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

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

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

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

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

By the same token, if Eq. (15.10-19) is valid for the object point at ðx0; y0; z0Þ and the reference wave with focal point at ðxc; yc; zcÞ, whose wavelength is l, it is also

where m is an integer. We note that m ¼ 0 corresponds to a wave that is the same as the reference wave and focuses at the focal point. If z0 zc, and assuming the initial and the final reference waves are the same, the image positions other than m ¼ 1 are

In other words, if the focal point of the lens is sufficiently close to the hologram, all the images other than the one for m ¼ 1 have to be so close to the hologram that they become completely defocused in the far distance where the desired image is.

However, it is possible to focus one order at a time. If the mth order is desired to be at the position ðx00; y00; z00Þ, the corresponding values of ðx0c; y0c; z0cÞ are simply found such that Eqs. (15.10-24), (15.10-25), and (15.10-26) are satisfied. Then, all the other images become completely defocused and out of view in the far distance where the desired image is, because they are located very close to the hologram.

The above analysis is not always reasonable because the paraxial approximation becomes difficult to justify as the focal point of the spherical reference wave approaches the hologram. Thus, the images other than the one desired may not be well defined. It is also possible that the encoding technique used loses its validity for very close distances to the hologram, especially due to the registration errors of the hologram apertures.

In order to support the above arguments, consider what happens when z0c ! 1 in Eqs. (15.10-24)–(15.10-26); in other words, a plane wave is used for reconstruction.