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

38	FUNDAMENTALS OF WAVE PROPAGATION

where the real parts are actually the physical solutions. They can be more generally written as

Eðr; tÞ ¼ E0 cosðk r wtÞ

ð3:7-2Þ

Hðr; tÞ ¼ H0 cosðk r wtÞ;

where E0 and H0 have components (Ex,Ey), (Hx,Hy), and k, r in this case are simply

k ¼ 2p=l and z, respectively.			@H
Substituting Eqs. (3.7-1) in to r E ¼ m					gives the following:
			@t
kEy^ex kEx^ey ¼ mw½Hx^ex þ Hy^ey&						ð3:7-3Þ
Hence,
Hx ¼	1	Ey
						ð3:7-4Þ

1

Hy ¼ Ex;

where is called the characteristic impedance of the medium. It is given by

	¼ k m ¼		m ¼ p		ð		Þ
	w	v		m=e		3:7-5
Note that
E H ¼ ðExHx þ EyHyÞejðkzþwtÞ ¼ 0					ð3:7-6Þ
Thus, the electric and magnetic fields are orthogonal to each other.
The Poynting vector S is defined by
	S ¼ E H;				ð3:7-7Þ

which has units of W/m2, indicating power flow per unit area in the direction of propagation.

Polarization indicates how the electric field vector varies with time. Again assuming the direction of propagation to be z, the electric field vector including time dependence can be written as

E ¼ Real½ðExex þ EyeyÞejðkzþwtÞ&	ð3:7-8Þ
Ex and Ey can be chosen relative to each other as
Ex ¼ Ex0	ð3:7-9Þ
Ey ¼ Ey0 ej ;	ð3:7-10Þ

PLANE EM WAVES

39

where Ex0 and Ey0 are positive scalars, and is the relative phase, which decides the direction of the electric field.

Linear polarization is obtained when ¼ 0 or p. Then, we get
E ¼ ðEx0 ex Ey0 eyÞ cosðkz þ wtÞ	ð3:7-11Þ

In this case, ðEx0 ex Ey0 eyÞ can be considered as a vector that does not change in direction with time or propagation distance.

Circular polarization is obtained when ¼ p=2; and E0 ¼ Ex0 ¼ Ey0 . Then, we get

E ¼ E0 cosðkz þ wtÞex E0 sinðkz þ wtÞey

ð3:7-12Þ

We note that jEj ¼ E0. For ¼ p=2, E describes a circle rotating clockwise during propagation. For ¼ þp=2, E describes a circle rotating counterclockwise during propagation.

Elliptic polarization corresponds to an arbitrary . Equation (3.7-1) for the electric field can be written as

E ¼ Exex þ Eyey

ð3:7-13Þ

where

Ex ¼ Ex0 cosðkz þ wtÞ

ð

3

:

7-14

Þ

Ey ¼ Ey0 cosðkz þ wt Þ

Equation (3.7-14) can be written as

Ey

¼ cosðkz þ wtÞ cos þ sinðkz þ wtÞ sin

Ey0

ð

3:7-15

Þ

cos þ "1

# sin

E

E

2

¼

x

x

Ex0

Ex0

This equation can be further written as

E02

2E0

E0 cos

þ

E02

¼

sin2

;

ð

3:7-16

Þ

x

x

y

y

where

Ex0

¼

Ex

Ey0 ¼

Ey

ð3:7-17Þ

;

Ex0

Ey0

Equation (3.7-17) is the equation of an ellipse.

40	FUNDAMENTALS OF WAVE PROPAGATION

Suppose the field is linearly polarized and E is along the x-direction. Then, we can write

E ¼ E0ejðkzþwtÞex					ð3:7-18Þ
	E0 j kz			wt
H ¼		e ð	þ		Þey;

where Real [ ] is assumed from the context. Intensity or irradiance I is defined as the time-averaged power given by

		2p=w		E2
	w	ð0
I ¼			jSjdt ¼ em	0	;	ð3:7-19Þ
	2p			2

where S is the Poynting vector. It is observed that the intensity is proportional to the square of the field magnitude. This will be assumed to be true in general unless otherwise specified.

4

Scalar Diffraction Theory

4.1INTRODUCTION

When the wavelength of a wave field is larger than the ‘‘aperture’’ sizes of the diffraction device used to control the wave, the scalar diffraction theory can be used. Even when this is not true, scalar diffraction theory has been found to be quite accurate [Mellin and Nordin, 2001]. Scalar diffraction theory involves the conversion of the wave equation, which is a partial differential equation, into an integral equation. It can be used to analyze most types of diffraction phenomena and imaging systems within its realm of validity. For example, Figure 4.1 shows the diffraction pattern from a double slit illuminated with a monochromatic plane wave. The resulting wave propagation can be quite accurately described with scalar diffraction theory.

In this chapter, scalar diffraction theory will be first derived for monochromatic waves with a single wavelength. Then, the results will be generalized to nonmonochromatic waves by using Fourier analysis and synthesis in the time direction.

This chapter consists of eight sections. In Section 4.2, the Helmholtz equation is derived. It characterizes the spatial variation of the wave field, by characterizing the time variation as a complex exponential factor. In Section 4.3, the solution of the Helmholtz equation in homogeneous media is obtained in terms of the angular spectrum of plane waves. This formulation also characterizes wave propagation in a homogeneous medium as a linear system. The FFT implementation of the angular spectrum of plane waves is discussed in Section 4.4.

Diffraction can also be treated by starting with the Helmholtz equation and converting it to an integral equation using Green’s theorem. The remaining sections cover this topic. In Section 4.5, the Kirchoff theory of diffraction results in one formulation of this approach. The Rayleigh–Sommerfeld theory of diffraction covered in Sections 4.6 and 4.7 is another formulation of the same approach. The Rayleigh–Sommerfeld theory of diffraction for nonmonochromatic waves is treated in Section 4.8.

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

Copyright # 2007 John Wiley & Sons, Inc.

41

42	SCALAR DIFFRACTION THEORY

Figure 4.1. Diffraction from a double slit [Wikipedia].

4.2HELMHOLTZ EQUATION

Monochromatic waves have a single time frequency f. As

w			¼	2pf	ð4:2-1Þ
k ¼
	c			c

the wavelength l is also fixed. For such waves, the plane wave solution discussed in Section 3.7 can be written as

uðr; tÞ ¼ AðrÞ cosðot þ ðrÞÞ

ð4:2-2Þ

where A(r) is the amplitude, and (r) is the phase at r. In phasor representation, this becomes

uðr; tÞ ¼ Re½UðrÞe jot&

ð4:2-3Þ

where the phasor U(r) also called the complex amplitude equals AðrÞej ðrÞ. Equation (4.2-3) is often written without explicitly writing ‘‘the real part’’ for the sake of simplicity as well as simplicity of computation as in the next equation.

Substituting u(r,t) into the wave equation (3.5-7) yields

ðr2 þ k2ÞUðrÞ ¼ 0

ð4:2-4Þ

where k ¼ o=c. This is called the Helmholtz equation. It is same as Eq. (3.6-9). Thus, in the case of EM plane waves, UðrÞ represents a component of the electric field or magnetic field phasor, as discussed in Chapter 3. The Helmholtz equation is valid for all waves satisfying the nondispersive wave equation. For example, with acoustical and ultrasonic waves, UðrÞ is the pressure or velocity potential.

If Uðr; tÞ is not monochromatic, it can be represented in terms of its time Fourier transform as

1ð

Uðr; tÞ ¼ Uf ðr; f Þe j2pftdf

ð4:2-5Þ

1

HELMHOLTZ EQUATION

43

where

	1
Uf ðr; f Þ ¼	ð	Uðr; tÞe j2pftdt	ð4:2-6Þ
	1

Substituting Eq. (4.2-6) into the wave equation (3.5-7) again results in the Helmholtz equation for Uf ðr; f Þ.

EXAMPLE 4.1 Show that a spherical wave, defined by e jkr=r where

pr

r ¼

2

2

2

x

þ y2

þ z

, is a solution of the Helmholtz equation.

Solution:

in spherical coordinates is given by

@2

1 @

1 @2

@2

r2 ¼

þ

þ

þ

@r2

r

@r

r2

@ 2

@y2

e jkr=r is spherically symmetric. In this case, r2 in polar coordinates is simplified to

r2	1 @				r2	@
	¼
		r2		@r		@r

The Helmholtz equation with a spherically symmetric wave function U(r) becomes

1 @			r2	@	UðrÞ þ k2UðrÞ ¼ 0
r @r				@r

Let U(r) be of the form f ðrÞ=r. It is not difficult to show by partial differentiation that

1 @

@

f

r

1 @2f r

r2

ð

Þ

¼

ð

Þ

r2 @r

@r

r

r2

@r2

The Helmholtz equation becomes

1

@2f ðrÞ

þ

K2

f ðrÞ

¼

0

@r2

r

r2

The solution of this equation is given by

f ðrÞ ¼ e jkr

Thus, UðrÞ ¼ e jkr=r:

44 SCALAR DIFFRACTION THEORY

EXAMPLE 4.2 Find a similar solution in cylindrical coordinates. Solution: In cylindrical coordinates, r2 is given by

r2

@2

1 @

1 @2

@2

¼

þ

þ

þ

@r2

r

@r

r2

@ 2

@z2

Assuming a solution with cylindrical symmetry, Uðr; ; zÞ becomes Uðr; zÞ. Cylindrical symmetry means the wavefronts are circles for constant z. In this case, there are no simple solutions. The exact solution has a Bessel-function type of dependence on r. It can also be shown that

C jkr

Uðr; zÞ p e r

approximately satisfies the wave equation. Such a wave is called a cylindrical wave.

4.3ANGULAR SPECTRUM OF PLANE WAVES

We will consider the propagation of the wave field Uðx; y; zÞ in the z-direction. The wave field is assumed to be at a wavelength l such that k ¼ 2p=l. Let z ¼ 0 initially. The 2-D Fourier representation of U(x,y,0) is given in terms of its Fourier transform

Aðfx; fy; 0Þ by

Uðx; y; 0Þ ¼ ðð		Að fx; fy; 0Þe j2pðfxxþfyyÞdfxdfy	ð4:3-1Þ
	1
	1
where	ðð	Uðx; y; 0Þe j2pðfxxþfy yÞdxdy	ð4:3-2Þ
Að fx; fy; 0Þ ¼
	1
	1
Including time variation, Að fx; fy; 0Þej2pðfxxþfy yþftÞ is a plane wave			at z ¼ 0,

propagating with direction cosines ai given by Eqs. (3.5-15)–(3.5-17). Að fx; fy; 0Þ is called the angular spectrum of Uðx; y; 0Þ.

Consider next the wave field Uðx; y; zÞ. Its angular spectrum Að fx; fy; zÞ is given by

Að fx; fy; zÞ ¼		ðð Uðx; y; zÞe j2pð fxxþfyyÞdxdy	ð4:3-3Þ
		1
		1
and its Fourier representation in terms of its angular spectrum is given by			ð4:3-4Þ
Uðx; y; zÞ ¼	ðð Að fx; fy; zÞe j2pð fxxþfyyÞdfxdfy
		1

1

ANGULAR SPECTRUM OF PLANE WAVES

45

Uðx; y; zÞ satisfies the Helmholtz equation at all points without sources, namely,

			r2Uðx; y; zÞ þ k2Uðx; y; zÞ ¼ 0		ð4:3-5Þ
Substitution of Uðx; y; zÞ from Eq. (4.3-4) into Eq. (4.3-5) yields
ðð	dz2 Að fx; fy; zÞ þ ðk2			4p2ð fx2 þ fy2ÞÞAð fx; fy; zÞ e j2pð fxxþfy yÞdfxdfy ¼ 0
1		d2
1

This is true for all waves only if the integrand is zero:

d22 Að fx; fy; zÞ þ ðk2 4p2ð fx2 þ fy2ÞÞAð fx; fy; zÞ ¼ 0 dz

This differential equation has the solution

Að fx; fy; zÞ ¼ Að fx; fy; 0Þe jmz

where

q

m ¼ k2 4p2ð fx2 þ fy2Þ ¼ kz

ð4:3-6Þ

ð4:3-7Þ

ð4:3-8Þ

ð4:3-9Þ

If 4p2ð fx2 þ fy2Þk2, m is real, and each angular spectrum component is just modified by a phase factor e jmz. Plane wave components satisfying this condition are known as homogeneous waves.

If 4p2ð fx2 þ fy2Þk2, then m can be written as

q
m ¼ j 4p2ð fx2 þ fy2Þ k2	ð4:3-10Þ
and Eq. (4.3-8 ) becomes
Að fx; fy; zÞ ¼ Að fx; fy; 0Þe mz	ð4:3-11Þ

This result indicates that the amplitudes of such plane wave components are strongly attenuated by propagation in the z-direction. They are called evanescent waves.

If 4p2ð fx2 þ fy2Þ ¼ k2, Að fx; fy; zÞ is the same as Að fx; fy; 0Þ. Such components correspond to plane waves traveling perpendicular to the z-axis.

Knowing Að fx; fy; zÞ in terms of Að fx; fy; 0Þ allows us to find the wave field at ðx; y; zÞ by using Eq. (4.3-11) in Eq. (4.3-4 ):

ðð1 p

Uðx; y; zÞ ¼

Að fx; fy; 0Þe jz k2 4p2ð fx2þfy2Þe j2pð fxxþfyyÞdfxdfy

ð4:3-12Þ

1

46	SCALAR DIFFRACTION THEORY

Thus, if Uðx; y; 0Þ is known, Að fx; fy; 0Þ can be computed, followed by the computation of Uðx; y; zÞ. The limits of integration in Eq. (4.3-12 ) can be limited to a circular region given by

4p2ð fx2 þ fy2Þ k2

ð4:3-13Þ

provided that the distance z is at least several wavelengths long so that the evanescent waves may be neglected.

Under these conditions, Eq. (4.3-8) shows that wave propagation in a homogeneous medium is equivalent to a linear 2-D spatial filter with the transfer function given by

(p

H	ð	fx; fy	Þ ¼	e jz k2 4p2ð fxþfy2Þ	4p2ð fx2 þ fy2Þk2	ð	4:3-14	Þ
				0	otherwise

This is schematically shown in Figure 4.1. The impulse response, which is the inverse Fourier transform of this transfer function, is discussed in Example 4.6.

Propagation of a wavefield in the z-direction in a source-free space is correctly described by the propagation of the angular spectrum in the near field as well as the far field. Two other ways to characterize such propagation are in terms of the Fresnel and Fraunhofer approximations, discussed in Chapter 5. However, they are valid only under certain constraints.

Let F[ ] and F 1[ ] denote the forward and inverse Fourier transform operators, respectively. In terms of these operators, Eq. (4.3-12) can be written as

hpi

Uðx; y; zÞ ¼ F 1 F½Uðx; y; 0Þ&ejkz	1 ax2	ay2	ð4:3-15Þ
Aðfx; fy; 0Þ ¼ F½Uðx; y; 0Þ& for particular values of	fx	and fy	is the complex

amplitude of a plane wave traveling in the direction specified by the direction cosines ax ¼ 2pfx, ay ¼ 2pfy, and az ¼ 2pfz, where fz ¼ 21p ½1 a2x a2y &1=2. The

effect of propagation is to modify the relative phases of the various plane waves by
e		p x y	without changing their amplitudes.
	jkz	1 a2 a2

EXAMPLE 4.3 Find the angular spectrum of a wave field at plane z ¼ a (constant)

if the wave field results from a plane wave Be jkx0 xe jkz0 z passing through a circular aperture of diameter d at z ¼ 0.

Solution: The wave at z ¼ 0 can be written as

Uðx; y; 0Þ ¼ Bejkx0 xcyl r

d	p
		!
¼ Bej2pfx0 xcyl	x2 þ y2
	d

FFT IMPLEMENTATION OF THE ANGULAR SPECTRUM OF PLANE WAVES

47

The FTs of the two factors above are

Be j2pfx0 x $ Bdð fx

fx0 ; fyÞ

r

d2

cyl

$

p

sombðd Þ

d

4

By convolution theorem, the angular spectrum of Uðx; y; 0Þ is given by

Að fx; fy; 0Þ ¼ Bdð fx

fx0 ; fyÞ

d2p

sombðd Þ

4

2

q

¼ Bdð fx

fx0 ; fyÞ

d p

somb d

fx2 þ fy2

4

d2pB

q

¼

4

somb d

ð fx

fx0 Þ2 þ fy2

It is observed that the single plane wave component before the circular aperture is changed to a spectrum with an infinite number of such components after the aperture. They are mostly propagating in the direction of the incident wave field, with a spread given by the sombrero function. This spread is inversely proportional to d, the diameter of the circular aperture.

At a distance z, the angular spectrum becomes

	d2pB		q
Að fx; fy; zÞ ¼	4	somb d	ð fx fx0 Þ2 þ fy2	e jzpk		4p	ð fx	þfy	Þ

4.4 FAST FOURIER TRANSFORM (FFT) IMPLEMENTATION OF THE ANGULAR SPECTRUM OF PLANE WAVES

The angular spectrum of plane waves relating Uðx; y; zÞ to Uðx; y; 0Þ can be implemented by the fast Fourier transform (FFT) algorithm [Brigham, 1974] after discretizing and truncating the space variables and the spacial frequency variables. For a discussion of the discrete Fourier transform and the FFT, see Appendix B. The discretized and truncated variables in the space domain and the spatial frequency domain are given by

x ¼ x n1

y ¼ y n2

ð4:4-1Þ

fx ¼ fx m1 fy ¼ fy m2