In the laser plasma interaction community, the related fields in one dimension along the z direction are often described by the normalized scalar potential φ(z, t) ≡ eΦ(z, t)/mc2 for the plasma electrostatic field (plasma wakefield) and the normalized vector potential a(z, t) ≡ eA(z, t)/mc2 for the laser field. We have Az = 0 if choosing the Coulomb gauge ∇ · A = 0. The peak of the normalized vector potentiala0, called laser strength parameter by the plasma community, is often used to determine the strength of the driving laser. Since the vector potential A is the spacial component of the 4-vector (ρ, A), the transverse components of A⊥ (Ax, Ay) are Lorentz invariant in any reference frame boosting along z direction. As a result, the a0 by definition is also a Lorentz invariant quantity. In MPWA study, we still follow the convention for laser case. The plasma fieldφ(z, t) is governed by the set of fluid equations
∂2φ with ignoring the ion motions. The influence of background magnetic field only takes place in the transverse momentum equation (Eq. (3.2c)), see Appendix A.
In these equations, we may neglect the thermal effect due to the assumptions:
(i) the electron quiver motion is much greater than the electron thermal motion (ii) the plasma temperature is so low that the thermal energy spread is not sufficient for the plasma to be trapped by the plasma wave. The Lorentzγ of plasma here defines (1−β⊥2−βz2)−1/2and theωcin Eq. (3.2c) gives the influence from the external magnetic field. While the normalized vector potential a(z, t)
satisfies the wave equation
Since the technology of high field laser has been well developed in a laboratory, laser wakefield acceleration is widely studied because of the possibility to the next generation of high accelerating gradient accelerators. When the driving pulse with frequencyω ωc (orωc → 0), the dispersion relation of the pulse approximates to that in unmagnetized plasma case. It is reasonable to study the LWFA mechanism under this limitation. Based on this consideration, the right hand side of Eq. (3.2c) can be ignored and Eq. (3.2c) is rewritten as
d
dt(γβ⊥− a) = 0. (3.4)
Therefore the transverse canonical momentum γβ⊥− a is conserved and the transverse velocity β⊥ = a/γ is easily obtained. Substituting the transverse velocity expression into Eq. (3.2d), we rewrite the Lorentz force in terms of the normalized vector potential a
where the second term on the right hand side is the ponderomotive force, the average of the second order Lorentz force. It is on the opposite direction to the gradient of laser intensity and is independent of the charge sign. Thus the electrons within the pulse are pushed away from the center and leave a positive region (ions are only barely moved by the same force), which generates the plasma wakefield.
3.2.1 Linear Regime
With the full set of fluid equations, we first study LWFA in the non-relativistic regime whereγ ∼ 1. In this regime, β⊥ is equal toa 1 so that the condition
for non-relativistic isa 1. Hence the complete set of fluid equations can be βz 1. It is convenient to write the equations in a co-moving coordinate system (ζ, τ) [44, 45], in which τ = t and ζ = z − ct. Then the derivatives
∂/∂z and ∂/∂t are replaced by ∂/∂ζ and ∂/∂τ − c∂/∂ζ respectively. If the laser pulse is sufficiently short, the field a andφ are expected to change very little during the transit time of the plasma through the pulse and the changes can be ignored in plasma reaction. Assuming that the laser envelop changes on a characteristic time scaleτe∼ 2|n0/n|(ω/ωp)/ωp, the quasistatic approximation (QSA) is applicable. In the QSA,∂/∂τ which determines the plasma response to the laser pulse are neglected in the plasma fluid equations . However,∂/∂τ is retained in the wave equation because it describes the evolution of the laser pulse [45, 32]. Thus for a short laser pulse, we can write Eq. (3.6a) to (3.6c) as
∂2φ
Substituting Eq. (3.7a) and (3.7b) into Eq. (3.7c), we arrive at
∂2
∂ζ2 +k2p
φ = k2p
2 a2 (3.8)
The solutions to the equation are easily calculated with the Green’s function such that[46]
φ ∼=kp
2 ζ
0 a2(ζ) sin[kp(ζ− ζ)]dζ (3.9) and the related axial field is obtained
Ez
Ewb =−1 kp
∂φ
∂ζ =χa20
whereχ = kp/(2a20) ζ
0 a2(ζ) cos[kp(ζ− ζ)]dζis the form factor which depends on the pulse shape. HereEwb≡ mcωp/e is the cold wavebreaking limit, charac-terizing the accelerating gradientG (G = eEz) of the plasma accelerator. Since kp=ωp/c, the solutions to Eq. (3.8) describe the plasma waves generated at the frequencyωp and are valid far from wavebreaking, Ez Ewb. Meanwhile the wakefields are generated sinusoidally and are efficient when the envelope scale length is on the order of the plasma wavelengthλp= 2πc/ωp[46].
3.2.2 Nonlinear Regime
When the laser power is extremely high such thata0 1, the plasma particle quiver motions become highly relativistic and a variety of nonlinear phenomena happens in the laser plasma interaction. It includes [45] (a) relativistic optical guiding of the laser beam[47, 48, 49], (b) the excitation of coherent radiation at harmonics of fundamental laser frequency, (c) the generation of large plasma wakefield, (d) frequency shift induced in the laser pulse by plasma waves[50, 51], (e) frequency amplification using an ionized front, and (f) the snow-plow acceleration[52, 53]. A full set of the fluid equations is required to describe the nonlinear phenomena. For the study of the generation of large plasma wakefield, we have the equations from Eq. (3.2a) to (3.2e) in terms ofa and φ
∂2φ
with neglectingωc. From Eq. (3.10c), the conservation of transverse canonical momentum gives γβ⊥ = a or γ = (1 + a2)1/2/(1 − βz2)1/2. So the pulse is described by the wave equation of a
∂2
which leads to the dispersion relation in the relativistic regime ω2=k2c2+ωp2
γ (3.12)
by assuming a plane wavea ∝ exp[i(kz−ωt)] and n = n0. We may combine this wave equation of a with the fluid equations to form a self-consistent equation set.
Insetting Eq. (3.10e) to Eq. (3.10d), we arrive at d
It is convenient to transform Eq. (3.13) into the new coordinate system (ζ, τ), then the second term on the right hand side vanishes. Together with the Poisson equation and the continuity equation, we obtain the complete equations forφ in (ζ, τ) coordinate with QSA applied
∂
in which Eq. (3.14c) expresses the potential of plasma wakefield. Equation (3.14a) and (3.14b) can be solved from the integration overζ. Since the plasma keeps stationary until the driving pulse passes through, the boundary conditions for the two equationsγ = 1, n = n0andβz= 0 are applied and give the solutions
and its quadratic form can be expressed as
n
withγ2= (1 +a2)/(1 − βz2). Hence the differential equation ofφ is written as
and the plasma wakefield normalized by E0 is obtained as ∂φ/∂ζ. We then express the plasma quantities in terms of the fieldsa and φ as
n/n0= 1 +1
2[(1 +a2)/(1 + φ)2− 1] (3.18a) γ = [1 + a2+ (1 +φ)2]/[2(1 + φ)] (3.18b) βz= [1 +a2− (1 + φ)2]/[1 + a2+ (1 +φ)2]. (3.18c) Considering the weak field limitφ 1, Eq. (3.17) becomes
∂2
∂ζ2 +k2p
φ = k2p
2 a2 (3.19)
after taking the Taylor expansion to the first power. This equation reduces to that in linear case (Eq. (3.8)).
Since Eq. (3.17) is fully nonlinear, its analytical solution only exists for a circularly polarized laser pulse with a square pulse profile as the laser envelop, aL=a0 for−L < ζ < 0 and aL = 0 otherwise [54, 55, 56]. For simplification, multiplying∂ϕ/∂ζ on both sides
∂2ϕ
where C1 = α2+ 1 is the integration constant determined by the boundary condition that ∂ϕ/∂ζ = 0 and ϕ(ζ) = 1 at ζ = 0. Within the pulse that
with
E(θ, k) here is the incomplete elliptic integration of the second kind and the second term on the right hand side of Eq. (3.22) indicates that ϕ is allowed to lie in the range 1 ≤ ϕ ≤ α2. Thus the maximum ϕ = α2 occurs at ζ = φ = 0. There is no wakefield excited behind the pulse.
Next with the laser pulse of lengthL = Lo, the equation of the plasma wakefield potential behind the pulse (ζ < −L and a = 0) is
Finally the axial electric fieldEz (wakefield) related toϕ is Ez
Ewb ≡ ˜Ez=−∂ϕ
∂ζ,
25 20 15 10 5 0
2
1 0 1 2 3
Ζ cΩp
EzEwb Δnn0
Δnn0
Ez
Figure 3.2: Density variationδn = n − n0 (dashed curve) and the axial electric fieldEznormalized byE0(solid curve). The Gray shaded region is the Gaussian pulse,a = a0exp[−(ζ + 5)2/22] witha0= 1.5.
and the field is given by Eq. (3.23) such that E˜z2 = −1
ϕ− ϕ + 1 α2 +α2
= a20− φ + 1
1 +a20 − 1 1 +φ.
Because ˜Ez is π/2 offset with φ, ˜Ez reaches the maximum when φ reaches 0.
So that the maximum ˜Ez is
E˜zmax= a20
1 +a20. (3.25)
We notice that in highly nonlinear regimea0 1, ˜Ezmax a0, the acceler-ating gradient is linearly proportional toa0, while in the linear regimea0 1, E˜z a20, the accelerating gradient is proportional to the quadratic ofa0, which is consistent with the result in the linear regime Eq. (3.10).
If in general cases with arbitrary laser pulse shapes, numerical calculations are essential to solve the equation Eq. (3.17). Assuming a circularly polarized gaussian pulse,a(ζ) = a0exp[−(ζ −ζ02/22)], we plot the solution of Eq. (3.17) in
25 20 15 10 5 0
0.010
0.005 0.000 0.005 0.010
Ζ cΩp
EzEwb Δnn0
Δnn0
Ez
Figure 3.3: Density variationδn = n − n0 (dashed curve) and the axial electric field Ez normalized by E0 (solid curve) for Gaussian pulse a = a0exp[−(ζ + 5)2/22] witha0= 0.1.
Fig. 3.2 and 3.3 fora0= 1.5 and 0.1. Here the solid curve presents the plasma wakefield and the dashed curve presents the density perturbationδn = n − n0. In the nonlinear case (a0= 1.5), the plasma wakefield exhibits a sawtooth-like shape and the plasma density piles up as a delta function. These were caused by the totally expelled electrons from a strong laser ponderomotive field. The plasma density piling up forms parallel charged plates which result in a linearly-varying electrostatic field between every two plates. As fora0 = 0.1, the plot as shown in Fig. 3.3 is purely sinusoidal, consistent with the result in the linear regime.
Chapter 4
Plasma Wakefield
Acceleration in Magnetized Plasma II
In Chap. 3 we have discussed the plasma wakefield underω ωc. In which the dispersion relation in Eq. (3.1) approaches a linear relation ofω to k, whose phase velocities are roughly equal to the speed of light. It is appropriately utilized as the driving pulse for plasma wakefield excitation. But whenω < ωc, the phase velocities of modes below the light curve (magnetowaves) are generally much less than c and vary with different k. Therefore a magnetowave pulse which is composed of different modes will quickly spread out during traveling.
Nevertheless, we will show that, under a special condition (MPWA condition), the magnetowave will behaves like the light in vacuum and can be considered as a new type of driving pulses. In this chapter, we will focus on the magnetowave modes and establish the general theory of MPWA.
4.1 MPWA Condition
The idea of MPWA is first working on Alfven modes[26]. Alfven wave is a magnetic tension wave, only existing in magnetized plasma (medium wave) and having a very low phase velocity. However for an effective plasma accelerator,
here we instead concentrate on that whistler modes (the higher frequency mode).
The dispersion relation of the whistler wave without considering the ion motion is given as
ω2=k2c2+ ωp2
1− ωc/ω, (4.1)
When the magnetic field is sufficiently strong such that ωc/ω 1 and ωcω/ω2p 1, the second term on the right hand side of Eq. (4.1) is negligi-ble. The whistler wave will have an approximately linear dispersion with phase velocity approachingc. It is instructive to combine the two linearity conditions into a chain inequality: (ωc/ωp)2 ωc/ω 1. Clearly, the range of ω com-patible with this chain inequality increases with the ratio ofωc/ωp. In other words, for a largerωc/ωp, the dispersion relation is approximately linear over a wider range of wavenumbers, as shown in Fig. 4.1. In the figure, there are three dispersion relation and phase velocity curves plotted withωc/ωp equal to 1, 6 and 12 respectively. It is obvious that the curves behave likely to a normal light wave over a wider wavenumber range while the ratioωc/ωp is sufficiently larger.
Thus, when ωc/ωp 1 is satisfied, the modes of whistler wave will contain coherent phase velocities which enables the whistler pulses to maintain their shape over a long distance, essential for an efficient plasma wakefield accelera-tor. Therefore, the requirement for MPWA isωc/ωp 1 where the dispersion relation is quasi-linear and the slope is near c. Such condition is referred to the
”MPWA condition”. In this chapter, the study of MPWA theory is under this condition.
4.2 Linear Theory
4.2.1 Ponderomotive Force
Once a whistler pulse is generated, the plasma wakefield will be sequentially excited by the ponderomotive force of the driving pulse. In Chap. 3, we have introduced the ponderomotive force as the gradient of the laser intensity. If ωc/ω is not negligible, the effect from background magnetic field should be taken into account. The non-relativistic ponderomotive force in magnetized plasma has been studied extensively in the past [57, 58]. Assuming an external
0 2 4 6 8 10 12 14 16 0.0
0.2 0.4 0.6 0.8 1.0
ωc/ ωp=6 ωc/ ωp=1 (b)
vph / c
ck/ ω’
p
0 2 4 6 8 10 12 14 16
ωc/ ωp=12 ωc/ ωp=1 ωc/ ωp=6 ω / ω p
(a)
ωc/ ωp=12
Figure 4.1: (a) Frequency and (b) phase velocity versus wavenumber for dif-ferent magnetic field strengths. When ωc/ωp 1, the dispersion relation is approximately linear over a wider range of wavenumbers with phase velocity approaching the speed of light.
magnetic field in z direction, according to [57], the longitudinal ponderomotive force acting on a unit volume is given as
fz= (ω) − 1 in-dex for waves propagation along the magnetic field in plasma (for whistler wave
= 1 − ωp2/ω(ω − ωc)). E⊥ is the slow-varying electric component of the wave E˜⊥ = 1/2(E⊥e−iωt+c.c.). Because whistler wave is a right-handed circularly polarized wave, it’s electric field can be written as E⊥ =E⊥(1, i)eikz. Substi-tuting(ω) and E⊥ into Eq. (4.2) and ignoring the effect of kinetic pressure, we arrive at
When taking theωc/ω = 0 limit, the expression (4.3) reduces to fz=−ωp2 which is the ponderomotive force in unmagnetized case.
4.2.2 Linear Formulation
With the ponderomotive force given above, we are able to calculate the plasma wakefield driven by the whistler pulse. Substituting Eq. (4.3) into the second term of the right hand side of Eq. (3.2d), the linear plasma wakefield can be formulated withγ ∼ 1 through the set of 1-D fluid equations (Eq. (3.2a),(3.2b) and (3.2d))
Under the MPWA condition whereωc/ωp 1, the phase velocity vph=ω/k ∼ c and the whistler pulse with central frequencyω would roughly travel at a group velocityvg ∼ vph∼ c. So that we can still rewrite the fluid equations in terms of the coordinate (ζ, τ) where ζ = z − vgt = z − ct and τ = t. Applying the QSA, we rearrange these equations as
∂ obtain the equation forφ
(∂ζ2+k2p)φ = 1
From Eq. (4.8), we notice that the denominator of the source term is quadratic so that this equation is applicable to waves with frequency both upper (R wave) and lower (whistler wave) branches. However there is a singularity forω = ωcin which the plasma will resonate with the cyclotron frequencyωc and eventually get heated. Therefore the wave propagation is forbidden. Whenω is extremely high compared toωc, the effect of ωc can be ignored and Eq. (4.8) reduces to the normal laser plasma wakefield equation in Eq. (3.8). With a fixed pulse frequency, the wakefield amplitude increases as the background magnetic field strength increases[43]. But for a whistler wave which has a frequency smaller than ωc, the plasma wakefield amplitude decreases as the magnetic field in-creases.
We can solve the analytic solution of Eq. (4.8) readily via the Green’s function with the boundary conditionsφ(ζ → ∞) = 0 and ∂φ(ζ → ∞)/∂ζ = 0 applied.
Hence Compared this result to that of the unmagnetized case in Eq. (3.10),Ez(ζ) has a multiplied factor 1/(1 − ωc/ω)2 from the influence of background mag-netic field. We may have an extra gain in the accelerating gradient withωc/ω approaching unity. For a circularly polarized Gaussian wavepacket of widthσ, i.e.,E⊥2 =E⊥02 exp(−ζ2/σ2) , the factorχ(ζ) can be calculated analytically as
4.3.1 MPWA Condition in Relativistic Regime
In astrophysical environment, the amplitude of magnetowave could be very in-tense and the plasma quiver motions become highly relativistic. In turn, the electron effective mass will be increased by a factor γ = (1 − β⊥2 − βz2)−1/2, causes the dispersion relation of whistler wave as
ω2=c2k2+ ωp2/γ
Considering theγ factor, the ponderomotive force from whistler waves in rela-tivistic regime becomes more complicated. It is not intuitive to write down the equations involving the ponderomotive force. So that we can only treat the full
fluid equations (Eq. (3.2a) to (3.2e)) such that,
Transforming the above equations into the (ζ,τ) coordinate and assuming the QSA condition, we obtain
∂2φ in which the total time derivatived/dt is replaced by −c(1−βz)∂/∂ζ. Under the MPWA condition wherevph∼ c, the Maxwell’s equation ∇×E= −(1/c)∂B/∂t claims
∂ζEx = ∂ζBy
∂ζEy = −∂ζBx.
Hence allB components in Eq. (4.15d) can be replaced by the E components,
−(1 − βz)∂
∂ζ(γβz) = ∂φ
∂ζ − e
mc2(βxEx− βyEy). (4.16) Substituting Eq. (4.15e) into Eq. (4.16), we obtain
∂
∂ζ(φ − γ(1 − βz)) = 0, (4.17) and together with Eq. (4.15b),
φ − γ(1 − βz) =−1, (4.18a)
n(1 − βz) =n0 (4.18b)
after integrating overζ and applying the boundary condition. They are in the same forms as those in the unmagnetized plasma case (Eq. (3.15a) and (3.15b)) [45]. The main difference between the two cases is on the Lorentz factorγ which defines (1− β2⊥− βz2)−1/2.
We can solve the transverse fluid velocityβ⊥directly from Eq. (4.15c). Under the QSA, the transverseβ⊥ is obtained as
β⊥= a
|γ − ωc
ω(1 − βz) |
. (4.19)
We note that in magnetized plasma the condition for non-relativistic case where γ ∼ 1, β⊥ 1 and βz 1 requires a ωc/ω − 1 (a 1 for unmagnetized plasma). Thus, the system could be still in non-relativistic regime even with a0> 1 so long as ωc/ω is much greater than a0.
Therefore by combining Eq. (4.20), (4.18a) and (4.18b) , the Poisson equation for the plasma wakefield (Eq. (4.15a)) becomes
∂2φ
which is also valid in all frequency ranges (Appendix B). We discuss the equation in two limits. For ω ωc (ωc/ω → 0), this equation reduces to that in the unmagnetized plasma (Eq. (3.17)) [45]; and forφ 1, it is easy to show that Eq. (4.21) returns to the non-relativistic MPWA equation in Eq. (4.8).
4.3.3 Numerical Results
Since Eq. (4.21) is fully nonlinear, there is no analytical solution found to the equation. Thus the only way to solve the equation is numerical calculation. As-sumingωc/ω = 5, we plot the solutions of plasma wakefield in Figs. 4.2 and 4.3 with a0 = 1 and a0 = 4 respectively. The plasma is driven by the whistler gaussian pulse with a width √
2(c/ωp). The solid curves denote the plasma
25 20 15 10 5 0
0.10
0.05 0.00 0.05 0.10
Ζ cΩp
EzEwb Δnn0
Δnn0
Ez
Figure 4.2: Density variationδn/n and axial field Ezfor whistler gaussian pulse located atζ = −5(c/ωp) anda0= 1.
wakefield amplitude normalized byEwband the dashed curves superimposed in the figures are the plasma density variationδn/n = n/n0− 1 in terms of fields a and φ derived from Eq. (4.18b), (4.18a) and (4.19). In a0 = 1 case, where a0 < ωc/ω − 1 = 4, the plasma wakefield behaves like sinusoidal. But in the other case, the plasma starts piling up and the associated axialEz(the plasma wakefield) becomes sawtooth-like whena0= 4.
4.4 Limitation of MPWA
By looking at the successful derivation of MPWA equation, we note that the right hand side of Eq. (4.21) becomes singular as 1 +φ → ωc/ω. In such a limit, both the slope of Ez and the plasma density become infinite, which indicates the occurrence of wavebreaking. Beyond this point, the development of plasma waves is expected to become turbulent due to the instability, and our fluid equation analysis will break down. The electric field is expected to remain finite since the amplitude of a relativistic plasma wave is proportional to √γ = (1 − βz,max2 )−1/4 where βz,max is the maximum electron velocity in
25 20 15 10 5 0
2 0 2 4 6
Ζ cΩp
EzEwb Δnn0
Δnn0
Ez
Figure 4.3: Density variationδn/n and axial field Ezfor whistler gaussian pulse located atζ = −5(c/ωp) anda0= 4.
the wave. The above infinite-density situation would not occur if the strength parametera0is smaller than an upper bound determined by the ratioωc/ω and the shape of the whistler pulse[60].
In order to study the sensitivity ofωc/ω ≡ b to the limit of a0, we compare three results of Eq. (4.21) corresponding tob = 0, 0 < b < 1 and b > 1.
4.4.1 Three Cases
1. b = 0
Whenb = 0, there is no background magnetic field, Eq. (4.21) can be reduced to
∂2ϕ
∂ζ2 = 1 2
α2 ϕ2 − 1
, (4.22)
whereϕ ≡ 1 + φ, α2≡ 1 + a20andζ ≡ kpζ. Assuming a circularly polarized square driving pulse with a(ζ) = a0 for −L ≤ ζ ≤ 0, and a(ζ) = 0 elsewhere, the equation within the pulse is integrated by multiplying∂ϕ/∂ζ on both sides
1 Α2
Ez'
(a)
1 Α2
1Ez'
(b)
Figure 4.4: The plots of (a)|Ez| and (b) |1/Ez| versus ϕ with b = 0 and a0= 3, whereϕ ≡ 1 + φ and α2≡ 1 + a20.
of Eq. (4.22). Subjecting the boundary conditionsϕ(ζ) = 1 and (∂ϕ/∂ζ) = 0 terms have to be both positive to satisfy the inequality andϕ is constrained to be 1≤ ϕ ≤ α2. Another possible solution to the inequality with both terms negative is ruled out because no overlapped ϕ for ϕ < 1 and ϕ > α2 exists.
Thus from the inequality solution ofϕ, the maximum ϕ is α2 at an optimized length where ∂ϕ/∂ζ = 0. There is no upper bound for α and neither is the plasma wakefield potential. We can clearly show that in the plots of |Ez| and
|1/Ez| versus ϕ (Fig. 4.4(a) and Fig. 4.4(a)).
2. 0< b < 1
Following the same strategy, we findEz2 from the integration of Eq. (4.21) in 0< b < 1 case,
Again, in order to satisfy the inequality, one requiresa20/(b−ϕ)(b−1)+1/ϕ−1 ≥ 0. It is easy to show that, forb < 1, the range of ϕ is solved as
where the square root is always real because of the reason, (1−ba20 +1+b)2−4b >
(1−ba20 )2+ (1− b)2> 0. Therefore the maximum of ϕ is determined by a0 andb and no upper bound on a0 exists. From Eq. (4.27), whenB0 (or b) increases, the maximumϕ as well as the maximum Ezalso enhance accordingly, consistent with the conclusion by P. K. Shukla in Ref. of [43].
0 1 b
Ez'
a
(a)
0 1 b
1Ez'
b
(b)
Figure 4.5: The plots of (a)|Ez| and (b) |1/Ez| versus ϕ with b = 5 and a0= 2.3 (<√
b − 1(√
b − 1) = 2.47).
3. b > 1
Finally we treat the MPWA case which has b > 1. Unlike the two previous cases, the range ofϕ is solved as
b > ϕ ≥
Mathematically, if the square root
(a20/(1 − b) + 1 + b)2− 4b in the nu-merator is real, then there exists two solutions for the range of ϕ. It can be traced back to the formula of β⊥ (Eq. (4.19)) in which a small a0 allows two possible solutions for β⊥s, i.e., β⊥ 1 or β⊥ ∼ 1 when γ → b/(1 − βz). For-tunately, the boundary conditions,β⊥(0) =βz(0) = 0 andγ(0) = 1, help us to
(a20/(1 − b) + 1 + b)2− 4b in the nu-merator is real, then there exists two solutions for the range of ϕ. It can be traced back to the formula of β⊥ (Eq. (4.19)) in which a small a0 allows two possible solutions for β⊥s, i.e., β⊥ 1 or β⊥ ∼ 1 when γ → b/(1 − βz). For-tunately, the boundary conditions,β⊥(0) =βz(0) = 0 andγ(0) = 1, help us to