Supplementary material
The supplementary material is organized into three parts. In Section A, we
experimentally demonstrate the separation of the polar and longitudinal procession
Kerr signal. In Section B, we provide numerical calculation of A p and A l based on
the two different thermal effects with different transient effect anisotropy fields. In
Section C, we perform numerical simulations of the laser induced magnetization
precession for the two different thermal effects with respective anisotropy recovery
time scales, and confirm the two relationships between the precession amplitude and
θoff .
Section A:
Method to separate the longitudinal and polar magnetization component
In the main text, we had shown that the TRMOKE signal in our measurement is
a combination of the contributions from the polar and longitudinal magnetization
components. The reflected probe beam intensity at applied field H can be expressed
as:
I  H   Aml  Bmp
(1)
where ml and m p represent the longitudinal and polar components of the
magnetization, respectively, and the magneto-optical coefficients A and B depend on
i and the refractive index and Voigt constants of the material.
Under opposite magnetic fields, the probe light intensity becomes
I  -H   Aml  Bmp
(2)
The longitudinal and polar signals can be separated by performing TRMOKE
measurements under reversed field directions:
m l  [I  H   I  -H ] / 2A
m p  [I  H   I  -H ] / 2B
We confirm the separation method experimentally.
(3)
Fig. S1 (a) Typical TRMOKE signals under opposite magnetic fields, and (b)-(d) the
separated polar, and longitudinal signals at different sample orientations θ s and field
orientations θ H with a fixed angle between the field and the Fe<110> axis. All data
were measured at H=1000 Oe.
Figure S1 (a) shows the typical TRMOKE signals under H  1000 Oe . The
major feature of the signal manifests as the periodic oscillations corresponding to the
uniform magnetization precessions. The precession frequency remains the same under
opposite fields as a result of the 4-fold symmetry. However, the precessions clearly
show different amplitudes and phases. This is because the polar signals remain
unchanged for reversed fields but the longitudinal signals are in opposite phase. We
use Eq. (3) to separate the two signals, and the results are shown in Fig. S1 (b). A  4
phase shift between the polar and longitudinal signals can be clearly identified,
consistent with the magnetization precession behavior.
To confirm the validity of the method to separate the polar and longitudinal
precession signals, we further performed measurements by rotating the sample to
different orientations ( θ s ) while keeping the angle between the [110] axis and the field
direction unchanged. Under such conditions, the directions of the effective anisotropy
fields of the Fe film before and after the laser excitation rotate the same angle as that
of the sample, whereas the magnitude of the effective fields remains unchanged. Thus,
the excited precessions are identical if observed in the coordinate system rotating with
the sample. We therefore expect that the polar signal remains unchanged for different
sample orientations, and this is confirmed by our experimental results, as shown in
Fig. S1. On the other hand, the longitudinal precession signal becomes smaller when
the magnetization rotates closer to the horizontal plane, also in consistent with the
experimental results. Hence, our experimentally confirm the successful separation of
precession signals contributed from different magnetization components.
Section B:
Model calculation of A p and A l with two excitation mechanism under different
transient anisotropy field H'4 .
As shown in the main text, in lattice thermalization mechanism, the polar and
longitudinal precession amplitude should follow as
A p  off R p  θ off f and
A l  off R l  off sin  M  , respectively. In electron thermalization model, the polar
and the longitudinal precession amplitude should follow as Ap  off f R p  θoff f 2 ,
A l  off f R l  off f sin  M  , respectively. All the field direction dependent
parameters off , f and sin  M  can be calculated based on two models, thus we
can compare the A p (H ) and A l (H ) curves with the experimental results shown
in Fig. 3 in main text, and judge which model can provide better results close to the
experimental value. Moreover, we note that the H'4 value has great impact on the
overall shape of the calculated A p and A l , so we also perform calculation to study
whether H'4 value will influence the validation of our conclusion.
Figures S2 (a)-(c) show the calculated A p  θ H  with different H'4 values by
assuming the in-plane precession amplitude is proportional to off . Figures S2(d)-(f)
show the calculated A p (H ) by assuming the in-plane precession amplitude
proportional to θoff f , which is clearly inconsistent with the experimental results
shown in Fig. 3 in main text. For example, the calculated precession amplitude at
H=500 Oe always shows a dip behavior regardless of the H'4 value in Fig. S2
(d)-(f) , which doesn’t exist in the experimental results in main text.
Fig. S2. The calculated θ H -dependent polar precession amplitude A p using (a-c)
the slow recovery model, and (d-f) the rapid recovery model with different strengths
of the reduced anisotropy field H'4 .
So, our calculation can confirm that the laser-exited in-plane precession
amplitude is proportional to θoff , thus we further draw the conclusion that only lattice
thermalization model agrees with the experiment result.
Section C:
Numerical simulations of the magnetization precession for the two different
thermal effectsto confirm the relationships between the precession amplitude and
θoff .
The ferromagnetic medium under interaction of an ultrafast laser pulse may
undergo: (1) a highly nonequilibrium state with very hot electron temperature which
persists a short time duration (<1 ps), or (2) a slightly delayed lattice thermalization
which has typical recovery time of ~100 ps. Both heating effects may drive the
magnetization precession, but they have different impacts on the precession amplitude.
For the lattice thermal effect, the in-plane precession amplitude scales linearly with
θoff [1] (referred as θoff model below), whereas for electron thermal effect, the
in-plane precession amplitude is proportional to θoff f [2] (referred as θoff f model
below). Here θoff is the in-plane tilting angle of the effective magnetic field caused
by the pump laser pulse, and f is the precession frequency. Our experimental data
strongly supports that the precession excitation mechanism in the Fe film is the lattice
thermal effect.
Here, we performed the numerical simulations of the laser induced magnetization
precession for the two different thermal effects to confirm the above two relationships
between the precession amplitude and θoff due to the different anisotropy recovery
time scales.
We performed numerical calculation to simulate the laser induced magnetization
precession based on the Landau-Lifshitz-Gilbert (LLG) equation:

dM
  M  H eff   M  M  H eff
dt

(4)
where    / Ms , and the effective field H eff contains the external field H, shape
anisotropy field 4πMS , and the cubic crystalline magnetic anisotropy field H 4 . In the
simulation, we used the Gilbert damping factor  =0.003, the gyromagnetic ratio  =
1.84 107 Hz Oe , and the static cubic anisotropy field H 4 =540Oe. In addition, under
the pump laser pulse interaction, H 4 is instantly reduced to H'4 =500 Oe at the
overlap time of the pump and probe pulses, and then exponentially recover to H 4
with the time constant Trec , as shown in Fig. S3. We set Trec =1ps for the electron
thermal effect, and Trec =100ps for the lattice thermal effect.
We neglected the ultrafast demagnetizationin the simulation since it is extremely
small in Fe [3] compared to that in Ni [4]. And on the other hand, the demagnetization
effect can be effectively included in the modulation of the anisotropy field.
Figure S3 shows the simulated polar magnetization dynamics for Trec =1ps and
Trec =100ps at H=800 Oe applied along θ H =20°. Here θ H is the angle between the
applied field and the Fe [110] direction, according to Fig. 1 in the main article. In both
cases, the precession periods are same, but clearly the precession amplitude is one
order smaller for the case of Trec =1ps.
Fig. S3 Simulated polar precession for (a) Trec =1ps and (b) Trec =100ps. Red lines
indicate the time evolution of H 4 .
Figure S4 shows the color contour of the polar magnetization component as a
function of the field orientation and delay time with different field strength and Trec .
Overall, the precessions show the increasing amplitude when H is rotated from the
easy axis ( θ H  45 and 135 ) to the hard axis ( θ H  0 , 90 , and 180 ). However,
for H=800 Oe, the amplitude shows a pronounced drop at field orientations nearly
along the hard axis.
Fig. S4. Field-orientation dependence of the simulated polar magnetization precession
with different Trec and field H.
We obtained the in-plane precession amplitude ( A / / ) and polar precession
amplitude ( A p ) by fitting the simulated precession dynamical data in Fig. S4 through
the Equ. (4) in main text, and the results are plot in Fig. S5. We compare them with
the calculated θ H -dependent θoff
and θoff f . Fig. S5(a) show the A / / and
calculated θoff f for Trec =1 ps, and the θoff f curves are multiplied by a fixed factor
for all the calculated curves with different applied fields. It is clear that the calculated
A / / curve with the short recovery time Trec agree well with the θoff f model. Fig.
S5(b) shows the A / / and calculated θoff for Trec =100 ps. It is clear that the
calculated A / / curve with the long recovery time Trec agree quantitatively with the
θoff model. The simulated polar precession amplitudes for Trec =1 ps and Trec =100 ps
also show good agreement with the respective models after considering the
polar/in-plane amplitude ratio R p , as can be seen in Figs. S5(c) and S5(d).
Fig. S5 Field orientation dependence of precession amplitude of the simulated result.
(a) in-plane amplitude for Trec =1ps, (b) in-plane amplitude for Trec =100ps, (c) polar
amplitude for Trec =1ps, and (d) polar amplitude for Trec =100ps. The solid lines
represent the calculated results according to θoff and θoff f models.
To directly compare the two thermal excitation models, the in-plane precession
amplitude obtained from the simulation for Trec =1ps and Trec =100ps at H=500 Oe are
plotted as a function of θoff and θoff f in Fig. S6. The amplitude for Trec =1ps is
magnified 30 times for better comparison. Clearly, we can see the linear dependence
of the precession amplitude on θoff for Trec =100 ps and θoff f for Trec =1 ps. In
contrast, the precession amplitude show a pronounced nonlinear dependence on θoff
for Trec =1ps and on θoff f for Trec =100 ps. Through this comparison, we confirmed
that for the lattice thermalization induced transient anisotropy field with slow recovery,
the precession amplitude should be linearly proportional to θoff f ; whereas for the hot
electron induced transient anisotropy field with rapid recovery, and the precession
amplitude should linearly scale with θoff f .
Fig. S6 Simulated in-plane precession amplitude for Trec =1ps and Trec =100 ps versus
(a) θoff and (b) θoff f at H=500 Oe. Amplitude for Trec =1ps is multiplied by 30 for
better comparison.
Furthermore, we explored the applicable time range of the θoff model and the
θoff f model. Figure S7 shows the Trec dependence of polar precession amplitude at
H=800 Oe with different field directions. For Trec <10ps, a linear increase of
amplitude with Trec can be observed, corresponding to the θoff f model. For
Trec >50ps, the amplitude almost reaches its saturation value, corresponding to the
θoff model. The transition occurs at Trec which is about a quarter of the precession
period (T/4). In Fig. S7, the time position of T/4 is indicated by a small vertical arrow,
where the precession amplitude reaches ~85% of its saturation value. Roughly
speaking, for Trec <T/4, the precession amplitude is proportional to θoff f , and for
Trec >T/4, the precession amplitude is just determined by θoff .
Fig. S7 Polar precession amplitude A p as a function of Trec , under H=800 Oe and
θ H =5°,15°,25°,35°, and 44°. The short vertical arrows denote the time
positions of T/4.
Through numerical stimulation using the LLG equation, we confirmed that the
precession amplitude scales linearly with θoff for the slow anisotropy recovery, and it
is proportional to θoff f for the fast anisotropy recovery.
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