Trapping of a microsphere pendulum resonator in an optical potential

We propose a method to spatially confine or corral the movements of a micropendulum via the optical forces produced by two simultaneously excited optical modes of a photonic molecule comprising two microspherical cavities. We discuss how the cavity-enhanced optical force generated in the photonic molecule can create an optomechanical potential of about 10 eV deep and 30 pm wide, which can be used to trap the pendulum at any given equilibrium position by a simple choice of laser frequencies. This result presents opportunities for very precise all-optical self-alignment of microsystems.

Figure 1

(Color online) Proposed experimental setup. Sphere 1 is stationary and optically coupled to a tapered optical fiber, while sphere 2 is free to move and optically coupled to sphere 1. The pumping laser field consists of two monochromatic field components with frequencies ${\nu }_1$ and ${\nu }_2$. Inset shows the transmission of a weak probe signal as a function of frequency for a given separation $x$ between the microspheres.

Figure 2

(Color online) (a) Qualitative dependences of the frequencies of the symmetric (lower line) and antisymmetric (upper line) optical modes in a two-sphere system as functions of the sphere separation; (b) nature of the optical force on sphere 2 as a function of sphere separation; (c) shape of the optical potential for sphere 2 as a function of sphere separation.

Figure 3

(Color online) Photonic molecule system with sphere radii $R=12.5\mu \text{m}$, $\gamma /2\pi =2\text{MHz}$, $n_s=1.46$, $Q={10}^8$, $\lambda =1568\text{nm}$, and $P=1\text{mW}$. (a) Frequency splitting of the antisymmetric and symmetric optical modes for an unshifted modal frequency ${\nu }_0=c/\lambda =191\text{THz}$ as a function of sphere separation. The equilibrium position of sphere 2 is $x_0=250\text{nm}$ and the applied laser frequencies ${\nu }_1$ and ${\nu }_2$ are blue-detuned from the equilibrium position frequencies $v_{0a}$ and $v_{0s}$ by $5\gamma$. The arrows show the direction of the partial forces $F_a$ and $F_s$. (b) Position dependences of the optical force on sphere 2 for three different detunings of ${\nu }_1$ and ${\nu }_2$ from $v_{0a}$ and $v_{0s}$. (c) Optical potential for sphere 2 as a function of position for the same detunings as (b).

Figure 4

(Color online) Optical potential in the $x\text{−}y$ plane. The color (gray) scale indicates the depth of the potential. The dashed white line represents the minimum of the potential. Point 1 shows the equilibrium position of sphere 2 and point 2 is chosen for illustration of the force components in Fig. 5.

Figure 5

Schematic of the forces on sphere 2 for the displacement from point 1 (the equilibrium position) to point 2 chosen on the $y$ axis. The dashed circular line centered on sphere 1 represents the minimum of the optical potential.