edge of-the image implies the FFT contains a frequency component at
the maximum frequency. A non-zero value on the corner of the image
implies the maximum frequency exits in the FFT pattern which is M/2 in
the x direction and N/2 in the y direction.
To record the complex Fourier transform as a hologram, the
function F(mAu,nAv) must be heterodyned to a spatial carrier frequency
so as to create a real non-negative pattern to record on film. To
prevent aliasing, the heterodyne frequency must be sufficiently high.
The frequency components in the hologram are shown in Figure 3.6 and
consist of the D.C. spike, the power spectral density of the function
F(u,v), and the two heterodyned terms. To record the function F(u,v)
on film without distortion from aliasing, the spatial carrier
frequency must be 3 times the highest frequency component of the FFT
pattern. This permits the power spectral density term to exist
adjacent to the heterodyned terms with no overlap. The frequencies in
the hologram extend to plus and minus 4B. Thus, the hologram has a
space-bandwidth product 4 times larger than the original image in the
heterodyne direction. When heterodyned in the v direction as implied
by equation 2.35, the resulting hologram matrix must be larger than
the original image by 4 times in the v direction and 2 times in the u
direction. The spectral content in two dimensions is shown in Figure
3.7. The space-bandwidth product is very large for this CGH to record
the information in H(u,v).
The requirement is even greater when the hologram is to be used as
a Vander Lugt filter. When used as a Vander Lugt filter, the CGH must
diffract the light sufficiently away from the origin and the
additional on-axis terms to prevent aliasing in the correlation plane.