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Rigorous coupled-wave analysis (RCWA), also known as Fourier modal method (FMM), [1] is a semi-analytical method in 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.
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 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, Maxwell's equations (in partial differential form) as well as the boundary conditions are expanded by the Floquet functions in Fourier space. This technique transforms the partial differential equation into a matrix valued ordinary differential equation as a function of height over the periodic media. The finite representation of these Floquet functions in Fourier space renders the matrices finite, thus allowing the method to be feasibly solved by computers.
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 [2] 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.
Finally, because the RCWA method usually requires the convolution of discontinuous functions represented in Fourier Space, very careful consideration must given to the how these discontinuities are treated in the formulation of Maxwell's equations in Fourier space. [3] A key contribution was made by Lifeng Li in understanding the properties of convolving the Fourier transforms of two functions with coincident discontinuities that nevertheless form a continuous function. This led to a reformulation of RCWA with a significant improvement in convergence when using truncated Fourier series. [4]
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 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. [5] Other methods exist, like the enhanced transmittance matrices (ETM), R matrices, and H matrices. ETM, for example, is considerably faster but less memory efficient.[ citation needed ]
RCWA can applied to aperiodic structures with appropriate use of perfectly matched layers. [1]
RCWA analysis applied to a polarized broadband reflectometry measurement is used within the 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.[ citation needed ]
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 Forouhi-Bloomer Dispersion relations for n and k). Studies have shown that the limited wavelength range of a standard reflectometer (375 - 750 nm) does not provide the sensitivity to accurately measure trench structures with small CD values (less than 200 nm). However, by using a reflectometer with the wavelength range extended from 190 - 1000 nm, it is possible to accurately measure these smaller structures. [6]
RCWA is also used to improve diffractive structures for high efficiency solar cells. For the simulation of the whole solar cell or solar module, RCWA can be efficiently combined with the OPTOS formalism.[ citation needed ]
The GRE physics test is an examination administered by the Educational Testing Service (ETS). The test attempts to determine the extent of the examinees' understanding of fundamental principles of physics and their ability to apply them to problem solving. Many graduate schools require applicants to take the exam and base admission decisions in part on the results.
Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of reflected radiation from the angle predicted by the law of reflection. Reflections of radiation that undergo scattering are often called diffuse reflections and unscattered reflections are called specular (mirror-like) reflections. Originally, the term was confined to light scattering. As more "ray"-like phenomena were discovered, the idea of scattering was extended to them, so that William Herschel could refer to the scattering of "heat rays" in 1800. John Tyndall, a pioneer in light scattering research, noted the connection between light scattering and acoustic scattering in the 1870s. Near the end of the 19th century, the scattering of cathode rays and X-rays was observed and discussed. With the discovery of subatomic particles and the development of quantum theory in the 20th century, the sense of the term became broader as it was recognized that the same mathematical frameworks used in light scattering could be applied to many other phenomena.
A photonic crystal is an optical nanostructure in which the refractive index changes periodically. This affects the propagation of light in the same way that the structure of natural crystals gives rise to X-ray diffraction and that the atomic lattices of semiconductors affect their conductivity of electrons. Photonic crystals occur in nature in the form of structural coloration and animal reflectors, and, as artificially produced, promise to be useful in a range of applications.
In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time, as in time series. Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how the signal is distributed within different frequency bands over a range of frequencies. A complex valued frequency-domain representation consists of both the magnitude and the phase of a set of sinusoids at the frequency components of the signal. Although it is common to refer to the magnitude portion as the frequency response of a signal, the phase portion is required to uniquely define the signal.
Finite-difference time-domain (FDTD) or Yee's method is a numerical analysis technique used for modeling computational electrodynamics. Since it is a time-domain method, FDTD solutions can cover a wide frequency range with a single simulation run, and treat nonlinear material properties in a natural way.
A superlens, or super lens, is a lens which uses metamaterials to go beyond the diffraction limit. The diffraction limit is a feature of conventional lenses and microscopes that limits the fineness of their resolution depending on the illumination wavelength and the numerical aperture (NA) of the objective lens. Many lens designs have been proposed that go beyond the diffraction limit in some way, but constraints and obstacles face each of them.
Computational electromagnetics (CEM), computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment using computers.
An optical waveguide is a physical structure that guides electromagnetic waves in the optical spectrum. Common types of optical waveguides include optical fiber waveguides, transparent dielectric waveguides made of plastic and glass, liquid light guides, and liquid waveguides.
The finite-difference frequency-domain (FDFD) method is a numerical solution method for problems usually in electromagnetism and sometimes in acoustics, based on finite-difference approximations of the derivative operators in the differential equation being solved.
Discrete dipole approximation (DDA), also known as coupled dipole approximation, is a method for computing scattering of radiation by particles of arbitrary shape and by periodic structures. Given a target of arbitrary geometry, one seeks to calculate its scattering and absorption properties by an approximation of the continuum target by a finite array of small polarizable dipoles. This technique is used in a variety of applications including nanophotonics, radar scattering, aerosol physics and astrophysics.
Plane wave expansion method (PWE) refers to a computational technique in electromagnetics to solve the Maxwell's equations by formulating an eigenvalue problem out of the equation. This method is popular among the photonic crystal community as a method of solving for the band structure of specific photonic crystal geometries. PWE is traceable to the analytical formulations, and is useful in calculating modal solutions of Maxwell's equations over an inhomogeneous or periodic geometry. It is specifically tuned to solve problems in a time-harmonic forms, with non-dispersive media.
Eigenmode expansion (EME) is a computational electrodynamics modelling technique. It is also referred to as the mode matching technique or the bidirectional eigenmode propagation method. Eigenmode expansion is a linear frequency-domain method.
Surface plasmon polaritons (SPPs) are electromagnetic waves that travel along a metal–dielectric or metal–air interface, practically in the infrared or visible-frequency. The term "surface plasmon polariton" explains that the wave involves both charge motion in the metal and electromagnetic waves in the air or dielectric ("polariton").
A. R. Forouhi and I. Bloomer deduced dispersion equations for the refractive index, n, and extinction coefficient, k, which were published in 1986 and 1988. The 1986 publication relates to amorphous materials, while the 1988 publication relates to crystalline. Subsequently, in 1991, their work was included as a chapter in “The Handbook of Optical Constants”. The Forouhi–Bloomer dispersion equations describe how photons of varying energies interact with thin films. When used with a spectroscopic reflectometry tool, the Forouhi–Bloomer dispersion equations specify n and k for amorphous and crystalline materials as a function of photon energy E. Values of n and k as a function of photon energy, E, are referred to as the spectra of n and k, which can also be expressed as functions of the wavelength of light, λ, since E = HC/λ. The symbol h represents Planck’s constant and c, the speed of light in vacuum. Together, n and k are often referred to as the “optical constants” of a material.
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JCMsuite is a finite element analysis software package for the simulation and analysis of electromagnetic waves, elasticity and heat conduction. It also allows a mutual coupling between its optical, heat conduction and continuum mechanics solvers. The software is mainly applied for the analysis and optimization of nanooptical and microoptical systems. Its applications in research and development projects include dimensional metrology systems, photolithographic systems, photonic crystal fibers, VCSELs, Quantum-Dot emitters, light trapping in solar cells, and plasmonic systems. The design tasks can be embedded into the high-level scripting languages MATLAB and Python, enabling a scripting of design setups in order to define parameter dependent problems or to run parameter scans.
Weng Cho Chew is a Malaysian-American electrical engineer and applied physicist known for contributions to wave physics, especially computational electromagnetics. He is a Distinguished Professor of Electrical and Computer Engineering at Purdue University.
In electromagnetism, surface equivalence principle or surface equivalence theorem relates an arbitrary current distribution within an imaginary closed surface with an equivalent source on the surface. It is also known as field equivalence principle, Huygens' equivalence principle or simply as the equivalence principle. Being a more rigorous reformulation of the Huygens–Fresnel principle, it is often used to simplify the analysis of radiating structures such as antennas.
The method of moments (MoM), also known as the moment method and method of weighted residuals, is a numerical method in computational electromagnetics. It is used in computer programs that simulate the interaction of electromagnetic fields such as radio waves with matter, for example antenna simulation programs like NEC that calculate the radiation pattern of an antenna. Generally being a frequency-domain method, it involves the projection of an integral equation into a system of linear equations by the application of appropriate boundary conditions. This is done by using discrete meshes as in finite difference and finite element methods, often for the surface. The solutions are represented with the linear combination of pre-defined basis functions; generally, the coefficients of these basis functions are the sought unknowns. Green's functions and Galerkin method play a central role in the method of moments.
