INTRODUCTION
Fast electron propagation in plasma is always accompanied with self generated
electric and magnetic fields which affect on plasma particles density and electron
beam propagational characteristics. Self electric and magnetic fields make an
electron beam collimated or focused and filamented (Hammer
and Rostoker, 1970; Robinson and Sherlock, 2007).
In some applications especially in Inertial Confinement Fusion (ICF) at fast
ignition scheme, it is necessary to keep beam intensity and to control beam
diverging or even increasing beam spot size intensity through focusing. The
requirement for fast ignition are currently to be an electron beam energy at
least 25 kJ, duration 1020 ps, radius ~20 μm and electron kinetic energy
of E_{b} ≈ 0.32 MeV. The beam current typically is larger than
3x10^{8} Amp and the maximum magnetic field associated with this huge
current is tens of mega Gauss (Li and Petrasso, 2006).
In subrelativistic regime self electric field in vacuum is dominant over the
magnetic field and the electron beam suffers naturally diverging but with increasing
beam energy in relativistic regime (>1 MeV) up to ultra relativistic limit
(v~c) magnetic force grows and is finally almost equal to electric force, and
beam may be collimated (Humphries, 1990). In plasma, there
is a different situation due to plasma response to beam presence, the magnetic
field can be so large that beam may be focused. Beam hollowing in very high
intensity and in time evolution may be generated and produces an annular pattern
in which varies the beam spot size.
SELF GENERATED FIELDS
Self generated fields in vacuum: When an electron beam moves in vacuum,
natural columbic repulsion expels the electrons off axes direction, the sSelf
electric field of the beam in vacuum is calculated as follows (Humphries,
1990):
If the electron beam density is considered as:
then the electron field will be obtained as:
The self magnetic field is then obtained as:
We can also determine the relation between two forces as; F_{rB} (magnetic)
= β^{2}F_{rE} (electric). Figure 1 shows
the electric and magnetic forces as defocusing and focusing forces. The forces
ratio, F_{B}/F_{E} approaches to unit with increasing beam velocity.

Fig. 1: 
Electric and magnetic forces act on electron with Gaussian
beam profile in vacuum (β = 0.6) 
In ultra relativistic limit (v~c) two forces balance and the electron beam
moves collimately.
Spacecharge potential and Wakefield due to beam arriving in plasma:
Beam behavior in plasma is very different from vacuum. When an external charge
arrives is arrives in plasma medium, the space charge potential is generated
and plasma particles (mostly electrons) distribution are perturbed. Plasma particles
are rearranged as the bulk of plasma keep and itself enclosed from this external
charge. Plasma shields the charge and after some Debye length the potential
in plasma returns back to its initial reference. If an electron beam arrives
in plasma, the spacecharge potential according to poison’s equation is
generated .This potential affects on plasma and beam particles distribution.
For an initial uniform plasma as; n_{e} = n_{i} = n_{o},
after beam arriving, the Poisson’s equation is written as (Humphries,
1990):
where, n_{b} = n_{bo}f(r) is beam profile. For the scale potential in terms of
and the length in terms of the Debye length, R = r/λ_{D}, the normalized form of Poisson’s equation is rewritten as:

Fig. 2: 
Normalized spacecharge potential variation (dashed line),
plasma electron variation (solid line) and electron beam density variation
(dotted line) due to Gaussian electron beam profile in unmagnetized uniform
plasma versus normalized radial distance 
Where:
and
therefore,
Now we can investigate the behavior of plasma and the beam particles after
spacecharge potential generation. In Fig. 2 we can see that
the spacecharge potential, after some Debye length, returns back to its initial
reference(zero). Also Fig. 2 shows that the plasma electrons
density is modulated as: n_{e} (r) = n_{e0} exp (ø)
(Humphries, 1990; Esarey et al.,
1996). To keep charge neutralization in plasma after beam arriving, δn_{e}
= n_{b} should be established. As we can see from Fig.
3, the electric field related to this plasma potential is directed offaxes,
therefore, the force on the electrons of beam is directed onaxes and this field
acts as a focusing force.

Fig. 3: 
Color map describes spacecharge potential but vectors indicate
the electric field related to this potential. This field is directed offaxes
therefore acts as a focusing force on the electrons of beam 
If electron beam terminates in a time short compared to ω_{pe}^{1},
in plasma Wakefield of the form, δn = n_{b} sin k_{p} (zct)
will be generated (Esarey et al., 1996). The
axial electric field in linear regime is given by,
so, we will get:
Figure 4 shows plasma Wakefield in two different beam densities with assuming a very long Gaussian beam:
arriving in plasma with
When beam density approaches to plasma density, the Wakefield and plasma electron density perturbation approaches to a beatlike pattern (Fig. 5). Beam and plasma electrons accelerate and decelerate due to this Wakefield. For n_{b0} = 10^{2}n_{0} we obtain,
and for the case n_{b0} = 10^{5}n_{0} we obtain:

Fig. 4: 
Wakefield in two beam density values. Higher value of the
beam density makes Wakefield beat like pattern 

Fig. 5: 
Wakefield and relative variation of plasma electron density
(shifted by 10^{5} times) versus beam propagating direction due
to a long Gaussian beam arriving in plasma. Wakefield with beat like pattern
makes density variation as beat like as 
which the later is negligible (Fig. 6).
Timeindependent self magnetic field of electron beam in plasma: The
magnetic field of an electron beam can affect on its propagational characteristics
and bends the particles trajectory. We consider an infinitely long cylindrical
fast electron beam with the Lorentz factor γ_{b}, the radius r_{b}
and the current of l_{b} propagating in plasma.

Fig. 6: 
Plasma electrons perturbation in two different beam density
magnitudes. The oscillating pattern in beam density closer to plasma density
is very different with the case of very lower 
According to Ampere’s law the time independent azimuthal magnetic field
is given by:
Where:
is the fraction of current contained within a radius, r. The equation of motion
for an electron beam in this field is given by Storm (2009):
recasting the Eq. 6 in terms of the dimensionless variables;
,
V_{z} = v_{z}/βc, τ = tv_{z}/r_{b}
and using c^{2} = 1/ε_{0}μ_{0}, the dimensionless
equation of motion is:
The quantity in bracket has units of current and is defined as the Alfven current I_{A}. After substituting the physical constants the Alfven current for electrons can be expressed as I_{A}≈17.1 β_{b}γ_{b}[KA].

Fig. 7: 
Trajectory of electrons in self generated magnetic field of
a uniform beam intensity profile in plasma medium (I_{b} = 0.5 I_{α}).
Only considering magnetic field there is a certain focal point 
If the electron beam current density is assumed uniform in radial direction
then current fraction becomes:
and the equation of motion becomes;
The numerical solution of the above differential equation leads to a curve that is shown in Fig. 7. The particles of the beam in different radius are focused on a certain point, but actually columbic repulsion makes impossible to reach a focused point. If we consider the current density profile as Gaussian;
then we will have;
Applying this profile with I_{b} = 0.51_{A} (in inertial confinement
fusion at fast ignition scheme I_{b}>>I_{A}) in Eq.
7 leads to Fig. 8. As we can see, the particles in distant
points experience larger magnetic field and therefore, have near focusing point
than particles in closer points to beam axes. For I_{b}>>I_{A}
the electrons trajectory so large bend in which the backward current is made
and return current will partially neutralize the magnetic field due to the forward
directed current.

Fig. 8: 
Trajectory of electrons in self generated magnetic field of
a non uniform beam intensity profile [j = j_{0}exp(r^{2}/2rb^{2})]
in plasma medium (with I_{b} = 0.5 I_{α}). With considering
only magnetic field we will have several focal points. Electrons in larger
radius have closer focal points 
As a consequence, the fastelectron current and the corresponding magnetic
field are reduced (Storm, 2009). Charged currents can
be transported in vacuum only up to a maximum current, the socalled Alfven
current (Atzeni and MeyerterVehn, 2009). The physical
reason for this limit is that currents larger than I_{A} generate a
magnetic field large enough that the Larmor radius of the electrons becomes
smaller than the beam radius. As a result the beam electrons are not further
transported in beam direction. Although the Alfven current limit does not apply
globally to beam transport in plasma, it appears that the current in each filament
cannot exceed I_{A} by a large factor (Atzeni and
MeyerterVehn, 2009). We should notice that fast ignition requires fast
electron beam with currents on the order of I_{b} ≈ 1000 I_{A}.
If electrical or magnetically neutrality is fractional the beam transportable
current in plasma becomes more than without them. If f_{E} is the fractional
electrical neutrality the limiting current becomes (Hammer
and Rostoker, 1970):
And if in addition f_{M} is the fractional degree of magnetic neutralization the limiting current becomes:
Time dependent self magnetic field of electron beam in plasma: The
magnetic field at the previous section is assumed time independent and therefore
Ampere’s law has been applied.

Fig. 9: 
Target resistivity variation due to beam arriving with Gaussian
profile according to Spitzer limit for hot electrons(solid line) and temperature
variation of target (dashed line)versus radial distance in t = 50 ps 
However, in more real situation for a non uniforms beam, plasma resistivity
is timespace dependent. Considering Faraday, s law for the rate of magnetic
field, we have (Robinson and Sherlock, 2007; Humphries,
1990; Davies et al., 2006):
where, η and j_{b} are plasma resistivity and beam current density,
respectively. Assuming rigid model beam (which assumes a static beam) and hot
plasma with Spitzer limit for resistivity we have (Davies
et al., 2006):
The coupling of the above equations gives hand:
where, C is a constant heat capacity and index zero indicates the initial situation.
Finally we can determine the resistivity gradient as below:

Fig. 10: 
Plasma resistivity contours [η (r, t)]. Each contour
shows an equivalent resistivity in different timespace 

Fig. 11: 
Plasma temperature contours [T(r, t) ]. Each contour shows
an equivalent temperature in different timespace 
Figure 9 shows plasma temperature and resistivity at a same
framework according to previous discussion. Figure 10 shows
that resistivity decreases at a certain radius with time increasing and at a
certain time is larger in distant point of the beam axes.

Fig. 12: 
Different self magnetic contributions and their resultant
in t = 100 ps. Positive part is ∇ηxj_{b} which defocuses
the beam and it is responsible to hollowing effect and negative part is
η ∇Hj_{b} which focuses the beam 

Fig. 13: 
Plasma resistivity gradient contours [∇ η(r, t)].
Each contour shows an equivalent resistivity gradient in different timespace 
As we can see from Fig. 11, plasma temperature increases
with time increasing and also at a certain time, temperature is maximum on the
beam propagating axes. The self magnetic of the beam related to terms η∇Hj_{b}
and ∇ηxj_{b} produce different behavior for electrons in
the beam (Solodov et al., 2008).

Fig. 14: 
Self magnetic field in different times versus radial distance.
In high intensity but not very long time there is no hollowing (typical
time duration of beam in fast ignition is 1020 ps). Hollowing effect becomes
significant with time increasing 

Fig. 15: 
Total self magnetic field contours B_{T} (r, t). Each
contour shows an equivalent magnetic field in different timespace 
η∇Hj_{b} Pushes the electrons to higher j_{b} and
acts as a onaxes force and focuses the beam but on the other hand ∇ηxj_{b}
pushes the electrons to higher η and acts as a offaxes force and defocuses
the beam (Robinson and Sherlock, 2007; Humphries,
1990; Storm, 2008; Kingham et
al., 2010). Figure 12 shows the effects of these
two parts which have been calculated numerically in high current intensity j_{bO}≥10^{15}
(A/m^{2}) and long pulse (t≥100 ps). The center part of the beam
is purely defocused and hollowing effect occurs and the beam experiences an
annular pattern (Davies et al., 2006). The effects
of two terms vary with time increasing. According to Fig. 13,
∇η increases with the increasing of time, therefore, the magnetic
field related to ∇ηxj_{b} dominates than η∇Hj_{b}.
As a result, if the pulse time to be long enough (in high intensity of beam)
the hollowing occurrence becomes more possible (Fig. 14)
(Norreys et al., 2006). Figure
15 shows magnetic field contours B_{T} (r,t) which Plus (minus)
contours indicate defocusing (focusing) part.
CONCLUSIONS
Columbic repulsion naturally expels the electrons of a the beam to an offaxes direction but in plasma medium, spacecharge potential acts as a focusing force. The plasma Wakefield, due to charge perturbation, generates an oscillating pattern field and therefore , the electrons accelerate and decelerate. The plasma Wakefield effect is negligible in the case of the low density. In the case of high intensity beam, the electric fields are less important than magnetic field. In time dependent situation, Fraday’s law determines the magnetic field. We showed that in high intensity and long pulse duration hollowing effect is obvious. In fast ignition, where duration is about 1020 ps the hollowing effect in intensity 10^{15} (A/m^{2}) is not significant but in time t>70 ps in our model an annular pattern is produced.