Nonlinear propagation analysis of few-optical-cycle pulses for subfemtosecond compression and carrier envelope phase effect

A numerical approach called Fourier direct method (FDM) is applied to nonlinear propagation of optical pulses with the central wavelength 800 nm, the width 2.67–12.00 fs, and the peak power 25–6870 kW in a fused-silica fiber. Bidirectional propagation, delayed Raman response, nonlinear dispersion (self-steepening, core dispersion), as well as correct linear dispersion are incorporated into “bidirectional propagation equations” which are derived directly from Maxwell’s equations. These equations are solved for forward and backward waves, instead of the electric-field envelope as in the nonlinear Schrödinger equation (NLSE). They are integrated as multidimensional simultaneous evolution equations evolved in space. We investigate, both theoretically and numerically, the validity and the limitation of assumptions and approximations used for deriving the NLSE. Also, the accuracy and the efficiency of the FDM are compared quantitatively with those of the finite-difference time-domain numerical approach. The time-domain size 500 fs and the number of grid points in time 2048 are chosen to investigate numerically intensity spectra, spectral phases, and temporal electric-field profiles up to the propagation distance 1.0 mm. On the intensity spectrum of a few-optical-cycle pulses, the self-steepening, core dispersion, and the delayed Raman response appear as dominant, middle, and slight effects, respectively. The delayed Raman response and the core dispersion reduce the effective nonlinearity. Correct linear dispersion is important since it affects the intensity spectrum sensitively. For the compression of femtosecond optical pulses by the complete phase compensation, the shortness and the pulse quality of compressed pulses are remarkably improved by the intense initial peak power rather than by the short initial pulse width or by the propagation distance longer than 0.1 mm. They will be compressed as short as 0.3 fs below the damage threshold of fused-silica fiber 6 MW. It is demonstrated that the carrier envelope phase (CEP) causes the difference on the temporal electric-field profile and the intensity spectrum for the initial peak power of the order of megawatts. At the propagation distance longer than the coherence length for third-order harmonics, the difference grows in the spectral components around the third-order and higher-order harmonics. The CEP can be a sensitive marker to monitor the evolution of nonlinear optical process by a few-optical-cycle electric-field wave-packet source.

Figure 1

Fourier-transformed nonlinear Raman response function $\tilde{R}\left(\omega \right)$, (a) real part and (b) imaginary part. The range $-1.5\mathrm{rad}∕\mathrm{fs}<\omega <1.5\mathrm{rad}∕\mathrm{fs}$ is omitted. A linear approximation to its imaginary part around ${\omega }_0$ is also shown. ${\omega }_{\mathrm{S}}=2.274\mathrm{rad}∕\mathrm{fs}$, ${\omega }_0=2.356\mathrm{rad}∕\mathrm{fs}$, ${\omega }_{\mathrm{A}}=2.438\mathrm{rad}∕\mathrm{fs}$, and $T_{\mathrm{R}}=2.44\mathrm{fs}$.

Figure 2

Intensity spectra used for the investigation of the core dispersion. $A$ or $B$: Calculated spectrum under the experimental or theoretical effective core radius as far as $z=2.5\mathrm{mm}$ where $B∕F=0.0$, ${\varphi }_1=0.0$, and $C_0=-12.0\mathrm{rad}{\mathrm{fs}}^2$. $C$: Output experimental spectrum $I_{\mathrm{out}}$ compared with $A$ or $B$. $D$: Input experimental spectrum $I_{\mathrm{in}}$ used as the initial condition. Inset: Variation of experimental or theoretical ($A$, $B$) effective core radius $w\left(\omega \right)∕a$ with the normalized frequency $V\left(\omega \right)\equiv N_{\mathrm{A}}a\omega ∕c$.

Figure 3

Dependence of (a) intensity spectrum $I\left(\nu \right)$, (b) spectral phase $\varphi \left(\nu \right)$, and (c) temporal electric-field profile $E\left(t\right)$ on the initial peak power and the pulse width at $z=0.1\mathrm{mm}$ (left) and $z=0.5\mathrm{mm}$ (right). Those calculated by the FDM, NLSE with the self-steepening, and NLSE without the self-steepening are shown by solid curves, dashed curves, and dotted curves, respectively.

Figure 4

Nonlinear length $L_{\mathrm{N}}$, shock length ${\mathrm{L}}_{\mathrm{S}}$, and dispersion length $L_{\mathrm{LD}}$ (full / half-full / empty circles) as functions of the initial peak power $P$ and the initial FWHM pulse width $t_1$ [$t_1=10.0\mathrm{fs}$ for (b)]. Below $L_{\mathrm{N}}$, the NLSE is applicable as well as the FDM. Refer to Table for headings and abbreviations in (a).

Figure 5

(a) Temporal electric-field profile of the compressed pulse $E_c\left(t\right)$ for $(P,t_1)=(2290\mathrm{kW},5.00\mathrm{fs})$ and $z_{\max }=0.1$, 0.5, and 1.0 mm (left, center, right) calculated by FDM, K-H, and NLSE ($A$, $B$, $C$). (b) Refer to Table for headings and abbreviations. (c) Dependence of $t_c$, the FWHM of the temporal intensity $I_c\left(t\right)\propto {\mid E_c\left(t\right)\mid }^2$, on the propagation distance. $E_c\left(t\right)$ in (a) are used, and $t_c$ by FDM, K-H, and NLSE are shown by full circles, half-full circles, and empty circles, respectively.

Figure 6

Dependence of the parameters in the temporal intensity of the compressed pulse $I_c\left(t\right)$ and the intensity spectrum $I\left(\nu \right)$ of Fig. 3 on the initial peak power $P$ (MW) and the initial pulse width $t_1$. (a) FWHM $t_c$ of $I_c\left(t\right)\left(\mathrm{fs}\right)$. (b) Mean ${\nu }_1$ of $I\left(\nu \right)\left(\mathrm{PHz}\right)$. (c) Height of first lobe $R_{\mathrm{l}}$ relative to main peak in $I_c\left(t\right)$. (d) RMS ${\nu }_2$ of $I\left(\nu \right)\left(\mathrm{PHz}\right)$. Those for the initial pulse width $t_1=10.00$, 5.00, and 2.67 fs are shown by full circles, half-full circles, and empty circles, respectively. They are fitted against $P$ as $t_c=e^{{\alpha }_{\mathrm{w}}}{P}^{{\beta }_{\mathrm{w}}}$, ${\nu }_1=e^{{\alpha }_1}{P}^{{\beta }_1}$, $R_{\mathrm{l}}=e^{{\alpha }_{\mathrm{r}}}{P}^{{\beta }_{\mathrm{r}}}$, and ${\nu }_2=e^{{\alpha }_2}{P}^{{\beta }_2}$ for $t_1=5.00\mathrm{fs}$ and $z_{\max }=0.1∕0.5∕1.0\mathrm{mm}$. Fit parameters are tabulated in (e).

Figure 7

Effect of the carrier envelope phase (CEP) for ${\varphi }_{\mathrm{I}}=0$, $\pi ∕2$, and $\pi$ ($A$, $B$, and $C$). (a) Initial temporal electric-field profile. (b) Temporal electric-field profile after propagation. $\left({\mathrm{b}}^{\prime }\right)$ Temporal electric-field profile difference $\Delta E\left(t\right)$. (c) Spectral phase. (d) Intensity spectrum. $\left({\mathrm{d}}^{\prime }\right)$ Intensity spectrum difference $\Delta I\left(\nu \right)$. In all cases, $(P,t_1)=(6870\mathrm{kW},3.40\mathrm{fs})$, $z=10\lambda$, $B∕F=0.0$, and $C_0=0.0\mathrm{rad}{\mathrm{fs}}^2$, respectively.

Figure 8

Difference caused by the CEP between ${\varphi }_{\mathrm{I}}=0$ and ${\varphi }_{\mathrm{I}}=\pi ∕2$ for the respective propagation distances $z$. Left: temporal electric-field profile difference $\Delta E\left(t\right)$. Right: spectral intensity difference $\Delta I\left(\nu \right)$.

Figure 9

Dependence of (a) relative error and (b) calculation time on the maximum step number $M_{\mathrm{stp}}$ to obtain $E(z,t)$ at $z_{\max }∕100$ from a set of one-dimensionalized Maxwell’s equations with neither the linear dispersion nor the nonlinear term: $A$, by the FDM with $M_{\mathrm{sam}}=2048$; $B$, by the FDM with $M_{\mathrm{sam}}=1024$; $C$, by the FDTD method with $M_{\mathrm{sam}}=2048$; and $D$, by the FDTD method with $M_{\mathrm{sam}}=1024$. In all cases, $t_{\max }=500\mathrm{fs}$ and $z_{\max }=1.0\mathrm{mm}$.

Figure 10

Effect of the linear dispersion keeping the soliton order 1.04 and $\xi =z∕L_{\mathrm{LD}}=35.2$ the same: $a$, by the modified Sellmeier equation [8] $(\mathrm{Ldis}.=\mathrm{S})$; $b$ or $c$, by the propagation constant $\beta \left(\nu \right)$ approximated up to the fifth order or the second order in $\nu -{\nu }_0$ ($\mathrm{Ldis}.=2345$ or 2). (a) Linear refractive index $n_{\mathrm{L}}^{\mathrm{R}}\left(\nu \right)$. (b) Difference of the propagation constant $\Delta \beta \left(\nu \right)=\beta \left(\nu \right)-\beta \left({\nu }_0\right)$. (c) Intensity spectrum (dotted curves in $A$, $D$, and $G$ are initial intensity spectra). (d) Spectral phase. (e) Refer to Table for headings and abbreviations.

Figure 11

Combined effects of the delayed Raman response (D), instantaneous response (I), core dispersion (SE), and the flat core dispersion (SF) shown by intensity spectrum (left) and spectral phase (right) at $z=1.0\mathrm{mm}$ under the condition of [8]: $A$, by (D)-(SE); $B$, by (I)-(SE); $C$, by (D)-(SF); and $D$, by (I)-(SF). The dotted curve with $D$ shows the initial spectrum.

Figure 12

Effects of nonlinear dispersions (self-steepening, core dispersion) shown by intensity spectrum (left) and spectral phase (right) at $z=1.0\mathrm{mm}$: $A$, by the experimental core dispersion and the self-steepening (SE); $B$, by the theoretical core dispersion and the self-steepening (ST); $C$, by the flat core dispersion and only the self-steepening (SF); and $D$, by neither the core dispersion nor the self-steepening (F). The dotted curve with $D$ shows the initial spectrum.