Finitedifference timedomain method
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Finitedifference timedomain (FDTD) is a popular computational electrodynamics modeling technique. It is considered easy to understand and easy to implement in software. Since it is a timedomain method, solutions can cover a wide frequency range with a single simulation run.
The FDTD method belongs in the general class of gridbased differential timedomain numerical modeling methods. The timedependent Maxwell's equations (in partial differential form) are discretized using centraldifference approximations to the space and time partial derivatives. The resulting finitedifference equations are solved in either software or hardware in a leapfrog manner: the electric field vector components in a volume of space are solved at a given instant in time; then the magnetic field vector components in the same spatial volume are solved at the next instant in time; and the process is repeated over and over again until the desired transient or steadystate electromagnetic field behavior is fully evolved.
The basic FDTD space grid and timestepping algorithm trace back to a seminal 1966 paper by Kane Yee in IEEE Transactions on Antennas and Propagation (Yee 1966). The descriptor "Finitedifference timedomain" and its corresponding "FDTD" acronym were originated by Allen Taflove in a 1980 paper in IEEE Transactions on Electromagnetic Compatibility (Taflove 1980). See "References" for these and other important journal papers in the development of FDTD techniques, as well as relevant textbooks and research monographs.
Since about 1990, FDTD techniques have emerged as primary means to computationally model many scientific and engineering problems dealing with electromagnetic wave interactions with material structures. As summarized in Taflove & Hagness (2005), current FDTD modeling applications range from nearDC (ultralowfrequency geophysics involving the entire Earthionosphere waveguide) through microwaves (radar signature technology, antennas, wireless communications devices, digital interconnects, biomedical imaging/treatment) to visible light (photonic crystals, nanoplasmonics, solitons, and biophotonics). In 2006, an estimated 2,000 FDTDrelated publications appeared in the science and engineering literature (see "Growth of FDTD publications"). At present, there are at least 27 commercial/proprietary FDTD software vendors; 8 freesoftware/opensourcesoftware FDTD projects; and 2 freeware/closedsource FDTD projects, some not for commercial use (see "External links").
Contents 
[edit] Workings of the FDTD method
When Maxwell's differential equations are examined, it can be seen that the change in the Efield in time (the time derivative) is dependent on the change in the Hfield across space (the curl). This results in the basic FDTD timestepping relation that, at any point in space, the updated value of the Efield in time is dependent on the stored value of the Efield and the numerical curl of the local distribution of the Hfield in space (Yee 1966).
The Hfield is timestepped in a similar manner. At any point in space, the updated value of the Hfield in time is dependent on the stored value of the Hfield and the numerical curl of the local distribution of the Efield in space. Iterating the Efield and Hfield updates results in a marchingintime process wherein sampleddata analogs of the continuous electromagnetic waves under consideration propagate in a numerical grid stored in the computer memory.
This description holds true for 1D, 2D, and 3D FDTD techniques. When multiple dimensions are considered, calculating the numerical curl can become complicated. Kane Yee's seminal 1966 paper in IEEE Transactions on Antennas and Propagation proposed spatially staggering the vector components of the Efield and Hfield about rectangular unit cells of a Cartesian computational grid so that each Efield vector component is located midway between a pair of Hfield vector components, and conversely. This scheme, now known as a Yee lattice, has proven to be very robust, and remains at the core of many current FDTD software constructs (Yee 1966).
Furthermore, Yee proposed a leapfrog scheme for marching in time wherein the Efield and Hfield updates are staggered so that Efield updates are conducted midway during each timestep between successive Hfield updates, and conversely (Yee 1966). On the plus side, this explicit timestepping scheme avoids the need to solve simultaneous equations, and furthermore yields dissipationfree numerical wave propagation. On the minus side, this scheme mandates an upper bound on the timestep to ensure numerical stability (Taflove & Brodwin 1975). As a result, certain classes of simulations can require many thousands of timesteps for completion.
[edit] Using the FDTD method
In order to use FDTD a computational domain must be established. The computational domain is simply the physical region over which the simulation will be performed. The E and H fields are determined at every point in space within that computational domain. The material of each cell within the computational domain must be specified. Typically, the material is either freespace (air), metal, or dielectric. Any material can be used as long as the permeability, permittivity, and conductivity are specified.
Once the computational domain and the grid materials are established, a source is specified. The source can be an 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 a point or a series of points within the computational domain. The simulation evolves the E and H fields forward in time.
Processing may be done on the E and H fields returned by the simulation. Data processing may also occur while the 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 neartofarfield transformations, as reported originally by Umashankar and Taflove (1982).
[edit] Strengths of FDTD modeling
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 easily understand how to use it and know what to expect from a given model.
FDTD is a timedomain 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. This is useful in applications where resonant frequencies are not exactly known, or anytime that a broadband result is desired.
Since FDTD calculates the E and H fields everywhere in the computational domain as they evolve in time, it lends itself to providing animated displays of the electromagnetic field movement through the model. This type of display is useful in understanding what is going on in the model, and to help ensure that the model is working correctly.
The FDTD technique allows the user to specify the material at all points within the computational domain. A wide 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 fields both 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 and H fields, it is convenient that no conversions must be made after the simulation has run to get these values.
[edit] Weaknesses of FDTD modeling
Since FDTD requires that the entire computational domain be gridded, and the grid spatial discretization must be sufficiently fine to resolve both the smallest electromagnetic wavelength and the smallest geometrical feature in the 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 large computational domain required.
FDTD finds the E/H fields directly everywhere in the computational domain. If the field values at some distance are desired, it is likely that this distance will force the computational domain to be excessively large. Farfield extensions are available for FDTD, but require some amount of postprocessing (Taflove & Hagness 2005).
Since FDTD simulations calculate the E and H fields at all points within the computational domain, the computational domain must be finite to permit its residence in the computer memory. In many cases this is achieved by inserting artificial boundaries into the simulation space. Care must be taken to minimize errors introduced by such boundaries. There are a number of available highly effective absorbing boundary conditions (ABCs) to simulate an infinite unbounded computational domain (Taflove & Hagness 2005). Most modern FDTD implementations instead use a special absorbing "material", called a perfectly matched layer (PML) to implement absorbing boundaries (Berenger 1994, Gedney 1996).
Because FDTD is solved by propagating the fields forward in the time domain, the electromagnetic time response of the medium must be modeled explicitly. For an arbitrary response, this involves a computationally expensive time convolution, although in most cases the time response of the medium (or Dispersion (optics)) can be adequately and simply modeled using either the recursive convolution (RC) technique, the auxiliary differential equation (ADE) technique, or the Ztransform technique. An alternative way of solving Maxwell's equations that can treat arbitrary dispersion easily is the Pseudospectral SpatialDomain method (PSSD), which instead propagates the fields forward in space.
[edit] Grid truncation techniques for openregion FDTD modeling problems
The most commonly used grid truncation techniques for openregion FDTD modeling problems are the Mur absorbing boundary condition (ABC) (Mur 1981), the Liao ABC (Liao et al. 1984), and various perfectly matched layer (PML) formulations (Berenger 1994, Gedney 1996, Taflove & Hagness 2005). The Mur and Liao techniques are simpler than PML. However, PML (which is technically an absorbing region rather than a boundary condition per se) can provide ordersofmagnitude lower reflections. The PML concept was introduced by J.P. Berenger in a seminal 1994 paper in the Journal of Computational Physics (Berenger 1994). Since 1994, Berenger's original splitfield implementation has been modified and extended to the uniaxial PML (UPML), the convolutional PML (CPML), and the higherorder PML. The latter two PML formulations have increased ability to absorb evanescent waves, and therefore can in principle be placed closer to a simulated scattering or radiating structure than Berenger's original formulation.
[edit] History of FDTD Techniques and Applications for Maxwell's Equations
We can begin to develop an appreciation of the basis, technical development, and possible future of FDTD numerical techniques for Maxwell’s equations by first considering their history. The following lists some of the key publications in this area, starting with Yee's seminal Paper #1 (1966), which has over 4000 citations according to the ISI Web of Science.
Partial Chronology of FDTD Techniques and Applications for Maxwell's Equations. (Adapted with permission from Taflove and Hagness (2005))
1966 — Yee (1966) described the basis of the FDTD numerical technique for solving Maxwell’s curl equations directly in the time domain on a space grid.
1975 — Taflove and Brodwin (1975a, 1975b) reported the correct numerical stability criterion for Yee’s algorithm; the first sinusoidal steadystate FDTD solutions of two and threedimensional electromagnetic wave interactions with material structures; and the first bioelectromagnetics models.
1977 — Holland (1977), and Kunz and Lee (1977) applied Yee’s algorithm to EMP problems.
1980 — Taflove (1980) coined the FDTD acronym and published the first validated FDTD models of sinusoidal steadystate electromagnetic wave penetration into a threedimensional metal cavity.
1981 — Mur (1981) published the first numerically stable, secondorder accurate, absorbing boundary condition (ABC) for Yee’s grid.
1982, '83 — Taflove and Umashankar (1982, 1983) developed the first FDTD electromagnetic wave scattering models computing sinusoidal steadystate nearfields, farfields, and radar crosssection for two and threedimensional structures.
1984 — Liao et al. (1984) reported an improved ABC based upon spacetime extrapolation of the field adjacent to the outer grid boundary.
1985 — Gwarek (1985) introduced the lumped equivalent circuit formulation of FDTD.
1986 — Choi and Hoefer (1986) published the first FDTD simulation of waveguide structures.
1987, '88 — Kriegsmann et al. (1987) and Moore et al. (1988) published the first articles on ABC theory in IEEE Trans. Antennas and Propagation.
1987, '88, '92 — Contourpath subcell techniques were introduced by Umashankar et al. (1987) to permit FDTD modeling of thin wires and wire bundles, by Taflove et al. (1988) to model penetration through cracks in conducting screens, and by Jurgens et al. (1992) to conformally model the surface of a smoothly curved scatterer.
1988 — Sullivan et al. (1988) published the first 3D FDTD model of sinusoidal steadystate electromagnetic wave absorption by a complete human body.
1988 — FDTD modeling of microstrips was introduced by Zhang et al. (1988).
1990, '91 — FDTD modeling of frequencydependent dielectric permittivity was introduced by Kashiwa and Fukai (1990), Luebbers et al. (1990), and Joseph et al. (1991).
1990, '91 — FDTD modeling of antennas was introduced by Maloney et al. (1990), Katz et al. (1991), and Tirkas and Balanis (1991).
1990 — FDTD modeling of picosecond optoelectronic switches was introduced by Sano and Shibata (1990), and ElGhazaly et al. (1990).
1992–94 — FDTD modeling of the propagation of optical pulses in nonlinear dispersive media was introduced, including the first temporal solitons in one dimension by Goorjian and Taflove (1992); beam selffocusing by Ziolkowski and Judkins (1993); the first temporal solitons in two dimensions by Joseph et al. (1993); and the first spatial solitons in two dimensions by Joseph and Taflove (1994).
1992 — FDTD modeling of lumped electronic circuit elements was introduced by Sui et al. (1992).
1993 — Toland et al. (1993) published the first FDTD models of gain devices (tunnel diodes and Gunn diodes) exciting cavities and antennas.
1994 — Thomas et al. (1994) introduced a Norton’s equivalent circuit for the FDTD space lattice, which permits the SPICE circuit analysis tool to implement accurate subgrid models of nonlinear electronic components or complete circuits embedded within the lattice.
1994 — Berenger (1994) introduced the highly effective, perfectly matched layer (PML) ABC for twodimensional FDTD grids, which was extended to three dimensions by Katz et al. (1994), and to dispersive waveguide terminations by Reuter et al. (1994).
1995, '96 — Sacks et al. (1995) and Gedney (1996) introduced a physically realizable, uniaxial perfectly matched layer (UPML) ABC.
1997 — Liu (1997) introduced the pseudospectral timedomain (PSTD) method, which permits extremely coarse spatial sampling of the electromagnetic field at the Nyquist limit.
1997 — Ramahi (1997) introduced the complementary operators method (COM) to implement highly effective analytical ABCs.
1998 — Maloney and Kesler (1998) introduced several novel means to analyze periodic structures in the FDTD space lattice.
1998 — Nagra and York (1998) introduced a hybrid FDTDquantum mechanics model of electromagnetic wave interactions with materials having electrons transitioning between multiple energy levels.
1998 — Hagness et al. (1998) introduced FDTD modeling of the detection of breast cancer using ultrawideband radar techniques.
1999 — Schneider and Wagner (1999) introduced a comprehensive analysis of FDTD grid dispersion based upon complex wavenumbers.
2000, '01 — Zheng, Chen, and Zhang (2000, 2001) introduced the first threedimensional alternatingdirection implicit (ADI) FDTD algorithm with provable unconditional numerical stability.
2000 — Roden and Gedney (2000) introduced the advanced convolutional PML (CPML) ABC.
2000 — Rylander and Bondeson (2000) introduced a provably stable FDTD  finiteelement timedomain hybrid technique.
2002 , '06 — Hayakawa et al. (2002) and Simpson and Taflove (2006) introduced FDTD modeling of the global Earthionosphere waveguide for extremely lowfrequency geophysical phenomena.
2003 — DeRaedt introduced the unconditionally stable, “onestep” FDTD technique (2003).
Interest in FDTD Maxwell’s equations solvers has increased nearly exponentially over the past 20 years. Increasingly, engineers and scientists in nontraditional electromagneticsrelated areas such as photonics and nanotechnology have become aware of the power of FDTD techniques. As shown in the figure on the left, an estimated 2,000 FDTDrelated publications appeared in the science and engineering literature in 2006, as opposed to fewer than 10 as recently as 1985. The current rate of growth (based upon a study of ISI Web of Science data) is approximately 5:1 over the period 1995 to 2006.
There are seven primary reasons for the tremendous expansion of interest in FDTD computational solution approaches for Maxwell’s equations:
1. FDTD uses no linear algebra. Being a fully explicit computation, FDTD avoids the difficulties with linear algebra that limit the size of frequencydomain integralequation and finiteelement electromagnetics models to generally fewer than 1e7 electromagnetic field unknowns. FDTD models with as many as 1e9 field unknowns have been run; there is no intrinsic upper bound to this number.
2. FDTD is accurate and robust. The sources of error in FDTD calculations are well understood, and can be bounded to permit accurate models for a very large variety of electromagnetic wave interaction problems.
3. FDTD treats impulsive behavior naturally. Being a timedomain technique, FDTD directly calculates the impulse response of an electromagnetic system. Therefore, a single FDTD simulation can provide either ultrawideband temporal waveforms or the sinusoidal steadystate response at any frequency within the excitation spectrum.
4. FDTD treats nonlinear behavior naturally. Being a timedomain technique, FDTD directly calculates the nonlinear response of an electromagnetic system. This allows natural hybriding of FDTD with sets of auxiliary differential equations that describe nonlinearities from either the classical or semiclassical standpoint. An exciting research frontier here is the development of hybrid algorithms which join FDTD classical electrodynamics models with phenomena arising from quantum electrodynamics, especially vacuum fluctuations.
5. FDTD is a systematic approach. With FDTD, specifying a new structure to be modeled is reduced to a problem of mesh generation rather than the potentially complex reformulation of an integral equation. For example, FDTD requires no calculation of structuredependent Green functions.
6. Parallelprocessing computer architectures have come to dominate supercomputing. FDTD scales with high efficiency on parallelprocessing CPUbased computers, and extremely well on recently developed GPUbased accelerator technology.
7. Computer visualization capabilities are increasing rapidly. While this trend positively influences all numerical techniques, it is of particular advantage to FDTD methods, which generate timemarched arrays of field quantities suitable for use in color videos to illustrate the field dynamics.
These factors combine to indicate that FDTD will likely remain one of the dominant computational electrodynamics techniques; and indeed may emerge as the dominant technique for mid21stcentury problems of surpassing volumetric complexity and/or multiphysics.
[edit] See also
[edit] References
[edit] Journal articles
 Kane Yee (1966). "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media". Antennas and Propagation, IEEE Transactions on 14: 302–307. doi:. http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1138693.
 A. Taflove and M. E. Brodwin (1975). "Numerical solution of steadystate electromagnetic scattering problems using the timedependent Maxwell's equations". Microwave Theory and Techniques, IEEE Transactions on 23: 623–630. doi:. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper2.pdf.
 A. Taflove and M. E. Brodwin (1975). "Computation of the electromagnetic fields and induced temperatures within a model of the microwaveirradiated human eye". Microwave Theory and Techniques, IEEE Transactions on 23: 888896. doi:. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper3.pdf.
 R. Holland (1977). "Threde: A freefield EMP coupling and scattering code". Nuclear Science, IEEE Transactions on 24: 24162421. doi: .
 K. S. Kunz and K. M. Lee (1978). "A threedimensional finitedifference solution of the external response of an aircraft to a complex transient EM environment". Electromagnetic Compatibility, IEEE Transactions on 20: 328341. doi: .
 A. Taflove (1980). "Application of the finitedifference timedomain method to sinusoidal steady state electromagnetic penetration problems". Electromagnetic Compatibility, IEEE Transactions on 22: 191–202. doi:. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper7.pdf.
 G. Mur (1981). "Absorbing boundary conditions for the finitedifference approximation of the timedomain electromagnetic field equations" (abstract). Electromagnetic Compatibility, IEEE Transactions on 23: 377–382. doi:. http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=4091487&arnumber=4091495.
 K. R. Umashankar and A. Taflove (1982). "A novel method to analyze electromagnetic scattering of complex objects". Electromagnetic Compatibility, IEEE Transactions on 24: 397–405. doi:. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper9.pdf.
 A. Taflove and K. R. Umashankar (1983). "Radar cross section of general threedimensional scatterers". Electromagnetic Compatibility, IEEE Transactions on 25: 433440. doi:. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper10.pdf.
 Z. P. Liao, H. L. Wong, B. P. Yang, and Y. F. Yuan (1984). "A transmitting boundary for transient wave analysis". Scientia Sinica a 27: 1063–1076.
 W. Gwarek (1985). "Analysis of an arbitrarily shaped planar circuit — A timedomain approach". Microwave Theory and Techniques, IEEE Transactions on 33: 10671072. http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=25151&arnumber=1133170&count=34&index=26.
 D. H. Choi and W. J. Hoefer (1986). "The finitedifference timedomain method and its application to eigenvalue problems". Microwave Theory and Techniques, IEEE Transactions on 34: 14641470. http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=25165&arnumber=1133564&count=56&index=35.
 G. A. Kriegsmann, A. Taflove, and K. R. Umashankar (1987). "A new formulation of electromagnetic wave scattering using an onsurface radiation boundary condition approach". Antennas and Propagation, IEEE Transactions on 35: 153161. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper14.pdf.
 T. G. Moore, J. G. Blaschak, A. Taflove, and G. A. Kriegsmann (1988). "Theory and application of radiation boundary operators". Antennas and Propagation, IEEE Transactions on 36: 17971812. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper20.pdf.
 K. R. Umashankar, A. Taflove, and B. Beker (1987). "Calculation and experimental validation of induced currents on coupled wires in an arbitrary shaped cavity". Antennas and Propagation, IEEE Transactions on 35: 12481257. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper16.pdf.
 A. Taflove, K. R. Umashankar, B. Beker, F. A. Harfoush, and K. S. Yee (1988). "Detailed FDTD analysis of electromagnetic fields penetrating narrow slots and lapped joints in thick conducting screens". Antennas and Propagation, IEEE Transactions on 36: 247257. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper17.pdf.
 T. G. Jurgens, A. Taflove, K. R. Umashankar, and T. G. Moore (1992). "Finitedifference timedomain modeling of curved surfaces". Antennas and Propagation, IEEE Transactions on 40: 357366. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper39.pdf.
 D. M. Sullivan, O. P. Gandhi, and A. Taflove (1988). "Use of the finitedifference timedomain method in calculating EM absorption in man models". Biomedical Engineering, IEEE Transactions on 35: 179186. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper18.pdf.
 X. Zhang, J. Fang, K. K. Mei, and Y. Liu (1988). "Calculation of the dispersive characteristics of microstrips by the timedomain finitedifference method". Microwave Theory and Techniques, IEEE Transactions on 36: 263267. doi:. http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=197&arnumber=3514&count=27&index=6.
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 R. M. Joseph, S. C. Hagness, and A. Taflove (1991). "Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses". Optics Letters 16: 14121414. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper35.pdf.
 J. G. Maloney, G. S. Smith, and W. R. Scott, Jr. (1990). "Accurate computation of the radiation from simple antennas using the finitedifference timedomain method". Antennas and Propagation, IEEE Transactions on 38: 10591065. doi:. http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=2009&arnumber=55618&count=30&index=14.
 D. S. Katz, A. Taflove, M. J. PiketMay, and K. R. Umashankar (1991). "FDTD analysis of electromagnetic wave radiation from systems containing horn antennas". Antennas and Propagation, IEEE Transactions on 39: 12031212. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper34.pdf.
 P. A. Tirkas and C. A. Balanis (1991). "Finitedifference timedomain technique for radiation by horn antennas". Antennas and Propagation Society International Symposium Digest, IEEE 3: 17501753. doi:. http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=4455&arnumber=175196&count=464&index=429.
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 P. M. Goorjian and A. Taflove (1992). "Direct time integration of Maxwell’s equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons". Optics Letters 17: 180182. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper37.pdf.
 R. W. Ziolkowski and J. B. Judkins (1993). "Fullwave vector Maxwell’s equations modeling of selffocusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time". Optical Society of America B, Journal of 10: 186198.
 R. M. Joseph, P. M. Goorjian, and A. Taflove (1993). "Direct time integration of Maxwell’s equations in 2D dielectric waveguides for propagation and scattering of femtosecond electromagnetic solitons". Optics Letters 18: 491493. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper44.pdf.
 R. M. Joseph and A. Taflove (1994). "Spatial soliton deflection mechanism indicated by FDTD Maxwell’s equations modeling". Photonics Technology Letters, IEEE 2: 12511254. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper54.pdf.
 W. Sui, D. A. Christensen, and C. H. Durney (1992). "Extending the twodimensional FDTD method to hybrid electromagnetic systems with active and passive lumped elements". Microwave Theory and Techniques, IEEE Transactions on 40: 724730. doi:. http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=3573&arnumber=127522&count=28&index=15.
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 V. A. Thomas, M. E. Jones, M. J. PiketMay, A. Taflove, and E. Harrigan (1994). "The use of SPICE lumped circuits as subgrid models for FDTD highspeed electronic circuit design". Microwave and Guided Wave Letters, IEEE 4: 141143. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper49.pdf.
 M. J. PiketMay, A. Taflove, and J. Baron (1994). "FDTD modeling of digital signal propagation in 3D circuits with passive and active loads". Microwave Theory and Techniques, IEEE Transactions on 42: 15141523. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper52.pdf.
 J. Berenger (1994). "A perfectly matched layer for the absorption of electromagnetic waves". Journal of Computational Physics 114: 185–200. doi:. http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHY45P0TJR1P&_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.
 D. S. Katz, E. T. Thiele, and A. Taflove (1994). "Validation and extension to three dimensions of the Berenger PML absorbing boundary condition for FDTD meshes". Microwave and Guided Wave Letters, IEEE 4: 268270. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper51.pdf.
 C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz, and A. Taflove (1994). "Ultrawideband absorbing boundary condition for termination of waveguiding structures in FDTD simulations". Microwave and Guided Wave Letters, IEEE 4: 344346. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper53.pdf.
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 O. M. Ramahi (1997). "The complementary operators method in FDTD simulations". Antennas and Propagation Magazine, IEEE 39: 3345. doi:. http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=14102&arnumber=646801&count=10&index=4.
 J. G. Maloney and M. P. Kesler (1998). "Analysis of Periodic Structures". Chap. 6 in Advances in Computational Electrodynamics: The FiniteDifference TimeDomain Method, A. Taflove, ed., Artech House, publishers.
 A. S. Nagra and R. A. York (1998). "FDTD analysis of wave propagation in nonlinear absorbing and gain media". Antennas and Propagation, IEEE Transactions on 46: 334340. doi:. http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=14516&arnumber=662652&count=23&index=5.
 S. C. Hagness, A. Taflove, and J. E. Bridges (1998). "Twodimensional FDTD analysis of a pulsed microwave confocal system for breast cancer detection: Fixedfocus and antennaarray sensors". Biomedical Engineering, IEEE Transactions on 45: 14701479. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper66.pdf.
 J. B. Schneider, and C. L. Wagner (1999). "FDTD dispersion revisited: Fasterthanlight propagation". Microwave and Guided Wave Letters, IEEE 9: 5456. doi:. http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=16327&arnumber=755044&count=12&index=3.
 F. Zhen, Z. Chen, and J. Zhang (2000). "Toward the development of a threedimensional unconditionally stable finitedifference timedomain method". Microwave Theory and Techniques, IEEE Transactions on 48: 15501558. doi:. http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=18818&arnumber=869007&count=27&index=17.
 F. Zheng and Z. Chen (2001). "Numerical dispersion analysis of the unconditionally stable 3D ADIFDTD method". Microwave Theory and Techniques, IEEE Transactions on 49: 10061009. doi:. 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 the CFSPML for arbitrary media". Microwave and Optical Technology Letters 27: 334339. doi:. http://www3.interscience.wiley.com/journal/73504513/abstract.
 T. Rylander and A. Bondeson (2000). "Stable FDTDFEM hybrid method for Maxwell’s equations". Computer Physics Communications 125: 7582. doi:. http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJ53YGKN255&_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.
 M. Hayakawa and T. Otsuyama (2002). "FDTD analysis of ELF wave propagation in inhomogeneous subionospheric waveguide models". ACES Journal 17: 239244. http://aces.ee.olemiss.edu/cgibin/Test_2.idc?pid=467.
 J. J. Simpson, R. P. Heikes, and A. Taflove (2006). "FDTD modeling of a novel ELF radar for major oil deposits using a threedimensional geodesic grid of the Earthionosphere waveguide". Antennas and Propagation, IEEE Transactions on 54: 17341741. http://www.ece.northwestern.edu/ecefaculty/taflove/Paper109.pdf.
 H. De Raedt, K. Michielsen, J. S. Kole, and M. T. Figge (2003). "Solving the Maxwell equations by the Chebyshev method: A onestep finite difference timedomain algorithm". Antennas and Propagation, IEEE Transactions on 51: 31553160. doi:. http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=27862&arnumber=1243513&count=22&index=14.
[edit] Universitylevel textbooks
 Karl S. Kunz and Raymond J. Luebbers (1993). The Finite Difference Time Domain Method for Electromagnetics. CRC Press. ISBN 0849386578. http://www.crcpress.com/shopping_cart/products/product_detail.asp?sku=8657&af=W1129.
 Allen Taflove and Susan C. Hagness (2005). Computational Electrodynamics: The FiniteDifference TimeDomain Method, 3rd ed.. Artech House Publishers. ISBN 1580538320. http://www.artechhouse.com/Detail.aspx?strBookId=1123.
 Wenhua Yu, Raj Mittra, Tao Su, Yongjun Liu, and Xiaoling Yang (2006). Parallel FiniteDifference TimeDomain Method. Artech House Publishers. ISBN 1596930853. http://www.artechhouse.com/default.asp?frame=book.asp&book=1596930853&Country=US&Continent=NO&State=.
[edit] External links
 Free software/Opensource software FDTD projects:
 JFDTD (2D/3D C++ FDTD codes developed for nanophotonics by Jeffrey M. McMahon)
 WOLFSIM (NCSU) (2D)
 Meep (MIT)
 (Geo) Radar FDTD
 bigboy (unmaintained, no release files. must get source from cvs)
 toyFDTD
 OpenGEMS (3D Parallel FDTD Package)
 FDTD codes in C++ (developed by Zs. Szabó)
 FDTD code in Fortran 90
 Freeware/Closed source FDTD projects (some not for commercial use):
 Commercial/proprietary FDTD software vendors:
 2COMU
 Acceleware Inc.
 Agilent EEsof EDA EMPro(accessed 2009 April 13)
 APLAC
 Apollo Photonics
 Applied Simulation Technology
 CFDRC
 Cray LC
 CST  Computer Simulation Technology
 Electro Magnetic Applications Inc.
 Emagware.com
 EM Photonics
 Empire
 EMS Plus
 ETHZ
 GdfidL
 Lumerical Solutions
 Nonlinear Control Strategies
 Optiwave
 Photon Design
 QuickWave
 Remcom
 RM Associates
 Rsoft
 SPEAG
 TafloveHagness book software
 Vector Fields
 Zeland
