Spin-splitting in a reflective beam off an antiferromagnetic surface

Shu-fang Fu; Xiang-Guang Wang; Yu-Qi Zhang; Sheng Zhou; Xuan-zhang Wang

doi:10.1364/OE.435243

1. Introduction

Interaction of an incident light beam with the interface between two media can cause spatial shifts of the secondary light beams [1], compared with their partners in geometric optics. The spatial shifts are well-known as the Goos-Hächen (GH) shift [2] and Imbert-Fedorov (IF) shift [3], respectively. The GH shift lies in the plane of incidence and originates from the dispersion of reflective or refractive coefficient (or transmitted coefficient), but the IF shift is normal to the plane of incidence and possesses a more complicated origination, the spin-orbit interaction or the spin-Hall effect [4]. A lot of works related to the two phenomena can be seen since both are important for micro-nano optical and detective technologies [5]. Besides the original works related to the isotropic and anisotropic dielectric interfaces [2,3,6], the relevant investigation has been extended to other interface structures, e.g., the surface of metals [7,8], photonic crystals [9–11], metamaterials or hyperbolic materials [12,13]. One also can see the relevant works related to the metasurfaces [14,15] and magnetic media [16,17].

Antiferromagnets (AFs) do not exhibit net magnetization in macroscopy so that they are practically nonmagnetic materials, but they are of magnetic-ordering below the AF Néel temperature [18]. For the simplest insulative AF, its crystal cell contains two staggered sublattices. Either sublattice is composed of atoms with the same magnetic moment, whereas the atomic magnetic moments of the two sublattices are the same in amplitude but opposite in direction. It is interesting that the AF is gyromagnetic in an external magnetic field along the easy axis, and meanwhile the response frequency is usually situated in the range of far-infrared or THz frequency [19–21]. The AF can be considered as a hyperbolic material in some specific case [22–24]. The GH shift of reflective beam from the AF surface has been investigated in the Voigt geometry where the easy axis and external magnetic field both lie in the AF surface and normal to the plane of incidence [25]. In the Voigt geometry, the transversely-electric-polarization beam (the s-incidence) excites only transversely- electric (TE) modes in the AF, and the transversely-magnetic-polarization beam (the p-incidence) excites only transversely magnetic (TM) modes in the AF. Therefore, one can separately solve the relevant reflective coefficients for the s- and p-incidences. The AF in the Voigt geometry supports surface magneton polaritons with TE-polarization [19,20], and it in another geometry supports the Dyakonov surface magnons and magnon-polaritons [24]. However, in another specific geometry where the anisotropic and external fields both are normal to the AF surface, either radiation mode in the AF is neither a TE mode nor a TM mode for an obliquely-incident wave with any polarization. Therefore, the mathematical process of solving the reflective electromagnetic fields becomes more complicated. In addition, the AF gyromagnetism and anisotropy can bring a substantially impact on features of the reflective beam. The reflective beam can be considered as the superposition of two spin components. The two spin components usually undergo different spatial shifts so that they are spatially separated. This spin-splitting can be decomposed into the out-plane one that is a spin-Hall effect and originates from the spin-orbit interaction, and the in-plane one that is the GH effect. The spin-splitting phenomenon first was investigated for the air-glass interface and the chiral metamaterial slab [26,27]. Recently, the spin-splitting of reflective beam was considered on the surface of chiral medium [28] and on the surface of the heterostructure composed of single black phosphorus layer and Si substrate [29]. More recently, an important result was reported [30], where the spin-splitting of both reflective beam and transmitted beam was investigated. For very small incident angles, the spin-splitting distance can reach 2.2 time the waist of beam, so the two photons with opposite spins were completely separated and possess a very high energy efficiency.

In this paper, we will investigate the spatial spin-splitting in the reflective beam off the AF surface for linearly-polarized incident beams.

2. Electromagnetic fields in secondary beams

The incidence-reflection geometry and the coordinate systems are illustrated in Fig. 1, where both the AF easy axis and external magnetic field are vertical to the AF surface (the x-z plane). The incident and reflective beams are in air for the air-AF interface, the central radiation in the incident beam lies in the x-y plane (the incident plane) and with incident angle $\theta $. We use the two solid lines with arrow to indicate the centroid position of the two spin-components in the reflective beam and the dashed line shows the geometrically optical position of the reflective beam. i and r show incidence and reflection. The dielectric constant of the AF is ${\varepsilon _a}$ and the AF permeability is a sector with nonzero elements that are [19–21]

(1a)$${\mu _{xx}} = {\mu _{zz}} = {\mu _0}{\mu _1} = {\mu _0}({1 + 2{\omega_a}{\omega_m}\{ {{[\omega_r^2 - {{(\omega + {\omega_0} + i\tau )}^2}]}^{ - 1}}\textrm{ + [}\omega_r^2 - {{(\omega - {\omega_0} + i\tau )}^2}{]^{ - 1}}\} } )$$

(1b)$${\mu _{yy}} = {\mu _0}, $$

(1c)$${\mu _{xz}} ={-} {\mu _{zx}} = i{\mu _0}{\mu _2} = i{\mu _0}\{ 2{\omega _a}{\omega _m}{[\omega _r^2 - {(\omega \textrm{ + }{\omega _0}\textrm{ + }i\tau )^2}]^{ - 1}} - {[\omega _r^2 - {(\omega - {\omega _0} + i\tau )^2}]^{ - 1}}\}$$

here ${\omega _m} = 4\pi \gamma {M_0}\textrm{/2}\pi c$, ${\omega _e} = \gamma {H_e}\textrm{/2}\pi c$, ${\omega _a} = \gamma {H_a}/2\pi c$,and $\omega _r^2 = {\omega _a}\textrm{(2}{\omega _e} + {\omega _a})$ with sublattice magnetization M₀, exchange field H_e, magnetically anisotropic field H_a, external magnetic field H₀, and damping τ. In addition, $\gamma = 1.97 \times {10^{10}}\textrm{rad}{\textrm{s}^{\textrm{ - 1}}}/\textrm{kG}$ and $c = 3 \times {10^{10}}\textrm{cm/s}$ are the gyromagnetic ratio and the vacuum light-velocity, respectively. Numerical calculations are based on the FeF₂ crystal with dielectric constant ${\varepsilon _a} = 5.5$. The other physical parameters are the sublattice magnetization 4πM₀=7.04kG (${\omega _m} = 0.736\textrm{c}{\textrm{m}^{ - 1}}$), the exchange field H_e=540.0kG (${\omega _e} = 56.44\textrm{c}{\textrm{m}^{ - 1}}$), and anisotropic field H_a=200.0kG (${\omega _a} = 20.9\textrm{c}{\textrm{m}^{ - 1}}$). We take τ=0.05cm⁻¹. The two AF resonant frequencies are ${\omega _\textrm{1}}\textrm{ = 52}\textrm{.36c}{\textrm{m}^{ - 1}}$ and ${\omega _\textrm{2}}\textrm{ = 53}\textrm{.37c}{\textrm{m}^{ - 1}}$ in the external magnetic field of H₀=5.0kG (${\omega _\textrm{0}}\textrm{ = 0}\textrm{.523c}{\textrm{m}^{ - 1}}$), but the resonant frequencies of effective permeability ${\mu _\nu } = ({\mu_1^2 - \mu_2^2} )/{\mu _1}$ are 52.578cm⁻¹ and 53.511cm⁻¹. We recognize from Eq. 1(c) that the external magnetic field produces the gyromagnetism in the AF. Therefore, the interactions of left and right circularly-polarized radiations with the AF surface are different. This gyromagnetism will impact the GH and IF shifts of secondary beams for any polarized-incident beam. All angular frequencies in Eq. (1) have been divided by 2πc in order to accord with those used in the relevant experiments [19–21]. At the AF resonant frequencies, the real part of any permeability-components varies rapidly with frequency, as shown in Fig. 1(b), and the imaginary part reaches its maximum.

Fig. 1. The left diagram shows used configuration and coordinate systems where the incident plane is the x-y plane, and the x-z plane is the surface and separates air and the AF. i indicates the incident beam with incident angle $\theta . {r_ + }$ and ${r_ - }$ represent the two separated spin-components in the reflective beam. The dashed line shows the position of reflective beam in the geometric optics. The XYZ frame is the coordinate system attached to the incident beam. The right graph illustrates the permeability of antiferromagnetic FeF₂, where ${k_0} = \omega /2\pi c$ is the vacuum wave-number or the reduced frequency.

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We use a paraxial radiation beam (Gauss beam) without intrinsic orbit angular momentum as the incident beam. The central-radiation electromagnetic fields of the incident, reflective or refractive beam include a factor $\textrm{exp} (i{k_x}x + i{k_j}y - i\omega t)$ with ${k_j}$ indicating the y-component of wave-vector. One can use ${{\textbf k}^i} = \left( {\begin{array}{{ccc}} {{k_0}\sin \theta }&{{k_0}\cos \theta }&0 \end{array}} \right)$ and ${{\textbf k}^r} = \left( {\begin{array}{{ccc}} {{k_0}\sin \theta }&{ - {k_0}\cos \theta }&0 \end{array}} \right)$ to represent the incident and reflective wave vectors, where ${k_0} = \omega /2\pi c$ is the reduced frequency or vacuum wave-number. We will apply the electromagnetic boundary conditions to solve the relevant reflective wave for the central radiation in the incident beam, and then to derive the shift expressions of the component with positive or negative spin in the reflective beam. The relations between electric field and magnetic field in either space should be first obtained. Above the AF, the relations between E and H are

(2a)$${H_x} = \frac{{{k_y}}}{{{\mu _0}\omega }}(E_z^i - E_z^r), $$

(2b)$${H_z} ={-} \frac{{k_0^2}}{{{\mu _0}\omega {k_y}}}(E_x^i - E_x^r), $$

with ${k_y} = {k_0}\cos \theta$, ${k_x} = {k_0}\sin \theta$ and the vacuum permeability ${\mu _0}$. In the AF, the projections of wave equation to the three coordinate axes are

(3a)$$- {k_x}{k_y}{H_y} + (k_y^2 - {\mu _1}{\varepsilon _a}k_0^2){H_x} - i{\mu _2}{\varepsilon _a}k_0^2{H_z} = 0 ,$$

(3b)$$- {k_x}{k_y}{H_x} + (k_x^2 - {\varepsilon _a}k_0^2){H_y} = 0, $$

(3c)$$(k_x^2 + k_y^2 - {\mu _1}{\varepsilon _a}k_0^2){H_z} + i{\mu _2}{\varepsilon _a}k_0^2{H_x} = 0.$$

Eqs. (3) imply two solutions of in the AF, i.e., $k_ \pm$ determined by

(4)$$k_ \pm ^\textrm{2} = \left( { - b \pm \sqrt {{b^2} - 4c} } \right)/2, $$

with

(5a)$$b = (k_x^2 - {\varepsilon _a}{\mu _1}k_0^2) + {\mu _1}(k_x^2 - {\varepsilon _a}k_0^2), $$

(5b)$$c = {\mu _1}(k_x^2 - {\varepsilon _a}k_0^2)(k_x^2 - {\varepsilon _a}k_0^2{\mu _\nu }), $$

and meanwhile offer the relation between the x- and z-components of H to be

(6)$$H_x^ \pm = {\Gamma }_x^ \pm H_z^ \pm = \frac{{i{\mu _2}(k_x^2 - {\varepsilon _a}k_0^2)}}{{{\varepsilon _a}{\mu _1}k_0^2 - k_ \pm ^2 - {\mu _1}k_x^2}}H_z^ \pm,$$

for either solution. From $\nabla \times {\textbf E} = i\omega \,{\kern 1pt} {\mathbf \mu } \cdot {\textbf H}$, we find the relations between H and E to be

(7a)$$H_x^ \pm{=} \frac{1}{{\omega {\mu _0}{\mu _\nu }}}\left( {{k_ \pm }E_z^ \pm{+} \frac{{i{\mu_2}(k_x^2 + k_ \pm^2)}}{{{\mu_1}{k_ \pm }}}E_x^ \pm } \right)$$

(7b)$$H_z^ \pm{=}{-} \frac{1}{{\omega {\mu _0}{\mu _\nu }}}\left( { - \frac{{i{\mu_2}}}{{{\mu_1}}}{k_ \pm }E_z^ \pm{+} \frac{{k_x^2 + k_ \pm^2}}{{{k_ \pm }}}E_x^ \pm } \right), $$

with ${\mu _\nu }\textrm{ = (}\mu _\textrm{1}^\textrm{2} - \mu _2^2)/{\mu _1}$, the Voigt permeability. Therefore, we can express the z- and x-components of total electric field in the AF as

(8)$${E_z} = {E_ + }{e^{i{k_ + }y}} + {E_ - }{e^{i{k_ - }y}},\;\;{E_x} = {\rho _ + }{E_ + }{e^{i{k_ + }y}} + {\rho _ - }{E_ - }{e^{i{k_ - }y}}$$

where the right two terms in Eq. (8) correspond to the two solutions of k_y, respectively, and

(9)$${\rho _ \pm }\textrm{ = }\frac{{k_ \pm ^2(i{\mu _2}\Gamma _x^ \pm{-} {\mu _1})}}{{(k_x^2 + k_ \pm ^2)({\mu _1}\Gamma _x^ \pm{+} i{\mu _2})}}, $$

found from Eqs. (6) and (7). On the above bases, the electromagnetic boundary conditions at the AF surface can be directly written as

(10a)$$E_z^i + E_z^r = {E_ + } + {E_ - }, $$

(10b)$$E_z^i - E_z^r = {\lambda _ + }{E_ + } + {\lambda _ - }{E_ - }, $$

(10c)$$E_x^i + E_x^r = {\rho _ + }{E_ + } + {\rho _ - }{E_ - }, $$

(10d)$$E_x^i - E_x^r = {\gamma _ + }{E_ + } + {\gamma _ - }{E_ - }, $$

where

(11)$${\lambda _ \pm }\textrm{ = }\frac{\textrm{1}}{{{\mu _\nu }{k_y}}}({k_ \pm } + i{\mu _2}{\rho _ \pm }\frac{{k_x^2 + k_ \pm ^2}}{{{\mu _\textrm{1}}{k_ \pm }}}),\;\;\;{\gamma _ \pm }\textrm{ = }\frac{{{k_y}}}{{{\mu _\nu }k_0^2}}( - \frac{{i{\mu _2}}}{{{\mu _1}}}{k_ \pm } + {\rho _ \pm }\frac{{k_x^2 + k_ \pm ^2}}{{{k_ \pm }}})$$

Obviously, the relation between the incident and refractive amplitudes can be shown as

(12)$$\left( {\begin{array}{{c}} {{E_ + }}\\ {{E_ - }} \end{array}} \right) = \frac{1}{\Lambda }\left( {\begin{array}{{cc}} {{\rho_ - } + {\gamma_ - }}&{ - (1 + {\lambda_ - })}\\ { - ({\rho_ + } + {\gamma_ + })}&{1 + {\lambda_ + }} \end{array}} \right)\left( {\begin{array}{{c}} {E_z^i}\\ {E_x^i} \end{array}} \right)\textrm{ = }A\left( {\begin{array}{{c}} {E_z^i}\\ {E_x^i} \end{array}} \right), $$

where $\Lambda \textrm{ = [(1 + }{\lambda _ + })({\rho _ - } + {\gamma _ - }) - (1 + {\lambda _ - })({\rho _ + } + {\gamma _ + })]/2$. After obtaining E₊ and E_-, we can express the reflective-field amplitudes at the AF surface as

(13)$$E_z^r = {E_ + }\textrm{ + }{E_ - } - E_z^i,\;\;\;E_x^r = {\rho _ + }{E_ + }\textrm{ + }{\rho _ - }{E_ - } - E_x^i, E_y^r = E_x^r\tan \theta.$$

One can use Eq. (13) to numerically calculate the reflective electric-field and further to derive the spatial shifts of the reflective beam. It should be noted that Eqs. (12) and (13) are not available outside the AF response-frequency range since $H_x^ \pm$ and $H_z^ \pm$ are mutually independent in this case.

3. Spatial shift of components with different spins in a reflective beam

In this section, we derive the in-plane and out-plane spatial shifts of two radiation components with different spins in the reflective beam. We neglect shape deformation of the reflective beam and propose that the incident beam possesses a finite Fourier spectrum in the k-space, where wave-vectors narrowly distribute around the central wave-vector. Except the central wave-vector, others include a small vertical component κⁱ, besides the parallel component, as described in Ref. [1]. In this respect, the reflective beam is similar to the incident beam. κ^j (j = i or r) is decomposed into one in-plane component and one out-plane component. The in-plane one is along the X-axis and varies $\theta $ by a small quantity $\eta$, but the out-plane component is parallel to the Z-axis or z-axis and rotates the incident plane by a small angle φ. The change of $\theta $ can induce the GH shift of the secondary beams, i.e., the in-plane shift. The small rotation of incident plane can bring the IF shift or the out-plane shift. It has been proven that the shifts are independent of the beam profile for an enough large beam waist in isotropic medium so that the relevant mathematic processes are reduced. In the above section, the central radiation of the reflective beam has been obtained from the incident central radiation. In order to derive the expressions of shifts, we are going to change Eqs. (13) into a three-dimensional matrix. Due to the second and third equations in Eqs. (13) and using Eq. (12), we find $E_y^r = tan\theta [{({{\rho_ + }{a_{11}} + {\rho_ - }{a_{21}}} )E_z^i + ({{\rho_ + }{a_{12}} + {\rho_ - }{a_{22}} - 1} )E_x^i} ]$ with ${a_{ij}}$ representing elements of matrix A in Eq. (12). Noting $E_y^i ={-} tan\theta E_x^i$, we achieve the relation between the centrally incident radiation and the relevant reflective radiation to be

(14)$$\left( {\begin{array}{{c}} {E_x^r}\\ {E_y^r}\\ {E_z^r} \end{array}} \right) = \left( {\begin{array}{{ccc}} { - {f_1}}&0&{{f_2}}\\ 0&{{f_1}}&{{f_2}\tan \theta }\\ {{f_3}}&0&{{f_4}} \end{array}} \right)\left( {\begin{array}{{c}} {E_x^i}\\ {E_y^i}\\ {E_z^i} \end{array}} \right), $$

where ${f_1} = 1 - {\rho _ + }{a_{12}} - {\rho _ - }{a_{22}}$, ${f_2} = {\rho _ + }{a_{1\textrm{1}}}\textrm{ + }{\rho _ - }{a_{2\textrm{1}}}$, ${f_\textrm{3}}\textrm{ = }{a_{12}} + {a_{22}}$ and ${f_\textrm{4}} = {a_{1\textrm{1}}}\textrm{ + }{a_{2\textrm{1}}} - 1$. However, Eq. (14) is given in the experimental coordinate system (xyz). In order to achieve the spatial shifts of the two spin-components, we have to transfer Eq. (14) into the beam coordinate-system (XYZ) with the Y-axis pointed along the relevant central wave-vector. Because the deviation (η, ν) of other wave-vectors from the central wave-vector in the incident beam will induce the corresponding deviation (-η, ν) in the reflective beam, where $\nu \textrm{ = }\varphi \textrm{/}\sin \theta$ with the azimuthal angle φ, Eq. (14) is written as

(15)$$\left( {\begin{array}{@{}c@{}} {E_\parallel^r}\\ \delta \\ {E_ \bot^r} \end{array}} \right) = \left( {\begin{array}{@{}ccc@{}} {\cos {\theta^r}}&{ - \sin {\theta^r}}&{ - \varphi \cos {\theta^r}}\\ {\sin {\theta^r}}&{\cos {\theta^r}}&{\varphi \sin {\theta^r}}\\ \varphi &0&1 \end{array}} \right)\left( {\begin{array}{@{}ccc@{}} { - {f_1}}&0&{{f_2}}\\ 0&{{f_1}}&{{f_2}\tan \theta }\\ {{f_3}}&0&{{f_4}} \end{array}} \right)\left( {\begin{array}{@{}ccc@{}} {\cos \theta }&{\sin \theta }&\varphi \\ { - \sin \theta }&{\cos \theta }&0\\ { - \varphi \cos \theta }&{\varphi \sin \theta }&1 \end{array}} \right)\left( {\begin{array}{@{}c@{}} {E_\parallel^i}\\ 0\\ {E_ \bot^i} \end{array}} \right), $$

where η and -η are included in the incident and reflective angles, and ${\theta ^r} = \pi - \theta$ [1]. Equation (15) is reduced to be

(16)$$\left( {\begin{array}{{c}} {E_\parallel^r}\\ {E_ \bot^r} \end{array}} \right) = \Gamma \left( {\begin{array}{{c}} {E_\parallel^i}\\ {E_ \bot^i} \end{array}} \right), $$

with

(17)$$\Gamma \textrm{ = }\left( {\begin{array}{{cc}} {{f_1} + \varphi ({f_2} + {f_3}{{\cos }^2}\theta )}&{ - {f_2}{{\cos }^{ - 1}}\theta + \varphi \cos \theta ({f_1} + {f_4})}\\ {\cos \theta {f_3} - \varphi \cos \theta ({f_1} + {f_4})}&{{f_4} + \varphi ({f_2} + {f_3})} \end{array}} \right). $$

${E_\parallel }$ and ${E_ \bot }$ are the field-amplitudes of the p- and s-waves in the incident or reflective beam, respectively. If the external magnetic field H₀=0, the permeability is diagonal and then both ${f_2}$ and ${f_3}$ are equal to 0. In this case, $\Gamma $ is

(18)$$\Gamma \textrm{ = }\left( {\begin{array}{{cc}} {{f_1}}&{\varphi \cos \theta ({f_1} + {f_4})}\\ { - \varphi \cos \theta ({f_1} + {f_4})}&{{f_4}} \end{array}} \right), $$

and we can achieve the known result [1] for the GH and IF shifts of the reflective beam from Eq. (18). We are really interested in the spatial shifts of the positive-spin component and negative-spin component in the reflective beam, or the spin splitting. Subsequently, we derive their expressions. A generally-polarized radiation in an isotropic medium can be decomposed into two components with positive and negative spins, respectively. Therefore, the reflective radiation can be written as the superposition of the positive-spin and negative-spin components, i.e.

(19)$${{\textbf E}^r} = ({F_ + }{\hat{e}_ + } + {F_ - }{\hat{e}_ - })/\sqrt 2, $$

where ${\hat{e}_ \pm } = [{\hat{e}_X} \mp i{\hat{e}_Z}]/\sqrt 2$ represents the positive or negative spin state and the superposition coefficients are determined by

(20)$${F_ \pm } = (E_\parallel ^r \pm iE_ \bot ^r)\textrm{/}\sqrt 2.$$

Thus, the in-plane shifts (GH shifts) of spin-components are determined [1] by

(21)$$\Delta _{GH}^ \pm{=} \frac{1}{{{Q_ \pm }}}{\mathop{\rm Im}\nolimits} (F_ \pm ^\ast \frac{\partial }{{{k_0}\partial \theta }}{F_ \pm }), $$

and the out-plane shifts (IF shifts) are defined [1] by

(22)$$\Delta _{IF}^ \pm \textrm{ = } - \frac{1}{{{Q_ \pm }}}{\mathop{\rm Im}\nolimits} \left( {F_ \pm^\ast \frac{\partial }{{\partial \kappa_o^r}}{F_ \pm }} \right), $$

with ${Q_ \pm } = F_ \pm ^{\ast }{F_ \pm }$ and $\kappa _o^r = {k_0}\nu \textrm{ = }{k_0}\varphi /\sin \theta$ that is the small k-component normal to the incident plane. ν is a small deviation angle to the central wave-vector of incidence and produces a small rotation (φ) of the incident plane. Obviously, $\Delta = {\Delta ^ + } - {\Delta ^ - }$ is the shift difference between the two spin-components, i.e., the spin splitting. For the s-incidence with $E_ \bot ^i = 1$ and $E_\parallel ^i = \textrm{0}$, the GH and IF shifts can be further reduced to be

(23)$$\Delta _{GH}^ \pm{=} \frac{1}{{{k_0}}}{\mathop{\rm Im}\nolimits} \left( {\frac{{ - ({{f^{\prime}}_2} + {f_2}\tan \theta ) \pm i{{f^{\prime}}_4}\cos \theta }}{{ - {f_2} \pm i{f_4}\cos \theta }}} \right), $$

(24)$$\Delta _{IF}^ \pm \textrm{ = } - \frac{1}{{{k_0}}}{\mathop{\rm Im}\nolimits} \left( {\frac{{({f_1} + {f_4}){{\tan }^{ - 1}}\theta \pm i({f_2} + {f_3}){{\sin }^{ - 1}}\theta }}{{ - {f_2}{{\cos }^{ - 1}}\theta \pm i{f_4}}}} \right). $$

For the p-incidence with $E_ \bot ^i = \textrm{0}$ and $E_\parallel ^i = \textrm{1}$, the shifts are specifically shown as

(25)$$\Delta _{GH}^ \pm{=} \frac{1}{{{k_0}}}{\mathop{\rm Im}\nolimits} \left( {\frac{{{{f^{\prime}}_1} \pm i({{f^{\prime}}_3}\cos \theta - {f_3}\sin \theta )}}{{{f_1} \pm i{f_3}\cos \theta }}} \right), $$

(26)$$\Delta _{IF}^ \pm \textrm{ = } - \frac{1}{{{k_0}}}{\mathop{\rm Im}\nolimits} \left( {\frac{{({f_2} + {f_3}{{\cos }^2}\theta ){{\sin }^{ - 1}}\theta \mp i({f_1} + {f_4}){{\tan }^{ - 1}}\theta }}{{{f_\textrm{1}} \pm i{f_\textrm{3}}\cos \theta }}} \right). $$

In the case of no external field (H₀=0), we discuss these results. One knows that Eq. (6) does not exist in this case since H_x and H_z are decoupled. It should be noted that according to Eq. (1c), the nondiagonal elements (${\mu _{xz}}\; \textrm{and}\; {\mu _{zx}}$) of the permeability are equal to 0 and the two branch waves in the AF are the TE wave (or s-wave) and TM wave (or p-wave), respectively. Therefore, the s-incidence on the surface produces only the s-wave of reflection, and in turn the p-incidence results in only the p-wave of reflection, i.e., $E_x^r ={-} {f_1}E_x^i$, $E_y^r = {f_1}E_y^i$ and $E_z^r = {f_4}E_z^i$, and ${f_1} = {f_p}$ and ${f_4} = {f_s}$ according to the concepts in Ref. [1], so ${f_2} = {f_3} = 0$, where ${f_s}$ and ${f_p}$ are the reflective coefficients of the central s-wave and p-wave in the incident beam. These results can be directly obtained from Eqs. (3) and the electromagnetic boundary conditions. As a result, we see

(27)$$\Delta _{GH}^ \pm{=} \frac{1}{{{k_0}}}{\mathop{\rm Im}\nolimits} \{ {f^{\prime}_s}/{f_s}\}, $$

(28)$$\Delta _{IF}^ \pm \textrm{ = } \mp \frac{1}{{{k_0}}}\textrm{Re} \left( {\frac{{{f_p} + {f_s}}}{{{f_s}\tan \theta }}} \right), $$

for the s-incidence, where ${f_s} = ({k_y^{\prime} - {k_ + }/{\mu_1}} )/({k_y^{\prime} + {k_ + }/{\mu_1}} )$ and ${f_p} = ({{k_ - } - {\varepsilon_a}{{k^{\prime}}_y}} )/({{k_ - } + {{k^{\prime}}_y}{\varepsilon_a}} )$. However,

(29)$$\Delta _{GH}^ \pm{=} \frac{1}{{{k_0}}}{\mathop{\rm Im}\nolimits} ({f^{\prime}_p}/{f_p})$$

(30)$$\Delta _{IF}^ \pm \textrm{ = } \pm \frac{1}{{{k_0}}}\textrm{Re} \left( {\frac{{{f_p} + {f_s}}}{{{f_p}\tan \theta }}} \right), $$

for the p-incidence. It is obvious that the in-plane shifts of the two spin components are the same, i.e., $\Delta _{GH}^ + - \Delta _{GH}^ -$ if no external magnetic field exists, so the in-plane spin splitting $(\Delta _{GH}^ + - \Delta _{GH}^ - )$ disappears. Equation (28) or (30) proves that the out-plane shifts of the two spin-components are equal in amplitude but opposite in the direction, so the spin-splitting distance is double the out-plane shift-distance of either spin-component. Equations (28) and (30) also exhibit another interesting phenomenon, which is that the out-plane shifts involve ${f_p}$ for the s-incidence {see (28)} and involve ${f_s}$ for the p-incidence {see (30)}. This additional contribution to the shifts results from the out-plane distribution of the incident beam, i.e., from $\kappa _o^r$. It is evident that ${f_p}/{f_s}$ in Eq. (28) or ${f_s}/{f_p}$ in Eq. (30) mainly determines the out-plane spin-splitting distance and dependent on the surface impedance mismatch [31].

4. Numerical results and discussions

Numerical calculations are based on the FeF₂ crystal with the physical parameters offered in the above section. The gyromagnetism originates from the external magnetic field. One has known that a linearly-polarized electromagnetic wave, i.e., the s-wave (the TE wave) or the p-wave (the TM wave), can be decomposed into two opposite-spin components (or a left and a right circularly-polarized waves). The two components are identical in amplitude and propagate along the same direction. An elliptically-polarized radiation also contains two spin components different in amplitude. For the air-AF interface, the reflective wave usually is an elliptically-polarized radiation for a linearly-polarized incident wave. The different interactions of the two incident spin components with the AF surface can cause different shifts of the two reflective spin-components, or say that the different interactions produce the spin splitting in the reflective beam. For better discussion on the shifts, we define the component-intensity as ${R_ \pm } = {|{E_ \pm^r} |^2}$. We first illustrate the intensity of either reflective spin-component for the p-incidence, as shown in Fig. 2. The reflective spectra exhibit some extraordinary features in the vicinities of some specific frequencies. We see that the two intensities are completely different in feature for the p-incidence. The maximum of ${R_ + }$ appears on the right side of the second resonant frequency but that of ${R_ - }$ is situated on the right side of the first resonant frequency, where ${\mu _1} < 0$. For some larger incident angles, the intensities are extraordinary. The reflective intensity of one spin-component reaches 0 at a value of k₀, but that of the other component is significant at the same value. It is obvious that the reflective beam is an approximately circularly-polarized beam in this case. This phenomenon occurs only for $\theta \ge {45^o}$ and the p-incidence. Figs. 3 show the reflective intensities of the two components for the s-incidence. The minimum of ${R_ - }$ approximately is at the first resonant frequency, but the maximum lies on the right side of this resonant frequency where ${\mu _1} < 0$. The minimum of ${R_ + }$ is situated about at the second resonant frequency and the maximum is situated on the right side of the resonant frequency, where ${\mu _1} < 0$.

Fig. 2. Reflective intensities of two spin-components in the reflective beam for different incident angles in the case of p-incidence. (a) For the component with positive spin and (b) for the component with negative spin, where the main peaks of two spin-components are situated near ${k_0} = 52.5$ and $53.5$, respectively.

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Fig. 3. Reflective intensities of two spin-components in the reflective beam for different incident angles in the case of p-incidence. (a) For the component with positive spin and (b) for the the component with negative spin, where the main peaks of two spin-components are situated the right side of ${k_0} = 52.5$ and $53.5$, respectively.

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Subsequently, combining Figs. 2 and 3 with Eqs. (27) to (30), we further discuss about the splitting distance of the two spin-components in the reflective beam for the linearly-polarized incidences. The amplitude of shift-difference between the two spin-components represents the splitting distance and the sign of shift-difference shows the relative positions of the two spin-components. Figs. 4 illustrate the in-plane shift-difference between the two spin components. The left graph shows the shift-difference for the p-incidence. Very large peaks or dips are seen. For example, the cyan curve exhibits two evident peaks and one large dip, but the blue curve shows two large dips and one small peak. Referring to Fig. 2, we find that the two cyan peaks correspond to the two minimums of ${R_ - }$ and the large cyan dip is related to the minimum of ${R_ + }$. The two blue dips correspond to the two minimums of ${R_ + }$, but the small peak corresponds to the minimum of ${R_ - }$. For the other curves, the similar corresponding relations also can be found. In general, the splitting distance is longer for bigger incident angles, and the two spin-components are obviously different in intensity. The maximum of the splitting distance can reach 70 ${\lambda _0}$. The right graph illustrates the shift-difference for the s-incidence and various incident angles. We see that every curve exhibits one dip and one peak. The dip is situated near the first AF resonant frequency (${\omega _1} = $ 52.35cm⁻¹) and the peak is localized near the second AF resonant frequency (${\omega _2} = $ 53.37cm⁻¹). The blue-shift effect of the dip or peak is found as the incident angle is enlarged. The maximum of splitting distance reaches about 2 ${\lambda _0}$. Compared with that for the p-incidence, the maximum of splitting distance is much shorter for the s-incidence.

Fig. 4. The in-plane shift-difference between the two components with different spins in the reflective beam for various incident angles. (a) For the p-incidence and (b) for the s-incidence.

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Figs. 5 illustrate the out-plane shift-difference between the two spin-components in the reflective beam. For the p-incidence, after carefully observing the curves in Fig. 5(a) and comparing them with those in Fig. 4(a), we recognize that adjacent peak and dip of each curve in Fig. 5(a) are situated on both sides of a peak or dip of the relevant curve in Fig. 4(a), respectively. For example, the first dip of the blue curve in Fig. 4(a) lies at ${k_0} = $ 52.28cm⁻¹, and meanwhile the first dip and first peak of the blue curve in Fig. 5(a) are localized at ${k_0} = $ 52.24cm⁻¹ and 52.33cm⁻¹, respectively. Likewise, while the first peak of the cyan curve in Fig. 4(a) is at ${k_0} = $ 52.58cm⁻¹, the first dip and first peak lie at ${k_0} = $ 52.73cm⁻¹ and 52.83cm⁻¹, respectively. This phenomenon is because that the in-plane shift-difference is mainly determined by the derivative of ${F_ \pm }$ but the out-plane splitting distance depends on only ${F_ \pm }$, as demonstrated by Eqs. (23) to (26). The maximum of the out-plane splitting distance can reaches 12.5 ${\lambda _0}$ for the p-incidence. For the s-incidence, Fig. 5(b) illustrates the out-plane splitting of the two spin components. In accordance with the above conclusion on Fig. 5(a), the peak or dip of any curve in Fig. 4(b) corresponds to a couple of peak and dip of the relevant curve in Fig. 5(b), and the peak or dip in Fig. 4(b) lies between the couple in Fig. 5(b). For example, the first peak and first dip of the green curve in Fig. 5(b) are at 52.27cm^{-}1 and 52.38cm^{-}1, and meanwhile the relevant green dip in Fig. 4(b) is localized at ${k_0} = $ 52.33cm⁻¹. In addition, we also realize that the largest-gradient of the out-plane splitting difference in Fig. 5 corresponds to the peak or dip in Fig. 4. The reason is similar to that for the p-incidence.

Fig. 5. The out-plane shift-difference between the two spin-components in the reflective beam for various incident angles. (a) For the p-incidence and (b) for the s-incidence.

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An external magnetic field along the y-axis produces the gyromagnetism in the AF, which plays a key role for the in-plane spin-splitting and substantively influences the out-plane spin-splitting. In order to examine the effect of the external magnetic field on the spin-splitting of the two components, we offer Figs. 6 for a fixed vacuum wave number. Figs.6(a) and (b) show the in-plane and out-plane spin-splitting of the two spin- components in the reflective beam for different incident angles, respectively. We find from Fig. 6(a) that the in-plane spin-splitting disappears for ${H_0} = 0$ and the maximum of the splitting distance presents at a definite external-field value. In addition, the splitting distance is much larger for the p-incidence than for the s-incidence. Figure 6(b) illustrates the external-field dependence of the out-plane spin-splitting. The two peak-values of the splitting distance are found for the p-incidence, or one is related to the peak and the other corresponds to the dip, as shown by the red, green and blue curves. The other curves exhibit a peak value of the splitting distance. Therefore, the gyromagnetism or external magnetic field plays a key role for the in-plane spin-splitting. The surface impedance mismatch is a critical factor for the existence of the out-plane spin-splitting since the splitting exists still without external magnetic field, but the gyromagnetism produces significant and complicated effects on the out-plane splitting of the two spin-components in the reflective beam.

Fig. 6. The in-plane and out-plane spin-splitting of the two components in the reflective beam for a fixed wave number and various incident angles. (a) The in-plane spin-splitting and (b) the out-plane spin-splitting, where the solid curves show the spin-splitting for the p-incidence and the dot curves represent the spin-splitting for the s-incidence.

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5. Summary

We have investigated the spatial splitting between the positive- and negative-spin components in the reflective beam for the two typical polarized beams incident on the air-AF interface, respectively. We used a special configuration where the AF anisotropic field and external magnetic field both are vertical to the AF surface. After obtaining the reflective beam from either incident beam, we derived the expressions of the spatial shifts of the two spin components in the reflective beam. Due to the difference between the spatial shifts, the two spin-components are spatially separated, defined as the spin splitting effect. The absolute value of the shift-difference represents the splitting-distance between the two spin-components and the sign indicates their relative positions in the xyz coordinate system. We found that the in-plane or out-plane splitting distance is much more essential for the p-incidence than the s-incidence. For the p-incidence, the in-plane splitting distance can reach about 70 ${\lambda _0}$ and the out-plane one can attain about 12.5 ${\lambda _0}$. However, for the s-incidence, both the in-plane and out-plane splitting distances can reach only about 2 ${\lambda _0}$. There is a definite relation between the in-plane and out-plane shift-difference, a peak value of the in-plane splitting distance is situated between two adjacent peak values of the out-plane splitting distance. It is because the in-plane shift is mainly determined by the derivative of ${F_ \pm }$, but the out-plane shift-difference is directly dependent on the reflective coefficients (${f_j}$). The gyromagnetism plays a key role for the in-plane spin-splitting. The in-plane spin-splitting cannot exist if the external magnetic field is removed or the gyromagnetism disappears. The out-plane spin-splitting is mainly attributed to the surface impedance mismatch, but the gyromagnetism can produce evident effect on the out-plane spin-splitting, i.e., the gyromagnetism significantly influences the orbit-spin interaction of secondary beams at the surface, but it cannot determine the existence or absence of the out-plane spin-splitting.

Funding

Natural Science Foundation of Heilongjiang Province (HL2020A014, ZD2009103).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.

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Spin-splitting in a reflective beam off an antiferromagnetic surface

Abstract

1. Introduction

2. Electromagnetic fields in secondary beams

3. Spatial shift of components with different spins in a reflective beam

4. Numerical results and discussions

5. Summary

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (40)

Optics Express