1
μ 0
∇ Â B ¼ j À ε 0
∂E
∂t
¼ 0 at ω ¼ ω pe
The electromagnetic field can propagate thanks to the induction of the magnetic field
by the time variation of the electric field. However, the induced current by the
electric field becomes equal to the displacement current at the critical point, and
the magnetic field is not induced any more.
In the situation where the plasma density gradually increases in space, consider
how the electromagnetic field propagates. This has been studied relating to the radio
wave communication in space in the early twentieth century. E. V. Appleton who
carried out comprehensive study was appreciated as a winner of Nobel Prize in 1947.
It is found that the electromagnetic wave approaching the cut-off density point is
reflected, and the reflecting point gets lower density for oblique incident to the
density gradient as seen in Fig. 2.15. In Fig. 2.15, trajectories of radio waves emitted
from an antenna are shown in red arrows, when radio waves are emitted from the
surface of the earth to the space. In space, the plasma layers are observed as the
ionosphere. For the case of oblique incidence to the ionosphere with angle θ, the
wave momentum in the perpendicular direction to the density gradient is conserved:
k sin θ : const:
Then, an effective wavenumber k x in the direction of the density gradient becomes:
c
2 k
2
x ¼ ω
2
À ω
2
pe À c
2 k
2 sin
2
θ
ð2:5:5Þ
This simple relation explains the trajectories of the radio waves in Fig. 2.15. The
waves are reflected at the point where k x ¼ 0 is satisfied. It is noted that the trajectory
of the wave is described by the same equation of a point mass in a potential in
Newton equation. In the past, one used such reflection by the ionosphere to send
radio waves far away.
For the convenience of further study, introduce the dielectric constant of the
plasmas. It is a function of k and ω in general; it is called dielectric function. In
Fig. 2.15 The schematic
picture showing the
propagation of radio waves
emitted from grand station
to the space. The plasmas in
the ionosphere reflect the
waves emitted obliquely.
Using this property, longdistance communication
was done before satellite
communication becomes
popular
2.5 Lasers in Plasmas
67
μ 0
∇ Â B ¼ j À ε 0
∂E
∂t
¼ 0 at ω ¼ ω pe
The electromagnetic field can propagate thanks to the induction of the magnetic field
by the time variation of the electric field. However, the induced current by the
electric field becomes equal to the displacement current at the critical point, and
the magnetic field is not induced any more.
In the situation where the plasma density gradually increases in space, consider
how the electromagnetic field propagates. This has been studied relating to the radio
wave communication in space in the early twentieth century. E. V. Appleton who
carried out comprehensive study was appreciated as a winner of Nobel Prize in 1947.
It is found that the electromagnetic wave approaching the cut-off density point is
reflected, and the reflecting point gets lower density for oblique incident to the
density gradient as seen in Fig. 2.15. In Fig. 2.15, trajectories of radio waves emitted
from an antenna are shown in red arrows, when radio waves are emitted from the
surface of the earth to the space. In space, the plasma layers are observed as the
ionosphere. For the case of oblique incidence to the ionosphere with angle θ, the
wave momentum in the perpendicular direction to the density gradient is conserved:
k sin θ : const:
Then, an effective wavenumber k x in the direction of the density gradient becomes:
c
2 k
2
x ¼ ω
2
À ω
2
pe À c
2 k
2 sin
2
θ
ð2:5:5Þ
This simple relation explains the trajectories of the radio waves in Fig. 2.15. The
waves are reflected at the point where k x ¼ 0 is satisfied. It is noted that the trajectory
of the wave is described by the same equation of a point mass in a potential in
Newton equation. In the past, one used such reflection by the ionosphere to send
radio waves far away.
For the convenience of further study, introduce the dielectric constant of the
plasmas. It is a function of k and ω in general; it is called dielectric function. In
Fig. 2.15 The schematic
picture showing the
propagation of radio waves
emitted from grand station
to the space. The plasmas in
the ionosphere reflect the
waves emitted obliquely.
Using this property, longdistance communication
was done before satellite
communication becomes
popular
2.5 Lasers in Plasmas
67