Optical Simulation of Grating Diffraction in MATLAB.pdf

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Optical Simulation of Grating Diffraction in MATLAB®

Part 1: Conceptual Overview

MATLAB development environment

Structure specification

Parameterization

Electromagnetic theory

Diffraction order selection

Part 2: GD-Calc Software Interface

GD-Calc usage overview

The grating struct

Stratum type 0: homogeneous

Stratum type 1: uniperiodic

Stratum type 2: biperiodic

Harmonic indices and stratum periods

Stratum type 3: coordinate break

Stratum type 4: replication module

Incident field and diffraction order selection

Going further

1 The “Grating Diffraction Calculator” (GD-Calc®) is a MATLAB-based, Optical Simulation of Grating Diffraction in MATLAB® by Ken Johnson electromagnetic simulation program that computes diffraction efficiencies of optical grating structures, including biperiodic gratings. The program’s capabilities include a general and flexible grating modeling facility, structure parameterization (with any number of parameters), and unrestricted control over diffraction order selection. Additionally, its implementation within the generic programming and application development framework of MATLAB provides a degree of flexibility and software interoperability that is not available with stand-alone diffraction analysis programs. Part 1 of this article provides a conceptual overview of GD-Calc, describing in general terms how grating structure is specified and how electromagnetic computations are carried out. The presentation is primarily concept-oriented, but a few simple code examples are provided to give the reader a sense of how the GD-Calc software interface works. Part 2 provides a more in-depth introduction to the software interface, using as an example a tungsten photonic crystal structure1 to illustrate how grating structure is specified. (The code listings from Part 2 are summarized in gdc_intro.m.) examples for electromagnetic computations are provided in an accompanying document, GD-Calc_Demo.pdf. (All of the code examples in this article and in GD-Calc_Demo.pdf can be run with the free demo/tutorial code from the GD-Calc website. The photonic crystal example is based on the demo script gdc_demo11.m.) The electromagnetic theory and algorithms underlying GD-Calc are detailed in GD-Calc.pdf. Part 1: Conceptual Overview The primary focus of this article is grating structure specification. Application An advantage of working in the MATLAB environment is that functional links MATLAB development environment into and out of GD-Calc can be created without having to rely on cumbersome data conversion and import/export procedures. For example, in a semiconductor lithography application, a photoresist grating’s thickness and refractive index might both be affected by exposure-related resist densification, so it would be natural to specify thickness and refractive index both as user-defined functions of exposure. This is especially useful with 1 The photonic crystal structure is described in S. Y. Lin, J. G. Fleming, and I. El-Kady, “Highly efficient light emission at by a three-dimensional tungsten photonic crystal,” Optics Letters 28(18), 1683-1685 (2003). GD-Calc_Intro.pdf, version 09/22/2006 Copyright 2006, Kenneth C. Johnson software.kjinnovation.com

2 Typically, a grating’s optical characteristics are not themselves of primary structure parameterization, e.g., exposure could be defined as a vectorized quantity, in which case all exposure-dependent quantities, including the resist thickness, refractive index, and calculated diffraction efficiencies, will be similarly vectorized. interest; what is of interest is the optical response of a complete system that includes the grating as one component. MATLAB’s generic programming capability makes it easy to functionally link GD-Calc into user-defined optical system models, which can themselves be incorporated into generic optimization routines to optimize design performance. GD-Calc is simply a MATLAB function (gdc.m), which can be incorporated into other MATLAB functions or scripts, and which takes arguments that can be instantiated to user-defined functions. MATLAB development environment, they can have the advantage of simplicity and ease-of-use. However, GD-Calc can be used in conjunction with MATLAB to create customized user interfaces that are optimally adapted for specific applications. Many of the functions and scripts associated with GD-Calc such as its plotting facility (gdc_plot.m), its output data conversion function (gdc_eff.m), and a number of demo scripts are distributed as public-domain software so that users can freely modify and adapt the code to best suit their own or their customers’ needs. Although stand-alone programs lack the generality and flexibility of the The primary limitation of GD-Calc’s grating modeling capability is that gratings Structure specification must be “block-structured” (or must be defined approximately in terms of a block- structured representation), meaning that the grating comprises optically homogeneous regions whose bounding surfaces are planes parallel to a set of primary coordinate planes. For example, a grating comprising a periodic array of pyramids would be represented using a “staircase approximation”, as illustrated in Figure 1. bounded by upper and lower planes parallel to the grating substrate. The grating has a height-independent lateral cross-section within each stratum. Figure 2 illustrates a particular stratum extracted from the pyramidal grating of Figure 1. Each stratum is partitioned into parallel “stripes”, which are further partitioned into rectangular “blocks” representing optically homogeneous regions. The grating is subdivided into a number of “strata”, wherein each stratum is GD-Calc_Intro.pdf, version 09/22/2006 Copyright 2006, Kenneth C. Johnson software.kjinnovation.com

3 stratum Figure 1 Pyramidal grating block stripe Figure 2 Grating stratum In terms of software representation, a structure such as that of Figure 1 would be represented as a nested hierarchy of cell arrays within structs. The top-level data structure, designated here as grating, is a struct containing a cell-array stratum field whose elements, grating.stratum{1}, grating.stratum{2}, … represent the grating strata (numbered from the bottom up). The i-th stratum’s stripes are represented as grating.stratum{i}.stripe{1}, grating.stratum{i}.stripe{2}, …, and the j-th such stripe’s blocks are represented as grating.stratum{i}.stripe{j}.block{1}, grating.stratum{i}.stripe{j}.block{2}, … . Other fields within these structs define the bounding plane locations and the optical material within each block. The optical materials are described in terms of their complex permittivities, which are GD-Calc_Intro.pdf, version 09/22/2006 Copyright 2006, Kenneth C. Johnson software.kjinnovation.com

4 enumerated in a top-level cell array grating.pmt, each element of which represents a particular material. A particular grating region’s optical material is specified as an integer index into this list (e.g. grating.stratum{i}.stripe{j}.block{k}.pmt_index); thus multiple regions can be constrained to represent the same material by giving them the same material index. grating. These are enumerated below: There are five different types of “stratum” objects that can be used to define the Stratum type homogeneous uniperiodic biperiodic coordinate break replication module Description homogeneous grating layer with no stripe boundaries stratum with homogeneous stripes, no block boundaries general biperiodic stratum (as illustrated in Figure 2) applies a translational shift to all strata above break plane for defining 3-dimensionally periodic grating regions Each stratum object has a “type” index indicating its type. For example, a stratum.type = 0; % homogeneous stratum.thick = 0.5; stratum.pmt_index=1; % index into grating.pmt grating.stratum{1} = stratum; homogeneous stratum is defined by three struct fields: the type index (zero), the stratum thickness, and the permittivity index, e.g., The homogeneous and uniperiodic stratum types are basically specializations of the more general biperiodic type. A “coordinate break” is a “stratum” in the abstract sense that it is associated with a lateral plane at a particular height in the grating, and it provides a simple mechanism for applying a lateral translational shift to all strata above the break plane without having to modify the individual stratum definitions. A “replication module” is a composite type of stratum object used to represent a structure pattern that repeats itself periodically in a direction transverse to the grating substrate. (The basic structure pattern is represented as a stack of strata, which can be of any type – including other replication modules.) Figures 3 and 4 illustrate conceptually how the above structuring elements could be combined to define a photonic crystal structure. First, the three-stratum configuration illustrated in Figure 3 is defined. The first two strata are uniperiodic with orthogonal stripe orientations, and the third stratum is a coordinate break, which applies a half-period translational shift in each of the two periodicity directions. (The translational shift is represented by the arrow in Figure 3.) Next, the three strata are combined into a replication module object, which has an associated replication count indicating how many times the pattern is to be repeated. Figure 4 illustrates the resulting structure with a replication count of four. The structure comprises four stacked copies of the basic bilayer pattern, with each copy laterally shifted by a half period in both periodicity directions relative to the underlying bilayer. GD-Calc_Intro.pdf, version 09/22/2006 Copyright 2006, Kenneth C. Johnson software.kjinnovation.com

translational shift stratum 3 stratum 2 stratum 1 Figure 3 Bilayer pattern (with coordinate break) for photonic crystal 5 Figure 4 Photonic crystal Rather than using a replication module in the above example, the grating could be defined by simply stacking four copies of the Figure 3 structure. However, the advantage of using a replication module is not just one of convenience. Whereas the GD-Calc computation time generally scales in proportion to the number of grating strata, the GD-Calc_Intro.pdf, version 09/22/2006 Copyright 2006, Kenneth C. Johnson software.kjinnovation.com

computation time for a replication module scales in proportion to the logarithm of the replication count. 6 The GD-Calc interface specification (defined in the gdc.m comment header) Parameterization identifies a number of grating attributes as “parameters”. A “parameter”, in this context, is a numeric quantity that can be vectorized as a multidimensional array. A parameter can be one of a number of basis parameters, each of which is associated with a particular array dimension (and which has only one non-singleton dimension), or it can be a function of other parameters. (A parameter’s non-singleton dimensions indicate its functional dependencies.) The only fundamental restriction on parameters is that they must all be size-matched, except that singleton dimensions are implicitly repmat- extended, if necessary, to match parameter sizes. A simple application example showing the use of parameterization is illustrated in Figure 5. 2 The figure shows a cross-section of an alignment-sensor grating comprising a substrate, a uniperiodic, surface-relief reflective grating (stratum 1), a homogeneous air space (stratum 2), a coordinate break (stratum 3), and a uniperiodic phase grating (stratum 4) on a transparent superstrate. A small, lateral translational shift between the two gratings will result in a measurable shift in the energy balance between the first- and minus-first-order diffraction efficiencies; thus the gratings can function as a sensitive positional transducer. Figure 5 Alignment-sensor grating In this example, two grating parameters are vectorized: the air space thickness, represented as grating.stratum{2}.thick, and the phase plate’s translational shift, represented as grating.stratum{3}.dx2. These could be defined, for example, as (The wavelength and d variables are scalars representing the illumination wavelength and grating period, respectively.) The air space is size-[3,1], and is vectorized in dimension 1, while the translational shift is size-[1,64], and is vectorized in dimension 2. Any quantities that are functionally dependent on both the air space and the translational grating.stratum{2}.thick = wavelength*[3;2;1]; grating.stratum{3}.dx2 = d*(0:63)/64; stratum 3 superstrate stratum 4 stratum 2 stratum 1 substrate 2 This example is based on the GD-Calc demo script, gdc_demo9.m. GD-Calc_Intro.pdf, version 09/22/2006 Copyright 2006, Kenneth C. Johnson software.kjinnovation.com

7 Parameterization can, in some instances, dramatically improve GD-Calc’s shift, including computed diffraction efficiency quantities, will be size-[3,64]. Based on the repmat-extension convention described above, these arrays are all size-compatible. computational performance. As explained in the next section, the GD-Calc algorithms are based primarily on two operations: first, calculating an “S matrix” for each stratum, and then combining the S matrices from bottom to top by means of a “stacking” operation to determine a composite S matrix for the entire grating. In the above example, there are 192 parameter combinations (3 air space thicknesses and 64 translational displacements), so if the parameters were iterated outside of GD-Calc all of the S-matrix computations would have to be repeated 192 times. But with parameterization, the S matrices for the grating layers (strata 1 and 4) will only have to be calculated once, the air space’s S matrix is calculated only three times, and the coordinate break’s S matrix is calculated only 64 times. Furthermore, the stacking operation for stratum 2 will be calculated just 3 times, and only the stratum 3 and 4 stacking will have to be done for all 192 parameter combinations. In the original formulation of RCW theory, the propagation equations were Electromagnetic theory GD-Calc’s algorithmic foundation is based on a generalized variant of rigorous coupled-wave (RCW) theory, which first came into use about three decades ago but has since undergone a couple of major improvements. In essence, the method represents both the electromagnetic field and the optical permittivity at any particular height in the grating in terms of their Fourier series in two lateral grating coordinates, and a set of differential equations are developed that describe the propagation of the field’s Fourier coefficients through the grating. numerically solved to determine a “transmission matrix” that defined the field amplitudes (Fourier coefficients) at the top of each stratum as a linear function of the amplitudes at the bottom, and these matrices were then multiplied to determine a composite transmission matrix for the entire grating. But numerical contamination from exponentially growing errors caused this method to be numerically unstable, especially with very deep or highly conducting gratings. The problem has been resolved by using an alternative “S-matrix” (scattering matrix) approach, which represents each stratum in terms of a linear mapping relating the outgoing field amplitudes to the incoming field amplitudes at both boundaries. The individual mappings (S matrices) for the strata are combined by means of a numerically stable “stacking” algorithm to determine the grating’s composite S matrix. with respect to the number of Fourier coefficients retained in the calculations was typically very slow. The main problem had to do with terms in the differential equations representing the product of the electromagnetic field and the permittivity. Each product’s Fourier series was determined from its factors’ truncated Fourier series, but severe The S-matrix method resolved the numerical stability problem, but convergence GD-Calc_Intro.pdf, version 09/22/2006 Copyright 2006, Kenneth C. Johnson software.kjinnovation.com

8 numerical convergence problems arose when the factors had concurrent discontinuities associated with optical boundaries. This problem has been resolved by using a “Fast Fourier Factorization” method that, in essence, rearranges the equations to avoid product factors with concurrent discontinuities (i.e., in each equation where a concurrent discontinuity occurs, the permittivity factor is brought to the other side of the equation before applying the Fourier decomposition). GD-Calc uses both the S-matrix method and Fast Fourier Factorization to optimize computational performance. (The numerical algorithms are detailed in GD-Calc.pdf.) One problem that remains with RCW methods, however, is the “staircase approximation” that must be used to describe sloped or curved surfaces in terms of the “block-structured” representation. Simply partitioning the grating into very small blocks does not ensure good accuracy, because the electromagnetic field can exhibit large spikes near the block corners, and the number of retained diffracted orders must be increased in proportion to the block partitioning density in order to adequately resolve the spikes. Thus, users of GD-Calc should be aware of convergence difficulties that can arise, particularly with highly-conducting gratings that are not inherently block-structured. (The GD-Calc demo scripts and GD-Calc_Demo.pdf provide examples of the program’s convergence behavior for a variety of test cases, including comparisons with published data.) GD-Calc gives the user complete freedom in selecting which diffracted orders Diffraction order selection (i.e., Fourier coefficients) to retain in calculations. Generally, computational data storage requirements scale in approximate proportion to the square, and runtime scales in proportion to the cube, of the number of retained orders, so optimizing the order selection can very significantly impact computational performance. The checkerboard grating illustrated in Figure 6 (in plan view) is one example where order selection is particularly useful. The grating is described in relation to two fundamental period vectors; for example vectors Ar and Br would be a natural choice. However, the grating has a periodic symmetry stronger than that described by vectors Ar and Br because, for example, the fundamental period vectors Ar and Cr define a unit cell whose area is half that of Ar and Br . Thus, if the grating’s optical permittivity is Fourier analyzed with respect to orthogonal coordinates represented by vectors Ar and Br , half of its Fourier orders will be identically zero, and similarly half of the electromagnetic field’s diffracted orders will be zero. diffraction order, has two associated Fourier order indices specify what set of In the Figure 6 example, using basis periods ; and the user must index pairs to retain in the electromagnetic field expansion. Each grating Fourier coefficient, and correspondingly each electromagnetic Ar and Br , all orders with ( 1 mm , 2 odd ) 1m and 2m 1 mm − 2 GD-Calc_Intro.pdf, version 09/22/2006 Copyright 2006, Kenneth C. Johnson software.kjinnovation.com

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