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Experimental and numerical study of short pulse effects in FELs
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Nuclear Instruments and Methods in Physics Research A 530 (2004) 194–204
Experimental and numerical study of short pulse effects in FELs S. Khodyachykha, M. Brunkena, H. Genza, C. Hesslera, A. Richtera,*, V. Asgekarb a
Institut fur . Kernphysik, Technische Universitat . Darmstadt, Schlossgartenstr. 9, D-64289 Darmstadt, Germany b Department of Physics, University of Pune, Pune 411 007, Maharashtra, India Received 27 January 2004; accepted 9 April 2004 Available online 15 June 2004
Abstract We report the experimental and numerical investigations of the influence of short pulse effects occurring in FELs in different operational regimes for electron bunch lengths which are of the order of the slippage distance. Several observables such as the small signal gain, the macropulse power and the spectral distribution of the FEL radiation were determined experimentally within the constraints of the stable focus regime at the infrared FEL at the S-DALINAC and for the limit cycle regime at the Dutch near infrared FEL FELIX. The experimental findings were compared to predictions of numerical simulations based on the 1D time dependent code FEL1D-OSC. The agreement between experiment and simulation is good. Furthermore, the simulations reveal a chaotic behavior of the macropulses for specific values of the slippage as well as period-doubling, two effects that are predicted to show up in the spectral distribution. r 2004 Elsevier B.V. All rights reserved. PACS: 41.60.Cr Keywords: Free electron laser; Short pulse effects; Numerical simulations
1. Introduction For the successful operation of the FEL the electron beam has to fulfil the following requirements. First, for the infrared or visible spectral regions the typical undulator period of a few centimeters and an undulator K-value of about *Corresponding author. Tel.: +49-6151-16-2116; fax: +496151-16-4321. E-mail address: [email protected] (A. Richter).
one require a relativistic electron beam energy. At the same time, since the FEL gain is proportional to the beam current density, a high value of the latter is required. Finally, the energy spread should be small compared to the width of the gain curve [1]. Based on the above specified criteria, most FEL facilities operating in the infrared region use radio-frequency (RF) linear accelerators. This type of accelerators produce a train of short picosecond pulses. The light wave produced
0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.04.215
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inside the resonator shows also this structure and thus consists also of a train of short micropulses. During the interaction of the electrons with the photon beam, the electrons slip back relative to the optical pulse for two reasons: (i) the electron velocity is always smaller than the velocity of light and (ii) the electrons travel along the oscillating trajectories inside the undulator. The resonance condition requires that the electron bunch slips back by one optical wavelength l per one undulator period. At the exit of the undulator, consisting of Nu periods, the electron has slipped back by a slippage distance of Nu l: If this quantity is comparable to the electron bunch length sz ; one defines such an FEL as short pulse FEL. In the present paper we investigate how the formation of short pulses, which occur under specific conditions in RF driven free electron lasers, affects the observables of an FEL. We report on the experimental determination of the small signal gain, the macropulse power and the spectral distribution of the FEL radiation for different values of the cavity desynchronization performed at the Darmstadt Infrared FEL [2] driven by the superconducting Darmstadt linear electron accelerator S-DALINAC [3]. The experimental results are compared to the predictions obtained from simulations using the 1D time dependent code FEL1D-OSC which had to be extended to become applicable for the current FEL configuration. We also compare the numerical results to the experiment we performed at the Dutch free electron laser FELIX [4] for several observables which have already been tested in former experiments [5,6] and which could be described correctly using an analytical model [5]. This paper is structured as follows. After the short theoretical description of the problem in Section 2, the numerical simulation algorithm and its limitations are discussed in Section 3. The FEL at the S-DALINAC and FELIX together with the related experimental areas are described in Section 4 and the experimental results obtained there are compared with the numerical results in Section 5. It is followed by a conclusion.
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2. Theoretical background With respect to the FEL operation one distinguishes between the small signal regime (when the power of the laser radiation grows) and the saturated or steady state regime. In the small signal regime the amplification of the radiation takes place towards the downstream end of the undulator where the electrons have slipped back relative to the front of the optical micropulse and therefore the optical pulse peaks up at its rear. This has the consequence that the effective optical group velocity is smaller than its vacuum value. In a perfectly synchronized resonator the micropulse will continue to narrow during the subsequent round trips. The reduced overlap between electron bunch and optical pulse will in turn reduce the gain. This effect is known as ‘‘laser lethargy’’. In order to restore the gain it is necessary to slightly reduce the cavity length from the value giving the perfect synchronization. For convenience the cavity-length change DL is expressed usually in units of the optical wavelength DL=l: Hahn and Lee [7,8] have shown that the various regimes of FEL operation can be identified for the set of parameters DL=l; Nu l=sz and Lc =sz : The last parameter contains the cooperation length Lc ¼ l=4pr which is equal to the slippage in one gain length, with r being the fundamental gain parameter. The phase diagram [7] for different regimes of the FEL operation is reproduced in Fig. 1. For a constant value of Lc =sz the regimes change with increase in Nu l=sz from the stable focus (when no subpulses develop) to the limit cycle region with sidebands in the spectrum. These sidebands double sequentially (period-doubling). Finally the chaotic regime starts. There is a number of papers which are dealing with chaos in FEL (see e.g. [9–16]). In the chaotic regime each macropulse evolves in a different way which can also be observed e.g. in the spectral distribution. To reach the limit-cycle and chaotic regime, the intracavity power has to saturate at a sufficiently higher level. In the present paper we report the experimental observation of various quantities in two different regimes, namely the stable focus and the limitcycle, carried out at the S-DALINAC and FELIX.
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Fig. 1. Phase diagram [7] for different regimes of operation of an FEL oscillator. The quantities Nu l and sz are the slippage distance and the electron bunch length, respectively, and Lc ¼ l=4pr: Lines corresponding to three different cavity desynchronizations separate three different regimes of operation namely stable focus, limit-cycle and chaotic regimes. The points a; b and c correspond to three cases studied in the present paper.
These experimental results are compared to the predictions of the numerical simulations. Furthermore, the chaotic behavior and period-doubling were observed in our simulations. The parameters of the experiments and the input parameters for the simulations correspond to the points a; b; and c plotted in the phase diagram (Fig. 1) for the experiments at the S-DALINAC FEL, at FELIX, and for the numerical simulations, respectively. Point a was obtained using the parameters for the simulation of the spectral distribution (see Section 5). For convenience, instead of r we use for our simulations the gain parameter for an FEL oscillator t defined in Ref. [17]. The relation between t and r is given by t ¼ 4prNu :
3. Numerical simulation algorithm To model the FEL oscillator and its light output, the equations of the electron motion have to be solved together with Maxwell’s equation taking into account the boundary conditions for the electron beam and for the electromagnetic radiation. It is assumed that the lasing process starts from the shot noise in the electron beam.
The linear regime of the FEL operation (the small signal regime) can be described analytically (see e.g. Ref. [17]) very accurately. For the description of the non-linear regime when the FEL reaches saturation, which is the subject of interest of the present work, there are no analytical solutions, that is why it was necessary to use numerical methods. Simulations have been performed with the onedimensional, time-dependent code FAST-OSC designed by Saldin, Schneidmiller and Yurkov [18]. This is a one-dimensional, time-dependent code which allows one to study the time evolution of intensity of the optical macropulse generated by the FEL oscillator, which is considered as an FEL amplifier with feedback, with variable undulator parameters from the shot noise up to saturation. Any longitudinal profile of the electron bunch can be simulated. The core of the code FAST-OSC is code FAST designed for simulation of an FEL amplifier starting from shot noise in the electron beam [19]. During the first pass of the electron beam through the undulator it produces a radiation pulse. The amplitude of the radiation pulse is reduced according to the value of round-trip losses (the phase change of the electromagnetic field after the reflection from the resonator mirrors is neglected because this effect does not influence the FEL-oscillator operation [17]). The radiation pulse, shifted by the according value of the cavity desynchronization and the next electron bunch with the initial noise distribution are fed to the undulator entrance, and the calculation procedure is repeated. The start-up from the shot noise in the electron beam has been thoroughly tested with analytical results given in Ref. [17]. Since the underlying model is one-dimensional, the interactions modifying the transverse Gaussian profile of the electron or optical beam are not taken into account. The code is valid only for an optical mode which has a waist larger than the electron beam radius, which is the case for both, the Darmstadt FEL and FELIX. For FEL oscillators with high gain the optical field is strongly deformed since the radiation of the electron beam is almost as strong as that inside the resonator. Thus, for the case of a high gain FEL a 3D model should be used to describe
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satisfactorily the FEL behavior. In case of the Darmstadt FEL, however, the small signal gain g is limited to about 5% which allows a 1D treatment. Furthermore, short-pulse effects are largely longitudinal in nature and that is why the FEL can be modelled adequately in a 1D approximation. The hole-coupling of the optical beam which is often used in FELs also deforms the optical field inside the resonator [20]. To study the FEL dynamics in a hole-coupled FEL a 3D code is preferable. The Darmstadt FEL uses multi-layered partially reflective mirrors which simplifies this task. However, the results presented in Section 5.2 show a satisfactory agreement between the results of the 1D simulation and the experiment carried out at FELIX which uses hole-coupling for the extraction of the optical beam.
4. Experiments In this section the experimental set ups at the SDALINAC FEL and FELIX are described briefly. Both FELs are operating in the infrared region of wavelength and have mostly similar parameters. However, one should mention two major differences between them. First, the peak current value in the case of the FELIX is about one order of magnitude higher than at the S-DALINAC FEL which also allows to operate FELIX in the limitcycle regime (Fig. 1). The second fundamental difference between two FELs is that contrary to FELIX, the S-DALINAC FEL is driven by a superconducting linear electron accelerator which allows to generate much longer optical macropulses and in general even to obtain an optical cw beam.
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beam with energies between 2.5 and 130 MeV can be produced [3]. Fig. 2 shows the layout of the S-DALINAC. The FEL is located in the space between the main linac and the first beam recirculation path. From the latter electron beams with an energy between 25 and 50 MeV can be bent over and passed through a 2:6 m long undulator. The latter is built up from 80 periods each having a length of 3:2 cm and has a variable gap width between 16 and 25 mm: The emitted photon pulses are stored in a 15 m long resonator [22] made out of two concave mirrors having reflectivities of 99.0% and 99.8%, respectively. Inside the undulator the amplification process takes place so that an infrared laser beam with a wavelength between 3 and 10 mm is generated. The tuning of the wavelength can be accomplished by varying the electron energy and/ or the undulator gap. The laser beam is extracted by the mirror at the downstream (DS) table of the FEL. Since the amplification of the optical pulse depends strongly on the peak current of the electron bunch inside the undulator, a system based on a 10 MHz sub-harmonic injection has been designed and built [23]. In this case only every 300th high frequency period is filled with electrons but with an increased bunch charge. The bunch length of 2 ps leads to a peak current of 2:7 A which, in turn implies a small signal gain of 2–5%. The time structure of the laser light has the
4.1. S-DALINAC and the FEL The superconducting Darmstadt linear electron accelerator S-DALINAC is since 1991 in operation at the Institute of Nuclear Physics of the Darmstadt University of Technology. It is used for experiments on nuclear and radiation physics and since 1996 also as a driver for the infrared FEL [2,21]. Using the high-frequency accelerating method and dual beam recirculation, a continuous
Fig. 2. The superconducting Darmstadt linear electron accelerator S-DALINAC with the FEL. (1) Electron gun, (2) superconducting 10 MeV injector, (3) superconducting 40 MeV accelerator, (4) undulator, (5) upstream and (6) downstream tables with mirror chambers, (7) transfer system to the optical laboratory located one floor above (not shown).
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structure of the electron beam and thus, short laser pulses of about 2 ps can be obtained.
designed. For the experiments in the optical laboratory the transfer system (6) is used. The results obtained with this experimental set up are presented in Section 5.1.
4.2. Experiment at the S-DALINAC FEL Fig. 3 shows the experimental set up mounted on the DS table in the accelerator hall. With this set up both, spontaneous and laser radiation can be diagnosed. The radiation, outcoupled through a multi-layer dielectric mirror (1) can either be transferred to the optics laboratory for further experiments or can be diagnosed directly at the DS-table. For the diagnostics on the DS-table a system for the measurement of the spectral distribution of the spontaneous and stimulated radiation consisting of a monochromator (2) and a nitrogen-cooled HgCdTe-detector (3) has been developed. The temporal resolution of the detector is about 100 ns so that the development of the intensity within the macropulse can be measured. It should be mentioned that the detector reaches its saturation at a power of about 100 mW: For the measurements above the lasing threshold additional attenuation is necessary. Alternatively the average power can be measured by a thermopile detector (4) which has a temporal resolution of about a few seconds. Two sets of filters (5) are used for the attenuation of the laser radiation or for the harmonics separation. Because of its lower power the spontaneous emission cannot be detected after passing through the outcoupling mirror. For this measurement a special bypass (7) has been
4.3. FELIX The Free Electron Laser for Infrared eXperiments (FELIX) in the Netherlands [4] went into operation in 1991 as the first infrared FEL that attained lasing in Europe [24]. Fig. 4 presents an overview of the FELIX layout. An electron beam with an energy of 3:8 MeV is produced by the injector. Two 3 GHz normal conducting linacs are used to increase the beam energy from 15 to 25 MeV and from 25 to 45 MeV; respectively. Each linac delivers an electron beam that consists of a train of 3–6 ps long bunches that can be introduced into an undulator. Both undulators have 38 periods of 6:5 cm length. Two gold-coated mirrors are placed at opposite sides of the undulator and define the 6 m long resonator. This allows to obtain 40 independent micropulses in the cavity simultaneously. An aperture in the mirror at the left side provides an outcoupling of a fraction of the optical radiation. The position of the mirror on the right side can be adjusted in order to synchronize the circulating optical pulses with electron bunches. After passing through the undulator, the electrons are bent out of the resonator and dumped. The laser typically provides several kW of average output power during the macropulse, with a maximum of 20 kW; corresponding to a micropulse energy of 20 mJ: Table 1 summarizes the main parameters of the S-DALINAC FEL and FELIX.
FEL-2
5 - 35 µm Injector
Fig. 3. Diagnostic elements on the DS table of the Darmstadt FEL: (1) resonator mirror, (2) monochromator, (3) HgCdTethermo resistor, (4) set up for measurement of the total intensity, (5) sets of filters, (6) transfer telescope, (7) bypass for the spontaneous emission measurements.
Linac 1 14 - 25 MeV
L inac 2 25 - 46 MeV
FEL 1
16 - 110 µm 6m
Fig. 4. Layout of FELIX.
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Table 1 Parameters of the S-DALINAC IR FEL and FELIX Parameter
S-DALINAC
FELIX
Electron energy (MeV) Energy spread (%) Electron peak current (A) Undulator period length (cm) Number of periods Undulator parameter K Wavelength ðmmÞ Optical resonator losses (%) Resonator length (m) Repetition rate (MHz) Macropulse duration ðmsÞ
25–50 o0:3 2.7 3.2 80 0.45–1.12 3–10 0.9 15.00 10 2000–cw
14–46 0.2 27 6.5 38 o1:9 5–110 5–10 6.15 1000 (25) o15
4.4. Experiment at FELIX For the experiment at FELIX IR light with wavelengths in the region between 5.7 and 9 mm was delivered to the experimental hall through an evacuated transfer system. The wavelength has been tuned by changing the undulator gap keeping the electron energy fixed at 43:6 MeV: The experimental set up is shown in Fig. 5. Collinearly propagating to the FEL beam (1), a HeNe-laser beam (2) was used for adjustment of the detection devices during the preparation of the experiment. Propagating along the optical table, the FEL beam can be directed either to a second-harmonic autocorrelator (3) described in Ref. [25] or to a monochromator (4) with the help of flip-mirrors (5). Using the monochromator and placing a HgCdTe-detector (6) behind it allows to measure the time-resolved spectral distributions of the FEL within the macropulse. Some results of the measurements are presented in Section 5.2.
5. Experimental and numerical observation of different regimes of operation In this section we report on the experimental determination of several observables which are compared to the results of numerical simulations. The comparison includes the investigation of the
Fig. 5. Experimental set up at FELIX. Collinearly propagating to the FEL beam (1) is a HeNe-laser beam (2) used for the adjustment of the measuring devices during the preparation of the experiment. Propagating along the optical table, the FEL beam can be directed either to a (3) second-harmonic autocorrelator or to a monochromator (4) with the help of flip-mirrors (5). Using the monochromator and placing a HgCdTe-detector (6) behind it allows to measure the timeresolved spectral distributions of the FEL within the macropulse.
macropulse power, the small signal gain and the spectral distribution. 5.1. Stable-focus regime The experimental data analyzed in this section were obtained [26] at the S-DALINAC. A pulsed electron beam with the macropulse length of 4 ms; a repetition rate of 31 Hz and an energy of 31:04 MeV was used. The electron beam radius inside the undulator was assumed to vary between 0.4 and 0:6 mm: Together with the undulator parameter K ¼ 1:106 this results in the resonant wavelength of lr ¼ 7:03 mm and in the dimensionless gain parameter t between 0.3 and 0.7. Fig. 6 shows an example of the development of the laser intensity within the macropulse. Saturation of the laser power was reached within the first 400 ms: These measurements were carried out for various cavity desynchronizations, five times for each setting. From the growth of the macropulse the value of the small signal gain g can be extracted. To reduce the error, the following procedure was chosen for each cavity setting. First, the time was converted
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Fig. 6. Typical development of the laser intensity measured within the macropulse at the S-DALINAC. The pulsed electron beam had an energy of 31:04 MeV; a pulse length of 4 ms and a repetition rate of 31 Hz:
into the number of the resonator round-trips. The leading edge of the macropulse was then plotted on a logarithmic scale and a linear function yi ¼ ai þ gi t; i ¼ 1; y; 5 was fitted to the linear regime. For the values of gi with the P corresponding P errors s2i the weighted mean g# ¼ 5i¼1 ðgi =s2i Þ= 5i¼1 ð1=s2i Þ # ¼ hasP been calculated with an error of s2 ðgÞ 1= 5i¼1 ð1=s2i Þ: The results are presented in Fig. 7 for different values of the cavity desynchronization. The circles with error bars represent the measured data. For the experimental parameters listed above the development of the intensity within the macropulses has been numerically simulated and the information about the small signal gain has been extracted using the same procedure as for the experimental data. The solid line is a fit to the values obtained from the simulations. Good agreement is observed between the experimental data and numerical results. Simultaneously to the time-resolved measurements, the average macropulse power was recorded by a thermopile detector for each value of the cavity desynchronization. In Fig. 8 the results of the numerical simulations are compared to the results of these measurements, and indeed both are in a good agreement. Finally, for a cavity desynchronization of DL=l ¼ 0:05; which corresponds to the max-
Fig. 7. Small signal gain as a function of the cavity desynchronization. Open circles represent the experimental results obtained at the Darmstadt FEL with the beam parameters listed in the caption of Fig. 6. The solid curve is a fit to the data from the simulation.
Fig. 8. Average macropulse power as a function of cavity desynchronization measured at the S-DALINAC (circles) compared with the results of the simulation (solid line).
imum value of the intensity the spectral distribution of the radiation was measured at the beginning of the macropulse with the help of the monochromator and the HgCdTe-detector. The spectral resolution was 25 nm: The time gate was set for 20 ms which is equal to 200 round trips. Each point was averaged over 31 macropulses. The measured spectral distribution (circles) is shown in Fig. 9. The spectral distributions obtained numerically were compared to the experimental results. The simulation was carried out like in the
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Fig. 10 (top) shows a sample of the timeresolved spectral distribution, obtained at the resonant wavelength lr ¼ 7:4 mm with a cavity desynchronization of DL=l ¼ 0:57: The intensity corresponds to the gray scale indicated in arbitrary units. The data acquisition system was synchronized with the accelerator triggering pulse. The zero position on the time-scale corresponds to the beginning of the macropulse. The electron beam was switched off after 7 ms:
Fig. 9. Spectral distribution of the laser radiation measured at the S-DALINAC (dashed line) and the results of simulations (solid line).
experiment for the first 200 round trips and the spectral distributions after each round trip were calculated. All spectral distributions were then averaged and the average is shown in Fig. 9 as a solid line. The results of the simulation show a smaller line width but are otherwise in fair agreement with the experiment. The small deviation e.g. at the longer wavelength might result from possible variations of the cavity length, since the measured spectral distribution is averaged over a large number of macropulses. There is also a possibility that the use of a 3D simulation code may explain this discrepancy. Summarizing, it is evident that the numerical simulation reproduce the following experimental data for the Darmstadt FEL with the uniform undulator well: the small signal gain vs. desynchronization, the power vs. desynchronization and the spectral distribution.
5.2. Limit-cycle regime The results presented in this section were obtained at FELIX with a pulsed electron beam having an energy of 43:6 MeV; a macropulse duration of 7 ms and a repetition rate of 5 Hz: The wavelength of the FEL radiation was set by changing the undulator gap. Time-resolved measurements of the laser spectrum were performed.
Fig. 10. A sample of a time-resolved spectral distribution measured at FELIX (top). The zero position on the time-scale corresponds to the beginning of the electron macropulse. The electron beam was switched off after 7 ms: The development of the intensity (middle) is obtained by integration of the timeresolved spectrum over the entire wavelength region. The spectral distribution averaged over time (bottom) is antisymmetric with a long tail at the longer wavelength side which together with the oscillation in the intensity points to the limitcycle regime.
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The time-resolved spectral distribution (Fig. 10, top) was measured in the wavelength range between 6600 and 7880 nm in steps of 20 nm: Laser generation started approximately after 3 ms and at 4 ms the FEL reached saturation. Vertical stripes presented on the time resolved spectral distribution were identified as absorption lines of water vapor. The integration of the time-resolved spectral distribution over the entire spectral range gives the time evolution of the total intensity which is shown in Fig. 10 (middle). One can see the oscillation of the intensity after saturation. This can be explained by the limit-cycle oscillation (Section 2) since at this wavelength FELIX operates in the short-pulse regime. The integration of the time-resolved spectral distribution over the entire macropulse duration gives the average spectral distribution of the FEL pulse, which is shown in Fig. 10 (bottom). The spectral distribution is asymmetric with the long tail at the longer-wavelength side which evidently also points to the presence of the limit-cycle oscillations. The experimentally observed development of the intensity within the macropulse, presented in Fig. 10 (middle) has been compared to the results of numerical simulations. As input parameters for the simulation the following values were chosen. The main technical parameters of the FELIX FEL are those listed in Table 1. The optical resonator length of the FELIX FEL amounts to 6:15 m which corresponds to the round-trip time of the optical pulse of 4:1 108 s: The macropulse with the length of 7 ms consists of 170 full round trips. The electron bunch length is equal to 0.4– 0:6 mm which corresponds to 57–85 lr : A comparison of the measured development of the intensity with its simulated results is presented in Fig. 11. The results of the simulation (dashed line) agree rather well with the experiment (solid line) with respect to the slope of the macropulse and the time necessary to reach the saturation. The period of the intensity oscillations obtained numerically also corresponds well to that observed experimentally. The more pronounced character of the oscillation in the case of the simulation can be explained by the fact that the simulation was performed
Fig. 11. Comparison of the macropulse evolution (solid line) measured at FELIX and its simulation (dashed line), wavelength lr ¼ 7 mm; cavity desynchronization Dl=l ¼ 0:57:
for an electron beam with the energy spread equal to zero. As mentioned in Section 2, the regime of the operation of the short pulse FEL influences not only the macropulse evolution but also the spectral distribution. In order to study this effect the spectral distributions were obtained for four different cavity desynchronizations similar to those presented in Fig. 10. The dependence of the spectral width (FWHM) from the cavity desynchronization is shown in Fig. 12. The circles represent the measured values. The solid line, corresponds to the values obtained by means of numerical simulations. A similar behavior of both experimental data and numerical results is observed. At the same time some discrepancy with respect to the absolute values of the spectral width is noticed. This difference might arise on the one hand from neglecting the energy spread of the electron beam within the simulations and on the other hand from the omission of 3D effects in the model chosen for the simulation code (see Section 3). 5.3. Other regimes of operation Moreover, the chaotic regime of the FEL operation, the limit-cycle oscillations as well as the transition to chaos via the period-doubling
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(c)
Fig. 12. Comparison of the measured (circles) relative spectral width (FWHM) at FELIX with the results of the simulation (solid line) for a cold electron beam for different values of cavity desynchronization.
cascade have been studied numerically. For this purpose the values of the electron beam current, the electron bunch length, the undulator parameters and the radiation wavelength were fixed. The value of the gain parameter has been set to t ¼ 2:9 which together with the 80 undulator periods and the radiation wavelength of 7 mm resulted in the branching parameters Nu l=sz ¼ 1:3 and Lc =sz ¼ 0:4: This point, marked (c) in Fig. 1, is located exactly on the border between the limitcycle and chaotic regime of an FEL operation on the Hahn–Lee phase diagram. The cavity desynchronization DL=l has been treated as the only variable. With these parameters the temporal evolution of the optical pulse intensity has been simulated. The results are shown in Fig. 13, where the intensity of the optical radiation is plotted as a function of time in units of the resonator roundtrips (RT). For the relatively strong cavity desynchronization DL=l ¼ 0:75 the intensity oscillates with the constant amplitude and period (Fig. 13a). These oscillations point to the limitcycle regime. With the decrease in the cavity desynchronization the nature of the oscillations changes significantly. The curve in Fig. 13b is an example of such an oscillation which corresponds to DL=l ¼ 0:5: Using the terminology of Hahn
(b)
(a)
Fig. 13. Different regimes of the FEL operation obtained numerically with the parameters corresponding to the point (c) in Fig. 1. For three different values of cavity desynchronization (i) DL=l ¼ 0:75 the limit-cycle regime (a), (ii) DL=l ¼ 0:5 period-two regime (b), and (iii) DL=l ¼ 0 chaotic behavior (c), are observed.
and Lee [8] these oscillations correspond to a period-two regime. Finally, at zero cavity desynchronization an irregular evolution of the optical intensity is observed. Higher optical intensities can be reached but at the expense of a low small-signal gain per pass and an unstable operation. This is the socalled chaotic regime of operation. To confirm this, simulations with different seedings for the modelling of the initial shot-noise in the electron beam have been performed. Three examples of the chaotic evolution of the optical intensity are shown in Fig. 13c. Apart from the initial seeding, all other input parameters for the simulation are equal. One can see that in this regime each
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macropulse develops independently, showing its own nature, which corresponds well to the theoretical predictions.
6. Conclusion Different regimes of a short pulse FEL were investigated experimentally (stable focus, limit cycle) and numerically (stable focus, limit cycle, chaotic and period-two). The measurements were carried out on two different short-pulse FELs, in Darmstadt and at FELIX, at almost the same electron bunch length and at the same wavelength, but with different electron beam currents. The results of the simulations with the 1D time dependent code are in a good agreement with the results observed. The dependencies of all main FEL characteristics such as small signal gain, average power and spectral width from the cavity desynchronization correspond well to the experimentally observed ones in the case of a uniform undulator.
Acknowledgements We are extremely grateful to E.L. Saldin, E.A. Schneidmiller, and M.V. Yurkov for providing us with the simulation code FAST-OSC and many useful discussions regarding the interpretation of the results. We also gratefully acknowledge the assistance of Dr. Lex van der Meer, Dr. Giel Berden and Dipl.-Phys. Maria Grigore during the measurements at FELIX. This work has been supported by the Graduiertenkolleg ‘‘Physik und Technik von Beschleunigern’’ under the Number 410/2-01 and the DFG Contract FOR 272/2-1 ‘‘Untersuchung von Vielteilcheneffekten in Atomkernen mit Elektronen und Photonen am S-DALINAC und die damit verbundene Weiterentwicklung des Beschleunigers’’. One of us (V.A.) would like to thank the Alexander von Humboldt Foundation for providing a fellowship.
References [1] W.B. Colson, Classical free electron laser theory, in: W.B. Colson, C. Pellegrini, A. Renieri (Eds.), Laser Handbook, Vol. 6, Elsevier Science Publishers B.V., Amsterdam, 1990. [2] A. Richter, Physikalische Bl.atter 54 (1998) 31. [3] A. Richter, Operational experience at the S-DALINAC, in: S. Meyers et al. (Ed.), Proceedings of the Fifth EPAC, IOP Publishers, Bristol, 1996. [4] www.rijnh.nl. [5] N. Piovella, P. Chaix, G. Shvets, D.A. Jaroszynski, Phys. Rev. E 52 (1995) 5470. [6] G.M.H. Knippels, A.F.G. van der Meer, R.F.X.A.M. Mols, D. Oepts, P.W. van Amersfoort, Phys. Rev. E 53 (1996) 2778. [7] S.J. Hahn, J.K. Lee, Phys. Rev. E 48 (1993) 2162. [8] S.J. Hahn, J.K. Lee, Phys. Lett. A 176 (1993) 339. [9] E.H. Park, J.K. Lee, T.H. Chung, Nucl. Instr. and Meth. A 358 (1995) 448. [10] S. Kawasaki, H. Ishizuka, T. Koarai, A. Watanabe, M. Shiho, Nucl. Instr. and Meth. A 358 (1995) 114. [11] P. Chaix, N. Piovella, G. Gr!egoire, Phys. Rev. E 59 (1999) 1136. [12] D.A. Jaroszynski, D. Oepts, A.F.G. van der Meer, P. Chaix, Nucl. Instr. and Meth. A 407 (1995) 407. [13] P. Chaix, P. Guimbal, Nucl. Instr. and Meth. A 358 (1995) 96. [14] W.P. Leemans, M.E. Conde, R. Govil, B. van der Geer, M. de Loos, H.A. Schwettman, T.I. Smith, R.L. Swent, Nucl. Instr. and Meth. A 358 (1995) 208. [15] R. Hajima, N. Nishimori, R. Nagai, E.J. Minehara, Nucl. Instr. and Meth. A 475 (2001) 270. [16] M.E. Couprie, Nucl. Instr. and Meth. A 507 (2003) 1 and references therein. [17] E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov, The Physics of Free Electron Lasers, Springer, Berlin, 2000. [18] E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov, FASTOSC—one-dimensional, time-dependent simulation code for simulations of an FEL oscillator (private communication). [19] E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov, Opt. Commun. 148 (1998) 383. [20] B. Faatz, R.W.B. Best, D. Oepts, P.W. van Amersfoort, IEEE J. Quantum Electron QE-29 (1993) 2229. . [21] M. Brunken, S. Dobert, R. Eichhorn, H. Genz, H.-D. Gr.af, H. Loos, A. Richter, B. Schweizer, A. Stascheck, T. Wesp, Nucl. Instr. and Meth. A 429 (1999) 21. [22] H. Genz, K. Gerstner, H.-D. Gr.af, R. Hahn, H. Jennewein, A. Richter, B. Schweizer, T. Tschudi, Nucl. Instr. and Meth. A 498 (2003) 22. [23] K. Alrutz-Ziemssen, J. Auerhammer, H. Genz, H.-D. . Gr.af, A. Richter, J. Topper, H. Weise, Nucl. Instr. and Meth. A 304 (1991) 300. [24] P.W. van Amersfoort, et al., Nucl. Instr. and Meth. A 318 (1992) 42. [25] M. Brunken, L. Casper, H. Genz, C. Hessler, S. Khodyachykh, A. Richter, Opt. Laser Technol. 35 (2003) 349. [26] T. Wesp, Dissertation D17, TU Darmstadt, 1998.
Nuclear Instruments and Methods in Physics Research A 530 (2004) 194–204
Experimental and numerical study of short pulse effects in FELs S. Khodyachykha, M. Brunkena, H. Genza, C. Hesslera, A. Richtera,*, V. Asgekarb a
Institut fur . Kernphysik, Technische Universitat . Darmstadt, Schlossgartenstr. 9, D-64289 Darmstadt, Germany b Department of Physics, University of Pune, Pune 411 007, Maharashtra, India Received 27 January 2004; accepted 9 April 2004 Available online 15 June 2004
Abstract We report the experimental and numerical investigations of the influence of short pulse effects occurring in FELs in different operational regimes for electron bunch lengths which are of the order of the slippage distance. Several observables such as the small signal gain, the macropulse power and the spectral distribution of the FEL radiation were determined experimentally within the constraints of the stable focus regime at the infrared FEL at the S-DALINAC and for the limit cycle regime at the Dutch near infrared FEL FELIX. The experimental findings were compared to predictions of numerical simulations based on the 1D time dependent code FEL1D-OSC. The agreement between experiment and simulation is good. Furthermore, the simulations reveal a chaotic behavior of the macropulses for specific values of the slippage as well as period-doubling, two effects that are predicted to show up in the spectral distribution. r 2004 Elsevier B.V. All rights reserved. PACS: 41.60.Cr Keywords: Free electron laser; Short pulse effects; Numerical simulations
1. Introduction For the successful operation of the FEL the electron beam has to fulfil the following requirements. First, for the infrared or visible spectral regions the typical undulator period of a few centimeters and an undulator K-value of about *Corresponding author. Tel.: +49-6151-16-2116; fax: +496151-16-4321. E-mail address: [email protected] (A. Richter).
one require a relativistic electron beam energy. At the same time, since the FEL gain is proportional to the beam current density, a high value of the latter is required. Finally, the energy spread should be small compared to the width of the gain curve [1]. Based on the above specified criteria, most FEL facilities operating in the infrared region use radio-frequency (RF) linear accelerators. This type of accelerators produce a train of short picosecond pulses. The light wave produced
0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.04.215
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inside the resonator shows also this structure and thus consists also of a train of short micropulses. During the interaction of the electrons with the photon beam, the electrons slip back relative to the optical pulse for two reasons: (i) the electron velocity is always smaller than the velocity of light and (ii) the electrons travel along the oscillating trajectories inside the undulator. The resonance condition requires that the electron bunch slips back by one optical wavelength l per one undulator period. At the exit of the undulator, consisting of Nu periods, the electron has slipped back by a slippage distance of Nu l: If this quantity is comparable to the electron bunch length sz ; one defines such an FEL as short pulse FEL. In the present paper we investigate how the formation of short pulses, which occur under specific conditions in RF driven free electron lasers, affects the observables of an FEL. We report on the experimental determination of the small signal gain, the macropulse power and the spectral distribution of the FEL radiation for different values of the cavity desynchronization performed at the Darmstadt Infrared FEL [2] driven by the superconducting Darmstadt linear electron accelerator S-DALINAC [3]. The experimental results are compared to the predictions obtained from simulations using the 1D time dependent code FEL1D-OSC which had to be extended to become applicable for the current FEL configuration. We also compare the numerical results to the experiment we performed at the Dutch free electron laser FELIX [4] for several observables which have already been tested in former experiments [5,6] and which could be described correctly using an analytical model [5]. This paper is structured as follows. After the short theoretical description of the problem in Section 2, the numerical simulation algorithm and its limitations are discussed in Section 3. The FEL at the S-DALINAC and FELIX together with the related experimental areas are described in Section 4 and the experimental results obtained there are compared with the numerical results in Section 5. It is followed by a conclusion.
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2. Theoretical background With respect to the FEL operation one distinguishes between the small signal regime (when the power of the laser radiation grows) and the saturated or steady state regime. In the small signal regime the amplification of the radiation takes place towards the downstream end of the undulator where the electrons have slipped back relative to the front of the optical micropulse and therefore the optical pulse peaks up at its rear. This has the consequence that the effective optical group velocity is smaller than its vacuum value. In a perfectly synchronized resonator the micropulse will continue to narrow during the subsequent round trips. The reduced overlap between electron bunch and optical pulse will in turn reduce the gain. This effect is known as ‘‘laser lethargy’’. In order to restore the gain it is necessary to slightly reduce the cavity length from the value giving the perfect synchronization. For convenience the cavity-length change DL is expressed usually in units of the optical wavelength DL=l: Hahn and Lee [7,8] have shown that the various regimes of FEL operation can be identified for the set of parameters DL=l; Nu l=sz and Lc =sz : The last parameter contains the cooperation length Lc ¼ l=4pr which is equal to the slippage in one gain length, with r being the fundamental gain parameter. The phase diagram [7] for different regimes of the FEL operation is reproduced in Fig. 1. For a constant value of Lc =sz the regimes change with increase in Nu l=sz from the stable focus (when no subpulses develop) to the limit cycle region with sidebands in the spectrum. These sidebands double sequentially (period-doubling). Finally the chaotic regime starts. There is a number of papers which are dealing with chaos in FEL (see e.g. [9–16]). In the chaotic regime each macropulse evolves in a different way which can also be observed e.g. in the spectral distribution. To reach the limit-cycle and chaotic regime, the intracavity power has to saturate at a sufficiently higher level. In the present paper we report the experimental observation of various quantities in two different regimes, namely the stable focus and the limitcycle, carried out at the S-DALINAC and FELIX.
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Fig. 1. Phase diagram [7] for different regimes of operation of an FEL oscillator. The quantities Nu l and sz are the slippage distance and the electron bunch length, respectively, and Lc ¼ l=4pr: Lines corresponding to three different cavity desynchronizations separate three different regimes of operation namely stable focus, limit-cycle and chaotic regimes. The points a; b and c correspond to three cases studied in the present paper.
These experimental results are compared to the predictions of the numerical simulations. Furthermore, the chaotic behavior and period-doubling were observed in our simulations. The parameters of the experiments and the input parameters for the simulations correspond to the points a; b; and c plotted in the phase diagram (Fig. 1) for the experiments at the S-DALINAC FEL, at FELIX, and for the numerical simulations, respectively. Point a was obtained using the parameters for the simulation of the spectral distribution (see Section 5). For convenience, instead of r we use for our simulations the gain parameter for an FEL oscillator t defined in Ref. [17]. The relation between t and r is given by t ¼ 4prNu :
3. Numerical simulation algorithm To model the FEL oscillator and its light output, the equations of the electron motion have to be solved together with Maxwell’s equation taking into account the boundary conditions for the electron beam and for the electromagnetic radiation. It is assumed that the lasing process starts from the shot noise in the electron beam.
The linear regime of the FEL operation (the small signal regime) can be described analytically (see e.g. Ref. [17]) very accurately. For the description of the non-linear regime when the FEL reaches saturation, which is the subject of interest of the present work, there are no analytical solutions, that is why it was necessary to use numerical methods. Simulations have been performed with the onedimensional, time-dependent code FAST-OSC designed by Saldin, Schneidmiller and Yurkov [18]. This is a one-dimensional, time-dependent code which allows one to study the time evolution of intensity of the optical macropulse generated by the FEL oscillator, which is considered as an FEL amplifier with feedback, with variable undulator parameters from the shot noise up to saturation. Any longitudinal profile of the electron bunch can be simulated. The core of the code FAST-OSC is code FAST designed for simulation of an FEL amplifier starting from shot noise in the electron beam [19]. During the first pass of the electron beam through the undulator it produces a radiation pulse. The amplitude of the radiation pulse is reduced according to the value of round-trip losses (the phase change of the electromagnetic field after the reflection from the resonator mirrors is neglected because this effect does not influence the FEL-oscillator operation [17]). The radiation pulse, shifted by the according value of the cavity desynchronization and the next electron bunch with the initial noise distribution are fed to the undulator entrance, and the calculation procedure is repeated. The start-up from the shot noise in the electron beam has been thoroughly tested with analytical results given in Ref. [17]. Since the underlying model is one-dimensional, the interactions modifying the transverse Gaussian profile of the electron or optical beam are not taken into account. The code is valid only for an optical mode which has a waist larger than the electron beam radius, which is the case for both, the Darmstadt FEL and FELIX. For FEL oscillators with high gain the optical field is strongly deformed since the radiation of the electron beam is almost as strong as that inside the resonator. Thus, for the case of a high gain FEL a 3D model should be used to describe
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satisfactorily the FEL behavior. In case of the Darmstadt FEL, however, the small signal gain g is limited to about 5% which allows a 1D treatment. Furthermore, short-pulse effects are largely longitudinal in nature and that is why the FEL can be modelled adequately in a 1D approximation. The hole-coupling of the optical beam which is often used in FELs also deforms the optical field inside the resonator [20]. To study the FEL dynamics in a hole-coupled FEL a 3D code is preferable. The Darmstadt FEL uses multi-layered partially reflective mirrors which simplifies this task. However, the results presented in Section 5.2 show a satisfactory agreement between the results of the 1D simulation and the experiment carried out at FELIX which uses hole-coupling for the extraction of the optical beam.
4. Experiments In this section the experimental set ups at the SDALINAC FEL and FELIX are described briefly. Both FELs are operating in the infrared region of wavelength and have mostly similar parameters. However, one should mention two major differences between them. First, the peak current value in the case of the FELIX is about one order of magnitude higher than at the S-DALINAC FEL which also allows to operate FELIX in the limitcycle regime (Fig. 1). The second fundamental difference between two FELs is that contrary to FELIX, the S-DALINAC FEL is driven by a superconducting linear electron accelerator which allows to generate much longer optical macropulses and in general even to obtain an optical cw beam.
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beam with energies between 2.5 and 130 MeV can be produced [3]. Fig. 2 shows the layout of the S-DALINAC. The FEL is located in the space between the main linac and the first beam recirculation path. From the latter electron beams with an energy between 25 and 50 MeV can be bent over and passed through a 2:6 m long undulator. The latter is built up from 80 periods each having a length of 3:2 cm and has a variable gap width between 16 and 25 mm: The emitted photon pulses are stored in a 15 m long resonator [22] made out of two concave mirrors having reflectivities of 99.0% and 99.8%, respectively. Inside the undulator the amplification process takes place so that an infrared laser beam with a wavelength between 3 and 10 mm is generated. The tuning of the wavelength can be accomplished by varying the electron energy and/ or the undulator gap. The laser beam is extracted by the mirror at the downstream (DS) table of the FEL. Since the amplification of the optical pulse depends strongly on the peak current of the electron bunch inside the undulator, a system based on a 10 MHz sub-harmonic injection has been designed and built [23]. In this case only every 300th high frequency period is filled with electrons but with an increased bunch charge. The bunch length of 2 ps leads to a peak current of 2:7 A which, in turn implies a small signal gain of 2–5%. The time structure of the laser light has the
4.1. S-DALINAC and the FEL The superconducting Darmstadt linear electron accelerator S-DALINAC is since 1991 in operation at the Institute of Nuclear Physics of the Darmstadt University of Technology. It is used for experiments on nuclear and radiation physics and since 1996 also as a driver for the infrared FEL [2,21]. Using the high-frequency accelerating method and dual beam recirculation, a continuous
Fig. 2. The superconducting Darmstadt linear electron accelerator S-DALINAC with the FEL. (1) Electron gun, (2) superconducting 10 MeV injector, (3) superconducting 40 MeV accelerator, (4) undulator, (5) upstream and (6) downstream tables with mirror chambers, (7) transfer system to the optical laboratory located one floor above (not shown).
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structure of the electron beam and thus, short laser pulses of about 2 ps can be obtained.
designed. For the experiments in the optical laboratory the transfer system (6) is used. The results obtained with this experimental set up are presented in Section 5.1.
4.2. Experiment at the S-DALINAC FEL Fig. 3 shows the experimental set up mounted on the DS table in the accelerator hall. With this set up both, spontaneous and laser radiation can be diagnosed. The radiation, outcoupled through a multi-layer dielectric mirror (1) can either be transferred to the optics laboratory for further experiments or can be diagnosed directly at the DS-table. For the diagnostics on the DS-table a system for the measurement of the spectral distribution of the spontaneous and stimulated radiation consisting of a monochromator (2) and a nitrogen-cooled HgCdTe-detector (3) has been developed. The temporal resolution of the detector is about 100 ns so that the development of the intensity within the macropulse can be measured. It should be mentioned that the detector reaches its saturation at a power of about 100 mW: For the measurements above the lasing threshold additional attenuation is necessary. Alternatively the average power can be measured by a thermopile detector (4) which has a temporal resolution of about a few seconds. Two sets of filters (5) are used for the attenuation of the laser radiation or for the harmonics separation. Because of its lower power the spontaneous emission cannot be detected after passing through the outcoupling mirror. For this measurement a special bypass (7) has been
4.3. FELIX The Free Electron Laser for Infrared eXperiments (FELIX) in the Netherlands [4] went into operation in 1991 as the first infrared FEL that attained lasing in Europe [24]. Fig. 4 presents an overview of the FELIX layout. An electron beam with an energy of 3:8 MeV is produced by the injector. Two 3 GHz normal conducting linacs are used to increase the beam energy from 15 to 25 MeV and from 25 to 45 MeV; respectively. Each linac delivers an electron beam that consists of a train of 3–6 ps long bunches that can be introduced into an undulator. Both undulators have 38 periods of 6:5 cm length. Two gold-coated mirrors are placed at opposite sides of the undulator and define the 6 m long resonator. This allows to obtain 40 independent micropulses in the cavity simultaneously. An aperture in the mirror at the left side provides an outcoupling of a fraction of the optical radiation. The position of the mirror on the right side can be adjusted in order to synchronize the circulating optical pulses with electron bunches. After passing through the undulator, the electrons are bent out of the resonator and dumped. The laser typically provides several kW of average output power during the macropulse, with a maximum of 20 kW; corresponding to a micropulse energy of 20 mJ: Table 1 summarizes the main parameters of the S-DALINAC FEL and FELIX.
FEL-2
5 - 35 µm Injector
Fig. 3. Diagnostic elements on the DS table of the Darmstadt FEL: (1) resonator mirror, (2) monochromator, (3) HgCdTethermo resistor, (4) set up for measurement of the total intensity, (5) sets of filters, (6) transfer telescope, (7) bypass for the spontaneous emission measurements.
Linac 1 14 - 25 MeV
L inac 2 25 - 46 MeV
FEL 1
16 - 110 µm 6m
Fig. 4. Layout of FELIX.
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Table 1 Parameters of the S-DALINAC IR FEL and FELIX Parameter
S-DALINAC
FELIX
Electron energy (MeV) Energy spread (%) Electron peak current (A) Undulator period length (cm) Number of periods Undulator parameter K Wavelength ðmmÞ Optical resonator losses (%) Resonator length (m) Repetition rate (MHz) Macropulse duration ðmsÞ
25–50 o0:3 2.7 3.2 80 0.45–1.12 3–10 0.9 15.00 10 2000–cw
14–46 0.2 27 6.5 38 o1:9 5–110 5–10 6.15 1000 (25) o15
4.4. Experiment at FELIX For the experiment at FELIX IR light with wavelengths in the region between 5.7 and 9 mm was delivered to the experimental hall through an evacuated transfer system. The wavelength has been tuned by changing the undulator gap keeping the electron energy fixed at 43:6 MeV: The experimental set up is shown in Fig. 5. Collinearly propagating to the FEL beam (1), a HeNe-laser beam (2) was used for adjustment of the detection devices during the preparation of the experiment. Propagating along the optical table, the FEL beam can be directed either to a second-harmonic autocorrelator (3) described in Ref. [25] or to a monochromator (4) with the help of flip-mirrors (5). Using the monochromator and placing a HgCdTe-detector (6) behind it allows to measure the time-resolved spectral distributions of the FEL within the macropulse. Some results of the measurements are presented in Section 5.2.
5. Experimental and numerical observation of different regimes of operation In this section we report on the experimental determination of several observables which are compared to the results of numerical simulations. The comparison includes the investigation of the
Fig. 5. Experimental set up at FELIX. Collinearly propagating to the FEL beam (1) is a HeNe-laser beam (2) used for the adjustment of the measuring devices during the preparation of the experiment. Propagating along the optical table, the FEL beam can be directed either to a (3) second-harmonic autocorrelator or to a monochromator (4) with the help of flip-mirrors (5). Using the monochromator and placing a HgCdTe-detector (6) behind it allows to measure the timeresolved spectral distributions of the FEL within the macropulse.
macropulse power, the small signal gain and the spectral distribution. 5.1. Stable-focus regime The experimental data analyzed in this section were obtained [26] at the S-DALINAC. A pulsed electron beam with the macropulse length of 4 ms; a repetition rate of 31 Hz and an energy of 31:04 MeV was used. The electron beam radius inside the undulator was assumed to vary between 0.4 and 0:6 mm: Together with the undulator parameter K ¼ 1:106 this results in the resonant wavelength of lr ¼ 7:03 mm and in the dimensionless gain parameter t between 0.3 and 0.7. Fig. 6 shows an example of the development of the laser intensity within the macropulse. Saturation of the laser power was reached within the first 400 ms: These measurements were carried out for various cavity desynchronizations, five times for each setting. From the growth of the macropulse the value of the small signal gain g can be extracted. To reduce the error, the following procedure was chosen for each cavity setting. First, the time was converted
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Fig. 6. Typical development of the laser intensity measured within the macropulse at the S-DALINAC. The pulsed electron beam had an energy of 31:04 MeV; a pulse length of 4 ms and a repetition rate of 31 Hz:
into the number of the resonator round-trips. The leading edge of the macropulse was then plotted on a logarithmic scale and a linear function yi ¼ ai þ gi t; i ¼ 1; y; 5 was fitted to the linear regime. For the values of gi with the P corresponding P errors s2i the weighted mean g# ¼ 5i¼1 ðgi =s2i Þ= 5i¼1 ð1=s2i Þ # ¼ hasP been calculated with an error of s2 ðgÞ 1= 5i¼1 ð1=s2i Þ: The results are presented in Fig. 7 for different values of the cavity desynchronization. The circles with error bars represent the measured data. For the experimental parameters listed above the development of the intensity within the macropulses has been numerically simulated and the information about the small signal gain has been extracted using the same procedure as for the experimental data. The solid line is a fit to the values obtained from the simulations. Good agreement is observed between the experimental data and numerical results. Simultaneously to the time-resolved measurements, the average macropulse power was recorded by a thermopile detector for each value of the cavity desynchronization. In Fig. 8 the results of the numerical simulations are compared to the results of these measurements, and indeed both are in a good agreement. Finally, for a cavity desynchronization of DL=l ¼ 0:05; which corresponds to the max-
Fig. 7. Small signal gain as a function of the cavity desynchronization. Open circles represent the experimental results obtained at the Darmstadt FEL with the beam parameters listed in the caption of Fig. 6. The solid curve is a fit to the data from the simulation.
Fig. 8. Average macropulse power as a function of cavity desynchronization measured at the S-DALINAC (circles) compared with the results of the simulation (solid line).
imum value of the intensity the spectral distribution of the radiation was measured at the beginning of the macropulse with the help of the monochromator and the HgCdTe-detector. The spectral resolution was 25 nm: The time gate was set for 20 ms which is equal to 200 round trips. Each point was averaged over 31 macropulses. The measured spectral distribution (circles) is shown in Fig. 9. The spectral distributions obtained numerically were compared to the experimental results. The simulation was carried out like in the
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Fig. 10 (top) shows a sample of the timeresolved spectral distribution, obtained at the resonant wavelength lr ¼ 7:4 mm with a cavity desynchronization of DL=l ¼ 0:57: The intensity corresponds to the gray scale indicated in arbitrary units. The data acquisition system was synchronized with the accelerator triggering pulse. The zero position on the time-scale corresponds to the beginning of the macropulse. The electron beam was switched off after 7 ms:
Fig. 9. Spectral distribution of the laser radiation measured at the S-DALINAC (dashed line) and the results of simulations (solid line).
experiment for the first 200 round trips and the spectral distributions after each round trip were calculated. All spectral distributions were then averaged and the average is shown in Fig. 9 as a solid line. The results of the simulation show a smaller line width but are otherwise in fair agreement with the experiment. The small deviation e.g. at the longer wavelength might result from possible variations of the cavity length, since the measured spectral distribution is averaged over a large number of macropulses. There is also a possibility that the use of a 3D simulation code may explain this discrepancy. Summarizing, it is evident that the numerical simulation reproduce the following experimental data for the Darmstadt FEL with the uniform undulator well: the small signal gain vs. desynchronization, the power vs. desynchronization and the spectral distribution.
5.2. Limit-cycle regime The results presented in this section were obtained at FELIX with a pulsed electron beam having an energy of 43:6 MeV; a macropulse duration of 7 ms and a repetition rate of 5 Hz: The wavelength of the FEL radiation was set by changing the undulator gap. Time-resolved measurements of the laser spectrum were performed.
Fig. 10. A sample of a time-resolved spectral distribution measured at FELIX (top). The zero position on the time-scale corresponds to the beginning of the electron macropulse. The electron beam was switched off after 7 ms: The development of the intensity (middle) is obtained by integration of the timeresolved spectrum over the entire wavelength region. The spectral distribution averaged over time (bottom) is antisymmetric with a long tail at the longer wavelength side which together with the oscillation in the intensity points to the limitcycle regime.
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The time-resolved spectral distribution (Fig. 10, top) was measured in the wavelength range between 6600 and 7880 nm in steps of 20 nm: Laser generation started approximately after 3 ms and at 4 ms the FEL reached saturation. Vertical stripes presented on the time resolved spectral distribution were identified as absorption lines of water vapor. The integration of the time-resolved spectral distribution over the entire spectral range gives the time evolution of the total intensity which is shown in Fig. 10 (middle). One can see the oscillation of the intensity after saturation. This can be explained by the limit-cycle oscillation (Section 2) since at this wavelength FELIX operates in the short-pulse regime. The integration of the time-resolved spectral distribution over the entire macropulse duration gives the average spectral distribution of the FEL pulse, which is shown in Fig. 10 (bottom). The spectral distribution is asymmetric with the long tail at the longer-wavelength side which evidently also points to the presence of the limit-cycle oscillations. The experimentally observed development of the intensity within the macropulse, presented in Fig. 10 (middle) has been compared to the results of numerical simulations. As input parameters for the simulation the following values were chosen. The main technical parameters of the FELIX FEL are those listed in Table 1. The optical resonator length of the FELIX FEL amounts to 6:15 m which corresponds to the round-trip time of the optical pulse of 4:1 108 s: The macropulse with the length of 7 ms consists of 170 full round trips. The electron bunch length is equal to 0.4– 0:6 mm which corresponds to 57–85 lr : A comparison of the measured development of the intensity with its simulated results is presented in Fig. 11. The results of the simulation (dashed line) agree rather well with the experiment (solid line) with respect to the slope of the macropulse and the time necessary to reach the saturation. The period of the intensity oscillations obtained numerically also corresponds well to that observed experimentally. The more pronounced character of the oscillation in the case of the simulation can be explained by the fact that the simulation was performed
Fig. 11. Comparison of the macropulse evolution (solid line) measured at FELIX and its simulation (dashed line), wavelength lr ¼ 7 mm; cavity desynchronization Dl=l ¼ 0:57:
for an electron beam with the energy spread equal to zero. As mentioned in Section 2, the regime of the operation of the short pulse FEL influences not only the macropulse evolution but also the spectral distribution. In order to study this effect the spectral distributions were obtained for four different cavity desynchronizations similar to those presented in Fig. 10. The dependence of the spectral width (FWHM) from the cavity desynchronization is shown in Fig. 12. The circles represent the measured values. The solid line, corresponds to the values obtained by means of numerical simulations. A similar behavior of both experimental data and numerical results is observed. At the same time some discrepancy with respect to the absolute values of the spectral width is noticed. This difference might arise on the one hand from neglecting the energy spread of the electron beam within the simulations and on the other hand from the omission of 3D effects in the model chosen for the simulation code (see Section 3). 5.3. Other regimes of operation Moreover, the chaotic regime of the FEL operation, the limit-cycle oscillations as well as the transition to chaos via the period-doubling
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(c)
Fig. 12. Comparison of the measured (circles) relative spectral width (FWHM) at FELIX with the results of the simulation (solid line) for a cold electron beam for different values of cavity desynchronization.
cascade have been studied numerically. For this purpose the values of the electron beam current, the electron bunch length, the undulator parameters and the radiation wavelength were fixed. The value of the gain parameter has been set to t ¼ 2:9 which together with the 80 undulator periods and the radiation wavelength of 7 mm resulted in the branching parameters Nu l=sz ¼ 1:3 and Lc =sz ¼ 0:4: This point, marked (c) in Fig. 1, is located exactly on the border between the limitcycle and chaotic regime of an FEL operation on the Hahn–Lee phase diagram. The cavity desynchronization DL=l has been treated as the only variable. With these parameters the temporal evolution of the optical pulse intensity has been simulated. The results are shown in Fig. 13, where the intensity of the optical radiation is plotted as a function of time in units of the resonator roundtrips (RT). For the relatively strong cavity desynchronization DL=l ¼ 0:75 the intensity oscillates with the constant amplitude and period (Fig. 13a). These oscillations point to the limitcycle regime. With the decrease in the cavity desynchronization the nature of the oscillations changes significantly. The curve in Fig. 13b is an example of such an oscillation which corresponds to DL=l ¼ 0:5: Using the terminology of Hahn
(b)
(a)
Fig. 13. Different regimes of the FEL operation obtained numerically with the parameters corresponding to the point (c) in Fig. 1. For three different values of cavity desynchronization (i) DL=l ¼ 0:75 the limit-cycle regime (a), (ii) DL=l ¼ 0:5 period-two regime (b), and (iii) DL=l ¼ 0 chaotic behavior (c), are observed.
and Lee [8] these oscillations correspond to a period-two regime. Finally, at zero cavity desynchronization an irregular evolution of the optical intensity is observed. Higher optical intensities can be reached but at the expense of a low small-signal gain per pass and an unstable operation. This is the socalled chaotic regime of operation. To confirm this, simulations with different seedings for the modelling of the initial shot-noise in the electron beam have been performed. Three examples of the chaotic evolution of the optical intensity are shown in Fig. 13c. Apart from the initial seeding, all other input parameters for the simulation are equal. One can see that in this regime each
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macropulse develops independently, showing its own nature, which corresponds well to the theoretical predictions.
6. Conclusion Different regimes of a short pulse FEL were investigated experimentally (stable focus, limit cycle) and numerically (stable focus, limit cycle, chaotic and period-two). The measurements were carried out on two different short-pulse FELs, in Darmstadt and at FELIX, at almost the same electron bunch length and at the same wavelength, but with different electron beam currents. The results of the simulations with the 1D time dependent code are in a good agreement with the results observed. The dependencies of all main FEL characteristics such as small signal gain, average power and spectral width from the cavity desynchronization correspond well to the experimentally observed ones in the case of a uniform undulator.
Acknowledgements We are extremely grateful to E.L. Saldin, E.A. Schneidmiller, and M.V. Yurkov for providing us with the simulation code FAST-OSC and many useful discussions regarding the interpretation of the results. We also gratefully acknowledge the assistance of Dr. Lex van der Meer, Dr. Giel Berden and Dipl.-Phys. Maria Grigore during the measurements at FELIX. This work has been supported by the Graduiertenkolleg ‘‘Physik und Technik von Beschleunigern’’ under the Number 410/2-01 and the DFG Contract FOR 272/2-1 ‘‘Untersuchung von Vielteilcheneffekten in Atomkernen mit Elektronen und Photonen am S-DALINAC und die damit verbundene Weiterentwicklung des Beschleunigers’’. One of us (V.A.) would like to thank the Alexander von Humboldt Foundation for providing a fellowship.
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