Morphology of femtosecond laser modification of bulk dielectrics

Morphology of femtosecond lasermodification of bulk dielectrics

K. I. Popov,∗ C. McElcheran, K. Briggs, S. Mack, and L. RamunnoDepartment of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario K1N 6N5,

Canada∗[email protected]

Abstract: Using 3D Finite-Difference-Time-Domain simulations, westudy the morphology of the laser-created damage inside fused silica.Among the competing effects limiting the intensity in the dielectric, wefind the most important is the pulse defocusing by the plasma lens, partiallybalanced by the Kerr effect. Less important are collisional energy dissipa-tion and laser depletion by multi-photon absorption. We also found that theprofile of generated plasma is asymmetrical in the transverse cross-section,with the plasma extended along the direction perpendicular to the laserpolarization.

© 2010 Optical Society of AmericaOCIS codes:(140.3330) Laser damage; (260.3230) Ionization.

References and links1. D. Homoelle, S. Wielandy, A. L. Gaeta, N. F. Borrelli, and C. Smith, “Infrared photosensitivity in silica glasses

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219–225 (2008).5. G. D. Valle, R. Osellame, P. Laporta, “Micromachining of photonic devices by femtosecond laser pulses,” J. Opt.

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laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett.89, 186601 (2002).7. J. R. Penano, P. Sprangle, B. Hafizi, W. Manheimer, and A. Zigler, “Transmission of intense femtosecond laser

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media,” Opt. Express13, 3208–3217 (2005).9. A. Q. Wu, I. H. Chowdhury, and X. Xu, “Femtosecond laser absorption in fused silica: numerical and experi-

mental investigation,” Phys. Rev. B72, 085128 (2005).10. C. L. Arnold , A. Heisterkamp, W. Ertmer, and H. Lubatschowski, “Computational model for nonlinear plasma

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#137563 - $15.00 USD Received 2 Nov 2010; revised 8 Dec 2010; accepted 13 Dec 2010; published 22 Dec 2010(C) 2011 OSA 3 January 2011 / Vol. 19, No. 1 / OPTICS EXPRESS 271

15. A. Taflove, S. C. Hagness,Computational Electrodynamics, 3rd. ed. (Artech House, 2005), pp. 58–79.16. K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic

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tightly focused laser pulses with nanoscale targets,” Phys. Plasmas16, 053106 (2009).22. J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev.56, 99–107 (1939).23. S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179,

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laser pulse,” Phys. Plasmas15, 013108 (2008).

1. Introduction

Recentadvances in laser science have enabled the precise control of the modification of dielec-tric materials via intense femtosecond pulses. An ultrafast pulse can be focused within the bulkof a dielectric, which is transparent in the linear regime, but the intensity can be made highenough that nonlinear interactions such as multiphoton ionization cause permanent modifica-tion of the target region near the laser focus. The damage can be used to tailor the structuralproperties of dielectrics in three dimensions on a sub-wavelength scale. Depending on the en-ergy deposited in the region, different types of damage have been observed, including changesto the index of refraction at intensities just above threshold [1], periodic nano-cracks inside thematerial above threshold [2], and the production of void-like structures at even higher inten-sities [3]. The numerous potential applications of this technology include fabrication of three-dimensional photonic crystals, microfluidic devices, volume gratings, optical memory devicesand optical waveguides [4, 5].

Precise control of the laser damage requires a detailed analysis of the dynamics of the laser-material interaction. There have been a number of numerical studies of the multi-photon ioniza-tion process and the resulting nonlinear electromagnetic wave propagation in the large-bandgapdielectrics such as water or fused silica [6] – [13]. Of particular interest is identification of theparticular mechanisms responsible for the morphology of the damaged region. Burakov et al.[11] use a two-dimensional nonlinear optical Schrodinger equation to analyze the ionizationpatterns produced by femtosecond and picosecond lasers. They attribute the elongated modi-fied region to a balance between the nonlinear defocusing due to the plasma dispersion and theself-focusing of the Kerr effect. Rayner et al. [8] and Grojo et al. [14] propose that the Kerreffect is not important and the shape arises from the depletion of the incident laser beam. Theyargue that since there are orders of magnitude more ionizable molecules in solids than photonsto excite these molecules, the depletion of the beam becomes an important factor in the nonlin-ear process. They hypothesize that the intensity will not exceed the threshold intensity because,once the intensity is high enough, ionization will occur and absorb photons, depleting the beam.In a focused laser, as the beam width narrows and the peak intensity increases, a larger portionof the intensity curve will be cut off. This causes the narrowing of the ionizing region. Rayner etal. model this effect by cutting the intensity at the threshold and assuming that the energy fromthe photons was used to ionize the material. Which of these models is correct, and in whichparameter regimes, is still up for debate.

In this paper, we introduce a 3D dynamic model for simulating nonlinear ionization in di-electrics that enables a systematic analysis of the underlying processes. In particular, we inves-tigate the time evolution of plasma creation to understand the microscopic mechanisms respon-sible for the morphology of the damage region in silica. The region is considered “damaged”


in the text below wherever there is an appreciable level of ionization happened. We extendedastandard Finite-Difference Time-Domain (FDTD) algorithm [20] to include multiphoton ion-ization (MPI), multiphoton absorption (MPA), the Kerr effect, and plasma dispersion, wherethe generation and optical response of the plasma is described via a plasma fluid model. Themodel introduced is intrinsically capable of modelling propagation of a tightly focused laserpulse in a dielectric medium with the possible over-dense plasma generation,i.e., it does notpossess limitations of the uni-directional nonlinear models presented in the majority of the pre-vious work. In this study we will concentrate on a relatively mild focusing with a numericalaperture (NA) equal to 0.65, as in Ref. [14].

In addition to the identification of the contribution of various mechanisms to the morphologyof the damaged region, our 3D model enabled us to study effects of laser polarization. In par-ticular, we have found that generated plasma pattern possesses an asymmetry in the transversecross-section that can be explained by the refraction index mismatch between the plasma andthe surrounding dielectric.

2. Model

2.1. Basic equations

In our 3D numerical model, we solve Maxwell equations,

∇×~E = −1c

∂~B∂ t

,

∇× ~H =1c

∂~D∂ t

+4πc

~J,

(1)

usingthe FDTD method [20]. In Eq. (1),~E and~B are the components of electromagnetic field,~D electric field displacement vector,~H the magnetic field auxiliary vector,t is the time, andc is the speed of light. We extend the Yee discretization algorithm [16] by introducing theconsitutive relations,

~H = ~B,

~D =(

1+4π(χl + χkE2))

~E,(2)

and the current density~J = ~Jp + ~JMPA. (3)

In (2), χl is the linear dispersionless susceptibility of the material andχk is the Kerr susceptibil-ity. The electromagnetic response of generated plasma is represented by the current density~Jp

whereas laser depletion due to multi-photon absorption is accounted for by the quantity~JMPA.The free particle current~Jp is calculated from the fluid equations for the electron component

of the generated plasma:

∂n∂ t

+∇(n~u) =∂nMPI

∂ t,

mn[∂~u

∂ t+(~u∇)~u

]

= −en~E −∇p−mΓn~u,

(4)

wheren and~u are electron particle density and fluid velocity, respectively,∇p is the pressuregradient, and the generation rate of free electrons via six-photon absorption is given by [7]

∂nMPI

∂ t=

σ6I6

ns(ns −n), (5)


whereI is laser intensity,σ = 2× 1013 cm−3psec−1(cm2/TW)6 is the six-photon absorptioncross-section [7, 17] andns is the saturation particle density. The quantityns can be estimatedas the electron density when every molecule of silica is singly-ionized:ns = ρmNA/Mµ , whereρm is the mass density of fused silica,Mµ its molar mass andNA Avogadro number. Substi-tution of the appropriate constants into this expression results inns ≈ 10ncr, wherencr is thecritical particle density for electrons,i.e., the density for which the electron plasma frequencyequals the laser frequency;ncr = 1.75×1021 cm−3 for the wavelength considered. In the actualcalculations, we approximate the laser intensity by the square of instantaneous electric field:

I =c

4πE2. (6)

Eq. (5) assumes that the ionization occurs on a time scale slower than a wave period. Theapproximation given by Eq. (6) introduces an inaccuracy of order〈E2〉/ 6

√

〈E12〉, where theangle brackets denote a cycle-average, which results in an under-estimation of the thresholdintensities by approximately 35%. Performing an accurate cycle averaging in 3D is numericallyunfeasible, given the large scale of our calculations as mentioned below, since each time pointover a wave period at each grid cell location would need to be stored in memory.

The quantityΓ in Eq. (4) is the damping factor representing collisional energy dissipation.In the present work, we consider it to be approximately constant. This constant is estimated asfollows. If we assume that the kinetic energy of an electron in the laser pulse is dissipated aftera few collisions, then

Γ ∼ νe, (7)

whereνe is the electron collision frequency. Neglecting collisions with neutrals, this frequencyis [18]

νe = 2.91×10−6ne[cm−3] lnΛ Te[eV]−3/2 sec−1, (8)

wherene is the electron particle density, lnΛ ≈ 10, andTe electron temperature. This tempera-ture is approximately equal to the energy of electron quiver motion,Te ≈ mv2

q/2, where thequiver velocity is

vq =eE

mω0; (9)

E is the electric field in the laser pulse,ω0 is laser frequency ande, m are electron charge andmass, respectively. The laser intensity in the dielectric is limited by the ionization threshold,equal to(1÷ 1.5)× 1013W/cm2 [14]. Substituting this into (9), one obtainsvq ≈ 3× 10−3c,wherec is the speed of light in vacuum. This velocity corresponds to an electron kinetic energyin eV range. Substituting this value into Eq. (8), one can obtain a pessimistic estimation for thedamping factor

Γ ∼ 1 fs−1. (10)

A more complete consideration with a temperature-dependent damping factor will be developedand published elsewhere.

To solve Eq. (4), we make further approximations. To achieve hydrodynamic closure we as-sume a cold plasma model, for whichp = 0. We also assume that the plasma fluid velocity~u inthe collision-dominated regime is small and thus(~u∇)~u ≈ 0. Finally, we assume that the domi-nant mechanism causing changes to the particle density of electrons is ionization rather than thehydrodynamic advance of the plasma,i.e., ∇(n~u) ≪ ∂nMPI/∂ t. Under these approximations,Eq. (4) simplifies to

∂n∂ t

=∂nMPI

∂ t,

∂~u∂ t

= − em

~E −Γ~u.

(11)


The last equation in (11) can be also identified with equation of motion for free particles in theDrudemodel for conductive media [19]. One may calculate the current of free plasma electrons

~Jp = −en~u. (12)

The effective current~JMPA is obtained by setting the laser energy density depletion rate equalto the product (~JMPA ·~E) [7]. Assuming that~JMPA is parallel to~E one obtains

~JMPA =~EE2Wion

∂nMPI

∂ t, (13)

whereWion is the ionization energy, equal to 9 eV for the case of fused silica.Equations (1) – (3), (5), (11) – (13) comprise a closed system of equations that is solved

numerically using FDTD method.A similar model was developed in Ref. [13] for a 2D plane geometry case.

2.2. Model benchmark

Now we demonstrate that the approximations made in Sec. 2.1 are reasonable and reproducethe basic physics of interaction.

Our first test is propagation of an electromagnetic field in an infinite cold plasma of prede-fined particle density with no ionization, multi-photon absorption or Kerr effect. The dispersionrelation for a plane wave in a cold plasma is

k(ω) =ωc

√

1−ω2

p

ω(ω − iΓ), (14)

whereωp =√

4πne2/m is the plasma frequency andi the imaginary unit. At the entrance ofthe plasma, the wave envelope is assumed to be Gaussian with the width∆T :

Eω(ω,0) =∆T√

2exp

(

− (ω −ω0)2∆T 2

4

)

. (15)

After propagation a distancex, the envelope becomes

Et(t,x) = F−1

(

Eω(ω,0)exp(

− ik(ω)x)

)

, (16)

whereF−1 is the inverse Fourier transformation andk(ω) is evaluated from Eq. (14).A comparison, not presented here, of Eq. (16) with our actual FDTD simulation results re-

vealed perfect matching. A similar test measuring the amount of reflection from underdenseand overdense semi-infinite plasmas has also shown perfect agreement with the appropriateanalytical formula.

Next we demonstrate the multi-photon ionization and absorption processes. For this test, weassumeJp ≡ 0, χk = 0, and propagate a plane electromagnetic wave through a 40-micron-thickslab of fused silica. Figure 1a shows the absolute value of Poynting vector of the pulse justbefore it enters the slab (att = 0) and right after it exits the slab (t = 350fs). The transmittedpulse is cut at intensity∼ 1013W/cm2. The dependence of the peak transmitted intensity onpeak incident pulse intensity is given in Fig. 1b. It is seen that as soon as the incident laser ex-ceeds the threshold intensity∼ 1013W/cm2, the pulses become depleted via ionization, and thetransmitted intensity is capped by the threshold value. Due to the reflections from the vacuum-dielectric and dielectric-vacuum interfaces, the actual intensity threshold inside the dielectric


20 40 60 80 100 120x, Μm

5.0´1013

1.0´1014

1.5´1014

2.0´1014ÈSÈ, W�cm2

t = 0

t = 350 fs

1012 1013 1014Iinc, W�cm21´1012

1´1013

3´1012

It , W�cm2HaL HbL

Fig. 1. Multi-photon absorption test. (a) 1D propagation of a 30-fs laser pulse withImax =〈|S|〉max/2 = 1014W/cm2 through a slab of fused silica. The slab is located betweenx =40µm andx = 80µm. (b) Maximum transmitted intensity vs. intensity of the incidentpulse.

should be slightly larger. The found value is consistent with the experimental measurements[8]. An additional energy conservation test (not shown) demonstrated that the total energy inthe systemEem +Eabs is conserved throughout the simulation withJp ≡ 0, whereEem is the totalelectromagnetic energy in the domain andEabs the absorbed energy.

Given the results of the demonstrated tests, we conclude that the code generates a reliableplasma response and correctly accounts for the multi-photon ionization and absorption pro-cesses.

2.3. Laser source excitation

In our 3D numerical model, the laser pulse is excited at the boundary of the box using the Total-Field-Scattered-Field approach [?]. This method generally requires an exact knowledge of theelectromagnetic field being solution to the Maxwell equations at the boundary. To evaluate thisfield we use the model of the focused laser pulse, previously employed in [21], which allows usto use high NA-optics, including NA> 1. We give only a short description of this model here.

Let the focusing optics be represented by a paraboloidal mirror (Fig. 2), with a wide Gaussian

Fig. 2. Scheme of the focusing optics.

beamincident onto the mirror. Right at the mirror surface, the field is evaluated using the properboundary conditions at the ideally reflecting surface. In any point in the space, the field is


evaluated using Stratton-Chu integrals [22]:

~E(~r) =1

4π

∫

A[ik(~n× ~H)G+(~n×~E)×∇G+(~n ·~E)∇G]dA,

~H(~r) =1

4π

∫

A[ik(~E ×~n)G+(~n× ~H)×∇G+(~n · ~H)∇G]dA,

(17)

whereA is the mirror surface,k the length of the wave vector,~n the inner normal to the integra-tion surface and

G(u) = exp(iku)/u, (18)

with u being the distance between the observer and the point at the surface:u = |~rp −~rs|. Theintegrals in (17) are evaluated numerically. When integration is performed over an unclosedsurface, as the surface of the mirror is, they approach a solution to Maxwell equations with thegiven boundary conditions asymptotically, asRm ≫ λ with NA kept constant, whereRm is themirror radius andλ the laser wavelength.

In the present paper we use NA = 0.65, for a Gaussian beam incident of radiusRg, withRg = 0.5Rm. The actual mirror radius used in the simulations was equal to 1 mm. The resultingfocused pulse had full width at intensity half-maximum (in silica, in the absence of nonlineareffects) equal to 1µm in the transverse direction and 7.5µm in the longitudinal direction.

3. Morphology of the generated plasma

In our analysis of the generated plasma morphology we will consider the following laser param-eters: laser wavelengthλ = 800nm, peak laser intensity at the focus (in the absence of nonlineareffects and dispersion)I = 1014W/cm2 and the laser pulse length (defined as the full width atthe half-maximum within a Gaussian envelope) equal to 50 fs. The corresponding laser pulseenergy is 4×10−7 J. In all the examples shown below we assume the medium possesses a linearrefraction indexn0 ≈ 1.45. The Kerr susceptibility was assumed to beχ3 = 1.9×10−15esu.

In our simulations we used a 50µm× 32µm× 32µm simulation domain, with 453 gridpoints perµm3. These simulations were performed on a 300-cpu computer cluster, with a singlesimulation requiring 650 Gb of RAM and approximately 24 hours of runtime. For all the resultsshown below, the laser pulse is propagating from left to right, with the laser axis passing troughthe center of the simulation domain.

As the laser propagates in the dielectric, it ionizes it and creates a plasma. As was estimatedabove, the energy of quiver motion of electrons in the laser pulse is on the∼eV scale. Thecorresponding sound velocity of plasma,

cs =

√

Te

M, (19)

whereM is the ion mass, is thus on the ordercs ∼ 103cm/s. The sound velocity defines thescale of hydrodynamic expansion of plasma. In this way, if recombination and subsequent so-lidification, that are not accounted for by our model, happen on the∼ns or a shorter time scale,the pattern of plasma, generated on the fs-scale, coincides with the shape of the laser-inflicteddamage.

Figure 3a shows the profile of the generated plasma in the polarization plane passing throughthe laser focus for the fused silica occupying the entire computational domain. The laser pulsepropagates in the+x direction. The profile is characterized by a narrowing shape in the directionof laser propagation (cf. [6], [14]), with the maximum particle density being before the actualgeometrical focus of the laser pulse.


Fig. 3. (Color online) Profile of the generated plasma. The plasma density is given in unitsof electron critical particle density. The position of the geometrical focal plane of the laserin the absence of nonlinear processes isx = 2.75µm. (a) All nonlinear propagation effects(plasma dispersion, multi-photon absorption, Kerr effect) accounted for; (b) no MPA; (c)MPA only; (d) all nonlinear effects, withΓ = 0.

To consider the effect of laser depletion, we have performed simulations with~JMPA ≡ 0, i.e.,with the effect of multi-photon absorption turned off, which is equivalent to assuming an infinitesupply of photons. The resulting plasma profile is shown in Fig. 3b. It is seen that in the absenceof multi-photon absorption, the maximum density of the generated plasma increases by∼ 50%.However, the shape itself is essentially unchanged. Further, contrary to our expectations, nocatastrophic ionization occurs. On the other hand, if all the nonlinear effects except for multi-photon absorption are switched off, the maximum density of plasma approaches saturationdensityns = 10ncr (Fig. 3c). The shape of the damaged region also changes considerably. Weconclude that although multi-photon absorption is an important effect, it is not the main limitingmechanism of the laser intensity inside the plasma, at least in our considered parameter regime.

Figure 3d shows the plasma pattern for all the effects accounted for but withΓ = 0, i.e., forcollisionless plasma. It will be shown below that collisional energy dissipation is responsiblefor a large part of absorbed energy by the medium. However, as follows from Fig. 3a,d, thisenergy dissipation has a little effect on the maximum plasma density, though it does increasethe total number of electrons.

If the Kerr effect is turned off (not shown), the plasma shape remains close to the one shownin Fig. 3a, however with the maximum density decreased by∼ 50%. This is consistent withwhat we would expect by removing self-focusing.

Figure 4 shows the details of interaction process. The left column of Fig. 4 shows the absolutevalue of the Poynting vector,|~S|, and the right column the plasma density att = 300 fs (top),t =320 fs (middle), andt =340 fs, wheret is the time after the start of the simulation. As the


laser focuses, its intensity grows to an above-threshold value already at the pulse leading edge,correspondingto |~S|max ≈ (2÷3)×1013W/cm2 (t = 300 fs in Fig. 4). As a result, plasma isgenerated. As the plasma density grows, the laser field appears to be expelled from the damagedregion, and thus the intensity profile forms a ring-like structure in the transverse cross-section(t = 320, 340 fs). Due to this modification the laser intensity decreases to approximately thethreshold value and as a result the ionization process stops. The ring-like intensity merges backto a spot behind the generated plasma (t = 340 fs), but intensity there is still below the ionizationthreshold, and almost no new plasma is generated.

Fig. 4. (Color online) Details of laser pulse propagation and plasma generation.

Thereare two possible reasons of the laser pulse being expelled from the region occupiedby plasma: 1) as plasma forms, it acts as a diverging lens for the remaining part of the pulse,and thus the pulse gets defocused; 2) due to the relatively large damping factorΓ the plasmaquiver motion is converted to heat at a fs time scale, and thus the laser energy dissipates insidethe underdense plasma.

To find out which mechanism is dominant, let us refer to the energy diagnostic shown in Fig.


5. The actual simulations for the energy diagnostic were done in a larger simulation domainthanthat of Figs. 3, 4 to fit the entire laser pulse into it. The diagnostic in Fig. 5 shows the elec-tromagnetic energy over the whole domainEem, total ionization-absorbed energyEabs, instan-taneous kinetic energy of all the free particles in the domainEk and their sumEem +Eabs +Ek.

0 100 200 300 400t, fs0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

Energy, a.u.

Eem+Eabs+Ek

Ek

Eabs

Eem

Fig. 5. (Color online) Energy diagnostic of the interaction process.

At t = 0, there is no electromagnetic field in the domain, the total energy is zero. Att > 0 thepulse starts entering the domain from its left boundary. Byt ≈ 260 fs, most of the pulse is insidethe domain. Att ≈ 290 fs, the ionization process starts. Eventually the total energy starts todecrease, indicating conversion of the particle kinetic energy into heat. At the end of simulation(t = 425 fs), the total electromagnetic energy in the domain has decreased approximately twotimes, with∼ 1/3 of this change caused by the multi-photon absorption and∼ 2/3 of thechange by the thermal energy dissipation.

Taking into consideration that for a collisionless plasma (Fig. 3d) the morphology of theionized medium is almost unchanged, we conclude that among the three competing effects themost important effect responsible for the shape of the damaged region is the defocusing ofthe laser pulse by the generated plasma, partially balanced by Kerr effect. An important effectresponsible for dissipation of a large portion of the laser energy is the collisional damping of thequiver motion of plasma particles, and the least important effect is the laser energy depletiondue to multi-photon absorption.

The interaction scenario is, in this way, temporally asymmetric. Whereas the leading edgeof the pulse propagates in the dielectric medium almost unchanged, the peak and the trailingedge of the pulse experience a strong defocusing by the generated plasma. The intensity of thedefocused peak of the laser pulse is still high enough to cause the broad low density plasma atback end, as seen in the the left sides of the plasma density profiles in Figure 4 att = 320 and340 fs. The rest of the defocused pulse does not contribute to additional plasma generation. Aportion of the first half of the pulse that has created the plasma and does not experience defocus-ing continues propagating undisturbed in the forward direction. This results in an asymmetricplasma profile, elongated and sharpened in the forward direction, as shown in Figs. 3a and 4 att = 340 fs.

Figure 6a shows the plasma density after the passage of the laser along its axis. The plasmadensity spans a distance∼ 15µm, a figure on the order of the laser Rayleigh length, with themaximum value before the geometrical focus of the focusing mirror. The longitudinal profileis characterized by a relatively sharp growth (x≈−7µm in Fig. 6a), a pedestal (−6µm . x .

−3µm), an extremum (x= 0) and a relatively slow decrease (x> 0). Of a practical interest is


-10 -5 0 5 10x, Μm

0.1

0.2

0.3

0.4

n�ncr

-6 -4 -2 0 2 4 60.0

0.2

0.4

0.6

0.8

1.0

x, Μm

Dy,Μ

m

Laser focus

HaL HbL

Laser focus

Fig. 6. Characterization of the profile from Fig. 3a. (a) Plasma density along the laser axis.(b) Transverse size, along the laser polarization, of the plasma, defined by leveln = 0.4ncr

(solid line),n = 0.2ncr (dashed line),n = 0.05ncr (dotted line)

the transverse size of the damaged region. This size vs. longitudinal coordinate is given in Fig.6b, where each plot gives the width defined for constant electron density values, as indicated inthe figure caption. This provides an effective contour plot of the plasma.

4. Laser polarization effect

A consequence of importance of the interaction with generated plasma is asymmetry that wehave observed in the transverse cross-section. Figure 7a shows the plasma density versus dis-tance, for the directions along the laser polarization and perpendicular to it, in the plane parallelto the laser focal plane and passing through the maximum plasma density point.

0.5 1.0 1.5r, Μm0.0

0.1

0.2

0.3

0.4

n�ncr

Perpendicular to the laser polarization

Along the laser polarization

0.5 1.0 1.5r, Μm0.0

0.2

0.4

0.6

0.8

1.0<E2>, a.u.

Perpendicular to the laser polarization

Along the laser polarization

HaL HbL

Fig. 7. (Color online) (a) Plasma density in the transverse-cross-section passing through themaximumplasma particle density. (b) Quantity〈E2〉 of the focused laser in the transversedirection, in the absence of nonlinear effects.

The plasma density along the laser polarization decreases faster with the radius than that inthe direction perpendicular to the laser polarization. The plasma patterns appear deformed, withmore plasma along the direction perpendicular to the laser polarization. There is an asymme-try of the focused laser spot intensity in the transverse direction due to the longitudinal fieldcomponents near the laser focus [23], [24]. However, this does not explain our observations,since this asymmetry is in the opposite direction, and for NA = 0.65 is nevertheless very small(see Fig. 7b). We conclude that the plasma pattern asymmetry is caused entirely by the field en-


hancement due to the plasma refraction index smaller than unity. We describe this below withasimple model.

Consider a dielectric cylinder of radiusr, having its axis along ˆx, located in space filledwith another dielectric. Let the cylinder be composed of dielectric with permittivityε2, and thebackground dielectric have permittivityε1. Let also the field far from the cylinder be parallel toy and the origin be located at a point at the cylinder axis. Then it can be shown that

E1

E2=

ε2

ε1. (20)

The quantityE1 is the magnitude of local field at the cylinder boundary in YZ cross-sectionat y = ±r, z = 0, and is perpendicular to the boundary.E2 is the local field magnitude at theboundary pointsy = 0, z = ±r, and is parallel to the boundary. As can be seen from Eq. (20),if ε2 < ε1, as is the case for a plasma in a dielectric background,E1 < E2. Thus there is a fieldenhancement along direction perpendicular to the external field. This is what we observe in oursimulations. Although the interaction problem is electromagnetic rather than electrostatic, thisreasoning should be approximately valid for a cylinder radius smaller than the laser wavelength.

In this way, the laser propagating in the dielectric creates plasma, and this plasma in turndiverts the laser field. Due to Eq. (20), the diverted field is asymmetric, with enhancementalong the direction perpendicular to laser polarization (i.e., ˆz for a laser field polarized alongy). This field enhancement produces more plasma in the ˆz direction. This process results in theasymmetry of the generated plasma pattern shown in Fig. 7a.

We have generally found that the extent of asymmetry increases with higher NA. We willquantitatively characterize this extent in a future study.

5. Conclusion

In the present study we have discussed the importance of particular effects for formation of adamaged region during propagation of a short intense laser in a bulk dielectric, for material pa-rameters corresponding to fused silica. We have developed a 3D numerical model that accountsfor multi-photon ionization, electromagnetic response of the resulting plasma, energy dissipa-tion due to the plasma electron collisions, laser energy depletion due to the photon absorption,and Kerr effect. The analysis given in the paper has demonstrated that, unlike 1D case, the mostimportant effect limiting the laser intensity in the 3D geometry is the refraction by the createdplasma lens. The energy absorbing processes (heating and photon absorption), although are re-sponsible for absorption of a considerable part of laser energy, do not have a drastic effect onthe morphology of the damaged region. We have also observed an asymmetry of the plasmapattern in the transverse direction. This effect was found to be caused by plasma dispersion.

6. Acknowledgments

We would like to acknowledge fruitful discussions with P. Corkum and M. Gertsvolf. Thiswork was supported in parts by Natural Sciences and Engineering Research Council of Canada,Ontario Ministry of Research and Innovation, Canada Research Chairs program and CanadaFoundation for Innovation.


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Morphology of femtosecond laser modification of bulk dielectrics

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