Collectively enhanced optomechanical coupling in periodic arrays of scatterers

We investigate the optomechanical properties of a periodic array of identical scatterers placed inside an optical cavity and extend our previous results [Xuereb, Genes, and Dantan, Phys. Rev. Lett. 109, 223601 (2012)]. We show that operating at the points where the array is transmissive results in linear optomechanical coupling strengths between the cavity field and collective motional modes of the array that may be several orders of magnitude larger than is possible with an equivalent reflective ensemble. We describe and interpret these effects in detail and investigate the nature of the scaling laws of the coupling strengths for the different transmissive points in various regimes.

Figure 1

Schematic diagram illustrating the basis of the model: $N$ scatterers interacting with four running waves; $B$ and $C$ represent incident fields, $A$ and $D$ outgoing fields. The four field amplitudes are connected by means of transfer matrices.

Figure 2

Optical properties of a stack of $N=6$ nonabsorbing elements. The intensity reflectivity of the stack (blue curve) varies strongly, from 0% to about 99%, as the distance between the elements is scanned; this curve is periodic with a period of $\lambda /2$. By way of comparison, we show the corresponding reflectivity for a single element as the dotted green line. We mark three important values of $d$ in this figure: $d=d_1$ and $d=d_5$, where the reflectivity is zero, and $d=d_0$, where the reflectivity attains its largest value.

Figure 3

A schematic drawing of the generic system we are considering. A periodic array of $N$ elements, each of which is independently harmonically bound, is positioned at, or very close to, the center of a Fabry-Pérot resonator. Throughout this paper, we shall consider only the case for which $L\gg Nd$.

Figure 4

Transmission through a cavity with $N=6$ immobile elements configured for maximal reflectivity. The dashed lines denote the bare-cavity resonances, which are shifted due to the presence of the ensemble. $x$ is normalized by a factor $\sqrt{N}$, such that the gradient of the bright curves directly gives the linear optomechanical coupling at that point ($\zeta =-0.5$, $L\approx 6.3\times {10}^4\lambda$, $d=d_1$, bare-cavity finesse $\approx 3\times {10}^4$, corresponding to cavity-mirror reflectivities of $99.99$%).

Figure 5

Individual coupling strengths for the case of six elements ($l=1,\cdots ,5$, top to bottom); the dashed curves are drawn as guides to the eye.

Figure 6

Transmission through a cavity with $N=6$ immobile elements close to the five transmission points; from top to bottom, the figures depict $l=1$ to $l=5$. This figure should be compared to Fig. 4. $x$ is normalized by a factor $\sqrt{N/2}$. (Parameters as in Fig. 4.)

Figure 7

Collective coupling strength $g_{\sin }^{\left(1\right)}$ (blue circles, closed red squares, orange triangles) for a cavity of length $L\approx 6.3\times {10}^4\lambda$ and several choices of scatterer reflectivity, compared to the center-of-mass coupling strength $g_{\mathrm{com}}$ (open red squares) and $g$ (green line). For the red, orange, and blue data points we choose a per-element intensity reflectivity of 20%, 50%, and $99.4$%, respectively. For the center-of-mass data we illustrate the $N^{-1/2}$ scaling law that applies for large $N$, whereas for the first sinusoidal mode we draw a curve through the data points as a guide to the eye. Throughout this plot we take $d=d_1$, except for the data denoted by the open blue circles, for which $d=d_1+20\lambda$.

Figure 8

(a) Optimal number of elements $N_{\mathrm{opt}}^{\left(1\right)}$ as a function of the single-element reflectivity. For reflectivities close to 100%, this number decreases rather quickly to $\lesssim$10. (b) The coupling of the first sinusoidal mode, normalized to $g$, optimized as $N=N_{\mathrm{opt}}^{\left(1\right)}$ (solid blue curve), compared to the case for $N=2$ (dashed-dotted red) and $g_{\mathrm{com}}$ (dotted orange). For these plots we used $L\approx 6.3\times {10}^4\lambda$ and $d=d_1$.

Figure 9

(a) Coupling strength $g_{\sin }^{\left(1\right)}$, normalized to $g$, as a function of the number of elements. Four sets of data are shown. The red diamonds (blue circles) correspond to a single-element reflectivity of 20% ($99.4$%). Closed (open) symbols represent an interelement spacing of $d=d_1$ ($d=d_1+20\lambda$). We note that the interelement separation has a much stronger effect for highly reflective elements ($L\approx 6.3\times {10}^4\lambda$, bare-cavity finesse $\approx 3\times {10}^4$). (b) A similar plot showing the effective cavity linewidth, normalized to the bare-cavity linewidth, obtained in each of the cases displayed in the upper plot. (c) Putting these two together, we can calculate the cooperativity, normalized to the single-element cooperativity, and demonstrate an enhancement by several orders of magnitude for the chosen parameters.

Figure 10

Coupling strength $g_{\sin }^{\left(l\right)}$, normalized to $g$, as a function of $\zeta$, for $N=6$. Three curves are shown: $d=d_1$ (solid red), $d=d_2$ (dashed green), and $d=d_3$ (dotted orange) ($L\approx 6.3\times {10}^4\lambda$, bare-cavity finesse $\approx 3\times {10}^4$).

Figure 11

(a) Coupling strength $g_{\sin }^{\left(l\right)}$, normalized to $g$; (b) effective linewidth, normalized to the bare-cavity linewidth; and (c) cooperativity, normalized to the single-element cooperativity, all plotted as functions of $N$, for $\zeta =-12.9$. Four curves are shown in each panel: $d=d_1$ (solid red), $d=d_3$ (dashed green), $d=d_5$ (dotted orange), and $d=d_7$ (long-dashed blue). The red curve in each part is to be compared to the corresponding solid blue data points in Fig. 9. See also Fig. 12 ($L\approx 6.3\times {10}^4\lambda$, bare-cavity finesse $\approx 3\times {10}^4$).

Figure 12

Similar to Fig. 11, but the four curves shown in each panel are for $d=20\lambda +d_1$ (solid red), $d=20\lambda +d_3$ (dashed green), $d=20\lambda +d_5$ (dotted orange), and $d=20\lambda +d_7$ (long-dashed blue). The red curve in each part is to be compared to the corresponding open blue data points in Fig. 9. (Other parameters as in Fig. 11.)

Figure 13

Reflectivity (red), transmission (green), and absorption (orange) for $N=6$ elements with an individual reflectivity of ca. 20% and an absorption of ca. $1.6$% per element. The dashed blue curve is identical to the solid blue curve in Fig. 2. Note that the absorption around the transmissive points is lowest close to $d_{N-1}=d_5$ and highest close to $d_1$.

Figure 14

Effect of absorption on the linewidth, shown normalized to the bare-cavity linewidth, in the case of a single-element reflectivity of $99.4$%. We show $d=D+d_1$ (solid red and dashed green curves) and $d=D+d_{N-1}$ for each $N$ (dotted orange curve), where (a) $D=0\lambda$ and (b) $D=20\lambda$. The solid red curve represents nonabsorbing scatterers ($\text{Im}\left\{\zeta \right\}=0$), and the other two curves $\text{Im}\left\{\zeta \right\}={10}^{-5}$. The curves for the two values of $d$ coincide in the absence of absorption. Larger interelement separations make the system more tolerant to higher levels of absorption. The linewidth of the bare cavity is represented by the horizontal dashed black line. (Bare-cavity finesse $\approx 3\times {10}^4$, other parameters as in Fig. 9.)