Modeling dispersive coupling and losses of localized optical and mechanical modes in optomechanical crystals

Matt Eichenfield,Kerry J Vahala,Oskar Painter,Amir H Safavi-Naeini,Jasper Chan

doi:10.1364/oe.17.020078

Abstract

Periodically structured materials can sustain both optical and mechanical excitations which are tailored by the geometry. Here we analyze the properties of dispersively coupled planar photonic and phononic crystals: optomechanical crystals. In particular, the properties of co-resonant optical and mechanical cavities in quasi-1D (patterned nanobeam) and quasi-2D (patterned membrane) geometries are studied. It is shown that the mechanical Q and optomechanical coupling in these structures can vary by many orders of magnitude with modest changes in geometry. An intuitive picture is developed based upon a perturbation theory for shifting material boundaries that allows the optomechanical properties to be designed and optimized. Several designs are presented with mechanical frequency approximately 1-10 GHz, optical Q-factor Qo > 107, motional masses meff approximately 100 femtograms, optomechanical coupling length LOM < 5 microm, and clampinig losses that are exponentially suppressed with increasing number of phononic crystal periods (radiation-limited mechanical Q-factor Qm > 107 for total device size less than 30 microm).

Highlights

It has previously been shown that “defects” in a planar periodic dielectric structure can simultaneously confine optical and mechanical resonances to sub-cubic-wavelength volumes[1]
In the experimental demonstration of a nanobeam optomechanical crystal[2], it was shown that the perturbation theory of Maxwell’s equations with shifting material boundaries [34] provides an accurate method of estimating the optomechanical coupling of these complex motions
As described in detail in previous work on “zipper” optomechanical resonators[38, 10] using general momentum-space design rules of photonic crystal cavities[39], the localized modes that come from this band-edge mode are as far as possible from the light line while having a minimal amount of momentum near kx = 0 when used with a defect that is symmetric about a hole in the center

Summary

Introduction

It has previously been shown that “defects” in a planar periodic dielectric structure can simultaneously confine optical and mechanical resonances to sub-cubic-wavelength volumes[1]. Unlike the simple motion of a mirror on a spring in more conventional cavity optomechanical systems[32, 33], the complex mechanics of optomechanical crystal structures makes it difficult to intuit the origin or strength of the optomechanical coupling. In the experimental demonstration of a nanobeam optomechanical crystal[2], it was shown that the perturbation theory of Maxwell’s equations with shifting material boundaries [34] provides an accurate method of estimating the optomechanical coupling of these complex motions. We describe how this perturbation theory can be used to create an intuitive, graphical picture of the optomechanical coupling of simultaneously localized optical and mechanical modes in periodic systems. We analyze the optomechanical coupling of a quasi-2D membrane structure, the well-known double-heterostructure photonic crystal cavity [35]. We show how the optical and mechanical modes and their coupling can be understood in terms of the quasi-one-dimensional nanobeam example

One-dimensional optomechanical crystal systems: an example

Modal cross-coupling and mechanical losses

Optomechanical coupling: definition and integral representation

Optomechanical coupling: visual representation and optimization

Optomechanical coupling in two-dimensional optomechanical crystals