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This paper reviews the current progress in mathematical modeling of antireflective subwavelength structures. Methods covered include effective medium theory (EMT), finitedifference timedomain (FDTD), transfer matrix method (TMM), the Fourier modal method (FMM)/rigorous coupledwave analysis (RCWA) and the finite element method (FEM). Timebased solutions to Maxwell’s equations, such as FDTD, have the benefits of calculating reflectance for multiple wavelengths of light per simulation, but are computationally intensive. Spacediscretized methods such as FDTD and FEM output field strength results over the whole geometry and are capable of modeling arbitrary shapes. Frequencybased solutions such as RCWA/FMM and FEM model one wavelength per simulation and are thus able to handle dispersion for regular geometries. Analytical approaches such as TMM are appropriate for very simple thin films. Initial disadvantages such as neglect of dispersion (FDTD), inaccuracy in TM polarization (RCWA), inability to model aperiodic gratings (RCWA), and inaccuracy with metallic materials (FDTD) have been overcome by most modern software. All rigorous numerical methods have accurately predicted the broadband reflection of ideal, gradedindex antireflective subwavelength structures; ideal structures are tapered nanostructures with periods smaller than the wavelengths of light of interest and lengths that are at least a large portion of the wavelengths considered.
Recent trends of global climate change and impending petroleum shortages have encouraged researchers to develop a variety of renewable energy production methods, solar electricity generation being among the most popular of solutions. Commercially available monocrystalline silicon solar cell efficiency is currently above 24%, a mere 5% below the theoretical maximum. As solar cell production becomes cheaper, the cost of installed solar modules is beginning to depend more on module production and installation. One way to reduce the cost of installed solar panels is to minimize losses due to light reflection at interfaces, including but not limited to the air/glass and adhesive/silicon interfaces. This paper addresses the current status of mathematical modeling of antireflective subwavelength structures (ARSWS), provides the background on the most popular of modeling techniques for ARSWS, and suggests appropriate applications for each technique.
This paper is intended to be a review of the most commonly used methods for optical modeling of antireflective subwavelength structures. Optical modeling methods have developed over time and, with the introduction of advanced computing resources, have largely discarded methods that include nonrigorous assumptions. Likewise, with the wide variety of optical modeling methods available, some methods have become more popular than others due to reasons other than their computational ability, such as availability of commercial software or abundant use in the literature. Currently, only four major modeling methods are commonly used in the field of ARSWS: finitedifference timedomain (FDTD), finite element method (FEM), transfer matrix method (TMM), and rigorous coupledwave analysis or Fourier modal method (RCWA/FMM). Of those, all but TMM are capable of describing the geometry of subwavelength structures; TMM relies on an effective media approximation for more complicated geometry. The mathematical approach for each of these methods is different, resulting in different advantages and disadvantages in modeling capabilities, which are the topic of this review. Although these methods are considered accurate and rigorous solutions to Maxwell’s equations, it is suggested that exploration of solutions through multiple modeling methods is most robust [
Some optical modeling methods are not covered here; these methods include method of moments (MoM; for background see Chapter 15 in reference [
Antireflective subwavelength structures are a type of biomimicry. It was found that the eyes of moths and butterflies include a surface layer of regular nipple arrays that reduced light reflection from the air/eye interface. Stavenga
Example images of motheye structures found in nature. The scale bars are (
ARSWSs can be tapered structures with a gradient index of refraction (GRIN), nontapered structures, sparse or densely packed, and/or made of the same material as one of the interface materials or a different material entirely. These characteristics are chosen as a balance between an ideal ARC for the situation at hand and the manufacturability of that ARC.
The least complicated ARCs are quarter wavelength, intermediate index thin films [
This review will use many terms common in the field to describe ARSWSs and optical theory. In the following pages, we will consistently refer to the
Orientation diagram of 2D grating (3D model,
ARSWS materials are chosen to be dielectrics to reduce reflections and absorption. Material parameters of concern in this review are index of refraction (RI) and permittivity (ε). The index of refraction is the square root of permittivity for materials with a relative permeability of one, and both RI and permittivity can be complex numbers. The real part of the permittivity describes how light slows down in a medium, which is described by Snell’s law, or the relationship between angles of incidence and angles of refraction. The imaginary part of permittivity describes the extinction coefficient and is related to light absorption. Both the real and imaginary parts of permittivity are found to be wavelength dependent.
The optical models covered in this review use plane waves as incident light, either polarized or unpolarized. Unpolarized plane waves are equivalent to the averaging of the two polarizations of the plane waves, transverse electric (TE) and transverse magnetic (TM). As shown in
Diagram of transverse electric (TE) and transverse magnetic (TM) incident light at a nonzero angle of incidence on an interface plane for angle of incidence (AOI) < Brewster’s angle.
The most promising broadband ARSWSs are gradientindex (GRIN) nanostructures. These materials provide a smooth gradient of index of refraction at the interface between two layers. For solar modules, those layers are commonly air:glass (
Inputs to ARSWS reflectivity simulations include the size, shape, period, and/or location of subwavelength structures, materials properties (real or complex permittivity and permeability) for both bulk materials and the SWS features. The simulation must include an input of appropriate EM radiation, usually a polarized plane wave, and a way to detect the power of reflected and transmitted EM energy. The simulation must have appropriate boundary conditions to describe the material being simulated, either periodic or absorbing conditions on
Although many optical modeling methods have been developed for diffraction gratings to handle higher order reflections and refractions, sufficiently small ARSWSs should primarily require consideration of zeroth order reflections and transmissions [
ARSWS reflectivity simulations can be performed over a range of wavelengths, angles of incidence (AOI), and polarization (TE or TM). For solar module applications ARSWS are often modeled between 400 and 800 nm or 400 and 1200 nm, with 400–800 nm being the peak solar input at AM1.5 and 400–1200 nm being the useful range of encapsulated solar modules based on the extinction coefficient of the lamination materials below 400 nm and the bandgap of silicon solar cells above 1200 nm.
For stationary (nontracking) solar modules the behavior of an ARC at a variety of angles of incidence is important. Chuang
Diffraction orders on a 1D grating at normal incidence.
There are commercial software packages available for many electromagnetic modeling techniques. Though some were originally intended to model EM situations such as antennas or EM interference in electric circuits, most can be used to model reflection and transmission of antireflective coatings as well. A nonexhaustive list of software for FDTD simulations is XF by Remcom, FDTD Solutions by Lumerical, Meep from MIT, OptiFDTD, EM Explorer, and FullWave by RSoft Design Group. CST Studio uses the Finite Integral Technique (FIT), Transmission Line Matrix method (TLM), and Finite Element Method (FEM) for frequency and time domain solvers. High Frequency Structure Simulator (HFSS) is another commercially available FEM software, as is COMSOL. Several software packages use the Rigorous Coupled Wave Analysis (RCWA), including RODIS, Unigit, GDCalc, and DiffractMOD by RSoft Design Group.
Several authors have described the exact size and shape of a theoretically optimal broadband antireflective coating [
There are many different mathematical models for treating the behavior of electromagnetic radiation through subwavelength antireflective structures. These methods include a numerical timebased approach called the finitedifference timedomain (FDTD) method, numerical frequencybased methods of the transfer matrix method (TMM, models thin films only), rigorous coupled wave analysis (also called Fourier modal method RCWA/FMM), coordinate transfer method (Cmethod), and finite element method (FEM), as well as exact approaches such as Knop’s or Sheng’s handling of 2D square grooves [
Unlike the other modeling methods reviewed in this paper, effective medium theory (EMT) is not a method for directly determining reflectance or transmittance of an ARSWS. Instead, this is a method that determines the effective index of refraction of a subwavelength structured geometry based on the volume fill factors of the multiple materials (see
Schematic of a graded index subwavelength structure (
EMT is used to predict the effective refractive index, and, in some cases unrelated to the subject at hand, the effective conductivity of a material. In 1956 Rytov derived an EMT solution for one dimensional periodic lamellar structures composed of two materials [
The effective index of refraction of a subwavelength mixture of materials falls between the upper and lower indices set by the bulk values for the constituent materials [see Equations (2) and (3) as well as
TEM images of moth eye nipple arrays (
The most common effective medium approximations (EMAs) are the Bruggeman’s model, the MaxwellGarnett Equation, and the LorentzLorentz model (See
Common effective medium approximations (EMA) methods.
Method  Model  Notes 

MaxwellGarnett [ 
Original model for effective index of refraction (RI), assumes homogenous mixture of low volume fraction of spherical subwavelength structures (SWS) for material 2  
Bruggeman [ 
Describes effective RI for any number, 

LorentzLorentz [ 
Can be extended to more than two constituents by adding more terms 
Lalanne and LemercierLalanne derived rigorous semianalytical effective medium approximations for normal incidence waves on one and twodimensional periodic structures [
Forberich
Simulation with effective medium theory (EMT) and measurement results of short circuit current from organic solar cells with no ARC, with a moth eye ARC, and with an idea GRIN structure ARC. Reprinted with permission from reference [
Brunner
TE and TM polarization hitting a 1D grating. The wavevector is indicated as
There are several commonly reported simple mathematical gradient index profiles found in the literature. Of these, the quintic and exponential sine profiles [Equations (6) and (7)] are considered to have the most ideal broadband antireflective properties [
Quintic index profile:
Exponential sine profile:
Other common GRIN profiles explored in the literature include the linear [Equation (8)] and cubic [Equation (9)]:
Linear index profile:
Cubic index profile:
SuperGaussian:
Xi
Reflectivity of several GRIN structures over a range of heights, wavelengths, and angles of incidence. Reprinted with permission from reference [
The finitedifference timedomain (FDTD) numerical modeling method is considered to be one of the most accurate and simple rigorous methods to model antireflective properties of subwavelength structures. Though it is computationally intensive, the FDTD method handles any arbitrarily shaped structure naturally using an explicit numerical solution to Maxwell’s curl equations.
The FDTD method was first introduced in 1966 by Yee and was furthered by Taflove [
Yee cell. Arrows indicate the direction of the
Prior to the turn of the millennium, lack of computing resources was a limiting factor to analyzing ARSWS using the FDTD method. Yamauchi
The FDTD method is derived from Faraday’s and Ampere’s laws [Equations (11) and (12)] as well as the relationships between the electric field (
Equations (17)–(22) can be discretized using the central difference approximation to produce six algebraic equations that describe the behavior of EM waves in three dimensions. These equations are solvable for the electric and magnetic fields in each dimension:
Appropriate boundary conditions must be applied to the computational boundaries to avoid artificial reflections within the domain. Implementing boundary conditions in FDTD for ARSWS analysis is normally done one of two ways: an absorbing boundary condition or a periodic boundary condition. Periodic boundary conditions are simple and allow for modeling of an infinitely large array of ARSWS features. Absorbing boundary conditions (ABCs), which are necessary for all nonperiodic boundaries, function to attenuate the EM signals at the interface. The most commonly used ABC is the perfectly matched layer (PML), which was introduced by Berenger in 1994 [
Introducing a plane wave into an FDTD simulation is done by setting the
Timebased
In FDTD materials are defined only by their permittivity and permeability properties. Most ARSWS FDTD studies are based on dielectric materials, thus, the relative permeability is one and absorptive losses can be ignored. This is especially true when analyzing dielectric materials that are optically thin. Most studies also ignore dispersion, or the effect of wavelength on the permittivity, as permittivity of the commonly studied materials is relatively constant over the AM1.5 range. FDTD does not handle dispersion naturally, as multiple wavelengths are input simultaneously, but modern computing programs have been improved to include dispersion.
Programming FDTD simulations manually is feasible and is fully supported by Taflove’s text on the subject [
Whether using inhouse or commercial software, several guidelines must be taken into account when setting up an FDTD simulation. To be accurate, the software requires at least ten computational cells per wavelength. The time step is usually chosen so that there are at least 20 timepoints per wavelength. There must also be at least three calculation points across any feature that one is expecting to model; failure to comply with this requirement often results in the improper modeling of the tips of pointy nanostructures [
Several groups have used the FDTD method as an accurate method to finetune the designs of ideal antireflective interfaces. FDTD has been used in conjunction with the transfer matrix method (TMM); Feng
Reflectivity calculated from the finitedifference timedomain (FDTD) and transfer matrix method (TMM) methods compared to experimental data shows that the FDTD method is more accurate than the TMM method. Reprinted with permission from reference [
Several groups used both FDTD and TMM, but refer to the TMM simulations as “effective medium theory” [
Reflectivity results from gradedindex films with an integral RI profile (
Comparisons between reflectivity calculated by EMT, FDTD, and experimental results for GRIN structures. Reprinted with permission from reference [
While effective medium theory requires features to be much smaller than the wavelengths of interest, geometrical optics or ray tracing is often used when feature sizes are much larger than the wavelengths [
Chen
As technology improvements have made 3D FDTD modeling faster and easier, more authors are performing several FDTD simulations to sweep across a range of feature properties in an attempt to design an optimal AR structure (Table S2). Over primarily the last decade researchers have used FDTD to model thin films [
The FDTD method has been verified by experimental results by several authors (Table S3). Deinega
FDTD has been used in literature to obtain reflectivity information about ARSWSs at nonzero angles of incidence. Deniz
SEM image of square packed silicon cones (
Angle of incidence FDTD simulations for nanorod arrays in TE and TM polarization [
The FDTD method has been shown repeatedly to be a versatile, simple, and accurate modeling method for 2D and 3D modeling of antireflective subwavelength structures. This method can be accurate over any wavelength and feature size combination and can be used with any structure, regular or irregular. The FDTD method does not naturally handle dispersion, though refractive indexes in the visible wavelengths are relatively constant, and thus the modeling results from FDTD have been shown to match experimental results satisfactorily well. This can be overcome by inputting one wavelength per simulation and assigning the appropriate wavelengthdependent optical properties to the material. Computational resources have, in the past, limited the utility of FDTD modeling, though with the introduction of newer computing technologies these limitations are becoming fewer.
FDTD has been found to have convergence problems when attempting to calculate dispersion, with some metal components [
The transfer matrix method (TMM) is a simple approach to modeling waves passing through layered media. Appropriate only for thin film ARC modeling, this method employs continuity boundary conditions between layers of material and wave equations to describe the electric fields or reflectance and transmittance values across each layer. Continuity requires that the fields at the interface between two materials be the same in each material. Then, if the electric field is known at the beginning of the layer a transfer matrix based on the wave equation can be used to determine the electric field at the other end of the layer (see Section 6.2.1 in reference [
Below is a summary of the mathematical derivation for TMM found in Condon and Odishaw’s Handbook of Physics, 2nd edition [
Equation (25) contains a 2 × 2 matrix called the transfer matrix; this equation calculates the input admittance
The boundary conditions applied are:
And overall yield for reflection is:
While overall transmission is:
In 1950 Abelès described a simple and fast method to determine the reflectivity from a thin film layered interface [
The transfer matrix method has been used in conjunction with the finitedifference timedomain method to converge on an optimal multilayered thin film ARC. Feng
Kuo
Geometry of the transfer matrix method. Each layer left to right represents a separate, homogeneous material.
SEM image of a seven layered thin film ARC produced and modeled by Kuo
In the application of measuring reflectivity for thin film ARCs the transfer matrix method is a fast and simple modeling method. TMM is capable of calculating reflectance and transmittance, can handle multiple wavelengths, dispersion, and multiple angles of incidence. This method can also handle absorption via complex refractive indexes. There are several commercially available software programs that utilize the TMM method, including FreeSnell, EMPy, Luxpop.com, and Thinfilm.
The rigorous coupled wave analysis has been described by several authors [
“The rigorous coupledwave analysis for grating diffraction was first applied to planar (volume) gratings [
Rigorous coupledwave analysis has also been applied to surfacerelief gratings [
The rigorous coupled wave analysis (RCWA) is a frequencybased, semianalytical optical simulation method that calculates the efficiencies of transmitted and reflected diffracted orders. Working in the subwavelength domain, generally only the zeroth order of diffracted waves must be considered. This method functions similarly to the transfer matrix method, except that it incorporates lateral periodic nonuniformities of material properties in the plane of the interface [
RCWA is also known as the Fourier Modal Method (FMM) [
The RCWA method begins by discretizing the model geometry into a superstrate, a grating region that might include many stairstepapproximated layers to implement the grating geometry, and a substrate. An example of a stairstepapproximated geometry can be found in
Stair step approximation of a periodic geometry as drawn in the commercially available rigorous coupledwave analysis (RCWA) software, GDCalc. Reprinted with permission from reference [
The RCWA method, its accuracy, and example calculations are provided by Hench and Strakos [
For a mathematical example of RCWA consider the early paper from Moharam and Gaylord that describes the 2D diffraction properties of a 1D grating with an arbitrarilyoriented sinusoidal permittivity function that is below a superstrate and above a substrate [
Geometry for planargrating diffraction. Region 1 (
Inside the diffraction grating one must consider all forward and backward diffracted waves from both bounding interfaces, which can be described by:
The permittivity of the grating region is described by an equation, either a Fourier expansion that describes the permittivity of one layer in a staircase approximation of a more complicated geometry, or, as in our example here, a simple equation that spatially describes the permittivity in the grating layer. Our grating follows the equation:
The amplitudes of the superimposed infinite sum of waves for each diffraction order are determined after considering the modulatedregion wave equation (Eigen function):
Once the permittivity and fields are appropriately described the Eigen function is solved using scattering matrices, similar to the transfer matrix used in the transfer matrix method, for the field strength of each diffracted wave.
RCWA is a highly accurate method for determining the reflectivity of a periodic grating. Numerical error comes from how well the grating can be described by a stairstep approximation, how many Fourier terms and diffraction orders are retained, and roundoff errors from the numerical calculations. Due to the Fourier and Flouquet expansions used in the derivation of this method, it is necessary to truncate the expansions appropriately in order to enable fast and efficient solution with a computer. The amount of truncation chosen is determined by the accuracy required and the computing resources available [
RCWA has been used to model a large variety of periodic antireflective structures in three dimensions (which are considered two dimensional gratings). Most of the structures modeled with this method are hexagonally or square packed, GRIN, motheyelike nipple arrays [
Any periodic structure can be modeled using RCWA, provided the geometry can be accurately described by a stack of slices that are homogeneous in the
Several motheyelike structures with different refraction index profiles and their associated reflectance
One of the requirements for the RCWA method is that the geometry of the model be periodic or unchanging in the
Klopfenstein structures have much better AR properties at shorter heights than do pyramids. Klopfenstein structures are shown on the (
(
Several alterations to the RCWA method have been made to improve the versatility of the method. When modeling onedimensional gratings using RCWA there are significant differences between TE and TM polarized light (see
RCWA simulations accurately predict experimental transmission results by modeling the SWS as a superGaussian profile with one dimensional height variations with a standard deviation of 15%. Reprinted with permission from reference [
One limitation of the RCWA method is that it does a poor job of modeling structures with very shallow slopes. This is because it requires significantly more layers to accurately describe a shallow slope than a steep slope. To overcome this problem the coordinate transfer method, also called the Chandezon method or Cmethod, can be used [
As one of the rigorous optical modeling methods, RCWA is known to be very accurate when applied appropriately, with sufficient discretization of layers, retention of Fourier terms and calculation of diffracted orders.
RCWA results have been compared to many other methods in the literature, including ray tracing [
Experimental and simulated results from flat (black) and etched nipple array (red) show good agreement for the RCWA method. Blue line is an experimental value from a commercial cSi solar cell. Reprinted with permission from reference [
The thin film multilayer model (TMM) and RCWA show very similar results for moth eye structures with 210 nm bases and 800 nm heights. Reprinted with permission from reference [
The finite element method (FEM) is a frequencybased optical modeling method that, like the finitedifference timedomain method, solves for the electric and magnetic field strengths throughout the spatial computational domain. The domain is divided into finite elements, or tetrahedral meshes, and the field strengths are calculated at the vertices of the mesh. The electric and magnetic fields are represented by timeharmonic complex vectors, with the time dependency described as exp(−
Maxwell’s equations in the harmonic regime are:
The FEM sets up these equations in matrix form as:
Many of the advantages of FEM come from the method’s meshing procedures. Meshing is the most important determinator of the accuracy of an FEM model [
FEM has been used to model 2D and 3D random triangular gratings [
Diagram of FEM simulation, including meshing scheme. Figure Reprinted with permission from reference [
The finite element method is effectively based on the transfer matrix method, except that the geometry of FEM is discretized to sufficiently describe the shape and size of subwavelength structures. This eliminates the need for effective media approximations, such as are used in the transfer matrix method. Lee
FEM has been used only sparingly in the literature to model the optics of antireflective coatings. Most commonly the method has been used to model the optics of thin film solar cells [
Comparison of reflectance calculations from FEM and TMM. Reprinted with permission from reference [
This article reviews the current popular modeling methods and their results for the antireflective properties of subwavelength structures. Characteristics of FDTD, FEM, TMM, and FMM/RCWA are summarized in
In general, these optical modeling techniques can be described by their spatial discretization techniques and their time or frequencybased treatment of Maxwell’s equations. Spatially discretized methods (FDTD and FEM) produce field strength results for each discretized point and naturally handle arbitrary geometries. Other methods provide only reflected or transmitted efficiency, though RCWA/FMM provides the efficiency of any diffracted order of interest. Timebased methods (FDTD) are capable of inputting a range of wavelengths into one simulation. Frequencybased approaches must be solved for each wavelength, which gives them the advantage of handling dispersion naturally. TMM, though intended for thin film simulations only, can be used in conjunction with effective medium approximations to model the optics of subwavelength structures. The accuracy of TMM suffers, though, when structures approach wavelength sizes.
Summary of features for the four main modeling methods for ARSWS.
Features  FDTD  FEM  TMM  FMM/RCWA 

Geometry Restrictions  None  None  Thin Films Only  Not efficient for aperiodic surfaces 
Time or Frequency Based  Time  Frequency  Frequency  Frequency 
Output  Field Strengths  Field Strengths  %R/%T  %R/%T 
Spatially discretized  Yes  Yes  No  No 
Models dispersion naturally  No  Yes  Yes  Yes 
Multiple wavelengths per simulation  Yes  No  No  No 
Rigorous  Yes  Yes  Yes  Yes 
Anisotropic gratings  Yes  Yes  No  Yes 
Computation Speed  Slow  Meshing slow, computation fast  Fast  Medium 
Source of Inaccuracies  Discretization of geometry and rounding error  Discretization of geometry and rounding error  EMT or slicing of geometry into layers  Truncation of Fourier series expansions for field values, permittivity and truncation of orders of diffracted light 
Numerical convergence  Difficult for some metals, dispersion, and wavelengthsized features  Good  Good  Difficult for TM polarization 
Maximum Dimensions  3D  3D  1D  3D 
Commercial modeling software is available for each of these modeling methods. Most modern software enables these methods to handle simulations for which the method is not normally appropriate, such as aperiodic structures (RCWA), dispersion (FDTD), and multiple wavelengths (frequencybased methods). For the most accurate optical simulation of antireflective subwavelength structures it is best to use a variety of modeling methods to account for the disadvantages of each.
Financial support from National Science Foundation STTR Phase II grant with CSD Nano Inc., Oregon BEST, the Microproducts Breakthrough Institute, Oregon State University College of Engineering, and the Rickert Engineering Fellowship. Special thanks to John Loeser, HaiYue Han, Lance Roy, Alan Wang, and Nick Wannenmacher for their essential discussions and patience.
The authors declare no conflict of interest.