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arXiv:2601.14645 · Physical Review A

Paraxial diffusion-field retrieval. II. Fokker-Planck generalization of the transport-of-intensity equation

Anonymous, Kaye Susannah Morgan

Published Aug 31, 2026• 1 citation
arXivHTMLPDF

Abstract

The transport-of-intensity equation (TIE), namely the continuity equation associated with a coherent paraxial optical wavefield, is widely used for phase retrieval. It is a second-order partial differential equation which may be solved for the phase of a coherent paraxial field such as a monochromatic scalar optical beam, given the intensity and longitudinal intensity derivative in a plane perpendicular to the optical axis. We show how the coherent flow associated with the TIE may be augmented by a diffusive flow associated with a scalar or tensor diffusion field. Such diffusive flow can arise via scattering from unresolved spatially random microstructure in an illuminated sample, blurring effects of an extended chaotic source that illuminates the sample, the resolution-reducing effect of shot noise in detected intensity images of the sample, and the sharpening effect (negative diffusion) associated with scattering from sharp sample edges. Augmenting the TIE's modeling of coherent flow with a diffuse-flow channel leads to a Fokker-Planck extension to this equation. Two different augmentations are obtained, using several complementary derivations. The inverse problems of phase retrieval and diffusion-field retrieval are then considered, for defocus-based imaging and mask-based imaging. When symmetric overfocus and underfocus images are used for phase retrieval, the diffusive term drops out and our Fokker-Planck formalism implies that any ensuing TIE-based phase-retrieval method needs no modification in light of our formalism. However, the same focal-series dataset---typically an infocus image, a weakly overfocused image, and a weakly underfocused image---may also be employed to access the additional channel of information associated with the Fokker-Planck diffusion field. Our formalism is applicable to visible light, x-ray, electron, and neutron imaging.

DOI
10.1103/zsg4-f26q

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The transport-of-intensity equation (TIE), namely the continuity equation associated with a coherent paraxial optical wavefield, is widely used for phase retrieval. It is a second-order partial differential equation which may be solved for the phase of a coherent paraxial field such as a monochromatic scalar optical beam, given the intensity and longitudinal intensity derivative in a plane perpendicular to the optical axis. We show how the coherent flow associated with the TIE may be augmented by a diffusive flow associated with a scalar or tensor diffusion field. Such diffusive flow can arise via scattering from unresolved spatially random microstructure in an illuminated sample, blurring effects of an extended chaotic source that illuminates the sample, the resolution-reducing effect of shot noise in detected intensity images of the sample, and the sharpening effect (negative diffusion) associated with scattering from sharp sample edges. Augmenting the TIE's modeling of coherent flow with a diffuse-flow channel leads to a Fokker-Planck extension to this equation. Two different augmentations are obtained, using several complementary derivations. The inverse problems of phase retrieval and diffusion-field retrieval are then considered, for defocus-based imaging and mask-based imaging. When symmetric overfocus and underfocus images are used for phase retrieval, the diffusive term drops out and our Fokker-Planck formalism implies that any ensuing TIE-based phase-retrieval method needs no modification in light of our formalism. However, the same focal-series dataset---typically an infocus image, a weakly overfocused image, and a weakly underfocused image---may also be employed to access the additional channel of information associated with the Fokker-Planck diffusion field. Our formalism is applicable to visible light, x-ray, electron, and neutron imaging.
Quantum entanglement and squeezing have significantly improved phase estimation and imaging in interferometric settings beyond the classical limits. However, for a wide class of non-interferometric phase imaging/retrieval methods vastly used in the classical domain, e.g., ptychography and diffractive imaging, a demonstration of quantum advantage is still missing. Here, we fill this gap by exploiting entanglement to enhance imaging of a pure phase object in a non-interferometric setting, only measuring the phase effect on the free-propagating field. This method, based on the so-called "transport of intensity equation", is quantitative since it provides the absolute value of the phase without prior knowledge of the object and operates in wide-field mode, so it does not need time-consuming raster scanning. Moreover, it does not require spatial and temporal coherence of the incident light. Besides a general improvement of the image quality at a fixed number of photons irradiated through the object, resulting in better discrimination of small details, we demonstrate a clear reduction of the uncertainty in the quantitative phase estimation. Although we provide an experimental demonstration of a specific scheme in the visible spectrum, this research also paves the way for applications at different wavelengths, e.g., X-ray imaging, where reducing the photon dose is of utmost importance.
The phase problem can be considered as one of the cornerstones of quantum mechanics intimately connected to the detection process and the uncertainty relation. The latter impose fundamental limits on the manifold phase reconstruction schemes invented to date, in particular, at small magnitudes of the quantum wave. Here, we show that a rigorous solution of the transport of intensity reconstruction (TIE) scheme in terms of a linear elliptic partial differential equation for the phase provides reconstructions even in the presence of wave zeros if particular boundary conditions are given. We furthermore discuss how partial coherence hampers phase reconstruction and show that a modified version of the TIE reconstructs the curl-free current density at arbitrary (in)coherence. Our results open the way for TIE-based phase retrieval of arbitrary wave fields, eventually containing zeros such as phase vortices.
Quantum correlation, such as entanglement and squeezing have shown to improve phase estimation in interferometric setups on one side, and non-interferometric imaging scheme of amplitude object on the other. In the last case, quantum correlation among a pair of beams leads to a sub-shot-noise readout of the image intensity pattern, where weak details, otherwise hidden in the noise, can be appreciated. In this paper we propose a technique which exploits entanglement to enhance quantitative phase retrieval of an object in a non-interferometric setting, i.e only measuring the propagated intensity pattern after interaction with the object. The method exploits existing technology, it operates in wide field mode, so does not require time consuming raster scanning and can operate with small spatial coherence of the incident field. This protocol can find application in optical microscopy and X-ray imaging, reducing the photon dose necessary to achieve a fixed signal-to-noise ratio.
A generalized second-order Fokker-Planck equation is derived for a single two-level atom in a cavity driven by an external classical field without using system size expansion or any truncation. Effects of detuning, as well as atomic and cavity decays, are incorporated in this Fokker-Planck equation. This equation is used to study atomic inversion and the second-order intensity correlation function of light in the bad-cavity limit by adiabatically eliminating the field variables. Several novel features in atomic inversion and intensity correlations that arise due to atomic and cavity detuning are discussed. Curves are presented to illustrate the behavior.
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A transformation of the linear second-order Fokker-Planck equations occurring in quantum optics is presented which overcomes difficulties connected with negative definiteness of the diffusion matrices. This transformation is based on a dynamical ordering of quantum quasiprobability distribution functions. The ordering depends on the parameters of the dynamical problem under consideration. An algorithm leading to exact solutions of the dynamically ordered Fokker-Planck equations with arbitrary initial conditions is given, and simple criteria, how to recognize squeezed light in arbitrary ordering, are presented. The problem of Fokker-Planck equations with time-dependent diffusion is also investigated.