- Email: [email protected]
An FEL design code running on Mathcad™
__ -_ BB
Nuclear Instruments and Methods in Physics Research A 358 (199.5)ABS 67-ABS 68
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH
--
Secrlon A
ELSJZVIER
An
FEL design code running on MathcadTM* D.C. Nguyen a,*, S.M. Gierman a, P.G. O’Shea b a Los Alamos National Laboratory, Los Alamos, NM 87545, USA b Duke University, Durham, NC 27708-0319, USA
different harmonic components of the wiggler magnetic field are calculated using the standard Halbach formula [4].
1. Introduction As free-electron lasers become increasingly a users’ tool rather than an accelerator physicists’ research facility, some users are getting involved in FEL physics for the first time. Many of them will understandably find the task of designing FEL experiments daunting. Some design decisions involve relatively straightforward issues such as the required wiggler period and electron beam energy for a given wavelength. Others such as the expected output energy and the macropulse length are a little more complicated. A few three-dimensional simulation codes already exist [1,2], but they are difficult to use effectively for designing FEL experiments. Furthermore, in comparing FEL performance, the ability to map out design parameters is highly desirable. It is thus important to obtain quick answers to basic FEL design questions without expensive numerical simulations. Toward this end, we wrote a program called FEL-CAD.
2. Description
of FEL-CAD
FEL-CAD runs on both the IBM PCTM and the Apple MacintoshTM with the help of MathcadTM. FEL-CAD is appropriate for Compton FELs that operate as low-gain oscillators driven by rf linac and permanent-magnet wigglers. We assume no sidebands and negligible slippage. We only treat linear resonators and cold-cavity optical mode. Parts of FEL-CAD were based on Dattoli’s FEL saturation dynamics [3]. In FEL-CAD, the electron beam traveling in the z direction is wiggled in the x direction by the wiggler magnetic field in the y direction. The magnitudes of
I’ Work supported by LANL Laboratory Directed Research and Development under the auspices of the US Department TMTrademarks of commercial hardware or software. * Corresponding author.
of Energy.
(1) where B,,, is the remanent field; E is the packing factor; n, and y are the wiggler harmonic and harmonic index, respectively; M is the number of magnets per period; k, is the wiggler wave number; L is the magnet height in the y direction; and G is the full gap. The factor S is - 1 for the case where M = 2 and the magnetization is along the z direction. S is 1 for M = 4, or M = 2 and the magnetization is along the y direction. The rms wiggler parameter a, is calculated from the magnitude of the fundamental field, a, = eB,,/(&mck,). In rf-linac driven FELs, the first portion of the rf macropulse is used to fill the linac cavity to the steady-state accelerating field. The cavity fill time is defined as the time needed for the electron energy to be within l/2 N,,, of the steady-state value. The difference between the rf macropulse and the cavity fill time is the usable macropulse length. Within the usable macropulse length, the electron micropulses have the right energy so that their emission will be amplified by subsequent electron micropulses. The FEL output is assumed to occur at the wavelength of maximum small-signal gain. This wavelength is given by
where A, is the wiggler period; n is the harmonic number; y is the relativistic factor; and - 1.303 is the value of the detuning parameter at which the small-signal gain is maximum. The electron beam is matched into the wiggler such that its y dimension is constant with respect to z. This constant
016%9002/95/$09.50 0 1995 El sevier Science B.V. AI1 rights reserved SSDI 016%9002(94)01567-S
dimension
is given
by T,, = /z
EXTENDED
SYNOPSES
L).C. Ngllyrn et al. / NM/. Instr. and M&.
ABS 68
where Ap is the betatron period; and E,, is the unnormalized emittance in the y direction. The x dimension of the beam follows the optical beam’s contour,
(3) We calculate the number of spontaneous photons at fundamental frequency using Colson’s formula N,,,, = 2 NeNwaf,a,’ [s]. The divergence angle of the TEM,,, cavity mode is used as the coherence angle. For a, smaller than 1, the number of spontaneous photons emitted into the coherence angle 0 is approximated by Nc=N,,,,
n
JJ’(n),
in Phys. Rex A 358 (1995) ABS 67-ABS 6H
We perform a multi-pass calculation to predict the FEL output energy. The number of passes is equal to the number of usable micropulses divided by the number of optical pulses in the cavity. The intracavity intensity builds up from the spontaneous emission within the coherence angle. The FEL gain is modified by the optical intensity in the wiggler as given by Eq. (5). When the number of round-trips exceeds the number of usable micropulses. the FEL gain is reset to 0 and the cavity power is allowed to decay over a few l/e ringdowns of the resonator. The optical energy in a micropulse is calculated by multiplying the optical intensity with the average optical mode area in the wiggler and the micropulse length.
x (3 + 3x + 2x2) (1+x)3
’
(4)
where JJ(n) is the usual difference in Bessel functions; and x is (y 0)‘. We use Dattoli’s FEL gain saturation model to calculate the optimum outcoupling from the small-signal gain g,, and total cavity loss r [4]. The total cavity loss is the sum of the outcoupling f’oc and dissipative loss r,,,, which includes vignetting, absorption and other loss. In Dattoli’s saturation model, the small signal gain is saturated according to
3. Conclusion We have written a simple code to predict the performance of rf-linac driven FEL oscillators. The advantages of FEL-CAD over numerical simulations are its simplicity and speed. Although we welcome any changes the users may have, we believe that FEL-CAD should be kept at this level of complexity. More elaborate, 3-D simulations codes [1,2] should be used to make accurate prediction of FEL performance.
References
where g(l) is the gain g,, is the small-signal fitting parameter which the saturation intensity the small-signal gain is
at a given gain; LY is falls in the defined as halved.
intracavity intensity I; the phenomenological range 0.14-0.18; I, is the intensity at which
[II J.C. Goldstein et al., Nucl. Instr. and Meth. A 285 (1989) 192. 121T.M. Tran and J.S. Wurtele. Phys. Rep. 195 11990) 1. [31 G. Dattoli et al., Nucl. Instr. and Meth. A 318 (1992) 495. [41 Ii. Halbach, Nucl. Instr. and Meth. 178 (1981) 109. [51W.B. Colson. in: Fret Electron Laser Handbook, rds. W.B.
Colson. C. Pellegrini and A. Rcnieri (North-Holland, Amsterdam 1990) Chap. 5.
Nuclear Instruments and Methods in Physics Research A 358 (199.5)ABS 67-ABS 68
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH
--
Secrlon A
ELSJZVIER
An
FEL design code running on MathcadTM* D.C. Nguyen a,*, S.M. Gierman a, P.G. O’Shea b a Los Alamos National Laboratory, Los Alamos, NM 87545, USA b Duke University, Durham, NC 27708-0319, USA
different harmonic components of the wiggler magnetic field are calculated using the standard Halbach formula [4].
1. Introduction As free-electron lasers become increasingly a users’ tool rather than an accelerator physicists’ research facility, some users are getting involved in FEL physics for the first time. Many of them will understandably find the task of designing FEL experiments daunting. Some design decisions involve relatively straightforward issues such as the required wiggler period and electron beam energy for a given wavelength. Others such as the expected output energy and the macropulse length are a little more complicated. A few three-dimensional simulation codes already exist [1,2], but they are difficult to use effectively for designing FEL experiments. Furthermore, in comparing FEL performance, the ability to map out design parameters is highly desirable. It is thus important to obtain quick answers to basic FEL design questions without expensive numerical simulations. Toward this end, we wrote a program called FEL-CAD.
2. Description
of FEL-CAD
FEL-CAD runs on both the IBM PCTM and the Apple MacintoshTM with the help of MathcadTM. FEL-CAD is appropriate for Compton FELs that operate as low-gain oscillators driven by rf linac and permanent-magnet wigglers. We assume no sidebands and negligible slippage. We only treat linear resonators and cold-cavity optical mode. Parts of FEL-CAD were based on Dattoli’s FEL saturation dynamics [3]. In FEL-CAD, the electron beam traveling in the z direction is wiggled in the x direction by the wiggler magnetic field in the y direction. The magnitudes of
I’ Work supported by LANL Laboratory Directed Research and Development under the auspices of the US Department TMTrademarks of commercial hardware or software. * Corresponding author.
of Energy.
(1) where B,,, is the remanent field; E is the packing factor; n, and y are the wiggler harmonic and harmonic index, respectively; M is the number of magnets per period; k, is the wiggler wave number; L is the magnet height in the y direction; and G is the full gap. The factor S is - 1 for the case where M = 2 and the magnetization is along the z direction. S is 1 for M = 4, or M = 2 and the magnetization is along the y direction. The rms wiggler parameter a, is calculated from the magnitude of the fundamental field, a, = eB,,/(&mck,). In rf-linac driven FELs, the first portion of the rf macropulse is used to fill the linac cavity to the steady-state accelerating field. The cavity fill time is defined as the time needed for the electron energy to be within l/2 N,,, of the steady-state value. The difference between the rf macropulse and the cavity fill time is the usable macropulse length. Within the usable macropulse length, the electron micropulses have the right energy so that their emission will be amplified by subsequent electron micropulses. The FEL output is assumed to occur at the wavelength of maximum small-signal gain. This wavelength is given by
where A, is the wiggler period; n is the harmonic number; y is the relativistic factor; and - 1.303 is the value of the detuning parameter at which the small-signal gain is maximum. The electron beam is matched into the wiggler such that its y dimension is constant with respect to z. This constant
016%9002/95/$09.50 0 1995 El sevier Science B.V. AI1 rights reserved SSDI 016%9002(94)01567-S
dimension
is given
by T,, = /z
EXTENDED
SYNOPSES
L).C. Ngllyrn et al. / NM/. Instr. and M&.
ABS 68
where Ap is the betatron period; and E,, is the unnormalized emittance in the y direction. The x dimension of the beam follows the optical beam’s contour,
(3) We calculate the number of spontaneous photons at fundamental frequency using Colson’s formula N,,,, = 2 NeNwaf,a,’ [s]. The divergence angle of the TEM,,, cavity mode is used as the coherence angle. For a, smaller than 1, the number of spontaneous photons emitted into the coherence angle 0 is approximated by Nc=N,,,,
n
JJ’(n),
in Phys. Rex A 358 (1995) ABS 67-ABS 6H
We perform a multi-pass calculation to predict the FEL output energy. The number of passes is equal to the number of usable micropulses divided by the number of optical pulses in the cavity. The intracavity intensity builds up from the spontaneous emission within the coherence angle. The FEL gain is modified by the optical intensity in the wiggler as given by Eq. (5). When the number of round-trips exceeds the number of usable micropulses. the FEL gain is reset to 0 and the cavity power is allowed to decay over a few l/e ringdowns of the resonator. The optical energy in a micropulse is calculated by multiplying the optical intensity with the average optical mode area in the wiggler and the micropulse length.
x (3 + 3x + 2x2) (1+x)3
’
(4)
where JJ(n) is the usual difference in Bessel functions; and x is (y 0)‘. We use Dattoli’s FEL gain saturation model to calculate the optimum outcoupling from the small-signal gain g,, and total cavity loss r [4]. The total cavity loss is the sum of the outcoupling f’oc and dissipative loss r,,,, which includes vignetting, absorption and other loss. In Dattoli’s saturation model, the small signal gain is saturated according to
3. Conclusion We have written a simple code to predict the performance of rf-linac driven FEL oscillators. The advantages of FEL-CAD over numerical simulations are its simplicity and speed. Although we welcome any changes the users may have, we believe that FEL-CAD should be kept at this level of complexity. More elaborate, 3-D simulations codes [1,2] should be used to make accurate prediction of FEL performance.
References
where g(l) is the gain g,, is the small-signal fitting parameter which the saturation intensity the small-signal gain is
at a given gain; LY is falls in the defined as halved.
intracavity intensity I; the phenomenological range 0.14-0.18; I, is the intensity at which
[II J.C. Goldstein et al., Nucl. Instr. and Meth. A 285 (1989) 192. 121T.M. Tran and J.S. Wurtele. Phys. Rep. 195 11990) 1. [31 G. Dattoli et al., Nucl. Instr. and Meth. A 318 (1992) 495. [41 Ii. Halbach, Nucl. Instr. and Meth. 178 (1981) 109. [51W.B. Colson. in: Fret Electron Laser Handbook, rds. W.B.
Colson. C. Pellegrini and A. Rcnieri (North-Holland, Amsterdam 1990) Chap. 5.