Numerical Analysis

The grating is composed of five interleaved subgratings with the different focal points and each focal point lies on the cross-sectional line of the ending facet for the output waveguide. Each subimage formed by the corresponding subgrating overlaps with each other as shown in Fig. 3-25, where E¹, E², E³, E⁴, and E⁵ denote the peak amplitudes of five subimages, a denotes the separation between subimage 1 (subimage 5) and subimage 3, and b denotes the separation between subimage 2 (subimage 4) and subimage 3. To obtain a symmetric spectral response, E¹and E⁵(E²and E⁴) are designed to be identical.

The spot sizes of all subimages are approximately the same when the subgratings are interleaved well [7].

If the subgratings are interleaved, the ratio of the peak amplitudes of the subimages is approximately equal to the ratio of the facet numbers of the corresponding subgratings [7].

We can adjust the ratio of the facet numbers of subimage 3 and subimage 1 (subimage 2 and subimage 1) to adjust the ratio E³/E¹ (E²/E¹). If the grating pole is located at the origin of the coordinates, after determining the arrayed sequence of the facets of the subgratings the x-axis coordinate position xi of the vertex fo the ith groove can be obtained from Eq. (3.11). The order m for each subgrating is identical while the grating period dk and the central wavelength λk for each subgrating are different as the multigrating method [3]. The y-axis coordinate position yi of the vertex of the ith groove can then be obtained from the solution of the root for Eq. (3.6).

The genetic algorithm is used to optimize the parameters in the five-focal-point method for a flat-top planar waveguide demultiplexer, and the parameters will be defined below. The genetic algorithm based on Darwin’s principle of “survival of the fittest” was invented by Holland in the 1960s [27]. By using operators of selection, crossover, and mutation, one population of “chromosomes”, strings of ones and zeros, can generate a new population. The fitter the chromosome, the more times it is likely to be selected to reproduce.

In our case, the parameters (E³/E¹, E²/E¹, a/w^outwg, b/w^outwg) need to be optimized and are encoded into the chromosome strings of the genetic algorithm for the fittest

Figure 3-25: Field distribution of the five subimages at the ending facet of the output waveguide.

solution. The problem involves two objectives, F1 and F2,:

F¹ = W₋^1dB

W₋^3dB, (3.18)

F² = 1

R^ripple, (3.19)

where W₋^1dB and W₋^3dB denote the −1-dB and −3-dB passband widths of the central channel, respectively, and R^ripple denotes the ripple, which is defined as the maximum difference among extremum points within −3-dB passband of the central channel. For the fittest solution, a higher value of W₋^1dB/W₋^3dB is obtained for a lower adjacent channel crosstalk and a higher value of 1/R^ripple is obtained for the lower ripple. So the multiobjective fitness function is required:

F^obj.= max (F¹, F²) . (3.20)

The multiobjective problem can be solved by combining all objectives into a single ob-jective:

F^obj. = w¹F¹+ w²F², (3.21) where w¹ and w² denote the weighting factors of the objectives and are chosen as 1/0.9 and 1/1000, respectively, in our case. The problem also involves some constraints. The values of E³/E¹ and E²/E¹ range between 0.0 and 1.0. The value of a/w^outwg is higher than the value of b/w^outwg.

The number n of the population for each generation is 100 and the total number of generations is 100. The crossover probability p^c is chosen as 0.8 with a mutation probability p^mof 0.1. At first, n chromosomes are randomly generated and the fitness F^obj.

of each chromosome is calculated. By using a method of “roulette-wheel sampling”, the operator of selection selects two chromosomes in proportion to the fitness for two parents.

With the crossover probability p^c, the operator of crossover randomly chooses a locus and exchanges the subsequences before and after that locus between two chromosomes (parents) to create two offspring [27]. And each bit (gene) in the chromosome will be mutated according to the mutation probability p^m, i.e., 0 to 1 and 1 to 0. The operator of mutation provides an efficient way to escape from a local maximum to a global maximum if the number of generations is big enough in a run. Until n chromosomes (offspring)

0 10 20 30 40 50 60 70 80 90 100

Figure 3-26: Maximal fitness in the individual generation.

are created, it is called one generation (iteration). Eventually, all the parameters will converge to the fittest results after a number of generations. All computer program codes for the simulation are written in Fortran 90.

It takes about 78 hours to finish a run with a CPU processor of Intel Core 2 Duo E6300 (1.87 GHz) and the 2-Gb DDR ram. Fig. 3-26 shows the maximal fitness in the population as the generation increases. It also shows that when the ratio E³/E¹ is assigned with the different values, the genetic algorithm will converge to the different results. A number of trials with the different initial populations have been carried out and give the similar results. When the ratio E³/E¹ increases from 0 to 1, the genetic algorithm will converge to the result with higher fitness but it needs more generations for a steady state. The optimal parameters (E³/E¹, E²/E¹, a/w^outwg, b/w^outwg) for the three cases are (1.0, 0.1, 2.11, 0.37), (0.5, 0.8, 2.11, 0.82), and (0.0, 0.9, 2.05, 0.61) with the values of fitness of 9.99, 9.52, and 2.14, respectively. The values of W₋^1dB/W₋^3dB for the three cases are all 0.72 and the values of 1/R^ripple are 9191, 8716, and 1342, respectively. The −1-dB passband widths for the three cases are 30.53, 28.10, and 27.92 GHz, respectively, with the ripples of 1.09 × 10⁻⁴, 1.15 × 10⁻⁴, and 7.45 × 10⁻⁴ dB, respectively. With the temperature coefficient of the refractive index for the silica being 1

-100 -50 0 50 100

Figure 3-27: Spectral responses and the chromatic dispersion characteristics for the three cases.

× 10⁻⁵ (1/^◦C) [17], the temperature tolerances for the intensity fluctuation of the central channel below −1 dB can be obtained as ±13.4, ±13.5, and ±13.7 ^◦C, respectively. It shows that the case of E³/E¹ = 1.0 gives a larger −1-dB passband width and a lower ripple [7].

With the Gaussian approximation, the three-focal-point method [7] is illustrated and verified with a design example in Section 3.1. With the numerical model of the scalar diffraction theory [35], the five-focal-point method is illustrated with the design example and optimized by using the genetic algorithm. The genetic algorithm is a search tech-nique used in the computer science to find the optimal solution even in a multiobjective problem. The −1-dB passband width and ripple for the three-focal-point method based on the Gaussian approximation are 62 GHz (62% of the channel spacing) and 0.013 dB [7]. Simulation results show that the five-focal-point method gives a near-equivalent ratio (61%) of the −1-dB passband width to the channel spacing and a lower ripple R^ripple (1.09 × 10⁻⁴ GHz) compared with the three-focal-point method based on the Gaussian approximation [7].

The chromatic dispersion characteristic D^C can be obtained from Eq. (3.12). Fig.

3-27 shows the spectral responses and the chromatic dispersion characteristics D^C of the central channel for the three cases. It shows that the case of E3/E1 = 0.0 has the lower

crosstalk and insertion loss, while the case of E3/E1 = 0.5 has the better chromatic dispersion characteristics. The crosstalks for the three cases are −35.74, −35.78, and

−36.31 dB, respectively, with the insertion losses of 4.87, 4.59, and 4.57 dB, respectively.