Laser dynamics in nonlinear transparent media with electron plasma generation: effects of electron-hole radiative recombinations
aa r X i v : . [ n li n . PS ] F e b Laser dynamics in nonlinear transparent media withelectron plasma generation: effects of electron-holeradiative recombinations
P. Kameni Nteutse , Alain M. Dikand´e and S. Zekeng Laboratory of Mechanics, Materials and Structures, Department of Physics, Facultyof Science, University of Yaounde I P.O. Box 812 Yaounde, Cameroon. Laboratory of Research on Advanced Materials and Nonlinear Science(LaRAMaNS), Department of Physics, Faculty of Science, University of Buea P.O.Box 63 Buea, Cameroon.E-mail: [email protected]
Resubmission date: January 2021
Abstract.
The performance of optical devices manufactured via laser micromachin-ing on nonlinear transparent materials, relies usually on three main factors whichare the characteristic laser parameters (i.e. the laser power, pulse duration and pulserepetition rate), characteristic properties of host materials (e.g. their chromatic disper-sions, optical nonlinearities or self-focusing features, etc.) and the relative importanceof physical processes such as the avalanche impact ionization, multiphoton ionizationand electron-hole radiative recombination processes. These factors act in conjunctionto impose the regime of laser operation, in particular their competition determines theappropriate laser operation regime. In this work a theoretical study is proposed toexplore the effects of the competition between multiphoton absorption, plasma ion-ization and electron-hole radiative recombination processes, on the laser dynamics intransparent materials with Kerr nonlinearity. The study rests on a model consisting ofa K-order nonlinear complex Ginzburg-Landau equation, coupled to a first-order equa-tion describing time variation of the electron plasma density. An analysis of stabilityof continuous waves, following the modulational-instability approach, reveals that thecombination of multiphoton absorption and electron-hole radiative recombination pro-cesses can be detrimental or favorable to continuous-wave operation, depending on thegroup-velocity dispersion of the host medium. Numerical simulations of the modelequations in the full nonlinear regime, reveal the eixstence of pulse trains the am-plitudes of which are enhanced by the radiative recombination processes. Numericalresults for the density of the induced electron plasma feature two distinct regimes oftime evolution, depending on the strength of the electron-hole radiative recombinationprocesses.
Keywords : Laser-matter interactions, Continuous waves, Nonlinear transparent media,Pulses, Plasma ionization aser interactions with nonlinear transparent media
1. Introduction
Femtosecond laser micromachining nowadays offers the most reliable and portable toolin a broad range of modern industrial material processing [1, 2, 3, 4], its applicationsextend from accurate manufacturing of electronic devices, to fine drilling and machiningof hard metals, ceramics and soft plastics into various micro textures for improvementof functions and properties of end products [2, 3, 4, 5]. In these applications laser pulsesfocused on a dielectric medium are absorbed via nonlinear photo-ionization mechanism[6, 7], leading to a permanent modification of material structure at scales on the orderof nanometers. In the specific context of transparent materials [6, 7, 8], at low pulsepowers the modification will be a smooth refractive index change which can be exploitedadvantageously in the fabrication of photonic devices [2, 9]. However, at higher pulsepowers the modification gives rise to more complex processes such as birefringences,periodic nanoplanes aligning themselves orthogonally to the laser polarization to formperiodic nanogratings, change in the electronic structure due to electron and holeproductions from charge ionization with the generation of electron plasma, electron-hole radiative recombination processes [2, 6, 9], etc..In accordance with the fineness required for the end product, optical fields usedin laser material processing can be grouped in two categories [2, 9, 10]: continuous-wave (CW) lasers, which usually extend up to several kilowatts, and pulsed lasers withan average power spanning well below one kilowatt thus providing a wide range ofwavelengths and pulse duration, as well as pulse repetition rates [10]. Due to theseattributes, pulse lasers can allow micromachining with high resolution in depth andtherefore offer a rich potential for applications in drilling [2], cutting [11], welding[12, 13], ablation [12], material surface texturing and scripting [12, 13]. We remark thatbesides their short duration and high powers, femtosecond pulse lasers have been mostattractive owing to their minimal thermal drawbacks [14, 15]. Indeed femtosecond lasersare able to accumulate heat such as to minimize defect-induced damages, as a matterof fact this heat accumulation prevents undesired physical casualties as for instance theformation of microcracks, material bending, etc. [16] notably during laser processinginvolving transparent materials [17, 18, 19].Theoretical investigations of femtosecond laser processing on transparent materialshave attracted a little attention in the past [7, 20, 21]. Yet theoretical studies areuseful for they provide fundamental knowledge relative to the global picture of the laserdynamics, relevant for a good understanding of its distinct possible operation regimesin the prospect of an optimization of the micromachining technology. Indeed, sincefemtosecond lasers are optical fields with duration far below picosecond, they belongto a specific class of lasers known as ultrashort lasers [22]. Lasers in this specific classoperate typically in pulsed modes of relatively high powers, nevertheless in some contextsthey can be tailored to operate in the CW regime [2, 9]. This is for instance the casewhen their input powers are below the typical power of a high-intensity optical pulse, orwhen the input field is of low power and is designed to grow upon propagation from CW aser interactions with nonlinear transparent media K . aser interactions with nonlinear transparent media
2. The model, CW solutions and modulational instability
Consider an optical field propagating along the z axis of a transparent medium withKerr nonlinearity. We assume that the energy stored by the propagating laser modifiesthe electronic structure of the bulk material, creating an electron plasma of variabledensity and multiphoton ionization processes. Instructively the problem was addressedexperimentally [7, 10, 20, 21] in the contexts of femtosecond laser pulses focused onsilica materials. It was established that for a strong focusing geometry the femtosecondlaser can cause bulk damages in the optical material, followed by a narrow track withsubmicron width indicating a filamentary propagation of the pulse laser.In refs. [7, 10, 20, 21] numerical simulations were carried out to investigate thedistribution of the energy stored in the bulk material by the propagating laser. Whilethe laser equation was exactly similar in all these previous studies [7, 10, 20, 21], thetime-evolution equations for the electron plasma density were different. The differenceresided in the nature of physical processes assumed to contribute to the electron plasmaionization in each of these studies, as a matter of fact the work of ref. [7] was mainlyinterested in the impact of a linear recombination process on the time evolution of theelectron plasma density, whereas in refs. [10, 20, 21] the linerar recombination processwas neglected in favor of radiative recombinations only [10], avalanche impact only [20]or a combination of avalanche impact ionization and linear recombination processes[21]. However in none of these studies a detailed analysis of the system dynamics wasconsidered.In the present study we pay interest to the system dynamics, considering a modelthat combines the effects of avalanche ionization process, the electron-hole radiativerecombination and multiphoton ionization processes. The model can be represented bya propagation equation for the laser field given in terms of the cubic complex Ginzburg-Landau equation with a K -order nonlinear term [7, 10, 20, 21], coupled to a first-order ordinary differential equation accounting for time evolution of the electron plasmadensity, i.e.: i ∂u∂z = δ ∂ u∂t − σ | u | u − iγ (1 − iω τ ) ρu − iµ | u | K − u, (1) ∂ρ∂t = ν | u | ρ + α | u | K − aρ . (2)Instructively, we assumed a paraxial approximation for the field propagationconsidering the propagation axis to be z , and ignored beam diffractions in the planetransverse to the propagation axis. The first term in the right-hand side of eq. (1)accounts for the group-velocity dispersion, the second term accounts for the intrinsic(i.e. Kerr) nonlinearity of the host material, the third term describes both the plasmaabsorption and laser defocusing, and the last term describes multiphoton absorptionprocesses with µ the K -photon ionization rate.Eq. (2) describes the temporal evolution of the electron plasma density ρ , weremark that this equation differs from those considered in refs. [7, 20, 21] where the same aser interactions with nonlinear transparent media • δ (in fs /m ) is the coefficient of group-velocity dispersion, • σ (in m /W ) is the Kerr nonlinearity coefficient, • ω (in MHz) and τ (in fs) are the frequency and characteristic relaxation timeof the electron plasma respectively, • γ (in m ) is the cross section for inverse Bremsstrahlung, • ν (in m /eV ) is the coefficient of avalanche impact ionization, • α is the multiphoton ionization coefficient in the dense medium, • and a (in m /s ) is the electron-hole radiative recombination coefficient.Typical values for these parameters are found in most experimental works dealingwith the problem, as for instance in the study of femtosecond laser filamentations intranspartent media [10], laser micro-modification of fused silica [7] and so on. However,for a theoretical study such as the present one, experimental values of these parametersare of no useful given that the mathematical model, represented by eqs. (1) and (2),involves normalized variables such as the propagation time t , the propagation distance z , the field amplitude u as well as some coefficients in these two equations [10, 7].Therefore in our study we shall select arbitrary but reasonable values for characteristicparameters of the model, but keeping track of their appropriate signs.Being two nonlinear equations, general solutions to the coupled set eqs. (1)-(2) arenonlinear waves. Nevertheless, provided specific conditions, linear solutions includingharmonic waves and CWs can also exist for the same set. Thus steady-state CWsolutions to eqs. (1)-(2) can be expressed: u ( z ) = q I p exp ( iP c z ) , ρ = ρ , (3)which upon substitution in (1)-(2) give: P c = σI p + γω τ ρ , I p = σµω τ ! K − , (4) ρ = ν a I p " − s aασν µω τ . (5)Here P c , the CW wave number, is fixed by the input power I p = | u | as well asthe equilibrium value ρ of the electron plasma density ρ . Given that these two lastquantities (i.e. I p and ρ ) depend on characteristic parameters of the model, they cannotbe arbitrary and hence can be tuned by varying characteristic parameters of the model. aser interactions with nonlinear transparent media f ( z, t ) of theCW amplitude q I p . Steady-state solutions to eqs. (1)-(2) thus become: u ( z, t ) = [ u + f ( z, t )] exp( iP c z ) , ρ ( t ) = ρ + δρ ( t ) . (6)Replacing in eqs. (1)-(2) and linearizing, we obtain: i ∂f∂z − δ ∂ f∂t + C (1) K ( f + f ∗ ) = − iγ (1 − iω τ ) u δρ ( t ) , (7) ∂δρ ( t ) ∂t − q δρ ( t ) = C (2) K ( f + f ∗ ) , (8)Where, C (1) K = σu + iµ ( K − u K − , C (2) K = 1 γ ( αγK − µν ) u K − ,q = νu + 2 aµγ u K − , (9) f ∗ in the linear equations (7) and (8) denotes the complex conjugate of f . The first-order inhomogeneous linear equation (8) can be solved by means of Green’s functiontechnique [25, 31], yielding: δρ ( t ) = C (2) K Z t −∞ ( f + f ∗ ) e − q ( t ′ − t ) dt ′ . (10)Because of the presence of f ∗ in eq. (7), we must consider its complex conjugate.With this consideration, as solution to eq. (7) we pick f ( z, t ) = A exp ( κz + i Ω t ) and f ∗ ( z, t ) = A exp ( κz + i Ω t ), where κ is the coefficient of spatial amplification of theperturbation and Ω is the modulation frequency for the perturbation. These solutionslead to the following matrix-form eigenvalue problem: κ A A ! = " M M M ∗ M ∗ ! − S ! A A ! − T N N N ∗ N ∗ ! A A ! , (11)where: M = i ( δ Ω + σu ) , M = iσu , (12) S = µ ( K − u K − , N = 1 − iω τ , (13) T = ( Kαγ − µν ) u K (cid:16) i Ω − νu − aµu K − /γ (cid:17) − . (14)The two possible eigenvalues of the above 2 × κ , = − µ ( K − u K − " γ ( Kαγ − µν ) u µ ( K − i Ω γ − νu γ − aµu K − ) ± q ( S + T ) − ( δ Ω + σu ) + σ u − T δ Ω ω τ , (15) aser interactions with nonlinear transparent media , κ , . Most generally the followingpossible situations are expected: • When the real part of κ is zero, the CW solution will be always stable irrespectiveof the sign of its imaginary part. • When the real part of κ is negative, the CW solution will be asymptotically stable(i.e. is stabilized after some roundtrips) irrespective of the sign of its imaginarypart. • When the real part of κ is positive, the CW regime will be always unstable.Given that the two eigenvalues are functions of the modulation frequency Ω, wefind it more appropriate to first consider the CW stability at zero modulation frequency.In this later case the eigenvalues are: κ = 0 , κ = − µ ( K − u K − + 2 γ ( Kαγ − µν ) u K νu γ + 2 aµu K − . (16)It turns out that laser self-starting (i.e. CW instability) will be favored provided κ > a < Kγ ( αγ − µν )2 µ ( K − I K − p . (17)Quantitatively, this condition implies two possible characteristic values of the radiativerecombination coefficient a above which laser self-starting can occur: One is negative for αγ < µν and hence is nonphysical, whereas the positive and physical one is conditionedby αγ > µν and is: a th = Kγ ( αγ − µν )2 µ ( K − I K − p . (18)In concrete terms the quantity a th sets a threshold value of the electron-ionrecombination coefficient, above which the laser will self-start.Due to the strong dependence of κ , in eq. (15) on the modulation frequency, discussingCW stability from the analytical expressions of κ , for arbitrary nonzero values of Ωis far from being an easy task. Therefore we resort to a global analysis, by mappingthe two eigenvalues onto a plane Re( κ )-Im( κ ) describing a two-dimensional complexparameter space, where Re( κ ) and Im( κ ) are real and imaginary parts respectivelyof the eigenvalue κ . In this parametric representation, the modulation frequency Ωplays the role of a parameter and so can span a broad range of values, which in ourcase will be the finite interval − ≤ Ω ≤
5. The first figures we consider are parametricrepresentations of Im( κ ) as a function of Re( κ ), for some selected combinations of valuesof key characteristic parameters of the model. To be more explicit, the four graphs infigs. 1 and 2 represent Im( κ ) as a function of Re( κ ) in the anomalous dispersion regime( δ <
0: fig. 1) and normal dispersion regime ( δ >
0: fig. 2) respectively, for K = 2, aser interactions with nonlinear transparent media − − Re [ κ (Ω)] − − I m [ κ ( Ω ) ] δ < , K = 2 a = 0 a = 0 a = 0 . a = 0 . a = 0 . a = 0 . − − − Re [ κ (Ω)] − − − − I m [ κ ( Ω ) ] δ < , K = 3 a = 0 a = 0 a = 0 . a = 0 . a = 0 . a = 0 . − − − − − Re [ κ (Ω)] − − − − I m [ κ ( Ω ) ] δ < , K = 4 a = 0 a = 0 a = 0 . a = 0 . a = 0 . a = 0 . − − − − − Re [ κ (Ω)] − − − I m [ κ ( Ω ) ] δ < , K = 5 a = 0 a = 0 a = 0 . a = 0 . a = 0 . a = 0 . Figure 1. (Color online) Imaginary versus real parts of κ (full curve) and κ (dashedcurve) for K = 2 , , ,
5. The radiative recombination coefficient a is varied as a = 0,0 . . . . α = 0 . ν = 0 . µ = 0 . I p = 2 . ω τ = 0 . σ = 0 . δ = − . γ = 0 .
3, 4 and 5. Values of model parameters are given in the captions, and different curvesin each graph correspond to different values of the radiative recombination coefficient a . Recall that the sign of δ determines the dispersion regime [24, 25, 34], indeed apositive δ corresponds to a normal group-velocity dispersion, whereas a negative δ willcorrespond to an anomalous group-velocity dispersion well known [24, 25, 34] to favorthe generation of pulse structures, of course provided the intrinsic refractive index ofthe host medium is of a self-focusing Kerr nonlinearity.Fig. 1 suggests that small values of K are expected to favor laser self-starting inthe anomalous dispersion regime. As K increases, the real part of the largest eigenvaluegradually shifts to the negative branch and consequently, the CW regime is stabilized.On the contrary fig. 2 indicates that in the normal dispersion regime, CW operationwill be favored for small values of the multiphoton absorption rate K . As K increases,CW modes become unstable thus favoring laser self-starting. aser interactions with nonlinear transparent media − . − . − . − . . . Re [ κ (Ω)] − − − − I m [ κ ( Ω ) ] δ > , K = 2 a = 0 a = 0 a = 0 . a = 0 . a = 0 . a = 0 . − − − − Re [ κ (Ω)] − − − − I m [ κ ( Ω ) ] δ > , K = 3 a = 0 a = 0 a = 0 . a = 0 . a = 0 . a = 0 . − − − − − Re [ κ (Ω)] − − − − I m [ κ ( Ω ) ] δ > , K = 4 a = 0 a = 0 a = 0 . a = 0 . a = 0 . a = 0 . − − − −
20 0 20 40 Re [ κ (Ω)] − − − − I m [ κ ( Ω ) ] δ > , K = 5 a = 0 a = 0 a = 0 . a = 0 . a = 0 . a = 0 . Figure 2. (Color online) Imaginary versus real parts of κ (full curve) and κ (dashedcurve) for K = 2 , , ,
5. The radiative recombination coefficient a is varied as a = 0,0 . . . . α = 0 . ν = 0 . µ = 0 . I p = 2 . ω τ = 0 . σ = 0 . δ = 0 . γ = 0 .
3. Nonlinear regime
In the full nonlinear regime, solutions to the laser equation eq. (1) are high-intensityfields which can be represented as real-amplitude pulses, undergoing spatio-temporalmodulations i.e. [33, 34]: u ( z, t ) = g ( z, t ) exp i [ φ ( z, t ) − ωz ] , (19)where g is the real amplitude and φ is the modulation phase. We introduce a reducedtime as τ = t − vz , in which v is the pulse inverse velocity such that ω in formula(19) emerges more explicitly as a nonlinear shift in the propagation constant. Letting g ( z, t ) ≡ g ( τ ) and φ ( z, t ) ≡ φ ( τ ), inserting eq. (19) into eq. (1) and eq. (2) andseparating real parts from imaginary parts, we obtain the following set of coupled first-order nonlinear ordinary differential equations: (cid:16) vM + ω + δM + γω τ ρ (cid:17) g − δy ′ + σg = 0 , (20) aser interactions with nonlinear transparent media δM ′ − γρ ) g + ( v + 2 δM ) y − µg K − = 0 , (21) ρ ′ − νg ρ − αg K + aρ = 0 , (22)with y = g ′ , M = φ ′ and the prime symbol refers to derivative with respect to τ . Let usfocus on the system dynamics in the particular case v = 0 [26, 33, 34]. For this value of v the set of coupled first-order nonlinear ordinary differential equations (20)-(22) reducesto: M ′ = γρδ − M yg + µg K − δ ,y ′ = ( ω + δM + γω τ ρ ) gδ + σg δ ,g ′ = y,ρ ′ = νg ρ + αg K − aρ . (23)Our first interest will be on the singular solutions to this system, which are their fixedpoints, with the aim to probe the effects of important characteristic parameters of themodel such as the radiative recombination coefficient a and the multiphoton absorptionrate K , on equilibrium solutions of the laser amplitude g and instantaneous frequency M , as well as of the electron plasma density ρ . Singular solutions to the set of first-order nonlinear ordinary differential equations (23),which are their fixed points, are the roots of the following nonlinear system: γρδ + µg K − δ = 0 , (24)( ω + δM + γω τ ρ ) gδ + σg δ = 0 , (25) y = 0 , (26) νg ρ + αg K − aρ = 0 . (27)From this system we derive: M = 1 δ h µω τ g K − − ω − σg i , ρ = g K vuut αγ − µνaγ ! . (28)Remarkable enough formula (28) suggests that irrespective of the value of K , the electronplasma density ρ will be zero when the laser amplitude g is zero.Fig. 3 represents the variations of the two fixed points of the laser amplitude g as afunction of ω , for K = 2 , , , g we understand its extrema, i.e. its maximumand minimum which are obtained by annihilating M in formula (28). According to fig.3, for K = 2 the laser dynamics is dominated by weakly nonlinear pulse trains withmaximum amplitudes for ω = 0. An increase of K (see graphs for K = 3, 4 and 5)favors strongly nonlinear pulse trains of larger amplitudes [33, 34]. We have also plotted aser interactions with nonlinear transparent media − . − . − . − . − . − . − . . ω − − g K = 2 − − − ω − − g K = 3 ω − − g K = 4 . . . . . . . . . ω − − g K = 5 Figure 3. (Color online) Fixed points of the laser amplitude g as a function of ω , for K = 2 , , , σ = 0 . ω τ = 0 . µ = 0 . the instantaneous frequency M as a function of the amplitude g of laser (fig. 4), andthe laser amplitude g as function of the electron plasma density ρ (fig. 5), for fourdifferent values of K . Remark that the expression of M given in formula (28) does notcontains the radiative recombination coefficient a but is controlled mainly by the laserpropagation constant ω , whereas g as a function of ρ extracted from eq. (28) depends on a but not on ω . Therefore, we have chosen to plot M as a function of g by consideringtwo cases i.e., the case ω = 0 and the case of finite nonzero value of ω as one sees in fig.4. The different curves in the graphs of fig. 4 show that M is enhanced by an increase of g , and that there is a threshold value of the amplitude beyond which the instantaneousfrequency is expected to decrease to zero. As it is apparent, this threshold value of g (and consequently of M ) is decreased with an increase of K . In fig. 5, the fixed pointof the electron plasma density is always zero at zero amplitude of the laser whateverthe value of K , consistently with what we learned from formula (28). However there isa drastic increase of ρ with an increase of K , meaning that for the same laser intensitythe stronger the multiphoton absorption processes the larger will be the electron plasma aser interactions with nonlinear transparent media g M K = ω = 0 ω = 0 . . . . . . . . . . g . . . . . . M K = ω = 0 ω = 0 . .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 .
75 2 . g . . . . . . . . M K = ω = 0 ω = 0 . . . . . . . . . . g . . . . . . . . M K = ω = 0 ω = 0 . Figure 4. (Color online) Variation of the instantaneous frequency M with theamplitude g of laser, for K = 2 , , , ω (Solid curvecorrespond to ω = 0, and dashed curve to ω = 0 . σ = 0 . ω τ = 0 . δ = − . µ = 0 . density. Considering the full nonlinear dynamics of the system, the set of first-order ordinarydifferential equations (23) was solved numerically using a sixth-order Runge-Kuttaalgorithm adapted from ref.. [32]. Because eqs. (1)-(2) involve several parameters, all ofwhich cannot be varied in this study, we fixed most parameters except two ones i.e. themultiphoton absorption rate K , which was given the four different values K = 2 , , , a which was varied in three distinct rangesof values where three distinct behaviors were noticed.Figs. 6 and 7 are time variations of the laser amplitude g ( t ) and of the electronplasma density ρ ( t ), for four distinct values of K and values of model parameters listedin the figure captions. Note that for each graph, we plotted dynamical quantities for aser interactions with nonlinear transparent media . . . . . . . . ρ g K = 2 a=0.001a=0.002a=0.003a=0.004 . . . . . . . . ρ g K = 3 a=0.001a=0.002a=0.003a=0.004 ρ g K = 4 a=0.001a=0.002a=0.003a=0.004 ρ g K = 5 a=0.001a=0.002a=0.003a=0.004 Figure 5. (Color online) Variation of the laser amplitude g with the electronplasma density ρ , for K = 2 , , , a indicated in the graphs. ν = 0 . α = 0 . γ = 0 . µ = 0 . different values of a starting with a = 0 (i.e. when there is no radiative recombination).Our objective in so doing was to highlight the qualitative and quantitative influences of a , on the system dynamics.In fig. 6, the laser amplitude g ( t ) is manifestly a pulse train with a maximumvarying only weakly with a for K = 2. However, as K is increased the maximum of g gets more and more large. Note the fall-off of g ( t ) with time, which is more and morepronounced as K gets larger and for a = 0. We attribute this fall-off to a dampingeffect induced by the density of electron plasma i.e. ρ ( t ), which in this context acts likea laser gain/loss.The electron plasma density ρ , plotted in fig. 7, is increasing with time for a = 0irrespective of the value of K . However, when a is increased in a relatively large rangeof values, ρ ( t ) decreases exponentially in time tending to its equilibrium value with anincreasingly sharp slope. Evidently, the nonzero values of a chosen for the numericalresults just discussed, are large enough and therefore do not enable one appreciate how aser interactions with nonlinear transparent media t . . . . . . g K = 2, ω = 1 . t . . . . . . . . . g K = 3, ω = 1 . t . . . . . . . . . g K = 4, ω = 1 . t . . . . . . . . . g K = 5, ω = 1 . Figure 6. (Color online) Time variation of the laser amplitude g , for K = 2 , , , a . ν = 0 . α = 0 . γ = 0 . µ = 0 .
25. Values of a are, from the smallest to the largest amplitudes: 0, 3.9, 5.6, 6. the electron plasma density ρ changes from its exponentially increasing feature for a = 0,to an exponentially decreasing feature when a = 0. In order to earn more insight ontothe effects of a on the time variation of ρ , in fig. 8 and fig. 9 we plot ρ versus timekeeping the same values of parameters used in fig. 7, but choosing smaller values of a .To be explicit, in fig. 8 values of a were chosen small but large enough such that ρ ( t )is already decreasing with time. On the contrary the values chosen for a in fig. 9, arecloser to zero. The physically relevant insight is that in the range of values of a for which ρ decreases, an increase of a sharpens the slope of the decrease of ρ . When a is verysmall such that ρ is an exponentially increasing function of time, an increase of a willsoften the variation of ρ . It is quite apparent that in the very small range of values of a , ρ is periodically oscillating as it increases exponentially with time (fig. 9). The quantity a th in the graphs of fig. 8, is the characteristic value of the radiative recombinationcoefficient for which the exponential variation of ρ with time is suppressed. For thisvalue of a , the electron plasma density ρ ( t ) constantly oscillates between two positive aser interactions with nonlinear transparent media t . . . . . . . . ρ K = 2, ω = 1 . a=0a=3.9a=5.6a=6 t . . . . . ρ K = 3, ω = 1 . a=0a=3.9a=5.6a=6 t . . . . . ρ K = 4, ω = 1 . a=0a=3.9a=5.6a=6 t . . . . . ρ K = 5, ω = 1 . a=0a=3.9a=5.6a=6 Figure 7. (Color online) Time variation of the electron plasma density ρ , for K = 2 , , , a indicated in the graphs. σ = 0 . ω τ = 0 . δ = − . µ = 0 . ν = 0 . γ = 0 . α = 0 . extrema and hence is always nonzero, in average.
4. Conclusion
We have investigated the dynamics of a model for femtosecond laser inscription in atransparent medium with Kerr nonlinearity. This model is an improvement of two recentones [20, 21], in which the contribution of electron-hole radiative recombination processesto the plasma generation was neglected. Mathematically the model is represented by acomplex Ginzburg-Landau equation with cubic nonlinearity plus a K -order nonlinearityaccounting for K photon absorption processes, coupled to a time first-order ordinarydifferential equation with a term quadratic in the electron plasma density, accountingfor radiative recombination processes.In view of the fact that the laser dynamics in the host material determines thespecific regime of operation of the laser during laser micromachining on the material, aser interactions with nonlinear transparent media t . . . . . . . . . ρ K = 2, ω = 1 . a=0a=0.002a=0.005a=0.008 a th =0.028 t . . . . . . . . ρ K = 3, ω = 1 . a=0a=0.002a=0.005a=0.008 a th =0.026 t . . . . . . . . ρ K = 4, ω = 1 . a=0a=0.002a=0.005a=0.008 a th =0.026 t . . . . . . . . ρ K = 5, ω = 1 . a=0a=0.002a=0.005a=0.008 a th =0.026 Figure 8. (Color online) Time variation of the electron plasma density ρ , for K = 2 , , , a indicated in the graphs. σ = 0 . ω τ = 0 . δ = − . µ = 0 . ν = 0 . γ = 0 . α = 0 . we considered two distinct regimes of laser dynamics namely CWs and pulses. Firstaddressing CWs and their stability, a linear-stability analysis was carried out followingthe modulational-instability theory. From this analysis we constructed a global stabilitymap, which enabled us explore parameter regimes in which CW operations could bestable.In the nonlinear regime, an ansatz was introduced which aimed at representingthe laser as a pulse field with a real amplitude and real phase. With the help of thisansatz the system dynamics was transformed into a set of four first-order ordinarydifferential equations. Numerical solutions to these equations established the pulse-train structure of the laser amplitude, the power of which was increased by an increaseof the radiative recombination coefficient a . The electron plasma density was found toincrease or decrease exponentially in time with an oscillating amplitude, depending onthe order of magnitude of a .Though the model treated in this work is rich, it remains limited for it concerns aser interactions with nonlinear transparent media t . . . . . . ρ K = 2, ω = 1 . a=0a=0.2a=0.8a=1.25 t . . . . . . ρ K = 3, ω = 1 . a=0a=0.2a=0.8a=1.25 t . . . . . . ρ K = 4, ω = 1 . a=0a=0.2a=0.8a=1.25 t . . . . . . ρ K = 5, ω = 1 . a=0a=0.2a=0.8a=1.25 Figure 9. (Color online) Time variation of the electron plasma density ρ , for K = 2 , , , a indicated in the graphs. σ = 0 . ω τ = 0 . δ = − . µ = 0 . ν = 0 . γ = 0 . α = 0 . only few out of a wide range of possible physical contexts. Indeed not all transparentmaterials oppose a nonlinear optical response of the Kerr type to field propagation,in fact the Kerr response is usually considered as a very weak nonlinearity. Thereexists transparent materials with stronger optical nonlinear responses, in general theirrefractive index is a series of powers of the optical field intensity extending beyondthe quadratic term. Usually this is also represented by saturable-nonlinearity functions[25]. Few examples of such materials include doped silica fiber glasses, where the dopingwith rare-earth ions provides means of controlling the strength of nonlinearity of thetransparent host. A study similar to the present one, but taking into consideration thesaturable nonlinearity of the host material, will surely enrich current knowledge on thefundamental physics of CW and pulse operations in femtosecond laser micromachininginvolving transparent materials with non-Kerr optical nonlinearity [25]. aser interactions with nonlinear transparent media Acknowledgments
A. M. Dikand´e thanks the Ministry of Higher Education of Cameroon (MINESUP) forfinancial assistance, within the framework of the presidential grant designated ”ResearchModernization Allowances”.
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