Abstract

In this paper we propose and demonstrate that the ultrafast nonlinear optical response of atoms may be accurately calculated in terms of metastable states obtained as solutions of the stationary Schrödinger equation including the quasi-static applied electric field. We first develop the approach in the context of an exactly soluble one-dimensional atomic model with delta-function potential, as this allows comparison between the exact ultrafast nonlinear optical response and our approximate approach, both in adiabatic approximation and beyond. These ideas are then applied to a three-dimensional hydrogen-like atom and yield similar excellent agreement between the metastable state approach and simulations of the Schrödinger equation for off-resonant excitation. Finally, our approach yields a model for the ultrafast nonlinear optical response with no free parameters. It can potentially replace the light–matter interaction treatment currently used in optical filamentation, and we present a numerical example of application to femtosecond pulse propagation.

© 2014 Optical Society of America

Corrections

M. Kolesik, J. M. Brown, A. Teleki, P. Jakobsen, J. V. Moloney, and E. M. Wright, "Metastable electronic states and nonlinear response for high-intensity optical pulses: erratum," Optica 2, 509-509 (2015)
https://www.osapublishing.org/optica/abstract.cfm?uri=optica-2-5-509

1. INTRODUCTION

Propagation of ultrashort, off-resonant optical pulses in atomic gases produces a broad range of extreme nonlinear optical effects including high-harmonic generation [1], synthesis of attosecond pulse forms [2], and optical filamentation [3,4]. While it is generally accepted that the origin of the first two examples resides in the quantum mechanical nature of the light–matter interaction, the standard model of the nonlinearity underlying filamentation, comprising third-order nonlinearity for bound electrons plus a Drude model for freed electrons, has met with some success. However, numerical calculations of the nonlinear optical response of atoms using the time-dependent Schrödinger equation (TDSE) [510] make it abundantly clear that the quantum mechanical nature of the extreme nonlinearity involved is also required for a more complete and deeper understanding of filamentation. In particular, for off-resonant pulse excitation the quantum coherent nature of the light–matter interaction becomes key, and the distinction between bound and freed electrons employed in the standard model becomes problematic. In addition to the numerical simulations based on the TDSE, some model systems have been explored to gain insight. For example, some of the present authors have studied the nonlinear optical response via an exactly soluble one-dimensional (1D) atomic model with delta-function potential [11], and Richter et al. studied the role of the Kramers–Hennenberger atom, which displays bound states of a free electron, to elucidate the higher order Kerr effect. Models employing separable interactions [12] have also been studied in this context.

The above discussion highlights the urgent need for microscopically based and computationally economical descriptions of extreme nonlinear interactions that lend themselves to incorporation into full space–time field simulations. In particular, numerical solution of the TDSE coupled to the field propagation equations over the space and time scales relevant to filamentation in gases poses a formidable computational task that will not be realistic in the foreseeable future, although forefront simulations over restricted spatial domains have appeared [13,14]. The standard model is easily incorporated into existing propagation schemes but the third-order nonlinear and plasma contributions to the model are not self-consistently and microscopically linked as they are in the present unified approach.

In this paper we propose and demonstrate that the ultrafast, off-resonant nonlinear optical response of atoms may be accurately calculated in terms of metastable states as opposed to the more common bound and continuum states of the free atom. In particular, we consider solutions of the atomic time-independent Schrödinger equation including the quasi-static applied electric field, these states having complex energies, the imaginary parts being related to the metastable state lifetime. Meta-stability refers to the fact that while these states ionize and thus “decay” over relatively long times, when observed on shorter time-scales in the vicinity of the atom, they are difficult to distinguish from the normal (i.e., field-free) electronic states. These metastable states, also termed resonance, Siegert, or Gamow states, encapsulate the quantum coherent coupling between the bound and continuum states of the free atom, and provide a natural time-domain description of an atom exposed to strong fields [15]. More specifically, although the single atom represents a closed system, we consider that the atomic wave function may be split into two components, ψ=ψM+ψF, in which ψM is expressed as a superposition of metastable states that is used to calculate the nonlinear optical response arising from the quantum coherent light–matter interaction. In contrast, ψF accounts for electrons that are to all intents and purposes freed from their parent ion and contribute a classical current density. This portion of the wavefunction is populated by the losses from the metastable states, and is subsequently driven by the applied electric field according to Ehrenfest’s theorem. The metastable portion of the wavefunction, ψM, gives rise to a complex, nonperturbative nonlinear response, which only reduces to the usual optical Kerr effect in the limit of a weak field. Using examples we demonstrate that our approach, which we term “the Metastable Electronic State Approach” or simply MESA, provides an extremely economical computational method of calculating the nonlinear optical response: already retaining only the ground metastable state in adiabatic approximation can provide a quantitative model for the nonlinear optical response and strong-field ionization, and further improvement is shown to result from retaining post-adiabatic corrections.

The remainder of this paper is organized as follows. We develop MESA in the context of an exactly soluble 1D atomic model with delta-function potential as a test bed as this allows for comparison between the exact ultrafast nonlinear optical response and our approximate approach, both in the adiabatic approximation and beyond. The method is then applied to a three-dimensional (3D) hydrogen-like atom, and this is key to showing that the approach can be employed for more realistic systems. We employ results for the nonlinear optical response from MESA to an example of optical filamentation as a demonstration of the computational economy and feasibility of the approach. Finally, a summary with conclusions and future research directions are given in Section 9.

2. EXACTLY SOLVABLE QUANTUM SYSTEM

An exactly solvable system proves to be of great utility for testing the method we propose. It also makes it easier to appreciate the structure of the theory, since all quantities we need can be evaluated analytically. We use a 1D, single-particle model with a Dirac-delta potential. Its time-domain Schrödinger equation can be written as

[it+12x2+Bδ(x)+xF(t)]ψ(x,t)=0,
with B controlling the ionization potential of the single bound state this model has, and F(t) being the time-dependent field strength of the optical pulse. An exact solution for the induced dipole moment and/or current has been recently derived [11], and this we use to test our approximate solutions. The metastable states (Stark resonances) can also be obtained exactly:
ψ(λ,x)={Ci(ξ0)Ai(ξ)x<0Ai(ξ0)Ci(ξ)x>0,
where we define Ci(z)=Bi(z)+iAi(z), with ξ=(2F)1/3(x+λ/F) and ξ0=(2F)1/3(λ/F). The spectral parameter λ must solve the eigenvalue equation for a given F:
1=2πB(2F)1/3Ai(2λ(2F)2/3)[iAi(2λ(2F)2/3)+Bi(2λ(2F)2/3)].

The infinitely many solutions to this equation, denoted by En(F),n=0,1,2,, represent the complex-valued meta-stable energies. The most important resonance, ψ0=ψ(λ=E0(F),x), converges to the ground state when F0. Besides ψ0, there are two distinct families of resonances. Energies of the first are located just below the positive real axis, while family-two energies extend along the ray with the complex argument of 2π/3 [16]. Meta-stable states constitute a bi-orthogonal system:

ψi|ψj=Cψi(z)ψj(z)dz=Ni2(F)δij,
where Ni2(F)=(2F)1/3π1[Ci(ξ0)Ai(ξ0)+Ci(ξ0)Ai(ξ0)]. The integration proceeds along a contour in the complex plane that follows the real axis and deviates into the upper half-plane for large positive real part of z (assuming F>0). Because far from the center the resonances are outgoing waves, the contour deformation ensures convergence of the integral.

The bi-orthogonality relations of Eq. (4) allow one to define a “norm,” and to generalize observables [17]. In particular, we use the dipole-moment expectation value in the metastable ground-state:

X0(F)=N02(F)Cψ0(z)zψ0(z)dz.
This quantity is shown in Fig. 1 as a function of F. Its real part is superlinear for weak fields (Kerr effect), and it saturates and decays in very strong fields, whereas the imaginary part is intimately related to the ionization losses. As noted in our previous work, the saturation and decay of the real part is accompanied by rapid growth of the freed electron population, which produces a defocusing nonlinear response. This generic behavior turns out to be the same for more realistic 3D quantum systems.

 

Fig. 1. Complex-valued expectation value of the dipole moment in the metastable ground state as a function of the external field strength. The fine dashed line indicates the linear susceptibility. The gap between the thin dashed and the thick black line represents the nonlinear response Eq. (11).

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3. GENERAL SCHEME

The central idea of the method we put forward, namely, MESA, is to abandon the traditional description of the quantum evolution in terms of Hamiltonian eigenstates. Instead, we argue that for a system exposed to a strong external field, such as due to a high-intensity optical pulse, it is more natural to utilize metastable states. They are superpositions of the (field-free) bound and free electronic states, reflecting the current strength of the field and the probability density “leakage” due to ionization [18].

We assume that it is possible to represent the wavefunction split into “metastable” and “free” components, ψ=ψM+ψF, as described in the introduction. Our method is motivated by Berggren’s completeness relations [19] employed in nuclear scattering theory:

1=n|ψnψn|+Ldλ|ϕλϕλ|,
where the sum projects onto several resonant states, and the integral involving scattering states ϕ proceeds along a contour L that deviates from the real axis in order to include the poles corresponding to the complex energies of the resonances included in the first term. In our method, one or more resonant states represent ψM, which we assume to be the part of the wave function interacting strongly with the atomic potential. The “remainder” of the wavefunction, ψF, is projected from the full state via the continuum component of the above completeness relation. Note that our approximation does not rely on completeness of the resonant states. Rather the key assumption is that the continuum part of the wavefunction, ψF, is so spread out in space that its interaction with the atom may be neglected. If it is, we do not need to know any specific properties of ψF, because the quantity of interest, namely, the current, can be obtained from the Ehrenfest theorem. Validity of this approximation obviously depends on the initial state, and it is not a priori obvious that a small number of resonant states can capture the “interacting part” of the wave function with sufficient accuracy. However, our numerical examples below demonstrate that even a single ground-state resonance provides an extremely good approximation.

We are specifically interested in the case of off-resonant excitation, meaning that the instantaneous frequency ωeff(t)=Φt of the applied field F(t)=A(t)cos(Φ(t)) should remain well below the ionization frequency I/ for all times, I being the ionization energy. Our method should therefore be more accurate for longer wavelengths, and we find this to be the case. Moreover, for the case of the exactly solvable 1D model there is an analytic result that states that the lowest Siegert state exactly captures the quantum atomic dynamics in the adiabatic or quasi-static approximation for which ωeff(t)/I1 [15].

Concentrating on the “metastable” component, we deal with an effectively open system. We write its evolution equation as an expansion into the system of metastable states slaved to the time-dependent field:

itψM=H^(F(t))ψM(x,t),ψM=ici(t)ψi(F(t),x),
where for any instantaneous value of F, ψi(F,x) solves the same differential equation as the proper Hamiltonian eigenstates,
H^(F)ψi(F,x)=Ei(F)ψi(F,x),
but these metastable states must behave as outgoing waves at infinity. One way to “select” such waves is to modify the domain of the differential operator representing the action of the Hamiltonian. We achieve this by replacing the real coordinate axis by a contour in the complex plane. The contour is chosen to follow the real axis in the inner domain, while in the positive (negative) “outside regions” it deviates into the upper (lower) half planes. While the outgoing waves decay exponentially at infinity along this “complexified” domain axis, they still represent the same solutions (see Ref. [20] for details of implementation). Such boundary conditions make this operator non-Hermitian, and the complex-valued metastable energies Ei(F) reflect the decaying nature of the resonant states. It can be shown that the adjoint H+ has eigenfunctions which are just complex-conjugates of ψi and that pairing Eq. (4) is nothing but a bi-orthogonality relation [21].

We use Eq. (4) to project out the evolution equations, normalizing all ψi to unity for all times t for simplicity of notation. The resulting evolution equations reflect the fact that ψj(F(t)) are effectively time dependent and, therefore, contain matrix elements that show the change of metastable states with field strength F:

cn(t)=icnEn(F(t))kck(t)F(t)ψn|Fψk(F(t),x).

We will first concentrate on the adiabatic approximation. For the initial condition in the ground state, the adiabatic wavefunction [15] becomes dominated by the ground-state resonance and the solution is

c0(a)(t)=exp{iEGtit[E0(F(τ))EG]dτ},
where EG is the field-free ground-state energy. If the field is not too strong, the ground-state resonance dominates at all times, and then the generalized dipole-moment expectation of Eq. (5) is the main contribution to the induced polarization [17], P(F(t))=|c0(t)|2X0(F(t)). In pulse propagation simulations the linear medium properties are captured exactly by a spectral solver. To avoid double counting, only the nonlinear part of the atom polarization is used and coupled to the Maxwell equations. This we obtain as
P(nl)(F(t))=P(F(t))limϵ01ϵP(ϵF(t)),
where the second term tends to the linear part of the response as the auxiliary parameter ϵ gets small. The polarization Eq. (11) is our first contribution to the nonlinear medium response. The second comes from ψF, in the form of a classical current, because we assume that this is the “distant part” of the wave function that is no longer interacting with the atom. Its norm grows as a result of the metastable decay, so that the total probability is conserved as it should be in a closed system. Thus, the ionization rate is given by the imaginary part of the ground-state resonance complex energy, and the ionized fraction of atoms obey
tρ(t)=[1ρ(t)]I{2E0(F(t))}.
The current induced by freed electrons is obtained from the Ehrenfest theorem, and is evaluated by integrating
tJ(t)=ρ(t)F(t).
Note that this induced current arises only after ionization, which is a highly nonlinear process, and it therefore does not contain a component linear in the field strength. Thus, the two functions X0(F) and E0(F) are needed to characterize the nonlinear optical response of the system. The proposed model has structure similar to the one used in filamentation simulations, with P(nl)(t) and J(t) coupled into pulse propagation equations. However, the meaning of its components is new: Kerr response is now contained within the nonlinear polarization of the metastable state, which now also includes contribution from the continuum states. Moreover, what used to be the Drude current is now solely due to electrons NOT in the resonant state(s) included in our treatment. Thus, the wavefunction split, and with it the relative contributions of polarization and current density, will depend on how many metastable states we can explicitly account for. As of now we include only the ground-state resonance, and next we look at how the higher order states can be accounted for in an approximate fashion.

4. POST-ADIABATIC CORRECTIONS

The purpose of this section is to describe three kinds of corrections that take MESA beyond the adiabatic approximation outlined above. In doing so, we will strive to formulate our theory in such a way that it will require the knowledge of only a single metastable state, namely, the one related to the ground state. This will greatly simplify practical applications, because it is relatively simple to characterize the ground-state resonance, while it may be more difficult to calculate properties of higher metastable states. The assumption underlying the following consideration is one of a weak field F(t), which is changing slowly. This is the case for many regimes in extreme nonlinear optics, where typical intensities attained in hot spots are still 2 orders of magnitude weaker than atomic fields. Moreover, the propagation dynamics dynamically adjusts the evolving pulse such that the peak intensity is clamped. As a result the quantum state is strongly dominated by the contribution from the metastable ground state, and we construct the post-adiabatic corrections under this assumption.

The first correction of the adiabatic solution is obtained by solving Eq. (9) for the expansion coefficients cn driven by the dominant c0. Feeding that intermediate result back into the equation for the ground-state resonance, c0(1)(t) is obtained in the same form as c0(a)(t), only with the resonant energy E0 replaced by the corrected one:

E0(R)(F(t))=E0(F(t))n0(F(t))2(ψ0|Fψn(F(t)))2[En(F(t))E0(F(t))].
Evaluation of this quantity requires the knowledge of higher order resonant states. Fortunately, an approximation in terms solely of ψ0 can be obtained as follows. First, since En(F) accumulate close to zero, at least in relatively weak fields, they are dominated by E0, which can approximate the denominator in Eq. (14). Next, because normalization to unity for all F ensures that ψ0|Fψn=Fψ0|ψn, the numerator in Eq. (14) can be rewritten, moving the action of F onto ψ0 and thus giving rise to n0|ψnψn|. Our second assumption is that, at least within the space of the solutions that evolve from the ground state, the system of {ψn} is complete, and this projection can be approximated by 1|ψ0ψ0|. Then we can simplify the above correction to
E0(R)(F(t))=E0(F(t))+(F(t))2E0(F(t))Fψ0(F(t))|Fψ0(F(t)).
This modifies both the real and imaginary parts of the complex metastable energy. It is the imaginary part that is more important for our purposes because it increases the ionization yield. Figure 2 illustrates this effect, and shows ionization yields caused by a driving pulse in the adiabatic and post-adiabatic [i.e., with Correction (15)] approximations, compared to the exact solution. It is evident that the correction becomes negligible for longer wavelengths. At shorter wavelengths, it significantly decreases the gap between the adiabatic and exact solutions, thus justifying the approximations adopted in the derivation. Perhaps the most important observation to be made here is that even the uncorrected adiabatic solution provides rather accurate ionization rates. For practical purposes in the field of optical filamentation, especially in the near and mid-infrared, the adiabatic treatment should therefore be sufficient.

 

Fig. 2. Ionization yield as a function of the intensity of the driving pulse. Exact, adiabatic, and corrected solutions are compared. The top and bottom panels correspond to wavelengths of λ=2400nm and λ=800nm, respectively.

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However, there is one post-adiabatic correction that should be included in such simulations; because it captures the losses, the optical field must suffer due to ionization. Interestingly, this correction is connected to the imaginary part of the metastable expectation value of the dipole moment. To derive the corresponding nonlinear polarization term, one has to include the first corrections cn(1) of the wavefunctions when evaluating the expectation value of the dipole moment. Then, under the same approximation as in the derivation of the corrected resonant energy in Eq. (15), one obtains the following estimate:

PNL(corr)(t)1EGI{tψ0(F(t))|X|ψ0(F(t))}.
This component of the nonlinear polarization turns out to be responsible for a major part of nonlinear losses. It is a microscopically motivated replacement for the purely phenomenological current, which is routinely introduced into the standard filamentation model [3] in order to salvage energy conservation. Our comparative simulations show that the amount of loss caused by P(corr) is actually quite similar to that obtained in the phenomenological treatment. It is fair to say that Eq. (16) is an approximation, yet it is a first step beyond the current method. Naturally, it could be improved provided one can calculate higher resonant states.

Finally, the third correction originates in the split between ψM and ψF, and the fact that the response of the latter is approximated by a Drude-like classical current as if this portion of the wavefunction was completely free. A generalized version of Eq. (13) should contain an additional term,

tJ(t)=ρ(t)F(t)+vitρ(t),
where vi stands for the initial velocity of freed electrons, which “disappear” from ψM and are “injected” into ψF. This correction can give rise to an important contribution to terahertz (THz) generation, and will be discussed in detail elsewhere. However, because it generates only a DC-like, low-frequency current, it has a negligible influence on the propagation of the driving pulse and can be safely ignored for that purpose.

5. COMPARISON WITH EXACT SOLUTIONS

The exactly solvable Dirac-delta system is an ideal test bed to assess accuracy of the proposed light–matter interaction description. We utilize the time-dependent exact solution for the nonlinear response given in Ref. [11]. This is then compared with the nonlinear response calculated as described in the previous sections.

The sole adjustment we make before comparing these results concerns the low-frequency part of the response described above. It shows up as a constant current after the driving pulse vanishes, reflecting the net average velocity imparted on the ionized electrons. While it is possible to capture this effect in the resonance-response model, at present an estimate of vi requires one adjustable parameter. We therefore filter out the very low-frequency part of the exact response and add this to the MESA result to compare the nonlinear response.

Figure 3 illustrates that very good agreement can be achieved between the exact and approximated nonlinear response for a 2.5 μm wavelength driving pulse. One can see that the approximate solution follows the exact one while it filters out the very-high-frequency components. These are due to higher order resonances and are therefore absent in our post-adiabatic approximation.

 

Fig. 3. Nonlinear response of the Dirac-delta system to a λ=2.5μm driving pulse indicated by the thin dashed line. The exact response is shown as a blue solid line, and the resonant-response model result is shown in thick red. The top panel zooms in to highlight that our response model “filters out” very-high-frequency components. The bottom panel demonstrates accurate overall agreement between approximate and exact solutions.

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The top panel shows that these response oscillations are too fast to affect the optical-frequency components of the driving pulse. For simulation of pulse propagation, it is therefore quite convenient that the model response does not follow them.

We thus can conclude that the proposed method captures precisely that frequency part of the total nonlinear response that is responsible for the back-reaction of the medium on the driving pulse. At the same time, even small-scale details in the nonlinear response shape are reproduced quite well.

6. 3D HYDROGEN-LIKE MODEL ATOM

Having seen that the nonlinear response of the exactly solvable model can be reproduced by the metastable-state response model quite accurately, the important question is if it still works for more realistic systems. The crucial difference is that one has to resort to approximate methods to find, and “measure,” properties of the ground-state resonance. In the following we describe the procedure that allows one to obtain parameterized nonlinear response from a series of TDSE simulations of a given system. The Hamiltonian represents a single-electron, hydrogen-like system,

H=12Δ1a2+x2+y2+z2xF(t),
with a “soft” Coulomb potential and a time-dependent external field (of the optical pulse) F(t).

To obtain the ground-state resonance as a function of the static-field strength F, we start a TDSE simulation in the numerical ground state, and add transparent boundary conditions realized as a perfectly matched layer (PML) to the computational domain. Because the ground-state resonance is the longest living state, the initial wavefunction is driven toward it during the simulated real-time evolution, while the “unwanted” states decay as they leak through the PML layers. Once the solution stabilizes, we characterize this ground-state resonance by calculating the metastable expectation value for its dipole moment. We also extract its complex-valued resonance energy. This process is repeated for a range of field strengths, and the relevant data are tabulated. We envision that this kind of procedure will be applied when working with realistic (single-active-electron) models of atoms and perhaps even molecules.

Figure 4 shows the real and imaginary parts of metastable dipole moment X0(F). We have determined this quantity as a function of the field strength on different-size grids, and with three different implementations of the PML boundary. Two data sets, obtained for grid sizes of L=100 a.u. and L=150 a.u., are shown in the figure to match closely, thus indicating that convergence is already achieved on relatively small computational domains. It should be noted that, if one attempts to determine the standard quantum-mechanical expectation value of the dipole moment, no convergence can be reached because metastable wavefunctions diverge at infinity. It is thus crucial that our method works with the metastable generalization of the dipole moment [Eq. (5)].

 

Fig. 4. Complex-valued dipole moment of a 3D hydrogen-like model atom, measured in the metastable state born from the ground-state as a function of the external field strength. Data obtained for two computational domain size L are shown, indicating fast convergence.

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Importantly, comparing Fig. 4 to Fig. 1, one can see that the behavior of the field-induced dipole is qualitatively the same in the 3D system as in the exactly solvable 1D model. This makes us believe that features of X0(F) are quite generic.

Together with the field-dependent metastable ground-state energy E0(F), tabulation of the nonlinear of the dipole moment X0(F) shown in Fig. 4 constitutes the core data that characterize the quantum system, and makes it possible to calculate its response to arbitrary pulsed excitation. Note that the TDSE simulations to obtain these data sets require significant, but only a one-time, numerical effort.

7. COMPARISON WITH TDSE SOLUTIONS

To demonstrate that the nonlinear response can be calculated the same way also for a more realistic system, we have generated time-dependent solutions of the Schrödinger equation for our hydrogen-like system, exposed to a near-infrared frequency pulse. For comparison purposes, we remove the linear polarization from our numerical solutions. We also remove a low-frequency background, which contributes only to the THz generation that would not affect dynamics of the driving infrared pulse.

Figure 5 shows an example in which both Kerr-like and plasma-like responses show up in the leading and trailing edge of the pulse, respectively. Regimes like this one, when counter-acting components of the nonlinear response manifest themselves on similar scales, are of utmost importance for extreme nonlinear optics, since the filamentation physics is naturally influenced by this kind of dynamic balance.

 

Fig. 5. Nonlinear response of a hydrogen-like system to a λ=2μm driving pulse indicated by the shaded area(s). The exact TDSE response is shown by the blue dashed line, and the resonant-response model result is shown in red.

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We have chosen the shape of the excitation pulse to have a flat middle portion in order to better visualize different processes that would control the dynamics of the propagating pulse. In the leading edge of the pulse, the polarization is in phase with the optical field. During this time, the self-focusing response dominates. In the trailing edge, in contrast, we see the response being out of phase, which is a sign that it acts mainly as a defocusing mechanism. In the temporal middle of the pulse one can see that the two compete. Importantly, the agreement between the numerically exact and the response calculated with the proposed model is rather good.

8. APPLICATION EXAMPLE: FEMTOSECOND FILAMENTATION

As an illustrative example of the utility of MESA, we consider a femtosecond filament created by a 30 fs optical pulse propagating in a model gaseous medium. The medium consists of atoms at atmospheric pressure, each responding to the optical field as described above. Besides the nonlinear atomic response, the medium has a linear susceptibility, which we choose to be the same as that of air. This is implemented as a part of the linear optical propagator [22]. To execute the simulation, we utilize the generalized unidirectional pulse propagation equations framework (gUPPEcore) [23] with a medium module that implements the MESA response.

The parameters of our illustration are chosen to create a situation in which several processes affecting the dynamics act simultaneously. In particular, we choose the wavelength λ=2μm, for which the generation of new harmonics, along with their subsequent “temporal walk-off,” are more relevant than for 800 nm pulses.

Furthermore, we assume relatively tight focusing geometry, with f=50cm. This is to verify that the defocusing properties of the model are sufficient to arrest the self-focusing collapse, which is made more severe by the external focusing. Figure 6, showing the on-axis energy fluence versus propagation distance for two different initial pulse intensities, demonstrates that it is indeed the case.

 

Fig. 6. On-axis energy fluence in a filament created by a 30 fs, λ=2.0μm pulse. The two curves represent simulations with the indicated initial intensity.

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The defocusing effects are mainly due to free electrons. Figure 7 shows the linear density of generated freed electrons are versus propagation distance. The model accounts for the corresponding energy losses via the imaginary part of the induced dipole moment.

 

Fig. 7. Free electrons generated per unit of propagation length.

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Figure 8 shows the spectrum for propagation distances just before and after the filament. Well-defined harmonic orders are obvious before the collapse: they give rise to an additional collapse regularization that is much stronger than for λ800nm. Eventually, an extremely broad supercontinuum is generated in the filament (full line). In contrast to the more studied near-infrared regime, at this and still longer wavelengths the spectral dynamics coupled with the dispersion properties of the medium become the most important physical mechanism controlling filamentation. It thus becomes crucial that properties of third- and fifth-harmonic generation, together with free-electron generation and accompanying energy losses, are all modeled in a unified manner, so that, as the wavelength is varied, all processes are included in correct proportion.

 

Fig. 8. Supercontinuum generation in a femtosecond filament. The spectrum before the collapse exhibits well-separated harmonic orders.

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Importantly for our demonstration, all relevant effects are captured in a self-consistent way, without the necessity and even possibility of parameter tuning. To our knowledge, this is the first demonstration of a filament simulation on an experimentally relevant scale in which the light–matter interaction description relies on first principles.

9. CONCLUSION

In summary, we have introduced the metastable state approach for calculating the ultrafast, off-resonant nonlinear optical response of gases, and demonstrated its accuracy and computational economy. As a test bed, we employed a 1D atomic model that allows for comparison of MESA against exact solutions, and excellent agreement between the exact and approximate results were obtained for the nonlinear optical response even using only a single metastable state in adiabatic approximation. We have also identified post-adiabatic corrections that are responsible for the slight increase of ionization rate and for losses that the optical field suffers as a consequence of ionization. The approach was also applied to a 3D hydrogen-like model with similarly good agreement between MESA and TDSE simulations. As a demonstration of the computational economy of MESA, an example of optical filamentation was presented.

In the present work, we required knowledge of the properties of only a single metastable state, namely, the one that arises from the system’s ground state. Higher order metastable states were included only indirectly with the help of additional approximations. It will be interesting to explore the feasibility of inclusion of several states in ψM. This should make it possible to capture even more noninstantaneous effects, and help to understand how a driven quantum system starts to exhibit behavior dependent on its own history.

In future work we plan to apply MESA to simulation of optical filamentation in current experiments, with argon, for example, and also to explore the application to molecules to allow simulation of air. While the initial TDSE-based characterization of the given model atom or molecule becomes much more demanding, the application of the method to such realistic systems remains conceptually the same. Recent progress in direct experimental characterization of optical nonlinearities at ultrafast time scales brought quantitative results concerning the self-focusing nonlinearity [24] as well as free-electron generation [25] in various gases. Importantly, spatiotemporal profiles of the nonlinear responses can now be obtained essentially free of propagation effects, which complicate interpretation of dynamics in other, for example, filamentation regimes. Comparison with these and similar future experiments should provide a practical way to validate and benchmark the nonlinear response models based on the approach proposed here.

To finish, we note that MESA yields a model for the ultrafast nonlinear optical response with no free parameters for a given gas. Importantly, from a practical simulation point of view, the approach has a computational complexity comparable to that of the standard light–matter interaction model used in optical filamentation. In this sense, the method provides a microscopically founded replacement.

FUNDING INFORMATION

Air Force Office of Scientific Research (AFOSR) (FA9550-10-1-0561, FA9550-11-1-0144, FA9550-13-1-0228).

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1. T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4, 822–832 (2010). [CrossRef]  

2. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009). [CrossRef]  

3. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007). [CrossRef]  

4. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007). [CrossRef]  

5. M. Nurhuda, A. Suda, and K. Midorikawa, “Generalization of the Kerr effect for high intensity, ultrashort laser pulses,” New J. Phys. 10, 053006 (2008). [CrossRef]  

6. E. A. Volkova, A. M. Popov, and O. V. Tikhonova, “Nonlinear polarization response of an atomic gas medium in the field of a high-intensity femtosecond laser pulse,” JETP Lett. 94, 519–524 (2011). [CrossRef]  

7. E. A. Volkova, A. M. Popov, and O. V. Tikhonova, “Polarization response of a gas medium in the field of a high-intensity ultrashort laser pulse: high order Kerr nonlinearities or plasma electron component?” Quantum Electron. 42, 680–686 (2012). [CrossRef]  

8. A. M. Popov, O. V. Tikhonova, and E. A. Volkova, “Polarization response of an atomic system in a strong mid-IR field,” Laser Phys. Lett. 10, 085303 (2013). [CrossRef]  

9. P. Béjot, E. Cormier, E. Hertz, B. Lavorel, J. Kasparian, J.-P. Wolf, and O. Faucher, “High-field quantum calculation reveals time-dependent negative Kerr contribution,” Phys. Rev. Lett. 110, 043902 (2013). [CrossRef]  

10. C. Köhler, R. Guichard, E. Lorin, S. Chelkowski, A. D. Bandrauk, L. Bergé, and S. Skupin, “Saturation of the nonlinear refractive index in atomic gases,” Phys. Rev. A 87, 043811 (2013). [CrossRef]  

11. J. M. Brown, A. Lotti, A. Teleki, and M. Kolesik, “Exactly solvable model for non-linear light-matter interaction in an arbitrary time-dependent field,” Phys. Rev. A 84, 063424 (2011). [CrossRef]  

12. T. C. Rensink, T. M. Antonsen, J. P. Palastro, and D. F. Gordon, “Model for atomic dielectric response in strong, time-dependent laser fields,” Phys. Rev. A 89, 033418 (2014). [CrossRef]  

13. E. Lorin, S. Chelkowski, and A. Bandrauk, “The WASP model: a micro–macro system of wave-Schrödinger-plasma equations for filamentation,” Commun. Comput. Phys. 9, 406–440 (2011).

14. E. Lorin, S. Chelkowski, and A. Bandrauk, “Maxwell-Schrödinger-Plasma (MASP) model for laser-molecule interactions: towards an understanding of filamentation with intense ultrashort pulses,” Physica D 241, 1059–1071 (2012). [CrossRef]  

15. O. I. Tolstikhin, T. Morishita, and S. Watanabe, “Adiabatic theory of ionization of atoms by intense laser pulses: one-dimensional zero-range-potential model,” Phys. Rev. A 81, 033415 (2010). [CrossRef]  

16. R. M. Cavalcanti, P. Giacconi, and R. Soldati, “Decay in a uniform field: an exactly solvable model,” J. Phys. A 36, 12065–12080 (2003). [CrossRef]  

17. T. Berggren, “Expectation value of an operator in a resonant state,” Phys. Lett. B 373, 1–4 (1996). [CrossRef]  

18. L. Hamonou, T. Morishita, O. I. Tolstikhin, and S. Watanabe, “Siegert-state method for ionization of molecules in strong field,” J. Phys. Conf. Ser. 388, 032030 (2012). [CrossRef]  

19. T. Berggren, “On the use of resonant states in eigenfunction expansions of scattering and reaction amplitudes,” Nucl. Phys. A 109, 265–287 (1968). [CrossRef]  

20. A. Bahl, A. Teleki, P. Jakobsen, E. M. Wright, and M. Kolesik, “Reflectionless beam propagation on a piecewise linear complex domain,” J. Lightwave Technol. 32, 3670–3676 (2014). [CrossRef]  

21. D. C. Brody, “Biorthogonal quantum mechanics,” J. Phys. A 47, 035305 (2014). [CrossRef]  

22. A. Couairon, E. Brambilla, T. Corti, D. Majus, O. J. Ramirez-Gongora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. J. Phys. Special Topics 199, 5–76 (2011).

23. J. Andreasen and M. Kolesik, “Nonlinear propagation of light in structured media: generalized unidirectional pulse propagation equations,” Phys. Rev. E 86, 036706 (2012). [CrossRef]  

24. J. K. Wahlstrand, Y.-H. Cheng, and H. M. Milchberg, “Absolute measurement of the transient optical nonlinearity in N2, O2, N2O, and Ar,” Phys. Rev. A 85, 043820 (2012).

25. Y.-H. Chen, S. Varma, T. M. Antonsen, and H. M. Milchberg, “Direct measurement of the electron density of extended femtosecond laser pulse-induced filaments,” Phys. Rev. Lett. 105, 215005 (2010). [CrossRef]  

References

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  1. T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4, 822–832 (2010).
    [Crossref]
  2. F. Krausz, M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
    [Crossref]
  3. A. Couairon, A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
    [Crossref]
  4. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
    [Crossref]
  5. M. Nurhuda, A. Suda, K. Midorikawa, “Generalization of the Kerr effect for high intensity, ultrashort laser pulses,” New J. Phys. 10, 053006 (2008).
    [Crossref]
  6. E. A. Volkova, A. M. Popov, O. V. Tikhonova, “Nonlinear polarization response of an atomic gas medium in the field of a high-intensity femtosecond laser pulse,” JETP Lett. 94, 519–524 (2011).
    [Crossref]
  7. E. A. Volkova, A. M. Popov, O. V. Tikhonova, “Polarization response of a gas medium in the field of a high-intensity ultrashort laser pulse: high order Kerr nonlinearities or plasma electron component?” Quantum Electron. 42, 680–686 (2012).
    [Crossref]
  8. A. M. Popov, O. V. Tikhonova, E. A. Volkova, “Polarization response of an atomic system in a strong mid-IR field,” Laser Phys. Lett. 10, 085303 (2013).
    [Crossref]
  9. P. Béjot, E. Cormier, E. Hertz, B. Lavorel, J. Kasparian, J.-P. Wolf, O. Faucher, “High-field quantum calculation reveals time-dependent negative Kerr contribution,” Phys. Rev. Lett. 110, 043902 (2013).
    [Crossref]
  10. C. Köhler, R. Guichard, E. Lorin, S. Chelkowski, A. D. Bandrauk, L. Bergé, S. Skupin, “Saturation of the nonlinear refractive index in atomic gases,” Phys. Rev. A 87, 043811 (2013).
    [Crossref]
  11. J. M. Brown, A. Lotti, A. Teleki, M. Kolesik, “Exactly solvable model for non-linear light-matter interaction in an arbitrary time-dependent field,” Phys. Rev. A 84, 063424 (2011).
    [Crossref]
  12. T. C. Rensink, T. M. Antonsen, J. P. Palastro, D. F. Gordon, “Model for atomic dielectric response in strong, time-dependent laser fields,” Phys. Rev. A 89, 033418 (2014).
    [Crossref]
  13. E. Lorin, S. Chelkowski, A. Bandrauk, “The WASP model: a micro–macro system of wave-Schrödinger-plasma equations for filamentation,” Commun. Comput. Phys. 9, 406–440 (2011).
  14. E. Lorin, S. Chelkowski, A. Bandrauk, “Maxwell-Schrödinger-Plasma (MASP) model for laser-molecule interactions: towards an understanding of filamentation with intense ultrashort pulses,” Physica D 241, 1059–1071 (2012).
    [Crossref]
  15. O. I. Tolstikhin, T. Morishita, S. Watanabe, “Adiabatic theory of ionization of atoms by intense laser pulses: one-dimensional zero-range-potential model,” Phys. Rev. A 81, 033415 (2010).
    [Crossref]
  16. R. M. Cavalcanti, P. Giacconi, R. Soldati, “Decay in a uniform field: an exactly solvable model,” J. Phys. A 36, 12065–12080 (2003).
    [Crossref]
  17. T. Berggren, “Expectation value of an operator in a resonant state,” Phys. Lett. B 373, 1–4 (1996).
    [Crossref]
  18. L. Hamonou, T. Morishita, O. I. Tolstikhin, S. Watanabe, “Siegert-state method for ionization of molecules in strong field,” J. Phys. Conf. Ser. 388, 032030 (2012).
    [Crossref]
  19. T. Berggren, “On the use of resonant states in eigenfunction expansions of scattering and reaction amplitudes,” Nucl. Phys. A 109, 265–287 (1968).
    [Crossref]
  20. A. Bahl, A. Teleki, P. Jakobsen, E. M. Wright, M. Kolesik, “Reflectionless beam propagation on a piecewise linear complex domain,” J. Lightwave Technol. 32, 3670–3676 (2014).
    [Crossref]
  21. D. C. Brody, “Biorthogonal quantum mechanics,” J. Phys. A 47, 035305 (2014).
    [Crossref]
  22. A. Couairon, E. Brambilla, T. Corti, D. Majus, O. J. Ramirez-Gongora, M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. J. Phys. Special Topics 199, 5–76 (2011).
  23. J. Andreasen, M. Kolesik, “Nonlinear propagation of light in structured media: generalized unidirectional pulse propagation equations,” Phys. Rev. E 86, 036706 (2012).
    [Crossref]
  24. J. K. Wahlstrand, Y.-H. Cheng, H. M. Milchberg, “Absolute measurement of the transient optical nonlinearity in N2, O2, N2O, and Ar,” Phys. Rev. A 85, 043820 (2012).
  25. Y.-H. Chen, S. Varma, T. M. Antonsen, H. M. Milchberg, “Direct measurement of the electron density of extended femtosecond laser pulse-induced filaments,” Phys. Rev. Lett. 105, 215005 (2010).
    [Crossref]

2014 (3)

T. C. Rensink, T. M. Antonsen, J. P. Palastro, D. F. Gordon, “Model for atomic dielectric response in strong, time-dependent laser fields,” Phys. Rev. A 89, 033418 (2014).
[Crossref]

D. C. Brody, “Biorthogonal quantum mechanics,” J. Phys. A 47, 035305 (2014).
[Crossref]

A. Bahl, A. Teleki, P. Jakobsen, E. M. Wright, M. Kolesik, “Reflectionless beam propagation on a piecewise linear complex domain,” J. Lightwave Technol. 32, 3670–3676 (2014).
[Crossref]

2013 (3)

P. Béjot, E. Cormier, E. Hertz, B. Lavorel, J. Kasparian, J.-P. Wolf, O. Faucher, “High-field quantum calculation reveals time-dependent negative Kerr contribution,” Phys. Rev. Lett. 110, 043902 (2013).
[Crossref]

C. Köhler, R. Guichard, E. Lorin, S. Chelkowski, A. D. Bandrauk, L. Bergé, S. Skupin, “Saturation of the nonlinear refractive index in atomic gases,” Phys. Rev. A 87, 043811 (2013).
[Crossref]

A. M. Popov, O. V. Tikhonova, E. A. Volkova, “Polarization response of an atomic system in a strong mid-IR field,” Laser Phys. Lett. 10, 085303 (2013).
[Crossref]

2012 (5)

J. Andreasen, M. Kolesik, “Nonlinear propagation of light in structured media: generalized unidirectional pulse propagation equations,” Phys. Rev. E 86, 036706 (2012).
[Crossref]

L. Hamonou, T. Morishita, O. I. Tolstikhin, S. Watanabe, “Siegert-state method for ionization of molecules in strong field,” J. Phys. Conf. Ser. 388, 032030 (2012).
[Crossref]

E. Lorin, S. Chelkowski, A. Bandrauk, “Maxwell-Schrödinger-Plasma (MASP) model for laser-molecule interactions: towards an understanding of filamentation with intense ultrashort pulses,” Physica D 241, 1059–1071 (2012).
[Crossref]

J. K. Wahlstrand, Y.-H. Cheng, H. M. Milchberg, “Absolute measurement of the transient optical nonlinearity in N2, O2, N2O, and Ar,” Phys. Rev. A 85, 043820 (2012).

E. A. Volkova, A. M. Popov, O. V. Tikhonova, “Polarization response of a gas medium in the field of a high-intensity ultrashort laser pulse: high order Kerr nonlinearities or plasma electron component?” Quantum Electron. 42, 680–686 (2012).
[Crossref]

2011 (4)

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. J. Ramirez-Gongora, M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. J. Phys. Special Topics 199, 5–76 (2011).

E. Lorin, S. Chelkowski, A. Bandrauk, “The WASP model: a micro–macro system of wave-Schrödinger-plasma equations for filamentation,” Commun. Comput. Phys. 9, 406–440 (2011).

J. M. Brown, A. Lotti, A. Teleki, M. Kolesik, “Exactly solvable model for non-linear light-matter interaction in an arbitrary time-dependent field,” Phys. Rev. A 84, 063424 (2011).
[Crossref]

E. A. Volkova, A. M. Popov, O. V. Tikhonova, “Nonlinear polarization response of an atomic gas medium in the field of a high-intensity femtosecond laser pulse,” JETP Lett. 94, 519–524 (2011).
[Crossref]

2010 (3)

Y.-H. Chen, S. Varma, T. M. Antonsen, H. M. Milchberg, “Direct measurement of the electron density of extended femtosecond laser pulse-induced filaments,” Phys. Rev. Lett. 105, 215005 (2010).
[Crossref]

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4, 822–832 (2010).
[Crossref]

O. I. Tolstikhin, T. Morishita, S. Watanabe, “Adiabatic theory of ionization of atoms by intense laser pulses: one-dimensional zero-range-potential model,” Phys. Rev. A 81, 033415 (2010).
[Crossref]

2009 (1)

F. Krausz, M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[Crossref]

2008 (1)

M. Nurhuda, A. Suda, K. Midorikawa, “Generalization of the Kerr effect for high intensity, ultrashort laser pulses,” New J. Phys. 10, 053006 (2008).
[Crossref]

2007 (2)

A. Couairon, A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
[Crossref]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[Crossref]

2003 (1)

R. M. Cavalcanti, P. Giacconi, R. Soldati, “Decay in a uniform field: an exactly solvable model,” J. Phys. A 36, 12065–12080 (2003).
[Crossref]

1996 (1)

T. Berggren, “Expectation value of an operator in a resonant state,” Phys. Lett. B 373, 1–4 (1996).
[Crossref]

1968 (1)

T. Berggren, “On the use of resonant states in eigenfunction expansions of scattering and reaction amplitudes,” Nucl. Phys. A 109, 265–287 (1968).
[Crossref]

Andreasen, J.

J. Andreasen, M. Kolesik, “Nonlinear propagation of light in structured media: generalized unidirectional pulse propagation equations,” Phys. Rev. E 86, 036706 (2012).
[Crossref]

Antonsen, T. M.

T. C. Rensink, T. M. Antonsen, J. P. Palastro, D. F. Gordon, “Model for atomic dielectric response in strong, time-dependent laser fields,” Phys. Rev. A 89, 033418 (2014).
[Crossref]

Y.-H. Chen, S. Varma, T. M. Antonsen, H. M. Milchberg, “Direct measurement of the electron density of extended femtosecond laser pulse-induced filaments,” Phys. Rev. Lett. 105, 215005 (2010).
[Crossref]

Arpin, P.

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4, 822–832 (2010).
[Crossref]

Bahl, A.

Bandrauk, A.

E. Lorin, S. Chelkowski, A. Bandrauk, “Maxwell-Schrödinger-Plasma (MASP) model for laser-molecule interactions: towards an understanding of filamentation with intense ultrashort pulses,” Physica D 241, 1059–1071 (2012).
[Crossref]

E. Lorin, S. Chelkowski, A. Bandrauk, “The WASP model: a micro–macro system of wave-Schrödinger-plasma equations for filamentation,” Commun. Comput. Phys. 9, 406–440 (2011).

Bandrauk, A. D.

C. Köhler, R. Guichard, E. Lorin, S. Chelkowski, A. D. Bandrauk, L. Bergé, S. Skupin, “Saturation of the nonlinear refractive index in atomic gases,” Phys. Rev. A 87, 043811 (2013).
[Crossref]

Béjot, P.

P. Béjot, E. Cormier, E. Hertz, B. Lavorel, J. Kasparian, J.-P. Wolf, O. Faucher, “High-field quantum calculation reveals time-dependent negative Kerr contribution,” Phys. Rev. Lett. 110, 043902 (2013).
[Crossref]

Bergé, L.

C. Köhler, R. Guichard, E. Lorin, S. Chelkowski, A. D. Bandrauk, L. Bergé, S. Skupin, “Saturation of the nonlinear refractive index in atomic gases,” Phys. Rev. A 87, 043811 (2013).
[Crossref]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[Crossref]

Berggren, T.

T. Berggren, “Expectation value of an operator in a resonant state,” Phys. Lett. B 373, 1–4 (1996).
[Crossref]

T. Berggren, “On the use of resonant states in eigenfunction expansions of scattering and reaction amplitudes,” Nucl. Phys. A 109, 265–287 (1968).
[Crossref]

Brambilla, E.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. J. Ramirez-Gongora, M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. J. Phys. Special Topics 199, 5–76 (2011).

Brody, D. C.

D. C. Brody, “Biorthogonal quantum mechanics,” J. Phys. A 47, 035305 (2014).
[Crossref]

Brown, J. M.

J. M. Brown, A. Lotti, A. Teleki, M. Kolesik, “Exactly solvable model for non-linear light-matter interaction in an arbitrary time-dependent field,” Phys. Rev. A 84, 063424 (2011).
[Crossref]

Cavalcanti, R. M.

R. M. Cavalcanti, P. Giacconi, R. Soldati, “Decay in a uniform field: an exactly solvable model,” J. Phys. A 36, 12065–12080 (2003).
[Crossref]

Chelkowski, S.

C. Köhler, R. Guichard, E. Lorin, S. Chelkowski, A. D. Bandrauk, L. Bergé, S. Skupin, “Saturation of the nonlinear refractive index in atomic gases,” Phys. Rev. A 87, 043811 (2013).
[Crossref]

E. Lorin, S. Chelkowski, A. Bandrauk, “Maxwell-Schrödinger-Plasma (MASP) model for laser-molecule interactions: towards an understanding of filamentation with intense ultrashort pulses,” Physica D 241, 1059–1071 (2012).
[Crossref]

E. Lorin, S. Chelkowski, A. Bandrauk, “The WASP model: a micro–macro system of wave-Schrödinger-plasma equations for filamentation,” Commun. Comput. Phys. 9, 406–440 (2011).

Chen, M.-C.

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4, 822–832 (2010).
[Crossref]

Chen, Y.-H.

Y.-H. Chen, S. Varma, T. M. Antonsen, H. M. Milchberg, “Direct measurement of the electron density of extended femtosecond laser pulse-induced filaments,” Phys. Rev. Lett. 105, 215005 (2010).
[Crossref]

Cheng, Y.-H.

J. K. Wahlstrand, Y.-H. Cheng, H. M. Milchberg, “Absolute measurement of the transient optical nonlinearity in N2, O2, N2O, and Ar,” Phys. Rev. A 85, 043820 (2012).

Cormier, E.

P. Béjot, E. Cormier, E. Hertz, B. Lavorel, J. Kasparian, J.-P. Wolf, O. Faucher, “High-field quantum calculation reveals time-dependent negative Kerr contribution,” Phys. Rev. Lett. 110, 043902 (2013).
[Crossref]

Corti, T.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. J. Ramirez-Gongora, M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. J. Phys. Special Topics 199, 5–76 (2011).

Couairon, A.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. J. Ramirez-Gongora, M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. J. Phys. Special Topics 199, 5–76 (2011).

A. Couairon, A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
[Crossref]

Faucher, O.

P. Béjot, E. Cormier, E. Hertz, B. Lavorel, J. Kasparian, J.-P. Wolf, O. Faucher, “High-field quantum calculation reveals time-dependent negative Kerr contribution,” Phys. Rev. Lett. 110, 043902 (2013).
[Crossref]

Giacconi, P.

R. M. Cavalcanti, P. Giacconi, R. Soldati, “Decay in a uniform field: an exactly solvable model,” J. Phys. A 36, 12065–12080 (2003).
[Crossref]

Gordon, D. F.

T. C. Rensink, T. M. Antonsen, J. P. Palastro, D. F. Gordon, “Model for atomic dielectric response in strong, time-dependent laser fields,” Phys. Rev. A 89, 033418 (2014).
[Crossref]

Guichard, R.

C. Köhler, R. Guichard, E. Lorin, S. Chelkowski, A. D. Bandrauk, L. Bergé, S. Skupin, “Saturation of the nonlinear refractive index in atomic gases,” Phys. Rev. A 87, 043811 (2013).
[Crossref]

Hamonou, L.

L. Hamonou, T. Morishita, O. I. Tolstikhin, S. Watanabe, “Siegert-state method for ionization of molecules in strong field,” J. Phys. Conf. Ser. 388, 032030 (2012).
[Crossref]

Hertz, E.

P. Béjot, E. Cormier, E. Hertz, B. Lavorel, J. Kasparian, J.-P. Wolf, O. Faucher, “High-field quantum calculation reveals time-dependent negative Kerr contribution,” Phys. Rev. Lett. 110, 043902 (2013).
[Crossref]

Ivanov, M.

F. Krausz, M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[Crossref]

Jakobsen, P.

Kapteyn, H. C.

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4, 822–832 (2010).
[Crossref]

Kasparian, J.

P. Béjot, E. Cormier, E. Hertz, B. Lavorel, J. Kasparian, J.-P. Wolf, O. Faucher, “High-field quantum calculation reveals time-dependent negative Kerr contribution,” Phys. Rev. Lett. 110, 043902 (2013).
[Crossref]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[Crossref]

Köhler, C.

C. Köhler, R. Guichard, E. Lorin, S. Chelkowski, A. D. Bandrauk, L. Bergé, S. Skupin, “Saturation of the nonlinear refractive index in atomic gases,” Phys. Rev. A 87, 043811 (2013).
[Crossref]

Kolesik, M.

A. Bahl, A. Teleki, P. Jakobsen, E. M. Wright, M. Kolesik, “Reflectionless beam propagation on a piecewise linear complex domain,” J. Lightwave Technol. 32, 3670–3676 (2014).
[Crossref]

J. Andreasen, M. Kolesik, “Nonlinear propagation of light in structured media: generalized unidirectional pulse propagation equations,” Phys. Rev. E 86, 036706 (2012).
[Crossref]

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. J. Ramirez-Gongora, M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. J. Phys. Special Topics 199, 5–76 (2011).

J. M. Brown, A. Lotti, A. Teleki, M. Kolesik, “Exactly solvable model for non-linear light-matter interaction in an arbitrary time-dependent field,” Phys. Rev. A 84, 063424 (2011).
[Crossref]

Krausz, F.

F. Krausz, M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[Crossref]

Lavorel, B.

P. Béjot, E. Cormier, E. Hertz, B. Lavorel, J. Kasparian, J.-P. Wolf, O. Faucher, “High-field quantum calculation reveals time-dependent negative Kerr contribution,” Phys. Rev. Lett. 110, 043902 (2013).
[Crossref]

Lorin, E.

C. Köhler, R. Guichard, E. Lorin, S. Chelkowski, A. D. Bandrauk, L. Bergé, S. Skupin, “Saturation of the nonlinear refractive index in atomic gases,” Phys. Rev. A 87, 043811 (2013).
[Crossref]

E. Lorin, S. Chelkowski, A. Bandrauk, “Maxwell-Schrödinger-Plasma (MASP) model for laser-molecule interactions: towards an understanding of filamentation with intense ultrashort pulses,” Physica D 241, 1059–1071 (2012).
[Crossref]

E. Lorin, S. Chelkowski, A. Bandrauk, “The WASP model: a micro–macro system of wave-Schrödinger-plasma equations for filamentation,” Commun. Comput. Phys. 9, 406–440 (2011).

Lotti, A.

J. M. Brown, A. Lotti, A. Teleki, M. Kolesik, “Exactly solvable model for non-linear light-matter interaction in an arbitrary time-dependent field,” Phys. Rev. A 84, 063424 (2011).
[Crossref]

Majus, D.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. J. Ramirez-Gongora, M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. J. Phys. Special Topics 199, 5–76 (2011).

Midorikawa, K.

M. Nurhuda, A. Suda, K. Midorikawa, “Generalization of the Kerr effect for high intensity, ultrashort laser pulses,” New J. Phys. 10, 053006 (2008).
[Crossref]

Milchberg, H. M.

J. K. Wahlstrand, Y.-H. Cheng, H. M. Milchberg, “Absolute measurement of the transient optical nonlinearity in N2, O2, N2O, and Ar,” Phys. Rev. A 85, 043820 (2012).

Y.-H. Chen, S. Varma, T. M. Antonsen, H. M. Milchberg, “Direct measurement of the electron density of extended femtosecond laser pulse-induced filaments,” Phys. Rev. Lett. 105, 215005 (2010).
[Crossref]

Morishita, T.

L. Hamonou, T. Morishita, O. I. Tolstikhin, S. Watanabe, “Siegert-state method for ionization of molecules in strong field,” J. Phys. Conf. Ser. 388, 032030 (2012).
[Crossref]

O. I. Tolstikhin, T. Morishita, S. Watanabe, “Adiabatic theory of ionization of atoms by intense laser pulses: one-dimensional zero-range-potential model,” Phys. Rev. A 81, 033415 (2010).
[Crossref]

Murnane, M. M.

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4, 822–832 (2010).
[Crossref]

Mysyrowicz, A.

A. Couairon, A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
[Crossref]

Nurhuda, M.

M. Nurhuda, A. Suda, K. Midorikawa, “Generalization of the Kerr effect for high intensity, ultrashort laser pulses,” New J. Phys. 10, 053006 (2008).
[Crossref]

Nuter, R.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[Crossref]

Palastro, J. P.

T. C. Rensink, T. M. Antonsen, J. P. Palastro, D. F. Gordon, “Model for atomic dielectric response in strong, time-dependent laser fields,” Phys. Rev. A 89, 033418 (2014).
[Crossref]

Popmintchev, T.

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4, 822–832 (2010).
[Crossref]

Popov, A. M.

A. M. Popov, O. V. Tikhonova, E. A. Volkova, “Polarization response of an atomic system in a strong mid-IR field,” Laser Phys. Lett. 10, 085303 (2013).
[Crossref]

E. A. Volkova, A. M. Popov, O. V. Tikhonova, “Polarization response of a gas medium in the field of a high-intensity ultrashort laser pulse: high order Kerr nonlinearities or plasma electron component?” Quantum Electron. 42, 680–686 (2012).
[Crossref]

E. A. Volkova, A. M. Popov, O. V. Tikhonova, “Nonlinear polarization response of an atomic gas medium in the field of a high-intensity femtosecond laser pulse,” JETP Lett. 94, 519–524 (2011).
[Crossref]

Ramirez-Gongora, O. J.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. J. Ramirez-Gongora, M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. J. Phys. Special Topics 199, 5–76 (2011).

Rensink, T. C.

T. C. Rensink, T. M. Antonsen, J. P. Palastro, D. F. Gordon, “Model for atomic dielectric response in strong, time-dependent laser fields,” Phys. Rev. A 89, 033418 (2014).
[Crossref]

Skupin, S.

C. Köhler, R. Guichard, E. Lorin, S. Chelkowski, A. D. Bandrauk, L. Bergé, S. Skupin, “Saturation of the nonlinear refractive index in atomic gases,” Phys. Rev. A 87, 043811 (2013).
[Crossref]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[Crossref]

Soldati, R.

R. M. Cavalcanti, P. Giacconi, R. Soldati, “Decay in a uniform field: an exactly solvable model,” J. Phys. A 36, 12065–12080 (2003).
[Crossref]

Suda, A.

M. Nurhuda, A. Suda, K. Midorikawa, “Generalization of the Kerr effect for high intensity, ultrashort laser pulses,” New J. Phys. 10, 053006 (2008).
[Crossref]

Teleki, A.

A. Bahl, A. Teleki, P. Jakobsen, E. M. Wright, M. Kolesik, “Reflectionless beam propagation on a piecewise linear complex domain,” J. Lightwave Technol. 32, 3670–3676 (2014).
[Crossref]

J. M. Brown, A. Lotti, A. Teleki, M. Kolesik, “Exactly solvable model for non-linear light-matter interaction in an arbitrary time-dependent field,” Phys. Rev. A 84, 063424 (2011).
[Crossref]

Tikhonova, O. V.

A. M. Popov, O. V. Tikhonova, E. A. Volkova, “Polarization response of an atomic system in a strong mid-IR field,” Laser Phys. Lett. 10, 085303 (2013).
[Crossref]

E. A. Volkova, A. M. Popov, O. V. Tikhonova, “Polarization response of a gas medium in the field of a high-intensity ultrashort laser pulse: high order Kerr nonlinearities or plasma electron component?” Quantum Electron. 42, 680–686 (2012).
[Crossref]

E. A. Volkova, A. M. Popov, O. V. Tikhonova, “Nonlinear polarization response of an atomic gas medium in the field of a high-intensity femtosecond laser pulse,” JETP Lett. 94, 519–524 (2011).
[Crossref]

Tolstikhin, O. I.

L. Hamonou, T. Morishita, O. I. Tolstikhin, S. Watanabe, “Siegert-state method for ionization of molecules in strong field,” J. Phys. Conf. Ser. 388, 032030 (2012).
[Crossref]

O. I. Tolstikhin, T. Morishita, S. Watanabe, “Adiabatic theory of ionization of atoms by intense laser pulses: one-dimensional zero-range-potential model,” Phys. Rev. A 81, 033415 (2010).
[Crossref]

Varma, S.

Y.-H. Chen, S. Varma, T. M. Antonsen, H. M. Milchberg, “Direct measurement of the electron density of extended femtosecond laser pulse-induced filaments,” Phys. Rev. Lett. 105, 215005 (2010).
[Crossref]

Volkova, E. A.

A. M. Popov, O. V. Tikhonova, E. A. Volkova, “Polarization response of an atomic system in a strong mid-IR field,” Laser Phys. Lett. 10, 085303 (2013).
[Crossref]

E. A. Volkova, A. M. Popov, O. V. Tikhonova, “Polarization response of a gas medium in the field of a high-intensity ultrashort laser pulse: high order Kerr nonlinearities or plasma electron component?” Quantum Electron. 42, 680–686 (2012).
[Crossref]

E. A. Volkova, A. M. Popov, O. V. Tikhonova, “Nonlinear polarization response of an atomic gas medium in the field of a high-intensity femtosecond laser pulse,” JETP Lett. 94, 519–524 (2011).
[Crossref]

Wahlstrand, J. K.

J. K. Wahlstrand, Y.-H. Cheng, H. M. Milchberg, “Absolute measurement of the transient optical nonlinearity in N2, O2, N2O, and Ar,” Phys. Rev. A 85, 043820 (2012).

Watanabe, S.

L. Hamonou, T. Morishita, O. I. Tolstikhin, S. Watanabe, “Siegert-state method for ionization of molecules in strong field,” J. Phys. Conf. Ser. 388, 032030 (2012).
[Crossref]

O. I. Tolstikhin, T. Morishita, S. Watanabe, “Adiabatic theory of ionization of atoms by intense laser pulses: one-dimensional zero-range-potential model,” Phys. Rev. A 81, 033415 (2010).
[Crossref]

Wolf, J.-P.

P. Béjot, E. Cormier, E. Hertz, B. Lavorel, J. Kasparian, J.-P. Wolf, O. Faucher, “High-field quantum calculation reveals time-dependent negative Kerr contribution,” Phys. Rev. Lett. 110, 043902 (2013).
[Crossref]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[Crossref]

Wright, E. M.

Commun. Comput. Phys. (1)

E. Lorin, S. Chelkowski, A. Bandrauk, “The WASP model: a micro–macro system of wave-Schrödinger-plasma equations for filamentation,” Commun. Comput. Phys. 9, 406–440 (2011).

Eur. J. Phys. Special Topics (1)

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. J. Ramirez-Gongora, M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. J. Phys. Special Topics 199, 5–76 (2011).

J. Lightwave Technol. (1)

J. Phys. A (2)

D. C. Brody, “Biorthogonal quantum mechanics,” J. Phys. A 47, 035305 (2014).
[Crossref]

R. M. Cavalcanti, P. Giacconi, R. Soldati, “Decay in a uniform field: an exactly solvable model,” J. Phys. A 36, 12065–12080 (2003).
[Crossref]

J. Phys. Conf. Ser. (1)

L. Hamonou, T. Morishita, O. I. Tolstikhin, S. Watanabe, “Siegert-state method for ionization of molecules in strong field,” J. Phys. Conf. Ser. 388, 032030 (2012).
[Crossref]

JETP Lett. (1)

E. A. Volkova, A. M. Popov, O. V. Tikhonova, “Nonlinear polarization response of an atomic gas medium in the field of a high-intensity femtosecond laser pulse,” JETP Lett. 94, 519–524 (2011).
[Crossref]

Laser Phys. Lett. (1)

A. M. Popov, O. V. Tikhonova, E. A. Volkova, “Polarization response of an atomic system in a strong mid-IR field,” Laser Phys. Lett. 10, 085303 (2013).
[Crossref]

Nat. Photonics (1)

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4, 822–832 (2010).
[Crossref]

New J. Phys. (1)

M. Nurhuda, A. Suda, K. Midorikawa, “Generalization of the Kerr effect for high intensity, ultrashort laser pulses,” New J. Phys. 10, 053006 (2008).
[Crossref]

Nucl. Phys. A (1)

T. Berggren, “On the use of resonant states in eigenfunction expansions of scattering and reaction amplitudes,” Nucl. Phys. A 109, 265–287 (1968).
[Crossref]

Phys. Lett. B (1)

T. Berggren, “Expectation value of an operator in a resonant state,” Phys. Lett. B 373, 1–4 (1996).
[Crossref]

Phys. Rep. (1)

A. Couairon, A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
[Crossref]

Phys. Rev. A (5)

O. I. Tolstikhin, T. Morishita, S. Watanabe, “Adiabatic theory of ionization of atoms by intense laser pulses: one-dimensional zero-range-potential model,” Phys. Rev. A 81, 033415 (2010).
[Crossref]

C. Köhler, R. Guichard, E. Lorin, S. Chelkowski, A. D. Bandrauk, L. Bergé, S. Skupin, “Saturation of the nonlinear refractive index in atomic gases,” Phys. Rev. A 87, 043811 (2013).
[Crossref]

J. M. Brown, A. Lotti, A. Teleki, M. Kolesik, “Exactly solvable model for non-linear light-matter interaction in an arbitrary time-dependent field,” Phys. Rev. A 84, 063424 (2011).
[Crossref]

T. C. Rensink, T. M. Antonsen, J. P. Palastro, D. F. Gordon, “Model for atomic dielectric response in strong, time-dependent laser fields,” Phys. Rev. A 89, 033418 (2014).
[Crossref]

J. K. Wahlstrand, Y.-H. Cheng, H. M. Milchberg, “Absolute measurement of the transient optical nonlinearity in N2, O2, N2O, and Ar,” Phys. Rev. A 85, 043820 (2012).

Phys. Rev. E (1)

J. Andreasen, M. Kolesik, “Nonlinear propagation of light in structured media: generalized unidirectional pulse propagation equations,” Phys. Rev. E 86, 036706 (2012).
[Crossref]

Phys. Rev. Lett. (2)

Y.-H. Chen, S. Varma, T. M. Antonsen, H. M. Milchberg, “Direct measurement of the electron density of extended femtosecond laser pulse-induced filaments,” Phys. Rev. Lett. 105, 215005 (2010).
[Crossref]

P. Béjot, E. Cormier, E. Hertz, B. Lavorel, J. Kasparian, J.-P. Wolf, O. Faucher, “High-field quantum calculation reveals time-dependent negative Kerr contribution,” Phys. Rev. Lett. 110, 043902 (2013).
[Crossref]

Physica D (1)

E. Lorin, S. Chelkowski, A. Bandrauk, “Maxwell-Schrödinger-Plasma (MASP) model for laser-molecule interactions: towards an understanding of filamentation with intense ultrashort pulses,” Physica D 241, 1059–1071 (2012).
[Crossref]

Quantum Electron. (1)

E. A. Volkova, A. M. Popov, O. V. Tikhonova, “Polarization response of a gas medium in the field of a high-intensity ultrashort laser pulse: high order Kerr nonlinearities or plasma electron component?” Quantum Electron. 42, 680–686 (2012).
[Crossref]

Rep. Prog. Phys. (1)

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[Crossref]

Rev. Mod. Phys. (1)

F. Krausz, M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[Crossref]

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Figures (8)

Fig. 1.
Fig. 1. Complex-valued expectation value of the dipole moment in the metastable ground state as a function of the external field strength. The fine dashed line indicates the linear susceptibility. The gap between the thin dashed and the thick black line represents the nonlinear response Eq. (11).
Fig. 2.
Fig. 2. Ionization yield as a function of the intensity of the driving pulse. Exact, adiabatic, and corrected solutions are compared. The top and bottom panels correspond to wavelengths of λ=2400nm and λ=800nm, respectively.
Fig. 3.
Fig. 3. Nonlinear response of the Dirac-delta system to a λ=2.5μm driving pulse indicated by the thin dashed line. The exact response is shown as a blue solid line, and the resonant-response model result is shown in thick red. The top panel zooms in to highlight that our response model “filters out” very-high-frequency components. The bottom panel demonstrates accurate overall agreement between approximate and exact solutions.
Fig. 4.
Fig. 4. Complex-valued dipole moment of a 3D hydrogen-like model atom, measured in the metastable state born from the ground-state as a function of the external field strength. Data obtained for two computational domain size L are shown, indicating fast convergence.
Fig. 5.
Fig. 5. Nonlinear response of a hydrogen-like system to a λ=2μm driving pulse indicated by the shaded area(s). The exact TDSE response is shown by the blue dashed line, and the resonant-response model result is shown in red.
Fig. 6.
Fig. 6. On-axis energy fluence in a filament created by a 30 fs, λ=2.0μm pulse. The two curves represent simulations with the indicated initial intensity.
Fig. 7.
Fig. 7. Free electrons generated per unit of propagation length.
Fig. 8.
Fig. 8. Supercontinuum generation in a femtosecond filament. The spectrum before the collapse exhibits well-separated harmonic orders.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

[it+12x2+Bδ(x)+xF(t)]ψ(x,t)=0,
ψ(λ,x)={Ci(ξ0)Ai(ξ)x<0Ai(ξ0)Ci(ξ)x>0,
1=2πB(2F)1/3Ai(2λ(2F)2/3)[iAi(2λ(2F)2/3)+Bi(2λ(2F)2/3)].
ψi|ψj=Cψi(z)ψj(z)dz=Ni2(F)δij,
X0(F)=N02(F)Cψ0(z)zψ0(z)dz.
1=n|ψnψn|+Ldλ|ϕλϕλ|,
itψM=H^(F(t))ψM(x,t),ψM=ici(t)ψi(F(t),x),
H^(F)ψi(F,x)=Ei(F)ψi(F,x),
cn(t)=icnEn(F(t))kck(t)F(t)ψn|Fψk(F(t),x).
c0(a)(t)=exp{iEGtit[E0(F(τ))EG]dτ},
P(nl)(F(t))=P(F(t))limϵ01ϵP(ϵF(t)),
tρ(t)=[1ρ(t)]I{2E0(F(t))}.
tJ(t)=ρ(t)F(t).
E0(R)(F(t))=E0(F(t))n0(F(t))2(ψ0|Fψn(F(t)))2[En(F(t))E0(F(t))].
E0(R)(F(t))=E0(F(t))+(F(t))2E0(F(t))Fψ0(F(t))|Fψ0(F(t)).
PNL(corr)(t)1EGI{tψ0(F(t))|X|ψ0(F(t))}.
tJ(t)=ρ(t)F(t)+vitρ(t),
H=12Δ1a2+x2+y2+z2xF(t),

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