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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 3, Iss. 2 — Jul. 20, 1998
  • pp: 81–88
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Spatial correlations of spontaneously down-converted photon pairs detected with a single-photon-sensitive CCD camera

Bradley M. Jost, Alexander V. Sergienko, Ayman F. Abouraddy, Bahaa E. A. Saleh, and Malvin C. Teich  »View Author Affiliations


Optics Express, Vol. 3, Issue 2, pp. 81-88 (1998)
http://dx.doi.org/10.1364/OE.3.000081


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Abstract

A single-photon-sensitive intensified charge-coupled-device (ICCD) camera has been used to simultaneously detect, over a broad area, degenerate and nondegenerate photon pairs generated by the quantum-optical process of spontaneous parametric down-conversion. We have developed a new method for determining the quantum fourth-order correlations in spatially extended detection systems such as this one. Our technique reveals the expected phase-matching-induced spatial correlations in a 2-f Fourier-transform system.

© Optical Society of America

1. Introduction

Photon pairs generated by the nonlinear quantum-optical process of spontaneous parametric down-conversion (SPDC) [1

1. D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,”Phys. Rev. Lett. 25, 84–87 (1970). [CrossRef]

] exhibit remarkable time and space correlations. The wavelengths and emission angles of the pairs, and hence the photon correlations, are governed by the interaction geometry and the phase-matching conditions within the nonlinear material via conservation of energy and momentum. The consequent quantum fourth-order temporal- and spatial-coherence properties of these twin beams are most simply characterized by an entanglement time Te and an entanglement area Ae (or equivalent polar- and azimuthal-entanglement angles) [2–7

2. D. N. Klyshko, “Transverse photon bunching and two-photon processes in the field of para-metrically scattered light,” Zh. Eksp. Teor. Phys. 83, 1313–1323 (1982) [Sov. Phys. JETP 56, 753–758 (1982)].

]. The entanglement time of the pairs has been determined using a variety of interferometric techniques [8

8. J. Peřina, Z. Hradil, and B. Jurčo, Quantum Optics and Fundamentals of Physics (Kluwer, Boston, 1994). [CrossRef]

, 9

9. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, New York, 1995), Ch. 22.

].

The spatial correlations were first observed using photomultiplier tubes behind pinholes that selected conjugate pair locations in a down-converted beam [10

10. A. A. Malygin, A. N. Penin, and A. V. Sergienko, “Spatiotemporal grouping of photons in spontaneous parametric scattering of light,” Dokl. Akad. Nauk SSSR 281, 308–313 (1985) [Sov. Phys. Dokl. 30, 227–229 (1985)].

]. More recently, using a pair of discrete detectors, the entanglement angles of degenerate photon pairs were measured [4

4. A. Joobeur, B. E. A. Saleh, T. S. Larchuk, and M. C. Teich, “Coherence properties of entangled light beams generated by parametric down-conversion: Theory and experiment,”Phys. Rev. A 53, 4360–4371 (1996). [CrossRef] [PubMed]

]. This paper reports a measurement of the overall spatial pair correlations using an intensified charge-coupled-device (ICCD) camera that simultaneously detects degenerate and nondegenerate photons from the down-converted beam over the broad area of the CCD array. Our results can be viewed as the spatial analog of the timing coincidence measurements that are found in the SPDC literature.

2. Experiments

We used an ICCD camera, which is similar to that shown in Fig. 1, as the photon-detection device throughout our experiments. It has as its first surface a photocathode from which an incident photon induces the ejection of an an electron; such electrons can be produced over the entire surface of the photocathode. The electrons are ejected into a microchannel-plate electron-multiplier stage where they are amplified in number. The resulting electron bunches impinge upon a phosphor screen that emits photons in response, thus providing intensification of the incident light distribution [11

11. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Ch. 16. [CrossRef]

]. These photons are transmitted into a fiber-optic bundle that is attached to the CCD-array input. Hence, each incident photon makes many photons available for detection by the CCD array. Of course, the process is not ideal—the quantum efficiency (electrons/photon) of the photocathode of our ICCD camera is η ~ 25% [12

12. Princeton Instruments Catalog of High Performance Digital CCD Cameras: September, 1995, Princeton Instruments, Inc., Trenton, NJ 08619; we used a slow scan model ICCD-576-G/RB-E detector with a third-generation intensifier for all experiments.

]. Further specifications of the ICCD camera are as follows: the array size is 576 × 384 pixels (12.7 × 8.5 mm; each pixel is 22 × 22 μm2), the dynamic range is 16 bits/pixel, the dark current at -45°C is > 16 electrons/(pixel s) (we operated our ICCD at -35°C), and the amplifier gain allows sensitivities to 100 counts (analog-to-digital units or digitized bits) per photoelectron.

Figure 1. The main components of an ICCD camera showing the photocathode front surface, microchannel-plate amplifier, fiber bundle, and CCD wafer package [12].

Down-converted light was generated by propagating an unpolarized multimode 40-mW He–Cd laser pump beam with a wavelength of 325 nm through a nonlinear SPDC crystal. As shown in the basic setup illustrated in Fig. 2, the laser beam was first directed through a prism to separate undesired plasma tube emissions at other wavelengths, which were blocked with an iris. A 650 ± 35 nm bandpass filter was used to reduce the irradiance of the residual pump light that remained mixed with the down-converted light and, after passing through a lens, the output photons were detected with the ICCD camera. The intensifier gain was set to its maximum value. Two different type-I (extraordinary-polarized pump, ordinary-polarized signal and idler) nonlinear down-conversion crystals were used in our experiments as described below.

Figure 2. Illustration of the basic spontaneous parametric down-conversion optical system.

We investigated two simple geometries. The first was a one-to-one 4-f imaging system (Fig. 2 with d = 2f) in which a 50-mm diameter lens with focal length f = 75 mm was used. Ideally, for a very thin crystal, even though they are emitted in different directions the down-converted signal and idler pair will appear at the same point at the surface of the ICCD since they are emitted from the same point within the crystal. For a thick crystal, the limited depth of field of the imaging system will lead to a range of pair separations at the ICCD surface. We used a 15 × 15 × 25 mm3 LiIO3 crystal (cut angles θc ≈ 59° and ϕc ≈ 45°) tilted ≈ 1° relative to the optic axis for noncollinear degenerate down-conversion in this experiment. An example of an image from this system is shown in Fig. 3(a). The region of larger spot density relative to that of the rest of the picture comprises the imaged down-converted light emitted from the crystal. It is difficult in this case to separate the residual pump photons from the down-converted photons. Autocorrelation analyses of such images did not provide useful results, possibly because of the spatial overlap of the undesired residual pump light with the desired down-converted light and because the thick crystal produced too wide a range of photon-pair separations.

Figure 3. Contrast-adjusted negative pictures obtained from (a) a 4-f one-to-one imaging system (4-μs exposure time) and (b) a 2-f angle-to-position optical system (100-μs exposure time). In addition to the features sought, there is a spatially random background-noise distribution.

A pair of pictures that appears similar to that shown in Fig. 3 was recently obtained in an investigation of SPDC classical spatial-irradiance patterns (Fig. 5 in Ref. [14

14. A. Gatti, L. A. Lugiato, G.-L. Oppo, R. Martin, P. Di Trapani, and A. Berzanskis, “From quantum to classical images,” Opt. Express 1, 21–30 (1997). http://epubs.osa.org/oearchive/source/1968.htm [CrossRef] [PubMed]

]). Those pictures, however, comprise classical (relatively high irradiance) measurements of only the signal beam. In distinction, our pictures contain combined signal and idler detections as pairs of correlated single-photon events.

3. Analysis

Our goal is to experimentally determine the form of the fourth-order spatial correlation (photon coincidence) of the down-converted light. That is, we desire to determine the uncertainties of the radius δr and angle δϕ from ideal, as shown in Fig. 4. The SPDC photon-emission wavelengths and angles are determined by the phase-matching conditions: ωp = ωs + ωi and k p = k s + k i where the pump, signal, and idler frequencies and wave vectors are ωp, ωs, ωi and k p, k s, k i, respectively. In the ideal case of an infinitely extended crystal and pump beam, each down-converted photon has a partner whose emission angle can be exactly determined.

Figure 4. Illustration of the spatial relationship between the signal and idler photons in a 2-f angle-to-position optical system. The signal and idler photon-detection points are specified by their radii rs,i and angles ϕs,i. In practice, an area of locations Ae is possible for the idler because of uncertainty in the radius δr and the angle δϕ.
Figure 5. Theoretical continuous-plane-wave pump phase-matching as a function of signal and idler radii (in ICCD camera pixels measured from the center of the rings) for a BBO crystal with cut angles θc = 36.2°, ϕc = 90°; tilted 1° with respect to the optic axis; and with a 325-nm pump wavelength. Results are shown for ideally pumped down-conversion to wavelengths in the range 600–700 nm for an infinitely long crystal (dashed curve) and for a 1-mm long crystal (solid curves defining a boundary within which pairs can be phase-matched).

In a system comprising finite crystals and beams, the correspondence between the signal- and idler-photon spatial locations is blurred because there are uncertainties in the energies and the transverse and longitudinal wave vectors [3–7

3. A. Joobeur, B. E. A. Saleh, and M. C. Teich, “Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion,” Phys. Rev. A 50, 3349–3361 (1994). [CrossRef] [PubMed]

] (see Fig. 5, for example). The principal origins of the spread in the spatial- and temporal-correlation functions are the finite crystal size, both transversely (when the crystal cross-section is smaller than that of the pump beam) and longitudinally, and the transverse structure of the pump beam. A less important contribution arises from the finite pump bandwidth. Yet another consideration is diffraction in the optical system. All of these factors contribute to the uncertainty in the location of the idler photon over a region defined as the entanglement area Ae, as shown in Fig. 4. In Fig. 5, we provide the theoretically expected ideal phase-matching curve (dashed) as well as the spread in possible values of the radii that arises from the longitudinal wavevector uncertainty associated with a finite crystal length lc: δrs,i ∝ 1/[kplcsin(ars,i)] where a is a proportionality constant associated with the SPDC photon-emission angles [15

15. D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach, New York, 1988), Ch. 6.

].

In our efforts to develop a feasible correlation analysis technique involving the 2-f configuration, we first considered an idealized situation in which a signal photon occurring at angle ϕs and radius rs is matched by one and only one idler photon at ϕi = ϕs + π and ri (see Fig. 4). In the degenerate case, the signal and idler wavelengths are equal and ri = rs. Suppose we observe the down-converted light with a noiseless CCD camera that detects each incident photon such that a single corresponding infinitesimal pixel is excited. If we had full knowledge of the remainder of the optical system, and if we were able to perfectly isolate the down-converted light from the residual pump light, we could use the phase-matching conditions to determine the exact position and wavelength of each idler photon that is associated with each detected (signal) photon. The correlation analysis of experimentally obtained data is straightforward in this case: one need simply compare each pixel with its diametrically opposed pixel along with its neighbors. However, this simple procedure cannot be gainfully deployed in non-ideal systems.

In practice, the arrivals of the photons are detected by intercepting the down-converted rings with an ICCD camera that amplifies the individual photons in such a way that each incident photon typically excites a small patch of pixels with various levels of illumination. Moreover, minor distortions of the rings are introduced by the optical system and the spatial discretization of the microchannel-plate amplifier, the fiber-optic coupler, and the CCD array. We initially attempted to use the idealized correlation analysis described above but failed to produce meaningful results primarily because of two practical difficulties: the center of the down-converted photon annulus is difficult to precisely identify (and is not very useful for full-ring analysis if the rings are distorted), and errors are introduced by the registration (mapping) of the rings in translating from a rectilinear (pixel) to a polar coordinate system. These problems introduce uncertainties into the analysis and degrade our ability to observe the correlations. Consequently, we devised a technique that provides a more robust measure of the correlation by limiting our angular ranges to small diametrically opposed sectors and comparing small patches instead of single pixels within these sectors.

Figure 6. Illustration of the mapping and correlation analysis scheme with the 2-f optical system. A pair of x’s mark examples of diametrically opposed patches with the same radii.

Our correlation analysis scheme is illustrated in Fig. 6. We first determined the center of the annulus of the down-converted light (within ≈ ±2 pixels), which defines the center of a polar coordinate system. We then chose a 17° sector and labeled it the signal sector. The intersection of the sector with the down-conversion annulus defined an annular segment. This was subdivided into seventeen subsegments 1° wide; each of these was further subdivided into patches three pixels thick along the radial direction. Thus, we obtained a set of concentric annular regions each of which contained seventeen patches.

The radius and polar angle identified each such patch. All of the values of the rectilinear pixels in each patch were summed to provide a set of numbers within our polar coordinate system. A radius-dependent threshold, the level of which was empirically determined by incremental adjustment and testing with the images, was implemented at this stage to reduce both background noise and effects due to the patch-size dependence on radius.

The cross-correlation was obtained by comparing the numbers associated with the set of seventeen signal-polar patches lying at a given radius with those associated with an equal number of idler-polar patches (obtained in the same manner) located at an arbitrary radius but at a complementary angle (the annular regions comprising the sets of seventeen patches are therefore opposite one another). This was repeated for annular regions that were symmetrically distributed about the horizontal and vertical axes, and at ±45° to them. Results from each of the four analyses were normalized to their own maximum and then averaged to produce a plot that displays the radial dependence of the correlation function. The correlations between non-diametrically opposed segments were subsequently calculated to determine the angular dependence of the correlation function. Standard correlation measurements were used.

Figure 7. Experimental spatial correlation (color plot) as a function of the signal and idler radii (1 pixel corresponds to an angular spread ≈ 0.021° in the radial direction) obtained from an analysis of 150 images taken with a 2-f system. The radii begin at slightly less than the minimum detected down-conversion radius and extend slightly beyond the maximum detected down-conversion radius as they appear in Fig. 3(b). Two ideal theoretical phase-matching curves for type-I BBO with a cut angle ϕc = 90° and a 325-nm pump beam incident on the crystal at an angle of 1° are provided: the θc = 36.2° result as in Fig. 5 (dashed curve) and a fit with θc = 36.7° and a -2° shift in the down-conversion angles (solid curve).

The result that obtains from an analysis of 150 2-f-system pictures taken with the BBO crystal and a 100-μs exposure time (we estimate that each such picture contains ~ 10–100 matched photon-pair detections) is shown in Fig. 7. The average correlation as a function of the signal and idler radii is shown in false color. The correlations in this figure were calculated as described above for each picture, the 150 results were averaged, and each data point was further averaged over its nearest neighbor’s values to smooth the results. The measured correlation function forms a “V”-like region tilted 45° with respect to vertical. The correlations are relative values; thus, for example, a signal-idler radii pair with correlation equal to 0.8 is eight times more likely to have matched illumination in the corresponding patches than a pair with correlation equal to 0.1. Points along the diagonal that extends from the origin correspond to the degenerate case.

In Fig. 7 we have also plotted curves corresponding to the theoretically expected locations of high correlation obtained under ideal SPDC phase-matching conditions. Two plots are provided. The first (dashed curve) is the plot from Fig. 5 which was obtained using an estimate of the crystal parameters and the geometry of our experimental configuration. The second (solid curve) is a fitted curve in which the crystal-cut angle and the deviation of the down-conversion detection angle from ideal provide two adjustable parameters. Our experimentally obtained regions of high correlation lie within the vicinity of both theoretical curves. We note that the parameters of the fitted curve, which provides a better match, might be obtained from more accurate evaluations of the refractive indices of the down-conversion crystal and the crystal cut. The measurement is generally well matched with both theoretical results yet we anticipate that further refinements in our experimental measurement and analysis techniques will provide even better correspondence.

It is of interest to compare our results with those obtained using a pair of discrete detectors in a prior investigation [4

4. A. Joobeur, B. E. A. Saleh, T. S. Larchuk, and M. C. Teich, “Coherence properties of entangled light beams generated by parametric down-conversion: Theory and experiment,”Phys. Rev. A 53, 4360–4371 (1996). [CrossRef] [PubMed]

]. Our measured width at degeneracy (along the diagonal) is δr ≈ 6 pixels which corresponds to 0.126° (2.2 mrad). To measure how the correlations were affected when the diametric opposition requirement was relaxed, we repeated the analysis using incremental 1° steps around 180°. We found that there is a very rapid dimunition of the high correlation regions. From our analysis, we determined that δϕ < 1° (17 mrad). The values of δϕ and δr obtained here are several times larger than those obtained in the earlier investigation [4

4. A. Joobeur, B. E. A. Saleh, T. S. Larchuk, and M. C. Teich, “Coherence properties of entangled light beams generated by parametric down-conversion: Theory and experiment,”Phys. Rev. A 53, 4360–4371 (1996). [CrossRef] [PubMed]

], most likely because we used a crystal with one-tenth the length and configured for one-tenth the degenerate photon-emission angle.

4. Discussion

We have described a method by means of which the quantum spatial-pairing correlations of both degenerate- and nondegenerate-SPDC photons can be measured over a broad spatial area using a single-photon-sensitive ICCD camera. This research is supportive of the development of entangled-photon imaging systems [16–21

16. A. V. Belinskii and D. N. Klyshko, “Two-photon optics: diffraction, holography, and transformation of two-dimensional signals,” Zh. Eksp. Teor. Fiz. 105, 487–493 (1994) [JETP 78, 259–262 (1994)].

]. Such systems provide some interesting capabilities. For example, the paired nature of the entangled photons means that if one photon is detected, there is also a partner photon at a corresponding position and time. Suppose one member of each pair is used to illuminate an object and the other member of each pair is simply detected. Knowledge of such a detection informs us that there should also be a photon that either passed through or was scattered by the object. This information can be used along with coincidence detection to significantly reduce background, detector, and quantum noise in the image, relative to conventional techniques, potentially to sub-shot-noise levels [22

22. E. Jakeman and J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmission measurements,” Opt. Commun. 59, 219–223 (1986). [CrossRef]

]. Furthermore, the signal and idler paths are functionally equivalent in terms of their optical properties [16

16. A. V. Belinskii and D. N. Klyshko, “Two-photon optics: diffraction, holography, and transformation of two-dimensional signals,” Zh. Eksp. Teor. Fiz. 105, 487–493 (1994) [JETP 78, 259–262 (1994)].

, 21

21. B. E. A. Saleh, S. Popescu, and M. C. Teich, “Generalized entangled-photon imaging,” in Proceedings of the IEEE LEOS 1996 Annual Meeting (Boston, MA, 1996), pp. 362–363.

]. Consequently, non-absorptive optical elements can be moved from the signal path to the idler path and vice versa, without affecting the overall imaging properties of the system. ICCD cameras are therefore viable compact alternatives to scanned small-area detectors in such experimental configurations.

We thank T. Perng for assistance with our numerical simulations. This work was supported in part by the Boston University Photonics Center.

References

1.

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,”Phys. Rev. Lett. 25, 84–87 (1970). [CrossRef]

2.

D. N. Klyshko, “Transverse photon bunching and two-photon processes in the field of para-metrically scattered light,” Zh. Eksp. Teor. Phys. 83, 1313–1323 (1982) [Sov. Phys. JETP 56, 753–758 (1982)].

3.

A. Joobeur, B. E. A. Saleh, and M. C. Teich, “Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion,” Phys. Rev. A 50, 3349–3361 (1994). [CrossRef] [PubMed]

4.

A. Joobeur, B. E. A. Saleh, T. S. Larchuk, and M. C. Teich, “Coherence properties of entangled light beams generated by parametric down-conversion: Theory and experiment,”Phys. Rev. A 53, 4360–4371 (1996). [CrossRef] [PubMed]

5.

M. H. Rubin, “Transverse correlation in optical spontaneous parametric down-conversion,”Phys. Rev. A 54, 5349–5360 (1996). [CrossRef] [PubMed]

6.

P. H. Souto Ribeiro, “Partial coherence with twin photons,” Phys. Rev. A 56, 4111–4117 (1997). [CrossRef]

7.

B. E. A. Saleh, A. Joobeur, and M. C. Teich, “Spatial effects in two- and four-beam interference of partially entangled biphotons,” Phys. Rev. A 57, 3991–4003 (1998). [CrossRef]

8.

J. Peřina, Z. Hradil, and B. Jurčo, Quantum Optics and Fundamentals of Physics (Kluwer, Boston, 1994). [CrossRef]

9.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, New York, 1995), Ch. 22.

10.

A. A. Malygin, A. N. Penin, and A. V. Sergienko, “Spatiotemporal grouping of photons in spontaneous parametric scattering of light,” Dokl. Akad. Nauk SSSR 281, 308–313 (1985) [Sov. Phys. Dokl. 30, 227–229 (1985)].

11.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Ch. 16. [CrossRef]

12.

Princeton Instruments Catalog of High Performance Digital CCD Cameras: September, 1995, Princeton Instruments, Inc., Trenton, NJ 08619; we used a slow scan model ICCD-576-G/RB-E detector with a third-generation intensifier for all experiments.

13.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Ch. 4. [CrossRef]

14.

A. Gatti, L. A. Lugiato, G.-L. Oppo, R. Martin, P. Di Trapani, and A. Berzanskis, “From quantum to classical images,” Opt. Express 1, 21–30 (1997). http://epubs.osa.org/oearchive/source/1968.htm [CrossRef] [PubMed]

15.

D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach, New York, 1988), Ch. 6.

16.

A. V. Belinskii and D. N. Klyshko, “Two-photon optics: diffraction, holography, and transformation of two-dimensional signals,” Zh. Eksp. Teor. Fiz. 105, 487–493 (1994) [JETP 78, 259–262 (1994)].

17.

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ‘ghost’ interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995). [CrossRef] [PubMed]

18.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995). [CrossRef] [PubMed]

19.

T. B. Pittman, A. V. Sergienko, D. V. Strekalov, Y. H. Shih, M. H. Rubin, and D. N. Klyshko, “Two-photon geometric optics,” Phys. Rev. A 53, 2804–2815 (1996). [CrossRef] [PubMed]

20.

P. H. Souto Ribeiro and G. A. Barbosa, “Direct and ghost interference in double-slit experiments with coincidence measurements,” Phys. Rev. A 54, 3489–3492 (1996). [CrossRef] [PubMed]

21.

B. E. A. Saleh, S. Popescu, and M. C. Teich, “Generalized entangled-photon imaging,” in Proceedings of the IEEE LEOS 1996 Annual Meeting (Boston, MA, 1996), pp. 362–363.

22.

E. Jakeman and J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmission measurements,” Opt. Commun. 59, 219–223 (1986). [CrossRef]

OCIS Codes
(190.4180) Nonlinear optics : Multiphoton processes
(190.4410) Nonlinear optics : Nonlinear optics, parametric processes
(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in
(270.4180) Quantum optics : Multiphoton processes
(270.5290) Quantum optics : Photon statistics

ToC Category:
Focus Issue: Quantum structures in nonlinear optics and atomic physics

History
Original Manuscript: April 8, 1998
Published: July 20, 1998

Citation
Bradley Jost, Alexander Sergienko, Ayman Abouraddy, Bahaa Saleh, and Malvin Teich, "Spatial correlations of spontaneously down-converted photon pairs detected with a single-photon-sensitive CCD camera," Opt. Express 3, 81-88 (1998)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-2-81


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References

  1. D. C. Burnham and D. L. Weinberg, "Observation of simultaneity in parametric production of optical photon pairs," Phys. Rev. Lett. 25, 84-87 (1970). [CrossRef]
  2. D. N. Klyshko, "Transverse photon bunching and two-photon processes in the field of parametrically scattered light," Zh. Eksp. Teor. Fiz. 83, 1313-1323 (1982) [Sov. Phys. JETP 56, 753-758 (1982)].
  3. A. Joobeur, B. E. A. Saleh, and M. C. Teich, "Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion," Phys. Rev. A 50, 3349-3361 (1994). [CrossRef] [PubMed]
  4. A. Joobeur, B. E. A. Saleh, T. S. Larchuk, and M. C. Teich, "Coherence properties of entangled light beams generated by parametric down-conversion: Theory and experiment,"Phys. Rev. A 53, 4360-4371 (1996). [CrossRef] [PubMed]
  5. M. H. Rubin, "Transverse correlation in optical spontaneous parametric down-conversion," Phys. Rev. A 54, 5349-5360 (1996). [CrossRef] [PubMed]
  6. B. E. A. Saleh, A. Joobeur, and M. C. Teich, "Spatial effects in two- and four-beam interference of partially entangled biphotons," Phys. Rev. A (to appear, May 1998). [CrossRef]
  7. P. H. Souto Ribeiro, "Partial coherence with twin photons," Phys. Rev. A 56, 4111-4117 (1997). [CrossRef]
  8. J. Perina, Z. Hradil, and B. Jurco, Quantum Optics and Fundamentals of Physics, (Kluwer, Boston, 1994). [CrossRef]
  9. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge, New York, 1995), Ch. 22.
  10. A. A. Malygin, A. N. Penin, and A. V. Sergienko, "Spatiotemporal grouping of photons in spontaneous parametric scattering of light," Dokl. Akad. Nauk SSSR 281, 308-313 (1985) [Sov. Phys. Dokl. 30, 227-229 (1985)].
  11. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (Wiley, New York, 1991), Ch. 16. [CrossRef]
  12. Princeton Instruments Catalog of High Performance Digital CCD Cameras: September, 1995, (Princeton Instruments, Inc., Trenton, NJ 08619). We used a slow scan model ICCD-576-G/RB-E detector with a third-generation intensifier for all experiments.
  13. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Ch. 4. [CrossRef]
  14. A. Gatti, L. A. Lugiato, G.-L. Oppo, R. Martin, P. Di Trapani, and A. Berzanskis, "From quantum to classical images," Opt. Express 1, 21-30 (1997). http://epubs.osa.org/oedocs/1968.pdf [CrossRef] [PubMed]
  15. D. N. Klyshko, Photons and Nonlinear Optics, (Gordon and Breach, New York, 1988), Ch. 6.
  16. A. V. Belinskii and D. N. Klyshko, "Two-photon optics: diraction, holography, and transformation of two-dimensional signals," Zh. Eksp. Teor. Fiz. 105, 487-493 (1994) [JETP 78, 259-262 (1994)].
  17. D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, "Observation of two-photon `ghost' interference and diraction," Phys. Rev. Lett. 74, 3600-3603 (1995). [CrossRef] [PubMed]
  18. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. Sergienko, "Optical imaging by means of two-photon quantum entanglement," Phys. Rev. A 52, R3429-R3432 (1995). [CrossRef] [PubMed]
  19. T. B. Pittman, A. V. Sergienko, D. V. Strekalov, Y. H. Shih, M. H. Rubin and D. N. Klyshko, "Two-photon geometric optics," Phys. Rev. A 53, 2804-2815 (1996). [CrossRef] [PubMed]
  20. P. H. Souto Ribeiro and G. A. Barbosa, "Direct and ghost interference in double-slit experiments with coincidence measurements," Phys. Rev. A 54, 3489-3492 (1996). [CrossRef] [PubMed]
  21. B. E. A. Saleh, S. Popescu, and M. C. Teich, "Generalized entangled-photon imaging," in Proceedings of the IEEE LEOS 1996 Annual Meeting (Boston, MA, 1996), pp. 362-363.

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