2 Narrow-Linewidth Laser Oscillators | 9
have used geometrical optics arguments in its derivation [39-42]. In addition,
recent work [1,38] indicates that the origin of Eq. (3) can be related to intra-
cavity interference as described using Dirac's notation [43].
6. BEAM DIVERGENCE
An expression for beam divergence including all the intracavity components
except the active region is given by [1,44]
[
A0-- ~- 1 + +
1/2
(7)
where w is the beam waist, L R =
a'w2/&
is the Rayleigh length, and A and B are
the corresponding propagation matrix elements. For propagation in free space A
= 1 and B = d so that A0 -- X f3-~row for d =
L r, AO ~ ~, 4~/%W(Lr/d )
for d <<L r,
and A0 = X/rtw for
d
>>L r.
AplSropriate ABCD matrices are given in Table 2. Matrices listed include those
for gratings, mirrors, etalons, and multiple-prism beam expanders. The matrices for
the multiple-prism beam expanders are general and enable a round-trip analysis.
Alternative 4 x 4 ray transfer matrices that include dispersion and other
optical parameters are discussed in [1,47,48]. The relation between the disper-
sion of multiple-prism arrays and 4 x 4 ray transfer matrices is discussed in the
Appendix.
7. INTRACAVITY DISPERSION
The return-pass intracavity dispersion for a multiple-prism grating assembly
(see Fig. 2) is given by
-•)p
, (8)
where the grating dispersion is given by [14]
2(sin 0 + sin 0') (9)
cos0
for a grating deployed in a grazing-incidence configuration and
20
F.J. Duarte
TABLE 2
ABCD Propagation Matrices
Optical element/system
ABCD propagation matrix
Reference
Distance L in free space
Flat grating
Flat mirror or grating in Littrow
configuration (0 = 0')
Slab of material with refractive
index n and parallel surfaces
Etalon
Thin convex (positive) lens
Thin concave (negative) lens
Galilean telescope
Newtonian telescope
[coso,,coso o ]
0 cos 0 / cos 0'
0 = angle of incidence
0' = angle of diffraction
0il
[ 1 1
I~ e -- angle of incidence
We = angle of refraction
I e
= optical path length
[lle/n 10 1
jl 0]
-1/f 1
f = focal length
1 7]
1/ f]
[sv o/, I ~-Is~ I
l i, lii~
--fdf~ A+ f~
o
-f~lf2
Multiple-prism beam expander
M1M 2 B
0 (M1M2) -1
ri
B=M1M 2 ~, L m kl k2,j
m=l 1 ' "=
+(MIIM2) ~ (lm/nm) fi k l'j Om k2'j
m=l j=l j
L m
= distance separating the prisms
l m
= optical path length of each individual prism
M 1, M2, k 1,j, k 2,j are defined in the text
Multiple prism beam expander [(M1M2 ) -1
B
(return pass) [ 0 M 1M2
[45]
[46]
[1]
[1]
[1]
[45]
[45]
[45]
[45]
[1,471
[1,44]
2 Narrow-Linewidth Laser Oscillators 2|
/90 ) 2 tan 0
c = k (10)
for a grating in Littrow configuration [13].
The generalized double-pass dispersion for any prismatic array composed of
r prisms (Fig. 6) is given by [1,2,34-36]
(n rim
)
c)(I) - 2MIM 2 1(4-
1 )ff-/l,m
k l,j ke,j
---~ p j=m "=
+ 2 (_+ 1
)"q/'2 m
k 1,/ k
2,j 0)~
= 1 ' = "=
-1 OH m
~k
(11)
where
M1 - n kl,/ (12a)
j=l
and
M2- I~I k2,j. 9 (12b)
j=l
Here,
kl, j -
cos tIJ1, j / cos (~l,j and k2, j -
COS ~)2,j/COS
~(]/2,j are the individual beam
expansion coefficients corresponding to the incidence and exit face of the prism,
respectively. Also H~, m = tan
~,m/nm,
H2, m = tan
(~2,m/nm ,
and (3nm//)A,) is charac-
teristic of the prism material. To estimate the single-pass dispersion (/)(l)2,r/~)M of
the multiple-prism beam expander, the return-pass dispersion given in Eq. (11)
should be multiplied by (2 M1M2 )-1 to obtain the expression [36]
( ) ~ (n './I--I m ) -1 ~F/m
b+Z'r = (-]-
1 )-q-/l,m k 1,j k
2,j
~k m=l
j=m "= Ok
+ k 1,/ k
2,j (__+
1)=q-/2
m
k l,j k 2,2 3X "
j=l '= m=l ' j=l "=
(13)
For multiple-prism assemblies composed of right-angle prisms (as shown in
Fig. 2) designed for orthogonal beam exit, Eq. (11) reduces to [2]
/9@ - 2M 1 (+
1).q-f l, m k
1,2 9
p m=l
j=m ~k
(14)
Further, if the prisms in the preceding expander are manufactured of the same
material and deployed so that the angle of incidence is the Brewster's angle, then
Eq. (14) reduces to

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