Padé approximant for refractive index and nonlocal envelope equations

Padé approximation is superior to Taylor expansion when functions contain poles. This is especially important for response functions in complex frequency domain, where singularities are present and intimately related to resonances and absorption. Therefore, we introduce a rational Padé approximant for the complex medium refractive index n(ω)n(ω). The approximant is calculated using only local information of medium dispersion properties close to a carrier frequency ω0ω0. In return it typically offers an accurate global representation of medium dispersion and absorption. Moreover, the fulfillment of the causality principle and the Kramers–Kronig relation can be established. In practice, our results are relevant if n(ω)n(ω) is known only for ω≃ω0ω≃ω0 whereas optical field is spectrally broad such that (i) the resonance absorption becomes important and (ii) a traditional polynomial dispersion operator diverges and induces huge errors. As an exemplary application we use the approximant to derive a nonlocal envelope model for ultrashort pulses. The model provides a natural bridge between the commonly used local envelope equations and the most general non-envelope models operating directly with the electric field.