Abstract

Fresnel amplitudes for specular reflection and refraction at the surface of an isotropic, intrinsically nonmagnetic chiral medium are derived for sets of constitutive relations that are invariant or noninvariant under a duality transformation of the electromagnetic fields. The invariant set leads to a differential reflection curve of incident left and right circularly polarized light that is null at normal incidence and peaks beyond Brewster’s angle; the noninvariant set leads to maximum differential reflection in the vicinity of normal incidence and extends over a wide range of incident angles. Both sets lead to effectively equivalent descriptions of standard optical rotation and circular dichroism.

© 1986 Optical Society of America

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References

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  1. L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge, New York, 1982).
  2. A. Fresnel, Bull. Soc. Philomat. 9, 150 (1824); cited in S. F. Mason, Molecular Optical Activity and the Chiral Discriminations (Cambridge, New York, 1982), p. 3.
  3. See, for example, L. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960), pp. 332–343.
  4. O. S. Heavens, “Fundamentals of electro- and magneto-optics and nonlinear optics,” in Progress in Electro-Optics: Reviews of Recent Developments, E. Camatini, ed. (Plenum, New York, 1974), p. 24.
  5. That this transformation is possible is alluded to in Ref. 3, although no explicit transformation is shown. One such transformation is given in R. M. Hornreich, S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev. 171, 1065–1074 (1968).
    [CrossRef]
  6. M. Born, Optik (Springer-Verlag, Heidelberg, 1972), p. 412.
  7. G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart & Winston, New York, 1975), pp. 185–189.
  8. O. S. Eritsyan, “Optical problems in the electrodynamics of gyrotropic media,” Sov. Phys. Usp. 25, 919–935 (1982).
    [CrossRef]
  9. B. V. Bokut, F. I. Federov, “Reflection and refraction of light in optically isotropic active media,” Opt. Spektrosk. 9, 334–336 (1960).
  10. B. V. Bokut, B. A. Sotski, “The passage of light through an optically active absorbing plate,” Opt. Spektrosk. 14, 117–120 (1963).
  11. F. I. Federov, B. V. Bokut, A. F. Konstantinova, “Optical activity of the planar classes of the middle groups,” Sov. Phys. Crystallogr. 7, 738–744 (1963).
  12. E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
    [CrossRef]
  13. Yu. Tsvirko, M. A. Tolmazina, “On the boundary conditions for electromagnetic waves at the surface of an optically active crystal,” Sov. Phys. Solid State 3, 1011–1015 (1961).
  14. R. M. Hornreich, S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev. 171, 1065–1074 (1968).
    [CrossRef]
  15. V. M. Agranovich, V. I. Ydson, “Transition layer effects in gyrotropic and nongyrotropic media,” Opt. Commun. 5, 422–424 (1975).
    [CrossRef]
  16. B. V. Bokut, A. N. Serdyukov, “On the phenomenological theory of natural optical activity,” Sov. Phys. JETP 34, 962–964 (1972).
  17. V. M. Agranovich, V. L. Ginzburg, “Phenomenological electrodynamics of gyrotropic media,” Sov. Phys. JETP 36, 440–443 (1973).
  18. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), pp. 17–22.
  19. To the exent that an analysis encompasses a broader theoretical framework than that of macroscopic electrodynamics alone, e.g., the use of coupled Maxwell–Schrödinger equations to investigate a particular microscopic polarization model, the complete specification of the characteristic waves of a medium may necessitate supplementary boundary conditions. See, for example, J. J. Hopfield, D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev. 132, 563–572 (1963).
    [CrossRef]
  20. M. P. Silverman, “Specular light scattering from a chiral medium: unambiguous test of gyrotropic constitutive relations,” Lett. Nuovo Cimento 43, 378–382 (1985).
    [CrossRef]
  21. M. P. Silverman, “Test of gyrotropic constitutive relations by specular light reflection,” J. Opt. Soc. Am. A 2(13), P99 (1985).
  22. See, for example, W. Kaufman, Quantum Chemistry (Academic, New York, 1957), pp. 616–636; H. Eyring, J. Walter, G. Kimball, Quantum Chemistry (Wiley, New York, 1944), pp. 332–347, and references therein.
  23. A. I. Mahan, C. V. Bitterli, “Total internal reflection: a deeper look,” Appl. Opt. 17, 509–519 (1978).
    [CrossRef] [PubMed]
  24. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 44.
  25. R. F. Cybulski, M. P. Silverman, “Enhanced internal reflection from an exponential amplifying region,” Opt. Lett. 8, 142–144 (1983).
    [CrossRef] [PubMed]
  26. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 252.
  27. C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), pp. 105–110.
  28. Application of the duality transformation, Eq. (35), to the symmetric constitutive relations given by Eqs. (1a) and (1b) leads to an additional term in Eq. (1a) proportional to H and to an additional term in Eq. (1b) proportional to E. Quantum mechanical analysis of a chiral medium does predict these terms; their contribution to the refractive index, however, is second order in the gyrotropic parameter. See H. Eyring, J. Walter, G. Kimball, Quantum Chemistry (Wiley, New York, 1944) pp. 342–347, and references therein.

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]

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

1983 (1)

1982 (1)

O. S. Eritsyan, “Optical problems in the electrodynamics of gyrotropic media,” Sov. Phys. Usp. 25, 919–935 (1982).
[CrossRef]

1978 (1)

1975 (1)

V. M. Agranovich, V. I. Ydson, “Transition layer effects in gyrotropic and nongyrotropic media,” Opt. Commun. 5, 422–424 (1975).
[CrossRef]

1973 (1)

V. M. Agranovich, V. L. Ginzburg, “Phenomenological electrodynamics of gyrotropic media,” Sov. Phys. JETP 36, 440–443 (1973).

1972 (1)

B. V. Bokut, A. N. Serdyukov, “On the phenomenological theory of natural optical activity,” Sov. Phys. JETP 34, 962–964 (1972).

1968 (2)

That this transformation is possible is alluded to in Ref. 3, although no explicit transformation is shown. One such transformation is given in R. M. Hornreich, S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev. 171, 1065–1074 (1968).
[CrossRef]

R. M. Hornreich, S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev. 171, 1065–1074 (1968).
[CrossRef]

1963 (3)

B. V. Bokut, B. A. Sotski, “The passage of light through an optically active absorbing plate,” Opt. Spektrosk. 14, 117–120 (1963).

F. I. Federov, B. V. Bokut, A. F. Konstantinova, “Optical activity of the planar classes of the middle groups,” Sov. Phys. Crystallogr. 7, 738–744 (1963).

To the exent that an analysis encompasses a broader theoretical framework than that of macroscopic electrodynamics alone, e.g., the use of coupled Maxwell–Schrödinger equations to investigate a particular microscopic polarization model, the complete specification of the characteristic waves of a medium may necessitate supplementary boundary conditions. See, for example, J. J. Hopfield, D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev. 132, 563–572 (1963).
[CrossRef]

1961 (1)

Yu. Tsvirko, M. A. Tolmazina, “On the boundary conditions for electromagnetic waves at the surface of an optically active crystal,” Sov. Phys. Solid State 3, 1011–1015 (1961).

1960 (1)

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

1937 (1)

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

1824 (1)

A. Fresnel, Bull. Soc. Philomat. 9, 150 (1824); cited in S. F. Mason, Molecular Optical Activity and the Chiral Discriminations (Cambridge, New York, 1982), p. 3.

Agranovich, V. M.

V. M. Agranovich, V. I. Ydson, “Transition layer effects in gyrotropic and nongyrotropic media,” Opt. Commun. 5, 422–424 (1975).
[CrossRef]

V. M. Agranovich, V. L. Ginzburg, “Phenomenological electrodynamics of gyrotropic media,” Sov. Phys. JETP 36, 440–443 (1973).

Barron, L. D.

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge, New York, 1982).

Bitterli, C. V.

Bokut, B. V.

B. V. Bokut, A. N. Serdyukov, “On the phenomenological theory of natural optical activity,” Sov. Phys. JETP 34, 962–964 (1972).

B. V. Bokut, B. A. Sotski, “The passage of light through an optically active absorbing plate,” Opt. Spektrosk. 14, 117–120 (1963).

F. I. Federov, B. V. Bokut, A. F. Konstantinova, “Optical activity of the planar classes of the middle groups,” Sov. Phys. Crystallogr. 7, 738–744 (1963).

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

Born, M.

M. Born, Optik (Springer-Verlag, Heidelberg, 1972), p. 412.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 44.

Condon, E. U.

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

Cybulski, R. F.

Eritsyan, O. S.

O. S. Eritsyan, “Optical problems in the electrodynamics of gyrotropic media,” Sov. Phys. Usp. 25, 919–935 (1982).
[CrossRef]

Eyring, H.

Application of the duality transformation, Eq. (35), to the symmetric constitutive relations given by Eqs. (1a) and (1b) leads to an additional term in Eq. (1a) proportional to H and to an additional term in Eq. (1b) proportional to E. Quantum mechanical analysis of a chiral medium does predict these terms; their contribution to the refractive index, however, is second order in the gyrotropic parameter. See H. Eyring, J. Walter, G. Kimball, Quantum Chemistry (Wiley, New York, 1944) pp. 342–347, and references therein.

Federov, F. I.

F. I. Federov, B. V. Bokut, A. F. Konstantinova, “Optical activity of the planar classes of the middle groups,” Sov. Phys. Crystallogr. 7, 738–744 (1963).

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

Fowles, G. R.

G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart & Winston, New York, 1975), pp. 185–189.

Fresnel, A.

A. Fresnel, Bull. Soc. Philomat. 9, 150 (1824); cited in S. F. Mason, Molecular Optical Activity and the Chiral Discriminations (Cambridge, New York, 1982), p. 3.

Ginzburg, V. L.

V. M. Agranovich, V. L. Ginzburg, “Phenomenological electrodynamics of gyrotropic media,” Sov. Phys. JETP 36, 440–443 (1973).

Heavens, O. S.

O. S. Heavens, “Fundamentals of electro- and magneto-optics and nonlinear optics,” in Progress in Electro-Optics: Reviews of Recent Developments, E. Camatini, ed. (Plenum, New York, 1974), p. 24.

Hopfield, J. J.

To the exent that an analysis encompasses a broader theoretical framework than that of macroscopic electrodynamics alone, e.g., the use of coupled Maxwell–Schrödinger equations to investigate a particular microscopic polarization model, the complete specification of the characteristic waves of a medium may necessitate supplementary boundary conditions. See, for example, J. J. Hopfield, D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev. 132, 563–572 (1963).
[CrossRef]

Hornreich, R. M.

R. M. Hornreich, S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev. 171, 1065–1074 (1968).
[CrossRef]

That this transformation is possible is alluded to in Ref. 3, although no explicit transformation is shown. One such transformation is given in R. M. Hornreich, S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev. 171, 1065–1074 (1968).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), pp. 17–22.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 252.

Kaufman, W.

See, for example, W. Kaufman, Quantum Chemistry (Academic, New York, 1957), pp. 616–636; H. Eyring, J. Walter, G. Kimball, Quantum Chemistry (Wiley, New York, 1944), pp. 332–347, and references therein.

Kimball, G.

Application of the duality transformation, Eq. (35), to the symmetric constitutive relations given by Eqs. (1a) and (1b) leads to an additional term in Eq. (1a) proportional to H and to an additional term in Eq. (1b) proportional to E. Quantum mechanical analysis of a chiral medium does predict these terms; their contribution to the refractive index, however, is second order in the gyrotropic parameter. See H. Eyring, J. Walter, G. Kimball, Quantum Chemistry (Wiley, New York, 1944) pp. 342–347, and references therein.

Konstantinova, A. F.

F. I. Federov, B. V. Bokut, A. F. Konstantinova, “Optical activity of the planar classes of the middle groups,” Sov. Phys. Crystallogr. 7, 738–744 (1963).

Landau, L.

See, for example, L. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960), pp. 332–343.

Lifshitz, E. M.

See, for example, L. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960), pp. 332–343.

Mahan, A. I.

Misner, C. W.

C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), pp. 105–110.

Serdyukov, A. N.

B. V. Bokut, A. N. Serdyukov, “On the phenomenological theory of natural optical activity,” Sov. Phys. JETP 34, 962–964 (1972).

Shtrikman, S.

R. M. Hornreich, S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev. 171, 1065–1074 (1968).
[CrossRef]

That this transformation is possible is alluded to in Ref. 3, although no explicit transformation is shown. One such transformation is given in R. M. Hornreich, S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev. 171, 1065–1074 (1968).
[CrossRef]

Silverman, M. P.

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, “Test of gyrotropic constitutive relations by specular light reflection,” J. Opt. Soc. Am. A 2(13), P99 (1985).

R. F. Cybulski, M. P. Silverman, “Enhanced internal reflection from an exponential amplifying region,” Opt. Lett. 8, 142–144 (1983).
[CrossRef] [PubMed]

Sotski, B. A.

B. V. Bokut, B. A. Sotski, “The passage of light through an optically active absorbing plate,” Opt. Spektrosk. 14, 117–120 (1963).

Thomas, D. G.

To the exent that an analysis encompasses a broader theoretical framework than that of macroscopic electrodynamics alone, e.g., the use of coupled Maxwell–Schrödinger equations to investigate a particular microscopic polarization model, the complete specification of the characteristic waves of a medium may necessitate supplementary boundary conditions. See, for example, J. J. Hopfield, D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev. 132, 563–572 (1963).
[CrossRef]

Thorne, K. S.

C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), pp. 105–110.

Tolmazina, M. A.

Yu. Tsvirko, M. A. Tolmazina, “On the boundary conditions for electromagnetic waves at the surface of an optically active crystal,” Sov. Phys. Solid State 3, 1011–1015 (1961).

Tsvirko, Yu.

Yu. Tsvirko, M. A. Tolmazina, “On the boundary conditions for electromagnetic waves at the surface of an optically active crystal,” Sov. Phys. Solid State 3, 1011–1015 (1961).

Walter, J.

Application of the duality transformation, Eq. (35), to the symmetric constitutive relations given by Eqs. (1a) and (1b) leads to an additional term in Eq. (1a) proportional to H and to an additional term in Eq. (1b) proportional to E. Quantum mechanical analysis of a chiral medium does predict these terms; their contribution to the refractive index, however, is second order in the gyrotropic parameter. See H. Eyring, J. Walter, G. Kimball, Quantum Chemistry (Wiley, New York, 1944) pp. 342–347, and references therein.

Wheeler, J. A.

C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), pp. 105–110.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 44.

Ydson, V. I.

V. M. Agranovich, V. I. Ydson, “Transition layer effects in gyrotropic and nongyrotropic media,” Opt. Commun. 5, 422–424 (1975).
[CrossRef]

Appl. Opt. (1)

Bull. Soc. Philomat. (1)

A. Fresnel, Bull. Soc. Philomat. 9, 150 (1824); cited in S. F. Mason, Molecular Optical Activity and the Chiral Discriminations (Cambridge, New York, 1982), p. 3.

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

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

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. (1)

V. M. Agranovich, V. I. Ydson, “Transition layer effects in gyrotropic and nongyrotropic media,” Opt. Commun. 5, 422–424 (1975).
[CrossRef]

Opt. Lett. (1)

Opt. Spektrosk. (2)

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

B. V. Bokut, B. A. Sotski, “The passage of light through an optically active absorbing plate,” Opt. Spektrosk. 14, 117–120 (1963).

Phys. Rev. (3)

That this transformation is possible is alluded to in Ref. 3, although no explicit transformation is shown. One such transformation is given in R. M. Hornreich, S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev. 171, 1065–1074 (1968).
[CrossRef]

R. M. Hornreich, S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev. 171, 1065–1074 (1968).
[CrossRef]

To the exent that an analysis encompasses a broader theoretical framework than that of macroscopic electrodynamics alone, e.g., the use of coupled Maxwell–Schrödinger equations to investigate a particular microscopic polarization model, the complete specification of the characteristic waves of a medium may necessitate supplementary boundary conditions. See, for example, J. J. Hopfield, D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev. 132, 563–572 (1963).
[CrossRef]

Rev. Mod. Phys. (1)

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

Sov. Phys. Crystallogr. (1)

F. I. Federov, B. V. Bokut, A. F. Konstantinova, “Optical activity of the planar classes of the middle groups,” Sov. Phys. Crystallogr. 7, 738–744 (1963).

Sov. Phys. JETP (2)

B. V. Bokut, A. N. Serdyukov, “On the phenomenological theory of natural optical activity,” Sov. Phys. JETP 34, 962–964 (1972).

V. M. Agranovich, V. L. Ginzburg, “Phenomenological electrodynamics of gyrotropic media,” Sov. Phys. JETP 36, 440–443 (1973).

Sov. Phys. Solid State (1)

Yu. Tsvirko, M. A. Tolmazina, “On the boundary conditions for electromagnetic waves at the surface of an optically active crystal,” Sov. Phys. Solid State 3, 1011–1015 (1961).

Sov. Phys. Usp. (1)

O. S. Eritsyan, “Optical problems in the electrodynamics of gyrotropic media,” Sov. Phys. Usp. 25, 919–935 (1982).
[CrossRef]

Other (11)

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge, New York, 1982).

M. Born, Optik (Springer-Verlag, Heidelberg, 1972), p. 412.

G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart & Winston, New York, 1975), pp. 185–189.

See, for example, L. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960), pp. 332–343.

O. S. Heavens, “Fundamentals of electro- and magneto-optics and nonlinear optics,” in Progress in Electro-Optics: Reviews of Recent Developments, E. Camatini, ed. (Plenum, New York, 1974), p. 24.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), pp. 17–22.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 252.

C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), pp. 105–110.

Application of the duality transformation, Eq. (35), to the symmetric constitutive relations given by Eqs. (1a) and (1b) leads to an additional term in Eq. (1a) proportional to H and to an additional term in Eq. (1b) proportional to E. Quantum mechanical analysis of a chiral medium does predict these terms; their contribution to the refractive index, however, is second order in the gyrotropic parameter. See H. Eyring, J. Walter, G. Kimball, Quantum Chemistry (Wiley, New York, 1944) pp. 342–347, and references therein.

See, for example, W. Kaufman, Quantum Chemistry (Academic, New York, 1957), pp. 616–636; H. Eyring, J. Walter, G. Kimball, Quantum Chemistry (Wiley, New York, 1944), pp. 332–347, and references therein.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 44.

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Figures (4)

Fig. 1
Fig. 1

Geometric configuration of the system. An incident linearly polarized wave in a nongyrotropic medium gives rise to two waves of opposite helicity in the gyrotropic medium. The unit vector ŝ normal to the plane of incidence; the unit vectors p ^ , p ^ , p ^ lie in the plane of incidence.

Fig. 2
Fig. 2

Differential reflection curves (solid lines) for incident LCP, RCP light for the case n± ≡ = n/n1 > 1. The individual LCP, RCP reflection curves (dashed line) are not distinguishable at this scale and are simply designated CP.

Fig. 3
Fig. 3

Individual LCP, RCP reflection curves (dashed lines) and differential reflection curves (solid lines) for the case n+ > 1 > n. (a) Calculations based on constitutive relations (I). (b) Calculations based on constitutive relations (II).

Fig. 4
Fig. 4

Individual LCP, RCP reflection curves (dashed lines) and differential reflection curves (solid lines) for the case 1 > n±. (a) Calculations based on constitutive relations (I). (b) Calculations based on constitutive relations (II).

Equations (90)

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D = E - g H / t ,
B = μ H + g E / t ,
D = [ E + ( c g / μ ) curl E + ( g 2 / μ ) 2 E / t 2 ] ,
B = μ [ H + ( c g / μ ) curl H + ( g 2 / μ ) 2 H / t 2 ] .
( I )             D = [ ( 1 - f 2 ) E + i f ( k × E ) / n k 0 ] ,
B = μ [ ( 1 - f 2 ) H + i f ( k × H ) / n k 0 ] ,
H = k × E / ( μ k 0 ) + i ( w g / μ ) E ,
E = - k × H / ( k 0 ) - i ( w g / ) H ,
[ ( k / n k 0 ) 2 - ( 1 - f 2 ) ] E - 2 i f ( k / k 0 ) × E = 0 ,
E ± ( r , t ) = E [ i β ± / n ± , 1 , ± i α ± / n ± ] × exp [ i k 0 ( α ± x + β ± z - w t ) ] ,
H ± ( r , t ) = i ( n / μ ) E ± ( r , t ) = i ( n ± / μ ± ) E ± ( r , t ) ,
k ± = k 0 ( α ± , 0 , β ± )
n ± = n ( 1 ± f ) .
D ± = ± E ± ,
B ± = μ ± H ± ,
± = ( 1 ± f ) ,
μ ± = μ ( 1 ± f ) .
n ± = ( ± μ ± ) 1 / 2 .
k ± = n ± k 0 ( sin ϕ ± , 0 , cos ϕ ± ) ;
β ± = i ( α ± 2 - n ± 2 ) 1 / 2 ;
( II )             D = [ E - i f ( k ^ × E ) ] ,
B = μ H ,
H = k × E / ( μ k 0 )
E = - μ k 0 k × H / k 2 - k × H / ( k 0 ) ,
[ ( k / n k 0 ) 2 - 1 ] E - i f ( k ^ × E ) = 0 ,
n ± = n [ 1 ± f ] 1 / 2 ,
± = ( 1 ± f ) ,
n ± = [ ± μ ] 1 / 2 .
H ± = i ( n ± / μ ) E ± ,
D ± = ± E ± ,
B ± = μ H ± .
β ± = i [ α ± 2 - n 2 ( 1 - ½ f 2 ) ½ n f ( 4 α ± 2 + n 2 f 2 ) 1 / 2 ] ,
˜ = r + i i ,             g ˜ = g r + i g i ;
n ˜ = n + i κ = ( ˜ μ ) 1 / 2 ,             f ˜ = f r + i f i = w g ˜ / n ˜ .
E ± ( r , t ) = E [ i ( β ± + i γ ± ) / n ± , 1 , ± i α ± / n ± ] exp ( - k 0 γ ± z ) × exp [ i k 0 ( α ± x + β ± z ) ] exp ( - i w t ) ,
H ± ( r , t ) = i ( n ˜ / μ ) E ± ( r , t ) ,
n ˜ ± = n ˜ ( 1 ± f ˜ ) = ( ˜ μ ) 1 / 2 ± w g ˜
β ± = { [ ( P ± - α ± 2 ) 2 + Q ± 2 ] 1 / 2 + ( P ± - α ± 2 ) } 1 / 2 / 2 ,
γ ± = { [ ( P ± - α ± 2 ) + Q ± 2 ] 1 / 2 - ( P ± - α ± 2 ) } 1 / 2 / 2 ,
P ± = ( n 2 - κ 2 ) [ ( 1 ± f r ) 2 - f i 2 ] 4 n κ f i ( 1 ± f r ) ,
Q ± = 2 [ n ( 1 ± f r ) κ f i ] [ κ ( 1 ± f r ) ± n f i ] .
α ± = n 1 sin θ ,
α ± = n 2 ± sin ϕ ± .
r i K = a i K e ^ + b i K e ^ ,
t i ± K = A i ± K E ± ,
a 1 ( I ) = [ x 2 - ½ ( z + + z - ) ( q - q - 1 ) x - z + z - ] / D ,
b 1 ( I ) = i ( z + - z - ) x / D ,
a 2 ( I ) = i ( z - - z + ) x / D ,
b 2 ( I ) = [ x 2 + ½ ( z + + z - ) ( q - q - 1 ) x - z + z - ] / D ,
A 1 + ( I ) = ( z - + q x ) x / q D ,
A 1 - ( I ) = ( z + + q x ) x / q D ,
A 2 + ( I ) = i ( q z - + x ) x / q D ,
A 2 - ( I ) = - i ( q z + + x ) x / q D ,
D = x 2 + ½ ( z + - z - ) ( q + q - 1 ) x + z + z - ,
x = cos θ ,
z ± = β ± / n 2 ± = [ 1 - ( n 1 sin θ / n 2 ± ) 2 ] 1 / 2 ;
q = n 2 μ 1 / n 1 μ 2 ;
a 1 ( II ) = [ x 2 - ( z + - z - ) ( q + q - - 1 q + + q - ) x - z + z - ] / D ,
b 1 ( II ) = 2 i x ( q - z + - q + z - ) / ( q + + q - ) D ,
a 2 ( II ) = 2 i x ( q - z - - q + z + ) / ( q + + q - ) D ,
b 2 ( II ) = [ x 2 + ( z + + z 1 ) ( q + q - - 1 q + + q - ) x - z + z - ] / D ,
A 1 + ( II ) = 2 x ( z - + q - x ) / ( q + + q - ) D ,
A 1 - ( II ) = 2 x ( z + + q + x ) / ( q + + q - ) D ,
A 2 + ( II ) = 2 i x ( q - z - + x ) / ( q + + q - ) D ,
A 2 - ( II ) = - 2 i x ( q + z + + x ) / ( q + + q - ) D ,
D = x 2 + ( z + + z - ) ( q + q - + 1 q + + q - ) x + z + z - ,
q ± = n 2 ± μ 1 / n 1 μ 2 .
R TE K = a 1 K 2 + b 1 K 2 ,
R TM K = a 2 K 2 + b 2 K 2
R LCP RCP K = ½ [ a 1 K i a 2 K 2 + b 2 K ± i b 1 K 2 ] .
ρ K = ( R LCP K - R RCP K ) / ( R LCP K + R RCP K ) .
ρ K 2 ( a 1 K α 2 K - b 2 K β 1 K ) / [ ( a 1 K ) 2 + ( b 2 K ) 2 ] .
ρ m ( I ) ~ 4 f [ m / ( m 2 - 1 ) ] s 4 [ ( 1 - s 2 ) / ( m 2 - s 2 ) ] 1 / 2 / × [ s 4 + ( 1 - s 2 ) ( m 2 - s 2 ) ]
s 2 = sin 2 θ m = [ ( 5 m 2 + 3 ) / 8 ] × { 1 - [ 1 - 64 m 2 / ( 5 m 2 + 3 ) 2 ] 1 / 2 }
m = n 2 / n 1 ,
ρ m ( II ) = [ 2 m / m 2 - 1 ] f
n = n ( 1 + f 2 ) ,
f = 2 f / ( 1 + f 2 ) .
T ( I ) = q [ F + I 2 Re ( z + ) + F - ( I ) 2 Re ( z - ) ] / cos θ ,
( TE )             F + ( I ) = 2 A 1 + ( I ) ,             F - ( I ) = 2 A 1 - ( I ) ;
( TM )             F + ( I ) = 2 A 2 + ( I ) ,             F - ( I ) = 2 A 2 - ( I ) ;
( LCP ) F + ( I ) = A 1 + ( I ) - A 2 + ( I ) ,             F - ( I ) = A 1 - ( I ) - i A 2 - ( I ) ;
( RCP ) F + ( I ) = A 1 + ( I ) + i A 2 + ( I ) ,             F - ( I ) = A 1 - ( I ) + i A 2 - ( I ) .
T ( II ) = { q + F + ( II ) 2 Re ( z + ) + q - F - ( II ) 2 Re ( z - ) - ½ ( q x + q - ) Re [ F - ( II ) F + ( II ) * ( z - - z + * ) ] } / cos θ ,
sin ϕ ± = α ± k 0 / k ± = 2 α ± { ( P ± + α ± 2 ) + [ ( P ± - α ± 2 ) 2 + Q ± 2 ] 1 / 2 } 1 / 2 .
z ± = ( β ± + i γ ± ) / n ˜ ± ,
q ˜ = n ˜ 2 μ 1 / n 1 μ 2 ,
P = P - μ curl M ,
M = M + ( u / c ) M / t ,
( F 1 F 2 ) = ( cos t sin t - sin t cos t ) ( F 1 F 2 ) ,

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