Abstract

Exact solutions are obtained for the reflected and transmitted fields resulting when an arbitrary electromagnetic field is incident on a plane interface separating an isotropic medium and a biaxially anisotropic medium in which one of the principal axes is along the interface normal. From our exact solutions for the reflected fields resulting when a plane TE or TM wave is incident on the plane interface, it can be inferred that the reflected field contains both a TE and a TM component. This gives a change in polarization that can be utilized to determine the properties of the biaxial medium. The time-harmonic solution for the reflected field is in the form of two quadruple integrals, one of which is a superposition of plane waves polarized perpendicular to the plane of incidence and the other a superposition of plane waves polarized parallel to the plane of incidence. The time-harmonic solution for the transmitted field is also in the form of two quadruple integrals. Each of these is a superposition of extraordinary plane waves with displacement vectors that are perpendicular to the direction of phase propagation.

© 2001 Optical Society of America

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References

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  1. C. E. Curry, Electromagnetic Theory of Light (Maclehose, London, 1905), pp. 356–369.
  2. R. E. Collins, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), pp. 101–106.
  3. A. M. Goncharenko, F. J. Federov, “Optical properties of crystalline plates,” Opt. Spectrosk. 14, 94–99 (1962).
  4. A. Wünsche, “Neue Formeln für die Reflexion und Brechung des Lichtes an anisotropen Medien” (“New formulas for reflection and refraction of light in anisotropic media”), Ann. Phys. (Leipzig) 25, 201–214 (1970).
    [CrossRef]
  5. J. Mac Cullagh, “On the dynamical theory of crystalline reflection and refraction,” Trans. R. Ir. Acad. (Dublin) 21, 17–50 (1848).
  6. G. Szivessy, “Licht als Wellenbewegung” (“Light as wave motion”), in Handbuch der Physik, H. Geiger, K. Scheel, eds. (Springer-Verlag, Berlin, 1928), Vol. 20, Chap. 11, pp. 635–904.
  7. J. Schesser, G. Eichmann, “Propagation of plane waves in biaxially anisotropic layered media,” J. Opt. Soc. Am. 62, 786–791 (1972).
    [CrossRef]
  8. H. Schopper, Handbuch der Physik (condensed by Heavens and Butterworths, London, 1928), Vol. 132, pp. 92–95.
  9. A. B. Winterbottom, “Optical studies of metal surfaces,” K. Norske Vidensk. Selsk. Skr. 1, 27,37 (1955).
  10. A. Wünsche, “Exacte Berechnung der Greenschen Tensor Funktionen zum Huygensschen Prinzip für optisch einachsige Medien” (“Exact calculation of the Green’s tensor function to obtain Huygens’s principle for optically uniaxial media”), Ann. Phys. (Leipzig) 25, 179–200 (1970).
  11. E. Lalor, “The angular-spectrum representation of electromagnetic fields in crystals. II. Biaxial crystals,” J. Math. Phys. 13, 443–449 (1972).
    [CrossRef]
  12. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, London, 1973).
  13. M. Lax, D. F. Nelson, “Linear and non linear electro dynamics in elastic anisotropic dielctrics,” Phys. Rev. A 13, 443–449 (1971).
  14. J. J. Stamnes, G. C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. 66, 780–788 (1976).
    [CrossRef]
  15. J. J. Stamnes, G. C. Sherman, “Radiation of electromagnetic fields in biaxially anisotropic media,” J. Opt. Soc. Am. 68, 502–508 (1978).
    [CrossRef]
  16. J. Gasper, G. C. Sherman, J. J. Stamnes, “Reflection and refraction of an arbitrary wave at a plane interface,” J. Opt. Soc. Am. 66, 955–961 (1976).
    [CrossRef]
  17. J. J. Stamnes, G. C. Sherman, “Reflection and refraction of an arbitrary wave at a plane interface separating two uniaxial crystals,” J. Opt. Soc. Am. 67, 683–695 (1977).
    [CrossRef]
  18. H. Ling, S. W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
    [CrossRef]
  19. P. Török, P. Varga, Z. Laczic, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
    [CrossRef]
  20. P. Török, P. Varga, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
    [CrossRef]
  21. P. Török, P. Varga, G. Nemeth, “Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. A 12, 2660–2671 (1995).
    [CrossRef]
  22. T. D. Visser, S. H. Wiersma, “Defocusing of a converging electromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. A 13, 320–325 (1996).
    [CrossRef]
  23. S. H. Wiersma, P. Török, T. D. Visser, P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997).
    [CrossRef]
  24. V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
    [CrossRef]
  25. J. J. Stamnes, D. Jiang, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
    [CrossRef]
  26. D. Jiang, J. J. Stamnes, “Theoretical and experimentalresults for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
    [CrossRef]
  27. V. Dhayalan, J. J. Stamnes, “Comparison of exact asymptotic results for the focusing of electromagnetic waves through a plane interface,” Appl. Opt. 39, 6332–6340 (2001).
    [CrossRef]
  28. J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
    [CrossRef]
  29. D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
    [CrossRef]
  30. D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into a uniaxial crystal,” Opt. Commun. 174, 321–334 (2000).
    [CrossRef]
  31. J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. Condens. Matter 3, 6121–6133 (1991).
    [CrossRef]
  32. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 15.2.2.
  33. Ref. 32, Eqs. (15.32b), (15.32c), (15.34c), and (15.34d) with sˆ=eˆz. Note that E˜i(kt) equals (2π)2 times the quantity Ei(ki,ω) in Eqs. (15.32d) and (15.32e).
  34. Ref. 32, Eqs. (15.32d) and (15.32e) with q=i and sˆ=eˆz.
  35. Ref. 32, Eqs. (15.37c) and (15.37d) with sˆ=eˆz and with a factor 1/π omitted, since we are considering a time-harmonic field.
  36. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  37. E. Lalor, “An analytical approach to the theory of internal conical refraction,” J. Math. Phys. 13, 449–454 (1972).
    [CrossRef]

2001 (1)

2000 (1)

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into a uniaxial crystal,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

1999 (1)

D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
[CrossRef]

1998 (4)

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
[CrossRef]

D. Jiang, J. J. Stamnes, “Theoretical and experimentalresults for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (3)

1991 (1)

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. Condens. Matter 3, 6121–6133 (1991).
[CrossRef]

1984 (1)

1978 (1)

1977 (1)

1976 (2)

1972 (3)

E. Lalor, “An analytical approach to the theory of internal conical refraction,” J. Math. Phys. 13, 449–454 (1972).
[CrossRef]

J. Schesser, G. Eichmann, “Propagation of plane waves in biaxially anisotropic layered media,” J. Opt. Soc. Am. 62, 786–791 (1972).
[CrossRef]

E. Lalor, “The angular-spectrum representation of electromagnetic fields in crystals. II. Biaxial crystals,” J. Math. Phys. 13, 443–449 (1972).
[CrossRef]

1971 (1)

M. Lax, D. F. Nelson, “Linear and non linear electro dynamics in elastic anisotropic dielctrics,” Phys. Rev. A 13, 443–449 (1971).

1970 (2)

A. Wünsche, “Exacte Berechnung der Greenschen Tensor Funktionen zum Huygensschen Prinzip für optisch einachsige Medien” (“Exact calculation of the Green’s tensor function to obtain Huygens’s principle for optically uniaxial media”), Ann. Phys. (Leipzig) 25, 179–200 (1970).

A. Wünsche, “Neue Formeln für die Reflexion und Brechung des Lichtes an anisotropen Medien” (“New formulas for reflection and refraction of light in anisotropic media”), Ann. Phys. (Leipzig) 25, 201–214 (1970).
[CrossRef]

1962 (1)

A. M. Goncharenko, F. J. Federov, “Optical properties of crystalline plates,” Opt. Spectrosk. 14, 94–99 (1962).

1955 (1)

A. B. Winterbottom, “Optical studies of metal surfaces,” K. Norske Vidensk. Selsk. Skr. 1, 27,37 (1955).

1848 (1)

J. Mac Cullagh, “On the dynamical theory of crystalline reflection and refraction,” Trans. R. Ir. Acad. (Dublin) 21, 17–50 (1848).

Booker, G. R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Collins, R. E.

R. E. Collins, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), pp. 101–106.

Curry, C. E.

C. E. Curry, Electromagnetic Theory of Light (Maclehose, London, 1905), pp. 356–369.

Dhayalan, V.

V. Dhayalan, J. J. Stamnes, “Comparison of exact asymptotic results for the focusing of electromagnetic waves through a plane interface,” Appl. Opt. 39, 6332–6340 (2001).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

Eichmann, G.

Federov, F. J.

A. M. Goncharenko, F. J. Federov, “Optical properties of crystalline plates,” Opt. Spectrosk. 14, 94–99 (1962).

Felsen, L. B.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, London, 1973).

Gasper, J.

Goncharenko, A. M.

A. M. Goncharenko, F. J. Federov, “Optical properties of crystalline plates,” Opt. Spectrosk. 14, 94–99 (1962).

Jiang, D.

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into a uniaxial crystal,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
[CrossRef]

D. Jiang, J. J. Stamnes, “Theoretical and experimentalresults for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

Laczic, Z.

Lalor, E.

E. Lalor, “The angular-spectrum representation of electromagnetic fields in crystals. II. Biaxial crystals,” J. Math. Phys. 13, 443–449 (1972).
[CrossRef]

E. Lalor, “An analytical approach to the theory of internal conical refraction,” J. Math. Phys. 13, 449–454 (1972).
[CrossRef]

Lax, M.

M. Lax, D. F. Nelson, “Linear and non linear electro dynamics in elastic anisotropic dielctrics,” Phys. Rev. A 13, 443–449 (1971).

Lee, S. W.

Lekner, J.

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. Condens. Matter 3, 6121–6133 (1991).
[CrossRef]

Ling, H.

Mac Cullagh, J.

J. Mac Cullagh, “On the dynamical theory of crystalline reflection and refraction,” Trans. R. Ir. Acad. (Dublin) 21, 17–50 (1848).

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, London, 1973).

Nelson, D. F.

M. Lax, D. F. Nelson, “Linear and non linear electro dynamics in elastic anisotropic dielctrics,” Phys. Rev. A 13, 443–449 (1971).

Nemeth, G.

Schesser, J.

Schopper, H.

H. Schopper, Handbuch der Physik (condensed by Heavens and Butterworths, London, 1928), Vol. 132, pp. 92–95.

Sherman, G. C.

Stamnes, J. J.

V. Dhayalan, J. J. Stamnes, “Comparison of exact asymptotic results for the focusing of electromagnetic waves through a plane interface,” Appl. Opt. 39, 6332–6340 (2001).
[CrossRef]

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into a uniaxial crystal,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

D. Jiang, J. J. Stamnes, “Theoretical and experimentalresults for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

J. J. Stamnes, G. C. Sherman, “Radiation of electromagnetic fields in biaxially anisotropic media,” J. Opt. Soc. Am. 68, 502–508 (1978).
[CrossRef]

J. J. Stamnes, G. C. Sherman, “Reflection and refraction of an arbitrary wave at a plane interface separating two uniaxial crystals,” J. Opt. Soc. Am. 67, 683–695 (1977).
[CrossRef]

J. Gasper, G. C. Sherman, J. J. Stamnes, “Reflection and refraction of an arbitrary wave at a plane interface,” J. Opt. Soc. Am. 66, 955–961 (1976).
[CrossRef]

J. J. Stamnes, G. C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. 66, 780–788 (1976).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 15.2.2.

Szivessy, G.

G. Szivessy, “Licht als Wellenbewegung” (“Light as wave motion”), in Handbuch der Physik, H. Geiger, K. Scheel, eds. (Springer-Verlag, Berlin, 1928), Vol. 20, Chap. 11, pp. 635–904.

Török, P.

Varga, P.

Visser, T. D.

Wiersma, S. H.

Winterbottom, A. B.

A. B. Winterbottom, “Optical studies of metal surfaces,” K. Norske Vidensk. Selsk. Skr. 1, 27,37 (1955).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Wünsche, A.

A. Wünsche, “Neue Formeln für die Reflexion und Brechung des Lichtes an anisotropen Medien” (“New formulas for reflection and refraction of light in anisotropic media”), Ann. Phys. (Leipzig) 25, 201–214 (1970).
[CrossRef]

A. Wünsche, “Exacte Berechnung der Greenschen Tensor Funktionen zum Huygensschen Prinzip für optisch einachsige Medien” (“Exact calculation of the Green’s tensor function to obtain Huygens’s principle for optically uniaxial media”), Ann. Phys. (Leipzig) 25, 179–200 (1970).

Ann. Phys. (Leipzig) (2)

A. Wünsche, “Exacte Berechnung der Greenschen Tensor Funktionen zum Huygensschen Prinzip für optisch einachsige Medien” (“Exact calculation of the Green’s tensor function to obtain Huygens’s principle for optically uniaxial media”), Ann. Phys. (Leipzig) 25, 179–200 (1970).

A. Wünsche, “Neue Formeln für die Reflexion und Brechung des Lichtes an anisotropen Medien” (“New formulas for reflection and refraction of light in anisotropic media”), Ann. Phys. (Leipzig) 25, 201–214 (1970).
[CrossRef]

Appl. Opt. (1)

J. Math. Phys. (2)

E. Lalor, “An analytical approach to the theory of internal conical refraction,” J. Math. Phys. 13, 449–454 (1972).
[CrossRef]

E. Lalor, “The angular-spectrum representation of electromagnetic fields in crystals. II. Biaxial crystals,” J. Math. Phys. 13, 443–449 (1972).
[CrossRef]

J. Opt. Soc. Am. (5)

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

J. Phys. Condens. Matter (1)

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. Condens. Matter 3, 6121–6133 (1991).
[CrossRef]

K. Norske Vidensk. Selsk. Skr. (1)

A. B. Winterbottom, “Optical studies of metal surfaces,” K. Norske Vidensk. Selsk. Skr. 1, 27,37 (1955).

Opt. Commun. (3)

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
[CrossRef]

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into a uniaxial crystal,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

Opt. Spectrosk. (1)

A. M. Goncharenko, F. J. Federov, “Optical properties of crystalline plates,” Opt. Spectrosk. 14, 94–99 (1962).

Phys. Rev. A (1)

M. Lax, D. F. Nelson, “Linear and non linear electro dynamics in elastic anisotropic dielctrics,” Phys. Rev. A 13, 443–449 (1971).

Pure Appl. Opt. (3)

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
[CrossRef]

D. Jiang, J. J. Stamnes, “Theoretical and experimentalresults for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

Trans. R. Ir. Acad. (Dublin) (1)

J. Mac Cullagh, “On the dynamical theory of crystalline reflection and refraction,” Trans. R. Ir. Acad. (Dublin) 21, 17–50 (1848).

Other (10)

G. Szivessy, “Licht als Wellenbewegung” (“Light as wave motion”), in Handbuch der Physik, H. Geiger, K. Scheel, eds. (Springer-Verlag, Berlin, 1928), Vol. 20, Chap. 11, pp. 635–904.

C. E. Curry, Electromagnetic Theory of Light (Maclehose, London, 1905), pp. 356–369.

R. E. Collins, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), pp. 101–106.

H. Schopper, Handbuch der Physik (condensed by Heavens and Butterworths, London, 1928), Vol. 132, pp. 92–95.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, London, 1973).

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 15.2.2.

Ref. 32, Eqs. (15.32b), (15.32c), (15.34c), and (15.34d) with sˆ=eˆz. Note that E˜i(kt) equals (2π)2 times the quantity Ei(ki,ω) in Eqs. (15.32d) and (15.32e).

Ref. 32, Eqs. (15.32d) and (15.32e) with q=i and sˆ=eˆz.

Ref. 32, Eqs. (15.37c) and (15.37d) with sˆ=eˆz and with a factor 1/π omitted, since we are considering a time-harmonic field.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

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Figures (1)

Fig. 1
Fig. 1

A time-harmonic current source in an isotropic medium radiates a field (Ei,Hi) that is incident on a plane interface separating the isotropic medium from a biaxial medium, giving rise to a reflected field (Er, Hr) and a transmitted field (Et, Ht).

Equations (118)

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

J0(r, t)=Re[J0(r)exp(-iωt)]
Vq(r, t)=Re[Vq(r)exp(-iωt)],
Vq(r)=VTEq(r)+VTMq(r),
Vpq(r)=12π2-V˜pq(kt)exp{i[kt·rt+kzq(z-z0)]}dkxdky(p=TE, TM),
kt=kxeˆx+kyeˆy,rt=xeˆx+yeˆy,
kzi=kz(1),kzr=-kz(1),
kz(1)={[k(1)]2-kt2}1/2,Im kz(1)0,
kt2=kx2+ky2,
[k(1)]2=ω2μ(1)c2(1)+i4πσ(1)ω.
E˜TEq(kt)=-cω[k(1)]2α˜TEq(kt)kt×eˆz,
H˜TEq(kt)=-cω2[k(1)]2μ(1)α˜TEq(kt)kq×(kt×eˆz),
E˜TMq(kt)=-α˜TMq(kt)kq×(kt×eˆz),
H˜TMq(kt)=cω[k(1)]2μ(1)α˜TMq(kt)kt×eˆz.
E˜i(kt)=E˜TEi(kt)+E˜TMi(kt),
α˜TEi(kt)=-ωc(kt×eˆz)·E˜ti(kt; z0)[k(1)]2kt2,
α˜TMi(kt)=-kt·E˜ti(kt; z0)kz(1)kt2,
E˜ti(kt; z0)=-Eti(rt)exp(-ikt·rt)dxdy.
α˜TEi(kt)=2πμ(1)ω2(kt×eˆz)·J˜0(ki)c3[k(1)]2kz(1)kt2exp[ikz(1)z0],
α˜TMi(kt)=2πμ(1)ω[ki×(kt×eˆz)]·J˜0(ki)c2[k(1)]2kz(1)kt2×exp[ikz(1)z0],
J˜0(k)=-J0(r)exp[-i(kt·rt+kzz)]dxdydz.
D=¯¯·E,
¯¯=110002200033.
V(r, t)=Re[V(r)exp(-iωt)],
V(r)=V1(r)+V2(r),
Vj(r)=-V˜j(kt)exp[i(kt·rt+kj,zz)]dkxdky.
E˜j(kt)=C(kj)[g(kj)g(kj)+Γ¯¯(kj)]·J˜0(kj),
H˜j(kt)=cωμkj×E˜j(kt),
Γ¯¯(kj)=(det Λ)[eˆxeˆx/(k2-ξ12)+eˆyeˆy/(k2-ξ22)+eˆzeˆz/(k2-ξ32)],
det Λ=-ξ32(kz2-k1,z2)(kz2-k2,z2),
C(kj)=ωμc2(-1)j+12ξ32kj,z(k1,z2-k2,z2),
g(kj)=cω3[kx(kj2-ξ22)(kj2-ξ32)eˆx+ky(kj2-ξ12)×(kj2-ξ32)eˆy+kj,z(kj2-ξ12)(kj2-ξ22)eˆz],
kj=kt+kj,zeˆz,kj2=kt2+kj,z2,
kj,z=(kj,z2)1/2,Im kj,z0,
kj,z2=12{A1+A2-(-1)j[(A1-A2)2+4B1B2]1/2},
A1=ξ12-(kx2ξ12/ξ32+ky2),
A2=ξ22-(ky2ξ22/ξ32+kx2),
B1=kxky(1-ξ12/ξ32),
B2=kxky(1-ξ22/ξ32),
ξj2=ω2μc2jj+4πiσω.
E˜j(kt)=C(kj)[g(kj)·J˜0(kj)]g(kj).
E˜j(kt)=f(kt)g(kj),
g(kj)=eˆx+LpKpeˆy-ξ12kx+ξ22ky(Lp/Kp)ξ32kp,zeˆz,
LpKp=kp,z2-A1B2.
(A1-A2)+4B1B2|ξ12-ξ22|+sgn|ξ1-ξ2|1ξ32×[(ξ32-ξ12)kx2-(ξ32-ξ22)ky2].
k1,z2ξ12-ξ12ξ32kx2+ky2,k2,z2ξ22-kx2+ξ22ξ32ky2.
k1,zξ1-12ξ1ξ12ξ32kx2+ky2,
k2,zξ2-12ξ2kx2+ξ22ξ32ky2.
k1,z2=ξ12-ky2,k2,z2=ξ22-ξ22ξ32ky2.
Γ¯¯(k)=-ξ32(kz2-k2,z2)eˆxeˆx+(kz2-k1,z2)×eˆyeˆyk2-ξ22+eˆzeˆzk2-ξ32,
Γ¯¯(k1)=-ξ32ξ12-ξ22+ky2ξ22ξ32-1eˆxeˆx.
E˜1(kt)=-C(k1)ξ32ξ12-ξ22+ky2ξ22ξ32-1J˜0x(k1)eˆx.
g(k2)=cω3(k22-ξ12)ky×(k22-ξ32)eˆy+k2,zky(k22-ξ22)eˆz.
g(k2)=cω3(k22-ξ12)(ky2-ξ32)ξ32-ξ22ξ32k2,z[k2,zk2-ξ22eˆz],
k1,z2=ξ12-ξ12ξ32kx2,k2,z2=ξ22-kx2,
Γ¯¯(k)=-ξ32(kz2-k1,z2)eˆyeˆy+(kz2-k2,z2)×eˆxeˆxk2-ξ12+eˆzeˆzk2-ξ32,
Γ¯¯(k2)=-ξ32ξ22-ξ12+kx2ξ12ξ32-1eˆyeˆy.
E˜2(kt)=-C(k2)ξ32ξ22-ξ12+kx2ξ12ξ32-1J˜0y(k2)eˆy.
g(k1)=cω3(k12-ξ22)kx×(k12-ξ32)eˆx+k1,zkx(k12-ξ12)eˆz.
g(k1)=cω3(k12-ξ22)(kx2-ξ32)ξ32-ξ12ξ32k1,z[k1,zk1-ξ12eˆz],
E˜j(kt)=f(kt)g(kj).
g(kj)=cω3[kx(kj2-ξ22)(kj2-ξ32)eˆx+ky(kj2-ξ12)×(kj2-ξ32)eˆy+kj,z(kj2-ξ12)(kj2-ξ22)eˆz]
g(kj)=cω3kx(kj2-ξ22)(kj2-ξ32)×eˆx+LpKpeˆy-ξ12kx+ξ22kyLpKpξ32kp,zeˆz,
LpKp=kp,z2-A1B2.
kj,z2=122ξ12-kt21+ξ12ξ32-(-1)jkt21-ξ12ξ32.
k1,z2=ξ12-ξ12ξ32kt2,k2,z2=ξ12-kt2,
g(k2)=1kykt×eˆz,
g(k1)=eˆx+kykxeˆy-ξ12ξ32kt2kxk1,zeˆz,
g(k1)=1kxk1,z[k1,zk1-ξ12eˆz].
E˜q=E˜TEq+E˜TMq,H˜q=H˜TEq+H˜TMq
(q=i, r),
Et(r)=E1t(r)+E2t(r),Ht(r)=H1t(r)+H2t(r),
Vjt(r)=12π2-V˜jt(kt)exp{i[kt·rt+kj,z(z-z0)]}dkxdky,
E˜jt(kt)=α˜jt(kt)g(kj),
H˜jt(kt)=cωμkj×E˜jt(kt).
eˆz×[Ei(r, t)+Er(r, t)-Et(r, t)]|z=z0=0,
eˆz×[Hi(r, t)+Hr(r, t)-Ht(r, t)]|z=z0=0.
-A˜j(kt)exp(ikt·rt)dkxdky=0,
A˜E(kt)=eˆz×-α˜TMi(kt)ki×(kt×eˆz)-cω[k(1)]2α˜TEi(kt)(kt×eˆz)-α˜TMr(kt)kr×(kt×eˆz)-cω[k(1)]2α˜TEr(kt)(kt×eˆz)-α˜1t(kt)g(k1)-α˜2t(kt)g(k2),
A˜H(kt)=eˆz×cω[k(1)]2μ(1)α˜TMi(kt)kt×eˆz-cω2[k(1)]2μ(1)α˜TEi(kt)ki×(kt×eˆz)+cω[k(1)]2μ(1)α˜TMr(kt)kt×eˆz-cω2[k(1)]2μ(1)α˜TEr(kt)kr×(kt×eˆz)-cωμα˜1t(kt)k1×g(k1)-cωμα˜2t(kt)k2×g(k2).
A˜E(kt)=A˜H(kt)=0.
eˆz×[kq×(kt×eˆz)]=kq·eˆz(eˆz×kt),
eˆz×(kt×eˆz)=kt
kz(1)(α˜TMr-α˜TMi)eˆz×kt-cω[k(1)]2(α˜TEr+α˜TEi)kt=eˆz×[α˜1tg(k1)+α˜2tg(k2)],
β(α˜TMr+α˜TMi)kt+cωkz(1)(α˜TEr-α˜TEi)eˆz×kt=eˆz×[α˜1tk1×g(k1)+α˜2tk2×g(k2)],
β=μμ(1)[k(1)]2.
h(kj)=kj×g(kj),
gy(k1)gy(k2)-cωkx[k(1)]2-kykz(1)gx(k1)gx(k2)cωky[k(1)]2-kxkz(1)hy(k1)hy(k2)-cωβkykz(1)βkxhx(k1)hx(k2)-cωβkxkz(1)-βkyα˜1tα˜2tα˜TErα˜TMr=α˜TEicωkx[k(1)]2-cωky[k(1)]2-βcωkykz(1)-βcωkxkz(1)+α˜TMi-kykz(1)-kxkz(1)-βkxβky,
α˜1t=α˜TEidet Ω1TEdet Ω+α˜TMidet Ω1TMdet Ω,
α˜2t=α˜TEidet Ω2TEdet Ω+α˜TMidet Ω2TMdet Ω,
α˜TEr=α˜TEidet Ω3TEdet Ω+α˜TMidet Ω3TMdet Ω,
α˜TMr=α˜TEidet Ω4TEdet Ω+α˜TMidet Ω4TMdet Ω,
det Ω1TM=det Ω2TE=det Ω3TM=det Ω4TE=0,
det Ω1TE=det Ω2TM=det Ω3TE=det Ω4TM=0,
Ei(r)=Ei exp{i[kxix+kyiy+kz(1)i(z-z0)]}=Ei exp(iki·r),
kz(1)i={[k(1)]2-(kxi)2-(kyi)2}1/2.
E˜i(kt; z0)=(2π)2δ(kx-kxi)δ(ky-kyi)Ei.
α˜TEi(kt)=-(2π)2ωc(kt×eˆz)·Eti[k(1)]2(kti)2×δ(kx-kxi)δ(ky-kyi),
α˜TMi(kt)=-(2π)2kt·Etikz(1)i(kti)2×δ(kx-kxi)δ(ky-kyi).
ETEi=(kti×eˆz)·Eti(kti)2(kti×eˆz)×exp{i[kti·rt+kz(1)i(z-z0)]},
ETMi=kti·Etikz(1)i(kti)2ki×(kti×eˆz)×exp{i[kti·rt+kz(1)i(z-z0)]}.
ETEr=(kti×eˆz)·Eti(kti)2det Ω3TEdet Ωkxi,kyi+cω[k(1)]2kti·Etikz(1)i(kti)2det Ω3TMdet Ωkxi,kyi×(kti×eˆz)exp{i[kti·rt-kz(1)i(z-z0)]},
ETMr=ωc(kti×eˆz)·Eti[k(1)]2(kti)2det Ω4TEdet Ωkxi,kyi+kti·Etikz(1)i(kti)2det Ω4TMdet Ωkxi,kyi×kr,i×(kti×eˆz)exp{i[ki·rt-kz(1)i(z-z0)]},
kr,i=kti-kz(1)ieˆz.
E1t(r)=-ωc(kti×eˆz)·Eti[k(1)]2(kti)2det Ω1TEdet Ωkxi,kyi+kti·Etikz(1)i(kti)2det Ω1TMdet Ωkxi,kyig(k1)×exp{i[kti·rt+k1,zi(z-z0)]},
E2t(r)=-ωc(kti×eˆz)·Eti[k(1)]2(kti)2det Ω2TEdet Ωkxi,kyi+kti·Etikz(1)i(kti)2det Ω2TMdet Ωkxi,kyig(k2)×exp{i[kti·rt+k2,zi(z-z0)]}.
det Ω1TE=det Ω2TM=det Ω3TM=det Ω4TE=0,
det Ω1TMdet Ω=-kz(1)ikxi2µk1,zμ(1)11kz(1)i+μ(1)k1,z,
det Ω2TEdet Ω=-kyi(c/ω)[k(1)]22µkz(1)iμkz(1)i+μ(1)k2,z,
det Ω3TEdet Ω=μkz(1)i-μ(1)k2,zμkz(1)i+μ(1)k2,z,
det Ω4TMdet Ω=μ(1)11kz(1)i-μ(1)k1,zμ(1)11kz(1)i+μ(1)k1,z.
RTE=|ETEr||ETEi|=μkz(1)i-μ(1)k2,zμkz(1)i+μ(1)k2,z,
RTM=|ETMr||ETMi|=μ(1)11kz(1)i-μ(1)k1,zμ(1)11kz(1)i+μ(1)k1,z,
TTE=|E2t||ETEi|=2µkz(1)iμkz(1)i+μ(1)k2,z,
TTM=|E1t||ETMi|=2µk1,zμ(1)11kz(1)i+μ(1)k1,z|k1,zk1-ξ12eˆz||kz(1)iki-(1)eˆz|.
(kx, ky, k1,z)=(kx, ky, k2,z).
(A1-A2)2+4B1B2=0.
s±=±22-1133-11331/2, 0 ,33-2233-22111/2.
tan β=22-1133-2233111/2.

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