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Resolution of Ghost Imaging for Non-degenerate Spontaneous Parametric
Down Conversion
Morton H. Rubin and Yanhua Shih
Department of Physics, University of Maryland, Baltimore County,
Baltimore, Maryland 21250
We examine the resolution of a ghost imaging system for the case that the entangled photons are non-degenerate.
Only the signal photon illuminates the object. We consider two cases, one in which the imaging lens is in the same
arm as the object and on in which it is in the idler arm. In the first case the Airy disk is shown to depend on the
wavelength of the idler photon; however, because the magnification depends on the two wavelengths, the resolution of
a pair of point scatters on the object depends on the wavelength both photons. In cases in which the position of the
object can be controlled, the minimum resolvable distance can be optimized. In the case that the object is in the far
field the resolution only depends on the signal wavelength. In the second case the Airy disc depends on the
wavelength of both photon but the resolution depends on the signal photon. While the ghost imaging system has
flexiblity for certain applications, the resolution of the non-degenerate ghost imaging system does not offer
improvement over the classical illumination of the object.
1. Introduction
Ghost imaging with entangled photon pairs (biphotons) has been extensively discussed in the
literature [1,2]. Recently, the question of whether the resolution of ghost imaging is improved
using non-degenerate biphotons (biphotons with pairs of photons of different frequency) has been
raised. In this paper we consider the ghost imaging systems shown in Fig.1. A pump from a laser
is incident on a crystal that produces entangled photon pairs (biphotons). One of the photons,
called the signal photon, scatters from the object and is detected by detector A. A point detector B
in a CCD array located in the image plane detects the second photon, called the idler photon. The
signals from the two detectors go to a coincidence counter. Detector A is called a bucket detector
and collects all the scattered light incident on it. The image plane is determined by a Gaussian lens
formula, Eq. (13) and the Airy disk depends on the idler wavelength, Eq. (18). However, the
resolution of two points on the object depends on the signal wavelength, Eq. (19). The imaging
lens may be placed either in the object arm as shown in Fig. 1 or in the idler arm as shown in Fig.
3. We also present the results of a second case illustrated in Fig. 3.
The paper is organized as follows: first we formulate the coincident counting rate in terms of the
fields at the detectors, then we compute these fields, finally we compute the minimum transverse
distance on the target that can be resolved.
1
Detector A
Object
Lens
Ls
ds
Pump
Coincidence
Counter
Di
di
Point
Detector B
Figure 1
2. Coincident Counting Rate
The coincident counting rate may be written as
1
C   dt A dt B S(t B ,t A )  d 2 A BGAB
T
A
S is the coincident time window that vanishes unless 0 ≤ tB-tA < T,
detector,
 B is the area of the point detector,
(1)
 A is the area of the bucket
GAB  tr  EA() EB() EB() EA()   ,
(2)
 is the quantum mechanical state of the electromagnetic field on the output face of the crystal,
r
and, for j=A or B, E ()
 E () (rj ,t j ) is the positive frequency part of the electric field at the point
j
 
†
rj evaluated at time tj, and E ()
. We will ignore the polarization of the photons but
 E ()
j
j
adding it in is not difficult. The electric field operator is defined with dimension of the square root
of photon flux. The detailed expression for the fields at the detectors will be given in the next
section.
We shall assume that the output of the crystal is a sequence of non-overlapping biphotons so that
2
we can write GAB  A AB in terms of the biphoton amplitude
2
A AB  0 EB( ) EA( ) 
(3)
where 0 is the vacuum state and  is the biphoton state vector given explicitly in Appendix II.
3. Detector Fields
We begin by computing the field at the point detector B in terms of photon destruction operators
at the surface of the crystal.
r
r
r
EB( )   g( ,  ,  B , zi )ai ( k )ei t B
(4)
r
k
r
where ai (k ) is the destruction operator for an idler photon of wave number k at the output surface
of the crystal, g is the optical transfer function that will be computed using classical optics [3], and
r
r
 an ( k ), am† ( k  )    kr , kr  n,m .
where VQ is the quantization volume. In writing out the transfer function, it is convenient to
r
introduce coordinates r  zêz   where the unit vector êz points along the path through the center
of the optical system and  is a two dimensional vector perpendicular to the path. Using the thin
lens formula, and Eqs. (AI.1), (AI.9), and (AI.5) we find
r

r
r
i(  k B )g L

r
r 
k
1
1
Di
EB()   ( B , )   d 2 L ( L , k(  ))e
PL ( L )  
r
Di 
Di f
k

(5)
r
1 i(kzi  t B )
di
e
 ( , )ai ( k )
i Di
k
where zi=di+Di. The physics of Eq. (5) may be understood by noting that the term in square
r
r
brackets expresses the scattering of a plane wave with wave number k  k 2   2 êz   ; kêz  
r
incident on a thin lens into a wave with transverse wave number k  B / Di . The output wave then
propagates to the detector which in the paraxial approximation is determined by Eq. (AI.1). The
remaining terms in (5) arise from the propagation of a plane wave created at the crystal surface to
the input face of the lens. Using a similar analysis, with the object described by a transparency
r
function, t(  ) , it is easy to show that
r
EA()
 r
r
k 
k r i a g(k LAS  ) 
  ( A , )   d a ( a , )t( a )e

r
Ls 
Ls
k

r
1 i(kzs  t A )
d
e
 ( ,  s )as (k )
i  Ls
k
(6)
4. Biphoton Imaging
We are now in a position to compute the biphoton amplitude. For the case of interest, the system
is constructed so that we use the slowly varying amplitude approximation,
r
i(K z  t )
E ()
 e j j j j e()
j  A, B
(7)
j
j (  j ,t j ),
where e()
is slowly varying on the length and time scales, 1/Kj and 1/j. This allows us to make
j
the several simplifying approximations. For the signal field in arm A of the system we take
3
  s   s
r r
r
( s   s )2

k s 
  s2 êz   s  (K s  s  kz )êz
2
c
c

Ks  s
c
kz 
s s
c s

 s2
2K s
(8)
K
where we can drop the term kz . Frequency filtering ensures that |  s | / s is sufficiently small so
that terms containing it can be ignored. Similarly, spatial filtering ensures that  s / K s is also
sufficiently small to be ignored. In arm B we make the same approximations where the subscript
becomes i for idler. We now can write
1
A AB 
Exp[i(K s zs  K i zi  st A  i t B )]A 
s i Ls Di
(9)
r
r
r
r
r
r
A    gA (A ,  s , s )gB (B ,  i , i ) 0 as (ks )ai ( ki ) 
r r
ks ki
where g A and g B are slowly varying functions determined from Eqs. (5) and (6).
We take the simplest model for the biphoton. A plane wave pump of angular frequency  p and
wave vector k p êz propagates in crystal of length L, then in Appendix II we show that
r
r
r
r
0 as ( s ,  s )a( i ,  i )   i(2 )3  ( p   s  i   s   i ) ( s   i )sinc(kz L / 2)
(10)
r
r
 i(2 )3  ( s   i ) ( s   i )sinc( s Dsi L / 2)
1
1
 , Us (Ui) is the group
where  is a dimensionless constant defined in Eq.(AII.8), Dsi 
U s Ui
velocity of the signal (idler) photon inside the crystal. The filters are chosen so that s  i   p
and s ns (s )  i ni (i )  ck p where nj is the index of refraction of the crystal for j=s and i. With
the assumptions used here the temporal and the transverse terms factor and we have
r r
A    i  d s ei s AB sinc( s Dsi L / 2)  V ( A , B )
(11)
To compute V, we first do the integrals over the  ' s . Evaluating the integral over  i using (5)
r
and (6) gives  i   s so that using (AI.4) gives
r
r K
ds di ir s g( r L  r a )
1
2
d


(

,
 ))e

 (| L  a |, s )
s
s

K s Ki
is Rs
Rs
(12)
i
Rs  ds  di
s
where Rs is the optical path length from the lens to the object in the sense that k s Rs is the phase
change that a plane wave would acquire in traveling from the lens to the object. Now substituting
(13) into the equation for V gives
4
r r
V (A , B ) 
r r
r
1
K
K
1
1
 ( A , s ) ( B , i )  d 2 at( a ) ( a , K s (  ))eiK s A ga / Ls 
is Rs
Ls
Di
Ls Rs
r
 f
r
 
r
r
Ks
1 1 iK s ( Ras f21  DBi si )gL
2
d


(

,

K
(
PL (  L )
 L L Rs i Di  f ))e
Defining the imaging condition
i 1
1
1


s Rs Di f
gives
r r
1
K
K
V (A , B ) 
 (  A , s ) (  B , i ) 
is Rs
Ls
Di
r
r

a  B  s  r
1
1 iK s r A gr a / Ls
2
%
d

P
K
(

)
t(

)

(

,
K
(

))e
a
L
s
a
a
s


Rs Di i 
Ls Rs
(13)
(14)
r
r
where PL ( ) is the Fourier transform of PL (  ) . This is the general result that we shall use to
discuss the resolution and field of view of the system.
First, let us make the simplifying assumption that the lens aperture is infinite, so
r
r
that P( )  (2 )2  ( ) , consequently
r 2
r r 2
1
B
V ( A , B )  2 2 t( )
(15)
s Rs
m
Di i
depends on the ratio of the wavelengths. This is the result
Rs s
that we would get from geometrical optics if the indeces of refraction on the two sides of the lens
were different and i / s  ns / ni .
In order to give some more insight in to these results, we consider an unfolded version of Fig. 1
developed by David Klyshko. In the Klyshko picture the source is shown to emphasize that the
ideal phase matching condition corresponds to transverse wave number conservation that can be
represented as a ray passing through the system. The object distance has been weighted with an
effective index of refraction and we omit the beam expander. It is now a simple matter to use
s
Di
.
geometrical optics to obtain Eq. (13) and the magnification m  i 
so s Rs / i
where the magnification m 
5
This picture is not limited to geometrical optics.
To compute the counting rate we need to sum over the surface of the bucket detector assuming
that each point on the surface detects the intensity of the light incident on it,
2
r
r


r r 2
r
1



2
2
a
B
s
 d A | V (A , B ) |  s2 Rs2  d a P%L  K s ( Rs  Di i ) t(a ) .
Therefore, the coincident counting rate is
2
1
C   dt A dt B S(t B ,t A )  d s ei s AB sinc( s Dsi L / 2)
T
2
r
r


r
2
1



2
a
B
s
 B 2 2 2 2 2  d a P%

) t(a )
L  Ks (
s i Ls Rs Di
Rs Di i 

(16)
5. Resolution
To discuss the resolution we consider a target made up of two point scatterers, one located at the
origin and the other at the point a in the target plane,
r
r
r
r
t( a )  t o ( a )  t1 ( a  a) .
From Eq. (4.8) we have
r r
1
K
K
V (A , B ) 
 ( A , s ) ( B , i ) 
is Rs
Ls
Di
.
(17)
r
r
r
r r





2

a


1
1
iK

g
a
/
L
B
)  t1P%
 B s )  (a, K s (  ))e s A s 
t 0 P%
L (
L  Ks (
D

R
Di i 
Ls Rs

i
i
s


From the first term in the square brackets of Eq. (4.9), we see that the point spread function is
determined by Fourier transform of the lens aperture function. For a circular aperture the radius of
the Airy disk has a radius
6
B  x
i Di
r
(18)
RL
where RL is the radius of the aperture and PL ( L ) negligible if  L  2 x / RL , x ; 1.22 . Note
that the radius of the Airy disk is proportional t o the idler wavelength. This is the standard result,
as can be seen by taking Rs ? Di in Eq. (12) so that Di  f and  B  x i / NA where
NA  RL / f is the numerical aperture of the lens. Referring to Fig. 2, we see that this is the
same result we would obtain from classical optics. We now use the Rayleigh criterion to determine
the resolution of the two image points. The Rayleigh criterion is not the ideal criterion for
coherent scattering, that is, for the case in which there is a definite phase relation between t0 and t1
[3]; however, it is simple and illustrative. The image of the second term in (17) is assumed to lie
on the edge of the Airy disk of the first term, so
amin 
s B

Rs  x s Rs .
i Di
RL
(19)
We see that the resolution depends on the signal photon wavelength for the case
Rs  ds ? i di / s . For example, if s  1 m, RL  10cm x  1, and Ds  1  10km , we
get amin  1  10cm . In the opposite case Rs  i di / s ? ds , we find that amin is prortional to
the idler wavelength. Using the relation between the pump, signal and idler wavelengths in free
space, we can minimize (19) we find that
i
ds

s
di
Rs  ds  ds di 
amin  x
s
d
p s
(20)
s2 ds
 p RL
If this result is compared to the classical case of an object a distance ds from the imaging lens
illuminated with light of wavelength s we get amin  s acl /  p  acl .
Finally we record the results for the set-up shown in Fig. 2. The Gauss lens equation is given by
1
1
1


Ri Ds f
(21)
i
Ri  ds  Di
s
Ri
, the Airy disk radius is
Ds
R
B  x s i
RL
and the minimum resolvable distance based on the Rayleigh criterion is
D
amin  x s s .
RL
the magnification is m 
(22)
(23)
7
Note that the in this case the radius of the Airy disk can be optimized but the minimum resolvable
distance cannot.
Detector A
Object
Lens
Pump
Ls
Ds
ds
Coincidence
Counter
Di
Point
Detector B
Figure 3
6. Conclusions
We have computed the resolution of non-degenerate ghost imaging. The precise results depend
on the particular set-up and for the two set-ups discussed here we get dual results, that is we can
either optimize the resolution or the minimum size of the Airy disk. In the case where the object
plane can be controlled, by placing the object close to the output face of the down conversion
crystal the minimimum resolvable distance is given by Eq. (19).
Acknowledgements
We wish to thank Jianming Wen, Michael Fitelson, and Doyle Nichols for helpful conversations.
After this work was completed we learned that some conclusions in this paper had been
independently derived by KW Chan, MN O'Sullivan, OD Herrera, and RW Boyd, University of
Rochester, unpublished and S Vinjanampathy, JP Dowling, LSU, unpublished. We thank Jonathan
Dowling for this information. This work was supported in part by U.S. ARO MURI Grant
W911NF-05-1-0197 and Air Force Research Laboratory under contract FA8750-07-C-0201 as part
of DARPA's Quantum Sensors Program.
8
References
[1] T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V Sergienko, Phys. Rev. A 52, R3429
(1995), T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V Sergienko and Y. H.
Shih, Phys. Rev. A 53, 2804,(1996), M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V Sergienko,
Phys. Rev. A50, 5122 (1994).
[2] For a review of quantum imaging see Y. H. Shih, IEEE J. of Selected Topics in Quantum
Electronics, 9, 1455 (2003).
[3] Joseph Goodman, Introduction to Fourier Optics (Roberts and Company Pub., Greenwood
Village, CO, 2005).
[4] Morton H. Rubin, Phys. Rev. A 54, 5349 (1996).
[5] A. VanderLugt, Optical Signal Processing (John Wiley & Sons, Inc., New York, 1992).
Appendix I
This appendix is based upon [4] and [5].
In the paraxial approximation the propagation of a component of a monochromatic electric field
r
r
along the positive z-axis from a point rX  zêz  X to a point rY  (z  d)êz  Y is given by
r
r
r
r k
1 ikd
h ( Y   X , d) 
e  ( Y   X , )
(AI.1)
i d
d
r
where   ck  c kz2   2 , k  kz êz   , k  2 /  , and
 (, kP)  eikP /2 .
2
(AI.2)
The following results for Gaussians [V] are useful:
r r
r r
r
r
 (    , kP)   (  , kP) (  , kP)eikP  g  ,
(AI.3)
and the two-dimensional Fourier transform is given by
r r
2
1
2
i g
d


(

,
kP)e
 i  ( ,  ) .
(AI.4)

kP
kP
r
A diffraction limited optical system can be represented in terms of an aperture function w(  )
where  is a two-dimensional vector in the plane of the aperture. We introduce the optical transfer
r
function for a plane wave incident on the plane X with wave vector k  kz ê   which goes
through the optical system to the output plane I, a distance z away. Taking the aperture plane to be
a distance d from the X plane, and D=z-d, we find
r r
r
r
r
r
r
r
r
(AI.5)
g( , , I , z)   d 2 a  d 2 X h (I  a , D)w(a )h (a  X ,d)ei gX ,
It is now a simple matter to show that Eq. (AI.5) may be written as
r
r
r
r
1 ikz
k
I k
d
%
g( ,  ,  I , z) 
e  ( I , )w(  k , ) ( ,  )
i D
D
D D
k
where
r r
r
r
w( , kP)   d 2 a (a , kP)ei ga w(a ) .
If a thin lens of focal length f is in the plane of the aperture so
r
r
w(  )   ( , kF)PL (  )
r
where F=1/f and PL (  ) is the aperture function for the lens, then
r r
r
r
w( , kP)   d 2 a (a , k(P  F))ei ga PL (a )
(AI.6)
(AI.7)
(AI.8)
(AI.9)
which becomes the Fourier transform of the aperture function if P=F, i.e. f=D.
9
Appendix II
In this appendix the units of the E fields are electric fields. The calculation of the biphoton wave
function is based on first order perturbation theory using the Hamiltonian
r
r
r
(AII.1)
H(t)   d 3r  0  (2) EP (r,t)Es (r )Ei (r )
where E P is the pump field which is taken to be classical, Es and Ei are the signal and idler field
each of which is taken to be quantized fields, and  is the second order electric susceptibility of
the crystal [5,6]. The integral is taken over the crystal. We have suppressed the tensor indices
(2 )
(2 )
since we shall assume that the fields are linearly polarized so that   2  psi êp ês êi . It is
(2 )
psi
most convenient to work in the interaction picture. Inside the crystal we set
r r
r
i ( k gr  
E p (r,t)  E p e
r
Es (r,t)  Es( )  hc
p
E
( )
s
pt )
 cc
(AII.2)
r i ( kr grr  t )
h s

bs ( ks )e
r
k
2 0 ns2VQ
s
s
s
where cc means complex conjugate, hc means Hermitian conjugate, VQ is the quantization volume,
r
and as ( k ) is the destruction operator for a photon of wave number k with polarization ês . The
expression for the idler field is the same as that for the signal with s->i. Using the rotating wave
approximation,
L /2
H (t)   d 2   dz  0  (2 ) 
 L /2
h  s i
 EP ei (
r
r
k k 2 n n V
0 s i Q
s
  i   p )t
r
ei (
s
r r
 i )g
s i
h  s i
 i (
L  (2 ) E p  e
k k 2V n n

Q s i
kz  (k p  ksz  kiz )
 rr
s i
s
  i   p )t
e
 i ( ksz  kiz  k p )z
r
r
bs† ( ks )bi† ( ki )  hc 
r
r
bs† ( ks )bi† ( ki )AQr
r r
s  i , 0
sinc(
(AII.3)
kz L

)  hc 
2

where we have assumed that the cross section of the beam is large so that the integral over the
transverse coordinates give (Kronecker) delta functions multiplied by the quantization area, the
crystal has a length L, and Ep is real.
Using first order perturbation theory, the biphoton state vector is given by

i
 
 dtH (t) 0
h 
.
(AII.4)
 s i
kz L † r † r
(2)
 i 
2 L  E p ( s   i   p )AQr s r i , 0r sinc(
)bs ( ks )bi ( ki ) 0
r r
2VQ
2
ks ki
We now convert from discrete to continuous variables. For the signal
10

ks

VQ
(2 )3
3
 d ks 
VQ
(2 )3
2
 d  s  d s
VQ
dksz

d 2 s  d s
3 
d s
(2 ) c
dksz
1

d s us ( s )
where u is the group velocity of the field. A similar expression holds for the idler. We also have
r
r
us
bs ( ks ) 
bs ( s ,  s )
VQ
.
r
r
r
r
bs ( s ,  s ),bs† ( s ,  s )   (2 )3 ( s   s ) ( s   s )
Consequently,
  i

k L
r
r
r
r

d 2 s d s  d 2 s d s ( s   i ) ( s   i )sinc( z )bs† ( s ,  s )bi† ( i ,  i ) 0
3 
(2 )
2
1
 s i
 (2) E p L
2 ns niU sU i
(AII.5)
dkz dk
1
;

, U  u()
d d u( )
dK s
dK i
 i
d s
di
where we have introduced the dimensionless coupling constant . For the case of interest to us we
can replace the frequency  ;  for the signal and idler and, similarly u( ) ; u()  U . The
phase matching condition is
1
k p  K s  K i   p n p ( p )   s ns ( s )  i n(i )  0
(AII.6)
c
and, consequently,
1
1
kz   s (  ) .
(AII.7)
U s Ui
We have ignored walk-off.
Finally, in order to ensure conservation of flux at the surface of the crystal we require
r
r
Us bs ( s , s )  Ts cas ( s , s )
where Ts is the transmission coefficient of the surface. We ignore the vacuum term since it does
not contribute to the final result. Introducing the dimensionless coupling
kz  k p  ksz  kiz ; (k p  K s  K i )   s

    TsTi

c2
U sU i
(AII.8)
we get the result given in Eq. (10).
9 ' :
48
11