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

58	SCALAR DIFFRACTION THEORY

	P0	r	01				r01	P0
				A

n

P1

Figure 4.9. Rayleigh–Sommerfeld modeling of diffraction by a plane aperture.

Green’s function:

	e	jkr01		e	jkr01
G2ðrÞ ¼							ð4:6-1Þ
	r01			r01

where r01 is the distance from P0 to P1, P0 being the mirror image of P0 with respect

to the initial plane. This is shown in Figure 4.9.

The use of the second Green’s function leads us to the first Rayleigh–Sommerfeld diffraction formula given by

Uðx0; y0; zÞ ¼ j1l		ðð	Uðx; y; 0Þ r01 er01				dxdy	ð4:6-2Þ
		1		z jkr01
		1

Another derivation of this equation by using the convolution theorem is given in the next section.

Equation (4.6-2) shows that Uðx0; y0; zÞ may be interpreted as a linear superposition of diverging spherical waves, each of which emanates from a point

ðx; y; 0Þ and is weighted by 1 z Uðx; y; 0Þ. This mathematical statement is also

jl r01

known as the Huygens–Fresnel principle.

It is observed that Eq. (4.6-2 ) is a 2-D linear convolution with respect to x,y:

Uðx0; y0; zÞ ¼ Uðx; y; 0Þ hðx; y; zÞ

ð4:6-3Þ

where the impulse response hðx; y; zÞ is given by

jkz 1

2

2

1=2

1

e

þ

x

þy

z2

hðx; y; zÞ ¼ jlz

þ

z2

y2

ð4:6-4Þ

1

x2

þ

The transfer function Hð f1; f2; zÞ, the 2-D Fourier transform of hðx; y; zÞ, is given by

Hð fx; fy; zÞ ¼ e jz½k2 4p2ð fx2þfyÞ&1=2

ð4:6-5Þ

ANOTHER DERIVATION OF THE FIRST RAYLEIGH–SOMMERFELD

59

This is the same as the transfer function for the propagation of angular spectrum of plane waves discussed in Section 4.5. Thus, there is complete equivalence between the Rayleigh–Sommerfeld theory of diffraction and the method of the angular spectrum of plane waves.

4.6.1The Kirchhoff Approximation

Incorporating the Kirchhoff approximation into the first Rayleigh–Sommerfeld integral yields

Uðx; y; zÞ ¼ jl ððA		Uðx; y; 0Þ r01		r01	dxdy	ð4:6-6Þ
1			z	e jkr01

z=r01 is sometimes written as cosðn; r01Þ, indicating the cosine of the angle between the z-axis and r01 shown in Figure 4.9.

4.6.2The Second Rayleigh–Sommerfeld Diffraction Formula

Another valid Green’s function that can be used instead of G2ðrÞ is given by

e jkr01

e jkr010

G3ðrÞ ¼

þ

ð4:6-7Þ

U2

r01

r010

Þ

r01

ðx; y; zÞ ¼ 2p ððA

d

ðd;n ;

dxdy

ð4:6-8Þ

1

U

x y

0

e jkr01

dUðx;y;0Þ

where

is the partial derivative of Uðx; y; 0Þ in the normal direction (z-direction).

dn

It can be shown that the Kirchoff solution discussed in Section 4.5 is the arithmetic average of the first and second Rayleigh–Sommerfeld solutions [Goodman, 2004].

4.7 ANOTHER DERIVATION OF THE FIRST RAYLEIGH–SOMMERFELD DIFFRACTION INTEGRAL

We assume that the field UðrÞ is due to sources in the half space z0 and that it is known in the plane z ¼ 0. We want to determine UðrÞ for z0. Uðx; y; 0Þ in terms of its angular spectrum is given by

Uðx; y; 0Þ ¼	ðð	Aðfx; fy; 0Þej2pðfxxþfyyÞdfxdfy	ð4:7-1Þ
	1
	1

In terms of the convolution theory, this equation can be interpreted as finding the output of a LSI system whose transfer function is unity. The impulse response

60 SCALAR DIFFRACTION THEORY

of the system is given by	ðð		ð4:7-2Þ
hðx; y; 0Þ ¼		ej2pðfxxþfyyÞdfxdfy
	1

1

What if z ¼6 0? hðx; y; zÞ consistent with Eq. (4.7-1) and other derivations of

diffraction integrals as in Section 4.6 is given by			ð4:7-3Þ
hðx; y; zÞ ¼ 4p2	ðð	e jk rdkxdky
1	1
	1

where k and r are given by Eqs. (3.2-28) and (3.2-30), respectively. Equation (4.7-3) can be written as

p

p

ðð

z

1

@

2

j

1 e jk

r

hðx; y; zÞ ¼

dkxdky&

ð4:7-4Þ

2 @z

2

1

k

4

The quantity inside the brackets can be shown to be the plane wave expansion of a spherical wave [Weyl]:

er

¼ 2jp

ðð ekz

r

dkxdky

ð4:7-5Þ

jkr

1

jk

1

Substituting this result in Eq. (4.7-4) yields

1

@

e jkr

hðx; y; zÞ ¼

ð4:7-6Þ

2p

@z

r

The above equation is the same as

hðx; y; zÞ ¼

1 z e jkr

ð4:7-7Þ

jl

r

r

The diffracted field Uðx; y; zÞ is interpreted as the convolution of Uðx; y; 0Þ with hðx; y; zÞ. Thus,

Uðx; y; zÞ ¼ jl		ðð	Uðx; y; 0Þ r01			r01	dxdy	ð4:7-8Þ
1		1		z	e jkr01
		1

Using Eq. (4.8-6), this result is sometimes written as

1 1					d		jkr
Uðx; y; zÞ ¼	1	ð	ð	Uðx; y; 0Þ			e	01	dxdy	ð4:7-9Þ
	2p				dz		r01
1			1

THE RAYLEIGH–SOMMERFELD DIFFRACTION INTEGRAL

61

4.8 THE RAYLEIGH–SOMMERFELD DIFFRACTION INTEGRAL FOR NONMONOCHROMATIC WAVES

In the case of nonmonochromatic waves, Uðr; tÞ is represented in terms of its Fourier transform Uf ðr; f Þ ¼ Uf ðx; y; z; f Þ as in Eq. (4.2-5). By substituting f 0 ¼ f , this equation can also be written as

	1
Uðx; y; z; tÞ ¼	ð	Uf ðx; y; z; f 0Þe j2pf 0tdf 0	ð4:8-1Þ
	1

Uf ðx; y; z; f 0Þ satisfies the Rayleigh–Sommerfeld integral at an aperture:

Uf ðx; y; z;

f 0Þ ¼ jl ððA

Uf ðx; y; 0;

f 0Þ r01

r01

dxdy

ð4:8-2Þ

1

z

e jkr01

Substituting this result in Eq. (4.8-1) yields

Uðx; y; z; tÞ ¼

ð

l

ðð

01

01

dxdy&e

ð4:8-3Þ

1

2 j1

Uf ðx; y; 0; f 0

Þ rz

erjkr01

j2pf 0tdf 0

1

4

A

Note that

lf ¼ c

ð4:8-4Þ

where c is the phase velocity of the wave. Using this relation and exchanging orders of integration in Eq. (4.9-3) results in

Uðx; y; z; tÞ ¼ ðð 2pcr012

2 ð

j2pf 0Uf ðx; y; 0;

f 0Þe j2pf 0ðt c Þdf 03dxdy ð4:8-5Þ

z

1

r01

A

4

1

5

As

1

d

d

ð

f 0Þe

j2pf 0tdf 0

Uðx; y; 0; tÞ ¼

Uf ðx; y; 0;

dt

dt

1

1

¼

ð

j2pf 0Uf ðx; y; 0;

f 0Þe j2pf 0tdf 0

1

62

SCALAR DIFFRACTION THEORY

Eq. (4.8-5) can also be written as

ð

Þ ¼

ðð

h

ð

ð

Þ

2pc

r012

dt

c

U x; y; z; t

z

A

i

4:8-6

1 d

U x; y; 0; t

r01 dxdy

Thus, the wave at ðx; y; zÞ, z0, is related to the time derivative of the wave at ðx; y; 0Þ with a time delay of r01=c, which is the time for the wave to propagate from P1 to P0.

5

Fresnel and Fraunhofer Approximations

5.1INTRODUCTION

Fresnel and Fraunhofer approximations of the scalar diffraction integral allow simpler Fourier integral computations to be used for wave propagation. They also allow different input and output plane window sizes. However, they are valid only in certain regions, not very close to the input aperture plane. The valid regions for the Rayleigh–Sommerfeld integral, Fresnel, and Fraunhofer approximations are shown in Figure 5.1.

The Rayleigh–Sommerfeld region is observed to be the entire half-space to the right of the input diffraction plane. The Fresnel and Fraunhofer regions are parts of the Rayleigh–Sommerfeld region. Approximate bounds indicating where they start will be derived below as Eqs. (5.25) and (5.27) for the Fresnel region and Eq. (5.41) for the Fraunhofer region. However, these bounds need to be interpreted with care. See Chapter 7 for further explanation.

The term far field usually refers to the Fraunhofer region. The term near field can be considered to be the region between the input diffraction plane and the Fraunhofer region.

This chapter consists of seven sections. Section 5.2 describes wave propagation in the Fresnel region. The FFT implementation of wave propagation in the Fresnel region is covered in Section 5.3. The fact that the Fresnel approximation is actually the solution of the paraxial wave equation is shown in Section 5.4. Wave propagation in the Fraunhofer region is discussed in Section 5.5.

Diffraction gratings are periodic optical devices that have many significant uses in applications. They also provide excellent examples of how to analyze waves emanating from such devices in the Fresnel and Fraunhofer regions. Section 5.6 discusses the fundamentals of diffraction gratings. Sections 5.7, 5.8, and 5.9 highlight Fraunhofer diffraction with a sinusoidal amplitude grating, Fresnel diffraction with a sinusoidal amplitude grating, and Fraunhofer diffraction with a sinusoidal phase grating, respectively.

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

Copyright # 2007 John Wiley & Sons, Inc.

63

64	FRESNEL AND FRAUNHOFER APPROXIMATIONS

Rayleigh–Sommerfeld

Integral Region

A

Fresnel Region

Fraunhofer Region

Figure 5.1. The three diffraction regions.

5.2DIFFRACTION IN THE FRESNEL REGION

Let the input wave field be restricted to a radial extent L1:

Uðx; y; 0Þ ¼ 0 if	p	ð5:2-1Þ
	x2 þ y2 > L1

Similarly, the radial extent of the observed wave field U(x,y,z) at the output plane is confined to a region L2 so that

q

Uðx0; y0; z0Þ ¼ 0 if x02 þ y02 > L2

ð5:2-2Þ

r01 in Eq. (4.6-3) is given by

r

¼

z 1

þ ð

x0

x

Þ

2 þ ðy0 yÞ2

#

1=2

5:2-3

Þ

z2

01

"

ð

where (x,y,0) are the coordinates of a point on the input plane and ðx0; y0; zÞ are the coordinates of a point on the observation plane. With the two restrictions discussed above, the following is true:

½ðx0 xÞ2 þ ðy0 yÞ2&max ðL1 þ L2Þ2

ð5:2-4Þ

The upper limit will be used below. If

jzj L1 þ L2;

ð5:2-5Þ

the term ði=jlÞðz=r012 Þ in Eq. (4.7-6) can be approximated by 1=jlz. However, more care is required with the phase. kr10 can be expanded in a binomial series as

kr10 ¼ kz þ 2kz ½ðx0 xÞ2 þ ðy0 yÞ2& 8kz3 ½ðx0 xÞ2 þ ðy0 yÞ2&2 þ . . .

ð5:2-6Þ

DIFFRACTION IN THE FRESNEL REGION

65

The third term has maximum absolute value equal to kðL1 þ L2Þ4=ð8jzj3Þ. This will be much less than 1 radian if

z	3			kðL1 þ L2Þ2	ð	5:2-7	Þ
j j				8
With this constraint on jzj, the phase can be approximated as
		k
kr10 kz þ			½ðx0 xÞ2 þ ðy0 yÞ2&		ð5:2-8Þ
		2z

The approximations made above are known as Fresnel approximations. The region decided by Eqs. (5.2-5) and (5.2-7) is known as the Fresnel region. In this region, Eq. (4.7-6) becomes

Uðx0	; y0	; zÞ ¼ jlz		ðð	Uðx; y; 0Þejlz½ðx0			xÞ þðy0	yÞ &dxdy	ð5:2-9Þ
			ejkz	1		p		2	2
				1

this is a 2-D convolution with respect to x and y and can be written as

Uðx0; y0; zÞ ¼ Uðx; y; 0Þ hðx; y; zÞ							ð5:2-10Þ
where the impulse response is given by
	ejkz		p 2		2
hðx; y; zÞ ¼		ej	lz	ðx	þy	Þ	ð5:2-11Þ
	jlz
The corresponding transfer function is given by
Hðfx; fy; zÞ ¼ e jkze jplzð fx2þfy2Þ							ð5:2-12Þ

The quadratic terms in Eq. (5.2-9) can be expanded such that Eq. (5.2-9) becomes

Uðx0

; y0

; zÞ ¼ jlz ej2zðx0

þy0Þ

ðð

U0ðx; y; 0Þe jlzðx0xþy0yÞdxdy

ð5:2-13Þ

ejkz

k 2

2

1

2p

1

where

U0ðx; y; 0Þ ¼ Uðx; y; 0Þej	k	ðx2þy2Þ	ð5:2-14Þ
	2z

66	FRESNEL AND FRAUNHOFER APPROXIMATIONS

Aside from multiplicative amplitude and phase factors, Uðx0; y0; zÞ is observed to be the 2-D Fourier transform of U0ðx; y; 0Þ at spatial frequencies fx ¼ x0=lz and fy ¼ y0=lz.

EXAMPLE 5.1 Determine the impulse response function for Fresnel diffraction by starting with Eq. (4.4-21) of the ASM method.

Solution: When z r in Eq. (4.4-21), the second and third terms on the right hand side can be approximated by 1. Also expanding ½z2 þ r2&1=2 in a Taylor’s series with two terms kept, Eq. (4.4-21) becomes

ejkz

hðx; y; zÞ ¼ jlz ejlpzðx2þy2Þ

which is the same as Eq. (5.2-11).

EXAMPLE 5.2 Determine the wave field in the Fresnel region due to a point source modeled by Uðx; y; 0Þ ¼ dðx 3Þdðy 4Þ.

Solution: The output wave field is given by

Uðx; y; zÞ ¼ Uðx; y; 0Þ hðx; y; zÞ

’ dðx			3Þdðy			1		ejkz 1þ	x2þy2
					4Þ				2z2
							jlz
1		ejkz 1þ		x 3 2þðy 4Þ2
’				ð Þ 2z2
	jlz

EXAMPLE 5.3 A circular aperture with a radius of 2 mm is illuminated by a normally incident plane wave of wavelength equal to 0.5 m. If the observation region is limited to a radius of 10 cm, find z in order to be in the Fresnel region.

Solution: With L1 ¼ 0:2 cm and L2 ¼ 10 cm, Eq. (5.2-4) gives

z 10:2 cm

If a factor of 10 is used to satisfy the inequality, we require z 1:02 m

The second condition given by Eq. (5.2-7) is

z3 2pð0:102Þ4m3 4 10 6

z 5:4 m

Using a factor of 10, we require

z 54 m

DIFFRACTION IN THE FRESNEL REGION

67

It is observed that the second condition given by Eq. (5.2-7) yields an excessive value of z. However, this condition does not have to be true for the Fresnel approximation to be valid [Goodman]. In general, kr01 is very large, causing the quadratic phase factor ejkr10 to oscillate rapidly. Then, the major contribution to the diffraction integral comes from points near x x0 and y y0, called points of stationary phase [Stamnes, 1986]. Consequently, the Fresnel approximation can be expected to be valid in a region whose minimum z value is somewhere between the two values given by Eqs. (5.2-4) and (5.2-7). As a matter of fact, the Fresnel approximation has been used in the near field to analyze many optical phenomena. The major reasons for this apparent validity are the subject matter of Chapter 7.

EXAMPLE 5.4 Determine the Fresnel diffraction pattern if a square aperture of width D is illuminated by a plane wave with amplitude 1.

Solution: The input wave can be written as

	x		y
Uðx; y; 0Þ ¼ rect		rect
	D		D

The Fresnel diffraction is written in the convolution form as

ejkz

D=2

ð ð ej

p

2

2

Uðx0; y0; zÞ ¼

½ðx x0Þ

þðy y0

Þ

&dxdy

lz

jlz

D=2

ejkz

2

1

D=2

ej

p

ðx x0Þ2 dx3 2

1

D=2

ð

lz

ð

¼ j

4

5 4

6

D=2

7 6

D=2

Consider the integral

Ix ¼ p1lz

Dð=2

ej

p

ðx x0Þ2 dx

lz

D=2

Let

r

n ¼

2

ðx x0Þ

lz

Then, Ix becomes

1

xe

p 2

Ix ¼

p2

ð ej2v dv

xb

			7	ð		Þ
ej	p	ðy y0	Þ2 dy3		5:2-15
	lz
			5

ð5:2-16Þ