Abstract

We present a theory of propagation of a partially coherent and partially polarized electromagnetic beam through a multilayered stratified medium. The analysis shows that spatial coherence and polarization properties of the beam change, in general, on propagation through such a medium. We illustrate the results by an example.

© 2013 Optical Society of America

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  1. F. Abelès, “Sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. 3, 504–520 (1948).
  2. F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. 5, 596–640 (1950).
  3. F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces (second part),” Ann. Phys. 5, 707–782 (1950).
  4. F. Abelès, “Methods for determining optical parameters of thin films,” Prog. Opt. 2, 249–288 (1963).
    [CrossRef]
  5. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, 1965).
  6. P. Yeh, A. Yariv, and C.-S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
    [CrossRef]
  7. A. Yariv and P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. 67, 438–448 (1977).
    [CrossRef]
  8. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  9. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  10. Another definition of spatial degree of coherence is proposed in [25]. There have been some discussions relating to this definition [26,27]. However, for the problem addressed in this paper, it does not matter which of the definitions is used.
  11. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  12. The formula is obtained as immediate generalization of the corresponding scalar version presented in Ref. [11], Section 5.6. In Eq. (5), we have neglected the contributions from evanescent waves. It is a reasonable assumption for this problem.
  13. In the case when the normal is not along the direction of stratification, one additional rotation matrix needs to be introduced in the analysis.
  14. M. Lahiri and E. Wolf, “Theory of refraction and reflection with partially coherent electromagnetic beams,” Phys. Rev. A 86, 043815 (2012).
    [CrossRef]
  15. It is to be noted that the matrix U↔(0) is, in general, not unitary. For a discussion on this see Ref. [14].
  16. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 2004).
  17. Formulas for s(M) and q(M) are obtained by following the same technique used in Ref. [14], Appendix D.
  18. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
    [CrossRef]
  19. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 232–237 (2005).
    [CrossRef]
  20. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004).
    [CrossRef]
  21. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
    [CrossRef]
  22. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25, 1016–1021 (2008).
    [CrossRef]
  23. M. Alonso, O. Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van Cittert–Zernike theorem for random electromagnetic fields,” J. Mod. Opt. 53, 969–978 (2006).
    [CrossRef]
  24. The formula used [see, for example, [28], Section 3.323, Eq. (2)]: ∫−∞∞dte−β2t2e±qt=(π/β)eq2/4β2,Re{β2}>0.
  25. T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328–330 (2004).
    [CrossRef]
  26. E. Wolf, “Comment on complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1712–1714 (2004).
    [CrossRef]
  27. T. Setälä, J. Tervo, and A. T. Friberg, “Reply to comment on complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1713 (2004).
    [CrossRef]
  28. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 2000).

2012 (1)

M. Lahiri and E. Wolf, “Theory of refraction and reflection with partially coherent electromagnetic beams,” Phys. Rev. A 86, 043815 (2012).
[CrossRef]

2008 (1)

2006 (1)

M. Alonso, O. Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van Cittert–Zernike theorem for random electromagnetic fields,” J. Mod. Opt. 53, 969–978 (2006).
[CrossRef]

2005 (2)

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 232–237 (2005).
[CrossRef]

2004 (4)

2001 (1)

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

1977 (2)

1963 (1)

F. Abelès, “Methods for determining optical parameters of thin films,” Prog. Opt. 2, 249–288 (1963).
[CrossRef]

1950 (2)

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. 5, 596–640 (1950).

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces (second part),” Ann. Phys. 5, 707–782 (1950).

1948 (1)

F. Abelès, “Sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. 3, 504–520 (1948).

Abelès, F.

F. Abelès, “Methods for determining optical parameters of thin films,” Prog. Opt. 2, 249–288 (1963).
[CrossRef]

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. 5, 596–640 (1950).

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces (second part),” Ann. Phys. 5, 707–782 (1950).

F. Abelès, “Sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. 3, 504–520 (1948).

Alonso, M.

M. Alonso, O. Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van Cittert–Zernike theorem for random electromagnetic fields,” J. Mod. Opt. 53, 969–978 (2006).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25, 1016–1021 (2008).
[CrossRef]

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

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Friberg, A. T.

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25, 1016–1021 (2008).
[CrossRef]

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

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 2000).

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, 1965).

Hong, C.-S.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 2004).

Korotkova, O.

M. Alonso, O. Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van Cittert–Zernike theorem for random electromagnetic fields,” J. Mod. Opt. 53, 969–978 (2006).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 232–237 (2005).
[CrossRef]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004).
[CrossRef]

Lahiri, M.

M. Lahiri and E. Wolf, “Theory of refraction and reflection with partially coherent electromagnetic beams,” Phys. Rev. A 86, 043815 (2012).
[CrossRef]

Mandel, L.

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

Mondello, A.

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

Piquero, G.

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

Ramírez-Sánchez, V.

Roychowdhury, H.

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 2000).

Salem, M.

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25, 1016–1021 (2008).
[CrossRef]

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

Setälä, T.

Shirai, T.

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 232–237 (2005).
[CrossRef]

Simon, R.

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

Tervo, J.

Wolf, E.

M. Lahiri and E. Wolf, “Theory of refraction and reflection with partially coherent electromagnetic beams,” Phys. Rev. A 86, 043815 (2012).
[CrossRef]

M. Alonso, O. Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van Cittert–Zernike theorem for random electromagnetic fields,” J. Mod. Opt. 53, 969–978 (2006).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 232–237 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004).
[CrossRef]

E. Wolf, “Comment on complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1712–1714 (2004).
[CrossRef]

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

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Yariv, A.

Yeh, P.

Ann. Phys. (3)

F. Abelès, “Sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. 3, 504–520 (1948).

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. 5, 596–640 (1950).

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces (second part),” Ann. Phys. 5, 707–782 (1950).

J. Mod. Opt. (1)

M. Alonso, O. Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van Cittert–Zernike theorem for random electromagnetic fields,” J. Mod. Opt. 53, 969–978 (2006).
[CrossRef]

J. Opt. A (2)

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

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 232–237 (2005).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004).
[CrossRef]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

M. Lahiri and E. Wolf, “Theory of refraction and reflection with partially coherent electromagnetic beams,” Phys. Rev. A 86, 043815 (2012).
[CrossRef]

Prog. Opt. (1)

F. Abelès, “Methods for determining optical parameters of thin films,” Prog. Opt. 2, 249–288 (1963).
[CrossRef]

Other (12)

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, 1965).

It is to be noted that the matrix U↔(0) is, in general, not unitary. For a discussion on this see Ref. [14].

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 2004).

Formulas for s(M) and q(M) are obtained by following the same technique used in Ref. [14], Appendix D.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Another definition of spatial degree of coherence is proposed in [25]. There have been some discussions relating to this definition [26,27]. However, for the problem addressed in this paper, it does not matter which of the definitions is used.

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

The formula is obtained as immediate generalization of the corresponding scalar version presented in Ref. [11], Section 5.6. In Eq. (5), we have neglected the contributions from evanescent waves. It is a reasonable assumption for this problem.

In the case when the normal is not along the direction of stratification, one additional rotation matrix needs to be introduced in the analysis.

The formula used [see, for example, [28], Section 3.323, Eq. (2)]: ∫−∞∞dte−β2t2e±qt=(π/β)eq2/4β2,Re{β2}>0.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 2000).

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