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OPTICAL QUALITY OF TILTED SPHERICAL
MIRROR UNSTABLE RESONATORS
C. Cason, R. Jones, J. Perkins
To cite this version:
JOURNAL
DE
PHYSIQUE CoZloque C9, suppl6ment au n O 1 l , Tome 41, novembre 1980, page C9-385OPTICAL QUALITY OF TILTED SPHERICAL MIRROR UNSTABLE RESONATORS
C.C. Cason, R.W. Jones and J.F. perkins*.
,Army Directed Energy Directorate,
U.
S . Army Missile Cononand, Redstone Arsenal, Alabama 35809,U.
S . A..
U n i v e r s i t y of AZabama i n HuntsviZZe, HuntsviZle, Alabama 35804, U.S.A..
REsumG.- Les lasers chimiques de grande puissance ont souvent des rzgions de gain de grand allonge- ment, pour lesquelles un rbsonateur instable avec diffgrentes amplifications dans les deux dimensions transversales est dssirable. Ce papier considsre une nouvelle classe de rbsonateurs instables qui rbalisent la propriEtb d'amplifications diffgrentielles avec des miroirs sph6riques en inclinant con- venablement les miroirs au moyen de grands angles. La qualit6 du rendement optique a bt6 recherchbe thsoriquement et exp6rimentalement. L'on a dbcouvert que des rssonateurs instables, produisant des faisceaux lasers presque uniphases, peuvent Stre fabriqubs en utilisant cette nouvelle approche. Une propristb spbciale permet aux amplifications transversales d'6tre continuellement modifibes, ce qui est une aide importante pour la dynamique des gaz de grande puissance ou pour les activitss de dbve- loppement expbrimental des lasers chimiques quand la longueur de la trajectoire de l'accroissement du flux est modifise.
Abstract.- High power chemical lasers often have high aspect ratio gain regions, for which an unsta- ble resonator with different magnifications in the two transverse dimensions is desirable. This pa- per considers a new class of unstable resonators which achieves the differential magnification pro- perty with spherical mirrors by suitably tilting the mirrors through large angles. Output optical quality has been investigated theoretically and experimentally. It was found that unstable resonators
producing near uniphase laser beams can be fabricated using this new approach.
A
special property al-lows the transverse magnifications to be continuously varied, a significant aid for high power gas dynamic or chemical laser experimental development activities when the flow gain path length is va- ried.
INTRODUCTION
In
various applications, including in particularhigh power gas dynamic and chemical lasers with rel- atively short gain lengths, compared to the gain height there is a need for unstable laser resonators which produce a collimated output beam but have asymmetric
magnification, i.e. Mx f M
.
Such resonators haveY
been achieved by others by use of mirrors with toroidal figuring, but there are substantial difficulties in- volved in fabrication and alignment of such mirrors.
Special cases of the toroidal resonator include the
cylindrical mirror system. Wave optics analyses of these special cases show that mode control difficul- ties exist. Fabrication of toroidal mirrors does not
permit the control of the surface radii, Rx and R of
Y
each mirror to much better than about 2%. This does not always allow optimum mirror separations to provide a uniphase beam in both transverse dimensions simulta- neously. Additionally some transverse mode control is sacrificed in the toroidal resonator concept. High energy laser research activities will require several sets of these complicated toroidal resonator designs, as the gain in the flow direction is varied. A need has evolved to develop a resonator that has the simul- taneous properties of producing a nearly uniphase beam, of allowing a continuously variable magnification in the two orthogonal directions and of accommodating a continuously variable aspect ratio gain medium. We have previously [11 proposed a new class of resonators which achieves all these properties without the dis-
JOURNAL DE PHYSIQUE
a d v a n t a g e s of n o n s p h e r i c a l m i r r o r f i g u r e r e q u i r e m e n t s . The d e s i g n s i n v o l v e t i l t i n g b o t h r e s o n a t o r s p h e r i c a l m i r r o r s by r a t h e r l a r g e a n g l e s . The p r e s e n t d e s i g n
c o n c e p t i s a p p l i c a b l e t o e i t h e r standing-wave o r r i n g Ct
bI TlLT ANGLES - OPPOSITE SENSE
r e s o n a t o r s . [ 2 , 3 ]
[
''1 The purpose of t h i s p a p e r i s t o r e p o r t on a
r e c e n t i n v e s t i g a t i o n of t h i s new r e s o n a t o r t o d e t e r - mine i t s u s e f u l n e s s . The approach i s based on an a n a l y s i s of t h e l a r g e d e g r e e of a s t i g m a t i s m i n t r o d u c e d by t h e t i l t of t h e convex m i r r o r and p r e d i c t i n g i t s c a n c e l l a t i o n by s u i t a b l e c h o i c e of m i r r o r s p a c i n g and t i l t of t h e concave m i r r o r . I n t h e p r e v i o u s work we have g i v e n t h e d e s i g n e q u a t i o n s and have confirmed t h e symmetric m a g n i f i c a t i o n p r o p e r t y . It was concluded t h a t t h e t i l t e d - s p h e r i c a l - m i r r o r r e s o n a t o r scheme o f f e r s c o n s i d e r a b l e p r a c t i c a l a d v a n t a g e s o v e r o t h e r schemes f o r a c h i e v i n g asymmetric m a g n i f i c a t i o n , b u t t h e c r i t i c a l problem of r e s i d u a l a b e r r a t i o n s i n t r o - d u c i n g d e t e r i o r a t i o n of o u t p u t o p t i c a l q u a l i t y needed f u r t h e r i n v e s t i g a t i o n . The p r e s e n t r e p o r t d e s c r i b e s a combined t h e o r e t i c a l - c o m p u t a t i o n a l s t u d y of beam- q u a l i t y p r o p e r t i e s of t i l t e d - s p h e r i c a l - m i r r o r r e s o - n a t o r c o n c e p t s t h a t was compared t o r e s u l t s of l a b o r a t o r y experiments. THEORETICAL DEVELOPMENT T h i s s t u d y c o n s i d e r s t h e e f f e c t of one complete p a s s a g e from convex t o concave m i r r o r on a n i n i t i a l l y c o l l i m a t e d beam, which i s e s s e n t i a l l y a r i n g - r e s o n a t o r c o n f i g u r a t i o n . Both c a s e s of r e l a t i v e s i g n of s p h e r - i c a l - m i r r o r t i l t a n g l e s a r e c o n s i d e r e d ; t h e s e a r e r e f e r r e d t o a s U and Z c o n f i g u r a t i o n s , a s i l l u s t r a t e d i n F i g u r e 1. It i s assumed t h r o u g h o u t t h a t c o o r d i n a t e s y s t e m s a r e chosen s u c h t h a t t h e x - a x i s i s p a r a l l e l t o t h e a x i s about which m i r r o r s a r e t i l t e d . The z-axis
i s a l o n g t h e nominal p r o p a g a t i o n d i r e c t i o n i n any r e g i o n .
(bl TlLT ANGLES -SAME SENSE
J
F i g u r e 1. Schematic drawing of t i l t e d - m i r r o r a r r a n g e - ments of U-type ( t o p drawing) and Z-type
(bottom drawing).
A two-dimensional approximate t r e a t m e n t was f i r s t pursued t h e o r e t i c a l l y . That i s , a t t e n t i o n i s c o n f i n e d t o a p l a n e p e r p e n d i c u l a r t o t h e m i r r o r - t i l t a x i s . From t h e closed-form a n a l y s i s one can o b t a i n p r e d i c - t i o n s of t h e magnitude of one of t h e major c o n t r i b u t o r s t o r e s i d u a l a b e r r a t i o n s , namely t h e y 3 term. For b r e v i t y , t h e d e r i v a t i o n i s o m i t t e d and o n l y t h e f i n a l r e s u l t i s q u o t e d . We o b t a i n e d
U s u a l l y v a l u e s of R R2, and a r e g i v e n . The v a l u e of O 2 c a n t h e n b e d e t e r m i n e d by t h e b a s i c d e s i g n equa- t i o n s . [11 I n t h e above e q u a t i o n f o r t h e y3 a b e r r a t i o n t h e upper s i g n on t h e second term i n s q u a r e b r a c k e t s c o r r e s p o n d s t o a Z c o n f i g u r a t i o n , w h i l e t h e lower s i g n c o r r e s p o n d s t o a U c o n f i g u r a t i o n . Thus t h e Z configu- r a t i o n i s p r e f e r r e d o v e r t h e U c o n f i g u r a t i o n a s r e g a r d s a b e r r a t i o n s .
beam. The r m s v a r i a t i o n s a r e c o n s i d e r a b l y s m a l l e r t h a n peak-to-peak o r center-to-peak v a r i a t i o n s , and t h e c h o i c e of r e f e r e n c e p l a n e i s i m p o r t a n t . The rms v a l u e f o r an u n t i l t e d r e f e r e n c e p l a n e is:
The r m s v a l u e of t h e OPD a f t e r choosing a n o p t i m a l t i l t f o r t h e r e f e r e n c e p l a n e can t h e n b e shown t o be:
Some n u m e r i c a l examples a r e g i v e n i n T a b l e I which i s r e p r e s e n t a t i v e of c h e m i c a l - l a s e r r e s o n a t o r d e s i g n s .
T a b l e I. Two-Dimensional Approximate OPD P r e d i c t i o n s ( i n Microns) f o r a Z C o n f i g u r a t i o n T r a v e l - ling-Wave R e s o n a t o r w i t h R =-I481 cm, R2=
1
3703 cm, Yay =7.5 cm, f o r Various Values of c o n v e x - ~ i ? % r T i l t Angle 8, From t h e s e e s t i m a t e s , t h e o p t i c a l q u a l i t y s h o u l d be q u i t e a c c e p t a b l e a t i n f r a r e d w a v e l e n g t h s , when r m s and r e f e r e n c e - p l a n e - t i l t e f f e c t s a r e accounted f o r , e x c e p t f o r t h e r a t h e r extreme c a s e of t h e m i r r o r s b e i n g used f o r t h e i n t e r f e r o m e t r i c t e s t i n g .
Such two-dimensional e s t i m a t e s would be consid- e r e d q u i t e u n c e r t a i n i f s t a n d i n g a l o n e . R e s u l t s of t h r e e - d i m e n s i o n a l r a y - t r a c i n g c a l c u l a t i o n s i n d i c a t e t h a t t h e two d i m e n s i o n a l e s t i m a t e s g i v e u s e f u l semi- q u a n t i t a t i v e guidance t o t h e e x p e c t e d d e g r e e of o p t i - c a l q u a l i t y d e g r a d a t i o n f o r a n i d e a l l y a l i g n e d system. On t h e o t h e r hand, t h e two-dimensional c o n s i d e r a t i o n s g i v e no i n d i c a t i o n whatever of t h e s e n s i t i v i t y of o p t i c a l q u a l i t y t o v a r i o u s s y s t e m s p a r a m e t e r s , i n c l u d - i n g i n c r e m e n t s t o m i r r o r s e p a r a t i o n L , and concave- m i r r o r - t i l t - a n g l e
e 2
f o r a s p e c i f i e d v a l u e of 0 1' Also t h e two-dimensional r e s u l t s g i v e n o i n f o r m a t i o n about expected s h a p e s of i n t e r f e r o g r a m s . The major p o r t i o n of t h e t h e o r e t i c a l - c o m p u t a t i o n - a 1 p a r t o f t h i s s t u d y h a s been concerned w i t h develop- i n g and a p p l y i n g t h r e e - d i m e n s i o n a l r a y - t r a c i n g compu- t a t i o n a l methods and w i t h c o r r e l a t i n g t h e r e s u l t s( p r i m a r i l y i n t h e form of q u a s i - i n t e r f e r o g r a m s ) w i t h t h e e x p e r i m e n t a l i n t e r f e r o g r a m s . The p a i r of m i r r o r s used f o r comparisons between e x p e r i m e n t a l and compu- t a t i o n a l r e s u l t s had r a d i i o f c u r v a t u r e o f R = - 2 9 0 cm,
1
R = 675 cm. These were d e l i b e r a t e l y chosen t o g i v e 2
r a t h e r l a r g e a b e r r a t i o n s i n o r d e r t o f a c i l i t a t e t h e comparisons.
E q u a t i o n s employed i n r a y t r a c i n g assume a c o l l i m a t e d beam i n c i d e n t on t h e convex m i r r o r . Given x and y c o o r d i n a t e s of a n i n c i d e n t r a y measured r e l a - t i v e t o t h e c e n t r a l r a y , t h e computer program c a l c u - l a t e s t h e p a t h of t h e r a y t h r o u g h t h e system and d e t e r m i n e s ( a ) d i r e c t i o n c o s i n e s a f t e r r e f l e c t i o n from t h e concave m i r r o r , measured i n a c o o r d i n a t e system w i t h t h e p o s i t i v e z - a x i s p a r a l l e l t o t h e c e n t r a l r a y and (b) t h e OPD between t h e r a y c o n s i d e r e d and t h e c e n t r a l r a y . The OPDs a r e of primary i n t e r e s t , and a r e c a l c u l a t e d f o r a p o s i t i o n c l o s e t o t h e concave m i r r o r . There i s a good d e a l of o r d e r i n t h e c a l c u - l a t e d OPDs; t h e i r f u n c t i o n a l dependence on x and y can be w e l l r e p r e s e n t e d by s e v e r a l t e r m s of a T a y l o r ' s
2
C9-388 JOURNAL DE PHYSIQUE
inteferograms do not immediately (without an inter- vening measurement and data-reduction process) lead to quantities predicted by the calculations. Therefore, it seemed useful to convert the calculated results into the form of computer-simulated quasi-interfero- grams, thereby facilitating visual comparison of experimental and computational results. One can obtain quantitative measures of optical quality from the computational results after confirming them by comparison with experimental interferograms. For this purpose the computer generates a fairly large array of OPDs obtained by evaluation of the truncated series expansion. The array of numerical values is converted to an array of alphabetic characters and blanks and used to generate a one-page printer plot whose general appearance simulates that of an interferogram.
Three-dimensional calculations were carrried out for the pair of mirrors used in the experimental stud- ies, using several values of the convex-mirror tilt
angle el. The three-dimensional predictions of the
optimized rms OPD range from some 25% to
50%
largerthan the corresponding two-dimensional predictions for these cases.
EXPERIMENTAL AND THEORETICAL RESULTS
The results of experiments agree satisfactorily with the computational results obtained in the three-dimensional calculations of wavefront OPDs. The set of mirrors was chosen to give a moderate number of fringes across the output aperture which could be compared directly to calculated quasi-inter- ferograms. This test represented an extreme case compared to the aberrations which would be present in the proposed high aspect ratio resonator mirrors but provided a direct correlation between theory and experiment.
Two types of resonator concepts were tested, the
Z
configuration and the U configuration. Most of theinterferograms were taken with the Z configuration
experimental setup. Schematics of the experimental
setup for the Z and U configurations are shown in
Figure 2.
A
plane wavefront is produced by passing ahelium-neon laser beam through a spatial filter (micro- scope objective plus pinhole) and collimating the resultant spherically divergent wavefront with a high quality objective lens of 6-in. diameter. This plane wavefront is then split into two components with a glass beamsplitter. One component (reference beam) is reflected directly into a focusing leris to a camera. The other portion (test beam) passes through the beam- splitter to the convex resonator mirror, reflects to the concave resonator mirror, and is directed back through the beamsplitter into the camera by means of appropriately positioned mirror flats.
FOCUSING LENS COLLIMATING LENS H..N. U S E R 'MIRROR CONCAVE RESONATOR MIRROR I*) Z CONFIGURATION
COLLIMATING FOCUSING LENS
T MIRROR
MIRROR
FLAT MIRROR C O N ~ A V E RESONATOR
MIRROR
(bl U CONFIGURATION
Figure 3 . Z Configuration, 0 = 45O,
RBT =
-0.074
1The interferograms were obtained by superimposing the focal points of the two beams from the beamsplit-
ter (by tilting the turning flat). There is a certain
arbitrariness in this adjustment - the interferogram obtained depends strongly upon the angle between the reference beam and the test beam (wavefront tilt). This variability made it difficult to compare exactly the quasi-interferograms obtained analytically with the experimental interferograms.
Some examples are given in Figures 3
-
8 of thecomparison of experimental interferograms and the
7s
-* - , a +
It..
.
Figure 4. Z Configuration,
O1
= 45O,RBT = -0.296
(approximately as judged by visual inspection) corres- ponding computer-generated quasi-interferograms. For each example, the experimental interferogram is shown in the top figure, while the computed quasi-interfer- ogram is shown at the bottom. The agreement between the two is acceptable. The relative beam tilt, RBT,
in units of radians, is the angle between the
C9-390 JOURNAL DE PHYSIQUE
Figure 5.
Z
Configuration, el, = 45V,RBT = -0.592
available only from the ray-tracing results. One can rather directly observe the change in interferogram shape with beam tilt in the experimental arrangement, however, and the change is of the same sort as found computationally. With increasing reference-beam tilt the pattern changes from a single oval to a double
oval, as can be seen in Figures 3-5, for which 8 =
1 45O.
Figure 6 shows the effect of a 7-milliradian
misadjustment of the tilt angle
e2
of the concave mir-ror, for
el
= 45'.Figure 7 shows a double-oval pattern for 6 = 70°,
1 x., E & = -
-
- - - . -- - -.
-
.,
..,-..
.
= s & . - . -"-.
, > A . , 4 a * . ' . . - & . . d , 4 - a - - a 2 d . ' . a - d .,a* .s z . . v. ez a-
".d ad-.'.. ,,,,,,:"
: c": 2. . a * + > * z g f : ! $ g g : g z g 8 =.
, a * I 2: I , r P - P 2: - I a * * a * = , :g: X.. : : '" e w l X I * l l *.*. *; g' iP z ~ ~ $ 5 $ z : ~ t ~ ~ - fgt:::""" :> , > a l l , * i* s z a - lllrL. ""Q: * * , 4 . . * r *..*.
F ?.!
..,
.! , 2 2 : ? : : :: I * . "- # a, e e . * . *-_
<...
., * w e.."
, > ' . , * t e , A .-
s ...> r < s-
.,, ' * . .*, ~.~
%.
. , > ' _ > n * A , * s ,<. a *,..,
...
," ~* ' _. I . j l. I .*" _I z .*- ~.
,- , * . * ~ . - * . " a . I . . I D r I # b.
e * x r i ~# . a * -, P I I I E ". ,-
<* e '...
'.
.,,.
b l S l L i l . . x < <' - ,?.'
t z e * * r n e P z , * 7 * , , a ~~ <. * S l l l l l b * * " r * * a r = * . . ,* m c ..*.
~, ~" > .- * % % % ! : * %e
: E = . * * # * * * * * v * . r l l t r * I X * , a 5 % '*
:: h * S , b 5 5 2::-
. + * - * a .*.+-a.4,4.,>*. * a S . " * , e'.
" . . , A - * . . . A a . . . . " ? " "..
n , c.. ."
. * * . * # * = = # * . . * - * e = w * - = * f a.
: a n 2 .-.
" - -,.
:
*
" ' *."
-
.
. ~ , - - * . w * - * . . - . .. .
- ."...--
.
; i n ::?
;
5;
,
*. -....-&.- ..*-'-. =M *,,...--
"....--XI-- _ * * * * + J * ? * *X#..
+ " . * . a , . -."__---LC _,_",....
-
* - ' < ? ::
:: - " . * z x 7 ~ = 5 ~ ? p ~ * z - - _ _ *- -
-
L = a / I * * - " ., 9 . - -'A ' " . * -Figure 6.
Z
Configuration, 8, = 45O, RBT = 0,~oncavei~irror ~ i l * Angle 8
2
Incremented by
7
mrad from ~ t sOptimal Position
and is somewhat similar to the patterns seen in Figures 4 and 5.
Figure8 shows aninterference pattern for a U-type configuration, which is qualitatively somewhat differ- ent than that of Z configurations.
SUMMARY AND CONCLUSIONS
The tilted spherical mirror resonator concept was validated theoretically and experimentally by choosing a convex/concave mirror set with small radii
. ~ f curvature (small f number). The deliberately
F i g u r e 7. 2 C o n f i g u r a t i o n ,
8
= 70°, RBT = -0.799 1F i g u r e 8. U C o n f i g u r a t i o n , 8 = 70°,
RBT = 0 1
d i f f e r e n c e s . The r e s u l t s of experiment and t h e o r y l a s e r development programs experiment w i t h v a r y i n g v a l - agreed q u a n t i t a t i v e l y . ues of xc, t h e r e b y r e q u i r i n g s e v e r a l s e t s of t o r o i d a l
The f o l l o w i n g c o n c l u s i o n s were drawn from t h i s m i r r o r s f o r beam e x t r a c t i o n . The p r e s e n t concept h a s work: t h e a b i l i t y t o v a r y M c o n t i n u o u s l y t o a c c o u n t f o r
X
(1) The s e n s e of r e l a t i v e t i l t of t h e s p h e r i c a l planned v a r i a t i o n s i n x of i n t e r e s t i n c h e m i c a l l a s e r m i r r o r s a f f e c t s t h e a b e r r a t i o n s , t h e a b e r r a t i o n s b e i n g development e x p e r i m e n t s , t h e r e b y r e q u i r i n g no more s m a l l e r f o r t h e Z c o n f i g u r a t i o n . t h a n one r e s o n a t o r .
( 2 ) The dominant terms c o n t r i b u t i n g t o a b e r r a t i o n s
2 3 REFERENCES
a r e of t h e form y3 and x y w i t h y predominating f o r
1. C h a r l e s Cason, R. W. J o n e s , and J . F. P e r k i n s , h i g h a s p e c t r a t i o r e s o n a t o r s . O p t i c s L e t t e r s
2 ,
145 (1978).(3) The concept h a s many advantages o v e r t o r o i d a l 2 . A. E. Siegman, Appl. Opt.
