Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf
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NUMERICAL METHODS FOR RIGOROUS DIFFRACTION THEORY |
20.5FINITE DIFFERENCE TIME DOMAIN METHOD
The FDTD method is an effective numerical method increasingly used in applications. In this method, the Maxwell’s equations are represented in terms of central-difference equations. The resulting equations are solved in a leapfrog manner. In other words, the electric field is solved at a given instant in time, then the magnetic field is solved at the next instant in time, and the process is repeated iteratively many times.
Consider Eqs. (3.3-14) and (3.3-15) of Chapter 3 for a source-free region (no electric or magnetic current sources) rewritten below for convenience:
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@D |
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@E |
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r H ¼ |
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¼ e |
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ð3:3-14Þ |
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@t |
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¼ m |
@H |
ð3:3-15Þ |
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r E ¼ |
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@t |
@t |
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Eq. (3.3-14) shows that the time derivative of the E field is proportional to the Curl of the H field. This means that the new value of the E field can be obtained from the previous value of the E field and the difference in the old value of the H field on either side of the E field point in space.
Equations (3.3-14) and (3.3-15) can be written in terms of the vector components as follows:
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@Hx |
1 |
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@Ey |
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@Ez |
r0Hx |
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@t |
¼ |
m |
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@z |
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@y |
ð20:5-1Þ |
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@Hy |
1 |
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@Ez |
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@Ex |
r0Hy |
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@t |
¼ |
m |
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@x |
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@z |
ð20:5-2Þ |
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@Hz |
1 |
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@Ex |
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@Ey |
r0Hz |
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@t |
¼ |
m |
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@y |
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@x |
ð20:5-3Þ |
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@Ex |
1 |
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@Hz |
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@Hy |
sEx |
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@t |
¼ |
e |
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@y |
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@z |
ð20:5-4Þ |
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@Ey |
1 |
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@Hx |
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@Hz |
sEy |
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@t |
¼ |
e |
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@z |
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@x |
ð20:5-5Þ |
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@Ez |
1 |
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@Hy |
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@Hx |
sEz |
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@t |
¼ |
e |
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@x |
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@y |
ð20:5-6Þ |
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These equations are next represented in terms of finite differences, using the Yee algorithm discussed below.
20.5.1Yee’s Algorithm
Yee’s algorithm solves Maxwell’s curl equations by using a set of finite-difference equations [Yee]. Using finite differences, each electric field component in 3-D is
FINITE DIFFERENCE TIME DOMAIN METHOD |
369 |
Figure 20.3. Electric and magnetic field vectors in a Yee cell [Yee].
surrounded by four circulating magnetic field components, and every magnetic field component is surrounded by four circulating electric field components as shown in Figure 20.3. The electric and magnetic field components are also centered in time in a leapfrog arrangement. This means all the electric components are computed and stored for a particular time using previously stored magnetic components. Then, all the magnetic components are determined using the previous electric field components [Kuhl, Ersoy].
With respect to the Yee cell, discretization is done as follows:
½i; j; k& ¼ ði x; j y; k zÞ where x; y; and z are the space increments in the x, y, and z directions, respectively, and i, j, k are integers.
tn ¼ n t
uði x; j y; k z; n tÞ ¼ uni;j;k
Yee’s centered finite-difference expression for the first partial space derivative of u in the x-direction, evaluated at time tn ¼ n t is given by [Yee]
@u |
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uin |
1=2;j;k uin |
1=2; j;k |
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2 |
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ði x; j y; k z; n tÞ ¼ |
þ |
x |
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þ Ohð xÞ |
i |
ð20:5-7Þ |
@x |
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It is observed that the data a distance x=2 away from the point in question is used, similarly to the Crank–Nicholson method. The first partial derivative of u with respect to time for a particular space point is also given by
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uin;þj;k1=2 uin; j;k1=2 |
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2 |
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ði x; j y; k z; n tÞ ¼ |
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þ Ohð tÞ |
i |
ð20:5-8Þ |
@t |
t |