Theory of radiation trapping by the accelerating solitons in optical fibers

We present a theory describing trapping of the normally dispersive radiation by the Raman solitons in optical fibers. Frequency of the radiation component is continuously blueshifting, while the soliton is redshifting. Underlying physics of the trapping effect is in the existence of the inertial gravitylike force acting on light in the accelerating frame of reference. We present analytical calculations of the rate of the opposing frequency shifts of the soliton and trapped radiation and find it to be greater than the rate of the redshift of the bare Raman soliton. Our findings are essential for understanding of the continuous shift of the high-frequency edge of the supercontinuum spectra generated in photonic crystal fibers toward higher frequencies.

Figure 1

(Color online) Frequency dependence of the group velocity normalized to speed of light in vacuum (group index) (a) and of the GVD (b) in the photonic crystal fiber as in Ref. [14]. Fiber dispersion like this is typical for numerous supercontinuum experiments.

Figure 2

(Color online) Left panel: spectral evolution along the fiber length for the dispersion profile as in Fig. 1. Pump pulse parameters: (a) wavelength 850 nm is in the anomalous GVD range, (c) 800 nm is in the normal GVD range. In both cases peak power is 6 kW, duration 200 fs; the radiation at the short wavelength edge of the continuum starts its continuous drift toward shorter frequencies at $z>0.2\text{m}$ for (a) and $z>1\text{m}$ for (c). Right panel: spectrograms showing simultaneous frequency and time domain images of the supercontinua in (a) and (c) for $z=1.5\text{m}$ and $z=3\text{m}$, respectively. The function plotted has been calculated using the cross-correlation frequency resolved optical gating (XFROG) integral $I\left(t,\omega \right)={\int }_{-\infty }^{\infty }d{t}^{\prime }{E}_{ref}\left(t^{\prime }-t\right)E\left(t^{\prime }\right)e^{-i\omega t^{\prime }}$. Here $E_{ref}$ is the 2 ps Gaussian pulse. See Ref. [28] for more details about the XFROG.

Figure 3

(Color online) Spectral (a) and time-domain (b) evolution along the fiber pumped with two pulses obtained from the numerical modeling of Eq. (2) with ${\beta }^{\left(m>3\right)}=0$ and $ϵ=+0.12$. Initial conditions are $E\left(z=0,t\right)=\sqrt{2q_1}\text{sech}\left(t∕{\tau }_1\right)\text{exp}\left(-i{\delta }_{1}t\right)+\sqrt{2q_2}\text{sech}\left[\left(t-T_0\right)∕{\tau }_2\right]\text{exp}\left(-i{\delta }_{2}t\right)$. with $q_1=2000$ (peak power $\sim 24\text{kW}$), $q_2=250$ (peak power $\sim 3\text{kW}$), ${\tau }_1=\sqrt{-D_s^{″}∕2q_1}=0.2\left(\sim 40\text{fs}\right)$, ${\tau }_2=\sqrt{-D_s^{″}∕2q_2}\approx 0.57\left(\sim 115\text{fs}\right)$, $T_0=0.25\left(\sim 50\text{fs}\right)$, and ${\delta }_1=-80\left(\sim 964\text{nm}\right)$, ${\delta }_2=80\left(\sim 684\text{nm}\right)$. Results in (b) are presented in the accelerating frame of reference: $t^{\prime }=t-g_0{z}^2∕2$, with $g_0$ defined by Eq. (15).

Figure 4

(Color online) Pulse propagation in the case of negative third-order dispersion, $ϵ=-0.12$. (a),(b) Two pulses pump similar to that in Fig. 3. Parameters of the pulses are $q_1=200$ (peak $\text{power}=\sim 2.4\text{kW}$), $q_2=25$ (peak power $\sim 0.3\text{kW}$, ${\tau }_1=\sqrt{-D_s^{″}∕2q_1}\approx 0.63\left(\sim 130\text{fs}\right)$, ${\tau }_2=\sqrt{-D_s^{″}∕2q_2}\approx 1.8\left(\sim 360\text{fs}\right)$, $T_0=1.5\left(\sim 300\text{fs}\right)$, and ${\delta }_1=80$, ${\delta }_2=-80$. (c),(d) XFROG spectrograms for the case of a two-pulse pump, calculated at different propagation distances. (e),(f) Soliton propagation with $q=200$ (peak power $\sim 2.4\text{kW}$). All time domain results are presented in the accelerating frame of reference: $t^{\prime }=t-g_0{z}^2∕2$, with $g_0$ defined by Eq. (15).

Figure 5

(Color online) (a) Full (dashed) line shows the potential $V$ $\left(V_b\right)$ for the soliton with $q=100$ and $T=0.0024$. (b)–(e) Selected quasibound eigenstates of the potential $V$, cf. horizontal levels in (a). Dashed lines show the corresponding modes of the potential $V_b$.

Figure 6

(Color online) Numerical propagation of the first three bound states within coupled equations (6, 7). Initial conditions are soliton for $A_1$ with $q=100$ and linear eigenstate of $V$ for $A_2$. Left column: spectrum of the total field $E$, Eq. (5), with ${\delta }_2=-{\delta }_1=20$. Right column: temporal profiles of ${\left|A_1\right|}^2$ (dashed lines) and ${\left|A_2\right|}^2$ (solid lines). Time domain results are presented in the accelerating frame of reference: $t^{\prime }=t-g_0{z}^2∕2$, with $g_0$ defined by Eq. (15).

Figure 7

(Color online) (a) Lowest trapped linear mode of the potential $V$ (solid line) and its variational approximation (dashed line), given by Eq. (19) with $\gamma$ and $w^2$ defined through Eqs. (22, 21), respectively. Soliton parameter is $q=100$. (b) ${\lambda }_0^{\left(b\right)}\left(q\right)$, solid line, ${\lambda }_0\left(q\right)$, circles, and ${\tilde{\lambda }}_0\left(q\right)$, dashed line.

Figure 8

(Color online) Numerically calculated acceleration $g$ for different branches of solutions of Eqs. (27, 28) with the potential $V$ being replaced by its asymptotic $V_b$. Circles correspond to the solutions in the full potential $V$. Approximate analytical solution (30) is indicated by a dashed line. Insets illustrate solution profiles (asymptotic potential) at different values of the parameter $\lambda$.

Figure 9

(Color online) Numerical propagation of the lowest bound state within coupled equations (11, 12) with acceleration $g=g_0$ (a) and (b), and $g=\tilde{g}$ (c) and (d). Left column: anomalous GVD component $\left(A_1\right)$, right column: normal GVD component $\left(A_2\right)$. Intensities are plotted in logarithmic scale. (e) and (f): Maximum of the intensity of $A_1$ and $A_2$ components, respectively, along propagation distance. Initial shape for $A_2$ was calculated with the approximate potential $V_b$ for the soliton parameter $q=100$ and $\lambda =70\left(\delta \lambda \approx 20\right)$. $A_1$ was initialized with the soliton (13) with $q=105$ in order to account for initial radiative losses, see text for more details.