In atomic physics, electromagnetic fields of optical frequencies are usually quite adequately described in terms of the dipole approximation, base on the inequality
1
0 <<
λ
a , (23)
where
a
0 is the Bohr radius, andλ
is the wavelength of the radiation. This means that the phase of a traveling plane wave can be approximated aswt − k v ⋅ r v ≈ wt
. In order to study the magnetic effect, an atom is subjected to a sufficiently intense fields, the dipole approximation may not valid, because the full relativity effect the spatial dependence is strong. To study the regime in which the effect of the magnetic field should be account for, but where the full relativity is not required. We need an electromagnetic gauge in which both electric and magnetic fields are present, and where both are retained only within the dipole approximation.
Electric and Magnetic fields in dipole approximation
In classical electrodynamics the scalar potential can be written as r
k wt E
rv v v v
⋅
−
≡
⋅
−
=
Φ (
ϕ
),ϕ
, (24) in addition, there is a vector potential of very similar appearance, given by[ ]
wc k k E
r k A
v v v v
≡
⋅
−
= ˆ (ϕ) , ˆ (25)
That is,
kˆ
is a unit vector in the direction of propagation. These potential describe completely the electric and magnetic field with full wt kv rv⋅
− phase dependence, A
B t A c E
v v v
v
×
∇
∂ =
− ∂ Φ
−∇
= 1 ,
(26) v
) dipole approximation limit ϕ→wt, the vector potential retains important spatial dependence. Now we take the dipole limit of the above potentials,
[
( )]
The extra term is seen to drop out in the dipole approximation, since 1 the magnetic field follows directly from the curl of vector potential,
)
These potential provide both electric and magnetic fields of a plane wave in the correct phase, amplitude, and vector relationships, but within the dipoleapproximation.
Interaction Hamiltonian
The Hamiltonian for a hydrogen atom in a plane-wave field is now,
)
Here, we consider ground state hydrogen under electric field in z-direction and magnetic field in x-direction, over 4-cycle laser pulse with w=0.057a.u .E0 =0.17a.u.
The blue line is the electric field only, and the red line is electric and magnetic fields
.
We can see from these figures, when the magnetic effects have to take into account, harmonics in the high radiation spectrum are changed strongly, but small for low order harmonic frequency. The intensities of plateau are lower than spectrum for electric field only, there are some peak intensity gone, the number of ionization is reduced by magnetic field, and the total spectrums also depend on laser phase.
Magnetic effect
In order to study the magnetic effects of atomic dynamics under intense laser pulse, we note that a free electron moving in a plane wave field follows a figure-8 motion caused by the combined action of the electron and magnetic field.
The free electron in plane wave
x
the Lorentz force is
B
zero order,
0
2 to simplify the problem, then we gain the trajectories,
∫
The 8-motion induced by the coupling of the electric and magnetic fields in a plane wave. The amplitude α0 is in the direction of the electric field, and β0 is in the direction of the propagation vector kv perpendicular to the electric and magnetic fields.
c w E w E
3 2 0 0
2 0 0
=8
= β α
(41)
Fig.5-7 the 8-motion
The magnetic component of the Lorentz force is perpendicular to the direction of motionvv
, it can do no work on the electron. That is the magnetic field of laser does not insert field energy into the electron, it couples to the electric field in such fashion as to distort the linear oscillatory motion.
Fig.5-8 The spectrum of free electron under electric field (blue line) and the free electron under both magnetic and electric fields (red line)
The spectrum of electric field and both electric and magnetic fields are identicalness.
That is there is no another frequency added by magnetic fields such as cyclotron frequency. From another point, the interaction Hamiltonian
) ( )
( ˆ 2
1 2 2
wt E r e r wt e
E r e c p k m H
v v v v
v − − ⋅
⋅
−
= , (42)
it can be rewrite into
The equation can be divided into three parts:
)
The first one is the Hamiltonian of hydrogen atom, second one is the Hamiltonian due to the electric field of laser, and third one is the Hamiltonian from the magnetic field of laser, in third one the second term’s magnitude is about ( c)2
v , where should be
dropped. The magnetic Hamiltonian, it roles as a c
v addition to electric field effect
and coupling into the propagation direction kˆ of the laser. This coupling underlies 8-motion.
The number of ionization is reduced by magnetic field, this is because the magnetic force changes the electron’s direction of motion and electrons are trapped by the magnetic effect. Even thought the number of trapped electron increase, some HHG plateau spectrum still gone. This can be treat as the consequence, that the magnetic force prevent the electron goes back to or near to their parent core, i.e., there is less interaction with the nucleus, the spectrum occurs only when the electronic motion is close to the nucleus that is sufficient acceleration to generate the harmonics.
In other hand, for lower harmonic orders there are no huge changes, because the electrons with lower velocity the magnetic effects are small, also the motions are strongly controlled by the nucleus, This results seen that the magnetic field serves
regime, the electrons with high velocity have strong magnetic force on them, and magnetic force change the direction of motion, it let electron’s motion more close to the nucleus than the electric force does, i.e., the dipole moment is decreased by magnetic force. After Fourier transform the intensity drop in plateau regime.
Chapter 6 Conclusion
In this thesis we use intense pulse on a classical hydrogen atom, and simulated it by Classical Trajectory Monte Carlo method to study atomic dynamic under intense short pulse and the magnetic effect. We find that for short pulse the total spectrum is phase dependence is strong, and the periodicity of the high harmonic generation process is completely suppressed, but kept for low order harmonic. From the above results, we find that for short pulse such as few-cycle laser, the high harmonic generation is governed directly by the intense pulse evolution. The magnetic part in laser field can trap the electrons in the region near to their parent core, and in high harmonic generation plateau regime, i.e., the high frequency, there are some intensity peaks were gone, also the intensity of plateau are lower than spectrums that caused by electric field only. Magnetic force can let electron miss their core and trap electron’s motion close to their core. For low order harmonic spectrum, i.e., perturbative regime, the spectrums are dominated by Coulomb potential, and the magnetic effect is small so that the spectrums don’t change too much. The magnetic effect serves to change high harmonic generation in plateau.
Reference
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[2] H. R. Reiss, "Dipole-approximation magnetic fields in strong laser beams", PRA, 63, 013409, 2000.
[3] James S. Cohen, "Comment on the classical-trajectory Monte Carlo method for ion-atom collisions", PRA, 26, 5, 1982.
[4] J. Cooper, "Harmonic generation by a classical hydrogen atom in the presence of an intense radiation field", PRA, 41, 3, 1990.
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Optimal control of high-harmonic generation", Reviews of Modem Physics, Vol.
80, No. 1, 2008
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