#### 6.1. Measurements of Femtosecond-LAES Singals

In 2010, we succeeded in recording the LAES signals induced by femtosecond laser pulses, where 1 keV electrons were scattered by Xe atoms in a femtosecond intense laser field (Δ

t = 200 fs,

λ = 800 nm, and

I = 1.8 × 10

^{12} W/cm

^{2}) [

9].

Figure 2 shows the raw images of the scattered electrons recorded in the femtosecond-LAES experiments. The net exposure time was around 83 h for each image.

Figure 2a shows the electron scattering signals obtained when laser pulses were introduced at the timing of the electron scattering by Xe atoms. The laser polarization was set as “vertical” (i.e., perpendicular to the electron beam axis). The intense signals that formed an arcuate line seen at the central area in

Figure 2a is

n = 0 scattering signals, and no other features could be recognized in

Figure 2a.

Figure 2b is an amplified image of

Figure 2a, which was obtained after adjusting the range of the signal intensity so that the weak LAES signals became visible. In

Figure 2b, weak arcuate lines indicated by the white arrows can be seen on both sides of the central arcuate line. On the other hand, such structures could not be seen in the background signals (

Figure 2c), which were obtained when the temporal delay of the electron pulse with respect to the laser pulse was set to be +100 ps.

The observed small difference between

Figure 2b,c becomes clear in the electron energy spectra, which were obtained through the integration of the signal of each pixel over the scattering angles along the arcuate isoenergetic coordinate. In the integration, the signals in the region of

y > 105 pixels in

Figure 2 were excluded from the analysis because the contributions of the stray electrons were significantly large in this region. The blue filled circles and the black filed squares in

Figure 3 show the energy spectra obtained from

Figure 2b,c, respectively. Unambiguous increases in the signal intensity appear at the kinetic energy shifts of ±

ħω (i.e., ±1.56 eV) in

Figure 3 (blue filled circles). This is clear experimental evidence of the

n = ± 1 transitions in the LAES process.

The blue filled circles in

Figure 4 represent the LAES signals obtained by subtracting the background signals from the scattering signals obtained with the laser field in

Figure 3. Both of the signals at the energies of ±

ħω can be recognized as distinct peaks, and the intensities of these peaks are around 3 × 10

^{−4} relative to the central

n = 0 scattering peak.

In order to confirm our assignment, we estimated the relative intensities of the LAES signals by a numerical simulation based on Equation (3). In the simulation of the observed LAES signals, denoted by

w^{(n)}(

E_{i};

**s**), the differential cross section in Equation (3) was averaged over the spatiotemporal distribution of the three beams (i.e., the electron beam, the laser beam, and the atomic beam). Because the d

σ_{el}(

Ẽ_{i};

**s**)/d

Ω ≈ d

σ_{el}(

E_{i};

**s**)/d

Ω is considered to be independent of the laser field, the

w^{(n)}(

E_{i};

**s**), can be factorized into two parts:

where

In Equation (11),

**ρ**(

**r**) is a density of the sample atoms and

j(

**r**,

t) is an electron flux density. The spatiotemporal distributions of

ρ(

**r**) and

j(

**r**,

t) can be determined experimentally [

63]. The vectorial quiver radius,

**α**_{0}(

**r**,

t), becomes a function of

**r** and

t because of the spatiotemporal distribution of

**F**, and can also be derived experimentally, as described in [

63] by using the laser field parameters such as the pulse energy, the temporal shape of the pulse envelope, and the spatial profile at the scattering point. Therefore,

w^{(n)}(

E_{i};

**s**) in Equation (10) can be calculated using

G_{n}(

E_{i};

**s**) obtained from Equation (11) using the differential cross section (d

σ_{el}(

E_{i};

**s**)/d

Ω) in the NIST database [

16]. The results of the simulation were plotted with a green solid line in

Figure 4. The calculated LAES signal intensities relative to the

n = 0 scattering signal intensities showed good agreement with the experimental results.

When the laser polarization vector is set to be “horizontal” (i.e., parallel to the direction of the incident electron beam), the factor of

**α**_{0}·

**s** in Equation (2) becomes close to zero because the polarization vector is nearly perpendicular to the scattering vector,

**s**, for the forward scattering of the high-energy electrons. Consequently, the LAES signal intensities are suppressed significantly, except for the

n = 0 scattering signal intensity. This polarization dependence provides further verification of our measurements of the LAES signals of

n = ±1. In

Figure 4, an energy spectrum with the horizontally polarized laser field is plotted with red filled triangles. In contrast to the corresponding spectrum obtained using the vertically polarized laser field, no distinguishable peaks are observed. This is consistent with the corresponding numerical calculation, showing that relative intensities for the

n = ±1 transitions were nearly zero (7 × 10

^{−6}).

The blue filled circles in

Figure 5 show the angular distribution of the background-subtracted LAES signals for the

n = +1 transition recorded using the vertically polarized laser field. The green solid line shows the results of the numerical calculations with KWA. The calculated angular distribution is in good agreement with the experimental angular distribution. The angular distribution of the

n = −1 transition is basically the same as that of the

n = +1 transition, and also shows agreement with the results of the numerical calculations.

After the first observation of the LAES signals induced by 200 fs laser pulses [

9], we report on the LAES experiments with laser pulses whose durations were 50 fs in 2011 [

63], 520 fs in 2014 [

58], 970 fs in 2015 [

10], and 100 fs in 2017 [

66]. In 2011, deHarak et al. [

67] reported the results of the measurements of the LAES processes occurring in the laser field whose intensity is of the order of 10

^{9} W/cm

^{2} generated using the Nd:YAG laser (Δ

t = 6 ns,

λ = 1064 nm). They also investigated the dependences of the LAES processes on the laser polarization [

68,

69] and target atomic species [

70], and determined the angular distribution of the scattered electrons [

71]. More recently, they reported preliminary results of laser-assisted inelastic electron-argon scattering induced by near-infrared nanosecond laser pulses [

72].

#### 6.2. Light-Dressing Effect in Laser-Assisted Elastic Electron Scattering

As discussed in

Section 2, the Kroll–Watson formula of the differential cross section of a LAES process with

n-photon absorption was derived under the assumptions that both of “the laser-electron interaction” and “the electron-atom interaction” are treated in a non-perturbative manner while “the laser-atom interaction” is neglected. However, when the laser field intensity becomes stronger, the laser-atom interaction cannot be neglected, and the spatiotemporal evolution of the electron distribution within the target atom influences the energy distribution and the angular distribution of the LAES signals. In 1984, Byron and Joachain [

23] calculated the differential cross section of LAES of a light-dressed hydrogen atom by considering the laser-atom interaction using the first-order perturbation theory, and predicted that an intense peak structure would appear at the small scattering angles of the LAES processes of

n = ±1. Their study showed that the small-angle LAES signals carries valuable information on the electron density distribution in the target atom influenced by an external laser field. In spite of this theoretical prediction, the light-dressing effect in LAES has not been identified experimentally for more than 30 years because the light-field intensities (<10

^{9} W/cm

^{2}) in the early LAES experiments, where mid-infrared cw- or pulsed-CO

_{2} lasers that were employed [

6,

7,

8], were not sufficiently high. In 2015, we reported the observation of LAES signals appearing through this light-dressing effect by measuring the scattering of 1 keV electrons by Xe in an intense laser field (

λ = 800 nm,

I = 1.5 × 10

^{12} W/cm

^{2}, Δ

t = 970 fs) [

10] and discussed the possibilities of probing the ultrafast evolution of electron distributions in atoms and molecules in intense laser fields by the LAES measurement.

Figure 6 shows the raw image of the scattered electrons when 1 keV electrons were scattered by Xe atoms in an intense laser field (

λ = 800 nm,

I = 1.5 × 10

^{12} W/cm

^{2}, Δ

t = 970 fs) [

10]. The intense arcuate line, seen in the center of the image, is the elastic scattering signal without energy shifts, and the weaker side arcuate lines are the LAES signals of

n = ±1 and ±2. Blue filled circles in

Figure 7a are the recorded energy spectrum of the LAES signals obtained after the integration of the electron signals over the angular range of 0.1° ≤

θ ≤ 10°. The LAES signals of

n = ±1 and

n = ±2 can clearly be recognized in the spectrum at the energy shifts of ±1.55 eV and ±3.10 eV, respectively, and the simulated energy spectrum based on the Kroll–Watson theory [

15] (green solid line) was in good agreement with the experimental data. In contrast, as shown in

Figure 7b, the intensity of the experimental energy spectrum of the

n = ±1 LAES signals within the small angular range of 0.1° ≤

θ ≤ 0.5° was one order of magnitude larger than the intensity of the corresponding energy spectrum obtained by the simulation based on the Kroll–Watson theory [

15].

Figure 8a,b show the angular distributions of the LAES signals of

n = +1 and

n = −1, respectively. In both of the angular distributions, a peak profile was recognized in the small scattering angle range (<0.5 ), which was not reproduced by the simulation based on the Kroll–Watson theory (green solid lines), as expected from the theoretical study on the light-dressing effect in the target atoms [

23]. In order to confirm that the observed peak profile originated from the light-dressing effect in Xe atoms, we performed a numerical simulation based on Zon’s model [

21].

In Zon’s model, the laser-atom interaction is treated as a polarization of an electron cloud in a target atom, creating a laser-induced dipole moment expressed as

where

a(

ω) is the frequency-dependent polarizability of the target atom, which can be described by the Unsöld expression [

73] expressed as

where

ω_{res} is the resonance frequency of the target atom and

ω_{res} >>

ω is assumed. Because the scattering process is affected by the interaction potential between the charge of the incident electron and the laser-induced dipole of the polarized atom, the Hamiltonian of the system is expressed as

Under the first Born approximation, the differential cross section of the LAES process can be derived analytically as

where

f_{FBA}(

**s**) is the scattering amplitude without laser fields expressed as

in the first Born approximation. The first term in the squared modulus in Equation (15) represents the scattering by the field-free potential,

V(

**r**), and the second term in the squared modulus represents the laser-induced polarization of the target atom. If the second term in Equation (15) is omitted, Equation (15) becomes identical to Equation (1).

As shown in

Figure 8a,b, the angular distributions of the LAES signals of

n = ±1 simulated using Zon’s model (red solid lines) qualitatively reproduce the observed peak profile at the small scattering angle range (<0.5°), showing that the observed peak profile originates from the light-dressing in Xe atoms induced by the intense laser field. From further theoretical analyses of the deviation of the observed angular distributions from the simulations based on Zon’s model, we will be able to extract information on the ultrafast evolution of the electron density distribution in the light-dressed Xe atoms.

#### 6.3. High-Order LAES Processes and Assignment of Collision Times

When LAES processes are induced by multi-cycle laser electric fields expressed as **F**_{0}cosωt, the mechanism of the LAES process can be described in terms of a dimensionless parameter, ξ, defined as

When |

ξ| >> |Δ

E| = |

nħω|, the LAES processes can be described in terms of the scattering trajectories of classical mechanical electrons in a laser field. Consequently, the collision time,

t_{c}, (i.e., the time when the electron-atom collision occurs) can be expressed as

where

m is an arbitrary integer and

T is the laser field period. Equation (10) shows that the collision time within the optical cycle can be estimated from the energy shift and the deflection angle of scattered electrons.

Figure 9a shows the energy-resolved angular distribution obtained from the measurements of LAES by Xe atoms in a multi-cycle near-infrared laser field (Δ

t = 100 fs,

λ = 800 nm,

I = 8.8 × 10

^{12} W/cm

^{2}) using a 1 keV electron beam [

66]. If it is assumed that the scattering occurs around the peak field intensity, the collision times can be estimated by Equation (18). For example, the collision times for the LAES signals at the scattering angles of ±11.8°, which are expressed as the square areas enclosed by the broken lines in

Figure 9a, are shown by the arrows in

Figure 9b for the respective harmonic orders,

n = Δ

E/(

ħω), showing that slight differences in the collision times of the order of 10 attoseconds can be discriminated.