A novel approach for the reflection equalization of a phase-only sampled fiber Bragg grating (FBG) is presented, where the grating is specially designed as a simultaneous dispersion and dispersion-slope compensator with channels up to 51. The sampling-function used is given with an analytical form with a linearly-chirped sampling period and is optimized by using the simulated annealing algorithm.
©2008 Optical Society of America
With the increasingly demands for the broad-band and high-speed fiber transmission link, the simultaneous chromatic dispersion and dispersion-slope compensation has become one of the critical issues to be resolved in a long-haul fiber communication system. [1–3] Various approaches have been proposed to manage the chromatic dispersion. Among them, the multichannel FBG, as one of the fiber-based broad-band promising components, has attracted a great interest due to the low cost, low insertion loss, and high performances for either wavelength filtering or chromatic dispersion management. [3–18] To date, several kinds of multi-channel FBGs have been proposed and demonstrated as the dispersion compensator, such as the sinc-sampled FBG,  the superimposed FBG,  amplitude and phase sampled FBG, [7–8] and the Talbot-effect based FBG  etc. In particular, the phase-only sampled FBG has attracted much more interest due to its lower index-modulation demanded and the smoother refractive-index profile which is especially compatible with the robust side-writing phase-mask technique. [10–15] With a diffraction-compensated phase mask, we have firstly demonstrated the phase-only sampled high channel-count FBG, in which almost identical dispersion through all the channels has been obtained, [11, 12] but the dispersion slope issues have not been addressed. Lee et al. [13, 14] numerically demonstrated a method for the simultaneous dispersion and dispersion-slope compensation, which is based on the utilization of a purely phase-sampled FBG while the sampling period is chirped. However, the proposed method is based on a discrete phase-only sampling (without an analytical form) and the channel number is limited less than 16, it is not available to the practical fabrication of the multi-channel FBG based on the phase-mask side-writing technique, because the split of phase shifts in the phase-mask caused by the diffraction effect have not been considered. [15,16] Moreover, their solution to address the non-identical bandwidth is based on an assumption that the grating is extremely strong and the reflection is saturated or changeless, which is unrealistic for a high channel-count FBG due to the limited index-change we can obtain at present. Most recently, full C-band slope-matched dispersion compensator based on a phase-only sampled FBG with 51channel has been experimentally demonstrated. [17, 18] Due to the utilization of the chirped sampling approach, the resulted reflection spectrum becomes a little distorted. Not only the channel-bandwidths are nonidentical, but also the channel-channel reflection becomes inclined. For the first one, it is generally known and has already been studied to date. [8, 19] Since the bandwidth is inversely changed with the magnitude of the dispersion, which in return limits the maximum dispersion slope one can obtain and may be neglected only as the dispersion slope demanded is small enough. In this paper, we concentrate our attentions on the later one although this phenomenon is rarely noticed and could be eliminated if the grating is extremely strong. We demonstrate a simultaneous dispersion and dispersion-slope compensator based on a phaseonly sampled FBG with channels up to 51, where the sampling function is continuous one given with an analytical form and thus the split of phase shifts in the phase-mask caused by the diffraction effect can be easily compensated. Moreover, a simple method is proposed to equalize the reflection spectrum distortion by using a specially designed sampling function.
2. Simultaneous dispersion and dispersion-slope compensation
As is generally known, the sampled FBG is the product of a single-channel seed grating with the sampling function in spatial domain. In general, the induced refractive index-modulation Δn can be expressed as
where Δn 1(z) is the index-modulation, z is the position along the grating, Λ is the local pitch of a seed grating and it can be expressed as Λ(z)=Λ 0(1-Cg×z) for a linearly chirped FBG, where Λ 0 is the period at the beginning position of grating and Cg is the chirp rate of the grating period. s(z) denotes a sampling function with period of P, in general, it can be expanded in a Fourier series
where m is the Fourier series, Sm is the complex-valued Fourier coefficient. To realize the simultaneous dispersion and dispersion-slope compensation, we introduce a chirp in the sampling period the same as the reported in Ref. 5, i.e., we make the local sampling period as P(z)=P 0(1+Cs·z), where P 0 is the initial sampling period, Cs is the linear variation coefficient. For a general case of Cs≪1, the sampling function may be expanded and approximately expressed as
where Ceff(=Cg-Cs·mΛ0/P 0) is the equivalent chirp rate of the grating period. It is obviously seen that the chirp in sampling period may be approximately equivalent to the chirp in grating period. Since the dispersion magnitude is inversely proportional to Ceff(which is a linear function of the channel number m), the dispersions in all the channels are no longer identical but changes according to the channel number m, and thus the required dispersion and dispersion slope may be approximately obtained by suitable choosing the value of Cg and Cs, respectively.  From Eq. (4), it is also seen that the absolute value of Δnm(z), i.e. the index modulation for the ghost grating m (channel m), is directly proportional to the Fourier coefficient Sm. Therefore, in order to create a FBG with multiple identical channels, in general, one needs to optimize the phase-only sampling function by making all the in-band (for a given channels N) Fourier coefficients Sm identical. 
To confirm the above proposal, a 51-channel linearly chirped FBG is designed by using a continuous phase-only sampling method. [11–12] The FBG is used for dispersion and dispersion-slope compensation of a conventional single-mode fiber (central wavelength=1545 nm; dispersion=16.5 ps/nm/km; dispersion slope=0.06 ps/nm2/km) with length of 110 km. Firstly, we write the phase-only sampling function s(z)with the initial sampling period P 0 as s(z)=sb(z)⊗∑mδ(z-mP 0), where sb(z) is the base sampling function in one period which is given as a continuous one with the analytical form: sb(z)=exp[iθg(z)]. We assume that θg(z) has the general form including many harmonic terms as:
where the number of terms M is minimized, 2M uniform channels could be achieved with M terms in this series since there are two free parameters for each term. By using the simulated annealing algorithm, the parameters αn and βn are optimized to make the channel spectrum flat within the band of interest.  In our case, we purposely eliminated the spatial frequencies from 17 to 21 to effectively avoid the phase-vanishing effect.  With the same cost function defined in Ref. , we obtain a set of αn, βn with n=1⋯16, 22⋯31 which are listed in table 1. Figure 1(a) shows the phase distribution of the optimized sampling function in a normalized period (1mm), which will be transferred into the phase mask with pre-compensation of the diffraction effect.  Figure 1(b) shows the obtained channel spectrum in which the non-uniformity over all 51 channels is less than 8.0×10-5, and the diffraction efficiency is larger than 80%.
Secondly, the sampling function is multiplied by a single channel FBG which is designed with the layer-peeling (inverse scattering) method, the reflection spectrum of the sampled FBG could be calculated with the transfer matrix method. Note that in order to match the dispersion (-1815 ps/nm) and dispersion slope (-6.6 ps/nm2) at the central wavelength of 1545 nm, parameters for the chirp rates Csa and Cg are optimally selected as -0.943×10-4 cm-1 and -1.723×10-5 cm-1, respectively. Figure 2 shows the calculation results without the reflection equalization. Dispersion about - 1815 ps/nm at wavelength of 1545 nm, dispersion slope of - 6.6 ps/nm2 have been successfully obtained, which are almost the same as what we expect. However, as shown in Fig. 2(b), the reflection between all the inter-channels become no longer identical, i.e., the reflections are linearly decreased with the wavelength decrement.
3. Equalization of the Inclined Reflection Spectrum
The inclined reflection-spectrum as shown in Fig. 2(b) will more or less affects the performance of optical transmission system especially for those grating with a reflection less than 90% and the Er-doped fiber amplifier is inserted at the same time. To equalize the reflection spectrum, a simple method is proposed and described as follows.
For a linearly chirped FBG, the relationship between the transmission loss TL and the grating parameters may be empirically expressed as 
where η denotes the confined coefficient of the electrical field in the fiber core, D (in a unit of ps/nm) denotes the grating dispersion, Δn is in the unit of 10-4. From Eq. (6), it can be seen that once the dispersion in each channel are linearly changed, the corresponding channel reflection will not be identical with each other even if all the Fourier coefficients Sm for a given channel number of N are the same. To equalize this distortion appeared in the reflection spectrum, the value of |Sm|2×|Dm| for each channel should be kept in a constant. Note that the proposal given in Ref. 14 for the dispersion-slope compensation with chirped Fourier coefficients Sm is only accurate when the FBG is extremely strong. i.e., the reflection is almost equal to 100%. As a matter of fact, it can be seen from Eq. (4) that any change of the amplitude of Sm will not affect the phase term and thus make no contributions to the dispersion.
To solve this inclination existed in the reflection spectrum, the value of |Sm|2×|Dm| has been kept in a constant. Firstly, the target Fourier coefficient STm is determined according to the dispersion spectrum shown in Fig. 2(a). Next, a new continuous phase-only sampling function is proposed and optimized with the simulated annealing algorithm. The cost function is defined as
Table 2 shows the optimized αn,βn used for the purpose of the reflection equalization. Figure 3(a) shows the phase distribution of the sampling function. Figure 3(b) shows the obtained channel spectrum in which the sum of the square difference between the target Fourier coefficient |STm|2 and the calculated one |SCm|2 over all 51 channels is less than 4.3×10-9. The Fourier coefficient SCm of the optimized sampling function is almost identical with the target one STm and the diffraction efficiency is larger than 80%. Figure 4(a) shows the effectively equalized reflection spectrum using the new sampling-function. To clearly show the equalization effect, the reflection spectra with and without reflection equalizations are illustrated in Fig. 4(b). It is seen that the maximum channel-channel reflection difference is decreased from 0.03 to 0.004 (i.e., about 1/10). There also exists some reflection fluctuations among the channels, which are attributed to the differences between the resulted Fourier coefficient and the target ones in the optimization process. Figure 5(a) shows the group delay spectra with and without reflection equalization. Seeing from a randomly selected channel as shown in the inset of Fig. 5(a), we can find that the reflection equalization process make no effect on the group delay spectrum (i.e., the dispersion), which in return means that the Fourier coefficient Sm has no relation with the dispersion slope obtained. Figure 5(b) illustrates the group delay ripples of the central channel with and without reflection equalization which indicates that the maximum ripples are both smaller than 0.2 ps.
Figure 6(a) shows the index modulations of the multi-channel FBGs with and without reflection equalization. It is seen that the index modulation are identical with each other which in turn means that the method for reflection equalization does not influences the index modulation. Figure 6(b) shows the spectra of the shortest- and the longest-wavelength channels. It is seen that nearly identical reflections have been obtained. However, it can also seen that the bandwidth of the first channel (i.e., 0.3319 nm) is larger than the 51th (i.e., 0.2884 nm). The bandwidths of all 51-channel are gradually decreased according to the equivalent chirp rate of the grating period influenced by the linearly increased sampling period. Noted that, to equalizing this non-identical channel bandwidth is certainly a critical issue to be resolved for DWDM transmission link with bit-rate above 10Gbs, however it is beyond this study. The purpose and main result of this study is to verify that the advanced phase-only sampling function proposed can be used to design a multi-channel FBG with non-identical characteristics but not limited to those ones with identical channels.
In conclusion, we theoretically and numerically demonstrate a simultaneous dispersion and dispersion-slope compensator based on a continuous phase-only sampled and sampling-period chirped FBG with channels up to 51. In particular, the inclined reflection spectrum due to the dispersion slope of FBG is successfully equalized by using a specially designed sampling-function in which the Fourier coefficient changes in accordance with the dispersion of each channel. It is believed that any other kind of multi-channel FBGs with non-identical channel-channel characteristics can also be realized with the proposed method.
This work was supported by the Grant-in-Aid for Scientific Research of the Ministry of Education, Culture, Sports, Science and Technology of Japan. This work was also partly supported by the Telecommunications Advancement Foundation and the Kurata Memorial Hitachi Science and Technology Foundation in Japan.
The authors would like to thank Dr. Yves Painchaud for his valuable comments and discussions.
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