Dynamical Drexhage Effect: Amplified Emission in Time-Modulated Electromagnetic Environments

Abstract

We investigate the effect of nonrelativistic motion on the emission dynamics of a dipole emitter moving next to a reflecting interface. Within the formalism of macroscopic QED, we obtain a general equation of motion for the dipole amplitude in terms of the dyadic Green's function, yielding a dynamical extension of the Drexhage effect. At short dipole-surface distances, the dipole can be described as a parametric oscillator featuring time-dependent dampings and Lamb shifts, both arising from the self-induced modulation of the surrounding electromagnetic environment. Importantly, these time-dependent parameters do not always average out, leading to amplification of the dipole amplitude and the radiated intensity when considering certain sinusoidal trajectories with specific modulation amplitudes and frequencies. We derive threshold modulation amplitudes as function of the relative permittivities at the interface. Qualitatively, in the vicinity of certain epsilon-near-zero materials, amplification is possible purely by modulation of the damping. Our findings open up avenues for the dynamic control of light-matter interaction in nanophotonic environments.

Figures (8)

• Figure 1: (Color online) a), b) Illustration of dipoles placed in front of infinite mirrors with complex permittivity $\varepsilon_2 = \varepsilon_2' + i \varepsilon_2"$. The dipole moment, depicted by an arrow, is oriented either a) perpendicular ($\perp$) or b) parallel ($\parallel$) to the mirror surface. The distance to the mirror, $h(t)$, is modulated periodically in time, as shown in the inset. The dipoles are embedded in a medium with permittivity $\varepsilon_1$. The corresponding image dipoles and axis orientations are also indicated. c), d) Static decay rates for perpendicular c) and parallel d) dipole orientations near a mirror, normalized to the homogeneous-medium result $\gamma_n$, as a function of distance to the mirror $R_0$ scaled by the wave number $k$. The gray dashed curves show the result for perfect mirrors ($\varepsilon_2 \to -\infty$), while the solid blue curves show the result for $\varepsilon_2 / \varepsilon_1 = -2^{-1/2} + i 2^{-1/2}$. e), f) Qualitative time dynamics of modulated dipoles: Parametric amplification is shown by the red curves with an amplitude of modulation of $A= 0.67$. The blue line corresponds to the case of a static position at $(kR_0)^{-1}=12$, while the black curve corresponds to decay in a homogeneous medium with refractive index $n$, $\exp(-\gamma_n t/2)$. Light-colored lines show the qualitative oscillations of the dipole moment (logarithmic scale), while dark-colored lines show the envelope of the absolute value.
• Figure 2: (Color online) a), b) Density plots of the quotient $R(\varepsilon_2 / \varepsilon_1)$ that determines the dynamical frequency shift [real part, a)] and damping [imaginary part, b)] coefficients, according to Eq. \ref{['eq_R_varepsilon']}, plotted as a function of the real and imaginary parts of $\varepsilon_2 / \varepsilon_1$. The dotted vertical yellow line in a) shows the ENZ regime $-1 \leq \text{Re}[\varepsilon_2 / \varepsilon_1] <0$. The dashed cyan curve indicates the permittivities where the dynamical Lamb shift vanishes. In panel a) exemplary points corresponding to different material regimes discussed in the text [lossy metal (LM), plasmonic metal (PM), ENZ material] are labeled for reference.
• Figure 3: (Color online) Threshold amplitude of modulation $A_{\mathrm{th} \perp, \parallel}$ required to achieve amplification at a distance of $(k R_0)^{-1}=12$, as given by Eq. \ref{['eq_thresh_amp_LM']} for a lossy metal. This amplitude defines the trajectory of the dipole given in Eqs. \ref{['eq_modulation_position']} and \ref{['eq_eta_func_t']}.
• Figure 4: (Color online) a), b) Decay rates for a lossy metal (LM) for the a) perpendicular and b) parallel orientations with $\varepsilon_2 / \varepsilon_1 = -10 + i$, shown with solid blue lines. For reference, we include the decays near a perfect mirror, shown with dashed lines. c), d) Qualitative behavior of the absolute value of the oscillations of a dipole on a logarithmic scale. The amplification case corresponds to a modulation amplitude of $A=0.67$ in Eq. \ref{['eq_eta_func_t']}. We chose a distance of $(k R_0)^{-1} = 12$. The blue line corresponds to the case of a static position, while the black curve corresponds to decay in the homogeneous medium.
• Figure 5: (Color online) Solid (orange) lines correspond to the coefficients for determining the threshold amplitude given in Eq. \ref{['eq_thresh_amp_IM']} for the case of an plasmonic metal with $\text{Re}[\varepsilon_2] <0$ and $\text{Im}[\varepsilon_2] =0$. Panel a) corresponds to the case of a dipole perpendicular to the mirror and b) to the parallel case. The dashed (gray) lines correspond to the asymptotic values of the decay rates at short distances $k R_0 \rightarrow 0$ as a function of $\varepsilon_2 / \varepsilon_1$.