Abstract

We consider the process of Faraday rotation in femtosecond laser direct-write waveguides. The birefringence commonly associated with such waveguides may be expected to impact the observable Faraday rotation. Here, we theoretically calculate and experimentally verify the competition between Faraday rotation and birefringence in two waveguides created by laser writing in a commercial magneto-optic glass. The magnetic field applied to induce Faraday rotation is nonuniform, and as a result, we find that the two effects can be clearly separated and used to accurately determine even weak birefringence. The birefringence in the waveguides was determined to be on the scale of Δn = 10−6 to 10−5. The reduction in Faraday rotation caused by birefringence of order Δn = 10−6 was moderate and we obtained approximately 9° rotation in an 11 mm waveguide. In contrast, for birefringence of order 10−5, a significant reduction in the polarization azimuth change was found and only 6° rotation was observed.

© 2014 Optical Society of America

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  1. S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
    [Crossref]
  2. Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon. 3, 91–94 (2009).
    [Crossref]
  3. H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 033901 (2012).
    [Crossref] [PubMed]
  4. X. Huang and S. Fan, “Complete all-optical silica fiber isolator via stimulated Brillouin scattering,” J. Lightwave Technol. 29, 2267–2275 (2011).
    [Crossref]
  5. C. G. Poulton, R. Pant, A. Byrnes, S. Fan, M. J. Steel, and B. J. Eggleton, “Design for broadband on-chip isolator using stimulated Brillouin scattering in dispersion-engineered chalcogenide waveguides,” Opt. Express 20, 21235–21246 (2012).
    [Crossref] [PubMed]
  6. J. Fujita, M. Levy, R. U. Ahmad, and R. M. Osgood, “Observation of optical isolation based on nonreciprocal phase shift in a Mach-Zehnder interferometer,” Appl. Phys. Lett. 75, 998–1000 (1999).
    [Crossref]
  7. J. Fujita, M. Levy, R. M. Osgood, L. Wilkens, and H. Dötsch, “Waveguide optical isolator based on Mach–Zehnder interferometer,” Appl. Phys. Lett. 76, 2158–2160 (2000).
    [Crossref]
  8. M. Levy, R. Scarmozzino, R. M. Osgood, R. Wolfe, F. J. Cadieu, H. Hegde, C. J. Gutierrez, and G. A. Prinz, “Permanent magnet film magnetooptic waveguide isolator,” J. App. Phys. 75, 6286–6288 (1994).
    [Crossref]
  9. M. Levy, R. Osgood, H. Hegde, F. Cadieu, R. Wolfe, and V. Fratello, “Integrated optical isolators with sputter-deposited thin-film magnets,” IEEE Photon. Technol. Lett. 8, 903–905 (1996).
    [Crossref]
  10. H. Dötsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, and P. Hertel, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B 22, 240–253 (2005).
    [Crossref]
  11. T. Suzuki, F. Sequeda, H. Do, T. C. Huang, and G. Gorman, “Magnetic and magneto-optic properties of Bi-substituted garnet films crystallized by rapid thermal processing,” J. App. Phys. 67, 4435–4437 (1990).
    [Crossref]
  12. R. Wolfe, R. A. Lieberman, V. J. Fratello, R. E. Scotti, and N. Kopylov, “Etch-tuned ridged waveguide magneto-optic isolator,” Appl. Phys. Lett. 56, 426–428 (1990).
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    [Crossref]
  16. T. Shih, R. Gattass, C. Mendonca, and E. Mazur, “Faraday rotation in femtosecond laser micromachined waveguides,” Opt. Express 15, 5809–5814 (2007).
    [Crossref] [PubMed]
  17. Y. Shimotsuma, P. G. Kazansky, J. R. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultrashort light pulses,” Phys. Rev. Lett. 91, 247405 (2003).
    [Crossref] [PubMed]
  18. Y. Shimotsuma, M. Sakakura, P. G. Kazansky, M. Beresna, J. Qiu, K. Miura, and K. Hirao, “Ultrafast manipulation of self-assembled form birefringence in glass,” Advanced Materials 22, 4039–4043 (2010).
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  20. C. Hnatovsky, R. S. Taylor, E. Simova, P. P. Rajeev, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Fabrication of microchannels in glass using focused femtosecond laser radiation and selective chemical etching,” Appl. Phys. A 84, 47–61 (2006).
    [Crossref]
  21. L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Writing of permanent birefringent microlayers in bulk fused silica with femtosecond laser pulses,” Opt. Comm. 171, 279–284 (1999).
    [Crossref]
  22. G. Corrielli, A. Crespi, R. Geremia, R. Ramponi, L. Sansoni, A. Santinelli, P. Mataloni, F. Sciarrino, and R. Osellame, “Rotated waveplates in integrated waveguide optics,” Nature Comm. 5, 1–6 (2014).
    [Crossref]
  23. B. Poumellec, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Femtosecond laser irradiation stress induced in pure silica,” Opt. Express 11, 1070–1079 (2003).
    [Crossref] [PubMed]
  24. A. M. Streltsov and N. F. Borrelli, “Study of femtosecond-laser-written waveguides in glasses,” J. Opt. Soc. Am. B 19, 2496–2504 (2002).
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  25. V. R. Bhardwaj, P. B. Corkum, D. M. Rayner, C. Hnatovsky, E. Simova, and R. S. Taylor, “Stress in femtosecond-laser-written waveguides in fused silica,” Opt. Lett. 29, 1312–1314 (2004).
    [Crossref] [PubMed]
  26. R. Heilmann, M. Gräfe, S. Nolte, and A. Szameit, “Arbitrary photonic wave plate operations on chip: Realizing Hadamard, Pauli-X, and rotation gates for polarisation qubits,” Sci. Rep. 4, 1–5 (2014).
    [Crossref]
  27. M. Lancry, B. Poumellec, J. Canning, K. Cook, J.-C. Poulin, and F. Brisset, “Ultrafast nanoporous silica formation driven by femtosecond laser irradiation,” Laser Photonics Rev. 7, 953–962 (2013).
    [Crossref]
  28. M. Beresna, M. Gecevicius, M. Lancry, B. Poumellec, and P. G. Kazansky, “Broadband anisotropy of femtosecond laser induced nanogratings in fused silica,” Appl. Phys. Lett. 103, 131903 (2013).
    [Crossref]
  29. F. Zimmermann, A. Plech, S. Richter, A. Tünnermann, and S. Nolte, “Ultrashort laser pulse induced nanogratings in borosilicate glass,” Appl. Phys. Lett. 104, 211107 (2014).
    [Crossref]
  30. W. J. Tabor, A. W. Anderson, and L. G. Van Uitert, “Visible and infrared faraday rotation and birefringence of single-crystal rare-earth orthoferrites,” J. Appl. Phys. 41, 3018–3021 (1970).
    [Crossref]
  31. R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, and L. E. Trimble, “Magneto-optic waveguide isolators based on laser annealed (Bi, Ga) YIG films,” IEEE Trans. Magnetics MAG-21, 1647–1650 (1985).
    [Crossref]
  32. R. Wolfe, V. J. Fratello, and M. McGlashanPowell, “Elimination of birefringence in garnet films for magnetooptic waveguide devices,” Appl. Phys. Lett. 51, 1221–1223 (1987).
    [Crossref]
  33. R. Wolfe, V. J. Fratello, and M. McGlashan-Powell, “Thin-film garnet materials with zero linear birefringence for magneto-optic waveguide devices,” J. App. Phys. 63, 3099–3103 (1988).
    [Crossref]
  34. H. Wen, M. A. Terrel, H. K. Kim, M. J. F. Digonnet, and S. Fan, “Measurements of the birefringence and Verdet constant in an air-core fiber,” J. Lightwave Technol. 27, 3194–3201 (2009).
    [Crossref]
  35. D. J. Rogers, C. J. K. Richardson, J. Goldhar, and C. W. Clark, “Measurement of small birefringence and loss in a nonlinear single-mode waveguide,” Rev. Sci. Inst. 80, 053107 (2009).
    [Crossref]
  36. Q. Liu, B. F. Johnston, S. Gross, M. J. Withford, and M. J. Steel, “A parametric study of laser induced-effects in terbium-doped borosilicate glasses: prospects for compact magneto-optic devices,” Opt. Mater. Express 3, 2096–2111 (2013).
    [Crossref]
  37. http://www.shanghai-optics.com/products/faraday-rotator-glass/ .
  38. Y. Ruan, R. A. Jarvis, A. V. Rode, S. Madden, and B. Luther-Davies, “Wavelength dispersion of Verdet constants in chalcogenide glasses for magneto-optical waveguide devices,” Opt. Comm. 252, 39–45 (2005).
    [Crossref]

2014 (4)

B. J. H. Stadler and T. Mizumoto, “Integrated magneto-optical materials and isolators: a review,” IEEE Photonics J. 6, 0600215 (2014).
[Crossref]

G. Corrielli, A. Crespi, R. Geremia, R. Ramponi, L. Sansoni, A. Santinelli, P. Mataloni, F. Sciarrino, and R. Osellame, “Rotated waveplates in integrated waveguide optics,” Nature Comm. 5, 1–6 (2014).
[Crossref]

R. Heilmann, M. Gräfe, S. Nolte, and A. Szameit, “Arbitrary photonic wave plate operations on chip: Realizing Hadamard, Pauli-X, and rotation gates for polarisation qubits,” Sci. Rep. 4, 1–5 (2014).
[Crossref]

F. Zimmermann, A. Plech, S. Richter, A. Tünnermann, and S. Nolte, “Ultrashort laser pulse induced nanogratings in borosilicate glass,” Appl. Phys. Lett. 104, 211107 (2014).
[Crossref]

2013 (3)

M. Lancry, B. Poumellec, J. Canning, K. Cook, J.-C. Poulin, and F. Brisset, “Ultrafast nanoporous silica formation driven by femtosecond laser irradiation,” Laser Photonics Rev. 7, 953–962 (2013).
[Crossref]

M. Beresna, M. Gecevicius, M. Lancry, B. Poumellec, and P. G. Kazansky, “Broadband anisotropy of femtosecond laser induced nanogratings in fused silica,” Appl. Phys. Lett. 103, 131903 (2013).
[Crossref]

Q. Liu, B. F. Johnston, S. Gross, M. J. Withford, and M. J. Steel, “A parametric study of laser induced-effects in terbium-doped borosilicate glasses: prospects for compact magneto-optic devices,” Opt. Mater. Express 3, 2096–2111 (2013).
[Crossref]

2012 (3)

C. G. Poulton, R. Pant, A. Byrnes, S. Fan, M. J. Steel, and B. J. Eggleton, “Design for broadband on-chip isolator using stimulated Brillouin scattering in dispersion-engineered chalcogenide waveguides,” Opt. Express 20, 21235–21246 (2012).
[Crossref] [PubMed]

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 033901 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (1)

Y. Shimotsuma, M. Sakakura, P. G. Kazansky, M. Beresna, J. Qiu, K. Miura, and K. Hirao, “Ultrafast manipulation of self-assembled form birefringence in glass,” Advanced Materials 22, 4039–4043 (2010).
[Crossref] [PubMed]

2009 (3)

D. J. Rogers, C. J. K. Richardson, J. Goldhar, and C. W. Clark, “Measurement of small birefringence and loss in a nonlinear single-mode waveguide,” Rev. Sci. Inst. 80, 053107 (2009).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon. 3, 91–94 (2009).
[Crossref]

H. Wen, M. A. Terrel, H. K. Kim, M. J. F. Digonnet, and S. Fan, “Measurements of the birefringence and Verdet constant in an air-core fiber,” J. Lightwave Technol. 27, 3194–3201 (2009).
[Crossref]

2008 (2)

R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nature Photon. 2, 219–225 (2008).
[Crossref]

M. Ams, G. Marshall, P. Dekker, M. Dubov, V. Mezentsev, I. Bennion, and M. Withford, “Investigation of ultrafast laser–photonic material interactions: Challenges for directly written glass photonics,” IEEE J. Sel. Top. Quantum Electron. 14, 1370–1381 (2008).
[Crossref]

2007 (1)

2006 (1)

C. Hnatovsky, R. S. Taylor, E. Simova, P. P. Rajeev, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Fabrication of microchannels in glass using focused femtosecond laser radiation and selective chemical etching,” Appl. Phys. A 84, 47–61 (2006).
[Crossref]

2005 (2)

H. Dötsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, and P. Hertel, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B 22, 240–253 (2005).
[Crossref]

Y. Ruan, R. A. Jarvis, A. V. Rode, S. Madden, and B. Luther-Davies, “Wavelength dispersion of Verdet constants in chalcogenide glasses for magneto-optical waveguide devices,” Opt. Comm. 252, 39–45 (2005).
[Crossref]

2004 (1)

2003 (2)

B. Poumellec, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Femtosecond laser irradiation stress induced in pure silica,” Opt. Express 11, 1070–1079 (2003).
[Crossref] [PubMed]

Y. Shimotsuma, P. G. Kazansky, J. R. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultrashort light pulses,” Phys. Rev. Lett. 91, 247405 (2003).
[Crossref] [PubMed]

2002 (1)

2000 (1)

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

1999 (2)

J. Fujita, M. Levy, R. U. Ahmad, and R. M. Osgood, “Observation of optical isolation based on nonreciprocal phase shift in a Mach-Zehnder interferometer,” Appl. Phys. Lett. 75, 998–1000 (1999).
[Crossref]

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Writing of permanent birefringent microlayers in bulk fused silica with femtosecond laser pulses,” Opt. Comm. 171, 279–284 (1999).
[Crossref]

1996 (1)

M. Levy, R. Osgood, H. Hegde, F. Cadieu, R. Wolfe, and V. Fratello, “Integrated optical isolators with sputter-deposited thin-film magnets,” IEEE Photon. Technol. Lett. 8, 903–905 (1996).
[Crossref]

1994 (1)

M. Levy, R. Scarmozzino, R. M. Osgood, R. Wolfe, F. J. Cadieu, H. Hegde, C. J. Gutierrez, and G. A. Prinz, “Permanent magnet film magnetooptic waveguide isolator,” J. App. Phys. 75, 6286–6288 (1994).
[Crossref]

1990 (2)

T. Suzuki, F. Sequeda, H. Do, T. C. Huang, and G. Gorman, “Magnetic and magneto-optic properties of Bi-substituted garnet films crystallized by rapid thermal processing,” J. App. Phys. 67, 4435–4437 (1990).
[Crossref]

R. Wolfe, R. A. Lieberman, V. J. Fratello, R. E. Scotti, and N. Kopylov, “Etch-tuned ridged waveguide magneto-optic isolator,” Appl. Phys. Lett. 56, 426–428 (1990).
[Crossref]

1988 (1)

R. Wolfe, V. J. Fratello, and M. McGlashan-Powell, “Thin-film garnet materials with zero linear birefringence for magneto-optic waveguide devices,” J. App. Phys. 63, 3099–3103 (1988).
[Crossref]

1987 (1)

R. Wolfe, V. J. Fratello, and M. McGlashanPowell, “Elimination of birefringence in garnet films for magnetooptic waveguide devices,” Appl. Phys. Lett. 51, 1221–1223 (1987).
[Crossref]

1985 (1)

R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, and L. E. Trimble, “Magneto-optic waveguide isolators based on laser annealed (Bi, Ga) YIG films,” IEEE Trans. Magnetics MAG-21, 1647–1650 (1985).
[Crossref]

1970 (1)

W. J. Tabor, A. W. Anderson, and L. G. Van Uitert, “Visible and infrared faraday rotation and birefringence of single-crystal rare-earth orthoferrites,” J. Appl. Phys. 41, 3018–3021 (1970).
[Crossref]

Ahmad, R. U.

J. Fujita, M. Levy, R. U. Ahmad, and R. M. Osgood, “Observation of optical isolation based on nonreciprocal phase shift in a Mach-Zehnder interferometer,” Appl. Phys. Lett. 75, 998–1000 (1999).
[Crossref]

Aitchison, J. S.

Ams, M.

M. Ams, G. Marshall, P. Dekker, M. Dubov, V. Mezentsev, I. Bennion, and M. Withford, “Investigation of ultrafast laser–photonic material interactions: Challenges for directly written glass photonics,” IEEE J. Sel. Top. Quantum Electron. 14, 1370–1381 (2008).
[Crossref]

Anderson, A. W.

W. J. Tabor, A. W. Anderson, and L. G. Van Uitert, “Visible and infrared faraday rotation and birefringence of single-crystal rare-earth orthoferrites,” J. Appl. Phys. 41, 3018–3021 (1970).
[Crossref]

Baets, R.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

Bahlmann, N.

Bennion, I.

M. Ams, G. Marshall, P. Dekker, M. Dubov, V. Mezentsev, I. Bennion, and M. Withford, “Investigation of ultrafast laser–photonic material interactions: Challenges for directly written glass photonics,” IEEE J. Sel. Top. Quantum Electron. 14, 1370–1381 (2008).
[Crossref]

Beresna, M.

M. Beresna, M. Gecevicius, M. Lancry, B. Poumellec, and P. G. Kazansky, “Broadband anisotropy of femtosecond laser induced nanogratings in fused silica,” Appl. Phys. Lett. 103, 131903 (2013).
[Crossref]

Y. Shimotsuma, M. Sakakura, P. G. Kazansky, M. Beresna, J. Qiu, K. Miura, and K. Hirao, “Ultrafast manipulation of self-assembled form birefringence in glass,” Advanced Materials 22, 4039–4043 (2010).
[Crossref] [PubMed]

Bhardwaj, V. R.

C. Hnatovsky, R. S. Taylor, E. Simova, P. P. Rajeev, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Fabrication of microchannels in glass using focused femtosecond laser radiation and selective chemical etching,” Appl. Phys. A 84, 47–61 (2006).
[Crossref]

V. R. Bhardwaj, P. B. Corkum, D. M. Rayner, C. Hnatovsky, E. Simova, and R. S. Taylor, “Stress in femtosecond-laser-written waveguides in fused silica,” Opt. Lett. 29, 1312–1314 (2004).
[Crossref] [PubMed]

Borrelli, N. F.

Brinkmeyer, E.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

Brisset, F.

M. Lancry, B. Poumellec, J. Canning, K. Cook, J.-C. Poulin, and F. Brisset, “Ultrafast nanoporous silica formation driven by femtosecond laser irradiation,” Laser Photonics Rev. 7, 953–962 (2013).
[Crossref]

Byrnes, A.

Cadieu, F.

M. Levy, R. Osgood, H. Hegde, F. Cadieu, R. Wolfe, and V. Fratello, “Integrated optical isolators with sputter-deposited thin-film magnets,” IEEE Photon. Technol. Lett. 8, 903–905 (1996).
[Crossref]

Cadieu, F. J.

M. Levy, R. Scarmozzino, R. M. Osgood, R. Wolfe, F. J. Cadieu, H. Hegde, C. J. Gutierrez, and G. A. Prinz, “Permanent magnet film magnetooptic waveguide isolator,” J. App. Phys. 75, 6286–6288 (1994).
[Crossref]

Canning, J.

M. Lancry, B. Poumellec, J. Canning, K. Cook, J.-C. Poulin, and F. Brisset, “Ultrafast nanoporous silica formation driven by femtosecond laser irradiation,” Laser Photonics Rev. 7, 953–962 (2013).
[Crossref]

Celler, G. K.

R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, and L. E. Trimble, “Magneto-optic waveguide isolators based on laser annealed (Bi, Ga) YIG films,” IEEE Trans. Magnetics MAG-21, 1647–1650 (1985).
[Crossref]

Clark, C. W.

D. J. Rogers, C. J. K. Richardson, J. Goldhar, and C. W. Clark, “Measurement of small birefringence and loss in a nonlinear single-mode waveguide,” Rev. Sci. Inst. 80, 053107 (2009).
[Crossref]

Cook, K.

M. Lancry, B. Poumellec, J. Canning, K. Cook, J.-C. Poulin, and F. Brisset, “Ultrafast nanoporous silica formation driven by femtosecond laser irradiation,” Laser Photonics Rev. 7, 953–962 (2013).
[Crossref]

Corkum, P. B.

C. Hnatovsky, R. S. Taylor, E. Simova, P. P. Rajeev, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Fabrication of microchannels in glass using focused femtosecond laser radiation and selective chemical etching,” Appl. Phys. A 84, 47–61 (2006).
[Crossref]

V. R. Bhardwaj, P. B. Corkum, D. M. Rayner, C. Hnatovsky, E. Simova, and R. S. Taylor, “Stress in femtosecond-laser-written waveguides in fused silica,” Opt. Lett. 29, 1312–1314 (2004).
[Crossref] [PubMed]

Corrielli, G.

G. Corrielli, A. Crespi, R. Geremia, R. Ramponi, L. Sansoni, A. Santinelli, P. Mataloni, F. Sciarrino, and R. Osellame, “Rotated waveplates in integrated waveguide optics,” Nature Comm. 5, 1–6 (2014).
[Crossref]

Crespi, A.

G. Corrielli, A. Crespi, R. Geremia, R. Ramponi, L. Sansoni, A. Santinelli, P. Mataloni, F. Sciarrino, and R. Osellame, “Rotated waveplates in integrated waveguide optics,” Nature Comm. 5, 1–6 (2014).
[Crossref]

Dekker, P.

M. Ams, G. Marshall, P. Dekker, M. Dubov, V. Mezentsev, I. Bennion, and M. Withford, “Investigation of ultrafast laser–photonic material interactions: Challenges for directly written glass photonics,” IEEE J. Sel. Top. Quantum Electron. 14, 1370–1381 (2008).
[Crossref]

Digonnet, M. J. F.

Dillon, J. F.

R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, and L. E. Trimble, “Magneto-optic waveguide isolators based on laser annealed (Bi, Ga) YIG films,” IEEE Trans. Magnetics MAG-21, 1647–1650 (1985).
[Crossref]

Do, H.

T. Suzuki, F. Sequeda, H. Do, T. C. Huang, and G. Gorman, “Magnetic and magneto-optic properties of Bi-substituted garnet films crystallized by rapid thermal processing,” J. App. Phys. 67, 4435–4437 (1990).
[Crossref]

Doerr, C. R.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
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Dötsch, H.

H. Dötsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, and P. Hertel, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B 22, 240–253 (2005).
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J. Fujita, M. Levy, R. M. Osgood, L. Wilkens, and H. Dötsch, “Waveguide optical isolator based on Mach–Zehnder interferometer,” Appl. Phys. Lett. 76, 2158–2160 (2000).
[Crossref]

Dubov, M.

M. Ams, G. Marshall, P. Dekker, M. Dubov, V. Mezentsev, I. Bennion, and M. Withford, “Investigation of ultrafast laser–photonic material interactions: Challenges for directly written glass photonics,” IEEE J. Sel. Top. Quantum Electron. 14, 1370–1381 (2008).
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Eggleton, B. J.

Eich, M.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
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Fan, S.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
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C. G. Poulton, R. Pant, A. Byrnes, S. Fan, M. J. Steel, and B. J. Eggleton, “Design for broadband on-chip isolator using stimulated Brillouin scattering in dispersion-engineered chalcogenide waveguides,” Opt. Express 20, 21235–21246 (2012).
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H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 033901 (2012).
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X. Huang and S. Fan, “Complete all-optical silica fiber isolator via stimulated Brillouin scattering,” J. Lightwave Technol. 29, 2267–2275 (2011).
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H. Wen, M. A. Terrel, H. K. Kim, M. J. F. Digonnet, and S. Fan, “Measurements of the birefringence and Verdet constant in an air-core fiber,” J. Lightwave Technol. 27, 3194–3201 (2009).
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Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon. 3, 91–94 (2009).
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Fernandes, L. A.

Franco, M.

B. Poumellec, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Femtosecond laser irradiation stress induced in pure silica,” Opt. Express 11, 1070–1079 (2003).
[Crossref] [PubMed]

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Writing of permanent birefringent microlayers in bulk fused silica with femtosecond laser pulses,” Opt. Comm. 171, 279–284 (1999).
[Crossref]

Fratello, V.

M. Levy, R. Osgood, H. Hegde, F. Cadieu, R. Wolfe, and V. Fratello, “Integrated optical isolators with sputter-deposited thin-film magnets,” IEEE Photon. Technol. Lett. 8, 903–905 (1996).
[Crossref]

Fratello, V. J.

R. Wolfe, R. A. Lieberman, V. J. Fratello, R. E. Scotti, and N. Kopylov, “Etch-tuned ridged waveguide magneto-optic isolator,” Appl. Phys. Lett. 56, 426–428 (1990).
[Crossref]

R. Wolfe, V. J. Fratello, and M. McGlashan-Powell, “Thin-film garnet materials with zero linear birefringence for magneto-optic waveguide devices,” J. App. Phys. 63, 3099–3103 (1988).
[Crossref]

R. Wolfe, V. J. Fratello, and M. McGlashanPowell, “Elimination of birefringence in garnet films for magnetooptic waveguide devices,” Appl. Phys. Lett. 51, 1221–1223 (1987).
[Crossref]

Freude, W.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

Fujita, J.

J. Fujita, M. Levy, R. M. Osgood, 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. U. Ahmad, and R. M. Osgood, “Observation of optical isolation based on nonreciprocal phase shift in a Mach-Zehnder interferometer,” Appl. Phys. Lett. 75, 998–1000 (1999).
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Gattass, R.

R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nature Photon. 2, 219–225 (2008).
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T. Shih, R. Gattass, C. Mendonca, and E. Mazur, “Faraday rotation in femtosecond laser micromachined waveguides,” Opt. Express 15, 5809–5814 (2007).
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Gecevicius, M.

M. Beresna, M. Gecevicius, M. Lancry, B. Poumellec, and P. G. Kazansky, “Broadband anisotropy of femtosecond laser induced nanogratings in fused silica,” Appl. Phys. Lett. 103, 131903 (2013).
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Geremia, R.

G. Corrielli, A. Crespi, R. Geremia, R. Ramponi, L. Sansoni, A. Santinelli, P. Mataloni, F. Sciarrino, and R. Osellame, “Rotated waveplates in integrated waveguide optics,” Nature Comm. 5, 1–6 (2014).
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Gerhardt, R.

Goldhar, J.

D. J. Rogers, C. J. K. Richardson, J. Goldhar, and C. W. Clark, “Measurement of small birefringence and loss in a nonlinear single-mode waveguide,” Rev. Sci. Inst. 80, 053107 (2009).
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Gorman, G.

T. Suzuki, F. Sequeda, H. Do, T. C. Huang, and G. Gorman, “Magnetic and magneto-optic properties of Bi-substituted garnet films crystallized by rapid thermal processing,” J. App. Phys. 67, 4435–4437 (1990).
[Crossref]

Gräfe, M.

R. Heilmann, M. Gräfe, S. Nolte, and A. Szameit, “Arbitrary photonic wave plate operations on chip: Realizing Hadamard, Pauli-X, and rotation gates for polarisation qubits,” Sci. Rep. 4, 1–5 (2014).
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Grenier, J. R.

Gross, S.

Gutierrez, C. J.

M. Levy, R. Scarmozzino, R. M. Osgood, R. Wolfe, F. J. Cadieu, H. Hegde, C. J. Gutierrez, and G. A. Prinz, “Permanent magnet film magnetooptic waveguide isolator,” J. App. Phys. 75, 6286–6288 (1994).
[Crossref]

Hammer, M.

Hegarty, J.

R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, and L. E. Trimble, “Magneto-optic waveguide isolators based on laser annealed (Bi, Ga) YIG films,” IEEE Trans. Magnetics MAG-21, 1647–1650 (1985).
[Crossref]

Hegde, H.

M. Levy, R. Osgood, H. Hegde, F. Cadieu, R. Wolfe, and V. Fratello, “Integrated optical isolators with sputter-deposited thin-film magnets,” IEEE Photon. Technol. Lett. 8, 903–905 (1996).
[Crossref]

M. Levy, R. Scarmozzino, R. M. Osgood, R. Wolfe, F. J. Cadieu, H. Hegde, C. J. Gutierrez, and G. A. Prinz, “Permanent magnet film magnetooptic waveguide isolator,” J. App. Phys. 75, 6286–6288 (1994).
[Crossref]

Heilmann, R.

R. Heilmann, M. Gräfe, S. Nolte, and A. Szameit, “Arbitrary photonic wave plate operations on chip: Realizing Hadamard, Pauli-X, and rotation gates for polarisation qubits,” Sci. Rep. 4, 1–5 (2014).
[Crossref]

Herman, P. R.

Hertel, P.

Hirao, K.

Y. Shimotsuma, M. Sakakura, P. G. Kazansky, M. Beresna, J. Qiu, K. Miura, and K. Hirao, “Ultrafast manipulation of self-assembled form birefringence in glass,” Advanced Materials 22, 4039–4043 (2010).
[Crossref] [PubMed]

Y. Shimotsuma, P. G. Kazansky, J. R. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultrashort light pulses,” Phys. Rev. Lett. 91, 247405 (2003).
[Crossref] [PubMed]

Hnatovsky, C.

C. Hnatovsky, R. S. Taylor, E. Simova, P. P. Rajeev, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Fabrication of microchannels in glass using focused femtosecond laser radiation and selective chemical etching,” Appl. Phys. A 84, 47–61 (2006).
[Crossref]

V. R. Bhardwaj, P. B. Corkum, D. M. Rayner, C. Hnatovsky, E. Simova, and R. S. Taylor, “Stress in femtosecond-laser-written waveguides in fused silica,” Opt. Lett. 29, 1312–1314 (2004).
[Crossref] [PubMed]

Huang, T. C.

T. Suzuki, F. Sequeda, H. Do, T. C. Huang, and G. Gorman, “Magnetic and magneto-optic properties of Bi-substituted garnet films crystallized by rapid thermal processing,” J. App. Phys. 67, 4435–4437 (1990).
[Crossref]

Huang, X.

Jalas, D.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

Jarvis, R. A.

Y. Ruan, R. A. Jarvis, A. V. Rode, S. Madden, and B. Luther-Davies, “Wavelength dispersion of Verdet constants in chalcogenide glasses for magneto-optical waveguide devices,” Opt. Comm. 252, 39–45 (2005).
[Crossref]

Joannopoulos, J. D.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

Johnston, B. F.

Kazansky, P. G.

M. Beresna, M. Gecevicius, M. Lancry, B. Poumellec, and P. G. Kazansky, “Broadband anisotropy of femtosecond laser induced nanogratings in fused silica,” Appl. Phys. Lett. 103, 131903 (2013).
[Crossref]

Y. Shimotsuma, M. Sakakura, P. G. Kazansky, M. Beresna, J. Qiu, K. Miura, and K. Hirao, “Ultrafast manipulation of self-assembled form birefringence in glass,” Advanced Materials 22, 4039–4043 (2010).
[Crossref] [PubMed]

Y. Shimotsuma, P. G. Kazansky, J. R. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultrashort light pulses,” Phys. Rev. Lett. 91, 247405 (2003).
[Crossref] [PubMed]

Kim, H. K.

Kopylov, N.

R. Wolfe, R. A. Lieberman, V. J. Fratello, R. E. Scotti, and N. Kopylov, “Etch-tuned ridged waveguide magneto-optic isolator,” Appl. Phys. Lett. 56, 426–428 (1990).
[Crossref]

Krause, M.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

Lancry, M.

M. Lancry, B. Poumellec, J. Canning, K. Cook, J.-C. Poulin, and F. Brisset, “Ultrafast nanoporous silica formation driven by femtosecond laser irradiation,” Laser Photonics Rev. 7, 953–962 (2013).
[Crossref]

M. Beresna, M. Gecevicius, M. Lancry, B. Poumellec, and P. G. Kazansky, “Broadband anisotropy of femtosecond laser induced nanogratings in fused silica,” Appl. Phys. Lett. 103, 131903 (2013).
[Crossref]

Levy, M.

J. Fujita, M. Levy, R. M. Osgood, 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. U. Ahmad, and R. M. Osgood, “Observation of optical isolation based on nonreciprocal phase shift in a Mach-Zehnder interferometer,” Appl. Phys. Lett. 75, 998–1000 (1999).
[Crossref]

M. Levy, R. Osgood, H. Hegde, F. Cadieu, R. Wolfe, and V. Fratello, “Integrated optical isolators with sputter-deposited thin-film magnets,” IEEE Photon. Technol. Lett. 8, 903–905 (1996).
[Crossref]

M. Levy, R. Scarmozzino, R. M. Osgood, R. Wolfe, F. J. Cadieu, H. Hegde, C. J. Gutierrez, and G. A. Prinz, “Permanent magnet film magnetooptic waveguide isolator,” J. App. Phys. 75, 6286–6288 (1994).
[Crossref]

Lieberman, R. A.

R. Wolfe, R. A. Lieberman, V. J. Fratello, R. E. Scotti, and N. Kopylov, “Etch-tuned ridged waveguide magneto-optic isolator,” Appl. Phys. Lett. 56, 426–428 (1990).
[Crossref]

Lipson, M.

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 033901 (2012).
[Crossref] [PubMed]

Lira, H.

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 033901 (2012).
[Crossref] [PubMed]

Liu, Q.

Luther, L. C.

R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, and L. E. Trimble, “Magneto-optic waveguide isolators based on laser annealed (Bi, Ga) YIG films,” IEEE Trans. Magnetics MAG-21, 1647–1650 (1985).
[Crossref]

Luther-Davies, B.

Y. Ruan, R. A. Jarvis, A. V. Rode, S. Madden, and B. Luther-Davies, “Wavelength dispersion of Verdet constants in chalcogenide glasses for magneto-optical waveguide devices,” Opt. Comm. 252, 39–45 (2005).
[Crossref]

Madden, S.

Y. Ruan, R. A. Jarvis, A. V. Rode, S. Madden, and B. Luther-Davies, “Wavelength dispersion of Verdet constants in chalcogenide glasses for magneto-optical waveguide devices,” Opt. Comm. 252, 39–45 (2005).
[Crossref]

Marques, P. V. S.

Marshall, G.

M. Ams, G. Marshall, P. Dekker, M. Dubov, V. Mezentsev, I. Bennion, and M. Withford, “Investigation of ultrafast laser–photonic material interactions: Challenges for directly written glass photonics,” IEEE J. Sel. Top. Quantum Electron. 14, 1370–1381 (2008).
[Crossref]

Mataloni, P.

G. Corrielli, A. Crespi, R. Geremia, R. Ramponi, L. Sansoni, A. Santinelli, P. Mataloni, F. Sciarrino, and R. Osellame, “Rotated waveplates in integrated waveguide optics,” Nature Comm. 5, 1–6 (2014).
[Crossref]

Mazur, E.

R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nature Photon. 2, 219–225 (2008).
[Crossref]

T. Shih, R. Gattass, C. Mendonca, and E. Mazur, “Faraday rotation in femtosecond laser micromachined waveguides,” Opt. Express 15, 5809–5814 (2007).
[Crossref] [PubMed]

McGlashanPowell, M.

R. Wolfe, V. J. Fratello, and M. McGlashanPowell, “Elimination of birefringence in garnet films for magnetooptic waveguide devices,” Appl. Phys. Lett. 51, 1221–1223 (1987).
[Crossref]

McGlashan-Powell, M.

R. Wolfe, V. J. Fratello, and M. McGlashan-Powell, “Thin-film garnet materials with zero linear birefringence for magneto-optic waveguide devices,” J. App. Phys. 63, 3099–3103 (1988).
[Crossref]

Melloni, A.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

Mendonca, C.

Mezentsev, V.

M. Ams, G. Marshall, P. Dekker, M. Dubov, V. Mezentsev, I. Bennion, and M. Withford, “Investigation of ultrafast laser–photonic material interactions: Challenges for directly written glass photonics,” IEEE J. Sel. Top. Quantum Electron. 14, 1370–1381 (2008).
[Crossref]

Miura, K.

Y. Shimotsuma, M. Sakakura, P. G. Kazansky, M. Beresna, J. Qiu, K. Miura, and K. Hirao, “Ultrafast manipulation of self-assembled form birefringence in glass,” Advanced Materials 22, 4039–4043 (2010).
[Crossref] [PubMed]

Mizumoto, T.

B. J. H. Stadler and T. Mizumoto, “Integrated magneto-optical materials and isolators: a review,” IEEE Photonics J. 6, 0600215 (2014).
[Crossref]

Mysyrowicz, A.

B. Poumellec, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Femtosecond laser irradiation stress induced in pure silica,” Opt. Express 11, 1070–1079 (2003).
[Crossref] [PubMed]

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Writing of permanent birefringent microlayers in bulk fused silica with femtosecond laser pulses,” Opt. Comm. 171, 279–284 (1999).
[Crossref]

Nolte, S.

F. Zimmermann, A. Plech, S. Richter, A. Tünnermann, and S. Nolte, “Ultrashort laser pulse induced nanogratings in borosilicate glass,” Appl. Phys. Lett. 104, 211107 (2014).
[Crossref]

R. Heilmann, M. Gräfe, S. Nolte, and A. Szameit, “Arbitrary photonic wave plate operations on chip: Realizing Hadamard, Pauli-X, and rotation gates for polarisation qubits,” Sci. Rep. 4, 1–5 (2014).
[Crossref]

Osellame, R.

G. Corrielli, A. Crespi, R. Geremia, R. Ramponi, L. Sansoni, A. Santinelli, P. Mataloni, F. Sciarrino, and R. Osellame, “Rotated waveplates in integrated waveguide optics,” Nature Comm. 5, 1–6 (2014).
[Crossref]

Osgood, R.

M. Levy, R. Osgood, H. Hegde, F. Cadieu, R. Wolfe, and V. Fratello, “Integrated optical isolators with sputter-deposited thin-film magnets,” IEEE Photon. Technol. Lett. 8, 903–905 (1996).
[Crossref]

Osgood, R. M.

J. Fujita, M. Levy, R. M. Osgood, 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. U. Ahmad, and R. M. Osgood, “Observation of optical isolation based on nonreciprocal phase shift in a Mach-Zehnder interferometer,” Appl. Phys. Lett. 75, 998–1000 (1999).
[Crossref]

M. Levy, R. Scarmozzino, R. M. Osgood, R. Wolfe, F. J. Cadieu, H. Hegde, C. J. Gutierrez, and G. A. Prinz, “Permanent magnet film magnetooptic waveguide isolator,” J. App. Phys. 75, 6286–6288 (1994).
[Crossref]

Pant, R.

Petrov, A.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

Plech, A.

F. Zimmermann, A. Plech, S. Richter, A. Tünnermann, and S. Nolte, “Ultrashort laser pulse induced nanogratings in borosilicate glass,” Appl. Phys. Lett. 104, 211107 (2014).
[Crossref]

Popovic, M.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

Poulin, J.-C.

M. Lancry, B. Poumellec, J. Canning, K. Cook, J.-C. Poulin, and F. Brisset, “Ultrafast nanoporous silica formation driven by femtosecond laser irradiation,” Laser Photonics Rev. 7, 953–962 (2013).
[Crossref]

Poulton, C. G.

Poumellec, B.

M. Beresna, M. Gecevicius, M. Lancry, B. Poumellec, and P. G. Kazansky, “Broadband anisotropy of femtosecond laser induced nanogratings in fused silica,” Appl. Phys. Lett. 103, 131903 (2013).
[Crossref]

M. Lancry, B. Poumellec, J. Canning, K. Cook, J.-C. Poulin, and F. Brisset, “Ultrafast nanoporous silica formation driven by femtosecond laser irradiation,” Laser Photonics Rev. 7, 953–962 (2013).
[Crossref]

B. Poumellec, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Femtosecond laser irradiation stress induced in pure silica,” Opt. Express 11, 1070–1079 (2003).
[Crossref] [PubMed]

Prade, B.

B. Poumellec, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Femtosecond laser irradiation stress induced in pure silica,” Opt. Express 11, 1070–1079 (2003).
[Crossref] [PubMed]

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Writing of permanent birefringent microlayers in bulk fused silica with femtosecond laser pulses,” Opt. Comm. 171, 279–284 (1999).
[Crossref]

Prinz, G. A.

M. Levy, R. Scarmozzino, R. M. Osgood, R. Wolfe, F. J. Cadieu, H. Hegde, C. J. Gutierrez, and G. A. Prinz, “Permanent magnet film magnetooptic waveguide isolator,” J. App. Phys. 75, 6286–6288 (1994).
[Crossref]

Qiu, J.

Y. Shimotsuma, M. Sakakura, P. G. Kazansky, M. Beresna, J. Qiu, K. Miura, and K. Hirao, “Ultrafast manipulation of self-assembled form birefringence in glass,” Advanced Materials 22, 4039–4043 (2010).
[Crossref] [PubMed]

Qiu, J. R.

Y. Shimotsuma, P. G. Kazansky, J. R. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultrashort light pulses,” Phys. Rev. Lett. 91, 247405 (2003).
[Crossref] [PubMed]

Rajeev, P. P.

C. Hnatovsky, R. S. Taylor, E. Simova, P. P. Rajeev, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Fabrication of microchannels in glass using focused femtosecond laser radiation and selective chemical etching,” Appl. Phys. A 84, 47–61 (2006).
[Crossref]

Ramponi, R.

G. Corrielli, A. Crespi, R. Geremia, R. Ramponi, L. Sansoni, A. Santinelli, P. Mataloni, F. Sciarrino, and R. Osellame, “Rotated waveplates in integrated waveguide optics,” Nature Comm. 5, 1–6 (2014).
[Crossref]

Rayner, D. M.

C. Hnatovsky, R. S. Taylor, E. Simova, P. P. Rajeev, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Fabrication of microchannels in glass using focused femtosecond laser radiation and selective chemical etching,” Appl. Phys. A 84, 47–61 (2006).
[Crossref]

V. R. Bhardwaj, P. B. Corkum, D. M. Rayner, C. Hnatovsky, E. Simova, and R. S. Taylor, “Stress in femtosecond-laser-written waveguides in fused silica,” Opt. Lett. 29, 1312–1314 (2004).
[Crossref] [PubMed]

Renner, H.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

Richardson, C. J. K.

D. J. Rogers, C. J. K. Richardson, J. Goldhar, and C. W. Clark, “Measurement of small birefringence and loss in a nonlinear single-mode waveguide,” Rev. Sci. Inst. 80, 053107 (2009).
[Crossref]

Richter, S.

F. Zimmermann, A. Plech, S. Richter, A. Tünnermann, and S. Nolte, “Ultrashort laser pulse induced nanogratings in borosilicate glass,” Appl. Phys. Lett. 104, 211107 (2014).
[Crossref]

Rode, A. V.

Y. Ruan, R. A. Jarvis, A. V. Rode, S. Madden, and B. Luther-Davies, “Wavelength dispersion of Verdet constants in chalcogenide glasses for magneto-optical waveguide devices,” Opt. Comm. 252, 39–45 (2005).
[Crossref]

Rogers, D. J.

D. J. Rogers, C. J. K. Richardson, J. Goldhar, and C. W. Clark, “Measurement of small birefringence and loss in a nonlinear single-mode waveguide,” Rev. Sci. Inst. 80, 053107 (2009).
[Crossref]

Ruan, Y.

Y. Ruan, R. A. Jarvis, A. V. Rode, S. Madden, and B. Luther-Davies, “Wavelength dispersion of Verdet constants in chalcogenide glasses for magneto-optical waveguide devices,” Opt. Comm. 252, 39–45 (2005).
[Crossref]

Sakakura, M.

Y. Shimotsuma, M. Sakakura, P. G. Kazansky, M. Beresna, J. Qiu, K. Miura, and K. Hirao, “Ultrafast manipulation of self-assembled form birefringence in glass,” Advanced Materials 22, 4039–4043 (2010).
[Crossref] [PubMed]

Sansoni, L.

G. Corrielli, A. Crespi, R. Geremia, R. Ramponi, L. Sansoni, A. Santinelli, P. Mataloni, F. Sciarrino, and R. Osellame, “Rotated waveplates in integrated waveguide optics,” Nature Comm. 5, 1–6 (2014).
[Crossref]

Santinelli, A.

G. Corrielli, A. Crespi, R. Geremia, R. Ramponi, L. Sansoni, A. Santinelli, P. Mataloni, F. Sciarrino, and R. Osellame, “Rotated waveplates in integrated waveguide optics,” Nature Comm. 5, 1–6 (2014).
[Crossref]

Scarmozzino, R.

M. Levy, R. Scarmozzino, R. M. Osgood, R. Wolfe, F. J. Cadieu, H. Hegde, C. J. Gutierrez, and G. A. Prinz, “Permanent magnet film magnetooptic waveguide isolator,” J. App. Phys. 75, 6286–6288 (1994).
[Crossref]

Sciarrino, F.

G. Corrielli, A. Crespi, R. Geremia, R. Ramponi, L. Sansoni, A. Santinelli, P. Mataloni, F. Sciarrino, and R. Osellame, “Rotated waveplates in integrated waveguide optics,” Nature Comm. 5, 1–6 (2014).
[Crossref]

Scotti, R. E.

R. Wolfe, R. A. Lieberman, V. J. Fratello, R. E. Scotti, and N. Kopylov, “Etch-tuned ridged waveguide magneto-optic isolator,” Appl. Phys. Lett. 56, 426–428 (1990).
[Crossref]

Sequeda, F.

T. Suzuki, F. Sequeda, H. Do, T. C. Huang, and G. Gorman, “Magnetic and magneto-optic properties of Bi-substituted garnet films crystallized by rapid thermal processing,” J. App. Phys. 67, 4435–4437 (1990).
[Crossref]

Shih, T.

Shimotsuma, Y.

Y. Shimotsuma, M. Sakakura, P. G. Kazansky, M. Beresna, J. Qiu, K. Miura, and K. Hirao, “Ultrafast manipulation of self-assembled form birefringence in glass,” Advanced Materials 22, 4039–4043 (2010).
[Crossref] [PubMed]

Y. Shimotsuma, P. G. Kazansky, J. R. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultrashort light pulses,” Phys. Rev. Lett. 91, 247405 (2003).
[Crossref] [PubMed]

Simova, E.

C. Hnatovsky, R. S. Taylor, E. Simova, P. P. Rajeev, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Fabrication of microchannels in glass using focused femtosecond laser radiation and selective chemical etching,” Appl. Phys. A 84, 47–61 (2006).
[Crossref]

V. R. Bhardwaj, P. B. Corkum, D. M. Rayner, C. Hnatovsky, E. Simova, and R. S. Taylor, “Stress in femtosecond-laser-written waveguides in fused silica,” Opt. Lett. 29, 1312–1314 (2004).
[Crossref] [PubMed]

Stadler, B. J. H.

B. J. H. Stadler and T. Mizumoto, “Integrated magneto-optical materials and isolators: a review,” IEEE Photonics J. 6, 0600215 (2014).
[Crossref]

Steel, M. J.

Streltsov, A. M.

Sudrie, L.

B. Poumellec, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Femtosecond laser irradiation stress induced in pure silica,” Opt. Express 11, 1070–1079 (2003).
[Crossref] [PubMed]

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Writing of permanent birefringent microlayers in bulk fused silica with femtosecond laser pulses,” Opt. Comm. 171, 279–284 (1999).
[Crossref]

Suzuki, T.

T. Suzuki, F. Sequeda, H. Do, T. C. Huang, and G. Gorman, “Magnetic and magneto-optic properties of Bi-substituted garnet films crystallized by rapid thermal processing,” J. App. Phys. 67, 4435–4437 (1990).
[Crossref]

Szameit, A.

R. Heilmann, M. Gräfe, S. Nolte, and A. Szameit, “Arbitrary photonic wave plate operations on chip: Realizing Hadamard, Pauli-X, and rotation gates for polarisation qubits,” Sci. Rep. 4, 1–5 (2014).
[Crossref]

Tabor, W. J.

W. J. Tabor, A. W. Anderson, and L. G. Van Uitert, “Visible and infrared faraday rotation and birefringence of single-crystal rare-earth orthoferrites,” J. Appl. Phys. 41, 3018–3021 (1970).
[Crossref]

Taylor, R. S.

C. Hnatovsky, R. S. Taylor, E. Simova, P. P. Rajeev, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Fabrication of microchannels in glass using focused femtosecond laser radiation and selective chemical etching,” Appl. Phys. A 84, 47–61 (2006).
[Crossref]

V. R. Bhardwaj, P. B. Corkum, D. M. Rayner, C. Hnatovsky, E. Simova, and R. S. Taylor, “Stress in femtosecond-laser-written waveguides in fused silica,” Opt. Lett. 29, 1312–1314 (2004).
[Crossref] [PubMed]

Terrel, M. A.

Trimble, L. E.

R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, and L. E. Trimble, “Magneto-optic waveguide isolators based on laser annealed (Bi, Ga) YIG films,” IEEE Trans. Magnetics MAG-21, 1647–1650 (1985).
[Crossref]

Tünnermann, A.

F. Zimmermann, A. Plech, S. Richter, A. Tünnermann, and S. Nolte, “Ultrashort laser pulse induced nanogratings in borosilicate glass,” Appl. Phys. Lett. 104, 211107 (2014).
[Crossref]

Van Uitert, L. G.

W. J. Tabor, A. W. Anderson, and L. G. Van Uitert, “Visible and infrared faraday rotation and birefringence of single-crystal rare-earth orthoferrites,” J. Appl. Phys. 41, 3018–3021 (1970).
[Crossref]

Vanwolleghem, M.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

Wen, H.

Wilkens, L.

H. Dötsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, and P. Hertel, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B 22, 240–253 (2005).
[Crossref]

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

Withford, M.

M. Ams, G. Marshall, P. Dekker, M. Dubov, V. Mezentsev, I. Bennion, and M. Withford, “Investigation of ultrafast laser–photonic material interactions: Challenges for directly written glass photonics,” IEEE J. Sel. Top. Quantum Electron. 14, 1370–1381 (2008).
[Crossref]

Withford, M. J.

Wolfe, R.

M. Levy, R. Osgood, H. Hegde, F. Cadieu, R. Wolfe, and V. Fratello, “Integrated optical isolators with sputter-deposited thin-film magnets,” IEEE Photon. Technol. Lett. 8, 903–905 (1996).
[Crossref]

M. Levy, R. Scarmozzino, R. M. Osgood, R. Wolfe, F. J. Cadieu, H. Hegde, C. J. Gutierrez, and G. A. Prinz, “Permanent magnet film magnetooptic waveguide isolator,” J. App. Phys. 75, 6286–6288 (1994).
[Crossref]

R. Wolfe, R. A. Lieberman, V. J. Fratello, R. E. Scotti, and N. Kopylov, “Etch-tuned ridged waveguide magneto-optic isolator,” Appl. Phys. Lett. 56, 426–428 (1990).
[Crossref]

R. Wolfe, V. J. Fratello, and M. McGlashan-Powell, “Thin-film garnet materials with zero linear birefringence for magneto-optic waveguide devices,” J. App. Phys. 63, 3099–3103 (1988).
[Crossref]

R. Wolfe, V. J. Fratello, and M. McGlashanPowell, “Elimination of birefringence in garnet films for magnetooptic waveguide devices,” Appl. Phys. Lett. 51, 1221–1223 (1987).
[Crossref]

R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, and L. E. Trimble, “Magneto-optic waveguide isolators based on laser annealed (Bi, Ga) YIG films,” IEEE Trans. Magnetics MAG-21, 1647–1650 (1985).
[Crossref]

Yu, Z.

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 033901 (2012).
[Crossref] [PubMed]

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon. 3, 91–94 (2009).
[Crossref]

Zhuromskyy, O.

Zimmermann, F.

F. Zimmermann, A. Plech, S. Richter, A. Tünnermann, and S. Nolte, “Ultrashort laser pulse induced nanogratings in borosilicate glass,” Appl. Phys. Lett. 104, 211107 (2014).
[Crossref]

Advanced Materials (1)

Y. Shimotsuma, M. Sakakura, P. G. Kazansky, M. Beresna, J. Qiu, K. Miura, and K. Hirao, “Ultrafast manipulation of self-assembled form birefringence in glass,” Advanced Materials 22, 4039–4043 (2010).
[Crossref] [PubMed]

Appl. Phys. A (1)

C. Hnatovsky, R. S. Taylor, E. Simova, P. P. Rajeev, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Fabrication of microchannels in glass using focused femtosecond laser radiation and selective chemical etching,” Appl. Phys. A 84, 47–61 (2006).
[Crossref]

Appl. Phys. Lett. (6)

M. Beresna, M. Gecevicius, M. Lancry, B. Poumellec, and P. G. Kazansky, “Broadband anisotropy of femtosecond laser induced nanogratings in fused silica,” Appl. Phys. Lett. 103, 131903 (2013).
[Crossref]

F. Zimmermann, A. Plech, S. Richter, A. Tünnermann, and S. Nolte, “Ultrashort laser pulse induced nanogratings in borosilicate glass,” Appl. Phys. Lett. 104, 211107 (2014).
[Crossref]

J. Fujita, M. Levy, R. U. Ahmad, and R. M. Osgood, “Observation of optical isolation based on nonreciprocal phase shift in a Mach-Zehnder interferometer,” Appl. Phys. Lett. 75, 998–1000 (1999).
[Crossref]

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

R. Wolfe, R. A. Lieberman, V. J. Fratello, R. E. Scotti, and N. Kopylov, “Etch-tuned ridged waveguide magneto-optic isolator,” Appl. Phys. Lett. 56, 426–428 (1990).
[Crossref]

R. Wolfe, V. J. Fratello, and M. McGlashanPowell, “Elimination of birefringence in garnet films for magnetooptic waveguide devices,” Appl. Phys. Lett. 51, 1221–1223 (1987).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

M. Ams, G. Marshall, P. Dekker, M. Dubov, V. Mezentsev, I. Bennion, and M. Withford, “Investigation of ultrafast laser–photonic material interactions: Challenges for directly written glass photonics,” IEEE J. Sel. Top. Quantum Electron. 14, 1370–1381 (2008).
[Crossref]

IEEE Photon. Technol. Lett. (1)

M. Levy, R. Osgood, H. Hegde, F. Cadieu, R. Wolfe, and V. Fratello, “Integrated optical isolators with sputter-deposited thin-film magnets,” IEEE Photon. Technol. Lett. 8, 903–905 (1996).
[Crossref]

IEEE Photonics J. (1)

B. J. H. Stadler and T. Mizumoto, “Integrated magneto-optical materials and isolators: a review,” IEEE Photonics J. 6, 0600215 (2014).
[Crossref]

IEEE Trans. Magnetics (1)

R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, and L. E. Trimble, “Magneto-optic waveguide isolators based on laser annealed (Bi, Ga) YIG films,” IEEE Trans. Magnetics MAG-21, 1647–1650 (1985).
[Crossref]

J. App. Phys. (3)

R. Wolfe, V. J. Fratello, and M. McGlashan-Powell, “Thin-film garnet materials with zero linear birefringence for magneto-optic waveguide devices,” J. App. Phys. 63, 3099–3103 (1988).
[Crossref]

T. Suzuki, F. Sequeda, H. Do, T. C. Huang, and G. Gorman, “Magnetic and magneto-optic properties of Bi-substituted garnet films crystallized by rapid thermal processing,” J. App. Phys. 67, 4435–4437 (1990).
[Crossref]

M. Levy, R. Scarmozzino, R. M. Osgood, R. Wolfe, F. J. Cadieu, H. Hegde, C. J. Gutierrez, and G. A. Prinz, “Permanent magnet film magnetooptic waveguide isolator,” J. App. Phys. 75, 6286–6288 (1994).
[Crossref]

J. Appl. Phys. (1)

W. J. Tabor, A. W. Anderson, and L. G. Van Uitert, “Visible and infrared faraday rotation and birefringence of single-crystal rare-earth orthoferrites,” J. Appl. Phys. 41, 3018–3021 (1970).
[Crossref]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. B (2)

Laser Photonics Rev. (1)

M. Lancry, B. Poumellec, J. Canning, K. Cook, J.-C. Poulin, and F. Brisset, “Ultrafast nanoporous silica formation driven by femtosecond laser irradiation,” Laser Photonics Rev. 7, 953–962 (2013).
[Crossref]

Nature Comm. (1)

G. Corrielli, A. Crespi, R. Geremia, R. Ramponi, L. Sansoni, A. Santinelli, P. Mataloni, F. Sciarrino, and R. Osellame, “Rotated waveplates in integrated waveguide optics,” Nature Comm. 5, 1–6 (2014).
[Crossref]

Nature Photon. (2)

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon. 3, 91–94 (2009).
[Crossref]

R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nature Photon. 2, 219–225 (2008).
[Crossref]

Opt. Comm. (2)

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Writing of permanent birefringent microlayers in bulk fused silica with femtosecond laser pulses,” Opt. Comm. 171, 279–284 (1999).
[Crossref]

Y. Ruan, R. A. Jarvis, A. V. Rode, S. Madden, and B. Luther-Davies, “Wavelength dispersion of Verdet constants in chalcogenide glasses for magneto-optical waveguide devices,” Opt. Comm. 252, 39–45 (2005).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Opt. Mater. Express (1)

Phys. Rev. Lett. (2)

Y. Shimotsuma, P. G. Kazansky, J. R. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultrashort light pulses,” Phys. Rev. Lett. 91, 247405 (2003).
[Crossref] [PubMed]

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 033901 (2012).
[Crossref] [PubMed]

Rev. Sci. Inst. (1)

D. J. Rogers, C. J. K. Richardson, J. Goldhar, and C. W. Clark, “Measurement of small birefringence and loss in a nonlinear single-mode waveguide,” Rev. Sci. Inst. 80, 053107 (2009).
[Crossref]

Sci. Rep. (1)

R. Heilmann, M. Gräfe, S. Nolte, and A. Szameit, “Arbitrary photonic wave plate operations on chip: Realizing Hadamard, Pauli-X, and rotation gates for polarisation qubits,” Sci. Rep. 4, 1–5 (2014).
[Crossref]

Science (1)

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

Other (1)

http://www.shanghai-optics.com/products/faraday-rotator-glass/ .

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Figures (7)

Fig. 1
Fig. 1 Polarization evolution in waveguides with Faraday rotation and birefringence on the Poincaré sphere for various values of γ. All curves have an initial polarization state of |↑〉 ≡ |ŷ〉 ≡ (0, 1).

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Fig. 2
Fig. 2 Schematic diagram of experimental setup for polarization examination. The orange arrow in the figure indicates the translation direction of the magnets. GL is a Glan-Laser polarizer and AL the aspheric lens.

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Fig. 3
Fig. 3 Magnetic field and the relative positions of the waveguide in the experiment. The orange line indicates the position when the entire waveguide passed the sign reversal point of the magnetic field. The gray box gives a reference for the length and position of the magnets. Note that in practice, the magnets were moved and the waveguide remained fixed.

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Fig. 4
Fig. 4 Illustrative calculations of polarization evolution within a waveguide (color curves) and output states (crosses) for discrete steps in the magnetic field position. Left: zoom-in view. Right: overview of the Poincaré sphere. The magnetic field and Verdet constant are those for the experiments and the waveguide birefringence is set at Δn = 2.0 × 10−5. The initial launch state of linear vertical polarization is marked with the red star.

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Fig. 5
Fig. 5 Polarization changes due to the Faraday effect and different levels of birefringence. The red star (S2 = 0, S3 = 0) indicates the launch polarization, which is vertically-oriented linear. Squares mark the start point for each curve and corresponding to zm = 0 in Fig. 3. Triangles correspond to the situation marked by the orange line in Fig. 3 when the entire waveguide passed the reversal point of the magnetic field.

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Fig. 6
Fig. 6 Polarization variations in waveguides and unmodified glass on a Poincaré sphere. Experimental data have uncertainties of ±0.008 on each axis.

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Fig. 7
Fig. 7 Experimental results of Faraday rotation in the waveguides.

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Equations (22)

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

Δ n 2 π λ L < 2 π .
∇ 2 E + k 2 ε ¯ E = 0 ,
ε ¯ = [ ε x x i α ( z ) 0 − i α ( z ) ε y y 0 0 0 ε z z ] ,
α ( z ) = V B ( z ) n λ π ,
E = 1 2 [ A f ( x , y ) e i ( β z − ω t ) + c . c . ] ,
[ − β 2 + ∇ t 2 + k 2 ε ( x , y ) ] f = 0 .
E = 1 2 [ ( A x ( z ) f x e i β x z + A y ( z ) f y e i β y z ) e − i ω t + c . c . ] ,
A ( z ) = [ A x ( z ) A y ( z ) ] .
[ ∂ 2 ∂ z 2 + k 2 ε ¯ eff ] A ( z ) = 0 ,
ε ¯ eff = [ n ¯ x 2 i α ( z ) − i α ( z ) n ¯ y 2 ] ,
ε ¯ eff = η ¯ η ¯ ,
( ∂ ∂ z + i k η ¯ ) ( ∂ ∂ z − i k η ¯ ) E = 0 .
∂ ∂ z E = i k η ¯ E .
S 1 = | E x | 2 − | E y | 2 S 2 = E x E y * + E x * E y S 3 = i ( E x E y * − E x * E y )
Ω = 2 k [ Δ η η x y r η x y i ] ,
d d z S = Ω × S ,
Ω = k ( ε + + α γ 2 + 1 − ε + − α γ 2 + 1 ) γ 2 + 1 [ γ 0 1 ] ,
Δ n = V B λ γ 2 π .
B ring ( z ) = μ 0 M 2 [ L m 2 − z b 2 + ( z − L m 2 ) 2 + L m 2 + z b 2 + ( z + L m 2 ) 2 − L m 2 − z a 2 + ( z − L m 2 ) 2 − L m 2 + z a 2 + ( z + L m 2 ) 2 ] ,
E 0 WG 1 = [ cos ϕ 1 sin ϕ 1     e i δ 1 ] = [ cos ( 84.2 ° ) sin ( 84.2 ° )     e i 0.66 π ] ,
E 0 WG 2 = [ cos ϕ 2 sin ϕ 2     e i δ 2 ] = [ cos ( 84.3 ° ) sin ( 84.3 ° )     e i 0.76 π ] ,
θ = V ∫ 0 L glass B ( z ) d z .

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