## Abstract

A double sampling method enabling to have excellent channel uniformity and high in-band energy efficiency is firstly proposed for the design of an ultrahigh-channel-count fiber Bragg grating (FBG), which is based on the simultaneously utilization of an amplitude-assisted phase sampling (AAPS) function and a phase-only sampling (POS) function. As
examples, two typical 10-dB FBGs with a length of 12 cm, dispersion of - 1360ps/nm, channel spacing of 0.8 nm, and a consecutive 135- and 405- channels are numerically designed. The maximum index-modulations required are about 0.8×10^{-3}, and 1.3×10^{-3}, respectively. Compared with the proposed method, the other two kinds of double sampling schemes by utilizing either the double AAPS (i.e., AAPS+AAPS) or the double POS (i.e., POS+POS) have also been introduced for the design of the multichannel FBGs. Fabrication tolerances to the designed 135-channel FBG obtained with the AAPS plus POS method are numerically investigated.

© 2009 Optical Society of America

## 1. Introduction

Recently, with the rapid development of the broad-band and high data-rate transmission link in fiber communication system, ultrahigh-channel-count and ultra-broadband fiber Bragg grating (FBG) has attracted great interests due to its excellent inter- and intra-channel performances for wavelength filtering used as either the chromatic dispersion compensator [1-13], the comb filter [14-19], or the wavelength readout [18]. The FBGs have many advantageous features such as small size, low insertion loss, high reliability, and the compatibility with other fiber components.

To date, several kinds of methods have been proposed to design and practically fabricate the high-channel-count FBG, such as the amplitude-only sampled FBG [1, 2], the superimposed FBG [3], the amplitude-phase sampled FBG [4-5], the phase-only sampled FBG [6-10], the Talbot-effect based FBG [14-19], the optimization algorithm based FBG [20-22], and the inverse-scattering method based FBG [23-24]. In particular, the phase-only sampled FBG has attracted more interests due to the minimum index-modulation requirement and the smooth index-modulation profile [4-8]. High-channel count FBG with the channel up to 81, used as either the dispersion compensator or the simultaneous dispersion and dispersion slope compensator have already been demonstrated [8-9, 25]. However, with further increasing the number of wavelength-division-multiplex (WDM) channels (e.g., ~120), optimization for the phase-only sampling function becomes extremely difficult or even impossible to be converged with the present nonlinear algorithms because too large number of the free-parameters need to be optimally decided. It’s too hard to realize a high-count-channel FBG covering broad telecom bands by using the traditional phase-only sampling method. M. Bernier et el. have recently demonstrated a significant results on the ultra-broadband FBG based on the utilization of a highly chirped phase mask and the infrared femtosecond pulses writing technique [12], where a high reflectivity filter providing a wavelength coverage of five telecom bands (E+S+C+L+U) is demonstrated. Note that, the resulted ultra-broadband FBG is non-channelized which is highly desired for the broadband chromatic dispersion compensation. Meanwhile it may largely increase the utilization efficiency of the wavelength band. However, a channelized ultra-broadband FBG as a function of wavelength filtering would also be attractive one for its wide applications to the multi-wavelength fiber laser, WDM add-drop multiplexers, and the interleavers etc.[26-27].

Most recently, we report a continuous phase-only sampling scheme enabling one to create an ultrahigh-channel-count FBG with excellent channel uniformity and high in-band energy efficiency based on two phase-only sampling functions. The resulted FBG has 135 channels covering three telecom bands (S+C+L) [10]. However, there exists a major drawback, i.e., the out-of-band channels produced by the 45-channel sampling function would inevitably overlap with the in-band channels, which in return brings a considerably distortion to the reflection and group delay spectra, and thus the ultra-high-count FBG with consecutive channels coving more than two bands are not available in practice.

In this paper, a consecutive high-channel-count FBG is successfully demonstrated based on the simultaneous utilization of the amplitude-assisted phase-only sampling (AAPS) function and a phase-only sampling (POS) function. The remainder of this paper is organized as follows. The principle of the AAPS+POS method is theoretically described in Section 2, which is based on the Fourier theory. In Section 3, two FBGs with the consecutive 135- and 405- channels are respectively designed by using the AAPS+POS method. In Section 4, the other two kinds of double sampling methods (i.e., AAPS+AAPS and POS+POS) are numerically demonstrated to compare with the proposed AAPS+POS method. In Section 5, the fabrication tolerances required for the double sampling based FBG are numerically investigated. Finally, the conclusions are given in Section 6.

## 2. Principle of the double sampling method

Unlike the traditional sampled FBG, a double sampled FBG is the product of a single-channel seed grating *Δn _{sg}* with two sampling functions

*s*

_{1}(

*z*) and

*s*

_{2}(

*z*). The induced refractive indexmodulation Δ

*n*can be expressed as

_{s}Here Δ*n*
_{1}(*z*) is the maximum index-modulation of the seed grating, z is the position along the grating, *Λ* represents the average uniform period of the seeding grating, *ϕ*
_{g}(*z*) denotes the phase-change related to the chirp properties of the seed grating. For convenience, the “dc”
part of the index-modulation is neglected in Eq. (1). In our case, *s*
_{1} (*z*) is assumed to be an AAPS function, and meanwhile the *s*
_{2}(*z*) is assumed to be a POS function which can be expressed as

where *α*
_{1}(*z*) and *θ*
_{1}(*z*) denote the amplitude and phase distribution of the function *s*
_{1}(*z*), and *θ*
_{2}(*z*) denotes the phase distribution of the POS function *s*
_{2}(*z*). The summation terms in Eqs.
(2) and (3) denotes the sampling functions expanded in Fourier series, respectively, where *P*
_{1} and *P*
_{2} are the periods of the sampling functions and the in-band channels considered are assumed to be 2*N*
_{1} + 1 and 2*N*
_{2} + 1, respectively. The channel spacings *Δv _{1}* and

*Δv*

_{2}(in frequency domain) are given as

*Δv*

_{1}=

*c*/ 2

*n*

_{eff}*P*

_{1}and

*Δv*

_{2}=

*c*/ 2

*n*

_{eff}*P*

_{2}, respectively.

*S*

_{1m}and

*S*

_{2m}are the complex Fourier coefficients,

*n*is the effective index of FBG and

_{eff}*c*is the velocity of light in vacuum. By using the simulated annealing algorithm and Gerchberg- Saxton iterative method [6, 28], this sampling functions

*s*

_{1}(

*z*) is optimized so that the absolute value of the Fourier coefficients are identical within the band of interest [6] and the other Fourier coefficients beyond |

*m*| ≤

*N*

_{1}are zero. Meanwhile the magnitude of |

*α*

_{1}(

*z*)| near to a unit is compulsively demanded. The figures of merit for the optimization of the Eq. (2) are given as follows:

- Same magnitudes of the Fourier coefficient |
*S*_{1m}| with*m*= -*N*_{1},….1,……,*N*_{1}for a given number of the channel 2*N*+1;_{1} - $\sum _{m=-{N}_{1}}^{m={N}_{1}}{\mid {S}_{1m}\mid}^{2}=1.$
- All values of the |
*α*_{1}(*z*)| are desired to be near to the unit.

So once the sampling function *s*
_{1}(*z*) is optimally established, the Eq. (2) can be rewritten as

For the phase-only sampling function *s*
_{2}(*z*) with in-band channels 2*N*
_{2}+1, it can be obtained by using the same method what we described and used in [6]. However, since it is phase-only one, the out-band channel cannot be eliminated. Figure 1 shows the principle of the double sampling function (AAPS+POS) based on the Fourier theory. All the designing steps are given as follows:

- As a seed grating, a single channel FBG is firstly designed by using the layer-peeling method.
- By utilizing the simulated annealing method and the Gerchberg-Saxton algorithm, optimization for the two sampling functions are accomplished. A (2
*N*_{1}+1) channels FBG can be generated by multiplying the AAPS function*s*_{1}(*z*) with the seed grating. Note that, there must exists an amplitude modulation with a period of*P*_{1}(2*N*_{1}+ 1) on the amplitude profile the AAPS function, so the channel number can not be too large, which is limited by a precision of the practical writing system utilized. - The phase-only sampling function
*s*_{2}(*z*) with a higher channel-count of 2*N*_{2}+ 1 is multiplied by the amplitude-assisted phase-only sampled FBG. To avoid overlapping among the generated channels, the condition*P*_{2}≤*P*_{1}/ (2*N*_{1}+ 1) should be satisfied.

Figure 1(d) shows the reflection spectrum of the realized FBG using the double sampling method (AAPS+POS), it can be seen that one can easily obtain consecutive (2*N*
_{1} + 1) · (2*N*
_{2} + 1) channels with a channel spacing of Δ*v*
_{1} in the frequency domain. Therefore, the channel number can be easily and considerably increased. Moreover, the outband channel resulted from the AAPS function *s*
_{1}(*z*) is null, so there exist no overlap between the channels. Note that the relationship between the maximum refractive index-modulation *Δn _{M}* required for the doubly sampled FBG and the single-channel one

*Δn*

_{1}can be expressed as

where *η*
_{1} and *η*
_{2} are the energy efficiency of the sampling functions *s*
_{1} (*z*) and *s*
_{2}(*z*), respectively. Since the energy efficiency of the AAPS function (*η*
_{1}) is 100%, the energy efficiency of the double sampling (AAPS+POS) is only determined by *η*
_{2} of the function *s*
_{2}(*z*).

## 3. Design results utilizing the double sampling method: (AAPS + POS)

To verify the above proposal, two typical designs for the FBG with consecutive 135 and 405 channels are implemented. Firstly, an AAPS function with 3-channel and a POS function with 45-channel are designed, which are shown in Figs. 2 and 3, respectively. Figures 2(a) and 2(b) show the amplitude and phase distribution of the 3-channel sampling function normalized in one period. It can be seen that there exists a small and low-frequency oscillation on the amplitude profile of the 3-channel sampling function. Figure 2(c) shows the obtained channel spectrum in which the in-band energy (diffraction) efficiency is 100%. Figure 3(a) shows the phase distribution of the 45-channel phase-only sampled function normalized in one period. Figure 3(b) shows the obtained channel spectrum in which the non-uniformity over all 45 channels is less than 0.5%, and the in-band energy (diffraction) efficiency is larger than 93%. The sampling period of the 3-channel sampling function is 1 mm (i.e., the channel spacing is 0.8 nm). To satisfy the condition *P*
_{2} ≤ *P*
_{1} / (2*N*
_{1} + 1), the sampling period of the 45-channel sampling functions is chosen as 1/3 mm (i.e., the channel spacing is 2.4 nm for wavelength at 1550 nm).

Secondly, a seed grating is designed by using the layer peeling method [29]. The seed grating is designed to have the chromatic dispersion compensation of -1360 ps/nm, a length of 12 cm, and the 0.5-dB bandwidth of 0.4 nm. The maximum refractive index-modulation required for the seed grating is about 6.5×10^{-5}. The central wavelength is 1545 nm. By multiplying the seed grating with the 3- and 45-channel sampling functions in spatial domain, an ultrahigh channel-count FBG with 45 sets of 3-channel (i.e., 135 consecutive channels) is realized. The design results of the 135-channel double sampled FBG (AAPS+POS) are illustrated in Fig. 4. Figure 4(a) shows the index-modulation profile, the inset shows the fine profile within 1mm region in the grating direction. It can be seen that the maximum index-modulation is less than 8×10^{-4}, which is easily obtainable with the current hydrogen-loaded
photosensitive fiber. There also exists a cosine-like oscillation on the grating profile, period of which is about 0.5 mm, which is determined by the 3-channel AAPS function. By using the advanced phase-mask writing technique incorporated with a narrowing writing beam [30], it is possible for one to write this kinds of low-frequency oscillation on the envelop of the FBG.

The calculated reflection and group delay spectra of the 135-channel FBG covering a wavelength range of 108 nm are shown in Fig. 4(c). To illustrate the spectra clearly, three channels locating at the central wavelengths of 1509 nm, 1545 nm, and 1581 nm are given in the insets of Fig. 4(c). Moreover, the group delay ripples for all in-band channels are smaller than ±1 ps. To demonstrate the strong ability of the double sampling method for designing ultrahigh-channel-count FBG, a 405-channel FBG covering a wavelength range of 324 nm is further designed by utilizing a 9-channel AAPS function and a 45-channel POS function. The amplitude and phase distribution of the 9-channel AAPS function normalized in one period are illustrated in Figs. 5(a), 5(b), respectively. The spectral response obtained is shown in Fig. 5(c). The corresponding reflection and group delay spectra are calculated with the transfer matrix method, and the results are shown in Fig. 6(c) in which three channels at the central wavelengths of 1420.5 nm, 1492.5 nm, and 1564.5 nm are illustrated, respectively. The group delay ripples (GDRs) of these three channels are smaller than ±1.5 ps. The index modulation and phase contribution of the 405-channel FBG are shown in Figs. 6(a) and 6(b), respectively.

The maximum index modulation is about 1.3×10^{-3} which is obtainable with the current hydrogen-loaded photosensitive fiber or could be very easily achieved by using the infrared femto-second laser technique [12]. There also exists a oscillation (with 8 periods in 1 mm) on the index-modulation profile, period of which is approximately equals to 0.125 mm, nearly four times small than the one shown in Fig. 4(a). Therefore a high-resolution translating stage and a UV laser with beam size at least smaller than 0.0625 mm are desired to produce this kind of index modulation.

## 4. Design results utilizing double sampling methods: (AAPS+AAPS) and (POS+POS)

There also exist the other two kinds of double sampling methods, i.e., AAPS+AAPS method and POS+POS method. To compare these two methods with the AAPS+POS method, two designs with AAPS+AAPS method and POS+POS method are given as follows.

#### 4.1. Design results of AAPS+AAPS method

The double AAPS method is implemented for the design of an 81-channel FBG by multiplying the seed grating with a double 9-channel AAPS, which may be the same one described in Section 3. Figure 7 shows the design results of the 81-channel FBG based on the AAPS+AAPS method. The grating index profile is shown in Fig. 7(a), the insets show the index-modulation profiles within a region of 0.1 and 1mm, respectively. The grating phase profile is also shown in Fig. 7(b), fine structures of which within a region of 0.1 mm and 1mm are shown in the insets. Figure 7(c) shows the reflection and group delay spectra, where the insets shows the spectra of the central three channels. The maximum group delay ripples within the three channels are smaller than ±1 ps. Just as is expected, the out-band of the 81- channel reflection spectrum is null, since the in-band energy efficiency is 100% for the AAPS function. The SNR of the reflection spectrum is larger than 120 dB. Nearly ideal performances for the multi-channel FBG can be obtained theoretically. However, from Fig. 7(a), it can be seen that there exists a high-frequency oscillation on the index-modulation profile compared with the one shown in Fig. 4(a). Period of this oscillation becomes considerably small, to be 0.0124 mm (with 81 periods in 1 mm), which is inversely proportional to the channel number once the double AAPS method is adopted. Therefore with the channel number is further increased up to 135, period of this oscillation then would be further decreased to 7.4 μm. In reality, this kind of fine structure on grating profile is nearly impossible to be created.

#### 4.2. Design results of POS+POS method

The double POS method is carried out for the design of a 135-channel FBG by multiplying the seed grating with a 3-channel POS function in [10] and the 45-channel POS function in Section 3. Figure 8 shows the design results of the 135-channel FBG based on the POS+POS method. The sampling periods of these two functions adopted are 1 mm and 1/3 mm, respectively. As is shown in Fig. 8(a), owing to the utilization of the double POS functions, the index-modulation profile of the grating desired is very smooth, which is the same as the one for the seed grating. The grating phase profile is shown in Fig. 8(b), where the insets illustrate the fine structures within the regions of 0.1mm and 1mm, respectively. Figure 8(c) shows the reflection and group delay spectra, where the insets show the spectra of typical two channels. The group delay ripple (GDR) within the channel of 1545 nm is about ±100ps. The signal-to-noise ratio (SNR) of the reflection spectrum is smaller than 6 dB. The GDR within the channel of 1587.1 nm is also about ±100 ps, the SNR of the reflection spectrum is smaller than 10 dB. Compared with the ones shown in Fig. 4(c), it is found that the GDR become rather large and the SNR obtained is considerably decreased down to 6 dB, which is obviously due to the inevitable overlap between the in-band channels with the out-band ones. In summary, the method utilizing AAPS+AAPS may exhibit the advantages of both high energy efficiency and high SNR. However, the index profile is too complex to be realized. The POS+POS method has a unique advantage, i.e., the desired index-modulation profile is smooth, which considerably facilitates the grating’s fabrication in reality. However, the ripples existed in the reflection and group delay spectra become rather large, meanwhile the SNR is too smaller to be employed practically in the optical communication system. The AAPS+POS based method may be the best selection by considering the trade-off between the fabrication feasibility and the performance of the FBG required.

## 5. Fabrication tolerances for the double sampled FBG

As an example, the fabrication tolerance of the double sampled 135-channel FBG (i.e.,
AAPS+POS) is numerically demonstrated. The influences of the perturbations of the two
general parameters (i.e., the amplitude and phase profiles of the index change) on the
reflection and group delay spectra are investigated. Firstly, *δn* and *δ _{φ}* as the perturbation factors are added to the amplitude

*Δn*and the phase

_{sa}*φ*profiles of the index-modulation, respectively. The renewed index-modulation

*Δn*'

_{s}is expressed as

The *δ _{n}* and

*δ*are generated by multiplying the tolerance values |

_{φ}*α*| and |

*β*|, respectively with a random function whose value is normally distributed with mean 0 and standard deviation

*σ*= 1. Figure 9 shows the reflection and group delay spectra of the central three channels when

*α*= 0, < 5% , or < 10% , and |

*β*| < 0.01. It can be seen from Fig. 9(a) that the ripples on the top of the reflection spectra are smaller than 0.15 dB and 0.30 dB as the tolerance values 5% and 10% are introduced, respectively. Figure 9(b) shows the group delay spectra where the group delay ripples are illustrated in the inset. The maximum ripples are smaller than 3 ps and 6 ps, respectively. Therefore, we can conclude that both 10% perturbation in the amplitude and a 0.01 variation in phase are acceptable for the practical fabrication of the 135-channel FBG.

## 6. Conclusion

A double sampling method enabling to have an excellent channel uniformity and high in-band energy efficiency is proposed for the design of a ultrahigh-channel-count fiber Bragg grating. Based on the proposed method, two typical FBGs with 10-dB strength and a consecutive channel of 135 and 405 are successfully demonstrated. The maximum index-modulations required are about 0.8×10^{-3} and 1.3×10^{-3}, respectively. Owing to the utilization of the low-channel count of AAPS function, there exist no distortions on the reflection and group delay spectra of the ultrahigh-channel-count FBG. Moreover the index profile of the AAPS+POS FBG may be realizable by using the well-established phase-mask writing technique. In addition, the other two kinds of double sampling methods (i.e., AAPS+AAPS and POS+POS) are numerically demonstrated to compare with the proposed AAPS+POS method. Fabrication tolerances required for the double sampling based FBG have been numerically investigated. The proposed scheme may be easily extended to any other kinds of the sampled FBGs with an ultrahigh-channel count.

## Acknowledgments

This work was also partly supported by Kurata Memorial Hitachi Science and Technology Foundation, the International Communications Foundation, and Hamamatsu Science and Technology Promotion in Japan. The authors would like to thank Dr. Yves Painchaud for his valuable comments. The authors would also like to thank the anonymous reviewer of the Ref. [10] and this paper for his valuable comments and suggestions.

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