Fig. 1: Problems in dynamic phase modulation and the usage of avoided crossing as a solution.
From: Full 2π tunable phase modulation using avoided crossing of resonances
a The trajectories of the complex reflection coefficient for under-coupled and over-coupled resonances at the operating frequency ω0 as the single control parameter α is varied (color change from green to blue). Over-coupling is critical for modulating phase with a large coverage, as well as maintaining uniform amplitude. Over-coupling alone however cannot close the 0–2π loop. b The reflectance (top) and the phase (bottom) distributions of a single resonance as the control parameter is varied from α1 through αres to α2. The trade-off due to the interdependence between the resonance shift and the spectral width results in a subpar figure of merit ∆ω/FWHM (full width at half maximum) insufficient for a complete 0–2π range. αres corresponds to the control parameter value that results in the resonance frequency coinciding with ω0. c The trajectory of the complex reflection coefficient when an avoided crossing is used. Use of an avoided crossing circumvents the trade-off issue, closes the phase loop and modulates phase with uniform amplitude with an upper bound of 4π. d The mode dynamics of the avoided crossing. The original resonance frequencies without the avoided crossing are shown with dashed lines, where resonance (i) is over-coupled and spectrally narrow while resonance (ii) has high resonance frequency tunability. The resonance frequencies of the coupled modes after the avoided crossing occurs are shown with solid red lines. The horizontal dashed line represents the operating frequency ω0. α1’ and α2’ are the values of the control parameter where the frequency of the coupled resonances coincide with ω0.
