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International Journal of Enhanced Research Publications, ISSN: XXXX-XXXXVol. 2 Issue 4, April-2013, pp: (1-4), Available online at: www.erpublications.com

111Equation Chapter 1 Section 1Self-fields effects on gain in a helical wiggler free electron laser with

ion-channel guiding and axial magnetic field

Hassan Ehsani Amri1 ; Taghi Mohsenpour2

1 Department of Physics, Islamic azad University,nour branch, nour, Iran 2Department of Physics, Faculty of Basic Sciences, University of Mazandaran, Babolsar, Iran

Abstract: In this paper, we have investigated the effects of self-fields on gain in a helical wiggler free electron laser with axial magnetic field and ion-channel guiding. The self-electric and self-magnetic fields of a relativistic electron beam passing through a helical wiggler are analyzed. The electron trajectories and the small signal gain are derived. Numerical investigation is shown that for group I orbits, gain decrement is obtained relative to the absence of the self-fields, while for group II orbit gain enhancement is obtained.

Keywords: Free-electron laser, self-fields, ion-channel, axial magnetic field, gain

Introduction

A Raman free-electron laser (FEL) produces coherent radiation by passage of a cold intense relativistic electron beam through a static magnetic (wiggler) field which is spatially periodic along the beam axis. In Raman regime, due to the high density and low energy of the electron beam, an axial magnetic field generated by current in a solenoid is usually employed to focus on the beam [1-4]. Also the ion-channel guiding is used to focus the electrons against the self-repulsive electrostatic force generated by the beam itself [5-9]. In Raman regime, equilibrium self-electric and self-magnetic fields, due to the charge and current densities of the beam, can have considerable effects on the equilibrium orbits. It has been shown that self-field can induce chaos in the single-particle trajectories [10]. The effects of self-fields on the stability of equilibrium trajectories and gain were studied in a FEL with a one-dimensional helical wiggler and axial magnetic field [11,12] or ion-channel guiding [13-14].In recent years, electron trajectories and gain in a planar and helical wiggler FEL with ion-channel guiding and axial magnetic field were studied, detailed analysis of the stability and negative mass regimes were considered [15-18]. The purpose of this paper is to study the effects of the self-fields on gain in a FEL with ion-channel guiding and axial magnetic field. In Sec. 2.1, steady-state trajectories are obtained in the absence of self-fields. In Sec. 2.2, self-fields are calculated using Poisson,s equation and Ampere,s law. Equilibrium orbits are found under the influence of self-fields. In Sec. 2.3, the derivation of the gain equation is presented. In Sec. 3, the numerical results are given. In Sec. 4, concluding remarks are made.

Theory Details

A. Basic AssumptionThe evolution of the motion of a single electron in a FEL is governed by the relativistic Lorentz equation

(1)

where Ei is the transverse electrostatic field of an ion-channel and it can be written as

(2)

and Bw is the idealized helical wiggler magnetic field and can be described byBw=Bw ( x coskw z+ y sin k w z ) (3)

Page | 1

zwi B

ce

tdmd eBvEv

01

yxE ˆˆ2 yxne ii


and B0 is the axial static magnetic field. Here, Bw denotes the wiggler amplitude, k w (¿2 π / λw ) is the wiggler wave

number, λw is the wiggler wavelength (period), ni is the density of positive ions having charge e. The steady-state velocity of an electron moving in this field is

v0=vw ( x cos kw z+ y sin kw z+v|| z ) (4)

where v|| is the component in the positive z direction and

vw=k w v||

2Ωw

ωi2+kw v||(Ω0−kw v||) (5)

is the signed magnitude of the wiggler-induced transverse velocity. ωi=(2 πni e

2 /γm )1/2 , is the ion-channel

frequency and Ω0 ,w=eB0 ,w/ γmc , are the axial guide and wiggler magnetic field frequencies.

B. Self-Field Calculation and Steady-State OrbitsThe self-electric and self-magnetic fields are induced by the steady-state charge density and current of the non-neutral electron beam. Solving Poisson’s equation

1r∂∂ r (rEr

( s) )=−4 π enb (r ) (6)

yields the self-electric field in the form, E( s )=−2πenbr r=−2πenb ( x x+ y y ) (7)

where nb is the number density of the electrons. The self-magnetic field is determined by Ampere,s law,

∇×B=4 πc

Jb (8)

where Jb=−enb (v¿+v||ez ) is the beam current density and v¿ is the transverse velocity. In cylindrical coordinates

(r , θ , z ) , Ampere,s law may be written in the form,

1r∂∂ θ

Bz(s )− ∂∂ z

Bθ( s )=−

4 πenb (r ) v¿c

cos (kw z−θ ) (9)

∂∂ z

Br(s )− ∂∂ r

B z(s )=−

4 πenb (r ) v¿c

sin (k w z−θ ) (10)

1r∂∂ r (rBθ

( s ) )−1r∂∂θ

B r(s )=−

4 πenb (r ) v||c (11)

where v¿=vw . The solution of Eqs. (9)-(11) can be obtained by using the methods used in reference 10 (or references 12 and 13). This yields

Bs(1 )=Bs||+Bsw

(1 ) (12)

where

Bsθ=−2πenb

v||c

r eθ=2πenb

v||c

( y x−x y ) (13)

Bsw(1 )=

4 πenb (r )kw

vw

c2 [er cos (kw z−θ )+eθ sin (kw z−θ ) ]

=2ωb

2 ( v||2/c2)ωi

2+k w v||(Ω0−k w v||)Bw

(14)

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and ωb=(2 πnb e2 /γm )1/2 . The first term in Eq. (12) is due to axial velocity; the second term is due to the transverse

velocity induced by the wiggler magnetic field. With inclusion of the self-magnetic field, the total magnetic field up to first-order correction, may be written as

B(1 )=Bw+B s(1 )+B0 ez=λ(1 ) Bw+2 πenb

v||c

( y x− x y )+B0 ez (15)

where

λ (1 )=1+2 ωb

2 (v||2 /c2)ωi

2+kw v||(Ω0−k w v||) (16) is the first-order correction factor for the wiggler magnetic field. The equation of electron motion in the presence of self-fields may be written (in the scalar form):d (γmv x )

dt=−2 πe2 (ni−nb) x−

2 πnb e2

c2 v||v z x+ev z

cλ (1 ) Bw sin k w z−

eB0

cvx

(17)d (γmv y )

dt=−2 πe2 (n i−nb) y−

2 πnb e2

c2 v||v z y−ev z

cλ(1 ) Bw cos kw z+

eB0

cv y

(18)d (γmv z)

dt=− e

c [ λ(1 ) Bw (vx sin k w z−v y cos kw z )−2 πnb ev||c (v x x+v y y ) ]

(19)

The steady-state solution of Eqs. (17) and (18) may be expressed in the form, vx=v w' cosk w z and v y=vw

' sin kw z , where

vw' =

k w v||2 λ(1 )Ωw

ωi2−ωb

2 γ||−2+kw v||(Ω0−kw v||) (20)

and vz≃v||=constant

. Equation (20) describes the transverse velocity in the presence of the self-electric and first order self-magnetic field. Substituting new electron velocity components into Eq. (8), and using the method applied in reference 10 (or references 12 and 13), we obtain the wiggler-induced self-magnetic field up to second-order correction as

Bsw(2 )=Kλ (1 ) Bw (21)

where

K=2ωb

2 (v||2 /c2)ωi

2+kw v||(Ω0−k w v||) (22)

and γ||=(1−v||

2 /c2 )−1/2

. Using Eq. (21), the total magnetic field up to second-order correction is given by

B(1 )=λ (2 ) Bw+2πenb

v||c

( y x−x y )+B0 ez (23)

where

λ (2 )=1+Kλ (1 ). (24)

This process may be continued to find the higher order terms. We can write Bsw

(n )=Kλ (n−1 ) Bw , n=3,4,5,⋯ (25)where

λ (n )=1+Kλ (n−1 ) , n=2,3,4 ,⋯ (26) The total wiggler-induced magnetic field becomes

Bsw= limn→∞

Bsw(n )=KλBw

(27) where

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λ= limn→∞

λ (n )= limn→∞∑i=0

n−1

K i+ limn→∞

2ωb2 (v||2 /c2)

ωi2+kw v||(Ω0−k w v||)

K n

(28)

If the absolute value of K is less than one, then the series in Eq. (28) will converge to 1/ (1−K ) , and the last term in the right-hand side will goes to zero. In this case, Eq. (28) may be expressed in the form

λ=ωi

2−ωb2 γ||

−2+kw v||(Ω0−kw v||)ωi

2−ωb2 [1+( v||2/c2) ]+k w v||(Ω0−k w v||) (29)

Substituting Eq. (29) into Eq. (27) and using Eq. (22), the total wiggler-induced self-magnetic field is given by

Bsw=2ωb

2 ( v||2 /c2 )

ωi2−ωb

2 [1+(v||2 /c2 )]+kw v||(Ω0−kw v||) (30)The transverse part of the steady-state helical trajectories of electrons, in the presence of self-fields, can be found as

vw=kw v||

2Ωw

ωi2−ωb

2 [1+(v||2 /c2) ]+k w v||(Ω0−k w v||) (31) Equation (31) shows resonant enhancement in the magnitude of the transverse velocity when

ωi2−ωb

2 [1+(v||2 /c2 )]+kw v||(Ω0−kw v||)≈0 (32)

Steady-state trajectories may be classified according to the type of guiding. There are three main categories:

1. When the value of axial magnetic field is zero (Ω0=0) ; that is, a FEL with a helical wiggler and an ion-channel guiding. In this type of guiding Eq. (31) becomes,

vw=k w v||

2Ωw

ωi2−ωb

2 [1+(v||2 /c2) ]−kw2 v||

2 (33)

2. When the density of positive ions is zero (ωi=0) ; that is, a FEL with a helical wiggler and an axial magnetic field. In this type of guiding Eq. (36) becomes,

vw=kw v||

2Ωw

kw v||(Ω0−k w v||)−ωb2 [1+(v||2 /c2 )] (34)

3. When both the ion-channel and the axial magnetic field are present; that is, a helical wiggler FEL with an ion-channel

and an axial magnetic field. When both types of guiding are present, vw is given by Eq. (31). This case contains three

types: (a) two guiding frequencies are taken to be equal (Ω0=ωi ), (b) the ion-channel frequency is taken to be constant, (c) the axial magnetic field frequency is taken to be constant.

C. Small Signal Gain Let electromagnetic radiation copropagate with the electron beam in the FEL interaction region, such as below

{E r=Er ( x cos ζ− y sin ζ ) ¿¿¿¿ (35)

where ζ=kr z−ωr t+φ , ωr (¿kr c ) is frequency, and Er is the amplitude of the wave. The electron equation of motion in the presence of the electromagnetic radiation and self-fields can be written in scalar form

γ βx+γ βx=−[2 πe2 (ni−nb )mc

+2πnb e2

mcv||

2

c2 ] x+ λeBw

mcβz sin kw z

−eB0

mcβx−

eE r

mc (1−βz )cosζ (36)

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γ β y+γ β y=−[2 πe2 (n i−nb)mc

+2πnb e2

mcv||

2

c2 ] y−λeBw

mcβ z cosk w z

+eB0

mcβ y+

eEr

mc (1−β z)sin ζ (37)

γ βz+γ β z=λeBw

mc (β y cosk w z−βx sin kw z )−2πnb e2

mcv||c (β x x+β y y )

+eEr

mc (β y sin ζ−β x cosζ )(38)

The last term in Eqs. (36) and (37), which represents the transverse optical force acting on the electron. Now, we consider radiation terms as a small external factor which produces small perturbation in the steady-state velocity components,

β x=βw coskw z+δβ x=β x0+δβ x (39) β y=βw sin kw z+δβ y=β y0+δβ y (40) βz=β||z+δβ z=β z0+δβ z (41)

Then, rewrite Eqs. (36)–(38) to first order in perturbed quantities and their derivatives,γ βx0+γ 0 δ βx+δ (γ ) β x 0=λγ 0Ωw δβ z sin kw z−γ 0Ωr (1−β z 0)cosζ

(42) γ β y 0+γ 0 δ β y+δ ( γ ) β y 0=−λγ0Ωw δβ z coskw z+γ 0Ωr (1−βz 0 )sin ζ

(43) γ βz 0+γ0 δ βz=λγ 0Ωw (δβ y cos kw z−δβ x sin kw z )+γ 0Ωr

×( β y 0 sin ζ−β x 0 cos ζ ) (44)

where Ωr=

eEr

γ mc , γ=γ 0+δ (γ ) , and γ=δ ( γ )=γ0

3 ( βx 0δ β x+β y 0 δ β y+ βz 0 δ β z+δβx β x0+δβ y β y0 ) (45)

Now, multiply Eqs. (42) and (43) by β x0 /γ0 and β y 0/ γ0 , respectively, and add the results to deduceγγ 0( βxo

2 + β yo2 )+ βx 0δ βx+ β y0 δ β y=−Ωr βw (1−βz 0)cos (kw z+ζ )

(46)

The relations β x 0 δ β x+β y 0 δ β y= γ / γ03−β z0 δ βz−δβ x β x0−δβ y β y0 , which can be obtained from Eq. (45), and

β x02 +β y0

2 =1−β z02 −1 /γ0

2, may be used in Eq. (46) to obtain

γγ0

(1−β z02 )−βz 0 δ βz−ck w βw β z0 (δβ x sin k w z−δβ y cosk w z )

=−Ωr βw (1−βz 0 )cos (k w z+ζ ) (47)

Now the term (δβ y cos kw z−δβ x sin kw z ) may be eliminated between Eqs. (44) and (47) to obtainγγ0[ λΩw (1− βz 0

2 )−ck w βw β z 02 ]−δ β z (λΩw β z0+ck w βw β z0 )=−Ωr βw

¿ [λΩw (1−βz 0)−ck w βw β z 0 ]cos (kw z+ζ ) ] (48)

This equation is an important result which relates γ to δ β z in the presence of radiation. Now a relation for γ can be derived as follows:

γ=− emc (β⋅Er )=−

eEr

mcβw cos (kw z+ζ )=−γ0Ωr βw cos (k w z+ζ )

(49)

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Substitution of this value in Eq. (48) with some algebra will deduce βz asβz=δ βz=−Ωr βw cos (k w z+ζ ) (50)

Using a first-order approximation for z=v||t=cβ||t , Eqs. (49) and (50) may be written in the following form:

γ=−γ 0Ωr βw cos (Ω t+φ ) (51)βz=δ βz=−Ωr βw cos (Ωt+φ ) (52)

where Ω=(k r+kw )v||−ωr .

The phase φ in the above equation determines the initial position of the electron relative to the optical wave. Averaging γ over all phases yields

¿ γ>φ ¿1

2 π∫0

2 πγ dφ=0

(53) Therefore, to first order there is no net transfer of power between the electron beam and the optical wave. The second-

order correction will consist of accounting for the fact that as an individual electron (with a phase φ ) gains or loses in

energy, its position relative to the unperturbed position (z=cβ||t ) is advanced or retarded. Therefore, the unperturbed

position, z=cβ||t , must be replaced by

z (t )=cβ||t+c∫0

tΔβ z (t

' )d t '

(54)

where Δβ z ( t )=∫0

tβz ( t

' )d t'

is the change of βz relative to the unperturbed state. The substitution of βz in Eq. (54) will yield

z (t )=cβ||t+cDΩ [Ω t sin φ+cos (Ω t+φ )−cosφ ]

(55)

where D=Ωr βw (1−β z0 )/Ω . Substitution of Eq. (55) in Eq. (49) will yield

γ=−Ωr γ0 βw cos{(Ωt+φ )+(ωr+ωw )D

Ω [Ωt sin φ+cos (Ω t+φ )−cosφ ]} (56)

where ωw=k w c . Comparison of Eq. (56) with Eq. (51) reveals that the perturbed state consists of a phase slippage,

Δφ=(ωw+ωr ) D

Ω [Ω t sin φ+cos (Ω t+φ )−cosφ ] (57)

Since D is proportional to γ−1

and Er , Δφ can be made arbitrarily small. Therefore, expanding the cosine term in Eq.

(55) for Δφ<< π leads to

γ=−Ωr γ0 βw {cos (Ωt+φ )−sin (Ω t+φ )(ωr+ωw )Ω

×[Ω t sin φ+cos (Ω t+φ )−cosφ ] } (58)

Averaging over phase φ yields

⟨ γ ⟩φ=Ωr γ 0 βw(ωr+ωw)

2(Ωt cosΩ t−sinΩ t )

(59) Now, integrating Eq. (59) over the electron transit time through the wiggler interaction length yields the average change in γ per electron:

¿ Δγ >φ¿∫0

T=L /cβ z ¿ γ>φ¿dt=−Ωr γ 0 βw(ωr+ωw )DΩT 3

2g (ΩT ) ¿

(60) where

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g (ΩT )= (2−2cosΩT−ΩT sinΩT )Ω3 T3

(61) and where L is the FEL interaction length. The change in radiation power in one transit is

ΔP=− Ie

mc2<Δγ>φ¿ ¿ (62)

where I=nb ev||πrb2

is the average electron beam current. By using Eqs. (60) and (62), under the assumption that the

electrons are near resonance with the wave, i.e., Ω=(k r+kw )cβ||−ωr≈0

, the gain equation in the presence of the self-fields can be rewritten as

Gs=ΔPP=ωb

2 ωw( Lcβ|| )

3

βw2 β||g (ΩT )

(63)

where P≡cπrb2 (Er

2 /4 π ) .In the limit eliminating self-fields, the maximum gain Eq. (63) reduces to

Gs 0=ωb2 ωw( L

cβ||0 )3

βw 02 β||0 g (ΩT )

(64)

where βw 0 and β||0 are the wiggler-induced transverse velocity and the axial velocity in the absence self-fields,

respectively. If ωi and Ω0 are both set equal to zero (eliminating the ion-channel and the axial magnetic field), Eq. (63) reduces to

G0=ωb2ωw ( L

cβ0)3

aw2 β0 g (ΩT )

(65)

where aw=−eBw /γmkw c2and

β0=(1−aw2−γ−2)1 /2 .

Numerical Results

A numerical study of the self-fields effects on gain in a helical wiggler free-electron laser with axial magnetic field and

ion-channel guiding have been made. The parameters that are used in this section are k w=3 .14 cm−1,

nb=8×1012 cm−3, Bw=1 kG . The Lorentz relativistic factor γ was taken to be 30, and the normalized axial

velocity β|| was taken to be close to 1. The FEL interaction length L was taken to be 100 λw .

Figure 1 shows the variation of the normalized gain (G s/G0 ) (solid lines) and value of the ratio of gain in the presence

of the self-fields to the gain in the absence of the self-fields Δ (¿Gs /G s0 ) (dotted lines) as a function of either

normalized guiding frequency with the two taken to be equal(ωi/k w c=Ω0/k w c ) , for group I orbits. Fig. 1 shows that the normalized gain increases with increase the normalized guiding frequency. As shown in Fig. 1, Δ is less than 1 and therefore the gain decrement is obtained due to the self-fields. Figure 2 shows the variation of the normalized gain (G s/G0 ) (solid lines) and the gain ratio

Δ (¿Gs /G s0 ) (dotted lines) as a function of either normalized guiding

frequency with the two taken to be equal(ωi/k w c=Ω0/k w c ) , for group II orbits. Fig. 2 shows that the normalized

gain decreases with increase the normalized guiding frequency. For group II orbits if (ωi/k w c=Ω0/k w c )<1, the

condition β|| close to 1, as required for derivation of the gain equation, breaks down. For group II orbits because Δ is

greater than 1, the gain enhancement is obtained due to the self-fields.

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Figure 1. The normalized gain (G s/G0 ) (solid lines) and the gain ratio Δ (¿Gs /G s0 ) (dotted lines) as a function of the

normalized axial magnetic field frequency that is equal to the ion-channel frequency, for group I orbits. The parameters are k w=3 .14 cm−1

, nb=8×1012 cm−3, and Bw=1 kG .


normalized axial magnetic field frequency that is equal to the ion-channel frequency, for group II orbits. The parameters are k w=3 .14 cm−1

, nb=8×1012 cm−3, and Bw=1 kG .

Page | 8



normalized axial magnetic field frequency Ω0 /kw c in the presence of an ion-channel, withωi/k w c=0 .2 , for group I

orbits. The parameters are k w=3 .14 cm−1, nb=8×1012 cm−3

, and Bw=1 kG .


normalized axial magnetic field frequency Ω0 /kw c in the presence of an ion-channel, withωi/k w c=0 .2 , for group II


, and Bw=1 kG .

Fig. 3 shows the normalized gain (G s/G0 ) (solid lines) and the gain ratio Δ (¿Gs /G s0 ) (dotted lines) as a function

of the normalized axial magnetic field frequency Ω0 /kw c in the presence of an ion-channel when it is held constant

with ωi/k w c=0 . 2 , for group I orbits. Fig. 3 shows that Δ is less than 1 and the gain decreases with considering the

self-fields. The normalized gain (G s/G0 ) (solid lines) and the gain ratio Δ (¿Gs /G s0 ) (dotted lines) as a function of

Page | 9


the normalized axial magnetic field frequency Ω0 /kw c in the presence of an ion-channel when it is held constant with ωi/k w c=0 . 2 , for group II orbits, are shown in Fig. 4.


normalized ion-channel frequency ωi/k w c in the presence of an axial magnetic field, with Ω0 /kw c=0 .2 , for group I


, and Bw=1 kG .


normalized ion-channel frequency ωi/k w c in the presence of an axial magnetic field, with Ω0 /kw c=0 .2 , for group II


, and Bw=1 kG .

Fig. 5 shows the normalized gain (G s/G0 ) (solid lines) and the gain ratio Δ (¿Gs /G s0 ) (dotted lines) as a function

of the normalized ion-channel frequency ωi/k w c in the presence of an axial magnetic field when it is held constant

Page | 10


with Ω0 /kw c=0 . 2 , for group I orbits. For group II orbits, the normalized gain (G s/G0 ) (solid lines) and the gain

ratio Δ (¿Gs /G s0 ) (dotted lines) as a function of the normalized ion-channel frequency ωi/k w c in the presence of

an axial magnetic field when it is held constant with Ω0 /kw c=0 .2 , are shown in Fig. 6. It should be noted that the foregoing result has been shown that the Budker condition eliminates the group II orbit in a FEL with ion-channel guiding. While, when both the ion-channel and the axial magnetic field are present, there are two groups of electron trajectories, because electron beam will be guided by the axial magnetic field.

Conclusion

In this work, the effects of self-fields on gain in a helical wiggler free electron laser with axial magnetic field and ion-channel guiding is investigated. The self-electric field is derived from Poisson ,s equation and the self-magnetic field is obtained from Ampere,s law by a self-consistent method. A detailed analysis of electron interaction with radiation field in the presence of self-fields is presented. It should be noted that the wiggler-induced self-magnetic field decreases the effective wiggler magnetic field for group I orbits, whereas for group II orbits it increases the effective wiggler magnetic field. Therefore, we expect that in the presence of self-fields the gain decreases for group I orbits and increases for group II orbits.

The results of the present one-dimensional analysis are valid for vw /v||<<1

. When vw is large the radial variation of the wiggler field can no longer be neglected and the present results are not valid. The radial variation of the wiggler field is also negligible for thin beams for which the beam radius is much smaller than the wiggler wavelength.

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