Transmissive optomechanical platforms with engineered spatial defects

Linear optomechanical photon-phonon couplings in the membrane-in-the-middle setup can be enhanced by taking a multielement approach as it was recently shown [A. Xuereb, C. Genes, and A. Dantan, Phys. Rev. Lett. 109, 223601 (2012)]. The particular example considered consists of a periodic array of membranes embedded in a high-finesse optical cavity and operating in the transmissive regime, i.e., around resonances of the compound cavity-membrane system. Here we propose further improvements in such a setup by breaking the translational invariance of the array, i.e., by considering quasiperiodic arrays with engineered quadratic spatial defects in the membrane positions. The localization of light modes induced by the defect combined with the access of the aforementioned transmissive regime window can lead to additional enhancement of the strength of both linear and quadratic optomechanical couplings.

Figure 1

Optomechanical platforms. (a) Cavity optomechanics with a transmissive equidistant membrane array shows localization of light within the middle region of the array when quadratic spacing defects are introduced. (b) A simple 1D model of an OM crystal divided into three regions: The side ones are modeled as effective dispersive mirrors while the middle one accommodates the quadratic defect.

Figure 2

Optical response: reflectivity of an array of seven immobile membranes (each with $\zeta =-5)$. From top to bottom the figures compare the behavior of the array with a defect (red solid curves) when $\alpha$ is scanned through values $9\times {10}^{-4},5\times {10}^{-3}$, and $5\times {10}^{-2}$ with the equidistant array (blue dashed curves). For large $\alpha$, single resonances are singled out inside the reflection band gap.

Figure 3

Field localization. Plot of the electric-field amplitude normalized to its equidistant case values along the array (vertical dashed lines show the membrane positions) for two situations in which the system is transparent: (i) no defect $\left(\alpha =0\right)$ (blue solid line) and (ii) with a defect $\left(\alpha =5\times {10}^{-3}\right)$ (red dashed line). While the equidistant array already shows localization of the field mode, the defect can enhance this effect. The field amplitude is normalized to the maximum value achieved for the equidistant case.

Figure 4

Optomechanical coupling strengths. Behavior of (a) linear and (b) quadratic OM couplings for an array of seven membranes in a cavity of length $L=6.3\times {10}^{-2}$ m with an intermembrane distance $d=525$ nm. The polarizability of the cavity mirrors is ${\zeta }_m=-20$ and every membrane has a polarizability $\zeta =-5$. The blue, green, and red curves correspond to $\alpha =0,5\times {10}^{-3}$, and $5\times {10}^{-2}$, respectively. For the numerical example considered, the introduction of the defect builds on the enhancement provided already by the access of the last transmission point by increasing $g^{\left(1\right)}/g_0^{\left(1\right)}$ by a factor of 23 and $g^{\left(2\right)}/g_0^{\left(2\right)}$ by a factor 434. Notice that we fixed $g^{\left(1\right)}/g_0^{\left(1\right)}$ for the equidistant case to the maximum value allowed, roughly equal to $\sqrt{2}/\pi {\zeta }^2{N}^{3/2}\simeq 217$. For an even larger defect $\left(\alpha =5\times {10}^{-2}\right)$, only one resonance survives and it is moved into the lower-energy band gap instead, with corresponding lower enhancement factors $1.73$ and $2.94$.

Figure 5

Optical response of an empty supercavity (solid red curve) overlapped to that of a single side mirror (dashed blue curve) for $N_m=6,{\zeta }_m=-0.5$, and $d_m=768$ nm.

Figure 6

Optomechanical crystal optical response (a) An empty supercavity shows resonances in the common band gaps of the individual extended side mirrors; in fact, the plot displays the reflectivity of the supercavity as a function of the wave vector $k$ (in units of $k_0=\pi /d)$. (b) A membrane array with an engineered quadratic defect can show, under certain conditions, field localization. (c) A supercavity with a quasiperiodic array inside simulates an OM crystal. The optical response is a convolution of the two previously plotted responses. To find transmissive regimes one finds the wave vectors at which both the top and middle reflectivity plots show zeros.

Figure 7

Tuning of resonances. Common resonances are obtained as the intersections of the empty supercavity resonances (dotted lines) with middle element resonances (dashed lines) as the intermembrane distance $d_m$ is varied. The first eight empty supercavity resonances are plotted and the black square indicates a common resonance operating point. Notice that the scattered plot regions correspond to the empty-cavity resonances between the band gaps where the effective finesse is small and the analysis is not valid. The red triangle shows the small shift of the crystal resonances with respect to the common resonance and is due to the phase shift ${\mu }_m$. The side mirrors consist of 20 elements while the middle region contains 7 membranes. We also chose $\alpha ={10}^{-3}$ and $\zeta ={\zeta }_m=-0.5$.

Figure 8

Light localization. The optical response is plotted in the top right plot as a function of the incoming wave vector for $\zeta =-4$ and $N=8$ membranes for both the equidistant (blue line) and defect cases with $\alpha =3\times {10}^{-3}$. For each of the $N-1$ transmissive points, we plot the electric-field amplitude normalized to its equidistant case values (the plot label corresponds to the transmission order) along the array with its defect-induced localization. The field amplitude is normalized to the maximum value achieved for the equidistant case.

Figure 9

Overlap between the transmission plot (blue) for a finite-size array of membranes and the band structure (red) of an infinite 1D photonic periodic crystal. Ten membranes are considered. Both plots have a polarizability $\zeta =-0.9$.