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1 1284 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 23, NO. 3, MARCH 2005 Optimal Design of Planar Wavelength Circuits Based on Mach Zehnder Interferometers and Their Cascaded Forms Qian Wang and Sailing He, Senior Member, IEEE Abstract Optimal design of planar wavelength circuits based on Mach Zehnder interferometers and their cascaded forms is presented with the wavelength dependence of the directional couplers considered. A simple approximate formula is presented to calculate quickly the wavelength-dependent phase difference for a directional coupler. By using this simple and effective formula and the genetic algorithm, a broadband coupler ( nm) based on a Mach Zehnder interferometer and a flat-top passband coarse wavelength division demultiplexer (with a channel spacing of 20 nm) based on cascaded Mach Zehnder interferometers are optimally designed. Index Terms Coarse wavelength-division demultiplexers, genetic algorithm, Mach Zehnder interferometers, optical directional couplers, wavelength dependence. I. INTRODUCTION AMACH Zehnder interferometer (MZI) based on the planar waveguide technology is constructed by connecting two directional couplers with an optical delay line. An MZI and cascaded MZIs can be used as couplers [1], interleavers [2], add-drop filters [3], [4], and dispersion compensators [5]. A directional coupler in a MZI performs the function of coupling, and this coupling is wavelength dependent. For a narrow-band device, such as an interleaver (based on cascaded MZIs) used in the dense wavelength-division multiplexing/delmultimplexing (DWDM), the wavelength dependence of the directional coupler affects little the performance of the device and thus can be ignored in the design. However, for some planar wavelength circuits based on a MZI or cascaded MZIs working in a relative broadband wavelength range, the wavelength dependence of the directional coupler has a significant effect on the performance of the device and thus should not be neglected. In order to take into account the wavelength dependence of the directional couplers, a parameter related to the accumulated phase difference between the two supermodes has to be calculated in advance numerically at each sampled wavelength in [6]. Such a method is time consuming. As a global optimization method, the genetic algorithm [7] Manuscript received October 8, 2003; revised September 30, This work was supported in part by the Wenner-Gren Foundation. The authors are with Centre for Optical and Electromagnetic Research, State Key Laboratory of Modern Optical Instrumentation, Hangzhou, China; the Laboratory of Photonics and Microwave Engineering, Department of Microelectronics and Information Technology, Royal Institute of Technology, S Kista, Sweden; and the Joint Research Center of Photonics, Royal Institute of Technology and Zhejiang University, Hangzhou , China ( Digital Object Identifier /JLT gives an effective way of finding out the optimal structure according to the desired spectral response. However, the slow speed for computing the wavelength dependence of the directional couplers limits the application of the genetic algorithm in the design of planar wavelength circuits when the wavelength dependence of the directional couplers has to be considered. In this paper, a simple formula for calculating quickly the wavelength dependence of a directional coupler is presented and used in the genetic algorithm to design some planar wavelength circuits with the wavelength dependence of the directional coupler considered. In Section II, the wavelength dependence of a directional coupler is studied and a simple formula for the accumulated phase differences between the two supermodes at different wavelengths is presented. With this simple formula the accumulated phase difference between the two supermodes of the directional coupler can be estimated quickly and the computational time is greatly reduced in the optimal design followed. In Section III, with the derived formula and the genetic algorithm, an MZI-based 3 db broadband coupler is optimally designed. The designed coupler has a flat wavelength response (50% 1.5%) in the wavelength range from 1450 to 1650 nm. In Section IV, we carry out the optimal design of a coarse wavelength-division demultimplexer (CWDM) based on cascaded MZIs, and a wide flat-top passband CWDM with the least number of cascaded MZIs is obtained. II. WAVELENGTH DEPENDENCE OF A DIRECTIONAL COUPLER The directional coupler used in an MZI consists of three parts: the input region, the central coupling region, and the output region (see Fig. 1). The coupling between the two single-mode waveguides occurs mainly in the central coupling region. However, the coupling in the input and output regions should not be ignored [8]. The transfer matrix for a single directional coupler is where, and,, and are the accumulated phase differences between two supermodes in the input region, central coupling region, and output region, respectively. If the geometrical parameters of the input and output regions are the same, one has. For a given directional coupler, the phase difference between the two supermodes is sensitive to the wavelength. Assume that the input wavelength is, the length of the central (1) /$ IEEE

2 WANG AND HE: PLANAR WAVELENGTH CIRCUITS BASED ON MACH ZEHNDER INTERFEROMETERS 1285 Fig. 1. Schematic structure of a directional coupler with the input and output regions. coupling region is. The separation between the two waveguides is at position (the separation is in the central coupling region), and the difference of the effective refractive indexes for the two supermodes is. Then we have Fig. 2. Accumulated phase differences calculated with different methods as the input wavelength varies. (2) In [6], this accumulated phase difference was calculated by an approximate formula containing a parameter which has to be calculated numerically at each sampled wavelength before designing a planar wavelength circuit consisting of directional couplers. If sampled wavelengths should be considered in the design, one has to calculate the accumulated phase differences times. In order to reduce the computational time, a simple approximate formula is derived in this paper to calculate the accumulated phase difference at different wavelengths quickly. It is known that the difference of the effective refractive indexes between the two supermodes attenuates exponentially as the distance between the two waveguides increases [9], i.e., Therefore, it follows from (2) that (4) Although the parameter in (4) is wavelength dependent, we find that its contribution to the wavelength dependence of can be neglected (as compared to the other terms) and this can simplify greatly our approximation. Therefore, there are two wavelength-dependent terms in (4), namely, 2 and. We can take the following first- and second-order Taylor series approximations for (4): (3) (5a) (5b) With (5), we can calculate the accumulated phase differences at different wavelengths quickly with the least computation effort. We only need to calculate and in advance with the central finite difference. In order to verify the above formula, we compare the results calculated from (5) and the simulation results obtained with an accurate numerical method (combining the beam propagation method and a local supermode method) developed in [10]. A 3 db directional coupler at wavelength 1550 nm is chosen as a numerical example. We assume the refractive indexes of the core and cladding of the waveguide are and 1.445, respectively, the cross-section of the waveguide is m m, and the separation distance between the two waveguides in the central coupling region is 6 m. The curve radius for the input and output waveguides is m. After considering the coupling effect in the input and output regions, the length of the central coupling region for a 3 db directional coupler at wavelength 1550 nm is m. For this directional coupler, the accumulated phase differences between the two supermodes at different wavelengths calculated with the accurate method (combining the beam propagation method and the local supermode method) are presented in Fig. 2 (marked as circles). The accumulated phase difference calculated with (5a) is shown in Fig. 2 with the dashed line and the one calculated with (5b) is shown with the solid line. From Fig. 2, we can see that in the wavelength range [1500 nm, 1600 nm], the results calculated with (5a) and (5b) agree well with the accurate results. In the whole wavelength range [1450 nm, 1650 nm], the results obtained with (5b) show a better overall accuracy as compared to those obtained with (5a). If the wavelength dependence of in (4) is neglected, the wavelength dependence of the accumu-

3 1286 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 23, NO. 3, MARCH 2005 Fig. 3. Schematic structure of an MZI. lated phase difference is approximated as (see [11]). The corresponding results are also given in Fig. 2 by the dotted line, which differs completely from the solid line (accurate results). Therefore, the wavelength dependence of has to be considered in the design. The approximated formula (5a) or (5b) reduces the computational time greatly while remaining a good accuracy. In the following sections, we carry out an optimal design of planar wavelength circuits based on MZIs and their cascaded forms using the approximated formula (5b). Fig. 4. coupler. Spectral responses of the designed MZI and a single directional III. OPTIMAL DESIGN OF A BROADBAND COUPLER From Fig. 2, one can see for a 3 db directional coupler at wavelength 1550 nm, the accumulated phase difference varies with the input wavelength and the coupling ratio varies consequently. In order to remain the required coupling ratio in a broadband range, an MZI formed by two directional couplers is employed (see Fig. 3). In this section, we design a broadband 3 db coupler based on such an MZI in the wavelength range of nm. The transfer matrix of the MZI is where and are the transfer matrices determined by (1) for the two directional couplers, is the length of the optical delay line, and is the effective index of the optical delay line (which is also wavelength dependent). Three parameters need to be designed for the broadband coupling: the length of the optical delay line, the accumulated phase differences, and for the two directional couplers. The objective function is defined as where ( for this design example), is the transmission at port 2, and is the total number of the sampled wavelengths (in the present design example, the sampled wavelength spacing is 4 nm and thus ). The optimal design is to find the appropriate parameters that give the minimal value of the objective function. Although there are only three parameters to be designed, a gradient-based optimization method may end at a local minimum easily. Therefore, the genetic algorithm (GA) is used to find the global minimum of the objective function. An MZI with the following structural parameters is considered as the (6) (7) prototype. At the center wavelength 1550 nm, the first directional coupler is a 3 db coupler (i.e., ), the accumulated phase difference for the second directional coupler is, and the length of the optical delay line is arbitrary. For such an MZI, the coupling ratio of the two outputs is 50%:50% at wavelength 1550 nm. We wish to obtain a broadband coupler by making an appropriate adjustment to this MZI. Therefore, the searching ranges of the three parameters are,, and m (1.07 m is the length of the optical delay line which can produce 2 phase difference between the two arms at wavelength 1550 nm). In our GA optimization, the number of encoding bits for each structural parameter is ten. The population size is fixed to 200 for each generation. The probability for the crossover is 0.7, and the mutation probability is chosen to be The genetic algorithm starts from an initial population, which is generated randomly in the search space. After the initial population is generated, the corresponding values of the objective function are calculated. In this paper, the proportional (roulette-wheel) selection scheme is used, and the parents are chosen with a probability proportional to their values of the objective function. The chosen pairs produce the new offspring by a crossover operator. There are many different ways to perform the crossover. In this paper a one-point crossover is used and the crossover point is chosen randomly for each pair of parents. Mutation is then applied to the offspring and a small percentage of the genes may be changed to the opposite values (1 to 0, or 0 to 1). After selection, crossover, and mutation in each iteration, the computer finds the global maximum after a number of iterations. In this design example, we obtain,, and m when the number of generation is 350. For this optimally designed MZI, the corresponding spectral response is indicated in Fig. 4. From Fig. 4, one can see that, for a single directional coupler, the coupling ratio varies from 0.26 to in the wavelength range nm, while the coupling ratio of the optimally designed MZI varies in a small range from to (which is much more flat as compared to the single directional coupler).

4 WANG AND HE: PLANAR WAVELENGTH CIRCUITS BASED ON MACH ZEHNDER INTERFEROMETERS 1287 Fig. 6. Design of cascaded MZI structure A without considering the wavelength dependence of directional couplers. Fig. 5. MZIs. Schematic structure of a four-channel CWDM based on cascaded IV. OPTIMAL DESIGN OF A CWDM BASED ON CASCADED MZIS The schematic structure of a four-channel CWDM is shown in Fig. 5 (see, e.g., [12]). Through cascaded MZI structure A, the input sequence is first interleaved into and at the two outputs of structure A. Through cascaded MZI structure B, and are separated at output 1 and output 2. Similarly, through cascaded MZI structure C, and are separated at output 3 and output 4 (see Fig. 5). In the present design, we choose [1530 nm, 1550 nm, 1570 nm, 1590 nm] as the input sequence and give a detailed description for the design of cascaded MZI structure A. For a structure of cascaded MZIs (with each delay line at the upper arm), the transfer matrix is where is the transfer matrix of the ith directional coupler. If the position of the optical delay line is at the bottom arm, we only need to represent the length of the delay line with.if the wavelength dependence of the directional coupler is ignored, the length of the delay line is determined by (8) method is to make the transfer function of the cascaded MZIs to approach the Fouries series of the desired spectral response through the optimal design of the structural parameters [9]. One may also apply the genetic algorithm directly to obtain the structure with a flat-top passband. Applying any of the above methods, we can obtain the design (shown in Fig. 6) for cascaded MZI structure A. The unit length of optical delay line is m (the center wavelength is 1550 and the channel spacing is 20 nm in this design). For this design, the spectral response calculated with neglecting the wavelength dependence of the directional couplers is indicated in Fig. 7(a), while the actual spectral response (with the wavelength dependence of directional couplers considered) is shown in Fig. 7(b). Comparing Fig. 7(a) and (b), one can see that the wavelength dependence of the directional couplers degrades the performance (particularly the channel cross talk) of this design for cascaded MZI structure A. In order to improve the actual spectral response of the cascaded MZI structure A, an optimal design with the wavelength dependence of directional couplers considered is carried out. First, we adjust the structural parameters shown in Fig. 6 and wish to find an appropriate structure with an improved spectral response through optimization. The objective function of the optimization can be defined as [see the spectral responses shown in Fig. 7(a) and (b)] (9) In the above equation, is the center wavelength and is the effective refractive index of the delay line. The design procedure for cascaded MZI structure A can be divided into two steps. First, the wavelength dependence of the directional coupler is neglected and a filter with a flat-top passband is obtained. Then, using this design results as the initial structure and considering the wavelength dependence of the directional coupler, we employ the genetic algorithm to make an appropriate adjustment to this structure and finally obtain the optimal structure. For the case of narrow-band devices, there are several methods for designing a filter with a flat-top passband based on the cascaded MZIs. One method is to treat the cascaded MZIs as a lattice filter and employ a digital signal processing method to synthesize the structural parameters [13]. The other (10) where are the channel wavelengths and is the width of the flat-top passband. Five parameters need to be determined in the optimization, and their search ranges are small (the design procedure is similar to the one for the broadband couplers described in the previous section). The genetic algorithm is a global optimization method, which does not have the problem of local maxima. However, it

5 1288 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 23, NO. 3, MARCH 2005 Fig. 8. (a) Optimized design for cascaded MZI structure A. (b) The corresponding spectral response. Fig. 7. Spectral response for the cascaded MZI structure A designed without considering the wavelength dependence of the directional couplers. (a) Calculated without considering the wavelength dependence of the directional couplers and (b) calculated with the wavelength dependence of directional couplers considered. has a slow convergence after many generations when the variables are close to the point of global maximum. At this stage, a gradient optimization can be used to speed up the convergence. This is an effective way to obtain optimal results with a relatively shorter computational time [14]. This design example of four-channel CWDM demultiplexer is more complex than the previous design example, e.g., it has more variables to be optimized. Therefore, we apply the genetic algorithm first (with the same GA parameters as in the previous design example) and obtain a set of optimized parameters. Using this set of parameters as the initial values, we obtain the final values for these design parameters with a gradient-based search method. For two-stage cascaded MZIs, the spectral response of the final structure obtained by our global optimization has little improvement compared to that shown in Fig. 7(b). Therefore, we add a third stage of MZI to the original structure. Seven parameters need to be optimized for the three-stage cascaded MZIs. The parameters for the first and second stages of the MZIs are searched in small ranges, and the search ranges of the parameters for the last stage are chosen as and. The final optimal parameters for this three-stage MZI structure are shown in Fig. 8(a). The actual spectral response of the designed structure is shown in Fig. 8(b). From Fig. 8(b), we can see that the performance of cascaded MZI structure A is improved compared to that of the initial structure shown in Fig. 6. Appropriate structural parameters can also be found for cascaded MZIs with more than three stages. However, we believe that this three-stage cascaded MZI is the most compact structure for the required spectral response. Cascaded MZI structures B and C are designed in a similar way. For cascaded MZI structure B, the optimal parameters are m,,,,,,, and, and the corresponding spectral response is shown in Fig. 9(a). For cascaded MZI structure C, the optimal parameters are m,,,,,,, and, and the corresponding spectral response is shown in Fig. 9(b). We then connect the three structures of cascaded MZI designed above in the way shown in Fig. 5. The transmission spectra of this four-channel CWDM are shown in Fig. 10. The

6 WANG AND HE: PLANAR WAVELENGTH CIRCUITS BASED ON MACH ZEHNDER INTERFEROMETERS 1289 passband of the designed CWDM is wide and has a flat-top, and the bandwidth of the 0.5 db passband is 12 nm. The center crosstalk is below 20 db. If we want to decrease the crosstalk further, we can connect the same cascaded MZIs to each structure as shown in [15] and obtain a CWDM with a lower crosstalk and sharp transitions while remaining a flat-top passband. V. CONCLUSION A simple approximated formula has been derived to calculate quickly the wavelength dependence of the accumulated phase difference of a directional coupler. Employing this simple formula and the genetic algorithm, optimal designs of broadband couplers and CWDMs based on MZIs and their cascaded forms have been carried out. The design procedure is simple and intuitive, and improved spectral responses have been obtained. This method can also be applied to the optimal design of other planar wavelength circuits based on the cascaded MZIs (with the least number of stages) according to the required spectral response. Fig. 9. (a) Spectral response of (a) the designed cascaded MZI structure B and (b) the designed cascaded MZI structure C. Fig. 10. Transmission spectra of the designed four-channel CWDM. REFERENCES [1] K. Jinguji, N. Takato, A. Sugita, and M. Kawachi, Mach-Zehnder interferometer type optical waveguide coupler with wavelength-flattened coupling ratio, Electron. Lett., vol. 26, no. 17, pp , [2] M. Oguma, K. Jinguji, T. Kitoh, T. Shibata, and A. Himeno, Flat-passband interleave filter with 200 GHz channel spacing based on planar lightwave circuit-type lattice structure, Electron. Lett., vol. 36, no. 15, pp , [3] M. Kuznetsov, Cascaded coupler Mach-Zehnder channel dropping filters for wavelength-division-multiplexed optical systems, J. Lightw. Technol., vol. 12, no. 2, pp , [4] H. H. Yaffe, C. H. Henry, M. R. Serbin, and L. G. Cohen, Resonant couplers acting as add-drop filters made with silica-on-silicon waveguide technology, J. Lightw. Technol., vol. 12, no. 6, pp , [5] K. Takiguchi, K. Okamoto, and K. Moriwaki, Planar lightwave circuit dispersion equalizer, J. Lightw. Technol., vol. 14, no. 9, pp , [6] K. Jinguji, N. Takato, Y. Hida, T. Kitoh, and M. Kawachi, Two-port optical wavelength circuits composed of cascaded Mach-Zehnder interferometers with point-symmetical configurations, J. Lightw. Technol., vol. 14, no. 10, pp , [7] D. S. Weile and E. Michielssen, Genetic algorithm optimization applied to electromagnetics: a review, IEEE Trans. Antennas Propag., vol. 45, no. 3, pp , [8] Q. Wang, S. He, and F. Chen, An effective and accurate method for the design of directional couplers, IEEE J. Sel. Topics Quantum Electron., vol. 8, no. 6, pp , [9] Y. P. Li and C. H. Henry, Silica-based optical integrated circuits, Proc. Inst. Elect. Eng. Optoelectron., vol. 143, no. 5, pp , [10] Q. Wang and S. He, A simple, fast and accurate method of designing directional couplers by evaluating the phase difference of local supermodes, J. Opt. A: Pure Appl. Opt., vol. 5, no. 5, pp , [11] B. E. Little and T. Murphy, Design rules for maximally flat wavelength-insensitive optical power dividers using Mach-Zehnder structures, IEEE Photon. Technol. Lett., vol. 9, no. 12, pp , [12] M. Oguma, T. Kitoh, T. Shibata, Y. Inoue, K. Jinguji, A. Himeno, and Y. Hibino, Four-channel flat-top and low-loss filter for wide passband WDM access network, Electron. Lett., vol. 37, no. 8, pp , [13] K. Jinguji and M. Kawachi, Synthesis of coherent two-port lattice-form optical delay-line circuit, J. Lightw. Technol., vol. 13, no. 1, pp , [14] Q. Wang, J. Lu, and S. He, Optimal design method of a low-loss broadband Y-branch with a multimode waveguide section, Appl. Opt., vol. 41, no. 36, pp , [15] Y. Inoue, M. Oguma, T. Kitoh, M. Ishii, T. Shibata, and Y. Hibino, Lowcrosstalk 4-channel coarse WDM filter using silica-based planar-lightwave-circuit, in OFC2002, 2002, pp

7 1290 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 23, NO. 3, MARCH 2005 Qian Wang received the diploma degree in optical engineering from Zhejiang University, Hangzhou, China, in His main research area is in simulation and optimal design of photonic integrated circuits and optical devices based on liquid crystals. He has published more than 20 journal papers in this research area. Sailing He (M 92 SM 98) received the Licentiate of Technology and the Ph.D. degree in electromagnetic theory from the Royal Institute of Technology, Stockholm, Sweden, in 1991 and 1992, respectively. He is a Professor in electromagnetic theory at the Royal Institute of Technology. He has also been with the Centre for Optical and Electromagnetic Research, Zhejiang University, China, since 1999 as a Special Professor appointed by the Ministry of Education of China. He is the Chief Scientist for the Joint Laboratory of Optical Communications of Zhejiang University, and a Chief Scientist for the Joint Research Center of Photonics of the Royal Institute of Technology and Zhejiang University. His current research interests are in the areas of photonic integration technologies, fiber optical communication technologies, metamaterials and photonic crystals, computational electromagnetics, biophotonics and -electromagnetics, and RF technologies. He has authored one monograph and about 200 papers in refereed international journals, and has received a dozen patents.

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