## Abstract

A novel continuous phase-only sampling function capable of producing up to 81-channel FBG with excellent channel uniformity and high in-band energy efficiency is presented and optimized by using the simulated annealing algorithm. In order to fabricate this kind of FBG with a conventional side-writing phase-mask technique, both the diffraction effects and fabrication tolerance of the phase-shifted phase mask have also been addressed. Compared with the numerical results, a 45-channel (spacing 100 GHz) and an 81-channel (spacing of 50 GHz) phase-only sampled linearly chirped FBG are successfully demonstrated.

© 2006 Optical Society of America

## 1. Introduction

Recently high channel-count fiber Bragg grating (FBG), as one of the fiber-based promising solutions to the broad-band chromatic dispersion compensation for the existing fiber link, has attracted great interests [1–12]. However, with an increasing number of wavelength-division-multiplex(WDM) channels to cover the full C-band, high channel-count FBG devices become extremely difficult to realize due to the requirements of a considerably high index-modulation and the tremendous precision of the FBG writing tools. To date, three main methods have been proposed trying to make FBG based broad-band devices realizable. One is the utilization of a long-length fiber as reported in [4], where a nonlinearly chirped FBG was demonstrated as a simultaneously chromatic dispersion and slope dispersion compensator, but the grating length is 1 meter too long to be used in a practical system due to the effects of the package, temperature and uniformity. The second one is to overwrite the gratings with different central wavelengths in a same length of fiber [5], but exact matching the channel central wavelengths to the ITU grids is a challenge, and the writing time for a high channel-count FBG is too long to be acceptable. The third one is the utilization of various advanced sampling functions [7–14] to construct a multiple identical FBG in a same length of fiber as that of one-channel FBG. Of all the sampling methods, the phase-only sampling seems to be the most prospective one, because it can deduce the index change required for a 40- or 80-channel sampled FBGs to a practical levels [10, 11]. Furthermore, phase-only sampling has no modulation in its amplitude such that the apodization profile for a multi-channel FBG is the same as that of the seed grating, which thus makes the multi-channel gratings particularly suitable to be fabricated with the robust side-writing phase-mask technique.

However, how to exactly write a varying sampling-phase in the profile of the FBG has been a critical issue in the practical processes. For a discrete phase-only sampling function, each period is divided into a number of discrete steps such that one can insert these phase steps into the profile of the seed grating directly by using the phase-shifted mask technique [15]. Although the phase-shifts can be imprinted into the FBGs by adjusting the relative transverse moving between fiber and the interference fringes in a direct writing system or by wiggling the phase mask in a scanning-writing system [16], the phase-shifted phase mask approach has advantages over the others since the phase shift is directly fabricated in an e-beam phase mask with very high accuracy and it has already been used to write a multichannel FBG with 5-9 channel [11, 17]. With increasing the number of channels up to 80 to cover full C-band (4THz), optimization for the discrete sampling function becomes too difficult to be realized, a continuous phase-only sampling (with it the phase profile can simply be encoded into every period of the seed grating by using the advanced stitch-error-free lithography mask) is strongly desired [18].

Furthermore, when a phase-shifted mask is used to write a grating, it was general considered that the phase-shifts in the phase-mask were completely replicated in the FBG [15].

However it is only correct when the distance between the mask and the fiber core is zero [19]. Due to the minimum spacing between the phase mask and the fiber core, which is the radius of the fiber cladding, the phase shift in mask is not replicated into the FBG but is split into two two-half amplitude phase-shifts, which are separated by a transverse distance between the +1 and -1 diffracted order. The split of the phase-shifts in the phase mask does occur not only for a single phase-step, but also for multiple phase-steps, which could be crossover one another. As a result, the phase information that is written into the mask is distorted when it is transferred to a piece of fiber core by proximity printing, which thus makes the channel-channel uniformity considerably distorted.

In this paper, a novel continuous phase-only sampling function is presented, where considerations and tolerance of the diffraction effects of the phase shifts have been taken into account. Optimization for 25-, 45-, and 81-channel phase-only sampling function with excellent channel uniformity and high in-band energy efficiency is carried out with using the simulated annealing algorithm. Finally, a 45-channel (spacing of 100 GHz) and 81-channel (spacing 50 of GHz) phase-only sampled FBG are demonstrated compared with the numerical results.

## 2. Theory of the diffraction compensation phase-mask for side-writing of FBG

It is known that phase steps are necessary in the phase-only sampled FBG, which introduce phase shifts in the profile of FBG. The phase-shifted phase-mask used for the fabrication of FBG is generally a surface relief binary phase grating with rectangle profile, where phase-steps *θ*_{m}
are inserted at certain locations as shown in Fig. 1. The diffraction of the incident UV light incident on the mask splits into two at angles of ±*φ*
_{0}. Their separation in the fiber core at a distance of *z* from the mask is ∆*x* = 2*z* tan *φ*
_{0}. Base on the fact that the phase in the
mask will be split into two two-half amplitude phase-shifts separated with a lateral distance between the +1 and -1 diffracted light [19], the relation between the phase in the mask and the resulted phase *θ*_{g}
(*x*) in the fiber core is expressed as

where ⊗ represents the convolution. Taking the Fourier transform to Eq. (1), one obtains the phase frequency transfer function as

where *$\tilde{\theta}$ _{g}* and

*$\tilde{\theta}$*are Fourier transforms the phase profiles and

_{m}*f*is the spatial frequency of the mask. Assuming that both

*θ*(

_{g}*x*) and

*θ*(

_{m}*x*) are changed periodically with period

*P*, the spatial frequency

*f*is then in the unit of 1/

*P*, and the channel frequency spacing in FBG is given by ∆

*ν*=

*c*/(2

*n*

_{group}

*P*) . Equation (2) shows that the diffraction effect can cause very sever distortion for large magnitude

*f*and made the grating phase vanishes at

*f*= 1/(2∆

*x*).

Now we propose one method for designing a mask to compensate for the diffractive propagation effects to the fiber core. We simply solve Eq. (2) for the mask phase

then the pre-compensation processes can be described in the following. First, we calculate the Fourier transform of the desired FBG phase *$\tilde{\theta}$ _{g}*, use Eq. (3) to find the spatial spectrum of the mask phase, and then transform it back to find the phase function in mask. Finally, with this diffraction-compensated phase mask, we can exactly imprint the sampling-phase

*θ*

_{g}(

*x*) in the profile of the FBG rather than the distorted one. Note that this compensated method is not suitable for any binary or multi-levels phase-only sampling functions [11] because of the existence of infinite spatial-frequency terms. Another limitation for this method is that the phase-sampling functions should be the one where its spatial frequencies is not too close to

*f*= 1/(2∆

*x*), otherwise it would be divided by a small number in Eq. (3) which would lead to inaccurate or unphysical results.

## 3. A continuous phase-only sampling for high channel-count FBG

#### 3.1. Phase-only sampling function with diffraction compensation

Sampling method is widely used to realize a multi-channel FBG in a limited length of photosensitive fiber. In a sampled FBG, the induced refractive index modulation ∆*n* can be expressed as

where ∆*n*
_{1} (*x*) is the maximum index modulation, *x* is the position along the grating, Λ is the central pitch of the grating, *ϕ*_{g}
(*x*) is the local phase for one channel grating which determines the dispersion of the grating. *s*(*x*) stands for a sampling function with period *P*.

Next, we consider the phase-only sampling function. As usual, we write the sampling function *s*(*x*) with a period *P* as

where *s*_{b}
(*x*) is the base sampling function in one period. In general, the base sampling function *s*_{b}
(*x*) discussed [10–11] is given as a discrete one. As matter of fact, it could be a continuous one with the analytical form:

We assume that *θ*_{g}
(*x*) has the general form including more harmonic terms as:

where the number of terms *M* is minimized, *α*_{n}
and *β*_{n}
are optimized such that the channel spectrum is flat within the band of interest. Considering the diffraction effect, the pre-compensated phase of the phase mask according to Eq. (3) is then obtained

When the diffraction effect is not taken into account for the design of the phase-mask, the grating phase becomes

#### 3.2 Optimization of the phase-only sampling function with simulated annealing algorithm

In Eq. (7), one would generally assume that at least 2*M* uniform channels could be achieved with *M* terms in this series since there are two free parameters for each term. In order to obtain optimal magnitudes of the series of *α*_{n}*,β*_{n}
, we exploit the simulated annealing algorithm, where the optimization criteria are the uniformity of the desired channels and the diffraction efficiency *η* [11]. Simulated annealing is a recursive Metropolis algorithm to reduce a cost function with a progressively decreasing control parameter called temperature *T*.

The cost function is defined as

where *S*_{m}
is the *mth* Fourier coefficient of the base sampling function *s*_{b}
(*x*), *η* (<1) is the target diffraction efficiency for all the in-band channel of 2*N* + 1. Depending on the desired channel numbers, a suitable value of *η* needs to be selected in order to remain the high channel uniformity as well. In our case, we choose it as 0.90. Obviously, the practical diffraction efficiency is equal to $\sum _{-N}^{N}}{\mid {S}_{m}({\alpha}_{1},{\alpha}_{2},\cdots {\alpha}_{M},{\beta}_{1},{\beta}_{2},\cdots ,{\beta}_{M})\mid}^{2},$. First, initial magnitudes of the set of *α*_{n}*,β*_{n}
with *n* = 1,…,*M* are chosen randomly. The cost function is evaluated by using Eqs. (6), (7), and (10). Then each of *α*
_{n},*β*
_{n}is shifted randomly with a uniform density of probability. If the shift of *α*_{n}*,β*_{n}
in the *k* iteration leads to a decrease of the cost, i.e., ∆*E*_{k}
=*E*_{k}
-*E*
_{k-1} ≤ 0, the new *α*_{n}*,β*_{n}
is accepted. Otherwise, if ∆*E*_{k}
> 0 the new *α*_{n}*,β*_{n}
will still be accepted with a probability

The simulated annealing algorithm is based on an analogy with statistical mechanics, which is often used to obtain a globally optimal solution for a nonlinear process with multi-variable parameters.

A number of designs were completed for the above continuous phase-only sampling function. The optimal results for a 25-, 45-, and 81-channels are shown in Fig. 2, Fig. 3 and Fig. 4, respectively, where the original phase profile (Eq. (7)) and the phase profiles under the diffraction effect (Eq. (9)) are both shown in Figs. (a), and correspondence of the two Fourier spectra are shown in Figs. (c), and (d), respectively. The spacing between the mask and the written fiber is assumed to be *z* = 10 μm. Considering the refraction happened in the fiber cladding (for cladding index of 1.45) and the radius of cladding, we have ∆*x* = 25 μm. For convenience, the period of the sampling function is normalized to one. In order to obtain the high diffraction efficiency *η*, 13, 23, and 41 harmonic terms were selected for 25-, 45-, and 81-channel, respectively. It can be seen that the original sampling design are pretty reasonable. The diffraction efficiencies obtained with the simulated annealing algorithm are 92%, 92% and 84%, respectively. Non-uniformity of the channel intensities are 0.1%, 0.8% and 1.0%, respectively. However, with the phase-split effect due to the mask diffraction, the resulted channels are considerably distorted (as shown in Fig. 2(c), Fig. 3(c), and Fig. 4(c)), which in return means that the compensation to the diffraction effect is necessary once a sampled FBG with high channel-count is written with the side-writing technique.

To verify the accomplished designs, the sampling functions were multiplied by a single channel FBG, which can be obtained by using the layer peeling method [20]. The reflection spectrum of the sampled FBG was computed with the transfer matrix method. The results for a linear chirped 45-channel with spacing 100 GHz, chromatic dispersion -1360 ps/nm and an 81-channel with spacing 50 GHz are shown in Figs. 5 and 6, respectively. The reflection spectra show high intra- and inter-channel uniformity within nearly whole C-band.

Note that, all the above results are based on the assumption that the separation between the mask and the fiber is *z* = 10 μm. However, in the practical fabrication, it is strongly desired to know in what extent the deviation of this separation could be allowable. We then investigated the effect of the spacing deviation on the inter-channel uniformity by following equation:

where ∆*x*^{R}
is the actual lateral distance between the +1 diffraction and -1 diffraction light at the fiber. ∆*x* (= 25 μm) is the nominal one for spacing *z* = 10 μm. ${\theta}_{g}^{R}$
is the resulted phase profile in a fiber core where a deviation to the nominal spacing is introduced. For convenience, the 45-channel phase-only sampling function shown in Fig. 3 was investigated. The simulation results are shown in Fig. 7. Compared with the original results shown in Fig. 3, it is found that the channel uniformity depends strongly on the magnitude of the spacing deviation, the non-uniformity of the channel intensities become 5%, 15%, and 22% when the deviation is ±1, ±3 , and ±5 μm respectively. In order to remain high channel-channel uniformity, it is known that precision of ±1 μm for the spacing is expected while the side-writing technique is used.

## 4. Experimental results

Next, the obtained diffraction-compensated phase function *θ*_{m}
(*x*) (as shown in Eq. (8)) and the phase *ϕ*_{g}
(*x*) for one seed grating (shown in Eq. (4)) with dispersion of -1360 ps/ns were encoded into the grating, whose local period becomes

where Λ is the central pitch and *φ*(*x*) = *θ*_{m}
(*x*) + *ϕ*_{g}
(*x*) is the local phase. There are two approaches to implement Λ_{M}(*x*). Ideally, one can change every period of the grating. In this case, the phase mask must be written with the required accuracy and written continuously without stitching. One can also divide the entire grating into a large number of steps. Each step is considered as a uniform grating with the local average period Λ_{M}(*x*). We used the former approach. The phase mask is designed and fabricated with a special “stitch-error-free” lithography tool. The sampled FBG was written with side-writing technique [11, 18]. To confirm our numerical results, several 45-channel and 81-channel FBGs were fabricated. Figure 8 shows the measured results of one typical FBG with 45-channel, where the grating length is 10 cm and the peak index-modulation required is about 5.6× 10^{-4}. It can be seen that nearly identical 45 channel with a channel spacing of 100 GHz, useable bandwidth of 0.21 nm, peak-peak group-delay near 20 ps, and chromatic dispersion of about -1374 ps/nm have been obtained. These results agree well with our design data. Noted that this kind of high-quality high channel-count FBG has already been successful used as a dispersion compensator through full C-band for transmission over 640-km SMF fiber [21]. Figure 9 shows the measured results of one typical FBG with 81-channel, where the grating length is 10 cm also and the peak index-modulation required is about 8×10^{-4}. It can also be seen that nearly identical 81 channel with a channel spacing of 50 GHz, useable bandwidth of 0.11 nm, and chromatic dispersion of about -1400 ps/nm have been obtained. Although the group delay ripple shown in Fig. 9(b) is a little bit large and the inter-channel uniformity is worse than that of 45-channel FBG shown in Fig. 8(b), which is most probably due to the demands of higher index-modulation and more strict precision control for the spacing between the mask and the fiber, this is the first reported for a sampled FBG with up to 81 channels and these results verify that the proposed continuous sampling with pre-diffraction compensation is reasonable.

## 4. Conclusion

A novel continuous phase-only sampling function has been proposed, where both the diffraction effects of the phase-shifted mask and the fabrication tolerance have been considered. Optimization for a 45- and an 81-channel sampling function with excellent channel uniformity and high in-band energy efficiency has been carried out with utilization of the simulated annealing algorithm. Finally, a 45-channel (spacing of 100 GHz) and 81-channel (spacing of 50 GHz) FBG for nearly whole C-band dispersion compensation have been demonstrated. The measured results for the fabricated FBGs agree well with the theoretical ones.

## Acknowledgments

H. Li acknowledges the financial supports from the Research Foundation for the Electrotechnology of Chubu (REFEC) and the Support Center for Advanced Telecommunications Technology Research (SCAT) in Japan.

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