Two-photon photoionization for second-order autocorrelation

and energy transfer processes. There are several ways to model the temporal evolution of a system upon excitation. Typically, the system’s phase memory is rather short. If it is much shorter than the excitation laser pulse, then the interaction with the electric field can be considered incoherent. In this case the time evolution can be described by rate equations in a form:

d

dt Nn DA.t / €nNn; (2.3.19) whereNn is the population of the statejni, A.t / is the pump rate and€n is the popula-tion decay rate. The time-dependent pump rate contains the electric field envelope and accounts for photon-induced transitions from the lower lying states. €nis the rate of the population loss as a result of various processes, e.g. inelastic collisions and spontaneous emission.

If the phase memory of the system is longer or comparable to the excitation pulse, the rate equation model for populations cannot give an accurate prediction of the system response. The interaction with light will depend on the relative phases of the excited electronic wavepackets evolving in time and the laser field in each particular moment, i.e. it becomes coherent. There is more than one way to model coherent processes.

The examples are solving the Schrödinger equation directly or using the density matrix formalism. The latter, in a form of Optical-Bloch equations, is an attractive approach for its lower computational costs. The detailed description of this method is given in AppendixB. In the following it will be briefly summarized. A quantum system withn states can be described by annndensity matrix. The elements of the density matrixO evolve in time according to the Liouville-von-Neumann equation [83]:

d

dt mn D i

„

hH ;O Oi

mn €mnmn: (2.3.20)

The diagonal elementsnnrepresent the populations of the corresponding states. The off-diagonal elementsmn are called coherences and are related to coherent superpositions of states. In contrast to the rate equations, the populationsnnare coupled to the electric field not directly but via the coherencesmn. Therefore, the probability of the atom to undergo a transition depends on the phase difference between the incident field and the oscillating average dipole momenth Odireflected in_mn[84].

As an example it is instructive to examine two-photon ionization of an electronic

ℏω ℏω

2ℏω IP

(a)

|2〉

|1〉

|0〉 (b)

T₁₂

T₀₁ T₀₂

T₁

Figure 2.3.9:Resonant two-photon ionization scheme of a three-level system with linear (a) and nonlinear (b) ionization pathways, with marked coherence and population decay timesT_mn and T₁accordingly.

three-level system by a sequence of two short coherent laser pulses with photon energy

„!L. The system contains the occupied ground statej0i, unoccupied transient statej1i and the unoccupied final statej2i located in the ionization continuum. The neighboring states are separated by the energy gap equal to„!L (Fig. 2.3.9). The ionization of this system may occur via two pathways: in a direct transitionj0i ^2„! j^! 2ior in a sequential transitionj0i ^„! j^! 1i ^„! j^! 2icreating a transient population on levelj1i. The two photons required for ionization may be absorbed either from the same pulse (pumpor probe) or from different pulses (pumpandprobe). The light field will couple all three energy levels and besides creating the transient population, it will induce polarizations (coherences) between the states. The coherences will oscillate at frequencies corresponding to the energy gaps between the levels. In the discussed resonant case these will be!L for one-photon coupling (j0iandj1i,j1iandj2i) and2!Lfor two-photon coupling (j0iandj2i).

The population and coherences will decay with characteristic timesT₁,T₀₁,T₁₂andT₀₂. If the excitation pulses are coherent, the population of the final statej2i(ionization yield) will oscillate as a function of the delay due to a combination of optical and quantum interferences. If the coherence times are long enough, the quantum interference will be manifested as oscillations in the ionization yield in the delay region free from the optical interference caused by the pulse overlap. The illustration is given in Fig. 2.3.10. The decaying population in statej1i manifests itself as broadening of the pump-probe trace (exponential wings) comparing to the FRIAC of the laser pulse.

delay [OC]

-20 -15 -10 -5 0 5 10 15 20

amplitute[a.u.]

0 1 2 3 4 5 6 7 8

FRIAC

delay [OC]

-20 -15 -10 -5 0 5 10 15 20

amplitute[a.u.]

0 1 2 3 4 5 6 7 8

T₀₁=30 fs

delay [OC]

-20 -15 -10 -5 0 5 10 15 20

amplitute[a.u.]

0 1 2 3 4 5 6 7 8

T1=30 fs

delay [OC]

-20 -15 -10 -5 0 5 10 15 20

amplitute[a.u.]

0 1 2 3 4 5 6 7 8

T02=30 fs

Figure 2.3.10: FRIAC of 800 nm, 15 fs FWHM Gaussian pulse and solutions of OBE for the three-level system ionized by this pulse for different values of the relaxation constantsT₁, T₀₁ andT₀₂. The thin gray lines mark the upper and lower envelopes of the FRIAC function. The pump-probe delay is shown in optical cycles=c.

Information on the population and phase decay rates can be obtained from the pump-probe delay scan if the temporal structure of the pulse is known. This can be done by fitting the solution of OBE to the measured data, or using a simpler method suggested by Nessleret al[85]. In short, the measured pump-probe delay scan is decomposed into the three frequency envelopes centered at zero,!_Land 2!_L in the Fourier domain, and then the envelopes are fitted with functions that are convolutions of the corresponding envelopes of the laser pulse FRIAC with symmetric exponentialse ^jj=Tmndescribing the phase and the population decays. The characteristic lifetimesTmn are found as free fit parameters. It must be noted that the mutual coherence of the two laser pulses in the

ionization volume is a prerequisite for distinguishing the phase and population relaxation dynamics. The phase integrated measurement, e.g. in noncollinear geometry, will display the overall broadening of the signal comparing to the laser pulse IAC, but will not allow to separate the different contributions.

3.1 Vacuum chamber

Experiments involving the ionization of gas-phase molecules by VUV/XUV laser pulses and subsequent detection of interaction products require a high vacuum environment with residual gas pressures below10 ⁶mbar. Design and commissioning of a vacuum setup was part of the present work.

The experimental apparatus comprises two main parts: a commercial CF160 cube and a CF250 cylindrical chamber of custom design. The cube serves as an ionization chamber, while the cylinder houses in-vacuum optics (detailed description is given in Section3.4) containing a split-and-delay unit (SDU) and focusing optics. The front side of the cube is used as the entrance for the laser beam. The back side is connected to the optics chamber leaving four other sides free for mounting a molecular beam source (Sec-tion3.2), a setup for detecting ionization products (Section3.5) and a vacuum pump. The laser beam passes through the cube into the optics chamber where it is split into a pump-probe sequence of two pulses and then back-focused into the center of the cube where the intersection of femtosecond VUV/XUV pulses with the molecular beam takes place.

The optics chamber is a 400 mm long CF250 tube equipped with five side flanges used for connection with the cube, attachment of electronic feedthroughs for in-vacuum manipu-lators and as viewports for diagnostics of the SDU by means of white light interferometry (Section3.4.2). The outline of the chamber is presented in Fig. 3.1.1. The chamber is mounted on a movable support. The support provides 100 mm of translation in horizontal and vertical directions, and enables rotation of the CF250 cylinder around its axis in a range of˙10°. The vacuum infrastructure consists of a scroll pump (Edwards nXDS10i) and a turbomolecular pump (Pfeiffer HiPace 700M). The SDU developed for the interfer-ometric experiments is extremely sensitive to vibrations, hence the turbomolecular pump was chosen to have a fully active magnetic bearing which excludes any contact of

ro-1

2 3

4

5

laser

Figure 3.1.1: Outline of the vacuum chamber and components of the experimental setup. In the drawing are shown: 1—CF160 cube, 2—CF250 cylinder chamber, 3—in-vacuum optics with manipulators, 4—molecular beam source, 5—charged particle detection setup. The dark blue line indicates the beam path of the short-wavelength pulses. A detailed description of each component is given in the corresponding section of the thesis.

tors with static parts of the pump during operation. The assembled setup, including the chamber, support, pumps and inner components weights200kg.