Circular polarization in A j = 1 → j = 0 - transition laser

Volume25A, number 4

CIRCULAR

PHYSICS LETTERS

POLARIZATION

IN

D. P O L D E R

A ] = 1 -~ j = 0 and W . V A N

28 August 1967

- TRANSITION

LASER

HAERINGEN

Philips Research Laboratories, N. V. Philips Gloeilampenfabriehen Eindhoven, The Netherlands Received 4 August 1967

The preference for circular polarization of an isotropic laser involving a j = 1 --~j = 0 transition is caused by the difference between the relaxation rates of the quadrupole moment and the angular momentum in the degenerate level.

A He-Ne l a s e r o p e r a t i n g in a single l o n g i t u d i nal mode at 1.153 ~ (a j = 1 ~ j = 2 t r a n s i t i o n ) shows a m a r k e d t e n d e n c y t o w a r d s l i n e a r optical p o l a r i z a t i o n . T h i s is an i m m e d i a t e c o n s e q u e n c e of the n o n - l i n e a r r e s p o n s e of the l a s e r m e d i u m to the h i g h - i n t e n s i t y field in the l a s e r cavity. T h i s p h e n o m e n o n can be d e s c r i b e d a s follows: If excited by e l l i p t i c a l l y p o l a r i z e d high i n t e n s i t y light the m e d i u m r e s p o n d s with a n o n - l i n e a r e l e c t r i c p o l a r i z a t i o n that has a slightly different e l liptic p o l a r i z a t i o n , which, through the d y n a m i c s of the cavity equations, c a u s e s the optical p o l a r i zation to change g r a d u a l l y in the d i r e c t i o n of l i n e a r p o l a r i z a t i o n . The e x i s t e n c e of this phenom e n o n of n s a t u r a t i o n induced a n i s o t r o p y " was f i r s t u n a m b i g u o u s l y d e m o n s t r a t e d by De Lang et al. [1-3] in a s e r i e s of e x p e r i m e n t s . A quantitative q u a n t u m - m e c h a n i c a l d e s c r i p t i o n of the s a t u r a t i o n - i n d u c e d a n i s o t r o p y in the a b sence of a m a g n e t i c field was f i r s t given by the p r e s e n t a u t h o r s [4]. We showed that the t e n d e n c y towards l i n e a r p o l a r i z a t i o n is typical of all t r a n s itions j *-*j + 1 ( j > 0) and the t r a n s i t i o n j = ½ - - ' j = = ½. We a l s o p r e d i c t e d that the t r a n s i t i o n s j --*j (j > 1) ought to show a tendency t o w a r d s c i r c u l a r p o l a r i z a t i o n , while the t r a n s i t i o n s j = 0 ~ j = 1 and j = 1 - - ' j = 1 w e r e expected to be n e u t r a l in this r e s p e c t . The p r e d i c t i o n of a j = 2 --*] = 2 t r a n s i tion was v e r i f i e d by De Lang [5], but c o n t r a r y to expectations a j = 1 --*] = 0 N e - t r a n s i t i o n at 1.523 ~t was found to have a r e l a t i v e l y s m a l l but definite tendency to c i r c u l a r p o l a r i z a t i o n [5]. The effect shows the n o r m a l p r o p o r t i o n a l i t y with i n t e n s i t y . It a p p e a r s to be s e n s i t i v e to the gas p r e s s u r e [6]. Incidentally the tendency to l i n e a r and c i r c u l a r p o l a r i z a t i o n can be s u g g e s t i v e l y called equatophily r e s p e c t i v e l y polarophily, in view of the fact that a l i n e a r p o l a r i z a t i o n is r e p r e s e n t e d by a point on

the equator and a c i r c u l a r p o l a r i z a t i o n as a point on a pole on the P o i n c a r ~ s p h e r e s o m e t i m e s used to r e p r e s e n t p o l a r i z a t i o n s t a t e s [7]. We now want to a r g u e that the s m a l l p o l a r o phily i n the j = 1 --*j = 0 c a s e can only have its o r i g i n in atomic r e l a x a t i o n p r o c e s s e s . To see this, let u s f i r s t identify the o r i g i n of the expected n e u t r a l i t y . The r e s p o n c e of a j = 1, j = 0 s y s t e m to an a r b i t r a r y e l l i p t i c a l l y p o l a r i z e d light wave of a r b i t r a r y i n t e n s i t y is actually that of a single t w o - l e v e l s y s t e m , a s long a s r e l a x a t i o n p r o c e s s e s b e t w e e n the d e g e n e r a t e l e v e l s a r e ignored. One of t h e s e l e v e l s is the j = 0 level, the other has as wave function a suitable l i n e a r c o m b i n a t i o n of the m = I, m - O , m = - l f u n c t i o n s f o r j = l . The s u i t a b l e l i n e a r c o m b i n a t i o n i s defined in such a way that m a t r i x e l e m e n t s with j = 0 for the e l l i p t i c a l l y p o l a r i z e d light exist for this wave function only and not f o r the other two orthogonal l i n e a r c o m b i n a t i o n s , at l e a s t in the r e s o n a n t wave app r o x i m a t i o u . It follows that the r e s p o n c e of the e l e c t r i c p o l a r i z a t i o n P has the s a m e p o l a r i z a t i o n state as that of the exciting wave. Also, for any given i n t e n s i t y the (complex) r a t i o of P and the exciting E is always the s a m e whatever the a c tual state of optical p o l a r i z a t i o n . T h i s i n v a r i a n c e e x p l a i n s the n e u t r a l behaviour. Note that the p o s s i b i l i t y of defining a s u i t a b l e l i n e a r c o m b i n a t i o n f o r any state of p o l a r i z a t i o n is typical of a j = 1 state. It t h e r e f o r e a p p e a r s that the only way to get r i d of the n e u t r a l i t y is to i n t r o d u c e a m e c h a n i s m that m i x e s the j = 1 s u b s t a t e s . I n t e r s t a t e r e l a x a tion, caused for i n s t a n c e by c o l l i s i o n s of the excited N e - a t o m s with H e - a t o m s , p r o v i d e s such a m e c h a n i s m . It r e m a i n s to be shown that i n t e r state r e l a x a t i o n c a n depend on the actual wave f u n c t i o n s of the s t a t e s involved and that this de337

Volume25A, number 4

PHYSICS LETTERS

p e n d e n c e l e a d s to the s m a l l p o l a r o p h i l y o b s e r v e d . In p a r t i c u l a r it m u s t be shown that this p o s s i b i l ity e x c i s t s f o r an i s o t r o p i c r e l a x a t i o n p r o c e s s . F o r this p u r p o s e c o n s i d e r the l i n e a r i n t e r s t a t e r e l a x a t i o n equation /5 = - Mp

(1)

w h e r e p is the 3 × 3 density m a t r i x and M the 9 × 9 r e l a x a t i o n m a t r i x . The m o s t g e n e r a l f o r m of M, s a t i s f y i n g the i s o t r o p y condition and l e a v ing the t r a c e T of p constant, can be e a s i l y w r i t ten down in the r e p r e s e n t a t i o n that has as b a s i s the t h r e e wave functions ~ x , ~ y , qJz which t r a n s f o r m a s x, y and z under a spatial r o t a t i o n of the x, y, z ax es . Eq. (1) then r e a d s :

~Sxx = - Cl Pxx + ~ Cl T /)xy + [:)yx = - Cl (Pxy + Pyx ) -

: -

(p

(2)

y- p

and s i m i l a r l y f o r the o t h e r c o m p o n e n t s . T h e s e e qua ti o n s a r e i n v a r i a n t f o r any r e a l orthogonal t r a n s f o r m a t i o n of the b a s i s , c o r r e s p o n d i n g to a spa ti a l r o t a t i o n . H o w e v e r , we a r e i n t e r e s t e d in the change of the shape of M f o r a u n i t a r y t r a n s f o r m a t i o n of the b a s i s , s in c e such a t r a n s f o r m a tion is r e q u i r e d f o r the c o n s t r u c t i o n of the s u i t able l i n e a r co m b i n at io n ~ a "on speaking t e r m s " with an a r b i t r a r y e l l i p t i c a l l y p o l a r i z e d wave as m e n t i o n e d above. It is e a s i l y shown that the shape is i n v a r i a n t f o r a u n i t a r y t r a n s f o r m a t i o n if c 1 = c2, in which c a s e n e u t r a l b e h a v i o u r is ma i ntai n ed . On the o t h e r hand if c 1 ¢ c2, the r e l a x a t i o n eq u at i o n s on the b a s i s of the functions ~a , ~b, ~ c , have a s t r u c t u r e d i f f e r e n t f r o m t h o s e in (2). T h i s has a profound effect on the e v a l u a t i o n o f the e l e c t r i c p o l a r i z a t i o n of the atom unde r the influence of an e l l i p t i c a l l y p o l a r i z e d wave, which now no l o n g e r shows the s a m e state of optical p o l a r i z a t i o n a s that of the e x c i t i n g field. The actual c a l c u l a t i o n shows the o c c u r r e n c e of p o l a r o p h i l y if c 2 > c 1 and equatophily if c 2 < c 1. The p h y s i c a l i n t e r p r e t a t i o n of the p o l a r o p h i l y if c 2 > c 1 is as follows. The m e a n i n g of the constant c 1 is the r e l a x a t i o n r a t e f o r the a v e r a g e l e v e l . The constant c2 is the r e l a x a t i o n r a t e f o r the a n g u l a r m o m e n t u m . The gain of e n e r g y of a light wave of given i n t e n s i t y in a l a s e r m e d i u m is d e t e r m i n e d among o t h e r things by the r a t e at which the upper l e v e l occupation is r e p l e n i s h e d while being depopulated by the l a s e r p r o c e s s . If a n g u l a r m o m e n t u m r e l a x a t i o n i s f a s t e r than quadr upole r e l a x a t i o n , the r e p l e n i s h m e n t f r o m the o t h e r s u b s t a t e s and thus the gain is h i g h e r the g r e a t e r the a n g u l a r m o m e n t u m c a r r i e d by the u p p e r state "on speaking t e r m s " with the wave. T h i s

338

28 August 1967

explains the p r e f e r e n c e for c i r c u l a r p o l a r i z a t i o n w h e r e the upper st at e l e v e l is 2-½ (~x ~ i ~ y ) . The e x i s t e n c e of d i f f e r e n t r e l a x a t i o n r a t e s in d e g e n e r a t e l e v e l s is well known f o r e x c i t e d s t a t e s in n o n - l a s i n g g a s e s , such as Hg, Cd or In. T h e o r e t i c a l cal cu l at i o n [8] b a s e d on c o l l i s i o n t h e o r y indicate that indeed c 2 > Cl, and this r e l a x a t i o n has v e r i f i e d in e x p e r i m e n t s [9]. One would indeed n e c e s s a r i l y have c 2 > c 1 if t h e r e a r e two i n d ependent r a n d o m p r o c e s s e s r e s p o n s i b l e f o r the r e l a x a t i o n , s i n c e a single i s o t r o p i c a l l y o p er at i n g quadrupole relaxation p r o c e s s already makes c 2 = c 1. A suggestion that c o l l i s i o n induced t r a n s i t i o n s may be r e s p o n s i b l e f o r the r e m a r k a b l e n o n - n e u t r a l b e h a v i o u r of a j = 1 ~ j = 0 l a s e r has been made by Culshaw and Kannelaud [10] and by T o m l i n s o n and F o r k [11]. We b e l i e v e , h o w e v e r , that the p r e c i s e connection b et w een p o l a r o p h l l y in this c a s e and the d i f f e r e n c e between quadrupole and m o m e n t u m r e l a x a t i o n as o b s e r v e d in o t h er g a s e s has not been e x p l i c i t l y e l u c i d a t e d b e f o r e . F i n a l l y one might d i s c u s s the p r e s s u r e d ep e n d e n c e of the effect f o r a given intensity. Due to c o l l i s i o n s the i n t e r s t a t e r e l a x a t i o n i t s e l f is p r o p o r t i o n a l to the p r e s s u r e . The t i m e d e r i v a t i v e of the optical p o l a r i z a t i o n (e.g. ellipticity) is, h o w e v e r , p r e s s u r e - d e p e n d e n t in a m o r e c o m p l i cated way, si n ce f o r a given intensity the s a t u r a tion is, roughly speaking, p r o p o r t i o n a l to the inv e r s e s q u a r e of the line width. As long a s the line width is the n a t u r a l line width only, we ex pect the p o l a r o p h i l y to be p r o p o r t i o n a l to the p r e s s u r e . If, on the o t h e r hand, c o l l i s i o n b r o a d ening d o m i n a t e s , the p o l a r o p h i l y will be divided again by a f a c t o r p r o p o r t i o n a l to the s q u a r e of the p r e s s u r e , r e s u l t i n g in a p - 1 dependence. T h e s e q u a l i t a t i v e p r e d i c t i o n s a r e not in d i s a g r e e m e n t with the o b s e r v a t i o n s by De Lang and Bouwhuis [6]. 1. H. de Lang, G. Bouwhuis and E.T. Ferguson, Phys. Letters 19 (1965) 482. 2. H. de Lang. Physics 33 (1967) 163. 3. H.de Lang, Thesis, Utrecht (1966). 4. D. Polder and W. van Haeringen, Phys. Letters 19 (1965) 380. 5. H.de Lang and G.Bouwhuis, Phys. Letters 20 (1966) 383. 6. H.de Lang and G.Bouwhuis, Phys. Letters to be published. 7. M.Born and E.Wolf, Principles of optics, (Pergamon Press, 1959). 8. A.Omont, J. de Phys. 26 (1965) 26. 9. W. Happer and E.B.Salomon, Phys. Rev. Letters 15 (1965) 441. 10. W. Culshaw and J.Kannelaud, Phys. Rev. 156 (1967) 308. 11. W.J. Tomlinson and R. L. Fork, Phys. Rev. to be published.