Abstract

Light propagation through an optically active Fabry–Perot interferometer, consisting of an isotropic chiral layer separating two homogeneous achiral media, is analyzed with an interest in the manifestations of chiral asymmetry in the reflected wave. It is shown that the differential reflection of circularly polarized light and the optical rotation and the ellipticity of incident linearly polarized light can be enhanced by several orders of magnitude (depending on reflection geometry, layer thickness, and optical parameters) under conditions of moderate to high levels of reflectance. This would provide a practical experimental approach to investigating chiral structure and interactions in systems not easily amenable to study, such as weakly chiral thin films and layers, both transparent and absorbing. (Previously proposed and confirmed enhancement techniques, based on multiple total external reflection, could be usefully applied only to absorbing chiral media.) Two experimental configurations of particular potential interest employ (1) total internal reflection and (2) metallic reflection at the second chiral–achiral interface. For a sufficiently thick chiral layer the reflectance of polarization orthogonal to that of the incident wave should be directly observable even under conditions of ordinary reflection. It is also shown that, in contrast to wide belief, reflection at normal incidence from an isotropic chiral medium can manifest optical activity (under appropriate circumstances when the ambient achiral media are birefringent).

© 1994 Optical Society of America

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  1. M. P. Silverman, J. Badoz, “Ellipsometric study of specular reflection from a naturally optically active medium,” presented at the First International Conference on Spectroscopic Ellipsometry, Paris, France, January 1993;Thin Solid Films 234491–495 (1993),reprinted in Spectroscopic Ellipsometry, A. C. Boccara, C. Pickering, J. Rivory, eds. (Elsevier Sequoia, Amsterdam, 1993), pp. 491–495.
  2. See, for example, J. C. Fabre, A. C. Boccara, “Circular dichroism microspectroscopy: evidence of broken symmetries in high-Tc superconducting films,” Opt. Commun. 93, 306–310 (1992);K. B. Lyons, J. Kwo, J. F. Dillon, G. P. Espinosa, M. McGlashan-Powell, A. P. Ramirez, L. F. Schneemeyer, “Search for circular dichroism in high-Tc superconductors,” Phys. Rev. Lett. 64, 2949–2952 (1990).
    [CrossRef] [PubMed]
  3. See, for example, A. Sihvola, ed., Proceedings of “Biisotropics ’93”: Workshop on Novel Microwave Materials, Helsinki University of Technology, Programme of the 1991 North American Radio Science Meeting (University of Western Ontario, London, Ontario, Canada, 1991), pp. 355–364.
  4. An analysis of these difficulties is given in J. Badoz, M. P. Silverman, J. C. Canit, “Wave propagation through a medium with static and dynamic birefringence: theory of the photoelastic modulator,” J. Opt. Soc. Am. A 7, 672–682 (1990).
    [CrossRef]
  5. M.-A. Bouchiat, L. Pottier, “Optical experiments and weak interactions,” Science 234, 1203–1210 (1986).
    [CrossRef] [PubMed]
  6. M. P. Silverman, J. Badoz, “Large enhancement of chiral asymmetry in light reflection near critical angle,” Opt. Commun. 74, 129–133 (1989).
    [CrossRef]
  7. M. P. Silverman, J. Badoz, “Multiple reflection from isotropic chiral media and the enhancement of chiral asymmetry,” J. Electromagn. Waves Appl. 6, 587–601 (1992).
  8. M. P. Silverman, J. Badoz, B. Briat, “Chiral reflection from a naturally optically active medium,” Opt. Lett. 17, 886–888 (1992).
    [CrossRef] [PubMed]
  9. S. Bassiri, C. H. 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]
  10. D. L. Jaggard, X. Sun, “Theory of chiral multilayers,” J. Opt. Soc. Am. A 9, 804–813 (1992).
    [CrossRef]
  11. M. Schmidt, K. Eidner, “Electromagnetic wave propagation through an isotropic chiral slab: solution for oblique incidence,” Optik (Stuttgart) 80, 43–46 (1990).
  12. I. J. Lalov, A. I. Miteva, “Optically active Fabry–Perot etalon,” J. Mod. Opt. 38, 395–411 (1990).
    [CrossRef]
  13. Summary reports of this research were given at the 1993 Annual Meeting of the Optical Society of America [M. P. Silverman and J. Badoz, in Multiple Reflection and Interference within a Chiral Medium, Vol. 16 of the OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 115]and in J. Badoz, M. P. Silverman, “Large chiral asymmetries in light reflected from an optically active Fabry–Perot interferometer,” Opt. Commun. 105, 15–21 (1994).
    [CrossRef]
  14. We adopt here the standard optical convention (in contrast to that used elsewhere in physics) that the electric field of a left (right) circularly polarized wave rotates to the left (right) of an observer facing the light source. The term helicity refers to the scalar projection of angular momentum of the wave on its linear momentum (or wave vector). Thus a positive helicity (with designated index n+) connotes a parallel orientation of angular and linear momenta and therefore a left circular (or, generally, elliptical) polarization. Correspondingly, right circular polarization (with designated index n−) is equivalent to a negative helicity.
  15. M. P. Silverman, R. B. Sohn, “Effects of circular birefringence on light propagation and reflection,” Am. J. Phys. 54, 69–76 (1986).
    [CrossRef]
  16. Fresnel exploited the difference in transmittance angles of circularly polarized light in optically active crystalline quartz to fabricate a compound prism for separating out pure LCP and RCP beams, thereby demonstrating for the first time the existence of circularly polarized light.See, for example, E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), pp. 311–312.The corresponding process of separating LCP and RCP components by differential reflection from a chiral medium has not, to our knowledge, been effected yet.
  17. M. P. Silverman, “Specular light scattering from a chiral medium: unambiguous test of gyrotropic constitutive relations,” Lett. Nuovo Cimento 43, 378–382 (1985).
    [CrossRef]
  18. M. P. Silverman, “Reflections 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]
  19. A. Laktakia, V. V. Varadan, V. K. Varadan, “A parametric study of microwave reflection characteristics of a planar achiral–chiral interface,” IEEE Trans. Electromagn. Compat. EMC-28, 90–95 (1986).
    [CrossRef]
  20. I. J. Lalov, A. I. Miteva, “Reflection optical activity of uniaxial media,” J. Chem. Phys. 85, 5505–5511 (1986).
    [CrossRef]
  21. M. P. Silverman, J. Badoz, “Light reflection from a naturally optically active birefringent medium,” J. Opt. Soc. Am. A 7, 1163–1173 (1990).
    [CrossRef]
  22. E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
    [CrossRef]
  23. M. Born, Optik, 3rd ed. (Springer-Verlag, Heidelberg, 1972), pp. 119–126.
  24. 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]
  25. 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]
  26. J. C. Kemp, “Piezo-optical birefringence modulators: new use for a long-known effect,” J. Opt. Soc. Am. 59, 950–954 (1969).
  27. 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]
  28. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), pp. 24–30.
  29. M. P. Silverman, R. F. Cybulski, “Investigation of light amplification by enhanced internal reflection. Part II. Experimental determination of the single-pass reflectance of an optically pumped gain region,” J. Opt. Soc. Am. 73, 1739–1743 (1983).
    [CrossRef]

1994 (1)

Summary reports of this research were given at the 1993 Annual Meeting of the Optical Society of America [M. P. Silverman and J. Badoz, in Multiple Reflection and Interference within a Chiral Medium, Vol. 16 of the OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 115]and in J. Badoz, M. P. Silverman, “Large chiral asymmetries in light reflected from an optically active Fabry–Perot interferometer,” Opt. Commun. 105, 15–21 (1994).
[CrossRef]

1992 (4)

M. P. Silverman, J. Badoz, “Multiple reflection from isotropic chiral media and the enhancement of chiral asymmetry,” J. Electromagn. Waves Appl. 6, 587–601 (1992).

See, for example, J. C. Fabre, A. C. Boccara, “Circular dichroism microspectroscopy: evidence of broken symmetries in high-Tc superconducting films,” Opt. Commun. 93, 306–310 (1992);K. B. Lyons, J. Kwo, J. F. Dillon, G. P. Espinosa, M. McGlashan-Powell, A. P. Ramirez, L. F. Schneemeyer, “Search for circular dichroism in high-Tc superconductors,” Phys. Rev. Lett. 64, 2949–2952 (1990).
[CrossRef] [PubMed]

D. L. Jaggard, X. Sun, “Theory of chiral multilayers,” J. Opt. Soc. Am. A 9, 804–813 (1992).
[CrossRef]

M. P. Silverman, J. Badoz, B. Briat, “Chiral reflection from a naturally optically active medium,” Opt. Lett. 17, 886–888 (1992).
[CrossRef] [PubMed]

1990 (4)

1989 (1)

M. P. Silverman, J. Badoz, “Large enhancement of chiral asymmetry in light reflection near critical angle,” Opt. Commun. 74, 129–133 (1989).
[CrossRef]

1988 (2)

1986 (5)

A. Laktakia, V. V. Varadan, V. K. Varadan, “A parametric study of microwave reflection characteristics of a planar achiral–chiral interface,” IEEE Trans. Electromagn. Compat. EMC-28, 90–95 (1986).
[CrossRef]

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

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

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

M. P. Silverman, “Reflections 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]

1985 (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]

1983 (1)

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]

1969 (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]

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

1937 (1)

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Badoz, J.

Summary reports of this research were given at the 1993 Annual Meeting of the Optical Society of America [M. P. Silverman and J. Badoz, in Multiple Reflection and Interference within a Chiral Medium, Vol. 16 of the OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 115]and in J. Badoz, M. P. Silverman, “Large chiral asymmetries in light reflected from an optically active Fabry–Perot interferometer,” Opt. Commun. 105, 15–21 (1994).
[CrossRef]

M. P. Silverman, J. Badoz, “Multiple reflection from isotropic chiral media and the enhancement of chiral asymmetry,” J. Electromagn. Waves Appl. 6, 587–601 (1992).

M. P. Silverman, J. Badoz, B. Briat, “Chiral reflection from a naturally optically active medium,” Opt. Lett. 17, 886–888 (1992).
[CrossRef] [PubMed]

An analysis of these difficulties is given in J. Badoz, M. P. Silverman, J. C. Canit, “Wave propagation through a medium with static and dynamic birefringence: theory of the photoelastic modulator,” J. Opt. Soc. Am. A 7, 672–682 (1990).
[CrossRef]

M. P. Silverman, J. Badoz, “Light reflection from a naturally optically active birefringent medium,” J. Opt. Soc. Am. A 7, 1163–1173 (1990).
[CrossRef]

M. P. Silverman, J. Badoz, “Large enhancement of chiral asymmetry in light reflection near critical angle,” Opt. Commun. 74, 129–133 (1989).
[CrossRef]

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. P. Silverman, J. Badoz, “Ellipsometric study of specular reflection from a naturally optically active medium,” presented at the First International Conference on Spectroscopic Ellipsometry, Paris, France, January 1993;Thin Solid Films 234491–495 (1993),reprinted in Spectroscopic Ellipsometry, A. C. Boccara, C. Pickering, J. Rivory, eds. (Elsevier Sequoia, Amsterdam, 1993), pp. 491–495.

Bassiri, S.

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]

Boccara, A. C.

See, for example, J. C. Fabre, A. C. Boccara, “Circular dichroism microspectroscopy: evidence of broken symmetries in high-Tc superconducting films,” Opt. Commun. 93, 306–310 (1992);K. B. Lyons, J. Kwo, J. F. Dillon, G. P. Espinosa, M. McGlashan-Powell, A. P. Ramirez, L. F. Schneemeyer, “Search for circular dichroism in high-Tc superconductors,” Phys. Rev. Lett. 64, 2949–2952 (1990).
[CrossRef] [PubMed]

Born, M.

M. Born, Optik, 3rd ed. (Springer-Verlag, Heidelberg, 1972), pp. 119–126.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), pp. 24–30.

Bouchiat, M.-A.

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

Briat, B.

Canit, J. C.

An analysis of these difficulties is given in J. Badoz, M. P. Silverman, J. C. Canit, “Wave propagation through a medium with static and dynamic birefringence: theory of the photoelastic modulator,” J. Opt. Soc. Am. A 7, 672–682 (1990).
[CrossRef]

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]

Condon, E. U.

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Cushman, G. M.

Cybulski, R. F.

Eidner, K.

M. Schmidt, K. Eidner, “Electromagnetic wave propagation through an isotropic chiral slab: solution for oblique incidence,” Optik (Stuttgart) 80, 43–46 (1990).

Engheta, N.

Fabre, J. C.

See, for example, J. C. Fabre, A. C. Boccara, “Circular dichroism microspectroscopy: evidence of broken symmetries in high-Tc superconducting films,” Opt. Commun. 93, 306–310 (1992);K. B. Lyons, J. Kwo, J. F. Dillon, G. P. Espinosa, M. McGlashan-Powell, A. P. Ramirez, L. F. Schneemeyer, “Search for circular dichroism in high-Tc superconductors,” Phys. Rev. Lett. 64, 2949–2952 (1990).
[CrossRef] [PubMed]

Fisher, B.

Hecht, E.

Fresnel exploited the difference in transmittance angles of circularly polarized light in optically active crystalline quartz to fabricate a compound prism for separating out pure LCP and RCP beams, thereby demonstrating for the first time the existence of circularly polarized light.See, for example, E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), pp. 311–312.The corresponding process of separating LCP and RCP components by differential reflection from a chiral medium has not, to our knowledge, been effected yet.

Jaggard, D. L.

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.

Laktakia, A.

A. Laktakia, V. V. Varadan, V. K. Varadan, “A parametric study of microwave reflection characteristics of a planar achiral–chiral interface,” IEEE Trans. Electromagn. Compat. EMC-28, 90–95 (1986).
[CrossRef]

Lalov, I. J.

I. J. Lalov, A. I. Miteva, “Optically active Fabry–Perot etalon,” J. Mod. Opt. 38, 395–411 (1990).
[CrossRef]

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

Miteva, A. I.

I. J. Lalov, A. I. Miteva, “Optically active Fabry–Perot etalon,” J. Mod. Opt. 38, 395–411 (1990).
[CrossRef]

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

Papas, C. H.

Pottier, L.

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

Ritchie, N.

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]

Schmidt, M.

M. Schmidt, K. Eidner, “Electromagnetic wave propagation through an isotropic chiral slab: solution for oblique incidence,” Optik (Stuttgart) 80, 43–46 (1990).

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.

Summary reports of this research were given at the 1993 Annual Meeting of the Optical Society of America [M. P. Silverman and J. Badoz, in Multiple Reflection and Interference within a Chiral Medium, Vol. 16 of the OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 115]and in J. Badoz, M. P. Silverman, “Large chiral asymmetries in light reflected from an optically active Fabry–Perot interferometer,” Opt. Commun. 105, 15–21 (1994).
[CrossRef]

M. P. Silverman, J. Badoz, “Multiple reflection from isotropic chiral media and the enhancement of chiral asymmetry,” J. Electromagn. Waves Appl. 6, 587–601 (1992).

M. P. Silverman, J. Badoz, B. Briat, “Chiral reflection from a naturally optically active medium,” Opt. Lett. 17, 886–888 (1992).
[CrossRef] [PubMed]

An analysis of these difficulties is given in J. Badoz, M. P. Silverman, J. C. Canit, “Wave propagation through a medium with static and dynamic birefringence: theory of the photoelastic modulator,” J. Opt. Soc. Am. A 7, 672–682 (1990).
[CrossRef]

M. P. Silverman, J. Badoz, “Light reflection from a naturally optically active birefringent medium,” J. Opt. Soc. Am. A 7, 1163–1173 (1990).
[CrossRef]

M. P. Silverman, J. Badoz, “Large enhancement of chiral asymmetry in light reflection near critical angle,” Opt. Commun. 74, 129–133 (1989).
[CrossRef]

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, R. B. Sohn, “Effects of circular birefringence on light propagation and reflection,” Am. J. Phys. 54, 69–76 (1986).
[CrossRef]

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

M. P. Silverman, R. F. Cybulski, “Investigation of light amplification by enhanced internal reflection. Part II. Experimental determination of the single-pass reflectance of an optically pumped gain region,” J. Opt. Soc. Am. 73, 1739–1743 (1983).
[CrossRef]

M. P. Silverman, J. Badoz, “Ellipsometric study of specular reflection from a naturally optically active medium,” presented at the First International Conference on Spectroscopic Ellipsometry, Paris, France, January 1993;Thin Solid Films 234491–495 (1993),reprinted in Spectroscopic Ellipsometry, A. C. Boccara, C. Pickering, J. Rivory, eds. (Elsevier Sequoia, Amsterdam, 1993), pp. 491–495.

Sohn, R. B.

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

Sun, X.

Varadan, V. K.

A. Laktakia, V. V. Varadan, V. K. Varadan, “A parametric study of microwave reflection characteristics of a planar achiral–chiral interface,” IEEE Trans. Electromagn. Compat. EMC-28, 90–95 (1986).
[CrossRef]

Varadan, V. V.

A. Laktakia, V. V. Varadan, V. K. Varadan, “A parametric study of microwave reflection characteristics of a planar achiral–chiral interface,” IEEE Trans. Electromagn. Compat. EMC-28, 90–95 (1986).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), pp. 24–30.

Am. J. Phys. (1)

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

IEEE Trans. Electromagn. Compat. (1)

A. Laktakia, V. V. Varadan, V. K. Varadan, “A parametric study of microwave reflection characteristics of a planar achiral–chiral interface,” IEEE Trans. Electromagn. Compat. EMC-28, 90–95 (1986).
[CrossRef]

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. Electromagn. Waves Appl. (1)

M. P. Silverman, J. Badoz, “Multiple reflection from isotropic chiral media and the enhancement of chiral asymmetry,” J. Electromagn. Waves Appl. 6, 587–601 (1992).

J. Mod. Opt. (1)

I. J. Lalov, A. I. Miteva, “Optically active Fabry–Perot etalon,” J. Mod. Opt. 38, 395–411 (1990).
[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. (2)

J. Opt. Soc. Am. A (6)

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. Commun. (3)

M. P. Silverman, J. Badoz, “Large enhancement of chiral asymmetry in light reflection near critical angle,” Opt. Commun. 74, 129–133 (1989).
[CrossRef]

Summary reports of this research were given at the 1993 Annual Meeting of the Optical Society of America [M. P. Silverman and J. Badoz, in Multiple Reflection and Interference within a Chiral Medium, Vol. 16 of the OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 115]and in J. Badoz, M. P. Silverman, “Large chiral asymmetries in light reflected from an optically active Fabry–Perot interferometer,” Opt. Commun. 105, 15–21 (1994).
[CrossRef]

See, for example, J. C. Fabre, A. C. Boccara, “Circular dichroism microspectroscopy: evidence of broken symmetries in high-Tc superconducting films,” Opt. Commun. 93, 306–310 (1992);K. B. Lyons, J. Kwo, J. F. Dillon, G. P. Espinosa, M. McGlashan-Powell, A. P. Ramirez, L. F. Schneemeyer, “Search for circular dichroism in high-Tc superconductors,” Phys. Rev. Lett. 64, 2949–2952 (1990).
[CrossRef] [PubMed]

Opt. Lett. (1)

Optik (Stuttgart) (1)

M. Schmidt, K. Eidner, “Electromagnetic wave propagation through an isotropic chiral slab: solution for oblique incidence,” Optik (Stuttgart) 80, 43–46 (1990).

Rev. Mod. Phys. (1)

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[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]

Science (1)

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

Other (6)

M. P. Silverman, J. Badoz, “Ellipsometric study of specular reflection from a naturally optically active medium,” presented at the First International Conference on Spectroscopic Ellipsometry, Paris, France, January 1993;Thin Solid Films 234491–495 (1993),reprinted in Spectroscopic Ellipsometry, A. C. Boccara, C. Pickering, J. Rivory, eds. (Elsevier Sequoia, Amsterdam, 1993), pp. 491–495.

See, for example, A. Sihvola, ed., Proceedings of “Biisotropics ’93”: Workshop on Novel Microwave Materials, Helsinki University of Technology, Programme of the 1991 North American Radio Science Meeting (University of Western Ontario, London, Ontario, Canada, 1991), pp. 355–364.

We adopt here the standard optical convention (in contrast to that used elsewhere in physics) that the electric field of a left (right) circularly polarized wave rotates to the left (right) of an observer facing the light source. The term helicity refers to the scalar projection of angular momentum of the wave on its linear momentum (or wave vector). Thus a positive helicity (with designated index n+) connotes a parallel orientation of angular and linear momenta and therefore a left circular (or, generally, elliptical) polarization. Correspondingly, right circular polarization (with designated index n−) is equivalent to a negative helicity.

M. Born, Optik, 3rd ed. (Springer-Verlag, Heidelberg, 1972), pp. 119–126.

Fresnel exploited the difference in transmittance angles of circularly polarized light in optically active crystalline quartz to fabricate a compound prism for separating out pure LCP and RCP beams, thereby demonstrating for the first time the existence of circularly polarized light.See, for example, E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), pp. 311–312.The corresponding process of separating LCP and RCP components by differential reflection from a chiral medium has not, to our knowledge, been effected yet.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), pp. 24–30.

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Figures (15)

Fig. 1
Fig. 1

Schematic diagram of reflector consisting of an isotropic chiral medium (indices n± with mean index n2) of thickness d separating achiral media (indices n1 and n3). The incident wave E0 originating in medium 1 is amplitude divided at the 1–2 interface, giving rise to a suite of coherent partial waves r0, r1, r2, etc. that superpose to form the total reflected wave and a corresponding suite of partial waves t1, t2, etc. that superpose to form the total transmitted wave in medium 3.

Fig. 2
Fig. 2

Single-pass (N = 0) reflection as a function of incident angle θ1 (deg) from the achiral–chiral interface: n1 = 1 and n± = 1.33 × (1 ± 10−5). (a) Variation of magnitudes a (×105) and b for an incident p-polarized wave, (b) DCR (×105) with reflectance curve superimposed.

Fig. 3
Fig. 3

Single-pass (N = 0) total reflection as a function of incident angle θ1 (deg) from the achiral–chiral interface: n1 = 1.46 and n± = 1.33 × (1 ± 10−5). The critical angle is 65.6°. (a) Variation of magnitudes a (×103) and b for an incident p-polarized wave, (b) DCR (×103) with reflectance curve superimposed, (c) (×103) with reflectance superimposed, (d) φ (×10 deg) with reflectance superimposed.

Fig. 4
Fig. 4

Variation of reflectance (of an incident p-polarized wave) with incident angle θ1 (deg) for different reflection orders N from the achiral–chiral–achiral system: n1 = 1.46, n± = 1.33 × (1 ± 10−5), n3 = 1; thickness parameter L = 1. The critical angles are θ12 = 65.6° and θ23 = 48.8°.

Fig. 5
Fig. 5

Reflection as a function of incident angle θ1 (deg) from the achiral–chiral–achiral system: n1 = 1, n± = 1.33 × (1 ± 10−5), n3 = 1.46, N = ∞, and L = 1. (a) Variation of magnitudes a (×105) and b for an incident p-polarized wave, (b) DCR (×104) with the reflectance curve superimposed, (c) (×102) and φ (deg) with reflectance superimposed.

Fig. 6
Fig. 6

Total reflection as a function of incident angle θ1 (deg) from the transparent achiral–chiral–achiral system: n1 = 1.46, n± = 1.33 × (1 ± 10−5), n3 = 1, N = ∞, and L = 1. The critical angles are θ12 = 65.6° and θ23 = 48.8°. (a) Variation of magnitudes a (×103) and b for an incident p-polarized wave, (b) DCR (×105) with the reflectance curve superimposed, (c) (×103) with reflectance superimposed, (d) φ (×10 deg) with reflectance superimposed.

Fig. 7
Fig. 7

Total reflection as a function of incident angle θ1 (deg) from the transparent achiral-chiral–achiral system with the same optical parameters as those in Fig. 6, except that L = 10. (a) Variation of magnitudes a (×102) and b for an incident p-polarized wave; (b) (×102) with reflectance superimposed; scan of region of total reflection with 0.75° angular step; (c) φ (deg) with reflectance superimposed; scan of the region of total reflection with a 0.75° angular step.

Fig. 8
Fig. 8

Total reflection as a function of incident angle θ1 (deg) from the transparent achiral–chiral–achiral system with the same optical parameters as those in Fig. 6, except that L = 100. (a) Magnitudes a and b for an incident p-polarized wave; scan of a 0.5° region between critical angles θ12 and θ23 with an angular step of 0.00125°; (b) (×10) with reflectance superimposed; same scan as that in (a); (c) φ (deg) with reflectance superimposed; same scan as that in (a); (d) s-polarized reflectance for an incident p-polarized wave.

Fig. 9
Fig. 9

Total reflection as a function of incident angle θ1 (deg) from an absorbing achiral–chiral–achiral system with optical parameters n1 = 1.46, n± = (1.33 + 5i × 10−4) × [1 ± (1 + 5i) × 10−5], n3 = 1, N = ∞, and L = 10. (a) Reflectance for an incident p-polarized wave; (b) DCR (×10) with reflectance curve superimposed; scan of a 5° sector in angular region between (approximate) critical angles θ23 and θ12; (c) DCR with reflectance curve superimposed; scan of a 1° sector close to critical angle θ12; (d) (×10) with reflectance superimposed; scan of the same range as that in (b).

Fig. 10
Fig. 10

Reflection from a transparent achiral–chiral–achiral system as a function of thickness parameter L with the same refractive indices (and critical angles) as those in Fig. 6. The sample step is ΔL = 10. (a) DCR at incident angle (40°) below θ23 with reflectance superimposed, (b) (×10) at incident angle (40°) below θ23 with reflectance superimposed, (c) φ (deg) at incident angle (40°) below θ23 with reflectance (×100) superimposed, (d) (×10) at incident angle (50°) above θ23 with reflectance superimposed, (e) φ (deg) at incident angle (50°) above θ23 with reflectance (×10) superimposed, (f) s-polarized reflectance for an incident p-polarized wave at angle (50°) above θ23 with reflectance superimposed.

Fig. 11
Fig. 11

s-polarized cross reflectance and total reflectance for a transparent achiral–chiral–achiral system with optical parameters n1 = n3 = 1, n± = 1.33 × (1 ± 10−5), N = ∞. (a) Variation as a function of incident angle θ1 (deg) for L = 5000, (b) variation as a function of thickness parameter L for incident angle 80°.

Fig. 12
Fig. 12

Metallic reflection (at the 2–3 interface) as a function of incident angle θ1 (deg) from an achiral–chiral‐achiral system: n1 = 1, n± = 1.33 × (1 ± 10−5), n3 = 0.18 + 3.64i, N = ∞, and L = 50. (a) Variation in a (×102) and reflectance for an incident p-polarized wave; angular step 0.09°; (b) DCR (×103) with reflectance curve superimposed, full scan; (c) DCR (×103) with reflectance curve superimposed, scan over a 10° range in the region of largest values of a [cf. (a)]; (d) (×103) with reflectance superimposed, full scan; (e) (×103) with reflectance superimposed; scan over the same 10° range of (c); (f) φ (×10°) with reflectance superimposed, full scan; (g) φ (× 10°) with reflectance superimposed, scan over the same 10° sector of (c).

Fig. 13
Fig. 13

Metallic reflection as a function of thickness parameter L from an achiral–chiral–chiral system with the same optical parameters as those in Fig. 12 and an incident angle of 60°; thickness step ΔL = 50. (a) DCR (×102) with reflectance superimposed, (b) (× 10) with reflectance superimposed, (c) φ (deg) with reflectance superimposed.

Fig. 14
Fig. 14

s-polarized cross reflectance (×10) and total reflectance from a system with the same refractive indices as those in Fig. 12. (a) Variation with incident angle for L = 5000, angular step 0.09°; (b) variation with thickness for θ1 = 70°, step ΔL = 10.

Fig. 15
Fig. 15

Alternative perspective (to that of Fig. 1) of interference at a plane-parallel chiral layer of thickness d. The relative phase of two waves issuing from incident waves E1 and E2 and contributing to the reflected wave r is determined from the optical path-length difference AB + BC, which lies entirely within the chiral layer.

Equations (83)

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n ± = n 2 ( 1 ± f ) ,
n 1 sin θ 1 = n ± sin θ ± = n 3 sin θ 3 ,
R = [ r + + r + r + r ] , T = [ t + + t + t + t ]
R = [ r + + r + r + r ] , T = [ t + + t + t + t ] ,
R = [ r + + r + r + r ] , T = [ t + + t + t + t ] ,
R 2 + T T = R 2 + T T = I ,
Δ = [ exp ( i δ + ) 0 0 exp ( i δ ) ] ,
δ ± = ( 2 π n ± d λ ) cos ( θ ± ) .
r [ N ] = r 0 + r 1 + r 2 + + r N 1 = M [ N ] E 0 ,
M [ N ] = R + T ( I + S + S 2 + S 3 + + S N 1 ) × ( Δ R Δ ) T ,
S Δ R Δ R .
M [ N ] = R + T Z ( Δ R Δ ) T ,
Z I S N I S = U Z D U 1 .
Z D = [ 1 s + N 1 s + 0 0 1 s N 1 s ] ;
s ± = ½ ( Tr ( S ) ± { [ Tr ( S ) ] 2 4 det ( S ) } 1 / 2 )
U = [ s + s 22 s 12 s 21 s s 11 ] .
t [ N ] = t 1 + t 2 + + t N 1 = M [ N ] E 0 ,
M [ N ] = T [ I + S + S 2 + S 3 + + ( S ) N 1 ] Δ T = T I S N I S Δ T ,
S = Δ R Δ R .
M [ ] = R + T ( I I S ) ( Δ R Δ ) T ,
M [ ] = T I I S Δ T ,
[ 1 1 1 1 1 1 0 0 β 1 β 1 β + β β + β 0 0 1 1 n 21 n 21 n 21 n 21 0 0 0 β 1 n 21 β + n 21 β n 21 β + n 21 β 0 0 0 0 Δ + Δ Δ + * Δ * Δ 3 Δ 3 0 0 β + Δ + β Δ β + Δ + * β Δ * β 3 Δ 3 β 3 Δ 3 0 0 n 23 Δ + n 23 Δ n 23 Δ + * n 23 Δ * Δ 3 Δ 3 0 0 n 23 β + Δ + n 23 β Δ n 23 β + Δ + * n 23 β Δ * β 3 Δ 3 β 3 Δ 3 ] [ r + r p + p q + q t + t ] = [ ( E + + E ) β 1 ( E + E ) ( E + E ) β 1 ( E + + E ) 0 0 0 0 ] ,
DCR = R + R R + + R ,
R [ N ] + = | M [ N ] ( 1 0 ) | 2 , R [ N ] = | M [ N ] ( 0 1 ) | 2
r = a exp ( i α ) ŝ + b exp ( i β ) p ̂ ,
= { ( a 2 + b 2 ) [ a 4 + b 4 + 2 a 2 b 2 cos ( 2 δ 0 ) 1 / 2 ] ( a 2 + b 2 ) + [ a 4 + b 4 + 2 a 2 b 2 cos ( 2 δ 0 ) 1 / 2 ] } 1 / 2 × sgn [ sin ( δ 0 ) ] ,
δ 0 = α β .
tan ( 2 φ ) = 2 a b cos ( δ 0 ) b 2 a 2 .
M ¯ [ N ] = [ 1 / 2 1 / 2 i / 2 i / 2 ] M [ N ] [ 1 / 2 i / 2 1 / 2 i / 2 ] ,
r [ N ] = ( R + T Z 0 T ) E 0 ,
Z 0 = exp ( i Δ ¯ ) × [ expl ( i δ ) ( Σ + + γ 0 Σ ) γ + Σ γ + Σ exp ( i δ ) ( Σ + γ 0 Σ ) ] ,
Δ ¯ = δ + + δ 2 , δ = δ + δ 2 ,
γ 0 = exp ( i δ ) r + + exp ( i δ ) r { [ exp ( i δ ) r + + exp ( i δ ) r ] 2 + 4 r + r + } 1 / 2 ,
γ + = r + { [ exp ( i δ ) r + + exp ( i δ ) r ] 2 + 4 r + r + } 1 / 2
± = 1 2 [ ( ρ + 1 ρ + 2 ) ± ( ρ 1 ρ 2 ) ] ;
ρ ± = exp ( i Δ ¯ ) 2 ( [ exp ( i δ ) r + + + exp ( i δ ) r ] ± { [ exp ( i δ ) r + + exp ( i δ ) r ] 2 + 4 r + r + } 1 / 2 )
r [ ] = ( 0 r p ) + 2 exp ( i Δ ¯ ) × [ i t 2 t 1 [ + sin δ + ½ γ 0 sin ( 2 δ ) ] i t 2 t 2 { + cos δ [ γ 0 sin 2 δ + γ 1 ] } ] ,
γ 0 = r d ( r n 2 r d 2 sin 2 δ ) 1 / 2 , γ 1 = r n ( r n 2 r d 2 sin 2 δ ) 1 / 2 ,
ρ ± = exp ( i Δ ¯ ) [ r d cos δ ± ( r n 2 r d 2 sin 2 δ ) 1 / 2 ] .
r + + = r = r + + = r = 0 , r + = r + = r + = r + r ; t + + = t t , t + + = t t , t + = t + = t + = t + = 0 ,
M [ ] = r [ 1 t t exp ( 2 i Δ ¯ ) 1 r 2 exp ( 2 i Δ ¯ ) ] [ 0 1 1 0 ] .
n ± > n 1 , n 3 b > n ± > n 3 a
r [ 1 ] = ( r s 0 ) + [ t t exp ( 2 i Δ ¯ ) 2 ] [ μ cos ( 2 δ ) + ν μ sin ( 2 δ ) ] ,
r s = ( n 2 n 1 n 2 + n 1 )
μ = ( r a + r b ) 2 = 1 2 [ ( n 2 n 3 a n 2 + n 3 a ) ( n 2 n 3 b n 2 + n 3 b ) ] ,
ν = ( r a r b ) 2 = 1 2 [ ( n 2 n 3 a n 2 + n 3 a ) + ( n 2 n 3 b n 2 + n 3 b ) ] .
n 3 a = n 3 b r a = r b ,
Δ δ = δ + δ 2 n 2 f L cos θ 2
r + + = 2 q 21 Δ [ ( β 1 2 β + β ) β 1 ( β + β ) ] ,
r + = r + = ( q 21 2 1 ) Δ ( β + + β ) ,
r = 2 q 21 Δ [ ( β 1 2 β + β ) + β 1 ( β + β ) ] ,
β 1 = cos θ 1 , β ± = cos θ ± ,
Δ = ( 2 q 21 ) ( β 1 2 + β + β ) + ( 1 + q 21 2 ) β 1 ( β + + β ) ,
q 21 = n 2 μ 1 n 1 μ 2 ,
r 11 = 2 Δ [ q 21 ( β 1 2 β + β ) β 1 ( β + + β ) ( q 21 2 1 2 ) ] ,
r 21 = r 12 = 2 i q 21 Δ β 1 ( β + + β ) ,
r 22 = 2 Δ [ q 21 ( β 1 2 β + β ) + β 1 ( β + + β ) ( q 21 2 1 2 ) ] .
r + + = ( 1 + q 12 2 ) β 1 ( β + β ) 2 q 12 ( β 1 2 β + β ) Δ ,
r + = 2 ( 1 q 12 2 ) β 1 β + Δ ,
r + = 2 ( 1 q 12 2 ) β 1 β Δ ,
r = [ ( 1 + q 12 2 ) β 1 ( β + β ) + 2 q 12 ( β 1 2 β + β ) ] Δ ,
Δ = 2 q 12 ( β 1 2 + β + β ) + ( 1 + q 12 2 ) β 1 ( β + + β ) = Δ q 21 2 ,
q 12 = n 1 μ 2 n 2 μ 1 = 1 q 21 .
t + + = ( 1 + q 21 ) β 1 ( β 1 + β ) Δ ,
t + = 2 ( q 21 1 ) β 1 ( β 1 β + ) Δ ,
t + = 2 ( q 21 1 ) β 1 ( β 1 β ) Δ ,
t = ( 1 + q 21 ) β 1 ( β 1 + β + ) Δ .
t 1 ± = 2 2 Δ β 1 ( β + q 21 β 1 ) ,
t 2 ± = ± i 2 2 Δ β 1 ( q 21 β + β 1 ) ,
t + + = ( 1 + q 21 ) β + ( β 1 + β ) Δ ,
t + = 2 ( 1 q 12 ) β + ( β 1 β ) Δ ,
t + = 2 ( 1 q 12 ) β ( β 1 β + ) Δ ,
t = 2 ( 1 + q 12 ) β ( β 1 + β + ) Δ .
t ± 1 = 2 2 Δ β ± ( β 1 + q 12 β ) ,
t ± 2 = ± i 2 2 Δ β ± ( β + q 12 β 1 ) ,
β 1 β 3 = cos ( θ 3 ) ,
q 12 q 32 = n 3 μ 2 n 2 μ 3 .
β 1 ( 1 | r + + | 2 | r + | 2 ) q 21 ( β + | t + + | 2 + β | t + | 2 ) = 0 ;
β 1 ( 1 | r + | 2 | r | 2 ) q 21 ( β + | t + | 2 + β | t | 2 ) = 0 .
β + ( 1 | r + + | 2 ) β | r + | 2 q 12 β 1 ( | t + + | 2 + | t + | 2 ) = 0 ;
β ( 1 | r | 2 ) β + | r + | 2 q 12 β 1 ( | t + | 2 + | t | 2 ) = 0 .
δ = 2 π d λ ( n + A B ¯ + n B C ¯ ) = 2 π d λ [ n + cos ( θ + + θ ) cos ( θ ) + n cos ( θ ) ] .
δ = 2 π d λ ( n + cos θ + + n cos θ ) = δ + + δ ,

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