Abstract

Complicated partially coherent beams (PCBs) are useful in many applications, such as free-space optical communications, particle trapping and optical imaging, while usually it is hard to derive analytical propagation formulae for such beams, and one has to fall back on numerical methods. The conventional numerical methods have some intrinsic drawbacks. In this paper, we introduce an efficient tensor approach (ETA) for simulating paraxial propagation of arbitrary PCBs. The ETA is a direct reconstruction of the propagated PCB without aliasing and rippling problems, and the algorithm is simple and robust with a tensor/matrix multiplication as the main calculation. The validity of ETA is verified through comparing simulation results with analytical results, numerical integration results and experimental results, respectively. The ETA provides a fast and reliable way for simulating paraxial propagation of arbitrary PCBs.

© 2017 Optical Society of America

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Wave optics simulation approach for partial spatially coherent beams

Xifeng Xiao and David Voelz
Opt. Express 14(16) 6986-6992 (2006)

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2017 (1)

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

2015 (6)

2014 (4)

2013 (1)

J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).

2012 (4)

2011 (4)

2010 (3)

S. Sahin, Z. Tong, and O. Korotkova, “Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams,” Opt. Commun. 282(22), 4512–4518 (2010).

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[PubMed]

F. Wang and Y. Cai, “Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010).
[PubMed]

2008 (2)

2007 (2)

2006 (2)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).

Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31(6), 685–687 (2006).
[PubMed]

2005 (1)

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5 Pt 2), 056607 (2005).
[PubMed]

2003 (1)

2002 (2)

1994 (1)

1993 (1)

1986 (1)

1982 (2)

A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72(7), 923–928 (1982).

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).

1980 (2)

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1980).

S. C. Som, C. Delisle, and M. Drouin, “Holography in partially coherent light,” Opt. Commun. 32(3), 370–374 (1980).

1979 (1)

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).

1970 (1)

Arce, G. R.

Bastiaans, M. J.

Baykal, Y.

Borghi, R.

Brennan, T. J.

Brown, D. P.

Brown, T. G.

Cai, Y.

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

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited Review),” Prog. Electromagnetics Res. 150, 123–143 (2015).

X. Liu, F. Wang, L. Liu, C. Zhao, and Y. Cai, “Generation and propagation of an electromagnetic Gaussian Schell-model vortex beam,” J. Opt. Soc. Am. A 32(11), 2058–2065 (2015).
[PubMed]

L. Liu, Y. Huang, Y. Chen, L. Guo, and Y. Cai, “Orbital angular moment of an electromagnetic Gaussian Schell-model beam with a twist phase,” Opt. Express 23(23), 30283–30296 (2015).
[PubMed]

Y. Cai and S. Zhu, “Orbital angular moment of a partially coherent beam propagating through an astigmatic ABCD optical system with loss or gain,” Opt. Lett. 39(7), 1968–1971 (2014).
[PubMed]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [Invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[PubMed]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).

F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012).
[PubMed]

F. Wang, S. Zhu, and Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36(16), 3281–3283 (2011).
[PubMed]

G. Wu and Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. 36(10), 1939–1941 (2011).
[PubMed]

F. Wang and Y. Cai, “Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010).
[PubMed]

Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007).
[PubMed]

Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31(6), 685–687 (2006).
[PubMed]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5 Pt 2), 056607 (2005).
[PubMed]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[PubMed]

Chen, Y.

Cheng, B.

J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).

Collins, S. A.

Davidson, F. M.

de Santis, P.

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).

Delisle, C.

S. C. Som, C. Delisle, and M. Drouin, “Holography in partially coherent light,” Opt. Commun. 32(3), 370–374 (1980).

Dolash, T. M.

Dong, Y.

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).

Drouin, M.

S. C. Som, C. Delisle, and M. Drouin, “Holography in partially coherent light,” Opt. Commun. 32(3), 370–374 (1980).

Eyyuboglu, H. T.

Fischer, D. G.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[PubMed]

Friberg, A. T.

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994).

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).

Fried, D. L.

Gbur, G.

Gori, F.

Guattari, G.

Guo, L.

He, S.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).

Hu, L.

Huang, Y.

Korotkova, O.

Lajunen, H.

Lin, Q.

J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[PubMed]

Liu, L.

Liu, X.

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

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited Review),” Prog. Electromagnetics Res. 150, 123–143 (2015).

X. Liu, F. Wang, L. Liu, C. Zhao, and Y. Cai, “Generation and propagation of an electromagnetic Gaussian Schell-model vortex beam,” J. Opt. Soc. Am. A 32(11), 2058–2065 (2015).
[PubMed]

Ma, L.

Ma, X.

Mukunda, N.

Palma, C.

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).

Pease, E. A.

Peschel, U.

Piquero, G.

Ponomarenko, S. A.

Ricklin, J. C.

Saastamoinen, T.

Sahin, S.

S. Sahin, Z. Tong, and O. Korotkova, “Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams,” Opt. Commun. 282(22), 4512–4518 (2010).

Santarsiero, M.

Shakir, S. A.

Shen, X.

J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).

Simon, R.

Som, S. C.

S. C. Som, C. Delisle, and M. Drouin, “Holography in partially coherent light,” Opt. Commun. 32(3), 370–374 (1980).

Starikov, A.

Sudol, R. J.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).

Tervonen, E.

Tong, Z.

Z. Tong and O. Korotkova, “Beyond the classical Rayleigh limit with twisted light,” Opt. Lett. 37(13), 2595–2597 (2012).
[PubMed]

Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012).
[PubMed]

S. Sahin, Z. Tong, and O. Korotkova, “Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams,” Opt. Commun. 282(22), 4512–4518 (2010).

Turunen, J.

van Dijk, T.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[PubMed]

Visser, T. D.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[PubMed]

Wang, F.

Wang, Q.

J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).

Wang, Z.

J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).

Wolf, E.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[PubMed]

A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72(7), 923–928 (1982).

Wu, B.

J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).

Wu, G.

Xu, Y.

J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).

Yu, J.

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

Zhang, J.

J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).

Zhao, C.

X. Liu, F. Wang, L. Liu, C. Zhao, and Y. Cai, “Generation and propagation of an electromagnetic Gaussian Schell-model vortex beam,” J. Opt. Soc. Am. A 32(11), 2058–2065 (2015).
[PubMed]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).

Zheng, L.

J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).

Zhu, S.

Zhu, S. Y.

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5 Pt 2), 056607 (2005).
[PubMed]

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (10)

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002).
[PubMed]

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20(1), 78–84 (2003).
[PubMed]

O. Korotkova and G. Gbur, “Angular spectrum representation for propagation of random electromagnetic beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 24(9), 2728–2736 (2007).
[PubMed]

G. Gbur, “Partially coherent beam propagation in atmospheric turbulence [Invited],” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014).
[PubMed]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [Invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[PubMed]

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994).

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3(8), 1227–1238 (1986).

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993).

Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012).
[PubMed]

X. Liu, F. Wang, L. Liu, C. Zhao, and Y. Cai, “Generation and propagation of an electromagnetic Gaussian Schell-model vortex beam,” J. Opt. Soc. Am. A 32(11), 2058–2065 (2015).
[PubMed]

Opt. Commun. (5)

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1980).

S. C. Som, C. Delisle, and M. Drouin, “Holography in partially coherent light,” Opt. Commun. 32(3), 370–374 (1980).

S. Sahin, Z. Tong, and O. Korotkova, “Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams,” Opt. Commun. 282(22), 4512–4518 (2010).

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).

Opt. Express (7)

Opt. Lett. (11)

Y. Cai and S. Zhu, “Orbital angular moment of a partially coherent beam propagating through an astigmatic ABCD optical system with loss or gain,” Opt. Lett. 39(7), 1968–1971 (2014).
[PubMed]

G. Wu and Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. 36(10), 1939–1941 (2011).
[PubMed]

F. Wang, S. Zhu, and Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36(16), 3281–3283 (2011).
[PubMed]

Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31(6), 685–687 (2006).
[PubMed]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[PubMed]

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[PubMed]

F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012).
[PubMed]

Z. Tong and O. Korotkova, “Beyond the classical Rayleigh limit with twisted light,” Opt. Lett. 37(13), 2595–2597 (2012).
[PubMed]

L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014).
[PubMed]

S. A. Ponomarenko, “Self-imaging of partially coherent light in graded-index media,” Opt. Lett. 40(4), 566–568 (2015).
[PubMed]

R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Twisted Schell-model beams with axial symmetry,” Opt. Lett. 40(19), 4504–4507 (2015).
[PubMed]

Phys. Rev. A (2)

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).

J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5 Pt 2), 056607 (2005).
[PubMed]

Phys. Rev. Lett. (1)

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[PubMed]

Prog. Electromagnetics Res. (1)

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited Review),” Prog. Electromagnetics Res. 150, 123–143 (2015).

Prog. Opt. (1)

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

Other (3)

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

X. Liu, L. Liu, Y. Chen, and Y. Cai, “Partially coherent vortex beam: from theory to experiment,” in Vortex Dynamics and Optical Vortices, H. Pérez-de-Tejada, ed. (InTech-open science, 2017), Chap.11, pp. 275–296.

Y. Cai, F. Wang, C. Zhao, S. Zhu, G. Wu, and Y. Dong, “Partially coherent vector beams: from theory to experiment,” in Vectorial Optical Fields: Fundamentals and Applications, Q. Zhan, ed. (World Scientific, 2013).

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Figures (4)

Fig. 1
Fig. 1 Amplitude of the CSD function |W(x,0,0,0)| of a TGSM beam in free space at (a)z = 0.5m, and (b)z = 1m for different values of twist factor μ . The twist factors are respectively μ = 1 × 1 0 3 mm−1(red curve & red dots), 5 × 1 0 4 mm−1(green curve & green dots), 2 × 1 0 4 mm−1(blue curve & blue dots). All the dots represent the data obtained by ETA. All the curves represent the data obtained by an analytic formula.
Fig. 2
Fig. 2 Intensity distribution of a nonuniformly correlated beam in free space at several propagation distances, (a) z = 0.2m, (b) z = 0.5m, (c) z = 1m. The blue solid curves are results obtained by ETA. The green dots are results obtained by direct numerical integration with adaptive Simpson algorithm.
Fig. 3
Fig. 3 Experimental setup for generating a GSMV beam and measuring its focused intensity. M, reflecting mirror; L1, L2, L3, Thin lenses; BE, beam expander; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; SPP, Spiral phase plate; BPA, beam profile analyzer; PC, personal computer.
Fig. 4
Fig. 4 Experimental results of the intensity distribution and the corresponding cross line (blue curve) of a focused GSMV beam at several propagation distances for different values of l. The first and second rows, l = 1; the third and fourth rows, l = 2. Red dashed lines denote the results calculated by ETA.

Equations (12)

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W ( u 1 , u 2 , v 1 , v 2 ) = 1 | λ B | 2 W 0 ( x 1 , x 2 , y 1 , y 2 ) × exp [ i π λ B * ( A * x 1 2 2 x 1 u 1 + D * u 1 2 ) + i π λ B ( A x 2 2 2 x 2 u 2 + D u 2 2 ) ] , × exp [ i π λ B * ( A * y 1 2 2 y 1 v 1 + D * v 1 2 ) + i π λ B ( A y 2 2 2 y 2 v 2 + D v 2 2 ) ] d x 1 d x 2 d y 1 d y 2
W = H y T H x T W 0 H x H y ,
[ H x ] j m H x ( x j , u m ) = w j i λ | B | exp [ i π λ B ( A x j 2 2 x j u m + D u m 2 ) ] ,
[ W ] m 1 m 2 n 1 n 2 = j 1 N 1 j 2 N 1 k 1 N 1 k 2 N 1 [ H y T ] n 1 k 1 [ H x T ] m 1 j 1 [ W 0 ] j 1 j 2 k 1 k 2 [ H x ] j 2 m 2 [ H y ] k 2 n 2 .
w j = Δ 1 exp [ i π B λ ] sin ( u m A x j ) Δ 1 / λ B ] π ( u m A x j ) Δ 1 / λ B .
W x = H x T W 0 x H x , W y = H y T W 0y H y .
[ W ] m 1 m 2 n 1 n 2 = [ W x ] m 1 m 2 [ W y ] n 1 n 2 ,
W 0 ( x 1 , x 2 , y 1 , y 2 ) = exp [ ( x 1 2 + x 2 2 + y 1 2 + y 2 2 ) 4 σ I 2 ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ g 2 i μ 2 π λ ( x 1 y 2 x 2 y 1 ) ] ,
( A B C D ) = ( 1 z 0 1 ) .
W 0 ( x 1 , x 2 ) = exp { ( x 1 2 + x 2 2 ) 2 σ I 2 [ ( x 2 x 0 ) 2 ( x 1 x 0 ) 2 ] 2 σ g 4 } ,
W 0 ( r 1 , ϕ 1 , r 2 , ϕ 2 ) = exp [ r 1 2 + r 2 2 σ I 2 ] exp [ ( r 1 r 2 ) 2 2 σ g 2 ] exp [ i l ( ϕ 2 ϕ 1 ) ] ,
W 0 ( x 1 , x 2 , y 1 , y 2 ) = ( x 1 i y 1 ) l ( x 2 + i y 2 ) l ( r 1 r 2 ) l exp [ ( x 1 2 + x 2 2 + y 1 2 + y 2 2 ) σ I 2 ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ g 2 ] .

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