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Physical Review A

Resolution and apodization in images generated by twin photons

Ivan F. Santos, M. A. Sagioro, Carlos Henrique Monken, Sebastião Pádua

Published Mar 26, 2003• 30 citations
Publisher

Abstract

We show that an image of an object is obtained when at least one of the photons of a parametric down-converted pair illuminates the object and the two photons are detected in coincidence after they have been transmitted by two lenses. The image is mathematically described by a quantum fourth-order correlation function that differs from the classical image description in two aspects. The quantum image is produced by a nonlocal effective lens whose aperture is described by a compressed convolution of the magnitude of lens transmission functions. The image is generated by the entangled state two-photon light source in which the effective wavelength is equal to the de Broglie wavelength $\ensuremath{\lambda}/2,$ where $\ensuremath{\lambda}$ is the wavelength associated to the individual photons. Images can be obtained with a resolution better than the obtained with coherent classical light sources. Better resolution and strong apodization effects are observed.

DOI
10.1103/physreva.67.033812

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We show that an image of an object is obtained when at least one of the photons of a parametric down-converted pair illuminates the object and the two photons are detected in coincidence after they have been transmitted by two lenses. The image is mathematically described by a quantum fourth-order correlation function that differs from the classical image description in two aspects. The quantum image is produced by a nonlocal effective lens whose aperture is described by a compressed convolution of the magnitude of lens transmission functions. The image is generated by the entangled state two-photon light source in which the effective wavelength is equal to the de Broglie wavelength \ensuremathλ/2,\ensuremath{\lambda}/2,\ensuremathλ/2, where \ensuremathλ\ensuremath{\lambda}\ensuremathλ is the wavelength associated to the individual photons. Images can be obtained with a resolution better than the obtained with coherent classical light sources. Better resolution and strong apodization effects are observed.
We show experimentally that quantum coincidence imaging with photon pairs generated by spontaneous parametric down conversion, in certain conditions, differs from the classical imaging in two aspects: the magnified image is produced by an ``apodized'' effective lens and its effective wavelength is the De Broglie wavelength of the photon pair $\ensuremath{\lambda}∕2$, where $\ensuremath{\lambda}$ is the central wavelength of the single photon wave packet. Better resolution and apodization effects (reduced intensity of the diffraction fringes around the image) are observed in the detected images.
Coincidence, or ghost, imaging is a technique that uses two correlated optical fields to form an image of an object. In this work we identify aspects of coincidence imaging which can be performed with classically correlated light sources and aspects which require quantum entanglement. We find that entangled photons allow high-contrast, high-resolution imaging to be performed at any distance from the light source. We demonstrate this fact by forming ghost images in the near and far fields of an entangled photon source, noting that the product of the resolutions of these images is a factor of 3 better than that which is allowed by classical diffraction theory.
Entangled-photon coincidence imaging is a method to nonlocally image an object by transmitting a pair of entangled photons through the object and a reference optical system, respectively. The image of the object can be extracted from the coincidence rate of these two photons. From a classical perspective, the image is proportional to the fourth-order correlation function of the wave field. Using classical statistical optics, we study a particular aspect of coincidence imaging with incoherent sources. As an application, we give a proposal to realize lensless Fourier-transform imaging, and discuss its applicability in x-ray diffraction.
Entangled photons, generated by spontaneous parametric downconversion from a second-order nonlinear crystal, present a rich potential for imaging and image-processing applications. Since this source is an example of a three-wave mixing process, there is more flexibility in the choices of illumination and detection wavelengths and in the placement of object(s) to be imaged. Moreover, this source is entangled, a fact that allows for imaging configurations and capabilities that cannot be achieved by use of classical sources of light. We examine a number of imaging and image-processing configurations that can be realized with this source. The formalism that we utilize facilitates the determination of the dependence of imaging resolution on the physical parameters of the optical arrangement.
The transverse correlation of triphoton entanglement generated within a single crystal is analyzed. Among many interesting features of the transverse correlation, they arise from the spectral function FFF of the triphoton state produced in the parametric processes. One consequence of transverse effects of entangled states is quantum imaging, which is theoretically studied in photon counting measurements. Klyshko's two-photon advanced-wave picture is found to be applicable to the multiphoton entanglement with some modifications. We found that in the two-photon coincidence counting measurement by using triphoton entanglement, although the Gaussian thin lens equation (GTLE) holds, the imaging shown in coincidences is obscure and has a poor quality. This is because of tracing the remaining transverse modes in the untouched beam. In the triphoton imaging experiments, two kinds of cases have been examined. For the case that only one object with one thin lens is placed in the system, we found that the GTLE holds as expected in the triphoton coincidences and the effective distance between the lens and imaging plane is the parallel combination of two distances between the lens and two detectors weighted by wavelengths, which behaves as the parallel combination of resistors in the electromagnetism theory. Only in this case, a point-point correspondence for forming an image is well-accomplished. However, when two objects or two lenses are inserted in the system, though the GTLEs are well-satisfied, in general a point-point correspondence for imaging cannot be established. Under certain conditions, two blurred images may be observed in the coincidence counts. We have also studied the ghost interference-diffraction experiments by using double slits as apertures in triphoton entanglement. It was found that when two double slits are used in two optical beams, the interference-diffraction patterns show unusual features compared with the two-photon case. This unusual behavior is a destructive interference between two amplitudes for two photons crossing two double slits.
High Resolution Image Download MS PowerPoint Slide The resolution of a classical imaging system is limited by diffraction. This limit can be overcome, for example, by implementing various forms of localization microscopy in which the center of a fluorescence distribution is estimated to an accuracy scaling with the square root of the number of detected photons, N . In quantum imaging the object can be illuminated using correlated photon-pairs, leading early work to suggest that a 2 improvement could be obtained as a result of averaging the position of N = 2 events. However, similar to quantum lithography, which relies upon quantum illumination using entangled photon-pairs and two-photon absorption, the minimum resolvable feature size is reduced by a factor of 2, not just 2 . Quantum imaging schemes can also lead to a factor of 2 improvement. By using a similar source of correlated photon-pairs to illuminate an object, a single-photon sensitive camera to detect the photon-pairs, and an image processing algorithm to record and sum the bisector positions of the transmitted photon-pairs, we realize a similar factor of ×2 improvement in image resolution, surpassing that of most earlier quantum imaging work.