Abstract

A new kind of partially coherent beam with non-conventional correlation function named generalized multi-Gaussian correlated Schell-model (GMGCSM) beam is proposed. The GMGCSM beam of the first or second kind is capable of producing dark hollow or flat-topped beam profile in the focal plane (or in the far field). Furthermore, we carry out experimental generation of a GMGCSM beam of the first or second kind, and measure its focused intensity. Our experimental results verify theoretical predictions. The GMGCSM beam will be useful for free-space optical communications, material thermal processing, particle or atom trapping.

© 2014 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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  6. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
    [Crossref] [PubMed]
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    [Crossref]
  8. 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).
    [Crossref] [PubMed]
  9. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
    [Crossref] [PubMed]
  10. 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).
    [Crossref] [PubMed]
  11. Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
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  12. 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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    [Crossref]
  21. S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
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    [Crossref]
  28. D. W. Coutts, “Double-pass copper vapor laser master-oscillator power-ampli- fier systems: generation of flat-top focused beams for fiber coupling and percussion drilling,” IEEE J. Quantum Electron. 38(9), 1217–1224 (2002).
    [Crossref]

2014 (9)

Y. Zhang and Y. Cai, “Random source generating far field with elliptical flat-topped beam profile,” J. Opt. 16(7), 075704 (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(1), 013801 (2014).
[Crossref]

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[Crossref] [PubMed]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

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

2013 (6)

2012 (3)

2011 (1)

2010 (1)

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

2009 (1)

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

2007 (1)

2002 (2)

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

D. W. Coutts, “Double-pass copper vapor laser master-oscillator power-ampli- fier systems: generation of flat-top focused beams for fiber coupling and percussion drilling,” IEEE J. Quantum Electron. 38(9), 1217–1224 (2002).
[Crossref]

1998 (1)

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1-3), 67–70 (1998).
[Crossref]

1979 (1)

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

1970 (1)

Cai, Y.

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

Y. Zhang and Y. Cai, “Random source generating far field with elliptical flat-topped beam profile,” J. Opt. 16(7), 075704 (2014).
[Crossref]

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).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

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(1), 013801 (2014).
[Crossref]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[Crossref] [PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

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).
[Crossref] [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).
[Crossref] [PubMed]

Chen, R.

Chen, Y.

Collins, S. A.

Coutts, D. W.

D. W. Coutts, “Double-pass copper vapor laser master-oscillator power-ampli- fier systems: generation of flat-top focused beams for fiber coupling and percussion drilling,” IEEE J. Quantum Electron. 38(9), 1217–1224 (2002).
[Crossref]

De Santis, P.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

Ding, B.

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

Du, S.

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

Eyyuboglu, H. T.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

Gbur, G.

Gori, F.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

Grimm, R.

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1-3), 67–70 (1998).
[Crossref]

Gu, Y.

Guattari, G.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

Korotkova, O.

Lajunen, H.

Liang, C.

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).
[Crossref] [PubMed]

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

Lin, Q.

Liu, L.

Liu, X.

Manek, I.

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1-3), 67–70 (1998).
[Crossref]

Mei, Z.

Ovchinnikov, Y. B.

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1-3), 67–70 (1998).
[Crossref]

Palma, C.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

Qu, J.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

Saastamoinen, T.

Sahin, S.

Sanchez, V. R.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Santarsiero, M.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

Shchepakina, E.

Shirai, T.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Suyama, T.

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

Tong, Z.

Wang, F.

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(1), 013801 (2014).
[Crossref]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[Crossref] [PubMed]

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

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).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

Wu, G.

Yuan, Y.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

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).
[Crossref] [PubMed]

Zhang, Y.

Y. Zhang and Y. Cai, “Random source generating far field with elliptical flat-topped beam profile,” J. Opt. 16(7), 075704 (2014).
[Crossref]

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

Zhao, C.

Zhu, S.

IEEE J. Quantum Electron. (1)

D. W. Coutts, “Double-pass copper vapor laser master-oscillator power-ampli- fier systems: generation of flat-top focused beams for fiber coupling and percussion drilling,” IEEE J. Quantum Electron. 38(9), 1217–1224 (2002).
[Crossref]

J. Opt. (1)

Y. Zhang and Y. Cai, “Random source generating far field with elliptical flat-topped beam profile,” J. Opt. 16(7), 075704 (2014).
[Crossref]

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

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (3)

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1-3), 67–70 (1998).
[Crossref]

Opt. Express (4)

Opt. Laser Technol. (1)

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

Opt. Lett. (11)

Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[Crossref] [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).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

Z. Tong and O. Korotkova, “Non-uniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[Crossref] [PubMed]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

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

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Phys. Rev. A (2)

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(1), 013801 (2014).
[Crossref]

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

Other (1)

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

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Figures (6)

Fig. 1
Fig. 1 Cross lines ( y 1 y 2 = 0 ) of the DOC of (a) the GMGCSM beam of the first kind and (b) the GMGCSM beam of the second kind versus ( x 1 x 2 ) / δ 0 for different values of the beam index N.
Fig. 2
Fig. 2 Normalized intensity distribution (cross line ρ y = 0 ) of the GMGCSM beam of the first kind (Figs. 2(a)-2(c)) or the second kind (Figs. 2(d)-2(f)) at several propagation distances in free space for different values of beam index N. The initial beam width and coherence length are set as σ 0 = 1 mm and δ 0 = 0.2 mm .
Fig. 3
Fig. 3 Experimental setup for generating a GMGCSM beam, measuring the square of the modulus of its DOC and its focused intensity. HP1, HP2, half-wave plates; PBS1, PBS2, polarization beam splitters; BE, beam expander; LP, linear polarizer; RGGD, rotating ground-glass disk; L1, L2, L3, thin lenses; GAF, Gaussian amplitude filter; BS, beam splitter; CCD, charge-coupled device; BPA, beam profile analyzer; PC, personal computer.
Fig. 4
Fig. 4 Experimental results of the square of the modulus of the DOC of (a) the generated GMGCSM beam of the first kind and (b) the generated GMGCSM beam of the second kind with N = 1 versus x1 with x2 = y1 = y2 = 0. The solid line denotes the theoretical fit of the experimental results.
Fig. 5
Fig. 5 Experimental results of the intensity distribution and the corresponding cross line (dotted curve) of the generated GMGCSM beam of the first kind in the source plane and at two propagation distances after passing through the thin lens L3. The solid line in Fig. 4(d) denotes Gaussian fit of the experimental results. The solid lines in Fig. 4(e) and Fig. 4(f) denote the theoretical results calculated by Eq. (13).
Fig. 6
Fig. 6 Experimental results of the intensity distribution and the corresponding cross line (dotted curve) of the generated GMGCSM beam of the second kind in the source plane and at two propagation distances after passing through the thin lens L3. The solid line in Fig. 5(d) denotes Gaussian fit of the experimental results. The solid lines in Fig. 4(e) and (f) denote the theoretical results calculated by Eq. (13).

Equations (20)

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J 0 ( r 1 , r 2 ) = exp [ r 1 2 + r 2 2 4 σ 0 2 ] γ ( r 1 , r 2 ) ,
γ ( r 1 , r 2 ) = 1 C α n = 1 2 N m = 1 2 N ( 1 ) n + m A m n α B m n α exp [ B m n α ( r 1 - r 2 ) 2 2 δ 0 2 ] ,
A m n D = ( 4 N 2 n 1 ) ( 4 N 2 m 1 ) , B m n D = 2 m n m + n ,
C D = m = 1 2 N n = 1 2 N ( 1 ) n + m A m n D B m n D .
A m n F = ( 2 N n ) ( 2 N m ) , B m n D = 2 / ( m + n ) ,
C F = m = 1 2 N n = 1 2 N ( 1 ) m + n A m n F B m n F .
J 0 ( r 1 , r 2 ) = I ( v ) H * ( r 1 , v ) H ( r 2 , v ) d 2 v ,
J 0 ( r 1 , r 2 ) = J i ( v 1 , v 2 ) H * ( r 1 , v 1 ) H ( r 2 , v 2 ) d 2 v 1 d 2 v 2 .
J i ( v 1 , v 2 ) = I ( v 1 ) I ( v 2 ) δ ( v 1 v 2 ) .
H ( r , v ) = i λ f 1 T ( r ) exp [ i π λ f 1 ( v 2 2 r v ) ] ,
I ( v ) = 1 π ω 0 2 [ n = 1 2 N ( 1 ) n 1 ( 4 N 2 n 1 ) exp ( v 2 / n ω 0 2 ) ] 2 ,
I ( v ) = 1 π ω 0 2 [ n = 1 2 N ( 1 ) n 1 ( 2 N n ) exp ( n v 2 ω 0 2 ) ] 2 ,
J ( ρ 1 , ρ 2 ) = 1 C α n = 1 2 N m = 1 2 N ( 1 ) n + m A m n α B m n α Δ m n 2 × exp ( ρ s 2 2 σ 0 2 Δ m n 2 ρ d 2 2 Ω m n 2 Δ m n 2 + i k ρ s ρ d R m n ) .
ρ s = ρ 1 + ρ 2 2 , ρ d = ρ 2 ρ 1 , Ω m n 2 = 1 4 σ 0 2 + B m n α δ 0 2 ,
Δ m n = A 2 + ( B k σ 0 Ω m n ) 2 , R m n = B Δ m n 2 D Δ m n 2 A .
E ( v , 0 ) = a x exp ( v 2 ω 0 2 ) e x + a y exp ( v 2 ω 1 2 ) e y .
I ( v ) = a x 2 | exp ( v 2 ω 0 2 ) cos θ + exp ( v 2 2 ω 0 2 ) sin θ | 2 .
I ( v ) = a x 2 2 | exp ( v 2 ω 0 2 ) exp ( v 2 2 ω 0 2 ) | 2 .
I ( v ) = 5 a x 2 4 | exp ( v 2 2 ω 0 2 ) 1 2 exp ( 2 v 2 2 ω 0 2 ) | 2 .
A = 1 z / f 3 , B = f 3 , C = 1 / f 3 , D = 0.

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