Closed-form expressions explicitly relating modulation bandwidth and active length in electro-optic traveling wave modulators are presented which fully account for skin-effect electrode loss and optical-electrical wave velocities mismatch. Four Optical modulators based on the electro-optic (EO) Pockels effect are basic building blocks of optical systems, with applications continuously growing up. As compared to lumped electrode configurations, bounded by a fundamental sensitivity-bandwidth limit, traveling wave (TW) configurations allow pushing the limit by optimization of the TW structure and provide extended modulation bandwidth with reduced drive power [1,2]. In EO-TW modulators (EO-TWM) the Pockels electrically induced optical phase shift accumulates with the copropagated distance (L) and therefore, the modulation drive voltage, usually quantified as the voltage required for a π phase shift, V π , is reduced proportionally to increases in L [3]. However, as it is well known, this comes at the expense of a corresponding reduction in operative bandwidth (B), which in typical EO-TWM based on coplanar waveguides (CPWs) over LiNbO 3 substrates stems mainly from the combined action of two basic mechanisms, namely the skin-effect electrode loss and the optical-electrical wave velocity mismatch. When trying to elucidate the expected B reduction that a specific L increase could have, a constant BL product rule proportional to the inverse of the velocity matching (VM) constant (ν) has been shown to govern the low-loss (LL) limit [4], while in the VM limit, a constant BL 2 rule proportional to the inverse of the square of the loss constant (α) has been found more appropriate [5]. To the best of our knowledge, no simple B-L rules have been derived for the intermediate ranges in which both α and ν are relevant, nor have the LL and VM limits been quantified in a general way.In this Letter we present closed-form expressions that fully account for the effects of skin-effect electrode loss and optical-electrical wave velocities mismatch and that explicitly and in a biunivocal way relate the operative bandwidth and the electrode length in EO-TWM. From these, four B-L trade-off figures of merit are identified with different validity ranges which are seen to depend both on the TW cross-sectional parameters (i.e., skineffect constant and optical-electrical velocity mismatch) and on the target B and L values.In order to analyze the B-L relationship, we begin by considering the electrical modulation frequency response of an impedance matched EO-TWM in the presence of skin-effect electrical loss in the conducting electrodes of length L and optical-electrical velocity mismatch [6] Mf e

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Compact closed-form mathematical expressions are presented for the effective permittivity, characteristic impedance and attenuation constant of coplanar waveguides with thick electrodes and high-frequency dispersion. Excellent agreements, for a wide range of frequencies, up to 100 GHz and for conductor thicknesses up to 40 m are found with full-wave finite-element results from commercial software (HFSS) in a margin of geometric parameter values and frequencies of interest for the design and optimisation of high-speed electro-optical modulators. The formulae are simple and specially tailored to the narrow gap widths and thick electrodes typical of electro-optical modulator applications. Other approaches to closed-form formulae for CPW parameters are also compared and analysed.Peer ReviewedPostprint (published version

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