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

268	DIFFRACTIVE OPTICS I

15.10.1Analysis of Image Formation

In order to analyze the method, the lengths of the radii will be expanded in the paraxial approximation [Meier, 1966]. Instead of a spherical reference wave, suppose that a plane reference wave given by

Ur ¼ AejkðaxþbyþgzÞ

ð15:10-17Þ

is used. Then, Eq. (15.10-5) can be written in the paraxial approximation as

x

x0

y

y0

xi2 þ yi2

n

f0l

15:10-18

i

a z0

þ

i b z0

þ

2z0

¼

l þ

2p

ð

Þ

where the sign of z0 should be chosen the same as the sign used in Eq. (15.10-2). If a spherical reference wave as determined by Eq. (15.9-3) is used instead,

Eq. (15.10-5) can be written as

i

zc

þ z0

i

zc

þ z0

þ

2

z0 þ zc

¼

l þ

2p

0c

x

xc

x0

y

yc

y0

xi2

þ yi2

1 1

n

f0l

F

ð15:10-19Þ

where

F

0c ¼

z

0

þ

x02 þ y02

þ

z

c þ

xc2 þ yc2

ð

15:10-20

Þ

2z0

2zc

Above the signs of z0 and zc should be chosen the same as the signs used in Eqs. (15.10-2) and (15.10-3).

An image forms whenever Eq. (15.10-18) or (15.10-19) is valid for arbitrary n, disregarding the constant terms f0l=2p and F0c. It is observed that if Eq. (15.10-18) is valid for the object point at ðx0; y0; z0Þ, the reference wave with direction cosines ða; b; gÞ, and wavelength l, it is also valid for an object point at ðx00; y00; z00Þ, the reference wave with direction cosines ða0; b0; g0Þ, and wavelength l0 such that

z00

¼

l z0

l0

m

x0

l

a0

ma

z

x

¼ l0

0

þ

0

0

b0

m

y0

l

mb

z

y

¼ l0

0

þ

0

0

m

ð15:10-21Þ

ð15:10-22Þ

ð15:10-23Þ

where m is an integer. m equal to 1 and –1 corresponds to real and virtual images while m equal to 0 corresponds to the zeroth order wave, a plane wave traveling in the same direction as the original reference wave. Other values of m correspond to higher order images.

ONE-IMAGE-ONLY HOLOGRAPHY

269

By the same token, if Eq. (15.10-19) is valid for the object point at ðx0; y0; z0Þ and the reference wave with focal point at ðxc; yc; zcÞ, whose wavelength is l, it is also

valid for the object point at ðx00 ; y00 ; z00 Þ and the reference wave with focal point at

ðxc0 ; yc0 ; zc0 Þ, whose wavelength is l0 such that

1

l0

1

1

¼

m

þ

ð15:10-24Þ

z00

l

z0

zc

l

xc

x0

x0

x00

¼ z00

m

þ

c

ð15:10-25Þ

l0

zc

z0

zc0

l

yc

y0

y0

y00

¼ z00

m

þ

c

ð15:10-26Þ

l0

zc

z0

zc0

where m is an integer. We note that m ¼ 0 corresponds to a wave that is the same as the reference wave and focuses at the focal point. If z0 zc, and assuming the initial and the final reference waves are the same, the image positions other than m ¼ 1 are

approximately given by			zc
z00	’								ð15:10-27Þ
		m 1				z0
x0	’	x	c þ	m		0	x	0	ð	15:10-28	Þ
0						z0
y0						z0
	’	y	c þ	m		0	y	0	ð	15:10-29	Þ
0						z0

In other words, if the focal point of the lens is sufficiently close to the hologram, all the images other than the one for m ¼ 1 have to be so close to the hologram that they become completely defocused in the far distance where the desired image is.

However, it is possible to focus one order at a time. If the mth order is desired to be at the position ðx00; y00; z00Þ, the corresponding values of ðx0c; y0c; z0cÞ are simply found such that Eqs. (15.10-24), (15.10-25), and (15.10-26) are satisfied. Then, all the other images become completely defocused and out of view in the far distance where the desired image is, because they are located very close to the hologram.

The above analysis is not always reasonable because the paraxial approximation becomes difficult to justify as the focal point of the spherical reference wave approaches the hologram. Thus, the images other than the one desired may not be well defined. It is also possible that the encoding technique used loses its validity for very close distances to the hologram, especially due to the registration errors of the hologram apertures.

In order to support the above arguments, consider what happens when z0c ! 1 in Eqs. (15.10-24)–(15.10-26); in other words, a plane wave is used for reconstruction.

	Again assuming z0 zc, and l equal to l0, we get
	z00	’	zc				ð15:10-30Þ
			m		zc
	x00	’ xc þ					ð15:10-31Þ
						x0
					z0
	y00	’ yc þ			zc		ð15:10-32Þ
						y0
					z0

270	DIFFRACTIVE OPTICS I

The above equations indicate that the images must focus close to the hologram. Any absence of images in the space of interest indicates that the interference between the harmonic images and the desired image is minimized.

It is also interesting to observe what happens to the positions of the higher order images as zc gets smaller. From Eq. (15.10-24), for equal wavelengths, we find

z00		zcz0
	’		c		ð15:10-33Þ
		mz0		zc
		c

As m increases, z00 approaches zc=m. This means that it becomes more difficult to observe higher order images as the hologram is designed closer to the focal point of the lens.

A very important consequence of Eq. (15.10-33) is that there is a variable focusing distance; z00 can be varied at will by a slight adjustment of the position of the lens.

15.10.2Experiments

In order to prove the above predictions experimentally, holograms were generated using the scanning electron microscope system, as discussed in Section 15.7.1. In all cases, the sizes of the holograms were 2 2 mm, and the sizes of the apertures were of the order of 1m.

Figure 15.25 shows the He–Ne laser reconstruction from a regular hologram of one object point with the plane-perpendicular reference wave. At the center, the overexposed main beam is observed; to the right of the main beam, there are the slightly overexposed real image of the object point and higher order images; to the left of the main beam, there are the virtual image and virtual higher order images.

Three holograms of an object point were generated using a spherical reference wave such that the focal point of the lens is 1, 3, and 5 cm to the left of the hologram plane, and its x- and y-coordinates are those of the center point of the hologram. Figure 15.26 shows the result with the 5-cm hologram. There is only one visible object point; the dark square is the enlarged picture of the hologram; the main beam covers the whole figure. Figure 15.27 shows the same object point with the main beam and zero-order image filtered out by putting a stop at the focal point of the lens. The results with the 1- and 3-cm holograms gave the same type of results supporting the arguments given above [Ersoy, 1979].

Figure 15.28 shows the reconstruction of a more complicated object. The hologram is designed to be 3 cm from the focal point of the lens, whose x- and

Figure 15.25. The He–Ne laser reconstruction from a regular hologram of one object point showing different orders.

ONE-IMAGE-ONLY HOLOGRAPHY

271

Figure 15.26. The single visible image from the 5-cm hologram without filtering.

y-coordinates are the same as those of the center point of the hologram. During the experiments, it was observed that it is a simple matter to focus the image at different distances, and thereby control the size of the image as well, by a slight adjustment of the position of the lens, in confirmation of Eq. (15.10-33).

Figure 15.27. The single visible image from the 5-cm hologram with filtering.

272	DIFFRACTIVE OPTICS I

Figure 15.28. The single reconstructed image of the number 3.

15.11FRESNEL ZONE PLATES

After covering more complicated designs for a DOE, we go back historically and discuss a classical DOE called Fresnel zone plate (FZP). Such a DOE serves as a flat lens, can be mass-manufactured and is commonly used in many technologies such as overhead projectors. In this section, we will discuss a binary FZP that consists of opaque and transparent circular zones. An example is shown in Figure 15.29.

A side view of a FZP is shown in Figure 15.30.

Figure 15.29. A binary Fresnel zone plate.

FRESNEL ZONE PLATES	273

Figure 15.30. Side view of a FZP.

The path difference between two rays traveling along SOP and SAP is given by

q p

ðrÞ ¼ ðr0 þ rÞ ðz0 þ zÞ ¼ r2 þ z20 þ r2 þ z2 ðz0 þ zÞ ð15:11-1Þ

The Fresnel-zone parameter, n, is defined such that the path difference is an integer multiple of half wavelengths [Hecht]:

n	l	¼ ðrÞ	ð15:11-2Þ
	2

The nth Fresnel zone is the area between the circles with radii rn 1 and rn. Note that the field at P coming from a point on the circle with radius rn is half a wavelength out of phase with the field from a point on the circle with radius rn 1. Using this fact, it is easy to show that the adjacent zones cancel each other out. Therefore, the total field at P will increase if either all even or all odd zones are blocked out, the remaining zones reinforce each other, thus creating a focal point at distance z. Assuming the zone plate is illuminated by a plane wave, z0 1, and setting z ¼ fo, Eq. (15.11-1) results in

		s
rn	¼	f0				1		ð2n 1Þ2
l			l	n		2	þ	16	ð	15:11-3	Þ

f0 is called the focal point. An FZP operates as a lens with focal length f0. It can be shown that there exist other focal points at f0=3, f0=5, f0=7, and so on [Hecht].

Let Rn be the radius of the nth circle on the FZP. It satisfies

2	¼ fo þ	nl		2	2
Rn					fo	ð15:11-4Þ
		2

274 DIFFRACTIVE OPTICS I

Using Eq. (15.11-3), this can be written as

2	2	nl			1			nl	2
Rn	¼ fo	"		þ						#	ð15:11-5Þ
			fo		4			fo

When n is large, the second term on the right-hand side above can be neglected, yielding

p	ð15:11-6Þ
Rn ’ nlfo

16

Diffractive Optics II

16.1INTRODUCTION

This chapter is a continuation of Chapter 15, covering more methods on diffractive optics. It consists of seven sections. Section 16.2 is on the method of virtual holography which makes DOEs easier to handle by accommodating to the requirements of technologies of implementation. It also brings closer holography and classical optics.

Section 16.3 discusses the method of POCS discussed in Section 14.8 to design binary DOEs. Section 16.4 incorporates the POCS method in a novel approach called iterative interlacing technique. In this method, subholograms are created, and the iterative design of subholograms is based on minimizing the remaining reconstruction error at each iteration.

Section 16.5 combines IIT and the decimation-in-frequency property used in FFT algorithms to come up with a better and faster strategy to design the subholograms in a method called ODIFIIT. Section 16.6 further generalizes the constant amplitude Lohmann method discussed in Section 15.4 in quantization of amplitude and phase in each Lohmann cell. The POCS method is used for the optimization of the subholograms. Section 16.7 combines the Lohmann and ODIFIIT methods, resulting in considerably higher accuracy in terms of reduced mean square error in the desired image region.

16.2VIRTUAL HOLOGRAPHY

It is known that most holograms are redundant, and only a small part of the hologram is sufficient to generate the desired information. If the hologram is to be viewed with the unaided eye, the resolution of image points is determined by the size of the eye lens aperture, which is of the order of several millimeters [Born and Wolf, 1969]. One desires to have as much information as possible in a hologram of this size.

When the hologram is recorded with a certain kind of wave, such as acoustical waves or microwaves, and reconstruction is done with another kind of wave, such as

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

Copyright # 2007 John Wiley & Sons, Inc.

275

276

DIFFRACTIVE OPTICS II

Figure 16.1. System diagram for virtual holography.

visible light, it is necessary to change the hologram size at the ratio of the two wavelengths in order to prevent image distortions [Smith, 1975]. However, this is a time-consuming and error-prone operation.

A virtual hologram is defined as the hologram that is not recorded in a medium but exists in space as the image of another hologram that is recorded, and which is called the real hologram [Ersoy, August 79]. The information coming from the virtual hologram is the desired information whereas the information coming from the real hologram consists of transformed information. The transformation between the real hologram and the virtual hologram is achieved with an optical system as shown in Figure 16.1.

It can be said that neither the real hologram nor the virtual hologram is exactly like regular holograms. If one looks through the real hologram under reconstruction, one sees transformed information that is probably unrecognizable. The virtual hologram is more like a regular hologram, but it is not registered in a physical medium.

16.2.1Determination of Phase

The rays coming from the real hologram parallel to the optical axis converge to point O as shown in Figure 16.1. Since the optical path lengths between a real hologram point and the corresponding virtual hologram point are the same, the phase at the virtual hologram point relative to the other virtual hologram points are determined by the radius vector length rc between O and the virtual hologram point as shown in Figure 16.1.

The system transfer matrix S that connects the input and the output in the form [Gerrard and Burch, 1975]

x2	¼	x1		ð16:2-1Þ
v2		S v1
can be determined as	C00	D00
S ¼				ð16:2-2Þ
	A	B

VIRTUAL HOLOGRAPHY		277
where
A0	¼ A þ CT2	ð16:2-3Þ
B0	¼ ðA þ CT2ÞT1 þ B þ DT2	ð16:2-4Þ
C0	¼ C	ð16:2-5Þ
D0	¼ CT1 þ D	ð16:2-6Þ

A, B, C, and D determine the optical system matrix; T1 and T2 are the distances shown in Figure 16.1. In order to have image generation, B0 ¼ 0 so that

M ¼	x2		¼ A0	1
				¼			ð16:2-7Þ
	x	1			D	0

A plane wave input to the optical system at an angle v1 focuses to a point at a distance

f ¼	A		ð16:2-8Þ
	C
with the lateral coordinate x2 given by
x2 ¼ ðB þ Df Þv1			ð16:2-9Þ

Using these equations, it is straightforward to calculate rc shown in Figure 16.1. rc determines the type of the reference wave on the virtual hologram. The phase

due to the reference wave on the real hologram is also to be transferred to the virtual hologram. Therefore, it makes sense to talk of the real reference wave and the virtual reference wave, respectively. This approach can be further extended by having several optical systems that give rise to several virtual holograms and reference waves. The end result would be to add all the reference waves on the last virtual hologram. However, the phase due to each reference wave would be determined by its position on the corresponding hologram.

Three examples will be considered. The first one is a single lens. This case

corresponds to
				1
A ¼ 1; B ¼ 0;				C ¼						; D ¼ 1	ð16:2-10Þ
								f
so that
1		1		1							ð16:2-11Þ
		þ		¼
T1			T2		f
			M ¼				T2				ð16:2-12Þ
							T1