800
C. Joshi et al.
12.1.2 Physical Concepts
We now discuss some of the important physical concepts in plasma accelerators.
These concepts will determine the necessary laser/beam and plasma conditions for
the efficient acceleration of electrons. Unless specified otherwise the expressions
given in this section are in the limit of small amplitude or sinusoidally varying
(linear) wakes.
12.1.2.1 Phase Velocity v φ
The phase velocity, v φ , of the accelerating wake structure is equal to the group
velocity of the driver. In the case of a LPA, v φ of the wakefield is equal to
v g = c
1 − ω 2
p /ω 2
o
1/2
of the laser pulse as it propagates through the plasma.
Often it is also useful to define a relativistic Lorentz factor of the wake γ φ =
1 − v 2
φ /c 2
−1/2 ω o /ω p . For a PWA driven by a highly relativistic electron
bunch v φ ≈ c.
12.1.2.2 Dephasing Length L d
In a LPA, electrons that are accelerated by the wakefield can gain enough
energy as to move faster than the phase velocity of the accelerating wake
structure. The difference in velocity causes a relative slip in position between
the electron beam and the accelerating phase of the wakefield. Eventually the
electron beam will slip out of the accelerating phase and into the decelerating
phase of the wakefield and begin to lose energy. The length over which the
electron beam can gain energy from the accelerating phase of the wakefield is
known as the dephasing length, L d . Within the wake, the accelerating phase
extends over approximately half a plasma wavelength and the dephasing length
is approximated by assuming that the velocity of the relativistic electrons
equals c. The dephasing length is then given by L d = cλ p /{2(c − v φ )}. This
length limits the maximum energy gain that an electron experiences in a
LPA.
In a PWA driven by an electron beam, where v φ ≈ c energy gain is not limited
by dephasing. However, in a proton driven PWA the dephasing between the drive
and witness bunch must be considered. The phase slippage between the drive and
witness bunch has been estimated as [5]
δ ≈
πL
λ p
1
γ f γ i
,
(12.3)
C. Joshi et al.
12.1.2 Physical Concepts
We now discuss some of the important physical concepts in plasma accelerators.
These concepts will determine the necessary laser/beam and plasma conditions for
the efficient acceleration of electrons. Unless specified otherwise the expressions
given in this section are in the limit of small amplitude or sinusoidally varying
(linear) wakes.
12.1.2.1 Phase Velocity v φ
The phase velocity, v φ , of the accelerating wake structure is equal to the group
velocity of the driver. In the case of a LPA, v φ of the wakefield is equal to
v g = c
1 − ω 2
p /ω 2
o
1/2
of the laser pulse as it propagates through the plasma.
Often it is also useful to define a relativistic Lorentz factor of the wake γ φ =
1 − v 2
φ /c 2
−1/2 ω o /ω p . For a PWA driven by a highly relativistic electron
bunch v φ ≈ c.
12.1.2.2 Dephasing Length L d
In a LPA, electrons that are accelerated by the wakefield can gain enough
energy as to move faster than the phase velocity of the accelerating wake
structure. The difference in velocity causes a relative slip in position between
the electron beam and the accelerating phase of the wakefield. Eventually the
electron beam will slip out of the accelerating phase and into the decelerating
phase of the wakefield and begin to lose energy. The length over which the
electron beam can gain energy from the accelerating phase of the wakefield is
known as the dephasing length, L d . Within the wake, the accelerating phase
extends over approximately half a plasma wavelength and the dephasing length
is approximated by assuming that the velocity of the relativistic electrons
equals c. The dephasing length is then given by L d = cλ p /{2(c − v φ )}. This
length limits the maximum energy gain that an electron experiences in a
LPA.
In a PWA driven by an electron beam, where v φ ≈ c energy gain is not limited
by dephasing. However, in a proton driven PWA the dephasing between the drive
and witness bunch must be considered. The phase slippage between the drive and
witness bunch has been estimated as [5]
δ ≈
πL
λ p
1
γ f γ i
,
(12.3)