[Paper Review] Experimental observation of fractal modes in unstable optical resonators

Javier A. Loaiza, E. R. Eliel+1 more|ArXiv.org|Apr 11, 2003

Photonic and Optical Devices2 references3 citations

TL;DR

This paper experimentally demonstrates the fractal nature of eigenmodes in unstable optical cavities using cavity ring-down spectroscopy, revealing a fractal dimension of 3.01 ± 0.04—confirming theoretical predictions that such modes exhibit maximum roughness. The result validates wave-optical models of open systems and suggests implications for laser design and matter-wave analogs.

ABSTRACT

We use a spatially resolved cavity ring-down technique to show that the 2D eigenmode of an unstable optical cavity has a fractal pattern, i.e. it looks the same at different length scales. In agreement with theory, we find that this pattern has the maximum conceivable roughness, i.e., its fractal dimension is 3.01 plus\minus 0.04. This insight in the nature of unstable cavity eigenmodes may lead to better understanding of wave dynamics in open systems, for both light and matter waves.

Motivation & Objective

• To experimentally verify the theoretical prediction that eigenmodes in unstable optical cavities exhibit fractal structure with maximum roughness.
• To test whether the fractal dimension of these modes reaches the theoretical upper limit of 3.0 under asymptotic conditions.
• To determine if the fractal nature of the mode is robust against variations in initial excitation conditions.
• To establish a link between the spatial power spectrum of the eigenmode and its fractal dimension via Fourier analysis.
• To explore the implications of fractal eigenmodes for quantum noise and device design in microlasers and matter-wave systems.

Proposed method

• A spatially resolved cavity ring-down technique was used to measure the transverse intensity profile of the lowest-loss eigenmode in an empty unstable cavity.
• Laser light with an arbitrary transverse profile was injected, then rapidly switched off, allowing the cavity to decay and the mode to emerge.
• The intensity distribution was recorded after a delay corresponding to multiple round trips to isolate the fundamental eigenmode.
• 1D cuts of the asymptotic intensity profile were Fourier-transformed to analyze the spatial power spectrum.
• The power spectrum was analyzed in logarithmic frequency bands to identify power-law behavior, with low- and high-frequency components separated.
• The fractal dimension was extracted from the high-frequency power-law slope using the relation D = D_cut + 1, where D_cut is the fractal dimension of the 1D cut.

Experimental results

Research questions

• RQ1 Does the eigenmode of an unstable optical cavity exhibit self-similar, fractal structure as predicted by wave-optical theory?
• RQ2 What is the measured fractal dimension of the 2D transverse intensity profile of the fundamental eigenmode?
• RQ3 Is the observed fractal structure robust to changes in the initial excitation profile?
• RQ4 Can the spatial power spectrum of the eigenmode be described by a power law, and does its slope correspond to a fractal dimension of approximately 3?
• RQ5 To what extent does the fractal nature of the mode persist despite diffraction and edge effects?

Key findings

• The fractal dimension of the 2D eigenmode was measured as 3.01 ± 0.04, in excellent agreement with the theoretical prediction of D = 3 for the asymptotic limit.
• The 1D cut of the eigenmode exhibited a power-law spatial power spectrum with a slope b = 0.98 ± 0.04, corresponding to a fractal dimension D_cut = 2.01 ± 0.04.
• The phase of the Fourier components of the 1D cut was found to be random, a key signature of fractal behavior.
• The asymptotic intensity pattern remained nearly identical regardless of whether the input beam was focused or diffused, indicating robust selection of a single eigenmode.
• The experimental results confirm that the eigenmode has the maximum possible roughness, as defined by a fractal dimension of 3.01, despite diffraction effects.
• The findings validate the analytical model of Berry et al. (2001) for fractal eigenmodes in unstable cavities, particularly in the limit of large Fresnel number.