Effect of strong wakes on waves in two-dimensional plasma crystals

We study the effects of the particle-wake interactions on the dispersion and polarization of dust lattice wave modes in two-dimensional plasma crystals. Most notably, the wake-induced coupling between the modes causes the branches to “attract” each other, and their polarizations become elliptical. Upon the mode hybridization the major axes of the ellipses (remaining mutually orthogonal) rotate by ${45}^{\circ }$. To demonstrate the importance of the obtained results for experiments, we plot representative particle trajectories and spectral densities of the longitudinal and transverse waves. These characteristics reveal distinct fingerprints of the mixed polarization. Furthermore, we show that at strong coupling the hybrid mode is significantly shifted towards smaller wave numbers, away from the border of the first Brillouin zone (where the hybrid mode is localized for a weak coupling).

Figure 1

(a) Elementary hexagonal lattice cell with the frame of reference. (b) The reciprocal lattice in $\mathbf{k}$ space with the basis vectors $|{\mathbf{b}}_1|=|{\mathbf{b}}_2|=\frac{4\pi }{\sqrt{3}}{\Delta }^{-1}$. Due to the lattice symmetry, it is sufficient to consider the wave vectors $\mathbf{k}$ at $0^{\circ }\le \theta \le {30}^{\circ }$ and from within the first Brillouin zone (gray region enclosed by dashed lines), so that $\left|\mathbf{k}\right|\Delta \le \frac{2\pi }{\sqrt{3}}$ for $\theta =0^{\circ }$ and $\left|\mathbf{k}\right|\Delta \le \frac{4\pi }{3}$ for $\theta ={30}^{\circ }$. (c) Enclosed regions where the particle-wake interactions cause the in-plane and out-of-plane wave modes to merge and form the hybrid mode [see also Fig. 2]. The smaller (larger) region corresponds to the effective dipole moment of wake ${\tilde{d}}_{\mathrm{eff}}=0.7\left(0.71\right)$ [Eq. (1)], the screening parameter is $\kappa =1$, the normalized eigenfrequency of the vertical confinement is ${\Omega }_{\mathrm{conf}}=5.66$.

Figure 2

Effect of the wake-induced coupling on the wave modes. (a)–(c) The colored lines show the (undamped) dispersion relations of the lower and upper modes ${\Omega }_{\mathrm{lo},\mathrm{up}}\left(k\right)$ [Eq. (5)], the dash-dotted lines represent the respective wake-free results ${\Omega }_{h,v}\left(k\right)$ [Eq. (A2) with $\tilde{q}=0$]. The curves are obtained for three different values of the vertical confinement frequency ${\Omega }_{\mathrm{conf}}$. In (a) the modes are attracted to each other, in (b) the gap between the modes is getting constricted, and in (c) they merge and form the hybrid mode (only the real part is shown). The color-coding shows the circularity of the mode polarization $\epsilon$, which is determined by the reduced coupling parameter [Eqs. (10) and (14) for the separate and hybrid modes, respectively]. (d) The coupling parameter $p\left(k\right)$ [Eq. (6)] for panels (a)–(c), the modes merge at $p=1$. The results are for $\kappa =1$, the dimensionless wake charge $\tilde{q}=0.7$, the wake length $\tilde{\delta }=0.3$ (${\tilde{d}}_{\mathrm{eff}}=0.7$).

Figure 3

Lissajous ellipses (a) of the lower and upper separate modes, ${\mathbf{w}}_{\mathrm{lo},\mathrm{up}}$ [Eq. (11)], and (b) of the lower and upper hybrid modes, ${\mathbf{w}}_{\mathrm{lo},\mathrm{up}}^{\left(\mathrm{hyb}\right)}$ [Eq. (13)]. For the separate modes the major axes of the ellipses are along the coordinate axes, for the hybrid modes the axes remain mutually orthogonal, but are rotated by ${45}^{\circ }$. The shape of the ellipses is determined by the circularity $\epsilon$: For $\epsilon =0$ the ellipses degenerate into lines, for $\epsilon =1$ they form a circle (the illustration is for $\epsilon =0.5$). The inset below shows the dependence of $\epsilon$ on the reduced coupling parameter $p$, which is determined by Eqs. (10) and (14) for the separate and hybrid modes, respectively.

Figure 4

Characteristic particle trajectories representing the lower and upper wave modes for the time interval of $50{\Omega }_{\mathrm{DL}}^{-1}$. All parameters of (a) and (b) correspond to Figs. 2 and 2, respectively, $x$ and $z$ are in arbitrary units. The solid-line ellipses depict “orbitals” (straight dotted lines indicate their major axes) inside which the particles mostly reside. In (a) the separated regime, the major axes are along the coordinate axes. In (b) the hybridized regime, the major axis for the (decaying) lower mode is practically horizontal, while for (growing) upper mode it is rotated by $\simeq {70}^{\circ }$. The dashed-line ellipse shows the contribution from the upper hybrid branch only.

Figure 5

Longitudinal (upper panel) and transverse (lower panel) mode spectral densities $I_{l,t}(k,\omega )$. Figures on the left show the results calculated from Eq. (17) for the separate modes, on the right the results from Eq. (18) for the hybrid modes are presented [all parameters correspond to Figs. 2 and 2, respectively]. In the separated regime, the lower mode (longitudinal without coupling) has a transverse component, the upper mode (transverse without coupling) has a longitudinal component. The longitudinal and transverse densities of the hybrid modes are equal, which follows from the orientation of their polarization ellipses [see Fig. 3]. The shown example is for the normalized damping rate $\nu =0.3$, the spectral density is in units of $I_0$.

Figure 6

The relative shift (a) of the critical wave number $\Delta k_{\mathrm{cr}}/k_{\mathrm{cr},0}$ and (b) of the critical confinement $\Delta {\Omega }_{\mathrm{cr}}/{\Omega }_{\mathrm{cr},0}$, as a function of the effective dipole moment of wake ${\tilde{d}}_{\mathrm{eff}}$. (c) Relation between the shifts, showing that it is very close to $\Delta {\Omega }_{\mathrm{cr}}\propto {\left(\Delta k_{\mathrm{cr}}\right)}^2$. The results are for $\tilde{q}=0.4$ and three different values of $\kappa$.