## Optical and mechanical design of a “zipper” photonic crystal optomechanical cavity

Optics Express, Vol. 17, Issue 5, pp. 3802-3817 (2009)

http://dx.doi.org/10.1364/OE.17.003802

Acrobat PDF (2600 KB)

### Abstract

Design of a doubly-clamped beam structure capable of localizing mechanical and optical energy at the nanoscale is presented. The optical design is based upon photonic crystal concepts in which patterning of a nanoscale-cross-section beam can result in strong optical localization to an effective optical mode volume of 0.2 cubic wavelengths ((*λ _{c}
*)

^{3}). By placing two identical nanobeams within the near field of each other, strong optomechanical coupling can be realized for differential motion between the beams. Current designs for thin film silicon nitride beams at a wavelength of

*λ*= 1.5 μ m indicate that such structures can simultaneously realize an optical

*Q*-factor of 7 × 10

^{6}, motional mass mu ~ 40 picograms, mechanical mode frequency Ω

_{ M }/2

*π*~ 170 MHz, and an optomechanical coupling factor (

*g*

_{OM}= dω

_{ c }/d

*x*= ω

_{ c }/Lom) with effective length

*L*

_{OM}~

*λ*= 1.5

*μ*m.

© 2009 Optical Society of America

## 1. Introduction

3. T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science **321**, 1172–1176 (2008). [PubMed]

3. T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science **321**, 1172–1176 (2008). [PubMed]

4. O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature **444**, 71–73 (2006). [PubMed]

5. S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. As-pelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature **444**, 67–70 (2006). [PubMed]

7. T. Corbitt, C. Wipf, T. Bodiya, D. Ottaway, D. Sigg, N. Smith, S. Whitcomb, and N. Mavalvala, “Optical Dilution and Feedback Cooling of a Gram-Scale Oscillator to 6.9 mK,” Phys. Rev. Lett . **99**, 160801 (2007). [PubMed]

8. J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to micromechanical membrane,” Nature **452**, 72–75 (2008). [PubMed]

9. T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett . **95**, 033901 (2005). [PubMed]

_{c}) with mechanical displacement amplitude (

*u*),

*g*

_{OM}= dω

_{c}/d

*u*= ω

_{c}/

*L*

_{OM}.

*L*

_{OM}is an effective length over which a cavity photon’s momentum can be exchanged with the mechanical system. In this work we describe a simple doubly clamped nanobeam system (a so-called

*zipper*cavity) which allows for the combined localization of optical and mechanical energy in a nanoscale structure so as to provide extremely large optomechanical coupling due to the gradient optical force. Optical energy is localized within the center of the nanobeam using a one-dimensional photonic crystal in combination with total internal reflection. Beyond the analysis provided here, future optimization of both the optical and mechanical properties of these chip-based structures should allow for a variety of new applications from precision metrology [11] to tunable photonics [12

12. M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett . **30**, 3042–3044 (2005). [PubMed]

13. M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, “Optomechanical Wavelength and Energy Conversion in High-Q Double-Layer Cavities of Photonic Crystal Slabs,” Phys. Rev. Lett . **97**, 023903 (2006). [PubMed]

## 2. Optical design and simulation

*Q*-factors in excess of 3 × 10

^{6}for whispering-gallery modes of microdisks formed from stoichiometric silicon nitride deposited by low-pressure-chemical-vapor-deposition (LPCVD). A second reason is that LPCVD-deposited stoichiometric silicon nitride films on silicon have a large internal tensile stress, which has been shown to be critical in producing high-

*Q*mechanical resonances in doubly-clamped nanobeams [22,23]. An additional concern is the two-photon absorption present in smaller bandgap semiconductors such as silicon and gallium arsenide, which results in additional free-carrier absorption (FCA), and which can result in significant parasitic effects such as thermo-optic and free-carrier dispersion in small-volume photonic crystal nanocavities [24

24. P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microcavities excited via an integrated waveguide and fiber taper,” Opt. Express **13**, 801–820 (2005). [PubMed]

*n*~ 2) of these films in comparison to semiconductor films (

*n*~ 3.4). As is shown below (and in Ref. [19]), with carefully chosen designs, high-

*Q*photonic crystal optical cavities of sub-cubic-wavelength mode volume can still be formed in silicon nitride thin films.

25. MIT Photonic Bands (MPB) is a free software package for the solution of the electromagnetic eigenmodes of periodic structures. MPB has been developed at MIT, http://ab-initio.mit.edu/wiki/index.php/MPB.

*t̄*=

*t*/

*a*= 2/3) and beam width (

*w̄*=

*w*/

*a*= 7/6). The refractive index of the silicon nitride beam is taken as

*n*= 2, and a resolution of 64 points per axial lattice period is used to ensure accurate band frequencies. Our coordinate convention is (see Fig. 3): (i)

*x*the in-plane coordinate along the long axis of the cavity, (ii)

*y*the in-plane transverse coordinate, and (iii)

*z*the out-of-plane transverse coordinate. Only the lower-lying bands with modes of even parity in the

*z̄*-direction and odd parity in the

*ȳ*-direction are shown, corresponding to the fundamental TE-like modes of the beam waveguide. As indicated by the electric field energy density plots of the two lowest lying band-edge modes (inset to Fig. 1(a)), the valence band lies predominantly in the region of the high refractive index silicon nitride beam, whereas the upper “conduction” band mode lies predominantly in the region of the air hole patterning.

*mirror*section (

*a*) to 90% of the nominal value at the center of the defect region. This was found to provide a good balance between axial-localization of the cavity modes and radiation loss into the

_{m}*y*-

*z*transverse directions of the nanobeams. In Fig. 1(c) we plot the

*local*lattice period, defined as

*a*=

_{nh}*x*(

*n*+ 1) -

_{h}*x*(

*n*), versus air hole number

_{h}*n*along the length of the cavity. In Fig. 1(d) we plot the corresponding lattice-normalized frequency of the local TE-like valence band-edge modes of the single beam cavity. The small variation in lattice-normalized frequency is a result of the distortion in the aspect ratio of the (perfectly periodic) structure as the lattice period is changed. More useful is the plot in Fig. 1(e) which shows the frequency of the local valence band-edge (blue solid curve) and conduction band-edge (red dashed curve) modes normalized to the valence band-edge mode frequency in the mirror section of the cavity. The quadratic grade in lattice constant results in a nearly-harmonic shift in the valence band-edge frequency versus position in the center of the cavity, with the band-edge frequency at the cavity center lying approximately mid-gap between the valence and conduction band-edges in the mirror section of the cavity.

_{h}*bonded*modes, and the odd parity supermodes,

*anti-bonded*modes [12

12. M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett . **30**, 3042–3044 (2005). [PubMed]

_{+,-}) being dependent upon the slot gap (

*s*) between the nanobeams. The electric field energy density plots of the bonded valence and conduction band-edge modes are shown in Figs. 2(b) and 2(c), respectively. The corresponding anti-bonded modes, with noticeably reduced energy density in the slot gap, are displayed in Figs. 2(d)–(e). Owing to the negative curvature of the valence band forming the localized cavity modes, the fundamental mode of the cavity for each manifold has the highest frequency, with the higher-order modes having reduced frequencies (schematically illustrated in Fig. 2(f)). In what follows, we will primarily be interested in the bonded mode manifold due to the larger electric field intensity of these modes in the slot gap.

26. COMSOL is a multiphysics software package for performing finite-element-method (FEM) simulations. See COMSOL AB, http://www.comsol.com/. We use the COMSOL multiphysics software package to perform both optical and mechanical numerical simulations of the zipper cavity.

*Q*-factor of the cavity modes, and the structure is simulated with all available symmetries taken into account allowing for 1/8th the simulation volume. Scattering boundary conditions are used in the outer axial and transverse boundaries to provide a nearly-reflectionless boundary for out-going radiation. In the transverse direction (

*y*-

*z*) a cylindrical outer boundary is also used to reduce the total simulation volume. Finally, a check of the accuracy of our FEM simulations was also performed through a series of equivalent simulations using finite-difference time-domain code (Lumerical [27

27. Lumerical FDTD Solutions is a finite-difference time-domain software package for performing electromagnetic simulations. See Lumerical Solutions, Inc., http://www.lumerical.com/.

_{+,0}) and anti-bonded (TE

_{-,0}) modes, respectively, of the double beam zipper cavity. Cross-sectional electric field profiles, displayed in Figs. 3(b) and 3(d), clearly show the even and odd parity of the modes. Also shown in Figs. 3(e)–3(g) are the electric field intensity of the lowest three bonded cavity mode orders, TE

_{+,0}, TE

_{+,1}, and TE

_{+,2}.

*w̄*=

*w*/

*a*= 700/600, (ii) beam thickness,

_{m}*t̄*=

*t*/

*a*= 400/600, (iii) axial hole length,

_{m}*h̄*=

_{x}*h*/

_{x}*a*= 267/600, (iv) axial hole width,

_{m}*h̄*=

_{y}*h*/

_{y}*a*= 400/600, and (v) slot gap,

_{m}*s̄*=

*s*/

*a*= 100/600, where

_{m}*a*is the lattice periodicity in the cavity mirror section. The length of the beam is set by the number of air hole periods in the cavity, which for the nominal structure is

_{m}*N*= 47 (23 holes to the left and right of the central hole, with the central 15 holes defining the defect region). The filling fraction of the air holes in the nominal structure is

_{h}*f*= 25.4%. For a fundamental bonded mode wavelength of λ ~ 1500 nm, the nominal lattice constant is

*a*= 600 nm, hence the normalization by 600 in the above expressions for normalized dimensions.

_{m}*Q*-factor of the fundamental bonded cavity mode (TE

_{+ 0}) versus the total number of periods

*N*of the cavity for the nominal structure. The normalized cavity resonance frequency is calculated to be

_{h}*a*/λ

_{m}_{c}= 0.3927. The radiation loss from the cavity is broken into two parts, yielding two effective

*Q*-factors: (i) the axial radiation loss out the ends of the nanobeams (yielding

*Q*

_{∥}), and (ii) the radiation loss transverse to the long axis of the zipper cavity and intercepted by the transverse boundary (yielding

*Q*

_{⊥}). In Figure 4,

*Q*

_{∥}is seen to rise exponentially as a function of

*N*, with an order of magnitude increase in

_{h}*Q*-factor for every 6 additional periods of air holes. The transverse

*Q*is seen to rise initially with hole periods, but then levels off and saturates at a value of

*Q*

_{⊥}= 7 × 10

^{6}. The variation in

*Q*

_{⊥}for structures with small

*N*, and low

_{h}*Q*

_{∥}, is a result of weak coupling between radiation loss into these (arbitrarily chosen) directions. Small reflections at the end of the nanobeams results in a small amount of axial radiation making its way to the transverse boundary. Nevertheless, a structure with

*N*< 47 results in a total radiation

_{h}*Q*-factor limited by the transverse

*Q*-factor of 7 × 10

^{6}. This value is very large given the modest refractive index of the silicon nitride film and small cross-section of the nanobeams (large air filling fraction).

*Q*-factor on the hole size, we have also simulated the nominal structure with varying axial and transverse hole size, as shown in Figs. 5 and 6. In each of the plots the axial

*Q*-factor increases with increasing hole size, but then saturates, as the hole size approaches that of the nominal structure. For hole sizes larger than the nominal structure, the

*Q*

_{∥}slightly drops, as does the transverse

*Q*-factor. This drop in transverse

*Q*-factor is a result of the increased normalized frequency of the mode (higher air filling fraction), which pushes the mode closer to the air-cladding light line, increasing radiation into the cladding (the drop in

*Q*

_{∥}is a result of low-angle transverse radiation making its way to the boundary at the end of the nanobeams). Therefore, the nominal structure represents an optimal structure in this regard, allowing for tight axial localization of the mode, without decreasing the transverse

*Q*-factor.

**r**) is the position dependent dielectric constant and

**r**

_{0}is the position of interest. In the case of the zipper cavity, the strength of the optomechanical coupling depends upon

*V*

_{eff}(

**r**

_{0}) evaluated in the slot gap of the nanobeams. Similarly, in the field of cavity-QED, the effective mode volume can be used to estimate the coherent coupling rate between an “atom” and the cavity field. In Fig. 7 we plot the effective mode volume versus slot gap size,

*s*, between the nanobeams for the TE

_{0,+}mode of the nominal structure. We plot two different effective mode volumes: (i)

*V*, the effective mode volume evaluated at the center of the nanobeam gap near the center of the cavity where the field is most intense, and (ii)

_{g}*V*, the minimum effective mode volume evaluated at the position of peak electric field energy density in the cavity. For normalized gap widths,

_{p}*s̄*< 1/6 the modes of the nanobeam are strongly coupled resulting in a peak electric field intensity in between the slot gap (hence the two effective mode volumes track each other). The effective mode volumes approach a value of

*V*

_{eff}=0.1(λ

_{c})

^{3}as the slot gap approaches zero. This small value is a result of the discontinuity in the dominant polarization of the TE mode (

*E*) as it crosses into the slot gap [28

_{y}28. J. T. Robinson, C. Manolatou, L. Chen, and M. Lipson, “Ultrasmall Mode Volumes in Dielectric Optical Micro-cavities,” Phys. Rev. Lett . **95**, 143901 (2005). [PubMed]

*s̄*> 1/6, the minimum effective mode volume saturates at a value of

*V*= 0.225(λ

_{p}_{c})3 corresponding to that of a single nanobeam (i.e., no enhancement from energy being pushed into the slot gap). The effective mode volume evaluated at the center of the slot gap, on the otherhand, continues to rise with slot gap due to the exponential decay of the field in the gap.

*Q*-factor increases with the beam width due to the increased effective index of the guided mode, and thus increased effecive contrast of the quasi-1D photonic crystal. Due to the reduced lateral localization of the cavity mode, the effective mode volumes also increase with beam width. In contrast, the transverse

*Q*-factor remains approximately constant.

## 3. Mechanical mode analysis

*h*) and out-of-plane motion (labeled

*v*) of the two nanobeams. In addition there are compression (labeled

*c*) and twisting (labeled

*t*) modes of the beams. In this work we focus on the in-plane differential modes,

*h*, as these modes are the most strongly opto-mechanically coupled due to the large change in the slot gap per unit (strain) energy. We use FEM numerical simulations to calculate the mechanical mode patterns and mode frequencies, the first few orders of which are shown in Fig. 9. The material properties of silicon nitride for the FEM simulations were obtained from a number of references. Where possible we have used parameters most closely associate with stoichiometric, low-pressure-chemical-vapor-deposition (LPCVD) silicon nitride deposited on 〈100〉 Si: mass density ρ = 3100 kg/m

_{qd}^{3}, Young’s modulus

*Y*~ 290 GPa, internal tensile stress σ ~ 1 GPa[22, 23], coefficient of thermal expansion η

_{TE}= 3.3 × 10

^{-6}K

^{-1}, thermal conductivity κ

_{th}, ~ 20 W/m/K, and specific heat

*c*= 0.7 J/g/K.

_{sh}*μ*m, corresponding to a geometry with

*a*= 600 nm,

_{m}*t*= 400 nm,

*s*= 100 nm,

*w*= 700 nm,

*h*= 267 nm, and

_{x}*h*= 400 nm. The total number of air holes is set to

_{y}*N*= 55, ensuring a theoretical

_{h}*Q*-factor dominated by transverse radiation (

*Q*

_{⊥}), yielding a cavity length of

*l*= 36

*μ*m. The zipper cavity is clamped at both ends using fixed boundary conditions at the far ends of the “clamping pads” shown in Fig. 9. This clamping scheme is suitable for estimating the mechanical mode eigenfrequencies, although more complex clamping schemes envisioned for real devices will likely introduce modified splittings betweeen nearly-degenrate common and differential modes. The effective spring constant listed in Table 1 for each mode is based upon a motional mass equal to that of the true physical mass of the patterned nanobeams,

*m*≈ 43 picograms (see below for self-consistent definition of motional mass).

_{u}*h*

_{1d}mode is Ω/2π ≈ 8 MHz. The in-plane mode frequency of a doubly clamped beam, with

*l*≫

*w,t*, is approximately given by [22]:

*q*is the mode index (approximately an integer),

*A*=

*tw*is the cross-sectional area of the beams, and

*I*=

_{y}*tw*

^{3}/12 is the cross-sectional moment of inertial about the in-plane axis (

*ŷ*) of the beam. This fits the numerical data reasonably accurately assuming an effective beam width of

*w′*= (1 -

*f*)

*w*. From the scaling in eq. (2), one finds that for mode number

*q* 3 (where internal stress can be neglected and

*μ*m wavelength of operation. Due to the air-filling-fraction of the etched holes in the zipper cavity nanobeams, the thermal conductivity of the patterned beams is approximately Γ

_{th}, = 75% of the bulk value. A simple estimate for the thermal resistance of the zipper cavity due to conduction of heat through the nitride nanobeams themselves is

*R*~

_{th}*l*/(8

*tw*Γ

_{th}^th) ≈ 1.15 × 10

^{6}K/W, where the factor of 1/8 comes from the ability for heat to escape out either end of the nanobeams and in either direction. The physical mass of the zipper cavity, taking into account the etched holes, is approximately

*m*= 43 picograms. The heat capacity of the zipper cavity is then

*C*≈ 3 × 10

_{h}^{-11}J/K. From the heat capacity and the conductive thermal resistance, the thermal decay rate (appropriate for high vacuum conditions in which convection is negligible) is estimated to be

*γ*= 1/

_{th}*R*~ 2.9 × 10

_{th}C_{h}^{4}s

^{-1}. Finite-element-method simulations of the thermal properties of the zipper cavity yield an effective thermal resistance of

*R*= 1.09 × 10

_{th}^{6}K/W and a thermal decay rate of

*γ*= 5.26 × 10

_{th}^{4}s

^{-1}for temperature at the center of the zipper cavity, in reasonable correspondence to the estimated values.

## 4. Optomechanical coupling

*g*

_{OM}≡ dω

_{c}/d

*u*, where ω

_{c}is the cavity resonance frequency and

*u*represents an amplitude of the mechanical displacement. In the case of the commonly studied Fabry-Perot cavity [8

8. J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to micromechanical membrane,” Nature **452**, 72–75 (2008). [PubMed]

*g*

_{OM}= ω

_{c}/

*L*, where

_{c}*L*is approximately the physical length of the cavity. A similar relation holds for whispering gallery structures, such as the recently studied microtoroid[2], in which the optomechanical coupling is proportional to the inverse of the radius of cavity (

_{c}*R*),

*g*

_{OM}= ω

_{c}/

*R*. Both these devices utilize the radiation pressurce, or scattering, force of light. By comparison, the zipper cavity operates using the gradient force for which the optomechanical coupling length can be on the scale of the wavelength of light,

*L*

_{OM}~

*λ*. Similar to the scaling found in the previous section for the effective mode volume, the optomechanical coupling length scales exponentially with the slot gap,

_{c}*L*~ λ

_{OM}_{c}

*e*

^{αs}, where α is proportional to the refractive index contrast between the nanobeams forming the zipper cavity and the surrounding cladding.

*s̄*≡

*s*/

*a*. Due to the strong intensity of the bonded mode in the center of the slot gap, it tunes more quickly than the anti-bonded mode. This tuning curve can be used to estimate the optomechanical coupling for the in-plane differential mode of motion of the zipper cavity nanobeams. The in-plane common mechanical modes and both types of vertical mechanical modes are expected to provide a much smaller level of optomechanical coupling due to the reduced change in slot gap with these types of motion. For complex geometries and motional patterns, one must use a consistent definition of displacement amplitude,

_{m}*u*, in determing

*g*OM,

*m*

_{eff}(motional mass), and

*k*

_{eff}(effective spring constant). In this work we use a convention in which

*u*(

*t*) represents the amplitude of motion for a normalized mechanical eigenmode displacement field pattern:

*n*is a mode label,

*l*is the length of the nanobeams, and for the simple two-beam geometry considered here,

*i*is an index indicating either the first or second nanobeam and we write the displacement vector only as a function of position along the long axis of the nanobeams (

*x*). With this definition of

*amplitude*, the effective motional mass is simply the total mass of the two nanobeams (

*m*= 43 picograms), and the effective spring constant is defined by the usual relation

_{u}*k*=

_{u}*m*Κ

_{u}^{2}

_{M}, with Ω

_{M}the mechanical eigenmode frequency. The amplitude associated with zero-point motion and used in the equipartition theorem to determine the thermal excitation of the mechanical mode is then

*u*(

_{n}*t*). Note, an alternative, but equally effective method, defines the amplitude first, and then adjusts the effective motional mass based upon the strain energy of the mechanical motion.

*q*th odd-order in-plane differential mechanical mode,

**u**

_{hqd}

^{o}(

*x,t*) ≈

*u*,(

_{hqd}*t*)(

*ŷ*

_{1}cos(

*qπx*/

*l*)+

*ŷ*

_{2}cos(

*qπx*/

*l*)), where

*ŷ*

_{1}and

*ŷ*

_{2}are (transverse) in-plane unit vectors associated with first and second nanobeams, respectively, and which point in opposite directions away from the center of the gap between the nanobeams. The even-order modes are anti-symmetric about the long axis of the cavity and are given approximately by,

**u**

_{hqd}

^{e}(

*z,t*) ≈

*u*(

_{hqd}*t*)(

*ŷ*

_{1}sin(

*q̂x*/

*l*) +

*ŷ*

_{2}sin(

*qπx*/

*l*)). To be consistent then, with this definition of mode displacement amplitude,

*g*

_{OM}must be defined in terms of the rate of change of cavity frequency with respect to

*half*the change in slot gap (

*g*

_{OM,0}≈ dω

_{c}/½dδ

*s*), as the amplitude

*u*(

_{hqd}*t*) corresponds to a (peak) change in slot gap of δ

*s*= 2

*u*(

_{hqd}*t*).

*s̄*= 0.1 (or roughly

*s*= 0.04λ

_{c}), the optomechanical coupling length to the fundamental bonded optical mode can be as small as

*L*

_{OM}/

*a*≈ 2. For the normalized frequency of the bonded mode (

_{m}*a*/λ

_{m}_{c}≈ 0.4), this corresponds to

*L*

_{OM}≈ 0.8λ

_{c}, as expected from the arguments laid out in the introduction. The TE

_{-,0}has a significantly smaller optomechanical coupling due to its reduced electric field energy in the slot.

_{+,0}optical mode is approximately given by,

**f̄**

_{hqd}(

*x;i*) is the normalized displacement vector of either of the nanobeams (=cos(

*qπx*/l) (sin(

*qπx*/

*l*)) for the odd(even)-order in-plane differential mode),

**Ẽ**

_{TE+,0}(

*x*) is the Gaussian-like envelope of the TE

_{+,0}optical cavity mode along the long-axis of the nanobeams, and

*g*

_{OM,0}is the optomechanical coupling factor as calculated from the tuning curve of Fig. 10.

*h*mechanical modes, and the even symmetry of the optical intensity for the zipper cavity optical modes, the optomechanical coupling to the even-order

_{qd}*h*modes is approximately zero. The odd-order in-plane differential modes, on the otherhand, have an anti-node of mechanical displacement at the optical cavity center and an even long-axis symmetry. For mode numbers small enough that the half-wavelength of the mechanical mode is roughly as large, or larger, than the effective length of the optical cavity mode along the axis of the beam, the optical mode will only sense the central half-wave displacement of the mechanical mode and the optomechanical coupling should still be quite large. As an example, from the intensity plot of the TE

_{qd}_{+,0}fundamental bonded optical mode in Fig. 3, the effective length of the optical mode along the long-axis of the nanobeams is

*L*

_{eff}~ 7

*a*= 4.2

_{m}*μ*m (for λ

_{c}= 1.5

*μ*m). The mechanical mode index

*q*is roughly equal to the number of half-wavelengths of the mechanical mode along the axis of the zipper cavity. Therefore, for the zipper cavity of length

*l*= 36

*μ*m studied above, the 9th order in-plane differential mechanical mode with Ω

_{h9d}/2π ≈ 170 MHz has a half-wavelength equal to 4

*μ*m ≈

*L*

_{eff}. The resulting optomechanical coupling of the

*h*

_{9d}mechanical mode to the TE

_{+,0}optical mode is then still relatively large, equal to approximately half that of the coupling to the fundamental

*h*

_{1d}mechanical mode. The optomechanical coupling factor to the TE

_{+,0}for each of the in-plane differential modes is tabulated along with the mechanical mode properties in Table 1.

*g*

_{OM}. The per-photon mechanical force is given by

*F*, =

_{ph}*h̄g*

_{OM}. For the zipper cavity, this yields a per-photon force of

*F*, ≈

_{ph}*h̄*ω

_{c}/λ

_{c}, which at near-infrared wavelengths corresponds to 0.2 pN/photon. Such a force could be measured using other, non-optical techniques, and could provide a means for detecting single photons in a non-demolition manner. Also, through the optomechanical coupling, intra-cavity light can stiffen [31, 7

7. T. Corbitt, C. Wipf, T. Bodiya, D. Ottaway, D. Sigg, N. Smith, S. Whitcomb, and N. Mavalvala, “Optical Dilution and Feedback Cooling of a Gram-Scale Oscillator to 6.9 mK,” Phys. Rev. Lett . **99**, 160801 (2007). [PubMed]

32. M. Hossein-Zadeh and K. J. Vahala, “Observation of optical spring effect in a microtoroidal optomechanical resonator,” Opt. Lett . **32**, 1611–1613 (2007). [PubMed]

4. O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature **444**, 71–73 (2006). [PubMed]

5. S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. As-pelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature **444**, 67–70 (2006). [PubMed]

7. T. Corbitt, C. Wipf, T. Bodiya, D. Ottaway, D. Sigg, N. Smith, S. Whitcomb, and N. Mavalvala, “Optical Dilution and Feedback Cooling of a Gram-Scale Oscillator to 6.9 mK,” Phys. Rev. Lett . **99**, 160801 (2007). [PubMed]

8. J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to micromechanical membrane,” Nature **452**, 72–75 (2008). [PubMed]

9. T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett . **95**, 033901 (2005). [PubMed]

_{M}≪ Γ) the effective mechanical frequency (Ω′

_{M}) and damping rate (γ′

_{M}) are given by the following relations (see Ref [2]):

_{M}and γ

_{M}are the bare mechanical properties of the zipper cavity, |

*a*

_{0}|

^{2}is the time-averaged stored optical cavity energy, ∆′

_{o}≡ ω

_{l}- ω

_{c}is the laser-cavity detuning, Γ is the waveguide-loaded optical cavity energy decay rate, and ∆′

_{o}= (∆′

_{o})

^{2}+ (Γ/2)

^{2}. The maxmimum “optical spring” effect occurs at a detuning point of ∆′

_{o}= Γ/2. Extrapolating equation (5) down to the single photon level (ignoring quantum fluctuations for arguments sake), for this laser-cavity detuning a single stored cavity photon introduces a shift in the mechanical frequency corresponding to ∆(Ω

_{M}

^{2})Ω

_{M}

^{2}= (2

*Q*

*h̄*ω

_{c})/(λ

_{c}

^{2}

*k*

_{eff}). For the zipper cavity, with

*Q*= 5 × 10

^{6}, λ

_{c}= 1

*μ*m, and

*k*

_{h1d}= 100 N/m, the resulting single-photon mechanical frequency shift is approximately ∆(Ω

_{M})/Ω

_{M}= 1%.

## 5. Summary and discussion

*g*

_{OM}~ ω

_{c}/λ

_{c}, where ω

_{c}and λ

_{c}are the optical resonant cavity frequency and wavelength, respectively. This coupling is several orders of magnitude larger than has been demonstrated in high-Finesse Fabry-Perot cavities, and is more than order of magnitude larger than for whispering-gallery micrototoid structures, both of which rely upon the radiation pressure force. Finite-element-method (FEM) simulations of the zipper cavity show that a structure with an optical

*Q*= 5 × 10

^{6}, mechanical resonance frequency of Ω

_{M}/2π ≈ 170 MHz, and motional mass of

*m*≈ 40 picograms is possible. In the future, further increase in the mechanical frequency and reduction in the motional mass may be attained by using planar phononic crystals [34] to form the mechanical resonator. The combination of phononic and photonic crystals would also provide an integrated, chip-scale platform for routing and coupling optical and mechanical energy.

_{u}28. J. T. Robinson, C. Manolatou, L. Chen, and M. Lipson, “Ultrasmall Mode Volumes in Dielectric Optical Micro-cavities,” Phys. Rev. Lett . **95**, 143901 (2005). [PubMed]

*V*

_{eff}~ 0.2(λ

_{c}

^{3}) for a slot gap of

*s*~ λ

_{c}/10. As an example, one can imagine placing nanoparticles of diamond (a popular solid-state system for quantum information processing [35, 36, 37

37. F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, “Observation of Coherent Oscillation of a Single Nuclear Spin and Realization of a Two-Qubit Conditional Quantum Gate,” Phys. Rev. Lett . **93**, 130501 (2004). [PubMed]

41. S. Kühn, C. Hettich, C. Schmitt, J-PH. Poizat, and V. Sandoghdar, “Diamond colour centres as a nanoscopic light source for scanning near field microscopy,” J. Microsc . **202**, 2–6 (2001). [PubMed]

42. S. Schietinger, T. Schroder, and O. Benson, “One-by-One Coupling of Single Defect Centers in Nanodiamonds to High-Q Modes of an Optical Microresonator,” Nano. Lett . **8**, 3911–3915 (2008). [PubMed]

*V*

_{eff}would produce a coherent coupling rate with the zero-phonon-line (ZPL) of the NV

^{-}transition of approximately,

*g*ZPL/2π ~ 3 GHz, even after accounting for the 3 – 5% branching ratio for the ZPL line. This is more than 100 times the radiative-limited linewidth measured for the NV

^{-}transition (12 MHz), and more than 10 times the theoretical zipper cavity decay rate (90 MHz), putting the coupled system deep within the strong coupling regime. The additional benefit provided by the zipper cavity is the ability to rapidly tune the cavity frequency into and out of resonance with the ZPL of the NV

^{-}transition. If mechanical resonance frequencies could be increased towards GHz values, using the suggested phononic crystal concepts for instance, then new approaches to photon-mediated quantum interactions and quantum state transfer can be envisioned for solid-state cavity QED systems.

## Acknowledgments

## References and links

1. | H. J. Metcalf and P. van der Straten, |

2. | T. J. Kippenberg and K. Vahala, “Cavity Optomechanics,” Opt. Express |

3. | T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science |

4. | O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature |

5. | S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. As-pelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature |

6. | A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation Pressure Cooling of a Micromechanical Oscillator Using Dynamical Backaction,” Phys. Rev. Lett . |

7. | T. Corbitt, C. Wipf, T. Bodiya, D. Ottaway, D. Sigg, N. Smith, S. Whitcomb, and N. Mavalvala, “Optical Dilution and Feedback Cooling of a Gram-Scale Oscillator to 6.9 mK,” Phys. Rev. Lett . |

8. | J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to micromechanical membrane,” Nature |

9. | T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett . |

10. | C. A. Regal, J. D. Tuefel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nature Phys . |

11. | T. D. Stowe, K. Yasumura, T. W. Kenny, D. Botkin, K. Wago, and D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett . |

12. | M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett . |

13. | M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, “Optomechanical Wavelength and Energy Conversion in High-Q Double-Layer Cavities of Photonic Crystal Slabs,” Phys. Rev. Lett . |

14. | J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-Bandgap microcavities in optical waveguides,” Nature |

15. | C. Sauvan, P. Lalanne, and J. P. Hugonin, “Slow-wave effect and mode-profile matching in photonic crystal microcavities,” Phys. Rev . B |

16. | A. R. M. Zain, N. P. Johnson, M. Solel, and R. M. D. L. Rue, “Ultra high quality factor one-dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI),” Opt. Express |

17. | P. Velha, J. C. Rodier, P. Lalanne, J. D. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, “Ultracompact silicon-on-insulator ridge-waveguide mirrors with high reflectance,” Appl. Phys. Lett . |

18. | M. Notomi, E. Kuramochi, and H. Taniyama, “Ultrahigh-Q Nanocavity with 1D Photonic Gap,” Opt. Express |

19. | M. W. McCutcheon and M. Loncar, “Design of a silicon nitride photonic crystal nanocavity with a Quality factor of one million for coupling to a diamond nanocrystal,” Opt. Express |

20. | O. Painter, K. Srinivasan, and P. E. Barclay, “A Wannier-like equation for photon states of locally perturbed photonic crystals,” Phys. Rev . B |

21. | P. E. Barclay, K. Srinivasan, O. Painter, B. Lev, and H. Mabuchi, “Integration of fiber-coupled high-Q SiNx microdisks with atom chips,” Appl. Phys. Lett . |

22. | S. S. Verbridge, J. M. Parpia, R. B. Reichenbach, L. M. Bellan, and H. G. Craighead, “High quality factor resonance at room temperature with nanostrings under high tensile stress,” J. Appl. Phys . |

23. | S. S. Verbridge, H. G. Craighead, and J. M. Parpia, “A megahertz nanomechanical resonator with room temperature quality factor over a million,” Appl. Phys. Lett . |

24. | P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microcavities excited via an integrated waveguide and fiber taper,” Opt. Express |

25. | MIT Photonic Bands (MPB) is a free software package for the solution of the electromagnetic eigenmodes of periodic structures. MPB has been developed at MIT, http://ab-initio.mit.edu/wiki/index.php/MPB. |

26. | COMSOL is a multiphysics software package for performing finite-element-method (FEM) simulations. See COMSOL AB, http://www.comsol.com/. We use the COMSOL multiphysics software package to perform both optical and mechanical numerical simulations of the zipper cavity. |

27. | Lumerical FDTD Solutions is a finite-difference time-domain software package for performing electromagnetic simulations. See Lumerical Solutions, Inc., http://www.lumerical.com/. |

28. | J. T. Robinson, C. Manolatou, L. Chen, and M. Lipson, “Ultrasmall Mode Volumes in Dielectric Optical Micro-cavities,” Phys. Rev. Lett . |

29. | M. Pinard, Y. Hadjar, and A. Heidmann, “Effective mass in quantum effects of radiation pressure,” Eur. Phys. J . D |

30. | S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory and Maxwell’s equations with shifting material boundaries,” Phys. Rev . E |

31. | P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes, “Theory of radiation-pressure-driven interferometers,” J. Opt. Soc. Am . B |

32. | M. Hossein-Zadeh and K. J. Vahala, “Observation of optical spring effect in a microtoroidal optomechanical resonator,” Opt. Lett . |

33. | B. S. Sheard, M. B. Gray, C. M. Mow-Lowry, D. E. McClelland, and S. E. Whitcomb, “Observation and characterization of an optical spring,” Phys. Rev . A |

34. | R. H. O. III and I. El-Kady, “Microfabricated phononic crystal devices and applications,” Meas. Sci. Technol . |

35. | A. Gruber, A. Dräbenstedt, C. Tietz, L. Fleury, J. Wratchtrup, and C. v. Borczyskowski, “Scanning confocal optical microscopy and magnetic resonance on single defect centers,” Science |

36. | C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D. Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A. D. Greentree, S. Prawer, F. Jelezko, and P. Hemmer, “Coherent Population Trapping of Single Spins in Diamond under Optical Excitation,” Phys. Rev. Lett . |

37. | F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, “Observation of Coherent Oscillation of a Single Nuclear Spin and Realization of a Two-Qubit Conditional Quantum Gate,” Phys. Rev. Lett . |

38. | M. V. G. Dutt, L. Childress, L. Jiang, E. Togan, J. Maze, F. Jelezko, A. S. Zibrov, P. R. Hemmer, and M. D. Lukin, “Quantum Register Based on Individual Electronic and Nuclear Spin Qubits in Diamond,” Science |

39. | L. Childress, M. V. Gurudev Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin, “Coherent Dynamics of Coupled Electron and Nuclear Spin Qubits in Diamond,” Science |

40. | A. Beveratos, R. Brouri, T. Gacoin, J.-P. Poizat, and P. Grangier, “Nonclassical radiation from diamond nanocrys-tals,” Phys. Rev . A |

41. | S. Kühn, C. Hettich, C. Schmitt, J-PH. Poizat, and V. Sandoghdar, “Diamond colour centres as a nanoscopic light source for scanning near field microscopy,” J. Microsc . |

42. | S. Schietinger, T. Schroder, and O. Benson, “One-by-One Coupling of Single Defect Centers in Nanodiamonds to High-Q Modes of an Optical Microresonator,” Nano. Lett . |

43. | P. E. Barclay, O. Painter, C. Santori, K.-M. Fu, and R. G. Beausoleil, “Coherent interference effects in a nano-assembled optical cavity-QED system,” ArXiv:0812.4505 (2008). |

**OCIS Codes**

(230.5750) Optical devices : Resonators

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: January 5, 2009

Revised Manuscript: February 25, 2009

Manuscript Accepted: February 25, 2009

Published: February 26, 2009

**Citation**

Jasper Chan, Matt Eichenfield, Ryan Camacho, and Oskar Painter, "Optical and mechanical design of a “zipper” photonic crystal
optomechanical cavity," Opt. Express **17**, 3802-3817 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-5-3802

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### References

- H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping, Graduate Texts in Contemporary Physics (Springer, New York, NY, 1999).
- T. J. Kippenberg and K. Vahala, "Cavity Optomechanics," Opt. Express 15, 17,172-17,205 (2007).
- T. J. Kippenberg and K. J. Vahala, "Cavity Optomechanics: Back-Action at the Mesoscale," Science 321, 1172-1176 (2008). [PubMed]
- O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, "Radiation-pressure cooling and optomechanical instability of a micromirror," Nature 444, 71-73 (2006). [PubMed]
- S. Gigan, H. R. B¨ohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. B¨auerle, M. Aspelmeyer, and A. Zeilinger, "Self-cooling of a micromirror by radiation pressure," Nature 444, 67-70 (2006). [PubMed]
- A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, "Radiation Pressure Cooling of a Micromechanical Oscillator Using Dynamical Backaction," Phys. Rev. Lett. 97, 243905 (2006).
- T. Corbitt, C. Wipf, T. Bodiya, D. Ottaway, D. Sigg, N. Smith, S. Whitcomb, and N. Mavalvala, "Optical Dilution and Feedback Cooling of a Gram-Scale Oscillator to 6.9 mK," Phys. Rev. Lett. 99, 160801 (2007). [PubMed]
- J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, "Strong dispersive coupling of a high-finesse cavity to micromechanical membrane," Nature 452, 72-75 (2008). [PubMed]
- T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, "Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity," Phys. Rev. Lett. 95, 033901 (2005). [PubMed]
- C. A. Regal, J. D. Tuefel, and K. W. Lehnert, "Measuring nanomechanical motion with a microwave cavity interferometer," Nat. Phys. 4, 555-560 (2008).
- T. D. Stowe, K. Yasumura, T. W. Kenny, D. Botkin, K. Wago, and D. Rugar, "Attonewton force detection using ultrathin silicon cantilevers," Appl. Phys. Lett. 71, 288-290 (1997).
- M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, "Evanescent-wave bonding between optical waveguides," Opt. Lett. 30, 3042-3044 (2005). [PubMed]
- M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, "Optomechanical Wavelength and Energy Conversion in High-Q Double-Layer Cavities of Photonic Crystal Slabs," Phys. Rev. Lett. 97, 023903 (2006). [PubMed]
- J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-Bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
- C. Sauvan, P. Lalanne, and J. P. Hugonin, "Slow-wave effect and mode-profile matching in photonic crystal microcavities," Phys. Rev. B 71, 165118 (2005).
- A. R. M. Zain, N. P. Johnson, M. Solel, and R. M. D. L. Rue, "Ultra high quality factor one-dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI)," Opt. Express 16, 12,084-12,089 (2008).
- P. Velha, J. C. Rodier, P. Lalanne, J. D. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, "Ultracompact silicon-on-insulator ridge-waveguide mirrors with high reflectance," Appl. Phys. Lett. 89, 171121 (2006).
- M. Notomi, E. Kuramochi, and H. Taniyama, "Ultrahigh-Q Nanocavity with 1D Photonic Gap," Opt. Express 16, 11095-11102 (2008).
- M.W. McCutcheon and M. Loncar, "Design of a silicon nitride photonic crystal nanocavity with a Quality factor of one million for coupling to a diamond nanocrystal," Opt. Express 16, 19,136-19,145 (2008).
- O. Painter, K. Srinivasan, and P. E. Barclay, "A Wannier-like equation for photon states of locally perturbed photonic crystals," Phys. Rev. B 68, 035214 (2003).
- P. E. Barclay, K. Srinivasan, O. Painter, B. Lev, and H. Mabuchi, "Integration of fiber-coupled high-Q SiNx microdisks with atom chips," Appl. Phys. Lett. 89, 131108 (2006).
- S. S. Verbridge, J. M. Parpia, R. B. Reichenbach, L. M. Bellan, and H. G. Craighead, "High quality factor resonance at room temperature with nanostrings under high tensile stress," J. Appl. Phys. 99, 124304 (2006).
- S. S. Verbridge, H. G. Craighead, and J. M. Parpia, "A megahertz nanomechanical resonator with room temperature quality factor over a million," Appl. Phys. Lett. 92, 013112 (2008).
- P. E. Barclay, K. Srinivasan, and O. Painter, "Nonlinear response of silicon photonic crystal microcavities excited via an integrated waveguide and fiber taper," Opt. Express 13, 801-820 (2005). [PubMed]
- MIT Photonic Bands (MPB) is a free software package for the solution of the electromagnetic eigenmodes of periodic structures. MPB has been developed at MIT, http://ab-initio.mit.edu/wiki/index.php/MPB.
- COMSOL is a multiphysics software package for performing finite-element-method (FEM) simulations. See COMSOL AB, http://www.comsol.com/We use the COMSOL multiphysics software package to perform both optical and mechanical numerical simulations of the zipper cavity.
- LumericalFDTD Solutions is a finite-difference time-domain software package for performing electromagnetic simulations. See Lumerical Solutions, Inc., http://www.lumerical.com/.
- J. T. Robinson, C. Manolatou, L. Chen, and M. Lipson, "Ultrasmall Mode Volumes in Dielectric Optical Microcavities," Phys. Rev. Lett. 95, 143901 (2005). [PubMed]
- M. Pinard and Y. Hadjar and A. Heidmann, "Effective mass in quantum effects of radiation pressure," Eur. Phys. J. D 7, 107-116 (1999).
- S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, "Perturbation theory and Maxwell’s equations with shifting material boundaries," Phys. Rev. E 65, 066611 (2002).
- P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes, "Theory of radiation-pressure-driven interferometers," J. Opt. Soc. Am. B 2, 1830-1840 (1985).
- M. Hossein-Zadeh and K. J. Vahala, "Observation of optical spring effect in a microtoroidal optomechanical resonator," Opt. Lett. 32, 1611-1613 (2007). [PubMed]
- B. S. Sheard, M. B. Gray, C. M. Mow-Lowry, D. E. McClelland, and S. E. Whitcomb, "Observation and characterization of an optical spring," Phys. Rev. A 69, 051801(R) (2004).
- R. H. O. III and I. El-Kady, "Microfabricated phononic crystal devices and applications," Meas. Sci. Technol. 20, 012002 (2008).
- A. Gruber, A. Drabenstedt, C. Tietz, L. Fleury, J. Wratchtrup, and C. v. Borczyskowski, "Scanning confocal optical microscopy and magnetic resonance on single defect centers," Science 276, 2012-2014 (1997).
- C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D. Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A. D. Greentree, S. Prawer, F. Jelezko, and P. Hemmer, "Coherent Population Trapping of Single Spins in Diamond under Optical Excitation," Phys. Rev. Lett. 97, 247401 (2006).
- F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, "Observation of Coherent Oscillation of a Single Nuclear Spin and Realization of a Two-Qubit Conditional Quantum Gate," Phys. Rev. Lett. 93, 130501 (2004). [PubMed]
- M. V. G. Dutt, L. Childress, L. Jiang, E. Togan, J. Maze, F. Jelezko, A. S. Zibrov, P. R. Hemmer, and M. D. Lukin, "Quantum Register Based on Individual Electronic and Nuclear Spin Qubits in Diamond," Science 316, 1312-1316 (2007).
- L. Childress, M. V. Gurudev Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin, "Coherent Dynamics of Coupled Electron and Nuclear Spin Qubits in Diamond," Science 314, 281-285 (2006).
- A. Beveratos, R. Brouri, T. Gacoin, J.-P. Poizat, and P. Grangier, "Nonclassical radiation from diamond nanocrystals," Phys. Rev. A 64, 061802 (2001).
- S. K¨uhn and C. Hettich and C. Schmitt and J-PH. Poizat and V. Sandoghdar, "Diamond colour centres as a nanoscopic light source for scanning near field microscopy," J. Microsc. 202, 2-6 (2001). [PubMed]
- S. Schietinger, T. Schroder, and O. Benson, "One-by-One Coupling of Single Defect Centers in Nanodiamonds to High-Q Modes of an Optical Microresonator," Nano. Lett. 8, 3911-3915 (2008). [PubMed]
- P. E. Barclay, O. Painter, C. Santori, K.-M. Fu, and R. G. Beausoleil, "Coherent interference effects in a nanoassembled optical cavity-QED system," ArXiv:0812.4505 (2008).

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