Fig. 1. Imaging: a lens produces an image of an object in its image plane, which is defined by the Gaussian thin-lens equation, 1/s_i + 1/s_o = 1/f. The object is illuminated by an incoherent light source. The concept of the optical image is based on the existence of a point-to-point relationship between the radiation field in the object plane and the radiation field in the image plane: any radiation starting from a point on the object plane must collapse or stop at a unique point on the image plane due to the constructive-destructive interferences.

Fig. 2. Projection: a light source illuminates an object, and no image forming system is present, no image plane is defined, and only projections, or shadows, of the object can be observed.

Fig. 3. A typical imaging system. A lens of finite size is used to produce a magnified or demagnified image of an object with limited spatial resolution. To simplify the notation, we set z_o = 0, t_o = 0.

Fig. 4. Schematic setup of the first ghost imaging experiment of 1995.

Fig. 5. (a) A reproduction of the actual aperture “UMBC” placed in the signal beam. (b) The image of “UMBC”: coincidence counts as a function of the fiber tip’s transverse plane coordinates. The step size is 0.25 mm. The image shown is a “slice” at the half-maximum value.

Fig. 6. An unfolded setup of the “ghost” imaging experiment, which is helpful for understanding the physics. Since the biphoton “light” propagates along “straight-lines,” it is not difficult to find that any geometrical light point on the subject plane corresponds to a unique geometrical light point on the image plane. Thus, a “ghost” image of the subject is made nonlocally in the image plane. Although the placement of the lens, object, and detector D₂ obeys the Gaussian thin-lens equation, it is important to remember that the geometric rays in the figure actually represent the biphoton amplitudes of an entangled signal–idler pair. The point-to-point correspondence is the result of the superposition of these biphoton amplitudes.

Fig. 7. In arm-1, the signal propagates freely over a distance d₁ from the output plane of the source to the imaging lens, then passes an object aperture at distance s_o, and then is focused onto a photon-counting detector D₁ by a collection lens. In arm-2, the idler propagates freely over distance d₂ from the output plane of the source to a point-like photon-counting detector D₂.

Fig. 8. Lensless Fresnel near-field ghost imaging with pseudo-thermal light demonstrated in 2006 by Scarcelli et al.^[11]. D₁ is a point-like photodetector that is scannable along the x₁-axis. The joint detection between D₁ and the bucket detector D₂ is realized either by a photon-counting coincidence counter or by a standard HBT linear multiplier (RF mixer). In this measurement, D₂ is fixed in the focal point of a convex lens, playing the role of a bucket detector. The counting rates or the photocurrents of D₁ and D₂, respectively, are measured to be constants. Surprisingly, an image of the 1-D object is observed in the joint detection between D₁ and D₂ by scanning D₁ in the plane of z₁ = z₂ along the x₁-axis. The image is blurred out when z₁ ≠ z₂. There is no doubt that thermal radiations propagate to any transverse plane in a random and chaotic manner. There is no lens applied to force the thermal radiation “collapsing” to a point or speckle either. What is the physical cause of the point-to-point image-forming correlation in coincidences?

Fig. 9. The experimentally observed ghost image of a double-slit. The lensless ghost image is observed to have equal size to that of the object. The ghost image has 50% contrast if measured by either a photon-counting coincidence circuit or by a standard HBT-type analog correlation circuit. If the measurement is for photon number fluctuation correlation or intensity fluctuation correlation, the visibility of the ghost image is close to 100%.

Fig. 10. Improved near-field lensless ghost imaging of pseudo-thermal light demonstrated by Meyers et al. The bucket detector collects the randomly scattered and reflected photons from an object. The CCD cannot “see” the object but is facing the light source.

Fig. 11. Ghost image of a toy soldier model.

Fig. 12. “Unfolded” schematic setup of lensless ghost imaging. A large number of randomly created and randomly distributed subfields from a disk-like thermal or pseudo-thermal light source with a relatively large angular diameter. A “bucket” photodetector, D₂, is placed behind the object plane of z_o = d to “collect” all transmitted or reflected-scattered light from the object. A scannable point-like photodetector, D₁, is placed on the ghost imaging plane of z₁ = d. Note: choosing z₁ = z_o = d, when ρ→1≈ρ→_o, the red and blue two-photon amplitudes of the (m − n)th pair of subfields superpose constructively with an equal path. Adding the contributions of all random pairs at ρ→1≈ρ→_o, the intensity fluctuation correlation, or the photon number fluctuation correlation, measurement yields a two-photon diffraction limited point-to-point correlation, which plays the role of an image-forming correlation to map the aperture function A(ρ→_o) onto the ghost image plane of z₁ = d.

Fig. 13. Unfolded schematic experimental setup of a secondary image measurement of the primary ghost image. By using a convex lens of focal length f, the primary lensless ghost image is imaged onto a secondary image plane, which is defined by the Gaussian thin-lens equation, 1/s_o + 1/s_i = 1/f, with magnification m = −s_i/s_o. This setup is useful for distant large scale ghost imaging applications. Bottom, a secondary image of the primary lensless ghost image of “UMBC” with a magnification factor of m = −s_i/s_o ∼ 2.9. Notice, the secondary image is “blurred” out quickly when the scanning photodetector D₁ is moved away from the image plane.

Fig. 14. The point-to-point correlation is made shot by shot by two co-rotating laser beams. A ghost shadow can be made in coincidences by “blocking-unblocking” of the correlated laser beams, or simply by “blocking-unblocking” two correlated gun shots.

Fig. 15. A ghost image is made by a man-made correlation of “speckles.” The two identical sets of speckles are the classical images of the speckles of the light source. The lens, which may be part of a CCD camera used for the joint measurement, reconstructs classical images of the speckles of the source onto the object plane and the image plane, respectively. s_o and s_i satisfy the Gaussian thin-lens equation, 1/s_o + 1/s_i = 1/f.

Fig. 16. Conceptual schematic of an Imaging Lidar.

Fig. 17. A typical histogram indicating a target at 750.14 ± 0.03 m distance with significant atmospheric turbulence and light background-noise.

Fig. 18. Typical measured histograms: number of coincidence counts versus temporal delay, t₁ − t₂ in nanoseconds. Upper left, without turbulence and background-noise; upper right, with significant turbulence and background-noise; bottom, two histograms, without and with significant turbulence, in one plot for comparison. The introduced atmospheric turbulence is strong enough to blur out the interference pattern of a classic Young’s double-slit interferometer. The background noise is 20 times stronger than the signal.

Fig. 19. Schematic setup of a typical thermal light ghost imaging experiment that captures the secondary image of the primary lensless ghost image. This experiment, however, added a set of powerful heating elements underneath the optical paths to produce laboratory atmospheric turbulence. The dashed line and arrows indicate the optical path of the “bucket” detector. The solid line and arrows indicate the optical path of the ghost image arm.

Fig. 20. Turbulence-resistant camera: an image of the target object, which is under the influence of atmospheric turbulence, is produced from the photon number fluctuation correlation measurement ⟨Δn(ρ→i1)Δn₂⟩. The classic image observed from ⟨n(ρ→i1)⟩ is completely “blurred” due to the influence of the atmospheric turbulence. However, a turbulence-resistant image is observed from the measurement of ⟨Δn(ρ→i1)Δn₂⟩, i.e., any atmospheric density, refractive index, or phase variations do not have any influence on this image. In this setup, the turbulence may appear either in the optical paths between the camera and the object or in the optical paths between the object and the light source, or appear in both.

Fig. 21. Experimental testing of a turbulence-resistant camera: turbulence-resistant image of group 0 of a 1951 USAF Resolution Testing Gauge from a demo unit of a turbulence-resistant camera. (a) shows the clear classical image in the measurement of ⟨n(ρ→i1)⟩ without atmospheric turbulence; (b) shows the “blurred” first-order classic image in the presence of atmospheric turbulence; (c) shows the image in the measurement of ⟨Δn(ρ→i1)Δn₂⟩. In (b) and (c), row (i) shows weak turbulence, row (ii) shows medium turbulence, and row (iii) shows strong turbulence. In this measurement, the incoherent light source was a 6.4 mm diameter 3200 Kelvin tungsten-halogen white light lamp. The atmospheric turbulence between the object and the camera was simulated by a propane camp stove with variable heat settings. The level of the turbulence can be easily adjusted by varying the heat settings. Due to the use of the white light thermal source, the control of the “exposure” time of the CCD (CMOS) is critical.

Fig. 22. X-ray ghost imaging. A beam splitter (most likely a crystal aligned to utilize Laue diffraction that would not provide 90° separation as depicted) creates two paths for the beam, one directed at the object followed by a bucket detector and the second directed at a pixel array (CCD or CMOS). Note : (1) the detectors could directly detect X-rays or be paired with a scintillator to convert the X-ray to visible; (2) different ghost image planes, at distance z = d, z = d^′, z = d^′′ …, correspond to a different “slice,” or cross section, of z = d, z = d^′, z = d^′′ … inside the object.

Fig. 23. Schematic of an X-ray ghost microscope using an X-ray lens system. A beam splitter (most likely a crystal aligned to utilize Laue diffraction) creates two paths for the beam, one directed at the object substance and the second directed at the primary ghost image plane. An X-ray lens system is placed behind the primary X-ray ghost image to reproduce a significantly magnified secondary ghost image that is resolvable by a standard CCD or CMOS. Note: (1) due to the “ghost” nature of the primary ghost image, s_o of the microscope can be placed as close as possible to the primary X-ray ghost image to “force” the angular separation of nanometer scale inside the object greater than the angular resolution of the secondary imaging system; (2) Δθ_min ≃ 1.22λ/D is the angular resolution of the secondary imaging device, where λ is the wavelength of the X-ray and D is the effective diameter of the X-ray lens.

Fig. 24. Schematic of an X-ray ghost microscope using a scintillator-visible-light-lens assembly to magnify the primary X-ray ghost image. This is nearly identical to the setup of Fig. 23, but now a scintillator is placed on the ghost image plane to convert the X-ray ghost image into the visible spectrum. Now an optical lens (or lens system) of visible-light produces a magnified secondary ghost image.