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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 5 — Feb. 27, 2012
  • pp: 5809–5816
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Storage and retrieval of ghost images in hot atomic vapor

Young-Wook Cho, Joo-Eon Oh, and Yoon-Ho Kim  »View Author Affiliations


Optics Express, Vol. 20, Issue 5, pp. 5809-5816 (2012)
http://dx.doi.org/10.1364/OE.20.005809


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Abstract

Ghost imaging is an imaging technique in which the image of an object is revealed only in the correlation measurement between two beams of light, whereas the individual measurements contain no imaging information. Here, we experimentally demonstrate storage and retrieval of ghost images in hot atomic rubidium vapor. Since ghost imaging requires (quantum or classical) multimode spatial correlation between two beams of light, our experiment shows that the spatially multimode correlation, a second-order correlation property of light, can indeed be preserved during the storage-retrieval process. Our work, thus, opens up new possibilities for quantum and classical two-photon imaging, all-optical image processing, and quantum communication.

© 2012 OSA

1. Introduction

Dynamic and reversible storage of the optical field has great potential both in classical (all-optical signal and image processing, etc.) and quantum information (quantum communication, photonic quantum computing, etc.) One promising approach to achieve coherent storage of the optical field is based on electromagnetically-induced transparency (EIT) in which the propagation of a weak signal field is coherently manipulated by a strong coupling field [1

1. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]

]. Since the quantum state of the signal field is preserved during the storage-retrieval process in the EIT medium [2

2. M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000). [CrossRef] [PubMed]

], the EIT-based light storage facilitates photonic quantum memory [3

3. D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783–786 (2001). [CrossRef] [PubMed]

8

8. Y.-W. Cho and Y.-H. Kim, “Storage and retrieval of thermal light in warm atomic vapor,” Phys. Rev. A 82, 033830 (2010). [CrossRef]

]. In recent years, other schemes for coherent light storage have also become available, including controlled reversible inhomogeneous broadening, atomic frequency comb, gradient echo memory, and off-resonant Raman memory [9

9. A. L. Alexander, J. J. Longdell, M. J. Sellars, and N. B. Manson, “Photon echoes produced by switching electric fields,” Phys. Rev. Lett. 96, 043602 (2006). [CrossRef] [PubMed]

14

14. K. F. Reim, J. Nunn, V. O. Lorenz, B. J. Sussman, K. C. Lee, N. K. Langford, D. Jaksch, and I. A. Walmsley, “Towards high-speed optical quantum memories,” Nat. Photonics 4, 218–221 (2010). [CrossRef]

].

Lately, there have been great interests in expanding the capacity of coherent light storage to spatially multimode fields for storing optical images [15

15. D. V. Vasilyev, I. V. Sokolov, and E. S. Polzik, “Quantum memory for images: a quantum hologram,” Phys. Rev. A. 77, 020302(R) (2008). [CrossRef]

18

18. G. Heinze, A. Rudolf, F. Beil, and T. Halfmann, “Storage of images in atomic coherences in a rare-earth-ion-doped solid,” Phys. Rev. A 81, 011401(R) (2010).

]. To date, coherent storage of optical images has been demonstrated only for images imprinted directly on the transverse profile of a laser pulse which is a first-order multimode property of light [16

16. P. K. Vudyasetu, R. M. Camacho, and J. C. Howell, “Storage and retrieval of multimode transverse images in hot atomic rubidium vapor,” Phys. Rev. Lett. 100, 123903 (2008). [CrossRef] [PubMed]

18

18. G. Heinze, A. Rudolf, F. Beil, and T. Halfmann, “Storage of images in atomic coherences in a rare-earth-ion-doped solid,” Phys. Rev. A 81, 011401(R) (2010).

]. Then, the question naturally arises: Can a second-order multimode correlation property of light be coherently stored and retrieved? One prominent and useful example of the second-order multimode correlation property of light is the transverse correlation between two beams of light which directly leads to ghost imaging.

In this paper, for the first time, we report storage and retrieval of ghost images in hot atomic rubidium vapor. By making use of the thermal ghost imaging scheme [20

20. F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005). [CrossRef] [PubMed]

, 21

21. A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005). [CrossRef] [PubMed]

] and the EIT light storage technique [3

3. D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783–786 (2001). [CrossRef] [PubMed]

], we demonstrate experimentally that the ghost image can still be revealed in the second-order correlation measurement of the retrieved fields. This result establishes clearly that the transverse multimode correlation can in fact survive the storage-retrieval process, enabling potential applications of quantum and classical correlation imaging.

2. Experimental setup

The experimental setup is shown in Fig. 1. Let us first describe the source of correlated twin speckle beams used in the experiment. The transverse correlated twin speckle beams were generated by splitting a pseudo-thermal light beam with a polarizing beam splitter PBS1 [20

20. F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005). [CrossRef] [PubMed]

, 21

21. A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005). [CrossRef] [PubMed]

]. The pseudo-thermal light source was prepared by focusing an external cavity diode laser beam, locked to one of the Rubidium 87 D1 transition lines 52S1/2F = 1 → 52P1/2F′ = 2, at a rotating ground disk (not shown in Fig. 1) [8

8. Y.-W. Cho and Y.-H. Kim, “Storage and retrieval of thermal light in warm atomic vapor,” Phys. Rev. A 82, 033830 (2010). [CrossRef]

]. To characterize the temporal properties of the pseudo-thermal light, we measured the second-order temporal coherence of the light by using the Hanbury-Brown–Twiss interferometer. The light reflected at PBS1 was collected at a single-mode fiber connected to a 3dB fiber beam splitter. The two outputs of the fiber beam splitter were connected to single-photon detectors for measuring the second-order temporal correlation function g(2) (τ) [8

8. Y.-W. Cho and Y.-H. Kim, “Storage and retrieval of thermal light in warm atomic vapor,” Phys. Rev. A 82, 033830 (2010). [CrossRef]

]. For thermal light, g(2) (τ) = 1 + exp[−π(τ/τc)2] and the measurement showed clear signature of photon bunching, which is a characteristic of thermal light, with the coherence time of τc = 7.26 ± 0.07 μs.

Fig. 1 (a) Two beams that have transverse spatial correlation are prepared by splitting a thermal light beam with a polarizing beam splitter PBS1. One beam (reference) goes through the mask with OCR-a character 5 and is detected by a bucket detector. The other beam (signal) is stored in and retrieved from the EIT medium (Rb vapor cell). Note that neither the bucket detector nor the CCD alone provide any information about the object (mask). The ghost image of the mask is revealed in the correlation measurement between the bucket detector and the CCD. (b) Schematic diagram of the experiment. See text for details.

Let us now describe the experimental setup shown in Fig. 1(a). Ghost imaging makes use of the transverse spatial correlation of the twin speckle beams prepared by splitting the thermal light beam into two with PBS1. The reflected beam at PBS1 (the reference beam; vertically polarized) goes through the mask (resolution target; Newport RES-1) with OCR-a character 5 and gets detected by a bucket detector. The transmitted beam at PBS1 (the signal beam; horizontally polarized) is stored to and retrieved from the EIT medium (Rubidium vapor cell). The vertically polarized coupling beam, locked to the Rubidium 87 D1 transition line 52S1/2F = 2 → 52P1/2F′ = 2, is spatially and temporally matched with the signal beam inside the vapor cell for preparation and manipulation of the EIT medium. Note that both the bucket detector and the CCD individually do not exhibit any imaging information. The ghost image of the mask is revealed in the correlation measurement between the bucket detector and the CCD.

A more detailed schematic of the experimental setup is shown in Fig. 1(b). At the detection plane (the bucket detector and the CCD), the light field can be written as Eout (r⃗i) = ∫ dr⃗iEin(r⃗i)hi(r⃗i,r⃗i), where r⃗i is the transverse position vector, Ein(r⃗i) refers to the fields at the plane immediately after PBS1, and hi(r⃗i,r⃗i) is the impulse response function of each optical system. In the reference beam, an object (Mask with OCR-a character 5) is placed immediately after PBS1. Thus, the impulse response function is given as h1(r⃗1, r⃗1) = T (r⃗1)δ(r⃗1r⃗1) where T (r⃗1) is the complex transmission function of the object. In the signal beam, the optical system consists of a 4f imaging system and the EIT medium (vapor cell). The impulse response function of the optical system without the EIT medium can be written as h2(r⃗2, r⃗2) = δ(r⃗2 + mr⃗2), where m = f2/ f1 is the magnification factor. Note that due to the lensless ghost imaging effect [27

27. G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?,” Phys. Rev. Lett. 96, 063602 (2006). [CrossRef] [PubMed]

], the sharp ghost image of the object placed in the reference beam at r⃗1 can be found by correlation measurement of the bucket detector and the CCD placed in the signal beam at the same distance from PBS1 at r⃗2. The ghost image plane r⃗2 is then relayed to r⃗2 using the 4f imaging system with m = 0.667

3. Construction of ghost images

The correlation of the intensity fluctuations measured at the two detectors is given as G(r⃗1, r⃗2) = 〈ΔI1(r⃗1I2(r⃗2)〉, where 〈...〉 is time averaging and, for thermal light, it becomes [28

28. A. Gatti, E. Brambilla, M. Bache, and A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70, 013802 (2004). [CrossRef]

] G(r1,r2)|dr1dr2h1*(r1,r1)h2(r2,r2)Γ(r1,r2)|2, where Γ(r1,r2)=Ein*(r1)Ein(r2) is the mutual spatial correlation function. Assuming that the thermal light is spatially incoherent with a uniform intensity distribution Γ(r⃗1, r⃗2) = I0δ(r1r⃗2) and this gives rise to G(r⃗1, r⃗2) ∝ |T (r⃗1)|2 |δ(r⃗1 + mr⃗2)|2. Since the reference beam is detected by a bucket detector which has no spatial resolution, integration of G(r⃗1, r⃗2) over r⃗1 is necessary and this yields ∫ dr⃗1G(r⃗1, r⃗2) ∝ |T (−mr⃗2)|2. Thus, in the absence of the EIT storage medium, we expect to observe an inverted ghost image with the magnification factor of m.

Fig. 2 Reconstruction of the ghost image. (a) A single shot image of the CCD shows the speckle pattern. (b) Normalized spatial intensity autocorrelation measured from the speckle pattern. (c) The reconstructed ghost image, averaged over 5,000 shots. An inverted image of the mask (OCR-a character 5) is revealed.

4. Storage and retrieval of ghost images

Having seen the ghost image, we now move on to discussing storage and retrieval of the ghost image. The 50 mm long natural isotopic abundant Rubidium vapor cell (filled with 49 Torr Ne buffer gas to enhance the storage time by reducing the diffusion velocity of the atoms) is placed at the Fourier plane of the 4f imaging system. The vapor cell was heated to 70 ∼ 80 °C, providing a sufficient rubidium vapor density of approximately 1012 cm−3. As discussed earlier, the thermal light source and the coupling beam were locked to 52S1/2F = 1 (|1〉) → 52P1/2F′ = 2 (|3〉) and 52S1/2F = 2 (|2〉) → 52P1/2F′ = 2 (|3〉) transitions of Rubidium 87 D1 line, respectively. To ensure better transmission, both beams are slightly blue-detuned by 60 MHz. The FWHM EIT linewidth of 188 kHz was observed by tuning the frequency of the coupling beam. Note that the bandwidth of the thermal light, 1/τc ≈ 138 kHz, fits within the EIT spectrum. The power and the beam diameter of the signal beam are approximately 250 μW and 2.5 mm, respectively, at the ghost object plane, r⃗2 in Fig. 1(b). The power of the coupling beam is 25 mW with a 5 mm beam diameter.

Figure 3(a) shows the synchronized timing sequence employed for storage and retrieval the ghost image [30

30. A delay/pulse generator (SRS, DG535) provides the synchronization pulses for the CCD (JAI, CM-030-GE), the digitizer (NI, PCI-5114), and the two acousto-optic modulators used in the experiment. Each “measurement” is then repeated at 1.5 Hz.

]. First, the thermal light source is shaped to a 10 μs rectangular pulse as mentioned earlier and the coupling beam is turned on to prepare the EIT medium for storage. After the signal pulse has completely entered the vapor cell, the coupling beam is turned off, storing the signal beam in the EIT medium. After some storage duration (4 μs ∼ 20 μs), the coupling beam is temporally turned back on for 4 μs and, during this time, the signal beam stored in the EIT medium is partially retrieved. The CCD is triggered so that the exposure window overlaps only with the retrieved signal.

Fig. 3 (a) Synchronized timing sequence for storage and retrieval of ghost images. (b) Ghost image storage and retrieval. The storage time is 4 μs ∼ 16 μs. The ghost images are reconstructed from 5,000 shots of bucket detector - CCD correlation measurements.

The ghost images reconstructed from the correlation measurement of the reference beam (with a bucket detector) and the stored signal beam (with a CCD) are shown in Fig. 3(b). The storage time varies between 4 μs to 16 μs and each ghost image is reconstructed from 5,000 shots of such measurements. It is apparent that transverse spatial multimode correlation between the twin speckle beams survives the storage-retrieval process. Also, the reconstructed ghost images are clearly identifiable without broadening by atomic diffusion.

To analyze the effect of storage time on the quality of the ghost image more clearly, we plot the vertical cross-sections of the ghost images in Fig. 4(a). The cross-sectional plots show that the magnitude of the correlation coefficient decays exponentially with the storage time. The visibility(contrast) values calculated from each cross-sectional data set do not vary much. However, the thermal light ghost images have the large background noise. This, then, suggests that the quality of the ghost image is more properly characterized not only with visibility but also with signal-to noise-ratio [31

31. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “Optimization of thermal ghost imaging: high-order correlation vs. background subtraction,” Opt. Express 18, 5562–5573 (2010). [CrossRef] [PubMed]

34

34. G. Brida, M. V. Chekhova, G. A. Fornaro, M. Genovese, E. D. Lopaeva, and I. Ruo Berchera, “Systematic analysis of signal-to-noise ratio in bipartite ghost imaging with classical and quantum light,” Phys. Rev. A 83, 063807 (2011). [CrossRef]

]. We will use the contrast-to-noise ratio (CNR) defined in Ref. [31

31. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “Optimization of thermal ghost imaging: high-order correlation vs. background subtraction,” Opt. Express 18, 5562–5573 (2010). [CrossRef] [PubMed]

] as the metric for quantifying quality of retrieved ghost images. Figure 4(b) shows the CNR and the partial retrieval efficiency (tsts+4μs|Eout(t)|2dt/|Ein(t)|2dt) as a function of storage time. Clearly, the quality of the retrieved ghost images is degraded with longer storage times and this is mainly due to the fact that, with longer storage times, reduced the retrieval efficiency is reduced. (the reduced intensity decreases both the contrast and the signal-to-noise ratio.) However, as the visibility is not severely degraded with longer storage times, it is possible to increase CNR by taking more measurements [31

31. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “Optimization of thermal ghost imaging: high-order correlation vs. background subtraction,” Opt. Express 18, 5562–5573 (2010). [CrossRef] [PubMed]

]. Additionally, computational algorithms can help to increase the ghost imaging CNR and resolution [35

35. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010). [CrossRef] [PubMed]

].

Fig. 4 (a) Vertical cross-sections, at x = −37 μm, of the ghost images shown in Fig. 3(b). (b) Contrast-to-Noise Ratio (CNR) of the ghost images and the partial retrieval efficiency with the storage time. The error bars denote statistical one standard deviation errors.

The effect of storage time on the image quality can be theoretically studied as follows. The signal field Ein(r⃗2) at the ghost object plane is Fourier transformed by the lens (L1) and stored as the coherence between the atomic ground states |1〉 and |2〉 [2

2. M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000). [CrossRef] [PubMed]

]. The atomic ground state coherence ρ12 evolves according to the diffusion equation
ρ12(r2,t)t=D(2x2+2y2)ρ12(r2,t)Γρ12(r2,t),
where D and Γ are the diffusion coefficient and the ground state decay rate, respectively. The field at the detection plane is then given as [16

16. P. K. Vudyasetu, R. M. Camacho, and J. C. Howell, “Storage and retrieval of multimode transverse images in hot atomic rubidium vapor,” Phys. Rev. Lett. 100, 123903 (2008). [CrossRef] [PubMed]

, 36

36. L. Zhao, T. Wang, Y. Xiao, and S. F. Yelin, “Image storage in hot vapors,” Phys. Rev. A 77, 041802(R) (2008). [CrossRef]

] Eout (r⃗2) = Ein(−mr⃗2) exp(−(β + Γ)ts), where m is the magnification of the 4f imaging system, ts is the storage time, and β = D(2π)2m2(x2 + y2)/(λf1)2. Here, f1 is the focal length of the first lens L1. The retrieved signal field at the CCD plane is then given as ∫ dr⃗2h2(r⃗2, r⃗2) Ein(r⃗2) = Ein(−mr⃗2) exp(−(β + Γ)ts). The second-order correlation function is therefore given as G(r⃗1,r⃗2) ∝ |T (r⃗1)|2 |δ (r⃗1 + mr⃗2)|2 exp (−2(β + Γ)ts). The ghost image is obtained by integrating over r⃗1
dr1G(r1,r2)|T(mr2)|2exp(2(β+Γ)ts).
This result shows that the ghost image can survive the storage-retrieval process while maintaining sharp edges although the overall “brightness” of the ghost image experiences exponential decay. Thus, as mentioned before, CNR of the ghost image can be improved by including more shots of measurements in ghost image reconstruction which is more or less equivalent to making a longer exposure in photography. We note that we could avoid the image degradation due to the atomic diffusion by storing the Fourier transformed image. Zero crossings in the Fourier transformed image is much insensitive to the atomic diffusion due to the destructive interference [16

16. P. K. Vudyasetu, R. M. Camacho, and J. C. Howell, “Storage and retrieval of multimode transverse images in hot atomic rubidium vapor,” Phys. Rev. Lett. 100, 123903 (2008). [CrossRef] [PubMed]

, 36

36. L. Zhao, T. Wang, Y. Xiao, and S. F. Yelin, “Image storage in hot vapors,” Phys. Rev. A 77, 041802(R) (2008). [CrossRef]

]. (The thermal light ghost imaging is a coherent imaging method although incoherent thermal light source is used. [37

37. A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. of Mod. Opt. 53, 739–760 (2006). [CrossRef]

])

5. Conclusion

We have demonstrated experimentally storage and retrieval of thermal light ghost images in hot atomic vapor. The results therefore clearly show that transverse spatial multimode correlation can be preserved during the storage-retrieval process. With the recent experimental results on the storage of quantum correlation in a single spatial mode, our result of preservation spatial multimode correlation in the EIT medium strongly implies the possibility to realize the storage of spatially multimode quantum correlation since storage-retrieval process is coherent. We therefore believe that the ghost imaging storage demonstrated in this work will open up important new applications of quantum and classical correlation imaging, all-optical image processing, remote sensing, quantum communication, and quantum information processing in high dimensions.

Acknowledgments

We would like to thank G. Scarcelli and J. Wen for helpful comments. This work was supported by National Research Foundation of Korea ( 2009-0070668 and 2009-0084473). YWC acknowledges the financial supported by the NRF ( 2011-0010895).

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The cross correlation for the ghost imaging is calculated by the following equation, G(r)=iNΔIiΔJi(r), where ΔIi=Ii1NiNIi and ΔJi(r)=Ji(r)1NiNJi(r) are fluctuations of photodetector and CCD output signals, respectively.

30.

A delay/pulse generator (SRS, DG535) provides the synchronization pulses for the CCD (JAI, CM-030-GE), the digitizer (NI, PCI-5114), and the two acousto-optic modulators used in the experiment. Each “measurement” is then repeated at 1.5 Hz.

31.

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K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-order thermal ghost imaging,” Opt. Lett. 34, 3343–3345 (2009). [CrossRef] [PubMed]

33.

B. I. Erkmen and J. H. Shapiro, “Signal-to-noise ratio of Gussian-state ghost imaging,” Phys. Rev. A 79, 023833 (2009). [CrossRef]

34.

G. Brida, M. V. Chekhova, G. A. Fornaro, M. Genovese, E. D. Lopaeva, and I. Ruo Berchera, “Systematic analysis of signal-to-noise ratio in bipartite ghost imaging with classical and quantum light,” Phys. Rev. A 83, 063807 (2011). [CrossRef]

35.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010). [CrossRef] [PubMed]

36.

L. Zhao, T. Wang, Y. Xiao, and S. F. Yelin, “Image storage in hot vapors,” Phys. Rev. A 77, 041802(R) (2008). [CrossRef]

37.

A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. of Mod. Opt. 53, 739–760 (2006). [CrossRef]

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(270.1670) Quantum optics : Coherent optical effects
(270.5585) Quantum optics : Quantum information and processing

ToC Category:
Quantum Optics

History
Original Manuscript: January 5, 2012
Revised Manuscript: February 12, 2012
Manuscript Accepted: February 13, 2012
Published: February 24, 2012

Citation
Young-Wook Cho, Joo-Eon Oh, and Yoon-Ho Kim, "Storage and retrieval of ghost images in hot atomic vapor," Opt. Express 20, 5809-5816 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-5-5809


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  29. The cross correlation for the ghost imaging is calculated by the following equation, G(r→)=∑iNΔIiΔJi(r→), where ΔIi=Ii−1N∑iNIi and ΔJi(r→)=Ji(r→)−1N∑iNJi(r→) are fluctuations of photodetector and CCD output signals, respectively.
  30. A delay/pulse generator (SRS, DG535) provides the synchronization pulses for the CCD (JAI, CM-030-GE), the digitizer (NI, PCI-5114), and the two acousto-optic modulators used in the experiment. Each “measurement” is then repeated at 1.5 Hz.
  31. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “Optimization of thermal ghost imaging: high-order correlation vs. background subtraction,” Opt. Express18, 5562–5573 (2010). [CrossRef] [PubMed]
  32. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-order thermal ghost imaging,” Opt. Lett.34, 3343–3345 (2009). [CrossRef] [PubMed]
  33. B. I. Erkmen and J. H. Shapiro, “Signal-to-noise ratio of Gussian-state ghost imaging,” Phys. Rev. A79, 023833 (2009). [CrossRef]
  34. G. Brida, M. V. Chekhova, G. A. Fornaro, M. Genovese, E. D. Lopaeva, and I. Ruo Berchera, “Systematic analysis of signal-to-noise ratio in bipartite ghost imaging with classical and quantum light,” Phys. Rev. A83, 063807 (2011). [CrossRef]
  35. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett.104, 253603 (2010). [CrossRef] [PubMed]
  36. L. Zhao, T. Wang, Y. Xiao, and S. F. Yelin, “Image storage in hot vapors,” Phys. Rev. A77, 041802(R) (2008). [CrossRef]
  37. A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. of Mod. Opt.53, 739–760 (2006). [CrossRef]

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