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
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 . 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) [6–9].
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 . This has evolved into an internationally widely adopted method for characterizing and quantifying biomolecular interactions . 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 . Since Hui and Stroud’s proposal , 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 . 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
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  as the high-index layer, SiO2 with nSiO2 = 1.44  as the low-index layer with an arbitrary thickness, t, and Fe with nFe = 3.62−5.56i  and an off-diagonal permittivity of γ = 3.12 + 1.8i  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 tensorEqs. (38) and (39) with the relative permittivity tensor
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.
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,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 criterionFig. 4(b)).
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.
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).
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.
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.
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).
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 . The PMOKE in complex form, , can be defined by
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.
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 (), 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.
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.
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.
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.
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% [34–40]. 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.
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.
In this Appendix, we describe how the transfer matrix method with a 4×4 scattering matrix is developed , 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 asEquations (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 using back substitution, the remaining four equations are in the x-y plane and can be summarized as the following 4 × 4 matrix: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 ‒iny ∂z’]T, where n and its subscripts represent the refractive index and corresponding axis on the indicatrix, respectively. Equation (9) can then be solved as followsEq. (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 c−i 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
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 . By letting the (i−1)-th and (i+1)-th layers be the same free space, namely, the same eigen-modes, Eq. (14) reduces to42]. If layers A and B are combined, one scattering matrix, SA SB, will be given by
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: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 = SrefS1S2…SnStrn. The electromagnetic field coming from the transmission region is typically assumed to be 0:Eq. (13) asEq. (28), the incident electric field can be introduced into c+inc with arbitrary polarization as
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:Eq. (34) is expressed in matrix form asEq. (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.
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.
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