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

The above equations indicate that the images must focus close to the hologram. Any absence of images in the space of interest indicates that the interference between the harmonic images and the desired image is minimized.

It is also interesting to observe what happens to the positions of the higher order images as zc gets smaller. From Eq. (15.10-24), for equal wavelengths, we find

As m increases, z00 approaches zc=m. This means that it becomes more difficult to observe higher order images as the hologram is designed closer to the focal point of the lens.

A very important consequence of Eq. (15.10-33) is that there is a variable focusing distance; z00 can be varied at will by a slight adjustment of the position of the lens.

15.10.2Experiments

In order to prove the above predictions experimentally, holograms were generated using the scanning electron microscope system, as discussed in Section 15.7.1. In all cases, the sizes of the holograms were 2 2 mm, and the sizes of the apertures were of the order of 1m.

Figure 15.25 shows the He–Ne laser reconstruction from a regular hologram of one object point with the plane-perpendicular reference wave. At the center, the overexposed main beam is observed; to the right of the main beam, there are the slightly overexposed real image of the object point and higher order images; to the left of the main beam, there are the virtual image and virtual higher order images.

Three holograms of an object point were generated using a spherical reference wave such that the focal point of the lens is 1, 3, and 5 cm to the left of the hologram plane, and its x- and y-coordinates are those of the center point of the hologram. Figure 15.26 shows the result with the 5-cm hologram. There is only one visible object point; the dark square is the enlarged picture of the hologram; the main beam covers the whole figure. Figure 15.27 shows the same object point with the main beam and zero-order image filtered out by putting a stop at the focal point of the lens. The results with the 1- and 3-cm holograms gave the same type of results supporting the arguments given above [Ersoy, 1979].

Figure 15.28 shows the reconstruction of a more complicated object. The hologram is designed to be 3 cm from the focal point of the lens, whose x- and

Figure 15.25. The He–Ne laser reconstruction from a regular hologram of one object point showing different orders.

Figure 15.30. Side view of a FZP.

The path difference between two rays traveling along SOP and SAP is given by

q p

ðrÞ ¼ ðr0 þ rÞ ðz0 þ zÞ ¼ r2 þ z20 þ r2 þ z2 ðz0 þ zÞ ð15:11-1Þ

The Fresnel-zone parameter, n, is defined such that the path difference is an integer multiple of half wavelengths [Hecht]:

The nth Fresnel zone is the area between the circles with radii rn 1 and rn. Note that the field at P coming from a point on the circle with radius rn is half a wavelength out of phase with the field from a point on the circle with radius rn 1. Using this fact, it is easy to show that the adjacent zones cancel each other out. Therefore, the total field at P will increase if either all even or all odd zones are blocked out, the remaining zones reinforce each other, thus creating a focal point at distance z. Assuming the zone plate is illuminated by a plane wave, z0 1, and setting z ¼ fo, Eq. (15.11-1) results in

f0 is called the focal point. An FZP operates as a lens with focal length f0. It can be shown that there exist other focal points at f0=3, f0=5, f0=7, and so on [Hecht].

Let Rn be the radius of the nth circle on the FZP. It satisfies

274 DIFFRACTIVE OPTICS I

Using Eq. (15.11-3), this can be written as

When n is large, the second term on the right-hand side above can be neglected, yielding

Figure 16.1. System diagram for virtual holography.

visible light, it is necessary to change the hologram size at the ratio of the two wavelengths in order to prevent image distortions [Smith, 1975]. However, this is a time-consuming and error-prone operation.

A virtual hologram is defined as the hologram that is not recorded in a medium but exists in space as the image of another hologram that is recorded, and which is called the real hologram [Ersoy, August 79]. The information coming from the virtual hologram is the desired information whereas the information coming from the real hologram consists of transformed information. The transformation between the real hologram and the virtual hologram is achieved with an optical system as shown in Figure 16.1.

It can be said that neither the real hologram nor the virtual hologram is exactly like regular holograms. If one looks through the real hologram under reconstruction, one sees transformed information that is probably unrecognizable. The virtual hologram is more like a regular hologram, but it is not registered in a physical medium.

16.2.1Determination of Phase

The rays coming from the real hologram parallel to the optical axis converge to point O as shown in Figure 16.1. Since the optical path lengths between a real hologram point and the corresponding virtual hologram point are the same, the phase at the virtual hologram point relative to the other virtual hologram points are determined by the radius vector length rc between O and the virtual hologram point as shown in Figure 16.1.

The system transfer matrix S that connects the input and the output in the form [Gerrard and Burch, 1975]

A, B, C, and D determine the optical system matrix; T1 and T2 are the distances shown in Figure 16.1. In order to have image generation, B0 ¼ 0 so that

A plane wave input to the optical system at an angle v1 focuses to a point at a distance

Using these equations, it is straightforward to calculate rc shown in Figure 16.1. rc determines the type of the reference wave on the virtual hologram. The phase

due to the reference wave on the real hologram is also to be transferred to the virtual hologram. Therefore, it makes sense to talk of the real reference wave and the virtual reference wave, respectively. This approach can be further extended by having several optical systems that give rise to several virtual holograms and reference waves. The end result would be to add all the reference waves on the last virtual hologram. However, the phase due to each reference wave would be determined by its position on the corresponding hologram.

Three examples will be considered. The first one is a single lens. This case