Relaxed and triggered photon counting distributions for laser and pseudo-thermal sources

Volume 31A number 8

P HYSI C S L E T T E R S

20 April 1970

RELAXED AND TRIGGERED PHOTON COUNTING DISTRIBUTIONS FOR LASER AND PSEUDO-THERMAL SOURCES C. BENDJABALLAH

Institut d' Electrique Fondamentale * , Batiraent 220 - Facult$ des Sciences, 91 - Orsay , France Received 24 February 1970

Experimental results for relaxed and triggered photon counting distributions of laser and pseudothermal sources are obtained. In these two cases there is a good agreement between theory and experiment.

We r e p o r t h e r e e x p e r i m e n t a l r e s u l t s on pho~ ton counting d i s t r i b u t i o n s p(n, T) and q(n, T, O) r e s p e c t i v e l y called r e l a x e d d i s t r i b u t i o n (the time i n t e r v a l o f o b s e r v a t i o n T is a r b i t r a r y ) and t r i g gered d i s t r i b u t i o n (the s t a r t of the o b s e r v a t i o n is delayed by 0 f r o m an o c c u r e n c e of the photoelect r o n point p r o c e s s ) [1]. The s t a t i s t i c a l d i s t r i b u t i o n s of p h o t o e l e c t r o n s r e g i s t e r e d with a single d e t e c t o r d u r i n g a counting time T v e r y s h o r t c o m p a r e d to the coherence t i m e ~'c of the field a r e given by the e x p r e s s i o n s

[1]

p(n, T) = (exp (- IT) (IT)n/n!)

(1)

p(n, 7") = (n)n/(1 +(n)) n + l and the g e n e r a l e x p r e s s i o n of q(n, T, O) is [1] (n) n

q(n, T, O) = (1 +(n)) n + l

I1 + i~,z(O)l 2 n1-+(n)J (n)]

,

(2)

where I is the r a n d o m i n t a n t a n e o u s light i n t e n s i ty at the t i m e i n s t a n t t and ( I ) i t s expectation value. A g e n e r a l i z a t i o n of eq. (2) for N-fold joint photon counting d i s t r i b u t i o n has b e e n obtained by B~dard [3]. a) L a s e r r a d i a t i o n case. F o r a monomode amplitude s t a b i l i z e d l a s e r light, I ( t ) is constant. T h e r e f o r e r e l a x e d and t r i g g e r e d d i s t r i b u t i o n s are identical

q ( n , T , O) = q(n, T) = p(n, T) = exp(- (n)) (n)n/n! (3) where (n) = (I). T. b) P s e u d o - t h e r m a l field case. In this case p (I) = exp (- I / ( I ) ) / ( I )

(4)

so that p(n, T) and q(n, T, O) a r e v e r y different. The d i s t r i b u t i o n p(n, T) i s given by * Laboratoire assocl~ au Centre National de la Recherche Seientiflque. Work supported in part by the I ~ l a gation Ggngrale ~ la Recherche Sclentiflque et Technique.

(6)

where yz(O) = ( Z ( t ) Z * ( t - 0))/(I Z I 2) is the n o r m alized coherence function of the complex r a d i a tion field and (n) is the m e a n value of the photoe l e c t r o n s - n u m b e r o b s e r v e d d u r i n g the t i m e - i n t e r v a l T. P a r t i c u l a r l y we have

q(n, T, O) = (n + 1)p(n+ 1, T ) / ( n ) . q(n, T,O) = ( l ( O ) . e x p ( - l T ) ( I T ) n / n ! ) / ( I )

(5)

(7)

We have o b s e r v e d the photoelectron counting d i s t r i b u t i o n produced by light from a m p l i t u d e s t a b i l i z e d single mode 6328 A He Ne gas l a s e r o p e r a t i n g above the threshold of o s c i l l a t i o n and by p s e u d o - t h e r m a l r a d i a t i o n obtained by s c a t t e r ing that light on a moving ground g l a s s disk [4]. These counting d i s t r i b u t i o n s have b e e n m e a s u r e d a 56TVP p h o t o m u l t i p l i e r , a 100 M c / s e c d i s c r i m i n a t o r , a 200 M c / s e c s c a l e r and a 800 channel pulse height a n a l y z e r . The n u m b e r n of p u l s e s o b s e r v e d d u r i n g a single 0.5 bts ( T / v c # 10 "2) counting period was c o n v e r t e d to height-voltage. The delay 0 is f a r l e s s i m p o r t a n t than the c o r r e l a t i o n time Vc(0/~-c # 2 × 10 -3 so that ]7(0)12 ¢ ¢ 1 ) . The a v e r a g e counting r a t e is about 2 × 106 counts p e r second and each d i s t r i b u t i o n c o n s i s t e d of about 104 s a m p l e s . Hence the d e a d - t i m e and other effects w e r e r e l a t i v e l y u n i m p o r t a n t . The e x p e r i m e n t a l r e s u l t s have b e e n r e p o r t e d in s e m i l o g a r i t h m i c c o o r d i n a t e s in fig. 1 and 2. In all c a s e s , a g r e e m e n t between theory and e x p e r i m e n t is very good along the l a r g e i n t e r v a l of p r o b a b i lity. 471

Volume 31A, number 8

PHYSICS LETTERS

20 April 1970

0.1

o.i aool

0

1

.

0.01

2

3

4

5

6

Fig. 1. Laser light. Probability distribution versus photoelectron count and experimental rasults for r e laxed (e) and triggered (*) distributions. Theoretical curve obtained from Poisson law with <,) '- 1.2.

Qoo 0

1

2

3

o

o

4

5

6

Fig. 2. Pseudo-thermal light. Probability distribution versus photoelectron count. (a) relaxed distribution: theoretical curve obtained by geometrical distribution with = 0.6. (I0)triggered distribution: theoretical curve obtained

from eq. (7) with = 0.7. This c o n s t i t u e s a new v e r i f i c a t i o n of P o i s s o n p r o c e s s produced by l a s e r light a c c o r d i n g to eq. (3) and compound P o i s s o n p r o c e s s produced by p s e u d o - t h e r m a l s o u r c e a c c o r d i n g to eq. (6). By v a r i a t i o n of the delay 0, we have obtained f r o m eq. (5) p r e l i m i n a r y r e s u l t s about the s p e c t r a p r o f i l e of the light. We wish to thank P r . B. Picinbono for useful d i s c u s s i o n s c o n c e r n i n g this r e s e a r c h .

472

References I. M. Rotmseau. J. de Phys. 30 (1969) 673. 2. F.T. Arecchl A. Berne and A. Sona. Phys. Rev. Letters 17 (1966) 260. 3. G. Bedard Phys. Rev. 161 (1967) 304.

4. W. Martlenssen and E. Spiller Phys. Rev. 145 (1966) 285.