An introduction to nonlinear fiber optics and optical analogues to gravitational phenomena

The optical fiber is a revolutionary technology of the past century. It enables us to manipulate single modes in nonlinear interactions with precision at the quantum level without involved setups. This setting is useful in the field of analogue gravity (AG), where gravitational phenomena are investigated in accessible analogue lab setups. These lecture notes provide an account of this AG framework and applications. Although light in nonlinear dielectrics is discussed in textbooks, the involved modelling often includes many assumptions that are directed at optical communications, some of which are rarely detailed. Here, we provide a self-contained and sufficiently detailed description of the propagation of light in fibers, with a minimal set of assumptions, which is relevant in the context of AG. Starting with the structure of a step-index fiber, we derive linear-optics propagating modes and show that the transverse electric field of the fundamental mode is well approximated as linearly polarized and of a Gaussian profile. We then incorporate a cubic nonlinearity and derive a general wave envelope propagation equation. With further simplifying assumptions, we arrive at the famous nonlinear Schrödinger equation, which governs fundamental effects in nonlinear fibers, such as solitons. As a first application in AG, we show how intense light in the medium creates an effective background spacetime for probe light akin to the propagation of a scalar field in a black hole spacetime. We introduce optical horizons and particle production in this effective spacetime, giving rise to the optical Hawking effect. Furthermore, we discuss two related light emission mechanisms. Finally, we present a second optical analogue model for the oscillations of black holes, the quasinormal modes, which are important in the program of black hole spectroscopy.

💡 Research Summary

The manuscript presents a self‑contained pedagogical treatment of light propagation in step‑index optical fibers and shows how this platform can be used to emulate key gravitational phenomena such as event horizons, Hawking radiation, and black‑hole quasinormal modes (QNMs). Starting from Maxwell’s equations in a loss‑free, non‑magnetic dielectric, the authors first derive the linear wave equation and, after Fourier transformation, obtain the Helmholtz equation with a frequency‑dependent refractive index (n(\omega)=\sqrt{1+\tilde\chi^{(1)}(\omega)}). By separating variables in cylindrical coordinates and imposing the standard continuity conditions at the core‑cladding interface, they recover the familiar Bessel‑function solutions for the longitudinal electric and magnetic field components. The fundamental LP(_{01}) mode emerges as a linearly polarized, Gaussian‑like transverse profile, which is the basis for all subsequent nonlinear analysis.

The nonlinear section introduces the third‑order Kerr response (\mathbf{P}^{(3)}=\varepsilon_{0}\chi^{(3)}|\mathbf{E}|^{2}\mathbf{E}) and adopts a slowly‑varying envelope ansatz (E(z,t)=\frac12