Abstract

Chirally asymmetric responses of a gyrotropic medium to left and right circularly polarized light as manifested in specular light reflection are systematically identified and interpreted. Experimental configurations using one or more photoelastic modulators and synchronous detection are described, by means of which the chiral asymmetries should be measurable with a sensitivity comparable with that of light-transmission techniques. Measurement of these chiral asymmetries would (a) provide experimental tests of recently proposed Fresnel scattering amplitudes for isotropic chiral media and (b) open up new spectroscopic possibilities for the investigation of nontransparent gyrotropic media.

© 1988 Optical Society of America

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References

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  1. Fresnel’s theory of optical rotation and his speculation on chirally inequivalent structures are given in, respectively, A. Fresnel, Ann. Chim. 28, 147 (1825); Bull. Soc. Philomath. (1824); Herschel’s identification of the two hemihedral forms of quartz is reported in J. F. W. Herschel, Trans. Cambridge Philos. Soc. 1, 43 (1822).
  2. L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge, London, 1982), p. xiv.
  3. M. Bouchiat, L. Pottier, “Optical experiments and weak interactions,” Science 234, 1203–1210 (1986).
    [CrossRef] [PubMed]
  4. L. D. Barron, M. P. Bogaard, A. D. Buckingham, “Raman scattering of circularly polarized light by optically active molecules,” J. Am. Chem. Soc. 95, 604–606 (1973).
    [CrossRef]
  5. W. Hug, S. Kint, G. Bailey, J. Scherer, “Raman circular intensity differential spectroscopy,” J. Am. Chem. Soc. 97, 5589–5590 (1975).
    [CrossRef]
  6. M. P. Silverman, “Specular light scattering from a chiral medium,” Lett. Nuovo Cimento 43, 378–382 (1985).
    [CrossRef]
  7. 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]
  8. M. P. Silverman, “Test of gyrotropic constitutive relations by specular light reflection,” Opt. News 11(9), 135 (1985); J. Opt. Soc. Am. A 2(13), P99 (1985).
  9. M. P. Silverman, “Coherent light scattering at the interface of an isotropic chiral medium and homogeneous inactive dielectric,” Bull. Am. Phys. Soc. 30, 798 (1985).
  10. M. P. Silverman, T. C. Black, “Test of the Fresnel relations for a chiral medium,” in Optics and the Information Age (14th Congress of the International Commission for Optics), H. H. Arsenault, ed. Proc. Soc. Photo-Opt. Instrum. Eng.813, 435–436 (1988).
  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. J. C. Canit, J. Badoz, “New design for a photoelastic modulator,” Appl. Opt. 22, 592–594 (1983), and references therein.
    [CrossRef] [PubMed]
  13. J. C. Canit, J. Badoz, “Photoelastic modulator for polarimetry and ellipsometry,” Appl. Opt. 23, 2861–2862 (1984).
    [CrossRef] [PubMed]
  14. See, for example, M. A. Islam, A. Kponou, B. Suleman, W. Happer, “Magnetic circular dichroism of excimer molecules,” Phys. Rev. Lett. 47, 643–646 (1981); J. C. Kemp, “Circular dichroism measurements with the photoelastic modulator,” PEM Application Note 2 (HINDS International, Portland, Ore., 1975).
    [CrossRef]
  15. J. C. Kemp, G. D. Henson, C. T. Steiner, E. R. Powell, “The optical polarization of the Sun measured at a sensitivity of parts in ten million,” Nature 326, 270–273 (1987).
    [CrossRef]
  16. 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. 8, 373–384 (1977).
    [CrossRef]
  17. This work was reported briefly in M. P. Silverman, N. Ritchie, G. M. Cushman, “Precision reflection spectroscopy by means of phase-modulated light,” Bull. Am. Phys. Soc. 33, 372 (1988).
  18. Several misprints in Ref. 7 were corrected in 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: errata,” J. Opt. Soc. Am. A 4, 1145 (1987).Equation (2k) corrects a further printing error.
    [CrossRef]
  19. H. Eyring, J. Walter, G. Kimball, Quantum Chemistry (Wiley, New York, 1944), pp. 332–347.
  20. The defining expression of dc used in this paper is the negative of that used in Ref. 7, in which the selection of the incident TM direction followed the convention eˆTM×eˆTE=kˆ. The reversal of the TM direction leads to interchange of right and left circular polarizations.
  21. The Fourier–Bessel series pertinent to this paper take the formsin(asinz)=2[J1(a)sin(z)+J3(a)sin(3z)+J5(a)sin(5z)+…],cos(asinz)=J0(a)+2[J2(a)cos(2z)+J4(a)cos(4z)+…].See, for example, M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965) p. 361.
  22. PEM remnant birefringence was measured (as described in Ref. 11) by setting the PEM between two crossed polarizers with the modulation axis at 45° to the transmission axes. The residual birefringence is determined from the relation I(f)/I(2f) = [J1(m)/J2(m)]tan(m′). For m′≪ 1, one has m′~ [J2(m)/J1(m)][(f)/I(2f)].

1988 (1)

This work was reported briefly in M. P. Silverman, N. Ritchie, G. M. Cushman, “Precision reflection spectroscopy by means of phase-modulated light,” Bull. Am. Phys. Soc. 33, 372 (1988).

1987 (3)

Several misprints in Ref. 7 were corrected in 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: errata,” J. Opt. Soc. Am. A 4, 1145 (1987).Equation (2k) corrects a further printing error.
[CrossRef]

J. C. Kemp, G. D. Henson, C. T. Steiner, E. R. Powell, “The optical polarization of the Sun measured at a sensitivity of parts in ten million,” Nature 326, 270–273 (1987).
[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]

1986 (2)

1985 (3)

M. P. Silverman, “Test of gyrotropic constitutive relations by specular light reflection,” Opt. News 11(9), 135 (1985); J. Opt. Soc. Am. A 2(13), P99 (1985).

M. P. Silverman, “Coherent light scattering at the interface of an isotropic chiral medium and homogeneous inactive dielectric,” Bull. Am. Phys. Soc. 30, 798 (1985).

M. P. Silverman, “Specular light scattering from a chiral medium,” Lett. Nuovo Cimento 43, 378–382 (1985).
[CrossRef]

1984 (1)

1983 (1)

1981 (1)

See, for example, M. A. Islam, A. Kponou, B. Suleman, W. Happer, “Magnetic circular dichroism of excimer molecules,” Phys. Rev. Lett. 47, 643–646 (1981); J. C. Kemp, “Circular dichroism measurements with the photoelastic modulator,” PEM Application Note 2 (HINDS International, Portland, Ore., 1975).
[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. 8, 373–384 (1977).
[CrossRef]

1975 (1)

W. Hug, S. Kint, G. Bailey, J. Scherer, “Raman circular intensity differential spectroscopy,” J. Am. Chem. Soc. 97, 5589–5590 (1975).
[CrossRef]

1973 (1)

L. D. Barron, M. P. Bogaard, A. D. Buckingham, “Raman scattering of circularly polarized light by optically active molecules,” J. Am. Chem. Soc. 95, 604–606 (1973).
[CrossRef]

1825 (1)

Fresnel’s theory of optical rotation and his speculation on chirally inequivalent structures are given in, respectively, A. Fresnel, Ann. Chim. 28, 147 (1825); Bull. Soc. Philomath. (1824); Herschel’s identification of the two hemihedral forms of quartz is reported in J. F. W. Herschel, Trans. Cambridge Philos. Soc. 1, 43 (1822).

Abramowitz, M.

The Fourier–Bessel series pertinent to this paper take the formsin(asinz)=2[J1(a)sin(z)+J3(a)sin(3z)+J5(a)sin(5z)+…],cos(asinz)=J0(a)+2[J2(a)cos(2z)+J4(a)cos(4z)+…].See, for example, M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965) p. 361.

Badoz, J.

J. C. Canit, J. Badoz, “Photoelastic modulator for polarimetry and ellipsometry,” Appl. Opt. 23, 2861–2862 (1984).
[CrossRef] [PubMed]

J. C. Canit, J. Badoz, “New design for a photoelastic modulator,” Appl. Opt. 22, 592–594 (1983), and references therein.
[CrossRef] [PubMed]

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. 8, 373–384 (1977).
[CrossRef]

Bailey, G.

W. Hug, S. Kint, G. Bailey, J. Scherer, “Raman circular intensity differential spectroscopy,” J. Am. Chem. Soc. 97, 5589–5590 (1975).
[CrossRef]

Barron, L. D.

L. D. Barron, M. P. Bogaard, A. D. Buckingham, “Raman scattering of circularly polarized light by optically active molecules,” J. Am. Chem. Soc. 95, 604–606 (1973).
[CrossRef]

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge, London, 1982), p. xiv.

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. 8, 373–384 (1977).
[CrossRef]

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]

M. P. Silverman, T. C. Black, “Test of the Fresnel relations for a chiral medium,” in Optics and the Information Age (14th Congress of the International Commission for Optics), H. H. Arsenault, ed. Proc. Soc. Photo-Opt. Instrum. Eng.813, 435–436 (1988).

Bogaard, M. P.

L. D. Barron, M. P. Bogaard, A. D. Buckingham, “Raman scattering of circularly polarized light by optically active molecules,” J. Am. Chem. Soc. 95, 604–606 (1973).
[CrossRef]

Bouchiat, M.

M. Bouchiat, L. Pottier, “Optical experiments and weak interactions,” Science 234, 1203–1210 (1986).
[CrossRef] [PubMed]

Buckingham, A. D.

L. D. Barron, M. P. Bogaard, A. D. Buckingham, “Raman scattering of circularly polarized light by optically active molecules,” J. Am. Chem. Soc. 95, 604–606 (1973).
[CrossRef]

Canit, J. C.

J. C. Canit, J. Badoz, “Photoelastic modulator for polarimetry and ellipsometry,” Appl. Opt. 23, 2861–2862 (1984).
[CrossRef] [PubMed]

J. C. Canit, J. Badoz, “New design for a photoelastic modulator,” Appl. Opt. 22, 592–594 (1983), and references therein.
[CrossRef] [PubMed]

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. 8, 373–384 (1977).
[CrossRef]

Cushman, G. M.

This work was reported briefly in M. P. Silverman, N. Ritchie, G. M. Cushman, “Precision reflection spectroscopy by means of phase-modulated light,” Bull. Am. Phys. Soc. 33, 372 (1988).

Eyring, H.

H. Eyring, J. Walter, G. Kimball, Quantum Chemistry (Wiley, New York, 1944), pp. 332–347.

Fresnel, A.

Fresnel’s theory of optical rotation and his speculation on chirally inequivalent structures are given in, respectively, A. Fresnel, Ann. Chim. 28, 147 (1825); Bull. Soc. Philomath. (1824); Herschel’s identification of the two hemihedral forms of quartz is reported in J. F. W. Herschel, Trans. Cambridge Philos. Soc. 1, 43 (1822).

Happer, W.

See, for example, M. A. Islam, A. Kponou, B. Suleman, W. Happer, “Magnetic circular dichroism of excimer molecules,” Phys. Rev. Lett. 47, 643–646 (1981); J. C. Kemp, “Circular dichroism measurements with the photoelastic modulator,” PEM Application Note 2 (HINDS International, Portland, Ore., 1975).
[CrossRef]

Henson, G. D.

J. C. Kemp, G. D. Henson, C. T. Steiner, E. R. Powell, “The optical polarization of the Sun measured at a sensitivity of parts in ten million,” Nature 326, 270–273 (1987).
[CrossRef]

Hug, W.

W. Hug, S. Kint, G. Bailey, J. Scherer, “Raman circular intensity differential spectroscopy,” J. Am. Chem. Soc. 97, 5589–5590 (1975).
[CrossRef]

Islam, M. A.

See, for example, M. A. Islam, A. Kponou, B. Suleman, W. Happer, “Magnetic circular dichroism of excimer molecules,” Phys. Rev. Lett. 47, 643–646 (1981); J. C. Kemp, “Circular dichroism measurements with the photoelastic modulator,” PEM Application Note 2 (HINDS International, Portland, Ore., 1975).
[CrossRef]

Kemp, J. C.

J. C. Kemp, G. D. Henson, C. T. Steiner, E. R. Powell, “The optical polarization of the Sun measured at a sensitivity of parts in ten million,” Nature 326, 270–273 (1987).
[CrossRef]

Kimball, G.

H. Eyring, J. Walter, G. Kimball, Quantum Chemistry (Wiley, New York, 1944), pp. 332–347.

Kint, S.

W. Hug, S. Kint, G. Bailey, J. Scherer, “Raman circular intensity differential spectroscopy,” J. Am. Chem. Soc. 97, 5589–5590 (1975).
[CrossRef]

Kponou, A.

See, for example, M. A. Islam, A. Kponou, B. Suleman, W. Happer, “Magnetic circular dichroism of excimer molecules,” Phys. Rev. Lett. 47, 643–646 (1981); J. C. Kemp, “Circular dichroism measurements with the photoelastic modulator,” PEM Application Note 2 (HINDS International, Portland, Ore., 1975).
[CrossRef]

Pottier, L.

M. Bouchiat, L. Pottier, “Optical experiments and weak interactions,” Science 234, 1203–1210 (1986).
[CrossRef] [PubMed]

Powell, E. R.

J. C. Kemp, G. D. Henson, C. T. Steiner, E. R. Powell, “The optical polarization of the Sun measured at a sensitivity of parts in ten million,” Nature 326, 270–273 (1987).
[CrossRef]

Ritchie, N.

This work was reported briefly in M. P. Silverman, N. Ritchie, G. M. Cushman, “Precision reflection spectroscopy by means of phase-modulated light,” Bull. Am. Phys. Soc. 33, 372 (1988).

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. 8, 373–384 (1977).
[CrossRef]

Scherer, J.

W. Hug, S. Kint, G. Bailey, J. Scherer, “Raman circular intensity differential spectroscopy,” J. Am. Chem. Soc. 97, 5589–5590 (1975).
[CrossRef]

Silverman, M. P.

This work was reported briefly in M. P. Silverman, N. Ritchie, G. M. Cushman, “Precision reflection spectroscopy by means of phase-modulated light,” Bull. Am. Phys. Soc. 33, 372 (1988).

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]

Several misprints in Ref. 7 were corrected in 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: errata,” J. Opt. Soc. Am. A 4, 1145 (1987).Equation (2k) corrects a further printing error.
[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, “Test of gyrotropic constitutive relations by specular light reflection,” Opt. News 11(9), 135 (1985); J. Opt. Soc. Am. A 2(13), P99 (1985).

M. P. Silverman, “Coherent light scattering at the interface of an isotropic chiral medium and homogeneous inactive dielectric,” Bull. Am. Phys. Soc. 30, 798 (1985).

M. P. Silverman, “Specular light scattering from a chiral medium,” Lett. Nuovo Cimento 43, 378–382 (1985).
[CrossRef]

M. P. Silverman, T. C. Black, “Test of the Fresnel relations for a chiral medium,” in Optics and the Information Age (14th Congress of the International Commission for Optics), H. H. Arsenault, ed. Proc. Soc. Photo-Opt. Instrum. Eng.813, 435–436 (1988).

Stegun, I. A.

The Fourier–Bessel series pertinent to this paper take the formsin(asinz)=2[J1(a)sin(z)+J3(a)sin(3z)+J5(a)sin(5z)+…],cos(asinz)=J0(a)+2[J2(a)cos(2z)+J4(a)cos(4z)+…].See, for example, M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965) p. 361.

Steiner, C. T.

J. C. Kemp, G. D. Henson, C. T. Steiner, E. R. Powell, “The optical polarization of the Sun measured at a sensitivity of parts in ten million,” Nature 326, 270–273 (1987).
[CrossRef]

Suleman, B.

See, for example, M. A. Islam, A. Kponou, B. Suleman, W. Happer, “Magnetic circular dichroism of excimer molecules,” Phys. Rev. Lett. 47, 643–646 (1981); J. C. Kemp, “Circular dichroism measurements with the photoelastic modulator,” PEM Application Note 2 (HINDS International, Portland, Ore., 1975).
[CrossRef]

Walter, J.

H. Eyring, J. Walter, G. Kimball, Quantum Chemistry (Wiley, New York, 1944), pp. 332–347.

Ann. Chim. (1)

Fresnel’s theory of optical rotation and his speculation on chirally inequivalent structures are given in, respectively, A. Fresnel, Ann. Chim. 28, 147 (1825); Bull. Soc. Philomath. (1824); Herschel’s identification of the two hemihedral forms of quartz is reported in J. F. W. Herschel, Trans. Cambridge Philos. Soc. 1, 43 (1822).

Appl. Opt. (2)

Bull. Am. Phys. Soc. (2)

This work was reported briefly in M. P. Silverman, N. Ritchie, G. M. Cushman, “Precision reflection spectroscopy by means of phase-modulated light,” Bull. Am. Phys. Soc. 33, 372 (1988).

M. P. Silverman, “Coherent light scattering at the interface of an isotropic chiral medium and homogeneous inactive dielectric,” Bull. Am. Phys. Soc. 30, 798 (1985).

J. Am. Chem. Soc. (2)

L. D. Barron, M. P. Bogaard, A. D. Buckingham, “Raman scattering of circularly polarized light by optically active molecules,” J. Am. Chem. Soc. 95, 604–606 (1973).
[CrossRef]

W. Hug, S. Kint, G. Bailey, J. Scherer, “Raman circular intensity differential spectroscopy,” J. Am. Chem. Soc. 97, 5589–5590 (1975).
[CrossRef]

J. Opt. (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. 8, 373–384 (1977).
[CrossRef]

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

Lett. Nuovo Cimento (1)

M. P. Silverman, “Specular light scattering from a chiral medium,” Lett. Nuovo Cimento 43, 378–382 (1985).
[CrossRef]

Nature (1)

J. C. Kemp, G. D. Henson, C. T. Steiner, E. R. Powell, “The optical polarization of the Sun measured at a sensitivity of parts in ten million,” Nature 326, 270–273 (1987).
[CrossRef]

Opt. News (1)

M. P. Silverman, “Test of gyrotropic constitutive relations by specular light reflection,” Opt. News 11(9), 135 (1985); J. Opt. Soc. Am. A 2(13), P99 (1985).

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. Lett. (1)

See, for example, M. A. Islam, A. Kponou, B. Suleman, W. Happer, “Magnetic circular dichroism of excimer molecules,” Phys. Rev. Lett. 47, 643–646 (1981); J. C. Kemp, “Circular dichroism measurements with the photoelastic modulator,” PEM Application Note 2 (HINDS International, Portland, Ore., 1975).
[CrossRef]

Science (1)

M. Bouchiat, L. Pottier, “Optical experiments and weak interactions,” Science 234, 1203–1210 (1986).
[CrossRef] [PubMed]

Other (6)

H. Eyring, J. Walter, G. Kimball, Quantum Chemistry (Wiley, New York, 1944), pp. 332–347.

The defining expression of dc used in this paper is the negative of that used in Ref. 7, in which the selection of the incident TM direction followed the convention eˆTM×eˆTE=kˆ. The reversal of the TM direction leads to interchange of right and left circular polarizations.

The Fourier–Bessel series pertinent to this paper take the formsin(asinz)=2[J1(a)sin(z)+J3(a)sin(3z)+J5(a)sin(5z)+…],cos(asinz)=J0(a)+2[J2(a)cos(2z)+J4(a)cos(4z)+…].See, for example, M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965) p. 361.

PEM remnant birefringence was measured (as described in Ref. 11) by setting the PEM between two crossed polarizers with the modulation axis at 45° to the transmission axes. The residual birefringence is determined from the relation I(f)/I(2f) = [J1(m)/J2(m)]tan(m′). For m′≪ 1, one has m′~ [J2(m)/J1(m)][(f)/I(2f)].

M. P. Silverman, T. C. Black, “Test of the Fresnel relations for a chiral medium,” in Optics and the Information Age (14th Congress of the International Commission for Optics), H. H. Arsenault, ed. Proc. Soc. Photo-Opt. Instrum. Eng.813, 435–436 (1988).

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge, London, 1982), p. xiv.

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Figures (7)

Fig. 1
Fig. 1

Generic schematic diagram of an experimental configuration with one PEM. Light (e.g., from a laser L) passes a mechanical chopper (LC) of low frequency F, a linear polarizer (LP), and a PEM (M) at high frequency f and is reflected from an optically flat sample attached to a rotatable reflection cell (RC); it then passes an analyzing polarizer (A) (if used) and is received at a detector (D). The reflected light is synchronously detected at F, f, and 2f by lock-in amplifiers (LA). The components of the light flux at desired frequencies (f, 2f, and F), or appropriate ratios, are displayed on a digital voltmeter (DVM).

Fig. 2
Fig. 2

Schematic diagram of polarization orientations. The PEM axis is at an angle w to the direction of TE polarization (upward normal to the plane of incidence). The transmission axes of the initial polarizer and analyzing polarizer (if used) are at angles u and υ, respectively, to the TE direction.

Fig. 3
Fig. 3

Variation of I(2f)/I(0) with PEM retardation in a measurement of the differential linear reflection from NaClO3 at 60°. The PEM modulation amplitude m is related to the retardation R by the expression m = πR/λ, where λ is the wavelength. The maximum occurs at the predicted value m2 ~ 3.1, for which J2(m2) is maximum.

Fig. 4
Fig. 4

Variation of residual PEM birefringence with retardation (i.e., modulation amplitude) for a fused-silica modulator with a 50-kHz modulation frequency. The residual birefringence is weak and fairly independent of retardation over the range of 100–500 nm (range of m, 0.5–2.48).

Fig. 5
Fig. 5

Differential linear reflection (DLR) of transparent single-crystal NaClO3 (refractive index 1.513) as a function of the angle of incidence measured by experimental configuration (A3) (■ curve) and compared with theory (solid curve).

Fig. 6
Fig. 6

Real part, s1, of the oblique dyadic projection (ODP) of transparent single-crystal NaClO3 as a function of the angle of incidence measured by experimental configuration (B3) (■ curve) and compared with theory (solid curve). s1 varies between −1 and +1 and crosses the abscissa at the Brewster angle.

Fig. 7
Fig. 7

Real part, s1, and imaginary part, s2, of the oblique dyadic projection (ODP) of a weakly absorbing glass (refractive index, 1.49; extinction coeffcient, 0.09) measured by experimental configuration (B3) and compared with theory (solid curves).

Equations (110)

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n ˜ ± = n ˜ ( 1 ± g ˜ ) ,
n ˜ = n + i k ,
g ˜ = g 1 + i g 2
n ˜ ± 2 = P ± + i Q ± = ˜ μ ( 1 ± g ˜ ) 2 ,
r ˜ i = a ˜ i e ˆ TE + b ˜ i e ˆ TM ( i = 1 , 2 ) ,
a ˜ 1 = [ x 2 ( 1 / 2 ) ( z + + z ) ( q q 1 ) x z + z ] / D ,
b ˜ 1 = i ( z + z ) x / D ,
a ˜ 2 = i ( z z + ) x / D ,
b ˜ 2 = [ x 2 + ( 1 / 2 ) ( z + + z ) ( q q 1 ) x z + z ] / D ,
D = x 2 + ( 1 / 2 ) ( z + + z ) ( q + q 1 ) x + z + z ,
x = cos ( θ i ) ,
z ± = ( B ± + i G ± ) / n ˜ ± ,
q = ( n ˜ 2 / n ˜ 1 ) ( μ 1 / μ 2 ) ,
B ± = ( 2 1 / 2 ) { [ ( P ± A ) 2 + Q ± 2 ] 1 / 2 + ( P ± A ) } 1 / 2 ,
G ± = ( 2 1 / 2 ) { [ ( P ± A ) 2 + Q ± 2 ] 1 / 2 ( P ± A ) } 1 / 2 ,
A = [ n 1 sin ( θ i ) ] 2 .
a ˜ 1 = a 1 + i α 1 ,
b ˜ 1 = b 1 + i β 1 ,
a ˜ 2 = a 2 + i α 2 ,
b ˜ 2 = b 2 + i β 2 .
r ˜ = 2 r ˜ 2 . r ˜ 1 * | r ˜ 1 | 2 + | r ˜ 2 | 2 = 2 ( a ˜ 2 a ˜ 1 * + b ˜ 2 b ˜ 1 * ) | a ˜ 1 | 2 + | b ˜ 1 | 2 + | a ˜ 2 | 2 + | b ˜ 2 | 2 ,
I = | r ˜ 1 | 2 + | r ˜ 2 | 2 = I TE + I TM = I R + I L
d c = Im ( r ˜ ) = I R I L I R + I L = 2 ( a 1 α 2 + b 1 β 2 a 2 α 1 b 2 β 1 ) | a ˜ 1 | 2 + | b ˜ 1 | 2 + | a ˜ 2 | 2 + | b ˜ 2 | 2 .
d c = 2 ( a 1 α 2 b 2 β 1 ) a 1 2 + b 2 2 .
s c = Re ( r ˜ ) = 2 Im ( r L · r R * ) I L + I R = 2 ( a 1 a 2 + α 1 α 2 + b 1 b 2 + β 1 β 2 ) I L + I R ,
r L = ( r 1 + i r 2 ) / 2 1 / 2 ,
r R = ( r 1 i r 2 ) / 2 1 / 2
p ˜ = 2 e ˆ TE · ( r ˜ 1 r ˜ 1 * + r ˜ 2 r ˜ 2 * ) · e ˆ TM | r ˜ 1 | 2 + | r ˜ 2 | 2 = 2 ( b ˜ 1 a ˜ 1 * + b ˜ 2 a ˜ 2 * ) | a ˜ 1 | 2 + | b ˜ 1 | 2 + | a ˜ 2 | 2 + | b ˜ 2 | 2 .
p 1 = Re ( p ˜ ) = 2 ( a 1 b 1 + α 1 β 1 + a 2 b 2 + α 2 β 2 ) | a ˜ 1 | 2 + | b ˜ 2 | 2 ,
p 2 = Im ( p ˜ ) = 2 ( a 1 β 1 b 1 α 1 + a 2 β 2 b 2 α 2 ) | a ˜ 1 | 2 + | b ˜ 2 | 2 ,
p 2 = 2 ( a 1 β 1 b 2 α 2 ) a 1 2 + b 2 2 .
e ˜ a = 2 e ˆ TE · r ˜ 1 r ˜ 2 * · e ˆ TE | e ˆ TE · r ˜ 1 | 2 + | r ˜ 2 · e ˆ TE | 2 = 2 a ˜ 1 a ˜ 2 * | a ˜ 1 | 2 + | a ˜ 2 | 2 ,
e ˜ b = 2 e ˆ TM · r ˜ 1 r ˜ 2 * · e ˆ TM | e ˆ TM · r ˜ 1 | 2 + | r ˜ 2 · e ˆ TM | 2 = 2 b ˜ 1 b ˜ 2 * | b ˜ 1 | 2 + | b ˜ 2 | 2 .
e a ( r ) = Re ( e ˜ a ) = 2 ( a 1 a 2 + α 1 α 2 ) | a ˜ 1 | 2 + | a ˜ 2 | 2 ,
e a ( i ) = Im ( e ˜ a ) = 2 ( a 2 α 1 a 1 α 2 ) | a ˜ 1 | 2 + | a ˜ 2 | 2 ,
e b ( r ) = Re ( e ˜ b ) = 2 ( b 1 b 2 + β 1 β 2 ) | b ˜ 1 | 2 + | b ˜ 2 | 2 ,
e b ( i ) = Im ( e ˜ b ) = 2 ( b 2 β 1 b 1 β 2 ) | b ˜ 1 | 2 + | b ˜ 2 | 2 .
e a ( r ) ~ 2 | a ˜ 2 / a ˜ 1 | cos ( E a ) ,
e a ( i ) ~ 2 | a ˜ 2 / a ˜ 1 | sin ( E a ) ,
e b ( r ) ~ 2 | b ˜ 1 / b ˜ 2 | cos ( E b ) ,
e b ( i ) ~ 2 | b ˜ 1 / b ˜ 2 | sin ( E b ) ,
E a = arg ( a ˜ 1 ) arg ( a ˜ 2 ) ,
E b = arg ( b ˜ 1 ) arg ( b ˜ 2 ) .
e a ( r ) = e b ( r ) = 0 ,
e a ( i ) = 2 a 1 α 2 / ( a 1 2 + α 2 2 ) ,
e b ( i ) = 2 b 2 β 1 / ( b 2 2 + β 1 2 ) .
r ˜ = 2 r ˜ L · r ˜ R * | r ˜ L | 2 + | r ˜ R | 2 .
d l = Re ( r ˜ ) = I TE I TM I TE + I TM = | r ˜ 1 | 2 | r ˜ 2 | 2 | r ˜ 1 | 2 + | r ˜ 2 | 2 = ( | a ˜ 1 | 2 + | b ˜ 1 | 2 ) ( | a ˜ 2 | 2 + | b ˜ 2 | 2 ) ( | a ˜ 1 | 2 + | b ˜ 1 | 2 ) + ( | a ˜ 2 | 2 + | b ˜ 2 | 2 ) .
d l = a 1 2 b 2 2 a 1 2 + b 2 2 .
s ˜ = 2 ( e ˆ 0 · r ˜ 1 r ˜ 2 * · e ˆ 0 ) | r ˜ 1 · e ˆ 0 | 2 + | r ˜ 2 · e ˆ 0 | 2 ,
e ˆ 0 = ( 2 1 / 2 ) ( e ˆ TE + e ˆ TM ) .
s 1 = Re ( s ˜ ) = 2 Re [ ( a ˜ 1 + b ˜ 1 ) ( a ˜ 2 + b ˜ 2 ) * ] | a ˜ 1 + b ˜ 1 | 2 + | a ˜ 2 + b ˜ 2 | 2 ,
s 2 = Im ( s ˜ ) = 2 Im [ ( a ˜ 1 + b ˜ 1 ) ( a ˜ 2 + b ˜ 2 ) * ] | a ˜ 1 + b ˜ 1 | 2 + | a ˜ 2 + b ˜ 2 | 2 .
s 1 = 2 ( a 1 b 2 + α 1 β 2 ) | a ˜ 1 | 2 + | b ˜ 2 | 2 2 a 1 b 2 a 1 2 + b 2 2 ( transparent medium ) ,
s 2 = 2 ( b 2 α 1 a 1 β 2 ) | a ˜ 1 | 2 + | b ˜ 2 | 2 0 ( transparent medium )
d TE = | a ˜ 1 | 2 | a ˜ 2 | 2 | a ˜ 1 | 2 + | a ˜ 2 | 2 ,
d TM = | b ˜ 2 | 2 | b ˜ 1 | 2 | b ˜ 1 | 2 + | b ˜ 2 | 2 .
E i = [ x ˆ cos ( u ) + y ˆ sin ( u ) ] exp ( i W t ) .
E m = [ x ˆ m cos ( u w ) exp ( i ϕ ) + y ˆ m sin ( u w ) ] exp ( i W t ) ,
ϕ = m sin ( 2 π f t ) + m ;
x ˆ a ˜ 1 x ˆ + b ˜ 1 y ˆ ,
y ˆ a ˜ 2 x ˆ + b ˜ 2 y ˆ ,
E r = x ˆ { a ˜ 1 [ cos ( u w ) cos ( w ) exp ( i ϕ ) sin ( u w ) sin ( w ) ] + a ˜ 2 [ cos ( u w ) sin ( w ) exp ( i ϕ ) + sin ( u w ) cos ( w ) ] } exp ( i W t ) + y ˆ { b ˜ 1 [ cos ( u w ) cos ( w ) exp ( i ϕ ) sin ( u w ) sin ( w ) ] + b ˜ 2 [ cos ( u w ) sin ( w ) exp ( i ϕ ) + sin ( u w ) cos ( w ) ] } exp ( i W t ) .
I d ( t ) = | E r | 2 ,
I d ( t ) = I TE [ cos 2 ( u w ) cos 2 ( w ) + sin 2 ( u w ) sin 2 ( w ) ] + I TM [ cos 2 ( u w ) cos 2 ( w ) + sin 2 ( u w ) sin 2 ( w ) ] + 2 Re ( a ˜ 1 a ˜ 2 * + b ˜ 1 b ˜ 2 * ) [ cos 2 ( u w ) sin 2 ( u w ) ] cos ( w ) sin ( w ) + 2 ( I R I L ) cos ( u w ) sin ( u w ) sin ( ϕ ) + 2 { ( I TM I TE ) cos ( w ) sin ( w ) + Re ( a ˜ 1 a ˜ 2 * + b ˜ 1 b ˜ 2 * ) cos [ ( w ) sin 2 ( w ) ] } cos ( u w ) sin ( u w ) cos ( ϕ ) .
I d ( t ) = ( I / 2 ) [ 1 + d c sin ( ϕ ) + s c cos ( ϕ ) ] ,
I = I TE + I TM .
I d ( t ) = I ( 0 ) + I ( f ) sin ( 2 π f t ) + I ( 2 f ) cos ( 4 π f t ) ,
I ( 0 ) = 1 + J 0 ( m ) [ d c sin ( m ) + s c cos ( m ) ] ,
I ( f ) = 2 J 1 ( m ) [ d c cos ( m ) s c sin ( m ) ] ,
I ( 2 f ) = 2 J 2 ( m ) [ d c sin ( m ) + s c cos ( m ) ] .
X = I ( f ) / I ( 0 ) = 2 J 1 ( m ) [ d c cos ( m ) s c sin ( m ) ] 1 + J 0 ( m ) [ d c sin ( m ) + s c cos ( m ) ] ,
Y = I ( 2 f ) / I ( 0 ) = 2 J 2 ( m ) [ d c sin ( m ) + s c cos ( m ) ] 1 + J 0 ( m ) [ d c sin ( m ) + s c cos ( m ) ] ,
d c = J 2 ( m ) X cos ( m ) + J 1 ( m ) Y sin ( m ) J 1 ( m ) [ 2 J 2 ( m ) J 0 ( m ) Y ] ,
s c = J 2 ( m ) X sin ( m ) + J 1 ( m ) Y cos ( m ) J 1 ( m ) [ 2 J 2 ( m ) J 0 ( m ) Y ] .
d c = X cos ( m ) / 2 J 1 ( m ) .
d c = X / [ 2 J 1 ( m 0 ) ] ,
s c = Y / [ 2 J 2 ( m 0 ) ] .
d c = J 2 ( m ) X J 1 ( m ) [ 2 J 2 ( m ) J 0 ( m ) Y ] ,
s c = Y [ 2 J 2 ( m ) J 0 ( m ) Y ] .
I d ( t ) = ( I / 2 ) [ 1 d c sin ( ϕ ) + s c cos ( ϕ ) ] ,
I d ( t ) = ( I / 2 ) [ 1 d c sin ( ϕ ) + d cos ( ϕ ) ] ,
I d ( t ) = ( I / 2 ) [ 1 + d c sin ( ϕ ) d cos ( ϕ ) ] .
E p = p ˆ { [ a ˜ 1 cos ( υ ) + b ˜ 1 sin ( υ ) ] [ cos ( u w ) cos ( w ) exp ( i ϕ ) sin ( u w ) sin ( w ) ] + [ a ˜ 2 cos ( υ ) + b ˜ 2 sin ( υ ) ] × [ cos ( u w ) sin ( w ) exp ( i ϕ ) + sin ( u w ) cos ( w ) ] } × exp ( i W t ) .
I d ( t ) = [ | a ˜ 1 | 2 cos 2 ( υ ) + | b ˜ 1 | 2 sin 2 ( υ ) + 2 Re ( a ˜ 1 b ˜ 1 * ) cos ( υ ) sin ( υ ) ] [ cos 2 ( u w ) cos 2 ( w ) + sin 2 ( u w ) sin 2 ( w ) ] + [ | a ˜ 2 | 2 cos 2 ( υ ) + | b ˜ 2 | 2 sin 2 ( υ ) + 2 Re ( a ˜ 2 b ˜ 2 * ) cos ( υ ) sin ( υ ) ] [ cos 2 ( u w ) sin 2 ( w ) + sin 2 ( u w ) cos 2 ( w ) ] + [ Re ( a ˜ 1 a ˜ 2 * ) cos 2 ( υ ) + Re ( b ˜ 1 b ˜ 2 * ) sin 2 ( υ ) + Re ( a ˜ 1 b ˜ 2 * + b ˜ 1 a ˜ 2 * ) cos ( υ ) sin ( υ ) ] [ 2 cos ( w ) sin ( w ) ] [ cos 2 ( u w ) sin 2 ( u w ) ] + { [ Re ( a ˜ 1 a ˜ 2 * ) cos 2 ( υ ) + Re ( b ˜ 1 b ˜ 2 * ) sin 2 ( υ ) + Re ( a ˜ 1 b ˜ 2 * + b ˜ 1 a ˜ 2 * ) cos ( υ ) sin ( υ ) ] [ cos 2 ( w ) sin 2 ( w ) ] [ ( | a ˜ 1 | 2 | a ˜ 2 | 2 ) cos 2 ( υ ) + ( | b ˜ 1 | 2 | b ˜ 2 | 2 ) sin 2 ( υ ) + 2 Re ( a ˜ 1 b ˜ 1 * a ˜ 2 b ˜ 2 * ) ] [ cos ( w ) sin ( w ) ] } cos ( ϕ ) [ Im ( a ˜ 1 a ˜ 2 * ) cos 2 ( υ ) + Im ( b ˜ 1 b ˜ 2 * ) sin 2 ( υ ) + Im ( a ˜ 1 b ˜ 2 * + b ˜ 1 a ˜ 2 * ) cos ( υ ) sin ( υ ) ] sin ( ϕ ) .
I d ( t ) = ( I a / 2 ) [ 1 + e a ( r ) cos ( ϕ ) + e a ( i ) sin ( ϕ ) ] ,
I a = | a ˜ 1 | 2 + | a ˜ 2 | 2 ~ | a ˜ 1 | 2 .
e a ( i ) = ( J 2 ( m ) / J 1 ( m ) ) X / [ 2 J 2 ( m ) J 0 ( m ) Y ] ,
e a ( r ) = Y / [ 2 J 2 ( m ) J 0 ( m ) Y ] ,
E a = tan 1 [ e a ( i ) / e a ( r ) ] ,
| a ˜ 2 | / | a ˜ 1 | = ( 1 / 2 ) { [ e a ( r ) ] 2 + [ e a ( i ) ] 2 } 1 / 2 ,
I d ( t ) = ( I b / 2 ) [ 1 + e b ( r ) cos ( ϕ ) + e b ( i ) sin ( ϕ ) ] ,
I b = | b ˜ 1 | 2 + | b ˜ 2 | 2 ~ | b ˜ 2 | 2 .
I d ( t ) = ( I a b / 2 ) [ 1 + s 1 cos ( ϕ ) + s 2 sin ( ϕ ) ] ,
I a b = | a ˜ 1 + b ˜ 1 | 2 + | a ˜ 2 + b ˜ 2 | 2 .
I d ( t ) = ( I a / 2 ) [ 1 + d TE cos ( ϕ ) + e a ( i ) sin ( ϕ ) ] ,
I d ( t ) = ( I b / 2 ) [ 1 + d TM cos ( ϕ ) + e b ( i ) sin ( ϕ ) ] ,
ϕ 1 = m 1 sin ( 2 π f 1 t ) ;
ϕ 2 = m 2 sin ( 2 π f 2 t ) .
E p = ( p ˆ / 2 ) [ a ˜ 1 exp [ i ( ϕ 1 + ϕ 2 ) + b ˜ 1 exp ( i ϕ 1 ) + a ˜ 2 exp ( i ϕ 2 ) + b ˜ 2 ] .
I d ( t ) = ( I / 4 ) [ 1 + s c cos ( ϕ 1 ) + d c sin ( ϕ 1 ) + p 1 cos ( ϕ 2 ) + p 2 sin ( ϕ 2 ) + s 1 cos ( ϕ 1 + ϕ 2 ) + s 2 sin ( ϕ 1 + ϕ 2 ) + 2 Re ( a ˜ 2 b ˜ 1 * ) cos ( ϕ 1 ϕ 2 ) + 2 Im ( a ˜ 2 b ˜ 1 * ) sin ( ϕ 1 ϕ 2 ) ,
I d ( t ) = ( I / 4 ) [ 1 + s c J 0 ( m 1 ) + p 1 J 0 ( m 2 ) + s 1 J 0 ( m 1 ) J 0 ( m 2 ) + 2 J 1 ( m 1 ) [ d c + s 2 J 0 ( m 2 ) sin ( 2 π f 1 t ) + 2 J 1 ( m 2 ) [ p 2 + s 2 J 0 ( m 1 ) ] sin ( 2 π f 2 t ) + 2 J 2 ( m 1 ) [ s c + s 1 J 0 ( m 2 ) ] cos ( 4 π f 1 t ) + 2 J 2 ( m 2 ) [ p 1 + s 1 J 0 ( m 1 ) ] cos ( 4 π f 2 t ) + 2 s 1 J 1 ( m 1 ) J 1 ( m 2 ) { cos [ 2 π ( f 1 + f 2 ) t ] cos [ 2 π ( f 1 f 2 ) t ] } + 2 s 2 ( J 1 ( m 1 ) J 2 ( m 2 ) { sin [ 2 π ( f 1 + 2 f 2 ) t ] + sin [ 2 π ( f 1 2 f 2 ) t ] } + J 1 ( m 2 ) J 2 ( m 1 ) { sin [ 2 π ( f 2 + 2 f 1 ) t ] + sin [ 2 π ( f 2 2 f 1 ) t ] } ] .
I d ( t ) = ( I / 4 ) ( 1 + 2 J 1 ( m 0 ) [ d c sin ( 2 π f 1 t ) + p 2 sin ( 2 π f 2 t ) ] + 2 J 2 ( m 0 ) [ s c sin ( 2 π f 1 t ) + p 1 sin ( 2 π f 2 t ) ] + 2 s 1 [ J 1 ( m 0 ) ] 2 { cos [ 2 π ( f 1 + f 2 ) t ] cos [ 2 π ( f 1 f 2 ) t ] } + 2 s 2 J 1 ( m 0 ) J 2 ( m 0 ) { sin [ 2 π ( f 1 + 2 f 2 ) t ] + sin [ 2 π ( f 1 2 f 2 ) t ] + sin [ 2 π ( f 2 + 2 f 1 ) t ] + sin [ 2 π ( f 2 2 f 1 ) t ] } ) .
I ( f 1 ) / I ( 0 ) = 2 J 1 ( m 0 ) d c ,
I ( f 2 ) / I ( 0 ) = 2 J 1 ( m 0 ) p 2 ,
I ( 2 f 1 ) / I ( 0 ) = 2 J 2 ( m 0 ) s c ,
I ( 2 f 2 ) / I ( 0 ) = 2 J 2 ( m 0 ) p 1 ,
I ( f 1 ± f 2 ) / I ( 0 ) = ± 2 [ J 1 ( m 0 ) ] 2 s 1 ,
I ( f 1 ± 2 f 2 ) / I ( 0 ) = I ( f 2 ± 2 f 1 ) / I ( 0 ) = 2 J 1 ( m 0 ) J 2 ( m 0 ) s 2 .
sin(asinz)=2[J1(a)sin(z)+J3(a)sin(3z)+J5(a)sin(5z)+],cos(asinz)=J0(a)+2[J2(a)cos(2z)+J4(a)cos(4z)+].

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