The Kramers-Kronig (KK) algorithm, useful for retrieving the phase of a spectrum based on the known spectral amplitude, is applied to reconstruct the impulse response of a diffusive medium. It is demonstrated by a simulation of a 1D scattering medium with realistic parameters that its impulse response can be generated from the KK method with high accuracy.

Reconstructing the impulse response of a diffusive medium with the Kramers-Kronig relations

We measure the sub-picosecond optical impulse response of a system consisting of a varying 1D diffusive medium and a stationary hidden object. It is shown that by averaging the temporal optical field response of a diffusive medium (as opposed to the optical intensity response) the signal-to-noise ratio of the object’s reflection can be improved considerably. The Spectral-Ballistic-Imaging technique is used to reconstruct the optical-field impulse response with a 200fs temporal resolution.

Quasi-ballistic imaging through a dynamic scattering medium with optical-field averaging using Spectral-Ballistic-Imaging

The differential multiply subtractive Kramers-Kronig (DMSKK) method is utilized for deriving the optical spectral response of a 1D scattering medium The technique is based on the multiply subtractive Kramers-Kronig technique, where the phase spectrum is reconstructed from the amplitude spectrum in a finite spectral range with the aid of one or more phase-anchoring values We employ a new phase-anchoring technique in the DMSKK method to partially mitigate the finite-range effects This method incorporates spectral ballistic imaging to anchor the phase difference (instead of the phase directly) at one or more reference wavelengths The simplicity of phase derivative measurements and its utilization in the DMSKK method are emphasized by a simple experiment

Spectral analysis of a one-dimensional scattering medium with the differential multiply subtractive Kramers-Kronig method

We suggest adopting an efficient digital signal processing algorithm, which includes several fast Fourier transforms, to efficiently reconstruct the impulse response of a diffusive medium from its amplitude spectrum. It is also demonstrated that the singularities, which appear in the phase spectrum reconstruction, can be easily eliminated through the implementation of at least two types of data padding.

Efficient impulse response reconstruction from the amplitude spectrum

We apply the multiply subtractive anchoring method for efficient phase spectrum retrieval, which is based on the fast Fourier transform and Lagrange polynomials. Because the polynomials eventually diverge, choosing the optimum anchoring points is crucial. It is demonstrated that, if more than two anchoring points are chosen, the algorithm’s performance can easily deteriorate.

Efficient phase spectrum reconstruction with the discrete multiply subtractive anchoring algorithm

We use the spectral ballistic imaging technique to measure the impulse response of a Fabry–Perot etalon with less than 0.2 ps temporal resolution. The results show excellent agreement with the theoretical predictions and negligible noise. Comparison to the Kramers–Kronig method along with its limitations is also presented.

D. Schermann

Papers

200 femtosecond impulse response of a Fabry–Pérot etalon with the spectral ballistic imaging technique