Page 640 - Introduction to Information Optics

624 11. Information Display with Optics

Eq. (11.8) is reduced to the following set of coupled differential equations:

and

(11.9)
at 2

with the boundary conditions E 0(£ — 0) = £ inc, and E^ = 0) = 0. Assuming a
being real positive for simplicity; i.e., a* = a = Ck 0\A\L/2 = a, and a is called
the peak phase delay of the sound through the medium, Eq. (11.9) may be
solved analytically by a variety of techniques; the solutions are given by the
well-known Phariseau formula [13]:

+J 2 2 1 2 (lu0a)
T [(<3e/4) +(a /2) ] / •?

(ll.lOb)
2-11/2

Equations (ll.lOa) and (ll.lOb) are similar to the standard two-wave solutions
found by Aggarwal and adapted by Kogelnik to holography. More recently, it
has been rederived with the Feynman diagram technique [14]. Equations
(ll.lOa) and (ll.lOb) represent the plane-wave solutions that are due to
oblique incidence, and by letting <5 = 0 we can reduce them to the following set
of well-known solutions for ideal Bragg diffraction:

£ 0 (£)=£ inc cos(af/2), (11.1 la)
E^) = -y£ lBcsin(a£/2). (ll.llb)

For a more general solution; i.e., assuming a* is complex, Eq. (ll.llb) becomes

£i(£) = -J 4 ^inc sin«/2) (11.1 lc)
1^1
with Eq. (11.1 la) remaining the same.