## Abstract

The observation of discrete lines in the white spectrum at the initial stage of filamentation of powerful femtosecond laser pulses, propagating in silica glasses, as well as the filamentation without plasma channels observed in the experiments in air, pushed us to look for other nonlinear mechanisms for describing these effects. In this paper, we present a new parametric conversion mechanism for asymmetric spectrum broadening of femtosecond laser pulses towards higher frequencies in isotropic media. This mechanism includes cascade generation with THz spectral shift for solids and GHz shift for gases. The process works simultaneously with the four-photon parametric wave mixing. The theoretical model proposed agrees well with the experimental data.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The propagation of a self-guided laser filament through gases and dense dielectric media leads to a rich variety of phenomena and applications [1–3]. For the first time filamentation was reported by A. Braun et al. [1]. They observed the channeled propagation of femtosecond pulses in air. The filamentation started with a nonlinear process of self-focusing of the high power femtosecond pulse. The field intensity in the filament core reached 5 × 10^{13} W/cm^{2}, which was high enough for the initiation of plasma generation process. Hence, the self-focusing effect was compensated by plasma defocusing, and wave-guiding effects were observed. That is why the first theoretical models were focused on the balance between self-focusing and plasma defocusing [4,5]. The filamentation process is accompanied by many other physical processes such as coherent and non-coherent GHz generation [6, 7], rotation of the polarization plane [8], merging and energy exchange between filaments [9–13], optical breakdown in fused silica [14, 15], filament wake and waveguide structures by changing the air density in the filamentation area [16, 17], etc. An important result in recent experiments is the observation of a stable post-filamentation regime observed at several meters from the laser source with intensity of the order of 10^{11}–10^{12} W/cm^{2}. Such intensities are enough for self-focusing but not so high as to ionize the medium [2, 18–21]. Channel-type propagation of the femtosecond pulse has been obtained also in glassy and crystalline transparent matrices [22–24]. For air, such process of high intensity femtosecond laser pulses propagation was characterized by white spectrum generation. The observed spectra in solid-state dielectrics were extended from the infrared to the UV region.

In the present work, we describe a series of experiments with femtosecond pulses, generated by a Ti-sapphire laser (798 nm), propagating through thin silica plates, where we observe blue shifted discrete lines up to the visible region, with 20-30 nm spectral separation. The power of the pulses is selected to be slightly above the self-focusing threshold (before the self-phase modulation to separate the spectral maximum). During the experiments, irreversible changes are not detected in dielectrics. This fact supposes that the ionization of solids in the irradiated area does not produce plasma-initiated defects. Thus, the following question arises:

What is the physical process that yields the discrete asymmetrical spectral broadening obtained for laser pulses propagating in glass, with power near the critical for self-focusing?

We propose an answer to this question in the frame of third order nonlinearity, beyond the Kerr type nonlinearity scope. We present a theoretical model for the nonlinear propagation of the ultrashort laser pulses in terms of cubic approximation (i.e., third harmonic generation (THG) is included). The theoretical model explains the genesis of the non-symmetrical white spectrum of the generated continuum.

## 2. Experimental part

We perform experiments with two different lasers and glass plates. The first one involves a 150 fs pulse propagating through a 5 mm BK7 glass plate, while the second experiment is performed with a 30 fs pulse and 10 mm fused silica glass. The experimental setup in both cases is similar and is presented in Fig. 1. It consists of a Ti-sapphire fs laser **1** (150 or 30 fs), a reflecting mirror **2**, attenuator **3**, focusing lens **4**, sample **5** (either BK7 glass sample or fused silica plate). A spectrometer **6** is situated behind the sample.

In both experiments the laser pulse intensity was of the order of $I={\left|{A}_{0}\right|}^{2}\approx {10}^{9}\text{\hspace{0.17em}}{\text{W/cm}}^{2}.$ The pulse energy was controlled by varying the transparency of the attenuator. Such intensity corresponds to a dimensionless nonlinear coefficient $\gamma ={n}_{2}{\left|{A}_{0}\right|}^{2}{k}_{0}^{2}{D}_{0}^{2}\ge 1$, where ${D}_{0}$ is the beam diameter, ${n}_{2}$ is the nonlinear refractive index, ${A}_{0}$ is the electric field amplitude and ${k}_{0}$ is the carrying wavenumber corresponding to the carrier frequency ${\omega}_{0}$.

#### 2.1. Experiments in BK7 glass

A pulse with λ_{0} = 798 nm (150 fs) propagates through the BK7 glass plate having 5 mm thickness, see Fig. 2(a). The pulse power is slightly above the critical for self-focusing, which corresponds to 450 nJ. The focusing lens has a 25 cm focal length and the diameter of the beam in the sample is 100 µm. This leads to observation of a white spot with a red ring. The spectrum of the propagating through the glass pulse is monitored in the range 400-950 nm and is plotted in Fig. 2(b).

Characteristic spectral peaks in the range from 450 to 870 nm are observed, and some of them meet the conditions for four-photon parametric mixing (FPPM). The high intensity peaks are observed mainly towards the short wavelengths compared to the laser wavelength.

#### 2.2 Experiments in fused silica

In order to characterize the observed peaks, and to evidence their origin mechanism, the experiment is performed in another type of medium – fused silica.

In the second experiment, pulse with λ_{0} = 800 nm (30 fs), propagates through the 10 mm thick fused silica plate. The pulse power is 130 nJ. The beam diameter is 5 mm and its divergence is almost diffraction-limited (1.25 M^{2}). The lens used has focal length of 30 cm. This corresponds to a beam diameter in the sample of 70 µm. The pulse power is slightly above the critical for self-focusing (P_{c} = 2.66 MW). Multifilamentation does not occur. White spot with intensive broad blue and red rings is observed, see Fig. 3(a). The initial pulse spectrum (in red) compared with the spectrum of the pulse propagating through the glass sample (in black) is presented in Fig. 3(b). A typical asymmetric conversion towards the short wavelengths is observed.

## 3. The nonlinear operator of cubic type ${P}_{i}={\chi}_{ijkl}^{\left(3\right)}{E}_{j}{E}_{k}{E}_{l}$

#### 3.1 Scalar approximation. One carrying frequency

In the experiments presented above, the pulses propagate in glass with thickness 5 mm and 10 mm, which is significantly smaller than the diffraction length. The diffraction length of both pulses, with ${d}_{0}\cong 100\text{\hspace{0.17em}}\text{\mu m}$ spot diameter and carrier frequency 800 nm is approximately ${z}_{dif}={k}_{0}{d}_{0}^{2}=7.85\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{cm}$. That is why we neglect the diffraction effects and use a 1 + 1 dimensional approximation of the Slowly Varying Nonlinear Amplitude Equation (SVNAE) to calculate the spectral and longitudinal spatial deformation of the laser pulse propagating through the glass (i.e. we neglect the diffraction and use a weak dispersion parameter only).

The propagation of narrow band (nano- and picosecond) pulses in isotropic materials is characterized by absence of condition for Third Harmonic generation (THG). In the scalar approximation, the nonlinearity is Kerr’s type ${P}_{nl}={n}_{2}{\left|E\right|}^{2}E$. If we use an envelope presentation of the electrical field $E(z,t)=A(z,t)\mathrm{exp}\left[i{k}_{0}\left(z-{v}_{ph}t\right)\right]$, the corresponding nonlinear envelope equation is the nonlinear Schrödinger equation (NSE):

It is well known that the NSE is Galilean invariant and phase independent, i.e. in Laboratory coordinate system the evolution of the solutions of the NSE is the same as the one obtained after transforming Eq. (1) in Galilean or Local time frames. In the case of nonlinear propagation of broad-band fs pulses, the third harmonics term should be taken into account. The scalar approximation $\overrightarrow{E}={E}_{x}\overrightarrow{x}$ of the tensor presentation of the cubic nonlinearity ${P}_{i}={\chi}_{ijkl}^{\left(3\right)}{E}_{j}{E}_{k}{E}_{l}$ on one carrying frequency, leads to only two components:

*Δω*has a THz spectral shift in solids [25]. If we look more carefully into the structure of Eq. (3), for arbitrary localized solution the term connected with the first order dispersion ${v}_{gr}\partial {A}_{x}/dz$ adds an additional phase shift proportional to the group velocity. In Laboratory coordinates this is seen in the third harmonic term, where the corresponding nonlinear operator will generate at frequency:

_{nl}This mathematical treatment has a simple physical explanation. The cubic type nonlinearity is connected with the amplitude maximum of the electrical field. This maximum propagates with the group velocity, therefore its frequency is not the pump frequency ${\omega}_{0}={k}_{0}{v}_{ph}$ but the carrier-to-envelope frequency ${\omega}_{CEF}={k}_{0}({v}_{ph}-{v}_{gr})$. Thus, the “third harmonic” of the absolute frequency ${\omega}_{CEF}={k}_{0}({v}_{ph}-{v}_{gr})$ is ${\omega}_{nl}=3{\omega}_{CEF}=3{k}_{0}({v}_{ph}-{v}_{gr})$ [25].

#### 3.2 Physical model of the avalanche parametric generation

The analysis of the spectra obtained from the experiments shows that some of the signal waves generated have a frequency shift proportional to $\Delta {\omega}_{nl}=3{k}_{0}({v}_{ph}-{v}_{gr})$, which is in the THz region for solids. For glass this THz shift corresponds to a wavelength shift of $\Delta {\lambda}_{10}=\frac{n{\lambda}_{0}^{2}}{2\pi c}\Delta {\omega}_{nl}^{}({\lambda}_{0})~20-30\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{nm}$ with respect to the carrying wavelength (*λ*_{0} = 800 nm) and depends on the glass matrices. The frequency emission lies in the wings of the spectrum of the 100-200 fs pulses, towards the shorter wavelengths (respectively higher frequencies). This leads to asymmetrical spectrum broadening towards the shorter wavelengths. The amplified wavelength *λ*_{1} = *λ*_{0} ‒ *∆λ*_{10} of the optical wave is near the spectral maximum, and if the intensity is high enough, new emission becomes possible at a different spectral distance $\Delta {\lambda}_{21}=\frac{n{\lambda}_{1}^{2}}{2\pi c}\Delta {\omega}_{nl}^{}({\lambda}_{1})\text{\hspace{0.17em}}.$ This difference is due to the fact that the basic wave for the new generation is the new frequency ${\omega}_{1}$. As a result, a next spectral line shifted at *λ*_{2} = *λ*_{1} ‒ ∆*λ*_{21} will appear. It is important to point out that the spectral shifts ∆*λ*_{21} and ∆*λ*_{10} are not equal. This observation plays a key role in the identification of the experimental spectra below. The joint action of this cascade process, along with the processes of four-photon mixing (FPP), would yield also the emission of weaker signals at frequencies ${\omega}_{3}=2{\omega}_{0}-{\omega}_{1}$ and ${\omega}_{4}=2{\omega}_{0}-{\omega}_{2}$. Since ${\omega}_{3}$ is shifted from ${\omega}_{0}$ towards the lower frequency (longer wavelengths), it can act as a basic wave in the process of generation of a signal with THz shift in the direction of the pump (${\omega}_{0}$) wave. Thus, the result is again an energy transfer from ${\omega}_{3}$ to a wave near${\omega}_{0}$ and emission towards the shorter wavelengths.

The above described mechanism of combined action of emission with THz shift and FPM processes could describe the asymmetrical ultra-broadening of the filament’s spectrum from the IR toward the Vis region. To prove this physical model, we need to use the dispersion $n(\lambda )$ of the materials [26] and to calculate first the group-phase velocity difference.

Figure 4(a) presents the nonlinear frequency shift for BK7 glass, which at 800 nm is calculated as:

To evidence the model proposed, we present on the same plot the experimental spectra and the calculated positions of the spectral peaks (which are determined by the shifts$\Delta {\lambda}_{ij}$) for both materials. The results for BK7 are shown in Fig. 5, and for the fused silica – in Fig. 6. It should be noted again that for the second case $\Delta {\lambda}_{10}$ is smaller, and the spectral width of the laser pulse is higher, so the characteristic peaks are less resolved. Nevertheless, the very good match between the experimental measurements and the theoretical calculations is also seen, which is a good demonstration supporting the validity of the proposed mechanism.

## 4. Mathematical model of the avalanche evolution

In order to understand the physical mechanisms governing the evolution of the process, we model theoretically the nonlinear dynamics of this avalanche formation.

#### 4.1 Two carrying frequencies

The nonlinear tensor in the case of two carrying frequencies (scalar approximation) has six components:

_{2}= λ

_{1}‒ ∆λ

_{21}. Taking into account that the glass samples have a finite length, this term can be neglected if we use thin enough samples. In this case the system (11), where the nonlinear term ${A}_{2}^{3}\mathrm{exp}(-i\delta Kz)$ is neglected, is transformed into:

#### 4.2 Numerical experiment with seven waves

If we denote the waves and their wavelengths in the following manner:

*nm*, in our numerical model we use equal normalized values$\Delta {\widehat{\lambda}}_{ij}=\Delta \lambda $. The waves with wavelengths shorter than ${\lambda}_{1}={\lambda}_{0}$ are located to the left and have even indices, while the waves at wavelengths longer than ${\lambda}_{1}={\lambda}_{0}$ are located to the right and have odd indices. The system of equations in the case of seven waves includes three different types of wave synchronisms:

- a) Six conditions for THz wave synchronism:$$\begin{array}{l}{\omega}_{1}+\Delta {\omega}_{nl}={\omega}_{2},\text{\hspace{1em}}{\omega}_{2}+\Delta {\omega}_{nl}={\omega}_{4},\text{\hspace{1em}}{\omega}_{4}+\Delta {\omega}_{nl}={\omega}_{6},\\ {\omega}_{3}+\Delta {\omega}_{nl}={\omega}_{1},\text{\hspace{1em}}{\omega}_{5}+\Delta {\omega}_{nl}={\omega}_{3},\text{\hspace{1em}}{\omega}_{7}+\Delta {\omega}_{nl}={\omega}_{5}.\end{array}$$

The system of seven equations, including nonlinear terms with the above synchronisms is:

Figure 7 presents the results from the numerical simulation of the spectrum evolution of the spectrally narrow 150 fs pulse with a power slightly above the critical for self-focusing over a distance of 5 mm. The numerical calculations are carried out for the following parameters $\gamma =0.7<1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta}_{j}=0.01$ and initial constants:

As seen in Fig. 7 the signal waves are spectrally shifted at a distance $\Delta \lambda =25.1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{nm}$ and are situated in the wings of the pump spectrum. That is why pulses with relatively narrow spectrum have well resolved and clearly seen spectral maxima of the signal waves. The nonlinear processes that lead to this spectral deformation can be better understood by plotting the evolution of each of the signal waves, see Fig. 8. The initial pulse A_{1} yields (by THz synchronism) the generation of a blue-shifted wave A_{2}. The combined action of A_{1} and A_{2} leads to the generation of a red-shifted A_{3} wave, by the FPP process. Further on, A_{2} yields a blue-shifted A_{4} wave due to the THz synchronism, while A_{3} is reduced, since it transfers its energy by the THz synchronism to a position, which in our model coincides with the position of A_{1}. The main result is that the total energy transfer will be towards the shorter wavelengths.

The calculated energies ${U}_{i}\sim {\displaystyle \int {\left|{A}_{i}\right|}^{2}},$ *i* = 2...7, of each signal wave in the process of interaction, presented in Fig. 9, show a significant energy transfer to the shorter wavelengths.

In the experiments with the relatively broadband 30 fs pulse, the spectral shifts of the generated signal waves are closer to the pump maximum. Therefore, the generated spectral lines overlap within the pump profile and are not well resolved. The result of the numerical simulation of the broadband pulse is presented in Fig. 10.

## 5. Conclusions

The presented experiments on femtosecond pulses with power slightly above the critical for self-focusing, propagating in two types of glass plates, demonstrate an effective transfer of energy from the IR to the Vis region. The absence of ionization in these processes pushed us to look for other nonlinear mechanisms for the origin of these effects. We present a new parametric conversion mechanism for asymmetric spectrum broadening of femtosecond laser pulses towards the higher frequencies in isotropic media. This mechanism includes cascade generation with THz spectral shift for solids, which is proportional to $\Delta {\omega}_{nl}=3{k}_{0}({v}_{ph}-{v}_{gr})$. This process works simultaneously with the four-photon parametric wave mixing. The proposed theoretical model shows excellent agreement with the experimental data.

## Funding

Bulgarian National Science Fund (Grant No. DN–18/11 and Grant No. DN–18/7).

## Acknowledgments

The authors would like to thank Dr. Christina Andrews for proofreading the manuscript.

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