Electrode Optimization for on Chip Resonant Electro-Optic Frequency Combs

The generation of electro-optic (EO) combs in Thin-Film Lithium Niobate (TFLN) based photonic integrated circuits (PICs) has been demonstrated in several publications [1]–[4]. Despite a solid understanding of the design of the optical resonator [3], the optimization of the Radio-Frequency (RF) electrodes is less explored. Here we inves-tigate how electrodes lengths and feeding points influence the EO comb modulation efficiency. In fact, contrary to EO fiber cavities, PIC based EO combs are often operated generating RF standing (and not traveling) waves. This results in a non-trivial spatial and time dependency of the local phase shift experienced by light along the ring. Here, we report an analytical model, based on transmission line theory, that allows to simulate the EO comb conversion efficiency for different electrode length and feeding point. The model only requires parameters (RF losses, RF index, characteristic impedance) that can be either simulated from the cross-section or directly measured on the sample. Fig. 1 a), b) and c) show the simulated accumulated phase by the optical wave-packet propagating over one round-trip in the resonator. The <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex> axis corresponds to the electrodes length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{e}$</tex>, normalized to the minimal resonator's length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(L_{\min})$</tex> that has free-spectral-range (FSR) equal to the driving RF frequency (fixed). The insets show the three different RF feeding configurations studied. The dotted line corresponds to RF frequency <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(\nu_{m})$</tex> being an even multiple <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(k)$</tex> of the ring's FSR, while the solid one corresponds to odd multiples. Grey area represents allowed electrode length choice for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k=3$</tex>. From the model we infer that: it is generally advantageous to use longer rings driven at a higher multiple of their FSR, while for standing-wave configuration (b) and c)), a well-defined electrode lengths maximize the phase shift, and this is shorter than the ring length (e.g. for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k=3,4L_{e}/L_{\min}=2\rightarrow L_{e}=L_{\min}/2)$</tex>, for central RF feeding (configuration a)), driving the resonator at even multiples of its FSR always gives zero phase accumulated during one round-trip. The model was validated experimentally by measuring 5 sets of 5 rings with the same FSR but different electrodes length (Fig 1 g)). The samples have been fabricated at ETH Zurich following the process described in [4]. The measured modulation efficiencies <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\beta$</tex> defined as in [5], are extracted from the EO comb bandwidth (Fig. 1 e)) and plotted after renormalization in Fig. 1f). The error bars come from measurements of different samples. A good agreement is present with the analytical model (dashed lines). In conclusion: we have developed and experimentally validated an analytical model to predict the EO comb bandwidth for different feeding configurations, electrodes length and excitation regimes. The model helps in optimizing design of the RF co-planar waveguide, to increase conversion efficiency for a given excitation frequency and feeding scheme. This work is funded by the Swiss-National-Science-Fundation: project number 194693.