Quantum limited detection of incoherent object with Lorentz spectru

Helstrom, C. W.

English

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Quantum-Limi ted  Detection of a n  Incoherent  
Object  wi th  a L o r e n t z  S p e c t r u m  
.I. 
C a r l  W .  ~ e l s t r o m " '  
D e p a r t m e n t  of Applied E l e c t r o p h y s i c s  
Univers i ty  of Cal i fornia ,  San Diego 
La J o l l a ,  Cal i fornia ,  92037 
CASE F 
A B S T R A C T  
When t h e  a p p r o p r i a t e  quantum th resho ld  
d e t e c t o r s  a r e  u s e d ,  an  object  emit t ing  incoheren t  Iight 
.with a E o r e n t z  s p e c t r u m  h a s  a Power probabi l i ty  oi' 
detect ion in the  p r e s e n c e  of t h e r m a l  backgrour~d  l ight  
L 
than an  object  emi t t ing  l ight  with a r e c t a n g u l a r  s p e c -  
t r u m  of the  s a m e  bandwidth. 
https://ntrs.nasa.gov/search.jsp?R=19700002890 2020-03-12T04:45:03+00:00Z
'The quantum threshold detector  f o r  an object emit t ing incoherent 
Light that is  received with t h e r m a l  background light was worked out in a 
1 
recent  papey. I t  m e a s u r e s  a quantum-mechanical  opera tor  U that  is a 
quadra t ic  functional of the field a t  the ape r tu re  of the observing ins t rurner t ,  
The outcome u of the measu remen t  i s  compared with a decision leve l  u 0 ;  
if u > aO, the decision that  the  object is presen t  i s  made .  
As a s t e p  toward calculating the probability Q of detection, the d 
moment-generat ing function (m. g. f .  ) 
o f  "&e d i s t r i b u t i o n  of u w a s  d e r i v e d .  H e r e  $ i s  t h e  
s t r e n g t h  of  t h e  o b j e c t  a c t u a l l y  p r e s e n t  r e l a t i v e  t o  
that of the  object fo r  which the  detector was designed, and P (B)  i s  the 
dewlsity opera tor  of the tota l  field made  up of object light and background. 
9"Lvas  a s sumed  that  the product of the observation in te rva l  T and the band- 
width Vg of the object  light i s  l a rge ,  TW > > 1, a s  i s  normal ly  the ca se .  
H e r e  we compare  the detectabil i t ies of two in'coherent objects with 
a rectangular  and a Loren tz  spec t rum,  respect ively .  F o r  simplicity we  
a s s u m e  that  the  light f r o m  both has f i r s t - o r d e r  spat ia l  coherence a t  the  
2 1 
ape r tu re ;  the spa t ia l  fac tor  i s  3 = 1. It was shown tha t  when the tempora l  
spect rum is rectangular ,  the outcome u, suitably normal ized ,  i s  a randorn 
var iab le  with a Poisson distr ibution,  permit t ing the probabil i ty ad(@) of 
de.i;ection to  be ea s i l y  calculated a s  a function of f3 fo r  a fixed false-alarms 
pyobability Q 
0 
When the object  emi t s  light with a Loren tz  spec t rum,  
t h e  m. g. f .  of the suitably no rma l i zed  s ta t i s t i c  u i s  
3 
where  N i s  the  ave rage  to ta l  number  of photons rece ived  f r o m  the object 
s 
dur ing the observat ion in te rva l ,  and N = JfTW, with 0 
;J. = PLanck's constant  h/2n, R = t h e  cen t r a l  angular f requency of the object 
spec t rum,  t o  which w in  Eq .  (2)  i s  r e f e r r e d ,  K = Boltzmann 's  constant ,  
and = the effective absolute t empe ra tu r e  of the  background. It is 
a s s u m e d  that  < < 1, but with TW >> 1, N i s  of the  o r d e r  of 1. 0 
The m. g. f. in Eq. ( 3 )  cannot be inver ted analytically t o  obtain the 
probabi l i ty  densi ty  function (p. d.  f .  ) p(u; $ )  of the outcome u. Numer ica l  
ca lcula t ions  of the  probabil i ty Q ( $ )  of detection w e r e  t he r e fo re  undertaken.  d 
The r e su l t s  a r e  shown in F ig .  1 ,  whe re  the  probabil i ty Q i s  plotted v e r s u s  d 
the  s ignal- to-noise  r a t i o  D = N jN 'I2 fo r  N = 15 and two values of t he  
s 0 0 
f a l s e  - a l a r m  probabil i ty Q The equivalent  bandwidths of the two s p e c t r a  0' 
were  taken equal, W = w e  The object with the Lorentz spec t rmn has the 
lower probability of detection. Calculations for  N = 5 revealed the s a m e  0 
relationship. 
The detection probabilities f o r  the rectangular spec t rum were caleu- 
4 lated as  described previously. F o r  the Lorentz spec t rum two methods were 
esed. At low values of the signal-to-noise rat io  the Laplace t r ans fo rm 
y, 6 - S ;  9 )IS of the cumulative distribution of u was inverted by the method 
-a 
5 
zteepest descents to  obtain the approximate formula 
I I - 112 1 " I 1  
C ~ U P  @)I  exp C P  ( t ) ~  ( I +  @ (t) ~ @ " ( t )  I-']
and t i s  the root of the equation @'  (t) = 0 ,  pr imes  denoting differentiation, 
Here  $ i s  proportional to  N . The equation q(u - 0 )  = Q O  was solved by 
s 0' 
Newton's method to obtain the decision leve l  u The, probability of detec- 0' 
tion i s  then Qd = q(u0; P ). 
At la rge  values of the signal-to-noise rat io  the probability Q ( P )  d 
was calculated by writing it first a s  
if; the rec.iangular function. R(x)  was expanded in a s e r i e s  of Laguarrce 
functions, which was substituted into Eq. ( 7 j  and integrated t e r m  by t e r m  
t o  obtain a s e r i e s  whose t e r m s  involve the m. g. f .  p ( s ;  $ ) arid i t s  derivatives 
U 
svaiua.ted at a cer tain value of s. Twenty t e r m s  of the Laguerre  s e r i e s  were  
6 
use3 .  Details of the method a r e  given elsewhere.  F o r  Q < 0 .  3 ,  the d - 
:suits of the two methods agreed closely. 
Footnotes 
J. *,. 
This r e s e a r c h  was  supported by g ran t  NGR-05-009-079 of the Nat io~la1  
-;;ei.-mav'tf c s  and Space Adminis t ra t ion,  
1. C ,  W ,  He l s t rom,  "Detection of Incoherent Objects by a Quantum- 
Li-mited Opt ical  Sys tem,  " submitted to  J.  Opt. Soc. Am. The  notation 
of that  paper  i s  used h e r e .  
.r, C ,  W. He l s t rom,  J. Opt. Soc. Am.  ...%-G~ 59 (1969); s e e  Eq. (3. 8 ) .  
3, See  r e f e r ence  1 ,  Eq, (5.22).  
4, See r e f e r e n c e  1 ,  Section V. 
5, C ,  Doetsch,  ( ~ i r k h a k k e r  
Ver lag,  Base1  and Stut tgar t ,  1955), vol. 2 ,  ch. 3, g 5 ,  pp. 83-68. 
6 ,  C, W e  He l s t rom,  P r o c .  IEEE 57, (1969). 
Figu re  Caption 
' i ~ l ~ ~ i x : ~  1: Probability of detection Q vs .  s ignal-to-noise r a t i o  
d 
1/ 2 - % 
=I >! /bj fo r  fa l se  - a l a r m  probabi l i t ies  (2 = 10 
s 0 0 
- .I? 
znd 10 R: rectxi?gular spec t rum;  L: Loren tz  spec t run-L~ 



Quantum-limited detection of an incoherent object with a Lorentz spectrum

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