Self-phase modulation and frequency generation with few-cycle optical pulses in nonlinear dispersive media

We analyze nonlinear effects associated with the spatiotemporal propagation of few-cycle optical pulses in nonlinear dispersive media, including nonlinearity-induced self-phase modulation, generation of higher harmonics, and the effects of diffraction. First, we discuss the nonlinear equations governing the spatiotemporal propagation of short pulses in dispersive and diffractive nonlinear media and demonstrate a link between the field equation and the cubic nonlinear Schrödinger equation employed for describing the evolution of the pulse envelope. Then, for several different regimes of competitive scales, we study the self-action effects of few-cycle pulses and describe, both analytically and numerically, the pulse self-modulation and self-focusing, including the transformation of the spectral density and harmonic generation in the cases of weak and strong dispersion. Finally, we analyze the effect of the beam diffraction on self-action of the Gaussian few-cycle pulses.

Figure 1

Example of the propagation of a single-cycle paraxial beam in a nonlinear medium. The beam undergoes strong temporal reshaping and diffracting in space.

Figure 2

Dispersion of stoichiometric MgO:LiNbO${}_3$ crystal: (a) real part of the refractive index $\mathrm{Re}\left(n_0\right)$ and (b) absorption coefficient $k_i$ vs the frequency. Solid line: experimental data according to Ref. [42]. Dashed blue line: approximate dependence based on Eq. (3) with parameters $N_0=4.734$, $a=2.224\times {10}^{-38}{\mathrm{s}}^3{\mathrm{cm}}^{-1}$, $K_0=1.55{\mathrm{cm}}^{-1}$, and $K_1=1.32\times {10}^{-26}{\mathrm{s}}^2{\mathrm{cm}}^{-1}$. Grey shading: characteristic spectrum ($\left|G\right|$) of a single-period pulse with the central frequency 1 THz.

Figure 3

(a) Electric-field profile of a single-cycle light pulse defined by Eq. (19) and (b) its spectrum according to Eq. (20).

Figure 4

Nonlinearly induced change of temporal and spectral structures of a single-cycle optical pulse. (a and b) Nonlinearly induced corrections to (a) field $(L_{\text{wave}}/L_{\text{nl}}){\tilde{E}}^{\left(1\right)}$ and (b) modulus of spectrum $(L_{\text{wave}}/L_{\text{nl}}){\tilde{G}}^{\left(1\right)}$ calculated based on Eqs. (22) and (23), respectively. (c and d) The total field and spectrum resulting field according to Eqs. (24) and (25), respectively. The propagation distance is $z^{\prime }=7$ and $L_{\text{wave}}/L_{\text{nl}}=0.5$.

Figure 5

(a and c) Field of a Gaussian optical pulse and (b and c) the corresponding optical spectrum calculated according to Eqs. (28) and (29), respectively. The normalized pulse duration ${\tau }_p/T_0$ is (a and b) 2 and (c and d) $0.3$. The Gaussian pulse envelope is shown by a dashed line in (a).

Figure 6

(a and b) The Gaussian pulse (a) field and (b) spectrum vs the normalized duration ${\tau }_p/T_0$ according to Eqs. (28) and (29), respectively. (c and d) Nonlinearly induced corrections to the pulse (c) field and (d) spectrum vs the normalized duration according to Eqs. (30) and (31), respectively. The white line in (d) marks the minimum of the spectral correction.

Figure 7

Numerical simulation of pulse propagation for different input durations ${\tau }_p/T_0$. (a and b) The output pulse (a) field and (b) spectrum. (c and d) Differences between the output and input (c) field and (d) spectrum. The white line in (d) marks the minimum of the spectral correction. Parameters are $L_{\text{nl}}=7$ mm and $L_{\text{disp}}=L_{\text{dif}}=\infty$. Propagation distance is $z=7$ mm.

Figure 8

Numerical simulation of pulse propagation for different input durations ${\tau }_p/T_0$. (a and b) The output pulse (a) field and (b) spectrum under the effects of nonlinear self-action and dispersion, $L_{\text{nl}}=L_{\text{disp}}=7$ mm. (c and d) Nonlinear corrections for the output (c) field and (d) spectrum calculated as a difference between results in (a and b) and linear simulations when $L_{\text{nl}}=\infty$. For all the plots $L_{\text{dif}}=\infty$, and the propagation distance is $z=7$ mm. The white lines in (d) mark the minima of the spectral correction.

Figure 9

(a and c) Electric fields and (b and d) spectra for the case $L_{\text{disp}}=7$ mm, $L_{\text{nl}}=$ 7 mm. (e–h) The nonlinear correction to the fields and output spectra for the cases $L_{\text{disp}}=7$ mm, $L_{\text{nl}}=$ 7 mm and $L_{\text{disp}}=7$ mm, $L_{\text{nl}}=\infty$. (a, b, e, and f) Output profiles at $z=$ 7 mm. (c, d, g, and h) Density plots showing evolution along the propagation direction $z$. For all the plots the input pulse duration is ${\tau }_p/T_0\cong$ 0.71.

Figure 10

Numerical simulation of pulse propagation for different dispersion strengths defined as inverse dispersion length $L_{\text{disp}}^{-1}$. (a and b) The output pulse (a) field and (b) spectrum under the effects of nonlinear self-action, $L_{\text{nl}}=7$ mm. (c and d) Nonlinear corrections for the output (c) field and (d) spectrum calculated as a difference between results in (a and b) and linear simulations when $L_{\text{nl}}=\infty$. For all the plots $L_{\text{dif}}=\infty$, the pulse duration is ${\tau }_p/T_0=0.3$, and the propagation distance is $z=7$ mm.

Figure 11

(a, c, and e) Spatiotemporal evolution of the electric-field and (b, d, and f) corresponding spatiospectral profiles of the spectrum modulus at distances (a and b) $z=0$, (c and d) $z=15$ mm, and (e and f) $z=30$ mm. Input pulse duration is ${\tau }_p/T_0=$0.3 and the width of the Gaussian transverse profile is $r_0=$ 8.3${\lambda }_0$. Parameters are $L_{\text{disp}}=L_{\text{nl}}=L_{\text{dif}}=$ 35 mm.

Figure 12

(a, c, and e) Spatiotemporal evolution of the electric-field and (b, d, and f) corresponding spatiospectral profiles of the spectrum modulus at distances (a and b) $z=0$, (c and d) $z=4$ mm, and (e and f) $z=8$ mm. Input pulse duration is ${\tau }_p/T_0=$0.3 and the width of the Gaussian transverse profile is $r_0=$ 8.3${\lambda }_0$. Parameters are $L_{\text{disp}}=L_{\text{dif}}=$ 35 mm and $L_{\text{nl}}=$7 mm.

Figure 13

(a, c, and e) Spatiotemporal evolution of the electric-field and (b, d, and f) corresponding spatiospectral profiles of the spectrum modulus at distances (a and b) $z=0$, (c and d) $z=2.5$ mm, and (e and f) $z=5$ mm. Input pulse duration is ${\tau }_p/T_0=$ 1.0 and the width of the Gaussian transverse profile is $r_0=$ 8.3${\lambda }_0$. Parameters are $L_{\text{disp}}=L_{\text{dif}}=$ 35 mm and $L_{\text{nl}}=7$ mm.