Abstract

We consider the field generated by a wavefront-folding interferometer which is illuminated by a stochastic electromagnetic beam. The specular property and anti-specular property are discussed in the vector case. Take electromagnetic Gaussian Schell-model beam as an example, we investigate the spectral density, the spectral degree of coherence, the spectral degree of polarization as well as the state of polarization of the polarized portion of the field on propagation. Results show that the polarization properties including the degree of polarization, the orientation angle and the degree of ellipse can be adjusted by the phase difference of the interferometer and the phase retardation introduced by the prism. The results may be applied in free-space optical communication.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  4. B. Lü and L. Pan, “Propagation of vector Gaussian–Schell-model beams through a paraxial optical ABCD system,” Opt. Commun. 205(1-3), 7–16 (2002).
    [Crossref]
  5. O. Korotkova, B. G. Hoover, V. L. Gamiz, and E. Wolf, “Coherence and polarization properties of far fields generated by quasi-homogeneous planar electromagnetic sources,” J. Opt. Soc. Am. A 22(11), 2547–2556 (2005).
    [Crossref] [PubMed]
  6. G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 (2000).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  20. S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69(3), 036604 (2004).
    [Crossref] [PubMed]
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    [Crossref]
  26. F. Roddier, C. Roddier, and J. Demarcq, “A rotation shearing interferometer with phase-compensated roof-prisms,” J. Opt. 9(3), 145–149 (1978).
    [Crossref]
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  29. 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(4-6), 225–230 (2004).
    [Crossref]
  30. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
    [Crossref]
  31. 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]

2017 (1)

2016 (2)

2015 (1)

2012 (1)

B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6(6), 355–359 (2012).
[Crossref] [PubMed]

2010 (1)

2009 (1)

2008 (1)

2007 (1)

2005 (3)

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (2005).
[Crossref]

O. Korotkova, B. G. Hoover, V. L. Gamiz, and E. Wolf, “Coherence and polarization properties of far fields generated by quasi-homogeneous planar electromagnetic sources,” J. Opt. Soc. Am. A 22(11), 2547–2556 (2005).
[Crossref] [PubMed]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
[Crossref]

2004 (5)

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, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69(3), 036604 (2004).
[Crossref] [PubMed]

T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21(10), 1907–1916 (2004).
[Crossref] [PubMed]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[Crossref]

2003 (2)

2002 (1)

B. Lü and L. Pan, “Propagation of vector Gaussian–Schell-model beams through a paraxial optical ABCD system,” Opt. Commun. 205(1-3), 7–16 (2002).
[Crossref]

2000 (1)

1996 (1)

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

1991 (1)

D. F. V. James and E. Wolf, “Determination of field correlations from spectral measurements with application to synthetic aperture imaging,” Radio Sci. 26(5), 1239–1243 (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 (2)

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988).
[Crossref]

1987 (2)

P. De Santis, F. Gori, G. Guattari, and C. Palma, “A space-time modulated field with specular coherence function,” Opt. Commun. 64(1), 9–14 (1987).
[Crossref]

A. S. Marathay, “Noncoherent-object hologram: its reconstruction and optical processing,” J. Opt. Soc. Am. A 4(10), 1861–1868 (1987).
[Crossref]

1978 (1)

F. Roddier, C. Roddier, and J. Demarcq, “A rotation shearing interferometer with phase-compensated roof-prisms,” J. Opt. 9(3), 145–149 (1978).
[Crossref]

Agrawal, G. P.

Cao, H.

B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6(6), 355–359 (2012).
[Crossref] [PubMed]

Choma, M. A.

B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6(6), 355–359 (2012).
[Crossref] [PubMed]

De Santis, P.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “A space-time modulated field with specular coherence function,” Opt. Commun. 64(1), 9–14 (1987).
[Crossref]

Demarcq, J.

F. Roddier, C. Roddier, and J. Demarcq, “A rotation shearing interferometer with phase-compensated roof-prisms,” J. Opt. 9(3), 145–149 (1978).
[Crossref]

Dogariu, A.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[Crossref]

Du, X.

Friberg, A. T.

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988).
[Crossref]

Gamiz, V. L.

Gori, F.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “A space-time modulated field with specular coherence function,” Opt. Commun. 64(1), 9–14 (1987).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “A space-time modulated field with specular coherence function,” Opt. Commun. 64(1), 9–14 (1987).
[Crossref]

Guo, M.

He, Q.

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988).
[Crossref]

Hoover, B. G.

James, D. F. V.

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

D. F. V. James and E. Wolf, “Determination of field correlations 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, and K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73(3), 169–172 (1989).
[Crossref]

Kandpal, H. C.

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]

Korotkova, O.

Z. Tong and O. Korotkova, “Stochastic electromagnetic beams in positive- and negative-phase materials,” Opt. Lett. 35(2), 175–177 (2010).
[Crossref] [PubMed]

O. Korotkova, B. G. Hoover, V. L. Gamiz, and E. Wolf, “Coherence and polarization properties of far fields generated by quasi-homogeneous planar electromagnetic sources,” J. Opt. Soc. Am. A 22(11), 2547–2556 (2005).
[Crossref] [PubMed]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
[Crossref]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (2005).
[Crossref]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[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(4-6), 225–230 (2004).
[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]

Lü, B.

B. Lü and L. Pan, “Propagation of vector Gaussian–Schell-model beams through a paraxial optical ABCD system,” Opt. Commun. 205(1-3), 7–16 (2002).
[Crossref]

Marathay, A. S.

Padovani, C.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

Palma, C.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “A space-time modulated field with specular coherence function,” Opt. Commun. 64(1), 9–14 (1987).
[Crossref]

Pan, L.

B. Lü and L. Pan, “Propagation of vector Gaussian–Schell-model beams through a paraxial optical ABCD system,” Opt. Commun. 205(1-3), 7–16 (2002).
[Crossref]

Partanen, H.

Ponomarenko, S. A.

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69(3), 036604 (2004).
[Crossref] [PubMed]

Redding, B.

B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6(6), 355–359 (2012).
[Crossref] [PubMed]

Roddier, C.

F. Roddier, C. Roddier, and J. Demarcq, “A rotation shearing interferometer with phase-compensated roof-prisms,” J. Opt. 9(3), 145–149 (1978).
[Crossref]

Roddier, F.

F. Roddier, C. Roddier, and J. Demarcq, “A rotation shearing interferometer with phase-compensated roof-prisms,” J. Opt. 9(3), 145–149 (1978).
[Crossref]

Roychowdhury, H.

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
[Crossref]

Salem, M.

M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33(11), 1180–1182 (2008).
[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]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[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(4-6), 225–230 (2004).
[Crossref]

Sharmin, N.

Shirai, T.

Tervo, J.

Tong, Z.

Turunen, J.

H. Partanen, N. Sharmin, J. Tervo, and J. Turunen, “Specular and antispecular light beams,” Opt. Express 23(22), 28718–28727 (2015).
[Crossref] [PubMed]

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988).
[Crossref]

Vaishya, J. S.

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]

Wang, H.

Wang, X.

Wolf, E.

M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33(11), 1180–1182 (2008).
[Crossref] [PubMed]

O. Korotkova, B. G. Hoover, V. L. Gamiz, and E. Wolf, “Coherence and polarization properties of far fields generated by quasi-homogeneous planar electromagnetic sources,” J. Opt. Soc. Am. A 22(11), 2547–2556 (2005).
[Crossref] [PubMed]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (2005).
[Crossref]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[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(4-6), 225–230 (2004).
[Crossref]

T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21(10), 1907–1916 (2004).
[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, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28(13), 1078–1080 (2003).
[Crossref] [PubMed]

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

G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 (2000).
[Crossref] [PubMed]

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

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

Yang, K.

Zeng, A.

Zhao, D.

Zhou, Z.

Appl. Opt. (1)

J. Opt. (1)

F. Roddier, C. Roddier, and J. Demarcq, “A rotation shearing interferometer with phase-compensated roof-prisms,” J. Opt. 9(3), 145–149 (1978).
[Crossref]

J. Opt. Soc. Am. A (4)

Nat. Photonics (1)

B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6(6), 355–359 (2012).
[Crossref] [PubMed]

Opt. Commun. (8)

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (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]

B. Lü and L. Pan, “Propagation of vector Gaussian–Schell-model beams through a paraxial optical ABCD system,” Opt. Commun. 205(1-3), 7–16 (2002).
[Crossref]

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “A space-time modulated field with specular coherence function,” Opt. Commun. 64(1), 9–14 (1987).
[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(4-6), 225–230 (2004).
[Crossref]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
[Crossref]

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988).
[Crossref]

Opt. Express (4)

Opt. Lett. (5)

Phys. Lett. A (1)

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

Phys. Rev. E (1)

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69(3), 036604 (2004).
[Crossref] [PubMed]

Radio Sci. (1)

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

Waves Random Media (1)

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[Crossref]

Other (2)

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

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

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Figures (6)

Fig. 1
Fig. 1 Wavefront-folding interferometer (a) and retroreflection by a compensated right-angle prism (b). S is the source, BS is a non-polarizing beam splitter, PRx and PRy are right-angle prisms, WP is a wave plate. The set of coordinate axes in (b) is used to describe the state of polarization of a light beam passing through a compensated right-angle prism.
Fig. 2
Fig. 2 Intensity distributions of the modulated EGSM beams with φ = π at z = 1000 m . S G max ( z ) = 2 C ( A x 2 + A y 2 ) / Δ x x represents the maximum value of the normal Gaussian beam. (a) θ = 0 ; (b) θ = π . We choose δ 0 = 1 mm in (c). The other parameters of the incident beam are chosen as follows: λ = 632.8 n m , A x = 2 , A y = 1 , B x y = 0.2 exp ( i π / 3 ) , σ 0 = 1 c m .
Fig. 3
Fig. 3 Changes with ρ / σ P0 of the spectral degree of polarization, the orientation angle, the degree of ellipticity and the spectral density for several selected values of φ and θ in the output plane of the WFI. σ P0 = 1 / 1 / σ 0 2 + 4 / δ m 2 with δ m = max { δ i j } represents the modulation width by θ and φ in the output plane of the interferometer. Rows 1-3 respectively correspond to φ = π / 6 , π / 4 , π / 2 . The values of θ are given beside the curves. The parameters are the same as Fig. 2(c), δ x y = 2 mm . The polarization of the incident EGSM beam is uniform with P = 0.621 , α =3 .797 and ε = 0.113 .
Fig. 4
Fig. 4 Changes with the propagation distance z of on-axis spectral degree of polarization, orientation angle and degree of ellipticity for several selected values of θ when φ = π / 4 . The values of θ are given beside the curves. The parameters are the same as Fig. 2(c), δ x y = 2 mm .
Fig. 5
Fig. 5 Changes with θ / π of on-axis spectral degree of polarization, orientation angle and degree of ellipticity for some selected values of φ in the far-zone. Black solid curve, φ = π / 6 ; red short dashed curve, φ = π / 4 ; blue short dotted curve, φ = π / 2 ; green short dashed-dotted curve, φ = π . The parameters are the same as Fig. 2(c), δ x y = 2 mm .
Fig. 6
Fig. 6 Changes with θ / π of the spectral degree of coherence between two symmetrical points ρ 1 = ( ρ / 2 , 0 ) and ρ 2 = ( ρ / 2 , 0 ) with ρ = 1 m m . (a) z = 0 ; (b) z = 200 m ; (c) z = 1000 m . Black solid curve, φ = 0 ; red short dashed curve, φ = π / 4 ; blue short dotted curve, φ = π / 2 ; green short dashed-dotted curve, φ = π .

Tables (2)

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Table 1 Expressions for the coefficients D i j , E i j , F i j and G i j

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Table 2 Expressions for the coefficients H i j and K i j

Equations (40)

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[ E 1 x ( x , y ) E 1 y ( x , y ) ] = M 1 × [ E 0 x ( x , y ) E 0 y ( x , y ) ] .
[ E 2 x ( x , y ) E 2 y ( x , y ) ] = M 2 × [ E 0 x ( x , y ) E 0 y ( x , y ) ] .
R = [ r x 0 0 r y ] , T = [ t x 0 0 t y ] , P = [ e i φ 0 0 1 ] , O = [ 0 1 1 0 ] ,
M 1 = [ r x t x e i φ 0 0 r y t y ] , M 2 = [ r x t x 0 0 r y t y e i φ ] .
[ E x ( x , y ) E y ( x , y ) ] = [ E 1 x ( x , y ) E 1 y ( x , y ) ] e i θ + [ E 2 x ( x , y ) E 2 y ( x , y ) ] .
E x ( x , y ) = r x t x e i φ e i θ E 0 x ( x , y ) r x t x E 0 x ( x , y ) ,
E y ( x , y ) = r y t y e i θ E 0 y ( x , y ) r y t y e i φ E 0 y ( x , y ) .
W ( r 1 , r 2 ; ω ) = [ W i j ( r 1 , r 2 ; ω ) ] = [ E i ( r 1 ; ω ) E j ( r 2 ; ω ) ] , ( i = x , y ; j = x , y ) .
W i j ( x 1 , y 1 , x 2 , y 2 ) = C i j [ D i j W 0 i j ( x 1 , y 1 , x 2 , y 2 ) + E i j W 0 i j ( x 1 , y 1 , x 2 , y 2 ) + F i j W 0 i j ( x 1 , y 1 , x 2 , y 2 ) + G i j W 0 i j ( x 1 , y 1 , x 2 , y 2 ) ] ,
W i j ( x 1 , y 1 , x 2 , y 2 ) = C i j [ W 0 i j ( x 1 , y 1 , x 2 , y 2 ) + W 0 i j ( x 1 , y 1 , x 2 , y 2 ) W 0 i j ( x 1 , y 1 , x 2 , y 2 ) e i θ W 0 i j ( x 1 , y 1 , x 2 , y 2 ) e i θ ] .
{ W i j ( x 1 , y 1 , x 2 , y 2 ) = W i j ( x 1 , y 1 , x 2 , y 2 ) when θ = 2 l π , W i j ( x 1 , y 1 , x 2 , y 2 ) = W i j ( x 1 , y 1 , x 2 , y 2 ) when θ = ( 2 l +1 ) π .
W i j ( x 1 , y 1 , x 2 , y 2 ) = L i j C i j [ W 0 i j ( x 1 , y 1 , x 2 , y 2 ) + W 0 i j ( x 1 , y 1 , x 2 , y 2 ) + W 0 i j ( x 1 , y 1 , x 2 , y 2 ) e i θ + W 0 i j ( x 1 , y 1 , x 2 , y 2 ) e i θ ] ,
L i j = { 1 when i = j , 1 when i j .
{ W i j ( x 1 , y 1 , x 2 , y 2 ) = W i j ( x 1 , y 1 , x 2 , y 2 ) when θ = 2 l π , W i j ( x 1 , y 1 , x 2 , y 2 ) = W i j ( x 1 , y 1 , x 2 , y 2 ) when θ = ( 2 l +1 ) π .
W i j ( ρ 1 , ρ 2 , z ; ω ) = ( k 2 π z ) 2 d 2 ρ 1 d 2 ρ 2 W i j ( 0 ) ( ρ 1 , ρ 2 ; ω ) × exp { i k 2 z [ ( ρ 1 2 ρ 2 2 ) 2 ( ρ 1 ρ 1 ρ 2 ρ 2 ) + ( ρ 1 2 ρ 2 2 ) ] } .
S ( ρ , z ; ω ) = T r W ( ρ , ρ , z ; ω ) ,
η ( ρ 1 , ρ 2 , z ; ω ) = T r W ( ρ 1 , ρ 2 , z ; ω ) S ( ρ 1 , z ; ω ) S ( ρ 2 , z ; ω ) .
P ( ρ , z ; ω ) = 1 4 Det W ( ρ , ρ , z ; ω ) [ T r W ( ρ , ρ , z ; ω ) ] 2 ,
α ( ρ , z ; ω ) = 1 2 arc tan ( 2 Re [ W x y ( ρ , z ; ω ) ] W x x ( ρ , z ; ω ) W y y ( ρ , z ; ω ) ) , π / 2 α π / 2 .
a 2 ( ρ , z ; ω ) = 1 2 [ ( W x x W y y ) 2 + 4 | W x y | 2 + ( W x x W y y ) 2 + 4 [ Re W x y ] 2 ] ,
b 2 ( ρ , z ; ω ) = 1 2 [ ( W x x W y y ) 2 + 4 | W x y | 2 ( W x x W y y ) 2 + 4 [ Re W x y ] 2 ] .
ε = b ( ρ , z ; ω ) / a ( ρ , z ; ω ) , 0 ε 1,
W 0 i j ( ρ 1 , ρ 2 ; ω ) = A i A j B i j exp ( ρ 1 2 4 σ i 2 ρ 2 2 4 σ j 2 ) exp ( | ρ 2 ρ 1 | 2 2 δ i j 2 ) , ( i = x , y ; j = x , y ) ,
B i j = 1 when i = j ,
| B i j | 1 when i j ,
B i j = B j i ,
δ j i = δ i j .
max { δ x x , δ y y } δ x y min { δ x x | B x y | , δ y y | B x y | } .
W i j ( ρ 1 , ρ 2 ; ω ) = 2 C i j A i A j B i j exp ( x 1 2 + y 1 2 4 σ i 2 x 2 2 + y 2 2 4 σ j 2 ) × { H i j exp [ ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 2 δ i j 2 ] + K i j exp [ ( x 2 + x 1 ) 2 + ( y 2 + y 1 ) 2 2 δ i j 2 ] } ,
W i j ( ρ 1 , ρ 2 , z ; ω ) = 2 C i j A i A j B i j Δ i j exp { [ i k z ( β i j + i k z ) 1 Δ i j ] ρ ρ } × { H i j exp ( 4 α i j ρ 2 Δ i j ) exp ( γ i j ρ 2 Δ i j ) + K i j exp ( α i j ρ 2 Δ i j ) exp ( 4 γ i j ρ 2 Δ i j ) } ,
ρ = ρ 1 + ρ 2 2 , ρ = ρ 2 ρ 1 ,
Δ i j = ( z k ) 2 [ 16 α i j γ i j ( β i j + i k z ) 2 ] ,
α i j = 1 16 ( 1 σ i 2 + 1 σ j 2 ) , β i j = 1 4 ( 1 σ i 2 1 σ j 2 ) , γ i j = 1 16 ( 1 σ i 2 + 1 σ j 2 ) + 1 2 δ i j 2 .
S ( ρ , z ; ω ) = 2 C Δ x x { ( A x 2 + A y 2 ) exp ( ρ 2 2 σ I 2 ( z ) ) [ A x 2 cos ( φ + θ ) + A y 2 cos ( φ θ ) ] exp ( ρ 2 2 σ A 2 ( z ) ) } ,
P ( ρ , 0 ; ω ) = { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] exp ( 2 ρ 2 δ 0 2 ) } 2 + 4 A x 2 A y 2 | B x y | 2 [ cos φ cos θ exp ( 2 ρ 2 δ x y 2 ) ] 2 { A x 2 + A y 2 [ A x 2 cos ( φ + θ ) + A y 2 cos ( φ θ ) ] exp ( 2 ρ 2 δ 0 2 ) } 2 .
α ( ρ , 0 ; ω ) = 1 2 arc tan ( 2 A x A y Re [ B x y ] [ cos φ cos θ exp ( 2 ρ 2 δ x y 2 ) ] A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] exp ( 2 ρ 2 δ 0 2 ) ) ,
ε ( ρ , 0 ; ω ) = { { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] exp ( 2 ρ 2 δ 0 2 ) } 2 + 4 A x 2 A y 2 | B x y | 2 [ cos φ cos θ exp ( 2 ρ 2 δ x y 2 ) ] 2 { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] exp ( 2 ρ 2 δ 0 2 ) } 2 + 4 A x 2 A y 2 Re 2 [ B x y ] [ cos φ cos θ exp ( 2 ρ 2 δ x y 2 ) ] 2 { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] exp ( 2 ρ 2 δ 0 2 ) } 2 + 4 A x 2 A y 2 | B x y | 2 [ cos φ cos θ exp ( 2 ρ 2 δ x y 2 ) ] 2 + { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] exp ( 2 ρ 2 δ 0 2 ) } 2 + 4 A x 2 A y 2 Re 2 [ B x y ] [ cos φ cos θ exp ( 2 ρ 2 δ x y 2 ) ] 2 } 1 / 2 .
P ( 0 , z ; ω ) = { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] } 2 + 4 A x 2 A y 2 | B x y | 2 χ 2 ( cos φ cos θ ) 2 { A x 2 + A y 2 [ A x 2 cos ( φ + θ ) + A y 2 cos ( φ θ ) ] } 2 ,
α ( 0 , z ; ω ) = 1 2 arc tan ( 2 A x A y χ Re [ B x y ] ( cos φ cos θ ) A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] ) ,
ε ( 0 , z ; ω ) = { { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] } 2 + 4 A x 2 A y 2 | B x y | 2 χ 2 ( cos φ cos θ ) 2 { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] } 2 + 4 A x 2 A y 2 Re 2 [ B x y ] χ 2 ( cos φ cos θ ) 2 { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] } 2 + 4 A x 2 A y 2 | B x y | 2 χ 2 ( cos φ cos θ ) 2 + { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] } 2 + 4 A x 2 A y 2 Re 2 [ B x y ] χ 2 ( cos φ cos θ ) 2 } 1 / 2 ,

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