Hyperbolic Metamaterials for Single-Photon Sources and Nanolasers

Abstract

Hyperbolic metamaterials are anisotropic media that behave as metals or as dielectrics depending on light polarization. These plasmonic materials constitute a versatile platform for promoting both spontaneous and stimulated emission for a broad range of emitter wavelengths. We analyze experimental realizations of a single–photon source and of a plasmonic laser based on two different architectures of hyperbolic metamaterials. At the heart of this material capability lies the high broadband photonic density of states originating from a rich structure of confined plasmonic modes.

Appendix: Semi-analytical Calculations of the Purcell Factor and Normalized Collected Emission Power

High LDOS provided by the HMM can lead to dramatic changes in fluorescence lifetimes of quantum emitters. In practice, one is also interested in the percentage of the dissipated optical power that can be collected by a far field detector. In order to describe effects of a realistic HMM on a single-dipole emitter placed in the vicinity of a planar HMM, we performed calculations of total power flow in the immediate vicinity of the emitter as well as through a far-field collection plane. A single emitter was modeled as an oscillating electric dipole with dipole moment p and angular frequency ω. The energy dissipation rate in an inhomogeneous environment is given by [41]

$$P = \frac{\omega }{2}\text{Im} \left[ {{\mathbf{p}}^{ * } \cdot \left( {{\mathbf{E}}_{0} ({\mathbf{r}}_{0} ) + {\mathbf{E}}_{\text{s}} ({\mathbf{r}}_{0} )} \right)} \right]$$

(A.1)

where E ₀(r ₀) and E _s(r ₀) are the primary dipole field and scattered field at the dipole position (r₀), respectively. These electric fields were calculated using the dyadic Green’s function formalism [41]. We analyzed the contribution of each spatial frequency mode using an angular spectrum representation of the Green’s functions. The Purcell factors F _P for in-plane (||) and perpendicular (⊥) oriented single-dipole emitters placed at a distance h above a multilayer planar structure were calculated using the following formulae [41]

$$F_{P}^{ \bot } = 1 + \frac{3}{2}\frac{1}{{\varepsilon_{\sup }^{3/2} }}\int_{0}^{\infty } {\text{Re} \left\{ {\frac{{s^{3} }}{{s_{ \bot ,\sup } (s)}}\tilde{r}^{\text{p}} (s)e^{{2ik_{0} s_{ \bot ,\sup } (s)h}} } \right\}} ds$$

(A.2)

$$F_{P}^{\parallel } = 1 + \frac{3}{4}\frac{1}{{\varepsilon_{\sup }^{1/2} }}\int_{0}^{\infty } {\text{Re} \left\{ {\frac{s}{{s_{ \bot ,\sup } (s)}}\left[ {\tilde{r}^{\text{s}} (s) - \frac{{s_{ \bot ,\sup }^{2} (s)}}{{\varepsilon_{\sup }^{{}} }}\tilde{r}^{p} (s)} \right]e^{{2ik_{0} s_{ \bot ,\sup } (s)h}} } \right\}} ds$$

(A.3)

The value of F _P for the isotropic (ave), statistically averaged dipole orientation is given by

$$F_{P}^{ave} = \frac{2}{3}F_{P}^{\parallel } + \frac{1}{3}F_{P}^{ \bot }$$

(A.4)

Normalized collected emission powers f _rad for the same dipole orientations are shown below

$$f_{rad}^{ \bot } = \frac{3}{4}\int\limits_{0}^{{\theta_{\hbox{max} } }} {\sin^{3} \theta \left| {e^{{ - i\varepsilon_{\sup }^{1/2} k_{0} h\,\cos \theta }} + \tilde{r}^{\text{p}} (\theta )e^{{i\varepsilon_{\sup }^{1/2} k_{0} h\,\cos \theta }} } \right|^{2} d\theta }$$

(A.5)

$$f_{rad}^{\parallel } = \frac{3}{8}\int\limits_{0}^{{\theta_{\hbox{max} } }} {\cos^{2} \theta \left| {e^{{ - i\varepsilon_{\sup }^{1/2} k_{0} h\,\cos \theta }} - \tilde{r}^{\text{p}} (\theta )e^{{i\varepsilon_{\sup }^{1/2} k_{0} h\,\cos \theta }} } \right|^{2} + \left| {e^{{ - i\varepsilon_{\sup }^{1/2} k_{0} h\,\cos \theta }} + \tilde{r}^{\text{s}} (\theta )e^{{i\varepsilon_{\sup }^{1/2} k_{0} h\,\cos \theta }} } \right|^{2} } \sin \theta d\theta$$

(A.6)

$$f_{rad}^{ave} = \frac{2}{3}f_{rad}^{\parallel } + \frac{1}{3}f_{rad}^{ \bot }$$

(A.7)

In equations A.2–A.7, $s = {{k_{\parallel } } \mathord{\left/ {\vphantom {{k_{\parallel } } {k_{0} }}} \right. \kern-0pt} {k_{0} }}$, $s_{ \bot ,\sup } (s) = {{k_{ \bot ,\sup } (s)} \mathord{\left/ {\vphantom {{k_{ \bot ,\sup } (s)} {k_{0} }}} \right. \kern-0pt} {k_{0} }} = \left( {\varepsilon_{\sup } - s^{2} } \right)^{1/2}$, $k_{0} = {\omega \mathord{\left/ {\vphantom {\omega c}} \right. \kern-0pt} c}$, θ is a polar angle measured from the ⊥ direction, the collection angle θ _max = 79.6°. k_|| is the in-plane component of the k-vector varying from 0 to infinity. $\tilde{r}^{p}$ and $\tilde{r}^{s}$ are generalized superlattice’s Fresnel reflection coefficients for p- and s-polarized light. The reflection coefficients were calculated utilizing the recursive imbedding method [42, 57]. The integrals were numerically evaluated by using an adaptive Gauss-Kronrod quadrature method [58]. In the formulae, we assumed that intrinsic quantum yield of NV centers is close to unity [59]. F _P and f _rad were normalized by the total radiation power and the power emitted into the collection angle, respectively. Both powers corresponded to the case of the emitter immersed into homogeneous medium with dielectric permittivity ε_sup, which models well the reference sample in the experiment.