HIGH ENERGY LASER OPTICS

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HIGH ENERGY LASER OPTICS

W. Christiansen

To cite this version:

W. Christiansen. HIGH ENERGY LASER OPTICS. Journal de Physique Colloques, 1980, 41 (C9),

pp.C9-371-C9-383. �10.1051/jphyscol:1980951�. �jpa-00220605�

JOURNAL DE PHYSIQUE CoZZoque C9, suppZ6ment au nO1l, tome 42, novernbre 1980, page C9-371

HIGH ENERGY LASER OPTICS

W.H. Christiansen

Department of Aeronautics and Astronautics, University o f Washinaton, SeattZe, Washington 98195 U. S.A.

Risumi -- Cet a r t i c l e propose une br&e r i c a p i t u l a t i o n des principes thgoriques e t des propri6t6s a e s k o n a t e u r s optiques.

A

p a r t i r de l a th6orie des r6sonateurs lasters s t a b l e s , on s 1 i n t 6 r e s s e aux

r k o n a t e u r s i n s t a b l e s en i n s i s t a n t sur l a compr6hension des propri6tes conduisant au d6veloppement de l a s e r s de grande puissance, proches de l a l i m i t e de d i f f r a c t i o n . On analyse aussi

1

'importance grandi ssante des optiques d6formabl e s dans 1 a transmission l a s e r en vue d1am61 i o r e r l e s performances en champ l o i n t a i n d'un l a s e r u t i l i s 6 dans des conditions dgfavorables. On prgsente enfin l e s am61 iorations dues 5 1 ' u t i l i s a t i o n d'optiques actives dans l e cas de l a turbulence e t de l a d6focal i s a t i on thermi que.

Abstract. -- This paper reviews b r i e f l y t h e theoretical understanding and properties of optical reson- a t o r s . Starting from the theory of stable l a s e r resonators, unstable resonators a r e discussed with emphasis on understanding t h e i r properties t h a t ensure near d i f f r a c t i o n limited performance of higb power l a s e r s . The growing f i e l d of adaptive optics f o r l a s e r transmission i s a l s o examined as an added means of improving f a r f i e l d performance of a l a s e r under adverse circumstances. The improve- ments obtained with active optics f o r turbulence and thermal blooming a r e given.

INTRODUCTION. -- The rapid development of l a s e r s has introduced the need f o r specialized optics to exploit the c a p a b i l i t i e s of t h i s new source of co- herent radiation. Of p a r t i c u l a r i n t e r e s t here a r e the o p t i c s related t o high power l a s e r s which a r e necessary f o r high efficiency and good f a r f i e l d performance. Oftentimes t h e requirement of minimum beam divergence c o n f l i c t s with t h e 1 imitations i n - troduced by t h e nature of high-power l a s e r c a v i t i e s and propagation phenomena. The structure of os- c i l l a t o r c a v i t i e s t o overcome these l i m i t a t i o n s , t h e search f o r optical materials, and design of mirrors and windows a b l e t o withstand t h e high- power d e n s i t i e s of l a s e r s represents formidable problems. In f a c t , aerodynamic windows were intro- duced as a method of withstanding very high l a s e r mwer i n wavelength regions f o r which there were no good optical materials available.

High power l a s e r s a r e s t i l l being developed a t a f a s t r a t e and i n recent years the' direction of these developments has tended t o introduce new op- t i c a l problems. For example, t h e high power chemi- cal l a s e r i s one type of l a s e r t h a t i s continually evolving and a s a r e s u l t has introduced unique geo- metrical problems of t h e cavity involving t h e flow

f i e l d . Then too, s a t u k t i o n power densities a r e increasing and wavelengths a r e shortening, which puts additional s t r a i n on e x i s t i n g technology; new technologies a r e developing. There is considerable i n t e r e s t i n f r e e electron l a s e r s a s e f f i c i e n t high

power l i g h t sources. Indications a r e t h a t e f f i c i - encies greater than 30% a r e possible provided cer- t a i n r e s t r i c t i v e properties can be met with the electron beam and radiation f i e l d . Among t h e re- quirements a r e the need f o r extremely high radia-

12 2

t i o n power d e n s i t i e s ( 1 0 ~ ~ - 1 0 w/cm

)

f o r electron trapping and no e l e c t r i c f i e l d reversals on the electron beam a x i s . Unusual d i f f r a c t i o n e f f e c t s a r e a l s o present because of t h e small cross section of t h e gain medium. Such requirements and prhper- t i e s will undoubtedlyaffectfuture resonator design.

This paper'will t r y t o review some of the'impor- t a n t aspects of o p t i c s f o r high energy systems and introduce some current subject material t h a t will be of use t o t h e symposium attendee interested i n continuous high power l a s e r s . Problems pertaining t o very high pulsed power l a s e r s such a s those used i n l a s e r fusion will not be covered i n t h i s paper.

Since t h e f i e l d has grown enormously, a l l aspects of t h e o p t i c s of high energy l a s e r s cannot be cov- ered in a s i n g l e paper. This paper covers only two topic areas. F i r s t t o be covered i s t h e theory of

&stab1 e l a s e r resonators a s these c a v i t i e s permit t h e e f f i c i e n t u t i l i z a t i o n of large mode volumes while maintaining good f a r f i e l d performance. Even so, propagation e f f e c t s which become especially im- portant f o r high power l a s e r s can l i m i t severely beam irradiance a t a distance. Fortunately, adap- t i v e optics i s an approach which helps t o circum- vent t h i s problem.and.thi3.i~ the. second t o p i c t o

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980951

C9-372 JOURNAL DE PHYSIQUE

be covered. By t a k i n g advantage o f t h e coherence p r o p e r t i e s o f t h e l a s e r , a c t i v e o p t i c s has been used t o improve o u r a b i l i t y t o d e l i v e r a h i g h i r r a - diance beam under adverse circumstances.

RESONATORS.

--

The l a s e r c a v i t y i s n o t simply a mi- crowave c a v i t y . The microwave c a v i t y i s a closed metal s t r u c t u r e a t which o n l y c e r t a i n wavelengths r e - sonate. Generally speaking, t h e dimensions o f t h e c a v i t y correspond t o these wavelengths. O f course, t h i s s i z e o f c a v i t y i s n o t u s e f u l ( t h e exception i s waveguide types which a r e n o t discussed here) f o r high power i a s e r s having wavelengths on t h e o r d e r o f microns. The h i g h power l a s e r c a v i t y i s l a r g e com- pared w i t h wavelength and operates i n a s l i g h t l y d i f f e r e n t way *than o r d i n a r y microwave c a v i t i e s . The o r d i n a r y l a s e r c a v i t y does n o t determine t h e f r e - quepcy

p f

t h e l s s e r ' i o much ( t h e c a v i t y dimension does determine t h e macrowave frequency) as t h e g a i n media .does. Rather, i t serves t o discriminate. among th$many pgssibl-e. frequencies' w i t h i n t h e g a i n band- w i d t h o f th'e malecular system. ,In so doing i t w i l l

pyovide opeical feedbatk to'maintai'n t h e g i v e n o s c i - l l a t i o 6 . p e r m i . t k i n g h i g h l y .monqchromatic deams and, r h o p e f u l l y , c d n t r o l t h e o p t i c a l phase )cros; t h e beam f r o h t .so t h a t low beam divergence and h i g h focus i n - t e n s i t i e s a r e .possible.

S t a b l e C a j i t i e s ,

--

As t h e r e i s a large.body o f 1 it e r a t u r e a v a i l a l i l e on',stable reson8tor.s

i:

^{i t i s the}

purpose:af t h i s sedtion. t o i n t r o d u c p those f e a t u r e s o f s t a b l e resonators which a r e h e l p f u l i n understan- -.

' d i n g t h e requirements

d f

high 'power l a s e r s o n l y . Re-,, sonators can be c l a s s i f i e d as s t a b l e o r ,unstable, dipending

o n

whether a r a y i s trapped between t h e m i r r o r & o r n o t .

~i~

o p t i c a n a l y s i s [I] shows t h a t t h e s t a b f l i t ; ' c o n d i t i o n " i s 0 < (1 ~ / R ~ ) ( I

-

L/R2) ,

<' l', where L ,is t h e d i s t a n c e betwe.en, t h e m i r r o r s and R1; R2 & f e r t o t h e r a d i u s o f c u r v e t u r e o f t h e m i r - r o r s 1 and 2. I f t h e value o f t h e product l i e s out- s i d e these numbers, t h e c a v i t y i s unstable. Because o f t h e repeated appearance o f these parameters and t h e i r importance, i t i s u s e f u l t o d e f i n e g1 = 1

- ^L/R,

and .g2 = 1

-

L/R2 so t h a t t h e c o n d i t i o n simply reads 0 < g1g2 < 1. Any resonator can be represented as a p o i n t on a s t a b i l i t y diagram such as shown i n Fig. 1.

Examples

b f

each t y p e o f c a v i t y a r e shown. The shaded r e g i o n corresponds t o s t a b l e c a v i t i e s . S t a b l e c a v i t i e s were used f i r s t and can be used f o r low o r high, g a i n media.

The beam q u a l i t y o f these c a v i t i e s may be meas- ured by t h e divergence o f t h e beam i n t h e f g r f i e l d .

Fig. 1 S t a b i l i t y Diagram

By a p p l i c a t i o n o f t h e d i f f r a c t i o n i i n t e g r a l , i t i s known t h a t f a r f i e 1 d patte,rns, o f l a s e r beams a r e merely F o u r i e r transforms

d f

t h i i r near f i e l d s [2].

Generally speaking, a .uniform f i e l d and pha:e out- p u t o f a l a s e r produces t h e smallest, divergence,and

'. ,

.

maximum i n t e n s i t y i n t h e c e n t r a l lobe. i n t $ e fa?

' f ie.1 d [3] : For a c i r c u l a r . a p e r t u r e t h e we1 1- knowh ,.

A i r y p a t t e r n w i t h t h e angular w i d t h o f 2.44 x/d i s a ' r e s u l t [2]. I f , however, t h e r e a r e ampl i tude,aid phase v a r i a t i o n s i n t h e near fiel'd, t h e n t h e d i v e r - gence i s l a r g e r and t h e c e n t r a l i n t e n s i t y i s redu- ced. Of, these, t h e phase d i s t r i b u t i o n i s t h e l a r - . ger e f f e c t and i t i s t h i s f a c t o r t H a t must be made as u n i f o r m as p o s s i b l e i n r e s o n a t o r , design. F:ig.- 2 shows s c h e m a t i c a l l y t h e e f f e c t s o f amplitude and phase v a r i a t i o n s across t h e beam.

The s o l u t i o n o f t h e wave equation w i l l g i v e f i e l d amplitude and phase t h a t i s n e c e s s a r i

td

^~pre;

d i c t o p t i c a l beam q u a l i t y . , For i n f i n i t e . s i z e m i r - r o r s w i t h a u n i f o r m l y passive inedium, m a t c h i n g ' t h e s o l u t i o n s o f t h e wave equation f o r n e a r l y ' plane waves gives a n a l y t i c r e s u l t s t h a t p r e d i c t t h e t r a n - sverse modes and frequency o f t h e l a s e r [I]. These r e s u l t s a r e i n t h e form o f Hermite

dr

~ ' a ~ u e r r e polynomials and can be found i n most t e x t s ah l a - sers [4]. The c o r r e c t s o l u t i o n which 'may corres- pond t o a l i n e a r s u p e r p o ~ i t i o n o f many o f ,th.e ,poly- nomials i s determined by t h e boli$ary c o n d i t i o n s . The important t h i n g t o n o t i c e i s , t h a t t h e r e i s a .sequence o f s o l u t i o n s , each more compli,cated t h a n

i t s predecessor. The fundamental mode' is.. a simple

NEAR- FIELD FAR-FIELD

I

NEAR-FIELD NEAR-

FIELD

FAR - FIELD

INTENSITY PHASE INTENSITY

L

Fig. 2 E f f e c t o f Near F i e l d I n t e n s i t y and Phase D i s t r i b u t i o n on F a r - F i e l d P a t t e r n

guassian beam and t h i s mode i s r e a l l y what one aims f o r i n designing a s t a b l e l a s e r c a v i t y as i t pro- duces t h e b e s t qua1 i t y beam.

These a r e t h e e s s e n t i a l f e a t u r e s o f s t a b l e c a v i - t i e s , but, o f course, t o be u s e f u l some l i g h t must be e x t r a c t e d and one must consider t h e e f f e c t s o f f i n i t e m i r r o r size. M i r r o r s o f f i n i t e s i z e l e a d t o

cated [I]. E s s e n t i a l l y one c a l c u l a t e s the f i e l d by a n a l y z i n g t h e steady s t a t e f i e l d d i s t r i b u t i o n t h a t can be sustained w i t h i n an empty resonator. This amounts t o c a l c u l a t i n g t h e f i e l d d i s t r i b u t i o n a t t h e second m i r r o r (2) from an assumed d i s t r i b u t i o n a t t h e f i r s t m i r r o r (1). The r e s u l t i n g d i s t r i b u t i o n can be o b t a i n e d v i a a n i n t e g r a l using t h e s c a l a r f o r m u l a t i o n o f Huygens' p r i n c i p l e [2,5]. T h i s must be repeated, f o r t h e reverse case, i .e., from ( 2 ) t o (1) and u l t i m a t e l y a r r i v e a t t h e i n i t i a l d i s t r i b u - t i o n . A f i e l d d i s t r i b u t i o n t h a t obeys t h i s condi- t i o n can be w r i t t e n as t h e i n t e g r a l equation

where 9 i s t h e f i e l d d i s t r i b u t i o n ,

6

i s t h e propa- g a t i o n i n t e g r a l o p e r a t o r [2,31, and y i s t h e complex eigenvalue f o r t h e given mode. More e x p l i c i t l y ,

= (1-p)ei$, where $ = phase s h i f t and 6 = round t r i p power l o s s . Therefore 8 i s t h e o u t p u t coup- l i n g f r a c t i o n and g i s used t o determine t h e exact frequency o f t h e l a s e r . E a r l y s o l u t i o n s o f these o s c i l l a t o r s was g i v e n by Fox and L i where t h e y showed t h a t c e r t a i n t y p e resonators and low o r d e r modes have very low d i f f r a c t i v e losses [5]. Higher o r d e r modes have h i g h e r losses and t h i s can be 'used as a means o f sel.ecting low o r d e r modes. A u s e f u l parameter i n d e s c r i b i n g t h e d i f f r a c t i o n e f f e c t s o f t h i s problem i s t h e Fresnel number, N = a /hL,where 2 a r e f e r s to: t h e r a d i u s of t h e m i r r o r s . F i g u r e 3 ^' shows how t h e losses vary w i t h N f o r t h e l o w e s t o r - der modes f o r a c a v i t y w i t h c i r c u l a r m i r r o r s having d i f f r a c t i o n losses and t h i s problem i s more compli-

- s

h -0

9 ^Y

Y

v m

m

m

m ⁰ A

S

... - -

0.1 1.0 10 100 0.1 1.0 , ^{10 I od}

N = a2/ ^{~d - N = a / ~ d}

Fig. 3 D i f f r a c t i o n Loss/Transit vs. Fresnel Number f o r S t a b l e Resonators w i t h C i r c u l a r M i r r o r s [I]

C9-374 JOURNAL DE PHYSIQUE

equal diameters and curvatures. J u s t as expected, the losses decrease with increases i n N.

Mode Volume.--

A

major problem i n the design of high power l a s e r s i s t o obtain a l a r g e mode volume while retaining good mode control. The point about mode volume cannot be over-emphasized in dealing w i t h high power l a s e r s . Not completely using a f u l l y energized cavity volume through poor mode f i l l i n g i s in r e a l i t y a large waste of energy and a loss in system efficiency. The lowest order and most desirable mode spot s i z e i s given by

where w i s t h e spot size.The multiplying f a c t o r

F

i s t y p i c a l l y of order unity f o r a wide range of gl and gz

[6].

Typically guassian s t a b l e resonator modes a r e slender and much-smaller than one would

l i k e , maybe only millimeters wide. The mode volume f o r t h i s case i s of order

v

^{% o(L.~W')} = L ~ X

.

For large volumes of a c t i v e media t h i s leads t o very long c a v i t i e s . Table 1 taken from 131 shows t h a t l a s e r s with volume measured i n l i t e r s have ca- vity lengths t h a t a r e very long f o r large N a t

1 = 3p.

Similar r e s u l t s e x i s t f o r other wavelengths.

I f we t r y t o use /a l a r g e r mode volume without in- creasing t h e length, then higher order modes develop i n the presence bf a gain medium

[3].

Efforts t o arrange t h e cavity geometry so t h a t e f f e c t i v e l y

F

i s large usually a r e d i f f i c u l t a s t h e s e n s i t i v i t y t o a.l

i

gnmen t becomes extreme and goes 1 i ke

[6]

As a practical matter t h e requirements on mirror f i - gure, a1 ignment, and gain homogeneity a r e too large.

Table 1. Laser-Cavity Length Requirements for

Various

Optical Fresnel Numbers

(A

= 3 pm)

Resonator Fresnel number

Resonator. a21N, L in em

m o d e voIume,

liters

1 10 100

(a) STABLE RESONATOR, N F >> ^I

1 \

(b) STABLE RESONATOR, NF >> ¹

- -

-.-. ...:.3v

STABLE RESONATOR, NF >> ^I

Fig. 4. Cross Coupling i n Various Resonator Geometries

C31

Unstable Resonators. --

A

solution t o t h e above problem i s the unstable resonator proposed by Sieg- man 173. In t h i s case both mode control and phase coherence e x i s t even with a l a r g e mpde volume.

Nearly a l l current high power l a s e r s use t h e unsta- ble resonator t o e x t r a c t power. This scheme i s practical f o r high gain l a s e r s only because of t h e large losses encountered. For high power l a s e r s with low gain t h e r e i s as y e t no good solution.

A

l a s e r can be made phase coherent over t h e beam f r o n t only i f the resonator i s designed so t h a t a l l p a r t s of t h e lasing region share an interaction through d i f f r a c t i o n . Then t h e optical phases of a l l the emitters a r e locked together through t h e cross- coupling furnished i n the cavity

[3]. A

low Fres- nel number s t a b l e cavity imp1 i e s s i g n i f i c a n t d i f f r a c - Ition e f f e c t s a r e present and a uniform phase can re-

s u l t . However, a high Fresnel number cavity only /has t h e central region strongly coupled. Other op- [tical paths with wide excursions across t h e cavity b r e not d i f f r a c t i v e l y coupled and t h i s leads to 'higher order modes.. The s i t u a t i o n i s i l l u s t r a t e d

i n Fig.

4

by the crosshatching. In an unstable

( a ) S Y M M E T R I C

( b ) A S Y M M E T R I C

( c l CONFOCAL

(COLLIMATED OUTPUT BEAM

Fig. 5 . Types o f Unstable Resonators

resonator a t l a r g e N, t h e l a c k o f d i f f r a c t i v e coup- l i n g i s compensated f o r by p r o v i d i n g a s o r t o f geo- m e t r i c a l crosscoupling. There i s a c e n t r a l r e g i o n i n t h e unstable o s c i l l a t o r w i t h i n which d i f f r a c t i o n i s strong and l a s e r o s c i l l a t i o n i s i n phase. The c a v i t y simply magnifies t h i s mode as t h e l i g h t bounces back and f o r t h and escapes. This i s shown as t h e crosshatching i n the t h i r d p i c t u r e i n F i g . 4 . I t can be viewed as an o s c i l l a t o r i n t h e c e n t r a l r e - gion and a multipass amp1 i f i e r as the beam enlarges.

As t h e m a g n i f i e d beam has t h e same phase p r o p e r t i e s o f the innermost beam, good beam c o n t r o l i s p o s s i b l e .

Types o f Unstable Resonators:- Figure 5 i l l u s - t r a t e s t h r e e types o f conmon unstable resonators.

I n t h e symmetric c a v i t y , l i g h t i s coupled from b o t h ends and so i s o f l i m i t e d usefulness. An asymmetric

c a v i t y i s used t o o b t a i n a s i n g l e beam and t h e m i r - r o r r a d i i can be chosen t o g i v e t h e d e s i r e d angular divergence o f t h e o u t p u t beam. A s p e c i a l case i s t h e confocal c o n f i g u r a t i o n where R1 +

R2

= 2L. I n t h i s case t h e beam o u t p u t i s p a r a l l e l i n t h e geome- t r i c approximation. Unstable resonators a r e c l a s s i - f i e d according t o t h e i r p o s i t i o n on t h e s t a b i l i t y diagram. The p o s i t i v e branch of t h e f i r s t type ( t h e f i r s t quadrant on t h e s t a b i l i t y diagram) i s used most commonly i n h i g h power l a s e r s s i n c e i t con- t a i n s no i n t r a c a v i t y f o c a l p o i n t s where n o n l i n e a r e f f e c t s may occur. This branch i s c h a r a c t e r i z e d by g1g2 > 1, g1 > 0. c31

A t f i r s t , i t would appear t h a t t h e presence o f t h e h o l e i n t h e o u t p u t beam would be an undesirable f e a t u r e . But except i n extreme cases i t was p o i n t e d o u t t h a t t h e f a r f i e l d d i s t r i b u t i o n depends mainly on t h e phase d i s t r i b u t i o n and t h e h o l e f i l l s i n t h e f a r f i e l d [8]. The main f e a t u r e i s t h a t t h e p a t t e r n has more energy i n t h e s i d e lobes than f o r t h e case w i t h o u t t h e hole. As l o n g as t h e h o l e accounts f o r l e s s than 50% o f t h e beam area t h e f a r f i e l d i s n e a r l y i d e a l .

As i n s t a b l e c a v i t y operation, t h e e s s e n t i a l as- pects o f an unstable o s c i l l a t o r o p e r a t i o n can be ob- t a i n e d using a geometric model [7,9]. The r e s u l t s o f t h i s a n a l y s i s shows t h a t t h e o u t p u t c o u p l i n g f o r c y l i n d r i c a l resonators i s approximately

6 = 1

- F

I

where M i s t h e c a v i t y round t r i p m a g n i f i c a t i o n and i s re1 ated t o t h e g parameters 191. The geometry o f t h e s i t u a t i o n i s i l l u s t r a t e d i n F i g . 6. N o t i c e t h e amount o f coupling i s independent o f the c a v i t y dia- meter and i t i s simple t o make t h e m i r r o r s as l a r g e as r e q u i r e d t o e x t r a c t energy from a given modevol- ume. This i s t r u e i r r e s p e c t i v e o f t h e type o f

F i g . 6. The M a g n i f i c a t i o n Factor

C9-376 JOURNAL DE PHYSIQUE

unstable c a v i t y . T y p i c a l l y , coupl i n g values o f 20%

t o 90% a r e usual. The r e s u l t s a r e a1 so t r u e f o r m i r r o r shapes ( o f s p h e r i c a l c u r v a t u r e ) o t h e r than c i r c u l a r . I t i s a l s o p o s s i b l e t o consider aspheric curvature m i r r o r s i n which M d i f f e r s depending on which a x i s o f t h e m i r r o r i s being described.

There i s a very u s e f u l equivalence p r i n c i p l e f o r unstable c a v i t i e s which a l l o w s one t o r e l a t e an un- symmetrical c a v i t y w i t h one l a r g e m i r r o r and one small m i r r o r (coupl i n g from one end) t o a symmetric unstable c a v i t y w i t h t h e same l e n g t h . The p r i n c i p l e i s given i n

[lo]. he

round t r i p m a g n i f i c a t i o n and losses o f an asymmetrical c a v i t y w i t h parameters

2 2

gl, g2 and Fresnel numbers N1 = al /hL, N2 = a2 / A L a r e t h e same as t h e one way losses and magnifica- t i o n o f a symmetric c a v i t y o f t h e same l e n g t h w i t h

I I

' 1

1

GEOMETRICAL 7

"%

Fig. 7. Power Loss vs. N and Neg [ll]

g values o f each m i r r o r given by g = 11

-

^2g1g2

1

^and

a m i r r o r Fresnel number N = a /xL = (N,/2g2( 2

.

^This

d e f i n e s t h e m i r r o r curvatures and diameters o f t h e e q u i v a l e n t symmetric resonator. This simp1 i f i e s t h i n g s considerably because one can d e r i v e a l l t h e p r o p e r t i e s o f one-sided resonators from t h e one-way r e s u l t s o f t h e simple symmetric resonator. This a p p l i e s n o t o n l y t o t h e simple geometric model b u t a l s o i n wave a n a l y s i s o f s t a b l e and unstable c a v i - t i e s . A very i m p o r t a n t parameter o f t h e symmetric unstable resonator which by t h e equivalence p r i n c i - p l e a p p l i e s t o one-sided c a v i t i e s too i s t h e so- c a l l e d e q u i v a l e n t Fresnel number given by Neq. = N v ( o r Neq. = N, [%(g1g2- 1) ] ' j 2 ) .

D i f f r a c t i o n E f f e c t s .

--

There a r e d i f f r a c t i o n e f f e c t s t h a t l e a d t o t r a n s v e r s e modes. To o p t i m i z e t h e f a r f i e l d p a t t e r n o f such a c a v i t y , t h e wave na- t u r e of t h e c a v i t y must be analyzed so t h a t one can d i s c r i m i n a t e a g a i n s t h i g h e r o r d e r modes and o b t a i n s i n g l e mode operation. Except f o r s p e c i a l cases,no a n a l y t i c sol u t i o n s have been found f o r unstable r e - sonator modes. One must r e s o r t t o numerical solu- t i o n of t h e i n t e g r a l equations o f t h e electromagne- t i c f i e l d as was done f o r s t a b l e c a v i t i e s , even i f t h e c a v i t y i s empty. The i d e a f o l l o w s as b e f o r e b u t t h e geometry i s s l i g h t l y d i f f e r e n t . The equivalence p r i n c i p l e s i m p l i f i e s t h i n g s considerably since by symmetry we o n l y have t o propagate waves i n one d i - r e c t i o n as t h e reverse d i r e c t i o n i s t h e same. Thus t h e r e i s j u s t a s i n g l e i n t e g r a l i n s t e a d o f a double i n t e g r a l f o r t h e round t r i p . I f one uses rectangu- l a r coordinates, t h e f i e l d amp1 i t u d e i s separable i n t o x and y components [5] which a r e independent f o r an unloaded o s c i l l a t o r (no gain medium). A t y p i - c a l r e s u l t from t h i s a n a l y s i s i s shown i n Fig. 7 along w i t h t h e geometrical model r e s u l t s [3,11].

Whereas t h e geometrical model shows a l o s s t h a t i s independent o f m i r r o r s i z e , t h e numerical wave s o l u- t i o n s show complicated o s c i l l a t o r y behavior having as a l i m i t t h e geometrical r e s u l t f o r l a r g e N. I f these r e s u l t s a r e p l o t t e d a g a i n s t N t h e o s c i l l a -

eg

t i o n s peak a t i n t e g r a l values o f N Further c a l - es

:

c u l a t i o n s have shown t h a t t h e cusps I n t h e l o s s curves i n d i c a t e mode crossings [12,13]. Thus i n un- s t a b l e resonators d i f f e r e n t modes can have lower l o s s e s a t l a r g e N which i s i n c o n t r a s t t o s t a b l e

eg

resonator r e s u l t s . The l o w e s t losses always e x i s t a t h a l f i n t e g e r values o f N and mode crossings occur a t i n t e g e r values of eg N This has been

eg'

i s s u f f i c i e n t so t h a t l i g h t i s d i r e c t e d back along

1 .oo

t h e r a y path t o t h e c e n t e r o f t h e c a v i t y , we f i n d h = a/N So d i f f r a c t i o n from t h i s g e n e r a l l y small

0.80

r e g i o n near t h e edge causes t h e complicated mode eg' s t r u c t u r e . This was recognized by Anan'ev [15,16].

By p r o p e r l y t a p e r i n g o r a d j u s t i n g t h e r e f 1 e c t i v i t y

0.60

-

o f t h e m i r r o r s i n t h i s small r e g i o n these d i f f r a c -

-

9 I 1

t i o n e f f e c t s can be e l i m i n a t e d and complete separa-

t i o n o f t h e low order modes e x i s t f o r any N as seen i n Fig. 9. This i d e a l s i t u a t i o n i s undoubtedly hard

i

t o implement i n p r a c t i c e . Another p o s s i b l e approach i s t o a l t e r t h e shape (curvature) o f a m i r r o r t o r e - duce edge e f f e c t s [I

71.

Also, i t i s known t h a t rec-

0 I ^{I I I I} I

t a n g u l a r m i r r o r s a r e n o t so s u s c e p t i b l e t o t h i s mode

0 0.20 0.40 0.60 0.80 1-00

C ~ u ~ l i n g as c i r c u l a r ones.

x/a

C y l i n d r i c a l l y A c t i v e Regions.

--

An important class o f resonators having a c y l i n d r i c a l l y a c t i v e Fig. 8. I n t e n s i t y D i s t r i b u t i o n f o r t h e Case

of Neq = 49.4. [ I 2 1

confirmed by experiment [14]. Figure 8 shows a t y p i c a l r e s u l t o f one o f these d i f f r a c t i o n c a l c u l a - t i o n s [12] showing amplitude v a r i a t i o n across the m i r r o r p r o f i l e f o r a symmetrical c a v i t y w i t h N = 49.5 and M=1.9. I n t h i s case t h e phase i s n e a r l y eg constant w h i l e t h e i n t e n s i t y v a r i e s q u i t e r a p i d l y . There i s a way o f d i s c r i m i n a t i n g a g a i n s t unwan- t e d transverse modes which r e s u l t s from an under- standing o f the complicated mode o s c i l l a t i o n s . I t t u r n s o u t t h a t t h e existence o f transverse modes a r i s e s from d i f f r a c t i o n o f t h e beam from t h e edges o f the m i r r o r s back i n t o t h e c a v i t y and as such couples t h e d i f f e r e n t p a r t s o f t h e modes together throughout t h e volume. Even w i t h l a r g e m i r r o r d i - mensions, waves o r i g i n a t i n g from an annular r e g i o n of dimension h a t t h e edge o f t h e m i r r o r have an an- g u l a r spread e = O(h/h)

.

I f t h e d i f f r a c t i o n angle

r e g i o n [18] and u s i n g special c o n i c a l o r cube r e - f l e c t o r s [19,20] poses a d d i t i o n a l problems n o t han- d l e d p r o p e r l y i n e a r l i e r t h e o r e t i c a l papers. See, f o r example, Fig. 10a. I n t h i s c l a s s o f resonators

r FLAT END MIRROR

rk

LEG

---

(a) OUTPUT BEAM

---

SrnE

( b ) FRONT

VIEW VIEW

Neq

Fig. 9. Mode Losses w i t h Guassian M i r r o r s

[ I 9 1 Fig.lO.a)A c y l i n d r i c a l Resonator and W-Axicon M i r r o r b)Geometric P o l a r i z a t i o n Scrambl i n g E f f e c t Produced by l i n e a r l y P o l a r i z e d I n p u t Beam on a W-Axicon M i r r o r .

C9-378 ^JOURNAL DE PHYSIQUE

i t has been shown e x p e r i m e n t a l l y t h a t p o l a r i z a t i o n e f f e c t s a r e important and t h a t m i x i n g o f t h e p o l a r - i z a t i o n s t a t e occurs due t o t h e W-axicon (see F i g . l o b ) . Thus the s c a l a r f i e l d t h e o r y described ear- l i e r i s n o t adequate i n t h i s instance. Instead, t h e v e c t o r f i e l d ' m u s t be analyzed. Recently Dente [21] has shown t h a t one can w r i t e two s c a l a r i n t e - g r a l equations which a r e coupled even i n t h e ab- sence o f a medium t o describe t h e e q u i v a l e n t v e c t o r f i e l d . Thus t o a l a r g e e x t e n t e x i s t i n g l a s e r codes can l a r g e l y be used t o study t h e case where p o l a r i - z a t i o n e f f e c t s a r e important [22]. These m o d i f i c a - t i o n s seem t o confirm experiment.

An a d d i t i o n a l problem w i t h these resonators comes from t h e use o f compactors, w-axicons, o r i n n e r ax- icons used t o condense t h e r a d i a t i o n i n t h e c a v i t y . These axicons a r e s u b j e c t t o h i g h r a d i a t i o n f i e l d s e s p e c i a l l y a t t h e t i p o f t h e o p t i c a l element [23].

This i s p r e c i s e l y where c o o l i n g i s d i f f i c u l t a n d t h e sharp edge and c o a t i n g s may be a b l a t e d o r melted away. One can imagine t h a t damage i f present i n such a c r i t i c a l area would have an adverse impact on c a v i t y performance.

S e n s i t i v i t y t o Alignment and M i r r o r Tolerance.-- Another important aspect o f u n s t a b l e resonators i s t h e i r s e n s i t i v i t y t o misalignment and m i r r o r f i g u r e . This i s o f p r a c t i c a l i n t e r e s t because o f mechanical v i b r a t i o n s which may be present and f i g u r e e r r o r s i n t h e manufacture o f r e a l m i r r o r s . The e f f e c t o f mis- a1 ignment o f c i r c u l a r m i r r o r s was i n v e s t i g a t e d by

3

2

TlVE BRANCH

I 2 3 4 5 6 7 8 9 1 0

G~OMETRIC MAGNIFICATION, M

I 1 I I I I I I

0 25% 50% 75% 90% 96% 98% 99%

GEOMETRIC OUTPUT COUPLING C = I-

M*

Fig. 11. M i r r o r Alignment E f f e c t s

Krupke and Sooy [24]. For a small angle misal ign- ment they p r e d i c t e d a change i n t h e o p t i c a l a x i s o f angle 4 versus M. T h e i r r e s u l t s t o g e t h e r w i t h solu- t i o n s [25] a t f i n i t e Fresnel number a r e shown i n F i g . 11. I n t h e 1 im i t o f l a r g e Fresnel ,number, i t was shown t h a t t h e r a t i o of g / X goes l i k e 2M/M-1 f o r t h e p o s i t i v e branch and 2M/M+1 f o r t h e negative branch. This r e s u l t shows t h a t t h e p o s i t i v e branch i s much more s e n s i t i v e t o t h i s form o f e r r o r than t h e l e s s used negative branch, e s p e c i a l l y as M -t 1 o r as t h e o u t p u t c o u p l i n g approaches zero. These trends a l s o apply t o m i r r o r f i g u r e t o l e r a n c e as we1 1.

Other forms o f e r r o r s i n c l u d i n g f i g u r e e r r o r s a r e c u r r e n t l y being i n v e s t i g a t e d . I t i s i n t e r e s t i n g t o n o t e t h a t o p t i c a l c a v i t i e s u s i n g phase-conjugate m i r r o r s (PCM) a r e being i n v e s t i g a t e d [26,27]. The PCM has t h e unusual p r o p e r t y o f r e f l e c t i n g waves t h a t are t h e complex conjugate o f t h e incidentwave.

It appears t h a t such c a v i t i e s have r e a l confined gaussian modes. The i m p l i c a t i o n f o r h i g h power l a - sers i s t o be determined, b u t t h i s may mean t h a t t h e conventional meaning o f t h e s t a b i 1 i t y of c a v i t i e s i s questionable.

E f f e c t o f Gain Medium & As.vmetries.

--

^{To a}

l a r g e e x t e n t t h e t h e o r y o f u n s t a b l e resonators has been borne o u t by numerous experiments [14,24,28].

These, f o r t h e most p a r t , have been cases where t h e gain medium i s f a i r l y u n i f o r m and does n o t a l t e r d r a s t i c a l l y t h e o p t i c a l p r o p e r t i e s o f t h e c a v i t y . However, i n many h i g h power l a s e r s t h e g a i n v a r i e s considerably i n t h e c a v i t y due t o n a t u r a l asymme- t r i e s and s a t u r a t i o n e f f e c t s . I n a d d i t i o n , r e f r a c - t i v e effects, whether by turbulence o r imposed non- u n i f o r m flow, w i l l cause phase e r r o r s l e a d i n g t o de- p a r t u r e s from t h e r e s u l t s described so f a r . These e f f e c t s must be included and, as one might expect, o n l y make t h e r e a l problem exceedingly complex. So- l u t i o n s t o such c a v i t i e s can o n l y be handled by l a r g e numerical codes. Some modeling of g a i n d i s - t r i b u t i o n has been c a r r i e d o u t by c o n s i d e r i n g t h e e f f e c t o f g a i n t o be represented by a gain sheet [29] o r sheets l o c a t e d i n t h e c a v i t y . The t e c h n i - que p e n n i t s decoupling o f t h e propagation from t h e change i n energy o f t h e wave. The method i s o f li- m i t e d use because o f t h e divergence o f t h e w a v e s i n t h e medium and probably n o t so good f o r l a r g e M.

C a l c u l a t i o n s have been made comparing t h e modes o f an u n s t a b l e resonator w i t h gain t o t h a t w i t h a u n i f o r m medium. The mode i n t e n s i t y d i s t r i b u t i o n be- comes more u n i f o r m i n t h e presence o f a s a t u r a b l e

medium because the gain i s lower a t high i n t e n s i t y than a t low i n t e n s i t y 1301. There i s l i t t l e e f f e c t on phase i f refraction e f f e c t s a r e excluded from the calculation. These e f f e c t s together with mirror heating and deformation can be extremely important when considering the s e n s i t i v i t y of the cavity to alignment e r r o r s . Some continuous l a s e r experiments Rave a c t u a l l y become pulsed with such interactions [31,32]. This i s referred t o as the mode media in- teraction problem and is. unique t o high power l a s e r s .

Of course, simple idealized models don't approach the real s i t u a t i o n s many times and more complicated codes must be used. For example, the

BLAZER

compu- t e r code i s used f o r wave analysis supplemented with RESALE codes f o r chemical processes i f present

[3]. The solutions a r e f a r too involved t o t r e a t in any analytic fashion and vary from device t o device.

This i s especially t r u e i f multiple spectral l i n e s a r e involved as in the case of an

HF

l a s e r . In the final scenario the only s u r e way t o analyze real ca- vity behavior i s by diagnostic experiments f o r beam power and energy [3].

ADAPTIVE OPTICS. -- Through design e r r o r s or unavoi- dable beam propagation problems, the ideal ampl itude and phase d i s t r i b u t i o n across a l a s e r beam may be l e s s than desired giving r i s e t o poor f a r f i e l d per- formance. Optical components whose c h a r a c t e r i s t i c s can be a l t e r e d a r e now being investigated and used as a means t o improve t h i s performance. The term active optics r e f e r s to an adaptive optical system t h a t operates i n real time i n order t o control wave-

uniform, a f f e c t s I(P) i n a deleterious way by in- creasing the energy i n the s i d e lobes. Nonuniform amplitude i s typical with large

N

l a s e r c a v i t i e s (Fig.

8 ) .

Finally, there i s the /nds which i s the optical path length. This integral i s affected by index of refraction variations caused e i t h e r by tur- bul ence o r thermal blooming ( i . e . , a change i n the medium by heating action of t h e beam i t s e l f ) . For- tunately, n i s almost one f o r many problems of in- t e r e s t . The value of n-1 may be written a s

^~p

where t h e value of

6

i s 0 (3.10-

4 )

t y p i c a l l y and the den- s i t y ,

p ,

i s measured in amagats. Thus the integral k/(n-1)ds represents phase changes due t o

p

varia- tions. I f

p

i s constant, there will be no e f f e c t o n I ( P ) , but a spatial variation i n p ( r ) gives a spa- t i a l phase variation which contributes to a degrada- tion i n f a r f i e l d performance. I t should be noted t h a t adaptive optics attempts to modify phase e r r o r s by i n t e r j e c t i n g a compensating optical element in the wave t r a i n . Appreciable success has already been achieved [34]. As y e t , to the a u t h o r ' s know- ledge t h e r e i s no known way t o correct f o r amplitude variations once they occur. Fortunately, the phase corrections are the most important as they a f f e c t the beam the most.

Suppose t h a t /(n-1)ds i s zero, t h a t the ampl itude of the f i e l d i s constant, and t h a t t h e i n i t i a l phase d i s t r i b u t i o n over the l a s e r beam i s nonzero and small. Then the reduction in peak 1 ight inten-' s i t y r e l a t i v e t o t h e beam i s given by [35]

-

11In _..

A

1 - 7 t ~ ~ = ^{1 -}

^A$

²

f r o n t s . Here we b r i e f l y discuss the r o l e of adap-

where I. i s the i n t e n s i t y f o r an unaberrated beam t i v e optics i n high power l a s e r operation. and am2 i s the mean square e r r o r of phase over the

The integral solution of the wave equation in a beam cross section ( a f t e r subtraction of t i l t and weakly r e f r a c t i v e medium shows t h a t the i n t e n s i t y a t f o c u s ) . T h i s f i g u r e of merit i s known as the Strehl a point P in t h e f a r f i e l d i s approximately 1331 r a t i o [36]. The r e s u l t shows t h a t t h i s r a t i o i s in-

1 i $ ikfndsdA

1 ) 1% ^{( ~ e e} _/ dependent of t h e nature of the aberration and demon-

& s t r a t e s the extreme s e n s i t i v i t y of the image peak where s represents the ray path from the source t o

P ,

E

i s the amplitude d i s t r i b u t i o n of the source,

^$

i s the i n i t i a l phase d i s t r i b u t i o n , and n represents the index of refraction along the ray path. Three things besides the overall 1 aser power level a f f e c t the i n t e n s i t y a t

P

and should be noted i n t h i s for- mula. Two of these were mentioned in the previous section. The f i r s t i s

$

which i s affected by t h e presence of many transverse modes, f i g u r e e r r o r s of t h e mirrors i n the cavity and other sources of phase e r r o r . The second i s E(1) which, i f not

i n t e n s i t y t o phase e r r o r .

Aberrations a r e wavefront deviations from a

spherical reference sphere. In a classical sense

these aberrations a r e described in terms of a s e t o f

c h a r a c t e r i s t i c functions depending on t h e wavefront

modification and then labeled as t i 1 t , astigmatism,

coma, spherical aberrations, e t c . [35]. The sim-

p l e s t a c t i v e systems a r e those t h a t control t i l t or

focus of a beam. In the former, t h i s i s j u s t a f l a t

mirror whereas the l a t t e r i s usually done by adjust-

ing the separation distance of two focusing elements

JOURNAL DE PHYSIQUE

as in a Cassegranian telescope. Most a c t i v e o p t i - cal systems a r e directed toward removal of t h e more complicated aberrations of wave fronts such a s as- tigmatism and coma and t h i s i s where the research currently 1 ie s f o r l a s e r application.

Basic Active Optics Systems f o r Laser Trans- mission.

--

Usually t h e a c t i v e optical system i s de- signed t o optimize a property of t h e system such as the energy dCnsi t y a t a s p e c i f i c location. Three basic components a r e required [36]:

a

wavefront modifying device; a measuring device which accepts l a s e r l i g h t and provides an output related t o the property being optimized; and an information pro- cessi ng device which through appropriate a1 gorithms converts the measured data i n t o control signals f o r t h e wavefront modifying device. Two basic systems which are used in l a s e r transmission a r e shown in Fig. 12. Such systems normally operate on a s i n g l e wavelength using the radiation generated by the la- s e r . I f more than one spectral l i n e i s involved such as in the chemical l a s e r , considerable addi- tional complication i s involved i n order t o phase lock t h e various spectral l i n e s [37]. Simpler sys- tems would be advantageous.

In Fig. 12a, the phase conjugation approach, the beam reaching the t a r g e t gives r i s e

t o

r e f l e c t i o n from small areas producing g l i n t s which generate spherical waves. These reflected waves traverse the same propagation path i n a reverse direction andare s p a t i a l l y modified by the same optical disturbance as the transmitted beam. The received wave i s com- pared t o a local reference wave and the required correction, which i s the phase conjugate of themea- sured wavefront d i s t o r t i o n , i s generated.

MODIFIER

OPTICAL PATH DISTURBANCE

IOPD)

o PHASE CONJUGATION

LASER TARGET

INTENSITY WAVEFRONT DETECTOR

PROCESSOR b APERTURE TAGGING

Fig. 1 2 . Active Optics Systems [36]

The second system (Fig. 12b) i s a multidither o r aperture tagging approach in which small perturba- tions in phase, A $ << IT, a r e made on each small seg- mented area on the outgoing beam. The optical power returned from a g l i n t i s analyzed to determinewhich part of the aperture i s to be adjusted. These per- turbations are always added t o the wavefront so t h a t there i s always some residual e r r o r , b u t t h i s i s small. In t h i s approach the return signals do

not

have t o propagate along t h e same optical path a s the transmi t t e d beam.

Depending on the physical s i t u a t i o n o r tracking mode one method may have an advantage over t h e o t h e r . Both approaches a r e being pursued f o r active l a s e r beam control. I t appears t h a t the phase conjugate system can operate quite f a s t (O(msec)) i f conver- gence of t h e loop has been obtained. Loop conver- gence times will be long (O(sec)), however, a t low signal to noise r a t i o s which occur a t large ranges.

In t h e case of t h e multidither system t h e conver- gence time can be short even a t rather large ranges [36]. Dither frequencies (sinusoidal perturbations) can be q u i t e high. In the case of one experiment

[38] these frequencies range from 8-32 kz with a spacing of 1.4

kHz

and gave an overall convergence time of 1.5 t o 3 msec.

Wavefront Modifying Devices.

--

Both r e f l e c t i v e and r e f r a c t i v e wavefront modifying devices a r e pos- s i b l e in principle.

A t

the present time r e f l e c t i v e devices (mirrors) a r e the most successfully used correctors and t h i s i s what i s described here. The regimes in which a c t i v e mirrors operate may be sepa- rated in terms of wavefront correction range and frequency response and a r e shown in Fig. 13. For l a r g e mirrors with large deflections t h e frequency response i s poor and t h e i r function i s primarily f o r figure control of a system. Curve B shows the oper- a t i n g range f o r mirrors t h a t require high frequency response b u t not necessarily large deflection as f o r atmospheric correction. Lastly the small rectangu- l a r area i s required f o r aperture tagging themirrors.

TO meet these needs four basic types of active mirrors have been developed and a r e shown in Fig.

14. Of these, only the f i r s t two seem t o be useful f o r high power application. The piston type mirrors use piezoelectric position actuators which under the r i g h t circumstances can be driven t o frequences on the order of 10 kHz and tend t o be small. From the optical point of view, the use of segments having both piston and t i l t capability i s preferable a s

Frequency, Hz

Fig. 13. Operating Ranges for Active Mirrors [36]

fewer segments are required t o match a given wave- f r o n t . Somewhat l a r g e r mirrors tend t o be made as

a

defomable thin-plate. This type

has

reached a high stage of development [36] and i s more e a s i l y cooled than others.

Another possible approach which i s j u s t being investigated i s a r e f r a c t i v e type device using f r e e gas j e t s [39]. This idea developed from our under- standing of t h e aerodynamic window as an optical e l - ement. I t may be possible t o improve the optical quality of a high power l a s e r beam by inducing phase changes in the aerodynamic window i t s e l f .

B u t

t h i s p o s s i b i l i t y has not y e t been proven in t h e laboratory.

Applications.

--

Turbulence: Much i n t e r e s t in adaptive optics centers on turbulence compensation a t the IR wavelengths and the question naturally a r i s e s a s t o how much compensation i s possible.

TO

a large extent t h i s depends on the turbulent scale dimension and the required frequency response [33, 401. Fortunately, in the case of atmospheric

t u r b u -

lence t h e required bandwidth i s only several hundred hertz [36] which, a1 though d i f f i c u l t t o imp1 ement, i s not an impossibly high bandwidth. I t i s extre- mely d i f f i c u l t to get a system with a bandwidth greater than 1 kHz. Intense turbulence and smaller scale turbulence requires increasing t h e number of degrees of freedom of the system sensibly. In f a c t , small s c a l e turbulence t h a t e x i s t s

i n

boundary la- yers in l a s e r c a v i t i e s or on a i r c r a f t probably

Piston only Piston plus tilt Segmented Mirrors

Discrete position actuators Discrete force actuators

Monolithic Mirror

\ 4

Membrane Mirror Fig. 14. Types of Active Mirrors c a n ' t be corrected f o r .

Atmospheric turbulence i s a d i s t o r t i o n which i s not a function of l a s e r power and i s referred t o as a l i n e a r interaction. As such, i t i s r e l a t i v e l y easy t o improve the Strehl r a t i o (provided the s c a l e length i s not too small )

.

Improvements with a mu1

-

t i d i t h e r system a r e detailed in [41,42].

Thermal blooming : Another and more d i f f i c u l t type of disturbance i s t h a t of thermal blooming.

This i s a nonlinear interaction caused by the inevi- table heating of the atmosphere by the l a s e r and creates an undesirable lensing action and defocusing of t h e beam. I t i s d i s t i n c t l y a high power l a s e r phenomenon and depends c r i t i c a l l y on the l a s e r po- wer. The lensing action takes place a l l along the beam path and so the e f f e c t i v e lens i s thick. Pre- liminary work on compensation f o r thermal blooming has been considerably l e s s successful than f o r

tur-

bulent compensation. Simply a l t e r i n g t h e phase pro- f i l e and thus t h e projected f a r f i e l d pattern j u s t s h i f t s t h e problem t o another region of t h e beam in the case of blooming. In some simple cases there i s no solution in principle [43]. In any case there i s l i t t l e o r no improvement, whereas f o r turbulence t h e

C 9 - 3 8 2 JOURNAL DE PHYSIQUE,

/

COAT CORRECTED

P = POWER I N T O CELL. mW

Fig. 15. Thermal Blooming Compensation [45]

improvement i s as good as one makes the wavefront corrector. Real time compensation f o r thermal blooming has been demonstrated i n the laboratory 1441 using a tagged aperture approach. The r e s u l t s f o r t h i s experiment a r e shown in Fig. 15 wherein i t i s seen t h a t the i n t e n s i t y on t a r g e t was increased 2.5 times over t h a t without using a c t i v e o p t i c s .

A

sensible improvement, but one which i s c l e a r l y 1 im- i t e d . One other promising approach i s the use of predictive phase compensation [45]. Here t h e opti- mum phase d i s t r i b u t i o n can be worked out in advance and applied with only minor feedback corrections.

Other a1 gorithms a r e being investigated too.

In many cases thermal blooming e x i s t s i n t h e pre- sence of a r e l a t i v e wind, e i t h e r due t o an actual wind, a movement of the transmitter, o r an apparent wind caused by the tracking of some moving t a r g e t . In some ways the l a t t e r i s an e a s i e r problem as the lensing action i s then strongest near the transmit- t e r , affording a b e t t e r opportunity t o improve the f a r f i e l d pattern. Thermal blooming with wind pro- duces crescent shaped f a r f i e l d patterns which move ( t i l t ) i n t o the wind. Figure 16 shows calculated r e s u l t s f o r the change in peak flux as a function of nondimensional slew r a t e NW ( r e l a t i v e wind) and dis- tortion number N D (beam power). One sees t h a t in- creasing t h e wind reduces the e f f e c t s of thermal blooming and so does the use of a c t i v e o p t i c s . For fixed slew r a t e we can increase the irradiance on t a r g e t by 2.5 again.

Herrmann [43] has shown t h a t the phase conjugate method leads to nearly complete compensation of

Fig. 16. Peak Flux Density in Wind with and without Correction [44]

atmospheric turbulence. Partial compensation f o r cases with moderate blooming with and without turbu- lence are possible i f t h e e f f e c t i v e thermal lenses i s near the transmitter as f o r slewed beams. For cases with substantial blooming, the method only gives 1

i m i

ted improvement.

Adaptive Optics in Resonators

:

Recent1 y there have been s t u d i e s concerning t h e use of adaptive op- t i c s i n resonators because of t h e i r known s e n s i t i v i i t y to a1 ignment. Computer studies [46] have shown t h a t azimuthal mirror misfigures i n cylindrical re- sonators as small as h/20 can lead t o severe degra- dation in t h e Strehl r a t i o . However, i t was shown t h a t t h i s e r r o r can be s u b s t a n t i a l l y compensated by using an adaptive optical element f o r t h e outer axi- con a s the r e s u l t s i n Table I 1 show. This element i s operated in a s t a t i c mode but the actuators a r e

Table 11. Computer Results for Distortions of t h e form

(X/N)

cos2g Applied t o the Rear Cone

1461

Misfigure

X/120

cos 2

1/40

cos 2

XI20

cos 2

~ / 1 0

cos

2

Strehl Ratio Without Compensation

0.901 0 0.6016 0.401

2

0.1056

Strehl Ratio With Cbmpensation

0.9996

0.9916

0.9916

0.9911

adjusted ( i n t h e ~0mptJtati0n) t o maximize t h e S t r e h l [28] Granek, H. and Morency, A. J . , ~~~l

. opt. 13

r a t i o . The improvement w i t h t h e proper a c t u a t o r (1974) 368.

[29] Reusch, D.B. and Chester, A.N., Appl

.

^Opt.

²

s e t t i n g i s e x c e l l e n t ; Other t r i a l cases using d i s - (1973) 997.

t o r t i o n s on o t h e r c o n i c a l elements showed s i m i l a r [30] w e i n e k , - ~ . ~ .

,

AVCO RR 425 (Jan. 1977).

[31] Yoder, M.J. and Ahouse, D.R., Appl

.

^Phys.

improvements. While these c a l c u l a t i o n s were c a r r i e d L e t t .

7

(1975) 673.

o u t w i t h o u t a g a i n medium, t h e b a s i c compensation [32]

m,

R.A. and Peressini, E.R., Gas-Flow and Chemical Lasers (Hemisphere Pub. Corp. ) 1979, method i s l i k e l y t o be successful under gain condi- 425,

t i o n s i n c l u d i n g t h e p o s s i b i l i t y of i n c l u d i n g an ac- [ 3 3 I L u t o m i r s k i

,

R.F. and Yura, H.T.9 A P P ~ . Opt.

10 (1971) 1652.

t i v e feedback system f o r s l o w l y v a r y i n g resonator [34] Hayes, C . I . , e t a l . , J. Opt. Soc. Am.

67

p r o p e r t i e s

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[35] Born, M. and Wolf, E., P r i n c i p l e s of Optics Acknowledgement. The a u t h o r wishes t o thank (Pergamon Press) 1959, 462.

[36] Hardy, J .W.

,

Proc. o f IEEE (1 978) 651

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Dr. A.T. M a t t i c k f o r t h e use of h i s notes on reson- 1371 Wang, C.P. Appl. Opt.

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a t o r theory and Professor A. Hertzberg f o r h e l p f u l

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Pearson, J . E . y et a1 _Opt.

15

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