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Effect of energy spread on micro-bunching from shot noise in SASE FELs
Physics Letters A 375 (2011) 1796–1802
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Physics Letters A www.elsevier.com/locate/pla
Effect of energy spread on micro-bunching from shot noise in SASE FELs M. Rezvani Jalal a,b , F.M. Aghamir a,∗ a b
Department of Physics, University of Tehran, North Kargar Avenue, Tehran 14399, Iran Department of Physics, Malayer University, 4th Kilometer of Arak road, Malayer, Iran
a r t i c l e
i n f o
Article history: Received 5 December 2010 Received in revised form 3 February 2011 Accepted 9 February 2011 Available online 16 February 2011 Communicated by F. Porcelli Keywords: Micro-bunching from shot noise Shot-noise micro-bunching Total energy spread Lienard–Wiechert field
a b s t r a c t The motion of N e electrons moving along a helical undulator and interacting with each other through Lienard–Wiechert fields is considered. The numerical solution for initially mono-energetic electrons shows the emergence and growth of micro-bunching. For the case of initially non-mono-energetic electrons, calculations show that the presence of energy spread results in partial suppression of the micro-bunching. The calculations also show that the initial energy spread would shift the microbunching maximum to deeper positions within the undulator. Analytical treatment of equations of motion demonstrates that the micro-bunching fall and shift is mainly due to the random motion of the interacting electrons. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Micro-bunching is a familiar and fundamental phenomenon in Free Electron Lasers (FELs). In this process, the smooth distribution of electrons in the injected electron beam is converted to a periodic (bunched) distribution. The period length is the fundamental wavelength of the undulator radiation. In fact, micro-bunching is an instability that converts weak and incoherent (spontaneous) emission of an oscillator FEL to a robust and coherent (induced) emission. As well, the weak input radiation of an amplifier FEL is converted to a powerful laser radiation through this instability. In brief, in FELs the electron beam is the active medium that transforms the spontaneous emission to an induced emission by means of micro-bunching instability. The reason behind the instability is the dispersion that is related to the ponderomotive phase of electrons. Electrons that have positive ponderomotive phase gain energy from the radiation. Conversely, electrons that have negative ponderomotive phase yield energy to the radiation. This energy distribution in the electron beam is converted to density distribution because of dispersive property of the undulator. In a special kind of FELs which cannot be operated in the oscillator mode due to the lack of appropriate cavity mirrors and also cannot be operated in the amplifier mode, because of the lack of coherent input radiation, the SASE instability plays a crucial role. In this kind of FEL instability, the spontaneous emission
*
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of electron beam remains overlapped with the electron beam for a long travel in the undulator. Because of this long effective interaction, special type of micro-bunching, which is called SASE micro-bunching, occurs within the electron beam. The startup and growth of such a micro-bunching lead to a laser-like radiation. There are two differences between SASE micro-bunching and typical micro-bunching: (1) typical micro-bunching is originated from weak coherent input radiation, while, SASE micro-bunching is originated from weak spontaneous emission of electron beam; (2) typical micro-bunching is a completely periodic pattern over the entire electron beam, while SASE micro-bunching is a periodic pattern over distinct sections of the electron beam where there is no phase relation between these sections. Therefore, the typical micro-bunching results in full longitudinal coherence for output radiation while SASE micro-bunching leads to an output with partial longitudinal coherence (i.e. some distinct portions of it are longitudinally coherent). SASE FEL is an amplifier FEL that its input radiation is the spontaneous emission of the electron beam occurring at the very beginning of the undulator. When such a spontaneous radiation is overlapped over the radiating electron beam, one of the following two conditions can be realized: if a well-matched overlap persists over a long enough length of the undulator and if the electron beam is of a perfect quality (low energy spread and small emittance) then the micro-bunching instability occurs. Contrary, if the overlap is imperfect or the energy spread of the electron beam is of significant amount then no micro-bunching happens. The discreteness of charge carriers plus their random distribution within the electron beam play essential role in the SASE micro-bunching. If
M. Rezvani Jalal, F.M. Aghamir / Physics Letters A 375 (2011) 1796–1802
the electron beam is considered to be infinitely long and homogeneous then micro-bunching instability never occurs. Such discreteness and randomness of electrons generate a noise in the electron density which is known as shot noise. The startup of SASE instability from shot noise has been studied in literature [1–4]. The discreteness of charges has been taken into account by means of Klimontovich distribution function. The paraxial Maxwell’s wave equation was then incorporated to form a set of coupled Maxwell–Klimontovich equations. By numerical solution of those coupled equations, many fundamental characteristics of SASE FELs can be obtained. The study of the SASE FELs by means of exact solutions of Maxwell wave equations (namely, Lienard–Wiechert fields) was first under taken by Tecimer and Elias [5,6]. Following their works, authors of the present article [7], presented a simpler form of longitudinal equations of motion and used them to find fundamental aspects of SASE FELs as a testbed for the validity of their set of equations. In the present study, the formulation of Ref. [7] is generalized to non-mono-energetic electrons and the effect of energy spread in the injected electron beam on micro-bunching is presented. It is shown, both numerically and analytically, that the presence of energy spread leads to partial suppression of micro-bunching and for a certain maximum energy spread, the micro-bunching is completely destroyed. Micro-bunching in electron beam is very similar to nucleation phenomenon in physics of crystals. A typical micro-bunched beam can be considered as a plasma single-crystal [8] and a SASE microbunched beam as a plasma poly-crystal. Temperature in physics of crystals usually leads to suppression of nucleation. We expect that in micro-bunching the temperature (or more precisely, energy spread and transverse emittance) act in a similar manner in degradation of micro-bunching process. Our numerical calculation shows that such suppression in micro-bunching does occur. The organization of the Letter is as follows: first the evolution of the longitudinal phase space of an injected mono-energetic electron beam within the undulator is considered. Then, some numerical calculations for the evolution of the longitudinal phase space of non-mono-energetic electron beams are carried out. This is followed by the section which is devoted to some analytical work to explain the observation of numerical results. Finally, the concluding remarks are presented in the last section.
der to study micro-bunching from shot noise, the bunching factor should be defined first. A quantity that can best explain the microbunching process is defined by [1,11]:
b≡
Consider an electron beam of N e relativistic electrons. The beam is injected into a helical undulator. The undulator is assumed to be semi-infinite meaning that its left end is at infinity and the right end is terminated at z = L U . The electrons are considered to be non-interacting when they travel from the left end up to the origin at z = 0; however, the electrons interaction is taken place in the region expanding from z = 0 up to z = L U . Due to the accelerating motion of electrons within the undulator, each electron radiates. The radiation field before the electrons reach the location z = 0 is completely spontaneous and covers the entire electron beam due to slippage effect. After electrons pass z = 0, they are allowed to interact with each other through spontaneous radiation field that they have generated in z < 0 region. For an ideal electron beam, i.e. low energy spread and small emittance, such an interaction can result in micro-bunching instability. Because, such a micro-bunching is originated from the interaction of spontaneous emission with the random distribution of the electrons (shot noise), it is called “micro-bunching from shot noise”. This should not be mistaken with “shot-noise micro-bunching”. For any electron beam “shot-noise micro-bunching” is a constant quantity equal to 1/ N e which cannot be affected by energy spread or emittance [9,10]. It is, in fact, the initial value of micro-bunching level for the study of “micro-bunching from shot noise”. In or-
Ne 1 −i θ j e Ne
(1)
j =1
It is indeed the mean value of the electrons phasors. Its modulus can be between 1 (full bunching) and 0 (null bunching). For cases in which the length of the electron beam (L e ) is much smaller than the undulator wavelength (λU ), and the radiation wavelength (λ) is very small compared to L e , the ponderomotive phase of the jth electron, θ j = (k − k U ) z j − ωt, can be written as θ j ≈ −ωt + kz j . Here z j is the longitudinal position of the jth electron within the undulator. When the longitudinal velocity of the electrons are nearly the same, t can be set equal to t ≈ z/ v 0z for all electrons, where z is the mean position of the electron beam in the undulator and v 0z is its mean longitudinal velocity. In this case, the bunching b, can only be a function of z j and can be written as Ne 1 −ikz j b= e Ne
(2)
j =1
where the constant phasor exp(i ω z/ v 0z ) is cancelled. In order to find z j for all the electrons we should find the equations of motion of the electrons. Electrons interact with each other through emitted radiation fields. The emitted radiation field of an accelerated relativistic electron can be best given by Lienard–Wiechert (LW) field [12]. This field consists of two parts. One is for the spacecharge field and the other is for the radiation field. In SASE FELs, the space-charge term can be neglected due to the smallness of the so-called Pierce parameter ρ (ρ ≈ 10−3 ). So the electrons interact through the radiation term of LW field. Following Refs. [5, 6], the differential equations of longitudinal phase space of the jth electron, i.e. [ z j , γ j ], can be derived as [7]
dz j dt dγ j
2. Longitudinal phase space and micro-bunching
1797
dt
= c βzj = c 1 −
=−
r0 a2U ck U
γj
z j > zi
1 + a2U
(3)
γi2
βzi γi (1 − βzi )2
2 sin 2γi 2 kU ( z j − zi )
1+a U
2γi2
1+a2U
k U ( z j − zi ) (4)
In derivation of these equations of motion, it was assumed that the transverse canonical momentum is conserved. Eqs. (3) and (4) do not imply conservation of total momentum and energy of the e-beam. This is due to the fact that in FELs some of the momentum and the energy of the e-beam are converted to radiation. To verify the conservation of total momentum and energy in FELs, the radiation dynamics as well as the dynamics of the electrons must be incorporated. Due to the Doppler effect, the jth electron only is interacted by the electrons that are placed behind it. These two equations for the jth electron are not only coupled to each other but are also coupled to equations of motion of the other electrons through the summation indicated in the equation for γ (Eq. (4)). Therefore, there are 2N e coupled equations which should be solved simultaneously. A computer code has been developed to solve these equations numerically [7]. It generates a Poisson distribution of N e electrons within the L e length of the electron beam and tracks their motion self-consistently within the undulator. It can also calculate the bunching parameter of the entire electron beam as well as the selected parts of the e-beam in any position within the undulator. The output of the code is the longitudinal phase space of
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all electrons. In the following subsections the code has been implemented for two separate cases: 1 – for the electrons that have the same energy at the injection time, 2 – for the case where electrons have different energies at the injection point.
Fig. 1. The evolution of bunching parameter, b, as a function of undulator length, Z .
2.1. Electron beam with no energy spread Consider an electron beam, extending through the length L e , containing N e electrons. The electrons are distributed within the e-beam by a Poisson generator and are injected into the undulator. In the present analysis, an electron beam of length L e = 70 μm, containing N e = 10000 electrons is injected into the left end (i.e. z = −∞) of a semi-infinite helical undulator. The undulator constant is taken to be aU = 1.12, with the undulator wavelength of λU = 1.5 cm, and the fundamental radiation wavelength of λ = λU (1 + a2U )/2γ02 = 3.5 μm. It is assumed that all the electrons have equal energy of E 0 = 35 MeV (γ0 ≈ 70) and the injection is completely aligned with the undulator axis. Throughout the journey from z = −∞ to z = 0 no interaction between electrons is considered, however it is assumed that they radiate and the radiation overlaps over the electrons and falls ahead of the e-beam front. Therefore, for z > 0 region the electron beam is completely immersed in the spontaneous emission that is generated earlier. The evolution of bunching parameter in terms of undulator length is illustrated in Fig. 1. It is evident that the bunching parameter at the beginning of the undulator is nearly zero and then starts growing until reaches a maximum. Fig. 2 shows the phase space evolution of the electron beam corresponding to small circles in Fig. 1. Fig. 2a is the phase space
Fig. 2. Longitudinal phase space of the electron beam in different positions within the undulator. a, b, c, d, e, and f refer to corresponding points in Fig. 1.
M. Rezvani Jalal, F.M. Aghamir / Physics Letters A 375 (2011) 1796–1802
plot of the electron beam at injection stage which shows that the electrons are totally mono-energetic. Fig. 2b is the phase space of the e-beam at the beginning of the exponential gain regime of micro-bunching. It shows the start of micro-bunching in longitudinal phase space. Fig. 2c is the phase space of the same beam at the end of the exponential regime and the beginning of the saturation regime. Fig. 2d represents the phase space plot at maximum micro-bunching (saturation). Finally, Figs. 2e and 2f show the phase space evolution at the start and at the end of post-saturation regime, respectively. The longitudinal phase space at the saturation point (Fig. 2d) shows that the maximum micro-bunching starts falling off when a rolling appears in the phase space (note the left part of saturation phase space plot). In the saturation regime, all the electrons within a segment of length λ approach each other in the bunching center. The electrons with positive ponderomotive phase gain energy and approach the bunching center from left where they encounter with other electrons with negative ponderomotive phase that lose energy and approach the bunching center from right. The rolling phenomenon occurs when they pass each other and bunching starts fading down. The maximum of induced energy spread (γ ) deposited within a mono-energetic electron beam in SASE process can be estimated through the rolling effect. Suppose that an electron with the maximum phase is placed at the extreme left end of the bunching center (at position = −λ/2) and another electron with the minimum phase is placed at the extreme right end of the bunching center (at position = λ/2). If the energy of the first electron is taken to be γ0 + γ /2 and the one of the second electron γ0 − γ /2, then the maximum γ at saturation can be found as
βz (γ0 + γ /2) − βz (γ0 − γ /2)
Ls
βz (γ0 )
=
λ 2
− −
λ 2
(5)
where L s is saturation position within the undulator and βz can be calculated from Eq. (3). This relation can be simplified by the fact that γ γ0 . The simplified form is
γ
γ0
=
1 L− s
4
2γ02 1 + a2U
λ
(6)
In SASE FELs, the saturation length is nearly 20 times of the 1D gain length L G which is defined as [13,14]
LG =
λU √
4π
3ρ
(7)
where ρ is the Pierce parameter in SASE FELs. If L s is replaced by 20L G then by substitution of λ−1 = 2γ02 /(1 + a2U )λU , Eq. (6) can be simplified as
γ
γ0
≈ρ
(8)
This is an important relationship which has been obtained through the foregoing simple assumptions. The same relation has been obtained in literature by rigorous analytical calculations [13]. It shows that the level of relative energy spread in the saturation length is of the order of SASE fundamental parameter. This can be considered as an important validation of the present study formalism. Another look at the phase space plots (Fig. 2) shows that at the end of the post-saturation regime a secondary rolling is taking place (note the left part of Fig. 2f). From this point on another regime of bunching starts. Fig. 1 shows that the first rolling at the saturation point is the onset of bunching degradation but the secondary rolling at the end of post-saturation regime is the sign of a new start of yet another amplification regime of bunching. The existence of other bunching maximums in addition to the first main
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Fig. 3. Evolution of the bunching parameter of four e-beams with different relative energy spreads: (a) δ = 0.09ρ , (b) δ = 0.28ρ , (c) δ = 0.55ρ , and (d) δ = 0.9ρ .
saturation is predicted by other simulation code [15]. The phenomenon of secondary rolling is not as simple as first rolling and its study needs more systematic work. 2.2. Electron beam with small energy spread In this subsection, the numerical studies of the effect of initial energy spread within the injected electrons on the phase space evolution are presented. The previously mentioned numerical code simultaneously integrates Eqs. (3) and (4). For the present analysis, the code is loaded with electrons that their position within the electron beam is Poissonian, however they are no longer monoenergetic. The electrons energy is a random number within the range of γ0 − γ0 /2 to γ0 + γ0 /2. Here γ0 is the initial energy spread of the injected electron beam and it is much smaller than γ0 . According to Eq. (8), in the saturation region, the induced energy spread in a mono-energetic electron beam is of the order ργ0 . This is due to the SASE process and it can be a good criterion for an acceptable initial energy spread level in the injected pulse. Moreover, it can be asserted that the energy spread in the injected beam must be smaller than the maximum induced energy spread in the mono-energetic case (γ0 < ργ0 ). Fig. 1 shows that the saturation length for the present analysis is approximately equal to L s ≈ 75 m. Considering that L s = 20L G and using Eq. (7), the Pierce parameter is found to be ρ = 2 × 10−4 . Based on this assumption, the bunching and the phase space evolution for different values of relative energy spread, namely δ = γ0 /γ0 = 0.09ρ , 0.28ρ , 0.55ρ , and 0.9ρ are numerically calculated. The plot of the bunching evolution in terms of beam position within the undulator is presented in Fig. 3. It is clearly evident that by increasing the relative energy spread from 0 to ρ the bunching maximum decreases and the saturation length increases. This fall and shift is a direct result of initial energy spread in the injected electron beam. The effective number of electrons that participate in the bunching process decreases if there is more spread in the e-beam energy. Thus, the bunching level falls off and a longer undulator length is needed to reach saturation. Fig. 3d shows that for δ ≈ ρ no bunching is realizable. Inspection of the phase space at different positions within the undulator reveals more information with respect to energy spread. In Fig. 4, the phase spaces of electron beams with different initial energy spread are plotted in terms of beam position within the undulator.
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Fig. 4. Typical diagrams of the phase space evolution for four injected electron beams with different initial energy spread. Left column is the initial phase space for relative energy spread of δ = 0.09ρ , 0.28ρ , 0.55ρ , and 0.9ρ from the top to the bottom, respectively. The middle column is corresponding phase space for typical position within the exponential regime (note Fig. 3). The right column is corresponding phase space for saturation points of bunching as can be seen from Fig. 3.
From Fig. 4, it is evident that an increase in the beam energy spread leads to a gradual disappearance of bunching in the phase space. Furthermore, the width of the total energy spread (initial + induced) at saturation is independent of initial energy spread and is nearly constant in all cases (note the width of phasespace curves at saturation for different initial energy spread). This shows that by increasing the initial energy spread contribution, the contribution of induced energy spread at saturation decreases. This is a new finding which has not been reported in the literature. For δ ≈ ρ there is no induced energy spread, therefore no bunching occurs. Thus, it can be concluded that the initial energy spread has a degrading effect on micro-bunching. The micro-bunching process is analogous to nucleation process in physics of crystal growth. In fact, a micro-bunched electron beam is a plasma crystal. It is well known that temperature has
destructive effect on crystal nucleation and growth in many of growth environment (e.g. growth from aqueous solution). The results of Fig. 4 suggest that, the undulator is an environment for growth of plasma crystal and temperature (energy spread) has negative effect on nucleation (micro-bunching) in much the same way as conventional crystal growth. Another noticeable feature of Fig. 4 is the unusual behavior of the e-beam leading edge which shows itself more through the increase of energy spread (note the three bottom plots of right column). This is related to those electrons that have energies higher than undulator resonance energy. These electrons escape from the beam head and radiate spontaneously. The spontaneous emission of these electrons leads to lower rate of energy decrease than the rest of the beam which experiences stimulated emission. However, the physics of beam end parts is complicated and needs
M. Rezvani Jalal, F.M. Aghamir / Physics Letters A 375 (2011) 1796–1802
more detailed study. For example, the SACSE (Self Amplified Coherent Spontaneous Emission) radiation, which must be distinguished from SASE radiation, is the stimulated emission of beam end parts [3,16,17]. 3. Analytical considerations In this section based on our main formulation, namely Eqs. (3) and (4), we present the effect of energy spread on micro-bunching from shot noise. Eq. (4) reveals that the rate of energy change of the jth electron is a sum over many interaction terms with electrons that are behind it. Each interaction term is represented by a Sinc function, i.e. − A i j sin(ki zi j )/ki zi j , where ki = 2γi2 k U /(1 + a2U ). This term is related to the energy of interacting (ith) and interacted ( jth) electrons through A i j and λi . In fact, A i j and λi are the amplitude and the wavelength of the radiation that is generated by the ith electron through its interaction with the jth electron. Numerical calculations show that in the acceptable range of energy spread to realize micro-bunching, i.e. δ < ρ , the effects of both A i j and ki are so small that cannot lead to observed suppression of micro-bunching in Fig. 4. By a simple analysis, it can be shown that the observed suppression in micro-bunching cannot be emanated from k j or A i j . Consider Eq. (4) and assume that A i j is nearly constant for all interacting electrons. The effect of energy spread can be realized through the change in λi . In this case, Eq. (4) is approximated by
dγ j dt
= −A
sin ki zi j
(9)
ki zi j
z j > zi
where γi takes a random value within the range [γ0 − γ /2, γ0 + γ /2]. Setting γi = γ0 + δi in Eq. (9), where γ0 δi , and neglecting δi2 yields:
dγ j dt
≈ −A
sin(k0 + δki ) zi j
(10)
k0 z i j
z j > zi
dγ j dt
≈ −A δk
sin k0 zi j sin k0 zi j /2 k0 z i j
z j > zi
≈ −A
respectively). This thermal motion leads to a continuous variation in the number of interacting neighbors of the interacted electron. In order to find the effect of random motion of the jth electron on the micro-bunching, a simple analytical work is done. In the presence of initial energy spread, the energy of the jth electron can be written as:
γ j (δ j , t ) = γ0 + δ j + h j (δ j , t )
where δ j is the random initial energy deviation in the range [−γ /2, γ /2] and h j (δ j , t ) is the induced energy deviation due to SASE. The above expression for the energy of the jth electron satisfies Eq. (4), and for γ = 0 the mono-energetic case is regained. Insertion of this expression into Eq. (3) and further assumption of the smallness of the deviations with respect to γ0 lead to the following longitudinal velocity:
v zj (δ j , t ) = v z0 +
v z0
γ
(11)
z j > zi
1
γ
3 0
δ j + h j (δ j , t )
(14)
z j (δ j , t ) = z0 j + v z0 t +
c2
1 + a2U
δ jt +
γ03
v z0
t h j (δ j , t ) dt 0
(15) In order to calculate the bunching factor, the longitudinal coordinate must be inserted in Eq. (2). The bunching in the presence of energy spread is denoted by uppercase “B” rather than the lowercase “b” which is assigned for mono-energetic beam. Thus, we have:
B=
Ne 1 −ikz j (δ j ,t ) e Ne j =1
Ne c 1 −ik[z0 j + v z0 t +( v z0 )( = e Ne
1+a2 U γ03
t
)(δ j t +
0
h j (δ j ,t ) dt )]
j =1
1+a2 U γ03
t
)(δ j t +
0
h j (δ j ,t ) dt )]
(16)
The parameter “B” was averaged over δ j in the range [−γ /2, γ /2] to estimate the total behavior of bunching:
Setting k0 = 4γ0 γ k U /(1 + in the last equation, and assuming that the two summations have nearly the same value, the following equation can be obtained:
The time integration of the above equation yields the longitudinal coordinate:
Ne c e ikv z0 t −ik[z0 j +( v z0 )( B δ = e Ne 2
j =1
a2U )
dγ j
1 + a2U
j =1
k0 z i j
24k0
dγ j
2
k20 A (k0 zi j sin k0 zi j ) − − 2
c2
Ne c e ikv z0 t −ik[z0 j +( v z0 )( = e Ne
k0 zi j /2
sin k0 zi j
z j > zi
(13)
2
The mean value of the rate of change of the energy with respect to δi over [−γ /2, γ /2] can be calculated as
1801
2
≈
e ikv z0 t Ne
ESinc
1+a2 U γ03
c 2 (1 + a2U )t γ 2v z0 03
γ
t
)(δ j t +
0
h j (δ j ,t ) dt )]
δ
Ne
t e −ik(z0 j + 0 h j (δ j ,t ) δ dt )
j =1
(12)
(17)
Since γ /γ0 < ρ ≈ 10−3 , the second term in the parenthesis is very small. Thus, for an energy spread smaller that ρ (which is necessary for micro-bunching build-up) the effects of the wavelength change on micro-bunching are completely negligible. Similar analysis can be done for A i j to show that the effect of energy spread on micro-bunching through A i j is not considerable. If λi and A i j both have very small effect on micro-bunching, what then causes the significant degrading of micro-bunching, as is observable in Fig. 4? The answer can be traced in the random motion of the jth electron rather than fluctuation of radiation wavelength and amplitude of interacting ith electron (i.e. λi and A i j ,
This is a very interesting result. The term ESinc is used to denote the “envelope of Sinc function” which appears due to the averaging. The multiplying factor appearing before the ESinc function represents a constant phasor and is of no interest. For γ = 0, the bunching for the case of mono-energetic beam is retrieved. For finite values of γ , the bunching is related to initial energy spread γ through the second factor (ESinc function) and the third factor (summation term). The ESinc function is a descending function; therefore, for a finite value of γ , the bunching parameter ‘B becomes smaller than ‘b as time goes by. In fact, the ESinc function is responsible for the fall of the bunching maximum, as was reflected in Fig. 3. The summation term in Eq. (17) is another term that
dt
≈ δk
dt
1−
6
γ0
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M. Rezvani Jalal, F.M. Aghamir / Physics Letters A 375 (2011) 1796–1802
has been affected by initial energy spread. This term is responsible for the shift in maximum bunching as is shown in Fig. 3. As was numerically deduced in previous section, the energy equation (Eq. (4)) behaves such that the sum of induced energy spread (due to SASE process) and initial energy spread is constant. This means that a high level of initial energy spread leads to a small level of induced energy spread. The summation term contains the mean value of the induced energy spread (h j (δ j , t )). The fact that the level of the induced energy spread is small in the presence of initial energy spread leads to “shallow bunching” in the phase space. The term “shallow bunching” is attributed to the process of slow micro-bunching. Hence, the micro-bunching maximum will occur over a period of a longer time and the bunching saturation will shift to longer distance positions in the undulator as can be seen form Fig. 3. Similar qualitative results obtained by other methods (theory or simulation) can be found in [18–20]. 4. Discussion and conclusion The equations of motion of N e radiating electrons which move along an undulator and interact through Lienard–Wiechert fields were used (derived before in Ref. [7]). Numerical solution of these 2N e coupled differential equations was obtained through an integrating code. In the case where, injected electrons are initially mono-energetic, the bunching of the phase space diagram with fundamental radiation wavelength is evident. The onset of the first rolling in the phase space plot is a sign of reaching to a saturation stage. The saturation can well be seen in the micro-bunching evolution within the undulator. The appearance of secondary rolling is another phenomenon which is a sign of yet another bunching process after saturation. With a simple assumption that, in the first rolling, the induced energy spread causes two electrons approach each other by fundamental wavelength λ, the maximum induced relative energy spread in the saturation was estimated by γ /γ0 ≈ ρ . This is consistent with what has been reported in Ref. [13]. The numerical solution for electrons that are not necessarily mono-energetic upon injection is also presented for different initial energy spread. It was shown that if the initial energy spread is of the order of Pierce parameter ρ , then no bunching would be feasible and no SASE occurs. The second finding is that the total energy width (initial + induced) is independent of initial energy spread and is constant. This implies that by increasing initial relative energy spread, the induced energy spread (which is the level of micro-bunching) decreases. Analytical investigation of the coupled longitudinal equations shows that the effect of energy spread on micro-bunching can be attributed to three factors: 1 – the amplitude A i j of the radiation of the ith interacting particle in interaction with the jth particle, 2 – wavelength λi of the radiation of the ith particle, 3 – thermal motion of the jth particle. Analytical treatment shows that the first two factors have very small contribution in degradation of micro-bunching and in fact it is the third factor that has the major role in micro-bunching suppression. It was shown that the random motion of particles causes the micro-bunching maximum to fall and its position is shifted to locations further downstream within
the undulator. It can also be argued that the sum of all radiations leads to ponderomotive wells when superposed on undulator magnetic field. Those electrons with sufficient energy for resonance, regardless of their phase, can fall into these wells and contribute in micro-bunching; however, the electrons with off-resonant energies hardly (depending on their phases and velocities) fall in the wells and do not have any contribution in micro-bunching. These results are in complete agreement with the analysis presented in literature which are based on Maxwell–Klimontovich coupled equations [13,21] rather that direct Lienard–Wiechert fields. The consistent results prove the validity of the obtained equations of motion. In this Letter the effect of initial energy spread (longitudinal temperature) on micro-bunching suppression has been considered by Lienard–Wiechert fields. The same procedure can also be undertaken for transverse emittance (transverse temperature). Analytical treatment by coupled Maxwell–Klimontovich equations shows that the fall and the shift of saturation can also be seen for transverse emittance. Gaining such results by Lienard–Wiechert field remains as our future work. Acknowledgements The first author would like to offer his sincere appreciation to Dr. Massimo Ferrario for accepting his request for sabbatical stay at INFN-LNF center and his scientific help during his stay. He also wishes to extend his gratitude to all members of SPARC project from INFN-LNF and ENEA centers. This work was supported by financial assistance of INFN-LNF center and University of Tehran, Research Council. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
R. Bonifacio, C. Pellegrini, L.M. Narducci, Opt. Commun. 50 (1984) 373. K.-J. Kim, Nucl. Instrum. Methods A 250 (1986) 396. B.W.J. McNeil, G.R.M. Robb, Phys. Rev. E 65 (2002) 046503. R. Bonifacio, L. DeSalvo, P. Pierini, N. Piovella, C. Pellegrini, Phys. Rev. Lett. 73 (1) (1994) 70. M. Tecimer, L. Elias, Nucl. Instrum. Methods A 375 (1996) 348. M. Tecimer, L. Elias, Nucl. Instrum. Methods A 393 (1997) 104. M. Rezvanijalal, F.M. Aghamir, Nucl. Inst. Phys. Res. A 601 (2009) 245. C. Pellegrini, S. Reiche, IEEE J. Selected Topics in Quantum Electron. 10 (6) (2004) 1393. M. Rezvani Jalal, F.M. Aghamir, Opt. Commun. 281 (2008) 3771. R. Bonifacio, Opt. Commun. 138 (1997) 99. B.W.J. McNeil, G.R.M. Robb, J. Phys. D: Appl. Phys. 30 (1997) 567; See also: B.W.J. McNeil, G.R.M. Robb, J. Phys. D: Appl. Phys. 31 (1998) 371. J.D. Jackson, Classical Electrodynamics, 2nd ed., Wiley & Sons, New York, 1975, p. 654. Z. Huang, K.-J. Kim, Phys. Rev. ST Accel. Beams 10 (2007) 034801. M. Boscolo, M. Ferrario, I. Boscolo, F. Castelli, S. Cialdi, V. Petrillo, R. Bonifacio, L. Palumbo, L. Serafini, Nucl. Instrum. Methods A 593 (2008) 137. C.B. Scherieder, W.M. Fawley, E. Esarey, Nucl. Instrum. Methods A 507 (2003) 110. B.W.J. McNeil, G.R.M. Robb, D.A. Jaroszynski, Opt. Commun. 165 (1999) 65. B.W.J. McNeil, M.W. Poole, G.R.M. Robb, Phys. Rev. ST Accel. Beams 6 (2003) 070701. E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov, Opt. Commun. 235 (2004) 415. B. Hafizi, C.W. Roberson, Phys. Rev. Lett. 68 (24) (1992) 3539. H.P. Freund, P.G. O’Shea, S.G. Biedron, Nucl. Instrum. Methods A 528 (2004) 44. G.T. Moore, Opt. Commun. 54 (2) (1985) 121.
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Physics Letters A www.elsevier.com/locate/pla
Effect of energy spread on micro-bunching from shot noise in SASE FELs M. Rezvani Jalal a,b , F.M. Aghamir a,∗ a b
Department of Physics, University of Tehran, North Kargar Avenue, Tehran 14399, Iran Department of Physics, Malayer University, 4th Kilometer of Arak road, Malayer, Iran
a r t i c l e
i n f o
Article history: Received 5 December 2010 Received in revised form 3 February 2011 Accepted 9 February 2011 Available online 16 February 2011 Communicated by F. Porcelli Keywords: Micro-bunching from shot noise Shot-noise micro-bunching Total energy spread Lienard–Wiechert field
a b s t r a c t The motion of N e electrons moving along a helical undulator and interacting with each other through Lienard–Wiechert fields is considered. The numerical solution for initially mono-energetic electrons shows the emergence and growth of micro-bunching. For the case of initially non-mono-energetic electrons, calculations show that the presence of energy spread results in partial suppression of the micro-bunching. The calculations also show that the initial energy spread would shift the microbunching maximum to deeper positions within the undulator. Analytical treatment of equations of motion demonstrates that the micro-bunching fall and shift is mainly due to the random motion of the interacting electrons. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Micro-bunching is a familiar and fundamental phenomenon in Free Electron Lasers (FELs). In this process, the smooth distribution of electrons in the injected electron beam is converted to a periodic (bunched) distribution. The period length is the fundamental wavelength of the undulator radiation. In fact, micro-bunching is an instability that converts weak and incoherent (spontaneous) emission of an oscillator FEL to a robust and coherent (induced) emission. As well, the weak input radiation of an amplifier FEL is converted to a powerful laser radiation through this instability. In brief, in FELs the electron beam is the active medium that transforms the spontaneous emission to an induced emission by means of micro-bunching instability. The reason behind the instability is the dispersion that is related to the ponderomotive phase of electrons. Electrons that have positive ponderomotive phase gain energy from the radiation. Conversely, electrons that have negative ponderomotive phase yield energy to the radiation. This energy distribution in the electron beam is converted to density distribution because of dispersive property of the undulator. In a special kind of FELs which cannot be operated in the oscillator mode due to the lack of appropriate cavity mirrors and also cannot be operated in the amplifier mode, because of the lack of coherent input radiation, the SASE instability plays a crucial role. In this kind of FEL instability, the spontaneous emission
*
Corresponding author. Tel.: +98 21 61118603. E-mail address: [email protected] (F.M. Aghamir).
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of electron beam remains overlapped with the electron beam for a long travel in the undulator. Because of this long effective interaction, special type of micro-bunching, which is called SASE micro-bunching, occurs within the electron beam. The startup and growth of such a micro-bunching lead to a laser-like radiation. There are two differences between SASE micro-bunching and typical micro-bunching: (1) typical micro-bunching is originated from weak coherent input radiation, while, SASE micro-bunching is originated from weak spontaneous emission of electron beam; (2) typical micro-bunching is a completely periodic pattern over the entire electron beam, while SASE micro-bunching is a periodic pattern over distinct sections of the electron beam where there is no phase relation between these sections. Therefore, the typical micro-bunching results in full longitudinal coherence for output radiation while SASE micro-bunching leads to an output with partial longitudinal coherence (i.e. some distinct portions of it are longitudinally coherent). SASE FEL is an amplifier FEL that its input radiation is the spontaneous emission of the electron beam occurring at the very beginning of the undulator. When such a spontaneous radiation is overlapped over the radiating electron beam, one of the following two conditions can be realized: if a well-matched overlap persists over a long enough length of the undulator and if the electron beam is of a perfect quality (low energy spread and small emittance) then the micro-bunching instability occurs. Contrary, if the overlap is imperfect or the energy spread of the electron beam is of significant amount then no micro-bunching happens. The discreteness of charge carriers plus their random distribution within the electron beam play essential role in the SASE micro-bunching. If
M. Rezvani Jalal, F.M. Aghamir / Physics Letters A 375 (2011) 1796–1802
the electron beam is considered to be infinitely long and homogeneous then micro-bunching instability never occurs. Such discreteness and randomness of electrons generate a noise in the electron density which is known as shot noise. The startup of SASE instability from shot noise has been studied in literature [1–4]. The discreteness of charges has been taken into account by means of Klimontovich distribution function. The paraxial Maxwell’s wave equation was then incorporated to form a set of coupled Maxwell–Klimontovich equations. By numerical solution of those coupled equations, many fundamental characteristics of SASE FELs can be obtained. The study of the SASE FELs by means of exact solutions of Maxwell wave equations (namely, Lienard–Wiechert fields) was first under taken by Tecimer and Elias [5,6]. Following their works, authors of the present article [7], presented a simpler form of longitudinal equations of motion and used them to find fundamental aspects of SASE FELs as a testbed for the validity of their set of equations. In the present study, the formulation of Ref. [7] is generalized to non-mono-energetic electrons and the effect of energy spread in the injected electron beam on micro-bunching is presented. It is shown, both numerically and analytically, that the presence of energy spread leads to partial suppression of micro-bunching and for a certain maximum energy spread, the micro-bunching is completely destroyed. Micro-bunching in electron beam is very similar to nucleation phenomenon in physics of crystals. A typical micro-bunched beam can be considered as a plasma single-crystal [8] and a SASE microbunched beam as a plasma poly-crystal. Temperature in physics of crystals usually leads to suppression of nucleation. We expect that in micro-bunching the temperature (or more precisely, energy spread and transverse emittance) act in a similar manner in degradation of micro-bunching process. Our numerical calculation shows that such suppression in micro-bunching does occur. The organization of the Letter is as follows: first the evolution of the longitudinal phase space of an injected mono-energetic electron beam within the undulator is considered. Then, some numerical calculations for the evolution of the longitudinal phase space of non-mono-energetic electron beams are carried out. This is followed by the section which is devoted to some analytical work to explain the observation of numerical results. Finally, the concluding remarks are presented in the last section.
der to study micro-bunching from shot noise, the bunching factor should be defined first. A quantity that can best explain the microbunching process is defined by [1,11]:
b≡
Consider an electron beam of N e relativistic electrons. The beam is injected into a helical undulator. The undulator is assumed to be semi-infinite meaning that its left end is at infinity and the right end is terminated at z = L U . The electrons are considered to be non-interacting when they travel from the left end up to the origin at z = 0; however, the electrons interaction is taken place in the region expanding from z = 0 up to z = L U . Due to the accelerating motion of electrons within the undulator, each electron radiates. The radiation field before the electrons reach the location z = 0 is completely spontaneous and covers the entire electron beam due to slippage effect. After electrons pass z = 0, they are allowed to interact with each other through spontaneous radiation field that they have generated in z < 0 region. For an ideal electron beam, i.e. low energy spread and small emittance, such an interaction can result in micro-bunching instability. Because, such a micro-bunching is originated from the interaction of spontaneous emission with the random distribution of the electrons (shot noise), it is called “micro-bunching from shot noise”. This should not be mistaken with “shot-noise micro-bunching”. For any electron beam “shot-noise micro-bunching” is a constant quantity equal to 1/ N e which cannot be affected by energy spread or emittance [9,10]. It is, in fact, the initial value of micro-bunching level for the study of “micro-bunching from shot noise”. In or-
Ne 1 −i θ j e Ne
(1)
j =1
It is indeed the mean value of the electrons phasors. Its modulus can be between 1 (full bunching) and 0 (null bunching). For cases in which the length of the electron beam (L e ) is much smaller than the undulator wavelength (λU ), and the radiation wavelength (λ) is very small compared to L e , the ponderomotive phase of the jth electron, θ j = (k − k U ) z j − ωt, can be written as θ j ≈ −ωt + kz j . Here z j is the longitudinal position of the jth electron within the undulator. When the longitudinal velocity of the electrons are nearly the same, t can be set equal to t ≈ z/ v 0z for all electrons, where z is the mean position of the electron beam in the undulator and v 0z is its mean longitudinal velocity. In this case, the bunching b, can only be a function of z j and can be written as Ne 1 −ikz j b= e Ne
(2)
j =1
where the constant phasor exp(i ω z/ v 0z ) is cancelled. In order to find z j for all the electrons we should find the equations of motion of the electrons. Electrons interact with each other through emitted radiation fields. The emitted radiation field of an accelerated relativistic electron can be best given by Lienard–Wiechert (LW) field [12]. This field consists of two parts. One is for the spacecharge field and the other is for the radiation field. In SASE FELs, the space-charge term can be neglected due to the smallness of the so-called Pierce parameter ρ (ρ ≈ 10−3 ). So the electrons interact through the radiation term of LW field. Following Refs. [5, 6], the differential equations of longitudinal phase space of the jth electron, i.e. [ z j , γ j ], can be derived as [7]
dz j dt dγ j
2. Longitudinal phase space and micro-bunching
1797
dt
= c βzj = c 1 −
=−
r0 a2U ck U
γj
z j > zi
1 + a2U
(3)
γi2
βzi γi (1 − βzi )2
2 sin 2γi 2 kU ( z j − zi )
1+a U
2γi2
1+a2U
k U ( z j − zi ) (4)
In derivation of these equations of motion, it was assumed that the transverse canonical momentum is conserved. Eqs. (3) and (4) do not imply conservation of total momentum and energy of the e-beam. This is due to the fact that in FELs some of the momentum and the energy of the e-beam are converted to radiation. To verify the conservation of total momentum and energy in FELs, the radiation dynamics as well as the dynamics of the electrons must be incorporated. Due to the Doppler effect, the jth electron only is interacted by the electrons that are placed behind it. These two equations for the jth electron are not only coupled to each other but are also coupled to equations of motion of the other electrons through the summation indicated in the equation for γ (Eq. (4)). Therefore, there are 2N e coupled equations which should be solved simultaneously. A computer code has been developed to solve these equations numerically [7]. It generates a Poisson distribution of N e electrons within the L e length of the electron beam and tracks their motion self-consistently within the undulator. It can also calculate the bunching parameter of the entire electron beam as well as the selected parts of the e-beam in any position within the undulator. The output of the code is the longitudinal phase space of
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all electrons. In the following subsections the code has been implemented for two separate cases: 1 – for the electrons that have the same energy at the injection time, 2 – for the case where electrons have different energies at the injection point.
Fig. 1. The evolution of bunching parameter, b, as a function of undulator length, Z .
2.1. Electron beam with no energy spread Consider an electron beam, extending through the length L e , containing N e electrons. The electrons are distributed within the e-beam by a Poisson generator and are injected into the undulator. In the present analysis, an electron beam of length L e = 70 μm, containing N e = 10000 electrons is injected into the left end (i.e. z = −∞) of a semi-infinite helical undulator. The undulator constant is taken to be aU = 1.12, with the undulator wavelength of λU = 1.5 cm, and the fundamental radiation wavelength of λ = λU (1 + a2U )/2γ02 = 3.5 μm. It is assumed that all the electrons have equal energy of E 0 = 35 MeV (γ0 ≈ 70) and the injection is completely aligned with the undulator axis. Throughout the journey from z = −∞ to z = 0 no interaction between electrons is considered, however it is assumed that they radiate and the radiation overlaps over the electrons and falls ahead of the e-beam front. Therefore, for z > 0 region the electron beam is completely immersed in the spontaneous emission that is generated earlier. The evolution of bunching parameter in terms of undulator length is illustrated in Fig. 1. It is evident that the bunching parameter at the beginning of the undulator is nearly zero and then starts growing until reaches a maximum. Fig. 2 shows the phase space evolution of the electron beam corresponding to small circles in Fig. 1. Fig. 2a is the phase space
Fig. 2. Longitudinal phase space of the electron beam in different positions within the undulator. a, b, c, d, e, and f refer to corresponding points in Fig. 1.
M. Rezvani Jalal, F.M. Aghamir / Physics Letters A 375 (2011) 1796–1802
plot of the electron beam at injection stage which shows that the electrons are totally mono-energetic. Fig. 2b is the phase space of the e-beam at the beginning of the exponential gain regime of micro-bunching. It shows the start of micro-bunching in longitudinal phase space. Fig. 2c is the phase space of the same beam at the end of the exponential regime and the beginning of the saturation regime. Fig. 2d represents the phase space plot at maximum micro-bunching (saturation). Finally, Figs. 2e and 2f show the phase space evolution at the start and at the end of post-saturation regime, respectively. The longitudinal phase space at the saturation point (Fig. 2d) shows that the maximum micro-bunching starts falling off when a rolling appears in the phase space (note the left part of saturation phase space plot). In the saturation regime, all the electrons within a segment of length λ approach each other in the bunching center. The electrons with positive ponderomotive phase gain energy and approach the bunching center from left where they encounter with other electrons with negative ponderomotive phase that lose energy and approach the bunching center from right. The rolling phenomenon occurs when they pass each other and bunching starts fading down. The maximum of induced energy spread (γ ) deposited within a mono-energetic electron beam in SASE process can be estimated through the rolling effect. Suppose that an electron with the maximum phase is placed at the extreme left end of the bunching center (at position = −λ/2) and another electron with the minimum phase is placed at the extreme right end of the bunching center (at position = λ/2). If the energy of the first electron is taken to be γ0 + γ /2 and the one of the second electron γ0 − γ /2, then the maximum γ at saturation can be found as
βz (γ0 + γ /2) − βz (γ0 − γ /2)
Ls
βz (γ0 )
=
λ 2
− −
λ 2
(5)
where L s is saturation position within the undulator and βz can be calculated from Eq. (3). This relation can be simplified by the fact that γ γ0 . The simplified form is
γ
γ0
=
1 L− s
4
2γ02 1 + a2U
λ
(6)
In SASE FELs, the saturation length is nearly 20 times of the 1D gain length L G which is defined as [13,14]
LG =
λU √
4π
3ρ
(7)
where ρ is the Pierce parameter in SASE FELs. If L s is replaced by 20L G then by substitution of λ−1 = 2γ02 /(1 + a2U )λU , Eq. (6) can be simplified as
γ
γ0
≈ρ
(8)
This is an important relationship which has been obtained through the foregoing simple assumptions. The same relation has been obtained in literature by rigorous analytical calculations [13]. It shows that the level of relative energy spread in the saturation length is of the order of SASE fundamental parameter. This can be considered as an important validation of the present study formalism. Another look at the phase space plots (Fig. 2) shows that at the end of the post-saturation regime a secondary rolling is taking place (note the left part of Fig. 2f). From this point on another regime of bunching starts. Fig. 1 shows that the first rolling at the saturation point is the onset of bunching degradation but the secondary rolling at the end of post-saturation regime is the sign of a new start of yet another amplification regime of bunching. The existence of other bunching maximums in addition to the first main
1799
Fig. 3. Evolution of the bunching parameter of four e-beams with different relative energy spreads: (a) δ = 0.09ρ , (b) δ = 0.28ρ , (c) δ = 0.55ρ , and (d) δ = 0.9ρ .
saturation is predicted by other simulation code [15]. The phenomenon of secondary rolling is not as simple as first rolling and its study needs more systematic work. 2.2. Electron beam with small energy spread In this subsection, the numerical studies of the effect of initial energy spread within the injected electrons on the phase space evolution are presented. The previously mentioned numerical code simultaneously integrates Eqs. (3) and (4). For the present analysis, the code is loaded with electrons that their position within the electron beam is Poissonian, however they are no longer monoenergetic. The electrons energy is a random number within the range of γ0 − γ0 /2 to γ0 + γ0 /2. Here γ0 is the initial energy spread of the injected electron beam and it is much smaller than γ0 . According to Eq. (8), in the saturation region, the induced energy spread in a mono-energetic electron beam is of the order ργ0 . This is due to the SASE process and it can be a good criterion for an acceptable initial energy spread level in the injected pulse. Moreover, it can be asserted that the energy spread in the injected beam must be smaller than the maximum induced energy spread in the mono-energetic case (γ0 < ργ0 ). Fig. 1 shows that the saturation length for the present analysis is approximately equal to L s ≈ 75 m. Considering that L s = 20L G and using Eq. (7), the Pierce parameter is found to be ρ = 2 × 10−4 . Based on this assumption, the bunching and the phase space evolution for different values of relative energy spread, namely δ = γ0 /γ0 = 0.09ρ , 0.28ρ , 0.55ρ , and 0.9ρ are numerically calculated. The plot of the bunching evolution in terms of beam position within the undulator is presented in Fig. 3. It is clearly evident that by increasing the relative energy spread from 0 to ρ the bunching maximum decreases and the saturation length increases. This fall and shift is a direct result of initial energy spread in the injected electron beam. The effective number of electrons that participate in the bunching process decreases if there is more spread in the e-beam energy. Thus, the bunching level falls off and a longer undulator length is needed to reach saturation. Fig. 3d shows that for δ ≈ ρ no bunching is realizable. Inspection of the phase space at different positions within the undulator reveals more information with respect to energy spread. In Fig. 4, the phase spaces of electron beams with different initial energy spread are plotted in terms of beam position within the undulator.
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Fig. 4. Typical diagrams of the phase space evolution for four injected electron beams with different initial energy spread. Left column is the initial phase space for relative energy spread of δ = 0.09ρ , 0.28ρ , 0.55ρ , and 0.9ρ from the top to the bottom, respectively. The middle column is corresponding phase space for typical position within the exponential regime (note Fig. 3). The right column is corresponding phase space for saturation points of bunching as can be seen from Fig. 3.
From Fig. 4, it is evident that an increase in the beam energy spread leads to a gradual disappearance of bunching in the phase space. Furthermore, the width of the total energy spread (initial + induced) at saturation is independent of initial energy spread and is nearly constant in all cases (note the width of phasespace curves at saturation for different initial energy spread). This shows that by increasing the initial energy spread contribution, the contribution of induced energy spread at saturation decreases. This is a new finding which has not been reported in the literature. For δ ≈ ρ there is no induced energy spread, therefore no bunching occurs. Thus, it can be concluded that the initial energy spread has a degrading effect on micro-bunching. The micro-bunching process is analogous to nucleation process in physics of crystal growth. In fact, a micro-bunched electron beam is a plasma crystal. It is well known that temperature has
destructive effect on crystal nucleation and growth in many of growth environment (e.g. growth from aqueous solution). The results of Fig. 4 suggest that, the undulator is an environment for growth of plasma crystal and temperature (energy spread) has negative effect on nucleation (micro-bunching) in much the same way as conventional crystal growth. Another noticeable feature of Fig. 4 is the unusual behavior of the e-beam leading edge which shows itself more through the increase of energy spread (note the three bottom plots of right column). This is related to those electrons that have energies higher than undulator resonance energy. These electrons escape from the beam head and radiate spontaneously. The spontaneous emission of these electrons leads to lower rate of energy decrease than the rest of the beam which experiences stimulated emission. However, the physics of beam end parts is complicated and needs
M. Rezvani Jalal, F.M. Aghamir / Physics Letters A 375 (2011) 1796–1802
more detailed study. For example, the SACSE (Self Amplified Coherent Spontaneous Emission) radiation, which must be distinguished from SASE radiation, is the stimulated emission of beam end parts [3,16,17]. 3. Analytical considerations In this section based on our main formulation, namely Eqs. (3) and (4), we present the effect of energy spread on micro-bunching from shot noise. Eq. (4) reveals that the rate of energy change of the jth electron is a sum over many interaction terms with electrons that are behind it. Each interaction term is represented by a Sinc function, i.e. − A i j sin(ki zi j )/ki zi j , where ki = 2γi2 k U /(1 + a2U ). This term is related to the energy of interacting (ith) and interacted ( jth) electrons through A i j and λi . In fact, A i j and λi are the amplitude and the wavelength of the radiation that is generated by the ith electron through its interaction with the jth electron. Numerical calculations show that in the acceptable range of energy spread to realize micro-bunching, i.e. δ < ρ , the effects of both A i j and ki are so small that cannot lead to observed suppression of micro-bunching in Fig. 4. By a simple analysis, it can be shown that the observed suppression in micro-bunching cannot be emanated from k j or A i j . Consider Eq. (4) and assume that A i j is nearly constant for all interacting electrons. The effect of energy spread can be realized through the change in λi . In this case, Eq. (4) is approximated by
dγ j dt
= −A
sin ki zi j
(9)
ki zi j
z j > zi
where γi takes a random value within the range [γ0 − γ /2, γ0 + γ /2]. Setting γi = γ0 + δi in Eq. (9), where γ0 δi , and neglecting δi2 yields:
dγ j dt
≈ −A
sin(k0 + δki ) zi j
(10)
k0 z i j
z j > zi
dγ j dt
≈ −A δk
sin k0 zi j sin k0 zi j /2 k0 z i j
z j > zi
≈ −A
respectively). This thermal motion leads to a continuous variation in the number of interacting neighbors of the interacted electron. In order to find the effect of random motion of the jth electron on the micro-bunching, a simple analytical work is done. In the presence of initial energy spread, the energy of the jth electron can be written as:
γ j (δ j , t ) = γ0 + δ j + h j (δ j , t )
where δ j is the random initial energy deviation in the range [−γ /2, γ /2] and h j (δ j , t ) is the induced energy deviation due to SASE. The above expression for the energy of the jth electron satisfies Eq. (4), and for γ = 0 the mono-energetic case is regained. Insertion of this expression into Eq. (3) and further assumption of the smallness of the deviations with respect to γ0 lead to the following longitudinal velocity:
v zj (δ j , t ) = v z0 +
v z0
γ
(11)
z j > zi
1
γ
3 0
δ j + h j (δ j , t )
(14)
z j (δ j , t ) = z0 j + v z0 t +
c2
1 + a2U
δ jt +
γ03
v z0
t h j (δ j , t ) dt 0
(15) In order to calculate the bunching factor, the longitudinal coordinate must be inserted in Eq. (2). The bunching in the presence of energy spread is denoted by uppercase “B” rather than the lowercase “b” which is assigned for mono-energetic beam. Thus, we have:
B=
Ne 1 −ikz j (δ j ,t ) e Ne j =1
Ne c 1 −ik[z0 j + v z0 t +( v z0 )( = e Ne
1+a2 U γ03
t
)(δ j t +
0
h j (δ j ,t ) dt )]
j =1
1+a2 U γ03
t
)(δ j t +
0
h j (δ j ,t ) dt )]
(16)
The parameter “B” was averaged over δ j in the range [−γ /2, γ /2] to estimate the total behavior of bunching:
Setting k0 = 4γ0 γ k U /(1 + in the last equation, and assuming that the two summations have nearly the same value, the following equation can be obtained:
The time integration of the above equation yields the longitudinal coordinate:
Ne c e ikv z0 t −ik[z0 j +( v z0 )( B δ = e Ne 2
j =1
a2U )
dγ j
1 + a2U
j =1
k0 z i j
24k0
dγ j
2
k20 A (k0 zi j sin k0 zi j ) − − 2
c2
Ne c e ikv z0 t −ik[z0 j +( v z0 )( = e Ne
k0 zi j /2
sin k0 zi j
z j > zi
(13)
2
The mean value of the rate of change of the energy with respect to δi over [−γ /2, γ /2] can be calculated as
1801
2
≈
e ikv z0 t Ne
ESinc
1+a2 U γ03
c 2 (1 + a2U )t γ 2v z0 03
γ
t
)(δ j t +
0
h j (δ j ,t ) dt )]
δ
Ne
t e −ik(z0 j + 0 h j (δ j ,t ) δ dt )
j =1
(12)
(17)
Since γ /γ0 < ρ ≈ 10−3 , the second term in the parenthesis is very small. Thus, for an energy spread smaller that ρ (which is necessary for micro-bunching build-up) the effects of the wavelength change on micro-bunching are completely negligible. Similar analysis can be done for A i j to show that the effect of energy spread on micro-bunching through A i j is not considerable. If λi and A i j both have very small effect on micro-bunching, what then causes the significant degrading of micro-bunching, as is observable in Fig. 4? The answer can be traced in the random motion of the jth electron rather than fluctuation of radiation wavelength and amplitude of interacting ith electron (i.e. λi and A i j ,
This is a very interesting result. The term ESinc is used to denote the “envelope of Sinc function” which appears due to the averaging. The multiplying factor appearing before the ESinc function represents a constant phasor and is of no interest. For γ = 0, the bunching for the case of mono-energetic beam is retrieved. For finite values of γ , the bunching is related to initial energy spread γ through the second factor (ESinc function) and the third factor (summation term). The ESinc function is a descending function; therefore, for a finite value of γ , the bunching parameter ‘B becomes smaller than ‘b as time goes by. In fact, the ESinc function is responsible for the fall of the bunching maximum, as was reflected in Fig. 3. The summation term in Eq. (17) is another term that
dt
≈ δk
dt
1−
6
γ0
1802
M. Rezvani Jalal, F.M. Aghamir / Physics Letters A 375 (2011) 1796–1802
has been affected by initial energy spread. This term is responsible for the shift in maximum bunching as is shown in Fig. 3. As was numerically deduced in previous section, the energy equation (Eq. (4)) behaves such that the sum of induced energy spread (due to SASE process) and initial energy spread is constant. This means that a high level of initial energy spread leads to a small level of induced energy spread. The summation term contains the mean value of the induced energy spread (h j (δ j , t )). The fact that the level of the induced energy spread is small in the presence of initial energy spread leads to “shallow bunching” in the phase space. The term “shallow bunching” is attributed to the process of slow micro-bunching. Hence, the micro-bunching maximum will occur over a period of a longer time and the bunching saturation will shift to longer distance positions in the undulator as can be seen form Fig. 3. Similar qualitative results obtained by other methods (theory or simulation) can be found in [18–20]. 4. Discussion and conclusion The equations of motion of N e radiating electrons which move along an undulator and interact through Lienard–Wiechert fields were used (derived before in Ref. [7]). Numerical solution of these 2N e coupled differential equations was obtained through an integrating code. In the case where, injected electrons are initially mono-energetic, the bunching of the phase space diagram with fundamental radiation wavelength is evident. The onset of the first rolling in the phase space plot is a sign of reaching to a saturation stage. The saturation can well be seen in the micro-bunching evolution within the undulator. The appearance of secondary rolling is another phenomenon which is a sign of yet another bunching process after saturation. With a simple assumption that, in the first rolling, the induced energy spread causes two electrons approach each other by fundamental wavelength λ, the maximum induced relative energy spread in the saturation was estimated by γ /γ0 ≈ ρ . This is consistent with what has been reported in Ref. [13]. The numerical solution for electrons that are not necessarily mono-energetic upon injection is also presented for different initial energy spread. It was shown that if the initial energy spread is of the order of Pierce parameter ρ , then no bunching would be feasible and no SASE occurs. The second finding is that the total energy width (initial + induced) is independent of initial energy spread and is constant. This implies that by increasing initial relative energy spread, the induced energy spread (which is the level of micro-bunching) decreases. Analytical investigation of the coupled longitudinal equations shows that the effect of energy spread on micro-bunching can be attributed to three factors: 1 – the amplitude A i j of the radiation of the ith interacting particle in interaction with the jth particle, 2 – wavelength λi of the radiation of the ith particle, 3 – thermal motion of the jth particle. Analytical treatment shows that the first two factors have very small contribution in degradation of micro-bunching and in fact it is the third factor that has the major role in micro-bunching suppression. It was shown that the random motion of particles causes the micro-bunching maximum to fall and its position is shifted to locations further downstream within
the undulator. It can also be argued that the sum of all radiations leads to ponderomotive wells when superposed on undulator magnetic field. Those electrons with sufficient energy for resonance, regardless of their phase, can fall into these wells and contribute in micro-bunching; however, the electrons with off-resonant energies hardly (depending on their phases and velocities) fall in the wells and do not have any contribution in micro-bunching. These results are in complete agreement with the analysis presented in literature which are based on Maxwell–Klimontovich coupled equations [13,21] rather that direct Lienard–Wiechert fields. The consistent results prove the validity of the obtained equations of motion. In this Letter the effect of initial energy spread (longitudinal temperature) on micro-bunching suppression has been considered by Lienard–Wiechert fields. The same procedure can also be undertaken for transverse emittance (transverse temperature). Analytical treatment by coupled Maxwell–Klimontovich equations shows that the fall and the shift of saturation can also be seen for transverse emittance. Gaining such results by Lienard–Wiechert field remains as our future work. Acknowledgements The first author would like to offer his sincere appreciation to Dr. Massimo Ferrario for accepting his request for sabbatical stay at INFN-LNF center and his scientific help during his stay. He also wishes to extend his gratitude to all members of SPARC project from INFN-LNF and ENEA centers. This work was supported by financial assistance of INFN-LNF center and University of Tehran, Research Council. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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