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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 9 — May. 5, 2003
  • pp: 1029–1038
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Three-step design optimization for multi-channel fibre Bragg gratings

Kazimir Y. Kolossovski, Rowland A. Sammut, Alexander V. Buryak, and Dmitrii Yu. Stepanov  »View Author Affiliations


Optics Express, Vol. 11, Issue 9, pp. 1029-1038 (2003)
http://dx.doi.org/10.1364/OE.11.001029


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Abstract

Methods to produce optimal designs for multi-channel fiber Bragg gratings (FBGs) with identical or close to identical channel-to-channel spectral characteristics are discussed. The proposed approach consists of three distinct steps. The first two steps (preliminary semi-analytic minimization and subsequent fine-tuning) do not depend on the grating design details, but on the number of channels only and can be readily applied to similar problems in other fields, e.g., in radio-physics and coding theory. The third step (spectral characteristic quality improvement) is FBG field specific. A comparison with other known optimization methods shows that the proposed approach yields generally superior results for small to moderate number of channels (N<60).

© 2003 Optical Society of America

1. Introduction

Fiber Bragg gratings (FBGs) are essential components in modern optical communication systems where they find applications as filters, dispersion compensators, laser source tuners and stabilizers, etc [1

1. A. Othonos and K. Kalli, Fiber Bragg Gratings (Boston, Artech House, 1999).

]. Multi-channel FBGs represent critical components for tunable lasers [2

2. H. Ishii, H. Tanobe, F. Kano, Y. Tohmori, Y. Kondo, and Y. Yoshikuni, “Quasicontinuous Wavelength Tuning in Super-Structure-Grating (SSG) DBR Lasers,” IEEE J. Quantum Electron. 32, 433–441 (1996). [CrossRef]

] and passive FBG-based devices [3

3. A. V. Buryak and D. Yu. Stepanov, “Novel multi-channel grating devices,” in proceedings of Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, vol. 60 of Top series, BThB3 (Washington DC, Optical Society of America, 2001).

, 4

4. A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, “Optimization of Refractive Index Sampling for Multi-channel Fiber Bragg Gratings,” IEEE J. Quantum Electron. 39, 91–98 (2003). [CrossRef]

, 5

5. J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Tech. Lett. 14, 1309–1311 (2002). [CrossRef]

]. Optical components based on multi-channel FBGs are of particular interest in wavelength division multiplexing (WDM) systems [6

6. M. Ibsen, A. Fu, H. Geiger, and R. I. Laming, “All-fibre 4×10Gbit/s WDM link with DFB fibre laser transmitters and single sinc-sampled fibre grating dispersion compensator,” Electron. Lett. 35, 982–983 (1999). [CrossRef]

, 7

7. Y. Painchaud, A. Mailloux, H. Chotard, E. Pelletier, and M. Guy, “Multi-channel fiber Bragg gratings for dispersion and slope compensaion,” in OSA Technical Digest of Optical Fiber Communication Conference, ThAA5, 581–582 (Washington DC, Optical Society of America, 2002).

] because of the benefits they offer in increasing system capacity. However, in comparison to a single-channel grating, manufacture of multi-channel FBG devices requires larger variation of the photo-induced refractive index change. Due to the saturation of fiber photosensitivity, the number of channels that can be recorded in a given fiber is limited. Additionally, the saturation of the photo-induced refractive index change can be the cause of phase and amplitude distortions, especially in the case of multi-channel FBGs [8

8. S. W. Lϕvseth and D. Yu. Stepanov, “Analysis of multiple wavelength DFB fiber lasers,” IEEE J. Quantum Electron. 37, 770–780 (2001). [CrossRef]

]. Hence, optimization of multi-channel FBG designs to reduce the peak value of the index change is a paramount issue before any practical implementation of the gratings. A conceptually similar problem arises in radio-physics in the context of low envelope variation of a multi-tone signal (see, e.g., [9

9. S. Narahashi, K. Kumagai, and T. Nojima, “Minimising peak to average power ratio of multitone signals using steepest descent method,” Electron. Lett. 31, 1552–1554 (1995). [CrossRef]

, 10

10. M. Friese, “Multitone signals with low crest factor,” IEEE Trans. Commun. 45, 1338–1344 (1997). [CrossRef]

]).

The most simplistic approach to fabricate a multi-channel FBG is to sequentially superimpose several single-channel FBGs at the same location within the fiber, (see, e.g., [11

11. A. Othonos, X. Lee, and R. M. Measures, “Superimposed multiple Bragg gratings,” Electron. Lett. 30, 1972–1974 (1994). [CrossRef]

, 12

12. G. Sarlet, G. Morthier, R. Baets, D. J. Robbins, and D. C. J. Reid, “Optimization of multiple exposure gratings for widely tunable laser,” IEEE Photon. Techn. Lett. 11, 21–23 (1999). [CrossRef]

]). This method has a few drawbacks making it quite impractical especially for a large number of channels. Firstly, due to complex saturation behaviour of the fiber photosensitivity, a UV exposure correction is needed prior to writing each grating. The channels written first need to be stronger initially to sustain further sharing of the induced refractive index change with the subsequent channels. Secondly, the local refractive index change in the UV exposed fiber area averaged over the grating period, Δn (av), accumulates with the number of UV exposures. As a result, the local Bragg wavelengths of the previously written channels become longer, thus requiring corresponding corrections to the period of the interference pattern to be made in advance. Finally, the major drawback of the FBG superimposing is the fact that this average index change grows approximately linearly with the number of UV exposures and, thus, with the number of channels, i.e., Δn (av)~N. As it will become clear from the rest of this section, a much better utilization of the fiber photosensitivity is possible using different methods of fabrication of the multi-channel FBGs.

Another, more widely accepted, approach to multi-channel FBG fabrication is based on sampling of a single-channel (“seeding”) FBG. In this approach the amplitude and/or phase of the seeding grating is periodically modulated. The resulting sampled design is then UV-written into a fibre in one go.

The first proposals for fabrication of multi-channel devices relied on amplitude-only modulation of a seeding grating. In the case of a comb-like sampling function [13

13. V. Jayaraman, Z.-M. Chuang, and L. A. Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled grating,” IEEE J. Quantum Electron. 29, 1824–1834 (1993). [CrossRef]

, 14

14. B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620–1622 (1994). [CrossRef]

], the spectral envelope of the channels is sinc-shaped. This approach is characterized by high level of the required peak index change Δn N that grows linearly with the number of channels N, Δn N~N. The efficiency of the comb-sampling approach is low because the fiber is utilized only partially, i.e., there are segments of the fibre without any grating written. If we introduce a fiber utilization figure of merit as F=(Δnenv(av)n N)2, where Δnenv(av) is the envelope of the index change averaged over a period of the sampling function, then for the comb-sampling Δnenv(av)~√N and F~1/N≪1, i.e., is prohibitively low. Modification of the comb-sampling by writing additional gratings into the unused parts of the fiber [15

15. W. H. Loh, F. Q. Zhou, and J. J. Pan, “Sampled Fiber Grating Based-Dispersion Slope Compensator,” IEEE Photon. Techn. Lett. 11, 1280–1282 (1999). [CrossRef]

] increases the fiber utilization, but does not lead to uniform channel-to-channel spectral characteristics.

A high level of variation of the index change, Δn N~N, is a characteristic property of another amplitude-modulation approach, so-called sinc-sampling [6

6. M. Ibsen, A. Fu, H. Geiger, and R. I. Laming, “All-fibre 4×10Gbit/s WDM link with DFB fibre laser transmitters and single sinc-sampled fibre grating dispersion compensator,” Electron. Lett. 35, 982–983 (1999). [CrossRef]

]. Although this method yields a uniform sequence of identical channels in the reflection spectrum, its inefficiency also relates to low utilization of extended parts of the fiber. The method can be modified using a general multiple-phase-shift technique when insertion of phase shifts at appropriate positions along the fiber [2

2. H. Ishii, H. Tanobe, F. Kano, Y. Tohmori, Y. Kondo, and Y. Yoshikuni, “Quasicontinuous Wavelength Tuning in Super-Structure-Grating (SSG) DBR Lasers,” IEEE J. Quantum Electron. 32, 433–441 (1996). [CrossRef]

, 5

5. J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Tech. Lett. 14, 1309–1311 (2002). [CrossRef]

, 16

16. H. Ishii, Y. Tohmori, Y. Yoshikuni, T. Tamamura, and Y. Kondo, “Multiple-Phase-Shift Super Structure Grating DBR Lasers for Broad Wavelength Tuning,” IEEE Photon. Techn. Lett. 5, 613–615 (1993). [CrossRef]

, 17

17. Y. Nasu and S. Yamashita, “Multiple phase-shift superstructure fibre Bragg gratings for DWDM systems,” Electron. Lett. 37, 1471–1472 (2001). [CrossRef]

] effectively modifies the sampling period and results in more channels due to increased channel density.

Summarizing, there are three major requirements for a practically valuable multi-channel FBG design optimization strategy: (1) the maximum needed refractive index change should be reduced close to the theoretical limit Δn N~√N (i.e., F~1); (2) quality of spectral characteristics should not be compromised; and (3) the optimization procedure should not require prohibitively long computer time.

In this paper, we considerably extend the analysis of [4

4. A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, “Optimization of Refractive Index Sampling for Multi-channel Fiber Bragg Gratings,” IEEE J. Quantum Electron. 39, 91–98 (2003). [CrossRef]

, 5

5. J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Tech. Lett. 14, 1309–1311 (2002). [CrossRef]

] and present a three-step procedure to obtain an efficiently optimized multi-channel FBG design satisfying all three criteria. These distinct steps involve: (1) semi-analytical searching of a phase profile to obtain a roughly optimized sampling function using the so-called functional approach, (2) fine tuning of this sampling function using certain iterative schemes, and (3) further improvement of spectral quality by applying inverse scattering-based iterative algorithms. Below we describe the whole procedure in more detail.

2. Formulation of the problem

The fundamental system of equations describing light propagation in FBGs is

Ebz+iδEbq(z)Ef=0,
EfziδEfq*(z)Eb=0,
(1)

where E f and E b are the amplitudes of the forward and backward propagating fields, respectively, δ is the normalized frequency detuning from the central Bragg reflection frequency, z is a local distance along FBG, q(z) is a spatial profile of the FBG coupling coefficient, and asterisk denotes complex conjugation. For a reciprocal and lossless (see, e.g., [23

23. C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (New York, Wiley & Sons, 1999).

] for definitions) FBG of length L we may find complex reflection and transmission coefficients from a transfer matrix, which relates field values at the grating ends

[Eb(0)Ef(0)]=[1t*r1tr2t1t][Eb(L)Ef(L)],
(2)

Following Refs. [2

2. H. Ishii, H. Tanobe, F. Kano, Y. Tohmori, Y. Kondo, and Y. Yoshikuni, “Quasicontinuous Wavelength Tuning in Super-Structure-Grating (SSG) DBR Lasers,” IEEE J. Quantum Electron. 32, 433–441 (1996). [CrossRef]

, 3

3. A. V. Buryak and D. Yu. Stepanov, “Novel multi-channel grating devices,” in proceedings of Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, vol. 60 of Top series, BThB3 (Washington DC, Optical Society of America, 2001).

, 4

4. A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, “Optimization of Refractive Index Sampling for Multi-channel Fiber Bragg Gratings,” IEEE J. Quantum Electron. 39, 91–98 (2003). [CrossRef]

], an N-channel grating design can be obtained by a dephasing approach, where the slowly varying envelope of a direct summation of N identical gratings, equally spaced in the frequency space, is taken with relative phases ϕl for each seeding grating,

q(z)=l=1Nκ(z)eiθ(z)ei[(2lN1)Δkz2+ϕl]=κ(z)eiθ(z)S(z),
(3)

where the phase of the complex sampling function S(z) is given by

arg{S(z)}=arctan[l=1Nsin([2lN1]Δkz2+ϕl)l=1Ncos([2lN1]Δkz2+ϕl)].
(4)

The amplitude of the sampling function can be presented in the following form,

S(z)=N(1+2NRep=1N1CpeipΔkz)12,
(5)

where i=√-1, Re stands for the real part, and

CP=l=1Npml+pml*,p=1,2,,N1,
(6)

is the aperiodic autocorrelation function (AACF) of a complex sequence associated with phases ϕl: m l=exp(iϕl). From Eq. (5) it follows that the amplitude modulation of the sampling function is small when autocorrelation of the sequence m l is low, i.e., when the amplitude of its AACF is small. If |Cp |≤1 for all p=1, 2, …, N-1, the corresponding sequence m l is called a generalized Barker sequence. Barker sequences have been reported for N up to 45 [25

25. L. Bömer and M. Antweiler, “Polyphase Barker sequences,” Electron. Lett. 25, 1577–1579 (1989); M. Friese and H. Zottmann, “Polyphase Barker sequences up to length 31,” Electron. Lett. 30, 1930–1931 (1994); M. Friese, “Polyphase Barker sequences up to length 36,” IEEE Trans. Inform. Theory 42, 1248–1250 (1996); A. R. Brenner, “Polyphase Barker sequences up to length 45 with small alphabets,” Electron. Lett. 34, 1576–1577 (1998).

].

It is easy to conclude from Eq. (5) that a two-channel design (N=2) cannot be optimized as the maximum of the sampling function is always 2 regardless of the values of the dephasingangles ϕ1, ϕ2. More generally, one might observe from Eq. (6) that |CN -1|=1 for an arbitrary N. This leads to a simple estimate from below for the peak of the sampling function (Eq. (5)),i.e., for the maximum amplitude of index variation (for more details see [10

10. M. Friese, “Multitone signals with low crest factor,” IEEE Trans. Commun. 45, 1338–1344 (1997). [CrossRef]

]),

ΔnN=maxzS(z)Δn1=N+2Δn1,
(7)

where Δn 1 is the maximum index change for the seeding grating design.

The scope of this paper is to develop a general approach to designing multi-channel FBGs with uniform or nearly uniform channel-to-channel spectral characteristics that are characterized by an index change with a low peak value.

3. Optimal design

Among the obvious strategies for optimizing the designs are minimization of the peak value of the sampling function or its contrast with respect to the dephasing angles ϕl, as in Ref. [4

4. A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, “Optimization of Refractive Index Sampling for Multi-channel Fiber Bragg Gratings,” IEEE J. Quantum Electron. 39, 91–98 (2003). [CrossRef]

]. These minimax problems inevitably involve scanning over z∈[0, 2π/Δk], which is numerically a burdensome task. A typical number of mesh points in z required to locate the min (max) to an appropriate accuracy is in the order of 102×N. Therefore, a single evaluation of the minimizing function only comprises in the order of 102×N 3 operations. Therefore it is highly desirable to formulate the optimization problem to avoid scanning over continuous variable z. This can be achieved via construction of a functional which comprises integration over z.

3.1 Functional approach

The most efficient optimization corresponds to the limit when the amplitude of the sampling function is constant. The natural measure for the fluctuations of envelope s(z)≡|S(z)|/√N is its standard deviation over the sampling period, z ∈ [0, 2π/Δk]. Hence, the functional to be minimized can be constructed as follows,

Δ(ϕ)=[s(z)s(z)]2=1s(z)2,
(8)

where the average over the sampling period 〈s(z)〉≡Δk/2π02π/Δk s(z)dz. To obtain the last expression in Eq. (8) we used the mean square value of Eq. (5), 〈s(z)2〉=1. Note that minimization of the standard deviation is equivalent to maximization of the mean value of s(z), i.e., Δnenv(av).

To calculate the mean value of s(z) explicitly, we assume that the second term in Eq. (5), x(z)≡(2/N)Rep=1N1 C p e ipΔkz, is much less than 1 for all z ∈ [0, 2π/Δk]. Truncating the Taylor series of s(z) with respect to x(z) at the third term and averaging the result over the period one obtains

s(z)=114N2p=1N1Cp2+O(x3).
(9)

Similar to theoretical limit (Eq. (7)), from Eq. (9) one can easily estimate the average index change from above,

Δnenv(av)=N(114N2)Δn1.
(10)

Therefore, the fiber utilization parameter of a fully optimized grating is

F=12N+O(1N2).
(11)

Finally, the function of Eq. (8) reduces to the following multi-variable function,

Δ(ϕ)=12Np=1N1Cp2+O(x3x2),
(12)

where ϕ=(ϕ1, ϕ2, …, ϕN).

Due to the invariance of the function in Eq. (12) with respect to a linear transformation, i.e., ϕl→ϕl+a 1+a 2 l, where a 1,2 are constants, dimension of the parameter space ϕ can be reduced by 2. But the most important advantage of the functional approach, which has been overlooked in the previously reported works, is that the gradient of the objective function in Eq. (12) can be calculated analytically,

Δ(ϕ)ϕl=Im{p=1l1Cpmlpml*+p=1NlCp*ml+pml*}2N2Δ(ϕ)+O(x3x2),
(13)

where Im stands for the imaginary part. A single evaluation of the objective function in Eq. (12) and the gradient in Eq. (13) requires O(N 2) operations that is a considerable improvement in comparison to the minimax strategies [4

4. A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, “Optimization of Refractive Index Sampling for Multi-channel Fiber Bragg Gratings,” IEEE J. Quantum Electron. 39, 91–98 (2003). [CrossRef]

].

To minimize the objective function in Eq. (12) we used the conjugate gradient method (Polak-Ribiere form) and the lagged Fibonacci generator with a Marsaglia shift [19

19. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, 2nd ed. (Cambridge England, Cambridge Univ. Press, 1992).

]. The later part of the algorithm generated random initial conditions for the minimization search, which typically comprised in the order of 106 tries.

Fig. 1. Normalized peak index change as a result of the first two steps in the three-step optimization process. Solid curve shows the analytic estimate (Eq. (7)).

3.2 Iterative schemes

Smallness of the envelope fluctuations in the mean-square sense does not necessarily imply the smallness of the envelope peak value. However, we assume that these two optimal cases correspond to close points in the parameter space ϕ . So the results of the functional approach might be used as good starting values for the subsequent iterative schemes of peak minimization. We found that among the rich variety of such schemes, the fast Fourier transform based algorithms, Gerchberg-Saxton reconstruction [4

4. A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, “Optimization of Refractive Index Sampling for Multi-channel Fiber Bragg Gratings,” IEEE J. Quantum Electron. 39, 91–98 (2003). [CrossRef]

, 18

18. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

] and clipping [10

10. M. Friese, “Multitone signals with low crest factor,” IEEE Trans. Commun. 45, 1338–1344 (1997). [CrossRef]

, 26

26. E. Van der Ouderaa, J. Schoukens, and J. Renneboog, “Peak Factor Minimization using a Time-Frequency Domain Swapping Algorithm,” IEEE Trans. Instr. Measur. 37, 145–147 (1988). [CrossRef]

], are the most efficient ones.

The Gerchberg-Saxton algorithm can be used in cases when a small out-of-band response is not too crucial and the requirements for the spectral resolution are not highly demanding. Swapping between time/direction and frequency domains under constraints that the amplitude of the complex sampling function is constant whereas the amplitude profile of the central part of its spectrum is uniform, one translates all amplitude modulation of S(z) into its phase. The peak of the sampling function is reduced significantly at the expense of small side-lobes in its spectrum. The integral size of the side-lobes is proportional to the mean-square deviation of |S(z)| [4

4. A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, “Optimization of Refractive Index Sampling for Multi-channel Fiber Bragg Gratings,” IEEE J. Quantum Electron. 39, 91–98 (2003). [CrossRef]

].

Another fast algorithm for reducing the highest peak in the profile of the sampling function is clipping. It is favorable in cases when absence of side-channels in the reflection spectrum is essential. Initially, one constructs an error function by clipping S(z) at some level S 0. By subtracting the Fourier transform of the complex error function from the finite spectrum of original S(z) and restoring the amplitude profile of the spectrum to the original form (that includes setting the out-of-band response to zero), one decreases the maximum peak value of S(z). Gradually increasing level S 0, one might significantly reduce the maximum peak of S(z) [10

10. M. Friese, “Multitone signals with low crest factor,” IEEE Trans. Commun. 45, 1338–1344 (1997). [CrossRef]

].

The results of the sampling function optimization (first two steps of the complete optimization procedure) are presented in Fig. 1. Output of the functional approach (hollow circles) has been used as initial data for the clipping procedure (filled circles). For comparison, we show initial peak values of S(z) obtained using generalized Barker sequences (see [25

25. L. Bömer and M. Antweiler, “Polyphase Barker sequences,” Electron. Lett. 25, 1577–1579 (1989); M. Friese and H. Zottmann, “Polyphase Barker sequences up to length 31,” Electron. Lett. 30, 1930–1931 (1994); M. Friese, “Polyphase Barker sequences up to length 36,” IEEE Trans. Inform. Theory 42, 1248–1250 (1996); A. R. Brenner, “Polyphase Barker sequences up to length 45 with small alphabets,” Electron. Lett. 34, 1576–1577 (1998).

]; we also found such sequences for N=47, 51 and 53) followed by the clipping (stars). It can be seen that the functional approach on its own yields sufficiently good results. Further optimization based on the clipping algorithm yields the same or better peak optimization than the corresponding two-step optimization of Barker sequences for all 5<N≤45 except N=12, 13, 15, 16 and 40. Functional approach is more simple and computationally more effective then the known algorithms of searching for generalized Barker sequences. Moreover, it yields optimized sampling functions characterized by zero-free profiles that avoids sometimes undesirable phase π-jumps.

Fig. 2. An illustration of three-stage optimization of a 9-channel dispersion compensator design. Non-trivially modulated phase profile and group delay characteristics are not shown. (a), (b) the amplitude profile and the transmission spectrum obtained after the first step of optimization; (c), (d) the same as (a), (b) but after the second step; (e), (f) result of the third step. The final result is presented in more detail in Fig. 3.

An example of the optimized 9-channel dispersion compensator design (-500 ps/nm dispersion within each channel) obtained after step 1 (functional minimization) and step 2 (Gerchberg-Saxton algorithm) is shown in Figs. 2(a,b) and Figs. 2(c,d) respectively.

3.3 Spectral fine-tuning

What would happen if one stops after the first two steps of the optimization procedure? The first two steps provide a significant reduction for the required Δn N, which, in turn, increases the fiber utilization parameter close to F≈1. In addition, they provide a universal optimization method, i.e., knowing a single N-channel optimal set of dephasing angles allows one to obtain the corresponding design for any given seeding grating by using a simple formula q N(z)=S N(z)q 1(z). However, it is important to note that all designs based on periodic sampling are inevitably non-ideal because the neighbouring channels (even well separated ones) may distort the spectra of each other. This is visible in Figs. 2(b,d) as small deviations of the transmission spectra from the square-like shape. For this particular example these deviations are relatively minor. However, for some other grating designs, especially for multi-channel filters with zero in band dispersion, the situation can be worse (see, e.g., [3

3. A. V. Buryak and D. Yu. Stepanov, “Novel multi-channel grating devices,” in proceedings of Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, vol. 60 of Top series, BThB3 (Washington DC, Optical Society of America, 2001).

]). In general, periodic sampling works rather well if the seeding grating has smooth, slowly-varying amplitude and phase, which is usually the case, e.g., for the conventional (i.e., second order only) dispersion compensating devices. In contrast, zero dispersion filters and third order (dispersion slope) dispersion compensators have abrupt jumps in the seeding grating phase and sinc-like seeding grating amplitude dependencies and are much less suited for the use of any pure periodic sampling methods. For such gratings a third optimization step is necessary. It is based on an observation that for weak gratings the first order Born approximation holds:

Fig. 3. Details of the 9-channel dispersion compensator design shown in Figs. 2(e,f). (a) amplitude and phase profiles; (b) enlarged (a); (c) central part of the reflection spectrum; (d) group delay. We note that, all fast oscillations of κ(z) profile in the vicinity of the main peak (z≈2.6) were completely eliminated after 30 iterations, though some unimportant weak modulation of the profile in the region z≈3.8 still presents.
12q(z2)=+r(δ)exp(iδz)dδ.
(14)

This scheme takes the multi-channel grating profile (obtained using the original Gerchberg-Saxton scheme at the 2nd step of optimization) as a zero approximation grating. One iterative step of the suggested generalized scheme includes: (1) replacing of the multi-channel grating amplitude κ(i)(z) (i is an iteration number) by A(i)κ1(z) , where κ1(z) is the seeding grating amplitude and the constant A (i) is defined by the normalization condition A (i)=0l(i)}2 dz/0lκ12 dz; (2) using the unchanged multi-channel grating phase and the modified grating amplitude as the input data for the direct scattering solver; (3) modifying the resulting spectrum data in the following way: within the central channel spectral domains the current spectrum data is replaced by pre-iteration data (fixed spectral domain), and outside the central channel spectral domains the current spectrum data is left intact (spectral domain allowed to evolve); (4) finally, solving the inverse scattering problem for the modified spectrum to obtain the optimized multi-channel grating amplitude and phase. That completes one iteration step which can be repeated until fast oscillations in κ(i)(z) profile are reduced close to an acceptable level.

Figures 2(e,f) and 3(a–d) show a completely optimized 9-channel dispersion compensator design. Figure 3(b) clearly demonstrates aperiodicity of the modulation of the grating amplitude κ(z).

A characteristic property of the third step is that it is specific for a particular grating design, i.e., it has to be completely redone for any new seeding grating shape or channel spacing. On the other hand, the new layer peeling inverse scattering algorithms (see, e.g., [24

24. J. Skaar, L. Wang, and T. Erdogan, “On the Synthesis of Fiber Bragg Gratings by Layer Peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001). [CrossRef]

]) are very fast and allow implementing the third step calculations in a reasonable time. In addition, the spectral fine-tuning allows easy generalizations for multi-channel designs with slightly non-identical channel-to-channel characteristics.

4. Conclusion

In this paper, we have presented an efficient method of optimization of multi-channel FBGs, which consists of three distinct steps. The first two steps (minimization of the sampling function fluctuations in the mean-square sense and iterative clipping of the highest peaks) have applications far beyond the FBG designs and can be readily applied for the conceptually similar problems in other fields of physics, e.g., in radio-physics and coding theory. A comparison of the sampling function minimization quality after the first two steps of the presented method with other known minimization methods demonstrated that the semi-analytical approach presented typically yields superior results in all the parameter domain where the data for other methods are available (for N≲45). Moreover, the method can readily be used to obtain optimization for the number of channels N up to 100 in a reasonable time using conventional personal computers. Finally the described algorithm incorporates a spectrum fine-tuning step, which is specific for FBGs, allowing one to get high quality spectral characteristics with little or no sacrifice of the achieved sampling function minimization level.

Our preliminary studies show that small deliberate variations of the channel-to-channel spectral characteristics do not seriously compromise the minimization level obtained for completely uniform N-channel FBGs. Thus, for example, broadband FBG devices compensating not only for the average dispersion, but for the dispersion slope as well, may be efficiently designed and implemented. To conclude, the efficient multi-channel optimization methods described in this paper open new exciting opportunities for FBG applications. For example, FBG-based dispersion compensators covering the whole communications’ C-band can replace dispersion compensating fibers as the leading dispersion management tool. The first practically usable experimental implementations of multi-channel FBG-based dispersion compensators with N≫1 have recently been presented [27

27. Y. Painchaud, H. Chotard, A. Mailloux, and Y. Vasseur, “Superposition of chirped fibre Bragg gratings for third-order dispersion compensation over 32 WDM channels,” Electron. Lett. 38, 1572–1573 (2002). [CrossRef]

, 28

28. A. V. Buryak, G. Edvell, A. Graf, K. Y. Kolossovski, and D. Yu. Stepanov, “Recent progress and novel directions in multi-channel FBG dispersion compensation,” in OSA Technical Digest of Conference on Lasers and Electro-Optics (Washington DC, Optical Society of America, 2003).

], with the described optimization methods applied in Ref. [28

28. A. V. Buryak, G. Edvell, A. Graf, K. Y. Kolossovski, and D. Yu. Stepanov, “Recent progress and novel directions in multi-channel FBG dispersion compensation,” in OSA Technical Digest of Conference on Lasers and Electro-Optics (Washington DC, Optical Society of America, 2003).

].

Acknowledgment

The authors are grateful to Malcolm Gourlay for stimulating discussions and many important references and to the Australian Research Council for financial support.

References and links

1.

A. Othonos and K. Kalli, Fiber Bragg Gratings (Boston, Artech House, 1999).

2.

H. Ishii, H. Tanobe, F. Kano, Y. Tohmori, Y. Kondo, and Y. Yoshikuni, “Quasicontinuous Wavelength Tuning in Super-Structure-Grating (SSG) DBR Lasers,” IEEE J. Quantum Electron. 32, 433–441 (1996). [CrossRef]

3.

A. V. Buryak and D. Yu. Stepanov, “Novel multi-channel grating devices,” in proceedings of Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, vol. 60 of Top series, BThB3 (Washington DC, Optical Society of America, 2001).

4.

A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, “Optimization of Refractive Index Sampling for Multi-channel Fiber Bragg Gratings,” IEEE J. Quantum Electron. 39, 91–98 (2003). [CrossRef]

5.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Tech. Lett. 14, 1309–1311 (2002). [CrossRef]

6.

M. Ibsen, A. Fu, H. Geiger, and R. I. Laming, “All-fibre 4×10Gbit/s WDM link with DFB fibre laser transmitters and single sinc-sampled fibre grating dispersion compensator,” Electron. Lett. 35, 982–983 (1999). [CrossRef]

7.

Y. Painchaud, A. Mailloux, H. Chotard, E. Pelletier, and M. Guy, “Multi-channel fiber Bragg gratings for dispersion and slope compensaion,” in OSA Technical Digest of Optical Fiber Communication Conference, ThAA5, 581–582 (Washington DC, Optical Society of America, 2002).

8.

S. W. Lϕvseth and D. Yu. Stepanov, “Analysis of multiple wavelength DFB fiber lasers,” IEEE J. Quantum Electron. 37, 770–780 (2001). [CrossRef]

9.

S. Narahashi, K. Kumagai, and T. Nojima, “Minimising peak to average power ratio of multitone signals using steepest descent method,” Electron. Lett. 31, 1552–1554 (1995). [CrossRef]

10.

M. Friese, “Multitone signals with low crest factor,” IEEE Trans. Commun. 45, 1338–1344 (1997). [CrossRef]

11.

A. Othonos, X. Lee, and R. M. Measures, “Superimposed multiple Bragg gratings,” Electron. Lett. 30, 1972–1974 (1994). [CrossRef]

12.

G. Sarlet, G. Morthier, R. Baets, D. J. Robbins, and D. C. J. Reid, “Optimization of multiple exposure gratings for widely tunable laser,” IEEE Photon. Techn. Lett. 11, 21–23 (1999). [CrossRef]

13.

V. Jayaraman, Z.-M. Chuang, and L. A. Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled grating,” IEEE J. Quantum Electron. 29, 1824–1834 (1993). [CrossRef]

14.

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620–1622 (1994). [CrossRef]

15.

W. H. Loh, F. Q. Zhou, and J. J. Pan, “Sampled Fiber Grating Based-Dispersion Slope Compensator,” IEEE Photon. Techn. Lett. 11, 1280–1282 (1999). [CrossRef]

16.

H. Ishii, Y. Tohmori, Y. Yoshikuni, T. Tamamura, and Y. Kondo, “Multiple-Phase-Shift Super Structure Grating DBR Lasers for Broad Wavelength Tuning,” IEEE Photon. Techn. Lett. 5, 613–615 (1993). [CrossRef]

17.

Y. Nasu and S. Yamashita, “Multiple phase-shift superstructure fibre Bragg gratings for DWDM systems,” Electron. Lett. 37, 1471–1472 (2001). [CrossRef]

18.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

19.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, 2nd ed. (Cambridge England, Cambridge Univ. Press, 1992).

20.

D. R. Gimlin and C. R. Patisaul, “On minimizing the Peak-to-Average Power Ration for the Sum of N inusoids,” IEEE Trans. Commun. 41, 631–635 (1993). [CrossRef]

21.

S. Narahashi and T. Nojima, “New phasing scheme of N-multiple carriers for reducing peak-to-average power ratio,” Electron. Lett. 30, 1382–1383 (1994). [CrossRef]

22.

J. Schoukens, Y. Rolain, and P. Guillaume, “Design of Narrowband, High-Resolution Multisines,” IEEE Trans. Instrument. Measur. 45, 750–753 (1996). [CrossRef]

23.

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (New York, Wiley & Sons, 1999).

24.

J. Skaar, L. Wang, and T. Erdogan, “On the Synthesis of Fiber Bragg Gratings by Layer Peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001). [CrossRef]

25.

L. Bömer and M. Antweiler, “Polyphase Barker sequences,” Electron. Lett. 25, 1577–1579 (1989); M. Friese and H. Zottmann, “Polyphase Barker sequences up to length 31,” Electron. Lett. 30, 1930–1931 (1994); M. Friese, “Polyphase Barker sequences up to length 36,” IEEE Trans. Inform. Theory 42, 1248–1250 (1996); A. R. Brenner, “Polyphase Barker sequences up to length 45 with small alphabets,” Electron. Lett. 34, 1576–1577 (1998).

26.

E. Van der Ouderaa, J. Schoukens, and J. Renneboog, “Peak Factor Minimization using a Time-Frequency Domain Swapping Algorithm,” IEEE Trans. Instr. Measur. 37, 145–147 (1988). [CrossRef]

27.

Y. Painchaud, H. Chotard, A. Mailloux, and Y. Vasseur, “Superposition of chirped fibre Bragg gratings for third-order dispersion compensation over 32 WDM channels,” Electron. Lett. 38, 1572–1573 (2002). [CrossRef]

28.

A. V. Buryak, G. Edvell, A. Graf, K. Y. Kolossovski, and D. Yu. Stepanov, “Recent progress and novel directions in multi-channel FBG dispersion compensation,” in OSA Technical Digest of Conference on Lasers and Electro-Optics (Washington DC, Optical Society of America, 2003).

OCIS Codes
(050.2770) Diffraction and gratings : Gratings
(060.2340) Fiber optics and optical communications : Fiber optics components

ToC Category:
Research Papers

History
Original Manuscript: March 24, 2003
Revised Manuscript: April 22, 2003
Published: May 5, 2003

Citation
Kazimir Kolossovski, Rowland Sammut, Alexander Buryak, and Dmitrii Stepanov, "Three-step design optimization for multi-channel fibre Bragg gratings," Opt. Express 11, 1029-1038 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-9-1029


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References

  1. A. Othonos and K. Kalli, Fiber Bragg Gratings (Boston, Artech House, 1999).
  2. H. Ishii, H. Tanobe, F. Kano, Y. Tohmori, Y. Kondo, and Y. Yoshikuni, �??Quasicontinuous Wavelength Tuning in Super-Structure-Grating (SSG) DBR Lasers,�?? IEEE J. Quantum Electron. 32, 433-441 (1996). [CrossRef]
  3. A. V. Buryak and D. Yu. Stepanov, �??Novel multi-channel grating devices,�?? in proceedings of Bragg Gratings, Photosensitivity, and Poling in GlassWaveguides, vol. 60 of Top series, BThB3 (Washington DC, Optical Society of America, 2001).
  4. A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, �??Optimization of Refractive Index Sampling for Multichannel Fiber Bragg Gratings,�?? IEEE J. Quantum Electron. 39, 91-98 (2003). [CrossRef]
  5. J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, �??Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,�?? IEEE Photon. Tech. Lett. 14, 1309-1311 (2002). [CrossRef]
  6. M. Ibsen, A. Fu, H. Geiger, and R. I. Laming, �??All-fibre 4 x 10Gbit/s WDM link with DFB fibre laser transmitters and single sinc-sampled fibre grating dispersion compensator,�?? Electron. Lett. 35, 982-983 (1999). [CrossRef]
  7. Y. Painchaud, A. Mailloux, H. Chotard, E. Pelletier, and M. Guy, �??Multi-channel fiber Bragg gratings for dispersion and slope compensaion,�?? in OSA Technical Digest of Optical Fiber Communication Conference, ThAA5, 581-582 (Washington DC, Optical Society of America, 2002).
  8. S. W. Løvseth and D. Yu. Stepanov, �??Analysis of multiple wavelength DFB fiber lasers,�?? IEEE J. Quantum Electron. 37, 770-780 (2001). [CrossRef]
  9. S. Narahashi, K. Kumagai, and T. Nojima, �??Minimising peak to average power ratio of multitone signals using steepest descent method,�?? Electron. Lett. 31, 1552-1554 (1995). [CrossRef]
  10. M. Friese, �??Multitone signals with low crest factor,�?? IEEE Trans. Commun. 45, 1338-1344 (1997). [CrossRef]
  11. A. Othonos, X. Lee, and R. M. Measures, �??Superimposed multiple Bragg gratings,�?? Electron. Lett. 30, 1972-1974 (1994). [CrossRef]
  12. G. Sarlet, G. Morthier, R. Baets, D. J. Robbins, and D. C. J. Reid, �??Optimization of multiple exposure gratings for widely tunable laser,�?? IEEE Photon. Techn. Lett. 11, 21-23 (1999). [CrossRef]
  13. V. Jayaraman, Z.-M. Chuang, and L. A. Coldren, �??Theory, design, and performance of extended tuning range semiconductor lasers with sampled grating,�?? IEEE J. Quantum Electron. 29, 1824-1834 (1993). [CrossRef]
  14. B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, �??Long periodic superstructure Bragg gratings in optical fibres,�?? Electron. Lett. 30, 1620- 1622 (1994). [CrossRef]
  15. W. H. Loh, F. Q. Zhou, and J. J. Pan, �??Sampled Fiber Grating Based-Dispersion Slope Compensator,�?? IEEE Photon. Techn. Lett. 11, 1280-1282 (1999). [CrossRef]
  16. H. Ishii, Y. Tohmori, Y. Yoshikuni, T. Tamamura, and Y. Kondo, �??Multiple-Phase-Shift Super Structure Grating DBR Lasers for Broad Wavelength Tuning,�?? IEEE Photon. Techn. Lett. 5, 613-615 (1993). [CrossRef]
  17. Y. Nasu and S. Yamashita, �??Multiple phase-shift superstructure fibre Bragg gratings for DWDM systems,�?? Electron. Lett. 37, 1471-1472 (2001). [CrossRef]
  18. R.W. Gerchberg andW. O. Saxton, �??A practical algorithm for the determination of phase from image and diffraction plane pictures,�?? Optik 35, 237-246 (1972).
  19. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, 2nd ed. (Cambridge England, Cambridge Univ. Press, 1992).
  20. D. R. Gimlin and C. R. Patisaul, �?? On minimizing the Peak-to-Average Power Ration for the Sum of N Sinusoids,�?? IEEE Trans. Commun. 41, 631-635 (1993). [CrossRef]
  21. S. Narahashi and T. Nojima, �??New phasing scheme of N-multiple carriers for reducing peak-to-average power ratio,�?? Electron. Lett. 30, 1382-1383 (1994). [CrossRef]
  22. J. Schoukens, Y. Rolain, and P. Guillaume, �??Design of Narrowband, High-Resolution Multisines,�?? IEEE Trans. Instrument. Measur. 45, 750-753 (1996). [CrossRef]
  23. C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (New York, Wiley & Sons, 1999).
  24. J. Skaar, L. Wang, and T. Erdogan, �??On the Synthesis of Fiber Bragg Gratings by Layer Peeling,�?? IEEE J. Quantum Electron. 37, 165-173 (2001). [CrossRef]
  25. L. B¨omer and M. Antweiler, �??Polyphase Barker sequences,�?? Electron. Lett. 25, 1577-1579 (1989); M. Friese and H. Zottmann, �??Polyphase Barker sequences up to length 31,�?? Electron. Lett. 30, 1930-1931 (1994); M. Friese, �??Polyphase Barker sequences up to length 36,�?? IEEE Trans. Inform. Theory 42, 1248-1250 (1996); A. R. Brenner, �??Polyphase Barker sequences up to length 45 with small alphabets,�?? Electron. Lett. 34, 1576-1577 (1998).
  26. E. Van der Ouderaa, J. Schoukens, and J. Renneboog, �??Peak Factor Minimization using a Time-Frequency Domain Swapping Algorithm,�?? IEEE Trans. Instr. Measur. 37, 145-147 (1988). [CrossRef]
  27. Y. Painchaud, H. Chotard, A. Mailloux, and Y. Vasseur, �??Superposition of chirped fibre Bragg gratings for thirdorder dispersion compensation over 32 WDM channels,�?? Electron. Lett. 38, 1572-1573 (2002). [CrossRef]
  28. A. V. Buryak, G. Edvell, A. Graf, K. Y. Kolossovski, and D. Yu. Stepanov, �??Recent progress and novel directions in multi-channel FBG dispersion compensation,�?? in OSA Technical Digest of Conference on Lasers and Electro- Optics (Washington DC, Optical Society of America, 2003).

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