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Spatial shifts of reflected beams from surface polaritons in antiferromagnets

Xiang-Guang Wang, Yu-Qi Zhang, and Xuan-Zhang Wang

Open Access Open Access

Abstract

Goos–Hänchen (GH) and Imbert–Fedorov (IF) shifts of reflected beams and the effects of two Dyakonov surface magnon polaritons (DSMPs) on these shifts are studied in the Otto configuration with an antiferromagnetic substrate. A modified dispersion equation of the DSMPs is proposed to recognize GH and IF shifts related to the DSMPs. Reflected beam shift spectra are simulated for a ${\rm{Fe}}{{\rm{F}}_2}$ substrate. Shift features around the two DSMPs are observed by frequency scanning with a fixed incident angle. The GH-shift amplitude corresponding to either DSMP is situated between the immediately adjacent maximum and minimum of a GH-shift curve. The shift-amplitude maximum of an IF-shift curve corresponds to the relevant DSMP accurately. GH and IF shifts also are given by frequency scanning along dispersion curves.

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1. INTRODUCTION

It is well known that secondary light beams are spatially shifted by small distances from their geometrically optical positions when total internal reflection (TIR) occurs for paraxial light beams incident at the interface between two dielectrics [1,2]. Such spatial shifts can be divided into two components: the in-plane component as the Goos–Hänchen (GH) shift [3] and the out-of-plane component as the Imbert–Fedorov (IF) shift [4]. The GH shift originates from dispersion of the reflected or refracted (transmitted) coefficients and was first observed for TIR from the internal surface of an ordinary dielectric [5] by the attenuated total reflection (ATR) technique. The GH-shift amplitude is generally comparable to the vacuum wavelength for an ordinary dielectric interface. The IF shift is an optical spin–Hall effect caused by spin–orbit interaction of light [6,7]; it is governed by the law of conservation of total angular momentum [1] and was first observed for TIR [4]. The IF-shift amplitude is only fractions of the vacuum wavelength and is closely related to the anisotropy of the corresponding media [8,9]. Numerous studies related to these two phenomena are available in literature since both are important for micro-nano optical and detection technologies [10]. In addition to the original works on isotropic and anisotropic dielectric interfaces [3,4,8,9], both GH and IF shifts have been widely discussed for the surfaces of various micro-structures, such as metal waveguides [11], photonic crystals [12], and metasurfaces [13].

Antiferromagnets (AFs) do not exhibit net macroscopic magnetization and are hence considered to be nonmagnetic materials; however, they have magnetic ordering in terms of microscopy below the AF Néel temperature [14]. AFs constitute a large class of materials, including transition metal fluorides, oxides, sulfides, etc. The magnetic crystal cells of the simplest AFs contain two mutually staggered magnetic sublattices, where both magnetic sublattices are composed of atoms with identical magnetic moments; however, a strong exchange field causes the atomic magnetic moments of the two sublattices to have opposite directions. Interestingly, the AF anisotropic field causes the atomic magnetic moments of the two sublattices to be parallel and antiparallel to the $c$ axis (easy axis). Such AFs are uniaxial hyperbolic crystals without external magnetic fields, and their permeabilities are negative in the plane normal to the easy axis and positive along the easy axis in the response frequency range, which is generally in the far-infrared or terahertz region [15,16]. The AFs support ordinary surface magnon polaritons with transverse electric (TE) polarization and magnetostatic modes in Voigt geometry [17–20]. Insulative AFs have thus attracted attention as a type of natural hyperbolic crystal [21–23]. Herein, we focus on a simple insulative AF, such as the transition metal fluoride ${\rm{Fe}}{{\rm{F}}_2}$ or ${\rm{Mn}}{{\rm{F}}_2}$, to determine the features of the spatial shifts corresponding to the Dyakonov surface magnon polaritons (DSMPs) in the AF [21].

AFs in an external magnetic field can also be considered as hyperbolic materials in specific cases [24–26]. The GH shift of a beam reflected from the surface of an AF has been investigated in Voigt geometry, where the easy axis and external magnetic field are along the AF surface and normal to the direction of polariton propagation [22]. In Voigt geometry, the TE incident beam ($s$ incidence) excites only the TE modes in the AF, and the transverse magnetic (TM)-polarized incident beam ($p$ incidence) excites only the TM modes in the AF. Therefore, one can separately solve the relevant reflected coefficients of $s$ and $p$ incidences. The AFs in Voigt geometry support conventional surface magneton polaritons with TE polarizations [15,22]. However, in another geometry without an external magnetic field, AFs support DSMPs that are two types of hybrid-polarization surface polaritons. In addition, AF gyromagnetism and anisotropy can impact the spatial shift features of the reflected beams substantially [27].

Because there are two opposite magnetic-moment-procession modes in AF anisotropic fields, AFs possess two opposite torques related to the two modes that act on a propagating electromagnetic wave via AF permeability, which evidently influences the GH or IF shifts of the secondary beams. The ATR technique is useful for exciting and detecting surface polaritons experimentally. Nonlinear GH shifts based on the ATR in the Raether–Kretschmann configuration have been reported [28], where the GH shifts related to surface plasmon polaritons were discussed; in this configuration, enhanced GH shifts are discussed in terms of surface plasmon polariton responses [29,30]. GH shifts related to surface plasmon polaritons have also been discussed in the Otto configuration [31].

Recently, physicists have made new progress in investigations of IF and GH shifts of secondary light beams from various material systems, including metal surfaces [32], compound grating waveguide structures [33,34], and metasurfaces [35–39].

The geometry used in this paper can be obtained from Voigt geometry by a rotation transformation. An AF in the case of no external magnetic field supports two DSMPs in the resulting geometry [21]. In contrast to the regular surface magnon polaritons at the AF surface in Voigt geometry, either DSMP is a hybrid-polarization surface magnon polariton composed of two branch waves in the AF. The AF is a naturally hyperbolic crystal in its reststrahlung frequency band, where the components of the primary permeability are opposite in sign. It has been proven that DSMP-I is a hyperbolic surface polariton situated inside the reststrahlung band, but DSMP-II lies outside the reststrahlung band, more like the Dyakonov surface wave [21]. We will focus on how GH and IF shifts relate to the excitation of the two DSMPs in the Otto configuration.

2. MODIFIED DISPERSION RELATION OF DSMPs

To determine the GH and IF shifts corresponding to the DSMPs exactly, we need to modify the dispersion equation of the DSMPs and then reconsider the GH and IF shifts. For the geometry and coordinate system shown in Fig. 1(a), the semi-infinite dielectric and AF are separated by an air spacer with thickness $d$, where no external magnetic field is applied, and the DSMPs propagate along the $z$ axis. The AF easy axis is in the $y - z$ plane and takes an angle $\beta$ with respect to the $z$ axis. The nonzero elements of relative AF permeability are ${\mu _{{xx}}} = 1$, ${\mu _{{yy}}} = {\mu _1}{\cos ^2}\beta + {\sin ^2}\beta $, ${\mu _{{zz}}} = {\mu _1}{\sin ^2}\beta + {\cos ^2}\beta $, and ${\mu _{{yz}}} = {\mu _{{zy}}} = ({{\mu _1} - 1})\sin \beta \cos \beta $, where $\beta$ also can be defined as the propagating angle of DSMPs to the easy axis. ${\mu _1} = 1 + 2{\omega _m}{\omega _a}/(\omega _r^2 - {\omega ^2} - i\tau \omega)$, with damping constant $\tau$, where ${\omega _m} = \gamma 4\pi {M_0}$, with the gyromagnetic ratio $\gamma$ and AF sublattice magnetization ${M_0}$, ${\omega _a} = \gamma {H_a}$, with the anisotropic field ${H_a}$, and $\omega _r^2 = {\gamma ^2}{H_a}({2{H_e} + {H_a}})$, with the exchange-interaction field ${H_e}$. We achieve the new dispersion relation (see Appendix A) as follows:

$$\left| {\begin{array}{*{20}{c}}{\chi ({{\Gamma _ +} - {\gamma _ +}k} ) - {\varepsilon _a}{f^2}}&{\chi ({{\Gamma _ -} - {\gamma _ -}k} ) - {\varepsilon _a}{f^2}}\\{{\lambda _ +}({\chi {\varepsilon _a} + {\Gamma _ +}} )}&{{\lambda _ -}({\chi {\varepsilon _a} + {\Gamma _ -}} )}\end{array}} \right| = 0,$$
where ${\varepsilon _a}$ represents the relative dielectric constant of the AF, $k$ is the polariton wavenumber, and $f = {\omega /2\pi c}$ is the reduced frequency. The attenuation constants of the DSMPs are ${\Gamma _ -} = ({k^2} - {\mu _1}{\varepsilon _a}{f^2})^{1/2}$, ${\Gamma _ +} = ({\mu _{zz}}{k^2}/{\mu _1} - {\varepsilon _a}{f^2})^{1/2}$, $\Gamma ^\prime = ({k^2} - {f^2})^{1/2}$, and $\chi = \Gamma ^\prime \tanh ({\Gamma ^\prime d})$, with spacer thickness $d$. Based on the AF crystal ${\rm{Fe}}{{\rm{F}}_2}$, we have shown [21] that DSMP-I is situated in the light-gray region with ${\mu _1} \lt - 1$, in which ${\rm{Fe}}{{\rm{F}}_2}$ is hyperbolic, and DSMP-II lies in the dark-gray region with $0 \lt {\mu _1} \lt 1/{\varepsilon _a}$, where ${\rm{Fe}}{{\rm{F}}_2}$ is an elliptical material. Therefore, DSMP-I is more like hyperbolic plasmon polaritons [40], and DSMP-II is more like the Dyakonov surface wave [41]. ${\mu _1}$ and the relevant regions are illustrated in Fig. 2.
 figure: Fig. 1.

Fig. 1. Configurations and coordinate system used in the theoretical derivations and numerical calculations, where the AF easy axis is in the $y - z$ plane or AF surface, and $\beta$ is the propagation angle of DSMPs with respect to the easy axis: (a) interface structure and geometry supporting DSMPs, where the bulk AF and bulk dielectric are separated by the air spacer with thickness $d$; (b) Otto configuration supporting the calculation of attenuated total reflection (ATR), which is composed of the prism, air spacer, and AF substrate, where $\theta$ is the incident angle, and ${\Delta _{{gh}}}$ and ${\Delta _{{if}}}$ represent the spatial shift components of the reflected beam; no external magnetic field is applied.

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 figure: Fig. 2.

Fig. 2. Permeability and separated frequency regions of ${\rm FeF}_2$, where the parameters of ${\rm FeF}_2$ are ${\varepsilon _a} = 5.5$, ${}{\omega _m} = 0.736\;{\rm{c}}{{\rm{m}}^{- 1}}$, ${\omega _e} = 56.44\;{\rm{c}}{{\rm{m}}^{- 1}}$, and ${\omega _a} = 20.9\;{\rm{c}}{{\rm{m}}^{- 1}}$, and the damping term is ignored in the permeability.

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It should be noted that both DSMPs are composed of two branch waves in the AF, i.e., the ${\pm}$ branches; ${\gamma _ \pm}$ and ${\lambda _ \pm}$ are used to relate the field components of both branch waves as follows:

$$\begin{split}H_x^ \pm &= \frac{{ik{\Gamma _ \pm}}}{{{k^2} - {\varepsilon _a}{\mu _1}{f^2}}}H_z^ \pm = i{\gamma _ \pm}H_z^ \pm , \\ H_y^ \pm &= \frac{{{\varepsilon _a}{\mu _{{yz}}}{f^2}}}{{{k^2} - \Gamma _ \pm ^2 - {\varepsilon _a}{\mu _{{yy}}}{f^2}}}H_z^ \pm = {\lambda _ \pm}H_z^ \pm .\end{split}$$
Due to ${{\textbf{E}}^ \pm} = i\nabla \times {{\textbf{H}}^ \pm}/{\varepsilon _0}{\varepsilon _a}\omega$ and Eq. (2), as well as the definition of spin-angular-momentum ${\textbf{S}} = - \hbar {\mathop{\rm Im}\nolimits} ({{{\textbf{E}}^*} \times {\textbf{E}}})$, we realize that either branch carries a spin angular momentum with two components ${S_y}$ and ${S_z}$, and further, either DSMP has a relevant two-component spin angular momentum. When numerically solving Eq. (1), the damping term in ${\mu _1}$ should be removed.

3. DERIVATION OF ATR COEFFICIENTS

The ATR method is effective for exciting and observing surface polaritons experimentally. We first derive expressions of the reflected coefficients from the bottom surface of the prism for the configuration shown in Fig. 1(b) according to the transfer matrix method. To reduce the electromagnetic field expressions in the derivations, we implicitly include the common factor $\exp ({ikz - i\omega t})$ in the expressions. In general, incident radiation is reflected completely at the bottom surface of the prism once the incident angle exceeds the critical angle, and a relevant evanescent wave is generated in the air spacer below the prism. However, this evanescent wave can excite DSMPs in the AF, which then partially absorb the energy of the incident wave to produce sharp dips on the reflection ratio curves. We used electromagnetic boundary conditions to solve the reflected coefficient matrix. The electric fields in various spaces are expressed as follows:

$${\textbf{E}} = \left\{{\begin{array}{l}{{{\textbf{E}}^i}{e^{i{k_x}x}} + {{\textbf{E}}^r}{e^{- i{k_x}x}}\quad ({{\rm{in\; the\; prism}}} )}\\{{\textbf{A}}{e^{\Gamma ^\prime x}} + {\textbf{B}}{e^{- \Gamma ^\prime x}} \quad ({{\rm{in\; the\; spacer}}} )}\\{{{\textbf{E}}^ -}{e^{- {\Gamma _ -}x}} + {{\textbf{E}}^ +}{e^{- {\Gamma _ +}x}}\quad ({{\rm{in\; the\; AF}}} )}\end{array}} \right.,$$
where ${k_x} = f\sqrt {{\varepsilon _p}} \cos \theta$, and $k = f\sqrt {{\varepsilon _p}} \sin \theta $, with dielectric constant ${\varepsilon _p}$ of the prism and incident angle $\theta$; superscripts $i$ and $r$ indicate the incident and reflected waves, respectively. The relevant magnetic fields are determined using $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over\mu} \cdot {\textbf{H}} = \nabla \times {\textbf{{\rm E}}}/i\omega$. In the prism and air spacer,
$${H_z} = \left\{{\begin{array}{l}{{k_x}(E_y^i{e^{i{k_x}x}} - E_y^r{e^{- i{k_x}x}})/{\mu _0}\omega \quad ({{\rm{in\; the\; prism}}} )}\\{\Gamma ^\prime ({A_y}{e^{\Gamma ^\prime x}} - {B_y}{e^{- \Gamma ^\prime x}})/i{\mu _0}\omega \quad ({{\rm{in\; the\; spacer}}} )}\end{array}} \right.,$$
$${H_y} = \left\{\!{\begin{array}{l}{- i{\varepsilon _p}{f^2}(E_z^i{e^{i{k_x}x}} - E_z^r{e^{- i{k_x}x}})/i{\mu _0}\omega {k_x}\;\; ({{\rm{in\; the\; prism}}} )}\\{{f^2}({A_z}{e^{\Gamma ^\prime x}} - {B_z}{e^{- \Gamma ^\prime x}})/i{\mu _0}\omega \Gamma ^\prime \;\; ({{\rm{in\; the\; spacer}}} )}\end{array}} \right.\!\!.$$

In the AF substrate, the expressions of the magnetic field components are complex, and we have

$${H_z} = \sum\limits_{j = -}^{j = +} {\frac{{{\mu _{{yz}}}({k^2} - \Gamma _j^2)E_z^j - {\mu _{{yy}}}\Gamma _j^2E_y^j}}{{i{\mu _0}{\mu _{{\rm eff},y}}{\mu _{{yy}}}\omega {\Gamma _j}}}{e^{- {\Gamma _j}x}}} = H_z^ - + H_z^ + ,$$
$${H_y} = \sum\limits_{j = -}^{j = +} {\frac{{{\mu _{{zz}}}(\Gamma _j^2 - {k^2})E_z^j + {\mu _{{yz}}}\Gamma _j^2E_y^j}}{{i{\mu _0}{\mu _{{\rm eff},z}}{\mu _{{zz}}}\omega {\Gamma _j}}}} {e^{- {\Gamma _j}x}} = H_y^ - + H_y^ + ,$$
where the effective permeabilities are ${\mu _{{\rm eff},y}} = ({\mu _{{yy}}}{\mu _{{zz}}} -\mu_{{yz}}^2)/{\mu _{{yy}}}$ and ${\mu _{{\rm eff},z}} = ({\mu _{{yy}}}{\mu _{{zz}}} -\mu_{{yz}}^2)/{\mu _{{zz}}}$. Owing to the relationships among the magnetic field components of both branch waves in the AF, i.e., Eq. (2), there should be relevant relationships among the electric field components as well. Combining Eqs. (5a) and (5b) with (2), we obtain the relationship between the two electric field components of both branches as
$$E_y^j = \frac{{({k^2} - \Gamma _j^2)({\mu _{{zz}}} + {\mu _{{yz}}}{\lambda _j})}}{{\Gamma _j^2({\mu _{{yz}}} + {\mu _{{yy}}}{\lambda _j})}}E_z^j = {\Lambda _j}E_z^j,$$
where $j = - , +$. The electromagnetic boundary conditions at the upper surface of the air spacer are the continuities of ${E_y}$, ${E_z}$, ${H_y}$, and ${H_z}$, which results in the following expressions:
$${k_x}(E_y^i - E_y^r) = - i\Gamma ^\prime ({A_y} - {B_y}),$$
$$E_y^i + E_y^r = {A_y} + {B_y},$$
$$\Gamma ^\prime {\varepsilon _p}(E_z^i - E_z^r) = i{k_x}({A_z} - {B_z}),$$
$$E_z^i + E_z^r = {A_z} + {B_z}.$$
From the above four equations, we have the following matrix relationships:
$$\begin{split}\left({\begin{array}{*{20}{c}}{{A_z}}\\{{B_z}}\end{array}} \right) &= \frac{1}{2}\left({\begin{array}{*{20}{c}}{{\gamma ^\prime_ +}}&{{\gamma ^\prime_ -}}\\{{\gamma ^\prime_ -}}&{{\gamma ^\prime_ +}}\end{array}} \right)\left({\begin{array}{*{20}{c}}{E_z^i}\\{E_z^r}\end{array}} \right), \\ \left({\begin{array}{*{20}{c}}{{A_y}}\\{{B_y}}\end{array}} \right)& = \frac{1}{2}\left({\begin{array}{*{20}{c}}{{\gamma _ +}}&{{\gamma _ -}}\\{{\gamma _ -}}&{{\gamma _ +}}\end{array}} \right)\left({\begin{array}{*{20}{c}}{E_y^i}\\{E_y^r}\end{array}} \right),\end{split}$$
with ${\gamma _ \pm} = 1 \pm i{k_x}/{\Gamma ^\prime}$ and ${\gamma ^\prime _ \pm} = 1 \pm {\varepsilon _p}\Gamma ^\prime /i{k_x}$. The electromagnetic boundary conditions at the lower surface of the air spacer are that ${E_y}$, ${E_z}$, ${H_y}$, and ${H_z}$ are continuous, which means that
$${A_y}{e^{\Gamma ^\prime d}} + {B_y}{e^{- \Gamma ^\prime d}} = {\Lambda _ -}E_z^ - + {\Lambda _ +}E_z^ + ,$$
$${A_z}{e^{\Gamma ^\prime d}} + {B_z}{e^{- \Gamma ^\prime d}} = E_z^ - + E_z^ + ,$$
$${f^2}({A_z}{e^{\Gamma ^\prime d}} - {B_z}{e^{- \Gamma ^\prime d}})/\Gamma ^\prime = \sum\limits_{j = -}^{j = +} {\frac{{{\mu _{{zz}}}(\Gamma _j^2 - {k^2}) + {\mu _{{yz}}}\Gamma _j^2{\Lambda _j}}}{{{\mu _{{\rm eff},z}}{\mu _{{zz}}}{\Gamma _j}}}} E_z^j,$$
$$\Gamma ^\prime ({A_y}{e^{\Gamma ^\prime d}} - {B_y}{e^{- \Gamma ^\prime d}}) = \sum\limits_{j = -}^{j = +} {\frac{{{\mu _{{yz}}}({k^2} - \Gamma _j^2) - {\mu _{{yy}}}\Gamma _j^2{\Lambda _j}}}{{{\mu _{{\rm eff},y}}{\mu _{{yy}}}{\Gamma _j}}}} E_z^j.$$
The above four equations are written as two matrix expressions, namely,
$$\left({\begin{array}{*{20}{c}}{{A_z}}\\{{B_z}}\end{array}} \right) = \frac{1}{2}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over S} \left({\begin{array}{*{20}{c}}{E_z^ -}\\{E_z^ +}\end{array}} \right),\quad \left({\begin{array}{*{20}{c}}{{A_y}}\\{{B_y}}\end{array}} \right) = \frac{1}{2}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over T} \left({\begin{array}{*{20}{c}}{E_z^ -}\\{E_z^ +}\end{array}} \right),$$
where the elements of matrix $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over S}$ are
$${S_{11}} = \left\{{\frac{{\Gamma ^\prime [{\mu _{{zz}}}(\Gamma _ - ^2 - {k^2}) + {\mu _{{yz}}}\Gamma _ - ^2{\Lambda _ -}]}}{{{f^2}{\mu _{{\rm eff},z}}{\mu _{{zz}}}{\Gamma _ -}}} + 1} \right\}{e^{- \Gamma ^\prime d}},$$
$${S_{12}} = \left\{{\frac{{\Gamma ^\prime [{\mu _{{zz}}}(\Gamma _ + ^2 - {k^2}) + {\mu _{{yz}}}\Gamma _ + ^2{\Lambda _ +}]}}{{{f^2}{\mu _{{\rm eff},z}}{\mu _{{zz}}}{\Gamma _ +}}} + 1} \right\}{e^{- \Gamma ^\prime d}},$$
$${S_{21}} = \left\{{\frac{{\Gamma ^\prime [{\mu _{{zz}}}({k^2} - \Gamma _ - ^2) - {\mu _{{yz}}}\Gamma _ - ^2{\Lambda _ -}]}}{{{f^2}{\mu _{{\rm eff},z}}{\mu _{{zz}}}{\Gamma _ -}}} + 1} \right\}{e^{\Gamma ^\prime d}},$$
$${S_{22}} = \left\{{\frac{{\Gamma ^\prime [{\mu _{{zz}}}({k^2} - \Gamma _ + ^2) - {\mu _{{yz}}}\Gamma _ + ^2{\Lambda _ +}]}}{{{f^2}{\mu _{{\rm eff},z}}{\mu _{{zz}}}{\Gamma _ +}}} + 1} \right\}{e^{\Gamma ^\prime d}},$$
and the elements of matrix $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over T}$ are
$${{\rm{T}}_{11}} = \left[{{\Lambda _ -} + \frac{{{\mu _{{yz}}}({k^2} - \Gamma _ - ^2) - {\mu _{{yy}}}\Gamma _ - ^2{\Lambda _ -}}}{{{\mu _{{\rm eff},y}}{\mu _{{yy}}}\Gamma ^\prime {\Gamma _ -}}}} \right]{e^{- \Gamma ^\prime d}},$$
$${{\rm{T}}_{12}} = \left[{{\Lambda _ +} + \frac{{{\mu _{{yz}}}({k^2} - \Gamma _ + ^2) - {\mu _{{yy}}}\Gamma _ + ^2{\Lambda _ +}}}{{{\mu _{{\rm eff},y}}{\mu _{{yy}}}\Gamma ^\prime {\Gamma _ +}}}} \right]{e^{- \Gamma ^\prime d}},$$
$${{\rm{T}}_{21}} = \left[{{\Lambda _ -} - \frac{{{\mu _{{yz}}}({k^2} - \Gamma _ - ^2) - {\mu _{{yy}}}\Gamma _ - ^2{\Lambda _ -}}}{{{\mu _{{\rm eff},y}}{\mu _{{yy}}}\Gamma ^\prime {\Gamma _ -}}}} \right]{e^{\Gamma ^\prime d}},$$
$${{\rm{T}}_{22}} = \left[{{\Lambda _ +} - \frac{{{\mu _{{yz}}}({k^2} - \Gamma _ + ^2) - {\mu _{{yy}}}\Gamma _ + ^2{\Lambda _ +}}}{{{\mu _{{\rm eff},y}}{\mu _{{yy}}}\Gamma ^\prime {\Gamma _ +}}}} \right]{e^{\Gamma ^\prime d}}.$$
Combining Eqs. (8) and (10), we obtain
$$\begin{split}\left({\begin{array}{*{20}{c}}{E_z^ -}\\{E_z^ +}\end{array}} \right)& = {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over S} ^{_{_{_{-1}}}}}\left({\begin{array}{*{20}{c}}{{\gamma ^\prime_ +}}&{{\gamma ^\prime_ -}}\\{{\gamma ^\prime_ -}}&{{\gamma ^\prime_ +}}\end{array}} \right)\left({\begin{array}{*{20}{c}}{E_z^i}\\{E_z^r}\end{array}} \right),\\ \left({\begin{array}{*{20}{c}}{E_z^ -}\\{E_z^ +}\end{array}} \right) &= {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over T} ^{_{_{_{_{\,-1}}}}}}\left({\begin{array}{*{20}{c}}{{\gamma _ +}}&{{\gamma _ -}}\\{{\gamma _ -}}&{{\gamma _ +}}\end{array}} \right)\left({\begin{array}{*{20}{c}}{E_y^i}\\{E_y^r}\end{array}} \right)\end{split}$$
and the relationship between the incident and reflected electric field amplitudes, i.e.,
$$\left({\begin{array}{*{20}{c}}{E_y^i}\\{E_y^r}\end{array}} \right) = {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over P} ^{_{_{_{-1}}}}}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over T} {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over S} ^{_{_{_{-1}}}}}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over P} ^{_{_{_\prime}}} \left({\begin{array}{*{20}{c}}{E_z^i}\\{E_z^r}\end{array}} \right),$$
where $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over P}$ is the $\gamma$ matrix in the second expression of Eq. (13), and $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over P} ^{_{_{_{\prime}}}} $ is the $\gamma ^\prime $ matrix in the first expression of Eq. (13). Equation (14) can also be rewritten in a more regular form as follows:
$$\left({\begin{array}{*{20}{c}}{E_y^r}\\{E_z^r}\end{array}} \right) = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over V} \left({\begin{array}{*{20}{c}}{E_y^i}\\{E_z^i}\end{array}} \right),$$
where $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over V}$ is the reflected coefficient matrix in the ATR method in the Otto configuration. Equations (14) and (15) are the main results in this section. It should be noted that the damping term in the permeability expression must be considered for calculating the ATR spectra because it is responsible for energy absorption from the incident wave when exciting the polaritons in the AF.

4. GH AND IF SHIFTS OF THE ATR SPECTRA

Herein, we derive the GH and IF shifts of the reflected beam off the interface between the prism and air spacer for the structure shown in Fig. 1(b) or in the Otto configuration. The ATR technique is useful for exciting and observing surface polaritons. GH and IF shifts are closely dependent on the polarization and dispersion of radiation at the interface. Hence, GH and IF shifts can indicate the properties of surface polaritons more completely. We emphasize the spatial shifts related to DSMPs as a new type of surface magnon polariton [21] based on the reflected coefficient matrix derived in Section 2. We illustrate the incidence–reflection geometry and coordinate system in Fig. 1(b), where the incident plane is always along the $x - z$ plane, and AF easy axis is along the AF surface ($y - z$ plane) at an angle $\beta$ with the $z$ axis. The two spatial shift components of the reflected beam are ${\Delta _{{gh}}}$ and ${\Delta _{{if}}}$, which are GH and IF shifts, respectively. The reflected coefficient matrix is obtained for plane radiation incidence; therefore, it is suitable only for the central radiation of the incident beam. To obtain the expression for the spatial shifts of the reflected beam, Eq. (15) is transformed to the beam frame as follows:

$$\left({\begin{array}{*{20}{c}}{E_p^r}\\{E_s^r}\end{array}} \right) = \left({\begin{array}{*{20}{c}}{- {V_{22}}}&{- {V_{21}}{{\cos}^{- 1}}\theta}\\{{V_{12}}\cos \theta}&{{V_{11}}}\end{array}} \right)\left({\begin{array}{*{20}{c}}{E_p^i}\\{E_s^i}\end{array}} \right),$$
where $E_s^{i,r}$ and $E_p^{i,r}$ are the two electric field components in the plane normal to the incident or reflected wave vector, respectively; the former is vertical to the incident plane, whereas the latter is in the incident plane. We assume that the incident beam is a Gauss paraxial radiation beam with a waist much larger than the radiation wavelength, and the reflected beam is also a Gauss paraxial beam. In the subsequent mathematical derivations, we follow the procedure in [1]. The beam expression for the normalized incident beam in the wave vector space is
$$G(\upsilon ,\nu) = \sqrt {\frac{{w_0^2}}{{2\pi}}} \exp \left({- {{[{k_p}{w_0}]}^2}\frac{{{\upsilon ^2} + {\nu ^2}}}{4}} \right),$$
under the condition of ${w_0} \gg k_p^{- 1}$, with the central wavenumber ${k_p} = \varepsilon _p^ {1/2}f$ and beam waist ${w_0}$. $\upsilon$ and $v$ are two small quantities related to the variation of the incident angle and rotation of the incident plane, respectively. For a noncentral plane wave in the incident Gauss beam, the relevant reflected wave amplitude is obtained as follows:
$$\begin{split}\left({\begin{array}{*{20}{c}}{E_p^r}\\{E_s^r}\end{array}} \right)& = {\left({\begin{array}{*{20}{c}}1&{{\nu ^r}{\rm{ctg}}{\theta ^r}}\\{- {\nu ^r}{\rm{ctg}}{\theta ^r}}&1\end{array}} \right)^{- 1}}\left({\begin{array}{*{20}{c}}{- {V_{22}}}&{- {V_{21}}/\cos \theta}\\{{V_{12}}\cos \theta}&{{V_{11}}}\end{array}} \right)\\[-3pt]&\quad\times\left({\begin{array}{*{20}{c}}1&{\nu {\rm{ctg}}\theta}\\{- \nu {\rm{ctg}}\theta}&1\end{array}} \right)\left({\begin{array}{*{20}{c}}{E_p^i}\\{E_s^i}\end{array}} \right).\end{split}$$
Because ${\theta ^r} = \pi - \theta$ and ${v^r} = v$, Eq. (18) is expressed as

$$\left({\begin{array}{*{20}{c}}{E_p^r}\\{E_s^r}\end{array}} \right) = \left({\begin{array}{*{20}{c}}{- {V_{22}} + \nu {\rm{ctg}}\theta ({V_{12}}\cos \theta + {V_{21}}/\cos \theta)}&{\nu {\rm{ctg}}\theta ({V_{11}} - {V_{22}}) - {V_{21}}/\cos \theta}\\{\nu {\rm{ctg}}\theta ({V_{22}} - {V_{11}}) + {V_{12}}\cos \theta}&{{V_{11}} + \nu {\rm{ctg}}\theta ({V_{12}}\cos \theta + {V_{21}}/\cos \theta)}\end{array}} \right)\left({\begin{array}{*{20}{c}}{E_p^i}\\{E_s^i}\end{array}} \right).$$
We note that if the AF is replaced by an isotropic medium, ${V_{12}} = {V_{21}} = 0,{V_{22}} = - {f_p}$, and ${V_{11}} = {f_s}$, then Eq. (19) should become [1]
$$\left({\begin{array}{*{20}{c}}{E_p^r}\\{E_s^r}\end{array}} \right) = \left({\begin{array}{*{20}{c}}{{f_p}}&{\nu {\rm{ctg}}\theta ({f_p} + {f_s})}\\{- \nu {\rm{ctg}}\theta ({f_s} + {f_p})}&{{f_s}}\end{array}} \right)\left({\begin{array}{*{20}{c}}{E_p^i}\\{E_s^i}\end{array}} \right).$$
It is noted that each element ${V_{{ij}}}$ is a function of quantities $\mu$ and $v$, so that we have ${V_{{ij}}} \approx {V_{{ij}}}({\theta ,\beta}) + \upsilon \partial {V_{{ij}}}/\partial \theta + v\partial {V_{{ij}}}/\partial \beta$. The in-plane component of the spatial shift or GH shift is expressed as
$${\Delta _{gh}} = {\mathop{\rm Im}\nolimits} \left[ {\left\langle {{E^r}} \left|\frac{\partial }{{{k_p}\partial \upsilon }}\right| {{E^r}} \right\rangle /\langle {E^r}|{E^r}\rangle } \right],$$
and the out-of-plane shift or IF shift is given by
$${\Delta _{if}} = - {\mathop{\rm Im}\nolimits} \left[ {\left\langle {{E^r}} \left|\frac{\partial }{{{k_p}\partial v}}\right| {{E^r}} \right\rangle /\langle {E^r}|{E^r}\rangle } \right],$$
with
$$| {{E^r}} \rangle = G(\upsilon ,v)\left({\begin{array}{*{20}{c}}{E_p^r}\\{E_s^r}\end{array}} \right).$$

In subsequent calculations, we ignore orders in the square matrix in Eq. (19) higher than the first-order terms of $\upsilon$ or $v$. In the integrating process of solving the GH or IF shift, the waist is eliminated [1], so the spatial shifts are independent of the waist or the profile of the incident beam. According to Eqs. (19) and (21)–(23), we obtain the final explicit expression of the GH shift as

$${\Delta _{gh}} = \frac{1}{{{k_p}\langle {E^r}|{E^r}\rangle }}{\mathop{\rm Im}\nolimits} \left\{ {{{\left( {\begin{array}{*{20}{c}}{E_p^i}&{E_s^i}\end{array}} \right)}^*}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over Q} \left( {\begin{array}{*{20}{c}}{E_p^i}\\{E_s^i}\end{array}} \right)} \right\},$$
with
$$\begin{split}\langle {E^r}|{E^r}\rangle &= \left( {{{| {{V_{22}}} |}^2} + {{| {{V_{12}}} |}^2}{{\cos }^2}\theta } \right){| {E_p^i} |^2} \\&\quad+ \left( {{{| {{V_{11}}} |}^2} + {{| {{V_{21}}} |}^2}{{\cos }^{ - 2}}\theta } \right){| {E_s^i} |^2}\\&\quad + 2{\mathop{\rm Re}\nolimits} \left[ {\left( {V_{12}^*{V_{11}}\cos \theta + V_{22}^*{V_{21}}{{\cos }^{ - 1}}\theta } \right)E{{_p^i}^*}E_s^i} \right],\end{split}$$
and the elements of $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over Q}$ are expressed as
$${Q_{11}} = V_{22}^*V_{22}^\prime + V_{12}^*(V_{12}^\prime - {V_{12}}{\rm{tg}}\theta){\cos ^2}\theta ,$$
$${Q_{12}} = V_{22}^*(V_{21}^\prime + {V_{21}}{\rm{tg}}\theta)/\cos \theta + V_{12}^*V_{11}^\prime \cos \theta ,$$
$${Q_{21}} = V_{11}^*(V_{12}^\prime - {V_{12}}{\rm{tg}}\theta)\cos \theta + V_{21}^*V_{22}^\prime /\cos \theta ,$$
$${Q_{22}} = V_{11}^*V_{11}^\prime + V_{21}^*(V_{21}^\prime + {V_{21}}{\rm{tg}}\theta)/{\cos ^2}\theta ,$$
where the prime represents $\partial /\partial \theta$. The IF shift is defined as
$${\Delta _{if}} = - \frac{1}{{{k_p}\langle {E^r}|{E^r}\rangle }}{\mathop{\rm Im}\nolimits} \left\{ {{{\left( {\begin{array}{*{20}{c}}{E_p^i}&{E_s^i}\end{array}} \right)}^*}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\rm T}} \left( {\begin{array}{*{20}{c}}{E_p^i}\\{E_s^i}\end{array}} \right)} \right\},$$
with the elements of $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over T}$ as follows:
$$\begin{split}{{\rm T}_{11}} &= V_{22}^*V_{22}^\prime + V_{12}^*V_{12}^\prime {\cos ^2}\theta + \cos \theta {\rm{ctg}}\theta [V_{12}^*({V_{22}} - {V_{11}}) \\&\quad- V_{22}^*({V_{21}}/{\cos ^2}\theta + {V_{12}})],\end{split}$$
$$\begin{split}{{\rm T}_{12}}& = {\rm{ctg}}\theta [V_{22}^*({V_{22}} - {V_{11}}) + V_{12}^*({V_{12}}{\cos ^2}\theta + {V_{21}})] \\&\quad+ V_{22}^*V_{21}^\prime {\cos ^{- 1}}\theta + V_{12}^*V_{11}^\prime \cos \theta ,\end{split}$$
$$\begin{split}{{\rm T}_{21}}& = {\rm{ctg}}\theta [V_{11}^*({V_{22}} - {V_{11}}) - V_{21}^*({V_{21}}{\cos ^{- 2}}\theta + {V_{12}})]\\&\quad + V_{21}^*V_{22}^\prime {\cos ^{- 1}}\theta + V_{11}^*V_{12}^\prime \cos \theta ,\end{split}$$
$$\begin{split}{{\rm T}_{22}}& = V_{11}^*V_{11}^\prime + V_{21}^*V_{21}^\prime {\cos ^{- 2}}\theta + {\sin ^{- 1}}\theta [V_{21}^*({V_{22}} - {V_{11}}) \\&\quad+ V_{11}^*({V_{21}} + {V_{12}}{\cos ^2}\theta)],\end{split}$$
where the prime denotes $\partial /\partial \beta$. Based on the above results, the GH and IF shifts of the reflected beam can be calculated numerically for different polarized incident beams. In numerical results for the spatial shifts, we will use the wavelength in the prism ${\lambda _p} = 1/{k_p}$ as the unity of shift.

5. NUMERICAL RESULTS AND DISCUSSION

Herein, we present the GH and IF shifts of the beam reflected from the bottom surface of the prism. In the ATR structure, the AF substrate was replaced by an ${\rm FeF}_2$ crystal. The physical parameters of the ${\rm FeF}_2$ crystal are well known [19,21], i.e., its dielectric constant ${\varepsilon _a} = 5.5$ and gyromagnetic ratio $\gamma = 1.97\;({{\rm{rad}} \cdot {{\rm{s}}^{- 1}}/{\rm{kG}}})$. The other physical parameters include sublattice magnetization $4\pi {M_0} = 7.04\;{\rm{kG}}$ $({{\omega _m} = 0.736\;{\rm{c}}{{\rm{m}}^{- 1}}})$, exchange field ${H_e} = 540.0\;{\rm{kG}}$ $({{\omega _e} = 56.44\;{\rm{c}}{{\rm{m}}^{- 1}}})$, and anisotropic field ${H_a} = 200.0\;{\rm{kG}}$ $({{\omega _a} = 20.9\;{\rm{c}}{{\rm{m}}^{- 1}}})$. The AF resonant frequency is $({{\omega _r} = 52.877\;{\rm{c}}{{\rm{m}}^{- 1}}})$ and zero-point frequency is ${\mu _1}$ is $53.165\;{\rm{c}}{{\rm{m}}^{- 1}}$. The AF is a magnetically hyperbolic material between the two special frequencies; otherwise, it is an elliptical material. For ATR calculations, damping must be considered in the AF permeability ${\mu _1}$ and is responsible for energy absorption in the ATR process. To discuss the spatial-shift features related to DSMPs, dispersion curves of the DSMPs are calculated for different values of $\beta$, where $d = 16\;{\rm{\unicode{x00B5}{\rm m}}}$ for DSMP-I, and $d = 108\;{\rm{\unicode{x00B5}{\rm m}}}$ for DSMP-II, as illustrated in Fig. 3. The reflection ratio curves corresponding to the dispersion curves, which are obtained by frequency scanning with a fixed incident angle, are shown in Figs. 4(a) and 6(a).

 figure: Fig. 3.

Fig. 3. Dispersion curves of the DSMPs for various propagation angles and fixed spacer thickness. (a) Dispersion curves of DSMP-I, where the vertical dashed line represents the scanning line for the numerical calculation of ATR spectra for $\theta = {75^\circ}$ and ${\varepsilon _p} = 31.3$ (TiCl prism). (b) Dispersion curves of DSMP-II, where the vertical dashed line representing the scanning line is obtained for $\theta = 17.5^\circ$ and ${\varepsilon _p} = 11.56$ (Si prism). The intersections of the scanning lines and green curves correspond to frequencies ${52.95}\;{\rm{c}}{{\rm{m}}^{- 1}}$ and ${53.20}\;{\rm{c}}{{\rm{m}}^{- 1}}$, as shown by the green solid circles.

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 figure: Fig. 4.

Fig. 4. ATR ratios as well as GH and IF shifts with frequency scanning for a fixed incident angle, TE incident beam, and parameters as shown in the diagrams: (a) ATR ratio, where the four sharp dips exactly correspond to the four intersections between the scanning line and dispersion curves of DSMP-I in Fig. 3(a); (b) GH shift of the reflected beam; (c) IF shift of the reflected beam.

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The two values of $d$ are optimum for the ATR spectra along the scanning lines related to two groups of dispersion curves, but not necessarily optimum for individual curves. The value of $d$ is generally taken according to the attenuating constant in the air spacer.

In the following paragraphs, we focus on the GH and IF shifts of the reflected radiation beam off the bottom surface of the prism. Because TIR occurs at the bottom surface, the reflection ratio is one without optical losses. However, owing to the existence of damping and excitation of magnon polaritons in the AF, the reflection ratio evidently decreases, especially for excitation of surface magnon polaritons. The bulk magnon polaritons occupy some continua in the $f - k$ space, but either DSMP is an isolated curve in the dispersion relation, so it corresponds to a sharper dip in the relevant ATR spectrum, where the dip exhibits the DSMP. In the geometry shown in Fig. 1(b), we note that only DSMPs and bulk polaritons can exist as eigen propagation modes in the AF. It should be noted that the ATR spectra in the Otto configuration based on an AF were comprehensively discussed in [42], where the AF substrate is put in an external magnetic field so that it does not support DSMPs. However, in the case of no external magnetic field, DSMPs can exist in the AF in the same geometry.

Either DSMP is a surface hybrid-polarization polariton with spin angular momentum. It is a mixture of TE and TM components, so they can in principle be excited by both TE and TM incident radiations. However, the proportions of the two components in either DSMP are different, and further, the proportions at the AF surface determine which polarized radiation of incidence is used to excite the relevant DSMP. We find that it is better for one to excite DSMP-I with TE incident radiation and excite DSMP-II with TM incident radiation.

In the Otto configuration with an isotropic material substrate for linearly polarized (TE or TM) incidence, the reflective light also is linearly polarized and does not carry spin, so the spin–Hall effect does not exist. When the AF is used as the substrate as shown in Fig. 1(b), the excitation of DSMPs and the AF anisotropic dispersion must influence the reflection through the boundary conditions of electromagnetic fields at the two interfaces. Thus, the reflective radiation generally becomes an elliptically polarized wave with spin angular momentum so that the spin–Hall effect of the reflective beam exists and is enhanced, i.e., the IF shift is enlarged by the excitation of DSMPs.

Figure 4 illustrates the reflection ratio, the GH and IF shifts along the scanning line in Fig. 3(a), for which we apply TE incident beams (i.e., $E_s^i = 1.0$ and $E_p^i = 0$). Owing to the large wavenumber of DSMP-I, we use a TiCl prism with a large dielectric constant ${\varepsilon _p} = 31.3$ and a large incident angle $\theta = 75^\circ$. We recognize that the $f$ and $k$ values related to each dip represent just the intersection of the relevant dispersion curve with the scanning line in Fig. 3(a). As shown in Fig. 4(a), the four dips clearly correspond to the four intersections of the scanning line and dispersion curves in Fig. 3(a). To examine the contribution of DSMP-I to the GH shift, Fig. 4(b) shows the GH shift of the reflected beam. We observe that the GH shift curves exhibit a jump phenomenon from maximum to minimum and that this phenomenon occurs when the scanning line crosses each dispersion curve. This demonstrates that DSMP-I does not correspond to the GH shift peak or dip. The maximum and minimum shifts are situated on either side of the dispersion curve, as indicated by the small solid circles on the green curves. Figure 4(c) illustrates the IF shift of the reflected beam. We see that the maximum (dip value) of the IF shift amplitude accurately corresponds with DSMP-I, where the shift is negative and approximately equal to ${-}{2.24}{\lambda _p}$. On either side of this sharp dip, the shift is smaller and positive. The red curves show that although the dip of the reflection ratio curve is very small and blunt, the dip for the IF shift is sharp.

The above reflection ratios and spatial shifts are achieved by frequency scanning at a fixed incident angle, where $k = \sqrt {{\varepsilon _p}} f{\sin}\theta$, and the scanning line is as shown in Fig. 3(a). The scanning line crosses the dispersion curves in the frequency range of DSMP-I, so Fig. 4 actually reflects the reflection ratios and spatial shifts in the vicinity of DSMP-I. It should be noted that the dip of the reflection ratio is not deep, as shown in Fig. 4(a). The dip depth is directly dependent on the thickness of the air spacer in the Otto configuration. One can use a thinner spacer to make the dip deeper, but the dip will become very blunt in this case, which will prevent us from exactly realizing the frequency position of the dip. To observe GH and IF shifts related to the dispersion curves in Fig. 3(a), we use every point ($k$, $f$) of the dispersion curve to calculate the shifts. However, it should be noted that because of the limitation of the prism dielectric constant, we cannot obtain them for the entire dispersion curves because $k$ is always smaller than $\sqrt {{\varepsilon _p}} f$. Therefore, we can obtain results only in the range of $k \lt \sqrt {{\varepsilon _p}} f$. Figure 5 shows these results. First, the GH shift changes rapidly with the propagation angle. For example, the majority of the green curve shows that the GH shift is positive, but the majority of the blue curve shows that the shift is negative. In addition, the green and blue curves exhibit completely different GH shift features, as seen in Fig. 5(a). Second, the IF shift also varies rapidly with the propagation angle. For example, it is very small for $\beta = 30^\circ$ but very large and negative for $\beta = 45^\circ$; it is also large but either positive or negative for $\beta = 60^\circ$ and 75º, as illustrated in Fig. 5(b). In contrast with the shifts at the Brewster or critical angle [43], the GH and IF shifts are not only large but also accompanied by higher reflection ratios. Interestingly, the shifts can be either positive or negative on a dispersion curve. The green solid circles on the green curves in Figs. 4 and 5 correspond exactly to those in Fig. 3(a).

 figure: Fig. 5.

Fig. 5. GH and IF shifts corresponding to the dispersion curves of DSMP-I in Fig. 3(a) for TE incidence and frequency scanning: (a) GH and (b) IF shifts.

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From Figs. 3 and 4, we note that each reflection ratio curve and each IF shift curve exhibits individual sharp dips, respectively, and these dips correspond accurately to the intersections of the scanning lines and relevant dispersion curves in Fig. 3(a), as indicated by the green solid circles and curves. It is thus proven that both the reflection ratio and IF shift reflect the existence of DSMP-I accurately. However, the peak or dip of the GH shift curve does not correspond with the relevant intersection in Fig. 3(a); instead, the GH shifts exhibit jumps from the maxima to minima as the frequency scanning is stepped across the dispersion curves in Fig. 3(a), as shown by the green solid circle and curve in Fig. 4(b).

Subsequently, we discuss the GH and IF shifts related to DSMP-II, which are situated in the range of $0 \lt {\mu _1} \lt 1/{\varepsilon _a}$ outside the reststrahlung band. The $k$ value of DSMP-II is comparable in amplitude to its frequency, so we used a Si prism with a dielectric constant of ${\varepsilon _p} = 11.56$ and air spacer of thickness $d = 108\;{\rm{\unicode{x00B5}{\rm m}}}$ in Otto geometry. Figure 6 illustrates the reflection ratio as well as GH and IF shifts of the reflected beam for TM incidence, where the frequency scanning is along the line in Fig. 3(b) at a fixed incident angle of $\theta = 17.5^\circ$. We first consider Fig. 6, where 6(a) shows the reflection ratio, 6(b) illustrates the GH shift, and 6(c) indicates the IF shift. The dips of the reflected curves are much smaller than those in Fig. 4(a) as the absorption by the AF substrate is higher. From the frequencies at the dips and relevant intersections in Fig. 3(b), we conclude that the dips accurately reflect DSMP-II. However, each GH shift curve exhibits a sharp peak and sharp dip that are immediately adjacent or a jump from the dip to the peak, as illustrated in Fig. 6(b). The GH shift value corresponding to DSMP-II is situated between the peak and dip, as shown by the green curve and solid green circle. This phenomenon is similar to that in DSMP-I [see Fig. 4(b)]. On the left side of DSMP-II (green solid circle), the shift reaches a higher value. The IF shift curves reflect that the shift peaks exactly correspond to DSMP-II, but these shifts are opposite in direction to those of DSMP-I and are positive, as shown in Fig. 6(c). In addition, we note that the shift values are negative outside the vicinities of the peaks. It is thus demonstrated that the effects of DSMP-II are opposite to those of DSMP-I on the IF shift.

 figure: Fig. 6.

Fig. 6. (a) Reflection ratio, (b) GH shift, and (c) IF shift related to DSMP-II with frequency scanning at a fixed incident angle of $\theta = 17.5^\circ$ for a TM incident beam, and other parameters as shown in the graphs.

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In Fig. 6, we realize the features of GH and IF shifts are related to DSMP-II along the scanning line in Fig. 3(b) for a fixed incident angle $\theta = 17.5^\circ$. However, Figs. 6(b) and 6(c) show only the shifts in the frequency vicinity of DSMP-II or around the intersections in Fig. 3(b). Here, we discuss GH and IF shifts of the reflected beam corresponding to the entire dispersion curve, where the frequency scanning goes along the dispersion curve. In this case, the wavenumber ($k$) was determined using Eq. (1). Because $k = \sqrt {{\varepsilon _p}} f{\sin}\theta$, the incident angle changes with the scanning frequency, and we can obtain the shifts in the range of $k \lt \sqrt {{\varepsilon _p}} f$. Figure 7 shows GH and IF shifts where the critical angle is 17.1º for TIR at the bottom surface of the prism. We first find that the GH shift is always negative and very large in amplitude. Each shift curve exhibits a V-style shape, except for the 75º curve, as illustrated in Fig. 7(a). Figure 7(b) shows IF shift curves. The IF shift corresponding to a dispersion curve can be either positive or negative, as indicated by the green and blue curves. However, it can also be positive on the whole dispersion curve, as indicated by the cyan curve, and it can be negative on another whole curve, as shown by the red curve. Therefore, the IF shift is immediately related to the propagating angle. In addition, the IF shift on the whole dispersion curve generally exhibits a Z-style shape, except the cyan curve, and the shift amplitude is much larger than those observed previously [43,44].

 figure: Fig. 7.

Fig. 7. GH and IF shifts corresponding to the dispersion curves of DSMP-II with frequency scanning along the dispersion curves in Fig. 3(b), where the TE incident beam is used, and the other parameters are shown in the graphs. (a) GH and (b) IF shifts.

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From Figs. 4 and 6, we also find that each reflection ratio curve and each IF-shift curve both have an individual sharp dip or peak, respectively, and the dip or peak accurately corresponds to the intersection of the scanning line and the relevant dispersion in Fig. 3(b). It is proven that both the reflection ratio and IF shift can accurately reflect the existence of DSMP-II. The peak or dip of a GH-shift curve does not correspond to the relevant intersection in Fig. 4(b), but the GH shift exhibits a jump from its dip to its peak as the frequency scanning steps across a dispersion curve along the scanning line in Fig. 3(b), as shown by the green solid circle and curve in Fig. 6(b).

We find that the dispersion relation obtained from the single interface between semi-infinite free space and the AF is not available for us to solve the GH and IF shifts corresponding to DSMPs because the air spacer is finite in thickness in Otto geometry. However, dispersion relation (1) is available to our objective. With frequency scanning for a fixed incident angle, the reflection ratio dips exactly correspond to the intersections between the scanning line and dispersion curves. The peak or dip of the IF shift can also accurately reflect DSMPs, but the GH shift cannot. We find that the two shifts rapidly change with frequency around the DSMPs and can be very large in amplitude.

If we apply a nonmagnetic medium with the same dielectric constant to replace the AF, under the same conditions, such as incident polarization, incident angle, and gap width, no IF shift exists, and the GH shift is nearly a constant in the frequency window in Fig. 4 or 6. For TE incidence, the reflective ratio is unity, and the GH shift is about $0.34{\lambda _p}$. For TM incidence, the reflective ratio is about 0.5, and the GH shift is equal to about ${-}1.03\;{\lambda _p}$, where there is a tunneling effect, i.e., the transmission into the nonmagnetic substrate. Comparing these results with Figs. 4 and 6, we clearly see the effect of DSMP excitation on the spatial shifts.

As a form of spin–Hall effect, the IF shift was observed in transmission spectra from an anisotropic crystal plate [45] or polymer film [46], where input beams directly incident on the relevant material surface and very large IF shift were found. A very large IF shift also was observed in transmission spectra from a vertical hyperbolic metamaterial [47]. The enlargement of IF shift was jointly caused by the different experiences of ordinary and extraordinary lights in the films and by the interferences of forward and backward waves in the films. In the present work, the IF shift was investigated in the Otto configuration and is the reflective-beam shift. The enlargement of the GH or IF shift results from the excitation of DSMPs, but it is smaller than those observed in the references mentioned above.

6. SUMMARY

We have investigated GH and IF shifts of a reflected radiation beam off an ATR device composed of a prism, air spacer, and AF substrate, where the AF substrate supports the two DSMPs. We focus on the effect of DSMP excitation on the GH and IF shifts of the reflected beam. To exactly reflect this effect, we obtained a new dispersion equation for DSMPs available to our objective and the ATR structure. The transfer-matrix method was used to achieve the coefficient matrix of reflection, and the expressions of GH and IF shifts were analytically obtained. The numerical calculations were based on the AF crystal ${\rm{Fe}}{{\rm{F}}_2}$. We found that DSMP excitation has fascinating effects on GH and IF shifts. With frequency scanning for a fixed incident angle, we observed the shift features around the two DSMPs. Each GH-shift curve demonstrates that the shift amplitude has the maximum (dip value) and minimum (peak value) that are immediately adjacent. However, the shift amplitude corresponding to either DSMP is neither the maximum nor minimum, but it is between the two values. This is completely different from the given conclusion [29,31]. The IF-shift curves demonstrate that the shift amplitude has a maximum (peak value or dip value) that accurately corresponds to the relevant DSMP. This is the first investigation of the IF shift related to surface polaritons. GH and IF shifts were also achieved by frequency scanning along any dispersion curve. The results show some interesting properties. The GH or IF shift can be either positive or negative on the whole dispersion curve of DSMP-I, and the shift distance is larger for some proportion of the dispersion curve. The GH-shift distance can reach $18{\lambda _p}$, and the IF shift distance is close to $24{\lambda _p}$. The GH shift on the dispersion curve of DSMP-II is always negative, and the shift distance can reach a very large value, especially on the curves of $\beta = 45^\circ$ and 60º, and the maximum is approximately $270{\lambda _p}$. The IF shift is either positive or negative on the dispersion curve of DSMPs. The shift distance can reach $7{\lambda _p}$. We also realize that, unlike the shifts near the Brewster angle [43], these large shifts are accompanied by high reflection ratios. The IF shift was first investigated in this study, which showed that the excitation of DSMPs can greatly enhance the IF shift of the reflected beam, and the IF shift can be used to detect surface polaritons.

APPENDIX A

We derive the dispersion equation for the DSMPs herein. In the geometry shown in Fig. 1(a), we assume that the magnetic field propagates along the $z$ axis and is expressed by $H_x^p = {A_p}{e^{{ikz}}}$, $H_y^p = {B_p}{e^{{ikz}}}$, and $H_z^p = 0$ in the dielectric, with dielectric constant ${\varepsilon _p}$. The magnetic field in the spacer is expressed as ${H^\prime _j} = ({{A_j}{e^{\Gamma ^\prime x}} + {B_j}{e^{- \Gamma ^\prime x}}}){e^{{ikz}}}$. It is noted that the relevant electric field can be obtained using ${\textbf{E}} = - \nabla \times {\textbf{H}}/i\varepsilon \omega$. The boundary conditions of the electromagnetic field at the top surface of the air spacer can be expressed by four equations as follows:

$${A_{\rm{z}}}{e^{- \Gamma ^\prime d}} + {B_z}{e^{\Gamma ^\prime d}} = 0,$$
$${A_y}{e^{- \Gamma ^\prime d}} + {B_y}{e^{\Gamma ^\prime d}} = {B_p},$$
$${\varepsilon _p}{f^2}\left({{A_z}{e^{- \Gamma ^\prime d}} - {B_z}{e^{\Gamma ^\prime d}}} \right) = ik\Gamma ^\prime {A_p},$$
$$\Gamma ^\prime \left({{A_y}{e^{- \Gamma ^\prime d}} - {B_y}{e^{\Gamma ^\prime d}}} \right) = 0.$$
At the bottom surface of the spacer, the application of the boundary conditions requires the expressions for the electric and magnetic field components in the AF. The expressions of the magnetic field are given in the text; here, we show the relevant electric field components to be
$${E_y} = \frac{1}{{- i{\varepsilon _0}{\varepsilon _a}\omega}}\left[{({{\Gamma _ +} - {\gamma _ +}k} )H_z^ + + ({{\Gamma _ -} - {\gamma _ -}k} )H_z^ -} \right],$$
$${E_z} = \frac{1}{{- i{\varepsilon _0}{\varepsilon _a}\omega}}\left[{- {\Gamma _ +}{\lambda _ +}H_z^ + - {\Gamma _ -}{\lambda _ -}H_z^ -} \right].$$
Therefore, at the interface between the air spacer and AF crystal, we obtain another four equations as
$${A_z} + {B_z} = H_z^ + + H_z^ - ,$$
$${A_y} + {B_y} = {\lambda _ +}H_z^ + + {\lambda _ -}H_z^ - ,$$
$${\varepsilon _a}\Gamma ^\prime ({{A_y} - {B_y}} ) = \left({- {\Gamma _ +}{\lambda _ +}H_z^ + - {\Gamma _ -}{\lambda _ -}H_z^ -} \right),$$
$${\varepsilon _a}{f^2}\left({{A_z} - {B_z}} \right) = \Gamma ^\prime \left[{({{\Gamma _ +} - {\gamma _ +}k} )H_z^ + + ({{\Gamma _ -} - {\gamma _ -}k} )H_z^ -} \right].$$
From Eqs. (A3a)–(A3d), we have
$$\begin{split}{A_z} &= \frac{1}{2}\left(H_z^ + + H_z^ - + \frac{{\Gamma ^\prime}}{{{\varepsilon _a}{f^2}}}\big[({{\Gamma _ +} - {\gamma _ +}k} )H_z^ + \right.\\&\quad+\left. ({{\Gamma _ -} - {\gamma _ -}k} )H_z^ - \big] \vphantom{\frac{{\Gamma ^\prime}}{{{\varepsilon _a}{f^2}}}}\right),\end{split}$$
$$\begin{split}{B_z} &= \frac{1}{2}\left(H_z^ + + H_z^ - - \frac{{\Gamma ^\prime}}{{{\varepsilon _a}{f^2}}}\big[({{\Gamma _ +} - {\gamma _ +}k} )H_z^ +\right.\\&\quad + \left.({{\Gamma _ -} - {\gamma _ -}k} )H_z^ - \big] \right),\end{split}$$
$${A_y} = \frac{1}{2}\left[{\lambda _ +}H_z^ + + {\lambda _ -}H_z^ - - \frac{1}{{\Gamma ^\prime {\varepsilon _a}}}\left({{\Gamma _ +}{\lambda _ +}H_z^ + + {\Gamma _ -}{\lambda _ -}H_z^ -} \right) \right],$$
$${B_y} = \frac{1}{2}\left[{{\lambda _ +}H_z^ + + {\lambda _ -}H_z^ - + \frac{1}{{\Gamma ^\prime {\varepsilon _a}}}\left({{\Gamma _ +}{\lambda _ +}H_z^ + + {\Gamma _ -}{\lambda _ -}H_z^ -} \right)} \vphantom{\frac{1}{{\Gamma ^\prime {\varepsilon _a}}}}\right].$$
Substituting them into Eqs. (A1a) and (A1d), we obtain
$$\begin{split}&{\varepsilon _a}{f^2}({H_z^ + + H_z^ -} )\cosh ({\Gamma ^\prime d} ) - \Gamma ^\prime \sinh ({\Gamma ^\prime d} )\\&\quad\times\left[{({{\Gamma _ +} - {\gamma _ +}k} )H_z^ + + ({{\Gamma _ -} - {\gamma _ -}k} )H_z^ -} \right] = 0,\end{split}$$
$$\begin{split}&{\varepsilon _a}\Gamma ^\prime \sinh ({\Gamma ^\prime d} )\left({{\lambda _ +}H_z^ + + {\lambda _ -}H_z^ -} \right) \\&\quad- \cosh ({\Gamma ^\prime d} )\left({{\Gamma _ +}{\lambda _ +}H_z^ + + {\Gamma _ -}{\lambda _ -}H_z^ -} \right) = 0.\end{split}$$
Equations (A5a) and (A5b) form a linear equation set of $H_z^ -$ and $H_z^ +$. The determinant of the coefficient matrix in this equation set must be equal to zero to guarantee the existence of a wave solution, that is,
$$\left| {\begin{array}{*{20}{c}}{\chi ({{\Gamma _ +} - {\gamma _ +}k} ) - {\varepsilon _a}{f^2}}&{\chi ({{\Gamma _ -} - {\gamma _ -}k} ) - {\varepsilon _a}{f^2}}\\{{\lambda _ +}\left({\chi {\varepsilon _a} + {\Gamma _ +}} \right)}&{{\lambda _ -}\left({\chi {\varepsilon _a} + {\Gamma _ -}} \right)}\end{array}} \right| = 0,$$
where $\chi = \Gamma ^\prime \tanh ({\Gamma ^\prime d})$. This dispersion equation demonstrates that the dispersion property of DSMPs changes with the thickness of the spacer in the Otto configuration. This dispersion equation can be reduced to a given relation [21] when $d \to \infty$.

Funding

Natural Science Foundation of Heilongjiang Province (ZD2009103); Doctoral Students’ Research Innovation Project of Harbin Normal University (HSDBSCX2020-14).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)
Fig. 1.
Fig. 1. Configurations and coordinate system used in the theoretical derivations and numerical calculations, where the AF easy axis is in the $y - z$ plane or AF surface, and $\beta$ is the propagation angle of DSMPs with respect to the easy axis: (a) interface structure and geometry supporting DSMPs, where the bulk AF and bulk dielectric are separated by the air spacer with thickness $d$ ; (b) Otto configuration supporting the calculation of attenuated total reflection (ATR), which is composed of the prism, air spacer, and AF substrate, where $\theta$ is the incident angle, and ${\Delta _{{gh}}}$ and ${\Delta _{{if}}}$ represent the spatial shift components of the reflected beam; no external magnetic field is applied.

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Fig. 2.
Fig. 2. Permeability and separated frequency regions of ${\rm FeF}_2$ , where the parameters of ${\rm FeF}_2$ are ${\varepsilon _a} = 5.5$ , ${}{\omega _m} = 0.736\;{\rm{c}}{{\rm{m}}^{- 1}}$ , ${\omega _e} = 56.44\;{\rm{c}}{{\rm{m}}^{- 1}}$ , and ${\omega _a} = 20.9\;{\rm{c}}{{\rm{m}}^{- 1}}$ , and the damping term is ignored in the permeability.

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Fig. 3.
Fig. 3. Dispersion curves of the DSMPs for various propagation angles and fixed spacer thickness. (a) Dispersion curves of DSMP-I, where the vertical dashed line represents the scanning line for the numerical calculation of ATR spectra for $\theta = {75^\circ}$ and ${\varepsilon _p} = 31.3$ (TiCl prism). (b) Dispersion curves of DSMP-II, where the vertical dashed line representing the scanning line is obtained for $\theta = 17.5^\circ$ and ${\varepsilon _p} = 11.56$ (Si prism). The intersections of the scanning lines and green curves correspond to frequencies ${52.95}\;{\rm{c}}{{\rm{m}}^{- 1}}$ and ${53.20}\;{\rm{c}}{{\rm{m}}^{- 1}}$ , as shown by the green solid circles.

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Fig. 4.
Fig. 4. ATR ratios as well as GH and IF shifts with frequency scanning for a fixed incident angle, TE incident beam, and parameters as shown in the diagrams: (a) ATR ratio, where the four sharp dips exactly correspond to the four intersections between the scanning line and dispersion curves of DSMP-I in Fig. 3(a); (b) GH shift of the reflected beam; (c) IF shift of the reflected beam.

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Fig. 5.
Fig. 5. GH and IF shifts corresponding to the dispersion curves of DSMP-I in Fig. 3(a) for TE incidence and frequency scanning: (a) GH and (b) IF shifts.

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Fig. 6.
Fig. 6. (a) Reflection ratio, (b) GH shift, and (c) IF shift related to DSMP-II with frequency scanning at a fixed incident angle of $\theta = 17.5^\circ$ for a TM incident beam, and other parameters as shown in the graphs.

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Fig. 7.
Fig. 7. GH and IF shifts corresponding to the dispersion curves of DSMP-II with frequency scanning along the dispersion curves in Fig. 3(b), where the TE incident beam is used, and the other parameters are shown in the graphs. (a) GH and (b) IF shifts.

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Equations (65)

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| χ ( Γ + − γ + k ) − ε a f 2 χ ( Γ − − γ − k ) − ε a f 2 λ + ( χ ε a + Γ + ) λ − ( χ ε a + Γ − ) | = 0 ,
H x ± = i k Γ ± k 2 − ε a μ 1 f 2 H z ± = i γ ± H z ± , H y ± = ε a μ y z f 2 k 2 − Γ ± 2 − ε a μ y y f 2 H z ± = λ ± H z ± .
E = { E i e i k x x + E r e − i k x x ( i n t h e p r i s m ) A e Γ ′ x + B e − Γ ′ x ( i n t h e s p a c e r ) E − e − Γ − x + E + e − Γ + x ( i n t h e A F ) ,
H z = { k x ( E y i e i k x x − E y r e − i k x x ) / μ 0 ω ( i n t h e p r i s m ) Γ ′ ( A y e Γ ′ x − B y e − Γ ′ x ) / i μ 0 ω ( i n t h e s p a c e r ) ,
H y = { − i ε p f 2 ( E z i e i k x x − E z r e − i k x x ) / i μ 0 ω k x ( i n t h e p r i s m ) f 2 ( A z e Γ ′ x − B z e − Γ ′ x ) / i μ 0 ω Γ ′ ( i n t h e s p a c e r ) .
H z = ∑ j = − j = + μ y z ( k 2 − Γ j 2 ) E z j − μ y y Γ j 2 E y j i μ 0 μ e f f , y μ y y ω Γ j e − Γ j x = H z − + H z + ,
H y = ∑ j = − j = + μ z z ( Γ j 2 − k 2 ) E z j + μ y z Γ j 2 E y j i μ 0 μ e f f , z μ z z ω Γ j e − Γ j x = H y − + H y + ,
E y j = ( k 2 − Γ j 2 ) ( μ z z + μ y z λ j ) Γ j 2 ( μ y z + μ y y λ j ) E z j = Λ j E z j ,
k x ( E y i − E y r ) = − i Γ ′ ( A y − B y ) ,
E y i + E y r = A y + B y ,
Γ ′ ε p ( E z i − E z r ) = i k x ( A z − B z ) ,
E z i + E z r = A z + B z .
( A z B z ) = 1 2 ( γ + ′ γ − ′ γ − ′ γ + ′ ) ( E z i E z r ) , ( A y B y ) = 1 2 ( γ + γ − γ − γ + ) ( E y i E y r ) ,
A y e Γ ′ d + B y e − Γ ′ d = Λ − E z − + Λ + E z + ,
A z e Γ ′ d + B z e − Γ ′ d = E z − + E z + ,
f 2 ( A z e Γ ′ d − B z e − Γ ′ d ) / Γ ′ = ∑ j = − j = + μ z z ( Γ j 2 − k 2 ) + μ y z Γ j 2 Λ j μ e f f , z μ z z Γ j E z j ,
Γ ′ ( A y e Γ ′ d − B y e − Γ ′ d ) = ∑ j = − j = + μ y z ( k 2 − Γ j 2 ) − μ y y Γ j 2 Λ j μ e f f , y μ y y Γ j E z j .
( A z B z ) = 1 2 S ↔ ( E z − E z + ) , ( A y B y ) = 1 2 T ↔ ( E z − E z + ) ,
S 11 = { Γ ′ [ μ z z ( Γ − 2 − k 2 ) + μ y z Γ − 2 Λ − ] f 2 μ e f f , z μ z z Γ − + 1 } e − Γ ′ d ,
S 12 = { Γ ′ [ μ z z ( Γ + 2 − k 2 ) + μ y z Γ + 2 Λ + ] f 2 μ e f f , z μ z z Γ + + 1 } e − Γ ′ d ,
S 21 = { Γ ′ [ μ z z ( k 2 − Γ − 2 ) − μ y z Γ − 2 Λ − ] f 2 μ e f f , z μ z z Γ − + 1 } e Γ ′ d ,
S 22 = { Γ ′ [ μ z z ( k 2 − Γ + 2 ) − μ y z Γ + 2 Λ + ] f 2 μ e f f , z μ z z Γ + + 1 } e Γ ′ d ,
T 11 = [ Λ − + μ y z ( k 2 − Γ − 2 ) − μ y y Γ − 2 Λ − μ e f f , y μ y y Γ ′ Γ − ] e − Γ ′ d ,
T 12 = [ Λ + + μ y z ( k 2 − Γ + 2 ) − μ y y Γ + 2 Λ + μ e f f , y μ y y Γ ′ Γ + ] e − Γ ′ d ,
T 21 = [ Λ − − μ y z ( k 2 − Γ − 2 ) − μ y y Γ − 2 Λ − μ e f f , y μ y y Γ ′ Γ − ] e Γ ′ d ,
T 22 = [ Λ + − μ y z ( k 2 − Γ + 2 ) − μ y y Γ + 2 Λ + μ e f f , y μ y y Γ ′ Γ + ] e Γ ′ d .
( E z − E z + ) = S ↔ − 1 ( γ + ′ γ − ′ γ − ′ γ + ′ ) ( E z i E z r ) , ( E z − E z + ) = T ↔ − 1 ( γ + γ − γ − γ + ) ( E y i E y r )
( E y i E y r ) = P ↔ − 1 T ↔ S ↔ − 1 P ↔ ′ ( E z i E z r ) ,
( E y r E z r ) = V ↔ ( E y i E z i ) ,
( E p r E s r ) = ( − V 22 − V 21 cos − 1 θ V 12 cos ⁡ θ V 11 ) ( E p i E s i ) ,
G ( υ , ν ) = w 0 2 2 π exp ⁡ ( − [ k p w 0 ] 2 υ 2 + ν 2 4 ) ,
( E p r E s r ) = ( 1 ν r c t g θ r − ν r c t g θ r 1 ) − 1 ( − V 22 − V 21 / cos ⁡ θ V 12 cos ⁡ θ V 11 ) × ( 1 ν c t g θ − ν c t g θ 1 ) ( E p i E s i ) .
( E p r E s r ) = ( − V 22 + ν c t g θ ( V 12 cos ⁡ θ + V 21 / cos ⁡ θ ) ν c t g θ ( V 11 − V 22 ) − V 21 / cos ⁡ θ ν c t g θ ( V 22 − V 11 ) + V 12 cos ⁡ θ V 11 + ν c t g θ ( V 12 cos ⁡ θ + V 21 / cos ⁡ θ ) ) ( E p i E s i ) .
( E p r E s r ) = ( f p ν c t g θ ( f p + f s ) − ν c t g θ ( f s + f p ) f s ) ( E p i E s i ) .
Δ g h = Im [ ⟨ E r | ∂ k p ∂ υ | E r ⟩ / ⟨ E r | E r ⟩ ] ,
Δ i f = − Im [ ⟨ E r | ∂ k p ∂ v | E r ⟩ / ⟨ E r | E r ⟩ ] ,
| E r ⟩ = G ( υ , v ) ( E p r E s r ) .
Δ g h = 1 k p ⟨ E r | E r ⟩ Im { ( E p i E s i ) ∗ Q ↔ ( E p i E s i ) } ,
⟨ E r | E r ⟩ = ( | V 22 | 2 + | V 12 | 2 cos 2 θ ) | E p i | 2 + ( | V 11 | 2 + | V 21 | 2 cos − 2 θ ) | E s i | 2 + 2 Re [ ( V 12 ∗ V 11 cos ⁡ θ + V 22 ∗ V 21 cos − 1 θ ) E p i ∗ E s i ] ,
Q 11 = V 22 ∗ V 22 ′ + V 12 ∗ ( V 12 ′ − V 12 t g θ ) cos 2 θ ,
Q 12 = V 22 ∗ ( V 21 ′ + V 21 t g θ ) / cos ⁡ θ + V 12 ∗ V 11 ′ cos ⁡ θ ,
Q 21 = V 11 ∗ ( V 12 ′ − V 12 t g θ ) cos ⁡ θ + V 21 ∗ V 22 ′ / cos ⁡ θ ,
Q 22 = V 11 ∗ V 11 ′ + V 21 ∗ ( V 21 ′ + V 21 t g θ ) / cos 2 θ ,
Δ i f = − 1 k p ⟨ E r | E r ⟩ Im { ( E p i E s i ) ∗ T ↔ ( E p i E s i ) } ,
T 11 = V 22 ∗ V 22 ′ + V 12 ∗ V 12 ′ cos 2 θ + cos ⁡ θ c t g θ [ V 12 ∗ ( V 22 − V 11 ) − V 22 ∗ ( V 21 / cos 2 θ + V 12 ) ] ,
T 12 = c t g θ [ V 22 ∗ ( V 22 − V 11 ) + V 12 ∗ ( V 12 cos 2 θ + V 21 ) ] + V 22 ∗ V 21 ′ cos − 1 θ + V 12 ∗ V 11 ′ cos ⁡ θ ,
T 21 = c t g θ [ V 11 ∗ ( V 22 − V 11 ) − V 21 ∗ ( V 21 cos − 2 θ + V 12 ) ] + V 21 ∗ V 22 ′ cos − 1 θ + V 11 ∗ V 12 ′ cos ⁡ θ ,
T 22 = V 11 ∗ V 11 ′ + V 21 ∗ V 21 ′ cos − 2 θ + sin − 1 θ [ V 21 ∗ ( V 22 − V 11 ) + V 11 ∗ ( V 21 + V 12 cos 2 θ ) ] ,
A z e − Γ ′ d + B z e Γ ′ d = 0 ,
A y e − Γ ′ d + B y e Γ ′ d = B p ,
ε p f 2 ( A z e − Γ ′ d − B z e Γ ′ d ) = i k Γ ′ A p ,
Γ ′ ( A y e − Γ ′ d − B y e Γ ′ d ) = 0.
E y = 1 − i ε 0 ε a ω [ ( Γ + − γ + k ) H z + + ( Γ − − γ − k ) H z − ] ,
E z = 1 − i ε 0 ε a ω [ − Γ + λ + H z + − Γ − λ − H z − ] .
A z + B z = H z + + H z − ,
A y + B y = λ + H z + + λ − H z − ,
ε a Γ ′ ( A y − B y ) = ( − Γ + λ + H z + − Γ − λ − H z − ) ,
ε a f 2 ( A z − B z ) = Γ ′ [ ( Γ + − γ + k ) H z + + ( Γ − − γ − k ) H z − ] .
A z = 1 2 ( H z + + H z − + Γ ′ ε a f 2 [ ( Γ + − γ + k ) H z + + ( Γ − − γ − k ) H z − ] Γ ′ ε a f 2 ) ,
B z = 1 2 ( H z + + H z − − Γ ′ ε a f 2 [ ( Γ + − γ + k ) H z + + ( Γ − − γ − k ) H z − ] ) ,
A y = 1 2 [ λ + H z + + λ − H z − − 1 Γ ′ ε a ( Γ + λ + H z + + Γ − λ − H z − ) ] ,
B y = 1 2 [ λ + H z + + λ − H z − + 1 Γ ′ ε a ( Γ + λ + H z + + Γ − λ − H z − ) 1 Γ ′ ε a ] .
ε a f 2 ( H z + + H z − ) cosh ⁡ ( Γ ′ d ) − Γ ′ sinh ⁡ ( Γ ′ d ) × [ ( Γ + − γ + k ) H z + + ( Γ − − γ − k ) H z − ] = 0 ,
ε a Γ ′ sinh ⁡ ( Γ ′ d ) ( λ + H z + + λ − H z − ) − cosh ⁡ ( Γ ′ d ) ( Γ + λ + H z + + Γ − λ − H z − ) = 0.
| χ ( Γ + − γ + k ) − ε a f 2 χ ( Γ − − γ − k ) − ε a f 2 λ + ( χ ε a + Γ + ) λ − ( χ ε a + Γ − ) | = 0 ,
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