Abstract

In ghost imaging schemes information about an object is extracted by measuring the correlation between a beam that passed the object and a reference beam. We present a spatial averaging technique that substantially improves the imaging bandwidth of such schemes, which implies that information about high-frequency Fourier components can be observed in the reconstructed diffraction pattern. In the many-photon regime the averaging can be done in parallel and we show that this leads to a much faster convergence of the correlations. We also consider the reconstruction of the object image, and discuss the differences between a pixel-like detector and a bucket detector in the object arm. Finally, it is shown how to non-locally make spatial filtering of a reconstructed image. The results are presented using entangled beams created by parametric down-conversion, but they are general and can be extended also to the important case of using classically correlated thermal-like beams.

© 2004 Optical Society of America

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References

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  1. D. N. Klyshko, �??Effect of focusing on photon correlation in parametric light scattering,�?? Zh. Eksp. Teor. Fiz. 94, 82�??90 (1988), [Sov. Phys. JETP 67, 1131-1135 (1988)].
  2. A. V. Belinskii and D. N. Klyshko, �??2-photon optics - diffraction, holography and transformation of 2- dimensional signals,�?? Zh. Eksp. Teor. Fiz. 105, 487�??493 (1994), [Sov. Phys. JETP 78, 259-262 (1994)].
  3. 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]
  4. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, �??Optical imaging by means of two-photon quantum entanglement,�?? Phys. Rev. A 52, R3429�??R3432 (1995).
    [CrossRef] [PubMed]
  5. P. H. Souto Ribeiro, S. Padua, J. C. Machado da Silva, and G. A. Barbosa, �??Controlling the degree of visibility of Young�??s fringes with photon coincidence measurements,�?? Phys. Rev. A 49, 4176�??4179 (1994).
    [CrossRef]
  6. B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, �??Duality between partial coherence and partial entanglement,�?? Phys. Rev. A 62, 043816 (2000).
    [CrossRef]
  7. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, �??Role of Entanglement in two photon imaging,�?? Phys. Rev. Lett. 87, 123602 (2001).
    [CrossRef] [PubMed]
  8. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, �??Entangled-photon Fourier optics,�?? J. Opt. Soc. Am. B 19, 1174�??1184 (2002).
    [CrossRef]
  9. A. Gatti, E. Brambilla, and L. A. Lugiato, �??Entangled Imaging andWave-Particle Duality: From the Microscopic to the Macroscopic Realm,�?? Phys. Rev. Lett. 90, 133603 (2003).
    [CrossRef] [PubMed]
  10. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, �??Correlated imaging, quantum and classical,�?? Phys. Rev. A 70, 013802 (2004); �??Ghost imaging with thermal light: comparing entanglement and classical correlation,�?? Phys. Rev. Lett. 93, 093602 (2004).
    [CrossRef]
  11. A. Gatti, E. Brambilla, and L. A. Lugiato, �??Correlated imaging with entangled light beams,�?? In Quantum Communications and Quantum Imaging, R. E. Meyers and Y. Shih, eds., SPIE Proceedings 5161, 192�??203 (2004).
  12. M. Bache, E. Brambilla, A. Gatti, and L. A. Lugiato, �??Ghost imaging using homodyne detection,�?? Phys. Rev. A 70, 023823 (2004).
    [CrossRef]
  13. F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, �??High-resolution ghost imaging and ghost diffraction experiments with classical thermal light,�?? submitted to Phys. Rev. Lett., quant-ph/0408021 (2004).
  14. O. Jedrkiewicz, Y. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, �??Detection of sub-shot-noise spatial correlation in high-gain parametric down-conversion,�?? to appear in Phys. Rev. Lett., quantph/ 0407211 (2004).
  15. E. Brambilla, A. Gatti, M. Bache, and L. A. Lugiato, �??Simultaneous near-field and far field spatial quantum correlations in the high gain regime of parametric down-conversion,�?? Phys. Rev. A 69, 023802 (2004).
    [CrossRef]
  16. J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, New York, 1968).
  17. E.-K. Tan, J. Jeffers, S. M. Barnett, and D. T. Pegg, �??Retrodictive states and two-photon quantum imaging,�?? Eur. Phys. J. D 22, 495�??499 (2003).
    [CrossRef]
  18. A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, �??Entangled-Photon Imaging of a Pure Phase Object,�?? Phys. Rev. Lett. 93, 213903 (2004)..
    [CrossRef] [PubMed]

Eur. Phys. J. D (1)

E.-K. Tan, J. Jeffers, S. M. Barnett, and D. T. Pegg, �??Retrodictive states and two-photon quantum imaging,�?? Eur. Phys. J. D 22, 495�??499 (2003).
[CrossRef]

J. Opt. Soc. Am. B (1)

Phys. Rev. A (6)

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

P. H. Souto Ribeiro, S. Padua, J. C. Machado da Silva, and G. A. Barbosa, �??Controlling the degree of visibility of Young�??s fringes with photon coincidence measurements,�?? Phys. Rev. A 49, 4176�??4179 (1994).
[CrossRef]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, �??Duality between partial coherence and partial entanglement,�?? Phys. Rev. A 62, 043816 (2000).
[CrossRef]

E. Brambilla, A. Gatti, M. Bache, and L. A. Lugiato, �??Simultaneous near-field and far field spatial quantum correlations in the high gain regime of parametric down-conversion,�?? Phys. Rev. A 69, 023802 (2004).
[CrossRef]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, �??Correlated imaging, quantum and classical,�?? Phys. Rev. A 70, 013802 (2004); �??Ghost imaging with thermal light: comparing entanglement and classical correlation,�?? Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef]

M. Bache, E. Brambilla, A. Gatti, and L. A. Lugiato, �??Ghost imaging using homodyne detection,�?? Phys. Rev. A 70, 023823 (2004).
[CrossRef]

Phys. Rev. Lett (1)

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, �??High-resolution ghost imaging and ghost diffraction experiments with classical thermal light,�?? submitted to Phys. Rev. Lett., quant-ph/0408021 (2004).

Phys. Rev. Lett. (5)

O. Jedrkiewicz, Y. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, �??Detection of sub-shot-noise spatial correlation in high-gain parametric down-conversion,�?? to appear in Phys. Rev. Lett., quantph/ 0407211 (2004).

A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, �??Entangled-Photon Imaging of a Pure Phase Object,�?? Phys. Rev. Lett. 93, 213903 (2004)..
[CrossRef] [PubMed]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, �??Role of Entanglement in two photon imaging,�?? Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, and L. A. Lugiato, �??Entangled Imaging andWave-Particle Duality: From the Microscopic to the Macroscopic Realm,�?? Phys. Rev. Lett. 90, 133603 (2003).
[CrossRef] [PubMed]

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]

Quantum Communications and Quantum Imagi (1)

A. Gatti, E. Brambilla, and L. A. Lugiato, �??Correlated imaging with entangled light beams,�?? In Quantum Communications and Quantum Imaging, R. E. Meyers and Y. Shih, eds., SPIE Proceedings 5161, 192�??203 (2004).

Zh. Eksp. Teor. Fiz. (2)

D. N. Klyshko, �??Effect of focusing on photon correlation in parametric light scattering,�?? Zh. Eksp. Teor. Fiz. 94, 82�??90 (1988), [Sov. Phys. JETP 67, 1131-1135 (1988)].

A. V. Belinskii and D. N. Klyshko, �??2-photon optics - diffraction, holography and transformation of 2- dimensional signals,�?? Zh. Eksp. Teor. Fiz. 105, 487�??493 (1994), [Sov. Phys. JETP 78, 259-262 (1994)].

Other (1)

J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, New York, 1968).

Supplementary Material (5)

» Media 1: AVI (450 KB)     
» Media 2: AVI (1305 KB)     
» Media 3: AVI (1423 KB)     
» Media 4: AVI (748 KB)     
» Media 5: AVI (1288 KB)     

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Figures (9)

Fig. 1.
Fig. 1.

System setup. In (a) the setup for reconstructing the diffraction pattern of the object is shown, where the reference arm is in the f-f setup. In (b) the reference arm is changed to a telescope setup, which is used to reconstruct the object near field. PBS: polarizing beam splitter. L: lens of focal length f. T obj: object. D 1, D 2: arrays of detectors.

Fig. 2.
Fig. 2.

The spatial averaging technique explained by the Klyshko picture. The lower sketch shows the unfolded version of Fig. 1(a). The red line represents the ray from the detector at position x10 as it travels through the system and reaches its conjugate point in the reference plane on the right side. The upper plot shows the source gain curve in Fourier space and the position of a typical diffraction pattern for the current value of x10; only the black part is amplified while the gray part is not amplified. As the test detector position is changed to x10x the dashed red ray comes into play, and the diffraction pattern in the upper plot moves a corresponding amount (also dashed). The gain instead remains the same, and hence a different part of the diffraction pattern is amplified. (Movie size 449 KB).

Fig. 3.
Fig. 3.

Reconstructing the diffraction pattern using the f-f setup in the reference arm. (a) shows the correlation from a fixed x1 after averaging over 10000 shots while (b) shows the correlation using the spatial averaging technique and averaging over 1000 shots. In (a) and (b) black curves are numerics, while dashed red curves are obj. In (a) the blue curve is the analytically calculated correlation in the SPWPA in arbitrary units [Eq. (6) including integration over time, as shown in Eq. (5)]. (c) shows ε (n) from Eq. (12) in a double-logarithmic plot using the spatial averaging technique (full) and a fixed detector (dashed), while the red curves are fits to Eq. (13). Double slit parameters: aperture 14 µm (5 pixels) and aperture center distance 87 µm (30 pixels). [Movie sizes: (a) 1 MB and (b) 933 KB].

Fig. 4.
Fig. 4.

Reconstructing the diffraction pattern of the object shown in (a) which has spatial Fourier components both inside and outside the PDC bandwidth. The real part of the inverse Fourier transform of the reconstructed diffraction pattern using the f-f setup is shown with (b) fixed x 1 and (c) using the spatial averaging technique. 35000 shots were used.

Fig. 5.
Fig. 5.

The reconstructed diffraction pattern of a phase object with the spatial averaging technique. (a) shows Re(T obj) consisting of 4×4 square holes (with the value T obj=-1 inside and T obj=+1 outside the holes) modulated by a Gaussian. (b) analytical |obj|2. (c) G f,SA after 2000 shots.

Fig. 6.
Fig. 6.

The reconstructed image using the telescope setup with (a) D 1 a pixel-like detector and (b) D 1 a bucket detector. The black curves are numerics, the dashed red curves are T obj, while the blue curves are the analytically calculated correlations in the SPWPA in arbitrary units [Eqs. (14) and (15)]. (c) shows ε T(n) (full: bucket detector and dashed: pixel detector), and the red curves are fits to Eq. (13). The same parameters and notation as in Fig. 3. [Movie sizes: (a) 748 KB and (b) 1.25 MB].

Fig. 7.
Fig. 7.

The reconstructed image from semi-analytical calculations with (a) D 1 a pixel-like detector and (b) D 1 a bucket detector, using different interferential filters (different temporal bandwidths). The blue curves are semi-analytical calculations in the SPWPA in arbitrary units [Eqs. (14) and (15)]. (c) shows |γ (qx ,Ω)| for the chosen PDC setup. (d) shows |Γ(x 1-x 2,Ω)| for different values of Ω. (e) shows ΓB(x 1-x 2) for different interferential filters.

Fig. 8.
Fig. 8.

The reconstructed near field of a mask with the letters “INFM” using the telescope setup. In the numerical simulations of (a) D 1 was a pixel-like detector, while in (b) D 1 was a bucket detector. (c) shows the convergence rate of the two cases (same notation as in Fig. 6.) 105 shots were used, as well as a larger pump waist than usual (corresponding to 800 µm). (d) shows semi-analytical calculations without time (<1 nm filter) and with time (22 nm filter).

Fig. 9.
Fig. 9.

Non-local filtering using the telescope setup with a bucket detector in the test arm. The setup (a) was as Fig. 1(b) but with a spatial filter inserted in the focal plane of the first lens in the reference arm. (b) shows the correlation using the filter, while (c) shows the correlation without filter. 15000 shots were used.

Equations (18)

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b j ( q , Ω ) = U j ( q , Ω ) a j in ( q , Ω ) + V j ( q , Ω ) a k in ( q , Ω ) , j k = 1 , 2 .
G ( x 1 , x 2 ) = I 1 ( x 1 ) I 2 ( x 2 ) I 1 ( x 1 ) I 2 ( x 2 ) .
G ( x 1 , x 2 ) = d x 1 d x 2 h 1 ( x 1 , x 1 ) h 2 ( x 2 , x 2 ) Γ ( x 1 x 2 , Ω = 0 ) 2 ,
Γ ( x 1 , x 2 , Ω ) d τ e i Ω τ b 1 ( x 1 , t 1 ) b 2 ( x 2 , t 1 + τ ) = d q ( 2 π ) 2 e i q · ( x 1 x 2 ) γ ( q , Ω ) ,
G ( x 1 , x 2 ) = 1 T D d Ω 2 π d x 1 d x 2 h 1 ( x 1 , x 1 ) h 2 ( x 2 , x 2 ) Γ ( x 1 x 2 , Ω ) 2 .
G f ( x 1 , x 2 ) γ ( x 2 k f , Ω = 0 ) T ˜ obj [ ( x 1 + x 2 ) k f ] 2 ,
G f , SA ( x ) d x 1 G f ( x 1 , x x 1 ) d x 1 γ [ ( x 1 x ) k f , Ω = 0 ] T ˜ obj ( x k f ) 2
= T ˜ obj ( x k f ) 2 d x 1 γ [ ( x 1 x ) k f , Ω = 0 ] 2 const × T ˜ obj ( x k f ) 2
ρ SA = δ q PDC δ q p .
G T ( x 1 , x 2 ) d x 1 Γ ( x 1 x 2 , Ω = 0 ) T obj ( x 1 ) e i x 1 · x 1 k f 2
T obj ( x 2 ) 2 γ ( x 1 k f , Ω = 0 ) 2 .
T obj ( x 2 ) 2 γ ( x 1 k f , Ω = 0 ) 2 .
G ̄ T ( x 2 ) = d x 1 G T ( x 1 , x 2 ) d x 1 Γ ( x 1 x 2 , Ω = 0 ) 2 T obj ( x 1 ) 2 ,
ε f ( n ) = ( x G f n ( x ) G f ( x ) 2 ) 1 2
ε fit ( n ) = ( d 0 n ) 1 2 + d 1
G T ( x 1 , x 2 ) d Ω d x 1 Γ ( x 1 x 2 , Ω ) T obj ( x 1 ) e i x 1 · x 1 k f 2 , coherent ( pixel-link D 1 )
G T ( x 2 ) d Ω d x 1 Γ ( x 1 x 2 , Ω ) 2 T obj ( x 1 ) 2 , incoherent ( bucket D 1 ) .
Γ B ( x 1 x 2 ) d Ω 2 π Γ ( x 1 x 2 , Ω ) 2 .

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