DE10112021C1 - Adjustable optical level adjuster - Google Patents

Abstract

In order to equalize different optical power levels in individual wavelength ranges of an optical WDM network, wavelength-dependent level adjusters are used. Various solutions based on Mach-Zehnder filters or AWG multiplexers in connection with phase shifters are known, but have switching power values that are too high and either too coarse filter characteristics or damping values that are too high. Therefore, in the level adjuster (GE) according to the invention, a first waveguide (WG¶1¶) with a second waveguide (WG¶2¶), which have different refractive indices (n¶1¶, n¶2¶), is over two 3 dB Filter coupler (GC¶1¶, GC¶2¶) connected, in their coupling areas (CA) each a grating (G¶1¶, G¶2¶) for the selection of a wavelength range (lambda¶Ci¶, i = 1 ... m) is integrated. In the second waveguide (WG¶2¶) an adjustable phase shifter (PS) is integrated between the two filter couplers (GC¶1¶, GC¶2¶) outside their coupling areas (CA). In a particularly preferred hybrid embodiment of the waveguide, the advantage of low attenuation in a silica waveguide is combined with the advantage of easily adjustable switchability in a polymer waveguide. For simultaneous level adjustment in several wavelength ranges, level adjusters according to the invention can be connected in series with different filter curves.

Description

The invention relates to an adjustable optical level equalizer for wavelength-dependent adjustment of the power level in WDM networks with two couplers for optically coupling a first waveguide with one in the coupling area, a second waveguide running parallel to it, one filter function element arranged between the couplers for selection of a wavelength range around a central wavelength and between the couplers adjustable outside their coupling areas Phase shifter.

A capacity increase in existing optical networks for data transmission, in the preferred communications wavelengths ranges around 1300 nm and 1550 nm can only be done by enlarging the transferable data bit rate or by parallelizing the transmission process using wavelength division multiplex (WDM) will be realized. Developments in the field of erbium doped fiber amplifiers (EDFA) make the second option special attractive. One of the problems with the implementation of WDM networks poses however, the different transmission characteristics of the individual WDM Components. Their properties are only ideal for all shafts length equal. Wavelength-dependent gain profiles, saturation properties EDFA and inconsistent attenuation on the multiplex and Demultiplex components lead to the fact that every channel in the WDM network has different transmission behavior. For a largely error-free However, the function of the WDM network is an even distribution of power necessary on the individual channels. Approximation of power levels in the individual transmission channels is therefore an urgent task especially since the number of parallel transmission channels due to progressive  Reduction of the channel spacing continues to increase (the minimum Channel bandwidth is currently around 10 GHz).

In particular, the optical amplifiers integrated into the lines point wavelength-dependent gain profiles and thus influence that Transmission behavior of the routes essential. The transmitted signal performance differs in the individual channels, the transmission profile is not smooth. This can be compensated for by adjustable optical level adjusters that are particularly important for an improvement in transmission behavior low additional attenuation values, low polarization dependence and have to be dynamic. They should also be simple be built and manufactured at the lowest possible cost. Known adjustable level adjuster for level adjustment over a wide range load spectrum are based on Mach-Zehnder filters, acoustic-optical Filters, holograms, micromechanically driven mirrors, multiplex Demultiplex pairs with variable dampers or phase shifters.

From the article [I] "Tunable gain equalization using a Mach-Zehnder optical filter in multistage fiber amplifiers" (K. Inoue, T. Kominato, and H. Toba, IEEE Photon. Technol. Lett., Vol. 3, pp. 718-720, 1991) an adjustable level is known on the basis of silica (silica: doped silicon dioxide SiO ₂ ), in which two adjustable optical couplers are connected to one another via two waveguide sections of different lengths, one of which is used as an adjustable phase shifter. In the article [II] "Dynamic wave length equalizer in silica using the single-filtered-arm interferometer" (CR Doerr, M. Cappuzzo, E. Laskowski, A. Paunescu, L. Gomez, LW Stulz and J. Gates, IEEE Photon. Technol. Lett., Vol. 11, pp. 581-583, 1999) describes an adjustable, wavelength-dependent level adjuster in silica technology for WDM networks. This is made up of a Mach-Zehnder interferometer with two couplers and a multiplex-demultiplex pair (arrayed waveguide gratings AWG) that acts as an optical filter function element. The multiplex-demultiplex pair is connected via a large number of parallel and equally long, thermo-optically controlled phase shifters, via which the transmission of the level adjuster can be adjusted depending on the wavelength. The known level adjusters either select the wavelength too coarse [I] or add a high attenuation to the transmission path [II]. Furthermore, relatively high switching powers are required for the phase shifters.

Other key optical components in WDM networks are add / drop filters, that introduce and filter out one or more wavelengths in or allow from the fiber. It is known to add / drop filters in the structure to form a directional coupler filter. In such a filter are at least two closely adjacent waveguides with different ones Refractive indices arranged, of which at least one waveguide Has grid as a filter function element. This is preferably on the Waveguide with the larger refractive index applied. Such one Add / drop filter is for example in IEEE Phot. Technol. Lett., Vol. 4, No. 12 December 1992, pp. 1386-1389. Purely fiber optic solutions for Add / drop filters using silica for these fibers are in IEEE Phot. Technol. Lett., Vol. 8, No. December 12, 1996, pp. 1656-1658 described. Among other things, this state of the art is an optical one lattice-assisted add / drop filter with a hybrid material structure Subject of an older German patent application 100 25 307.5 des same applicant as for the present invention.

The object of the invention is one over closest prior art improved optical level equalizer with the lowest possible attenuation and sufficient wavelengths selectivity and the lowest possible switching capacity. In doing so this also has low polarization dependency, high dynamics and have good tunability and is simple and inexpensive be constructed.  

The solution to this problem is at the level according to the invention of the type described in the introduction therefore provided that everyone Coupler as a grid-supported 3 dB filter coupler between the two waves conductors with different refractive indices and the filter function element is in the form of two assigned grids, each in the coupling range of a 3 dB filter coupler are arranged, and that of the phases slider is integrated directly into one of the two waveguides. At a advantageous continuation of the level adjuster according to the invention is above also provided that the first waveguide as a silica waveguide and the second waveguide designed as a polymer waveguide and the adjustable Phase shifter is integrated in the polymer waveguide. Furthermore, can according to a next continuation of the polymer waveguides nonlinear have optical properties.

The level adjuster according to the invention solves the problem of improving optical power adjustment in WDM networks through an advantageous Combination of filter couplers, their filter functions through optical gratings be trained with a single phase shifter. According to the preferred embodiment, the particularly advantageous station wagon nation of silica technology with polymer technology (hybrid techno logie, compare DE 198 49 862 C1 "Thermo-optical switch") used become. The advantages can be seen here in particular in that the Very high refractive index difference between silica and polymer waveguides can be set high, creating a conception of a special compact level adjuster is possible. In addition, the good optical Tunability of the polymer technology combined with the low optical Attenuation of silica technology. The silica waveguide is called optical transmittive, passive layer, the polymer waveguide as an active layer for the Switching function and for the phase or wavelength shift function used. In particular, the thermo-optical effect in polymer waveguides is very pronounced, only low switching capacities at a correspond the control required. The amount to be discharged is correspondingly low  Amount of heat. Typical switching times are in the ms range. By Utilization of nonlinear optical effects can still significantly be undercut. All in all, the planned ones Measures in the level adjuster according to the invention are therefore too good tunable grid-assisted filter couplers with a phase shifter low electrical power consumption.

In the particularly advantageous embodiment of the invention wavelength-dependent level equalizer, the optical power in the Silica waveguide coupled. The two grid-supported 3 dB filters Couplers are set via the geometry so that half of the optical power of an optical wavelength defined by the grating range around a central wavelength in the polymer waveguide is coupled (coupling factor R = 0.5). The high associated with the bars Wavelength selectivity is in the level equalizer for exploited the filter function. Depending on how strong the wavelength band is to be damped, the phase of the signals in the phase shifter shifted between 0 and π between the two waveguides. By doing the second filter coupler overlap the portions of the filtered out Wavelength band again to complement each other in their amplitudes dampen or completely extinguish. The filter couplers therefore have two Functions that result from their special structure. You select for a predetermined wavelength band through the impressed filter curve and on the other hand divide or unite the optical power in it Wavelength range.

In a next advantageous embodiment of the level according to the invention it is also provided that the assigned grids are periodic Have refractive index variation caused by geometric or optical Measures is defined. The geometric measures are to those that, for example, by material removal to a comb-like Lead structure. The resulting gaps can be created with a  second material to be filled, which has a different refractive index having. With the optical methods the refractive index differences are for example inscribed by means of UV exposure. The grids can be on be located anywhere in the coupling area of the coupler, for example in the space between the two waveguides. Another one Design of the level adjuster according to the invention is advantageous provided that the assigned grids each directly in one of the two Waveguides are inscribed. There can be one on all the outside Waveguide can be arranged, especially if it is a planar Has cross section. More details on the different grids definition options and further configurations and dimensions Information on the waveguides can be of the older ones mentioned above German patent application 100 25 307.5 after its publication be removed.

In a further advantageous continuation of the level according to the invention it is also provided that the two waveguides are vertical to one another are arranged and a thin between the two waveguides Interface and one above and one below the two waveguides Passivation layer is provided. The separating layer ("gap") serves defined optical coupling between the two waveguides, whereas the two passivation layers remove the waveguide from that Isolate the substrate and the environment. With such a layer structure can then be provided according to a next continuation of the invention, that the central wavelength of the wavelength range to be selected is also adjustable by means of a control electrode, which is based on the passive tion layer, in particular arranged above the polymer waveguide is.

According to a next embodiment of the invention, it is possible that the Phase shifter is adjustable via a control that is on the Passivation layer, especially above the polymer waveguide,  is arranged. Good thermo-optical adjustability of the phase shifter can be realized by simply adding heat. The thermo-optical Effect is with many polymer materials that are considered for a molding Waveguides are particularly strong. That's why it is particularly useful if after a further training of the Level shifter according to the invention of the phase shifter thermo-optical is adjustable via a heating electrode as a control element. The electro-optical Effect also has a good effect and enables particularly low switching times. In the case of alternative further training is therefore provided that the phase shifter electro-optically via a control electrode is adjustable as a control element. In both cases, a good one local influence on one of the two waveguides, in particular of the polymer waveguide, for the exact generation of preselected phase ver shifts between the two light waves in the waveguides.

In the known silica level adjuster, of which the present invention goes out, the two couplers are AWG multiplexers with phase shifters switched between for each wavelength range to be selected. In order to there is a large amount in one arm of the Mach-Zehnder interferometer rich circuit design. At the hybrid level according to the invention however, the same is possible after a next invention continuation, that a series arrangement together with additional level equalizers of the same type are provided, each of which Level equalizer in a different wavelength range around another Central wavelength works. The result is a very clear, cascaded arrangement that is easy to manufacture. Every level adjuster is assigned a predetermined wavelength range, which by the Embossed filter function of the associated grid-supported filter coupler is specified. Each phase shifter can be localized, exactly adjustable heat supply then to the required phase shift can be set. A particular advantage of this arrangement is that that in a hybrid construction with a silica and a polymer waveguide  each wavelength only once over a short distance the polymer Waveguide must pass, but the longest distance in the low damping silica waveguide remains. Such a cascaded structure from several level equalizers of the type according to the invention is ent speaking of the number of wavelength ranges to be adjusted expandable.

Forms of the invention are described below with reference to schematic figures explained in more detail. It shows:

Fig. 1 is a diagram for varying the grating length,

Fig. 2 is a diagram for varying the thickness of the separating layer,

Fig. 3 is a table for the coupling-dependent damping behavior of the Pegelangleichers according to the invention,

Fig. 4 angleichers the schematic diagram of the level according to the invention,

Fig. 5 shows an integrated planar structure of the level adjuster according to the invention in spatial longitudinal section and

Fig. 6 shows a cascaded arrangement of several inventive level equalizers for several central wavelengths in longitudinal section in the front view.

First, some basic mathematical relationships with the further understanding of the mode of operation of the level adjuster according to the invention in the particularly preferred embodiment with a silica waveguide as the first waveguide and a polymer waveguide as the second Given waveguide.

The level adjuster according to the invention basically consists of two components: the grid-supported coupler and the phase shifter. The grid-supported coupler is used both as a 3 dB coupler and as a filter. The central wavelength to be filtered out is determined by the refractive indices that occur, by the geometry of the grating and by the structure of the waveguide. The grating period Λ results in

with λ _ci as the central wavelength in a wavelength range Δλ _i with i = 1.. .m (3 dB filter bandwidth), n _silica (= n ₁ ) as the refractive index of the silica waveguide and n _polymer (= n ₂ ) as the refractive index of the polymer waveguide. Each grating-supported coupling filter has a 3 dB coupling coefficient at the central wavelength λ _ci (R = 0.5). The coupling coefficient of the grating-assisted coupler depends on the distance between the waveguides ("gap") if the grating period Λ and the total length of the grating L are fixed. The 3 dB filter bandwidth Δλ _i initially depends on the length of the grating in inverse proportionality. If one designates the number of individual grid elements with N _G , the following approximation can be established:

From this context it can be deduced that the filter bandwidth Δλ _i decreases with increasing lattice length L. In FIG. 1, a simulation for a grating-assisted 3 dB is Kopplerfilter microns with a grating period Λ of 64 and a separation layer thickness (gap) of 4 micrometers shown at a variable grating length L. It can be seen that both the coupled optical power P and the 3 dB filter bandwidth Δλ _i change when the grating length L is varied.

In FIG. 2, a simulation for a grating-assisted filter coupler with a constant grating period Λ of 64 microns and a grating length L of 10 mm shown in which the thickness of the separating layer (gap) is varied. This illustration shows that the peak optical power coupling depends on the thickness of the interface. The thicker it is, the smaller the coupling coefficient R. If it is 4.4 µm, the coupling coefficient R is exactly 3 dB. It can clearly be seen in FIG. 2 that the 3 dB filter bandwidth Δλ _{i of} the coupler does not fluctuate for different thicknesses of the interface.

The phase shifter connects the two grid-supported filter couplers. The transfer function of the grid-assisted filter coupler can be specified with Φ as its phase shift to

The following applies to a 3 dB coupler: Φ (λ) = π / 4 at λ _ci . The transfer function for the phase shifter can be specified as

The transfer function of the combination of grid-assisted filter couplers - phase shifters - grid-assisted filter couplers is

The coupling optical power P _polymer at the output of the polymer waveguide is therefore:

P _polymer = sin ² (2Φ) × cos ² (Δϕ / 2) (6)

The power at the output of the silica waveguide, however, is:

P _silica = cos ² (2Φ) × cos ² (Δϕ / 2) + sin ² (Δϕ / 2) (7)

If: Δϕ / 2 = 2mπ (m = whole positive number), cos (Δϕ / 2) = 1, sin (Δϕ / 2) = 0, there is a complete overcoupling ("cross-state"):

P _polymer = sin ² (2Φ) × cos ² (Δϕ / 2) = sin ² (2Φ) and P _silica = cos ² (2Φ).

Therefore, for a 3 dB coupling with Φ = π / 4, the optimal cross-state is obtained: P _polymer = 1 and P _silica = 0.

If the coupling coefficient deviates from the ideal state: Φ = π / 4 + ΔΦ, they are calculated

P _polymer = sin ² (2Φ) = sin ² (π / 2 + 2ΔΦ)
= Cos ² (2ΔΦ) ~ {1 - (2ΔΦ) ^{2/2} 2} ~ 1 - 4 (ΔΦ) ²

P _silica = cos ² (2Φ) = cos ² (π / 2 + 2ΔΦ)
= sin ² (2ΔΦ) ~ 4 (ΔΦ) ²

If: Δϕ / 2 = (2m - 1) π, cos (Δϕ / 2) = 0, sin (Δϕ / 2) = 1, there is no overcoupling, but transmission in the silica waveguide ("bar-state" ):

P _silica = cos ² (2Φ) × cos ² (Δϕ / 2) + sin ² (Δϕ / 2) = 1 and

P _polymer = sin ² (2Φ) × cos ² (Δϕ / 2) = 0.

The coupling coefficient R has no influence here, it is always a factor perfect bar-state before.  

The damping for a real 3 dB coupling behavior in the cross-state can be estimated from equation (6). The results are shown in the table according to FIG. 3. Generally speaking, with a coupling coefficient R of 0.5 ± 0.2 (or 3 dB ± 1.5 dB) only additional attenuation below 0.8 dB is caused. This shows that the level adjuster according to the invention solves the sub-task of the lowest possible additional attenuation in a particularly suitable manner.

The structure of the adjustable optical level adjuster GE according to the invention is shown in principle in FIG. 4. It consists of a first waveguide WG ₁ with a refractive index n ₁ and a second waveguide WG ₂ with a refractive index n ₂ running parallel to it in coupling areas CA and has two grating-supported filter couplers GC ₁ , GC ₂ . The two filter couplers GC ₁ , GC ₂ have a coupling coefficient R = 0.5, that is, they divide a signal power P into two halves (50% or 3 dB) and also combine them again. In their coupling areas CA, both filter couplers GC ₁ , GC ₂ each have a grating G ₁ , G ₂ . These can preferably be written into one of the two waveguides WG ₁ , WG ₂ and filter a wavelength range Δλ _i by a central wavelength λ _Ci from the transmission spectrum due to their geometric relationships. An additional setting can be made via an electrical control electrode (not shown). Between the two filter couplers GC ₁ , GC ₂ , the second waveguide WG _{2 has} an adjustable phase shifter PS, the setting of which is used to set a phase shift ϕ between the power components of the coupled wavelength range Δλ _i in the two waveguides WG ₁ , WG ₂ . Depending on the size of the phase shift ϕ in the second filter coupler GC ₂ , complete or partial damping or supplementation is then brought about by superimposition, so that the wavy transmission curve of the line can be compensated.

A planar embodiment of the wavelength-dependent level adjuster GE according to the invention is spatially shown in FIG. 5 in a longitudinal section. The level adjuster GE is optically integrated and consists of two grid-supported filters GC ₁ , GC ₂ and a thermo-optically adjustable phase shifter PS connected between them. The technology used is a combination of silica technology with low damping and polymer technology with good, especially thermal, tunability. The grating-supported filter couplers GC ₁ , GC ₂ consist of a silica waveguide WG _silica and a polymer waveguide WG _polymer lying parallel to it in the coupling regions CA, which are constructed vertically in the exemplary embodiment shown. Both waveguides WG _silica , WG _polymer are separated from one another by a silica or polymer separating layer SL and isolated from a substrate S and from the environment by passivation layers PL made of silica or polymer. In the selected exemplary embodiment, _polymer grids G ₁ , G ₂ are written into the polymer waveguide WG, which determine the property of the filter function. The combination of WG _Silica and WG _{Polymer silica} and polymer waveguides enables a high refractive index difference to be set, which greatly reduces the overall length of the GE level adjuster. The second component of the level adjuster GE according to the invention is the phase shifter PS. Here, the polymer waveguide WG _polymer is arranged in such a way that it cannot couple with the silica waveguide WG _silica . In addition, a heating electrode HE is applied above the polymer waveguide WG _polymer outside the coupling areas CA.

In FIG. 6 is a cascaded arrangement of a plurality of series-connected GEC Pegelangleichern GE _1, GE _2. , ., GE _m (GE _i , i = 1.. .M, in the example shown m = 3) with different filter functions (the gratings are inscribed in the silica waveguide WG _silica in this embodiment) for simultaneous level adjustment in different wavelength ranges Δλ _i (i = 1.. .m, m = 3 in the example shown) shown in longitudinal section in the front view (spatial representation see FIG. 5). This structure can be expanded as required depending on the number of signal-carrying wavelengths in the WDM network. By separately adjustable phase shifters PS ₁ , PS ₂ ,. , ., PS _m (PS _i , i = 1.. .M, in the example shown m = 3), the wavelength ranges Δλ ₁ , Δλ ₂ ,. , ., Δλ _m are damped independently of one another and thus their amplitude adjusted. Each wavelength range Δλ ₁ , Δλ ₂ ,. , ., Δλ _m only has to pass through the polymer waveguide WG _polymer once. For the largest part, it remains in the low-attenuation silica waveguide WG _Silica . As a result, such a cascaded arrangement GEC from a plurality of level adjusters GE _i according to the invention produces only a small additional attenuation in optical WDM networks.

The entire structure of the level adjuster according to the invention can be by means of the known silica-polymer hybrid integration technology become. The silica waveguide can be combined by flame Hydrolysis deposition (FHD) fabricated using reactive ion beam etching (RIE) become. The grid height can be precisely controlled in the RIE process. After the manufacture of the silica waveguide, the separation layer applied. In principle, silica or polymer materials can be used for this be used. The polymeric interface, the polymer waveguide and the passivation layer can be produced by means of a spin coating process be put. The waveguide core can be shaped by RIE. To the Finally, the control electrodes above the grid-supported Filter coupler and the phase shifter arranged.  

Reference and formula symbol list

ϕ phase shift in PS
Φ phase shift in GC
Λ grating period
Δλ _i

; Wavelength range (3 dB filter bandwidth)
λ _ci

; central wavelength in the wavelength range Δλ _i

CA coupling area
SL separation layer
G ₁

, G ₂

; grid
gap separation layer thickness
GC ₁

, GC ₂

; grid-supported filter coupler
GE level adjuster
GE _i

; Level equalizer in GEC
GEC combined component for level adjustment
i running index, i = 1.. .m
HE heating electrode
L grid length
N _G

; Number of grid elements
n ₁

; Refractive index of the first waveguide
n ₂

Refractive index of the second waveguide
n _polymer

Refractive index of the polymer waveguide
n _silica

Refractive index of the silica waveguide
P optical power
PL passivation layer
PS phase shifter
PS _i

; Phase shifter in GEC
R coupling coefficient
S substrate
WG ₁

; first waveguide
WG ₂

; second waveguide
Flat share _polymer

Polymer waveguide
Flat share _silica

Silica waveguide

Claims (11)

1. Adjustable optical level adjuster for wavelength-dependent equalization of the power levels in WDM networks with two couplers for optically coupling a first waveguide with a second waveguide running in the coupling area, a filter function element arranged between the couplers for selecting a wavelength range around a central wavelength and one adjustable phase shifters arranged between the couplers outside their coupling areas, characterized in that each coupler as a grating-supported 3 dB filter coupler (GC ₁ , GC ₂ ) between the two waveguides (WG ₁ , WG ₂ ) with different refractive indices (n ₁ , n ₂ ) and the filter function element is designed in the form of two assigned grids (G ₁ , G ₂ ), which are each arranged in the coupling area (CA) of a 3 dB filter coupler (GC ₁ , GC ₂ ), and that the phase shifter (PS) is directly in one of the two waveguides (WG ₁ , WG ₂ ) is integrated t. 2. Adjustable optical level adjuster according to claim 1, characterized in that the first waveguide as a silica waveguide (WG _silica ) and the second waveguide formed as a polymer waveguide (WG _polymer ) and the adjustable phase shifter (PS) in the polymer waveguide (WG _polymer ) is integrated. 3. Adjustable optical level adjuster according to claim 2, characterized in that the polymer waveguide (WG _polymer ) has non-linear optical properties. 4. Adjustable optical level adjuster according to one of claims 1 to 3, characterized in that the associated grating (G ₁ , G ₂ ) have a periodic refractive index variation, which is defined by geometric or optical measures. 5. Adjustable optical level adjuster according to one of claims 1 to 4, characterized in that the associated grating (G ₁ , G ₂ ) are each directly written into one of the two waveguides (WG ₁ , WG _silica ; WG ₂ , WG _polymer ). 6. Adjustable optical level adjuster according to one of claims 1 to 5, characterized in that the two waveguides (WG _silica , WG _polymer ) are arranged vertically to one another and between the two waveguides (WG _silica , WG _polymer ) a thin separation layer (SL) and A passivation layer (PL) is provided above and below the two waveguides (WG _silica , WG _polymer ). 7. Adjustable optical level adjuster according to claim 6, characterized in that the central wavelength of the wavelength range to be selected by means of a control electrode can also be set on the passivation layer, in particular above the polymer waveguide, is arranged. 8. Adjustable optical level adjuster according to one of claims 1 to 7, characterized in that the phase shifter (PS) is adjustable via a control element which is arranged on the passivation layer (PL), in particular above the polymer waveguide (WG _polymer ). 9. Adjustable optical level adjuster according to claim 8, characterized in that the phase shifter (PS) thermo-optically via a heating electrode (HE) Control is adjustable. 10. Adjustable optical level adjuster according to claim 8, characterized in that the phase shifter electro-optically via a control electrode as a control element is adjustable. 11. Adjustable optical level adjuster according to one of claims 1 to 10, characterized in that a series arrangement (GEC) connected in series is provided together with further level adjusters (GE _i , i = 1.. .M) of the same type, each of which level adjuster ( GE _i ) works in a different wavelength range (Δλ _i ) around a different central wavelength (λ _Ci ).