1546
IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 15, NO. 11, NOVEMBER 2003
Superimposed Fiber Bragg Grating Simulation by the Method of Single Expression for Optical CDMA Systems David M. Meghavoryan and Ara V. Daryan
Index Terms—Bandpass filters, fiber Bragg gratings (FBGs), optical code-division multiple-access (CDMA), optical fiber communication.
of superimposed fiber Bragg gratings (SFBGs). MSE is a convenient tool for analyzing one-dimensional wave propagation trough modulated medium with complicated refractive index modulation functions. It is based on solving the wave equation without any approximations (such as slowly varying envelope approximation in coupled mode theory) and has no limitations on the depth of modulation index and refractive index modulation law [11]–[14]. Besides the spectral characteristics, the MSE also permits us to easily obtain the electrical field distribution and time delay response [14], which is very useful for analyzing FBGs for different applications.
I. INTRODUCTION
II. MSE FORMALISM
N-LINE multiwavelength optical Bragg grating filters have numerous applications in fiber-optic communication systems. In particular, coder–decoder of optical code-division multiple-access (CDMA) [1]–[4] and optical pulse train generation [5] has been reported recently. Several fiber Bragg grating (FBG) structures with multiple passbands have been shown such as concatenation of several Bragg gratings [6], chirped moiré gratings [7], phase-shifted [8], and superimposed FBGs [9], [10]. Several experiments have been reported on the superimposition of several Bragg gratings on the same location of fiber. For the first time, the inscription of seven FBG has been performed [9], where reflectivity of the existing gratings decreased considerably with the inscription of each new grating, and after the inscription of all seven gratings the reflectivity decreased to 45%–60%. Additionally, each time a new grating was superimposed, center wavelength of the existing gratings shifted to longer wavelength and the linewidths of the existing gratings were reduced. On the other hand, [10] reported inscription of ten gratings, where each new grating leads to the increase of the linewidths of the existing gratings, while keeping their reflectivities unaffected. In this letter, we perform a numerical analysis of the superimposed structures by the method of single expression (MSE). We propose a model that gives an accurate qualitative explanation of the impact of ultraviolet (UV) writing process to the properties
For an isotropic dielectric medium, normal incidence and linear polarization, the wave equation for the electrical field component has the form
Abstract—Superimposed fiber Bragg gratings are very attractive as multiwavelength filters for many applications. Particularly, their spectral parameters can be specially tailored to have different passband and linewidth characteristics as a coding–decoding element for optical code-division multiple-access systems. In this letter, we perform a simulation of such structure by implementation of an advanced method of single expression, which accurately describes the wave propagation problem in the stratified media by numerical solving of wave equation.
I
Manuscript received June 3, 2003, revised July 4, 2003. The authors are with the Research and Development Department, Epygi Labs AM LLC, 375026 Yerevan, Armenia, and also with the Fiber-Optics Communication Laboratory, State Engineering University of Armenia, 375009 Yerevan, Armenia (e-mail:
[email protected]). Digital Object Identifier 10.1109/LPT.2003.818638
(1) is the relative dielectric permittivity and is light where velocity in vacuum. In contrast to traditional approach, where the solution of the wave equation is searched as a sum of two counterpropagating waves, the solution in MSE is searched in the form of single expression [11]–[14] as (2) and are amplitude and phase of electromagwhere netic field respectively. After substitution of (2) in (1) and some modifications, the following MSE equation for the lossless media can be written [11]–[14]: (3) and is a quantity proportional to where the power flow density along axis. and its derivative satisfy The the continuity conditions at the boundaries. Let us consider the modulated medium of length , bounded on both sides with unperturbed media , and set the illuminated end at the . Taking into account that beyond the perturbed point ) only one outgoing wave exists, and after medium ( applying boundary conditions, the following initial conditions for solving differential equation (3) are obtained:
1041-1135/03$17.00 © 2003 IEEE
(4)
MEGHAVORYAN AND DARYAN: SFBG SIMULATION BY THE MSE FOR OPTICAL CDMA SYSTEMS
1547
where – is the amplitude of outgoing traveling wave. The reflection coefficient can be found by following expression [12]–[14]: (5) Hence, direct integration approach is straightforward and an adaptive step-size Runge–Kutta numerical integration method can be successfully applied. III. SIMULATION RESULTS AND DISCUSSIONS The permittivity (or refractive index) modulation for one grating can be represented as a harmonic oscillating function (6)
Fig. 1. Reflectivity of one grating (solid lines), two (dashed lines) and six (dotted lines) superimposed gratings (only three of the six peaks are shown). M = M = = M = 0:005, L = 0:5 mm.
111
is the initial refractive index of unperturbed fiber, where is the modulation index, and is the grating period. In the case of superimposition of several gratings (7) is the number of superimposed gratings. Such reprewhere sentation follows from the physical property of interaction of UV light with photosensitive media. This representation is in agreement with experimental results [9], [10], where the process of writing a new grating onto already exposed fiber shifts the center wavelengths of the existing grating to the longer wavelengths range due to the increase in the effective refractive index. According to the well-known Bragg condition the central wavelength of each grating is equal to
where is the effective permittivity change during the superimposition of several gratings. For the case (7)
Spectral characteristics of superimposed Bragg gratings (BGs) according to (7) are depicted in Fig. 1. The reflection reflection peaks, which spectrum of SFBG consists of have the same parameters as single BGs except for central wavelength. Until now it was assumed that the photosensitive fiber reacts linearly to UV radiation. In practice, the recording process of a single grating, as well as the superposition of several gratings, is nonlinear process due to the nonlinear dependence of the UV-induced index increase in photosensitive media [15]. To take into account this effect, the permittivity modulation can be represented in the following form: (8)
Fig. 2. Reflectivity of one grating (solid lines), two (dashed lines) and six (dotted lines) superimposed gratings (only three of the six peaks are shown). g = 0:2.
0
where is the parameter that describes the nonlinear interaction corresponds of UV radiation with photosensitive fiber [ and —weighing coeffito the linear case (7)], cient for th grating. For energies that approach the saturation condition, an exponential function provides better approximation [15]. However, (8) is valid for UV flux far from saturation and allows us to consider two types of nonlinear functions, i.e., ) and concave ( ) characteristics. The calconvex ( and with culation results for the cases when other parameters remaining the same as in Fig. 1 are depicted in the Figs. 2 and 3, respectively. As it is seen, depending on the sign of , either a decrease or an increase of reflectivities and linewidths is observed. In the case of negative , the reflection, and correspondingly the linewidth, is reduced with new grating inscription. The central wavelengths exhibits smaller shift compared with the linear case. On the other hand, the positive sign of makes the gratings stronger and increases the wavelengths shift when new gratings are superimposed. Based on this analysis, the experimental results reported in [9] and [10] may be interpreted as an influence of different types of and ) during UV light interaction nonlinearities ( with photosensitive fiber on the properties of SFBGs.
1548
IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 15, NO. 11, NOVEMBER 2003
adjacent 0s in the codeword, respectively. Other parameters of the simulation are the following: mm, . IV. SUMMARY We have performed an accurate analysis of SFBGs and demonstrate the possibility of obtaining multiple passbands with steep slopes and low ripples when appropriate apodization is applied. These filters can be designed with desired stop and passband characteristics by proper choice of modulation index and central wavelength of each grating and taking into account the wavelength shift and linewidths change during the writhing process. Their implementation as a coding–decoding element of spectrally encoded optical CDMA system is proposed. Fig. 3. Reflectivity of one grating (solid lines), two (dashed lines) and six (dotted lines) superimposed gratings (only three of the six peaks are shown). g = 0:2.
Fig. 4. Transmission spectrum of SFBGs that corresponds the Walsh codes of length 16: (a) 1 100 110 011 001 100 (M = 0:01), (b) 1 001 100 110 011 001 (M = 0:01), and (c) 1010010110100101, (M = M = M = M = 0:005, M = M = 0:01).
The presented structure is well suited as an encoding and decoding element to implement wavelength encoded optical CDMA system. For this purpose, we need a multiwavelength transmission filter set that corresponds to a particular code passbands can be obtained by superimposition family. FBGs. The blocks of zeros in the code sequence can be of easily formed by proper choice of modulation index of one grating in superimposed structure. As an example, in Fig. 4, transmission spectral characteristics of SFBGs are presented, which are specially tailored to meet different Walsh code specifications of length 16 with 24-nm spectral range coding. In these calculations, a Gaussian apodization with full-width at half-maximum of apodization function equal to 0.75 L is used to eliminate side lobes of each grating, which will cause large ripples in the passbands of the superimposed structure. and are used to Two modulation indices form the reflection bands, which correspond to single and two
REFERENCES [1] H. Fathallah, L. A. Rusch, and S. LaRochelle, “Passive optical fast frequency-hop CDMA communication system,” J. Lightwave Technol., vol. 17, pp. 397–405, Mar. 1999. [2] A. Grunnet-Jepsen, A. E. Johnson, E. S. Maniloff, T. W. Mossberg, M. J. Munroe, and J. N. Sweetser, “Fiber Bragg grating based spectral encoder/decoder for lightwave CDMA,” Electron. Lett., vol. 35, pp. 1096–1097, June 1999. [3] L. R. Chen, P. W. E. Smith, and C. M. de Sterke, “Wavelength-encoding/temporal-spreading optical code division multiple access system with in-fiber chirped moiré gratings,” Appl. Opt., vol. 38, pp. 4500–4508, July 1999. [4] H. V. Baghdasaryan, D. M. Meghavoryan, and N. K. Uzunoglu, “Optical CDMA system based on amplifying fiber Bragg gratings chain,” in Proc. ICTON Conf., 2000, pp. 169–172. [5] J. Azana, P. Kockaert, R. Slavik, L. R. Chen, and S. LaRochelle, “Generation of a 100-GHz optical pulse train by pulse repetition-rate multiplication using superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett., vol. 15, pp. 413–415, Mar. 2003. [6] K. Sugden, L. Zhang, J. A. R. Williams, R. W. Fallon, L. A. Everall, K. E. Chisholm, and I. Bennion, “Fabrication and characterization of bandpass filters based on concatenated chirped fiber gratings,” J. Lightwave Tecnol., vol. 15, pp. 1424–1432, Aug. 1997. [7] L. R. Chen, D. J. F. Cooper, and P. W. E. Smith, “Transmission filters with multiple flattened passbands based on chirped moiré gratings,” IEEE Photon. Technol. Lett., vol. 10, pp. 1283–1285, Sept. 1998. [8] G. P. Agrawal and S. Radic, “Phase-shifted fiber Bragg gratings and their application for wavelength demultiplexing,” IEEE Photon. Technol. Lett., vol. 6, pp. 995–997, Aug. 1994. [9] A. Othonos, X. Lee, and R. M. Measures, “Superimposed multiple Bragg gratings,” Electron. Lett., vol. 30, pp. 1972–1974, Nov. 1994. [10] A. Arigiris, M. Konstantaki, A. Ikades, D. Chronis, P. Florias, K. Kallimani, and G. Pagiatakis, “Fabrication of high-reflectivity superimposed multiple-fiber Bragg gratings with unequal wavelength spacing,” Opt. Lett., vol. 27, pp. 1306–1308, Aug. 2002. [11] H. V. Baghdasaryan, A. V. Daryan, T. M. Knyazyan, and N. K. Uzunoglu, “Modeling of a nonlinear enhanced performance Fabry–Pérot interferometer filter,” Microw. Opt. Technol. Lett., vol. 14, pp. 105–108, Feb. 1997. [12] H. V. Baghdasaryan and T. M. Knyazyan, “Problem of plane EM wave self-action in multiplayer structure: An exact solution,” Opt. Quantum Electron., vol. 31, pp. 1059–1072, 1999. [13] H. V. Baghdasaryan, “Method of backward calculation,” in Photonics Devices for Telecommunications. How to Model and Measure, G. Guekos, Ed. New York: Springer-Verlag, 1999, pp. 56–65. [14] H. V. Baghdasaryan and T. M. Knyazyan, “Modeling of linearly chirped Bragg gratings by the method of single expression,” Opt. Quantum Electron., vol. 34, pp. 481–492, 2002. [15] D. Johlen, H. Renner, A. Ewald, and E. Brinkmeyer, “Fiber Bragg grating Fabry–Pérot interferometer for a precise measurement of the UV-induced index change,” in Proc. ECOC’98 Conf., 1998, pp. 393–394.
