View source for Rigorous coupled-wave analysis

{{Short description|Semi-analytic method of computational electromagnetism}}
[[File:Analysis of a plasmonic grating with RCWA method.jpg|thumb|Analysis of [[Physics:Plane wave|plane wave]] scattering from a subwavelength plasmonic grating with RCWA method]]
'''Rigorous coupled-wave analysis (RCWA)''', also known as '''Fourier modal method (FMM)''', is a semi-analytical method in [[Physics:Computational electromagnetics|computational electromagnetics]] that is most typically applied to solve scattering from periodic dielectric structures.  It is a Fourier-space method so devices and fields are represented as a sum of spatial harmonics.  

==Floquet's theorem==
The method is based on Floquet's theorem that the solutions of periodic differential equations can be expanded with Floquet functions (or sometimes referred as a [[Bloch wave]], especially in the [[Physics:Solid-state physics|solid-state physics]] community).  A device is divided into layers that are each uniform in the z direction.  A staircase approximation is needed for curved devices with properties such as dielectric permittivity graded along the z-direction.  The electromagnetic modes in each layer are calculated and analytically propagated through the layers.  The overall problem is solved by matching boundary conditions at each of the interfaces between the layers using a technique like scattering matrices.  To solve for the electromagnetic modes, which are decided by the wave vector of the incident plane wave, in periodic dielectric medium, [[Physics:Maxwell's equations|Maxwell's equations]] (in partial differential form) as well as the boundary conditions are expanded by the Floquet functions and turned into infinitely large algebraic equations. With the cutting off of higher order Floquet functions, depending on the accuracy and convergence speed one needs, the infinitely large algebraic equations become finite and thus solvable by computers.

==Fourier factorization==
Being a Fourier-space method it suffers several drawbacks.  [[Gibbs phenomenon]] is particularly severe for devices with high dielectric contrast.  Truncating the number of spatial harmonics can also slow convergence and techniques like fast Fourier factorization (FFF) should be used.  FFF<ref>{{Cite journal|last=Popov|first=Evgeny|title=Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media|journal=Journal of the Optical Society of America A|volume=18|issue=11|pages=2886–94|doi=10.1364/JOSAA.18.002886|pmid=11688878|bibcode=2001JOSAA..18.2886P|year=2001}}</ref> is straightforward to implement for 1D gratings, but the community is still working on a straightforward approach for crossed grating devices.  The difficulty with FFF in crossed grating devices is that the field must be decomposed into parallel and perpendicular components at all of the interfaces.  This is not a straightforward calculation for arbitrarily shaped devices.

==Boundary conditions==
Boundary conditions must be enforced at the interfaces between all the layers. When many layers are used, this becomes too large to solve simultaneously.  Instead, RCWA borrows from [[Network theory|network theory]] and calculates scattering matrices.  This allows the boundary conditions to be solved one layer at a time.  Almost without exception, however, the scattering matrices implemented for RCWA are inefficient and do not follow long standing conventions in terms of how S11, S12, S21, and S22 are defined.<ref>{{Cite journal |last=Rumpf |first=Raymond C. |title=Improved Formulation of Scattering Matrices for Semi-Analytical Methods That is Consistent with Convention |date=2011 |url=http://www.jpier.org/PIERB/pier.php?paper=11083107 |journal=Progress in Electromagnetics Research B |language=en |volume=35 |pages=241–261 |doi=10.2528/PIERB11083107 |issn=1937-6472|doi-access=free }}</ref>  Other methods exist, like the enhanced transmittance matrices (ETM), R matrices, and H matrices. ETM, for example, is considerably faster but less memory efficient.{{Cn|date=June 2023}}

RCWA can applied to aperiodic structures with appropriate use of [[Physics:Perfectly matched layer|perfectly matched layer]]s.<ref>{{cite journal |last1=Hugonin |first1=Jean Paul |last2=Lalanne |first2=Philippe |title=Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization |journal=Journal of the Optical Society of America A |date=2005 |volume=22 |issue=9 |pages=1844-1849 |doi=10.1364/JOSAA.22.001844}}</ref>

== Applications==
RCWA analysis applied to a polarized broadband reflectometry measurement is used within the [[Engineering:Power semiconductor device|semiconductor power device]] industry as a measurement technique to obtain detailed profile information of periodic trench structures. This technique has been used to provide trench depth and critical dimension (CD) results comparable to cross-section SEM, while having the added benefit of being both high-throughput and non-destructive.{{Cn|date=June 2023}}

In order to extract critical dimensions of a trench structure (depth, CD, and sidewall angle), the measured polarized reflectance data must have a sufficiently large wavelength range and analyzed with a physically valid model (for example: RCWA in combination with the [[Physics:Forouhi–Bloomer model|Forouhi-Bloomer Dispersion relations for ''n'' and ''k'']]). Studies have shown that the limited wavelength range of a standard reflectometer (375 - 750&nbsp;nm) does not provide the sensitivity to accurately measure trench structures with small CD values (less than 200&nbsp;nm). However, by using a reflectometer with the wavelength range extended from 190 - 1000&nbsp;nm, it is possible to accurately measure these smaller structures.<ref>{{cite journal|last1=Heider|first1=F.|last2=Roberts|first2=J.|last3=Huang|first3=J.|last4=Lam|first4=J.|last5=Forouhi|first5=A.R.|title=Effects of measured spectral range on accuracy and repeatability of OCD analysis|journal=Solid State Technology|date=July 2013|volume=56|issue=5|page=21}}</ref>

RCWA is also used to improve diffractive structures for high efficiency [[Physics:Solar cell|solar cells]]. For the simulation of the whole solar cell or [[Physics:Solar panel|solar module]], RCWA can be efficiently combined with the [[Physics:OPTOS formalism|OPTOS formalism]].{{Cn|date=June 2023}}

== References ==
{{Reflist}}
*{{cite journal |doi=10.1364/JOSA.71.000811|title=Rigorous coupled-wave analysis of planar-grating diffraction|journal=Journal of the Optical Society of America|volume=71|issue=7|pages=811|year=1981|last1=Moharam|first1=M. G.|last2=Gaylord|first2=T. K.|bibcode=1981JOSA...71..811M}}
*See Chapter 6 in [http://purl.fcla.edu/fcla/etd/CFE0001159 Design and Optimization of Nano-Optical Elements by Coupling Fabrication to Optical Behavior]

== External links ==
* [http://photonics.intec.ugent.be/research/facilities/design/rodis/default.htm RODIS]
* [https://github.com/lbolla/EMpy EMpy]
* [https://github.com/facebookresearch/fmmax FMMAX]
* [http://mrcwa.sourceforge.net/ MRCWA]
* [http://www.stanford.edu/group/fan/S4/ S4]
* [http://www.unigit.net/ Unigit] (RCWA, Rayleigh–Fourier & C-method)
* [http://rawdogapp.weebly.com/ RawDog]

[[Category:Mathematical physics]]
[[Category:Computational electromagnetics]]
[[Category:Fourier analysis]]
[[Category:Holography]]

{{Sourceattribution|Rigorous coupled-wave analysis}}