- Email: [email protected]
Electron beam adaptation measurement of an infrared FEL
Nuclear Instruments and Methods in Physics Research A 707 (2013) 69–72
Contents lists available at SciVerse ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Electron beam adaptation measurement of an infrared FEL J.-M. Ortega n, B. Rieul, J.-P. Berthet, F. Glotin, R. Prazeres LCP, Bat 201 P. 2 Orsay 91405, France
a r t i c l e i n f o
abstract
Article history: Received 9 October 2012 Received in revised form 3 December 2012 Accepted 25 December 2012 Available online 3 January 2013
We have studied the electron beam transverse adaptation in an FEL oscillator in a very large interval of wavelength (3 to 150 mm) and electron energy range (12 to 45 MeV). Beam dimensions are measured by a moving wire whose temperature dependent resistivity is monitored. By a fast motion of the wire we measure its temperature increase at each transverse position before thermal equilibrium takes place. The beam profile is then directly proportional to the series of the measured values. The results fits well the analytical theory of FEL beam adaptation, even when the undulator vacuum chamber is used as an optical waveguide. & 2012 Elsevier B.V. All rights reserved.
Keywords: Free-electron laser Undulator beam adaptation Beam size measurement
1. Introduction In a free electron laser, the optimized optical power, Popt, depends on the so-called optical SSG (Small Signal Gain) at the start of the amplification process. A simplified small gain formula shows the electron beam parameters that can be varied to optimize the optical gain: Glaser
Ipp
Se þ So
F inh sg sx sy syx syy
ð1Þ
where Finh is a function depending on the energy spread, sg, the beam transverse dimensions, sx, sy and respective divergence, syx and syy. Ipp is the peak current and Se and So are respectively the electron and the optical beam cross-section areas, assuming Gaussian beams. The laser power is optimized [1] by improving as much as possible the peak current, the emittance (acting on Finh) and the beam sizes. For low (optical) gain FELs, as much of the infrared ones, the optimization of the gain at short wavelengths is achieved by minimizing the surface (Se) of the electron beam along the undulator [2–4] (the surface of the optical beam being determined by the cavity mirrors). The results of this wellknown optimization is displayed in Section 3 with the numerical values of interest. For high gain, in particular single pass (SASE), FELs, beam matching is different and has been studied by many authors [5,6]. Here, we are interested in the beam matching in low gain infrared FELs. At relatively short wavelength (typically o30 to 40 mm), the above optimization is expected to work. However, since, to our knowledge, there are no published measurements, it is part of our study to demonstrate experimentally this approach in this range. The other part deals with the long wavelength n
Corresponding author. Tel.: þ33 1 6915 3294; fax.: þ 33 1 6915 3328. E-mail address: [email protected] (J.-M. Ortega).
0168-9002/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2012.12.115
range. A particularity of these lasers is that, at longer wavelengths, diffraction losses take place since the diffraction limited optical beam becomes larger than the undulator vacuum chamber. To alleviate partially this effect we use the undulator vacuum chamber as an optical waveguide by focusing the beam at its ends [7]. The guiding occurs only in this vacuum chamber and free space propagation occurs between it and the cavity mirrors. The output laser power exhibit then spectral holes that are characteristics of this configuration. They were reproduced satisfactorily by numerical simulations [7]. Therefore, it could happen that a standard adaptation of the electron beam could not fit this configuration and that the optimized size can be different. In order to address this problem, we have measured the electron beam profiles at different energies and compared with the standard analytical values. The CLIO mid-infrared free electron laser (FEL) is a user facility since 1992. It is based on a 3 GHz RF LINAC with a thermo-ionic gun [8]. It has given rise to many FEL developments [9] and applications [http://clio.lcp.u-psud.fr/clio_eng/topics.html]. CLIO is designed to operate in a large spectral range. This has enabled us to lase from 3 mm at 45 MeV to 150 mm at 12 MeV [10]. Fig. 1 displays the typical average power emitted from the FEL operating at 25 Hz macro-pulse rate. Each macro-pulse is 10 ms long and contain about 600 micro-pulses, a few ps long, separated by 16 ns. The average power up to 20 mm at 32 MeV is about 1 W. It decreases strongly at longer wavelengths due to a lower transmitted electron current and diffraction losses both in the optical cavity and transport beam line.
2. Wire scanner principle There are many imaging techniques [11] of transverse beam profiles used worldwide. In our case, the average beam current is
70
J.-M. Ortega et al. / Nuclear Instruments and Methods in Physics Research A 707 (2013) 69–72
Fig. 1. Spectral range of the infrared FEL CLIO at different energies.
the wires were not conclusive, since the shape of the electron beam is not known a priori. Therefore, we implemented a new measurement method that we describe in this study: By fast moving the wire we measure its temperature-induced resistance increase for each macro-pulse at a different transverse position. This produces an easily measurable signal, by the fact we have a high current accelerator. Each macro-pulse contains approximately 3 1012 particles, i.e., 0.5 mC and produces a resistance variation between typically 0.1 to 0.01 O. These variations appear as a series of peaks superimposed on a global warming of the wire (Figs. 3 and 4). The beam profile is then directly proportional to the series of recorded values, as we show below. There are 2 quadrupoles used to adapt the beam in the 2 m long CLIO undulator. These quadrupoles are located respectively at 80 and 35 cm from the undulator entrance. After a first adjustment calculated to match roughly the standard matching, they are adjusted at each energy in order to produce the maximum FEL power, without any other consideration. The resulting beam sizes are displayed on Fig. 5. In order to evaluate the magnitude of resistance variations, we assumed first that the beam has a Gaussian elliptical shape whose axis is oriented along the horizontal and vertical direction. The parameters of the tungsten wires are shown on Table I. A beam of N particles, centered at coordinate (0,0), crossing the wire at coordinate (x,y) looses an energy: ! Nar dE x2 y2 dwd ðx,yÞ ¼ dxdy exp þ ð2Þ 2psx sy dx 2s2x 2s2y When the wire is located at the center of the beam, the energy deposited increases locally the temperature of the wire by a maximum of: ! dwd ð0,yÞ N dE y2 exp ð3Þ ¼ DT ð0,yÞ ¼ rC p adxdy 2psx sy C p dx 2s2y For one CLIO macro-pulse at 44 MeV, N E3 1012, sx ¼0.7 mm and sy ¼1.2 mm, it comes:
DTð0,0Þ ¼ 95K
ð4Þ
The local resistance of the portion of wire of length dy is: ð5Þ r ¼ RT =LÞdy
Fig. 2. Photography of the 2 orthogonal tungsten moving wires.
high (20 mA), leading to almost 1 kW of average power. Therefore we chose a technique intercepting the smallest possible part of the beam. We use wires of diameter of 20 mm. A set of two orthogonal movable wires is installed at entry of the 2 m undulator (Fig. 2). During displacement, the intercepted part of the beam produces a current of secondary electrons [12] and a variation of the wire resistance [13]. The accelerator having notable parasitic beam losses, which produce noise in the secondary electron emission, it appeared impossible to use phenomenon. Therefore, we opted for the second method, i.e., measuring the wire resistance variations. In a previous paper [12] we discussed measurements made by moving slowly the wire across the beam, so as to have thermal equilibrium, and recording its resistance increase. In this case, in order to correlate the measurements with the size of the beam, we had to make assumptions on the cooling processes of the wire. For simplicity, we assumed that black body radiation was dominant, and it appeared that the results cannot be considered as reliable and, indeed, did not fit the theory. Attempts to perform more sophisticated calculations including thermal cooling along
where RT (21 O) is the total resistance of the wire of length L (6 cm) The local variation of resistance is then: ! RT dy aN dE y2 exp r ð0,yÞ ¼ r aDT ð0,yÞ ¼ ð6Þ L 2psx sy C p dx 2s2y where a is the W temperature coefficient Integrating over the wire length: Z aNR dE DRT ðx ¼ 0Þ ¼ rð0,yÞ ¼ pffiffiffiffiffiffi T 2psx LC p dx y and
DRT ðxÞ ¼ DRT ðx ¼ 0Þexp
x2 2s2x
ð7Þ
One finds
DRT ðx ¼ 0Þ 0:45O
ð8Þ
The experimental value is close to 0.25 O. The difference seems to be due to a beam halo, visible on the curves (Figs. 3 and 4). This produces wings on the distribution so that 40% of the particles are not located in a Gaussian envelope. The maximum resistance variation, due to the interplay between accumulated macro-pulses and cooling, is approximately 4 O.
J.-M. Ortega et al. / Nuclear Instruments and Methods in Physics Research A 707 (2013) 69–72
71
Fig. 3. Display of the series of wire resistance increments at 44 MeV when moving the wire (left) and best fit with a Gaussian distribution (right, gray line). The series of peaks on the left side is for internal use.
Fig. 4. Same as Fig. 3 at an electron energy of 15 MeV.
Table I Physical properties of the tungsten wires. Parameters
Symbol
Unit
Value
Wire diameter Energy loss Specific heat capacity Melting temp. Density
A dE/dx Cp T
M J m2/kg J kg 1 K 1 1C kg m 3
2.10 5 2.2 10 14 130 3410 19300
r
the maximum resistance variation of the wire would be obtained (from Eq. (9)) for a temperature variation of only 40 K, if we assume a uniform temperature along its length. This temperature is certainly not uniform, but it is certain that the radiative cooling and thermal propagation through the wire, occurring during a measurement of 5 s duration, limit the temperature increase at any point to less than 100 K. Therefore, we can consider than the assumption of linearity between the resistance variation and beam size is well verified within a few percent. This is confirmed by the fact that the measured profiles are quite symmetric despite the average temperature increase during the measurement.
Fig. 5. Beam sizes at the undulator ends from the wire resistance variations in horizontal (black points) and vertical (gray points) directions and comparison with analytical theory (dotted lines).
The resistivity of the tungsten wire is given by: Rw =Rw ðT ¼ 0Þ ¼ 1þ 4:8297 103 T þ1:663 106 T 2
ð9Þ
And dRw =dT 1 þ0:687 103 T
3. Comparison with optimization The analytical standard minimized values [2–4] are: In the undulator focusing (vertical) plane, the undulator focusing makes a flat beam profile if the value at the entrance (wire position) is: sffiffiffiffiffiffiffiffiffiffiffiffiffi
sy ¼ ð10Þ
Thus, the variation of resistance has a tendency of being nonlinear due to temperature increase. However, let us point out that
e l
pnffiffiffi o ffi 0:7 mm 2pK
ð11Þ
where en ( ¼ gex,y with ex,y ¼ ssy) is the normalized emittance, which is the same in both planes (40 mm at CLIO [1]), lo ¼50 mm
72
J.-M. Ortega et al. / Nuclear Instruments and Methods in Physics Research A 707 (2013) 69–72
and K is close to 1, where K is the well-known undulator parameter. In the horizontal plane the minimum average size result from minimization of the radius along the undulator. In a planar undulator with no horizontal focusing it comes: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2 L=2 z2 e2 ð12Þ sx ¼ s2xo þ 2 dz L 0 sxo sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50 EðMeVÞ
w x ðmmÞ ffi 1:1
)s
at wire position
ð13Þ
Fig. 5 displays the beam measurements at energy ranging from 12 to 45 MeV. It can be seen that the measured value are closed to the ones predicted by analytical theory. This agreement justifies our assumption that the axis of our elliptical beam is roughly oriented along the horizontal and vertical directions. In conclusion, this article reports measurements of an FEL beam matching in a very large range of wavelengths: This corresponds to a laser propagation going progressively from a purely free space propagation to a partially guided mode. The result justify quite well the simple analytical analysis that assume that the best conditions are a beam of constant size in the magnetic field direction and of average minimum size in the perpendicular one. This is not straightforward considering that the optical gain can be non linear with respect to the undulator length and that diffraction effects vary with wavelength. It can be attributed to— the fact that the standard optimization is the one that also minimizes the inhomogeneous broadening effects [4], since these reduce the gain—the overlap integral (see Eq. (1)) is
always maximized for the smallest possible transverse size of the electron beam.
References [1] G. Datolli, T. Letardi, J.M.J Madey, A. Renieri, IEEE Journal Quantum Electron. QE-20 (6) (1984) 63. [2] M.W. Poole, CAS—CERN accelerator school and Daresbury laboratory: course on Synchrotron radiation and free-electron lasers, 6–13 Apr 1989, Chester, UK, pp. 306–334, http://cdsweb.cern.ch/record/340449?ln=fr. [3] C. Rippon, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 445 (2000) 399. [4] G. Dattoli, A. Renieri, A. Torre, Lectures on the Free Electron Laser Theory and Related Topics, World Scientific Press, Singapore, 1993. [5] H.P. Freund, P.G. O’Shea, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 429 (1999) 219. [6] M. Quattromini, et al., Physical Review STAB 15 (2012) 080704. [7] R. Prazeres, F. Glotin, J.-M. Ortega, Physical Review STAB 12 (2009) 010701. [8] R. Chaput, B. Kergosien, P. Joly, J. Lesrel, O. Marcouille´, in: Proceedings of EPAC 94, 1, 728 (World Scientific Publishers, 1995), http://accelconf.web. cern.ch/AccelConf/e94/PDF/EPAC1994_0728.PDF. [9] R. Prazeres, F. Glotin, C. Insa, D.A. Jaroszynski, J.M. Ortega, European Physical Journal D3 (1998) 87. [10] J.M. Ortega, F. Glotin, R. Prazeres, Infrared Physics and Technology 49 (2006) 133. [11] A. Lumpkin, in: Proceedings of BIW2012, Newport News, Virginia, USA, http://accelconf.web.cern.ch/AccelConf/BIW2012/papers/tubp02.pdf. [12] P. Bourgeois, J.P. Carlos, M.E. Couprie, J.L. Fallou, P. Garganne, A. Leprˆetre, J. Martin, J.M. Ortega, C. Re´, Nuclear Instrument and Methods A301 (1991) 150. [13] J.P. Berthet, F. Glotin, J.-M. Ortega, W. Salah, in: Proceedings of EPAC 2006, Edinburgh, Scotland, http://accelconf.web.cern.ch/AccelConf/e06/PAPERS/ MOPCH006.PDF.
Contents lists available at SciVerse ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Electron beam adaptation measurement of an infrared FEL J.-M. Ortega n, B. Rieul, J.-P. Berthet, F. Glotin, R. Prazeres LCP, Bat 201 P. 2 Orsay 91405, France
a r t i c l e i n f o
abstract
Article history: Received 9 October 2012 Received in revised form 3 December 2012 Accepted 25 December 2012 Available online 3 January 2013
We have studied the electron beam transverse adaptation in an FEL oscillator in a very large interval of wavelength (3 to 150 mm) and electron energy range (12 to 45 MeV). Beam dimensions are measured by a moving wire whose temperature dependent resistivity is monitored. By a fast motion of the wire we measure its temperature increase at each transverse position before thermal equilibrium takes place. The beam profile is then directly proportional to the series of the measured values. The results fits well the analytical theory of FEL beam adaptation, even when the undulator vacuum chamber is used as an optical waveguide. & 2012 Elsevier B.V. All rights reserved.
Keywords: Free-electron laser Undulator beam adaptation Beam size measurement
1. Introduction In a free electron laser, the optimized optical power, Popt, depends on the so-called optical SSG (Small Signal Gain) at the start of the amplification process. A simplified small gain formula shows the electron beam parameters that can be varied to optimize the optical gain: Glaser
Ipp
Se þ So
F inh sg sx sy syx syy
ð1Þ
where Finh is a function depending on the energy spread, sg, the beam transverse dimensions, sx, sy and respective divergence, syx and syy. Ipp is the peak current and Se and So are respectively the electron and the optical beam cross-section areas, assuming Gaussian beams. The laser power is optimized [1] by improving as much as possible the peak current, the emittance (acting on Finh) and the beam sizes. For low (optical) gain FELs, as much of the infrared ones, the optimization of the gain at short wavelengths is achieved by minimizing the surface (Se) of the electron beam along the undulator [2–4] (the surface of the optical beam being determined by the cavity mirrors). The results of this wellknown optimization is displayed in Section 3 with the numerical values of interest. For high gain, in particular single pass (SASE), FELs, beam matching is different and has been studied by many authors [5,6]. Here, we are interested in the beam matching in low gain infrared FELs. At relatively short wavelength (typically o30 to 40 mm), the above optimization is expected to work. However, since, to our knowledge, there are no published measurements, it is part of our study to demonstrate experimentally this approach in this range. The other part deals with the long wavelength n
Corresponding author. Tel.: þ33 1 6915 3294; fax.: þ 33 1 6915 3328. E-mail address: [email protected] (J.-M. Ortega).
0168-9002/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2012.12.115
range. A particularity of these lasers is that, at longer wavelengths, diffraction losses take place since the diffraction limited optical beam becomes larger than the undulator vacuum chamber. To alleviate partially this effect we use the undulator vacuum chamber as an optical waveguide by focusing the beam at its ends [7]. The guiding occurs only in this vacuum chamber and free space propagation occurs between it and the cavity mirrors. The output laser power exhibit then spectral holes that are characteristics of this configuration. They were reproduced satisfactorily by numerical simulations [7]. Therefore, it could happen that a standard adaptation of the electron beam could not fit this configuration and that the optimized size can be different. In order to address this problem, we have measured the electron beam profiles at different energies and compared with the standard analytical values. The CLIO mid-infrared free electron laser (FEL) is a user facility since 1992. It is based on a 3 GHz RF LINAC with a thermo-ionic gun [8]. It has given rise to many FEL developments [9] and applications [http://clio.lcp.u-psud.fr/clio_eng/topics.html]. CLIO is designed to operate in a large spectral range. This has enabled us to lase from 3 mm at 45 MeV to 150 mm at 12 MeV [10]. Fig. 1 displays the typical average power emitted from the FEL operating at 25 Hz macro-pulse rate. Each macro-pulse is 10 ms long and contain about 600 micro-pulses, a few ps long, separated by 16 ns. The average power up to 20 mm at 32 MeV is about 1 W. It decreases strongly at longer wavelengths due to a lower transmitted electron current and diffraction losses both in the optical cavity and transport beam line.
2. Wire scanner principle There are many imaging techniques [11] of transverse beam profiles used worldwide. In our case, the average beam current is
70
J.-M. Ortega et al. / Nuclear Instruments and Methods in Physics Research A 707 (2013) 69–72
Fig. 1. Spectral range of the infrared FEL CLIO at different energies.
the wires were not conclusive, since the shape of the electron beam is not known a priori. Therefore, we implemented a new measurement method that we describe in this study: By fast moving the wire we measure its temperature-induced resistance increase for each macro-pulse at a different transverse position. This produces an easily measurable signal, by the fact we have a high current accelerator. Each macro-pulse contains approximately 3 1012 particles, i.e., 0.5 mC and produces a resistance variation between typically 0.1 to 0.01 O. These variations appear as a series of peaks superimposed on a global warming of the wire (Figs. 3 and 4). The beam profile is then directly proportional to the series of recorded values, as we show below. There are 2 quadrupoles used to adapt the beam in the 2 m long CLIO undulator. These quadrupoles are located respectively at 80 and 35 cm from the undulator entrance. After a first adjustment calculated to match roughly the standard matching, they are adjusted at each energy in order to produce the maximum FEL power, without any other consideration. The resulting beam sizes are displayed on Fig. 5. In order to evaluate the magnitude of resistance variations, we assumed first that the beam has a Gaussian elliptical shape whose axis is oriented along the horizontal and vertical direction. The parameters of the tungsten wires are shown on Table I. A beam of N particles, centered at coordinate (0,0), crossing the wire at coordinate (x,y) looses an energy: ! Nar dE x2 y2 dwd ðx,yÞ ¼ dxdy exp þ ð2Þ 2psx sy dx 2s2x 2s2y When the wire is located at the center of the beam, the energy deposited increases locally the temperature of the wire by a maximum of: ! dwd ð0,yÞ N dE y2 exp ð3Þ ¼ DT ð0,yÞ ¼ rC p adxdy 2psx sy C p dx 2s2y For one CLIO macro-pulse at 44 MeV, N E3 1012, sx ¼0.7 mm and sy ¼1.2 mm, it comes:
DTð0,0Þ ¼ 95K
ð4Þ
The local resistance of the portion of wire of length dy is: ð5Þ r ¼ RT =LÞdy
Fig. 2. Photography of the 2 orthogonal tungsten moving wires.
high (20 mA), leading to almost 1 kW of average power. Therefore we chose a technique intercepting the smallest possible part of the beam. We use wires of diameter of 20 mm. A set of two orthogonal movable wires is installed at entry of the 2 m undulator (Fig. 2). During displacement, the intercepted part of the beam produces a current of secondary electrons [12] and a variation of the wire resistance [13]. The accelerator having notable parasitic beam losses, which produce noise in the secondary electron emission, it appeared impossible to use phenomenon. Therefore, we opted for the second method, i.e., measuring the wire resistance variations. In a previous paper [12] we discussed measurements made by moving slowly the wire across the beam, so as to have thermal equilibrium, and recording its resistance increase. In this case, in order to correlate the measurements with the size of the beam, we had to make assumptions on the cooling processes of the wire. For simplicity, we assumed that black body radiation was dominant, and it appeared that the results cannot be considered as reliable and, indeed, did not fit the theory. Attempts to perform more sophisticated calculations including thermal cooling along
where RT (21 O) is the total resistance of the wire of length L (6 cm) The local variation of resistance is then: ! RT dy aN dE y2 exp r ð0,yÞ ¼ r aDT ð0,yÞ ¼ ð6Þ L 2psx sy C p dx 2s2y where a is the W temperature coefficient Integrating over the wire length: Z aNR dE DRT ðx ¼ 0Þ ¼ rð0,yÞ ¼ pffiffiffiffiffiffi T 2psx LC p dx y and
DRT ðxÞ ¼ DRT ðx ¼ 0Þexp
x2 2s2x
ð7Þ
One finds
DRT ðx ¼ 0Þ 0:45O
ð8Þ
The experimental value is close to 0.25 O. The difference seems to be due to a beam halo, visible on the curves (Figs. 3 and 4). This produces wings on the distribution so that 40% of the particles are not located in a Gaussian envelope. The maximum resistance variation, due to the interplay between accumulated macro-pulses and cooling, is approximately 4 O.
J.-M. Ortega et al. / Nuclear Instruments and Methods in Physics Research A 707 (2013) 69–72
71
Fig. 3. Display of the series of wire resistance increments at 44 MeV when moving the wire (left) and best fit with a Gaussian distribution (right, gray line). The series of peaks on the left side is for internal use.
Fig. 4. Same as Fig. 3 at an electron energy of 15 MeV.
Table I Physical properties of the tungsten wires. Parameters
Symbol
Unit
Value
Wire diameter Energy loss Specific heat capacity Melting temp. Density
A dE/dx Cp T
M J m2/kg J kg 1 K 1 1C kg m 3
2.10 5 2.2 10 14 130 3410 19300
r
the maximum resistance variation of the wire would be obtained (from Eq. (9)) for a temperature variation of only 40 K, if we assume a uniform temperature along its length. This temperature is certainly not uniform, but it is certain that the radiative cooling and thermal propagation through the wire, occurring during a measurement of 5 s duration, limit the temperature increase at any point to less than 100 K. Therefore, we can consider than the assumption of linearity between the resistance variation and beam size is well verified within a few percent. This is confirmed by the fact that the measured profiles are quite symmetric despite the average temperature increase during the measurement.
Fig. 5. Beam sizes at the undulator ends from the wire resistance variations in horizontal (black points) and vertical (gray points) directions and comparison with analytical theory (dotted lines).
The resistivity of the tungsten wire is given by: Rw =Rw ðT ¼ 0Þ ¼ 1þ 4:8297 103 T þ1:663 106 T 2
ð9Þ
And dRw =dT 1 þ0:687 103 T
3. Comparison with optimization The analytical standard minimized values [2–4] are: In the undulator focusing (vertical) plane, the undulator focusing makes a flat beam profile if the value at the entrance (wire position) is: sffiffiffiffiffiffiffiffiffiffiffiffiffi
sy ¼ ð10Þ
Thus, the variation of resistance has a tendency of being nonlinear due to temperature increase. However, let us point out that
e l
pnffiffiffi o ffi 0:7 mm 2pK
ð11Þ
where en ( ¼ gex,y with ex,y ¼ ssy) is the normalized emittance, which is the same in both planes (40 mm at CLIO [1]), lo ¼50 mm
72
J.-M. Ortega et al. / Nuclear Instruments and Methods in Physics Research A 707 (2013) 69–72
and K is close to 1, where K is the well-known undulator parameter. In the horizontal plane the minimum average size result from minimization of the radius along the undulator. In a planar undulator with no horizontal focusing it comes: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2 L=2 z2 e2 ð12Þ sx ¼ s2xo þ 2 dz L 0 sxo sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50 EðMeVÞ
w x ðmmÞ ffi 1:1
)s
at wire position
ð13Þ
Fig. 5 displays the beam measurements at energy ranging from 12 to 45 MeV. It can be seen that the measured value are closed to the ones predicted by analytical theory. This agreement justifies our assumption that the axis of our elliptical beam is roughly oriented along the horizontal and vertical directions. In conclusion, this article reports measurements of an FEL beam matching in a very large range of wavelengths: This corresponds to a laser propagation going progressively from a purely free space propagation to a partially guided mode. The result justify quite well the simple analytical analysis that assume that the best conditions are a beam of constant size in the magnetic field direction and of average minimum size in the perpendicular one. This is not straightforward considering that the optical gain can be non linear with respect to the undulator length and that diffraction effects vary with wavelength. It can be attributed to— the fact that the standard optimization is the one that also minimizes the inhomogeneous broadening effects [4], since these reduce the gain—the overlap integral (see Eq. (1)) is
always maximized for the smallest possible transverse size of the electron beam.
References [1] G. Datolli, T. Letardi, J.M.J Madey, A. Renieri, IEEE Journal Quantum Electron. QE-20 (6) (1984) 63. [2] M.W. Poole, CAS—CERN accelerator school and Daresbury laboratory: course on Synchrotron radiation and free-electron lasers, 6–13 Apr 1989, Chester, UK, pp. 306–334, http://cdsweb.cern.ch/record/340449?ln=fr. [3] C. Rippon, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 445 (2000) 399. [4] G. Dattoli, A. Renieri, A. Torre, Lectures on the Free Electron Laser Theory and Related Topics, World Scientific Press, Singapore, 1993. [5] H.P. Freund, P.G. O’Shea, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 429 (1999) 219. [6] M. Quattromini, et al., Physical Review STAB 15 (2012) 080704. [7] R. Prazeres, F. Glotin, J.-M. Ortega, Physical Review STAB 12 (2009) 010701. [8] R. Chaput, B. Kergosien, P. Joly, J. Lesrel, O. Marcouille´, in: Proceedings of EPAC 94, 1, 728 (World Scientific Publishers, 1995), http://accelconf.web. cern.ch/AccelConf/e94/PDF/EPAC1994_0728.PDF. [9] R. Prazeres, F. Glotin, C. Insa, D.A. Jaroszynski, J.M. Ortega, European Physical Journal D3 (1998) 87. [10] J.M. Ortega, F. Glotin, R. Prazeres, Infrared Physics and Technology 49 (2006) 133. [11] A. Lumpkin, in: Proceedings of BIW2012, Newport News, Virginia, USA, http://accelconf.web.cern.ch/AccelConf/BIW2012/papers/tubp02.pdf. [12] P. Bourgeois, J.P. Carlos, M.E. Couprie, J.L. Fallou, P. Garganne, A. Leprˆetre, J. Martin, J.M. Ortega, C. Re´, Nuclear Instrument and Methods A301 (1991) 150. [13] J.P. Berthet, F. Glotin, J.-M. Ortega, W. Salah, in: Proceedings of EPAC 2006, Edinburgh, Scotland, http://accelconf.web.cern.ch/AccelConf/e06/PAPERS/ MOPCH006.PDF.