# AMACH Zehnder interferometer (MZI) based on the

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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).

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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