Abstract

Giant enhancement of the magneto-optical Kerr effect (MOKE) by surface plasmon polaritons (SPPs) is theoretically shown in a trilayer structure consisting of double-layer dielectrics and a ferromagnetic metal (Al2O3/SiO2/Fe). We calculated the resonant enhancement of the transverse MOKE (TMOKE) and polar MOKE (PMOKE) using the attenuated total reflection (ATR) configuration with the transfer matrix method using a 4 × 4 scattering matrix. At a specific film thickness of the low-index SiO2 layer, where confinement of the SPPs on the Fe surface becomes close to the cutoff condition, the incident light from the Al2O3 couples with the SPPs at the SiO2/Fe boundary most efficiently, resulting in resonant enhancement of the MOKE at an incident angle corresponding to the wave vector of the SPPs. The calculated PMOKE showed orthogonal transformation (90°-rotation) and almost full-orbed deformation (44°-ellipticity) of the polarization, and the TMOKE showed a change in reflectance of about 34 dB upon magnetization reversal.

© 2015 Optical Society of America

1. Introduction

The interdisciplinary study of magnetic/magneto-optic and plasmonic functionalities, referred to as magnetoplasmonics, has been under development to take advantage of unique features from both fields. The magneto-optical (MO) effect is one of the rare phenomena that offer the possibility of time-reversal symmetry breaking, and therefore, it has been employed in a variety of industrial applications. For example, a well-known application is MO recording, which uses the magneto-optical Kerr effect (MOKE) to read data from a magnetic disc [1,2]. The MO effect is the operating principle of essential components such as optical isolators and circulators used in optical telecommunications in order to guide light in only one direction. To realize an optical isolator, Lorentz reciprocity must be broken to prevent backward propagating light [3]. Hence, the most common isolator relies on the Faraday effect and crossed polarizers. Some proposed integrated isolators for photonic/optoelectronic integrated circuits (PICs/OEICs) also employ the Faraday effect [4,5], whereas others use the transverse magneto-optical Kerr effect (TMOKE) [69].

Surface plasmon polaritons (SPPs) are quasi-particles in which an electromagnetic wave is coupled to collective oscillations of free electron gas at an interface between materials having positive and negative permittivities, typically a dielectric and a metal. The surface plasmons can propagate only when the resonance condition with incident light is satisfied and allow breaking of the diffraction limit for the localization of light into subwavelength dimensions, enabling strong field enhancement. So far, numerous studies utilizing these features of SPPs have been conducted. For example, Nylander et al. first reported using surface plasmon resonance (SPR) for gas detection [10]. This has evolved into an internationally widely adopted method for characterizing and quantifying biomolecular interactions [11]. Another famous application is surface-enhanced Raman scattering (SERS), which is regarded as enhanced Raman scattering due to electromagnetic field enhancement caused by plasmon excitation and a charge transfer process between an adsorbed analyte and a metal surface. SERS allows highly sensitive molecular spectroscopy and has therefore been employed in surface science, electrochemistry, biology, and materials science [12,13].

Magnetoplasmonics combines the features of both magneto-optics and plasmonics and offers the possibility of interesting new applications [14]. Since Hui and Stroud’s proposal [15], it has been known that the MO effect can be enhanced by SPPs. For example, Ni nanowire arrays, which contain less Ni than a continuous film, show comparable polar MOKE (PMOKE) to a continuous film at the resonance frequency of localized SPPs on Ni [16]. Similar enhancement has been observed in nanodisks [17,18], core-shells [19,20], and nanoparticles [21,22] composed of noble metals, ferromagnetic metals, and/or ferromagnetic dielectrics. Such MO enhancement would allow SPR biosensors to be made more sensitive if it employed the TMOKE signal instead of the reflective intensity as the sensing parameter [23,24]. The quest for enhancement of the MO effect with SPPs is also a worthwhile challenge in the fields of information communication and sensing. Recently, Zayets et al. theoretically proposed significant enhancement of the transverse MO effect in a structure consisting of double-layer dielectrics and a ferromagnetic metal [25,26]. This trilayer structure could achieve very small propagation loss of SPPs along with a large nonreciprocal loss change by exploiting the transverse MO effect in the proximity of the cutoff condition of the propagating SPP mode, and therefore, it is expected to be used in integrated optical isolators. What is worthy of attention is that the proposed configuration is the same as the Otto configuration, in which light comes from the outside, even though it was proposed as a waveguide structure, in which light is originally confined inside. In this paper, we follow this work and investigate MO enhancement of the TMOKE and PMOKE in the same trilayer structure consisting of double-layer dielectrics and a ferromagnetic metal, using the attenuated total reflection (ATR) configuration.

2. Calculation model

The structure proposed in the previous studies [25,26] is composed of a ferromagnetic metal, a dielectric of low refractive index, and a dielectric of high refractive index, as shown in Fig. 1.

 

Fig. 1 A trilayer structure consisting of double-layer dielectrics and a ferromagnetic metal for nonreciprocal plasmonic propagation. The yellow curve represents the distribution of the SPPs.

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There is a specific range of thicknesses of the low-index layer at which a propagating SPP mode cannot be supported because of the different refractive indices of the two dielectrics. When the low-index layer is very thin, the SPPs’ effective index is close to the refractive index of the high-index layer, and in contrast, when the low-index layer is very thick, the refractive index is close to that of the low-index layer. When there is a big difference of the effective index, it cannot continuously change. As a result, there is no supported mode at intermediate thicknesses of the low-index layer, and the propagating SPP mode is thus cut off. Near the cutoff condition, the small change of the refractive index of the metal or dielectric causes a significant change of the propagation constant of SPPs. Therefore, MO effect produces enhancement on the nonreciprocal change of the propagation constant in this condition.

We calculated the enhancement of the TMOKE and PMOKE in such a trilayer structure, specifically an Al2O3/SiO2/Fe trilayer, with the ATR configuration. Figure 2 shows the geometry used for our calculation,which assumes the following optical properties, from the top to the bottom: Al2O3 with a refractive index of nAl2O3 = 1.746 [27] as the high-index layer, SiO2 with nSiO2 = 1.44 [28] as the low-index layer with an arbitrary thickness, t, and Fe with nFe = 3.62−5.56i [29] and an off-diagonal permittivity of γ = 3.12 + 1.8i [30] as the ferromagnetic metal with a thickness of 300 nm, which is enough to isolate the structure from the SiO2 substrate. Light with p-polarization and a wavelength of 1550 nm is incident from the Al2O3 side and is reflected by the SiO2/Fe layers. The amount of light reflected at the SiO2 surface is calculated while varying the thickness of the SiO2 layer, t, and the incident angle, θ, at a resolution of 0.01°. The MO effect is expected to be enhanced at an incident angle corresponding to the surface plasmon resonance, where the reflectance falls sharply. Because of the PMOKE, the polarization of light is rotated after reflection; therefore, the calculations cannot be performed with separate orthogonal polarizations, such as TE and TM modes. Also, it is essential to handle a lossy ferromagnetic metal. For these reasons, we employ the transfer matrix method using a full vector 4 × 4 scattering-matrix to calculate the light reflected by the magnetized lossy multilayer structure. A detailed explanation of this method is given in the Appendix. The TMOKE is calculated using Eq. (32) in the Appendix with the relative permittivity tensor

εr=(n20γ0n20γ0n2),
and the PMOKE is calculated using Eqs. (38) and (39) with the relative permittivity tensor
εr=(n2γ0γn2000n2).
The relative permeability tensor is always defined as μr = I since the optical regime is assumed.

 

Fig. 2 A schematic diagram of the Al2O3/SiO2/Fe trilayer deposited on a SiO2 substrate. The incident light, defined by orthogonal unit vectors ÂTM and ÂTE, comes from the Al2O3 side with p-polarization (ϕ = 0, CTE = 0) and an arbitrary incident angle, θ. Arrows M indicate the positive magnetization directions of the PMOKE and TMOKE.

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3. Results and discussion

3.1 Enhancement of TMOKE

Let us begin our investigation with the TMOKE. Figure 3 shows the calculated ATR curve of the Al2O3/SiO2/Fe trilayer.

 

Fig. 3 Reflectance, R, and change in reflectance, ΔR, of the TMOKE as a function of incident angle, θ, at SiO2 thicknesses, t, of (a) 645 nm and (b) 661 nm. Enlarged views depict the reflectance in decibel units. Red, blue, and black curves denote positive, negative, and no magnetization, respectively. The inset in (a) shows the case of s-polarization.

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The incident light is assumed to be p-polarized, and magnetization is set along the y-axis in this case. The change in reflectance due to the TMOKE,

ΔR=R(M)R(+M) [dB],
is largest around the minimum reflectance. The extreme values of ΔR are calculated to be −32.4 dB (θ = 55.07°) and 34.6 dB (θ = 54.93°) at the SiO2 thicknesses of 645 nm (Fig. 3(a)) and 661 nm (Fig. 3(b)), respectively. As can be seen in the close-ups, ΔR is largest when the plasmon resonance with either + or − magnetization direction occurs. The TMOKE changes the permittivity of Fe depending on the magnetization direction. Thus, the wave vector of the SPPs on the Fe also changes, and the incident angle corresponding to the SPPs’ wave vector differs with respect to the magnetization. At the thickness of 645 nm, the SPPs on the Fe magnetized in the −y direction (−M) are coupled with the incident light having the wave vector corresponding to θ = 55.07° more effectively than those magnetized in the + y direction ( + M); therefore, the blue curve (−M) has the minimum value due to plasmon excitation and deviates more from the red curve ( + M). On the other hand, strong coupling of the SPPs with magnetization in the + y direction ( + M) occurs at the thickness of 661 nm with the light having a wave vector corresponding to θ = 54.93°. In the case of the incident light having s-polarization, the light cannot interact with the transversally magnetized Fe and its SPPs. Thus, no TMOKE and no sharp drop in the reflectance are observed as shown in the inset of Fig. 3(a). In Fig. 4(a), the maximum ΔR (black line) and the often-used criterion
ΔRR=R(M)R(+M) [%](R(M)+R(+M) [%])/2,
(green line) are plotted as a function of the SiO2 thickness from 620 nm to 680 nm, and are compared with the minimum reflectance ( + M in red, and −M in blue in Fig. 4(b)).

 

Fig. 4 SiO2 thickness dependence of (a) maximum ΔR (black) and ΔR/R (green) and (b) minimum reflectance, R, ( + M in red, −M in blue) of the TMOKE. The values of (a) and (b) are obtained at incident angles shown in (c) and (d), respectively.

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Note that ΔR/R (Eq. (4)) is a function of both the MO effect (ΔR) and plasmonic absorption (R). Therefore, we prefer to mainly use ΔR (Eq. (3)) in this paper in order to clear the contribution from the MO effect. ΔR and ΔR/R are maximized at 645 nm and 661 nm, as already depicted in Fig. 3. These peaks coincide very closely with the thicknesses at which the minimum reflectances occur, as shown in Figs. 4(a) and 4(b). This means that enhancement of TMOKE can be maximized at the SiO2 thickness where the light is coupled with SPPs in the +/− magnetization direction most efficiently. It should be noted that the ATR curves for + and − magnetizations are not perfectly overlapped at any thicknesses; therefore, ΔR and ΔR/R can always be obtained without having zero even when the minimum R is comparable for + and − magnetizations. That is why a gap is shown in ΔR and ΔR/R around t = 653 nm in Fig. 4(a). We also examined the SiO2 thickness dependence from the point of view of the propagation constant of the SPPs in the Al2O3/SiO2/Fe trilayer by the effective index method (EIM). Figure 5 shows the effective refractive index, Neff, as a function of the SiO2 film thickness, t.

 

Fig. 5 (a) Real part and (b) imaginary part of the effective refractive index, Neff, of the SPPs with the TMOKE as a function of the SiO2 thickness. Red, blue, and black curves denote positive, negative, and no magnetization, respectively.

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As the SiO2 film thickness decreases, the Neff of the SPPs decreases, and eventually no SPP mode is supported. This is because the light tends to be distributed into the higher-index material; therefore, with decreasing t, the influence of the high-index Al2O3 layer becomes more significant, and finally the confinement of SPPs at the SiO2/Fe boundary reaches the cutoff condition, as explained in Sect. 2. This cutoff thickness of the SiO2 varies nonreciprocally, because the TMOKE changes the Neff of the SPPs depending on the magnetization. Neff of + M becomes the cutoff condition at 661 nm, where Neff = 1.4290 and θ = arcsin(Neff/1.746) = 54.93°. On the other hand, Neff of −M becomes the cutoff condition at 645 nm, where Neff = 1.4314 and θ = 55.07°. These thicknesses and angles are quite consistent with those obtained in the ATR configuration. It is worth mentioning that the Neff of the SPPs is usually greater than that of the adjacent dielectric; however this Al2O3/SiO2/Fe trilayer has a smaller Neff than that of SiO2 (n = 1.44), and that is why the incident angles at the minimum reflectance are also smaller than the critical angle of the SiO2/Al2O3 interface, namely, arcsin(1.44/1.746) = 55.56° (see Fig. 3). To show what happens at the cutoff thickness, Fig. 6 shows the optical field distribution (magnetic field, H) assuming that the Al2O3/SiO2/Fe trilayer is a plasmonic waveguide and is negatively magnetized (−M).

 

Fig. 6 Optical field distribution of TM mode in the Al2O3/SiO2/Fe trilayer waveguide with t = ∞, 1645, 1145, and 645 nm. The curves are normalized at 0 nm.

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In the case of t = ∞ (black curve indicated by arrow ∞), the field profile shows the form of a typical surface wave. When the SiO2 thickness reduces along the broken red arrow in Fig. 6, the field confinement on the Fe surface leaks and forms a hybrid mode of light and SPPs. The light exponentially decays inside the SiO2 (SPPs-like) and oscillates inside the Al2O3 (guided-like). It is not pure SPPs which should decay exponentially away from the metal-dielectric interface. The Al2O3/SiO2/Fe trilayer with finite thickness of SiO2 (particularly, less than t ≈1000 nm, where Re[Neff] < 1.44) is a kind of a metal-clad waveguide of leaky mode instead of a plasmonic waveguide; therefore, the effective index, Neff, shows smaller index than that of SiO2 which can be regarded as a core. To couple a standard light wave on to SPPs, conversion from a light wave to an evanescent wave is necessary using such as Kretschmann or Otto configuration. In the case of t = ∞ in Fig. 6, the field forms a surface wave which is always evanescent throughout the structure; therefore, a light wave cannot directly couple to this mode. On the other hand, in the case of t = 645 nm, the field is converted from a light wave inside the Al2O3 to an evanescent wave inside the SiO2. This is what happens in the Otto configuration. The thickness corresponding to the best conversion was revealed to be the cutoff thickness, where the reflectance falls to the minimum, increasing the separation between the opposite magnetization directions.

3.2 Enhancement of PMOKE

Now, we discuss the PMOKE. Figure 7 shows the calculated ATR curve for the magnetization along the z-axis (PMOKE) with p-polarized incident light for a thickness t = 652 nm.

 

Fig. 7 Reflectance, R, and change in reflectance, ΔR, of the PMOKE as a function of incident angle, θ, at t = 652 nm. The enlarged view depicts the reflectance in decibel units. The curve for positive magnetization (red) is perfectly overlapped with that for negative magnetization (blue). The black curve denotes the no magnetization case.

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A reflectance drop indicating the SPP resonance is observed at θ = 54.98°. The PMOKE produces nonreciprocal modification only in the polarization, i.e., rotation and ellipticity. Therefore, nonreciprocal reflectance is not obtained; that is, ΔR = 0. However, the reflectance reciprocally changes by a maximum of 18.7 dB depending on whether the Fe is magnetized ( ± M) or not (M = 0), because the permittivity of the Fe is changed by the MO effect in any magnetization direction; as a result, the Neff of the SPPs can be different from that in the demagnetized state. In Fig. 8, the Kerr rotation (red curve) and ellipticity (blue curve) at t = 652 nm are plotted as a function of the incident angle.

 

Fig. 8 The Kerr rotation (red), ϕK, and ellipticity (blue), ηK, at t = 652 nm as a function of the incident angle with positive magnetization, + M. Insets show the polarization normalized by the maximum norm at each angle indicated by arrows.

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Note that the PMOKE is an odd function of magnetization, and −M is perfectly opposite in sign to + M of Fig. 8. The PMOKE is maximized around θ = 55° and almost reaches the upper limits (|ϕK| ≤ 90°, |ηK| ≤ 45°), as shown in the insets. Figure 9(a) shows how the maximum values of the rotation and ellipticity (|ϕK|, |ηK|) in the PMOKE depend on the SiO2 film thickness, compared with the minimum reflectance in Fig. 9(b).

 

Fig. 9 SiO2 thickness dependence of (a) the maximum Kerr rotation, |ϕK| (red), ellipticity, |ηK| (blue), and (b) the minimum reflectance, R (black). The values of (a) and (b) are obtained at incident angles shown in (c) and (d), respectively.

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The minimum reflectance is obtained at θ = 54.98° for t = 652 nm, which is in good agreement with Neff = 1.4299 (θ = 54.98°) at the cutoff thickness of the SiO2 without magnetization (black curve in Fig. 5). Unlike the TMOKE, the Neff of the SPPs varies very little from the demagnetized state by the PMOKE, and thus, the cutoff thickness is almost the same as that without magnetization. That is why the PMOKE is enhanced and reaches 90°-rotation and 44°-ellipticity in the vicinity of this thickness. As discussed in the latter part of this section, the large rotation of ϕK is attributed to almost perfect absorption of the p-polarization component of the incident light while a little amount of s-polarization remains. Consequently, the rotation cannot exceed 90°. In addition, our calculation is based on arctangent (Eq. (38)) which limits the rotation to 90°.

The mechanism by which the PMOKE is enhanced by SPPs can generally be explained as follows [31]. The PMOKE in complex form, ϕ˜K, can be defined by

ϕ˜K=rsprpp
where rsp is the reflection component of the s-polarization generated from the p-polarization (i.e., pure MO contribution), and rpp is the reflection component of the p-polarization generated from the p-polarization (i.e., pure optical contribution). When the SPP resonance occurs, rpp is obviously decreased, enhancing ϕ˜K. As for rsp, the polarization conversion by the MO effect is proportional to the electromagnetic field inside the MO material, according to the expression
|rsp|Es(z) ΔεM Ep(z)dz,
where Es(z) and Ep(z) are the orthogonal electric fields (p-polarization, Ep, and s-polarization, Es) inside a magnetized anisotropic layer that is homogeneous in the x-y plane, and ΔεM is a permittivity tensor of the anisotropic part caused by magnetization. The enhancement of the electromagnetic field is one of the notable features of SPPs; therefore, Es(z) and Ep(z) increase and then rsp can also be increased. Both rpp and rsp contribute to enhancing the PMOKE under SPP resonance.

In order to analyze the PMOKE enhancement in our case, the reflected electric field components of |Es|, |Ep|, arg(Es)/π, arg(Ep)/π, Δφ = (arg(Ep) ‒ arg(Es))/π, and the PMOKE of + M around the cutoff thicknesses (645, 652, and 660 nm) are illustrated in Fig. 10.

 

Fig. 10 Incident angle dependence at t = 645, 652 and 660 nm: (a) The absolute amplitude of the p-polarization, Ep (red), and induced s-polarization, Es (blue). (b) The normalized phase shift of each argument, arg(Ep) (red) and arg(Es)(blue), and phase difference, Δφ (black). (c) The Kerr rotation (red) and ellipticity (blue).

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It is revealed that the amplitude of the reflected s-polarization, |Es| (blue curve in Fig. 10(a)), and its phase shift, arg(Es) (blue curve in Fig. 10(b)), which arise from the pure MO effect, are not very dependent on the incident angle and the thickness. This means that the SPP resonance in our trilayer system does not contribute to the MO effect by means of enhancing the s-polarization even at the cutoff thickness. Let us discuss the enhancement of the Kerr ellipticity first. It should be noted that the ellipticity can be 45° when both |Ep| = |Es| and |arg(Ep) ‒ arg(Es)| = π/2 are satisfied simultaneously. This occurs twice when |Ep| reduces and approaches |Es| around the SPP resonance near θ = 55°, where |Δφ| ≈π/2 is also satisfied, as indicated by the vertical broken lines in Figs. 10(a)-10(c). With regard to the enhancement of the Kerr rotation, the reductions of |Ep| due to the SPP resonance makes the proportion of |Es| equal or even surpass the proportion of |Ep|, and the rotation increases over 45°. At θ ≈55° of t = 652 nm, |Ep| almost vanishes in conjunction with Δφ = 0, and thus the Kerr rotation reaches 90°. The enhancement of the Kerr ellipticity and rotation is a mainly consequence of the strong absorption of the p-polarized component of the incident light, |Ep|, at the SPP resonance.

It is worth mentioning that the PMOKE in Fig. 10(c) shows a typical resonance curve, which is commonly observed in the spectra of optical rotatory dispersion (ORD) and the circular dichroism dispersion relations and is governed by Kramers-Kronig relations [32,33]. It is known that they exhibit resonance enhancement at a characteristic frequency of polarization charge in a material. Since sweeping of the incident angle, θ, in the ATR measurement corresponds to sweeping the wave vector, k, and the angular frequency, ω (θkω), Fig. 10(c) clearly shows that the collective oscillation of the SPPs at the resonance works as the characteristic frequency and enhances the MO effect.

3.3 Barriers to implementation

In this section, we focus interest on a deviation from the ideal condition which causes barriers to practical implementation. In practice, the interfaces of deposited layers are not perfectly smooth. First, we consider the case that the SiO2 and Fe layers are fluctuating, and they partially coexist at the interface. We assume the interface layer between the SiO2 and Fe layers, which yields blurred optical constants, namely, (nsio2 + nFe)/2 and γ/2. Figure 11 shows the ATR curves of the TMOKE, assuming that the thickness of the interface layer is tint = 0, 1 and 3 nm with the SiO2 layer of t = 661 − tint/2 nm and the Fe layer of 300 − tint/2 nm.

 

Fig. 11 Reflectance, R (positive (red), negative (blue) and no (black) magnetizations), and change in reflectance, ΔR (broken curve), of The TMOKE as a function of the incident angle. The thickness of the interface layer is assumed to be (a) tint = 0 nm, (b) 1 nm and (c) 3 nm with the SiO2 layer of t = 661 − tint/2 nm and the Fe layer of 300 − tint/2 nm.

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Only tint = 1 nm insertion of the interface layer decreases ΔR to less than half of tint = 0 nm (The interface is ideally smooth.). Figure 12 shows the PMOKE assuming the same interface layer as the TMOKE and the SiO2 layer of t = 652 − tint/2 nm.

 

Fig. 12 The Kerr (a) rotation and (b) ellipticity as a function of the incident angle. The thickness of the interface layer is assumed to be tint = 0, 1, 3 and 5 nm as written beside the curves, and the SiO2 layer is set to t = 652 − tint/2 nm and the Fe layer of 300 − tint/2 nm.

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With increasing the interface layer up to only tint = 5 nm, the rotation and ellipticity of the PMOKE are significantly reduced and broadened. According to Figs. 11 and 12, the smooth interface between the SiO2 and Fe layers without the interface layer is necessary to obtain the huge MO effect.

Second, a deviation in the polarization of incident light deserves attention. The aforementioned MO enhancement always assumed ideally p-polarized incident light. However, it is hardly achieved in a practical setup. We assume the incident light having a deviation from ideal p-polarization. Figure 13 shows the ATR curves of the TMOKE, assuming a deviation angle of 0°, 1°, and 3° from the p-polarization to the s-polarization at a thickness of SiO2 layer of t = 661 nm.

 

Fig. 13 Reflectance, R (positive (red), negative (blue) and no (black) magnetizations), and change in reflectance, ΔR (broken curve), of The TMOKE as a function of the incident angle. The deviation angle from p-polarization is assumed to be (a) tint = 0°, (b) 1° and (c) 3° with the SiO2 layer of t = 661 nm.

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A deviation angle of 1° implies only 1.7% worth of incident light is s-polarized. However, Fig. 13(b) shows ~30-dB reduction in ΔR from ideal p-polarization shown in Fig. 13(a). This is because the incident s-polarization is not absorbed by the SPP resonance and moderates steep reduction of reflectance at the SPP resonance. Figure 14 shows the same case for the PMOKE assuming a deviation angle up to 10° at a thickness of SiO2 layer of t = 652 nm.

 

Fig. 14 The Kerr (a) rotation and (b) ellipticity as a function of the incident angle. The deviation angle from p-polarization is assumed to be 0° up to 10°.as written beside the curves, and the SiO2 layer is set to t = 652 nm.

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In this case, the rotation of the PMOKE in Fig. 14(a) increases and broadens with respect to the incident angle as the deviation angle increases. This can be explained as follows: The deviation angle adds the s-polarization component and makes the s-polarization more significant when the p-polarization component reduces at the SPP resonance. This leads to more rotation than without initial s-polarization component. The reason of the reduction shown in the ellipticity of the PMOKE in Fig. 14(b) with increasing the deviation angle can also be due to increased s-polarization component. The incident angle at which the amplitude of p-polarization is comparable to that of the s-polarization becomes away from the angle of the SPP resonance, θ ≈55°, with increasing the s-polarization component. However, the phase shift, |Δφ|, decreases from π/2 as the incident angle separates from θ ≈55°, and then |Ep| = |Es| and |Δφ| = π/2 cannot be satisfied simultaneously.

3.4 Summary of the MO effect in the Al2O3/SiO2/Fe trilayer

The preceding investigation of the TMOKE and PMOKE is summarized as follows. The TMOKE shows a reflectance change of more than ΔR = 10 dB within the thickness range 620 nm ≤ t ≤ 680 nm, and maximum values of ΔR = 34 dB and ΔR/R = 200% are obtained at t = 661 nm and 645 nm, which correspond to the cutoff thickness of the low-index SiO2 layer with magnetizations of + M and –M, respectively. The PMOKE shows rotation and ellipticity of more than |ϕK| = 20° and |ηK| = 10° within the thickness range 620 nm ≤ t ≤ 680 nm, and the maximum values are the rotation of |ϕK| = 90° in the range 650 nm ≤ t ≤ 655 nm and the ellipticity of |ηK| = 44° at t = 650 nm and 655 nm. The center thickness, t = 652 nm, corresponds to the cutoff thickness of the SiO2 layer without magnetization. Previous studies using SPP enhancement have reported that the PMOKE usually shows a rotation of around 1° and the TMOKE shows a ΔR/R value of about 1−70% [3440]. In the case of s-polarization, light cannot couple with the SPPs and cannot interact with the magnetization transverse to the plane of incidence. Therefore, the ATR curve shows total reflection above θ ≈60° and no TMOKE is obtained throughout incident angles at every thickness. The PMOKE only shows maximum values of |ϕK| ≈1° and |ηK| ≈0.4° near the normal incident angle at every thickness.

These huge enhancements on the TMOKE and PMOKE are, however, achieved only in the ideal condition, where the SiO2 and Fe layers are deposited without interface roughness, and pure p-polarization incident light is projected.

4. Conclusions

We have demonstrated giant enhancement of the transverse and polar magneto-optical Kerr effects (TMOKE and PMOKE) by surface plasmon polaritons (SPPs) in a trilayer structure composed of double-layer dielectrics and a ferromagnetic metal (Al2O3/SiO2/Fe). We investigated the behavior of the enhancement of TMOKE and PMOKE using the attenuated total reflection (ATR) configuration by the transfer matrix method with a 4 × 4 scattering matrix. The calculated TMOKE showed a reflectance change of about 34 dB upon magnetization reversal, and the PMOKE showed orthogonal transformation (90°-rotation) and almost full-orbed deformation (44°-ellipticity) with just a single reflection at a SiO2 thickness around the cutoff thickness. Based on our analysis, this enhancement can be accounted for by the SPPs in the Al2O3/SiO2/Fe trilayer, which transform an incoming light wave to an evanescent wave most effectively at the cutoff thickness of the low-index SiO2 layer and are thus most easily intertwined with the SPPs on the Fe surface. The enhancement of the TMOKE is mainly due to the larger difference in the effective index between the opposite magnetization directions at the cutoff thicknesses. The enhancement of the PMOKE is not due to optical field enhancement by the SPPs, but is mainly due to the strong absorption of the incident p-polarized light caused by the SPP resonance at the cutoff thickness. Although the condition to achieve this huge enhancement is very strict, the structure proposed here is simpler than those such as nanowires, nanodisks, core-shells or perforated membranes, and needs only two dielectrics of high and low refractive indices deposited continuously on one ferromagnetic metal, while still giving a huge reflectance change and almost perfect polarization conversion.

Appendix

In this Appendix, we describe how the transfer matrix method with a 4×4 scattering matrix is developed [41], and how the Kerr rotation, ϕK, and ellipticity, ηK, are determined. Given that multiple layers are stacked along the z-direction and are homogeneous in the x-y plane, the formulation begins with Maxwell's curl equations describing the field inside each layer. Assuming that the forward-propagating light wave is given by ei(ωt‒k·r), Maxwell's curl equations can be written as

'×E=μrH˜
'×H˜=εrE
where the magnetic field is normalized according to H˜ = ‒iz0 H, where z0 is the impedance in vacuum, and both sides of the equations are divided by the wave number in vacuum, k0, and defined as /k0 = = [∂x’y’z’]T. Equations (7) and (8) are composed of three-dimensional vectors of the x-, y- and z-axes and are expanded into a set of six partial differential equations. By eliminating the longitudinal field components Ez and H˜z using back substitution, the remaining four equations are in the x-y plane and can be summarized as the following 4 × 4 matrix:
z'ψ=Ωψ
Ω=(i(nxε31ε33+nxμ23μ33)inx(ε32ε33μ23μ33)nxnyε33+μ21μ23μ31μ33nx2ε33+μ22μ23μ32μ33iny(ε31ε33μ13μ33)i(nyε32ε33+nxμ13μ33)ny2ε33μ11+μ13μ31μ33nxnyε33μ21+μ23μ32μ33ε21ε23ε31ε33+nxnyμ33ε22ε23ε32ε33nx2μ33i(nyε23ε33+nxμ31μ33)inx(ε23ε33μ32μ33)ε11+ε13ε31ε33+ny2μ33ε12+ε13ε32ε33nxnyμ33iny(ε13ε33μ31μ33)i(nxε13ε33+nyμ32μ33))
ψ=[ExEyH˜xH˜y]T
In Eq. (10), εij and μij respectively stand for the element of the relative permittivity and relative permeability tensors in the i-th row and j-th column, and spatial derivative operators were replaced with the refractive indices inside each layer by using the relationship = [‒inx ‒inyz’]T, where n and its subscripts represent the refractive index and corresponding axis on the indicatrix, respectively. Equation (9) can then be solved as follows
ψ(z)=WeλzW1ψ(0)
where W is the eigenvector matrix formed by all column eigenvectors, and eλ is the matrix formed by all diagonally sorted eigenvalues, λ, of Ω and sorted according to the forward or backward propagation. By defining c = W−1·Ψ(0), Ψ inside the i-th layer can be written in block matrix form as
ψ(z')=Wieλiz'ci=(WEi+WEiWHi+WHi)(eλiz'00eλiz')(ci+ci).
In this equation, W ±E and W ±H respectively describe the eigen-modes of the electric and magnetic fields for the forward ( + ) and backward (−) propagations. Note that each element of the matrix in Eq. (13) is actually a 2 × 2 matrix, and the total 4 × 4 matrix is in Ψ. The transfer matrix method is formulated by obeying the boundary conditions. Let us assume that the (i−1)-th layer to the (i + 1)-th layer across i-th layer has thickness Li. By eliminating c+i and ci of the amplitude coefficients of the eigen-modes in the i-th layer, we can connect the (i−1)-th and (i + 1)-th layers by

(WEi+WEiWHi+WHi)1(WEi1+WEi1WHi1+WHi1)(ci1+ci1)=(eλik0Li00eλik0Li)(WEi+WEiWHi+WHi)1(WEi+1+WEi+1WHi+1+WHi+1)(ci+1+ci+1)

The scattering matrix has been widely popular since the introduction of microwave network analyzers capable of swept amplitude and phase measurements. However, the elements of a scattering matrix are dependent on the materials in the surroundings. This prevents the matrix of each layer from being defined separately and interchanged arbitrarily. To improve this, Rumpf proposed surrounding each layer with free space gaps with zero thickness [42]. By letting the (i−1)-th and (i+1)-th layers be the same free space, namely, the same eigen-modes, Eq. (14) reduces to

(WEi+'WEi'WHi+'WHi')(ci1+ci1)=(eλik0Li00eλik0Li)(WEi+'WEi'WHi+'WHi')(ci+1+ci+1)
where
(WEi+'WEi'WHi+'WHi')=(WEi+WEiWHi+WHi)1(WEi±1+WEi±1WHi±1+WHi±1).
In this setting, the scattering matrices can be virtually independent of the adjacent layers and are interchangeable. After some algebra and transposition, we have the improved scattering matrix of the i-th layer:
(ci1ci+1+)=(S11iS12iS21iS22i)(ci1+ci+1)
S11i=(A1i)1((WHi')1λiWHi+'(WEi+')1λiWEi+'(WHi')1WHi+')
S12i=(A1i)1((WHi')1λiWHi'(WHi')1λiWHi'(WEi+')1WEi')
S21i=(A2i)1((WEi+')1λiWEi'(WHi')1WHi+'+(WEi+')1λiWEi+')
S22i=(A2i)1((WEi+')1WEi'+(WEi+')1λiWEi'(WHi')1λiWHi')
A1i=I(WHi')1λiWHi+'(WEi+')1λiWEi'
A2i=I(WEi+')1λiWEi'(WHi')1λiWHi+'
where λi is the eigen value matrix of the matrix Ω for the i-th layer. For a multilayer structure, this scattering matrix is defined individually, and the individual matrices are concatenated into one global scattering matrix using Redheffer’s star product [42]. If layers A and B are combined, one scattering matrix, SA SB, will be given by
SASB=(S11a+S12a(IS11bS22a)1S11bS21aS12a(IS11bS22a)1S12bS21b(IS22aS11b)1S21aS21b(IS22aS11b)1S22aS12b+S22b)
where

SA/B=(S11a/bS12a/bS21a/bS22a/b).

The scattering matrices for the reflection region, Sref, and the transmission region, Strn, are separately defined as follows. At the interface between the reflection region and the 1st layer, in-plane components should be continuous without phase accumulation. This can be formulated based on Eq. (14) without any thickness, that is, L = 0. Then we have another scattering matrix, Sref, which connects the reflection/incident coefficients of the reflection region, cref/inc, and the 1st layer, c1:

(crefc1+)=(S11refS12refS21refS22ref)(cinc+c1),
The scattering matrix between the n-th (last) layer and the transmission region, Strn, can be obtained from Eq. (26) by replacing the reflection region with the n-th layer and the 1st layer with the transmission region. The global scattering matrix associates the electromagnetic fields from the reflection region through the transmission region: Sglobal = SrefS1S2SnStrn. The electromagnetic field coming from the transmission region is typically assumed to be 0:
(crefctrn+)=(S11globalS12globalS21globalS22global)(cinc+ctrn=0).
Then, the reflected light at z = 0 is calculated from Eq. (13) as
ψ(0)=(WEref+WErefWHref+WHref)(cinc+cref)(Exref(0)Eyref(0)H˜xref(0)H˜yref(0))=(WErefcrefWHrefcref)=(WErefS11globalcinc+WHrefS11globalcinc+)
and the transmitted light at the end of the stack, z = Ltotal, is calculated by
ψ(Ltotal)=(WEtrn+WEtrnWHtrn+WHtrn)(ctrn+ctrn=0)(Extrn(Ltotal)Eytrn(Ltotal)H˜xtrn(Ltotal)H˜ytrn(Ltotal))=(WEtrn+ctrn+WHtrn+ctrn+)=(WEtrn+S21globalcinc+WHtrn+S21globalcinc+)
The normal components are given by
Ezref/trn=i(H˜yref/trnnx+H˜xref/trnny)Exref/trnε31ref/trnEyref/trnε32ref/trnε33ref/trnH˜zref/trn=i(Eyref/trnnx+Exref/trnny)H˜xref/trnμ31ref/trnH˜yref/trnμ32ref/trnμ33ref/trn
From Eq. (28), the incident electric field can be introduced into c+inc with arbitrary polarization as
cinc+=(WEref+)1(Exinc(0)Eyinc(0)).
After all, the reflectance of the multilayer stack, R, can be calculated as

R=(ErefEinc)2.

In what follows, we describe how the rotation and ellipticity are determined. Assuming that reflected light having complex amplitude is projected onto the x-y plane, the x and y components draw a Lissajous figure:

[xy]T=[(Exr+iExi)eiωt(Eyr+iEyi)eiωt]T=[Exrcos(ωt)Exisin(ωt)Eyrcos(ωt)Eyisin(ωt)]T
This can be written in the standard form of a quadratic curve as
cos2(ωt)+sin2(ωt)=(Eyi2+Eyr2)x2(ExrEyiExiEyr)22(ExiEyi+ExrEyr)xy(ExrEyiExiEyr)2+(Exi2+Exr2)y2(ExrEyiExiEyr)2=1
By defining the symmetric matrix, Q, Eq. (34) is expressed in matrix form as
XTQX=1Q=(Eyi2+Eyr2(ExrEyiExiEyr)2ExiEyi+ExrEyr(ExrEyiExiEyr)2ExiEyi+ExrEyr(ExrEyiExiEyr)2Exi2+Exr2(ExrEyiExiEyr)2)X=[xy]T
By using eigenvector matrix, P, of Q, the matrix Q can be diagonalized as B = P−1 Q P. And Eq. (35) can then be expressed as X’T B X’ = 1, where the diagonal matrix, B, implies an elliptic curve in a certain coordinate system, X’ = [x’ y’]T. Based on linear algebra, X’ can be related to X as follows.
X'TBX'=1(X'TP1)Q(PX')=1(PX')TQ(PX')=1X=PX'
It should be noted that P describes the rotation matrix associating the coordinates X and X’,
P=(cos(φK)sin(φK)sin(φK)cos(φK))
As a result, the Kerr rotation, ϕK, and ellipticity, ηK, will be given by
φK=arctan[2(E'xiEyi+E'xrEyr)E'xi2+E'xr2Eyi2Eyr2+4(E'xrEyiE'xiEyr)2+(Eyi2+Eyr2+E'xi2+E'xr2)2]
ηK=arctan[Eyi2+Eyr2+E'xi2+E'xr24(E'xrEyiE'xiEyr)2+(Eyi2+Eyr2+E'xi2+E'xr2)2Eyi2+Eyr2+E'xi2+E'xr2+4(E'xrEyiE'xiEyr)2+(Eyi2+Eyr2+E'xi2+E'xr2)2]
In our calculation, the reflection amplitude is corrected by taking account of the angle of reflection as E’x = Ex sec(θ), and the sign of ηK is determined by the phase difference between the orthogonal electric fields. In the case where the incident light is p-polarized, ηK is multiplied by −1 when the phase difference between the s- and p-polarizations, Δφ = arg(Es) − arg(Ep), satisfies −π ≤ Δφ < 0 or π ≤ Δφ.

Acknowledgments

This work was supported by a Grant-in-Aid for Scientific Research (No. 24686045), from the Ministry of Education, Culture, Sports, Science, and Technology, Japan.

References and links

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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]  

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6. Y. Shoji, Y. Shirato, and T. Mizumoto, “Silicon Mach–Zehnder interferometer optical isolator having 8 nm bandwidth for over 20 dB isolation,” Jpn. J. Appl. Phys. 53(2), 022202 (2014). [CrossRef]  

7. Y. Sobu, Y. Shoji, K. Sakurai, and T. Mizumoto, “GaInAsP/InP MZI waveguide optical isolator integrated with spot size converter,” Opt. Express 21(13), 15373–15381 (2013). [CrossRef]   [PubMed]  

8. H. Shimizu and Y. Nakano, “First Demonstration of TE Mode Nonreciprocal Propagation in an InGaAsP/InP Active Waveguide for an Integratable Optical Isolator,” Jpn. J. Appl. Phys. 43(12A12A), L1561–L1563 (2004). [CrossRef]  

9. H. Shimizu and Y. Nakano, “Fabrication and characterization of an InGaAsP/InP active waveguide optical isolator with 14.7 dB/mm TE mode nonreciprocal attenuation,” J. Lightwave Technol. 24(1), 38–43 (2006). [CrossRef]  

10. C. Nylander, B. Liedberg, and T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators 3, 79–88 (1982). [CrossRef]  

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12. K. Hering, D. Cialla, K. Ackermann, T. Dörfer, R. Möller, H. Schneidewind, R. Mattheis, W. Fritzsche, P. Rösch, and J. Popp, “SERS: a versatile tool in chemical and biochemical diagnostics,” Anal. Bioanal. Chem. 390(1), 113–124 (2008). [CrossRef]   [PubMed]  

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15. P. M. Hui and D. Stroud, “Theory of Faraday rotation by dilute suspensions of small particles,” Appl. Phys. Lett. 50(15), 950–952 (1987). [CrossRef]  

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References

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  1. H. J. Williams, R. C. Sherwood, F. G. Foster, and E. M. Kelley, “Magnetic Writing on Thin Films of MnBi,” J. Appl. Phys. 28(10), 1181–1184 (1957).
    [Crossref]
  2. R. J. Gambino and T. Suzuki, Magneto-Optical Recording Materials (IEEE, 2000).
  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]
  4. T. Shintaku and T. Uno, “Optical waveguide isolator based on nonreciprocal radiation,” J. Appl. Phys. 76(12), 8155–8159 (1994).
    [Crossref]
  5. T. Shintaku, “Integrated optical isolator based on efficient nonreciprocal radiation mode conversion,” Appl. Phys. Lett. 73(14), 1946–1948 (1998).
    [Crossref]
  6. Y. Shoji, Y. Shirato, and T. Mizumoto, “Silicon Mach–Zehnder interferometer optical isolator having 8 nm bandwidth for over 20 dB isolation,” Jpn. J. Appl. Phys. 53(2), 022202 (2014).
    [Crossref]
  7. Y. Sobu, Y. Shoji, K. Sakurai, and T. Mizumoto, “GaInAsP/InP MZI waveguide optical isolator integrated with spot size converter,” Opt. Express 21(13), 15373–15381 (2013).
    [Crossref] [PubMed]
  8. H. Shimizu and Y. Nakano, “First Demonstration of TE Mode Nonreciprocal Propagation in an InGaAsP/InP Active Waveguide for an Integratable Optical Isolator,” Jpn. J. Appl. Phys. 43(12A12A), L1561–L1563 (2004).
    [Crossref]
  9. H. Shimizu and Y. Nakano, “Fabrication and characterization of an InGaAsP/InP active waveguide optical isolator with 14.7 dB/mm TE mode nonreciprocal attenuation,” J. Lightwave Technol. 24(1), 38–43 (2006).
    [Crossref]
  10. C. Nylander, B. Liedberg, and T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators 3, 79–88 (1982).
    [Crossref]
  11. J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108(2), 462–493 (2008).
    [Crossref] [PubMed]
  12. K. Hering, D. Cialla, K. Ackermann, T. Dörfer, R. Möller, H. Schneidewind, R. Mattheis, W. Fritzsche, P. Rösch, and J. Popp, “SERS: a versatile tool in chemical and biochemical diagnostics,” Anal. Bioanal. Chem. 390(1), 113–124 (2008).
    [Crossref] [PubMed]
  13. M. Fleischmann, P. J. Hendra, and A. J. McQuillan, “Raman spectra of pyridine adsorbed at a silver electrode,” Chem. Phys. Lett. 26(2), 163–166 (1974).
    [Crossref]
  14. G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: combining magnetic and plasmonic functionalities,” Adv. Opt. Mater. 1(1), 10–35 (2013).
    [Crossref]
  15. P. M. Hui and D. Stroud, “Theory of Faraday rotation by dilute suspensions of small particles,” Appl. Phys. Lett. 50(15), 950–952 (1987).
    [Crossref]
  16. J. B. González-Díaz, A. García-Martín, G. Armelles, D. Navas, M. Vázquez, K. Nielsch, R. B. Wehrspohn, and U. Gösele, “Enhanced Magneto-Optics and Size Effects in Ferromagnetic Nanowire Arrays,” Adv. Mater. 19(18), 2643–2647 (2007).
    [Crossref]
  17. J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, G. Armelles, B. Sepúlveda, Y. Alaverdyan, and M. Käll, “Plasmonic Au/Co/Au Nanosandwiches with Enhanced Magneto-Optical Activity,” Small 4(2), 202–205 (2008).
    [Crossref] [PubMed]
  18. D. Meneses-Rodríguez, E. Ferreiro-Vila, P. Prieto, J. Anguita, M. U. González, J. M. García-Martín, A. Cebollada, A. García-Martín, and G. Armelles, “Probing the Electromagnetic Field Distribution within a Metallic Nanodisk,” Small 7(23), 3317–3323 (2011).
    [Crossref] [PubMed]
  19. G. Armelles, A. Cebollada, A. García-Martín, J. M. Montero-Moreno, M. Waleczek, and K. Nielsch, “Magneto-optical Properties of Core-Shell Magneto-plasmonic Au-CoxFe3 - xO4 Nanowires,” Langmuir 28(24), 9127–9130 (2012).
    [Crossref] [PubMed]
  20. L. Wang, C. Clavero, Z. Huba, K. J. Carroll, E. E. Carpenter, D. Gu, and R. A. Lukaszew, “Plasmonics and Enhanced Magneto-Optics in Core-Shell Co-Ag Nanoparticles,” Nano Lett. 11(3), 1237–1240 (2011).
    [Crossref] [PubMed]
  21. H. Uchida, Y. Masuda, R. Fujikawa, A. V. Baryshev, and M. Inoue, “Large enhancement of Faraday rotation by localized surface plasmon resonance in Au nanoparticles embedded in Bi:YIG film,” J. Magn. Magn. Mater. 321(7), 843–845 (2009).
    [Crossref]
  22. H. Uchida, Y. Mizutani, Y. Nakai, A. A. Fedyanin, and M. Inoue, “Garnet composite films with Au particles fabricated by repetitive formation for enhancement of Faraday effect,” J. Phys. D Appl. Phys. 44(6), 064014 (2011).
    [Crossref]
  23. B. Sepúlveda, A. Calle, L. M. Lechuga, and G. Armelles, “Highly sensitive detection of biomolecules with the magneto-optic surface-plasmon-resonance sensor,” Opt. Lett. 31(8), 1085–1087 (2006).
    [Crossref] [PubMed]
  24. D. Regatos, B. Sepúlveda, D. Fariña, L. G. Carrascosa, and L. M. Lechuga, “Suitable combination of noble/ferromagnetic metal multilayers for enhanced magneto-plasmonic biosensing,” Opt. Express 19(9), 8336–8346 (2011).
    [Crossref] [PubMed]
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2014 (3)

Y. Shoji, Y. Shirato, and T. Mizumoto, “Silicon Mach–Zehnder interferometer optical isolator having 8 nm bandwidth for over 20 dB isolation,” Jpn. J. Appl. Phys. 53(2), 022202 (2014).
[Crossref]

S. M. Hamidi, M. A. Oskuei, S. Sadeghi, and M. M. Tehranchi, “Enhanced polar magneto-optical Kerr rotation in cobalt thin film incorporating Ag nanoparticles,” J. Supercond. Novel Magn. 27(3), 867–870 (2014).
[Crossref]

H.-T. Huang, P.-J. Chen, T.-R. Ger, Y.-J. Chi, C.-W. Huang, K.-T. Liao, J.-Y. Lai, J.-Y. Chen, W.-Y. Peng, Q. Zhang, T.-F. Hsieh, W.-J. Sheu, and Z.-H. Wei, “Magneto-optical Kerr effect enhanced by surface plasmon resonance and its application on biological detection,” IEEE Trans. Magn. 50(1), 1001604 (2014).
[Crossref]

2013 (5)

S. Zhang, S. Tang, J. Gao, X. Luo, W. Xia, and Y. Du, “Theoretical calculation of magneto-optical properties in cobalt nanotube array with hexagonal symmetry,” Solid State Commun. 170, 19–23 (2013).
[Crossref]

L. E. Kreilkamp, V. I. Belotelov, J. Y. Chin, S. Neutzner, D. Dregely, T. Wehlus, I. A. Akimov, M. Bayer, B. Stritzker, and H. Giessen, “Waveguide-plasmon polaritons enhance transverse magneto-optical Kerr effect,” Phys. Rev. X 3, 041019 (2013).

Y. Sobu, Y. Shoji, K. Sakurai, and T. Mizumoto, “GaInAsP/InP MZI waveguide optical isolator integrated with spot size converter,” Opt. Express 21(13), 15373–15381 (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]

G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: combining magnetic and plasmonic functionalities,” Adv. Opt. Mater. 1(1), 10–35 (2013).
[Crossref]

2012 (4)

G. Armelles, A. Cebollada, A. García-Martín, J. M. Montero-Moreno, M. Waleczek, and K. Nielsch, “Magneto-optical Properties of Core-Shell Magneto-plasmonic Au-CoxFe3 - xO4 Nanowires,” Langmuir 28(24), 9127–9130 (2012).
[Crossref] [PubMed]

B. Caballero, A. García-Martín, and J. C. Cuevas, “Generalized scattering-matrix approach for magneto-optics in periodically patterned multilayer systems,” Phys. Rev. B 85(24), 245103 (2012).
[Crossref]

V. Zayets, H. Saito, S. Yuasa, and K. Ando, “Enhancement of the transverse non-reciprocal magneto-optical effect,” J. Appl. Phys. 111(2), 023103 (2012).
[Crossref]

V. Zayets, H. Saito, K. Ando, and S. Yuasa, “Optical isolator utilizing surface plasmons,” Materials (Basel) 5(5), 857–871 (2012).
[Crossref]

2011 (5)

H. Uchida, Y. Mizutani, Y. Nakai, A. A. Fedyanin, and M. Inoue, “Garnet composite films with Au particles fabricated by repetitive formation for enhancement of Faraday effect,” J. Phys. D Appl. Phys. 44(6), 064014 (2011).
[Crossref]

D. Meneses-Rodríguez, E. Ferreiro-Vila, P. Prieto, J. Anguita, M. U. González, J. M. García-Martín, A. Cebollada, A. García-Martín, and G. Armelles, “Probing the Electromagnetic Field Distribution within a Metallic Nanodisk,” Small 7(23), 3317–3323 (2011).
[Crossref] [PubMed]

D. Regatos, B. Sepúlveda, D. Fariña, L. G. Carrascosa, and L. M. Lechuga, “Suitable combination of noble/ferromagnetic metal multilayers for enhanced magneto-plasmonic biosensing,” Opt. Express 19(9), 8336–8346 (2011).
[Crossref] [PubMed]

R. C. Rumpf, “Improved formulation of scattering matrices for semi-analytical methods that is consistent with convention,” Prog. Electromagn. Res. B 35, 241–261 (2011).
[Crossref]

L. Wang, C. Clavero, Z. Huba, K. J. Carroll, E. E. Carpenter, D. Gu, and R. A. Lukaszew, “Plasmonics and Enhanced Magneto-Optics in Core-Shell Co-Ag Nanoparticles,” Nano Lett. 11(3), 1237–1240 (2011).
[Crossref] [PubMed]

2010 (1)

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, “Giant transversal Kerr effect in magneto-plasmonic heterostructures: The scattering-matrix method,” Sov. Phys. JETP 110(5), 816–824 (2010).
[Crossref]

2009 (1)

H. Uchida, Y. Masuda, R. Fujikawa, A. V. Baryshev, and M. Inoue, “Large enhancement of Faraday rotation by localized surface plasmon resonance in Au nanoparticles embedded in Bi:YIG film,” J. Magn. Magn. Mater. 321(7), 843–845 (2009).
[Crossref]

2008 (3)

J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, G. Armelles, B. Sepúlveda, Y. Alaverdyan, and M. Käll, “Plasmonic Au/Co/Au Nanosandwiches with Enhanced Magneto-Optical Activity,” Small 4(2), 202–205 (2008).
[Crossref] [PubMed]

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108(2), 462–493 (2008).
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K. Hering, D. Cialla, K. Ackermann, T. Dörfer, R. Möller, H. Schneidewind, R. Mattheis, W. Fritzsche, P. Rösch, and J. Popp, “SERS: a versatile tool in chemical and biochemical diagnostics,” Anal. Bioanal. Chem. 390(1), 113–124 (2008).
[Crossref] [PubMed]

2007 (2)

J. B. González-Díaz, A. García-Martín, G. Armelles, D. Navas, M. Vázquez, K. Nielsch, R. B. Wehrspohn, and U. Gösele, “Enhanced Magneto-Optics and Size Effects in Ferromagnetic Nanowire Arrays,” Adv. Mater. 19(18), 2643–2647 (2007).
[Crossref]

J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. García-Martín, C. Clavero, A. Cebollada, R. A. Lukaszew, J. R. Skuza, D. P. Kumah, and R. Clarke, “Surface-magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76(15), 153402 (2007).
[Crossref]

2006 (2)

2004 (1)

H. Shimizu and Y. Nakano, “First Demonstration of TE Mode Nonreciprocal Propagation in an InGaAsP/InP Active Waveguide for an Integratable Optical Isolator,” Jpn. J. Appl. Phys. 43(12A12A), L1561–L1563 (2004).
[Crossref]

2001 (1)

P. Bertrand, C. Hermann, G. Lampel, J. Peretti, and V. I. Safarov, “General analytical treatment of optics in layered structures: Application to magneto-optics,” Phys. Rev. B 64(23), 235421 (2001).
[Crossref]

1998 (1)

T. Shintaku, “Integrated optical isolator based on efficient nonreciprocal radiation mode conversion,” Appl. Phys. Lett. 73(14), 1946–1948 (1998).
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1994 (1)

T. Shintaku and T. Uno, “Optical waveguide isolator based on nonreciprocal radiation,” J. Appl. Phys. 76(12), 8155–8159 (1994).
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1987 (1)

P. M. Hui and D. Stroud, “Theory of Faraday rotation by dilute suspensions of small particles,” Appl. Phys. Lett. 50(15), 950–952 (1987).
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1982 (1)

C. Nylander, B. Liedberg, and T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators 3, 79–88 (1982).
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1976 (1)

1974 (1)

M. Fleischmann, P. J. Hendra, and A. J. McQuillan, “Raman spectra of pyridine adsorbed at a silver electrode,” Chem. Phys. Lett. 26(2), 163–166 (1974).
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1957 (1)

H. J. Williams, R. C. Sherwood, F. G. Foster, and E. M. Kelley, “Magnetic Writing on Thin Films of MnBi,” J. Appl. Phys. 28(10), 1181–1184 (1957).
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Ackermann, K.

K. Hering, D. Cialla, K. Ackermann, T. Dörfer, R. Möller, H. Schneidewind, R. Mattheis, W. Fritzsche, P. Rösch, and J. Popp, “SERS: a versatile tool in chemical and biochemical diagnostics,” Anal. Bioanal. Chem. 390(1), 113–124 (2008).
[Crossref] [PubMed]

Akimov, I. A.

L. E. Kreilkamp, V. I. Belotelov, J. Y. Chin, S. Neutzner, D. Dregely, T. Wehlus, I. A. Akimov, M. Bayer, B. Stritzker, and H. Giessen, “Waveguide-plasmon polaritons enhance transverse magneto-optical Kerr effect,” Phys. Rev. X 3, 041019 (2013).

Alaverdyan, Y.

J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, G. Armelles, B. Sepúlveda, Y. Alaverdyan, and M. Käll, “Plasmonic Au/Co/Au Nanosandwiches with Enhanced Magneto-Optical Activity,” Small 4(2), 202–205 (2008).
[Crossref] [PubMed]

Ando, K.

V. Zayets, H. Saito, S. Yuasa, and K. Ando, “Enhancement of the transverse non-reciprocal magneto-optical effect,” J. Appl. Phys. 111(2), 023103 (2012).
[Crossref]

V. Zayets, H. Saito, K. Ando, and S. Yuasa, “Optical isolator utilizing surface plasmons,” Materials (Basel) 5(5), 857–871 (2012).
[Crossref]

Anguita, J.

D. Meneses-Rodríguez, E. Ferreiro-Vila, P. Prieto, J. Anguita, M. U. González, J. M. García-Martín, A. Cebollada, A. García-Martín, and G. Armelles, “Probing the Electromagnetic Field Distribution within a Metallic Nanodisk,” Small 7(23), 3317–3323 (2011).
[Crossref] [PubMed]

Armelles, G.

G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: combining magnetic and plasmonic functionalities,” Adv. Opt. Mater. 1(1), 10–35 (2013).
[Crossref]

G. Armelles, A. Cebollada, A. García-Martín, J. M. Montero-Moreno, M. Waleczek, and K. Nielsch, “Magneto-optical Properties of Core-Shell Magneto-plasmonic Au-CoxFe3 - xO4 Nanowires,” Langmuir 28(24), 9127–9130 (2012).
[Crossref] [PubMed]

D. Meneses-Rodríguez, E. Ferreiro-Vila, P. Prieto, J. Anguita, M. U. González, J. M. García-Martín, A. Cebollada, A. García-Martín, and G. Armelles, “Probing the Electromagnetic Field Distribution within a Metallic Nanodisk,” Small 7(23), 3317–3323 (2011).
[Crossref] [PubMed]

J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, G. Armelles, B. Sepúlveda, Y. Alaverdyan, and M. Käll, “Plasmonic Au/Co/Au Nanosandwiches with Enhanced Magneto-Optical Activity,” Small 4(2), 202–205 (2008).
[Crossref] [PubMed]

J. B. González-Díaz, A. García-Martín, G. Armelles, D. Navas, M. Vázquez, K. Nielsch, R. B. Wehrspohn, and U. Gösele, “Enhanced Magneto-Optics and Size Effects in Ferromagnetic Nanowire Arrays,” Adv. Mater. 19(18), 2643–2647 (2007).
[Crossref]

J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. García-Martín, C. Clavero, A. Cebollada, R. A. Lukaszew, J. R. Skuza, D. P. Kumah, and R. Clarke, “Surface-magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76(15), 153402 (2007).
[Crossref]

B. Sepúlveda, A. Calle, L. M. Lechuga, and G. Armelles, “Highly sensitive detection of biomolecules with the magneto-optic surface-plasmon-resonance sensor,” Opt. Lett. 31(8), 1085–1087 (2006).
[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]

Baryshev, A. V.

H. Uchida, Y. Masuda, R. Fujikawa, A. V. Baryshev, and M. Inoue, “Large enhancement of Faraday rotation by localized surface plasmon resonance in Au nanoparticles embedded in Bi:YIG film,” J. Magn. Magn. Mater. 321(7), 843–845 (2009).
[Crossref]

Bayer, M.

L. E. Kreilkamp, V. I. Belotelov, J. Y. Chin, S. Neutzner, D. Dregely, T. Wehlus, I. A. Akimov, M. Bayer, B. Stritzker, and H. Giessen, “Waveguide-plasmon polaritons enhance transverse magneto-optical Kerr effect,” Phys. Rev. X 3, 041019 (2013).

Belotelov, V. I.

L. E. Kreilkamp, V. I. Belotelov, J. Y. Chin, S. Neutzner, D. Dregely, T. Wehlus, I. A. Akimov, M. Bayer, B. Stritzker, and H. Giessen, “Waveguide-plasmon polaritons enhance transverse magneto-optical Kerr effect,” Phys. Rev. X 3, 041019 (2013).

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, “Giant transversal Kerr effect in magneto-plasmonic heterostructures: The scattering-matrix method,” Sov. Phys. JETP 110(5), 816–824 (2010).
[Crossref]

Bertrand, P.

P. Bertrand, C. Hermann, G. Lampel, J. Peretti, and V. I. Safarov, “General analytical treatment of optics in layered structures: Application to magneto-optics,” Phys. Rev. B 64(23), 235421 (2001).
[Crossref]

Bykov, D. A.

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, “Giant transversal Kerr effect in magneto-plasmonic heterostructures: The scattering-matrix method,” Sov. Phys. JETP 110(5), 816–824 (2010).
[Crossref]

Caballero, B.

B. Caballero, A. García-Martín, and J. C. Cuevas, “Generalized scattering-matrix approach for magneto-optics in periodically patterned multilayer systems,” Phys. Rev. B 85(24), 245103 (2012).
[Crossref]

Calle, A.

Carpenter, E. E.

L. Wang, C. Clavero, Z. Huba, K. J. Carroll, E. E. Carpenter, D. Gu, and R. A. Lukaszew, “Plasmonics and Enhanced Magneto-Optics in Core-Shell Co-Ag Nanoparticles,” Nano Lett. 11(3), 1237–1240 (2011).
[Crossref] [PubMed]

Carrascosa, L. G.

Carroll, K. J.

L. Wang, C. Clavero, Z. Huba, K. J. Carroll, E. E. Carpenter, D. Gu, and R. A. Lukaszew, “Plasmonics and Enhanced Magneto-Optics in Core-Shell Co-Ag Nanoparticles,” Nano Lett. 11(3), 1237–1240 (2011).
[Crossref] [PubMed]

Cebollada, A.

G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: combining magnetic and plasmonic functionalities,” Adv. Opt. Mater. 1(1), 10–35 (2013).
[Crossref]

G. Armelles, A. Cebollada, A. García-Martín, J. M. Montero-Moreno, M. Waleczek, and K. Nielsch, “Magneto-optical Properties of Core-Shell Magneto-plasmonic Au-CoxFe3 - xO4 Nanowires,” Langmuir 28(24), 9127–9130 (2012).
[Crossref] [PubMed]

D. Meneses-Rodríguez, E. Ferreiro-Vila, P. Prieto, J. Anguita, M. U. González, J. M. García-Martín, A. Cebollada, A. García-Martín, and G. Armelles, “Probing the Electromagnetic Field Distribution within a Metallic Nanodisk,” Small 7(23), 3317–3323 (2011).
[Crossref] [PubMed]

J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, G. Armelles, B. Sepúlveda, Y. Alaverdyan, and M. Käll, “Plasmonic Au/Co/Au Nanosandwiches with Enhanced Magneto-Optical Activity,” Small 4(2), 202–205 (2008).
[Crossref] [PubMed]

J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. García-Martín, C. Clavero, A. Cebollada, R. A. Lukaszew, J. R. Skuza, D. P. Kumah, and R. Clarke, “Surface-magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76(15), 153402 (2007).
[Crossref]

Chen, J.-Y.

H.-T. Huang, P.-J. Chen, T.-R. Ger, Y.-J. Chi, C.-W. Huang, K.-T. Liao, J.-Y. Lai, J.-Y. Chen, W.-Y. Peng, Q. Zhang, T.-F. Hsieh, W.-J. Sheu, and Z.-H. Wei, “Magneto-optical Kerr effect enhanced by surface plasmon resonance and its application on biological detection,” IEEE Trans. Magn. 50(1), 1001604 (2014).
[Crossref]

Chen, P.-J.

H.-T. Huang, P.-J. Chen, T.-R. Ger, Y.-J. Chi, C.-W. Huang, K.-T. Liao, J.-Y. Lai, J.-Y. Chen, W.-Y. Peng, Q. Zhang, T.-F. Hsieh, W.-J. Sheu, and Z.-H. Wei, “Magneto-optical Kerr effect enhanced by surface plasmon resonance and its application on biological detection,” IEEE Trans. Magn. 50(1), 1001604 (2014).
[Crossref]

Chi, Y.-J.

H.-T. Huang, P.-J. Chen, T.-R. Ger, Y.-J. Chi, C.-W. Huang, K.-T. Liao, J.-Y. Lai, J.-Y. Chen, W.-Y. Peng, Q. Zhang, T.-F. Hsieh, W.-J. Sheu, and Z.-H. Wei, “Magneto-optical Kerr effect enhanced by surface plasmon resonance and its application on biological detection,” IEEE Trans. Magn. 50(1), 1001604 (2014).
[Crossref]

Chin, J. Y.

L. E. Kreilkamp, V. I. Belotelov, J. Y. Chin, S. Neutzner, D. Dregely, T. Wehlus, I. A. Akimov, M. Bayer, B. Stritzker, and H. Giessen, “Waveguide-plasmon polaritons enhance transverse magneto-optical Kerr effect,” Phys. Rev. X 3, 041019 (2013).

Cialla, D.

K. Hering, D. Cialla, K. Ackermann, T. Dörfer, R. Möller, H. Schneidewind, R. Mattheis, W. Fritzsche, P. Rösch, and J. Popp, “SERS: a versatile tool in chemical and biochemical diagnostics,” Anal. Bioanal. Chem. 390(1), 113–124 (2008).
[Crossref] [PubMed]

Clarke, R.

J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. García-Martín, C. Clavero, A. Cebollada, R. A. Lukaszew, J. R. Skuza, D. P. Kumah, and R. Clarke, “Surface-magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76(15), 153402 (2007).
[Crossref]

Clavero, C.

L. Wang, C. Clavero, Z. Huba, K. J. Carroll, E. E. Carpenter, D. Gu, and R. A. Lukaszew, “Plasmonics and Enhanced Magneto-Optics in Core-Shell Co-Ag Nanoparticles,” Nano Lett. 11(3), 1237–1240 (2011).
[Crossref] [PubMed]

J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. García-Martín, C. Clavero, A. Cebollada, R. A. Lukaszew, J. R. Skuza, D. P. Kumah, and R. Clarke, “Surface-magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76(15), 153402 (2007).
[Crossref]

Cuevas, J. C.

B. Caballero, A. García-Martín, and J. C. Cuevas, “Generalized scattering-matrix approach for magneto-optics in periodically patterned multilayer systems,” Phys. Rev. B 85(24), 245103 (2012).
[Crossref]

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]

Dörfer, T.

K. Hering, D. Cialla, K. Ackermann, T. Dörfer, R. Möller, H. Schneidewind, R. Mattheis, W. Fritzsche, P. Rösch, and J. Popp, “SERS: a versatile tool in chemical and biochemical diagnostics,” Anal. Bioanal. Chem. 390(1), 113–124 (2008).
[Crossref] [PubMed]

Doskolovich, L. L.

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, “Giant transversal Kerr effect in magneto-plasmonic heterostructures: The scattering-matrix method,” Sov. Phys. JETP 110(5), 816–824 (2010).
[Crossref]

Dregely, D.

L. E. Kreilkamp, V. I. Belotelov, J. Y. Chin, S. Neutzner, D. Dregely, T. Wehlus, I. A. Akimov, M. Bayer, B. Stritzker, and H. Giessen, “Waveguide-plasmon polaritons enhance transverse magneto-optical Kerr effect,” Phys. Rev. X 3, 041019 (2013).

Du, Y.

S. Zhang, S. Tang, J. Gao, X. Luo, W. Xia, and Y. Du, “Theoretical calculation of magneto-optical properties in cobalt nanotube array with hexagonal symmetry,” Solid State Commun. 170, 19–23 (2013).
[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]

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]

Fariña, D.

Fedyanin, A. A.

H. Uchida, Y. Mizutani, Y. Nakai, A. A. Fedyanin, and M. Inoue, “Garnet composite films with Au particles fabricated by repetitive formation for enhancement of Faraday effect,” J. Phys. D Appl. Phys. 44(6), 064014 (2011).
[Crossref]

Ferreiro-Vila, E.

D. Meneses-Rodríguez, E. Ferreiro-Vila, P. Prieto, J. Anguita, M. U. González, J. M. García-Martín, A. Cebollada, A. García-Martín, and G. Armelles, “Probing the Electromagnetic Field Distribution within a Metallic Nanodisk,” Small 7(23), 3317–3323 (2011).
[Crossref] [PubMed]

Fleischmann, M.

M. Fleischmann, P. J. Hendra, and A. J. McQuillan, “Raman spectra of pyridine adsorbed at a silver electrode,” Chem. Phys. Lett. 26(2), 163–166 (1974).
[Crossref]

Foster, F. G.

H. J. Williams, R. C. Sherwood, F. G. Foster, and E. M. Kelley, “Magnetic Writing on Thin Films of MnBi,” J. Appl. Phys. 28(10), 1181–1184 (1957).
[Crossref]

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]

Fritzsche, W.

K. Hering, D. Cialla, K. Ackermann, T. Dörfer, R. Möller, H. Schneidewind, R. Mattheis, W. Fritzsche, P. Rösch, and J. Popp, “SERS: a versatile tool in chemical and biochemical diagnostics,” Anal. Bioanal. Chem. 390(1), 113–124 (2008).
[Crossref] [PubMed]

Fujikawa, R.

H. Uchida, Y. Masuda, R. Fujikawa, A. V. Baryshev, and M. Inoue, “Large enhancement of Faraday rotation by localized surface plasmon resonance in Au nanoparticles embedded in Bi:YIG film,” J. Magn. Magn. Mater. 321(7), 843–845 (2009).
[Crossref]

Gao, J.

S. Zhang, S. Tang, J. Gao, X. Luo, W. Xia, and Y. Du, “Theoretical calculation of magneto-optical properties in cobalt nanotube array with hexagonal symmetry,” Solid State Commun. 170, 19–23 (2013).
[Crossref]

García-Martín, A.

G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: combining magnetic and plasmonic functionalities,” Adv. Opt. Mater. 1(1), 10–35 (2013).
[Crossref]

B. Caballero, A. García-Martín, and J. C. Cuevas, “Generalized scattering-matrix approach for magneto-optics in periodically patterned multilayer systems,” Phys. Rev. B 85(24), 245103 (2012).
[Crossref]

G. Armelles, A. Cebollada, A. García-Martín, J. M. Montero-Moreno, M. Waleczek, and K. Nielsch, “Magneto-optical Properties of Core-Shell Magneto-plasmonic Au-CoxFe3 - xO4 Nanowires,” Langmuir 28(24), 9127–9130 (2012).
[Crossref] [PubMed]

D. Meneses-Rodríguez, E. Ferreiro-Vila, P. Prieto, J. Anguita, M. U. González, J. M. García-Martín, A. Cebollada, A. García-Martín, and G. Armelles, “Probing the Electromagnetic Field Distribution within a Metallic Nanodisk,” Small 7(23), 3317–3323 (2011).
[Crossref] [PubMed]

J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, G. Armelles, B. Sepúlveda, Y. Alaverdyan, and M. Käll, “Plasmonic Au/Co/Au Nanosandwiches with Enhanced Magneto-Optical Activity,” Small 4(2), 202–205 (2008).
[Crossref] [PubMed]

J. B. González-Díaz, A. García-Martín, G. Armelles, D. Navas, M. Vázquez, K. Nielsch, R. B. Wehrspohn, and U. Gösele, “Enhanced Magneto-Optics and Size Effects in Ferromagnetic Nanowire Arrays,” Adv. Mater. 19(18), 2643–2647 (2007).
[Crossref]

J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. García-Martín, C. Clavero, A. Cebollada, R. A. Lukaszew, J. R. Skuza, D. P. Kumah, and R. Clarke, “Surface-magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76(15), 153402 (2007).
[Crossref]

García-Martín, J. M.

D. Meneses-Rodríguez, E. Ferreiro-Vila, P. Prieto, J. Anguita, M. U. González, J. M. García-Martín, A. Cebollada, A. García-Martín, and G. Armelles, “Probing the Electromagnetic Field Distribution within a Metallic Nanodisk,” Small 7(23), 3317–3323 (2011).
[Crossref] [PubMed]

J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, G. Armelles, B. Sepúlveda, Y. Alaverdyan, and M. Käll, “Plasmonic Au/Co/Au Nanosandwiches with Enhanced Magneto-Optical Activity,” Small 4(2), 202–205 (2008).
[Crossref] [PubMed]

J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. García-Martín, C. Clavero, A. Cebollada, R. A. Lukaszew, J. R. Skuza, D. P. Kumah, and R. Clarke, “Surface-magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76(15), 153402 (2007).
[Crossref]

Ger, T.-R.

H.-T. Huang, P.-J. Chen, T.-R. Ger, Y.-J. Chi, C.-W. Huang, K.-T. Liao, J.-Y. Lai, J.-Y. Chen, W.-Y. Peng, Q. Zhang, T.-F. Hsieh, W.-J. Sheu, and Z.-H. Wei, “Magneto-optical Kerr effect enhanced by surface plasmon resonance and its application on biological detection,” IEEE Trans. Magn. 50(1), 1001604 (2014).
[Crossref]

Giessen, H.

L. E. Kreilkamp, V. I. Belotelov, J. Y. Chin, S. Neutzner, D. Dregely, T. Wehlus, I. A. Akimov, M. Bayer, B. Stritzker, and H. Giessen, “Waveguide-plasmon polaritons enhance transverse magneto-optical Kerr effect,” Phys. Rev. X 3, 041019 (2013).

González, M. U.

G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: combining magnetic and plasmonic functionalities,” Adv. Opt. Mater. 1(1), 10–35 (2013).
[Crossref]

D. Meneses-Rodríguez, E. Ferreiro-Vila, P. Prieto, J. Anguita, M. U. González, J. M. García-Martín, A. Cebollada, A. García-Martín, and G. Armelles, “Probing the Electromagnetic Field Distribution within a Metallic Nanodisk,” Small 7(23), 3317–3323 (2011).
[Crossref] [PubMed]

González-Díaz, J. B.

J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, G. Armelles, B. Sepúlveda, Y. Alaverdyan, and M. Käll, “Plasmonic Au/Co/Au Nanosandwiches with Enhanced Magneto-Optical Activity,” Small 4(2), 202–205 (2008).
[Crossref] [PubMed]

J. B. González-Díaz, A. García-Martín, G. Armelles, D. Navas, M. Vázquez, K. Nielsch, R. B. Wehrspohn, and U. Gösele, “Enhanced Magneto-Optics and Size Effects in Ferromagnetic Nanowire Arrays,” Adv. Mater. 19(18), 2643–2647 (2007).
[Crossref]

J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. García-Martín, C. Clavero, A. Cebollada, R. A. Lukaszew, J. R. Skuza, D. P. Kumah, and R. Clarke, “Surface-magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76(15), 153402 (2007).
[Crossref]

Gösele, U.

J. B. González-Díaz, A. García-Martín, G. Armelles, D. Navas, M. Vázquez, K. Nielsch, R. B. Wehrspohn, and U. Gösele, “Enhanced Magneto-Optics and Size Effects in Ferromagnetic Nanowire Arrays,” Adv. Mater. 19(18), 2643–2647 (2007).
[Crossref]

Gu, D.

L. Wang, C. Clavero, Z. Huba, K. J. Carroll, E. E. Carpenter, D. Gu, and R. A. Lukaszew, “Plasmonics and Enhanced Magneto-Optics in Core-Shell Co-Ag Nanoparticles,” Nano Lett. 11(3), 1237–1240 (2011).
[Crossref] [PubMed]

Hamidi, S. M.

S. M. Hamidi, M. A. Oskuei, S. Sadeghi, and M. M. Tehranchi, “Enhanced polar magneto-optical Kerr rotation in cobalt thin film incorporating Ag nanoparticles,” J. Supercond. Novel Magn. 27(3), 867–870 (2014).
[Crossref]

Hendra, P. J.

M. Fleischmann, P. J. Hendra, and A. J. McQuillan, “Raman spectra of pyridine adsorbed at a silver electrode,” Chem. Phys. Lett. 26(2), 163–166 (1974).
[Crossref]

Hering, K.

K. Hering, D. Cialla, K. Ackermann, T. Dörfer, R. Möller, H. Schneidewind, R. Mattheis, W. Fritzsche, P. Rösch, and J. Popp, “SERS: a versatile tool in chemical and biochemical diagnostics,” Anal. Bioanal. Chem. 390(1), 113–124 (2008).
[Crossref] [PubMed]

Hermann, C.

P. Bertrand, C. Hermann, G. Lampel, J. Peretti, and V. I. Safarov, “General analytical treatment of optics in layered structures: Application to magneto-optics,” Phys. Rev. B 64(23), 235421 (2001).
[Crossref]

Homola, J.

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108(2), 462–493 (2008).
[Crossref] [PubMed]

Hsieh, T.-F.

H.-T. Huang, P.-J. Chen, T.-R. Ger, Y.-J. Chi, C.-W. Huang, K.-T. Liao, J.-Y. Lai, J.-Y. Chen, W.-Y. Peng, Q. Zhang, T.-F. Hsieh, W.-J. Sheu, and Z.-H. Wei, “Magneto-optical Kerr effect enhanced by surface plasmon resonance and its application on biological detection,” IEEE Trans. Magn. 50(1), 1001604 (2014).
[Crossref]

Huang, C.-W.

H.-T. Huang, P.-J. Chen, T.-R. Ger, Y.-J. Chi, C.-W. Huang, K.-T. Liao, J.-Y. Lai, J.-Y. Chen, W.-Y. Peng, Q. Zhang, T.-F. Hsieh, W.-J. Sheu, and Z.-H. Wei, “Magneto-optical Kerr effect enhanced by surface plasmon resonance and its application on biological detection,” IEEE Trans. Magn. 50(1), 1001604 (2014).
[Crossref]

Huang, H.-T.

H.-T. Huang, P.-J. Chen, T.-R. Ger, Y.-J. Chi, C.-W. Huang, K.-T. Liao, J.-Y. Lai, J.-Y. Chen, W.-Y. Peng, Q. Zhang, T.-F. Hsieh, W.-J. Sheu, and Z.-H. Wei, “Magneto-optical Kerr effect enhanced by surface plasmon resonance and its application on biological detection,” IEEE Trans. Magn. 50(1), 1001604 (2014).
[Crossref]

Huba, Z.

L. Wang, C. Clavero, Z. Huba, K. J. Carroll, E. E. Carpenter, D. Gu, and R. A. Lukaszew, “Plasmonics and Enhanced Magneto-Optics in Core-Shell Co-Ag Nanoparticles,” Nano Lett. 11(3), 1237–1240 (2011).
[Crossref] [PubMed]

Hui, P. M.

P. M. Hui and D. Stroud, “Theory of Faraday rotation by dilute suspensions of small particles,” Appl. Phys. Lett. 50(15), 950–952 (1987).
[Crossref]

Inoue, M.

H. Uchida, Y. Mizutani, Y. Nakai, A. A. Fedyanin, and M. Inoue, “Garnet composite films with Au particles fabricated by repetitive formation for enhancement of Faraday effect,” J. Phys. D Appl. Phys. 44(6), 064014 (2011).
[Crossref]

H. Uchida, Y. Masuda, R. Fujikawa, A. V. Baryshev, and M. Inoue, “Large enhancement of Faraday rotation by localized surface plasmon resonance in Au nanoparticles embedded in Bi:YIG film,” J. Magn. Magn. Mater. 321(7), 843–845 (2009).
[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]

Joannopoulos, J. 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]

Kalish, A. N.

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, “Giant transversal Kerr effect in magneto-plasmonic heterostructures: The scattering-matrix method,” Sov. Phys. JETP 110(5), 816–824 (2010).
[Crossref]

Käll, M.

J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, G. Armelles, B. Sepúlveda, Y. Alaverdyan, and M. Käll, “Plasmonic Au/Co/Au Nanosandwiches with Enhanced Magneto-Optical Activity,” Small 4(2), 202–205 (2008).
[Crossref] [PubMed]

Kelley, E. M.

H. J. Williams, R. C. Sherwood, F. G. Foster, and E. M. Kelley, “Magnetic Writing on Thin Films of MnBi,” J. Appl. Phys. 28(10), 1181–1184 (1957).
[Crossref]

Kreilkamp, L. E.

L. E. Kreilkamp, V. I. Belotelov, J. Y. Chin, S. Neutzner, D. Dregely, T. Wehlus, I. A. Akimov, M. Bayer, B. Stritzker, and H. Giessen, “Waveguide-plasmon polaritons enhance transverse magneto-optical Kerr effect,” Phys. Rev. X 3, 041019 (2013).

Kumah, D. P.

J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. García-Martín, C. Clavero, A. Cebollada, R. A. Lukaszew, J. R. Skuza, D. P. Kumah, and R. Clarke, “Surface-magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76(15), 153402 (2007).
[Crossref]

Lai, J.-Y.

H.-T. Huang, P.-J. Chen, T.-R. Ger, Y.-J. Chi, C.-W. Huang, K.-T. Liao, J.-Y. Lai, J.-Y. Chen, W.-Y. Peng, Q. Zhang, T.-F. Hsieh, W.-J. Sheu, and Z.-H. Wei, “Magneto-optical Kerr effect enhanced by surface plasmon resonance and its application on biological detection,” IEEE Trans. Magn. 50(1), 1001604 (2014).
[Crossref]

Lampel, G.

P. Bertrand, C. Hermann, G. Lampel, J. Peretti, and V. I. Safarov, “General analytical treatment of optics in layered structures: Application to magneto-optics,” Phys. Rev. B 64(23), 235421 (2001).
[Crossref]

Lechuga, L. M.

Liao, K.-T.

H.-T. Huang, P.-J. Chen, T.-R. Ger, Y.-J. Chi, C.-W. Huang, K.-T. Liao, J.-Y. Lai, J.-Y. Chen, W.-Y. Peng, Q. Zhang, T.-F. Hsieh, W.-J. Sheu, and Z.-H. Wei, “Magneto-optical Kerr effect enhanced by surface plasmon resonance and its application on biological detection,” IEEE Trans. Magn. 50(1), 1001604 (2014).
[Crossref]

Liedberg, B.

C. Nylander, B. Liedberg, and T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators 3, 79–88 (1982).
[Crossref]

Lind, T.

C. Nylander, B. Liedberg, and T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators 3, 79–88 (1982).
[Crossref]

Lukaszew, R. A.

L. Wang, C. Clavero, Z. Huba, K. J. Carroll, E. E. Carpenter, D. Gu, and R. A. Lukaszew, “Plasmonics and Enhanced Magneto-Optics in Core-Shell Co-Ag Nanoparticles,” Nano Lett. 11(3), 1237–1240 (2011).
[Crossref] [PubMed]

J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. García-Martín, C. Clavero, A. Cebollada, R. A. Lukaszew, J. R. Skuza, D. P. Kumah, and R. Clarke, “Surface-magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76(15), 153402 (2007).
[Crossref]

Luo, X.

S. Zhang, S. Tang, J. Gao, X. Luo, W. Xia, and Y. Du, “Theoretical calculation of magneto-optical properties in cobalt nanotube array with hexagonal symmetry,” Solid State Commun. 170, 19–23 (2013).
[Crossref]

Masuda, Y.

H. Uchida, Y. Masuda, R. Fujikawa, A. V. Baryshev, and M. Inoue, “Large enhancement of Faraday rotation by localized surface plasmon resonance in Au nanoparticles embedded in Bi:YIG film,” J. Magn. Magn. Mater. 321(7), 843–845 (2009).
[Crossref]

Mattheis, R.

K. Hering, D. Cialla, K. Ackermann, T. Dörfer, R. Möller, H. Schneidewind, R. Mattheis, W. Fritzsche, P. Rösch, and J. Popp, “SERS: a versatile tool in chemical and biochemical diagnostics,” Anal. Bioanal. Chem. 390(1), 113–124 (2008).
[Crossref] [PubMed]

McQuillan, A. J.

M. Fleischmann, P. J. Hendra, and A. J. McQuillan, “Raman spectra of pyridine adsorbed at a silver electrode,” Chem. Phys. Lett. 26(2), 163–166 (1974).
[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]

Meneses-Rodríguez, D.

D. Meneses-Rodríguez, E. Ferreiro-Vila, P. Prieto, J. Anguita, M. U. González, J. M. García-Martín, A. Cebollada, A. García-Martín, and G. Armelles, “Probing the Electromagnetic Field Distribution within a Metallic Nanodisk,” Small 7(23), 3317–3323 (2011).
[Crossref] [PubMed]

Mizumoto, T.

Y. Shoji, Y. Shirato, and T. Mizumoto, “Silicon Mach–Zehnder interferometer optical isolator having 8 nm bandwidth for over 20 dB isolation,” Jpn. J. Appl. Phys. 53(2), 022202 (2014).
[Crossref]

Y. Sobu, Y. Shoji, K. Sakurai, and T. Mizumoto, “GaInAsP/InP MZI waveguide optical isolator integrated with spot size converter,” Opt. Express 21(13), 15373–15381 (2013).
[Crossref] [PubMed]

Mizutani, Y.

H. Uchida, Y. Mizutani, Y. Nakai, A. A. Fedyanin, and M. Inoue, “Garnet composite films with Au particles fabricated by repetitive formation for enhancement of Faraday effect,” J. Phys. D Appl. Phys. 44(6), 064014 (2011).
[Crossref]

Möller, R.

K. Hering, D. Cialla, K. Ackermann, T. Dörfer, R. Möller, H. Schneidewind, R. Mattheis, W. Fritzsche, P. Rösch, and J. Popp, “SERS: a versatile tool in chemical and biochemical diagnostics,” Anal. Bioanal. Chem. 390(1), 113–124 (2008).
[Crossref] [PubMed]

Montero-Moreno, J. M.

G. Armelles, A. Cebollada, A. García-Martín, J. M. Montero-Moreno, M. Waleczek, and K. Nielsch, “Magneto-optical Properties of Core-Shell Magneto-plasmonic Au-CoxFe3 - xO4 Nanowires,” Langmuir 28(24), 9127–9130 (2012).
[Crossref] [PubMed]

Nakai, Y.

H. Uchida, Y. Mizutani, Y. Nakai, A. A. Fedyanin, and M. Inoue, “Garnet composite films with Au particles fabricated by repetitive formation for enhancement of Faraday effect,” J. Phys. D Appl. Phys. 44(6), 064014 (2011).
[Crossref]

Nakano, Y.

H. Shimizu and Y. Nakano, “Fabrication and characterization of an InGaAsP/InP active waveguide optical isolator with 14.7 dB/mm TE mode nonreciprocal attenuation,” J. Lightwave Technol. 24(1), 38–43 (2006).
[Crossref]

H. Shimizu and Y. Nakano, “First Demonstration of TE Mode Nonreciprocal Propagation in an InGaAsP/InP Active Waveguide for an Integratable Optical Isolator,” Jpn. J. Appl. Phys. 43(12A12A), L1561–L1563 (2004).
[Crossref]

Navas, D.

J. B. González-Díaz, A. García-Martín, G. Armelles, D. Navas, M. Vázquez, K. Nielsch, R. B. Wehrspohn, and U. Gösele, “Enhanced Magneto-Optics and Size Effects in Ferromagnetic Nanowire Arrays,” Adv. Mater. 19(18), 2643–2647 (2007).
[Crossref]

Neutzner, S.

L. E. Kreilkamp, V. I. Belotelov, J. Y. Chin, S. Neutzner, D. Dregely, T. Wehlus, I. A. Akimov, M. Bayer, B. Stritzker, and H. Giessen, “Waveguide-plasmon polaritons enhance transverse magneto-optical Kerr effect,” Phys. Rev. X 3, 041019 (2013).

Nielsch, K.

G. Armelles, A. Cebollada, A. García-Martín, J. M. Montero-Moreno, M. Waleczek, and K. Nielsch, “Magneto-optical Properties of Core-Shell Magneto-plasmonic Au-CoxFe3 - xO4 Nanowires,” Langmuir 28(24), 9127–9130 (2012).
[Crossref] [PubMed]

J. B. González-Díaz, A. García-Martín, G. Armelles, D. Navas, M. Vázquez, K. Nielsch, R. B. Wehrspohn, and U. Gösele, “Enhanced Magneto-Optics and Size Effects in Ferromagnetic Nanowire Arrays,” Adv. Mater. 19(18), 2643–2647 (2007).
[Crossref]

Nylander, C.

C. Nylander, B. Liedberg, and T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators 3, 79–88 (1982).
[Crossref]

Oskuei, M. A.

S. M. Hamidi, M. A. Oskuei, S. Sadeghi, and M. M. Tehranchi, “Enhanced polar magneto-optical Kerr rotation in cobalt thin film incorporating Ag nanoparticles,” J. Supercond. Novel Magn. 27(3), 867–870 (2014).
[Crossref]

Peng, W.-Y.

H.-T. Huang, P.-J. Chen, T.-R. Ger, Y.-J. Chi, C.-W. Huang, K.-T. Liao, J.-Y. Lai, J.-Y. Chen, W.-Y. Peng, Q. Zhang, T.-F. Hsieh, W.-J. Sheu, and Z.-H. Wei, “Magneto-optical Kerr effect enhanced by surface plasmon resonance and its application on biological detection,” IEEE Trans. Magn. 50(1), 1001604 (2014).
[Crossref]

Peretti, J.

P. Bertrand, C. Hermann, G. Lampel, J. Peretti, and V. I. Safarov, “General analytical treatment of optics in layered structures: Application to magneto-optics,” Phys. Rev. B 64(23), 235421 (2001).
[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]

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]

Popp, J.

K. Hering, D. Cialla, K. Ackermann, T. Dörfer, R. Möller, H. Schneidewind, R. Mattheis, W. Fritzsche, P. Rösch, and J. Popp, “SERS: a versatile tool in chemical and biochemical diagnostics,” Anal. Bioanal. Chem. 390(1), 113–124 (2008).
[Crossref] [PubMed]

Prieto, P.

D. Meneses-Rodríguez, E. Ferreiro-Vila, P. Prieto, J. Anguita, M. U. González, J. M. García-Martín, A. Cebollada, A. García-Martín, and G. Armelles, “Probing the Electromagnetic Field Distribution within a Metallic Nanodisk,” Small 7(23), 3317–3323 (2011).
[Crossref] [PubMed]

Regatos, D.

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]

Rösch, P.

K. Hering, D. Cialla, K. Ackermann, T. Dörfer, R. Möller, H. Schneidewind, R. Mattheis, W. Fritzsche, P. Rösch, and J. Popp, “SERS: a versatile tool in chemical and biochemical diagnostics,” Anal. Bioanal. Chem. 390(1), 113–124 (2008).
[Crossref] [PubMed]

Rumpf, R. C.

R. C. Rumpf, “Improved formulation of scattering matrices for semi-analytical methods that is consistent with convention,” Prog. Electromagn. Res. B 35, 241–261 (2011).
[Crossref]

Sadeghi, S.

S. M. Hamidi, M. A. Oskuei, S. Sadeghi, and M. M. Tehranchi, “Enhanced polar magneto-optical Kerr rotation in cobalt thin film incorporating Ag nanoparticles,” J. Supercond. Novel Magn. 27(3), 867–870 (2014).
[Crossref]

Safarov, V. I.

P. Bertrand, C. Hermann, G. Lampel, J. Peretti, and V. I. Safarov, “General analytical treatment of optics in layered structures: Application to magneto-optics,” Phys. Rev. B 64(23), 235421 (2001).
[Crossref]

Saito, H.

V. Zayets, H. Saito, S. Yuasa, and K. Ando, “Enhancement of the transverse non-reciprocal magneto-optical effect,” J. Appl. Phys. 111(2), 023103 (2012).
[Crossref]

V. Zayets, H. Saito, K. Ando, and S. Yuasa, “Optical isolator utilizing surface plasmons,” Materials (Basel) 5(5), 857–871 (2012).
[Crossref]

Sakurai, K.

Schneidewind, H.

K. Hering, D. Cialla, K. Ackermann, T. Dörfer, R. Möller, H. Schneidewind, R. Mattheis, W. Fritzsche, P. Rösch, and J. Popp, “SERS: a versatile tool in chemical and biochemical diagnostics,” Anal. Bioanal. Chem. 390(1), 113–124 (2008).
[Crossref] [PubMed]

Sepúlveda, B.

Sherwood, R. C.

H. J. Williams, R. C. Sherwood, F. G. Foster, and E. M. Kelley, “Magnetic Writing on Thin Films of MnBi,” J. Appl. Phys. 28(10), 1181–1184 (1957).
[Crossref]

Sheu, W.-J.

H.-T. Huang, P.-J. Chen, T.-R. Ger, Y.-J. Chi, C.-W. Huang, K.-T. Liao, J.-Y. Lai, J.-Y. Chen, W.-Y. Peng, Q. Zhang, T.-F. Hsieh, W.-J. Sheu, and Z.-H. Wei, “Magneto-optical Kerr effect enhanced by surface plasmon resonance and its application on biological detection,” IEEE Trans. Magn. 50(1), 1001604 (2014).
[Crossref]

Shimizu, H.

H. Shimizu and Y. Nakano, “Fabrication and characterization of an InGaAsP/InP active waveguide optical isolator with 14.7 dB/mm TE mode nonreciprocal attenuation,” J. Lightwave Technol. 24(1), 38–43 (2006).
[Crossref]

H. Shimizu and Y. Nakano, “First Demonstration of TE Mode Nonreciprocal Propagation in an InGaAsP/InP Active Waveguide for an Integratable Optical Isolator,” Jpn. J. Appl. Phys. 43(12A12A), L1561–L1563 (2004).
[Crossref]

Shintaku, T.

T. Shintaku, “Integrated optical isolator based on efficient nonreciprocal radiation mode conversion,” Appl. Phys. Lett. 73(14), 1946–1948 (1998).
[Crossref]

T. Shintaku and T. Uno, “Optical waveguide isolator based on nonreciprocal radiation,” J. Appl. Phys. 76(12), 8155–8159 (1994).
[Crossref]

Shirato, Y.

Y. Shoji, Y. Shirato, and T. Mizumoto, “Silicon Mach–Zehnder interferometer optical isolator having 8 nm bandwidth for over 20 dB isolation,” Jpn. J. Appl. Phys. 53(2), 022202 (2014).
[Crossref]

Shoji, Y.

Y. Shoji, Y. Shirato, and T. Mizumoto, “Silicon Mach–Zehnder interferometer optical isolator having 8 nm bandwidth for over 20 dB isolation,” Jpn. J. Appl. Phys. 53(2), 022202 (2014).
[Crossref]

Y. Sobu, Y. Shoji, K. Sakurai, and T. Mizumoto, “GaInAsP/InP MZI waveguide optical isolator integrated with spot size converter,” Opt. Express 21(13), 15373–15381 (2013).
[Crossref] [PubMed]

Skuza, J. R.

J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. García-Martín, C. Clavero, A. Cebollada, R. A. Lukaszew, J. R. Skuza, D. P. Kumah, and R. Clarke, “Surface-magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76(15), 153402 (2007).
[Crossref]

Smith, D. Y.

Sobu, Y.

Stritzker, B.

L. E. Kreilkamp, V. I. Belotelov, J. Y. Chin, S. Neutzner, D. Dregely, T. Wehlus, I. A. Akimov, M. Bayer, B. Stritzker, and H. Giessen, “Waveguide-plasmon polaritons enhance transverse magneto-optical Kerr effect,” Phys. Rev. X 3, 041019 (2013).

Stroud, D.

P. M. Hui and D. Stroud, “Theory of Faraday rotation by dilute suspensions of small particles,” Appl. Phys. Lett. 50(15), 950–952 (1987).
[Crossref]

Tang, S.

S. Zhang, S. Tang, J. Gao, X. Luo, W. Xia, and Y. Du, “Theoretical calculation of magneto-optical properties in cobalt nanotube array with hexagonal symmetry,” Solid State Commun. 170, 19–23 (2013).
[Crossref]

Tehranchi, M. M.

S. M. Hamidi, M. A. Oskuei, S. Sadeghi, and M. M. Tehranchi, “Enhanced polar magneto-optical Kerr rotation in cobalt thin film incorporating Ag nanoparticles,” J. Supercond. Novel Magn. 27(3), 867–870 (2014).
[Crossref]

Uchida, H.

H. Uchida, Y. Mizutani, Y. Nakai, A. A. Fedyanin, and M. Inoue, “Garnet composite films with Au particles fabricated by repetitive formation for enhancement of Faraday effect,” J. Phys. D Appl. Phys. 44(6), 064014 (2011).
[Crossref]

H. Uchida, Y. Masuda, R. Fujikawa, A. V. Baryshev, and M. Inoue, “Large enhancement of Faraday rotation by localized surface plasmon resonance in Au nanoparticles embedded in Bi:YIG film,” J. Magn. Magn. Mater. 321(7), 843–845 (2009).
[Crossref]

Uno, T.

T. Shintaku and T. Uno, “Optical waveguide isolator based on nonreciprocal radiation,” J. Appl. Phys. 76(12), 8155–8159 (1994).
[Crossref]

Vanwolleghem, 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]

Vázquez, M.

J. B. González-Díaz, A. García-Martín, G. Armelles, D. Navas, M. Vázquez, K. Nielsch, R. B. Wehrspohn, and U. Gösele, “Enhanced Magneto-Optics and Size Effects in Ferromagnetic Nanowire Arrays,” Adv. Mater. 19(18), 2643–2647 (2007).
[Crossref]

Waleczek, M.

G. Armelles, A. Cebollada, A. García-Martín, J. M. Montero-Moreno, M. Waleczek, and K. Nielsch, “Magneto-optical Properties of Core-Shell Magneto-plasmonic Au-CoxFe3 - xO4 Nanowires,” Langmuir 28(24), 9127–9130 (2012).
[Crossref] [PubMed]

Wang, L.

L. Wang, C. Clavero, Z. Huba, K. J. Carroll, E. E. Carpenter, D. Gu, and R. A. Lukaszew, “Plasmonics and Enhanced Magneto-Optics in Core-Shell Co-Ag Nanoparticles,” Nano Lett. 11(3), 1237–1240 (2011).
[Crossref] [PubMed]

Wehlus, T.

L. E. Kreilkamp, V. I. Belotelov, J. Y. Chin, S. Neutzner, D. Dregely, T. Wehlus, I. A. Akimov, M. Bayer, B. Stritzker, and H. Giessen, “Waveguide-plasmon polaritons enhance transverse magneto-optical Kerr effect,” Phys. Rev. X 3, 041019 (2013).

Wehrspohn, R. B.

J. B. González-Díaz, A. García-Martín, G. Armelles, D. Navas, M. Vázquez, K. Nielsch, R. B. Wehrspohn, and U. Gösele, “Enhanced Magneto-Optics and Size Effects in Ferromagnetic Nanowire Arrays,” Adv. Mater. 19(18), 2643–2647 (2007).
[Crossref]

Wei, Z.-H.

H.-T. Huang, P.-J. Chen, T.-R. Ger, Y.-J. Chi, C.-W. Huang, K.-T. Liao, J.-Y. Lai, J.-Y. Chen, W.-Y. Peng, Q. Zhang, T.-F. Hsieh, W.-J. Sheu, and Z.-H. Wei, “Magneto-optical Kerr effect enhanced by surface plasmon resonance and its application on biological detection,” IEEE Trans. Magn. 50(1), 1001604 (2014).
[Crossref]

Williams, H. J.

H. J. Williams, R. C. Sherwood, F. G. Foster, and E. M. Kelley, “Magnetic Writing on Thin Films of MnBi,” J. Appl. Phys. 28(10), 1181–1184 (1957).
[Crossref]

Xia, W.

S. Zhang, S. Tang, J. Gao, X. Luo, W. Xia, and Y. Du, “Theoretical calculation of magneto-optical properties in cobalt nanotube array with hexagonal symmetry,” Solid State Commun. 170, 19–23 (2013).
[Crossref]

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]

Yuasa, S.

V. Zayets, H. Saito, K. Ando, and S. Yuasa, “Optical isolator utilizing surface plasmons,” Materials (Basel) 5(5), 857–871 (2012).
[Crossref]

V. Zayets, H. Saito, S. Yuasa, and K. Ando, “Enhancement of the transverse non-reciprocal magneto-optical effect,” J. Appl. Phys. 111(2), 023103 (2012).
[Crossref]

Zayets, V.

V. Zayets, H. Saito, S. Yuasa, and K. Ando, “Enhancement of the transverse non-reciprocal magneto-optical effect,” J. Appl. Phys. 111(2), 023103 (2012).
[Crossref]

V. Zayets, H. Saito, K. Ando, and S. Yuasa, “Optical isolator utilizing surface plasmons,” Materials (Basel) 5(5), 857–871 (2012).
[Crossref]

Zhang, Q.

H.-T. Huang, P.-J. Chen, T.-R. Ger, Y.-J. Chi, C.-W. Huang, K.-T. Liao, J.-Y. Lai, J.-Y. Chen, W.-Y. Peng, Q. Zhang, T.-F. Hsieh, W.-J. Sheu, and Z.-H. Wei, “Magneto-optical Kerr effect enhanced by surface plasmon resonance and its application on biological detection,” IEEE Trans. Magn. 50(1), 1001604 (2014).
[Crossref]

Zhang, S.

S. Zhang, S. Tang, J. Gao, X. Luo, W. Xia, and Y. Du, “Theoretical calculation of magneto-optical properties in cobalt nanotube array with hexagonal symmetry,” Solid State Commun. 170, 19–23 (2013).
[Crossref]

Zvezdin, A. K.

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, “Giant transversal Kerr effect in magneto-plasmonic heterostructures: The scattering-matrix method,” Sov. Phys. JETP 110(5), 816–824 (2010).
[Crossref]

Adv. Mater. (1)

J. B. González-Díaz, A. García-Martín, G. Armelles, D. Navas, M. Vázquez, K. Nielsch, R. B. Wehrspohn, and U. Gösele, “Enhanced Magneto-Optics and Size Effects in Ferromagnetic Nanowire Arrays,” Adv. Mater. 19(18), 2643–2647 (2007).
[Crossref]

Adv. Opt. Mater. (1)

G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: combining magnetic and plasmonic functionalities,” Adv. Opt. Mater. 1(1), 10–35 (2013).
[Crossref]

Anal. Bioanal. Chem. (1)

K. Hering, D. Cialla, K. Ackermann, T. Dörfer, R. Möller, H. Schneidewind, R. Mattheis, W. Fritzsche, P. Rösch, and J. Popp, “SERS: a versatile tool in chemical and biochemical diagnostics,” Anal. Bioanal. Chem. 390(1), 113–124 (2008).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

P. M. Hui and D. Stroud, “Theory of Faraday rotation by dilute suspensions of small particles,” Appl. Phys. Lett. 50(15), 950–952 (1987).
[Crossref]

T. Shintaku, “Integrated optical isolator based on efficient nonreciprocal radiation mode conversion,” Appl. Phys. Lett. 73(14), 1946–1948 (1998).
[Crossref]

Chem. Phys. Lett. (1)

M. Fleischmann, P. J. Hendra, and A. J. McQuillan, “Raman spectra of pyridine adsorbed at a silver electrode,” Chem. Phys. Lett. 26(2), 163–166 (1974).
[Crossref]

Chem. Rev. (1)

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108(2), 462–493 (2008).
[Crossref] [PubMed]

IEEE Trans. Magn. (1)

H.-T. Huang, P.-J. Chen, T.-R. Ger, Y.-J. Chi, C.-W. Huang, K.-T. Liao, J.-Y. Lai, J.-Y. Chen, W.-Y. Peng, Q. Zhang, T.-F. Hsieh, W.-J. Sheu, and Z.-H. Wei, “Magneto-optical Kerr effect enhanced by surface plasmon resonance and its application on biological detection,” IEEE Trans. Magn. 50(1), 1001604 (2014).
[Crossref]

J. Appl. Phys. (3)

H. J. Williams, R. C. Sherwood, F. G. Foster, and E. M. Kelley, “Magnetic Writing on Thin Films of MnBi,” J. Appl. Phys. 28(10), 1181–1184 (1957).
[Crossref]

T. Shintaku and T. Uno, “Optical waveguide isolator based on nonreciprocal radiation,” J. Appl. Phys. 76(12), 8155–8159 (1994).
[Crossref]

V. Zayets, H. Saito, S. Yuasa, and K. Ando, “Enhancement of the transverse non-reciprocal magneto-optical effect,” J. Appl. Phys. 111(2), 023103 (2012).
[Crossref]

J. Lightwave Technol. (1)

J. Magn. Magn. Mater. (1)

H. Uchida, Y. Masuda, R. Fujikawa, A. V. Baryshev, and M. Inoue, “Large enhancement of Faraday rotation by localized surface plasmon resonance in Au nanoparticles embedded in Bi:YIG film,” J. Magn. Magn. Mater. 321(7), 843–845 (2009).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. D Appl. Phys. (1)

H. Uchida, Y. Mizutani, Y. Nakai, A. A. Fedyanin, and M. Inoue, “Garnet composite films with Au particles fabricated by repetitive formation for enhancement of Faraday effect,” J. Phys. D Appl. Phys. 44(6), 064014 (2011).
[Crossref]

J. Supercond. Novel Magn. (1)

S. M. Hamidi, M. A. Oskuei, S. Sadeghi, and M. M. Tehranchi, “Enhanced polar magneto-optical Kerr rotation in cobalt thin film incorporating Ag nanoparticles,” J. Supercond. Novel Magn. 27(3), 867–870 (2014).
[Crossref]

Jpn. J. Appl. Phys. (2)

Y. Shoji, Y. Shirato, and T. Mizumoto, “Silicon Mach–Zehnder interferometer optical isolator having 8 nm bandwidth for over 20 dB isolation,” Jpn. J. Appl. Phys. 53(2), 022202 (2014).
[Crossref]

H. Shimizu and Y. Nakano, “First Demonstration of TE Mode Nonreciprocal Propagation in an InGaAsP/InP Active Waveguide for an Integratable Optical Isolator,” Jpn. J. Appl. Phys. 43(12A12A), L1561–L1563 (2004).
[Crossref]

Langmuir (1)

G. Armelles, A. Cebollada, A. García-Martín, J. M. Montero-Moreno, M. Waleczek, and K. Nielsch, “Magneto-optical Properties of Core-Shell Magneto-plasmonic Au-CoxFe3 - xO4 Nanowires,” Langmuir 28(24), 9127–9130 (2012).
[Crossref] [PubMed]

Materials (Basel) (1)

V. Zayets, H. Saito, K. Ando, and S. Yuasa, “Optical isolator utilizing surface plasmons,” Materials (Basel) 5(5), 857–871 (2012).
[Crossref]

Nano Lett. (1)

L. Wang, C. Clavero, Z. Huba, K. J. Carroll, E. E. Carpenter, D. Gu, and R. A. Lukaszew, “Plasmonics and Enhanced Magneto-Optics in Core-Shell Co-Ag Nanoparticles,” Nano Lett. 11(3), 1237–1240 (2011).
[Crossref] [PubMed]

Nat. Photonics (1)

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]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (3)

B. Caballero, A. García-Martín, and J. C. Cuevas, “Generalized scattering-matrix approach for magneto-optics in periodically patterned multilayer systems,” Phys. Rev. B 85(24), 245103 (2012).
[Crossref]

J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. García-Martín, C. Clavero, A. Cebollada, R. A. Lukaszew, J. R. Skuza, D. P. Kumah, and R. Clarke, “Surface-magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76(15), 153402 (2007).
[Crossref]

P. Bertrand, C. Hermann, G. Lampel, J. Peretti, and V. I. Safarov, “General analytical treatment of optics in layered structures: Application to magneto-optics,” Phys. Rev. B 64(23), 235421 (2001).
[Crossref]

Phys. Rev. X (1)

L. E. Kreilkamp, V. I. Belotelov, J. Y. Chin, S. Neutzner, D. Dregely, T. Wehlus, I. A. Akimov, M. Bayer, B. Stritzker, and H. Giessen, “Waveguide-plasmon polaritons enhance transverse magneto-optical Kerr effect,” Phys. Rev. X 3, 041019 (2013).

Prog. Electromagn. Res. B (1)

R. C. Rumpf, “Improved formulation of scattering matrices for semi-analytical methods that is consistent with convention,” Prog. Electromagn. Res. B 35, 241–261 (2011).
[Crossref]

Sens. Actuators (1)

C. Nylander, B. Liedberg, and T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators 3, 79–88 (1982).
[Crossref]

Small (2)

J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, G. Armelles, B. Sepúlveda, Y. Alaverdyan, and M. Käll, “Plasmonic Au/Co/Au Nanosandwiches with Enhanced Magneto-Optical Activity,” Small 4(2), 202–205 (2008).
[Crossref] [PubMed]

D. Meneses-Rodríguez, E. Ferreiro-Vila, P. Prieto, J. Anguita, M. U. González, J. M. García-Martín, A. Cebollada, A. García-Martín, and G. Armelles, “Probing the Electromagnetic Field Distribution within a Metallic Nanodisk,” Small 7(23), 3317–3323 (2011).
[Crossref] [PubMed]

Solid State Commun. (1)

S. Zhang, S. Tang, J. Gao, X. Luo, W. Xia, and Y. Du, “Theoretical calculation of magneto-optical properties in cobalt nanotube array with hexagonal symmetry,” Solid State Commun. 170, 19–23 (2013).
[Crossref]

Sov. Phys. JETP (1)

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, “Giant transversal Kerr effect in magneto-plasmonic heterostructures: The scattering-matrix method,” Sov. Phys. JETP 110(5), 816–824 (2010).
[Crossref]

Other (7)

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig relations in optical materials research (Springer, 2005), Chap. 4.

F. Gervais, “Aluminum Oxide (Al2O3),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, 1991).

H. R. Philipp, “Silicon Dioxide (SiO2) (Glass),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, 1991).

D. W. Lynch and W. R. Hunter, “An introduction to the Data for Several Metals,” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, 1991).

M. B. Stearns, “Optical constants, magneto-optic Kerr or Faraday effect,” in Landolt-Börnstein - Group III Condensed Matter, H. P. J. Wijn, ed. (Springer, 1986).

R. J. Gambino and T. Suzuki, Magneto-Optical Recording Materials (IEEE, 2000).

EM Lab - University of Texas at El Paso, “News 2012 | EM Lab,” http://emlab.utep.edu/ee5390cem.htm

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Figures (14)

Fig. 1
Fig. 1 A trilayer structure consisting of double-layer dielectrics and a ferromagnetic metal for nonreciprocal plasmonic propagation. The yellow curve represents the distribution of the SPPs.
Fig. 2
Fig. 2 A schematic diagram of the Al2O3/SiO2/Fe trilayer deposited on a SiO2 substrate. The incident light, defined by orthogonal unit vectors ÂTM and ÂTE, comes from the Al2O3 side with p-polarization (ϕ = 0, CTE = 0) and an arbitrary incident angle, θ. Arrows M indicate the positive magnetization directions of the PMOKE and TMOKE.
Fig. 3
Fig. 3 Reflectance, R, and change in reflectance, ΔR, of the TMOKE as a function of incident angle, θ, at SiO2 thicknesses, t, of (a) 645 nm and (b) 661 nm. Enlarged views depict the reflectance in decibel units. Red, blue, and black curves denote positive, negative, and no magnetization, respectively. The inset in (a) shows the case of s-polarization.
Fig. 4
Fig. 4 SiO2 thickness dependence of (a) maximum ΔR (black) and ΔR/R (green) and (b) minimum reflectance, R, ( + M in red, −M in blue) of the TMOKE. The values of (a) and (b) are obtained at incident angles shown in (c) and (d), respectively.
Fig. 5
Fig. 5 (a) Real part and (b) imaginary part of the effective refractive index, Neff, of the SPPs with the TMOKE as a function of the SiO2 thickness. Red, blue, and black curves denote positive, negative, and no magnetization, respectively.
Fig. 6
Fig. 6 Optical field distribution of TM mode in the Al2O3/SiO2/Fe trilayer waveguide with t = ∞, 1645, 1145, and 645 nm. The curves are normalized at 0 nm.
Fig. 7
Fig. 7 Reflectance, R, and change in reflectance, ΔR, of the PMOKE as a function of incident angle, θ, at t = 652 nm. The enlarged view depicts the reflectance in decibel units. The curve for positive magnetization (red) is perfectly overlapped with that for negative magnetization (blue). The black curve denotes the no magnetization case.
Fig. 8
Fig. 8 The Kerr rotation (red), ϕK, and ellipticity (blue), ηK, at t = 652 nm as a function of the incident angle with positive magnetization, + M. Insets show the polarization normalized by the maximum norm at each angle indicated by arrows.
Fig. 9
Fig. 9 SiO2 thickness dependence of (a) the maximum Kerr rotation, |ϕK| (red), ellipticity, |ηK| (blue), and (b) the minimum reflectance, R (black). The values of (a) and (b) are obtained at incident angles shown in (c) and (d), respectively.
Fig. 10
Fig. 10 Incident angle dependence at t = 645, 652 and 660 nm: (a) The absolute amplitude of the p-polarization, Ep (red), and induced s-polarization, Es (blue). (b) The normalized phase shift of each argument, arg(Ep) (red) and arg(Es)(blue), and phase difference, Δφ (black). (c) The Kerr rotation (red) and ellipticity (blue).
Fig. 11
Fig. 11 Reflectance, R (positive (red), negative (blue) and no (black) magnetizations), and change in reflectance, ΔR (broken curve), of The TMOKE as a function of the incident angle. The thickness of the interface layer is assumed to be (a) tint = 0 nm, (b) 1 nm and (c) 3 nm with the SiO2 layer of t = 661 − tint/2 nm and the Fe layer of 300 − tint/2 nm.
Fig. 12
Fig. 12 The Kerr (a) rotation and (b) ellipticity as a function of the incident angle. The thickness of the interface layer is assumed to be tint = 0, 1, 3 and 5 nm as written beside the curves, and the SiO2 layer is set to t = 652 − tint/2 nm and the Fe layer of 300 − tint/2 nm.
Fig. 13
Fig. 13 Reflectance, R (positive (red), negative (blue) and no (black) magnetizations), and change in reflectance, ΔR (broken curve), of The TMOKE as a function of the incident angle. The deviation angle from p-polarization is assumed to be (a) tint = 0°, (b) 1° and (c) 3° with the SiO2 layer of t = 661 nm.
Fig. 14
Fig. 14 The Kerr (a) rotation and (b) ellipticity as a function of the incident angle. The deviation angle from p-polarization is assumed to be 0° up to 10°.as written beside the curves, and the SiO2 layer is set to t = 652 nm.

Equations (39)

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ε r =( n 2 0 γ 0 n 2 0 γ 0 n 2 ),
ε r =( n 2 γ 0 γ n 2 0 0 0 n 2 ).
ΔR=R( M )R( +M ) [ dB ],
ΔR R = R( M )R( +M ) [ % ] ( R( M )+R( +M ) [ % ] ) /2 ,
ϕ ˜ K = r sp r pp
| r sp | E s ( z ) Δ ε M   E p ( z )dz,
'×E= μ r H ˜
'× H ˜ = ε r E
z' ψ=Ωψ
Ω=( i( n x ε 31 ε 33 + n x μ 23 μ 33 ) i n x ( ε 32 ε 33 μ 23 μ 33 ) n x n y ε 33 + μ 21 μ 23 μ 31 μ 33 n x 2 ε 33 + μ 22 μ 23 μ 32 μ 33 i n y ( ε 31 ε 33 μ 13 μ 33 ) i( n y ε 32 ε 33 + n x μ 13 μ 33 ) n y 2 ε 33 μ 11 + μ 13 μ 31 μ 33 n x n y ε 33 μ 21 + μ 23 μ 32 μ 33 ε 21 ε 23 ε 31 ε 33 + n x n y μ 33 ε 22 ε 23 ε 32 ε 33 n x 2 μ 33 i( n y ε 23 ε 33 + n x μ 31 μ 33 ) i n x ( ε 23 ε 33 μ 32 μ 33 ) ε 11 + ε 13 ε 31 ε 33 + n y 2 μ 33 ε 12 + ε 13 ε 32 ε 33 n x n y μ 33 i n y ( ε 13 ε 33 μ 31 μ 33 ) i( n x ε 13 ε 33 + n y μ 32 μ 33 ) )
ψ= [ E x E y H ˜ x H ˜ y ] T
ψ( z )=W e λ z W 1 ψ( 0 )
ψ( z' )= W i e λ i z' c i =( W Ei + W Ei W Hi + W Hi )( e λ i z' 0 0 e λ i z' )( c i + c i ).
( W Ei + W Ei W Hi + W Hi ) 1 ( W Ei1 + W Ei1 W Hi1 + W Hi1 )( c i1 + c i1 ) =( e λ i k 0 L i 0 0 e λ i k 0 L i ) ( W Ei + W Ei W Hi + W Hi ) 1 ( W Ei+1 + W Ei+1 W Hi+1 + W Hi+1 )( c i+1 + c i+1 )
( W Ei + ' W Ei ' W Hi + ' W Hi ' )( c i1 + c i1 )=( e λ i k 0 L i 0 0 e λ i k 0 L i )( W Ei + ' W Ei ' W Hi + ' W Hi ' )( c i+1 + c i+1 )
( W Ei + ' W Ei ' W Hi + ' W Hi ' )= ( W Ei + W Ei W Hi + W Hi ) 1 ( W Ei±1 + W Ei±1 W Hi±1 + W Hi±1 ).
( c i1 c i+1 + )=( S 11 i S 12 i S 21 i S 22 i )( c i1 + c i+1 )
S 11 i = ( A 1 i ) 1 ( ( W Hi ' ) 1 λ i W Hi + ' ( W Ei + ' ) 1 λ i W Ei + ' ( W Hi ' ) 1 W Hi + ' )
S 12 i = ( A 1 i ) 1 ( ( W Hi ' ) 1 λ i W Hi ' ( W Hi ' ) 1 λ i W Hi ' ( W Ei + ' ) 1 W Ei ' )
S 21 i = ( A 2 i ) 1 ( ( W Ei + ' ) 1 λ i W Ei ' ( W Hi ' ) 1 W Hi + '+ ( W Ei + ' ) 1 λ i W Ei + ' )
S 22 i = ( A 2 i ) 1 ( ( W Ei + ' ) 1 W Ei '+ ( W Ei + ' ) 1 λ i W Ei ' ( W Hi ' ) 1 λ i W Hi ' )
A 1 i =I ( W Hi ' ) 1 λ i W Hi + ' ( W Ei + ' ) 1 λ i W Ei '
A 2 i =I ( W Ei + ' ) 1 λ i W Ei ' ( W Hi ' ) 1 λ i W Hi + '
S A S B =( S 11 a + S 12 a ( I S 11 b S 22 a ) 1 S 11 b S 21 a S 12 a ( I S 11 b S 22 a ) 1 S 12 b S 21 b ( I S 22 a S 11 b ) 1 S 21 a S 21 b ( I S 22 a S 11 b ) 1 S 22 a S 12 b + S 22 b )
S A/B =( S 11 a/b S 12 a/b S 21 a/b S 22 a/b ).
( c ref c 1 + )=( S 11 ref S 12 ref S 21 ref S 22 ref )( c inc + c 1 ),
( c ref c trn + )=( S 11 global S 12 global S 21 global S 22 global )( c inc + c trn =0 ).
ψ( 0 )=( W Eref + W Eref W Href + W Href )( c inc + c ref ) ( E x ref ( 0 ) E y ref ( 0 ) H ˜ x ref ( 0 ) H ˜ y ref ( 0 ) )=( W Eref c ref W Href c ref )=( W Eref S 11 global c inc + W Href S 11 global c inc + )
ψ( L total )=( W Etrn + W Etrn W Htrn + W Htrn )( c trn + c trn =0 ) ( E x trn ( L total ) E y trn ( L total ) H ˜ x trn ( L total ) H ˜ y trn ( L total ) )=( W Etrn + c trn + W Htrn + c trn + )=( W Etrn + S 21 global c inc + W Htrn + S 21 global c inc + )
E z ref/trn = i( H ˜ y ref/trn n x + H ˜ x ref/trn n y ) E x ref/trn ε 31 ref/trn E y ref/trn ε 32 ref/trn ε 33 ref/trn H ˜ z ref/trn = i( E y ref/trn n x + E x ref/trn n y ) H ˜ x ref/trn μ 31 ref/trn H ˜ y ref/trn μ 32 ref/trn μ 33 ref/trn
c inc + = ( W Eref + ) 1 ( E x inc ( 0 ) E y inc ( 0 ) ).
R= ( E ref E inc ) 2 .
[ xy ] T = [ ( E xr +i E xi ) e iωt ( E yr +i E yi ) e iωt ] T = [ E xr cos( ωt ) E xi sin( ωt ) E yr cos( ωt ) E yi sin( ωt ) ] T
cos 2 ( ωt )+ sin 2 ( ωt ) = ( E yi 2 + E yr 2 ) x 2 ( E xr E yi E xi E yr ) 2 2 ( E xi E yi + E xr E yr )xy ( E xr E yi E xi E yr ) 2 + ( E xi 2 + E xr 2 ) y 2 ( E xr E yi E xi E yr ) 2 =1
X T QX=1 Q=( E yi 2 + E yr 2 ( E xr E yi E xi E yr ) 2 E xi E yi + E xr E yr ( E xr E yi E xi E yr ) 2 E xi E yi + E xr E yr ( E xr E yi E xi E yr ) 2 E xi 2 + E xr 2 ( E xr E yi E xi E yr ) 2 ) X= [ xy ] T
X ' T BX'=1( X ' T P 1 )Q( PX' )=1 ( PX' ) T Q( PX' )=1 X=PX'
P=( cos( φ K ) sin( φ K ) sin( φ K ) cos( φ K ) )
φ K =arctan[ 2( E ' xi E yi +E ' xr E yr ) E ' xi 2 +E ' xr 2 E yi 2 E yr 2 + 4 ( E ' xr E yi E ' xi E yr ) 2 + ( E yi 2 + E yr 2 +E ' xi 2 +E ' xr 2 ) 2 ]
η K =arctan[ E yi 2 + E yr 2 +E ' xi 2 +E ' xr 2 4 ( E ' xr E yi E ' xi E yr ) 2 + ( E yi 2 + E yr 2 +E ' xi 2 +E ' xr 2 ) 2 E yi 2 + E yr 2 +E ' xi 2 +E ' xr 2 + 4 ( E ' xr E yi E ' xi E yr ) 2 + ( E yi 2 + E yr 2 +E ' xi 2 +E ' xr 2 ) 2 ]

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