Polarization-Dependent Formation of Extremely Compressed Femtosecond Wave Packets and Supercontinuum Generation in Fused Silica

Abstract

Previous studies of formation of extremely compressed wave packets during femtosecond filamentation in the region of anomalous group velocity dispersion in solid dielectrics mostly considered the case of linearly polarized laser pulses. However, recent results suggest potential applications of polarization state manipulation for ultrafast laser writing of optical structures in bulk solid-state media. In the present work, evolution of radiation polarization parameters during formation of such extreme wave packets at the pump wavelength of 1900 nm in fused silica is studied numerically on the basis of the carrier-resolved unidirectional pulse propagation equation (UPPE). It was revealed that initial close-to-circular polarization leads to higher intensity of the anti-Stokes wing in the spectrum of the generated supercontinuum. Numerical simulations indicate a complex, space–time variant polarization state, and the resulting spatiotemporal electric field distribution exhibits a strong dependence on the initial polarization of the femtosecond pulse. At the same time, electric field polarization tends to linear one in the region with the highest field strength regardless of the initial parameters. The origin of this behavior is attributed to the properties of the supercontinuum components generation during filament-induced plasma formation.

1. Introduction

Nonlinear self-action effects arising during propagation of high-power femtosecond laser pulses in bulk transparent dielectrics lead to the high spatial and temporal localization of electric field energy and filament formation, accompanied by the generation of a broadband supercontinuum and a plethora of other nonlinear optical phenomena [1,2]. Under the conditions of anomalous group velocity dispersion, the interplay between self-phase modulation, self-steepening and dispersive effects causes effective temporal self-compression of a filamenting pulse. The resulting wave packets, extremely compressed both in space and time, are conventionally called “light bullets” [3,4,5,6]. The spectrum of a supercontinuum generated by a light bullet (LB) is marked by the characteristic isolated maximum in the anti-Stokes region (the so-called anti-Stokes wing), which is separated from the pump wavelength by a pronounced dip [7]. The coherent broadband supercontinuum generated by filaments and LBs has many promising applications in parametric light generation, advanced time-resolved spectroscopy, and other areas of ultrafast optics (e.g., see [8,9]).

Previous studies of the formation of filaments and LBs in bulk solid dielectrics primarily focused on the case of linearly polarized laser pulses [10]. However, research on femtosecond filamentation of visible and near-infrared radiation in gases demonstrated the potential to manipulate spectral characteristics of the resulting supercontinuum by altering the polarization state of the initial radiation [11,12]. Moreover, generation of optical harmonics by circularly polarized laser pulses is forbidden in optically isotropic media [13], which leads to significant changes in the pulse spectrum dynamics during filamentation in air [14]. A recent study of polarization dependence of femtosecond pulse spectrum broadening in fused silica showed the possibility of more efficient supercontinuum generation with the circularly polarized pulses under certain conditions [15]. Formation of elliptically polarized LBs in lithium fluoride was considered in [16]—experimental and numerical findings show that transitioning from linear to circular polarization reduces the depth of spatial modulation of long-lived color centers induced by LBs to zero. These results suggest potential real-world applications of LB polarization state manipulation for ultrafast laser writing of optical structures in solid dielectrics. The current consideration of initiated supercontinua from elliptically polarized femtosecond pulses is justified by the recent advances in the experimental technique for full spatiotemporal polarization measurement in ultrafast vector beams [17].

In the present work, we numerically study the evolution of radiation polarization parameters during LB formation in fused silica and investigate the influence of the polarization state of an LB on spectral dynamics of supercontinuum generation.

2. Materials and Methods

Numerical simulation of LB formation and filamentation of radiation with arbitrary ellipticity of polarization is carried out on the basis of the carrier-resolved unidirectional pulse propagation equation (UPPE) [18]. Unlike the slowly evolving wave approximation (SEWA), commonly used for computer simulation of filamentation, the UPPE model does not impose any restrictions on the pulse duration and correctly describes the evolution of all electric field harmonics. The numerical algorithm for solving the UPPE was adapted in order to take into account the temporal lag of high-frequency radiation components due to the significant group velocity dispersion over broadband spectral range [19]. For a nonlinear dispersive isotropic medium, the radially symmetric equations written for the two nonlinearly coupled polarization components of the electric field in the frequency-angular domain ${\widehat{E}}_X\left(\omega ,k_r,z\right),{\widehat{E}}_Y\left(\omega ,k_r,z\right)$ have the following form in the traveling coordinate system comoving with the pulse (here and in what follows, $\widehat{F}\left(\omega ,k_r,z\right)$ denotes the spatiotemporal Fourier transform of the function $F\left(t,r,z\right)$):

$$\begin{array}{l}\frac{\partial \widehat{E_{X|Y}}}{\partial z}=i\left(k_z\left(\omega \right)-\frac{\omega }{v_g\left({\omega }_0\right)}\right)\widehat{E_{X|Y}}\\ +i\frac{{\mu }_0{\omega }^2}{2k_z\left(\omega \right)}\left[{\epsilon }_0{\chi }^{\left(3\right)}\left({\omega }_0\right)\widehat{I{E}_{X|Y}}+\frac{i}{\omega }\left\{\frac{e^2\widehat{N_e{E}_{X|Y}}}{m^{\ast }\left({\nu }_c-i\omega \right)}+U_i\widehat{\left(\frac{\partial N_e}{\partial t}\frac{E_{X|Y}}{I}\right)}\right\}\right],\end{array}$$

(1)

where $I=E_X^2+E_Y^2$, $k_z^2\left(\omega \right)={\left(\omega n\right(\omega )/c)}^2-k_r^2$; $n\left(\omega \right)$ describes the dispersion of fused silica according to the Sellmeier formula, $v_g\left({\omega }_0\right)$ and ${\chi }^{\left(3\right)}\left({\omega }_0\right)$ are the group velocity and the cubic nonlinearity coefficient of the medium at the center frequency of the pulse ${\omega }_0$. Correspondingly, $e$ is the elementary charge, $m^{\ast }$ is the effective reduced electron mass, $N_e$ is the concentration of free electrons, ${\nu }_c$ is the electron-neutral collision frequency, and $U_i$ is the dielectric bandgap width. Equation (1) accounts for diffraction and dispersion of radiation, Kerr and plasma nonlinearities, radiation attenuation due to photoionization, and absorption in the laser plasma.

The following kinetic equation which takes into account the optical field-induced and avalanche ionization as well as the recombination process used to find the concentration of free electrons $N_e$:

$$\frac{\partial N_e}{\partial t}=W\left(E\right)\left(N_0-N_e\right)+{\nu }_i\left(E\right)N_e-\frac{N_e}{{\tau }_{relax}},$$

(2)

where ${\tau }_{relax}$ is the relaxation time of laser plasma, ${\nu }_i=e^2{\left|E\right|}^2{\nu }_c/\left({2U}_i{m}^{\ast }\left({\omega }_0^2+{\nu }_c^2\right)\right)$ is the avalanche ionization rate, and $W\left(E\right)$ is the field ionization rate, which is given by the Keldysh formula in the case of linearly polarized radiation [20]. Recent experimental findings confirmed good quantitative description of the plasma yield based on the Keldysh formula [21]. Unfortunately, a satisfactory generalization of this formula for the case of elliptical polarization has not yet been proposed. However, following the authors of [14], we refer to the description of polarization ellipticity $\epsilon$ influence on the ionization rate from the work [22]:

$$\ln W\left(E\right)~\left(-\frac{2}{3}{r}_H^{1.5}\frac{E_{at}}{E}\left[1-\left(1-\frac{{\epsilon }^2}{3}\right)\frac{{\gamma }^2}{10}\right]\right),$$

(3)

where $r_H=U_i/U_H$, $U_H$ is the ionization potential of a hydrogen atom (used for normalization), $E_{at}\approx 5.2\cdot {10}^{11}\mathrm{V}/\mathrm{m}$ is the atomic unit of the electric field and $\gamma$ is the Keldysh parameter [20]. For LBs with pump wavelengths in near- and mid-infrared, the Keldysh parameter is much less than unity for intensities more than ${10}^{18}\mathrm{W}/{\mathrm{m}}^2$, which are steadily achieved during filamentation in fused silica. Thus, we can neglect terms proportional to ${\gamma }^2$ and describe the field ionization in the tunneling regime using the same formula for all values of the polarization ellipticity.

The following parameters of fused silica were used in the numerical simulations: the cubic nonlinear susceptibility coefficient ${\chi }^{\left(3\right)}\left({\omega }_0\right)=1.3\cdot {10}^{-22}{\mathrm{m}}^2/{\mathrm{V}}^2$, the electron-neutral collision frequency ${\nu }_c={10}^{14}{\mathrm{s}}^{-1}$, the band gap $U_i=9\mathrm{e}\mathrm{V}$, the plasma relaxation time ${\tau }_{relax}=200\mathrm{f}\mathrm{s}$ [20]. The frequency dependence of the nonlinear susceptibility ${\chi }^{\left(3\right)}$ was considered negligible in the infrared range according to the experiments [23]. The recent ab initio calculations of ${\chi }^{\left(3\right)}\left(\omega \right)$ in a wider wavelength range from the mid-infrared to the near ultraviolet allow us to assume weak dispersion of the third-order nonlinearity through the overall supercontinuum extension [24,25]. The initial condition $E_{X|Y}\left(r,t,z=0\right)$ was set as a collimated transform-limited elliptically polarized femtosecond wave packet with a Gaussian spatiotemporal distribution of the electric field amplitude:

$$E_{X|Y}\left(r,t,z=0\right)=E_{0_{X|Y}}\exp \left(-\frac{r^2}{2r_0^2}-\frac{t^2}{2t_0^2}\right)\cos \left(\frac{2\pi c}{{\lambda }_0}t+{\varphi }_{X|Y}\right),$$

(4)

where $E_{0_Y}={{\epsilon E}_0}_X,{\varphi }_X=0,{\varphi }_Y=\pi /2.$ Equation (4) can be viewed as a Gaussian wave packet focused on the front face of the fused silica sample. The duration of the pulse with the central wavelength ${\lambda }_0=1900\mathrm{n}\mathrm{m}$ at the $e^{-1}$ intensity level was $2t_0=84\mathrm{f}\mathrm{s}$, the beam radius at the $e^{-1}$ intensity level—${\mathrm{r}}_0=80\mathsf{\mu }\mathrm{m}$, energy of the pulse—$W=5.7 \mathsf{\mu }\mathrm{J}$, and its peak power—$P=5P_{cr}$, where $P_{cr}=15.2\mathrm{M}\mathrm{W}$ is the critical power of self-focusing. It should be emphasized that we choose the phase shift between the field components $E_X$ and $E_Y$ to be equal to $\pi /2$ and adjust the ellipticity only by varying the parameter $\epsilon$ without any loss of generality, as the optical isotropic properties of fused silica allow us to define the initial rotation of the XY-plane about the Z-axis arbitrarily. Similarly, initial right- and left-handed states of polarization lead to the same electric field dynamics due to the optical isotropy of fused silica.

The simulations were performed on a multiprocessor computer (parallel 12 threads on two Intel Xeon E5-2630 processors, manufactured by Intel Corporation and obtained from a local distributor in Moscow, Russia in 2015) using parallel programming code initially developed by the authors of [26] and adapted by us for the current work.

3. Results

3.1. Polarization-Dependent Light Bullet Supercontinuum

The pulse spectrum evolution during LB formation is shown in Figure 1 for the pulses with close-to-linear $(\epsilon =0.1)$ and close-to-circular $(\epsilon =0.9)$ polarizations. The presented pulse spectra are integrated over transverse spatial coordinates. Note that the self-focusing and the concomitant spectral broadening occur at a larger distance for the close-to-circularly polarized pulse due to the increase in critical power for self-focusing as the electric field polarization deviates from the linear one [27]. The scenario of the spectrum transformation and short-wavelength cutoff shift [28] is the same for both cases, but the partially suppressed odd harmonics generation leads to an increase in both maximal $S_{AS}^{\max }$ and integrated ${S_{\mathsf{\Sigma }}}_{AS}$ spectral intensity of the anti-Stokes wing for the pulse with close-to-circular polarization (Figure 1c,d). In Figure 1, spectral intensity at the central wavelength is designated as $S_0$, and ${S_{\mathsf{\Sigma }}}_{AS}$ is the intensity integrated over the spectral range from 200 to 500 nm, which approximately corresponds to the position of the anti-Stokes wing.

As a result, the spectrum of supercontinuum generated by an LB formed from a close-to-circularly polarized pulse in fused silica can be characterized as having not only a “smoother” shape due to the absence of odd harmonics peaks, which was previously shown for filament propagation of circularly polarized pulses in gases [14], but also higher spectral intensity of the short-wavelength region.

3.2. Dynamics of Polarization Parameters

The polarization state of radiation is commonly characterized by the Stokes parameters [29]. Alternatively, to describe fully polarized states, one can use a set of another three parameters [30]: the intensity, the ellipticity degree of the polarization ellipse $M=2\epsilon \sin \mathsf{\Delta }/(1+{\epsilon }^2),$ and the angle of the polarization ellipse orientation $\Psi =-0.5\arctan [2\epsilon \cos \mathsf{\Delta }/(1-{\epsilon }^2\left)\right]$, where $\mathsf{\Delta }$ is the phase shift between the two components of the electric field with orthogonal polarizations and $\epsilon$ is the ratio of their amplitudes, as already introduced above. “Local” polarization parameters can be defined if the slowly varying envelope approximation can be applied to the description of the ultrafast processes [17]. Nevertheless, as we use the carrier-resolved UPPE model in order to correctly describe the formation of extremely compressed LBs and supercontinuum generation, we face a problem with defining the conventionally used polarization parameters. The classic concept of a polarization ellipse is applicable only in case of a well-defined slowly varying envelope of the electric field, which cannot be properly introduced for the LB. However, following the work [30], it is possible to define integral parameters of the ellipticity degree $M(z,r,t)$ and the angle of polarization ellipse orientation $\Psi (z,r,t)$ as the following functions of discrete time:

$$\left|M\left(z,r,{\tilde{t}}_n\right)\right|=\frac{\sqrt{2I\left(z,r,{\tilde{t}}_n\right)\left[I\left(z,r,{\overline{t}}_n\right)+I\left(z,r,{\overline{t}}_{n+1}\right)\right]}}{I\left(z,r,{\tilde{t}}_n\right)+0.5\left[I\left(z,r,{\overline{t}}_n\right)+I\left(z,r,{\overline{t}}_{n+1}\right)\right]},$$

(5)

$$\Psi \left(z,r,{\tilde{t}}_n\right)=\arctan [E_x\left(z,r,{\tilde{t}}_n\right)/E_y\left(z,r,{\tilde{t}}_n\right)];$$

(6)

which are defined in a set of points ${t=\tilde{t}}_n$, where function $I\left(z,r,t\right)=E_x^2\left(z,r,t\right)+E_y^2\left(z,r,t\right)$ achieves its local maxima. In Equations (5) and (6), ${\overline{t}}_n$ and ${\overline{t}}_{n+1}$ are the local minimum points of $I\left(z,r,t\right)$, chosen in such a way that ${\overline{t}}_n\le {\tilde{t}}_n\le {\overline{t}}_{n+1}$. In the limit of long pulses, the interpolants of these functions tend toward common parameters $M$ and $\Psi$. The sign of $M$ is determined by the direction of the electric field vector rotation, so that $M=+1(-1)$ corresponds to the right (left) circular polarization state, and $M=0$ corresponds to linear polarization.

The hodograph of the electric field on the axis $(r=0)$ of the formed LB (i.e., at a fixed propagation distance corresponding to the LB formation), as well as its non-closed polarization ellipses defined for a quasi-period of electric field oscillations, which are graphically representing the changes in calculated values of $M\left(z,r,{\tilde{t}}_n\right)$ and $\Psi \left(z,r,{\tilde{t}}_n\right)$, are shown in Figure 2 for the initial ellipticity degree of $M_0=0.5$ ($\epsilon =0.27$ in Equation (4)).

It can be seen in Figure 2b–e that the polarization state of radiation trends toward the linear in the vicinity of the region with the strongest electrical field in the LB. At the same time, the polarization ellipse starts to rotate and changes its orientation angle. These dependencies are also shown in Figure 3a for pulses with different initial ellipticity degrees $M$.

Such behavior can be interpreted as breaking of the initial phase relations between orthogonal electric field components $E_x$ and $E_y$ (see Equation (4)). The hodographs of the electric field as well as the orthogonal field components for the initial degrees of ellipticity of $M_0=0.5$ and $M_0=1.0$ are presented in Figure 4. One of the possible reasons for the polarization state transformation lies in the properties of the electric field evolution on the trailing edge of the forming LB, where steep spatiotemporal gradients of Kerr and plasma nonlinearities induced in the medium lead to generation of the anti-Stokes components spanning the ultraviolet spectral region. The short-wavelength spectral components emerge for both orthogonal polarizations simultaneously in the vicinity of the points in the ($r,t$)-space where the filament-induced plasma is generated. This process leads to the effective phase-matching of $E_x$ and $E_y$ (Figure 3b), resulting in the close-to-linear polarization of the electric field with the angle of polarization ellipse rotation approximately equal to a multiple of $\pi /4$. The observed effect is similar to the phase-locking between the fundamental and the third harmonic pulses in a two-color filament, which was reported in [31]. Therefore, it can be said that the large drop in ellipticity M shown in Figure 3a is explained by polarization transformation from the elliptical to the linear due to the efficient phase-locking in the high-intensity LB emerging at the trailing part of the pulse.

The whole picture of spatiotemporal transformation of the electric field and polarization parameters in a formed LB with the initial degree of ellipticity $M_0=0.5$ is shown in Figure 5. The space–time variant polarization state and its convergence to a linear one in the core of the LB are illustrated by the non-closed polarization ellipses presented for several positions in the space-time domain. It should be noted that the region with the highest electric field strength in the LB is shifted in time since the radiation group velocity is intensity-dependent and gradually becomes lower than the constant velocity of the traveling coordinate system.

4. Discussion

Spectral analysis of LB formation in bulk fused silica reveals a clear dependence of the generated supercontinuum on the initial polarization state of radiation. Partially suppressed odd harmonic generation in an LB with close-to-circular polarization leads to higher intensity of the anti-Stokes wing in the supercontinuum spectrum. At the same time, all other features of LB spectral dynamics—i.e., rapid asymmetrical spectral broadening in the vicinity of the main nonlinear focus, the shift of the short-wavelength cutoff to the ultraviolet region, and the emergence of a pronounced spectral dip separating the anti-Stokes wing from the central wavelength—remain the same for pulses with various initial polarization parameters.

Numerical simulations indicate that a formed LB has a complex, space–time variant polarization state, and the resulting electric field distribution exhibits a strong dependence on the initial polarization of the femtosecond pulse—compare Figure 2a with Figure 4a,b. The main revealed feature of the electric field evolution in an LB is that its polarization tends toward the linear in the vicinity of the region with the highest field strength regardless of the initial radiation parameters. As highlighted above, the origin of this behavior should be attributed to the properties of the short-wavelength supercontinuum components generation during filament-induced plasma formation. Generation of the anti-Stokes components arises first of all from the limitation of infinite intensity growth due to the plasma production (see [1] for the detailed review). This process is universal, whether the initial pump pulse radiation state is linear or elliptical. The difference is that in the case of elliptical initial polarization, harmonic generation is strongly reduced, however, self-phase modulation, self-focusing and the plasma production leading together to the supercontinuum generation still persist. All these three latter processes are nonlinear functions of the intensity. Since the electric field components $E_X$ and $E_Y$ influence each other through the intensity (see Equation (1)), that is, the instant field magnitudes are connected through the average one, the efficient phase-locking or phase-matching occurs leading to polarization state transformation from elliptical to linear one.

Interestingly, previous related works on self-focusing and femtosecond filamentation in gases report contrasting results regarding the polarization state evolution in the filament core. According to the findings of [32], both linear and circular polarization states are stable, and any deviations from them disappear in the regions of high radiation intensity in the filament. At the same time, the results of numerical modeling shown in [33] indicate that the center of the filament always tends to be circularly polarized, and it is emphasized that this finding stands in contrast to the classic work of Marburger [27] predicting linear polarization in the center of the self-focusing beam, which is explained by the authors as the impact of the extremely dynamic character of pulse evolution in the process of femtosecond filamentation. The source of such discrepancies with our findings should be sought in the principal difference between the numerical model based on slowly varying envelopes of the electric field used in [32,33] and the carrier-resolved UPPE model that we use in our work. The UPPE equation correctly accounts for LB formation and broadband supercontinuum generation, which allows us to accurately reproduce the effect of phase-locking between orthogonal electric field components $E_x$ and $E_y$ in the LB core.

5. Conclusions

Our findings show that electric field distribution in an LB and the spectrum of a generated supercontinuum exhibit complex polarization-dependent dynamics, which prove the viability of using the polarization state of a femtosecond pulse as a new degree of freedom to control the LB parameters crucial for the prospective practical applications. Moreover, the spectrum of the supercontinuum generated by an LB formed from a close-to-circularly polarized pulse was found to have higher spectral intensity of the short-wavelength region in comparison with the case of linearly polarized radiation. At the same time, the resulting linear polarization state in the core of the LB develops in the same way whatever ellipticity of the initial radiation was chosen. It was revealed that electric field polarization tends toward the linear in the region with the highest field strength due to phase locking of the two orthogonally polarized electric field components. The resulting almost linearly polarized electric field forms the LB and in the spectral domain ensures the maximum conversion efficiency to the supercontinuum. Thus, our research opens the possibility for future experimental investigations of controlled LB formation in bulk solid-state dielectric media.