Abstract

Passive linear nonreciprocal devices, such as isolators or circulators, require gyrotropic media. If there is a contribution to the dielectric permittivity tensor of odd order in the magnetization, the modes propagate differently in the forward and the backward directions. We investigate dielectric waveguides that are formed by a rib of rectangular cross section on top of a planar structure. The rib or planar structure may consist of layers, each of which may be gyrotropic. We extend the spectral-index method for calculating differences between forward- and backward-propagation constants. A new design for an efficient nonreciprocal phase shifter is proposed.

© 1994 Optical Society of America

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  1. T. L. Koch, U. Koren, “Photonic integrated circuits,” AT&T Tech. J. 71, 63–74 (1992).
  2. P. Paroli, “Magnetooptic devices based on garnet films,” Thin Solid Films 114, 187–219 (1984).
    [CrossRef]
  3. H. J. Schmitt, “Magneto-optic devices,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1274, 208–219 (1990).
  4. H. Dötsch, P. Hertel, B. Lührmann, S. Sure, H. P. Winkler, M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. 28, 2979–2984 (1992).
    [CrossRef]
  5. S. Jackel, S. Atzmon, R. Lallouz, S. Sternklar, P. Shalev, “Nonlinear optical isolators based on high-reflectivity Brillouin mirrors and their applications to advanced lasers,” Opt. Eng. 31, 328–334 (1992).
    [CrossRef]
  6. L. Solymar, T. Wilson, “Controllable nonreciprocal phase shifter,” Electron. Lett. 21, 234–235 (1985).
    [CrossRef]
  7. T. Mizumoto, K. Oochi, T. Harada, Y. Naito, “Measurement of optical nonreciprocal phase shift in a Bi-substituted Gd3Fe5O12 film and application to waveguide-type optical circulator,” J. Lightwave Technol. 4, 347–352 (1986).
    [CrossRef]
  8. J. P. Castera, G. Hepner, “Isolator in integrated optics using the Faraday and Cotton–Mouton effects,” IEEE Trans. Magn. MAG-13, 1583–1585 (1977).
    [CrossRef]
  9. H. Dammann, E. Pross, G. Rabe, W. Tolksdorf, “45° waveguide isolator with phase mismatch,” Appl. Phys. Lett. 56, 1302–1304(1990).
    [CrossRef]
  10. H. Dammann, E. Pross, G. Rabe, W. Tolksdorf, M. Zinke, “Phase matching in symmetrical single-mode magneto-optic waveguides by application of stress,” Appl. Phys. Lett. 49, 1755–1757 (1986).
    [CrossRef]
  11. R. Wolfe, J. F. Dillon, R. A. Lieberman, V. J. Fratello, “Broadband magneto-optic waveguide isolator,” Appl. Phys. Lett. 57, 960–962 (1990).
    [CrossRef]
  12. S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, M. Bridenbaugh, P. Grabbe, “Semileaky thin-film optical isolator,” J. Appl. Phys. 52, 3190–3199 (1981).
    [CrossRef]
  13. F. Auracher, H. H. Witte, “A new design for an integrated optical isolator,” Opt. Commun. 13, 435–438 (1975).
    [CrossRef]
  14. Y. Okamura, T. Negami, S. Yamamoto, “Integrated optical isolator and circulator using nonreciprocal phase shifters: a proposal,” Appl. Opt. 23, 1886–1889 (1984).
    [CrossRef] [PubMed]
  15. C. L. Chen, A. Kumarswami, “Nonreciprocal TM-mode thin-film phase shifters,” Appl. Opt. 25, 3664–3670 (1986).
    [CrossRef] [PubMed]
  16. D. Marcuse, “Influence of position of magnetooptic layer on differential phase shift of slab waveguide,” IEEE J. Quantum Electron. QE-23, 1268–1272 (1987).
    [CrossRef]
  17. P. N. Robson, P. C. Kendall, Rib Waveguide Theory by the Spectral Index Method (Wiley, New York, 1990), pp. 1–193.
  18. M. S. Stern, P. C. Kendall, P. W. A. McIlroy, “Analysis of the spectral index method for vector modes of rib waveguides,” Proc. Inst. Electr. Eng. Part J 137, 21–26 (1990).
  19. C. Vassalo, Y. H. Wang, “A new semirigorous analysis of rib waveguides,” J. Lightwave Technol. 8, 56–65 (1990).
    [CrossRef]
  20. A. Erdmann, M. Shamonin, P. Hertel, H. Dötsh, “Finite difference analysis of gyrotropic waveguides,” Opt. Commun. 102, 25–30 (1993).
    [CrossRef]
  21. Y. Okamura, T. Negami, S. Yamamoto, “A design for a nonreciprocal phase shifter,” Opt. Quantum Electron. 17, 195–199 (1985).
    [CrossRef]
  22. C. Vasallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991), Chap. 1, pp. 7–8.
  23. J. B. Davies, F. A. Fernandez, Y. Fang, “Finite-difference solution of inhomogeneous waveguide modes using a fast direct solver routine,” IEEE Trans. Magn. 27, 4028–4031 (1991).
    [CrossRef]
  24. M. Koshiba, K. Hayata, M. Suzuki, “On accuracy of approximate scalar finite-element analysis of dielectric optical waveguides,” Trans. Inst. Electron. Commun. Eng. Jpn. E66, 157–158 (1983).
  25. S. Yamamoto, T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys. 45, 882–888 (1974).
    [CrossRef]
  26. R. Gerhardt, S. Sure, H. Dötsch, T. Linkewitz, W. Tolksdorf, “Optical properties of bismuth and gallium substituted thulium iron garnet films,” Opt. Commun. 102, 31–35 (1993).
    [CrossRef]
  27. M. Shamonin, A. Erdmann, P. Hertel, H. Dötsch, “A note on the analysis of nonrecirpocal phase shifters by the spectral index method,” Microwave Opt. Technol. Lett. 6, 790–792 (1993).
    [CrossRef]

1993

A. Erdmann, M. Shamonin, P. Hertel, H. Dötsh, “Finite difference analysis of gyrotropic waveguides,” Opt. Commun. 102, 25–30 (1993).
[CrossRef]

R. Gerhardt, S. Sure, H. Dötsch, T. Linkewitz, W. Tolksdorf, “Optical properties of bismuth and gallium substituted thulium iron garnet films,” Opt. Commun. 102, 31–35 (1993).
[CrossRef]

M. Shamonin, A. Erdmann, P. Hertel, H. Dötsch, “A note on the analysis of nonrecirpocal phase shifters by the spectral index method,” Microwave Opt. Technol. Lett. 6, 790–792 (1993).
[CrossRef]

1992

T. L. Koch, U. Koren, “Photonic integrated circuits,” AT&T Tech. J. 71, 63–74 (1992).

H. Dötsch, P. Hertel, B. Lührmann, S. Sure, H. P. Winkler, M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. 28, 2979–2984 (1992).
[CrossRef]

S. Jackel, S. Atzmon, R. Lallouz, S. Sternklar, P. Shalev, “Nonlinear optical isolators based on high-reflectivity Brillouin mirrors and their applications to advanced lasers,” Opt. Eng. 31, 328–334 (1992).
[CrossRef]

1991

J. B. Davies, F. A. Fernandez, Y. Fang, “Finite-difference solution of inhomogeneous waveguide modes using a fast direct solver routine,” IEEE Trans. Magn. 27, 4028–4031 (1991).
[CrossRef]

1990

H. Dammann, E. Pross, G. Rabe, W. Tolksdorf, “45° waveguide isolator with phase mismatch,” Appl. Phys. Lett. 56, 1302–1304(1990).
[CrossRef]

M. S. Stern, P. C. Kendall, P. W. A. McIlroy, “Analysis of the spectral index method for vector modes of rib waveguides,” Proc. Inst. Electr. Eng. Part J 137, 21–26 (1990).

C. Vassalo, Y. H. Wang, “A new semirigorous analysis of rib waveguides,” J. Lightwave Technol. 8, 56–65 (1990).
[CrossRef]

R. Wolfe, J. F. Dillon, R. A. Lieberman, V. J. Fratello, “Broadband magneto-optic waveguide isolator,” Appl. Phys. Lett. 57, 960–962 (1990).
[CrossRef]

1987

D. Marcuse, “Influence of position of magnetooptic layer on differential phase shift of slab waveguide,” IEEE J. Quantum Electron. QE-23, 1268–1272 (1987).
[CrossRef]

1986

C. L. Chen, A. Kumarswami, “Nonreciprocal TM-mode thin-film phase shifters,” Appl. Opt. 25, 3664–3670 (1986).
[CrossRef] [PubMed]

H. Dammann, E. Pross, G. Rabe, W. Tolksdorf, M. Zinke, “Phase matching in symmetrical single-mode magneto-optic waveguides by application of stress,” Appl. Phys. Lett. 49, 1755–1757 (1986).
[CrossRef]

T. Mizumoto, K. Oochi, T. Harada, Y. Naito, “Measurement of optical nonreciprocal phase shift in a Bi-substituted Gd3Fe5O12 film and application to waveguide-type optical circulator,” J. Lightwave Technol. 4, 347–352 (1986).
[CrossRef]

1985

L. Solymar, T. Wilson, “Controllable nonreciprocal phase shifter,” Electron. Lett. 21, 234–235 (1985).
[CrossRef]

Y. Okamura, T. Negami, S. Yamamoto, “A design for a nonreciprocal phase shifter,” Opt. Quantum Electron. 17, 195–199 (1985).
[CrossRef]

1984

1983

M. Koshiba, K. Hayata, M. Suzuki, “On accuracy of approximate scalar finite-element analysis of dielectric optical waveguides,” Trans. Inst. Electron. Commun. Eng. Jpn. E66, 157–158 (1983).

1981

S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, M. Bridenbaugh, P. Grabbe, “Semileaky thin-film optical isolator,” J. Appl. Phys. 52, 3190–3199 (1981).
[CrossRef]

1977

J. P. Castera, G. Hepner, “Isolator in integrated optics using the Faraday and Cotton–Mouton effects,” IEEE Trans. Magn. MAG-13, 1583–1585 (1977).
[CrossRef]

1975

F. Auracher, H. H. Witte, “A new design for an integrated optical isolator,” Opt. Commun. 13, 435–438 (1975).
[CrossRef]

1974

S. Yamamoto, T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys. 45, 882–888 (1974).
[CrossRef]

Atzmon, S.

S. Jackel, S. Atzmon, R. Lallouz, S. Sternklar, P. Shalev, “Nonlinear optical isolators based on high-reflectivity Brillouin mirrors and their applications to advanced lasers,” Opt. Eng. 31, 328–334 (1992).
[CrossRef]

Auracher, F.

F. Auracher, H. H. Witte, “A new design for an integrated optical isolator,” Opt. Commun. 13, 435–438 (1975).
[CrossRef]

Biolsi, W. A.

S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, M. Bridenbaugh, P. Grabbe, “Semileaky thin-film optical isolator,” J. Appl. Phys. 52, 3190–3199 (1981).
[CrossRef]

Blank, S. L.

S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, M. Bridenbaugh, P. Grabbe, “Semileaky thin-film optical isolator,” J. Appl. Phys. 52, 3190–3199 (1981).
[CrossRef]

Bridenbaugh, M.

S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, M. Bridenbaugh, P. Grabbe, “Semileaky thin-film optical isolator,” J. Appl. Phys. 52, 3190–3199 (1981).
[CrossRef]

Castera, J. P.

J. P. Castera, G. Hepner, “Isolator in integrated optics using the Faraday and Cotton–Mouton effects,” IEEE Trans. Magn. MAG-13, 1583–1585 (1977).
[CrossRef]

Chen, C. L.

Dammann, H.

H. Dammann, E. Pross, G. Rabe, W. Tolksdorf, “45° waveguide isolator with phase mismatch,” Appl. Phys. Lett. 56, 1302–1304(1990).
[CrossRef]

H. Dammann, E. Pross, G. Rabe, W. Tolksdorf, M. Zinke, “Phase matching in symmetrical single-mode magneto-optic waveguides by application of stress,” Appl. Phys. Lett. 49, 1755–1757 (1986).
[CrossRef]

Davies, J. B.

J. B. Davies, F. A. Fernandez, Y. Fang, “Finite-difference solution of inhomogeneous waveguide modes using a fast direct solver routine,” IEEE Trans. Magn. 27, 4028–4031 (1991).
[CrossRef]

Dillon, J. F.

R. Wolfe, J. F. Dillon, R. A. Lieberman, V. J. Fratello, “Broadband magneto-optic waveguide isolator,” Appl. Phys. Lett. 57, 960–962 (1990).
[CrossRef]

Dötsch, H.

R. Gerhardt, S. Sure, H. Dötsch, T. Linkewitz, W. Tolksdorf, “Optical properties of bismuth and gallium substituted thulium iron garnet films,” Opt. Commun. 102, 31–35 (1993).
[CrossRef]

M. Shamonin, A. Erdmann, P. Hertel, H. Dötsch, “A note on the analysis of nonrecirpocal phase shifters by the spectral index method,” Microwave Opt. Technol. Lett. 6, 790–792 (1993).
[CrossRef]

H. Dötsch, P. Hertel, B. Lührmann, S. Sure, H. P. Winkler, M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. 28, 2979–2984 (1992).
[CrossRef]

Dötsh, H.

A. Erdmann, M. Shamonin, P. Hertel, H. Dötsh, “Finite difference analysis of gyrotropic waveguides,” Opt. Commun. 102, 25–30 (1993).
[CrossRef]

Erdmann, A.

A. Erdmann, M. Shamonin, P. Hertel, H. Dötsh, “Finite difference analysis of gyrotropic waveguides,” Opt. Commun. 102, 25–30 (1993).
[CrossRef]

M. Shamonin, A. Erdmann, P. Hertel, H. Dötsch, “A note on the analysis of nonrecirpocal phase shifters by the spectral index method,” Microwave Opt. Technol. Lett. 6, 790–792 (1993).
[CrossRef]

Fang, Y.

J. B. Davies, F. A. Fernandez, Y. Fang, “Finite-difference solution of inhomogeneous waveguide modes using a fast direct solver routine,” IEEE Trans. Magn. 27, 4028–4031 (1991).
[CrossRef]

Fernandez, F. A.

J. B. Davies, F. A. Fernandez, Y. Fang, “Finite-difference solution of inhomogeneous waveguide modes using a fast direct solver routine,” IEEE Trans. Magn. 27, 4028–4031 (1991).
[CrossRef]

Fratello, V. J.

R. Wolfe, J. F. Dillon, R. A. Lieberman, V. J. Fratello, “Broadband magneto-optic waveguide isolator,” Appl. Phys. Lett. 57, 960–962 (1990).
[CrossRef]

Gerhardt, R.

R. Gerhardt, S. Sure, H. Dötsch, T. Linkewitz, W. Tolksdorf, “Optical properties of bismuth and gallium substituted thulium iron garnet films,” Opt. Commun. 102, 31–35 (1993).
[CrossRef]

Grabbe, P.

S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, M. Bridenbaugh, P. Grabbe, “Semileaky thin-film optical isolator,” J. Appl. Phys. 52, 3190–3199 (1981).
[CrossRef]

Harada, T.

T. Mizumoto, K. Oochi, T. Harada, Y. Naito, “Measurement of optical nonreciprocal phase shift in a Bi-substituted Gd3Fe5O12 film and application to waveguide-type optical circulator,” J. Lightwave Technol. 4, 347–352 (1986).
[CrossRef]

Hayata, K.

M. Koshiba, K. Hayata, M. Suzuki, “On accuracy of approximate scalar finite-element analysis of dielectric optical waveguides,” Trans. Inst. Electron. Commun. Eng. Jpn. E66, 157–158 (1983).

Hepner, G.

J. P. Castera, G. Hepner, “Isolator in integrated optics using the Faraday and Cotton–Mouton effects,” IEEE Trans. Magn. MAG-13, 1583–1585 (1977).
[CrossRef]

Hertel, P.

A. Erdmann, M. Shamonin, P. Hertel, H. Dötsh, “Finite difference analysis of gyrotropic waveguides,” Opt. Commun. 102, 25–30 (1993).
[CrossRef]

M. Shamonin, A. Erdmann, P. Hertel, H. Dötsch, “A note on the analysis of nonrecirpocal phase shifters by the spectral index method,” Microwave Opt. Technol. Lett. 6, 790–792 (1993).
[CrossRef]

H. Dötsch, P. Hertel, B. Lührmann, S. Sure, H. P. Winkler, M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. 28, 2979–2984 (1992).
[CrossRef]

Jackel, S.

S. Jackel, S. Atzmon, R. Lallouz, S. Sternklar, P. Shalev, “Nonlinear optical isolators based on high-reflectivity Brillouin mirrors and their applications to advanced lasers,” Opt. Eng. 31, 328–334 (1992).
[CrossRef]

Kendall, P. C.

M. S. Stern, P. C. Kendall, P. W. A. McIlroy, “Analysis of the spectral index method for vector modes of rib waveguides,” Proc. Inst. Electr. Eng. Part J 137, 21–26 (1990).

P. N. Robson, P. C. Kendall, Rib Waveguide Theory by the Spectral Index Method (Wiley, New York, 1990), pp. 1–193.

Kirsch, S. T.

S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, M. Bridenbaugh, P. Grabbe, “Semileaky thin-film optical isolator,” J. Appl. Phys. 52, 3190–3199 (1981).
[CrossRef]

Koch, T. L.

T. L. Koch, U. Koren, “Photonic integrated circuits,” AT&T Tech. J. 71, 63–74 (1992).

Koren, U.

T. L. Koch, U. Koren, “Photonic integrated circuits,” AT&T Tech. J. 71, 63–74 (1992).

Koshiba, M.

M. Koshiba, K. Hayata, M. Suzuki, “On accuracy of approximate scalar finite-element analysis of dielectric optical waveguides,” Trans. Inst. Electron. Commun. Eng. Jpn. E66, 157–158 (1983).

Kumarswami, A.

Lallouz, R.

S. Jackel, S. Atzmon, R. Lallouz, S. Sternklar, P. Shalev, “Nonlinear optical isolators based on high-reflectivity Brillouin mirrors and their applications to advanced lasers,” Opt. Eng. 31, 328–334 (1992).
[CrossRef]

Lieberman, R. A.

R. Wolfe, J. F. Dillon, R. A. Lieberman, V. J. Fratello, “Broadband magneto-optic waveguide isolator,” Appl. Phys. Lett. 57, 960–962 (1990).
[CrossRef]

Linkewitz, T.

R. Gerhardt, S. Sure, H. Dötsch, T. Linkewitz, W. Tolksdorf, “Optical properties of bismuth and gallium substituted thulium iron garnet films,” Opt. Commun. 102, 31–35 (1993).
[CrossRef]

Lührmann, B.

H. Dötsch, P. Hertel, B. Lührmann, S. Sure, H. P. Winkler, M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. 28, 2979–2984 (1992).
[CrossRef]

Makimoto, T.

S. Yamamoto, T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys. 45, 882–888 (1974).
[CrossRef]

Marcuse, D.

D. Marcuse, “Influence of position of magnetooptic layer on differential phase shift of slab waveguide,” IEEE J. Quantum Electron. QE-23, 1268–1272 (1987).
[CrossRef]

Martin, R. J.

S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, M. Bridenbaugh, P. Grabbe, “Semileaky thin-film optical isolator,” J. Appl. Phys. 52, 3190–3199 (1981).
[CrossRef]

McIlroy, P. W. A.

M. S. Stern, P. C. Kendall, P. W. A. McIlroy, “Analysis of the spectral index method for vector modes of rib waveguides,” Proc. Inst. Electr. Eng. Part J 137, 21–26 (1990).

Mizumoto, T.

T. Mizumoto, K. Oochi, T. Harada, Y. Naito, “Measurement of optical nonreciprocal phase shift in a Bi-substituted Gd3Fe5O12 film and application to waveguide-type optical circulator,” J. Lightwave Technol. 4, 347–352 (1986).
[CrossRef]

Naito, Y.

T. Mizumoto, K. Oochi, T. Harada, Y. Naito, “Measurement of optical nonreciprocal phase shift in a Bi-substituted Gd3Fe5O12 film and application to waveguide-type optical circulator,” J. Lightwave Technol. 4, 347–352 (1986).
[CrossRef]

Negami, T.

Y. Okamura, T. Negami, S. Yamamoto, “A design for a nonreciprocal phase shifter,” Opt. Quantum Electron. 17, 195–199 (1985).
[CrossRef]

Y. Okamura, T. Negami, S. Yamamoto, “Integrated optical isolator and circulator using nonreciprocal phase shifters: a proposal,” Appl. Opt. 23, 1886–1889 (1984).
[CrossRef] [PubMed]

Okamura, Y.

Y. Okamura, T. Negami, S. Yamamoto, “A design for a nonreciprocal phase shifter,” Opt. Quantum Electron. 17, 195–199 (1985).
[CrossRef]

Y. Okamura, T. Negami, S. Yamamoto, “Integrated optical isolator and circulator using nonreciprocal phase shifters: a proposal,” Appl. Opt. 23, 1886–1889 (1984).
[CrossRef] [PubMed]

Oochi, K.

T. Mizumoto, K. Oochi, T. Harada, Y. Naito, “Measurement of optical nonreciprocal phase shift in a Bi-substituted Gd3Fe5O12 film and application to waveguide-type optical circulator,” J. Lightwave Technol. 4, 347–352 (1986).
[CrossRef]

Paroli, P.

P. Paroli, “Magnetooptic devices based on garnet films,” Thin Solid Films 114, 187–219 (1984).
[CrossRef]

Pross, E.

H. Dammann, E. Pross, G. Rabe, W. Tolksdorf, “45° waveguide isolator with phase mismatch,” Appl. Phys. Lett. 56, 1302–1304(1990).
[CrossRef]

H. Dammann, E. Pross, G. Rabe, W. Tolksdorf, M. Zinke, “Phase matching in symmetrical single-mode magneto-optic waveguides by application of stress,” Appl. Phys. Lett. 49, 1755–1757 (1986).
[CrossRef]

Rabe, G.

H. Dammann, E. Pross, G. Rabe, W. Tolksdorf, “45° waveguide isolator with phase mismatch,” Appl. Phys. Lett. 56, 1302–1304(1990).
[CrossRef]

H. Dammann, E. Pross, G. Rabe, W. Tolksdorf, M. Zinke, “Phase matching in symmetrical single-mode magneto-optic waveguides by application of stress,” Appl. Phys. Lett. 49, 1755–1757 (1986).
[CrossRef]

Robson, P. N.

P. N. Robson, P. C. Kendall, Rib Waveguide Theory by the Spectral Index Method (Wiley, New York, 1990), pp. 1–193.

Schmitt, H. J.

H. J. Schmitt, “Magneto-optic devices,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1274, 208–219 (1990).

Shalev, P.

S. Jackel, S. Atzmon, R. Lallouz, S. Sternklar, P. Shalev, “Nonlinear optical isolators based on high-reflectivity Brillouin mirrors and their applications to advanced lasers,” Opt. Eng. 31, 328–334 (1992).
[CrossRef]

Shamonin, M.

A. Erdmann, M. Shamonin, P. Hertel, H. Dötsh, “Finite difference analysis of gyrotropic waveguides,” Opt. Commun. 102, 25–30 (1993).
[CrossRef]

M. Shamonin, A. Erdmann, P. Hertel, H. Dötsch, “A note on the analysis of nonrecirpocal phase shifters by the spectral index method,” Microwave Opt. Technol. Lett. 6, 790–792 (1993).
[CrossRef]

Solymar, L.

L. Solymar, T. Wilson, “Controllable nonreciprocal phase shifter,” Electron. Lett. 21, 234–235 (1985).
[CrossRef]

Stern, M. S.

M. S. Stern, P. C. Kendall, P. W. A. McIlroy, “Analysis of the spectral index method for vector modes of rib waveguides,” Proc. Inst. Electr. Eng. Part J 137, 21–26 (1990).

Sternklar, S.

S. Jackel, S. Atzmon, R. Lallouz, S. Sternklar, P. Shalev, “Nonlinear optical isolators based on high-reflectivity Brillouin mirrors and their applications to advanced lasers,” Opt. Eng. 31, 328–334 (1992).
[CrossRef]

Sure, S.

R. Gerhardt, S. Sure, H. Dötsch, T. Linkewitz, W. Tolksdorf, “Optical properties of bismuth and gallium substituted thulium iron garnet films,” Opt. Commun. 102, 31–35 (1993).
[CrossRef]

H. Dötsch, P. Hertel, B. Lührmann, S. Sure, H. P. Winkler, M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. 28, 2979–2984 (1992).
[CrossRef]

Suzuki, M.

M. Koshiba, K. Hayata, M. Suzuki, “On accuracy of approximate scalar finite-element analysis of dielectric optical waveguides,” Trans. Inst. Electron. Commun. Eng. Jpn. E66, 157–158 (1983).

Tien, P. K.

S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, M. Bridenbaugh, P. Grabbe, “Semileaky thin-film optical isolator,” J. Appl. Phys. 52, 3190–3199 (1981).
[CrossRef]

Tolksdorf, W.

R. Gerhardt, S. Sure, H. Dötsch, T. Linkewitz, W. Tolksdorf, “Optical properties of bismuth and gallium substituted thulium iron garnet films,” Opt. Commun. 102, 31–35 (1993).
[CrossRef]

H. Dammann, E. Pross, G. Rabe, W. Tolksdorf, “45° waveguide isolator with phase mismatch,” Appl. Phys. Lett. 56, 1302–1304(1990).
[CrossRef]

H. Dammann, E. Pross, G. Rabe, W. Tolksdorf, M. Zinke, “Phase matching in symmetrical single-mode magneto-optic waveguides by application of stress,” Appl. Phys. Lett. 49, 1755–1757 (1986).
[CrossRef]

Vasallo, C.

C. Vasallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991), Chap. 1, pp. 7–8.

Vassalo, C.

C. Vassalo, Y. H. Wang, “A new semirigorous analysis of rib waveguides,” J. Lightwave Technol. 8, 56–65 (1990).
[CrossRef]

Wang, Y. H.

C. Vassalo, Y. H. Wang, “A new semirigorous analysis of rib waveguides,” J. Lightwave Technol. 8, 56–65 (1990).
[CrossRef]

Wilson, T.

L. Solymar, T. Wilson, “Controllable nonreciprocal phase shifter,” Electron. Lett. 21, 234–235 (1985).
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Figures (8)

Fig. 1
Fig. 1

Planar single magneto-optic film phase shifter. The magneto-optic film of thickness d is characterized by ∊ xz = iξ.

Fig. 2
Fig. 2

Planar composite magneto-optic film phase shifter. The magneto-optic film consists of two layers (thicknesses h and dh, respectively) with opposite magnetization, hence ξ.

Fig. 3
Fig. 3

Optimization of a planar composite film phase shifter. Nonreciprocal differential phase shift Δβ versus layer thickness d is shown for various ratios of top to total film thickness (see Fig. 2). Note the flat, hence uncritical, optimum for h/d = 0.6. The parameters are ∊ c = 1.00 (air), ∊ s = 3.80 (GGG), ∊ f = 5.30, ξ = 5 × 10−3 (YIG:Bi), and λ = 1.5 μm.

Fig. 4
Fig. 4

Finite-difference method versus spectral-index method. A magneto-optic rib (YIG) of width 2w = 2.0 μm on top of a substrate (GGG) is analyzed by the use of the finite-difference method (filled squares) and the spectral-index method (solid curve). The differential nonreciprocal phase shift Δβ is plotted versus the height d of the rib. The parameters correspond to those in Fig. 3, but λ = 1.3 μm.

Fig. 5
Fig. 5

High-refractive-index top layer. The nonreciprocal phase shifter was proposed by Okamura et al.21 Note the high permittivity, ∊ t = 7.64, of ZnTe.

Fig. 6
Fig. 6

Effective-index method versus spectral-index method. The length of the device proposed by Okamura et al.21 is calculated for different values of rib height h and width 2w. The dashed curves represent the results of the spectral-index method, and the solid curves refer to the results of the effective-index method. The two L curves for h = 0.01 μm are indistinguishable. See the text for the parameter values.

Fig. 7
Fig. 7

Composite magneto-optic rib-waveguide phase shifter. A rib consisting of two magneto-optic layers of opposite magnetization on top of an unstructured substrate is shown.

Fig. 8
Fig. 8

Optimization of a composite-rib phase shifter. The non-reciprocal differential phase shift Δβ is plotted versus the top to total layer thickness ratio for various rib widths. See Fig. 7 for a view of the waveguide cross section. The parameters are the same as in Fig. 3, and the total rib height is d 1 + d 2 = 0.68 μm. The parameters are as in Fig. 3. Note that the optimum value does not depend critically on the waveguide dimensions.

Equations (35)

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= [ x x 0 i ξ 0 y y 0 - i ξ 0 z z ]             or             η = [ η x x 0 - i ζ 0 η y y 0 i ζ 0 η z z ] ,
curl η curl H = k 0 2 H ,
[ - x η z z x - η x x y 2 + β 2 η x x - β ( x ζ ) ] H y = k 0 2 H y ,
[ - η y y x y + y η z z x + ( β y ζ ) H y = k 0 2 H x ,
Q = d x d y η H x 2 d x d y η ( H x 2 + H y 2 )
( η z z x 2 + η x x y 2 + k 0 2 - β 2 η x x ) H y = 0
( x x z z x 2 + y 2 + k 0 2 x x - β 2 ) H y = 0.
g ( y ) = cos π y / 2 W
H y ( x , y ) = d s 2 π H y ( x , u ) , exp ( i u y ) .
n j ( u ) = ( j - u 2 ) 1 / 2
ϒ = x H y ( x , y ) H y ( x , y ) | x = - 0 = f ( 0 ) f ( 0 )
Γ ( u ) = x H y ( x , u ) H y ( x , u ) | x = + 0
ϒ = 4 u 1 3 π 3 - + d u Γ ( u ) cos 2 u W ( u 2 - u 1 2 ) 2
- i E z = η x H y + β ζ H y
( - i E z H y ) x = x j + D j = [ T 11 ( j ) T 12 ( j ) T 21 ( j ) T 22 ( j ) ] ( - i E z H y ) x = x j
γ j ( u ) = ( j - u 2 - β 2 ) 1 / 2 and Γ j ( u ) = ( - j + u 2 + β 2 ) ,
T 11 ( j ) = cos γ j D j + β ζ j η j Γ j sin γ j D j ,
T 12 ( j ) = η j γ j sin γ j D j ,
T 21 ( j ) = - 1 η j γ j sin γ j D j ,
T 22 ( j ) = cos γ j D j - β ζ j η j Γ j sin γ j D j .
T 11 ( j ) = cosh Γ j D j + β ζ j η j Γ j sinh Γ j D j ,
T 12 ( j ) = - η j Γ j sinh Γ j D j ,
T 21 ( j ) = - 1 η j Γ j sinh Γ j D j ,
T 22 ( j ) = cosh Γ j D j - β ζ j η j Γ j sinh Γ j D j .
η r Γ = ( P 11 - β ζ r P 21 ) ( η s Γ s - β ζ s ) ( P 12 + β ζ r P 22 ) P 22 + ( η s Γ s - β ζ s ) P 21 ,
η r ϒ = - R 11 R 21 - β ζ r .
Δ β = 2 Re 0 d d x i ξ E x * E z = 2 Re 0 d d x β ζ H y * x H y H y 2 fs - H y 2 cf ,
Re d x E x * H y = β d x η H y 2 = 1.
Γ = γ t γ t B 1 sin γ t t - Γ s B 2 cos γ t t γ t B 1 cos γ t t + Γ s B 1 sin γ t t
Γ = - Γ t Γ t B 1 sinh Γ t t + Γ s B 2 cosh Γ t t Γ t B 1 cosh Γ t t + Γ s B 2 sinh Γ t t
B 1 = cos γ f d + ( f Γ s s γ f + β ζ γ f ) sin γ f d ,
s c B 2 = cos γ f d + ( - s γ f f Γ s - β ζ γ f ) sin γ f d ,
B 1 = cosh Γ f d + ( f Γ s s Γ f + β ζ Γ f ) sinh Γ f d ,
s c B 2 = cosh Γ f d + ( s Γ f f Γ s - β ζ Γ f ) sinh Γ f d ,
ϒ = r cos γ ( D 1 + D 2 ) - ζ sin γ ( D 2 - D 1 ) r sin γ ( D 1 + D 2 ) + 2 ζ sin γ D 1 sin γ D 2 + r ζ ,

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