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Journal of the Optical Society of America A

Analysis of the average photon path length between two consecutive reflections from the cavity walls

Lazo M. Manojlović

Published Dec 14, 2016• 0 citations
Publisher

Abstract

In this paper a general theoretical analysis of the average photon path length between two consecutive photon reflections from the cavity walls has been presented. The presented analysis relies only on the energy conservation law and the assumption of the uniform photon distribution within the cavity volume. The obtained general result coincides with the solution previously obtained but under more stringent conditions.

DOI
10.1364/josaa.34.000097

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In this paper a general theoretical analysis of the average photon path length between two consecutive photon reflections from the cavity walls has been presented. The presented analysis relies only on the energy conservation law and the assumption of the uniform photon distribution within the cavity volume. The obtained general result coincides with the solution previously obtained but under more stringent conditions.
The laws of reflection and refraction are typically derived using a dispersion relation that assumes that the angular frequency stays the same across a vacuum--dielectric interface and that the angular wave number increases. At the single photon level, this means that the total energy of the photon is conserved but the linear momentum is not. Here I derive the laws of reflection and refraction for a single photon using a dispersion relation that is consistent with the conservation of both total energy and linear momentum. To ensure that both conservation laws are upheld, the optical path length must be considered to be more fundamental than the geometrical path length when reckoning optical distances. As long as the medium is transparent, the optical path length traveled by a photon in a given duration of time is constant. By contrast, the geometrical path length traveled in a given duration of time is inversely proportional to the index of refraction. A photon propagating across a vacuum--dielectric interface is considered to be an indivisible package whose total energy (and angular frequency) and linear momentum (and angular wave number) remain constant, but whose velocity decreases with refractive index as a result of unspecified electromagnetic interactions.
So far quantum-mechanical laser theories used approaches based on different but equivalent formulations either relying on a density matrix description or on the Fokker-Planck equation [1], [2]. In all these descriptions an important approximation and assumption have been used since: The outcoupling of photons out of the laser resonator was modeled by a loss term in analogy to what an equivalent intra-cavity absorption of photons would cause to the laser quantum state, i.e., an uncorrelated loss of single photons, thereby incorporating the finite cavity Q-factor into the description of the laser. Although this procedure is consistent in terms of energy and average photon number it lacks the fact that a typical dielectric laser output mirror is not just a uncorrelated photon extracting device like an absorbing or scattering object. As the output coupler is a beam splitter it has a complex action on the incoming intra-cavity photon state. The photon state, therefore, reflected back into the laser resonator is not just shifted to a lower average photon number but is a rather complex combination of various number states that are binomially distributed. The goal of this paper is to investigate the consequences of these findings on the laser photon statistics of continuous-wave lasers, now described by the new master equation including the beam-splitter effect [3]. Therefrom, the detailed balance of probability fluxes (Fig. 1) obtains as and the results with the common laser theory based on absorption-like outcoupling used so far is compared.
Physical conservation laws, such as those of energy and momentum, are generally believed to hold exactly and locally in spacetime, including in quantum phenomena. Yet Aharonov, Popescu, and Rohrlich (APR) recently argued, on the basis of a thought experiment, that individual quantum events, unlike ensemble averages, may occasionally violate local conservation laws. Their argument relies on the wave phenomenon known as "superoscillations", which APR themselves discovered more than 30 years ago. Here we provide experimental evidence for such a violation. We extract photons from a small superoscillatory region near the core of an optical vortex and show that their mean transverse momentum is statistically incompatible with a general bound implied by local momentum conservation. The derivation of this bound requires only the theoretically well-supported assumption that the extraction mechanism does not alter the photons' mean transverse momentum. We also detect photons with high transverse momentum at a rate significantly exceeding that predicted by a model assuming local momentum conservation. Because this violation can be established only counterfactually and through postselection, it does not conflict with relativistic causality. Our results may represent the first example of a distinct form of quantum nonlocality that does not explicitly rely on entanglement.
We study domain-wall excitations in two-dimensional random-bond Ising spin systems on a square lattice with side length LLL, subject to two different continuous disorder distributions. In both cases an adjustable parameter allows tuning of the disorder so as to yield a transition from a spin-glass ordered ground state to a ferromagnetic ground state. We formulate an auxiliary graph-theoretical problem in which domain walls are given by undirected shortest paths with possibly negative distances. Due to the details of the mapping, standard shortest-path algorithms cannot be applied. To solve such shortest-path problems we have to apply minimum-weight perfect-matching algorithms. We first locate the critical values of the disorder parameters, where the ferromagnet to spin-glass transition occurs for the two types of the disorder. For certain values of the disorder parameters close to the respective critical point, we investigate the system size dependence of the width of the average domain-wall energy $(\ensuremath{\sim}{L}^{\ensuremath{\theta}})$ and the average domain-wall length $(\ensuremath{\sim}{L}^{{d}_{f}})$. Performing a finite-size scaling analysis for systems with a side length up to L=512L=512L=512, we find that both exponents remain constant in the spin-glass phase, i.e., $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.28$ and ${d}_{f}\ensuremath{\approx}1.275$. This is consistent with conformal field theory, where it seems to be possible to relate the exponents from the analysis of stochastic Loewner evolutions via ${d}_{f}\ensuremath{-}1=3/[4(3+\ensuremath{\theta})]$. Finally, we characterize the transition in terms of ferromagnetic clusters of spins that form, as one proceeds from spin-glass ordered to ferromagnetic ground states.