Nonreciprocal optical solitons in a spinning Kerr resonator

We propose a spinning nonlinear resonator as an experimentally accessible platform to achieve nonreciprocal control of optical solitons. Nonreciprocity here results from the relativistic Sagnac-Fizeau optical drag effect, which is different for pump fields propagating in the spinning direction or in the direction opposite to it. We show that in a spinning Kerr resonator, different soliton states appear for the input fields in different directions. These nonreciprocal solitons are more stable against losses induced by intermodal coupling between clockwise and counterclockwise modes of the resonator. Our work builds a bridge between nonreciprocal physics and soliton science, providing a promising route towards achieving soliton-wave optical isolators and one-way soliton communications.

Figure 1

Schematic illustration of nonreciprocal solitons. Different Sagnac frequency shifts ${\mathrm{\Delta }}_{\mathrm{F}}$ are induced for countercirculating modes in a spinning Kerr microresonator. (a) and (b) Solitons appear for the input laser in one direction but not the other. (c) Sagnac-Fizeau shift ${\mathrm{\Delta }}_{\mathrm{F}}$ versus angular velocity $\mathrm{\Omega }$. Increasing the angular velocity results in linear opposite frequency shifts for the counterpropagating modes, leading to the formation of nonreciprocal solitons.

Figure 2

Optical solitons with different numbers in different directions. (a)–(g) Three-dimensional representations of optical field energy evolution of the soliton mode $|{\psi }_i{|}^2$ in the azimuthal direction at $\tau =200$ and (h) corresponding Kerr combs in a static and a spinning Kerr microresonator without backscattering. For ${\mathrm{\Delta }}_{\mathrm{F}}=0$, the solitons emerge with the same numbers in different driving directions. For ${\mathrm{\Delta }}_{\mathrm{F}}\ne 0$, the detuning ${\mathrm{\Delta }}_{\mathrm{p}}$ for the countercirculating modes is modified by the opposite Sagnac shifts, resulting in solitons with different numbers in different directions. Here we show three different types of nonreciprocal solitons in different parameter ranges. The initial conditions of the soliton optical fields are Gaussian pulses (a)–(c), (g) and (h) ${\psi }_0=0.5+\exp [-{(\theta /0.1)}^2]$ and (d)–(f) ${\psi }_0=0.5+\exp [-{(\theta /0.55)}^2]$; the rotation velocity (b) and (c) $\mathrm{\Omega }=2.0\mathrm{kHz}$, (e) and (f) $\mathrm{\Omega }=350\mathrm{Hz}$; and (g) and (h) $\mathrm{\Omega }=2.2\mathrm{kHz}$; and the dimensionless detuning (a)–(f) ${\mathrm{\Delta }}_{\mathrm{p}}=2$ and (g) and (h) ${\mathrm{\Delta }}_{\mathrm{p}}=2.3$. We used normalized dispersion $\beta =-0.004$ and scaled pump intensity $F_i=1.37$ in the simulations. The figures do not display the $xyz$ coordinate axis for convenience since the numbers of solitons can be seen from the waveform.

Figure 3

Soliton suppression due to random-defect-induced backscattering and its revival resulting from the rotation-induced compensation. (a)–(f) Three-dimensional representations of the optical field energy evolution of the soliton mode $|{\psi }_i{|}^2$ in the azimuthal direction at $\tau =200$ in a static and a spinning microresonator when backscattering ($J=1$) is considered. (g) Transient dynamics of the soliton mode $|{\psi }_i{|}^2$ at $\tau =200$ as a function of azimuthal angle $\theta$ and the detuning of the counterclockwise mode ${\alpha }_2$ in the dual pump system as reported in Ref. [12] when backscattering ($J=1$) is considered. For ${\mathrm{\Delta }}_{\mathrm{F}}=0$, solitons are suppressed for increased values of backscattering $J=1$. For $\mathrm{\Omega }=60\mathrm{kHz}$, the soliton can exist stably. The initial conditions of the soliton optical fields are Gaussian pulses (a)–(c) and (g) ${\psi }_0=0.5+\exp [-{(\theta /0.1)}^2]$ and (d)–(f) ${\psi }_0=0.5+\exp [-{(\theta /0.55)}^2]$; the rotation velocity $\mathrm{\Omega }=60\mathrm{kHz}$; and the normalized detuning (a) and (d) ${\mathrm{\Delta }}_{\mathrm{p}}=2.0$, (b) and (e) ${\mathrm{\Delta }}_{\mathrm{p}}=-6.9$, and (c) and (f) ${\mathrm{\Delta }}_{\mathrm{p}}=11$. The other parameters are the same as in Fig. 2 and we have set the dimensionless driving strength $F_i=F_j=1.37$ and the scaled detuning of the clockwise mode ${\alpha }_1=2.2$ in the dual pump system simulations.

Figure 4

Competitive relationship between the backscattering strength and the spinning velocity in nonreciprocal solitons. The intracavity field energy of the soliton mode $|{\psi }_i{|}^2$ is plotted as a function of the azimuthal angle $\theta$ at $\tau =200$ and the backscattering strength $J$ with various values of the spinning velocity $\mathrm{\Omega }$: (a) $\mathrm{\Omega }=0$, (b) $\mathrm{\Omega }=60$ kHz, (c) $\mathrm{\Omega }=120$ kHz, and (d) $\mathrm{\Omega }=180$ kHz. The input signal of the optical soliton field is a Gaussian pulse ${\psi }_0=0.5+\exp [-{(\theta /0.1)}^2]$, the dimensionless detuning is ${\mathrm{\Delta }}_{\mathrm{p}}=2.1+{\mathrm{\Delta }}_{\mathrm{Sag}}$, and the other parameters are the same as in Fig. 2.