Chapter 4
Nonlinear Physics of Laser-Plasma
Interaction
4.1 Ponderomotive (PM) Force
Consider the time-averaged force to electrons by the laser force. This time-averaged
force stems from the nonuniform profile of the electromagnetic energy density.
Intuitively speaking, the electrons or any charged particles are affected by nonzero
force after the time averaging the electromagnetic force, when the electromagnetic
energy is localized spatially. In other words, the electrons obtain an effective
pressure stemming from the quivering motions, and this pressure gradient provides
a new force to the electrons. Since the electron energy density of the quivering
motion is proportional to the energy density of the electromagnetic fields, the timeaveraged force to the electrons is regarded as the repelling force to the electrons from
localized field energy density. In order to confirm this explanation, derive
mathematically the form of this force. Note that this force is called ponderomotive
force and is written as PM force in short.
Consider the case of non-relativistic electron motion governed by (2.3.1) in an
electromagnetic field. In order to use Taylor expansion, it is assumed that the spatial
variation length of the laser amplitude is much longer than the electron quivering
distance ξ. The spatial dependence of the electric field structure can be approximated
with Taylor expansion form:
F ¼ ÀeE À e ξ Á ∇
ð
ÞE À ev  B
ð4:1:1Þ
It is clear that the second term of RHS in (4.1.1) remains for the case of the electric
field is not constant in space. Since the non-relativistic case is considered, the third
term in RHS of (4.1.1) is small enough. Taking the time average of (4.1.1) yields:
© Springer Nature Switzerland AG 2020
H. Takabe, The Physics of Laser Plasmas and Applications - Volume 1, Springer
Series in Plasma Science and Technology,
https://doi.org/10.1007/978-3-030-49613-5_4
131
Nonlinear Physics of Laser-Plasma
Interaction
4.1 Ponderomotive (PM) Force
Consider the time-averaged force to electrons by the laser force. This time-averaged
force stems from the nonuniform profile of the electromagnetic energy density.
Intuitively speaking, the electrons or any charged particles are affected by nonzero
force after the time averaging the electromagnetic force, when the electromagnetic
energy is localized spatially. In other words, the electrons obtain an effective
pressure stemming from the quivering motions, and this pressure gradient provides
a new force to the electrons. Since the electron energy density of the quivering
motion is proportional to the energy density of the electromagnetic fields, the timeaveraged force to the electrons is regarded as the repelling force to the electrons from
localized field energy density. In order to confirm this explanation, derive
mathematically the form of this force. Note that this force is called ponderomotive
force and is written as PM force in short.
Consider the case of non-relativistic electron motion governed by (2.3.1) in an
electromagnetic field. In order to use Taylor expansion, it is assumed that the spatial
variation length of the laser amplitude is much longer than the electron quivering
distance ξ. The spatial dependence of the electric field structure can be approximated
with Taylor expansion form:
F ¼ ÀeE À e ξ Á ∇
ð
ÞE À ev  B
ð4:1:1Þ
It is clear that the second term of RHS in (4.1.1) remains for the case of the electric
field is not constant in space. Since the non-relativistic case is considered, the third
term in RHS of (4.1.1) is small enough. Taking the time average of (4.1.1) yields:
© Springer Nature Switzerland AG 2020
H. Takabe, The Physics of Laser Plasmas and Applications - Volume 1, Springer
Series in Plasma Science and Technology,
https://doi.org/10.1007/978-3-030-49613-5_4
131