Observation of Self-Amplified Spontaneous Emission in the Near-Infrared and Visible*
M. Babzien a, I. Ben-Zvi a, P. Catravas b, J-M. Fang c, T.C. Marshall c, X.J. Wang a, J.S. Wurtele d, V. Yakimenko a, L.H. Yu
a. National Synchrotron Light Source, Brookhaven National Laboratory, Upton, NY 11973
b. Massachusetts Institute of Technology, Cambridge, MA 02139
c. Department of Applied Physics, Columbia University, New York, NY 10027
d. Department of Physics, University of California at Berkeley, Berkeley, CA 94720
We report evidence of self-amplified spontaneous emission at 1064 nm and 633 nm. Single pass, on-axis microwiggler emissions into a 25 nm bandwidth have been recorded as a function of beam charge, and show a clear enhancement over spontaneous emission after threshold. These are the first measurements of SASE at such a short wavelength, and employ the smallest period wiggler, 8.8 mm, used to date in a successful SASE experiment. The experiments are being performed with the MIT microwiggler at the Accelerator Test Facility at BNL. For the 1064 nm measurement, a single micropulse at 34 MeV with a variable charge of 0 to 1 nC and <5 ps FWHM bunch length is passed through the microwiggler and emissions into a limited solid angle and bandwidth, selected by an aperture and interference filter, are focused onto a silicon photodiode. Enhancement of the emissions from 2 to 6 times the spontaneous emission level is observed at the highest charges. In addition, we observe SASE gain at a wavelength of 633 nm at a beam energy of 48 MeV, without detailed measurements.
PACS numbers: 41.60.Cr
* This work performed under the auspices of the U.S. Department of Energy, under contracts DE-AC02-76CH00016 and 02-91ER40669, and by the Office of Naval Research, Grant N00014-90-J-4130.
Understanding the physics of self amplified spontaneous emission (SASE) in a free-electron laser (FEL) has been a subject of considerable theoretical, and of late experimental, effort in recent years. This research has been motivated by the possibility of using an FEL operating in the SASE mode to produce a high-brightness x-rays. In the SASE mode of operation, a high-current electron beam propagates through a long wiggler and amplifies its own spontaneous emission. The physics of a SASE FEL is not sufficiently characterized and has only been demonstrated at wavelengths separated by up to four orders of magnitude from proposed devices [1,2].
Initial experiments were conducted at microwave frequencies and more recent experiments have been reported at 12 to 5 microns [2-8]. Moving studies toward shorter wavelengths is technically challenging because of tighter demands on the quality of the system. Longer wavelength experiments are favored by higher gain and less sensitivity to emittance and energy spread. Experiments at longer wavelength have noted that the amplified noise is greater than would be expected from shot-noise theory. In the microwave region this not surprising since the frequency separation between microwaves, the beam pipe cut-off (characteristic of wakefields), and beam plasma frequency is not great. In far infrared wavelength experiments the ratio of wavelength to bunchlength can provide coherent enhancement of spontaneous emission without SASE gain. Levels of coherent enhancement of four orders of magnitude have been reported by several groups [9,10] working in the far infrared. These experiments, therefore, cannot study the growth from spontaneous emission---which is the regime that the x-ray experiments are expected to operate startup requirements for their experiments.
We report a first demonstration of the startup of SASE at 1 micron and preliminary results at 632 nm, the first observation of its kind in the visible wavelengths. We have performed a check, using transition radiation, for coherent spontaneous emission and find none. Furthermore, while previous work generally made use of long period wigglers, having periods on the order of 5 cm, we have used a microwiggler in these experiments. This permitted the measurement at 632 nm to occur at a lower beam energy than the recent studies  at 5 microns. Our experimental results are consistent with theoretical and numerical predictions.
This paper is organized as follows. In Section 2 we describe the experimental set-up, in Section 3 we present the measurements, in Section 4 we briefly describe the theory and compare with experiments and in Section 5 we present conclusions.
2. Experimental Setup
The electron beam at the Brookhaven Accelerator Test Facility is produced by a high-brightness photocathode RF gun  and accelerated up to 60 MeV by two S-band linac sections. The drive laser illuminating the magnesium cathode has a pulse duration of 10 ps FWHM. The transport line consists of three 20° dipole magnets and multiple quadrupole magnets. The dispersive section between the first two dipoles enables measurement of beam energy spread, and using a collimating slit, control of beam charge. The nominal beam parameters are 0.15% rms energy spread and 2 ´ 10-6 m× rad rms beam emittance, (the exact values depend on other beam parameters and will be given below). Diagnostics include phosphor coated flags, stripline beam position monitors, and Faraday cups. A more complete discussion of the ATF can be found in Ref. 5.
The MIT microwiggler is a tunable, pulsed electromagnet with a period of 8.8 mm, which provides a peak on-axis field of 0.45 Tesla. Sixty one periods were used for these experiments. Each of the half periods is individually adjustable, and a novel tuning regimen is employed which consistently provides rms spreads in the peak field of better than 0.1%, thus providing the field quality essential for research in short wavelength generation. Further details of the MIT microwiggler construction and capabilities have been published elsewhere.[13,14]
A schematic of the experiment components is shown in Fig. 1. Located on either side of the wiggler are two multifunction diagnostic ports. Designed to provide a means for precision co-alignment to the wiggler axis and electron beam trajectory, these ports employ three-position pneumatic translators. One position places a phosphor screen on axis where it can be imaged by a CCD camera to visualize the electron beam or helium neon alignment laser distribution. A second position inserts a 45° pellicle beamsplitter. In addition to coupling out on-axis light from the alignment laser or wiggler, the pellicle acts as a transition radiation screen. A CCD camera also images the pellicle, or the light may be collected by a silicon photodiode.
The wiggler emission is normally collected after the third dipole separates the photons and electrons. The emission is passed through a variable diameter iris diaphragm to limit collection angle, and an interference filter which limits bandwidth. A focusing lens concentrates all optical emission onto a silicon avalanche photodiode that has enhanced infrared quantum efficiency. The diode signal is measured through an amplifier with a digitizing oscilloscope. The responsivity of this optoelectronic system was calibrated with the photocathode drive laser at 1064 nm, and agrees well with expectations based on specifications of the individual components.
The entire optical system is very robust. The ATF provides optical pulses from the photocathode drive laser which mimic the wiggler emission very closely in both wavelength and temporal structure, allowing good calibration and preparation to be performed. Additionally, the visible wavelength regime has the broadest availability of optics and detectors.
3. Experimental Results
3.1. Electron beam Measurements
In order to provide data which can be inserted into theoretical models, the electron beam must be characterized. Important beam parameters measured are the longitudinal current distribution, emittance, and energy spread. These measurements are needed to check that the wiggler emissions are consistent with theoretical and numerical predictions.
The current distribution is measured by changing the RF phase of the second linac section to produce a linear dependence of the particle energy (relative to a nominal energy) on longitudinal position or arrival time at the slit . The collimating slit in the dispersive region after the first dipole acts as a filter which passes only a narrow slice in time. The precision of this method is limited by the stability of the RF system, which is approximately ±1 ps, and the intrinsic energy spread of the bunch, which at <0.3% corresponds to 0.7 ps. A Faraday cup and charge sensitive ADC are used to measure transmitted charge, and the RF phase of the second linac section is scanned. The absolute phase and relative phase shift are used to calibrate the time scale.
The result of the SASE calculation depends sensitively on the beam density in six dimensional phase-space. In particular, the transverse emittances are used, in simulation and theory, to parametrize the beam distribution in the two transverse phase spaces, (px,x) and (py,y). We measured the emittance using two methods, the quadrupole magnet scan and a two-screen method. Since we cannot make a complete mapping of the beam distribution in phase space, we follow the common practice and approximate the distribution by an ellipse in each transverse phase space. The measured parameters (ellipse area, aspect ratio and orientation) are represented by the symmetric beam matrix
where s 12 =s 21 is the correlation, is the beam size, and is the beam divergence (the beam line is designed so that coupling among different dimensions is negligible at the measurements region). For the emittance measurement using the quadrupole magnet scan technique, the measured beam size is related to the beam matrix at the entrance of the varying quadrupole magnet by
s 11m = R112s 11 + 2 R11 R12s 12 + R122 s 12
Where Rij are the elements of the beam transfer matrix for the measurements region (from quadrupole magnet entrance to the beam profile monitor). The beam matrix at the entrance of the quadrupole magnet represents three parameters that are the variables in a best fit procedure. Typical quadrupole scan data corresponding to the normalized rms emittance eN = 3.2 mm mrad for 0.8 nC charge are shown in Fig. 2.
That method gives information about the second moment only. The greatest uncertainty is related to the calculation of the measured beam size from the camera image. We used, for those measurements, a camera with a dynamic range of 8 bits. The natural noise of the camera (with closed iris) was measured about 23 units. The value 23 was subtracted as a background before the second moment was calculated. As the beam size changed during the scan, the image intensity varied too, and with it the sensitivity to the background.
A second method was also used to characterize the second moment of the beam distribution - the two-screen method. The distance between the two monitors (located at a beam line nearly identical to the wiggler beam line) was 4 m. The last quadrupole magnet was used to minimize the beam size at the second screen. The emittance is related to the measured beam sizes by:
where s 1 and s 2 are measured beam sizes at first and second beam profile monitors correspondingly, L is the distance between the two screens and g is the average beam energy in units of the electrons rest mass. The results of the two-screen beam measurements are presented in Table 1 for two cases, the full beam and a beam slice:
Beam sizes s 1/s 2 [mm]
1.02 / 0.12
2.4 [mm mrad]
0.73 / 0.09
1.2 [mm mrad]
Table 1. Emittance measurement for the complete beam and for a longitudinal slice.
Because the beam intensity variation did not affect our measurements (irises on both monitors were adjusted to use the whole dynamic range of the cameras) we may conclude that the two-screens method is less sensitive to the limited dynamic range of the camera. However, it is important to remember that both emittance measurement methods involve a-priori assumptions on the beam distribution in phase space. Furthermore, these two techniques may be measuring dissimilar attributes of the electron beam distribution. Since we do not have access to the actual beam distribution in phase space, this is the best characterization that can be done. A single number (the emittance) can not represent faithfully an unknown distribution function and thus one need not expect complete agreement with measurement by another technique. Likewise, a theoretical prediction for a result (such as FEL gain), or a simulation, that requires assumptions on the beam distribution in phase space need not agree with the measured result, i.e. gain, even though the measured emittance is used as input into the theory.
In the SASE measurements the charge of the beam is varied using the collimating slit after the first dipole. The beam tune after the linac can be set so that the beam size at the slit is dominated by betatron distribution, not by the very low energy spread produced at ATF. This is verified using additional screens after the collimating slit. An undesirable effect however, is some reduction in both energy spread and horizontal emittance with smaller slit openings. Therefore, if these effects are significant, the beam quality for SASE may improve with decreasing charge, making SASE harder to detect with a charge scan. On the other hand, variation of emittance and energy spread with charge can shift and broaden the spontaneous emission spectrum., this can, when optical filters and apertures are used, reduce the expected (proportional to charge) spontaneous emission signal, although this effect is less than the effect of emittance or energy spread on SASE gain.
A longitudinal scan taken after SASE was observed is shown in Fig. 3. The full beam charge delivered to the Faraday cup for this measurement is 0.8 nC. The short pulse duration, almost 1/3 the drive laser duration, and corresponding high beam current is caused in part by RF compression in the gun. The very high quantum efficiency of the magnesium photocathode used is also important, although the measured distribution is not well explained by present theory of pulse evolution in an RF gun. Several parameters contributing to the high peak current are not easily measured directly or controlled, therefore day to day variations in current were observed. This makes it necessary to measure the longitudinal pulse distribution at least once during every run.
3.2 Optical measurements
The charge dependence of emissions from the wiggler in a 25 nm bandwidth around 1064 nm and an opening of 1.2 ´ 10-3 rad (half-angle) about the central axis of the wiggler is shown in Fig. 4. The straight line represents the expected spontaneous emission dependence, as extrapolated from the low charge points, if variation of the beam distribution, such as energy spread and emittance, are not included. A detailed discussion of such dependence will be presented elsewhere .
Previous experiments (e.g. reference ) have seen signals from coherent spontaneous emission. We have strong evidence that our signal was not due to coherent spontaneous emission. One test used transition radiation from the pellicle near the wiggler. The transition radiation charge dependence shown in Fig. 5 is measured using the same photodiode and interference filter as the wiggler emission, but with a collection angle large enough to include all the transition radiation. The charge was varied, as in the SASE studies, using the collimating slit. For transition radiation, the emission depends linearly on charge. The form factor governing the contribution from coherent transition radiation involves the same Fourier components of the electron beam distribution as does the form factor for coherent enhancement of spontaneous emission. Therefore, any coherent enhancement of the wiggler emission that scales with the square of the number of electrons should also be evident in the transition radiation measurement. The lack of this behavior demonstrates that the observed enhancement of spontaneous emission is not related to coherent enhancement. In addition, electron beam structure on the micron scale is unlikely. This is supported by slice measurement of the wiggler emission shown in Fig. 3. Again using the time slice technique and transporting the portion of the electron bunch selected by the collimating slit through the beamline and wiggler, the optical emission of each slice is recorded. The emission is again limited to 1.2 ´ 10-3 rad and 25 nm around 1064 nm. The proportionality between wiggler emission and charge in each slice demonstrates that no individual slice is significantly enhanced, and there is no evidence of any small scale structure in the bunch.
Additional preliminary evidence of SASE at visible wavelengths is observed by increasing the electron beam energy. Fig. 6 shows the charge dependence of wiggler emission at 633 nm. Again, only on-axis emission was collected, and the bandwidth measured was 1 nm. This emission displays enhancement similar to that observed at 1064 nm, therefore any coherent enhancement at both wavelengths would require even smaller structure to be present in the electron beam distribution.
The data in Figs. 4 and 6 also indicate another characteristic of SASE, the increase in fluctuation of the emission with charge. This is expected because the total radiation emitted depends not only on the gain, which varies linearly with charge, but also on the spontaneous emission which is to be amplified. Fluctuation theory predicts that for this pulse duration the intensity fluctuation should be 20% rms for 1064 nm emission, which agrees qualitatively with the measurement.
4. Theoretical analysis and comparisons
The ratio of SASE radiation spectrum over spontaneous radiation spectrum is calculated approximately by two methods to compare with the experiment. One is an analytical estimate, the other is an approximation by numerical simulation  using a three dimensional version of the code TDA  based on a newly derived scaling relation between the output power and the number of simulation particles used in the code. These two methods agree with each other very well, and provide an effective tool to analyze the experimental results.
4.1 The Analytical Estimate
The analytical estimate for start-up noise is based on a three-dimensional linear theory [19,20] for an electron beam with a step-function profile, zero energy spread and zero angular spread. This idealized model for start-up noise is justified because the geometrical emittance is much smaller than the wavelength divided by 2p , and the betatron wavelength (3m) is much longer than two power gain lengths (» 0.2 m), resulting in negligible betatron motion during the start-up process. The ratio of SASE radiation spectral power in the guided mode, labeled by index, n to the spontaneous radiation spectral power is given by :
where and are the power and gain length in the guided mode n respectively. The label n used here actually represents an index, which could be a set of several discrete indices. As explained in , the power is a sum over ''diagonal'' terms and ''cross'' terms . However the cross terms are usually negligible. The measured ratio corresponds to a sum over all the modes, and since the gain is not high, includes the oscillating and decaying components as well. The factor in the parenthesis is the coupling factor of the radiation from the first two power gain lengths into the guided mode n, and, along with the other factors, is calculated in Ref. .
Since the result is sensitive to the power gain length of the fundamental mode, it is calculated by the universal scaling gain function [21,22] for a waterbag model, which is close to the experiment. The difference for waterbag model and Gaussian model is negligible for our case, as long as we take same rms emittance for both models. We assume the step-function distribution has the same rms beam size and peak current density as the waterbag model. The gain length for higher modes are estimated from the fundamental mode using the ratio of the scaled growth rate l n for different modes. This serves as a good approximation because the contribution from higher modes drops rapidly due to larger gain length, and contributes little, as shown in the following.
4.2 The Numerical Simulation
In order to provide a numerical check of the SASE magnitude, the numerical simulation code TDA3D has been run for a wide range of emittances and currents. In a separate paper  we show how a single frequency code TDA3D can be used to simulate a phenomenon with finite bandwidth, such as the SASE process. Here we try to simulate the experiment at a fixed line by a scaling relation between power and the number of simulation particles.
In the linear regime, i.e., during the exponential growth before saturation, the average output power, which arises from shot noise (modeled by random loading), is inversely proportional to the number of simulation particles in an optical wavelength. Therefore, we estimate the time average output power <P> using the simulation output power <P>:
where and are the number of electrons and simulation particles within one optical wavelength respectively. This conclusion is verified, by direct simulation, after averaging over many runs with different initial random electron distributions. We know, by comparing with the universal gain function  that 1200 simulation particles per cell are sufficient to predict the correct gain length for an amplifier. The azimuthal modes used in the calculation are from m=-2 to m=2 , i.e., 5 modes are used. The number of electrons within is 6.7× 106, so the output power is 9× 103 watt, given by TDA3D with I=320 amperes and emittance of 0.7× 10-6 m× rad, is multiplied by 1200/6.7× 106 to get the corrected simulation power of 1.5 watt. As a further check, the theoretical spontaneous radiation power is obtained as follows. The brightness is
where Z0=377W is the vacuum impedance, e0 is the electrons charge, K the wiggler parameter, NW is the number of wiggler periods, and is the Bessel factor.
At I=320 amperes. The power for the wigglers opening angle of and the bandwidth , is estimated as
. The extrapolated simulation spontaneous power is 0.75 watt, in qualitative agreement with the spontaneous radiation theory value of 1.1 watt. With more modes in the simulation the agreement can be made better.
Using this method, varying the current from 0 to 320 ampere, we plot the output power as a function of current in Fig.7, where each point is an average over 30 runs. (30 is roughly the number of coherence lengths in a pulse length see discussion below). The result shows that the power linearly increases with the current until approximately 100A, corresponding to spontaneous radiation without gain, and then deviates from linear dependence at larger current. At 320 ampere the power is a factor 2 above the linear extrapolation from the spontaneous emission regime. Notice that the factor 2 also agrees with the analytical estimate 2.1 (see Section 4.4).
Similar calculations have been done for various wiggler lengths, currents and emittances. The results all approximately agree with the analytical estimates. When we increase the number of modes and correspondingly the number of simulation particles to achieve the correct growth rate, there is better agreement between spontaneous radiation theory and the simulation results. Using larger numbers of particles did not substantially change our results.
4.3 The Intensity Fluctuation
We also carried out a 3D analytical analyses of the intensity fluctuation . One particular result of our analysis is that in the 1D limit our 3D fluctuation formula is significantly simplified to:
where s W is the rms fluctuation of <W>, the average output SASE energy per pulse, l is the length of a flat-top pulse and lc is a correlation length characterizing SASE coherence given by ,
where NW is the number of wiggler periods, l s is the radiation wavelength, LW the wiggler length, and LG is the power e-folding length. The relative fluctuation level is anticipated since we expect SASE to produce pulselets each with the same average spectral features and random phases.
The gain length, the electron beam pulse width and the intensity fluctuations are simply related and can provide a useful consistency check for the theory and experiment. The comparison between the measured fluctuation level, beam peak current and the theoretical gain length will be done in the next section.
Very similar spectral features are predicted for random backscatter of short laser pulses in a plasma .
4.4 Comparison with the Experiment
Based on the approximations discussed in the analytical estimate section, the ratio for our case with a current of 320 ampere and assumed normalized emittance 0.8 ´ 10-6 m× rad is calculated in the high gain limit,
Since the wiggler is much shorter than the measured betatron wavelength (3m), even though there is no horizontal focusing, we can approximately assume the beam size as constant, and apply this formula given in , rather than the treatment of Kim , which includes the angular spread of the beam. The gain length of 0.11 m is calculated using the formula given in , i.e., using the fact that the focusing is different from the natural focusing of the wiggler.
Each term is a contribution from one mode. The first term represents the fundamental mode with the azimuthal mode number m=0, and radial mode number j=1. The second term is for m=1, j=1. The factor in front of the parenthesis for each term is the gain for that mode. For higher modes these factors drops rapidly to near one or even smaller than one: the exponential growth terms in these modes are not dominating yet, so the gain is negligible, and the above formula is not valid. However, we know that if the gain is very small the sum must be nearly unity because the output must equal the spontaneous radiation power. The fractional power into the decaying modes of the fundamental is lost, of course. Based on this, we estimate that a low gain correction of about 1 should be added, so that the expected ratio is ~2. The error due to the neglect of higher order modes is estimated at ± 10%.
This ratio is in agreement with the experimental result, as shown in Figure 4. If we assume a larger emittance, the theory predicts a smaller ratio. The current of 320 amperes is as measured, however the measured emittance is 50% larger than the measured slice emittance. The difference between the measured slice emittance and the emittance value that produces the best fit with the theory was discussed in the electron beam measurement section. The difference is in part due to the experimental electron distribution and the assumed theoretical distribution. The experimental distribution has non-Gaussian tails that increase the measured emittance value without a significant effect on the gain. As remarked earlier, the emittance is an incomplete descriptor of a complicated phase space distribution. In addition, as can be seen from Table 1, the measured emittance improves when a slice smaller than the whole bunch is measured. The local emittance of a slice smaller than what we can achieve with our resolution is expected to be smaller, in better agreement with the measured gain. Thus, the uncertainties in the inputs to the model easily account for the differences between theory and experiment.
Another comparison can be made between the optical energy measurement and the simulation, that is, between the absolute values in Figures 4 and 7. To do that we must correct for the bandwidth and angular opening in the measurement. The simulation is done for a bandwidth 1/NW~1/60 and a full solid angle. For the small gain measured, the radiation bandwidth and opening angle should be almost the same as spontaneous radiation. Therefore, with the bandwidth of 1/NW, the radiation angle should be 1.9 · 10-3 rad. Now, the measurement was made with a 25 nm filter, giving a bandwidth of 25nm/1m m=1/40>1/NW but the angular acceptance, set by a known aperture, is 1.2 · 10-3 rad. Thus the bandwidth is retained as 1/NW, but the solid angle is reduced by a factor . The simulated power total power at 320 amperes is a factor of 2 above spontaneous, which is extrapolated to 1.1 watts at this current. There is some uncertainty in converting the simulated power to energy. As can be seen in Figure 3, the current falls fast as a function of time. Slices with a current less than 320 amperes will produce less SASE gain. If we divide the peak current by the measured pulse charge of 0.8 nC, we get an effective pulse width of 2.5 ps. Because of the gain dependence on the peak current it would be wrong to use this number for the pulse width, and we estimate that 2 ps would be more appropriate. The pulse energy, with a pulse length of approximately 2 ps is then . This is to be compared with the measured energy at the peak charge (corresponding to the peak current of 320 amperes) of 1.1 pJ. The shortfall of about 30% in the absolute value of the measured radiation energy is surprisingly good in view of the uncertainties discussed above.
Lastly, we can compare the fluctuations in the shot-to-shot optical signal energy between the experiment and the theory outlined above. For our experiment, l ~1m m, l W~8.8 mm, I~320A, e ~0.8× 10-6 m× rad, E~34 MeVand Lg~0.11m. The slippage is 60 m m, and the coherence length is reduced to
The pulse length is measured to be about 3.5 ps FWHM, i.e. , about 1050 m m, hence the fluctuation s W/<W> is calculated to be about . This is consistent with the measured fluctuation of about 15%, considering that the pulse shape is actually not a step function and the calculated beam size is not really large enough to be near the one dimensional limit. Nevertheless, the calculation serves as a rough estimate and a check that a description based on random noise is applicable.
In summary, we have extended the shortest wavelength SASE measurements an order of magnitude using the MIT Microwiggler, which provides high field quality at a period of 8.8 mm, and a 34 MeV high-brightness electron beam at the Accelerator Test Facility. The charge dependence of wiggler emissions and optical transition radiation, longitudinal structure, and electron beam parameters required to theoretically predict SASE gain have been measured. Experimental measurement of transition radiation show no evidence of coherent enhancement of spontaneous emission, and numerical and theoretical predictions are consistent with measured quantities. The favorable comparison of our measurements with theory based on particle noise startup suggests that the SASE model is valid.
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Fig. 1: Experimental configuration.
Fig. 2: Beam size vs. quadrupole current for a quadrupole scan emittance measurement.
Fig. 3: Longitudinal slice charge (· ), wiggler emission (Ñ ), and current (solid line) as a function of time along the electron bunch.
Fig. 4: Charge dependence of wiggler emission at 1064 nm. Each point is an independent measurement pair of optical energy and beam charge.
Fig. 5: Charge dependence of transition radiation at 1064 nm.
Fig. 6: Charge dependence of wiggler emission at 633 nm. Each point is an independent measurement pair of optical energy and beam charge.
Fig. 7: Numerical simulation of wiggler emission as a function of beam current.