3 Bichromatic Control of Free Electron Wave Packets
53
dipole selection rules
2
= 1 and m = ±1 for σ
± transitions, the bichromatic
OLP pulse gives rise to numerous interfering ionization pathways. Three-photon
ionization connects the ground state |4s, 0 to different target states | f, m j , with
m j = 3, 1, −1 and −3, in the continuum. In the perturbative limit, the kinetic energy
distribution of the created photoelectron wave packets is essentially determined by
the third-order spectrum of the ionizing laser pulse [22, 25]. The third-order spectrum ˜
E
(3)
olp (ω) = F[E
3
olp (t)](ω) of the OLP field E olp (t) = |E olp (t)| in (3.4) exhibits
four contributions centered around ω
n = (3 − n) ω 1 + n ω 2 , with n = 0, . . . , 3. As
a result, we obtain four energetically separated ionization channels centered around
the photoelectron kinetic energies ε n = ω
n − I P, where I P denotes the atomic
ionization potential. The two outermost channels at ε 0 and ε 3 correspond to singlecolor three-photon ionization by the red and the blue pulse, respectively. The inner
channels at ε 1 and ε 2 correspond to intra-pulse frequency mixing of both colors. In
[22] it was shown, that the general photoelectron wave function of the n-th ionization
channel can be expressed as
ψ n (ε, θ, φ) ∝ γ (ε − ε n )
3
j=0
a nj Y 3,m j (θ, φ),
(3.5)
which describes a superposition of the angular momentum target states | f, m j . The
corresponding angular distributions are given by the spherical harmonics Y ,m (θ, φ).
The shape function γ (ε) determines the photoelectron kinetic energy distribution in
dependence of the bichromatic amplitude profile. In the case of two identically shaped
colors, each channel has the same distribution centered around the corresponding
energy ε n and given by the third-order spectrum of the envelope function ˜
E(ω) =
F[E(t)](ω), with ω = ε/. The amplitudes a nj are determined by the interference
of all ionization pathways leading to the respective target state | f, m j at energy ε n .
These amplitudes are given by the coherent sum over the pathway weights which
are products of the dipole couplings along each pathway and the frequency mixing
amplitude of the associated third order optical spectrum. Thus, the final target state
population results from an interplay of dipole selection rules, addressed via the
polarization state of the bichromatic field, and intra-pulse frequency mixing.
Figure 3.5b shows the measured bichromatic amplitude profile of the OLP field
used in the experiment. The amplitude ratio was set to A 1 /A 2 ≈ 2 to promote the nonresonant pathways starting with a red photon. The tomographically reconstructed
3D PMD created by REMPI of K atoms using this field is shown in Fig. 3.5c in
momentum ( p) representation. Four nested photoelectron wave packets are observed
in different radial shells centered around the momenta p n , corresponding to the four
ionization channels (ε n ). For better visibility, all four contributions are displayed
separately in the central insets (right column) where the experimental results are
compared to numerical calculations based on (3.5) (left column). The inner- ( p 0 ) and
2 Transitions with = −1 are possible as well. However, for simplicity we focus on the = 1
case which, in addition, is favored by propensity rules [52].
53
dipole selection rules
2
= 1 and m = ±1 for σ
± transitions, the bichromatic
OLP pulse gives rise to numerous interfering ionization pathways. Three-photon
ionization connects the ground state |4s, 0 to different target states | f, m j , with
m j = 3, 1, −1 and −3, in the continuum. In the perturbative limit, the kinetic energy
distribution of the created photoelectron wave packets is essentially determined by
the third-order spectrum of the ionizing laser pulse [22, 25]. The third-order spectrum ˜
E
(3)
olp (ω) = F[E
3
olp (t)](ω) of the OLP field E olp (t) = |E olp (t)| in (3.4) exhibits
four contributions centered around ω
n = (3 − n) ω 1 + n ω 2 , with n = 0, . . . , 3. As
a result, we obtain four energetically separated ionization channels centered around
the photoelectron kinetic energies ε n = ω
n − I P, where I P denotes the atomic
ionization potential. The two outermost channels at ε 0 and ε 3 correspond to singlecolor three-photon ionization by the red and the blue pulse, respectively. The inner
channels at ε 1 and ε 2 correspond to intra-pulse frequency mixing of both colors. In
[22] it was shown, that the general photoelectron wave function of the n-th ionization
channel can be expressed as
ψ n (ε, θ, φ) ∝ γ (ε − ε n )
3
j=0
a nj Y 3,m j (θ, φ),
(3.5)
which describes a superposition of the angular momentum target states | f, m j . The
corresponding angular distributions are given by the spherical harmonics Y ,m (θ, φ).
The shape function γ (ε) determines the photoelectron kinetic energy distribution in
dependence of the bichromatic amplitude profile. In the case of two identically shaped
colors, each channel has the same distribution centered around the corresponding
energy ε n and given by the third-order spectrum of the envelope function ˜
E(ω) =
F[E(t)](ω), with ω = ε/. The amplitudes a nj are determined by the interference
of all ionization pathways leading to the respective target state | f, m j at energy ε n .
These amplitudes are given by the coherent sum over the pathway weights which
are products of the dipole couplings along each pathway and the frequency mixing
amplitude of the associated third order optical spectrum. Thus, the final target state
population results from an interplay of dipole selection rules, addressed via the
polarization state of the bichromatic field, and intra-pulse frequency mixing.
Figure 3.5b shows the measured bichromatic amplitude profile of the OLP field
used in the experiment. The amplitude ratio was set to A 1 /A 2 ≈ 2 to promote the nonresonant pathways starting with a red photon. The tomographically reconstructed
3D PMD created by REMPI of K atoms using this field is shown in Fig. 3.5c in
momentum ( p) representation. Four nested photoelectron wave packets are observed
in different radial shells centered around the momenta p n , corresponding to the four
ionization channels (ε n ). For better visibility, all four contributions are displayed
separately in the central insets (right column) where the experimental results are
compared to numerical calculations based on (3.5) (left column). The inner- ( p 0 ) and
2 Transitions with = −1 are possible as well. However, for simplicity we focus on the = 1
case which, in addition, is favored by propensity rules [52].