Abstract

The surface mode that propagates along the interface between isotropic and uniaxial materials, as first suggested by M. I. D’Yakonov [Sov. Phys. JETP 67, 714 (1988)], is quantitatively characterized in terms of (1) the range of crystallographic orientations for which the mode propagates, (2) its propagation constant β, and (3) its field profiles. Previous studies have considered only uniaxial materials whose optic axis is in the plane of the interface. We show that a surface mode can also propagate along the interface between isotropic and arbitrarily oriented uniaxial or biaxial materials. This mode is also quantitatively characterized. For the biaxial material oriented so that its optic axes lie in the plane of the interface, it is shown that this surface mode is guided over a greater range of propagation directions and that the light is confined more tightly than for any isotropic–uniaxial interface of comparable birefringence. In addition, it is shown that the surface modes that occur at isotropic–uniaxial interfaces combine to form a new type of hybrid mode in uniaxial slab waveguides (two interfaces). The resulting modes differ from conventional slab waveguide modes in that (1) they are composed entirely of inhomogeneous waves and (2) at most two of these modes can exist regardless of the waveguide thickness.

© 1998 Optical Society of America

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  1. J. R. Wait, “Electromagnetic surface waves,” in Advances in Radio Research, J. A. Saxton, ed. (Academic, New York, 1964), Vol. 1, pp. 157–218.
  2. E. Burstein, W. P. Chen, Y. J. Chen, A. Hartstein, “Surface polaritons—propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004–1019 (1974).
    [CrossRef]
  3. K. Welford, “Surface plasmon-polaritons and their uses,” Opt. Quantum Electron. 23, 1–27 (1991).
    [CrossRef]
  4. D. E. N. Brancus, “Polaritons in uniaxial crystals,” Rev. Roum. Phys. 29, 815–822 (1984).
  5. D. E. N. Brancus, “Polaritons in uniaxial crystals. II: the energy propagation,” Rev. Roum. Phys. 35, 385–394 (1990).
  6. V. N. Lyubimov, D. G. Sannikov, “Surface electromagnetic waves in a uniaxial crystal,” Sov. Phys. Solid State 14, 575–579 (1972).
  7. D. F. Baı̆sa, A. Y. Zarenin, S. P. Makarenko, S. V. Strizhevskiı̆, “Surface polaritons in biaxial crystals,” Sov. Phys. Solid State 27, 631–633 (1985).
  8. D. Mihalache, D. M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Guided waves in anisotropic antiguide structures,” Opt. Commun. 108, 239–242 (1994).
    [CrossRef]
  9. M. I. D’Yakonov, “New type of electromagnetic wave propagating at an interface,” Sov. Phys. JETP 67, 714–716 (1988).
  10. N. S. Averkiev, M. I. D’Yakonov, “Electromagnetic waves localized at the interface of transparent anisotropic media,” Sov. Phys. JETP 68, 653–655 (1990).
  11. L. Torner, J. P. Torres, C. Ojeda, D. Mihalache, “Hybrid waves guided by ultra-thin films,” J. Lightwave Technol. 13, 2027–2033 (1995).
    [CrossRef]
  12. L. Torner, C. Santos, J. P. Torres, D. Mihalache, “New waveguide modes in anisotropic structures,” Fiber Integr. Opt. 13, 271–280 (1994).
    [CrossRef]
  13. L. Torner, J. P. Torres, D. Mihalache, “New type of guided waves in birefringent media,” IEEE Photonics Technol. Lett. 5, 201–203 (1993).
    [CrossRef]
  14. M. Ciumac, D. Mihalache, “Properties of Bragg reflectors composed of isotropic dielectric layers cladded with birefringent media,” IEEE J. Quantum Electron. 32, 513–518 (1996).
    [CrossRef]
  15. M. Ciumac, D. Mihalache, “Hybrid modes in asymmetric periodic stratified dielectric waveguides,” J. Opt. Soc. Am. A 12, 1695–1701 (1995).
    [CrossRef]
  16. M. Ciumac, D.-M. Baboiu, D. Mihalache, “Hybrid surface modes in periodic stratified media: transfer matrix technique,” Opt. Commun. 111, 548–555 (1994).
    [CrossRef]
  17. A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083–1104 (1988).
    [CrossRef]
  18. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).
  19. T. A. Maldonado, T. K. Gaylord, “Hybrid guided modes in biaxial planar waveguides,” J. Lightwave Technol. 14, 486–499 (1996).
    [CrossRef]
  20. D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  21. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1988).
  22. S. T. Lagerwall, N. A. Clark, “Ferroelectric liquid crystals: the development of devices,” Ferroelectrics 94, 3–62 (1989).
    [CrossRef]
  23. D. B. Walker, E. N. Glytsis, T. K. Gaylord, “Ferroelectric liquid crystal waveguide modulation based on switchable uniaxial–uniaxial interface,” Appl. Opt. 35, 3016–3030 (1996).
    [CrossRef] [PubMed]
  24. E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarva, J. DuCroz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, LAPACK Users’ Guide, 2nd ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1995).

1996 (3)

M. Ciumac, D. Mihalache, “Properties of Bragg reflectors composed of isotropic dielectric layers cladded with birefringent media,” IEEE J. Quantum Electron. 32, 513–518 (1996).
[CrossRef]

T. A. Maldonado, T. K. Gaylord, “Hybrid guided modes in biaxial planar waveguides,” J. Lightwave Technol. 14, 486–499 (1996).
[CrossRef]

D. B. Walker, E. N. Glytsis, T. K. Gaylord, “Ferroelectric liquid crystal waveguide modulation based on switchable uniaxial–uniaxial interface,” Appl. Opt. 35, 3016–3030 (1996).
[CrossRef] [PubMed]

1995 (2)

L. Torner, J. P. Torres, C. Ojeda, D. Mihalache, “Hybrid waves guided by ultra-thin films,” J. Lightwave Technol. 13, 2027–2033 (1995).
[CrossRef]

M. Ciumac, D. Mihalache, “Hybrid modes in asymmetric periodic stratified dielectric waveguides,” J. Opt. Soc. Am. A 12, 1695–1701 (1995).
[CrossRef]

1994 (3)

M. Ciumac, D.-M. Baboiu, D. Mihalache, “Hybrid surface modes in periodic stratified media: transfer matrix technique,” Opt. Commun. 111, 548–555 (1994).
[CrossRef]

L. Torner, C. Santos, J. P. Torres, D. Mihalache, “New waveguide modes in anisotropic structures,” Fiber Integr. Opt. 13, 271–280 (1994).
[CrossRef]

D. Mihalache, D. M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Guided waves in anisotropic antiguide structures,” Opt. Commun. 108, 239–242 (1994).
[CrossRef]

1993 (1)

L. Torner, J. P. Torres, D. Mihalache, “New type of guided waves in birefringent media,” IEEE Photonics Technol. Lett. 5, 201–203 (1993).
[CrossRef]

1991 (1)

K. Welford, “Surface plasmon-polaritons and their uses,” Opt. Quantum Electron. 23, 1–27 (1991).
[CrossRef]

1990 (2)

D. E. N. Brancus, “Polaritons in uniaxial crystals. II: the energy propagation,” Rev. Roum. Phys. 35, 385–394 (1990).

N. S. Averkiev, M. I. D’Yakonov, “Electromagnetic waves localized at the interface of transparent anisotropic media,” Sov. Phys. JETP 68, 653–655 (1990).

1989 (1)

S. T. Lagerwall, N. A. Clark, “Ferroelectric liquid crystals: the development of devices,” Ferroelectrics 94, 3–62 (1989).
[CrossRef]

1988 (2)

M. I. D’Yakonov, “New type of electromagnetic wave propagating at an interface,” Sov. Phys. JETP 67, 714–716 (1988).

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

1985 (1)

D. F. Baı̆sa, A. Y. Zarenin, S. P. Makarenko, S. V. Strizhevskiı̆, “Surface polaritons in biaxial crystals,” Sov. Phys. Solid State 27, 631–633 (1985).

1984 (1)

D. E. N. Brancus, “Polaritons in uniaxial crystals,” Rev. Roum. Phys. 29, 815–822 (1984).

1974 (1)

E. Burstein, W. P. Chen, Y. J. Chen, A. Hartstein, “Surface polaritons—propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004–1019 (1974).
[CrossRef]

1972 (2)

V. N. Lyubimov, D. G. Sannikov, “Surface electromagnetic waves in a uniaxial crystal,” Sov. Phys. Solid State 14, 575–579 (1972).

D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
[CrossRef]

Anderson, E.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarva, J. DuCroz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, LAPACK Users’ Guide, 2nd ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1995).

Averkiev, N. S.

N. S. Averkiev, M. I. D’Yakonov, “Electromagnetic waves localized at the interface of transparent anisotropic media,” Sov. Phys. JETP 68, 653–655 (1990).

Baboiu, D. M.

D. Mihalache, D. M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Guided waves in anisotropic antiguide structures,” Opt. Commun. 108, 239–242 (1994).
[CrossRef]

Baboiu, D.-M.

M. Ciumac, D.-M. Baboiu, D. Mihalache, “Hybrid surface modes in periodic stratified media: transfer matrix technique,” Opt. Commun. 111, 548–555 (1994).
[CrossRef]

Bai, Z.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarva, J. DuCroz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, LAPACK Users’ Guide, 2nd ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1995).

Bai?sa, D. F.

D. F. Baı̆sa, A. Y. Zarenin, S. P. Makarenko, S. V. Strizhevskiı̆, “Surface polaritons in biaxial crystals,” Sov. Phys. Solid State 27, 631–633 (1985).

Berreman, D. W.

Bischof, C.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarva, J. DuCroz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, LAPACK Users’ Guide, 2nd ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1995).

Brancus, D. E. N.

D. E. N. Brancus, “Polaritons in uniaxial crystals. II: the energy propagation,” Rev. Roum. Phys. 35, 385–394 (1990).

D. E. N. Brancus, “Polaritons in uniaxial crystals,” Rev. Roum. Phys. 29, 815–822 (1984).

Burstein, E.

E. Burstein, W. P. Chen, Y. J. Chen, A. Hartstein, “Surface polaritons—propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004–1019 (1974).
[CrossRef]

Chen, W. P.

E. Burstein, W. P. Chen, Y. J. Chen, A. Hartstein, “Surface polaritons—propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004–1019 (1974).
[CrossRef]

Chen, Y. J.

E. Burstein, W. P. Chen, Y. J. Chen, A. Hartstein, “Surface polaritons—propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004–1019 (1974).
[CrossRef]

Ciumac, M.

M. Ciumac, D. Mihalache, “Properties of Bragg reflectors composed of isotropic dielectric layers cladded with birefringent media,” IEEE J. Quantum Electron. 32, 513–518 (1996).
[CrossRef]

M. Ciumac, D. Mihalache, “Hybrid modes in asymmetric periodic stratified dielectric waveguides,” J. Opt. Soc. Am. A 12, 1695–1701 (1995).
[CrossRef]

M. Ciumac, D.-M. Baboiu, D. Mihalache, “Hybrid surface modes in periodic stratified media: transfer matrix technique,” Opt. Commun. 111, 548–555 (1994).
[CrossRef]

D. Mihalache, D. M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Guided waves in anisotropic antiguide structures,” Opt. Commun. 108, 239–242 (1994).
[CrossRef]

Clark, N. A.

S. T. Lagerwall, N. A. Clark, “Ferroelectric liquid crystals: the development of devices,” Ferroelectrics 94, 3–62 (1989).
[CrossRef]

D’Yakonov, M. I.

N. S. Averkiev, M. I. D’Yakonov, “Electromagnetic waves localized at the interface of transparent anisotropic media,” Sov. Phys. JETP 68, 653–655 (1990).

M. I. D’Yakonov, “New type of electromagnetic wave propagating at an interface,” Sov. Phys. JETP 67, 714–716 (1988).

Demmel, J.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarva, J. DuCroz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, LAPACK Users’ Guide, 2nd ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1995).

Dongarva, J.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarva, J. DuCroz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, LAPACK Users’ Guide, 2nd ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1995).

DuCroz, J.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarva, J. DuCroz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, LAPACK Users’ Guide, 2nd ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1995).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

Gaylord, T. K.

T. A. Maldonado, T. K. Gaylord, “Hybrid guided modes in biaxial planar waveguides,” J. Lightwave Technol. 14, 486–499 (1996).
[CrossRef]

D. B. Walker, E. N. Glytsis, T. K. Gaylord, “Ferroelectric liquid crystal waveguide modulation based on switchable uniaxial–uniaxial interface,” Appl. Opt. 35, 3016–3030 (1996).
[CrossRef] [PubMed]

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

Glytsis, E. N.

Greenbaum, A.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarva, J. DuCroz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, LAPACK Users’ Guide, 2nd ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1995).

Hammarling, S.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarva, J. DuCroz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, LAPACK Users’ Guide, 2nd ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1995).

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

Hartstein, A.

E. Burstein, W. P. Chen, Y. J. Chen, A. Hartstein, “Surface polaritons—propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004–1019 (1974).
[CrossRef]

Knoesen, A.

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

Lagerwall, S. T.

S. T. Lagerwall, N. A. Clark, “Ferroelectric liquid crystals: the development of devices,” Ferroelectrics 94, 3–62 (1989).
[CrossRef]

Lyubimov, V. N.

V. N. Lyubimov, D. G. Sannikov, “Surface electromagnetic waves in a uniaxial crystal,” Sov. Phys. Solid State 14, 575–579 (1972).

Makarenko, S. P.

D. F. Baı̆sa, A. Y. Zarenin, S. P. Makarenko, S. V. Strizhevskiı̆, “Surface polaritons in biaxial crystals,” Sov. Phys. Solid State 27, 631–633 (1985).

Maldonado, T. A.

T. A. Maldonado, T. K. Gaylord, “Hybrid guided modes in biaxial planar waveguides,” J. Lightwave Technol. 14, 486–499 (1996).
[CrossRef]

McKenney, A.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarva, J. DuCroz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, LAPACK Users’ Guide, 2nd ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1995).

Mihalache, D.

M. Ciumac, D. Mihalache, “Properties of Bragg reflectors composed of isotropic dielectric layers cladded with birefringent media,” IEEE J. Quantum Electron. 32, 513–518 (1996).
[CrossRef]

L. Torner, J. P. Torres, C. Ojeda, D. Mihalache, “Hybrid waves guided by ultra-thin films,” J. Lightwave Technol. 13, 2027–2033 (1995).
[CrossRef]

M. Ciumac, D. Mihalache, “Hybrid modes in asymmetric periodic stratified dielectric waveguides,” J. Opt. Soc. Am. A 12, 1695–1701 (1995).
[CrossRef]

M. Ciumac, D.-M. Baboiu, D. Mihalache, “Hybrid surface modes in periodic stratified media: transfer matrix technique,” Opt. Commun. 111, 548–555 (1994).
[CrossRef]

D. Mihalache, D. M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Guided waves in anisotropic antiguide structures,” Opt. Commun. 108, 239–242 (1994).
[CrossRef]

L. Torner, C. Santos, J. P. Torres, D. Mihalache, “New waveguide modes in anisotropic structures,” Fiber Integr. Opt. 13, 271–280 (1994).
[CrossRef]

L. Torner, J. P. Torres, D. Mihalache, “New type of guided waves in birefringent media,” IEEE Photonics Technol. Lett. 5, 201–203 (1993).
[CrossRef]

Moharam, M. G.

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

Ojeda, C.

L. Torner, J. P. Torres, C. Ojeda, D. Mihalache, “Hybrid waves guided by ultra-thin films,” J. Lightwave Technol. 13, 2027–2033 (1995).
[CrossRef]

Ostrouchov, S.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarva, J. DuCroz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, LAPACK Users’ Guide, 2nd ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1995).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

Sannikov, D. G.

V. N. Lyubimov, D. G. Sannikov, “Surface electromagnetic waves in a uniaxial crystal,” Sov. Phys. Solid State 14, 575–579 (1972).

Santos, C.

L. Torner, C. Santos, J. P. Torres, D. Mihalache, “New waveguide modes in anisotropic structures,” Fiber Integr. Opt. 13, 271–280 (1994).
[CrossRef]

Sorensen, D.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarva, J. DuCroz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, LAPACK Users’ Guide, 2nd ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1995).

Strizhevskii?, S. V.

D. F. Baı̆sa, A. Y. Zarenin, S. P. Makarenko, S. V. Strizhevskiı̆, “Surface polaritons in biaxial crystals,” Sov. Phys. Solid State 27, 631–633 (1985).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

Torner, L.

L. Torner, J. P. Torres, C. Ojeda, D. Mihalache, “Hybrid waves guided by ultra-thin films,” J. Lightwave Technol. 13, 2027–2033 (1995).
[CrossRef]

L. Torner, C. Santos, J. P. Torres, D. Mihalache, “New waveguide modes in anisotropic structures,” Fiber Integr. Opt. 13, 271–280 (1994).
[CrossRef]

D. Mihalache, D. M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Guided waves in anisotropic antiguide structures,” Opt. Commun. 108, 239–242 (1994).
[CrossRef]

L. Torner, J. P. Torres, D. Mihalache, “New type of guided waves in birefringent media,” IEEE Photonics Technol. Lett. 5, 201–203 (1993).
[CrossRef]

Torres, J. P.

L. Torner, J. P. Torres, C. Ojeda, D. Mihalache, “Hybrid waves guided by ultra-thin films,” J. Lightwave Technol. 13, 2027–2033 (1995).
[CrossRef]

L. Torner, C. Santos, J. P. Torres, D. Mihalache, “New waveguide modes in anisotropic structures,” Fiber Integr. Opt. 13, 271–280 (1994).
[CrossRef]

D. Mihalache, D. M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Guided waves in anisotropic antiguide structures,” Opt. Commun. 108, 239–242 (1994).
[CrossRef]

L. Torner, J. P. Torres, D. Mihalache, “New type of guided waves in birefringent media,” IEEE Photonics Technol. Lett. 5, 201–203 (1993).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

Wait, J. R.

J. R. Wait, “Electromagnetic surface waves,” in Advances in Radio Research, J. A. Saxton, ed. (Academic, New York, 1964), Vol. 1, pp. 157–218.

Walker, D. B.

Welford, K.

K. Welford, “Surface plasmon-polaritons and their uses,” Opt. Quantum Electron. 23, 1–27 (1991).
[CrossRef]

Zarenin, A. Y.

D. F. Baı̆sa, A. Y. Zarenin, S. P. Makarenko, S. V. Strizhevskiı̆, “Surface polaritons in biaxial crystals,” Sov. Phys. Solid State 27, 631–633 (1985).

Appl. Opt. (1)

Ferroelectrics (1)

S. T. Lagerwall, N. A. Clark, “Ferroelectric liquid crystals: the development of devices,” Ferroelectrics 94, 3–62 (1989).
[CrossRef]

Fiber Integr. Opt. (1)

L. Torner, C. Santos, J. P. Torres, D. Mihalache, “New waveguide modes in anisotropic structures,” Fiber Integr. Opt. 13, 271–280 (1994).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Ciumac, D. Mihalache, “Properties of Bragg reflectors composed of isotropic dielectric layers cladded with birefringent media,” IEEE J. Quantum Electron. 32, 513–518 (1996).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

L. Torner, J. P. Torres, D. Mihalache, “New type of guided waves in birefringent media,” IEEE Photonics Technol. Lett. 5, 201–203 (1993).
[CrossRef]

J. Lightwave Technol. (3)

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

T. A. Maldonado, T. K. Gaylord, “Hybrid guided modes in biaxial planar waveguides,” J. Lightwave Technol. 14, 486–499 (1996).
[CrossRef]

L. Torner, J. P. Torres, C. Ojeda, D. Mihalache, “Hybrid waves guided by ultra-thin films,” J. Lightwave Technol. 13, 2027–2033 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Vac. Sci. Technol. (1)

E. Burstein, W. P. Chen, Y. J. Chen, A. Hartstein, “Surface polaritons—propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004–1019 (1974).
[CrossRef]

Opt. Commun. (2)

D. Mihalache, D. M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Guided waves in anisotropic antiguide structures,” Opt. Commun. 108, 239–242 (1994).
[CrossRef]

M. Ciumac, D.-M. Baboiu, D. Mihalache, “Hybrid surface modes in periodic stratified media: transfer matrix technique,” Opt. Commun. 111, 548–555 (1994).
[CrossRef]

Opt. Quantum Electron. (1)

K. Welford, “Surface plasmon-polaritons and their uses,” Opt. Quantum Electron. 23, 1–27 (1991).
[CrossRef]

Rev. Roum. Phys. (2)

D. E. N. Brancus, “Polaritons in uniaxial crystals,” Rev. Roum. Phys. 29, 815–822 (1984).

D. E. N. Brancus, “Polaritons in uniaxial crystals. II: the energy propagation,” Rev. Roum. Phys. 35, 385–394 (1990).

Sov. Phys. JETP (2)

M. I. D’Yakonov, “New type of electromagnetic wave propagating at an interface,” Sov. Phys. JETP 67, 714–716 (1988).

N. S. Averkiev, M. I. D’Yakonov, “Electromagnetic waves localized at the interface of transparent anisotropic media,” Sov. Phys. JETP 68, 653–655 (1990).

Sov. Phys. Solid State (2)

V. N. Lyubimov, D. G. Sannikov, “Surface electromagnetic waves in a uniaxial crystal,” Sov. Phys. Solid State 14, 575–579 (1972).

D. F. Baı̆sa, A. Y. Zarenin, S. P. Makarenko, S. V. Strizhevskiı̆, “Surface polaritons in biaxial crystals,” Sov. Phys. Solid State 27, 631–633 (1985).

Other (4)

J. R. Wait, “Electromagnetic surface waves,” in Advances in Radio Research, J. A. Saxton, ed. (Academic, New York, 1964), Vol. 1, pp. 157–218.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarva, J. DuCroz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, LAPACK Users’ Guide, 2nd ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1995).

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Figures (12)

Fig. 1
Fig. 1

The structure analyzed consists of the interface between an isotropic cover (index nc) and a biaxial substrate (principal indices nx, ny, and nz). The direction of propagation is given by z, and the normal to the interface is given by x. The coordinates x˜, y˜, and z˜ are the principal axes of the biaxial crystal. The coordinate systems are related to each other through the Euler angles θ, ϕ, and ψ.

Fig. 2
Fig. 2

Hypothetical surface mode profiles for (a) an isotropic–isotropic interface and (b) an isotropic–uniaxial interface. For a mode to be confined along the interface, all waves must be inhomogeneous, decaying evanescently in the x direction. For the isotropic–isotropic interface (a), the slope of the field component at the interface must change sign. This can occur for TM waves if one of the materials is a good conductor. For the anisotropic–isotropic interface, the wave in the uniaxial medium is made up of ordinary and extraordinary components with differing decay constants. When added, these waves can form a wave whose derivative of the field amplitude is continuous at the interface.

Fig. 3
Fig. 3

For a surface mode to exist, all waves in the cover and the substrate must be inhomogeneous. To ensure this for a given crystallographic orientation angle θ, the propagation constant β must be greater than some minimum value βmin. When βmin is computed for all orientations, the βmin surface is formed. (a) For isotropic media the surface is simply a sphere; the x˜z˜ cross section is the circle. (b) For uniaxial materials with the optic axis (o.a.) constrained in the plane of the interface, the βmin surface is elliptically shaped. (c) For biaxial materials with both optic axes constrained in the plane of the interface, the βmin surface is formed by the exterior of a circle and an ellipse.

Fig. 4
Fig. 4

The propagation constant for the surface mode is found for the case of an isotropic cover and a uniaxial substrate (optic axis in the plane of the interface). The βmin surface for the isotropic cover is depicted by the circle (dashed curves), and the βmin surface for the uniaxial substrate is depicted by the ellipse (solid curves). The propagation constant β of the surface mode was computed by using the full-wave analysis of Subsection 2.B and is shown by the bold solid curves. The lower cutoff occurs at θl, where βl intersects the isotropic βmin surface. The upper cutoff occurs at θu, where βu intersects the uniaxial βmin surface. The angular range where the surface mode propagates is defined as Δθθu-θl. The surface mode can always be found in the direction θm, where the two βmin surfaces intersect.

Fig. 5
Fig. 5

Profile of the surface mode. The effective thickness is given by heff=- Re[Sz(x)]dx/12max[Sz(x)].

Fig. 6
Fig. 6

The effective thickness heff and the upper and lower cutoff propagation angles, θu and θl, are shown. The striped regions represent orientations that cannot support the surface mode. As ϕ0°, the upper and lower cutoffs converge (θuθl) and heff. The maximum angular range (Δθ =θu-θl=0.52°) and the minimum effective thickness (heff =7.67λ0) are observed when ϕ=90° (the optic axis is in the plane of the interface).

Fig. 7
Fig. 7

The real part of the transverse electric field in the y direction, Re(Ey), is shown for the cases of ϕ=10°, 40°, 70°, and 90°. The y component of the electric field must be continuous in both its value and its derivative across the interface. As ϕ decreases, the oscillating component in the uniaxial substrate increases in both spatial frequency and amplitude.

Fig. 8
Fig. 8

The range of possible propagation directions in arbitrarily oriented potassium titanyl phosphate (KTP) is shown when nc=1.7760. The maximum angular range is observed for y-cut KTP.

Fig. 9
Fig. 9

The angular range Δθ achieves a maximum and the effective thickness heff achieves a minimum for a cover index of refraction (nc) value close to ny, providing the best possible containment of the surface mode. For x-cut material the minimum effective thickness and the maximum angular range are found for an intermediate value of nc. The confinement as well as the angular range is always greatest for the y-cut case.

Fig. 10
Fig. 10

For thick films the doubly inhomogeneous pure guided mode (DIPGM) consists of two coupled surface modes. This leads to a symmetric mode (solid curves) and an antisymmetric mode (dashed curves). Unlike the case for other waveguide modes, increasing the thickness of the structure does not add an increasing number of modes. As the film thickness decreases, the surface modes interact more strongly until the antisymmetric mode cuts off. This occurs at h=10.64λ0.

Fig. 11
Fig. 11

The ferroelectric liquid-crystal (FLC) modulator is composed of two waveguides, labeled Region 1 and Region 2, abutting each other. The symmetric waveguide has an identical isotropic cover and substrate index of refraction nc. The cover and substrate material is identical for both regions. The FLC material is oriented with the optic axis at angle θ1 in region 1 and θ2 in region 2. For the example presented, an inhomogeneous pure guided mode is able to propagate in region 1, and a DIPGM propagates in region 2.

Fig. 12
Fig. 12

The boundary error BE in the tangential fields at the boundary between the region 1 and 2 waveguides as a function of the number of modes used in the analysis is shown excluding and including the DIPGM. Without DIPGM’s the BE does not go to zero. Inclusion of DIPGM’s causes the BE to approach zero.

Tables (1)

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Table 1 Mode Classification for Uniaxial Slab Waveguides

Equations (47)

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Ey1(x)|x=0=Ey2(x)|x=0,Ey1(x)xx=0=Ey2(x)xx=0,
Hy1(x)|x=0=Hy2(x)|x=0,
1ε1Hy1(x)xx=0=1ε2Hy2(x)xx=0,
βmin=max(βmincov, βminsub),
Eysubη0HysubEzsubη0Hzsubx=0=Eycovη0HycovEzcovη0Hzcovx=0,
ddxEyη0HyEzη0Hzν=-jΔ(β)Eyη0HyEzη0Hzν,
Eyη0HyEzη0Hzν=(v1v2v3v4)exp(-jκ1x)0000exp(-jκ2x)0000exp(-jκ3x)0000exp(-jκ4x)A1A2A3A4,
Eycovη0HycovEzcovη0Hzcovν=(v1covv2cov)A1covA2cov,
Eysubη0HysubEzsubη0Hzsubν=(v3subv4sub)A3subA4sub.
(v1covv2covv3subv4sub)A1covA2cov-A3sub-A4sub=V(β)A1covA2cov-A3sub-A4sub=0.
P=-Re[Sz(x)]dx=12max[Re(Sz)]heff,
η0Hxν=-βEyν/k0,
Exν=k0η0Hyνβ+κnEzν/β.
heff=2P/max[Re(Sz)].
BE=-(|E1x-E2x|2+|E1y-E2y|2+|H1x-H2x|2+|H1y-H2y|2)|z=0 dx,
Δ=0001εxyεzxεxx-εzy-βεzxεxxεxzεzxεxx-εzz0-βεxyεxx(β)2εxx-1-βεxzεxx0εyy-εxy2εxx-(β)2βεxyεxxεyz-εxyεxzεxx0,
εxxεyxεzxεxyεyyεzyεxzεyzεzz=Rz(ϕ)Ry(θ)Rz(ψ)nx2000ny2000nz2×RzT(ψ)RyT(θ)RzT(ϕ),
Rz(ξ)=cos ξsin ξ0-sin ξcos ξ0001,
Ry(ξ)=cos ξ0-sin ξ010sin ξ0cos ξ.
κ=±[n2-(β)2]1/2,
vTE=100κ,vTM=0n2-κ0.
κO=±[nO2-(β)2]1/2,
κE=±[nE2-(β)2(nE2 cos2 θ+nO2 sin2 θ)/nO2]1/2,
vO=κO cos θnO2 sin θ-κO sin θ(κO)2 cos θ,vE=-nO2 sin θκEnO2 cos θ[(β)2-nO2]cos θ-κEnO2 sin θ.
κO=±[nO2-(β)2]1/2,
κE=-β εxzεxx±nOεxx×[nE2εxx-(nE2+nO2-εyy)(β)2]1/2,
vO=κO cos θ-β cos ϕ sin θnO2 sin ϕ sin θ-κO sin ϕ sin θ(κO)2 cos θ-βκO cos ϕ sin θ,
vE=-nO2 sin ϕ sin θκEnO2 cos θ-βnO2 cos ϕ sin θβκE cos ϕ sin θ-[nO2-(β)2]cos θ-κEnO2 sin ϕ sin θ.
Δ=0001-a0-b00c00d0a0,
a=(nz2-nx2)sin θ cos θ,
b=nx2 sin2 θ+nz2 cos2 θ,
c=(β)2/ny2-1,
d=nx2 cos2 θ+nz2 sin2 θ-(β)2.
(κ)4+(bc-d)(κ)2+a2c-bcd=0,
v=-κ[a2-(κ)2b-b2c]a(a2-bd)-κa[(κ)2+bc-d]-(a2-bd)[(κ)2+bc].
βmin=k0nc.
ne(θ)=nEnO/(nE2 cos2 θ+nO2 sin2 θ)1/2.
βmin=k0ne(θ).
(β)2>nEεxxnE2+nO2-εyy=nE2[(nE2-nO2)cos2 ϕ sin2 θ+nO2]nE2-(nE2-nO2)sin2 ϕ sin2 ϕ,
βmin=nE2[(nE2-nO2)cos2 ϕ sin2 θ+nO2]nE2-(nE2-nO2)sin2 ϕ sin2 θ1/2.
(κ)4+(bc-d)(κ)2+a2c-bcd=0.
(κ)2=d=nx2 cos2 θ+nz2 sin2 θ-ny2.
θ<θ1=tan-1ny2-nx2nz2-ny21/2.
(β)2=nx2nz2nz2 cos2 θ+nx2 sin2 θ
(κ)2=d-bc=(ny2-nx2)nz4-[(nz2-ny2)nx4+(ny2-nx2)nz4]sin2 θny2(nx2 sin2 θ+nz2 cos2 θ).
θ>θ2=tan-1nz2nx2tan θ1.
βmin=k0×ny,θθ1ny/cos(θ-θ1),θ1θθ2nxnz/(nz2 cos2 θ+nx2 sin2 θ)1/2,θ2<θ.

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