Figure 1

Experimental spectra of the 3PPE resonance as a function of the central energy of the excitation pulse, $p$-polarized excitation (logarithmic scale). For comparison, the 2PPE spectrum from cesiated Cu(001) is shown shifted upward by $h\nu$ and the 4PPE peak is shifted downward by $h\nu$. Feature C in the 2PPE spectrum near the vacuum edge is due a background of secondary electrons which could not be further decreased. The spectra are displaced vertically for better visibility. They are not normalized with respect to the laser intensity.Reuse & Permissions

Figure 2

(Color online) Relativistic band structure (Ref. 20) of Cu(001) with the proposed three-photon resonance for a photon energy near 3 eV. The final photoemitted states correspond to time-reversed low-energy electron diffraction states in the direction of the detector, with complex wave vectors in the band gap (Ref. 21).Reuse & Permissions

Figure 3

(Color online) Estimation of resonance conditions for $h\nu =2.97\text{eV}$ (left) and $h\nu =3.08\text{eV}$ (right) by shifting initial and intermediate states according to the number of necessary photons to reach the final-state energy. The bands are numbered according to Fig. 2. The crossing of initial (solid black), first intermediate (dashed red), and second intermediate (dash-dot-dot, blue) state bands indicates simultaneous one-photon and two-photon resonances between one of the two $d$ bands, the $sp$ band, and the $n=1$ image-potential state. Due to the broad pulse spectrum $\left(\approx 170\text{meV}\right)$, the separate resonances connected to the two $d$ bands are simultaneously observed in the experimental spectrum (solid gray, experimental data for 3.08 eV central photon energy of the laser pulse, compare with Fig. 1).Reuse & Permissions

Figure 4

(Color online) Simulations of one-color 2PPE from two closely spaced initial states via the same intermediate state for a continuous flat excitation spectrum with single-photon energies from 1.5 to 4.5 eV. The panels show the final-state 2PPE yield (logarithmic scale) as a function of the photon energy and final-state energy. The initial and intermediate-state peaks can be distinguished by the change in their final-state energy as a function of the photon energy $h\nu$ (initial state: $2\Delta h\nu$ and intermediate state: $1\Delta h\nu$). The two vertical lines (best visible in the lower panels) indicate the presence of two energy-filtering resonances A and B that increase the absolute intensity in the photoelectron spectra for the respective photon energies. The corresponding peak positions in the photoelectron spectrum are shown at A and B. The total photoelectron spectrum for the 3 eV broad continuum of incident light is obtained by integrating over the horizontal axes. With increasing dephasing time, the total photoelectron spectrum evolves from a single broad peak (5 fs) to two separate peaks $\left(>10\text{fs}\right)$ (see Fig. 5). In each panel, the assumed initial-state pure dephasing times are indicated. For the intermediate state, a population lifetime of 35 fs and a pure dephasing of 1 meV was assumed. In the final state, we took an infinite population lifetime and pure dephasing rate of 1 meV. The separation of the initial states is 150 meV.Reuse & Permissions

Figure 5

(Color online) Summation of the spectra from Fig. 4 for all photon energies of the excitation continuum to simulate the total photoelectron spectrum. The presence of two resonances can be clearly distinguished. While the underlying initial and intermediate-state features disperse with photon energy, they strongly change their intensity (logarithmic scale in Fig. 4) to result in the resonance peaks in the total sum spectra shown above. An instrumental broadening of 50 meV was assumed.Reuse & Permissions

Figure 6

(Color online) Principle of detecting coherent bulk band-structure resonances in 3PPE via an image-potential state on Cu(001). Left: nonresonant incoherent excitation of the IP state from unspecific bulk bands at normal emission $\left(k_{\parallel }=0\right)$. The initial states are averaged over $k_{\perp }$ illustrating the loss of $k_{\perp }$ information in the sequential one-photon pathways due to the total lack of a $k_{\perp }$ selection rule in the IP to final-state one-photon transition ($k_{\perp }$ is undefined for surface states). The colors of the arrows correspond to photon frequencies provided by the excitation pulse, their thickness corresponds to the respective spectral weight. The photons can act in any combination for sequential incoherent excitations. The peak position of the IP state is determined by the central photon energy of the pulse (thickest arrow) and the measured width of the IP state is determined by the width of the pulse spectrum. Middle: coherent 3PPE resonances in the bulk band structure coupled to the image-potential state produce specific peaks A and B in the spectrum, providing $k_{\perp }$ information. A fixed photon energy defines each resonant pathway. The width of features A and B is not limited by the spectral pulse width. Right: the corresponding 1PPE process for the same level scheme using a single-photon energy of $\approx 3h{\nu }_{3\text{PPE}}$ is less $k$ selective if no resonance to a dispersing final-state band is present (e.g., off-resonant transition to final band broadened by absorption).Reuse & Permissions