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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
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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
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transmittance:
ðð
gðx; yÞ ¼ Gðnx; nyÞe2pi½xnxþyny&dnxdny
X X
¼ nx ny sin c½ nxx& sin c½ nyy& |
Hðk; lÞ exp½2piðkx nx þ ly nyÞ& |
ð16:3-2Þ
By ignoring the two constants and the two sin c factors outside the sums, the reconstructed image is approximated by the two-dimensional inverse discrete Fourier transform (2D-IDFT) of the transmittance values.
The POCS method discussed in Section 14.8 is used to optimize the design of the hologram. Letting the dimensions of both the observation and CGH planes be M N, the relationship between the wave fronts at the observation plane hðm; nÞ and the CGH plane Hðk; lÞ is given by the following discrete Fourier transform pair:
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where 0 m M 1, 0 n N 1, and
MX1 XN 1
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where 0 k M 1, l N 1, and
The goal of the POCS method is to generate the CGH whose reconstructed image most resembles the desired image.
Given a desired image f ðm; nÞ in a region R of the observation plane, the POCS method works as follows:
1.Using Eq. (16.3-4), compute Fðm; nÞ from f ðm; nÞ.
2.Generate the binary transmittance values Hðk; lÞ from Fðk; lÞ as follows:
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ð |
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3. Using Eq. (16.3-2), find the reconstructed image hðm; nÞ. The accuracy of the reconstructed image is measured based on the mean square error (MSE)
ITERATIVE INTERLACING TECHNIQUE (IIT) |
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between f ðm; nÞ and hðm; nÞ within R, the region of the desired image. The MSE is defined as [Seldowitz]
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MSE ¼ |
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jf ðm; nÞ lhðm; nÞj2 |
ð16:3-7Þ |
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where l is a scaling factor. The minimum MSE for hðm; nÞ is achieved if,
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4.Define a new input image f 0ðm; nÞ such that
(a)Outside R, f 0ðm; nÞ equals hðm; nÞ.
(b)Inside R, f 0ðm; nÞ has the amplitude of the original image f ðm; nÞ and the phase of hðm; nÞ.
5.Letting f ðm; nÞ ¼ f 0ðm; nÞ, go to step 1.
6.Repeat steps 1–5 until the MSE converges or specified conditions are met.
16.4ITERATIVE INTERLACING TECHNIQUE (IIT)
The IIT technique discussed in this section can be incorporated into any existing DOE synthesis method in order to improve its performance [Ersoy, Zhuang, Brede]. The interlacing technique (IT) will first be introduced, and then it will be generalized to the IIT. The IT divides the entire hologram plane into a set of subholograms. A subhologram consists of a set of cells, or points, referred to as a ‘‘block.’’ All the subholograms are designed separately and then interlaced, or entangled, to create one hologram. Two examples of interlacing schemes with two subholograms are shown in Figures 16.8 and 16.9.
Figure 16.8. Interlacing scheme 1 with two subholograms.
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Figure 16.9. Interlacing scheme 2 with two subholograms.
In the IT method, once the entire hologram is divided into smaller subholograms, the first subhologram is designed to reconstruct the desired image f ðm; nÞ. The reconstructed image due to the first subhologram is h1ðm; nÞ. Because the subhologram cannot perfectly reconstruct the desired image, there is an error image e1ðm; nÞ defined as
e1ðm; nÞ ¼ f ðm; nÞ l1h1ðm; nÞ |
ð16:4-1Þ |
In order to eliminate this error, the second subhologram is designed with e1ðm; nÞ=l1 as the desired image. Since the Fourier transform is a linear operation, the total reconstruction due to both subholograms is simply the sum of the two individual reconstructions. If the second subhologram was perfect and its scaling factor matched l1, the sum of the two reconstructed images would produce f ðm; nÞ. However, as with the first subhologram, there will be error. So, the third subhologram serves to reduce the left over error from the first two subholograms. Therefore, each subhologram is designed to reduce the error between the desired image and the sum of the reconstructed images of all the previous blocks. This procedure is repeated until each subhologram has been designed.
Each subhologram is generated suboptimally by the POCS algorithm (other methods can also be used). However, the total CGH may not yet reach the optimal result even after all the subholograms are utilized once. To overcome this problem, the method is generalized to the IIT.
The IIT is an iterative version of the IT method, which is designed to achieve the minimum MSE [Ersoy, Zhuang and Brede, 1992]. The reconstruction image of the ith subhologram at the jth iteration will be written as hjiðm; nÞ. After each subhologram has been designed using the IT method, the reconstruction due to the entire hologram hf ðm; nÞ has a final error ef ðm; nÞ. To apply the iterative interlacing technique, a new sweep through the subholograms is generated. In the new sweep,
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the new desired image f 0ðm; nÞ for the first subhologram is chosen as |
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1ð |
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ef ðm; nÞ ¼ f ðm; nÞ lf hf ðm; nÞ |
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lf is the scaling factor used after the last subhologram. Once the first subhologram is redesigned, the error image due to the entire hologram is calculated, including the new reconstruction created by the first subhologram. Similarly, the second subhologram is designed to reconstruct
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being the updated error. This process is continued until convergence which is achieved when the absolute difference between successive reconstructed images
MX1 XN 1
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reaches a negligible value or remains steady for all the subholograms. By using the IIT method, the convergence tends to move away from the local-minimum MSE and moves towards the global-minimum MSE or at least a very deep minimum MSE.
16.4.1Experiments with the IIT
In all the experiments carried out with the IIT, it was observed that the IIT can improve reconstruction results when used with another algorithm such as the POCS. A particular set of experiments were carried out with the image shown in Figure 16.10.
The associated iterative optimization algorithm used was the POCS. Table 16.1 shows the way the reconstruction MSE is reduced as the number of subholograms is increased.
Figure 16.10. The cat brain image used in IIT experiments.