Abstract

Using Fresnel reflection amplitudes, the Jones and Mueller matrices for reflection from a nonabsorbing gyrotropic medium are presented. Some basic chiral parameters are defined by using the elements of the Mueller matrix; experimental configurations are described.

© 1991 Optical Society of America

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References

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  1. M. P. Silverman, “Reflection and refraction at the surface of a chiral medium: comparison of gyrotropic constitutive relations invariant or noninvariant under a duality transformation,” J. Opt. Soc. Am. A 3, 830–837 (1986).
    [CrossRef]
  2. A. Lakhtakia, V. V. Varadan, V. K. Varadan, “Field equations, Huygens's principle, integral equations, and theorems for radiation and scattering of electromagnetic waves in isotropic chiral media,” J. Opt. Soc. Am. A 5, 175–183 (1988).
    [CrossRef]
  3. S. Bassiri, C. Papas, N. Engheta, “Electromagnetic wave propagation through a dielectric chiral interface and through a chiral slab,” J. Opt. Soc. Am. A 5,1450–1459 (1988).
    [CrossRef]
  4. M. P. Silverman, “Effect of circular birefringence on light propagation and reflection,” Am. J. Phys. 54, 69–76 (1986).
    [CrossRef]
  5. M. P. Silverman, “Specular light scattering from a chiral medium: unambiguous test of gyrotropic constitutive relations,” Lett. Nuovo Cimento 43, 378–382 (1985).
    [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. A 5, 1852–1862 (1988).
    [CrossRef]
  7. I. Lalov, “Effects of vibrational optical activity in the reflection spectra of crystals for the frequency regions of nondegenerate vibrations,” Phys. Rev. B 32(10), 6884–6891 (1985).
    [CrossRef]
  8. B. V. Bokut, F. I. Fedorov, “Reflection and refraction of light in optically isotropic active media, Opt. Spectrosk. (USSR) 9, 334–336 (1960).
  9. W. R. Hunter, Effects of component imperfections on ellipsom eter calibration, J. Opt. Soc. Am. 63, 951 (1973).
    [CrossRef]
  10. See, for example, A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).
  11. 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]
  12. 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]

1988 (3)

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)

1985 (2)

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

I. Lalov, “Effects of vibrational optical activity in the reflection spectra of crystals for the frequency regions of nondegenerate vibrations,” Phys. Rev. B 32(10), 6884–6891 (1985).
[CrossRef]

1973 (1)

1969 (1)

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]

1960 (1)

B. V. Bokut, F. I. Fedorov, “Reflection and refraction of light in optically isotropic active media, Opt. Spectrosk. (USSR) 9, 334–336 (1960).

Bassiri, S.

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]

Bokut, B. V.

B. V. Bokut, F. I. Fedorov, “Reflection and refraction of light in optically isotropic active media, Opt. Spectrosk. (USSR) 9, 334–336 (1960).

Burch, J. M.

See, for example, A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

Cushman, G. M.

Engheta, N.

Fedorov, F. I.

B. V. Bokut, F. I. Fedorov, “Reflection and refraction of light in optically isotropic active media, Opt. Spectrosk. (USSR) 9, 334–336 (1960).

Fisher, B.

Gerrard, A.

See, for example, A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

Hunter, W. R.

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]

Lakhtakia, A.

Lalov, I.

I. Lalov, “Effects of vibrational optical activity in the reflection spectra of crystals for the frequency regions of nondegenerate vibrations,” Phys. Rev. B 32(10), 6884–6891 (1985).
[CrossRef]

Papas, C.

Ritchie, N.

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]

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. A 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: comparison of gyrotropic constitutive relations invariant or noninvariant under a duality transformation,” J. Opt. Soc. Am. A 3, 830–837 (1986).
[CrossRef]

M. P. Silverman, “Effect of circular birefringence on light propagation and reflection,” Am. J. Phys. 54, 69–76 (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]

Varadan, V. K.

Varadan, V. V.

Am. J. Phys. (1)

M. P. Silverman, “Effect of circular birefringence on light propagation and reflection,” Am. J. Phys. 54, 69–76 (1986).
[CrossRef]

J. Opt. Soc. Am. (1)

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

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]

Opt. Spectrosk. (USSR) (1)

B. V. Bokut, F. I. Fedorov, “Reflection and refraction of light in optically isotropic active media, Opt. Spectrosk. (USSR) 9, 334–336 (1960).

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]

Phys. Rev. B (1)

I. Lalov, “Effects of vibrational optical activity in the reflection spectra of crystals for the frequency regions of nondegenerate vibrations,” Phys. Rev. B 32(10), 6884–6891 (1985).
[CrossRef]

Rev. Sci. Instrum. (1)

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]

Other (1)

See, for example, A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

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Figures (5)

Fig. 1
Fig. 1

(a) Amplitude E of the incident light is resolved into Es (perpendicular to the plane of incidence) and Ep (parallel to the plane of incidence) components, (b) Incident light Es gives rise to the reflected light E″ss = rssEss and E″ps = rpsEs. (c) Incident light Ep generates the reflected light E″sp = rspEp and E″pp = rppEp.

Fig. 2
Fig. 2

Configuration I: X is perpendicular to the plane of incidence, Y is parallel to the plane of incidence, the light is along the Z axis, P is at 45° with respect to X, and the modulator fast axis is at 90° with respect to X.

Fig. 3
Fig. 3

Configuration II: X is perpendicular to the plane of incidence, Y is parallel to the plane of incidence, the light is along the Z axis, P is at 45° with respect to X, the modulator fast axis is at 90° with respect to X, and A is at 45° with respect to X.

Fig. 4
Fig. 4

Configuration III: X is perpendicular to the plane of incidence, Y is parallel to the plane of incidence, the light is along the Z axis, P is at 45° with respect to X, the modulator fast axis is at 90° with respect to X, the fast axis of the λ/4 plate is at 90° with respect to X, and A is at 45° with respect to X.

Fig. 5
Fig. 5

Configuration IV: X is perpendicular to the plane of incidence, Y is parallel to the plane of incidence, the light is along the Z axis, P is at 0° with respect to X, the modulator fast axis is at 45° with respect to X.

Equations (27)

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D = ( E + β × E ) , B = μ ( H + β × H ) ,
E s = E s s + E s p = r s s E s + r s p E p , E p = E p p + E p s = r p s E s + r p p E p .
[ E s E p ] = [ r s s r s p r p s r p p ] [ E s E p ] = [ r ssr + i r ssi r spr + i r spi r psr + i r psi r ppr + i r ppi ] [ E s E p ] ,
r ssr a 1 , r spr a 2 , r psr b 1 , r ppr b 2 , r ssi α 1 , r spi α 2 , r psi β 1 , r ppi β 2 .
M 11 = ( r ssr 2 + r ssi 2 + r psr 2 + r psi 2 + r spr 2 + r spi 2 + r ppr 2 + r ppi 2 ) / 2 , M 12 = ( r ssr 2 + r ssi 2 + r psr 2 + r psi 2 r spr 2 r spi 2 r ppr 2 r ppi 2 ) / 2 , M 13 = r ssr r spr + r ssi r spi + r psr r ppr + r psi r ppi , M 14 = r spr r ssi + r ppr r psi r ssr r spi r psr r ppi , M 21 = ( r ssr 2 + r ssi 2 + r spr 2 + r spi 2 r psr 2 r psi 2 r ppr 2 r ppi 2 ) / 2 , M 22 = ( r ssr 2 + r ssi 2 + r ppr 2 + r ppi 2 r psr 2 r psi 2 r spr 2 r spi 2 ) / 2 , M 23 = r spr r ssr + r spi r ssi r ppr r psr r ppi r psi , M 24 = r ssr r spi + r ssi r spr r ppr r psi + r ppi r psr , M 31 = r ssr r psr + r ssi r psi + r spr r ppr + r spi r ppi , M 32 = r ssr r psr + r ssi r psi r spr r ppr r spi r ppi , M 33 = r ssr r ppr + r ssi r ppi + r psr r spr + r psi r spi , M 34 = r ssr r ppi + r ssi r ppr r psr r spi + r spr r psi , M 41 = r ssr r psi + r spr r ppi r psr r ssi r ppr r spi , M 42 = r ssr r psi + r ppr r spi r psr r ssi r spr r ppi , M 43 = r spr r psi + r ssr r ppi r psr r spi r ppr r ssi , M 44 = r ssr r ppr + r ppi r ssi r spr r psr r spi r psi .
d c = Im 2 ( E s p + E p p ) ( E s s + E p s ) * | E s s + E p s | 2 + | E s p + E p p | 2 = I L I R I R + I L = 2 r ssr r spi + r psr r ppi r spr r ssi r ppr r psi r ssr 2 + r ssi 2 + r psr 2 + r psi 2 + r spr 2 + r spi 2 + r ppr 2 + r ppi 2 = M 14 M 11 .
s c = Re 2 ( E s p + E p p ) ( E s s + E p s ) * | E s s + E p s | 2 + | E s p + E p p | 2 = 2 r ssr r spr + r ssi r spi + r psr r ppr + r psi r ppi r ssr 2 + r ssi 2 + r psr 2 + r psi 2 + r spr 2 + r spi 2 + r ppr 2 + r ppi 2 = M 13 M 11 .
d l = I s I p I s + I p = | E s s + E p s | 2 | E s p + E p p | 2 | E s s + E p s | 2 + | E s p + E p p | 2 = | r s s | 2 + | r p s | 2 ( | r s p | 2 + | r p p | 2 ) | r s s | 2 + | r p s | 2 + | r s p | 2 + | r p p | 2 = ( r ssr 2 + r ssi 2 + r psr 2 + r psi 2 ) ( r spr 2 + r spi 2 + r ppr 2 + r ppi 2 ) r ssr 2 + r ssi 2 + r psr 2 + r psi 2 + r spr 2 + r spi 2 + r ppr 2 + r ppi 2 = M 12 M 11 .
[ M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 ] [ 1 0 0 0 0 1 0 0 0 0 cos δ sin δ 0 0 sin δ cos δ ] [ 1 0 1 0 ] ,
M 11 + M 13 cos δ + M 14 sin δ .
sin δ ( t ) = sin ( A sin w t ) = 2 J 1 ( A ) sin w t + 2 J 3 ( A ) sin 3 w t + 2 J 5 ( A ) sin 5 w t + , cos δ ( t ) = cos ( A sin w t ) = J 0 ( A ) + 2 J 2 ( A ) cos 2 w t + 2 J 4 ( A ) cos 4 w t + ,
M 11 + 2 M 13 J 2 ( A ) cos 2 w t + 2 J 1 ( A ) M 14 sin w t ,
I 1 = M 11 + 2 M 13 J 2 ( A ) cos 2 w t + 2 M 14 J 1 ( A ) sin w t , U 1 = p 1 [ M 11 + 2 M 13 J 2 ( A ) cos 2 w t + 2 M 14 J 1 ( A ) sin w t ] ,
U 1 = C 0 + 2 C 0 ( M 14 / M 11 ) J 1 ( A ) sin w t + 2 C 0 ( M 13 / M 11 ) J 2 ( A ) cos 2 w t ,
| U ( w ) U d υ | I = 2 J 1 ( A ) ( M 14 / M 11 ) = 2 J 1 ( A ) d c ,
| U ( 2 w ) U d υ | I = 2 J 2 ( A ) ( M 13 / M 11 ) = 2 J 2 ( A ) s c .
[ M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 ] [ 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 ] [ 1 0 0 0 0 1 0 0 0 0 cos δ sin δ 0 0 sin δ cos δ ] [ 1 0 1 0 ] .
I 2 = ( M 11 + M 13 ) ( 1 + cos δ ) = ( M 11 + M 13 ) [ 1 + 2 J 2 ( A ) cos 2 w t ] .
U 2 = p 2 ( M 11 + M 13 ) [ 1 + 2 J 2 ( A ) cos 2 w t ] = C 0 + 2 C 0 J 2 ( A ) cos 2 w t ,
| U ( 2 w ) U d υ | I I = 2 J 2 ( A ) .
[ M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 ] [ 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 ] [ 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ] [ 1 0 0 0 0 1 0 0 0 0 cos δ sin δ 0 0 sin δ cos δ ] [ 1 0 1 0 ] ,
I 3 = ( M 11 + M 13 ) ( 1 sin δ ) , U 3 = p 3 ( M 11 + M 13 ) ( 1 sin δ ) = p 3 ( M 11 + M 13 ) p 3 ( M 11 + M 13 ) 2 J 1 ( A ) sin w t = C 0 2 C 0 J 1 ( A ) sin w t
| U ( w ) U d υ | III = 2 J 1 ( A ) .
[ M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 ] [ 1 0 0 0 0 cos δ 0 sin δ 0 0 1 0 0 sin δ 0 cos δ ] [ 1 1 0 0 ] .
I 4 = ( M 11 + M 12 cos δ + M 14 sin δ ) , U 4 = p 4 ( M 11 + 2 J 2 ( A ) M 12 cos 2 w t + 2 J 1 ( A ) M 14 sin w t ) = C 0 + 2 C 0 J 2 ( A ) ( M 12 / M 11 ) cos 2 w t + 2 C 0 J 1 ( A ) ( M 14 / M 11 ) sin w t .
| U ( w ) U d υ | I V = 2 J 1 ( A ) ( M 14 / M 11 ) = 2 J 1 ( A ) d c ,
| U ( 2 w ) U d υ | I V = 2 J 2 ( A ) ( M 12 / M 11 ) = 2 J 2 ( A ) d l .

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