Abstract

Using a combination of resist reflow to form a highly circular etch mask pattern and a low-damage plasma dry etch, high-quality-factor silicon optical microdisk resonators are fabricated out of silicon-oninsulator (SOI) wafers. Quality factors as high as Q=5×106 are measured in these microresonators, corresponding to a propagation loss coefficient as small as α~0.1 dB/cm. The different optical loss mechanisms are identified through a study of the total optical loss, mode coupling, and thermally-induced optical bistability as a function of microdisk radius (5-30 µm). These measurements indicate that optical loss in these high-Q microresonators is limited not by surface roughness, but rather by surface state absorption and bulk free-carrier absorption.

©2005 Optical Society of America

1. Introduction

Silicon (Si) photonics has received renewed interest of late due to the rapidly approaching limits of “Moore’s Law” scaling in Si microelectronics, and the potential to leverage the near half-century of processing development in the microelectronics industry [1, 2]. There has followed several recent advances in Si optoelectronics, among them the demonstrations of a high-speed Si optical modulator (>1 Gbit/sec [3]), an all-optical high-speed switch [4], and a nonlinear optical Si laser source based on the stimulated Raman effect [5, 6]. Aiding in these and previous developments of integrated optical and electronic Si circuits is the availability of high index contrast silicon-on-insulator (SOI) wafers, which provide the tight optical confinement of light necessary for high-density optoelectronic integration and nonlinear optics, and exceptional photonic and electronic isolation through the high quality underlying thermal oxide. As Si microphotonic device functionality and integration advances, and light is more often routed into the Si, it will be important to develop low-loss Si microphotonic circuits in addition to the already low-loss glass-based Planar Lightwave Circuits (PLCs) [7, 8]. One key element in such circuits is the microresonator, where light can be distributed by wavelength or localized to enhance nonlinear interactions. Here we report advances in silicon microfabrication which have allowed the creation of SOI-based microdisk optical resonators [9] with extremely smooth high-index-contrast etched sidewalls. These microdisks provide tight optical confinement down to microdisk radii of 1.5 µm, while maintaining the low loss of the high-purity crystalline silicon. Resonant mode quality factors as high as Q~5×106 are measured, corresponding to an effective propagation loss as small as α~0.1 dB/cm.

Inspired by the ultra-smooth glass microspheres [10, 11] and microtoroids [12] formed under surface tension, in this work an electron-beam resist reflow technique is used to significantly reduce surface imperfections in the edge of the microdisk resonator. A fiber-based, wafer-scale probing technique is used to rapidly and non-invasively test the optical properties of these fabricated microdisks. After a brief description of the fabrication and measurement technique, a comprehensive analysis of the different optical loss mechanisms in these structures is presented below, with a detailed theoretical treatment in the appendices. These measurements indicate that with the reduced level of microdisk surface roughness, we are in fact quantitatively probing the absorption of the surface-states at the edge of the disk (for smaller microdisks) and the bulk Si free-carrier absorption due to ionized dopants (for larger microdisks). Owing to the high finesse of the Si microdisks, the input pump powers required to observe even extremely weak nonlinearities such as two-photon absorption (TPA) and Raman scattering is reduced to the microwatt level. In what follows we study the low power thermally-induced optical bistability, including the distortion effects of TPA and free-carrier absorption (FCA), as a second, more direct, measure of the linear absorption component of the optical loss. These measurements indicate that for further reduction in the optical loss of high-index contrast Si structures, such as the microdisks studied here, new passivation and annealing techniques, analogous to those used in the manufacture of high-quality Si CMOS devices [13], will need to be developed.

2. Fabrication and measurement technique

The silicon microdisks characterized in this work are fabricated from a silicon-on-insulator (SOI) wafer with 344 nm thick p-doped Si device layer on a 2 µm SiO 2 layer (Fig. 1(c)). Processing of the microdisks begins with the deposition of a 20 nm SiO 2 protective cap layer using plasma-enhanced chemical-vapor-deposition. An electron-beam resist, Zeon®ZEP520A, is spin-coated onto the wafer at 6000 rpm for 60 sec, resulting in a 400 nm thick film. Disks of radii ranging from 5–30 µm are defined in the electron-beam resist. The wafer is then subjected to a post-lithography bake. By suitable choice of temperature and duration, this bake can significantly reduce imperfections in the electron-beam resist pattern. Temperatures too low do not result in resist reflow, while temperatures too high can cause significant loss of resist to sublimation. A temperature high enough to allow the resist to reflow must be reached and maintained for the imperfections in the resist pattern to be reduced. The appropriate duration and temperature for the resist prepared as described above was empirically determined to be 5 minutes at 160°C. After the reflow process, the roughness in the patterns is greatly reduced, and the sidewall angle is reduced from 90° to approximately 45°. The resulting angled mask is prone to erosion during the etch process, and so the inductively-coupled-plasma reactive-ion etch is optimized to minimize roughness caused by mask erosion.

The patterns are then transferred into the Si device layer using a low DC-bias, inductively-coupled-plasma reactive-ion etch with SF6/C4F8 chemistry [14, 15]. To enable optical fiber probing of the devices (as described below), an etch-mask surrounding the disks is photolitho-graphically defined and the wafer surrounding the disks etched down several microns, leaving the devices isolated on a mesa. Following a Piranha etch to remove organic materials, a dilute hydrofluoric acid solution is used to remove the protective SiO 2 layer and partially undercut the disk (Fig. 1(c)). The undercut pedestal takes on its angular hour-glass shape due to a higher etch rate on the wafer Unibond®[2] versus the bulk silica. The wafer is then rinsed in deionized water and dried with clean, dry N2. Upon completion of the processing, the wafer is immediately removed to an N2 purged enclosure for characterization.

To characterize the microdisk resonators an evanescent fiber taper coupling technique is employed [9, 14]. In this process, an optical fiber is adiabatically drawn to a 1–2 µm diameter using a hydrogen torch so that its evanescent field is made accessible to the environment. Using DC motor stages with 50 nm encoded resolution, the fiber taper can be accurately positioned within the microdisk near-field so as to evanescently couple power into the microdisks. Measurements of the taper transmission as a function of the taper-microdisk gap and input power are performed using swept wavelength tunable laser sources (λ=1410–1625 nm, linewidth <5 MHz). A set of paddle wheels is used to adjust the polarization state of the fiber taper mode in the microdisk coupling region, providing selective coupling to the TE-like (TM-like) whispering gallery modes (WGMs) with dominant electric field parallel (normal) to the plane of the microdisk. Further details of the fiber taper apparatus and measurement techniques can be found in Refs. [16, 14].

 figure: Fig. 1.

Fig. 1. Schematic representation of a fabricated silicon microdisk. (a) Top view showing ideal disk (red) against disk with roughness. (b) Top view close-up illustrating the surface roughness, Δr(s), and surface reconstruction, ξ. Also shown are statistical roughness parameters, σ r and Lc , of a typical scatterer. (c) Side view of a fabricated SOI microdisk highlighting idealized SiO2 pedestal.

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3. Experimental results and analysis

In order to study the optical loss mechanisms within the Si microdisk resonators, a series of disks with radii, R=5,10,15,20,30 µm, was created. The intrinsic optical loss in these structures can be quantified by four different components of a modal quality factor, Q i , according to

1Qi=1Qr+1Qss+1Qsa+1Qb,

where Qr , Qss , Qsa , Qb are related to optical loss due to radiation, surface scattering, surface absorption, and absorption in the bulk Si, respectively. Due to the near ideal nature of the optical confinement of the silicon whispering-gallery-mode (WGM) resonator, the radiation losses become increasingly negligible as the disk radius is increased (Qr ≳108 for R >1.5 µm [17]). While all of the disks considered in this work possess inconsequential radiation losses, the better radial confinement of larger radii microdisks pulls the WGMs away from the disk edge, as described in Appendix E. Thus, varying the disk radius provides a means to separate sidewall surface effects, quantified by Qss and Qsa , from bulk effects, Qb .

A typical taper transmission spectrum of an R=30 µm microdisk is shown in Fig. 2. The observed double resonance dip (doublet) is a result of Rayleigh scattering from disk surface roughness, illustrated in Fig. 1(b) as Δr(s). The surface imperfections created during fabrication lift the degeneracy of clockwise (cw) and counter-clockwise (ccw) propagating WGMs in the microdisk, creating instead standing wave modes [18, 19, 20, 21]. As described in Ref. [9], the highest Q WGMs in these microdisks are found to be of TM-like polarization and of radial mode number n=1, where the field interaction with the disk-edge surface is minimized (see Appendices). For the wavelengths studied here, the corresponding azimuthal number is M~60 for the R=5 µm disks, and scales approximately linearly with radius for larger microdisks. All of the WGMs studied in this work were confirmed to be of TM-like polarization and of radial number n~1 through studies of their polarization and position dependent coupling to the fiber taper [9].

 figure: Fig. 2.

Fig. 2. Taper transmission versus wavelength showing a high-Q doublet mode for the R=30 µm disk. Qcλ 0/δλ c and Qsλ 0/δλ s are the unloaded quality factors for the long and short wavelength modes respectively, where δλ c and δλ s are resonance linewidths. Also shown is the doublet splitting, Δλ, and normalized splitting quality factor, Qβλ 0λ.

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A statistical model for the doublet splitting is constructed using the effective index model from Appendix A and the time dependent perturbation theory described in Appendix C. The resulting model [20, 21] is used to fit the data in Fig. 2 with linewidth parameters, δλ c and δλs , and doublet splitting, Δλ. Two independent linewidth parameters must be used because the resulting orthogonal standing wave modes sense different regions of the disk surface [21]. A normalized measure of the mode splitting is defined to be the free-spacewavelength, λ 0, divided by the total resonance splitting (Δλ), given here by Qβ . To illustrat the utility of this defintion, limiting cases are taken: QβQi would indicate no doublet splitting and pure cw and ccw traveling WGMs; conversely, QβQi would indicate large doublet splitting and consequently widely separated standing wave WGMs. In the latter case, Qβ gives a better measure of the required coupling strength and the useful bandwidth at critical coupling, where all power is transferred into the resonant cavity from a bus waveguide (see inset to Fig. 3).

Figure 3 plots Qβ for each of the measured microdisk radii, where for each microdisk we plot the results for the four highest Qi doublet modes in the 1410–1500 nm wavelength range. Combining the results of Appendix C with an approximate form for disk edge energy density (see Appendix E), we get the following equation for the normalized doublet splitting parameter:

Qβλ0Δλ=12π34ξ(VdVs),

where Vd is simply the physical volume of the microdisk and ξ is the relative dielectric contrast constant defined as

ξ=n̅2(nd2n02)nd2(n̅2n02).

nd ~3.55, n 0=1.0, and are the indices of refraction for the silicon disk, cladding, and 2-D effective slab, respectively [22, 23]. The key parameter from this analysis is the effective volume of a typical scatterer, defined as VsRLchσr, where Lc is the correlation length of the roughness, h is the disk height, and σ r is the standard deviation of the roughness amplitude (illustrated in Fig. 1(b)). Fitting the doublet splitting versus disk radius in Fig. 3 (solid blue curve) to the Qβ ~R 3/2 dependence shown in Eq. (2), gives a value of Lcσr=2.7nm32. This parameter is also useful in estimating optical loss because the same Rayleigh scattering mechanism responsible for lifting the azimuthal degeneracy couples the unperturbed microdisk modes to radiation modes [24, 20].

 figure: Fig. 3.

Fig. 3. Normalized doublet splitting (Qβ ) versus disk radius. (inset) Taper transmission data and fit of deeply coupled doublet demonstrating 14 dB coupling depth.

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From an analysis similar to that used for the mode coupling, the surface scattering quality factor, Qss , is approximated as (see Appendix B):

Qss=3λ038π72n0δn2ξ(VdVs2),

where δn 2nd2 -n02. Figure 4 plots the measured linewidths (δλ c,s) of each of the microdisk modes represented in Fig. 3 as a dimensionless quality factor, Qiλ 0/δλ c,s (shown as black circles, one for each mode of a doublet pair). From Eq. (4), we see that the model of Appendix B yields a linear dependence versus disk radius of the surface scattering quality factor, Qss . The dash-dotted blue curve shown in Fig. 4 represents the resulting surface scattering component of the total loss as predicted by the fit to Lcσr from the observed doublet splitting. In comparison to recently reported results not incorporating the e-beam resist reflow technique [9] (shown as an * in Figs. 3 and 4), the doublet splitting has been reduced by nearly a factor of 2.5. This results in an increase in the predicted Qss by more than a factor of 6. Given that the measured quality factor of the current microdisk resonators has only doubled, this suggests that our new work is limited by loss mechanisms not significant in previous work [9].

 figure: Fig. 4.

Fig. 4. Measured intrinsic quality factor, Qi , versus disk radius and resulting breakdown of optical losses due to: surface scattering (Qss ), bulk doping and impurities (Qb ), and surface absorption (Qsa ).

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Subtracting the fit to Qss from the data, a strong linear dependence with radius still remains. Since the optical losses decrease as the modes are pulled in toward the center of the microdisk, we assume that neither the pedestal nor the top and bottom surfaces are significant sources of optical loss. As shown in Appendix D, a disk edge surface absorption component of optical loss would have a linear dependence with microdisk radius, similar to that of surface scattering. The approximate dependence of the surface absorption quality factor, Qsa , is found in the analysis from Appendix D to be given as:

Qsa=πc(n̅2n02)Rλ0n̅2γsaζ,

where γsa is the bulk absorption rate of a material consisting entirely of that at the microdisk surface. As schematically illustrated in Fig. 1(b), we propose that the dominant form of surface absorption occurs at the edge of the microdisk along the etched sidewalls, where reactive ion etch damage of the Si lattice can result in a reconstruction depth, ζ, for many monolayers [15]. Fitting the remaining unaccounted for optical loss (i.e., subtracting out the predicted surface scattering component) versus microdisk radius with a linear (Qsa , solid red curve) and constant (Qb , dashed green curve) component, we arrive at the plot shown in Fig. 4. From this fit we find that our microdisks are limited to a Qb ≅8.5×106. Using resistivity measurements from the manufacturer of the boron doped SOI material used in this work (1–3 Ω·cm) and silicon absorption studies [25], residual free carriers in our material should limit Qb ≲(3.7–8.8×106), consistent with our analysis.

 figure: Fig. 5.

Fig. 5. Plot showing absorbed power versus intra-cavity energy for a R=5 µm disk to deduce linear, quadratic, and cubic loss rates. (inset) normalized data selected to illustrate bistability effect on resonance.

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To corroborate the above surface absorption conclusions, a series of power dependent experiments was performed. The small SiO2 pedestal (r max≅2.5 µm, r min≅1 µm, h ped≅1.2 µm) in the R=5 micron microdisk presented here provides relatively poor thermal contact as compared to the larger radii disks. As a result, thermally-induced shifts in resonant mode wavelengths (inset to Fig. 5) provide a sensitive measure of internal cavity absorption [26, 27, 16]. As the input power to the microdisk is increased, the change in resonance wavelength from the “cold-cavity” position can be related to absorbed power inside the microdisk, P abs. Since the thermal conductivity of silicon is roughly 100 times greater than that of silica, the steady-state temperature inside the silicon disk is constant (disregarding the neglible heat flow to the ambient N2 environment). Furthermore, the remaining buried oxide and silicon substrate, being several orders of magnitude larger in cross-section than the pedestal, can be assumed to act as a thermal bath of constant temperature. These assumptions reduce the thermal behavior of the microresonator to a time-independent, one-dimensional heat flow problem, depending primarily on the geometry of the pedestal.

The thermal resistance, Rth , of the narrow pedestal can be calculated from the thermal conductivity of thermal oxide (κ SiO2~1.38W·K-1m-1) and the geometry of the pedestal (shown schematically in Fig. 1(c)):

Rth=hped(κSiO2πrmaxrmin).

The change in resonance wavelength can then be approximately related to the optical power absorbed via the linear relationships

Δλ0nSiλ0Δn(dndT)Si1ΔTRth1Pabs,

where the thermo-optic coefficient of Si is denoted by (dn/dT) Si~1.5×10-4 K-1[16]. We finally arrive at the working equation relating the wavelength shift seen in the inset to Fig. 5 to the absorbed power:

Pabs=κSiO2πrmaxrminnSihpeddndTSiλ0Δλ0.

To calculate the linear absorption rate, the internal cavity energy must also be found from the minima in the normalized transmission spectra, T min, and input power, Pin . Under steady-state conditions, conservation of power requires that the power dropped into the cavity, Pd =(1-T min)Pin , be equal to the power lost out of the cavity-waveguide system, PL . From the formal definition of intrinsic quality factor, Qi , we can relate the cavity energy to the power loss by

Uc=QiωPL=Qiω(1Tmin)Pin,

where ω is the optical frequency and Uc is the internal cavity energy. Taking into account the external loading from the fiber taper, the total observed quality factor (linewidth) can be written as 1/Qt =1/Qe +1/Qi , where Qe relates to the coupling of optical power from the fiber taper to the microdisk. Qi in this case includes all intrinsic loss of the microdisk resonator and any additional parasitic loading from the fiber taper. As the resonant mode linewidth cannot be directly measured at high powers (due to nonlinear distortion), it must be inferred from the depths of transmission and the “cold-cavity” linewidth [16]. Using a simple model for the waveguide-resonator system [28], we can find Qe by measuring the total linewidth at low powers and using the relation Qe=2Qt(1Tmin)). One can then show that Qi=Qe(1Tmin)(1+Tmin)) for all powers, assuming that the taper loading does not change with input power. With complete knowledge of Qe and Qi , we can finally determine the internal cavity energy versus input power.

Figure 5 shows a typical data set for the power dependent absorption effects for the R=5 µm microdisk modes previously described for Figs 3 and 4. At low powers, P abs scales linearly with Uc , while at higher powers, the onset of two-photon absorption (quadratic term) and free-carrier absorption (cubic term) become readily apparent. A cubic polynomial fit (P abs~aUc3 +bUc2 +γ lin Uc ) for the four R=5 µm modes yields a linear absorption quality factor of Q linω/γ lin=(1.5±0.3)×106. This value is in very good agreement with the estimated Qsa from the analysis above (solid red curve in Fig. 4). Also, as the measurements of the modes in the larger microdisks have significantly higher measured Qi than the measured Q lin, this is a strong indication that we are indeed quantitatively probing the surface state absorption and not the bulk absorption for the smaller R=5 µm microdisks.

4. Conclusion

Using a combination of resist reflow to form a surface-tension limited smooth etch mask and a low DC bias dry etch to reduce roughness and damage in the etched Si sidewalls, we have fabricated high-index contrast Si microdisk microresonators with strong optical confinement and losses as low as 0.1 dB/cm (Q>5×106). Passive fiber optic measurements of the scaling of optical loss with microdisk radius, along with power dependent measurement of the thermooptic properties of the microdisks, provide evidence supporting the view that optical loss is dominated in these structures by surface (5<R<20 µm) and bulk free-carrier absorption (R>20 µm), as opposed to surface roughness on the microdisk. Applications of these devices to nonlinear light scattering, all-optical switching, Si laser resonators, or highly sensitive Si optical sensors are envisioned, where the low loss and tight optical confinement results in extremely large circulating intensities for low optical input powers.

A. Analytic approximation for the modes of a microdisk

Starting with Maxwell’s equations for a linear, non-dispersive medium with no free charges or currents,

·D(r,t)=0×E(r,t)=B(r,t)t·B(r,t)=0×H(r,t)=D(r,t)t,

where D(r,t)=ε (r)E(r,t) and H(r,t)=1μoB(r,t) are the linear constitutive relations. Using the well known series of vector identities, we can rewrite Maxwell’s equations for a piecewise homogeneous medium as wave equations:

2Fn2(r)c22Ft2=0,

where F={E,H}, c2=1μ0ε0, and n2(r)=ε(r)ε0. Furthermore, looking for oscillatory solutions of the form: F(r,t)=F(r)exp(-iωt), we can write the cylindrical form of the time-independent Maxwell’s equations as

(2ρ2+1ρρ+1ρ22ϕ2+2z2+(ωc)2n2(r))F(r)=0.

Note, the real physical fields are given by Re[F(r,t)].

In a thin semiconductor microdisk, the high index contrast provides large vertical confinement. This large confinement admits a powerful approximation by effectively reducing the problem to a two-dimensional one. In this case, there are two dominant polarizations labeled as TE (E field parallel to the disk plane) and TM (E field perpendicular to the disk plane), where Eq. (12) becomes scalar in the zẑ direction. As a consequence, Fz corresponds to Hz (Ez ) for TE(TM) modes. For ρ<R, where R is the disk radius, separation of variables can be used to rewrite Eq. (12) as

1W(2Wρ2+1ρWρ+1ρ22Wϕ2)+1Zd2Zdz2+k02n2(r)=0,

where Fz =W (ρ, ϕ)Z(z) and k 0=ω/c. Thus we have two differential equations, to self-consistently solve for the effective index, . The solution of Eq. (15) follows the standard slab mode calculations as in [23] taking note that Fz is continuous(discontinuous) across the interfaces for the TE(TM) modes. Equation (14) can be solved by again using separation of variables,W(ρ,ϕ)=Ψ(ρ)Ω(ϕ), to rewrite it as:

(2Wρ2+1ρWρ+1ρ22Wϕ2)+k02n̅2(ρ)W=0
d2Zdz2+k02(n2(z)n̅2)Z=0,
2Ψρ2+1ρΨρ+(k02n̅2(ρ)m2ρ2)Ψ=0
2Ωϕ2+m2Ω=0.

While the solution to Eq. (17) is simply Ω(ϕ)~exp(imϕ), the solution to Eq. (16) is most tractably solved approximately [22]. Inside the disk the radial solutions are Bessel functions, Ψ(ρ)~Jm (ko ρ ). Outside the disk, the solutions are Hankel functions which can be approximated by a decaying exponential with decay constant α=k 0(2-n02)1/2. The free-space wavelength can then be found by matching the boundary conditions at ρ=R using Eq. (10). Matching Hz (Ez ) and Eϕ for TE(TM) leads to the transcendental equation,

k0n̅(k0)Jm+1(k0n̅(k0)R)=(mR+ηα)Jm(k0n̅(k0)R),

where η=2/n02 for TE and unity for TM modes. Using the self-consistent solution of Eq. (15) and Eq. (18), the unnormalized radial mode dependence is given by

Ψ(ρ){Jm(k0n̅ρ)ρRJm(k0n̅R)exp(α(ρR))ρ>R.

B. Derivation of Qss from the Volume Current Method

Optical losses in microresonators are often times limited by index perturbations,δε, on the surfaces. These index perturbations are sourced approximately by the unperturbed field solutions, E 0, to create polarization currents,

J=iωδεE0.

In analogy with microwave electronics, the polarization currents drive new electromagnetic fields which radiate into space. Optical losses due to the perturbations can be calculated from the far field solutions setup by J [24]. In most microdiskwork, the dominant deviations from the ideal disk geometry occur on the sidewalls during fabrication. Etch-induced sidewall roughness typically runs the height of the disk, providing a nearly one-dimensional way of representing this roughness along the arclength. In this way, the index perturbation (Fig. 1) can be approximated by

δε=ε0δn2hΔr(ϕ)δ(rR)δ(z),

where ε 0 is the free space permittivity, δn 2=nd2 -n02, nd is the disk refractive index, n 0 is the index of the surrounding medium, h is the disk height, and Δr(ϕ) is the radial surface roughness relative to the unperturbed disk radius [19]. The parameters ρ, ϕ, and z correspond to the radius from the disk center, angle along the disk perimeter, and height along the disk edge, respectively. The far field vector potential sourced by J is given by [24]

Arad(r)=μ04π(eik1rr)J(r)eik1r̂·rdr,

where k 1n 0 k 0 is the wave vector in the surrounding medium. Writing the unperturbed electric field at the disk edge as E 0=E m (ρ, z)exp(imϕ), Eq. (22) simplifies to

Arad(r)=iμ0ωhδn2ε0Em(R,0)R4π(eik1rr)02πΔr(ϕ)eimϕexp(ik1Rsinθcosϕ)dϕ.

For low loss microdisks, the direct solution to Eq. (23) becomes increasingly sensitive to the limitations on the measurement of Δr(s), where sϕR, the arclength along the disk edge perimeter. However, a simple statistical solution becomes possible once surface roughness becomes much smaller than the wavelength in the material [19]. In effect, this simplification assumes that roughness separated by much more than a characteristic correlation length, L c , is statistically independent. This allows each infinitesimal arclength along the perimeter to be treated as an ensemble member of all the possible processing outcomes under the same fabrication conditions. Ensemble averaging over Eq. (23) yields

Arad(r)·Arad(r)*=μ0ωhδn2ε0Em(R,0)4π2(Rr)2Θ
Θ02π02πC(ϕϕ)exp(imϕϕ)exp[ik1Rsinθ(cosϕcosϕ)]dϕdϕ,

where C(|ϕ -ϕ |)=〈Δr(ϕ r(ϕ )〉, the correlation function for the etch roughness. With the substitutions, t=ϕ -ϕ and z=(ϕ +ϕ )/2, the integral is evaluated to be

Θ=2π02πC(t)exp(imt)J0[2k1Rsinθsin(t2)]dt.

With a Gaussian model for the correlation function given by C(s)=σR2 exp(-s 2/Lc2 ), this integral becomes

Θ=2πRπRπRσR2exp(s2Lc2+imRs)J0[2k1Rsinθsin(sR2)]ds2π32σR2LcR,

where σ R is the standard deviation for the surface roughness. Combining Eqs. (27) and (24), the far field Poynting vector is given by

Srad=r̂ωk02μ0r̂×Arad2=r̂ωk13n0(δn2)2Vs2ε0Em(R,0)216πr̂×ê2r2,

where ê is the polarization of the electric field and VsRLchσr is the effective volume for a typical scatterer. The total power radiated, Prad , far from the disk can be found by integrating the outward intensity over a large sphere and summing the polarization components to obtain

Prad=(S·r̂)r2dΩ=η̂π72ωn0(δn2)2Vs2ε0Em(R,0;η̂)2G(η̂)λ03,

where η̂={ρ̂,ϕ̂,} and G(η̂)={2/3,2,4/3} is a geometrical radiation factor for the different electric field polarizations. Since the quality factor of a cavity mode is given by Q=ωUc /Prad , where Uc=12ε0(r)E02dr dr is the time-averaged stored energy in the cavity, we can rewrite Eq. (29) as a surface scattering quality factor

Qss=λ03π72n0(δn2)2Vs2Ση̂us¯(η̂)G(η̂),

where ū s (η̂) is the normalized, spatially-averaged (see Appendix E) η̂-polarized electric field energy density at the disk edge given by

u¯s(η̂)=ε0E0(η̂)s,avg212ε0(r)E02dr.

Numerical calculations of ū s (η̂) show that TE modes couple to far field radiation modes roughly 50 % more strongly than corresponding TM modes, mainly due to geometrical considerations through G(η̂). Note that this and the following section has corrected minor errors present in reference [9].

C. Derivation of Qβ via time-dependent perturbation theory

In addition to coupling to radiation modes, surface roughness on microdisks will also couple the degenerate clockwise (cw) and counterclockwise (ccw) modes [18]. Lifting this degeneracy creates a downshifted-and upshifted-frequency standing wave mode. Based on the work of Gorodetsky, et al.[20], a time-dependent perturbation theory can be formulated to quantify this doublet splitting. Then a statistical approach similar to that of Little and Laine [19] can be used to relate the doublet frequency splitting to measured surface roughness. The same polarization currents from Eq. (20) can be added to Eq. (11) to arrive at

2Eμ0(ε0+δε)2Et2=0,

where ε 0(r) is the dielectric structure for the ideal disk. The unperturbed modes with an assumed harmonic time dependence, denoted by Ej0 (r,t)=Ej0 (r)exp(jt), are a solution of

2Ej0(r)+μ0ε0(r)ωj2Ej0(r)=0.

Now assuming slowly varying envelopes aj (t), the perturbed modes are given by

E(r,t)=exp(iω0t)jaj(t)Ej0(r).

Substituting Eq. (34) into Eq. (32) and keeping terms to first order yields

j(2iω0ε0dajdt+δεω02ajε0(ωj2ω02)aj)Ej0(r)=0.

Using the fact that ∫ε 0(Ej0 (r))*Ek0 (r)d r=0 for jk (shown in Ref. [29]), Eq. (35) can be multiplied by (Ej0 (r))* and integrated over all space to obtain

dakdt+iΔωkak(t)=ijβjkaj,
withβjk=ω02δε(Ej0(r))*Ek0(r)drε0Ek0(r)2dr,

where Δωk =ωk -ω 0. The radius of the ideal disk in ε 0(r) is chosen as the average radius of the perturbed disk so as to work in a basis where the strength of the perturbation does not create a self-term frequency shift (i.e., βjj =0). For the case of initially degenerate cw and ccw modes, we can take this coupled mode formalism and resonantly pump the system with an external waveguide. For small perturbations, we can assume that the backscattering rate for each mode is identical, β cw,ccw=β ccw,cwβ. When all other modes are far off-resonance, the coupled mode equations then become symmetrically

dacwdt=iΔωacw+iβaccw
daccwdt=iΔωaccw+iβacw,

where Δωcwωccw ≡Δω. A functional form for β can be analytically derived for small amounts of etch roughness. Substituting the delta function approximation for the dielectric perturbation from Eq. (21), we obtain

β=ω04Ucε0δn2hΔr(s)δ(rR)δ(z)(Ecw(r))*Eccw(r)dr.

This integral is easily reduced to a Fourier-type integral by noting that at the disk edge E cw =E m (R,0)exp(-imϕ) and E ccw =E m (R,0)exp(imϕ). Integrating over the δ-functions yields

β=ω0δn2hRε0Em(R,0)24Ucϒ
ϒ=02πΔr(ϕ)ei2mϕdϕ.

As in Eq. (23), the integral, γ, becomes statistically solvable when surface roughness becomes much smaller than the wavelength in the material. Following the same method of solution and definitions as above,

ϒ2=2πRπRπRσR2exp(s2Lc2+i2mRs)2π32σR2LcR.

Plugging Eq. (42) into Eq. (40) and solving gives β2=(π4)34ω0δn2Vsus¯, where ū s ≡∑η̂ū s (η̂). To model coupling power into this resonant system, a phenomenological loss rate, γtω/Qt , and a coupling coefficient, κ, can be added to obtain, [21]

dacwdt=(γt2+iΔω)acw+iβaccw+κs
daccwdt=(γt2+iΔω)accw+iβacw,

where αj are normalized energy amplitudes, |s|2 is the normalized input power, and γt =γei . The loss rate can been broken down into loss from the cavity back into the waveguide, γe, and all other loss, γi . Assuming the coupling itself is lossless and obeys time reciprocity, a scattering matrix formalism can further show that κ 2=γe [30]. Additionally, the transmitted and reflected powers are given by [21] |t|2=|-s+κ acw |2 and |r|2=|κaccw |2. Steady-state solutions of Eqs. (43a) and (43b) show transmission dips at ω=ω 0±β. Thus, a normalized measure of the backscattering rate is defined to be the natural frequency divided by the total resonance splitting, Qβω 0/(2β):

Qβ=2π34δn2Vsus¯.

D. Derivation of Qsa

In contrast to surface scattering, surface absorption resulting from lattice reconstructions is also present as a source of optical loss. Since great care is taken to preserve the quality of the top and bottom surfaces of the silicon microdisk, it is reasonable to assume that the dominant lattice damage occurs at the disk sidewalls during the etch. While efforts are taken to minimize etch damage, it remains clear that reactive ion etching locally damages the lattice during the material ablation allowing for the incorporation of various lattice impurities and defects [15]. The local surface absorption rate coefficient, γsa (r), is used to calculate a spatially-averaged loss coefficient according to:

γ̅sa=γsa(r)n2(r)E(r)2drn2(r)E(r)2dr,

where the appropriate weighting function is proportional to the electric field energy density of the optical mode [16]. An approximate model for γsa (r) would be to assume that their exists a reconstruction depth, ζ, where the loss rate is a constant γsa and zero elsewhere. Thus, the electric field in the numerator of Eq. (45) is approximately constant over a cylindrical shell with volume δVsa =2πRζh. By using Eq. (31) and assuming that the surface reconstruction has approximately the same index of refraction as that of the undisturbed lattice [15], we have

γ̅sa=γsanSi2δVsaE(r)2drn2(r)E(r)2dr=12γsanSi2us¯δVsa.

Defining Qsa =ω/γ̄ sa , we can then write for the surface-absorption quality factor

Qsa=4πcλ0γsanSi2us¯δVsa.

E. Approximate form for the TM case

Considering the TM case only, the normalized disk-edge energy density, ū s (zẑ) (defined in Eq. (31)), may be estimated analytically for low radial numbers [20]. Having an approximate form for ū s (zẑ) is extremely important if we are to develop intuition from the previous results. As an approximation to the actual (unperturbed) normalized disk-edge energy density for all points within the disk-edge perturbations, an average value over a thin cylindrical shell is used. Defining the peak radial amplitude of the roughness to be δr, the averaged intensity is given by

Es,avg2=1δVRδrR+δrh2h202πρdρdϕdzE2,

where δV=4πδrRh. Note that while surface absorption reconstruction depth does not necessarily correlate with surface roughness amplitude, this mathematical formalism is identical with ζ=2δr. For high index contrast structures, the cavity mode energy can be approximated as existing only inside the cavity. Therefore we have,

Uc=12ε0(r)E2dr12ε0nd20Rh2h202πρdρdϕdzE2.

Since the ϕ and z integrations in Eqs. (48) and (49) are identical, only the radial integrals need to be considered. Since E~Jm (k 0 ρ), we can write

us̅(Ẑ)=2nd2δVRδrR+δrρdρ[Jm(k0n̅ρ)]20Rρdρ[Jm(k0n̅p)]2.

As an alternative to solving Eq. (50) numerically, a closed form expression can be obtained by reintroducing the δ-function approximation for the radial integral over the thin shell. Using several Bessel identities along with the continuity conditions for Maxwell’s equations, Eq. (50) can be reduced to

us̅(Ẑ)=[Jm(k0n̅R)]2πhnd20Rρdρ[Jm(k0n̅ρ)]22n̅2Vdnd2(n̅2n02),

where Vd =πR 2 h. This approximate form for ū s () shows that the sensitivity to the disk edge decreases as the radius of the disk is increased. Substituting Eq. (51) into Eqs. (30,44,47) gives the working equations for the analysis presented in the main text of this paper:

Qss=3λ038π72n0δn2ξ(VdVs2),
Qβ=12π34ξ(VdVs),
Qsa=πc(n̅2n02)Rλ0n̅2γsaζ,

where a relative dielectric contrast constant ξ is defined to be

ξ=n̅2(nd2n02)nd2(n̅2n02).

Acknowledgements

This work was supported by DARPA through the EPIC program, and by the Charles Lee Powell Foundation. The authors would like to thank Paul Barclay and Kartik Srinivasan for useful discussions. M.B. thanks the Moore Foundation, NPSC, and HRL Laboratories, and T.J. thanks the Powell Foundation for their graduate fellowship support.

References and links

1. L. Pavesi and D. Lockwood, Silicon Photonics (Springer-verlag, New York, 2004).

2. G. Reed and A. Knights, Silicon Photonics: An Introduction (John Wiley, West Sussex, 2004). [CrossRef]  

3. A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004). [CrossRef]   [PubMed]  

4. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004). [CrossRef]   [PubMed]  

5. O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269–5273 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5269. [CrossRef]   [PubMed]  

6. H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005). [CrossRef]   [PubMed]  

7. K. Okamoto, Fundamentals of optical waveguides (Academic Press, San Diego, 2000).

8. B. E. Little, “A VLSI Photonics Platform,” in Optical Fiber Communication Conference (2003).

9. M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693–3695 (2004). [CrossRef]  

10. D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. 23, 247–249 (1998). [CrossRef]  

11. M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000). [CrossRef]   [PubMed]  

12. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003). [CrossRef]   [PubMed]  

13. E. Yablonovitch, D. L. Allara, C. C. Chang, T. Gmitter, and T. B. Bright, “Unusually Low Surface-Recombination Velocity on Silicon and Germanium Surfaces,” Phys. Rev. Lett. 57, 249–252 (1986). [CrossRef]   [PubMed]  

14. K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, “Optical-fiber-based measurement of an ultrasmall volume, high-Q photonic crystal microcavity,” Phys. Rev. B 70, 081,206(R) (2004). [CrossRef]  

15. M. Haverlag, D. Vender, and G. S. Oehrlein, “Ellipsometric study of silicon surface damage in electron cyclotron resonance plasma etching using CF4 and SF6,” Appl. Phys. Lett. 61, 2875–2877 (1992). [CrossRef]  

16. P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and a fiber taper,” Opt. Express 13, 801–820 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-801. [CrossRef]   [PubMed]  

17. S. Spillane (2004). Private communication.

18. D. Weiss, V. Sandoghdar, J. Hare, V. Lefévre-Seguin, J. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced light backscattering in silica microspheres,” Opt. Lett. 22, 1835–1837 (1995). [CrossRef]  

19. B. E. Little, J.-P. Laine, and S. T. Chu, “Surface-Roughness-Induced Contradirectional Coupling in Ring and Disk Resonators,” Opt. Lett. 22, 4–6 (1997). [CrossRef]   [PubMed]  

20. M. Gorodetsky, A. Pryamikov, and V. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17, 1051–1057 (2000). [CrossRef]  

21. T. J. Kippenburg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. 27, 1669–1671 (2002). [CrossRef]  

22. B. E. Little and S. T. Chu, “Estimating surface-roughness loss and output coupling in microdisk resonators,” Opt. Lett. 21, 1390–1392 (1996). [CrossRef]   [PubMed]  

23. A. Yariv, Optical Electronics, 4th ed. (Saunder College Publishing, a division of Holt, Rinehart and Winston, Inc., Orlando, Florida, 1991).

24. M. Kuznetsov and H. A. Haus, “Radiation Loss in Dielectric Waveguide Structures by the Volume Current Method,” IEEE J. Quan. Elec. 19, 1505–1514 (1983). [CrossRef]  

25. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quan. Elec. 23, 123–129 (1987). [CrossRef]  

26. V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29, 2387–2389 (2004). [CrossRef]   [PubMed]  

27. H. Rokhsari, S. M. Spillane, and K. J. Vahala, “Loss characterization in microcavities using the thermal bistability effect,” Appl. Phys. Lett. 85, 3029–3031 (2004). [CrossRef]  

28. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043,902 (2003). [CrossRef]  

29. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, New Jersey, 1995).

30. H. A. Haus, Waves and Fields in Optoelectronics, 1st ed. (Prentice-Hall, Englewood Cliffs, New Jersey 07632, 1984).

References

  • View by:
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  • |

  1. L. Pavesi and D. Lockwood, Silicon Photonics (Springer-verlag, New York, 2004).
  2. G. Reed and A. Knights, Silicon Photonics: An Introduction (John Wiley, West Sussex, 2004).
    [Crossref]
  3. A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
    [Crossref] [PubMed]
  4. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
    [Crossref] [PubMed]
  5. O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269–5273 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5269.
    [Crossref] [PubMed]
  6. H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
    [Crossref] [PubMed]
  7. K. Okamoto, Fundamentals of optical waveguides (Academic Press, San Diego, 2000).
  8. B. E. Little, “A VLSI Photonics Platform,” in Optical Fiber Communication Conference (2003).
  9. M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693–3695 (2004).
    [Crossref]
  10. D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. 23, 247–249 (1998).
    [Crossref]
  11. M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
    [Crossref] [PubMed]
  12. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
    [Crossref] [PubMed]
  13. E. Yablonovitch, D. L. Allara, C. C. Chang, T. Gmitter, and T. B. Bright, “Unusually Low Surface-Recombination Velocity on Silicon and Germanium Surfaces,” Phys. Rev. Lett. 57, 249–252 (1986).
    [Crossref] [PubMed]
  14. K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, “Optical-fiber-based measurement of an ultrasmall volume, high-Q photonic crystal microcavity,” Phys. Rev. B 70, 081,206(R) (2004).
    [Crossref]
  15. M. Haverlag, D. Vender, and G. S. Oehrlein, “Ellipsometric study of silicon surface damage in electron cyclotron resonance plasma etching using CF4 and SF6,” Appl. Phys. Lett. 61, 2875–2877 (1992).
    [Crossref]
  16. P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and a fiber taper,” Opt. Express 13, 801–820 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-801.
    [Crossref] [PubMed]
  17. S. Spillane (2004). Private communication.
  18. D. Weiss, V. Sandoghdar, J. Hare, V. Lefévre-Seguin, J. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced light backscattering in silica microspheres,” Opt. Lett. 22, 1835–1837 (1995).
    [Crossref]
  19. B. E. Little, J.-P. Laine, and S. T. Chu, “Surface-Roughness-Induced Contradirectional Coupling in Ring and Disk Resonators,” Opt. Lett. 22, 4–6 (1997).
    [Crossref] [PubMed]
  20. M. Gorodetsky, A. Pryamikov, and V. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17, 1051–1057 (2000).
    [Crossref]
  21. T. J. Kippenburg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. 27, 1669–1671 (2002).
    [Crossref]
  22. B. E. Little and S. T. Chu, “Estimating surface-roughness loss and output coupling in microdisk resonators,” Opt. Lett. 21, 1390–1392 (1996).
    [Crossref] [PubMed]
  23. A. Yariv, Optical Electronics, 4th ed. (Saunder College Publishing, a division of Holt, Rinehart and Winston, Inc., Orlando, Florida, 1991).
  24. M. Kuznetsov and H. A. Haus, “Radiation Loss in Dielectric Waveguide Structures by the Volume Current Method,” IEEE J. Quan. Elec. 19, 1505–1514 (1983).
    [Crossref]
  25. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quan. Elec. 23, 123–129 (1987).
    [Crossref]
  26. V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29, 2387–2389 (2004).
    [Crossref] [PubMed]
  27. H. Rokhsari, S. M. Spillane, and K. J. Vahala, “Loss characterization in microcavities using the thermal bistability effect,” Appl. Phys. Lett. 85, 3029–3031 (2004).
    [Crossref]
  28. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043,902 (2003).
    [Crossref]
  29. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, New Jersey, 1995).
  30. H. A. Haus, Waves and Fields in Optoelectronics, 1st ed. (Prentice-Hall, Englewood Cliffs, New Jersey 07632, 1984).

2005 (2)

2004 (7)

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693–3695 (2004).
[Crossref]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[Crossref] [PubMed]

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[Crossref] [PubMed]

O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269–5273 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5269.
[Crossref] [PubMed]

K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, “Optical-fiber-based measurement of an ultrasmall volume, high-Q photonic crystal microcavity,” Phys. Rev. B 70, 081,206(R) (2004).
[Crossref]

V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29, 2387–2389 (2004).
[Crossref] [PubMed]

H. Rokhsari, S. M. Spillane, and K. J. Vahala, “Loss characterization in microcavities using the thermal bistability effect,” Appl. Phys. Lett. 85, 3029–3031 (2004).
[Crossref]

2003 (2)

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043,902 (2003).
[Crossref]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[Crossref] [PubMed]

2002 (1)

2000 (2)

M. Gorodetsky, A. Pryamikov, and V. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17, 1051–1057 (2000).
[Crossref]

M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[Crossref] [PubMed]

1998 (1)

1997 (1)

1996 (1)

1995 (1)

D. Weiss, V. Sandoghdar, J. Hare, V. Lefévre-Seguin, J. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced light backscattering in silica microspheres,” Opt. Lett. 22, 1835–1837 (1995).
[Crossref]

1992 (1)

M. Haverlag, D. Vender, and G. S. Oehrlein, “Ellipsometric study of silicon surface damage in electron cyclotron resonance plasma etching using CF4 and SF6,” Appl. Phys. Lett. 61, 2875–2877 (1992).
[Crossref]

1987 (1)

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quan. Elec. 23, 123–129 (1987).
[Crossref]

1986 (1)

E. Yablonovitch, D. L. Allara, C. C. Chang, T. Gmitter, and T. B. Bright, “Unusually Low Surface-Recombination Velocity on Silicon and Germanium Surfaces,” Phys. Rev. Lett. 57, 249–252 (1986).
[Crossref] [PubMed]

1983 (1)

M. Kuznetsov and H. A. Haus, “Radiation Loss in Dielectric Waveguide Structures by the Volume Current Method,” IEEE J. Quan. Elec. 19, 1505–1514 (1983).
[Crossref]

Allara, D. L.

E. Yablonovitch, D. L. Allara, C. C. Chang, T. Gmitter, and T. B. Bright, “Unusually Low Surface-Recombination Velocity on Silicon and Germanium Surfaces,” Phys. Rev. Lett. 57, 249–252 (1986).
[Crossref] [PubMed]

Almeida, V. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[Crossref] [PubMed]

V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29, 2387–2389 (2004).
[Crossref] [PubMed]

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[Crossref] [PubMed]

Barclay, P. E.

P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and a fiber taper,” Opt. Express 13, 801–820 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-801.
[Crossref] [PubMed]

K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, “Optical-fiber-based measurement of an ultrasmall volume, high-Q photonic crystal microcavity,” Phys. Rev. B 70, 081,206(R) (2004).
[Crossref]

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693–3695 (2004).
[Crossref]

Barrios, C. A.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[Crossref] [PubMed]

Bennett, B. R.

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quan. Elec. 23, 123–129 (1987).
[Crossref]

Borselli, M.

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693–3695 (2004).
[Crossref]

K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, “Optical-fiber-based measurement of an ultrasmall volume, high-Q photonic crystal microcavity,” Phys. Rev. B 70, 081,206(R) (2004).
[Crossref]

Boyraz, O.

Bright, T. B.

E. Yablonovitch, D. L. Allara, C. C. Chang, T. Gmitter, and T. B. Bright, “Unusually Low Surface-Recombination Velocity on Silicon and Germanium Surfaces,” Phys. Rev. Lett. 57, 249–252 (1986).
[Crossref] [PubMed]

Cai, M.

M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[Crossref] [PubMed]

Chang, C. C.

E. Yablonovitch, D. L. Allara, C. C. Chang, T. Gmitter, and T. B. Bright, “Unusually Low Surface-Recombination Velocity on Silicon and Germanium Surfaces,” Phys. Rev. Lett. 57, 249–252 (1986).
[Crossref] [PubMed]

Chu, S. T.

Cohen, O.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[Crossref] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[Crossref] [PubMed]

Fang, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[Crossref] [PubMed]

Gmitter, T.

E. Yablonovitch, D. L. Allara, C. C. Chang, T. Gmitter, and T. B. Bright, “Unusually Low Surface-Recombination Velocity on Silicon and Germanium Surfaces,” Phys. Rev. Lett. 57, 249–252 (1986).
[Crossref] [PubMed]

Gorodetsky, M.

Hak, D.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[Crossref] [PubMed]

Hare, J.

D. Weiss, V. Sandoghdar, J. Hare, V. Lefévre-Seguin, J. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced light backscattering in silica microspheres,” Opt. Lett. 22, 1835–1837 (1995).
[Crossref]

Haroche, S.

D. Weiss, V. Sandoghdar, J. Hare, V. Lefévre-Seguin, J. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced light backscattering in silica microspheres,” Opt. Lett. 22, 1835–1837 (1995).
[Crossref]

Haus, H. A.

M. Kuznetsov and H. A. Haus, “Radiation Loss in Dielectric Waveguide Structures by the Volume Current Method,” IEEE J. Quan. Elec. 19, 1505–1514 (1983).
[Crossref]

H. A. Haus, Waves and Fields in Optoelectronics, 1st ed. (Prentice-Hall, Englewood Cliffs, New Jersey 07632, 1984).

Haverlag, M.

M. Haverlag, D. Vender, and G. S. Oehrlein, “Ellipsometric study of silicon surface damage in electron cyclotron resonance plasma etching using CF4 and SF6,” Appl. Phys. Lett. 61, 2875–2877 (1992).
[Crossref]

Ilchenko, V.

Ilchenko, V. S.

Jalali, B.

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, New Jersey, 1995).

Jones, R.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[Crossref] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[Crossref] [PubMed]

Kimble, H. J.

Kippenberg, T. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[Crossref] [PubMed]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043,902 (2003).
[Crossref]

Kippenburg, T. J.

Knights, A.

G. Reed and A. Knights, Silicon Photonics: An Introduction (John Wiley, West Sussex, 2004).
[Crossref]

Kuznetsov, M.

M. Kuznetsov and H. A. Haus, “Radiation Loss in Dielectric Waveguide Structures by the Volume Current Method,” IEEE J. Quan. Elec. 19, 1505–1514 (1983).
[Crossref]

Laine, J.-P.

Lefévre-Seguin, V.

D. Weiss, V. Sandoghdar, J. Hare, V. Lefévre-Seguin, J. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced light backscattering in silica microspheres,” Opt. Lett. 22, 1835–1837 (1995).
[Crossref]

Liao, L.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[Crossref] [PubMed]

Lipson, M.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[Crossref] [PubMed]

V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29, 2387–2389 (2004).
[Crossref] [PubMed]

Little, B. E.

Liu, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[Crossref] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[Crossref] [PubMed]

Lockwood, D.

L. Pavesi and D. Lockwood, Silicon Photonics (Springer-verlag, New York, 2004).

Mabuchi, H.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, New Jersey, 1995).

Nicolaescu, R.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[Crossref] [PubMed]

Oehrlein, G. S.

M. Haverlag, D. Vender, and G. S. Oehrlein, “Ellipsometric study of silicon surface damage in electron cyclotron resonance plasma etching using CF4 and SF6,” Appl. Phys. Lett. 61, 2875–2877 (1992).
[Crossref]

Okamoto, K.

K. Okamoto, Fundamentals of optical waveguides (Academic Press, San Diego, 2000).

Painter, O.

P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and a fiber taper,” Opt. Express 13, 801–820 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-801.
[Crossref] [PubMed]

K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, “Optical-fiber-based measurement of an ultrasmall volume, high-Q photonic crystal microcavity,” Phys. Rev. B 70, 081,206(R) (2004).
[Crossref]

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693–3695 (2004).
[Crossref]

M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[Crossref] [PubMed]

Painter, O. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043,902 (2003).
[Crossref]

Panepucci, R. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[Crossref] [PubMed]

Paniccia, M.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[Crossref] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[Crossref] [PubMed]

Pavesi, L.

L. Pavesi and D. Lockwood, Silicon Photonics (Springer-verlag, New York, 2004).

Pryamikov, A.

Raimond, J.

D. Weiss, V. Sandoghdar, J. Hare, V. Lefévre-Seguin, J. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced light backscattering in silica microspheres,” Opt. Lett. 22, 1835–1837 (1995).
[Crossref]

Reed, G.

G. Reed and A. Knights, Silicon Photonics: An Introduction (John Wiley, West Sussex, 2004).
[Crossref]

Rokhsari, H.

H. Rokhsari, S. M. Spillane, and K. J. Vahala, “Loss characterization in microcavities using the thermal bistability effect,” Appl. Phys. Lett. 85, 3029–3031 (2004).
[Crossref]

Rong, H.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[Crossref] [PubMed]

Rubin, D.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[Crossref] [PubMed]

Samara-Rubio, D.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[Crossref] [PubMed]

Sandoghdar, V.

D. Weiss, V. Sandoghdar, J. Hare, V. Lefévre-Seguin, J. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced light backscattering in silica microspheres,” Opt. Lett. 22, 1835–1837 (1995).
[Crossref]

Soref, R. A.

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quan. Elec. 23, 123–129 (1987).
[Crossref]

Spillane, S.

S. Spillane (2004). Private communication.

Spillane, S. M.

H. Rokhsari, S. M. Spillane, and K. J. Vahala, “Loss characterization in microcavities using the thermal bistability effect,” Appl. Phys. Lett. 85, 3029–3031 (2004).
[Crossref]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043,902 (2003).
[Crossref]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[Crossref] [PubMed]

T. J. Kippenburg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. 27, 1669–1671 (2002).
[Crossref]

Srinivasan, K.

P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and a fiber taper,” Opt. Express 13, 801–820 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-801.
[Crossref] [PubMed]

K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, “Optical-fiber-based measurement of an ultrasmall volume, high-Q photonic crystal microcavity,” Phys. Rev. B 70, 081,206(R) (2004).
[Crossref]

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693–3695 (2004).
[Crossref]

Streed, E. W.

Vahala, K.

M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[Crossref] [PubMed]

Vahala, K. J.

H. Rokhsari, S. M. Spillane, and K. J. Vahala, “Loss characterization in microcavities using the thermal bistability effect,” Appl. Phys. Lett. 85, 3029–3031 (2004).
[Crossref]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043,902 (2003).
[Crossref]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[Crossref] [PubMed]

T. J. Kippenburg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. 27, 1669–1671 (2002).
[Crossref]

Vender, D.

M. Haverlag, D. Vender, and G. S. Oehrlein, “Ellipsometric study of silicon surface damage in electron cyclotron resonance plasma etching using CF4 and SF6,” Appl. Phys. Lett. 61, 2875–2877 (1992).
[Crossref]

Vernooy, D. W.

Weiss, D.

D. Weiss, V. Sandoghdar, J. Hare, V. Lefévre-Seguin, J. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced light backscattering in silica microspheres,” Opt. Lett. 22, 1835–1837 (1995).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, New Jersey, 1995).

Yablonovitch, E.

E. Yablonovitch, D. L. Allara, C. C. Chang, T. Gmitter, and T. B. Bright, “Unusually Low Surface-Recombination Velocity on Silicon and Germanium Surfaces,” Phys. Rev. Lett. 57, 249–252 (1986).
[Crossref] [PubMed]

Yariv, A.

A. Yariv, Optical Electronics, 4th ed. (Saunder College Publishing, a division of Holt, Rinehart and Winston, Inc., Orlando, Florida, 1991).

Appl. Phys. Lett. (3)

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693–3695 (2004).
[Crossref]

M. Haverlag, D. Vender, and G. S. Oehrlein, “Ellipsometric study of silicon surface damage in electron cyclotron resonance plasma etching using CF4 and SF6,” Appl. Phys. Lett. 61, 2875–2877 (1992).
[Crossref]

H. Rokhsari, S. M. Spillane, and K. J. Vahala, “Loss characterization in microcavities using the thermal bistability effect,” Appl. Phys. Lett. 85, 3029–3031 (2004).
[Crossref]

IEEE J. Quan. Elec. (2)

M. Kuznetsov and H. A. Haus, “Radiation Loss in Dielectric Waveguide Structures by the Volume Current Method,” IEEE J. Quan. Elec. 19, 1505–1514 (1983).
[Crossref]

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quan. Elec. 23, 123–129 (1987).
[Crossref]

J. Opt. Soc. Am. B (1)

Nature (4)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[Crossref] [PubMed]

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[Crossref] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[Crossref] [PubMed]

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (6)

Phys. Rev. B (1)

K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, “Optical-fiber-based measurement of an ultrasmall volume, high-Q photonic crystal microcavity,” Phys. Rev. B 70, 081,206(R) (2004).
[Crossref]

Phys. Rev. Lett. (3)

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043,902 (2003).
[Crossref]

M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[Crossref] [PubMed]

E. Yablonovitch, D. L. Allara, C. C. Chang, T. Gmitter, and T. B. Bright, “Unusually Low Surface-Recombination Velocity on Silicon and Germanium Surfaces,” Phys. Rev. Lett. 57, 249–252 (1986).
[Crossref] [PubMed]

Other (8)

S. Spillane (2004). Private communication.

A. Yariv, Optical Electronics, 4th ed. (Saunder College Publishing, a division of Holt, Rinehart and Winston, Inc., Orlando, Florida, 1991).

K. Okamoto, Fundamentals of optical waveguides (Academic Press, San Diego, 2000).

B. E. Little, “A VLSI Photonics Platform,” in Optical Fiber Communication Conference (2003).

L. Pavesi and D. Lockwood, Silicon Photonics (Springer-verlag, New York, 2004).

G. Reed and A. Knights, Silicon Photonics: An Introduction (John Wiley, West Sussex, 2004).
[Crossref]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, New Jersey, 1995).

H. A. Haus, Waves and Fields in Optoelectronics, 1st ed. (Prentice-Hall, Englewood Cliffs, New Jersey 07632, 1984).

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Figures (5)

Fig. 1.
Fig. 1. Schematic representation of a fabricated silicon microdisk. (a) Top view showing ideal disk (red) against disk with roughness. (b) Top view close-up illustrating the surface roughness, Δr(s), and surface reconstruction, ξ. Also shown are statistical roughness parameters, σ r and Lc , of a typical scatterer. (c) Side view of a fabricated SOI microdisk highlighting idealized SiO2 pedestal.
Fig. 2.
Fig. 2. Taper transmission versus wavelength showing a high-Q doublet mode for the R=30 µm disk. Qcλ 0/δλ c and Qsλ 0/δλ s are the unloaded quality factors for the long and short wavelength modes respectively, where δλ c and δλ s are resonance linewidths. Also shown is the doublet splitting, Δλ, and normalized splitting quality factor, Qβλ 0λ.
Fig. 3.
Fig. 3. Normalized doublet splitting (Qβ ) versus disk radius. (inset) Taper transmission data and fit of deeply coupled doublet demonstrating 14 dB coupling depth.
Fig. 4.
Fig. 4. Measured intrinsic quality factor, Qi , versus disk radius and resulting breakdown of optical losses due to: surface scattering (Qss ), bulk doping and impurities (Qb ), and surface absorption (Qsa ).
Fig. 5.
Fig. 5. Plot showing absorbed power versus intra-cavity energy for a R=5 µm disk to deduce linear, quadratic, and cubic loss rates. (inset) normalized data selected to illustrate bistability effect on resonance.

Equations (57)

Equations on this page are rendered with MathJax. Learn more.

1 Q i = 1 Q r + 1 Q s s + 1 Q s a + 1 Q b ,
Q β λ 0 Δ λ = 1 2 π 3 4 ξ ( V d V s ) ,
ξ = n ̅ 2 ( n d 2 n 0 2 ) n d 2 ( n ̅ 2 n 0 2 ) .
Q s s = 3 λ 0 3 8 π 7 2 n 0 δ n 2 ξ ( V d V s 2 ) ,
Q s a = π c ( n ̅ 2 n 0 2 ) R λ 0 n ̅ 2 γ s a ζ ,
R th = h ped ( κ Si O 2 π r max r min ) .
Δ λ 0 n Si λ 0 Δ n ( d n d T ) Si 1 Δ T R th 1 P abs ,
P abs = κ SiO 2 π r max r min n S i h ped d n d T Si λ 0 Δ λ 0 .
U c = Q i ω P L = Q i ω ( 1 T min ) P in ,
· D ( r , t ) = 0 × E ( r , t ) = B ( r , t ) t · B ( r , t ) = 0 × H ( r , t ) = D ( r , t ) t ,
2 F n 2 ( r ) c 2 2 F t 2 = 0 ,
( 2 ρ 2 + 1 ρ ρ + 1 ρ 2 2 ϕ 2 + 2 z 2 + ( ω c ) 2 n 2 ( r ) ) F ( r ) = 0 .
1 W ( 2 W ρ 2 + 1 ρ W ρ + 1 ρ 2 2 W ϕ 2 ) + 1 Z d 2 Z d z 2 + k 0 2 n 2 ( r ) = 0 ,
( 2 W ρ 2 + 1 ρ W ρ + 1 ρ 2 2 W ϕ 2 ) + k 0 2 n ̅ 2 ( ρ ) W = 0
d 2 Z d z 2 + k 0 2 ( n 2 ( z ) n ̅ 2 ) Z = 0 ,
2 Ψ ρ 2 + 1 ρ Ψ ρ + ( k 0 2 n ̅ 2 ( ρ ) m 2 ρ 2 ) Ψ = 0
2 Ω ϕ 2 + m 2 Ω = 0 .
k 0 n ̅ ( k 0 ) J m + 1 ( k 0 n ̅ ( k 0 ) R ) = ( m R + η α ) J m ( k 0 n ̅ ( k 0 ) R ) ,
Ψ ( ρ ) { J m ( k 0 n ̅ ρ ) ρ R J m ( k 0 n ̅ R ) exp ( α ( ρ R ) ) ρ > R .
J = i ω δ ε E 0 .
δ ε = ε 0 δ n 2 h Δ r ( ϕ ) δ ( r R ) δ ( z ) ,
A rad ( r ) = μ 0 4 π ( e i k 1 r r ) J ( r ) e i k 1 r ̂ · r d r ,
A rad ( r ) = i μ 0 ω h δ n 2 ε 0 E m ( R , 0 ) R 4 π ( e i k 1 r r ) 0 2 π Δ r ( ϕ ) e i m ϕ exp ( i k 1 R sin θ cos ϕ ) d ϕ .
A rad ( r ) · A rad ( r ) * = μ 0 ω h δ n 2 ε 0 E m ( R , 0 ) 4 π 2 ( R r ) 2 Θ
Θ 0 2 π 0 2 π C ( ϕ ϕ ) exp ( i m ϕ ϕ ) exp [ i k 1 R sin θ ( cos ϕ cos ϕ ) ] d ϕ d ϕ ,
Θ = 2 π 0 2 π C ( t ) exp ( i m t ) J 0 [ 2 k 1 R sin θ sin ( t 2 ) ] d t .
Θ = 2 π R π R π R σ R 2 exp ( s 2 L c 2 + i m R s ) J 0 [ 2 k 1 R sin θ sin ( s R 2 ) ] d s 2 π 3 2 σ R 2 L c R ,
S rad = r ̂ ω k 0 2 μ 0 r ̂ × A rad 2 = r ̂ ω k 1 3 n 0 ( δ n 2 ) 2 V s 2 ε 0 E m ( R , 0 ) 2 16 π r ̂ × e ̂ 2 r 2 ,
P rad = ( S · r ̂ ) r 2 d Ω = η ̂ π 7 2 ω n 0 ( δ n 2 ) 2 V s 2 ε 0 E m ( R , 0 ; η ̂ ) 2 G ( η ̂ ) λ 0 3 ,
Q ss = λ 0 3 π 7 2 n 0 ( δ n 2 ) 2 V s 2 Σ η ̂ u s ¯ ( η ̂ ) G ( η ̂ ) ,
u ¯ s ( η ̂ ) = ε 0 E 0 ( η ̂ ) s , avg 2 1 2 ε 0 ( r ) E 0 2 d r .
2 E μ 0 ( ε 0 + δ ε ) 2 E t 2 = 0 ,
2 E j 0 ( r ) + μ 0 ε 0 ( r ) ω j 2 E j 0 ( r ) = 0 .
E ( r , t ) = exp ( i ω 0 t ) j a j ( t ) E j 0 ( r ) .
j ( 2 i ω 0 ε 0 d a j d t + δ ε ω 0 2 a j ε 0 ( ω j 2 ω 0 2 ) a j ) E j 0 ( r ) = 0 .
d a k d t + i Δ ω k a k ( t ) = i j β j k a j ,
with β j k = ω 0 2 δ ε ( E j 0 ( r ) ) * E k 0 ( r ) d r ε 0 E k 0 ( r ) 2 d r ,
d a c w d t = i Δ ω a c w + i β a c c w
d a c c w d t = i Δ ω a c c w + i β a c w ,
β = ω 0 4 U c ε 0 δ n 2 h Δ r ( s ) δ ( r R ) δ ( z ) ( E c w ( r ) ) * E c c w ( r ) d r .
β = ω 0 δ n 2 h R ε 0 E m ( R , 0 ) 2 4 U c ϒ
ϒ = 0 2 π Δ r ( ϕ ) e i 2 m ϕ d ϕ .
ϒ 2 = 2 π R π R π R σ R 2 exp ( s 2 L c 2 + i 2 m R s ) 2 π 3 2 σ R 2 L c R .
d a c w d t = ( γ t 2 + i Δ ω ) a c w + i β a c c w + κ s
d a c c w d t = ( γ t 2 + i Δ ω ) a c c w + i β a c w ,
Q β = 2 π 3 4 δ n 2 V s u s ¯ .
γ ̅ s a = γ s a ( r ) n 2 ( r ) E ( r ) 2 d r n 2 ( r ) E ( r ) 2 d r ,
γ ̅ s a = γ s a n S i 2 δ V s a E ( r ) 2 d r n 2 ( r ) E ( r ) 2 d r = 1 2 γ s a n S i 2 u s ¯ δ V s a .
Q s a = 4 π c λ 0 γ s a n S i 2 u s ¯ δ V s a .
E s , avg 2 = 1 δ V R δ r R + δ r h 2 h 2 0 2 π ρ d ρ d ϕ d z E 2 ,
U c = 1 2 ε 0 ( r ) E 2 d r 1 2 ε 0 n d 2 0 R h 2 h 2 0 2 π ρ d ρ d ϕ d z E 2 .
u s ̅ ( Z ̂ ) = 2 n d 2 δ V R δ r R + δ r ρ d ρ [ J m ( k 0 n ̅ ρ ) ] 2 0 R ρ d ρ [ J m ( k 0 n ̅ p ) ] 2 .
u s ̅ ( Z ̂ ) = [ J m ( k 0 n ̅ R ) ] 2 π h n d 2 0 R ρ d ρ [ J m ( k 0 n ̅ ρ ) ] 2 2 n ̅ 2 V d n d 2 ( n ̅ 2 n 0 2 ) ,
Q s s = 3 λ 0 3 8 π 7 2 n 0 δ n 2 ξ ( V d V s 2 ) ,
Q β = 1 2 π 3 4 ξ ( V d V s ) ,
Q s a = π c ( n ̅ 2 n 0 2 ) R λ 0 n ̅ 2 γ s a ζ ,
ξ = n ̅ 2 ( n d 2 n 0 2 ) n d 2 ( n ̅ 2 n 0 2 ) .

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