Abstract

Magneto-optical garnets combine high Faraday rotation with low optical losses in the near-infrared region where optical communication through glass fiber is established. In this spectral range, garnets are the only materials discussed to realize nonreciprocal devices as optical isolators and circulators. Although such devices are available as micro-optical components, practical versions of their integrated counterparts are still lacking. Numerous concepts have been developed theoretically, many of which are tested experimentally. We present an overview of the state of the art of the applications of garnet films in integrated optics. Also, the technique of combining garnets with semiconductor materials is discussed.

© 2005 Optical Society of America

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2004 (1)

2003 (2)

H. Yokoi, T. Mizumoto, and Y. Shoji, "Optical nonreciprocal devices with a silicon guiding layer fabricated by wafer bonding," Appl. Opt. 42, 6605-6612 (2003).
[Crossref] [PubMed]

L. Wilkens, D. Träger, H. Dötsch, A. M. Alexeev, A. F. Popkov, and V. I. Korneev, "Compensation walls in gallium and aluminium substituted gadolinium-bismuth-iron garnet films created by laser annealing: measurements and simulations," J. Appl. Phys. 93, 2839-2847 (2003).
[Crossref]

2002 (3)

B. Stegmueller, E. Baur, and M. Kicherer, "15-GHz modulation performance of integrated DFB laser diode EA modulator with identical multiple-quantum-well double-stack active layer," IEEE Photonics Technol. Lett. 14, 1647-1649 (2002).
[Crossref]

S. Kahl and A. M. Grishin, "Pulsed laser deposition of Y3Fe5O12 and Bi3Fe5O12 films on garnet substrates," J. Appl. Phys. 93, 6945-6947 (2002).
[Crossref]

T. Izuhara, J. Fujita, and M. Levy, "Integration of magneto-optical waveguides onto a III-V semiconductor surface," IEEE Photonics Technol. Lett. 14, 167-169 (2002).
[Crossref]

2001 (3)

M. Lohmeyer, L. Wilkens, O. Zhuromskyy, H. Dötsch, and P. Hertel, "Integrated magneto-optic cross strip isolator," Opt. Commun. 189, 251-259 (2001).
[Crossref]

L. Wilkens, D. Träger, A. F. Popkov, A. Alexeev, and H. Dötsch, "Nonreciprocal phase shift of TE modes induced by a compensation wall in a magneto-optic rib waveguide," Appl. Phys. Lett. 79, 4292-4294 (2001).
[Crossref]

O. Zhuromskyy, H. Dötsch, M. Lohmeyer, L. Wilkens, and P. Hertel, "Magneto-optical waveguides with polarization-independent nonreciprocal phase shift," J. Lightwave Technol. 19, 214-221 (2001).
[Crossref]

2000 (3)

J. Fujita, M. Levy, R. M. Osgood, Jr., L. Wilkens, and H. Dötsch, "Polarization-independent waveguide optical isolator based on nonreciprocal phase shift," IEEE Photonics Technol. Lett. 12, 1510-1512 (2000).
[Crossref]

J. Fujita, M. Levy, R. M. Osgood, Jr., L. Wilkens, and H. Dötsch, "Waveguide optical isolator based on Mach-Zehnder interferometer," Appl. Phys. Lett. 76, 2158-2160 (2000).
[Crossref]

O. Zhuromskyy, M. Lohmeyer, N. Bahlmann, P. Hertel, H. Dötsch, and A. F. Popkov, "Analysis of nonreciprocal light propagation in multimode imaging devices," Opt. Quantum Electron. 32, 885-897 (2000).
[Crossref]

1999 (7)

N. Bahlmann, M. Lohmeyer, O. Zhuromskyy, H. Dötsch, and P. Hertel, "Nonreciprocal coupled waveguides for integrated optical isolators and circulators for TM modes," Opt. Commun. 161, 330-337 (1999).
[Crossref]

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Börger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dötsch, "Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magneto-optic rib waveguide," Appl. Phys. Lett. 74, 2918-2920 (1999).
[Crossref]

O. Zhuromskyy, M. Lohmeyer, N. Bahlmann, H. Dötsch, P. Hertel, and A. F. Popkov, "Analysis of polarization independent Mach-Zehnder-type integrated optical isolator," J. Lightwave Technol. 17, 1200-1205 (1999).
[Crossref]

M. Lohmeyer, N. Bahlmann, O. Zhuromskyy, H. Dötsch, and P. Hertel, "Unidirectional magneto-optic polarization converters," J. Lightwave Technol. 17, 2605-2611 (1999).
[Crossref]

R. Gerhardt, J. Kleine-Börger, L. Beilschmidt, M. Frommeier, B. Gather, and H. Dötsch, "Efficient channel-waveguide laser in Nd:GGG at 1.062 mm wavelength," Appl. Phys. Lett. 75, 1210-1212 (1999).
[Crossref]

N. Bahlmann, M. Wallenhorst, L. Wilkens, V. Backherms, A. Josef, P. Hertel, and H. Dötsch, "Reduction of the temperature dependence of the nonreciprocal effect of magneto- optic channel waveguides," Appl. Opt. 38, 5747-5751 (1999).
[Crossref]

N. Bahlmann, M. Lohmeyer, H. Dötsch, and P. Hertel, "Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall," IEEE J. Quantum Electron. 35, 250-253 (1999).
[Crossref]

1998 (7)

M. Lohmeyer, "Vectorial wave-matching mode analysis of integrated optical waveguides," Opt. Quantum Electron. 30, 385-396 (1998).
[Crossref]

A. F. Popkov, M. Fehndrich, M. Lohmeyer, and H. Dötsch, "Nonreciprocal TE mode phase shift by domain walls in magneto-optic rib waveguides," Appl. Phys. Lett. 72, 2508-2510 (1998).
[Crossref]

M. Lohmeyer, N. Bahlmann, O. Zhuromskyy, H. Dötsch, and P. Hertel, "Phase matched rectangular magneto-optic waveguides for applications in integrated optical isolators: numerical assessment," Opt. Commun. 158, 189-200 (1998).
[Crossref]

N. Bahlmann, V. Chandrasekhara, A. Erdmann, R. Gerhardt, P. Hertel, R. Lehmann, D. Salz, F. J. Schröteler, M. Wallenhorst, and H. Dötsch, "Improved design of magneto-optic rib waveguides for optical isolators," J. Lightwave Technol. 16, 818-823 (1998).
[Crossref]

J. Yamauchi, G. Takahashi, and H. Nakano, "Full-vectorial beam-propagation method based on the McKee-Mitchell scheme with improved finite-difference formulas," J. Lightwave Technol. 16, 2458-2464 (1998).
[Crossref]

T. Shintaku, "Integrated optical isolator based on efficient nonreciprocal radiation mode conversion," Appl. Phys. Lett. 73, 1946-1948 (1998).
[Crossref]

N. Bahlmann, M. Lohmeyer, H. Dötsch, and P. Hertel, "Integrated magneto-optic Mach-Zehnder isolator for TE modes," Electron. Lett. 34, 2122-2123 (1998).
[Crossref]

1997 (4)

A. A. Anderson, C. L. Bonner, D. P. Shepherd, R. W. Eason, C. Grivas, D. S. Gill, and N. Vainos, "Low loss (0.5 dB/cm) Nd:Gd3Ga5O12 waveguide layers grown by pulsed laser deposition," Opt. Commun. 144, 183-186 (1997).
[Crossref]

C. T. A. Brown, C. L. Bonner, T. J. Warburton, D. P. Shepherd, A. C. Tropper, and D. C. Hanna, "Thermally bonded planar waveguide lasers," Appl. Phys. Lett. 71, 1139-1141 (1997).
[Crossref]

C. L. Bonner, A. A. Anderson, R. W. Eason, D. P. Shepherd, D. S. Gill, C. Grivas, and N. Vainos, "Performance of a low-loss pulsed-laser-deposited Nd:Gd3Ga5O12 waveguide laser at 1.06 and 0.94 µm," Opt. Lett. 22, 988-990 (1997).
[Crossref] [PubMed]

M. Lohmeyer, "Wave-matching method for mode analysis of dielectric waveguides," Opt. Quantum Electron. 29, 907-922 (1997).
[Crossref]

1996 (3)

K. S. Chiang, "Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides," IEEE Trans. Microwave Theory Tech. 44, 692-700 (1996).
[Crossref]

M. Shimokozono, N. Sugimoto, A. Tate, and Y. Katoh, "Room-temperature operation of an Yb-doped Gd3Ga5O12 buried channel waveguide laser at 1.025 µm wavelength," Appl. Phys. Lett. 68, 2177-2179 (1996).
[Crossref]

N. Sugimoto, H. Terui, A. Tate, Y. Katoh, Y. Yamada, A. Sugita, A. Shibukawa, and Y. Inoue, "A hybrid integrated waveguide isolator on a silica-based planar lightwave circuit," J. Lightwave Technol. 14, 2537-2546 (1996).
[Crossref]

1995 (2)

H. Yokoi, T. Mizumoto, K. Maru, and Y. Naito, "Direct bonding between InP and rare-earth iron-garnet grown on Gd3Ga5O12 substrate by liquid-phase epitaxy," Electron. Lett. 31, 1612-1613 (1995).
[Crossref]

M. Wallenhorst, M. Niemöller, H. Dötsch, P. Hertel, R. Gerhardt, and B. Gather, "Enhancement of the nonreciprocal magneto-optic effect of TM modes using iron garnet double layers with opposite Faraday rotation," J. Appl. Phys. 77, 2902-2905 (1995).
[Crossref]

1994 (2)

F. Wijnands, H. J. W. M. Hoekstra, G. J. M. Krijnen, and R. M. de Ridder, "Modal fields calculation using the finite difference beam propagation method," J. Lightwave Technol. 12, 2066-2072 (1994).
[Crossref]

H. P. Winkler, H. Dötsch, B. Lührmann, and S. Sure, "Dynamic conversion of optical modes in magnetic garnet films induced by resonance of periodic stripe domains," J. Appl. Phys. 76, 3272-3278 (1994).
[Crossref]

1993 (1)

A. S. Sudbo, "Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides," Pure Appl. Opt. 2, 211-233 (1993).
[Crossref]

1992 (2)

1991 (3)

L. Zhang, P. J. Chandler, P. D. Townsend, S. J. Field, D. C. Hanna, D. P. Shepherd, and A. C. Tropper, "Characterization of ion-implanted wave-guides in Nd:YAG," J. Appl. Phys. 69, 3440-3446 (1991).
[Crossref]

S. J. Field, D. C. Hanna, A. C. Large, D. P. Shepherd, A. C. Tropper, P. J. Chandler, P. D. Townsend, and L. Zhang, "An efficient, diode-pumped, ion-implanted Nd:GGG planar waveguide laser," Opt. Commun. 86, 161-166 (1991).
[Crossref]

S. J. Field, D. C. Hanna, D. P. Shepherd, A. C. Tropper, P. J. Chandler, P. D. Townsend, and L. Zhang, "Ion-implanted Nd:YAG waveguide lasers," IEEE J. Quantum Electron. 27, 428-433 (1991).
[Crossref]

1990 (3)

H. Dammann, E. Pross, G. Rabe, and W. Tolksdorf, "45° waveguide isolators with phase mismatch," Appl. Phys. Lett. 56, 1302-1304 (1990).
[Crossref]

R. Wolfe, J. F. Dillon, Jr., R. A. Liebermann, and V. J. Fratello, "Broadband magneto-optic waveguide isolator," Appl. Phys. Lett. 57, 960-962 (1990).
[Crossref]

C. S. Tsai and D. Young, "Magnetostatic-forward-volume-wave-based guided-wave magneto-optic Bragg cells and applications to communications and signal processing," IEEE Trans. Microwave Theory Tech. 38, 560-570 (1990).
[Crossref]

1989 (1)

C. S. Tsai and D. Young, "Wideband scanning of a guided-light beam and spectrum analysis using magnetostatic waves in an yttrium iron garnet-gadolinium gallium garnet wave-guide," Appl. Phys. Lett. 54, 196-198 (1989).
[Crossref]

1988 (6)

H. Tamada, M. Kaneko, and T. Okamoto, "TM-TE optical-mode conversion induced by a transversely propagating magnetostatic wave in a (BiLu)3Fe5O12 film," J. Appl. Phys. 64, 554-559 (1988).
[Crossref]

S. H. Talisa, "The collinear interaction between forward volume magnetostatic waves and guided light in YIG films," IEEE Trans. Magn. 24, 2811-2813 (1988).
[Crossref]

M. Gomi, K. Satoh, and M. Abe, "Giant Faraday rotation of Ce-substituted YIG films epitaxially grown by RF sputtering," Jpn. J. Appl. Phys. 27, L1536-L1538 (1988).
[Crossref]

M. S. Stern, "Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles," IEE Proc. Optoelectron. 135, 56-63 (1988).
[Crossref]

M. S. Stern, "Semivectorial polarized H field solutions for dielectric waveguides with arbitrary index profiles," IEE Proc.-J 135, 333-338 (1988).

K. Ando, T. Okoshi, and N. Koshizuka, "Waveguide magneto-optic isolator fabricated by laser annealing," Appl. Phys. Lett. 53, 4-6 (1988).
[Crossref]

1987 (3)

R. Wolfe, V. J. Fratello, and M. NcGlashan-Powell, "Elimination of birefringence in garnet films for magneto-optic wave-guide devices," Appl. Phys. Lett. 51, 1221-1223 (1987).
[Crossref]

B. Neite and H. Dötsch, "Dynamical conversion of optical modes in garnet films induced by ferrimagnetic resonance," J. Appl. Phys. 62, 648-652 (1987).
[Crossref]

J. Haisma, A. M. W. Cox, B. H. Koek, D. Mateika, J. A. Pistorius, and E. T. J. M. Smeets, "Heteroepitaxial growth of InP on garnet," J. Cryst. Growth 87, 180-184 (1987).
[Crossref]

1986 (4)

M. Razeghi, P. L. Meunier, and P. Maurel, "Growth of GaInAs-InP multiquantum wells on garnet (GGG=Gd3Ga5O12) substrate by metalorganic chemical vapor deposition," J. Appl. Phys. 59, 2261-2263 (1986).
[Crossref]

J. P. Krumme, V. Doormann, and B. Strocka, "Selected-area sputter epitaxy of iron-garnet films," J. Appl. Phys. 60, 2065-2068 (1986).
[Crossref]

H. Dammann, E. Pross, and G. Rabe, "Phase matching in symmetrical single-mode magneto-optic waveguides by application of stress," Appl. Phys. Lett. 49, 1755-1757 (1986).
[Crossref]

T. Mizumoto, Y. Kawaoka, and Y. Naito, "Waveguide-type optical isolator using the Faraday and Cotton-Mouton effects," Trans. Inst. Electron. Commun. Eng. Jpn., Sect. E 69, 968-972 (1986).

1985 (2)

R. Wolfe, J. Hegarty, J. F. Dillon, Jr., L. C. Luther, G. K. Celler, L. E. Trimble, and C. S. Dorsey, "Thin-film waveguide magneto-optic isolator," Appl. Phys. Lett. 46, 817-819 (1985).
[Crossref]

P. Hansen, C. P. Klages, J. Schuldt, and K. Witter, "Magnetic and magneto-optical properties of bismuth-substituted lutetium iron garnet films," Phys. Rev. B 31, 5858-5864 (1985).
[Crossref]

1984 (1)

B. M. A. Rahmann and J. B. Davies, "Finite-element solution of integrated optical waveguides," J. Lightwave Technol. 2, 682-687 (1984).
[Crossref]

1982 (1)

A. D. Fisher, J. N. Lee, E. S. Gaynor, and A. B. Tveten, "Optical guided-wave interactions with magnetostatic waves at microwave-frequencies," Appl. Phys. Lett. 41, 779-781 (1982).
[Crossref]

1975 (1)

F. Auracher and H. H. Witte, "A new design for an integrated optical isolator," Opt. Commun. 13, 435-438 (1975).
[Crossref]

1974 (1)

S. Yamamoto and 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]

1972 (1)

S. L. Blank and J. W. Nielsen, "The growth of magnetic garnets by liquid phase epitaxy," J. Cryst. Growth 17, 302-311 (1972).
[Crossref]

1965 (1)

G. J. Gabriel and M. E. Brodwin, "The solution of guided waves in inhomogeneous anisotropic media by perturbation and variational methods," IEEE Trans. Microwave Theory Tech. 13, 364-370 (1965).
[Crossref]

Abe, M.

M. Gomi, K. Satoh, and M. Abe, "Giant Faraday rotation of Ce-substituted YIG films epitaxially grown by RF sputtering," Jpn. J. Appl. Phys. 27, L1536-L1538 (1988).
[Crossref]

Alexeev, A.

L. Wilkens, D. Träger, A. F. Popkov, A. Alexeev, and H. Dötsch, "Nonreciprocal phase shift of TE modes induced by a compensation wall in a magneto-optic rib waveguide," Appl. Phys. Lett. 79, 4292-4294 (2001).
[Crossref]

Alexeev, A. M.

L. Wilkens, D. Träger, H. Dötsch, A. M. Alexeev, A. F. Popkov, and V. I. Korneev, "Compensation walls in gallium and aluminium substituted gadolinium-bismuth-iron garnet films created by laser annealing: measurements and simulations," J. Appl. Phys. 93, 2839-2847 (2003).
[Crossref]

Anderson, A. A.

A. A. Anderson, C. L. Bonner, D. P. Shepherd, R. W. Eason, C. Grivas, D. S. Gill, and N. Vainos, "Low loss (0.5 dB/cm) Nd:Gd3Ga5O12 waveguide layers grown by pulsed laser deposition," Opt. Commun. 144, 183-186 (1997).
[Crossref]

C. L. Bonner, A. A. Anderson, R. W. Eason, D. P. Shepherd, D. S. Gill, C. Grivas, and N. Vainos, "Performance of a low-loss pulsed-laser-deposited Nd:Gd3Ga5O12 waveguide laser at 1.06 and 0.94 µm," Opt. Lett. 22, 988-990 (1997).
[Crossref] [PubMed]

Ando, K.

K. Ando, T. Okoshi, and N. Koshizuka, "Waveguide magneto-optic isolator fabricated by laser annealing," Appl. Phys. Lett. 53, 4-6 (1988).
[Crossref]

Auracher , F.

F. Auracher and H. H. Witte, "A new design for an integrated optical isolator," Opt. Commun. 13, 435-438 (1975).
[Crossref]

Backherms, V.

Bahlmann, N.

O. Zhuromskyy, M. Lohmeyer, N. Bahlmann, P. Hertel, H. Dötsch, and A. F. Popkov, "Analysis of nonreciprocal light propagation in multimode imaging devices," Opt. Quantum Electron. 32, 885-897 (2000).
[Crossref]

M. Lohmeyer, N. Bahlmann, O. Zhuromskyy, H. Dötsch, and P. Hertel, "Unidirectional magneto-optic polarization converters," J. Lightwave Technol. 17, 2605-2611 (1999).
[Crossref]

O. Zhuromskyy, M. Lohmeyer, N. Bahlmann, H. Dötsch, P. Hertel, and A. F. Popkov, "Analysis of polarization independent Mach-Zehnder-type integrated optical isolator," J. Lightwave Technol. 17, 1200-1205 (1999).
[Crossref]

N. Bahlmann, M. Lohmeyer, O. Zhuromskyy, H. Dötsch, and P. Hertel, "Nonreciprocal coupled waveguides for integrated optical isolators and circulators for TM modes," Opt. Commun. 161, 330-337 (1999).
[Crossref]

N. Bahlmann, M. Wallenhorst, L. Wilkens, V. Backherms, A. Josef, P. Hertel, and H. Dötsch, "Reduction of the temperature dependence of the nonreciprocal effect of magneto- optic channel waveguides," Appl. Opt. 38, 5747-5751 (1999).
[Crossref]

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Börger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dötsch, "Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magneto-optic rib waveguide," Appl. Phys. Lett. 74, 2918-2920 (1999).
[Crossref]

N. Bahlmann, M. Lohmeyer, H. Dötsch, and P. Hertel, "Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall," IEEE J. Quantum Electron. 35, 250-253 (1999).
[Crossref]

N. Bahlmann, V. Chandrasekhara, A. Erdmann, R. Gerhardt, P. Hertel, R. Lehmann, D. Salz, F. J. Schröteler, M. Wallenhorst, and H. Dötsch, "Improved design of magneto-optic rib waveguides for optical isolators," J. Lightwave Technol. 16, 818-823 (1998).
[Crossref]

M. Lohmeyer, N. Bahlmann, O. Zhuromskyy, H. Dötsch, and P. Hertel, "Phase matched rectangular magneto-optic waveguides for applications in integrated optical isolators: numerical assessment," Opt. Commun. 158, 189-200 (1998).
[Crossref]

N. Bahlmann, M. Lohmeyer, H. Dötsch, and P. Hertel, "Integrated magneto-optic Mach-Zehnder isolator for TE modes," Electron. Lett. 34, 2122-2123 (1998).
[Crossref]

Baur, E.

B. Stegmueller, E. Baur, and M. Kicherer, "15-GHz modulation performance of integrated DFB laser diode EA modulator with identical multiple-quantum-well double-stack active layer," IEEE Photonics Technol. Lett. 14, 1647-1649 (2002).
[Crossref]

Beilschmidt, L.

R. Gerhardt, J. Kleine-Börger, L. Beilschmidt, M. Frommeier, B. Gather, and H. Dötsch, "Efficient channel-waveguide laser in Nd:GGG at 1.062 mm wavelength," Appl. Phys. Lett. 75, 1210-1212 (1999).
[Crossref]

Blank , S. L.

S. L. Blank and J. W. Nielsen, "The growth of magnetic garnets by liquid phase epitaxy," J. Cryst. Growth 17, 302-311 (1972).
[Crossref]

Bonner, C. L.

C. L. Bonner, A. A. Anderson, R. W. Eason, D. P. Shepherd, D. S. Gill, C. Grivas, and N. Vainos, "Performance of a low-loss pulsed-laser-deposited Nd:Gd3Ga5O12 waveguide laser at 1.06 and 0.94 µm," Opt. Lett. 22, 988-990 (1997).
[Crossref] [PubMed]

A. A. Anderson, C. L. Bonner, D. P. Shepherd, R. W. Eason, C. Grivas, D. S. Gill, and N. Vainos, "Low loss (0.5 dB/cm) Nd:Gd3Ga5O12 waveguide layers grown by pulsed laser deposition," Opt. Commun. 144, 183-186 (1997).
[Crossref]

C. T. A. Brown, C. L. Bonner, T. J. Warburton, D. P. Shepherd, A. C. Tropper, and D. C. Hanna, "Thermally bonded planar waveguide lasers," Appl. Phys. Lett. 71, 1139-1141 (1997).
[Crossref]

Brodwin, M. E.

G. J. Gabriel and M. E. Brodwin, "The solution of guided waves in inhomogeneous anisotropic media by perturbation and variational methods," IEEE Trans. Microwave Theory Tech. 13, 364-370 (1965).
[Crossref]

Brown, C. T. A.

C. T. A. Brown, C. L. Bonner, T. J. Warburton, D. P. Shepherd, A. C. Tropper, and D. C. Hanna, "Thermally bonded planar waveguide lasers," Appl. Phys. Lett. 71, 1139-1141 (1997).
[Crossref]

Celler, G. K.

R. Wolfe, J. Hegarty, J. F. Dillon, Jr., L. C. Luther, G. K. Celler, L. E. Trimble, and C. S. Dorsey, "Thin-film waveguide magneto-optic isolator," Appl. Phys. Lett. 46, 817-819 (1985).
[Crossref]

Chandler, P. J.

S. J. Field, D. C. Hanna, A. C. Large, D. P. Shepherd, A. C. Tropper, P. J. Chandler, P. D. Townsend, and L. Zhang, "Ion-implanted Nd:GGG channel waveguide laser," Opt. Lett. 17, 52-54 (1992).
[Crossref] [PubMed]

S. J. Field, D. C. Hanna, D. P. Shepherd, A. C. Tropper, P. J. Chandler, P. D. Townsend, and L. Zhang, "Ion-implanted Nd:YAG waveguide lasers," IEEE J. Quantum Electron. 27, 428-433 (1991).
[Crossref]

S. J. Field, D. C. Hanna, A. C. Large, D. P. Shepherd, A. C. Tropper, P. J. Chandler, P. D. Townsend, and L. Zhang, "An efficient, diode-pumped, ion-implanted Nd:GGG planar waveguide laser," Opt. Commun. 86, 161-166 (1991).
[Crossref]

L. Zhang, P. J. Chandler, P. D. Townsend, S. J. Field, D. C. Hanna, D. P. Shepherd, and A. C. Tropper, "Characterization of ion-implanted wave-guides in Nd:YAG," J. Appl. Phys. 69, 3440-3446 (1991).
[Crossref]

Chandrasekhara, V.

Chartier, I.

Chiang, K. S.

K. S. Chiang, "Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides," IEEE Trans. Microwave Theory Tech. 44, 692-700 (1996).
[Crossref]

Cox, A. M. W.

J. Haisma, A. M. W. Cox, B. H. Koek, D. Mateika, J. A. Pistorius, and E. T. J. M. Smeets, "Heteroepitaxial growth of InP on garnet," J. Cryst. Growth 87, 180-184 (1987).
[Crossref]

Dammann, H.

H. Dammann, E. Pross, G. Rabe, and W. Tolksdorf, "45° waveguide isolators with phase mismatch," Appl. Phys. Lett. 56, 1302-1304 (1990).
[Crossref]

H. Dammann, E. Pross, and G. Rabe, "Phase matching in symmetrical single-mode magneto-optic waveguides by application of stress," Appl. Phys. Lett. 49, 1755-1757 (1986).
[Crossref]

Davies, J. B.

B. M. A. Rahmann and J. B. Davies, "Finite-element solution of integrated optical waveguides," J. Lightwave Technol. 2, 682-687 (1984).
[Crossref]

de Ridder, R. M.

F. Wijnands, H. J. W. M. Hoekstra, G. J. M. Krijnen, and R. M. de Ridder, "Modal fields calculation using the finite difference beam propagation method," J. Lightwave Technol. 12, 2066-2072 (1994).
[Crossref]

Dillon, J. F.

R. Wolfe, J. F. Dillon, Jr., R. A. Liebermann, and V. J. Fratello, "Broadband magneto-optic waveguide isolator," Appl. Phys. Lett. 57, 960-962 (1990).
[Crossref]

R. Wolfe, J. Hegarty, J. F. Dillon, Jr., L. C. Luther, G. K. Celler, L. E. Trimble, and C. S. Dorsey, "Thin-film waveguide magneto-optic isolator," Appl. Phys. Lett. 46, 817-819 (1985).
[Crossref]

Doormann, V.

J. P. Krumme, V. Doormann, and B. Strocka, "Selected-area sputter epitaxy of iron-garnet films," J. Appl. Phys. 60, 2065-2068 (1986).
[Crossref]

Dorsey, C. S.

R. Wolfe, J. Hegarty, J. F. Dillon, Jr., L. C. Luther, G. K. Celler, L. E. Trimble, and C. S. Dorsey, "Thin-film waveguide magneto-optic isolator," Appl. Phys. Lett. 46, 817-819 (1985).
[Crossref]

Dötsch, H.

R. L. Espinola, T. Izuhara, M.-C. Tsai, R. M. Osgood, Jr., and H. Dötsch, "Magneto-optical nonreciprocal phase shift in garnet/silicon-on-insulator waveguides," Opt. Lett. 29, 941-943 (2004).
[Crossref] [PubMed]

L. Wilkens, D. Träger, H. Dötsch, A. M. Alexeev, A. F. Popkov, and V. I. Korneev, "Compensation walls in gallium and aluminium substituted gadolinium-bismuth-iron garnet films created by laser annealing: measurements and simulations," J. Appl. Phys. 93, 2839-2847 (2003).
[Crossref]

O. Zhuromskyy, H. Dötsch, M. Lohmeyer, L. Wilkens, and P. Hertel, "Magneto-optical waveguides with polarization-independent nonreciprocal phase shift," J. Lightwave Technol. 19, 214-221 (2001).
[Crossref]

M. Lohmeyer, L. Wilkens, O. Zhuromskyy, H. Dötsch, and P. Hertel, "Integrated magneto-optic cross strip isolator," Opt. Commun. 189, 251-259 (2001).
[Crossref]

O. Zhuromskyy, M. Lohmeyer, N. Bahlmann, P. Hertel, H. Dötsch, and A. F. Popkov, "Analysis of nonreciprocal light propagation in multimode imaging devices," Opt. Quantum Electron. 32, 885-897 (2000).
[Crossref]

J. Fujita, M. Levy, R. M. Osgood, Jr., L. Wilkens, and H. Dötsch, "Waveguide optical isolator based on Mach-Zehnder interferometer," Appl. Phys. Lett. 76, 2158-2160 (2000).
[Crossref]

J. Fujita, M. Levy, R. M. Osgood, Jr., L. Wilkens, and H. Dötsch, "Polarization-independent waveguide optical isolator based on nonreciprocal phase shift," IEEE Photonics Technol. Lett. 12, 1510-1512 (2000).
[Crossref]

N. Bahlmann, M. Wallenhorst, L. Wilkens, V. Backherms, A. Josef, P. Hertel, and H. Dötsch, "Reduction of the temperature dependence of the nonreciprocal effect of magneto- optic channel waveguides," Appl. Opt. 38, 5747-5751 (1999).
[Crossref]

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Börger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dötsch, "Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magneto-optic rib waveguide," Appl. Phys. Lett. 74, 2918-2920 (1999).
[Crossref]

N. Bahlmann, M. Lohmeyer, H. Dötsch, and P. Hertel, "Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall," IEEE J. Quantum Electron. 35, 250-253 (1999).
[Crossref]

N. Bahlmann, M. Lohmeyer, O. Zhuromskyy, H. Dötsch, and P. Hertel, "Nonreciprocal coupled waveguides for integrated optical isolators and circulators for TM modes," Opt. Commun. 161, 330-337 (1999).
[Crossref]

O. Zhuromskyy, M. Lohmeyer, N. Bahlmann, H. Dötsch, P. Hertel, and A. F. Popkov, "Analysis of polarization independent Mach-Zehnder-type integrated optical isolator," J. Lightwave Technol. 17, 1200-1205 (1999).
[Crossref]

M. Lohmeyer, N. Bahlmann, O. Zhuromskyy, H. Dötsch, and P. Hertel, "Unidirectional magneto-optic polarization converters," J. Lightwave Technol. 17, 2605-2611 (1999).
[Crossref]

R. Gerhardt, J. Kleine-Börger, L. Beilschmidt, M. Frommeier, B. Gather, and H. Dötsch, "Efficient channel-waveguide laser in Nd:GGG at 1.062 mm wavelength," Appl. Phys. Lett. 75, 1210-1212 (1999).
[Crossref]

N. Bahlmann, M. Lohmeyer, H. Dötsch, and P. Hertel, "Integrated magneto-optic Mach-Zehnder isolator for TE modes," Electron. Lett. 34, 2122-2123 (1998).
[Crossref]

A. F. Popkov, M. Fehndrich, M. Lohmeyer, and H. Dötsch, "Nonreciprocal TE mode phase shift by domain walls in magneto-optic rib waveguides," Appl. Phys. Lett. 72, 2508-2510 (1998).
[Crossref]

N. Bahlmann, V. Chandrasekhara, A. Erdmann, R. Gerhardt, P. Hertel, R. Lehmann, D. Salz, F. J. Schröteler, M. Wallenhorst, and H. Dötsch, "Improved design of magneto-optic rib waveguides for optical isolators," J. Lightwave Technol. 16, 818-823 (1998).
[Crossref]

M. Lohmeyer, N. Bahlmann, O. Zhuromskyy, H. Dötsch, and P. Hertel, "Phase matched rectangular magneto-optic waveguides for applications in integrated optical isolators: numerical assessment," Opt. Commun. 158, 189-200 (1998).
[Crossref]

M. Wallenhorst, M. Niemöller, H. Dötsch, P. Hertel, R. Gerhardt, and B. Gather, "Enhancement of the nonreciprocal magneto-optic effect of TM modes using iron garnet double layers with opposite Faraday rotation," J. Appl. Phys. 77, 2902-2905 (1995).
[Crossref]

H. P. Winkler, H. Dötsch, B. Lührmann, and S. Sure, "Dynamic conversion of optical modes in magnetic garnet films induced by resonance of periodic stripe domains," J. Appl. Phys. 76, 3272-3278 (1994).
[Crossref]

B. Neite and H. Dötsch, "Dynamical conversion of optical modes in garnet films induced by ferrimagnetic resonance," J. Appl. Phys. 62, 648-652 (1987).
[Crossref]

Dötsch, H.

L. Wilkens, D. Träger, A. F. Popkov, A. Alexeev, and H. Dötsch, "Nonreciprocal phase shift of TE modes induced by a compensation wall in a magneto-optic rib waveguide," Appl. Phys. Lett. 79, 4292-4294 (2001).
[Crossref]

Eason, R. W.

C. L. Bonner, A. A. Anderson, R. W. Eason, D. P. Shepherd, D. S. Gill, C. Grivas, and N. Vainos, "Performance of a low-loss pulsed-laser-deposited Nd:Gd3Ga5O12 waveguide laser at 1.06 and 0.94 µm," Opt. Lett. 22, 988-990 (1997).
[Crossref] [PubMed]

A. A. Anderson, C. L. Bonner, D. P. Shepherd, R. W. Eason, C. Grivas, D. S. Gill, and N. Vainos, "Low loss (0.5 dB/cm) Nd:Gd3Ga5O12 waveguide layers grown by pulsed laser deposition," Opt. Commun. 144, 183-186 (1997).
[Crossref]

Erdmann, A.

Espinola, R. L.

Fehndrich, M.

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Börger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dötsch, "Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magneto-optic rib waveguide," Appl. Phys. Lett. 74, 2918-2920 (1999).
[Crossref]

A. F. Popkov, M. Fehndrich, M. Lohmeyer, and H. Dötsch, "Nonreciprocal TE mode phase shift by domain walls in magneto-optic rib waveguides," Appl. Phys. Lett. 72, 2508-2510 (1998).
[Crossref]

Ferrand, B.

Field, S. J.

S. J. Field, D. C. Hanna, A. C. Large, D. P. Shepherd, A. C. Tropper, P. J. Chandler, P. D. Townsend, and L. Zhang, "Ion-implanted Nd:GGG channel waveguide laser," Opt. Lett. 17, 52-54 (1992).
[Crossref] [PubMed]

I. Chartier, B. Ferrand, D. Pelenc, S. J. Field, D. C. Hanna, A. C. Large, D. P. Shepherd, and A. C. Tropper, "Growth and low-threshold laser oscillation of an epitaxially grown Nd:YAG waveguide," Opt. Lett. 17, 810-812 (1992).
[Crossref] [PubMed]

S. J. Field, D. C. Hanna, D. P. Shepherd, A. C. Tropper, P. J. Chandler, P. D. Townsend, and L. Zhang, "Ion-implanted Nd:YAG waveguide lasers," IEEE J. Quantum Electron. 27, 428-433 (1991).
[Crossref]

L. Zhang, P. J. Chandler, P. D. Townsend, S. J. Field, D. C. Hanna, D. P. Shepherd, and A. C. Tropper, "Characterization of ion-implanted wave-guides in Nd:YAG," J. Appl. Phys. 69, 3440-3446 (1991).
[Crossref]

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M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Börger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dötsch, "Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magneto-optic rib waveguide," Appl. Phys. Lett. 74, 2918-2920 (1999).
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N. Bahlmann, M. Wallenhorst, L. Wilkens, V. Backherms, A. Josef, P. Hertel, and H. Dötsch, "Reduction of the temperature dependence of the nonreciprocal effect of magneto- optic channel waveguides," Appl. Opt. 38, 5747-5751 (1999).
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M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Börger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dötsch, "Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magneto-optic rib waveguide," Appl. Phys. Lett. 74, 2918-2920 (1999).
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T. Izuhara, J. Fujita, and M. Levy, "Integration of magneto-optical waveguides onto a III-V semiconductor surface," IEEE Photonics Technol. Lett. 14, 167-169 (2002).
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J. Fujita, M. Levy, R. M. Osgood, Jr., L. Wilkens, and H. Dötsch, "Waveguide optical isolator based on Mach-Zehnder interferometer," Appl. Phys. Lett. 76, 2158-2160 (2000).
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M. Lohmeyer, L. Wilkens, O. Zhuromskyy, H. Dötsch, and P. Hertel, "Integrated magneto-optic cross strip isolator," Opt. Commun. 189, 251-259 (2001).
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N. Bahlmann, M. Lohmeyer, H. Dötsch, and P. Hertel, "Integrated magneto-optic Mach-Zehnder isolator for TE modes," Electron. Lett. 34, 2122-2123 (1998).
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H. Yokoi, T. Mizumoto, K. Maru, and Y. Naito, "Direct bonding between InP and rare-earth iron-garnet grown on Gd3Ga5O12 substrate by liquid-phase epitaxy," Electron. Lett. 31, 1612-1613 (1995).
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T. Mizumoto, Y. Kawaoka, and Y. Naito, "Waveguide-type optical isolator using the Faraday and Cotton-Mouton effects," Trans. Inst. Electron. Commun. Eng. Jpn., Sect. E 69, 968-972 (1986).

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K. Ando, T. Okoshi, and N. Koshizuka, "Waveguide magneto-optic isolator fabricated by laser annealing," Appl. Phys. Lett. 53, 4-6 (1988).
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C. T. A. Brown, C. L. Bonner, T. J. Warburton, D. P. Shepherd, A. C. Tropper, and D. C. Hanna, "Thermally bonded planar waveguide lasers," Appl. Phys. Lett. 71, 1139-1141 (1997).
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Warburton, T. J.

C. T. A. Brown, C. L. Bonner, T. J. Warburton, D. P. Shepherd, A. C. Tropper, and D. C. Hanna, "Thermally bonded planar waveguide lasers," Appl. Phys. Lett. 71, 1139-1141 (1997).
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O. Zhuromskyy, H. Dötsch, M. Lohmeyer, L. Wilkens, and P. Hertel, "Magneto-optical waveguides with polarization-independent nonreciprocal phase shift," J. Lightwave Technol. 19, 214-221 (2001).
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J. Fujita, M. Levy, R. M. Osgood, Jr., L. Wilkens, and H. Dötsch, "Waveguide optical isolator based on Mach-Zehnder interferometer," Appl. Phys. Lett. 76, 2158-2160 (2000).
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J. Fujita, M. Levy, R. M. Osgood, Jr., L. Wilkens, and H. Dötsch, "Polarization-independent waveguide optical isolator based on nonreciprocal phase shift," IEEE Photonics Technol. Lett. 12, 1510-1512 (2000).
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N. Sugimoto, H. Terui, A. Tate, Y. Katoh, Y. Yamada, A. Sugita, A. Shibukawa, and Y. Inoue, "A hybrid integrated waveguide isolator on a silica-based planar lightwave circuit," J. Lightwave Technol. 14, 2537-2546 (1996).
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[Crossref]

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[Crossref]

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S. J. Field, D. C. Hanna, A. C. Large, D. P. Shepherd, A. C. Tropper, P. J. Chandler, P. D. Townsend, and L. Zhang, "Ion-implanted Nd:GGG channel waveguide laser," Opt. Lett. 17, 52-54 (1992).
[Crossref] [PubMed]

S. J. Field, D. C. Hanna, D. P. Shepherd, A. C. Tropper, P. J. Chandler, P. D. Townsend, and L. Zhang, "Ion-implanted Nd:YAG waveguide lasers," IEEE J. Quantum Electron. 27, 428-433 (1991).
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[Crossref]

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[Crossref]

Zhuromskyy, O.

M. Lohmeyer, L. Wilkens, O. Zhuromskyy, H. Dötsch, and P. Hertel, "Integrated magneto-optic cross strip isolator," Opt. Commun. 189, 251-259 (2001).
[Crossref]

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O. Zhuromskyy, M. Lohmeyer, N. Bahlmann, P. Hertel, H. Dötsch, and A. F. Popkov, "Analysis of nonreciprocal light propagation in multimode imaging devices," Opt. Quantum Electron. 32, 885-897 (2000).
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O. Zhuromskyy, M. Lohmeyer, N. Bahlmann, H. Dötsch, P. Hertel, and A. F. Popkov, "Analysis of polarization independent Mach-Zehnder-type integrated optical isolator," J. Lightwave Technol. 17, 1200-1205 (1999).
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Appl. Opt. (2)

Appl. Phys. Lett. (16)

C. T. A. Brown, C. L. Bonner, T. J. Warburton, D. P. Shepherd, A. C. Tropper, and D. C. Hanna, "Thermally bonded planar waveguide lasers," Appl. Phys. Lett. 71, 1139-1141 (1997).
[Crossref]

M. Shimokozono, N. Sugimoto, A. Tate, and Y. Katoh, "Room-temperature operation of an Yb-doped Gd3Ga5O12 buried channel waveguide laser at 1.025 µm wavelength," Appl. Phys. Lett. 68, 2177-2179 (1996).
[Crossref]

R. Gerhardt, J. Kleine-Börger, L. Beilschmidt, M. Frommeier, B. Gather, and H. Dötsch, "Efficient channel-waveguide laser in Nd:GGG at 1.062 mm wavelength," Appl. Phys. Lett. 75, 1210-1212 (1999).
[Crossref]

H. Dammann, E. Pross, and G. Rabe, "Phase matching in symmetrical single-mode magneto-optic waveguides by application of stress," Appl. Phys. Lett. 49, 1755-1757 (1986).
[Crossref]

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Figures (21)

Fig. 1
Fig. 1

Schematic of the bulk optical isolator.

Fig. 2
Fig. 2

Simple rib waveguide typically used in integrated optics.

Fig. 3
Fig. 3

Contour plots of the main field components |Ey| (top) and |Hy| (center) of the fundamental TE0 and TM0 modes, respectively. The contour levels are spaced at 10% of the maximum amplitude. The waveguide parameters (bottom) are d=0.5 µm, h=0.3 µm, w=2.0 µm, and λ=1.3 µm, and the respective refractive indices are ns=1.9, nf=2.2, and nc=1.0.

Fig. 4
Fig. 4

Calculated nonreciprocal TM0 mode phase shift |Δβ| for a double- and single-layer planar waveguide with a Faraday rotation of 1152°/cm at a wavelength of λ=1.3 µm. The refractive indices are nc=1, nf=2.33, and ns=1.95 for the cover region, film, and substrate, respectively. The ratio d1/d2 is kept fixed at 0.4.29

Fig. 5
Fig. 5

Calculated and measured nonreciprocal TM0 mode phase shift for a double- and single-layer monomode waveguide versus total film thickness (width 2 µm, height 40 nm).30

Fig. 6
Fig. 6

Principle of the nonreciprocal TE mode phase shift with a stripe domain lattice (top) and an experimental arrangement (bottom).31

Fig. 7
Fig. 7

Top: Principle of the nonreciprocal TE mode phase shift by use of a compensation wall located inside the rib waveguide. Bottom: Calculated nonreciprocal phase shift as a function of the location ycomp of the compensation wall. Parameters are rib width w=1.4 µm, film thickness d=0.8 µm, and Faraday rotation -821°/cm at λ=1.3 µm; the refractive indices of film, cover, and substrate are nf=2.237, nc=1.0, and ns=1.95, respectively.32

Fig. 8
Fig. 8

(a) and (b) Geometries of two different waveguides each consisting of two layers with horizontal and vertical boundaries of discontinuity of the Faraday rotation. (c) Calculated nonreciprocal phase shift for both TE and TM modes in the two waveguides versus the thickness hb of the bottom layer. Waveguide parameters are ns=1.95, nm=2.2, ne=2.3, nc=1.0, hr=0.7 µm, hl=0.5 µm, W=1.0 µm, wd=0.5 µm, and λ=1.3 µm. 35

Fig. 9
Fig. 9

Principle of a nonreciprocal MZI for TM modes. Top: One nonreciprocal interferometer arm. Bottom: Two nonreciprocal interferometer arms with an opposite sign of the phase shift.47

Fig. 10
Fig. 10

Principle of a polarization-independent Mach–Zehnder isolator with a nonreciprocal TE and TM mode phase shift separated in the two interferometer arms.50

Fig. 11
Fig. 11

Principle of a four-port nonreciprocal X coupler.51

Fig. 12
Fig. 12

Calculated performance of a four-port nonreciprocal X coupler for TM modes showing (a) one of the forward connections (straight, even number of couplings) and (b) one of the backward connections (crossing, odd number of couplings).51

Fig. 13
Fig. 13

Principle of a two-port MMI structure.53

Fig. 14
Fig. 14

Calculated intensity distribution of a two-port MMI isolator.

Fig. 15
Fig. 15

Principle of a four-port MMI structure.53

Fig. 16
Fig. 16

Calculated performance of a four-port nonreciprocal MMI circulator. The ratio between output and input power in decibels is plotted versus the length L of the multimode section. The material parameters are nc=1.0, nf=2.2, ns=1.95, ӨF,sat=2880°/cm, and λ=1.3 µm. The geometric parameters (see Fig. 15) are h=0.7 µm, a=0.5 µm, W=4.45 µm, and S=1.1 µm. 53

Fig. 17
Fig. 17

Principle of a cross strip isolator.54

Fig. 18
Fig. 18

Simulation of the operation of a cross strip isolator. Forward propagation is shown in the top two graphs; backward propagation is shown in the bottom two graphs.54

Fig. 19
Fig. 19

Calculated performance of a cross strip isolator. The top graph shows isolation; the bottom graph shows forward loss.54

Fig. 20
Fig. 20

Differential efficiency of a Nd:GGG channel waveguide laser.72

Fig. 21
Fig. 21

Laser emission of an Er:GGG channel waveguide laser.73

Equations (23)

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

Ms(T)=Md(T)-Ma(T),
ӨF,sat(λ, T)=D(λ)Md(T)+A(λ)Ma(T),
E(x, y)H(x, y)exp[i(βz-ωt)],
k02+x1x+y2x1y-xyy1x-yxk02+x2+y1yExEy=β2ExEy,
k02+x2+y1yxy-y1xyx-x1yk02+x1x+y2HxHy=β2HxHy.
div(E)=0,div(μH)=0,
quasi-TE:E=0EyEz,H=HxHyHz
k02+x2+y1yEy=βTE2Ey,
quasi-TM:E=ExEyEz,H=0HyHz
k02+x1x+y2Hy=βTM2Hy.
ˆ=n02100010001+K0Mz-My-Mz0MxMy-Mx0=n02100010001+Δˆ.
ӨF,sat=-k0KMs2n0.
βforward=β+δβ,βbackward=β-δβ,
δβ=ω0NE*ΔˆEdxdy
N=E×H*+E*×Hzdxdy.
ΔβTE=2ω0βTEN(KMx)EyyEydxdy;
ΔβTM=-2βTMω0N(KMy)n04HyxHydxdy.
ΔβTE=2ω0βTEN m(KMx)|Ey2|dxm,v-0-(KMx)|Ey2|dxm,v+0,
ΔβTM=2βTMω0N m (KMy)n04|Hy2|dym,h-0- (KMy)n04|Hy2|dym,h+0.
ATM(z)ATE(z)=exp(iβ¯z)×cos Γz-iΔβ2Γ sin Γz-iκΓ sin Γz-iκ*Γ sin Γzcos Γz+iΔβ2Γ sin Γz×ATM(0)ATE(0),
κ=ω04E*TEΔˆETMdxdy,
|ATM(L)|2|ATE(L)|2=|κ|2|κ|2+Δβ2/4sin2[L(|κ|2+Δβ2/4)1/2].
ACBDA.

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