Abstract

The emerging polarization states from a linearly polarized monochromatic light passing through a rotating linear quarter-wave plate have been characterized as the intersection curve of a cylinder and the Poincaré sphere. But in the cases where the input polarization states are in general elliptical or circular and pass through a rotating linear retarder, the emerging polarization states produce trajectories that do not correspond to the intersection of a sphere with one cylinder. Hence, in this work, we present a full characterization of the trajectories on the Poincaré sphere for monochromatic input beams with an arbitrary polarization state passing through a rotating linear retarder as the intersection curve of the Poincaré sphere with a cone. Moreover, it is shown that these trajectories are characterized by their projection on the equator plane, having the form of limaçon of Pascal (Pascal’s snails).

© 2017 Optical Society of America

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References

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  1. D. H. Goldstein, Polarized Light (CRC Press, 2016).
  2. H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007).
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    [Crossref]
  5. N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012).
    [Crossref]
  6. R. Azzam, “Stokes-vector and Mueller-matrix polarimetry,” J. Opt. Soc. Am. A 33, 1396–1408 (2016).
    [Crossref]
  7. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
    [Crossref]
  8. W. S. Bickel, J. Davidson, D. Huffman, and R. Kilkson, “Application of polarization effects in light scattering: a new biophysical tool,” Proc. Natl. Acad. Sci. USA 73, 486–490 (1976).
    [Crossref]
  9. T. Yasui, Y. Tohno, and T. Araki, “Determination of collagen fiber orientation in human tissue by use of polarization measurement of molecular second-harmonic-generation light,” Appl. Opt. 43, 2861–2867 (2004).
    [Crossref]
  10. V. Sankaran, J. T. Walsh, and D. J. Maitland, “Comparative study of polarized light propagation in biologic tissues,” J. Biomed. Opt. 7, 300–306 (2002).
    [Crossref]
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    [Crossref]
  12. J. Ladd, A. D. Taylor, M. Piliarik, J. Homola, and S. Jiang, “Label-free detection of cancer biomarker candidates using surface plasmon resonance imaging,” Anal. Bioanal. Chem. 393, 1157–1163 (2009).
    [Crossref]
  13. C. Browne and F. Zerban, Physical and Chemical Methods of Sugar Analysis: A Practical and Descriptive Treatise for Use in Research, Technical, and Control Laboratories (Wiley, 1941).
  14. P. A. Williams, “Rotating-wave-plate Stokes polarimeter for differential group delay measurements of polarization-mode dispersion,” Appl. Opt. 38, 6508–6515 (1999).
    [Crossref]
  15. J. De Feijter, D. J. Benjamins, and F. Veer, “Ellipsometry as a tool to study the adsorption behavior of synthetic and biopolymers at the air-water interface,” Biopolymers 17, 1759–1772 (1978).
    [Crossref]
  16. F. Xu, C.-C. Cheng, A. Scherer, R.-C. Tyan, P.-C. Sun, and Y. Fainman, “Fabrication, modeling, and characterization of form-birefringent nanostructures,” Opt. Lett. 20, 2457–2459 (1995).
    [Crossref]
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    [Crossref]
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    [Crossref]
  19. J. M. López-Téllez, N. C. Bruce, and O. G. Rodrguez-Herrera, “Characterization of optical polarization properties for liquid crystal-based retarders,” Appl. Opt. 55, 6025–6033 (2016).
    [Crossref]
  20. S. G. Reddy, S. Prabhakar, P. Chithrabhanu, R. Singh, and R. Simon, “Polarization state transformation using two quarter wave plates: application to Mueller polarimetry,” Appl. Opt. 55, B14–B19 (2016).
    [Crossref]
  21. R. M. A. Azzam, “Poincaré sphere representation of the fixed-polarizer rotating-retarder optical system,” J. Opt. Soc. Am. A 17, 2105–2107 (2000).
    [Crossref]
  22. A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34, 1651 (1995).
    [Crossref]
  23. A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng. 34, 1656–1658 (1995).
    [Crossref]
  24. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006).
    [Crossref]
  25. D. Sabatke, M. Descour, E. Dereniak, W. Sweatt, S. Kemme, and G. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802–804 (2000).
    [Crossref]
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  27. S. G. Reddy, S. Prabhakar, A. Aadhi, A. Kumar, M. Shah, R. Singh, and R. Simon, “Measuring the Mueller matrix of an arbitrary optical element with a universal SU(2) polarization gadget,” J. Opt. Soc. Am. A 31, 610–615 (2014).
    [Crossref]
  28. G. Monge, Géométrie descriptive: leçons données aux écoles normales, l’an 3 de la république (Baudouin, imprimeur du Corps Legislatif et de l’Institut National, 1798).
  29. R. Ferréol and J. Mandonnet, “Courbe de Viviani = Viviani’s curve = Vivianische Kurve,” in Encyclopédie des formes mathématiques remarquables, 2016, http://www.mathcurve.com/courbes3d/viviani/viviani.shtml .
  30. J. Oprea, Differential Geometry and Its Applications (MAA, 2007).
  31. E. Abbena, S. Salamon, and A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed., Textbooks in Mathematics (Taylor & Francis, 2006).
  32. P. Baudoin, Les ovales de Descartes et le limaçon de Pascal, Mathématiques elémentaires (Vuibert, 1938).
  33. J. Lawrence, A Catalog of Special Plane Curves, Dover Books on Advanced Mathematics (Dover, 1972).

2016 (3)

2014 (1)

2012 (2)

A. Roberts and L. Lin, “Plasmonic quarter-wave plate,” Opt. Lett. 37, 1820–1822 (2012).
[Crossref]

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012).
[Crossref]

2009 (1)

J. Ladd, A. D. Taylor, M. Piliarik, J. Homola, and S. Jiang, “Label-free detection of cancer biomarker candidates using surface plasmon resonance imaging,” Anal. Bioanal. Chem. 393, 1157–1163 (2009).
[Crossref]

2007 (1)

2006 (1)

2004 (1)

2002 (1)

V. Sankaran, J. T. Walsh, and D. J. Maitland, “Comparative study of polarized light propagation in biologic tissues,” J. Biomed. Opt. 7, 300–306 (2002).
[Crossref]

2000 (2)

1999 (1)

1998 (1)

1995 (4)

1992 (1)

1978 (1)

J. De Feijter, D. J. Benjamins, and F. Veer, “Ellipsometry as a tool to study the adsorption behavior of synthetic and biopolymers at the air-water interface,” Biopolymers 17, 1759–1772 (1978).
[Crossref]

1976 (1)

W. S. Bickel, J. Davidson, D. Huffman, and R. Kilkson, “Application of polarization effects in light scattering: a new biophysical tool,” Proc. Natl. Acad. Sci. USA 73, 486–490 (1976).
[Crossref]

Aadhi, A.

Abbena, E.

E. Abbena, S. Salamon, and A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed., Textbooks in Mathematics (Taylor & Francis, 2006).

Aieta, F.

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012).
[Crossref]

Ambirajan, A.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng. 34, 1656–1658 (1995).
[Crossref]

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34, 1651 (1995).
[Crossref]

Araki, T.

Azzam, R.

Azzam, R. M. A.

R. M. A. Azzam, “Poincaré sphere representation of the fixed-polarizer rotating-retarder optical system,” J. Opt. Soc. Am. A 17, 2105–2107 (2000).
[Crossref]

R. M. A. Azzam and N. Bashara, Ellipsometry and Polarized Light, North-Holland Personal Library (North-Holland, 1987).

Bashara, N.

R. M. A. Azzam and N. Bashara, Ellipsometry and Polarized Light, North-Holland Personal Library (North-Holland, 1987).

Baudoin, P.

P. Baudoin, Les ovales de Descartes et le limaçon de Pascal, Mathématiques elémentaires (Vuibert, 1938).

Benjamins, D. J.

J. De Feijter, D. J. Benjamins, and F. Veer, “Ellipsometry as a tool to study the adsorption behavior of synthetic and biopolymers at the air-water interface,” Biopolymers 17, 1759–1772 (1978).
[Crossref]

Bickel, W. S.

W. S. Bickel, J. Davidson, D. Huffman, and R. Kilkson, “Application of polarization effects in light scattering: a new biophysical tool,” Proc. Natl. Acad. Sci. USA 73, 486–490 (1976).
[Crossref]

Browne, C.

C. Browne and F. Zerban, Physical and Chemical Methods of Sugar Analysis: A Practical and Descriptive Treatise for Use in Research, Technical, and Control Laboratories (Wiley, 1941).

Bruce, N. C.

Capasso, F.

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012).
[Crossref]

Chenault, D. B.

Cheng, C.-C.

Chithrabhanu, P.

Colston, B.

Da Silva, L.

Davidson, J.

W. S. Bickel, J. Davidson, D. Huffman, and R. Kilkson, “Application of polarization effects in light scattering: a new biophysical tool,” Proc. Natl. Acad. Sci. USA 73, 486–490 (1976).
[Crossref]

De Feijter, J.

J. De Feijter, D. J. Benjamins, and F. Veer, “Ellipsometry as a tool to study the adsorption behavior of synthetic and biopolymers at the air-water interface,” Biopolymers 17, 1759–1772 (1978).
[Crossref]

Dereniak, E.

Descour, M.

Everett, M.

Fainman, Y.

Fujiwara, H.

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007).

Gaburro, Z.

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012).
[Crossref]

Genevet, P.

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012).
[Crossref]

Goldstein, D. H.

Goldstein, D. L.

Gray, A.

E. Abbena, S. Salamon, and A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed., Textbooks in Mathematics (Taylor & Francis, 2006).

Homola, J.

J. Ladd, A. D. Taylor, M. Piliarik, J. Homola, and S. Jiang, “Label-free detection of cancer biomarker candidates using surface plasmon resonance imaging,” Anal. Bioanal. Chem. 393, 1157–1163 (2009).
[Crossref]

Huffman, D.

W. S. Bickel, J. Davidson, D. Huffman, and R. Kilkson, “Application of polarization effects in light scattering: a new biophysical tool,” Proc. Natl. Acad. Sci. USA 73, 486–490 (1976).
[Crossref]

Jiang, S.

J. Ladd, A. D. Taylor, M. Piliarik, J. Homola, and S. Jiang, “Label-free detection of cancer biomarker candidates using surface plasmon resonance imaging,” Anal. Bioanal. Chem. 393, 1157–1163 (2009).
[Crossref]

Kats, M. A.

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012).
[Crossref]

Kawakami, S.

Kemme, S.

Kilkson, R.

W. S. Bickel, J. Davidson, D. Huffman, and R. Kilkson, “Application of polarization effects in light scattering: a new biophysical tool,” Proc. Natl. Acad. Sci. USA 73, 486–490 (1976).
[Crossref]

Kumar, A.

Ladd, J.

J. Ladd, A. D. Taylor, M. Piliarik, J. Homola, and S. Jiang, “Label-free detection of cancer biomarker candidates using surface plasmon resonance imaging,” Anal. Bioanal. Chem. 393, 1157–1163 (2009).
[Crossref]

Lawrence, J.

J. Lawrence, A Catalog of Special Plane Curves, Dover Books on Advanced Mathematics (Dover, 1972).

Lin, L.

Look, D. C.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34, 1651 (1995).
[Crossref]

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng. 34, 1656–1658 (1995).
[Crossref]

López-Téllez, J. M.

Maitland, D. J.

V. Sankaran, J. T. Walsh, and D. J. Maitland, “Comparative study of polarized light propagation in biologic tissues,” J. Biomed. Opt. 7, 300–306 (2002).
[Crossref]

Monge, G.

G. Monge, Géométrie descriptive: leçons données aux écoles normales, l’an 3 de la république (Baudouin, imprimeur du Corps Legislatif et de l’Institut National, 1798).

Oprea, J.

J. Oprea, Differential Geometry and Its Applications (MAA, 2007).

Phipps, G.

Piliarik, M.

J. Ladd, A. D. Taylor, M. Piliarik, J. Homola, and S. Jiang, “Label-free detection of cancer biomarker candidates using surface plasmon resonance imaging,” Anal. Bioanal. Chem. 393, 1157–1163 (2009).
[Crossref]

Prabhakar, S.

Reddy, S. G.

Richter, I.

Roberts, A.

Rodrguez-Herrera, O. G.

Sabatke, D.

Salamon, S.

E. Abbena, S. Salamon, and A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed., Textbooks in Mathematics (Taylor & Francis, 2006).

Sankaran, V.

V. Sankaran, J. T. Walsh, and D. J. Maitland, “Comparative study of polarized light propagation in biologic tissues,” J. Biomed. Opt. 7, 300–306 (2002).
[Crossref]

Sasaki, Y.

Sato, T.

Scherer, A.

Schoenenberger, K.

Shah, M.

Shaw, J. A.

Shurcliff, W.

W. Shurcliff, Polarized Light: Production and Use (Harvard University, 2013).

Simon, R.

Singh, R.

Sun, P.-C.

Sweatt, W.

Tadokoro, T.

Taylor, A. D.

J. Ladd, A. D. Taylor, M. Piliarik, J. Homola, and S. Jiang, “Label-free detection of cancer biomarker candidates using surface plasmon resonance imaging,” Anal. Bioanal. Chem. 393, 1157–1163 (2009).
[Crossref]

Tohno, Y.

Tsuru, T.

Tyan, R.-C.

Tyo, J. S.

Veer, F.

J. De Feijter, D. J. Benjamins, and F. Veer, “Ellipsometry as a tool to study the adsorption behavior of synthetic and biopolymers at the air-water interface,” Biopolymers 17, 1759–1772 (1978).
[Crossref]

Walsh, J. T.

V. Sankaran, J. T. Walsh, and D. J. Maitland, “Comparative study of polarized light propagation in biologic tissues,” J. Biomed. Opt. 7, 300–306 (2002).
[Crossref]

Williams, P. A.

Xu, F.

Yasui, T.

Yu, N.

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012).
[Crossref]

Zerban, F.

C. Browne and F. Zerban, Physical and Chemical Methods of Sugar Analysis: A Practical and Descriptive Treatise for Use in Research, Technical, and Control Laboratories (Wiley, 1941).

Anal. Bioanal. Chem. (1)

J. Ladd, A. D. Taylor, M. Piliarik, J. Homola, and S. Jiang, “Label-free detection of cancer biomarker candidates using surface plasmon resonance imaging,” Anal. Bioanal. Chem. 393, 1157–1163 (2009).
[Crossref]

Appl. Opt. (8)

P. A. Williams, “Rotating-wave-plate Stokes polarimeter for differential group delay measurements of polarization-mode dispersion,” Appl. Opt. 38, 6508–6515 (1999).
[Crossref]

I. Richter, P.-C. Sun, F. Xu, and Y. Fainman, “Design considerations of form birefringent microstructures,” Appl. Opt. 34, 2421–2429 (1995).
[Crossref]

T. Sato, T. Araki, Y. Sasaki, T. Tsuru, T. Tadokoro, and S. Kawakami, “Compact ellipsometer employing a static polarimeter module with arrayed polarizer and wave-plate elements,” Appl. Opt. 46, 4963–4967 (2007).
[Crossref]

J. M. López-Téllez, N. C. Bruce, and O. G. Rodrguez-Herrera, “Characterization of optical polarization properties for liquid crystal-based retarders,” Appl. Opt. 55, 6025–6033 (2016).
[Crossref]

S. G. Reddy, S. Prabhakar, P. Chithrabhanu, R. Singh, and R. Simon, “Polarization state transformation using two quarter wave plates: application to Mueller polarimetry,” Appl. Opt. 55, B14–B19 (2016).
[Crossref]

D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
[Crossref]

T. Yasui, Y. Tohno, and T. Araki, “Determination of collagen fiber orientation in human tissue by use of polarization measurement of molecular second-harmonic-generation light,” Appl. Opt. 43, 2861–2867 (2004).
[Crossref]

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006).
[Crossref]

Biopolymers (1)

J. De Feijter, D. J. Benjamins, and F. Veer, “Ellipsometry as a tool to study the adsorption behavior of synthetic and biopolymers at the air-water interface,” Biopolymers 17, 1759–1772 (1978).
[Crossref]

J. Biomed. Opt. (1)

V. Sankaran, J. T. Walsh, and D. J. Maitland, “Comparative study of polarized light propagation in biologic tissues,” J. Biomed. Opt. 7, 300–306 (2002).
[Crossref]

J. Opt. Soc. Am. A (3)

Nano Lett. (1)

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012).
[Crossref]

Opt. Eng. (2)

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34, 1651 (1995).
[Crossref]

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng. 34, 1656–1658 (1995).
[Crossref]

Opt. Lett. (4)

Proc. Natl. Acad. Sci. USA (1)

W. S. Bickel, J. Davidson, D. Huffman, and R. Kilkson, “Application of polarization effects in light scattering: a new biophysical tool,” Proc. Natl. Acad. Sci. USA 73, 486–490 (1976).
[Crossref]

Other (11)

D. H. Goldstein, Polarized Light (CRC Press, 2016).

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007).

W. Shurcliff, Polarized Light: Production and Use (Harvard University, 2013).

C. Browne and F. Zerban, Physical and Chemical Methods of Sugar Analysis: A Practical and Descriptive Treatise for Use in Research, Technical, and Control Laboratories (Wiley, 1941).

R. M. A. Azzam and N. Bashara, Ellipsometry and Polarized Light, North-Holland Personal Library (North-Holland, 1987).

G. Monge, Géométrie descriptive: leçons données aux écoles normales, l’an 3 de la république (Baudouin, imprimeur du Corps Legislatif et de l’Institut National, 1798).

R. Ferréol and J. Mandonnet, “Courbe de Viviani = Viviani’s curve = Vivianische Kurve,” in Encyclopédie des formes mathématiques remarquables, 2016, http://www.mathcurve.com/courbes3d/viviani/viviani.shtml .

J. Oprea, Differential Geometry and Its Applications (MAA, 2007).

E. Abbena, S. Salamon, and A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed., Textbooks in Mathematics (Taylor & Francis, 2006).

P. Baudoin, Les ovales de Descartes et le limaçon de Pascal, Mathématiques elémentaires (Vuibert, 1938).

J. Lawrence, A Catalog of Special Plane Curves, Dover Books on Advanced Mathematics (Dover, 1972).

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Figures (9)

Fig. 1.
Fig. 1. Trajectories on the Poincaré sphere are characterized by the curve of intersection of the Poincaré sphere with a cone, where the axis of symmetry is defined by the points representing the polarization state of input beam p and its enantiogyre states p, the state associated with the vertex of the cone.
Fig. 2.
Fig. 2. Computational simulation of Eq. (18). Locus of the emerging polarization states (a) from a linearly polarized input beam; (b) with asymmetrical eight-shape.
Fig. 3.
Fig. 3. Scheme of asymmetrical eight-curve, for an input state Sinput; the output states at the same meridian are Supper, Scross, and Slower.
Fig. 4.
Fig. 4. Scheme of the plane of the meridian where the states Sinput, Supper, Slower, and Scross are located. The states Supper and Slower are diametrically opposed.
Fig. 5.
Fig. 5. Emerging states for (a) input polarized beam with ellipticity 2χ=π/4; (b) input polarized beam with ellipticity π/4<2χ<π/2.
Fig. 6.
Fig. 6. Projections on the plane (e1,e2) corresponding to the curves. (a) Projection of Fig. 2(a); (b) projection of Fig. 2(b); (c) projection of Fig. 5(a); (d) projection of Fig. 5(b).
Fig. 7.
Fig. 7. Emerging states from a half-wave plate rotating.
Fig. 8.
Fig. 8. Schematization of the optical system for the recording of experimental data.
Fig. 9.
Fig. 9. Red paths correspond to the data calculated from Eq. (18), and the blue paths correspond to the experimental data. The trajectories are characterized by the curve of intersection of the Poincaré sphere and a cone. (a) Case 1, input linear polarization state S(π/2,0); (b) Case 2, input polarization state S(π/2,0.1311π); (c) Case 3; input polarization state S(π/2,π/4); (d) Case 4, input polarization state S(π/2,0.3120π).

Equations (46)

Equations on this page are rendered with MathJax. Learn more.

(S0S1S2S3)=(1cos2χcos2α+[cos2χsin(2α2θ)(1cosδ)+sin2χsinδ]sin2θcos2χsin2α[cos2χsin(2α2θ)(1cosδ)+sin2χsinδ]cos2θsinδcos2χsin(2α2θ)+sin2χcosδ).
x2+y2+z2=1,
(xa)2+(yb)2=tan2(δ2)(zc)2,
(xa)2+(yb)2=(1cosδsinδ)2(zc)2.
x=a+(1cosδsinδ)(zc)sin2t,
y=b(1cosδsinδ)(zc)cos2t.
c2[(1cosδsinδ)21]+2c(1cosδsinδ)(bcos2tasin2t)2z(1cosδsinδ)[bcos2tasin2t+c(1cosδsinδ)]+z2[1+(1cosδsinδ)2]=0,
C=c2[(1cosδsinδ)21]+2c(1cosδsinδ)(bcos2tasin2t),
B=2(1cosδsinδ)[bcos2tasin2t+c(1cosδsinδ)],
A=[1+(1cosδsinδ)2].
z±=B±B24AC2A.
a,b,c=cos2αcos2χ,sin2αcos2χ,sin2χ,
B24AC=12(1+(1cosδsinδ)2)(4(1cosδsinδ)2cos22χsin2(2α2t)8sin(2α2t)sin2χcos2χ(1cosδsinδ)+4sin2  2χ)=4[(1cosδsinδ)cos2χsin(2α2t)sin2χ]2.
z±=2(1cosδsinδ)[cos2χsin(2α2t)+sin2χ(1cosδsinδ)]2(1+(1cosδsinδ)2)±2[(1cosδsinδ)cos2χsin(2α2t)sin2χ]2(1+(1cosδsinδ)2),
z+=sinδcos2χsin(2α2t)sin2χcosδ,
x+=cos2χcos2α+[cos2χsin(2α2θ)(1cosδ)sin2χsinδ]sin2θ,
y+=cos2χsin2α[cos2χsin(2α2θ)(1cosδ)sin2χsinδ]cos2θ.
S=(1(cos2χcos2α+[cos2χsin(2α2θ)+sin2χ]sin2θ)(cos2χsin2α[cos2χsin(2α2θ)+sin2χ]cos2θ)cos2χsin(2α2θ)).
S(2α,0)=(1cos(2α2θ)cos2θcos(2α2θ)sin2θsin(2α2θ)).
±cos2χcos2α=cos2χcos2α+[cos2χsin(2α2θ)+sin2χ]sin2θ,
±cos2χsin2α=cos2χsin2α[cos2χsin(2α2θ)+sin2χ]cos2θ,
sin2χ=cos2χsin(2α2θ).
[cos2χsin(2α2θ)+sin2χ]cos(2θ2α)=0.
2θ1=2αarcsin(tan2χ),
2θ2=2α+arcsin(tan2χ)π,
2θ=2απ/2,
2θ+=2α+π/2,
Scross(2α,2χ)=(1cos2χcos2αcos2χsin2αsin2χ),
Slower(2α,2χ)=(1sin2χcos2αsin2χsin2αcos2χ),
Supper(2α,2χ)=(1sin2χcos2αsin2χsin2αcos2χ);
Supper=S(2α+π,π/22χ),
Slower=S(2α,2χπ/2),
S=(112[cos2α+(sin(2α2θ)+1)sin2θ]12[sin2α(sin(2α2θ)+1)cos2θ]12sin(2α2θ)).
S=(1sin2θcos2θ0),
(S1,S2)=(1,cos2χcos2α+cos2χsin2αcos2θsin2θcos2χcos2αsin22θ+sin2χsin2θ,cos2χsin2α+cos2χsin2αcos22θcos2χcos2αsin2θcos2θ+sin2χcos2θ).
(S1,S2)=(1,a+bcos2θsin2θasin22θ+csin2θ,bbcos22θ+asin2θcos2θccos2θ),
(S1,S2)=(1,a+bsin4θ2a(1cos4θ2)+csin2θ,bb(1+cos4θ2)+asin4θ2ccos2θ),
S1=a2+bsin4θ2+acos4θ2+csin2θ,
S2=b2bcos4θ2+asin4θ2ccos2θ,
V(2α,2χ)=S(2α,±2χ),
sin(2α2θ1)=tan2χ,
sin(2α2θ2)=tan2χ.
Scross=(1cos2χcos2αcos2χsin2αsin2χ),
S±=(1,cos2χcos2α+[cos2χsin(2α2απ/2)+sin2χ]sin(2α±π/2),cos2χsin2α[cos2χsin(2α2απ/2)+sin2χ]cos(2α±π/2),cos2χsin(2α2απ/2)),
S±=(1,cos2χcos2α(cos2χsin2χ)cos2α,cos2χsin2α(cos2χsin2χ)sin2α,cos2χ).
S±=(1±sin2χcos2α±sin2χsin2αcos2χ),

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