Abstract

This article presents a method to simulate a three-dimensional (3D) electromagnetic Gaussian-Schell model (EGSM) source with desired characteristics. Using the complex screen method, originally developed for the synthesis of two-dimensional stochastic electromagnetic fields, a set of equations is derived which relate the desired 3D source characteristics to those of the statistics of the random complex screen. From these equations and the 3D EGSM source realizability conditions, a single criterion is derived, which when satisfied guarantees both the realizability and simulatability of the desired 3D EGSM source. Lastly, a 3D EGSM source, with specified properties, is simulated; the Monte Carlo simulation results are compared to the theoretical expressions to validate the method.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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2017 (4)

2016 (3)

A. T. Friberg and T. Setälä, “Electromagnetic theory of optical coherence,” J. Opt. Soc. Am. A 33, 2431–2442 (2016).
[Crossref]

I. Iliopoulos, M. Casaletti, R. Sauleau, P. Pouliguen, P. Potier, and M. Ettorre, “3-D shaping of a focused aperture in the near field,” IEEE Trans. Antennas Propag. 64, 5262–5271 (2016).
[Crossref]

M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Applied 6, 064030 (2016).
[Crossref]

2015 (2)

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Generating partially coherent Schell-model sources using a modified phase screen approach,” Opt. Eng. 54, 120501 (2015).
[Crossref]

D. Voelz, X. Xiao, and O. Korotkova, “Numerical modeling of Schell-model beams with arbitrary far-field patterns,” Opt. Lett. 40, 352–355 (2015).
[Crossref] [PubMed]

2014 (6)

2013 (2)

2011 (1)

2010 (2)

2006 (1)

2005 (2)

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[Crossref]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 7, 232–237 (2005).
[Crossref]

2004 (1)

2003 (1)

2002 (2)

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002).
[Crossref]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

Abramochkin, E.

Ahad, L.

Alieva, T.

Avramov-Zamurovic, S.

S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, and O. Korotkova, “Polarization-induced reduction in scintillation of optical beams propagating in simulated turbulent atmospheric channels,” Waves Random Complex Media 24, 452–462 (2014).
[Crossref]

Basu, S.

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Generating partially coherent Schell-model sources using a modified phase screen approach,” Opt. Eng. 54, 120501 (2015).
[Crossref]

S. Basu, M. W. Hyde, X. Xiao, D. G. Voelz, and O. Korotkova, “Computational approaches for generating electromagnetic Gaussian Schell-model sources,” Opt. Express 22, 31691–31707 (2014).
[Crossref]

Borghi, R.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002).
[Crossref]

Bose-Pillai, S.

M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Applied 6, 064030 (2016).
[Crossref]

Bose-Pillai, S. R.

Cai, Y.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review,” J. Opt. Soc. Am. A 31, 2083–2096 (2014).
[Crossref]

Casaletti, M.

I. Iliopoulos, M. Casaletti, R. Sauleau, P. Pouliguen, P. Potier, and M. Ettorre, “3-D shaping of a focused aperture in the near field,” IEEE Trans. Antennas Propag. 64, 5262–5271 (2016).
[Crossref]

Castro, I.

Chang, C.

Chen, H.

Chen, Y.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review,” J. Opt. Soc. Am. A 31, 2083–2096 (2014).
[Crossref]

Chen, Z.

Ding, J.

Dogariu, A.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[Crossref]

Ellis, J.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[Crossref]

Ettorre, M.

I. Iliopoulos, M. Casaletti, R. Sauleau, P. Pouliguen, P. Potier, and M. Ettorre, “3-D shaping of a focused aperture in the near field,” IEEE Trans. Antennas Propag. 64, 5262–5271 (2016).
[Crossref]

Friberg, A. T.

Gao, Y.

Gbur, G.

G. Gbur, “Partially coherent beam propagation in atmospheric turbulence,” J. Opt. Soc. Am. A 31, 2038–2045 (2014).
[Crossref]

G. Gbur and T. Visser, “The structure of partially coherent fields,” Prog. Opt. 55, 285–341 (2010).
[Crossref]

Gori, F.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002).
[Crossref]

Hanson, S. G.

Hao, J.

Hyde, M. W.

X. Xiao, D. G. Voelz, S. R. Bose-Pillai, and M. W. Hyde, “Modeling random screens for predefined electromagnetic Gaussian-Schell model sources,” Opt. Express 25, 3656–3665 (2017).
[Crossref] [PubMed]

M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Applied 6, 064030 (2016).
[Crossref]

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Generating partially coherent Schell-model sources using a modified phase screen approach,” Opt. Eng. 54, 120501 (2015).
[Crossref]

S. Basu, M. W. Hyde, X. Xiao, D. G. Voelz, and O. Korotkova, “Computational approaches for generating electromagnetic Gaussian Schell-model sources,” Opt. Express 22, 31691–31707 (2014).
[Crossref]

Iliopoulos, I.

I. Iliopoulos, M. Casaletti, R. Sauleau, P. Pouliguen, P. Potier, and M. Ettorre, “3-D shaping of a focused aperture in the near field,” IEEE Trans. Antennas Propag. 64, 5262–5271 (2016).
[Crossref]

Kaivola, M.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

Korotkova, O.

O. Korotkova, L. Ahad, and T. Setälä, “Three-dimensional electromagnetic Gaussian Schell-model sources,” Opt. Lett. 42, 1792–1795 (2017).
[Crossref] [PubMed]

D. Voelz, X. Xiao, and O. Korotkova, “Numerical modeling of Schell-model beams with arbitrary far-field patterns,” Opt. Lett. 40, 352–355 (2015).
[Crossref] [PubMed]

S. Basu, M. W. Hyde, X. Xiao, D. G. Voelz, and O. Korotkova, “Computational approaches for generating electromagnetic Gaussian Schell-model sources,” Opt. Express 22, 31691–31707 (2014).
[Crossref]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, and O. Korotkova, “Polarization-induced reduction in scintillation of optical beams propagating in simulated turbulent atmospheric channels,” Waves Random Complex Media 24, 452–462 (2014).
[Crossref]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 7, 232–237 (2005).
[Crossref]

O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21, 2382–2385 (2004).
[Crossref]

O. Korotkova, Random Light Beams: Theory and Applications (CRC, 2014).

Liu, L.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Liu, X.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Mack, C. A.

Malek-Madani, R.

S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, and O. Korotkova, “Polarization-induced reduction in scintillation of optical beams propagating in simulated turbulent atmospheric channels,” Waves Random Complex Media 24, 452–462 (2014).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Mei, Z.

Mondello, A.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002).
[Crossref]

Nelson, C.

S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, and O. Korotkova, “Polarization-induced reduction in scintillation of optical beams propagating in simulated turbulent atmospheric channels,” Waves Random Complex Media 24, 452–462 (2014).
[Crossref]

Nie, S.

Piquero, G.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002).
[Crossref]

Ponomarenko, S.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[Crossref]

Potier, P.

I. Iliopoulos, M. Casaletti, R. Sauleau, P. Pouliguen, P. Potier, and M. Ettorre, “3-D shaping of a focused aperture in the near field,” IEEE Trans. Antennas Propag. 64, 5262–5271 (2016).
[Crossref]

Pouliguen, P.

I. Iliopoulos, M. Casaletti, R. Sauleau, P. Pouliguen, P. Potier, and M. Ettorre, “3-D shaping of a focused aperture in the near field,” IEEE Trans. Antennas Propag. 64, 5262–5271 (2016).
[Crossref]

Rodrigo, J. A.

Romanini, P.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002).
[Crossref]

Santarsiero, M.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002).
[Crossref]

Sauleau, R.

I. Iliopoulos, M. Casaletti, R. Sauleau, P. Pouliguen, P. Potier, and M. Ettorre, “3-D shaping of a focused aperture in the near field,” IEEE Trans. Antennas Propag. 64, 5262–5271 (2016).
[Crossref]

Setälä, T.

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

Shirai, T.

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 7, 232–237 (2005).
[Crossref]

Tervo, J.

Visser, T.

G. Gbur and T. Visser, “The structure of partially coherent fields,” Prog. Opt. 55, 285–341 (2010).
[Crossref]

Voelz, D.

Voelz, D. G.

X. Xiao, D. G. Voelz, S. R. Bose-Pillai, and M. W. Hyde, “Modeling random screens for predefined electromagnetic Gaussian-Schell model sources,” Opt. Express 25, 3656–3665 (2017).
[Crossref] [PubMed]

M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Applied 6, 064030 (2016).
[Crossref]

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Generating partially coherent Schell-model sources using a modified phase screen approach,” Opt. Eng. 54, 120501 (2015).
[Crossref]

S. Basu, M. W. Hyde, X. Xiao, D. G. Voelz, and O. Korotkova, “Computational approaches for generating electromagnetic Gaussian Schell-model sources,” Opt. Express 22, 31691–31707 (2014).
[Crossref]

Wang, F.

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review,” J. Opt. Soc. Am. A 31, 2083–2096 (2014).
[Crossref]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Wang, H.-T.

Wolf, E.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[Crossref]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 7, 232–237 (2005).
[Crossref]

O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21, 2382–2385 (2004).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Xia, J.

Xiao, X.

Yu, J.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Yu, Z.

Yura, H. T.

Zhao, C.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

I. Iliopoulos, M. Casaletti, R. Sauleau, P. Pouliguen, P. Potier, and M. Ettorre, “3-D shaping of a focused aperture in the near field,” IEEE Trans. Antennas Propag. 64, 5262–5271 (2016).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 7, 232–237 (2005).
[Crossref]

J. Opt. Soc. Am. A (5)

Opt. Commun. (2)

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[Crossref]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002).
[Crossref]

Opt. Eng. (1)

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Generating partially coherent Schell-model sources using a modified phase screen approach,” Opt. Eng. 54, 120501 (2015).
[Crossref]

Opt. Express (6)

Opt. Lett. (3)

Phys. Rev. A (1)

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Phys. Rev. Applied (1)

M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Applied 6, 064030 (2016).
[Crossref]

Phys. Rev. E (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

Prog. Opt. (2)

G. Gbur and T. Visser, “The structure of partially coherent fields,” Prog. Opt. 55, 285–341 (2010).
[Crossref]

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Waves Random Complex Media (1)

S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, and O. Korotkova, “Polarization-induced reduction in scintillation of optical beams propagating in simulated turbulent atmospheric channels,” Waves Random Complex Media 24, 452–462 (2014).
[Crossref]

Other (3)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

O. Korotkova, Random Light Beams: Theory and Applications (CRC, 2014).

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Figures (5)

Fig. 1
Fig. 1 Complex screen 3D EGSM source simulatability condition given in Eq. (20)—(a) view of volume’s surface with interior removed and (b) different view of surface.

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Fig. 2
Fig. 2 Single realization spectral density volume of the complex screen 3D EGSM source whose parameters are given in Table 1—(a) surface view and (b) x–y, x–z, and y–z planar slices.

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Fig. 3
Fig. 3 Spectral densities theory (solid traces) versus simulation (circles) in the (a) x, (b) y, and (c) z directions.

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Fig. 4
Fig. 4 Spectral degrees of correlation theory (solid traces) versus simulation (circles)—(a) real and imaginary parts of μxx, (b) real and imaginary parts of μyy, and (c) real and imaginary parts of μzz.

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Fig. 5
Fig. 5 Spectral degrees of correlation theory (solid traces) versus simulation (circles and squares)—(a) real and imaginary parts of μxy and μ y x *, (b) real and imaginary parts of μxz and μ z x *, and (c) real and imaginary parts of μyz and μ z y *.

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Tables (1)

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Table 1 3D EGSM Source Parameters

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Equations (30)

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W i j ( r 1 , r 2 ) = A i A j B i j exp ( − r 1 2 4 σ i 2 ) exp ( − r 2 2 4 σ j 2 ) exp ( − | r 1 − r 2 | 2 2 δ i j 2 ) ,
δ i j = δ j i
B i j = B j i *
δ i i 2 + δ j j 2 ≤ 2 δ i j 2
| B x y | 2 ( δ x y 2 δ x x δ y y ) 3 + | B x z | 2 ( δ x z 2 δ x x δ z z ) 3 + | B y z | 2 ( δ y z 2 δ y y δ z z ) 3 ≤ 1 − 2 | B x y | | B x z | | B y z | ( δ x y δ x z δ y z δ x x δ y y δ z z ) 3 .
E i ( r ) = C i exp ( − r 2 4 σ i 2 ) T i ( r ) ,
〈 E i ( r 1 ) E j * ( r 1 ) 〉 = C i C j * exp ( − r 1 2 4 σ i 2 ) exp ( − r 2 2 4 σ j 2 ) 〈 T i ( r 1 ) T j * ( r 2 ) 〉 .
| C i | = A i arg ( C i C j * ) = arg ( B i j ) 〈 T i ( r 1 ) T i * ( r 2 ) 〉 = exp ( − | r 1 − r 2 | 2 2 δ i i 2 ) 〈 T i ( r 1 ) T j * ( r 2 ) 〉 = | B i j | exp ( − | r 1 − r 2 | 2 2 δ i j 2 ) ⋅
( 1 − 1 0 1 0 − 1 0 1 − 1 ) ( α x α y α z ) = ( θ x y θ x z θ y z ) ,
T i [ i , j , k ] = ∑ l , m , n r i [ l , m , n ] Φ i [ l , m , n ] 2 L x L y L z exp ( j 2 π L x l i ) exp ( j 2 π L y m j ) exp ( j 2 π L z n k ) ,
Φ i i ( f ) = ∭ − ∞ ∞ exp ( − r 2 2 δ i i 2 ) exp ( − j 2 π f ⋅ r ) d 3 r = ( 2 π δ i i 2 ) 3 / 2 exp ( − 2 π 2 δ i i 2 f 2 ) .
〈 T i [ i 1 , j 1 , k 1 ] T j * [ i 2 , j 2 , k 2 ] 〉 = ∑ l 1 , m 1 , n 1 ∑ l 2 , m 2 , n 2 〈 r i [ l 1 , m 1 , n 1 ] r j * [ l 2 , m 2 , n 2 ] 〉 × Φ i i [ l 1 , m 1 , n 1 ] Φ j j [ l 2 , m 2 , n 2 ] 2 L x L y L z exp [ j 2 π L x ( l 1 i 1 − l 2 i 2 ) ] exp [ j 2 π L y ( m 1 j 1 − m 2 j 2 ) ] × exp [ j 2 π L z ( n 1 k 1 − n 2 k 2 ) ] .
〈 T i [ i 1 , j 1 , k 1 ] T j * [ i 2 , j 2 , k 2 ] 〉 = ∑ l , m , n { Γ i j ( 2 π δ i i δ j j ) 3 / 2 exp [ − π 2 ( δ i i 2 + δ j j 2 ) ( l 2 + m 2 + n 2 ) ] } exp [ j 2 π L x l ( i 1 − i 2 ) ] exp [ j 2 π L y m ( j 1 − j 2 ) ] exp [ j 2 π L z n ( k 1 − k 2 ) ] .
Φ i i ( f ) = ∭ − ∞ ∞ | B i j | exp ( − r 2 2 δ i i 2 ) exp ( − j 2 π f ⋅ r ) d 3 r = | B i j | ( 2 π δ i i 2 ) 3 / 2 exp ( − 2 π 2 δ i i 2 f 2 ) ,
δ i j = δ i i 2 + δ j j 2 2 | B i j | = Γ i j ( δ i i δ j j δ i j 2 ) 3 / 2 = Γ i j ( 2 δ i i δ j j δ i i 2 + δ i j 2 ) 3 / 2 .
A i = | C i |
arg ( B i j ) = arg ( C i C j * )
δ i j = δ i i 2 + δ j j 2 2
| B i j | = Γ i j 2 δ i i δ j j δ i i 2 + δ j j 2 2 δ i i δ j j δ i i 2 + δ j j 2 .
Γ x y 2 + Γ x z 2 + Γ y z 2 + 2 Γ x y Γ x z Γ y z − 1 ≤ 0 .
S i ( r ) = W i i ( r , r ) = 〈 | E i ( r ) | 2 〉 μ i j ( r 1 , r 2 ) = W i j ( r 1 , r 2 ) S i ( r 1 ) S j ( r 2 ) ,
E i ( r ) = C i exp ( − r 2 4 σ i 2 ) exp [ j ϕ i ( r ) ] ,
A i = | C i |
arg ( B i j ) = arg ( C i C j * )
δ i i = ℓ ϕ i ϕ i 2 σ ϕ i
δ i j = 1 2 ℓ ϕ i ϕ i 2 + ℓ ϕ j ϕ j 2 4 Γ ϕ i ϕ j σ ϕ i σ ϕ j ℓ ϕ i ϕ j ℓ ϕ j ϕ j ( ℓ ϕ i ϕ i 2 + ℓ ϕ j ϕ j 2 2 ℓ ϕ i ϕ i ℓ ϕ j ϕ j ) 1 / 4
| B i j | = exp [ − 1 2 ( σ ϕ i 2 − 4 Γ ϕ i ϕ j σ ϕ i σ ϕ j ℓ ϕ i ϕ i ℓ ϕ j ϕ j ℓ ϕ i ϕ i 2 + ℓ ϕ j ϕ j 2 2 ℓ ϕ i ϕ i ℓ ϕ j ϕ j ℓ ϕ i ϕ i 2 + ℓ ϕ j ϕ j 2 + σ ϕ j 2 ) ]
σ ϕ i , σ ϕ j ≥ π ,
ϕ i [ i , j , k ] = Re { ∑ l , m , n r i [ l , m , n ] Φ ϕ i ϕ i [ l , m , n ] L x L y L z exp ( j 2 π L x l i ) exp ( j 2 π L y m j ) exp ( j 2 π L z n k ) } ,
Φ ϕ i ϕ i ( f ) = ∭ − ∞ ∞ σ ϕ i 2 exp ( − r 2 ℓ ϕ i ϕ i 2 ) exp ( − j 2 π f ⋅ r ) d 3 r = σ ϕ i 2 ( π ℓ ϕ i ϕ i 2 ) 3 / 2 exp ( − π 2 ℓ ϕ i ϕ i 2 f 2 )

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