AZIMUTHAL DISTRIBUTION

NONUNIFORMITIES

IN THE P U M P I N G

IN A N E L L I P T I C A L

E . S. K o v a l e n k o , L. a n d T . N. B a b c h e n k o

[.

HOUSING

Shangtna,

UDC 621.378.32:535.89

It is usual to pump a s o l i d - s t a t e l a s e r via the side s u r f a c e ; this r e s u l t s in an uneven d i s tribution of the pumping energy over the c r o s s section, which affects the energy, spectrum, and spatial c h a r a c t e r i s t i c s of the emission. Much attention has been given to the distribution of the pumping energy in the rod; a solution has been given [1-5] for the radial nonuniformtty for two-dimensional and t h r e e dimensional models. It is more complicated to take into account the azimuthal variation in the pumping distribution, because in this case one has to consider the actual g e o m e t r y of the lamp, r e f l e c t o r , and c r y s tal. Numerical calculations by the Monte Carlo method [6] have shown that the azimuthal variations can be 20-30%, which is quite sufficient to alter substantially the mode s t r u c t u r e of the radiation. P a r t i c u l a r importance attaches to such nonuniformities for continuous-wave l a s e r s , because the t h e r m a l strains a r i s ing during pumping may result in a b e r r a t i o n s large enough to halt the lasing. These a b e r r a t i o n s cannot be balanced out by lenses, and it is t h e r e f o r e important to estimate azimuthal variations in the pumping. Here we give equations for the azimuthal distribution of the pumping in a cylinder within an elliptical lamphouse; the r e s u l t s are c o m p a r e d with experiments. The total field g at point M within the rod may be found as the sum of the r a y s arising at this point f r o m v a r i o u s p a r t s of the c r y s t a l surface; we take into account not only r a y s reaching M d i r e c t l y but also those f r o m M reflections within the c r y s t a l , and then in the two-dimensional case 2~

g = ~ Rn~

f ~ i ( 1 - - r ) e - ~ L ~ r m e _ m k D G(r - -c- ~ r~ 0

(1)

+ Zmcp, ~ )da,

ttz=O

where 1R is the rod radius, n is the r e f r a c t i v e index, k is absorption index, and r is the F r e s n e l reflection coefficient for a r a y falling on the s u r f a c e at an angle fix and r e f r a c t e d at an angle L.

Glu,v).\~%lJ~.x ~ 0

As the choice of polarization only slightly affects the final result, we a s s u m e that the electric field v e c t o r is parallel to the cylinder axis; then r=

[r

Fig. I.

Ray paths within a cylinder.

~

(~ cos i -

.~o~p,

"I'.

(2)

n cos i + cos p j )

The symbols used in (1) are shown in Fig. 1; the position of point H (where the r a y strikes the surface) is determined by angle ce, while the point M at which one calculates the field is d e f i n e d b y angle ~ and radius p. The distance between M and H is denoted by L. It is convenient also to introduce the angle 0, which defines the direction to the point in the illuminator to which the ray is reflected after s t r i k ing H at an angle fix, and we also introduce the angle ~obetween the rays OM and OH.

Translated f r o m Zhurna[ Prikladnoi Spektroskopii, Vol. 16, No. 2, pp. 274-278, F e b r u a r y , 1972. Original article submitted May 8, 1970; revision submitted D e c e m b e r 13, 1971.

@ 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 F('est 17th Street, New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. ,t copy of this article is available from the publisher for $15.00.

203

F i g . 2. R e g i o n of v a l u e of ~, fl• ~, P w h e r e G(c~, /3• 4, P) = 1 for p = c o n s t . /3.-< 1 [n the h a t c h e d r e g i o n s . The b o u d a r y c u r v e s c o r r e s p o n d to/3• = 1 and define the l i m i t s of i n t e g r a t i o n in (1). C a l c u l a t e d p a r a m e t e r s e = 0.4; r l / R = 0.3; p/R = 0.5 (a) and 0.1 (b). F u n c t i o n G(c~, /3•

p, ~) e q u a l s u n i t y if the r a y with the p a r a m e t e r s oz and /3• r e a c h the point p, 4, b e -

tag z e r o if the r a y does not r e a c h that point. T h e n the i n t r o d u c t i o n of G into (1) i n v o l v e s the a s s u m p t i o n that t h e r e is a r e l a t i o n b e t w e e n ~, p, fl~, ce, s i n c e for a given p, 4, a t h e r e is only one 3• f o r which G -- 1. Secondly, t h e r e is a r a n g e in the v a r i a b l e s p, ~, o~ in which none of the r a y s i n c i d e n t at the r e a l angle p• r e a c h e s M, so i n t r o d u c t i o n of G n a r rows the l i m i t s of i n t e g r a t i o n in (1), and t h e s e l i m i t s m a y be found as follows. We c o n s i d e r the r a y paths in t h e l a m p h o u s e (the light s o u r c e h e r e and s u b s e q u e n t l y is a s s u m e d to lie to the left of the rod) and c o n s i d e r only the c a s e w h e r e the r a d i u s r 1 of the l a m p ks much l e s s than the l i n e a r d i m e n s i o n s of the e l l i p s e (r l << a, b, w h e r e a and b a r e the s e m i a x e s of the e l l i p s e ) ; then we r e a d i l y get a s y s t e m of e q u a t i o n s f o r 3_L as a f u n c t i o n of a , p, and 4: c~ -----0 -- I]~, (3.1) ~-- a -- T,

sin ~, = + r-L

1 -- e ~

(3.3)

1We 2-2esin0'

--R

sin i --

(3.2)

p sin q~ ] / R ~ + 92 - - 2Rp cos q0 '

(3.4)

sin i = j - sin ~ a.

(3.5)

n

H e r e e is the e c c e n t r i c i t y of the e l l i p s e . The s o l u t i o n of (3) is of p h y s i c a l s i g n i f i c a n c e only ff sin fi• < 1; h o w e v e r , we get for a c e r t a i n r a n g e in p, ~, c~ that (3) gives s i n 3 • > 1, which m e a n s that in this r e g i o n t h e r e a r e no r a y s that r e a c h M f r o m H. Then G = 0 for t h e s e c e , p, and ~, We put sin/3• = l to get the b o u n d a r y v a l u e s a --~ f (~, 9),

which

correspond

tion in (I).

The

to transition behavior

from

(4)

G = 1 to G = 0, and these values

of the curves

of (4) ks determined

of c~ determine

the limits of integra-

by np/r|.

If

[1+~ r]R}

(5)

1 - - e
!

then c~ = f(~) is a p a i r of c l o s e d c u r v e s ~142; ~1~2 as shown in Fig. 2a;

if on the o t h e r hand

(6) fl

the branches

204

of the curves

do not close

(Fig.

2b).

--8 ' q

~

9 ~

z

F i g . 3. A z i m u t h a l d i s t r i b u t i o n of the p u m p i n g e n e r g y in the s p e c i e = 0.3;

q S [ ~ S t !

r l / t l = 0.465;

R = 3.25;

p : 0.5 (1), 1.0 (2), 1.5 (3), 2 (4),

1,0

3 (5); b) o b s e r v e d p u m p i n g e n e r g i e s f o r p = 3.

F i n a l l y , if np< 1 --8 r1

(7)

l+e

one can g e t any o~ f o r any ~; the h a t c h e d r e g i o n s in the d i a g r a m c o r r e s p o n d to G = 1. All the c o m p u t a t i o n s w e r e done w i t h an M-20 c o m p u t e r t a k i n g a c c o u n t only of t e r m s h a v i n g m = 0 in (1), and with g put in the f o r m eP,

g =

"J'f(~)d ~ ,

where naRS (R - - p cos ~p)2 f (~P) = n ( ~ + p~n~sin z $) iS + n (R - - p cos r

e"~L, (8)

3 z = R2 -k p2 (i - - n 2 sin 2 (p) - - 2 Rp cos qD). T h e l i m i t s of i n t e g r a t i o n in (8) w e r e taken f r o m d i a g r a m s of the t y p e of F i g . 2a and b; the r e s u l t s f o r e = 0.3. The c a l c u l a t i o n s w e r e t e s t e d b y e x p e r i m e n t . ~v

F i g . 3 shows

T h e l u m i n e s c e n c e i n t e n s i t y W m a y b e put in the f o r m n2

= c --av, no

w h e r e C i s a c o n s t a n t and n2,/a 0 i s the r e l a t i v e p o p u l a t i o n of the u p p e r l e v e l , in w h i c h c a s e W f o r low e x c i t a tion l e v e l s m a y be c o n s i d e r e d a s p r o p o r t i o n a l to the i n t e n s i t y of the e x c i t i n g light. One t h e r e f o r e m e a s u r e s the d i s t r i b u t i o n W o v e r the c r o s s s e c t i o n to j u d g e the d i s t r i b u t i o n of the light e n e r g y , and t h i s m a y b e c a r r i e d out in two d i s t i n c t w a y s . In the f i r s t m e t h o d , the light e m e r g i n g f r o m a g i v e n p o i n t on the end of the r o d i s t a k e n b y a light g u i d e t o an Ft~U-22 p h o t o m u l t t p l i e r , and t h i s light guide is s c a n n e d o v e r the end of the r o d in the a x i m u t h a l d i r e c t i o n . The t r e n d s w e r e r e a d i l y r e p r o d u c i b l e in r e p e a t m e a s u r e m e n t s . In the s e c o n d m e t h o d , w e e x a m i n e d the d i s t r i b u t i o n of the a b s o r b e d e n e r g y b y m e a n s of n e a r - f i e l d p i c t u r e s r e c o r d e d b y the m e t h o d of [7]; in t h i s c a s e , the b l a c k e n i n g at e a c h p o i n t on the i m a g e c o r r e s p o n d s to the l u m i n e s c e n c e e n e r g y e m i t t e d f r o m the c o r r e s p o n d i n g p o i n t on the end f a c e . A z i m u t h a l m i c r o p h o t o m e t r y gave a c u r v e t h a t is d i r e c t l y p r o p o r t i o n a l to the p u m p i n g e n e r g y d i s t r i b u t i o n in the r o d . T h e t e s t s w e r e done in s y s t e m s w i t h e l l i p t i c a l r e f l e c t o r s h a v i n g e = 0.4 (a = 16.5 mm) and e = 0.3 (a = 49 mm) with p u l s e and c o n t i n u o u s - w a v e c o n d i t i o n s . The p u m p i n g w a s p r o v i d e d b y l a m p s t y p e s R K L 25 [8, 9] and [ F P - 8 0 0 . F i g u r e 3 s h o w s the r e s u l t s f o r one of the r u b y r o d s , and the r e s u l t s f o r o t h e r s were generally similar. The p u m p i n g i n t e n s i t y is then l e a s t on the l a m p side; n e a r the c e n t e r of the c r y s t a l t h e r e is a r e gion a l s o w h e r e the p u m p i n g d i s t r i b u t i o n is i n d e p e n d e n t o f a z i m u t h . T h e r e w a s a l s o no m a r k e d d i s c r e p a n c y b e t w e e n the n o n u n i f o r m i t i e s in the c o n t i n u o u s - w a v e and p u l s e d s t a t e , and the c a l c u l a t e d a z i m u t h a l d i s t r i b u t i o n s a g r e e d s a t i s f a c t o r i l y with the m e a s u r e d o n e s . T h e p u m p i n g i n t e n s i t y d e v i a t e d f r o m the m e a n b y up to 25% a s one t r a v e r s e s the a z i m u t h a l c u r v e , as a g r e e s with the c a l c u l a t i o n . C o m p l e t e a g r e e m e n t is not to b e e x p e c t e d on a c c o u n t of the m a n y a s s u m p t i o n s m a d e in the c a l c u l a t i o n and a l s o the u n c e r t a i n t y of c e r t a i n q u a n t i t i e s a s m e a s u r e d .

205

LITERATURE I.

2. 3. 4. 5. 6. 7. 8. 9.

206

CITED

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