- Email: [email protected]
Limitations of the transverse coherence in the self-amplified spontaneous emission FEL
Nuclear Instruments and Methods in Physics Research A 475 (2001) 92–96
Limitations of the transverse coherence in the self-amplified spontaneous emission FEL E.L. Saldina,*, E.A. Schneidmillera, M.V. Yurkovb a
Deutches Elektronen Synchrotron (DESY), Notkestrasse 85, 22607 Hamburg, Germany b Joint Institute for Nuclear Research, Dubna, 141980 Moscow Region, Russia
Abstract In this paper we analyze the process of the formation of transverse coherence of radiation from a self-amplified spontaneous emission (SASE) FEL. It is shown that in the high-gain linear regime the degree of transverse coherence approaches unity asymptotically as z1 ; but not exponentially, as one would expect from a simple physical assumption that the transverse coherence is established due to the transverse mode selection. It has been found that even after finishing the transverse mode selection process the degree of transverse coherence of the radiation from SASE FEL visibly differs from unity. This is a consequence of the interdependence of the longitudinal and transverse coherence. The SASE FEL has poor longitudinal coherence which develops slowly with the undulator length thus preventing a full transverse coherence. r 2001 Elsevier Science B.V. All rights reserved. PACS: 41.60.Cr; 52.75.M Keywords: Free electron laser; High-gain FEL; Initial-value problem
1. Introduction A description of the properties of the output radiation from the SASE FELs is of great practical importance in view of the X-ray SASE FEL projects under development [1–4]. In paper [5] we developed a technique for an analytical description of the start-up from shot noise. The solution of the initial-value problem is based on an analytical approach originally developed by Krinsky and Yu [6,7] and a multilayer approximation method for *Corresponding author. Tel.: +49-40-8998-2676; fax: +4940-8998-4475. E-mail address: [email protected] (E.L. Saldin).
calculations of the eigenfunctions developed in Ref. [8]. This systematic approach allows us to calculate the average radiation power, radiation spectrum envelope, the angular distribution of the radiation intensity in the far zone, and the degree of transverse coherence. Analytical results serve as a primary standard for testing the three-dimensional, time-dependent simulation code FAST [9]. Comparison with analytical results shows that in the high-gain linear limit there is a good agreement between the numerical and analytical results. In this paper we analyze the process of the formation of the transverse coherence in the SASE FEL. Both analytical and numerical results show that, in the high-gain linear regime, the degree of
0168-9002/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 5 5 0 - 9
93
E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 475 (2001) 92–96 1.0
0.8
<(dW)/(dC)>
transverse coherence approaches unity asymptotically as z1 ; but not exponentially, as one would expect from the assumption that transverse coherence is established due to the transverse mode selection. Even after finishing the transverse mode selection process the degree of transverse coherence of the radiation from SASE FEL visibly differs from unity. Since the gain in a SASE FEL always has a finite value, this effect imposes a limit on the maximum value of the degree of transverse coherence.
0.6
0.4
0.2
0.0 -2
-1
0
1
2
C/ Γ
2. Analytical description of start-up from shot noise We consider an FEL amplifier with a helical undulator and an axisymmetric electron beam having a transverse current density j0 ðrÞ ¼ I0 S RN ðr=r0 Þ=½2p 0 rSðr=r0 Þ dr; where r0 is the beam profile parameter (typical transverse size of the beam) and I0 is the beam current. It is assumed that the electron beam has a Gaussian energy spread, FðE E0 Þ ¼ ð2p/ðDEÞ2 SÞ1=2 exp½ðE E0 Þ2 =ð2/ðDEÞ2 SÞ; where /ðDEÞ2 S is r.m.s. energy spread. We neglect the transverse variation of the undulator field and assume that electrons move along constrained helical trajectories in parallel with the z-axis. The electron rotation angle is considered to be small and the longitudinal electron velocity vz is close to the velocity of light c: Initial-value problem for start-up from shot noise is solved for the case of long rectangular electron bunch [5]. At a sufficiently large undulator length, the spectrum of the SASE radiation is concentrated within the narrow band near the resonance frequency o0 (see Fig. 1). Therefore, the electric field of the wave can be presented as
Fig. 1. Averaged spectrum of the radiation from the FEL amplifier starting from shot noise at the undulator length z# ¼ 15: Here B ¼ 1; L2p -0; L2T ¼ 0; and Nc ¼ 7 107 : Solid curve represents analytical results calculated for nine beam radiation modes (m; n ¼ 0; 1; 2). The circles are the results obtained with linear simulation code FAST.
r; zÞ is effective transverse correlation where geff ð~ function: R n ~~ ~ ~þ~ *R r=2; z; tÞS dR /Eð r=2; z; tÞE* ðR geff ð~ r; zÞ ¼ : R 2 ~ ~; z; tÞj S dR *R /jEð Here, the averaging symbol /yS means the ensemble average over bunches. In the axisymmetric case the angular spectrum can be written in the following dimensionless form [5]: " Z N X Z N ðnÞ # # rÞ#r d#r # # ¼ hðy; zÞ dC Okj Fnk ð#rÞJn ðy#
Z
N
"
2p
0
0
X Z n;k;j
Z
0
# r Þ#r d#r Fnj ð#r ÞJn ðy# n
0
* r> ; z; tÞ exp½io0 ðz=c tÞ þ C:C: Ex þ iEy ¼ Eð~ where E* is the slowly varying complex amplitude. Taking into account Parseval’s theorem, and using ~ ¼ ð~ the notation ~ r ¼~ r> ~ r 0> and R r> þ ~ r 0> Þ=2; we can write the average angular spectrum of the radiation in the far zone in the following form: Z 1 ~ hðk> ; zÞ ¼ geff ð~ r; zÞ expðik~>~ rÞ d~ r ð2pÞ2
N
n;k;j
N
N
N
0
0
dC# OðnÞ kj
1 Fnk ð#rÞFnj ð#rÞ#r d#r n
0
where n o ðnÞ ðnÞ n ðnÞ ðnÞ n # OðnÞ ¼ u ðu Þ exp ½l þ ðl Þ z j j kj k k Z N Fnk ð#rÞFnnj ð#rÞSð#rÞ#r d#r 0
ð1Þ
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E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 475 (2001) 92–96
" Z ðnÞ # ¼ D0 ðlj Þ B
F2nj ð#rÞ#r d#r
0
Z
! ¼
1
N
# dD dp p¼lðnÞ
Z
N 0
ix 1
!
1
Σ TEM0n
j
F2nj ð#rÞSð#rÞ#r d#r
0
D# D# 0
N
0.01
/( ρW b )
uðnÞ j
exp½L#2T x2 =2
Σ TEM1n 1E-4
Σ TEM2n 1E-6
# dx ðp þ iCÞx
3. Transverse coherence We illustrate the process of formation of transverse coherence in the SASE FEL for specific example of diffraction parameter B ¼ 1 and number of electrons in the volume of coherence Nc ¼ 7 107 which is typical for a VUV FEL [10]. Transverse distribution of the beam current density is Gaussian. Fig. 2 shows the evolution along the undulator of the partial contributions of different beam radiation modes into the total power. It is seen that analytical and simulation # results agree well at z\7: We also see that at large # undulator length, zB15; contribution of the fundamental F00 mode to the total power is close to 99.9%. Analytical predictions for the averaged angular distribution of the radiation power are given by Eq. (1). Similar characteristic has also been calculated with numerical simulation code. It is seen from Fig. 3 that both approaches agree well in the high-gain linear regime. It is important to stress that even after finishing the transverse mode
1E-8 6
8
10
12
14
Γz Fig. 2. Partial contributions to the total power of three azimuthal modes with m ¼ 0; 1, and 2. Here B ¼ 1; L2p -0; L2T ¼ 0; and Nc ¼ 7 107 : Solid curves represent analytical results for sum of three radial modes (n ¼ 0; 1; 2). The circles are the results obtained with linear simulation code FAST.
1.0
0.8
h(θ)/h(0)
with C# ¼ C=G ¼ ½kw o=ð2cg2z Þ=G being the detuning parameter, B ¼ r20 Go=c the diffraction # 2 ¼ /ðDEÞ2 S=ðr2 E2 Þ the energy parameter, L T 0 spread parameter, r ¼ cg2z G=o Rthe saturation paN rameter, G ¼ ½I0 o2 y2s ð2IA c2 g2z g 0 zSðzÞ dzÞ1 1=2 ; y# ¼ yo0 r0 =c; o is frequency, ys ¼ K=g; K ¼ eHw =ðE0 kw Þ is undulator parameter, Hw and kw are the field and wavenumber of the undulator, 2 respectively, g2 þ y2s ; g ¼ E0 =ðme c2 Þ; z# ¼ z ¼ g Gz; r# ¼ r=r0 ; and IA C17 kA is the Alfven current. The eigenvalues and eigenfunctions of the beam radiation modes, lðnÞ rÞ; are calculated k and Fnk ð# using multilayer approximation method [8].
0.6
0.4
Γz = 8
Γ z = 15
0.2
0.0 0.0
0.5
1.0
θσω /c
1.5
2.0
Fig. 3. Averaged angular distribution of the radiation intensity in the far zone for the FEL amplifier starting from shot noise. Here B ¼ 1; L2p -0; and L2T ¼ 0: Solid curves are the results of analytical calculations with (1), and the circles are the results obtained with linear simulation code FAST. Dashed line represents angular distribution of the fundamental F00 mode for maximum growth rate calculated in the framework of the steady-state theory.
# selection process (which takes place after z\10 for the considered numerical example) the intensity distribution in the far zone differs visibly from the angular distribution of the fundamental F00 mode for maximum growth rate calculated in the framework of the steady-state theory (dotted line in Fig. 3).
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E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 475 (2001) 92–96
In the case of axisymmetric electron beam the radiation field is statistically isotropic. For such a field the effective correlation function depends only on the modulus j~ rj and the angular spectrum depends on the modulus jk~> j: It is natural to define the area of coherence in this case as Z N # zÞj # 2 r# dr; # p#r2coh ¼ 2p jgeff 1 ðr; 0 Z N # yÞ # y# dy# # # ð r; zÞ ¼ 2p J0 ðr# yÞhð ð2Þ geff 1 0
where r# ¼ j~ rj=r0 : The degree of coherence, z; may be defined as z ¼ r#2coh =#r2max
ð3Þ
where r#max is the radius of coherence r#max for the fully coherent radiation which is represented by the fundamental F00 ð#rÞ mode for maximum growth rate. Using angular distributions of the radiation field in the far diffraction zone we can trace the dependence of the degree of transverse coherence versus undulator length. Solid line in Fig. 4 is the result of analytical calculations, and the circles are the results obtained with numerical simulation code. It is clearly seen that the degree of coherence differs visibly from the unity in the highgain linear regime, zC0:9 at z# ¼ 15:
Another possible way to define the degree of coherence is based on the statistical analysis of fluctuations of the instantaneous power [10,11]. Since in the linear regime we deal with a Gaussian random process, the power density at fixed point in space fluctuates in accordance with the negative exponential distribution [10]. If there is full transverse coherence, then the same refers to the instantaneous power W equal to the power density integrated over cross-section of the radiation pulse. If the radiation is partially coherent, then we have a more general law for instantaneous power fluctuations, namely the gamma distribution [10,11]: M1 MM W pðWÞ ¼ GðMÞ /WS 1 W exp M ð4Þ /WS /WS where GðMÞ is the gamma function and M ¼ /WS2 =/ðW /WSÞ2 S: The parameter M of this distribution can be considered as the number of transverse modes. Then, the degree of coherence in the linear regime, may be defined as z ¼ 1=M: The value of M should be calculated with numerical simulation code producing time-dependent results for the radiation power. In Fig. 5 we
1.0
6 0.8
1 4
2
ζ
1/ζ
0.6
0.4
2 0.2
0 6
0.0 6
8
10
12
14
16
18
Γz Fig. 4. Degree of transverse coherence of the radiation from the FEL amplifier versus the undulator length. Solid curve represents analytical results, and the circles are the results obtained with linear simulation code FAST. Here B ¼ 1; L2p -0; L2T ¼ 0; and Nc ¼ 7 107 :
8
10
12
14
Γz Fig. 5. Inverse value of transverse coherence versus undulator length. Here B ¼ 1; L2p -0; L2T ¼ 0; and Nc ¼ 7 107 : Calculations have been performed with linear simulation code FAST. Curve 1 is calculated using instantaneous fluctuations of the radiation power. Curve 2 is calculated using angular distribution of the radiation power in far zone.
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present the dependence of the number of transverse modes on the undulator length for the specific value of the diffraction parameter B ¼ 1: It is seen that both definitions for the degree for the transverse coherence are consistent in the highgain linear regime. Let us discuss asymptotical behaviour of the degree of transverse coherence. At a large value of the undulator length it approaches unity asymptotically as ð1 zÞp1=z; but not exponentially, as one can expect from simple physical assumption that transverse coherence establishes due to the transverse mode selection. Due to the latter effect the degree of coherence grows quickly at an early stage of amplification only. Starting from some undulator length the contribution to the total power of the fundamental mode starts to be dominant (see Fig. 2). However, one should take into account that the spectrum width has always finite value (see Fig. 1). Actually, this means that in the high-gain linear regime the radiation of the SASE FEL is formed by many fundamental F00 modes with different frequencies. The transverse distribution of the radiation field of the mode is also different for different frequencies. As a result of interference of these modes we do not have full transverse coherence. Taking into account this consideration, we can simply explain asymptotical behaviour of the degree of transverse coherence F this is reflection of the slow evolution of the width of the radiation spectrum as z1=2 with the undulator length. All the results presented above have been obtained in the framework of the linear theory. Simulations with nonlinear code shows that for the considered numerical example the saturation # occurs at zC18: Using the plot presented in Fig. 4 we find that the value of the transverse
coherence is o0:9 in the end of the linear regime. A typical range of the values of Nc ¼ I0 =ðeo0 rÞ is 106 –109 for the SASE FEL of wavelength range from X-ray up to infrared. The numerical example presented in this paper is calculated for Nc ¼ 7 107 which is typical for a VUV FEL. It is worth to mention that the dependence of the saturation length of the SASE FEL on the value of Nc is rather weak, in fact logarithmic. Therefore, we can state that obtained effect limiting the value of transverse coherence might be important for practical SASE FELs.
References [1] A VUV Free Electron Laser at the TESLA Test Facility: Conceptual Design Report, DESY Print TESLA-FEL 9503, Hamburg, DESY, 1995. [2] J. Rossbach, Nucl. Instr. and Meth. A 375 (1996) 269. [3] R. Brinkmann, et al. (Ed.), Conceptual Design of 500 GeV eþ e Linear Collider with Integrated X-ray Facility, DESY 1997-048, ECFA 1997-182, Hamburg, May 1997. [4] Linac Coherent Light Source (LCLS) Design Study Report, The LCLS Design Study Group, Stanford Linear Accelerator Center (SLAC) Report No. SLAC-R-521, 1998. [5] E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov, Opt. Commun. 186 (2000) 185. [6] K.J. Kim, Phys. Rev. Lett. 57 (1986) 1871. [7] S. Krinsky, L.H. Yu, Phys. Rev. A 35 (1987) 3406. [8] E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov, Opt. Commun. 97 (1993) 272. [9] E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov, Nucl. Instr. and Meth. A 429 (1999) 233. [10] E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov, The Physics of Free Electron Laser, Springer, Berlin, Heidelberg, New York, 1999. [11] E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov, Nucl. Instr. Meth. A 429 (1999) 229.
Limitations of the transverse coherence in the self-amplified spontaneous emission FEL E.L. Saldina,*, E.A. Schneidmillera, M.V. Yurkovb a
Deutches Elektronen Synchrotron (DESY), Notkestrasse 85, 22607 Hamburg, Germany b Joint Institute for Nuclear Research, Dubna, 141980 Moscow Region, Russia
Abstract In this paper we analyze the process of the formation of transverse coherence of radiation from a self-amplified spontaneous emission (SASE) FEL. It is shown that in the high-gain linear regime the degree of transverse coherence approaches unity asymptotically as z1 ; but not exponentially, as one would expect from a simple physical assumption that the transverse coherence is established due to the transverse mode selection. It has been found that even after finishing the transverse mode selection process the degree of transverse coherence of the radiation from SASE FEL visibly differs from unity. This is a consequence of the interdependence of the longitudinal and transverse coherence. The SASE FEL has poor longitudinal coherence which develops slowly with the undulator length thus preventing a full transverse coherence. r 2001 Elsevier Science B.V. All rights reserved. PACS: 41.60.Cr; 52.75.M Keywords: Free electron laser; High-gain FEL; Initial-value problem
1. Introduction A description of the properties of the output radiation from the SASE FELs is of great practical importance in view of the X-ray SASE FEL projects under development [1–4]. In paper [5] we developed a technique for an analytical description of the start-up from shot noise. The solution of the initial-value problem is based on an analytical approach originally developed by Krinsky and Yu [6,7] and a multilayer approximation method for *Corresponding author. Tel.: +49-40-8998-2676; fax: +4940-8998-4475. E-mail address: [email protected] (E.L. Saldin).
calculations of the eigenfunctions developed in Ref. [8]. This systematic approach allows us to calculate the average radiation power, radiation spectrum envelope, the angular distribution of the radiation intensity in the far zone, and the degree of transverse coherence. Analytical results serve as a primary standard for testing the three-dimensional, time-dependent simulation code FAST [9]. Comparison with analytical results shows that in the high-gain linear limit there is a good agreement between the numerical and analytical results. In this paper we analyze the process of the formation of the transverse coherence in the SASE FEL. Both analytical and numerical results show that, in the high-gain linear regime, the degree of
0168-9002/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 5 5 0 - 9
93
E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 475 (2001) 92–96 1.0
0.8
<(dW)/(dC)>
transverse coherence approaches unity asymptotically as z1 ; but not exponentially, as one would expect from the assumption that transverse coherence is established due to the transverse mode selection. Even after finishing the transverse mode selection process the degree of transverse coherence of the radiation from SASE FEL visibly differs from unity. Since the gain in a SASE FEL always has a finite value, this effect imposes a limit on the maximum value of the degree of transverse coherence.
0.6
0.4
0.2
0.0 -2
-1
0
1
2
C/ Γ
2. Analytical description of start-up from shot noise We consider an FEL amplifier with a helical undulator and an axisymmetric electron beam having a transverse current density j0 ðrÞ ¼ I0 S RN ðr=r0 Þ=½2p 0 rSðr=r0 Þ dr; where r0 is the beam profile parameter (typical transverse size of the beam) and I0 is the beam current. It is assumed that the electron beam has a Gaussian energy spread, FðE E0 Þ ¼ ð2p/ðDEÞ2 SÞ1=2 exp½ðE E0 Þ2 =ð2/ðDEÞ2 SÞ; where /ðDEÞ2 S is r.m.s. energy spread. We neglect the transverse variation of the undulator field and assume that electrons move along constrained helical trajectories in parallel with the z-axis. The electron rotation angle is considered to be small and the longitudinal electron velocity vz is close to the velocity of light c: Initial-value problem for start-up from shot noise is solved for the case of long rectangular electron bunch [5]. At a sufficiently large undulator length, the spectrum of the SASE radiation is concentrated within the narrow band near the resonance frequency o0 (see Fig. 1). Therefore, the electric field of the wave can be presented as
Fig. 1. Averaged spectrum of the radiation from the FEL amplifier starting from shot noise at the undulator length z# ¼ 15: Here B ¼ 1; L2p -0; L2T ¼ 0; and Nc ¼ 7 107 : Solid curve represents analytical results calculated for nine beam radiation modes (m; n ¼ 0; 1; 2). The circles are the results obtained with linear simulation code FAST.
r; zÞ is effective transverse correlation where geff ð~ function: R n ~~ ~ ~þ~ *R r=2; z; tÞS dR /Eð r=2; z; tÞE* ðR geff ð~ r; zÞ ¼ : R 2 ~ ~; z; tÞj S dR *R /jEð Here, the averaging symbol /yS means the ensemble average over bunches. In the axisymmetric case the angular spectrum can be written in the following dimensionless form [5]: " Z N X Z N ðnÞ # # rÞ#r d#r # # ¼ hðy; zÞ dC Okj Fnk ð#rÞJn ðy#
Z
N
"
2p
0
0
X Z n;k;j
Z
0
# r Þ#r d#r Fnj ð#r ÞJn ðy# n
0
* r> ; z; tÞ exp½io0 ðz=c tÞ þ C:C: Ex þ iEy ¼ Eð~ where E* is the slowly varying complex amplitude. Taking into account Parseval’s theorem, and using ~ ¼ ð~ the notation ~ r ¼~ r> ~ r 0> and R r> þ ~ r 0> Þ=2; we can write the average angular spectrum of the radiation in the far zone in the following form: Z 1 ~ hðk> ; zÞ ¼ geff ð~ r; zÞ expðik~>~ rÞ d~ r ð2pÞ2
N
n;k;j
N
N
N
0
0
dC# OðnÞ kj
1 Fnk ð#rÞFnj ð#rÞ#r d#r n
0
where n o ðnÞ ðnÞ n ðnÞ ðnÞ n # OðnÞ ¼ u ðu Þ exp ½l þ ðl Þ z j j kj k k Z N Fnk ð#rÞFnnj ð#rÞSð#rÞ#r d#r 0
ð1Þ
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E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 475 (2001) 92–96
" Z ðnÞ # ¼ D0 ðlj Þ B
F2nj ð#rÞ#r d#r
0
Z
! ¼
1
N
# dD dp p¼lðnÞ
Z
N 0
ix 1
!
1
Σ TEM0n
j
F2nj ð#rÞSð#rÞ#r d#r
0
D# D# 0
N
0.01
uðnÞ j
exp½L#2T x2 =2
Σ TEM1n 1E-4
Σ TEM2n 1E-6
# dx ðp þ iCÞx
3. Transverse coherence We illustrate the process of formation of transverse coherence in the SASE FEL for specific example of diffraction parameter B ¼ 1 and number of electrons in the volume of coherence Nc ¼ 7 107 which is typical for a VUV FEL [10]. Transverse distribution of the beam current density is Gaussian. Fig. 2 shows the evolution along the undulator of the partial contributions of different beam radiation modes into the total power. It is seen that analytical and simulation # results agree well at z\7: We also see that at large # undulator length, zB15; contribution of the fundamental F00 mode to the total power is close to 99.9%. Analytical predictions for the averaged angular distribution of the radiation power are given by Eq. (1). Similar characteristic has also been calculated with numerical simulation code. It is seen from Fig. 3 that both approaches agree well in the high-gain linear regime. It is important to stress that even after finishing the transverse mode
1E-8 6
8
10
12
14
Γz Fig. 2. Partial contributions to the total power of three azimuthal modes with m ¼ 0; 1, and 2. Here B ¼ 1; L2p -0; L2T ¼ 0; and Nc ¼ 7 107 : Solid curves represent analytical results for sum of three radial modes (n ¼ 0; 1; 2). The circles are the results obtained with linear simulation code FAST.
1.0
0.8
h(θ)/h(0)
with C# ¼ C=G ¼ ½kw o=ð2cg2z Þ=G being the detuning parameter, B ¼ r20 Go=c the diffraction # 2 ¼ /ðDEÞ2 S=ðr2 E2 Þ the energy parameter, L T 0 spread parameter, r ¼ cg2z G=o Rthe saturation paN rameter, G ¼ ½I0 o2 y2s ð2IA c2 g2z g 0 zSðzÞ dzÞ1 1=2 ; y# ¼ yo0 r0 =c; o is frequency, ys ¼ K=g; K ¼ eHw =ðE0 kw Þ is undulator parameter, Hw and kw are the field and wavenumber of the undulator, 2 respectively, g2 þ y2s ; g ¼ E0 =ðme c2 Þ; z# ¼ z ¼ g Gz; r# ¼ r=r0 ; and IA C17 kA is the Alfven current. The eigenvalues and eigenfunctions of the beam radiation modes, lðnÞ rÞ; are calculated k and Fnk ð# using multilayer approximation method [8].
0.6
0.4
Γz = 8
Γ z = 15
0.2
0.0 0.0
0.5
1.0
θσω /c
1.5
2.0
Fig. 3. Averaged angular distribution of the radiation intensity in the far zone for the FEL amplifier starting from shot noise. Here B ¼ 1; L2p -0; and L2T ¼ 0: Solid curves are the results of analytical calculations with (1), and the circles are the results obtained with linear simulation code FAST. Dashed line represents angular distribution of the fundamental F00 mode for maximum growth rate calculated in the framework of the steady-state theory.
# selection process (which takes place after z\10 for the considered numerical example) the intensity distribution in the far zone differs visibly from the angular distribution of the fundamental F00 mode for maximum growth rate calculated in the framework of the steady-state theory (dotted line in Fig. 3).
95
E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 475 (2001) 92–96
In the case of axisymmetric electron beam the radiation field is statistically isotropic. For such a field the effective correlation function depends only on the modulus j~ rj and the angular spectrum depends on the modulus jk~> j: It is natural to define the area of coherence in this case as Z N # zÞj # 2 r# dr; # p#r2coh ¼ 2p jgeff 1 ðr; 0 Z N # yÞ # y# dy# # # ð r; zÞ ¼ 2p J0 ðr# yÞhð ð2Þ geff 1 0
where r# ¼ j~ rj=r0 : The degree of coherence, z; may be defined as z ¼ r#2coh =#r2max
ð3Þ
where r#max is the radius of coherence r#max for the fully coherent radiation which is represented by the fundamental F00 ð#rÞ mode for maximum growth rate. Using angular distributions of the radiation field in the far diffraction zone we can trace the dependence of the degree of transverse coherence versus undulator length. Solid line in Fig. 4 is the result of analytical calculations, and the circles are the results obtained with numerical simulation code. It is clearly seen that the degree of coherence differs visibly from the unity in the highgain linear regime, zC0:9 at z# ¼ 15:
Another possible way to define the degree of coherence is based on the statistical analysis of fluctuations of the instantaneous power [10,11]. Since in the linear regime we deal with a Gaussian random process, the power density at fixed point in space fluctuates in accordance with the negative exponential distribution [10]. If there is full transverse coherence, then the same refers to the instantaneous power W equal to the power density integrated over cross-section of the radiation pulse. If the radiation is partially coherent, then we have a more general law for instantaneous power fluctuations, namely the gamma distribution [10,11]: M1 MM W pðWÞ ¼ GðMÞ /WS 1 W exp M ð4Þ /WS /WS where GðMÞ is the gamma function and M ¼ /WS2 =/ðW /WSÞ2 S: The parameter M of this distribution can be considered as the number of transverse modes. Then, the degree of coherence in the linear regime, may be defined as z ¼ 1=M: The value of M should be calculated with numerical simulation code producing time-dependent results for the radiation power. In Fig. 5 we
1.0
6 0.8
1 4
2
ζ
1/ζ
0.6
0.4
2 0.2
0 6
0.0 6
8
10
12
14
16
18
Γz Fig. 4. Degree of transverse coherence of the radiation from the FEL amplifier versus the undulator length. Solid curve represents analytical results, and the circles are the results obtained with linear simulation code FAST. Here B ¼ 1; L2p -0; L2T ¼ 0; and Nc ¼ 7 107 :
8
10
12
14
Γz Fig. 5. Inverse value of transverse coherence versus undulator length. Here B ¼ 1; L2p -0; L2T ¼ 0; and Nc ¼ 7 107 : Calculations have been performed with linear simulation code FAST. Curve 1 is calculated using instantaneous fluctuations of the radiation power. Curve 2 is calculated using angular distribution of the radiation power in far zone.
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present the dependence of the number of transverse modes on the undulator length for the specific value of the diffraction parameter B ¼ 1: It is seen that both definitions for the degree for the transverse coherence are consistent in the highgain linear regime. Let us discuss asymptotical behaviour of the degree of transverse coherence. At a large value of the undulator length it approaches unity asymptotically as ð1 zÞp1=z; but not exponentially, as one can expect from simple physical assumption that transverse coherence establishes due to the transverse mode selection. Due to the latter effect the degree of coherence grows quickly at an early stage of amplification only. Starting from some undulator length the contribution to the total power of the fundamental mode starts to be dominant (see Fig. 2). However, one should take into account that the spectrum width has always finite value (see Fig. 1). Actually, this means that in the high-gain linear regime the radiation of the SASE FEL is formed by many fundamental F00 modes with different frequencies. The transverse distribution of the radiation field of the mode is also different for different frequencies. As a result of interference of these modes we do not have full transverse coherence. Taking into account this consideration, we can simply explain asymptotical behaviour of the degree of transverse coherence F this is reflection of the slow evolution of the width of the radiation spectrum as z1=2 with the undulator length. All the results presented above have been obtained in the framework of the linear theory. Simulations with nonlinear code shows that for the considered numerical example the saturation # occurs at zC18: Using the plot presented in Fig. 4 we find that the value of the transverse
coherence is o0:9 in the end of the linear regime. A typical range of the values of Nc ¼ I0 =ðeo0 rÞ is 106 –109 for the SASE FEL of wavelength range from X-ray up to infrared. The numerical example presented in this paper is calculated for Nc ¼ 7 107 which is typical for a VUV FEL. It is worth to mention that the dependence of the saturation length of the SASE FEL on the value of Nc is rather weak, in fact logarithmic. Therefore, we can state that obtained effect limiting the value of transverse coherence might be important for practical SASE FELs.
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