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Nonlinear Bessel vortex beams for applications

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2015 J. Phys. B: At. Mol. Opt. Phys. 48 094006

(http://iopscience.iop.org/0953-4075/48/9/094006)

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Nonlinear Bessel vortex beams forapplications

C L Arnold1,2, S Akturk1,3, A Mysyrowicz1, V Jukna4,5, A Couairon4, T Itina5,R Stoian5, C Xie6, J M Dudley6, F Courvoisier6, S Bonanomi7,O Jedrkiewicz8 and P Di Trapani7

1 Laboratoire d’Optique Appliquée, ENSTA ParisTech, Ecole Polytechnique, CNRS, F-91762 Palaiseau,France2Division of Atomic Physics, Department of Physics, Lund University, Sweden3 Istanbul Technical University, Engineering Physics Department, Turkey4Centre de Physique Théorique, CNRS, Ecole Polytechnique, F-91128 Palaiseau, France5 Laboratoire Hubert Curien, CNRS, Université de Lyon, Saint-Etienne, France6Département d’Optique PM Duffieux, Institut FEMTO-ST CNRS, Université de Franche-Comté,Besançon, France7Dipartimento di Scienza e Alta Technologia, University of Insubria, Como, Italy8 Istituto di Fotonica e Nanotecnologie, CNR and CNISM, Como, Italy

E-mail: couairon@cpht.polytechnique.fr

Received 3 November 2014Accepted for publication 16 December 2014Published 10 April 2015

AbstractWe investigate experimentally and numerically the nonlinear propagation of intense Bessel–Gauss vortices in transparent solids. We show that nonlinear Bessel–Gauss vortices preserve allproperties of nonlinear Bessel–Gauss beams while their helicity provides an additional controlparameter for single-shot precision micro structuring of transparent solids. For sufficiently largecone angle, a stable hollow tube of intense light is formed, generating a plasma channel whoseradius and density are increasing with helicity and cone angle, respectively. We assess thepotential of intense Bessel vortices for applications based on the generation of hollow plasmachannels.

Keywords: filamentation, Bessel beams, optical vortices, ablation

(Some figures may appear in colour only in the online journal)

1. Introduction

Bessel beams are interesting for several applications in dif-ferent disciplines, from quantum communication and infor-mation systems to resonant self-trapping in plasmas, to glassprecision drilling [1–6]. For instance in micromachiningapplications, the main advantage of Bessel beams over con-ventionally used Gaussian beams lies in their propagationinvariance, i.e., their ability to retain their shape and peakintensity over an extended focal line, the Bessel zone.Translation of the sample is no longer required in contrastwith the use of tightly focused Gaussian beams. Even mul-tiple laser shots can be avoided as shown in recent experi-ments on long aspect-ratio drilling, with high potential gain inthe processing time for transparent samples [7–9]. This is

possible due to the high intensity in the main lobe of theBessel beam, leading to the generation of a thin column ofplasma by optical field ionization and to efficient plasmamediated laser energy deposition in the bulk of the glass.Physical mechanisms occuring between laser energy deposi-tion and the extrusion of the material are complex and not yetfully understood. However, material ablation is ultimatelygoverned by the shape of the plasma channel where laserenergy is deposited, which in turn is controlled by a fewparameters.

In the case of bulk ablation with Bessel beams, the coneangle θ of the Bessel beam in the material determines the

diameter of the light channel =π

λθ

d 2.405

sin0 . Here, λ0 is the

wavelength of the light beam and 2.405 is the first zero of thezero-order Bessel function. In the regime of multiphoton

Journal of Physics B: Atomic, Molecular and Optical Physics

J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 094006 (10pp) doi:10.1088/0953-4075/48/9/094006

0953-4075/15/094006+10$33.00 © 2015 IOP Publishing Ltd Printed in the UK1

ionization, the width of the plasma channel generated with aGaussian beam is expected to be smaller by a factor of K ,where K is the number of photons involved in the multi-photon ionization process. Similarly, for a Bessel beam, thewidth of the plasma channel is smaller by a factor of

− − − −J J(2 ) (2 )K0

1 1 20

1 1 2 . The extent of the plasma channel isexpected to cover the Bessel zone θw tan0 , where w0 is thewidth of the Gaussian beam transformed to a Bessel beam bye.g. use of a conical lens (axicon). There are multiple ways ofgenerating Bessel beams in practice, such as an annular slittogether with a lens, reflective and refractive axicons, planarcircular diffraction gratings, tunable acoustic gradients, andspatial light modulators (SLM) [10].

Motivations for this paper come from all applications thatrequire in depth laser energy deposition. They could not bebetter stated than by citing [3]: non-conventional beam shapeshave the advantage that they can be explicitly designed tomeet the requirements of a given material configuration orapplication that could not be feasible with either Gaussian ortop-hat beams. For Bessel beams a single parameter, the coneangle, determines the width of energy deposition and itslength simultaneously (figure 1b). We investigate here thenonlinear propagation of a different kind of annular beams,Bessel beams carrying helicity, called for this reason Besselvortices. These are ring-shaped and deposit energy in a tub-ular focal region (figure 1c). Examples of using these beamsin applications include photoinscription of tubular wave-guides in multiple shots [11] and surface ablation [12, 13].

In this paper, we asses the potential of nonlinear Besselvortices for specific applications that require generation ofhollow cylindrical plasma channels, deposition of laserenergy in a cylindrical focal volume, or micro-structuring oftransparent materials with long aspect ratio. We demonstrateexperimentally that a nonlinear Bessel vortex is generated byfocusing a Gaussian beam carrying helicity with an axicon ina Kerr medium, in both regimes of small and large coneangles. By comparing measurements and numerical simula-tions, we show how the Gaussian beam is reshaped into apropagation invariant nonlinear beam identified in [14] as anonlinear Bessel beam carrying angular momentum. Forsmall cone angles, we find that the occurrence of the non-linear regime coincides with the appearance of super-continuum generation and a typical contrast reduction. Wepresent a method to evaluate the peak intensity in the tubularfilament and the electron density of the tubular plasmachannel, indicating that refractive index changes or damage tothe material can be obtained for cone angles larger than∼10 deg. We show results of material structuring tests insingle shot with Bessel vortices.

2. Experimental setup

The experimental setup is shown in figure 2. Two differentschemes based on different ways to generate the Bessel vortexbeams were used. Cone angles smaller than 2 deg wereinvestigated with scheme A whereas higher cone angles were

Figure 1. (a) A Gaussian beam focused by a lens (focal length f)generates a localized plasma in the focal region of volume π k0 wf

4

where the width = + −w fw k w f( )f 014 0

204 2 1 2. (b) A Bessel–Gauss

beam, obtained by focusing a Gaussian beam of width w0 by anaxicon, generates a plasma channel in the Bessel zone of length

θw tan0 , where θ is the cone angle of the Bessel beam in thepropagation medium. (c) A Bessel vortex, obtained by focusing aGaussian beam carrying helicity m with an axicon, generates ahollow plasma cylinder over the Bessel zone. Its diameter Dincreases with helicity.

Figure 2. Setup to observe nonlinear propagation of Bessel beamsand Bessel vortices. Scheme A uses a combination of spiral phaseplate and liquid immersion axicon and permits moderate cone angleswith helicity m = 1. Scheme B uses a spatial light modulator and ademagnifying telescope giving access to higher order Bessel beamsat large cone angles. P—pinhole. SPP—spiral phase plate. M—

mirror. CCD—charged coupled device. L.i. axicon—liquid immer-sion axicon. SLM—spatial light modulator.

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J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 094006 C L Arnold et al

achieved with scheme B. In a first set of experiments (schemeA), we used an amplified Ti:Sapphire laser system (Thales,alpha 100), which provides up to 15 mJ pulses at 800 nm witha duration of 45 fs, at the repetition rate of 100 Hz. The laseroutput was first strongly attenuated and afterwards spatiallyfiltered by means of a pinhole in the focal plane of an imagingsystem. The beam was sent through a liquid-immersion axi-con, aiming at easily modifying the cone angle of theresulting Bessel beam [15]. Bessel vortices are generated byinserting a spiral phase plate (SPP) (RPC Photonics) beforethe axicon. We characterized linear and nonlinear propagationof Bessel vortices. For linear propagation, the beam propa-gates in air and is imaged at different distances from theaxicon back surface. For nonlinear propagation, samples of10, 20, and 40 mm of fused silica were placed almost incontact (a gap of ≈100 μm remained) with the exit surface ofthe liquid-immersion axicon. The beam at the exit surface ofthe sample was imaged on the CCD camera, as we varied theinput power.

3. Experimental results

3.1. Linear propagation

Figure 3 shows a comparison of propagation features forBessel beams and Bessel-vortices in air, for the same coneangle θ = 2.2 deg. Very regular beam profiles were observedalong the focal region, both for the Bessel beam and theBessel vortex. From the spacing between the rings( μ∼r2 15.70 m), the Bessel cone angle can be calculated:

θ = k rsin 2.405 ( )0 0 , leading to θ ∼ 2.24 deg, Focusing aGaussian beam with an axicon is well known to result in aBessel–Gauss beam that remains quasi propagation invariantalong a focal region, the Bessel zone, of length

θ=L w tanB 0 . Adding helicity (here m = 1) to the Gaussianbeam results in a higher-order Bessel–Gauss beam thatremains quasi-propagation invariant over the same focalregion. The highest intensity is reached on the surface of ahollow cylinder corresponding to the maximum of the

θJ k r( sin )m 0 function. More precisely, in the linear propa-gation regime, the intensity profile is determined by the

relation [16]

θθ= −

⎛⎝⎜

⎞⎠⎟( )I r z

Pk

w

z

zJ k r

z

L( , )

4 sinsin exp 2 , (1)

Bm

B

0

0

20

2

2

where P denotes the input power of the Gaussian beam and zdenotes the propagation distance from the tip of the axicon.With w0 = 4.4 mm, we obtain a Bessel zone LB = 11.3 cm.The highest intensity is reached at =z L˜ 2B and equals

θ= −I P w k M2( ) sin exp ( 1 2) mmax 0 0 , where Mm = (1, 0.34,0.24, 0.19, …) denotes the maximum of the squared Besselfunction of order m, for = …m (0, 1, 2, 3, ). We estimatethat the beam enters a nonlinear propagation regime when themaximum intensity is sufficient to ionize the medium. With athreshold of 1012W cm−2 in glass and 1013W cm−2 in air, weneed a minimum beam power of ∼240MW in fused silica(corresponding to an energy of 13 μJ for a 45 fs pulse)and 3 GW in air (0.2 mJ). With pulse energies smaller than100 μJ, we could explore the nonlinear propagation regimesin glass.

3.2. Nonlinear propagation

Propagation in the nonlinear regime was investigated byplacing samples of fused silica with thicknesses 10, 20, and40 mm close to the exit surface of the liquid immersion axi-con. Figure 4 shows the fluence patterns at the exit of the20 mm long fused silica sample when the input energy of theGaussian beam is increased, for the Bessel beam (m = 0) and

Figure 3. Images of the intensity profiles for the Bessel and theBessel vortex beam taken at different distances behind the liquid-immersion axicon.

Figure 4. Bessel and Bessel vortex beams in nonlinear propagationconditions imaged at the exit of a 20 mm long sample of fused silica.The cone angle is θ = 1.5 deg in fused silica. First row: nonlinearBessel beams with input pulse energies of (from left to right) 6.9,12.1, 17.4, 22.2 and 26.4 μJ. Second row: nonlinear Bessel vorticeswith input pulse energies of 14.8, 27.8, 39.9, 59.7, 66.6 μJ. Thirdrow: line cuts (along x-direction) through the Bessel beam (left) andthe Bessel vortex (right) at different energies.

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J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 094006 C L Arnold et al

the Bessel vortex (m = 1). Again in both cases, clean Bessel-like beams are obtained, which look like the ideal Besselbeams θJ k r( sin )m

20 (with m = 0 or m = 1). We checked that

these fluence patterns remain nearly identical for all samplelengths, i.e., nonlinear propagation is also featured by pro-pagation invariance (data not shown). However, it is knownthat propagation invariant nonlinear Bessel beams exhibit ringcompression (decrease of the ring spacing close to the mainlobe with respect to the ring spacing of Bessel beams) and aloss of ring contrast, as the optical Kerr effect and multi-photon absorption (MPA) increase, respectively [17, 18].Moreover, nonlinear propagation of Bessel beams with peakintensity above a certain threshold no longer leads to propa-gation invariant beams [19, 20]. As can be seen from theprofiles in figure 4, ring compression and ring contrast loss isobserved for the Bessel vortices as well, though higher pulseenergy is required to observe the same ring spacing or con-trast decrease as for Bessel beams with the same cone angle.Experimentally, white-light generation, another indicator fornonlinear propagation, was observed at 22.2 μJ for the Besselbeam and 36 μJ for the Bessel vortex, respectively.

3.3. Interpretation of observations

These observations are reminiscent of similar features pre-viously reported for nonlinear Bessel beams: Porras and co-workers showed in 2004 the existence of Bessel-likepropagation-invariant solutions to the nonlinear Schrödingerequation in the presence of Kerr effect and MPA [19]:

β∂∂

= + −⊥−

E

z kE k

n

nE E E E

i

2i

2, (2)K K

0

20

2

0

2 2 2

where in polar coordinates = + +ϑ⊥

∂∂

∂∂

∂∂

Er

E

r

E

r r

E2 1 12

2 2

2

2 . Theparameters K0, n0, n2 and βK are provided in table 1. Recently,weakly localized propagation invariant solutions toequation (2) were found in the form of nonlinear high-orderBessel beams [14, 21]. These solutions have a stationary (z-independent) intensity profile and a linear dependence of thephase profile upon z:

∫ϑ = δ ϑ− + +( )E r z a r( , , ) ( )e , (3)z q r r mi ( )d

where the wave vector shift is positive δ > 0, and theamplitude a(r) and phase gradient q(r) only depend on thetransverse variable r. By introducing this amplitude and phasedecomposition into equation (2), we find that a(r) and q(r) aresolutions to Newton-like equations for a pseudo-particle withradial and angular positions a, ϕ, (with ϕ =r q( ) ˙), angularmomentum Jr ≡ k0

−1q(r)a2(r) and pseudo-time r:

− = − + − ∂∂

a q aa

r

m

ra

V

a¨

˙, (4)2

2

2

β= − −JJ

ra˙ , (5)r

rK

K2

where dots indicate differentiation with respect to r and thepseudo-potential is defined as δ≡ +V a k a k n a( ) 0

202

24

n(2 ).0 Vortex solutions must satisfy the boundary conditions

a(0) = 0, q(0) = 0. The last boundary condition is governed bythe decay rate as → ∞r which corresponds to that of a Besselbeam carrying helicity m: as →a r( ) 0, nonlinearity is neg-ligible and V(a) ∼ k0δa

2, therefore, the solution tail can beviewed as a superposition of the two linearly independentHankel functions that are solutions to the linearizedequation (4):

α δ α δ= +⎡⎣ ⎤⎦( ) ( )a ra

H k r H k r( )2

2 2 . (6)m m0

out(1)

0 in(2)

0

δH k r( 2 )m(1,2)

0 represent solutions carrying an outward orinward power flux, respectively. Their respective amplitudesαout,in differ, α α∣ ∣ > ∣ ∣in out , leading to a fundamental feature ofnonlinear Bessel vortices: a helical and inward power flux

= + =ϑ ϑ−J J kJ u ur r 0

1 + ϑa q ma ru u[ ( ) ]r2 2 transports

energy from the tail of the beam to the most intense lobewhere it is dissipated by MPA. In this respect, integration ofequation (5) along the radial direction is equivalent to theenergy conservation equation

∫π β π= −rJ r a r r2 ( ) 2 d , (7)r K

rK

0

2

which states that power losses due to MPA within a disk ofradius r are compensated by the power flux flowing throughits boundary (circle of radius r). This ensures propagationinvariance of the nonlinear Bessel vortex [14, 21].

Examples of nonlinear Bessel beams m = 0 and vorticesm = 1 integrated numerically are shown in figure 5 for a givencone angle and given arbitrary nonlinear parameter chosen toillustrate the main features of nonlinear Bessel vortices. For agiven medium (n0, n2, βK), cone angle (δ or θ) and helicity m,a continuum of solutions of this type exists for a range ofintensities I[0, ]max , where Imax can be found only numericallyand is usually two orders of magnitude larger than the

Figure 5. Nonlinear Bessel beam (first column) and Bessel vortexcarrying helicity m = 1 (second column). First row: amplitude incontinuous curve. The red curve shows the linear solution with thesame cone angle and normalized to the same maximum. Second row:phase gradient. Third row: the radial power flux rJr = k0

−1 rqa2

represents power losses for a beam cross section of radius r, perlength unit.

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J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 094006 C L Arnold et al

breakdown threshold of the material. For example forCorning0211 glass, >I 1max PW cm−2 [14]. As the helicity isincreasing, the position of the most intense lobe is foundat a larger radius. This radius is however slightly smallerthan the radius of the linear solution with the same coneangle, i.e., the Kerr effect manifests itself in a compression ofthe most intense rings and a decrease of the ring spacingclose to the center. Another manifestation of nonlinearity isthe ring contrast which is decreasing as MPA increases.Asymptotically, this contrast can be readily obtained froman expansion of equation (6) as → ∞r and reads:

α α α α= ∣ ∣ ∣ ∣ + ∣ ∣C 2 ( )in out in2

out2 . Therefore the highest con-

trast C = 1 is reached for linear propagation, i.e., whenα α∣ ∣ = ∣ ∣in out , the inward and outward flux are balanced andthere is no absorption (β = 0K ). In contrast, the ring dis-appear, leading to zero contrast C = 0 if the power flux isdirected only inward and the outward component is zeroα∣ ∣ = 0out , corresponding to the highest possible MPA in apropagation invariant regime. The link between losses and thecontrast ≡ +C R R2 (1 )2 , where α α≡ ∣ ∣R out in , can beinferred from an introduction of the asymptotic expansion ofequation (6) at large r into the energy conservationequation (7): ∫α β∣ ∣ − = ∞

a R k a r r(1 ) dKK

0 in2 2

0 02 . Since

Bessel vortex beams are computed numerically, the linkbetween losses and contrast can only be entirely specifiednumerically. The reader is referred to [21] for further detailson this point.

We interpret our observations as a manifestation of thereshaping of the beam into quasi-stationary nonlinear Besselvortices. Propagation invariant nonlinear Bessel vortices areideal beams carrying infinite power as shown by the r1decay rate of their tail. Without this feature, MPA wouldunavoidably prevent stationarity. Rather than exact statio-narity, quasi-stationarity arise over a finite length, the Besselzone, because our Bessel vortices are shaped as the idealpropagation invariant solutions and apodized by a Gaussianbeam, of width much larger than the most intense lobe of theBessel-like solutions where MPA occurs. All features ofnonlinear Bessel vortices are therefore observed within theBessel zone. To complete characterization of the beamreshaping process during propagation, we investigated thedecay of the contrast of the rings of the Bessel vortex beamduring nonlinear propagation in fused silica samples of dif-ferent thickness (10, 20, and 40 mm). We take line-cuts (in x-direction) through the beam center, being imaged at the exitof the samples. The line-cuts are normalized and Fouriertransformed. We registered the relative amplitude of theFourier peak associated with the transverse wave number ofthe Bessel vortex. Results are shown in figure 6 for a range ofpulse energies up to 90 μJ. The relative amplitude of theFourier peak decreases as nonlinear propagation becomesmore prominent. In each case, we reported the onset ofsupercontinuum generation, which is a standard signature ofnonlinear propagation. Supercontinuum generation wasexperimentally observed, when the Fourier contrast haddecreased to about half the initial value. We mark the ener-gies, when experimentally supercontinuum generation was

observed by dashed lines in figure 6 for the different samplelengths.

By using equation (1), we can give estimations ofintensity levels at the exit face of the samples. For the threeenergies marked in figure 6, we obtain

= = ×I z( 10 mm) 1.2 1012 Wcm−2 for Ein = 66 μJ,= = ×I z( 20 mm) 1.4 1012 Wcm−2 for Ein = 50 μJ, and= = ×I z( 40 mm) 1.3 1012 Wcm−2 for Ein = 32 μJ. In other

words, the point when the contrast is lost due to nonlinearpropagation can be well identified with the same intensitylevels on the back surfaces for the three different samplelengths. We can therefore conclude that the nonlinear pro-pagation we observe, when the contrast is lost, happens atabout the same intensity and in the vicinity of the respectiveback surface and not within the sample. As shown in the nextsection, this would be possibly different for longer sampleswhich would include the peak of the pulse. The onset ofnonlinear propagation arises for quite low intensity levels of1012W cm−2. At this intensity, no significant plasma densitycan be formed for a 45 fs pulse. For this cone angle(θ = 1.5 deg in fused silica), the Bessel vortex beam does notallow for the generation of hollow plasma cylinders.

3.4. Evaluation of the peak intensity and electron density onthe hollow plasma cylinder

The maximum plasma density in the tubular plasma channelcan be evaluated from the peak intensity, which in turn can beevaluated by energy conservation arguments similar to thoseapplied for a Bessel beam [22]. The nonlinear Bessel vortex isassumed to have an invariant peak intensity Ip and shape overthe Bessel zone. A fraction x of the input beam energy Ein islost nearly exclusively in the main lobe to generate the plasmachannel, by multiphoton ionization. This energy balance reads

∫ ∫β π=xE

LI r t r r t( , )2 d d , (8)

BK

Kin

where the intensity distribution is approximated by the pro-duct of a high-order Bessel beam profile by a Gaussian pulse:

θ∼ −( ) ( )I r t I t T J k r M( , ) exp 4 ln 2 sin . (9)p m m2 2 2

0

By introducing this profile in equation (8), the peak intensity

Figure 6. Fourier contrast as function of pulse energy measured atthe exit face of fused silica samples of lengths 10, 20 and 40 mm.

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J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 094006 C L Arnold et al

is explicitly determined by

θ θβ λ

=g K m

n

Tw I

xEtan sin

( , )(10)K p

K2

02

0 02

in

where ∫π≡ − −g K m M K J u u u( , ) ( ln 2) ( ) dmK u

mK1 2

02

m0( )

.

Figure 7 shows the peak intensity and electron densityevaluated as functions of the cone angle from equation (10)for different helicities of the Bessel vortex. Note that electrondensity is evaluated by considering only multiphoton ioni-zation processes. When it approaches the density of neutralatoms, depletion of the valence band should lead to satura-tion. Depletion remains however negligible for electrondensities smaller than 1021 cm−3. This simple scaling lawpredicts that a cone angle of 1.5 deg would lead to a peakintensity of 4 × 1012W cm−2 and an electron density of1.3 × 1018 cm−3 for the Bessel vortex of lowest helicity m = 1.Peak intensity and electron density decrease with increasinghelicity. An increase of the cone angle in the range10–15 deg, is sufficient to reach peak electron densities in therange 4 × 1020–1021 cm−3, for which energy transfer to thematerial can be deemed sufficiently efficient for inducingrefractive index change or damage [23, 24].

4. Numerical simulations

4.1. Model

The model used for numerical simulations consists in a (2+1)-dimensional unidirectional propagation equation for thenonlinear envelope E(x, y, z), which takes a canonical form[25]. As a nonparaxial equation, it is more easily written forthe Fourier components of the electric field envelope

≡ k k z E x y z( , , ) [ ( , , )]x y , where denotes Fourier

transform in the transverse plane:

ωϵ ϵ

∂∂

= + − ( )z

k kc n n c

i , i2 2

. (11)x y0

0 0 0 0

Nonparaxial diffraction is described in the propagation con-

stant ≡ − − − k k k k k k( , )x y x y02 2 2

0. The Kerr effect isaccounted for in the nonlinear polarization

≡ k k z P x y z( , , ) [ ( , , )]x y , with ϵ≡P n n IE2 0 0 2 , where

≡I (1 2)ϵ ∣ ∣cn E0 02. The current ≡ k k z( , , )x y

J x y z[ ( , , )], where ≡ +J J JMPA PL comprises two con-tributions due to MPA and plasma-induced effects (PL). Thecontribution of MPA reads ϵ β= −J cn I EK

KMPA 0 0

1 , withcoefficient βK and number of photons K.

Plasma-induced effects, i.e., plasma absorption andplasma defocusing are included in a single expression for thecurrent, derived from the Drude model:

ϵω τ

ωτρρ

=−

J E1 i

, (12)c

c cPL 0

02

where τc denotes a phenomenological collision time in themedium and ρc, the critical plasma density beyond which themedium is no longer transparent.

The evolution of the electron plasma density is modeledby a rate equation modeling multiphoton and avalancheionization with cross section σ, as well as electron plasmarelaxation with typical time τr:

ρ βω

σρ ρρ

ρτ

∂∂

= + − −⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟t K

II

U1 . (13)K K

i r0 nt

We consider large cone angles, giving rise to a short Besselzone over which dispersion is negligible. The plasma densityis therefore evaluated in the frozen time approximation thatconsists in using a Gaussian pulse with the nominal durationof the laser.

In table 1, we provide the model parameters used forfused silica and BK7 at the pulse central wavelength of

Figure 7. Maximum Intensity and plasma density for a nonlinearBessel vortex as a function of its cone angle for different helicities.Parameters correspond to fused silica and experimental scheme Awith T = 45 fs, λ0 = 800 nm, n0 = 1.45, w0 = 4.4 mm, Ein = 30 μJand x = 100%.

Table 1. Parameters uses for modeling the response of fused silicaand BK7. Refraction index: n0. Kerr index coefficient: n2 [23, 26].Number of photons: ≡ ⟨ + ⟩

ωK 1Ui

0. Coefficient for multiphoton

absorption: βK [27]. Bandgap: Ui collision time: τc. Plasmarelaxation time: τr . Neutral density: ρnt. Critical plasma density:ρ ω ϵ≡ m qc e e0

20

2.

Fused silica BK7

n0 1.45 1.51n2 (cm

2 W−1) 3.54 × 10−16 3.45 × 10−16

K 5 3βK (cm2K−3 W−(K − 1)) 6.8 × 10−50 3.9 × 10−28

Ui (eV) 7.1 4.2τc (fs) 1.3 1τr (fs) 150 2000ρnt (cm

−3) 2.1 × 1022 2.1 × 1022

ρc (cm−3) 1.8 × 1021 1.8 × 1021

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J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 094006 C L Arnold et al

800 nm, together with references from which the values wereobtained.

4.2. Results

4.2.1. Nonlinear Bessel vortex in fused silica at small coneangles. Figure 8 shows simulation results for theexperimental scheme A with a cone angle of θ = 1.5 deg infused silica. First, we note that the sample thickness wasapproximately half the length of the Bessel zone. For a pulseenergy of 7 μJ, the beam forms a Bessel vortex with its typicalmain ring surrounded by lower intensity rings. The highestintensity reaches 400 GW cm−2 and does not lead to asignificant plasma density. The highest calculated plasmadensity is about 1010 cm−3 (not shown). This regime thereforeshows the formation of a quasi linear Bessel vortex. If theinput energy is increased, there is however no abruptthreshold but a continuous transition between this regimeand the nonlinear regime featured by the formation ofpropagation invariant nonlinear Bessel vortices, as foundsemi-analytically in section 3.3. By further increasing theinput energy above a certain threshold, the propagationinvariant regime is no longer found. Figure 8 shows that at66 μJ, the uniform ring formed in the beginning of the Besselzone undergoes instability, forming a high intensity peak thatrotates around the propagation axis at a fairly regular rate,reflecting the angular momentum carried by the beam. Thehighest intensity of 5 TW cm−2 is sufficient to generate aplasma of density of 1019 cm−3. However, this density mightbe still too low to induce damage in the material. These

experimental conditions might be just below threshold forrefractive index change in fused silica. As a final comparisonwith the experimental patterns, we note that the rotating peakappears above 32 μJ, similarly to the results in [14]. At higherenergy the contrast gets smoothed out, and the rotating peak isnot so pronounced as in the simulation. Dispersion of the45 fs pulse was neglected in the simulation but could stretchthe pulse by a factor of ∼2.3 over 6 cm in the experiment andtherefore decrease the peak intensity and shift the instabilitythreshold towards larger pulse energies.

4.2.2. Nonlinear Bessel vortex in BK7 at large cone angles.Figure 9 shows simulation results for experimental parametersin scheme B, performed at higher cone angle of 11 deg inBK7 (16.95 deg in air). The experimental scheme shown infigure 2 is described in more detail in section 5. Thesimulations allow us to anticipate a high electron density(1021 cm−3) distributed uniformly on a hollow cylinderextending over the entire thickness of the BK7 sample(100 μm). Uniformity appears after a regularization stage inthe first microns after the entrance face of the sample wherethe intensity is sufficiently high for secondary lobes of theBessel vortex to also generate plasma. However, it is clearthat a propagation invariant regime takes place. The peakintensity is a factor of 2 smaller than in the simulation for thesmall-cone-angle, 45 fs pulse. Among the reasons leading to ahigher electron density, not only has BK7 a slightly smallerbandgap than fused silica but also the picosecond pulse usedfor BK7 facilitates the generation of electrons by avalanchewhile requiring a lower intensity to reach the same level asthat obtained with a fs pulse. Finally, propagation invariantnonlinear Bessel vortices at a given peak intensity are morestable for large cone angles and can therefore attract the beamdynamics [14].

Figure 8. (a) and (b) Numerically simulated intensity profile alongthe propagation axis for the experiment (scheme A) with small coneangle. The pulse duration is 45 fs and pulse energies are (a) 7 μJ and(b) 66 μJ. (c) Plasma density obtained for intensity shown in (b). Thecolorbar indicates ρlog10 when the plasma density ρ is expressedin cm−3.

Figure 9. Numerically simulated plasma density and intensity profilealong the propagation axis for the experiment (scheme B) with largecone angle. The pulse duration is 1 ps and pulse energy is 37 μJ.

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J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 094006 C L Arnold et al

5. Material structuring with single shot picosecondlaser Bessel vortex beams

In order to assess the potential of intense Bessel vortex beamsfor microstructuring the bulk of glass materials, we performedmicrofabrication tests in single shot with picosecond laserpulses for different cone angles. In this case (scheme B infigure 2) the laser source was a 20 Hz regeneratively ampli-fied Ti:Sapphire laser system delivering λ = 800 nm, 40 fspulses with about 30 nm bandwidth, that could be chirped upto picosecond duration. The Gaussian laser output beam wasspatially filtered, demagnified and sent to a spatial lightmodulator (SLM, (PLUTO, Holoeye) for the beam shaping. A10 nm bandwidth interference filter was used to reduce thelaser bandwidth in order to improve the efficiency and qualityof the phase modulated beam reflected by the SLM. The pulseenergy was adjusted by means of a half-waveplate combinedwith a linear polarizer, and a mechanical shutter on the beamline allowed the selection of single pulses for single shotmachining. High-order Bessel-Beams (Jm) are generated byapplying onto the SLM a spatial phase modulation corre-sponding to that of an axicon superimposed with a phasemask imposing a helicity of order m (or topological charge),which determines the radius of the innermost ring. By chan-ging the axicon angle parameter (linked to the Bessel coneangle) we can adjust the Bessel zone length and the Besselcore size or ring thickness. The spatial quality of the gener-ated beam is guaranteed by a far-field spatial filtering after theSLM. After demagnification by means of a telescope con-stituted by a f5 = 300 mm focal length plano-convex lens anda 0.45 N.A. 20 X microscope objective, the resulting high-order Bessel beam Jm is then incident onto 100 μm thin glasssamples. The sample is placed on a computer controlled three-axes motorized stage for precise position adjustment duringthe microfabrication process. Before machining, the evolutionof the intensity beam profile after the demagnification set-upis monitored by means of an imaging system in combinationwith a CCD camera. Figure 10 shows the imaging of theBessel vortices taken exactly at the sample position but in air.For machining, we added the glass sample. The Bessel zonewas 200 μm, the ring thickness and diameter were about 1 and8 μm, respectively.

Our tests showed that no clean tubular structure writingin glass was obtained below a certain cone angle (7 deg inglass for our experimental conditions). We attribute this to thefact that propagation invariant Bessel vortex beams with anintensity sufficient to ionize glass significantly only exist forcone angles exceeding a certain threshold [14]. For smallcone angles, propagation invariant Bessel vortex beams canundergo instability since nonlinear effects are predominant(nonlinear length is much shorter than the length over whichthe Bessel core regenerates due to the refilling from anexternal ring). In order to cure this issue whithout increasingcone angles to extremely large values, we increased the pulseduration above 1 ps with the goal of simultaneously (i)reducing the peak intensity of the propagation invariantBessel vortex beam, thus increasing the nonlinear length andthe stability of the Bessel vortex beam, and (ii) increasing thenonlinear absorption associated with the generation of elec-trons by avalanche ionization. We also note that similarly tothe propagation of intense Bessel beams, picosecond pulsesare associated with weaker electron carrier defocusing, whichmay facilitate reaching a propagation invariant regime and amore efficient laser energy deposition [28].

Figures 11 and 12 show pictures of clean high aspectratio tubular microstructures obtained in single shot for a J8Bessel beam with cone angle θ = 11 deg in BK7 glass,obtained with 1 ps, 800 nm pulses. Figure 11 shows the inputand exit surfaces of the glass sample (respectively denoted astop and bottom in the figure) for different pulse energies.Clean ring structures matching the radius of the J8 beam areobserved, indicating the presence of a tubular refractive indexchange over the entire thickness of the sample. This structureis clearly shown in figure 12 for a pulse energy of 35 μJ.Similar results were obtained with fused silica (not shown)with lower pulse energy. In our geometry, when using pulseduration much shorter than 500 fs, the tubular structure wasweaker and traces on the surfaces were not so evident as thoseshown in figure 11.

Figure 10. Images of the Bessel vortices at different propagationdistances in air. Cone angle in air θ = 16.95 deg.

Figure 11. Optical microscope images of top and bottom glasssample surfaces micromachined with J8 Bessel vortex beam in singleshot, and for three different energy values E = 22 μJ (a), E = 37 μJ(b) and E = 56 μJ (c). The cone angle is θ = 11 deg in BK7.

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6. Conclusions

Clean Bessel–Gauss beams and Bessel–Gauss vortices weregenerated by various methods: a combination of a liquid-immersion axicon and a SPP allowed flexibility in the regimeof small cone angles. Through samples of 10, 20, and 40 mmof fused silica, Bessel–Gauss vortices were shown to propa-gate as quasi invariant beams with pulse energy of up to 30 μJbefore nonlinear propagation degraded the contrast of thehollow beam. This is much more than the critical energy(associated with the critical power of ∼200MW) in fusedsilica, which is below 1 μJ. However, the intensity at whichthe ring structure starts to undergo symmetry breaking in ourexperiments at small cone-angles does not seem to be suffi-cient to generate a dense plasma. The same propagationregime in air would lead to an underdense hollow plasmachannel with ideal density and dimensions for the generationof a transient waveguide for microwaves [29, 30]. In thefemtosecond regime, we have proposed a simple scaling lawto link the laser and material parameters to the density ofelectrons generated by multiphoton ionization.

The regime of larger cone angles was tackled using aSLM and a demagnification system (telescope and micro-scope objective). Quasi propagation invariant annular Besselbeams were shown to lead to tubular light filaments, which inturn generate a hollow plasma channel in a BK7 glass withdensity evaluated by numerical simulations about 1021 cm−3.Although the physical mechanism at the origin of materialdensification was not investigated here, observation of the

sample after single shot illumination by a picosecond pulseshows that pulses of a few microjoules with picoseconddurations and Bessel cone angles larger than 7 deg lead to atubular refractive index change over the entire Bessel zone(sample thickness of 100 μm).

Acknowledgments

We acknowledge funding from Deutsche Akademie derNaturforscher Leopoldina (grant no. BMBF-LPD 9901/8-181), the French National Research Agency, ANR contract2011-BS04-010-01 NANOFLAM, Region Franche-Comteand the International Program for Scientific Cooperation(PICS) at CNRS.

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