1. Introduction

Since the early days of lasers it has been known that when high power laser beams propagate in bulk transparent dielectric media they can give rise to hot spots or filaments in the plane transverse to the laser propagation axis, where the field intensity would become very large compared to the average intensity. The pioneeringworks of Chiao, Garmire, and Townes [1, 2], Kelley [3], Lallemand and Bloembergen [4], and Shen and Shaham [5] established that the hot spots arose from nonlinear optical self-focusing, whereby an incident field contracts spatially thereby increasing its intensity, and for long pulse durations, say a nanosecond or more, the hot spots were accompanied by optical breakdown in which the free electron density in the vicinity of the filament grows exponentially due to avalanche ionization. This rapid growth of the electron plasma in turn depletes the energy of the incident light pulse, so that the filament propagation is terminated, and as the generated electron plasma evolves it produces a number of byproducts or remnants such as a flash of light, as the plasma recombines, and possibly an audible signal. In the last twelve years or so the study of filamentation in bulk dielectric media has entered a new regime in which the incident pulses are ultrashort, say picosecond and shorter durations, and the filamentation process can persist over anomalously long distances [6, 7]. The key physics underlying this new regime is the ultrashort laser pulse durations employed are shorter than the time needed for electron number growth due to avalanche ionization to occur, so that much weaker electron plasmas are generated by ultrashort pulses via multi-photon ionization. The net result is that the laser filaments produced by ultrashort laser pulses are much less depleted by the plasma generation process, and the plasma rather acts as a mechanism that arrests self-focusing collapse but does not terminate the filament [8], thereby allowing long distance propagation of filaments, also called light strings for the case of air propagation. A number of byproducts also accompany the filaments as they propagate, including conical emission [9] (CE) or colored light emission off-axis from the filament, white light or supercontinuum [10, 11] (SC) generation resulting in a spectrum covering the full optical range, third-harmonic (TH) generation [12], Tera-Hertz radiation [13, 14], and X-waves [15, 16, 17] and O-waves [18, 19]. In general terms, the byproducts account for the broad and varied spectrum of the propagating pulse, and span several orders of magnitude in strength. For the example of air, light strings have been proposed as a broad bandwidth source for remote sensing of chemical and biological agents, and for this a thorough understanding of the spectrum of the propagating pulse is required.

 

Fig. 1. Femtosecond pulse’s self-similar collapse in water. On the left, the maximal amplitude of the pulse is shown as a function of the propagation distance in the water sample. The symbols show the simulated increase of amplitude as the collapse distance is approached. The line represents the analytic prediction of the self-similar collapsing solution in the pure NLS equation. The scaling plot on the right shows comparison of the Townes profile, i.e. of the expected transverse shape of the collapsing beam, with the simulation data obtained in the region indicated in the left panel. Different symbol colors correspond to different propagation distances. The fact that the scaled data collapses closely on the Townes profile corroborates the expectation that even in the non-idealized system, the collapse as predicted by the NLS equation still governs the evolution of the filament core.

Download Full Size | PDF

The area of femtosecond filamentation in transparent bulk media has now matured and the physics of this phenomenon has recently been reviewed in depth in the papers by Couairon and Mysyrowicz [20] and Kasparian and Wolf [21]. A review of the literature makes it evident that there are different interpretations of what constitutes a filament. This is not surprising in view of the very different physical scenarios in which filaments arise. In all cases some additional physics has to intervene to prevent uncontrolled growth of the light intensity during the self-focusing process. For example, in gases where group velocity dispersion is weak, electron generation through multiphoton ionization acts as an optical limiting process. In condensed media, group velocity dispersion will typically dominate over ionization processes and the long term asymptotic behavior of the resultant pulse will depend on the sign of the dispersion co-efficient - this depends on the incident pulse carrier wavelength. As our goal is to provide a self-consistent interpretation of filamentation that captures the physical aspects that are universal, or common to all physical scenarios, we appeal to a substantial body of mathematical results of the underlying Nonlinear Schrüdinger equation (NLSE) that may not be generally known to the optics community.

Following the seminal work of Chiao, Garmire and Townes [1] there were significant developments in the theoretical study of NLSE in the early seventies and subsequently. The first major breakthrough by Zakharov and Shabat [22] in 1972 showed that the 1D NLS was an integrable system, supporting stable soliton solutions. Numerical simulations of 2D NLSE in the context of laser beam propagation in the sixties showed that this added spatial dimension led to an explosively unstable solution - this supported experiments that showed the onset of damage hot spots in transparent glasses. It was first established by Vlasov, Petrishschev, and Talanov [23] that 2D NLSE supported an unstable mode that had the self-similar form [24]

where R is the finite-norm solution to

This solution shows that the transverse cross-section of the solution (laser beam) assumes a fixed functional form that continuously rescales as the cross-section collapses and eventually the peak goes to infinity. This mathematical singularity was referred to as “blow-up in finite time” which, in the optics context should read as “blow-up in a finite distance” as NLSE reflects propagation of a slowly-varying envelope in space. There is a considerable literature (see e.g. [25] and extensive literature review within) on this solution of 2D NLSE and it will play a central role in our definition of a filament. Figure 1 shows the evolution of the peak of a realistic initial Gaussian pulse and the analytic result corresponding to Eq. (1). We see that these curves coincide in the asymptotic regime where the cross-section of the pulse has begun to collapse or self-focus. We remark here that the part of the solution in Eq. (1) that depends on transverse space coordinate is colloquially referred to as the Townes soliton (soliton here is a misnomer). Moll, Gaeta and Fibich have recently done very precise experiments [26] to verify the existence of this solution. As a final remark, we note that the onset of deviation of the peak of the numerical solution is an indicator that the pulse is beginning to split due to normal GVD. Luther et al. [27] used the nonlinear mode, represented by Eq. (1) as a zero order solution in a perturbation analysis taking into account normal GVD and accurately predicted this onset of pulse splitting. They showed that the onset of splitting and short term subsequent evolution could be accurately predicted using a relatively simple dynamical systems picture. We remark here that, although the singular perturbation analysis assumes that GVD is a small O(ε) perturbation, the results extend to strong GVD (O(1)) as shown in [27]. This is a characteristic feature of these perturbation techniques. The analytical structure contained in the self-similar collapse solution and Luther’s pulse splitting analysis gives significant insight into what we refer to as the nonperturbative part of the evolution in this paper. In a recent work, Porras et al. [28] further analyzed the geometry of unstable perturbations of the collapsing Townes profile, obtaining information on their spectral and temporal properties. Of course, neither of these approaches can fully capture the internal structure of the asymptotically evolved individual split pulse waveforms - for this we rely on numerical computation.

Our goal in this paper is to take a step toward a unified approach to the theory and experimental interpretation in ultrashort pulse filamentation. By unified we mean a picture that spans a range of physical situations varying in both, their characteristic scales as well as in the relative roles of different effect governing the evolution of the filament core. We want to show that a useful way to tackle this complex behavior is to, roughly speaking, split all the processes responsible for, and accompanying the filamentation into non-perturbative and perturbative. The non-perturbative component in filamentation is responsible for the creation of high optical intensities, subsequent optical collapse, and its arrest in the core of the filament. Much of this structure is captured by the self-similar collapse and pulse splitting analysis discussed above. In general this evolution, but not always, is accompanied by creation of what we like to call by-products of filamentation. These include white light generation, third and fifth harmonic radiation and Tera-Hertz radiation. We call these manifestations of filamentation perturbative, because the core of the filament is barely affected by their presence. In other words, there is only very weak feedback that the by-products show on the evolution (with propagation distance) of the filament core. To avoid confusion, it should be emphasized that there is no sharply defined divide that could identify any component as either strong or weak. For example, parts of supercontinuum generated in the femtosecond collapse, especially close to the pump wavelength, may be quite strong and definitely contribute to the pulse evolution. At the same time, a large part of supercontinuum will be in general weak enough to be considered perturbative in the above sense. The practical consequence of this splitting into perturbative and non-perturbative components is that we can infer a lot about the dynamics inside the filament from observation of the perturbative by-products.

In our analysis, we often use the notion of an X-wave. Similarly to the situation around the notion of filament, the research community has not yet accepted any well-defined definition of these objects. For the purpose of this work, we will call X-wave any waveform that exhibits concentration of its energy in spectral space along lines that correspond to a constant group velocity in the propagation direction. An X-wave may posses two “arms” as shown in [16], or only one of them may be excited, depending on concrete conditions. Here, we do not make a distinction between linear and nonlinear X-waves, as these objects can propagate in both regimes. In fact, as we pointed out in [16], it is the very property of X-waves that nonlinearity always tends to concentrate the wave energy around the same spectral locus corresponding to a constant group velocity, and this is responsible for the long-distance propagation in filaments. Therefore, we can encounter X-waves both in a perturbative and nonperturbative regimes and we show such examples in this work.

A key diagnostic tool in our approach is to employ the angularly resolved spectrum, or K - Ω spectrum, for the propagating pulse, a notion that has been used to great effect in the area of conical waves [29]. The unified approach uses two tools: First, the effective three-wave mixing [16, 30] (ETWM) picture is introduced that, given one has some knowledge of the space-time structure of the filament, allows one to predict the locus of points in the K - Ω plane where phase-matched nonlinear mixing can occur and generate the range of byproducts. Second, we introduce an approximation to the spectrum that is formally akin to the first-Born approximation [31] of scattering theory to produce a quantitative approach to calculating the angularly resolved spectrum, and we validate this against full numerical simulations. A key idea we are advancing here is that knowledge of the space-time structure of the optical filament, even on a relatively restricted range of temporal and spatial frequencies, can be converted into quantitative information about the byproducts over a much broader range of frequencies using the ETWM picture and first-Born approximation, and this underpins their utility. The validity of this view is borne out by comparison between full numerical simulations and the first-Born approximation results for the angularly resolved spectrum.

Before continuing into the subject proper, it may be useful to make certain connections to related fields. Short pulse propagation and its wave-form evolution, and spectral transformation bears many common features between fiber and microstructured media, on one hand and propagation in gases and in condensed matter samples on the other. It is probably correct to say that the underlying nonlinear processes are the same, though entering with varying degrees of importance in different nonlinear media. These include Kerr nonlinearity, chromatic dispersion, and ionization and plasma defocusing. Consequently, such notions as phase-matching, and importance of chromatic dispersion landscape are encountered across these fields. However, there is a theme or aspect in what follows that makes our treatment specific to bulk media. This is, of course, the fact that bulk allows for “additional”, namely transverse degrees of freedom compared to the essentially one-dimensional fiber propagation medium. It is taking advantage of these degrees of freedom that makes it possible to obtain a lot of information about the underlying nonlinear interactions. Consequently, one can say that the higher dimensionality in bulk media makes it actually easier to understand complex nonlinear processes in relatively simple terms.

The remainder of this paper is organized as follows: In Sec. 2 we set up our basic model for pulse propagation in bulk transparent media and apply it to the propagation of optical filaments in air as representative examples of gaseous and condensed matter. We shall also introduce the angularly resolved spectrum or K - Ω spectrum and use it to explain specifically what we mean by byproducts accompanying optical filaments. Section 3 will introduce the ideas underlying the ETWM picture and the first-Born approximation, and we apply them to the examples of X-and O-waves in water, and conical emission and third-harmonic radiation in air and its relation to SC generation. Section 4 will give our summary and conclusions.

2. Optical filament propagation

In this section we describe the basic propagation model used for our study and give examples of simulations of filamentation process in water and air as representative examples of long distance propagation in condensed and gaseous media at atmospheric pressure.We also introduce the angularly resolved spectrum and explain how it relates to the byproducts that accompany filamentation.

2.1. Basic propagation and medium response equations

We work within the scalar field approximation and consider propagation in a transparent and isotropic nonlinear dielectric medium. Then the real electric field for the pulse may be represented as a plane-wave superposition

where the field propagates dominantly along the z-axis, and r⃗={x, y} is the transverse position vector. The effects of linear medium dispersion on the propagating pulse are fully taken into account here through the dispersion relation $K(\omega ,k)=\sqrt{\frac{{\omega }^2\epsilon \left(\omega \right)}{c^2-k^2}}$ , which gives the z-component of the wave-vector for an angular frequency ω, transverse wavevector k⃗, and for a medium of linear permittivity ε(ω). The spectral amplitudes A(z,k⃗, ω) of the field evolve as functions of the propagation distance z solely due to the medium nonlinearities, and obey the z-propagated unidirectional pulse propagating equation (UPPE) equation [32, 33] written here in terms of spectral amplitudes entering the field expansion in (3)

The nonlinear polarization P(r⃗, z, t) implicitly depends on the propagating optical field through the medium nonlinearity.We write it in the form

where Δχ is local modification of the medium’s optical susceptibility caused by a combined action of the Kerr and stimulated Raman effects together with the free electrons generated by ionization in the high intensity optical field:

Here, the first term represents the Kerr and Raman effects, with f the fraction of the delayed nonlinear response, and ℜ is the memory function often approximated by ℜ(τ)~sin (Ωτ)e ^-Γτ as in [8]. The second term in the local susceptibility modification is due to free electron with Δ_χe standing for susceptibility modification per one electron in a unit volume, and the free electron density is denoted ρ. In femtosecond pulses, plasma diffusion and ion motion are neglected, and the free-electron density ρ is obtained from the evolution equation including the avalanche and the Multi Photon Ionization (MPI) free-electron generation [8, 34, 35]. In general, it is better to treat the plasma-related nonlinear response in terms of current, in order to be able to include chromatic dispersion in the plasma-induced modification of refractive index, but these effects are mostly very small.

2.2. (K - Ω) spectrum & byproducts

We now turn to our main diagnostic tool, namely the far-field spectrum. Also called angularly resolved spectrum, it is a quantity that can be both measured and accessed easily in the simulation. That is why it is no surprise that the far-field spectra contributed a lot to understanding dynamics of filamentation in dielectric media.

The experimental technique for measuring the angularly resolved spectra in the axially symmetric geometries were described in some detail [29]. The experimental set-up is simple in principle; First, one obtains the transverse spatial spectrum in the focal plane of a lens positioned after the nonlinear sample. This transverse Fourier spectrum is then fed into an imaging spectrometer which produces a two-dimensional spatial-temporal Fourier spectrum on its output. From the simulation point of view the situation is even simpler. When using the Unidirectional Pulse Propagation Equation, the angularly resolved spectral amplitude A(z, k⃗, ω) is actually the native representation of the ultrashort pulse (see Eqn.(3,4)). But even if using other implementations of the femtosecond pulse propagation model, the far field spectrum is straightforward to obtain since it is a mere Fourier transform of the real-space representation of the pulse at any given propagation distance z:

The angularly resolved spectrum thus yields the distribution of the pulse energy over both the temporal frequency and transverse spatial frequency domains, the transverse spatial frequency spectrum being directly related to the angular spectrum. The spectral power for a given z may be obtained from the angularly resolved spectrum by integration over the transverse spatial frequency giving

It is clear that the angle-integrated spectrum |S(ω)|², or just on-axis spectrum |A(z, k⃗=0,ω)|² that is also frequently measured in experiments contain much less information about the optical waveform than the angularly resolved spectrum. However, it is instructive to illustrate this difference on the following example to emphasize the utility of far-field spectra.

Figure 2 compares how supercontinuum and third harmonic generation in femtosecond pulses propagating in air looks in the conventional spectrum and in the angularly resolved spectrum. In the normal spectrum the third harmonic contributions appears to merge with the supercontinuum. This was experimentally observed in [36] and it was later argued [37] that it was the third harmonic that helped to extend the width of the supercontinuum. However, as we pointed out in [38], the angularly resolved spectrum shows clear separation of the two components. This is an example where having access to angular spectral information crucially helps correct interpretation.

3. Effective Three-Wave Mixing & First Born Approximation

This section presents the mathematical background for the Effective Three-Wave Mixing and then extends it into the first Born approximation for angularly resolved spectra. This approach was developed in a series of works [39, 40, 16, 30, 31] dealing with filamentation in both, gases and condensed media. Here we want to summarize it into a single coherent picture.

 

Fig. 2. Supercontinuum and third-harmonic generation in air. The right panel shows angle-integrated spectrum. The weak feature above the high-frequency background is due to third harmonic generation. The angularly resolved spectrum on the left reveals angular separation between the supercontinuum and third harmonic radiation parts of the spectrum.

Download Full Size | PDF

Central to the unified approach we want to present is the notion of separating the entirety of femtosecond pulse dynamics into what might be called non-perturbative and perturbative components. The non-perturbative aspect emerges in our theory in the form of a “scattering potential,” which is nothing but the nonlinear modification of the optical properties of the medium by the pulse. From a mathematical perspective, much of the structure of this scattering potential can be appreciated from prior asymptotic analysis of 2D NLSE. The physical interpretation can be represented in several ways, be it the modification of the refractive index, or its equivalent susceptibility change, or an induced current of free electrons. The perturbative aspect comes in when the non-perturbative core of the filament produces a variety of relatively weak by-products such as supercontinuum or third-harmonic and Tera-Hertz radiation. These byproducts cause little feedback to the core of the filament and are in this sense perturbative.

Figure 3 illustrates the typical properties of the nonlinear response in bulk nonlinear media. It shows both the intensity and susceptibility modification profiles as functions of radial coordinate and of the local pulse time. There is a close correspondence between the two. The main difference is due to plasma that causes the dips between the peaks and behind the trailing peak in the susceptibility profile. Thus, the nonlinear response portrait closely reflects the shape of the splitting pulse in time and space.

It is clear that Δχ is nonperturbative, because it controls the nonlinear evolution of the core of the femtosecond filament. Therefore, only a full solution of the femtosecond pulse propagation problem can give us detailed information on this quantity. However, our approach goes in the opposite direction: Suppose we can estimate or measure Δχ, what does it tell us then? Indeed, Δχ is the most complete “map” of what happened in the interaction volume; it reveals its dimensions, intensities within, and the temporal evolution. Roughly speaking, the spatial and temporal shape of Δχ reflects the pulse intensity in time and space because the main contribution to the nonlinear index change is usually proportional to n¯ ₂ E ²(z, r⃗, t). Next we will see how the non-perturbative Δχ leaves its perturbative signatures in the far field spectrum. Understanding the signatures will give us a tool for interpreting measured K-Ω spectra and thus infer Δχ with all the information about the pulse-medium interaction it carries.

 

Fig. 3. Profiles of the intensity (left) and nonlinear modification of the local susceptibility (right) as functions of radius and time in the frame co-moving with the pulse. This is a typical picture shortly after pulse splitting that creates two daughter pulses that propagate with velocities somewhat larger (front) and smaller (back) than the group velocity of the original pulse. Both pulses leave strong signature in the nonlinear response. The dip in the susceptibility located between the peaks and behind the trailing peak is due to the generated plasma.

Download Full Size | PDF

3.1. Effective Three-Wave Mixing

The first approximation we make here consists in replacing the “incident field” in Eq. (5) by the initial carrier plane-wave:

Second, we use the fact that usually there are well pronounced peaks in the nonlinear medium response (see Fig. 3), and these propagate with group velocities which are in general different from the original pulse group velocity. We denote by v_p the velocity of the p-th peak in the nonlinear response, and approximate the total nonlinear change in susceptibility Δχ as a sum of contributions from the main peaks:

The purpose of this decomposition is to achieve that each of Δ^χp(z, r⃗, t - z/v_p) becomes a slowly evolving function of its first argument z. That is the only property we use, while the concrete shape of these response peaks is unimportant on the ETWM level. When (9,10) are inserted into (4), we change the time integration variable for each of the peak contributions to τ=→t-z/v_p to get

If v_p is chosen appropriately, then the function Δ_χp(z, r⃗, τ) changes slower than the z-dependent phase factor in the integrand. Consequently, the |A(k⃗, ω)|² will be highest for those k⃗, ω which ensure that the latter vanishes:

This equation can be interpreted as a phase matching condition for the process in which a “material wave” with frequency ω - ω ₀ and the transverse wavenumber k scatters an incident optical wave with frequency ω ₀ and the wave-vector {k⃗=0, k_z=K(ω ₀ ,0)} to produce a scattered or output wave of frequency ω, transverse wavenumber k, and the z component of the wavevector K(ω, k). We term this Effective Three-Wave Mixing, since the material wave is a “composite excitation” that arises in the core of the filament as a result of complex nonlinear processes. As such, it doesn’t obey any photon-like dispersion relation, and its velocity v _p can in principle be “arbitrary.”

3.2. ETWM picture of X- & O-waves in water

In this section, we illustrate the Effective Three Wave Mixing approach by analyzing a rather complex far-field spectra generated in a sample of water trough two-color filamentation. The example we choose involves the interaction of two femtosecond pulses centered at 1100 (pump) and 527 nm (seed pulse) wavelengths, respectively. Pulses are loosely focused into the water cell, and their relative delay is adjusted such that the maximal temporal overlap is attained just when the filament forms. This situation is similar to that studied in [41] and [42], but in the present case we have no resonant interactions such as SRS. The energy and duration of the 1100 nm wavelength pulse are chosen to be 0.4µJ and 20 fs, while the other pulse is weaker, 0.2µJ, and longer, 100 fs, with the initial intensity ten times less. The reason for having a week seed pulse is to use the strong pulse as as reshaping tool to act on the weaker pulse that will be transformed into an X-wave such that the group velocities of the ensuing waveforms will be equal.

 

Fig. 4. Angularly resolved spectrum generated in water by combination of two ultrashort pulses centered at 1100 (ω≈1.7×10¹⁵s^-1) and 527 nm (ω≈3.7 ×10¹⁵s^-1) wavelengths. The resulting spectrum spans both the region of anomalous and normal GVD. The anomalous regime is manifested through the oval-shaped core of the spectrum (so called O-wave) around ω≈1.7×10¹⁵s^-1. The structure characteristic for the normal GVD regime is the X-wave “arm” (conical emission) originating at ω ≈2.5×10¹⁵s^-1 and extending to wide angles and high frequencies. The narrow, weak, extended structures are due to four-wave mixing processes in which supercontinuum beats against either pump or seed.

Download Full Size | PDF

The simulated angularly resolved spectra, recorded after the exit from the water sample, are shown in Fig. 4 and Fig. 5 together with lines indicating various phase-matched loci which we explain in the following. To our knowledge, far-field spectra as shown in these figures have not been observed yet. However, each of the prominent features we see here can be related to their counterparts observed in an experiment [42] in which the seed pulse was amplified through the resonant SRS effect. In the present case, the dynamics range required for the experimental verification would be significantly larger, but all the spectral structures shown should be observable in principle.

The spectrum in Fig. 4 exhibits two strong peaks which are remnants of the initial pump and seed pulses. The fact that even after supercontinuum is generated, the pump and seed wavelength show strongest in the spectrum reflects the perturbative nature of supercontinuum generated in this numerical experiment. In the following we use wave mixing arguments to explain the general shape of this spectrum.

 

Fig. 5. Angularly resolved spectrum generated in water by combination of two ultrashortpulses centered at 1100 (ω≈1.7×10¹⁵s^-1) and 527 nm (ω≈3.7×10¹⁵s^-1) wavelengths. White lines mark loci of effective phase matching for three and four wave mixing processes described in the text. In the left panel, the top white line is given by Eq. (14) and represents scattering of the seed-pulse photon off the nonlinear response peak. The lower white line represents the process of pump-photon scattering on the nonlinear response peak according to Eq. (13). The right hand side panel shows phase matching loci corresponding to Eq. (18) (upper line) and Eq. (17) (lower line), respectively. Note that all the phase-matched loci are controlled by a single parameter, namely the propagation speed of the nonlinear response peak.

Download Full Size | PDF

Let us start with the first panel of Fig. 5, which illustrates the use of phase matching relation (12). Since in this case there are two strong sources of “incident waves,” namely the carrier frequencies of the pump and seed pulses, we have two effective phase matching relations. One for scattering of the pump wave,

and one for scattering of the seed wave:

These are implicit relations for the loci of maximal energy concentration in the far-field spectrum. The response peak velocity v_p is a fitting parameter that can be determined by fitting the resulting curves to the spectrum. In the left panel of Fig. 5 the upper and lower curve (the latter including two disjoint components) correspond to Eq.(14) and Eq.(13), respectively, for the response peak velocity v_p=c/1.3423 (Note: the group index at pump wavelength is 1.3416). This value is in agreement with the velocity obtained directly from the simulated response temporal and spatial profiles. Thus, the three-wave mixing phase matching relation allowed us to measure the group velocity of the response peak and thus also the group velocity of the optical pulse inside the filament, and identified the long “arms” in the spectrum as due to a scattering process in which the original carrier wave scatters off the response peak potential. Also, it shows that the seed pulse is transformed due to its scattering on the potential created by the pump pulse. The intensity of the seed pulse can be arbitrary small, and this is an example of a purely “perturbative” X-wave.

An alternative way to view the process that creates the seed-frequency centered X-wave is that of cross-phase modulation, when the seed-pulse phase is modulated by the strong pump pulse, and can also be interpreted in terms of four-wave mixing; Indeed, one could argue, that the scattering picture takes us away from first-principles explanation because the scattering potential is not a priori known, and that the four-wave mixing picture is more fundamental. However, the four-wave mixing interpretation requires introduction of a modified, intensity dependent, dispersion relation, a questionable step that can’t be justified for bulk media the same way as it is done, correctly, in fibers. The three-wave mixing paradigm, on the other hand, goes in the opposite direction; It acknowledges that there is a whole family or a “re-summation” of four-wave mixing processes that together manifest themselves as if a material wave causing the scattering was propagating with velocity v_p. Thus, the effective three-wave mixing picture gives us actually some quantitative information about the filament, namely the pulse group velocity inside the filament core.

Next, let us examine the unusual slender structures in the far-field spectra in Fig. 5. They might appear puzzling at first, but can be explained in the similar way as above.

Let us consider a four-wave mixing process in which a pump wave scatters from the beating between X-wave generated from the seed pulse as described above and the pump-wave. In the language of four-wave mixing, we have two pump waves mixing with a single X-wave centered at the seed wavelength. Because of small transverse dimensions of the filament core, only the longitudinal phase matching and energy conservation need to be explicitly taken into account:

Using the three-wave mixing equation (14) for K(ω_X, k_X):

where v_p is the velocity of the response peak that generated the X-wave from the seed radiation. After elimination of K(ω _X, k_X) and ω _X, we obtain this phase-matching relation:

Of course, we get a similar condition for the process in which the role of pump and seed waves are exchanged:

Here, it is the seed pulse wave that scatters from the beat between X-wave (generated from the pump) and the seed. Of course, both relations are parametrized by the same response peak velocity “measured” by fitting the ETWM spectral features above.

The right hand side panel of Fig. 5 shows two white curves, corresponding to Eq. (17) (lower) and Eq. (18) (upper), respectively. The fit is perfect, and leaves no doubt that the proposed processes are indeed responsible for these unusual spectral features.

We have thus illustrated as the ETWM picture lets us to understand the structure of an at first sight quite complicated far field spectrum. It in fact shows that the prominent features can be all related to the group velocity of the response peaks created inside filament by the pump pulse. The fact that the same scenario works for the strong X-wave centered around the pump frequency as well as for the much weaker X-wave centered around the seed frequency indicates that we have a separation into non-perturbative scattering potential and perturbative (linear) scattering byproduct in the form of well-defined structure in the far field.

In the next Section, we show that one can extend this approach even further by relating the whole spectrum shape to the spatial and temporal properties of the filament core.

3.3. First Born approximation

The approximation we make here is the same as above; we replace the “incident field” in Eq. (5) by the pulse carrier plane-wave:

Then, we integrate the evolution equation (4) over z to obtain an expression for the far-field spectrum A(k⃗, ω)≡A(z→∞,k⃗, ω)

We have conjectured earlier [30] and shown recently [31] that Eq.(20) is actually quite good approximation for the far-field spectrum. The formula (20) closely resembles the structure of the first Born approximation from scattering theory: If Δχ is interpreted as a scattering potential, and the exponential phase factor is read a difference of the incoming and outgoing (four) wavevectors, we have the first term in the Born series for the scattering amplitude. This is the reason we decided to call this the first Born approximation.

We have demonstrated [31] that the first Born calculation can give accurate representation of the far-field spectrum in anomalous GVD regime in water and in the normal GVD in air. We have also shown that it reproduces not only the supercontinuum component of the entire spectrum, but also the third-harmonic radiation part. Here we supplement our illustration of first Born approximation with the case of normal GVD in water. Figure 6 shows the comparison of the K-Ω spectrum obtained from the full simulation and from the first Born approximation (20). The agreement is evidently quite good, all structures in the spectrum are reproduced.

Figure 7 shows the comparison of the K-Ω spectrum obtained from the full simulation and from the first Born approximation (20) for SC and TH generation in air at relatively high energy, when supercontinuum extends into the third harmonic frequency range. Again, the first Born approximation reproduces all features in the angularly resolved spectrum, including thirdharmonic generation.

To illustrate the common and different aspects of the SC and TH origins, Fig. 8 shows a snapshot example of the on-axis E ² profile of a pulse that has just gone through splitting. This represents the shape of the first-Born scattering potential at a fixed propagation distance. It can be decomposed into the “envelope,” or DC part (shown in Fig. 8 in the middle panel) and into a component oscillating at about second-harmonic frequency (shown in Fig. 8 in the right panel). The former is responsible for the SC production. The high-frequency of the latter causes the scattering waves to acquire additional frequency shift roughly twice the fundamental and thus produces the third-harmonic radiation. Thus, SC and TH are generated through the same scattering process, but they each respond to a different aspect of the nonlinear response.

 

Fig. 6. Log plot of the angularly resolved power spectrum |A(k, ω)|² for SC generation in the normal GVD regime in water. Left panel: full simulation. Right panel: first Born approximation.

Download Full Size | PDF

The nature of approximation we adopted in derivation of the first Born approximation indicates that if it holds well for far-field spectra measured after the pulse exits the sample, it should be accurate also for intermediate propagation distances and, in fact, also inside the interaction zone. This is indeed the case, and we illustrate the evolution of the first Born approximation with the propagation distance in Animation 9. Similarly as for the final spectra, the agreement between full and approximated spectra is good for all propagation distances.

Thus, the first Born approximation takes the ETWM picture to a quantitative level. While ETWM allowed us to understand the locus of maximal spectral energy concentration in the far-field spectrum, and estimate several quantities describing the pulse evolution (splitting) in the interaction zone, the first Born approximation describes correctly also the whole shape of the spectrum.

The physical consequences of the fact that the first Born approximation can be taken as a good approximation are twofold. First, it is an expression of the fact that supercontinuum and third-harmonic radiation are both perturbative byproducts of the femtosecond collapse and filamentation; They are much weaker than the central filament, and do not feed back to influence the pulse propagation significantly. Consequently, also their mutual interaction is negligible. Second, the first Born approximation connects the experimentally accessible quantity, namely the spectral power in the far field |A(k⃗, ω)| to the “scattering potential” Δχ through a simple linear formula. This make it possible, in principle, to obtain knowledge about the pulse evolution inside the filament from a relatively simple measurement of the angularly resolved spectrum.

 

Fig. 7. Log plot of the angular resolved power spectrum for SC and TH generation in air. Left panel: full simulation. Right panel: first Born approximation.

Download Full Size | PDF

4. Conclusions

In this paper, we have attempted to provide a unified picture of ultra-fast high power femtosecond self-focusing propagation phenomena within a broad class of bulk materials including gases and condensed media. Physical interpretation of experimental data, such a white light generation, higher harmonic generation, nonlinear X- and O-wave generation, is often blurred by the myriad of competing processes occurring within the nonlinear self-focal region for example, optical limitation can occur through plasma generation in gases or normal GVD in condensed media.We would like to propose an interpretation that does not attempt explanation by dcomposition into contributions from specific physical processes, but rather identifies their universal features. The central idea that we employ is that the onset of critical self-focusing (2D critical collapse) defines a relatively robust nonlinear core or filament that acts as a scattering potential for the remaining energy within the pulse that is not captured within the core. Many current experimental measurements of white-light generation, nonlinear X and O-waves and third harmonic generation in air and condensed media can then be interpreted within a framework where they are generated by scattering from this nonlinear potential. Much of the analytical structure of this nonlinear core can be understood in terms of known mathematical results on the established self-similar form of the critical collapse singularity of 2D NLSE and on a perturbation analysis about this self-similar solution that accounts for the onset of pulse splitting. Of course NLSE at this level is a spectrally-local, slowly varying envelope description that cannot account for an asymptotic behavior that includes pulse splitting into individually evolving pulses with subsequent cascades of pulse splittings. A powerful tool for visualizing complex interactions within the nonlinear core is the spectrally-resolved far-field. This readily available experimental data provides deep insight into the nature of the many scattering processes. Our numerical simulations with the UPPE model have demonstrated quantitative agreement with experimentally measured spectrally-resolved far-field spectra over many decades of spectral intensities.

We feel additionally, that our simulation capability and semi-analytic interpretation of the results will provide a powerful tool for extracting three dimensional space and time information on the internal pulse propagating within a bulk nonlinear material. We have shown that the full numerical simulations of these spectra can be reproduced by assuming that the interaction can be described as a dynamical three-wave mixing process described by a simple first Born scattering approximation. For example, the DC-component of the scattering potential accounts primarily for supercontinuum generation whereas the high-frequency 2ω component acts as a scattering center for generating third harmonic waves. The fact that a first-order scattering amplitude approximates the observed spectra so well opens up an intriguing question, namely whether we can solve the inverse problem i.e. knowing the measured spectrally resolved farfield, can we infer the induced material modification and hence the 3D pulse shape at any point as it propagates within the bulk medium? Strictly speaking this would require that we have access to both amplitude (spectral intensity) and phase simultaneously. The latter is generally unavailable, and therefore the problem is numerically ill-posed. On the other hand, one can use strong constraints on the shape of the pulse within the nonlinear medium, such as compact support, simple bell-like (though time-dependent) transverse profile to obtain an “optimized” approximate solution.

 

Fig. 8. “Scattering potentials” for supercontinuum and third-harmonic generation. On the left, the on-axis E ₂ profile of a pulse is shown shortly after pulse splitting. Its shape is essentially the shape of the scattering potential that enters the first-Born approximation formula. The scattering potential is decomposed into AC (right) and DC (middle) components that cause production of third-harmonic and supercontinuum spectral components, respectively.

Download Full Size | PDF

 

Fig. 9. Animation comparing evolution, with the propagation distance in the water sample,of the far-field spectrum (left) and its first-Born approximation (right) in a femtosecond pulse propagating in water. [Media 1]

Download Full Size | PDF