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

TL;DR

The paper develops a self-contained framework showing how nonlinear fiber optics can simulate aspects of gravitational physics, notably optical horizons and Hawking-like radiation, through a Kerr-nonlinear, dispersive medium. Beginning with linear Maxwell theory in a step-index fiber, it derives guided modes and a nonlinear extension that yields the generalized NLSE, whose solitons establish a moving background metric for probe light. It then constructs a robust analogue of event horizons via horizon conditions, analyzes mode mixing and quantum particle creation using a Hopfield-type dispersion for the medium, and discusses the optical analogues of black-hole ringdown through soliton perturbations and quasinormal modes. Together, these sections illustrate how optical systems can model curved spacetimes, enabling experimental-accessible investigations of quantum-gravity phenomena and beyond. The work also highlights resonant radiation and negative-frequency modes as additional, observable analogue effects with potential quantum signatures and practical photonic applications.

Abstract

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.

Figures (17)

• Figure 1: Left panel: Cross-section of a step-index fiber, consisting of a core and a cladding surrounding it. Right panel: Index of refraction, modeled as a step-function with a value $n_1$ in the core and $n_2<n_1$ in the cladding. Typical value of the difference are $n_1-n_2\sim \mathcal{O}(10^{-3})$.
• Figure 2: Vector plot of the electric field, for $m=1$, $\ell=1$ (mode $HE_{11}$), on the transverse $x-y$ plane of a typical fiber. We use the following values: $a=5\text{$\,\mu$m}$, $\lambda_o=1\text{$\,\mu$m}$, (resulting in $\omega=1.884 \,\text{fs}^{-1}$), and fused silica as the medium for the cladding, resulting in $n_2=1.440$, and germanium doped fused silica for the core, resulting in $n_1=1.445$.
• Figure 3: $HE_{11}$ mode. Upper-left panel: 3D plot of the $E_x(x,y,0)$ component. Upper-right panel: Cross-section of the $E_x$ component with the $y=0$ and $z=0$ planes and comparison with the Gaussian function $|\tilde{E}_x(x,0,0)|/|\tilde{E}_x(0,0,0)|=\exp(-x^2/w^2)$. The best-fit parameter is $w=0.93 a$. Bottom panel: Cross-section of $E_x(x)$ with the $y=0$ and $z=0$ planes for different vacuum wavelengths. The vertical axis is expressed on the same units as the freely chosen amplitude $\tilde{E}_{o1}$. The Gaussian character is preserved as the frequency (or, equivalently, vacuum wavelength) varies, but with a different Gaussian peak and effective width. We use the same model parameters as in Fig.\ref{['fig.vector.plot']}.
• Figure 4: Soliton formation: The effects of group-velocity dispersion and nonlinearity conspire to generate opposite chirps that cancel to produce an unchirped stable pulse.
• Figure 5: Rays interacting with a soliton: (a) Spatial domain: The grey shading represents soliton intensity for illustration. Rays that experience a black hole (red) and a white hole (blue) horizon at the front ($\tau<0$) and the back ($\tau>0$) of the soliton, respectively. (b) Frequency domain: Rays of (a) are blue shifting at the white hole (blue) and red shifting at the black hole (red). (c-d) Spatial and frequency domain interactions for (i) the typical white hole interaction from (a,b), (ii) a faster ray entering further into the soliton, (iii) a ray riding on top of the soliton, (iv) a fast ray passing through the soliton. All rays propagate under normal dispersion. $\omega_{\text{gvm}}$: a frequency which is group velocity matched to the soliton.