Avalanche parametric conversion and white spectrum generation from infrared femtosecond pulses in glasses

Daniela A. Georgieva; Todor. S. Petrov; Hitoki Yoneda; Rakish Shikne; Nikolay N. Nedyalkov; Lubomir M. Kovachev

doi:10.1364/OE.26.017649

Avalanche parametric conversion and white spectrum generation from infrared femtosecond pulses in glasses

Daniela A. Georgieva, Todor. S. Petrov, Hitoki Yoneda, Rakish Shikne, Nikolay N. Nedyalkov, and Lubomir M. Kovachev

Optics Express
Vol. 26,
Issue 13,
pp. 17649-17661
(2018)
•https://doi.org/10.1364/OE.26.017649

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Daniela A. Georgieva, Todor. S. Petrov, Hitoki Yoneda, Rakish Shikne, Nikolay N. Nedyalkov, and Lubomir M. Kovachev, "Avalanche parametric conversion and white spectrum generation from infrared femtosecond pulses in glasses," Opt. Express 26, 17649-17661 (2018)

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Related Topics
Table of Contents Category
- Nonlinear Optics
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Nonlinear optics, four-wave mixing (190.4380)
- Nonlinear optics, parametric processes (190.4410)
- Nonlinear wave mixing (190.4223)

About this Article
History
- Original Manuscript: February 28, 2018
- Revised Manuscript: June 16, 2018
- Manuscript Accepted: June 18, 2018
- Published: June 22, 2018

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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.

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 × 10¹³ W/cm², 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 10¹¹–10¹² W/cm². 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.

Fig. 1 Experimental setup for observation of the initial filamentation in glass. 1- Ti-sapphire fs laser, 2- reflecting mirror, 3- attenuator, 4- focusing lens, 5- BK7 glass sample or fused silica plate, 6- spectrometer.

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In both experiments the laser pulse intensity was of the order of $I = {| A_{0} |}^{2} \approx 10^{9} {W/cm}^{2} .$ The pulse energy was controlled by varying the transparency of the attenuator. Such intensity corresponds to a dimensionless nonlinear coefficient $γ = n_{2} {| A_{0} |}^{2} k_{0}^{2} D_{0}^{2} \geq 1$ , 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 $ω_{0}$ .

2.1. Experiments in BK7 glass

A pulse with λ₀ = 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).

Fig. 2 (a) A white spot with a red ring obtained with the infrared pulse propagating through the BK7 glass sample. (b) The spectrum of the narrow-band pulse propagating through the glass. Discrete spectral components from 450 to 870 nm are seen: the typical asymmetric conversion towards the short wavelengths shows that we observe the initial stage of the filamentation process.

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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 λ₀ = 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 M²). 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.

Fig. 3 (a) The observed white spot with intensive broad blue and red rings of the pulse propagating through the fused silica sample. (b) The initial pulse spectrum (in red) compared with the spectrum of the pulse propagating through the material (in black). The spectrum modification significantly smoother with small and less resolved spectral maxima, see also Fig. 6.

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3. The nonlinear operator of cubic type $P_{i} = χ_{i j k l}^{(3)} E_{j} E_{k} E_{l}$

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 $d_{0} ≅ 100 μm$ spot diameter and carrier frequency 800 nm is approximately $z_{d i f} = k_{0} d_{0}^{2} = 7.85 cm$ . 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 $P_{n l} = n_{2} {| E |}^{2} E$ . If we use an envelope presentation of the electrical field $E (z, t) = A (z, t) \exp [i k_{0} (z - v_{p h} t)]$ , the corresponding nonlinear envelope equation is the nonlinear Schrödinger equation (NSE):

- 2 i k_{0} (\frac{\partial A}{\partial z} + \frac{1}{v_{g r}} \frac{\partial A}{\partial t}) = - k_{0} k " \frac{\partial^{2} A}{\partial t^{2}} + k_{0}^{2} {\tilde{n}}_{2} {| A |}^{2} A,

where

v_{g r}

is the group velocity,

{\tilde{n}}_{2} = \frac{3}{8} n_{2}

is the nonlinear refractive index and

{k^{″}}_{0}

is the dispersion of the group velocity.

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 $\vec{E} = E_{x} \vec{x}$ of the tensor presentation of the cubic nonlinearity $P_{i} = χ_{i j k l}^{(3)} E_{j} E_{k} E_{l}$ on one carrying frequency, leads to only two components:

χ_{x x x x}^{(3)} (3 ω = ω + ω + ω) and χ_{x x x x}^{(3)} (ω = ω + ω - ω) .

The corresponding non-paraxial equation in Laboratory frame (neglecting the diffraction) is:

2 i k_{0} [\frac{\partial A}{\partial z} + \frac{1}{v_{g r}} \frac{\partial A}{\partial t}] = \frac{\partial^{2} A}{\partial z^{2}} - (\frac{1 + β}{v_{g r}^{2}}) \frac{\partial^{2} A}{\partial t^{2}} + k_{0}^{2} n_{2} ({| A |}^{2} A + 3 A^{3} e^{- 2 i (k_{0} z - ω_{0} t)}) .

The same equation written in a frame moving with the group velocity

(z' = z - v_{g r} t; t' = t)

and neglecting the non-paraxial terms, becomes as follows (the primes are removed for clarity):

- 2 i \frac{k_{0}}{v_{g r}} \frac{\partial A_{x}}{\partial t} = - β \frac{\partial^{2} A_{x}}{\partial z^{2}} + k_{0}^{2} {\tilde{n}}_{2} [\frac{1}{3} A_{x}^{3} \exp (2 i k_{0} (z - (v_{p h} - v_{g r}) t)) + {| A_{x} |}^{2} A_{x}],

where

v_{p h}

is the phase velocity,

β = k_{0} v_{g r}^{2} {k^{″}}_{0}

and

z_{0} = v_{g r} t_{0}

. These equations do not contain the self-steepening term, because for such short distances the shock term is negligibly small. Comparing Eq. (3) and Eq. (4), it can be supposed that in this case the Galilean invariance and phase independence are broken. In Eq. (3), beside the self-action term, there is a nonlinear operator describing third harmonic generation

3 ω_{0} = 3 k_{0} v_{p h}

, while in Eq. (4) the new frequency is equal to

Δ ω_{n l} = 3 ω_{C E F} = 3 k_{0} (v_{p h} - v_{g r})

, with

ω_{C E F}

being the Carrier-to-Envelope Frequency (CEF). This frequency Δω_nl 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_{g r} \partial A_{x} / d z

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:

ω_{n l} = 3 k_{0} (v_{p h} - v_{g r}) = 3 ω_{C E F} .

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 $ω_{0} = k_{0} v_{p h}$ but the carrier-to-envelope frequency $ω_{C E F} = k_{0} (v_{p h} - v_{g r})$ . Thus, the “third harmonic” of the absolute frequency $ω_{C E F} = k_{0} (v_{p h} - v_{g r})$ is $ω_{n l} = 3 ω_{C E F} = 3 k_{0} (v_{p h} - v_{g r})$ [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 $Δ ω_{n l} = 3 k_{0} (v_{p h} - v_{g r})$ , which is in the THz region for solids. For glass this THz shift corresponds to a wavelength shift of $Δ λ_{10} = \frac{n λ_{0}^{2}}{2 π c} Δ ω_{n l}^{} (λ_{0}) ~ 20 - 30 nm$ with respect to the carrying wavelength (λ₀ = 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 λ₁ = λ₀ ‒ ∆λ₁₀ 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 $Δ λ_{21} = \frac{n λ_{1}^{2}}{2 π c} Δ ω_{n l}^{} (λ_{1}) .$ This difference is due to the fact that the basic wave for the new generation is the new frequency $ω_{1}$ . As a result, a next spectral line shifted at λ₂ = λ₁ ‒ ∆λ₂₁ will appear. It is important to point out that the spectral shifts ∆λ₂₁ and ∆λ₁₀ 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 $ω_{3} = 2 ω_{0} - ω_{1}$ and $ω_{4} = 2 ω_{0} - ω_{2}$ . Since $ω_{3}$ is shifted from $ω_{0}$ 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 ( $ω_{0}$ ) wave. Thus, the result is again an energy transfer from $ω_{3}$ to a wave near $ω_{0}$ 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 $n (λ)$ 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:

Δ ω_{n l}^{B K 7} (800 nm) = 3 k_{0} (v_{p h} - v_{g r}) = 48.78 THz .

This corresponds to a spectral shift towards the short wavelengths of:

Δ λ_{10}^{B K 7} = \frac{n λ_{0}^{2}}{2 π c} Δ ω_{n l}^{} (λ_{0}) = 25 .1 nm .

Figure 4(b) presents the nonlinear frequency shift for fused silica, which at 800 nm is:

Δ ω_{n l}^{s i l i c a} (800 nm) = 3 k_{0} (v_{p h} - v_{g r}) = 45.8 THz,

and the spectral distance towards the short wavelengths is:

Fig. 4 (a) The nonlinear frequency shift for BK7 glass $Δ ω_{n l}^{B K 7} = 3 ω_{c e p} = 3 k_{0} (v_{p h} - v_{g r})$ in the region 780 nm to 2 µm. The frequency shift at 800 nm is $Δ ω_{n l}^{B K 7} (800 nm) = 48.78 THz .$ (b) The nonlinear frequency shift for fused silica in the region 780 nm to 2 µm. The frequency shift at 800 nm is $Δ ω_{n l}^{s i l i c a} (800 nm) = 45.8 THz .$

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Δ λ_{10}^{s i l i c a} = \frac{n λ_{0}^{2}}{2 π c} Δ ω_{n l}^{} (λ_{0}) = 22 .6 nm .

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 $Δ λ_{i j}$ ) 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 $Δ λ_{10}$ 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.

Fig. 5 Comparison of the calculated shifts $Δ λ_{i j} = \frac{n λ_{j}^{2}}{2 π c} Δ ω_{n l}^{} (λ_{j})$ towards the short wavelengths obtained from the physical model (colored vertical lines) with the experimental spectrum for BK7 glass (black line).

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Fig. 6 Comparison between the calculated spectral shifts $Δ λ_{i j}$ obtained from the physical model (colored vertical lines) with the experimental spectrum for fused silica (black line).

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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:

\begin{array}{l} χ_{x x x x}^{(3)} (3 ω_{1} = ω_{1} + ω_{1} + ω_{1}), χ_{x x x x}^{(3)} (ω_{1} = ω_{1} + ω_{2} - ω_{2}), χ_{x x x x}^{(3)} (ω_{1} = ω_{1} + ω_{1} - ω_{1}), \\ χ_{x x x x}^{(3)} (3 ω_{2} = ω_{2} + ω_{2} + ω_{2}), χ_{x x x x}^{(3)} (ω_{2} = ω_{2} + ω_{1} - ω_{1}), χ_{x x x x}^{(3)} (ω_{2} = ω_{2} + ω_{2} - ω_{2}) . \end{array}

The system of equations written in a frame moving with the group velocity in normalized coordinate becomes:

\begin{array}{l} i \frac{\partial A_{1}}{\partial t} = - β_{1} \frac{\partial^{2} A_{1}}{\partial z^{2}} + γ_{1} ({| A_{1} |}^{2} A_{1} + 2 {| A_{2} |}^{2} A_{1} + A_{1}^{* 2} A_{2} e x p (- i δ k z)), \\ i \frac{\partial A_{2}}{\partial t} = - β_{2} \frac{\partial^{2} A_{2}}{\partial z^{2}} + γ_{2} ({| A_{2} |}^{2} A_{2} + 2 {| A_{1} |}^{2} A_{2} + A_{1}^{3} e x p (i δ k z) + A_{2}^{3} e x p (- i δ K z)), \end{array}

where

δ k = (k_{2} - 3 k_{1}) / 2 k_{1}

and

δ K = (k_{3} - 3 k_{2}) / 2 k_{2}

denote the normalized wave mismatch vectors,

β_{i} = k_{0}^{"} v_{g r}^{2} / z_{0}

are the dimensionless dispersion parameters and

γ_{i} = n_{2} {| A_{0} |}^{2} k_{i} z_{0}^{2}

are the nonlinear coefficients. As the waves are spectrally close, the mismatch wave numbers are practically negligible

δ k ≅ δ K \approx 10^{- 2} - 10^{- 3}

. The above presented physical model, as well as Eqs. (11), show that the system is not conservative. The nonlinear term

A_{2}^{3} \exp (- i δ K z)

generates at a new shorter wavelength λ₂ = λ₁ ‒ ∆λ₂₁. 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} \exp (- i δ K z)

is neglected, is transformed into:

\begin{array}{l} i \frac{\partial A_{1}}{\partial t} = - β_{1} \frac{\partial^{2} A_{1}}{\partial z^{2}} + γ_{1} ({| A_{1} |}^{2} A_{1} + 2 {| A_{2} |}^{2} A_{1} + A_{1}^{* 2} A_{2} e x p (- i δ k z)) \\ i \frac{\partial A_{2}}{\partial t} = - β_{2} \frac{\partial^{2} A_{2}}{\partial z^{2}} + γ_{2} ({| A_{2} |}^{2} A_{2} + 2 {| A_{1} |}^{2} A_{2} + A_{1}^{3} e x p (i δ k z)), \end{array}

and obeys the Manley-Raw conservation relations.

4.2 Numerical experiment with seven waves

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

λ_{6} \overset{Δ}{\leftarrow} λ_{4} \overset{Δ}{\leftarrow} λ_{2} \overset{Δ}{\leftarrow} λ_{1} \overset{Δ}{\leftarrow} λ_{3} \overset{Δ}{\leftarrow} λ_{5} \overset{Δ}{\leftarrow} λ_{7},

with

λ_{1} = λ_{0}

being the wavelength of the pump wave, then the normalized spectral shift between the waves is

Δ {\hat{λ}}_{i j} = (\frac{n λ_{0}^{2}}{2 π c} Δ ω_{n l}^{} (λ_{j})) / λ_{j}

. Since the spectral differences between

Δ λ_{i j}

are less than 1 nm, in our numerical model we use equal normalized values

Δ {\hat{λ}}_{i j} = Δ λ

. The waves with wavelengths shorter than

λ_{1} = λ_{0}

are located to the left and have even indices, while the waves at wavelengths longer than

λ_{1} = λ_{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: $\begin{array}{l} ω_{1} + Δ ω_{n l} = ω_{2}, ω_{2} + Δ ω_{n l} = ω_{4}, ω_{4} + Δ ω_{n l} = ω_{6}, \\ ω_{3} + Δ ω_{n l} = ω_{1}, ω_{5} + Δ ω_{n l} = ω_{3}, ω_{7} + Δ ω_{n l} = ω_{5} . \end{array}$
b) Nine FPP mismatch conditions between three waves of the kind: $ω_{i} + ω_{j} = 2 ω_{k} .$
c) Eight FPP mismatch conditions between four waves of the kind: $ω_{i} + ω_{j} = ω_{k} + ω_{l} .$

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

i \frac{\partial A_{j}}{\partial t} = - β_{j} \frac{\partial^{2} A_{j}}{\partial z^{2}} + γ_{j} P_{n l}^{j}, j = 1 \dots 7,

with polarization components:

\begin{array}{l} P_{n l}^{j} = ({| A_{j} |}^{2} + \sum_{k \neq j}^{7} 2 {| A_{k} |}^{2}) A_{j} + \frac{1}{3} A_{j - Δ}^{3} \exp (- i δ k z) + A_{j}^{*}^{2} A_{j + Δ} \exp (i δ k z) \\ + 2 A_{j}^{*} A_{j + Δ} A_{j - Δ} \exp (i δ k z) + 2 A_{j}^{*} A_{j + 2 Δ} A_{j - 2 Δ} \exp (2 i δ k z) + 2 A_{j}^{*} A_{j + 3 Δ} A_{j - 3 Δ} \exp (3 i δ k z) \\ + A_{j \pm 1 Δ}^{2} A_{j \pm 2 Δ}^{*} \exp (- i δ k z) + A_{j \pm 2 Δ}^{2} A_{j \pm 4 Δ}^{*} \exp (- i 2 δ k z) + A_{j + Δ}^{*} A_{j - Δ} A_{j + 2 Δ} \exp (i δ k z) \\ + A_{j - Δ}^{*} A_{j + Δ} A_{j - 2 Δ} \exp (i δ k z) + A_{j + 3 Δ}^{*} A_{j + Δ} A_{j + 2 Δ} \exp (- i δ k z) + A_{j - 3 Δ}^{*} A_{j - Δ} A_{j - 2 Δ} \exp (- i δ k z) \\ + A_{j + 2 Δ}^{*} A_{j - 2 Δ} A_{j + 4 Δ} \exp (i δ k z) + A_{j - 2 Δ}^{*} A_{j + 2 Δ} A_{j - 4 Δ} \exp (i δ k z) + A_{j + 6 Δ}^{*} A_{j + 4 Δ} A_{j + 2 Δ} \exp (- i δ k z) \\ + A_{j - 6 Δ}^{*} A_{j - 4 Δ} A_{j - 2 Δ} \exp (- i δ k z), j = 1 \dots 7. \end{array}

We use localized initial conditions in the form:

A_{j} = A_{0 j} \exp (i ϕ_{j}) \exp (i Δ k_{j} z) \exp (- z^{2} / 2 z_{0 j}^{2}), j = 1 \dots 7,

where

A_{0 j}

are the amplitude constants,

ϕ_{j}

are the initial phases,

Δ k_{j} = 2 π / Δ λ_{j}

are the spectral shifts between the carrying wave numbers of the waves, and

z_{0 j}

are the initial widths of the pulses. The numerical experiments are performed by using the split-step Fourier method.

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 $γ = 0.7 < 1, β_{j} = 0.01$ and initial constants:

Fig. 7 Evolution of the spectrum and generation of signal waves from an initial 150 fs pulse in BK7 glass. The results are obtained by numerical calculation of the system of Eqs. (17)-(18) under initial conditions (19)-(20). The total spectrum of the pulse becomes asymmetrical with spectral maxima at distances $Δ λ \approx 25 nm$ . The spectral maxima are well resolved, compare with Fig. 2(b) and Fig. 5.

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\begin{array}{l} A_{01} = 1, A_{02} = A_{03} = 0.01, A_{04} = A_{05} = 0.01, A_{06} = A_{07} = 0.005, \\ z_{01} = 1, z_{02} = z_{03} = 15, δ k = 0.001 \\ Δ λ_{1} = 0, Δ λ_{2} = 2.5, Δ λ_{3} = - 2.5; Δ λ_{4} = 5, Δ λ_{5} = 5, Δ λ_{6} = 7.5, Δ λ_{7} = - 7.5. \end{array}

As seen in Fig. 7 the signal waves are spectrally shifted at a distance $Δ λ = 25.1 nm$ 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 A₁ yields (by THz synchronism) the generation of a blue-shifted wave A₂. The combined action of A₁ and A₂ leads to the generation of a red-shifted A₃ wave, by the FPP process. Further on, A₂ yields a blue-shifted A₄ wave due to the THz synchronism, while A₃ is reduced, since it transfers its energy by the THz synchronism to a position, which in our model coincides with the position of A₁. The main result is that the total energy transfer will be towards the shorter wavelengths.

Fig. 8 Evolution of the spectra of each of the seven pulses in the process of nonlinear interaction in BK7 glass. The initial pulse A₁ yields (by THz synchronism) the generation of a blue-shifted wave A₂. The combined action of A₁ and A₂ leads to the generation of a red-shifted A₃ wave, by the FPP process. Further on, A₂ yields a blue-shifted A₄ wave, due to the THz synchronism, while A₃ is reduced, since it transfers its energy by the THz synchronism to a position, which in our model coincides with the position of A₁. The generation of signal waves with THz shift combined with the FPP processes leads to significant increase of the pulses situated to the short wavelengths.

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The calculated energies $U_{i} \sim \int {| A_{i} |}^{2},$ 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.

Fig. 9 The calculated energies of each signal wave during the process of interaction: (a) even indexed waves, i.e., the waves generated to the shorter wavelengths with respect to the pump; (b) energy transfer of the odd indexed waves, i.e. the waves generated to the longer wavelengths with respect to the pump. A significant energy transfer to the short wavelengths is observed:

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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.

Fig. 10 The spectral evolution and generation of signal waves from the broadband 30 fs pulse at different distances for fused silica. The total spectrum of the pulse becomes asymmetrical. The shape of the spectrum at 10 mm reveals the same overlapping of the spectral lines as in the real experiment, compare with Fig. 3(b) and Fig. 6.

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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 $Δ ω_{n l} = 3 k_{0} (v_{p h} - v_{g r})$ . 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.

References and links

1. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995). [CrossRef] [PubMed]

2. S. L. Chin, T.-J. Wang, C. Marceau, J. Wu, J. S. Liu, O. Kosareva, N. Panov, Y. P. Chen, J.-F. Daigle, S. Yuan, A. Azarm, W. W. Liu, T. Seideman, H. P. Zeng, M. Richardson, R. Li, and Z. Z. Xu, “Advances in Intense Femtosecond Laser Filamentation in Air,” Laser Phys. 22(1), 1–53 (2012). [CrossRef]

3. L. Berge, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007). [CrossRef]

4. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007). [CrossRef]

5. M. Kolesik and J. V. Moloney, “Perturbative and non-perturbative aspects of optical filamentation in bulk dielectric media,” Opt. Express 16(5), 2971–2988 (2008). [CrossRef] [PubMed]

6. S. Tzortzakis, G. Méchain, G. Patalano, Y.-B. André, B. Prade, M. Franco, A. Mysyrowicz, J.-M. Munier, M. Gheudin, G. Beaudin, and P. Encrenaz, “Coherent Subterahertz Radiation from Femtosecond Infrared Filaments in Air,” Opt. Lett. 27(21), 1944–1946 (2002). [CrossRef] [PubMed]

7. C. D’Amico, A. Houard, M. Franco, B. Prade, and A. Mysyrowicz, “Coherent and incoherent radial THz radiation emission from femtosecond filaments in air,” Opt. Express 15(23), 15274–15279 (2007). [CrossRef] [PubMed]

8. O. Kosareva, N. Panov, V. Makarov, I. Perezhogin, C. Marceau, Y. Chen, S. Yuan, T. Wang, H. Zeng, A. Savel’ev, and S. L. Chin, “Polarization rotation due to femtosecond filamentation in an atomic gas,” Opt. Lett. 35(17), 2904–2906 (2010). [CrossRef] [PubMed]

9. S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Breakup and Fusion of Self-Guided Femtosecond Light Pulses in Air,” Phys. Rev. Lett. 86(24), 5470–5473 (2001). [CrossRef] [PubMed]

10. L. Bergé, M. R. Schmidt, J. Juul Rasmussen, P. L. Christiansen, and K. Rasmussen, “Amalgamation of interacting light beamlets in Kerr-type media,” J. Opt. Soc. Am. B 14(10), 2550–2562 (1997). [CrossRef]

11. Y. Liu, M. Durand, S. Chen, A. Houard, B. Prade, B. Forestier, and A. Mysyrowicz, “Energy exchange between femtosecond laser filaments in air,” Phys. Rev. Lett. 105(5), 055003 (2010). [CrossRef] [PubMed]

12. P. Ding, Z. Guo, X. Wang, Y. Cao, M. Sun, P. Zhao, Y. Shi, S. Sun, X. Liu, and B. Hu, “Energy exchange between two noncollinear filament-forming laser pulses in air,” Opt. Express 21(23), 27631–27640 (2013). [CrossRef] [PubMed]

13. L. M. Kovachev, D. A. Georgieva, and A. M. Dakova, “Influence of the four-photon parametric processes and cross-phase modulation on the relative motion of optical filaments,” Laser Phys. 25(10), 105402 (2015). [CrossRef]

14. N. T. Nguyen, A. Saliminia, S. L. Chin, and R. Vallée, “Control of femtosecond laser written waveguides in silica glass,” Appl. Phys. B 85(1), 145–148 (2006). [CrossRef]

15. E. Toratani, M. Kamata, and M. Obara, “Self-fabrication of void array in fused silica by femtosecond laser processing,” Appl. Phys. Lett. 87(17), 171103 (2005). [CrossRef]

16. O. Lahav, L. Levi, I. Orr, R. A. Nemirovsky, J. Nemirovsky, I. Kaminer, M. Segev, and O. Cohen, “Long-lived waveguides and sound-wave generation by laser filamentation,” Phys. Rev. A 90(2), 021801 (2014). [CrossRef]

17. J. K. Wahlstrand, N. Jhajj, E. W. Rosenthal, S. Zahedpour, and H. M. Milchberg, “Direct imaging of the acoustic waves generated by femtosecond filaments in air,” Opt. Lett. 39(5), 1290–1293 (2014). [CrossRef] [PubMed]

18. S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E. 70(4), 046602 (2004). [CrossRef] [PubMed]

19. G. Méchain, A. Couairon, Y.-B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new regime without ionization,” Appl. Phys. B 79(3), 379–382 (2004). [CrossRef]

20. S. Champeaux and L. Bergé, “Postionization regimes of femtosecond laser pulses self-channeling in air,” Phys. Rev. E. 71(4), 046604 (2005). [CrossRef] [PubMed]

21. J.-F. Daigle, O. Kosareva, N. Panov, T. -J. Wang, S. Hosseini, S. Yuan, G. Roy, and S. L. Chin, “Formation and evolution of intense, postfilamentation, ionization-free low divergence beams,” Opt. Commun. 284(14), 3601–3606 (2011). [CrossRef]

22. S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-Guided Propagation of Ultrashort IR Laser Pulses in Fused Silica,” Phys. Rev. Lett. 87(21), 213902 (2001). [CrossRef] [PubMed]

23. S. Skupin and L. Bergé, “Self-guiding of femtosecond light pulses in condensed media: Plasma: generation versus chromatic dispersion,” Physica D 220(1), 14–30 (2006). [CrossRef]

24. L. Bergé, S. Mauger, and S. Skupin, “Multifilamentation of powerful optical pulses in silica,” Phys. Rev. A 81(1), 013817 (2010). [CrossRef]

25. D. Georgieva, L. Kovachev, and N. Nedyalkov, “Avalanche parametric conversion in the initial moment of filamentation,” Proc. SPIE 10226, 102261G (2017). [CrossRef]

26. www.refractiveindex.info

References

View by:
Article Order
|
Year
|
Author
|
Publication

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
[Crossref] [PubMed]
S. L. Chin, T.-J. Wang, C. Marceau, J. Wu, J. S. Liu, O. Kosareva, N. Panov, Y. P. Chen, J.-F. Daigle, S. Yuan, A. Azarm, W. W. Liu, T. Seideman, H. P. Zeng, M. Richardson, R. Li, and Z. Z. Xu, “Advances in Intense Femtosecond Laser Filamentation in Air,” Laser Phys. 22(1), 1–53 (2012).
[Crossref]
L. Berge, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[Crossref]
A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007).
[Crossref]
M. Kolesik and J. V. Moloney, “Perturbative and non-perturbative aspects of optical filamentation in bulk dielectric media,” Opt. Express 16(5), 2971–2988 (2008).
[Crossref] [PubMed]
S. Tzortzakis, G. Méchain, G. Patalano, Y.-B. André, B. Prade, M. Franco, A. Mysyrowicz, J.-M. Munier, M. Gheudin, G. Beaudin, and P. Encrenaz, “Coherent Subterahertz Radiation from Femtosecond Infrared Filaments in Air,” Opt. Lett. 27(21), 1944–1946 (2002).
[Crossref] [PubMed]
C. D’Amico, A. Houard, M. Franco, B. Prade, and A. Mysyrowicz, “Coherent and incoherent radial THz radiation emission from femtosecond filaments in air,” Opt. Express 15(23), 15274–15279 (2007).
[Crossref] [PubMed]
O. Kosareva, N. Panov, V. Makarov, I. Perezhogin, C. Marceau, Y. Chen, S. Yuan, T. Wang, H. Zeng, A. Savel’ev, and S. L. Chin, “Polarization rotation due to femtosecond filamentation in an atomic gas,” Opt. Lett. 35(17), 2904–2906 (2010).
[Crossref] [PubMed]
S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Breakup and Fusion of Self-Guided Femtosecond Light Pulses in Air,” Phys. Rev. Lett. 86(24), 5470–5473 (2001).
[Crossref] [PubMed]
L. Bergé, M. R. Schmidt, J. Juul Rasmussen, P. L. Christiansen, and K. Rasmussen, “Amalgamation of interacting light beamlets in Kerr-type media,” J. Opt. Soc. Am. B 14(10), 2550–2562 (1997).
[Crossref]
Y. Liu, M. Durand, S. Chen, A. Houard, B. Prade, B. Forestier, and A. Mysyrowicz, “Energy exchange between femtosecond laser filaments in air,” Phys. Rev. Lett. 105(5), 055003 (2010).
[Crossref] [PubMed]
P. Ding, Z. Guo, X. Wang, Y. Cao, M. Sun, P. Zhao, Y. Shi, S. Sun, X. Liu, and B. Hu, “Energy exchange between two noncollinear filament-forming laser pulses in air,” Opt. Express 21(23), 27631–27640 (2013).
[Crossref] [PubMed]
L. M. Kovachev, D. A. Georgieva, and A. M. Dakova, “Influence of the four-photon parametric processes and cross-phase modulation on the relative motion of optical filaments,” Laser Phys. 25(10), 105402 (2015).
[Crossref]
N. T. Nguyen, A. Saliminia, S. L. Chin, and R. Vallée, “Control of femtosecond laser written waveguides in silica glass,” Appl. Phys. B 85(1), 145–148 (2006).
[Crossref]
E. Toratani, M. Kamata, and M. Obara, “Self-fabrication of void array in fused silica by femtosecond laser processing,” Appl. Phys. Lett. 87(17), 171103 (2005).
[Crossref]
O. Lahav, L. Levi, I. Orr, R. A. Nemirovsky, J. Nemirovsky, I. Kaminer, M. Segev, and O. Cohen, “Long-lived waveguides and sound-wave generation by laser filamentation,” Phys. Rev. A 90(2), 021801 (2014).
[Crossref]
J. K. Wahlstrand, N. Jhajj, E. W. Rosenthal, S. Zahedpour, and H. M. Milchberg, “Direct imaging of the acoustic waves generated by femtosecond filaments in air,” Opt. Lett. 39(5), 1290–1293 (2014).
[Crossref] [PubMed]
S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E. 70(4), 046602 (2004).
[Crossref] [PubMed]
G. Méchain, A. Couairon, Y.-B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new regime without ionization,” Appl. Phys. B 79(3), 379–382 (2004).
[Crossref]
S. Champeaux and L. Bergé, “Postionization regimes of femtosecond laser pulses self-channeling in air,” Phys. Rev. E. 71(4), 046604 (2005).
[Crossref] [PubMed]
J.-F. Daigle, O. Kosareva, N. Panov, T. -J. Wang, S. Hosseini, S. Yuan, G. Roy, and S. L. Chin, “Formation and evolution of intense, postfilamentation, ionization-free low divergence beams,” Opt. Commun. 284(14), 3601–3606 (2011).
[Crossref]
S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-Guided Propagation of Ultrashort IR Laser Pulses in Fused Silica,” Phys. Rev. Lett. 87(21), 213902 (2001).
[Crossref] [PubMed]
S. Skupin and L. Bergé, “Self-guiding of femtosecond light pulses in condensed media: Plasma: generation versus chromatic dispersion,” Physica D 220(1), 14–30 (2006).
[Crossref]
L. Bergé, S. Mauger, and S. Skupin, “Multifilamentation of powerful optical pulses in silica,” Phys. Rev. A 81(1), 013817 (2010).
[Crossref]
D. Georgieva, L. Kovachev, and N. Nedyalkov, “Avalanche parametric conversion in the initial moment of filamentation,” Proc. SPIE 10226, 102261G (2017).
[Crossref]
www.refractiveindex.info

2017 (1)

D. Georgieva, L. Kovachev, and N. Nedyalkov, “Avalanche parametric conversion in the initial moment of filamentation,” Proc. SPIE 10226, 102261G (2017).
[Crossref]

2015 (1)

L. M. Kovachev, D. A. Georgieva, and A. M. Dakova, “Influence of the four-photon parametric processes and cross-phase modulation on the relative motion of optical filaments,” Laser Phys. 25(10), 105402 (2015).
[Crossref]

2014 (2)

O. Lahav, L. Levi, I. Orr, R. A. Nemirovsky, J. Nemirovsky, I. Kaminer, M. Segev, and O. Cohen, “Long-lived waveguides and sound-wave generation by laser filamentation,” Phys. Rev. A 90(2), 021801 (2014).
[Crossref]

J. K. Wahlstrand, N. Jhajj, E. W. Rosenthal, S. Zahedpour, and H. M. Milchberg, “Direct imaging of the acoustic waves generated by femtosecond filaments in air,” Opt. Lett. 39(5), 1290–1293 (2014).
[Crossref] [PubMed]

2013 (1)

P. Ding, Z. Guo, X. Wang, Y. Cao, M. Sun, P. Zhao, Y. Shi, S. Sun, X. Liu, and B. Hu, “Energy exchange between two noncollinear filament-forming laser pulses in air,” Opt. Express 21(23), 27631–27640 (2013).
[Crossref] [PubMed]

2012 (1)

S. L. Chin, T.-J. Wang, C. Marceau, J. Wu, J. S. Liu, O. Kosareva, N. Panov, Y. P. Chen, J.-F. Daigle, S. Yuan, A. Azarm, W. W. Liu, T. Seideman, H. P. Zeng, M. Richardson, R. Li, and Z. Z. Xu, “Advances in Intense Femtosecond Laser Filamentation in Air,” Laser Phys. 22(1), 1–53 (2012).
[Crossref]

2011 (1)

J.-F. Daigle, O. Kosareva, N. Panov, T. -J. Wang, S. Hosseini, S. Yuan, G. Roy, and S. L. Chin, “Formation and evolution of intense, postfilamentation, ionization-free low divergence beams,” Opt. Commun. 284(14), 3601–3606 (2011).
[Crossref]

2010 (3)

L. Bergé, S. Mauger, and S. Skupin, “Multifilamentation of powerful optical pulses in silica,” Phys. Rev. A 81(1), 013817 (2010).
[Crossref]

O. Kosareva, N. Panov, V. Makarov, I. Perezhogin, C. Marceau, Y. Chen, S. Yuan, T. Wang, H. Zeng, A. Savel’ev, and S. L. Chin, “Polarization rotation due to femtosecond filamentation in an atomic gas,” Opt. Lett. 35(17), 2904–2906 (2010).
[Crossref] [PubMed]

Y. Liu, M. Durand, S. Chen, A. Houard, B. Prade, B. Forestier, and A. Mysyrowicz, “Energy exchange between femtosecond laser filaments in air,” Phys. Rev. Lett. 105(5), 055003 (2010).
[Crossref] [PubMed]

2008 (1)

M. Kolesik and J. V. Moloney, “Perturbative and non-perturbative aspects of optical filamentation in bulk dielectric media,” Opt. Express 16(5), 2971–2988 (2008).
[Crossref] [PubMed]

2007 (3)

L. Berge, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[Crossref]

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007).
[Crossref]

C. D’Amico, A. Houard, M. Franco, B. Prade, and A. Mysyrowicz, “Coherent and incoherent radial THz radiation emission from femtosecond filaments in air,” Opt. Express 15(23), 15274–15279 (2007).
[Crossref] [PubMed]

2006 (2)

N. T. Nguyen, A. Saliminia, S. L. Chin, and R. Vallée, “Control of femtosecond laser written waveguides in silica glass,” Appl. Phys. B 85(1), 145–148 (2006).
[Crossref]

S. Skupin and L. Bergé, “Self-guiding of femtosecond light pulses in condensed media: Plasma: generation versus chromatic dispersion,” Physica D 220(1), 14–30 (2006).
[Crossref]

2005 (2)

S. Champeaux and L. Bergé, “Postionization regimes of femtosecond laser pulses self-channeling in air,” Phys. Rev. E. 71(4), 046604 (2005).
[Crossref] [PubMed]

E. Toratani, M. Kamata, and M. Obara, “Self-fabrication of void array in fused silica by femtosecond laser processing,” Appl. Phys. Lett. 87(17), 171103 (2005).
[Crossref]

2004 (2)

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E. 70(4), 046602 (2004).
[Crossref] [PubMed]

G. Méchain, A. Couairon, Y.-B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new regime without ionization,” Appl. Phys. B 79(3), 379–382 (2004).
[Crossref]

2002 (1)

S. Tzortzakis, G. Méchain, G. Patalano, Y.-B. André, B. Prade, M. Franco, A. Mysyrowicz, J.-M. Munier, M. Gheudin, G. Beaudin, and P. Encrenaz, “Coherent Subterahertz Radiation from Femtosecond Infrared Filaments in Air,” Opt. Lett. 27(21), 1944–1946 (2002).
[Crossref] [PubMed]

2001 (2)

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Breakup and Fusion of Self-Guided Femtosecond Light Pulses in Air,” Phys. Rev. Lett. 86(24), 5470–5473 (2001).
[Crossref] [PubMed]

S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-Guided Propagation of Ultrashort IR Laser Pulses in Fused Silica,” Phys. Rev. Lett. 87(21), 213902 (2001).
[Crossref] [PubMed]

1997 (1)

L. Bergé, M. R. Schmidt, J. Juul Rasmussen, P. L. Christiansen, and K. Rasmussen, “Amalgamation of interacting light beamlets in Kerr-type media,” J. Opt. Soc. Am. B 14(10), 2550–2562 (1997).
[Crossref]

1995 (1)

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
[Crossref] [PubMed]

André, Y.-B.

Azarm, A.

Beaudin, G.

Berge, L.

Bergé, L.

L. Bergé, S. Mauger, and S. Skupin, “Multifilamentation of powerful optical pulses in silica,” Phys. Rev. A 81(1), 013817 (2010).
[Crossref]

S. Skupin and L. Bergé, “Self-guiding of femtosecond light pulses in condensed media: Plasma: generation versus chromatic dispersion,” Physica D 220(1), 14–30 (2006).
[Crossref]

S. Champeaux and L. Bergé, “Postionization regimes of femtosecond laser pulses self-channeling in air,” Phys. Rev. E. 71(4), 046604 (2005).
[Crossref] [PubMed]

Bourayou, R.

Braun, A.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
[Crossref] [PubMed]

Cao, Y.

Champeaux, S.

S. Champeaux and L. Bergé, “Postionization regimes of femtosecond laser pulses self-channeling in air,” Phys. Rev. E. 71(4), 046604 (2005).
[Crossref] [PubMed]

Chen, S.

Chen, Y.

Chen, Y. P.

Chin, S. L.

N. T. Nguyen, A. Saliminia, S. L. Chin, and R. Vallée, “Control of femtosecond laser written waveguides in silica glass,” Appl. Phys. B 85(1), 145–148 (2006).
[Crossref]

Christiansen, P. L.

Cohen, O.

Couairon, A.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007).
[Crossref]

D’Amico, C.

Daigle, J.-F.

Dakova, A. M.

Ding, P.

Du, D.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
[Crossref] [PubMed]

Durand, M.

Encrenaz, P.

Forestier, B.

Franco, M.

Georgieva, D.

D. Georgieva, L. Kovachev, and N. Nedyalkov, “Avalanche parametric conversion in the initial moment of filamentation,” Proc. SPIE 10226, 102261G (2017).
[Crossref]

Georgieva, D. A.

Gheudin, M.

Guo, Z.

Hosseini, S.

Houard, A.

Hu, B.

Jhajj, N.

Juul Rasmussen, J.

Kamata, M.

E. Toratani, M. Kamata, and M. Obara, “Self-fabrication of void array in fused silica by femtosecond laser processing,” Appl. Phys. Lett. 87(17), 171103 (2005).
[Crossref]

Kaminer, I.

Kasparian, J.

Kolesik, M.

M. Kolesik and J. V. Moloney, “Perturbative and non-perturbative aspects of optical filamentation in bulk dielectric media,” Opt. Express 16(5), 2971–2988 (2008).
[Crossref] [PubMed]

Korn, G.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
[Crossref] [PubMed]

Kosareva, O.

Kovachev, L.

D. Georgieva, L. Kovachev, and N. Nedyalkov, “Avalanche parametric conversion in the initial moment of filamentation,” Proc. SPIE 10226, 102261G (2017).
[Crossref]

Kovachev, L. M.

Lahav, O.

Lederer, F.

Levi, L.

Li, R.

Liu, J. S.

Liu, W. W.

Liu, X.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
[Crossref] [PubMed]

Liu, Y.

Makarov, V.

Marceau, C.

Mauger, S.

L. Bergé, S. Mauger, and S. Skupin, “Multifilamentation of powerful optical pulses in silica,” Phys. Rev. A 81(1), 013817 (2010).
[Crossref]

Méchain, G.

Méjean, G.

Milchberg, H. M.

Moloney, J. V.

M. Kolesik and J. V. Moloney, “Perturbative and non-perturbative aspects of optical filamentation in bulk dielectric media,” Opt. Express 16(5), 2971–2988 (2008).
[Crossref] [PubMed]

Mourou, G.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
[Crossref] [PubMed]

Munier, J.-M.

Mysyrowicz, A.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007).
[Crossref]

Nedyalkov, N.

D. Georgieva, L. Kovachev, and N. Nedyalkov, “Avalanche parametric conversion in the initial moment of filamentation,” Proc. SPIE 10226, 102261G (2017).
[Crossref]

Nemirovsky, J.

Nemirovsky, R. A.

Nguyen, N. T.

N. T. Nguyen, A. Saliminia, S. L. Chin, and R. Vallée, “Control of femtosecond laser written waveguides in silica glass,” Appl. Phys. B 85(1), 145–148 (2006).
[Crossref]

Nuter, R.

Obara, M.

E. Toratani, M. Kamata, and M. Obara, “Self-fabrication of void array in fused silica by femtosecond laser processing,” Appl. Phys. Lett. 87(17), 171103 (2005).
[Crossref]

Orr, I.

Panov, N.

Patalano, G.

Perezhogin, I.

Peschel, U.

Prade, B.

Rasmussen, K.

Richardson, M.

Rodriguez, M.

Rosenthal, E. W.

Roy, G.

Saliminia, A.

N. T. Nguyen, A. Saliminia, S. L. Chin, and R. Vallée, “Control of femtosecond laser written waveguides in silica glass,” Appl. Phys. B 85(1), 145–148 (2006).
[Crossref]

Salmon, E.

Sauerbrey, R.

Savel’ev, A.

Schmidt, M. R.

Segev, M.

Seideman, T.

Shi, Y.

Skupin, S.

L. Bergé, S. Mauger, and S. Skupin, “Multifilamentation of powerful optical pulses in silica,” Phys. Rev. A 81(1), 013817 (2010).
[Crossref]

S. Skupin and L. Bergé, “Self-guiding of femtosecond light pulses in condensed media: Plasma: generation versus chromatic dispersion,” Physica D 220(1), 14–30 (2006).
[Crossref]

Squier, J.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
[Crossref] [PubMed]

Sudrie, L.

Sun, M.

Sun, S.

Toratani, E.

E. Toratani, M. Kamata, and M. Obara, “Self-fabrication of void array in fused silica by femtosecond laser processing,” Appl. Phys. Lett. 87(17), 171103 (2005).
[Crossref]

Tzortzakis, S.

Vallée, R.

N. T. Nguyen, A. Saliminia, S. L. Chin, and R. Vallée, “Control of femtosecond laser written waveguides in silica glass,” Appl. Phys. B 85(1), 145–148 (2006).
[Crossref]

Wahlstrand, J. K.

Wang, T.

Wang, T. -J.

Wang, T.-J.

Wang, X.

Wolf, J. P.

Wolf, J.-P.

Wöste, L.

Wu, J.

Xu, Z. Z.

Yu, J.

Yuan, S.

Zahedpour, S.

Zeng, H.

Zeng, H. P.

Zhao, P.

Appl. Phys. B (2)

N. T. Nguyen, A. Saliminia, S. L. Chin, and R. Vallée, “Control of femtosecond laser written waveguides in silica glass,” Appl. Phys. B 85(1), 145–148 (2006).
[Crossref]

Appl. Phys. Lett. (1)

E. Toratani, M. Kamata, and M. Obara, “Self-fabrication of void array in fused silica by femtosecond laser processing,” Appl. Phys. Lett. 87(17), 171103 (2005).
[Crossref]

J. Opt. Soc. Am. B (1)

Laser Phys. (2)

Opt. Commun. (1)

Opt. Express (3)

M. Kolesik and J. V. Moloney, “Perturbative and non-perturbative aspects of optical filamentation in bulk dielectric media,” Opt. Express 16(5), 2971–2988 (2008).
[Crossref] [PubMed]

Opt. Lett. (4)

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
[Crossref] [PubMed]

Phys. Rep. (1)

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007).
[Crossref]

Phys. Rev. A (2)

L. Bergé, S. Mauger, and S. Skupin, “Multifilamentation of powerful optical pulses in silica,” Phys. Rev. A 81(1), 013817 (2010).
[Crossref]

Phys. Rev. E. (2)

S. Champeaux and L. Bergé, “Postionization regimes of femtosecond laser pulses self-channeling in air,” Phys. Rev. E. 71(4), 046604 (2005).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

Physica D (1)

S. Skupin and L. Bergé, “Self-guiding of femtosecond light pulses in condensed media: Plasma: generation versus chromatic dispersion,” Physica D 220(1), 14–30 (2006).
[Crossref]

Proc. SPIE (1)

D. Georgieva, L. Kovachev, and N. Nedyalkov, “Avalanche parametric conversion in the initial moment of filamentation,” Proc. SPIE 10226, 102261G (2017).
[Crossref]

Rep. Prog. Phys. (1)

Other (1)

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Fig. 4 (a) The nonlinear frequency shift for BK7 glass

Δ ω_{n l}^{B K 7} = 3 ω_{c e p} = 3 k_{0} (v_{p h} - v_{g r})

in the region 780 nm to 2 µm. The frequency shift at 800 nm is

Δ ω_{n l}^{B K 7} (800 nm) = 48.78 THz .

(b) The nonlinear frequency shift for fused silica in the region 780 nm to 2 µm. The frequency shift at 800 nm is

Δ ω_{n l}^{s i l i c a} (800 nm) = 45.8 THz .

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Fig. 5 Comparison of the calculated shifts

Δ λ_{i j} = \frac{n λ_{j}^{2}}{2 π c} Δ ω_{n l}^{} (λ_{j})

towards the short wavelengths obtained from the physical model (colored vertical lines) with the experimental spectrum for BK7 glass (black line).

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Fig. 6 Comparison between the calculated spectral shifts

Δ λ_{i j}

obtained from the physical model (colored vertical lines) with the experimental spectrum for fused silica (black line).

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Δ λ \approx 25 nm

. The spectral maxima are well resolved, compare with Fig. 2(b) and Fig. 5.

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Equations (20)

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- 2 i k_{0} (\frac{\partial A}{\partial z} + \frac{1}{v_{g r}} \frac{\partial A}{\partial t}) = - k_{0} k " \frac{\partial^{2} A}{\partial t^{2}} + k_{0}^{2} {\tilde{n}}_{2} {| A |}^{2} A,

χ_{x x x x}^{(3)} (3 ω = ω + ω + ω) and χ_{x x x x}^{(3)} (ω = ω + ω - ω) .

2 i k_{0} [\frac{\partial A}{\partial z} + \frac{1}{v_{g r}} \frac{\partial A}{\partial t}] = \frac{\partial^{2} A}{\partial z^{2}} - (\frac{1 + β}{v_{g r}^{2}}) \frac{\partial^{2} A}{\partial t^{2}} + k_{0}^{2} n_{2} ({| A |}^{2} A + 3 A^{3} e^{- 2 i (k_{0} z - ω_{0} t)}) .

- 2 i \frac{k_{0}}{v_{g r}} \frac{\partial A_{x}}{\partial t} = - β \frac{\partial^{2} A_{x}}{\partial z^{2}} + k_{0}^{2} {\tilde{n}}_{2} [\frac{1}{3} A_{x}^{3} \exp (2 i k_{0} (z - (v_{p h} - v_{g r}) t)) + {| A_{x} |}^{2} A_{x}],

ω_{n l} = 3 k_{0} (v_{p h} - v_{g r}) = 3 ω_{C E F} .

Δ ω_{n l}^{B K 7} (800 nm) = 3 k_{0} (v_{p h} - v_{g r}) = 48.78 THz .

Δ λ_{10}^{B K 7} = \frac{n λ_{0}^{2}}{2 π c} Δ ω_{n l}^{} (λ_{0}) = 25 .1 nm .

Δ ω_{n l}^{s i l i c a} (800 nm) = 3 k_{0} (v_{p h} - v_{g r}) = 45.8 THz,

Δ λ_{10}^{s i l i c a} = \frac{n λ_{0}^{2}}{2 π c} Δ ω_{n l}^{} (λ_{0}) = 22 .6 nm .

\begin{array}{l} χ_{x x x x}^{(3)} (3 ω_{1} = ω_{1} + ω_{1} + ω_{1}), χ_{x x x x}^{(3)} (ω_{1} = ω_{1} + ω_{2} - ω_{2}), χ_{x x x x}^{(3)} (ω_{1} = ω_{1} + ω_{1} - ω_{1}), \\ χ_{x x x x}^{(3)} (3 ω_{2} = ω_{2} + ω_{2} + ω_{2}), χ_{x x x x}^{(3)} (ω_{2} = ω_{2} + ω_{1} - ω_{1}), χ_{x x x x}^{(3)} (ω_{2} = ω_{2} + ω_{2} - ω_{2}) . \end{array}

\begin{array}{l} i \frac{\partial A_{1}}{\partial t} = - β_{1} \frac{\partial^{2} A_{1}}{\partial z^{2}} + γ_{1} ({| A_{1} |}^{2} A_{1} + 2 {| A_{2} |}^{2} A_{1} + A_{1}^{* 2} A_{2} e x p (- i δ k z)), \\ i \frac{\partial A_{2}}{\partial t} = - β_{2} \frac{\partial^{2} A_{2}}{\partial z^{2}} + γ_{2} ({| A_{2} |}^{2} A_{2} + 2 {| A_{1} |}^{2} A_{2} + A_{1}^{3} e x p (i δ k z) + A_{2}^{3} e x p (- i δ K z)), \end{array}

\begin{array}{l} i \frac{\partial A_{1}}{\partial t} = - β_{1} \frac{\partial^{2} A_{1}}{\partial z^{2}} + γ_{1} ({| A_{1} |}^{2} A_{1} + 2 {| A_{2} |}^{2} A_{1} + A_{1}^{* 2} A_{2} e x p (- i δ k z)) \\ i \frac{\partial A_{2}}{\partial t} = - β_{2} \frac{\partial^{2} A_{2}}{\partial z^{2}} + γ_{2} ({| A_{2} |}^{2} A_{2} + 2 {| A_{1} |}^{2} A_{2} + A_{1}^{3} e x p (i δ k z)), \end{array}

λ_{6} \overset{Δ}{\leftarrow} λ_{4} \overset{Δ}{\leftarrow} λ_{2} \overset{Δ}{\leftarrow} λ_{1} \overset{Δ}{\leftarrow} λ_{3} \overset{Δ}{\leftarrow} λ_{5} \overset{Δ}{\leftarrow} λ_{7},

\begin{array}{l} ω_{1} + Δ ω_{n l} = ω_{2}, ω_{2} + Δ ω_{n l} = ω_{4}, ω_{4} + Δ ω_{n l} = ω_{6}, \\ ω_{3} + Δ ω_{n l} = ω_{1}, ω_{5} + Δ ω_{n l} = ω_{3}, ω_{7} + Δ ω_{n l} = ω_{5} . \end{array}

ω_{i} + ω_{j} = 2 ω_{k} .

ω_{i} + ω_{j} = ω_{k} + ω_{l} .

i \frac{\partial A_{j}}{\partial t} = - β_{j} \frac{\partial^{2} A_{j}}{\partial z^{2}} + γ_{j} P_{n l}^{j}, j = 1 \dots 7,

\begin{array}{l} P_{n l}^{j} = ({| A_{j} |}^{2} + \sum_{k \neq j}^{7} 2 {| A_{k} |}^{2}) A_{j} + \frac{1}{3} A_{j - Δ}^{3} \exp (- i δ k z) + A_{j}^{*}^{2} A_{j + Δ} \exp (i δ k z) \\ + 2 A_{j}^{*} A_{j + Δ} A_{j - Δ} \exp (i δ k z) + 2 A_{j}^{*} A_{j + 2 Δ} A_{j - 2 Δ} \exp (2 i δ k z) + 2 A_{j}^{*} A_{j + 3 Δ} A_{j - 3 Δ} \exp (3 i δ k z) \\ + A_{j \pm 1 Δ}^{2} A_{j \pm 2 Δ}^{*} \exp (- i δ k z) + A_{j \pm 2 Δ}^{2} A_{j \pm 4 Δ}^{*} \exp (- i 2 δ k z) + A_{j + Δ}^{*} A_{j - Δ} A_{j + 2 Δ} \exp (i δ k z) \\ + A_{j - Δ}^{*} A_{j + Δ} A_{j - 2 Δ} \exp (i δ k z) + A_{j + 3 Δ}^{*} A_{j + Δ} A_{j + 2 Δ} \exp (- i δ k z) + A_{j - 3 Δ}^{*} A_{j - Δ} A_{j - 2 Δ} \exp (- i δ k z) \\ + A_{j + 2 Δ}^{*} A_{j - 2 Δ} A_{j + 4 Δ} \exp (i δ k z) + A_{j - 2 Δ}^{*} A_{j + 2 Δ} A_{j - 4 Δ} \exp (i δ k z) + A_{j + 6 Δ}^{*} A_{j + 4 Δ} A_{j + 2 Δ} \exp (- i δ k z) \\ + A_{j - 6 Δ}^{*} A_{j - 4 Δ} A_{j - 2 Δ} \exp (- i δ k z), j = 1 \dots 7. \end{array}

A_{j} = A_{0 j} \exp (i ϕ_{j}) \exp (i Δ k_{j} z) \exp (- z^{2} / 2 z_{0 j}^{2}), j = 1 \dots 7,

\begin{array}{l} A_{01} = 1, A_{02} = A_{03} = 0.01, A_{04} = A_{05} = 0.01, A_{06} = A_{07} = 0.005, \\ z_{01} = 1, z_{02} = z_{03} = 15, δ k = 0.001 \\ Δ λ_{1} = 0, Δ λ_{2} = 2.5, Δ λ_{3} = - 2.5; Δ λ_{4} = 5, Δ λ_{5} = 5, Δ λ_{6} = 7.5, Δ λ_{7} = - 7.5. \end{array}

Abstract

1. Introduction

2. Experimental part

2.1. Experiments in BK7 glass

2.2 Experiments in fused silica

3. The nonlinear operator of cubic type Pi=χijkl(3)EjEkEl

3.1 Scalar approximation. One carrying frequency

3.2 Physical model of the avalanche parametric generation

4. Mathematical model of the avalanche evolution

4.1 Two carrying frequencies

4.2 Numerical experiment with seven waves

5. Conclusions

Funding

Acknowledgments

References and links

References

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2006 (2)

2005 (2)

2004 (2)

2002 (1)

2001 (2)

1997 (1)

1995 (1)

André, Y.-B.

Azarm, A.

Beaudin, G.

Berge, L.

Bergé, L.

Bourayou, R.

Braun, A.

Cao, Y.

Champeaux, S.

Chen, S.

Chen, Y.

Chen, Y. P.

Chin, S. L.

Christiansen, P. L.

Cohen, O.

Couairon, A.

D’Amico, C.

Daigle, J.-F.

Dakova, A. M.

Ding, P.

Du, D.

Durand, M.

Encrenaz, P.

Forestier, B.

Franco, M.

Georgieva, D.

Georgieva, D. A.

Gheudin, M.

Guo, Z.

Hosseini, S.

Houard, A.

Hu, B.

Jhajj, N.

Juul Rasmussen, J.

Kamata, M.

Kaminer, I.

Kasparian, J.

Kolesik, M.

Korn, G.

Kosareva, O.

Kovachev, L.

Kovachev, L. M.

Lahav, O.

Lederer, F.

Levi, L.

Li, R.

Liu, J. S.

Liu, W. W.

3. The nonlinear operator of cubic type $P_{i} = χ_{i j k l}^{(3)} E_{j} E_{k} E_{l}$