1. Introduction
Silicon microring resonators are promising building blocks for highly compact optoelectronic integrated circuits (OEICs) that are compatible with mature silicon microelectronics [
11. R. Soref, “The past, present, and future of silicon photonics,” IEEE J. of Sel. Top. Quantum Electron. 12, 1678–1687 (2006). [CrossRef]

33. B. Jalali, “Silicon photonics,” J. Lightwave Technol. 24, 4600–4615 (2006). [CrossRef]
]. Silicon microring OEICs are potentially relevant to a wide scope of applications including telecommunications, computer interconnects, and biochemical sensing. While conventional microringbased notch and adddrop filters comprise simple waveguide coupling to the microring resonator, in which resonance dips appear in the throughputport and resonance peaks appear in the dropport, it is conceivable that resonance control and thereby advanced wavelengthagile functionalities can be attained by means of external feedback. Several research groups [
44. G. Lenz and C. K. Madsen, “General optical allpass filter structures for dispersion control in WDM systems,” J. Lightwave Technol. 17, 1248–1254 (1999). [CrossRef]

99. C. Li, L. Zhou, and A. W. Poon, “Silicon microring carrierinjectionbased modulators/switches with tunable extinction ratios and ORlogic switching by using waveguide crosscoupling,” Opt. Express 15, 5069–5076 (2007). [CrossRef] [PubMed]
] have proposed and investigated such interferometric approach to resonance control by means of waveguide crosscoupling to a microresonator, in which a crosscoupled waveguide provides an external feedback to the microresonator. Various resonance characteristics and functionalities for such feedbackcoupled microresonators have been studied including transmission dispersion [
44. G. Lenz and C. K. Madsen, “General optical allpass filter structures for dispersion control in WDM systems,” J. Lightwave Technol. 17, 1248–1254 (1999). [CrossRef]
], interferometers with slowlight [
55. G. T. Paloczi, Y. Huang, A. Yariv, and S. Mookherjea, “Polymeric MachZehnder interferometer using serially coupled microring resonators,” Opt. Express 11, 2666–2671 (2003). [CrossRef] [PubMed]
], resonance extinction ratio (ER) control [
66. W. Green, R. Lee, G. DeRose, A. Scherer, and A. Yariv, “Hybrid InGaAsPInP MachZehnder Racetrack Resonator for Thermooptic Switching and Coupling Control,” Opt. Express 13, 1651–1659 (2005). [CrossRef] [PubMed]
], clockwise and counterclockwisetravelingwave modes mutual coupling [
77. S. Mookherjea, “Mode cycling in microring optical resonators,” Opt. Lett. 30, 2751–2753 (2005). [CrossRef] [PubMed]
], resonance freespectral range (FSR) expansion [
88. M. R. Watts, T. Barwicz, M. Popovic, P. T. Rakich, L. Socci, E. P. Ippen, H. I. Smith, and F. Kaertner, “Microringresonator filter with doubled freespectralrange by twopoint coupling,” in proceedings of Conference on Lasers and ElectroOptics (Optical Society of America, Washington, DC, 2005), CMP3.
], and our previous work on electrooptic logicswitching [
99. C. Li, L. Zhou, and A. W. Poon, “Silicon microring carrierinjectionbased modulators/switches with tunable extinction ratios and ORlogic switching by using waveguide crosscoupling,” Opt. Express 15, 5069–5076 (2007). [CrossRef] [PubMed]
]. However, to the best of our knowledge, the resonance tuning characteristics by using freecarrier injection and their dependence on the feedbackwaveguidemicroring coupling have yet been investigated in detail.
Here, we report our experimental and theoretical study on the resonance characteristics of electrically reconfigurable silicon microring resonatorbased filters with waveguidecoupled feedback. We tune the feedback phase and amplitude through the freecarrier plasma dispersion and absorption effect [
1010. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon”, IEEE J. Quantum Electron 23, 123–129 (1987). [CrossRef]
] by forward biasing a laterally embedded pin diode. We show that the transmission resonances exhibit various attributes depending on the attenuated feedback coupling. We model our devices using scatteringmatrixbased approach and find good agreement with the measurements.
3. Model
where
E_{i} and
E
_{o} are the input and outputcoupled electricfields in the buswaveguide,
E
_{1} and
E
_{2} are the electricfields inside the microring just prior to the inputcoupler and just after the outputcoupler,
κ and
τ (real numbers) are the coupling and transmission coefficients of the symmetric input/outputcouplers (for lossless coupling,
τ
^{2}+
κ
^{2}=1),
φ is the transmission phase change of the couplers (for circular microring,
φ=0),
a is the halfcircular ring amplitude transmission factor,
θ is the halfcircular ring phase change,
b is the feedbackwaveguide amplitude transmission factor,
γ is the amplitude transmission factor under freecarrier absorption (
γ=1 means no carrier injection), and
ϕ=
ϕ
_{0}+
δϕ is the feedbackwaveguide phase change including a passive pathlength phase change
ϕ
_{0} and a carrierinjection induced phase shift
δϕ. We note that
ϕ
_{0} and
θ are related with the feedbackwaveguide length
L_{b} and the halfcircular ring length
L_{a} as
ϕ
_{0}=
n_{eff}L_{b}(2
π/
λ) and
θ=
n_{eff}L_{a}(2
π/
λ), where
n_{eff} is the waveguide effective refractive index and
λ is the freespace wavelength. The two tuning parameters,
δϕ and
γ, are related by freecarrier plasma dispersion effect [
1010. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon”, IEEE J. Quantum Electron 23, 123–129 (1987). [CrossRef]
].
Figure 2(a) depicts the modeling parameters.
The electricfields E
_{1} and E
_{2} are related as follows,
We define the resonance phasematching condition between the feedbackcoupled roundtrip field A and the inputcoupled field at a point just prior to the inputcoupler in terms of
where Φ is the feedbackcoupled roundtrip phase change. Resonances occur at Φ=2mπ (m=1, 2, 3…).
Fig. 2. (a). Modeling schematic. (b), (c) Physical interpretations of the terms in
Eqs. (3) and
(4). The corresponding terms of the electricfield transmissions through various paths are labelled. In (b), M: startingpoint in the feedbackwaveguide just prior to the inputcoupler. N: endingpoint in the feedbackwaveguide just after the outputcoupler. In (c), P: starting and endingpoints in the microring just prior to the inputcoupler.
3.1 Two limiting cases
The microring and the feedbackloop ring each imposes a different roundtrip phase change (i.e. 2
θ+2
φ and
θ+2
φ+
ϕ+
π).
Equation (4) suggests that in case
τ
^{2}
a
^{2}≫
κ
^{2}
abγ, the resonances can be approximated by the microring resonances given by Φ≈2
θ+2
φ=2
mπ. Whereas in case
τ
^{2}
a
^{2}≪
κ
^{2}
abγ, the resonances can be approximated by the feedbackloop ring resonances given by Φ≈
θ+2
φ+
ϕ+
π=2
mπ. In general the resonances are given by the Φ phasematching (
Eq. (4)). Here, we further analyze the two limiting cases.
3.1.1 Microring approximation
For the ease of analysis, we assume that the transmission factors a and b are of the same order of magnitude, given both are largely determined by the waveguide propagation and bending losses. We then examine the microring approximation under two scenarios: (i) weak coupling τ
^{2} >> κ
^{2}, and (ii) excess carrier absorption loss γ ≈ 0.
For weak coupling (
τ
^{2}≫
κ
^{2}) and under low injections (
γ≈1), we simplify
Eq. (3) as
For excess carrier absorption loss (
γ≈0), the feedback is largely attenuated and thereby the transmission is like the “drop” function as in a conventional adddrop filter. We simplify
Eq. (3) as
which suggests a Lorentzian resonance line shape.
3.1.2 Feedbackloop ring approximation
Here, the roundtrip phase change Φ becomes an approximately linear function of the feedback phase change
ϕ, suggesting that the resonance wavelength approximately linearly depends on
ϕ. The feedbackloop ring approximation is satisfied under strong coupling
τ
^{2}≪
κ
^{2} and low carrier absorption loss
γ≈1. We simplify
Eq. (3) as
3.2 Phasematched feedback
Here, we show that each resonance exhibits the same transmission intensity in case the relative feedback phase change Δϕ≡ϕ  θ satisfies
where λ_{m} is the mth resonance wavelength, n_{m} is an integer, and c is a constant phase angle.
From
Eqs. (4) and
(8), we express field
A upon phasematched feedback as
The term in the square bracket is a constant complex number (independent of wavelength), and thus we rewrite it as τ2a2−κ2abγe−ic=∣A∣eiΦc. The resonance condition becomes
The righthand side of
Eq. (11) is independent of resonance mode number
m. Hence,
Eq. (11) suggests that the transmission is identical at each resonance wavelength upon satisfying the phasematched feedback condition.
In order to determine the proper L_{b} and L_{a} for phasematched feedback, we express Δϕ as
We note that
δϕ≪
n_{eff} 2
π(
L_{b} −
L_{a})/
λ. It is thus possible to choose a particular (
L_{b} −
L_{a}) such that
Eq. (8) is approximately satisfied for consecutive resonance wavelengths over a narrow spectral range.
We can write the resonance condition in general form as,
where L_{res} is the effective roundtrip length for resonance phase matching.
From
Eqs. (12) and
(13) and considering Δ
ϕ(
λ_{m}
_{+1}) −
Δϕ(
λ_{m})=2
nπ, where
n is an integer, we obtain the phasematched feedback condition as
For devices close to the microring approximation,
L_{res}≈2(
L_{a}+
L_{c}), we can choose
L_{b} values to satisfy
Eq. (14) as follows
In case
n=1, we have
L_{b}+
L_{a}+2
L_{c}≈2(2
L_{a}+2
L_{c}), which means that the feedbackloop ring roundtrip length needs to be approximately twice the microring roundtrip length for nearly phasematched feedback. However, for devices close to the feedbackloop ring approximation,
L_{res}≈
L_{b}+
L_{a}+2
L_{c}>
L_{b}−
L_{a}, there are no
L_{b}’s to satisfy
Eq. (14).
Fig. 3. (a). Illustration of the resonancedependent line shapes in general cases. Resonances at wavelengths λ_{m}
_{+1} and λ_{m} see different feedback phase values Δϕ_{m}
_{+1} and Δϕ_{m}. (b) and (c) Modeled feedback phase Δϕ (λ) and the corresponding transmission spectra with (b) Δϕ(λ_{m}
_{+1})  Δϕ(λ_{m})≈1.4π, and (c) Δϕ(λ_{m}
_{+1})  Δϕ(λ_{m})≈2π.
4. Device design and fabrication
We design three different devices which address different waveguidemicroring coupling regimes and different feedback path lengths. Device (I) comprises a circular microring of radius of 25 µm, L_{b}=180 µm, and an integrated pin diode spanning 130 µm. Device (II) comprises a circular microring of radius of 25 µm, L_{b}=230 µm, and an integrated pin diode spanning 180 µm. Device (III) comprises a racetrack microring with a curved waveguide radius of 25 µm and a straight interaction length L_{c}=10 µm, L_{b}=230 µm, and an integrated pin diode spanning 130 µm. All the devices have the same design parameters except those specified above. For devices (I) and (II), we determine the power coupling ratio κ
^{2} is ~10% between the straight bus waveguide and the curved ring waveguide (according to numerical beampropagation method (BPM) simulations). This suggests that the circular microring is weakly coupled with the feedbackwaveguide. For device (III), we determine κ
^{2} is ~50% between the straight bus waveguide and the straight waveguide section of the racetrack. This suggests that the racetrack microring is strongly coupled with the feedbackwaveguide.
Fig. 4. (a).4(c). Optical micrographs of the three fabricated devices (I), (II), and (III). (d) and (e) Zoomin view SEMs of (d) the waveguidecircularmicroring coupling region, and (e) the waveguideracetrackmicroring coupling region without oxide uppercladding. (f) Crosssectional view SEM of the singlemode waveguide without oxide uppercladding.
5. Results
We first examine device (I) transmission characteristics under low injection levels (
V_{d}≤1 V), where the freecarrier absorption loss is small.
Figures 5(a)
5(c) show the measured and modeled TEpolarized (electric field parallel to the chip) singlemode transmission spectra of device (I) under forward biases of
V_{d}=0 V, 0.9 V, and 1.0 V. The modeled transmission spectra (using
Eq. (3)) assume
a=0.96 and
b=0.95. We choose
κ=0.34 (satisfying
κ
^{2}≪
τ
^{2}). We determine the wavelengthdependent feedback phase
ϕ
_{0} by BPM simulation (thus incorporating the waveguide dispersion). We then choose
δϕ and
γ values (labeled in the Figs.) to bestfit the measured transmission spectra. In particular, we choose
δϕ=0 and
γ=1 for
V_{d}=0 V, and
γ remains ≈1 for the small
V_{d}’s. We also assume
δϕ and
γ are independent to wavelength within the spectral range of interest. The modeled spectra show good agreement with the experimental results.
We notate three resonances as resonances A, B, and C. The FSR is ~3.8 nm, which is consistent with the microring resonator roundtrip path length. As the microring resonator is weakly coupled with the feedbackwaveguide, the device is close to the microring approximation and the resonances are mainly given by the microring resonator.
We observe that the resonances ER and Q values display quasiperiodic oscillations over multiple FSRs. Resonance A exhibits the maximum Q of ~10^{4}. Moreover, while the resonance wavelengths are almost fixed within ~0.05nm shift under the three V_{d}’s, the ER values vary significantly (most pronounced for resonance C) yet with only slight variations in the Q values. Thus, we see that in the case of low injection levels with low carrier absorption loss, V_{d} (essentially δϕ) primarily controls the resonance ER. We attribute the background intensity fluctuations in the measured spectra to modulations due to the Ubend waveguide endface backreflections.
Fig. 5. (a)(c) Measured (solid grey lines) and modeled (dashed red lines) TEpolarized transmission spectra of device (I) upon low injection levels with bias voltages of V_{d}=(a) 0 V, (b) 0.9 V, and (c) 1.0 V. (d)(f) Measured (solid grey lines) and modeled (dashed red lines) TEpolarized transmission spectra of device (I) upon high injection levels with bias voltages of V_{d}=(a) 1.8 V, (b) 2.3 V, and (c) 2.9 V.
We also examine device (I) transmission characteristics upon high injection levels, in which the freecarrier absorption loss becomes significant.
Figures 5(d)
5(f) show the measured and modeled TEpolarized singlemode transmission spectra of device (I) under three high forward biases of
V_{d}=1.8 V, 2.3 V, and 2.9 V. In the modeling we again choose
δφ and
γ values (labeled in the Figs.) to bestfit the measured transmission spectra, while other modeling parameters remain the same as those in the low injection cases. Interestingly, we find asymmetric resonance line shapes [
1212. L. Zhou and A. W. Poon, “Fano resonancebased electrically reconfigurable adddrop filters in silicon microring resonatorcoupled MachZehnder interferometers,” Opt. Lett. 32, 781–783(2007). [CrossRef] [PubMed]
] (Fano resonances [
1313. U. Fano, “Effect of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961). [CrossRef]
]) quasiperiodically appear over a wide range of wavelength. Besides, the Fano line shapes evolve with Vd. Thus, we find that upon certain high injections to the feedbackwaveguide, the resulting phase shifts and attenuations can controllably tune the asymmetric resonance line shapes. We also observe that upon high Vd, the measured resonance spectra display a slight redshift relative to those at low
V_{d} (~0.6 nm redshift at
V_{d}=2.9 V), which we attribute to the thermooptic effect in silicon (estimated ~5°C temperature rise). The measured transmission spectra also display insertion loss increases with
V_{d}, which is consistent with an increased freecarrier absorption loss. We also note that the measured transmission intensity fluctuations with the high
V_{d} are less pronounced than those with the low
V_{d}, which again can be attributed to the enhanced freecarrier absorption loss in the Ubend waveguide.
Fig. 6. (a).(d). Measured (solid grey lines) and modeled (dashed red lines) TEpolarized transmission spectra of device (II) under various bias voltages of V_{d}=(a) 1.2 V, (b) 1.4 V, (c) 2.0 V, and (d) 3.0 V.
Figures 6(a)6(d) show the measured and modeled TEpolarized transmission spectra of device (II) under various bias voltages of
V_{d}=1.2 V, 1.4 V, 2.0 V, and 3.0 V. Except for the
δϕ and
γ values labeled in the Figs., the modeling parameters are the same as those in device (I). The resonances under the different
V_{d}’s exhibit four main characteristics: resonance dips with ER >20 dB [
Fig. 6(a)], modulated allpass transmission [
Fig. 6(b)], asymmetric Fano line shapes [
Fig. 6(c)], and resonance peaks (
Fig. 6(d)). Like device (I), device (II) displays an FSR of ~3.8 nm, which is consistent with the microring roundtrip length. This suggests that the resonances are also mainly given by the microring resonator. Yet, unlike device (I), device (II) displays nearly uniform resonance line shapes over multiple FSRs. This is because device (II) design parameters (see Sec. 4) satisfy the phasematched feedback condition (
Eq. (15) with
n=1).
Figures 7(a)
7(d) show the measured and modeled TEpolarized multimode transmission spectra of device (III) under various bias voltages of
V_{d}=0.7 V (subthreshold), 1.5 V, 2.0 V, and 2.9 V. We choose
κ=0.7 in the modeling while other parameters are the same as those in device (I). As
κ
^{2}≈
τ
^{2}, the microring approximation no longer holds. Instead, we discern two sets of resonances that can be attributed to the mutual coupling between the microring and the feedbackloop ring. We denote two resonance modes D and E [
Figs. 7(b)
7(d)], which display different ER values and line shapes under various bias voltages. We remark that although device (III) is designed to satisfy the phasematched feedback condition (
Eq. (15) with
n=1), the significant mutual coupling violates the microring approximation and thus forbids the resonance line shapes to remain uniform across multiple FSRs.
Fig. 7. Measured (solid grey lines) and modeled (dashed red lines) TEpolarized transmission spectra of device (III) under various bias voltages of V_{d}=(a) 0.7 V, (b) 1.5 V, (c) 2.0 V, and (d) 2.9 V.
Fig. 8. (a).(e). Modeled transmission spectra under various attenuated feedback conditions for a racetrack microring device with strong coupling (κ≈0.94).
Under low attenuations (see
Figs. 8(a)
8(c)), we find that the modeled resonances are mainly given by the feedbackloop ring. The modeled FSR of 1.54 nm is consistent with the feedbackloop ring path length of 394 µm, suggesting that the resonances are close to the feedbackloop ring approximation. We also discern resonance wavelength blueshifts by ~0.8 nm with
δϕ=
π. This wavelength tuning with δ
ϕ is again consistent with the feedbackloop ring approximation. In fact, among the four device configurations studied here, the feedbackloop ring approximation under strong coupling and low attenuation exhibits the most efficient wavelength tuning by phaseshifting the feedbackwaveguide. The modeling also shows that the resonance ER and line shape vary with
δϕ and
γ, yet the resonance ER and line shape are nonuniform across the spectrum as the feedbackloop ring approximation does not have the phasematched feedback condition.
Under high attenuations (
Figs. 8(d) and
8(e)), however, we see that the modeled resonances are essentially attenuated microring resonances. The modeled FSR is expanded to 3.1 nm, which is consistent with the microring roundtrip path length of 197 µm. This suggests that the strongly attenuated feedback, regardless of its strong coupling, yields attenuated microring resonances.