OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 8 — Apr. 22, 2013
  • pp: 9563–9572
« Show journal navigation

Guided modes in magneto-optical waveguides and the role in resonant transmission

Teng-Fei Li, Tian-Jing Guo, Hai-Xu Cui, Mu Yang, Ming Kang, Qing-Hua Guo, and Jing Chen  »View Author Affiliations


Optics Express, Vol. 21, Issue 8, pp. 9563-9572 (2013)
http://dx.doi.org/10.1364/OE.21.009563


View Full Text Article

Acrobat PDF (2174 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Magneto-optical (MO) effect can break the reciprocal propagation of an optical wave along a MO-metal interface. We show that this nonreciprocal property also influences the guided modes in metal-MO-metal waveguides. Especially, the field profiles of the guided modes are neither symmetric nor anti-symmetric, but asymmetric. We then study the resonant optical transmission through a thin metal film with subwavelength MO slits. Magnetic field changes the transmission spectra of the structure, and a MO-induced transparent window is open, where the MO medium becomes extremely anisotropic. The guided-mode mediated high transmission is associated with an asymmetric field distribution and a circling energy flux.

© 2013 OSA

1. Introduction

The advances of metamaterials enable people to manipulate optical resonances in artificial nanostructures toward various practical applications. Although the optical property of a metamaterial can be controlled by changing the shapes, positions and compositions of its constituent elements, it is still highly desired that it can be manipulated by a fast and versatile method. Magneto-optical (MO) effect is then utilized for this purpose, in which the permittivity tensor of a MO medium is tunable by applying an external magnetic field. Great efforts have been devoted to the novel optical phenomena in subwavelength nanostructures with MO constituent [1

1. A. Battula, S. Chen, Y. Lu, R. J. Knize, and K. Reinhardt, “Tuning the extraordinary optical transmission through subwavelength hole array by applying a magnetic field,” Opt. Lett. 32, 2692–2694 (2007) [CrossRef] [PubMed] .

8

8. H. Zhu and C. Jiang, “Nonreciprocal extraordinary optical transmission through subwavelength slits in metallic film,” Opt. Lett. 36, 1308–1310 (2011) [CrossRef] [PubMed] .

]. It is shown that the MO effect can manipulate the characteristics of the extraordinary optical transmission (EOT) in metamaterials [1

1. A. Battula, S. Chen, Y. Lu, R. J. Knize, and K. Reinhardt, “Tuning the extraordinary optical transmission through subwavelength hole array by applying a magnetic field,” Opt. Lett. 32, 2692–2694 (2007) [CrossRef] [PubMed] .

6

6. 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,” Nature Nanotechnology 6, 370–376 (2011) [CrossRef] [PubMed] .

], not only change the wavelengthes of resonant transmission peaks, but also enhance the MO Faraday and Kerr effects. Nonreciprocal properties, including nonreciprocal dispersion and one-way EOT, have also been discussed [7

7. A. B. Khanikaev, S. H. Mousavi, G. Shvets, and Y. S. Kivshar, “One-way extraordinary optical transmission and nonreciprocal spoof plasmons,” Phys. Rev. Lett. 105, 126804 (2010) [CrossRef] [PubMed] .

, 8

8. H. Zhu and C. Jiang, “Nonreciprocal extraordinary optical transmission through subwavelength slits in metallic film,” Opt. Lett. 36, 1308–1310 (2011) [CrossRef] [PubMed] .

].

Fig. 1 (a) A single MO-metal interface, and (b) a metal-MO-metal waveguide. With a y-direction applied magnetic field B, the right-handed orthogonal vector triplet frames of (n±, B, k±) of the two MO-metal interfaces are plotted.

Comparing to a single MO-metal interface, the configuration of a metal-MO-metal waveguide has not attracted much attention. This configuration has widely applications, for example, in EOT when the subwavelength waveguides support guided mode resonance [13

13. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999) [CrossRef] .

15

15. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature (London) 445, 39–46 (2007) [CrossRef] .

]. A metal-MO-metal waveguide contains two metal-MO interfaces, see Fig. 1(b). Assuming that the static magnetic field B is applied in the y direction, a nonzero B then breaks the degeneracy of these two interfaces via the induced non-reciprocality. To understand this effect, let us label the two interfaces by ′+′ for x = +d/2 and ′−′ for x = −d/2, respectively, and still assume that the normal n of each MO-metal interface is from the metal to the MO medium. We can see now the normals n± of the two MO-metal interfaces are antiparallel with each other, i.e. n+ = −n. With the requirement of forming right-handed orthogonal vector triplet, now the wavevector k+ in the triplet frame (n+, B, k+) of the upper interface at x = +d/2 is antiparallel to the wavevector k in the triplet frame (n, B, k) of the lower interface at x = −d/2, i.e. k+ = −k, see Fig. 1(b). For a forwardly propagating guided mode with +z-directed wavevector k, the light properties of the independent upper and lower metal-MO interfaces, if we neglect the coupling effect now, are just identical to these two counter-propagating light modes along a single MO-metal interface. It is well known that the MO-induced nonreciprocal effect renders different field profiles to the two counter-propagating light modes [9

9. J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett. 89, 251115 (2006) [CrossRef] .

]. Consequently, the field profiles of the guided modes in the metal-MO-metal configuration, by considering the mutual coupling between the lower and upper interfaces, are neither symmetric nor anti-symmetric, but asymmetric. In other words, due to the MO-induced non-reciprocality, the field intensities I at the two metal-MO interfaces are not equal, that I+I.

Since the waveguide resonance by the interference between the forwardly and backwardly propagating guided modes is of great importance for EOT in subwavelength metal gratings [13

13. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999) [CrossRef] .

15

15. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature (London) 445, 39–46 (2007) [CrossRef] .

], it is expected that above mentioned broken symmetry of field distribution would, in principle, influence the resonant transmission spectrum, forming some interesting nonreciprocal phenomena such as MO-induced light transparency and circling energy flux.

2. Guided modes in magneto-optical waveguides

Let us discuss the dispersion and field distribution of a guided mode in a metal-MO-metal waveguide. A schematic of the waveguide structure is shown in Fig. 1(b). Assume the MO medium occupies the region of −d/2 < x < +d/2, and consider a transverse magnetic mode.

The field component Hy can be expressed as
Hy=eikz+iωt{H1eα(xd2),x>+d2H2eβx+H3e+βx,+d2>x>d2H4e+α(x+d2),x<d2
(1)

Equation (1) can be solved by using the Maxwell’s equations. Assume the dielectric constant of the metal is εmetal, and the permittivity tensor of the MO medium reads [9

9. J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett. 89, 251115 (2006) [CrossRef] .

, 10

10. 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, 023902 (2008) [CrossRef] [PubMed] .

]
ε¯¯=(ε0jγ0ε||0jγ0ε).
(2)
By using ∇×H = D we can develop the expressions of Ex and Ez. With the requirement that ∇ × E = −iωμ0H still gives the expression of Eq. (1), we can find the constants in Eq. (1) are given, in terms of k, by
α2=k2εmetalk02,
(3)
β2=k2ε2γ2εk02,
(4)
where k0 = ω/c = 2π/λ, λ is the free-space wavelength.

From the boundary continuum conditions of Hy, as
H1=H2eβd2+H3e+βd2,
(5)
H4=H2e+βd2+H3eβd2,
(6)
and Ez, as
αεmetalH1=γk+βεε2γ2H2eβd2+γkβεε2γ2H3e+βd2,
(7)
αεmetalH4=γk+βεε2γ2H2e+βd2+γkβεε2γ2H3eβd2,
(8)
we can find the dispersion (ω, k) of the guided mode, which is given by the existence of nontrivial solutions of the equation set
[(Ck+B)eβd2(CkA)e+βd2(Ck+A)e+βd2(CkB)eβd2][H2H3]=0,
(9)
where
A=βεε2γ2+αεmetal,
(10)
B=βεε2γ2αεmetal,
(11)
C=γε2γ2.
(12)

Equation (9) generally gives
e2βd=B2C2k2A2C2k2.
(13)
We can see it depends on the value of k2, thus the dispersion relation is reciprocal, i.e. when the propagation direction is reversed (k changes to −k), the solution of (ω, k) does not vary.

However, the asymmetric degree of the field distribution, indicated by the ratio of field intensity I ∝ |Hy|2 at the two MO-metal interfaces at x = −d/2 and x = +d/2, as
I+I=A+CkACkBCkB+Ck,
(14)
becomes dependent on the sign of k. To get a symmetric distribution of field intensity, i.e. I+/I = 1, the requirement of C(BA) = 0 should be satisfied. This generally means that γ = 0, i.e. no magnetic field is applied, and the MO medium is isotropic.

From above analysis we prove that, in general, the field profile of the guided mode in a metal-MO-metal waveguide is indeed asymmetric, that I+I. Further more, when a static magnetic field B is applied in the y direction, when k reverses its direction to −k, the profile of the field distribution also reverses with respect to x = 0. Because the asymmetric profile of the guided mode depends on the propagating wavevector k, this effect can also be understood as a MO-induced nonreciprocal characteristic.

In realistic applications, a MO response can be generally achieved at GHz frequency region [10

10. 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, 023902 (2008) [CrossRef] [PubMed] .

, 12

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

]. This frequency is much lower than that of free electron resonance in metal, usually at 3×1014 Hz (around 5 orders larger). Consequently, it is a good approximation to treat the metal as a perfect electron conductor (PEC). In this approximation, because εmetal is much smaller than zero, i.e. εmetal → −, we can approximately write α=εmetalk0. With A = B, Eq. (9) can give two seemly-independent nontrivial solutions, as
k=εk0,β=k0γ/ε,H2=1,H3=0,
(15)
for B +Ck = A +Ck = 0, and
k=εk0,β=+k0γ/ε,H2=0,H3=1,
(16)
for BCk = ACk = 0, respectively. However, these two solutions are in fact degenerated, because they all describe a field distribution of
Hy=H0eiεk0z+iωtexL
(17)
inside the MO medium. Here a characteristic decay length L is defined to represent the degree of field asymmetry so that I+/I = exp(2d/L), where
L=εk0γ.
(18)
A positive (negative) L implies that I+ > I (I+ < I). A strong nonreciprocal effect is expected when L is small, which requires a small diagonal element ε and a great off-diagonal element γ, i.e. when the MO medium is extremely anisotropic.

Above we discuss how a magnetic field induces anisotropy into the MO media, thus modifies the properties of a guided mode. It is shown that although the mode dispersion (ω, k) does not change when the wavevector k reverses its direction, the field distribution Hy(x) is k-dependent and becomes asymmetric. This phenomenon can be considered to of nonreciprocal nature. Note that magnetic field has also been shown to change the symmetry of the wave profile as well as the symmetry of the current distribution in the magneto-transport problems [16

16. D. J. Bergman and Y. M. Strelniker, “Calculation of strong-field magnetoresistance in some periodic composites,” Phys. Rev. B 49, 16256–16268 (1994) [CrossRef] .

, 17

17. Y. M. Strelniker and D. J. Bergman, “Optical transmission through metal films with a subwavelength hole array in the presence of a magnetic field,” Phys. Rev. B 59, R12763–R12766 (1999) [CrossRef] .

]. The possibility in breaking the symmetry of the guided mode profile and its role in EOT have not been noticed before. Below, let us show how this phenomenon influences EOT in subwavelength MO gratings.

3. Influence on extraordinary optical transmission

Let us consider a MO subwavelength grating as shown in the inset of Fig. 2, which contains a periodically subwavelength slit array in a thin metal (Au) film. With an applied magnetic field B in the y direction, we assume that the permittivity tensor of the MO medium has the form of [4

4. Y. M. Strelniker and D. J. Bergman, “Transmittance and transparency of subwavelength-perforated conducting films in the presence of a magnetic field,” Phys. Rev. B 77, 205113 (2008) [CrossRef] .

, 5

5. H. Yin and P. M. Hui, “Controlling enhanced transmission through semiconductor gratings with subwavelength slits by a magnetic field: Numerical and analytical results,” Appl. Phys. Lett. 95, 011115 (2009) [CrossRef] .

, 10

10. 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, 023902 (2008) [CrossRef] [PubMed] .

]
ε=1ωp2ω2ωB2+jω2Γ,
(19)
γ=ωBωp2ω(ω2ωB2+jω2Γ),
(20)
ε||=1ωp2ω2+jω2Γ,
(21)
where ωp is the bulk plasmon frequency, ωB = eB/m* is the cyclotron frequency, e and m* are the charge and effective mass of carrier, and Γ represents damping, respectively.

Fig. 2 Transmission spectra at different magnetic field B, for ωB = 0 (solid line), 0.1ωp (short dashed line) and 0.2ωp (short dotted line), respectively. Inset is the structure under investigation.

Consider the situation with d = 0.6 cm, h = 2.4 cm, a = 2 cm, and ωp = 2π × 12.5 GHz, i.e. λp = 2.4 cm. By using the standard parameter for carriers in semiconductors, as m* = 0.2 ∼ 0.5me, we can find that it is relatively easy to achieve the required magnetic field B of 0.05 ∼ 0.02 tesla for ωB = 0.2ωp in experiments [5

5. H. Yin and P. M. Hui, “Controlling enhanced transmission through semiconductor gratings with subwavelength slits by a magnetic field: Numerical and analytical results,” Appl. Phys. Lett. 95, 011115 (2009) [CrossRef] .

, 12

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

]. Because the frequency in our interests is around 12.5 GHz, the metal Au can be assumed to be PEC for simplification. Although here we are interested in an in-principle demonstration of the new mechanism in manipulating EOT by using the asymmetric field distribution in the subwavelength metal-MO-metal resonator, to represent the realistic situation we choose Γ = 0.001 for all the simulations throughout this article. The transmission properties of the structure are investigated by full-field three-dimensional finite element optical simulations (COMSOL Multiphysics 4.3a).

The dependency of the transmission spectra on the applied magnetic field B under a normal incidence is shown in Fig. 2. We can see in the absence of external magnetic field B, there exists only one transmission peak around 2.065 cm. As the magnetic field B increases to 0.2ωp, this transmission peak slightly shifts to shorter wavelength. The transmittance T could not reach 1 due to the presence of absorption. In particular, a new MO-induced transparent peak appears in the long-wavelength region, around 2.360 cm, which also blue-shifts with the increased magnitude of B. By studying how the distributions of field intensity and energy flux vary at these transmission peaks, we can briefly understand the physical mechanism of their appearance and variation with the magnetic field B.

First, we pay attention to the transmission peak with a shorter wavelength, at 2.065 cm and 2.051 cm when ωB = 0 and 0.2ωp, respectively. The distributions of field intensity and energy flux (Poynting vectors) are shown in Fig. 3.

Fig. 3 Distributions of field intensity and energy flux (Poynting vectors) at the transmission peaks of (a) 2.065 cm when ωB = 0, and (b) 2.051 cm when ωB = 0.2ωp, respectively.

From Fig. 3(a) we can see when no magnetic field is applied, the field pattern is uniformly distributed inside the MO slits, and does not vary with x. Obviously, this EOT peak is associated with a guided mode resonance [13

13. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999) [CrossRef] .

]. The MO medium is isotropic, with ε = 0.259 − 0.741 × 10−3i and γ = 0. According to Eqs. (15) to (18), now the profile of the guided mode is uniformly distributed in the x direction because L approaches infinite. Interference between the forwardly and backwardly propagating guided modes, with wavevectors k and −k, leads to the intensity node at the middle of the structure. The energy flux inside the MO medium is also uniformly distributed, and is z directed. Outside the slit apertures, energy flux forms two counter-rotating vortices at each opening, which can be explained by considering the excitation of high diffractive orders that are evanescent in nature [13

13. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999) [CrossRef] .

].

When a nonzero magnetic field B is applied, the transmission wavelength and its field profile are modified. They can be understood as a manifestation of the nonreciprocal characteristic from the non-diagonal elements in the permittivity tensor. The larger the applied magnetic field B be, the stronger distortion the field distribution is, which can be easily understood by considering the B-dependent values of tensor elements, see Eqs. (17) to (21). For the transmission peak at 2.062 cm when ωB = 0.1ωp, ε = 0.256−0.749×10−3i and γ = 0.064+0.644×10−4i. The characteristic length L equals 2.602 cm. For the transmission peak at 2.051 cm when ωB = 0.2ωp, ε = 0.248 − 0.775 × 10−3i and γ = 0.129 + 0.133 × 10−3i, and L equals 1.263 cm.

From the distributions of field intensity and energy flux shown in Fig. 3(b), we can see they are all asymmetrically distributed inside the MO medium, unlike that in Fig. 3(a). Now they become to concentrate to one side of the metal-MO-metal waveguide. This phenomenon can be understood by considering the asymmetric field distribution following Eq. (17), and the non-100% reflection of guided wave at the rear aperture. To be more explicitly, because now L is positive, according to Eq. (17) the field of forwardly (backwardly) guided mode with positive (negative) k prefers the upper (lower) MO-metal interface at x = d/2 (x = −d/2). The forwardly and backwardly guided modes thus dominates different MO-metal interfaces. Interference between these two guided modes produces an elliptically sharped node in the middle of the structure, because their spatial overlapping is x-dependent. The energy flux S is proportional to the local field intensity and directs to the same direction as that of the wavevector k, thus clockwisely rotated energy flux can be observed inside the MO resonator. Because the reflection from the rear aperture is not 100%, the field intensity and energy flux at x = −d/2 is relatively weaker than these at x = d/2.

Although nonreciprocal property of the guided mode in the metal-MO-metal waveguide plays a role in the modification of the short-wavelength transmission peak, the contribution from the nonzero off-diagonal element is limited, because the characteristic decay length L is usually much greater than the width d of the MO medium. On the contrary, the MO-induced transparent window in the long-wavelength regime belongs to the situation of extreme anisotropy, with a much smaller L, i.e. a greater degree of asymmetry. For example, at the transmission peak of 2.387 cm for ωB = 0.1ωp, ε = 10−2 × (0.124 − 0.101i), γ = 0.099 + 0.100 × 10−3i, and L =0.144 cm. At the transmission peak of 2.350 cm for ωB = 0.2ωp, ε = 10−2 × (0.284 − 0.104i), γ = 0.195 + 0.203 × 10−3i, and L =0.104 cm. Now the MO medium is extremely anisotropic. The diagonal element ε is very small, thus the wavevector k=εk0 is much smaller than k0. On the other hand, the off-diagonal element γ becomes much larger. As a result of this extreme anisotropy, the characteristics decay length L is much smaller than d. The spatial overlapping of the forwardly and backwardly propagating guided modes is thus very poor.

Figure 4 plots the distributions of field intensity and energy flux for this transmission peak, where the MO medium is extremely anisotropic. Because the energy flux inside the MO medium is much larger than that outsides, here the size of arrow is logarithmically proportional to the magnitude of the energy flux. We can see that, in sharp contrast with Fig. 3, the H field and the energy flux is strongly localized to the metal-MO interfaces. This is in consistence with above numerical analysis about the value of L that Ld. The forwardly propagating energy flux is mainly concentrated at the upper metal-MO interface at x = d/2, while the backwardly propagating one is localized at the lower x = −d/2 interface, forming a obvious vortex-like circling flux loop. Now the metal-MO-metal waveguide is more likely to be a traveling-wave resonator. Longitudinal resonance that contributes to the node in Fig. 3(a) is destroyed. On the other hand, a huge void of energy intensity and field intensity in the middle of the MO media is formed, due to the concentration of energy flux to the boundaries. Obviously, it is the extreme anisotropy that contributes to this MO-induced transparent window.

Fig. 4 Distributions of field intensity and energy flux at the MO-induced transparent peak of 2.350 cm, for ωB = 0.2ωp.

Recently, epsilon-near-zero metamaterials have attracted much attention, see [18

18. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008) [CrossRef] [PubMed] .

, 19

19. E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n = 0 structures for visible light,” Phys. Rev. Lett. 110, 013902 (2013) [CrossRef] [PubMed] .

] and references therein. The MO-induced extreme anisotropy, in fact, represents an approach toward the realization of epsilon-near-zero media. Now the guided mode has a effective refractive index of ε. From Eq. (19) it is evident that the diagonal element ε approaches zero when ω is around (ωB2+ωp2)0.5, if we can neglect the damping. For ωB = 0.1ωp and 0.2ωp, the corresponding wavelengthes are around 2.388 cm and 2.353 cm, respectively, close to the wavelengthes of the MO-induced new transparent peaks shown in Fig. 2 and Fig. 4. As a future potential subject of investigation, it might be of great interest to investigate the relevancy of the MO-induced transparency here and the other documented properties of epsilon-near-zero metamaterials [18

18. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008) [CrossRef] [PubMed] .

, 19

19. E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n = 0 structures for visible light,” Phys. Rev. Lett. 110, 013902 (2013) [CrossRef] [PubMed] .

], with an emphasis on the induced extreme anisotropy.

Before ending this section, we would like to provide a more detailed discussion about the mechanism of the MO-modified resonant transmission. Usually it is believed that the resonance transmission appears due to the existence of the SPP resonance [13

13. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999) [CrossRef] .

15

15. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature (London) 445, 39–46 (2007) [CrossRef] .

], which exists also in the absence of the applied magnetic field. However, here the metal is PEC, so all the SPP resonances are forbidden. Although the wavelength variation of the short-wavelength transmission peak (∼2.06 cm) can be briefly understood as a MO-induced shift of the transparency peaks, the peak with a longer wavelength (∼2.36 cm), which is the emphasis of this investigation, must be associated with the unique anisotropic properties of the MO medium. In the absence of the applied magnetic field, this transmission feature could not be observed. When a magnetic field is applied, the MO medium becomes anisotropic, which, in turn, enables the existence of asymmetric field distribution. This effect is especially strong near the resonance at (ωB2+ωp2)0.5, which can be termed the effective cyclotron resonance [4

4. Y. M. Strelniker and D. J. Bergman, “Transmittance and transparency of subwavelength-perforated conducting films in the presence of a magnetic field,” Phys. Rev. B 77, 205113 (2008) [CrossRef] .

]. Although now the field profile is localized at the MO-metal boundaries, see Fig. 4, it is not associated with any SPP resonances, and a strong MO-induced anisotropy is evidently the deeply physical mechanism of this transmission feature.

4. Conclusion

In summary, we provide a theoretic interpretation and analysis about the existence of asymmetric field profiles for guided modes in metal-MO-metal waveguides, due to the MO-induced non-reciprocality. We study the resonant optical transmission through thin metal film with sub-wavelength MO slits, and show that magnetic field changes the transmission spectra of the structure. MO-induced transparent windows are observed, where the MO medium becomes extremely anisotropic. From the analysis of field distribution, we prove that the resonant transmission is associated with an asymmetric field distribution and a circling energy flux, which implies that the mechanism discussed in this paper is experimentally observable.

Acknowledgments

The authors acknowledge the support from the National Natural Science Foundation of China (NSFC) under grants 11174157 and 11074131, and the Specialized Research Fund for the Doctoral Program (SRFDP) under grant 20110031110005.

References and links

1.

A. Battula, S. Chen, Y. Lu, R. J. Knize, and K. Reinhardt, “Tuning the extraordinary optical transmission through subwavelength hole array by applying a magnetic field,” Opt. Lett. 32, 2692–2694 (2007) [CrossRef] [PubMed] .

2.

A. B. Khanikaev, A. V. Baryshev, A. A. Fedyanin, A. B. Granovsky, and M. Inoue, “Anomalous Faraday effect of a system with extraordinary optical transmittance,” Opt. Express 15, 6612–6622 (2007) [CrossRef] [PubMed] .

3.

V. I. Belotelov, L. L. Doskolovich, and A. K. Zvezdin, “Extraordinary magneto-optical effects and transmission through metal-dielectric plasmonic systems,” Phys. Rev. Lett. 98, 077401 (2007) [CrossRef] [PubMed] .

4.

Y. M. Strelniker and D. J. Bergman, “Transmittance and transparency of subwavelength-perforated conducting films in the presence of a magnetic field,” Phys. Rev. B 77, 205113 (2008) [CrossRef] .

5.

H. Yin and P. M. Hui, “Controlling enhanced transmission through semiconductor gratings with subwavelength slits by a magnetic field: Numerical and analytical results,” Appl. Phys. Lett. 95, 011115 (2009) [CrossRef] .

6.

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,” Nature Nanotechnology 6, 370–376 (2011) [CrossRef] [PubMed] .

7.

A. B. Khanikaev, S. H. Mousavi, G. Shvets, and Y. S. Kivshar, “One-way extraordinary optical transmission and nonreciprocal spoof plasmons,” Phys. Rev. Lett. 105, 126804 (2010) [CrossRef] [PubMed] .

8.

H. Zhu and C. Jiang, “Nonreciprocal extraordinary optical transmission through subwavelength slits in metallic film,” Opt. Lett. 36, 1308–1310 (2011) [CrossRef] [PubMed] .

9.

J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett. 89, 251115 (2006) [CrossRef] .

10.

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, 023902 (2008) [CrossRef] [PubMed] .

11.

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, 013904 (2008) [CrossRef] [PubMed] .

12.

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

13.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999) [CrossRef] .

14.

F. J. García de Abajo, “Colloquium: light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1289 (2007) [CrossRef] .

15.

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature (London) 445, 39–46 (2007) [CrossRef] .

16.

D. J. Bergman and Y. M. Strelniker, “Calculation of strong-field magnetoresistance in some periodic composites,” Phys. Rev. B 49, 16256–16268 (1994) [CrossRef] .

17.

Y. M. Strelniker and D. J. Bergman, “Optical transmission through metal films with a subwavelength hole array in the presence of a magnetic field,” Phys. Rev. B 59, R12763–R12766 (1999) [CrossRef] .

18.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008) [CrossRef] [PubMed] .

19.

E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n = 0 structures for visible light,” Phys. Rev. Lett. 110, 013902 (2013) [CrossRef] [PubMed] .

OCIS Codes
(160.3820) Materials : Magneto-optical materials
(260.5740) Physical optics : Resonance
(310.2790) Thin films : Guided waves

ToC Category:
Physical Optics

History
Original Manuscript: February 1, 2013
Revised Manuscript: March 28, 2013
Manuscript Accepted: March 28, 2013
Published: April 10, 2013

Citation
Teng-Fei Li, Tian-Jing Guo, Hai-Xu Cui, Mu Yang, Ming Kang, Qing-Hua Guo, and Jing Chen, "Guided modes in magneto-optical waveguides and the role in resonant transmission," Opt. Express 21, 9563-9572 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-8-9563


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. A. Battula, S. Chen, Y. Lu, R. J. Knize, and K. Reinhardt, “Tuning the extraordinary optical transmission through subwavelength hole array by applying a magnetic field,” Opt. Lett.32, 2692–2694 (2007). [CrossRef] [PubMed]
  2. A. B. Khanikaev, A. V. Baryshev, A. A. Fedyanin, A. B. Granovsky, and M. Inoue, “Anomalous Faraday effect of a system with extraordinary optical transmittance,” Opt. Express15, 6612–6622 (2007). [CrossRef] [PubMed]
  3. V. I. Belotelov, L. L. Doskolovich, and A. K. Zvezdin, “Extraordinary magneto-optical effects and transmission through metal-dielectric plasmonic systems,” Phys. Rev. Lett.98, 077401 (2007). [CrossRef] [PubMed]
  4. Y. M. Strelniker and D. J. Bergman, “Transmittance and transparency of subwavelength-perforated conducting films in the presence of a magnetic field,” Phys. Rev. B77, 205113 (2008). [CrossRef]
  5. H. Yin and P. M. Hui, “Controlling enhanced transmission through semiconductor gratings with subwavelength slits by a magnetic field: Numerical and analytical results,” Appl. Phys. Lett.95, 011115 (2009). [CrossRef]
  6. 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,” Nature Nanotechnology6, 370–376 (2011). [CrossRef] [PubMed]
  7. A. B. Khanikaev, S. H. Mousavi, G. Shvets, and Y. S. Kivshar, “One-way extraordinary optical transmission and nonreciprocal spoof plasmons,” Phys. Rev. Lett.105, 126804 (2010). [CrossRef] [PubMed]
  8. H. Zhu and C. Jiang, “Nonreciprocal extraordinary optical transmission through subwavelength slits in metallic film,” Opt. Lett.36, 1308–1310 (2011). [CrossRef] [PubMed]
  9. J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett.89, 251115 (2006). [CrossRef]
  10. 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, 023902 (2008). [CrossRef] [PubMed]
  11. 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, 013904 (2008). [CrossRef] [PubMed]
  12. Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačic̀, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett.100, 013905 (2008). [CrossRef] [PubMed]
  13. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett.83, 2845–2848 (1999). [CrossRef]
  14. F. J. García de Abajo, “Colloquium: light scattering by particle and hole arrays,” Rev. Mod. Phys.79, 1267–1289 (2007). [CrossRef]
  15. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature (London)445, 39–46 (2007). [CrossRef]
  16. D. J. Bergman and Y. M. Strelniker, “Calculation of strong-field magnetoresistance in some periodic composites,” Phys. Rev. B49, 16256–16268 (1994). [CrossRef]
  17. Y. M. Strelniker and D. J. Bergman, “Optical transmission through metal films with a subwavelength hole array in the presence of a magnetic field,” Phys. Rev. B59, R12763–R12766 (1999). [CrossRef]
  18. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett.100, 033903 (2008). [CrossRef] [PubMed]
  19. E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n = 0 structures for visible light,” Phys. Rev. Lett.110, 013902 (2013). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited