## Abstract

We have developed a rigorous unidirectional method for designing finite-aperture diffractive optical elements (DOE’s) that employs a micro-genetic algorithm (µGA) for global optimization in conjunction with a 2-D Finite-Difference Time-Domain (FDTD) method for rigorous electromagnetic computation. The theory and implementation of this µGA-FDTD design method for normally incident TE illumination are briefly discussed. Design examples are presented, including a micro-lens, a 1-to-2 beam-fanner and a 1-to-3 beam-fanner.

©2000 Optical Society of America

## 1. Introduction

Finite aperture diffractive optical elements (DOE’s) with feature sizes on the order of or less than a wavelength are useful for a wide variety of applications [1–3]. However, the assumptions of typical scalar-based design methods are violated in this feature size regime [4]. A general, fully rigorous design tool is therefore needed to enable the direct synthesis of such elements. One approach to develop a suitable tool is to combine a rigorous electromagnetic computational engine for diffraction calculations with an optimization method. This was done in Refs. [3] and [5] with the boundary element method (BEM) [6] and simulated quenching (which is suitable for finding locally optimum solutions) and applied to the design of 2-phase level subwavelength structures. In this paper we introduce an alternate approach that is appropriate for finding global or near-global solutions for a wide variety of finite aperture DOE’s with feature sizes on the order of or smaller than a wavelength. It consists of a rigorous electromagnetic diffraction analysis kernel based on the 2-D finite-difference time-domain (FDTD) [7] method combined with global optimization using a micro-genetic algorithm (µGA) [8]. Our µGA-FDTD method can be classified as a unidirectional design approach [9] for DOE’s in that it requires a search of the possible solution space for a given problem, which involves evaluating many potential candidate solutions. The use of efficient (both in terms of number of operations and memory usage) diffraction analysis and global search algorithms are therefore critical to obtain a practical design tool. In this paper we briefly introduce such a tool for the case of normally incident TE illumination, and present three design examples to illustrate its use.

## 2. Implementation of the µGA-FDTD design method

Due to its flexibility and relative computational efficiency [10], we selected the FDTD method as the rigorous diffraction computational kernel. The geometry of the grid region in our FDTD calculations is shown schematically in Fig. 1, in which the finite aperture DOE is totally embedded in the FDTD grid. The refractive indices of the incident and exit media are n_{1} and n_{2}, respectively. The computation region is truncated along its outside faces by implementing perfectly matched layer (PML) absorbing boundary conditions (ABCs) [11] bounded by a perfect electric conductor (PEC). Because of the negligible reflection error caused by the PML ABC, this boundary can be placed only a few cells from the DOE boundaries, which significantly reduces the size of the computation region and hence the computational load [12]. To excite the entire FDTD grid, a total/scattered field algorithm [7] is employed to generate a normally incident TE plane wave. Once steady state has been reached, the electromagnetic plane wave spectrum approach [13] is used to propagate the electromagnetic field from the near field output plane to the observation plane.

For global optimization, the genetic algorithm (GA) has been found to be useful for DOE design for the case of scalar diffraction [14]. GA simulates the natural evolutionary process, that is, survival of the fittest. However, conventional GA uses a relatively large population size (~100), which results in a very large computational burden for rigorous diffraction analysis due to the intensive computational nature of tools such as FDTD. To avoid this, we employ a modified GA, called micro-GA (µGA) [8], in which the population size is 5. A micro-GA involves the same genetic operations as in conventional GA, which include chromosomal coding of the problem parameters, evaluation of the fitness function, selection methods for survival of the fittest individuals, and breeding of a new generation of individuals [15]. Convergence to a globally non-optimal result typically occurs within four to five generations because of the small population size. This result, however, can be treated as a local optimum. To achieve a global optimum, the GA search is restarted from this local optimum by retaining the best individual found so far and randomly initializing the others. This new population serves as the starting point for the next GA process. The µGA algorithm therefore searches for a global optimum by successively finding local optima through a series of conventional GA processes executed with small populations.

## 3. Numerical design examples

To illustrate the utility of our µGA-FDTD design method, we examine several DOE design examples. The physical geometry for all cases is shown in Fig. 2. The DOE profiles are assumed to be etched into a Si substrate. A monochromatic plane wave with a free space wavelength of λ_{0}=5 µm is normally incident from a silicon substrate (n_{1}=3.4) onto the DOE profile. The exit medium is air (n_{2}=1.0). The desired field distribution is observed in a plane 100 µm from the DOE. The aperture of the DOE is 50 µm. Since the minimum feature size is 2 µm, the DOE aperture is divided into 25 feature cells which means that the associated optimization problem has 25 variables, each of which is the etch depth of a particular feature cell. The etch depth at each cell position is coded as a real variable, which spans the range of [0, 1.2 µm].

In our first example we use a simple analytical, quadratic profile, micro-lens to test our µGA-FDTD design method to determine whether it can reproduce the analytical micro-lens profile. The electrical field distribution E_{analy} of the analytical micro-lens profile is used as a target function for µGA. We therefore specify the fitness function for the DOE profile as

in which xi refers to the position of the *i*
^{th} sample point in the observation plane. The design task is to minimize the fitness function. We start from a randomly chosen DOE profile and run 500 µGA generations. The convergence curve (i.e., *f* as a function of the µGA generation) is shown in Fig. 3(a). It rapidly converges to a final value of the fitness function of three, which is very close to the global minimum (in this case zero). The µGA-optimized field distribution along with the field distribution for the analytical micro-lens profile is shown in Fig. 3(b), while the corresponding DOE profiles are shown in Fig. 3(c). The field distribution for the optimized DOE profile is nearly identical to the target field distribution. However, the optimized DOE profile, while close to the analytical profile, nevertheless exhibits some small differences. This illustrates a general result that we have found in our work with µGA-FDTD in which nearly the same diffraction pattern can be generated by a number of differing DOE profiles. This is also an example of the strength of genetic algorithms for global searches, while at the same time they can be weak at local searches in the neighborhood of the global solution.

The second design example is a 1-to-2 beamfanner for an imaging polarimetry system, in which the objective is to design a DOE profile that creates two focused beams separated by 25 µm in the observation plane. Since the desired electric field profile is not easily specified, a Gaussian weighting function is used for µGA-FDTD to control the field distribution at the desired peak positions. We also use a constant penalty function (i.e., negative weighting) to control the side-lobes. The fitness function (which in this case must be maximized by µGA) is defined as

in which σ is a peak width parameter and *rect*() is a rectangular window function. Our design method found several profiles that can achieve the desired beamfanning function. Fig. 4 shows two resultant DOE profiles and the corresponding field distributions in the observation plane. Note that the field distributions are very similar, while the DOE profiles are quite different.

The last design example is a wide-angle 1-to-3 beamfanner. The desired peak separation is 50 µm, which corresponds to a 53.2° anglular separation between the outermost peaks. For this case, we used the same type of weighting function as in the previous example along with another penalty function to control the peak uniformity. Without this additional constraint, the µGA-FDTD algorithm would converge to a micro-lens profile. The optimized DOE profile and resultant electric field distribution in the observation plane are shown in Fig. 5. Reasonable uniformity in the peak heights (within 5%) has clearly been achieved. Equalizing the amount of optical power diffracted into each peak would require an additional constraint in the fitness function.

## 4. Summary and Discussion

We have developed a completely rigorous design method for finite-aperture diffractive optical elements that have minimum feature sizes on the order of or less than a wavelength. The method combines a micro-genetic algorithm (µGA) for global optimization and a 2-D FDTD method for rigorous electromagnetic computation. Based on the presented design examples, µGA-FDTD appears to be an attractive approach for rigorous DOE design. We implemented µGA-FDTD in Fortran 90 running on a PC with a 500 MHz CPU and 256 Mbyte RAM. This is sufficient for the design examples considered herein, in which the width of the DOE’s is small (10λ0). Computation of a single µGA generation takes only 20 seconds. However, for electrically large DOE’s, significantly greater computational resources are needed. Currently, we are working to parallelize our code and port it to an 8-processor system.

## Acknowledgements

G. P. Nordin acknowledges support by National Science Foundation CAREER Award ECS-9625040 and grant EPS-9720653.

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