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Gain measurements versus theory for the ACO storage ring laser
Volume 40, number 5
OPTICS COMMUNICATIONS
1 February 1982
GAIN MEASUREMENTS VERSUS THEORY FOR THE AC0 STORAGE RING LASER
D.A.G. DEACON a, K.E. ROBINSON a, J.M.J. MADEY a, C. BAZIN b, M. BILLARDON ‘, P. ELLEAUME d, Y. FARGE, J.M. ORTEGA ‘, Y. PETROFF, M.F. VELGHE e LURE, Batiment 209 C, Universite*de Paris&d,
91405 Orsay. France
Received 10 November 1981
We discuss the gain measurements made on the Orsay storage ring free electron laser, and show that the peak measured gain is identical to that predicted by a modified theory which takes into account the electron trajectory distortions. This is the fist free electron laser experiment in which the quality of the data is sufficiently good to permit a detailed verification of the Madey theorem.
The free electron laser was first operated [ 1,2] and characterized [3,4] in the high energy regime with the single-pass electron beam provided by the superconducting accelerator at Stanford. A number of approaches [5-71 have since been proposed to utilize the high quality electron beams available in electron storage rings to drive the free electron laser interaction. The present collaboration between Orsay and Stanford has been formed to create a storage ring laser on the existing storage ring AC0 and to verify the theories of its operation. The initial goals of the project have been to build and investigate the operation of an undulator on the storage ring, to measure the gain produced on an external laser, and to examine the effects of this laser on the stored electron beam. The characteristics of the spontaneous emission of the undulator [8] have been published in ref. [9], and the preliminary measurements of the bunch lengthening induced on the electron beam in ref. [lo]. In this paper we report the
a High energy Physics Lab., Stanford University, Stanford CA 93305, USA. b Laboratoire de l’A&lerateur Lineaire, Universite de ParisSud, 91405 Orsay, France. c Ecole Supkrieure de Physique et de Chimie, 75231 Paris Cedex 05, France. d Departement de Physico-Chimie, Service de Photophysique, CEN Saclay, 91190 Gif-sur-Yvette, France. e Laboratoire de Photophysique Moleculaire, Universite de Paris-Sud, 91405 Orsay, France.
0 030-4018/82/0000-0000/$02.75
0 1982 North-Holland
measurements of the magnitude of the gain and its dependence on the electron energy in the AC0 storage ring laser. The comparison of the gain data with the theory confirms the applicability of the weak signal, single particle classical theory. In addition, a comparison of the derivative of the measured spontaneous signal to that of the gain verifies the Madey theorem in a situation even more general than that originally demonstrated [ 111. The gain is measured on the vertically polarized beam from a CW argon laser operating on the blue or the green line (4880 A or 5 145 A). The energy of AC0 is lowered to permit interaction at the wavelength of interest, and the laser beam is aligned coaxially to the electron beam in the undulator with the aid of the concentric ring structure of the spontaneous radiation [9]. The bunched electron beam amplifies a portion of the laser wave train every 36.7 ns when two bunches are stored in the ring. The gain is observed through a diode detector and a 27 MHz phase sensitive amplifier. The spontaneous radiation, which produces a large signal compared to the gain, is eliminated by chopping the argon laser beam at 1 kHz and demodulating the signal again at this frequency. A more detailed discussion of the apparatus can be found in ref. [ 121. Calibration of the apparatus is performed with the synchrotron radiation emitted in the fringing fields of the main bending magnets. If this radiation is 373
Volume 40. number 5
chopped, it has time dependence similar to the gain signal. The total current drain in the diode detector is proportional to the total intercepted power, while the output of the double demodulation system is proportional to the power in the gain signal. The ratio of these two proportionality constants is obtained from the calibration run in which it is known that the average power in the gain signal is equal to the total intercepted power. Once the phases are reset for the experiment, a measurement of the laser power seen on the diode is enough to permit the calculation [ 121 of the time averaged gain from the demodulated signal . The data shown as fig. 1 were taken by recording the output of the RF demodulation system as the energy of AC0 was slowly swept across the resonance energy. The largest average gain for fig. 1 is G = (6.5 _+1.6) X 10p6, which corresponds to an instantaneous peak gain averaged over the transverse area of the laser mode of G = (3.3 f 1.2) X lop4 where we have used the measured value of the bunch CJ= 290 ps to calculate the peak gain. As is shown below, this is the same value as can be calculated from the low field, single electron theory. The energy dependence of the gain curve is 38% wider than would be predicted for an ideal 23 period undulator. There are three processes which can produce a broadening of the gain spectrum. The finite volume of phase space occupied by the electron beam produces inhomogeneous broadening [ 131; the variation of the magnetic field amplitude in the plane of the field causes the trajectories to oscillate slowly in this plane about the axis thereby shifting the phase of the inter-
Fig. 1. The gain measured on AC0 with the superconducting undulator. The current decay during the data taking is superimposed.
374
1 February
OPTICS COMMUNICATIONS Table 1 Experimental Electron
parameters
beam characteristics: r= 12 mA or = 290 ps +2o’j: ox = 0.26 mm uY=0.15 mm i de/E = 6.5 x lo+
Average current/bunch Bunch length
Transverse dimensions Energy
1982
spread
Free electron
a)
amplifier
Magnetic field amplitude Magnetic period Number of periods Pole face full width Argon laser beam waist Argon laser wavelength ___
characteristics: B = 4.0 kG (K = 1.5) he = 4 cm N= 23 2w = 5 cm we = 0.64 mm t 5%’ h = 4880 A
a) This is the theoretical energy spread calculated from the bunch length under the assumption that the anomalous bunch lengthening is negligible.
action [ 141; and the deviation of the on axis field amplitude from the perfect sinusoidal configuration produces an average electron motion perpendicular to the magnetic field [ 151, which also disturbs the phase of the emitted light. The inhomogeneous broadening is negligible (see table I), so we can conclude that the curve shapes are modified by non rectilinear electron motion. We also know from studies [ 151 of the spontaneous emission that the electron beam was centered in the field of the undulator. The broadening due to the focussing characteristics of the undulator [14] is therefore only on the order of a few percent. We can conclude that the broadening apparent in fig. 1 is primarily due to magnetic field imperfections. We have measured the magnetic field produced by the late undulator at low excitation current and find [IO] that the orbit deviations are considerably larger than the normal short period motion, as shown in fig. 2. The emission is modified by this additional component of the motion so that the spectrum depends on the vertical angle of observation. This behavior has been observed experimentally [lo]. The gain, which by the Madey theorem is the derivative of the spontaneous spectrum, is broadened and distorted by this effect as shown in fig. 3. Unfortunately the magnetic field distribution at high excitation, which is modified by the saturation of the magnet iron, is unknown. So we
Volume 40, number 5
IW’/K(mm)
1 February 1982
OPTICS COMMUNICATIONS
tx
helical undulator
/
predicts a gain of:
Ghelical(v) = PrOi; X 16n2K2N3 ?3
1 -cost-kvsinv (\
“3
Ff,
(2)
where p is the peak electron density, r. = e2/rnc2, K = eB/mc2q, q = 274X0 and v is the resonance parameter v = 47rNCyly. Ff is a filling factor which can be derived from the overlap of the laser mode and the electron beam cross section. For parallel gaussian beams, Ff = [(l + w2,/4&(1
Fig. 2. The electron trajectory in the x-z plane calculated
2
from the magnetic field of the superconducting undulator which was measured [lo] at low excitation current.
(
X exp -
cannot make a direct comparison of the high field data (fig. 1) with the low field measurements. We can however, get an idea of the size of the effects from the low field data. It is evident in fig. 3 that as the angle of observation increases, the curves become wider and more asymmetric, and the peak gain drops. None of these curves matches the observed data perfectly. While the left hand curve has the same form as the measured gain, the widths of the spontaneous curves match for the curve on the right. We therefore make the conservative estimate that the reduction of the peak gain produced by errors in the saturated magnetic field lies in the range calculated for the outer curves of fig. 3. G,,(distorted
+ w;/4r~;)]-~~ 2 EY
Ex 20; + w;/2 1
exp 9 ( 20; + w;/2 )
(3)
where we have assumed the undulator is short compared to the two beam divergences (MO < 2nw$h and MO Q 2fl*) and where /3* is the beta function at the undulator, w. is the beam waist parameter of the TEMOO argon probe laser, ax and u,, are the horizontal and vertical electron beam dimensions, and ex and e,, are the horizontal and vertical beam displacements. This expression is valid for the experiments reported here in which the beam divergences change the beam size by less than two percent, and the angular alignment tolerance is much smaller than the laser mode divergence. The expression for the gain on axis in a perfect linear undulator at the nth harmonic is similar to (2):
field)
Cm&sinusoidal
(1)
field) = OS ’ “1
The single particle weak field theory [ 161 for the &i=.15
&S,.20
&%.235
Fig. 3. Several gain curves calculated from the trajectory of fig. 2 for different angles of observation. The angle 0 is deflned in fig. 2.
G&&u = Ghelid(nv) X (n/2)[++1)/2(E)
-
4n+1y2(o12~
(4)
where g = nK2/4 t 2K2 and n is odd. We can calculate the gain for the conditions of the experiment (table 1) using the expressions (2 j(4). Assuming that the two beams were coparallel so that e, = E,, = 0, and that the electron trajectory effects are described as in fig. 3 and eq. (l), we find the theoretical value for the peak gain to be G = 2.7 + 1.2 X 10e4, once the uncertainties in the bunch dimensions and the current are taken into account. This value is the same as the experimental value within the error bars. A severe experimental problem limited the amount 375
Volume
40, number
5
of data which could be taken. The low energy at which it was necessary to run AC0 with the superconducting undulator made the stored electron beam extremely sensitive to a number of instabilities among which we have been able to identify the head-tail effect, trapped ion instabilities, and resonant radio frequency instabilities in the vacuum chamber. These effects conspired to produce a situation in which it was impossible to store a large amount of current for an appreciable length of time. The experiments were done on a stored beam with a lifetime on the order of 10 minutes. As a consequence, no time was available to work on the alignment of the laser beam once the gain signal had been acquired. The prealignment technique was able to produce parallel laser and electron beams, but with a transverse displacement uncertainty on the order of the electron beam radius. The measured values for the gain therefore reflect the presence of an unknown filling factor since eX and eY of eq. (3) are undetermined. The sequence of measurements [ 1 l] reflects this problem in that a number of the gain curves are considerably reduced in amplitude. Since the data of fig. 1 is among the largest curves taken, we believe the alignment of the laser beam to be good in this case. We note that the gain data are taken in the presence of a constant noise background so that the precise zero level is poorly determined. So long as the Madey theorem remains valid, the baseline can be determined by matching the gain curve with the derivative of the spontaneous emission curve taken under exactly the same conditions of alignment. This procedure has been used to locate the zero level for the gain curve shown in fig. 1. A number of authors [ 11,17,18] have now derived the Madey theorem, which states that under certain conditions, the gain of the free electron laser is related to the spontaneous emission spectrum through the derivative relationship: G(y) = _ 87rpc d_ dI+‘(-r) ,,2 dy dodSl ’ 4)
(5)
This type of behavior has been observed qualitatively in all three of the gain experiments that have been made to data [ 1 ,12,19]. Here we present the first detailed comparison of the theorem to experimental results. 376
1 I;ebruary
OPTICS COMMUNICATIONS
........
derivative
01
1982
spontaneous radtat,on
Fig. 4. Two measured gain curves superimposed on the derivative of the measured spontaneous emission (dotted curve). The data is corrected for the decay of the stored current, and the unknown zero level is adjusted to optimize the fit.
In fig. 4 we show superimposed three curves taken on the storage ring laser at Orsay. The dotted curve is the derivative of the spontaneous power spectrum, which is superimposed on two gain curves taken one after the other in the ascending and the descending energy directions. The hysterestis of the storage ring bending magnets produced an uncontrollable energy offset with a dispersion of about 0.3 MeV each time the direction of the energy sweep was reversed. This effect has been eliminated from the data in the figure by adding an offset to each data sweep so that the zero crossing points all occur at the mean value of the measured resonance energies. All three traces were taken after the same injection, so there is no possibility of an alignment change altering the line shapes. The gain curves have been corrected for the decay of the stored current, and the uncalibrated spontaneous emission spectrum has been normalized to the same scale. The closeness of the fit is remarkable. Three fundamental assumptions are common to all the proofs of the theorem: the electromagnetic field strength must not saturate the interaction, the electromagnetic field must be well approximated by a plane wave, and the transverse canonical momentum is assumed to be conserved. Only the first two restrictions have been shown [ 11,201 to be necessary. The small signal requirement is satisfied in the gain experiment where the probe laser intensity was on the order of 50 W/cm2. This intensity is nine orders of
Volume 40, number 5
OPTICS COMMUNICATIONS
magnitude too small to saturate the laser by overbunching (fit Q 2.6 where a2 = $CB/y2rn2c2 and $ is the electric field amplitude). The effects of the finite size of the laser mode were also negligible. The divergence of the laser mode over the length of the undulator was too small to distort the gain curve and produced only a slight shift in the resonance parameter (6~ = m,h/nwi = 0.36) which was unobservable in the presence of the storage ring hysteresis. The transverse momentum induced in an undulator depends on the magnetic field strength, which has both longitudinal and transverse functional dependence. The anomalous transverse motion shown in fig. 2 is due to the longitudinal field dependence, and therefore conserves transverse canonical momentum. In spite of the fact that the change in the angle of emission is substantial (TAO Z=Z 0.7) the magnetic field changed little across the trajectory (AB/B5 0.5%)due to the axial alignment. A substantial transverse misalignment of the electron beam would have to be introduced to probe regions of the magnetic field with a transverse gradient. A test of the importance to the Madey theorem of the constancy of the transverse canonical momentum must await future experimental results. In summary, we have shown that the gain data taken with the late superconducting undulator are in close agreement with the small signal, single particle classical theory. We are able to account for the effects of a substantial deviation of the orbit from the ideal one and arrive at a numerical figure for the gain which is identical to the measured one within the parameter uncertainties. The comparison of the gain curves with the derivative of the spontaneous emission spectrum show detailed agreement with the Madey theorem [5], even though the spectra are broadened and distorted by the trajectory deviations (fig. 2). Although the gain is extremely small in the AC0 storage ring laser, the important conclusion is that the theory is verified. The size of the gain in this experiment depends specifically on the small number of periods N which enters to the third power in the gain expression (2), and on the low current density available in ACO. A more optimal storage ring laser system can be constructed with theoretically predicted gains on the order of unity per pass, and respectable power outputs. The purpose of the AC0 experiments is to
1 February 1982
establish a body of experimental knowledge on an existing storage ring in preparation for the next generation of experiments. We are at present constructing an optical klystron which should raise the gain [21] into the range of 10e3. This would permit investigation of the behavior of an oscillator in the threshold region. This work has been supported by the Delegation G&r&-ale a la Recherche Scientifique et Technique, the Direction des Recherches, Etudes et Techniques France and the United States Air Force Office of Scientific Research.
of
References [l] L.R. Elias, W.M. Fairbank, J.M.J. Madey, H.A. Schwettman and T.I. Smith, Phys. Rev. Lett. 36 (1976) 717. [2] D.A.G. Deacon, L.R. Elias, J.M.J. Madey, G.J. Ramian, H.A. Schwettman and T.I. Smith, Phys. Rev. Lett. 38 (1977) 892. [3] J.M.J. Madey, Final Technical Report to ERDA, Contract EY 76-S-03-0326”, (1977) available from High Energy Physics Lab., Stanford University, Stanford CA 93305. [4] J.N. Eckstein, J.M.J. Madey, K. Robinson, T.I. Smith, , S. Benson, D. Deacon and R. Taber, Additional experimental results from the Stanford FEL, in: Physics of quantum electronics, Vols. 8-9, eds. S.F. Jacobs, G.T. Moore, H.S. PiIIoff, M. Sargent III, M.O. Scully and R. Spitzer (Addison-Wesley, 1982). [5] A. Renieri, 11Nuevo Cimento 53B (1979) 160. [6] T.I. Smith, J.M.J. Madey, L.R. El&and D.A.G. Deacon, J. Appl. Phys. 50 (1979) 4580. [7] D.A.G. Deacon and J.M.J. Madey, Phys. Rev. Lett. 44 (1980) 449. [8] C. Bazin, Y. Farge, M. Lemonnier, J. Perot and Y. Petroff, Nucl. Inst. Meth. 172 (1980) 61. [9] C. Bazin, M. Billardon, D.A.G. Deacon, Y. Farge, J.M. Ortega, J. Perot, Y. Petroff and M. Velghe, J. de Phys. Lettres 41 (1980) 547. [lo] C. Bazin, M. Billardon, D.A.G. Deacon, P. Elleaume, Y. Farge, J.M.J. Madey, J.M. Ortega, Y. Petroff, K.E. Robinson and M. Velghe, Results of the first phase of the AC0 storage Ring Laser Experiment, in: ref. [4]. [ll] J.M.J. Madey, 11Nuevo Cimento SOB (1979) 64. [12] D.A.G. Deacon, J.M.J. Madey, K.E. Robinson,C. Bazin, M. Billardon, P. Elleaume, Y. Farge, J.M. Ortega, Y. Petroff and M.F. Velghe, IEEE Trans. NucI. Sci. NS-28 (1981) 3142. [ 131 W.B. Colson and S.K. Ride, Physics of quantum electronics Vol. 7, eds. S.F. Jacobs, H.S. PilIoff, M. Sargent III, M.O. Scully and R. Spitzer (Addison-Wesley, 1981) p. 377.
377
Volume
[14] [15]
[16] [17]
378
40, number
5
OPTICS COMMUNICATIONS
B. Moussalam, M. Billardon, manuscript in preparation. M. Billardon, J.M. Ortega, C. Bazin, D.A.G. Deacon, P. Elleaume, Y. Petroff and M.F. Velghe, manuscript in preparation. W.B. Colson, Phys. Lett. 64A (1977) 190. N.A. Vinokurov, On the classical analog of the Einstein relations between spontaneous emission, stimulated emission and absorption (1981, unpublished), preprint 81-02, Institute of Nuclear Physics 630090, Novossibirsk 90, USSR.
1 February
1982
[ 181 N.M. Kroll, A note on the Madey Gain-Spread Theorem, in: ref. (41. [ 191 G.A. Kornyukhin, G.N. Kulipanov, V.N. Litvinenko, N.A. Mezentsev, A.N. Skrinsky, N.A. Vinokurov and P.D. Voblyi, Experiments with an optical klystron on the VEPP-3 storage ring (unpublished), Institute of Nuclear Physics, 630090 Novosibirsk, USSR. [20] W.B. Colson and P. Elleaume, in preparation. [21] P. Elleaume, Optical klystron spontaneous emission and gain, in: ref. [4].
OPTICS COMMUNICATIONS
1 February 1982
GAIN MEASUREMENTS VERSUS THEORY FOR THE AC0 STORAGE RING LASER
D.A.G. DEACON a, K.E. ROBINSON a, J.M.J. MADEY a, C. BAZIN b, M. BILLARDON ‘, P. ELLEAUME d, Y. FARGE, J.M. ORTEGA ‘, Y. PETROFF, M.F. VELGHE e LURE, Batiment 209 C, Universite*de Paris&d,
91405 Orsay. France
Received 10 November 1981
We discuss the gain measurements made on the Orsay storage ring free electron laser, and show that the peak measured gain is identical to that predicted by a modified theory which takes into account the electron trajectory distortions. This is the fist free electron laser experiment in which the quality of the data is sufficiently good to permit a detailed verification of the Madey theorem.
The free electron laser was first operated [ 1,2] and characterized [3,4] in the high energy regime with the single-pass electron beam provided by the superconducting accelerator at Stanford. A number of approaches [5-71 have since been proposed to utilize the high quality electron beams available in electron storage rings to drive the free electron laser interaction. The present collaboration between Orsay and Stanford has been formed to create a storage ring laser on the existing storage ring AC0 and to verify the theories of its operation. The initial goals of the project have been to build and investigate the operation of an undulator on the storage ring, to measure the gain produced on an external laser, and to examine the effects of this laser on the stored electron beam. The characteristics of the spontaneous emission of the undulator [8] have been published in ref. [9], and the preliminary measurements of the bunch lengthening induced on the electron beam in ref. [lo]. In this paper we report the
a High energy Physics Lab., Stanford University, Stanford CA 93305, USA. b Laboratoire de l’A&lerateur Lineaire, Universite de ParisSud, 91405 Orsay, France. c Ecole Supkrieure de Physique et de Chimie, 75231 Paris Cedex 05, France. d Departement de Physico-Chimie, Service de Photophysique, CEN Saclay, 91190 Gif-sur-Yvette, France. e Laboratoire de Photophysique Moleculaire, Universite de Paris-Sud, 91405 Orsay, France.
0 030-4018/82/0000-0000/$02.75
0 1982 North-Holland
measurements of the magnitude of the gain and its dependence on the electron energy in the AC0 storage ring laser. The comparison of the gain data with the theory confirms the applicability of the weak signal, single particle classical theory. In addition, a comparison of the derivative of the measured spontaneous signal to that of the gain verifies the Madey theorem in a situation even more general than that originally demonstrated [ 111. The gain is measured on the vertically polarized beam from a CW argon laser operating on the blue or the green line (4880 A or 5 145 A). The energy of AC0 is lowered to permit interaction at the wavelength of interest, and the laser beam is aligned coaxially to the electron beam in the undulator with the aid of the concentric ring structure of the spontaneous radiation [9]. The bunched electron beam amplifies a portion of the laser wave train every 36.7 ns when two bunches are stored in the ring. The gain is observed through a diode detector and a 27 MHz phase sensitive amplifier. The spontaneous radiation, which produces a large signal compared to the gain, is eliminated by chopping the argon laser beam at 1 kHz and demodulating the signal again at this frequency. A more detailed discussion of the apparatus can be found in ref. [ 121. Calibration of the apparatus is performed with the synchrotron radiation emitted in the fringing fields of the main bending magnets. If this radiation is 373
Volume 40. number 5
chopped, it has time dependence similar to the gain signal. The total current drain in the diode detector is proportional to the total intercepted power, while the output of the double demodulation system is proportional to the power in the gain signal. The ratio of these two proportionality constants is obtained from the calibration run in which it is known that the average power in the gain signal is equal to the total intercepted power. Once the phases are reset for the experiment, a measurement of the laser power seen on the diode is enough to permit the calculation [ 121 of the time averaged gain from the demodulated signal . The data shown as fig. 1 were taken by recording the output of the RF demodulation system as the energy of AC0 was slowly swept across the resonance energy. The largest average gain for fig. 1 is G = (6.5 _+1.6) X 10p6, which corresponds to an instantaneous peak gain averaged over the transverse area of the laser mode of G = (3.3 f 1.2) X lop4 where we have used the measured value of the bunch CJ= 290 ps to calculate the peak gain. As is shown below, this is the same value as can be calculated from the low field, single electron theory. The energy dependence of the gain curve is 38% wider than would be predicted for an ideal 23 period undulator. There are three processes which can produce a broadening of the gain spectrum. The finite volume of phase space occupied by the electron beam produces inhomogeneous broadening [ 131; the variation of the magnetic field amplitude in the plane of the field causes the trajectories to oscillate slowly in this plane about the axis thereby shifting the phase of the inter-
Fig. 1. The gain measured on AC0 with the superconducting undulator. The current decay during the data taking is superimposed.
374
1 February
OPTICS COMMUNICATIONS Table 1 Experimental Electron
parameters
beam characteristics: r= 12 mA or = 290 ps +2o’j: ox = 0.26 mm uY=0.15 mm i de/E = 6.5 x lo+
Average current/bunch Bunch length
Transverse dimensions Energy
1982
spread
Free electron
a)
amplifier
Magnetic field amplitude Magnetic period Number of periods Pole face full width Argon laser beam waist Argon laser wavelength ___
characteristics: B = 4.0 kG (K = 1.5) he = 4 cm N= 23 2w = 5 cm we = 0.64 mm t 5%’ h = 4880 A
a) This is the theoretical energy spread calculated from the bunch length under the assumption that the anomalous bunch lengthening is negligible.
action [ 141; and the deviation of the on axis field amplitude from the perfect sinusoidal configuration produces an average electron motion perpendicular to the magnetic field [ 151, which also disturbs the phase of the emitted light. The inhomogeneous broadening is negligible (see table I), so we can conclude that the curve shapes are modified by non rectilinear electron motion. We also know from studies [ 151 of the spontaneous emission that the electron beam was centered in the field of the undulator. The broadening due to the focussing characteristics of the undulator [14] is therefore only on the order of a few percent. We can conclude that the broadening apparent in fig. 1 is primarily due to magnetic field imperfections. We have measured the magnetic field produced by the late undulator at low excitation current and find [IO] that the orbit deviations are considerably larger than the normal short period motion, as shown in fig. 2. The emission is modified by this additional component of the motion so that the spectrum depends on the vertical angle of observation. This behavior has been observed experimentally [lo]. The gain, which by the Madey theorem is the derivative of the spontaneous spectrum, is broadened and distorted by this effect as shown in fig. 3. Unfortunately the magnetic field distribution at high excitation, which is modified by the saturation of the magnet iron, is unknown. So we
Volume 40, number 5
IW’/K(mm)
1 February 1982
OPTICS COMMUNICATIONS
tx
helical undulator
/
predicts a gain of:
Ghelical(v) = PrOi; X 16n2K2N3 ?3
1 -cost-kvsinv (\
“3
Ff,
(2)
where p is the peak electron density, r. = e2/rnc2, K = eB/mc2q, q = 274X0 and v is the resonance parameter v = 47rNCyly. Ff is a filling factor which can be derived from the overlap of the laser mode and the electron beam cross section. For parallel gaussian beams, Ff = [(l + w2,/4&(1
Fig. 2. The electron trajectory in the x-z plane calculated
2
from the magnetic field of the superconducting undulator which was measured [lo] at low excitation current.
(
X exp -
cannot make a direct comparison of the high field data (fig. 1) with the low field measurements. We can however, get an idea of the size of the effects from the low field data. It is evident in fig. 3 that as the angle of observation increases, the curves become wider and more asymmetric, and the peak gain drops. None of these curves matches the observed data perfectly. While the left hand curve has the same form as the measured gain, the widths of the spontaneous curves match for the curve on the right. We therefore make the conservative estimate that the reduction of the peak gain produced by errors in the saturated magnetic field lies in the range calculated for the outer curves of fig. 3. G,,(distorted
+ w;/4r~;)]-~~ 2 EY
Ex 20; + w;/2 1
exp 9 ( 20; + w;/2 )
(3)
where we have assumed the undulator is short compared to the two beam divergences (MO < 2nw$h and MO Q 2fl*) and where /3* is the beta function at the undulator, w. is the beam waist parameter of the TEMOO argon probe laser, ax and u,, are the horizontal and vertical electron beam dimensions, and ex and e,, are the horizontal and vertical beam displacements. This expression is valid for the experiments reported here in which the beam divergences change the beam size by less than two percent, and the angular alignment tolerance is much smaller than the laser mode divergence. The expression for the gain on axis in a perfect linear undulator at the nth harmonic is similar to (2):
field)
Cm&sinusoidal
(1)
field) = OS ’ “1
The single particle weak field theory [ 161 for the &i=.15
&S,.20
&%.235
Fig. 3. Several gain curves calculated from the trajectory of fig. 2 for different angles of observation. The angle 0 is deflned in fig. 2.
G&&u = Ghelid(nv) X (n/2)[++1)/2(E)
-
4n+1y2(o12~
(4)
where g = nK2/4 t 2K2 and n is odd. We can calculate the gain for the conditions of the experiment (table 1) using the expressions (2 j(4). Assuming that the two beams were coparallel so that e, = E,, = 0, and that the electron trajectory effects are described as in fig. 3 and eq. (l), we find the theoretical value for the peak gain to be G = 2.7 + 1.2 X 10e4, once the uncertainties in the bunch dimensions and the current are taken into account. This value is the same as the experimental value within the error bars. A severe experimental problem limited the amount 375
Volume
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of data which could be taken. The low energy at which it was necessary to run AC0 with the superconducting undulator made the stored electron beam extremely sensitive to a number of instabilities among which we have been able to identify the head-tail effect, trapped ion instabilities, and resonant radio frequency instabilities in the vacuum chamber. These effects conspired to produce a situation in which it was impossible to store a large amount of current for an appreciable length of time. The experiments were done on a stored beam with a lifetime on the order of 10 minutes. As a consequence, no time was available to work on the alignment of the laser beam once the gain signal had been acquired. The prealignment technique was able to produce parallel laser and electron beams, but with a transverse displacement uncertainty on the order of the electron beam radius. The measured values for the gain therefore reflect the presence of an unknown filling factor since eX and eY of eq. (3) are undetermined. The sequence of measurements [ 1 l] reflects this problem in that a number of the gain curves are considerably reduced in amplitude. Since the data of fig. 1 is among the largest curves taken, we believe the alignment of the laser beam to be good in this case. We note that the gain data are taken in the presence of a constant noise background so that the precise zero level is poorly determined. So long as the Madey theorem remains valid, the baseline can be determined by matching the gain curve with the derivative of the spontaneous emission curve taken under exactly the same conditions of alignment. This procedure has been used to locate the zero level for the gain curve shown in fig. 1. A number of authors [ 11,17,18] have now derived the Madey theorem, which states that under certain conditions, the gain of the free electron laser is related to the spontaneous emission spectrum through the derivative relationship: G(y) = _ 87rpc d_ dI+‘(-r) ,,2 dy dodSl ’ 4)
(5)
This type of behavior has been observed qualitatively in all three of the gain experiments that have been made to data [ 1 ,12,19]. Here we present the first detailed comparison of the theorem to experimental results. 376
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derivative
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spontaneous radtat,on
Fig. 4. Two measured gain curves superimposed on the derivative of the measured spontaneous emission (dotted curve). The data is corrected for the decay of the stored current, and the unknown zero level is adjusted to optimize the fit.
In fig. 4 we show superimposed three curves taken on the storage ring laser at Orsay. The dotted curve is the derivative of the spontaneous power spectrum, which is superimposed on two gain curves taken one after the other in the ascending and the descending energy directions. The hysterestis of the storage ring bending magnets produced an uncontrollable energy offset with a dispersion of about 0.3 MeV each time the direction of the energy sweep was reversed. This effect has been eliminated from the data in the figure by adding an offset to each data sweep so that the zero crossing points all occur at the mean value of the measured resonance energies. All three traces were taken after the same injection, so there is no possibility of an alignment change altering the line shapes. The gain curves have been corrected for the decay of the stored current, and the uncalibrated spontaneous emission spectrum has been normalized to the same scale. The closeness of the fit is remarkable. Three fundamental assumptions are common to all the proofs of the theorem: the electromagnetic field strength must not saturate the interaction, the electromagnetic field must be well approximated by a plane wave, and the transverse canonical momentum is assumed to be conserved. Only the first two restrictions have been shown [ 11,201 to be necessary. The small signal requirement is satisfied in the gain experiment where the probe laser intensity was on the order of 50 W/cm2. This intensity is nine orders of
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magnitude too small to saturate the laser by overbunching (fit Q 2.6 where a2 = $CB/y2rn2c2 and $ is the electric field amplitude). The effects of the finite size of the laser mode were also negligible. The divergence of the laser mode over the length of the undulator was too small to distort the gain curve and produced only a slight shift in the resonance parameter (6~ = m,h/nwi = 0.36) which was unobservable in the presence of the storage ring hysteresis. The transverse momentum induced in an undulator depends on the magnetic field strength, which has both longitudinal and transverse functional dependence. The anomalous transverse motion shown in fig. 2 is due to the longitudinal field dependence, and therefore conserves transverse canonical momentum. In spite of the fact that the change in the angle of emission is substantial (TAO Z=Z 0.7) the magnetic field changed little across the trajectory (AB/B5 0.5%)due to the axial alignment. A substantial transverse misalignment of the electron beam would have to be introduced to probe regions of the magnetic field with a transverse gradient. A test of the importance to the Madey theorem of the constancy of the transverse canonical momentum must await future experimental results. In summary, we have shown that the gain data taken with the late superconducting undulator are in close agreement with the small signal, single particle classical theory. We are able to account for the effects of a substantial deviation of the orbit from the ideal one and arrive at a numerical figure for the gain which is identical to the measured one within the parameter uncertainties. The comparison of the gain curves with the derivative of the spontaneous emission spectrum show detailed agreement with the Madey theorem [5], even though the spectra are broadened and distorted by the trajectory deviations (fig. 2). Although the gain is extremely small in the AC0 storage ring laser, the important conclusion is that the theory is verified. The size of the gain in this experiment depends specifically on the small number of periods N which enters to the third power in the gain expression (2), and on the low current density available in ACO. A more optimal storage ring laser system can be constructed with theoretically predicted gains on the order of unity per pass, and respectable power outputs. The purpose of the AC0 experiments is to
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establish a body of experimental knowledge on an existing storage ring in preparation for the next generation of experiments. We are at present constructing an optical klystron which should raise the gain [21] into the range of 10e3. This would permit investigation of the behavior of an oscillator in the threshold region. This work has been supported by the Delegation G&r&-ale a la Recherche Scientifique et Technique, the Direction des Recherches, Etudes et Techniques France and the United States Air Force Office of Scientific Research.
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References [l] L.R. Elias, W.M. Fairbank, J.M.J. Madey, H.A. Schwettman and T.I. Smith, Phys. Rev. Lett. 36 (1976) 717. [2] D.A.G. Deacon, L.R. Elias, J.M.J. Madey, G.J. Ramian, H.A. Schwettman and T.I. Smith, Phys. Rev. Lett. 38 (1977) 892. [3] J.M.J. Madey, Final Technical Report to ERDA, Contract EY 76-S-03-0326”, (1977) available from High Energy Physics Lab., Stanford University, Stanford CA 93305. [4] J.N. Eckstein, J.M.J. Madey, K. Robinson, T.I. Smith, , S. Benson, D. Deacon and R. Taber, Additional experimental results from the Stanford FEL, in: Physics of quantum electronics, Vols. 8-9, eds. S.F. Jacobs, G.T. Moore, H.S. PiIIoff, M. Sargent III, M.O. Scully and R. Spitzer (Addison-Wesley, 1982). [5] A. Renieri, 11Nuevo Cimento 53B (1979) 160. [6] T.I. Smith, J.M.J. Madey, L.R. El&and D.A.G. Deacon, J. Appl. Phys. 50 (1979) 4580. [7] D.A.G. Deacon and J.M.J. Madey, Phys. Rev. Lett. 44 (1980) 449. [8] C. Bazin, Y. Farge, M. Lemonnier, J. Perot and Y. Petroff, Nucl. Inst. Meth. 172 (1980) 61. [9] C. Bazin, M. Billardon, D.A.G. Deacon, Y. Farge, J.M. Ortega, J. Perot, Y. Petroff and M. Velghe, J. de Phys. Lettres 41 (1980) 547. [lo] C. Bazin, M. Billardon, D.A.G. Deacon, P. Elleaume, Y. Farge, J.M.J. Madey, J.M. Ortega, Y. Petroff, K.E. Robinson and M. Velghe, Results of the first phase of the AC0 storage Ring Laser Experiment, in: ref. [4]. [ll] J.M.J. Madey, 11Nuevo Cimento SOB (1979) 64. [12] D.A.G. Deacon, J.M.J. Madey, K.E. Robinson,C. Bazin, M. Billardon, P. Elleaume, Y. Farge, J.M. Ortega, Y. Petroff and M.F. Velghe, IEEE Trans. NucI. Sci. NS-28 (1981) 3142. [ 131 W.B. Colson and S.K. Ride, Physics of quantum electronics Vol. 7, eds. S.F. Jacobs, H.S. PilIoff, M. Sargent III, M.O. Scully and R. Spitzer (Addison-Wesley, 1981) p. 377.
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B. Moussalam, M. Billardon, manuscript in preparation. M. Billardon, J.M. Ortega, C. Bazin, D.A.G. Deacon, P. Elleaume, Y. Petroff and M.F. Velghe, manuscript in preparation. W.B. Colson, Phys. Lett. 64A (1977) 190. N.A. Vinokurov, On the classical analog of the Einstein relations between spontaneous emission, stimulated emission and absorption (1981, unpublished), preprint 81-02, Institute of Nuclear Physics 630090, Novossibirsk 90, USSR.
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[ 181 N.M. Kroll, A note on the Madey Gain-Spread Theorem, in: ref. (41. [ 191 G.A. Kornyukhin, G.N. Kulipanov, V.N. Litvinenko, N.A. Mezentsev, A.N. Skrinsky, N.A. Vinokurov and P.D. Voblyi, Experiments with an optical klystron on the VEPP-3 storage ring (unpublished), Institute of Nuclear Physics, 630090 Novosibirsk, USSR. [20] W.B. Colson and P. Elleaume, in preparation. [21] P. Elleaume, Optical klystron spontaneous emission and gain, in: ref. [4].