Abstract

In a recent publication (Appl. Phys. Lett. 100 (2012) 051108), a monochromatic partially coherent radially polarized (RP) beam was generated experimentally. In this paper, we analyze the spectral changes of a polychromatic partially coherent RP beam focused by a thin lens for the first time, and compare with that of a focused scalar polychromatic GSM beam. Furthermore, we report experimental generation of a polychromatic partially coherent RP beam and carry out experimental measurement of the spectral changes of such beam focused by a thin lens. Our results show that the behavior of the spectral changes of a focused polychromatic partially coherent RP beam is different from that of a focused scalar polychromatic GSM beam. Our experimental results are consistent with the theoretical predictions.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett.56(13), 1370–1372 (1986).
    [CrossRef] [PubMed]
  2. E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature326(6111), 363–365 (1987).
    [CrossRef]
  3. E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun.62(1), 12–16 (1987).
    [CrossRef]
  4. E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett.58(25), 2646–2648 (1987).
    [CrossRef] [PubMed]
  5. E. Wolf and D. F. James, “Correlation-induced spectral changes,” Rep. Prog. Phys.59(6), 771–818 (1996).
    [CrossRef]
  6. H. C. Kandpal, A. Wasan, J. S. Vaishya, E. S. R. Gopal, M. Singh, B. B. Sanwal, and R. Sagar, “Application of spatial coherence spectroscopy for determining the angular diameters of stars: feasibility experiment,” Indian J. Pure Appl. Phys.36, 665–674 (1988).
  7. D. F. James, H. C. Kandpal, and E. Wolf, “A new method for determining the angular separation of double stars,” Astrophys. J.445, 406–410 (1995).
    [CrossRef]
  8. D. F. V. James and E. Wolf, “Determination of field corrections from spectral measurements with application to synthetic aperture imaging,” Radio Sci.26(5), 1239–1243 (1991).
    [CrossRef]
  9. H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, and K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt.42(2), 455–464 (1995).
    [CrossRef]
  10. H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun.73(3), 169–172 (1989).
    [CrossRef]
  11. E. Wolf, T. Shirai, H. Chen, and W. Wang, “Coherence filters and their uses: 1. Basic theory and examples,” J. Mod. Opt.44, 1345–1353 (1997).
  12. T. Shirai, E. Wolf, H. Chen, and W. Wang, “Coherence filters and their uses: 2. One-dimensional realizations,” J. Mod. Opt.45(4), 799–816 (1998).
    [CrossRef]
  13. D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett.32(24), 3483–3485 (2007).
    [CrossRef] [PubMed]
  14. B. K. Yadav, S. A. M. Rizvi, S. Raman, R. Mehrotra, and H. C. Kandpal, “Information encoding by spectral anamolies of spatially coherent light diffracted by an annular aperture,” Opt. Commun.269(2), 253–260 (2007).
    [CrossRef]
  15. F. Gori, G. L. Marcopoli, and M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun.81(1–2), 123–130 (1991).
    [CrossRef]
  16. C. Palma, G. Cincotti, and G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron.34(2), 378–383 (1998).
    [CrossRef]
  17. J. Pu and S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron.36(12), 1407–1411 (2000).
    [CrossRef]
  18. L. Pan and B. Lu, “The spectral switch of partially coherent light in Young’s experiment,” IEEE J. Quantum Electron.36, 1407–1411 (2001).
  19. Y. Cai, Y. Huan, and Q. Lin, “Spectral shift of partially coherent twisted anisotropic Gaussian–Schell-model beams focused by a thin lens,” J. Opt. A, Pure Appl. Opt.5(4), 397–401 (2003).
    [CrossRef]
  20. O. Korotkova and E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence,” Opt. Lett.35(22), 3772–3774 (2010).
    [CrossRef] [PubMed]
  21. Z. Tong and O. Korotkova, “Spectral shifts and switches in random fields upon interaction with negative-phase materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82, 013829 (2010).
  22. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312(5–6), 263–267 (2003).
    [CrossRef]
  23. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt.3(1), 1–9 (2001).
    [CrossRef]
  24. O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett.29(11), 1173–1175 (2004).
    [CrossRef] [PubMed]
  25. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A25(5), 1016–1021 (2008).
    [CrossRef] [PubMed]
  26. O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett.30(2), 198–200 (2005).
    [CrossRef] [PubMed]
  27. A. Luis, “Degree of polarization for three-dimensional fields as a distance between correlation matrices,” Opt. Commun.253(1–3), 10–14 (2005).
    [CrossRef]
  28. T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett.34(21), 3394–3396 (2009).
    [CrossRef] [PubMed]
  29. M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett.33(19), 2266–2268 (2008).
    [CrossRef] [PubMed]
  30. B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett.33(20), 2410–2412 (2008).
    [CrossRef] [PubMed]
  31. F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett.36(14), 2722–2724 (2011).
    [CrossRef] [PubMed]
  32. J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.75(5), 056610 (2007).
    [CrossRef] [PubMed]
  33. J. Pu, O. Korotkova, and E. Wolf, “Invariance and noninvariance of the spectra of stochastic electromagnetic beams on propogation,” Opt. Lett.31(14), 2097–2099 (2006).
    [CrossRef] [PubMed]
  34. O. Korotkova, J. Pu, and E. Wolf, “Spectral changes in electromagnetic stochastic beams propagating through turbulent atmosphere,” J. Mod. Opt.55(8), 1199–1208 (2008).
    [CrossRef]
  35. F. Zhou, S. Zhu, and Y. Cai, “Spectral shift of an electromagnetic Gaussian Schell-model beam propagating through tissue,” J. Mod. Opt.58(1), 38–44 (2011).
    [CrossRef]
  36. L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett.95(18), 181112 (2009).
    [CrossRef]
  37. S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B99(1–2), 317–323 (2010).
    [CrossRef]
  38. M. Yao, Y. Cai, and O. Korotkova, “Spectral shift of a stochastic electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Commun.283(22), 4505–4511 (2010).
    [CrossRef]
  39. S. Joshi, B. K. Yadav, M. Verma, M. S. Khan, and H. C. Kandpal, “Effect of polarization on spectral anomalies of diffracted stochastic electromagnetic beams,” J. Opt.15(3), 035405 (2013).
    [CrossRef]
  40. Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express19(7), 5979–5992 (2011).
    [CrossRef] [PubMed]
  41. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics1(1), 1–57 (2009).
    [CrossRef]
  42. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7(2), 77–87 (2000).
    [CrossRef] [PubMed]
  43. D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express9(10), 490–497 (2001).
    [CrossRef] [PubMed]
  44. Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express10(7), 324–331 (2002).
    [CrossRef] [PubMed]
  45. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91(23), 233901 (2003).
    [CrossRef] [PubMed]
  46. P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett.102(18), 183902 (2009).
    [CrossRef] [PubMed]
  47. H. Wang, L. Shi, B. Lukyanchuk, C. J. R. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
    [CrossRef]
  48. W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett.9(12), 4320–4325 (2009).
    [CrossRef] [PubMed]
  49. Y. Dong, F. Feng, Y. Chen, C. Zhao, and Y. Cai, “Statistical properties of a nonparaxial cylindrical vector partially coherent field in free space,” Opt. Express20(14), 15908–15927 (2012).
    [CrossRef] [PubMed]
  50. R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B112(2), 247–259 (2013).
    [CrossRef]
  51. F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett.100(5), 051108 (2012).
    [CrossRef]
  52. G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express20(27), 28301–28318 (2012).
    [CrossRef] [PubMed]
  53. 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. A86(1), 013840 (2012).
    [CrossRef]
  54. F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett.103(9), 091102 (2013).
    [CrossRef]
  55. 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]
  56. H. Mashaal, A. Goldstein, D. Feuermann, and J. M. Gordon, “First direct measurement of the spatial coherence of sunlight,” Opt. Lett.37(17), 3516–3518 (2012).
    [CrossRef] [PubMed]
  57. F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A25(8), 2001–2010 (2008).
    [CrossRef] [PubMed]

2013 (3)

S. Joshi, B. K. Yadav, M. Verma, M. S. Khan, and H. C. Kandpal, “Effect of polarization on spectral anomalies of diffracted stochastic electromagnetic beams,” J. Opt.15(3), 035405 (2013).
[CrossRef]

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B112(2), 247–259 (2013).
[CrossRef]

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett.103(9), 091102 (2013).
[CrossRef]

2012 (5)

2011 (3)

2010 (4)

O. Korotkova and E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence,” Opt. Lett.35(22), 3772–3774 (2010).
[CrossRef] [PubMed]

S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B99(1–2), 317–323 (2010).
[CrossRef]

M. Yao, Y. Cai, and O. Korotkova, “Spectral shift of a stochastic electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Commun.283(22), 4505–4511 (2010).
[CrossRef]

Z. Tong and O. Korotkova, “Spectral shifts and switches in random fields upon interaction with negative-phase materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82, 013829 (2010).

2009 (5)

L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett.95(18), 181112 (2009).
[CrossRef]

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett.9(12), 4320–4325 (2009).
[CrossRef] [PubMed]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics1(1), 1–57 (2009).
[CrossRef]

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett.102(18), 183902 (2009).
[CrossRef] [PubMed]

T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett.34(21), 3394–3396 (2009).
[CrossRef] [PubMed]

2008 (6)

2007 (3)

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.75(5), 056610 (2007).
[CrossRef] [PubMed]

B. K. Yadav, S. A. M. Rizvi, S. Raman, R. Mehrotra, and H. C. Kandpal, “Information encoding by spectral anamolies of spatially coherent light diffracted by an annular aperture,” Opt. Commun.269(2), 253–260 (2007).
[CrossRef]

D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett.32(24), 3483–3485 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (2)

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett.30(2), 198–200 (2005).
[CrossRef] [PubMed]

A. Luis, “Degree of polarization for three-dimensional fields as a distance between correlation matrices,” Opt. Commun.253(1–3), 10–14 (2005).
[CrossRef]

2004 (1)

2003 (3)

Y. Cai, Y. Huan, and Q. Lin, “Spectral shift of partially coherent twisted anisotropic Gaussian–Schell-model beams focused by a thin lens,” J. Opt. A, Pure Appl. Opt.5(4), 397–401 (2003).
[CrossRef]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312(5–6), 263–267 (2003).
[CrossRef]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91(23), 233901 (2003).
[CrossRef] [PubMed]

2002 (2)

2001 (3)

D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express9(10), 490–497 (2001).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt.3(1), 1–9 (2001).
[CrossRef]

L. Pan and B. Lu, “The spectral switch of partially coherent light in Young’s experiment,” IEEE J. Quantum Electron.36, 1407–1411 (2001).

2000 (2)

J. Pu and S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron.36(12), 1407–1411 (2000).
[CrossRef]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7(2), 77–87 (2000).
[CrossRef] [PubMed]

1998 (2)

T. Shirai, E. Wolf, H. Chen, and W. Wang, “Coherence filters and their uses: 2. One-dimensional realizations,” J. Mod. Opt.45(4), 799–816 (1998).
[CrossRef]

C. Palma, G. Cincotti, and G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron.34(2), 378–383 (1998).
[CrossRef]

1997 (1)

E. Wolf, T. Shirai, H. Chen, and W. Wang, “Coherence filters and their uses: 1. Basic theory and examples,” J. Mod. Opt.44, 1345–1353 (1997).

1996 (1)

E. Wolf and D. F. James, “Correlation-induced spectral changes,” Rep. Prog. Phys.59(6), 771–818 (1996).
[CrossRef]

1995 (2)

D. F. James, H. C. Kandpal, and E. Wolf, “A new method for determining the angular separation of double stars,” Astrophys. J.445, 406–410 (1995).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, and K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt.42(2), 455–464 (1995).
[CrossRef]

1991 (2)

D. F. V. James and E. Wolf, “Determination of field corrections from spectral measurements with application to synthetic aperture imaging,” Radio Sci.26(5), 1239–1243 (1991).
[CrossRef]

F. Gori, G. L. Marcopoli, and M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun.81(1–2), 123–130 (1991).
[CrossRef]

1989 (1)

H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun.73(3), 169–172 (1989).
[CrossRef]

1988 (1)

H. C. Kandpal, A. Wasan, J. S. Vaishya, E. S. R. Gopal, M. Singh, B. B. Sanwal, and R. Sagar, “Application of spatial coherence spectroscopy for determining the angular diameters of stars: feasibility experiment,” Indian J. Pure Appl. Phys.36, 665–674 (1988).

1987 (3)

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature326(6111), 363–365 (1987).
[CrossRef]

E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun.62(1), 12–16 (1987).
[CrossRef]

E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett.58(25), 2646–2648 (1987).
[CrossRef] [PubMed]

1986 (1)

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett.56(13), 1370–1372 (1986).
[CrossRef] [PubMed]

Abeysinghe, D. C.

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett.9(12), 4320–4325 (2009).
[CrossRef] [PubMed]

Antosiewicz, T. J.

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett.102(18), 183902 (2009).
[CrossRef] [PubMed]

Baykal, Y.

Biss, D. P.

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A25(5), 1016–1021 (2008).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt.3(1), 1–9 (2001).
[CrossRef]

Brown, T. G.

Cai, Y.

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B112(2), 247–259 (2013).
[CrossRef]

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett.103(9), 091102 (2013).
[CrossRef]

Y. Dong, F. Feng, Y. Chen, C. Zhao, and Y. Cai, “Statistical properties of a nonparaxial cylindrical vector partially coherent field in free space,” Opt. Express20(14), 15908–15927 (2012).
[CrossRef] [PubMed]

G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express20(27), 28301–28318 (2012).
[CrossRef] [PubMed]

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett.100(5), 051108 (2012).
[CrossRef]

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. A86(1), 013840 (2012).
[CrossRef]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express19(7), 5979–5992 (2011).
[CrossRef] [PubMed]

F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett.36(14), 2722–2724 (2011).
[CrossRef] [PubMed]

F. Zhou, S. Zhu, and Y. Cai, “Spectral shift of an electromagnetic Gaussian Schell-model beam propagating through tissue,” J. Mod. Opt.58(1), 38–44 (2011).
[CrossRef]

S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B99(1–2), 317–323 (2010).
[CrossRef]

M. Yao, Y. Cai, and O. Korotkova, “Spectral shift of a stochastic electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Commun.283(22), 4505–4511 (2010).
[CrossRef]

F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A25(8), 2001–2010 (2008).
[CrossRef] [PubMed]

M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett.33(19), 2266–2268 (2008).
[CrossRef] [PubMed]

Y. Cai, Y. Huan, and Q. Lin, “Spectral shift of partially coherent twisted anisotropic Gaussian–Schell-model beams focused by a thin lens,” J. Opt. A, Pure Appl. Opt.5(4), 397–401 (2003).
[CrossRef]

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, H.

T. Shirai, E. Wolf, H. Chen, and W. Wang, “Coherence filters and their uses: 2. One-dimensional realizations,” J. Mod. Opt.45(4), 799–816 (1998).
[CrossRef]

E. Wolf, T. Shirai, H. Chen, and W. Wang, “Coherence filters and their uses: 1. Basic theory and examples,” J. Mod. Opt.44, 1345–1353 (1997).

Chen, R.

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B112(2), 247–259 (2013).
[CrossRef]

Chen, W.

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett.9(12), 4320–4325 (2009).
[CrossRef] [PubMed]

Chen, Y.

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. J. R. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Cincotti, G.

C. Palma, G. Cincotti, and G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron.34(2), 378–383 (1998).
[CrossRef]

Ding, C.

L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett.95(18), 181112 (2009).
[CrossRef]

Dong, Y.

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B112(2), 247–259 (2013).
[CrossRef]

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett.100(5), 051108 (2012).
[CrossRef]

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. A86(1), 013840 (2012).
[CrossRef]

Y. Dong, F. Feng, Y. Chen, C. Zhao, and Y. Cai, “Statistical properties of a nonparaxial cylindrical vector partially coherent field in free space,” Opt. Express20(14), 15908–15927 (2012).
[CrossRef] [PubMed]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express19(7), 5979–5992 (2011).
[CrossRef] [PubMed]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91(23), 233901 (2003).
[CrossRef] [PubMed]

Eyyuboglu, H. T.

Feng, F.

Feuermann, D.

Friberg, A. T.

Goldstein, A.

Gopal, E. S. R.

H. C. Kandpal, A. Wasan, J. S. Vaishya, E. S. R. Gopal, M. Singh, B. B. Sanwal, and R. Sagar, “Application of spatial coherence spectroscopy for determining the angular diameters of stars: feasibility experiment,” Indian J. Pure Appl. Phys.36, 665–674 (1988).

Gordon, J. M.

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A25(5), 1016–1021 (2008).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt.3(1), 1–9 (2001).
[CrossRef]

F. Gori, G. L. Marcopoli, and M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun.81(1–2), 123–130 (1991).
[CrossRef]

Guattari, G.

C. Palma, G. Cincotti, and G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron.34(2), 378–383 (1998).
[CrossRef]

Huan, Y.

Y. Cai, Y. Huan, and Q. Lin, “Spectral shift of partially coherent twisted anisotropic Gaussian–Schell-model beams focused by a thin lens,” J. Opt. A, Pure Appl. Opt.5(4), 397–401 (2003).
[CrossRef]

James, D. F.

E. Wolf and D. F. James, “Correlation-induced spectral changes,” Rep. Prog. Phys.59(6), 771–818 (1996).
[CrossRef]

D. F. James, H. C. Kandpal, and E. Wolf, “A new method for determining the angular separation of double stars,” Astrophys. J.445, 406–410 (1995).
[CrossRef]

James, D. F. V.

D. F. V. James and E. Wolf, “Determination of field corrections from spectral measurements with application to synthetic aperture imaging,” Radio Sci.26(5), 1239–1243 (1991).
[CrossRef]

Joshi, K. C.

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, and K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt.42(2), 455–464 (1995).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun.73(3), 169–172 (1989).
[CrossRef]

Joshi, S.

S. Joshi, B. K. Yadav, M. Verma, M. S. Khan, and H. C. Kandpal, “Effect of polarization on spectral anomalies of diffracted stochastic electromagnetic beams,” J. Opt.15(3), 035405 (2013).
[CrossRef]

Kandpal, H. C.

S. Joshi, B. K. Yadav, M. Verma, M. S. Khan, and H. C. Kandpal, “Effect of polarization on spectral anomalies of diffracted stochastic electromagnetic beams,” J. Opt.15(3), 035405 (2013).
[CrossRef]

B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett.33(20), 2410–2412 (2008).
[CrossRef] [PubMed]

B. K. Yadav, S. A. M. Rizvi, S. Raman, R. Mehrotra, and H. C. Kandpal, “Information encoding by spectral anamolies of spatially coherent light diffracted by an annular aperture,” Opt. Commun.269(2), 253–260 (2007).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, and K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt.42(2), 455–464 (1995).
[CrossRef]

D. F. James, H. C. Kandpal, and E. Wolf, “A new method for determining the angular separation of double stars,” Astrophys. J.445, 406–410 (1995).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun.73(3), 169–172 (1989).
[CrossRef]

H. C. Kandpal, A. Wasan, J. S. Vaishya, E. S. R. Gopal, M. Singh, B. B. Sanwal, and R. Sagar, “Application of spatial coherence spectroscopy for determining the angular diameters of stars: feasibility experiment,” Indian J. Pure Appl. Phys.36, 665–674 (1988).

Kanseri, B.

Khan, M. S.

S. Joshi, B. K. Yadav, M. Verma, M. S. Khan, and H. C. Kandpal, “Effect of polarization on spectral anomalies of diffracted stochastic electromagnetic beams,” J. Opt.15(3), 035405 (2013).
[CrossRef]

Korotkova, O.

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett.100(5), 051108 (2012).
[CrossRef]

Z. Tong and O. Korotkova, “Spectral shifts and switches in random fields upon interaction with negative-phase materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82, 013829 (2010).

M. Yao, Y. Cai, and O. Korotkova, “Spectral shift of a stochastic electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Commun.283(22), 4505–4511 (2010).
[CrossRef]

O. Korotkova and E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence,” Opt. Lett.35(22), 3772–3774 (2010).
[CrossRef] [PubMed]

O. Korotkova, J. Pu, and E. Wolf, “Spectral changes in electromagnetic stochastic beams propagating through turbulent atmosphere,” J. Mod. Opt.55(8), 1199–1208 (2008).
[CrossRef]

M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett.33(19), 2266–2268 (2008).
[CrossRef] [PubMed]

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.75(5), 056610 (2007).
[CrossRef] [PubMed]

D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett.32(24), 3483–3485 (2007).
[CrossRef] [PubMed]

J. Pu, O. Korotkova, and E. Wolf, “Invariance and noninvariance of the spectra of stochastic electromagnetic beams on propogation,” Opt. Lett.31(14), 2097–2099 (2006).
[CrossRef] [PubMed]

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett.30(2), 198–200 (2005).
[CrossRef] [PubMed]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett.29(11), 1173–1175 (2004).
[CrossRef] [PubMed]

Leger, J. R.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91(23), 233901 (2003).
[CrossRef] [PubMed]

Lin, Q.

Lindfors, K.

Liu, L.

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett.103(9), 091102 (2013).
[CrossRef]

Liu, X.

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett.103(9), 091102 (2013).
[CrossRef]

F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett.36(14), 2722–2724 (2011).
[CrossRef] [PubMed]

Lu, B.

L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett.95(18), 181112 (2009).
[CrossRef]

L. Pan and B. Lu, “The spectral switch of partially coherent light in Young’s experiment,” IEEE J. Quantum Electron.36, 1407–1411 (2001).

Luis, A.

A. Luis, “Degree of polarization for three-dimensional fields as a distance between correlation matrices,” Opt. Commun.253(1–3), 10–14 (2005).
[CrossRef]

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. J. R. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Marcopoli, G. L.

F. Gori, G. L. Marcopoli, and M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun.81(1–2), 123–130 (1991).
[CrossRef]

Mashaal, H.

Mehrotra, R.

B. K. Yadav, S. A. M. Rizvi, S. Raman, R. Mehrotra, and H. C. Kandpal, “Information encoding by spectral anamolies of spatially coherent light diffracted by an annular aperture,” Opt. Commun.269(2), 253–260 (2007).
[CrossRef]

Mehta, D. S.

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, and K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt.42(2), 455–464 (1995).
[CrossRef]

Mondello, A.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt.3(1), 1–9 (2001).
[CrossRef]

Nelson, R. L.

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett.9(12), 4320–4325 (2009).
[CrossRef] [PubMed]

Nemoto, S.

J. Pu and S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron.36(12), 1407–1411 (2000).
[CrossRef]

Palma, C.

C. Palma, G. Cincotti, and G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron.34(2), 378–383 (1998).
[CrossRef]

Pan, L.

L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett.95(18), 181112 (2009).
[CrossRef]

L. Pan and B. Lu, “The spectral switch of partially coherent light in Young’s experiment,” IEEE J. Quantum Electron.36, 1407–1411 (2001).

Piquero, G.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt.3(1), 1–9 (2001).
[CrossRef]

Pniewski, J.

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett.102(18), 183902 (2009).
[CrossRef] [PubMed]

Pu, J.

O. Korotkova, J. Pu, and E. Wolf, “Spectral changes in electromagnetic stochastic beams propagating through turbulent atmosphere,” J. Mod. Opt.55(8), 1199–1208 (2008).
[CrossRef]

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.75(5), 056610 (2007).
[CrossRef] [PubMed]

J. Pu, O. Korotkova, and E. Wolf, “Invariance and noninvariance of the spectra of stochastic electromagnetic beams on propogation,” Opt. Lett.31(14), 2097–2099 (2006).
[CrossRef] [PubMed]

J. Pu and S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron.36(12), 1407–1411 (2000).
[CrossRef]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91(23), 233901 (2003).
[CrossRef] [PubMed]

Raman, S.

B. K. Yadav, S. A. M. Rizvi, S. Raman, R. Mehrotra, and H. C. Kandpal, “Information encoding by spectral anamolies of spatially coherent light diffracted by an annular aperture,” Opt. Commun.269(2), 253–260 (2007).
[CrossRef]

Ramírez-Sánchez, V.

Rizvi, S. A. M.

B. K. Yadav, S. A. M. Rizvi, S. Raman, R. Mehrotra, and H. C. Kandpal, “Information encoding by spectral anamolies of spatially coherent light diffracted by an annular aperture,” Opt. Commun.269(2), 253–260 (2007).
[CrossRef]

Sagar, R.

H. C. Kandpal, A. Wasan, J. S. Vaishya, E. S. R. Gopal, M. Singh, B. B. Sanwal, and R. Sagar, “Application of spatial coherence spectroscopy for determining the angular diameters of stars: feasibility experiment,” Indian J. Pure Appl. Phys.36, 665–674 (1988).

Salem, M.

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A25(5), 1016–1021 (2008).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt.3(1), 1–9 (2001).
[CrossRef]

F. Gori, G. L. Marcopoli, and M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun.81(1–2), 123–130 (1991).
[CrossRef]

Sanwal, B. B.

H. C. Kandpal, A. Wasan, J. S. Vaishya, E. S. R. Gopal, M. Singh, B. B. Sanwal, and R. Sagar, “Application of spatial coherence spectroscopy for determining the angular diameters of stars: feasibility experiment,” Indian J. Pure Appl. Phys.36, 665–674 (1988).

Saxena, K.

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, and K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt.42(2), 455–464 (1995).
[CrossRef]

Setälä, T.

Shchepakina, E.

Sheppard, C. J. R.

H. Wang, L. Shi, B. Lukyanchuk, C. J. R. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. J. R. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Shirai, T.

T. Shirai, E. Wolf, H. Chen, and W. Wang, “Coherence filters and their uses: 2. One-dimensional realizations,” J. Mod. Opt.45(4), 799–816 (1998).
[CrossRef]

E. Wolf, T. Shirai, H. Chen, and W. Wang, “Coherence filters and their uses: 1. Basic theory and examples,” J. Mod. Opt.44, 1345–1353 (1997).

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt.3(1), 1–9 (2001).
[CrossRef]

Singh, M.

H. C. Kandpal, A. Wasan, J. S. Vaishya, E. S. R. Gopal, M. Singh, B. B. Sanwal, and R. Sagar, “Application of spatial coherence spectroscopy for determining the angular diameters of stars: feasibility experiment,” Indian J. Pure Appl. Phys.36, 665–674 (1988).

Szoplik, T.

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett.102(18), 183902 (2009).
[CrossRef] [PubMed]

Tong, Z.

Z. Tong and O. Korotkova, “Spectral shifts and switches in random fields upon interaction with negative-phase materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82, 013829 (2010).

Vaishya, J. S.

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, and K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt.42(2), 455–464 (1995).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun.73(3), 169–172 (1989).
[CrossRef]

H. C. Kandpal, A. Wasan, J. S. Vaishya, E. S. R. Gopal, M. Singh, B. B. Sanwal, and R. Sagar, “Application of spatial coherence spectroscopy for determining the angular diameters of stars: feasibility experiment,” Indian J. Pure Appl. Phys.36, 665–674 (1988).

Verma, M.

S. Joshi, B. K. Yadav, M. Verma, M. S. Khan, and H. C. Kandpal, “Effect of polarization on spectral anomalies of diffracted stochastic electromagnetic beams,” J. Opt.15(3), 035405 (2013).
[CrossRef]

Wang, F.

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett.103(9), 091102 (2013).
[CrossRef]

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B112(2), 247–259 (2013).
[CrossRef]

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett.100(5), 051108 (2012).
[CrossRef]

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. A86(1), 013840 (2012).
[CrossRef]

G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express20(27), 28301–28318 (2012).
[CrossRef] [PubMed]

F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett.36(14), 2722–2724 (2011).
[CrossRef] [PubMed]

F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A25(8), 2001–2010 (2008).
[CrossRef] [PubMed]

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. J. R. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Wang, W.

T. Shirai, E. Wolf, H. Chen, and W. Wang, “Coherence filters and their uses: 2. One-dimensional realizations,” J. Mod. Opt.45(4), 799–816 (1998).
[CrossRef]

E. Wolf, T. Shirai, H. Chen, and W. Wang, “Coherence filters and their uses: 1. Basic theory and examples,” J. Mod. Opt.44, 1345–1353 (1997).

Wasan, A.

H. C. Kandpal, A. Wasan, J. S. Vaishya, E. S. R. Gopal, M. Singh, B. B. Sanwal, and R. Sagar, “Application of spatial coherence spectroscopy for determining the angular diameters of stars: feasibility experiment,” Indian J. Pure Appl. Phys.36, 665–674 (1988).

Wolf, E.

O. Korotkova, J. Pu, and E. Wolf, “Spectral changes in electromagnetic stochastic beams propagating through turbulent atmosphere,” J. Mod. Opt.55(8), 1199–1208 (2008).
[CrossRef]

D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett.32(24), 3483–3485 (2007).
[CrossRef] [PubMed]

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.75(5), 056610 (2007).
[CrossRef] [PubMed]

J. Pu, O. Korotkova, and E. Wolf, “Invariance and noninvariance of the spectra of stochastic electromagnetic beams on propogation,” Opt. Lett.31(14), 2097–2099 (2006).
[CrossRef] [PubMed]

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett.30(2), 198–200 (2005).
[CrossRef] [PubMed]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett.29(11), 1173–1175 (2004).
[CrossRef] [PubMed]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312(5–6), 263–267 (2003).
[CrossRef]

T. Shirai, E. Wolf, H. Chen, and W. Wang, “Coherence filters and their uses: 2. One-dimensional realizations,” J. Mod. Opt.45(4), 799–816 (1998).
[CrossRef]

E. Wolf, T. Shirai, H. Chen, and W. Wang, “Coherence filters and their uses: 1. Basic theory and examples,” J. Mod. Opt.44, 1345–1353 (1997).

E. Wolf and D. F. James, “Correlation-induced spectral changes,” Rep. Prog. Phys.59(6), 771–818 (1996).
[CrossRef]

D. F. James, H. C. Kandpal, and E. Wolf, “A new method for determining the angular separation of double stars,” Astrophys. J.445, 406–410 (1995).
[CrossRef]

D. F. V. James and E. Wolf, “Determination of field corrections from spectral measurements with application to synthetic aperture imaging,” Radio Sci.26(5), 1239–1243 (1991).
[CrossRef]

E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett.58(25), 2646–2648 (1987).
[CrossRef] [PubMed]

E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun.62(1), 12–16 (1987).
[CrossRef]

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature326(6111), 363–365 (1987).
[CrossRef]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett.56(13), 1370–1372 (1986).
[CrossRef] [PubMed]

Wróbel, P.

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett.102(18), 183902 (2009).
[CrossRef] [PubMed]

Wu, G.

Yadav, B. K.

S. Joshi, B. K. Yadav, M. Verma, M. S. Khan, and H. C. Kandpal, “Effect of polarization on spectral anomalies of diffracted stochastic electromagnetic beams,” J. Opt.15(3), 035405 (2013).
[CrossRef]

B. K. Yadav, S. A. M. Rizvi, S. Raman, R. Mehrotra, and H. C. Kandpal, “Information encoding by spectral anamolies of spatially coherent light diffracted by an annular aperture,” Opt. Commun.269(2), 253–260 (2007).
[CrossRef]

Yao, M.

Youngworth, K. S.

Yuan, Y.

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett.103(9), 091102 (2013).
[CrossRef]

Zhan, Q.

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett.9(12), 4320–4325 (2009).
[CrossRef] [PubMed]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics1(1), 1–57 (2009).
[CrossRef]

Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express10(7), 324–331 (2002).
[CrossRef] [PubMed]

Zhao, C.

Zhao, D.

Zhao, Z.

L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett.95(18), 181112 (2009).
[CrossRef]

Zhou, F.

F. Zhou, S. Zhu, and Y. Cai, “Spectral shift of an electromagnetic Gaussian Schell-model beam propagating through tissue,” J. Mod. Opt.58(1), 38–44 (2011).
[CrossRef]

Zhu, S.

F. Zhou, S. Zhu, and Y. Cai, “Spectral shift of an electromagnetic Gaussian Schell-model beam propagating through tissue,” J. Mod. Opt.58(1), 38–44 (2011).
[CrossRef]

F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett.36(14), 2722–2724 (2011).
[CrossRef] [PubMed]

S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B99(1–2), 317–323 (2010).
[CrossRef]

Adv. Opt. Photonics (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics1(1), 1–57 (2009).
[CrossRef]

Appl. Phys. B (2)

S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B99(1–2), 317–323 (2010).
[CrossRef]

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B112(2), 247–259 (2013).
[CrossRef]

Appl. Phys. Lett. (3)

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett.100(5), 051108 (2012).
[CrossRef]

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett.103(9), 091102 (2013).
[CrossRef]

L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett.95(18), 181112 (2009).
[CrossRef]

Astrophys. J. (1)

D. F. James, H. C. Kandpal, and E. Wolf, “A new method for determining the angular separation of double stars,” Astrophys. J.445, 406–410 (1995).
[CrossRef]

IEEE J. Quantum Electron. (3)

C. Palma, G. Cincotti, and G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron.34(2), 378–383 (1998).
[CrossRef]

J. Pu and S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron.36(12), 1407–1411 (2000).
[CrossRef]

L. Pan and B. Lu, “The spectral switch of partially coherent light in Young’s experiment,” IEEE J. Quantum Electron.36, 1407–1411 (2001).

Indian J. Pure Appl. Phys. (1)

H. C. Kandpal, A. Wasan, J. S. Vaishya, E. S. R. Gopal, M. Singh, B. B. Sanwal, and R. Sagar, “Application of spatial coherence spectroscopy for determining the angular diameters of stars: feasibility experiment,” Indian J. Pure Appl. Phys.36, 665–674 (1988).

J. Mod. Opt. (5)

E. Wolf, T. Shirai, H. Chen, and W. Wang, “Coherence filters and their uses: 1. Basic theory and examples,” J. Mod. Opt.44, 1345–1353 (1997).

T. Shirai, E. Wolf, H. Chen, and W. Wang, “Coherence filters and their uses: 2. One-dimensional realizations,” J. Mod. Opt.45(4), 799–816 (1998).
[CrossRef]

O. Korotkova, J. Pu, and E. Wolf, “Spectral changes in electromagnetic stochastic beams propagating through turbulent atmosphere,” J. Mod. Opt.55(8), 1199–1208 (2008).
[CrossRef]

F. Zhou, S. Zhu, and Y. Cai, “Spectral shift of an electromagnetic Gaussian Schell-model beam propagating through tissue,” J. Mod. Opt.58(1), 38–44 (2011).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, and K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt.42(2), 455–464 (1995).
[CrossRef]

J. Opt. (1)

S. Joshi, B. K. Yadav, M. Verma, M. S. Khan, and H. C. Kandpal, “Effect of polarization on spectral anomalies of diffracted stochastic electromagnetic beams,” J. Opt.15(3), 035405 (2013).
[CrossRef]

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

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt.3(1), 1–9 (2001).
[CrossRef]

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

Y. Cai, Y. Huan, and Q. Lin, “Spectral shift of partially coherent twisted anisotropic Gaussian–Schell-model beams focused by a thin lens,” J. Opt. A, Pure Appl. Opt.5(4), 397–401 (2003).
[CrossRef]

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

Nano Lett. (1)

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett.9(12), 4320–4325 (2009).
[CrossRef] [PubMed]

Nat. Photonics (1)

H. Wang, L. Shi, B. Lukyanchuk, C. J. R. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Nature (1)

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature326(6111), 363–365 (1987).
[CrossRef]

Opt. Commun. (6)

E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun.62(1), 12–16 (1987).
[CrossRef]

B. K. Yadav, S. A. M. Rizvi, S. Raman, R. Mehrotra, and H. C. Kandpal, “Information encoding by spectral anamolies of spatially coherent light diffracted by an annular aperture,” Opt. Commun.269(2), 253–260 (2007).
[CrossRef]

F. Gori, G. L. Marcopoli, and M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun.81(1–2), 123–130 (1991).
[CrossRef]

M. Yao, Y. Cai, and O. Korotkova, “Spectral shift of a stochastic electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Commun.283(22), 4505–4511 (2010).
[CrossRef]

A. Luis, “Degree of polarization for three-dimensional fields as a distance between correlation matrices,” Opt. Commun.253(1–3), 10–14 (2005).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun.73(3), 169–172 (1989).
[CrossRef]

Opt. Express (6)

Opt. Lett. (11)

H. Mashaal, A. Goldstein, D. Feuermann, and J. M. Gordon, “First direct measurement of the spatial coherence of sunlight,” Opt. Lett.37(17), 3516–3518 (2012).
[CrossRef] [PubMed]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett.29(11), 1173–1175 (2004).
[CrossRef] [PubMed]

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett.30(2), 198–200 (2005).
[CrossRef] [PubMed]

J. Pu, O. Korotkova, and E. Wolf, “Invariance and noninvariance of the spectra of stochastic electromagnetic beams on propogation,” Opt. Lett.31(14), 2097–2099 (2006).
[CrossRef] [PubMed]

D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett.32(24), 3483–3485 (2007).
[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. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett.36(14), 2722–2724 (2011).
[CrossRef] [PubMed]

M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett.33(19), 2266–2268 (2008).
[CrossRef] [PubMed]

B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett.33(20), 2410–2412 (2008).
[CrossRef] [PubMed]

T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett.34(21), 3394–3396 (2009).
[CrossRef] [PubMed]

O. Korotkova and E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence,” Opt. Lett.35(22), 3772–3774 (2010).
[CrossRef] [PubMed]

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312(5–6), 263–267 (2003).
[CrossRef]

Phys. Rev. A (1)

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. A86(1), 013840 (2012).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.75(5), 056610 (2007).
[CrossRef] [PubMed]

Z. Tong and O. Korotkova, “Spectral shifts and switches in random fields upon interaction with negative-phase materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82, 013829 (2010).

Phys. Rev. Lett. (4)

E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett.58(25), 2646–2648 (1987).
[CrossRef] [PubMed]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett.56(13), 1370–1372 (1986).
[CrossRef] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91(23), 233901 (2003).
[CrossRef] [PubMed]

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett.102(18), 183902 (2009).
[CrossRef] [PubMed]

Radio Sci. (1)

D. F. V. James and E. Wolf, “Determination of field corrections from spectral measurements with application to synthetic aperture imaging,” Radio Sci.26(5), 1239–1243 (1991).
[CrossRef]

Rep. Prog. Phys. (1)

E. Wolf and D. F. James, “Correlation-induced spectral changes,” Rep. Prog. Phys.59(6), 771–818 (1996).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (16)

Fig. 1
Fig. 1

Focusing geometry.

Fig. 2
Fig. 2

Spectral intensities (contour graphs) of a focused polychromatic partially coherent RP beam and its corresponding composition components W xx ( u,u,ω ) and W yy ( u,u,ω ) at several propagation distances.

Fig. 3
Fig. 3

Normalized on-axis spectrum of a focused polychromatic partially coherent RP beam in the focal plane (z = 2f) and the normalized spectrum of such beam in the source plane (z = 0).

Fig. 4
Fig. 4

On-axis relative spectral shift of a focused polychromatic partially coherent RP beam versus the propagation distance.

Fig. 5
Fig. 5

On-axis relative spectral shift of a focused scalar polychromatic GSM beam versus the propagation distance.

Fig. 6
Fig. 6

Relative spectral shift of a focused polychromatic partially coherent RP beam versus the transverse coordinate u x with u y =0 at several propagation distances.

Fig. 7
Fig. 7

Relative spectral shift of a focused polychromatic partially coherent RP beam versus the propagation distance and the transverse coordinate u x with u y =0 .

Fig. 8
Fig. 8

Relative spectral shift of a focused scalar polychromatic GSM beam versus the transverse coordinate u x with u y =0 at several propagation distances.

Fig. 9
Fig. 9

Relative spectral shift of a focused scalar polychromatic GSM beam versus the propagation distance and the transverse coordinate u x with u y =0 .

Fig. 10
Fig. 10

Experimental setup for generating a polychromatic partially coherent RP beam and measuring its focused spectral intensity and spectrum. LED, light-emitting diode; CA1, CA2, circular apertures; L1, L2, thin lenses; RM, reflecting mirror; GAF, Gaussian amplitude filter; BE, beam expander; LP, linear polarizer; RPC, radial polarization converter; CCD, charge-coupled device; PC, personal computer.

Fig. 11
Fig. 11

Experimental results of (a) the spectral intensity of the generated polychromatic GSM beam just behind the LP and (b) its normalized spectrum for different values of the transverse coordinate x with y = 0.

Fig. 12
Fig. 12

Experimental results of (a) the spectral intensity and (b) the corresponding cross line (y = 0, dotted curve) of the generated polychromatic partially coherent RP beam just behind the RPC. The solid curve denotes the theoretical fit of the experimental data with σ 0 =0.68 .

Fig. 13
Fig. 13

Experimental results of the normalized spectrum of the generated polychromatic partially coherent RP beam at z = 2cm for different values of the transverse coordinate x with y = 0.

Fig. 14
Fig. 14

Experimental results of the spectral intensities and its corresponding composition components W xx ( u,u,ω ) and W yy ( u,u,ω ) of the generated polychromatic partially coherent RP beam focused by a thin lens at several propagation distances.

Fig. 15
Fig. 15

Experimental results of the normalized on-axis spectrum of the generated polychromatic partially coherent RP beam focused by a thin lens at two propagation distances.

Fig. 16
Fig. 16

Experimental results of the relative spectral shift of the generated polychromatic partially coherent RP beam focused by a thin lens versus the transverse coordinate u x with u y =0 at several propagation distances.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

W ( r 1 , r 2 ,ω )=( W xx ( r 1 , r 2 ,ω ) W xy ( r 1 , r 2 ,ω ) W yx ( r 1 , r 2 ,ω ) W yy ( r 1 , r 2 ,ω ) ),
W αβ ( r 1 , r 2 ,ω)= E α * ( r 1 ,ω) E β ( r 2 ,ω) , (α=x,y;β=x,y),
W αβ ( r 1 , r 2 ,ω )= Γ 0 2 α 1 β 2 σ 0 2 [ ( ω ω 0 ) 2 + Γ 0 2 ] exp( r 1 2 + r 2 2 σ 0 2 )exp( ( r 1 r 2 ) 2 2 δ 0 2 ), ( α,β=x,y ),
W αβ ( u 1 , u 2 ,ω)= ω 2 4 π 2 c 2 B 2 W αβ ( r 1 , r 2 ,ω)exp[ ikA 2B ( r 1 2 r 2 2 ) ] ×exp[ ik B ( r 1 u 1 r 2 u 2 ) ikD 2B ( u 1 2 u 2 2 ) ]d r 1 2 d r 2 2 ,
W αα ( u 1 , u 2 ,ω )= Γ 0 2 V( u 1 , u 2 ) 16 σ 0 2 [ ( ω ω 0 ) 2 + Γ 0 2 ] [ δ 0 2 + k 2 B 2 ( u α2 u α1 2Δ δ 0 2 )( u α1 + u α1 4ΔΠ δ 0 4 u α2 2Π δ 0 2 ) ],
W αβ ( u 1 , u 2 ,ω )= k 2 Γ 0 2 V( u 1 , u 2 ) 16 σ 0 2 B 2 [ ( ω ω 0 ) 2 + Γ 0 2 ] ( u β2 u β1 2Δ δ 0 2 )( u α1 + u α1 4ΔΠ δ 0 4 u α2 2Π δ 0 2 ), ( αβ, α,β=x,y ),
V( u 1 , u 2 )= k 2 Δ 2 Π 2 B 2 exp[ k 2 u 1 2 4Δ B 2 k 2 4Π B 2 ( u 2 u 1 2Δ δ 0 2 ) 2 ikD 2B ( u 1 2 u 2 2 ) ],
Δ= 1 σ 0 2 + 1 2 δ 0 2 + ikA 2B , Π= 1 σ 0 2 + 1 2 δ 0 2 ikA 2B 1 4Δ δ 0 4 .
S( u,ω )= W xx ( u,u,ω )+ W yy ( u,u,ω ).
W( r 1 , r 2 ,ω )= Γ 0 2 ( ω ω 0 ) 2 + Γ 0 2 exp[ r 1 2 + r 2 2 σ 0 2 ( r 1 r 2 ) 2 2 δ 0 2 ].
W( r ˜ ,ω )= Γ 0 2 ( ω ω 0 ) 2 + Γ 0 2 exp( ik 2 r ˜ T M 0 1 r ˜ ),
M 0 1 =( ( i 2k σ 0 2 i k δ 0 2 )I i k δ 0 2 I i k δ 0 2 I ( i 2k σ 0 2 i k δ 0 2 )I ),
W( u ˜ ,ω )= Γ 0 2 [ ( ω ω 0 ) 2 + Γ 0 2 ] [ det( A ˜ + B ˜ M 0 1 ) ] 1/2 ×[ ik 2 u ˜ T ( C ˜ + D ˜ M 0 1 ) ( A ˜ + B ˜ M 0 1 ) 1 u ˜ ],
A ˜ =( AI 0I 0I AI ), B ˜ =( BI 0I 0I BI ), C ˜ =( CI 0I 0I CI ), D ˜ =( DI 0I 0I DI ).
( A B C D )=( 1 zf 0 1 )( 1 0 1/f 1 )( 1 f 0 1 )=( 2z/f f 1/f 0 ).
S θ ( u,ω )= W xx ( u,u,ω ) cos 2 θ+ W yy ( u,u,ω ) sin 2 θ+ W xy ( u,u,ω )sin2θ.
η=( ω m ω 0 )/ ω 0 .

Metrics