Page 63 - Tunable Lasers Handbook

44 R. C. Sze and D. G. Harris

The exact double-pass multiple-prism dispersion for any geometry can be
estimated using Duarte's equations [ 121. Note that the double-pass dispersion
can also be calculated by multiplying the single-pass dispersion by 2M, where M
is the overall beam expansion factor [12]. For the case of incidence at the Brew-
ster angle, the individual beam expansion at the mth right-angle prism (kl nz) can
be written

where 11 is the refractive index. Also, for an angle of incidence equal to the
Brewster angle we have tan c$l,nl = n. Under these conditions, for a prism
sequence of r prisms, the overall beam expansion becomes M = nr. Sze et al.
[ 151 write an expression for the dispersive (passive) linewidth of the form

where N is the number of round trips (R in Chapter 2). In this equation the initial
beam divergence is expressed as the ratio of the cavity aperture (a) and the cav-
ity length (I). Under the preceding interpretation where the spectral linewidth is
estimated through a convergence of the beam divergence, the narrowing of the
linewidth cannot proceed indefinitely but must stop as A0 reaches the diffraction
limit [ 151
($I& + 7 (4)
1.22h

Hence. the linewidth expression has the form

Figure 6a shows the situation for a number of initial geometric beam diver-
gence's versus the number of round-trips as calculated using Eq. (3) with the
straight line given as the diffraction limit. The corrected curves are given in Fig.
6(b). Therefore, in many situations where the cavity length is long and the aper-
ture is made very small, the diffraction limit can be reached in one or two round-
trips, implying that there is therefore no need to go to long-pulse lasers. In fact,
however, what was observed in flashlamp-pumped dye lasers [62] and what is
observed in long-pulse excimer lasers [64] is that when a large number of cavity
round-trip times are available the linewidth is generally one-tenth that calculated
by Eq. (5). It was argued in [64] that a frequency-selective aperture transfer
function needs to be incorporated into the general formula in Eq. (3).