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640 11. Information Display with Optics

11.3.3. SYNTHETIC APERTURE HOLOGRAPHY

This section discusses synthetic aperture holography (SAH), one promising
technique for reconstructing computer-generated holographic information for
real-time 3-D display. Synthetic aperture holography, proposed by Benton
[26], showed the display of images 150mm x 75mm with a viewing angle of
30" [27]. The system is enough to be considered a real-time display for the
human visual system [28].
In the holographic display, a large amount of information is required to
display a large area with a wide viewing angle. In fact, for a given spatial
resolution capability, / 0 of a spatial light modulator, it determines the viewing
angle, 0. The situation is illustrated in Fig. 11.18 for on-axis Fresnel zone plate
reconstruction.
2
For intensity pattern given of the form as bias + cos(fc 0x /2z 0), an instan-
2
taneous spatial frequency f ins = (\./2n)(d/dx)(k 0x /2z 0) = x/Az 0. By setting / insl =
f Q, we solve for the size x max = Az 0/ 0 of the limiting aperture of the hologram,
which determines the reconstructed image resolution. Defining the numerical
aperture of the system as NA = sin 8 = x majz 0, we have NA = A/o- Now,
according to the Nyquist sampling theorem, the sampling interval Ax ^ l/2/ 0;
hence, in terms of NA, we have Ax ^ 2./2NA. Assuming the size of the SLM' is
2
/ x /, the number of samples (or resolvable pixels) is N = (//Ax) . In terms of
2
NA, we have N = (I x 2NA/1) . Hence, for a full parallax 100 mm x 100 mm
on-axis hologram that is presented on a SLM, A = 0.6 /mi, and a viewing angle
of 60°, i.e., 8 = 30°, the required number of resolvable pixels is about 6.7 billion
on the SLM. Because the human visual system normally extracts depth
information of objects through their horizontal parallax, vertical parallax can
be eliminated in order to reduce the amount of information to be stored in the
SLM. For 256 vertical lines, the number of pixels required is 256 x (//Ax) ~ 21
million if vertical parallax is eliminated. This technique of information reduc-
tion is the well-known principle of rainbow holography invented by Benton

Fig. 11.18. Viewing angle for on-axis Fresnel-zone plate reconstruction.