Abstract

A theoretical analysis is given of the propagation of a linearly polarized electromagnetic wave through a medium with a harmonically oscillating birefringence and a static birefringence whose principal axis is inclined to the oscillation axis. The theory is applied to the photoelastic modulator (PEM) to predict the transmitted light flux in experimental configurations of particular spectroscopic and polarimetric importance. Included is the general configuration of a modulator between two polarizers (applicable to nearly all PEM-based light-transmission experiments) and the single-polarizer configuration (characteristic of recent experiments to measure natural optical activity by light reflection). Good agreement between observed results not only provides a test of the theory but accounts for recently observed anomalies that are not explicable on the basis of the standard description of the photoelastic modulator.

© 1990 Optical Society of America

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References

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  1. M. Billardon, J. Badoz, “Modulateur de birefringence,” C. R. 262B, 1672–1675 (1966).
  2. An excellent summary of the characteristics and applications of the photoelastic modulator is provided in the application notes of Hinds International, Inc., Portland, Oregon.See, for example, the note of J. C. Kemp, R. L. Jacob, “The photoelastic birefringent modulator: a versatile tool for modulation of light, strain birefringence, and dichroism measurements,” therein.
  3. J. C. Kemp, “Piezo-optical birefringence modulators: new use for a long-known effect,” J. Opt. Soc. Am. 59, 950–954 (1969).
  4. S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
    [CrossRef]
  5. M. P. Silverman, T. C. Black, “Experimental method to detect chiral asymmetry in specular light scattering from a naturally optically active medium,” Phys. Lett. A 126, 171–176 (1987).
    [CrossRef]
  6. M. P. Silverman, N. Ritchie, G. M. Cushman, B. Fisher, “Experimental configurations using optical phase modulation to measure chiral asymmetries in light specularly reflected from a naturally gyrotropic medium,” J. Opt. Soc. Am. 5, 1852–1862 (1988).
    [CrossRef]
  7. M. P. Silverman, “Specular light scattering from a chiral medium: unambiguous test of gyrotropic constitutive relations,” Lett. Nuovo Cimento 43, 378–382 (1985).
    [CrossRef]
  8. M. P. Silverman, “Reflection and refraction at the surface of a chiral medium,” J. Opt. Soc. Am. A 3, 830–837 (1986).
    [CrossRef]
  9. I. J. Lalov, A. I. Miteva, “Reflection optical activity of uniaxial media,” J. Chem. Phys. 85, 5505–5511 (1986).
    [CrossRef]
  10. J. Badoz, M. P. Silverman, J. C. Canit, “A new model of the photoelastic modulator,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166 (to be published).
  11. J. Badoz, M. Billardon, J. C. Canit, M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light beam using a birefringence modulator,” J. Opt. (Paris) 8, 373–384 (1977).
    [CrossRef]
  12. Y. Shindo, M. Nakagawa, “Circular dichroism measurements. I. Calibration of a circular dichroism spectrometer,” Rev. Sci. Istrum. 56, 32–39 (1985).
    [CrossRef]
  13. Y. Shindo, M. Nakagawa, Y. Ohmi, “On the problems of CD spectropolarimeters. II: Artifacts in CD spectrometers,” Appl. Spectrosc. 39, 860–868 (1985).
    [CrossRef]
  14. L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon, London, 1958), p. 191.
  15. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 360.
  16. O. Acher, E. Bigan, B. Drévillon, “Improvements of phasemodulated ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
    [CrossRef]
  17. M. P. Silverman, J. Badoz, “Light reflection from a naturally optically active birefringent medium and implications for spectroscopy of chiral materials,” submitted to J. Opt. Soc. Am. A.
  18. M. P. Silverman, J. Badoz, “Large enhancement of chiral asymmetry in light reflection near critical angle,” Opt. Commun. (to be published).
  19. M. P. Silverman, J. Badoz, “Differential amplification of circularly polarized light by enhanced internal reflection from an active chiral medium,” Opt. Commun. (to be published).
  20. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 67, 148.
  21. R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974), pp. 47–52.
  22. R. C. O’Handley, “Modified Jones calculus for the analysis of errors in polarization-modulation ellipsometry,” J. Opt. Soc. Am. 63, 523–528 (1973).
    [CrossRef]

1989 (1)

O. Acher, E. Bigan, B. Drévillon, “Improvements of phasemodulated ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
[CrossRef]

1988 (1)

M. P. Silverman, N. Ritchie, G. M. Cushman, B. Fisher, “Experimental configurations using optical phase modulation to measure chiral asymmetries in light specularly reflected from a naturally gyrotropic medium,” J. Opt. Soc. Am. 5, 1852–1862 (1988).
[CrossRef]

1987 (1)

M. P. Silverman, T. C. Black, “Experimental method to detect chiral asymmetry in specular light scattering from a naturally optically active medium,” Phys. Lett. A 126, 171–176 (1987).
[CrossRef]

1986 (2)

M. P. Silverman, “Reflection and refraction at the surface of a chiral medium,” J. Opt. Soc. Am. A 3, 830–837 (1986).
[CrossRef]

I. J. Lalov, A. I. Miteva, “Reflection optical activity of uniaxial media,” J. Chem. Phys. 85, 5505–5511 (1986).
[CrossRef]

1985 (3)

M. P. Silverman, “Specular light scattering from a chiral medium: unambiguous test of gyrotropic constitutive relations,” Lett. Nuovo Cimento 43, 378–382 (1985).
[CrossRef]

Y. Shindo, M. Nakagawa, “Circular dichroism measurements. I. Calibration of a circular dichroism spectrometer,” Rev. Sci. Istrum. 56, 32–39 (1985).
[CrossRef]

Y. Shindo, M. Nakagawa, Y. Ohmi, “On the problems of CD spectropolarimeters. II: Artifacts in CD spectrometers,” Appl. Spectrosc. 39, 860–868 (1985).
[CrossRef]

1977 (1)

J. Badoz, M. Billardon, J. C. Canit, M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light beam using a birefringence modulator,” J. Opt. (Paris) 8, 373–384 (1977).
[CrossRef]

1973 (1)

1969 (2)

J. C. Kemp, “Piezo-optical birefringence modulators: new use for a long-known effect,” J. Opt. Soc. Am. 59, 950–954 (1969).

S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
[CrossRef]

1966 (1)

M. Billardon, J. Badoz, “Modulateur de birefringence,” C. R. 262B, 1672–1675 (1966).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 360.

Acher, O.

O. Acher, E. Bigan, B. Drévillon, “Improvements of phasemodulated ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 67, 148.

Badoz, J.

J. Badoz, M. Billardon, J. C. Canit, M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light beam using a birefringence modulator,” J. Opt. (Paris) 8, 373–384 (1977).
[CrossRef]

M. Billardon, J. Badoz, “Modulateur de birefringence,” C. R. 262B, 1672–1675 (1966).

J. Badoz, M. P. Silverman, J. C. Canit, “A new model of the photoelastic modulator,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166 (to be published).

M. P. Silverman, J. Badoz, “Light reflection from a naturally optically active birefringent medium and implications for spectroscopy of chiral materials,” submitted to J. Opt. Soc. Am. A.

M. P. Silverman, J. Badoz, “Large enhancement of chiral asymmetry in light reflection near critical angle,” Opt. Commun. (to be published).

M. P. Silverman, J. Badoz, “Differential amplification of circularly polarized light by enhanced internal reflection from an active chiral medium,” Opt. Commun. (to be published).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 67, 148.

Bigan, E.

O. Acher, E. Bigan, B. Drévillon, “Improvements of phasemodulated ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
[CrossRef]

Billardon, M.

J. Badoz, M. Billardon, J. C. Canit, M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light beam using a birefringence modulator,” J. Opt. (Paris) 8, 373–384 (1977).
[CrossRef]

M. Billardon, J. Badoz, “Modulateur de birefringence,” C. R. 262B, 1672–1675 (1966).

Black, T. C.

M. P. Silverman, T. C. Black, “Experimental method to detect chiral asymmetry in specular light scattering from a naturally optically active medium,” Phys. Lett. A 126, 171–176 (1987).
[CrossRef]

Canit, J. C.

J. Badoz, M. Billardon, J. C. Canit, M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light beam using a birefringence modulator,” J. Opt. (Paris) 8, 373–384 (1977).
[CrossRef]

J. Badoz, M. P. Silverman, J. C. Canit, “A new model of the photoelastic modulator,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166 (to be published).

Cushman, G. M.

M. P. Silverman, N. Ritchie, G. M. Cushman, B. Fisher, “Experimental configurations using optical phase modulation to measure chiral asymmetries in light specularly reflected from a naturally gyrotropic medium,” J. Opt. Soc. Am. 5, 1852–1862 (1988).
[CrossRef]

Drévillon, B.

O. Acher, E. Bigan, B. Drévillon, “Improvements of phasemodulated ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
[CrossRef]

Fisher, B.

M. P. Silverman, N. Ritchie, G. M. Cushman, B. Fisher, “Experimental configurations using optical phase modulation to measure chiral asymmetries in light specularly reflected from a naturally gyrotropic medium,” J. Opt. Soc. Am. 5, 1852–1862 (1988).
[CrossRef]

Gilmore, R.

R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974), pp. 47–52.

Jacob, R. L.

An excellent summary of the characteristics and applications of the photoelastic modulator is provided in the application notes of Hinds International, Inc., Portland, Oregon.See, for example, the note of J. C. Kemp, R. L. Jacob, “The photoelastic birefringent modulator: a versatile tool for modulation of light, strain birefringence, and dichroism measurements,” therein.

Jasperson, S. N.

S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
[CrossRef]

Kemp, J. C.

J. C. Kemp, “Piezo-optical birefringence modulators: new use for a long-known effect,” J. Opt. Soc. Am. 59, 950–954 (1969).

An excellent summary of the characteristics and applications of the photoelastic modulator is provided in the application notes of Hinds International, Inc., Portland, Oregon.See, for example, the note of J. C. Kemp, R. L. Jacob, “The photoelastic birefringent modulator: a versatile tool for modulation of light, strain birefringence, and dichroism measurements,” therein.

Lalov, I. J.

I. J. Lalov, A. I. Miteva, “Reflection optical activity of uniaxial media,” J. Chem. Phys. 85, 5505–5511 (1986).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon, London, 1958), p. 191.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon, London, 1958), p. 191.

Miteva, A. I.

I. J. Lalov, A. I. Miteva, “Reflection optical activity of uniaxial media,” J. Chem. Phys. 85, 5505–5511 (1986).
[CrossRef]

Nakagawa, M.

Y. Shindo, M. Nakagawa, Y. Ohmi, “On the problems of CD spectropolarimeters. II: Artifacts in CD spectrometers,” Appl. Spectrosc. 39, 860–868 (1985).
[CrossRef]

Y. Shindo, M. Nakagawa, “Circular dichroism measurements. I. Calibration of a circular dichroism spectrometer,” Rev. Sci. Istrum. 56, 32–39 (1985).
[CrossRef]

O’Handley, R. C.

Ohmi, Y.

Ritchie, N.

M. P. Silverman, N. Ritchie, G. M. Cushman, B. Fisher, “Experimental configurations using optical phase modulation to measure chiral asymmetries in light specularly reflected from a naturally gyrotropic medium,” J. Opt. Soc. Am. 5, 1852–1862 (1988).
[CrossRef]

Russel, M. F.

J. Badoz, M. Billardon, J. C. Canit, M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light beam using a birefringence modulator,” J. Opt. (Paris) 8, 373–384 (1977).
[CrossRef]

Schnatterly, S. E.

S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
[CrossRef]

Shindo, Y.

Y. Shindo, M. Nakagawa, “Circular dichroism measurements. I. Calibration of a circular dichroism spectrometer,” Rev. Sci. Istrum. 56, 32–39 (1985).
[CrossRef]

Y. Shindo, M. Nakagawa, Y. Ohmi, “On the problems of CD spectropolarimeters. II: Artifacts in CD spectrometers,” Appl. Spectrosc. 39, 860–868 (1985).
[CrossRef]

Silverman, M. P.

M. P. Silverman, N. Ritchie, G. M. Cushman, B. Fisher, “Experimental configurations using optical phase modulation to measure chiral asymmetries in light specularly reflected from a naturally gyrotropic medium,” J. Opt. Soc. Am. 5, 1852–1862 (1988).
[CrossRef]

M. P. Silverman, T. C. Black, “Experimental method to detect chiral asymmetry in specular light scattering from a naturally optically active medium,” Phys. Lett. A 126, 171–176 (1987).
[CrossRef]

M. P. Silverman, “Reflection and refraction at the surface of a chiral medium,” J. Opt. Soc. Am. A 3, 830–837 (1986).
[CrossRef]

M. P. Silverman, “Specular light scattering from a chiral medium: unambiguous test of gyrotropic constitutive relations,” Lett. Nuovo Cimento 43, 378–382 (1985).
[CrossRef]

J. Badoz, M. P. Silverman, J. C. Canit, “A new model of the photoelastic modulator,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166 (to be published).

M. P. Silverman, J. Badoz, “Light reflection from a naturally optically active birefringent medium and implications for spectroscopy of chiral materials,” submitted to J. Opt. Soc. Am. A.

M. P. Silverman, J. Badoz, “Large enhancement of chiral asymmetry in light reflection near critical angle,” Opt. Commun. (to be published).

M. P. Silverman, J. Badoz, “Differential amplification of circularly polarized light by enhanced internal reflection from an active chiral medium,” Opt. Commun. (to be published).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 360.

Appl. Spectrosc. (1)

C. R. (1)

M. Billardon, J. Badoz, “Modulateur de birefringence,” C. R. 262B, 1672–1675 (1966).

J. Chem. Phys. (1)

I. J. Lalov, A. I. Miteva, “Reflection optical activity of uniaxial media,” J. Chem. Phys. 85, 5505–5511 (1986).
[CrossRef]

J. Opt. (Paris) (1)

J. Badoz, M. Billardon, J. C. Canit, M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light beam using a birefringence modulator,” J. Opt. (Paris) 8, 373–384 (1977).
[CrossRef]

J. Opt. Soc. Am. (3)

J. C. Kemp, “Piezo-optical birefringence modulators: new use for a long-known effect,” J. Opt. Soc. Am. 59, 950–954 (1969).

M. P. Silverman, N. Ritchie, G. M. Cushman, B. Fisher, “Experimental configurations using optical phase modulation to measure chiral asymmetries in light specularly reflected from a naturally gyrotropic medium,” J. Opt. Soc. Am. 5, 1852–1862 (1988).
[CrossRef]

R. C. O’Handley, “Modified Jones calculus for the analysis of errors in polarization-modulation ellipsometry,” J. Opt. Soc. Am. 63, 523–528 (1973).
[CrossRef]

J. Opt. Soc. Am. A (1)

Lett. Nuovo Cimento (1)

M. P. Silverman, “Specular light scattering from a chiral medium: unambiguous test of gyrotropic constitutive relations,” Lett. Nuovo Cimento 43, 378–382 (1985).
[CrossRef]

Phys. Lett. A (1)

M. P. Silverman, T. C. Black, “Experimental method to detect chiral asymmetry in specular light scattering from a naturally optically active medium,” Phys. Lett. A 126, 171–176 (1987).
[CrossRef]

Rev. Sci. Instrum. (2)

S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
[CrossRef]

O. Acher, E. Bigan, B. Drévillon, “Improvements of phasemodulated ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
[CrossRef]

Rev. Sci. Istrum. (1)

Y. Shindo, M. Nakagawa, “Circular dichroism measurements. I. Calibration of a circular dichroism spectrometer,” Rev. Sci. Istrum. 56, 32–39 (1985).
[CrossRef]

Other (9)

J. Badoz, M. P. Silverman, J. C. Canit, “A new model of the photoelastic modulator,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166 (to be published).

M. P. Silverman, J. Badoz, “Light reflection from a naturally optically active birefringent medium and implications for spectroscopy of chiral materials,” submitted to J. Opt. Soc. Am. A.

M. P. Silverman, J. Badoz, “Large enhancement of chiral asymmetry in light reflection near critical angle,” Opt. Commun. (to be published).

M. P. Silverman, J. Badoz, “Differential amplification of circularly polarized light by enhanced internal reflection from an active chiral medium,” Opt. Commun. (to be published).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 67, 148.

R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974), pp. 47–52.

An excellent summary of the characteristics and applications of the photoelastic modulator is provided in the application notes of Hinds International, Inc., Portland, Oregon.See, for example, the note of J. C. Kemp, R. L. Jacob, “The photoelastic birefringent modulator: a versatile tool for modulation of light, strain birefringence, and dichroism measurements,” therein.

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon, London, 1958), p. 191.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 360.

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the light-transmission and -reflection configurations employing the photoelastic modulator. The principal axes of dynamic and static birefringence are along x1 and x2 respectively, at angles w and w + γ to the vertical. The transmission axes of the first and second polarizers are at angles u and υ to the vertical. For the measurement of differential circular reflection a second polarizer is not employed. (The process of reflection itself serves as the analysing polarizer.) For a light-transmission configuration there is no reflecting sample, and the light beam (designated by wave vector k) is undeviated.

Fig. 2
Fig. 2

Schematic diagram of the apparatus for examining the PEM between two polarizers. Light from a Xe lamp (S), filtered by a monochromater (M), passes through the incident polarizer (P1), the PEM, and an analysing polarizer (P2) and is synchronously detected by means of a photodetector (D) and a lock-in amplifier (LA). The amplifier output at the modulation frequency f = 50 kHz and first harmonic 2f is registered on a galvanometer (G).

Fig. 3
Fig. 3

Variation of I(f)/I(0) with modulation amplitude ϕm as determined A, experimentally by means of light reflection from an achiral glass prism in configuration Al of Ref. 6 (equivalent to the π/4-polarizer configuration); B, by computer Fourier analysis of (exact) Eq. (11a); and C, by analytical approximation of relation (16b). The index of refraction is 1.466, and the angle of reflection is 70°. D, The comparable variation of J1(ϕm) is also shown (ϕm is expressed in degrees).

Fig. 4
Fig. 4

Variation with modulation amplitude ϕm of I(f)/I(2f) for the (π/2)-polarizer configuration as determined by A, experiment and B, computer Fourier analysis of Eq. (17a) (ϕm is expressed in degrees).

Fig. 5
Fig. 5

Experimental variation (points designated by open circles) of I(f)/I(0) with angle of incidence for configuration Al of Ref. 6 corresponding to the measurement of differential circular reflection. The reflecting sample is an achiral prism of index n ≃ 1.466 (Brewster angle θB ≃ 55.7°). The modulation amplitude ϕm is 220°. The solid curve shows the theoretical differential linear reflection (normalized to maximum value at θB) (θ is expressed in degrees).

Equations (72)

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

ϕ 1 ( t ) = ϕ m sin ( 2 π f t )
tan ϕ 2 ϕ 2 = J 2 ( ϕ m ) J 1 ( ϕ m ) I ( f ) I ( 2 f ) ,
ϕ = ϕ 1 + ϕ 2 .
2 β = [ ( ϕ 1 + ϕ 2 cos 2 γ ) 2 + ( ϕ 2 sin 2 γ ) 2 ] 1 / 2 .
E 0 = [ cos u sin u ] ,
E m = M 5 M 4 M 3 M 2 M 1 E 0 ,
M 1 = R ( w ) = [ cos w sin w sin w cos w ]
M 2 = T ( ϕ 1 ) = [ e i ϕ 1 0 0 1 ]
M 3 = R ( γ )
M 4 = T ( ϕ 2 )
M 5 = R ( Γ )
M 21 = M 4 ( ϕ 2 ) M 3 ( γ ) M 2 ( ϕ 1 )
M 21 ( ϕ , δ , γ ) = e i ϕ [ e i ϕ cos γ e i δ sin γ e i δ sin γ e i ϕ cos γ ] ,
ϕ = ( ϕ 1 + ϕ 2 ) / 2 ,
δ = ( ϕ 1 ϕ 2 ) / 2 ,
M 12 ( ϕ , δ , γ ) = M 2 ( ϕ 1 ) M 3 ( γ ) M 4 ( ϕ 2 ) = M 21 ( ϕ , δ , γ ) ,
[ R ( γ ) T ( ϕ 2 ) R ( γ ) , T ( ϕ 1 ) ] = 0
M 21 = R ( γ ) + idG ,
d G = [ ( d ϕ ) cos γ ( d δ ) sin γ ( d δ ) sin γ ( d ϕ ) cos γ ] .
E 0 + d E = [ 1 + i R ( Γ ) d G R ( w ) ] E 0 ,
σ i σ j = 1 δ i j + i k = 1 3 ijk σ k ,
σ 1 = [ 0 1 1 0 ] ,
σ 2 = [ 0 i i 0 ] ,
σ 3 = [ 1 0 0 1 ] .
E ( β + d β ) = ( 1 + i d β · σ ) E ( β ) ,
d β = ( d β 1 , d β 2 , d β 3 )
d β 1 = ( d ϕ 1 sin 2 w + d ϕ 2 sin 2 Γ ) / 2 ,
d β 2 = 0 ,
d β 3 = ( d ϕ 1 cos 2 w + d ϕ 2 cos 2 Γ ) / 2
σ = ( σ 1 , σ 2 , σ 3 ) .
E m ( β ) = ( 1 cos β + i β · σ β sin β ) E 0 ,
β = ( β 1 2 + β 2 2 + β 3 2 ) 1 / 2 = ( ϕ 1 2 + ϕ 2 2 + 2 ϕ 1 ϕ 2 cos 2 γ ) 1 / 2 .
E m = ( cos β cos u + i sin β 2 β { [ ϕ 1 cos ( 2 w ) + ϕ 2 cos ( 2 Γ ) ] cos u + [ ϕ 1 sin ( 2 w ) + ϕ 2 sin ( 2 Γ ) ] sin u } cos β sin u + i sin β 2 β { [ ϕ 1 cos ( 2 w ) + ϕ 2 cos ( 2 Γ ) ] sin u + [ ϕ 1 sin ( 2 w ) + ϕ 2 sin ( 2 Γ ) ] cos u } ) .
I m = | E m | 2 = 1 ,
I d = cos 2 ( u υ ) 1 + cos ( 2 β ) 2 + { 1 + A cos [ 2 ( u + υ ) ] + B sin [ 2 ( u + υ ) ] } 1 cos ( 2 β ) 2 ,
A = ( ϕ 1 + ϕ 2 cos 2 γ ) 2 ( ϕ 2 sin 2 γ ) 2 ( ϕ 1 + ϕ 2 cos 2 γ ) 2 + ( ϕ 2 sin 2 γ ) 2 cos 4 w 2 ( ϕ 2 sin 2 γ ) ( ϕ 1 + ϕ 2 cos 2 γ ) ( ϕ 1 + ϕ 2 cos 2 γ ) 2 + ( ϕ 2 sin 2 γ ) 2 sin 4 w ,
B = 2 ( ϕ 2 sin 2 γ ) ( ϕ 1 + ϕ 2 cos 2 γ ) ( ϕ 1 + ϕ 2 cos 2 γ ) 2 + ( ϕ 2 sin 2 γ ) 2 cos 4 w + ( ϕ 1 + ϕ 2 cos 2 γ ) 2 ( ϕ 2 sin 2 γ ) 2 ( ϕ 1 + ϕ 2 cos 2 γ ) 2 + ( ϕ 2 sin 2 γ ) 2 sin 4 w
A cos 4 w 2 ( ϕ 2 sin 2 γ ) ( ϕ 1 + ϕ 2 cos 2 γ ) sin 4 w ,
B sin 4 w + 2 ( ϕ 2 sin 2 γ ) ( ϕ 1 + ϕ 2 cos 2 γ ) cos 4 w ,
I d ½ { 1 + cos [ 2 ( w u ) ] cos [ 2 ( w υ ) ] + sin [ 2 ( w υ ) ] sin [ 2 ( w υ ) ] cos 2 β + ( ϕ 2 sin 2 γ ) × sin [ 2 ( u + υ 2 w ) ] 1 cos 2 β 2 β } .
2 β = ϕ m [ ½ b 0 + a 1 sin ( 2 π f t ) + b 1 cos ( 2 π f t ) + a 2 sin ( 4 π f t ) + b 2 cos ( 4 π f t ) + ]
½ b 0 ϕ m ϕ 2 cos 2 γ ,
a 1 ϕ m 1 ,
b 2 ϕ m 2 ( ϕ 2 sin 2 γ ) 2 / π ϕ m .
2 β ϕ 1 + ϕ 2 cos 2 γ ,
cos 2 β J 0 ( ϕ m ) cos ϕ 2 2 J 1 ( ϕ m ) ( sin ϕ 2 ) sin ( 2 π f t ) + 2 J 2 ( ϕ m ) cos ϕ 2 cos ( 4 π f t ) ,
1 cos 2 β 2 β 2 [ 1 J 0 ( ϕ m ) ϕ m ] sin ( 2 π f t ) ,
ϕ 2 = ϕ 2 cos 2 γ
I ( t ) = I ( 0 ) + I ( f ) sin ( 2 π f t ) + I ( 2 f ) cos ( 4 π f t ) +
I ( 0 ) 1 + cos [ 2 ( w u ) ] cos [ 2 ( w υ ) ] + J 0 ( ϕ m ) cos ϕ 2 sin [ 2 ( w υ ) ] sin [ 2 ( w υ ) ] ,
I ( f ) 2 [ 1 J 0 ( ϕ m ) ϕ m ] ( ϕ 2 sin 2 γ sin [ 2 ( u + υ 2 w ) ] J 1 ( ϕ m ) sin ϕ 2 sin [ 2 ( w u ) ] sin [ 2 ( w υ ) ] ,
I ( 2 f ) 2 J 2 ( ϕ m ) cos ( ϕ 2 cos 2 γ ) sin [ 2 ( w u ) ] × sin [ 2 ( w υ ) ] .
I d ½ [ 1 ( ϕ 2 sin 2 γ ) ( 1 cos 2 β 2 β ) ] ,
I ( f ) / I ( 0 ) 2 ( ϕ 2 sin 2 γ ) [ 1 J 0 ( ϕ m ) ϕ m ] .
I d = ½ ( 1 cos 2 β ) ,
I ( f ) / I ( 2 f ) J 1 ( ϕ m ) J 2 ( ϕ m ) tan ( ϕ 2 cos 2 γ ) ,
I ( f ) I ( 2 f ) = Q 11 Q 12 tan ϕ 2 Q 21 Q 22 tan ϕ 2 ,
Q 11 = J 1 ( a 1 ϕ m ) J 1 ( b 2 ϕ m ) J 3 ( a 1 ϕ m ) J 1 ( b 2 ϕ m ) ,
Q 12 = J 1 ( a 1 ϕ m ) J 0 ( b 2 ϕ m ) + J 3 ( a 1 ϕ m ) J 2 ( b 2 ϕ m ) ,
Q 21 = J 2 ( a 1 ϕ m ) J 0 ( b 2 ϕ m ) J 2 ( a 1 ϕ m ) J 2 ( b 2 ϕ m ) ,
Q 22 = J 0 ( a 1 ϕ m ) J 1 ( b 2 ϕ m ) J 4 ( a 1 ϕ m ) J 3 ( b 2 ϕ m ) ,
J 2 ( ϕ m ) I ( f ) J 1 ( ϕ m ) I ( 2 f ) ϕ 2 cos 2 γ + J 1 ( ϕ m ) J 3 ( ϕ m ) J 1 ( ϕ m ) ( ϕ 2 sin 2 γ ) 2 π ϕ m ,
E r = [ a 1 b 1 a 2 b 2 ] E m ,
I d = I 0 ( 1 + σ 2 ( 1 + cos 2 β ) cos u + σ 2 { ϕ 1 2 cos 4 w + ϕ 2 2 cos 4 Γ + 2 ϕ 1 ϕ 2 cos [ 2 ( Γ + w ) ] } 1 cos β 4 β 2 cos 2 u + σ 2 { ϕ 1 2 sin 4 w + ϕ 2 2 sin 4 Γ + 2 ϕ 1 ϕ 2 × sin [ 2 ( Γ + w ) ] } 1 cos β 4 β 2 sin 2 u + ρ { ϕ 1 sin [ 2 ( u w ) ] + ϕ 2 sin [ 2 ( u Γ ) ] } sin 2 β 2 β ) ,
I 0 = a 1 2 + b 2 2 2
σ = I TE I TM I TE + I TM = a 1 2 b 2 2 a 1 2 + b 2 2
ρ = I RCP I LCP I RCP + I LCP = a 1 α 2 b 2 β 1 a 1 2 + b 2 2
I d = I 0 [ 1 + σ ( ϕ 2 sin 2 γ )( ϕ 1 + ϕ 2 cos 2 γ ) 1 cos 2 β 4 β 2 + ρ ( ϕ 1 + ϕ 2 cos 2 γ ) sin 2 β 2 β ] ,
I d = I 0 [ 1 + σ ( ϕ 2 sin 2 γ ) 1 cos 2 β 2 β + ρ sin 2 β ] .
I ( f ) / I ( 0 ) = 2 σ ( ϕ 2 sin 2 γ ) 1 J 0 ( ϕ m ) ϕ m 2 ρ J 1 ( ϕ m ) cos ϕ 2 ,
I ( 2 f ) / I ( 0 ) = 2 ρ J 2 ( ϕ m ) sin ϕ 2 .
I d θ = I 0 [ 1 + ρ ( ϕ 1 sin δ 1 + ϕ 2 sin δ 2 ) sin 2 β 2 β ] ,

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