Photoelectric detection of coherent light in filtered background ligh

Helstrom, C. W.

English

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P h o t o e l e c t r i c  Detec t ion  of Coherent Light  
i n  F i l t e r e d  Background Light  
Jx C a r l  W. Helstrom 
Department of Applied Phys ics  and Information Sc ience  
/" 
Unive r s i ty  of C a l i f o r n i a ,  San Diego 
La J o l l a ,  C a l i f o r n i a  92037 
ABSTRACT 
Calcula ted  i s  t h e  p r o b a b i l i t y  of d e t e c t i n g  coherent  
l i g h t  i n  t h e  presence  of background l i g h t  t h a t  has  passed 
through a  narrowband f i l t e r  of e i t h e r  r e c t a n g u l a r  o r  Lorentz  
frequency c h a r a c t e r i s t i c .  Both s i g n a l  and background f a l l  
on a  p h o t o e l e c t r i c  s u r f a c e  whose emit ted e l e c t r o n s  a r e  
counted and subjec ted  t o  a  randomized d e c i s i o n  procedure 
t h a t  y i e l d s  a pre-assigned fa l se-a la rm p r o b a b i l i t y .  
,/ 
* This  r e sea rch  was c a r r i e d  ou t  under Research Grant NGL-05-009-079 
from t h e  National  Aeronaut ics  and Space Adminis t ra t ion  
https://ntrs.nasa.gov/search.jsp?R=19700032598 2020-03-11T23:06:13+00:00Z
A p h o t o e l e c t r i c  counter  i s  t o  d e t e c t  t h e  s i g n a l  from an i d e a l  
l a s e r  i n  t h e  presence  of incoherent  background l i g h t .  The i n c i d e n t  l i g h t  
pas ses  through a  f i l t e r  whose passband, of width W, i s  centered  a t  t h e  
frequency R of t h e  s i g n a l ,  and through a  p o l a r i z e r  t h a t  removes a l l  
background l i g h t  po la r i zed  i n  a  p lane  perpendicular  t o  t h a t  of t h e  
s i g n a l ,  a f t e r  which t h e  l i g h t  f a l l s  on a  p h o t o e l e c t r i c  su r f ace .  The 
s i g n a l ,  when p r e s e n t ,  has  cons t an t  ampli tude dur ing  t h e  e n t i r e  i n t e r v a l  
(-112 T ,  1 / 2  T) of observa t ion .  
When t h e  passband of t h e  f i l t e r  i s  so  broad t h a t  WT >> 1, t h e  
d i s t r i b u t i o n  of t h e  number n  of pho toe l ec t rons  i s  approximately Poisson ,  
and t h i s  i s  t h e  u s u a l  assumption i n  ana lyz ing  t h e  performance of l a s e r  
1 
r a d a r  r e c e i v e r s .  I f  t h e  passband could b e  made so  narrow t h a t  WT << 1, 
t h e  number n  would have a Bose o r  a  Laguerre  d i s t r i b u t i o n ,  depending on 
whether t h e  s i g n a l  is  absent  o r  p r e s e n t .  Detec t ion  p r o b a b i l i t i e s  under 
2 
t h i s  c o n d i t i o n  have been publ i shed .  So narrow a  passband i s  d i f f i c u l t  
t o  a t t a i n  i n  p r a c t i c e ,  e s p e c i a l l y  when t h e  frequency of t h e  s i g n a l  may 
be  u n c e r t a i n  because of Doppler s h i f t .  The p r o b a b i l i t y  of d e t e c t i o n  
should be worked o u t  f o r  i n t e rmed ia t e  v a l u e s  of WT i n  o r d e r  t o  a s s e s s  
t h e  accuracy of c a l c u l a t i o n s  based on t h e  s imp l i fy ing  assumptions 
WT << 1 o r  WT >> 1. Some r e s u l t s  of such a n  a n a l y s i s  a r e  presented  he re .  
Let  V ( t )  r e p r e s e n t  t he  complex envelope of t h e  plane-polar ized 
l i g h t  f i e l d  a t  t h e  p h o t o e l e c t r i c  s u r f a c e .  The p r o b a b i l i t y  P(n)  t h a t  
e x a c t l y  n  pho toe l ec t rons  a r e  emit ted dur ing  t h e  i n t e r v a l  (-112 T ,  112 T )  
is  given by a  modified Poisson d i s t r i b u t i o n ,  
n  -x P(n)  = (x  e In!) ,  
TI 2  
= i T / 2  
l v ( t )  1 d t ,  
where a i s  a  cons t an t  p r o p o r t i o n a l  t o  t h e  quantum e f f i c i e n c y  of t h e  su r -  
f a c e  and t o  i t s  a rea .3  The average  i n d i c a t e d  by ( ) i s  taken  over  t h e  
ensemble of c i rcular-complex Gaussian p roces se s  V ( t ) .  
When t h e  s i g n a l  i s  p r e s e n t  (hypothes i s  HI) ,  
when i t  i s  n o t  (hypothes i s  Ho),  V( t )  = N( t ) .  Here S ( t )  = A ei' r e p r e s e n t s  
t h e  laser f i e l d  of cons t an t  ampl i tude  A and random phase $; N(t )  i s  t h e  
complex envelope of t h e  l i n e a r l y  po l a r i zed  background f i e l d .  The complex 
au tocovar iance ,  o r  temporal coherence f u n c t i o n ,  of t h e  background 
Re  N( t )  eint i s  
t h e  s p e c t r a l  d e n s i t y  
PW 
of t h e  background i s  p r o p o r t i o n a l  t o  1 (w) 1 2, where y (w) is  t h e  t r a n s f e r  
f u n c t i o n  of  t h e  f i l t e r ,  angular  f r equenc i e s  w being r e f e r r e d  t o  t h e  s i g n a l  
f requency R a s  o r i g i n .  
By expanding t h e  f i e l d  V ( t )  i n  terms of t h e  orthonormal e igen-  
f u n c t i o n s  f j ( t )  of t h e  i n t e g r a l  equa t ion  
i t  can be shown t h a t  t he  number n of pho toe l ec t rons  can be w r i t t e n  a s  4 
where t h e  n 's a r e  s t a t i s t i c a l l y  independent ,  in tegra l -va lued  random j 
v a r i a b l e s  having under hypothesis  Ho Bose d i s t r i b u t i o n s  p j O ) ( n j )  and 
(1)  under hypothes is  H1 Laguerre d i s t r i b u t i o n s  p ( n j ) .  These d i s t r i b u t i o n s  j 
a r e  
where 
N is  t h e  t o t a l  average  number of e l e c t r o n s  i n  (-112 T, 112 T) due t o  t h e  
background, S i s  t h e  t o t a l  average number t h a t  would be e j e c t e d  by t h e  
s i g n a l  S ( t )  a lone ,  and Lm(x) is  t h e  m-th Laguerre  polynomial. Here 
The d i s t r i b u t i o n s  Po(n)  and P l (n )  of t h e  t o t a l  number n of e l e c t r o n s  a r e  
e a s i l y  computed d i g i t a l l y  by convolving t h e  component d i s t r i b u t i o n s  i n  
(6) and ( 7 )  f o r  which p j  and a a r e  s i g n i f i c a n t l y  g r e a t e r  than  zero .  j 
The d e t e c t i o n  p r o b a b i l i t i e s  have been c a l c u l a t e d  f o r  a  randomized 
d e t e c t o r  t h a t  chooses HI ( s i g n a l  p r e s e n t )  whenever t h e  number n  of 
e l e c t r o n s  exceeds a c e r t a i n  i n t e g r a l  d e c i s i o n  l e v e l  8 .  When n  < 8 ,  t h e  
d e t e c t o r  chooses Ho ( s i g n a l  absen t ) .  When n  = 8 ,  hypothes is  H1 i s  
chosen wi th  p r o b a b i l i t y  f ,  0  5 f  < 1. The va lues  of 8 and f  a r e  s e l e c t e d  
s o  t h a t  t h e  fa l se-a la rm p r o b a b i l i t y  
t a k e s  on a  pre-assigned va lue .  To f i n d  f  and 8 ,  Po(k)  i s  summed from 
k=O u n t i l  t h e  sum C exceeds 1 - Q O .  The va lue  of k  a t  which t h i s  f i r s t  
happens i s  8 ,  and 
The p r o b a b i l i t y  Qd of d e t e c t i o n  i s  then  g iven  by 
(a )  Rectangular Spectrum 
When t h e  f i l t e r  has  a  r ec t angu la r  f requency c h a r a c t e r i s t i c ,  t h e  
complex au tocovar iance  of t h e  background l i g h t  s t r i k i n g  t h e  p h o t o s e n s i t i v e  
s u r f a c e  i s  
and t h e  i n t e g r a l  equat ion  (4) i s  t h e  one t r e a t e d  by S lep ian  and Po l l ak .  5 
The e igen func t ions  a r e  t h e  angular  p r o l a t e  s p h e r o i d a l  wave func t ions  
n  
% S (c ,  2t /T)  f n ( t )  = u - ' ( 2 / ~ )  On (15) 
w i t h  c  = n WT/2 and un t h e  normal iza t ion  f a c t o r  g iven  by 
un2 = IsOn(cY x )  l 2  dx.  
I n  t h e  n o t a t i o n  of S t r a t t o n  e t  aZ., 6 
i n  terms of t h e  Legendre polynomials Pn(x) ,  and by t h e i r  o r thogona l i t y  
over (-1, 1 )  , 
The e igenvalues  of (4 )  a r e ,  i n  t h e  n o t a t i o n  of S l ep ian  and P o l l a k ,  5  
The q u a n t i t i e s  a i n  (10) a r e  k 
f o r  wi th  S ( t )  cons t an t  over  (-112 T, 112 T),  w e  need only  t o  eva lua t e  
t h e  F o u r i e r  t ransform 7 
% s ( c ,  W/TIW) (21) f n ( t )  eiwt d t  = u n - ' [ 2 h n ( c ) / ~ l  On 
a t  w = 0. Here ok = 0 f o r  k odd. 
By us ing  t h e s e  formulas  we c a l c u l a t e d  t h e  d i s t r i b u t i o n s  i n  (6) and 
( 7 ) ,  convolved them t o  determine t h e  d i s t r i b u t i o n s  Po(k)  and P l ( k ) ,  and 
used (12) and (13) t o  work o u t  t h e  p r o b a b i l i t i e s  of d e t e c t i o n ,  which a r e  
p l o t t e d  i n  F igs .  1 - 3 f o r  v a r i o u s  va lues  of fa l se-a la rm p r o b a b i l i t y  Qo, 
mean number N of background pho toe l ec t rons ,  and time-bandwidth product  
c = a WT/2. For a f i x e d  background i l luminance ,  t h e  number N i n c r e a s e s ,  
and t h e  p r o b a b i l i t y  Qd of d e t e c t i o n  dec reases ,  as t h e  bandwidth W of t h e  
f i l t e r  i n c r e a s e s .  For our  p r e s e n t  purposes i t  seemed more i n s t r u c t i v e  
t o  avoid t h i s  t r i v i a l  e f f e c t  and keep t h e  t o t a l  average  number N of 
e l e c t r o n s  f i x e d .  The d e t e c t i o n  p r o b a b i l i t y  then  i n c r e a s e s  a s  t h e  f i l t e r  
passband widens and t h e  d i s t r i b u t i o n s  become more n e a r l y  Poisson.  
(b)  Lorentz  Spectrum 
When t h e  f i l t e r  has  t h e  pas s  c h a r a c t e r i s t i c  of a  simply resonant  
l i n e a r  system, t h e  s p e c t r a l  d e n s i t y  of t h e  background l i g h t  f a l l i n g  on 
t h e  photosur face  has  t h e  Lorentz  form 
and i t s  temporal coherence func t ion  i s  
The i n t e g r a l  equa t ion  (4) i s  then  e a s i l y  so lved;  t h e  e igen func t ions  
a r e  s i n u s o i d a l  . 8 
The e igenvalues  uk do n o t  drop  o f f  w i th  i n c r e a s i n g  k  r a p i d l y  
enough t o  make i t  f e a s i b l e  t o  c a l c u l a t e  t h e  d i s t r i b u t i o n  Po(k)  by convo- 
l u t i o n .  In s t ead  w e  determined i t  by an  ingenious r e c u r r e n t  equat ion  due 
t o  G. ~ d d a r d .  The gene ra t ing  f u n c t i o n  of t h e  d i s t r i b u t i o n  {Pl (k) 1 has  
t h e  form 4 
m 
where g o ( s )  i s  t h e  gene ra t ing  func t ion  of { P o ( k ) ) .  We can  determine 
{ p l ( k ) ) ,  t h e r e f o r e ,  by convolving t h e  sequence ( P o ( k ) ) ,  k  = 0,  1, 2 ,  ..., 
wi th  t h e  q u a s i - d i s t r i b u t i o n s  ( q j ( k ) )  whose gene ra t ing  func t ions  a r e  t h e  
exponent ia l  f a c t o r s  i n  ( 2 4 ) .  These q u a s i - d i s t r i b u t i o n s  a r e  given by 
q . ( k )  = vjk  exp[-(1 - v . )  s j ]  
~k  J J ( - l ) [ - ( l  - v j I 2  S j / v j ] ,  (25) 
where LL-I) (x) i s  a  gene ra l i zed  Laguerre  It can e a s i l y  be 
computed from i t s  r e c u r r e n t  equat ion ,  
(k  + 1 )  L:;:)(X) = (2k - x )  L;-')(X) - (k  - 1) Lk-l (-I1 (x) , 
~ 6 - l )  (x) = 1, L ! - ~ ) ( ~ )  = - X .  (26) 
I n  t h i s  c a l c u l a t i o n  only  t h e  even e igen func t ions  f  (t) need t o  be k 
inc luded ,  
t h e  odd ones y i e l d  a = 0 i n  (10) .  The e igenvalues  pk a r e  now 8 j 
where ek i s  t h e  k-th r o o t  of t h e  t r anscenden ta l  equat ion  
0 t a n  B k  = m,  k  6, = wk T/2, (29) 
which is  e a s i l y  solved by Newton's method. The normal iza t ion  cons t an t  
u is  g iven  by k 
and 
The d e t e c t i o n  p r o b a b i l i t i e s f o r  v a r i o u s  va lues  of WT are p l o t t e d  i n  
F i g s .  4-6. The curves have t h e  same gene ra l  behavior  a s  t hose  f o r  t he  
r e c t a n g u l a r  spectrum. 
References 
J .  W .  Goodman, "Comparative Performance of Optical-Radar Detec t ion  
Techniques", Trans. IEEE, v o l .  AES-2, pp. 526-535; Sept .  1966. 
C .  W. Helstrom, "Performance of an  I d e a l  Quantum Receiver of a 
Coherent S igna l  of Random Phase", Trans .  IEEE, v o l .  AES-5, pp. 
562-564; May, 1969. 
L. Mandel, "F luc tua t ions  of Photon Beams and t h e i r  Corre la t ions" ,  
Proc.  Phys. Soc. (London), v o l .  72, pp. 1037-1048; December, 1958. 
S. Karp, J .  R.  Clark ,  "Photon Counting: A Problem i n  C l a s s i c a l  
Noise Theory", Trans. IEEE, v o l .  IT-16, pp. ; November, 1970. 
D.  S l ep i an ,  H. 0. Po l l ak ,  "P ro la t e  Sphero ida l  Wave Funct ions ,  
Four i e r  Analys is  and Uncer ta in ty ,  I", B e l l  System Tech. J . ,  v o l .  40, 
pp. 43-63; January ,  1961. 
J .  A. S t r a t t o n ,  P. M. Morse, L. J. Chu, J. D. C. L i t t l e ,  F. J .  
Corbatd, Sphero ida l  'wave Funct ions;  M. I .T. P r e s s ,  1956. 
Reference 5 ,  Eq. (25) ,  p. 57. 
C. W. Helstrom, S t a t i s t i c a l  Theory of S igna l  Detec t ion ,  2nd. ed . ;  
Pergamon P r e s s ,  London, 1968. See Ch. I V ,  Sec. 5 ,  pp. 138-141. 
G.  Bgdard, "Photon Counting S t a t i s t i c s  of Gaussian Light" ,  Phys. 
Rev., v o l .  151,  pp. 1038-1039; Nov. 25, 1966. 
A. E rd6 ly i  e t  al., Higher Transcendental  Funct ions;  McGraw-Hill 
Book Co., New York, 1953. See v o l .  2,  Sec. 10.12,  pp. 188-192. 
Figu re  Capt ions  
F ig .  I. P r o b a b i l i t y  of d e t e c t i o n  v s .  average number S of s i g n a l  
coun t s  f o r  r e c t a n g u l a r  background spectrum; QO = N = 0.5. 
The curves  a r e  indexed w i t h  t h e  v a l u e  of c  = II WT/2. 
F ig .  2.  P r o b a b i l i t y  of d e t e c t i o n  v s .  average  number S of s i g n a l  
counts  f o r  r e c t a n g u l a r  background spectrum; Q0  = N = 2.0. 
The cu rves  a r e  indexed w i t h  t h e  v a l u e  of c  = II WT/2. 
F ig .  3 .  P r o b a b i l i t y  of d e t e c t i o n  v s .  average  number S of s i g n a l  
coun t s  f o r  r e c t a n g u l a r  background spectrum; QO = N = 0.5. 
The curves  a r e  indexed w i t h  t h e  v a l u e  of c = II WT/2. 
F ig .  4 .  P r o b a b i l i t y  of d e t e c t i o n  v s .  average number S of s i g n a l  
coun t s  f o r  Lorentz  background spectrum; QO = N = 0.5. The 
cu rves  a r e  indexed w i t h  t h e  v a l u e  of WT. 
F ig .  5 .  P r o b a b i l i t y  of d e t e c t i o n  v s .  average  number S of s i g n a l  
coun t s  f o r  Lorentz  background spectrum; Q O  = N = 2.0 .  
The curves  a r e  indexed wi th  t h e  v a l u e  of WT. 
P i g .  6 .  P r o b a b i l i t y  of d e t e c t i o n  v s .  average  number S of s i g n a l  
coun t s  f o r  Lorentz  background spectrum; QO = N = 0.5.  
The curves  a r e  indexed wi th  t h e  v a l u e  of WT. 

P r o b a b i l i t y  of d e t e c t i o n  v s .  ave rage  number S of  s i g n a l  
coun t s  f o r  r e c t a n g u l a r  background spectrum; Q o  = N = 2.0. 
The cu rves  a r e  indexed w i t h  t h e  v a l u e  of c  = T WT/2. 

99.99 
99,9 
99 
98 
95 
90 
Q (Ole) 
50 
10 
5 
2 
I 
P r o b a b i l i t y  of d e t e c t i o n  vs. average  number S of s i g n a l  
coun t s  f o r  Lorentz  background spectrum; QO = N = 0.5. The 
0.1 cu rves  a r e  indexed w i t h  t h e  v a l u e  of WT. 
0 2 4 6 8 10 12 14 16 18 
S 




Photoelectric detection of coherent light in filtered background light

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