2 Mechanism of Population Inversion in N 2
+
31
2.3 Population Inversion in Aligned N 2
+
In most of the analysis of the results of the experiments in which the 391 nm coherent emission from N 2
+ was recorded, rotational motion of N 2
+ has been neglected
because the rotational period (∼ 8 ps) [43] is much longer than the laser pulse duration (∼ 40 fs) [6]. Therefore, it can be assumed that N 2
+ interacts with the laser
field as if N 2
+ is fixed in the space. As mentioned in Sect. 2.2.3, the polar angle θ ,
can be regarded as the alignment angle between the molecular axis and polarization
direction of the laser pulse.
In the simulation of the population transfer among the vibronic levels of N 2
+
interacting with a laser pulse, N 2
+ is assumed to be created by the tunneling ionization
of N 2 at the moment when the electric field of the near-IR pulse takes the maximum
amplitude during on optical cycle, so that the excitation is calculated from t = 0 with
an envelope function defined in (2.25).
The total wave function of the resultant N 2
+ is expressed using the field-free
vibrational basis set {ψ αv (r )} obtained by the same method in Sect. 2.2.1 without
the rotational motion as
(r, t) =
α=X,A,B
V max
v=0
c αv (t)ψ αv (r ).
(2.31)
Upon the creation of N 2
+ at t = 0, the vibrational wave packet of N 2
+ is assumed
to be the same as that in the vibrational ground state of the electronic ground state
X
2
g
+ of neutral N 2 according to the Franck-Condon principle as
c αv (t = 0) =
ψ α,v
ψ
N 2
v=0
δ α X ,
(2.32)
where α = X, A, and B.
2.3.1 Time-Dependent Population Transition
When N 2
+ is created by the laser pulse, the initial states are calculated by the projection of the wave function in the vibrational ground state of neutral N 2 onto the three
electronic states in N 2
+
. By solving the time-dependent Schrödinger equation, with
the total wavefunction given by (2.31) and the experimental conditions reported in
[6], the post-ionization coupled-state dynamics is obtained as shown in Fig. 2.3. The
simulation is done by employing a laser pulse with a full-width at half-maximum
(FWHM) of 20 fs, the field intensity of 2 × 10
14 Wcm
−2 , and the alignment angle of
θ = 45
◦ . In Fig. 2.3, populations in the respective electronic states oscillate fast and
become constants after the laser pulse vanishes, that is, after the interaction with the
+
31
2.3 Population Inversion in Aligned N 2
+
In most of the analysis of the results of the experiments in which the 391 nm coherent emission from N 2
+ was recorded, rotational motion of N 2
+ has been neglected
because the rotational period (∼ 8 ps) [43] is much longer than the laser pulse duration (∼ 40 fs) [6]. Therefore, it can be assumed that N 2
+ interacts with the laser
field as if N 2
+ is fixed in the space. As mentioned in Sect. 2.2.3, the polar angle θ ,
can be regarded as the alignment angle between the molecular axis and polarization
direction of the laser pulse.
In the simulation of the population transfer among the vibronic levels of N 2
+
interacting with a laser pulse, N 2
+ is assumed to be created by the tunneling ionization
of N 2 at the moment when the electric field of the near-IR pulse takes the maximum
amplitude during on optical cycle, so that the excitation is calculated from t = 0 with
an envelope function defined in (2.25).
The total wave function of the resultant N 2
+ is expressed using the field-free
vibrational basis set {ψ αv (r )} obtained by the same method in Sect. 2.2.1 without
the rotational motion as
(r, t) =
α=X,A,B
V max
v=0
c αv (t)ψ αv (r ).
(2.31)
Upon the creation of N 2
+ at t = 0, the vibrational wave packet of N 2
+ is assumed
to be the same as that in the vibrational ground state of the electronic ground state
X
2
g
+ of neutral N 2 according to the Franck-Condon principle as
c αv (t = 0) =
ψ α,v
ψ
N 2
v=0
δ α X ,
(2.32)
where α = X, A, and B.
2.3.1 Time-Dependent Population Transition
When N 2
+ is created by the laser pulse, the initial states are calculated by the projection of the wave function in the vibrational ground state of neutral N 2 onto the three
electronic states in N 2
+
. By solving the time-dependent Schrödinger equation, with
the total wavefunction given by (2.31) and the experimental conditions reported in
[6], the post-ionization coupled-state dynamics is obtained as shown in Fig. 2.3. The
simulation is done by employing a laser pulse with a full-width at half-maximum
(FWHM) of 20 fs, the field intensity of 2 × 10
14 Wcm
−2 , and the alignment angle of
θ = 45
◦ . In Fig. 2.3, populations in the respective electronic states oscillate fast and
become constants after the laser pulse vanishes, that is, after the interaction with the