Transverse magnetooptic effect in multilayers applied to mapping of microwave currents

Štefan Višňovský; Eva Lišková-Jakubisová; Ian Harward; Zbigniew Celinski

doi:10.1364/OME.7.002368

1. Introduction

It has been proposed to employ magneto-optical (MO) effects in ferro- or ferri-magnetic films for mapping microwave (mw) current distribution in integrated circuits (chip inspection). The mw currents generate fringing magnetic fields, which in turn cause the magnetization (M) in an adjacent magnetic film to precess [1–6]. The precessional angle, θ, is proportional to the mw current, I_mw. It is advantageous that detection is contact-free and practically non-invasive. The precession-induced magnetization component, $M_{⊥},$ generates MO effects. The focus is on those MO effects which are odd in M. The MO polar Kerr effect detects the M component perpendicular to the film plane, while the MO effects at transverse or longitudinal magnetization detect the in-plane components. The configuration with polar M detects mw currents in the interface plane. It gives the strongest MO signal and the sensing can be performed with a polarized focused laser beam at arbitrary angles of incidence, φ⁽⁰⁾. Usually, φ⁽⁰⁾≈0 are preferred for a good lateral resolution. The configurations with longitudinal or transverse M require φ⁽⁰⁾>0. The MO sensor employs in-plane static magnetic flux density field, B_appl, biasing the film close to the ferromagnetic resonance (FMR) at a desired mw frequency, f. The design must enable operation at moderate B_appl. Other sensor requirements include low mw power absorption with a strong MO response, and sufficient lateral resolution. The sensor should also be resistant against ambient effects and deposited on a thin substrate to reduce the distance between the sensor element and the semiconductor chip surface.

In present-day semiconductor circuits, the bandwidth may exceed 100 GHz. The requirement to keep B_appl reasonably low limits the material choices for the MO probe to either cubic Fe, for its high saturation magnetization, μ₀M_s, or to Ba hexagonal ferrite, BaFe₁₂O₁₉ (BaM), with in-plane c-axis and a strong effective anisotropy field, B_A. In an Fe film, the required B_appl can be evaluated from [7] $f_{Fe} = \frac{γ}{2 π} {[B_{appl} (B_{appl} + μ_{0} M_{s})]}^{1 / 2},$ where $γ \approx 2 π \times 28 \begin{matrix} \end{matrix} GHz/Tesla$ and μ₀M_s = 2.158 Tesla.

The estimate for BaM films with μ₀M_s = 0.4 Tesla and B_A≈1.7 Tesla using the formula [8,9], $f_{BaM} = \frac{γ}{2 π} {[(B_{a p p l} + B_{A}) (B_{a p p l} + B_{A} + μ_{0} M_{s})]}^{1 / 2},$ shows that for a given f the required B_appl is even lower in BaM (Fig. 1).

Fig. 1 Ferromagnetic resonance frequency, f, as a function of the applied magnetic flux density, B_appl, in thin films of iron (Fe) and hexagonal ferrite (BaM).

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The present work deals with MO sensors using transverse MO effect (TMOE) designed for mapping I_mw distribution in semiconductor circuits. Compared to the polar and longitudinal configurations, the operation at transverse M enables the easiest optical access, and may be done using a simpler optical element sequence which does not necessarily require a polarizer-analyzer pair. This potentially extends the working spectral range to the extreme ultraviolet regions where the lack of adequate polarizers presents difficulties [10]. The sensor operates in reflection mode, with direct laser irradiation of its surface. The sensor configuration along with the chosen Cartesian coordinate system is shown in Fig. 2. TMOE is proportional to $M_{⊥}$ perpendicular to the plane of incidence and detects selectively I_mw flowing in the adjustable plane of incidence. An efficient MO sensor performance requires multilayer structures with optimized layer thicknesses. The problem is illustrated on systems with Fe nanolayers as active MO elements and sandwiched between AlN dielectrics. The role of the AlN layers is to protect the Fe nanolayer(s) against ambient, and to maximize the MO response by tuning their thickness [6]. The approach can be applied to other systems including those incorporating hexagonal ferrite nanolayers.

Fig. 2 Interpretation of geometry for transverse magnetization magnetooptic probe. The microwave current, I_mw, the applied magnetic flux density field, B_appl, and the magnetization, M, are parallel to the y-axis. $M_{⊥}$ and B_mw denote the microwave magnetization and field components parallel to the x-axis and normal to the plane yz of light incidence.

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The performance is evaluated in terms of two characteristics: One is a complex perturbation, Δr_pp, to the p-amplitude reflection coefficient, r_pp; this effect is induced by the transverse magnetization, $M_{⊥},$ proportional to the microwave magnetic flux density, B_mw, $| Δ r_{p p} | ≪ | r_{p p} | .$ There is a proportionality, $| Δ r_{p p} | \propto M_{⊥} \propto I_{mw} .$ |Δr_pp|, characterizes the TMOE figure of merit. In the experiment, the complex Δr_pp enters the ellipsometric ratio [11]. Secondly, the experimentally observable difference in the irradiation reflectance, R_pp, at $\pm M_{⊥}$ , i.e., $Δ R_{p p} = [R_{p p} (+ M_{⊥}) - R_{p p} (- M_{⊥})] / 2,$ represents another useful sensor characteristic, which is also proportional to $M_{⊥} \propto I_{mw} .$ The parameters |Δr_pp| and ΔR_pp give the amplitude and power of the MOsignal delivered to the photodetector, the quantities important from the point of view of signal to noise ratio. TMOE is often evaluated in terms of the observable ratio ΔR_pp/R_pp [12–16], though this parameter does not provide useful information without the knowledge of either ΔR_pp or R_pp individually [17,18]. The TMOE sensor performance will be optimized for an incident plane wave polarized in the plane of incidence (p-polarized) at a convenient laser wavelength, λ = 410 nm. Here, |Δr_pp| at an unprotected Fe surface peaks in the region of the principal angle of incidence, $φ_{P}^{(0)}$ ≈72 deg, i.e., at an angle of incidence, $φ^{(0)}$ ≈75 deg, |Δr_pp|≈3.23 × 10^‒3, ΔR_pp peaks at $φ^{(0)}$ ≈58 deg, ΔR_pp≈1.125 × 10^‒3. At the same λ, the maximum of ΔR_pp/R_pp is situated at $φ^{(0)}$ ≈68 deg and has the value ΔR_pp/R_pp≈5.31 × 10^‒3 [19–22]. A trade-off between lateral resolution of the polarized focused laser beam and MO signal amplitude requires smaller angles of incidence, $φ^{(0)} .$ The TMOE sensor performance will therefore be optimized at a still acceptable $φ^{(0)}$ = 45 deg.

In multilayers, Δr_pp can be determined using a transfer matrix approach. Here, it is sufficient to consider a multilayer structure consisting of five layers, indexed 1-5, surrounded by an isotropic ambient (0) and a substrate (6). By restricting Δr_pp to terms linear in magnetization, a common approximation, Δr_pp can be expressed analytically with a satisfactory precision. Then, the TMOE represents an independent component in the MO vector magnetometry affecting the diagonal elements of the Jones reflection matrix, r_pp, while the polar and longitudinal components manifest themselves in the off-diagonal Jones reflection matrix elements [23–33].

In Section 2, the transfer matrix approach based on previous studies is briefly summarized [34–39], while Section 3 lists the analytical expressions for Δr_pp and ΔR_pp components. Section 4 provides practical sensor layer sequences with Fe and AlN on a Si substrate, leading to maxima in either |Δr_pp| or ΔR_pp. The discussion of results is given in the last section, Section 5. The conclusions are given in Section 6. Finally, the Appendix lists isotropic reflection and transmission coefficients required in the computing of Δr_pp.

2. Transfer matrix approach

The specification of electromagnetic plane waves in multilayers at transverse magnetization employs the coordinate system with the z-axis normal to the interfaces where the plane of incidence is normal to the x-axis and to the magnetization, $M_{⊥} = M_{x} .$ At the optical angular frequency, $ω,$ a uniformly magnetized medium (m) with the magnetization vector parallel to the x-axis is characterized by a relative permittivity tensor

ε^{(m)} (ω) = (\begin{matrix} ε_{x x}^{(m)} (ω) & 0 & 0 \\ 0 & ε_{y y}^{(m)} (ω) & ε_{y z}^{(m)} (ω) \\ 0 & ε_{z y}^{(m)} (ω) & ε_{z z}^{(m)} (ω) \end{matrix}) .

With the restrictions to the terms of zero and first order in magnetization in the media of at least cubic symmetry,

ε_{x x}^{(m)} = ε_{y y}^{(m)} = ε_{z z}^{(m)} = ε_{0}^{(m)} .

Here the off-diagonal element

ε_{y z}^{(m)} (M_{x}) = - ε_{z y}^{(m)} (M_{x}) = j ε_{1}^{(m)} (M_{x}) = - j ε_{1}^{(m)} (- M_{x}) .

It is reasonable to assume (ε₁⁽^m⁾<<ε₀⁽^m⁾) and

ε_{1}^{(m)} (M_{x}) \propto M_{x} .

The magnetic permeability, μ, is assumed equal to its vacuum value, μ₀, a valid approximation for laser light at λ = 410 nm, even though μ changes significantly in the GHz frequency range, where I_mw induces the precessional motion of M in the sensor.

The Helmholtz equation for the electric field vector $E^{(m)} = E_{0}^{(m)} exp [j (ω τ - γ^{(m)} \cdot r)]$ of the plane waves propagating in the plane perpendicular to the x-axis with the propagation vector, $γ^{(m)} = (ω / c) (N_{y} \hat{y} + N_{z}^{(m)} \hat{z}),$ becomes

\begin{array}{l} (ε_{0}^{(m)} - N_{y}^{2} - N_{z}^{(m) 2}) E_{x}^{(m)} \hat{x} + (ε_{0}^{(m)} - j ε_{1}^{(m)} + N_{y} N_{z}^{(m) 2} - N_{z}^{(m) 2}) E_{y}^{(m)} \hat{y} \\ + (ε_{0}^{(m)} + j ε_{1}^{(m)} - N_{y}^{2} + N_{y} N_{z}^{(m) 2}) E_{z}^{(m)} \hat{z} = 0. \end{array}

The propagation vector component,

(ω / c) N_{y} \hat{y},

is an invariant. Here

r,

τ, and c denote the position vector, time and light velocity in a vacuum,

\hat{x}, \begin{matrix}  \end{matrix} \hat{y},

and

\hat{z}

are the unit vectors. Equation (2) provides four eigen-values of

N_{z}^{(m)},

i.e.,

N_{z 1}^{(m)} = N_{z 0}^{(m)} = {(ε_{0}^{(m)} - N_{y}^{2})}^{1 / 2} = N^{(m)} α^{(m)}, \begin{matrix}  \end{matrix} N^{(m)} = ε_{0}^{(m) 1 / 2},

N_{z 2}^{(m)} = - N_{z 0}^{(m)}

for the waves with the eigen-polarizations parallel to

\hat{x},

e_{1, 2}^{(m)} = \hat{x},

i.e., perpendicular to the plane of incidence (TE or s-polarized waves) and

N_{z 3}^{(m)} = N_{z 0}^{(m)} (1 - \frac{ε_{1}^{(m) 2}}{2 ε_{0}^{(m)} N_{z 0}^{(m) 2}}) \approx N_{z 0}^{(m)}, \begin{matrix}  \end{matrix} N_{z 4}^{(m)} = - N_{z 0}^{(m)} (1 - \frac{ε_{1}^{(m) 2}}{2 ε_{0}^{(m)} N_{z 0}^{(m) 2}}) \approx - N_{z 0}^{(m)},

for the waves with the eigen-polarization

e_{3, 4}^{(m)} \propto N_{z 0}^{(m) 2} \hat{y} - (N_{y} N_{z 3, 4}^{(m)} - j ε_{1}^{(m)}) \hat{z},

in the plane perpendicular to the x-axis, i.e., parallel to the plane of incidence (TM or p-polarized waves).

The multilayer problem can be dealt with rigorously using the 4 × 4 matrix formalism [34]. The case of $N_{z 1, 2}^{(m)}$ and $e_{1, 2}^{(m)}$ reduce to that of isotropic multilayers. It is therefore sufficient to consider the cases of $e_{3, 4}^{(m)}$ and $N_{z 3, 4}^{(m)}$ affected by the magnetization via $ε_{1}^{(m)}$ , and to reduce the formalism to that of 2 × 2 submatrices. The reduced dynamical matrix becomes

D^{(\underline{m})} = (\begin{matrix} N_{z 0}^{(m)} ε_{0}^{(m) - 1 / 2} & N_{z 0}^{(m)} ε_{0}^{(m) - 1 / 2} \\ - ε_{0}^{(m) 1 / 2} + j ε_{1}^{(m)} N_{y} N_{z 0}^{(m) - 1} ε_{0}^{(m) - 1 / 2} & ε_{0}^{(m) 1 / 2} + j ε_{1}^{(m)} N_{y} N_{z 0}^{(m) - 1} ε_{0}^{(m) - 1 / 2} \end{matrix}) (\begin{matrix} C_{3}^{(m)} & 0 \\ 0 & C_{4}^{(m)} \end{matrix}),

where

C_{3}^{(m)}

and

C_{3}^{(m)}

normalize the eigen-polarizations

e_{3, 4}^{(m)}

to unity. The underlined superscript (m) indicates the medium with transverse magnetization. In the layer m, the dynamical matrix,

D^{(\underline{m})},

transforms the amplitudes of forward,

E_{03}^{(\underline{m})},

and backward,

E_{04}^{(\underline{m})},

propagating eigen-mode amplitudes at the front layer interface plane,

z = z^{(\underline{m} - 1)},

into the tangential field amplitudes,

E_{0 y}^{(\underline{m})}

and

H_{0 x}^{(\underline{m})},

at the same plane,

z = z^{(\underline{m} - 1)},

(\begin{matrix} E_{0 y}^{(\underline{m})} (z^{(\underline{m} - 1)}) \\ H_{0 x}^{(\underline{m})} (z^{(\underline{m} - 1)}) \end{matrix}) = D^{(\underline{m})} (\begin{matrix} E_{03}^{(\underline{m})} (z^{(\underline{m} - 1)}) \\ E_{04}^{(\underline{m})} (z^{(\underline{m} - 1)}) \end{matrix}) .

For a layer, the characteristic matrix relates the in-plane wave field components on the front layer boundary, z⁽^m^-1), to those on the back layer boundary, z⁽^m⁾,

(\begin{matrix} E_{0 y}^{(\underline{m})} (z^{(\underline{m} - 1)}) \\ H_{0 x}^{(\underline{m})} (z^{(\underline{m} - 1)}) \end{matrix}) = S^{(\underline{m})} (\begin{matrix} E_{0 y}^{(\underline{m})} (z^{(\underline{m})}) \\ H_{0 x}^{(\underline{m})} (z^{(\underline{m})}) \end{matrix}) .

The characteristic matrix becomes

S^{(\underline{m})} = D^{(\underline{m})} P^{(m)} D^{(\underline{m}) - 1} = (\begin{matrix} \cos β^{(m)} & - j \frac{N_{z 0}^{(m)}}{ε_{0}^{(m)}} \sin β^{(m)} \\ - j \frac{ε_{0}^{(m)}}{N_{z 0}} \sin β^{(m)} & \cos β^{(m)} \end{matrix}) + q^{(m)} \sin β^{(m)} (\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix}),

where the TMOE parameter [20,21] is given by

q^{(m)} (M_{x}) = \frac{ε_{1}^{(m)} (M_{x}) N_{y}}{ε_{0}^{(m)} N_{z 0}^{(m)}} = - q^{(m)} (- M_{x}) .

The product contains the propagation matrix

P^{(\underline{m})} = P^{(m)} = (\begin{matrix} e^{j β^{(m)}} & 0 \\ 0 & e^{- j β^{(m)}} \end{matrix}),

where

β^{(\underline{m})} = β^{(m)} = \frac{ω}{c} N_{z 0}^{(m)} d^{(m)} .

Here

d^{(m)}

denotes the layer thickness. With the restriction to the terms of zero and first order in

ε_{1}^{(m)},

the transverse magnetization affects

D^{(\underline{m})}

while

P^{(\underline{m})} = P^{(m)}

and

β^{(\underline{m})} = β^{(m)}

take the values of isotropic layers.

Magneto-optic effects in multilayers consisting of N layers at transverse magnetization sandwiched between isotropic half-space (0) and the half space (N + 1) follow from 2 × 2 matrix product represented by

M = (\begin{matrix} M_{33} & M_{34} \\ M_{43} & M_{44} \end{matrix}) = D^{(0) - 1} \prod_{\underline{m} = 1}^{N} S^{(\underline{m})} D^{(\underline{N + 1})} .

The reduced M matrix relates the electric fields of the incident

(E_{p \to}^{(0)})

, reflected

(E_{p \leftarrow}^{(0)}),

and transmitted

(E_{p \to}^{(N + 1)})

p-polarized waves

(\begin{matrix} E_{p \to}^{(0)} \\ E_{p \leftarrow}^{(0)} \end{matrix}) = (\begin{matrix} M_{33} & M_{34} \\ M_{43} & M_{44} \end{matrix}) (\begin{matrix} E_{p \to}^{(N + 1)} \\ E_{p \leftarrow}^{(N + 1)} \end{matrix}),

with either

E_{p \leftarrow}^{(N + 1)} = 0

or

E_{p \to}^{(0)} = 0.

In some reflection problems, it is useful to account for the transverse magnetization in the half space (N + 1) represented by

D^{(\underline{N + 1})}

(the case of the magnetic substrate at

M_{⊥}

). The form of

D^{(\underline{N + 1})}

is given in Eq. (4). The inverse dynamical matrix of the entrance isotropic medium (0)

D^{(0) - 1} = {(2 N^{(0)} α^{(0)})}^{- 1} (\begin{matrix} N^{(0)} & - α^{(0)} \\ N^{(0)} & α^{(0)} \end{matrix})

with

α^{(0)} = \cos φ^{(0)},

φ^{(0)}

denotes the angle of incidence. Then

N_{y} = N^{(0)} \sin φ^{(0)} .

The dynamical matrix of the isotropic exit medium (N + 1) is given by

D^{(N+1)} = (\begin{matrix} α^{(N+1)} & α^{(N+1)} \\ - N^{(N+1)} & N^{(N+1)} \end{matrix}),

where the cosine of the refraction angle is given by

α^{(N+1)} = N_{z 0}^{(N+1)} ε_{0}^{(N+1) - 1 / 2} .

The reflection coefficient for the p-polarized wave incident from the medium (0) on a stack of layers 1 to N and the substrate N + 1 at

M_{⊥}

is given by

r_{p p}^{(0 \underline{1} \dots N, \underline{N + 1})},

r_{p p}^{(0 \underline{1} \dots N, \underline{N + 1})} = \frac{E_{p \leftarrow}^{(0)}}{E_{p \to}^{(0)}} = {[M^{(0 \underline{1} \dots N, \underline{N + 1})}]}_{43} {[M^{(0 \underline{1} \dots N, \underline{N + 1})}]}_{33}^{- 1} .

3. Analytical expressions

The formulae for a general multilayer at transverse magnetization, $M_{⊥},$ will be now applied to a five magnetic layer system on a magnetic substrate. The reflection coefficient follows from $M^{(0 \underline{123456})}$ represented by

M^{(0 \underline{123456})} = D^{(0) - 1} S^{(\underline{1})} S^{(\underline{2})} S^{(\underline{3})} S^{(\underline{4})} S^{(\underline{5})} D^{(\underline{6})} .

Thanks to the restriction to terms linear in

M_{x} \equiv M_{⊥},

the total reflection coefficient,

r_{p p}^{(0 \underline{123456})},

of the structure can be expressed as a sum of components from individual layers

r_{p p}^{(0 \underline{123456})} = r_{p p}^{(0123456)} + Δ r_{p p}^{(0 \underline{1} 23456)} + Δ r_{p p}^{(01 \underline{2} 3456)} + ... + Δ r_{p p}^{(012345 \underline{6})},

where

r_{p p}^{(0123456)}

is the reflection coefficient of the corresponding isotropic system

(M_{⊥} = 0) .

The TMOE components are defined as

\begin{array}{l} Δ r_{p p}^{(0 \underline{1} 23456)} = \frac{1}{2} [r_{p p}^{(0 \underline{1} 23456)} (M_{x}) - r_{p p}^{(0 \underline{1} 23456)} (- M_{x})], \\ Δ r_{p p}^{(01 \underline{2} 3456)} = \frac{1}{2} [r_{p p}^{(01 \underline{2} 3456)} (M_{x}) - r_{p p}^{(01 \underline{2} 3456)} (- M_{x})], \\ \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \\ Δ r_{p p}^{(012345 \underline{6})} = \frac{1}{2} [r_{p p}^{(012345 \underline{6})} (M_{x}) - r_{p p}^{(012345 \underline{6})} (- M_{x})], \end{array}

where

\begin{array}{l} Δ r_{p p}^{(0 \underline{1} 23456)} = \frac{j}{2} q^{(1)} (1 - e^{- j2 β^{(1)}}) (1 - r_{p p}^{(123456) 2} e^{- 2 j β^{(1)}}) t_{p p}^{(01)} t_{p p}^{(10)} {(1 - r_{p p}^{(10)} r_{p p}^{(123456)} e^{- 2 j β^{(1)}})}^{- 2}, \\ Δ r_{p p}^{(01 \underline{2} 3456)} = \frac{j}{2} q^{(2)} (1 - e^{- j2 β^{(2)}}) (1 - r_{p p}^{(23456) 2} e^{- 2 j β^{(2)}}) t_{p p}^{(012)} t_{p p}^{(210)} {(1 - r_{p p}^{(210)} r_{p p}^{(23456)} e^{2j β^{(2)}})}^{- 2}, \\ Δ r_{p p}^{(012 \underline{3} 456)} = \frac{j}{2} q^{(3)} (1 - e^{- j2 β^{(3)}}) (1 - r_{p p}^{(3456) 2} e^{- 2 j β^{(3)}}) t_{p p}^{(0123)} t_{p p}^{(3210)} {(1 - r_{p p}^{(3210)} r_{p p}^{(3456)} e^{- 2 j β^{(3)}})}^{- 2}, \end{array}

\begin{array}{l} Δ r_{p p}^{(0123 \underline{4} 56)} = \frac{j}{2} q^{(4)} (1 - e^{- j2 β^{(4)}}) (1 - r_{p p}^{(456) 2} e^{- 2 j β^{(4)}}) t_{p p}^{(01234)} t_{p p}^{(43210)} {(1 - r_{p p}^{(43210)} r_{p p}^{(456)} e^{- 2 j β^{(4)}})}^{- 2}, \\ Δ r_{p p}^{(01234 \underline{5} 6)} = \frac{j}{2} q^{(5)} (1 - e^{- j2 β^{(5)}}) (1 - r_{p p}^{(56) 2} e^{- 2 j β^{(5)}}) t_{p p}^{(012345)} t_{p p}^{(543210)} {(1 - r_{p p}^{(543210)} r_{p p}^{(56)} e^{- 2 j β^{(5)}})}^{- 2}, \\ Δ r_{p p}^{(012345 \underline{6})} = j \frac{1}{2} q^{(6)} t_{p p}^{(0123456)} t_{p p}^{(6543210)} . \end{array}

They are given as a product of a corresponding TMOE parameter q⁽ⁱ⁾, i = 1,…6, a factor

j (1 - e^{- j2 β^{(i)}}) / 2

, and the expressions constructed from reflection and transmission coefficients in isotropic multilayers (Appendix). The reflectance difference,

Δ R_{p p},

is defined as

\begin{array}{l} Δ R_{p p}^{(0 \underline{123456})} = \frac{1}{2} [R_{p p} (+ M_{⊥}) - R_{p p} (- M_{⊥})] = \frac{1}{2} {[r_{p p}^{(0 \underline{123456})} r_{p p}^{(0 \underline{123456}) *}]}_{+} \\ - \frac{1}{2} {[r_{p p}^{(0 \underline{123456})} r_{p p}^{(0 \underline{123456}) *}]}_{-} \end{array}

The sum is approximated by

R_{p p}^{(0 \underline{123456})} = \frac{1}{2} {[r_{p p}^{(0 \underline{123456})} r_{p p}^{(0 \underline{123456}) *}]}_{+} + \frac{1}{2} {[r_{p p}^{(0 \underline{123456})} r_{p p}^{(0 \underline{123456}) *}]}_{-} \approx R_{p p}^{(0123456)}

where

R_{p p}^{(0123456)}

is the isotropic reflectance

(M_{⊥} = 0) .

It enters the expression for the ratio ΔR_pp/R_pp. The total

Δ R_{p p}^{(0 \underline{123456})}

can be approximated as a sum of components originating from the layers 1-5 and the substrate (6)

\begin{array}{l} Δ R_{p p}^{(0 \underline{123456})} \approx \frac{1}{2} r_{p p}^{(0123456)} (Δ r_{p p}^{(0 \underline{1} 23456) *} + Δ r_{p p}^{(01 \underline{2} 3456) *} + ... + Δ r_{p p}^{(012345 \underline{6}) *}) \\ + \frac{1}{2} r_{p p}^{(0123456) *} (Δ r_{p p}^{(0 \underline{1} 23456)} + Δ r_{p p}^{(01 \underline{2} 3456)} + ... + Δ r_{p p}^{(012345 \underline{6})}) . \end{array}

4. Optimized sensor response

In this section, realistic configurations of TMOE sensors working in reflection mode will be evaluated at λ = 410 nm of the laser source and at $φ^{(0)}$ = 45 deg. The model applied to find an optimized sensor response will be demonstrated on multilayer systems with Fe and AlN layers on Si or Au/Si. The material optical and MO parameters for Fe are collected in Table 1 [19–22]. They include ε_xx^(Fe) and ε_yz^(Fe) defined in Eq. (1), the penetration depth, d^(Fe)_penetr, and q^(Fe) defined in Eq. (6). For AlN, the real index of refraction is taken as N^(AlN) = 1.9264 [6]. The thickness, d(λ/2) = 114.404 nm, corresponds to β^(AlN) = π, Eq. (8). For Au, the relative permittivity assumes the value of ε^(Au) = ‒1.70216‒5.71736j with ε^(Au)1/2 = 1.46-1.958j_, and the penetration depth, d^(Au)_penetr = 16.6633 nm [40]. The complex index of refraction is of Si, N^(Si) = 5.289‒0.292j for the optically thick silicon wafer employed as a substrate [41]. The high resistivity Si is transparent for mw radiation up to terahertz frequencies [42].

Table 1. Optical and magneto-optical parameters of Fe

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The performance of sensor multilayers is evaluated in terms of their MO characteristics, |Δr_pp| and ΔR_pp. The maxima of |Δr_pp| and ΔR_pp are found by varying the thickness of layers forming the sensor. For the optimal sensor multilayers, it is interesting to record a set of additional parameters. These include the total reflectivity, R_pp, total absorbed power, A_pp, the distribution of power dissipated in the absorbing layers, $A_{p p}^{(i)}$ (i = 1,2,…,5), and the distribution among the Fe layers of $Δ r_{p p}^{(\underline{i})}$ (including their phase relations) and $Δ R_{p p}^{(\underline{i})},$ where $Δ R_{p p}^{(\underline{i})} = \frac{1}{2} [{(Δ r_{p p}^{(\underline{i})})}^{*} (r_{p p}^{(0123456)}) + (Δ r_{p p}^{(\underline{i})}) {(r_{p p}^{(0123456)})}^{*}] .$ The conditions for maximal ΔR_pp/R_pp are included.

4.1. Single Fe layer

The analysis starts with simpler structures. The thickness dependence of $| Δ r_{p p} |$ and $Δ R_{p p}$ in an uncovered Fe film on Si substrate are not monotonous, but show slightly displaced flat maxima (Fig. 3). Table 2 gives the Fe thickness, d_Fe, required for the maximal $| Δ r_{p p} |,$ $Δ R_{p p},$ and ΔR_pp/R_pp. The reflectance of the Fe film on Si as a function of d_Fe displays a pronounced maximum at d_Fe = 16 nm and a flat minimum at d_Fe = 61 nm.

Fig. 3 Effect of the thickness, d^(Fe), of the Fe layer on a Si substrate on the reflectance, R_pp, and the magneto-optic parameters, |∆r_pp|, ∆R_pp, ∆R_pp/R_pp (See Table 2).

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Table 2. Fe film on Si

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A modest |Δr_pp| and ΔR_pp enhancement is achieved with a dielectric film on an optically thick Fe (Table 3).

Table 3. Thick Fe film, d_Fe = 10⁴ nm, with AlN cover, d_AlN denotes the thickness of AlN cover

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Both of |Δr_pp| and ΔR_pp are improved for optimized Fe and AlN thicknesses. The results are listed in Table 4. The optimized parameters are printed in bold.

Table 4. AlN(d₁)Fe(d₂)Si structure

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An attempt was made to further improve both |Δr_pp| and ΔR_pp by an AlN layer between Fe and Si. The results for the AlN(d₁)Fe(d₂)AlN(d₃)Si structure are collected in Table 5. The optimization was achieved for the Fe thickness, d₂ = 32 nm (Table 5). However, nosignificant improvement was achieved with respect to the simpler AlN(d₁)Fe(d₂)Si. Also, the replacement of the AlN layer with an Au reflector on Si was not very efficient (Table 6). The structure may be nevertheless of some interest as it avoids a more complicated Fe deposition on AlN. The value of A_pp(Fe) shows that most power is absorbed in Fe and only a small part in Au layer.

Table 5. AlN(d₁)Fe(d₂)AlN(d₃)Si structure.

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Table 6. AlN(d₁)Fe(d₂)Au(d₃)Si structure, d₃ = 2 nm

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Table 7 illustrates the addition of both lower AlN and Au reflector. Compared to the AlN(d₁)Fe(d₂)Si structure of Table 4, there is some improvement in ΔR_pp, only. At the maximal ΔR_pp/R_pp, the structure becomes anti-reflecting (R_pp≈0) with a low |Δr_pp| and negligible ΔR_pp.

Table 7. AlN(d₁)Fe(d₂) AlN(d₃)Au(d₄)Si structure, d₄ = 2 nm

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In the optimized structures with a single ultrathin Fe, |Δr_pp| and ΔR_pp were enhanced with respect to an uncovered thick Fe film by the factors of 1.46 and 1.16, respectively.

4.2 Fe multilayers

Further increases in |Δr_pp| and ΔR_pp can be achieved by using structures with two or more ultrathin Fe layers. The effect is illustrated on Fe(d₁)AlN(d₂)Fe(d₃)AlN(d₄)Fe(d₅)Si. Table 8 demonstrates the increase of the optimized |Δr_pp| with number of Fe layers. In the optimized 3Fe system, the three Fe layers separated by AlN are of different thicknesses, the first Fe layer absorbs most power, A_pp⁽¹⁾, and has the strongest MO components, i.e., |∆r_pp⁽¹⁾|>|∆r_pp⁽³⁾|>|∆r_pp⁽⁵⁾|, and |∆R_pp⁽¹⁾|>|∆R_pp⁽³⁾|>|∆R_pp⁽⁵⁾|. The small difference between |∆r_pp⁽¹⁾| + |∆r_pp⁽³⁾| + |∆r_pp⁽⁵⁾|≈2.33 × 10^‒3 and |∆r_pp| = |∆r_pp⁽¹⁾ + ∆r_pp⁽³⁾ + ∆r_pp⁽⁵⁾|≈2.30 × 10^‒3 indicates that the three components are almost in phase.

Table 8. Fe(d₁)AlN(d₂)Fe(d₃)AlN(d₄)Fe(d₅) structure on Si

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4.3. Fe bilayers

In what follows, however, the analysis is confined to structures with Fe bilayers, as they give reasonable |Δr_pp| and ΔR_pp while remaining relatively easy to fabricate. The results of optimization for AlN(d₁)Fe(d₂)AlN(d₃)Fe(d₄)AlN(d₅) structure on Si are collected in Table 9. The structure can be made anti-reflecting (R_pp = 4.09 × 10⁻⁵), but with |Δr_pp| and ΔR_pp too small for the practical sensor use. The replacement of the AlN layer on Si with an Au reflector on Si simplifies the deposition without any significant changes inperformance (Table 10). In the optimized structures with two Fe layers, |Δr_pp| and ΔR_pp were enhanced with respect to an uncovered thick Fe film (Table 2) by the factors of 1.78 (Table 10) and 1.55 (Table 9), respectively. The rates of decrease of |Δr_pp| and ΔR_pp when d₁ through d₅ deviate from their values found for |Δr_pp|_max and (ΔR_pp)_max follow from Eqs. (50) and (53). All characteristics are the functions of thicknesses of AlN layers, d_AlN, with the period d(λ/2). In the following figures, the effect of d_AlN is illustrated on the structure of Table 10 optimized for |∆r_pp|_max with varying d₃, the thickness of the AlN spacer separating the Fe layers, i.e. on AlN(45 nm)Fe(17 nm) AlN(d₃)Fe(31 nm)Au(2 nm) on Si. Figure 4 shows |∆r_pp| = |∆r_pp⁽²⁾| + |∆r_pp⁽⁴⁾| and its components, |∆r_pp⁽²⁾| and |∆r_pp⁽⁴⁾|, originating from the second and fourth layers, Fe(17 nm) and Fe(31 nm), respectively. The complex ∆r_pp⁽²⁾ and ∆r_pp⁽⁴⁾ are exactly in phase at d₃ = 93 nm (Fig. 5). The effect of d⁽³⁾ on ∆R_pp and its components is shown in Fig. 6. Figures 7 and 8 display ΔR_pp(d⁽³⁾)/R_pp(d⁽³⁾)and the optical parameters, i.e., the fractions of reflected power, R_pp(d⁽³⁾), transmitted power, T_pp(d⁽³⁾), and absorbed power, A_pp(d⁽³⁾) and its distribution among the absorbing layers, A_pp⁽²⁾(d⁽³⁾) and A_pp⁽⁴⁾(d⁽³⁾). Inspection of Figs. 4, and 6-8 shows that the maxima of|∆r_pp(d⁽³⁾)| and ΔR_pp(d⁽³⁾) are situated close to the maxima of R_pp(d⁽³⁾) and T_pp(d⁽³⁾) and close to the minimal A_pp(d⁽³⁾). The maximal ΔR_pp(d⁽³⁾)/R_pp(d⁽³⁾) in the AlN(45 nm)Fe(17 nm)AlN(d⁽³⁾)Fe(31 nm) Au(2 nm) multilayer (Fig. 7) shifts towards the minimum of R_pp(d⁽³⁾). The trends are almost the same in the structure optimized for (∆R_pp)_max, i.e., in the AlN(26 nm)Fe(18 nm)AlN(d⁽³⁾)Fe(29 nm)Au(2 nm) multilayer on Si, as illustrated in Fig. 9 for ΔR_pp(d⁽³⁾). The ΔR_pp/R_pp maxima are situated close to the R_pp minima and A_pp maxima. As expected, the optimized $| Δ r_{p p} |,$ $Δ R_{p p},$ and ΔR_pp/R_pp require low T_pp.

Table 9. AlN(d₁)Fe(d₂)AlN(d₃)Fe(d₄)AlN(d₅) structure on Si

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Table 10. AlN(d₁)Fe(d₂)AlN(d₃)Fe(d₄)Au(d₅)Si structure, d₅ = 2 nm

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Fig. 4 Absolute values of the magnetoooptic amplitude reflection coefficient, |∆r_pp| = |∆r_pp⁽²⁾ + ∆r_pp⁽⁴⁾|, and its components, |∆r_pp⁽²⁾| and |∆r_pp⁽⁴⁾| as functions of the thickness, d⁽³⁾, of the AlN(d⁽³⁾) layer in the AlN(45 nm)Fe(17 nm)AlN(d⁽³⁾)Fe(31 nm) Au(2 nm) multilayer on a Si substrate. See Table 10.

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Fig. 5 Effect of the thickness, d⁽³⁾, of the AlN(d⁽³⁾) layer, on |∆r_pp⁽²⁾| + |∆r_pp⁽⁴⁾|, |∆r_pp⁽²⁾ + ∆r_pp⁽⁴⁾|, and their difference in the AlN(45 nm)Fe(17 nm)AlN(d⁽³⁾)Fe(31 nm)Au(2 nm) multilayer on a Si substrate. See Table 10.

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Fig. 6 Transverse magneto-optic reflectance difference, ∆R_pp = ∆R_pp⁽²⁾ + ∆R_pp⁽⁴⁾, and its components as functions of the thickness, d⁽³⁾, of the AlN(d⁽³⁾) layer in the AlN(45 nm) Fe(17 nm) AlN(d⁽³⁾) Fe(31 nm) Au(2 nm) multilayer on a Si substrate. See Table 10.

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Fig. 7 Transverse magneto-optic reflectance difference ratio, ∆R_pp/R_pp, reflectance, R_pp, and the power transmitted into the Si substrate, T_pp, in the AlN(45 nm) Fe(17 nm) AlN(d⁽³⁾) Fe(31 nm) Au(2 nm) multilayer. See Table 10.

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Fig. 8 Power, A_pp, absorbed in the AlN(45 nm) Fe(17 nm) AlN(d⁽³⁾) Fe(31 nm) Au(2 nm) multilayer on Si substrate and its distribution among the absorbing layers, Fe(d⁽²⁾), A_pp⁽²⁾, and Fe(d⁽⁴⁾), A_pp⁽⁴⁾, as functions of the thickness, d⁽³⁾, of the AlN(d⁽³⁾). The power absorbed in the Au reflector, Au(d⁽⁵⁾) was below 0.0023 in the whole d⁽³⁾ range. See Table 10.

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Fig. 9 Transverse magneto-optic reflectance difference, ∆R_pp = ∆R_pp⁽²⁾ + ∆R_pp⁽⁴⁾, and its components as functions of the thickness, d⁽³⁾, of the AlN(d⁽³⁾) layer in the AlN(26 nm) Fe(18 nm) AlN(d⁽³⁾) Fe(29 nm) Au(2 nm) multilayer on a Si substrate. See Table 10.

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5. Discussion

Tables 4-10 provide data required in the design of TMOE sensors with MO active Fe nanolayers operating at the wavelength of 410 nm at an angle of incidence of 45 deg. Both |∆r_pp| and ∆R_pp increase in multilayers with two or more Fe nanolayers. The microwave energy losses in the sensor are expected to remain low as the total Fe thickness falls in a ten nm range. For practical reasons, the choice is restricted to structures with only one or two Fe layers, as the fabrication of structures with higher number of layers of defined thickness and low interface roughness requires more effortand cost. The simplest and cheapest solutions are represented by the structures deposited on a Si substrate with a single AlN protected Fe nanolayer, i.e., the AlN(40 nm)Fe(33 nm) bilayer optimized for |∆r_pp|_max≈2.77 × 10^‒3 and the AlN(14 nm)Fe(31 nm) bilayer optimized for (∆R_pp)_max≈1.19 × 10^‒3 (Table 4). Cost effective solutions with an improved performance will employ structures on Si with the two Fe nanolayers of Table 10, i.e., the AlN(45 nm)Fe(17 nm)AlN(93 nm) Fe(31 nm)Au(2 nm) multilayer optimized for |∆r_pp|_max≈3.37 × 10^‒3 (Fig. 4)and the AlN(26 nm)Fe(18 nm)AlN(92 nm)Fe(29 nm)Au(2 nm) multilayer optimized for (∆R_pp)_max≈1.63 × 10^‒3 in Fig. 9 (Table 9).

The choice of the short visible λ = 410 nm (E = 3.024 eV) of the laser source enables reasonable lateral resolution. Unfortunately, at E = 3.024 eV the function |q(E)| for Fe computed from the literature data [19–22] is close to its minimum (situated at E≈3.4 eV). In BaM, the situation is even less favorable, as shown in Fig. 10. The BaM spectrum of |q| was computed from the data by Atkinson et al. [43]. The shift of the operating wavelength to λ = 380 nm (E = 3.25 eV) would considerably improve the sensor performance. In addition to lower B_appl requirements, BaM is distinguished by a low FMR linewidth [8]. This, in principle, would enable a mw frequency selective operation. An order in magnitude lower FMR linewidth in BaM compared to Fe [9,44,45] may produce a more efficient coupling between I_mw and $M_{⊥},$ and compensate for lower |q| in BaM. A practical realization of the MO sensor employing BaM with in-plane c-axis and high B_A depends on the progress in technology. In the multilayer design of optimized structures, it is advantageous that BaM is compatible with oxide dielectrics and needs no protection against ambient.

Fig. 10 Absolute value of the transverse magneto-optic parameter, q, for iron (Fe) and hexagonal ferrite (BaF) as a function of photon energy in eV at an angle of incidence of 45 deg.

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The analytical formulae for ∆r_pp and ∆R_pp employed here in the design of TMOE sensor multilayers can be applied in other situations. These include MO magnetometry of exchange coupled layers (adjacent or coupled across a spacer), systems with coupling of pairs of magnetic layers across dielectric (AlN) or metallic (Au, Ag, Cu, Pt, Pd) spacers [31,46]. Here the system response can be compared with the model and separated into the TMOE contributions from individual layers. From them, the magnetic state of each layer can be deduced. In modeling non-uniform layers, the TMOE response can be approximated by that produced by a stack of uniform layers, which is much easier to compute. The approach may also be extended to multilayers with sandwiching layers made of plasmonic and double negative media [47–49]. The five layer system employed here can serve as a useful limiting case in the computer analysis of systems with higher number of magnetic layers.

6. Conclusions

Analytical expressions for the optical response in multilayers at transverse magnetization with up to five magnetic layers are applied to the evaluation of magnetooptic characteristics of sensors with several Fe nanolayers applied to mapping the microwave current distribution in integrated circuits with diffraction limited lateral resolution at a laser wavelength of 410 nm. The optimized sensor characteristics are expressed in terms of magnetooptic contributions to the reflected electric field amplitude, Δr_pp and the reflected power ΔR_pp, the parameter proportional to the energy delivered to the photodetector.

Appendix Reflection and transmission coefficients in isotropic multilayers

The reflection and transmission p-coefficients for isotropic multilayers can be obtained using transfer matrix formalism with M=0 or q⁽ⁱ⁾=0 (i=1,…6) [50–53]. Alternatively, they can be expressed analytically using the following parameters

A_{01} = 1, B_{01} = r_{p p}^{(01)}, C_{01} = r_{p p}^{(01)}, D_{01} = 1,

\begin{array}{l} A_{02} = 1 + r_{p p}^{(01)} r_{p p}^{(12)} e^{- 2 j β^{(1)}},_{}^{} B_{02} = r_{p p}^{(01)} e^{- 2 j β^{(1)}} + r_{p p}^{(12)},_{}^{} \\ C_{02} = r_{p p}^{(01)} + r_{p p}^{(12)} e^{- 2 j β^{(1)}},_{}^{} D_{02} = r_{p p}^{(01)} r_{p p}^{(12)} + e^{- 2 j β^{(1)}} . \end{array}

Then

\begin{array}{l} A_{03} = [(1 + r_{p p}^{(01)} r_{p p}^{(12)} e^{- 2 j β^{(1)}}) + (r_{p p}^{(01)} e^{- 2 j β^{(1)}} + r_{p p}^{(12)}) r_{p p}^{(23)} e^{- 2 j β^{(2)}}] = A_{02} + B_{03} r_{p p}^{(23)} e^{- 2 j β^{(2)}}, \\ B_{03} = [(1 + r_{p p}^{(01)} r_{p p}^{(12)} e^{- 2 j β^{(1)}}) r_{p p}^{(23)} + (r_{p p}^{(01)} e^{- 2 j β^{(1)}} + r_{p p}^{(12)}) e^{- 2 j β^{(2)}}] = A_{02} r_{p p}^{(23)} + B_{03} e^{- 2 j β^{(2)}}, \end{array}

\begin{array}{l} C_{03} = [(r_{p p}^{(01)} + r_{p p}^{(12)} e^{- 2 j β^{(1)}}) + (r_{p p}^{(01)} r_{p p}^{(12)} + e^{- 2 j β^{(1)}}) r_{p p}^{(23)} e^{- 2j β^{(2)}}] = C_{02} + D_{02} r_{p p}^{(23)} e^{- 2j β^{(2)}}, \\ D_{03} = [(r_{p p}^{(01)} + r_{p p}^{(12)} e^{- 2 j β^{(1)}}) r_{p p}^{(23)} + (r_{p p}^{(01)} r_{p p}^{(12)} + e^{- 2 j β^{(1)}}) e^{- 2 j β^{(2)}}] = C_{02} r_{p p}^{(23)} + D_{02} e^{- 2j β^{(2)}}, \end{array}

and

\begin{array}{l} A_{04} = A_{03} + B_{03} r_{p p}^{(34)} e^{- 2 j β^{(3)}}, B_{04} = A_{03} r_{p p}^{(34)} + B_{03} e^{- 2 j β^{(3)}}, \\ C_{04} = C_{03} + D_{03} r_{p p}^{(34)} e^{- 2 j β^{(3)}}, D_{04} = D_{03} e^{- 2 j β^{(3)}} + C_{03} r_{p p}^{(34)}, \end{array}

\begin{array}{l} A_{05} = A_{04} + B_{04} r_{p p}^{(45)} e^{- 2 j β^{(4)}}, B_{05} = A_{04} r_{p p}^{(45)} + B_{04} e^{- 2 j β^{(4)}}, \\ C_{05} = C_{04} + D_{04} r_{p p}^{(45)} e^{- 2 j β^{(4)}}, D_{05} = D_{04} e^{- 2 j β^{(4)}} + C_{04} r_{p p}^{(45)}, \\ A_{06} = A_{05} + B_{05} r_{p p}^{(56)} e^{- 2 j β^{(5)}}, B_{06} = A_{05} r_{p p}^{(56)} + B_{05} e^{- 2 j β^{(5)}}, \\ C_{06} = C_{05} + D_{05} r_{p p}^{(56)} e^{- 2 j β^{(5)}}, D_{06} = D_{05} e^{- 2 j β^{(5)}} + C_{05} r_{p p}^{(56)}, \end{array}

and so on. The Fresnel reflection coefficients are defined as,

(α^{(i)} = \cos φ^{(i)})

r_{p p}^{(i - 1, i)} = - r_{p p}^{(i, i - 1)} = \frac{N^{(i - 1)} α^{(i)} - N^{(i)} α^{(i - 1)}}{N^{(i - 1)} α^{(i)} + N^{(i)} α^{(i - 1)}}, \begin{matrix}  \end{matrix} i = 1, ..., 6.

The reflection coefficient for the wave incident from the medium (0)

r_{p p}^{(0, 1, ... n, n + 1)} = C_{0, n + 1} A_{0, n + 1}^{- 1},

and for the wave incident from the medium (n + 1)

r_{p p}^{(n + 1, n, ..., 1, 0)} = - B_{0, n + 1} A_{0, n + 1}^{- 1} .

The transmission coefficient for the wave incident from the medium (0)

t_{p p}^{(0, 1, ... n, n + 1)} = t_{p p}^{(01)} e^{- j β^{(1)}} t_{p p}^{(12)} e^{- j β^{(2)}} ... t_{p p}^{(n - 1, n)} e^{- j β^{(n - 1)}} t_{p p}^{(n, n + 1)} A_{0, n + 1}^{- 1},

and for the wave incident from the medium (n + 1)

t_{p p}^{(n + 1, n, ..., 1, 0)} = t_{p p}^{(10)} e^{- j β^{(1)}} t_{p p}^{(21)} e^{- j β^{(2)}} ... t_{p p}^{(n, n - 1)} e^{- j β^{(n - 1)}} t_{p p}^{(n + 1, n)} A_{0, n + 1}^{- 1},

where the Fresnel transmission coefficients are given by

t_{p p}^{(i - 1, i)} = \frac{2 N^{(i - 1)} α^{(i - 1)}}{N^{(i - 1)} α^{(i)} + N^{(i)} α^{(i - 1)}}, \begin{matrix}  \end{matrix} i = 1, ..., 6, and t_{p p}^{(i, i - 1)} = \frac{2 N^{(i)} α^{(i)}}{N^{(i - 1)} α^{(i)} + N^{(i)} α^{(i - 1)}}, \begin{matrix}  \end{matrix} i = 1, ..., 6.

The reflection and transmission s-coefficients for isotropic multilayers are obtained by replacing the Fresnel reflection and transmission p-coefficients by the corresponding s-coefficients

r_{s s}^{(i - 1, i)} = - r_{s s}^{(i, i - 1)} = \frac{N^{(i - 1)} α^{(i - 1)} - N^{(i)} α^{(i)}}{N^{(i - 1)} α^{(i - 1)} + N^{(i)} α^{(i)}}, \begin{matrix}  \end{matrix} i = 1, ..., 6,

t_{s s}^{(i - 1, i)} = \frac{2 N^{(i - 1)} α^{(i - 1)}}{N^{(i - 1)} α^{(i - 1)} + N^{(i)} α^{(i)}}, \begin{matrix}  \end{matrix} i = 1, ..., 6,

and

t_{s s}^{(i, i - 1)} = \frac{2 N^{(i)} α^{(i)}}{N^{(i - 1)} α^{(i - 1)} + N^{(i)} α^{(i)}}, \begin{matrix}  \end{matrix} i = 1, ..., 6.

Funding

The work at Charles University was supported by Czech Science Foundation (15-21547S).

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Fe	λ = 410 nm
ε_xx^(Fe)	‒1.52873‒12.2736j
ε_xx^(Fe)1/2	2.32806‒2.63602j
ε_yz^(Fe)	0.246865‒0.022733j
d^(Fe)_penetr	12.3773 nm
q^(Fe)	0.00249978 + 0.00314619j
\|q^(Fe)\|	0.00401838

Optimized parameter	d_Fe nm	\|∆r_pp\| × 10³	∆R_pp × 10³	∆R_pp/R_pp × 10³	R_pp	A_pp
\|∆r_pp\|	10⁴	1.89703	1.08387	3.00243	0.360994	0.63006
\|∆r_pp\|_max	36	1.98718	1.14032	2.96248	0.384915	0.560606
(∆R_pp)_max	35	1.98618	1.14082	2.94064	0.387946	0.552931
(∆R_pp/R_pp)_max	48	1.95066	1.11225	3.06244	0.363189	0.616639
(R_pp)_max	16	1.50566	0.88932	1.97055	0.451303	0.298336
(R_pp)_min	61	1.90914	1.08877	3.04190	0.358832	0.634313

Optimized parameter	d_AlN nm	∆r_pp × 10³	∆R_pp × 10³	∆R_pp/R_pp × 10³	R_pp	A_pp
\|∆r_pp\|_max	40*	2.60349	0.812747	6.60579	0.123029	0.876971
(∆R_pp)_max	10	2.07518	1.108200	3.68184	0.300986	0.699014
(∆R_pp/R_pp)_max	37	2.59364	0.850985	6.73510	0.126344	0.873656

Optimized parameter	AlN	Fe	\|∆r_pp\| × 10³	∆R_pp × 10³	∆R_pp/R_pp × 10³	R_pp	A_pp
\|∆r_pp \|_max	40	33	2.77359	0.915960	6.02467	0.152027	0.750533
(∆R_pp)_max	14	31	2.26502	1.194320	3.8332	0.311567	0.594349
(∆R_pp/R_pp)_max	37	55	2.62745	0.856438	6.82359	0.125505	0.859102
(R_pp)_max	100	18	1.53438	0.865211	1.81533	0.476611	0.315987
(R_pp)_min	40	62	2.61078	0.810463	6.66462	0.121600	0.870441

Optimized parameter	d₁ nm	d₂ nm	d₃ nm	\|∆r_pp\| × 10³	∆R_pp × 10³	∆R_pp/R_pp × 10³	R_pp	A_pp
\|∆r_pp \|_max	40	32	2	2.77428	0.909143	6.05490	0.150142	0.749612
(∆R_pp)_max	14	32	1	2.27599	1.191540	3.91060	0.304689	0.610763
(∆R_pp/R_pp)_max	37	32	22	2.67936	0.859952	6.97016	0.123369	0.815034

Transverse magnetooptic effect in multilayers applied to mapping of microwave currents

Abstract

1. Introduction

2. Transfer matrix approach

3. Analytical expressions

4. Optimized sensor response

4.1. Single Fe layer

4.2 Fe multilayers

4.3. Fe bilayers

5. Discussion

6. Conclusions

Appendix Reflection and transmission coefficients in isotropic multilayers

Funding

References and links

Cited By

Figures (10)

Tables (10)

Equations (38)

Optical Materials Express

Optimized parameter	d_AlN nm	d_Fe nm	\|∆r_pp\| × 10³	∆R_pp × 10³	∆R_pp/R_pp × 10³	R_pp	A_pp	A_pp(Fe)
\|∆r_pp\|_max	40	32	2.77704	0.903768	6.08451	0.148528	0.75860	0.754185
∆R_pp_max	14	31	2.27912	1.187890	3.91886	0.303116	0.614017	0.610078
(∆R_pp/R_pp)_max	37	52	2.63945	0.861499	6.83638	0.126010	0.856621	0.855796

Optimized parameter	d₁ nm	d₂ nm	d₃ nm	\|∆r_pp\| × 10³	∆R_pp × 10³	∆R_pp/R_pp × 10³	R_pp	A_pp	A_pp(Fe)
\|∆r_pp \|_max	43	33	76	2.7787	0.99842	6.50205	0.15355	0.7597	0.7555
∆R_pp_max	17	31	84	2.2475	1.26157	3.90748	0.32286	0.5752	0.5703
(∆R_pp/R_pp)_max	36	9	48	1.3081	‒3.70× 10⁻³	‒337.70	9.2 × 10⁻⁶	0.7104	0.6966

	(a) MO amplitudes
System	d1 nm	d2 nm	d3 nm	d4 nm	d5 nm	\|∆rpp\| × 10³		\|∆rpp⁽¹⁾\| × 10³	\|∆rpp⁽³⁾\| × 10³		\|∆rpp⁽⁵⁾\| × 10³
1Fe	0	0	0	0	36	0.9872		0	0		1.9872
2Fe	0	0	21	93	33	2.2100		0	1.62496		0.60964
3Fe	18	100	11	98	33	2.2976		1.433754	0.46025		0.43847
	(b) MO reflectance
System		∆Rpp × 10³		∆Rpp⁽¹⁾ × 10³			∆Rpp⁽³⁾ × 10³		∆Rpp⁽⁵⁾ × 10³		∆Rpp/Rpp × 10³
1Fe		1.14032		0			0		1.14032		2.96251
2Fe		1.42542		0			1.00318		0.422243		2.89429
3Fe		1.54086		0.89786			0.324082		0.318917		2.86783
	(c) Optical parameters
System		Rpp		Tpp			App		App⁽¹⁾	App⁽³⁾		App⁽⁵⁾
1Fe		0.384915		0.054480			0.560606		0	0		0.560606
2Fe		0.492491		0.021417			0.486091		0	0.321124		0.164967
3Fe		0.53729		0.015404			0.447306		0.253139	0.0755185		0.118649

	(a) MO amplitudes
Optimized parameter		d1 (nm)	d2 (nm)	d3 (nm)	d4 (nm)	d5 (nm)	\|∆rpp\| × 10³	\|∆rpp⁽²⁾\| × 10³	\|∆rpp⁽⁴⁾\| × 10³
\|∆rpp \|max		46	18	92	33	79	3.3764	2.17328	1.20405
(∆Rpp)max		28	18	92	31	87	2.96647	1.89711	1.07033
(∆Rpp/Rpp)max		38	11	39	32	92	0.895269	1.45513	0.679669
(b) MO reflectance
		Optimized parameter		∆Rpp × 103	∆Rpp(2) × 103		∆Rpp(4) × 103	∆Rpp/Rpp × 103
		\|∆rpp \|max		1.50184	0.974865		0.526972	6.09254
		(∆Rpp)max		1.67755	1.07822		0.599328	4.88404
		(∆Rpp/Rpp)max		0.011304	0.0182925		‒6.99 × 10−3	276.702
(c) Optical parameters
	Optimized parameter		Rpp		Tpp	App	App⁽²⁾		App⁽⁴⁾
	\|∆rpp \|max		0.246504		0.0400805	0.713415	0.386434		0.326981
	(∆Rpp)max		0.343476		0.0513362	0.605187	0.33347		0.27172
	(∆Rpp/Rpp)max		4.09 × 10^-5		0.0335743	0.966385	0.794695		0.17169

(a) MO amplitudes
Optimized parameter		d₁ (nm)	d₂ (nm)	d₃ (nm)	d₄ (nm)	\|∆r_pp\| × 10³	\|∆r_pp⁽²⁾\| × 10³	\|∆r_pp⁽⁴⁾\| × 10³
\|∆r_pp \|_max		45	17	93	31	3.37781*	2.16313	1.21589
(∆R_pp)_max		26	18	92	29	2.93902	1.9249	1.01412
(∆R_pp/R_pp)_max		36	10	36	27	0.942318	1.36735	0.759629
(\|∆r_pp⁽²⁾\| + \|∆r_pp*⁽⁴⁾\|) × 10³ = 3.37902
(b) MO reflectance
	Optimized parameter	∆R_pp × 10³		∆R_pp⁽²⁾ × 10³		∆R_pp⁽⁴⁾ × 10³		∆R_pp/R_pp × 10³
	\|∆r_pp \|_max	1.45296		0.919634		0.533322		5.93622
	(∆R_pp)_max	1.6326		1.06937		0.56323		4.73056
	(∆R_pp/R_pp)_max	‒3.053 × 10⁻³		‒3.962 × 10⁻³		0.909 × 10⁻³		‒288.504
(c) Optical parameters
Optimized parameter		R_pp		T_pp		A_pp	A_pp⁽²⁾	A_pp⁽⁴⁾	A_pp⁽⁵⁾
\|∆r_pp \|_max		0.244761		0.0442091		0.71103	0.383456	0.325472	0.00210146
(∆R_pp)_max		0.345118		0.043665		0.611217	0.345824	0.263317	0.0020756
(∆R_pp/R_pp)_max		1.06 × 10⁻⁵		0.0388494		0.96114	0.768345	0.190948	0.00184669