Complete Electromagnetic Dyadic Green Function Characterization in a Complex Environment—Resonant Dipole-Dipole Interaction and Cooperative Effects

The Green function completely encapsulates a system’s linear response to external sources, and plays a central role in optics, electromagnetism, and acoustics. In electromagnetism, a broad range of phenomena are connected to the Green function, including the local density of optical states, superradiance, and the cooperative Lamb shift. Therefore, knowing the Green function is important for progress in fields as diverse as cavity quantum electrodynamics, plasmonics, metamaterials, and photovoltaics. However, experimentally characterizing the full complex Green function is challenging, as it requires amplitude and phase sensitive measurements with deep-subwavelength spatial resolution. Here, we report a method to characterize the full complex Green function with a resolution of $\lambda /100$ by measuring the mutual impedance between two dipoles at microwave frequencies. We apply it to a resonant planar cavity, with both parallel and nonparallel sides, and also explore the effects of modal resonances in a dielectric cube on dipole-dipole interactions. The ability to characterize the Green function with high spatial resolution provides a unique way to investigate cooperative effects in complex photonic systems.

Popular Summary

The Green function plays a central role in wave propagation, as it describes the response of a system to an arbitrary impulse. However, determining it experimentally is very challenging, since it is a complex quantity that needs to be measured with deep subwavelength spatial resolution. Therefore, Green functions are determined analytically or numerically for systems with relatively simple geometries, but this is only practical for a limited number of systems. Here, we describe a method to fully measure and characterize both the real and imaginary parts of the Green function by recording the mutual impedance between two dipoles at microwave frequencies.

The effectiveness of our approach is demonstrated by the full characterization of the complex Green function inside a resonant planar cavity of parallel or nonparallel mirrors at an ultrahigh resolution one-hundredth the size of the wavelength. With this data, we are able to investigate various aspects of resonant dipole-dipole interactions and cooperative effects inside a photonic cavity.

Our experimental results are in excellent agreement with classical electrodynamics simulations and illustrate the power of the Green formalism. Moreover, this experimental approach provides an efficient way to solve problems related to open cavities or configurations for which no analytic solution exists and where numerical simulations would demand excessive computational resources.

Figure 1

Complex-valued Green function in vacuum describing the RDDI versus the dipole-dipole separation $R$. (a) Imaginary part of the Green function, which is related to the cooperative decay rate (CDR) and cross density of states (CDOS). It converges to the LDOS as $R\to 0$. The inset describes the dipoles’ mutual orientations considered here. (b) Real part of the Green function, which determines the cooperative Lamb shift (CLS). In contrast to the imaginary part, the real part of the Green function diverges as $R\to 0$. (c) Square modulus of the Green function, which is linked to the energy transfer FRET. $G_{DA}^0$ is the projection of the Green tensor on the dipole direction in homogeneous space which is related to the mutual impedance $Z_{21}^0$ between the two dipoles. The inset, showing the two-port network model of dipole-dipole interaction, illustrates how the mutual impedance may be measured.

Figure 2

Complex-valued Green function in a planar cavity compared to that in vacuum. (a) Configuration under study: the source donor dipole is located at 10 mm from the nearest mirror and is parallel to the mirror plane. The acceptor dipole is parallel to the donor dipole at $R=10\mathrm{mm}$ distance. The frequency varies between 2.5 and 5.25 GHz. (b) Measured LDOS enhancement versus cavity length $L$ at 5 GHz. The thin colored curves are measurements of the impedance $Z_{11}$, whereas the thick black curve is the analytical prediction. (c)–(e) CDR or CDOS enhancement over vacuum versus cavity length for different $kR$ values (indicated on top). The colored thin curves indicate measurements, whereas the thick black curves are the analytic results. (f)–(h) CLS enhancement, i.e., enhancement of the real part, over vacuum. (i)–(k) FRET enhancement, i.e., enhancement of the square modulus, over vacuum. These measurements provide a complete characterization of the complex Green function in a planar cavity.

Figure 3

Effect of a nonplanar cavity on the Green function, corresponding to a case for which no analytic solution exists. (a) One mirror forming the cavity is tilted by an angle θ; the rest of the configuration is similar to Fig. 2. The distance between the dipoles is $kR=0.9$. (b) CDR-CDOS, (c) CLS, and (d) FRET enhancement factors versus the cavity length for different tilt angles θ. The left-hand column shows the experimental results, whereas the right-hand column presents the numerical simulation results using the finite domain time difference method (CST Studio suite 2019).

Figure 4

Dipole-dipole interaction mediated by a dielectric cube scatterer. (a) Schematic of the experiment: the source dipole is on the left, the receiving dipole is on the right, and a 16 mm cube of dielectric permittivity 16 is placed between them. The distance between the dipole and the surface of the cube is 3 mm ($\lambda /26$) or 6 mm ($\lambda /13$). (b) Numerical simulations of the scattering cross section; the magnetic dipole (MD), electric dipole (ED), and magnetic quadrupole (MQ) resonances are clearly visible. The insets show the electric field amplitude at the MD and ED resonance frequencies, respectively, for plane wave illumination incoming from the left. (c) Numerical simulations of the electric field amplitude in the presence and absence of the dielectric cube at the MD or ED resonance frequencies. The dipoles are placed 6 mm from the cube. (d)–(f) Measurements of the FRET enhancement for different configurations. In (d) the dipoles are parallel and on opposite faces of the cube with 3 and 6 mm distances to the cube, respectively. The gray dashed lines show the result of numerical simulations. In (e) the dipoles are on adjacent faces of the cubes with 6 mm distance. In (f) the dipoles have a perpendicular orientation to each other. The dipole distance to the cube surface is again 6 mm.