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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 )ei 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 ir s g( r L r a ) 1 2 d ( , ))e (| L a |, s ) s s K s Ki is 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 ( ))eiK s A ga / Ls is Rs Ls Di Ls Rs r f r r r Ks 1 1 iK s ( Ras f21 DBi si )gL 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 ) is Rs Ls Di r r a B s r 1 1 iK s r A gr 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 ) is 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)eikP 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 gX , 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 ga 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 ga 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 ei ( 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 ) rr 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 )AQr 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 )AQr 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 di 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