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We observe theoretically and experimentally electromagnetically induced transparency (EIT)-like effect in a single microdisk resonator (MDR) evanescently coupled with two bus waveguides. This structure is modeled using transfer matrix method, and it is revealed that the EIT-like spectrum originates from the coherent interference between two nearby low-order whispering-gallery modes (WGMs) with comparable quality factors. The EIT-like properties have been investigated analytically with respect to coupling efficiency, round-trip power attenuation, as well as phase spacing between two resonances. The resonance spacing and mode coupling are adjustable by varying the effective indices of WGMs and waveguide mode. Consequently, fully integrated MDRs were fabricated in silicon. Resonant modes and coupling efficiency are studied in one-bus waveguide coupled MDRs. Finally, EIT-like resonance is observed in a two-bus waveguides coupled MDR of 3 μm in radius with a quality factor of 4,200 and central transmission larger than 0.65. The experimental results agree with our modeling well and show good internal consistency, confirming that two WGMs coupled in a point-to-point manner are required for EIT-like effect.
© 2014 Optical Society of America
1. Introduction
Electromagnetically induced transparency (EIT) has attracted considerable attentions in the past decades, due to its wide applications in slowing/stopping light, nonlinear optics, and quantum information processing [1–3]. Similar to EIT effect caused by quantum interference in multi-level atomic systems, EIT-like spectrum, having a narrow transparency peak residing in a broader absorption valley, also can be generated by coherent interference between coupled resonant modes [4]. All-optical analogies to EIT have been demonstrated in various configurations comprising coupled two resonators, including microsphere [5,6], microtoroid [7], microring [8–15], self-coupled optical waveguide resonator [16,17], and photonic crystal microcavity [18,19].
Recently, without employment of additional optical resonator, EIT-like phenomenon has been demonstrated in a single microsphere/microtoroid coupled with a single fiber taper waveguide [20–22]. The EIT-like effect is induced by the indirect interaction between two whispering-gallery modes (WGMs), which are simultaneously triggered by a coupled fiber waveguide. In this case, it is necessary that the two WGM resonances are fully overlapped and differ by no less than two orders of magnitude in quality factor to obtain EIT-like spectral response. However, these structures are either non-planar or hardly integrated with bus waveguide, since they are fabricated by reflowing the resonator material.
In this paper, we have demonstrated EIT-like effect in a microdisk resonator (MDR) coupled with two bus waveguides. A theoretical model is given using transfer matrix method [23], and then applied to study the mechanism and analyze the influencing factors of this device. It is found that the EIT-like resonance in a planar MDR stems from the coherent interference between two low-order WGMs with comparable quality factors [23–25]. Unlike microsphere/microtoroid, it is not easy for an on-chip MDR to achieve two WGM resonances of tremendously different quality factor. Using nanofabrication processing, we realized silicon based 3-μm radius MDRs with one bus and two buses coupled. One-bus waveguide coupled MDRs are characterized to study the properties of resonant modes and coupling efficiency. Finally, EIT-like resonance is observed in a two-bus waveguides coupled MDR with a quality factor of 4,200 and central transmission larger than 0.65. The experimental results fit our modeling well and show good internal consistency.
2. Model and mechanism
Figure 1(a) shows the schematic structure of a two-bus waveguides coupled MDR, or a one-bus waveguide coupled MDR as one bus is removed. It is supposed that the first-order radial WGM (WGM1) and second-order radial WGM (WGM2) are excited simultaneously and indirectly coupled through the 3 × 3 couplers.
Fig. 1 (a) Schematic of a two-mode MDR coupled with two bus waveguides. (b) A SEM image of the fabricated device.
Using transfer matrix method, the input-output relations for a 3 × 3 coupler can be expressed by
where a0,1,2 and b0,1,2 are the optical fields (the subscripts 0, 1, and 2 represent the waveguide mode, WGM1, and WGM2, respectively) at the input and output ports; -jk1,2 and -kc are the field coupling coefficients between these modes; t0,1,2 is the field transmission coefficient.We use θ = 2π2R (neff1 + neff2)/λ to describe the phase shift of two-bus waveguides coupled MDR, where R, neff1,2 and λ denote the MDR radius, effective mode index of WGM1,2, and vacuum wavelength, respectively. The optical fields of WGMs are phase shifted and cross-coupled in the resonator, and the relations are described as for (2m-0.5) π<θ< (2m + 0.5) π, where m is an integer,
for (2m-1) π<θ≤ (2m-0.5) π or (2m + 0.5) π≤θ< (2m + 1) π,where α1,2 and φ1,2 are the round-trip field attenuation and phase shift for WGM1,2, respectively, satisfying φ1,2 = (4π2Rneff1,2)/λ.For the one-bus waveguide coupled MDR, Eqs. (2) and (3) can be simplified as
In both cases, the field transmission in the through channel is given by
We use C01 and C02 to denote the fields coupled out from WGM1 and WGM2, corresponding to the second and third terms on the right-hand side of Eq. (5), respectively.
To obtain general performance of this structure, we simulate the power and phase responses of St via solving the previous equations, assuming k12 = k22 = 0.15, α12 = 0.99, α22 = 0.98, and phase spacing Δψ = 0.08π (determined by the difference in effective round-trip optical path lengths of two WGMs). As shown in Fig. 2(a), the power transmission exhibits a narrow transparent window residing between two broader dips caused by WGM resonances, analogous to an EIT spectrum. According to Eq. (5), the transfer function essentially consists of three parts: waveguide transmission term (t0), WGM1 and WGM2 coupling-out terms (C01 and C02). Here we study the amplitude and phase properties of C01 and C02 to gain physical insight into the EIT-like phenomenon. Meanwhile, the optical responses (T1 and T2) for individual WGM1 and WGM2 resonances are also presented for comparison. When the two WGMs are twice-coupled through 3 × 3 couplers, the asymmetric resonant peaks of C01 and C02 get closer and a resonant dip for one WGM occurs on the side of the other WGM resonance, as seen in Fig. 2(a). It is attributable to the competition between two WGMs, meaning that the field of an off-resonant WGM is strongly suppressed by the other on-resonant WGM. In addition, the coupling-out fields (C01 and C02) have a ~π phase difference in the overlapping region of two resonances as seen in Fig. 2(b), which causes the two resonances to be in destructive interference. Hence, a narrow transparent window is generated in the middle of two resonant dips, since |C01| and |C02| are comparable and t0 approaches unity. The inset of Fig. 2(a) shows the field distribution simulated by finite-difference time-domain (FDTD) method in the MDR at the EIT-like resonant wavelength. It is observed that the input field mostly goes through without field tunneled to the drop channel and the structure thereby shows optical transparency. In the microdisk, the EIT-like resonant mode is a mixture of WGM1 and WGM2, and we can see the destructive interference between the fields coupled out from WGMs at the drop port, which reduces the dropping field severely.
Fig. 2 (a) Normalized amplitude/power and (b) phase transmissions of the physical quantities St, C01, C02, T1 and T2 for our proposed structure, inset: FDTD-simulated field distribution in the MDR corresponding to the EIT-like transparency peak.
3. Analysis of the EIT-like transmission
In this section, we provide a parametric analysis of the EIT-like transmission in two-bus waveguides coupled MDR with respect to coupling efficiency (k12 and k22), round-trip power attenuation (α12 and α22), and phase spacing between two WGM resonances (Δψ). Figures 3(a), 3(d) and 3(g) show the contour plots of power transmission (|St|2) as functions of phase detuning (Δθ/π) and coupling efficiency (k12 = k22) with a phase spacing Δψ/π = 0, 0.03, and 0.08, respectively, assuming α12 = 0.99 and α22 = 0.98. For clear observation, power transmissions with k12 = k22 = 0.05, 0.10, 0.15, 0.20, and 0.25 are illustrated in Figs. 3(b), 3(e) and 3(h) corresponding to the five dashed lines depicted in Figs. 3(a), 3(d) and 3(g), respectively. It is seen that the two WGM resonances combine and form a single valley as they are fully overlapped, i.e. Δψ/π = 0. When the two resonances are detuned by Δψ/π = 0.03, and 0.08, EIT-like phenomenon emerges, as shown in Figs. 3(e)-3(h). As the coupling efficiency increases, the central transmission of transparency window decreases evidently, while the central bandwidth (ΔθFWHM) becomes smaller slightly, indicating a higher quality factor. That is to say, the quality factor increases with the coupling efficiency at the cost of central transmission for the EIT-like resonance in this structure. It is noteworthy that the inbuilt EIT-like resonance evolves into two separated WGM resonances when either the coupling efficiency is too weak or the phase spacing is too large, as observed in Figs. 3(e) and 3(h). Moreover, the phase transmissions are simulated and presented in Figs. 3(c), 3(f) and 3(i), corresponding to the power transmissions in Figs. 3(b), 3(e) and 3(h), respectively. As seen, the phase response of an EIT-like resonance in the MDR is similar to that of a traditional EIT-like resonance. It is interesting that, as the two WGMs are fully overlapped, the phase response for k12 = k22 = 0.25 is quite different from that for weaker coupling efficiency, because of the increased coupling between two WGMs (kc2>0.02). If the two resonances are seriously different in linewidth, our structure possibly can provide an EIT-like resonance with other types of phase response, as predicted previously in the coupled microring resonators [15].
Fig. 3 (a) (d) (g) Contour plots of power transmission as functions of phase detuning and coupling efficiency with Δψ/π = 0, 0.03, 0.08, respectively, (b) (e) (h) power transmissions and (c) (f) (i) phase transmissions as a function of phase detuning corresponding to the five dashed lines in the left panel.
Figure 4(a) illustrates the influence of difference in k12 and k22 on the EIT-like transmission with a variety of k22, assuming a fixed k12 of 0.10 and Δψ/π = 0.03. We find that the right dip for WGM2 is broadened more rapidly as both dips become wider simultaneously, and the EIT-like resonance is shifted leftward as k22 increases. As seen, the line shape of EIT-like resonance achieves almost symmetrical when k12 = k22. The influence of α12 and α22 in the MDR is also examined under k12 = k22 = 0.10, and Δψ/π = 0.03, as shown in Fig. 4(b). It is observed that the central transmission increases distinctly and the central peak becomes sharper as α12 and α22 increase, representing a higher quality factor, while the bandwidths of resonant dips are almost unchanged. That is because the optical transparency and intrinsic quality factors of WGMs grow higher as the light propagates in a lower-loss MDR.
Fig. 4 Power transmissions as a function of phase detuning under various coupling efficiencies (a) and round-trip power attenuations (b) of WGMs, insets: detailed spectra of EIT-like resonance.
We use central bandwidth and transmission to describe the performance of EIT-like effect. Figure 5 presents the central bandwidth and transmission as a function of phase spacing under two sets of coupling efficiency: k12 = k22 = 0.08 (solid lines) and k12 = k22 = 0.15 (dot dash lines). The bandwidths of an individual WGM1 resonance with k12 = 0.08 (red solid line) and 0.15 (red dot dash line) are also calculated and shown in Fig. 5. It is seen that the bandwidth and transmission of EIT-like resonance both increase and start from zero with the increasing of phase spacing, and higher coupling efficiency brings a larger central bandwidth, as well as lower central transmission. In the calculation domain, the EIT-like resonance exhibits a smaller bandwidth than individual WGM1 for coupling efficiency of 0.15, while it begins to have a larger bandwidth when Δψ/π>0.13 for coupling efficiency of 0.08. Exactly, in the case of Δψ/π>0.13 and k12 = k22 = 0.08, the EIT-like resonance no longer exists and it evolves into two separated WGM resonances.
Fig. 5 The central bandwidth and transmission of EIT-like resonance under k12 = k22 = 0.08 (solid lines) and 0.15 (dot dash lines), and the resonant bandwidth of WGM1 under k12 = 0.08 (red solid line) and 0.15 (red dot dash line), assuming α12 = 0.99 and α22 = 0.98.
4. Resonance spacing and mode coupling
As analyzed previously, EIT-like phenomenon is sensitive to the resonance spacing between two WGMs and the coupling efficiency between WGM and waveguide mode. Here, we use resonance spacing in wavelength instead of phase spacing, while they are intrinsically the same. The resonance position and spacing are mostly determined by the effective index of WGM, and slightly influenced by the coupling induced phase shift. The simulation of WGM effective index is performed for silicon-on-insulator (SOI) based MDRs with a 340-nm-thick top silicon layer, 3-μm MDR radius, and refractive indices of Si and SiO2 referred from [26]. The effective indices of transverse magnetic (TM) polarized WGM1 and WGM2 for slab thicknesses of 40 nm and 80 nm are calculated respectively using finite mode matching method, as shown in Fig. 6. It is seen that the mode effective index decreases with the wavelength, while it increases with the slab height. According to the relation 2πRneff = mλ0, where m is the azimuthal order of WGM, the resonant wavelengths (λ0) are obtained for WGM1 and WGM2, as denoted by the black dots in Fig. 6. Obviously, the resonance spacing between WGM1 and WGM2 varies with the slab height and wavelength. It is expected that the resonance spacing is also affected by the thickness of top silicon layer and MDR radius, and even can be tuned by local heating [21] or carrier injection [27] in the MDR. With respect to the coupling, mode matching is very essential for enhancing the coupling between two modes. Hence, with the aim of higher k12 and k22, the effective index of waveguide mode should fall in between the effective indices of WGM1 and WGM2. The inset of Fig. 6 shows that the effective index of TM0 mode in the waveguide increases monotonously with the waveguide width, indicating that optimal waveguide width is achievable to approach mode matching between WGM and waveguide mode. The gap between MDR and waveguide, as another important factor, also has an impact on the coupling efficiency between WGM and waveguide mode, and the coupling efficiency becomes lower as the gap enlarged.
Fig. 6 Effective indices of WGM1 (red lines) and WGM2 (blue lines) as a function of wavelength, insets: effective index of TM0 mode (black lines) in the waveguide as a function of width at the wavelength of 1550 nm, and the Ey field profiles of WGM1, WGM2, and TM0 mode. The solid and dashed lines are for slab thicknesses of 40 nm and 80 nm, respectively.
5. Experiment and discussion
MDRs with one bus and two buses coupled were fabricated on a SOI wafer with a 340-nm-thick top silicon layer and a 2-μm-thick buried oxide layer. The fine pattern was defined by electron beam lithography, followed by inductively coupled plasma etching with a depth of 300 nm. The radius of MDR is 3 μm, and the waveguide width is 290 nm. On the basis of measured structure dimensions, the effective indices of TM-polarized WGM1, WGM2 and TM0 mode in waveguide are evaluated to be about 2.35, 2.00, and 2.18 at 1550-nm wavelength, respectively, indicating comparable k12 and k22 due to mode matching.
For speculating the power attenuation and coupling efficiency, transmissions of one-bus waveguide coupled MDRs are measured and analyzed for different gaps. Figure 7(a) and 7(b) show the measured power transmissions for gaps of 180 nm and 270 nm, respectively. It is seen that two low-order WGMs are excited by the bus waveguide and exhibit comparable resonance linewidths. The influence of roughness-induced backscattering is neglectable here, since the resonance splitting is hardly observed in the spectra, except the resonance of WGM1 near 1488 nm for the gap of 180 nm. Using the proposed model, the fitting curves are given and agree well with the experimental results, as seen in the insets of Fig. 7(a) and 7(b). The fitting parameters are obtained as follows: for a gap of 180 nm, we have k12 = 0.031, k22 = 0.048, α12 = 0.990 and α22 = 0.981; and for a gap of 270 nm, we have k12 = 0.004, k22 = 0.011, α12 = 0.989 and α22 = 0.981. As the quality factors of two WGMs are on the same order of magnitude, the one-bus waveguide coupled MDRs do not behavior as reported in literatures [20,21].
Fig. 7 (a) The measured power transmission for a gap of 180 nm, inset: experimental results (blue circles) and theoretical fitting (red line), and a SEM image. (b) The measured power transmission for a gap of 270 nm, inset: experimental results (blue circles) and theoretical fitting (red line).
As shown in Fig. 1(b), a two-bus waveguides coupled MDR with a gap of 180 nm was fabricated and characterized. It is observed that a narrow EIT-like transparency window appears between two broader dips around the wavelength of 1546 nm, with central transmission larger than 0.65, and a central bandwidth of 0.37 nm, corresponding to a quality factor of 4,200. The theoretical fitting in Fig. 8 exhibits good agreement with the experiment, when we set k12 = 0.031, k22 = 0.048, α12 = 0.990, α22 = 0.981, neff1 = 2.37917 and neff2 = 2.04998. It shows a good internal consistency with the result of one-bus waveguide coupled MDR with the same dimensions. Note that a blueshift of the resonant spectrum occurs as compared with the spectrum in the inset of Fig. 7 (a), due to the additional phase shift in the 3 × 3 coupler at the drop channel [28]. The experimental results confirm that two-bus waveguides coupled MDR with two WGMs coupled in a point-to-point manner is required for EIT-like effect. Making use of the two modes in a MDR, this structure offers us another way to achieve EIT-like effect on a chip, and it is more compact than conventional coupled double resonators, as the resonator number is decreased by a half.
Fig. 8 The measured power transmission (blue circles) and theoretical fitting (red line) of the fabricated two-bus waveguides coupled MDR.
6. Conclusion
In summary, EIT-like effect has been demonstrated theoretically and experimentally in a fully integrated MDR coupled with two buses. The structure is modeled and EIT-like spectral response is found to originate from the destructive interference between two nearby resonances of low-order WGMs with comparable quality factors. The influencing factors of EIT-like effect have been studied, including coupling efficiency, round-trip power attenuation, and phase spacing. EIT-like resonance is experimentally observed in an ultra-compact and fully integrated MDR of 3 μm in radius on a SOI platform with a quality factor of 4,200 and central transmission larger than 0.65. The experimental result agrees with our modeling well. It is approved that two buses coupled are required for a two-mode MDR to obtain EIT-like effect. Due to the compactness and integratability, the proposed device is promising for applications in on-chip time delay lines and nonlinear signal processing.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grants 61006045 and 61177049, by the Major State Research Program of China under Grant 2013CB933303, and by the Major State Basic Research Development Program of China under Grants 2013CB632104 and 2010CB923204.
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