We report transmission spectra and mode-field distributions of a waveguide-coupled spiral-shaped microdisk resonator on a silicon nitride-on-silica substrate. Our measured and simulated transmission spectra reveal reciprocal transmissions between clockwise and counterclockwise traveling-waves of such microcavity that lacks mirror symmetry. Our measured out-of-plane scattering intensity distributions and simulated steady-state mode-field patterns, however, indicate asymmetric modal distributions that depend on the sense of lightwave circulations and the input-coupling mechanisms. We discuss implications of the observed reciprocal transmissions with asymmetric modal distributions to unidirectional lasing from spiral-shaped microcavities reported in the literature.
© 2007 Optical Society of America
Micrometer-scale ring or disk resonators, with their key merits of high-Q modes, compact size, and accessibility with optical waveguides, have long been regarded as promising microlaser cavities and wavelength-selective filter components in large-scale-integrated photonic circuits [1–3]. However, one major shortcoming of microring/disk laser cavities is that clockwise (CW) and counterclockwise (CCW) traveling-wave modes are degenerate. The degeneracy between CW and CCW traveling-wave modes results in undesirable bidirectional emissions from waveguide-coupled microresonator lasers, as recently reported by Fang et al.,  on racetrack-shaped microring lasers. In order to obtain unidirectionality from ring lasers, conventional wisdom employs selective feedback to only one traveling-wave mode, which consequently lases and unidirectionally emits . For waveguide ring lasers, selective feedback approach for unidirectional lasing has been demonstrated by using asymmetric waveguide structures including S-waveguides [6–7], tapered waveguides , and spiral waveguides .
Alternatively, Chern et al.  in their pioneering work demonstrated unidirectional lasing from a spiral-shaped micropillar, which was believed to lift the degeneracy between CW and CCW traveling-wave modes due to the lack of mirror symmetry in the cavity shape.
Unidirectional lasing emission was observed to be non-evanescently out-coupled from the spiral notch. Later experiments by a number of independent research groups confirmed the unidirectional lasing from spiral-shaped micropillar/disk lasers [11–17] using various material systems (III-V semiconductors [11, 16], polymer [12–15, 17]), quantum well structures, and pumping techniques. Recently, optical switching mechanism based on high-Q-preserving direct non-evanescent coupling between a spiral-shaped microdisk laser and a semicircle microdisk amplifier has also been proposed and demonstrated [21–22].
Meanwhile, theoretical calculations of lasing modes in a spiral-shaped microcavity [10, 18–20] revealed quasi-bound states with complicated modal structures, which can be whispering-gallery (WG)-like or quasiscarred modes. Theoretical analysis in Ref.  showed that WG-like traveling-wave modes in both CW and CCW circulations exist in spiral-shaped microcavites, albeit the circulation that tends to directly out-couple at the spiral notch exhibits much weaker amplitude. Other independent theoretical calculations  also showed strongly localized WG-like traveling-wave modes in only one sense of circulation that does not favor direct out-coupling at the spiral notch. Furthermore, calculations [18–19] suggested that quasiscarred modes with three-bounce orbits only propagate in one sense of circulation and do not tend to spatially overlap with the spiral notch. Hence, to our knowledge, theoretical calculations on lasing modes in a spiral-shaped microcavity have yet identified resonances that explain the experimentally observed unidirectionality of spiral-shaped microcavity lasers.
Although the detailed unidirectional lasing mechanism in spiral-shaped microcavities is yet clearly understood , it is generally believed that the non-degenerate CW/CCW traveling-wave modes encounter different total cavity losses due to their different tendencies to out-couple at the spiral notch. Thus, CW/CCW traveling-wave modes of different quality factors (Q’s) can lase at different thresholds, resulting in unidirectional lasing for only one sense of lightwave circulations. Specifically, the traveling-wave mode that preferentially transmit at the spiral notch every round trip are expected to exhibit a higher cavity loss (lower Q), and thus a higher lasing threshold. However, to our knowledge, CW/CCW traveling-wave modes Q’s (and lasing thresholds) in spiral-shaped microcavity lasers have yet been directly measured. Therefore, it is much needed to devise a way to separately access CW and CCW traveling-wave modes in a spiral-shaped microcavity and investigate their modal characteristics.
Inspired by Chern et al.’s work , we previously proposed  and demonstrated [24–26] spiral-shaped microdisk resonator-based filters on a silicon nitride-on-silica chip, with waveguides in/out-coupling non-evanescently at the spiral notch and evanescently at the cavity sidewall. By input-coupling from each of the three waveguide ports, we can separately access CW or CCW traveling-wave modes, with direct and/or evanescent coupling. Our previous work thus provided an initial direct measurement of the spiral-shaped microdisk transmission spectra of both CW and CCW traveling-waves, without complications of active media.
Here, we report our systematic experiments and numerical simulations of waveguide-coupled spiral-shaped microdisk resonators in silicon nitride. We separately measure the transmission spectra from CW and CCW traveling-waves by in-coupling from each of the three waveguide ports. Moreover, we gain insights to the cavity mode-field distributions by measuring the out-of-plane scattering from the cavity by using a scanning lensed-fiber. Our experimental measurements and two-dimensional (2-D) finite-difference time-domain (FDTD) numerical simulations reveal reciprocal transmission spectra accompanied with input-port-dependent asymmetric modal distributions. Yet, we find no convincing evidence for CW and CCW traveling-wave modes having different Q’s. We interpret our observations in light of reciprocity relations, and discuss their implications regarding unidirectional lasing from spiral-shaped microcavities.
2. Wavelength-selective filter configurations, device fabrications, and experimental setup
Figures 1(a)–(c) depict the schematics of the waveguide-coupled spiral-shaped microdisk resonator-based wavelength-selective filter configurations. The filter comprises a spiral-shaped microdisk, a single-mode waveguide that is seamlessly butt-coupled to the spiral notch, and another single-mode waveguide that is evanescently side-coupled to the cavity [23–26]. Lightwave can be in/out-coupled to the microdisk non-evanescently via the notch-waveguide or evanescently from the side-coupled waveguide. The spiral shape is defined as :
where r 0 is the spiral radius at azimuthal angle ϕ = 0, and e is the deformation parameter. The radius mismatch of r 0 ε at ϕ = 2π defines the spiral notch width and the notch-waveguide width. The notch-waveguide and the side-coupled waveguide do not necessarily have the same width, thus enabling the non-evanescent and evanescent coupling to be separately optimized.
The filter functionality depends on from which waveguide-port the lightwave is input-coupled. The evanescently in-coupled CCW traveling-waves do not favor out-coupling to the notch-waveguide, and thus the microdisk can act as a notch filter (Fig. 1(a)). In contrast, the evanescently in-coupled CW traveling-waves favor partial transmission to the notch-waveguide, and thus the microdisk can act as a drop filter (Fig. 1(b)). Moreover, the non-evanescently in-coupled CCW traveling-waves also do not favor out-coupling to the notch-waveguide upon a cavity round trip, yet the cavity mode field can be evanescently out-coupled to the side-coupled waveguide. We refer to this configuration as add-only filter. The non-evanescent coupling via the notch junction offers a potentially efficient coupling to the spiral-shaped microdisk resonances, without imposing the technologically-challenging constraint for fabricating an evanescent coupling gap.
We fabricate the spiral-shaped microdisk filters on a silicon nitride-on-silica substrate using standard silicon microelectronics processes. In brief, a bulk silicon wafer is first thermally oxidized to form a 1.5-μm-thick silica under-cladding layer. A 1.0-μm-thick low-stress silicon nitride device layer is then deposited using low-pressure chemical-vapor deposition (LPCVD). The device structures are defined by photolithography (i-line, 365 nm) and CF4-based reactive ion plasma etching (RIE). The etch depth is ~0.9 μm.
Figure 1(d) shows the scanning electron micrograph (SEM) of the spiral-shaped microdisk filter. We adopt ro = 25 μm and ε = 0.016. The small ε value gives a notch size of 0.4 μm for single-mode notch-waveguide coupling. Insets show the zoom-in view SEMs of the evanescent-coupling region and the notch junction. The side-coupled waveguide width is ~450 nm while the notch-waveguide width is ~400 nm. The gap spacing between the cavity sidewall and the side-coupled waveguide (and between the cavity sidewall and the notch-waveguide) is ~300 nm limited by our lithography.
Figure 1(e) schematically depicts the experimental setup. We use an external-cavity wavelength-tunable diode laser with wavelength tuning range between 1505 nm and 1585 nm (spectral resolution of 20 pm). Laser light is coupled into a polarization-maintaining (PM) single-mode (SM) lensed-fiber for end-firing to the waveguide. The lensed-fiber output spot diameter is ~2.5 μm, which approximately matches the waveguide tapered end-face. The waveguide is out-coupled to another lensed-fiber. We lock-in detect the transmission spectra of notch/drop/add-only filters by separately in-coupling to the three waveguide ports. The inset shows the optical micrograph of the filter. The notch-waveguide has a 180°-bend (with a large 50-μm radius of curvature) towards the throughput-port direction. This enables throughput measurements for add-only filter, and enables measurements of both throughput-and drop-port spectra in the same output direction for drop filter. For each in-coupling configuration, we also collect the out-of-plane scattering intensities near the spiral notch junction and the evanescent-coupling region by linearly scanning a third lensed-fiber at a height of a few μm above the cavity top surface.
3. Transmission spectra measurements
Figure 2(a) shows the measured TE-polarized (electric field in plane) throughput-port transmission spectra of notch (CCW) and drop (CW) filters. The measured multimode transmission spectra are essentially identical with nearly the same resonance attributes. This suggests that the transmission spectra of evanescently in-coupled CCW and CW traveling-waves are reciprocal related with the same Q’s.
We identify a free spectral range (FSR) of ~7.2 nm, which is consistent with the spiral microdisk circumference. This suggests that the resonance lightwave is guided along the microcavity rim in round-trips as in WG modes. The observed highest Q value is ~12,000 with an extinction ratio (ER) exceeding 5 dB. We estimate Q values from the measured spectra using the relationship Q = λ0/∆λ, where λ0 is the resonance wavelength and Δλ, is the estimated 3-dB-linewidth. However, as resonances in waveguide-coupled microresonator transmission spectra typically exhibit asymmetric line shapes , our Q estimations from the transmission spectra can be skewed by the line shapes.
Figure 2(b) shows the measured TE-polarized drop-port transmission spectrum of drop filter and throughput-port transmission spectrum of the add-only filter configuration. Most of the resonance peaks are relatively pronounced (with ER as high as ~20 dB) and find corresponding resonances in the throughput-port transmission spectra in Fig. 2(a). The two transmission spectra are nearly matched with each other, again suggesting a reciprocal pair between CW and CCW traveling-waves. We attribute the slight mismatches at most of the modal features in part to the different degrees of misalignment between the input/output lensed-fiber and the waveguide end-faces. However, additional mechanisms may be needed in order to account for large mismatches in ER at particular resonances (see Sec. 6 Discussion).
The observation of reciprocal transmissions in a spiral-shaped microdisk resonator is significant, though may not be surprising in hindsight. The observed reciprocal transmissions can be expected from reciprocity relations [27–30]. In essence, reciprocity relations mean that light transmissions are preserved by interchanging positions of the source and the detector in a linear dielectric media with symmetric permittivity tensors . Reciprocity relations have been proven in stratified media , in scattering from dielectric objects of arbitrary shapes [28, 29], and in one-dimensional Fabry-Perot cavities . Hence, it is conceivable that the transmissions of the passive silicon nitride waveguide-coupled spiral microdisk resonator follow reciprocity relations. To our knowledge, this is the first report on reciprocity relations in two-dimensional microdisk resonators in a linear dielectric medium.
Furthermore, the concept of reciprocity relations provides a new insight in understanding the spiral-shaped microdisk cavity CW and CCW traveling-waves. Reciprocal transmissions with identical resonance spectra and identical quality factors imply that CW and CCW traveling-wave modes encounter identical total cavity loss. The total cavity loss in a spiral-shaped microdisk with direct-coupled waveguide comprise (i) out-coupling via the notch-waveguide, (ii) diffraction and scattering at the notch junction, (iii) distributed cavity loss along the cavity sidewall (including curved sidewall diffraction, refraction, and roughness-induced scattering), and (iv) material absorption. Given CW traveling-wave modes can preferentially out-couple via the notch-waveguide, the loss from (ii) and (iii) should therefore be relatively lower. In contrast, CCW traveling-wave modes can see relatively higher loss from (ii) and/or (iii) in order to balance the relatively lower loss from (i). We remark that scattering at the notch junction and along the cavity sidewall may result in cross-coupling between CCW and CW traveling-wave modes. While a large distributed cavity loss can be originated from a less confined traveling-wave orbit (e.g. a quasiscarred three-bounce mode ).
4. Out-of-plane scattering spectra measurements
In order to gain insights to the cavity modal structures that underlie the reciprocal transmissions, we collect the out-of-plane scattering spectra from both CW and CCW traveling-waves by a scanning lensed-fiber positioned at ~3 μm above the chip.
In detail (see Fig. 1(e)), the fiber-to-device vertical separation is calibrated by means of in-coupling a separate light source to the fiber and measuring the etalon modulations due to the fiber-end and chip-surface multiple reflections. By repeating the vertical separation calibration at two other positions on the chip, we estimate the chip is tilted by less than 1.5° in both the x- and y-axes. We scan the lensed-fiber along the x-axis using a closed-loop piezoelectric transducer stage in a step of 0.8 μm, and position the lensed-fiber along the y-axis by using an open-loop transducer stage. At every fiber-probe position, we scan the wavelength for all three input configurations over the spectral window of 1551.7 nm - 1553.2 nm, which is wide enough to span at least two resonances according to the transmission measurements. In order to minimize uncertainties due to positioning in the y-direction, we average over three adjacent profiles stepped in 1 μm in the y-direction. We estimate the lensed-fiber field-of-view is ~3 μm in diameter, which is consistent with the fiber focused spot diameter. Although our scanning fiber spatial resolution is limited by the lensed-fiber field-of-view, such scattering light probing technique is sufficient to indicate the trends of CW/CCW traveling-wave modal distributions of the relatively large-sized microdisk and their corresponding Q’s, without complications from a typical scanning near-field probe.
Figures 3(a)–(c) show the averaged out-of-plane scattering spectra near the notch junction in the x-direction (centered at y = -5 μm) for notch, drop, and add-only filters. The spatial scanning spans from x = 39 μm (11 μm inward from the notch at x = 50 μm) to slightly outside the notch junction. The scattering intensity is normalized to the input-coupling lensed-fiber transmission intensity. For all three filter configurations, the spatial profiles reveal similarly pronounced scattering intensities in the neighborhood of the notch junction (x ≈ 50 μm). This suggests that CCW and CW traveling-waves encounter similar out-of-plane scattering at the notch junction, regardless of the evanescent or non-evanescent input-coupling.
Figure 3(d) depicts the measured light scattering spectra at the notch junction (x = 50 μm). Figure 3(e) shows the measured throughput- and drop-port transmission spectra of drop filter for reference (see Fig. 2). The light scattering spectra and the transmission spectra exhibit the same resonance wavelengths at 1552.06 nm and 1553.02 nm. The estimated Q values from the scattering spectra of notch filter marginally differ from those estimated for drop filter. We note that the difference in the estimated Q values here is similar to that between CW and CCW traveling-wave modes in circular microdisk resonators in our control experiments (data not shown). Thus, our light scattering measurements near the notch junction reveal no evidence beyond tolerance that CW and CCW traveling-wave modes see different Q’s or total cavity loss. We are not able to estimate the Q values from the add-only filter scattering spectrum due to the asymmetric broadened line shapes.
Figures 3(f) and 3(g) show the light scattering intensity profiles near the notch junction at resonance wavelengths of 1552.06 nm and 1553.02 nm. Each filter configuration displays scattering intensity profiles that are consistent between the two resonance wavelengths. Nonetheless, the scattering intensity profiles vary among the three filter configurations. The profiles of notch filter extend relatively inward to the cavity bulk, in comparison with those of drop and add-only filters, suggesting that the cavity modal distributions depend on the input-coupling port. In order to further quantify the intensity profiles, we measure the profile width by the distance between the 5% of the maximum intensity and the cavity rim (labeled in Figs. 3(f) and 3(g)).
Similarly, Figs. 4 show the measured out-of-plane scattering near the evanescent-coupling region for the three filter configurations. Again, averaged out-of-plane scattering spectra (Figs. 4(a)–(c)) and on-resonance light scattering profiles (Figs. 4(f)–(g)) suggest that the cavity modal distributions highly depend on the input-coupling port. From the light scattering spectra at the evanescent-coupled waveguide position (Fig. 4(d)), the estimated Q’s from the light scattering spectrum of drop filter are consistent with those estimated from the transmission spectra (Fig. 4(e)). It is, however, difficult to estimate Q’s from the light scattering spectra of notch and add-only filters due to the multimodal features and the broadened line shapes.
We therefore conclude from the out-of-plane scattering intensity profiles that the waveguide-coupled spiral-shaped microdisk modal distributions are input-port-dependent. Comparing the scattering intensity profiles of notch and drop filters, we see that the reciprocal CCW/CW traveling-wave transmissions (Fig. 2(a)) do not imply symmetric modal distributions between the two traveling-waves. Likewise, the scattering intensity profiles of drop and add-only filters do not suggest symmetric modal distributions despite their reciprocal transmissions (Fig. 2(b)). In addition, comparing the scattering intensity profiles of notch and add-only filters, we find that even the traveling-wave modal distributions in the same sense of CCW circulation can depend on the evanescent/non-evanescent in-coupling mechanisms.
5. FDTD simulations
In order to provide numerical guidance to interpreting the measured transmissions and modal structures, we simulate by using a commercial FDTD tool  the waveguide-coupled spiral-shaped microdisk (i) transmission spectra, (ii) cavity internal-field spectra near the notch junction and the evanescent-coupling region, (iii) input-coupling field distributions, and (iv) steady-state modal distributions. We adopt 2-D simulations and a relatively small-sized cavity of radius r0 = 10 μm for convenience of computation, and ε = 0.04 in order to preserve a notch size of 0.4 μm for the notch-coupled single-mode waveguide. The evanescently coupled waveguide width is 0.3 μm, and the gap separation is 0.3 μm. We choose the refractive index n = 2 to represent silicon nitride. It should be emphasized that the objective here is to compare the qualitative trends measured with those simulated, while the details are of lesser concern.
5.1 Simulated transmission spectra
Figure 5(a) shows the simulated TE-polarized throughput-port transmission spectra of notch and drop filters. The multimode spectra overlap with each other, revealing reciprocal transmissions that are consistent with our measurements (Fig. 2(a)). We identify an FSR of ~22.6 nm, which is consistent with the spiral microdisk circumference. Figure 5(b) shows the simulated TE-polarized drop-filter drop-port transmission spectrum and add-only filter throughput-port transmission spectrum. Again, the multimode spectra are exactly matched with each other, indicating reciprocal transmissions that are consistent with our measurements (Fig. 2(b)). We find corresponding modal features between the simulated spectra in (a) and (b).
5.2 Simulated cavity internal-field spectra near the notch junction and the evanescent-coupling region
We simulate the cavity internal-field spectra near the notch junction and the evanescent-coupling region. We integrate the cavity fields over a square area of 1.2 μm × 1.2 μm, which is approximately in scale with the lensed-fiber field-of-view relative to the actual cavity size. We choose the wavelength range of 1531 nm – 1537 nm, spanning resonances H1 and A2. The integrated internal-field intensity is normalized to the input-field intensity.
Figures 6(a)–(c) show the simulated cavity internal-field spectra near the notch junction in the x-direction (at y = 0 μm) for the three filter configurations. The profiles span from the cavity bulk at x = 12.9 μm to the notch junction at x = 19 μm, in steps of 0.32 μm. The drop and add-only filters show similar spatial intensity profiles with relatively high intensities spanning the cavity rim region. Whereas, notch filter displays relatively low internal-field intensities near the cavity rim region, yet relatively high internal-field intensities extended into the cavity bulk. This indicates that the cavity modal structures depend on the input-coupling port. Specifically, the cavity internal-field intensity profiles at resonance wavelengths (Figs. 6(f)–(g) for resonances H1 and A2) reveal distinct field intensity profiles among the three configurations.
Figure 6(d) depicts the simulated cavity internal-field spectra at the notch junction (x = 19 μm). We find the same resonance H1 (1532.72 nm) as in the transmission spectra shown in Fig. 6(e). Yet, resonance A2 (1535.13 nm) cannot be discerned from the cavity internal-field spectra due to the pulse-excitation-induced multimode line-shape broadening. The estimated Q values at resonance H1 are essentially identical for drop and add-only filters, and thus suggest no evidence that CW and CCW traveling-wave modes see non-reciprocal total cavity losses.
Same conclusion can also be obtained from the simulated cavity internal-field spectra near the evanescent-coupling region. In Figs. 7(a)–(c), we observe that the intensity profiles of drop and add-only filters exhibit similar patterns, with relatively high intensities spanning the cavity rim region. Yet, notch-filter intensity profiles display a local maximum in close proximity to the cavity rim, and another intensity maximum extended relatively inward into the cavity bulk. This again suggests that the cavity modal structures depend on the input-coupling port. Furthermore, the cavity internal-field intensity profiles at resonance wavelengths (Figs. 7(f)–(g)) also indicate asymmetric intensity profiles.
We therefore conclude that our simulated reciprocal transmission spectra, along with input-port-dependent cavity internal-field spectra near the notch junction and the evanescent-coupling region, are consistent with our measurements. We find no evidence that CW and CCW traveling-wave modes display different Q’s.
5.3 Simulated input-coupling field patterns
In order to gain further insights to the input-coupling dependence, we simulate the input-coupling field patterns. Figures 8(a)–(c) show the FDTD-simulated TE-polarized input-coupling H-field patterns at wavelength of 1512.52 nm (resonance A1) for each filter configuration. We consider only the input-coupling side of the cavity in the computation window by truncating the other half by a perfectly matched layer (PML). The evanescently in-coupled field patterns for notch and drop filters display essentially identical Gaussian beam-like profiles (due to the curved sidewall focusing effect). While the non-evanescently in-coupled field pattern for add-only filter shows that the lightwave is diffracted at the notch junction.
We apply spatial Fourier transform to the simulated input-coupling field patterns over the dashed rectangular windows as shown in Figs. 8(a)–(c). Figures 8(d)–(f) show the corresponding transformed k-vector distributions (up to half-maximum of the Fourier transform amplitude). We define kx and ky components in units of π/r 0. Our Fourier analysis reveals that notch and drop filters display identical in-coupling k-vector distributions with central angle ϕ = 43.5° (relative to the x-axis) and full-width-half-maximum ∆ϕ = 27.3°. Whereas, add-only filter displays a different in-coupling k-vector distribution with ϕ = 60.5° and ∆ϕ = 20.1°. Therefore, we see that although the modal structures depend on the input-coupling port, it is not solely the input-coupling k-vector distributions that govern the resonance modal structures.
It is worth mentioning that the simulated non-evanescently coupled field amplitude in add-only filter is marginally larger than that of the evanescently coupled field amplitude in notch/drop filters. This hints that the non-evanescent coupling via the notch can be more efficient than the conventional evanescent coupling. However, in order to attain highly efficient non-evanescent coupling at the notch, proper modal matching between the cavity mode and the waveguide mode is of essence. This imposes detailed designs for the spiral cavity shape, the notch shape , and the notch-coupling waveguide.
5.4 Simulated mode-field patterns
Figures 9 show the FDTD-simulated TE-polarized mode-field (H-field) patterns at three resonances A1, H1, and A2 for notch (left column), drop (center column), and add-only (right column) filters. For these three resonances (with resonances A1 and A2 related by a FSR), notch filter consistently exhibits three-bounce traveling-wave mode-field patterns that are less confined to the cavity rim and do not spatially overlap with the spiral notch. Whereas, drop and add-only filters consistently exhibit WG-like traveling-wave mode-field patterns that are better confined to the cavity rim and have good spatial overlap with the spiral notch. We remark that the mode-field patterns discussed here (at resonances H1 and A2) are consistent with the cavity internal-field spectra discussed in Sec. 5.2.
Thus, we confirm that the evanescently in-coupled CCW and CW traveling-wave modes display asymmetric modal distributions, despite their reciprocal transmissions (Fig. 5(a)) and identical in-coupling k-vector distributions (Figs. 8(d), 8(e)). Interestingly, the evanescently in-coupled CCW traveling-wave three-bounce modes that do not favor out-coupling to the notch-waveguide also do not significantly build up their internal-field amplitudes. In contrast, the evanescently in-coupled CW traveling-wave modes that do favor out-coupling to the notch-waveguide display relatively enhanced WG-like mode-field amplitudes.
However, the largely similar WG-like modal distributions of the evanescently in-coupled CW traveling-wave modes and the non-evanescently in-coupled CCW traveling-wave modes can be arguably expected from their reciprocal transmissions. In each case, the traveling-wave modes either output-couple or input-couple via the spiral notch, which spatially overlaps with the WG-like modes but not the three-bounce modes.
We note that the simulated three-bounce and WG-like traveling-wave modes here largely resemble the theoretically calculated quasiscarred and WG-like resonances in a spiral-shaped microdisk without waveguide coupling [10, 18–20]. Although we expect that the waveguide direct coupling can perturb the cavity boundary conditions, and in particular perturb the modes from seeing a Fresnel reflection at the spiral notch end-face, our simulations suggest that the effect of the perturbation is not severe. For the WG-like modes that spatially overlap with the notch-waveguide, we reason that the modal mismatch between the WG-like modes and the waveguide mode induces significant reflection at the notch-waveguide junction much like the Fresnel reflection at the notch end-face. Thereby, the waveguide perturbation to the WG-like modes is effectively minimized. Whereas, for the three-bounce modes that do not spatially overlap with the spiral notch, the perturbation from the waveguide direct coupling should be negligible.
We also confirm that the reciprocal transmissions remain valid in both experimental measurements (within tolerance) and numerical simulations for large-ε waveguide-coupled spiral-shaped microdisks, with up to ε = 0.16 for experiments and ε = 0.20 for simulations (data not shown). We conduct the experiments on the same chip as the ε = 0.016 devices reported here.
Relatively large mismatches in resonance ER in the measured reciprocal transmissions between drop and add-only filters (e.g. near 1557 nm in Fig. 2(b)) may in part be attributed to possible asymmetric modal distributions between CW and CCW traveling-waves. In practice, CW and CCW traveling-waves of different spatial distributions and field amplitudes can encounter different roughness-induced scattering losses along the cavity circumference or from the cavity top surface. Thus, it is possible to see individual resonances display pronounced mismatches in ER’s or linewidth’s (due to inhomogeneous broadening) between CW and CCW traveling-waves in otherwise reciprocal transmissions.
Hence, based on this work, we offer our interpretations to unidirectional lasing from spiral-shaped microcavities as follows: Given the same total cavity losses (the same Q’s) between CCW and CW traveling-wave modes, we expect CCW traveling-wave modes see larger distributed losses along the cavity rim and/or diffraction/scattering losses at the notch junction than CW traveling-wave modes in order to balance CW traveling-wave mode direct out-coupling losses (see Sec. 3). As scattering along the cavity rim and diffraction/scattering at the notch junction can result in cross-coupling between CCW and CW traveling-wave modes, it is thus conceivable that the CCW-to-CW cross-coupling can exceed the CW-to-CCW cross-coupling, resulting in selective feedback to unidirectional lasing in CW traveling-wave modes that enable out-coupling via the spiral notch. Our proposal is much similar to the conventional selective feedback approach to obtaining unidirectionality from ring lasers . Further experiments on silicon waveguide-coupled spiral-shaped microdisks exploiting silicon Raman gain  are in progress in order to test our hypothesis.
Our experiments and numerical simulations of a waveguide-coupled spiral-shaped microdisk resonator have revealed reciprocal throughput-port transmissions between the evanescently in-coupled clockwise (CW) and counterclockwise (CCW) traveling-waves, and also between the evanescently in-coupled CW traveling-wave drop-port transmission and the non-evanescently in-coupled CCW traveling-wave throughput-port transmission. The reciprocal transmissions suggest that CW and CCW traveling-wave modes see the same cavity Q’s, and thus reciprocal total cavity losses. This observation is, however, inconsistent with the general belief that CW and CCW traveling-wave modes in a spiral-shaped microdisk see different total cavity losses.
Our linearly-scanned out-of-plane scattering measurements near the spiral notch junction and the evanescent-coupling region have also revealed no clear evidence on CW and CCW traveling-wave modes having different total cavity losses. Nonetheless, our scattering measurements have indicated asymmetric modal distributions that are consistent with our simulated cavity internal-field spectra near the notch junction and the evanescent-coupling region. Furthermore, our simulations have revealed three-bounce and whispering-gallery like traveling-wave modes that depend on the sense of the lightwave circulations and the input/output-coupling mechanisms.
We therefore conclude that spiral-shaped microdisk resonators by their cavity-shape chirality result in asymmetric modal distributions that depend on the sense of the lightwave circulations and the input/output-coupling mechanisms, yet support the same total cavity losses (same Q) for CW and CCW traveling-wave modes, and thereby preserve reciprocal transmissions.
Spiral-shaped microdisks with non-evanescent resonance coupling points to a possible paradigm shift in microresonator-based device designs from conventional microresonators that impose evanescent coupling. One implication is that spiral-shaped microdisk silicon Raman lasers can be non-evanescently pumped through the spiral notch from a seamlessly butt-coupled waveguide, and the unidirectional lasing emission can then be out-coupled evanescently or non-evanescently .
We thank Prof. Richard K. Chang of Yale University for his foresight on spiral-shaped microresonators; Prof. Vahid Sandoghdar of Eldgenössische Technische Hochschule (ETH) Zürich for his suggestion on probing the out-of-plane scattering by a scanning fiber. X. Luo acknowledges the fellowship support from the NANO program of HKUST. This work was substantially supported by a grant from the Research Grants Council of The Hong Kong Special Administrative Region, China (Project No. 618506).
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