Dispersion theory and phase retrieval of meromorphic total susceptibility

Abstract

Meromorphic susceptibility is a material parameter that appears usually in pumpprobe experiments, where so-called degenerate susceptibilities, as a function of angular frequency, describe the radiation and material interaction. Kramers-Kronig dispersion relations are invalid for meromorphic susceptibilities. Phase retrieval by the maximum-entropy model from the modulus of total susceptibility, with poles and zeros in the complex plane, is considered. Complex analysis is employed in the description of the nature of the meromorphic total susceptibility.

Kramers—Kronig Relations and Sum Rules in Nonlinear Optical Spectroscopy

“…6,7 The advantage of MEM relies on the fact that there is no need for data extrapolation beyond the measured spectral range, but it requires knowledge of the phase at a few discrete wavelengths that are called anchor points. MEM has been successfully applied for phase retrieval in linear optical spectroscopy, [47][48][49][50][51][52] in nonlinear spectroscopy of holomorphic susceptibilities, 53,54 and also for phase retrieval of meromorphic nonlinear susceptibility of homogenous media 55 and nanostructures. 46,56 It should be noted that there is no causality breaking with meromorphic nonlinear susceptibilities.…”

Model-independent maximum-entropy method for the analysis of sum-frequency vibrational spectroscopy

“…The error e m is a zero-mean random variable, and the a k denotes the same coefficients as those in Eq. (5). Let the z transform of f n and f nϪi be F(z) and z Ϫi F(z), so that the z transform of Eq.…”

“…Recently, a phase-retrieval procedure based on the maximum-entropy principle has been proposed by Vartiainen et al [1][2][3][4] and Peiponen. 5,6 The maximumentropy phase-retrieval procedure (MEPRP) is superior to that based on the Kramers-Kronig relations 7,8 in two aspects. First, the MEPRP is applicable to degenerate cases 3 such as second-harmonic generation and degenerate four-wave mixing, for which the Kramers-Kronig relations do not hold.…”

Dispersion theory of effective meromorphic nonlinear susceptibilities of nanocomposites

“…However, the assumption of holomorphicity of the effective nonlinear susceptibility of nanocomposites, crucial for the validity of the Kramers-Kronig relations, is no longer valid for χ (3) eff (ω; ω, ω, −ω). We have recently studied a simple model for the total meromorphic susceptibility of two-level atoms, and suggested the solving of the phaseretrieval problem by means of an analysis based on the maximum-entropy model [17]. It turns out that χ (3) eff (ω; ω, ω, −ω) is also a meromorphic function in the complex plane.…”

“…Finally we remark that, in principle, the existence of both poles and zeros can be established and their numbers can be estimated using the argument theory of complex analysis. Another method would be to apply Jensen's formula for meromorphic functions [17].…”