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Using Microwave and Macroscopic Samples of Dielectric Solids to Study the Photonic Properties of Disordered Photonic Bandgap Materials

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Sampling distributions of random electromagnetic fields in mesoscopic or dynamical systems.

¹Time, Quantum and Electromagnetics Division, National Physical Laboratory, Teddington, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|November 13, 2009

Summary

This study derives sampling probability density functions (pdfs) for random electromagnetic fields. Results reveal distinct pdfs for dimensioned versus standardized fields, impacting statistical uncertainty estimations in small-sample scenarios.

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Area of Science:

Physics
Electromagnetism
Statistical Mechanics

Background:

Understanding random electromagnetic fields is crucial for analyzing mesoscopic systems and signal detection.
Existing models often rely on asymptotic approximations that may not hold for small sample sizes (degrees of freedom, ν < 40).

Purpose of the Study:

To derive the sampling probability density functions (pdfs) for localized random electromagnetic fields, their amplitudes, and intensities.
To analyze how field normalization and standardization affect sampling distributions and statistical uncertainties.
To provide accurate statistical estimation methods for small sample sizes in electromagnetic environments.

Main Methods:

Derivation of sampling pdfs for dimensioned and standardized electromagnetic fields (electric field, amplitude, intensity).
Analysis of coherent and incoherent detection methods for Cartesian, planar, and full-vectorial fields.
Comparison of derived pdfs (Bessel K, Student t, Fisher-Snedecor F) with asymptotic distributions (Gauss, chi-squared).

Main Results:

Dimensioned fields exhibit Bessel K sampling pdfs, differing from asymptotic Gauss and chi-squared distributions for small ν.
Standardized fields yield Student t, Fisher-Snedecor F, and root-F pdfs with heavier tails than Bessel K pdfs.
Statistical uncertainties for dimensionless quantities are overestimated compared to dimensioned quantities in classical small-sample theory.

Conclusions:

The functional form of the sampling pdf critically depends on whether the random variable is dimensioned or standardized.
Accurate statistical estimations require using appropriate pdfs based on the field's representation (dimensioned vs. standardized).
This work provides a more precise framework for analyzing electromagnetic field statistics, especially in small-sample regimes.