Tunable Lasers Handbook

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