Abstract

We introduce a new kind of partially coherent vortex (PCV) beam with fractional topological charge named partially coherent fractional vortex (PCFV) beam and derive the propagation formula for such beam passing through a stigmatic ABCD optical system with the help of the convolution method. We calculate numerically the propagation properties of a PCFV beam focused by a thin lens, and we find that the PCFV beam exhibits unique propagation properties. The opening gap of the intensity pattern and the rotation of the beam spot disappear gradually and the cross-spectral density (CSD) distribution becomes more symmetric and more recognizable with the decrease of the spatial coherence width, being qualitatively different from those of the PCV beam with integral topological charge. Furthermore, we carry out experimental generation of a PCFV beam with controllable spatial coherence, and measure its focusing properties. Our experimental results are consistent with the theoretical predictions.

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

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References

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2018 (2)

A. V. Volyar and Y. A. Egorov, “Super pulses of orbital angular momentum in fractional-order spiroid vortex beams,” Opt. Lett. 43(1), 74–77 (2018).
[Crossref] [PubMed]

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018).
[Crossref]

2017 (5)

Y. Fang, Q. Lu, X. Wang, W. Zhang, and L. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A (Coll. Park) 95(2), 023821 (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]

A. S. Ostrovsky, J. García-García, C. Rickenstorff-Parrao, and M. A. Olvera-Santamaría, “Partially coherent diffraction-free vortex beams with a Bessel-mode structure,” Opt. Lett. 42(24), 5182–5185 (2017).
[Crossref] [PubMed]

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

X. Liu, T. Wu, L. Liu, C. Zhao, and Y. Cai, “Experimental determination of the azimuthal and radial mode orders of a partially coherent LGpl beam,” Chin. Opt. Lett. 15(3), 030002 (2017).
[Crossref]

2016 (8)

R. Liu, F. Wang, D. Chen, Y. Wang, Y. Zhou, H. Gao, P. Zhang, and F. Li, “Measuring mode indices of a partially coherent vortex beam with Hanbury Brown and Twiss type experiment,” Appl. Phys. Lett. 108(5), 051107 (2016).
[Crossref]

B. Perez-Garcia, A. Yepiz, R. I. Hernandez-Aranda, A. Forbes, and G. A. Swartzlander, “Digital generation of partially coherent vortex beams,” Opt. Lett. 41(15), 3471–3474 (2016).
[Crossref] [PubMed]

J. Gao, Y. Zhu, D. Wang, Y. Zhang, Z. Hu, and M. Cheng, “Bessel–Gauss photon beams with fractional order vortex propagation in weak non-Kolmogorov turbulence,” Photon. Res. 4(2), 30–34 (2016).
[Crossref]

G. Gbur, “Fractional vortex Hilbert’s hotel,” Optica 3(3), 222–225 (2016).
[Crossref]

J. Strohaber, Y. Boran, M. Sayrac, L. Johnson, F. Zhu, A. Kolomenskii, and H. Schuessler, “Nonlinear mixing of optical vortices with fractional topological charge in Raman sideband generation,” J. Opt. 19(1), 015607 (2016).
[Crossref]

R. Ni, Y. Niu, L. Du, X. Hu, Y. Zhang, and S. Zhu, “Topological charge transfer in frequency doubling of fractional orbital angular momentum state,” Appl. Phys. Lett. 109(15), 151103 (2016).
[Crossref]

P. Li, Y. Zhang, S. Liu, C. Ma, L. Han, H. Cheng, and J. Zhao, “Generation of perfect vectorial vortex beams,” Opt. Lett. 41(10), 2205–2208 (2016).
[Crossref] [PubMed]

X. Wang, G. Rui, L. Gong, B. Gu, and Y. Cui, “Manipulation of resonant metallic nanoparticle using 4Pi focusing system,” Opt. Express 24(21), 24143–24152 (2016).
[Crossref] [PubMed]

2015 (4)

Y. Yuan and Y. Yang, “Propagation of anomalous vortex beams through an annular apertured paraxial ABCD optical system,” Opt. Quantum Electron. 47(7), 2289–2297 (2015).
[Crossref]

Z. Dai, Z. Yang, S. Zhang, and Z. Pang, “Propagation of anomalous vortex beams in strongly nonlocal nonlinear media,” Opt. Commun. 350, 19–27 (2015).
[Crossref]

D. Zhang and Y. Yang, “Radiation forces on Rayleigh particles using a focused anomalous vortex beam under paraxial approximation,” Opt. Commun. 336, 202–206 (2015).
[Crossref]

X. L. Ge, B. Y. Wang, and C. S. Guo, “Evolution of phase singularities of vortex beams propagating in atmospheric turbulence,” J. Opt. Soc. Am. A 32(5), 837–842 (2015).
[Crossref] [PubMed]

2014 (5)

2013 (4)

2012 (2)

C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[Crossref]

Z. Mei, Z. Tong, and O. Korotkova, “Electromagnetic non-uniformly correlated beams in turbulent atmosphere,” Opt. Express 20(24), 26458–26463 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (3)

Y. Gu and G. Gbur, “Measurement of atmospheric turbulence strength by vortex beam,” Opt. Commun. 283(7), 1209–1212 (2010).
[Crossref]

C. S. Guo, Y. N. Yu, and Z. Hong, “Optical sorting using an array of optical vortices with fractional topological charge,” Opt. Commun. 283(9), 1889–1893 (2010).
[Crossref]

M. A. Molchan, E. V. Doktorov, and R. A. Vlasov, “Propagation of vector fractional charge Laguerre-Gaussian light beams in the thermally nonlinear moving atmosphere,” Opt. Lett. 35(5), 670–672 (2010).
[Crossref] [PubMed]

2009 (3)

2008 (2)

2007 (2)

2006 (2)

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[Crossref] [PubMed]

S. Oemrawsingh, J. de Jong, X. Ma, A. Aiello, E. Eliel, and J. Woerdman, “High-dimensional mode analyzers for spatial quantum entanglement,” Phys. Rev. A 73(3), 032339 (2006).
[Crossref]

2005 (4)

S. Tao, X. C. Yuan, J. Lin, X. Peng, and H. Niu, “Fractional optical vortex beam induced rotation of particles,” Opt. Express 13(20), 7726–7731 (2005).
[Crossref] [PubMed]

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

F. Flossmann, U. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. 250(4–6), 218–230 (2005).
[Crossref]

F. Flossmann, U. Schwarz, and M. Maier, “Optical vortices in a Laguerre-Gaussian LG01 beam,” J. Mod. Opt. 52(13), 1009–1017 (2005).
[Crossref]

2004 (4)

Z. Bouchal and J. Courtial, “The connection of singular and nondiffracting optics,” J. Opt. A 6(5), S184–S188 (2004).
[Crossref]

W. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239(1–3), 129–135 (2004).
[Crossref]

M. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6(2), 259–268 (2004).
[Crossref]

Y. Cai and S. Y. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29(23), 2716–2718 (2004).
[Crossref] [PubMed]

2003 (4)

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1–6), 117–125 (2003).
[Crossref]

G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28(11), 878–880 (2003).
[Crossref] [PubMed]

S. H. Tao, W. M. Lee, and X. C. Yuan, “Dynamic optical manipulation with a higher-order fractional bessel beam generated from a spatial light modulator,” Opt. Lett. 28(20), 1867–1869 (2003).
[Crossref] [PubMed]

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[Crossref] [PubMed]

2002 (1)

1998 (1)

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45(3), 539–554 (1998).
[Crossref]

1995 (1)

I. Basistiy, M. Soskin, and M. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119(5–6), 604–612 (1995).
[Crossref]

1984 (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984).
[Crossref]

Aiello, A.

S. Oemrawsingh, J. de Jong, X. Ma, A. Aiello, E. Eliel, and J. Woerdman, “High-dimensional mode analyzers for spatial quantum entanglement,” Phys. Rev. A 73(3), 032339 (2006).
[Crossref]

Anzolin, G.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[Crossref] [PubMed]

Arinaga, S.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984).
[Crossref]

Banerji, J.

Barbieri, C.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[Crossref] [PubMed]

Basistiy, I.

I. Basistiy, M. Soskin, and M. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119(5–6), 604–612 (1995).
[Crossref]

Berry, M.

M. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6(2), 259–268 (2004).
[Crossref]

Bianchini, A.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[Crossref] [PubMed]

Bogatyryova, G. V.

Boran, Y.

J. Strohaber, Y. Boran, M. Sayrac, L. Johnson, F. Zhu, A. Kolomenskii, and H. Schuessler, “Nonlinear mixing of optical vortices with fractional topological charge in Raman sideband generation,” J. Opt. 19(1), 015607 (2016).
[Crossref]

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45(3), 539–554 (1998).
[Crossref]

Bouchal, Z.

Z. Bouchal and J. Courtial, “The connection of singular and nondiffracting optics,” J. Opt. A 6(5), S184–S188 (2004).
[Crossref]

Cai, Y.

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018).
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Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
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X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
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X. Liu, T. Wu, L. Liu, C. Zhao, and Y. Cai, “Experimental determination of the azimuthal and radial mode orders of a partially coherent LGpl beam,” Chin. Opt. Lett. 15(3), 030002 (2017).
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C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
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Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
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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(9), 2083–2096 (2014).
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F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
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X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
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Y. Yang, Y. Dong, C. Zhao, and Y. Cai, “Generation and propagation of an anomalous vortex beam,” Opt. Lett. 38(24), 5418–5421 (2013).
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F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
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Li, F.

R. Liu, F. Wang, D. Chen, Y. Wang, Y. Zhou, H. Gao, P. Zhang, and F. Li, “Measuring mode indices of a partially coherent vortex beam with Hanbury Brown and Twiss type experiment,” Appl. Phys. Lett. 108(5), 051107 (2016).
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Li, X.

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Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
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X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
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R. Liu, F. Wang, D. Chen, Y. Wang, Y. Zhou, H. Gao, P. Zhang, and F. Li, “Measuring mode indices of a partially coherent vortex beam with Hanbury Brown and Twiss type experiment,” Appl. Phys. Lett. 108(5), 051107 (2016).
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Liu, X.

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Y. Fang, Q. Lu, X. Wang, W. Zhang, and L. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A (Coll. Park) 95(2), 023821 (2017).
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Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018).
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Ma, X.

S. Oemrawsingh, J. de Jong, X. Ma, A. Aiello, E. Eliel, and J. Woerdman, “High-dimensional mode analyzers for spatial quantum entanglement,” Phys. Rev. A 73(3), 032339 (2006).
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F. Flossmann, U. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. 250(4–6), 218–230 (2005).
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F. Flossmann, U. Schwarz, and M. Maier, “Optical vortices in a Laguerre-Gaussian LG01 beam,” J. Mod. Opt. 52(13), 1009–1017 (2005).
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R. Ni, Y. Niu, L. Du, X. Hu, Y. Zhang, and S. Zhu, “Topological charge transfer in frequency doubling of fractional orbital angular momentum state,” Appl. Phys. Lett. 109(15), 151103 (2016).
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X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
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Z. Dai, Z. Yang, S. Zhang, and Z. Pang, “Propagation of anomalous vortex beams in strongly nonlocal nonlinear media,” Opt. Commun. 350, 19–27 (2015).
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Yu, J.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
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C. S. Guo, Y. N. Yu, and Z. Hong, “Optical sorting using an array of optical vortices with fractional topological charge,” Opt. Commun. 283(9), 1889–1893 (2010).
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Yuan, Y.

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Y. Fang, Q. Lu, X. Wang, W. Zhang, and L. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A (Coll. Park) 95(2), 023821 (2017).
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Appl. Phys. Lett. (4)

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Figures (10)

Fig. 1
Fig. 1 Normalized intensity distribution I( ρ )/ I max ( ρ ) of a PCIV beam ( l=1) focused by a thin lens at several propagation distances z for different values of σ g .
Fig. 2
Fig. 2 Normalized intensity distribution I( ρ )/ I max ( ρ ) of a PCFV beam ( l=1.5) focused by a thin lens at several propagation distances z for different values of σ g .
Fig. 3
Fig. 3 Normalized intensity distribution I( ρ )/ I max ( ρ ) of a PCV beam with different l (both integral and fractional) focused by a thin lens at focal plane ( z=f) for different values of σ g .
Fig. 4
Fig. 4 Density plot of the modulus of the CSD | W(ρ,0) | of a PCIV beam ( l=1) focused by a thin lens at several propagation distances z for different values of σ g .
Fig. 5
Fig. 5 Density plot of the modulus of the CSD | W(ρ,0) |of a PCFV beam ( l=1.5) focused by a thin lens at several propagation distances z for different values of σ g .
Fig. 6
Fig. 6 Density plot of the modulus of the CSD | W(ρ,0) |of a PCV beam with different l (both integral and fractional) focused bya lens at focal plane ( z=f) for different values of σ g .
Fig. 7
Fig. 7 Experimental setup for generating PCFV beam, measuring the focused intensity distribution and the modulus of the CSD distribution. Laser, Nd: YAG laser; BE, beam expander; L1, L2 and L3, thin lenses; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; BS, beam splitter; SLM, spatial light modulator; CGH, computer-generated holograms; CCD, charge-coupled device.
Fig. 8
Fig. 8 Experimental results of the normalized intensity distribution of the generated PCFV beam ( l=1.5) focused by a thin lens with focal length f = 400mm at several propagation distances z for different values of σ g .
Fig. 9
Fig. 9 Experimental results of the normalized intensity distribution of the generated PCV beam with different l focused by a thin lens with focal length f = 400mm at focal plane (z = f) for different values of σ g .
Fig. 10
Fig. 10 Experimental results of the modulus of the CSD | W(ρ,0) | distribution of the generated PCV beam with different l (both integral and fractional) focused by a thin lens with focal length f = 400mm at focal plane (z = f) for different values of σ g .

Equations (22)

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W 0 ( r 1 , r 2 )= E * ( r 1 )E( r 2 ) ,
W 0 ( r 1 , r 2 )= p(v) H * ( r 1 ,v)H( r 2 ,v) d 2 v,
p(v)=exp(2 v 2 / σ g 2 ),
H(r,v)= ( 2 r w 0 ) l exp( r 2 w 0 2 )exp( ilφ )exp(ikrv),
W 0 ( r 1 , r 2 )= ( 2 r 1 r 2 w 0 2 ) l exp( r 1 2 + r 2 2 w 0 2 )exp[ il( φ 1 φ 2 ) ]exp( ( r 2 r 1 ) 2 2 σ g 2 ).
W( ρ 1 , ρ 2 )= k 2 4 π 2 B 2 W 0 ( r 1 , r 2 )exp( ikA 2B r 1 2 + ik B r 1 ρ 1 ikD 2B ρ 1 2 ) ×exp( ikA 2B r 2 2 ik B r 2 ρ 2 + ikD 2B ρ 2 2 ) d 2 r 1 d 2 r 2 ,
r s =( r 2 + r 1 )/2, r d = r 2 r 1 ,
ρ s =( ρ 2 + ρ 1 )/2, ρ d = ρ 2 ρ 1 .
W( ρ 1 , ρ 2 )= k 2 4 π 2 B 2 exp( ikD ρ s ρ d B ) A * ( r s r d /2)A( r s + r d /2)p(v) ×exp( ik B ( r s ρ d + r d ρ s ) )exp( ik r d v ) d 2 r s d 2 r d d 2 v,
A(r)= ( 2 r w 0 ) l exp( r 2 w 0 2 )exp( ilφ )exp( ikA r 2 /2B ).
A * ( r s r d /2)= ( k 2π ) 2 A ˜ * ( u )exp( iku( r s r d /2) ) d 2 u ,
A( r s + r d /2)= ( k 2π ) 2 A ˜ ( u )exp( iku( r s + r d /2) ) d 2 u .
W( ρ 1 , ρ 2 )= k 2 4 π 2 B 2 exp( ikD ρ s ρ d B ) A ˜ * ( v+(2 ρ s ρ d )/2B ) × A ˜ ( v+(2 ρ s + ρ d )/2B )p(v) d 2 v.
I(ρ)=W(ρ,ρ)= k 2 4 π 2 B 2 | A ˜ ( v+ρ/B ) | 2 p(v) d 2 v.
I=f(ρ/B)p(ρ/B),
A ˜ ( v )= A(r)exp( ikvr ) d 2 r .
F ˜ (ε)= F(x)exp( i2πεx )dx .
I(ρ)= λ 2 k 2 4 π 2 B 2 | A ˜ ( v λ + ρ λB ) | 2 p( v λ ) d 2 v λ .
W(ρ,0)= k 2 4 π 2 B 2 exp( ikD ρ 2 2B ) A ˜ * ( v )p(v) A ˜ ( v+ρ/B ) d 2 v.
F Ap (v)= A ˜ * ( v )p(v).
W(ρ,0)= 1 B 2 exp[ iπλBD ( ρ λB ) 2 ] F AP ( v λ ) A ˜ ( v λ + ρ λB ) d 2 v λ .
( A B C D )=( 1 z 0 1 )( 1 0 1/f 1 )=( 1z/f z 1/f 1 ).

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