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

258	DIFFRACTIVE OPTICS I

and was the algorithm which demonstrated for the first time the feasibility of implementing DOEs with scanning electron microscopy [Ersoy, 76]. Even though initially very simple, the algorithm was later developed in more complex ways to generate 3-D images in arbitrary locations in space [Bubb, Ersoy]. The methodology is quite different from other methods since it does not explicitly depend on the Fourier transform. It also leads to one-image only holography discussed in Section 15.9, which is further used in optical phased arrays in Chapter 18.

If (xo, yo, zo) represents the position of an object point to be reconstructed, and (xi, yi, zi) represents the position of a phase-shifting aperture on the hologram, the Huygens–Fresnel principle for a collection of N apertures on a plane (z ¼ 0) leads to

ð o o oÞ ¼	i	ðð	ð i i	iÞ jl		roi		d i i	ð		Þ
	X
	N				1	expðjkroiÞ
U x ; y ; z			U x ; y ; z				cos	dx dy		15:8-1

where d is the angle between the z-axis and the vector from the center of the aperture to the object point, whose length is roi, l is the wavelength, and k is the wave number. Uðxi; yi; ziÞ will be assumed to be a plane wave equal to unity below. For small hologram dimensions, cos(d) can be assumed to be constant. If the phase variations on the hologram plane are also small compared to the phase variations of expðjkroiÞ, the above equation can be approximated by

ð o o oÞ ¼	jlR x y		i	ðð	roi	i i	ð		Þ
			X
	cos d		N		expðjkroiÞ
U x ; y ; z		d d				dx dy		15:8-2

where yi is the phase shift of the reference wave at the ith aperture.

Assume that each aperture is rectangular on the x-y plane with dimensions dx and dy, and has a central point (xsi, ysi, 0) whose radial distance roi from the observation

point satisfies
kroi ¼ 2pn þ yi n ¼ integer	ð15:8-3Þ

Using the Fraunhofer approximation, the source field at the apertures can be considered as point sources approximated by narrow sinc functions. Thus, the integral in Eq. (15.8-2) becomes a double summation:

	cos			X dx	Yidy
Uðxo; yo; zoÞ ¼	d	dxdy	X X expðjyiÞ sin c	i	sin c			ð15:8-4Þ
	jlR			lroi		lroi
where
			Xi ¼ xo xsi
			Yi ¼ yo ysi					ð15:8-5Þ
			R ¼ average value of roi

A SIMPLE ALGORITHM FOR CONSTRUCTION

259

if all yi are set equal to y, and xi, yi roi, the sinc functions can be replaced by 1, and

Uðxo; yo; zoÞ ¼	cos d	dxdyNejy	ð15:8-6Þ
	jlR
Therefore, the amplitude of the field will be proportional to			dxdyN, and

its phase will be y for a plane wave incident on the hologram at right angle, namely, an on-axis plane wave. If the incoming wave has phase variations on the hologram, its phase being i at each hologram point, then Eq. (15.8-2) should be written as

kroi þ i ¼ 2pn þ yi n ¼ integer

ð15:8-7Þ

The aperture locations (xi, yi) can be chosen such that the resulting yi will be a constant. An object point is then obtained since all wave fronts generated by the hologram apertures will add up in phase at the specified object point location. Thus, the amplitude of the field will be proportional to dxdyN, and its phase will be 0 for a plane incoming wave incident on the hologram at right angle. If there are groups of such apertures that satisfy Eq. (15.8-6) at different points in space, a sampled wavefront is essentially created with a certain amplitude and phase at each object point. We note that the modulation of Eq. (15.8-6) is very simple. We vary dx and/or dy and/or N for amplitude and y for phase. The fact that N is normally a large number means that it can be varied almost continuously so that amplitude modulation can be achieved very accurately.

If amplitude modulation accuracy does not need to be very high, the method is valid in the near field as well because roi is exactly computed for each object point. If the apertures are circular, the sinc functions are replaced by a first-order Bessel function, but Eq. (15.8-6) essentially remains the same.

In practice, each point-aperture on the hologram plane is first chosen randomly and then moved slightly in the x- and/or y-direction so that its center coordinates satisfy Eq. (15.8-7) with constant y. Overlapping of the apertures is considered negligible so that there is no need for using memory.

15.8.1Experiments

The method described above was used to test implementation of a DOE with a scanning electron microscope [Ersoy, 1976]. The working area for continuous exposure was 2 2 mm. The number of point apertures with the smallest possible diameter of about 1 m was 4096 4096. In the experiments performed, the hologram material used was either KPR negative photoresist or PMMA positive photoresist.

Figures 15.17 and 15.18 show the reconstructions from two holograms that were produced. They were calculated at the He–Ne laser wavelength of 0. 6328 m. In Figure 15.17, eleven points were chosen on a line 3 cm long, satisfying z ¼ 60 cm, x ¼ 4 cm, 0 y 3 cm on the object plane. The number of hologram

260	DIFFRACTIVE OPTICS I

Figure 15.17. Reconstruction of eleven points on a line 3 cm long in space from a DOE generated with a scanning electron microscope.

apertures used were 120,000, each being 8 8 adjacent points in size. The image points were chosen of equal intensity. The picture was taken at approximately 60 cm from the hologram, namely the focal plane. The main beam was blocked not to overexpose the film. It is seen that there are both the real and the conjugate images.

In order to show the effect of three-dimensionality, the object shown in Figure 15.18 was utilized. Each of the four letters was chosen on a different plane in space. The distances of the four planes from the hologram plane were 60, 70, 80, and 90 cm, respectively. If all the letters were on a single plane, the distance in the x- direction between them would be 1 cm. In the picture it is seen that this distance is decreasing from the first to the last letter because of the depth effect. The picture was taken at approximately 90 cm from the hologram, namely, the focal plane of the letter E. That is why E is most bright, and L is least bright. The number of hologram apertures used were 100,000, each being 4 4 adjacent points in size. The image points were chosen of equal intensity.

Figure 15.18. Reconstruction of the word LOVE in 3-D space. Each letter is on a different plane in space.

THE FAST WEIGHTED ZERO-CROSSING ALGORITHM

261

y

Locations of zerocrossings closest to origin

∆x1

∆y1

x

Figure 15.19. Zero-crossings closest to the origin.

15.9THE FAST WEIGHTED ZERO-CROSSING ALGORITHM

The algorithm discussed in Section 5.7 corresponds to choosing a number of zero crossings of phase for each spherical wave at random positions on the hologram. A disadvantage of this approach is that the hologram quickly saturates as the number of object points increases. Additionally, it is also computationally intensive. The fast weighted zero-crossing algorithm (FWZC) is devised to combat these problems [Bubb, Ersoy].

For each object point to be generated with coordinates ðxo; yo; zoÞ, the following procedure is used in the FWZC algorithm:

1.Calculate the zero-crossings x1; y1 of phase closest to the origin on the x-axis and the y-axis, respectively, as shown in Figure 15.19.

Let ð x1; 0:0Þ be the location of the zero-crossing on the x-axis. Then,

roi

ðxo x1

Þ2

þ yo2

þ zo2

nl

15:9-1

¼ q

¼

ð

Þ

r0

x2

þ

y2

z2

¼

n

l þ

B

ð

15:9-2

Þ

oi

o

o þ

o

¼ q

Solving these equations for x1 yields

q

x1 ¼ xo xo2 þ ðB2 2Broi0 Þ

ð15:9-3Þ

An entirely similar expression can be derived for y1.

2.Using Heron’s expression [Ralston], calculate rx and ry, the radial distances from these zero-crossings to the object point. For example, rx is

262

DIFFRACTIVE OPTICS I

derived as

rx ¼ z1 3 þ x12

þ y12

1

ð15:9-4Þ

ðx12 þ y12 þ 1Þ

zo

ð15:9-5Þ

z1 ¼

2

1

x1

x1

z2

15:9-7

xo

o

z2

¼ p2z

ð15:9-6Þ

¼ ð

Þ

ð

Þ

y1 ¼ yozo

ð15:9-8Þ

3. Determine all the zero-crossings on the x-axis. These are calculated by starting at an arbitrary zero-crossing on the x-axis, say, x ¼ a, and calculating

the movement required to step to the next zero-crossing, say, x ¼ b. This is q

done by approximating the distance function rx ¼

ðx xoÞ2 þ yo2 þ zo2 with

the first two terms of its Taylor series. The result is given by

x ¼ b a ¼ xoð

Þla þ

2 xo1

a

"

xoð

Þla

2

ð15:9-9Þ

a

a

l2#

r

r

ð

Þ

When doing the same procedure in the negative x-direction such that rðbÞ rðaÞ ¼ l, a similar analysis shows that the first term on the right-hand side of Eq. (15.9-9). changes sign.

4.An identical procedure can be used to find all of the zero-crossings for both the positive and the negative y-axis.

5.It is straightforward to show that for any x ¼ x11 and y ¼ y11, if ðx11; 0Þ and ð0; y11Þ are zero-crossings, then ðx11; y11Þ is also zero-crossing if the Fresnel approximation is valid.

Utilizing all of the ‘‘fast’’ zero-crossings on the x- and y-axis, we form a grid of zero-crossings on the hologram plane. It is also possible to generate the remaining zero-crossings by interpolation between the fast zero-crossings.

6.The grid of zero-crossings generated as shown in Figure 15.20 indicate the centers of the locations of the apertures to be generated in the recording medium to form the desired hologram.

Once the grid of zero-crossings for each object point is generated, their locations are noted by assigning a ‘‘hit’’ (for example, one) to each location. After zero-crossings for all the object points have been calculated, each aperture location will have accumulated a number of hits, ranging from zero to the number of object points. This is visualized in Figure 15.21.

For object scenes of more than trivial complexity, the great majority of aperture locations will have accumulated a hit. Next, we set a threshold number of hits, and only encode apertures for which the number of hits exceeds the assigned threshold.

THE FAST WEIGHTED ZERO-CROSSING ALGORITHM								263

denotes an original on axis zero-crossing

denotes the new zero-crossings at the intersection points

Figure 15.20. The grid of zero-crossings generated.

Increment counter		Number of apertures used at
value		least this many times
.		.
.		.
.		.
.		.
30		46,748
29		64,816
28		71,903
27		86,274
26		95,816
25		106,892
24		145,981
.		.
.		.
.		.
.		.

Figure 15.21. Accumulation of zero-crossings at hologram points.

264	DIFFRACTIVE OPTICS I

In this way, we choose the most important zero-crossings, for example, the ones that contribute the most to the object to be reconstructed. One scheme that works well in practice is to set the threshold such that the apertures generated cover roughly 50% of the hologram plane.

One problem does exist with this thresholding scheme. Consider the spacing between zero-crossings on the x-axis given by Eq. (15.9-9). As xo increases, the spacings between zero-crossings decrease. This causes object points distant from the center of the object to have a greater number of zero-crossings associated with them. To attempt to correct for this, the number of hits assigned for each zero-crossing can be varied so that the total number of hits per object point is constant. One possible way to implement this approach is to use

A

no hits per zero-crossing ¼ no zero-crossings for the current object point

ð15:9-10Þ

where A is a suitably large constant.

15.9.1Off-Axis Plane Reference Wave

An off-axis plane-wave tilted with respect to the x-axis can be written as

Uðx; yÞ ¼ e j2pax			ð15:9-11Þ
where
a ¼	sin y		ð15:9-12Þ
	l

y being the angle with respect to the optical axis. Using this expression, Eq. (15.9-1) becomes

roi ¼ nl þ x sin y

ð15:9-13Þ

The rest of the procedure is similar to the procedure with the on-axis plane wave method.

15.9.2Experiments

We generated a number of holograms with the methods discussed above [Bubb, Ersoy]. A particular hologram generated is shown in Figure 15.22 as an example. The object reconstruction from it using a He–Ne laser shining on the hologram transparency of reduced size is shown in Figure 15.23.

The reconstruction in Figure 15.23 consists of seven concentric circles, ranging in radius from 0.08 to 0.15 m. The circles are on different z-planes in space: the smallest at 0.75 m, and the largest at 0.6 m. The separation in the z-direction can clearly be seen, in that the effects of perspective and foreshortening are evident. In all the experiments carried out, no practical limit on the number of object points was discovered.

ONE-IMAGE-ONLY HOLOGRAPHY

265

Figure 15.22. A hologram generated with the FWZC method.

Figure 15.23. The reconstruction of concentric circles from the hologram of Figure 15.22.

15.10ONE-IMAGE-ONLY HOLOGRAPHY

For a plane perpendicular reference wave incident on the hologram, it is easy to observe that the existence of the twin images with various encoding techniques is due to the symmetry of the physical spaces on either side of the plane hologram. If we think of the object wave as a sum of spherical waves coming from individual object points, points that are mirror images of each other with respect to the hologram plane, correspond to the object waves on the hologram with the same

266 DIFFRACTIVE OPTICS I

amplitude and the opposite phase. Thus, when we choose a hologram aperture i that corresponds to the phase fi of the virtual image, the same aperture corresponds to the phase fi of the real image.

The conclusion is that the symmetry of the two physical spaces with respect to the hologram needs to be eliminated to possibly get rid of one of the images. This symmetry can be changed either by choosing a hologram surface that is not planar, or a reference wave that is not a plane-perpendicular wave. However, choosing another simple geometry such as an off-axis plane wave distorts only the symmetry and results in images that are at different positions than before.

An attractive choice is a spherical reference wave because it can easily be achieved with a lens [Ersoy, 1979]. If it becomes possible to reconstruct only the real image, the focal point of the lens can be chosen to be past the hologram so that the main beam and the zero-order wave can be filtered out by a stop placed at the focal point. In the following sections, we are going to evaluate this scheme, as shown in Figure 15.24.

In the encoding technique discussed in Section 15.7, the position of each

hologram aperture was chosen according to
jðxi; yiÞ þ kroi ¼ 2np þ f0	ð15:10-1Þ

In the present case, j(xi, yi) is the phase shift caused by the wave propagation from the origin of the reference wave front at (xc; yc; zc) to the hologram aperture at (xi, yi); kroi is the phase shift caused by the wave propagation from the aperture at (xi, yi) on hologram to an object point located at (xo, yo, zo). The radial distance roi is given by

	¼ q	ð		Þ
roi	ðxo xiÞ2 þ ðyo yiÞ2 þ zo2		15:10-2

þ ( ) sign is to be used if the object is desired to be real (virtual).

For a spherical reference wave with its focal point at the position (xc, yc, zc), the phase of the reference wave jðxi; yiÞ can be written as

where	jðxi; yiÞ ¼ krci	ð15:10-3Þ
	¼ q
		ð		Þ
rci	ðxc xiÞ2 þ ðyc yiÞ2 þ zc2		15:10-4

Laser	Object
	Hologram
Lens	Stop

Figure 15.24. The setup for one-image- only holography.

ONE-IMAGE-ONLY HOLOGRAPHY

267

where þ ( ) sign is to be used if the focal point of the lens is in the front (back) of the hologram.

Equation (15.10-1) can be written as

rci þ roi ¼ nl þ	f0l	ð15:10-5Þ
	2p

Suppose that the position of the aperture is to be moved a distance in a defined direction, say the x-direction, such that Eq. (15.10-5) is satisfied. Then, the new lengths of the radius vectors roi0 and rci0 are given by

r0

¼ q

r2

2 x

0

x

i

Þ þ

2

oi

oi

ð

r0

¼ q

r2

2 x

x

iÞ þ

2

ci

ci

ð c

rci0 þ roi0 satisfy

rci0 þ roi0 ¼ rci þ roi B

Solving Eq. (15.10-8) for yields

p&

¼ C1

½

1

1 C2=C1

where

C1 ¼

F1

F2

C2 ¼

F3

F1

F1 ¼ F4ðX þ XcÞ þ Xrci2 þ Xcroi2

F2 ¼ ðrci þ roiÞ2 ðX XcÞ2 2F5

F3 ¼ 2F5roirci F52

F4 ¼ roirci F5

B2

F5 ¼

þ Bðrci þ roiÞ

2

ð15:10-6Þ

ð15:10-7Þ

ð15:10-8Þ

ð15:10-9Þ

ð15:10-10Þ

ð15:10-11Þ

ð15:10-12Þ

ð15:10-13Þ

ð15:10-14Þ

ð15:10-15Þ

ð15:10-16Þ

We note that amplitude modulation of the object points can still be achieved by varying the number of hologram apertures for each object point proportional to the desired amplitude at the object point. Even though the amplitude of the spherical wave varies slowly at the hologram, the average effect from randomly distributed apertures can be considered to be constant.