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

278	DIFFRACTIVE OPTICS II

Figure 16.2. The telescopic system.

The second example is the telescopic system shown in Figure 16.2. Here f equals 1 so that the virtual reference wave is planar. Using two lenses with focal lengths f1 and f2 yields

A0

1

¼

f2

¼

ð16:2-13Þ

F

f1

B0

¼ f1 þ f2

ð16:2-14Þ

C0

¼ 0

ð16:2-15Þ

D0

¼ F

ð16:2-16Þ

such that

1

¼

f2

M ¼

ð16:2-17Þ

F

f1

T

f1 þ f2

T1

16:2-18

2

¼

F2

ð

Þ

F

It is seen that M is independent of the real hologram coordinates, and, if F is large, T2 is insensitive to T1.

The third example is the lensless Fourier arrangement discussed in Section 13.3. It is obtained when the point O of Figure 16.1 lies on the image plane. If z0 is the

distance from the virtual hologram to the image plane, we have
A	¼ ðT2 þ z0Þ	ð16:2-19Þ
C

16.2.2Aperture Effects

In order to obtain effective diffraction from the virtual hologram, it is desirable that virtual hologram apertures spread waves as much as possible. An interesting observation here is that aperture size may not be important in this context, since the virtual hologram is not to be recorded. In other words, even if virtual hologram

VIRTUAL HOLOGRAPHY

279

apertures are overlapping, diffraction effects at a distance can be explained in terms of waves coming from point sources on the virtual hologram to interfere with each other in the volume of interest.

Spreading of waves can be discussed in terms of v1 and v2, the angles a ray makes at two reference planes. From Eq. (16.2-2) we find

v2 ¼ C0x1 þ D0v1						ð16:2-20Þ
In the imaging planes, the following is true:
C0	1					ð16:2-21Þ
	¼
				f
D0	1					ð16:2-22Þ
	¼
		M

It follows that spreading of waves increases as f and M are reduced. In the telescopic system, C0 ¼ 0, and the input spreading of waves coming from an aperture is increased at the output by a factor of 1/M.

In order to find the size of the virtual hologram apertures, both magnification and diffraction effects need to be considered. The size of a virtual hologram aperture dv can be written as

dv ¼ Mdr þ D

ð16:2-23Þ

where dr is the size of the real hologram aperture, and D is the additional size obtained due to diffraction coming from the limited size of the optical system. For example, in the telescopic system, D can be approximated by [Gerrard and Burch, 1975]

D ¼	2:44f l	ð16:2-24Þ
	dA

where dA is the diameter of the telescope objective, and f is its focal length.

16.2.3Analysis of Image Formation

Analysis of image formation can be done in a similar way to the analysis of image formation in Section 15.8 on one-image-only holography. We assume that the contributions of various reference waves result in an effective reference wave coming from the point ðxc; yc; zcÞ. This can always be done within the paraxial approximation. If ðxi; yi; 0Þ are the coordinates of a sample point on the virtual hologram, and ðx0; y0; z0Þ are the coordinates of an object point, the equation that determines the positions of various image harmonics is given by

i

rc

þ r0

i

yc

þ y0

þ

2

r0 þ rc

¼

l þ

2p

0c

x

xc

x0

y

yc

y0

xi2

þ yi2

1 1

n

fl

F

ð16:2-25Þ

280	DIFFRACTIVE OPTICS II
where f is some constant phase, and
F0c ¼ r0 þ rc	ð16:2-26Þ
r0 ¼ ðx02 þ y02 þ z02Þ21	ð16:2-27Þ

where þð Þ sign is used if the object point is real (virtual) with respect to the virtual hologram, and

1

rc ¼ ðx2c þ y2c þ z2c Þ2 ð16:2-28Þ

where þð Þ sign is used if the focal point of the reference wave comes before (after) the virtual hologram.

If Eq. (16.2-25) is divided by M, we find that the real hologram is designed for the object coordinates

r00	¼	r0			ð16:2-29Þ
		M
x00	¼	x0			ð16:2-30Þ
		M
y00	¼	y0			ð16:2-31Þ
		M
z00	¼ ðr002 x002 y002Þ				ð16:2-32Þ
and the reference wave originating from
rc0	¼	rc			ð16:2-33Þ
		M
xc0	¼	xc			ð16:2-34Þ
		M
yc0	¼	yc			ð16:2-35Þ
		M

zc0 ¼ ðrc02 xc02 yc02Þ				ð16:2-36Þ
and the wavelength
l0	¼	l		ð16:2-37Þ
		M

If a reconstruction wave coming from ðx00c ; y00c ; z00c Þ and with wavelength l is used, the mth harmonic without the optical system will be reconstructed at

1

1

1

m 1

r000

¼

M2

r0

þ

rc

rc00

ð16:2-38Þ

x000

¼ r000 mM

x0

þ

xc

xc00

ð16:2-39Þ

r0

rc

rc00

y000

¼ r000 mM

y0

þ

yc

yc00

ð16:2-40Þ

r0

rc

rc00

VIRTUAL HOLOGRAPHY

281

Equations (16.2-38)–(16.2-40) indicate that there is also another object distorted with respect to the desired object. In analogy with the concept of virtual hologram, it will be called the virtual object. This means that we can start either from the virtual hologram to generate the real object or from the real hologram to generate the virtual object. It looks like it is easier to use the virtual hologram concept in most cases. However, there are occasions when it is more advantageous to use the virtual object

concept.					wave (xc ¼ yc ¼ 0), Eqs.
Assuming plane perpendicular		reconstruction
(16.2-38)–(16.2-40) for the first harmonic reduce to
r000	¼	R			ð16:2:41Þ
			r0
		M
x000	¼ Rx0				ð16:2:42Þ
y000	¼ Ry0				ð16:2:43Þ
R ¼			rc		ð16:2:44Þ
		Mðz0 þ rcÞ

The last four equations indicate one computational advantage that can be achieved, namely, the use of convolution to determine the hologram. In digital holography, normally the hologram is much smaller than the image. Convolution requires the image and hologram sizes to be the same. With this method, one can obtain equal sizes by choosing R properly. For the case of a single lens, this is satisfied by

f
R ¼ r0 þ rc	ð16:2:45Þ

The appearance of various holograms and objects in this case is shown in Figure 16.3.

Figure 16.3. Virtual and real holograms and objects.

282	DIFFRACTIVE OPTICS II

16.2.4Information Capacity, Resolution, Bandwidth, and Redundancy

Considering the virtual hologram, the resolution requirements set the lower limit of its size. For a rectangular aperture, the minimum resolvable distance between object points, defined according to Rayleigh criterion, is given by [Born and Wolf]

h ¼	z0l	ð16:2-46Þ
	Lv

where Lv is the size of the virtual hologram. Accordingly, the virtual hologram size Lv should be large enough to give a desired h.

An equally important consideration is the spatial frequency limit of the recording medium. The distance between the fringes in the x direction is approximately given by

		l	xi xc		xi xo		1		16:2-47
	F ¼ jMj rc			þ	ro			ð		Þ

Equations (16.2-46) and (16.2-47) show that, if jMj is made small enough, F can be increased to such a magnitude that almost any recording medium can be used to make a hologram. However, this is accompanied with an increase in h. Conversely, in order to increase image resolution, the virtual hologram can be made larger than the real hologram, provided that the frequency limit of the recording medium is not exceeded.

Given a certain hologram size, it is desirable to record information in such a way that redundancy is reduced to a desired degree, especially if multiplexing is to be used. It seems to be advantageous to discuss redundancy in terms of the number of fringes. If N fringes are needed to obtain a desired image resolution, the virtual hologram size can be chosen to cover N fringes. The real hologram will also have N fringes. In this way, the recording medium is used as efficiently as desired. If the recording medium has a space-bandwidth product SB given by [Caulfield and Lu, 1970]

hologram area
SB ¼ area of minimum resolution element	ð16:2-48Þ

and the average space-bandwidth product used per signal is SBs, the number of signals that can be recorded is of the order of SB=SBs times the capacity for linear addition of signals in the recording medium. Direct recording of the hologram would cause SB=SBs to be of the order of 1. Thus, the information capacity of the real hologram can be used effectively via this method at the cost of reduced resolution. However, this may require a nonlinear recording technique such as hard clipping. If SBs is reasonably small for a signal, coarse recording devices such as a laser printer can be used to make a digital hologram. Another advantage that can be cited is that it

VIRTUAL HOLOGRAPHY

283

r	(x0,z0)
1

x1

r2

x2

d

Figure 16.4. Interference of two spherical waves coming from points on two planes separated by distance d.

should be much faster to calculate the virtual hologram of reduced size because of a few numbers of fringes.

16.2.5Volume Effects

Equations (16.2-17) and (16.2-18) describe the transformation obtained with the telescopic system. Lateral magnification is independent of real hologram coordinates, whereas T2 versus T1 varies as 1=F2. If F is reasonably large, and if several real hologram planes are separated by certain distances, these distances will be reduced by F2 in the virtual hologram space. Thus, any errors made in positioning the real holograms would be reduced F times in the lateral direction and F2 times in the vertical direction in the virtual hologram space. This means that it should be quite simple to obtain interference between various holograms and/or optical elements using the virtual holography concept and the telescopic system.

As a simple example, consider the interference at point O of two spherical waves coming from points at x1 and x2 on two planes separated by a plane d as shown in Figure 16.4.

Fraunhofer approximation will be assumed to be valid since the quadratic terms can always be removed with a lens if d is sufficiently small. Then, the difference between the two optical path lengths including the effect of a plane perpendicular reference wave can be approximated by

¼

x0

ð

z0

þ 2

z0

2

Þ

ð

x2 x1

Þ

x2

16:2-49

If x0 is much larger than x2, this can be simplified to

x0

x2 x1

Þ

d

x0

2

16:2-50

¼

ð

z0

þ 2

z0

ð

Þ

If d ¼ 0, the second term in Eq. (16.2-50) disappears, and the expression used by Lohmann to determine the position of a synthetic aperture is obtained [ Lohmann, 1970]. For different object points, one needs to assume that x0=z0 remains

284 DIFFRACTIVE OPTICS II

approximately constant. However, this assumption can be relaxed to a large extent by finding the stationary point of with respect to x0=z0. This is given by

	x2 x1 ¼	x0					ð16:2-51Þ
				d
		z0
		d			x0		2
	¼						ð16:2-52Þ
		2			z0

It is possible to extend this approach further to make digital volume holograms. For example, the phase can be quantized by using several hologram planes separated by distance d, which is determined by Eq. (16.2-52). Then, the conjugate image problem would also disappear. The reason why it is practical now to do so is that d and ðx2 x1Þ can be easily controlled by adjustments in the real hologram space where any errors are transformed to the virtual hologram space on a much reduced scale.

16.2.6 Distortions Due to Change of Wavelength and/or Hologram Size Between Construction and Reconstruction

It is well known in holographic microscopy and acoustical holography that lateral and vertical magnifications differ when wavelength and/or hologram size are changed between construction and reconstruction if these two changes are not done in the same ratio. Various techniques are proposed to get around this problem such as a phase plate [Firth, 1972]. With virtual holography, it is possible to match hologram size to wavelength change without actually matching the physical hologram size. However, distortions can be reduced without matching if a spherical reference wave other than the one used in construction is used on the virtual or the real hologram in reconstruction.

If N is the number of times the hologram size is changed between construction and reconstruction, k1 and k2 are the wave numbers during construction and reconstruction, respectively, the equation governing image formation can be written, similar to Eq. (16.2-25), as

k

x01

Nx

y01

Ny

N2

ð

x2 þ y2Þ

r01

þ

1

r01

2r01

þ

x2

r02

rc

ð16:2-53Þ

¼

2

r02 rc

r02

rc

2

k

x

x02

xc

y

y02

yc

þ y2

1

1

where x01, y01, and z01 are the object coordinates, x02, y02, and z02 are the image coordinates, and

r01 ¼ ½x012	þ y012	þ z012	&21	ð16:2-54Þ
r02 ¼ ½x022	þ y022	þ z022	&21	ð16:2-55Þ

VIRTUAL HOLOGRAPHY

285

Equating corresponding terms, we obtain

1

k1N

2

1

1

¼

þ

ð16:2-56Þ

r02

r01

rc

k2

k Nx

01

xc

1

x02 ¼ r02

1

þ

ð16:2-57Þ

r01

rc

k2

k Ny

01

yc

1

y02 ¼ r02

1

þ

ð16:2-58Þ

r01

rc

k2

Various magnifications will be equal if

r01

ð16:2-59Þ

r ¼

N

x01

ð16:2-60Þ

x02 ¼

N

y01

ð16:2-61Þ

y02 ¼

N

z01

ð16:2-62Þ

z02 ¼

N

It is also possible to use an effective virtual reference wave which is a sum of a number of reference waves corresponding to different holograms, as discussed in Section 16.2.3, in order to reduce distortions and to scan over the different parts of the image field.

16.2.7Experiments

The first digital holograms using this method were made with a scanning electron microscope system discussed previously in Chapter 15. All the calculations for encoding the hologram were done for the virtual hologram. The virtual hologram was then transformed in the computer using the optical system parameters into the real hologram that was physically generated. The one-image-only holography technique was used to encode the holograms. Here the position of each hologram aperture is chosen according to

fðxi; yiÞ þ kroi ¼ 2pn þ f0

ð16:2-63Þ

where fðxi; yiÞ is the phase of the reference wave at the virtual aperture position ðxi; yi; 0Þ, n is an integer, f0 is the desired phase at the object point with position coordinates ðx0; y0; z0Þ, and

1
roi ¼ hðx0 xiÞ2 þ ðy0 yiÞ2 þ z02i2	ð16:2-64Þ

286	DIFFRACTIVE OPTICS II

Figure 16.5. Reconstruction with the first virtual hologram.

where fðxi; yi; 0Þ is given by
fðxi; yi; 0Þ ¼ krci	1	ð16:2-65Þ
rci ¼ hðxc xiÞ2 þ ðyc yiÞ2 þ zc2i
	2	ð16:2-66Þ

where ðxc; yc; zcÞ are the coordinates of point O in Figure 16.1. For example, for an on-axis beam with a single lens, they can be chosen as

xc ¼ 0	ð16:2-67Þ
yc ¼ 0	ð16:2-68Þ
zc ¼ fM	ð16:2-69Þ

where M is the desired magnification, and f is the focal length of the lens system used. The aperture positions on the real hologram are xi=M and yi=M.

The reconstruction obtained from the first hologram generated in this way is shown in Figure 16.5. The object was chosen to be a circle slanted in the Z direction. This is why it looks slightly elliptic in the picture indicating the 3-D nature of the object points. The real hologram is 2 mm 2 mm in size. This is reduced 4 with a 20 mm lens to give the virtual hologram. The distance T2 f in Figure 16.1 was chosen to be 5 mm.

It was possible with the SEM system to generate holograms, 2 mm 2 mm in size each, side by side up to a total size of 7:4 cm 7:4 cm. At each shift there was an uncertainty of 5 m in positioning. Because each hologram is a window by itself, the light coming from each individual hologram will be directed in a different direction, which is similar to the problem of nondiffuse illumination. Even if the uncertainty in positioning was negligible and a large single hologram was made, this would not increase the information density, since only a small part of the total information coming from an area of the order of 2 mm 2 mm is visible to the human eye at a time.

With this method it is possible to reduce the total virtual hologram set to a size of several millimeters on a side so that all the information coming from different

THE METHOD OF POCS FOR THE DESIGN OF BINARY DOE

287

Figure 16.6. Reconstruction with the second virtual hologram generating part of a 3-D cube.

Figure 16.7. Reconstruction with the hologram set generating the 3-D cube.

holograms is made visible to the human eye, even though this is done at the expense of image resolution.

In order to show this, sixteen adjacent holograms arranged in a 4 4 matrix were generated. The virtual hologram set 2 mm 2 mm in size was obtained by reducing the real hologram set 4 with a 50-mm objective. The distance T2 f in Figure 16.1 was chosen as 12.5 mm. Each hologram generated one side or a diagonal, all the corner points, and the midpoint of a 3-D cube. The output from one such hologram is shown in Figure 16.6. When all the holograms were illuminated, the total image was obtained, as shown in Figure 16.7. Because all the holograms contribute to the corner points and midpoint, these object points appear much more intense than the other object points in Figure 16.7. Looking through the area where the virtual hologram set was supposed to be located, the whole cube in space could be seen.

16.3THE METHOD OF POCS FOR THE DESIGN OF BINARY DOE

In the methods discussed in this section and the following sections of this chapter, the Fourier transform system shown in Figure 15.1 is used. The sampled hologram consists of an array of discrete points. The transmittance function of the hologram consisting of an M N array of nx ny sized pixels can be represented by the sum

X X

G ;

M 1 N 1

H

k; l

rect

nx k nx

rect

ny l ny

16:3-1

ðnx

nyÞ ¼ k

0 l

0

ð

Þ

nx

ny

ð

Þ

¼

¼

where Hðk; lÞ is the binary transmittance of the ðk; lÞth point. The reconstructed image in the observation plane is given by the Fourier transform of the