Vortices Formed by Two Counter-Propagating Waves for Surface Phonon Polaritons with Material Losses

Abstract

We examine the surface phonon polaritons established on resonance between lossy silicon carbide and vacuum over the Reststrahlen frequency range. To this goal, we investigate two waves counter-propagating along a planar material interface in terms of the orbital and spin parts of the Poynting vector. Vortices of the spin part are thus found to be generated near the collision point of the two waves. In addition, vortex pairs of opposite circulations are identified near the material interface. The locality of the electric permittivity plays an essential role in enabling our simple analysis by neglecting the spatial propagations of phonons while keeping the material damping. The resulting dynamics of the total displacement field turns out to be loss-free in the middle of the lossy dynamics of the other field variables, thus exhibiting a superfluid-like feature in an otherwise normal fluid.

Appendices

Appendix A: Uncoupled Elastic Plane Strain Problem in Half-Space Solid

For the uncoupled elasticity problem with $g \equiv \sqrt {\varepsilon_{\infty } \left( {\omega_{LO}^{2} - \omega_{TO}^{2} } \right)} = 0$, $\omega_{LO} = \omega_{TO}$. Hence, Eq. (6) is reduced to the following pair.

$$\left\{ \begin{gathered} \left[ {\omega_{TO}^{2} - \omega \left( {\omega + {{\text{i}}}\gamma } \right) + \left( {\lambda + 2\mu } \right)\nabla^{2} } \right]Y = 0 \hfill \\ \left[ {\omega_{TO}^{2} - \omega \left( {\omega + {{\text{i}}}\gamma } \right) + \mu \nabla^{2} } \right]\Theta_{y} = 0 \hfill \\ \end{gathered} \right.,\begin{array}{*{20}c} {} \\ \end{array} \left\{ \begin{gathered} Y \equiv \left( {\nabla \cdot \vec{u}} \right) \hfill \\ \Theta \equiv \left( {\nabla \times \vec{u}} \right) \hfill \\ \end{gathered} \right.$$

(17)

Here, $\left\{ {\lambda ,\mu_{sh} } \right\}$ are the dimensionless Lamé parameters defined by Eq. (5). Assuming evanescent waves in the depth-wise $z$-direction, all field variables are proportional to, say, $\exp \left( { - \kappa_{L} z} \right)$ over $z \ge 0$ in silicon carbide. Besides, $\kappa_{L}$ is defined below. The solutions to this equation are straightforward although they require careful treatments in many steps. Let us list below only the final formulae for the sake of saving space.

$$\begin{gathered} \left\{ \begin{gathered} \kappa_{T} = \sqrt {k^{2} - \frac{{\omega_{TO}^{2} - \omega \left( {\omega + {{\text{i}}}\gamma } \right)}}{{\mu_{sh} }}} \hfill \\ \kappa_{L} = \sqrt {k^{2} - \frac{{\omega_{TO}^{2} - \omega \left( {\omega + {{\text{i}}}\gamma } \right)}}{{\lambda + 2\mu_{sh} }}} \hfill \\ \end{gathered} \right.,\begin{array}{*{20}c} {} \\ \end{array} \vec{V}_{{{\text{fun}}}} \equiv \left\{ {\begin{array}{*{20}c} {u_{H0} {\text{e}}^{ - kz} } \\ {\left( {\kappa_{T}^{2} - k^{2} } \right)^{ - 1} \Theta_{y,0} {\text{e}}^{{ - \kappa_{T} z}} } \\ {\left( {\kappa_{L}^{2} - k^{2} } \right)^{ - 1} Y_{0} {\text{e}}^{{ - \kappa_{L} z}} } \\ \end{array} } \right\} \hfill \\ \tilde{T}_{{\text{int}}} \equiv \left( {\begin{array}{*{20}c} 1 & { - \kappa_{T} } & {{{\text{i}}}k} \\ {{\text{i}}} & { - {{\text{i}}}k} & { - \kappa_{L} } \\ \end{array} } \right),\begin{array}{*{20}c} {} \\ \end{array} \left\{ {\begin{array}{*{20}c} {u_{x} \left( z \right)} \\ {u_{z} \left( z \right)} \\ \end{array} } \right\} = \tilde{T}_{{\text{int}}} \vec{V}_{{{\text{fun}}}} \hfill \\ \end{gathered}$$

(18)

Here, $\Theta_{y,0} \equiv \left. {\Theta_{y} } \right|_{z = 0}$ and $Y_{0} \equiv \left. Y \right|_{z = 0}$ are the interface values evaluated at $z = 0 -$. Besides, $u_{H0}$ is an integration constant standing for homogeneous solutions. Furthermore, the three-vector $\vec{V}_{{{\text{fun}}}}$ represents the three independent fundamental solutions, whereas the three-by-two matrix $\tilde{T}_{{\text{int}}}$ stands for the interactions between photons and phonons.

Both the normal stress $\sigma_{zz}$ and the shear stress $\sigma_{xz}$ introduced in Eq. (16) vanish at the interface [8] so that $\left. {\sigma_{zz} } \right|_{z = 0} = \left. {\sigma_{xz} } \right|_{z = 0} = 0$. As an additional interface condition, a generic surface admittance $\Gamma {\text{e}}^{{{{\text{i}}}\beta }}$ is employed to give ${{\text{i}}}Y_{0} = \Gamma {\text{e}}^{{{{\text{i}}}\beta }} \Theta_{y,0}$, where $\Gamma$ and $\beta$ denote, respectively, the magnitude and the phase [3].

These three constraints are just enough to eliminate the three parameters of $\left\{ {u_{H0} ,\Theta_{y,0} ,Y_{0} } \right\}$ in Eq. (18) on resonance. Therefore, there arises a wave–dispersion relation for $k\left( \omega \right)$ that have been employed in the combined propagative phase factor $\exp \left( {{{\text{i}}}kx - {{\text{i}}}\omega t} \right)$.

$$\Gamma {\text{e}}^{{{{\text{i}}}\beta }} \left[ {\lambda \left( {\kappa_{T} + k} \right) + 2\mu \kappa_{L} } \right] + \mu \left( {\kappa_{T} - k} \right) = 0$$

(19)

We proved that there are no surface waves for the two extreme cases $\Gamma = \infty$ and $\Gamma = 0$ both with $\beta = 0$. A suitable parameter regime would certainly ensure the existence of propagating surface waves. Results from numerically solving Eq. (19) would be presented elsewhere.

Appendix B: Dynamics of Total Displacement Field in Silicon Carbide

Let us now find what kind of dynamics is satisfied by $\vec{D}^{{{\text{SiC}}}}$. To this goal, we make use of various formulae in Eq. (1) to proceed in the following way.

$$\begin{gathered} \nabla \times \vec{H}^{{{\text{SiC}}}} = - {{\text{i}}}\omega \vec{D}^{{{\text{SiC}}}} ,\begin{array}{*{20}c} {} \\ \end{array} \vec{D}^{{{\text{SiC}}}} \equiv \varepsilon_{\infty } \vec{E}^{{{\text{SiC}}}} + g\vec{u} \hfill \\ \nabla \cdot \vec{D}^{{{\text{SiC}}}} = 0\begin{array}{*{20}c} {} \\ \end{array} \Rightarrow \begin{array}{*{20}c} {} \\ \end{array} \nabla \cdot \vec{E}^{{{\text{SiC}}}} = - \left( {{g \mathord{\left/ {\vphantom {g {\varepsilon_{\infty } }}} \right. \kern-\nulldelimiterspace} {\varepsilon_{\infty } }}} \right)\nabla \cdot \vec{u} \hfill \\ \end{gathered}$$

(20)

Furthermore, consider $\nabla \times \left( {\nabla \times \vec{E}^{{{\text{SiC}}}} } \right)$ in two different ways.

$$\left\{ \begin{gathered} \nabla \times \left( {\nabla \times \vec{E}^{{{\text{SiC}}}} } \right) = {{\text{i}}}\omega \nabla \times \vec{H}^{{{\text{SiC}}}} = {{\text{i}}}\omega \left( { - {{\text{i}}}\omega \vec{D}^{{{\text{SiC}}}} } \right) = \omega^{2} \vec{D}^{{{\text{SiC}}}} \hfill \\ \nabla \times \left( {\nabla \times \vec{E}^{{{\text{SiC}}}} } \right) = \nabla \left( {\nabla \cdot \vec{E}^{{{\text{SiC}}}} } \right) - \nabla^{2} \vec{E}^{{{\text{SiC}}}} = - \left( {{g \mathord{\left/ {\vphantom {g {\varepsilon_{\infty } }}} \right. \kern-\nulldelimiterspace} {\varepsilon_{\infty } }}} \right)\nabla^{2} \vec{u} - \nabla^{2} \vec{E}^{{{\text{SiC}}}} \hfill \\ \end{gathered} \right.$$

(21)

By equating the two expressions, we obtain the following.

$$\begin{gathered} - \left( {{\alpha \mathord{\left/ {\vphantom {\alpha {\varepsilon_{\infty } }}} \right. \kern-\nulldelimiterspace} {\varepsilon_{\infty } }}} \right)\nabla^{2} \vec{u} - \nabla^{2} \vec{E}^{{{\text{SiC}}}} = \omega^{2} \vec{D}^{{{\text{SiC}}}} = \omega^{2} \left( {\varepsilon_{\infty } \vec{E}^{{{\text{SiC}}}} + \alpha \vec{u}} \right) \hfill \\ \Rightarrow \begin{array}{*{20}c} {} \\ \end{array} \left( {\varepsilon_{\infty } \omega^{2} + \nabla^{2} } \right)\vec{D}^{{{\text{SiC}}}} = \vec{0} \hfill \\ \end{gathered}$$

(22)

Let us find the governing equation for $\vec{E}^{\alpha }$ in the case of incompressible liquids. In fact, this can be achieved for both the dielectric and silicon carbide on equal footing by the use of Eq. (1). It is only necessary to implement $\vec{D}^{D} = \varepsilon^{D} \vec{E}^{D}$ and $\vec{D}^{{{\text{SiC}}}} = \varepsilon^{{{\text{SiC}}}} \left( \omega \right)\vec{E}^{{{\text{SiC}}}}$. Hence,

$$\begin{gathered} \nabla \times \vec{H}^{\alpha } = - {{\text{i}}}\omega \vec{D}^{\alpha } \begin{array}{*{20}c} {} \\ \end{array} \Rightarrow \begin{array}{*{20}c} {} \\ \end{array} \nabla \times \left( {\nabla \times \vec{H}^{\alpha } } \right) = - {{\text{i}}}\omega \nabla \times \vec{D}^{\alpha } \hfill \\ \begin{array}{*{20}c} {} \\ \end{array} \Rightarrow \begin{array}{*{20}c} {} \\ \end{array} \nabla \left( {\nabla \cdot \vec{H}^{\alpha } } \right) - \nabla^{2} \vec{H}^{\alpha } = - {{\text{i}}}\omega \nabla \times \vec{D}^{\alpha } = - {{\text{i}}}\omega \nabla \times \left( {\varepsilon^{\alpha } \vec{E}^{\alpha } } \right) \hfill \\ \nabla \cdot \vec{H}^{\alpha } = 0\begin{array}{*{20}c} {} \\ \end{array} \Rightarrow \begin{array}{*{20}c} {} \\ \end{array} \nabla^{2} \vec{H}^{\alpha } = {{\text{i}}}\omega \varepsilon^{\alpha } \nabla \times \vec{E}^{\alpha } \hfill \\ \nabla \times \vec{E}^{\alpha } = {{\text{i}}}\omega \vec{H}^{\alpha } \begin{array}{*{20}c} {} \\ \end{array} \Rightarrow \begin{array}{*{20}c} {} \\ \end{array} \nabla^{2} \vec{H}^{\alpha } = {{\text{i}}}\omega \varepsilon^{\alpha } \left( {{{\text{i}}}\omega \vec{H}^{\alpha } } \right) = - \varepsilon^{\alpha } \omega^{2} \vec{H}^{\alpha } \hfill \\ \begin{array}{*{20}c} {} \\ \end{array} \Rightarrow \begin{array}{*{20}c} {} \\ \end{array} \left( {\varepsilon^{\alpha } \omega^{2} + \nabla^{2} } \right)\vec{H}^{\alpha } = \vec{0} \hfill \\ \begin{array}{*{20}c} {} \\ \end{array} \Rightarrow \begin{array}{*{20}c} {} \\ \end{array} \left( {\varepsilon^{\alpha } \omega^{2} + \nabla^{2} } \right)\vec{E}^{\alpha } = \vec{0},\begin{array}{*{20}c} {} \\ \end{array} \left( {\varepsilon^{\alpha } \omega^{2} + \nabla^{2} } \right)\vec{u}^{\alpha } = \vec{0} \hfill \\ \end{gathered}$$

(23)

Appendix C: Photon–Plasmon Polaritons

The formulae for the photon plasmon polaritons can be easily recovered by setting $\omega_{TO} = 0$ and $v_{L} = v_{T} = 0$ in the appropriate formulae of Eqs. (6)-(8). In consultation of the Lorentz–Drude model in Eq. (102) of [2], we can set the longitudinal optical frequency $\omega_{LO}$ to the plasma frequency $\omega_{pl}$ of free electrons, i.e., $\omega_{LO} \equiv \omega_{pl}$. The coupling constant in Eq. (2) is reduced then to $g = \sqrt {\varepsilon_{\infty } } \omega_{pl}$ for $\omega_{TO} = 0$. Both Eqs. (2) and (8) give rise thus to the effective electric permittivity $\varepsilon_{\infty } \left\{ {1 - \left[ {\omega \left( {\omega + {{\text{i}}}\gamma } \right)} \right]^{ - 1} \omega_{LO}^{2} } \right\}$, thus recovering the Drude model for lossy and local plasmas [3,4,5]. In addition, $\vec{u} = - \left[ {\omega \left( {\omega + {{\text{i}}}\gamma } \right)} \right]^{ - 1} \sqrt {\varepsilon_{\infty } } \omega_{pl} \vec{E}^{{{\text{SiC}}}}$ is easily deduced.

From physical viewpoints, $\omega_{TO} = 0$ corresponds to vanishing restoring force on bound electrons [1], which then feel free to move around to become free electron plasma. These free electrons swim around in an elevated background environment with $\varepsilon_{\infty } > \varepsilon^{D}$, which sets the Thomas–Fermi screening length on free electrons and the attendant spatial coherence length [3]. This photon–plasmon interaction has been the key ingredient of our recent publication [19]. As discussed in the last subject of Sect. 3, $- {{\text{i}}}\omega \vec{u}$ represents now the electron velocity for this phonon-plasmon interactions. Resultantly, $- {{\text{i}}}\omega g\vec{u} = - {{\text{i}}}\omega \sqrt {\varepsilon_{\infty } } \omega_{pl} \vec{u}$ serves then as a genuine electric current [24]. Moreover, the electron plasma fulfills the simple divergence-free $\nabla \cdot \vec{u} = 0$ [11], thus signifying the usual charge neutrality assumed for the electron plasma.

Appendix D: Coherence

Recall the Reststrahlen band summarized by the data in Eq. (4) and shown in Fig. 1b. In addition, let us consult the degree of polarization (DOP) defined by [1]. Their DOP refers to the degree of the two-component TM wave (as in this study) in comparison with a generic three-component EM wave of equal intensity. This DOP is plotted in Fig. 1 of [1] against a normalized distance measured from the planar SiC-D interface. In Fig. 1 of [1] presented are two curves, respectively, for $\overline{\lambda } = 11.36{\mu m}$ (within the Reststrahlen band) and $\overline{\lambda } = 9.1{\mu m}$ (outside the Reststrahlen). It is concluded there that the coherent degree of polarization is greater within the band than outside. Therefore, the TM waves within the band is well correlated especially along the longitudinal direction [2], except for the close sub-wavelength vicinity of the SiC-D interface as displayed in Fig. 2 of [1]. This coherence property found along the longitudinal direction assures us of the validity of the coherent combination of two colliding waves along the SiC-D interface. Such an increased coherence is closely linked to the evanescent confinement by the surface waves taking place within the Reststrahlen band [2].

The silicon carbide under this study serves as an energy sink, since the Poynting energy is absorbed from vacuum into silicon carbide, while a heated silicon carbide serves as an energy source wherefrom the thermal energy is emitted into vacuum [1,2,3, 7, 30]. A difference is that the loss-induced absorption in this study takes place on resonance, whilst the thermal emission is more likely to take place through non-equilibrium processes because of the requisite external heating. In this respect, the spatial correlations of a stationary thermal current are finite for $\varepsilon_{i}^{{{\text{SiC}}}} \ne 0$ [1].

Consider the Casimir configuration [21], where two silicon carbide parallel plates of differing temperatures and of semi-infinite extents are separated by a vacuum gap. This configuration is considered to include both thermal source and sink [3, 30]. In this connection, it is no surprise to find that the configuration of the sink of counter-propagating waves is entirely opposite to that of the source. In our study, two coherent waves counter-propagating along the material interface are colliding with each other. In case with a source [7], waves emanating from a certain point on the material interface are somehow separated out into two counter-propagating waves which creep along the material interface.

In both source and sink configurations, the imaginary part $\varepsilon_{i}^{{{\text{SiC}}}}$ is essential in establishing the energy transfer across the material interface [2]. Since the fluctuations as phonons within polarizable materials such as silicon carbide are governed by the Bose–Einstein statistical factor in addition to the zero-point level, they decrease with lower temperatures. Therefore, both the divergence-free condition $\nabla\cdot \vec{u} = 0$ and the locality discussed in Sect. 3 are more likely to hold true with decreasing temperatures. In this regard, $\nabla \cdot \vec{u} = 0$ imposes the harshest condition on the fluctuations in the mechanical displacement, viz., vanishing fluctuations.

As regards coherences, the thermal sources involve the fluctuations in the polarizable solids, whereas the sink problem under this study is accompanied by coherent light illumination on resonance. Resultantly, the incoherence of the illuminating light, say, of laser fields, is something to be avoided. Hence, the polarizable solid that absorbs the electromagnetic energy is less vulnerable to fluctuations for coherent light illuminations. After all, the coherence of the photon–phonon coupling stays robust by the depth-wise evanescence enabled by the frequency falling in the Reststrahlen band [2].

In the interactions between a particle and EM waves, any fluctuations possibly destroying a coherent observation of spinning motions would be decreased with lower temperatures [11, 14, 20]. See also [11] for the properties at low temperatures employed for numerical computations.