Results

In Fig. 2.7, we show the photoelectron spectra of the TDSE as well as two versions of SFA results of hydrogen atom under a laser pulse with 5 cycles, wavelength 800 nm, peak intensity 1× 10¹⁴ W/cm², and ϕ = 0. Left sub-figure is from Ref. [59] using R-space TDSE and KFR theory, while right sub-figure is our results using P-space TDSE and Lewenstein model. ”SFA1” and ”SFA2” in the left sub-figure means the first and second-order KFR theory respectively. ”SUM” means the summation of ”SFA1” and ”SFA2”.

”SFA1” in the right sub-figure means the results of solving Eq. (2.74) of Lewenstein

Figure 2.7: Photoelectron spectra of the TDSE as well as two versions of SFA results of a hydrogen atom with a linear polarized laser pulse with 5 cycles, wavelength 800 nm, peak intensity 1× 10¹⁴ W/cm², and ϕ = 0. Left sub-figure is from Ref. [59] using R-space TDSE and KFR theory, ”SFA1” and ”SFA2” mean the first and second order of the KFR theory, and ”SUM” is the summation of ”SFA1” and ”SFA2”. Right sub-figure is our results using P-space TDSE and Lewenstein model, ”SFA1” is the result of Eq. (2.70) without scattering term and corresponding to ”SFA1” of the KFR theory, and ”SFA all order” contain Coulomb scattering term and corresponding to the summation of all order of the KFR theory.

model without the scattering term⟨p|V (r)|p^′⟩ which is just the SFA1 in the KFR theory.

”SFA all order” is the result of solving Eq. (2.74). Since we don’t do any expansion, this results is corresponding the summation of all order of KFR theory. ”SUM” (blue-dotted line) in the left sub-figure is consistent with ”SFA all order” (blue-dotted line) in the right sub-figure except after electron energy greater then 12U_p. The probability drop rapidly in

”SUM” while there exist another plateau in ”SFA all order”. Just as ”SFA2” contributes a plateau at 3-9Up, the plateau from 13Up in ”SFA all order” can be understood as the contribution from the third and higher order terms corresponding to the KFR theory. After the third-order terms in the KFR theory, it is very difficult to evaluate due to at least 9-fold integration. Our P-space TDSE result consistent with ”SFA all order” (right sub-figure) but the R-space TDSE result exhibit a rapidly drop as ”SUM” after 12U_p(left sub-figure).

We have known that the drop in ”SUM” is because only the first two orders of KFR theory

are carried out. How about the drop in the R-space TDSE result? In the R-space TDSE calculation, a finite box in the coordinate space will be set. And the wave function will be filtered out when it reaches the box edge. More higher energy the electron possesses, more larger distant it can travel and hence be filtered out at the box edge. This should be the reason why the photoelectron spectra from R-space TDSE drop rapidly after 13U_p.

We note that SFA only give a qualitative agreement of photoelectron spectra. ”SFA1”

describe the tunneling process and contribute to the low energy part of photoelectron spec-tra (below 2U_p). Coulomb scattering of the ionized electron and ion core contribute to the higher energy plateau. However, the magnitude is smaller than the TDSE result by about 2 orders. The magnitude can be improved by forcing the final state to be orthonormal to the initial state which is called orthonormalized strong-field approximation (OSFA) [52, 68].

In Fig. 2.8, we present the two-dimensional momentum distribution of TDSE and SFA1 (Since Coulomb scattering wouldn’t affect low energy distribution in SFA, we only com-pare SFA1 to TDSE.). Up row are plotted according to Eq. (2.29) which emphasize electron distribution in the direction perpendicular to polarization axis while bottom row plotted according to Eq. (2.30) which emphasize that along the polarization axis. p_||and p_perp denote the momentum parallel and perpendicular to polarization axis respectively.

In Fig. 2.9, we present the energy-angular momentum distribution of TDSE and SFA1 which shows each angular momentum contribution to the photoelectron spectra. From Fig. 2.8 and 2.9, we find SFA give a totally different prediction from TDSE at low energy.

Further, the breakdown of SFA is also pointed out in the recent interested mid-infrared regime [22, 23]. It is believed that the low energy electron would be affected by atomic potential significantly. The SFA can be improved by adding the Coulomb correction to Volkov wave function called Coulomb-Volkov wave function [69, 70].

Figure 2.8: Two-dimensional momentum distribution of TDSE and SFA1 of the same system as Fig. 2.7. Up row are plotted according to Eq. (2.29) which emphasize electron distribution in the direction perpendicular to polarization axis while bottom row plotted according to Eq. (2.30)which emphasize that along the polarization axis. p_|| and p_⊥ denote the momentum parallel and perpendicular to polarization axis, respectively.

Figure 2.9: Energy-angular momentum distribution of TDSE and SFA1 of the same sys-tem as Fig. 2.7 which shows each angular momentum contribution to the photoelectron spectra.

Chapter 3

Strong-field Ionization of a Lithium Atom

In the section 3.1, we compare our calculation results to experimental results for cali-bration. To compare to experimental results, we need to average the signals of ionized electron from different atoms in the different region of the laser-focal volume (hence feel different laser intensity). Since the volume-averaged results contain so many signals, it is not convenient to analyze the underlying mechanism inside. In the section 3.2, we present the results at a single peak intensity (or without laser-focal volume average). In the sec-tion 3.3, we will discuss the multiphoton ionizasec-tion (MPI) at relatively small intensities, including nonresonant multiphoton ionization (NRMPI), dynamical resonant multipho-ton ionization (DRMPI), and ponderomotive shift. In the section 3.4, we will discuss the generation of Rydberg states in the lithium atom. In the section 3.5, we will discuss the fan structure in the direction perpendicular to polarization axis.

3.1 Compare with experimental results

Before comparing to experimental results [51], we need to discuss laser-focal volume av-erage first. The focus of a actual laser beam is not a spot but has a volume. When we talk about the laser peak intensity I₀of a laser beam, we always mean the value at the cental of the laser-focal volume. And, laser intensity decays gradually outward. The spatial distri-bution of the laser intensity can be formulated as Lorentzian in the propagation direction (z) and Gaussian in the transverse direction (ρ) [71, 72]:

I(r, z) = I₀w²₀

where w₀ is the radius of the focal spot and z_Ris the Rayleigh range of the focus. In Fig.

3.1, we show the Iso-intensity surface plot of a laser-focal volume. We assume the gas

Figure 3.1: Iso-intensity surface plot of a laser-focal volume, z is the propagation direc-tion. This figure is from Ref. [72].

volume is filled over the laser-focal volume, then the ejected electron signals are the sum of electrons ionized from atom at different intensity region of the laser-focal volume. For a peak intensity I₀, the ejected electron signals with momentum p is given by [71]:

S(P, I₀) = D

where D is the density of the target atoms, PI(P) is the ionization probability for a par-ticular intensity I, and⁽−^∂V_∂I⁾dI is the volume element between I and I+dI iso-intensity surface. The volume element for the laser beam of Eq. (3.1) is given as

−∂V

The trapezoidal rule are used for the integration over intensity.

Now, we return to compare with experiment results. The experiment was carried out with a linear polarized laser pulse of wavelength 785nm, FWHM 30fs, and peak intensity I0 ranging form 4× 10¹¹W/cm² (γ = 11.6) to 7× 10¹³W/cm²(γ = 0.8) which is from multiphoton ionization regime (γ > 1) to tunneling ionization regime (γ < 1).

We adopt the following model potential for a lithium atom [73].

V (r) =−1

r −a₁e^−a2r+ a₃re^−a4r

r (3.5)

where a₁ = 2, a₂ = 3.395, a₃ = 3.212, and a₃ = 3.207. In our calculation, we use 1024 grid points, lmax = 14, ∆t = 0.2 (a.u.) and the laser-focal volume average is carried out as: For peak intensity I0 = 4× 10¹¹W/cm²and I0 = 8× 11¹¹W/cm², we integrate from 5%× I0 to I₀ and ∆I = 0.2× 10¹¹W/cm². For peak intensity I₀ = 7× 10¹³W/cm², we integrate from 2.8%× I0to I₀ and ∆I = 0.1× 10¹³W/cm². All others are integrated from 5%× I0 to I₀and ∆I = 0.1× 10¹²W/cm².

In Fig. 3.2 and 3.3, we show the two-dimensional momentum distribution of our results (left column) and experiment results (left column) for seven different peak inten-sity ranging from multiphoton ionization regime (γ > 1) to tunneling ionization regime (γ < 1). In Fig. 3.4 and 3.5, we show the photoelectron spectra with laser parameters corresponding to Fig. 3.2 and 3.3. Our results agree well with experiment results.

Figure 3.2: Two-dimensional momentum distribution of a lithium atom with a lin-ear polarized laser pulse with FWHM 30fs, wavelength 785nm, and peak intensity 4× 10¹¹W/cm², 8× 10¹¹W/cm², 2× 10¹²W/cm², and 4 × 10¹²W/cm² (from up to bottom). Left column is our results and right column is experimental results from Ref.

[51]. p_|| and p_⊥ denote the momentum parallel and perpendicular to polarization axis, respectively.

Figure 3.3: The same as Fig. 3.2 but now laser peak intensity 8× 10¹²W/cm², 2× 10¹³W/cm²,and 7 × 10¹³W/cm² (from up to bottom). Left column is our results and right column is experimental results from Ref. [51]. p_|| and p_⊥ denote the momentum parallel and perpendicular to polarization axis, respectively.

Figure 3.4: Photoelectron spectra of a lithium atom with a linear polarized laser pulse with FWHM 30fs, wavelength 785nm, and peak intensity 4× 10¹¹W/cm², 8× 10¹¹W/cm², 2× 10¹²W/cm², and 4× 10¹²W/cm² (from up to bottom). Left column is our results and right column is experimental results from Ref. [51].

Figure 3.5: The same as Fig. 3.4 but now laser peak intensity 8× 10¹²W/cm², 2× 10¹³W/cm²,and 7 × 10¹³W/cm² (from up to bottom). Left column is our results and right column is experimental results from Ref. [51].