Abstract

The polarization characteristics of unpolarized light passing through a double wedge depolarizer are studied. It is found that the degree of polarization of the radiation propagating after the depolarizer is uniform across transverse planes after the depolarizer, but it changes from one plane to another in a periodic way giving, at different distances, unpolarized, partially polarized, or even perfectly polarized light. An experiment is performed to confirm this result. Measured values of the Stokes parameters and of the degree of polarization are in complete agreement with the theoretical predictions.

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  6. C. Vena, C. Versace, G. Strangi, and R. Bartolino, “Light depolarization by non-uniform polarization distribution over a beam cross section,” J. Opt. A: Pure Appl. Opt.11, 125704–10 (2009).
    [CrossRef]
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  9. F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun.163, 159–163 (1999).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett.31, 688–690 (2006).
    [CrossRef] [PubMed]
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  14. T. D. Visser, D. Kuebel, M. Lahiri, T. Shirai, and E. Wolf, “Unpolarized light beams with different coherence properties,” J. Mod. Opt.56, 1369–1374 (2009).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  24. G. Piquero, R. Borghi, and M. Santarsiero, “Gaussian Schell-model beams propagating through polarization gratings,” J. Opt. Soc. Am. A18, 1399–1405 (2001).
    [CrossRef]
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    [CrossRef]
  27. J. Tervo and J. Turunen, “Transverse and longitudinal periodicities in fields produced by polarization gratings,” Opt. Commun.190, 51–57 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]

2012 (1)

2010 (1)

R. Martínez-Herrero and P. M. Mejías, “On the propagation of random electromagnetic fields with position-independent stochastic behavior,” Opt. Commun.283, 4467–4469 (2010).
[CrossRef]

2009 (5)

T. D. Visser, D. Kuebel, M. Lahiri, T. Shirai, and E. Wolf, “Unpolarized light beams with different coherence properties,” J. Mod. Opt.56, 1369–1374 (2009).
[CrossRef]

F. Gori, J. Tervo, and J. Turunen, “Correlation matrices of completely unpolarized beams,” Opt. Lett.34, 1447–1449 (2009).
[CrossRef] [PubMed]

V. A. Bagan, B. L. Davydov, and I. E. Samartsev, “Characteristics of Cornu depolarisers made from quartz and paratellurite optically active crystals,” Quant. Electron.39, 73–78 (2009).
[CrossRef]

C. Vena, C. Versace, G. Strangi, and R. Bartolino, “Light depolarization by non-uniform polarization distribution over a beam cross section,” J. Opt. A: Pure Appl. Opt.11, 125704–10 (2009).
[CrossRef]

V. Kuznetsov, D. Faleiev, E. Savin, and V. Lebedev, “Crystal-based device for combining light beams,” Opt. Lett.34, 2856–2857 (2009).
[CrossRef] [PubMed]

2008 (1)

2006 (1)

2003 (2)

2002 (2)

L. V. Alekseeva, I. V. Povkh, V. I. Stroganov, B. I. Kidyarov, and P. G. Pasko, “Four-ray splitting in optical crystals,” J. Opt. Technol.39, 441–443 (2002).
[CrossRef]

Z. Bomzon, A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Polarization Talbot self-imaging with computer-generated, space-variant subwavelength dielectric gratings,” Appl. Opt.41, 5218–5222 (2002).
[CrossRef] [PubMed]

2001 (2)

J. Tervo and J. Turunen, “Transverse and longitudinal periodicities in fields produced by polarization gratings,” Opt. Commun.190, 51–57 (2001).
[CrossRef]

G. Piquero, R. Borghi, and M. Santarsiero, “Gaussian Schell-model beams propagating through polarization gratings,” J. Opt. Soc. Am. A18, 1399–1405 (2001).
[CrossRef]

2000 (1)

1999 (3)

F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun.163, 159–163 (1999).
[CrossRef]

F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett.24, 584–586 (1999).
[CrossRef]

G. Ghosh, “Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals,” Opt. Commun.163, 95–102 (1999).
[CrossRef]

1998 (1)

F. Gori, M. Santarsiero, S. Vicalvi, and R. Borghi, “Beam coherence-polarization matrix,” Pure and Appl. Opt.7, 941–951 (1998).
[CrossRef]

1996 (3)

M. El Sherif, M.S. Khalil, S. Khodeir, and N. Nagib, “Simple depolarizers for spectrophotometric measurements of anisotropic samples,” Opt. & Laser Technol.28, 561–563 (1996).
[CrossRef]

Z. Zhang and H. J. Caufield, “Reflection and refraction by interfaces of uniaxial crystals,” Opt. & Laser Technol.28, 549–553 (1996).
[CrossRef]

V. Arrizón, E. Tepichin, M. Ortíz-Gutiérrez, and A.W. Lohmann, “Fresnel diffraction at l/4 of the Talbot distance of an anisotropic grating,” Opt. Commun.127, 171–175 (1996).
[CrossRef]

1993 (2)

1992 (1)

1990 (1)

J. P. McGuire and R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” Opt. Eng.29, 1478–1484 (1990).
[CrossRef]

1985 (1)

1836 (1)

H. F. Talbot, “Facts relating to optical science,” Phil. Mag.9, 401–407 (1836).

Agrawal, G. P.

Alekseeva, L. V.

L. V. Alekseeva, I. V. Povkh, V. I. Stroganov, B. I. Kidyarov, and P. G. Pasko, “Four-ray splitting in optical crystals,” J. Opt. Technol.39, 441–443 (2002).
[CrossRef]

Arrizón, V.

V. Arrizón, E. Tepichin, M. Ortíz-Gutiérrez, and A.W. Lohmann, “Fresnel diffraction at l/4 of the Talbot distance of an anisotropic grating,” Opt. Commun.127, 171–175 (1996).
[CrossRef]

Bagan, V. A.

V. A. Bagan, B. L. Davydov, and I. E. Samartsev, “Characteristics of Cornu depolarisers made from quartz and paratellurite optically active crystals,” Quant. Electron.39, 73–78 (2009).
[CrossRef]

Bartolino, R.

C. Vena, C. Versace, G. Strangi, and R. Bartolino, “Light depolarization by non-uniform polarization distribution over a beam cross section,” J. Opt. A: Pure Appl. Opt.11, 125704–10 (2009).
[CrossRef]

Biener, G.

Bomzon, Z.

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett.31, 688–690 (2006).
[CrossRef] [PubMed]

G. Piquero, R. Borghi, and M. Santarsiero, “Gaussian Schell-model beams propagating through polarization gratings,” J. Opt. Soc. Am. A18, 1399–1405 (2001).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun.163, 159–163 (1999).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, and R. Borghi, “Beam coherence-polarization matrix,” Pure and Appl. Opt.7, 941–951 (1998).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 7th expanded, Cambridge, 1999).

Caufield, H. J.

Z. Zhang and H. J. Caufield, “Reflection and refraction by interfaces of uniaxial crystals,” Opt. & Laser Technol.28, 549–553 (1996).
[CrossRef]

Chipman, R. A.

Davydov, B. L.

V. A. Bagan, B. L. Davydov, and I. E. Samartsev, “Characteristics of Cornu depolarisers made from quartz and paratellurite optically active crystals,” Quant. Electron.39, 73–78 (2009).
[CrossRef]

de Sande, J. C. G.

El Sherif, M.

M. El Sherif, M.S. Khalil, S. Khodeir, and N. Nagib, “Simple depolarizers for spectrophotometric measurements of anisotropic samples,” Opt. & Laser Technol.28, 561–563 (1996).
[CrossRef]

Faleiev, D.

Ghosh, G.

G. Ghosh, “Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals,” Opt. Commun.163, 95–102 (1999).
[CrossRef]

Gori, F.

Guattari, G.

F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun.163, 159–163 (1999).
[CrossRef]

Hasman, E.

Hillman, L. W.

Khalil, M.S.

M. El Sherif, M.S. Khalil, S. Khodeir, and N. Nagib, “Simple depolarizers for spectrophotometric measurements of anisotropic samples,” Opt. & Laser Technol.28, 561–563 (1996).
[CrossRef]

Khodeir, S.

M. El Sherif, M.S. Khalil, S. Khodeir, and N. Nagib, “Simple depolarizers for spectrophotometric measurements of anisotropic samples,” Opt. & Laser Technol.28, 561–563 (1996).
[CrossRef]

Kidyarov, B. I.

L. V. Alekseeva, I. V. Povkh, V. I. Stroganov, B. I. Kidyarov, and P. G. Pasko, “Four-ray splitting in optical crystals,” J. Opt. Technol.39, 441–443 (2002).
[CrossRef]

Kleiner, V.

Kuebel, D.

T. D. Visser, D. Kuebel, M. Lahiri, T. Shirai, and E. Wolf, “Unpolarized light beams with different coherence properties,” J. Mod. Opt.56, 1369–1374 (2009).
[CrossRef]

Kuznetsov, V.

Lahiri, M.

T. D. Visser, D. Kuebel, M. Lahiri, T. Shirai, and E. Wolf, “Unpolarized light beams with different coherence properties,” J. Mod. Opt.56, 1369–1374 (2009).
[CrossRef]

Lebedev, V.

Lohmann, A.W.

V. Arrizón, E. Tepichin, M. Ortíz-Gutiérrez, and A.W. Lohmann, “Fresnel diffraction at l/4 of the Talbot distance of an anisotropic grating,” Opt. Commun.127, 171–175 (1996).
[CrossRef]

Lotem, H.

Martínez-Herrero, R.

R. Martínez-Herrero and P. M. Mejías, “On the propagation of random electromagnetic fields with position-independent stochastic behavior,” Opt. Commun.283, 4467–4469 (2010).
[CrossRef]

McClain, S. C.

McGuire, J. P.

J. P. McGuire and R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” Opt. Eng.29, 1478–1484 (1990).
[CrossRef]

Mejías, P. M.

R. Martínez-Herrero and P. M. Mejías, “On the propagation of random electromagnetic fields with position-independent stochastic behavior,” Opt. Commun.283, 4467–4469 (2010).
[CrossRef]

Nagib, N.

M. El Sherif, M.S. Khalil, S. Khodeir, and N. Nagib, “Simple depolarizers for spectrophotometric measurements of anisotropic samples,” Opt. & Laser Technol.28, 561–563 (1996).
[CrossRef]

Niv, A.

Ortíz-Gutiérrez, M.

V. Arrizón, E. Tepichin, M. Ortíz-Gutiérrez, and A.W. Lohmann, “Fresnel diffraction at l/4 of the Talbot distance of an anisotropic grating,” Opt. Commun.127, 171–175 (1996).
[CrossRef]

Pasko, P. G.

L. V. Alekseeva, I. V. Povkh, V. I. Stroganov, B. I. Kidyarov, and P. G. Pasko, “Four-ray splitting in optical crystals,” J. Opt. Technol.39, 441–443 (2002).
[CrossRef]

Piquero, G.

Povkh, I. V.

L. V. Alekseeva, I. V. Povkh, V. I. Stroganov, B. I. Kidyarov, and P. G. Pasko, “Four-ray splitting in optical crystals,” J. Opt. Technol.39, 441–443 (2002).
[CrossRef]

Salem, M.

Samartsev, I. E.

V. A. Bagan, B. L. Davydov, and I. E. Samartsev, “Characteristics of Cornu depolarisers made from quartz and paratellurite optically active crystals,” Quant. Electron.39, 73–78 (2009).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett.31, 688–690 (2006).
[CrossRef] [PubMed]

G. Piquero, R. Borghi, and M. Santarsiero, “Gaussian Schell-model beams propagating through polarization gratings,” J. Opt. Soc. Am. A18, 1399–1405 (2001).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun.163, 159–163 (1999).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, and R. Borghi, “Beam coherence-polarization matrix,” Pure and Appl. Opt.7, 941–951 (1998).
[CrossRef]

Savin, E.

Shirai, T.

T. D. Visser, D. Kuebel, M. Lahiri, T. Shirai, and E. Wolf, “Unpolarized light beams with different coherence properties,” J. Mod. Opt.56, 1369–1374 (2009).
[CrossRef]

Strangi, G.

C. Vena, C. Versace, G. Strangi, and R. Bartolino, “Light depolarization by non-uniform polarization distribution over a beam cross section,” J. Opt. A: Pure Appl. Opt.11, 125704–10 (2009).
[CrossRef]

Stroganov, V. I.

L. V. Alekseeva, I. V. Povkh, V. I. Stroganov, B. I. Kidyarov, and P. G. Pasko, “Four-ray splitting in optical crystals,” J. Opt. Technol.39, 441–443 (2002).
[CrossRef]

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science,” Phil. Mag.9, 401–407 (1836).

Taor, U.

Teijeiro, C.

Tepichin, E.

V. Arrizón, E. Tepichin, M. Ortíz-Gutiérrez, and A.W. Lohmann, “Fresnel diffraction at l/4 of the Talbot distance of an anisotropic grating,” Opt. Commun.127, 171–175 (1996).
[CrossRef]

Tervo, J.

F. Gori, J. Tervo, and J. Turunen, “Correlation matrices of completely unpolarized beams,” Opt. Lett.34, 1447–1449 (2009).
[CrossRef] [PubMed]

J. Tervo and J. Turunen, “Transverse and longitudinal periodicities in fields produced by polarization gratings,” Opt. Commun.190, 51–57 (2001).
[CrossRef]

Turunen, J.

F. Gori, J. Tervo, and J. Turunen, “Correlation matrices of completely unpolarized beams,” Opt. Lett.34, 1447–1449 (2009).
[CrossRef] [PubMed]

J. Tervo and J. Turunen, “Transverse and longitudinal periodicities in fields produced by polarization gratings,” Opt. Commun.190, 51–57 (2001).
[CrossRef]

Vena, C.

C. Vena, C. Versace, G. Strangi, and R. Bartolino, “Light depolarization by non-uniform polarization distribution over a beam cross section,” J. Opt. A: Pure Appl. Opt.11, 125704–10 (2009).
[CrossRef]

Versace, C.

C. Vena, C. Versace, G. Strangi, and R. Bartolino, “Light depolarization by non-uniform polarization distribution over a beam cross section,” J. Opt. A: Pure Appl. Opt.11, 125704–10 (2009).
[CrossRef]

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, and R. Borghi, “Beam coherence-polarization matrix,” Pure and Appl. Opt.7, 941–951 (1998).
[CrossRef]

Visser, T. D.

T. D. Visser, D. Kuebel, M. Lahiri, T. Shirai, and E. Wolf, “Unpolarized light beams with different coherence properties,” J. Mod. Opt.56, 1369–1374 (2009).
[CrossRef]

Wolf, E.

T. D. Visser, D. Kuebel, M. Lahiri, T. Shirai, and E. Wolf, “Unpolarized light beams with different coherence properties,” J. Mod. Opt.56, 1369–1374 (2009).
[CrossRef]

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

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett.31, 688–690 (2006).
[CrossRef] [PubMed]

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

G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A17, 2019–2023 (2000).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 7th expanded, Cambridge, 1999).

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

Zhang, Z.

Z. Zhang and H. J. Caufield, “Reflection and refraction by interfaces of uniaxial crystals,” Opt. & Laser Technol.28, 549–553 (1996).
[CrossRef]

Appl. Opt. (3)

J. Mod. Opt. (1)

T. D. Visser, D. Kuebel, M. Lahiri, T. Shirai, and E. Wolf, “Unpolarized light beams with different coherence properties,” J. Mod. Opt.56, 1369–1374 (2009).
[CrossRef]

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

C. Vena, C. Versace, G. Strangi, and R. Bartolino, “Light depolarization by non-uniform polarization distribution over a beam cross section,” J. Opt. A: Pure Appl. Opt.11, 125704–10 (2009).
[CrossRef]

J. Opt. Soc. Am. A (5)

J. Opt. Technol. (1)

L. V. Alekseeva, I. V. Povkh, V. I. Stroganov, B. I. Kidyarov, and P. G. Pasko, “Four-ray splitting in optical crystals,” J. Opt. Technol.39, 441–443 (2002).
[CrossRef]

Opt. & Laser Technol. (2)

Z. Zhang and H. J. Caufield, “Reflection and refraction by interfaces of uniaxial crystals,” Opt. & Laser Technol.28, 549–553 (1996).
[CrossRef]

M. El Sherif, M.S. Khalil, S. Khodeir, and N. Nagib, “Simple depolarizers for spectrophotometric measurements of anisotropic samples,” Opt. & Laser Technol.28, 561–563 (1996).
[CrossRef]

Opt. Commun. (5)

R. Martínez-Herrero and P. M. Mejías, “On the propagation of random electromagnetic fields with position-independent stochastic behavior,” Opt. Commun.283, 4467–4469 (2010).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun.163, 159–163 (1999).
[CrossRef]

G. Ghosh, “Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals,” Opt. Commun.163, 95–102 (1999).
[CrossRef]

V. Arrizón, E. Tepichin, M. Ortíz-Gutiérrez, and A.W. Lohmann, “Fresnel diffraction at l/4 of the Talbot distance of an anisotropic grating,” Opt. Commun.127, 171–175 (1996).
[CrossRef]

J. Tervo and J. Turunen, “Transverse and longitudinal periodicities in fields produced by polarization gratings,” Opt. Commun.190, 51–57 (2001).
[CrossRef]

Opt. Eng. (1)

J. P. McGuire and R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” Opt. Eng.29, 1478–1484 (1990).
[CrossRef]

Opt. Lett. (6)

Phil. Mag. (1)

H. F. Talbot, “Facts relating to optical science,” Phil. Mag.9, 401–407 (1836).

Phys. Lett. A (1)

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

Pure and Appl. Opt. (1)

F. Gori, M. Santarsiero, S. Vicalvi, and R. Borghi, “Beam coherence-polarization matrix,” Pure and Appl. Opt.7, 941–951 (1998).
[CrossRef]

Quant. Electron. (1)

V. A. Bagan, B. L. Davydov, and I. E. Samartsev, “Characteristics of Cornu depolarisers made from quartz and paratellurite optically active crystals,” Quant. Electron.39, 73–78 (2009).
[CrossRef]

Other (2)

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

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 7th expanded, Cambridge, 1999).

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Figures (6)

Fig. 1
Fig. 1

(a) Schematic sketch of a DWD; (b) notations used throughout the paper.

Fig. 2
Fig. 2

Theoretical behaviors of the Stokes parameters s1 and s3, normalized to the input intensity I0, and the degree of polarization pout across the xz plane at the exit of a DWD for an unpolarized input plane wave (s0(x, z) = I0 and s2(x, z) = 0).

Fig. 3
Fig. 3

Experimental setup: Mi, mirrors; F, neutral density filter; HWP, half wave plate; PBS polarizing beam splitter; MO, microscope objective; L, lens. Blue arrow and dots represent polarization directions

Fig. 4
Fig. 4

Measured values of the DOP as a function of transverse displacement for several planes after the DWD when an unpolarized plane wave impinges on the DWD.

Fig. 5
Fig. 5

Experimental and theoretical DOP as a function of the free space propagation distance after the DWD when an unpolarized plane wave impinges on the DWD.

Fig. 6
Fig. 6

Measured Stokes parameters (symbols) of the exiting light at several z-planes when an unpolarized plane wave impinges on the DWD. Calculated values (solid lines) are also represented at z = 0.50, z = 1.50, m and at z = 2.75 m. All values are normalized to the input intensity I0.

Equations (41)

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E ( r ) = ( E x ( r ) E y ( r ) ) ,
P ^ ( r ) = E ( r ) E ( r ) ,
p ( r ) = 1 4 Det { P ^ ( r ) } Tr 2 { P ^ ( r ) } .
s ( r ) = ( s 0 ( r ) s 1 ( r ) s 2 ( r ) s 3 ( r ) ) = ( P x x ( r ) + P y y ( r ) P x x ( r ) P y y ( r ) 2 Re { P y x ( r ) } 2 Im { P y x ( r ) } ) ,
E in ( x ) = A ( 1 0 ) ,
E out ( x ) ( x , z ) = A 2 t x o a o o ( z ) [ t o o ( t o x ( x ) t o y ( x ) ) + t o e ( t e x ( x ) cos γ o e t e y ( x ) ) exp [ i δ o ( x , z ) ] ] ,
a o o ( z ) = exp ( i k o d i k z )
δ o ( x , z ) = d 2 ( k o e cos β o e k o ) + x k sin γ o e z k ( 1 cos γ o e ) .
1 k o e 2 = sin 2 φ k e 2 + cos 2 φ k o 2 ,
T x t x o t o o t o x ( x ) = t x o t o o t o y ( x ) t x o t o e t e x ( x ) cos γ o e t x o t o e t e y ( x ) ,
E out ( x ) ( x , z ) A 2 T x a o o ( z ) [ ( 1 1 ) + ( 1 1 ) exp [ i δ o ( x , z ) ] ] .
E in ( y ) = A ( 0 1 ) .
E out ( y ) ( x , z ) = A 2 t y e a e e ( x , z ) [ t e e ( t e x ( y ) cos γ e e t e y ( y ) ) + t e o ( t o x ( y ) cos γ e o t o y ( y ) ) exp [ i δ e ( x , z ) ] ]
a e e ( x , z ) = exp ( i k e d 1 i k e e d 2 cos β e e i k x sin γ e e i k z cos γ e e )
δ e ( x , z ) = d 2 ( k o cos β e o k e e cos β e e ) + x k ( sin γ e o sin γ e e ) + z k ( cos γ e o cos γ e e ) .
T y t y e t e e t e x ( y ) cos γ e e t y e t e e t e y ( y ) t y e t e o t o x ( y ) cos γ e o t y e t e o t o y ( y ) ,
E out ( y ) ( x , z ) A 2 T y a e e ( x , z ) [ ( 1 1 ) + ( 1 1 ) exp [ i δ e ( x , z ) ] ] .
P ^ in = I 0 2 ( 1 0 0 0 ) + I 0 2 ( 0 0 0 1 ) = I 0 2 ( 1 0 0 1 ) ,
P out , x x ( x , z ) = I 0 4 [ 2 + cos δ o ( x , z ) cos δ e ( x , z ) ] ,
P out , x y ( x , z ) = P out , y x * ( x , z ) = I 0 4 i [ sin δ o ( x , z ) + sin δ e ( x , z ) ] ,
P out , y y ( x , z ) = I 0 4 [ 2 cos δ o ( x , z ) + cos δ e ( x , z ) ] ,
s ( x , z ) = I 0 2 ( 2 cos δ o ( x , z ) cos δ e ( x , z ) 0 sin δ o ( x , z ) sin δ e ( x , z ) ) ,
p out ( x , z ) = | sin [ δ o ( x , z ) + δ e ( x , z ) 2 ] | .
δ o ( x , z ) k d 2 ( n e n o ) k x ( n e n o ) α k z ( n e n o ) 2 α 2 / 2 ,
δ e ( x , z ) k d 2 ( n e n o ) + k x ( n e n o ) α k z ( n e n o ) 2 α 2 / 2 ,
s ( x , z ) I 0 ( 1 sin [ k z ( n e n o ) 2 α 2 / 2 ] sin [ k ( n e n o ) ( d 2 α x ) ] 0 sin [ k z ( n e n o ) 2 α 2 / 2 ] cos [ k ( n e n o ) ( d 2 α x ) ] ) .
p out ( x , z ) | sin [ k z ( n e n o ) 2 α 2 / 2 ] | ,
Z m λ ( 2 m + 1 ) 2 ( n e n o ) 2 α 2 ,
z m λ m ( n e n o ) 2 α 2 ,
E in ( x ) ( x i , d ) = A ( 1 0 0 ) ,
x i = x z tan γ o e d 2 tan β o e 1 tan α tan β o e .
E x o ( x ) ( x i , d 2 + x i tan α ) = A t x o ( 1 0 0 ) exp [ i k o ( d 1 + x i tan α ) ] .
k o sin α = k o sin ( β o o + α ) ,
k o sin α = k o e sin ( β o e + α ) .
E o o ( x ) ( x i , z = 0 ) = A t x o t o o ( cos ( π / 4 ) sin ( π / 4 ) 0 ) exp ( i k o d ) ,
E o o ( x ) ( x , z > 0 ) = A 2 t x o t o o ( t o x ( x ) t o y ( x ) 0 ) exp ( i k o d i k z ) .
E o e ( x ) ( x o e , z = 0 ) = A t x o t o e ( e o e , x e o e , y e o e , z ) exp [ i k o ( d 1 + x i tan α ) i k o e ( d 2 x i tan α ) cos β o e ] ,
E o e ( x ) ( x , z > 0 ) = A 2 t x o t o e ( t e x ( x ) cos γ o e t e y ( x ) t e x ( x ) sin γ o e ) × exp [ i k o ( d 1 + x i tan α ) i k o e ( d 2 x i tan α ) cos β o e i k z cos γ o e ] ,
E o e ( x ) ( x , z > 0 ) = A 2 t x o t o e ( t e x ( x ) cos γ o e t e y ( x ) t e x ( x ) sin γ o e ) × exp ( i k o d 1 i d 2 k o e cos β o e i x k sin γ o e i z k cos γ o e ) .
E e o ( y ) ( x , z > 0 ) = A 2 t y e t e o ( t o x ( y ) cos γ e o t o y ( y ) t o x ( y ) sin γ e o ) × exp ( i k e d 1 i k o d 2 cos β e o i k x sin γ e o i k z cos γ e o ) ,
E e e ( y ) ( x , z > 0 ) = A 2 t y e t e e ( t e x ( y ) cos γ e e t e y ( y ) t e x ( y ) sin γ e e ) × exp ( i k e d 1 i k e e d 2 cos β e e i k x sin γ e e i k z cos γ e e ) ,

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