Abstract

Unconventional holography called photon correlation holography is proposed and experimentally demonstrated. Using photon correlation, i.e. intensity correlation or fourth order correlation of optical field, a 3-D image of the object recorded in a hologram is reconstructed stochastically with illumination through a random phase screen. Two different schemes for realizing photon correlation holography are examined by numerical simulations, and the experiment was performed for one of the reconstruction schemes suitable for the experimental proof of the principle. The technique of photon correlation holography provides a new insight into how the information is embedded in the spatial as well as temporal correlation of photons in the stochastic pseudo thermal light.

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References

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  1. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52(10), 1123–1128 (1962).
    [CrossRef]
  2. M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005).
    [CrossRef] [PubMed]
  3. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express 17(13), 10633–10641 (2009).
    [CrossRef] [PubMed]
  4. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express 18(13), 13782–13787 (2010).
    [CrossRef] [PubMed]
  5. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Phase-shift coherence holography,” Opt. Lett. 35(10), 1728–1730 (2010).
    [CrossRef] [PubMed]
  6. R. Hanbury Brown and R. Q. Twiss, “Correlations between photons in 2 coherent beams of light,” Nature 177(4497), 27–29 (1956).
    [CrossRef]
  7. R. Q. Twiss, “Applications of intensity interferometry in physics and astronomy,” J. Mod. Opt. 16(4), 423–451 (1969).
  8. 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(5), R3429–R3432 (1995).
    [CrossRef] [PubMed]
  9. R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-Photon” coincidence imaging with a classical source,” Phys. Rev. Lett. 89(11), 113601 (2002).
    [CrossRef] [PubMed]
  10. A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53(5-6), 739–760 (2006).
    [CrossRef]
  11. J. W. Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985), Chap. 6.
  12. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006), Chap. 4.
  13. J. W. Dalle Molle and M. V. Hinich, “Trispectral analysis of stationary random time series,” J. Acoust. Soc. Am. 97(5), 2963–2978 (1995).
    [CrossRef]
  14. B. Picinbono, “Ergodicity and fourth-order spectral moments,” IEEE Trans. Inf. Theory 43(4), 1273–1276 (1997).
    [CrossRef]

2010

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express 18(13), 13782–13787 (2010).
[CrossRef] [PubMed]

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Phase-shift coherence holography,” Opt. Lett. 35(10), 1728–1730 (2010).
[CrossRef] [PubMed]

2009

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express 17(13), 10633–10641 (2009).
[CrossRef] [PubMed]

2006

A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53(5-6), 739–760 (2006).
[CrossRef]

2005

M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005).
[CrossRef] [PubMed]

2002

R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-Photon” coincidence imaging with a classical source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[CrossRef] [PubMed]

1997

B. Picinbono, “Ergodicity and fourth-order spectral moments,” IEEE Trans. Inf. Theory 43(4), 1273–1276 (1997).
[CrossRef]

1995

J. W. Dalle Molle and M. V. Hinich, “Trispectral analysis of stationary random time series,” J. Acoust. Soc. Am. 97(5), 2963–2978 (1995).
[CrossRef]

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(5), R3429–R3432 (1995).
[CrossRef] [PubMed]

1969

R. Q. Twiss, “Applications of intensity interferometry in physics and astronomy,” J. Mod. Opt. 16(4), 423–451 (1969).

1962

E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52(10), 1123–1128 (1962).
[CrossRef]

1956

R. Hanbury Brown and R. Q. Twiss, “Correlations between photons in 2 coherent beams of light,” Nature 177(4497), 27–29 (1956).
[CrossRef]

Bache, M.

A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53(5-6), 739–760 (2006).
[CrossRef]

Bennink, R. S.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-Photon” coincidence imaging with a classical source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[CrossRef] [PubMed]

Bentley, S. J.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-Photon” coincidence imaging with a classical source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[CrossRef] [PubMed]

Boyd, R. W.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-Photon” coincidence imaging with a classical source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[CrossRef] [PubMed]

Brambilla, E.

A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53(5-6), 739–760 (2006).
[CrossRef]

Dalle Molle, J. W.

J. W. Dalle Molle and M. V. Hinich, “Trispectral analysis of stationary random time series,” J. Acoust. Soc. Am. 97(5), 2963–2978 (1995).
[CrossRef]

Duan, Z.

M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005).
[CrossRef] [PubMed]

Ezawa, T.

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express 18(13), 13782–13787 (2010).
[CrossRef] [PubMed]

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Phase-shift coherence holography,” Opt. Lett. 35(10), 1728–1730 (2010).
[CrossRef] [PubMed]

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express 17(13), 10633–10641 (2009).
[CrossRef] [PubMed]

Ferri, F.

A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53(5-6), 739–760 (2006).
[CrossRef]

Gatti, A.

A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53(5-6), 739–760 (2006).
[CrossRef]

Hanbury Brown, R.

R. Hanbury Brown and R. Q. Twiss, “Correlations between photons in 2 coherent beams of light,” Nature 177(4497), 27–29 (1956).
[CrossRef]

Hinich, M. V.

J. W. Dalle Molle and M. V. Hinich, “Trispectral analysis of stationary random time series,” J. Acoust. Soc. Am. 97(5), 2963–2978 (1995).
[CrossRef]

Leith, E. N.

E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52(10), 1123–1128 (1962).
[CrossRef]

Lugiato, L. A.

A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53(5-6), 739–760 (2006).
[CrossRef]

Magatti, D.

A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53(5-6), 739–760 (2006).
[CrossRef]

Miyamoto, Y.

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Phase-shift coherence holography,” Opt. Lett. 35(10), 1728–1730 (2010).
[CrossRef] [PubMed]

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express 18(13), 13782–13787 (2010).
[CrossRef] [PubMed]

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express 17(13), 10633–10641 (2009).
[CrossRef] [PubMed]

M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005).
[CrossRef] [PubMed]

Naik, D. N.

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express 18(13), 13782–13787 (2010).
[CrossRef] [PubMed]

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Phase-shift coherence holography,” Opt. Lett. 35(10), 1728–1730 (2010).
[CrossRef] [PubMed]

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express 17(13), 10633–10641 (2009).
[CrossRef] [PubMed]

Picinbono, B.

B. Picinbono, “Ergodicity and fourth-order spectral moments,” IEEE Trans. Inf. Theory 43(4), 1273–1276 (1997).
[CrossRef]

Pittman, T. B.

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(5), R3429–R3432 (1995).
[CrossRef] [PubMed]

Sergienko, A. V.

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(5), R3429–R3432 (1995).
[CrossRef] [PubMed]

Shih, Y. H.

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(5), R3429–R3432 (1995).
[CrossRef] [PubMed]

Strekalov, D. V.

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(5), R3429–R3432 (1995).
[CrossRef] [PubMed]

Takeda, M.

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express 18(13), 13782–13787 (2010).
[CrossRef] [PubMed]

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Phase-shift coherence holography,” Opt. Lett. 35(10), 1728–1730 (2010).
[CrossRef] [PubMed]

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express 17(13), 10633–10641 (2009).
[CrossRef] [PubMed]

M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005).
[CrossRef] [PubMed]

Twiss, R. Q.

R. Q. Twiss, “Applications of intensity interferometry in physics and astronomy,” J. Mod. Opt. 16(4), 423–451 (1969).

R. Hanbury Brown and R. Q. Twiss, “Correlations between photons in 2 coherent beams of light,” Nature 177(4497), 27–29 (1956).
[CrossRef]

Upatnieks, J.

E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52(10), 1123–1128 (1962).
[CrossRef]

Wang, W.

M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory

B. Picinbono, “Ergodicity and fourth-order spectral moments,” IEEE Trans. Inf. Theory 43(4), 1273–1276 (1997).
[CrossRef]

J. Acoust. Soc. Am.

J. W. Dalle Molle and M. V. Hinich, “Trispectral analysis of stationary random time series,” J. Acoust. Soc. Am. 97(5), 2963–2978 (1995).
[CrossRef]

J. Mod. Opt.

R. Q. Twiss, “Applications of intensity interferometry in physics and astronomy,” J. Mod. Opt. 16(4), 423–451 (1969).

A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53(5-6), 739–760 (2006).
[CrossRef]

J. Opt. Soc. Am.

E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52(10), 1123–1128 (1962).
[CrossRef]

Nature

R. Hanbury Brown and R. Q. Twiss, “Correlations between photons in 2 coherent beams of light,” Nature 177(4497), 27–29 (1956).
[CrossRef]

Opt. Express

M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005).
[CrossRef] [PubMed]

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express 17(13), 10633–10641 (2009).
[CrossRef] [PubMed]

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express 18(13), 13782–13787 (2010).
[CrossRef] [PubMed]

Opt. Lett.

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Phase-shift coherence holography,” Opt. Lett. 35(10), 1728–1730 (2010).
[CrossRef] [PubMed]

Phys. Rev. A

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(5), R3429–R3432 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-Photon” coincidence imaging with a classical source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[CrossRef] [PubMed]

Other

J. W. Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985), Chap. 6.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006), Chap. 4.

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Figures (9)

Fig. 1
Fig. 1

A conceptual diagram for photon correlation holography.

Fig. 2
Fig. 2

Geometry for generation of a Fourier-transform hologram in photon correlation holography.

Fig. 3
Fig. 3

Geometry for reconstruction of hologram in photon correlation holography.

Fig. 4
Fig. 4

(a) Hologram generated by Eq. (3) for simulated reconstruction using time average; (b), (c) and (d) are the 3-D object images reconstructed from the time average cross-covariance of the intensity variations between the reference point at the origin (for which r = 0 and z=0mm ) and other points on the three planes for which z= 2mm , z=0mm and z=2mm , respectively.

Fig. 5
Fig. 5

(a) Hologram generated by Eq. (3) for simulated reconstruction using space average; (b), (c) and (d) are the 3-D object images reconstructed from the space average cross-covariance of the intensity distributions between the two planes for which Δ z = 2 mm , Δ z = 0 mm and Δ z = 2 mm , respectively.

Fig. 6
Fig. 6

The experimental setup for photon correlation holography.

Fig. 7
Fig. 7

Hologram generated by Eq. (3) for experiment using space average.

Fig. 8
Fig. 8

(a)-(l) Magnified images of the 128x128 pixel area selected from the center of the full-field speckle intensity image having resolution of 2048x2048 pixels, recorded at the planes from z = 6 mm to z = 5mm with steps of 1mm. (m) The full-field speckle intensity recorded at z = 0 plane with the CCD camera having total image resolution of 2048x2048 pixels. The yellow square in the center indicates the location of the magnified image area shown in (a)-(l).

Fig. 9
Fig. 9

(a)-(l) The 3-D object images reconstructed from the space average cross-covariance of the intensity distributions C ( Δ x , Δ y , Δ z ) with Δ z varied from Δ z = 6 mm to Δ z = 5 mm with steps of 1mm. (b) represents the reconstructed image O, (g) represents the reconstructed image Γ with its conjugate image and (l) represents the reconstructed conjugate image of O. The focusing function is achieved by intensity correlation.

Equations (21)

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  g ˜ ( x , y ) = [ { g ( x ˜ , y ˜ , z ) exp [ i 2 π λ f ( x ^ x ˜ + y ^ y ˜ ) ] d x ˜ d y ˜ } exp [ i k z ( x ^ , y ^ ) z ] d z ] × exp [ i 2 π λ f ( x x ^ + y y ^ ) ] d x ^ d y ^   .
G ( r ^ ) = G ( x ^ , y ^ ) = g ˜ ( x , y ) exp [ i 2 π λ f ( x x ^ + y y ^ ) ] d x d y
H ( r ^ ) | G ( r ^ ) | + 1 2 G ( r ^ ) + 1 2 G * ( r ^ ) = | G ( r ^ ) | { 1 + cos [ Φ G ( r ^ ) ] }
u ( r ; t ) = H ( r ^ ) exp [ i Φ R ( r ^ , t ) ] exp ( i 2 π λ f r r ^ ) d r ^
I ( r , t ) = u * ( r , t ) u ( r , t ) = | u ( r , t ) | 2
m 4 ( r , r + Δ r , t ) = I ( r , t ) I ( r + Δ r , t ) = u * ( r , t ) u ( r , t ) u ( r + Δ r , t ) u * ( r + Δ r , t )
C ( r , r + Δ r , t ) = Δ I ( r , t ) Δ I ( r + Δ r , t ) = m 4 ( r , r + Δ r , t ) I ¯ 2    
m 4 ( r , r + Δ r , t ) = H ( r ^ 1 ) H ( r ^ 2 ) H ( r ^ 3 ) H ( r ^ 4 ) × exp { i [ Φ R ( r ^ 2 , t ) Φ R ( r ^ 1 , t ) + Φ R ( r ^ 4 , t ) Φ R ( r ^ 3 , t ) ] } × exp [ i 2 π λ f ( r ^ 2 r ^ 1 + r ^ 4 r ^ 3 ) r ] exp [ i 2 π λ f ( r ^ 4 r ^ 3 ) Δ r ] d r ^ 1 d r ^ 2 d r ^ 3 d r ^ 4
m 4 ( r , r + Δ r ) = H ( r ^ 1 ) H ( r ^ 2 ) H ( r ^ 3 ) H ( r ^ 4 ) × exp { i [ Φ R ( r ^ 2 , t ) Φ R ( r ^ 1 , t ) + Φ R ( r ^ 4 , t ) Φ R ( r ^ 3 , t ) ] } T × exp [ i 2 π λ f ( r ^ 2 r ^ 1 + r ^ 4 r ^ 3 ) r ] exp [ i 2 π λ f ( r ^ 4 r ^ 3 ) Δ r ] d r ^ 1 d r ^ 2 d r ^ 3 d r ^ 4 ,
exp { i [ Φ R ( r ^ 2 , t ) Φ R ( r ^ 1 , t ) + Φ R ( r ^ 4 , t ) Φ R ( r ^ 3 , t ) ] } T = exp { i [ Φ R ( r ^ 2 , t ) Φ R ( r ^ 1 , t ) ] } T exp { i [ Φ R ( r ^ 4 , t ) Φ R ( r ^ 3 , t ) ] } T + exp { i [ Φ R ( r ^ 4 , t ) Φ R ( r ^ 1 , t ) ] } T exp { i [ Φ R ( r ^ 2 , t ) Φ R ( r ^ 3 , t ) ] } T = δ ( r ^ 2 r ^ 1 ) δ ( r ^ 4 r ^ 3 ) + δ ( r ^ 4 r ^ 1 ) δ ( r ^ 2 r ^ 3 )
r ^ 2 r ^ 1 = 0 and r ^ 4 r ^ 3 = 0
r ^ 4 r ^ 1 = 0 and r ^ 2 r ^ 3 = 0
m 4 ( Δ r ) = [ H ( r ^ ) d r ^ ] 2 + H ( r ^ 1 ) exp [ i 2 π λ f r ^ 1 Δ r ] d r ^ 1 × H ( r ^ 2 ) exp [ i 2 π λ f r ^ 2 Δ r ] d r ^ 2 = I ¯ 2 + | H ( r ^ ) exp [ i 2 π λ f r ^ Δ r ] d r ^ | 2
C ( Δ r ) = | H ( r ^ ) exp [ i 2 π λ f r ^ Δ r ] d r ^ | 2
m 4 ( r , r + Δ r ) = H ( r ^ 1 ) H ( r ^ 2 ) H ( r ^ 3 ) H ( r ^ 4 ) × exp { i [ Φ R ( r ^ 2 , t ) Φ R ( r ^ 1 , t ) + Φ R ( r ^ 4 , t ) Φ R ( r ^ 3 , t ) ] } × exp [ i 2 π λ f ( r ^ 2 r ^ 1 + r ^ 4 r ^ 3 ) r ] S exp [ i 2 π λ f ( r ^ 4 r ^ 3 ) Δ r ] d r ^ 1 d r ^ 2 d r ^ 3 d r ^ 4 ,
m 4 ( Δ r , t ) = H ( r ^ 1 ) H ( r ^ 2 ) H ( r ^ 3 ) H ( r ^ 4 ) × exp { i [ Φ R ( r ^ 2 , t ) Φ R ( r ^ 1 , t ) + Φ R ( r ^ 4 , t ) Φ R ( r ^ 3 , t ) ] } × δ ( r ^ 2 r ^ 1 + r ^ 4 r ^ 3 ) exp [ i 2 π λ f ( r ^ 4 r ^ 3 ) Δ r ] d r ^ 1 d r ^ 2 d r ^ 3 d r ^ 4
r ^ 2 r ^ 1 = 0 and r ^ 4 r ^ 3 = 0
r ^ 1 + r ^ 3 = 0 and r ^ 2 + r ^ 4 = 0
r ^ 4 r ^ 1 = 0   and r ^ 2 r ^ 3 = 0
m 4 ( Δ r ) = I ¯ 2 + | H ( r ^ ) exp [ i 2 π λ f r ^ Δ r ] d r ^ | 2
C ( Δ r ) = | H ( r ^ ) exp [ i 2 π λ f r ^ Δ r ] d r ^ | 2

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