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Physical Review A

Existence of a threshold for second-harmonic generation inside high-confinement microresonators as a consequence of the generalized creation and annihilation operators

Marc Collette, Serge Gauvin

Published Jul 29, 2022• 1 citation
Publisher

Abstract

This work presents a quantum theory of the nonlinear optical process of second-harmonic generation (SHG) in one-dimensional microresonators. More specifically, we show how the manipulation of vacuum field fluctuations in high-confinement systems, leading to a spectrally (and spatially) modulated commutation relation for the photon's generalized formulations of their creation and annihilation operators, deeply affects SHG behavior and gives rise to a threshold level. The two main effects the modulated commutator has on this optical process are an inhibition of the SHG process at low pumping level and a significant (cubic) amplification of the second-harmonic signal production rate once the threshold is overcome (finally reaching the usual quadratic dependence at sufficiently high pumping level). Our predictions, which represent a concrete picture of a fractional quantum system, could be used to probe vacuum field fluctuations present in high-confinement microresonators and emphasize the fundamental importance of vacuum field fluctuations.

DOI
10.1103/physreva.106.013721

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This work presents a quantum theory of the nonlinear optical process of second-harmonic generation (SHG) in one-dimensional microresonators. More specifically, we show how the manipulation of vacuum field fluctuations in high-confinement systems, leading to a spectrally (and spatially) modulated commutation relation for the photon's generalized formulations of their creation and annihilation operators, deeply affects SHG behavior and gives rise to a threshold level. The two main effects the modulated commutator has on this optical process are an inhibition of the SHG process at low pumping level and a significant (cubic) amplification of the second-harmonic signal production rate once the threshold is overcome (finally reaching the usual quadratic dependence at sufficiently high pumping level). Our predictions, which represent a concrete picture of a fractional quantum system, could be used to probe vacuum field fluctuations present in high-confinement microresonators and emphasize the fundamental importance of vacuum field fluctuations.
The interest toward strong electromagnetic confinement structures through the use of microcavities is of great interest, from both the fundamental understanding and the technological application sides. Here, we focus our attention to second order nonlinear processes inside such structures. From a basic point of view, electromagnetic confinement leads, physically, to the manipulation of vacuum field fluctuations and, formally, to an anomalous commutation relation [Ueda, M. and Imoto, N., "Anomalous commutation relation and modified spontanenous emission inside a microcavity," Phys. Rev. A, 50(1), 89-92 (1994)]. However, an attempt to probe the resulting anomaly with a beam splitter located inside a cavity is theoretically proved inadequate for this task. For this reason, parametric fluorescence (PF, parametric down conversion) and second harmonic generation (SHG) are considered as potential tools to probe vacuum field fluctuations in confinement structures. Indeed, it was recently showed that PF can be strongly intensified, by two orders of magnitude or more, when it occurs inside a high confinement (open) cavity. Therefore, PF could efficiently be used to probe the expected anomaly. Additionally, in consideration of the simplest possible experimental scheme to probe the anomaly and because it is generally admitted that SHG is the time reversal of degenerate parametric fluorescence, here our attention is also directed toward SHG. Formally, we show that the equations describing SHG and PF are not symmetrical regarding quantum noise. It turns out that, conversely to the case of PF, the intensification of vacuum field fluctuations tends to inhibit SHG. We interpret this result as due to the reduction of waves coherence induced by stronger quantum noise. In conclusion, albeit both processes could be used as probes to prove or disprove the realm of quantum anomalies, PF appears as a more convenient tool.
In this work we develop a quantum theory of second-order nonlinear optical processes such as parametric fluorescence and second-harmonic generation (SHG), generated by a strong electromagnetic field in an active medium placed in a microcavity. Fields are quantized and expanded in terms of a suitable set of cavity normal modes. In the first part of this work we consider a single many-level quantum system (an atom or molecule), which interacts with all the radiation field modes (spontaneous emission). We show how vacuum fluctuations affect both SHG and parametric processes. For SHG, we demonstrate that the presence of the microcavity allows the introduction of the concept of coherence length, even for a medium made of a single molecule. In the second part of this paper we discuss the case of a uniform distribution of emitting dipoles. For this configuration we calculate the differential extinction coefficient, and discuss the dependence of the emitting power on the microcavity's parameters. Finally we suggest the possibility of realizing a micro-optical parametric oscillator.
In this paper we study the quantum-theoretical description of second-harmonic generation (SHG) in optical waveguides. The interaction operator of three waveguide modes, by means of which the equations for spatially slowly varying annihilation and creation operators of the field strength amplitudes can be determined, is obtained by using the mode fields in the waveguides. The equations for spatially slowly varying annihilation and creation operators of the field are derived. We seek approximate solutions of the operator equations, and obtain the conversion efficiency of SHG.
We analyze squeezing of both the fundamental and harmonic fields undergoing traveling-wave, second-harmonic generation (SHG) in second-order (${\mathrm{\ensuremath{\chi}}}^{(2)}$) nonlinear media. We take into account depletion of the fundamental field as well as the phase mismatch between the fundamental and harmonic fields. The behavior of the quantum noises on the propagating fields is studied by linearizing the nonlinear operator equations around the mean-field values. We first consider the degenerate case that is applicable to type-I phase-matching geometries, obtaining expressions for squeezing in both the fundamental and harmonic fields under the conditions of perfect phase matching and large phase mismatch. We show that in the case of a large phase mismatch, the intensity-dependent self-phase shift of the fundamental field, arising due to cascading of the ${\mathrm{\ensuremath{\chi}}}^{(2)}$ nonlinearity, is responsible for the squeezing generation. We also numerically solve the linearized quadrature-operator equations together with the nonlinear mean-field equations. We find that in the case of a finite phase mismatch the harmonic field can be highly squeezed. This is in contrast to the perfectly phase-matched case where the maximum squeezing is limited to 50%. Finally, we analyze the nondegenerate case that applies to type-II phase-matching geometries. Here we show that the commonly used type-II phase-matched SHG process with the input harmonic field in the vacuum state is equivalent to a type-I SHG process in parallel with a degenerate optical-parametric process. The latter causes squeezing in the mode that is polarized orthogonal to the fundamental beam. In the perfect phase-matching case, the squeezing in the orthogonally polarized mode follows the simple expression S=1-\ensuremath{\gamma}, where \ensuremath{\gamma} is the harmonic conversion frequency.
We investigate the spatial behavior of nonclassical light produced by type I second-harmonic generation in the traveling-wave configuration. An input-output transformation for the system of fundamental and second-harmonic field is derived in the framework of a linearization approach and used to investigate the properties of an optical system, which consists of a crystal with a quadratic nonlinearity pumped at frequency $\ensuremath{\omega}$ and enclosed in a two-lens telescopic system. If a faint input image at frequency $2\ensuremath{\omega}$ is injected into this device, for a sufficiently large interaction length, the output displays a pair of symmetric amplified versions of the input image at both fundamental and second-harmonic frequency. The analysis of the quantum fluctuations in the output images shows that under certain conditions, this optical device operates noiselessly with respect to the output at frequency $\ensuremath{\omega},$ whereas the output images at frequency $2\ensuremath{\omega}$ are affected by a slight degradation of the signal-to-noise ratio.