Nonlinear Fourier transform for analysis of optical spectral combs

The nonlinear Fourier transform (NFT) is used to characterize the optical combs in the Lugiato-Lefever equation with both anomalous and normal dispersion. We demonstrate that the NFT signal processing technique can simplify analysis of the formation of dissipative dark solitons and regimes exploiting modulation instability for a generation of coherent structures, by approximating the comb with several discrete eigenvalues, providing a platform for the analytical description of dissipative coherent structures.

Figure 1

Dissipative dark soliton on the stable cw background: (a) evolution with $T$ of the intensity, ${\left|\mathrm{\Psi }(T,\tau )\right|}^2$, (b) the spectral power density ${\left|\mathrm{\Psi }(T=12,\omega )\right|}^2$, (c) intensity of the field without background, $|{\mathrm{\Psi }}_1{(T,\tau )|}^2$, (d) the nonlinear discrete spectrum at $T=12$, (e) the field reconstructed from the discrete spectrum only, $|{\mathrm{\Psi }}_1^{\left(DS\right)}{(T,\tau )|}^2$, (f) the blue line shows the evolution with $T$ of the fraction of energy in the discrete spectrum, $E_d\left(T\right)/E_t\left(T\right)$; the red line shows the relative integral $L_2$-norm of the difference between the field ${\mathrm{\Psi }}_1$ and the field ${\mathrm{\Psi }}_1^{\left(DS\right)}$ reconstructed from the nonlinear discrete spectrum. Here, $\mathrm{\Psi }(T=0,\tau )=2-1.8exp[-{(\tau /3.1)}^2]$, $\beta =1$, ${\zeta }_0=6$, $f^2=8.5$.

Figure 2

Formation of the comb through the modulation instability with switched detuning: (a) full field with background, ${\left|\mathrm{\Psi }(T,\tau )\right|}^2$, (b) comb spectral power density ${\left|\mathrm{\Psi }(T=12,\omega )\right|}^2$, (c) $|{\mathrm{\Psi }}_1{(T,\tau )|}^2$, (d) dynamics of the nonlinear spectrum with $T$ shown in the complex plane $\lambda =\xi +i\eta$: the discrete spectrum (upper part) and the logarithm of ${\left|r\left(\xi \right)\right|}^2$ for the continuous spectrum (contour plot), (e) the field reconstructed from the discrete spectrum only, $|{\mathrm{\Psi }}_1^{\left(DS\right)}{(T,\tau )|}^2$, (f) the blue line shows the evolution with $T$ of $E_d\left(T\right)/E_t\left(T\right)$; the red line shows the relative integral $L_2$-norm of the difference between the field ${\mathrm{\Psi }}_1$ and the field ${\mathrm{\Psi }}_1^{\left(DS\right)}$ reconstructed from the nonlinear discrete spectrum. Here, ${\zeta }_0=2$ for $T<5$ and ${\zeta }_0=8.7666$ for $T\ge 5$. Also, $\beta =-1$, $f^2=3$, $\mathrm{\Psi }(T=0,\tau )=1.8exp(-2{\tau }^2)cos(0.3+5\tau )$.

Figure 3

Comb generation with a pulsed pumping wave $\left[f\right(\tau )=1.9\text{sech}(\tau /20\left)\right]$ generated from the Gaussian pulse $3exp(-{\tau }^2/2)$: (a) dynamics of intensity, ${\left|\mathrm{\Psi }(T,\tau )\right|}^2$, (b) formed comb shown at $T={10\left|\mathrm{\Psi }(T=10,\tau )\right|}^2$, (c) the field $|{\mathrm{\Psi }}^{\left(DS\right)}{(\tau ,T)|}^2$ reconstructed from the discrete spectrum shown in (e), (d) the spectral power density of the comb, ${\left|\mathrm{\Psi }(T=12,\omega )\right|}^2$, (e) dynamics of the nonlinear spectrum shown as the evolution with $T$ in the complex plane $\lambda =\xi +i\eta$: the discrete spectrum (upper part) and the logarithm of ${\left|r\left(\xi \right)\right|}^2$ for the continuous spectrum (contour plot), (f) the blue line shows the evolution with $T$ of $E_d\left(T\right)/E_t\left(T\right)$; the red line shows the relative integral $L_2$-norm of the difference between the field $\mathrm{\Psi }$ and ${\mathrm{\Psi }}^{\left(DS\right)}$ reconstructed only from the nonlinear discrete spectrum. Here, ${\zeta }_0=4$, $\beta =-1$.

Figure 4

Dissipative dark soliton: $\beta =1$, ${\zeta }_0=2.5$, $f^2=2.61$, $\mathrm{\Psi }(T=0,\tau )=1.7-exp[-{(\tau /4.4721)}^2]$. (a) ${\left|\mathrm{\Psi }(\tau ,T)\right|}^2$, (b) the spectral power density ${\left|\mathrm{\Psi }(T=30,\omega )\right|}^2$.

Figure 5

Analytical approximation of dissipative dark soliton, with the same parameters as in Fig. 4. (a) Dynamics in $T$ of the nonlinear spectrum in the complex plane $\lambda =\xi +i\eta$: the discrete spectrum (upper part) and the logarithm of ${\left|r\left(\xi \right)\right|}^2$ for the continuous spectrum (contour plot), (b) solution of Eq. (5) $|{\mathrm{\Psi }}_1{(T,\tau )|}^2$, (c) the field $|{\mathrm{\Psi }}_1^{\left(DS\right)}{(T,\tau )|}^2$ reconstructed from the discrete spectrum using the $L_2$-optimization procedure, (d) relative integral $L_2$-norm of the difference between the field ${\mathrm{\Psi }}_1$ and the field ${\mathrm{\Psi }}_1^{\left(DS\right)}$ reconstructed from the nonlinear discrete spectrum using the general procedure (purple line) and using the $L_2$-optimization procedure (red line).