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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 7 — Mar. 29, 2010
  • pp: 6914–6921
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Broadband unidirectional electromagnetic mode at interface of anti-parallel magnetized media

Haibin Zhu and Chun Jiang  »View Author Affiliations


Optics Express, Vol. 18, Issue 7, pp. 6914-6921 (2010)
http://dx.doi.org/10.1364/OE.18.006914


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Abstract

We report a kind of broadband electromagnetic boundary mode at an interface of anti-parallel magnetized media, which can only propagate in one direction perpendicular to the magnetization and parallel to the interface. The unidirectionality of this mode originates from the permeability or permittivity tensor introduced by magnetization. We theoretically and numerically analyze the existence of the unidirectional mode, and point out that this mode can exist in both gyromagnetic and gyroelectric medium. We also propose a one-way waveguide based on this unidirectional mode, which may realize a new kind of electromagnetic isolation differing from those existing ones.

© 2010 OSA

1. Introduction

Here we present and demonstrate a novel mechanism with a simple straight waveguide, in which broadband unidirectional mode exists along a magnetic domain wall formed in MO materials. Utilizing this effect, broadband isolation can be achieved with a simple structure. This nonreciprocal MO effect originates from the off-diagonal elements in permittivity or permeability tensor induced by magnetization. Different from the unidirectional mode mentioned in Ref [30

30. 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]

], unidirectionality of this mode only depends on the signs of off-diagonal elements in the tensor. Thus, this mode can support broadband unidirectional frequencies, consequently.

2. Theoretical analysis

We start by analyzing the additional effects introduced by an external static magnetic field, as in this paper along the z axis in Cartesian coordinates, applied on homogenous MO materials. The external field changes the media’s response to an electromagnetic wave, by changing the permittivity or permeability into a tensor form, and making the media anisotropic. In frequencies near optical range, the change usually takes place in the permittivity, while in permeability near frequencies of microwave range, as below

ε=ε0[ε1iε20iε2ε10001],μ=μ0[μ1iμ20iμ2μ10001].
(1)

Where ε0 and μ0 are the permittivity and permeability of vacuum, respectively. Here the elements of these two tensors, i.e. ε1,2 and μ1,2, usually fluctuate with working frequency, but in a relatively large frequency range, the signs of them keep unchanged. We will show in this paper, the signs of these tensor’s elements will determine the propagating direction of boundary mode at domain walls, across which the signs of off-diagonal elements, i.e. ε2 or μ2 change rapidly while signs of diagonal elements, i.e. ε1 or μ1, keep unchanged. Technologically, this can be realized by applying anti-parallel external magnetic field along a fixed plane [35

35. H. Takeda and S. John, “Compact optical one-way waveguide isolators for photonic-band-gap microchips,” Phys. Rev. A 78(2), 023804 (2008). [CrossRef]

,36

36. J. R. Goldman, T. D. Ladd, F. Yamaguchi, and Y. Yamamoto, “Magnet designs for a crystal-lattice quantum computer,” Appl. Phys., A Mater. Sci. Process. 71, 11–17 (2000).

].

For detail, we focus on the case of permittivity, which means the working frequency is near optical range, and the media is gyroelectronic material. The tensor form of ε in Eq. (1) shows gyrotropic, which will only give chirality on the TM mode (with non-zero Hz, Ex and Ey), and leave the TE mode (with non-zero Ez, Hx and Hy) nonchiral. Then for the TE mode, the media is still an isotropic one, and there is no boundary mode at domain wall. For TM mode localized at the boundary, we want their field to exponentially decay on both sides of the boundary. Either side of the domain wall is composed of MO material, with reverse sign of ε2. Since ε2 is not homogeneous in the system, we consider ε2 as function of position r. Assuming the domain wall is at zx plane in Cartesian coordinate system, as shown in Fig. (1a)
Fig. 1 Magnetic domain wall and steady-state field pattern for unidirectional mode. The external magnetic field for y>0 and y<0 are applied along + z and -z, respectively. A TE polarized (Ez) point source, indicated by the big arrow in (a), is located at the domain wall. Ez field pattern is indicated with blue-red color map in (a)-(e), while H field pattern is shown in (a) with arrows grid. At very boundary of the domain wall we can observe a unidirectional boundary mode, which is exactly a TEM mode. Different frequencies 0.050, 0.075, 0.100, 0.125 and 0.150 (×2πc/a) are utilized to show the unidirectional phenomenon, which are shown in (a), (b), (c), (d) and (e), respectively.
, then for y>0 and y<0 we have ε2(r)=ε2 and ε2(r)=ε2, respectively. For simplicity, we neglect the material loss. We therefore take
H(>)(r;t)=A(0,0,1)exp[ikxαyiωt]
(2a)
Ε(>)(r;t)=Aiω[ε˜1α+ε˜2k, i(ε˜2α+ε˜1k), 0]exp[ikxαyiωt]
(2b)
in the region y>0, and
H(<)(r;t)=B(0,0,1)exp[ikx+βyiωt]
(3a)
Ε(<)(r;t)=Aiω[(ε˜1β+ε˜2k), i(ε˜2β+ε˜1k), 0]exp[ikx+βyiωt]
(3b)
in the region y<0. Here ε˜1=ε1/(ε12ε22) and ε˜2=ε2/(ε12ε22), α and β are positive decay parameters, k is the wave number along the domain wall. Then the sign of k can determine propagating direction of the mode. A positive k means that the mode propagates along + x axis, while a negative k means the mode propagates along the -x axis. In order to complete the problem, we must match the solutions in each region by the use of boundary conditions at y = 0 plane, that the tangential components of E and the normal components of B are continuous. These two conditions reduce to the results that A=B and α=β. Also, we finally get to a novel result that

k=ε˜1ε˜2α=αQE.
(4)

Here QE=ε˜2/ε˜1=ε2/ε1, is the voigt parameter, which fluctuates with external field and working frequency, but whose sign keeps unchanged in a relative wide range of frequency. Thus, given the positive α and assuming positive ε1 and ε2, we therefore have positive QE and negative k, which determine the mode can only propagate along -x direction. On the other hand, with the reversed magnetization in y<0 and y>0 regions, respectively, ε2(r) changes signs and we can also get a boundary mode which can only propagate in + x direction. With the relation k=neffω/c, we conclude that the transverse decay parameter α, which defines the intensity of localization for boundary mode, is determined by frequency ω, effective refractive index neff and voigt parameter QE, with the relation α=|QEk|. Therefore, the larger QE, the larger α, and the stronger unidirectional phenomenon will be. Also we can conclude by taking Eq. (4) into Eq. (2) and Eq. (3) that this boundary mode is actually a TEM mode, with only non-zero Hz and Ey.

The situation is similar with gyromagnetic media, in which permeability is a tensor as μ in Eq. (1). The domain wall in gyromagnetic media can support unidirectional boundary mode which is proven to be TEM mode with non-zero Hy and Ez. The voigt parameter QM=μ2/μ1 determines the decay parameter by α=|QMk|, then the localization of the boundary mode at domain wall. And also, with signs of μ1 and μ2 unchanged, we have fixed-sign k with relation k=α/QM. Thus we have a broadband unidirectional boundary mode at the interface.

3. Numerical calculation

3.1 Unidirectional boundary mode at domain wall

As shown above, the unidirectional property of this boundary mode only depends on the signs reversing of off-diagonal elements in permittivity or permeability tensors. Therefore it is robust even concerning the loss and frequency dependency of real materials. To verify our theoretical analysis about this unidirectional boundary mode, we utilize a 2D finite-difference time-domain (FDTD) method [37

37. A. Taflove, and S. C. Hagness, Computational Electrodynamics:The Finite-Difference Time-Domain Method (Artech House, Inc., Norwood, MA, 2005).

] to demonstrate the propagation of the unidirectional mode. As discussed above, larger voigt parameter can show greater unidirectional boundary phenomenon, we choose gyromagnetic materials mentioned in Ref [28

28. Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008). [CrossRef] [PubMed]

], which show a large voigt parameter. For detail, we choose μ1=14 and μ2=12.4, while the relative permittivity is ε=15. Here, the signs of these parameters keep unchanged in a wide frequency range, and for simplicity, we assume values of these parameters unchanged in our demonstration. As shown in Fig. 1, the nonreciprocal boundary mode can only propagate along + x direction, while the backward counterpart is absent. Magnetic field H is shown as arrows in Fig. 1(a), which is confirmed to be a TEM mode boundary at the interface. We choose five different working frequencies normalized to a unit length a, which are 0.050, 0.075, 0.100, 0.125 and 0.150 (×2πc/a), respectively. Here we use the normalized frequency instead of real frequency, for the idiomatic use in integrated optics. It is easy to transform to real frequencies, if only a is provided. All these frequencies can possess a unidirectional mode, which prove that this unidirectional boundary mode can work with a broadband frequency range. The localization intensity of boundary mode's energy increases with the frequency, as shown in Fig. 1(b-e). Simultaneously in our demonstration, there are bulk modes which can propagate off the domain wall, and which are reciprocal. With proper designation and with the help of a proper absorbing boundary, this reciprocal affection can be inhibited maximally.

Our discussion above bases on the conditions that magnetization over the domain wall reverses rapidly. In some occasions magnetization reversion may experience a thin layer of transition region. With transition layer, the unidirectional mode discussed above transforms into a nonreciprocal mode, which propagates alone the layer. It can be proved that with this layer’s thickness increasing, the unidirectionality of this mode decreases. Furthermore, in our discussion, the unidirectional effect relies on the anti-symmetrical profile of the domains. If the profile of the domain deviates from this situation, e.g. y>0 half part is partly demagnetized while y<0 half part keeps unchanged, the unidirectionality of boundary mode also decreases. In the extreme occasion, when y>0 half part is unmagnetized, the unidirectional mode disappears. The simulation results for deviation from anti-symmetrical profile of domain wall are shown in Fig. 2
Fig. 2 Deviation cases from anti-symmetrical profile of domain wall. Partly demagnetization case with μ1=6 and μ2=2for y>0 is shown in (a); and unmagnetization case with μ1=1 and μ2=0 for y>0 is shown in (b). Operating frequency is chosen as 0.2×2πc/a for clearly demonstration, where a is the normalized length.
. For a partly demagnetization, we assume the parameters as μ1=6and μ2=2 for y>0, and keep parameters the same as above for y<0. Different from the anti-symmetrical case, more energy radiates into bulk modes, while still keeping unidirectional boundary mode observable. For this case the result is shown in Fig. 2(a). Another extreme occasion, when y>0 half part is unmagnetized, we have μ1=1andμ2=0. The result is shown in Fig. 2(b). In this case we cannot find any boundary mode at the interface, and all the radiation energy transforms into bulk modes. In conclusion, the nonreciprocal effect and the unidirectionality mainly depend on the rapid reversion of the magnetization.

3.2 Broadband isolator based on unidirectional boundary mode

The introduction of this unidirectional mode at domain wall will provide us a potential alternative method to achieve compact nonreciprocal components such as broadband isolators, with a very simple structure. As an example, we consider a one-way waveguide composed of a straight domain wall, with anti-parallel magnetization on each side, which is shown in Fig. 3(a)
Fig. 3 Isolator based on unidirectional boundary mode at the domain wall. The isolator is connected with two reciprocal waveguide, which work as input and output ports. The mechanism of this isolator is shown in (a), and the forward and backward transmitting steady-state field patterns are shown in (b) and (c). Operating frequency is 0.05×2πc/a, where a is the waveguide's width.
. The material’s parameters are chosen the same as above in Fig. 1. In order to prevent perturbation of bulk modes, and to achieve high isolation ratio, we clad the isolating component with absorbing layers. We here use two reciprocal waveguides as input and output ports. In order to minimize the reflection between reciprocal waveguides and one-way waveguide, we assume the reciprocal waveguide with same relative permittivityε=15, and a relative permeabilityμ=18.7, for the purpose of impedance matching with one-way waveguide. The length of domain wall is chosen as 16a, while the input and output reciprocal waveguide’s width is a. As discussed above, this waveguide possess broadband unidirectional boundary mode at the interface. The forward and backward propagating transient field pattern are shown in Fig. 3(b) and Fig. 3(c) respectively, with forward and backward transmission, and isolation ratio shown in Fig. 4(a)
Fig. 4 Forward and backward transmission (a), and the isolation ratio (b) for our example isolator. In (a), the blue line represents the forward transmission and the red one represents the backward counterpart. The magnified backward transmission is plotted in the inside box. (b) shows the high isolation ratio in a wide frequency range.
and Fig. 4(b), respectively. We can observe a great isolation ratio in this simple isolation based on unidirectional boundary mode, and in a wide frequency range, the isolation is still robust. In Fig. 4, the fluctuating of transmissions and isolation ratio is induced by bulk mode which is reciprocal. Also we can observe ripples in both forward and backward transmissions, which are induced by reflection interference. Due to the unidirectionality of the boundary mode, backward propagating energy along the boundary is suppressed, and the only backward transmission of energy comes from bulk mode, which is reciprocal. This makes the backward transmission very low, as shown in the inside box in Fig. 4(a). The maximum of forward transmission is about 73%, due to the loss on reflections at interface with reciprocal waveguides, and also due to the loss on bulk modes which propagate off the domain wall deviating from the output port. The peak of forward transmission occurs at frequency 0.07 (×2πc/a), at which frequency the mode matching allows most energy from reciprocal waveguide coupled into unidirectional boundary mode. Considering the backward transmission, which is only affected by reciprocal bulk mode, the isolation ratio possesses a peak value of 25 dB at frequency 0.093 (×2πc/a).

4. Conclusion

In summary, we report the theoretical and numerical demonstration of a unidirectional boundary mode at domain wall. This unidirectional property relies on the signs reversing of off-diagonal elements in permittivity or permeability tensors, therefore it is robust in a wide frequency range. This broadband nonreciprocal behavior may be used to design compact and integratable nonreciprocal component, such as isolators and circulators. The intensity of this boundary mode is proportional to the voigt parameter, which is limited in real MO materials. Metamaterials [38

38. J. B. Pendry, “A chiral route to negative refraction,” Science 306(5700), 1353–1355 (2004). [CrossRef] [PubMed]

,39

39. B. Bai, Y. Svirko, J. Turunen, and T. Vallius, “Optical activity in planar chiral metamaterials: Theoretical study,” Phys. Rev. A 76(2), 023811 (2007). [CrossRef]

] may provide a more tunable way to achieve higher level unidirectional performance.

Acknowledgments

This work was supported by the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (Grant No. 708038), and Thanks are given to Xiaofei Zang and Cai Huang for their helpful discussion.

References and links

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2.

C. Brosseau, S. Mallégol, P. Quéffelec, and J. Ben Youssef, “Nonreciprocal electromagnetic properties of nanocomposites at microwave frequencies,” Phys. Rev. B 70(9), 092401 (2004). [CrossRef]

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P. K. Amiri, B. Rejaei, Z. Yan, M. Vroubel, L. Dok Won, and S. X. Wang, “Nonreciprocal Spin Waves in Co-Ta-Zr Films and Multilayers,” IEEE Trans. Magn. 45(10), 4215–4218 (2009). [CrossRef]

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A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320(5876), 646–649 (2008). [CrossRef] [PubMed]

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C. J. Barrelet, A. B. Greytak, and C. M. Lieber, “Nanowire Photonic Circuit Elements,” Nano Lett. 4(10), 1981–1985 (2004). [CrossRef]

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8.

B. L. Johnson and R. E. Camley, “Nonreciprocal propagation of surface waves in quasiperiodic superlattices,” Phys. Rev. B 44(3), 1225–1231 (1991). [CrossRef]

9.

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A. Figotin and I. Vitebsky, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(6), 066609 (2001). [CrossRef] [PubMed]

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R. L. Espinola, T. Izuhara, M.-C. Tsai, R. M. Osgood Jr, and H. Dötsch, “Magneto-optical nonreciprocal phase shift in garnet/silicon-on-insulator waveguides,” Opt. Lett. 29(9), 941–943 (2004). [CrossRef] [PubMed]

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W. V. Parys, B. Moeyersoon, D. V. Thourhout, R. Baets, M. Vanwolleghem, B. Dagens, J. Decobert, O. L. Gouezigou, D. Make, R. Vanheertum, and L. Lagae, “Transverse magnetic mode nonreciprocal propagation in an amplifying AlGaInAs/InP optical waveguide isolator,” Appl. Phys. Lett. 88(7), 071115 (2006). [CrossRef]

13.

Y. Shoji, I. W. Hsieh, J. R. M. Osgood, and T. Mizumoto, “Polarization-Independent Magneto-Optical Waveguide Isolator Using TM-Mode Nonreciprocal Phase Shift,” J. Lightwave Technol. 25(10), 3108–3113 (2007). [CrossRef]

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A. Potts, W. Zhang, and D. M. Bagnall, “Nonreciprocal diffraction through dielectric gratings with two-dimensional chirality,” Phys. Rev. A 77(4), 043816 (2008). [CrossRef]

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T. R. Zaman, X. Guo, and R. J. Ram, “Semiconductor Waveguide Isolators,” J. Lightwave Technol. 26(2), 291–301 (2008). [CrossRef]

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A. F. Popkov, M. Fehndrich, M. Lohmeyer, and H. Dötsch, “Nonreciprocal TE-mode phase shift by domain walls in magnetooptic rib waveguides,” Appl. Phys. Lett. 72(20), 2508–2510 (1998). [CrossRef]

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O. Zhuromskyy, H. Dötsch, M. Lohmeyer, L. Wilkens, and P. Hertel, “Magnetooptical Waveguides with Polarization-Independent Nonreciprocal PhaseShift,” J. Lightwave Technol. 19(2), 214–221 (2001). [CrossRef]

35.

H. Takeda and S. John, “Compact optical one-way waveguide isolators for photonic-band-gap microchips,” Phys. Rev. A 78(2), 023804 (2008). [CrossRef]

36.

J. R. Goldman, T. D. Ladd, F. Yamaguchi, and Y. Yamamoto, “Magnet designs for a crystal-lattice quantum computer,” Appl. Phys., A Mater. Sci. Process. 71, 11–17 (2000).

37.

A. Taflove, and S. C. Hagness, Computational Electrodynamics:The Finite-Difference Time-Domain Method (Artech House, Inc., Norwood, MA, 2005).

38.

J. B. Pendry, “A chiral route to negative refraction,” Science 306(5700), 1353–1355 (2004). [CrossRef] [PubMed]

39.

B. Bai, Y. Svirko, J. Turunen, and T. Vallius, “Optical activity in planar chiral metamaterials: Theoretical study,” Phys. Rev. A 76(2), 023811 (2007). [CrossRef]

OCIS Codes
(230.3810) Optical devices : Magneto-optic systems
(240.6690) Optics at surfaces : Surface waves

ToC Category:
Optics at Surfaces

History
Original Manuscript: December 2, 2009
Revised Manuscript: January 15, 2010
Manuscript Accepted: January 18, 2010
Published: March 19, 2010

Citation
Haibin Zhu and Chun Jiang, "Broadband unidirectional electromagnetic mode at interface of anti-parallel magnetized media," Opt. Express 18, 6914-6921 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-6914


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References

  1. P. A. Belov, S. A. Tretyakov, and A. J. Viitanen, “Nonreciprocal microwave band-gap structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016608 (2002). [CrossRef] [PubMed]
  2. C. Brosseau, S. Mallégol, P. Quéffelec, and J. Ben Youssef, “Nonreciprocal electromagnetic properties of nanocomposites at microwave frequencies,” Phys. Rev. B 70(9), 092401 (2004). [CrossRef]
  3. P. K. Amiri, B. Rejaei, Z. Yan, M. Vroubel, L. Dok Won, and S. X. Wang, “Nonreciprocal Spin Waves in Co-Ta-Zr Films and Multilayers,” IEEE Trans. Magn. 45(10), 4215–4218 (2009). [CrossRef]
  4. B. K. Kuanr, V. Veerakumar, R. Marson, S. R. Mishra, R. E. Camley, and Z. Celinski, “Nonreciprocal microwave devices based on magnetic nanowires,” Appl. Phys. Lett. 94(20), 202505 (2009). [CrossRef]
  5. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320(5876), 646–649 (2008). [CrossRef] [PubMed]
  6. C. J. Barrelet, A. B. Greytak, and C. M. Lieber, “Nanowire Photonic Circuit Elements,” Nano Lett. 4(10), 1981–1985 (2004). [CrossRef]
  7. Y. Okamura, H. Inuzuka, T. Kikuchi, and S. Yamamoto, “Nonreciprocal propagation in magnetooptic YIG rib waveguides,” J. Lightwave Technol. 4(7), 711–714 (1986). [CrossRef]
  8. B. L. Johnson and R. E. Camley, “Nonreciprocal propagation of surface waves in quasiperiodic superlattices,” Phys. Rev. B 44(3), 1225–1231 (1991). [CrossRef]
  9. A. F. Popkov, M. Fehndrich, O. Zhuromskyy, and H. Dötsch, “Nonreciprocal light channeling in a film by a magnetic nonuniformity akin to a Néel domain wall,” J. Appl. Phys. 84(6), 3020 (1998). [CrossRef]
  10. A. Figotin and I. Vitebsky, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(6), 066609 (2001). [CrossRef] [PubMed]
  11. R. L. Espinola, T. Izuhara, M.-C. Tsai, R. M. Osgood, and H. Dötsch, “Magneto-optical nonreciprocal phase shift in garnet/silicon-on-insulator waveguides,” Opt. Lett. 29(9), 941–943 (2004). [CrossRef] [PubMed]
  12. W. V. Parys, B. Moeyersoon, D. V. Thourhout, R. Baets, M. Vanwolleghem, B. Dagens, J. Decobert, O. L. Gouezigou, D. Make, R. Vanheertum, and L. Lagae, “Transverse magnetic mode nonreciprocal propagation in an amplifying AlGaInAs/InP optical waveguide isolator,” Appl. Phys. Lett. 88(7), 071115 (2006). [CrossRef]
  13. Y. Shoji, I. W. Hsieh, J. R. M. Osgood, and T. Mizumoto, “Polarization-Independent Magneto-Optical Waveguide Isolator Using TM-Mode Nonreciprocal Phase Shift,” J. Lightwave Technol. 25(10), 3108–3113 (2007). [CrossRef]
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