Molecular dynamics simulations of coherent optical photon emission from shock waves in crystals

We have previously predicted that coherent electromagnetic radiation in the $1–100\mathrm{THz}$ frequency range can be generated in crystalline polarizable materials when subject to a shock wave or solitonlike propagating excitation [E. J. Reed et al., Phys. Rev. Lett. 96, 013904 (2006)]. In this work, we present analysis and molecular dynamics simulations of shock waves in crystalline NaCl which expand upon this prediction. We demonstrate that the coherent polarization currents responsible for the effect are generated by a nonresonant, nonlinear effect that occurs at the shock front. We consider the effect of thermal noise and various shock pressures on the coherent polarization currents and find that the amplitude generally increases with increasing shock pressure and decreasing material temperature. Finally, we present calculations of the amplitude and distribution of emitted radiation showing that the radiation can be directed or undirected under various realistic conditions of the shape of the shock front.

Figure 1

(Color online) Schematic depicting a fashion in which a temporally periodic polarization can arise in a shocked polarizable crystal. Red and blue atoms on the left represent positively and negatively charged atoms. Atom positions within the shocked crystal are shown at four instances in time on the left with a gray area highlighting the shock front region. As the shock propagates over each lattice plane (left to right), the symmetry of atoms in that lattice plane can be temporarily broken yielding a nonzero static polarization of the atoms at the shock front. This process yields a temporally periodic polarization current which can potentially emit coherent radiation.

Figure 2

(Color online) Material speed as a function of position in the shock propagation direction in a NaCl molecular dynamics simulation of a shock propagating in the [100] direction. A “piston” exists on the left side of the computational cell consisting of atoms with infinite mass fixed in the crystal structure (shown at top). The piston is given a velocity of $200\mathrm{m}∕\mathrm{s}$ to the right which drives a shock to the right. The dimensions of the computational cell shown at the top are reduced from those employed in this work for clarity.

Figure 3

(Color online) Fourier transform of the electric polarization surface current component in the shock propagation direction for molecular dynamics simulations of a shock propagating through NaCl in the [111] direction (left) and [100] direction (right). Narrow bandwidth, coherent peaks exist in the shocked simulations (black) that do not exist in the simulations without shocks. Gray arrows are emission frequencies predicted by Eq. (3). Thermal noise gives rise to an incoherent background.

Figure 4

(Color online) Electric polarization surface current component in the shock propagation direction for molecular dynamics simulations of a shock propagating through NaCl in the [100] direction with an initial temperature of $4.2\mathrm{K}$. The shocked simulation (black) contains a number of excitations that are not present in the unshocked simulation (red). These include the longitudinal-optical (LO) phonons, harmonics of lower-frequency phonons, and the higher-frequency narrow band excitations discussed in this work. Transverse-optical (TO) phonons are also visible in both the shocked and unshocked simulations.

Figure 5

(Color online) Electric polarization surface current component in the shock propagation direction for molecular dynamics simulations of a shock propagating through NaCl in the [100] direction with an initial temperature of $4.2\mathrm{K}$. The left plot shows that the $22\mathrm{THz}$ signal (nonresonant) polarization current occurs for the duration of the shock propagation with roughly constant amplitude. On the right is a plot of frequency as a function of position in the computational cell in the shock propagation direction. During the time over which this Fourier transform is taken, the shock front propagates between the white dotted lines. This plot shows that the $22\mathrm{THz}$ current occurs in the region where the shock front is located rather than behind it or in front of it.

Figure 6

Comparison of electric polarization surface current component in the shock propagation direction for molecular dynamics simulation shocks propagating through a NaCl model in the [100] direction with an initial temperature of $4.2\mathrm{K}$. The shocks were generated using piston speeds of 200, 600, and $1000\mathrm{m}∕\mathrm{s}$. The 600 and $1000\mathrm{m}∕\mathrm{s}$ curves are shifted up by factors of 100 and 10000, respectively. Gray arrows are emission frequencies predicted by Eq. (3). As the piston speeds increase, the peaks broaden (becoming less coherent) and increase in amplitude. The broadening is due in part to unsteady shock speeds.

Figure 7

(Color online) Electric polarization surface current component in the shock propagation direction for molecular dynamics simulations of a shock propagating through NaCl in the [100] direction with a piston speed of $200\mathrm{m}∕\mathrm{s}$ for various temperatures. The coherent peak amplitude at $22\mathrm{THz}$ decreases with increasing temperature and the background increases with increasing temperature.

Figure 8

Comparison of electric polarization surface current component in the shock propagation direction for shocks propagating through a NaCl in the [100] direction with an initial temperature of $300\mathrm{K}$. The shocks were generated using piston speeds of 200, 600, and $1000\mathrm{m}∕\mathrm{s}$. The 600 and $1000\mathrm{m}∕\mathrm{s}$ curves are multiplied by factors of 100 and 10 000, respectively, for clarity. This plot is equivalent to Fig. 6 but with a $300\mathrm{K}$ initial temperature. Gray arrows are emission frequencies predicted by Eq. (3). Coherent peaks are not observed with a $200\mathrm{m}∕\mathrm{s}$ piston speed, but they are observed for higher piston speeds.

Figure 9

(Color online) Comparison of Fourier transforms of the electric polarization surface current component in the shock propagation direction for shocks propagating through a NaCl model in the [100] direction with initial temperatures of 4.2 and $300\mathrm{K}$. The shocks were generated using piston speeds of $1000\mathrm{m}∕\mathrm{s}$. The peak height above the incoherent background is lower and the bandwidth is narrower in the $300\mathrm{K}$ case.

Figure 10

Schematic depicting variations of shock front position $\left[z\left(r\right)\right]$ superimposed on periodic units of the crystal (e.g., lattice planes). The phase of the polarization current $j\left({\vec{r}}^{\prime }\right)=j_0{e}^{i\omega [z\left({\vec{r}}^{\prime }\right)∕v_s]}$ depends very sensitively on the flatness of the shock front on the length scale of $2\pi v_s∕\omega$ which is $0.282\mathrm{nm}$ in the case of the $22\mathrm{THz}$ peak observed in NaCl shocked along the [100] direction. Variations in phase across the surface of the shock front determine, in part, the emission characteristics through Eq. (6).

Figure 11

(Color online) Electric-field magnitude at $22\mathrm{THz}$ from a $100\times 100\mu {\mathrm{m}}^2$ square shock front for $j_0=0.18\mathrm{C}∕\mathrm{m}\mathrm{s}$ for the $22\mathrm{THz}$ peak on the right side of Fig. 3 and a curved shock front based on the experimentally observed shock front in Ref. 17.

Figure 12

(Color online) Electric-field magnitude at $22\mathrm{THz}$ from a $1\times 1{\mathrm{mm}}^2$ square shock front for $j_0=0.18\mathrm{C}∕\mathrm{m}\mathrm{s}$ for the $22\mathrm{THz}$ peak on the right side of Fig. 3 and a curved, tilted shock front. Emission of directed radiation can be achieved for this sufficiently planar shock wave.

Figure 13

(Color online) Electric-field magnitude at $22\mathrm{THz}$ from a $100\times 100\mu {\mathrm{m}}^2$ square shock front for $j_0=0.18\mathrm{C}∕\mathrm{m}\mathrm{s}$ for the $22\mathrm{THz}$ peak on the right side of Fig. 3 and a shock front with random spatial fluctuations in the phase of the polarization current with $k_{\max }=2\pi ∕\left(5\mu \mathrm{m}\right)$. The radiation is not directed but has nodes in the direction of shock propagation.

Figure 14

(Color online) Angular dependence of the electric-field magnitude $1\mathrm{mm}$ away from a $100\times 100\mu {\mathrm{m}}^2$ square shock front for $j_0=0.18\mathrm{C}∕\mathrm{m}\mathrm{s}$ for the $22\mathrm{THz}$ peak on the right side of Fig. 3. Random spatial variations in the phase of the polarization current are considered with various values of $k_{\max }$. As the deviation from planarity of the shock becomes more pronounced (i.e., $2\pi ∕k_{\max }$ decreases), the emission field amplitudes decrease.