[Paper Review] Shannon entropy as a measure of structure in optical beams

Maria Solyanik‐Gorgone, Jiachi Ye+2 more|arXiv (Cornell University)|Nov 16, 2020

Orbital Angular Momentum in Optics4 citations

TL;DR

This paper proposes a novel method to quantify classical Shannon information in optical beams by defining it through the Wigner distribution function, which captures structure in both real and reciprocal space while respecting the Heisenberg uncertainty principle. The approach enables measurement of information content as a function of beam complexity, experimentally validated for Gaussian, Hermite-Gaussian, and Laguerre-Gaussian modes, offering a general framework independent of communication alphabet or topology.

ABSTRACT

While information is ubiquitously generated, shared, and analyzed in a modern-day life, there is still some controversy around the ways to asses the amount and quality of information inside a noisy channel. A number of theoretical approaches based on, e.g., conditional Shannon entropy and Fisher information have been developed, along with some experimental validations. Some of these approaches are limited to a certain alphabet, while others tend to fall short when considering optical beams with non-trivial wavefront topology, such as orbital momentum laser modes. Here, we introduce the concept of expressing information as a measure of structure in a laser beam. We propose a new definition of classical Shannon information via the Wigner distribution function, while respecting the Heisenberg inequality. Following this, we calculate the amount of information in a Gaussian, Hermite-Gaussian, and Laguerre-Gaussian laser modes in juxtaposition and experimentally validate it by reconstruction of the Wigner distribution function from the intensity distribution of structured laser beams. We experimentally demonstrate measuring structure of the laser beams in singular optics to assess the amount of contained information. Given the generality of this approach of defining information via analyzing the beam complexity is applicable to laser modes of any topology that can be described by 'well-behaved' functions. Classical Shannon information defined in this way is detached from a particular alphabet, i.e. communication scheme, and scales with the structural complexity of the system. Such a synergy between the Wigner distribution function encompassing the information in both real and reciprocal space, and information being a measure of disorder, can contribute into future coherent detection algorithms and remote sensing.

Motivation & Objective

• To address the lack of a general, alphabet-independent measure of information in optical beams with complex wavefronts.
• To overcome limitations of existing information measures that fail for beams with non-trivial topology, such as orbital angular momentum modes.
• To define classical Shannon information in terms of structural complexity using the Wigner distribution function.
• To experimentally validate the proposed information measure through reconstruction of the Wigner distribution from intensity measurements.
• To establish a scalable, general framework applicable to any laser mode described by well-behaved functions.

Proposed method

• Defining classical Shannon information via the Wigner distribution function to simultaneously represent position and momentum (spatial frequency) content.
• Applying the Heisenberg uncertainty principle as a constraint to ensure physical consistency of the information measure.
• Calculating the Shannon entropy of the Wigner distribution to quantify the information content as a measure of structural complexity.
• Using experimental intensity measurements to reconstruct the Wigner distribution function for various laser modes.
• Comparing information content across Gaussian, Hermite-Gaussian, and Laguerre-Gaussian modes to assess structural differences.
• Validating the theoretical framework by demonstrating consistency between reconstructed Wigner functions and predicted information values.

Experimental results

Research questions

• RQ1 How can classical Shannon information be defined in optical beams without relying on a specific communication alphabet or modulation scheme?
• RQ2 To what extent does the Wigner distribution function enable a physically consistent measure of information that respects quantum mechanical limits like the Heisenberg uncertainty principle?
• RQ3 How does the information content correlate with the structural complexity of laser beams, particularly in modes with non-trivial wavefronts such as Laguerre-Gaussian beams?
• RQ4 Can the Wigner distribution be reliably reconstructed from intensity-only measurements to enable experimental validation of the information measure?
• RQ5 What is the relationship between beam topology (e.g., orbital angular momentum) and the resulting information content as defined by this new framework?

Key findings

• The proposed information measure based on the Wigner distribution function successfully quantifies the structural complexity of optical beams without dependence on a specific alphabet or modulation scheme.
• The method respects the Heisenberg uncertainty principle, ensuring physical consistency in the information estimation across all beam types.
• Experimental reconstruction of the Wigner distribution from intensity data confirmed the theoretical predictions for Gaussian, Hermite-Gaussian, and Laguerre-Gaussian modes.
• Laguerre-Gaussian beams, which possess non-trivial wavefront topology, were shown to carry higher information content due to their increased structural complexity.
• The framework is general and applicable to any laser mode described by well-behaved functions, regardless of topology or wavefront structure.
• The synergy between the Wigner distribution and Shannon entropy enables a robust, physically grounded measure of information that can inform future coherent detection and remote sensing systems.