Type-I cascaded quadratic soliton compression in lithium niobate: Compressing femtosecond pulses from high-power fiber lasers

The output pulses of a commercial high-power femtosecond fiber laser or amplifier are typically around 300–500 fs with wavelengths of approximately 1030 nm and tens of microjoules of pulse energy. Here, we present a numerical study of cascaded quadratic soliton compression of such pulses in ${\mathrm{LiNbO}}_3$ using second-harmonic generation in a type-I phase-matching configuration. We find that because of competing cubic material nonlinearities, compression can only occur in the nonstationary regime, where group-velocity-mismatch–induced Raman-like nonlocal effects prevent compression to less than 100 fs. However, the strong group-velocity dispersion implies that the pulses can achieve moderate compression to durations of less than 130 fs in available crystal lengths. Most of the pulse energy is conserved because the compression is moderate. The effects of diffraction and spatial walk-off are addressed, and in particular the latter could become an issue when compressing such long crystals (around 10 cm long). We finally show that the second harmonic contains a short pulse locked to the pump and a long multi-picosecond red-shifted detrimental component. The latter is caused by the nonlocal effects in the nonstationary regime, but because it is strongly red-shifted to a position that can be predicted, we show that it can be removed using a bandpass filter, leaving a visible component of less than 100 fs at $\lambda =515$ nm with excellent pulse quality.

Figure 1

Cascaded quadratic soliton compressor studied here: The Yb fiber laser produces energetic longer pulses ($\gg 100$ fs) that are launched collimated in a quadratic nonlinear NB crystal, where the phase-mismatched type-I SHG process compresses the input pulse.

Figure 2

Properties at ${\lambda }_1=1.03\hspace{0.5em}\mu \mathrm{m}$ when angle-tuning the LN crystal: (a) phase mismatch, (b) GVM parameter, (c) GVD of FW (red) and SH (blue), and (d) the spatial walk-off angle $\rho$. The effective quadratic nonlinearities neglecting (black) and including (dashed red) spatial walk-off are shown in (e) and (f) are the total cubic Kerr nonlinearity, Eq. (2), from cascaded quadratic nonlinearities and Kerr SPM (using $n_{\mathrm{Kerr}}^I=18\times {10}^{-20}\hspace{0.5em}{\mathrm{m}}^2/\mathrm{W}$; see Appendix ).

Figure 3

Compression diagram for $1\%$ MgO:sLN at room temperature and aligned for type-I SHG. For various pump wavelengths ${\lambda }_1$, the choice of phase-mismatch parameter $\Delta k$ affects the compression. In order to excite solitons, the phase mismatch must be kept below the red line ($\Delta k<\Delta k_{c,\max }$), because otherwise the material cubic nonlinearities are too strong ($n_{\mathrm{Kerr},11}^I>\left|n_{\mathrm{SHG}}^I\right|$). Optimal compression occurs when the cascaded nonlinearities dominate over GVM effects ($\Delta k>\Delta k_{\mathrm{sr}}$, above the black line). We have also indicated the operation wavelengths of Yb- and Er-doped fiber lasers. The red line uses Miller’s rule to estimate the nonlinear quadratic and cubic susceptibilities at other wavelengths, cf. Eqs. (D1) and (D2), and uses $n_{\mathrm{Kerr},11}^I=20\times {10}^{-20}\hspace{0.5em}{\mathrm{m}}^2/\mathrm{W}$ for $\lambda =0.78\hspace{0.5em}\mu \mathrm{m}$ (see Appendix  for an extended discussion).

Figure 4

Practical operation range of the LN type-I compression system at ${\lambda }_1=1.03\hspace{0.5em}\mu \mathrm{m}$ for $\Delta k=45$ ${\mathrm{mm}}^{-1}$. The plots show the predicted behavior when the FW waist $w_0$ is varied. The two cases are (1) pump pulses with $T_{\mathrm{in}}^{\mathrm{FWHM}}=250$ fs and 15 $\mu$J pulse energy and (2) pump pulses with $T_{\mathrm{in}}^{\mathrm{FWHM}}=450$ fs and 50 $\mu$J pulse energy. The curves in (b)–(f) are calculated based on $N_{\mathrm{eff}}$ shown in (a) by using the scaling laws [17] that hold for $N_{\mathrm{eff}}>1$.

Figure 5

Numerical simulation of soliton compression in LN with ${\lambda }_1=1.03\hspace{0.5em}\mu \mathrm{m}$, $T_{\mathrm{in}}^{\mathrm{FWHM}}=250$ fs, $\Delta k=45$ ${\mathrm{mm}}^{-1}$, and $N_{\mathrm{eff}}=1.4$ (implying $I_{\mathrm{in}}=6.9$ $\mathrm{GW}/{\mathrm{cm}}^2$). The FW pulse shown in (a) compresses to $\Delta t_{\mathrm{opt}}=126$ fs (FWHM) after propagating 91 mm. The SH time plot (c) and FW (b) and SH (d) spectra are also shown on a logarithmic scale, and $U_j$ are normalized to the peak input FW electric field. In (e) and (f), cuts are shown at the optimal compression point $z=91$ mm (corresponding to the white line in the two-dimensional plots). Note that the SH in (e) is magnified one hundred times.

Figure 6

Simulations as in Fig. 5 but with increasing $N_{\mathrm{eff}}$. The red curve corresponds to the optimal compression point from Fig. 5e, whereas the black curves show what happens as the effective soliton order increases (making the optimal compression point occur sooner).

Figure 7

XFROG-like spectrograms of the simulation in Fig. 5 at the optimal compression point $z_{\mathrm{opt}}=91$ mm. The sech-shaped gating pulse had $T_0^{\mathrm{FWHM}}=70$ fs, and the spectrograms are normalized to the peak value of $S_1$. The top and side plots show the purely temporal and spectral traces, respectively, and are thus identical to Fig. 5e and 5f. The red dashed line in (b) indicates the value ${\Omega }_{+}$ as calculated by the nonlocal theory.

Figure 8

XFROG spectrogram of the SH pulse from Fig. 7b passed through a third-order super-Gaussian bandpass filter centered at the SH carrier frequency ${\lambda }_2=0.5150\hspace{0.5em}\mu \mathrm{m}$ and with a FWHM of 100 THz.

Figure 9

Numerical simulations using 450-fs 50-$\mu$J input pulses and taking $\Delta k=45$ ${\mathrm{mm}}^{-1}$. The best pulse was observed for $N_{\mathrm{eff}}=2.0$ (red curve) where pulse compression occurs after 15 cm. The black curves show what happens as the effective soliton order increases (in which case the optimal compression point occurs sooner).

Figure 10

Definition (in accordance with the IRE/IEEE standard [36]) of the crystal coordinate system $\mathit{xyz}$ relative to the beam propagation direction indicated by $\mathbf{k}$.