Linearity of silicon ring modulators for analog optical links

doi:10.1364/OE.20.013115

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Linearity of silicon ring modulators for analog optical links

Ali Ayazi, Tom Baehr-Jones, Yang Liu, Andy Eu-Jin Lim, and Michael Hochberg »View Author Affiliations

Optics Express, Vol. 20, Issue 12, pp. 13115-13122 (2012)
http://dx.doi.org/10.1364/OE.20.013115

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Abstract

We study the nonlinear distortions of a silicon ring modulator based on the carrier depletion effect for analog links. Key sources of modulation nonlinearity are identified and modeled. We find that the most important source of nonlinearity is from the pn junction itself, as opposed to the nonlinear wavelength response of the ring modulator. Spurious free dynamic range for intermodulation distortion of as high as 84 dB.Hz^2/3 is obtained.

1. Introduction

Ring resonator modulators feature great advantages like compact footprint, low drive-voltages, and the ability to drive as lumped RF elements, which make them promising candidates for a number of applications in silicon photonics fro example switches [1

H. L. R. Lira, S. Manipatruni, and M. Lipson, “Broadband hitless silicon electro-optic switch for on-chip optical networks,” Opt. Express 17(25), 22271–22280 (2009). [CrossRef] [PubMed]

], tunable filters [2

P. Rabiei and W. H. Steier, “Tunable polymer double micro-ring filters,” Photon. Technol. Lett. 15(9), 1255–1257 (2003). [CrossRef]

], and modulators [3

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]

–5

M. Gould, T. Baehr-Jones, R. Ding, S. Huang, J. Luo, A. K.-Y. Jen, J.-M. Fedeli, M. Fournier, and M. Hochberg, “Silicon-polymer hybrid slot waveguide ring-resonator modulator,” Opt. Express 19(5), 3952–3961 (2011). [CrossRef] [PubMed]

]. There are applications to analog optical links as well [6

R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microw. Theory Tech. 54(2), 832–846 (2006). [CrossRef]

–9

J. Yao, “Microwave photonics,” J. Lightwave Technol. 27(3), 314–335 (2009). [CrossRef]

], such as radio-over-fiber, antenna remoting, subcarrier transmission, and phase array antenna control. For an analog optical link, nonlinearity is one of the key performance metrics [10

W. B. Bridges and J. H. Schaffner, “Distortion in linearized electrooptic modulators,” IEEE Trans. Microw. Theory Tech. 43(9), 2184–2197 (1995). [CrossRef]

–13

T. Ismail, C.-P. Liu, J. E. Mitchell, and A. J. Seeds, “High-dynamic- range wireless-over-fiber link using feed forward linearization,” J. Lightwave Technol. 25(11), 3274–3282 (2007). [CrossRef]

]. There are some prior works in the literature on the modeling of the nonlinearity in the ring resonator [14

M. Song, L. Zhang, R. G. Beausoleil, and A. E. Willner, “Nonlinear distortion in a silicon microring-based electro-optic modulator for analog optical links,” IEEE J. Sel. Top. Quantum Electron. 16, 185–191 (2010).

,15

B. Dingel, A. Prescod, N. Madamopoulos, and R. Madabhushi, “Performance of ring resonator-based linear optical modulator (IMPACC) under Critical Coupling (CC), Over Coupling (OC), and Under Coupling (UC) conditions,” IEEE Photonics Conference (PHO) 260–261 (2011).

]. Here we also develop a model for the nonlinearities of silicon ring modulators with observations that are beyond what were in the prior works. This model is then matched with experimental results for a reverse-biased pn junction modulator. We find that the most important source of nonlinearity is the pn junction itself, as opposed to the nonlinear wavelength response of the ring modulator. This suggests minimizing pn junction nonlinearities, in addition to maximizing modulation response and lower optical losses, may be an important area for future investigation.

2. Principle

The ring resonator is a cavity working on the principle of the constructive interference of light inside the resonator, resulting in a Lorentzian transfer function. For the maximum modulation efficiency, the modulator is typically biased at the largest slope of the transfer curve. Signal modulation and the bias point are illustrated in Fig. 1 . The curvature of the transfer function at the bias point introduces nonlinear distortion to the output. It is also well known that a reverse-biased pn junction, which we utilize in our device, will exhibit a capacitance that is relatively nonlinear as a function of voltage, having the approximate functional form of an inverse square root [16

C. Kittel, Introduction to Solid State Physics, 5th ed.(Wiley), (1976).

]. A similar nonlinearity will be seen on the refractive index shift. As a result of these nonlinearities, in the output RF signal, in addition to the carrier, we will have a second harmonic distortion (SHD), third harmonic distortion (THS), third-order inter-modulation (IMD), and other higher order nonlinear terms. This is conceptually shown in Fig. 1 where two tones at frequencies f₁ and f₂ are input to the modulator. 2f₁-f₂ and 2f₂-f₁ components are called the third-order intermodulation distortions (IMD). If the modulator is used in a broadband application, the second harmonic distortion is the most dominant distortion that needs to be studied. Many of the analog microwave applications however, have bandwidths less than an octave, in which second order distortion is not a problem and the third-order intermodulation distortion has to be taken into account.

Fig. 1 Steady state transfer function of the silicon ring modulator.

There are a number of measures to evaluate the nonlinear distortion performance of an analog system. Spurious-Free-Dynamic-Range (SFDR) is one of the most widely used ones for this purpose. It is defined as the difference in the RF input in the link between the signal level that produces an output equal to the noise level and the signal level that would produce distortion products equal to the noise level [14

3. Design and fabrication

Fabrication occurred at the Institute of Microelectronics (IME)/ASTAR [17

http://www.ime.a-star.edu.sg/PPSSite/index.asp

]. The starting material was an 8” Silicon-on-Insulator (SOI) wafer from SOITEC, with a Boron-doped top silicon layer of around 10 ohm-cm resistivity and 220 nm thickness, a 2 μm bottom oxide thickness, and a 750 ohm-cm handle silicon wafer, needed for RF performance. A 60 nm anisotropic dry etch was first applied to form the trenches of the grating couplers. Next, the rib waveguides for the ring were formed using additional etch steps. In all cases 248 nm photolithography was utilized. The p + + , p, n + + , and n implants for the modulator were performed on the exposed silicon, prior to any oxide fill. The peak doping density for N side is 4.2 × 10¹⁷ cm⁻³, and 6.8 × 10¹⁷ cm⁻³ for the P side. This was followed by a rapid thermal anneal at 1030 °C for 5s for Si dopant activation. It was followed by the formation of contact vias and two levels of aluminum interconnects. Chemical-mechanical planarization (CMP) was not utilized. The schematic cross-section is shown in Fig. 2 .

Fig. 2 Cross-section of the rib waveguides of the ring modulator.

Rings with a 30 um radius and a lateral pn junction centered in a rib waveguide were fabricated. Figure 3 shows the device layout. Other than the through port which is used for the modulated signal to come from, a drop waveguide is also placed in order to intentionally spoil the Q. This gives us greater control over the modulator's performance and allows a larger extinction ratio.

Fig. 3 Device Layout. Optical micrograph of the ring modulator. Note that the insets with the highest magnification are renderings of the mask layout, instead of micrographs.

4. Theory

Carrier depletion effect that is used in this silicon ring modulator to shift the resonance is not linear, which is not a problem for digital signals but is very important in an analog application. In order to analyze the nonlinear distortion of the modulator, it has to be taken into account.

Electrical signal that is driving the modulator varies the carrier density and hence the refractive index in the ring cavity. The induced change in the refractive index due to the free-carrier plasma dispersion effect at the wavelength of 1.55um are calculated as [18

R. A. Soref and B. R. Bennet, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]

Δ n = Δ n_{e} + Δ n_{h} = - [8.8 \times 10^{- 22} Δ N + 8.5 \times 10^{- 18} {(Δ P)}^{0.8}]

(1)

Where Δn_e and Δn_h are the refractive index changes due to electron and hole concentration changes, ΔN and ΔP (in cm⁻³).

In order to find the relation between the change in the refractive index and the input voltage, thorough simulation of the exact structures in this paper has been implemented in Sentaurus software (from Synopsys). At first, Sentaurus-process was used to model the fabrication sequence of the pn phase shifter, including waveguide etching, ion implantation, thermal annealing, and contact metallization. After the process simulation, the distribution of ion dopants on a cross section of the silicon waveguide was obtained (Fig. 4 ).

Fig. 4 Distribution of doping concentration in the cross-section of the device.

Then, the electron and hole distribution in the virtual modulator was solved by Sentaurus-device under different bias voltage. Once the carrier concentration at each spatial point was known, one can readily calculate the local refractive index change

Δ n (x, y)

, and do the overlap with the optical mode to obtain the effective index change

Δ n_{e f f}

of the waveguide as a function of the bias voltage. These results are compared to the measured values in Fig. 5 . Measured values are derived from the measured phase change in a Mach-Zehnder structure with the same waveguide characteristic as the ones in the ring which are fabricated in the same platform.

Fig. 5 Measured and simulated results of the effective refractive index change vs. input voltage.

This data can then be used to approximate the effective index in the following form:

n_{e f f} (t) = n_{e o} + \frac{d n_{e f f}}{d V} \cdot V_{i n} (t) + \frac{d^{2} n_{e f f}}{d V^{2}} \cdot V_{i n}^{2} (t) + \dots

(2)

n_eff (t) can be incorporated in the Lorentzian transfer function which will then account for both nonlinear sources.

The transfer function of the ring can be written as [19

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000). [CrossRef]

,20

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron. 40(6), 726–730 (2004). [CrossRef]

];

P_{o p} (t) = P_{o u t, m a x} \cdot [1 - \frac{1}{1 + \frac{4 F^{2}}{π^{2}} s i n^{2} (\frac{2 π^{2} \cdot r \cdot n_{e f f} (t) \cdot f_{o}}{c})}]

(3)

In which F is the finesse of the ring, r is the radius of the ring and f_o is the optical frequency of operation. This equation models the steady-state property of the resonator and does not take cavity dynamics into account. But since the Q-factor and the modulation speed are low, time-dependent effects can be ignored. It should also be noted that free-carrier refraction in the pn junction is also not included in this model but again at this level of Q-factor and modulation speed, its contribution to the modulation nonlinearity is minimal.

The detected RF current is,

I_{R F} = R_{P D} \cdot P_{o p}

(4)

In which R_PD is the responsivity of the photodetector.

Hence using Eq. (2), Eq. (3) and Eq. (4), I_RF can be calculated as a function of time for any input voltage.

In order to calculate the SFDR based on both second, two tones are applied to the modulator and the output at corresponding frequencies are observed.

A voltage in the form of

V_{i n} (t) = V_{o 1} \cdot s i n (2 π f_{1} t) + V_{o 2} \cdot s i n (2 π f_{2} t)

(5)

is then substituted in the transfer function equation and is Taylor expanded around the bias voltage of use. Taking the first and second derivative of the transfer function with respect to the two frequency components, the output power at different harmonics can be calculated.

With the optical power of 20mW at the detector, biased at the quadrature point (25% of the maximum), a quality factor of around 5000 and with FSR of 3.2m, according to this calculation SFDR_SHD is predicted to be about 75dB.Hz^1/2. As we’ll see later in the result section, the results of this model successfully follows the trends that we observed in measurements. There is about 10dB difference in terms of the absolute values between what the model predicts and the measured results that is the uncertainty in the .

The more important impact of this modeling is its capability in predicting the dominant source of nonlinearity. As was previously mentioned, the major sources of nonlinearity responsible for the overall SFDR are the nonlinearity in the Lorentizian transfer function and the pn junction nonlinearity. Current modeling takes care of both of these sources. If we intentionally take out the nonlinearities of the pn junction from the model, we end up getting 109dB.Hz^1/2 as compared to 75dB.Hz^1/2 when both effects are considered. This is quite interesting as it suggests that it is indeed the pn junction nonlinearity that is the limiting source.

This finding can be incorporated in the future designs in order to further improve the SFDR by carefully engineered junctions that would be more linear.

5. Results

The ring FSR is around 3.2 nm, and a typical Q was 5,000. It has a tunability of about 10.6 pm/V and its RF bandwidth is measured to be 18.8 GHz at 0 V dc bias. The analog signal used in the nonlinearity measurements is at around 1 GHz. Since the Q-factor and modulation speed are low, free-carrier refraction in the pn junction becomes the major source of modulation nonlinearity.

With typical link parameters, CW laser power of 20mW detected and the overall detected noise level of −168 dBm, load impedance of 50 Ω, and detector responsivity of 0.6 A/W, the output at the main carrier frequency and second harmonic frequency has been measured. The bias point is optimized to get the maximum SFDR. Since the nonlinearity is dictated by the nonlinear junction, the distortion from the transfer function can be ignored and hence the optimum bias point would be where the slope of the transfer function curve is maximum. This condition occurs at the quadrature point on the curve.

The measured curve that shows both the carrier and SHD for a −24dBm input RF power is shown in Fig. 6 where the noise floor is at −129 dBm. At the 10kHz resolution bandwidth of our measurement, this would correspond to −169 dBm/Hz for the noise power per unit bandwidth. This noise floor is used in calculating SFDR.

Fig. 6 Detected RF power of the carrier frequency (1030 MHz) and SHD (at 2060 MHz) at 10 kHz resolution bandwidth.

Having two tones at 1030MHz and 1050MHz applied to the input of the system, the power of the carriers, second harmonic and intermodulation distortion are detected at several input powers which is illustrated in Fig. 7 .

Fig. 7 SFDR of the ring due to second harmonic and IMD distortion. Dots represent measured values.

The measurements in this figure would correspond to SFDR_SHD = 63 dB.Hz^1/2 and SFDR_IMD = 80 dB.Hz^2/3. There is however about 6 dB loss in the link after the device that we can safely take out. This loss is coming from the output grating coupler. Hence we need to normalize the output optical power to make up for it.

Taking this factor into account, we’d have to scale up the optical power and noise floor and hence the SFDR. Carrier signal, SHD and IMD will improve by 12 dB, the noise will grow by 6dB and therefore the SFDR_SHD = 64.5 dB.Hz^1/2 and SFDR_IMD = 84 dB.Hz^2/3.

6. Conclusion

The silicon microring modulator is proposed and analyzed for analog RF electro-optic modulation. The nonlinear refractive index and Lorentzian-shaped transfer function, play a key role in determining the nonlinearity of the modulated signal. We show 84 dB.Hz^2/3 SFDR on these modulators. We find that the most important source of nonlinearity is from the pn junction itself, as opposed to the nonlinear wavelength response of the ring modulator.

Acknowledgments

The authors would like to thank Gernot Pomrenke, of the Air Force Office of Scientific Research, for his support under the OPSIS and PECASE programs, and would like to thank Mario Panniccia and Justin Rattner, of Intel, for their support of the Institute for Photonic Integration. The authors would also like to thank Mentor Graphics for their support of the OPSIS project.

References and links

1.	H. L. R. Lira, S. Manipatruni, and M. Lipson, “Broadband hitless silicon electro-optic switch for on-chip optical networks,” Opt. Express 17(25), 22271–22280 (2009). [CrossRef] [PubMed]
2.	P. Rabiei and W. H. Steier, “Tunable polymer double micro-ring filters,” Photon. Technol. Lett. 15(9), 1255–1257 (2003). [CrossRef]
3.	Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]
4.	F. Y. Gardes, A. Brimont, P. Sanchis, G. Rasigade, D. Marris-Morini, L. O’Faolain, F. Dong, J. M. Fedeli, P. Dumon, L. Vivien, T. F. Krauss, G. T. Reed, and J. Martí, “High-speed modulation of a compact silicon ring resonator based on a reverse-biased pn diode,” Opt. Express 17(24), 21986–21991 (2009). [CrossRef] [PubMed]
5.	M. Gould, T. Baehr-Jones, R. Ding, S. Huang, J. Luo, A. K.-Y. Jen, J.-M. Fedeli, M. Fournier, and M. Hochberg, “Silicon-polymer hybrid slot waveguide ring-resonator modulator,” Opt. Express 19(5), 3952–3961 (2011). [CrossRef] [PubMed]
6.	R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microw. Theory Tech. 54(2), 832–846 (2006). [CrossRef]
7.	C. H. Cox III, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006). [CrossRef]
8.	I. Gasulla and J. Capmany, “Analysis of the harmonic and intermodulation distortion in a multimode fiber optic link,” Opt. Express 15(15), 9366–9371 (2007). [CrossRef] [PubMed]
9.	J. Yao, “Microwave photonics,” J. Lightwave Technol. 27(3), 314–335 (2009). [CrossRef]
10.	W. B. Bridges and J. H. Schaffner, “Distortion in linearized electrooptic modulators,” IEEE Trans. Microw. Theory Tech. 43(9), 2184–2197 (1995). [CrossRef]
11.	S. Dubovitsky, W. H. Steier, S. Yegnanarayanan, and B. Jalali, “Analysis and improvement of Mach-Zehnder modulator linearity performance for chirped and tunable optical carriers,” J. Lightwave Technol. 20(5), 886–891 (2002). [CrossRef]
12.	H. Tazawa and W. H. Steier, “Linearity of ring resonator-based electrooptic polymer modulator,” Electron. Lett. 41(23), 1297–1298 (2005). [CrossRef]
13.	T. Ismail, C.-P. Liu, J. E. Mitchell, and A. J. Seeds, “High-dynamic- range wireless-over-fiber link using feed forward linearization,” J. Lightwave Technol. 25(11), 3274–3282 (2007). [CrossRef]
14.	M. Song, L. Zhang, R. G. Beausoleil, and A. E. Willner, “Nonlinear distortion in a silicon microring-based electro-optic modulator for analog optical links,” IEEE J. Sel. Top. Quantum Electron. 16, 185–191 (2010).
15.	B. Dingel, A. Prescod, N. Madamopoulos, and R. Madabhushi, “Performance of ring resonator-based linear optical modulator (IMPACC) under Critical Coupling (CC), Over Coupling (OC), and Under Coupling (UC) conditions,” IEEE Photonics Conference (PHO) 260–261 (2011).
16.	C. Kittel, Introduction to Solid State Physics, 5th ed.(Wiley), (1976).
17.	http://www.ime.a-star.edu.sg/PPSSite/index.asp
18.	R. A. Soref and B. R. Bennet, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]
19.	A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000). [CrossRef]
20.	J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron. 40(6), 726–730 (2004). [CrossRef]

OCIS Codes
(040.6040) Detectors : Silicon
(130.0130) Integrated optics : Integrated optics
(130.2790) Integrated optics : Guided waves
(230.2090) Optical devices : Electro-optical devices
(250.7360) Optoelectronics : Waveguide modulators
(130.4110) Integrated optics : Modulators

ToC Category:
Integrated Optics

History
Original Manuscript: March 9, 2012
Revised Manuscript: May 21, 2012
Manuscript Accepted: May 21, 2012
Published: May 25, 2012

Citation
Ali Ayazi, Tom Baehr-Jones, Yang Liu, Andy Eu-Jin Lim, and Michael Hochberg, "Linearity of silicon ring modulators for analog optical links," Opt. Express 20, 13115-13122 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13115

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References

H. L. R. Lira, S. Manipatruni, and M. Lipson, “Broadband hitless silicon electro-optic switch for on-chip optical networks,” Opt. Express17(25), 22271–22280 (2009). [CrossRef] [PubMed]
P. Rabiei and W. H. Steier, “Tunable polymer double micro-ring filters,” Photon. Technol. Lett.15(9), 1255–1257 (2003). [CrossRef]
Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature435(7040), 325–327 (2005). [CrossRef] [PubMed]
F. Y. Gardes, A. Brimont, P. Sanchis, G. Rasigade, D. Marris-Morini, L. O’Faolain, F. Dong, J. M. Fedeli, P. Dumon, L. Vivien, T. F. Krauss, G. T. Reed, and J. Martí, “High-speed modulation of a compact silicon ring resonator based on a reverse-biased pn diode,” Opt. Express17(24), 21986–21991 (2009). [CrossRef] [PubMed]
M. Gould, T. Baehr-Jones, R. Ding, S. Huang, J. Luo, A. K.-Y. Jen, J.-M. Fedeli, M. Fournier, and M. Hochberg, “Silicon-polymer hybrid slot waveguide ring-resonator modulator,” Opt. Express19(5), 3952–3961 (2011). [CrossRef] [PubMed]
R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microw. Theory Tech.54(2), 832–846 (2006). [CrossRef]
C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech.54(2), 906–920 (2006). [CrossRef]
I. Gasulla and J. Capmany, “Analysis of the harmonic and intermodulation distortion in a multimode fiber optic link,” Opt. Express15(15), 9366–9371 (2007). [CrossRef] [PubMed]
J. Yao, “Microwave photonics,” J. Lightwave Technol.27(3), 314–335 (2009). [CrossRef]
W. B. Bridges and J. H. Schaffner, “Distortion in linearized electrooptic modulators,” IEEE Trans. Microw. Theory Tech.43(9), 2184–2197 (1995). [CrossRef]
S. Dubovitsky, W. H. Steier, S. Yegnanarayanan, and B. Jalali, “Analysis and improvement of Mach-Zehnder modulator linearity performance for chirped and tunable optical carriers,” J. Lightwave Technol.20(5), 886–891 (2002). [CrossRef]
H. Tazawa and W. H. Steier, “Linearity of ring resonator-based electrooptic polymer modulator,” Electron. Lett.41(23), 1297–1298 (2005). [CrossRef]
T. Ismail, C.-P. Liu, J. E. Mitchell, and A. J. Seeds, “High-dynamic- range wireless-over-fiber link using feed forward linearization,” J. Lightwave Technol.25(11), 3274–3282 (2007). [CrossRef]
M. Song, L. Zhang, R. G. Beausoleil, and A. E. Willner, “Nonlinear distortion in a silicon microring-based electro-optic modulator for analog optical links,” IEEE J. Sel. Top. Quantum Electron.16, 185–191 (2010).
B. Dingel, A. Prescod, N. Madamopoulos, and R. Madabhushi, “Performance of ring resonator-based linear optical modulator (IMPACC) under Critical Coupling (CC), Over Coupling (OC), and Under Coupling (UC) conditions,” IEEE Photonics Conference (PHO) 260–261 (2011).
C. Kittel, Introduction to Solid State Physics, 5th ed.(Wiley), (1976).
http://www.ime.a-star.edu.sg/PPSSite/index.asp
R. A. Soref and B. R. Bennet, “Electrooptical effects in silicon,” IEEE J. Quantum Electron.23(1), 123–129 (1987). [CrossRef]
A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett.36(4), 321–322 (2000). [CrossRef]
J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron.40(6), 726–730 (2004). [CrossRef]

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