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]. Multi-channel FBGs represent critical components for tunable lasers [2] and passive FBG-based devices [3, 4, 5]. Optical components based on multi-channel FBGs are of particular interest in wavelength division multiplexing (WDM) systems [6, 7] 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]. 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, 10]).

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, 12]). 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, 14], 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=(Δ$n_{\mathit{\text{env}}}^{\left(\mathit{\text{av}}\right)}$/Δn _N)², where Δ$n_{\mathit{\text{env}}}^{\left(\mathit{\text{av}}\right)}$ is the envelope of the index change averaged over a period of the sampling function, then for the comb-sampling Δ$n_{\mathit{\text{env}}}^{\left(\mathit{\text{av}}\right)}$~√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] 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]. 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, 5, 16, 17] effectively modifies the sampling period and results in more channels due to increased channel density.

Phase-only sampling is a preferable method for multi-channel FBG designs as it utilizes the available Δn _N most efficiently resulting in only a square root growth with the number of channels, Δn _N~√N, for optimized gratings. However, to the best of our knowledge, the phase profile of the sampling function leading to a uniform sequence of quasi-identical channels in the reflection spectrum has not been derived analytically. Therefore different numerical optimization strategies have been invoked (see, e.g., [2]). In this context, the related problem of finding the phase profile which corresponds to the minimal amplitude variation of the sampling function is more attractive as it allows for some analytical or semi-analytical calculations to be invoked in optimization (see, e.g., [20, 21, 22]) and leads to uniform channel profiles. The analytical calculations, in turn, may dramatically increase the efficiency of numerical schemes searching for the optimal solutions.

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, 5] 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

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] 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

where r ₁ is the reflection measured at z=0, r ₂ is the reflection measured at z=L, and t is the complex transmission coefficient (which is independent of the light propagation direction for reciprocal FBGs). Conventional reflection R _i and group delay τ_i (i=1, 2 depending on the grating side) are related to the complex reflection coefficients as R _i=|r_i |², τ_i=-∂Arg(r_i )/∂ω, where ω is the light wave frequency. Similar relations exist for transmission T, group delay in transmission τ_t and the complex transmission coefficient t. The calculation of the reflection data r(δ) from the grating profile q(z) is usually referred to as the direct scattering transform, whereas the inverse operation of recovering q(z) profile from the spectral data r(δ) is called the inverse scattering transform.

Before one designs any multi-channel grating with equal inter-channel separations and identical in-band specifications, a corresponding single-channel grating design should be constructed. For any physically viable spectral response in reflection of a single-channel FBG device the corresponding inverse scattering problem should be solved, e.g., by applying a layer-peeling algorithm (LPA; see, e.g., [24]) to Eq. (1). As a result of solving the inverse scattering problem, we obtain a grating design q(z)≡κ(z)exp(iθ(z)), where κ(z) is the envelope grating amplitude and θ(z) is the grating phase. The grating amplitude κ is measured in cm^-1 and is related to the induced refractive index modulation Δn as κ(z)=πΔn(z)/[2Λ(n _eff+Δn ^(av))], where Λ is the FBG period and n _eff is the effective refractive index prior to grating writing. In the literature, FBG grating amplitude and phase are often referred to as apodization and chirp profiles, respectively.

Following Refs. [2, 3, 4], 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 ϕ_l for each seeding grating,

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

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

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

is the aperiodic autocorrelation function (AACF) of a complex sequence associated with phases ϕ_l: m _l=exp(iϕ_l). 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 |C_p |≤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].

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 |C_{N -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]),

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 ϕ_l, as in Ref. [4]. 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,

where the average over the sampling period 〈s(z)〉≡Δk/2π${\int }_0^{2\mathrm{\pi }/\mathrm{\Delta }k}$ 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., Δ$n_{\mathit{\text{env}}}^{\left(\mathit{\text{av}}\right)}$.

To calculate the mean value of s(z) explicitly, we assume that the second term in Eq. (5), x(z)≡(2/N)Re${\sum }_{p=1}^{N-1}$ 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

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

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

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

where $\overrightarrow{\varphi }$=(ϕ₁, ϕ₂, …, ϕ_N).

Due to the invariance of the function in Eq. (12) with respect to a linear transformation, i.e., ϕ_l→ϕ_l+a ₁+a ₂ l, where a _1,2 are constants, dimension of the parameter space $\overrightarrow{\varphi }$ 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,

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].

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]. 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)).

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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 $\overrightarrow{\varphi }$ . 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, 18] and clipping [10, 26], 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].

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].

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]; 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.

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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]). 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.

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The Fourier transform (Eq. (14)) is a linear operation with a major property F(a ₁ r ⁽¹⁾+a ₂ r ⁽²⁾)=a ₁ F(r ⁽¹⁾)+a ₂ F(r ⁽²⁾), where a _1,2 are constants. Hence, in this approximation, dephasing of partial gratings is equivalent to dephasing of partial spectral channels. Formally the last statement does not hold beyond a weak grating limit. However, the dephasing of only weakly overlapping spectral channels is still approximately equivalent to the dephasing of the corresponding partial gratings. The most important change related to the sampling function S(z) is that it becomes slightly aperiodic. The analogy between the direct and inverse Fourier transform and the direct and inverse scattering transform allows one to generalize both Gerchberg-Saxton and iterative clipping algorithm into the spectral domain of linear FBGs retaining both the minimization of grating amplitude level and the quality of spectral characteristics. As an example, we describe one of the possible generalizations of Gerchberg-Saxton iterative scheme in detail.

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 $\sqrt{A^{\left(i\right)}}{\kappa }_1\left(z\right)$ , where κ₁(z) is the seeding grating amplitude and the constant A ⁽ⁱ⁾ is defined by the normalization condition A ⁽ⁱ⁾=${\int }_0^l${κ⁽ⁱ⁾}² dz/${\int }_0^l$${\mathrm{\kappa }}_1^2$ 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]) 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, 28], with the described optimization methods applied in Ref. [28].