We demonstrate that ultrashort optical pulses propagating in a nonlinear dispersive medium are naturally described through incorporation of analytic signal for the electric field. To this end a second-order nonlinear wave equation is first simplified using a unidirectional approximation. Then the analytic signal is introduced, and all nonresonant nonlinear terms are eliminated. The derived propagation equation accounts for arbitrary dispersion, resonant four-wave mixing processes, weak absorption, and arbitrary pulse duration. The model applies to the complex electric field and is independent of the slowly varying envelope approximation. Still the derived propagation equation posses universal structure of the generalized nonlinear Schr？dinger equation (NSE). In particular, it can be solved numerically with only small changes of the standard split-step solver or more complicated spectral algorithms for NSE. We present exemplary numerical solutions describing supercontinuum generation with an ultrashort optical pulse. 1. Introduction Complex envelope adequately describes linear and nonlinear propagation of a wave packet with many field cycles [1]. A slowly varying envelope approximation (SVEA) reduces the full set of Maxwell equations for the pulse field to a much more simple first-order nonlinear Schr？dinger equation (NSE) for the complex envelope [2–4]. On the other hand, SVEA lacks precision when the relevant time scales are comparable to a single cycle period. Nonenvelope pulse propagation regimes include self-focusing [5, 6], optical shocks [7, 8], supercontinuum (SC) generation [9], and dynamics of ultrashort pulses [10–15]. In such situations NSE should be replaced by a more general propagation model. Several simplified unidirectional propagation equations have been derived for special dispersion profiles. Such models do not use the pulse envelope and apply directly to the pulse field (see [16–20] and a review paper [21]). For a general dispersion profile, pulse propagation is commonly described by a generalized NSE [2, 4] in which a polynomial approximation of dispersion in the frequency domain is used. An arbitrary dispersion is then accounted for by a local dispersion operator in the time domain. To resolve convergence problems [22] also rational approximations and nonlocal dispersion operators may be considered [18, 23]. The nonlinear term in the generalized NSE is also modified to capture an arbitrary pulse duration [8, 24], Raman scattering [25, 26], and diffraction effect [24, 27, 28]. Being an envelope model, the generalized NSE was
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