Purcell effect in hyperbolic metamaterial resonators

The radiation dynamics of optical emitters can be manipulated by properly designed material structures modifying local density of photonic states, a phenomenon often referred to as the Purcell effect. Plasmonic nanorod metamaterials with hyperbolic dispersion of electromagnetic modes are believed to deliver a significant Purcell enhancement with both broadband and nonresonant nature. Here, we have investigated finite-size resonators formed by nanorod metamaterials and shown that the main mechanism of the Purcell effect in such resonators originates from the supported hyperbolic modes, which stem from the interacting cylindrical surface plasmon modes of the finite number of nanorods forming the resonator. The Purcell factors delivered by these resonator modes reach several hundreds, which is up to 5 times larger than those in the ε-near-zero regime. It is shown that while the Purcell factor delivered by the Fabry-Pérot modes depends on the resonator size, the decay rate in the ε-near-zero regime is almost insensitive to geometry. The presented analysis shows a possibility to engineer emission properties in structured metamaterials, taking into account their internal composition.

Figure 1

(a) Schematic view of the hyperbolic metamaterial resonator with the transverse dimensions $L_x$ and $L_y$. (b) Schematics of the numerical setup. An emitting dipole is inserted in the center of the resonator. (c) The effective permittivity of the metamaterial calculated for an infinite array of nanorods with $L_z=350\mathrm{nm},a=60\mathrm{nm},r=15\mathrm{nm}$ (Au permittivity was taken from [32]).

Figure 2

Comparison of the numerical (red and blue lines) and analytical (green and black lines) of the Purcell factor in the case of real losses (solid lines) and reduced losses $\mathrm{Im}\left(\varepsilon \right)/10$ (dashed lines). The metamaterial parameters are as in Fig. 1. The resonator size is $16\times 16$ nanorods with $L_x=L_y=900\mathrm{nm}$. In the numerical simulations we have used a small dipole source placed in the centre of the resonator and oriented along x direction.

Figure 3

Dependence of the eigenmode wavelengths on (a) $L_z$ for the fixed $L_x=L_y=900\mathrm{nm}$ and (b) $L_x$ for fixed $L_z=350\mathrm{nm}$. Bars indicate the width of the resonance.

Figure 4

The Purcell factor dependence on the number of nanorods in (a) and (b) square and (c) rectangular resonators for an emitting dipole (a) and (c) perpendicular and (b) parallel to the nanorods. The dipole is located in the center of the array.

Figure 5

(a) The Purcell factor dependence on the height of the hyperbolic metamaterial resonator $(16\times 16$ nanorod array, $L_x=L_y=900\mathrm{nm})$. (b), (c) The electric field $E_x$ distributions excited by the dipole positioned at the center of the resonator with $L_z=350\mathrm{nm}$ at the wavelength of (b) 1000 nm and (c) 600 nm (nonresonant wavelength).

Figure 6

(a) Comparison of the Purcell factor for different numbers of plasmonic rods forming a resonator. (b)–(e) The electric fields $E_x$ of the dipole (b) and (c) near a single nanorod and (d) and (e) near a four-nanorod array at the wavelengths of (b) 856 nm, (c) 611 nm, (d) 937 nm. and (e) 637 nm.

Figure 7

(a) The Purcell factor dependence on an emitter position inside the metamaterial resonator with $L_z=350\mathrm{nm}$ and $L_x=L_y=900\mathrm{nm}$. The coordinate $z=0$ corresponds to the edge of the hyperbolic medium. (b)–(e) The electric field $E_x$ distributions excited by the dipole positioned at $z=0$ for the wavelengths of (b) 1363 nm $(\mathrm{T}{\mathrm{M}}_{130}$ mode, first Fabry-Pérot resonance $n=0$, with two electric field maxima, one magnetic field maximum, no magnetic field zero crossing), (c) 1000 nm $(\mathrm{T}{\mathrm{M}}_{551}$ mode, second Fabry-Pérot resonance $n=1$ with three electric field maxima, two magnetic field maxima, one magnetic field zero crossing), (d) 750 nm $(\mathrm{T}{\mathrm{M}}_{552}$ mode, third Fabry-Pérot resonance $n=2$ with four electric field maxima, three magnetic field maxima, two magnetic field zero crossings), and (e) 600 nm (nonresonant wavelength). All other parameters are as in Fig. 3.