SOVIET PHYSICS JOURNAL 49

FIELD DISTRIBUTION IN A FABRY-PEROT INTERFEROMETER WITH INC LINED MIRRORS

Mahmoud Hassan

Izvestiya VUZ. Fizika, Vol. i0, No. 5, pp. 89-93, 1967

The field is localized in planes perpendicular to the principal section of the wedge formed by the mirrors; the positions of these are defined by kiry's function, with secondary components accessible to observa- tion.

i. INTRODUCTION

Serious losses arise if the laser mirrors are not

parallel; not only do the emergent beams diverge (the

directions of the maxima of waves diffracted at the

exit aperature differ), but each is displaced relative

to the previous in the reflection plane, which is ac-

companied by vignetting. The latter is inevitable if

the mirrors are not strictly parallel, even if the in-

cidence is initially normal. Both effects reduce Q in proportion to the angle between the mirrors. The loss

of Q due to divergence alone has been considered [i];

the maximum angle at which Herriot [2] still observed

oscillation was about 1/4 of the diffraction angle at

the angle at the exit aperture for a He-Ne laser. Here

I consider the field distribution in such a cavity.

2. INTERFERENCE WITHIN THE CAVITY

Different parts of a given wave travel different dis-

tances within such a cavity. The coordinate system

has mirror I in the yz plane (Fig. i), extending a

distance b along the y axis and a distance a along the

z axis. The normal at one corner of this mirror is

taken as the x axis, which lies in the plane

x = h + ytg~., (t)

in which c~ is the angle be tween the p lanes of the m i r - r o r s in the xy plane. Let the p r i m a r y p lane wave p r o - pagat ing towards the inc l ined m i r r o r have phase ~0 at x = 0. An a r b i t r a r y r ay (hatehed in Fig. 1) af ter j -- - i r e f l ec t ions f rom the f i r s t m i r r o r p roduces in the xy plane the wave e l emen t

(d&)j = c (7'2 9,) s-I e* <*;)y .a dy , (2)

and af ter j r e f l ec t i ons f rom the second,

(dsii)j = c f,,, (~, p2) j - I ei(r .ad% (3)

in which (c" ady) and (e" ad~) a re the ampl i tudes of the e l e m e n t a r y wave su r faces at the f i r s t and second m i r - r o r s respec t ive ly ; c is a eonstant ; d~ = dy. sec c~ (see f igure), and pt and pa a re the re f lec t ion coeff ic ients of the m i r r o r s for the ampl i tudes . The phases of the wave e l emen t s a re given by

j - I

O';)j = % - ,~ {2cos =~.~. cos [~ + rt--1

j - - I

+ ( 2 n - - t ) ~ ] - - 2 s i n ~ ~ y ~ c o s [ ? ,'-1-

+ (2 n -- 1) a] + Yj (sin ~j - - sin 9) + x cos ,:?+

+ y sin '4;}, (4)

] - I

(q);i)]=r {2 Cos a Z x,,cosI,~+(2n-- 1)a]-- n=!

] - - i

- - 2 s i n ~ Z y ~ c o s [~ + (2n ~- 1)~.] q - x j ( c o s ~ j + c o s g ) + n= l

+ y j ( s i n ~ , i - - s i n ~ , ) - - x c o s ~ + y s i n , ~ } , (5)

in which k = 27r/X is wave n u m b e r ; (0, Yj) is the point at which the r a y me e t s the f i r s t m i r r o r ; (x n, Yn) is the point at which the r a y mee t s the second m i r r o r ; ~0j = = [q) + 2(j - 1)c~] is the angle at which the diff racted wave of m a x i m u m ampl i tude propaga tes (i. e . , dif- f r ac t ion l o s s e s at each r e f l ec t ion a re neglected) , and

is the angle at which a wave e l emen t propaga tes in acco rdance with Huygen ' s p r i nc ip l e .

We subs t i tu te q) = 0 in (1) and r e p l a c e the t r i g o - n o m e t r i c funct ions by the i r a r g u m e n t s for a sma l l c~,

}, = q - g , q>~O (6)

whereupon i n t e r g r a t i o n of (2) and (3) gives the o s c i l - la t ions r e f l ec ted f rom both m i r r o r s as

i _1 k (b- y)) ,~j SIj = C a ( p l o2)j._ I eiCVj e 2

i k (b - Y)) ~j sin -~

1 - - k~j 2

su/ = ca.e2 (91 ,%)]-i ei'hrj e

(7)

in which

2

! k~j 2

(8)

(])u- = ,'1,o-- k {2 ( j - - l ) h + [ 2 ( j - - 1)a + } / ] y + x } , (9)

O & i ] = ~ o - - k { 2 j h + [ 2 ( / - - 1 ) ~ . - , L O j ] y - - x } , (10)

(b - Y'j), (b - y'j) are the widths of the beams re-

fleeted from both mirrors, as

~1 = sin ~ (cos ~j + cos 9) +

+ sin ? j - - sin 9 ~ 2 a q- ?a - - ~. (11)

The p r e s e n c e of (7) and (8) together gives r i s e to the total ose i l l a t ion

(X sty + Z s.j) d~ dO, (12)

J

50 IZVESTIYA VUZ. FIZIKA

which takes account of the angle of d i f f rac t ion 0 in the xz p lanes . At the points of obse rva t ion (x, y, z), where we have

Io'l > -2 '

I ;.I >> b - - y ; . ' . >>

(13)

we can r ep l ace the l im i t s of i n t eg ra t i on in (12) by infinity.

Y ' 1

I f

f~

In tegra t ion and s u m m a t i o n of (12) gives the f ina l fo rmula :

in which

S = cV- e i (,~0-kl0 [e*k (h-x) + he e-*k (1L-x) ]x

1 - - (?102) N e - " N * x

p e -iq' , 1--?1 (14)

r = 2 k (h + my + ~3), (15)

and 6 is the c o r r e c t i o n for phase shift on re f lec t ion . F o r m u l a (14) d e s c r i b e s a s tanding wave, whose

in tens i ty d i s t r i bu t ion is

I = d).4. [(1 - - p 2 ) 2 + 4 p 2 c o s 2 t e ( h - - x ) ] T N , (16)

pxx 4p~ r ?~ sin 2 N (1-r ' (o) + T:~: = 2 (17)

(1 - - p, 9,)~ + 4 ~ ~a sin2 - ~ 2

We see f rom (16) that the in t ens i ty d i s t r i bu t i on in any p lane x = cons tant is given by T N, which for l a rge N is a lmos t p r e c i s e l y A i r y ' s funct ion if Pl and P2 a re l e s s than one. This funct ion has equal va lues in the p lanes yp = cons tan t with �9 = 2p~ in (17), p be ing an o rder number . The in t ens i ty d i s t r i bu t ion in the x d i - r ec t ion in any of these p lanes is s inuso ida l be tween l imi t s (1 + p2) 2 and (1 - p2) 2 as mul t ip l i ed by the c o r - r e spond ing va lue of the Airy function.

To cons ider the side lobes of T N it is b e s t to ex- p r e s s the a rgumen t s of the s ines in t e r m s of v in such a way that n~ = 2PN~ and �9 = 2pv, so

P,v = p N . (18)

Here p and PN may or may not be in tege r s . The n u m - b e r of i n t e r f e r i n g b e a m s is

As N m u s t be an in teger , it c o r r e sponds to d i s c r e t e va lues of y, name ly YN" Hence, for y be tween YN and YN+I we should cons ide r T N as a funct ion of y only, with N a constant . Then the d i s t r i bu t ion of s ide f r i n - ges is given by (18) as

dpw = N. (20) dp

The s e q u e n c e of these bands will be the longer the n e a r e r the point of obse rva t ion to the base of the wedge, where N has i ts m a x i m u m value:

_ / b Nm~ ~ I / 2 ~ h " (21)

V

The i n t e r v a l s Ay be tween YN and YN+I may be found f rom (19) by d i f fe ren t ia t ion and putt ing AN = 1:

Ay ~ 2 lfl2~hy N . (22)

The n u m b e r of s ide f r inges in Ay is

dpN hy 4 = YN" (23)

dy ),

F r o m (6) we may r e w r i t e (23) as

dp~v by = 4q YN . (23r) dy b

The n u m b e r of these f r inges is shown by (23) or (23') to i n c r e a s e with (~ and as the base is approached, the va lue be ing roughly one if q is sma l l in (23'). Osc i l - la t ion can be obse rved only for q < 1/2, so the n u m b e r of these f r inges is l e s s than Nma x, s ince the ape r tu re of the cavi ty is b = NmaxAy; for example, if b = 1 era, h = 1 0 2 e m , q = 1 / 4 , a n d ~ = 5 . 1 0 -5 cm, t h e n N m a x = = 20, so the side f r inges in that case a re acces s ib l e to observa t ion .

3. SUMMARY

1. The field s t r u c t u r e within a m u l t i p l e - b e a m i n - t e r f e r o m e t e r is cons idered , with a l lowance for d i f - f r ac t ion of the i n t e r f e r ing beams .

2. E x p r e s s i o n s a re de r ived that show that the f ield is loca l ized in p lanes pe r pe nd i c u l a r to the p r inc ipa l sec t ion of the wedge fo rmed by the m i r r o r s ; the pos i - t ions of these p l anes a re defined by A i r y ' s function, and it is found that the side f r inges a re acces s ib l e to obse rva t ion for the i n t e r f e r o m e t e r p a r a m e t e r s , com- monly used with l a s e r s .

I am indebted to P r o f e s s o r F. A. Korolev for d i s - cussions.

REFERENCES

1. M. Hassan , Izv. VUZ. Fiz ika (in p re s s ) .

2. D. R. Her r io t t , J. Opt. Soc. Am, 52, no. 1, 31, 1962.

| / Y (19) N ~ V 2~h t June 1966 Moscow State Un ive r s i t y