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

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

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

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.

y

Locations of zerocrossings closest to origin

∆x1

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.

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

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

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.

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

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

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.