Open Access
Add to BibSonomyAbstract
A novel technique to implement a Bragg grating with multi-channel and high channel-count response based on nonuniform amplitude-only sampling is proposed. Thanks to the nonuniform sampling, a sophisticated phase modulation for the generation of multi-channel spectral response can be equivalently achieved, while the pitch of the Bragg grating is maintained uniform. The principle is presented. Two design examples with two Bragg gratings for multi-channel filtering and for multi-channel chromatic dispersion compensation are provided.
©2008 Optical Society of America
1. Introduction
Bragg gratings, either fiber- or waveguide-based, have found important applications in optical communications, sensors, and other optical systems, thanks to the many advantages such as small size, low loss, high reliability and compatibility with other fiber or waveguide components or devices. Among the many applications, a grating with multi-channel spectral response is highly desirable in a wavelength-division-multiplexing (WDM) system, to perform the functions such as wavelength filtering or chromatic dispersion management.Therefore, the design and realization of a multi-channel, high-channel-count Bragg grating has been a topic of interest recently, with many approaches having been proposed [1].
One approach to realizing a multi-channel Bragg grating is to write multiple sub-gratings with different central wavelengths at the same location of a fiber or waveguide. In theory, sub-gratings written at the same location will not interfere with each other if their spectra are not overlapped [2]. However, the fabrication of the multiple sub-gratings involves a multi-inscription process, which requires a long fabrication time, a sophisticated fabrication facility and a large refractive index modulation. Another approach to realizing a multi-channel Bragg grating is to use a composite multi-channel grating [3–6], achieved by summing the sub-grating functions with an optimized phase relationship to minimize the index modulation.However, such a grating has a profile with complicated amplitude and phase modulations,which may increase the complexity of fabrication. A third approach to realizing a multi-channel Bragg is to sample a regular grating, where a periodic modulation is imposed on the envelope of a seed grating in the spatial domain, which would result in the presence of multiple and identical channels in the spectral domain. Of all the sampling methods, the amplitude sampling [7] is the simplest, but the grating has a very small duty cycle and intolerably large index modulation when the channel count is high. Frequency sampling [8] and phase-only sampling [9–12] are two other sampling methods to generate multi-channel Bragg gratings, which have the advantage of a decreased index modulation compared to the amplitude sampling. In addition, phase sampling has no modulation in its amplitude so that the apodization profile for the multi-channel grating is the same as that for the seed grating, which would reduce the realization difficulty [13, 14]. In addition to the above approaches, a multi-channel Bragg grating can also be designed by an inverse scattering algorithm such as the layer-peeling method [15], and some optimization techniques [16, 17].
To decrease the required index modulation in a Bragg grating, phase modulation should be employed. Since the grating period is in the order of several hundreds of nanometers, the realization of a precise phase modulation requires a line-by-line control with a fabrication system to have a nanometer precision, or one may use a customer-designed phase mask with the required phase modulation incorporated in the phase mask at a higher cost.
In this paper, we propose a novel technique to implement a Bragg grating with multi-channel and high channel-count response based on nonuniform amplitude-only sampling. We show that the required phase modulation to achieve multi-channel operation can be obtained equivalently by nonuniform amplitude-only sampling. The nonuniform sampling technique has been successfully applied to the design of single-channel gratings for applications in tunable dispersion compensation [19], distributed feedback lasers [20, 21], and optical code division multiple access (OCDMA) [22]. Here this technique is used, for the first time to the best of our knowledge, to design Bragg gratings with multi-channel and high channel-count response. The key advantage of the proposed technique is that the control precision is significantly reduced from the order of nanometers to sub-microns.
The remainder of the paper is organized as follows. In Section 2, the principle of the proposed multi-channel Bragg grating is presented in detail. Then, in Section 3 two design examples are presented with two Bragg gratings having 45 channels and 100-GHz channel spacing for multi-channel filtering and for multi-channel chromatic dispersion compensation are presented. A conclusion is drawn in Section 4.
2. Principle
Figure 1 shows the principle of a multi-channel Bragg grating realized based on nonuniform amplitude-only sampling.
Fig. 1. The principle of a multi-channel Bragg grating implemented based on the nonuniform sampling technology. (a) Conventional sampling (FSR ≈ 40 nm). (b) If the nonuniform sampling is used, the desired multi-channel response is realized in the m=-1 channel (marked in the dashed box).
It is known that a Bragg grating will have a multi-channel spectral response if it is spatially sampled. Figure 1(a) shows a Bragg grating with uniform sampling. It has a multi-channel spectral response, with the spectral responses in different channels being identical, and separated by a free spectrum range (FSR) determined by the sampling period. However, if a proper nonuniform sampling function is employed, as shown in Fig. 1(b), a multi-channel spectral response with the spectral responses in different channels being different would be generated. Usually, the -1st-order channel is the channel of interest. The spectral response at the -1st-order channel can be controlled by designing a proper nonuniform sampling function,with the required phase modulation being provided through the nonuniform amplitude-only sampling. In the following, the relationship between the multi-channel spectral response and the nonuniform sampling function is developed.
Assume s 0(z) is a periodic amplitude sampling function with a period of P, a nonuniform sampling function s(z) is then obtained by introducing a spatial modulation function f(z),
where A(z) is the apodization profile used to suppress the sidelobes. To simplify the implementation, s 0(z) is usually chosen as a quare-wave with a duty cycle of 0.5. If the sampling function in Eq. (1) is used to modulate a uniform Bragg grating with a period of λ,the index modulation is then
Based on the Fourier series expansion, we have
where Fm is the coefficient of the mth-order Fourier component. Then, based on Eqs. (1–3) the index modulation of the proposed sampled Bragg grating (SBG) is
where
Equation (4) shows that the SBG is actually a superposition of many sub-gratings with different grating periods, λm ∙ If f(z) is zero, then the grating is simply a conventional uniformly sampled Bragg grating, and the sub-gratings are identical except the index modulation depth. For nonuniform sampling, although the grating period is uniform, additional phase modulations, mf(z) in Eq. (4), are introduced to the sub-gratings.Obviously, the phase modulation function, πPS(z), which was used in the phase-only sampling technique [9–12], can be achieved in one of the sub-gratings if
where m=-1 is selected [18] because 1) there is no phase modulation in the m = 0 sub-grating; 2) the index modulation depth, Fm, is smaller for higher-order channels; 3) the m = 1 channel suffers from the cladding-mode loss. Based on Eq. (6), an equivalently generated phase-only sampled grating for multi-channel operation is then obtained in the m = -1 sub-grating of the nonuniformly-sampled Bragg grating in Eq. (4). If the sampling period P is small enough, the SBG would have a large FSR such that the spectra of sub-gratings are not overlapped, we can then achieve the desired spectral response at the m = -1 channel, as shown in Fig. 1(b). Obviously, the bandwidth of the desired multi-channel response, B, should be less than two FSRs. Then based on Eq. (5) we get
where neff is the effective refractive index of the fiber or waveguide. For example, neff=1.455 for a fiber Bragg grating (FBG), then based on Eq. (7) the available bandwidth as large as 40 nm can be achieved if P≤40 μm. Such a bandwidth is wide enough for most of the applications.
The index modulation is an important parameter in a multi-channel Bragg grating. Based on Eq. (4) we can see that the index modulation of the sub-gratings should be the actual index modulation of the nonuniformly sampled grating weighted by Fm. In the fabrication of a SBG, s 0(z) is usually a square-wave with a duty cycle of 0.5, and then F -1 is about 1/3.Therefore, the required index modulation for the proposed nonuniform sampling technique is three times that of the true phase-only sampling technique. In general, for the proposed nonuniform amplitude sampling technique, the required index modulation is proportional to 3√M, where M is the total channel count in the frequency domain, which is practically achievable. However, for a regular amplitude-sampled grating, the required index modulation is proportional to M, which is high and may not be practically achievable.
3. Design examples for multi-channel filtering and chromatic dispersion
Bragg gratings for multi-channel, high-channel-count filtering and chromatic dispersion are designed using the proposed technique. Based on Eq. (6), a phase-only sampling function, πPS(z), is first designed. The design process has been reported in [1].
In the first example, a Bragg grating for multi-channel filtering is designed. The phase modulation πPS(z) is then a period function with a period of P 0
where N is the sample number of the phase-only SBG, P 0 is the sampling period which is determined by the desired channel spacing of the multi-channel response, Δλ, which is given by
where λ is the center wavelength of the desired multi-channel response, which is 1550 nm in our design. π0(z) is the phase modulation within one sample of the phase-only SBG. If the desired channel count is M=45, based on the design process in [1], π0(z) is calculated, which is shown in Fig. 2(a). In our design neff is 1.455, Δλ is 0.8 nm. Then based on Eq. (9),P 0 is calculated to be 1.03 mm. If λ=532.6 nm, N=30, the apodization profile, A(z), is a raise cosine function, and the index modulation depth is 6.5×10-4, the reflection spectrum of the phase-only SBG is calculated, which is shown in Fig. 2(b).
The amplitude-only sampled grating can then be designed based on Eqs. (6) and (1). In our design s 0(z) is a square-wave with a duty cycle of 0.5 and its period, P, is determined based on Eq. (7) which should be less than 45 μm. Here P=P 0/34, about 30.4 μm. Based on Eq. (1) the new amplitude-only sampling function is then a periodic function with a period of P 0.Within one such sample the amplitude sampling function is calculated and plotted in Fig. 3. Clearly this is a nonuniformly-spaced amplitude sampling function, with the period changing from 16.3 to 88.8 μm. To control such sampling period change, a sub-micron precision is required instead of a nano precision control in the true phase-only sampling technique.
Fig. 2. (a). The phase-only sampling function, π0(z), within one sample. (b) The reflection spectrum of the phase-only sampled grating. The transmission loss in each channel is -20 dB.
Fig. 4. (a). The reflection spectrum of the amplitude-only sampled grating for multi-channel filtering. (b). The spectral response in the m=-1 channel. A 45-channel response is achieved without any true phase modulation. The transmission loss in each channel is -20 dB.
In our design λ=523.63 nm, N=30, the apodization profile is a raise cosine function,and the index modulation depth is 2×10-3, which is three times that of the grating designed based on the true phase-only sampling technique to keep the same reflectivity. The reflection spectrum of the amplitude-only sampled grating is calculated and shown in Fig. 4. Clearly in the m=-1 channel the desired multi-channel spectral response for filtering application is achieved.
In the second example, a Bragg grating for chromatic dispersion compensation is designed. For a given chromatic dispersion in the fiber link, the Bragg grating should be designed to have an opposite chromatic dispersion in each channel to compensate for the chromatic dispersion of the fibre link. For a linearly chirped Bragg grating, its phase response is quadratic. Mathematically, the phase-only sampling function that incorporates the quadratic phase response is given by
where C is the chirp rate determined by the required dispersion. In our design C=-0.033 nm/cm, corresponding to a chromatic dispersion of about -1020 ps/nm. N=70, and the apodization profile is a 4th-order super-Gaussian function with a full-width at half-maximum (FWHM) of 55 mm. Other parameters are the same as those used in the first example. The Bragg grating with amplitude-only sampling for the multi-channel dispersion compensation is thus designed. The amplitude sampling function is very similar to that shown in Fig. 3, and the spectral response is plotted in Fig. 5. The desired chromatic dispersion response in the multiple channels is then achieved.
Fig. 5. Spectral response of the 45-channel dispersion compensating grating with amplitude-only sampling. The chromatic dispersion in each channel is -1020 ps/nm. The transmission loss in each channel is -20 dB. (a). The whole band spectrum; (b) and (c) the channels at the left side and at the center,respectively.
4. Conclusion
As a conclusion, we have proposed a novel technique to realize a Bragg grating with a multi-channel and high-channel-count response by nonuniform amplitude-only sampling. The required phase modulation can be achieved equivalently by nonuniform amplitude-only sampling. The principle of a multi-channel Bragg grating was detailed. Two design examples were presented with two Bragg gratings being designed for applications in multi-channel filtering and for multi-channel dispersion compensation. The key significance of the technique is that a sampling period in the order of sub-micron is required, which would simplify significantly the fabrication process.
Acknowledgment
The work was supported by The Natural Sciences and Engineering Research Council of Canada (NSERC).
References and links
1. H. Li, M. Li, Y. Sheng, and J. E. Rothenberg, “Advances in the design and fabrication of high-channel-count fiber Bragg gratings,” J. Lightw. Technol. 25, 2739–2750 (2007). [CrossRef]
2. Y. Painchaud, H. Chotard, A. Mailloux, and Y. Vasseur, “Superposition of chirped fiber Bragg grating for third-order dispersion compensation over 32 WDM channels,” Electron. Lett. 38, 1572–1573 (2002). [CrossRef]
3. A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multi-channel fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 91–98 (2003). [CrossRef]
4. K. Kolossovski, R. Sammut, A. Buryak, and D. Stepanov, “Three-step design optimization for multi-channel fibre Bragg gratings,” Opt. Express 11, 1029–1038 (2003). [CrossRef] [PubMed]
5. Q. Wu, C. Yu, K. Wang, X. Wang, Z. Pu, H. P. Chan, and P. L. Chu, “New sampling-based design of simultaneous compensation of both dispersion and dispersion slope for multi-channel fiber Bragg gratings,” IEEE Photon. Technol. Lett. 17, 381–383 (2005). [CrossRef]
6. Q. Wu, P. L. Chu, and H. P. Chan, “General design approach to multi-channel fiber Bragg grating,” J. Lightw. Technol. 24, 1571–1580 (2006). [CrossRef]
7. M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, “Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation,” IEEE Photon. Technol. Lett. 10, 842–844 (1998). [CrossRef]
8. L. Xia, X. Li, X. Chen, and S. Xie, “A novel dispersion compensating fiber grating with a large chirp parameter and period sampled distribution,” Opt. Commun. 227, 311–315 (2003). [CrossRef]
9. 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. Technol. Lett. 14, 1309–1311 (2002). [CrossRef]
10. H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, “Phase-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion compensation,” J. Lightw. Technol. 21, 2074–2083 (2003). H. Lee and G. P. Agrawal, “Purely phase-sampled fiber Bragg gratings for broad-band dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 15,1091–1093 (2003). [CrossRef]
11. H. Li, M. Li, K. Ogusu, Y. Sheng, and J. E. Rothenberg, “Optimization of a continuous phase-only sampling for high channel-count fiber Bragg gratings,” Opt. Express 14, 3152–3160 (2006). [CrossRef] [PubMed]
12. M. Poulin, Y. Vasseur, F. Trepanier, M. Guy, M. Morin, Y. Painchaud, and J. Rothenberg, “Apodization of a multi-channel dispersion compensator by phase modulation coding of a phase mask,” OFC 2005, paper OME17.
13. Y. Painchaud, M. Poulin, M. Morin, and M. Guy, “Fiber Bragg grating based dispersion compensator slope-matched for LEAF fiber,” OFC 2006, paper OThE2.
14. H. Li, T. Kumagai, and K. Ogusu, “Advanced design of a multi-channel fiber Bragg grating based on a layer-peeling method,” J. Opt. Soc. Am. B , 21, 1929–1938 (2004). [CrossRef]
15. N. Plougmann and M. Kristensen, “Efficient iterative technique for designing Bragg gratings,” Opt. Lett. 29, 23–25 (2004). [CrossRef] [PubMed]
16. C. Lee, R. Lee, and Y. Kao, “Design of multichannel DWDM fiber Bragg grating filters by Lagrange multiplier constrained optimization,” Opt. Express , 14, 11002–11011 (2006). [CrossRef] [PubMed]
17. Y. Dai, X. Chen, L. Xia, Y. Zhang, and S. Xie, “Sampled Bragg grating with desired response in one channel by use of a reconstruction algorithm and equivalent chirp,” Opt. Lett. 29, 1333–1335 (2004). [CrossRef] [PubMed]
18. J. Sun, Y. Dai, X. Chen, Y. Zhang, and S. Xie, “Thermally tunable dispersion compensator in 40-Gh/b system using FBG fabricated with linearly chirped phase mask,” Opt. Express , 14, 44–49 (2006). [CrossRef] [PubMed]
19. D. Jiang, X. Chen, Y. Dai, H. Liu, and S. Xie, “A novel distributed feedback fiber laser based on equivalent phase shift,” IEEE Photon. Technol. Lett. 16, 2598–2600 (2004). [CrossRef]
20. Y. Dai and X. Chen, “DFB semiconductor lasers based on reconstruction-equivalent-chirp technology,” Opt. Express , 15, 2348–2353 (2007). [CrossRef] [PubMed]
21. Y. Dai, X. Chen, J. Sun, Y. Yao, and S. Xie, “High-performance, high-chip-count optical code division multiple access encoders-decoders based on a reconstruction equivalent-chirp technique,” Opt. Lett. 31, 1618–1620 (2006). [CrossRef] [PubMed]
