Spin angular momentum and nonreciprocity of ghost surface polariton in antiferromagnets

Yuqi Zhang; Xiangguang Wang; Shaopeng Hao; Haoyuan Song; Xuan-Zhang Wang

doi:10.1364/OE.466066

1. Introduction

Some magnetic materials or magnetic metamaterials were demonstrated to be a kind of hyperbolic materials in specific situations [1–5]. In the case of no external magnetic field, the principal components of the permeability tensor in an insulative antiferromagnet (AF) have opposite signs in the reststrahlen frequency band, so the dispersion equation is hyperbolic in this material. This hyperbolicity is supported by its permeability tensor rather than its permittivity (a positive scalar quantity), different from that of hBN and MoO₃ [6–7]. Naturally insulative AFs form a large family including transition metal oxides, fluorides and sulfides (e.g., NiO, MnO, FeF₂ and MnF₂) [8–10]. Due to their hyperbolism, they have been attracting the attention of physicists again [11–13]. Engineering surface magnon polaritons were investigated in an external magnetic field orthogonal to the AF easy axis [11], where the effect of external field on surface magnon polaritons was focused on. The Goos-Hänchen shift of reflected light beam from the AF surface was investigated [12] and hyperbolic dispersion and negative refraction of AF were discussed [13]. Recently, we predicted Dyakonov surface magnons and magnon polaritons [14]. Surface electromagnetic waves possess a common feature, i.e., they are localized at the surface of the materials supporting them and their electromagnetic fields exponentially decay with the distance away from the surface. Surface polaritons are a kind of surface electromagnetic waves. A conventional surface polariton propagates along the surface and has a definite polarization, e.g., surface magnon polaritons are of TE-polarization in antiferromagnets or ferromagnets [9–10,15–18], surface phonon or plasmon polaritons are of TM polarization in polar crystals [10], metals or graphene systems [19–20].

The ghost electromagnetic waves were firstly introduced by Narimanov [21], representing the optical analogue of the ghost orbits in the semiclassical theory of non-integrable systems [22–24]. The combination of exponential decay and oscillatory behavior of their electromagnetic fields is the characteristic [21]. A ghost surface phonon polariton with TM polarization was predicted [25] in a metamaterial and this kind of surface polariton was experimentally observed in the anisotropic polar crystal recently [26]. Its electromagnetic fields attenuate and oscillate with distance away from the material surface. More recently, we predicted another ghost surface polariton at the surface of an AF in the normal geometry [27]. It is a hybrid-polarization surface polariton, composed of two coherent branch-waves with both the exponential decay and oscillatory behavior. A series of interferent fringes can be seen on the plane normal to the propagation direction, where the main fringe lies at or close to the surface and the others lying deep in the AF are very weak. An important work has revealed that the two-wave interference in vacuum brings about some surprising spin and momentum properties, where the two coherent electromagnetic waves can be of any polarization [28]. Therefore, this ghost surface polariton [27] is formed by the interference of two branch waves in the AF may have unique properties of spin angular momentum (SAM), where the polarization is the inherent feature of the ghost surface polariton.

The study of optical angular momentum has grown into a great research field with numerous potential applications in optical manipulations, quantum information, photonics, plasmonics, and astrophysics [29–30]. The spin angular momentum (SAM) and orbital angular momentum of light are separately observable properties in optics and form a very meaningful research subject. It has been well-known that conventional surface polaritons or evanescent waves have a transverse spin angular momentum [31–32] that forms a universal right-handed triplet with wavevector and attenuation-direction together [32–33]. Our previous work shows that the Dyakonov-like surface phonon polaritons in the hexagonal boron nitride has a unique SAM with a transverse component and a longitudinal component [34]. In this paper, we will investigate the spin angular momentum and nonreciprocity of the ghost surface polariton (GSP) in the AF [27].

2. Analytical results and discussion

The geometry and coordinate system are shown in Fig. 1, where both the external magnetic field (${H_0}$) and AF easy axis are vertical to the surface. The permeability tensor of AF is a nondiagonal matrix, namely

(1)$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over \mu } = {\mu _0}\left( {\begin{array}{{ccc}} 1&0&0\\ 0&{{\mu_1}}&{i{\mu_2}}\\ 0&{ - i{\mu_2}}&{{\mu_1}} \end{array}} \right),$$

where ${\mu _1} = 1 + {\psi _ + } + {\psi _ - }$ and ${\mu _2} = {\psi _ + } - {\psi _ - }$ with ${\psi _ \pm } = {\omega _m}{\omega _a}/[{\omega_r^2 - {{({\omega \pm {\omega_0} + i\tau } )}^2}} ]$ with damping constant τ [9–10]. We should remind that the specific frequencies included in the formulae are defined with AF physical parameters. For the FeF₂ crystal, its sublattice magnetization 4πM₀= 7.04 kG is converted to ${\omega _m} = 0.74\,{\rm{c}}{{\rm{m}}^{ - 1}}$, exchange field H_e= 540.0 kG to ${\omega _e} = 56.44\,{\rm{c}}{{\rm{m}}^{ - 1}}$, and anisotropic field H_a= 200.0 kG to ${\omega _a} = 20.9\,{\rm{c}}{{\rm{m}}^{ - 1}}$ [12,16]. The zero-field resonant frequency ${\omega _r} = {\{ {\omega _a}({2{\omega_e} + {\omega_a}} )\} ^{1/2}} \approx 52.88\;{\rm{c}}{{\rm{m}}^{ - 1}}$ and ${\omega _0} = \gamma {H_0}$ and gyromagnetic ratio γ=0.1024 cm^-1/kG. The AF has two resonant frequencies linearly changing with ${H_0}$, ${\omega _r} - {\omega _0}$ and ${\omega _r} + {\omega _0}$. Its permittivity is ${\varepsilon _a} = 5.5$. For convenience, we ignore the damping term and assume that the GSP propagates along the y-axis and its fields attenuate with the distance away from the surface. Leaving out the common factor ${\rm{exp}}({2\pi iky - i\omega t} )$ in the expressions of GSP electromagnetic fields, the magnetic field is written as

(2)$${\bf{h^{\prime}}} = {\bf{H^{\prime}}}\exp (2\pi \varGamma ^{\prime}x), $$

with attenuating constant ${\rm{\varGamma ^{\prime}}} = {({{k^2} - {f^2}} )^{1/2}}$ above the AF. $f = \omega /2\pi c\;\;$ is defined as the reduced frequency and $k$ is the reduced wavevector or wavenumber. In the AF, the magnetic field consists of two branch-fields, so it is written as

(3)$${\bf{h}} = \exp ( - 2\pi \alpha x)[{{\bf{H}}^ + }\exp ( - 2i\pi \beta x) + {{\bf{H}}^ - }\exp (2i\pi \beta x)].$$

Fig. 1. Geometry and coordinate system used in theory, where both the AF easy axis and external magnetic field are vertical to the surface and the surface polariton propagates along the y-axis. The y-z plane is the surface or the air-AF interface. Two dashed arrows with ${\pm} {k_x}$ indicate the propagating directions of two branches in the AF.

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The right-side vectors in Eqs. (2) and (3) are the magnetic-field amplitudes at the surface. We found in our previous work [27] that the attenuation constant is

(4a)$$\alpha = {[{a^{\prime} + {{({{{a^{\prime}}^2} + {{b^{\prime}}^2}} )}^{1/2}}} ]^{1/2}}/\sqrt 2,$$

and oscillating constant,

(4b)$$\beta = {[{ - a^{\prime} + {{({{{a^{\prime}}^2} + {{b^{\prime}}^2}} )}^{1/2}}} ]^{1/2}}/\sqrt 2.$$

The two quantities in Eqs. (4a) and (4b) are expressed as

(5a)$$a^{\prime} = [{\mu _1}({k^2} - {\varepsilon _a}{f^2}) + ({k^2} - {\varepsilon _a}{\mu _1}{f^2})]/2,$$

and

(5b)$$b^{\prime} = \sqrt {{{({2a^{\prime}} )}^2} - 4c^{\prime}}/2,$$

where $c^{\prime}$ is determined by

(6)$$c^{\prime} = {\mu _1}({\varepsilon _a}{\mu _v}{f^2} - {k^2})({\varepsilon _a}{f^2} - {k^2}),$$

with the Voigt permeability ${\mu _v} = {\mu _1} - \mu _2^2/{\mu _1}$. $a^{\prime}$ and $b^{\prime}$ both are positive real quantities. Therefore, the two branches have the same attenuating constant $\alpha $ and opposite-sign oscillatory constants ${\pm} \beta $. We directly introduce some results in Ref. [27], i.e.,

(7a)$$\lambda = k(\alpha + i\beta )/({k^2} - {\varepsilon _a}{f^2}),$$

(7b)$$\eta = {\varepsilon _a}{\mu _2}{f^2}/[{\varepsilon _a}{\mu _1}{f^2} - {k^2} + {(\alpha + i\beta )^2}],$$

and

(8)$$\varLambda ={-} \eta ({\varepsilon _a}\varGamma ^{\prime} + \alpha + i\beta )/{\eta ^\ast }({\varepsilon _a}\varGamma ^{\prime} + \alpha - i\beta ), $$

with ${\rm{\varLambda }}{{\rm{\varLambda }}^\ast } = 1$ and ${[{{\rm{\varLambda }}/({1 + {\rm{\varLambda }}} )} ]^\ast } = 1/({1 + {\rm{\varLambda }}} )$. The y-component of GSP magnetic field is

(9a)$${h_y} = 2{H^{\prime}_y}{e^{ - 2\pi \alpha x}}[{R_y}\cos (2\pi \beta x) + {I_y}\sin (2\pi \beta x)],$$

where R_y and I_y are the real and imaginary parts of $1/({1 + {\rm{\varLambda }}} )$. Similarly, the other two components can be achieved, i.e.,

(9b)$${h_{x,z}} = 2i{H^{\prime}_y}{e^{ - 2\pi \alpha x}}[{R_{x,z}}\cos (2\pi \beta x) + {I_{x,z}}\sin (2\pi \beta x)],$$

where ${R_x}\;$ and ${I_x}$ are the real and imaginary parts of $\lambda /({1 + {\rm{\varLambda }}} )$, but ${R_z}\;$ and ${I_z}$ are the real and imaginary parts of $\eta /({1 + {\rm{\varLambda }}} )$. $H{^{\prime}_y}$ is the magnetic-amplitude component on the surface. Due to $e = i\delta \nabla \times h$ with $\delta = \sqrt {{\mu _0}/{\varepsilon _0}} /{\varepsilon _a}f$, the electric field of GSP in the AF is obtained to be

(10)$$e = \delta \left[ { - k{h_z},\;\; - i\frac{{\partial {h_z}}}{{\partial x}},\;\;i\left( {\frac{{\partial {h_y}}}{{\partial x}} - ik{h_x}} \right)} \right]. $$

Equations (9) and (10) express the electromagnetic fields of GSP in the AF. However, its electromagnetic fields are directly given above the surface and they are

(11)$${\bf{h^{\prime}}} = ( - ik/\varGamma ^{\prime},\,\;1,\;\,2i{R_z}){H^{\prime}_y}\exp (2\pi \varGamma ^{\prime}x)\;, $$

and

(12)$${\bf{e^{\prime}}} = ( - 2ik{R_z},\,\,2\varGamma ^{\prime}{R_z},\,\, - i{f^2}/\varGamma ^{\prime}){H^{\prime}_y}\delta ^{\prime}\exp (2\pi \varGamma ^{\prime}x), $$

with $\delta ^{\prime} = {\varepsilon _a}\delta $. The dispersion equation of GSP is

(13)$${\mathop{\rm Im}\nolimits} \{ \eta ({\varepsilon _a}\varGamma ^{\prime} + \alpha + i\beta )[{\varepsilon _a}{f^2} + \varGamma ^{\prime}(k{\lambda _ - } - \alpha + i\beta )]\} = 0. $$

The GSP moves along the y-axis since its fields have a common factor ${\rm{exp}}[{i({2\pi {k_y}y - \omega t} )} ]$. We realize that the main features of the GSP as follows. (i) There are very simple phase relations among its electromagnetic-field components. (ii) Its electromagnetic fields oscillate and attenuate with the distance from the surface. (iii) no GSP exists when there is no external magnetic field.

Subsequently, we discuss the changes of GSP from the inversion of H₀ or k. For the transformation of ${H_0} \to - {H_0}$, dispersion Eq. (13) is invariant as the inversion varies only the sign of $\eta $, and meanwhile ${h_z},{e_x}\;{\rm{and}\;}{e_y}$ are changed in sign. It proves that dispersion property is invariant, but the polarization is changed under the inversion of external magnetic field. For the transformation of $k \to - k$, the dispersion equation also is invariant since the transformation changes only the sign of $\lambda $, but ${h_x}$ and ${e_x}$ are changed in sign. Therefore, the dispersion equation is invariant but the polarization is changed for the inversion of k.

A circularly-polarized light with electric field $e = \sqrt 2 ({\hat x + i\hat y} )/2$ possesses a spin angular momentum (SAM) $\hbar $ along the propagation direction. It has been known that a conventional surface polariton or evanescent wave carry a transverse SAM that is of k-locking [32–33], and the SAM, wave-vector and direction of attenuation form a universal right-handed triplet. Due to the localization of surface polariton, its SAM may be larger at the material surface. An exception is the SAM of Dyakonov surface polariton at the surface of an anisotropic material, i.e., the SAM is not perpendicular to its propagation direction, where one component is transverse and the other is longitudinal [34]. The GSP is a unique surface polariton, so its SAM may be unusual. The (SAM) of an electromagnetic wave is defined with [35]

(14)$${\bf{S}} = \hbar {\mathop{\rm Im}\nolimits} ({{\bf{e}}^\ast } \times {\bf{e}} + {{\bf{h}}^\ast } \times {\bf{h}}), $$

where $\hslash$ is the reduced Planck constant. Its first part originates from the polarization of electric field and the second part results from the polarization of magnetic field. Generally speaking, the electric part is much larger in amplitude than the magnetic part, and the difference is about $\delta ^{\prime 2}$ times in the air or vacuum space. We have known that a conventional surface polariton with TM polariton carries only the electric part but one with TE polarization possesses only the magnetic part. In addition, the first part acts on electric dipole moments, but the second part produces torque on magnetic dipole moments in materials. According to the simple phase relations among the electromagnetic-field components, we find that S has two components. One is the in-plane component that lies in the propagation plane and is perpendicular to the surface, expressed by

(15)$${S_x} = 2\hbar {\mathop{\rm Im}\nolimits} (e_y^\ast {e_z} + h_y^\ast {h_z}) = S_x^e + S_x^h. $$

The other is the out-plane component, normal to the propagation plane, indicated with

(16)$${S_z} = 2\hbar {\mathop{\rm Im}\nolimits} (e_x^\ast {e_y} + h_x^\ast {h_y}) = S_z^e + S_z^h. $$

It is interesting to discuss the SAM nonreciprocity of GSP according to the phase relations among the field components. Firstly, the inversion of external field changes the signs of ${h_z}$, ${e_x}\;$ and ${e_y}$, and meanwhile those of ${h_x}$, ${h_y}$ and ${e_z}$ are invariant, so the in-plane component S_x is changed in sign for the inversion of external field and the out-plane component S_z is invariant. It shows that the in-plane component is locked to the external magnetic field. Secondly, the inversion of wavenumber vector ($k \to - k$) varies only the sign of ${h_x}$ and ${e_x}$, so the out-plane component is changed in sign but the in-plane component is not changed. It proves that the out-plane component is locked to the wavevector or is of k-locking. In Ref. 28, the SAM is determined by the polarization parameters of the two coherent waves and the angle between their propagation directions, but the SAM of the GSP is fixed by the GSP state and is inherent, but it is magnetically tunable since this polariton is controlled by the external magnetic field [27].

3. Numerical calculation and results

Numerical calculations are based on the antiferromagnetic FeF₂ with the physical parameters given previously. It is necessary that the dispersion curves of GSP should be first offered. For various values of external magnetic field, Fig. 2(a) illustrates the dispersion curves of GSP and all these dispersion curves are finite. We see that the GSP is situated in the region of $f < {\omega _r} = 52.877\;{\rm{c}}{{\rm{m}}^{ - 1}}$ and $k > 150\;{\rm{c}}{{\rm{m}}^{ - 1}}$ and is magnetic-field-tunable. Attenuating constant α and oscillating constant $\beta $ are very important to characterize the GSP. α shows the localization at the surface and β determines the oscillating behavior of GSP. Figure 2(b) shows the two constants.

Fig. 2. (a) Dispersion curves of GSP for various values of ${H_0}$. (b) The attenuating and oscillating constants as functions of wave number, where the solid curves show α and the dotted curves represent β.

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The oscillating phenomenon is obviously visible only if β is larger than α, so this phenomenon is weaker for a larger value of ${H_0}$. Because the GSP is defined in the case of $\alpha > 0$ and $\beta > 0$, Fig. 2(b) reflects that every dispersion curve is terminated by the condition of $\beta > 0$ so that its left and right endpoints both correspond to β$= 0$.

The electromagnetic fields of GSP not only decay but also oscillate with the distance away from the surface. In order to intuitionally reflect this feature, we offer Fig. 3. We see that the electromagnetic fields are highly localized at the surface and they obviously oscillate with the distance from the surface. Figure 3 also demonstrates the phase differences among the electric-field or magnetic-field components in the AF. The electric field is larger in amplitude than the magnetic field by about 300 times. Although the electromagnetic boundary conditions demand only the continuity of tangential components of electromagnetic fields, the normal component of magnetic field (${h_x}$) is continuous sill due to the easy axis vertical to the surface. It proves that the magnetic part of SAM should be continuous at the surface.

Fig. 3. The distributions of GSP electromagnetic fields along the x-axis, where $x = 0$ shows the position of the surface and the curves correspond to a specific point on the black curve in Fig. 2(a) and $i = \sqrt { - 1} $. (a) For the electric field and (b) for the magnetic field.

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The SAM of GSP can produce magnetic and electric torques, interacts with electric dipole moments and with magnetic dipole moments in materials, respectively. Therefore, they can be used to manipulate different micron and nano particles. In the following numerical calculations for the SAM of GSP, we select three points on dispersion curves, i.e., the two endpoints and one middle point.

Figure 4(a) shows the electric part of SAM. We see a very large SAM near the surface, especially on the surface in the vacuum or air, which results from the high localization of electromagnetic fields at the surface and the interference of two branch-waves in the AF. The evident oscillatory phenomenon is seen in the AF and rapidly decays since the field amplitudes rapidly declines. The out-plane component of SAM is not continuous at the surface and the great difference is found, e.g., the difference is about 6.2${\times} {10^5}$ as shown by the black-dot curve, corresponding to the left endpoint of the dispersion curve. The in-plane component is continuous, as shown by the solid curves. The oscillatory behavior originates from the interference of two branch-waves in the AF. Very large peak- and dip-values appear inside the AF, except large values at the surface. In the space above the surface, the in-plane component, k and the out-plane component form the left-handed triplet. The picture is different in the AF, where the components of the SAM either positive or negative, related to the distance x. Figure 4(b) shows the magnetic part of SAM. In amplitude, it is much smaller than the electric part. Its two components both are continuous at the surface since the three components of the magnetic field are continuous in our geometry. Above the surface, both the two components are negative, so the two components and k bult a right-handed triplet. The picture is different in the AF. Either component has one obvious peak-value or dip-value in the AF. $S_x^h$ and $S_z^h$ are either positive or negative in the AF, which are completely different from those above the surface. Similar to Fig. 4(a), there is an oscillatory behavior here and we can clearly see it only if the relevant ${S^h} - x$ region is enlarged.

Fig. 4. The SAM distribution of GSP at three specific points on the black curve in Fig. 2(a). (a) illustrates the electric part of the SAM, where the number above the arrow is the value of $S_z^e$ on the surface. (b) indicates its magnetic part. $x = 0$ represents the position of the surface. The SAM is measured in $\hbar {|{H_y^{\prime}} |^2}$ and electromagnetic fields is measured in $H_y^{\prime}$.

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The external magnetic field ${H_0}$ is important for the existence and dispersion properties of the GSP. We use Fig. 5 to illustrate the distributions of the two SAM-components for ${H_0} = 2.0\;{\rm{kG}}$. Figure 5(a) is qualitatively similar to Fig. 4(a), but the quantitative difference is evident. Figure 5(b) is quantitatively different from Fig. 4(b). The two components have the peak or dip value in the AF. Obviously, the increase of external magnetic field leads to the obvious decrease of SAM.

Fig. 5. The SAM distribution of GSP at three specific points on the green curve in Fig. 2(a) for ${H_0} = 2.0\;{\rm{kG}}$. (a) illustrates the electric part of the SAM, where the number above the arrow is the value of $S_z^e$ on the surface. (b) indicates its magnetic part. $x = 0$ represents the position of the surface. The SAM is measured in $\hbar {|{H_y^{\prime}} |^2}$ and electromagnetic fields is measured in $H_y^{\prime}$.

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4. Conclusions

We have discussed the SAM of the GSP in the normal geometry where the AF easy axis and external magnetic field both are normal to the AF surface. The SAM consists of two components vertical to the propagation direction. The out-plane component is normal to the propagation plane and the in-plane component lies in the propagation plane. We find that the inversion of external magnetic field leads to the sign-change of the in-plane component, but the out-plane component is invariant. The inversion of wavevector (k) results in that the out-plane component is changed in sign but the in-plane one is invariant. Therefore, the in-plane component is locked to the external field but the out-plane component is locked to the wavevector. It is very interesting that the SAM is very large above the AF surface, especially on the surface. Its two components monotonously and exponentially decay above the surface with the distance from the surface. Either SAM component contains two parts, or the electric part and the magnetic part, which can interact with electric dipole moments and magnetic dipole moments in media, respectively. The two parts are numerically calculated separately. The results shows that the electric part is not continuous at the surface but the magnetic part is continuous at the surface. They are highly localized near or at the surface, especially the electric part. The electric part is much larger in amplitude than the magnetic part. The out-plane component is positive but the in-plane one is negative above the surface, or say that $S_x^e$, k and $S_z^e$ form the left-handed triplet above the surface [see Figs. 4(a) and 5(a)]. For the magnetic part above the surface, the right-handed triplet is formed in Fig. 4(b) and Fig. 5(b). It is very useful for one to manipulate micron or nano particles with dipole moments on or above the AF surface. Inside the AF, the two parts oscillate and exponentially attenuate with distance away from the surface. The oscillatory behavior is more obvious near the surface, and the two parts are either positive or negative, depending on the distance.

Funding

Natural Science Foundation of Heilongjiang Province (ZD2009103).

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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Spin angular momentum and nonreciprocity of ghost surface polariton in antiferromagnets

Abstract

1. Introduction

2. Analytical results and discussion

3. Numerical calculation and results

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (20)

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