Abstract

We develop a method to synthesize any partially coherent source (PCS) with a genuine cross-spectral density (CSD) function using complex transmittance screens. Prior work concerning PCS synthesis with complex transmittance screens has focused on generating Schell-model (uniformly correlated) sources. Here, using the necessary and sufficient condition for a genuine CSD function, we derive an expression, in the form of a superposition integral, that produces stochastic complex screen realizations. The sample autocorrelation of the screens is equal to the complex correlation function of the desired PCS. We validate our work by generating, in simulation, three PCSs from the literature—none has ever been synthesized using stochastic screens before. Examining planar slices through the four-dimensional CSD functions, we find the simulated results to be in excellent agreement with theory, implying successful realization of all three PCSs. The technique presented herein adds to the existing literature concerning the generation of PCSs and can be physically implemented using a simple optical setup consisting of a laser, spatial light modulator, and spatial filter.

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2019 (9)

X. Zhu, F. Wang, C. Zhao, Y. Cai, and S. A. Ponomarenko, “Experimental realization of dark and antidark diffraction-free beams,” Opt. Lett. 44, 2260–2263 (2019).
[Crossref]

A. Bhattacharjee, R. Sahu, and A. K. Jha, “Generation of a Gaussian Schell-model field as a mixture of its coherent modes,” J. Opt. 21, 105601 (2019).
[Crossref]

M. W. Hyde and S. Avramov-Zamurovic, “Generating dark and antidark beams using the genuine cross-spectral density function criterion,” J. Opt. Soc. Am. A 36, 1058–1063 (2019).
[Crossref]

M. W. Hyde, “Partially coherent sources generated from the incoherent sum of fields containing random-width Bessel functions,” Opt. Lett. 44, 1603–1606 (2019).
[Crossref]

M. W. Hyde, “Generating electromagnetic Schell-model sources using complex screens with spatially varying auto- and cross-correlation functions,” Results Phys. 15, 102663 (2019).
[Crossref]

M. W. Hyde, X. Xiao, and D. G. Voelz, “Generating electromagnetic nonuniformly correlated beams,” Opt. Lett. 44, 5719–5722 (2019).
[Crossref]

A. Efimov, “Effects of residual coherence on the scintillation of a partially coherent beam,” Opt. Express 27, 26874–26881 (2019).
[Crossref]

M. Santarsiero, F. Gori, and M. Alonzo, “Higher-order twisted/astigmatic Gaussian Schell-model cross-spectral densities and their separability features,” Opt. Express 27, 8554–8565 (2019).
[Crossref]

H. Wang, X. Peng, L. Liu, F. Wang, Y. Cai, and S. A. Ponomarenko, “Generating bona fide twisted Gaussian Schell-model beams,” Opt. Lett. 44, 3709–3712 (2019).
[Crossref]

2018 (6)

2017 (6)

A. Fathy, Y. M. Sabry, and D. A. Khalil, “Quasi-homogeneous partial coherent source modeling of multimode optical fiber output using the elementary source method,” J. Opt. 19, 105605 (2017).
[Crossref]

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

T. D. Visser, G. P. Agrawal, and P. W. Milonni, “Fourier processing with partially coherent fields,” Opt. Lett. 42, 4600–4602 (2017).
[Crossref]

T. Wu, C. Liang, F. Wang, and Y. Cai, “Shaping the intensity and degree of coherence of a partially coherent beam by a 4f optical system with an amplitude filter,” J. Opt. 19, 124010 (2017).
[Crossref]

M. W. Hyde, S. R. Bose-Pillai, and R. A. Wood, “Synthesis of non-uniformly correlated partially coherent sources using a deformable mirror,” Appl. Phys. Lett. 111, 101106 (2017).
[Crossref]

Z. Mei and O. Korotkova, “Random sources for rotating spectral densities,” Opt. Lett. 42, 255–258 (2017).
[Crossref]

2016 (3)

A. Efimov, “Gigabit per second modulation and transmission of a partially coherent beam through laboratory turbulence,” Proc. SPIE 9739, 97390L (2016).
[Crossref]

M. W. Hyde, S. Bose-Pillai, X. Xiao, and D. G. Voelz, “A fast and efficient method for producing partially coherent sources,” J. Opt. 19, 025601 (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. Appl. 6, 064030 (2016).
[Crossref]

2015 (3)

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]

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118, 093102 (2015).
[Crossref]

2014 (3)

2013 (2)

2011 (1)

2010 (1)

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

2009 (2)

2007 (1)

2006 (2)

2005 (1)

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

2001 (1)

1999 (1)

1994 (1)

1993 (3)

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[Crossref]

P. Hlubina, “Spatial and temporal coherence of light in a fibre waveguide,” J. Mod. Opt. 40, 1893–1907 (1993).
[Crossref]

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, M. Nakatsuka, and S. Nakai, “Partially coherent light generated by using single and multimode optical fibers in a high-power Nd:glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
[Crossref]

1986 (1)

1982 (1)

R. Grella, “Synthesis of generalized Collett-Wolf sources,” J. Opt. 13, 127–131 (1982).
[Crossref]

1973 (1)

C. Pask and A. W. Snyder, “The Van Cittert-Zernike theorem for optical fibres,” Opt. Commun. 9, 95–97 (1973).
[Crossref]

Agrawal, G. P.

Alonzo, M.

Avramov-Zamurovic, S.

Basu, S.

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118, 093102 (2015).
[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]

Bhattacharjee, A.

A. Bhattacharjee, R. Sahu, and A. K. Jha, “Generation of a Gaussian Schell-model field as a mixture of its coherent modes,” J. Opt. 21, 105601 (2019).
[Crossref]

Borghi, R.

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. Appl. 6, 064030 (2016).
[Crossref]

M. W. Hyde, S. Bose-Pillai, X. Xiao, and D. G. Voelz, “A fast and efficient method for producing partially coherent sources,” J. Opt. 19, 025601 (2016).
[Crossref]

Bose-Pillai, S. R.

M. W. Hyde, S. R. Bose-Pillai, and R. A. Wood, “Synthesis of non-uniformly correlated partially coherent sources using a deformable mirror,” Appl. Phys. Lett. 111, 101106 (2017).
[Crossref]

Cai, Y.

Chen, X.

Chen, Y.

Efimov, A.

Fathy, A.

A. Fathy, Y. M. Sabry, and D. A. Khalil, “Quasi-homogeneous partial coherent source modeling of multimode optical fiber output using the elementary source method,” J. Opt. 19, 105605 (2017).
[Crossref]

Friberg, A. T.

Gbur, G.

Goodman, J. W.

J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015).

Gori, F.

Grella, R.

R. Grella, “Synthesis of generalized Collett-Wolf sources,” J. Opt. 13, 127–131 (1982).
[Crossref]

Gu, Y.

Guattari, G.

Hanson, S. G.

Hlubina, P.

P. Hlubina, “Spatial and temporal coherence of light in a fibre waveguide,” J. Mod. Opt. 40, 1893–1907 (1993).
[Crossref]

Hyde, M. W.

M. W. Hyde, “Generating electromagnetic Schell-model sources using complex screens with spatially varying auto- and cross-correlation functions,” Results Phys. 15, 102663 (2019).
[Crossref]

M. W. Hyde and S. Avramov-Zamurovic, “Generating dark and antidark beams using the genuine cross-spectral density function criterion,” J. Opt. Soc. Am. A 36, 1058–1063 (2019).
[Crossref]

M. W. Hyde, “Partially coherent sources generated from the incoherent sum of fields containing random-width Bessel functions,” Opt. Lett. 44, 1603–1606 (2019).
[Crossref]

M. W. Hyde, X. Xiao, and D. G. Voelz, “Generating electromagnetic nonuniformly correlated beams,” Opt. Lett. 44, 5719–5722 (2019).
[Crossref]

M. W. Hyde, “Controlling the spatial coherence of an optical source using a spatial filter,” Appl. Sci. 8, 1465 (2018).
[Crossref]

M. W. Hyde, S. R. Bose-Pillai, and R. A. Wood, “Synthesis of non-uniformly correlated partially coherent sources using a deformable mirror,” Appl. Phys. Lett. 111, 101106 (2017).
[Crossref]

M. W. Hyde, S. Bose-Pillai, X. Xiao, and D. G. Voelz, “A fast and efficient method for producing partially coherent sources,” J. Opt. 19, 025601 (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. Appl. 6, 064030 (2016).
[Crossref]

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118, 093102 (2015).
[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]

Jha, A. K.

A. Bhattacharjee, R. Sahu, and A. K. Jha, “Generation of a Gaussian Schell-model field as a mixture of its coherent modes,” J. Opt. 21, 105601 (2019).
[Crossref]

Kanabe, T.

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, M. Nakatsuka, and S. Nakai, “Partially coherent light generated by using single and multimode optical fibers in a high-power Nd:glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
[Crossref]

Khalil, D. A.

A. Fathy, Y. M. Sabry, and D. A. Khalil, “Quasi-homogeneous partial coherent source modeling of multimode optical fiber output using the elementary source method,” J. Opt. 19, 105605 (2017).
[Crossref]

Korotkova, O.

Li, J.

Liang, C.

T. Wu, C. Liang, F. Wang, and Y. Cai, “Shaping the intensity and degree of coherence of a partially coherent beam by a 4f optical system with an amplitude filter,” J. Opt. 19, 124010 (2017).
[Crossref]

Liu, L.

H. Wang, X. Peng, L. Liu, F. Wang, Y. Cai, and S. A. Ponomarenko, “Generating bona fide twisted Gaussian Schell-model beams,” Opt. Lett. 44, 3709–3712 (2019).
[Crossref]

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

Lohmann, A. W.

Mack, C. A.

Mandel, L.

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

Martínez-Herrero, R.

Mei, Z.

Mejías, P. M.

Mendlovic, D.

Milonni, P. W.

Miyanaga, N.

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, M. Nakatsuka, and S. Nakai, “Partially coherent light generated by using single and multimode optical fibers in a high-power Nd:glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
[Crossref]

Mukunda, N.

Nakai, S.

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, M. Nakatsuka, and S. Nakai, “Partially coherent light generated by using single and multimode optical fibers in a high-power Nd:glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
[Crossref]

Nakano, H.

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, M. Nakatsuka, and S. Nakai, “Partially coherent light generated by using single and multimode optical fibers in a high-power Nd:glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
[Crossref]

Nakatsuka, M.

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, M. Nakatsuka, and S. Nakai, “Partially coherent light generated by using single and multimode optical fibers in a high-power Nd:glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
[Crossref]

Palma, C.

Pask, C.

C. Pask and A. W. Snyder, “The Van Cittert-Zernike theorem for optical fibres,” Opt. Commun. 9, 95–97 (1973).
[Crossref]

Peng, X.

Ponomarenko, S. A.

Rafsanjani, S. M. H.

Ramírez-Sánchez, V.

Sabry, Y. M.

A. Fathy, Y. M. Sabry, and D. A. Khalil, “Quasi-homogeneous partial coherent source modeling of multimode optical fiber output using the elementary source method,” J. Opt. 19, 105605 (2017).
[Crossref]

Sahu, R.

A. Bhattacharjee, R. Sahu, and A. K. Jha, “Generation of a Gaussian Schell-model field as a mixture of its coherent modes,” J. Opt. 21, 105601 (2019).
[Crossref]

Santarsiero, M.

Santis, P. D.

Shabtay, G.

Shirai, T.

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

Simon, R.

Snyder, A. W.

C. Pask and A. W. Snyder, “The Van Cittert-Zernike theorem for optical fibres,” Opt. Commun. 9, 95–97 (1973).
[Crossref]

Stahl, C. S. D.

Tervonen, E.

Tsubakimoto, K.

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, M. Nakatsuka, and S. Nakai, “Partially coherent light generated by using single and multimode optical fibers in a high-power Nd:glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
[Crossref]

Turunen, J.

Visser, T.

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

Visser, T. D.

Voelz, D.

Voelz, D. G.

M. W. Hyde, X. Xiao, and D. G. Voelz, “Generating electromagnetic nonuniformly correlated beams,” Opt. Lett. 44, 5719–5722 (2019).
[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. Appl. 6, 064030 (2016).
[Crossref]

M. W. Hyde, S. Bose-Pillai, X. Xiao, and D. G. Voelz, “A fast and efficient method for producing partially coherent sources,” J. Opt. 19, 025601 (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]

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118, 093102 (2015).
[Crossref]

Wang, F.

Wang, H.

Watkins, D. S.

D. S. Watkins, Fundamentals of Matrix Computations, 2nd ed. (Wiley, 2002).

Wolf, E.

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Supplementary Material (1)

NameDescription
» Code 1       MATLAB simulation code.

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Figures (6)

Fig. 1.
Fig. 1. $ {I_m} $ -Bessel correlated field realization: (a)  $ | U | $ and (b)  $ \arg ( U ) $ .
Fig. 2.
Fig. 2. $ {I_m} $ -Bessel correlated source simulation results: (a)  $ S $ theory, (b)  $ S $ simulation, (c) 2D correlation coefficient $ C $ for $ S $ simulation versus trial number, (d) real (top) and imaginary (bottom) parts of $ W( {{x_1},{y_1},\gamma ,\gamma } ) $ theory, and (e) real (top) and imaginary (bottom) parts of $ W( {{x_1},{y_1},\gamma ,\gamma } ) $ simulation.
Fig. 3.
Fig. 3. RHNUC field realization: (a)  $ | U | $ and (b)  $ \arg ( U ) $ .
Fig. 4.
Fig. 4. RHNUC source simulation results: (a)  $ S $ theory, (b)  $ S $ simulation, (c) 2D correlation coefficient $ C $ for $ S $ simulation versus trial number, (d) real (top) and imaginary (bottom) parts of $ W( {{x_1},0,{x_2},0} ) $ theory, and (e) real (top) and imaginary (bottom) parts of $ W( {{x_1},0,{x_2},0} ) $ simulation.
Fig. 5.
Fig. 5. TAGSM field realization: (a)  $ | U | $ and (b)  $ \arg ( U ) $ .
Fig. 6.
Fig. 6. TAGSM source simulation results: (a)  $ S $ theory, (b)  $ S $ simulation, (c) 2D correlation coefficient $ C $ for $ S $ simulation versus trial number, (d) real (top) and imaginary (bottom) parts of $ W( {{x_1},0,0,{y_2}} ) $ theory, and (e) real (top) and imaginary (bottom) parts of $ W( {{x_1},0,0,{y_2}} ) $ simulation.

Equations (33)

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W ( ρ 1 , ρ 2 ) = τ ( ρ 1 ) τ ( ρ 2 ) μ ( ρ 1 , ρ 2 ) ,
W ( ρ 1 , ρ 2 ) = p ( v ) H ( ρ 1 , v ) H ( ρ 2 , v ) d 2 v ,
H ( ρ , v ) = τ ( ρ ) h ( ρ , v ) .
μ ( ρ 1 , ρ 2 ) = p ( v ) h ( ρ 1 , v ) h ( ρ 2 , v ) d 2 v .
U ( ρ ) = τ ( ρ ) T ( ρ ) ,
μ ( ρ 1 , ρ 2 ) = T ( ρ 1 ) T ( ρ 2 ) .
T ( ρ 1 ) T ( ρ 2 ) = p ( v ) h ( ρ 1 , v ) h ( ρ 2 , v ) d 2 v ;
T ( ρ ) = r ( f ) [ 1 2 Φ ( f ) ] 1 / 2 exp ( j 2 π f ρ ) d 2 f ,
Φ ( f ) = μ ( ρ d ) exp ( j 2 π f ρ d ) d 2 ρ d ,
Φ ( f ) exp ( j 2 π f ρ d ) d 2 f = p ( v ) exp ( j v ρ d ) d 2 v .
r ( f 1 ) r ( f 2 ) = r r ( f 1 ) r r ( f 2 ) + r i ( f 1 ) r i ( f 2 ) + j [ r i ( f 1 ) r r ( f 2 ) r r ( f 1 ) r i ( f 2 ) ] = 2 δ ( f 1 f 2 ) ,
T ( ρ ) = r ( f ) [ 1 2 ( 2 π ) 2 p ( 2 π f ) ] 1 / 2 h ( ρ , 2 π f ) d 2 f ,
W ( ρ 1 , ρ 2 ) = ξ m / 2 1 ξ exp ( 1 + ξ 1 ξ ρ 1 2 + ρ 2 2 σ 2 ) × exp [ j m ( ϕ 1 ϕ 2 ) ] I m ( 4 ξ 1 ξ ρ 1 ρ 2 σ 2 ) ,
τ ( ρ ) = ξ m / 2 1 ξ exp ( j m ϕ ) exp [ ( 1 ξ ) 2 1 ξ ρ 2 σ 2 ] , p ( v ) = 1 π σ 2 8 1 ξ ξ exp ( σ 2 8 1 ξ ξ v 2 ) , h ( ρ , v ) = J m ( ρ v ) ,
T ( ρ ) = ( 2 π ) 3 / 2 σ 2 8 1 ξ ξ 0 f r ( f ) exp × [ 1 2 ( 2 π ) 2 σ 2 8 1 ξ ξ f 2 ] J m ( 2 π ρ f ) d f .
T ( ρ 1 ) T ( ρ 2 ) = ( 2 π ) 3 σ 2 8 1 ξ ξ 0 f 1 f 2 r ( f 1 ) r ( f 2 ) × exp [ 1 2 ( 2 π ) 2 σ 2 8 1 ξ ξ ( f 1 2 + f 2 2 ) ] × J m ( 2 π ρ 1 f 1 ) J m ( 2 π ρ 2 f 2 ) d f 1 d f 2 .
T ( ρ 1 ) T ( ρ 2 ) = μ ( ρ 1 , ρ 2 ) = ( 2 π ) 3 1 π σ 2 8 1 ξ ξ × 0 f exp [ ( 2 π ) 2 σ 2 8 1 ξ ξ f 2 ] × J m ( 2 π ρ 1 f ) J m ( 2 π ρ 2 f ) d f .
T ( ρ ) = ( 2 π ) 3 / 2 σ 2 8 1 ξ ξ 0 r ( f ) f × exp [ 1 2 ( 2 π ) 2 σ 2 8 1 ξ ξ f 2 ] J m ( 2 π ρ f ) d f .
T [ i ] = 2 π σ 2 8 1 ξ ξ n = 0 N 1 r [ n ] n Δ f × exp [ 1 2 ( 2 π ) 2 σ 2 8 1 ξ ξ ( n Δ f ) 2 ] × J m ( 2 π i n Δ s Δ f ) Δ f ,
γ = σ 2 8 1 ξ ξ = 211.7 µ m .
W ( ρ 1 , ρ 2 ) = exp ( x 1 2 + x 2 2 4 σ x 2 ) 1 H 2 m ( 0 ) H 2 m [ f ( x 1 , x 2 ) δ x 2 ] × exp [ f 2 ( x 1 , x 2 ) δ x 4 ] exp ( y 1 2 + y 2 2 4 σ y 2 ) × 1 H 2 n ( 0 ) H 2 n [ g ( y 1 , y 2 ) δ y 2 ] exp [ g 2 ( y 1 , y 2 ) δ y 4 ] ,
f ( x 1 , x 2 ) = ( x 1 x 0 ) 2 ( x 2 x 0 ) 2 , g ( y 1 , y 2 ) = ( y 1 y 0 ) 2 ( y 2 y 0 ) 2 ,
τ ( ρ ) = exp ( x 2 4 σ x 2 ) exp ( y 2 4 σ y 2 ) , p ( v ) = ( δ x 4 / 4 ) m + 1 / 2 ( δ y 4 / 4 ) n + 1 / 2 Γ ( m + 1 / 2 ) Γ ( n + 1 / 2 ) v x 2 m v y 2 n × exp ( δ x 4 4 v x 2 ) exp ( δ y 4 4 v y 2 ) , h ( ρ , v ) = exp [ j v x ( x x 0 ) 2 ] exp [ j v y ( y y 0 ) 2 ] ,
T ( ρ ) = ( δ x / 2 ) 2 m + 1 ( δ y / 2 ) 2 n + 1 2 Γ ( m + 1 / 2 ) Γ ( n + 1 / 2 ) ( 2 m ) m + 1 / 2 ( 2 n ) n + 1 / 2 × r ( f ) | f x | m | f y | n exp ( 1 2 π 2 δ x 4 f x 2 ) × exp ( 1 2 π 2 δ y 4 f y 2 ) exp [ j 2 π f x ( x x 0 ) 2 ] × exp [ j 2 π f y ( y y 0 ) 2 ] d f x d f y .
ξ = ( x x 0 ) 2 , η = ( y y 0 ) 2 .
L ξ η = 2 [ L xy 2 + max ( | x 0 | , | y 0 | ) ] 2 ,
T [ i , j ] = ( δ x / 2 ) 2 m + 1 ( δ y / 2 ) 2 n + 1 2 Γ ( m + 1 / 2 ) Γ ( n + 1 / 2 ) ( 2 m ) m + 1 / 2 ( 2 n ) n + 1 / 2 × k = N / 2 N / 2 1 l = N / 2 N / 2 1 r [ k , l ] | k L ξ η | m | l L ξ η | n × exp [ 1 2 π 2 δ x 4 ( k L ξ η ) 2 ] exp [ 1 2 π 2 δ y 4 ( l L ξ η ) 2 ] × exp ( j 2 π N k i ) exp ( j 2 π N l j ) 1 L ξ η ,
W ( ρ 1 , ρ 2 ) = exp ( x 1 2 + x 2 2 4 σ x 2 ) exp [ ( x 1 x 2 ) 2 2 δ x 2 ] × exp ( y 1 2 + y 2 2 4 σ y 2 ) exp [ ( y 1 y 2 ) 2 2 δ y 2 ] × exp [ j u ( x 1 y 2 x 2 y 1 ) ] ,
τ ( ρ ) = exp ( σ ρ 2 ) , p ( v ) = α β π exp ( α v x 2 ) exp ( β v y 2 ) , h ( ρ , v ) = exp [ j ( x j α u y ) v x ] exp [ j ( y + j β u x ) v y ] ,
1 4 σ x 2 = σ β u 2 2 , 1 2 δ x 2 = β u 2 4 + 1 4 α , 1 4 σ y 2 = σ α u 2 2 , 1 2 δ y 2 = α u 2 4 + 1 4 β .
T ( ρ ) = 2 π ( α β ) 1 / 4 r ( f ) × exp [ 1 2 ( 2 π ) 2 ( α f x 2 + β f y 2 ) ] × exp ( 2 π α u y f x ) exp ( 2 π β u x f y ) × exp ( j 2 π ρ f ) d 2 f .
T [ i , j ] = 2 π ( α β ) 1 / 4 m = N / 2 N / 2 1 n = N / 2 N / 2 1 r [ m , n ] × exp [ 1 2 ( 2 π ) 2 α ( m L x ) 2 ] exp ( 2 π α u j L y N y m L x ) × exp [ 1 2 ( 2 π ) 2 β ( n L y ) 2 ] exp ( 2 π β u i L x N x n L y ) × exp ( j 2 π N x m i ) exp ( j 2 π N y n j ) 1 L x L y ,
T [ i , j ] = T [ i , j , N ( j 1 ) + i ] ,