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

Figure 16.2. The telescopic system.

The second example is the telescopic system shown in Figure 16.2. Here f equals 1 so that the virtual reference wave is planar. Using two lenses with focal lengths f1 and f2 yields

It is seen that M is independent of the real hologram coordinates, and, if F is large, T2 is insensitive to T1.

The third example is the lensless Fourier arrangement discussed in Section 13.3. It is obtained when the point O of Figure 16.1 lies on the image plane. If z0 is the

16.2.2Aperture Effects

In order to obtain effective diffraction from the virtual hologram, it is desirable that virtual hologram apertures spread waves as much as possible. An interesting observation here is that aperture size may not be important in this context, since the virtual hologram is not to be recorded. In other words, even if virtual hologram

apertures are overlapping, diffraction effects at a distance can be explained in terms of waves coming from point sources on the virtual hologram to interfere with each other in the volume of interest.

Spreading of waves can be discussed in terms of v1 and v2, the angles a ray makes at two reference planes. From Eq. (16.2-2) we find

It follows that spreading of waves increases as f and M are reduced. In the telescopic system, C0 ¼ 0, and the input spreading of waves coming from an aperture is increased at the output by a factor of 1/M.

In order to find the size of the virtual hologram apertures, both magnification and diffraction effects need to be considered. The size of a virtual hologram aperture dv can be written as

where dr is the size of the real hologram aperture, and D is the additional size obtained due to diffraction coming from the limited size of the optical system. For example, in the telescopic system, D can be approximated by [Gerrard and Burch, 1975]

where dA is the diameter of the telescope objective, and f is its focal length.

16.2.3Analysis of Image Formation

Analysis of image formation can be done in a similar way to the analysis of image formation in Section 15.8 on one-image-only holography. We assume that the contributions of various reference waves result in an effective reference wave coming from the point ðxc; yc; zcÞ. This can always be done within the paraxial approximation. If ðxi; yi; 0Þ are the coordinates of a sample point on the virtual hologram, and ðx0; y0; z0Þ are the coordinates of an object point, the equation that determines the positions of various image harmonics is given by

ð16:2-25Þ

where þð Þ sign is used if the object point is real (virtual) with respect to the virtual hologram, and

1

rc ¼ ðx2c þ y2c þ z2c Þ2 ð16:2-28Þ

where þð Þ sign is used if the focal point of the reference wave comes before (after) the virtual hologram.

If Eq. (16.2-25) is divided by M, we find that the real hologram is designed for the object coordinates

If a reconstruction wave coming from ðx00c ; y00c ; z00c Þ and with wavelength l is used, the mth harmonic without the optical system will be reconstructed at