Research · Publication & note
Oscillatory Vertical Coupling between a Whispering-Gallery Resonator and a Bus Waveguide
Physical Review Letters · Journal article · 15 April 2013
Abstract
We report on a theoretical and experimental study of the optical coupling between a whispering-gallery type resonator and a waveguide lying on different planes. In contrast to the usual in-plane geometry, the present vertical one is characterized by an oscillatory behavior of the effective coupling as a function of the vertical gap. This behavior manifests itself as oscillations in both the resonance peak waveguide transmission and the mode quality factor. An analytical description based on coupled-mode theory and a two-port beam-splitter model of the waveguide-resonator vertical coupling is developed for arbitrary phase-matching conditions and is successfully used to interpret the experimental observations.
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Figures
Figure 1. In-plane and vertical resonator-waveguide coupling geometries, an optical image of the vertically coupled device, and the corresponding separation profiles.
From M. Ghulinyan et al., Physical Review Letters 110, 163901 (2013), DOI 10.1103/PhysRevLett.110.163901. © 2013 American Physical Society.
Figure 2. Coupled-mode description of the vertical geometry, including the flat-zone length, beam-splitter model, resonant transmission and quality factor.
From M. Ghulinyan et al., Physical Review Letters 110, 163901 (2013), DOI 10.1103/PhysRevLett.110.163901. © 2013 American Physical Society.
Figure 3. Experimental transmission spectrum for a device with a nominal vertical gap of 814 nm, compared with the theoretical response of two radial-mode families.
From M. Ghulinyan et al., Physical Review Letters 110, 163901 (2013), DOI 10.1103/PhysRevLett.110.163901. © 2013 American Physical Society.
Figure 4. Measured and calculated transmission and quality factor as functions of the vertical gap, with representative resonance spectra across the coupling regimes.
From M. Ghulinyan et al., Physical Review Letters 110, 163901 (2013), DOI 10.1103/PhysRevLett.110.163901. © 2013 American Physical Society.
Research fields
Top-down
The resonator and bus waveguide are fabricated in separate vertical planes, creating an extended co-propagation region instead of a local side-coupling point. Film thickness and gap set the phase accumulated between the two guided fields.
Simulations & fits
Coupled-mode theory and a two-port beam-splitter model predict that the net transfer oscillates with vertical separation. The same description accounts for periodic changes in transmission depth and loaded quality factor for different radial mode families.
Characterization
Spectra from devices spanning nominal coupling gaps resolve alternating over-, critical- and under-coupled conditions. Comparing broad and narrow radial families confirms that the oscillation follows their distinct propagation constants rather than a monotonic evanescent-gap law.
RESEARCH TOPICS
CouplingIn a vertical geometry the guide and resonator run parallel over an extended interaction length rather than touching at a single local point. The coupling region can therefore be modelled as two evanescently coupled waveguides followed by a two-port beam splitter. As the vertical gap changes, the coupled supermodes accumulate different phases before recombining. The effective transfer consequently oscillates between undercoupling, critical coupling and overcoupling instead of increasing monotonically as it does for a conventional lateral gap.
InterferenceThe oscillation is a phase-matching effect. At 1.58 μm the fabricated guide and cavity modes have a finite mismatch of about 0.142 μm⁻¹, so both separation and interaction length determine the phase accumulated by the two supermodes. Interferometric gap measurements with roughly ±30 nm accuracy let the theoretical curve be compared directly with experiment. Resonance depth and linewidth oscillate together, confirming that the repeated critical-coupling points arise from coherent exchange rather than fabrication scatter.
MultimodeThe measurement follows more than one radial family, and Figure 3 shows that the second family has its own gap-dependent transmission curve. Each family has a different propagation constant and overlap with the buried guide, so its oscillation period and critical gaps differ. This family dependence provides an additional test of the coupled-mode model and warns against assigning a single monotonic coupling coefficient to every resonance in a vertically integrated disk.
ResonatorsWhispering-gallery modes circulate around the planar resonator while the buried guide passes beneath a flattened portion of the rim. A cavity round-trip condition is combined with the extended coupler model to predict both transmission minima and loaded Q. The measured repeated changes between under- and overcoupling agree with that description. The result turns vertical separation into a precise design parameter, but also shows that choosing the smallest gap does not necessarily give the strongest loading.