using the optical filter of FIG.

Next, FIG. 54 shows the shape of the optical filter used to generate bright rectangular pulses in the eighth embodiment of the present invention. Here, T = 12.5 ps and M _PM = 2. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that band-limits F _Fpb (ω) (see Figure 17(a)) in equation (51) at 560 GHz, and the dashed line is a filter that is approximated by a step function every 20 GHz and implemented in the LCoS element. It has a filter shape.

FIG. 55 shows the waveform and optical spectrum of a bright rectangular pulse generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that bright rectangular pulses can be generated as designed using the optical filter of FIG.

FIG. 56 shows the shape of the optical filter used to generate the dark Nyquist pulse in the ninth embodiment of the present invention. Here, the roll-off rate is set to α = 0.5, T = 6.25 ps, and M _PM = 1. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that band-limits F _Fpd (ω) (see Figure 19(b)) in Equation (56) to 180 GHz, and the dashed line is a filter that is approximated by a step function every 20 GHz and implemented in the LCoS element. It has a filter shape.

FIG. 57 shows the waveform and optical spectrum of a dark Nyquist pulse generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking its √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that the dark Nyquist pulse can be generated as designed using the optical filter of FIG.

Next, FIG. 58 shows the shape of the optical filter used to generate bright Nyquist pulses in the ninth embodiment of the present invention. Here, the roll-off rate is set to α = 0.5, T = 6.25 ps, and M _PM = 1. In the same figure, (a) shows the amplitude characteristics, and (b) shows the phase characteristics. The gray line is a filter that band-limits F _Fpb (ω) (see Figure 19(a)) in Equation (55) to 180 GHz, and the dashed line is a filter that is approximated by a step function every 20 GHz and implemented in the LCoS element. It has a filter shape.

FIG. 59 shows the waveform and optical spectrum of a bright Nyquist pulse generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that bright Nyquist pulses can be generated as designed using the optical filter of FIG.

FIG. 60 shows the shape of the optical filter used to generate dark solitons in the tenth embodiment of the present invention. Here, T = 1.25 ps and M _PM = 2.4. In the figure, (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line is a filter that limits the band of F _tanh (ω) in Equation (59) (see Figure 21) to 540 GHz, and the dashed line is a filter shape that approximates this with a step function every 20 GHz and implements it in the LCoS element. be.

FIG. 61 shows the waveform and optical spectrum of a dark rectangular pulse generated using this filter. In the same figure, (a) is the intensity waveform a ² (t) observed with an optical sampling oscilloscope, (b) is the intensity waveform a 2 (t) obtained by taking √ and converted to the amplitude a(t), and (c) is the optical spectrum observed with an optical spectrum analyzer. A(ω) is expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the broken line is the result of computer analysis. The two agree well, and it can be seen that dark solitons can be generated as designed using the optical filter of FIG. 60.

As explained in detail above, the present invention can generate positive or negative bright or dark pulses with arbitrary time waveforms by appropriately designing the amplitude and phase characteristics of the optical filter inserted into the laser resonator. It can be easily generated from a mode-locked laser. The positive or negative bright or dark pulse train obtained by the present invention can be used in a wide range of applications, equipment, and systems such as ultrahigh-speed optical communication, optical measurement, and optical signal processing.

As a specific use of the positive or negative bright or dark pulse of the present invention, by superimposing the bright pulse and the dark pulse, the pulse signal can be erased, so it can be used to modify communication signals. There is sex. Furthermore, since the pulse signal can be erased by using bright pulses and dark pulses, there is a possibility that it can be applied to complex and robust encryption technology that is not available in conventional encryption technology. Furthermore, the use of bright pulses and dark pulses may open up new possibilities for encoding that have not existed in conventional communications.

1 Optical fiber 2 Optical amplifier 3 Optical phase modulator 4 Optical filter 5 Polarization-maintaining erbium fiber 6 Semiconductor laser 7 Wavelength division multiplexing (WDM) coupler 8 Phase modulator 9 Wavelength filter (etalon)
10 PZT element 11 Coupler 12 Isolator 13 LCoS element 14 Amplifier 15 Phase shifter 16 Frequency synthesizer

Claims (14)

It has a frequency modulation mode-locked laser that includes an optical phase modulator and an optical filter in a laser resonator,
By setting the amplitude and phase characteristics of the optical filter according to the desired optical pulse shape and continuous wave offset amount, the amplitude and the phase characteristics are variable, and the desired light having the continuous wave offset amount is A pulsed light source configured to generate dark pulses and bright pulses having a pulse shape. The optical phase modulator is driven by a sine wave with a repetition angular frequency Ω _m ,
The amplitude and phase characteristics of the optical filter are determined by the modulation degree of the optical phase modulator, the spectrum A(ω) of the pulse to be output, and the sinc function S(ω) = sin(ωτ/2 )/ω (τ= 2π/Ω _m ), and functions A(ω-nΩ _m ), A in which A(ω) and S(ω) are shifted positively or negatively by an integer multiple of the repetition angular frequency Ω _m The pulsed light source according to claim 1, wherein the pulsed light source is given using (ω+nΩ _m ), S(ω-nΩ _m ), S(ω+nΩ _m ), (n: integer). The optical filter generates a positive bright pulse and a positive dark pulse depending on the combination by taking the sign of the spectrum A(ω) and the sinc function S(ω) giving the continuous wave offset amount to be positive or negative. 3. The pulsed light source according to claim 2, wherein the pulsed light source is configured to generate four types of pulses: , negative bright pulse, and negative dark pulse. 4. The pulsed light source according to claim 3, wherein the optical filter is configured to generate a positive or negative bright or dark Gaussian pulse by making the spectrum A(ω) a Gaussian function. 4. The pulsed light source according to claim 3, wherein the optical filter is configured to generate a positive or negative bright or dark sech pulse by making the spectrum A(ω) a sech function. The optical filter is characterized in that it is configured to generate a positive or negative bright or dark pulse having a shape of the biexponential functions by subjecting the spectrum A(ω) to Fourier transform of the biexponential functions. 4. The pulsed light source according to claim 3. The optical filter is characterized in that it is configured to generate a positive or negative bright or dark pulse having a triangular electric field amplitude by making the spectrum A(ω) the square of a sinc function. The pulsed light source according to claim 3. The optical filter is configured such that the spectrum A(ω) is a series of Bessel functions of the first kind J _2k (ω) (k: an integer), so that the intensity as the square of the electric field has a positive or negative triangular shape. 4. The pulsed light source according to claim 3, wherein the pulsed light source is configured to generate a bright or dark pulse. The optical filter generates a positive or negative bright or dark pulse with a parabolic electric field amplitude by making the spectrum A(ω) the sum or difference of the functions sin ω/ω ³ and cos ω/ω ² . 4. The pulsed light source according to claim 3, wherein the pulsed light source is configured to generate a pulsed light source. The optical filter makes the spectrum A(ω) a function J ₁ (ω)/ω (J ₁ (ω): first-order Bessel function of the first kind), so that the intensity as the square of the electric field becomes parabolic. 4. The pulsed light source according to claim 3, wherein the pulsed light source is configured to generate a positive or negative bright or dark pulse having a (parabolic) shape. The optical filter is configured to generate a positive or negative bright or dark pulse having a rectangular shape by setting the spectrum A(ω) to sin(ωT)/(ωT) given by a pulse width T. 4. The pulsed light source according to claim 3, characterized in that: The optical filter is configured to generate a positive or negative bright or dark pulse having a shape of a Nyquist pulse by using the spectrum A(ω) as a transfer function of a Nyquist filter. Item 3. Pulsed light source according to item 3. The optical filter has a function A(ω) in which the spectrum A(ω) is cosech ω, cos ω/ω, sin ω, and the spectrum A(ω) is shifted in positive and negative directions by an integral multiple of the repetition angular frequency Ω _m . 2. The pulsed light source according to claim 1, wherein the pulsed light source is configured to generate dark solitons by giving −nΩ _m ) and A(ω+nΩ _m ). A system using the pulsed light source according to any one of claims 1 to 13.

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