Memetic Algorithm Approach to Thin-Film Optical Coating Design

Algorithms” by Moscato (Moscato, 1989). They have also been referred to in the literature by other names, such as Hybrid Genetic Algorithms, Genetic Local Searchers, or Lamarkian Genetic Algorithms.
In this section we outline the Memetic Evolutionary Algorithm which we have employed for the optical thinfilm design problem. The approach involves a selection of recombination and mutation algorithms together with local searches that are performed with varying levels of probability. We will see that the addition of the local search improves the rate of convergence of the search for optimal performance. It also shows that less effort needs to be placed into the optimization of the Evolutionary Algorithm in order to ensure that it’s parameters and structure yield the best results. The actual algorithm we employ is very similar to the Family Competition Evolutionary Algorithm (FCEA) (Yang and Kao, 2000) that was used to treat a similar problem. The initial population of M individuals is chosen at random. Each individual is composed of a series of real-valued chromosomes made up of the thickness values of the M layers of the optical coating, together with 3M parameters which control the step sizes associated with the 3 different types of mutation algorithms applied. These latter parameters are expressed as M-component vectors σ, v and ψ. Their meaning will be described later. Finally, the value of the Merit Function is also incorporated as part of each individual. The merit function employed here is the basic one that is used to represent the difference between the design reflectance and the computed reflectance over a range of wavelengths.
(2.2)
3
The Memetic Algorithm
and yj, nj, and dj are the optical admittance, refractive index, and physical thickness, respectively of the jth thin film. In addition ysub is the optical admittance of the substrate. As long as there is no absorption in the medium, the admittance and refractive indices can be taken to be the same.
2 Statement of the Problem
The calculation of the interference effects of a system of thin films can be accomplished simply by considering a ray approach in which beams are reflected backwards and forwards between the various interfaces. In the steady state these rays combine to form the resultant reflected or transmitted ray. This is very straightforward to implement numerically. The technique usually employed (Abelès, 1950) involves a two-stage process in which the electromagnetic waves in each film are reduced to two partial waves, one positive and the other negative-going. These partial waves transfer the total electric and magnetic fields at the rear interface of each layer to the forward interface. This transfer operation reduces to a matrix multiplication, and the calculation of the multilayer properties reduces to a series of matrix operations. We consider a series of thin films, each having an index of refraction nj and a thickness dj . Incident light is incident normally from an absorption-free medium (usually air) and strikes the top layer (layer M) of an Mlayer film. The last layer (layer 1) is adjacent to the substrate. (See Figure 1) The matrix expression from which the total reflectance is computed, is
Memetic Algorithm Approach to Thin-Film Optical Coating Design
R.J.W. Hodgson
Department of Physics University of Ottawa Ottawa, Ontario, Canada rhodgson@physics.uottawa.ca
B M cos δ j C = ∏ j =1 iy j sin δ j
where
i sin δ j 1 yj y (2.1) sub cos δ j
1
δj =
2π n j d jλ源自1Introduction
Almost all optical instruments consist of a series of optical surfaces that reflect and refract light. In practice optical coatings are required to modify the reflectance and transmission of the surfaces involved. The performance of these coatings is determined by the interference that occurs due to the multiple reflections within the coating. Several approaches have been developed for the design of thin film multilayer coatings. (Macleod, 1995) Among these approaches the use of digital design methods offer a number of advantages. These in turn are generally classified as either refinement or synthesis methods. In the former one starts with an approximate design and uses various tools to refine it to better achieve the design criteria. In the latter, one frequently generates the initial design. This work centers on the application of evolutionary algorithms to the design of a thin film multilayer coating to achieve a particular reflectance over a defined wavelength region. In addition we incorporate local searches on a random basis in order to try to speed up the search for a global minimum. The combination of Evolutionary Algorithms with a local search provides a very powerful metaheuristic to search complex continuous spaces (Hart, 1994; Land, 1998). These categories of search algorithms were labeled “Memetic
Abstract
The problem of optimizing a series of thin-film layers in order to achieve a desired reflectivity of light over a range of frequencies is studied. The approach studied here is to employ an evolutionary algorithm combined with random local searches to obtain the best fit. This Memetic Algorithm is found to result in faster convergence than in the case when no local searches are employed.