Abstract

We present a rigorous derivation of the weak-form formulation of the Helmholtz equation for electromagnetic structures incorporating general nonreciprocal, dispersive materials such as magnetized ferrites or magnetized free-carrier plasmas. This formulation allows an efficient self-consistent treatment using finite elements of a variety of problems involving magnetic or magneto-optical materials biased by an external DC field where the eigenmodes become nonreciprocal or even unidirectional. The possibilities of this method are illustrated with several examples of TE-polarized modes at microwave frequencies and TM-polarized modes at optical and infrared wavelengths.

© 2017 Optical Society of America

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  16. L. Halagačka, K. Postava, M. Vanwolleghem, F. Vaurette, J. Ben Youssef, B. Dagens, and J. Pištora, “Mueller matrix optical and magneto-optical characterization of Bi-substituted gadolinium iron garnet for application in magnetoplasmonic structures,” Opt. Mater. Express 4(9), 1903–1919 (2014).
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2016 (1)

J. Chochol, K. Postava, M. Čada, M. Vanwolleghem, L. Halagačka, J.-F. Lampin, and J. Pištora, “Magneto-optical properties of InSb for terahertz applications,” AIP Adv. 6(11), 115021 (2016).
[Crossref]

2014 (4)

M. Mruczkiewicz and M. Krawczyk, “Nonreciprocal dispersion of spin waves in ferromagnetic thin films covered with a finite-conductivity metal,” J. Appl. Phys. 115(11), 113909 (2014).
[Crossref]

L. Halagačka, K. Postava, M. Vanwolleghem, F. Vaurette, J. Ben Youssef, B. Dagens, and J. Pištora, “Mueller matrix optical and magneto-optical characterization of Bi-substituted gadolinium iron garnet for application in magnetoplasmonic structures,” Opt. Mater. Express 4(9), 1903–1919 (2014).
[Crossref]

A. Degiron and D. R. Smith, “One-way glass for microwaves using nonreciprocal metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(5), 053203 (2014).
[Crossref] [PubMed]

L. Lu, J. D. Joannopoulos, and M. Soljacic, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

2013 (3)

2012 (2)

V. Kuzmiak, S. Eyderman, and M. Vanwolleghem, “Controlling surface plasmon polaritons by a static and/or time-dependent external magnetic field,” Phys. Rev. B 86(4), 045403 (2012).
[Crossref]

W. Śmigaj, L. Magdenko, J. Romero-Vivas, S. Guenneau, B. Dagens, B. Gralak, and M. Vanwolleghem, “Compact optical circulator based on a uniformly magnetized ring cavity,” Photonics Nano. Fund. Appl. 10(1), 83–101 (2012).
[Crossref]

2011 (2)

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
[Crossref] [PubMed]

2009 (1)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

2008 (3)

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100(2), 023902 (2008).
[Crossref] [PubMed]

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[Crossref]

2007 (2)

2004 (1)

I. Žutić, J. Fabian, and S. Das Sarma, “Spintronics: Fundamentals and applications,” Rev. Mod. Phys. 76(2), 323–410 (2004).
[Crossref]

1987 (1)

R. E. Camley, “Nonreciprocal surface waves,” Surf. Sci. Rep. 7(3-4), 103–187 (1987).
[Crossref]

1980 (1)

K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

1974 (1)

T. J. Gerson and J. S. Nadan, “Surface Electromagnetic Modes of a Ferrite Slab,” IEEE Trans. Microw. Theory Tech. 22(8), 757–763 (1974).
[Crossref]

1973 (1)

P. de Santis, “Dispersion Characteristics for a Ferrimagnetic Plate,” Appl. Phys. (Berl.) 2(4), 197–200 (1973).
[Crossref]

1972 (1)

L. Courtois, G. Declercq, and M. Peurichard, “On the non-reciprocal aspect of gyromagnetic surface waves,” AIP Conf. Proc. 5, 1541–1545 (1972).
[Crossref]

1970 (1)

E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys. 33(3), 1193–1322 (1970).
[Crossref]

1961 (1)

R. W. Damon and J. R. Eshbach, “Magnetostatic modes of a ferromagnet slab,” J. Phys. Chem. Solids 19(3-4), 308–320 (1961).
[Crossref]

1931 (1)

L. Onsager, “Irreversible processes,” Phys. Rev. 37(4), 237–241 (1931).
[Crossref]

1930 (1)

L. Landau, “Diamagnetismus der Metalle,” Z. Phys. 64(9-10), 629–637 (1930).
[Crossref]

Akimov, I. A.

V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
[Crossref] [PubMed]

Atwater, H. A.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007).
[Crossref] [PubMed]

Baets, R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “what is – and what is not – an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

S. Ghosh, S. Keyvaninia, W. Van Roy, T. Mizumoto, G. Roelkens, and R. Baets, “Adhesively bonded Ce:YIG/SOI integrated optical circulator,” Opt. Lett. 38(6), 965–967 (2013).
[Crossref] [PubMed]

Bayer, M.

V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
[Crossref] [PubMed]

Belotelov, V. I.

V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
[Crossref] [PubMed]

Ben Youssef, J.

Bi, L.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

Cada, M.

J. Chochol, K. Postava, M. Čada, M. Vanwolleghem, L. Halagačka, J.-F. Lampin, and J. Pištora, “Magneto-optical properties of InSb for terahertz applications,” AIP Adv. 6(11), 115021 (2016).
[Crossref]

Camley, R. E.

R. E. Camley, “Nonreciprocal surface waves,” Surf. Sci. Rep. 7(3-4), 103–187 (1987).
[Crossref]

Chochol, J.

J. Chochol, K. Postava, M. Čada, M. Vanwolleghem, L. Halagačka, J.-F. Lampin, and J. Pištora, “Magneto-optical properties of InSb for terahertz applications,” AIP Adv. 6(11), 115021 (2016).
[Crossref]

Chong, Y.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

Courtois, L.

L. Courtois, G. Declercq, and M. Peurichard, “On the non-reciprocal aspect of gyromagnetic surface waves,” AIP Conf. Proc. 5, 1541–1545 (1972).
[Crossref]

Dagens, B.

Damon, R. W.

R. W. Damon and J. R. Eshbach, “Magnetostatic modes of a ferromagnet slab,” J. Phys. Chem. Solids 19(3-4), 308–320 (1961).
[Crossref]

Das Sarma, S.

I. Žutić, J. Fabian, and S. Das Sarma, “Spintronics: Fundamentals and applications,” Rev. Mod. Phys. 76(2), 323–410 (2004).
[Crossref]

Davanco, M.

de Santis, P.

P. de Santis, “Dispersion Characteristics for a Ferrimagnetic Plate,” Appl. Phys. (Berl.) 2(4), 197–200 (1973).
[Crossref]

Declercq, G.

L. Courtois, G. Declercq, and M. Peurichard, “On the non-reciprocal aspect of gyromagnetic surface waves,” AIP Conf. Proc. 5, 1541–1545 (1972).
[Crossref]

Degiron, A.

A. Degiron and D. R. Smith, “One-way glass for microwaves using nonreciprocal metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(5), 053203 (2014).
[Crossref] [PubMed]

Dionne, G. F.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

Dionne, J. A.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007).
[Crossref] [PubMed]

Doerr, C. R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “what is – and what is not – an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Dorda, G.

K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

Eich, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “what is – and what is not – an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Eshbach, J. R.

R. W. Damon and J. R. Eshbach, “Magnetostatic modes of a ferromagnet slab,” J. Phys. Chem. Solids 19(3-4), 308–320 (1961).
[Crossref]

Eyderman, S.

V. Kuzmiak, S. Eyderman, and M. Vanwolleghem, “Controlling surface plasmon polaritons by a static and/or time-dependent external magnetic field,” Phys. Rev. B 86(4), 045403 (2012).
[Crossref]

Fabian, J.

I. Žutić, J. Fabian, and S. Das Sarma, “Spintronics: Fundamentals and applications,” Rev. Mod. Phys. 76(2), 323–410 (2004).
[Crossref]

Fan, S.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “what is – and what is not – an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100(2), 023902 (2008).
[Crossref] [PubMed]

Freude, W.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “what is – and what is not – an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Furdyna, J. K.

E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys. 33(3), 1193–1322 (1970).
[Crossref]

Gerson, T. J.

T. J. Gerson and J. S. Nadan, “Surface Electromagnetic Modes of a Ferrite Slab,” IEEE Trans. Microw. Theory Tech. 22(8), 757–763 (1974).
[Crossref]

Ghosh, S.

Gopal, A. V.

V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
[Crossref] [PubMed]

Gralak, B.

W. Śmigaj, L. Magdenko, J. Romero-Vivas, S. Guenneau, B. Dagens, B. Gralak, and M. Vanwolleghem, “Compact optical circulator based on a uniformly magnetized ring cavity,” Photonics Nano. Fund. Appl. 10(1), 83–101 (2012).
[Crossref]

Guenneau, S.

W. Śmigaj, L. Magdenko, J. Romero-Vivas, S. Guenneau, B. Dagens, B. Gralak, and M. Vanwolleghem, “Compact optical circulator based on a uniformly magnetized ring cavity,” Photonics Nano. Fund. Appl. 10(1), 83–101 (2012).
[Crossref]

Halagacka, L.

Haldane, F. D. M.

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[Crossref]

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

Hu, J.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

Jalas, D.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “what is – and what is not – an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Jiang, P.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

Joannopoulos, J. D.

L. Lu, J. D. Joannopoulos, and M. Soljacic, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “what is – and what is not – an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

Kasture, S.

V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
[Crossref] [PubMed]

Keyvaninia, S.

Kim, D. H.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

Kimerling, L. C.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

Klitzing, K. V.

K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

Kotov, V. A.

V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
[Crossref] [PubMed]

Krawczyk, M.

M. Mruczkiewicz and M. Krawczyk, “Nonreciprocal dispersion of spin waves in ferromagnetic thin films covered with a finite-conductivity metal,” J. Appl. Phys. 115(11), 113909 (2014).
[Crossref]

Kuzmiak, V.

V. Kuzmiak, S. Eyderman, and M. Vanwolleghem, “Controlling surface plasmon polaritons by a static and/or time-dependent external magnetic field,” Phys. Rev. B 86(4), 045403 (2012).
[Crossref]

Lampin, J.-F.

J. Chochol, K. Postava, M. Čada, M. Vanwolleghem, L. Halagačka, J.-F. Lampin, and J. Pištora, “Magneto-optical properties of InSb for terahertz applications,” AIP Adv. 6(11), 115021 (2016).
[Crossref]

Landau, L.

L. Landau, “Diamagnetismus der Metalle,” Z. Phys. 64(9-10), 629–637 (1930).
[Crossref]

Lezec, H. J.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007).
[Crossref] [PubMed]

Lu, L.

L. Lu, J. D. Joannopoulos, and M. Soljacic, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Magdenko, L.

W. Śmigaj, L. Magdenko, J. Romero-Vivas, S. Guenneau, B. Dagens, B. Gralak, and M. Vanwolleghem, “Compact optical circulator based on a uniformly magnetized ring cavity,” Photonics Nano. Fund. Appl. 10(1), 83–101 (2012).
[Crossref]

Melloni, A.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “what is – and what is not – an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Mizumoto, T.

Mruczkiewicz, M.

M. Mruczkiewicz and M. Krawczyk, “Nonreciprocal dispersion of spin waves in ferromagnetic thin films covered with a finite-conductivity metal,” J. Appl. Phys. 115(11), 113909 (2014).
[Crossref]

Nadan, J. S.

T. J. Gerson and J. S. Nadan, “Surface Electromagnetic Modes of a Ferrite Slab,” IEEE Trans. Microw. Theory Tech. 22(8), 757–763 (1974).
[Crossref]

Onsager, L.

L. Onsager, “Irreversible processes,” Phys. Rev. 37(4), 237–241 (1931).
[Crossref]

Palik, E. D.

E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys. 33(3), 1193–1322 (1970).
[Crossref]

Pepper, M.

K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

Petrov, A.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “what is – and what is not – an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Peurichard, M.

L. Courtois, G. Declercq, and M. Peurichard, “On the non-reciprocal aspect of gyromagnetic surface waves,” AIP Conf. Proc. 5, 1541–1545 (1972).
[Crossref]

Pištora, J.

Pohl, M.

V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
[Crossref] [PubMed]

Popovic, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “what is – and what is not – an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Postava, K.

Raghu, S.

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[Crossref]

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

Renner, H.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “what is – and what is not – an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Roelkens, G.

Romero-Vivas, J.

W. Śmigaj, L. Magdenko, J. Romero-Vivas, S. Guenneau, B. Dagens, B. Gralak, and M. Vanwolleghem, “Compact optical circulator based on a uniformly magnetized ring cavity,” Photonics Nano. Fund. Appl. 10(1), 83–101 (2012).
[Crossref]

Ross, C. A.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

Shvets, G.

Smigaj, W.

W. Śmigaj, L. Magdenko, J. Romero-Vivas, S. Guenneau, B. Dagens, B. Gralak, and M. Vanwolleghem, “Compact optical circulator based on a uniformly magnetized ring cavity,” Photonics Nano. Fund. Appl. 10(1), 83–101 (2012).
[Crossref]

Smith, D. R.

A. Degiron and D. R. Smith, “One-way glass for microwaves using nonreciprocal metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(5), 053203 (2014).
[Crossref] [PubMed]

Soljacic, M.

L. Lu, J. D. Joannopoulos, and M. Soljacic, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

Urzhumov, Y.

Van Roy, W.

Vanwolleghem, M.

J. Chochol, K. Postava, M. Čada, M. Vanwolleghem, L. Halagačka, J.-F. Lampin, and J. Pištora, “Magneto-optical properties of InSb for terahertz applications,” AIP Adv. 6(11), 115021 (2016).
[Crossref]

L. Halagačka, K. Postava, M. Vanwolleghem, F. Vaurette, J. Ben Youssef, B. Dagens, and J. Pištora, “Mueller matrix optical and magneto-optical characterization of Bi-substituted gadolinium iron garnet for application in magnetoplasmonic structures,” Opt. Mater. Express 4(9), 1903–1919 (2014).
[Crossref]

L. Halagačka, M. Vanwolleghem, K. Postava, B. Dagens, and J. Pištora, “Coupled mode enhanced giant magnetoplasmonics transverse Kerr effect,” Opt. Express 21(19), 21741–21755 (2013).
[Crossref] [PubMed]

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “what is – and what is not – an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

W. Śmigaj, L. Magdenko, J. Romero-Vivas, S. Guenneau, B. Dagens, B. Gralak, and M. Vanwolleghem, “Compact optical circulator based on a uniformly magnetized ring cavity,” Photonics Nano. Fund. Appl. 10(1), 83–101 (2012).
[Crossref]

V. Kuzmiak, S. Eyderman, and M. Vanwolleghem, “Controlling surface plasmon polaritons by a static and/or time-dependent external magnetic field,” Phys. Rev. B 86(4), 045403 (2012).
[Crossref]

Vaurette, F.

Vengurlekar, A. S.

V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
[Crossref] [PubMed]

Veronis, G.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100(2), 023902 (2008).
[Crossref] [PubMed]

Wang, Z.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100(2), 023902 (2008).
[Crossref] [PubMed]

Yakovlev, D. R.

V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
[Crossref] [PubMed]

Yu, Z.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “what is – and what is not – an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100(2), 023902 (2008).
[Crossref] [PubMed]

Žutic, I.

I. Žutić, J. Fabian, and S. Das Sarma, “Spintronics: Fundamentals and applications,” Rev. Mod. Phys. 76(2), 323–410 (2004).
[Crossref]

Zvezdin, A. K.

V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
[Crossref] [PubMed]

AIP Adv. (1)

J. Chochol, K. Postava, M. Čada, M. Vanwolleghem, L. Halagačka, J.-F. Lampin, and J. Pištora, “Magneto-optical properties of InSb for terahertz applications,” AIP Adv. 6(11), 115021 (2016).
[Crossref]

AIP Conf. Proc. (1)

L. Courtois, G. Declercq, and M. Peurichard, “On the non-reciprocal aspect of gyromagnetic surface waves,” AIP Conf. Proc. 5, 1541–1545 (1972).
[Crossref]

Appl. Phys. (Berl.) (1)

P. de Santis, “Dispersion Characteristics for a Ferrimagnetic Plate,” Appl. Phys. (Berl.) 2(4), 197–200 (1973).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

T. J. Gerson and J. S. Nadan, “Surface Electromagnetic Modes of a Ferrite Slab,” IEEE Trans. Microw. Theory Tech. 22(8), 757–763 (1974).
[Crossref]

J. Appl. Phys. (1)

M. Mruczkiewicz and M. Krawczyk, “Nonreciprocal dispersion of spin waves in ferromagnetic thin films covered with a finite-conductivity metal,” J. Appl. Phys. 115(11), 113909 (2014).
[Crossref]

J. Phys. Chem. Solids (1)

R. W. Damon and J. R. Eshbach, “Magnetostatic modes of a ferromagnet slab,” J. Phys. Chem. Solids 19(3-4), 308–320 (1961).
[Crossref]

Nat. Nanotechnol. (1)

V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
[Crossref] [PubMed]

Nat. Photonics (3)

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “what is – and what is not – an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljacic, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Nature (1)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Opt. Mater. Express (1)

Photonics Nano. Fund. Appl. (1)

W. Śmigaj, L. Magdenko, J. Romero-Vivas, S. Guenneau, B. Dagens, B. Gralak, and M. Vanwolleghem, “Compact optical circulator based on a uniformly magnetized ring cavity,” Photonics Nano. Fund. Appl. 10(1), 83–101 (2012).
[Crossref]

Phys. Rev. (1)

L. Onsager, “Irreversible processes,” Phys. Rev. 37(4), 237–241 (1931).
[Crossref]

Phys. Rev. A (1)

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[Crossref]

Phys. Rev. B (1)

V. Kuzmiak, S. Eyderman, and M. Vanwolleghem, “Controlling surface plasmon polaritons by a static and/or time-dependent external magnetic field,” Phys. Rev. B 86(4), 045403 (2012).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

A. Degiron and D. R. Smith, “One-way glass for microwaves using nonreciprocal metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(5), 053203 (2014).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100(2), 023902 (2008).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys. 33(3), 1193–1322 (1970).
[Crossref]

Rev. Mod. Phys. (1)

I. Žutić, J. Fabian, and S. Das Sarma, “Spintronics: Fundamentals and applications,” Rev. Mod. Phys. 76(2), 323–410 (2004).
[Crossref]

Science (1)

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007).
[Crossref] [PubMed]

Surf. Sci. Rep. (1)

R. E. Camley, “Nonreciprocal surface waves,” Surf. Sci. Rep. 7(3-4), 103–187 (1987).
[Crossref]

Z. Phys. (1)

L. Landau, “Diamagnetismus der Metalle,” Z. Phys. 64(9-10), 629–637 (1930).
[Crossref]

Other (8)

S. Visnovsky, Optics in Magnetic Multilayers and Nanostructures (CRC, 2010).

L. D. Landau and E. M. Liffschitz, “The symmetry of the kinetic coefficients” in Statistical Physics, vol. 5 Course in Theoretical Physics (Pergamon, 1980).

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).

D. D. Stancil, Theory of Magnetostatic Waves (Springer-Verlag, 1993).

D. M. Pozar, Microwave Engineering (John Wiley & Sons, 2012).

A. K. Zvezdin and V. A. Kotov, Modern magnetooptics and magnetooptical materials (CRC, 1997).

J. D. Jackson, Classical Electrodynamics (Wiley,1999).

Note also how inside the plasma the energy flux is locally reversed as is known from plasmonic systems.

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Figures (5)

Fig. 1
Fig. 1 The problem geometry.

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Fig. 2
Fig. 2 The grounded ferrite slab. (a) View of the computational domain. The ground plane is applied on top of the ferrite. We use a YIG slab with permittivity ε = 14 and permeability components μr = 1 + ω0ωm/(ω02−ω2) and κ = ωωm/(ω02−ω2). Here ω0 = μ0γH0 = 1.4 GHz is the frequency of the gyromagnetic resonance, γ is the gyromagnetic ratio, H0 = 39789 A/m (i.e. 500 Oe) is the bias field, ωm = μ0γMs is the saturation magnetization frequency and Ms = 139260 A/m (i.e. 4πMs = 1750 Gauss) is the saturation magnetization. (b) Dispersion relations of the reverse M mode (violet), the forward M mode (red), the reverse FD mode (yellow) and the forward FD mode (green). The continuous curves are the theoretical predictions [Eq. (11)] and the open triangles are the finite-element simulations. Also shown are the light lines in gray. Note that the frequency scale is not the same for the M modes and the FD modes. (c) and (d) Theoretical field pattern of the different modes with the same color scheme as in (b). The gray dashed curves that are superimposed on them are the fields generated by Comsol Multiphysics. The FD modes have been evaluated at 25 GHz and the M modes at 3.3 GHz.

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Fig. 3
Fig. 3 The YIG slab decorated with a periodic array of Cu cylinders. (a) View of the computational domain. The material parameters of the YIG slab as the same as before, except that we have introduced magnetic losses by imposing a finite linewidth of 2387 A/m (i.e. 30 Oe) to the gyromagnetic resonance. The permittivity of the Cu cylinders is of the form εr - jσ/ωε0, with εr = 1 and σ = 5.998e7 S/m. (b) Dispersion relations of the reverse M mode (violet), the forward M mode (red), the reverse FD mode (yellow) and the forward FD mode (green). Other branches have been found by our solver but they are not discussed because they are extremely lossy [Im(k)>800]. The light lines are represented in gray. Note that the frequency scale is not the same for the M modes and the FD modes. (c) Propagation length 0.5/Im(k) of the different solutions with the same color code.

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Fig. 4
Fig. 4 Nonreciprocal TM-polarized surface magnetoplasmon polaritons (Ez = 0) on a magnetoplasma slab in the same configuration as in Fig. 2(a) (with YIG replaced by transversely magnetized InSb). The Drude-Lorentz parameters of Eq. (12) are obtained by fitting to MO-FTIR measurements on undoped InSb wafers [32]. Only the Drude collision frequency Γ has been hypothetically set to 0. The dispersion relations of the reverse SPP mode (blue) and the forward SPP mode (green) are plotted by a continuous line for the FEM model and by markers for the analytical model. The theoretically predicted forward and reverse limiting surface plasmon frequencies ωSPP, ± are indicated by the dashed lines. Other bands found by the FEM solver are located in the gray-shaded light cones containing the radiation modes in air or in the semi-infinite magnetized InSb. The bulk TM dispersion in a transverse magnetoplasma is given by ( ε ⊥ 2 − g 2 )/ ε ⊥ [33]. It explains the narrow leaky band just below the reduced plasma frequency ω P / ε ∞ .

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Fig. 5
Fig. 5 (a) Schematic representation of the studied magnetoplasmonic periodic one-way guide. The gyro-electric InSb plasma layers are described by the model of Eq. (12) and parameters as in Fig. 4. (b) TM band structure using the FEM of section 2 in units reduced to the plasma wavelength and frequency. Only the non-evanescent bands are plotted. As a guide to the eye, the surface plasmon frequency limits ωSPP, ± , as obtained for the uniform interface in the previous paragraph, are repeated in dotted lines. Dashed (resp. continuous) lines represent bands with negative (resp. positive) group velocity. The grayed zone is the TM radiation band of the InSb cladding; (c) Sx Poynting component along the cut lines shown in part (d). (d) H-field maps and Sx Poynting maps at the band points indicated by the red and black dot, proving the one-way regime: i.e. backward modes on the bottom while forward modes only on the top.

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Equations (15)

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

μ=( μ r iκ 0 −iκ μ r 0 0 0 1 ).
∇ → ∧[ μ ˜ (x,y) ∇ → ∧ E → ]−ε(x,y) ω 2 c 2 E → = 0 → ,
μ ˜ (x,y)=( η(x,y) −ξ(x,y) 0 ξ(x,y) η(x,y) 0 0 0 1 ) , with η(x)= μ r (x) μ r 2 (x)− κ 2 (x) and ξ(x)= i κ r (x) μ r 2 (x)− κ 2 (x) .
∂ξ ∂x ∂ ∂y E z +ξ ∂ 2 ∂x∂y E z − ∂η ∂x ∂ ∂x E z −η ∂ 2 ∂ x 2 E z − ∂η ∂y ∂ ∂y E z −η ∂ 2 ∂ y 2 E z − ∂ξ ∂y ∂ ∂x E z −ξ ∂ 2 ∂y∂x E z −ε ω 2 c 2 E z =0
E → = z ^ .u(x,y)exp(−i. k → . x ^ ) with u(x,y)=u(x+P,y),
∂ξ ∂x ∂u ∂y w− ∂η ∂x ∂u ∂x w+iku ∂η ∂x w−η ∂ 2 u ∂ x 2 w+ikη ∂u ∂x w+ikη ∂u ∂x w + k 2 uηw− ∂η ∂y ∂u ∂y w−η ∂ 2 u ∂ y 2 w− ∂ξ ∂y ∂u ∂x w+ik ∂ξ ∂y uw−ε ω 2 c 2 uw=0.
∫ ik ∂ξ ∂y uw dΩ= [ ∫ ikξuwdx ] y min y max −ik ∫ ξ( u ∂w ∂y +w ∂u ∂y ) dΩ
∫ [ iκ μ r 2 − κ 2 ( ∂u ∂x ∂w ∂y − ∂u ∂y ∂w ∂x )+ μ r μ r 2 − κ 2 ( ∂u ∂x ∂w ∂x + ∂u ∂y ∂w ∂y )−ε ω 2 c 2 uw ] dΩ =− k 2 ∫ μ r μ r 2 − κ 2 uw dΩ+ik ∫ μ r μ r 2 − κ 2 ( u ∂w ∂x − ∂u ∂x w ) dΩ−k ∫ κ μ r 2 − κ 2 ( ∂u ∂y w+u ∂w ∂y ) dΩ
∫ iκ μ r 2 − κ 2 det( ∇ → u ∇ → w)dΩ+ ∫ μ r μ r 2 − κ 2 ∇ → u ∇ → wdΩ − ∫ ε ω 2 c 2 uw dΩ= − k 2 ∫ μ r μ r 2 − κ 2 uwdΩ +ik{ ∫ μ r μ r 2 − κ 2 u k → ‖ k ‖ ( ∇ → w)dΩ − ∫ μ r μ r 2 − κ 2 k → ‖ k ‖ ( ∇ → u)wdΩ } −k{ ∫ μ r μ r 2 − κ 2 udet( k → ‖ k ‖ ∇ → w)dΩ + ∫ μ r μ r 2 − κ 2 det( k → ‖ k ‖ ∇ → u)wdΩ }
ε=( ε ⊥ ig 0 −ig ε ⊥ 0 0 0 ε )
β air = kκ−i β mag μ r coth(i β mag d) ( μ r 2 − κ 2 ) k 2 + β air 2 = ω 2 / c 2 k 2 + β mag 2 =( ω 2 / c 2 ) ε f ( μ r 2 − κ 2 ) μ r
ε ↔ = ε ∞ I ↔ − ω P 2 ω( ω+iΓ ) ( ( ω+iΓ ) 2 ( ω+iΓ ) 2 − ω B 2 +i ω B ( ω+iΓ ) ( ω+iΓ ) 2 − ω B 2 0 −i ω B ( ω+iΓ ) ( ω+iΓ ) 2 − ω B 2 ( ω+iΓ ) 2 ( ω+iΓ ) 2 − ω B 2 0 0 0 1 )≡( ε ⊥ +ig 0 −ig ε ⊥ 0 0 0 ε )
( n eff 2 − ε d )( i.g. n eff − ε ⊥ ε ⊥ ( 1− g 2 ε ⊥ 2 )− n eff 2 )− ε d ε d − n eff 2 ( n eff 2 − ε ⊥ )=0,
ω fw,bw SPP = ω P 2 ε d + ε ∞ + ω B 2 4 ∓ ω B 2
S x = 1 2 Re( E y H z * ) = 1 2 c Z 0 ω Re[ 1 ε ⊥ 2 − g 2 ( j ε ⊥ ( ∂ x H z ) H z * +g( ∂ y H z ) H z * ) ] = exp[ 2Im( k B )x ] 2 c Z 0 ω Re[ 1 ε ⊥ 2 − g 2 ( j ε ⊥ u x u * + ε ⊥ k B | u | 2 +g u y u * ) ],

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