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

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]

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

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]

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

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

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

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

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

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

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

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

Figure 16.5. Reconstruction with the first virtual hologram.

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

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