Abstract

Liquid crystals have effective electro-optic coefficients that are orders of magnitude larger than other integrated optical materials such as lithium niobate. However, previous studies of liquid-crystal waveguides have mainly focused on nematic liquid crystals, which exhibit impractically large scattering losses as waveguides. Studies of smectic liquid crystals and liquid crystals under strong confinement suggest the losses in these materials may be more manageable. In this study, the possibility of using ferroelectric liquid crystals in active waveguide modulators is explored through the analysis of several modulator configurations: a cutoff modulator, a deflection modulator, and an input coupler. As a way to study these structures, a mode-matching technique was developed to analyze the effects of a discontinuity in a uniaxial slab waveguide whose optic axis is in the plane of the waveguide. The results from the mode-matching technique were compared with those from simple bulk models. The analysis shows that ferroelectric liquid-crystal modulators have many desirable performance characteristics and could form the basis for practical waveguide modulators.

© 1996 Optical Society of America

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1993 (5)

M. Lawrence, “Lithium niobate integrated optics,” Rep. Prog. Phys. 56, 363–429 (1993).
[CrossRef]

K. M. Johnson, D. J. McKnight, I. Underwood, “Smart spatial light modulators using liquid crystals on silicon,” IEEE J. Quantum Electron. 29, 699–714 (1993).
[CrossRef]

N. F. Hartman, T. K. Gaylord, T. J. Drabik, M. A. Handschy, “Phase stability of ferroelectric liquid crystals upon repeated switching and static temperature characteristics,” Appl. Opt. 32, 3720–3725 (1993).
[CrossRef] [PubMed]

L. Torner, J. Recolons, J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab waveguides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

M. Lu, M. M. Fejer, “Anisotropic dielectric waveguides,” J. Opt. Soc. Am. 10, 246–261 (1993).
[CrossRef]

1992 (4)

E. Anemogiannis, E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
[CrossRef]

S. Sawa, M. Geshiro, M. Hotta, H. Kanetake, “Coupling efficiency of butt-joined isotropic and anisotropic single-mode slab waveguides,” IEEE Trans. Microwave Theory Tech. 40, 338–345 (1992).
[CrossRef]

A. Jákli, A. Saupe, “Method to obtain uniform bookshelf textures in smectic C* liquid crystals,” Appl. Phys. Lett. 60, 2622–2624 (1992).
[CrossRef]

Y. Sadohara, M. Ozaki, K. Yoshino, “Waveguide modulator using ferroelectric liquid crystal,” Tech. Rep. Osaka Univ. 42, 137–143 (1992).

1991 (4)

S. K. Lo, L. M. Galarneau, D. J. Rogers, S. R. Flom, “Smectic liquid crystal waveguides with cylindrical geometry,” Mol. Cryst. Liq. Cryst. 201, 137–145 (1991).
[CrossRef]

A. Donaldson, “Candidate materials and technologies for integrated optics: fast and efficient electro-optic modulation,” J. Phys. D 24, 785–802 (1991).
[CrossRef]

E. V. Tomme, P. P. V. Daele, R. G. Baets, P. E. Lagasse, “Integrated optic devices based on nonlinear optical polymers,” IEEE J. Quantum Electron. 27, 778–786 (1991).
[CrossRef]

S. Sawa, M. Geshiro, F. Takeda, “Low-loss optical branching waveguides consisting of anisotropic materials,” IEEE Trans. Microwave Theory Tech. 39, 1140–1147 (1991).
[CrossRef]

1990 (4)

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

W. Biehlig, U. Langbein, “Three-dimensional step discontinuities in planar waveguides: angular-spectrum representation of guided wavefields and generalized matrix-operator formalism,” Opt. Quantum Electron. 22, 319–333 (1990).
[CrossRef]

N. A. Clark, M. A. Handschy, “Surface-stabilized ferroelectric liquid-crystal electro-optic waveguide switch,” Appl. Phys. Lett. 57, 1852–1854 (1990).
[CrossRef]

M. Ozaki, Y. Sadohara, T. Hatai, K. Yoshino, “Fast optical switching in polymer waveguide using ferroelectric liquid crystal,” Jpn. J. Appl. Phys. 29, L843–L845 (1990).
[CrossRef]

1989 (4)

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

M. Green, S. J. Madden, “Low loss nematic liquid crystal cored fiber waveguides,” Appl. Opt. 28, 5202–5203 (1989).
[CrossRef] [PubMed]

M. Kawaida, T. Yamaguchi, T. Akahane, “Measurement of refractive indices of ferroelectric SmC* liquid crystal by the Fabry–Perot interference method,” Jpn. J. Appl. Phys. 28, L1602–L1605 (1989).
[CrossRef]

H. J. Caulfield, J. Kinser, S. K. Rogers, “Optical neural networks,” Proc. IEEE 77, 1573–1583 (1989).
[CrossRef]

1988 (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]

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

T. H. Wood, “Multiple quantum well (MQW) waveguide modulators,” J. Lightwave Technol. 6, 743–757 (1988).
[CrossRef]

1986 (5)

Y. Okamura, K. Kitatani, S. Yamamoto, “Low-voltage driving in nematic liquid crystal overlayered waveguide,” J. Lightwave Technol. 4, 360–363 (1986).
[CrossRef]

W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities: part 1. Operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
[CrossRef]

W. Biehlig, “Light propagation in a planar dielectric slab waveguide with step discontinuities: part 2. Numerical analysis of TE-polarized fields,” Opt. Quantum Electron. 18, 229–238 (1986).
[CrossRef]

W. Biehlig, C. Wächter, “Light propagation in a planar dielectric slab waveguide with step discontinuities: part 3. TM-polarized fields,” Opt. Quantum Electron. 18, 239–246 (1986).
[CrossRef]

P. G. Suchoski, V. Ramaswamy, “Exact numerical technique for the analysis of step discontinuities and tapers in optical dielectric waveguides,” J. Opt. Soc. Am. A 3, 194–203 (1986).
[CrossRef]

1984 (3)

Y. Okamura, K. Kitatani, S. Yamamoto, “Electrooptic leaky anisotropic waveguides using nematic liquid crystal overlayers,” J. Lightwave Technol. 2, 292–295 (1984).
[CrossRef]

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Band-edge electroabsorption in quantum well structures: the quantum confined Stark effect,” Phys. Rev. Lett. 53, 2173–2176 (1984).
[CrossRef]

M. Geshiro, Y. Kaihara, S. Sawa, “Analysis of wave propagation in anisotropic film waveguides with bent optical axes,” IEEE Trans. Microwave Theory Tech. 32, 339–347 (1984).
[CrossRef]

1983 (1)

N. A. Clark, M. A. Handschy, S. T. Laggerwall, “Ferroelectric liquid crystal electro-optics using the surface stabilized structure,” Mol. Cryst. Liq. Cryst. 94, 213–234 (1983).
[CrossRef]

1982 (2)

M. Kobayashi, H. Terui, M. Kawachi, J. Noda, “2 × 2 optical waveguide matrix switch using nematic liquid crystal,” IEEE J. Quantum Electron. QE-18, 1603–1609 (1982).
[CrossRef]

G. H. Brooke, M. M. Z. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
[CrossRef]

1980 (2)

T. J. M. Boyd, I. Moshkun, I. M. Stephenson, “Radiation losses due to discontinuities in asymmetric three-layer optical waveguides,” Opt. Quantum Electron. 12, 143–158 (1980).
[CrossRef]

N. A. Clark, S. T. Lagerwall, “Submicrosecond bistable electro-optic switching in liquid crystals,” Appl. Phys. Lett. 36, 899–901 (1980).
[CrossRef]

1979 (2)

P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
[CrossRef]

D. Marcuse, I. P. Kaminow, “Modes of a symmetric slab optical waveguide in birefringent media, part II: slab with coplanar optical axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
[CrossRef]

1978 (1)

T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. 26, 738–746 (1978).
[CrossRef]

1977 (2)

B. Rulf, “On the matching of two optical waveguides,” Rad. Sci. 12, 593–601 (1977).
[CrossRef]

J. R. Whinnery, C. Hu, Y. S. Kwon, “Liquid-crystal waveguides for integrated optics,” IEEE J. Quantum Electron. QE-13, 363–429 (1977).

1976 (2)

T. G. Giallorenzi, J. A. Weiss, J. P. Sheridan, “Light scattering from smectic liquid-crystal waveguides,” J. Appl. Phys. 47, 1820–1826 (1976).
[CrossRef]

G. H. Brooke, M. M. Z. Kharadly, “Step discontinuities on dielectric waveguides,” Electron. Lett. 12, 473–475 (1976).
[CrossRef]

1975 (2)

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54, 985–995 (1975).

T. G. Giallorenzi, J. P. Sheridan, “Light scattering from nematic liquid crystal waveguides,” J. Appl. Phys. 46, 1271–1282 (1975).
[CrossRef]

1973 (2)

D. J. Channin, “Optical waveguide modulation using nematic liquid crystal,” Appl. Phys. Lett. 22, 365–366 (1973).
[CrossRef]

J. S. Cook, W. L. Mammel, R. J. Grow, “Effect of misalignments on coupling efficiency of single-mode optical fiber butt joints,” Bell Syst. Tech. J. 52, 1439–1448 (1973).

1972 (2)

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

P. J. B. Clarricoats, A. B. Sharpe, “Modal matching applied to a discontinuity in a planar surface waveguide,” Electron. Lett. 8, 28–29 (1972).
[CrossRef]

1967 (1)

P. J. B. Clarricoats, K. R. Slinn, “Numerical solution of waveguide-discontinuity problems,” Proc. IEE 114, 878–886 (1967).

Akahane, T.

M. Kawaida, T. Yamaguchi, T. Akahane, “Measurement of refractive indices of ferroelectric SmC* liquid crystal by the Fabry–Perot interference method,” Jpn. J. Appl. Phys. 28, L1602–L1605 (1989).
[CrossRef]

Anemogiannis, E.

E. Anemogiannis, E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
[CrossRef]

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).

Baets, R. G.

E. V. Tomme, P. P. V. Daele, R. G. Baets, P. E. Lagasse, “Integrated optic devices based on nonlinear optical polymers,” IEEE J. Quantum Electron. 27, 778–786 (1991).
[CrossRef]

Berreman, D. W.

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

Biehlig, W.

W. Biehlig, U. Langbein, “Three-dimensional step discontinuities in planar waveguides: angular-spectrum representation of guided wavefields and generalized matrix-operator formalism,” Opt. Quantum Electron. 22, 319–333 (1990).
[CrossRef]

W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities: part 1. Operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
[CrossRef]

W. Biehlig, “Light propagation in a planar dielectric slab waveguide with step discontinuities: part 2. Numerical analysis of TE-polarized fields,” Opt. Quantum Electron. 18, 229–238 (1986).
[CrossRef]

W. Biehlig, C. Wächter, “Light propagation in a planar dielectric slab waveguide with step discontinuities: part 3. TM-polarized fields,” Opt. Quantum Electron. 18, 239–246 (1986).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970) Chap. 14, p. 665.

Boyd, T. J. M.

T. J. M. Boyd, I. Moshkun, I. M. Stephenson, “Radiation losses due to discontinuities in asymmetric three-layer optical waveguides,” Opt. Quantum Electron. 12, 143–158 (1980).
[CrossRef]

Brooke, G. H.

G. H. Brooke, M. M. Z. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
[CrossRef]

G. H. Brooke, M. M. Z. Kharadly, “Step discontinuities on dielectric waveguides,” Electron. Lett. 12, 473–475 (1976).
[CrossRef]

Burrus, C. A.

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Band-edge electroabsorption in quantum well structures: the quantum confined Stark effect,” Phys. Rev. Lett. 53, 2173–2176 (1984).
[CrossRef]

Caulfield, H. J.

H. J. Caulfield, J. Kinser, S. K. Rogers, “Optical neural networks,” Proc. IEEE 77, 1573–1583 (1989).
[CrossRef]

Channin, D. J.

D. J. Channin, “Optical waveguide modulation using nematic liquid crystal,” Appl. Phys. Lett. 22, 365–366 (1973).
[CrossRef]

Chemla, D. S.

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Band-edge electroabsorption in quantum well structures: the quantum confined Stark effect,” Phys. Rev. Lett. 53, 2173–2176 (1984).
[CrossRef]

Chen, H. C.

H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, New York, 1983), Chap. 2, p. 58.

Clark, N. A.

N. A. Clark, M. A. Handschy, “Surface-stabilized ferroelectric liquid-crystal electro-optic waveguide switch,” Appl. Phys. Lett. 57, 1852–1854 (1990).
[CrossRef]

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

N. A. Clark, M. A. Handschy, S. T. Laggerwall, “Ferroelectric liquid crystal electro-optics using the surface stabilized structure,” Mol. Cryst. Liq. Cryst. 94, 213–234 (1983).
[CrossRef]

N. A. Clark, S. T. Lagerwall, “Submicrosecond bistable electro-optic switching in liquid crystals,” Appl. Phys. Lett. 36, 899–901 (1980).
[CrossRef]

Clarricoats, P. J. B.

P. J. B. Clarricoats, A. B. Sharpe, “Modal matching applied to a discontinuity in a planar surface waveguide,” Electron. Lett. 8, 28–29 (1972).
[CrossRef]

P. J. B. Clarricoats, K. R. Slinn, “Numerical solution of waveguide-discontinuity problems,” Proc. IEE 114, 878–886 (1967).

Cook, J. S.

J. S. Cook, W. L. Mammel, R. J. Grow, “Effect of misalignments on coupling efficiency of single-mode optical fiber butt joints,” Bell Syst. Tech. J. 52, 1439–1448 (1973).

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).

Daele, P. P. V.

E. V. Tomme, P. P. V. Daele, R. G. Baets, P. E. Lagasse, “Integrated optic devices based on nonlinear optical polymers,” IEEE J. Quantum Electron. 27, 778–786 (1991).
[CrossRef]

Damen, T. C.

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Band-edge electroabsorption in quantum well structures: the quantum confined Stark effect,” Phys. Rev. Lett. 53, 2173–2176 (1984).
[CrossRef]

de Doncker-Kapenga, E.

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, D. K. Kahaner, QUADPACK: A Subroutine Package for Automatic Integration (Springer-Verlag, New York, 1983).

Donaldson, A.

A. Donaldson, “Candidate materials and technologies for integrated optics: fast and efficient electro-optic modulation,” J. Phys. D 24, 785–802 (1991).
[CrossRef]

Drabik, T. J.

Fejer, M. M.

M. Lu, M. M. Fejer, “Anisotropic dielectric waveguides,” J. Opt. Soc. Am. 10, 246–261 (1993).
[CrossRef]

Flom, S. R.

S. K. Lo, L. M. Galarneau, D. J. Rogers, S. R. Flom, “Smectic liquid crystal waveguides with cylindrical geometry,” Mol. Cryst. Liq. Cryst. 201, 137–145 (1991).
[CrossRef]

Galarneau, L. M.

S. K. Lo, L. M. Galarneau, D. J. Rogers, S. R. Flom, “Smectic liquid crystal waveguides with cylindrical geometry,” Mol. Cryst. Liq. Cryst. 201, 137–145 (1991).
[CrossRef]

Gaylord, T. K.

Geshiro, M.

S. Sawa, M. Geshiro, M. Hotta, H. Kanetake, “Coupling efficiency of butt-joined isotropic and anisotropic single-mode slab waveguides,” IEEE Trans. Microwave Theory Tech. 40, 338–345 (1992).
[CrossRef]

S. Sawa, M. Geshiro, F. Takeda, “Low-loss optical branching waveguides consisting of anisotropic materials,” IEEE Trans. Microwave Theory Tech. 39, 1140–1147 (1991).
[CrossRef]

M. Geshiro, Y. Kaihara, S. Sawa, “Analysis of wave propagation in anisotropic film waveguides with bent optical axes,” IEEE Trans. Microwave Theory Tech. 32, 339–347 (1984).
[CrossRef]

Giallorenzi, T. G.

T. G. Giallorenzi, J. A. Weiss, J. P. Sheridan, “Light scattering from smectic liquid-crystal waveguides,” J. Appl. Phys. 47, 1820–1826 (1976).
[CrossRef]

T. G. Giallorenzi, J. P. Sheridan, “Light scattering from nematic liquid crystal waveguides,” J. Appl. Phys. 46, 1271–1282 (1975).
[CrossRef]

Glytsis, E. N.

E. Anemogiannis, E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
[CrossRef]

Gossard, A. C.

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Band-edge electroabsorption in quantum well structures: the quantum confined Stark effect,” Phys. Rev. Lett. 53, 2173–2176 (1984).
[CrossRef]

Green, M.

Grow, R. J.

J. S. Cook, W. L. Mammel, R. J. Grow, “Effect of misalignments on coupling efficiency of single-mode optical fiber butt joints,” Bell Syst. Tech. J. 52, 1439–1448 (1973).

Handschy, M. A.

N. F. Hartman, T. K. Gaylord, T. J. Drabik, M. A. Handschy, “Phase stability of ferroelectric liquid crystals upon repeated switching and static temperature characteristics,” Appl. Opt. 32, 3720–3725 (1993).
[CrossRef] [PubMed]

N. A. Clark, M. A. Handschy, “Surface-stabilized ferroelectric liquid-crystal electro-optic waveguide switch,” Appl. Phys. Lett. 57, 1852–1854 (1990).
[CrossRef]

N. A. Clark, M. A. Handschy, S. T. Laggerwall, “Ferroelectric liquid crystal electro-optics using the surface stabilized structure,” Mol. Cryst. Liq. Cryst. 94, 213–234 (1983).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961) Chap. 3, p. 95.

Hartman, N. F.

Hatai, T.

M. Ozaki, Y. Sadohara, T. Hatai, K. Yoshino, “Fast optical switching in polymer waveguide using ferroelectric liquid crystal,” Jpn. J. Appl. Phys. 29, L843–L845 (1990).
[CrossRef]

Hehl, K.

W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities: part 1. Operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
[CrossRef]

Hotta, M.

S. Sawa, M. Geshiro, M. Hotta, H. Kanetake, “Coupling efficiency of butt-joined isotropic and anisotropic single-mode slab waveguides,” IEEE Trans. Microwave Theory Tech. 40, 338–345 (1992).
[CrossRef]

Hu, C.

J. R. Whinnery, C. Hu, Y. S. Kwon, “Liquid-crystal waveguides for integrated optics,” IEEE J. Quantum Electron. QE-13, 363–429 (1977).

Jákli, A.

A. Jákli, A. Saupe, “Method to obtain uniform bookshelf textures in smectic C* liquid crystals,” Appl. Phys. Lett. 60, 2622–2624 (1992).
[CrossRef]

Johnson, K. M.

K. M. Johnson, D. J. McKnight, I. Underwood, “Smart spatial light modulators using liquid crystals on silicon,” IEEE J. Quantum Electron. 29, 699–714 (1993).
[CrossRef]

Kahaner, D. K.

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, D. K. Kahaner, QUADPACK: A Subroutine Package for Automatic Integration (Springer-Verlag, New York, 1983).

Kaihara, Y.

M. Geshiro, Y. Kaihara, S. Sawa, “Analysis of wave propagation in anisotropic film waveguides with bent optical axes,” IEEE Trans. Microwave Theory Tech. 32, 339–347 (1984).
[CrossRef]

Kaminow, I. P.

D. Marcuse, I. P. Kaminow, “Modes of a symmetric slab optical waveguide in birefringent media, part II: slab with coplanar optical axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
[CrossRef]

Kanetake, H.

S. Sawa, M. Geshiro, M. Hotta, H. Kanetake, “Coupling efficiency of butt-joined isotropic and anisotropic single-mode slab waveguides,” IEEE Trans. Microwave Theory Tech. 40, 338–345 (1992).
[CrossRef]

Kawachi, M.

M. Kobayashi, H. Terui, M. Kawachi, J. Noda, “2 × 2 optical waveguide matrix switch using nematic liquid crystal,” IEEE J. Quantum Electron. QE-18, 1603–1609 (1982).
[CrossRef]

Kawaida, M.

M. Kawaida, T. Yamaguchi, T. Akahane, “Measurement of refractive indices of ferroelectric SmC* liquid crystal by the Fabry–Perot interference method,” Jpn. J. Appl. Phys. 28, L1602–L1605 (1989).
[CrossRef]

Kharadly, M. M. Z.

G. H. Brooke, M. M. Z. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
[CrossRef]

G. H. Brooke, M. M. Z. Kharadly, “Step discontinuities on dielectric waveguides,” Electron. Lett. 12, 473–475 (1976).
[CrossRef]

Kinser, J.

H. J. Caulfield, J. Kinser, S. K. Rogers, “Optical neural networks,” Proc. IEEE 77, 1573–1583 (1989).
[CrossRef]

Kitatani, K.

Y. Okamura, K. Kitatani, S. Yamamoto, “Low-voltage driving in nematic liquid crystal overlayered waveguide,” J. Lightwave Technol. 4, 360–363 (1986).
[CrossRef]

Y. Okamura, K. Kitatani, S. Yamamoto, “Electrooptic leaky anisotropic waveguides using nematic liquid crystal overlayers,” J. Lightwave Technol. 2, 292–295 (1984).
[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]

Kobayashi, M.

M. Kobayashi, H. Terui, M. Kawachi, J. Noda, “2 × 2 optical waveguide matrix switch using nematic liquid crystal,” IEEE J. Quantum Electron. QE-18, 1603–1609 (1982).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Theory of dielectric waveguides,” in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, New York, 1988), Chap. 2, p. 7.
[CrossRef]

Kwon, Y. S.

J. R. Whinnery, C. Hu, Y. S. Kwon, “Liquid-crystal waveguides for integrated optics,” IEEE J. Quantum Electron. QE-13, 363–429 (1977).

Lagasse, P. E.

E. V. Tomme, P. P. V. Daele, R. G. Baets, P. E. Lagasse, “Integrated optic devices based on nonlinear optical polymers,” IEEE J. Quantum Electron. 27, 778–786 (1991).
[CrossRef]

Lagerwall, S. T.

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

N. A. Clark, S. T. Lagerwall, “Submicrosecond bistable electro-optic switching in liquid crystals,” Appl. Phys. Lett. 36, 899–901 (1980).
[CrossRef]

Laggerwall, S. T.

N. A. Clark, M. A. Handschy, S. T. Laggerwall, “Ferroelectric liquid crystal electro-optics using the surface stabilized structure,” Mol. Cryst. Liq. Cryst. 94, 213–234 (1983).
[CrossRef]

Langbein, U.

W. Biehlig, U. Langbein, “Three-dimensional step discontinuities in planar waveguides: angular-spectrum representation of guided wavefields and generalized matrix-operator formalism,” Opt. Quantum Electron. 22, 319–333 (1990).
[CrossRef]

W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities: part 1. Operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
[CrossRef]

Lawrence, M.

M. Lawrence, “Lithium niobate integrated optics,” Rep. Prog. Phys. 56, 363–429 (1993).
[CrossRef]

Lederer, F.

W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities: part 1. Operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
[CrossRef]

Lo, S. K.

S. K. Lo, L. M. Galarneau, D. J. Rogers, S. R. Flom, “Smectic liquid crystal waveguides with cylindrical geometry,” Mol. Cryst. Liq. Cryst. 201, 137–145 (1991).
[CrossRef]

Lu, M.

M. Lu, M. M. Fejer, “Anisotropic dielectric waveguides,” J. Opt. Soc. Am. 10, 246–261 (1993).
[CrossRef]

Madden, S. J.

Mammel, W. L.

J. S. Cook, W. L. Mammel, R. J. Grow, “Effect of misalignments on coupling efficiency of single-mode optical fiber butt joints,” Bell Syst. Tech. J. 52, 1439–1448 (1973).

Marcuse, D.

D. Marcuse, I. P. Kaminow, “Modes of a symmetric slab optical waveguide in birefringent media, part II: slab with coplanar optical axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
[CrossRef]

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54, 985–995 (1975).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974) Chap. 1, p. 47.

McKnight, D. J.

K. M. Johnson, D. J. McKnight, I. Underwood, “Smart spatial light modulators using liquid crystals on silicon,” IEEE J. Quantum Electron. 29, 699–714 (1993).
[CrossRef]

Miller, D. A. B.

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Band-edge electroabsorption in quantum well structures: the quantum confined Stark effect,” Phys. Rev. Lett. 53, 2173–2176 (1984).
[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]

Moshkun, I.

T. J. M. Boyd, I. Moshkun, I. M. Stephenson, “Radiation losses due to discontinuities in asymmetric three-layer optical waveguides,” Opt. Quantum Electron. 12, 143–158 (1980).
[CrossRef]

Noda, J.

M. Kobayashi, H. Terui, M. Kawachi, J. Noda, “2 × 2 optical waveguide matrix switch using nematic liquid crystal,” IEEE J. Quantum Electron. QE-18, 1603–1609 (1982).
[CrossRef]

Okamura, Y.

Y. Okamura, K. Kitatani, S. Yamamoto, “Low-voltage driving in nematic liquid crystal overlayered waveguide,” J. Lightwave Technol. 4, 360–363 (1986).
[CrossRef]

Y. Okamura, K. Kitatani, S. Yamamoto, “Electrooptic leaky anisotropic waveguides using nematic liquid crystal overlayers,” J. Lightwave Technol. 2, 292–295 (1984).
[CrossRef]

Ozaki, M.

Y. Sadohara, M. Ozaki, K. Yoshino, “Waveguide modulator using ferroelectric liquid crystal,” Tech. Rep. Osaka Univ. 42, 137–143 (1992).

M. Ozaki, Y. Sadohara, T. Hatai, K. Yoshino, “Fast optical switching in polymer waveguide using ferroelectric liquid crystal,” Jpn. J. Appl. Phys. 29, L843–L845 (1990).
[CrossRef]

Piessens, R.

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, D. K. Kahaner, QUADPACK: A Subroutine Package for Automatic Integration (Springer-Verlag, New York, 1983).

Ramaswamy, V.

Recolons, J.

L. Torner, J. Recolons, J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab waveguides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

Rogers, D. J.

S. K. Lo, L. M. Galarneau, D. J. Rogers, S. R. Flom, “Smectic liquid crystal waveguides with cylindrical geometry,” Mol. Cryst. Liq. Cryst. 201, 137–145 (1991).
[CrossRef]

Rogers, S. K.

H. J. Caulfield, J. Kinser, S. K. Rogers, “Optical neural networks,” Proc. IEEE 77, 1573–1583 (1989).
[CrossRef]

Rozzi, T. E.

T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. 26, 738–746 (1978).
[CrossRef]

Rulf, B.

B. Rulf, “On the matching of two optical waveguides,” Rad. Sci. 12, 593–601 (1977).
[CrossRef]

Sadohara, Y.

Y. Sadohara, M. Ozaki, K. Yoshino, “Waveguide modulator using ferroelectric liquid crystal,” Tech. Rep. Osaka Univ. 42, 137–143 (1992).

M. Ozaki, Y. Sadohara, T. Hatai, K. Yoshino, “Fast optical switching in polymer waveguide using ferroelectric liquid crystal,” Jpn. J. Appl. Phys. 29, L843–L845 (1990).
[CrossRef]

Saupe, A.

A. Jákli, A. Saupe, “Method to obtain uniform bookshelf textures in smectic C* liquid crystals,” Appl. Phys. Lett. 60, 2622–2624 (1992).
[CrossRef]

Sawa, S.

S. Sawa, M. Geshiro, M. Hotta, H. Kanetake, “Coupling efficiency of butt-joined isotropic and anisotropic single-mode slab waveguides,” IEEE Trans. Microwave Theory Tech. 40, 338–345 (1992).
[CrossRef]

S. Sawa, M. Geshiro, F. Takeda, “Low-loss optical branching waveguides consisting of anisotropic materials,” IEEE Trans. Microwave Theory Tech. 39, 1140–1147 (1991).
[CrossRef]

M. Geshiro, Y. Kaihara, S. Sawa, “Analysis of wave propagation in anisotropic film waveguides with bent optical axes,” IEEE Trans. Microwave Theory Tech. 32, 339–347 (1984).
[CrossRef]

Sharpe, A. B.

P. J. B. Clarricoats, A. B. Sharpe, “Modal matching applied to a discontinuity in a planar surface waveguide,” Electron. Lett. 8, 28–29 (1972).
[CrossRef]

Sheridan, J. P.

T. G. Giallorenzi, J. A. Weiss, J. P. Sheridan, “Light scattering from smectic liquid-crystal waveguides,” J. Appl. Phys. 47, 1820–1826 (1976).
[CrossRef]

T. G. Giallorenzi, J. P. Sheridan, “Light scattering from nematic liquid crystal waveguides,” J. Appl. Phys. 46, 1271–1282 (1975).
[CrossRef]

Shih, Y. C.

Y. C. Shih, “The mode-matching method,” in Numerical Techniques for Microwave and Millimeter-Wave Passive Structures, T. Itoh, ed. (Wiley, New York, 1989), Chap. 9, p. 592.

Slinn, K. R.

P. J. B. Clarricoats, K. R. Slinn, “Numerical solution of waveguide-discontinuity problems,” Proc. IEE 114, 878–886 (1967).

Stephenson, I. M.

T. J. M. Boyd, I. Moshkun, I. M. Stephenson, “Radiation losses due to discontinuities in asymmetric three-layer optical waveguides,” Opt. Quantum Electron. 12, 143–158 (1980).
[CrossRef]

Suchoski, P. G.

Takeda, F.

S. Sawa, M. Geshiro, F. Takeda, “Low-loss optical branching waveguides consisting of anisotropic materials,” IEEE Trans. Microwave Theory Tech. 39, 1140–1147 (1991).
[CrossRef]

Terui, H.

M. Kobayashi, H. Terui, M. Kawachi, J. Noda, “2 × 2 optical waveguide matrix switch using nematic liquid crystal,” IEEE J. Quantum Electron. QE-18, 1603–1609 (1982).
[CrossRef]

Tomme, E. V.

E. V. Tomme, P. P. V. Daele, R. G. Baets, P. E. Lagasse, “Integrated optic devices based on nonlinear optical polymers,” IEEE J. Quantum Electron. 27, 778–786 (1991).
[CrossRef]

Torner, L.

L. Torner, J. Recolons, J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab waveguides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

Torres, J. P.

L. Torner, J. Recolons, J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab waveguides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

Überhuber, C. W.

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, D. K. Kahaner, QUADPACK: A Subroutine Package for Automatic Integration (Springer-Verlag, New York, 1983).

Underwood, I.

K. M. Johnson, D. J. McKnight, I. Underwood, “Smart spatial light modulators using liquid crystals on silicon,” IEEE J. Quantum Electron. 29, 699–714 (1993).
[CrossRef]

Wächter, C.

W. Biehlig, C. Wächter, “Light propagation in a planar dielectric slab waveguide with step discontinuities: part 3. TM-polarized fields,” Opt. Quantum Electron. 18, 239–246 (1986).
[CrossRef]

Weiss, J. A.

T. G. Giallorenzi, J. A. Weiss, J. P. Sheridan, “Light scattering from smectic liquid-crystal waveguides,” J. Appl. Phys. 47, 1820–1826 (1976).
[CrossRef]

Whinnery, J. R.

J. R. Whinnery, C. Hu, Y. S. Kwon, “Liquid-crystal waveguides for integrated optics,” IEEE J. Quantum Electron. QE-13, 363–429 (1977).

Wiegmann, W.

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Band-edge electroabsorption in quantum well structures: the quantum confined Stark effect,” Phys. Rev. Lett. 53, 2173–2176 (1984).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970) Chap. 14, p. 665.

Wood, T. H.

T. H. Wood, “Multiple quantum well (MQW) waveguide modulators,” J. Lightwave Technol. 6, 743–757 (1988).
[CrossRef]

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus, “Band-edge electroabsorption in quantum well structures: the quantum confined Stark effect,” Phys. Rev. Lett. 53, 2173–2176 (1984).
[CrossRef]

Yamaguchi, T.

M. Kawaida, T. Yamaguchi, T. Akahane, “Measurement of refractive indices of ferroelectric SmC* liquid crystal by the Fabry–Perot interference method,” Jpn. J. Appl. Phys. 28, L1602–L1605 (1989).
[CrossRef]

Yamamoto, S.

Y. Okamura, K. Kitatani, S. Yamamoto, “Low-voltage driving in nematic liquid crystal overlayered waveguide,” J. Lightwave Technol. 4, 360–363 (1986).
[CrossRef]

Y. Okamura, K. Kitatani, S. Yamamoto, “Electrooptic leaky anisotropic waveguides using nematic liquid crystal overlayers,” J. Lightwave Technol. 2, 292–295 (1984).
[CrossRef]

Yeh, P.

Yoshino, K.

Y. Sadohara, M. Ozaki, K. Yoshino, “Waveguide modulator using ferroelectric liquid crystal,” Tech. Rep. Osaka Univ. 42, 137–143 (1992).

M. Ozaki, Y. Sadohara, T. Hatai, K. Yoshino, “Fast optical switching in polymer waveguide using ferroelectric liquid crystal,” Jpn. J. Appl. Phys. 29, L843–L845 (1990).
[CrossRef]

Appl. Phys. Lett. (1)

A. Jákli, A. Saupe, “Method to obtain uniform bookshelf textures in smectic C* liquid crystals,” Appl. Phys. Lett. 60, 2622–2624 (1992).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

N. A. Clark, S. T. Lagerwall, “Submicrosecond bistable electro-optic switching in liquid crystals,” Appl. Phys. Lett. 36, 899–901 (1980).
[CrossRef]

Appl. Phys. Lett. (2)

N. A. Clark, M. A. Handschy, “Surface-stabilized ferroelectric liquid-crystal electro-optic waveguide switch,” Appl. Phys. Lett. 57, 1852–1854 (1990).
[CrossRef]

D. J. Channin, “Optical waveguide modulation using nematic liquid crystal,” Appl. Phys. Lett. 22, 365–366 (1973).
[CrossRef]

Bell Syst. Tech. J. (2)

J. S. Cook, W. L. Mammel, R. J. Grow, “Effect of misalignments on coupling efficiency of single-mode optical fiber butt joints,” Bell Syst. Tech. J. 52, 1439–1448 (1973).

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54, 985–995 (1975).

Electron. Lett. (1)

P. J. B. Clarricoats, A. B. Sharpe, “Modal matching applied to a discontinuity in a planar surface waveguide,” Electron. Lett. 8, 28–29 (1972).
[CrossRef]

Electron. Lett. (1)

G. H. Brooke, M. M. Z. Kharadly, “Step discontinuities on dielectric waveguides,” Electron. Lett. 12, 473–475 (1976).
[CrossRef]

Ferroelectrics (1)

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

IEEE Trans. Microwave Theory Tech. (2)

M. Geshiro, Y. Kaihara, S. Sawa, “Analysis of wave propagation in anisotropic film waveguides with bent optical axes,” IEEE Trans. Microwave Theory Tech. 32, 339–347 (1984).
[CrossRef]

S. Sawa, M. Geshiro, M. Hotta, H. Kanetake, “Coupling efficiency of butt-joined isotropic and anisotropic single-mode slab waveguides,” IEEE Trans. Microwave Theory Tech. 40, 338–345 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Kobayashi, H. Terui, M. Kawachi, J. Noda, “2 × 2 optical waveguide matrix switch using nematic liquid crystal,” IEEE J. Quantum Electron. QE-18, 1603–1609 (1982).
[CrossRef]

IEEE J. Quantum Electron. (2)

K. M. Johnson, D. J. McKnight, I. Underwood, “Smart spatial light modulators using liquid crystals on silicon,” IEEE J. Quantum Electron. 29, 699–714 (1993).
[CrossRef]

E. V. Tomme, P. P. V. Daele, R. G. Baets, P. E. Lagasse, “Integrated optic devices based on nonlinear optical polymers,” IEEE J. Quantum Electron. 27, 778–786 (1991).
[CrossRef]

IEEE J. Quantum Electron. (2)

J. R. Whinnery, C. Hu, Y. S. Kwon, “Liquid-crystal waveguides for integrated optics,” IEEE J. Quantum Electron. QE-13, 363–429 (1977).

D. Marcuse, I. P. Kaminow, “Modes of a symmetric slab optical waveguide in birefringent media, part II: slab with coplanar optical axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. 26, 738–746 (1978).
[CrossRef]

S. Sawa, M. Geshiro, F. Takeda, “Low-loss optical branching waveguides consisting of anisotropic materials,” IEEE Trans. Microwave Theory Tech. 39, 1140–1147 (1991).
[CrossRef]

G. H. Brooke, M. M. Z. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
[CrossRef]

J. Lightwave Technol. (1)

Y. Okamura, K. Kitatani, S. Yamamoto, “Electrooptic leaky anisotropic waveguides using nematic liquid crystal overlayers,” J. Lightwave Technol. 2, 292–295 (1984).
[CrossRef]

J. Appl. Phys. (2)

T. G. Giallorenzi, J. P. Sheridan, “Light scattering from nematic liquid crystal waveguides,” J. Appl. Phys. 46, 1271–1282 (1975).
[CrossRef]

T. G. Giallorenzi, J. A. Weiss, J. P. Sheridan, “Light scattering from smectic liquid-crystal waveguides,” J. Appl. Phys. 47, 1820–1826 (1976).
[CrossRef]

J. Lightwave Technol. (3)

Y. Okamura, K. Kitatani, S. Yamamoto, “Low-voltage driving in nematic liquid crystal overlayered waveguide,” J. Lightwave Technol. 4, 360–363 (1986).
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E. Anemogiannis, E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
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Figures (11)

Fig. 1
Fig. 1

Operation of the deflection modulator. In state 1, both regions have the same optic axis orientation. The direction of power flow must be normal to the allowed wave-vector surface (the ellipse) and so the power is deflected down in both regions as shown, resulting in a net displacement of the beam at the output. In state 2, the optic axis in region B is in a complementary orientation. The beam is deflected in the opposite direction in region B and no net displacement is observed at the output.

Fig. 2
Fig. 2

Operation of the cutoff modulator. In the on state, both regions A and B support at least one guided mode. Because the optic axis is in the same orientation for both regions, the interface is transparent. In the off state, region B has its optic axis rotated so all modes in the waveguide are cut off. Light no longer propagates across the interface and is instead radiated away from the structure.

Fig. 3
Fig. 3

Operation of the input coupler. A simple scheme for coupling light into an FLC waveguide (W.G.) initially launches light into an isotropic polymer waveguide, and then power is coupled from the isotropic waveguide into the FLC waveguide. The process is reversed to couple light out.

Fig. 4
Fig. 4

Cross section of (a) a practical FLC waveguide and (b) a simplified model. The structure is symmetrical about the z axis. For the practical FLC waveguide to be simplified, the indices of the cover, ITO contact layer, and buffer layer are lumped together with a single index n c , and the FLC layer is modeled by a uniaxial material with the optic axis in the yz plane at angle u with respect to z, and ordinary index n O and extraordinary index n E ; t1 is the half-width of the waveguide and t B is the location of the PEC boundary.

Fig. 5
Fig. 5

Discontinuity in the waveguide occurs when two waveguides are butted together at z = 0. An incident mode in waveguide A interacts with the interface and forms a set of reflected modes in waveguide A and transmitted modes in waveguide B.

Fig. 6
Fig. 6

Waveguide effective index of hybrid modes in a FLC waveguide. Because n c > n O , as θ becomes small, all modes in the waveguide are cut off. At θ = 90° the modes decouple into pure TE as marked. The top trace is the TE0 hybrid mode and the lower trace is the TE2 hybrid mode.

Fig. 7
Fig. 7

Deflection angle and conversion efficiency of the TE0 hybrid mode in a FLC deflection modulator. Data were computed by the use of both a simple bulk plane-wave analysis and the mode-matching techniques.

Fig. 8
Fig. 8

Coupling efficiency of the TE0 mode in an isotropic waveguide to the TE0 hybrid mode in a FLC waveguide versus the isotropic waveguide film index (n i ) for the deflection (θ = 67.5°) and cutoff (θ = 45°) modulators. Results for a simple bulk, plane-wave model are also shown.

Fig. 9
Fig. 9

Power in the direction of propagation as a function of position in the cutoff modulator. The dense grid represents the power in the film. Power in the output region is radiating away from the film in the output region. The inset shows a small peak in the power in the output region, indicating a focusing of the beam in the output region.

Fig. 10
Fig. 10

Convergence of the boundary error versus the number of modes for various boundary locations for a deflection modulator. The boundary error generally reduces as the number of modes increases. The rate of convergence is the same for each boundary location.

Fig. 11
Fig. 11

Convergence of the transmission coefficient for the lowest order mode of a deflection modulator as a function of boundary position, using a different number of modes, N. The sudden breaks in the transmission coefficient occur at the boundary positions where N modes specify all of the modes corresponding to the guided and radiation modes.

Tables (1)

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Table 1 Rate of Attenuation for the Cutoff Modulator

Equations (51)

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˜ = [ x x 0 0 0 y y y z 0 z y z z ] .
× E = - i 2 π η 0 H , × η 0 H = i 2 π ˜ E ,
E x ( x , z ) = - E x ( x , - z ) , H x ( x , z ) = - H x ( x , - z ) , E y ( x , z ) = E y ( x , - z ) , H y ( x , z ) = H y ( x , - z ) , E z ( x , z ) = E z ( x , - z ) , H z ( x , z ) = H z ( x , - z ) .
E ( x , z ) = E * ( x , z ) ,             H ( x , z ) = - H * ( x , z ) ,
x x = n O 2 , y y = n E 2 sin 2 θ + n O 2 cos 2 θ , z z = n O 2 sin 2 θ + n E 2 cos 2 θ ,
y z = z y = ( n E 2 - n O 2 ) sin θ cos θ .
E x ν ( x ) = N ν / n O 2 η 0 H y ν ( x ) , E y ν ( x ) = A O ν κ o ν cos θ Ψ s c ( κ o ν x ) + A E ν n O 2 sin θ Ψ s c ( κ e ν x ) , E z ν ( x ) = - A O ν κ o ν sin θ Ψ s c ( κ o ν x ) + A E ν ( κ o ν ) 2 cos θ Ψ s c ( κ e ν x ) , η 0 H x ν ( x ) = - N ν E y ν ( x ) , η 0 H y ν ( x ) = - i A O ν n O 2 sin θ Ψ - c s ( κ o ν x ) + i A E ν n O 2 κ e ν cos θ Ψ - c s ( κ e ν x ) , η 0 H z ν ( x ) = - i A O ν ( κ o ν ) 2 cos θ Ψ - c s ( κ o ν x ) - i A E ν n O 2 κ e ν sin θ Ψ - c s ( κ e ν x ) ,
Ψ s c ( ξ ) = { cos ( ξ ) , even modes sin ( ξ ) , odd modes , Ψ - c s ( ξ ) = { sin ( ξ ) , even modes - cos ( ξ ) , odd modes ,
( κ o ν / 2 π ) 2 = ( κ o ν ) 2 = n O 2 - N ν 2 , ( κ e ν / 2 π ) 2 = ( κ e ν ) 2 = n E 2 - N ν 2 ( n E 2 cos 2 θ + n O 2 sin 2 θ ) / n O 2 .
E y ν ( x ) = - A TE ν sin [ κ c ν ( t B - x ) ] , E z ν ( x ) = A TM ν κ c ν sin [ κ c ν ( t B - x ) ] , η 0 H y ν ( x ) = i A TM ν n c 2 cos [ κ c ν ( t B - x ) ] , η 0 H z ν ( x ) = i A TE ν κ c ν cos [ κ c ν ( t B - x ) ] ,
C 1 C 4 κ o ν cos 2 θ + C 2 C 3 n O 2 sin 2 θ = 0 ,
C 1 = n c 2 ( κ o ν ) 2 Ψ s c ( κ e ν t 1 ) cos [ ( t B - t 1 ) κ c ν ] - n O 2 κ c ν κ e ν Ψ - c s ( κ e ν t 1 ) sin [ ( t B - t 1 ) κ e ν ] , C 2 = n c 2 κ o ν Ψ s c ( κ o ν t 1 ) cos [ ( t B - t 1 ) κ c ν ] - n O 2 κ c ν Ψ - c s ( κ o ν t 1 ) sin [ ( t B - t 1 ) κ c ν ] , C 3 = κ c ν Ψ s c ( κ e ν t 1 ) cos [ ( t B - t 1 ) κ c ν ] - κ e ν Ψ - c s ( κ e ν t 1 ) sin [ ( t B - t 1 ) κ c ν ] , C 4 = κ c ν Ψ s c ( κ o ν t 1 ) cos [ ( t B - t 1 ) κ c ν ] - κ o ν Ψ - c s ( κ o ν t 1 ) sin [ ( t B - t 1 ) κ c ν ] .
N max = n E n O / ( n E 2 cos 2 θ + n O 2 sin 2 θ ) 1 / 2 .
N trans evan max = ( n E 2 + n O 2 - n E 2 n O 2 / n c 2 ) 1 / 2 .
N max = max ( N max , N trans evan max ) .
A E ν n O 2 sin θ C 3 = - A O ν κ o ν cos θ C 4 ,
A TE ν = - [ A O κ o ν cos θ Ψ s c ( κ o ν t 1 ) + A E n O 2 sin θ Ψ s c ( κ e ν t 1 ) ] / sin [ ( t B - t 1 ) κ c ν ] ,
A TM ν = ( n O 2 / n c 2 ) [ - A O sin θ Ψ - c s ( κ o ν t 1 ) + A E κ e ν cos θ Ψ - c s ( κ e ν t 1 ) ] / cos [ ( t B - t 1 ) κ c ν ] .
- d x d y ( E ν × H μ * + E μ * × H ν ) · z ^ = 0 ,             β μ β ν ,
- d x d y ( E x ν H y μ * - E y μ * H x ν ) = 0 ,             β μ β ν .
E x ν , H y μ - E y ν , H x μ * = δ ν μ P ν ,
E x i a - n = 1 a n E x n a = m = 1 b m E x m b ,
E y i a + n = 1 a n E y n a = m = 1 b m E y m b ,
H x i a - n = 1 a n H x n a = m = 1 b m H x m b ,
H y i a + n = 1 a n H y n a = m = 1 b m H y m b ,
E x i a , H y ν a - n = 1 N a n E x n a , H y ν a = m = 1 M b m E x m b , H y ν a ,
E y ν a , H x i a * - n = 1 N a n E y ν a , H x n a * = m = 1 M b m E y ν a , H x m b * ,
δ i n P i a - a n P n a = m = 1 M b m [ E x m b , H y ν a - E y ν a , H x m b * ] .
E x ν b , H y i a * + n = 1 N a n E x ν b , H y n a * = m = 1 M b m E x ν b , H y m b * ,
E y i a , H x ν b + n = 1 N a n E y n a , H x ν b = m = 1 M b m E y m b , H x ν b .
n = 1 N ( a n + δ i n ) [ E x m b , H y n a - E y n a , H x m b * ] * = b m ( P m b ) * .
a = u - Qb ,
b = V ( u + a ) ,
u j = δ i j , Q j k = [ E x k b , H y j a - E y j a , H x k b * ] / P j a , V j k = [ E x j b , H y k a - E y k a , H x j b * ] * / ( P j b ) * .
b = ( I + VQ ) - 1 2 Vu ,
R n i = a n 2 P n a / P i a ,
T m i = b m 2 P m b / P i a .
γ bulk = θ - arctan [ n O 2 / n E 2 tan ( θ ) ] .
γ = arctan { 0 t B Re [ S y ( x ) ] d x 0 t B Re [ S z ( x ) ] d x } ,
T = 4 n i n e / ( n i + n e ) 2 ,
BE = 0 t B ( E x a - E x b 2 + E y a - E y b 2 + H x a - H x b 2 + H y a - H y b 2 ) d x .
O j k film = 0 t 1 [ E x t b ( H y j a ) * - ( E y j a ) * H x k b ] d x .
I = 0 t 1 [ A 1 cos ( q 1 x ) + A 2 cos ( q 2 x ) ] × [ B 1 cos ( r 1 x ) + B 2 cos ( r 2 x ) ] d x ,
I = [ A 1 A 2 ] M qr [ B 1 B 2 ] ,
M qr = 0 t 1 [ cos ( q 1 x ) cos ( r 1 x ) cos ( q 1 x ) cos ( r 2 x ) cos ( q 2 x ) cos ( r 1 x ) cos ( q 2 x ) cos ( r 2 x ) ] d x .
M n m qr = t 1 2 { sinc [ ( q n + r m ) t 1 ] + sinc [ ( q n - r m ) t 1 ] } .
J = 0 t 1 [ A 1 sin ( q 1 x ) + A 2 sin ( q 2 x ) ] × [ B 1 sin ( r 1 x ) + B 2 sin ( r 2 x ) ] d x = [ A 1 A 2 ] L qr [ B 1 B 2 ] ,
L n m qr = t 1 2 { sinc [ ( q n - r m ) t 1 ] - sinc [ ( q n + r m ) t 1 ] } .
O j k film = A T { L rq * M rq * } B * - C * T { M q * r L q * r } D ,
q = [ κ o j a , κ e j a ] , r = [ κ o k b , κ e k b ] , A = i N k b [ - A O k b sin θ b , A E k b κ e k b cos θ b ] , B = i N j a [ - A O j a sin θ a , A E j a κ e j a cos θ a ] , C = [ A O j a κ o j a sin θ a , A E j a n O a 2 sin θ a ] , D = - N k b [ A O k b κ o k b sin θ b , A E k b n O b 2 sin θ b ] .
O j k cover = N k b Δ B 2 ( A TE k b A TE j a * { sinc [ ( κ c k b - κ c j a * ) Δ B ] - sinc [ ( κ c k b + κ c j a * ) Δ B ] } + n c a A TM k b A TM j a * × { sinc [ ( κ c k b - κ c j a * ) Δ B ] + sinc [ ( κ c k b + κ c j a * ) Δ B ] } )

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