Page 636 - Introduction to Information Optics

620 11. Information Display with Optics

applied, respectively, to obtain relationships similar to those expressed in Eqs.
(1.1. 2) and (11.3); i.e.,
k_ l=k 0-K, (11.4)
and
«,_! = (« 0 -Q, (11. 5}
where the — 1 subscripts indicate the interaction is downshifted. Note that the
closed triangles in Figs, ll.l(a) and 11.2(a) stipulate that there are certain
critical angles of incidence for plane waves of light and sound to interact. The
angle </> B is called the Bragg angle, and it is given by

.

where /, is the wavelength of light inside the acoustic medium, and A is the
wavelength of sound. Note that the diffracted beams differ in direction by an
angle equal to 2<p B, as shown in Figs. 11.1 and 11.2.
In actual experiments, scattering happens even though the direction of
incident light is not exactly at the Bragg angle. However, the maximum
diffracted intensity occurs at the Bragg angle. The reason is that a finite length
transducer does not produce ideal plane waves, and the sound waves actually
spread as they propagate inside the acoustic medium. As the length of the
transducer decreases, the sound column will act less and less like a single plane
wave; in fact, it is now more appropriate to consider it an angular spectrum of
plane waves.
For a transducer with an aperture L, sound waves spread out over an angle
± A/L, as shown in Fig. 11.3. Considering the upshifted Bragg interaction and,
referring to Fig. 11.3, we see that the K-vector can be orientated through an
angle + A/L due to the spread of sound. In order to have only one diffracted
order of light generated (i.e., fc + 1), we have to impose the condition that
//A » A/L,

2
L»A //. (11.7)
This is because for £_ t to be generated, for example, a pertinent sound wave
vector must lie along K'\ however, this either is not present, or is present in
negligible amounts in the angular spectrum of sound, if condition in Eq. (11.7)
is satisfied. If L satisfies this condition, the acousto-optic device is said to
operate in the Bragg regime and the device is commonly known as a Bragg
cell. Thus, physical reality dictates that a complete energy transfer between the
two diffracted beams is impossible since there always exists more than two dif-