Theory of filter-induced modulation instability in driven passive optical resonators

We present the theory of modulation instability induced by spectrally dependent losses (optical filters) in passive driven nonlinear fiber ring resonators. Starting from an Ikeda map description of the propagation equation and boundary conditions, we derive a mean-field model—a generalized Lugiato-Lefever equation—which reproduces with great accuracy the predictions of the map. The effects on instability gain and comb generation of the different control parameters such as dispersion, cavity detuning, filter spectral position, and bandwidth are discussed.

Figure 1

(a) Amplitude and phase of the filter described by Eqs. (9) and (10). (b) Instability gain $g_{\text{MAP}}\left(\omega \right)$ calculated from Eq. (21) revealing the effect of magnitude and phase of the filter transfer function. The following parameters have been used: ${\beta }_2=0.5{\mathrm{ps}}^2/\mathrm{km}$, $\gamma =2.5{\mathrm{W}}^{-1}{\mathrm{km}}^{-1}$, $L=100\mathrm{m}$, $\rho =\sqrt{0.9}$, $\theta =\sqrt{0.1}$, ${\varphi }_0=-\psi \left(0\right)$ (zero global cavity detuning), ${\omega }_{\text{f}}=200·2\pi$ rad/ns, $b=-1$, $a=400$ rad/ns, intracavity power $P=1.18\mathrm{W}$, input power $P_{\text{IN}}=1\mathrm{W}$. The vertical dashed line indicates the phase-matching frequency calculated from Eq. (32).

Figure 2

The positive part of the MI gain is depicted a function of $P_{IN}$, calculated from (a) the map and (b) the LLE. The dashed lines denote filter position ${\omega }_f/\left(2\pi \right)$. Parameters used are ${\beta }_2=0.5{\mathrm{ps}}^2{\mathrm{km}}^{-1}$, $\gamma =2.5{\mathrm{W}}^{-1}{\mathrm{km}}^{-1}$, $L=0.1\mathrm{km}$, $\rho =\sqrt{0.9}$, ${\varphi }_0=-\psi \left(0\right)$, $\theta =\sqrt{0.1}$, $a=500$ rad/ns, $b=-3.2$, ${\omega }_f=2\pi ·400$ rad/ns.

Figure 3

The positive part of the MI gain is depicted for map and LLE as a function of the filter frequency shift with respect to the pump ${\omega }_f/\left(2\pi \right)$ [(a), (b)], of the filter width $a$ [(c), (d)]; and of the filter strength $b$ [(e), (f)] . The dashed black lines denote filter position, whereas the dotted red lines denote the filter width at half maximum. Parameters used are ${\beta }_2=0.5{\mathrm{ps}}^2{\mathrm{km}}^{-1}$, $P_{IN}=0.5\mathrm{W}$, $\gamma =2.5{\mathrm{W}}^{-1}{\mathrm{km}}^{-1}$, $L=0.1\mathrm{km}$, $\rho =\sqrt{0.95}$, ${\varphi }_0=-\psi \left(0\right)$, $\theta =\sqrt{0.05}$, $a=400$ rad/ns in (a), (b), (e), and (f); $b=-1$ in (a)–(d); ${\omega }_f=2\pi ·400$ rad/ns in (c)–(f).

Figure 4

The positive part of the MI gain is depicted as a function of the phase shift ${\varphi }_0$ for the map without filter (a), with filter positively detuned with respect to the pump (${\omega }_f/\left(2\pi \right)=400\mathrm{GHz}$) (c), and with filter negatively detuned with respect to the pump (${\omega }_f/\left(2\pi \right)=-400\mathrm{GHz}$) (e); the corresponding LLE cases are plotted in (b)–(e). The dashed lines denote filter position. Parameters used are ${\beta }_2=0.5{\mathrm{ps}}^2{\mathrm{km}}^{-1}$, $\gamma =2.5{\mathrm{W}}^{-1}{\mathrm{km}}^{-1}$, $L=0.1\mathrm{km}$, $\rho =\sqrt{0.9}$, $\theta =\sqrt{0.1}$, $a=800$ rad/ns, $b=-3.2$ and intracavity power 1 W.

Figure 5

The evolution of the field power profile over round trips from (a) the Ikeda map and (b) the LLE. The output power pattern and the corresponding power spectrum (normalized to its maximum) are shown in panels (c) and (d). Solid red curves denotes results from the Ikeda map and blue circles from the LLE. The dashed black curve in (c) represents the initial condition, whereas in (d) it represents the filter amplitude $\left|H\right(\omega \left)\right|$. The parameters are ${\beta }_2=0.5{\mathrm{ps}}^2{\mathrm{km}}^{-1}$, $\gamma =2.5{\mathrm{W}}^{-1}{\mathrm{km}}^{-1}$, $L=0.1\mathrm{km}$, $\rho =\sqrt{0.95}$, ${\varphi }_0=-\psi \left(0\right)$, $\theta =\sqrt{0.05}$, $a=400$ rad/ns, $b=-1$, ${\omega }_f/\left(2\pi \right)=200\mathrm{GHz}$, $P_{\text{IN}}=0.0149\mathrm{W}$ for the Ikeda map and $P_{\text{IN}}=0.015\mathrm{W}$ for the LLE (intracavity power of stationary solution is 0.2 W in both cases). Simulations initial conditions were $A(0,t)=\sqrt{0.2}[1+0.001cos\left(2\pi {\nu }_{\text{max}}t\right)]$ (${\nu }_{\text{max}}=455\text{GHz}$).