Abstract

A theory of refraction and reflection of partially coherent electromagnetic beams has been recently developed. In this paper, we apply it to study the change in spatial coherence caused by refraction and by reflection more fully. By considering a Gaussian Schell-model beam, we show that the change is, in general, dependent on the angle of incidence.

© 2013 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  2. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  3. M. Lahiri and E. Wolf, “Theory of refraction and reflection with partially coherent electromagnetic beams,” Phys. Rev. A 86, 043815 (2012).
    [CrossRef]
  4. In the quantum theory of coherence these correlation properties are referred to as first-order ones.
  5. T. Setala, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328–330 (2004).
    [CrossRef]
  6. E. Wolf, “Comment on complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1712 (2004).
    [CrossRef]
  7. T. Setala, J. Tervo, and A. T. Friberg, “Reply to comment on complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1713–1714 (2004).
    [CrossRef]
  8. L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
    [CrossRef]
  9. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 2004).
  10. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  11. In the case of the total internal reflection, contributions from the plane wave components in the transmitted field would be minimal, and therefore the effects due to evanescent waves cannot be neglected.
  12. The treatment provided in [10], Section 5.6, is based on scalar theory, but it can be readily generalized to vector theory for the case of optical beams, as we did here for our purpose.
  13. 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]
  14. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 232–237 (2005).
    [CrossRef]
  15. 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]
  16. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
    [CrossRef]
  17. 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]
  18. Such curves were also previously produced in Figs. 5 and 6 of [3], for n′=1.62 and for a Gaussian Schell-model beam characterized by different parameters. However, because of a minor error in the computation, the curves presented in [3] are not accurate.
  19. E. Baleine and A. Dogariu, “Variable coherence tomography,” Opt. Lett. 29, 1233–1235 (2004).
    [CrossRef]
  20. M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13, 9629–9635 (2005).
    [CrossRef]

2012 (2)

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

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[CrossRef]

2008 (1)

2005 (3)

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

M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13, 9629–9635 (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 (5)

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]

Baleine, E.

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).

Dogariu, A.

Duan, Z.

Fleischer, J. W.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[CrossRef]

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]

Jackson, J. D.

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

Korotkova, O.

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).

Miyamoto, Y.

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]

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]

Setala, 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]

Situ, G.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[CrossRef]

Takeda, M.

Tervo, J.

Waller, L.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[CrossRef]

Wang, W.

Wolf, E.

M. Lahiri and E. Wolf, “Theory of refraction and reflection with partially coherent electromagnetic beams,” Phys. Rev. A 86, 043815 (2012).
[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 (2004).
[CrossRef]

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).

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

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. A (1)

Nat. Photonics (1)

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[CrossRef]

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. Express (1)

Opt. Lett. (4)

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]

Other (8)

In the quantum theory of coherence these correlation properties are referred to as first-order ones.

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).

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

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

In the case of the total internal reflection, contributions from the plane wave components in the transmitted field would be minimal, and therefore the effects due to evanescent waves cannot be neglected.

The treatment provided in [10], Section 5.6, is based on scalar theory, but it can be readily generalized to vector theory for the case of optical beams, as we did here for our purpose.

Such curves were also previously produced in Figs. 5 and 6 of [3], for n′=1.62 and for a Gaussian Schell-model beam characterized by different parameters. However, because of a minor error in the computation, the curves presented in [3] are not accurate.

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Figures (8)

Fig. 1.
Fig. 1.

Illustrating the geometry relating to refraction and reflection of a monochromatic plane wave at an interface; the h and v directions are chosen to be parallel and perpendicular to the plane of incidence.

Fig. 2.
Fig. 2.

(Adapted from [3]) Illustrating the coordinate system (xv(i),xh(i),xp(i)) of the incident beam.

Fig. 3.
Fig. 3.

(Adapted from [3]) Illustrating the geometry relating to the plane of incidence A of the plane wave component with wave vector k(i). The plane B indicated by the dotted line is the plane formed by the axis of the incident beam, and the normal n to the interface.

Fig. 4.
Fig. 4.

Modulus of the spectral degree of coherence of the transmitted beams (solid curves), at two fixed points on the interface, plotted against the angle of incidence for ethanol (n=1.36), for flint glass (n=1.62), and for diamond (n=2.42), at frequency ω3.2×1015s1, when the parameters δvv=5×104m, δhh=5.5×104m, δvh=6×104m, σ=5×103m, Bhv=9/16, and Av/Ah=0.7. The dashed curve represents the modulus of the spectral degree of coherence of the incident beam at the same pair of points.

Fig. 5.
Fig. 5.

Modulus of the spectral degree of coherence of the reflected beams (solid curves), at two points located on the interface, plotted against the angle of incidence, for ethanol (n=1.36), for flint glass (n=1.62), and for diamond (n=2.42), with the same choice of parameters as used in Fig. 4. Slight differences between the curves for the three different media can be noted. The dashed curve represents the modulus of the spectral degree of coherence of the incident beam for the same pair of points.

Fig. 6.
Fig. 6.

Enlarged version of the curves shown in Fig. 5 when 35°θi60°.

Fig. 7.
Fig. 7.

Modulus |η| of the spectral degree of coherence of the transmitted beam (solid curves) plotted as function of ρ, for θi=50°, for ethanol (n1.36), for flint glass (n1.62), and for diamond (n2.42); the other parameters are the same as before. The dashed curve represents |η| of the incident beam under the same circumstances.

Fig. 8.
Fig. 8.

Modulus |η| of the spectral degree of coherence of the reflected beams (for the three media) plotted as functions of ρ=|rr|/2 for θi=50°; the other parameters are taken to be the same as used in Fig. 4. The dashed curve represents |η| of the incident beam at the same pairs of points.

Equations (18)

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E(r,ω)=(Ev(r,ω)Eh(r,ω))=(Ev(r,ω)Eh(r,ω))T,
W(r1,r2;ω)=E*(r1;ω)·ET(r2;ω).
η(r1,r2;ω)TrW(r1,r2;ω)TrW(r1,r1;ω)TrW(r2,r2;ω),
E0(t)=T·E0(i),E0(r)=R·E0(i).
T=(Tv00Th),R=(Rv00Rh)
Tv=2ncosθincosθi+μμn2n2sin2θi,
Th=2nncosθiμμn2cosθi+nn2n2sin2θi,
Rv=ncosθiμμn2n2sin2θincosθi+μμn2n2sin2θi,
Rh=μμn2cosθinn2n2sin2θiμμn2cosθi+nn2n2sin2θi,
W(l)(r,r;ω)=d2k(l)d2k(l)WA(l)(k(l),k(l);ω)×exp[i(k(l)·rk(l)·r)],
WA(t)=UT*·WA(i)·UTT,
WA(r)=UR*·WA(i)·URT.
UT={U(t)}·T·U(i),
UR={U(r)}·R·U(i).
T(Tv00Th),R(Rv00Rh)
U(i)={U(r)}=cosθi(cosα/cosθisinαsinαcosαcosθ˜i+sinθ˜itanθi),
U(t)=cosθt(cosα/cosθtsinαsinαcosαcosθ˜t+sinθ˜ttanθt).
Wβγ(ρ0,ρ0;ω)=AβAγBβγexp[(ρ02+ρ02)/(4σ2)]×exp[(ρ0ρ0)2/(2δβγ2)],

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