Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers

We develop a theory of the generation of new spectral components in optical fibers pumped with a solitonic pulse and a weak continuous wave (cw). We derive the wave number matching conditions for the above process and present an analytical method of finding the amplitudes of the generated waves. We discuss related effects of the depletion of the cw pump and spectral recoil on the soliton. We also point out examples of the generation of supercontinuum spectra in fibers, where mixing between solitons and dispersive waves plays an important role.

Figure 1

(Color online) Wave number matching diagrams (top) and XFROG spectrograms (bottom), when the fiber with the negative GVD slope, $ϵ=-0.015$, is pumped with the soliton and cw; see Eqs. (14). $w=0.4$, $a=1$, and ${\delta }_s=0$ in all cases. ${\delta }_{\mathit{cw}}=-18$ for (a), $-27$ for (b), and $-22.2$ for (c). Soliton parameter $q=18$ for (a),(b) and 3 for (c). Propagation distance $z$ is 1 for $\left({\mathrm{a}}_2\right)$, 6 for $\left({\mathrm{b}}_2\right)$, and 12 for $\left({\mathrm{c}}_2\right)$.

Figure 2

(Color online) Wave number matching diagrams (top) and XFROG spectrograms (bottom), when the fiber with the positive GVD slope, $ϵ=0.015$, is pumped with the soliton and cw; see Eqs. (14). $w=0.4$, $a=1$, $q=18$, and ${\delta }_s=0$ in all cases. ${\delta }_{\mathit{cw}}=-18$ for (a), 27 for (b), and 17.5 for (c). Propagation distance $z$ is 1 for $\left({\mathrm{a}}_2\right)$ and 6 for $\left({\mathrm{b}}_2\right)$, $\left({\mathrm{c}}_2\right)$.

Figure 3

(Color online) $\mid A\mid$ as a function of $t$ for $z=0,3,6$ obtained by solving Eq. (1) with the initial conditions (14). Parameters are $ϵ=0.015$, $a=1$, $q=18$, ${\delta }_s=0$, $w=0.4$, and ${\delta }_{\mathit{cw}}=17.5$.

Figure 4

(Color online) (a) The rate $\Gamma$ of the energy change at the frequency of the FWM signal as a function of the frequency of the cw pump, ${\delta }_{\mathit{cw}}$. The dashed line is obtained using an analytical approximation for $\mathcal{J}$; see Eq. (31). The solid line corresponds to the numerical calculation of $\mathcal{J}$ using Eq. (30), and the dots are obtained from the numerical solution of Eq. (1). Parameters are $ϵ=-0.015$, $a=1$, $q=3$, ${\delta }_s=0$, and $w=0.02$. (b) is the same as (a) but plotted in the logarithmic scale.

Figure 5

(Color online) Change of the soliton frequency ${\delta }_s$ with propagation due to the spectral recoil effect, as calculated from Eq. (1) with $ϵ=0.015$. Initial parameters for the ${\delta }_{\mathit{cw}}=27$ case are the same as for Fig. 2b and for the ${\delta }_{\mathit{cw}}=17.5$ case are the same as for Fig. 2c. The dashed line demonstrates the spectral recoil effect due to Cherenkov radiation only—i.e., in the absence of the cw pump.

Figure 6

(Color online) Modeling of supercontinuum generation in a fiber with $\gamma =0.05∕\left[\mathrm{Wm}\right]$ pumped with femtosecond optical pulses. $\left({\mathrm{a}}_{1,2}\right)$ Pump wavelength coincides with the zero GVD wavelength at $638\mathrm{nm}$, GVD slope ${\partial }_{\omega }{\beta }_2=0.9{\mathrm{ps}}^3∕\mathrm{km}$, $\tau =200\mathrm{fs}$, peak pump power $3.6\mathrm{kW}$, and propagation distances are $0.24\mathrm{m}$ $\left({\mathrm{a}}_1\right)$ and $1.12\mathrm{m}$ $\left({\mathrm{a}}_2\right)$. Corresponding dimensionless parameters are $a=4.6$, $q=3.4$, $ϵ=0.015$, ${\delta }_s=11$, $z_1=0.3$, and $z_2=1.4$. $\left({\mathrm{b}}_{1,2}\right)$ Pump wavelength coincides with the zero GVD wavelength at $1570\mathrm{nm}$, GVD slope ${\partial }_{\omega }{\beta }_2=-0.9{\mathrm{ps}}^3∕\mathrm{km}$, $\tau =200\mathrm{fs}$, peak pump power $3.6\mathrm{kW}$, and propagation distances are $0.4\mathrm{m}$ $\left({\mathrm{a}}_1\right)$ and $1.44\mathrm{m}$ $\left({\mathrm{a}}_2\right)$. Corresponding dimensionless parameter are $a=4.6$, $q=3.4$, $ϵ=-0.015$, ${\delta }_s=-11$, $z_1=0.5$, and $z_2=1.8$. $\left(c_{1,2}\right)$ Pump wavelength is in the anomalous GVD region at $1432\mathrm{nm}$, fiber has the zero GVD wavelength at $1651\mathrm{nm}$, ${\beta }_2=-50{\mathrm{ps}}^2∕\mathrm{km}$, GVD slope ${\partial }_{\omega }{\beta }_2=-0.68{\mathrm{ps}}^3∕\mathrm{km}$, $\tau =150\mathrm{fs}$, peak pump power $3.6\mathrm{kW}$ $\left({\mathrm{c}}_1\right)$ and $4.4\mathrm{kW}$ $\left({\mathrm{c}}_2\right)$, and propagation distance $1.12\mathrm{m}$. Corresponding dimensionless parameters are $a=3.47$ $\left({\mathrm{c}}_1\right)$, $a=3.85$ $\left({\mathrm{c}}_2\right)$, $ϵ=-0.015$, ${\delta }_s=15$, and $z=2.5$.