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320:(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.
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385:). 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.
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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,
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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.
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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 is straightforward to implement for 1D gratings, but the community is still working on a
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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. A key contribution was made by
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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
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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
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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.
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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.
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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
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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.
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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
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Other methods exist, like the enhanced transmittance matrices (ETM), R matrices, and H matrices. ETM, for example, is considerably faster but less memory efficient.
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Heider, F.; Roberts, J.; Huang, J.; Lam, J.; Forouhi, A.R. (July 2013). "Effects of measured spectral range on accuracy and repeatability of OCD analysis".
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Popov, Evgeny (2001). "Maxwell equations in
Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media".
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Please help improve this article by looking for better, more reliable sources. Unreliable citations may be challenged and removed.
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that the solutions of periodic differential equations can be expanded with
Floquet functions (or sometimes referred as a
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599:"Improved Formulation of Scattering Matrices for Semi-Analytical Methods That is Consistent with Convention"
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Moharam, M. G.; Gaylord, T. K. (1981). "Rigorous coupled-wave analysis of planar-grating diffraction".
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Li, Lifeng (1996). "Use of
Fourier series in the analysis of discontinuous periodic structures".
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538:"The RCWA Method - A case study with open questions and perspectives of algebraic computations"
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Design and
Optimization of Nano-Optical Elements by Coupling Fabrication to Optical Behavior
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RCWA analysis applied to a polarized broadband reflectometry measurement is used within the
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RCWA is also used to improve diffractive structures for high efficiency
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RCWA can applied to aperiodic structures with appropriate use of
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Being a
Fourier-space method it suffers several drawbacks.
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Semi-analytic method of computational electromagnetism
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396:, RCWA can be efficiently combined with the
495:Journal of the Optical Society of America A
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53:Learn how and when to remove these messages
658:Journal of the Optical Society of America
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376:Forouhi-Bloomer Dispersion relations for
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225:Learn how and when to remove this message
163:Learn how and when to remove this message
110:Learn how and when to remove this message
73:This article includes a list of general
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603:Progress in Electromagnetics Research B
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536:Hench, John; Strakoš, Zdeněk (2008).
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597:Rumpf, Raymond C. (2011).
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609:: 241–261.
390:solar cells
140:may not be
92:introducing
759:Holography
738:Categories
664:(7): 811.
551:: 331–357.
417:References
309:Bloch wave
263:plane wave
199:improve it
75:references
39:improve it
625:1937-6472
267:plasmonic
203:verifying
45:talk page
646:(5): 21.
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472:16211811
404:See also
142:reliable
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572:Bibcode
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480:8001894
452:Bibcode
270:grating
197:Please
88:improve
729:RawDog
723:Unigit
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621:ISSN
519:PMID
468:PMID
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