Finite-difference time-domain method - Wikipedia, … Method Finite-difference time-domain method -...

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Finite-difference time-domain (FDTD) is a popular computational electrodynamics modeling technique. It isconsidered easy to understand and easy to implement in software. Since it is a time-domain method, solutionscan cover a wide frequency range with a single simulation run.

The FDTD method belongs in the general class of grid-based differential time-domain numerical modelingmethods. The time-dependent Maxwell's equations (in partial differential form) are discretized using central-difference approximations to the space and time partial derivatives. The resulting finite-difference equations aresolved in either software or hardware in a leapfrog manner: the electric field vector components in a volume ofspace are solved at a given instant in time; then the magnetic field vector components in the same spatialvolume are solved at the next instant in time; and the process is repeated over and over again until the desiredtransient or steady-state electromagnetic field behavior is fully evolved.

The basic FDTD space grid and time-stepping algorithm trace back to a seminal 1966 paper by Kane Yee in

IEEE Transactions on Antennas and Propagation.[1] The descriptor "Finite-difference time-domain" and itscorresponding "FDTD" acronym were originated by Allen Taflove in a 1980 paper in IEEE Transactions on

Electromagnetic Compatibility.[2]

Since about 1990, FDTD techniques have emerged as primary means to computationally model many scientificand engineering problems dealing with electromagnetic wave interactions with material structures. CurrentFDTD modeling applications range from near-DC (ultralow-frequency geophysics involving the entire Earth-ionosphere waveguide) through microwaves (radar signature technology, antennas, wireless communicationsdevices, digital interconnects, biomedical imaging/treatment) to visible light (photonic crystals, nanoplasmonics,

solitons, and biophotonics).[3] In 2006, an estimated 2,000 FDTD-related publications appeared in the scienceand engineering literature (see Popularity). At present (2008), there are at least 27 commercial/proprietaryFDTD software vendors; 8 free-software/ open-source-software FDTD projects; and 2 freeware/closed-sourceFDTD projects, some not for commercial use (see External links).

1 Workings of the FDTD method2 Using the FDTD method3 Strengths of FDTD modeling4 Weaknesses of FDTD modeling5 Grid truncation techniques for open-region FDTD modeling problems6 History of FDTD techniques and applications for Maxwell's equations7 Popularity8 See also9 References10 Further reading11 External links

Finite-difference time-domain method - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Finite-difference_time-domain_method

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Illustration of a standard Cartesian Yee cell usedfor FDTD, about which electric and magnetic

field vector components are distributed.[1]

Visualized as a cubic voxel, the electric fieldcomponents form the edges of the cube, and themagnetic field components form the normals tothe faces of the cube. A three-dimensional space

lattice consists of a multiplicity of such Yeecells. An electromagnetic wave interaction

structure is mapped into the space lattice byassigning appropriate values of permittivity to

each electric field component, and permeabilityto each magnetic field component.

When Maxwell's differential equations are examined, it can be seen that the change in the E-field in time (thetime derivative) is dependent on the change in the H-field across space (the curl). This results in the basic FDTDtime-stepping relation that, at any point in space, the updated value of the E-field in time is dependent on the

stored value of the E-field and the numerical curl of the local distribution of the H-field in space.[1]

The H-field is time-stepped in a similar manner. At any point in space, the updated value of the H-field in time isdependent on the stored value of the H-field and the numerical curl of the local distribution of the E-field inspace. Iterating the E-field and H-field updates results in a marching-in-time process wherein sampled-dataanalogs of the continuous electromagnetic waves under consideration propagate in a numerical grid stored in thecomputer memory.

This description holds true for 1-D, 2-D, and 3-D FDTDtechniques. When multiple dimensions are considered,calculating the numerical curl can become complicated.Kane Yee's seminal 1966 paper proposed spatiallystaggering the vector components of the E-field and H-fieldabout rectangular unit cells of a Cartesian computationalgrid so that each E-field vector component is locatedmidway between a pair of H-field vector components, and

conversely.[1] This scheme, now known as a Yee lattice,has proven to be very robust, and remains at the core ofmany current FDTD software constructs.

Furthermore, Yee proposed a leapfrog scheme for marchingin time wherein the E-field and H-field updates arestaggered so that E-field updates are conducted midwayduring each time-step between successive H-field updates,

and conversely.[1] On the plus side, this explicittime-stepping scheme avoids the need to solve simultaneousequations, and furthermore yields dissipation-free numericalwave propagation. On the minus side, this scheme mandatesan upper bound on the time-step to ensure numerical

stability.[4] As a result, certain classes of simulations canrequire many thousands of time-steps for completion.

In order to use FDTD a computational domain must be established. The computational domain is simply thephysical region over which the simulation will be performed. The E and H fields are determined at every point inspace within that computational domain. The material of each cell within the computational domain must bespecified. Typically, the material is either free-space (air), metal, or dielectric. Any material can be used as longas the permeability, permittivity, and conductivity are specified.

Once the computational domain and the grid materials are established, a source is specified. The source can bean impinging plane wave, a current on a wire, or an applied electric field, depending on the application.

Since the E and H fields are determined directly, the output of the simulation is usually the E or H field at apoint or a series of points within the computational domain. The simulation evolves the E and H fields forwardin time.


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Processing may be done on the E and H fields returned by the simulation. Data processing may also occur whilethe simulation is ongoing.

While the FDTD technique computes electromagnetic fields within a compact spatial region, scattered and/or

radiated far fields can be obtained via near-to-far-field transformations.[5]

Every modeling technique has strengths and weaknesses, and the FDTD method is no different.

FDTD is a versatile modeling technique used to solve Maxwell's equations. It is intuitive, so users can easilyunderstand how to use it and know what to expect from a given model.

FDTD is a time-domain technique, and when a broadband pulse (such as a Gaussian pulse) is used as the source,then the response of the system over a wide range of frequencies can be obtained with a single simulation. Thisis useful in applications where resonant frequencies are not exactly known, or anytime that a broadband result isdesired.

Since FDTD calculates the E and H fields everywhere in the computational domain as they evolve in time, itlends itself to providing animated displays of the electromagnetic field movement through the model. This typeof display is useful in understanding what is going on in the model, and to help ensure that the model is workingcorrectly.

The FDTD technique allows the user to specify the material at all points within the computational domain. Awide variety of linear and nonlinear dielectric and magnetic materials can be naturally and easily modeled.

FDTD allows the effects of apertures to be determined directly. Shielding effects can be found, and the fieldsboth inside and outside a structure can be found directly or indirectly.

FDTD uses the E and H fields directly. Since most EMI/EMC modeling applications are interested in the E andH fields, it is convenient that no conversions must be made after the simulation has run to get these values.

Since FDTD requires that the entire computational domain be gridded, and the grid spatial discretization must besufficiently fine to resolve both the smallest electromagnetic wavelength and the smallest geometrical feature inthe model, very large computational domains can be developed, which results in very long solution times.Models with long, thin features, (like wires) are difficult to model in FDTD because of the excessively largecomputational domain required.

FDTD finds the E/H fields directly everywhere in the computational domain. If the field values at some distanceare desired, it is likely that this distance will force the computational domain to be excessively large. Far-field

extensions are available for FDTD, but require some amount of postprocessing.[3]

Since FDTD simulations calculate the E and H fields at all points within the computational domain, thecomputational domain must be finite to permit its residence in the computer memory. In many cases this isachieved by inserting artificial boundaries into the simulation space. Care must be taken to minimize errorsintroduced by such boundaries. There are a number of available highly effective absorbing boundary conditions

(ABCs) to simulate an infinite unbounded computational domain.[3] Most modern FDTD implementations


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instead use a special absorbing "material", called a perfectly matched layer (PML) to implement absorbing

boundaries.[6][7]

Because FDTD is solved by propagating the fields forward in the time domain, the electromagnetic timeresponse of the medium must be modeled explicitly. For an arbitrary response, this involves a computationallyexpensive time convolution, although in most cases the time response of the medium (or Dispersion (optics)) canbe adequately and simply modeled using either the recursive convolution (RC) technique, the auxiliarydifferential equation (ADE) technique, or the Z-transform technique. An alternative way of solving Maxwell'sequations that can treat arbitrary dispersion easily is the Pseudospectral Spatial-Domain method (PSSD), whichinstead propagates the fields forward in space.

The most commonly used grid truncation techniques for open-region FDTD modeling problems are the Mur

absorbing boundary condition (ABC),[8] the Liao ABC,[9] and various perfectly matched layer (PML)

formulations.[6][7][3] The Mur and Liao techniques are simpler than PML. However, PML (which is technicallyan absorbing region rather than a boundary condition per se) can provide orders-of-magnitude lower reflections.The PML concept was introduced by J.-P. Berenger in a seminal 1994 paper in the Journal of Computational

Physics.[6] Since 1994, Berenger's original split-field implementation has been modified and extended to theuniaxial PML (UPML), the convolutional PML (CPML), and the higher-order PML. The latter two PMLformulations have increased ability to absorb evanescent waves, and therefore can in principle be placed closerto a simulated scattering or radiating structure than Berenger's original formulation.

An appreciation of the basis, technical development, and possible future of FDTD numerical techniques forMaxwell’s equations can be developed by first considering their history. The following lists some of the keypublications in this area, starting with Yee's seminal Paper #1 (1966), which has over 4,000 citations accordingto the ISI Web of Science.

Partial chronology of FDTD techniques and applications for Maxwell's equations.[10]

year event

1966Yee described the basis of the FDTD numerical technique for solving Maxwell’s curl

equations directly in the time domain on a space grid.[1]

1975

Taflove and Brodwin reported the correct numerical stability criterion for Yee’salgorithm; the first sinusoidal steady-state FDTD solutions of two- and three-

dimensional electromagnetic wave interactions with material structures;[4] and the first

bioelectromagnetics models.[11]

1977 Holland and Kunz & Lee applied Yee’s algorithm to EMP problems.[12][13]

1980Taflove coined the FDTD acronym and published the first validated FDTD models ofsinusoidal steady-state electromagnetic wave penetration into a three-dimensional metal

cavity.[2]


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1981Mur published the first numerically stable, second-order accurate, absorbing boundary

condition (ABC) for Yee’s grid.[8]

1982–83Taflove and Umashankar developed the first FDTD electromagnetic wave scatteringmodels computing sinusoidal steady-state near-fields, far-fields, and radar cross-section

for two- and three-dimensional structures.[5][14]

1984Liao et al reported an improved ABC based upon space-time extrapolation of the field

adjacent to the outer grid boundary.[9]

1985 Gwarek introduced the lumped equivalent circuit formulation of FDTD.[15]

1986 Choi and Hoefer published the first FDTD simulation of waveguide structures.[16]

1987–88Kriegsmann et al and Moore et al published the first articles on ABC theory in IEEE

Transactions on Antennas and Propagation.[17][18]

1987–88,1992

Contour-path subcell techniques were introduced by Umashankar et al to permit FDTD

modeling of thin wires and wire bundles,[19] by Taflove et al to model penetration

through cracks in conducting screens,[20] and by Jurgens et al to conformally model the

surface of a smoothly curved scatterer.[21]

1988Sullivan et al published the first 3-D FDTD model of sinusoidal steady-state

electromagnetic wave absorption by a complete human body.[22]

1988 FDTD modeling of microstrips was introduced by Zhang et al.[23]

1990–91FDTD modeling of frequency-dependent dielectric permittivity was introduced by

Kashiwa and Fukai,[24] Luebbers et al,[25] and Joseph et al.[26]

1990–91FDTD modeling of antennas was introduced by Maloney et al,[27] Katz et al,[28] and

Tirkas and Balanis.[29]

1990FDTD modeling of picosecond optoelectronic switches was introduced by Sano and

Shibata,[30] and El-Ghazaly et al.[31]

1992–94

FDTD modeling of the propagation of optical pulses in nonlinear dispersive media wasintroduced, including the first temporal solitons in one dimension by Goorjian and

Taflove;[32] beam self-focusing by Ziolkowski and Judkins;[33] the first temporal

solitons in two dimensions by Joseph et al;[34] and the first spatial solitons in two

dimensions by Joseph and Taflove.[35]

1992 FDTD modeling of lumped electronic circuit elements was introduced by Sui et al.[36]

1993Toland et al published the first FDTD models of gain devices (tunnel diodes and Gunn

diodes) exciting cavities and antennas.[37]

1994Thomas et al introduced a Norton’s equivalent circuit for the FDTD space lattice, whichpermits the SPICE circuit analysis tool to implement accurate subgrid models of

nonlinear electronic components or complete circuits embedded within the lattice.[38]

1994

Berenger introduced the highly effective, perfectly matched layer (PML) ABC for

two-dimensional FDTD grids,[6] which was extended to three dimensions by Katz et

al,[39] and to dispersive waveguide terminations by Reuter et al.[40]


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1995–96Sacks et al and Gedney introduced a physically realizable, uniaxial perfectly matched

layer (UPML) ABC.[41][7]

1997Liu introduced the pseudospectral time-domain (PSTD) method, which permits

extremely coarse spatial sampling of the electromagnetic field at the Nyquist limit.[42]

1997Ramahi introduced the complementary operators method (COM) to implement highly

effective analytical ABCs.[43]

1998Maloney and Kesler introduced several novel means to analyze periodic structures in

the FDTD space lattice.[44]

1998Nagra and York introduced a hybrid FDTD-quantum mechanics model ofelectromagnetic wave interactions with materials having electrons transitioning between

multiple energy levels.[45]

1998Hagness et al introduced FDTD modeling of the detection of breast cancer using

ultrawideband radar techniques.[46]

1999Schneider and Wagner introduced a comprehensive analysis of FDTD grid dispersion

based upon complex wavenumbers.[47]

2000–01Zheng, Chen, and Zhang introduced the first three-dimensional alternating-direction

implicit (ADI) FDTD algorithm with provable unconditional numerical stability.[48][49]

2000 Roden and Gedney introduced the advanced convolutional PML (CPML) ABC.[50]

2000Rylander and Bondeson introduced a provably stable FDTD - finite-element

time-domain hybrid technique.[51]

2002Hayakawa et al and Simpson and Taflove independently introduced FDTD modeling ofthe global Earth-ionosphere waveguide for extremely low-frequency geophysical

phenomena.[52][53]

2003 DeRaedt introduced the unconditionally stable, “one-step” FDTD technique.[54]


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Interest in FDTD Maxwell’s equationssolvers has increased nearly exponentiallyover the past 20 years. Increasingly,engineers and scientists in nontraditionalelectromagnetics-related areas such asphotonics and nanotechnology have becomeaware of the power of FDTD techniques. Asshown in the figure on the right, anestimated 2,000 FDTD-related publicationsappeared in the science and engineeringliterature in 2006, as opposed to fewer than10 as recently as 1985. The current rate ofgrowth (based upon a study of ISI Web ofScience data) is approximately 5:1 over theperiod 1995 to 2006.

Notwithstanding both the general increase inacademic publication throughput during the same period and the overall expansion of interest in all CEMtechniques, there are seven primary reasons for the tremendous expansion of interest in FDTD computationalsolution approaches for Maxwell’s equations:

FDTD uses no linear algebra. Being a fully explicit computation, FDTD avoids the difficulties with linearalgebra that limit the size of frequency-domain integral-equation and finite-element electromagneticsmodels to generally fewer than 1e7 electromagnetic field unknowns. FDTD models with as many as 1e9field unknowns have been run; there is no intrinsic upper bound to this number.

1.

FDTD is accurate and robust. The sources of error in FDTD calculations are well understood, and can bebounded to permit accurate models for a very large variety of electromagnetic wave interaction problems.

2.

FDTD treats impulsive behavior naturally. Being a time-domain technique, FDTD directly calculates theimpulse response of an electromagnetic system. Therefore, a single FDTD simulation can provide eitherultrawideband temporal waveforms or the sinusoidal steady-state response at any frequency within theexcitation spectrum.

3.

FDTD treats nonlinear behavior naturally. Being a time-domain technique, FDTD directly calculates thenonlinear response of an electromagnetic system. This allows natural hybriding of FDTD with sets ofauxiliary differential equations that describe nonlinearities from either the classical or semi-classicalstandpoint. An exciting research frontier here is the development of hybrid algorithms which join FDTDclassical electrodynamics models with phenomena arising from quantum electrodynamics, especiallyvacuum fluctuations.

4.

FDTD is a systematic approach. With FDTD, specifying a new structure to be modeled is reduced to aproblem of mesh generation rather than the potentially complex reformulation of an integral equation. Forexample, FDTD requires no calculation of structure-dependent Green functions.

5.

Parallel-processing computer architectures have come to dominate supercomputing. FDTD scales withhigh efficiency on parallel-processing CPU-based computers, and extremely well on recently developedGPU-based accelerator technology.

6.

Computer visualization capabilities are increasing rapidly. While this trend positively influences allnumerical techniques, it is of particular advantage to FDTD methods, which generate time-marched arraysof field quantities suitable for use in color videos to illustrate the field dynamics.

7.

These factors combine to indicate that FDTD will likely remain one of the dominant computationalelectrodynamics techniques; and indeed may emerge as the dominant technique for mid-21st-century problemsof surpassing volumetric complexity and/or multiphysics.


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Computational electromagneticsFinite-difference frequency-domainScattering Matrix Method

^ a b c d e f Kane Yee (1966). "Numerical solution of initial boundary value problems involving Maxwell'sequations in isotropic media" (http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1138693) (). IEEETransactions on Antennas and Propagation 14: 302–307. doi:10.1109/TAP.1966.1138693 (http://dx.doi.org/10.1109%2FTAP.1966.1138693) . http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1138693.

1.

^ a b A. Taflove (1980). "Application of the finite-difference time-domain method to sinusoidal steady stateelectromagnetic penetration problems" (http://www.ece.northwestern.edu/ecefaculty/taflove/Paper7.pdf) . IEEETransactions on Electromagnetic Compatibility 22: 191–202. doi:10.1109/TEMC.1980.303879 (http://dx.doi.org/10.1109%2FTEMC.1980.303879) . http://www.ece.northwestern.edu/ecefaculty/taflove/Paper7.pdf.

2.

^ a b c d Allen Taflove and Susan C. Hagness (2005). Computational Electrodynamics: The Finite-DifferenceTime-Domain Method, 3rd ed. (http://www.artechhouse.com/Detail.aspx?strBookId=1123) . Artech HousePublishers. ISBN 1-58053-832-0. http://www.artechhouse.com/Detail.aspx?strBookId=1123.

3.

^ a b A. Taflove and M. E. Brodwin (1975). "Numerical solution of steady-state electromagnetic scatteringproblems using the time-dependent Maxwell's equations" (http://www.ece.northwestern.edu/ecefaculty/taflove/Paper2.pdf) . IEEE Transactions on Microwave Theory and Techniques 23: 623–630.doi:10.1109/TMTT.1975.1128640 (http://dx.doi.org/10.1109%2FTMTT.1975.1128640) .http://www.ece.northwestern.edu/ecefaculty/taflove/Paper2.pdf.

4.

^ a b K. R. Umashankar and A. Taflove (1982). "A novel method to analyze electromagnetic scattering of complexobjects" (http://www.ece.northwestern.edu/ecefaculty/taflove/Paper9.pdf) . IEEE Transactions on ElectromagneticCompatibility 24: 397–405. doi:10.1109/TEMC.1982.304054 (http://dx.doi.org/10.1109%2FTEMC.1982.304054) .http://www.ece.northwestern.edu/ecefaculty/taflove/Paper9.pdf.

5.

^ a b c d J. Berenger (1994). "A perfectly matched layer for the absorption of electromagnetic waves"(http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHY-45P0TJR-1P&_coverDate=10%2F31%2F1994&_alid=375031649&_rdoc=1&_fmt=&_orig=search&_qd=1&_cdi=6863&_sort=d&view=c&_acct=C000051292&_version=1&_urlVersion=0&_userid=1072239&md5=9afa9291d22a44614de567dd9427c63d) . Journal of Computational Physics 114: 185–200.doi:10.1006/jcph.1994.1159 (http://dx.doi.org/10.1006%2Fjcph.1994.1159) . http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHY-45P0TJR-1P&_coverDate=10%2F31%2F1994&_alid=375031649&_rdoc=1&_fmt=&_orig=search&_qd=1&_cdi=6863&_sort=d&view=c&_acct=C000051292&_version=1&_urlVersion=0&_userid=1072239&md5=9afa9291d22a44614de567dd9427c63d.

6.

^ a b c S. D. Gedney (1996). "An anisotropic perfectly matched layer absorbing media for the truncation of FDTDlattices" (http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=11941&arnumber=546249&count=18&index=14) .IEEE Transactions on Antennas and Propagation 44: 1630–1639. doi:10.1109/8.546249 (http://dx.doi.org/10.1109%2F8.546249) . http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=11941&arnumber=546249&count=18&index=14.

7.

^ a b G. Mur (1981). "Absorbing boundary conditions for the finite-difference approximation of the time-domainelectromagnetic field equations" (http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=4091487&arnumber=4091495) . IEEE Transactions on Electromagnetic Compatibility 23: 377–382.doi:10.1109/TEMC.1981.303970 (http://dx.doi.org/10.1109%2FTEMC.1981.303970) . http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=4091487&arnumber=4091495.

8.

^ a b Z. P. Liao, H. L. Wong, B. P. Yang, and Y. F. Yuan (1984). "A transmitting boundary for transient waveanalysis". Scientia Sinica a 27: 1063–1076.

9.

^ Adapted with permission from Taflove and Hagness (2005).10.^ A. Taflove and M. E. Brodwin (1975). "Computation of the electromagnetic fields and induced temperatureswithin a model of the microwave-irradiated human eye" (http://www.ece.northwestern.edu/ecefaculty/taflove

11.


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/Paper3.pdf) . IEEE Transactions on Microwave Theory and Techniques 23: 888–896.doi:10.1109/TMTT.1975.1128708 (http://dx.doi.org/10.1109%2FTMTT.1975.1128708) .http://www.ece.northwestern.edu/ecefaculty/taflove/Paper3.pdf.^ R. Holland (1977). "Threde: A free-field EMP coupling and scattering code". IEEE Transactions on NuclearScience 24: 2416–2421. doi:10.1109/TNS.1977.4329229 (http://dx.doi.org/10.1109%2FTNS.1977.4329229) .

12.

^ K. S. Kunz and K. M. Lee (1978). "A three-dimensional finite-difference solution of the external response of anaircraft to a complex transient EM environment". IEEE Transactions on Electromagnetic Compatibility 20:328–341. doi:10.1109/TEMC.1978.303727 (http://dx.doi.org/10.1109%2FTEMC.1978.303727) .

13.

^ A. Taflove and K. R. Umashankar (1983). "Radar cross section of general three-dimensional scatterers"(http://www.ece.northwestern.edu/ecefaculty/taflove/Paper10.pdf) . IEEE Transactions on ElectromagneticCompatibility 25: 433–440. doi:10.1109/TEMC.1983.304133 (http://dx.doi.org/10.1109%2FTEMC.1983.304133) .http://www.ece.northwestern.edu/ecefaculty/taflove/Paper10.pdf.

14.

^ W. Gwarek (1985). "Analysis of an arbitrarily shaped planar circuit — A time-domain approach"(http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=25151&arnumber=1133170&count=34&index=26) . IEEETransactions on Microwave Theory and Techniques 33: 1067–1072. doi:10.1109/TMTT.1985.1133170(http://dx.doi.org/10.1109%2FTMTT.1985.1133170) . http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=25151&arnumber=1133170&count=34&index=26.

15.

^ D. H. Choi and W. J. Hoefer (1986). "The finite-difference time-domain method and its application to eigenvalueproblems" (http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=25165&arnumber=1133564&count=56&index=35). IEEE Transactions on Microwave Theory and Techniques 34: 1464–1470. doi:10.1109/TMTT.1986.1133564(http://dx.doi.org/10.1109%2FTMTT.1986.1133564) . http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=25165&arnumber=1133564&count=56&index=35.

16.

^ G. A. Kriegsmann, A. Taflove, and K. R. Umashankar (1987). "A new formulation of electromagnetic wavescattering using an on-surface radiation boundary condition approach" (http://www.ece.northwestern.edu/ecefaculty/taflove/Paper14.pdf) . IEEE Transactions on Antennas and Propagation 35: 153–161.doi:10.1109/TAP.1987.1144062 (http://dx.doi.org/10.1109%2FTAP.1987.1144062) .http://www.ece.northwestern.edu/ecefaculty/taflove/Paper14.pdf.

17.

^ T. G. Moore, J. G. Blaschak, A. Taflove, and G. A. Kriegsmann (1988). "Theory and application of radiationboundary operators" (http://www.ece.northwestern.edu/ecefaculty/taflove/Paper20.pdf) . IEEE Transactions onAntennas and Propagation 36: 1797–1812. doi:10.1109/8.14402 (http://dx.doi.org/10.1109%2F8.14402) .http://www.ece.northwestern.edu/ecefaculty/taflove/Paper20.pdf.

18.

^ K. R. Umashankar, A. Taflove, and B. Beker (1987). "Calculation and experimental validation of inducedcurrents on coupled wires in an arbitrary shaped cavity" (http://www.ece.northwestern.edu/ecefaculty/taflove/Paper16.pdf) . Antennas and Propagation, IEEE Transactions on 35: 1248–1257. doi:10.1109/TAP.1987.1144000(http://dx.doi.org/10.1109%2FTAP.1987.1144000) . http://www.ece.northwestern.edu/ecefaculty/taflove/Paper16.pdf.

19.

^ A. Taflove, K. R. Umashankar, B. Beker, F. A. Harfoush, and K. S. Yee (1988). "Detailed FDTD analysis ofelectromagnetic fields penetrating narrow slots and lapped joints in thick conducting screens"(http://www.ece.northwestern.edu/ecefaculty/taflove/Paper17.pdf) . Antennas and Propagation, IEEE Transactionson 36: 247–257. doi:10.1109/8.1102 (http://dx.doi.org/10.1109%2F8.1102) . http://www.ece.northwestern.edu/ecefaculty/taflove/Paper17.pdf.

20.

^ T. G. Jurgens, A. Taflove, K. R. Umashankar, and T. G. Moore (1992). "Finite-difference time-domain modelingof curved surfaces" (http://www.ece.northwestern.edu/ecefaculty/taflove/Paper39.pdf) . Antennas and Propagation,IEEE Transactions on 40: 357–366. doi:10.1109/8.138836 (http://dx.doi.org/10.1109%2F8.138836) .http://www.ece.northwestern.edu/ecefaculty/taflove/Paper39.pdf.

21.

^ D. M. Sullivan, O. P. Gandhi, and A. Taflove (1988). "Use of the finite-difference time-domain method incalculating EM absorption in man models" (http://www.ece.northwestern.edu/ecefaculty/taflove/Paper18.pdf) .IEEE Transactions on Biomedical Engineering 35 (3): 179–186. doi:10.1109/10.1360 (http://dx.doi.org/10.1109%2F10.1360) . PMID 3350546 (http://www.ncbi.nlm.nih.gov/pubmed/3350546) .http://www.ece.northwestern.edu/ecefaculty/taflove/Paper18.pdf.

22.

^ X. Zhang, J. Fang, K. K. Mei, and Y. Liu (1988). "Calculation of the dispersive characteristics of microstrips bythe time-domain finite-difference method" (http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=197&arnumber=3514&count=27&index=6) . IEEE Transactions on Microwave Theory and Techniques 36: 263–267.doi:10.1109/22.3514 (http://dx.doi.org/10.1109%2F22.3514) . http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=197&arnumber=3514&count=27&index=6.

23.


9 of 14 12/14/2010 10:05 AM

Visited 12/14/10

^ T. Kashiwa and I. Fukai (1990). "A treatment by FDTD method of dispersive characteristics associated withelectronic polarization". Microwave and Optics Technology Letters 3: 203–205. doi:10.1002/mop.4650030606(http://dx.doi.org/10.1002%2Fmop.4650030606) .

24.

^ R. Luebbers, F. Hunsberger, K. Kunz, R. Standler, and M. Schneider (1990). "A frequency-dependent finite-difference time-domain formulation for dispersive materials" (http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=2069&arnumber=57116&count=12&index=5) . Electromagnetic Compatibility, IEEETransactions on 32: 222–227. doi:10.1109/15.57116 (http://dx.doi.org/10.1109%2F15.57116) .http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=2069&arnumber=57116&count=12&index=5.

25.

^ R. M. Joseph, S. C. Hagness, and A. Taflove (1991). "Direct time integration of Maxwell’s equations in lineardispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses"(http://www.ece.northwestern.edu/ecefaculty/taflove/Paper35.pdf) . Optics Letters 16 (18): 1412–1414.doi:10.1364/OL.16.001412 (http://dx.doi.org/10.1364%2FOL.16.001412) . PMID 19776986(http://www.ncbi.nlm.nih.gov/pubmed/19776986) . http://www.ece.northwestern.edu/ecefaculty/taflove/Paper35.pdf.

26.

^ J. G. Maloney, G. S. Smith, and W. R. Scott, Jr. (1990). "Accurate computation of the radiation from simpleantennas using the finite-difference time-domain method" (http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=2009&arnumber=55618&count=30&index=14) . Antennas and Propagation, IEEETransactions on 38: 1059–1065. doi:10.1109/8.55618 (http://dx.doi.org/10.1109%2F8.55618) .http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=2009&arnumber=55618&count=30&index=14.

27.

^ D. S. Katz, A. Taflove, M. J. Piket-May, and K. R. Umashankar (1991). "FDTD analysis of electromagneticwave radiation from systems containing horn antennas" (http://www.ece.northwestern.edu/ecefaculty/taflove/Paper34.pdf) . Antennas and Propagation, IEEE Transactions on 39: 1203–1212. doi:10.1109/8.97356(http://dx.doi.org/10.1109%2F8.97356) . http://www.ece.northwestern.edu/ecefaculty/taflove/Paper34.pdf.

28.

^ P. A. Tirkas and C. A. Balanis (1991). "Finite-difference time-domain technique for radiation by horn antennas"(http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=4455&arnumber=175196&count=464&index=429) .Antennas and Propagation Society International Symposium Digest, IEEE 3: 1750–1753.doi:10.1109/APS.1991.175196 (http://dx.doi.org/10.1109%2FAPS.1991.175196) . http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=4455&arnumber=175196&count=464&index=429.

29.

^ E. Sano and T. Shibata (1990). "Fullwave analysis of picosecond photoconductive switches"(http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=1701&arnumber=44970&count=26&index=24) . QuantumElectronics, IEEE Journal of 26: 372–377. doi:10.1109/3.44970 (http://dx.doi.org/10.1109%2F3.44970) .http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=1701&arnumber=44970&count=26&index=24.

30.

^ S. M. El-Ghazaly, R. P. Joshi, and R. O. Grondin (1990). "Electromagnetic and transport considerations insubpicosecond photoconductive switch modeling" (http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=1982&arnumber=54932&count=33&index=19) . Microwave Theory and Techniques, IEEE Transactions on 38: 629–637.doi:10.1109/22.54932 (http://dx.doi.org/10.1109%2F22.54932) . http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=1982&arnumber=54932&count=33&index=19.

31.

^ P. M. Goorjian and A. Taflove (1992). "Direct time integration of Maxwell’s equations in nonlinear dispersivemedia for propagation and scattering of femtosecond electromagnetic solitons" (http://www.ece.northwestern.edu/ecefaculty/taflove/Paper37.pdf) . Optics Letters 17: 180–182. doi:10.1364/OL.17.000180 (http://dx.doi.org/10.1364%2FOL.17.000180) . http://www.ece.northwestern.edu/ecefaculty/taflove/Paper37.pdf.

32.

^ R. W. Ziolkowski and J. B. Judkins (1993). "Full-wave vector Maxwell’s equations modeling of self-focusing ofultra-short optical pulses in a nonlinear Kerr medium exhibiting a finite response time". Journal of the OpticalSociety of America B 10: 186–198. doi:10.1364/JOSAB.10.000186 (http://dx.doi.org/10.1364%2FJOSAB.10.000186) .

33.

^ R. M. Joseph, P. M. Goorjian, and A. Taflove (1993). "Direct time integration of Maxwell’s equations in 2-Ddielectric waveguides for propagation and scattering of femtosecond electromagnetic solitons"(http://www.ece.northwestern.edu/ecefaculty/taflove/Paper44.pdf) . Optics Letters 18 (7): 491–493.doi:10.1364/OL.18.000491 (http://dx.doi.org/10.1364%2FOL.18.000491) . PMID 19802177(http://www.ncbi.nlm.nih.gov/pubmed/19802177) . http://www.ece.northwestern.edu/ecefaculty/taflove/Paper44.pdf.

34.

^ R. M. Joseph and A. Taflove (1994). "Spatial soliton deflection mechanism indicated by FDTD Maxwell’sequations modeling" (http://www.ece.northwestern.edu/ecefaculty/taflove/Paper54.pdf) . IEEE PhotonicsTechnology Letters 2: 1251–1254. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper54.pdf.

35.

^ W. Sui, D. A. Christensen, and C. H. Durney (1992). "Extending the two-dimensional FDTD method to hybridelectromagnetic systems with active and passive lumped elements" (http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=3573&arnumber=127522&count=28&index=15) . IEEE Transactions on Microwave

36.


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Theory and Techniques 40: 724–730. doi:10.1109/22.127522 (http://dx.doi.org/10.1109%2F22.127522) .http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=3573&arnumber=127522&count=28&index=15.^ B. Toland, B. Houshmand, and T. Itoh (1993). "Modeling of nonlinear active regions with the FDTD method"(http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=6289&arnumber=244870&count=17&index=14) . IEEEMicrowave and Guided Wave Letters 3: 333–335. doi:10.1109/75.244870 (http://dx.doi.org/10.1109%2F75.244870) . http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=6289&arnumber=244870&count=17&index=14.

37.

^ V. A. Thomas, M. E. Jones, M. J. Piket-May, A. Taflove, and E. Harrigan (1994). "The use of SPICE lumpedcircuits as sub-grid models for FDTD high-speed electronic circuit design" (http://www.ece.northwestern.edu/ecefaculty/taflove/Paper49.pdf) . IEEE Microwave and Guided Wave Letters 4: 141–143.http://www.ece.northwestern.edu/ecefaculty/taflove/Paper49.pdf.

38.

^ D. S. Katz, E. T. Thiele, and A. Taflove (1994). "Validation and extension to three dimensions of the BerengerPML absorbing boundary condition for FDTD meshes" (http://www.ece.northwestern.edu/ecefaculty/taflove/Paper51.pdf) . IEEE Microwave and Guided Wave Letters 4: 268–270. doi:10.1109/75.311494 (http://dx.doi.org/10.1109%2F75.311494) . http://www.ece.northwestern.edu/ecefaculty/taflove/Paper51.pdf.

39.

^ C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz, and A. Taflove (1994). "Ultrawideband absorbing boundarycondition for termination of waveguiding structures in FDTD simulations" (http://www.ece.northwestern.edu/ecefaculty/taflove/Paper53.pdf) . IEEE Microwave and Guided Wave Letters 4: 344–346. doi:10.1109/75.324711(http://dx.doi.org/10.1109%2F75.324711) . http://www.ece.northwestern.edu/ecefaculty/taflove/Paper53.pdf.

40.

^ Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee (1995). "A perfectly matched anisotropic absorber for use asan absorbing boundary condition" (http://ieeexplore.ieee.org/xpl/tocresult.jsp?isnumber=10144) . IEEETransactions on Antennas and Propagation 43: 1460–1463. doi:10.1109/8.477075 (http://dx.doi.org/10.1109%2F8.477075) . http://ieeexplore.ieee.org/xpl/tocresult.jsp?isnumber=10144.

41.

^ Q. H. Liu (1997). "The pseudospectral time-domain (PSTD) method: A new algorithm for solutions of Maxwell’sequations" (http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=13688&arnumber=630102&count=142&index=30). IEEE Antennas and Propagation Society International Symposium Digest 1: 122–125.doi:10.1109/APS.1997.630102 (http://dx.doi.org/10.1109%2FAPS.1997.630102) . http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=13688&arnumber=630102&count=142&index=30.

42.

^ O. M. Ramahi (1997). "The complementary operators method in FDTD simulations" (http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=14102&arnumber=646801&count=10&index=4) . IEEE Antennas and PropagationMagazine 39: 33–45. doi:10.1109/74.646801 (http://dx.doi.org/10.1109%2F74.646801) . http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=14102&arnumber=646801&count=10&index=4.

43.

^ J. G. Maloney and M. P. Kesler (1998). "Analysis of Periodic Structures". Chap. 6 in Advances inComputational Electrodynamics: the Finite-Difference Time-Domain Method, A. Taflove, ed., Artech House,publishers.

44.

^ A. S. Nagra and R. A. York (1998). "FDTD analysis of wave propagation in nonlinear absorbing and gain media"(http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=14516&arnumber=662652&count=23&index=5) . IEEETransactions on Antennas and Propagation 46: 334–340. doi:10.1109/8.662652 (http://dx.doi.org/10.1109%2F8.662652) . http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=14516&arnumber=662652&count=23&index=5.

45.

^ S. C. Hagness, A. Taflove, and J. E. Bridges (1998). "Two-dimensional FDTD analysis of a pulsed microwaveconfocal system for breast cancer detection: Fixed-focus and antenna-array sensors"(http://www.ece.northwestern.edu/ecefaculty/taflove/Paper66.pdf) . IEEE Transactions on Biomedical Engineering45 (12): 1470–1479. doi:10.1109/10.730440 (http://dx.doi.org/10.1109%2F10.730440) . PMID 9835195(http://www.ncbi.nlm.nih.gov/pubmed/9835195) . http://www.ece.northwestern.edu/ecefaculty/taflove/Paper66.pdf.

46.

^ J. B. Schneider, and C. L. Wagner (1999). "FDTD dispersion revisited: Faster-than-light propagation"(http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=16327&arnumber=755044&count=12&index=3) . IEEEMicrowave and Guided Wave Letters 9: 54–56. doi:10.1109/75.755044 (http://dx.doi.org/10.1109%2F75.755044) .http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=16327&arnumber=755044&count=12&index=3.

47.

^ F. Zhen, Z. Chen, and J. Zhang (2000). "Toward the development of a three-dimensional unconditionally stablefinite-difference time-domain method" (http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=18818&arnumber=869007&count=27&index=17) . Microwave Theory and Techniques, IEEE Transactions on 48:1550–1558. doi:10.1109/22.869007 (http://dx.doi.org/10.1109%2F22.869007) . http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=18818&arnumber=869007&count=27&index=17.

48.

^ F. Zheng and Z. Chen (2001). "Numerical dispersion analysis of the unconditionally stable 3-D ADI-FDTD49.


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Visited 12/14/10

method" (http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=19889&arnumber=920165&count=26&index=25) .Microwave Theory and Techniques, IEEE Transactions on 49: 1006–1009. doi:10.1109/22.920165(http://dx.doi.org/10.1109%2F22.920165) . http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=19889&arnumber=920165&count=26&index=25.^ J. A. Roden and S. D. Gedney (2000). "Convolution PML (CPML): An efficient FDTD implementation of theCFS-PML for arbitrary media" (http://www3.interscience.wiley.com/journal/73504513/abstract) . Microwave andOptical Technology Letters 27: 334–339. doi:10.1002/1098-2760(20001205)27:5<334::AID-MOP14>3.0.CO;2-A(http://dx.doi.org/10.1002%2F1098-2760%2820001205%2927%3A5%3C334%3A%3AAID-MOP14%3E3.0.CO%3B2-A) . http://www3.interscience.wiley.com/journal/73504513/abstract.

50.

^ T. Rylander and A. Bondeson (2000). "Stable FDTD-FEM hybrid method for Maxwell’s equations"(http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJ5-3YGKN25-5&_user=10&_coverDate=03%2F31%2F2000&_alid=821821401&_rdoc=1&_fmt=high&_orig=search&_cdi=5301&_sort=d&_docanchor=&view=c&_ct=1&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=03651a2f768b72648ad40c38e21072f8) . Computer Physics Communications 125: 75–82.doi:10.1016/S0010-4655(99)00463-4 (http://dx.doi.org/10.1016%2FS0010-4655%2899%2900463-4) .http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJ5-3YGKN25-5&_user=10&_coverDate=03%2F31%2F2000&_alid=821821401&_rdoc=1&_fmt=high&_orig=search&_cdi=5301&_sort=d&_docanchor=&view=c&_ct=1&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=03651a2f768b72648ad40c38e21072f8.

51.

^ M. Hayakawa and T. Otsuyama (2002). "FDTD analysis of ELF wave propagation in inhomogeneoussubionospheric waveguide models" (http://handle.dtic.mil/100.2/ADP013476) . ACES Journal 17: 239–244.http://handle.dtic.mil/100.2/ADP013476.

52.

^ J. J. Simpson and A. Taflove (2002). "Two-dimensional FDTD model of antipodal ELF propagation andSchumann resonance of the Earth" (http://www.ece.unm.edu/~simpson/Paper1.pdf) . IEEE Antennas and WirelessPropagation Letters 1 (2): 53–56. doi:10.1109/LAWP.2002.805123 (http://dx.doi.org/10.1109%2FLAWP.2002.805123) . http://www.ece.unm.edu/~simpson/Paper1.pdf.

53.

^ H. De Raedt, K. Michielsen, J. S. Kole, and M. T. Figge (2003). "Solving the Maxwell equations by theChebyshev method: A one-step finite difference time-domain algorithm" (http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=27862&arnumber=1243513&count=22&index=14) . IEEE Transactions on Antennasand Propagation 51: 3155–3160. doi:10.1109/TAP.2003.818809 (http://dx.doi.org/10.1109%2FTAP.2003.818809). http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=27862&arnumber=1243513&count=22&index=14.

54.

The following article in Nature Milestones: Photons illustrates the historical significance of the FDTD methodas related to Maxwell's equations:

David Pile (May 2010). "Milestone 2 (1861) Maxwell's equations" (http://www.nature.com/milestones/milephotons/full/milephotons02.html) . Nature Milestones: Photons. Nature. doi:10.1038/nmat2639(http://dx.doi.org/10.1038%2Fnmat2639) . http://www.nature.com/milestones/milephotons/full/milephotons02.html. Retrieved 17 June 2010.

The following university-level textbooks provide a good general introduction to the FDTD method:

Karl S. Kunz and Raymond J. Luebbers (1993). The Finite Difference Time Domain Method forElectromagnetics (http://www.crcpress.com/shopping_cart/products/product_detail.asp?sku=8657&af=W1129) . CRC Press. ISBN 0-8493-8657-8. http://www.crcpress.com/shopping_cart/products/product_detail.asp?sku=8657&af=W1129.

Allen Taflove and Susan C. Hagness (2005). Computational Electrodynamics: The Finite-DifferenceTime-Domain Method, 3rd ed. (http://www.artechhouse.com/Detail.aspx?strBookId=1123) . ArtechHouse Publishers. ISBN 1-58053-832-0. http://www.artechhouse.com/Detail.aspx?strBookId=1123.


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Wenhua Yu, Raj Mittra, Tao Su, Yongjun Liu, and Xiaoling Yang (2006). Parallel Finite-DifferenceTime-Domain Method (http://www.artechhouse.com/default.asp?frame=book.asp&book=1-59693-085-3&Country=US&Continent=NO&State=) . Artech House Publishers.ISBN 1-59693-085-3. http://www.artechhouse.com/default.asp?frame=book.asp&book=1-59693-085-3&Country=US&Continent=NO&State=.

Free software/Open-source software FDTD projects:JFDTD (http://www.thecomputationalphysicist.com) (2D/3D C++ FDTD codes developed fornanophotonics by Jeffrey M. McMahon)WOLFSIM (http://www.ece.ncsu.edu/oleg/wiki/WOLFSIM) (NCSU) (2-D)Meep (http://ab-initio.mit.edu/meep/) (MIT, 2D/3D/cylindrical parallel FDTD)(Geo-) Radar FDTD (http://freshmeat.net/projects/radarfdtd/)bigboy (http://sourceforge.net/projects/bigboy) (unmaintained, no release files. must get sourcefrom cvs)toyFDTD (http://www.cemtach.com/reference/software/toyFDTD/)FDTD codes in C++ (http://www.evtsz.bme.hu/web/staff/szabo/web_fdtd/FDTD_main.html)(developed by Zs. Szabó)FDTD code in Fortran 90 (http://cs.tu-berlin.de/~peutetre/sfdtd/)FDTD code in C for 2D EM Wave simulation (http://code.google.com/p/emwave2d/)

Freeware/Closed source FDTD projects (some not for commercial use):GprMax (http://www.gprmax.org) (Last updated on 12 May 2004 - unmaintained, no source code)EMTL (Electromagnetic Template Library) (http://fdtd.kintechlab.com/) (Free С++ library forelectromagnetic simulations. The current version implements mainly the FDTD).

Commercial/proprietary FDTD software vendors:2COMU (http://www.2comu.com/index.html)Acceleware Inc. (http://www.acceleware.com/)Agilent EEsof EDA EMPro (http://www.agilent.com/find/eesof-empro) (accessed 2009,April 13)Apollo Photonics (http://www.apollophoton.com/apollo/page.php?id=4)Applied Simulation Technology (http://www.apsimtech.com/)Cray LC (http://lc.cray.com/)CST - Computer Simulation Technology (http://www.cst.com/)Electro Magnetic Applications Inc. (http://electromagneticapplications.com/ema3d_main.html)Emagware.com (http://www.emagware.com)EM Explorer (http://www.emexplorer.net/)EM Photonics (http://www.emphotonics.com/)Empire (http://www.empire.de/)EMS Plus (http://www.ems-plus.com/ezfdtd.html)ETHZ (http://www.iis.ee.ethz.ch/research/bioemc/em_simulation_platform.en.html)GdfidL (http://www.gdfidl.de/)Lumerical Solutions (http://www.lumerical.com/)Nonlinear Control Strategies (http://www.nlcstr.com/sim3d.htm)Optiwave (http://www.optiwave.com/)Photon Design (http://www.photond.com)QuickWave (http://www.qwed.eu)Remcom XFdtd 7 (http://www.remcom.com/xf7)RM Associates (http://www.rm-associates.biz/)Rsoft (http://rsoftdesign.com/products.php?sub=Component+Design&itm=FullWAVE)


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SPEAG (http://www.speag.com/)Taflove-Hagness book software (http://www.artechhouse.com/Detail.aspx?strBookId=1123)Vector Fields (http://www.vectorfields.com/content/view/47/70/)Zeland (http://www.zeland.com/)

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