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To the theory of a free-electron laser
Volume 65A, number 4
PHYSICS LETTERS
20 March 1978
TO THE THEORY OF A FREE-ELECTRON LASER V.N. BAlER and A.I. MILSTEIN Institute of Nuclear Physics, Novosibirsk 90, USSR Received 1 July 1977
The interaction of an electron beam, moving inside a magnetic lattice, with an electromagnetic wave is considered. It is shown that under certain conditions generation (amplification) of coherent electromagnetic radiation occurs. The gain is calculated.
Now a free-electron laser (FEL) seems to become a reality [1—4].The model of a free-electron laser [1—4](see ref. [5] as well) presented by the authors is based on the quantum induced magnetic bremsstrahlung, although in a realistic situation Planck’s constant is not included in the gain expressions. This circumstance is not accidental. The FEL action is based on stimulated radiation when an electron moves within a magnetic lattice. Here one deals with high excitations of the field oscillators which lie in the domain of applicability of classical theory. Moreover, even at spontaneous radiation in the lattice, at which the quanturn properties of radiation are revealed first of all, the radiation is classical if the energy of the quanta[6]. is much smaller than the energy of the electrons Therefore, to describe the FEL action one can use a classical approach which, as will be shown below, has an essential advantage. We consider the motion of a relativistic particle in superposition of a transverse magnetic field and the field of a plane electromagnetic wave propagating along the direction of particle motion. The z-axis is chosen along the lattice axis and the magnetic lattice is taken to be helical [3,4] : F31 =Hsin (v
23 =Hcos(~ 0z/c), F 0z/c). (1) The wave is assumed to be circularly polarized. The wave field strength is represented as follows p2X
=
Hw(K~~àX(v,) gXaI~)) —
(2)
where ~C.Z= (1, 0, 0, 1), p = v(t z/c) + ~i, à~ da~’/dp,a1 = cos p, a2 = sin The equations of motion in this field have the following form: —
~•
du0/dr = &1(u1 sin p ~/dr
u2cos p)
3sin (v 0u 0z/c), 2/dr = cos p(u0 u3) ~Z 3cos (v du 0u 0z/c), 3ldr = du0ldr ~ (u1 sin (v 2cos (v du 0z/c) u 0z/c)), 1 sin p u2cosp), (3) d&l/dr = w~(u where &2 = eH ~/mc, ~ = eH/mc, w~= 4lTe2Ne/mV, r is the proper time, V is the volume of the electromagnetic radiation bunch,Ne is the number of electrons in this volume, u /~L is the electron four-velocity, c is the velocity of light. The bar in the last of eqs. (3) de=
&~sin ~(u0
—
—
—~
u3)
—
+
~Z
—
—
—
____________—
—~
notes averaging over all the electrons in the interaction region. This equation describes the change of the wave field because of the interaction with all the electrons and follows from the energy conservation law:
E~°
2 = —(d/dr)(4~)~H~V. (4) (d/dr)mc Assuming that &2 is a smooth function of time, we integrate the second and third equations of the set (3): U’
=
—
[(a/v) cos ~ + (~
1, 0/v0)cos (v0z/c)]
+a
319
Volume 65A, number 4
[(~2/v)sin
=
PHYSICS LETTERS
~ + (I2~/v
0)sin (P0z/c)j +
(5)
2
~
2~2~2~ sin 0.
=
(&2
0)sinc5,
p
0y/~~~ (ö 1) 0 w 2)(l + &2~j/r~).In the
(8) centre of the res-
here 6 =(v/2v0’y onance line ~ = 1. i.e.
2)(1 +
(9)
&2~/v~).
X = (X0/2y When ispassing through the lattice the “time” merement equal to —
2irAT
(10) (6) If,< ~ 1, SN/K ~ 1, which is valid when the wave field is weak, the solution of the set (6’) may be found
where =
(v
2/2v 0w
0
~‘(0) =
a
As will be seen below, near the resonance the wave frequency r’ 2y2v 0 and of interest is the situation when ~2/t ~ &2~/v~. In order that a transverse displacement of theamplitude particle init the lattice does exceed oscillation is necessary thatnot a1’2 ~f the (~7 0/v0)(l/2irN)(here Nis —~o1~o~ the number of lattice 1(0) u2(0) = 0. wavelengths), Substitutingthen the usolutions (5) into eqs. (3) we get the set of equations =
20 March 1978
-—
v0z/c.
(7)
in analytical form by using the perturbation series expansion in inverse powers of K. In this case 3RL~~s), (I 1) (s) = 1 + Ps where
The terms I /~2are discarded in eq. (6). We carry out the substitution (6):these &2 = equations ~y, T~= Note thaty(0) = in1.eq. Then take the s. form “-j
0”= 2vsinØ,
~
sinz sinz R(z)=~— —~~cosz~.
(6’)
~2
______
where 0 = ~ The initial condition for 0’
(~
(12)
In deriving eq. (11) the electrons are assumed to be dO/ds (see eq. (7)) is: uniformly distributed over the initial phases at s = 0.
I, /5,
LI,
1~5
£1 —ii
.5
-2
-1
1
-
2
3
5
~
qva
-17/
-17/3
Fig. I. Function R(z), z = —~gs.(1) Theoretical curve at weak fields of the wave (formula (12)), (2) function for a strong field (17 0.1I,Hw/H 7 X 102,s= 2.25).
320
Volume 65A, number 4
PHYSICS LETTERS
The function (see fig. 1) describes the dependence of the gain on the distance of the resonance (cf. fig. lb in ref. [3]), the maximum valueR being 0.13 5 at z = 1.3. Taking account of eqs. (8) and (10) we have for passing maximum R the function 1 the 6 isresonance l.3/~N.At athat single (amplification mode) line width is determined by the function R (fIg. 1). In this case, at lie half-width L~~/v = 2.2/irN, this gives = 0.4% in agreement with experiment [3]. In the generation mode an additional narrowing of the line takes place due to multiple passings. Estimates show that for experiment [4] the line width should be twice as narrow, what is in agreement with the observation as well. The wave is naturally amplified with the same polarization as the initial one. Moreover, this polarization should correspond to the lattice symmetry. From eq. (11) we have for the gain —
G ~y2
—
1
=
2~(l+~-~),
(13)
where
20 March 1978
ç~2 0 1
~2 +
0
—‘
l6R(z)(~)~(NX3 )3 ~ > K~ 1-~(14) ______
~ 2/mc2 is the classical electron radius, I is where r0 = e current. This expression determines, parthe electron ticularly, a threshold magnitude of the current. As the wave field strength is increasing, the inequalities determining the applicability of eqs. (11) cease to be fulfilled. In this case the set (6’) has been solved numerically for the parameter region close to those cited in ref. [4]. This solution shows that the gain G which for a weak field (eq. (13)) was independent of H~,begins to fall down with increasing Hw. In fig. 2 the dependence of the gain on Hw/H is shown. For strong fields the spectrum function is also distorted (see fig. 1). The decrease of the gain finally stops the wave-field increase. After that the generation becomes stationary. All the presented results concern electron bunches with uniform initial phase distribution. In the course of the electron interaction with the electromagnetic
R(z) O.)~~.2 =
2~
~3~4
0NX0
___________
z
=
mv(1
—
wave and the magnetic lattice theenergies phase distribution transforms such that electron higher than the resonance one (seeat eq. (9))the radiation of an
~).
The formulae presented above may be applied directly to the description of the wave amplification [3] In the case of small amplification, when y 1 ~ 1, in eq. (13) it is possible to confine ourselves to the term linear in2 N,InXthis particular case, the dependence of G on w 0, ‘y is in agreement with that found in refs. [3~~5]~ Thus, we have obtained a very simple and instructive description of the weak-signal domain in FEL, for which both classical and quantum approaches gave consistent results. However, it is very essential that our approach is also valid in the strong-signal domain. In the case of generation, a bunch of electromagnetic radiation moving in an optical cavity successively interacts with the electron bunches. In the beginning of the generation the wave field is quite small, i.e. i~~‘ 1 (see eq. (8)), then eqs. (1 1)—(13) are1ef applicable. where The generation is possible if the gain G > ~ ~ef is the total loss coefficient (transmission of mirrors, diffractional losses, etc.). For a confocal cavity when the transverse cross section of the radiation bunch is LX/4 (L is the distance between mirrors) this inequality may be rewritten as follows —
~.
electron bunch exceeds the absorption, and at electron energies lower than the resonance one the absorption dominates. This means that in the weak-signal domain a specific phase stability arises. In the strong-signal domain phase oscillations appear which disturb this phase stability and lead to a significant decrease of resonance energy transfer from particles to wave, if one makes
5 C )‘ ~ ~ ~ ~‘ S Fig. 2. Dependence of the gain on the wave field strength (ps3 = 1.78).
321
Volume 65A, number 4
PHYSICS LETTERS
no special efforts to suppress these oscillations. Since the frequency of the phase oscillations is ~ there exists a limiting power which may be stored in the FEL resonator for every lattice. The most efficient transfer of the electron energy into the wave field would take place in the case when at the beginning all the electrons were phased in an appropriate manner, and the increment of the phase when the electron passes through the lattice were 60 1. In such a situation the gain is ~
G—
(~)
W~ p 0
~
2irN sin ~,
plays a dominant role), since in the neighbourhood of the resonance compensation occurs in it, so that an inaccuracy in determining the longitudinal components of the field results in an invalid expression for the phase in the WW method. In addition, in ref. [7] oversimplifications are committed and as a result, the factor (1 + Iv~)has not been taken into account, which has a purely kinematic nature and is equal to 1.5 under experimental conditions. We wish to thank A.N. Skrinsky for many invalua-
(15)
so that at 0 < i,ti
322
20 March 1978
G.N. ble conversations, Kulipanov and weE.L are Saldin indebted fortodiscussions. M.M. Karliner, The authors wish to express their gratitude to M.S. Obrekht for the help with numerical calculations.
References [1] J.M.J. Madey, J. Appi. Phys. 42 (1971) 1906. [21 J.M.J. Madey, HA. Schwettman and W.M. Fairbank, IEEE Trans. Nuci. Sd. 20(1973)980. (31 L.R. Elias et al., Phys. Rev. Lett. 36 (1976) 717. [4] D.A. Deacon et al., Phys. Rev. Lett. 38 (1977) 892. [5] W.B. Colson, Phys. Lett. 59A (1976) 187. [61V.N. Baier, V.M. Katkov and V.M. Strakhovenko, Soy. Phys. JETP 36 (1973) 1120. [7] F.A. Hopf, P. Meystre, MO. Scully and N.H. Louisell, Phys. Rev. Lett. 37 (1976) 1215, 1342.
PHYSICS LETTERS
20 March 1978
TO THE THEORY OF A FREE-ELECTRON LASER V.N. BAlER and A.I. MILSTEIN Institute of Nuclear Physics, Novosibirsk 90, USSR Received 1 July 1977
The interaction of an electron beam, moving inside a magnetic lattice, with an electromagnetic wave is considered. It is shown that under certain conditions generation (amplification) of coherent electromagnetic radiation occurs. The gain is calculated.
Now a free-electron laser (FEL) seems to become a reality [1—4].The model of a free-electron laser [1—4](see ref. [5] as well) presented by the authors is based on the quantum induced magnetic bremsstrahlung, although in a realistic situation Planck’s constant is not included in the gain expressions. This circumstance is not accidental. The FEL action is based on stimulated radiation when an electron moves within a magnetic lattice. Here one deals with high excitations of the field oscillators which lie in the domain of applicability of classical theory. Moreover, even at spontaneous radiation in the lattice, at which the quanturn properties of radiation are revealed first of all, the radiation is classical if the energy of the quanta[6]. is much smaller than the energy of the electrons Therefore, to describe the FEL action one can use a classical approach which, as will be shown below, has an essential advantage. We consider the motion of a relativistic particle in superposition of a transverse magnetic field and the field of a plane electromagnetic wave propagating along the direction of particle motion. The z-axis is chosen along the lattice axis and the magnetic lattice is taken to be helical [3,4] : F31 =Hsin (v
23 =Hcos(~ 0z/c), F 0z/c). (1) The wave is assumed to be circularly polarized. The wave field strength is represented as follows p2X
=
Hw(K~~àX(v,) gXaI~)) —
(2)
where ~C.Z= (1, 0, 0, 1), p = v(t z/c) + ~i, à~ da~’/dp,a1 = cos p, a2 = sin The equations of motion in this field have the following form: —
~•
du0/dr = &1(u1 sin p ~/dr
u2cos p)
3sin (v 0u 0z/c), 2/dr = cos p(u0 u3) ~Z 3cos (v du 0u 0z/c), 3ldr = du0ldr ~ (u1 sin (v 2cos (v du 0z/c) u 0z/c)), 1 sin p u2cosp), (3) d&l/dr = w~(u where &2 = eH ~/mc, ~ = eH/mc, w~= 4lTe2Ne/mV, r is the proper time, V is the volume of the electromagnetic radiation bunch,Ne is the number of electrons in this volume, u /~L is the electron four-velocity, c is the velocity of light. The bar in the last of eqs. (3) de=
&~sin ~(u0
—
—
—~
u3)
—
+
~Z
—
—
—
____________—
—~
notes averaging over all the electrons in the interaction region. This equation describes the change of the wave field because of the interaction with all the electrons and follows from the energy conservation law:
E~°
2 = —(d/dr)(4~)~H~V. (4) (d/dr)mc Assuming that &2 is a smooth function of time, we integrate the second and third equations of the set (3): U’
=
—
[(a/v) cos ~ + (~
1, 0/v0)cos (v0z/c)]
+a
319
Volume 65A, number 4
[(~2/v)sin
=
PHYSICS LETTERS
~ + (I2~/v
0)sin (P0z/c)j +
(5)
2
~
2~2~2~ sin 0.
=
(&2
0)sinc5,
p
0y/~~~ (ö 1) 0 w 2)(l + &2~j/r~).In the
(8) centre of the res-
here 6 =(v/2v0’y onance line ~ = 1. i.e.
2)(1 +
(9)
&2~/v~).
X = (X0/2y When ispassing through the lattice the “time” merement equal to —
2irAT
(10) (6) If,< ~ 1, SN/K ~ 1, which is valid when the wave field is weak, the solution of the set (6’) may be found
where =
(v
2/2v 0w
0
~‘(0) =
a
As will be seen below, near the resonance the wave frequency r’ 2y2v 0 and of interest is the situation when ~2/t ~ &2~/v~. In order that a transverse displacement of theamplitude particle init the lattice does exceed oscillation is necessary thatnot a1’2 ~f the (~7 0/v0)(l/2irN)(here Nis —~o1~o~ the number of lattice 1(0) u2(0) = 0. wavelengths), Substitutingthen the usolutions (5) into eqs. (3) we get the set of equations =
20 March 1978
-—
v0z/c.
(7)
in analytical form by using the perturbation series expansion in inverse powers of K. In this case 3RL~~s), (I 1) (s) = 1 + Ps where
The terms I /~2are discarded in eq. (6). We carry out the substitution (6):these &2 = equations ~y, T~= Note thaty(0) = in1.eq. Then take the s. form “-j
0”= 2vsinØ,
~
sinz sinz R(z)=~— —~~cosz~.
(6’)
~2
______
where 0 = ~ The initial condition for 0’
(~
(12)
In deriving eq. (11) the electrons are assumed to be dO/ds (see eq. (7)) is: uniformly distributed over the initial phases at s = 0.
I, /5,
LI,
1~5
£1 —ii
.5
-2
-1
1
-
2
3
5
~
qva
-17/
-17/3
Fig. I. Function R(z), z = —~gs.(1) Theoretical curve at weak fields of the wave (formula (12)), (2) function for a strong field (17 0.1I,Hw/H 7 X 102,s= 2.25).
320
Volume 65A, number 4
PHYSICS LETTERS
The function (see fig. 1) describes the dependence of the gain on the distance of the resonance (cf. fig. lb in ref. [3]), the maximum valueR being 0.13 5 at z = 1.3. Taking account of eqs. (8) and (10) we have for passing maximum R the function 1 the 6 isresonance l.3/~N.At athat single (amplification mode) line width is determined by the function R (fIg. 1). In this case, at lie half-width L~~/v = 2.2/irN, this gives = 0.4% in agreement with experiment [3]. In the generation mode an additional narrowing of the line takes place due to multiple passings. Estimates show that for experiment [4] the line width should be twice as narrow, what is in agreement with the observation as well. The wave is naturally amplified with the same polarization as the initial one. Moreover, this polarization should correspond to the lattice symmetry. From eq. (11) we have for the gain —
G ~y2
—
1
=
2~(l+~-~),
(13)
where
20 March 1978
ç~2 0 1
~2 +
0
—‘
l6R(z)(~)~(NX3 )3 ~ > K~ 1-~(14) ______
~ 2/mc2 is the classical electron radius, I is where r0 = e current. This expression determines, parthe electron ticularly, a threshold magnitude of the current. As the wave field strength is increasing, the inequalities determining the applicability of eqs. (11) cease to be fulfilled. In this case the set (6’) has been solved numerically for the parameter region close to those cited in ref. [4]. This solution shows that the gain G which for a weak field (eq. (13)) was independent of H~,begins to fall down with increasing Hw. In fig. 2 the dependence of the gain on Hw/H is shown. For strong fields the spectrum function is also distorted (see fig. 1). The decrease of the gain finally stops the wave-field increase. After that the generation becomes stationary. All the presented results concern electron bunches with uniform initial phase distribution. In the course of the electron interaction with the electromagnetic
R(z) O.)~~.2 =
2~
~3~4
0NX0
___________
z
=
mv(1
—
wave and the magnetic lattice theenergies phase distribution transforms such that electron higher than the resonance one (seeat eq. (9))the radiation of an
~).
The formulae presented above may be applied directly to the description of the wave amplification [3] In the case of small amplification, when y 1 ~ 1, in eq. (13) it is possible to confine ourselves to the term linear in2 N,InXthis particular case, the dependence of G on w 0, ‘y is in agreement with that found in refs. [3~~5]~ Thus, we have obtained a very simple and instructive description of the weak-signal domain in FEL, for which both classical and quantum approaches gave consistent results. However, it is very essential that our approach is also valid in the strong-signal domain. In the case of generation, a bunch of electromagnetic radiation moving in an optical cavity successively interacts with the electron bunches. In the beginning of the generation the wave field is quite small, i.e. i~~‘ 1 (see eq. (8)), then eqs. (1 1)—(13) are1ef applicable. where The generation is possible if the gain G > ~ ~ef is the total loss coefficient (transmission of mirrors, diffractional losses, etc.). For a confocal cavity when the transverse cross section of the radiation bunch is LX/4 (L is the distance between mirrors) this inequality may be rewritten as follows —
~.
electron bunch exceeds the absorption, and at electron energies lower than the resonance one the absorption dominates. This means that in the weak-signal domain a specific phase stability arises. In the strong-signal domain phase oscillations appear which disturb this phase stability and lead to a significant decrease of resonance energy transfer from particles to wave, if one makes
5 C )‘ ~ ~ ~ ~‘ S Fig. 2. Dependence of the gain on the wave field strength (ps3 = 1.78).
321
Volume 65A, number 4
PHYSICS LETTERS
no special efforts to suppress these oscillations. Since the frequency of the phase oscillations is ~ there exists a limiting power which may be stored in the FEL resonator for every lattice. The most efficient transfer of the electron energy into the wave field would take place in the case when at the beginning all the electrons were phased in an appropriate manner, and the increment of the phase when the electron passes through the lattice were 60 1. In such a situation the gain is ~
G—
(~)
W~ p 0
~
2irN sin ~,
plays a dominant role), since in the neighbourhood of the resonance compensation occurs in it, so that an inaccuracy in determining the longitudinal components of the field results in an invalid expression for the phase in the WW method. In addition, in ref. [7] oversimplifications are committed and as a result, the factor (1 + Iv~)has not been taken into account, which has a purely kinematic nature and is equal to 1.5 under experimental conditions. We wish to thank A.N. Skrinsky for many invalua-
(15)
so that at 0 < i,ti
322
20 March 1978
G.N. ble conversations, Kulipanov and weE.L are Saldin indebted fortodiscussions. M.M. Karliner, The authors wish to express their gratitude to M.S. Obrekht for the help with numerical calculations.
References [1] J.M.J. Madey, J. Appi. Phys. 42 (1971) 1906. [21 J.M.J. Madey, HA. Schwettman and W.M. Fairbank, IEEE Trans. Nuci. Sd. 20(1973)980. (31 L.R. Elias et al., Phys. Rev. Lett. 36 (1976) 717. [4] D.A. Deacon et al., Phys. Rev. Lett. 38 (1977) 892. [5] W.B. Colson, Phys. Lett. 59A (1976) 187. [61V.N. Baier, V.M. Katkov and V.M. Strakhovenko, Soy. Phys. JETP 36 (1973) 1120. [7] F.A. Hopf, P. Meystre, MO. Scully and N.H. Louisell, Phys. Rev. Lett. 37 (1976) 1215, 1342.