## Abstract

We report on the experimental observations of on-axis spectral broadening arising from self-focusing of the axicon-generated femtosecond Bessel beam in water. The observed spectral broadening is interpreted by a nonlinearly phase-matched four-wave mixing process involving the intense conical pump, the axial signal and a conical idler wave.

©2007 Optical Society of America

## 1. Introduction

Since the first experimental realization of so-called non-diffracting light beam in 1987 [1], Bessel beams (BB) continuously attract a broad interest from the scientific community inspired by expanding area of applications in atom optics, acoustics and plasma physics [2]. Pulsed BB are of particular interest in nonlinear optics providing new routes for frequency mixing and conversion [3, 4, 5], material processing [6] and device applications [7]. In spite of this evidence, self-focusing of pulsed BB in Kerr media still remains a poorly investigated phenomenon, especially in the femtosecond regime. In the presence of self-focusing nonlinearity, stationary monochromatic Bessel-like solutions were predicted, featured by a ”compressed” transverse beam profile [8], but shown to be unstable, owing to radial collapse [9]. Nonlinear losses (NLL) were predicted to counteract collapse, leading to radially stable unbalanced BB solutions, featured by a ring-like angular spectrum an by a reduction in near-field modulation contrast that was also detected in experiments in fused silica [9, 10]. Far-field detection in case of propagation in benzene with quasi-monochromatic pump, however, has outlined the appearance of an intense axial component, accompanied by an outer ring at the angle fulfilling longitudinally phase matched four-wave mixing (FWM) interaction [11]. Note that the interference between the produced axial component and the conical one has been proposed as responsible for the creation of periodic modulation of the peak fluence during BB nonlinear propagation, which results in a series of regularly spaced damage spots in fused silica [12] or in instabilities in plasma waveguides generated in gasses [13]. As to modifications in the time-domain spectrum, Stokes-shifted axial emission due to stimulated Raman scattering (SRS) was also reported in earlier experiments with nonlinear Bessel beams in the femtosecond regime [14]. To the best of our knowledge, the space-time dynamics of pulsed BB has not been investigated so far.

In this Letter we report on the first observation of axial broadband radiation excited by Bessel beam pumping. Differently from the Gaussian pump case, spectral broadening appears free from the intense-pump central peak, the pump beam propagating here out of axis. The largest conversion occurs at frequencies far from the pump, where the FWM interaction is shown to experience large gain due to optimum (longitudinal and transversal) phase matching. The process depends critically on the nonlinear-sample position in the beam path, which can be adjusted in order to excite either SRS or broadband light centered at the carrier frequency. The results are interpreted assuming a key role of BB collapse as the triggering mechanism for axial-component amplification.

## 2. Experiment

In the experiment, we have investigated the nonlinear propagation of femtosecond BB in water. The 200 fs, 527 nm pump pulse has been generated by a chirped-pulse-amplification Nd:glass laser system (Twinkle, Light Conversion Ltd) followed by a frequency-doubling compressor. A spatially filtered Gaussian beam with 1/*e*
^{2} radius of *w*
_{0} = 1.02 mm has been launched into BK7-glass axicon with base angle of *γ*= 5.17 deg. In this configuration, the length of the Bessel zone (diffraction free path for the central spot) in water is *z _{F}* =

*w*

_{0}/

*θ*= 29 mm, where

*θ*= (

*n*- 1)

_{a}*γ*/

*n*= 35 mrad is the cone half-angle,

_{w}*n*= 1.52 is the BK7-glass refractive indexes and

_{a}*n*= 1.33 the refractive indexes of water. The resulting profile leads to a central interference spike with diameter of 9

_{w}*μm*(between the zeros of Bessel function), which grows in intensity during propagation reaching its maximum at a distance

*z*≈ 9.5 mm and

*z*≈ 7.5 mm from the axicon for propagation in water and in air, respectively. The cuvette input-facet position was varied from close contact to the axicon to

*z*= 7.5 mm. The first setting allows the spike to grow in intensity inside water, thus supporting the smoothest possible transition from linear to nonlinear regime. In what follows, this setting is referred therefore as

*smooth transition*case. The second setting, in contrast, provides the most abrupt transition from linear to nonlinear regime, and is referred therefore as

*sharp transition*case. In all measurements, the input-pulse energy was kept to

*E*= 19.5

_{in}*μ*J.

## 3. Results

Figures 1(a) and 1(b) show typical axial spectra obtained for the smooth and sharp-transition configuration, respectively, as measured by means of fiber spectrometer (QE6500, Ocean Optics). Here cuvette lengths have been set to *L* = 29 mm and *L* = 22 mm, respectively, so to ensure the same output-facet position (at the very end of the Bessel zone), in both cases. The corresponding angular distributions of the generated radiation are shown in Figs. 2(b) and 2(c), where far fields detected by means of digital photo-camera (Fujifilm Finepix S7000) are reported. Figure 3 shows the dependence of the axial-beam energy on cuvette length, both for the smooth and sharp transition regimes. Sample lengths have been here modified by tuning the output-facet position, while keeping the input as specified above. Figure 4 proposes few examples of sharp-transition axial spectra relative to different sample lengths.

The results shown in Figures 1–3 clearly outline a substantial difference between the nonlinear dynamics developed in the smooth and sharp transition regimes. When the BB peak intensity is made to grow smoothly inside the sample, a clean SRS axial emission takes place, with pulse energy growing exponentially with propagation and reaching the value of *E _{Stokes}* ≃ 750 nJ at the end of the Bessel zone. We note that SRS axial emission is peaked at

*λ*= 650 nm,

*i.e*. at a Δλ ≈ 13 mm red-shifted value respect to what is expected for quasi plane and monochromatic pump waves [15]. The Stokes axial emission is here accompanied by an off-axis anti-Stokes radiation, emitted over a cone with half-aperture angle

*α*≈ 6°. Further measurements (data not shown) revealed the bandwidth of the Stokes-shifted axial radiation to increase on increasing pump energy, in line with the result reported in Ref. [14].

_{aS}For the sharp-transition setting, in contrast, no SRS is detected for the entire range of investigated sample lengths. The axial radiation appears initially (*i.e*. for short samples) around the pump wavelength. The axial-spectrum keeps broadening with further propagation, developing a sharply modulated structure featured by large peaks at the external sidebands (Fig. 4). A spectral width of > 4000 cm^{-1} is measured for sample length *L* = 18 mm. The axial-pulse energy increases linearly with propagation, reaching the value *E _{cont}* ≃ 250 nJ for the same

*L*= 18 mm that is after 25 mm from the axicon (Fig. 3), which corresponds to 1.3 % energy conversion from the pump pulse. The axial component is accompanied by the appearance of an outer idler ring. Further measurements (data not shown) reveal the idler to be composed by a broadband radiation featured by angular dispersion, with minimum half-aperture cone angle

*α*, ≈ 3.3° at degeneracy and with quadratically increasing angles on increasing the frequency shift. As to the axial beam quality, near-field detection revealed the mode to be shaped as a well-confined Gaussian beam of 17

_{i}*μ*m waist radius (at 1/

*e*

^{2}). At higher pump energy

*E*= 30

_{in}*μ*J an explosive broadening of the axial spectrum due to self-focusing of the axial beam itself was observed, accompanied by a beam-quality deterioration and by the appearance of multiple colored rings (conical emission) inside the cone of the input BB. We mention that any intermediate position of the cuvette yields the generation of both SRS and broadband axial emission components.

## 4. Discussion

In order to provide a consistent scenario for interpreting the observed phenomenology we consider in what follows a generic, Kerr-driven, FWM process which involves: (i) two pump waves at *ω*
_{0}, whose wavevectors **k _{p}** =

**k**coincide with those of the Bessel beam; (ii) a signal wave

_{0}*ω*=

_{s}*ω*

_{0}+ Δ

*ω*, with

**k**oriented on axis; (iii) an idler wave, with frequency

_{s}*ω*, =

_{i}*ω*

_{0}- Δ

*ω*fulfilling energy conservation 2

*ω*

_{0}=

*ω*+

_{s}*ω*and with

_{i}**k**, oriented at the angle which permits the longitudinal phase matching relation

_{i}**2k**

_{0}_{∥}=

**k**

_{s}_{∥}+

**k**

_{i}_{∥}to be fulfilled. We define

*k*

_{0}=

*k*(

*ω*

_{0}),

*k*=

_{s}*k*(

*ω*

_{0}+ Δ

*ω*) and

*k*=

_{i}*k*(

*ω*

_{0}-Δ

*ω*), where Δ

*ω*is the frequency shift relative to carrier frequency. The angle, at which the idler wave is longitudinally phase-matched is frequency dependent and, in the low intensity limit, reads as

where *k*(*ω*) = *n*(*ω*)*ω*/*c* obeys the dispersion relation for water as given in Ref. [16]. In small angle and second-order dispersion approximation the cone angle for the idler wave is

where *k*” = *∂ ^{2}k*/

*∂ω*|

^{2}_{ω0}denotes the group-velocity dispersion coefficient.

Assuming that in the case of BB one has to account for various azimuthal components giving their contribution to the generated axial beam, the transverse phase matching is expressed by the transverse phase-matching integral [17], which in the low-intensity limit is given by:

where *r* is the radial coordinate and *r*
_{0} = 1.5 mm is the upper integration limit set by BB apodization. By substituting *α _{i}* from Eq. (1) to Eq. (3) and assuming that an intense BB pump modifies by cross-phase modulation the refraction index seen by the weak axial signal, i.e

*n*(

*ω*

_{0}+ Δ

*ω*) =

*n*

_{0}(

*ω*

_{0}+ Δ

*ω*)+2

*n*

_{2}

*I*, we obtain

with

where *n*
_{2} = 4.1 × 10^{-16} cm^{2}/W is the nonlinear refraction index, *I* is the peak intensity and *n*
_{0} = *n*
_{0}(ω_{0}). Here we assumed that both pump and idler waves propagate linearly, owing to expected negligible off-axis nonlinear phase accumulation. Figure 5 shows transverse phase matching integral as computed from Eq. (4). In FWM process, the gain experienced by the spectral component *ω*
_{0} + Δ*ω* of the signal is proportional to the product of Kerr nonlinearity, *I* and *T*(Δ*ω*,*I*). The full black line shows the case of low-intensity limit, for a given pump cone angle *θ* = 35 mrad. The lines in color represent modifications of the gain profile in high-intensity propagation regime. Note how phase-matched peaks shift toward the center in the presence of the intense pump wave. And finally, the peak position of the transverse phase matching integral strongly depends of the cone angle *θ*, as shown by a dashed line, which represents the computed *T*(Δ*ω*,*I*) in the case of a 40 % larger BB cone angle.

In interpreting the SRS process, we note that: (i) owing to the chosen BB cone angle, and to the related tilt in the pump-pulse amplitude front, the effective pump group-velocity component in the axial direction matches with good accuracy that of the axial Stokes pulse, leading to pump-to-Stokes splitting length of 8 mm for 200 fs pulse duration (see Ref. [14] for derivation); (ii) the SRS under plane-monochromatic pump leads to Stokes *λ _{S}* = 637 nm [15], very close
to optimum condition for linearly phase matched FWM process. The observed red shift of the Stokes wavelength might be attributed to peculiarities of nonlinear propagation regime as suggested in Ref. [18]. The smooth-transient configuration provides therefore the ideal setting for efficient SRS and subsequent parametric FWM amplification, this being the configuration that ensures: (i) the large interaction path for the vacuum-state fluctuations to be readily amplified and (ii) the condition for the signal to be amplified without severe pump-phase distortion, these being expected to be detrimental for the FWM parametric process.

We assume the triggering mechanism for the occurrence of broadband radiation to be the production of axial seed at degenerate frequency, which in turn is a consequence of self-action experienced by intense BB as discussed in Ref. [11]. In accordance to the results of Fig. 5, this seed is not expected to experience degenerate amplification, in the smooth-transient regime. The key factor which we believe characterizes the sharp-transient regime is the occurrence of Kerr-driven BB collapse-type transverse-beam dynamics [9], which enhances peak intensity and relaxes constrain for transverse momentum conservation. In accordance to our calculations, these two elements should greatly enhance the degenerate axial-signal gain, thus permitting the instability to begin. As axial intensity descends with propagation, as illustrated in Fig. 1 (b), the occurrence of larger gain at larger frequency shift is then expected to contribute in settling a coherent spectral broadening of the signal, where axial sidebands are produced and are boosted away from degeneracy with propagation, in the presence of conjugated off-axis idler components. The experimental observation of spectral width dynamics versus z presented in Fig. 4 is in qualitative agreement with calculation results outlined in Fig. 5. We mention that the outlined scenario does not consider the further impact of the temporal modification of the pump-pulse profile. Indeed, in analogy to what recently described for the spatio-temporal instability of the Townes beam, pulse splitting is expected to occur also for the present pulsed BB, seeded by interference between the pump and the amplified axial and conical waves [19]. Notably, a recursive temporal splitting of the pulse during propagation has to be forecasted in this case, owing to the replenishment feature of the conical propagation geometry. In analogy to what described by Agrawal [20] for the case of spectral broadening in fibers, the pulse splitting dynamics might substantially impact (via cross-phase modulation) the instability gain profile, leading to the appearance of sidebands in normal dispersion as well. The interplay between this effect and what we have described above deserves therefore to be investigated.

## 5. Conclusion

In conclusion, we have demonstrated that differently from the Gaussian beam case, the broadband radiation excited by the BB appears here free from the exponential-like decay of intensity with increasing frequency shift, and free from the intense-pump component. Therefore it could be particularly useful for many applications demanding spectrally broadened (continuum) light pulses without strong residual radiation at the central frequency. The key factor which allows spectral broadening to overcome SRS was found to be important modification of transverse phase-matching condition by the nonlinearity that accompanies the abrupt transition between BB linear and nonlinear regimes. In fact, spectral broadening is quenched where the nonlinear dynamics is approached by focusing smoothly in the sample. The possible contribution of spatiotemporal modification of the pump pulse profile is also considered.

## Acknowledgment

Authors thank E. Kučinskas for his valuable contribution in the early stages of the experiment. P. P. acknowledges the support from Sixth EU Framework Programme Contract No. MEST-CF-2004-008048 (ATLAS), P.D.T. acknowledges the support from Marie Curie Chair project STELLA, Contract No. MEXC-CT-2005-025710. This work is performed in the framework of VINO (the Virtual Institute for Nonlinear Optics) collaboration: www.vino-stella.eu.

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