3 Bichromatic Control of Free Electron Wave Packets
49
Different types of polarization-shaped bichromatic pulse sequences are illustrated
in Fig. 3.3a–h and compared to their single-color counterparts in frames (i)–(p). In
the case of temporally overlapping pulses (τ = 0), bichromatic fields (first column) exhibit unusual polarization characteristics and vectorial field symmetries
while single-color fields (third column) become linearly polarized (PLP, CRCP,
OLP) or circularly polarized (COCP). For example, bichromatic (N 1 ω:N 2 ω) OLP
fields with commensurable center frequencies ω 2 = (N 2 /N 1 )ω 1 (frame (c)) exhibit
Lissajous-type polarization profiles, whereas the corresponding COCP and CRCP
fields (frames (b) and (d)) feature cycloidal heart- and propeller-shaped polarization
profiles, respectively [18, 35, 36]. The bichromatic polarization profiles are highly
sensitive to both the relative phases ϕ i and the CEP ϕ ce , as shown in the insets. The
cycloidal pulses (CRCP and COCP) are rotated by any kind of phase variation [cf.
(3.3)] and also the Lissajous-pattern of the OLP pulses is strongly phase-dependent.
The polarization profiles of single-color pulses are mainly affected by the relative
phase which controls the spatial rotation in the CRCP case and the ellipticity in
the OLP case. Only in the COCP case, the pulse is rotated by both the relative
phase and the CEP. These differences between bichromatic and single-color fields
has profound consequences for light-matter interactions and the optical parameters
available for control, as will be discussed in Sects. 3.3.2 and 3.3.4. In the case of temporally separated pulses, the respective polarization states are meaningfully defined
for bichromatic and also for single-color fields. Such pulse sequences are used e.g.
in Sect. 3.3.3 for the creation of single-color electron vortices and further in Sect.
3.3.5 and Sect. 3.3.6 for bichromatic pump-probe studies on Rydberg and spin-orbit
wave packets, respectively.
Traditionally, commensurable bichromatic fields are generated by superposition
of harmonic beams using an interferometer [37, 38]. These fields are inherently CEPinsensitive, leaving the relative phase between the colors as the central parameter for
coherent control. In contrast, shaper-generated bichromatic fields depend on both the
relative phases ϕ i and the CEP ϕ ce , each with different sensitivities. For example, in
the case of cycloidal (N 1 ω:N 2 ω) fields, any phase variation translates into a spatial
rotation of the laser field about its propagation axis by the angles
α
cr
=
N 2 − N 1
N 2 + N 1
ϕ ce +
N 2
N 2 + N 1
ϕ 1 −
N 1
N 2 + N 1
ϕ 2
α
co
= ϕ ce +
N 2
N 2 − N 1
ϕ 1 −
N 1
N 2 − N 1
ϕ 2 ,
(3.3)
where α
cr and α
co denote the rotation angles of CRCP and COCP fields, respectively
[18]. On the one hand this implies that all optical phases are utilizable as control
parameters in the shaper-based scheme. On the other hand, stabilization of all phases –
in particular the CEP – is crucial if shaper-generated bichromatic fields are employed
for the spatial control of quantum dynamics. Otherwise phase fluctuations, leading to
random spatial rotations of the cycloidal fields, will average out the phase-sensitive
interferences.
49
Different types of polarization-shaped bichromatic pulse sequences are illustrated
in Fig. 3.3a–h and compared to their single-color counterparts in frames (i)–(p). In
the case of temporally overlapping pulses (τ = 0), bichromatic fields (first column) exhibit unusual polarization characteristics and vectorial field symmetries
while single-color fields (third column) become linearly polarized (PLP, CRCP,
OLP) or circularly polarized (COCP). For example, bichromatic (N 1 ω:N 2 ω) OLP
fields with commensurable center frequencies ω 2 = (N 2 /N 1 )ω 1 (frame (c)) exhibit
Lissajous-type polarization profiles, whereas the corresponding COCP and CRCP
fields (frames (b) and (d)) feature cycloidal heart- and propeller-shaped polarization
profiles, respectively [18, 35, 36]. The bichromatic polarization profiles are highly
sensitive to both the relative phases ϕ i and the CEP ϕ ce , as shown in the insets. The
cycloidal pulses (CRCP and COCP) are rotated by any kind of phase variation [cf.
(3.3)] and also the Lissajous-pattern of the OLP pulses is strongly phase-dependent.
The polarization profiles of single-color pulses are mainly affected by the relative
phase which controls the spatial rotation in the CRCP case and the ellipticity in
the OLP case. Only in the COCP case, the pulse is rotated by both the relative
phase and the CEP. These differences between bichromatic and single-color fields
has profound consequences for light-matter interactions and the optical parameters
available for control, as will be discussed in Sects. 3.3.2 and 3.3.4. In the case of temporally separated pulses, the respective polarization states are meaningfully defined
for bichromatic and also for single-color fields. Such pulse sequences are used e.g.
in Sect. 3.3.3 for the creation of single-color electron vortices and further in Sect.
3.3.5 and Sect. 3.3.6 for bichromatic pump-probe studies on Rydberg and spin-orbit
wave packets, respectively.
Traditionally, commensurable bichromatic fields are generated by superposition
of harmonic beams using an interferometer [37, 38]. These fields are inherently CEPinsensitive, leaving the relative phase between the colors as the central parameter for
coherent control. In contrast, shaper-generated bichromatic fields depend on both the
relative phases ϕ i and the CEP ϕ ce , each with different sensitivities. For example, in
the case of cycloidal (N 1 ω:N 2 ω) fields, any phase variation translates into a spatial
rotation of the laser field about its propagation axis by the angles
α
cr
=
N 2 − N 1
N 2 + N 1
ϕ ce +
N 2
N 2 + N 1
ϕ 1 −
N 1
N 2 + N 1
ϕ 2
α
co
= ϕ ce +
N 2
N 2 − N 1
ϕ 1 −
N 1
N 2 − N 1
ϕ 2 ,
(3.3)
where α
cr and α
co denote the rotation angles of CRCP and COCP fields, respectively
[18]. On the one hand this implies that all optical phases are utilizable as control
parameters in the shaper-based scheme. On the other hand, stabilization of all phases –
in particular the CEP – is crucial if shaper-generated bichromatic fields are employed
for the spatial control of quantum dynamics. Otherwise phase fluctuations, leading to
random spatial rotations of the cycloidal fields, will average out the phase-sensitive
interferences.