Spontaneous parity-time-symmetry breaking in moving media

Optical instabilities in moving media are linked to a spontaneous parity-time-symmetry breaking of the system. It is shown that in general the time evolution of the electromagnetic waves in moving media is determined by a non-Hermitian parity-time-symmetric operator. For lossless systems the frequency spectrum of the time evolution operator may be complex valued, and has a mirror symmetry with respect to the real-frequency axis. The possibility of optical amplification of a light pulse in the broken parity-time-symmetry regime is demonstrated.

Figure 1

Geometry of the system: two infinitely extended (in the $x$ and $y$ directions) material slabs separated by an air gap with thickness $d$ are in relative motion.

Figure 2

Top row: Dispersion diagrams for $p$-polarized waves. Bottom row: Dispersion diagrams for $s$-polarized waves. First column: Dispersion diagram of the guided modes for a single material slab calculated in the comoving frame. Second and third columns: Dispersion ${\omega }^{\prime }+i{\omega }^{\prime \prime }$ vs $k_x$ in the lab frame for $k_y=0$. In these plots the solid lines represent the nonrelativistic calculation and the dashed lines the exact relativistic calculation. The distance between the slabs is $d=0.75h_s$ and $v_1=0$, $v_2=c/5$, and $n_1=n_2=14$. The normalization factor for the transverse wave vector is $k_{x0}=1.60/h_s$ for $p$-polarized waves, and $k_{x0}=2.41/h_s$ for $s$-polarized waves.

Figure 3

Top row (unities $\left[\mathrm{V}/\mathrm{m}\right]$): electromagnetic fields distribution for a $p$-polarized guided mode associated with a system instability ($\omega h_s/c=0.1580+i0.00018$ and $k_x{h}_s=1.56$) and for the same configuration as in Fig. 2. The field amplitudes are normalized such that $\langle \tilde{\mathbf{F}}|\mathbf{F}\rangle ={\varepsilon }_0{A}_0{h}_s\times \left[1{\mathrm{V}}^2/{\mathrm{m}}^2\right]$. Green solid lines: absolute value; black dashed lines: real part; blue dot-dashed lines: imaginary part. Bottom row (unities $\left[{\mathrm{V}}^2/{\mathrm{m}}^2\right]$): profiles of the normalized wave momentum density, electromagnetic momentum density, and wave energy density.

Figure 4

Top row (unities $\left[\mathrm{V}/\mathrm{m}\right]$): electromagnetic fields profile for a $s$-polarized guided mode associated with a system instability ($\omega h_s/c=0.235+0.007i$ and $k_x{h}_s=2.33$) and for the same configuration as in Fig. 2. The field amplitudes are normalized such that $\langle \tilde{\mathbf{F}}|\mathbf{F}\rangle ={\varepsilon }_0{A}_0{h}_s\times \left[1{\mathrm{V}}^2/{\mathrm{m}}^2\right]$. Green solid lines: absolute value; black dashed lines: real part; blue dot-dashed lines: imaginary part. Bottom row (unities $\left[{\mathrm{V}}^2/{\mathrm{m}}^2\right]$): profiles of the normalized wave momentum density, electromagnetic momentum density, and wave energy density.

Figure 5

(a) Normalized wave momentum (in unities $\left[{\mathrm{V}}^2/{\mathrm{m}}^2\right]$) as a function of the normalized velocity of slab 1, $v_1$, in the lab frame. The velocity of slab 2 in the frame of slab 1 is kept invariant and equal to $v_{\Delta }=c/5$. The structural parameters are as in Fig. 2 and the fields are normalized as $\langle \tilde{\mathbf{F}}|\mathbf{F}\rangle ={\varepsilon }_0{A}_0{h}_s\times \left[1{\mathrm{V}}^2/{\mathrm{m}}^2\right]$. The wave is $p$ polarized. Green solid line: wave momentum in slab 1; blue dashed line: wave momentum in slab 2; black dot-dashed line: analytical formula [Eq. (21)]. (b) Normalized force acting on slab 1 as a function of the normalized velocity of slab 1 in the lab frame with $v_{\Delta }=c/5$. The fields are normalized as $\langle \tilde{\mathbf{F}}|\mathbf{F}\rangle =\hslash \left|\omega \right|/2$. Green solid line: $p$ polarization; blue dashed line: $s$ polarization.

Figure 6

(a) A line source in between two moving dielectric slabs is excited by a current pulse. (b) The current pulse used in the numerical simulations.

Figure 7

Time evolution of the field ($E_y$) radiated by a current pulse with finite duration placed at the origin. The distance between the slabs is $d=0.75h_s$, $v_1=0$, and $n_1=n_2=14$. The field $E_y$ is calculated at the interface with the second slab and for (a) $x=-10h_s$, (b) $x=h_s$, (c) $x=10h_s$, and (d) $x=19h_s$. The results of the first row are for the case (i) $v_2=c/5$ whereas the results of the second row are for (ii) $v_2=0$. The development of a field instability due to the spontaneous parity-time-symmetry breaking is evident in case (i).

Figure 8

Two silver semispaces ($h_1=h_2\to \infty$) separated by a vacuum gap with thickness $d=10\mathrm{nm}$ [${\omega }_{sp}d/c=0.135$] are in relative motion. (a) Real part and (b) imaginary part of oscillation frequencies of two natural modes (black and green lines) as a function of the normalized $k_x$ for $\Gamma =0$, and $v_2=2{\omega }_{sp}d$. The transverse wave number is normalized to $k_{x0}=2{\omega }_{sp}/v_2$. In all the plots, $k_y=0$, and the solid lines represent the nonrelativistic calculation and the dashed lines the exact relativistic calculation. (c) and (d) Similar to (b) but for $\Gamma =0.1{\omega }_{sp}$ and $\Gamma =0.5{\omega }_{sp}$, respectively.

Figure 9

Similar to Fig. 8, but ${\omega }^{\prime \prime }$ is calculated as a function of $v_2$ for $k_x=2{\omega }_{sp}/v_2$, $\Gamma =0$, and $k_y=0$. The calculation is nonrelativistic and only the mode with ${\omega }^{\prime \prime }>0$ is represented.