Theory of Kerr frequency combs in Fabry-Perot resonators

We derive a spatiotemporal equation describing nonlinear optical dynamics in Fabry-Perot (FP) cavities containing a Kerr medium. This equation is an extension of the equation that describes dynamics in Kerr-nonlinear ring resonators, referred to as the Lugiato-Lefever equation (LLE) due to its formulation by Lugiato and Lefever in 1987. We use the equation to study the generation and properties of Kerr frequency combs in FP resonators. The derivation of the equation starts from the set of Maxwell-Bloch equations that govern the dynamics of the forward and backward propagating envelopes of the electric field coupled to the atomic polarization and population difference variables in a FP cavity. The final equation is formulated in terms of an auxiliary field $\psi (z,t)$ that evolves over a slow time $t$ on the domain $-L\le z\le L$ with periodic boundary conditions, where $L$ is the cavity length. We describe how the forward and backward propagating field envelopes can be obtained after solving the equation for $\psi$. This formulation makes the comparison between the FP and ring geometries straightforward. The FP equation includes an additional nonlinear integral term relative to the LLE for the ring cavity, with the effect that the value of the detuning parameter $\alpha$ of the ring LLE is increased by an amount equal to twice the spatial average of ${\left|\psi \right|}^2$. This feature establishes a general connection between the stationary phenomena in the two geometries. For the FP-LLE, we discuss the linear stability analysis of the flat stationary solutions, analytic approximations of solitons, Turing patterns, and nonstationary patterns. We note that Turing patterns with different numbers of rolls may exist for the same values of the system parameters. We then discuss some implications of the nonlinear integral term in the FP-LLE for the kind of experiments that have been conducted in Kerr-nonlinear ring resonators.

Figure 1

Experimental geometries for microresonator frequency combs. (a) The ring geometry. A pump laser is coupled into the ring resonator through a bus waveguide, and the intracavity intensity envelope (here a soliton pulse) is coupled out once per round trip. (b) The Fabry-Perot geometry. A pump laser is coupled in through a cavity mirror, and the intensity envelope is coupled out at each reflection. Not shown is the background standing wave in the Fabry-Perot cavity, which is made up of forward-propagating and backward-propagating field components.

Figure 2

The relationship between the FP-LLE quantity $\psi$ and the electric field. (a) Soliton-shaped stationary solution of the FP-LLE, Eq. (33), shown in a frame moving in the direction of the forward field (the mirrors move in this frame) so that the field $\psi$ is stationary. (b) Forward-propagating field $F_F$ in the laboratory frame for $t=0$ [see Eq. (27)]. (c) Backward-propagating field $F_B(z,t)=F_F(-z,t)$ in the laboratory frame for $t=0$ [see Eq. (28)]. The fields $F_F(z,t)$ and $F_B(z,t)$ obey periodic boundary conditions in the interval $-L\le z\le L$. (d, e) Fields $F_F$ and $F_B$ after propagation for half a round-trip time. The soliton in the forward-propagating field has entered the unphysical region $-L\le z\le 0$, and the soliton in the backward-propagating field appears in the physical region $0\le z\le L$. (f) The quantity $|F_F{|}^2+{\left|F_B\right|}^2$ corresponding to panels (d) and (e), proportional to the intensity averaged over fast temporal and spatial oscillations associated with the optical frequency. (g) The physical intensity ${\left|E\right|}^2$, calculated from Eq. (14) and shown here at a particular time $t_1$. Cavity parameters have been chosen to facilitate depiction of the standing wave and the soliton on the same scale.

Figure 3

Analytical curves depicting behavior of the FP-LLE. (a) Stationary curves $\rho$ of normalized transmitted intensity as a function of normalized input intensity $F^2$ for the indicated values of the cavity detuning parameter $\alpha$. (b) Important curves in the $\alpha -F^2$ plane, as discussed in the text. Shown in dashed red is the line obtained from Eq. (41) by setting $\rho ={\rho }_{\text{inst}}\left(\alpha \right)$, where ${\rho }_{\text{inst}}\left(\alpha \right)=1$ for $\alpha \le 4$ and for $\alpha >4$ is given by the solution of the equation ${\mu }_{-}\left(\rho \right)=0$ with respect to $\rho$, where ${\mu }_{-}$ is defined by Eq. (54). The dotted red curve is the continuation of the curve obtained from Eq. (41) by setting $\rho =1$ for $\alpha >4$. Multiple real values of $\rho$ exist in the region bounded by the solid blue curves, which trace out the local extrema values of the curves $F^2\left(\rho \right)$ as a function of $\alpha$. (c) Analogous curves in the ring cavity. For direct comparison we also plot the FP-cavity curves in thick light gray.

Figure 4

Exploration of the flat solutions and their stability. (a) Curves ${\mu }_{\text{max}}$ where the gain $\mathrm{\Gamma }$ is greatest, for $\beta =-0.02$ and various values of $\alpha$, with shading indicating the region between ${\mu }_{-}$ and ${\mu }_{+}$ where $\mathrm{\Gamma }>1$. Of note is that the region where $\mathrm{\Gamma }>1$ does not extend to $\rho =1$ for $\alpha =6$. Regardless of the value of $\beta$, the curve ${\mu }_{\text{max}}$ for $\alpha =4$ passes through the point $(\rho =1,\mu =0)$. (b) Plots of the Kerr-tilted intensity resonance profiles $\rho (\alpha ,F^2)$ for three values of $F^2$ (thin black): $F^2=0.5$ (smallest peak), 1 (middle peak), and 4 (largest peak). The thick orange dashed line shows ${\rho }_{\text{inst}}\left(\alpha \right)$, and the thick blue solid lines indicate the values ${\rho }_{+}\left(\alpha \right)$ and ${\rho }_{-}\left(\alpha \right)$ that bound the region where multiple flat solutions exist. Resonance curves for $F^2>1$ are qualitatively similar to the curve for $F^2=4$.

Figure 5

Soliton solutions to the FP-LLE. Analytical approximations are shown by thin black curves (dotted, for spectra); numerically calculated solutions are shown in thick color. Here $\beta =-0.02$ and the resonator FSR is 16.5 GHz. (a, b) $F^2=3$, $\alpha =4.37$, time domain curves in (a) with inset showing the deviation between analytic approximation and numerical solution in the level of the c.w. background near the pulse, optical spectrum in (b). (c, d) $F^2=12$, $\alpha =8.68$.

Figure 6

Simulations of Turing patterns in the FP-LLE. (a) Three Turing patterns, with 18, 17, and 16 rolls, all stable against perturbations at the point $(\alpha =2.5,F^2=6)$. The Turing pattern with 17 rolls (middle, blue) arises most frequently from vacuum fluctuations. (b) Plots of $|F_F{|}^2+{\left|F_B\right|}^2$ in the physical domain $0<z<L$ at two different times for the 17-roll Turing pattern plotted in (a), demonstrating how a stationary solution of the FP-LLE relates to a time-varying intensity pattern. (c) Optical spectrum of the 17-roll Turing pattern from (a) and (b), assuming a cavity with 16.5 GHz FSR. (d) Summary of multistabilility of Turing patterns as revealed by simulations for $F^2=6$. Data points indicate Turing patterns that can be excited from appropriate initial conditions. Data points enclosed by a circle indicate Turing patterns stable against perturbations, and blue (gray) data points indicate Turing patterns that arise from vacuum fluctuations.

Figure 7

Numerical investigations of nonstationary solutions to the FP-LLE. (a) Maximum and minumum amplitudes of an oscillating breather soliton at the point $(\alpha =5.6,F^2=8)$ for $\beta =-0.02$. (b) A snapshot of the time-varying, aperiodic intensity profile of spatiotemporal chaos at the point $(\alpha =5.3,F^2=8)$ for $\beta =-0.02$. (c) Time-averaged optical spectrum of spatiotemporal chaos under the conditions given in (b) for a cavity with 16.5 GHz FSR. (d) Top: A histogram of local maxima values of spatiotemporal chaos for the FP LLE at the point $(\alpha =5.3,F^2=8)$, $\beta =-0.02$ (blue, behind), and at the corresponding point $(\alpha =1.1,F^2=8)$ for the ring LLE (orange, in front), recorded over simulations with the same duration. Bottom: Fractional difference between the two histograms.

Figure 8

Effects of nonlinear integral term in the FP-LLE, Eq. (33). (a) Approximate existence bounds of single solitons in the zero-dispersion limit (green, furthest left) and for $\beta =-0.001$ (blue, second from left), $\beta =-0.02$ (purple, third), and $\beta =-0.3$ (red, furthest right), calculated using the analytical approximation to the single soliton. (b) Comparisons between the approximate existence bounds and the bounds as determined numerically. Solid points indicate the maximum and minimum values of $F^2$ at which solitons have been simulated for a given value of $\alpha$. Values for the dispersion parameters are as shown in (a): $\beta =-0.001$ (blue, top), $\beta =-0.02$ (purple, middle), and $\beta =-0.3$ (red, bottom). (c) Simulated spatiotemporal chaos (blue, extended pattern) and single soliton solution (purple, localized near $\theta =0$), either of which can exist at the point $(\alpha =8,F^2=8)$. The amplitude of the soliton is larger than the characteristic amplitude of the features in the chaos because the effective detuning ${\alpha }^{\prime }$ is larger for the soliton. (d) Analytical and numerical soliton existence limits (purple) for $\beta =-0.02$ from panel (a) and the upper bound in $\alpha$ for the existence of spatiotemporal chaos and/or Turing patterns (black with error bars), estimated as described in the text.