[Paper Review] Optical Production of the Husimi Function of Two Gaussian Functions

Jesús R. Moya-Cessa, L. R. Berriel‐Valdos+1 more|arXiv (Cornell University)|Jun 19, 2014

Photonic and Optical Devices3 references3 citations

TL;DR

This paper demonstrates the experimental optical generation of the Husimi function and squared Wigner function for a superposition of two Gaussian functions with different variances and a spatial separation. Using an astigmatic optical processor with a He-Ne laser, the authors detect intensity distributions in phase space that match numerically calculated results, successfully bridging quantum optical distribution functions with classical optics through direct experimental realization.

ABSTRACT

The intensity distribution of the Husimi function (HF) and the squared modulus of the Wigner function (WF) are detected in the phase space of an astigmatic optical processor. These results, obtained in the laboratory, are compared against numerical results generated by using analytical calculation for the HF and WF. The signal function is the superposition of two Gaussian functions with a separation between them, having the same amplitude but a different variance.

Motivation & Objective

• To experimentally demonstrate the optical production of the Husimi function, a quasi-probability distribution used in quantum mechanics, within a classical optical system.
• To compare experimentally measured intensity distributions of the Husimi function and squared Wigner function against analytically derived numerical results.
• To validate the use of astigmatic optical processors for detecting bilinear phase-space distributions in classical optics.
• To extend the applicability of quantum mechanical distribution functions—specifically the Husimi Q-function—to classical optical signal processing.

Proposed method

• An astigmatic optical processor composed of a cylindrical lens is used to perform a 1D Fourier transform on a signal, enabling detection of phase-space distributions.
• The input signal is a superposition of two Gaussian functions with equal amplitude, different variances (controlled by parameter b), and a separation of 2q₀, with q₀ = 1.5 in the experiment.
• Theoretical expressions for the Husimi function and squared Wigner function are derived analytically using the signal's wavefunction and the definitions of the Q- and Wigner functions.
• Photographic transparencies of the numerically calculated bilinear functions (Husimi and Wigner) are created via 170× photo-reduction on Kodak Technical Pan film and developed with D-19 solution.
• The optical setup uses a He-Ne laser (632.8 nm) to illuminate the transparencies, and the intensity distribution in the Fourier plane is recorded as the output.
• Experimental results are compared with numerically generated plots of the Husimi and Wigner functions to validate the optical implementation.

Experimental results

Research questions

• RQ1 Can the Husimi function of a two-Gaussian superposition be experimentally generated and detected in a classical optical system?
• RQ2 How do the intensity distributions of the Husimi function and squared Wigner function compare between experimental measurements and theoretical calculations?
• RQ3 To what extent do optical aberrations, film nonlinearity, and spatial coherence affect the fidelity of the detected phase-space distributions?
• RQ4 Can the astigmatic optical processor accurately reproduce the interference and spatial structure of non-classical-like distributions such as the Wigner function?
• RQ5 Does the optical detection of the Husimi function confirm its role as a directly measurable, non-negative phase-space distribution in classical optics?

Key findings

• The experimental intensity distribution of the Husimi function shows good qualitative agreement with the numerically calculated function, including the correct spatial separation and intensity modulation.
• The squared Wigner function exhibits negative values in the phase space, as expected from its quantum mechanical origin, and these features are experimentally observable through interference lobes.
• The interference term in the Wigner function, which causes negativity, is clearly visible in the experimental data as oscillatory structures in the intensity pattern.
• Despite noise from film development, vignetting, and speckle, the reconstructed images preserve the main features of the theoretical functions, especially the central and side lobes.
• Mechanical misalignment during film reduction causes displacement of secondary intensity lobes, confirming the sensitivity of the system to alignment precision.
• The successful detection of the Husimi function in the optical domain validates its use as a directly measurable, non-negative phase-space distribution in classical optics, bridging quantum and classical signal analysis.