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Free-space Optical Diffraction and the Fisher Information

Jonathan M. Nichols, Frank Bucholtz

TL;DR

This work reframes free-space diffraction through the transport-of-intensity approach, showing that diffraction is intimately tied to the Fisher information (FI) of the beam’s transverse intensity, when intensity is treated as a photon-position probability density. As beams propagate, their intensity distributions flatten, and the associated FI with respect to propagation-distance parameters monotonically decreases, approaching zero in the far field. The authors illustrate this with three Gaussian-based examples—circularly symmetric Gaussian, Hermite-Gaussian HG10, and a bivariate Gaussian—deriving explicit FI expressions (e.g., ${\mathcal{F}}_{00}=4/\tilde{σ}_z^2$, ${\mathcal{F}}_{10}=8/\tilde{σ}_z^2$) and showing that FI decays at rates tied to beam shape. The results offer a principled FI-based criterion for assessing and designing diffraction properties in free-space optics, with potential implications for communications and directed energy applications.

Abstract

Using the transport-of-intensity approach for free-space optical propagation in the paraxial regime, we show that diffraction is fundamentally related to the Fisher information associated with models of the transverse intensity distribution. By interpreting intensity as a probability density, we show that a) free-space diffraction will always act to flatten the intensity distribution as the beam propagates, and consequently b) diffraction will monotonically minimize the Fisher Information with respect to any parameterization of the intensity distribution model that depends on propagation distance.

Free-space Optical Diffraction and the Fisher Information

TL;DR

This work reframes free-space diffraction through the transport-of-intensity approach, showing that diffraction is intimately tied to the Fisher information (FI) of the beam’s transverse intensity, when intensity is treated as a photon-position probability density. As beams propagate, their intensity distributions flatten, and the associated FI with respect to propagation-distance parameters monotonically decreases, approaching zero in the far field. The authors illustrate this with three Gaussian-based examples—circularly symmetric Gaussian, Hermite-Gaussian HG10, and a bivariate Gaussian—deriving explicit FI expressions (e.g., , ) and showing that FI decays at rates tied to beam shape. The results offer a principled FI-based criterion for assessing and designing diffraction properties in free-space optics, with potential implications for communications and directed energy applications.

Abstract

Using the transport-of-intensity approach for free-space optical propagation in the paraxial regime, we show that diffraction is fundamentally related to the Fisher information associated with models of the transverse intensity distribution. By interpreting intensity as a probability density, we show that a) free-space diffraction will always act to flatten the intensity distribution as the beam propagates, and consequently b) diffraction will monotonically minimize the Fisher Information with respect to any parameterization of the intensity distribution model that depends on propagation distance.
Paper Structure (13 sections, 61 equations, 9 figures, 1 table)

This paper contains 13 sections, 61 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Physics of Optical Beam Propagation
  3. The Fisher Information for Optical Intensity
  4. Optical Intensity as a Probability Model for Photons
  5. Fisher Information
  6. Diffraction as a Minimizer of Fisher Information
  7. Examples
  8. Uniformly-polarized, Circularly-symmetric Gaussian Beam
  9. Uniformly-polarized Hermite-Gaussian Beam
  10. Uniformly-polarized Bivariate Gaussian Beam
  11. Plane Wave and Paraboloidal Beam
  12. Discussion
  13. Summary

Figures (9)

  • Figure 1: The three beam examples studied in this Section. CG = circularly-symmetric Gaussian beam, HG$_{10}$ = the $\{1,0\}-$Hermite-Gaussian beam, BV = bivariate Gaussian beam.
  • Figure 2: Propagation of the intensity of a circularly-symmetric Gaussian beam at $\lambda=1.55\,\mu m$ for $\tilde{\sigma}_{0}=1.25\,mm$. (a) Intensity profiles in the $\tilde{x}\tilde{z}-$plane at $\tilde{y}=0$. (b) Intensity profiles in the $\tilde{y}\tilde{z}-$plane at $\tilde{x}=0$.
  • Figure 3: (a) Density plots of the intensity pdf for the circularly-symmetric Gaussian beam (\ref{['eq:BVDensity']}) at the indicated $z-$ positions for model parameters $\lambda=1.55\mu m, \sigma_{0}=1.25\,mm$. Here each intensity plot has been normalized to the value at the origin. (b) Corresponding acceleration vector field depicted as streamlines. Each plot has dimensions $-4000\,m/m\leq\tilde{x},\tilde{y}\leq4000\,m/m$
  • Figure 4: Propagation of a Hermite-Gaussian beam at $\lambda=1.55\,\mu m$ for $\sigma_{0}=1.25\,mm$. (a) Intensity profiles in the $\tilde{x}\tilde{z}-$plane at $\tilde{y}=0$. (b) Intensity profiles in the $\tilde{y}\tilde{z}-$plane at $\tilde{x}=0$. Note that in this case the initial intensity vanishes along the $\tilde{y}-$axis and the intensity remains zero along this axis as the beam propagates.
  • Figure 5: (a) Density plots of the intensity pdf for the 10-Hermite Gaussian beam (\ref{['eq:BVDensity']}) at the indicated $z-$ positions for model parameters $\lambda=1.55\mu m, \sigma_{0}=1.25\,mm$. Here each intensity plot has been normalized to the value at the origin. (b) Corresponding acceleration vector field depicted as streamlines. Each plot has dimensions $-4000\,m/m\leq\tilde{x},\tilde{y}\leq4000\,m/m$
  • ...and 4 more figures