Optimization of photonic crystal structures - OSA Publishing

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Vol. 21, No. 11 / November 2004 / J. Opt. Soc. Am. A

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Optimization of photonic crystal structures Jasmin Smajic, Christian Hafner, and Daniel Erni Laboratory for Electromagnetic Fields and Microwave Electronics, Swiss Federal Institute of Technology, ETH-Zentrum, CH-8092 Zurich, Switzerland Received March 25, 2004; revised manuscript received June 10, 2004; accepted June 21, 2004 We report on the numerical structural optimization of two-dimensional photonic crystal (PhC) power dividers by using two different classes of optimization algorithms, namely, a modified truncated Newton (TN) gradient search as deterministic local optimization scheme and an evolutionary optimization representing the probabilistic global search strategies. Because of the severe accuracy requirements during optimization, the proper PhC device has been simulated by using the multiple-multipole program that is contained in the MaX-1 software package. With both optimizer classes, we found reliable and promising solutions that provide vanishing power reflection and perfect power balance at any specified frequency within the photonic bandgap. This outcome is astonishing in light of the discrete nature inherent in the underlying PhC structure, especially when the optimizer is allowed to intervene only within a very small volume of the device. Even under such limiting constraints structural optimization is not only feasible but has proven to be highly successful. © 2004 Optical Society of America OCIS codes: 130.2790, 060.1810, 260.2030.

1. INTRODUCTION The inhibition of light propagation in bulk photonic crystals (PhCs)1 turns out to be a very promising feature with respect to tight light channeling in high-density integrated optics devices.2 Numerical simulations have proved that PhC defects may provide an efficient lightguiding scheme for various compact PhC devices such as sharp waveguide bends, power splitters, multiplexers and demultiplexers, high-Q filters, and switches.3–8 In general the simulation proceeds as follows: First, the band diagram of the underlying perfect crystal is computed to retrieve the proper characteristics of the photonic bandgap (PBG).9–11 Second, the introduction of defects—i.e., vacancies or substitutional defects consisting of lattice sites with modified shape or different material properties—is much more demanding, because it requires an efficient eigenmode computation to yield structures such as waveguides12 or cavities with the desired dispersion relations or localization properties.13,14 Third, since we know that functionality may emerge only at the device level, the simulation (and optimization) of complete PhC building blocks has inevitably to discharge into the engineering process. Unfortunately, we are not quite ready for that. Even compact functional PhC devices, e.g., a diplexer based on a filtering T-junction,14 turned out to be extremely demanding to analyze (not to mention to optimize). Thus large model size with respect to the wavelength and therefore large memory requirements and long simulation times are the major challenges to be met by any known numerical method for computational optics. Design tools based on structural optimization usually evaluate from hundreds to millions of possible device structures until a reasonable solution is found. These structures must be computed accurately and reliably with a field solver (i.e., the forward solver) that associates a figure of merit (i.e., a fitness value) with each evaluated device topology. Note that accuracy becomes an impor1084-7529/2004/112223-10$15.00

tant issue—because inaccuracies in the results heavily disturb the optimization process—and that reliability is important because a crash of the field simulation caused by some exceptional structure would have a severe impact on the optimization process. This means that both the computational optics tool and the optimization method should be as efficient as possible and be accurate and robust as well. Structural optimization based on global search strategies has already proved to be highly qualified to solve ‘‘real-world’’ inverse problems in various applications in the field of planar integrated optics and optical communication technology.15 It is well known that such probabilistic optimizers (evolutionary strategies,16,17 genetic algorithms,18,19 particle swarm optimization,20,21 and simulated annealing,22,23 etc.) promise to find global optima, whereas deterministic optimizers (e.g., downhill simplex24,25 and conjugate gradients26) usually fail while being trapped in local optima. Although the former is preferable, probabilistic optimizers require many more fitness evaluations than do deterministic optimizers. Therefore deterministic optimizers are currently the first choice, provided that an experienced user can find a good structure as initial guess for the subsequent optimization scenario. In the remainder of this paper, we shall apply both types of optimizers to the design of a PhC power splitter. Recent approaches in the optimization of PhC structures are based on rather intuitive numerical experiments in which PhC properties are formed by introducing defects,27 by changing the geometry of defects,14 or by means of a sensitivity analysis.28 Such empirical approaches are reasonable for experienced researchers with good intuition, but they lack generality and are not applicable for complex device designs. To obtain a more general procedure, an appropriate optimization scheme and an appropriate field solver must be linked. Formally, © 2004 Optical Society of America

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this is very easy. The main problems thus arise from implementation where appropriate codes have to be developed and even carried down to the hardware level to achieve the required efficiency. Since we know that inaccuracies of the forward solver (i.e., in the fitness computation) heavily disturb all kinds of optimizers, we prefer accurate field solvers, namely, boundary methods that are close to analytic solutions and provide an explicit error measure as well. For that reason we focus on the multiple-multipole method29 that is contained in the MaX-1 software package.30 The remainder of the paper is organized as follows. In Section 2 we present a deterministic optimization algorithm (i.e., a gradient scheme) that numerically approximates the gradient by means of a sensitivity analysis. This gradient scheme is then adapted in Section 3 to the optimal design of a PhC power splitter. An evolutionary search heuristic is pursued in Section 4, where improved genetic and microgenetic algorithms are applied to optimize the same PhC structure toward a most compact power divider topology.

2. DETERMINISTIC OPTIMIZATION SCHEME The PhC power divider that is subject to our optimization is illustrated in Fig. 1. Its underlying PhC may consist of many lattice sites that are arranged according to a square or a hexagonal two-dimensional (2D) crystal symmetry. These sites are formed either as dielectric posts or as holes in a corresponding background material. For our conceptual device studies, the proper lattice type is actually of minor importance. For the sake of simplicity we therefore focus on a generic structure, i.e., the square lattice made of dielectric rods. The characteristic texture of

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PhC devices entails structural optimization schemes that operate directly on individual lattice sites. The schemes comprise interventions such as changing shape, size, position, or material properties of some specific lattice sites. To reduce complexity of the posed problem, one may favor a combinatorial optimization scenario with a binary type of intervention. Here, whether a specific lattice site is present becomes a binary decision (i.e., a vacancy is introduced) to make it possible to cash in on some potentially emergent functionality. Combinatorial optimization based on such binary representations typically calls for genetic algorithms,17,18 which will be described below. In Ref. 28 we have already presented a simple optimization procedure for a PhC 90°-waveguide bend by using a sensitivity analysis. The sensitivity analysis essentially considers the influence of small variations of the PhC, such as shape, position, or material properties of individual lattice sites, on the characteristic device performance in terms of power transmission or reflection spectra. In the case of the PhC lattice with circular dielectric rods, the variable parameters are the radius r and the position (x, y) of each rod. Sensitivity analysis revealed that only the rods in the proper branching region of the T-junction play an important role; hence we can focus on only a few rods in order to keep the search space as lowdimensional as possible. Nevertheless, the dimension of the search space is rather high. For example with ten circular rods, where each is characterized by the three variables (r, x, y), we already obtain a 30-dimensional search space. Deterministic optimizers may be subdivided into two essential categories depending on whether gradient information is present or not. Optimization schemes with explicit gradient information are known to be much more efficient with respect to local search capabilities. A typical

Fig. 1. Left, top view of the initial PhC power divider topology with input and output ports. The crystal area (i.e., the proper branching region) that is subject to optimization is contained within the indicated rectangle near the center. Right, the performance of the initial device is given by the spectral response of the power reflection (R) and the power transmission with respect to the upper (T u ) and lower (T d ) output port. The spectral constraints relevant to the various optimization schemes are indicated by labels 1, 2 and 3.

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⳵F

example is the popular truncated-Newton (TN) method,31,32 which is based on the following equations:

⳵xi

H 共 x k 兲 p k ⫽ ⫺g 共 x k 兲 ,

⳵F (1)

⳵yi

where H(x k ) is the Hessian matrix of the objective function F(x k ) (i.e., the fitness function) and g(x k ) is the gradient of F(x k ). Furthermore, x k denotes the kth approximation of the search vector in the N-dimensional search space, and p k is the kth search step. By referring to the local interventions along with the optimization of a PhC structure we call p k also the movement vector. The TN method computes each p k as a solution of linear system (1) while iteratively running toward a (local) maximum of F. It is obvious that both the gradient and the Hessian matrix need to be calculated for every iteration. Most of the familiar details of the convergence conditions, stopping criteria, and suggestions for an efficient solution of linear system (1) can be found in Refs. 31 and 32. Since the evaluation of the fitness function F for our PhC application is tantamount to the very timeconsuming computation of the electromagnetic field, we want to explicitly calculate neither the gradient nor the Hessian matrix associated with F ( g involves first-order and H second-order partial derivatives of F). Thus we must approximate numerically the partial derivatives of the fitness function. Let us assume that the fitness function F is deduced from a specific output characteristic of the PhC structure, e.g., F ⫽ 1/(R ⫹ ⑀ ), where R is the power reflectivity at some frequency and ⑀ Ⰶ 1 prevents F from becoming singular. As a generic expression we may write

⳵F

x k⫹1 ⫽ x k ⫹ p k ,

F ⫽ F 共 x 1 ,..., x N ; y 1 ,..., y N ; r 1 ,..., r N 兲 ,

(2)

where N corresponds to the number of selected significant posts, (x i , y i ) stands for the position of the post, and r i stands for the radius. The partial derivatives can be approximated, for example, by the central finite differences:

⳵F ⳵xi ⳵F ⳵yi ⳵F ⳵ri

⬇

⬇

⬇

F 共 x i ⫹ h x /2兲 ⫺ F 共 x i ⫺ h x /2兲 hx

,

F 共 y i ⫹ h y /2兲 ⫺ F 共 y i ⫺ h y /2兲 hy F 共 r i ⫹ h r /2兲 ⫺ F 共 r i ⫺ h r /2兲 hr

.

,

(3)

It is worth noting that for the central difference quotient, highly accurate solutions of the field calculations (i.e., fitness evaluations) become indispensable. These different solutions are directly associated with small changes in the underlying PhC structure such as variations in size and position of each subset of posts. The central difference quotient as a second-order approximation of gradient (3) is very time-consuming because it requires 6N fitness evaluations, whereas a first-order scheme such as the forward finite difference [relations (4)] reduces the number of fitness evaluations to 3N ⫹ 1:

⳵ri

⬇

⬇

⬇

F共 xi ⫹ hx兲 ⫺ F共 xi兲 hx

,

F共 yi ⫹ hy兲 ⫺ F共 yi兲 hy F共 ri ⫹ hr兲 ⫺ F共 ri兲 hr

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,

.

(4)

While the gradient has 3N components that entail approximations of single derivatives, the Hessian matrix has as many as (3N) 2 elements that require the approximations of second-order derivatives. Hence we are forced to work without the Hessian matrix (except for some extremely simple, low-dimensional cases) necessitating some very specific modifications regarding the general TN scheme [Eqs. (1)]. The simplest way to avoid the Hessian matrix is simply to run in the direction of the gradient. In the framework of the aforementioned sensitivity analysis28 we already dispose of reasonable approximations for the partial derivatives involved and hence for the gradient. Since this is still time-consuming, we want to minimize the number of gradient evaluations by using reasonable step sizes while tracking down the fitness optimum. Referring to the distinct search direction, one may rewrite the TN scheme as a one-dimensional (1D) optimization procedure with the optimization variable ␣:

冉

max: F X k ⫹ ␣ ␣

ⵜF k 兩 ⵜF k 兩

冊

,

X k⫹1 ⫽ X k ⫹ ␣

ⵜF k 兩 ⵜF k 兩

, (5)

r N 兴 (Tk )

where X k ⫽ 关 x 1 ,..., x N ; y 1 ,..., y N ; r 1 ,..., is the search vector for the kth iteration. The solution in terms of ␣ is easy to obtain. As soon as the optimum of system (5) is found, the gradient computation (i.e., the sensitivity analysis) is restarted and the search procedure of system (5) is performed along the direction of the new gradient. For each iteration step, we therefore need at least 3N ⫹ 1 fitness evaluations for the gradient approximations and M fitness evaluations for the numerical solution of system (5). Typically the value of M is considerably smaller than 3N ⫹ 1, i.e., the computational effort is dominated by the gradient approximation. Even though the proposed optimization procedure looks very primitive compared with the standard TN scheme of system (1), it turns out to be very efficient for the structural optimization of PhC devices, and in particular for the optimization of the PhC power splitter described in Section 3.

3. DETERMINISTIC OPTIMIZATION OF A POWER DIVIDER In the following we consider a PhC power divider topology as displayed in Fig. 1, which is based on W1-defect waveguides (one line of vacancies). The underlying structure consists of a 2D PhC with circular dielectric rods (gray posts) arranged in a square lattice and embedded in air. The lattice parameters are as follows: The radius of each dielectric rod is r ⫽ 0.18a (with a ⫽ 1 ␮m being the lattice constant), and the rod’s dielectric constant is ⑀ ⫽ 11.56. For TM polarized light the

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PhC’s first photonic bandgap (PBG) appears in the normalized frequency range of ␻ a/(2 ␲ c) 苸关 0.30, 0.44兴 , whereas the fundamental W1-defect waveguide mode shows a cutoff at ␻ a/(2 ␲ c) ⫽ 0.312.13 As depicted in Fig. 1 the initial power divider obtains a spectral response of the power reflection (or reflectance R) and power transmission (or transmittance T) over nearly the entire PBG. The reflection minimum R ⫽ 17.22% occurs at the normalized frequency ␻ a/(2 ␲ c) ⫽ 0.380 (indicated by the label 2), which corroborates that the power divider is quite far from being effective and should therefore undergo further optimization.

Fig. 2. Outcome of the sensitivity analysis: The movement toward an improved power divider performance is depicted for every single rod by a corresponding displacement vector. This vector is also assigned to the gradient vector in 2D real space. The rod’s radius variations for a better performance are encoded according to the corresponding filling color (see details in the text).

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First, one has to select a subset of posts in the proper branching area that will become subject to the optimizer’s interventions. The size of the subset is chosen according to a sensitivity argument28 and is marked by a corresponding rectangular box (as indicated in Fig. 1) comprising 20 lattice points. Because of the specific topology and the underlying symmetry of the structure, the number of posts with variable radius and position is reduced to seven. Thus our search space contains 3N ⫽ 21 dimensions. Furthermore, we need 22 fitness evaluations for the first-order gradient approximation of relations (4) and 42 fitness evaluations for the second-order gradient approximation of relations (3). The underlying sensitivity analysis is visualized in Fig. 2. To check the accuracy of the retrieved gradient information both types of approximations have been used. In the following we start our optimization procedure of the power divider in such a way that the power reflection R ⫽ 19% at ␻ a/(2 ␲ c) ⫽ 0.36 (label 1) is minimized; therefore the fitness function is defined simply as the reciprocal value of the power reflection at this specific frequency. After evaluating the gradient, we execute the 1D search procedure inherent to the modified TN scheme of system (5) where an accuracy of 0.0001 has already been obtained within 19 search steps (i.e., 19 fitness evaluations in addition to those of the prior gradient approximation). After this first optimization step, we already achieve R ⫽ 0.04%. This means that our power divider is almost perfect and no further iteration (with a further gradient evaluation) is needed. The resulting power divider and its corresponding frequency characteristics are depicted in Fig. 3. On a personal computer running under Windows XP with a 2-GHz Intel Pentium 4 processor, we require 408 s per fitness evaluation, which sums up to 6 h 54 min for the entire optimization (i.e., one iteration step) with the second-order gradient approximation, or 4 h 25 min for the first-order gradient approximation

Fig. 3. Left, top view of an optimal PhC power divider topology, which was achieved after a single optimization step. The step size (as a result of the 1D search procedure inherent in the modified TN scheme) in the direction of the gradient is 250 nm ⫽ 0.25a, i.e., 25% of the lattice constant. Right, the frequency response of the PhC power divider already yields vanishing power reflection around the specified normalized frequency of ␻ a/(2 ␲ c) ⫽ 0.36.

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Fig. 4. Left, top view of the resulting PhC power divider for the optimization procedure that minimizes power reflection at a normalized operation frequency of ␻ a/(2 ␲ c) ⫽ 0.38. Three iteration steps were needed to reach the optimum, whereas for the last one a 1D search step size of 50 nm ⫽ 0.05a, i.e., 5% of the lattice constant has been applied. Right, the frequency response of the PhC power divider yields a residual power reflection of R ⫽ 0.06% at the specified frequency.

Fig. 5. Left, top view of the resulting PhC power divider for the optimization procedure that minimizes power reflection at a normalized operation frequency of ␻ a/(2 ␲ c) ⫽ 0.40. For the 1D search in the last iteration step a step size of 12 nm ⫽ 0.012a, i.e., 1.2% of the lattice constant was required. Right, the frequency response of the PhC power divider yields a residual power reflection of R ⫽ 0.25% at the specified frequency.

scheme. For both approximations, almost identical solutions have been achieved. Hence the first-order approximation turns out to be sufficient for any further optimization. If the same optimization procedure is repeated for a higher normalized operation frequency, namely ␻ a/(2 ␲ c) ⫽ 0.38 (label 2), the preliminary power reflection to be minimized has a value of R ⫽ 17%. After a first iteration, one obtains R ⫽ 4.19%. Since this value is considered too high, the optimization is continued with

further iterations yielding R ⫽ 0.42% after the second iteration step and R ⫽ 0.06% after the third iteration step, which is certainly sufficient to terminate the optimization procedure. For this optimization scenario, we need 13 h 2 min on the aforementioned machine, and the final results are illustrated in Fig. 4. As expected, even more iteration steps are required for optimizations at higher normalized operation frequencies, such as ␻ a/(2 ␲ c) ⫽ 0.40 (label 3). Here we obtain a final value for the power reflection of R ⫽ 0.25% (Fig. 5) after four iterations whose

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computational effort amounts to 17 h 7 min of simulation time. Obviously, each optimization procedure associated with its corresponding operation frequency resulted in a slightly different PhC topology; therefore the optimal PhC power splitter topology becomes slightly frequency dependent. Provided that the structural differences with respect to the different operation frequencies are sufficiently small, one may apply the parameter estimation technique (PET)29 to considerably speed up the optimization procedure. This efficient technique provides adapted initial parameter guesses for the field solver being estimated from preceding simulations. Under such

Fig. 6. Frequency responses of the PhC power divider after several consecutive optimization schemes associated with the different normalized operation frequencies ␻ a/(2 ␲ c) ⫽ 兵 0.36, 0.37, 0.38, 0.39, 0.40其 labeled 1, 2,...5. Nearly vanishing power reflection was achieved for each of the specified operation frequencies.

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circumstances the optimum is always tracked down within one single iteration per frequency point. To underpin this statement we have performed several singlestep optimizations at the following normalized frequen␻ a/(2 ␲ c) ⫽ 兵 0.36, 0.37, 0.38, 0.39, 0.40其 . The cies: corresponding normalized frequency step of 0.01 is sufficiently small to obtain accurate results within a single optimization step (per frequency) even with a zero-order parameter estimation technique29; i.e., the starting point for the subsequent optimization is simply retrieved from the previous operation frequency. Figure 6 depicts the resulting spectral responses of the various PhC power dividers with each of them being optimized according to the given operation frequency. To give an impression of the resulting optical field, Fig. 7 depicts the time-averaged Poynting vector fields for two optimal solutions at the corresponding normalized operation frequency of 0.36 and 0.40. It may seem surprising that the simplified optimization procedure has proved to be so successful. This success is due mainly to the fact that the well-behaved fitness function allows the application of the very cost effective, zero-order parameter estimation technique. Note that the extension to a broadband power divider design is straightforward. One simply has to reformulate the fitness function taking into account an appropriate set of frequencies within the desired spectral band. However, special care is needed because the computational effort may scale up by the same amount as for the number of frequency points. Furthermore, the resulting fitness landscape may become more complicated, resulting in an additional slow down of the broadband optimization. Up to now our optimization has led to distinct PhC topologies in which an assortment of posts must retain very precise sizes and very specific positions as well. This may inevitably result in some impractical constraints regarding the proper fabrication process. However, when referring to the sensitivity analysis as elucidated in Ref. 28, the sensitivity, which is always inherent in our optimization scheme, may itself become an optimization issue when properly introduced into the fitness function. Thus

Fig. 7. Poynting vector field distributions within two optimized PhC power dividers. normalized operation frequency of (left) ␻ a/(2 ␲ c) ⫽ 0.36, (right) ␻ a/(2 ␲ c) ⫽ 0.40.

Structure with minimized power reflection at a

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structural optimization bears a wide potential stemming from its obvious design adaptability, which may be directed toward the design of either highly sensitive structures such as sensors or rugged designs insensitive to all kinds of environmental influences including fabrication tolerances. Structural optimization also allows discovering some promising means for local interventions such as trimming and tuning, and even for switching purposes.

4. EVOLUTIONARY OPTIMIZATION OF A POWER DIVIDER Most solutions stemming from deterministic optimizations depend considerably on the intuitive initial design. Up to now our final device has deviated very little from its initial topology, which seems reasonable when regarding the evaluation of local deterministic optimization schemes. A more general optimizer should not depend on a starting guess but should rather enable drastic interventions such as changing the number of involved rods within the PhC power divider’s branching area. Thus one may favor the aforementioned combinatorial optimization with a binary type of intervention where it becomes a binary decision, whether a specific rod is present or not (i.e., a vacancy is introduced). A promising candidate among such probabilistic global search schemes is the standard genetic algorithm (GA)18,19 with its straightforward implementation. Again we define a rectangular section encompassing the most significant lattice sites within the proper branching area of the T-junction, as displayed in Fig. 8. To keep the search space tractable, we select a relatively small area of 4 ⫻ 5 lattice sites. With symmetry considerations, only 12 independent lattice sites were at the GA’s disposal, leading to a search space of 4096 different designs. Each design (i.e., each individual) is therefore labeled by a 12-bit string (i.e., the chromosome). A brute-force evaluation of all these configurations at one single operation frequency takes ⬇2 weeks on a state-of-the-art personal computer. It is worth mentioning that because of the discrete nature of the problem representation, each (binary) intervention fully renders the highly counterintuitive nature of the structure of the underlying PhC power divider, which is barely intelligible by simple trial-and-error manipulation. A distinct flavor of this unique feature may be experienced by operating the Java applet provided for educational purposes by Ref. 33. As a probabilistic global search strategy, the GA is likely to find one among the best solutions, we hope within reasonable time. Preliminary experience has revealed that the standard GA may somehow get trapped, and it then usually requires quite a long time until a better solution is found. It sometimes takes even more iterations than the full-grid search. For practical reasons we therefore limit the number of generations for the standard GA that is simultaneously run on several machines using different initializations. After some numerical experiments we have found reasonable (suboptimal) values for the basic parameters of the GA (size of population, mutation rate, mating rate, and elitism rate) in order to achieve an acceptable convergence rate of the GA, i.e., a reasonable computational time. It is also important to


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mention that the crossover point for mating of parents was always generated randomly. Having a population size of 45, a mutation rate of 0.005, a mating rate of 0.6, and an elitism rate of 0.3, we tracked down several suboptimal solutions after 35 generations, i.e., after 1575 fitness evaluations. The global optimum (see Fig. 9) was found in approximately 20% of the runs. Since the search space contains 4096 potential solutions, we were disillusioned by the poor performance of the standard GA apparent within the 1575 fitness evaluations. Note that a purely random search with a simple additional procedure to avoid duplicate fitness evaluations finds the global optimum within an average of 2048 fitness evaluations. As a countermeasure we implemented a micro-GA that works with small population sizes of, e.g., only five individuals. Our proposed scheme follows an elitist strategy in keeping the best individual for the next generation, and it uses the four best-performing individuals for mating, using single-point crossover. When all individuals of a generation have become identical, the population is reinitialized randomly. Furthermore, all structures and their corresponding fitness values are logged into a database to avoid the re-evaluation of identical individuals, which are likely to reappear during this kind of optimization procedure. The resulting speedup is significant: When 1000 micro-GA optimization runs were compared, the global optimum was always found within considerably fewer than 2048 fitness evaluations. On average, 636 fitness evaluations were sufficient to trace the optimal solution, which renders the micro-GA approximately three times faster than random search. Even more sophisticated procedures have been implemented by using a statistical analysis of the database. One of them is still based on a micro-GA but now has the random reinitialization process of the population being biased according to a kind of ‘‘bit-fitness.’’ In this reinitialization scheme each bit of the individual’s bit string is varied according to a

Fig. 8. PhC power divider structure adapted for the application of a standard GA. For symmetry reasons the resulting binary representation of the proper branching area (i.e., the rectangle including the most significant lattice sites) is encoded into 12 bits spanning a search space that consists of 4096 different solutions.

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Fig. 9. Spectral response and device topology of two optimal PhC power dividers: left, globally optimal solution and right, second-best solution with respect to the discrete search space. These solutions have been used to evaluate the performance of various proposed evolutionary optimization schemes as elucidated in the text.

Fig. 10. Poynting vector field pattern of the second-best power splitter at the critical normalized operation frequency ␻ a/(2 ␲ c) ⫽ 0.416 where strong reflection and relatively low accuracy are obtained. This frequency coincides with the upper resonance peak in the spectral response presented in Fig. 9.

certain probability function (i.e., according to its bitfitness), which mimics the bit’s potential for a successful influence on the overall fitness. The bit-fitness is a product of the aforementioned statistical analysis and therefore is always updated during the overall optimization scheme. Here, best-search scenarios have computational costs of ⬇388 fitness evaluations and are therefore five times more efficient than random search. It is characteristic of this modified scheme that the initialization becomes more pronounced at the expense of the crossover, rendering the latter obsolete. In this respect the similarity to evolutionary strategies (ES) becomes striking, and hence we applied an additional optimization algorithm based on a (1 ⫹ 6)-ES with a population size of seven in-

dividuals in which the best one is always kept (elitism), and six individuals are generated according to a mutation operator similar to the scheme for the bit-fitness. This simplest optimization strategy has proved to be the most successful. The optimal solution has been found within only 284 fitness evaluations in an average of 1000 test runs. The two best of the 4096 possible solutions are shown in Fig. 9. Both PhC power divider structures provide an excellent performance superior to the deterministic optimization outcomes, even showing tendencies toward broadband operation. The topologies essentially consist of two resonators, one having a resonance frequency below and one above the target frequency at which power reflection is to be minimized. Near the two resonance frequencies our numerical model lacks slightly in accuracy, but we still obtain a reliable field pattern (shown in Fig. 10) at the upper resonance peak as provided by the corresponding spectral response of Fig. 9. Given the discrete (binary) nature of the underlying PhC structure, the very small volume in which the optimizer is allowed to intervene, and hence the resulting highly discontinuous behavior, it seems quite astonishing that under such limiting constraints structural optimization is yet feasible, and even proved highly successful.

5. CONCLUSIONS We have described the numerical structural optimization of a 2D PhC power divider using two different classes of optimization algorithms: the modified TN method as a deterministic local optimization scheme working within a high-dimensional, real-valued search space and four types of evolutionary optimization schemes representing probabilistic global search strategies, each of them operating on a binary search space. Deterministic optimizers usually need an initial starting topology, whereas our probabilistic ones build on a self-generated initial random population of potential PhC structures. With both

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classes, we found reliable and promising solutions with almost zero-power reflection at the given frequency specification within the PBG. From these results we find that perfect PhC power dividers may exist for nearly arbitrary specifications, even supporting almost perfect broadband performance within a reasonably large portion of the PBG. To pursue, e.g., the broadband design, one should consider hybrid optimization schemes in which an ‘‘optimal’’ solution stemming from an evolutionary search scenario is used as the initial guess for a subsequent deterministic optimization. Here the main difficulty lies in providing a reliable state variable that denotes the evolutionary optimization’s inherent potential to initiate the turnover to more rapid deterministic search procedures. Some preliminary examples regarding state variables and hybrid structural optimization are discussed in Ref. 15. From the fabrication point of view, combinatorial optimization that deals with identical rods may become the most favorable procedure, because it does not impose a variety of rod shapes and positions that have to meet the fabrication tolerances. Our proposed evolutionary search strategies even allow the inclusion of production constraints during optimization. Even if we are among the very first to perform numerical structural optimization of PhC devices, a question still remains, namely, whether a 2D device design is reliable enough for 3D planar PhC realizations. A contemporary, real-world design based on planar PhCs therefore has to address the following considerations: First, device concepts may be explored by using efficient 2D computational optics tools such as MMP, where promising PhC device topologies emerge from 2D structural optimizations as proposed in this paper. Here, corresponding phenomenological models34 have already proved to be best suited to bridge the gap between a realistic (planar) PhC structure and its proper 2D representation. Second, the simulation of realistic PhC devices requires true 3D modeling capabilities that are numerically much more demanding. Emergent planar PhC device topologies are thus evaluated by, e.g., 3D-finite-difference-time-domain methods, or better, more efficient, and highly accurate 3D PhC simulation tools that are yet to be developed. Despite the severe challenges posed by such methods for realistic device design, we are prone to propose 2D structural optimization as highly appropriate and valuable for exploiting the peculiar nature of PhCs and pushing them toward more compact device topologies.

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ACKNOWLEDGMENTS This work was supported by the Swiss National Science Foundation in the framework of project NFP-2000065102.01/1 and the research initiative National Centre of Competence in Research (NCCR) Quantum Photonic. Corresponding author J. Smajic’s e-mail address is [email protected]. He may also be reached by phone, 41-1-632-6960 and fax, 41-1-632-1647.

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