Two-photon geometric optics T. B. Pittman, D. V. Strekalov, D. N. Klyshko,* M. H. Rubin, A. V. Sergienko, and Y. H. Shih Department of Physics, University of Maryland Baltimore County, Baltimore, Maryland 21228 ~Received 10 October 1995! We report two-photon correlation experiments using spontaneous parametric down-conversion under a se- vere manipulation of the input pump field. Considering the case of passing the laser beam through a focusing lens before the down-conversion crystal, theoretical calculations and a series of imaging experiments demon- strate two-photon geometric optics effects. In particular, the imaging in coincidence counts of an aperture placed in one of the down-conversion beams is found to be the analog of a simple spherical mirror system, which displays a ‘‘vacuum dispersion’’ effect in that the object and image distances are wavelength weighted. PACS number~s!: 42.50.Dv, 03.65.Bz I. INTRODUCTION When a laser pump beam is incident on a noncentrosym- metric crystal, the nonlinear process of spontaneous paramet- ric down-conversion ~SPDC! of pump photons into pairs of correlated photons may occur @1#. Recently, the inherently quantum mechanical two-photon states produced in SPDC have been used in a number of interesting experiments. The well known correlations @2–4# between the signal and idler photons constituting a down-converted pair have been clev- erly used in a variety of situations ranging from the very practical matter of absolute calibration of single-photon de- tectors @5–9#, to the intuition-challenging studies of photon tunneling time @10# and interaction-free measurement @11#. Furthermore, a variety of two-photon entangled states have been produced by taking advantage of the polarization cor- relations of the signal and idler photons, as well as the fre- quency and momentum correlations expressed through the well known phase-matching conditions @1,12#. These en- tangled states have proved to be an extremely useful experi- mental source for investigating the concepts of reality and locality found in the Einstein-Podolsky-Rosen ~EPR! gedankenexperiment @13#. The convenience of SPDC has led to many experiments ~see, for example, @14–22#! concerning tests of Bell’s inequalities @23# and has no doubt influenced the thinking in recent proposals and theoretical papers con- cerning some of the foundations of quantum mechanics. There have been numerous other observations of nonclas- sical states @24–26# and two-photon interference ~see, for example, @27–52#! and SPDC has even been proposed and used in the relatively new field of quantum cryptography @53–55#. SPDC has also been found useful in the character- ization of optical materials @56#, and various communication schemes @57#. In general, most of the above experiments have used pin- holes or other means to subselect certain transverse spatial modes of the down-conversion spectrum. Very roughly speaking, in these configurations the SPDC process has been thought of as an intense plane-wave pump input into a simple ‘‘black box’’ source whose output is two correlated photons in well defined directions. The interesting physics in these setups occurs ‘‘down stream,’’ so to speak, where the photons are manipulated with polarizers, interferometers, de- tectors, etc. to see the desired effects. Recently, however, there have been several good studies @58–60# of the spatial distributions of the down-converted photons, and their correlations, with respect to several pa- rameters affecting the interaction inside the crystal. In other words, still considering a plane-wave pump input, it is useful to examine the rigorous validity of the phase-matching con- ditions for various pump spectral widths, crystal lengths, etc. In fact, the transverse spatial coherence properties of the down-converted radiation have been used to observe two- photon physical optics by means of interference and diffrac- tion in the coincidence counting rate when slits are intro- duced into the down-converted beams @61–63#. Therefore one of the remaining considerations concerning SPDC is what will happen to the correlations of the signal and idler photons if there is a controlled manipulation of the pump beam. In this paper, we pass the pump through a fo- cusing lens so that the wave fronts entering the crystal can no longer be taken as plane-wave approximations. Rather, the wave-vector distribution allows the pump to be more accu- rately thought of as having spherical wave fronts. What we find, through a theoretical model and a series of imaging experiments, is a dramatic restructuring of the momentum correlations that can be interpreted through a simple model based on geometric optics. In particular, we observe two- photon effects that are analogous to standard imaging with a spherical mirror. The basic idea of the imaging experiment is shown in Fig. 1, which is a topologically equivalent cartoon of the actual experimental setup. The plane wave fronts of the pump beam are weakly focused through the crystal, producing pairs of diverging signal and idler photons. The signal beam travels a distance Z 1 and encounters a detailed aperture in front of a large detector, D 1 . Because it is spatially insensitive, there is no image or shadow of the aperture recorded by D 1 . The idler beam travels a distance Z 2 and is met by a very tiny detector D 2 , which is scanned around in the transverse plane. Therefore, by recording coincidence counts as a func- tion of the transverse spatial coordinates of D 2 , we see an image of the aperture placed in the signal beam, even though * Permanent address: Quantum Radiophysics Division, Depart- ment of Physics, Moscow State University, Moscow 119899, Rus- sia. PHYSICAL REVIEW A APRIL 1996 VOLUME 53, NUMBER 4 53 1050-2947/96/53~4!/2804~12!/$10.00 2804 © 1996 The American Physical Society