Nonlinear Optical Probes and Processes in Polymers and Liquid ...

32
that of writing beams. If these wavelengths are different, the experimental geometry
should be modified since the reading beam angle of incidence θ ′ has to be calculated
from the Bragg condition sin θ ′ = λ ′ /(2nΛ) for a new wavelength λ ′ .
The coupled wave theory for thick holograms was worked out in detail by Kogelnik
[10]. Similar to the approach we used in the previous section to derive the coupled
equations for the optical fields of the incident beams (Eqs. 2.18), the wave equation
(Eq. 2.11) was solved in the presence of dielectric (refractive) grating described by
Eq. 2.9 and absorption grating described by an equation analogous to Eq. 2.9 for
conductivity. Since for all the PR composites we used for the PR measurements,
the only relevant gratings are dielectric ones, we will consider here only dielectric
gratings. Then, the diffraction efficiency observed in the transmission geometry shown
in Figure 2.4 with an s-polarized reading beam is given by [10]
η = e −αd(1/cR+1/cS) sin2 √ υ 2 − ς 2
√ υ 2 − ς 2
(2.24)
where cR and cS are geometrical factors in terms of the appropriate angles in the
notations of Figure 2.4 as
α is the absorption coefficient,
cR = cos ˜ θ1 , cS = cos ˜ θ2 (2.25)
υ = π∆nd
λ √ cRcS
, ς = αd
2
1
cR
− 1
cS
(2.26)