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We study coupled spiral-shaped microdisk resonators with non-evanescent asymmetric inter-cavity coupling via seamlessly jointed notches. Our finite-difference time-domain numerical simulations reveal that the throughput-port transmissions are reciprocal between counterclockwise (CCW) and clockwise (CW) traveling-wave modes, while the drop-port transmissions and modal field distributions are input-port dependent. By introducing a slight mismatch in radii between two coupled microdisks while preserving their seamlessly jointed notches, we are able to show selectively enhanced extinction ratio for one of the split modes while suppressing the other. Our experiments using coupled spiral-shaped microdisk resonators in silicon nitride-on-silica suggest split resonances with an extinction ratio of ~20 dB using identical coupled microdisks, and an enhanced resonance extinction ratio of ~24 dB using slightly mismatched coupled microdisks. The non-evanescent coupling preserves high-Q resonances.
©2007 Optical Society of America
1. Introduction
Coupled microresonators, e.g., coupled microdisk or microring resonators, are widely used to realize high-order filter response [1], optical delay line [2], and coupled resonator induced transparency [3]. The microresonators are evanescently coupled via submicrometer gap separation. Recently [4], our group also studied the feasibility of electro-optic logic switching in coupled dual-microring resonator configuration. However, due to the high structural symmetry, the transmissions are independent on the waveguide input-port, implying limited device configurations and switching functionality.
Most recently [5], we proposed a new configuration of coupled microdisk resonators with high structural asymmetry based on spiral-shaped microdisk resonators [6–18]. Spiral-shaped microresonators are totally rotational asymmetric microresonators that enable non-evanescent coupling [17–23], and asymmetric modal distributions [23] distinct from conventional whispering-gallery microresonators. Our coupled cavity structure comprises two spiral-shaped microresonators that are seamlessly jointed at their notches without interfaces, thus enabling direct non-evanescent coupling between two cavities. Due to the structural asymmetry at the notch junction, the inter-cavity coupling between the two microdisk resonators is asymmetric between two senses of circulation modes. Our previous two-dimensional (2-D) finite-difference time-domain (FDTD) simulations of such coupled microdisk resonators suggested that only the throughput-port transmissions preserve the reciprocity relations, whereas the drop-port transmissions and the mode-field distributions strongly depend on from which waveguide port the light is input-coupled.
Here, we present our systematic study on coupled spiral-shaped microdisk resonator-based filters. Our numerical simulations reveal modal field distributions that are highly asymmetric between two senses of circulation modes. By introducing a slight mismatch in radii while preserving seamlessly joint notches, we observe in both simulations and experiments selectively enhanced resonance extinction ratio (ER) for one of the split resonances while suppressing the other.
2. Non-evanescent asymmetric inter-cavity coupling
Fig. 1. Schematics of the coupled spiral-shaped microdisk resonator-based filter configurations with lightwave input-coupled to the first cavity (a) counterclockwise (CCW) circulation mode, and (b) clockwise (CW) circulation mode. Insets: zoom-in schematics of the non-evanescent coupling via the joint notches. CCW circulation is preferentially confined in the first cavity and only enables a weak tail-to-tail mode spatial overlap with the second cavity. CW circulation is preferentially coupled to the second cavity due to significant mode spatial overlap [5].
Figures 1(a) and 1(b) show the schematics of the coupled spiral-shaped microdisk resonator-based filter in two input-coupling configurations, with lightwave launched from two different input-ports exciting counterclockwise (CCW) and clockwise (CW) traveling-wave modes of the first cavity. For simplicity, we refer to them as CCW and CW configurations hereafter. The structure comprises two identical spiral-shaped microdisk resonators that are seamlessly jointed at their notches, with singlemode waveguides that are evanescently side-coupled to the cavities. The spiral shape is defined in terms of the azimuthal angle dependent radius r(ϕ) as [20–23]: r(ϕ)=r 0(1-εϕ/2π), where r 0=r(ϕ=0) and ε is a deformation parameter giving a notch junction width of r 0 ε. We wrap the side-coupled waveguides along an arc length of the microdisk in order to enhance the waveguide evanescent coupling.
The cavity light can partially transmit between the two microcavities via the notch junction. Due to the structural asymmetry, the mode spatial overlaps between the two microdisks at the notch junction are not identical between CCW and CW configurations. Insets of Fig. 1 illustrate the tail-to-tail mode spatial overlap for CCW configuration, and the more significant mode spatial overlap for CW configuration. Thus, the light transmissions from the first microdisk to the second microdisk (and likewise feedback from the second microdisk to the first microdisk) are asymmetric between CCW and CW configurations. Specifically, CCW configuration only enables weak transmissions to the second microdisk (blue dashed arrows), yet light from the second microdisk preferentially feeds back to the first microdisk CCW orbit upon traveling a round trip. This results in an asymmetric mode-field distribution between the coupled microdisks, with more intense mode-field distributions in the first microdisk. Whereas, for CW configuration, the lightwave can be preferentially transmitted to the second microdisk (red solid arrow), yet light from the second cavity is weakly coupled back (red dashed arrow). This enables more evenly distributed field intensity in the coupled microdisks.
Fig. 2. Schematic illustrations of (a)–(c) coupled spiral-shaped microresonators and (d)–(f) coupled circular-shaped microresonators upon three input-coupling configurations. I, I’, I”: input, D, D’, D”: drop-port transmissions, and T, T’, T”: throughput-port transmissions.
Figure 2 illustrates the transmission relations between CCW and CW configurations for coupled identical spiral-shaped microdisks and coupled identical circular-shaped microdisks. We observe that the throughput-port transmissions of CCW and CW configurations, namely I-to-T and I”-to-T”, for both structures are related by reciprocity [23] and thus are always identical. In contrast, we note that for both structures the drop-port transmissions, namely I-to-D and I”-to-D”, are not related by reciprocity and thus can depend on the input-coupling configuration. Based on the scattering theory, reciprocity relation means that light transmissions are preserved by interchanging positions of the source and the detector in a linear dielectric media with symmetric permittivity tensors [24]. Therefore, it is conceivable that the throughput-port transmissions in our passive devices follow the reciprocity relation, while in general the drop-port transmissions do not necessarily follow the reciprocity relation.
For further analysis, we observe that I-to-D and I’-to-D’ for both structures are related by reciprocity and thus are always identical. In case of coupled spiral-shaped microdisks, I’-to-D’ and I”-to-D” are not related by mirror-reflection symmetry due to structural asymmetry, thus in general D and D” are not identical. Yet, in the case of coupled circular-shaped microdisks, I’-to-D’ and I”-to-D” are related by mirror-reflection symmetry due to structural symmetry, thus D and D” are identical.
Hence, we see that the coupled spiral-shaped microdisk resonators have two key merits: 1) non-evanescent inter-cavity coupling is not constraint by fabricating submicrometer coupling gap; and 2) asymmetric inter-cavity coupling between different input-coupling configurations offers unique transmission characteristics that are distinct from conventional coupled microresonators.
3. 2-D FDTD simulations
Here, we numerically simulate the coupled spiral-shaped microdisk resonators using a commercial FDTD tool [25]. We adopt identical dimensions for the coupled microdisks with r 0=5 µm, and ε=0.16. The waveguide width is 0.4 µm, with a side interaction length defined by a 36° arc, via a gap separation of 0.3 µm. We use an effective refractive index of 1.92 in order to effectively account for the vertical dimension in a silicon nitride-on-silica substrate.
Fig. 3. FDTD-simulated TE-polarized multimode (a) throughput- and (b) drop-port transmission spectra of coupled spiral-shaped microdisk filters with r 0=5 µm, ε=0.16 for CCW (blue dashed line) and CW (red solid line) configurations.
Figures 3(a)–3(b) show the simulated TE-polarized (electric field in plane) multimode throughput- and drop-port transmission spectra of the coupled identical spiral-shaped microdisk resonators for CCW and CW configurations. The two throughput-port transmission spectra overlap with each other, as expected from reciprocity relations. The corresponding drop-port transmission spectra only show identical resonance wavelengths, yet with distinct relative resonance peak heights and ERs. We attribute the lower drop-port transmission for CCW configuration to an additional radiation loss induced at the notch junction in order to balance the preferential light out-coupling for CW configuration [23]. Moreover, we identify a free-spectral range (FSR) of ~45.3 nm, which is consistent with the single microdisk circumference. The resonances are split due to the strong inter-cavity coupling.
Fig. 4. FDTD-simulated steady-state mode-field patterns of the coupled spiral-shaped microdisk resonators for CCW and CW configurations at resonance wavelength of (a), (b) 1544.5 nm (resonance A) and (c), (d) 1554.1 nm (resonance B).
Figures 4(a)–4(b) show the FDTD-simulated steady-state mode-field patterns of the coupled identical spiral-shaped microdisk resonators for CCW and CW configurations at split resonance wavelength of 1544.5 nm (resonance A in Fig. 3(a)). For both configurations, the modal field distributions at the notch junction exhibit an odd parity. Yet, while CCW configuration suggests a preferred light field confinement in the first cavity (Fig. 4(a)), CW configuration reveals relatively pronounced and evenly distributed mode-field pattern along both microdisk rims and through the notch junction (Fig. 4(b)). This is consistent with the circulation-mode-dependent mode spatial overlap at the notch junction schematically depicted in Fig. 1. Figures 4(c)–4(d) show the corresponding steady-state mode-field patterns at the other split resonance wavelength of 1554.1 nm (resonance B in Fig. 3(a)). The modal field distributions here show even parity at the notch junction. The odd- and even-parity mode field distributions for the two adjacent resonances further confirm the resonance split due to the strong inter-cavity coupling. Again, we observe asymmetric mode-field distributions between CCW and CW configurations, which is consistent with Figs. 4(a) and 4(b).
Here, we simulate coupled non-identical spiral-shaped microdisk resonators with radii r 1 and r 2 at the notch. In order to preserve the notch junction, the two microdisks also have slightly different spiral shapes (i.e. a slight mismatch in both their radii and ε’s). We vary the radius of the second microdisk r 2 while fixing the radius of the first microdisk. Figures 5(a)–(d) show the simulated transmission spectra of structures with (a), (b) r 2=4.8 µm (ε=0.167) and (c), (d) r 2=5.4 µm (ε=0.148), while fixing r 1=5 µm. Still, the throughput-port transmissions are reciprocal for CCW and CW configurations, while the drop-port transmissions are non-identical. The FSR in both cases are slightly differed, which is consistent with a change of average round trip path length of the two microdisks. Compared with those corresponding resonances in coupled identical spiral-shaped microdisk resonators (Fig. 3), we observe that one of the split modes is selectively enhanced while the other one is nearly suppressed. Specifically, for r 2=4.8 µm, the enhanced ER is ~25 dB for resonance C compared with ~10 dB for resonance A. Whereas, for r 2=5.4 µm, the enhanced ER is ~27 dB for resonance F compared with ~9 dB for resonance B.
Fig. 5. FDTD-simulated TE-polarized multimode transmission spectra with radius perturbations in the second microdisk for CCW and CW configurations. (a) Throughput-port, and (b) drop-port transmission spectra with r 1=5 µm and r 2=4.8 µm. (c) Throughput-port, and (d) drop-port transmission spectra with r 1=5 µm and r 2=5.4 µm.
Fig. 6. Variations of (a) resonance wavelengths, (b) extinction ratios, and (c) quality factors as a function of the radii mismatch Δr=r 2-r 1 for both split modes.
We also simulate the transmission spectra of coupled non-identical spiral-shaped microdisk resonators with r 2=4.6 µm (ε=0.174) and r 2=5.2 µm (ε=0.154). Figure 6 show the variations of resonance wavelengths, extinction ratios, and quality factors as a function of the radii mismatch Δr=r 2-r 1 for both split modes. Figure 6(a) shows that the wavelengths for both split modes are blueshifted as Δr increases. Figure 6(b) shows that for Δr < 0, the split resonance with shorter wavelength is ER enhanced, while the other split resonance is suppressed. In contrast, for Δr>0, the split resonance with longer wavelength is ER enhanced. We also see that the ER values for each split resonance are optimized at different Δr values, implying the size and shape mismatches should be carefully designed. Figure 6(c) shows that resonance A exhibits an optimized Q value at Δr=0. We attribute the reduced Q values to a possible modal mismatch induced loss at the notch junction between the two non-identical microdisk resonators. However, for resonance B, the change in Q values is not significant. We are not able to estimate the Q values for some of the resonances due to the line shape broadening.
Fig. 7. FDTD-simulated steady-state mode-field patterns at resonance wavelengths of (a), (b) 1549.1 nm (resonance C), and (c), (d) 1561.9 nm (resonance D) for CCW and CW configurations with r 1=5 µm and r 2=4.8 µm.
Figures 7(a) and 7(b) show the simulated steady-state mode-field patterns at the ER enhanced resonance wavelength of 1549.1 nm (resonance C in Fig. 5(a)) for CCW and CW configurations with r 1=5 µm and r 2=4.8 µm. For CCW configuration, the mode-field intensity in the second microdisk is significantly lower than that in coupled identical microdisk resonators (see Fig. 4(a)). For CW configuration, the mode intensity distribution is less even compared with that in coupled identical microdisk resonators, with lower intensity in the second microdisk. We attribute the lower intensity in the second microdisk in part to the cavity mode mismatch between two non-identical microdisk resonators.
Similarly, Figs. 7(c) and 7(d) show the simulated steady-state mode-field patterns at the suppressed resonance wavelength of 1561.9 nm (resonance D in Fig. 5(a)). For CCW and CW configurations, we observe only weak cavity internal field distributions with the input-coupled lightwave essentially out-couple to the throughput-port. We also simulate the steady-state modal field distributions for CCW and CW configurations with r 1=5 µm and r 2=5.4 µm at both split resonance wavelengths (resonances E and F in Fig. 5(c)), and obtain similar field patterns as in resonances C and D (not shown here).
We see that the change in size and shape of the second microdisk modify the coupled cavity mode characteristics such as cavity loss, which affects the balance between cavity loss and waveguide coupling [26], thus results in the change in ERs and Q values. It is also difficult to distinguish the field parities at the notch junction in coupled non-identical spiral-shaped microdisk resonators.
4. Experiments
We fabricate the coupled spiral-shaped microdisk resonator-based filters on a silicon nitrideon-silica substrate using standard silicon microelectronics processes. We use a 1.1-µm-thick silicon nitride device layer on a 1.5-µm-thick silica under-cladding layer. The device structures are defined by photolithography (i-line, 365 nm) and CF4-based reactive ion plasma etching (RIE). Figure 8(a) shows the top-view scanning electron micrograph (SEM) of the fabricated device. The measured radii for both spirals are ~20 µm. Figure 8(b) shows the cross-section of the waveguide evanescent coupling region, denoted as dash line in Fig. 8(a). The measured etch depth h is ~0.93 µm. Figures 8(c)–8(d) show the zoom-in view SEMs of the notch-coupling and waveguide evanescent-coupling regions. The notch width is ~0.4 µm. The width of the side-coupled waveguide is ~0.38 µm, and the gap spacing between the cavity sidewall and the waveguide is ~0.46 µm.
Fig. 8. (a) Top-view SEM of our fabricated device on a silicon nitride-on-silica substrate. The waveguide coupling length spans an angle of 36°. (b) Cross-section view SEM of the evanescent coupling region. h~0.93 µm. (c)–(d) Zoom-in view SEMs of the notch-coupling region and the lateral evanescent-coupling region. r 0~20 µm, ε~0.02, w~0.38 µm, and g~0.46 µm.
Figure 9 shows the measured TE-polarized transmission spectra for coupled identical and non-identical spiral-shaped microdisk resonators. The experimental setup follows standard passive wavelength scanning measurements and has been detailed elsewhere [23]. Figure 9(a) shows the measured throughput-port multimode transmission spectra for CCW and CW configurations. We observe essentially identical throughput-port transmission spectra, suggesting reciprocal transmissions. The transmission spectra suggest pronounced split resonances with large ERs, although it is difficult to differentiate a split mode in a multimode cavity. We label two of the possible split resonances as modes A and B, each displays a Q ~6,500. We measure a FSR of ~9.1 nm, which is consistent with a single microcavity circumference. The measured highest Q is ~15,000, which means our coupled spiral-shaped microdisk resonators preserve high-Q modes [18].
Fig. 9. Measured TE-polarized transmission spectra for (a)–(b) coupled identical spiral-shaped microdisk resonators, and (c)–(d) coupled non-identical spiral-shaped microdisk resonators. (a) Throughput-port and (b) drop-port multimode transmission spectra for fabricated device with r 0=20 µm, ε=0.02. (c) Throughput-port and (d) drop-port multimode transmission spectra for fabricated device with r 1=20 µm, ε=0.02 and r 2=19.8 µm, ε=0.0202.
Figure 9(b) show the measured drop-port multimode transmission spectra. We note that most of the modal features find corresponding resonances in the throughput-port transmission spectra. However, the drop-port transmission spectra for CCW and CW configurations are substantially different in resonance ERs, indicating that the drop-port transmissions are non-identical as expected from the asymmetric inter-cavity coupling.
We also fabricate coupled non-identical spiral-shaped microdisk resonators with radius and shape perturbations in the second microdisk. Figures 9(c)–9(d) show the measured transmission spectra for a device with r 1=20 µm, r 2=19.8 µm for CCW and CW configurations. We observe that the throughput-port transmission spectra are essentially identical as expected from reciprocity relations, whereas the drop-port transmission spectra exhibit essentially the same set of resonance modes but display more pronounced variations in their resonance ERs. The measured highest Q is ~13,500, which is slightly lower than the case of coupled identical microdisks.
The transmission spectra here suggest that one of the split resonances is selectively enhanced in ER. For instance, it is conceivable that split resonances A and B in coupled identical spiral-shaped microdisk resonators (Fig. 9(a)) become resonance C in coupled non-identical spiral-shaped microdisk resonators (Fig. 9(c)), with reduced Q (reduced from ~6,500 to ~3,500) and enhanced ER (enhanced from ~20 dB to ~24 dB). There is, however, no significant FSR expansion or resonance wavelength shifts as compared with the case of coupled identical microdisks, due to the small difference between the two cavities.
5. Conclusion
We systematically investigated the characteristics of coupled spiral-shaped microdisk resonators which lack structural symmetry. The non-evanescent asymmetric inter-cavity coupling mechanism is distinct from that in conventional coupled microdisk/microring resonators. Our 2-D FDTD simulations revealed reciprocal throughput-port transmissions between counterclockwise (CCW) and clockwise (CW) configurations, yet drop-port transmissions and modal field distributions are input-port dependent. By introducing slight mismatches in size and shape between the two coupled microdisks, we observed enhanced resonance with ~27 dB extinction ratio compared with an extinction ratio of ~9 dB in coupled identical microdisks. Our experimental demonstrations confirmed our numerical simulations and revealed that our fabricated devices were of high-Q. We envision that by integrating p-i-n diodes on such structures [4,27,28], we can actively tune the cavity characteristics (e.g., Q values and ERs). On potential applications, we envision that by applying the structure as electro-optic switch [4] and utilizing the different drop-port transmissions of CCW and CW traveling-wave mode, we can expand switching functionality. Furthermore, this structure can also be functioned as optical switch as the existing demonstrations [17,18] between a spiral-shaped microdisk laser and a semicircle microdisk amplifier.
Acknowledgements
This work was substantially supported by a grant from the Research Grants Council of The Hong Kong Special Administrative Region, China (Project No. 618506). X. Luo acknowledges the fellowship support from the NANO program of HKUST.
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