Abstract

Determination of the intrinsic quality factor of a loaded whispering gallery mode microcavity can be important for many applications where the coupling conditions cannot be tuned. We propose a single-scan technique based on a Stokes parameters analysis to extract the intrinsic quality factor and therefore determine the coupling regime. We propose a simple model for this analysis and present experimental measurements, which are in very good agreement with the model.

© 2011 OSA

1. Introduction

Whispering gallery mode (WGM) optical microcavities are widely known and used in many applications such as single molecule detection [1], nanoparticle sizing [2], compact narrow band filters [3], microlasers [4], optomechanical systems [5] and light switching [6]. Several materials are used in all these applications: Si, doped and undoped SiO2, fluoride materials, III–V and II–VI semiconductors, polymers, chalcogenide glass, etc. WGM microcavities exist mostly in the form of microspheres [7], microdisks [4], microtoroids [8] and integrated ring resonators [9]. Main features of these microcavities include their high intrinsic quality factor (Q0), their compactness and for some, their ease for on-chip integration.

Their optical characterization is usually done from their radiative emission spectrum collected by a free space setup or by measuring the transmission spectrum using the evanescent coupling to a waveguide. In the former case, the mode Q0, related to the losses caused by scattering, absorption and radiative process, is directly obtained from the resonance peak full width half maximum (FWHM or Δλ). In the latter case, Δλ is linked to the loaded or total quality factor (QT = λλ). It is related to Q0 and the coupling quality factor Qc by QT1=Q01+Qc1. As the coupling is increased, the light tends to leave the cavity sooner via the waveguide, thus increasing the losses due to the coupling and decreasing Qc. Depending on the dominating loss processes, the coupling regime can be identified as undercoupled (Q0 < Qc), critically coupled (Q0 = Qc = 2QT) or overcoupled (Q0 > Qc). Knowing precisely the contribution of the intrinsic losses and the coupling losses can be important in order to optimize the fabrication process or for pratical applications. For example, for sensing applications, the optimum sensitivity is achieved using a Qc = 2Q0 configuration [10]. In the case of filters or dispersion compensation devices, the maximization of Q0 is desirable [11].

The simplest approach to measure Q0 and Qc is done by using QT and the normalized transmission value at the resonance wavelength λr [12]. Unfortunately, due to their symmetric contribution to QT, the coupling regime has to be known to distinguish between the Q0 and Qc contributions, except when critically coupled where the normalized transmission is zero. The coupling regime can be identified by varying the gap between the waveguide and the cavity. Since sensors or telecommuncations devices are mainly designed to be on-chip with fixed waveguide configuration, a correct determination can be difficult. Most Fabry-Perot characterization techniques cannot be used since a modulated signal in the frequency range of the free spectral range (FSR) is needed [13, 14]. Since WGM cavities usually have FSR above 100 GHz, instruments operating at these frequencies are not only very expensive, but they make the experimental setup bulky and cumbersome as well. Dumeige et al. [15] showed that Q0 and Qc extraction is possible if a laser line is swept fast enough across the resonance, because the resulting ringing phenomenom does not depend on Q0 and Qc in the same way. Unfortunately, this technique requires a high speed sweep and thus, cannot be applied to QT below 107. Finally, Ito et al. [16] also proposed a method limited to the case of two busline waveguides using the drop port and throughput port responses where both waveguides have the same coupling coefficient.

In this article, we propose a simple method to extract both Q0 and Qc and therefore the coupling regime of WGM and ring-type cavities. Based on a Stokes parameters analysis near the resonance wavelength, this single scan technique can be used to determine Q0 and Qc when QT is above 1 × 105 with < 1% estimation error. Also, the method does not depend on the coupling regime nor on the input polarisation states. First, the theoretical model used to describe the Stokes parameters analysis is explained for different coupling regimes. The estimation of both quality factors is described with simple relations and the minimum QT where the technique is valid is determined based on the estimation errors. Then, we present the experimental setup and the measurements for different coupling gaps. The very good agreement of the experimental Stokes parameters curves supports the reliability of the proposed technique.

2. Theoretical model

The model used to extract Q0 and Qc is based on a polarization analysis (Stokes parameters) where the phase change in the WGM cavity gives, along with QT, an additional information about the coupling regime. In this section, we present the theoretical basis of how Q0 and Qc are determined and the conditions for which this technique is applicable.

The coupling arrangement is described in Fig. 1. An input polarization state a⃗x + a⃗y = axx⃗ + ayey⃗ enters the coupling region where ax and ay are the modulus of the x and y components, and ϕ, their phase difference. Only one axis is coupled to the cavity mode, we chose a⃗y in this article.

 

Fig. 1 Lossless coupling scheme between a single mode waveguide and an optical resonator. Polarization states axx⃗ + ayey⃗ and bxx⃗ + byy⃗ are entering and exiting the coupling region respectively. An amplitude transmission coefficient of |T| and a phase difference of θ are added by the cavity. The power fraction κ2 = 1 – t2 coupled to the cavity and the losses α characterize the waveguide-resonator system.

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Thus, the x and y axes correspond to the quasi-TE and quasi-TM modes of the cavity. The coupled (κ2) and uncoupled (t2) power fractions are related to each other as κ2 + t2 = 1 in the lossless case. They are related to Qc by Qc = k0nmL/(1 – t2) [17], where k0 = 2π/λ, nm and L are the vacuum wavenumber, the mode refractive index and the round-trip path of the mode in the cavity respectively. The losses inside the cavity are described by the attenuation factor α via exp(−αL). When the losses are small (exp(−αL) ≈ 1 – αL), α is related to the intrinsic quality factor by Q0 = k0nm/(2α). The exiting polarization state is described by b⃗x + b⃗y and it can be expressed using a Jones matrices formulation [18]:

[bxby]=[100|T|eiθ][axayeiϕ].
We assume that the coupling region is lossless and the resonant frequencies of the TE/TM modes are sufficiently separated to leave a⃗x unperturbed by the cavity presence. If a mode were not sufficiently TE or TM, it would imply that a⃗x would be perturbed by the cavity response as well and this proposed model would no longer describe properly the polarization changes. We experimentally validated that the mode is quasi-TE(TM) by making sure that the measured resonance peak can be fully turned off (flat transmission) and back on by rotating the input linear polarization using the first half-wave plate HWP1 (see Fig. 3). The cavity’s Jones matrix represents the complex response of a cavity mode coupled to a waveguide mode as follows [17]:
|T|2=t2+e2αL2teαLcos(β0L)1+t2e2αL2teαLcos(β0L)
θ=tan1([t21]eαLsin(β0L)t(1+e2αL)eαL[1+t2]cos(β0L))
where β0 = k0nm, |T|2 is the power transmission coefficient and θ is the phase change due to the interaction with the cavity. We assume the cavity is passive (Q0 > 0) and that t, κ, ϕ and α are constant over the spectral range of interest, typically twice the FWHM.

 

Fig. 3 Experimental setup used for S0 and S2 parameters measurements: TLS - Tunable laser source, OSC - Oscilloscope, TRIG - Trigger signal, CAV - Microcavity, TAP - Tapered fiber, C - 50:50 non-polarizing beamsplitter cube, CO1, CO2 et CO3 - Collimators, HWP1 et HWP2 - λ/2 wave plates, POL - Polarizer, L1 et L2 - Lens, D1 et D2 - Detectors. A SMF-28 fiber is used up to CO3.

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The Stokes parameters can now be expressed in terms of the cavity parameters (T, θ) and the system parameters (ax, ay and ϕ) as follows [19]:

S0=|bx|2+|by|2=ax2+|T|2ay2S1=|bx|2|by|2=ax2|T|2ay2S2=2|bx||by|cos(δ)=2axay|T|cos(θ+ϕ)S3=2|bx||by|sin(δ)=2axay|T|sin(θ+ϕ)
where δ is the accumulated phase difference between axis. These expressions reveal that a spectral characterization of the Stokes parameters provides, along with the normalized transmission S0, an additional information about the phase θ considering a non-zero value of ax and ay. Since S0 and S2 are similar to S1 and S3 respectively, the following equations are developed for S2 but one can use S3 as well since it only has a π/2 phase difference.

It is possible to represent S2 as a function of the system parameters t, α, L, β0 and ϕ by inserting Eq. (1) and Eq. (2) into the definition for S2 using trigonometric identities:

S2=2axaycos(ϕ)[t(1+e2αL)eαL(1+t2)cos(β0L)]+sin(ϕ)eαL(1t2)sin(β0L)1+t2e2αL2teαLcos(β0L).

In Fig. 2, we present the Stokes parameters spectra calculated from Eq. (3) for different coupling regimes and different ϕ values. The spectra of S0 in blue, S1 in purple, S2 in green and S3 in red are shown for each case. It can be seen that the S1 does not provide additional information compared to S0 since it is only its complementary response.

 

Fig. 2 Calculated spectra of the Stokes parameters across a resonance for the undercoupled regime 2(a)–2(b) (Q0 = 1 × 106 and Qc = 3 × 106), the critically coupled regime 2(c)–2(d) (Q0 = Qc = 1×106) and the overcoupled regime 2(e)–2(f) (Q0 = 1×106 and Qc = 3×105). The case where ϕ = 0 and ϕ = −π/5 are shown on the left and right side respectively. The black and red dots represent the extrema of the S3 spectra and the position of the FWHM values of S0 respectively. ax and ay are set to 1/2.

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The increasing amplitude of the S2 and S3 spectra as Qc decreases can be understood considering that more photons are entering the cavity and are then recollected by the waveguide. Consequently, the polarization state of a higher amount of light is changed by the cavity. Since the S0 spectrum tends to flatten as Qc decreases, the S2 spectrum can be used to spot collapsed peaks, mainly in the overcoupled regime (Fig. 2(e) and 2(f)).

The black dots indicate the extrema of the S3 curves and the red dots show the points that define the FWHM of the S0 curves. For ϕ = 0, the wavelength positions of the black and red dots match for any coupling regime. In the case where ϕ ≠ 0, the positions no longer agree and the distance between black dots increases as the coupling increases. However, it can be seen that ΔS2=S2maxS2min and ΔS3=S3maxS3min do not change when ϕ is modified but increase monotonically when the coupling increases. Their values go from 0 to 1 as the coupling regime passes from undercoupled to critically coupled and from 1 to 2 as it passes from critically coupled to overcoupled. Compared to the resonance transmission value which can give the same value in the undercoupled regime and in overcoupled regime [12], ΔS2 or ΔS3 can be used to determine the coupling regime.

An additional feature of the S3 spectrum (or S2 spectrum depending on ϕ value) is its steep slope near the resonance, which is a consequence of its phase response dependency (Eq. (3)). This attribute can be exploited in applications where high sensitivity is required, such as biosensing or laser stabilization.

Using a normalized S2 definition, S2N=S2/(2axay), it is possible to estimate Q0 and Qc. This can be done with S3N as well. The S2 extrema are found by inserting their respective β0L values, β0L(1) and β0L(2),

β0L(1)=arctan((1t2e2αL)sin(ϕ)cos(ϕ)(1+t2e2αL)+2teαL)
β0L(2)=arctan((1t2e2αL)sin(ϕ)cos(ϕ)(1+t2e2αL)2teαL)+π
in Eq. (4). Using trigonometric identities, the resulting extrema, S2N,max and S2N,min, are
S2N,max=S2max2axay=t(1e2αL)cos(ϕ)+eαL(1t2)1t2e2αL
S2N,min=S2min2axay=t(1e2αL)cos(ϕ)eαL(1t2)1t2e2αL
where it can be seen that both have similar ϕ dependence. Thus, the difference ΔS2N=S2N,maxS2N,min can be written as:
ΔS2N=ΔS22axay=2eαL(1t2)1t2e2αL2Q0β0LQ0+Qcβ0L
using Q0 and Qc definitions and exp(−αL) ≈ 1 – αL. This approximation represents a 0.5% deviation from the equality for Q0 > 104, λ ∼ 1550 nm, nm ∼ 1.45 and L ∼ 100πμm.

Furthermore, the total quality factor QT quantifies the intrinsic and coupling optical losses as:

QT=λΔλ=Q0QcQ0+Qc.
Finally, both Q0 and Qc can be estimated from Eq. (5) and Eq. (6) as:
Q0(e)=4axayQT4axayΔS2=2QT1ΔS2NandQc(e)=4axayQTΔS2=2QTΔS2N
when β0L can be neglected compared to Q0 and Qc. Thus, knowing ax and ay, both Qc(e) and Q0(e) can be extracted using a single measurement of Δλ and ΔS2. In order to be precise, these estimations require that the real Q0 and Qc be high enough. This condition is usually achieved for WGM and ring-type resonators. Also, since both are independent of ϕ, phase fluctuations do not change the results.

For a real measurement, however, we need to know within which limits these approximations hold based on a measurable quantity. We shall now define the lowest QT value needed in order to estimate Q0 and Qc within an acceptable error range. The relative errors of Qc and Q0, ΔQc/Qc and ΔQ0/Q0, are calculated from ΔS2 and QT for each (Q0, Qc) combination, for Q0 and Qc above 104. We limit the analysis to 104 since previous assumptions (t, κ, α and ϕ constant) are not guaranteed below this limit. This lower value of Qc implies a power coupling coefficient κ2 up to 0.18 which includes realistic values below 0.05 or Qc > 3.7 × 104 for a WGM cavity-to-waveguide evanescent coupling [20]. Finally, we set ax=ay=1/2 where S2 amplitude is maximized.

Using the calculations above, the relative errors in Q0 and Qc can be expressed as:

ΔQ0Q0=14axayQT(4axayΔS2)Q0,ΔQcQc=14axayQTQcΔS2.
For a relative error smaller than 1%, in the case of Q0(e), it requires a Q0 ≥ 104 and a Qc ≥ 105. For Qc(e), the combination (Q0, Qc) must be higher than (1 × 104,7.5 × 104) or (9 × 104,1 × 104). From Eq. (6), for a given QT value, Q0 and Qc are higher than QT. Consequently, the measured QT has to be above 105 in order to be within the 1% error limit of Q0(e) and Qc(e). For an error below 5%, a measured QT ≥ 2 × 104 is required.

3. Measurements and discussion

In order to relate the detected output signal of the cavity and the Stokes parameters, simple optical analysis can be used. The intensity of a signal passing through a wave plate with a dephasing angle of ϕl and a polarizer rotated of αp compared to the wave plate axis is [19]:

I(αp,ϕl)=12[S0+S1cos(2αp)+S2cos(ϕl)sin(2αp)+S3sin(ϕl)sin(2αp)].
The Stokes parameters can be related to different configurations of this optical system as follows:
S0=I(0°,0°)+I(90°,0°)S1=I(0°,0°)I(90°,0°)S2=2I(45°,0°)I(0°,0°)I(90°,0°)=2I(45°,0°)S0S3=2I(45°,90°)I(0°,0°)I(90°,0°)=2I(45°,90°)S0.

It should be noted that S2 is easier to measure than S3 since it does not need a quarter wave plate. S0 and S2 can be obtained by measuring the total power, I(0°, 0°) + I(90°,0°), and the intensity after the polarizer rotated by 45° with respect to the chosen reference axis, I(45°,0°). The experimental setup used to simultaneously determine S0 and S2 is shown in Fig. 3. A tunable laser source (TLS) sweeps across the resonance wavelength while an oscilloscope (OSC) records the signal from the detectors D1 and D2. Both are synchronized via a trigger signal (TRIG). The first half wave plate HWP1 is used to rotate the linearly polarized output of the source and therefore, to control the input power ratio ax2/ay2. To ensure that the polarization state did not change along the SMF-28 fiber, the optical fiber length and curvature were minimized. Furthermore, before measurement and without any coupling to the cavity, the output polarization was measured after the collimator CO3 to ensure that it stayed linear along the fiber. To do so, a linear polarizer and a free space detector were used to verify the flatness of the polarization ellipse. We did not notice any significant change compared to the polarization ellipse measured right after the collimator CO1. Thus, we conclude that the polarization along the optical fiber and the tapered fiber stays linear. The second half wave plate HWP2 is adjusted such as the x and y axis are turned 45° compared to the horizontal and vertical reference axis. This allows to write the detector intensities as a function of the cavity parameters in a simple form (Eq. (7)). Finally, the polarizer axis is set along the horizontal axis or 45° compared to ax. Both signals are focused on the detectors. This simple configuration can be used to mesure the transmission spectrum via D1 without any modification. Since both signals are recorded simultaneously, any fluctuation in the resonance wavelength due to external parameters such as temperature does not affect the result.

A 1.2 μm diameter tapered optical fiber is used to couple the light to a silica toroidal micro-cavity. The microcavities are formed from a 0.8 μm thick thermal SiO2 layer. Using a standard photolithographic process, the disk shapes are transfered to the silica layer. An SF6 isotropic etch of the subjacent silicon follows. The toroidal shapes are obtained using laser reflow process [8]. The microtoroids have a 5 μm minor diameter and are formed out of a 100 μm diameter disk. A micrography of the coupling region is shown in Fig. 4. The gap between the tapered fiber and the cavity is controlled using a piezoelectric stage.

 

Fig. 4 Micrography of the resonator and waveguide. A 1.2 μm diameter tapered optical fiber is brought near to a toroidal silica microtoroid resonator using a piezoeletric stage.

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Using this experimental setup, the intensities detected at D1 and D2, I1 and I2, can be written as:

I1=|T45|2ax2+|T45|2|T|2ay2|T45|2(ax2+|T|2ay2)I2px2|R45|22[ax2+|T|2ay2+2axay|T|cos(θ+ϕ)].
where the |T45|2 and |T−45|2 cube beamsplitter transmission coefficients are considered equal for a 45° and a −45° linear input polarization with respect to the horizontal axis. The reflection coefficients, |R45|2 and |R−45|2, are also considered equal. px2 is the polarizer transmission coefficient. Using these relations, the experimental normalized Stokes parameters S0N(e) and S2N(e) are retrieved as:
S0N(e)=I1|T45|2Ptot=I1I1offS2N(e)=2I2px2|R45|2I1|T45|22Ptotf(1f)
where f is the power fraction on the ay axis. The total power, Ptot, is written as:
Ptot=ax2+ay2=I1off|T45|2ax=(1f)Ptotanday=fPtot.
I1off is the intensity measurement at D1 off the resonance where the coupling can be neglected (|T|2 = 1).

In Figs. 5(a)–5(c), the normalized S0N(e) (red curve) and S2N(e) (green curve) are shown for three coupling conditions (CC). The black dotted lines show the extrema of S2N(e). The measured parameters, ΔS2N, QT and their corresponding Q0(e) and Qc(e) are presented in Table 1 for the three CC. The QT values were obtained by fitting a lorentzian curve to S0N(e). This can be done because the FWHM and the center wavelength are the same for both S0 and the transmission |T|2. As it can be seen from the increasing amplitude of S2N(e) from Fig. 5(a) to Fig. 5(c), the coupling coefficient is also increased, meaning a smaller gap betwen the waveguide and the cavity.

 

Fig. 5 Experimental and calculated S0 and S2 parameters obtained for three coupling conditions giving (a) Q0(e)=(2.273±0.017)×106 and Qc(e)=(1.58±0.12)×108 (b) Q0(e)=(2.333±0.010)×106 and Qc(e)=(3.81±0.08)×107 and (c) Q0(e)=(2.311±0.009)×106 and Qc(e)=(5.11±0.03)×106.

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Tables Icon

Table 1. Extracted parameters from S0N(e) (red curve) and S2N(e) (green curve) presented in Fig. 5(a)–5(c).

Using the extracted values of Q0(e) and Qc(e) in the proposed model (Eq. (1)(3)), the calculated S0 (black curve) and S2 (blue curve) are drawn using f-parameters equal to 0.4, 0.435 and 0.445 for the first, second and third coupling condition respectively. These changes show the rotation of the polarization between different measurements. The ϕ parameter is extracted using the off resonance value of S2N(e) where cos(θ + ϕ) → cos(ϕ). There is a very good agreement between the experimental and the calculated curves, which shows that the proposed model, despite its simplicity, represents well the polarization changes inside the cavity.

It is worth noting the quasi constant value of Q0(e) for the three coupling conditions. An additional measurement was taken for a very low coupling coefficient where QTQ0 gave a QT = (2.26 ± 0.05) × 106 which fits well with the presented data. This anticipated behavior shows that only the coupling losses are changed if the coupling conditions change. Thus, the intrinsic losses are only slightly changed by the fiber presence.

4. Conclusion

In this paper, we proposed a single scan method to quantitatively extract the intrinsic Q-factor Q0 and the coupling Q-factor Qc of a WGM or ring-type microcavity, regardless of the coupling regime. This technique is based on a simple Stokes parameters measurement. The theoretical model has been detailed and experimentally verified with a very good agreement. The determination of Q0 and Qc is accurate within 1 percent if QT is higher than 1 × 105. Compared to the laser sweeping technique [15] which is limited to QT ∼ 107, our method can be used for lower QT resonators such as integrated microrings. The detection setup requires only simple optical components excluding usually needed fast electronics [13, 14]. This method provides a direct way to determine the relevant quality factors for waveguide-coupled microresonators, and is particularly useful for integrated systems.

Acknowledgments

We thank Dr. Bianucci, S. Virally and Dr. Saidi for useful discussions. This work was supported by the Natural Sciences and Engineering Research Council of Canada, Strategic Grant 365207-08 and the Fonds Québécois de la Recherche sur la Nature et les Technologies, Equip Grant PR-119043.

References and links

1. F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008). [CrossRef]   [PubMed]  

2. J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009). [CrossRef]  

3. A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804(R) (2004). [CrossRef]  

4. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992). [CrossRef]  

5. 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). [CrossRef]   [PubMed]  

6. 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]  

7. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996). [CrossRef]   [PubMed]  

8. 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]  

9. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998). [CrossRef]  

10. M. Sumetsky, “Optimization of optical ring resonator devices for sensing applications,” Opt. Lett. 32, 2577–2579 (2007). [CrossRef]   [PubMed]  

11. M. Hossein-Zadeh and K. J. Vahala, “Importance of intrinsic-Q in microring-based optical filters and dispersion-compensation devices,” IEEE Photonics Technol. Lett. 19, 1045–1047 (2007). [CrossRef]  

12. 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, 043902 (2003). [CrossRef]   [PubMed]  

13. B. J. J. Slagmolen, M. B. Gray, K. G. Baigent, and D. E. McClelland, “Phase-sensitive reflection technique for characterization of a Fabry-Perot interferometer,” Appl. Opt. 39, 3638–3643 (2000). [CrossRef]  

14. C. R. Locke, D. Stuart, E. N. Ivanov, and A. N. Luiten, “A simple technique for accurate and complete characterisation of a Fabry-Perot cavity,” Opt. Express 17, 21935–21943 (2009). [CrossRef]   [PubMed]  

15. Y. Dumeige, S. Trebaol, L. Ghişa, T. K. N. Nguyên, H. Tavernier, and P. Féron, “Determination of coupling regime of high-Q resonators and optical gain of highly selective amplifiers,” J. Opt. Soc. Am. B 25, 2073–2080 (2008). [CrossRef]  

16. T. Ito and Y. Kokubun, “Nondestructive measurement of propagation loss and coupling efficiency in microring resonator filters using filter responses,” Jpn. J. Appl. Phys. 43, 1002–1005 (2004). [CrossRef]  

17. G. Griffel, S. Arnold, D. Taskent, A. Serpengüzel, J. Connolly, and N. Morris, “Morphology-dependent resonances of a microsphere-optical fiber system,” Opt. Lett. 21, 695–697 (1996). [CrossRef]   [PubMed]  

18. P. Bianucci, C. R. Fietz, J. W. Robertson, G. Shvets, and C.-K. Shih, “Whispering gallery mode microresonators as polarization converters,” Opt. Lett. 32, 2224–2226 (2007). [CrossRef]   [PubMed]  

19. D. Goldstein, Polarized Light (Marcel Dekker, Inc., New York, 2003), 2nd ed. [CrossRef]  

20. M. J. Humphrey, E. Dale, A. T. Rosenberger, and D. K. Bandy, “Calculation of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled microsphere,” Opt. Commun. 271, 124–131 (2007). [CrossRef]  

References

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  1. F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
    [CrossRef] [PubMed]
  2. J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
    [CrossRef]
  3. A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804(R) (2004).
    [CrossRef]
  4. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
    [CrossRef]
  5. 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).
    [CrossRef] [PubMed]
  6. 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]
  7. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
    [CrossRef] [PubMed]
  8. 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]
  9. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
    [CrossRef]
  10. M. Sumetsky, “Optimization of optical ring resonator devices for sensing applications,” Opt. Lett. 32, 2577–2579 (2007).
    [CrossRef] [PubMed]
  11. M. Hossein-Zadeh and K. J. Vahala, “Importance of intrinsic-Q in microring-based optical filters and dispersion-compensation devices,” IEEE Photonics Technol. Lett. 19, 1045–1047 (2007).
    [CrossRef]
  12. 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, 043902 (2003).
    [CrossRef] [PubMed]
  13. B. J. J. Slagmolen, M. B. Gray, K. G. Baigent, and D. E. McClelland, “Phase-sensitive reflection technique for characterization of a Fabry-Perot interferometer,” Appl. Opt. 39, 3638–3643 (2000).
    [CrossRef]
  14. C. R. Locke, D. Stuart, E. N. Ivanov, and A. N. Luiten, “A simple technique for accurate and complete characterisation of a Fabry-Perot cavity,” Opt. Express 17, 21935–21943 (2009).
    [CrossRef] [PubMed]
  15. Y. Dumeige, S. Trebaol, L. Ghişa, T. K. N. Nguyên, H. Tavernier, and P. Féron, “Determination of coupling regime of high-Q resonators and optical gain of highly selective amplifiers,” J. Opt. Soc. Am. B 25, 2073–2080 (2008).
    [CrossRef]
  16. T. Ito and Y. Kokubun, “Nondestructive measurement of propagation loss and coupling efficiency in microring resonator filters using filter responses,” Jpn. J. Appl. Phys. 43, 1002–1005 (2004).
    [CrossRef]
  17. G. Griffel, S. Arnold, D. Taskent, A. Serpengüzel, J. Connolly, and N. Morris, “Morphology-dependent resonances of a microsphere-optical fiber system,” Opt. Lett. 21, 695–697 (1996).
    [CrossRef] [PubMed]
  18. P. Bianucci, C. R. Fietz, J. W. Robertson, G. Shvets, and C.-K. Shih, “Whispering gallery mode microresonators as polarization converters,” Opt. Lett. 32, 2224–2226 (2007).
    [CrossRef] [PubMed]
  19. D. Goldstein, Polarized Light (Marcel Dekker, Inc., New York, 2003), 2nd ed.
    [CrossRef]
  20. M. J. Humphrey, E. Dale, A. T. Rosenberger, and D. K. Bandy, “Calculation of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled microsphere,” Opt. Commun. 271, 124–131 (2007).
    [CrossRef]

2009 (2)

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

C. R. Locke, D. Stuart, E. N. Ivanov, and A. N. Luiten, “A simple technique for accurate and complete characterisation of a Fabry-Perot cavity,” Opt. Express 17, 21935–21943 (2009).
[CrossRef] [PubMed]

2008 (2)

2007 (4)

P. Bianucci, C. R. Fietz, J. W. Robertson, G. Shvets, and C.-K. Shih, “Whispering gallery mode microresonators as polarization converters,” Opt. Lett. 32, 2224–2226 (2007).
[CrossRef] [PubMed]

M. Sumetsky, “Optimization of optical ring resonator devices for sensing applications,” Opt. Lett. 32, 2577–2579 (2007).
[CrossRef] [PubMed]

M. J. Humphrey, E. Dale, A. T. Rosenberger, and D. K. Bandy, “Calculation of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled microsphere,” Opt. Commun. 271, 124–131 (2007).
[CrossRef]

M. Hossein-Zadeh and K. J. Vahala, “Importance of intrinsic-Q in microring-based optical filters and dispersion-compensation devices,” IEEE Photonics Technol. Lett. 19, 1045–1047 (2007).
[CrossRef]

2005 (1)

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).
[CrossRef] [PubMed]

2004 (3)

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]

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804(R) (2004).
[CrossRef]

T. Ito and Y. Kokubun, “Nondestructive measurement of propagation loss and coupling efficiency in microring resonator filters using filter responses,” Jpn. J. Appl. Phys. 43, 1002–1005 (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, 043902 (2003).
[CrossRef] [PubMed]

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]

2000 (1)

1998 (1)

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

1996 (2)

1992 (1)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

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]

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]

Arnold, S.

Baigent, K. G.

Bandy, D. K.

M. J. Humphrey, E. Dale, A. T. Rosenberger, and D. K. Bandy, “Calculation of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled microsphere,” Opt. Commun. 271, 124–131 (2007).
[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]

Bianucci, P.

Carmon, T.

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).
[CrossRef] [PubMed]

Chen, D.-R.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

Chu, S. T.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Connolly, J.

Dale, E.

M. J. Humphrey, E. Dale, A. T. Rosenberger, and D. K. Bandy, “Calculation of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled microsphere,” Opt. Commun. 271, 124–131 (2007).
[CrossRef]

Dumeige, Y.

Féron, P.

Fietz, C. R.

Foresi, J. S.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Ghisa, L.

Goldstein, D.

D. Goldstein, Polarized Light (Marcel Dekker, Inc., New York, 2003), 2nd ed.
[CrossRef]

Gorodetsky, M. L.

Gray, M. B.

Greene, W.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Griffel, G.

Haus, H. A.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

He, L.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

Hossein-Zadeh, M.

M. Hossein-Zadeh and K. J. Vahala, “Importance of intrinsic-Q in microring-based optical filters and dispersion-compensation devices,” IEEE Photonics Technol. Lett. 19, 1045–1047 (2007).
[CrossRef]

Humphrey, M. J.

M. J. Humphrey, E. Dale, A. T. Rosenberger, and D. K. Bandy, “Calculation of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled microsphere,” Opt. Commun. 271, 124–131 (2007).
[CrossRef]

Ilchenko, V. S.

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804(R) (2004).
[CrossRef]

M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
[CrossRef] [PubMed]

Ippen, E. P.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Ito, T.

T. Ito and Y. Kokubun, “Nondestructive measurement of propagation loss and coupling efficiency in microring resonator filters using filter responses,” Jpn. J. Appl. Phys. 43, 1002–1005 (2004).
[CrossRef]

Ivanov, E. N.

Kimerling, L. C.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Kippenberg, T. J.

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).
[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, 043902 (2003).
[CrossRef] [PubMed]

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]

Kokubun, Y.

T. Ito and Y. Kokubun, “Nondestructive measurement of propagation loss and coupling efficiency in microring resonator filters using filter responses,” Jpn. J. Appl. Phys. 43, 1002–1005 (2004).
[CrossRef]

Levi, A. F. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Li, L.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

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]

Little, B. E.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Locke, C. R.

Logan, R. A.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Luiten, A. N.

Maleki, L.

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804(R) (2004).
[CrossRef]

Matsko, A. B.

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804(R) (2004).
[CrossRef]

McCall, S. L.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

McClelland, D. E.

Morris, N.

Nguyên, T. K. N.

Ozdemir, S. K.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

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, 043902 (2003).
[CrossRef] [PubMed]

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]

Pearton, S. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Robertson, J. W.

Rokhsari, H.

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).
[CrossRef] [PubMed]

Rosenberger, A. T.

M. J. Humphrey, E. Dale, A. T. Rosenberger, and D. K. Bandy, “Calculation of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled microsphere,” Opt. Commun. 271, 124–131 (2007).
[CrossRef]

Savchenkov, A. A.

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804(R) (2004).
[CrossRef]

M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
[CrossRef] [PubMed]

Scherer, A.

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).
[CrossRef] [PubMed]

Serpengüzel, A.

Shih, C.-K.

Shvets, G.

Slagmolen, B. J. J.

Slusher, R. E.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Spillane, S. M.

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, 043902 (2003).
[CrossRef] [PubMed]

Steinmeyer, G.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Stuart, D.

Sumetsky, M.

Taskent, D.

Tavernier, H.

Thoen, E. R.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Trebaol, S.

Vahala, K. J.

M. Hossein-Zadeh and K. J. Vahala, “Importance of intrinsic-Q in microring-based optical filters and dispersion-compensation devices,” IEEE Photonics Technol. Lett. 19, 1045–1047 (2007).
[CrossRef]

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).
[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, 043902 (2003).
[CrossRef] [PubMed]

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]

Vollmer, F.

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[CrossRef] [PubMed]

Xiao, Y. F.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

Yang, L.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

Zhu, J.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

IEEE Photonics Technol. Lett. (2)

M. Hossein-Zadeh and K. J. Vahala, “Importance of intrinsic-Q in microring-based optical filters and dispersion-compensation devices,” IEEE Photonics Technol. Lett. 19, 1045–1047 (2007).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

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

Jpn. J. Appl. Phys. (1)

T. Ito and Y. Kokubun, “Nondestructive measurement of propagation loss and coupling efficiency in microring resonator filters using filter responses,” Jpn. J. Appl. Phys. 43, 1002–1005 (2004).
[CrossRef]

Nat. Methods (1)

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[CrossRef] [PubMed]

Nat. Photonics (1)

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

Nature (2)

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]

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]

Opt. Commun. (1)

M. J. Humphrey, E. Dale, A. T. Rosenberger, and D. K. Bandy, “Calculation of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled microsphere,” Opt. Commun. 271, 124–131 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (1)

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804(R) (2004).
[CrossRef]

Phys. Rev. Lett. (2)

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).
[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, 043902 (2003).
[CrossRef] [PubMed]

Other (1)

D. Goldstein, Polarized Light (Marcel Dekker, Inc., New York, 2003), 2nd ed.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Lossless coupling scheme between a single mode waveguide and an optical resonator. Polarization states axx⃗ + ayey⃗ and bxx⃗ + byy⃗ are entering and exiting the coupling region respectively. An amplitude transmission coefficient of |T| and a phase difference of θ are added by the cavity. The power fraction κ2 = 1 – t2 coupled to the cavity and the losses α characterize the waveguide-resonator system.

Fig. 3
Fig. 3

Experimental setup used for S0 and S2 parameters measurements: TLS - Tunable laser source, OSC - Oscilloscope, TRIG - Trigger signal, CAV - Microcavity, TAP - Tapered fiber, C - 50:50 non-polarizing beamsplitter cube, CO1, CO2 et CO3 - Collimators, HWP1 et HWP2 - λ/2 wave plates, POL - Polarizer, L1 et L2 - Lens, D1 et D2 - Detectors. A SMF-28 fiber is used up to CO3.

Fig. 2
Fig. 2

Calculated spectra of the Stokes parameters across a resonance for the undercoupled regime 2(a)–2(b) (Q0 = 1 × 106 and Qc = 3 × 106), the critically coupled regime 2(c)–2(d) (Q0 = Qc = 1×106) and the overcoupled regime 2(e)–2(f) (Q0 = 1×106 and Qc = 3×105). The case where ϕ = 0 and ϕ = −π/5 are shown on the left and right side respectively. The black and red dots represent the extrema of the S3 spectra and the position of the FWHM values of S0 respectively. ax and ay are set to 1 / 2.

Fig. 4
Fig. 4

Micrography of the resonator and waveguide. A 1.2 μm diameter tapered optical fiber is brought near to a toroidal silica microtoroid resonator using a piezoeletric stage.

Fig. 5
Fig. 5

Experimental and calculated S0 and S2 parameters obtained for three coupling conditions giving (a) Q 0 ( e ) = ( 2.273 ± 0.017 ) × 10 6 and Q c ( e ) = ( 1.58 ± 0.12 ) × 10 8 (b) Q 0 ( e ) = ( 2.333 ± 0.010 ) × 10 6 and Q c ( e ) = ( 3.81 ± 0.08 ) × 10 7 and (c) Q 0 ( e ) = ( 2.311 ± 0.009 ) × 10 6 and Q c ( e ) = ( 5.11 ± 0.03 ) × 10 6.

Tables (1)

Tables Icon

Table 1 Extracted parameters from S 0 N ( e ) (red curve) and S 2 N ( e ) (green curve) presented in Fig. 5(a)–5(c).

Equations (18)

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[ b x b y ] = [ 1 0 0 | T | e i θ ] [ a x a y e i ϕ ] .
| T | 2 = t 2 + e 2 α L 2 t e α L cos ( β 0 L ) 1 + t 2 e 2 α L 2 t e α L cos ( β 0 L )
θ = tan 1 ( [ t 2 1 ] e α L sin ( β 0 L ) t ( 1 + e 2 α L ) e α L [ 1 + t 2 ] cos ( β 0 L ) )
S 0 = | b x | 2 + | b y | 2 = a x 2 + | T | 2 a y 2 S 1 = | b x | 2 | b y | 2 = a x 2 | T | 2 a y 2 S 2 = 2 | b x | | b y | cos ( δ ) = 2 a x a y | T | cos ( θ + ϕ ) S 3 = 2 | b x | | b y | sin ( δ ) = 2 a x a y | T | sin ( θ + ϕ )
S 2 = 2 a x a y cos ( ϕ ) [ t ( 1 + e 2 α L ) e α L ( 1 + t 2 ) cos ( β 0 L ) ] + sin ( ϕ ) e α L ( 1 t 2 ) sin ( β 0 L ) 1 + t 2 e 2 α L 2 t e α L cos ( β 0 L ) .
β 0 L ( 1 ) = arctan ( ( 1 t 2 e 2 α L ) sin ( ϕ ) cos ( ϕ ) ( 1 + t 2 e 2 α L ) + 2 t e α L )
β 0 L ( 2 ) = arctan ( ( 1 t 2 e 2 α L ) sin ( ϕ ) cos ( ϕ ) ( 1 + t 2 e 2 α L ) 2 t e α L ) + π
S 2 N , max = S 2 max 2 a x a y = t ( 1 e 2 α L ) cos ( ϕ ) + e α L ( 1 t 2 ) 1 t 2 e 2 α L
S 2 N , min = S 2 min 2 a x a y = t ( 1 e 2 α L ) cos ( ϕ ) e α L ( 1 t 2 ) 1 t 2 e 2 α L
Δ S 2 N = Δ S 2 2 a x a y = 2 e α L ( 1 t 2 ) 1 t 2 e 2 α L 2 Q 0 β 0 L Q 0 + Q c β 0 L
Q T = λ Δ λ = Q 0 Q c Q 0 + Q c .
Q 0 ( e ) = 4 a x a y Q T 4 a x a y Δ S 2 = 2 Q T 1 Δ S 2 N and Q c ( e ) = 4 a x a y Q T Δ S 2 = 2 Q T Δ S 2 N
Δ Q 0 Q 0 = 1 4 a x a y Q T ( 4 a x a y Δ S 2 ) Q 0 , Δ Q c Q c = 1 4 a x a y Q T Q c Δ S 2 .
I ( α p , ϕ l ) = 1 2 [ S 0 + S 1 cos ( 2 α p ) + S 2 cos ( ϕ l ) sin ( 2 α p ) + S 3 sin ( ϕ l ) sin ( 2 α p ) ] .
S 0 = I ( 0 ° , 0 ° ) + I ( 90 ° , 0 ° ) S 1 = I ( 0 ° , 0 ° ) I ( 90 ° , 0 ° ) S 2 = 2 I ( 45 ° , 0 ° ) I ( 0 ° , 0 ° ) I ( 90 ° , 0 ° ) = 2 I ( 45 ° , 0 ° ) S 0 S 3 = 2 I ( 45 ° , 90 ° ) I ( 0 ° , 0 ° ) I ( 90 ° , 0 ° ) = 2 I ( 45 ° , 90 ° ) S 0 .
I 1 = | T 45 | 2 a x 2 + | T 45 | 2 | T | 2 a y 2 | T 45 | 2 ( a x 2 + | T | 2 a y 2 ) I 2 p x 2 | R 45 | 2 2 [ a x 2 + | T | 2 a y 2 + 2 a x a y | T | cos ( θ + ϕ ) ] .
S 0 N ( e ) = I 1 | T 45 | 2 P tot = I 1 I 1 off S 2 N ( e ) = 2 I 2 p x 2 | R 45 | 2 I 1 | T 45 | 2 2 P tot f ( 1 f )
P tot = a x 2 + a y 2 = I 1 off | T 45 | 2 a x = ( 1 f ) P tot and a y = f P tot .

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