Abstract

This article reports a study on a turbulence-free Young’s double-slit interferometer. When the environmental turbulence blurs out the classic Young’s double-slit interference completely, a two-photon interference pattern is still observable from the measurement of intensity or photon number fluctuation correlation. This two-photon interferometer always produces a turbulence-free interference pattern, when the double-slit interferometer is utilizing both first-order spatially incoherent light and spatially coherent light. This type of two-photon interferometer establishes new capabilities in optical observations and sensing measurements that require high sensitivity and stability.

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References

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  1. E. Hecht, Optics (Addison Wesley, 2002).
  2. A. R. Thompson, J. M. Moran, and G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (John Wiley & Sons, 2001).
  3. P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors (World Scientific, 1994).
  4. D. C. Williams, Optical Methods in Engineering Metrology (Chapman and Hall, 1993).
  5. D. D. Nolte, Optical Interferometry for Biology and Medicine (Springer, 2011).
  6. M. E. Brezinski, Optical Coherence Tomography: Principles and Applications (Academic, 2006).
  7. Z. Y. Ou and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987).
    [Crossref]
  8. Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61(26), 2921–2924 (1988).
    [Crossref]
  9. J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. 62(19), 2205–2208 (1989).
    [Crossref]
  10. R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177(4497), 27–29 (1956).
    [Crossref]
  11. R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178(4541), 1046–1048 (1956).
    [Crossref]
  12. G. Scarcelli, A. Valencia, and Y. H. Shih, “Two-photon interference with thermal light,” EPL 68(5), 618–624 (2004).
    [Crossref]
  13. Y. S. Ihn, Y. Kim, V. Tamma, and Y. H. Kim, “Second-order temporal interference with thermal light: Interference beyond the coherence time,” Phys. Rev. Lett. 119(26), 263603 (2017).
    [Crossref]
  14. V. I. Tatarski, Wave Propagation in a Turbulent Medium, Translated by R. A. Silverman, (Mcgraw-Hill, 1961).
  15. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
    [Crossref]
  16. The LIGO Scientific Collaboration and Virgo Collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116(6), 061102 (2016).
    [Crossref]
  17. T. A. Smith and Y. H. Shih, “Turbulence-free double-slit interferometer,” Phys. Rev. Lett. 120(6), 063606 (2018).
    [Crossref]
  18. T. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts, (Printed for J. Johnson by W. Savage, 1807).
  19. Y. H. Shih, An Introduction to Quantum Optics: Photon and Biphoton Physics (CRC press, Taylor & Francis, 1st edition, 2011).
  20. G. Scarcelli, V. Berardi, and Y. H. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. 96(6), 063602 (2006).
    [Crossref]
  21. A. Einstein, “Concerning an heuristic point of view toward the emission and transformation of light,” Ann. Phys. 322(6), 132–148 (1905).
    [Crossref]
  22. R. J. Glauber, “The quantum theory of optical coherenece,” Phys. Rev. 130(6), 2529–2539 (1963).
    [Crossref]
  23. M. O. Scully and M. S. Zubairy, Quantum Optics, (Cambridge University, 1997).
  24. W. Martienssen and E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32(12), 919–926 (1964).
    [Crossref]
  25. H. Chen, T. Peng, and Y. H. Shih, “100% correlation of chaotic thermal light,” Phys. Rev. A 88(2), 023808 (2013).
    [Crossref]

2018 (1)

T. A. Smith and Y. H. Shih, “Turbulence-free double-slit interferometer,” Phys. Rev. Lett. 120(6), 063606 (2018).
[Crossref]

2017 (1)

Y. S. Ihn, Y. Kim, V. Tamma, and Y. H. Kim, “Second-order temporal interference with thermal light: Interference beyond the coherence time,” Phys. Rev. Lett. 119(26), 263603 (2017).
[Crossref]

2016 (1)

The LIGO Scientific Collaboration and Virgo Collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116(6), 061102 (2016).
[Crossref]

2013 (1)

H. Chen, T. Peng, and Y. H. Shih, “100% correlation of chaotic thermal light,” Phys. Rev. A 88(2), 023808 (2013).
[Crossref]

2006 (1)

G. Scarcelli, V. Berardi, and Y. H. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. 96(6), 063602 (2006).
[Crossref]

2004 (1)

G. Scarcelli, A. Valencia, and Y. H. Shih, “Two-photon interference with thermal light,” EPL 68(5), 618–624 (2004).
[Crossref]

1989 (1)

J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. 62(19), 2205–2208 (1989).
[Crossref]

1988 (1)

Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61(26), 2921–2924 (1988).
[Crossref]

1987 (1)

Z. Y. Ou and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987).
[Crossref]

1981 (1)

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[Crossref]

1964 (1)

W. Martienssen and E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32(12), 919–926 (1964).
[Crossref]

1963 (1)

R. J. Glauber, “The quantum theory of optical coherenece,” Phys. Rev. 130(6), 2529–2539 (1963).
[Crossref]

1956 (2)

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

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178(4541), 1046–1048 (1956).
[Crossref]

1905 (1)

A. Einstein, “Concerning an heuristic point of view toward the emission and transformation of light,” Ann. Phys. 322(6), 132–148 (1905).
[Crossref]

Alley, C. O.

Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61(26), 2921–2924 (1988).
[Crossref]

Berardi, V.

G. Scarcelli, V. Berardi, and Y. H. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. 96(6), 063602 (2006).
[Crossref]

Brezinski, M. E.

M. E. Brezinski, Optical Coherence Tomography: Principles and Applications (Academic, 2006).

Chen, H.

H. Chen, T. Peng, and Y. H. Shih, “100% correlation of chaotic thermal light,” Phys. Rev. A 88(2), 023808 (2013).
[Crossref]

Einstein, A.

A. Einstein, “Concerning an heuristic point of view toward the emission and transformation of light,” Ann. Phys. 322(6), 132–148 (1905).
[Crossref]

Franson, J. D.

J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. 62(19), 2205–2208 (1989).
[Crossref]

Glauber, R. J.

R. J. Glauber, “The quantum theory of optical coherenece,” Phys. Rev. 130(6), 2529–2539 (1963).
[Crossref]

Hanbury Brown, R.

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

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178(4541), 1046–1048 (1956).
[Crossref]

Hecht, E.

E. Hecht, Optics (Addison Wesley, 2002).

Ihn, Y. S.

Y. S. Ihn, Y. Kim, V. Tamma, and Y. H. Kim, “Second-order temporal interference with thermal light: Interference beyond the coherence time,” Phys. Rev. Lett. 119(26), 263603 (2017).
[Crossref]

Kim, Y.

Y. S. Ihn, Y. Kim, V. Tamma, and Y. H. Kim, “Second-order temporal interference with thermal light: Interference beyond the coherence time,” Phys. Rev. Lett. 119(26), 263603 (2017).
[Crossref]

Kim, Y. H.

Y. S. Ihn, Y. Kim, V. Tamma, and Y. H. Kim, “Second-order temporal interference with thermal light: Interference beyond the coherence time,” Phys. Rev. Lett. 119(26), 263603 (2017).
[Crossref]

Mandel, L.

Z. Y. Ou and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987).
[Crossref]

Martienssen, W.

W. Martienssen and E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32(12), 919–926 (1964).
[Crossref]

Moran, J. M.

A. R. Thompson, J. M. Moran, and G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (John Wiley & Sons, 2001).

Nolte, D. D.

D. D. Nolte, Optical Interferometry for Biology and Medicine (Springer, 2011).

Ou, Z. Y.

Z. Y. Ou and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987).
[Crossref]

Peng, T.

H. Chen, T. Peng, and Y. H. Shih, “100% correlation of chaotic thermal light,” Phys. Rev. A 88(2), 023808 (2013).
[Crossref]

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[Crossref]

Saulson, P. R.

P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors (World Scientific, 1994).

Scarcelli, G.

G. Scarcelli, V. Berardi, and Y. H. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. 96(6), 063602 (2006).
[Crossref]

G. Scarcelli, A. Valencia, and Y. H. Shih, “Two-photon interference with thermal light,” EPL 68(5), 618–624 (2004).
[Crossref]

Scully, M. O.

M. O. Scully and M. S. Zubairy, Quantum Optics, (Cambridge University, 1997).

Shih, Y. H.

T. A. Smith and Y. H. Shih, “Turbulence-free double-slit interferometer,” Phys. Rev. Lett. 120(6), 063606 (2018).
[Crossref]

H. Chen, T. Peng, and Y. H. Shih, “100% correlation of chaotic thermal light,” Phys. Rev. A 88(2), 023808 (2013).
[Crossref]

G. Scarcelli, V. Berardi, and Y. H. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. 96(6), 063602 (2006).
[Crossref]

G. Scarcelli, A. Valencia, and Y. H. Shih, “Two-photon interference with thermal light,” EPL 68(5), 618–624 (2004).
[Crossref]

Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61(26), 2921–2924 (1988).
[Crossref]

Y. H. Shih, An Introduction to Quantum Optics: Photon and Biphoton Physics (CRC press, Taylor & Francis, 1st edition, 2011).

Smith, T. A.

T. A. Smith and Y. H. Shih, “Turbulence-free double-slit interferometer,” Phys. Rev. Lett. 120(6), 063606 (2018).
[Crossref]

Spiller, E.

W. Martienssen and E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32(12), 919–926 (1964).
[Crossref]

Swenson, G. W.

A. R. Thompson, J. M. Moran, and G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (John Wiley & Sons, 2001).

Tamma, V.

Y. S. Ihn, Y. Kim, V. Tamma, and Y. H. Kim, “Second-order temporal interference with thermal light: Interference beyond the coherence time,” Phys. Rev. Lett. 119(26), 263603 (2017).
[Crossref]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium, Translated by R. A. Silverman, (Mcgraw-Hill, 1961).

Thompson, A. R.

A. R. Thompson, J. M. Moran, and G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (John Wiley & Sons, 2001).

Twiss, R. Q.

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178(4541), 1046–1048 (1956).
[Crossref]

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

Valencia, A.

G. Scarcelli, A. Valencia, and Y. H. Shih, “Two-photon interference with thermal light,” EPL 68(5), 618–624 (2004).
[Crossref]

Williams, D. C.

D. C. Williams, Optical Methods in Engineering Metrology (Chapman and Hall, 1993).

Young, T.

T. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts, (Printed for J. Johnson by W. Savage, 1807).

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum Optics, (Cambridge University, 1997).

Am. J. Phys. (1)

W. Martienssen and E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32(12), 919–926 (1964).
[Crossref]

Ann. Phys. (1)

A. Einstein, “Concerning an heuristic point of view toward the emission and transformation of light,” Ann. Phys. 322(6), 132–148 (1905).
[Crossref]

EPL (1)

G. Scarcelli, A. Valencia, and Y. H. Shih, “Two-photon interference with thermal light,” EPL 68(5), 618–624 (2004).
[Crossref]

Nature (2)

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

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178(4541), 1046–1048 (1956).
[Crossref]

Phys. Rev. (1)

R. J. Glauber, “The quantum theory of optical coherenece,” Phys. Rev. 130(6), 2529–2539 (1963).
[Crossref]

Phys. Rev. A (1)

H. Chen, T. Peng, and Y. H. Shih, “100% correlation of chaotic thermal light,” Phys. Rev. A 88(2), 023808 (2013).
[Crossref]

Phys. Rev. Lett. (7)

Z. Y. Ou and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987).
[Crossref]

Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61(26), 2921–2924 (1988).
[Crossref]

J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. 62(19), 2205–2208 (1989).
[Crossref]

Y. S. Ihn, Y. Kim, V. Tamma, and Y. H. Kim, “Second-order temporal interference with thermal light: Interference beyond the coherence time,” Phys. Rev. Lett. 119(26), 263603 (2017).
[Crossref]

G. Scarcelli, V. Berardi, and Y. H. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. 96(6), 063602 (2006).
[Crossref]

The LIGO Scientific Collaboration and Virgo Collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116(6), 061102 (2016).
[Crossref]

T. A. Smith and Y. H. Shih, “Turbulence-free double-slit interferometer,” Phys. Rev. Lett. 120(6), 063606 (2018).
[Crossref]

Prog. Opt. (1)

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[Crossref]

Other (10)

V. I. Tatarski, Wave Propagation in a Turbulent Medium, Translated by R. A. Silverman, (Mcgraw-Hill, 1961).

T. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts, (Printed for J. Johnson by W. Savage, 1807).

Y. H. Shih, An Introduction to Quantum Optics: Photon and Biphoton Physics (CRC press, Taylor & Francis, 1st edition, 2011).

E. Hecht, Optics (Addison Wesley, 2002).

A. R. Thompson, J. M. Moran, and G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (John Wiley & Sons, 2001).

P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors (World Scientific, 1994).

D. C. Williams, Optical Methods in Engineering Metrology (Chapman and Hall, 1993).

D. D. Nolte, Optical Interferometry for Biology and Medicine (Springer, 2011).

M. E. Brezinski, Optical Coherence Tomography: Principles and Applications (Academic, 2006).

M. O. Scully and M. S. Zubairy, Quantum Optics, (Cambridge University, 1997).

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Figures (4)

Fig. 1.
Fig. 1. Overlap of two-photon amplitudes. In the figure, the superposed two different yet indistinguishable two-photon amplitudes are indicated by red and blue colors. When the detectors are scanning in the neighborhood of $x_1 \approx x_2$, the red amplitude and the blue amplitude “overlap” which means the pair experience the same phase variations, resulting in an interference pattern insensitive to turbulence.
Fig. 2.
Fig. 2. Experimental setup. Similar to the turbulence-free double-slit interferometer, a pseudo-thermal light source is directed at a pair of slits, except now the source is far enough away to be partially coherent, $l_c > d$. Beyond the slits, point-like tips of single mode fibers collect light and direct it to a pair of single-photon detectors (Perkin-Elmer SPCM-AQR). A Photon Number Fluctuation Correlation (PNFC) circuit then measures the photon number fluctuation correlation $\langle \Delta n(x_1) \Delta n(x_2) \rangle$.
Fig. 3.
Fig. 3. Typical measurement of $\langle \Delta n(x_1) \Delta n(x_2) \rangle$ when scanning $x_1$ and $x_2$ is stationary. (a) Without turbulence present, the interference pattern has a large amplitude and approximately 100% visibility. (b) With turbulence present, certain terms no longer contribute to the measurement, thus lowering the amplitude and decreasing the visibility.
Fig. 4.
Fig. 4. Typical measurement of $\langle n(x_1) \rangle$ when scanning $x_1$. (a) As expected with a partially coherent source, the interference pattern does not have 100% visibility, but an interference pattern is still clearly visible. (b) After introducing turbulence, the interference pattern is blurred completely.

Equations (51)

Equations on this page are rendered with MathJax. Learn more.

E ( r j , t j ) = m E m ( r j , t j ) = m E m g m ( r j , t j ) ,
I ( r j , t j ) = m E m ( r j , t j ) n E n ( x j ) = m | E m ( r j , t j ) | 2 + m n E m ( r j , t j ) E n ( r j , t j ) = m | E m ( r j , t j ) | 2 + 0.
Δ I ( r 1 , t 1 ) Δ I ( r 2 , t 2 ) = m n E m ( r 1 , t 1 ) E n ( r 1 , t 1 ) p q E p ( r 2 , t 2 ) E q ( r 2 , t 2 ) = m n E m ( r 1 , t 1 ) E n ( r 1 , t 1 ) E n ( r 2 , t 2 ) E m ( r 2 , t 2 ) .
m n | E m ( r 1 , t 1 ) E n ( r 2 , t 2 ) + E n ( r 1 , t 1 ) E m ( r 2 , t 2 ) | 2 ,
I ( r 1 , t 1 ) I ( r 2 , t 2 ) = m n | 1 2 [ E m ( r 1 , t 1 ) E n ( r 2 , t 2 ) + E n ( r 1 , t 1 ) E m ( r 2 , t 2 ) ] | 2         = m | E m ( r 1 , t 1 ) | 2 n | E n ( r 2 , t 2 ) | 2 + m n E m ( r 1 , t 1 ) E n ( r 1 , t 1 ) E n ( r 2 , t 2 ) E m ( r 2 , t 2 )         = I ( r 1 , t 1 ) I ( r 2 , t 2 ) + Δ I ( r 1 , t 1 ) Δ I ( r 2 , t 2 ) .
E ^ ( + ) ( r j , t j ) = m d k a ^ m ( k ) g m ( k ; r j , t j ) E ^ ( ) ( r j , t j ) = m d k a ^ m ( k ) g m ( k ; r j , t j )
G ( 1 ) ( r 1 , t 1 ) = Ψ | E ^ ( ) ( r 1 , t 1 ) E ^ ( + ) ( r 1 , t 1 ) | Ψ E s = Ψ | m E ^ m ( ) ( r 1 , t 1 ) n E ^ n ( + ) ( r 1 , t 1 ) | Ψ E s = m = n | ψ m ( r 1 , t 1 ) | 2 E s + m n ψ m ( r 1 , t 1 ) ψ n ( r 1 , t 1 ) E s = n ( r 1 , t 1 ) + 0 ,
ψ m ( r 1 , t 1 ) = d k α m ( k ) g m ( k ; r 1 , t 1 ) .
G ( 2 ) ( r 1 , t 1 ; r 2 , t 2 ) = Ψ | m E ^ m ( ) ( r 1 , t 1 ) n E ^ n ( ) ( r 2 , t 2 ) p E ^ p ( + ) ( r 2 , t 2 ) q E ^ q ( + ) ( r 1 , t 1 ) | Ψ E s = m | ψ m ( r 1 , t 1 ) | 2 E s n | ψ n ( r 2 , t 2 ) | 2 E s + m n ψ m ( r 1 , t 1 ) ψ n ( r 1 , t 1 ) ψ n ( r 2 , t 2 ) ψ m ( r 2 , t 2 ) E s = m , n | ψ m ( r 1 , t 1 ) ψ n ( r 2 , t 2 ) + ψ n ( r 1 , t 1 ) ψ m ( r 2 , t 2 ) | 2 = n ( r 1 , t 1 ) n ( r 2 , t 2 ) + Δ n ( r 1 , t 1 ) Δ n ( r 2 , t 2 ) .
g m ( r j , t j ) = d k g m ( k ; r j , t j ) d k e i [ k ( r j r m ) ω ( t j t m ) ] ,
g m ( x j ) = e i k z e i k 2 z ( x j x m ) 2 = e i 2 π n 0 z λ e i π n 0 λ z ( x j x m ) 2
g m ( x j ) = 1 2 [ g m A ( x j ) + g m B ( x j ) ] ,
I ( x 1 ) I ( x 2 ) = I 0 2 cos 2 π n 0 d λ z x 1 cos 2 π n 0 d λ z x 2 Δ I ( x 1 ) Δ I ( x 2 ) = I 0 2 cos 2 π n 0 d λ z x 1 cos 2 π n 0 d λ z x 2 ,
I ( x 1 ) I ( x 2 ) = 2 I 0 2 cos 2 π n 0 d λ z x 1 cos 2 π n 0 d λ z x 2 .
I ( x j ) = 1 2 m | E m | 2 | g m A ( x j ) + g m B ( x j ) | 2 = I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ m | g m A ( x j ) + g m B ( x j ) | 2 = I 0 2 [ 1 + sinc π n 0 d Δ θ s λ cos 2 π n 0 d λ z x j ] ,
V I m a x I m i n I m a x + I m i n = sinc π n 0 d Δ θ s λ .
I ( x 1 ) I ( x 2 ) = I 0 2 4 [ 1 + sinc π n 0 d Δ θ s λ cos 2 π n 0 d λ z x 1 ] [ 1 + sinc π n 0 d Δ θ s λ cos 2 π n 0 d λ z x 2 ] .
V = 2 sinc [ π n 0 d Δ θ s / λ ] 1 + sinc 2 [ π n 0 d Δ θ s / λ ] .
V = sinc π n 0 d Δ θ s λ .
Δ I ( x 1 ) Δ I ( x 2 ) = 1 4 m n | E m | 2 | E n | 2 [ g m A ( x 1 ) + g m B ( x 1 ) ] [ g n A ( x 1 ) + g n B ( x 1 ) ] × [ g n A ( x 2 ) + g n B ( x 2 ) ] [ g m A ( x 2 ) + g m B ( x 2 ) ] .
Δ I ( x 1 ) Δ I ( x 2 ) = i j k l Δ I ( x 1 ) i j Δ I ( x 2 ) k l ,
Δ I ( x 1 ) i j Δ I ( x 2 ) k l = 1 4 m n | E m | 2 | E n | 2 g m i ( x 1 ) g n j ( x 1 ) g n k ( x 2 ) g m l ( x 2 ) ,
Δ I ( x 1 ) Δ I ( x 2 ) = I 0 2 [ cos π n 0 d λ z ( x 1 x 2 ) + sinc π n 0 d Δ θ s λ cos π n 0 d λ z ( x 1 + x 2 ) ] 2 ,
Δ I ( x 1 ) Δ I ( x 2 ) = I 0 2 cos 2 π n 0 d λ z ( x 1 x 2 ) .
n 0 = 1 V V d r   1 τ τ d t   n ( r , t ) .
n i j = 1 | r j r i | r i r j d r   1 | t j t i | t i t j d t   n ( r , t ) .
g i T ( x j ) = e i 2 π n i j z λ e i π n i j λ z ( x j x i ) 2 = e i 2 π n 0 z λ e i π n 0 λ z ( x j x i ) 2 e i 2 π δ n i j z λ e i π δ n i j λ z ( x j x i ) 2
g i T ( x j ) = e i 2 π n 0 z λ e i π n 0 λ z ( x j x i ) 2 e i δ ϕ i j = g i ( x j ) e i δ ϕ i j ,
I ( x j ) = I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ m 1 Δ ϕ Δ ϕ / 2 Δ ϕ / 2 d ϕ j | g m A T ( x j ) + g m B T ( x j ) | 2 = I 0 2 [ 1 + sinc Δ ϕ   sinc π n 0 d Δ θ s λ cos 2 π n 0 d λ z x j ] ,
Δ I ( x 1 ) i j Δ I ( x 2 ) k l = 1 4 m n | E m | 2 | E n | 2 g m i ( x 1 ) e i δ ϕ i 1 g n j ( x 1 ) e i δ ϕ j 1 g n k ( x 2 ) e i δ ϕ k 2 g m l ( x 2 ) e i δ ϕ l 2 ,
I ( x 1 ) i j I ( x 2 ) k l = m n | E m i ( x 1 ) e i δ ϕ i 1 E n k ( x 2 ) e i δ ϕ k 2 + E n j ( x 1 ) e i δ ϕ j 1 E m l ( x 2 ) e i δ ϕ l 2 | 2 .
Δ I ( x 1 ) Δ I ( x 2 ) = I 0 2 [ 1 2 sinc 2 π n 0 d Δ θ s λ + cos 2 π n 0 d λ z ( x 1 x 2 ) ] .
V = 1 1 + sinc 2 [ π n 0 d Δ θ s / λ ] .
Δ I ( x 1 ) Δ I ( x 2 ) = i j k l Δ I ( x 1 ) i j Δ I ( x 2 ) k l ,
Δ I ( x 1 ) i j Δ I ( x 2 ) k l = 1 4 m n | E m | 2 | E n | 2 g m i ( x 1 ) g n j ( x 1 ) g n k ( x 2 ) g m l ( x 2 ) ,
Δ I ( x 1 ) A A Δ I ( x 2 ) A A + Δ I ( x 1 ) B B Δ I ( x 2 ) B B = I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ m I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ n × [ g m A ( x 1 ) g n A ( x 1 ) g n A ( x 2 ) g m A ( x 2 ) + g m B ( x 1 ) g n B ( x 1 ) g n B ( x 2 ) g m B ( x 2 ) ] = 1 2 I 0 2 .
Δ I ( x 1 ) A A Δ I ( x 2 ) B B + Δ I ( x 1 ) B B Δ I ( x 2 ) A A = I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ m I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ n × [ g m A ( x 1 ) g n A ( x 1 ) g n B ( x 2 ) g m B ( x 2 ) + g m B ( x 1 ) g n B ( x 1 ) g n A ( x 2 ) g m A ( x 2 ) ] = 1 2 I 0 2 sinc 2 π d Δ θ s λ .
Δ I ( x 1 ) A B Δ I ( x 2 ) A A + Δ I ( x 1 ) B A Δ I ( x 2 ) B B + Δ I ( x 1 ) A B Δ I ( x 2 ) B B + Δ I ( x 1 ) B A Δ I ( x 2 ) A A = I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ m I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ n × [ g m A ( x 1 ) g n B ( x 1 ) g n A ( x 2 ) g m A ( x 2 ) + g m B ( x 1 ) g n A ( x 1 ) g n B ( x 2 ) g m B ( x 2 ) + g m A ( x 1 ) g n B ( x 1 ) g n B ( x 2 ) g m B ( x 2 ) + g m B ( x 1 ) g n A ( x 1 ) g n A ( x 2 ) g m A ( x 2 ) ] = I 0 2 sinc π d Δ θ s λ cos 2 π d λ z x 1 .
Δ I ( x 1 ) A A Δ I ( x 2 ) A B + Δ I ( x 1 ) B B Δ I ( x 2 ) B A + Δ I ( x 1 ) A A Δ I ( x 2 ) B A + Δ I ( x 1 ) B B Δ I ( x 2 ) A B = I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ m I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ n × [ g m A ( x 1 ) g n A ( x 1 ) g n A ( x 2 ) g m B ( x 2 ) + g m B ( x 1 ) g n B ( x 1 ) g n B ( x 2 ) g m A ( x 2 ) + g m A ( x 1 ) g n A ( x 1 ) g n B ( x 2 ) g m A ( x 2 ) + g m B ( x 1 ) g n B ( x 1 ) g n A ( x 2 ) g m B ( x 2 ) ] = I 0 2 sinc π d Δ θ s λ cos 2 π d λ z x 2 .
Δ I ( x 1 ) A B Δ I ( x 2 ) B A + Δ I ( x 1 ) B A Δ I ( x 2 ) A B = I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ m I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ n × [ g m A ( x 1 ) g n B ( x 1 ) g n B ( x 2 ) g m A ( x 2 ) + g m B ( x 1 ) g n A ( x 1 ) g n A ( x 2 ) g m B ( x 2 ) ] = 1 2 I 0 2 cos 2 π d λ z ( x 1 x 2 ) .
Δ I ( x 1 ) A B Δ I ( x 2 ) A B + Δ I ( x 1 ) B A Δ I ( x 2 ) B A = I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ m I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ n × [ g m A ( x 1 ) g n B ( x 1 ) g n A ( x 2 ) g m B ( x 2 ) + g m B ( x 1 ) g n A ( x 1 ) g n B ( x 2 ) g m A ( x 2 ) ] = 1 2 I 0 2 sinc 2 π d Δ θ s λ cos 2 π d λ z ( x 1 + x 2 ) .
Δ I ( x 1 ) Δ I ( x 2 ) = 1 2 I 0 2 [ 1 + sinc 2 π d Δ θ s λ + 2 sinc π d Δ θ s λ cos 2 π d λ z x 1 + 2 sinc π d Δ θ s λ cos 2 π d λ z x 2 + cos 2 π d λ z ( x 1 x 2 ) + sinc 2 π d Δ θ s λ cos 2 π d λ z ( x 1 + x 2 ) ] ,
Δ I ( x 1 ) Δ I ( x 2 ) = i j k l Δ I ( x 1 ) i j Δ I ( x 2 ) k l ,
Δ I ( x 1 ) i j Δ I ( x 2 ) k l = 1 4 m n | E m | 2 | E n | 2 g m i ( x 1 ) e i δ ϕ i 1 g n j ( x 1 ) e i δ ϕ j 1 g n k ( x 2 ) e i δ ϕ k 2 g m l ( x 2 ) e i δ ϕ l 2 ,
Δ I ( x 1 ) A A Δ I ( x 2 ) A A + Δ I ( x 1 ) B B Δ I ( x 2 ) B B = I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ m I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ n 1 Δ ϕ Δ ϕ / 2 Δ ϕ / 2 d ϕ 1 1 Δ ϕ Δ ϕ / 2 Δ ϕ / 2 d ϕ 2 × [ g m A ( x 1 ) e i δ ϕ A 1 g n A ( x 1 ) e i δ ϕ A 1 g n A ( x 2 ) e i δ ϕ A 2 g m A ( x 2 ) e i δ ϕ A 2 + g m B ( x 1 ) e i δ ϕ B 1 g n B ( x 1 ) e i δ ϕ B 1 g n B ( x 2 ) e i δ ϕ B 2 g m B ( x 2 ) e i δ ϕ B 2 ] = 1 2 I 0 2 .
Δ I ( x 1 ) A A Δ I ( x 2 ) B B + Δ I ( x 1 ) B B Δ I ( x 2 ) A A = I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ m I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ n 1 Δ ϕ Δ ϕ / 2 Δ ϕ / 2 d ϕ 1 1 Δ ϕ Δ ϕ / 2 Δ ϕ / 2 d ϕ 2 × [ g m A ( x 1 ) e i δ ϕ A 1 g n A ( x 1 ) e i δ ϕ A 1 g n B ( x 2 ) e i δ ϕ B 2 g m B ( x 2 ) e i δ ϕ B 2 + g m B ( x 1 ) e i δ ϕ B 1 g n B ( x 1 ) e i δ ϕ B 1 g n A ( x 2 ) e i δ ϕ A 2 g m A ( x 2 ) e i δ ϕ A 2 ] = 1 2 I 0 2 sinc 2 π d Δ θ s λ .
Δ I ( x 1 ) A B Δ I ( x 2 ) A A + Δ I ( x 1 ) B A Δ I ( x 2 ) B B + Δ I ( x 1 ) A B Δ I ( x 2 ) B B + Δ I ( x 1 ) B A Δ I ( x 2 ) A A = I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ m I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ n 1 Δ ϕ Δ ϕ / 2 Δ ϕ / 2 d ϕ 1 1 Δ ϕ Δ ϕ / 2 Δ ϕ / 2 d ϕ 2 × [ g m A ( x 1 ) e i δ ϕ A 1 g n B ( x 1 ) e i δ ϕ B 1 g n A ( x 2 ) e i δ ϕ A 2 g m A ( x 2 ) e i δ ϕ A 2 + g m B ( x 1 ) e i δ ϕ B 1 g n A ( x 1 ) e i δ ϕ A 1 g n B ( x 2 ) e i δ ϕ B 2 g m B ( x 2 ) e i δ ϕ B 2 + g m A ( x 1 ) e i δ ϕ A 1 g n B ( x 1 ) e i δ ϕ B 1 s g n B ( x 2 ) e i δ ϕ B 2 g m B ( x 2 ) e i δ ϕ B 2 + g m B ( x 1 ) e i δ ϕ B 1 g n A ( x 1 ) e i δ ϕ A 1 g n A ( x 2 ) e i δ ϕ A 2 g m A ( x 2 ) e i δ ϕ A 2 ] = I 0 2 sinc Δ ϕ   sinc π d Δ θ s λ cos 2 π d λ z x 1 ,
Δ I ( x 1 ) A A Δ I ( x 2 ) A B + Δ I ( x 1 ) B B Δ I ( x 2 ) B A + Δ I ( x 1 ) A A Δ I ( x 2 ) B A + Δ I ( x 1 ) B B Δ I ( x 2 ) A B = I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ m I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ n 1 Δ ϕ Δ ϕ / 2 Δ ϕ / 2 d ϕ 1 1 Δ ϕ Δ ϕ / 2 Δ ϕ / 2 d ϕ 2 × [ g m A ( x 1 ) e i δ ϕ A 1 g n A ( x 1 ) e i δ ϕ A 1 g n A ( x 2 ) e i δ ϕ A 2 g m B ( x 2 ) e i δ ϕ B 2 + g m B ( x 1 ) e i δ ϕ B 1 g n B ( x 1 ) e i δ ϕ B 1 g n B ( x 2 ) e i δ ϕ B 2 g m A ( x 2 ) e i δ ϕ A 2 + g m A ( x 1 ) e i δ ϕ A 1 g n A ( x 1 ) e i δ ϕ A 1 g n B ( x 2 ) e i δ ϕ B 2 g m A ( x 2 ) e i δ ϕ A 2 + g m B ( x 1 ) e i δ ϕ B 1 g n B ( x 1 ) e i δ ϕ B 1 g n A ( x 2 ) e i δ ϕ A 2 g m B ( x 2 ) e i δ ϕ B 2 ] = I 0 2 sinc Δ ϕ   sinc π d Δ θ s λ cos 2 π d λ z x 2 .
Δ I ( x 1 ) A B Δ I ( x 2 ) B A + Δ I ( x 1 ) B A Δ I ( x 2 ) A B = I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ m I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ n 1 Δ ϕ Δ ϕ / 2 Δ ϕ / 2 d ϕ 1 1 Δ ϕ Δ ϕ / 2 Δ ϕ / 2 d ϕ 2 × [ g m A ( x 1 ) e i δ ϕ A 1 g n B ( x 1 ) e i δ ϕ B 1 g n B ( x 2 ) e i δ ϕ B 2 g m A ( x 2 ) e i δ ϕ A 2 + g m B ( x 1 ) e i δ ϕ B 1 g n A ( x 1 ) e i δ ϕ A 1 g n A ( x 2 ) e i δ ϕ A 2 g m B ( x 2 ) e i δ ϕ B 2 ] = 1 2 I 0 2 cos 2 π d λ z ( x 1 x 2 ) .
Δ I ( x 1 ) A B Δ I ( x 2 ) A B + Δ I ( x 1 ) B A Δ I ( x 2 ) B A = I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ m I 0 2 Δ θ s Δ θ s / 2 Δ θ s / 2 d θ n 1 Δ ϕ Δ ϕ / 2 Δ ϕ / 2 d ϕ 1 1 Δ ϕ Δ ϕ / 2 Δ ϕ / 2 d ϕ 2 × [ g m A ( x 1 ) e i δ ϕ A 1 g n B ( x 1 ) e i δ ϕ B 1 g n A ( x 2 ) e i δ ϕ A 2 g m B ( x 2 ) e i δ ϕ B 2 + g m B ( x 1 ) e i δ ϕ B 1 g n A ( x 1 ) e i δ ϕ A 1 g n B ( x 2 ) e i δ ϕ B 2 g m A ( x 2 ) e i δ ϕ A 2 ] = 1 2 I 0 2 sinc 2 Δ ϕ   sinc 2 π d Δ θ s λ cos 2 π d λ z ( x 1 + x 2 ) .
Δ I ( x 1 ) Δ I ( x 2 ) = 1 2 I 0 2 [ 1 + sinc 2 π d Δ θ s λ + 2 sinc Δ ϕ   sinc π d Δ θ s λ cos 2 π d λ z x 1 + 2 sinc Δ ϕ   sinc π d Δ θ s λ cos 2 π d λ z x 2 + cos 2 π d λ z ( x 1 x 2 ) + sinc 2 Δ ϕ   sinc 2 π d Δ θ s λ cos 2 π d λ z ( x 1 + x 2 ) ] ,

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