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

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.

Figure 1

Illustration of a one-dimensional universe conceived as a lossless closed cavity, bounded by perfectly conducting walls ($R=1$ at $z=-L$ and $z=D$) including a second-order nonlinear resonator. The pump wave originates from the rest of the universe, between $z=0$ and $z=D$. In the event of a high-finesse resonator, the semireflecting mirror at $z=0$ is assumed to have a reflectance close to unity. The operators $\widehat{a}$ represent the annihilation operators for each electric field component $E$. The subscripts “in” and “ex” refer to the resonator and the rest of the universe, respectively, while the superscripts “-” and “$+$” refer to the propagation direction.

Figure 2

Examples of the modulation function $\mathrm{\Lambda }$ for various confinement levels, which are determined by the value of $R$ ($\omega$ is expressed in units of $\pi c/L$). The ${\mathrm{\Lambda }}_{\max }$ value is calculated from Eq. (5).