## Theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam |

Optics Express, Vol. 21, Issue 23, pp. 27682-27696 (2013)

http://dx.doi.org/10.1364/OE.21.027682

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### 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

## 1. Introduction

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,” Nature **326**(6111), 363–365 (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]

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

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]

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]

22. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A **312**(5–6), 263–267 (2003). [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]

22. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A **312**(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]

22. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A **312**(5–6), 263–267 (2003). [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]

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]

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]

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. Express **19**(7), 5979–5992 (2011). [CrossRef] [PubMed]

41. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics **1**(1), 1–57 (2009). [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. Express **20**(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. B **112**(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]

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. A **86**(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]

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]

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]

## 2. Spectral changes of a polychromatic partially coherent RP beam: Theory

*x*and

*y*directions, perpendicular to the z-axis. The asterisk denotes the complex conjugate and the angular brackets denote ensemble average.

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. Express **20**(27), 28301–28318 (2012). [CrossRef] [PubMed]

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]

*c*being the speed of light in vacuum. Note Eq. (4) represents a special case of Eq. (10) in Ref [55

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]

**312**(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]

52. G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express **20**(27), 28301–28318 (2012). [CrossRef] [PubMed]

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]

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]

**I**is a

**27**(4), 216–218 (2002). [CrossRef] [PubMed]

*f*is located at

*z*=

*f*and the exit plane is located at

*z*. The transfer matrix for the optical system between the source plane and the exit plane reads as

*x*-axis. The spectral intensity of the transmitted beam reads asFor the case of

_{10}beam. For the case of

_{01}beam. Thus the spectral intensity of a polychromatic partially coherent RP beam can be expressed as the superposition of those of orthogonally polarized partially coherent TEM

_{10}and TEM

_{01}beams (see Eq. (9)).

**100**(5), 051108 (2012). [CrossRef]

**100**(5), 051108 (2012). [CrossRef]

*f*). For the convenience of comparison, the normalized spectrum in the source plane (z = 0) is also shown in Fig. 3. Note that the spectral intensity of the beam center of the polychromatic partially coherent RP beam at z = 0 is zero, thus the dark solid curve in Fig. 3 in fact denotes the off-axis normalized spectrum, which is independent of the transverse position across the source plane. One finds from Fig. 3 that the normalized on-axis spectrum of the focused polychromatic partially coherent RP beam is similar to the normalized off-axis spectrum at z = 0, but its peak position is blue-shifted, which is similar to that of the focused scalar polychromatic GSM beam as reported in [16

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]

## 3. Spectral changes of a polychromatic partially coherent RP beam: Experiment

_{1}) with radius

_{2}) with radius

_{2}can be regarded as a polychromatic partially coherent beam, and it becomes a polychromatic GSM beam after passing through the collimation lens L

_{1}and the Gaussian amplitude filter (GAF). After passing through a beam expander and a linear polarizer, the generated polychromatic GSM beam becomes linearly polarized, and then it illuminates a radial polarization converter (RPC) produced by the company Arcoptix which is used to convert a linearly polarized polychromatic GSM beam into a polychromatic partially coherent RP beam. The transmitted beam just behind the RPC is regarded as the light source for a polychromatic partially coherent RP beam. Part II of Fig. 10 shows our setup for measuring the spectral intensity and the spectrum of the generated polychromatic partially coherent RP beam focused by the thin lens L

_{2}with focal length

*f*= 100mm. In our experiment, we use a charge-coupled device to measure the spectral intensity and a spectrometer to measure the spectrum. The distance between the RPC and the L

_{2}equals to

*f*, and the transfer matrix of the optical system between the source plane and the receiver plane is given by Eq. (15).

*x*with

*y*= 0. One finds from Fig. 11 (a) that the spectral intensity of the generated polychromatic GSM beam has a Gaussian beam profile as expected. From Fig. 11(b), we find that the normalized on-axis spectrum and the normalized off-axis spectrum is almost the same.

*y*= 0, dotted curve) of the generated polychromatic partially coherent RP beam just behind the RPC. The corresponding theoretical fit (solid curve) of the experimental data is also shown in Fig. 12(b). One finds that the spectral intensity of the generated partially coherent RP beam can be approximately characterized by Eq. (3) with

**100**(5), 051108 (2012). [CrossRef]

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. A **25**(8), 2001–2010 (2008). [CrossRef] [PubMed]

_{2}is chosen to be comparable to the width of the incident beam, thus the coherence function of the generated polychromatic partially coherent RP beam can be approximately characterized by Eq. (3).

*x*with

*y*= 0. From Fig. 13, we find that the profile of the normalized spectrum of the generated polychromatic partially coherent RP beam is not as smooth as that of the generated polychromatic GSM beam due to the influence of the RPC, while the peak position of the normalized on-axis or off-axis spectrum is the same with that of the generated polychromatic GSM beam. Furthermore, the difference between the normalized on-axis spectrum and the normalized off-axis spectrum is quite small.

## 4. Summary

## Acknowledgments

## References and links

1. | E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. |

2. | E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature |

3. | E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun. |

4. | E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. |

5. | E. Wolf and D. F. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. |

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. |

7. | D. F. James, H. C. Kandpal, and E. Wolf, “A new method for determining the angular separation of double stars,” Astrophys. J. |

8. | D. F. V. James and E. Wolf, “Determination of field corrections from spectral measurements with application to synthetic aperture imaging,” Radio Sci. |

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. |

10. | H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. |

11. | E. Wolf, T. Shirai, H. Chen, and W. Wang, “Coherence filters and their uses: 1. Basic theory and examples,” J. Mod. Opt. |

12. | T. Shirai, E. Wolf, H. Chen, and W. Wang, “Coherence filters and their uses: 2. One-dimensional realizations,” J. Mod. Opt. |

13. | D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. |

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. |

15. | F. Gori, G. L. Marcopoli, and M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. |

16. | C. Palma, G. Cincotti, and G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron. |

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. |

18. | L. Pan and B. Lu, “The spectral switch of partially coherent light in Young’s experiment,” IEEE J. Quantum Electron. |

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. |

20. | O. Korotkova and E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence,” Opt. Lett. |

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. |

22. | E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A |

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. |

24. | O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. |

25. | F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A |

26. | O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. |

27. | A. Luis, “Degree of polarization for three-dimensional fields as a distance between correlation matrices,” Opt. Commun. |

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. |

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. |

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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. |

33. | J. Pu, O. Korotkova, and E. Wolf, “Invariance and noninvariance of the spectra of stochastic electromagnetic beams on propogation,” Opt. Lett. |

34. | O. Korotkova, J. Pu, and E. Wolf, “Spectral changes in electromagnetic stochastic beams propagating through turbulent atmosphere,” J. Mod. Opt. |

35. | F. Zhou, S. Zhu, and Y. Cai, “Spectral shift of an electromagnetic Gaussian Schell-model beam propagating through tissue,” J. Mod. Opt. |

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. |

37. | S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B |

38. | M. Yao, Y. Cai, and O. Korotkova, “Spectral shift of a stochastic electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Commun. |

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. |

40. | Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express |

41. | Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics |

42. | K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express |

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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. |

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. Photonics |

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. |

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. Express |

50. | R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B |

51. | F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. |

52. | G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express |

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. A |

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. |

55. | Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. |

56. | H. Mashaal, A. Goldstein, D. Feuermann, and J. M. Gordon, “First direct measurement of the spatial coherence of sunlight,” Opt. Lett. |

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. A |

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(260.5430) Physical optics : Polarization

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: September 25, 2013

Revised Manuscript: October 16, 2013

Manuscript Accepted: October 30, 2013

Published: November 4, 2013

**Citation**

Shijun Zhu, Xianglong Zhu, Lin Liu, Fei Wang, and Yangjian Cai, "Theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam," Opt. Express **21**, 27682-27696 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-27682

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### References

- E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett.56(13), 1370–1372 (1986). [CrossRef] [PubMed]
- 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]
- E. Wolf and D. F. James, “Correlation-induced spectral changes,” Rep. Prog. Phys.59(6), 771–818 (1996). [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).
- 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]
- 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]
- 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]
- 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]
- 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]
- 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).
- 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]
- 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]
- 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).
- E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312(5–6), 263–267 (2003). [CrossRef]
- 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]
- 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]
- 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]
- O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett.30(2), 198–200 (2005). [CrossRef] [PubMed]
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