Page 655 - Introduction to Information Optics

11.3. 3-D Holographic Display 639
where A and B are constants, representing the amplitude of the plane wave and
the diverging spherical wave, respectively. The frequencies co 0 + Q and co 0 are
the temporal frequencies of the plane wave and the spherical wave, respectively.
The parameter z is a depth parameter measured away from the focal plane of
lens L (which is on the surface of the 2-D scanning mirrors) toward the 3-D
object. Equation (11.25) can be expanded and written as

2
2
2
I(x, y; Q) = A + C + 2AC sin + v )+n r (11.26)
where C = Bk 0/2izz. This is a familiar Fresnel zone plate (FZP) expression
(see Eq. [11.21]) but has a time-dependent variable. We shall call it a time-
dependent Fresnel zone plate (TDFZP). This TDFZP is now used to scan the
3-D object in two dimensions and the photodetector collects all the transmitted
light. For the sake of explaining the concept, let us assume that a single off-axis
point object 8(x — x 0, y — y 0) is located Z Q away from the focal plane of lens
L. The scanning of the TDFZP on the point object will cause the photodetec-
tor to deliver a heterodyne-scanned current z(x, y; z, t) given by

2
1 27
- Vo) ] + «4- d - )
After electronic multiplying with sin(iQf) and low-pass filtering, the scanned
demodulated electrical signal i d, as indicated in Fig. 11.17, is given by

2
2
i d(x, y) ~ cos \^- [(x - x 0) + (y - >' 0) ] j , ( H -28)
z
C^ o J
where x and y in the above equation are determined by the motion of the
scanning mechanism. Note that the electrical signal i d contains the location
(x 0, y 0) as well as the depth (z 0) information of the off-axis point object. If this
scanned demodulated signal is stored in synchronization with the x — y scan
signals of the scanning mechanism, then what is stored is the 3-D information
of the object or a hologram and the transmission function of the hologram is
t s(x, y) oc i d(x, y). Hence t d(x, y) is the hologram of 6(x — x 0, y — y 0) when it is
located z 0 from the focus of the spherical wave which is used to generate the
scanning time-dependent Fresnel zone plate. This way of generating holo-
graphic information is nowadays commonly known as electronic holography
[25].