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A fast Fourier transform (FFT) is an algorithm that computes the Discrete Fourier Transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse factors. As a result, it manages to reduce the complexity of computing the DFT from , which arises if one simply applies the definition of DFT, to , where n is the data size. The difference in speed can be enormous, especially for long data sets where n may be in the thousands or millions. In the presence of round-off error, many FFT algorithms are much more accurate than evaluating the DFT definition directly or indirectly. There are many different FFT algorithms based on a wide range of published theories, from simple complex-number arithmetic to group theory and number theory.
A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. The DCT, first proposed by Nasir Ahmed in 1972, is a widely used transformation technique in signal processing and data compression. It is used in most digital media, including digital images, digital video, digital audio, digital television, digital radio, and speech coding. DCTs are also important to numerous other applications in science and engineering, such as digital signal processing, telecommunication devices, reducing network bandwidth usage, and spectral methods for the numerical solution of partial differential equations.
The Cooley–Tukey algorithm, named after J. W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size in terms of N1 smaller DFTs of sizes N2, recursively, to reduce the computation time to O(N log N) for highly composite N (smooth numbers). Because of the algorithm's importance, specific variants and implementation styles have become known by their own names, as described below.
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.
Photonic-crystal fiber (PCF) is a class of optical fiber based on the properties of photonic crystals. It was first explored in 1996 at University of Bath, UK. Because of its ability to confine light in hollow cores or with confinement characteristics not possible in conventional optical fiber, PCF is now finding applications in fiber-optic communications, fiber lasers, nonlinear devices, high-power transmission, highly sensitive gas sensors, and other areas. More specific categories of PCF include photonic-bandgap fiber, holey fiber, hole-assisted fiber, and Bragg fiber. Photonic crystal fibers may be considered a subgroup of a more general class of microstructured optical fibers, where light is guided by structural modifications, and not only by refractive index differences. Hollow-core fibers are a related type of optical fiber which bears some resemblance to holey optical fiber.
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.
Optical computing or photonic computing uses light waves produced by lasers or incoherent sources for data processing, data storage or data communication for computing. For decades, photons have shown promise to enable a higher bandwidth than the electrons used in conventional computers.
The Fastest Fourier Transform in the West (FFTW) is a software library for computing discrete Fourier transforms (DFTs) developed by Matteo Frigo and Steven G. Johnson at the Massachusetts Institute of Technology.
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.
A perfectly matched layer (PML) is an artificial absorbing layer for wave equations, commonly used to truncate computational regions in numerical methods to simulate problems with open boundaries, especially in the FDTD and FE methods. The key property of a PML that distinguishes it from an ordinary absorbing material is that it is designed so that waves incident upon the PML from a non-PML medium do not reflect at the interface—this property allows the PML to strongly absorb outgoing waves from the interior of a computational region without reflecting them back into the interior.
Allen Taflove was a full professor in the Department of Electrical and Computer Engineering of Northwestern's McCormick School of Engineering, since 1988. Since 1972, he pioneered basic theoretical approaches, numerical algorithms, and applications of finite-difference time-domain (FDTD) computational solutions of Maxwell's equations. He coined the descriptors "finite difference time domain" and "FDTD" in the 1980 paper, "Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic penetration problems." In 1990, he was the first person to be named a Fellow of the Institute of Electrical and Electronics Engineers (IEEE) in the FDTD area. Taflove was the recipient of the 2014 IEEE Electromagnetics Award with the following citation: "For contributions to the development and application of finite-difference time-domain (FDTD) solutions of Maxwell's equations across the electromagnetic spectrum." He was a Life Fellow of the IEEE and a Fellow of the Optical Society (OSA). His OSA Fellow citation reads: "For creating the finite-difference time-domain method for the numerical solution of Maxwell's equations, with crucial application to the growth and current state of the field of photonics."
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.
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.
The James H. Wilkinson Prize for Numerical Software is awarded every four years to honor outstanding contributions in the field of numerical software. The award is named to commemorate the outstanding contributions of James H. Wilkinson in the same field.
Coupled mode theory (CMT) is a perturbational approach for analyzing the coupling of vibrational systems in space or in time. Coupled mode theory allows a wide range of devices and systems to be modeled as one or more coupled resonators. In optics, such systems include laser cavities, photonic crystal slabs, metamaterials, and ring resonators.
John D. Joannopoulos is an American physicist, focused in condensed matter theory. He is currently the Francis Wright Davis Professor of Physics at Massachusetts Institute of Technology, an Elected Member of the National Academy of Sciences (NAS), an Elected Member of the American Academy of Arts and Sciences (AAA&S), and an Elected Fellow of the American Association for the Advancement of Science (AAAS) and American Physical Society (APS).
Shanhui Fan is a Chinese-born American electrical engineer and physicist, with a focus on theoretical, computational and numerical aspects of photonics and electromagnetism. He is a professor of electrical engineering, and a professor of applied physics at Stanford University. He is the director of the Edward L. Ginzton Lab and Senior Fellow at the Precourt Institute for Energy.
The Bailey's FFT is a high-performance algorithm for computing the fast Fourier transform (FFT). This variation of the Cooley–Tukey FFT algorithm was originally designed for systems with hierarchical memory common in modern computers. The algorithm treats the samples as a two dimensional matrix and executes short FFT operations on the columns and rows of the matrix, with a correction multiplication by "twiddle factors" in between.
Shawn Yu Lin is a Taiwanese American physicist, researcher, and educator who made pioneering contributions to the field of photonics and photonic crystals. He authored more than 250 technical papers.
Meep is an free and open-source software package for electromagnetic simulations, developed by ab initio research group at Massachusetts Institute of Technology in 2006. Operating under Unix-like systems, it uses finite-difference time-domain method with perfectly matched layer or periodic boundary conditions for field computation.
References
- ↑ "Steven Johnson | MIT Mathematics". math.mit.edu. Retrieved 27 February 2020.
- ↑ "Johnson, Steven G., 1973-". viaf.org. Retrieved 27 February 2020.
- 1 2 Frigo M, Johnson SG (February 2005). "The design and implementation of FFTW3" (PDF). Proceedings of the IEEE. 93 (2): 216–231. CiteSeerX 10.1.1.66.3097 . doi:10.1109/JPROC.2004.840301. S2CID 6644892.
- ↑ Frigo M, Johnson SG (1998). "FFTW: An adaptive software architecture for the FFT". Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181). Vol. 3. pp. 1381–1384. CiteSeerX 10.1.1.47.8661 . doi:10.1109/ICASSP.1998.681704. ISBN 978-0-7803-4428-0. S2CID 12560207.
- ↑ Johnson SG, Frigo M (September 2008). "ch.11: Implementing FFTs in practice". In C. S. Burrus (ed.). Fast Fourier Transforms. Houston TX: Connexions: Rice University.
- ↑ "Steven Johnson | MIT Mathematics". math.mit.edu. Retrieved 27 February 2020.
- ↑ Frigo M, Johnson SG (September 11, 1997). "The Fastest Fourier Transform in the West" (PDF). MIT Labroratory for Computer Science.
- ↑ "THE WILKINSON PRIZE FOR NUMERICAL SOFTWARE". Numerical Algorithms Group. Retrieved 22 November 2017.
- ↑ SIAM. "James H. Wilkinson Prize for Numerical Software". Society for Industrial and Applied Mathematics. Retrieved 22 November 2017.
- ↑ Steven G. Johnson, The NLopt nonlinear-optimization package, https://nlopt.readthedocs.io/en/latest/
- ↑ Oskooi, Ardavan F.; Roundy, David; Ibanescu, Mihai; Bermel, Peter; Joannopoulos, J.D.; Johnson, Steven G. (March 2010). "Meep: A flexible free-software package for electromagnetic simulations by the FDTD method". Computer Physics Communications . 181 (3): 687–702. doi:10.1016/j.cpc.2009.11.008. hdl: 1721.1/60946 .
- ↑ Johnson, Steven G.; Joannopoulos, J. D. (2001). "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis". Optics Express . 8 (3): 173–190. doi:10.1364/OE.8.000173.
- ↑ Herriman, Jane (29 March 2019). "Steven Johnson as a JuliaCon 2019 keynote speaker!". Julia Discourse. Retrieved 29 March 2019.
