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THE DEGREE OF PHASE COMPENSATION OF
LASER BEAMS USING GAS JETS
W. Christiansen, E. Wasserstrom, A. Hertzberg
To cite this version:
THE DEGREE
OF
PHASE COMPENSATION OF LASER BEAMS USING GAS JETS
W.H. Christiansen, E. ~asserstrom* and A. Hertzberg
Department o f Aeronautics and Astronautics, University of Washington, S e a t t l e , Washington 98195 U.S.A.
RCsum6.
--
Lors de l a propagation d'un faisceau l a s e r dans l'atmosphGre, des distorsions de phase
peuvent apparaitre, s o i t de fason passive
b
cause de l a turbulence, s o i t de fafon a c t i v e 3 causede l a dC-
focalisation thermique.
I1 a e t 6 prouv6 q u ' i l e s t possible de compenser partiellement ces Ccarts de
phase en modifiant physiquement 1 'optique de transmission. L ' u t i l i s a t i o n de j e t s de gaz avec des
gradients de densitd adgquats, convenablement interposEs dans l e faisceau, e s t pr6sent6e i c i comme
une a u t r e mdthode d'introduction de facteurs c o r r e c t i f s . Le degrE de compensation de phase susceptible
d 1 6 t r e a t t e i n t en principe a 6t6 6tudi6 pour des j e t s id6aux. On montre que 1 'd c a r t quadratique moyen
peut d t r e r6duit consid6rablement dans l e cas de l a dMocalisation thermique.
Abstract.
--
When a l a s e r beam propagates in the atmosphere, i t s phase can be distorted e i t h e r
passively, due t o turbulence, or actively, due t o thermal blooming.
I t has been demonstrated t h a t i t
i s possible t o p a r t i a l l y compensate f o r these phase e r r o r s by physically d i s t o r t i n g the transmitting
optics. The use of gas j e t s , w i t h controllable density variation, t h a t a r e suitably interposed i n the
beam
is investigated as another method of adding the correcting elements. The degree of phase compen-
sation, which may be achieved in principle, has been studied using ideal j e t s .
I t i s shown t h a t f o r
t h e thermal blooming case t h e mean square phase e r r o r can be s u b s t a n t i a l l y reduced.
1. Introduction.
--
Active optics r e f e r s t o
a detector or t a r g e t . Usually t h i s i s done by
optical components whose c h a r a c t e r i s t i c s a r e con-
physically d i s t o r t i n g mirrors in the transmitting
t r o l l e d during actual operation, in order
t o
modify
optics
[2]and thus varying the optical paths of
wavefronts.
While the concept of a c t i v e optics
the various rays of the beam.
Good improvements
has been known f o r some time, only recently i s t h e
have been achieved with low-power v i s i b l e l a s e r
technology emerging t h a t will influence the design
propagation in the atmosphere. Correction f o r
of high performance l a s e r systems
[I
,2].Phase
high-power infrared l a s e r s i s more d i f f i c u l t be-
variations produced by internal cavity d i s t u r -
cause of the need f o r larger amplitude mirror
bances, atmospheric turbulence, or thermal
deflections and heating of the a c t i v e mirror
blooming impose severe
1imitations on del ivering
surfaces.
a high irradlance l a s e r beam a t a distance. One
Anew method of phase compensation using
of t h e main applications of a c t i v e optics i s the
gaseous optics i s described here as an a l t e r n a t i v e
compensation of these wavefront d i s t o r t i o n s in
method f o r phase f r o n t control. Gaseous optics
order t o enhance the i n t e n s i t y of a l a s e r beam on
o f f e r s a potentially e f f i c i e n t technique f o r phase
a d i s t a n t t a r g e t . I t i s possible t o correct t o a
control of the beam because of small l i g h t losses,
substantial degree f o r these d i s t o r t i o n s by pas-
tolerance of extremely high power l e v e l s , and a
sing the beam i n question through an array of
c h a r a c t e r i s t i c rapid response.
I f many gas j e t s
elements which impose controlled phase variations
w i t hd i f f e r e n t optical properties and with s u f f i -
which compensate f o r the d i s t o r t i o n . This involves
c i e n t optical depth a r e placed in the path of the
the measurement and control of wavefronts often in
l a s e r , then the gas j e t s produce phase s h i f t s in
real time i n order to concentrate the energy on t o
the beam i t s e l f . The geometry required t o bring
+
Present address : Technion, Israel Institute ofTechnology, Haifa, Israel.
C9-432 JOURNAL DE PHYSIQUE
about l o c a l i z e d phase compensation i n t h e l a s e r beam i s p o s s i b l e by using independent j e t s o f gas. A c t i v e l y changing t h e gas index o f r e f r a c t i o n using f l o w
w i l l
p e r m i t t h e c o n t r o l necessary t o achieve phase compensation.In
o r d e r t o i l l u s t r a t e t h e concept, consider a u n i f o r m l y i l l u m i n a t e d l a s e r beam propagating i n t h ez d i r e c t i o n as shown i n F i g . 1. Fig. l a shows t h r e e views o f an a r r a y o f f i v e r e c t a n g u l a r phase- s h i f t i n g elements assembled t o form an a p e r t u r e through which a l a s e r beam i s passed. Each phase- s h i f t i n g element can be independently c o n t r o l 1 ed t o produce any d e s i r e d phase s h i f t (up t o a c e r t a i n maximum value). The p h a s e - s h i f t i n g elements a r e gas j e t s w i t h v a r i a b l e r e f r a c t i v e i n d i c e s . Fig. l ( b ) shows t h e a r r a y i n o p e r a t i o n w i t h t h e phase- s h i f t i n g elements producing l i n e a r l y decreasing phase delays from element 1 t o element 5. The approximate r e s u l t o f o p e r a t i n g t h e a r r a y i n t h i s manner would be t o t i l t t h e o u t p u t beam as shown. The f i g u r e i l l u s t r a t e s one p a r t i c u l a r type o f l a s e r beam c o n t r o l , chosen as an i n t r o d u c t o r y example because o f i t s s i m p l i c i t y . However, t h e same technique can be a p p l i e d using crossed j e t arrays t o a d j u s t the phase f r o n t s o f a l a s e r beam d i f - f e r e n t l y .
,AREA O r LASER BEAM
-
APPROXIMATE OUTPUT LASER BEAU- PHASE-SHIFTED UAYE FRONTS PHASE-SHIFTING ELEHENTI (GAS OF DIFFERENT INDICLS OF REFPACTION1
3 V I E U S 01 PHASE-SHIFTING ARRAY EACH RECTANGLE REPRESENTS A FREE JET
l n p m LUIER BEAH (YAVEFIIONTI NOT
ra S c a L r l
Fig. 1. Beam T i l t i n g Using Free J e t Phase- S h i f t i n g Elements.
Suppose now t h a t t h e phase d i s t r i b u t i o n over t h e l a s e r beam i s o r i g i n a l l y $(x,y). The r e d u c t i o n i n l i g h t i n t e n s i t y on t h e c e n t e r l i n e o f t h e undis- turbed beam i n the f a r f i e l d i s given by [31
where I.i s t h e i n t e n s i t y f o r a p e r f e c t (unaber- r a t e d ) beam and
?
i s t h e mean square e r r o r o f phase over t h e beam cross section. I n d e r i v i n g Eq. ( I ) , i t has been assumed t h a t t h e a b e r r a t i o n s a r e small, say7
j (,/5l2. I n order t o reduce t h i s phase d i s t o r t i o n (and thus increase t h e l i g h t i n t e n s i t y ) , gas j e t s are passed through t h e beam i n two orthogonal d i r e c t i o n s , as shown i n F i g . 2. I ft h e d e n s i t y v a r i a t i o n o f an i d e a l j e t ( d e f i n e d as
a
j e t i n which d i f f u s i o n and entrainment a r e absent) i n the y d i r e c t i o n i s denoted by p(x,z), then t h e r e l a t i v e phase change o f t h e l a s e r beam t r a v e r s i n g t h e j e t i s given by [4]
0
where A i s t h e wavelength o f the l a s e r l i g h t , t i s t h e thickness of the j e t , pr i s some reference den- s i t y and B i s t h e Gladstone-Dale constant. Simi- l a r l y , t h e phase change due t o t h e j e t i n t h e x
LASER- &X BEAM
changes due t o t h e two j e t s a r e a d d i t i v e , t h e phase o f t h e l a s e r beam a f t e r passing t h e two crossed j e t s i s
$(X,Y) = $(X,Y)
-
f ( x )-
g ( y ) . ( 3 )The main problem addressed i n t h i s paper i s t h e determination o f t h e f u n c t i o n s f ( x ) and g ( y ) such t h a t
7,
t h e mean square o f t h e new phase func- t i o n , i s minimized. For purposes o f simp1 i c i t y ,the j e t s a r e assumed t o be i n v i s c i d f o r these c a l c u l a - t i o n s . Both d i s c r e t e and continuous j e t cases have been studied. I t i s recognized t h a t f e a l j e t s w i l l have an e f f e c t on t h e performance o f t h e f l u i d a r l a p - t i v e o p t i c s scheme since j e t s can cause o p t i c a l ab- e r r a t i o n s o f t h e i r own. Thus, what i s analyzed here i s the best phase compensation which may be achieved using i d e a l j e t s .2. A Simple Minimization.
- -
I n t h i s s e c t i o n t h e cross s e c t i o n o f t h e l a s e r beam i s assumed t o be a r e c t a n g l e D. The phase c o r r e c t i o n i s per- formed by M constant d e n s i t y j e t s i n t h e x d i r e c - t i o n and N constant d e n s i t y j e t s i n t h e y d i r e c t i o n (see Fig. 3 ) . The i n t e r s e c t i o n o f t h e boundaries o fminimizes J, where
2 J =
1
A . . ( $ . . - f . - 9 . ),
1J 1 J 1 J ( 5 )
i ,j
The Aij a r e t h e areas of t h e r e c t a n g l e s .
To minimize J we must have aJ/afi = aJ/ag. = 0,
J
where a l l Aij were assumed t o be equal. Thus, t h e a n a l y s i s i s r e s t r i c t e d t o equal w i d t h j e t s . J e t s o f d i f f e r e n t widths a r e a p o s s i b i l i t y , b u t have n o t been considered i n t h i s o r insubsequent c a l c u l a t i o n s .
The M+N equations (6,7) f o r t h e unknowns fi and g. have a non-unique s o l u t i o n , s i n c e i f fi and g. i s
J J
a s o l u t i o n so i s fi + C, g
.
-
C, where C i s an a r b i -J
t r a r y constant. I n o r d e r t o make t h e s o l u t i o n u n i - que we r e q u i r e t h e m i n i m i z a t i o n o f t h e f u n c t i o n
s u b j e c t t o t h e c o n s t r a i n t s given by eqs. ( 6 ) and ( 7 ) . This problem i s solved by the method o f La- grange mu1 t i p 1 ie r s by d e f i n i n g
these j e t s as viewed by t h e l a s e r subdivide D i n t o
s
2= Efi t zgj2 + C x i ( ~ f i + ,7gj - I $ i j ) t
MxN s m a l l e r r e c t a n g l e s i n each o f which t h e phase 1 3 1
J
J$ . . i s assumed constant. The problem i s t o d e t e r -
lj
pj(Mgj +Ifi
-
1 1
( 9 )
1
J
mine f i and g j so t h a t where
xi
and p . a r e t h e Lagrange mu1 t i p l i e r s . HenceJ
GAS JETS
C9-434 JOURNAL DE PHYSIQUE
3. A Continuous Case.-- Here t h e cross sec- t i o n o f the l a s e r beam i s assumed t o be a convex domain D i n which i t s phase e r r o r $(x,y) i s c o n t i n - uously given. This phase e r r o r i s t o be decreased by two crossed j e t s o f v a r i a b l e index o f r e f r a c - t i o n i n t h e x and y d i r e c t i o n s , as described i n th e i n t r o d u c t i o n , so t h a t
2
J =
jj
[$(x,y)-
f ( x )-
g ( y ) ] dxdy = min. (14)D
Equation (14) i s solved by standard v a r i a t i o n a l methods [5]. One o b t a i n s two i n t e g r a l equations which a r e analogous t o eqs. (6) and ( 7 ) f o r t h e f i - n i t e d i f f e r e n c e s scheme [6].
Again, t h e s o l u t i o n o f these two i n t e g r a l equ- a t i o n s i s n o t unique, and t o make i t so we r e q u i r e t h e m i n i m i z a t i o n o f t h e f u n c t i o n a l
As i n t h e previous section, t h e problem i s solved by u s i n g Lagrange mu1 t i p l i e r s h ( x ) and ~ ( y ) . A f t e r some manipulations one can show t h a t
b d
1
f ( x ) d x =I
g(y)dyi
.
(16)a c
where a, b, c, d a r e t h e extreme l i m i t s o f t h e con- vex domain. A d d i t i o n a l d e t a i l s r e q u i r e t h e speci- f i c a t i o n o f t h e geometry o f t h e domain.
For t h e s p e c i a l case where t h e domain i s a rec- tangle, t h e equation can be solved e x p l i c i t l y t o
y i e l d d 1 f ( x ) = ;FT (
1
$ ( x , Y ) ~ Y-
i
.
b ~ ( Y I = ( j $ ( x y y ) d x-
i ) , (18) a where (19) cawhich i s s i m i l a r t o t h e s o l u t i o n obtained previous- l y i n eqs. ( l l ) , (12), and (13).
Unfortunately, i t i s n o t p o s s i b l e t o w r i t e an e x p l i c i t s o l u t i o n f o r t h e general case, where t h e
domain i s n o t a rectangle. One way t o proceed i s t o s o l v e t h e equation i t e r a t i v e l y [6]. This method has been used s u c c e s s f u l l y f o r t h e case o f t h e c i r c u l a r a p e r t u r e and t h e r e s u l t s w i l l be described i n t h e n e x t s e c t i o n .
4. A Numerical Example
Recently, Bushnell and Skogh [7] have computed t h e phase compensation which can be achieved f o r a thermal l y bloomed l a s e r beam by m i r r o r deformation. A s i m i l a r problem has been solved, except t h a t now t h e compensation i s achieved by means o f d i s c r e t e gas j e t s , as described above. Here we w i l l g i v e r e - s u l t s which a r e a p p l i c a b l e t o a square a r r a y o f f r e e j e t s .
The phase d i s t o r t i o n o f a t h e r m a l l y bloomed beam i s given i n p o l a r coordinates [8] by
3
$(r,0) =
1
f n ( r ) c o s n(O - 8,)n=O (20)
where r = E 1, 0 = tan-' [ y l and Qo i s t h e
d i r e c t i o n o f t h e wind. The f u n c t i o n s f n ( r ) a r e given by
where C1 i s a non-dimensional c o e f f i c i e n t equal t o 3/4 i n t h e c a l c u l a t i o n s . This c o e f f i c i e n t i s equal t o t h e constant C (eq. (11) o f [7]) d i v i d e d by t h e wave number f o r C02 l a s e r r a d i a t i o n . The z ( i ) a r e Zernike polynomials and a r e d e f i n e d i n [3], [7] o r [8]. F i n a l l y , t h e c o e f f i c i e n t s A.. i n eq. (21)
1 J
t h a t were used i n t h e c a l c u l a t i o n a r e given i n [7] as AZ0 = -0.053 A4, = -0.050 AsO = t0.021
All = t0.440 Agl z t 0 . 0 7 2 A5] = -0.061 (22) The f i r s t exercise considered was t o reduce t h e phase e r r o r $(r,Q) o f a c i r c u l a r l a s e r beam by
I
tfl
I f 2 I f 3If,
GAS JETS
Fig.
4
A Circular Laser Beam Covered by Discrete J e t swidth j e t s i n t h e y direction in an optimal manner ( s e e Fig. 4 ) . The solution obtained in Sec. 2 can- not be applied i n a straightforward manner since t h e domain
D
i s now c i r c u l a r r a t h e r than rectangular, and, secondly, the phase i s not constant over the sub-rectangles generated by the intersection o f t h e j e t s . The problem was solved by rewriting the func- tionalJ
i n eq. (14) a swhere $(x,y) i s the cartesian equivalent of eq. (20). I t should be noted t h a t
3
used i n eq. (1) i s equal t o (3/0j21
$2 dxdy divided by the area. Hence,= (9/16)JN/*, where the area of a c i r c u l a r beam of normalized radius i s s . The minimization of J N ( f i g j ) was carried out by using t h e general pur- pose minimization program MIN written by Professor Carl E. Pearson of the University of Washington. The r e s u l t s of these calculations for
Qo
= 0 and N = 1 ,--
J 7 . 0 0131able
1 . Exact Numerical Results f o rNxN
J e t s with Q o = O . N varies from 1 t o 7 .2,.. . , 7 a r e shown in Table 1 and f o r 8, = ~ / 4 and N = 1,2,.
. .
,5 i n Table 2. As was noted i n Sections 2 and 3 , the solution of t h i s problem i s not uni- que, and the data presented in Tables 1 and 2 i s obtained by s h i f t i n g the computed data so t h a t eq.(13) i s s a t i s f i e d . The main conclusion t o be drawn from these calculations i s about the r a t e of decrease of J N with
N ,
where J N i s a measure o f t h e loss in the maximum i n t e n s i t y , which can be achie- ved by focusing the l a s e r beam. I t i s seen t h a t most of the compensation i s achieved by using a small matrix, say of 2x2 o r 3x3 j e t s . The varia- tion of JN/J1 as function ofN
i s shown in Fig.5.
The asymptotic value of JN/J1+0.040 was computed by substituting the solutions f ( x ) and g(y) of the continuous case in eq. (14).The phase d i s t o r t i o n has been expressed in terms of Zernike c i r c l e polynomials (eq. ( 2 0 ) ) , each
C9-436 JOURNAL DE PHYSIQUE
term o f which corresponds t o i n d i v i d u a l abberations such as t i l t , focus, coma, and so f o r t h [7,81. I n order t o compare t h e r e s u l t s o f those o f [7], t h e f u n c t i o n a l J has been recomputed f o r t h e case o f no t i l t a b e r r a t i o n (Al1 s e t equal t o zero i n eq.(21)). A c t u a l l y , a p r a c t i c a l adaptive o p t i c s system may be one i n which t i 1 t (and focus too) can be c o r r e c t e d by t h e t r a n s m i t t e r i t s e l f . I n p r a c t i c e , t h i s may be done by simply p o i n t i n g t h e l a s e r i n a s l i g h t l y d i f f e r e n t d i r e c t i o n . Since t h e polynomials a r e o r - thonormal over t h e range o f i n t e r e s t , 0 _< r _< 1, 0 <_
Q 5 2 ~ , t h i s does n o t a f f e c t t h e o t h e r h i g h e r order a b e r r a t i o n s and i s e q u i v a l e n t t o using a plane m i r - r o r t o c o r r e c t f o r t i l t . These c a l c u l a t i o n s were c a r r i e d o u t and t h e r e s u l t s f o r JN a r e given i n Table 3 f o r Elo =O. The value o f the rms phase
Table 3. JN versus NxN Jets w i t h T i l t A b e r r a t i o n Set Equal t o Zero f o r Elo = 0.
e r r o r
m=
3/4v,
versus N i s shown i n F i g . 6 f o r Qo = 0, I t i s seen t h a t t h e r e l a t i v e improve- ments, w h i l e s u b s t a n t i a l , a r e n o t as good as those which i n c l u d e tilt ( F i g . 5). I n f a c t , f o r a small m a t r i x o f 2x2 j e t s , t h e mean square o f t h e e r r o r i s n o t decreased a t a l l . This i s because o n l y a t i l tSINGLE ARRAY DOURLE ARRAY o LIMITING VALUE 0 035
e,
= 0 A,, = 0 0 : 2 3 4 5 6 7 8 9 ' ! 3 I I I I I I I I NFig. 6 RMS Phase Compensation E r r o r Versus t h e Number o f Crossed J e t s f o r a S i n g l e & Double Array. T i l t A b e r r a t i o n Set Equal t o Zero. Q o = 0.
c o r r e c t i o n i s p o s s i b l e w i t h a small number o f j e t s . Figure 6 may be used t o compare w i t h F i g . 9 i n [7], where a s i m i l a r e f f e c t o f reducing the phase e r r o r i s achieved by deforming t h e m i r r o r . By m u l t i p l y - i n g t h e rms phase e r r o r by X/2a = 10.6p/2a, the o r - d i n a t e s o f t h e two f i g u r e s a r e made i d e n t i c a l .While a comparison o f the r e l a t i v e improvement between [7) and t h i s work i s d i f f i c u l t due t o the d i f f e r e n t methods o f approach, i t should be remembered t h a t t h e number o f a c t u a t o r s w i t h the f r e e j e t system i s 2N. The rms phase e r r o r i s reduced by 28% f o r N =7,
N
Fig. 7 R e l a t i v e Improvement i n RMS E r r o r f o r Aberrations a t 8, = 0.
While t h i s i s an approximated r e s u l t s u b j e c t t o a number o f p o s s i b l e e r r o r s such as t h e curve f i t t i n g procedure and non-optimized r e s u l t s , the c a l c u l a- t i o n shows t w i c e t h e improvement o f a s i n g l e a r r a y .
5. Improvement o f I n d i v i d u a l Abberation E r r o r s
I n t h i s s e c t i o n i n d i v i d u a l a b e r r a t i o n s a r e ex- amined t o see how each o f them i s minimized by a s i n g l e crossed j e t a r r a y . A s e r i e s o f c a l c u l a t i o n s was c a r r i e d o u t i n which t h e phase e r r o r was taken t o be a s i n g l e c l a s s i c a l a b e r r a t i o n . That i s , o n l y one Ai c o e f f i c i e n t i n eq. (20) was taken t o be non- zero and then t h e o p t i m i z a t i o n scheme was a p p l i e d f o r d i f f e r e n t numbers o f j e t s . The r e l a t i v e im- provement i n was noted and i s p l o t t e d i n Fig. 7 as a f u n c t i o n o f t h e a b e r r a t i o n and N f o r 8, =
0.
The numbers f o r each l i n e r e f e r t o thesub- s c r i p t on t h e A..
f a c t o r s . The lowest o r d e r aber-1 J
r a t i o n s are 11 ( t i l t ) , 20 ( f o c u s ) , 22 (astigmatism), 40 ( s p h e r i c a l ) , and 31 (coma), f o r example. Except f o r some minor i r r e g u l a r i t i e s a t low values o f N, a l l a b e r r a t i o n s showed improvement w i t h t h e c r o s s e d j e t p r i n c i p l e . The most dramatic improvement was i n tilt, as expected, although a l l t h e lowest o r - der a b e r r a t i o n s a r e improved g r e a t l y . There i s a systematic progression i n l o s s o f a b i l i t y t o remove modal e r r o r s as t h e o r d e r o f t h e r a d i a l coordinate
F i g . 8 Basis o f T i l t Correction Using I d e a l J e t s f o r 8, = 0.
( r ) increases. That i s , simple f u n c t i o n s can be minimized e a s i l y w i t h small values o f N whereas complex f u n c t i o n s having many maxima and minima across t h e beam r e q u i r e l a r g e r values o f N t o re- duce t h e phase e r r o r . Somewhat s i m i l a r r e s u l t s e x i s t
far
t h e case o f 9, = 4 5 O .The case o f t i l t about t h e y ( o r x ) a x i s f o r a square l a s e r beam i s easy t o examine i n d e t a i l . The phase e r r o r i s given as
4, = CA 11 rcose = CAl lx
.
(24) The phase c o r r e c t i o n s a r e discontinuous changes i n @a, each of which i s 2CA11/N d i f f e r e n t from i t s neighbor and s e t a t the l o c a l mean value o f $c. A sketch o f t h e idea i s given i n Fig. 8. Equation (23) becomes f o r t h i s case1 1/2N
2
JN = 2N
1 1
(2CAllx) dxdy.
(25)0 0
C9-438 JOURNAL DE PHYSIQUE
