Maximum Refractive Index of an Atomic Medium

It is interesting to observe that all optical materials with a positive refractive index have a value of index that is of order unity. Surprisingly, though, a deep understanding of the mechanisms that lead to this universal behavior seems to be lacking. Moreover, this observation is difficult to reconcile with the fact that a single isolated atom is known to have a giant optical response, as characterized by a resonant scattering cross section that far exceeds its physical size. Here, we theoretically and numerically investigate the evolution of the optical properties of an ensemble of ideal atoms as a function of density, starting from the dilute gas limit, including the effects of multiple scattering and near-field interactions. Interestingly, despite the giant response of an isolated atom, we find that the maximum index does not indefinitely grow with increasing density but rather reaches a limiting value of $n\approx 1.7$. This limit arises purely from electrodynamics, as it occurs at densities far below those where chemical processes become important. We propose an explanation based upon strong-disorder renormalization group theory, in which the near-field interaction combined with random atomic positions results in an inhomogeneous broadening of atomic resonance frequencies. This mechanism ensures that, regardless of the physical atomic density, light at any given frequency only interacts with at most a few near-resonant atoms per cubic wavelength, thus limiting the maximum index attainable. Our work is a promising first step to understand the limits of the refractive index from a bottom-up, atomic physics perspective, and it also introduces the renormalization group as a powerful tool to understand the generally complex problem of multiple scattering of light overall.

Popular Summary

The ability to confine, guide, and bend light has led to astonishing technological achievements in diverse fields such as microscopy, photochemistry, and telecommunications. The key enabling property of materials that allows for the control of light is its refractive index. Notably, all materials are characterized by a small index of refraction, of order unity, at optical frequencies. Yet, we have no clear understanding of what governs this seemingly universal value. Here, we address this question, showing that even in a minimal model of an atomic medium, multiple scatterings of light between atoms impose a maximum value of order unity.

A single isolated atom has a giant optical response, scattering resonant light as if it were an object much bigger than its actual physical size. This would naively suggest that the index of a dense medium should be huge, in contradiction with natural observations. Instead, we theoretically show that the refractive index is limited by the combination of random atomic positions and the physics of each atom interacting with its nearest neighbor. The importance of granularity illustrates why conventional macroscopic theories of the refractive index, based upon the assumption of a smooth medium, inevitably fail.

Our approach represents a promising step toward the development of a fundamental, bottom-up theory of the refractive index. Understanding the mechanisms that bound the refractive index could constitute a first step toward the development of ultrahigh index materials, with game-changing implications in all optical technologies.

Figure 1

Optical response of an atomic medium. (a) Illustration of a single atom with a dipole-allowed optical transition between ground and excited states, $|g⟩$ and $|e⟩$, characterized by a transition wavelength ${\lambda }_0$ and spontaneous emission rate ${\mathrm{\Gamma }}_0$. Such an atom exhibits a scattering cross section (illustrated by the shaded region) of ${\sigma }_{\mathrm{sc}}\sim {\lambda }_0^2$ for a single resonant photon (wavy green arrows). (b) In a dense ensemble with many atoms per cubic wavelength ${\lambda }_0^3$, the scattering of an incident photon can involve multiple scattering and interference between atoms. (c) In conventional theories of macroscopic optical response, the atoms are approximated by a smooth medium, and the index is derived from the product of single-atom polarizability and density. The maximum index $n$ near the atomic resonance then scales with atomic density like $n\sim \sqrt{N{\lambda }_0^3/V}$. (d) In our renormalization group theory, we retain multiple scattering and granularity, showing that the optical properties of the ensemble are determined by a hierarchy of nearby atomic pairs that strongly interact via their near fields. These interactions effectively produce an inhomogeneously broadened ensemble, where the amount of broadening scales with density (with the different colors of atoms representing the different resonance frequencies in the figure). An incident photon of a given frequency thus sees only about 1 near-resonant atom per reduced cubic wavelength to interact with, regardless of atomic density, resulting in a maximum index of $n\approx 1.7$.

Figure 2

Simulated physical system. (a) Cylindrical ensemble of randomly distributed atoms (green points) illuminated by a z-directed Gaussian beam, whose beam waist $w(z)\gg {\lambda }_0$ is represented in orange. The transverse radius of the cylinder is chosen to be much larger than the beam waist to avoid edge diffraction. (b) Color-coded 3D representation of the forward scattered intensity $I(\mathbf{r},{\omega }_0)=|\mathbf{E}(\mathbf{r},{\omega }_0){|}^2/(2{\mu }_{0}c)$ (with the value indicated in the color bar) over a hemispherical surface far from the ensemble (the radius of this hemisphere is $35{\lambda }_0$), given an incident resonant Gaussian beam. The intensity is calculated for a single random atomic configuration. The system parameters used are as follows: beam waist $w_0=3{\lambda }_0$, cylinder radius $l_{\mathrm{cyl}}=7{\lambda }_0$, and thickness $d=2{\lambda }_0$.

Figure 3

Frequency-dependent refractive index for different atomic densities. The solid lines portray the imaginary (a) and real (b) parts of the refractive index versus dimensionless detuning $\mathrm{\Delta }$, obtained through Eq. (6), while the dotted lines show the MB predictions. The colors denote different atomic densities (color bar on the right), with the specific values indicated by the dotted white lines. The refractive index is inferred by averaging the complex transmission coefficient $t(\mathrm{\Delta })$ over about ${10}^3–{10}^4$ atomic configurations. Other system parameters are as follows: thickness $d=0.4{\lambda }_0$, transverse radius $5\le l_{\mathrm{cyl}}/{\lambda }_0\le 7$, and beam waist $2.5\le w_0/{\lambda }_0\le 3$. The insets show the curves at the three highest densities as a function of the rescaled detuning $\mathrm{\Delta }/\eta$.

Figure 4

Renormalization group analysis. (a) Representative optical response of two identical atoms separated by a distance ${\rho }_{12}\ll 1$. Here, we plot the absorption spectrum (blue curve), which consists of two well-separated Lorentzians. The positions of the resonances are given approximately by $\mp G_{12}^{\mathrm{near}}$, where $G_{12}^{\mathrm{near}}\propto 1/{\rho }_{12}^3$ is the near-field component of the Green’s function. To compare, we also plot twice the response of a single isolated atom (green dashed line). (b) Pictorial representation of the RG scheme for a many-atom system. At each step of the RG flow, the nearby pairs (identified by orange circles) that most strongly interact via their near fields are identified and replaced with atoms with different resonance frequencies (indicated by different colors) in such a way as to produce an equivalent optical response. At the end of the RG process (last panel), the overall system is equivalent to an inhomogeneously broadened ensemble of weakly interacting atoms. (c) Comparison between the maximum real refractive index predicted by the full coupled-dipole simulations of identical atoms (blue points), and the index of the equivalent, inhomogeneously broadened ensemble predicted by the RG (green). For each value of density, the maximum index is obtained by optimizing over detuning. For comparison, the MB and LL models both predict a maximum index given by the orange curve. The inset compares the rescaled spectra $\mathrm{Re}n(\mathrm{\Delta }/\eta )$ of the RG (green) and full coupled-dipole (blue) simulations, given the points at densities $\eta \gtrsim 2$. (d) Rescaled probability distribution of effective, inhomogeneously broadened, resonance frequencies $P({\omega }_{\mathrm{eff}}/\eta )$ obtained from the application of the RG scheme. Given nine different values of the density $\eta$ (ranging from $\eta \approx 2.5$ up to $\eta \approx 80$), the distributions of effective resonance frequencies are plotted with different colors, according to the bar on the right. The exact values chosen for the curves are emphasized by dotted white lines in the color bar. The curves at $\eta \approx 2.5$ and $\eta \approx 3$ are calculated using the cylindrical system studied in Fig. 3 (with thickness $d=0.4{\lambda }_0$ and transverse radius $l_{\mathrm{cyl}}=5{\lambda }_0$), while the distributions $P({\omega }_{\mathrm{eff}}/\eta )$ at densities $\eta >3$ are evaluated using a spherical geometry of radius $r_{\mathrm{sph}}=0.55{\lambda }_0$. Finally, for the case of density $\eta \approx 3$, we plot (black dashed curve) the many-atom distribution of the eigenvalues of the near-field matrix $-G_{\mathrm{near}}$ (also rescaled by a factor of $1/\eta$ for consistency), as discussed further in Sec. 5. All distributions are obtained by accumulating results from about 100 different configurations of atomic positions.

Figure 5

Radiation pattern of a single dipole, compared to that of two in-phase or out-of-phase dipoles. (a) Given an isolated dipole $\mathbf{d}$ of fixed dipole amplitude $d_0$ and direction $\widehat{\mathbf{x}}$, which is placed at $\mathit{\rho }=0$ and radiates light at the frequency ${\omega }_0$, we plot the intensity of the radiated field $I_{\mathrm{sc}}(\mathbf{r})=|{\mu }_0{\omega }_0^2\overline{\overline{\mathbf{G}}}(\mathbf{r},{\omega }_0)·\mathbf{d}{|}^2/(2{\mu }_{0}c)$ in the $\widehat{\mathbf{x}}-\widehat{\mathbf{z}}$ plane, with the value indicated in the color bar. (b,c) Radiation pattern $I_{\mathrm{sc}}(\mathbf{r})=|{\mu }_0{\omega }_0^2{\sum }_{j=1,2}\overline{\overline{\mathbf{G}}}(\mathbf{r}-{\mathbf{r}}_j,{\omega }_0)·{\mathbf{d}}_j{|}^2/(2{\mu }_{0}c)$ for two near-positioned dipoles of fixed amplitude $d_0$ and direction $\widehat{\mathbf{x}}$, oscillating either in phase (b) or out of phase (c) with one another (i.e., ${\mathbf{d}}_1=\pm {\mathbf{d}}_2=d_0\widehat{\mathbf{x}}$), and placed at positions ${\mathit{\rho }}_1=-{\mathit{\rho }}_2=0.1(\widehat{\mathbf{x}}+\widehat{\mathbf{z}})/\sqrt{2}$. (d) Intensity radiated by two out-of-phase dipoles averaged over all possible interatomic orientations, keeping the mutual distance $|{\mathit{\rho }}_1-{\mathit{\rho }}_2|=0.2$ fixed. This pattern closely resembles that of a single oscillating dipole.

Figure 6

Microscopic analysis of the renormalization of the antisymmetric modes. (a) Pictorial representation of the near-field interaction between the antisymmetric mode of a pair (represented by two out-of-phase dipoles, pink circles labeled 1 and 2) and a third atom (green circle, with label 3), which may sit very far from the pair (characterized by the distances ${\rho }_{12}<{\rho }_{13}\sim {\rho }_{23}$). In this case, the interaction strength scales like $1/{\rho }_{13}^4$, reflecting the higher-order multipole nature of the out-of-phase dipoles. (b) Similar illustration for the case where the third atom sits closer to one atom of the pair (say, atom 1) than the pair separation itself (${\rho }_{13}<{\rho }_{12}\sim {\rho }_{23}$). The interaction strength then scales like $1/{\rho }_{13}^3$. (c) System properties during the RG flow. The horizontal axis quantifies how many pairs $N_{\mathrm{RG}}$ have been renormalized, from the beginning ($N_{\mathrm{RG}}=0$) towards the end ($N_{\mathrm{RG}}=N_{\mathrm{RG}}^{\text{final}}$) of the algorithm. As one atom can be renormalized more than once, typically $N_{\mathrm{RG}}^{\text{final}}>N/2$. The blue circles represent the average interatomic distance ${\rho }_{\mathrm{RG}}$ of those pairs that get renormalized, while the orange squares show the average distance ${\rho }_{\text{nearest}}$ between the atoms of those pairs and their respective nearest atom, chosen among those that are still allowed to interact (i.e., with ${\mathcal{L}}_{ij}=1$). The green triangles display the fraction of atoms $N_0/N$ that have never been renormalized up to that moment of the flow. The data represent the average over around 300 runs over different random atomic positions, uniformly sampled inside a sphere of radius $r_{\mathrm{sph}}=0.55{\lambda }_0$ and density $\eta \approx 32$. The bars show 1 standard deviation in the accumulated statistics.

Figure 7

Independence of the refractive index from the thickness of the ensemble. Given the physical system of Fig. 2 of the main text (with $w_0=2.5{\lambda }_0$, $l_{\mathrm{cyl}}=5{\lambda }_0$), we compare the resonant ($\mathrm{\Delta }=0$) refractive index as a function of the density for various ensemble thicknesses: $d=0.4{\lambda }_0$ (as in the main text, shown here in blue), $d=0.6{\lambda }_0$ (in green), and $d=0.8{\lambda }_0$ (in orange). Panels (a) and (b) illustrate the real and imaginary parts of the index, respectively. The insets show the full spectra $n(\mathrm{\Delta })$ at a fixed density $\eta \simeq 0.28$. All data are obtained by averaging $⟨t(\mathrm{\Delta })⟩$ over more than 1000 configurations.

Figure 8

RG scheme based upon repositioning atoms. Because of the finite size of the sample, if one defines the positions of the new effective atoms as being at the midpoint between the original pair, then, at each RG step, the cloud effectively shrinks, resulting in a distortion of the ensemble.