Bending and focusing effects in an FEL oscillator II: numerical simulations

158

BENDING AND FOCUSING EFFECTS NUMERICAL SIMULATIONS *

Nuclear Instruments and Methods in Physics Research A259 (1987) 158-162 North-Holland, Amsterdam

IN AN FEL OSCILLATOR

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B.D. M c V E Y a n d R.W. W A R R E N Los Alamos National Laboratory, Los Alamos, N M 87545, USA

Bending and focusing of the optical beam by the electron beam is investigated for an FEL oscillator configuration. Numerical calculations are performed with the simulation code, FELEX.

1. Introduction The free-electron laser interaction results in both a gain and a phase shift of the optical field [1]. In an F E L oscillator, the phase shift manifests itself in bending and focusing of the optical beam relative to the optical axis. Bending and focusing effects are the topic of this paper. As discussed in part I of this work, bending and tilt of the optical beam were observed experimentally in the Los Alamos F E L experiment [2]. The observations were made when there was a misalignment between the electron beam and the optical beam axis. This suggested the electron beam had a prism effect on the optical beam. These physics considerations also predicted a focusing of the optical beam when there was alignment with the electron beam. The electron beam was both a gain media and could be modeled as a lens. We investigate these models in more detail by using the F E L oscillator simulation code F E L E X [3]. In section 2, we evaluate the effective index of refraction of the electron beam, and discuss its characteristics. In section 3, we observe bending and focusing effects in an F E L oscillator simulation with parameters characteristic of the Los Alamos ERX experiment. In section 4, we discuss our results.

2. The effective refractive index In this and the following section, we present results from the three-dimensional F E L simulation code, FELEX. The code has been described in detail elsewhere [3]. Relevant to the topic of this paper, the simulation code models the self-consistent interaction between the electron beam and the optical field. At the end of the wiggler, the optical field is then propagated through a resonant cavity back to the wiggler entrance where the * Work performed under the auspices of the U.S. Department of energy and supported by the Department of Defense U.S. Army Strategic Defense Command. 0168-9002/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

F E L interaction is reinitiated. The optical properties of the electron beam can be characterized by a complex refractive index [4], 2~rj a w c o s ~ + 00) rb - 1 = T k ~ 7 0 F" y 2~r rJi = -- T

a w - sin(~p + 0o) J k~o

PU

-.,f-

(1)

(2)

where J is the current density, a w = eBw/(rnc2kw), E o and 00 are the optical field strength and phase, k 0 is the optical wave number, and F u is the difference of Bessel functions. The energy of the electron is y m c 2 and the phase of the electron in the optical wave is +. The brackets indicate an average over the electron distribution. A few observations can be made from the refractive index formula. Both the gain O1i) and the additional phase shift (~r) are dependent upon bunching of the electron beam (nonzero values for the sine and cosine terms). Examples of the evolution of this electron bunching process along the length of the wiggler is seen in figs. l a and lb. Fig. l a is for the small signal regime, /in = 107 W / c m 2 , and fig. l b for the saturated regime, /in = 101° W / c m 2 . At saturation, the gain (~i) is much lower, G = 2.5%, compared to G = 100% for fig. la. For both signal levels, the average value of ~r is approximately the same but the variation, or electron bunching along the length of the wiggler is quite different. Further the radial variation of ~r is quite different for the two signal levels. At saturation, oscillations in nr - 1 follow the synchrotron oscillations of the gain ~i. The gain and refractive properties of the electron beam are seen to be nonlinear functions of the optical field strength and in general quite complicated. We first estimate the magnitude of the bending and focusing by performing single pass simulations, and observing the bend and focusing in the output optical wave. The electron beam and optical beam parameters are listed in table 1. The electron beam profile is assumed to be Gaussian, and the width of the electron

B.D. McVey, R.W. Warren / Bending and focusing effects in an FEL oscillator II Table 1 Electron and optical beam parameters

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-3 - b e a m is approximately one half the w i d t h of the G a u s sian optical mode. T h e waist of the initial optical m o d e was assumed to be at the center of the wiggler. The optical intensity was assumed to be 10 l° W / c m 2 a n d the gain was typically a few percent. F o r an aligned system, the effective focal strength of the electron b e a m m a y be estimated from the curvature of the optical b e a m in the far field. The curvature was f o u n d to b e less t h a n the v a c u u m p r o p a g a t i o n case, indicating that the waist m o v e d from the center toward the e n d of the wiggler. F o r the electron b e a m p a r a m e t e r s of table 1, the effective focal length of the lens at the center of the wiggler was 1140 cm. In the next set of simulations, the axis of the initial optical b e a m was translated a distance Ay from the electron b e a m axis. The resultant b e n d angle (defined in part I) of the optical b e a m is plotted in fig. 2a as a function of Ay. The b e n d angle increases almost linearly with displacement a n d then saturates w h e n the displacement equals the radius of the electron beam. T h e b e n d angle is larger t h a n would be predicted from a lens of the focal strength calculated for the aligned case which is plotted as the dashed fine. T h e reason for this is illustrated in fig. 2b where radial profiles of ~r - 1 are plotted for three displacements of the optical beam. At each radial location, the value of T/r- 1 plotted has b e e n averaged over the length of the wiggler. F o r the aligned optical b e a m (solid line), the refractive index

0 -60

/ -40

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I 20

I 40

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60

z(cm) (SATURATED) Fig. 1. Refractive index along the wiggler axis, (a) small signal, (b) large signal. variation models a n ideal lens (parabolic variation in r ) near the center of the electron beam. N e a r the edges the variation with r is linear, and the lens is generally smaller t h a n the optical b e a m with a radius indicated in table 1. The refractive index or b e n d i n g power is dependent u p o n the location of center of the optical field. The refractive index tends to follow b u t lag b e h i n d the displacement of the optical beam. It also increases in m a g n i t u d e a n d focusing strength. T h e effective lens that models the electron b e a m is clearly a n o n l i n e a r m e d i u m d e p e n d e n t u p o n the local optical field strength. W e also note that w h e n the ~ r - 1 term is set equal to zero in the simulations a n d only gain is present, the b e n d angle is less than 3/~rad a n d has a negative sign.

3. FEL oscillator simulations In this section, we describe simulation results for an F E L oscillator with parameters summarized in table 1. C o m p u t e r runs were m a d e to simulate the startup of the IV. BEAM GUIDING THEORY

B.D. McVey, R.W. Warren / Bending and focusing effects in an F E L oscillator I I

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PASS NUMBER Fig. 3. Optical centroid on the fight hand mirror during startup and decay (walking mode oscillation) on the FEL oscillator.

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-1.2-0.9-0.6-o.a

0 ELECTRON BEAM

system. The oscillations are observed in most diagnostics, of which fig. 3 is typical. In fig. 3 the centroid of the optical beam on the right mirror is plotted as a function of pass number. When the electron beam is turned off (pass number = 121), the optical beam oscillates about the optical axis with a characteristic frequency consistent with the ray tracing analysis of part I. Both tilt and and centroid displacement are

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F E L oscillator from the small signal regime to the saturated state. For the parameters of table 1, the signal reached saturation in about 50 passes and the electron beam was turned off, after 120 passes. The optical signal then freely decayed. Three sets of simulations were performed. The first two illustrate the bending or refractive properties of the electron beam and the last the focusing power of the beam. In the first set, the optical axis was translated from the electron beam axis, and in the second set, the electron beam current was varied for a fixed optical axis displacement of 0.6 ram. The final set was a variation of electron beam current for a perfectly aligned electron beam and optical axis. The simulations were performed at a single frequency. The frequency was selected to be at the peak of the small signal gain curve. The gain curve was calculated by a series of single pass calculations assuming an incident Gaussian mode. The walking mode oscillations (described in part I) are readily observed in the simulations for misaligned

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O P T I C A L A X I S D I S P L A C E M E N T (mm) Fig. 4. For the oscillator, bend and tilt angles as a function of optical axis displacement for I=150 A, (a) small signal, (b) large signal.

161

B.D. McVey, R. W. Warren / Bending and focusing effects in an FEL oscillator 11

larger in the small signal regime (before pass number 30) than in the saturated regime (pass numbers 60-120). The oscillations in the small signal regime are due to the fact that the initial starting conditions of the optical field (which match the empty cavity solution) are not an equilibrium solution when the electron beam is present. Figs. 4a and 4b plot the bend and tilt angles as defined in fig. 2 of part I as a function of the displacement of the optical axis. Fig. 4a is for the small signal regime, and fig. 4b is for the saturated state. An immediate observation is that the bend angles are comparable, but the tilt angles are quite different for the two signal levels. We attribute the larger tilt in the small signal regime to higher gain ( - 1 0 0 % ) compared to a gain of 2.5% at saturation. This is also illustrated in the refractive index plots of figs. l a and l b where ~ - 1 values are comparable, but the ~i are quite different. Further, when ~ r - 1 is set equal to zero in the simulations, the tilt angle in the small signal gain regime is quite large ( - 5 0 0 mrad for Ay = 0.6 mm) while at saturation the tilt angle was approximately zero. For displacements of the optical axis greater than 0.6 m m higher order optical modes (TEM01) develop, and the optical field solution has two peaks. Generation of higher order modes is observed when the optical axis

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displacement is comparable to the size ( 1 / e point) of the electron beam. Figs. 5a and 5b illustrate the bend and tilt angles as a function of electron beam current for a fixed optical axis displacement of 0.6 ram. Fig. 5a is at small signal and fig. 5b is at saturation. In the small signal regime, the tilt angle increases along with an increase in gain, however the bend angle remains nearly constant and then decreases. At saturation, the bend angle increases with current and the tilt angle remains nearly constant. The linear increase of bend angle, suggests an increase in focal strength of the lens as given by eq. (1). For currents greater than 250 A, the optical field is unstable to higher order mode generation. In the last set of simulations, we investigate focusing of the electron beam for a perfectly aligned system. Breathing mode oscillations are observed in the simulations as seen in fig. 6. The width of the optical b e a m on the right mirror is seen to oscillate at twice the frequency of the walking mode of fig. 3. The optical mode size in the presence of the electron beam is smaller than the empty cavity cases which is indicative of focusing of the electron beam. The effect is much larger in the small signal regime. An estimate of the focal power of the electron beam observed in the simulations can be made by comparing the radius of curvature of the beam on pass 120 to that on pass 121. The comparison is made in the far field and pass 120 is the last pass with the electron beam present. The estimate places a lens of focal power f at the geometric center of the resonant cavity which is the waist of the optical beam. The difference in far field curvatures with an without the lens is equal to the change in the position of the optical waist, L = R12a - Ra20. F r o m Gaussian optics, L - Z { / f , so we obtain the formula for the focal power of the lens.

I

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50

100

150

200

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ELECTRON

CURRENT

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1.5

I ~g 20

I 40

I 60

PASS

t 80

I 100

I 120

1.6 140

NUMBER

Fig. 6. Optical width on the right hand mirror during startup and decay (breathing mode oscillation) of the FEL oscillator. IV. BEAM GUIDING THEORY

162

B.D. McVey, R. W. Warren / Bending and focusing effects in an FEL oscillator II STABILITY L I M I T

result has been predicted previously [6]. In our case, the gain is low ( - 2.5%), and it appears higher order mode generation is a result of the focal properties of the electron beam. The performance was degraded due to higher order mode generation. The output power and extraction efficiency were degraded by a factor of 2.

1.0

0.8

4. Discussion

of results

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,p0.4

0.2

v

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SATURATED

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0 0

J 200

I

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i

400

600

800

100

I (amp) Fig. 7. Focusing power, l / f , of the electron beam for an aligned system at small signal and saturation.

Fig. 7 illustrates the effective focal strength, l / f , of the electron beam as defined by eq. (3) as a function of current for small signal and for saturation where the gain is fixed at 2.5%. It is observed that the focal length tends to flatten out at larger currents and, for small signal, approaches the stability limit. This flattening is probably due to a greater number of synchrotron oscillations due to larger optical intensity as the current is increased. The refractive index oscillates as in fig. l b which was at 150 A and shows one synchrotron oscillation. The stability limit in fig. 7 is for a lens of the focal strength indicated centered in the cavity with parameters listed in table 1. This would be a stability limit on the focal strength of the lens to sustain a Gaussian mode of a finite transverse size. It assumes the effective lens has a diameter much larger than the Gaussian mode waist. This is clearly violated in the simulations where the electron beam is smaller than the optical beam. Thus, the stability limit is renamed a mode stability limit suggesting the higher order modes will be generated when the limit is exceeded. A simulation was performed with a concentric cavity L = 67 m, and I = 500 A. F r o m the focal strength indicated in fig. 7, the electron beam cavity configuration should be unstable. At saturation, there was a significant content of higher order modes present. For the concentric cavity this

In the context of numerical simulations and corroborated by experiment (part I), we have shown that an electron beam can bend and focus an optical beam. The underlying physics involved is identical to that of optical guiding by an elongated high current electron beam [4,5]. The bending angle is toward the electron beam axis indicating a guiding effect. In a F E L oscillator, the guiding of the electron beam is balanced by the requirement that the optical beam be aligned with the optical axis. In the simulations~ we have observed an instability boundary for the TEM00 mode. For a given current, an increasing displacement of the optical axis will generate the higher order modes. For a given misalignment, higher currents will be unstable to higher order mode generation. The boundaries are indicated in figs. 4 and 5, and at even higher currents (1000 A) the displacement boundary for fig. 4 is moved in to 0.3 mm. The bounded volume of parameter space for a stable TEM00 mode will decreases in concentric and confocal resonators [6], and we have observed a decrease when the output coupling is increased above the 2.5% assumed here.

References

[1] W.B. Colson and S.K. Ride, Physics of Quantum Electronics, Vol. 5, eds., S.F. Jacobs, M. Sargent and M.O. Scully (Addison-Wesley, Reading MA, 1978) p. 157. [2] B.E. Newnam et al., IEEE J. Quantum Electron. QE-21 (1985) 867. [3] B.D. McVey, Proc. Lake Tahoe FEL Workshop, Nucl. Instr. and Meth. A250 (1986) 449. [4] E.T. Scharlemann, A.M. Sessler and J.S. Wurtele, Phys. Rev. Lett. 54 (1985) 1925. [5] G.T. Moore, Proceedings of the International Workshop on Coherent and Collective Properties of Electron and Electron Radiation, Como, Italy, 1984, Nucl. Instr. and Meth. A239 (1985) 19. [6] D.C. Quimby and J. Slater, IEEE J. Quantum Electron. QE-19 May (1983) 800.