Abstract

The efficiency of second-harmonic generation from thin films by use of two input beams at the fundamental frequency depends sensitively on the polarization states of the fundamental beams. This dependence allows precise measurement of the retardation induced by optical elements. We present a theoretical analysis of the technique and discuss its advantages and limitations with regard to retardation measurements. We demonstrate our technique by measuring the retardation of a commercial half-wave plate to a precision and repeatability of better than λ/104. The technique is remarkably insensitive to misalignments of the optical components and of the fundamental beams for the retardation range investigated (δ=180±10°). The extension of the technique to measure low values of retardation (δ0°) is straightforward.

© 2003 Optical Society of America

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References

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  1. S. Huard and G. Vacca, Polarization of Light (Wiley, New York, 1997).
  2. C. Brosseau, Fundamentals of Polarized Light: a Statistical Optics Approach (Wiley, New York, 1998).
  3. E. W. Thulstrup and J. Michl, Elementary Polarization Spectroscopy (Wiley, New York, 1997).
  4. J. Michl, Spectroscopy with Polarized Light (Wiley, New York, 1995).
  5. J. Tinbergen, Astronomical Polarimetry (Cambridge University, Cambridge, England, 1996).
  6. F. T. Ulaby and C. Elachi, Radar Polarimetry for Geoscience Applications (Remote Sensing) (Artech House, Norwood, Mass., 1990).
  7. See, for example, J. Hecht, Understanding Fiber Optics (Prentice-Hall, Englewood Cliffs, N.J., 1999).
  8. M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1980).
  9. A. R. Bungay, S. V. Popov, and N. I. Zheludev, “Specular nonlinear anisotropic polarization effect along fourfold crystal symmetry axes,” Opt. Lett. 20, 356–358 (1995).
    [CrossRef] [PubMed]
  10. S. C. Read, M. Lai, T. Cave, S. W. Morris, D. Shelton, A. Guest, and A. D. May, “Intracavity polarimeter for measuring small optical anisotropies,” J. Opt. Soc. Am. B 5, 1832–1837 (1988).
    [CrossRef]
  11. F. Brandi, E. Polacco, and G. Ruoso, “Stress-optic modulator: a novel device for high sensitivity linear birefringence measurements,” Meas. Sci. Technol. 12, 1503–1508 (2001).
    [CrossRef]
  12. J. Y. Lee, H-W. Lee, J. W. Kim, Y. S. Yoo, and J. W. Hahn, “Measurement of ultralow supermirror birefringence by use of the polarimetric differential cavity ringdown technique,” Appl. Opt. 39, 1941–1945 (2000).
    [CrossRef]
  13. Polarization Solutions, 2001–2002 Catalog (Meadowlark Optics, Frederick, Colo., 2001).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  16. K. B. Rochford and C. M. Wang, “Accurate interferometric retardance measurements,” Appl. Opt. 36, 6473–6479 (1997).
    [CrossRef]
  17. Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency splitting technology,” Opt. Eng. 40, 1071–1075 (2001).
    [CrossRef]
  18. B. Wang and T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
    [CrossRef]
  19. B. Wang and W. Hellman, “Accuracy assessment of a linear birefringence measurement system using a Soleil–Babinet compensator,” Rev. Sci. Instrum. 72, 4066–4070 (2001).
    [CrossRef]
  20. T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383–1388 (1993).
    [CrossRef]
  21. M. Kauranen, T. Verbiest, and A. Persoons, “Second-order nonlinear optical signatures of surface chirality,” J. Mod. Opt. 45, 403–423 (1998).
    [CrossRef]
  22. M. Kauranen, S. V. Elshocht, T. Verbiest, and A. Persoons, “Tensor analysis of the second-order nonlinear optical susceptibility of chiral anisotropic thin films,” J. Chem. Phys. 112, 1497–1502 (2000).
    [CrossRef]
  23. S. M. Saltiel, P. D. Yankov, and N. I. Zheludev, “Second harmonic generation as a method for polarizing and analyzing laser light,” Appl. Phys. B 42, 115–119 (1987).
    [CrossRef]
  24. S. Cattaneo, O. Zehnder, P. Günter, and M. Kauranen, “Nonlinear optical technique for precise retardation measurements,” Phys. Rev. Lett. 88, 243901 (2002).
    [CrossRef] [PubMed]
  25. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  26. Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984).
  27. T. Verbiest, M. Kauranen, and André Persoons, “Light-polarization-induced optical activity,” Phys. Rev. Lett. 82, 3601–3604 (1999).
    [CrossRef]
  28. J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
    [CrossRef]
  29. Ch. Bosshard, K. Sutter, Ph. Prêtre, J. Hulliger, M. Flörsheimer, P. Kaatz, and P. Günter, Organic Nonlinear Optical Materials (Gordon & Breach, Basel, Switzerland, 1995).
  30. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1989), p. 488.
  31. P. D. Hale and G. W. Day, “Stability of birefringent linear retarders (waveplates),” Appl. Opt. 27, 5146–5153 (1988).
    [CrossRef] [PubMed]
  32. Yu. P. Svirko and N. I. Zheludev, Polarization of Light in Nonlinear Optics (Wiley, New York, 1998).

2002 (1)

S. Cattaneo, O. Zehnder, P. Günter, and M. Kauranen, “Nonlinear optical technique for precise retardation measurements,” Phys. Rev. Lett. 88, 243901 (2002).
[CrossRef] [PubMed]

2001 (3)

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency splitting technology,” Opt. Eng. 40, 1071–1075 (2001).
[CrossRef]

B. Wang and W. Hellman, “Accuracy assessment of a linear birefringence measurement system using a Soleil–Babinet compensator,” Rev. Sci. Instrum. 72, 4066–4070 (2001).
[CrossRef]

F. Brandi, E. Polacco, and G. Ruoso, “Stress-optic modulator: a novel device for high sensitivity linear birefringence measurements,” Meas. Sci. Technol. 12, 1503–1508 (2001).
[CrossRef]

2000 (2)

M. Kauranen, S. V. Elshocht, T. Verbiest, and A. Persoons, “Tensor analysis of the second-order nonlinear optical susceptibility of chiral anisotropic thin films,” J. Chem. Phys. 112, 1497–1502 (2000).
[CrossRef]

J. Y. Lee, H-W. Lee, J. W. Kim, Y. S. Yoo, and J. W. Hahn, “Measurement of ultralow supermirror birefringence by use of the polarimetric differential cavity ringdown technique,” Appl. Opt. 39, 1941–1945 (2000).
[CrossRef]

1999 (2)

T. Verbiest, M. Kauranen, and André Persoons, “Light-polarization-induced optical activity,” Phys. Rev. Lett. 82, 3601–3604 (1999).
[CrossRef]

B. Wang and T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

1998 (1)

M. Kauranen, T. Verbiest, and A. Persoons, “Second-order nonlinear optical signatures of surface chirality,” J. Mod. Opt. 45, 403–423 (1998).
[CrossRef]

1997 (1)

1995 (1)

1993 (1)

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383–1388 (1993).
[CrossRef]

1990 (1)

1988 (3)

1987 (2)

J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
[CrossRef]

S. M. Saltiel, P. D. Yankov, and N. I. Zheludev, “Second harmonic generation as a method for polarizing and analyzing laser light,” Appl. Phys. B 42, 115–119 (1987).
[CrossRef]

Brandi, F.

F. Brandi, E. Polacco, and G. Ruoso, “Stress-optic modulator: a novel device for high sensitivity linear birefringence measurements,” Meas. Sci. Technol. 12, 1503–1508 (2001).
[CrossRef]

Bungay, A. R.

Byers, J. D.

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383–1388 (1993).
[CrossRef]

Cattaneo, S.

S. Cattaneo, O. Zehnder, P. Günter, and M. Kauranen, “Nonlinear optical technique for precise retardation measurements,” Phys. Rev. Lett. 88, 243901 (2002).
[CrossRef] [PubMed]

Cave, T.

Day, G. W.

Elshocht, S. V.

M. Kauranen, S. V. Elshocht, T. Verbiest, and A. Persoons, “Tensor analysis of the second-order nonlinear optical susceptibility of chiral anisotropic thin films,” J. Chem. Phys. 112, 1497–1502 (2000).
[CrossRef]

Guest, A.

Günter, P.

S. Cattaneo, O. Zehnder, P. Günter, and M. Kauranen, “Nonlinear optical technique for precise retardation measurements,” Phys. Rev. Lett. 88, 243901 (2002).
[CrossRef] [PubMed]

Hahn, J. W.

Hale, P. D.

Han, Y.

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency splitting technology,” Opt. Eng. 40, 1071–1075 (2001).
[CrossRef]

Hellman, W.

B. Wang and W. Hellman, “Accuracy assessment of a linear birefringence measurement system using a Soleil–Babinet compensator,” Rev. Sci. Instrum. 72, 4066–4070 (2001).
[CrossRef]

Hicks, J. M.

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383–1388 (1993).
[CrossRef]

Kauranen, M.

S. Cattaneo, O. Zehnder, P. Günter, and M. Kauranen, “Nonlinear optical technique for precise retardation measurements,” Phys. Rev. Lett. 88, 243901 (2002).
[CrossRef] [PubMed]

M. Kauranen, S. V. Elshocht, T. Verbiest, and A. Persoons, “Tensor analysis of the second-order nonlinear optical susceptibility of chiral anisotropic thin films,” J. Chem. Phys. 112, 1497–1502 (2000).
[CrossRef]

T. Verbiest, M. Kauranen, and André Persoons, “Light-polarization-induced optical activity,” Phys. Rev. Lett. 82, 3601–3604 (1999).
[CrossRef]

M. Kauranen, T. Verbiest, and A. Persoons, “Second-order nonlinear optical signatures of surface chirality,” J. Mod. Opt. 45, 403–423 (1998).
[CrossRef]

Kim, J. W.

Lai, M.

Lee, H-W.

Lee, J. Y.

Li, Y.

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency splitting technology,” Opt. Eng. 40, 1071–1075 (2001).
[CrossRef]

May, A. D.

Morris, S. W.

Nakadate, S.

Oakberg, T. C.

B. Wang and T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

Persoons, A.

M. Kauranen, S. V. Elshocht, T. Verbiest, and A. Persoons, “Tensor analysis of the second-order nonlinear optical susceptibility of chiral anisotropic thin films,” J. Chem. Phys. 112, 1497–1502 (2000).
[CrossRef]

M. Kauranen, T. Verbiest, and A. Persoons, “Second-order nonlinear optical signatures of surface chirality,” J. Mod. Opt. 45, 403–423 (1998).
[CrossRef]

Persoons, André

T. Verbiest, M. Kauranen, and André Persoons, “Light-polarization-induced optical activity,” Phys. Rev. Lett. 82, 3601–3604 (1999).
[CrossRef]

Petralli-Mallow, T.

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383–1388 (1993).
[CrossRef]

Polacco, E.

F. Brandi, E. Polacco, and G. Ruoso, “Stress-optic modulator: a novel device for high sensitivity linear birefringence measurements,” Meas. Sci. Technol. 12, 1503–1508 (2001).
[CrossRef]

Popov, S. V.

Read, S. C.

Rochford, K. B.

Runwen, W.

Ruoso, G.

F. Brandi, E. Polacco, and G. Ruoso, “Stress-optic modulator: a novel device for high sensitivity linear birefringence measurements,” Meas. Sci. Technol. 12, 1503–1508 (2001).
[CrossRef]

Saltiel, S. M.

S. M. Saltiel, P. D. Yankov, and N. I. Zheludev, “Second harmonic generation as a method for polarizing and analyzing laser light,” Appl. Phys. B 42, 115–119 (1987).
[CrossRef]

Shelton, D.

Sipe, J. E.

Verbiest, T.

M. Kauranen, S. V. Elshocht, T. Verbiest, and A. Persoons, “Tensor analysis of the second-order nonlinear optical susceptibility of chiral anisotropic thin films,” J. Chem. Phys. 112, 1497–1502 (2000).
[CrossRef]

T. Verbiest, M. Kauranen, and André Persoons, “Light-polarization-induced optical activity,” Phys. Rev. Lett. 82, 3601–3604 (1999).
[CrossRef]

M. Kauranen, T. Verbiest, and A. Persoons, “Second-order nonlinear optical signatures of surface chirality,” J. Mod. Opt. 45, 403–423 (1998).
[CrossRef]

Wang, B.

B. Wang and W. Hellman, “Accuracy assessment of a linear birefringence measurement system using a Soleil–Babinet compensator,” Rev. Sci. Instrum. 72, 4066–4070 (2001).
[CrossRef]

B. Wang and T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

Wang, C. M.

Wong, T. M.

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383–1388 (1993).
[CrossRef]

Xu, X.

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency splitting technology,” Opt. Eng. 40, 1071–1075 (2001).
[CrossRef]

Yankov, P. D.

S. M. Saltiel, P. D. Yankov, and N. I. Zheludev, “Second harmonic generation as a method for polarizing and analyzing laser light,” Appl. Phys. B 42, 115–119 (1987).
[CrossRef]

Yao, L.

Yee, H. I.

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383–1388 (1993).
[CrossRef]

Yoo, Y. S.

Zehnder, O.

S. Cattaneo, O. Zehnder, P. Günter, and M. Kauranen, “Nonlinear optical technique for precise retardation measurements,” Phys. Rev. Lett. 88, 243901 (2002).
[CrossRef] [PubMed]

Zhang, S.

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency splitting technology,” Opt. Eng. 40, 1071–1075 (2001).
[CrossRef]

Zhang, Y.

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency splitting technology,” Opt. Eng. 40, 1071–1075 (2001).
[CrossRef]

Zheludev, N. I.

A. R. Bungay, S. V. Popov, and N. I. Zheludev, “Specular nonlinear anisotropic polarization effect along fourfold crystal symmetry axes,” Opt. Lett. 20, 356–358 (1995).
[CrossRef] [PubMed]

S. M. Saltiel, P. D. Yankov, and N. I. Zheludev, “Second harmonic generation as a method for polarizing and analyzing laser light,” Appl. Phys. B 42, 115–119 (1987).
[CrossRef]

Zhiyao, Z.

Appl. Opt. (4)

Appl. Phys. B (1)

S. M. Saltiel, P. D. Yankov, and N. I. Zheludev, “Second harmonic generation as a method for polarizing and analyzing laser light,” Appl. Phys. B 42, 115–119 (1987).
[CrossRef]

J. Chem. Phys. (1)

M. Kauranen, S. V. Elshocht, T. Verbiest, and A. Persoons, “Tensor analysis of the second-order nonlinear optical susceptibility of chiral anisotropic thin films,” J. Chem. Phys. 112, 1497–1502 (2000).
[CrossRef]

J. Mod. Opt. (1)

M. Kauranen, T. Verbiest, and A. Persoons, “Second-order nonlinear optical signatures of surface chirality,” J. Mod. Opt. 45, 403–423 (1998).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. Chem. (1)

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383–1388 (1993).
[CrossRef]

Meas. Sci. Technol. (1)

F. Brandi, E. Polacco, and G. Ruoso, “Stress-optic modulator: a novel device for high sensitivity linear birefringence measurements,” Meas. Sci. Technol. 12, 1503–1508 (2001).
[CrossRef]

Opt. Eng. (1)

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency splitting technology,” Opt. Eng. 40, 1071–1075 (2001).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (2)

S. Cattaneo, O. Zehnder, P. Günter, and M. Kauranen, “Nonlinear optical technique for precise retardation measurements,” Phys. Rev. Lett. 88, 243901 (2002).
[CrossRef] [PubMed]

T. Verbiest, M. Kauranen, and André Persoons, “Light-polarization-induced optical activity,” Phys. Rev. Lett. 82, 3601–3604 (1999).
[CrossRef]

Rev. Sci. Instrum. (2)

B. Wang and T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

B. Wang and W. Hellman, “Accuracy assessment of a linear birefringence measurement system using a Soleil–Babinet compensator,” Rev. Sci. Instrum. 72, 4066–4070 (2001).
[CrossRef]

Other (14)

Polarization Solutions, 2001–2002 Catalog (Meadowlark Optics, Frederick, Colo., 2001).

S. Huard and G. Vacca, Polarization of Light (Wiley, New York, 1997).

C. Brosseau, Fundamentals of Polarized Light: a Statistical Optics Approach (Wiley, New York, 1998).

E. W. Thulstrup and J. Michl, Elementary Polarization Spectroscopy (Wiley, New York, 1997).

J. Michl, Spectroscopy with Polarized Light (Wiley, New York, 1995).

J. Tinbergen, Astronomical Polarimetry (Cambridge University, Cambridge, England, 1996).

F. T. Ulaby and C. Elachi, Radar Polarimetry for Geoscience Applications (Remote Sensing) (Artech House, Norwood, Mass., 1990).

See, for example, J. Hecht, Understanding Fiber Optics (Prentice-Hall, Englewood Cliffs, N.J., 1999).

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1980).

Ch. Bosshard, K. Sutter, Ph. Prêtre, J. Hulliger, M. Flörsheimer, P. Kaatz, and P. Günter, Organic Nonlinear Optical Materials (Gordon & Breach, Basel, Switzerland, 1995).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1989), p. 488.

Yu. P. Svirko and N. I. Zheludev, Polarization of Light in Nonlinear Optics (Wiley, New York, 1998).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984).

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Figures (6)

Fig. 1
Fig. 1

Geometry of nonlinear retardation measurements. Two beams (target and probe) at the fundamental frequency ω are applied to the same spot on a poled thin film and coherent second-harmonic light in the sum direction is detected. All beams are on the same plane of incidence. Note that angles θ1 and θ2 are drawn as positive and negative, respectively.

Fig. 2
Fig. 2

Elliptically polarized target beam. The principal axes of the polarization ellipse are oriented along ξ and η. Azimuth ψ is the angle between the major axis and the p direction (0<ψπ). The lengths of major and minor axes of the ellipse are 2a and 2b, respectively.

Fig. 3
Fig. 3

Simulations for the case of unpolarized detection assuming a nominal half-wave plate with a retardation error of λ/1000. (a) Variation of the circular-difference (CD) response (%) on incident angles θ1 and θ2 (degrees) of target and probe beams, respectively. (b) Second-harmonic generation (SHG) intensity (a.u.) for right-hand circular probe polarization.

Fig. 4
Fig. 4

Experimental setup for retardation measurements. Laser light at 1064 nm is split into two beams (target and probe). After the beam splitter (BS), the beams are p polarized by Glan polarizers (P). The polarization of the probe beam is varied by a zero-order quarter-wave plate (QWP). The nominal half-wave plate (HWP) to be investigated is placed in the target beam. The beams are applied to the same spot of a poled polymer film and second-harmonic light at 532 nm is detected by a photomultiplier (PM) in the sum direction.

Fig. 5
Fig. 5

Second-harmonic generation (SHG) intensity recorded continuously as the probe quarter-wave plate (QWP) is rotated. Left- (LHC) and right-hand (RHC) circularly polarized probe beams correspond to rotation angles of -45° and +45°, respectively. For this measurement, the target retardation was determined as 178.45±0.03°.

Fig. 6
Fig. 6

Normalized circular-difference response as a function of the target retardation (simulation). The angle between the fast axis of the wave plate and the p direction was assumed to be 45°. The incident angles of the target and the probe beams were assumed to be θ1=24° and θ2=-4.5°, respectively; the refractive indices of all materials were equal and unity; and the poling ratio of the film was 3.

Equations (36)

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

Pi=χijk(2)EjEk,
E3[n×P]×n.
Eix=Eipcos θi,Eiy=-Eis,Eiz=Eipsin θi.
E3x=(Pxcos θ3+Pzsin θ3)cos θ3,
E3y=Py,
E3z=(Pxcos θ3+Pzsin θ3)sin θ3.
E3p=fpE1pE2p+gpE1sE2s.
E3s=fsE1pE2s+gsE1sE2p.
fp=2χxxz(2)sin θ1cos θ2cos θ3+2χxxz(2)cos θ1sin θ2cos θ3+2χzxx(2)cos θ1cos θ2sin θ3+2χzzz(2)sin θ1sin θ2sin θ3,
gp=2χzxx(2)sin θ3,
fs=2χxxz(2)sin θ1,
gs=2χxxz(2)sin θ2,
ΔII=Ileft-Iright(Ileft+Iright)/2,
|E3p|2=[|fp|2|E1p|2+|gp|2|E1s|2±i(fp*gpE1sE1p*-fpgp*E1s*E1p)]|E2p|2,
|E3s|2=[|fs|2|E1p|2+|gs|2|E1s|2±i(fsgs*E1s*E1p-fs*gsE1sE1p*)]|E2p|2,
E1ξ(t)=a exp(-iωt),E1η(t)=ib exp(-iωt).
E1p=E1ξcos ψ-E1ηsin ψ=(cos ψ+ie sin ψ)E1ξ,
E1s=E1ξsin ψ+E1ηcos ψ=(sin ψ-ie cos ψ)E1ξ.
Im(E1sE1p*)=-e|Eξ|2(cos2 ψ+sin2 ψ)=-ea2.
EpEs=TwpEp0Es0,
Twp
=cos δ/2-i cos 2ϕ sin δ/2-i sin 2ϕ sin δ/2-i sin 2ϕ sin δ/2cos δ/2+i cos 2ϕ sin δ/2,
 
E1p=cos(δ/2)E1p0,E1s=-i sin(δ/2)E1p0,
|E3p|2=|fpcos(δ/2)±gpsin(δ/2)|2|E2p|2|E1p0|2,
|E3s|2=|fscos(δ/2)gssin(δ/2)|2|E2p|2|E1p0|2.
I=|E3p|2+|E3s|2=[(|fp|2+|fs|2)cos2(δ/2)+(|gp|2+|gs|2)sin2(δ/2)±(fpgp*-fsgs*+fp*gp-fs*gs)×cos(δ/2)sin(δ/2)]|E2p|2|E1p0|2.
Ileft-Iright=(fpgp*-fsgs*+fp*gp-fs*gs)×|E2p|2|E1p0|2sin δ.
|E3p|2=|fpσ/2gp|2|E2p|2|E1p0|2,
|E3s|2=|fsσ/2±gs|2|E2p|2|E1p0|2.
sin 2ε=sin 2ϕ sin δ,
E2p=[1-i cos(2ϕ)]E2p0,
E2s=-i sin(2ϕ)E2p0,
I[fp2cos2(δ/2)+gs2sin2(δ/2)](1+cos2 2ϕ)+[fs2cos2(δ/2)+gp2sin2(δ/2)]sin2 2ϕ+(fsgs-fpgp)sin δ sin 2ϕ.
Ia|E2p|2+b|E2s|2+(c+id)E2pE2s*+(c-id)E2p*E2s.
Ia[1+cos2(2ϕ)]+b sin2(2ϕ)+c sin(4ϕ)-2d sin(2ϕ).

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