Page 35 - Tunable Lasers Handbook

16 F. J. Duarte

use less than 5 cm of intracavity space to illuminate a 5-cm-long grating [l].
Certainly, further intracavity space is necessary for a multiple-prism beam
expander designed to provide M = 100 in a MPL oscillator.

4. GENERALIZED INTERFERENCE EQUATION

Consider a generalized transmission grating illuminated by a dispersionless
multiple-prism beam expander as illustrated in Fig. 4. Using the notation of Dirac
the probability amplitude for the propagation of electromagnetic radiation from
the beam expander (s) to a total reflector (x) via a grating of N slits is given by

N
(1)

Using (xlj) = Y (r. ) e-@j and (jI s) = Y (rJ eeiej, where Y (r$ and
4”
Y (rsj) are appropriate diffraction functions, we can write [1,38]

where Y (rj) = Y (r$ Y (T-J and f2; = (0; + $J.
This generalized equation enables the prediction of interference and/or dif-
fraction intensity patterns produced by the interaction of electromagnetic radia-
tion with N-slit gratings of any geometry and/or dimensions. An important appli-
cation of Eq. (2) is the prediction of the transverse-mode structure produced by
an intracavity slit [37]. In this case the intracavity slit is represented by an array
of a large number of small individual slits [37]. For instance, in Fig. 5 the trans-
verse-mode structures corresponding to Fresnel numbers of 0.86 and 0.25, at X =
580 nm, are illustrated.
The interference term of Eq. (2) can be used, in conjunction with the geom-
etry related to the path differences I Lm - Lm-I 1, to establish the expression Ak
= Ae(ae/ah)-1.
For a two-dimensional slit array the equation for the probability amplitude
becomes

Using (XI&,) = Y [rjzyx]e-’*a and (j&) = ‘I’(I+~,,~~) , the two-dimensional
probability can be written as