VECTOR INTERPRETATION OF DEFORMABLE MIRROR C.O.A.T. SYSTEM

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VECTOR INTERPRETATION OF DEFORMABLE

MIRROR C.O.A.T. SYSTEM

G. Roger

To cite this version:

JOURNAL DE PHYSIQUE CoZZoque C9, suppldment au n o l l , Tome 41, novembre 1980, page ~ 9 - 3 9 9

f

VECTOR I N T E R P R E T A T I O N O F DEFORMABLE MIRROR

C ,

0

, A , T ,

SYSTEM

G. Roger.

Laboratoires de Marcoussis, Centre de Recherches de Za C.G.E., Route de IITozay, 91460 Marcoussis, France.

R6sum6.- L'exposd concerne une optique auto-adaptative munie de deux miroirs dgformables soumis cha- cun

B

N actuateurs. Le premier de ces miroirs a pour r61e de corriger les variations de chemin opti- que qui rdsultent soit des d6formations de la surface de l'onde produite par un laser gmetteur, soit des variations d'indice du milieu de transmission que traverse le faisceau pour atteindre un point cible. Le second a pour seul r81e d'introduire les perturbations p6riodiques de chemin optique des- tinses P mettre en 6vidence les erreurs de phase qui permettent de commander les actuateurs du pre- mier miroii-. On traite le rdgime d76quilibre de l'asservissement. Les erreurs rgsiduelles de phase sont donc faibles. On peut alors remplacer l'ensemble des surfaces qui interviennent dans les cal- culs par un systSme vectoriel 6quivalent et en ddduire de fason simple les principales caract6ris- tiques du systSme.

Abstract.- This study concerns a self-adaptive optics fitted with two deformable mirrors, each of them subjected to N actuators. The first mirror corrects the beam variations that result from either the deformations of the emitted wave-front or the variations of refractive index of the transmitting medium the beam passes through to reach the target. The second mirror's only role is to introduce the recurrent disturbances of the optical path that are necessary to produce thephase error signals which are to control the actuators of the first mirror. We deal with the stationary state of the servo-system, that is the residual phase errors are small. It is then possible to replace all the surfaces that are involved in the calculations by an equivalent vectorial system and to obtain very simply from it the main characteristics of this system.

I

-

INTRODUCTION (FIGURE

I )

The goal i s t o d e l i v e r f r o m a l a s e r system a maximal power on a r e v o t e t a r g e t , b u t t h e shape o f t h e l a s e r e m i t t e d w a v e - f r o n t may vary, and more- o v e r t h e o p t i c a l p a t h between t h e system a p e r t u r e and t h e t a r g e t i s s u b j e c t t o d e f o r m a t i o n s due t o a t ~ o s p h e r i c p e r t u r b a t i o n s . We must i n t r o d u c e i n t h e o p t i c a l system two deformable m i r r o r s and

M2.

The f i r s t one has t o compensate t h e o p t i c a l p a t h v a r i a t i o n s and i t s shape i s a d j u s t e d b y N a c t u a t o r s . The second one i s s u b m i t t e d through N o t h e r a c t u a t o r s t o v e r y s l ~ a l l p r e s c r i b e d a m p l i - tude v i b r a t i o n s a t v a r i o u s frequencies and p e r m i t s t o measure t h e r e s i d u a l phase e r r o r s . An o p t i c a l r e c e i v e r , f o l l o w e d by a l o c k - i n amp1 i f i e r , g i v e s t h e phase e r r o r s i g n a l s t o a c o n t r o l system which

*

Sponsored by DRET. feeds t h e MI a c t u a t o r s . I4!e c a l l E (m) t h e a m p l i t u d e o f t h e o p t i c a l f i e l d a t t h e p o i n t m o f t h e a p e r t u r e , t h e t o t a l area o f which i s S, and (m) t h e o p t i c a l l e n g t h o f t h e r a y g o i n g t h r o u g h m f r o m t h e l a s e r w a v e - f r o n t t o t h e t a r g e t .

I 1

-

RELATIVE POb!ER LOSS

l<e assume t h a t t h e phase v a r i a t i o n s do n o t m o d i f y t h e a m p l i t u d e d i s t r i b u t i o n on S and we t a k e as q u a l i t y c r i t e r i o n t h e r a t i o :

where would be o b t a i n e d i f

'P

would be a

JOURNAL DE PHYSIQUE c o n s t a n t f o r every m. We w r i t e :

Is

E ej'f'

cis

= R e j y and

n

=

Is

E

.

,j(Y-.y) ds then (2, r e a l )

We assume now t h a t t h e system i s c o r r e c t l y per- forming, so t h e d i f f e r e n c e between and y i s small everywhere on S. He w r i t e :

E - _{- er}

_,

_then - dS - - do

L J E ~ S

E

s

and we o b t a i n :

I t i s v e r y easy t o v e r i f y t h a t these surfaces s a t i s f y the vector spaces axioms, and we w r i t e

-+ J e , 0

.(c)

= \p

(c)

Ke d e f i n e t h e i n n e r ( o r d o t ) product o f two -+ + vectors and $* as : -+

q 2

= em q 2 do i s the squared norm o f $ and t h e

J o ' -f -+

RPL q i s t h e squared norm o f

( q

-

y ) :

+ -+ +

I f $ 2 = 1

,

11) i s o f norm 1. Every v e c t o r J, may be considered as the product o f a s c a l a r nurber w i t h

+

a vector Q ' o f norm one :

$

=

0.

_$'

-+ -+

_-+

If

q l

.

$* = _er )2 do = 0,

jl

and iy2 a r e "orthogonal" (which does n o t mean t h a t t h e corres- ponding wave-fronts are perpendicular). A s e t o f surfaces

qi

such t h a t :

c o n s t i t u t e s an orthonormal basis and a surface can be represented by :

q i s t h e r e l a t i v e power l o s s (R.P.L.). For example,

-+

-+ -+

i f q = .25, the l o s s i s 1 dB.

t

= z g i G i w i t h g i = @

.\Pi

I 1 1

-

SURFACE VECTOR SPACE

The l a s t r e l a t i o n shows t h a t we s h a l l have t o consider expressions l i k e

5

'P

and y r e s u l t from the a d d i t i o n o f several s i m i l a r f u n c t i o n s r e p r e s e n t i n g wave-front, o r m i r r o r defor- mations.

He

s h a l l c a l l weighted surfaces o r "surfaces" expressions l i k e

J

'

)

.

'f

(m)

.

y i s the weighted mean value o f $

.

I f y = 0, t h e subspace d e f i n e d by t h e ai as a basis. I f not, we surface $ i s centered. We s h a l l assume i n the w r i t e :

f o l l o w i n g t h a t every surface i s centered, thus :

-+ 3 3 -+ -+

(c8

-

Z.

a. C L . ) ~ = c 2

e 2

-

2 c C . ai 0 ai

1 1 1 1

I

V

-

_MINIMAL POlrfER LOSS

The phase

?

(m)

r e s u l t s from the a d d i t i o n o f seve- r a l terms :

-+

8 i s the shape o f a d i s t o r s i o n o f t h e wave-front created by t h e l a s e r , o r the atmosphere. c i s i t s amplitude. We admit t h a t t h e small d i s t o r t i o n s

-+

created by t h e a c t u a t o r s add l i n e a r l y . Xi ai ai corresponds t o t h e shape o f t h e c o r r e c t i n g m i r r o r

PIl

and r e s u l t s from t h e a d d i t i o n o f t h e deforma-

-+

t i o n s caused by the a c t u a t o r s . ai i s t h e shape o f t h e deformation created by t h e a c t u a t o r i, w i t h amp1 i t u d e ai

.

C . a .

;.

corresponds t o t h e e f f e c t

J J J

+ - + -+

8, ai, 6 . a r e centered and o f norm 1. ci, ai, bi

J

a r e time dependent.

The terms bi a r e sinuso5dally time dependent and small, so we n e g l e c t the term c o n t a i n i n g products bi b

..

The second term i s o n l y a small c o n t r i b u t i o n

J

t o the mean value o f q, b u t i t contains t h e phase

-+ 3 3

e r r o r c0

-

Ci ai ai, p r o j e c t e d on each o f the 6

j and

w i l l

be u s e f u l l a t e r . which i n t r o d u c e s t h e v e c t o r o f components : and t h e m a t r i x o f element T h i s m a t r i x convert s t h e c o n t r a v a r i a n t components -+ o f t h e v e c t o r space d e f i n e d by t h e ai i n t o cova- r i a n t coordinates. I t i s easy, by d i f f e r e n c i a t i o n , t o determine the s e t o f the values ai, such t h a t q i s minima7. I t i s t h e vector

a

given by :

The components ai o f t h i s v e c t o r

a

are t h e contra- v a r i a n t coordinates o f the orthogonal p r o j e c t i o n o f

3

the v e c t o r ce on the subspace d e f i n e d by the vectors ai ( f i g . 2 ) .

The minimal l o s s i s :

3 I t i s t h e squared d i s t a n c e between t h e p o i n t c8 and the subspace a, as shown on f i g u r e 2.

So, t o minimize q, i t i s necessary t o minimize

3 3

(c8

-

Ci ai ai) which i s equal t o zero o n l y i f

-+ 3

C9-402 JOURNAL DE PHYSIQUE

V

-

ERROR SIFNALS

-

SUBOPTI?qRL LOSS which may be obtained by a p p l i c a t i o n o f The frequency-1 ocked d e t e c t i o n o f the terms c o n t a i - Pythagorus Theorem.

n i n g b . enables us t o o b t a i n the e r r o r s i g n a l s :

J

V I

-

SERVO SYSTEMS

b!ith the e r r o r s i ~ n a l v e c t o r :

-

The servo l o o p system tends t o cancel these e r r o r s ,

5

= c F

. -

I D . . I a

-f -+ -+

OJ 1 J

which w i l l occur i f co = Ci ai ai ( 9 i n t h e

-+

subspace a), o r i f t h e orthogonal p r o j e c t i o n of t h e i t i s p o s s i b l e t o compute a new e r r o r v e c t o r :

3

phase e r r o r on each

6

i s zero, which means t h a t

3 3 -f - - 1

-

1 -

ce and Xi ai ai have the same p r o j e c t i o n on B.

El

= IDij[

.

E = c ID. IJ

.I-

F ~ j - a T h i s i s shown by f i g u r e 3, where the number o f

dimensions o f the vector space, which c o u l d be and t o apply i t t o t h e a c t u a t o r s o f Ell w i t h a g a i n i n f i n i t e , i s reduced t o two. The best choice f o r

G

( p ) which rakes :

ai corresponds t o A, the orthogonal p r o j e c t i o n o f

3

T on a,' b u t t h e servo system w i l l choose A ' , such Z = F ( p )

it

and gives :

3

t h a t vector

fin

i s orthogonal t o 6.

3 -+

So we see t h a t i t i s advisable t h a t a and 8

.. .

s y s t e m be about the same, t h a t i s , t h a t the

d i s t o r s i o n s impressed on r i r r o r s bll and

F?*

by the and we see t h a t B assumes t h e suboptimal value when a c t u a t o r s have s i m i l a r shapes. This c o n s t r a i n t G tends t o i n f i n i t y . To a t t a i n s t a b i l i t y r e q u i r e s

-+ -+

may be released i f 0 i s close t o the subspace a . t h a t the r e a l p a r t s o f the r o o t s o f 1

+

G ( p ) be To compute

c;,

we must i n t r o d u c e the m a t r i x negative. It i s p o s s i b l e t o dispense w i t h c i r c u i t r y

= Bj ai do and the v e c t o r

1

needed f o r implementing ID..

1 -

,

by a p p l y i n g

3 1

J

d i r e c t l y E t o

a

w i t h a g a i n 6 ( p ) :

( 'II = u n i t y m a t r i x ) , we see t h a t

a

takes a l s o t h e suboptimal value for i n f i n i t e F, b u t t o be s t a b l e , The b e s t values o f ai are given by : the system must stay away from values o f G such

1

t h a t

-

-E

i s an eigen value o f ID..

I,

which l i m i t s

1 J

V I I

-

CONCLUSION

T h i s vector space i n t e r p r e t a t i o n o f a COAT systen imp1 i e s several simp1 i f y i n a hypotheses and i s valuable o n l y f o r small phase t r a c k i n g e r r o r s . It must be considered as a f i r s t order theory and may be u s e f u l f o r a simple understanding o f the behaviour o f such a system and t o i t s primary design. Phase mo F i r r o r (I 0 wave-front

/

Target Laser wave-fron t Phase c o r r e c t i o n !?i r r o r

( 5 )

FIGURE 1

-

C.O.A.T. SCHEI!ATIC