Abstract

We derive integral representations that are suitable for studying the transmission of an electromagnetic Gaussian beam through a plane interface that lies between an isotropic medium and a biaxially anisotropic crystal for the case in which the interface normal is along one of the principal axes of the crystal. To that end, we use recently developed exact solutions for the transmitted fields to derive explicit expressions for the corresponding dyadic Green’s functions as well as integral representations that are suitable for asymptotic analysis and efficient numerical evaluation.

© 2001 Optical Society of America

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References

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  1. E. O. Ammann, “Optical birefringent networks,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Vol. 10.
  2. D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart & Winston, New York, 1969).
  3. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976).
  4. J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), Sec. 15.2.2.
  5. S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998).
    [CrossRef]
  6. S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
    [CrossRef]
  7. C. J. R. Sheppard, S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
    [CrossRef]
  8. C. E. Curry, Electromagnetic Theory of Light (Maclehose, London, 1905), pp. 356–369.
  9. R. E. Collins, Field Theory of Guided Waves, 2nd ed. (Institute of Electrical and Electronics Engineers, New York, 1991), Vol. 12.
  10. A. M. Goncharenko, F. J. Federov, “Optical properties of crystalline plates,” Opt. Spectrosk. 14, 94–99 (1961).
  11. A. Wünsche, “Neue Formeln fur die Reflexion und Brechung des Lichtes an anisotropen Medien,” Ann. Phys. (Leipzig) 25, 201–214 (1970).
    [CrossRef]
  12. J. MacCullagh, “On the dynamical theory of crystalline reflection and refraction,” Trans. R. Irish Acad. (Dublin) 21, 17–50 (1848).
  13. G. Szivessy, “Licht als Wellenbewegung,” in Handbuch der Physik (Springer-Verlag, Berlin, 1928), Vol. 20, Chap. 11, pp. 635–904.
  14. J. Schesser, G. Eichmann, “Propagation of plane waves in biaxially anisotropic layered media,” J. Opt. Soc. Am. 62, 786–791 (1972).
    [CrossRef]
  15. H. Schopper, Handbuch der Physik, condensed (Heavens and Butterworths, London, 1928), Vol. 132, pp. 92–95.
  16. A. Wünsche, “Exacte Berechnung der Greenschen Tensor Funktionen zum Huygensschen Prinzip für optisch einachsige Medien,” Ann. Phys. (Leipzig) 25, 179–200 (1970).
    [CrossRef]
  17. E. Lalor, “The angular-spectrum representation of electromagnetic fields in crystals. II,” J. Math. Phys. 13, 443–449 (1972).
    [CrossRef]
  18. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves, IEEE Press Series on Electromagnetic Waves (Institute of Electrical and Electronics Engineers, New York1994).
  19. M. Lax, D. F. Nelson, “Linear and nonlinear electrodynamics in elastic anisotropic dielectrics,” Phys. Rev. B 4, 3694–3731 (1971).
    [CrossRef]
  20. J. J. Stamnes, G. C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. 66, 780–788 (1976).
    [CrossRef]
  21. J. J. Stamnes, G. C. Sherman, “Radiation of electromagnetic fields in biaxially anisotropic media,” J. Opt. Soc. Am. 68, 502–508 (1978).
    [CrossRef]
  22. 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]
  23. 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]
  24. H. Ling, S. W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
    [CrossRef]
  25. 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]
  26. 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]
  27. 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]
  28. 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]
  29. 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]
  30. 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]
  31. J. J. Stamnes, D. Jiang, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
    [CrossRef]
  32. D. Jiang, J. J. Stamnes, “Theoretical and experimental results for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
    [CrossRef]
  33. 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 (2000).
    [CrossRef]
  34. J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
    [CrossRef]
  35. 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]
  36. 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]
  37. J. J. Stamnes, G. Sithambaranathan have submitted the following paper for publication: “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface separating an isotropic and a biaxial medium.”
  38. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 9.1.1.
  39. J. J. Stamnes, B. Spelkjavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
    [CrossRef]
  40. In addition to Ref. 39, see also J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 7.2.
  41. J. J. Stamnes, V. Dhayalan, “Transmission of a two-dimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18, 1662–1669 (2001).
    [CrossRef]

2001

2000

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 (2000).
[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]

1999

1998

S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998).
[CrossRef]

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

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 experimental results for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

1997

1996

1995

1984

1983

J. J. Stamnes, B. Spelkjavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

1978

1977

1976

1972

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,” J. Math. Phys. 13, 443–449 (1972).
[CrossRef]

1971

M. Lax, D. F. Nelson, “Linear and nonlinear electrodynamics in elastic anisotropic dielectrics,” Phys. Rev. B 4, 3694–3731 (1971).
[CrossRef]

1970

A. Wünsche, “Exacte Berechnung der Greenschen Tensor Funktionen zum Huygensschen Prinzip für optisch einachsige Medien,” Ann. Phys. (Leipzig) 25, 179–200 (1970).
[CrossRef]

A. Wünsche, “Neue Formeln fur die Reflexion und Brechung des Lichtes an anisotropen Medien,” Ann. Phys. (Leipzig) 25, 201–214 (1970).
[CrossRef]

1961

A. M. Goncharenko, F. J. Federov, “Optical properties of crystalline plates,” Opt. Spectrosk. 14, 94–99 (1961).

1848

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

Ammann, E. O.

E. O. Ammann, “Optical birefringent networks,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Vol. 10.

Bell, W. E.

D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart & Winston, New York, 1969).

Booker, G. R.

Collins, R. E.

R. E. Collins, Field Theory of Guided Waves, 2nd ed. (Institute of Electrical and Electronics Engineers, New York, 1991), Vol. 12.

Curry, C. E.

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

Dhayalan, V.

Eichmann, G.

Federov, F. J.

A. M. Goncharenko, F. J. Federov, “Optical properties of crystalline plates,” Opt. Spectrosk. 14, 94–99 (1961).

Felsen, L. B.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves, IEEE Press Series on Electromagnetic Waves (Institute of Electrical and Electronics Engineers, New York1994).

Gasper, J.

Goncharenko, A. M.

A. M. Goncharenko, F. J. Federov, “Optical properties of crystalline plates,” Opt. Spectrosk. 14, 94–99 (1961).

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 experimental results 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,” J. Math. Phys. 13, 443–449 (1972).
[CrossRef]

Lax, M.

M. Lax, D. F. Nelson, “Linear and nonlinear electrodynamics in elastic anisotropic dielectrics,” Phys. Rev. B 4, 3694–3731 (1971).
[CrossRef]

Lee, S. W.

Ling, H.

MacCullagh, J.

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

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves, IEEE Press Series on Electromagnetic Waves (Institute of Electrical and Electronics Engineers, New York1994).

Nelson, D. F.

M. Lax, D. F. Nelson, “Linear and nonlinear electrodynamics in elastic anisotropic dielectrics,” Phys. Rev. B 4, 3694–3731 (1971).
[CrossRef]

Nemeth, G.

Pedersen, H. M.

J. J. Stamnes, B. Spelkjavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Saghafi, S.

Schesser, J.

Schopper, H.

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

Seshadri, S. R.

Sheppard, C. J. R.

Sherman, G. C.

Sinclair, D. C.

D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart & Winston, New York, 1969).

Sithambaranathan, G.

J. J. Stamnes, G. Sithambaranathan have submitted the following paper for publication: “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface separating an isotropic and a biaxial medium.”

Spelkjavik, B.

J. J. Stamnes, B. Spelkjavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, V. Dhayalan, “Transmission of a two-dimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18, 1662–1669 (2001).
[CrossRef]

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 (2000).
[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]

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

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

J. J. Stamnes, B. Spelkjavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[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, London, 1986), Sec. 15.2.2.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 9.1.1.

In addition to Ref. 39, see also J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 7.2.

J. J. Stamnes, G. Sithambaranathan have submitted the following paper for publication: “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface separating an isotropic and a biaxial medium.”

Szivessy, G.

G. Szivessy, “Licht als Wellenbewegung,” in Handbuch der Physik (Springer-Verlag, Berlin, 1928), Vol. 20, Chap. 11, pp. 635–904.

Török, P.

Varga, P.

Visser, T. D.

Wiersma, S. H.

Wünsche, A.

A. Wünsche, “Neue Formeln fur die Reflexion und Brechung des Lichtes an anisotropen Medien,” Ann. Phys. (Leipzig) 25, 201–214 (1970).
[CrossRef]

A. Wünsche, “Exacte Berechnung der Greenschen Tensor Funktionen zum Huygensschen Prinzip für optisch einachsige Medien,” Ann. Phys. (Leipzig) 25, 179–200 (1970).
[CrossRef]

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976).

Ann. Phys. (Leipzig)

A. Wünsche, “Neue Formeln fur die Reflexion und Brechung des Lichtes an anisotropen Medien,” Ann. Phys. (Leipzig) 25, 201–214 (1970).
[CrossRef]

A. Wünsche, “Exacte Berechnung der Greenschen Tensor Funktionen zum Huygensschen Prinzip für optisch einachsige Medien,” Ann. Phys. (Leipzig) 25, 179–200 (1970).
[CrossRef]

Appl. Opt.

J. Math. Phys.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

H. Ling, S. W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
[CrossRef]

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]

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]

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]

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]

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]

J. J. Stamnes, V. Dhayalan, “Transmission of a two-dimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18, 1662–1669 (2001).
[CrossRef]

S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998).
[CrossRef]

S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
[CrossRef]

C. J. R. Sheppard, S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
[CrossRef]

Opt. Acta

J. J. Stamnes, B. Spelkjavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Opt. Commun.

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.

A. M. Goncharenko, F. J. Federov, “Optical properties of crystalline plates,” Opt. Spectrosk. 14, 94–99 (1961).

Phys. Rev. B

M. Lax, D. F. Nelson, “Linear and nonlinear electrodynamics in elastic anisotropic dielectrics,” Phys. Rev. B 4, 3694–3731 (1971).
[CrossRef]

Pure Appl. Opt.

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 experimental results for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

Trans. R. Irish Acad. (Dublin)

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

Other

G. Szivessy, “Licht als Wellenbewegung,” in Handbuch der Physik (Springer-Verlag, Berlin, 1928), Vol. 20, Chap. 11, pp. 635–904.

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

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves, IEEE Press Series on Electromagnetic Waves (Institute of Electrical and Electronics Engineers, New York1994).

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

R. E. Collins, Field Theory of Guided Waves, 2nd ed. (Institute of Electrical and Electronics Engineers, New York, 1991), Vol. 12.

E. O. Ammann, “Optical birefringent networks,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Vol. 10.

D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart & Winston, New York, 1969).

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976).

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

J. J. Stamnes, G. Sithambaranathan have submitted the following paper for publication: “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface separating an isotropic and a biaxial medium.”

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 9.1.1.

In addition to Ref. 39, see also J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 7.2.

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Figures (5)

Fig. 1
Fig. 1

Time-harmonic current source in an isotropic medium that radiates a field (Ei, Hi) that is incident upon 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).

Fig. 2
Fig. 2

Electromagnetic Gaussian beam with its main propagation direction along the interface normal is incident upon a plane interface between an isotropic medium and a biaxially anisotropic crystal. The interface normal is along one of the principal axes of the crystal.

Fig. 3
Fig. 3

Axial intensities of the transmitted beam when a normally incident Gaussian beam polarized in the x direction in air is refracted at the interface that separates an isotropic medium and a biaxially anisotropic crystal. The interface normal is along one of the principal axes of the crystal. Here σ0=1 mm. The aperture radius is (a) 5 mm, (b) 2 mm, (c) 1 mm, (d) 0.5 mm.

Fig. 4
Fig. 4

Intensity distribution of the transmitted beam in planes z=constant for the case in which a normally incident Gaussian beam polarized in the x direction in air is refracted at the interface that separates an isotropic medium and an biaxially anisotropic crystal. The interface normal is along one of the principal axes of the crystal. Here the beam waist in the aperture plane is σ0=1 mm. The parameters are (a) an aperture radius of 0.5 mm and z=0.5 mm, (b) an aperture radius of 0.5 mm and z=2 m.

Fig. 5
Fig. 5

Comparisons of the transverse intensities of the transmitted extraordinary beam for the case in which a normally incident Gaussian beam polarized in the x direction in air is refracted at the interface that separates an isotropic medium and a biaxially anisotropic crystal. The solid and the dotted curves were obtained for the transverse intensities along the x and the y axes, respectively, for a constant value of z that corresponds to the maximum intensity shown in Fig. 3(d).

Equations (78)

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

Et(r, t)=Re[Et(r)exp(-iωt)],
Et(r)=E1t(r)+E2t(r),
Ejt(r)=12π2- α˜jt(kt; z0)g(kj)
×exp{i[kj,z(z-z0)+kt  rt]}dkxdky,
j=1, 2,
kt={kx, ky, 0},rt={x,y,0},
g(kj)=cω3[kx(kj2-ξ22)(kj2-ξ32)e^x+ky(kj2-ξ12)(kj2-ξ32)e^y+kj,z(kj2-ξ12)(kj2-ξ22)e^z].
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.
α˜jt(kt; z0)=TαjTEα˜TEi(kt; z0)+TαjTMα˜TMi(kt; z0),
α˜TEi(kt; z0)=-ωc(kt×e^z)  E˜ti(kt; z0)[k(1)]2kt2,
α˜TMi(kt; z0)=-kt  E˜ti(kt)kz(1)kt2.
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,
Eti(kt; z0)=E˜xi(kt; z0)e^x+E˜yi(kt; z0)e^y,
E˜ti(kt; z0)=exp[ikz(1)z0]- Eti(rt, 0)×exp(-ikt  rt)dxdy.
Tαjp=det Ωjpdet Ω,p=TE, TM,
Ω=gy(k1)gy(k2)-cω kx[k(1)]2-kykz(1)gx(k1)gx(k2)cω ky[k(1)]2-kxkz(1)hy(k1)hy(k2)-cω βkykz(1)βkxhx(k1)hx(k2)-cω βkxkz(1)-βky,
β=μμ(1) [k(1)]2
h(kj)=kj×g(kj).
cω kx[k(1)]2-cω ky[k(1)]2-β cω kykz(1)-β cω kxkz(1),
-kykz(1)-kxkz(1)-βkxβky.
Ejt(r)=- G¯¯jEEt(rt-rt)  Eti(rt, 0)dxdy,
G¯¯jEEt(rt-rt)=12π2- G˜¯¯jEEt(kt)×exp[ikt  (rt-rt)]dkxdky,
G˜¯¯jEEt(kt)exp[ikt  (rt-rt)]
=-ωckt2g(kj)TαjTE[k(1)]2kt × e^z+cωTαjTMkz(1)ktexp[ihj(kt)].
hj(kt)=kt  (rt-rt)+kz(1)z0+kj,z(z-z0).
Ejt(r)= - [Exi(rt, 0)FjEx(rt-rt)+Eyi(rt, 0)FjEy(rt-rt)]dxdy,
FjEl(rt-rt)=G¯¯jEEt(rt-rt)  e^l
=12π2- fjEl(kt)×exp[ihj(kt)]dkxdky,
fjEl(kt)=[G¯¯jEEt(kt)  e^l]×exp{-i[kz(1)z0+kj,z(z-z0)]}.
Et(r)=E1t(r)+E2t(r),
fjEl(kt)=-ωckt2g(kj)×TαjTE[k(1)]2kt × e^z+cωTαjTMkz(1)kt  e^l.
Ht(r)=H1t(r)+H2t(r),
Hjt(r)=- [Exi(rt, 0)FjHx(rt-rt)+Eyi(rt, 0)FjHy(rt-rt)]dxdy,
FjHl(rt-rt)=12π2- fjHl(kt)exp[ihj(kt)]dkxdky,
fjHl(kt)=cωμkj × fjEl(kt)
FjEl12πσj|Hj|1/2fjEl(kts)exp[ihj(kts)],
Hj=2hjkx22hjky2-2hjkxky2kj,xs,kj,ys,
σj=1ifHj<0iifHj>0,2hjkx2kj,xs,kj,ys>0-iifHj>0,2hjkx2kj,xs,kj,ys<0.
hjkxkj,xs,kj,ys=hjkykj,xs,kj,ys=0.
Exi(rt, 0)=E0exp[-(x2+y2)/2σ02]x2+y2<a20otherwise.
Ejt(r)E02π Aexp[-(x2+y2)/2σ02]×σj|Hj|1/2fjEx(kts)exp[ihj(kts)]dxdy,
hj(kx, ky)=kx(x-x)+ky(y-y)+kz(1)z0
+kj,z(z-z0),
hjkx=(x-x)+kz(1)kx z0+kj,zkx (z-z0),
hjky=(y-y)+kz(1)ky z0+kj,zky (z-z0),
kz(1)kx=-kxkz(1),kz(1)ky=-kykz(1).
kj,z2=12 {f(kx, ky)-(-1)j[g(kx, ky)]1/2},
f(kx, ky)=p0+p1kx2+p2ky2,
g(kx, ky)=q0+q1kx2+q2ky2
+q3kx4+q4ky4+q5kx2ky2,
p0=ξ12+ξ22,
p1=-ξ22ξ32+1,
p2=-ξ22ξ32+1,
q0=(ξ12-ξ22)2,
q1=2(ξ12-ξ22)1-ξ12ξ32,
q2=2(ξ12-ξ22)ξ22ξ32-1,
q3=1-ξ12ξ322,
q4=ξ22ξ32-12,
q5=21-ξ12ξ321-ξ22ξ32.
kj,zkx=14kj,zfkx-(-1) j12ggkx,
kj,zky=14kj,zfky-(-1) j12ggky.
2hjkx2=2kz(1)kx2 z0+2kj,zkx2 (z-z0),
2hjky2=kz(1)ky2 z0+2kj,zky2 (z-z0),
2hjkxky=2kz(1)kxky z0+2kj,zkxky (z-z0),
2kz(1)kx2=-[kz(1)]2+kx2[kz(1)]3,
kz(1)ky2=-[kz(1)]2+ky2[kz(1)]3,
2kz(1)kxky=-kxky[kz(1)]3,
2kj,zkx2=14kj,z-4kj,zkx2+2fkx2+(-1) j14g1ggkx2-2 2gkx2,
2kj,zky2=14kj,z-4kj,zky2+2fky2+(-1) j14g1ggky2-2 2gky2,
2kj,zkxky=14kj,z-4 kj,zkxkj,zky+2fkxky+(-1)j14g1ggkxgky-2 2gkxky,

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