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Here one can find introductory description of the Finite-Difference Time-Domain (FDTD) method.

FDTD (Finite-Difference Time-Domain)

FDTD is one of the most popular numerical methods in computational electrodynamics. Since introduction in 70th years of the previous century this method became popular due to it certain advantages:

  • simplicity of explicit numerical scheme
  • high parallel efficiency
  • easiness of complex geometry generation
  • ability to handle dispersive and nonlinear media
  • natural description of impulsive regimes

FDTD includes various numerical techniques and options, such as algorithm for dispersive and nonlinear media modeling, different mesh types, simulation results postprocessing etc.

Real optical applications often require extensive parallel FDTD calculations. One can use existing commercial solutions for this purpose, but they do not provide open code that can be modified. To cover this gap we developed Electromagnetic Template Library.

Yee algorithm

FDTD numerical experiment

Typical scenario of FDTD experiment includes following steps:

  • User specifies calculated volume and mesh resolution, optical properties and geometry of the structure, boundary conditions (typically, periodic or absorbing), wave source and set of points where field values should be recorded (we call it detectors).
  • Source generates finite time width impulse impinging on structure. Its propagation and scattering is recorded by detectors and possibly transformed to the frequency domain. Total exit of the radiation through absorbing boundaries determines the simulation time.
  • Recorded field values are processed (for example, energy flux integrating through the chosen surface) to get optical characteristics of the structure.

Using FDTD

FDTD can be used for varios type of simulations: light scattering from arbitrary shaped objects, modeling of source radiation in specified electromagnetic environment, optical properties of resonators and waveguides. In this section we consider these and other possible examples in details.

Preliminary notes

Solution of Maxwell's equations tex:{\bf F}({\bf r},t) (tex:\bf F is tex:\bf E or tex:\bf H), in absence of free charges, current sources and any nonlinearities, can be represented as a superposition of harmonic fields:

tex:{\bf F}({\bf r},t)={\bf A}({\bf r},\omega)\cos{\omega t}+{\bf B}({\bf r},\omega)\sin{\omega t}.

It is convenient to look at tex:{\bf F} as a real part of complex vector tex:{\bf F_c}={\bf C}\exp(-i\omega t), where tex:{\bf C}={\bf A}+i{\bf B}:

tex:{\bf F}={\rm Re}({\bf F_c}).

Complex dependency tex:\exp(-i\omega t) is introduced for convenience purposes only and does not have any physical meaning.

The Poynting vector tex:{\bf P} = {\bf E} \times {\bf H} specifies the magnitude and direction of the rate of electromagnetic energy transfer. The instantaneous Poynting vector is rapidly varying function of time for frequencies that are usually of interest. Most instruments are not capable of following the rapid oscillations of the instantaneous Poynting vector, but respond to some time average tex:{\bf S}:

tex:{\bf S}=<{\bf P}>=\frac{1}{\tau}\int_t^{t+T}{\bf P}(\tau) d\tau,

where tex:T is a time interval long compared with tex:1/\omega.

It can be shown that for harmonic field

tex:{\bf S}=\frac{1}{2}{\rm Re}({\bf E_c}\times{\bf H^*_c}).

Light intensity tex:I is absolute value of tex:{\bf S}.

Absorption and scattering by arbitrary object

Consider arbitrary object illuminated by harmonic incident wave. Field in the medium surrounding the object can be represented as superposition of incident and scattered fields:

tex:{\bf E}={\bf E_i}+{\bf E_{s}}, \qquad
{\bf H}={\bf H_i}+{\bf H_{s}}.

Here we consider how to estimate energy scattered and absorbed by object. We construct imaginary closed surface around the object tex:A; the net rate at which electromagnetic energy crosses this surface is

tex:W_{abs}=-\int{\bf S}\cdot{\bf n} dA,

where tex:{\bf n} is normal to the surface.

If tex:W_{abs}>0, energy is absorbed within the volume confined by surface. If object is embedded in nonabsorbing environment, tex:W_{abs} is the rate at which energy is absorbed by object.

Time averaged Poynting vector tex:{\bf S} can be represented as (we omit here index tex:c for complex vectorstex:{\bf E} and tex:{\bf H})

tex:{\bf S}=\frac{1}{2}{\rm Re}({\bf E}\times{\bf H^*})={\bf S_i}+{\bf S_{sca}}+{\bf S_{ext}},

where

tex:{\bf S_i}=\frac{1}{2}{\rm Re}({\bf E_i}\times{\bf H_i^*}), \qquad
{\bf S_{sca}}=\frac{1}{2}{\rm Re}({\bf E_{s}}\times{\bf H_{s}^*}), \qquad
{\bf S_{ext}}=\frac{1}{2}{\rm Re}({\bf E_i}\times{\bf H_{s}^*}+{\bf E_{s}}\times{\bf H_i^*}).

Last term tex:{\bf S_{ext}} is a consequence of an interference between incident and scattered fields.

After integrating over surface we have

tex:W_{abs}=W_i-W_{sca}+W_{ext},

tex:W_i=-\int{\bf S_i}\cdot{\bf n} dA, \qquad
W_{sca}=\int{\bf S_{sca}}\cdot{\bf n} dA, \qquad
W_{ext}=-\int{\bf S_{ext}}\cdot{\bf n} dA.

For nonabsorbing environment tex:W_i=0, and

tex:W_{ext}=W_{abs}+W_{sca}.

These energy flow rates are linearly dependent on incident wave intensity tex:I. Their normalized values

tex:C_{ext}=\frac{W_{ext}}{I}, \qquad
C_{abs}=\frac{W_{abs}}{I}, \qquad
C_{sca}=\frac{W_{sca}}{I},

are extinction, absorption and scattering cross sections with dimensions of area.

We may define efficiencies for extinction, scattering and absorption

tex:Q_{ext}=\frac{C_{ext}}{G}, \qquad
Q_{abs}=\frac{C_{abs}}{G}, \qquad
Q_{sca}=\frac{C_{sca}}{G},

where tex:G is the object cross-sectional area projected onto a plane perpendicular to the incident beam (e.g. tex:G=\pi r^2 for a sphere of radius tex:r). Particles can scatter and absorb more light that is geometrically incident upon them (corresponding efficiencies are greater than unity) if their sizes are comparable or smaller than the incident wavelength.

The amplitude scattering matrix

This matrix is used to characterize angular distribution of scattered light.

Consider object that is illuminated by a harmonic wave.

The direction of propagation of the incident light defines tex:z axis, the forward direction. Any point in object may be chosen as the origin tex:O of a rectangular coordinate system, where tex:x and tex:y axes are orthogonal to tex:z axis and to each other but otherwise arbitrary. The orthonormal basis vectors tex:{\bf e_x}, tex:{\bf e_y}, tex:{\bf e_z} are in direction of positive tex:x, tex:y and tex:z axes.

The scattering direction tex:{\bf e_r} and forward direction tex:{\bf e_z} define a scattering plane. This plane is uniquely determined by the azimuthal angle tex:\phi, except when tex:{\bf e_r} is parallel to the tex:z axis. In this two instances (tex:{\bf e_r}=\pm{\bf e_z}) any plane containing tex:z axis is a suitable scattering plane.

It is convenient to resolve the incident electric field tex:{\bf E_i}, which lies in the tex:xy plane, into components parallel and perpendicular to the scattering plane

tex:{\bf E_i}=\left( E_{0\parallel}{\bf e_{\parallel{}i}}+E_{0\perp}{\bf e_{\perp{}i}} \right)
\exp(i\vec{k}\vec{z}-i\omega{}t)=
E_{\parallel{}i}{\bf e_{\parallel{}i}}+E_{\perp{}i}{\bf e_{\perp{}i}}

The orthonormal basis vectors

tex:{\bf e_{\perp{}i}}=\sin\phi{\bf e_x}-\cos\phi{\bf e_y},

tex:{\bf e_{\parallel{}i}}=\cos\phi{\bf e_x}+\sin\phi{\bf e_y}

form a right-handed triad with tex:{\bf e_z}:

tex:{\bf e_{\perp{}i}}\times{\bf e_{\parallel{}i}}={\bf e_z}

We also have

tex:{\bf e_{\perp{}i}}=-{\bf e_{\phi}}, \qquad {\bf e_{\parallel{}i}}=\sin\theta{\bf e_r}+\cos\theta{\bf e_{\theta}},

where tex:{\bf e_r}, tex:{\bf e_{\theta}}, tex:{\bf e_{\phi}} are orthonormal basis vectors associated with the spherical polar coordinate system tex:(r, \theta, \phi).

If the x and y components of the incident field are denoted by tex:E_{xi} and tex:E_{yi}, then

tex:E_{\parallel{}i}=\cos\phi{}E_{xi}+\sin\phi{}E_{yi},

tex:E_{\perp{}i}=\sin\phi{}E_{xi}-\cos\phi{}E_{yi}.

At sufficient distances from the origin (tex:{\bf kr}\gg{}1), in the far-field region, the scattered field is approximately transverse (tex:{\bf e_r} \cdot {\bf E_s} \sim 0) and has the asymptotic form

tex:{\bf E_s}=E_{\parallel{}s}{\bf e_{\parallel{}s}}+E_{\perp{}s}{\bf e_{\perp{}s}},

where

tex:{\bf e_{\parallel{}s}}={\bf e_{\theta}}, \qquad
{\bf e_{\perp{}s}}=-{\bf e_{\phi}}, \qquad
{\bf e_{\perp{}s}} \times {\bf e_{\parallel{}s}} = {\bf e_r}.

The basis vector tex:{\bf e_{\parallel{}s}} is parallel and tex:{\bf e_{\perp{}s}} is perpendicular to the scattering plane. Note, however, that tex:{\bf E_s} and tex:{\bf E_i} are specified relative to different set of basis vectors. Because of the linearity of Maxwell's equations, the relation between them can be written in matrix form

,

where tex:S_j (tex:j=1,2,3,4) are elements of the amplitude scattering matrix, and depend in general on scattering angle tex:\theta and azimuthal angle tex:\phi.

Experimental measurement of elements tex:S_j is difficult. However, the amplitude scattering matrix is related to elements of so called scattering matrix, the measurement of which poses considerable fewer experimental problems. Scattering matrix is real matrix tex:4\times{}4, with 7 independent elements, which can be expressed using absolute values tex:\left|S_j\right| (j=1,2,3,4) and phase differences between tex:S_j. One can find more detailed information in chapter 3.3 of book 1).

Transmission and reflection for planar layers of scatterers

Dipole radiative decay

Applications

EMTL was applied for various range of optical applications. Here is chosen publications list:

  • A. Deinega, I. Valuev, B. Potapkin and Yu. Lozovik, “Minimizing light reflection from dielectric textured surfaces,” JOSA A 28, 770 (2011) httpPDF
  • S. Zalyubovskiy et. al., “Theoretical limit of localized surface plasmon resonance sensitivity to local refractive index change and its comparison to conventional surface plasmon resonance sensor”, JOSA A 29, 994 (2012) httpPDF
  • A. Deinega, S. John, “Solar power conversion efficiency in modulated silicon nanowire photonic crystals”, J. Appl. Phys. 112, 074327 (2012) httpPDF
  • S. Belousov et. al., “Using metallic photonic crystals as visible light sources”, Phys. Rev. B 86, 174201 (2012) httpPDF
  • A. Deinega, S. Eyderman, S. John, “Coupled optical and electrical modeling of solar cell based on conical pore silicon photonic crystals”, J. Appl. Phys. 113, 224501 (2013) httpPDF
1) C. F. Bohren and D. R. Huffman: Absorption and Scattering of Light by Small Particles, Wiley-Interscience, New York (1983)
 
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