Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf

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372

NUMERICAL METHODS FOR RIGOROUS DIFFRACTION THEORY

Figure 20.4. 1-D FZP with m ¼ 3.

When the modeled region extends to infinity, absorbing boundary conditions (ABCs) are used at the boundary of the grid. This allows all outgoing waves to leave the region with negligible reflection. The region of interest can also be enclosed by a perfect electrical conductor.

Once the fields are calculated for the specified number of time steps, near zone transient and steady state fields can be visualized as color intensity images, or a field component at a specific point can be plotted versus time. When the steady-state output is desired, observing a specific point over time helps to indicate whether a steady-state has been reached.

In the computer experiments performed, a cell size of l=20 was used [Kuhl, Ersoy]. The excitation was a y-polarized sinusoidal plane wave propagating in the z-direction. All diffracting structures were made of perfect electrical conductors. All edges of the diffracting structures were parallel to the x-axis to avoid canceling the y-polarized electric field.

A 1-D FZP is the same as the FZP discussed in Section 15.10 with the x-variable dropped. The mode m is defined as the number of even or odd zones which are opaque.

An example with three opaque zones (m ¼ 3) is shown in Figure 20.4.

Using XFDTD, a 1-D FZP with a thickness of 0:1l and focal length 3l was simulated. Its output was analyzed for the first three modes. The intensity along the z-axis passing through the center of the FZP is plotted as a function of distance from the FZP in Figure 20.5. The plot shows that the intensity peaks near 3l behind the plate, and gets higher and narrower as the mode increases. The peak also gets closer to the desired focal length of 3l for higher modes.

The plot of the intensity along the y-axis at the focal line is shown in Figure 20.6. The plot shows that the spot size decreases with increasing mode and side lobe intensity is reduced for higher modes.

These experiments show that the FDTD method provides considerable freedom in generating the desired geometry and specifying material parameters. It is especially useful when the scalar diffraction theory cannot be used with sufficient accuracy due to reasons such as difficult geometries and/or diffracting apertures considerably smaller than the wavelength.

COMPUTER EXPERIMENTS

373

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 20.5. FDTD results for intensity along the z-axis.

Figure 20.6. FDTD results for intensity along the y-axis on the focal line.

374

NUMERICAL METHODS FOR RIGOROUS DIFFRACTION THEORY

20.7FOURIER MODAL METHODS

Fourier modal methods are among the several most versatile rigorous methods to analyze surface relief DOEs. The initial work in this area is known as rigorous coupled-wave analysis (RCWA) [Moharam, 82]. It was mostly used initially with diffraction gratings, but they were later generalized to aperiodic structures as well [Lalanne–Silberstein]. In the introductory discussion in this section, we will assume a grating structure.

The FMM is a frequency domain method involving computations of the grating modes as eigenvectors, Fourier expansions of the permittivity and of the EM fields inside the grating region. The Fourier expansion of the field inside the grating generates a system of differential equations. After finding the eigenvalues and eigenvectors of this system, the boundary conditions at the grating interfaces are matched to compute the diffraction efficiencies [Lalagne and Morris 1996].

The geometry to be used to explain the method in 2-D is shown in Figure 20.7. The periodic grating with period is along the x-axis with a permittivity function eðxÞ. The TM mode is considered such that the magnetic field is polarized in the y- direction, and there are no variations along the y-direction. The incident plane wave makes an angle y with the z-direction. In addition, the following Fourier series

definitions are made:

 

eðxÞ

1

 

 

e jKkx

 

20:7-1

 

 

¼ k¼X1 ek

ð

Þ

 

eo

 

 

 

 

 

o

1

 

 

 

 

 

 

 

e

 

¼ k¼X1 ake jKkx

ð20:7-2Þ

eðxÞ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 20.7. The geometry used in the FMM method [Lalanne and Morris 1996].

FOURIER MODAL METHODS

375

where K ¼ 2p= . The field components can be written as [Lalanne and Morris 1996]

X

 

 

Ex ¼

 

m

 

SmðzÞe jðKmþbÞx

ð20:7-3Þ

 

 

 

 

 

 

 

 

 

 

 

Ez ¼

X

 

ð20:7-4Þ

 

 

 

m

 

fmðzÞe jðKmþbÞx

 

 

 

 

 

 

 

 

 

 

 

Hy ¼

X

 

ð20:7-5Þ

 

 

 

m

 

UmðzÞe jðKmþbÞx

 

 

 

 

 

 

 

 

 

where b ¼ k sin y ¼

2p

y.

 

 

 

 

 

 

 

l

 

 

 

 

 

 

 

Maxwell’s curl equations in this case are given by

 

 

 

 

dEz

þ

dEx

¼ jwmoHy

ð20:7-6Þ

 

 

dx

 

dx

 

 

 

 

 

 

 

dHy

¼ jweEx

ð20:7-7Þ

 

 

 

 

 

 

 

 

dz

 

 

 

 

1 dHy

¼ jwEz

ð20:7-8Þ

 

 

 

 

 

e

 

 

dx

Prime and double prime quantities will be used to denote the first and second partial derivatives with respect to the z-variable. Using Eqs. (20.7-3)–(20.7-5) in Eqs. (20.7-6)–(20.7-8) results in

jðKm þ bÞfm þ Sm0 ¼ jkoUm

 

ð20:7-9Þ

 

 

Um0 ¼ jko Xp

em pSp

ð20:7-10Þ

1

Xp ðpK þ bÞam pUp

 

 

 

fm ¼

 

ð20:7-11Þ

ko

Using Eq. (20.7-11) in Eq. (20.7-9) yields

 

 

 

Sm0 ¼ jkoUm þ

j

ðKm þ bÞ Xp

ðpK þ bÞam pUp

ð20:7-12Þ

ko

In practice, the sums above are truncated with jpj M. Then, Eqs. (20.7-10) and (20.7-12) make up an eigenvalue problem of size 2ð2M þ 1Þ. However, it turns out to be more advantageous computationally to compute the second partial derivative of Um with Eq. (20.7-10) and obtain [Caylord, Moharam, 85], [Peng]

X

 

 

 

X

1

 

Um00 ¼ ko2( p

1

 

)( r

ðrK þ bÞap rUr) ð20:7-13Þ

em p Up

ko

ðpK þ bÞ

ko

376

 

72

 

70

(%)

68

intensity

66

Transmitted

64

 

 

62

60

580

NUMERICAL METHODS FOR RIGOROUS DIFFRACTION THEORY

69.822

69.229

50

100

150

200

250

300

350

400

Number of retained orders

Figure 20.8. Diffraction efficiency of the transmitted zeroth order beam from a metallic grating with TM polarized light [Courtesy of Lalanne and Morris, 1996].

Equation (20.7-13) can be written in matrix form as

 

1

U00 ¼ EðKxE 1Kx IÞU

ð20:7-14Þ

 

 

k2

 

o

 

where all the bold upper case letters represent matrices; I is the identity matrix, E is the permittivity harmonic coefficients matrix, and Kx is a diagonal matrix whose ith diagonal element equals ðiK þ bÞ=ko.

A more efficient form of Eq. (20.7-14) computationally is given by [Lalanne

1996]

 

 

 

1

U00 ¼ EðEKxE 1Kx IÞU

ð20:7-15Þ

 

 

 

k2

 

o

 

As an example, Figure 20.8 shows the diffraction efficiency of the zeroth order beam of a metallic grating with TM polarized light with respect to the number of Fourier coefficients kept [Lalagne, Morris, 1996]. The solid curve and the curve with circles were obtained with Eqs. (20.7-14) and (20.7-15), respectively.

The coverage above is introductory at best. There are currently a number of refinements of the Fourier modal method for different applications, and further refinements are expected

Appendix A

The Impulse Function

The impulse (delta or Dirac delta) function dðtÞ can be regarded as the idealization of a very narrow pulse with unit area. Consider the finite pulse shown in Figure A.1. It is defined by

 

8

1

 

a

< t <

a

 

 

 

x t

 

 

 

 

ð

A:1-1

Þ

a

2

2

 

:

0

otherwise

 

 

ð Þ ¼ <

 

 

The area under the pulse is 1 and remains as 1 for all values of a. The impulse function can be defined as

dðtÞ ¼ lim xðtÞ

ðA:1-2Þ

a!0

dðtÞ satisfies

1ð

dðtÞdt ¼ 1

ðA:1-3Þ

1

dðtÞ ¼ 0 t 6¼ 0

In mathematics, dðtÞ is considered to be not an ordinary function, but a generalized function or a distribution, as discussed in the following sections. dðtÞ has the sampling (sifting) property given by

t2

t2

 

tð1 dðt tÞhðtÞdt ¼ tð1 dðt tÞhðtÞdt ¼ hðtÞ

ðA:1-4Þ

where h(t), a given function, is continuous at t ¼ t, and t1 < t < t2.

It is observed that if h(t) is the impulse response of a LTI system, application of dðtÞ to the system leads to the identification of h(t).

Diffraction, Fourier Optics and Imaging, by Okan K. Ersoy

Copyright # 2007 John Wiley & Sons, Inc.

377

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