Intense few-cycle pulses are in the process of becoming available as research tools [1, 2, 3, 4, 5, 6]. Such pulses may currently have as few as two or three cycles with a peak intensity around or exceeding 10¹⁴ Wcm^-2. While an infinitely extended monochromatic plane wave is, in principle, fully characterized by its frequency and intensity, at least two additional parameters are required to specify a few-cycle pulse: besides its duration, this is the carrier-envelope relative phase (or, briefly, the “absolute phase”), that is, the relative phase between the maximum of the pulse envelope and the nearest maximum of the carrier wave. The actual shape of a few-cycle pulse crucially depends on the value of the absolute phase, and so do the physical processes that such a pulse may induce. This is reminiscent of the relative phase between the two components of an infinitely extended two-color plane wave with commensurate frequencies [7, 8, 9]. However, in contrast to the former, the arrival of a few-cycle pulse can be arbitrarily timed, thus allowing pump-probe spectroscopy with sub-femtosecond resolution.

Recently, few-cycle pulses with stable absolute phase were produced for the first time. In order to verify that the phase is fixed and in order to determine its value, one has to have the pulse interact with matter and to study the response. High-order harmonic generation (HHG) was suggested for this purpose [10, 11, 12, 13, 14], and the first experimental implementation was recently carried out in this context [15]. However, HHG, being insensitive to spatial inversion, only admits determination of the absolute phase modulo π. The fact that atomic ionization depends on the absolute phase was emphasized in Refs. [16, 17], while in Refs. [18, 19, 20] a measurement of the angular distribution of the ionized electrons was proposed as a method to determine the absolute phase. The left-right asymmetry of above-threshold ionization (ATI) yields was used to obtain the first experimental evidence of absolute-phase phenomena in a “stereo-ATI” experiment with circularly polarized few-cycle laser pulses [21]. A theoretical analysis of this experiment was presented in Refs. [22, 23]. The left-right asymmetry in atomic ionization by combined infrared femtosecond and ultraviolet attosecond laser pulses as a tool for directly measuring the absolute phase was suggested in Ref. [24]. This is similar to the attosecond cross-correlation technique for measuring the attosecond pulse duration, which was proposed in Ref. [25] and experimentally realized in Refs. [26, 27, 28]. Molecular ionization with ultrashort pulses was considered in [29]. The recollision dynamics of electron pulses have been suggested as a tool for probing molecular dynamics [30]. In this paper, we suggest to use the process of atomic high-order above-threshold ionization as a “phasemeter”.

Here is a brief outline of the paper. We start with a sketch of the formalism that we will employ: this is the Keldysh-Faisal-Reiss ionization amplitude improved so that it incorporates rescattering, also known as the strong-field approximation. We apply it to the calculation of angle-resolved photoelectron energy spectra generated by a few-cycle pulse with a sine-square envelope. As pioneered in the first demonstration of absolute-phase effects in few-cycle pulses where the phase was unstable [21, 22, 23], we concentrate on the left-right asymmetry of the energy spectrum in the direction of the laser polarization, which here we assume to be linear.We present explicit results for a four-cycle pulse (two cycles FWHM) with a peak intensity of 6×10¹³ Wcm^-2 and discuss their potential as a phasemeter. The results of the quantum-mechanical calculation are reconsidered from the point of view of a classical simple-man picture. We use atomic units (h̄=e=|e|=m=1) throughout the paper.

Let us consider an atom initially in its ground state subject to a few-cycle pulse of duration T_p impinging during the interval 0≤t≤T_p . An exact expression for the probability amplitude of detecting an ATI electron with momentum p set free by this pulse from the atom is (see Ref. [31] and references therein)

Here U(t, t′) is the time-evolution operator of the complete Hamiltonian

where r·E(t) is the laser-field-electron interaction in the length gauge and the dipole approximation and V(r) is the atomic binding potential. The states ψ _p and ψ ₀ are a scattering state with asymptotic momentum p and the ground state, respectively, of the atomic Hamiltonian H _A=-∇²/2+V(r). The time-evolution operator U(t, t′) satisfies the Dyson equation

where U _F(t, t′) is the time-evolution operator that corresponds to the Hamiltonian H _F(t)=-∇²/2+r·E(t) of a free electron in the laser field. The eigenstates of the time-dependent Schrödinger equation with the Hamiltonian H _F(t) are the Volkov states

with the action

The vector potential of the laser field E(t) is denoted by A(t), and |q〉 is a plane-wave state [〈r|q〉=(2π)^-3/2 exp(i q·r)]. The Volkov time-evolution operator is

Before going on, we have to mention a subtlety. The Volkov state given by Eq. (4) corresponds to the time-dependent velocity v(t)≡p+A(t) where p is a constant. If we choose a vector potential A(t) such that A(t)=0 for t>T_p , then p has the physical meaning of the kinetical momentum at the detector. Below, however, we will employ a vector potential [cf. Eq. (12)] such that A(0)=A(T_p )≠0. Then, in order that the Volkov state given by Eq. (4) really describe an electron with the final momentum p at the position of the detector, we have to replace p by

everywhere in Eqs. (4) and (5); cf. Ref. [23].

With this in mind, we shall now evaluate the matrix element given by Eq. (1) by means of the strong-field approximation. This consists in replacing 〈ψ _p(t)|U(t, t′) by 〈χ _p̃(t)|U(t, t′) in Eq. (1) and U by U _F on the right-hand side of Eq. (3). This yields

where

The amplitude $M_{\mathbf{p}}^{\left(0\right)}$ is the Keldysh-Faisal-Reiss amplitude [32, 33, 34], generalized to the case of a few-cycle laser pulse [22, 23], which describes direct ionization where the electron once ionized does not interact with the binding potential again. The amplitude $M_{\mathbf{p}}^{\left(1\right)}$ is the rescattering amplitude, which accounts for one such additional interaction. The amplitude $M_{\mathbf{p}}^{\left(1\right)}$ is responsible for the high-energy plateau in the electron energy spectrum. In between ionization (at time t) and rescattering (at time t′), the electron does not feel the binding potential so that its propagation is governed by the Volkov propagator U _F(t′, t). In the rescattering amplitude $M_{\mathbf{p}}^{\left(1\right)}$ , the rescattering time t′ may be arbitrarily late (the integral over t′ extends from t to infinity). However, if t′≥T_p , then the laser pulse has passed through already at the time of rescattering. Rescattering itself is elastic, so the electron may only change its direction but not its energy in the process. The energy gain that does result from rescattering is due to subsequent acceleration by the laser field. Since the maximal energy of the electron before rescattering is rather low¹, rescattering in the absence of the laser field makes no contribution to the high-energy part of the spectrum. Hence, we will ignore it, by restricting in the amplitude given by Eq. (10) the integration over t′ by the upper limit T_p in place of infinity.

For the computation of the amplitudes given by Eqs. (9) and (10), we choose a linearly polarized few-cycle pulse with a sine-square envelope, specified by the electric field

for t∈[0,T_p ] and zero outside this interval. We assume an integer number of cycles so that T_p =n_pT with T=2π/ω. Moreover, we introduced the abbreviations ω_p =ω/n_p , ω ₀=ω, ω _1,2=ω±ω_p , ℰ₀=E ₀/2, and ℰ _i =-ℰ₀/2(i=1, 2). At the center of the pulse at t=T_p /2, we have $\cos \left(\frac{\omega T_p}{2+\varphi }\right)={\left(-1\right)}^{n_p}\cos \varphi$ . Hence, the phase ϕ is the absolute phase (carrier-envelope relative phase) as verbally defined above. Notice that specification of the absolute phase within the interval 0°≤ϕ≤360° requires us to define one spatial direction as positive. HHG only allows us to read off the value of the absolute phase modulo 180°. The field given by Eq. (11) is trichromatic comprising the frequencies ω and ω±ω_p . This allows us to employ the formalism developed earlier in Refs. [36, 37, 38].

For the evaluation of the action given by Eq. (5) we need a vector potential such that -(d/dt)A(t)=E(t), and we choose

The electric field given by Eq. (11) is, of course, zero before and after the pulse, E(0)=E(T_p )=0. However, the vector potential given by Eq. (12) is, in general, different from zero for t≤0 and t≥T_p , so that we have to make the substitution given in Eq. (7). Still, we have A(0)=A(T_p ) so that the integral over the electric field is zero, ${\int }_0^{T_p}d\tau \mathbf{E}\left(\tau \right)=-\left[\mathbf{A}\left(T_p\right)-\mathbf{A}\left(0\right)\right]=\mathbf{0}$ , as should be so that the electric field given by Eq. (11) has no dc component.

We will now calculate the differential ionization probability for the emission of an electron with energy E _p=p ²/2 in the direction θ (cosθ=ê·p̂), which is

 

Fig. 1. Angle-resolved ATI energy spectrum generated by the 4-cycle pulse (11) with intensity I=$E_0^2$=6×10¹³ Wcm^-2 and λ=800 nm in krypton (I_P =14 eV) for an absolute phase ϕ=0°, for the two opposite detector positions θ=0° and θ=180° along the laser-field polarization. For comparison, the envelope of the ATI spectrum for the same atomic and laser parameters but for an infinitely long pulse with flat envelope (amplitude E ₀) is also shown (green dashed line with circles). For this case, the high-energy cutoff is at 10U_P =35.9 eV. The high-energy part of the 4-cycle spectrum for θ=0° virtually agrees with the envelope of the infinitely-long-pulse spectrum. The explanation will be found in the classical model considered below.

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where the matrix element M_p is approximated by Eqs. (8)–(10). The explicit calculations will be done for a hydrogen-like model atom, for which

For the rescattering potential we use the Yukawa potential V(r)=-(a/r)exp(-λr) with a=1 a.u. and λ=1 a.u. Notice that we take different potentials for binding and rescattering; for a justification see Refs. [36, 37, 38].

Figure 1 shows two spectra in the direction of the laser polarization, that is, for θ=0° and for θ=180°, for a four-cycle pulse. The absolute phase is ϕ=0°. In this case, the field E(t) is symmetric and the vector potential A(t)-A(T_p ) is antisymmetric with respect to t=T_p /2. This implies that the spectrum of the direct electrons obeys left-right symmetry, as can easily be shown with the help of the matrix element $M_{\mathbf{p}}^{\left(0\right)}$. This is not so for the rescattered electrons, whose spectrum is governed by $M_{\mathbf{p}}^{\left(1\right)}$, as it is determined by two times, the ionization time t and the rescattering time t′. This is borne out in Fig. 1: for low energies, the small difference between the spectra in the two directions that is visible is due to the contribution of the rescattered electrons. In contrast, in the rescattering regime where the direct electrons do not contribute, the two spectra are completely different.

In what follows, we will concentrate on this high-energy regime where the contribution of the direct electrons is not dominant or even entirely negligible. For such energies, the left-right asymmetry is particularly pronounced. Moreover, the strong-field approximation generally furnishes a better description of the high-energy (rescattered) electrons than of the low-energy (direct) ones. The reason is its neglecting the Coulomb potential in the Volkov state. Two direct electrons that both contribute to the same final momentum but are set free at different times are differently affected by the Coulomb field: one may revisit the ion and be subject to refocusing while the other is not. This is different for two electrons rescattered into the same final state: their ionization times are very close, and so is the effect of the Coulomb field on their orbits.

 

Fig. 2. The same as Fig. 1, but for nine different absolute phases between ϕ=0° and ϕ=180°. The uppermost left panel with ϕ=0° shows the same spectra as Fig. 1. The spectra for θ=0° are designated by black lines, those for θ=180° by thin red (gray) lines.

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In Fig. 2 we scan the absolute phase over nine values between ϕ=0° and ϕ=180°, for otherwise the same parameters as in Fig. 1. Absolute phases -180°≤ϕ≤0° yield the same spectra except that the left and the right direction are interchanged.

We focus our attention on two features of the associated spectra: (1) for phases 0°≤ϕ≲110° (and again for 310°≲ϕ≤360°), emission into the direction θ=0° is stronger than into θ=180° for sufficiently high energies, while for 130°≲ϕ≲290° more electrons go in the direction θ=180°. Insomuch as there is an experimental lower detection threshold, detectors at θ=0° and at θ=180° will collect roughly the same number of high-energy electrons for absolute phases in the ranges 110°≲ϕ≲130° and 290°≲ϕ≲310°. (2) For all phases ϕ outside the afore-mentioned two ranges, there exists a well-defined cross-over energy where left-hand emission becomes more prominent than right-hand emission (or vice versa). In the plots, this point moves from E _p=26 eV for ϕ=135° via E _p=31 eV for ϕ=180° and ϕ=0° to E _p=43 eV for ϕ=90°. These features could be employed to establish the value of the absolute phase in an experiment by comparison with a phase scan of calculated left-versus-right energy spectra such as Fig. 2.

Many of the properties just discussed can be understood from a simple classical model [35] (see also [8, 31, 39]), which can easily be generalized to few-cycle pulses. We suppose that the electron is “born” at the origin, that is, at the position of the ion, with zero velocity at some time t ∈[0,T_p ]. Solving Newton’s equation of motion v̇(t)=-E(t) in the presence of only the laser field, we obtain the electron momentum at some arbitrary later time t ₁>t:

If at some time t′ the electron returns to the origin, i.e., if

where α(t)=∫ ^t dt′A(t′), then it may elastically rescatter off its parent ion, so that its kinetic energy immediately after the rescattering is v ²(t′₊)/2=v ²(t′_)/2 (where t′_±≡t′±0⁺). Thereafter, the electron moves in the laser field only, up to the time T_p , when the laser pulse has passed through. For t>T_p the electron does not change its momentum anymore. The final electron energy, registered at the detector, is

with

where v̂(t′₊) is the unit vector in the direction of the electron momentum immediately after the rescattering. In this paper, we only consider the case when v̂(t′₊)·ê=cosθ=±1.

Figure 3 exhibits results of a numerical solution of Eqs. (16)–(18) for the parameters underlyingFig. 2 and an absolute phase of ϕ=0°, in the two directions θ=0° and θ=180°. It corresponds to Fig. 1 or to the left uppermost panel of Fig. 2. Figure 3 allows one to find the energy E _p at the detector of an electron that is born at the ionization time t. Such an electron may return not at all, once, or several times to its parent ion. Each scenario leads to a different energy at the detector. This is why in the figure, in general, more than one energy corresponds to each ionization time.

Of the solutions displayed in Fig. 3, owing to the highly nonlinear field dependence of the ionization process, only those have practical significance, whose ionization time t corresponds to a reasonably high electric field, say |E(t)|≳0.6E ₀. Moreover, high rescattering energies E _p are only reached (i) if the electron rescatters close to a zero of the electric field and (ii) if the subsequent electric field still is sufficiently strong. This last condition marginalizes all solutions with ionization times much later than 2 optical cycles (o.c.). All of these conditions together have the consequence that high-energy electrons are only generated in one or two ultrashort bursts. In a very similar fashion, high-order harmonic generation by few-cycle pulses generates very few ultrashort bursts of uv radiation. In omparison, the number of high-energy-electron bursts in high-order ATI is smaller, since the above condition (ii) is absent for HHG.

 

Fig. 3. Solutions of the classical rescattering model formulated in Eqs. (16)–(18) for the parameters of Fig. 1, absolute phase ϕ=0°, and θ=0° (upper panel) and θ=180° (lower panel). The various black curves specify the energy E _p at the detector as a function of the ionization time t. The dashed magenta curve represents the electric field given by Eq. (11) with the scale given on the right-hand ordinate. The solid blue curves that are identical in both panels, being symmetric with respect to t=2 o.c. and having two maxima at E _p=7.1 eV refer to direct ionization. They are trivial solutions of Eq. (16) for t=t′.

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Fig. 4. Rescattering times t′ as a function of the ionization time t for the solutions of the classical model, given by Eqs. (16)–(18). The parameters are those of Fig. 1 and the absolute phase is ϕ=0°. Rescattering times that give rise to E _p>10 eV, E _p>20 eV, and E _p>30 eV, are marked in red (medium gray), blue (dark gray), and cyan (light gray), respectively.

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We shall discuss the two directions θ=0° and θ=180° separately.

(1) θ=0°:

The upper panel of Fig. 3 shows three pairs of solutions with energies E _p≳15 eV. Of these, the pair of solutions with ionization times below 0.5 o.c. plays absolutely no role, since for these times the electric field is still very weak. The second pair of solutions with ionization times t≈1 o.c. has its cutoff at E _p=27 eV. No qualitative effect of this cutoff is visible in the energy spectrum of Fig. 1 (or the uppermost left panel of Fig. 2).We conclude that this is due to the still rather low value (0.45E ₀) of the electric field at the ionization time. Hence, the energy spectrum is dominated by the remaining pair of solutions with cutoff E _p=35.5 eV, which start at t≈1.5 o.c. The corresponding rescattering times t′ can be read off from Fig. 4 and found to be in the interval 2.1 o.c.≤t′≤2.35 o.c. for E _p>20 eV. Consequently, these electrons are emitted in one single burst with a temporal width of 0.68 fs.

The sharp dip visible in Fig. 1 at the energy 28.5 eV is due to a destructive interference of the two solutions of the pair. Such dips are an ubiquitous feature of high-order ATI spectra; see, e.g. Ref. [31]. The dip will be hard to see in an experiment: Its position strongly depends on the intensity, so that it will not normally survive focal averaging. In general, in viewing the spectra of Fig. 2 one should mentally smooth over such dips.

The one pair of solutions that accounts for the high-energy spectrum is not very different from the corresponding solution in the case of an infinitely long pulse with constant envelope. In particular, their cutoffs are practically identical. This is the origin of the close agreement of the associated spectra for high energies, which can be seen in Fig. 1. Below E _p=30 eV, the two spectra begin to separate. This is probably due to the increasing effect of the above-mentioned pair of solutions with their cutoff at 27 eV.

(2) θ=180°:

Here we encounter a more involved situation even though the spectrum, which continuously drops after its maximum at 22 eV, looks simpler than for θ=0°. In the lower panel of Fig. 3, we can identify four pairs of solutions that might be significant, which we characterize by their approximate ionization times and cutoffs: (t(o.c.), E _max(eV)). They are (1, 31.1), (1, 15.7), (1.5, 12.6), and (2, 21.4). The last two have the highest |E(t)|, but the first has the highest cutoff. For 24 eV≲E _p≲32 eV, the solution (2, 21.4) strongly dominates the spectrum, but owing to its cutoff at 21.4 eV its magnitude continuously decreases for increasing energy. Therefore, with increasing energy, the quantitatively smaller contribution of (1, 31.1) becomes more and more competitive. Indeed, for E _p≳36 eV, the interference of the contributions of these two pairs of solutions, whose ionization and rescattering times lie one cycle apart, produces an ATI spectrum with well developed peaks. This is in contrast to the completely smooth spectrum for θ=0°, which is generated by just one pair of solutions. For energies below about 25 eV, the spectrum again displays ATI peaks. These are probably due to interference of the dominant solution (2, 21.4) with (1.5, 12.6).

We just found that electron emission at just one well-defined instant of time generates a smooth spectrum whereas emission at two times causes the appearance of peaks whose energy separation reflects the temporal distance of the emission events. This is the temporal analog of two-slit interference. The same phenomenon was observed in calculations of direct ATI by a few-cycle pulse with circular polarization [22, 23]. In the closely related case of high-order harmonic generation, this effect - the presence or absence of interference peaks - was exploited in the determination (modulo π) of the absolute phase [15]. The analysis of the spectrum, viz. the energy separation of the peaks, the contrast, and the variation of the peak separation as a function of energy, allow us to reconstruct an image of electron emission in the time domain, in other words, a movie with sub-femtosecond resolution.

For example, Figs. 3 and 4 show that both the relevant ionization times and the rescattering times each lie about one cycle apart. This is why the spacing between the peaks is roughly the energy of one photon of the carrier wave. Closer inspection of the spectra of Fig. 2 reveals, however, small deviations from this pattern, which can be traced to differences of the corresponding rescattering times deviating from the period T. Also, the positions of the peaks depend on the value of the absolute phase.

For energies above 40 eV, the spectrum of Fig. 1 for θ=180° shows peak structures with an energy separation around h̄ω/2. Better examples of the same effect can be found in Fig. 2, in particular for ϕ=112.5° and θ=0°. This spectrum exhibits high-contrast peaks with a precise separation of h̄ω for E _p≲30 eV followed by a sequence of peaks with an approximate spacing of h̄ω/2 and lesser but still good contrast for E _p≳30 eV. The h̄ω/2 spacing requires contributions with rescattering times 2T apart, in analogy to three-slit interference. A classical analysis analogous to Figs. 3 and 4 yields several contributions all of which have very low electric fields at the time of ionization. We have not been able to identify those that are responsible for the h̄ω/2 peak structure. However, it seems worthwhile to notice that few-cycle pulses are capable of producing peaks with a separation of one half of the carrier frequency.

For high-order ATI to make a useful phase meter, it must be applicable for a wide variety of few-cycle-pulse specifications, such as pulse duration, peak intensity, and various shapes of the pulse envelope. As for the intensity, we have checked that the electron spectrum does not change very much up to about 10¹⁴ Wcm^-2, except that its cutoff moves to higher and higher energy, roughly proportional to 10U_p , cf. Fig. 1. For higher intensity, the fixed-intensity long-pulse spectrum develops a lot of structure, mostly due to beating between the contributions of the shortest two orbits [31]. For few-cycle pulses, we expect a much less violent beating pattern such that the left-right asymmetry should persist. As for the pulse duration, the left-right asymmetry quickly decreases with increasing pulse length. For intensities below 10¹⁴ Wcm^-2, we found that it will no longer be useful to measure the absolute phase when T_p ≳7T. We do not expect a pronounced effect of the detailed pulse shape on the energy spectrum and its asymmetry.

In summary, we have shown that high-order above-threshold ionization by a few-cycle pulse is very sensitive to its absolute phase. In particular, we suggest that comparing the two high-energy ATI spectra along the direction of the laser polarization (left and right) permits a rather precise determination of the value of the absolute phase. High-order ATI has several intrinsic advantages over low-order ATI or high-order harmonic generation for this purpose. Compared with the former, the left-right asymmetry is stronger and its analysis is simpler owing to the reduced significance of the Coulomb field. Regarding the latter, the asymmetry effects are not folded with propagation effects, and they allow determination of the absolute phase modulo 2π.

D.B.M. gratefully acknowledges support by the VolkswagenStiftung.