Basic theory of the isochronous storage ring laser

BASIC THEORY OF THE ISOCHRONOUS STORAGE RING LASER

David A.G. DEACON High Energy Physics Laboratory, Stanford University, Stanford, California 94305, U.S.A.

50

I93~

NJ.}j 1981

(IA.~~C NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

PHYSICS REPORTS (Review Section of Physics Letters) 76, No. 5 (1981) 349—391. North-Holland Publishing Company

BASIC THEORY OF THE ISOCHRONOUS STORAGE RING LASER David A.G. DEACON High Energy Physics Laboratory, Stanford University, Stanford, California 94305, U.S.A. Received November 1980

Contents: 1. Introduction 2. Analysis of electron motion in the optical potential wells 2.1. Equations of motion for the coupled laser-ring system 2.2. The linearized analysis 2.3. Nonlinear effects 2.4. Transverse motion 2.5. Resonances with the transverse motion 2.6. Numerical simulations 2.7. Laser power and the trap population 2.8. Starting and operating the laser

351 354 354 358 363 364 366 368 372 374

2.9. Nonlinear effects on the RF trajectories 2.10. Parameter optimization 2.11. A numerical example 3. The general isochronism conditions in a storage ring 3.1. Derivation of the longitudinal coordinate shift 3.2. The effects of transverse motion 3.3. The effects of energy deviations 4. Summary Appendix A. List of symbols References

376 378 381 382 383 385 387 389 390 391

Abstract: The use of an isochronous storage ring as the electron source for a higher power, high efficiency free electron laser is proposed. Electrons can be trapped in the optical potential wells in such a system, where they couple energy directly from the radio frequency acceleratingcavity into the optical frequency laser beam with negligible losses. The conditions for the existence of such traps are derived, as well as the dependence of their size and shape on the laser parameters and the deviations from isochronism in the arc of the storage ring. The results of a numerical stimulation extend the analytic results to the general case of large phase advance. The optimization of the laser parameters is discussed with the aid of a numerical example of the performance assuming a design which can be realized in practice.

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David A.G. Deacon. Basic theory of the isochronous storage ring laser

351

1. Introduction Free electron lasers convert a portion of the enormous kinetic energy of a relativistic electron beam directly into coherent light. This kind of laser is attractive due to the simplicity of the interaction mechanism: if the electron beam source can be tailored appropriately, the interaction region can be designed to produce high power output with excellent efficiency. While such a device has a comparatively elevated minimum cost and size set by the requirement for an electron beam generator, the projected high power performance is excellent when evaluated by average power per dollar. Along with the output power, the intrinsic tunability within the far-infrared through the ultraviolet qualifies the free electron laser as a strong candidate for future industrial use (isotope separation, nuclear reprocessing, chemical synthesis, etc.) [1]. Both linear accelerators and storage rings are being considered as electron sources for free electron lasers. Although the question of which technology to use in a given application will require a number of years of experimentation, it is clear that a stored beam has a number of advantages. The available current and duty cycle are high, and the characteristics of the beam are independent of the source. In this paper, I discuss a technique for utilizing the beam stored in a storage ring to drive the laser mechanism, in which the efficiency of the energy transfer is preserved by making the electron optics nearly isochronous; hence the name isochronous storage ring laser. The problem with the use of stored beams to drive free electron lasers lies in the fact that the energy modulation induced during the bunching process [2]can accumulate from pass to pass, interfering with the energy extractionto the laser. Renieri [3]has shown that in a system in which the energy spread is allowed to build up, the laser power is limited to a small fraction of the spontaneous power emitted by the electrons as synchrotron radiation, so that the efficiency would be low. However, since the laser interaction conserves the density of the electron distribution in the six dimensional phase space [4],it follows that growth of the energy spread need not occur. The isochronous storage ring laser [5]is a device designed to eliminate this problem by conserving the initial electron energy spread. The principle of operation of an isochronous storage ring laser is illustrated by the phase space diagrams in fig. 1. The upper panel shows the orbits followed by the electrons in the free electron laser interaction region [2].If a uniform distribution as shown by the dotted lines in the central panel is injected into the laser, the distribution will evolve into the complicated form shown in solid lines by the time it leaves the interaction region. The phase space transformation produced by the storage ring on bringing these electrons around again for reinjection operates on a scale much larger than that of the laser, which is the wavelength of light. The electron distribution cannot be untangled by such a transformation, and is reinjected into the laser with the same energy spread it acquired on the previous pass. The process repeats, and the energyspread accumulates, giving rise to the efficiency limitation discussed by Renieri. However, if the initial electron distribution is bunched, as shown in the lower panel, the microbunches can be injected into the closed orbit regions where the phase space evolution is simple. The bunching of the beam is retained after the interaction, and an acceleration by the RF cavity suffices to bring the distribution close to its original form for reinjection into the laser. This process can be repeated indefinitely without excitation of the electron energy spread. This type of system has a number of unique properties. The circulating electrons pickup energyfrom the RF cavity on each pass, only to give it up again on the average to the laser field. The microbunched electrons act as a frequency transformer, absorbing energy from the radio frequency accelerating cavity, and emitting energy into the high frequency laser field. The phase relationship between the RF cavity and the laser radiation is maintained by the isochronism of the electron transport system.

352

David A.G. Deacon, Basic theory of the isochronous storage ring laser

-

ELECTRON PHASE SPACE TRAJECTORIES

~

•0•O•

’

UNIFORM INPUT DISTRIBUTION

~INAL~

Fig. 1. The evolution of an electron distribution is demonstrated by tracing its phase space motion inside the laser. The trajectories defined by the interaction are shown in the top panel. The central panel shows the beginning of the filamentation of an initially uniform distribution through one transit of an operating laser. The lower panel shows how this problem is eliminated with a microbunched initial distribution.

David AG. Deacon, Basic theory of the isochronous storage ring laser

353

The emission into the laser is strongly enhanced by the bunching of the electrons. The mean energy transfer now occurs through the first order interaction, so that the extracted energyis linearly proportional to the laser electric field. Gain is no longer an appropriate concept. Both the starting and saturation mechanisms are novel: the laser starts oscillation through a progressive creation of laser potential wells and redistribution of the electron charge, and saturates by equilibrating the linear energy transfer to the field with the quadratic energy loss through the mirrors. It should be recognized at the outset that difficulties will be encountered in making such a system work. The ratio of the frequencies involved is on the order of kcIwRF io~ which implies that the tolerances on the isochronism will be tight. For this reason, it was long considered impossible to take advantage of the electron bunching in a storage ring. Indeed, this work was initially undertaken with some expectation that the idea would rapidly be shown to be hopeless. I therefore attempt to select parameters in the examples I use to demonstrate that an isochronous storage ring laser can be built with known technology [6,7]. Although the ideas discussed here are specifically directed at the storage ring system, the same analysis also applies to a linear system in which a number of free electron lasers are strung together with accelerating cavities to replenish the extracted energy as in fig. 2. Each energy extraction module is equivalent to one circuit of the storage ring, and the entire system may include as many modules as desired. The linear system is easier to realize in some respects since the momentum compaction is zero in the absence of bending fields. However, the quality of electron beams available from linear electron accelerators is orders of magnitude poorer than that available in storage rings. For this reason, a linear device would require a much higher laser intensity. In addition, synchrotron damping is no longer present, so that a special interaction region would be desirable at the start to establish the bunching of the uniform input beam. The modular linear configuration could also be incorporated in one long straight section of a storage ring so that a high quality stored beam could be used to drive the laser. While my discussion will focus on the storage ring system with a single laser interaction region, the reader may wish to keep in mind that the results also apply to these other variations of an electron trapping machine. I begin the paper with the heart of the problem, deriving the conditions under which the optical traps exist, and characterizing their dependence on the various deviations from isochronism in the arc of the storage ring. These effects are illustrated with the results of a numerical simulation of the one dimensional system. Included here is a discussion of the peculiar operational characteristics of the laser. Following a discussion of how the optical traps fit into the large scale electron motion in the storage ring as a whole, I discuss the optimization of the system and give two numerical examples of its performance. ELECTRON BEAM

LASER AXIS (AMPLIFIER OR OSCILLATOR)

L

ENERGY

FREE ELECTRON LASER INTERACTION REGION

EXTRACTION MODULE

Fig. 2. A linear configuration embodying the essential ideas of electron trapping and periodic reacceleration.

354

David AG. Deacon, Basic theory of the isochronous storage ring laser

The second section treats the isochronism of the storage ring. The implications on the storage ring design of the isochronism conditions are stated after a brief derivation of the storage ring effects on the laser. For reference, note the list of symbols provided as appendix A. 2. Analysis of electron motion in the optical potential wells A description of the motion of electrons as a function of time across the optical phase is crucial in an understanding of an isochronous storage ring laser. The distribution of the electrons determines the rate of energy extraction into the laser field, and the range of acceptable parameter magnitudes in the arc of the isochronous ring. It is the purpose of this section to detail the dynamics of the device so that we can obtain a grasp of the considerations which go into the answers to these questions. Of the three degrees of freedom of the particle, the longitudinal is the most important. The electrons undergo transverse oscillations with a magnitude determined by the focussing provided in the magnet lattice. If the transport system has been properly designed, these oscillations will be stable against perturbations, and we can turn our attention to the longitudinal motions. The longitudinal effects of the transverse motions can be incorporated in the equations of motion so that the basic aspects of the performance of the laser can be obtained from a study of the longitudinal motions of the electrons. The analytical technique one chooses to describe the electron motion must take into account both the nonlinear dependence of the energy loss on the azimuthal phase, and the discrete nature of the successive interactions in the ring. As there is no known way of solving the general problem analytically, one must be satisfied with a number of approximate solutions, each valid in a different region of interest. In this section I present a linearized discrete analysis which describes the fundamental characteristics of the electron trajectories in the traps. This analysis is extended to the nonlinear phase region by assuming that another parameter, the laser phase advance, is small. The complete problem is solved by a numerical simulation, which I use to illustrate the analytical results. Armed with a knowledge of the conditions on the existence of the traps, we can progress to the next problem. The interrelationship between the electron motion in the traps and the large scale motion around the RF bucket determines the operational characteristics of the system. I discuss this relationship with the view of illuminating the available degrees of freedom which influence the laser output power. At this point, the preparatory material is integrated in a consideration of the optimization of the laser parameters, and a numerical example of its performance. 2.1. Equations of motion for the coupled laser-ring system Colson [8] has shown that the laser can be characterized most simply with a classical description in which the electron coordinates obey a pendulum equation. The electrons travel through the spatially periodic laser magnet B(z) = Re{B(I + if) exp(ikoz)} which induces a periodic transverse velocity. Energy exchange with the laser field occurs when the longitudinal electron velocity is adjusted so that the force produced by the electric field passing over the electron maintains a constant phase relationship to the transverse velocity. The tranverse velocity can be obtained from the constancy of the transverse canonical momentum p.L = ymv1 + eAjc along the axis of the magnetic structure. The energy exchange is then given by the fourth component of the Lorentz force dE/dt = e~S v. For a helically polarized

David A.G. Deacon, Basic theory of the isochronous storage ring laser

355

magnet and electromagnetic wave, these two equations can be combined to show that the electron motion obeys a pendulum equation d20/dt2= ~~~f22 sin 0

(1)

if the effects of the radiation field on the trajectory [9] are neglected. The coordinate is 0 k ôz where ôz is the longitudinal displacement of the electron from the phase stable resonant particle, and the frequency is 112 2e2~’B/y2m2c2,where ~ is the electric field of the radiation, and B is the magnetic field strength in the wiggler. The energy has been related to the phase slip by the approximate relation (y Yr)IYr (dO/dt)/2coo, where ~ is the effective frequency of the laser magnet field wo 2lTc/Ao. Terms of order 1/y2 have been dropped whenever they add to a term of order unity, and will be dropped in all results quoted in this work. For later convenience, I also define the laser phase advance (1L,Ic, which is the angle of rotation in phase space experienced by an electron near the stable fixed point in a complete pass through the laser magnet. This expression is valid in a restricted region, but is applicable in most of the configurations which can be proposed as realizations for this kind of device. It treats the electromagnetic field classically, an approximation which Madey and I [10] have shown to be accurate at 1 ~ for intensities of more than 0.16 W/cm2. At high current densities, the pendulum equation must be modified by the presence of the space charge interaction. Above a current density of about 1 kA/cm2, the plasma frequency in the lab frame becomes comparable to the interaction frequency fl2, and space charge effects begin to dominate. If one operates below this level —

(W~)Iab=

4irne2/y3m

(2)

4112,

so that the space charge modifications of the gain [11] are small, to a good approximation the particles exchange energy with the radiation field independent of their neighbors. The derivation of (1) assumes that the radiation field strength is constant throughout a single transit, which means that the net gain is small. Over most of the expected operating range, the low gain approximation holds, and I will use it here. But it is well to keep in mind that it is possible to produce amplification factors of the order of unity in devices such as this one which ulilizes the first order interaction with the field. Under these circumstances the analysis based on (1) breaks down, and a numerical calculation becomes necessary. Finally, the pendulum equation holds true for a system in which the transverse variation of the fields is small. To take advantage of the simplicity this equation affords, I restrict consideration to systems in which the optical mode size is larger than the electron beam size, and where the undulator magnet fields are independent of transverse position to first order. Provided the filling factor is not critical for reaching threshold, and that distributed focussing is not required within the laser, it is easy to comply with these conditions in any experiment. In order to describe an orbit around the system, we must add to (1) a description of the electron motion in the arc of the storage ring outside the laser. The linear equations of motion in a storage ring are presented in a very accessible way by Sands [12].In general, the RF cavity accelerates the electrons according to their phase =

e V[sin w~(r+

To)

—

sin W~sj~To],

(3)

356

David AG. Deacon, Basic theory of the isochronous storage ring laser

where V is the cavity potential, co~its frequency, r the time displacement of an electron from bunch center, and eV sin(w~ro)is the mean energy lost due to synchrotron radiation. For low energy rings, the radiated synchrotron power is very low and may be neglected to simplify the equations. Electrons in a 100 MeV ring emit roughly 10 eV per pass so that sin WRFTO is typically smaller than 10k, and can be dropped without omitting any essential physics. It is simple to resuscitate this term when it becomes important. In treating the motions of the electrons in the laser, it is most convenient to refer the coordinates of the problem to the laser reference values: that is, to the resonant energy and stable phase at which an electron exchanges no energy with the radiation field. In doing this, we must keep in mind that the laser reference variables can be shifted with respect to the synchronous phase and energy of the ring, which serve as the fundamental references for the system, independent of the laser parameters. I therefore choose as dimensionless coordinates the optical phase 0 to describe the longitudinal position, and the normalized energy Y

(4)

y~(YYr~

4) \

I’

7r

where N is the number of periods in the undulator. Theeffect of the cavity on an electron, in these variables, is 2ck0eV LIE

I

.

0\ (5)

slnwRF~Ti+k)

where r1 is the RF phase of the 21T optical centerchange underin examination. Since so energy small, causephase negligible the acceleration. Towp.j~/kcis describe the variations of 0 through its range of shift produced by the ring on a trapped electron, it is sufficient to replace the right hand side with a constant indexed to the trap number (6) This is the first equation for the motion around the arc of the storage ring. Here, 4ITNeV. Ui

—

E

~

(7)

~0RFTi.

We will find that those electrons trappedin a region of acceleration (positive U 1) will transfer energy into the radiation field, and those trapped in a region of deceleration (negative U1) will absorb from the laser. It will be our goal to trap a large fraction of the electrons in regions of large acceleration so that the power transferred to the laser is maximized. The electrons also experience a longitudinal displacement as they travel around the arc of the storage ring. As will be seen later in (76), every coordinate deviation from the central particlein the bunch produces some kind of displacement. The most important terms are c L~T=

~

4

(~+~) I3~ f3~

aaLa~~

a2aLa ~

Y

(8)

David AG. Deacon, Basic theory of the isochronous storage ring laser

357

where La is the length of the trajectory between two successive laser sections, a~,a~,f3~and 13~ are the horizontal and vertical amplitudes and mean beta functions [12],and aa and a2a are the first and second order momentum compaction factorsin the arc of the storage ring defined in (92) and (93). Let us once again recast this expression into a form more directly related to the laser. The change in optical phase on each successive pass is related to the electron’s longitudinal deviation through the relation ~

(9)

where ~ describes the phase shift produced in one pass by the mismatch between the central orbit and the laser lengths. If T0 is the storage ring period, and T is the laser round trip time, then ~ is defined by the deviation of the two times from perfect alignment T=T0+~/kc.

(10)

If ~ is a multiple of 2ir, the electrons will be shiftedfrom one opticalpotential well to another with no other effects on the motion of the electrons in the traps. Rewriting the phase shift in terms of the deviation from the resonant energy (E — Er)IEr instead of from the synchronous energy (E E~)/E.results in the second equation for the motion in the arc of the storage ring —

~9=—D—A4)Y,

(11)

where the two constants D and A have been defined as 2~

D

~~+~)_~+

kLa{aaE~+a2a

[Er

.}

(12)

E~]

and kLa

I

IErEsl

A~~—~1taa+2a2a[

E.

~

(13)

The parameter D describes the phase shift experienced from pass to pass by a particlewith transverse amplitude a. The parameter A is the normalized momentum compaction in the arc of the storage ring. When A = 0, the total momentum compaction of the storage ring is that produced by the laser section only. When A = 1 [see eq. (94)], the contribution to the momentum compaction from the storage ring exactly cancels that of the laser so that the total momentum compaction of the system is zero, and the particle dynamics become unstable. The laser and the ring act sequentially on the electrons. Although the motion in either part separately can be easily integrated to find a constant of the motion,the combined system does not have this property. Due to the motion of the electrons alternately through the two different sections, H

—

JHLASER ]~HRING

0< z
14

358

David AG. Deacon, Basic theory of the isochronous storage ring laser

the Hamiltonian for their motion is time dependent. If the phase space motion induced by either separate Hamiltonian is very small compared to the total motion in phase space, this time dependence can be ignored, and a single Hamiltonian can be adopted. But in the general case, the time dependence produces quite remarkable effects, as will be seen in two of the examples. In order to handle this kind of discrete problem analytically, a matrix operator approach has been adopted. However, a matrix approach is inherently linear, therebyomitting the important nonlinear character of the solution. To handle these latter effects, I will return to a Hamiltonian formulation, bearing in mind that the discrete effects have been ignored. 2.2. The linearized analysis In the linear approximation, the laser induces a simple rotation on the electron coordinates about 0=0

(°\

=1 cos4) sin4)\j’O\ \Y)~+1 \—sin4) cos4)AY)~

15)

where the laser phase advance has been defined as I1L,/c =4). A specific particle, described by a given energy deviation and transverse amplitude or emittance, undergoes the linearized transformation M=—D—AçbY,

~YU/çb

(16)

on a transit through the arc of the ring. The transformation provided by the entire system is then (°\

(1

—A4)\Jf cos4)

sin4)\(0”~ ~f—D\~

17

\Y)~÷1\0 1 )l\—sin4) cos4)AY)~ \U/4))J’ where the laser acts first, the cavity second, and the ring last. Defining the coordinate vector T, the transformation matrix M, and the source vector S, one can rewrite this as (18)

Tn+1MTn+S.

The static solution is obtained by inverting the matrix M, a simple procedure for two by two matrices T~=(I—M)’S.

(19)

Doing the algebra gives the result T= 1 (—AU+ Usin4)/4)—D(1—2cos4)) S

2(1—cos4))—A4)sin4) \,

Ucos4)/4)+Dsin4)

20 .

(

For small phase advance 4) s 0.1 this becomes T5

((U + 2D)/2~(1 A)) -

(small 4))

(21)

David AG. Deacon, Basic theory of the isochronous storage ring laser

359

which can also be derived from an intuitive consideration of the requirements of stasis. Note that both the coordinate 0 and the energy Y differ from zero at the center of the trap, a characteristic of the discrete system not duplicated in a continuous one. The coordinate of the center increases as the ratio U/4)2, moving into the nonlinear region of (1) as this ratio becomes large. We will see that for large enough values, the system becomes unstable. The energy of the center also has a resonant behavior at A = 1. As this value, which corresponds to the transition energy of the ring, is approached, the trap center disappears, moving rapidly off towards infinity. The stability of the orbits around the center is the key question, and can be determined from the general solution of (18) T~=(I—M)~S+Re~a 1A7P1.

(22)

The eigenvectors F1, and the eigenvalues A, of M define the behavior of small deviations from the center. For these motions to be stable, we must require that the eigenvalues not exceed unity IAII 1. From its construction the determinant of M is unity. Since the product of the eigenvalues equals the determinant, detM=flA1=1,

(23)

it follows that the eigenvalues must be pure phases in order to achieve stability. Evaluating the trace of M and defining A exp(iifr) TrM=~A1=2cosçfnS2

(24)

implies the restriction ITrMI=I2cos4)+A4)sin4)I2.

(25)

The requirement that electrons in the neighborhood of the fixed points remain near it restricts the acceptable parameter region to the area between the boundaries A = (2/4)) tan 4)12,

(26)

A = (—2/4)) cot 4)/2.

(27)

and

The stable region is shown in fig. 3, where the product A4) is plotted against 4) to produce symmetric stability regions. A is the slope of a line drawn through the origin. If the intensity builds up, 4) increases, and the operating point moves out along the line with slope characterized by the momentum compaction of the system. The system is stable for low intensity, but goes unstable for 4) greater than a critical value. Higher order stable regions also exist, but if they are to be useful, a means must be found to bring the system into these regions and maintain it there stably. When the system parameters are located in the stable region, traps exist near 0 = 0modulo 2ir, and electrons can be stored in these closed orbit regions. It is apparent from the figure that the preferred operating point lies in the neighborhood of IAI ~ 1.

360

David A.G. Deacon, Basic theory of the isochronous storage ring laser 44’

I I

I

:1

10-

c

4

.1

~t

•

4.4’!. S’~

21T

I STABLE

5

/“UNSTABLE OPERATING LINE

—0

I1:..:

(CONSTANT A)

\~ Fig. 3. The region of stability of the optical traps is shown as a function ofthe normalized momentum compaction A and ~,which is proportional to the fourth root of the laser intensity. If the intensity changes, the operating point moves along a line of constant slope A.

For negative A, some stable region always exists at low intensity even for very large momentum compaction, although the available operating range is reduced, and the trap size, as we shall see, becomes very small. This is an important characteristic of the system because it allows operation on some level far from the optimum operating conditions. Here, the operating point hits the first unstable boundary at 4’max = 2/VA.

(28)

The maximum laser intensity varies inversely as the momentum compaction. If the ring is designed isochronous in the arc, the laser intensity can be increased without bound, provided the system is not operated near the unstable points at integer multiples of ir. For A > 1, stability is achieved around the 0 = 0 point only at high intensity, where the phase advance must remain near 4) = nir, n 0. Here, the system is unstable to fluctuations in the power density, and is likely to be unsuitable for operation. The phase advance per pass in the overall phase space 4’ is determined from (24) and (25) as cos4i=cos4)+~A4)sin4).

(29)

On the edges of the stability region 4’ is a multiple of ir. As the intensity increases, 4’ increases monotonically in steps of ir as the operating point passes through each successive stability region. The character of the orbits can be obtained from the eigenvectors of the transfer matrix M Diagonalizing yields the unnonnalized eigenvector P = (x’~1), where

x=

—A4)/2

—

iVi

—

A4) cot 4) A24)2/4 —

(30)

and the eigenvalue with the positive exponential has been chosen. The transient portion of the response (22) therefore behaves like

David A. G. Deacon, Basic theory of the isochronous storage ring laser

Re ~I.

(x”~T cos(n4i) + sin(n4’)V1 \l/J = (—~A4) \ cos(n4r)— Açb cot 4)

—

~A2ç62”~ /

361

(31)

For a given Y, the coordinate 0 has both in- and out-of-phase components, producing the skewed motion shown in fig. 4. Both the angle of the axis of symmetry and the area of the orbit are functions of the operating point in the stability diagram. The area of the entire curve is equal to the area of the out-of-phase component, or —ir Im(k’). In the linear approximation, all orbits have the same shape and the rotation frequency is independent of amplitude. In the complete problem, the nonlinearity of the interaction (1) reduces the rotation frequency and distorts the orbits progressively as the amplitude is increased. There exists a maximum amplitude for closed orbits determined by the trajectory which passes through the nearest saddle point in the phase plane. This curve encloses the region of stable orbits and determines the area of the optical trap. Even though the phase extent of this curve is constant, its area depends on the orbit shape through 4) and A. The functional dependence of this area variation can be approximated with the results of the linear analysis. The area of the traps in the nonlinear system will scale in the same way as the area of a linear orbit which has the same phase extent. Normalizing the linear trajectory to unit phase extent gives the functional dependence of the orbit area Area = ir V1—A4)cot4)—A24)2/4 1 — A4) cot 4~

phase extent).

(unlt

(32)

While this expression includes the nonlinear effects in only an approximate way, it contains the discrete effects which cannot be obtained from a Hamiltonian approach. It is desirable to maximize the area in order to fit the largest possible charge into each trap. As the operating point moves out towards larger 4), the area within the phase space curve goes to zero. The region in the neighborhood of the lower boundary is therefore unfavorable for operation, and sets a limit on the laser phase advance 4). The value of A should be boosted up toward zero in order to maximize the phase advance available to the system. At A = 0, the trap area is independent of the phase advance, and no limit need be placed on 4). The region near the upper boundary is somewhat different. The determinant Det(I M) vanishes on the boundary so that the center (2) moves away to infinity, producing an unstable system. On the other hand, we find that the area of an orbit with unit phase —

Y 0/b

O~Vc-A~CO~ 4I _A2#2/4 b=’JI-AI~

cot 4’

Fig. 4. The electron orbit shape is constructed from the eigenvectors of the transfer matrix. This description is valid near the phase stable point in the center of the optical trap.

-

362

David AG. Deacon, Basic theory of the isochronotss storage ring laser

extent expands without bound as the transition is approached, Area~ir/V1—A

(4)41).

(33)

This means additional trapping area can be obtained if one is willing to reduce the stability of the system. This possibility is likely to be of limited value in a real structure. From these two considerations, we see that the optimal momentum compaction lies in the region around A 0, which corresponds to isochronism in the arc of the storage ring. A region of stability also exists near the unstable fixed point of (1) at 0 = ir. We can examine this behavior by linearizing the laser equation about the “ir” fixed point. The analysis of the motion here goes through in the same way, with the exception that the laser transformation now is composed of hyperbolic functions instead of circular (0’\ (cosh4) \YL~+i \sinh4)

sinh4)\(0’\ cosh4))\Y)~

(0’=O—ir)

(34)

As before, the stability criterion is that the magnitude of the trace of the transformation matrix be less than two TrMI= 2cosh4)—A4)sinh4)I<2.

(35)

The stable operating region is presented in fig. 5 where it is compared with that of the center we considered previously. The extent is much smaller, as might have been expected, but for small 4), the region completes the filling of the plane above the transition energy. On the far left of the stability diagram, the stable operating regions about the two fixed points fill a symmetric region Al 4/4)2• This means some operation is possible at any energy, no matter how far from transition, and no matter in which direction.

I~~

~5.//TI(
Fig. 5. The additional stability region for orbits about the “ir” fixed point is superimposed on the larger ‘0” stability region.

David AG. Deacon, Basic theory of the isochronousstorage ring laser

363

2.3. Nonlinear effects The nonlinearity of the laser interaction reduces the frequency of rotation in the traps as the amplitude grows. Even for motion which is stable in the linear region, there exists a maximum amplitude within which stability is retained. The most direct way of describing this effect is with the Hamiltonian approximation discussed earlier in this section. Restricting consideration to systems with small phase advance per turn allows the generation of an approximate constant of the motion, neglecting all discrete phenomena. Combining the laser with ring effects (6) and (11), gives the constant of the motion for the complete system lU\ 1—Al D \2 H~0~=—cos0—(,~~)0+ 2 k~4)(1—A))

.

(36)

The first two terms represent a potential in which the particle travels. Figure 6 shows the potential and the phase space trajectories it establishes for D = 0. The laser produces the periodic modulation which is responsible for the electron trapping. The voltage U imposes a slope in the periodic potential, separating the closed orbit regions of the successive traps. The extrema of the potential 2, (37) sin 0 = U/4) define the fixed points of the system. The separatrix crosses itself at the unstable fixed point, and encircles the stable fixed point. The saddle and center points are symmetrically placed about 0 = ir/2 modulo 2ir. Clearly there is a limit on the magnitude of the RF acceleration imposed by (37). As the voltage is increased, the slope imposed on the potential grows larger, the two fixed points move closer together, and the region of phase space enclosed by the separatrix eventually will vanish. The voltage, which is proportional to the energy extraction rate, must be kept smaller than

U/4)21

(38)

in order to retain trapping area. V

POTENT I AL

ORBITS

Fig. 6. The effective potential in which the electrons travel is sketched above the phase space orbits defined by the potential. This picture is valid only when the discrete effects are negligible; that is when the laser phase advance .j, is small.

364

David AG. Deacon, Basic theory of the isochronous storage ring laser

The normalized energy half aperture of the trap can be calculated from the Hamiltonian: 2

_____

/

V1—A\v

I

2

TI

4)4

4,2

TI

TT~1/2

(39)

4,2/

The parameters influencing the aperture are the momentum compaction A and the energy extraction rate U/4)2. For small U/4)2, the dependence is the same as derived from the linear analysis. The additional effects of the nonlinearity of (1) limit the longitudinal extent of the traps to a value depending only on the voltage through the ratio U/çb2. Since it is the voltage which is operationally fixed, the size of a trap will be a strong function of its RF phase and the power level in the cavity. Given a fixed operating power level, traps exist out to a maximum RF phase, each having a size according to its value of U/4)2. Before the system reaches its operating power level, 4, can be very small so that only a few traps exist near r = r 0. As the power grows, more traps are established, and a larger proportion of the electrons contribute to emission through the first order interaction. 2.4. Transverse motion The transverse motion of a particlemoving around the arc outside the laser alters the length of the path it travels, and induces a phase shift on injection into the subsequent laser section. This effect is either constant or oscillatory; the more important constant phase shift is dealt with first. The constant phase slip contributed by the emittance e and the laser-ring misalignment ~ produces two effects. The center of the trap is displaced from the resonant energy an an amount linear in D (20), and the energy lost to the radiation drops as the square of the displacement as will be shown. The phase slip produced in the ring is compensated in the laser for electrons with the right energy offset from resonance; they experience an equal and opposite phase slip there, and are trapped around a new phase stable point with an energy increment (21) Y=D/4)(1—A).

(40)

The different populations of electrons occupy traps vertically displaced from one another by an amount characterized by the transverse motion. An electron injected into the laser with an energy deviation sweeps over a certain range of phase during the transit of the laser. Its interaction with the laser is therefore averaged over this phase range. Integrating (1) for small 4,, the energy exchanged with the laser becomes ~Y=—4)sin(0+ y4,12)SiflY4)/2

(41)

where 0 and Y are the coordinates of the electron on entering the laser. This energy exchange is reduced from the on-resonance result by the sin x/x factor, and phase shifted by Y4)/2. Electrons moving in the vicinity of the phase stable point (40) therefore interact less strongly with the field by virtue of their energy displacement. Figure 7 illustrates this effect for a fixed RF voltage. In order to ensure the existence of a closed orbit region for a population with a given transverse amplitude, the maximum energy loss in the laser must exceed that gained in the RF cavity

David A.G. Deacon, Basic theory of the isochronotes storage ring laser

sin[D/2(1— A)] >U [D/2(1—A)]

365

42

4’2

Particles with a larger transverse amplitude find that the acceleration produced by the RF cavity is always dominant, and they are swept away: no possibility exists for trapping. The limiting value for Y produced by this effect is sketched in as the dashed line in fig. 7 for a trap experiencing the voltage U/4)2 = 1/2. As the traps are displaced upward, their size diminishes quadratically, disappearing entirely at the limiting displacement (42). In order to produce an equilibrium situation in which all of the particles are trapped, the emittance in each plane and the length mismatch must be maintained sufficiently small. The requirement depends on the local RF voltage, but if one requires the emittance to remain less than one sixth of the upper limit for a trap with a moderately large acceleration, say, U/4s2 = 1/2, the restriction is D~3.8(1—A).

(43)

Any displacement of the traps will have another deleterious effect which must be kept small. When the laser is operating, the electrons must be able to find their way into the traps from the equilibrium distribution in the storage ring. Should the displacement become large enough so that the trap aperture no longer encloses a significant portion of the electron distribution, it becomes difficult to fill the traps when the laser is turned on. A phase shift ~or a resonant energy deviation Er E 5 0 can cause the trapped particles to move away from the synchronous energy. The emittance is not operative here because the synchronous energy itself depends on the energy of the particle. The deviation of the trap center from the synchronous energy is —

‘V — Vs

‘Vs

—

(‘V —_‘Vr\ + (Yr —_‘Ys\ ‘Vs ) k ‘Vs )(1A$irN

+

1 (Er E. 1—Ak E. —

If the trap is small, we must require that it not be displaced by more than the natural energy spread, and if it is large, that it not be displaced by more than its aperture. These two restrictions imply —

~ + 4irN

(Er

Es)

<4irN(1 A) ~ —

(small traps)

(45)

~mox

2. Fig. 7. Thedependence is shown of the maximum electron energy loss in the laser asa function ofthe displacement of the electron injection energy from the resonance Since transverse energy.For motion displaces trappingthe to be trap possible, centerthis to larger energyenergy loss must deviations, remain larger this effect than limits the acceleration the acceptable produced amplitude by theof RFtransverse cavity, in motion. units of U14i

3~

David AG. Deacon, Basic theory of the isochronous storage ring laser

—~+

4irN (Er_Es) <4,V1 A

(large traps)

—

(46)

whichever is less stringent. If D is large, there may be a number of different closed orbit regions for a given population in transverse motion. As D becomes larger than ir, it becomes possible for the electrons to hop from one trap to the next at a smaller value of Y. Apart from the trap hopping, the motion of these particles is identical to that of particles with aD value reduced by 2ir or a multiple thereof. Particles which move in this fashion will be transported along the chain of traps until finally moving beyond the trapping region of RF phase. At this point, they maybe lost to the walls of the ring or be recirculated in the RF bucket to become retrapped. If the large D value is caused by the emittance, particles will spill out of the line of traps in both directions. To avoid this situation, the emittance must be kept well within the limit (43). If the largeD value is caused by the path length effect ~, the entire trapped charge may be induced to slip optical phase together, slowly shifting RF phase. This procedure does not appear to have great utility as other means exist to shift the RF phase. Again, it appears that (43) is a limit which should be obeyed. 2.5. Resonances with the transverse motion

The storage ring also produces phase shifts with a magnitude that oscillates due to the phase advance of the transverse motion. Provided that the magnitude of the oscillatory shifts is at least as small as the constant phase shifts discussed earlier, the particle will remain trapped and the average contribution is zero. However, it is possible for the oscillating contributions to resonate with the particle’s motion around the trap (fig. 8), amplifying the effect of the displacements. This effect, combined with the nonlinear nature of the interaction, gives rise to a characteristic limit cycle behavior near resonance. The various pieces of the longitudinal displacement contain oscillatory terms whose phase differs by 2iw, 4irv, 6iw, and so on, one for each order in (77), on each successive pass through the laser. Each of these terms can resonate with the particle motion in the trap, which progresses with a phase advance of V SE PARATRIX

/

/

/

/ I,

//

6

~ RESONANT DISPLACEMENTS

Fig. 8. Thesuccessive longitudinal displacements induced on a particle as a result ofits transverse motion are superimposed on the orbit ofthe particle in phase space at the instants they occur. If, as in the sketch, the direction ofthese displacements oscillates at the same rate asthe particle on itsphase space trajectory, resonance occurs, and the amplitude of the particle’s motion in the phase space increases. Asdiscussed in the text, Landau damping limits the amplitude of motion induced by the resonance.

David AG. Deacon, Basic theory of the isochronous storage ring laser

367

if’ (29) on each pass. If the system is near resonance, n 2irv

if, modulo 2ir

(47)

the resonant term will increase the amplitude of the particle motion in the trap until the relative phase shifts by ir whereupon the amplitude decreases again. So long as the particle remains within the linear orbit region, the resonance causes the particle orbit to “breathe” at the beat frequency. At n 2irv = mi/I, m > 1, the effect averages out in a few cycles around the trap. For larger amplitude excursions, the nonlinear nature of the interaction acts in a self-limiting way on the size of the breathing oscillations. As the amplitude of the particle oscillation grows, the laser phase advance is progressively reduced from the linear value of 4, to an effective value of zero at the separatrix. Therefore when the amplitude is pumped up by the transverse resonance, the relativephase of the perturbation shifts by ir, and the particle amplitude is again reduced. Even if the particleis directly on resonance with one of the oscillating displacements at small amplitude, the resonance will be stabilized at some mean amplitude smaller than the separatrix where a limit cycle behavior will develop. Instead of depopulating the trap, this phenomenon tends to pump up the electron distribution to a larger amplitude motion in the trap, with magnitude dependent on the detuning from resonance and the size of the perturbation. Although ahollow distribution will be produced within the trap, this effect does not appear to degrade the performance of the system. There is a very similar effect in the energy dimension. The angular deviation of the particle at the start of the laser reduces its longitudinal velocity and shifts its resonance energy for interaction with the laser ‘V~=~f[1+K2+y202].

(48)

The angle of a particle oscillates with a phase shift of 2in’ on each pass due to the focussing of the magnetic transport system. The resonant energy seen by that particle therefore oscillates with a phase shift of 4iw, and a magnitude y202

~!—

Yr

ya

49

2(1+K2)4I3*(1+K2)~

( )

Seen from the point of view of the trap, the effective energy deviation per pass of the particle oscillates with a magnitude t,

)eff

—

Niry’2a2

50

Provided that the energy kick per pass is small compared to the trap size Niry2a2i/i ~ 1 2) Vt—A 4,/3*(1+J(

1

(4)4 )

51

( )

the particle will remain within the trap for many circuits of the ring. As before, a limit cycle behavior

368

David AG. Deacon, Basic theory of the isochronous storage ring laser

will be seen near resonance, with the only difference being that the driving force is in the energy dimension. If (51) is satisfied, this phenomenon should not be detrimental to the performance of the device. The effect becomes more important for small traps, but if necessary in any specific case, (51) can always be satisfied by enlarging the beam over the interaction region to reduce /3111• 2.6. Numerical simulations

To verify the foregoing analytic results and extend them to the general nonlinear discrete system, a numerical simulation has been performed. These results are not hampered by the small 0 or the small 4’ approximations. We will see that the analytical techniques discussed above are adequate to the task of describing the behavior of the system through the main part of the accessible operating region. Figure 9 shows the most simple realization of the optical trap which is physically interesting. With the dimensionless energy and phase on the vertical and horizontal axes, this figure shows the trajectories of the electrons in their one dimensional phase space. To obtain this figure, I chose several initial conditions and calculated the subsequent motion of the electrons using the full equations of motion for the system (1), (6) and (11), with A = 0 and D = 0, but with U/4i2 = 1/3. Each dot in the figure thus represents the electron coordinates at the entrance to the laser on one of 200 passes. The successive coordinates are not adjacent to each other because the system phase advance is quite large and several complete oscillations around the trap have been made during the simulation. It is clear that a large region of stability can exist in the presence of substantial RF acceleration. For reference, the solid line is the separatrix for the motion in an infinite, unperturbed laser, as described in fig. 1. The separatrix for the motion in the perturbed system is very close to the outermost particletrajectory shown. As expected, this orbit comes to a point at the unstable fixed point.

2~

~

-‘H

~

(“f ~ ~

~>N~ -H

OPTICAL PHASE ® A~O, 4,rO.3, U/4 2~O.33 D=O 1 Fig. 9. Resultsof a numerical simulation of an electron’s longitudinal motion in an optical trap. The coordinatesof anelectron at the entrance to the laser are plotted on manysubsequent passes,tracing out aclosed curve in phase space. Severaltrajectories have been chosen todemonstrate theshape and area of the trap. The outermost curve is traced by an electron which travels very near the separatrix of the trap, as indicated by the sharpness of the corner formed by this trajectory near the unstable fixed point.

David AG. Deacon, Basic theory of the isochronou.s storage ring laser

369

The center of gravity of the orbits is shifted to the right so that emission, which occurs in the positive 0 region, begins to dominate over the absorption in the negative region. The particles oscillate about the phase stable point (20) at which the energy transfer is always in balance, with maximum amplitude determined by the rolloff of the nonlinear potential. The separatrix is traced out by the particle at the unstable fixed point Omax = IT — sin1(U/4)2). If the cavity voltage is increased, the nonlinear effects become more important, effectively reducing the potential well depth as discussed below eq. (37). In fig. 10, the orbits are shown for a system with twice the acceleration as in fig. 9, which I treat as a reference point in the discussion to come. As expected, the two fixed points have moved closer to each other, reducing the trapping area by about a factor of three. The ratio U/4,2 in this case is 2/3, close to the absolute maximum value (38) at which the area disappears altogether. Reducing the value of 4, directly reduces the depth of the optical potential well, and identical trajectories to those in the figure are produced by cutting 4) and U down by a factor of two from the baseline value. The effect of nonzero momentum compaction is demonstrated in fig. 11. On the left, the momentum compaction has been chosen negative so that A = —5. The optical phase shift produced by the ring adds to that of the laser so that the orbits are squeezed down, while the locations of the fixed points remain essentially unchanged. On the right, aa is positive with A = 0.7, but the system remains below transition and in the stable region of fig. 2. Here, the longitudinal displacement induced by the ring subtracts from that of the laser, expanding the energy extent of the orbits. If the slope A is pushed further toward unity, the orbit size blows up and the stable fixed point moves off to infinity as the trap becomes unstable. Figure 12 demonstrates the effects of transverse motion on the trap positions and size. Superimposed on the same graph are shown the motion of five different populations with progressively increasing phase shift D. The quadratic reduction of the trap size is apparent, as is the limiting displacement indicated by the dotted line. U/4,2 has been chosen large (0.83) for clarity. Smaller values of acceleration, of course, produce larger traps and a wider range of phase shift over which the traps exist.

N

O -2

H -2

~

H

~.

0

2

OPTICAL PHASE ® A~O,4~O.3,U/q2~O.67,D~O Fig. 10. Calculation of trajectories for a larger voltage.

370

I’

~~

ii:

-~

-~

OPTICAL PHASE

e

~

OPTICAL PHASE lEt A~O.7

4’r 0.3 U/4’2 r033

Dr 0

Fig. 11. The effect of a positive and a negative value for A is calculated; compare with fig. 9.

8

‘‘H’’H’’ Y’6.85

—

6—

—

1-

-2

I

0

2I

OPTICAL PHASE 9 A~O, 4~O.3, U/4’2~O.83 D~2.O,1.5, 1.0, 0.5, 0.0

Fig. 12. The electron trajectories are calculated for a series offive values ofthe phase shift D.This figure demonstrates that ifany of the components ofD become large (the transverse amplitude a, the orbit length mismatch ~,or the energy deviation F, — E,), the trap is displaced from the resonance energy, and shrinks from its optimum size.

David A.G. Deacon, Basic theory of the isochronous storage ring laser

371

While the above figures describe the behavior of the system that the “0” fixed point, the smaller stability region about the “ir” fixed point also supports large aperture closed orbits. Figure 13 shows an example of the behavior in this region. The two fixed points have interchanged roles and the direction of rotation has reversed. Increasing the momentum compaction reduces the size of the traps as before, until the unstable boundary (27) is reached, at which point the system again goes unstable. The behavior of the system in this region is very similar to that in the larger region of orbits around the “0” fixed point, so I concentrate henceforth on the larger stable region. Figure 14 shows the very complex and interesting behavior which occurs for large system phase advance 4, where the time dependence of the Hamiltonian becomes important. The dots incicate the electron coordinates on successive passes as before. Near the central region of the plot, the electron traces out a complete, closed orbit as it passes many times through the system. Two successive points may lie across from each other on the curve but in many passes the coordinates “precess” around the complete curve.

-3 0

2

4

OPTICAL PHASE

6

e

2~O.33,D~O

A’2, 4~O.3,U/4’

Fig. 13. The shape of the trap produced in the “ir” stability region of fig. 5. The trap is located near 0

=

3

I

IH,~N~o!

Hw~2H

W2~

~

3 OPTICAL PHASE 9

4r

H

0.83, 4’rI.2, U/4’2rO.083, D~O

OPTICAL PHASE 9 4rQ~59,4~rI.7,U/42r0.059,

Fig. 14. Quasi-stochastic orbit behavior observed for large system phase advance 4’.

DrO

ir.

372

David AG. Deacon, Basic theory of the isochronous storage ring laser

Farther from the central point, the curve traced by the electrons breaks up into several pieces, four in the left plot, and three in the right, producing a phase space footprint with several toes. The central point in each of these subsidiary regions is also a fixed point, but maps into its fraternal fixed points on each pass through the system, only returning after a characteristic number of passes. These are higher order fixed points, but it is clear from the simulation that they act as centers for stable oscillation much like the first order fixed point they surround. The oscillating electrons also jump from toe to toe on consecutive passes, but the coordinate in any single toe traces out a single closed curve. Surrounding the four toes on the left side of fig. 14 is another single closed curve, itself surrounded by a seemingly stochastic layer traced out by a single electron. The simulation illustrated here was not continued long enough to define the boundaries of the stochastic layer more precisely. On close examination, several voids can be discerned, imbedded within the perimeter of the stochastic layer. In a subsequent check, it was found that these areas contain additional closed orbit regions. Fixed points of order five and nine were found on the first try, and I am sure many others exist as well for these parameters. This example demonstrates in a striking way the difference between a true discrete system and the approximate time independent Hamiltonian system. The time independent conservative system evolves along smooth trajectories defined by the constant of the motion H and never deviates from these orbits. On the other hand, the discrete system is similar to a driven Hamiltonian system. The perturbation of the storage ring on the electron trajectories is switched on and off as the electron cycles out of the laser and back again. This time dependent perturbation can be Fourier analyzed into its components, each of which creates a resonance in the nonlinear system. The nonlinearity stabilizes the resonances, but if the nonlinearity is strong enough, it mixes them together, producing higher order resonances, and the phase space trajectories become exceedingly complicated. The nonlinearity in this problem is produced by the laser field. For small intensity, simple curves are generated such as we have just observed (figs. 9—13). However, as the intensity is raised enough to move the operating point in fig. 3 near the unstable boundary, the resonances become dominant and behavior like that shown in fig. 14 appears. When the particle moves under the influence of a number of interacting resonances, it can wander about the phase space in an apparently aimless fashion completely filling the region of phase space dominated by the resonant terms. This is the origin of the stochastic layer. A good introduction to the theory of this type of behavior is presented in refs. [13, 14]. The importance of this phenomenon to the storage ring laser lies in the presence of the stochastic layers. The nonlinear system becomes unstable in a way closely analogous to the beam-beam effect [15] which limits the current in colliding beams. If the particle can move freely in a band which extends outside the stable region of the entire system, then it can get lost from the RF bucket and run into the vacuum chamber. The RF bucket leaks as charge diffuses along these channels. While the diffusion is restrained by the synchrotron damping, its rate cannot be allowed to exceed a certain maximum value if the laser is to operate on a continuous basis. As a minimum, the lifetime of the storage ring must be maintained above the injection time, which is minimally on the order of minutes. The stability of the system at high power density is an important problem remaining to be resolved. 2.7. Laser power and the trap population In a storage ring, the ever present synchrotron radiation both excites and damps the energy distribution of the stored electrons. The natural energy spread o-~is determined [12] by the balance between these two effects, and depends on the operating energy and the storage ring design parameters.

David AG. Deacon, Basic theory of the isochronous storage ring laser

373

The distribution of charge in phase space then depends on the shape of the trajectories. In the absence of the laser, the trajectories are large, smooth, concentric ellipses which are filled with electrons up to the energy spread 0e. The aspect ratio of the trajectories combined with the known value of 0e therefore set the length of the bunch. With the laser on, the electron trajectories describe a long series of traps approximately one laser wavelength wide, and with an energy dimension as described above. When the system is at equilibrium the presence of these traps greatly distorts the distribution from its earlier single-bunch form. If the energy aperture of the traps is made larger than the energy spread of the electron beam, at equilibrium all of the electrons will have been captured in one or other of the optical traps, and the initially uniform distribution will have redistributed itself into a long chain of microbunches. This will be a stable situation if the trap aperture is sufficiently large — say, 6 0e — to produce a long lifetime against diffusive depopulation. This is the most interesting isochronous storage ring laser configuration, and an example of the parameter magnitudes required to produce this state of affairs will be given shortly [16]. If the great-majority of the electrons have been trapped, the expression for the laser power becomes very simple. On the average, each trapped electron delivers as much energy to the field as it picks up from the RF cavity (neglecting synchrotron radiation). The power emitted is therefore equal to the product of the charge in each trap and the rate of absorption of energy from the RF cavity, integrated over all traps: Pextracseci

I

eV.

J d0p(0)~~5ln w~.j~r(0).

J

(52)

10

The integral extends over all of phase space, and p(O) is the equilibrium electron density per unit laser phase.

There may also be regions of operation in which the system sets up optical traps considerably smaller than the energy spread o~.In this case, some electrons are trapped, and some continue to circulate in the RF bucket, following perturbed trajectories around the traps, and sampling all laser phases in each orbit. These two groups of electrons behave very differently. The trapped population contributes strongly to the extracted power as in (52). The untrapped population however, suffers all the ills the isochronous system was designed to eliminate. Its energy spread is blown up by filamentation [3,4, 17—19], and, being uniform over optical phase, it can only contribute to the extracted power through the second order mechanism. Under these conditions, the expression for the power becomes Pextracte~j’

J trapped charge

d0p(0)~sinw1~~r(0)+I

J

dYp(Y)G(Y,I)~

(53)

trapped charge

where in the second term, I is the intracavity laser intensity, p(Y) is the charge density per unit normalized energy, G(Y, I) is the saturated gain as a function of energy , and .X is the transverse electron beam area. The second term will in most cases be quite small, and is almost zero unless the synchronous energy of the storage ring is set somewhat below the resonance energy. To determine the form of p (Y) requires a self consistent calculation along the lines of that performed by Renieri [3],but including the orbit distortions produced by the laser, and remembering that the partition fraction between the trappedand the untrapped charge also depends on the energyspread of the untrapped charge.

374 2.8.

David A.G. Deacon, Basic theory of the isochronous storage ring laser

Starting and operating the laser

Imagine for simplicity’s sake that one suddenly introduces an external laser beam into the optical cavity of the laser, making large traps that completely enclose a~. It is not necessary to initiate the system in this manner, and we will return to the more general case but let us consider this situation as a simple illustration of the laser process. Some of the electrons will be trapped immediately. That portion of the initial distribution which falls between the traps is then forced to follow the perturbed trajectories leading around them. This charge now moves on a higher energy orbit, and begins to damp back down to its equilibrium energy spread. Since the lowest energy orbits lie inside the traps, the distribution forms into microbunches in a few damping times (fig. 15). No power can yet be extracted from the system as the charge is symmetrically distributed about the synchronous RF phase. Some of this charge must be transported from the absorbing regions where it is being decelerated by the RF cavity, to the emitting regions. This can be made to happen automatically if the system is aligned so that the relative path length deviation ~ is zero. Then the same electrons see the same photons on each pass, the absorbing traps disappear, and the emitting traps grow or sustain themselves. The absorbing electrons are released as their traps collapse, and in another few damping times relocate in one of the emitting traps. In the time required to absorb the initial radiation plus a few damping times, the electrons have redistributed themselves into microbunches along the leading edge of the bunch where they emit strongly into the laser. Once the electrons are stably trapped, the frequency of the RF cavity can be shiftedmomentarily to move the microbunched beam to a different RF phase, thereby adjusting the output power (52) of the laser. Care must be taken during this process that the path length deviation ~ does not grow too large,

l~I=

~ Il—Al.

(54) ELECTRON

LASER OFF

DISTRIBUTION

y~

—

WRF

(r—r

0)

LASER ON (t ~TD) —

I

II

—

____I

Fig. 15. The electron distribution is sketched with the laser off (top),and one damping time after the laser turns on (bottom). The charge has redistributed itself into the traps created by the laser field, which are assumed in this figure to be large compared to the equilibrium energy spread.

David AG. Deacon, Basic theory of the isochronous storage ring laser

375

Under optimum conditions, the induced phase slip can be a fraction of an optical wavelength per cycle. The time required for a large change in the output power by this method would be on the order of tens of seconds. The amplitude of the RF can also be changed to modify the laser power. The only restriction here is that the voltage be increased adiabatically with respect to the total phase advance per pass in the trap 4,, (55) If the voltage is changed any more rapidly, charge may be lost from the trap on increasing the voltage, and coherent oscillations in the trap will be excited. Of course, care must be taken to remain within the maximum acceleration boundary set by (38) and the minimum acceleration boundary set by the requirement that the traps be self-sustaining in the presence of the fixed losses of the cavity. If the ring parameters are not optimal, the size of the traps is constrained by (39) and (40) to a value which may be substantially smaller than 0e. Alternatively, if the laser is to start as an oscillator, the field must build up from a small initial value which is insufficient to make large traps. In either situation, only a portion of the electron distribution is trapped, and the remainder circulates outside on trajectories scalloped or made stochastic by the optical field, depending on the energy deviation. The charge distribution under these circumstances must be calculated by a rather involved self-consistent procedure, as mentioned above. However, some simple observations can be made at the outset. Any radiation field in the cavity, such as the spontaneous radiation, will establish a set of small closed orbit regions since the energy aperture of these regions only drops as the square root of the intensity. The synchrotron damping tends to enhance the charge in these regions, increasing the field which produces them. The lasing instability is produced if the cavity losses absorb less energy than is radiated by the charge enhancement in the incipient optical traps, P(I)’Z>yI

(56)

where P(I) is the power radiated according to (53), and ‘V is (in this equation only) the cavity loss per pass. The power emitted by the trapped charge scales linearly with the trap volume until the charge density begins to change appreciably, and with the number of traps times the RF voltage at each trap. The longitudinal extent of a given trap is roughly constant once established while its energy extent varies linearly with 4,. The number of traps depends on the RF phase extent within the limit (38) and therefore varies as 4,2 The mean accelerating voltage will also vary as 4,2~The emitted power therefore varies as 4)5 or P(I) cc I~”~ so that once above threshold, the intensity will grow faster than exponentially. As the laser field grows, two contrary processes takes place. The trapped charge grows as noted above, increasing the total emission. But in addition, the energy spread of the untrapped charge is progressively augmented as the intensity grows, reducing the partition fraction between the trapped and the untrapped charge. This latter effect becomes important after the laser induced energy spread becomes comparable to the natural energy spread. Beyond this point the untrapped energy spread grows as the square root of the intensity so that the trap size and the untrapped energy spread grow at the same rate. For a certain period of time, the total trapped charge will remain almost constant. Further charge flow into the traps will now occur only as the size of the traps approaches the unheated energy spread of the electron beam. The reduction of the charge density of the colder trapped

376

David AG. Deacon, Basic theory of the isochronous storage ring laser

distribution at the interior of the trap boundary allows net diffusion out of the hotter distribution surrounding it. If the energy aperture of the system is not large enough, permanent charge loss will occur as the hotter particles in the untrapped distribution are driven into the sides of the storage ring. For a given set of parameters, the system will saturate somewhere along this path at a point determined by the total current initially stored in the ring. The laser intensity and the trapped charge will grow until the total emission equals the losses to the laser cavity. A stable situation is established if, as discussed previously, the traps grow large enough to complete enclose the equilibrium distribution. Under these circumstances, the untrapped electrons will disappear by diffusion either into the traps, or into the walls of the storage ring. At a smaller trap size, another stable situation can be established below the intensity level necessary to excite the particles beyond the energy acceptance of the storage ring. Here, the interior and exterior populations are connected by diffusion across the trap boundary, and the laser power is determined by (53). If the system parameters, the initial stored current, and the cavity losses do not allow the establishment of a completely trapped distribution, charge leakage from the two connected distributions will eventually produce the stable small trap situation with a laser intensity insufficient to cause further charge loss.

2.9. Nonlinear effects on the RF trajectories

Up to now I have considered the motion of the electron on the scale of the optical wavelength, and assumed that if particles escaped from the optical traps, they circulate in the RF potential. In this subsection, I consider the effects of an isochronous design on the large scale electron motion. With our goal of creating optical traps, we have found it necessary to reduce a by several orders of magnitude over that of existing rings. It is important to consider the effects of the higher order energy terms because they are no longer swamped beneath a large first order term. In general, the presence of higher order terms in (8) create additional lines of fixed points in phase space. The equations ~‘V eV = E [sinw~(r— ro) sin u~r0] —

(57) föy\ T~l~_ra2~_)

16’V\2

/e5’V\~ -ask--)

describe the motion of the electrons in the ring with the laser off, and have fixed points at I

~

Wpj4,7 — To)

= 1o~,~ro+n2IT . tsin (1r—w~ro)+n 2ir

(58)

and 8y y

~ 0 zzrj_a/a2.

Here, a and a2 are the total momentum compaction factors of the storage ring.

(59)

David AG. Deacon, Basic theory of the isochronous storage ring laser

377

V-

<
Vs

-)~~z~x~ ~‘“~4~’~

C>>I

6 2

-

~

W~

2 6

Fig. 16. Theeffect of ~on the orbits in the storage ring phase space.When ~is small, the two bands of bucketsare well separated. When ~is large, the two bands come together, changing the form of the buckets and reducing the bunch length and energy aperture.

The so-called buckets formed around these fixed points are the RF stable regions. At the top of fig. 16, the buckets formed by two lines of fixed points are shown for the case of a large separation between the two lines. The presence of the second order term introduces the new set of buckets displaced in energy from the original set by an amount proportional to the momentum compaction a. If the ring is designed to be nearly isochronous, a becomes small, the buckets approach each other, and as their separation becomes smaller than their aperture, they begin to interfere with each other. From the Hamiltonian approximation (almost always valid for storage rings), the half aperture of the principal line of buckets would be V(2ev’/hiraE5) if the adjacent lines were removed. But the energy displacement —a/a2 to the nearest secondary line can be very small. The number which characterizes the operating regime of the system is the ratio ~ of the aperture of the unperturbed bucket to the separation of the fixed points lines =

—

height . of separatrix (a2 . =. 0) —— separation of fixed point lines

~v~rsITz~5 ~, a2 ~...e 3/2 a

.~,,

~

If is small, then the two lines of fixed points are each unchanged by the presence of the other, as illustrated at the top of fig. 16. However, if ~is large, a center point on one line of fixed points is closer to the saddle point on the next line than to its former mate. The separatrix surrounding the center is then reduced sufficiently in size to pass through the adjacent saddle point, as shown in the bottom half of the figure. Since a appears in the denominator of (60), ~becomes very large for isochronous rings. We are then forced to operate in the region where the RF buckets are much smaller than their normal size, both in phase extent, and in energy aperture. If we keep only the two most important energy terms in the Hamiltonian and again neglect the small synchronous phase 2 fo’V\~ H = ~-eV cos ORFT ira I,,—) (2ir/3)a Iö’V\ 2 1,,-;;-) (61) ~‘

—

—

the phase extent and the energy aperture can be found in the limit of large ~to be

David A.G. Deacon, Basfc theory of the isochronous storage ring laser

378

(62)

Wpj~Tmax =

fôy\

J—a/a2

“~‘V)max

[a/2a2

(63) The presence of Coulomb scattering among the electrons in the bunch (Touschek effect) implies that an energy aperture which is too small will reduce the lifetime of the current stored in the ring. We must therefore require that the aperture stay larger than about (ÔYIY)min 0.005. If we were to choose the optimum value of a = iO~corresponding to A = 1, the second order term would have to be kept smaller than about a2 = iO~ a

(64) 2.A large tune is therefore required to The second order momentum compaction lI~ is v 34, the energy aperture will be maintain the lifetime in an isochronous ring. [20]varies If the tune asofa2 the-~ ring 0.5% as specified above. Special techniques must be utilized to keep the tune stableenough to operate aring of large tune because of the variation of tune with the magnetic field in the quadrupoles as ~si -~i’ t~BIB. Clearly any design technique which enablesthe production of a small a 2 with a lower tune carries with it the rewards of a simpler magnetic structure and lower requirements on the stability of the focussing elements. An example will also be given of a laser system with a tune of 11, implying a = 10~and A = —28. If the numerical relationship between the tune and a2 can be improved by appropriate design techniques, it should be possible to obtain a performance intermediate between the two examples to be given, but in the lower tune machine. The relation (62) makes the phase aperture of the RF trap quite small. A typical value [21] for ~is 11, for which the total phase aperture is w~r 0.1 rad. The small bunch length produced in such a system will result in a small Touschek lifetime athigh currents [16].The bunch length may be stretched by applying another accelerating element to the beam so that the RF potential well is flattened out. A pair of nonresonant electric accelerating plates driven by a programmed amplifier could be expected to flatten the RF slope over a range in the fundamental of perhaps a kilovolt. This would greatly increase the current which could be stored in the system for a given Touschek lifetime. There remains an attribute of the laser system which can profitably be utilized in this situation. Once the laser is operating, the laser traps are stable even if the RF bucket is not. It is possible, for example, to establish operation with the charge stored in the RF bucket, and then to shift the phase of the full optical buckets so that they lie outside of the RF bucket. If the traps are large enough to contain the electron equilibrium energy spread for a significant time, the traps themselves can be used to maintain the stored charge outside of the RF stable region. Further, if particle injection could be accomplished without disturbing the isochronous character of the ring, one could even put additional charge into the bucket and repeat the process. Of course, if the laser power were interrupted, any extra charge stored in this way would be instantly lost. Given sufficient stability, this technique shows promise for significantly enhancing the power output of an isochronous storage ring laser. 2(&y/y)min

2.10. Parameter optimization There are four important parameters which characterize the storage ring laser system: the cavity voltage U, the optical phase advance 4,, the momentum compaction A, and the phase shift per pass D.

David A.G. Deacon, Basic theory of the isochronous storage ring laser

379

Table 1 summary of key relations stability:

fA <(2/4’) tan 412 IA>—(2/4’)cot4’/2 (4’<~r) optimum IA~< 1 — maximum phase advance 4’rax 2/VA (A ~‘ 1) 24’2/4 Vi—A4’cot4’—A area: Area = ~ I — Açi’ cot 4, (linear) fixed points: sin 0, = U/4i2 trap size: emittance: power:

~y =

{vi

~

sin[D/2(1—A)] D/2(1 — A)

>

—

U2/44

+

(-v) cos~

(4,41)

U

P~~tr~oai=Jd0p(0)~sinwasr(0)

bucket size: a 2<

a

y_41TN(Y~Yr —

4, ‘

yr

0 E kôz 4’ DL,Ic A kLaaj4irN U (4 irNeVIE,) 2kL,j4/3)—sin~ +~)RFI~ kL,aa[(Er— E,)/E,l D (a

Let us discuss the considerations which enter into a choice of magnitudes for these parameters. For reference, a summary of the key results which have been obtained so far is provided in table 1. The RF voltage is the source for the energy transferred to the laser cavity, and should be made large to increase the available power. On the other hand, the value of U for a given RF phase must lie within the range (38) for an optical trap to exist there at all. Clearly an optimum value exists for this quantity for any specific trap. A good figure of merit is provided by the product of the area of the trap and the applied voltage U. If the trap is uniformly filled with electrons, this number will be proportional to the power radiated by the trapped electrons. The trap area at constant A and 4, scales as the product of the aperture (39) and the phase extent, derived from (36). Figure 17 shows the behavior of the figure of merit is a function of U/4,2 for small 4,. The curve has a broad maximum in the range UI4,2 = 0.3 to 0.6 which defines the optimum operating point. In the large trap case where the central region of the trap is populated more densely than the boundary region, one can more easily afford to reduce the area. As a result, the optimum voltage will

I

4,2

Fig. 17. The figure of merit U (trap area) is plotted against U/4,2 for constant A and 4’.

380

David A.G. Deacon, Basic theory of the isochronous storage ring laser

turn out to be somewhat higher. The best value for U/4,2 cannot be obtained for these cases in the absence of a knowledge of the equilibrium distribution. The optical phase advance is determined by a number of factors including the intensity I in the radiation field and the number of magnet periods N 4) = I1L,/c

2irNK

(65)

where .1~ cB3/4rr is the effective intensity in the static helical magnetic field (I~= 1.87 x iO~W/cm2 for our assumed 2.8 kg field). However, it depends only on the fourth root of the power density, so that it is almost insensitive to changes in I. The most effective parameters for fixed magnet geometry are number of magnet periods and electron energy. The system should be designed with a large phase advance. If U/4i2 is held constant, the voltage can be increased as the square of 4). Since the extracted energy per electron is proportional to the voltage and v cc 4’2/N cc N\/I, the reward for higher phase advance per pass is higher output power. The phase advance is limited for any system by the stability region in fig. 2. As 4) is increased, the shrinking area of the traps limits the number of electrons which can participate in the first order energy exchange process. And as we have seen, for the RF voltage at constant 4,, there is an optimum operating point. Figure 18 shows the dependence of the figure of merit on the phase advance. The solid curve is scaled from the linear result (32) multiplied by the voltage U. As the phase advance increases from zero, the system performance increase quadratically until the unstable boundary is reached. Here, the area drops precipitously and the extracted energy drops to zero. The points are the results of the simulation, which follows the linear prediction very well until close to the edge of the stability zone. The deviation from the theory here marks the onset of the complex quasi-stochastic behavior noted previously in fig. 14. The momentum compaction must be reduced from its values in typical rings, as we have discussed. The system goes unstable at A = 1, which forms the upper boundary of the stable region for small 4,. Depressing A far into the negative region also impedes performance as the range of phase advance available to the system reduces and the trap size shrinks. Both the phase advance and the dimensionless aperture shrink by the square root according to (28) and (39) so that the real energy aperture scales as

Fig. 18. The figure of merit U (trap area) is plotted against 4’, holding UI4’2 and A constant.

David AG. Deacon, Basic theory of the isochronous storage ring laser

381

57/’V or 1/a. After a certain point, the traps will no longer be able to contain the electron distribution, and charge is progressively lost as the traps shrink. If U/4,2 is held constant so that the phase extent of the traps remains constant, and RF voltage applied to the group of occupied traps must also be reduced as V or cc 1/a. The optimal extracted power therefore drops as the square of the momentum compaction, effectively setting an upper limit on A,

IAI<1.

(66)

If the system is limited by the size of a 2 through the reduction in RF aperture, then the size of the tune of the storage ring is very important. Further reductions in a2 by increasing allow a to for be 2. the Thetune output power reduced via (64) so that the momentum compaction can be scaled as a 1/v traps smaller than the equilibrium spread is then proportional to the fourth power of the tune. The emittance of the beam is not excited by the interaction in the laser section if the dispersion function of the storage ring is made zero there. It is therefore the natural emittance in the storage ring which must be made to satisfy the requirement (43) with the transverse amplitude a equal to its RMS value a = Ve/ir. The requirement (46) can then be easily satisfied. 2.11. A numerical example Having established the limits on the four critical parameters, we can now unravel the dimensionless definitions and calculate the numbers implied by these restrictions. I will try to stay close to known technology in this effort, with the goal of generating the description of a device which can be built. The wavelength sets the scale of the tolerances in the system, so that long wavelengths are easier to achieve. Since a major part of the interest in high power lasers lies in the infrared and the visible, I choose for my example a wavelength of A = 1 p.. An electron energy of 100 MeV can produce this radiation from real magnets, and is not so large that the equilibrium energy spread cannot be contained within the traps. Choosing the momentum compaction factor within the optimum region A —1, and the number of magnet periods at a large but acceptably long N = 175, we find the momentum compaction coefficient in the arc becomes —0.35 X iO~,so that for the entire ring (91), a = —0.7 X 10~. This small value for a is the distinctive requirement of the isochronous storage ring laser. We known that, following the design criteria established in the next subsection and ref. [16], such a ring can be designed. The design for the conventional storage ring BESSY, for example, is sufficiently flexible that without taking extraordinary measures, an a = 1.6 x i0~can be achieved [22].I have shown in ref. [7]that the tolerances one must hold in order to build an isochronous ring are also within our capabilities. The stabilization of the electron bunch against excitation by fluctuations in the RF, gas scattering, and so on also appears to be possible. For the time being, it is assumed that a ring with such a small a can be designed and operated for the free electron laser. Continuing with the reduction of the dimensionless specifications into real numbers, I find from (27) that the optical phase advance must be kept smaller than 4’max = 1.72. If we choose a power density in the cavity of 1 MW/cm2, the phase advance 4, = 1.2 lies close to the optimum shown in fig. 18. To obtain the optimum energyexchange, the electrons should be concentrated into traps in the neighborhood of the RF phase corresponding to U/4)2 0.5. This requires that the peak voltage be larger than something like 100 kV. A ring of this energy and size typically operates with an energy spread at equilibrium of about = 4 x iO~.This translates into a spread in Y of about 0.07, which compares to the calculated

382

David AG. Deacon, Basic theory of the isochronous storage ring laser

aperture of z~Y = ±0.83,leaving a 12 o free aperture. If the ring is designed with a beam emittance of = 4 x 108 m rad, we find from (43) that traps exist out to seven standard deviations in the transverse motion. The distribution is well contained within the traps for both transverse and longitudinal motion. From (46) we see that the misalignment tolerance is about a quarter of a wavelength on either side. It is now a standard procedure to stabilize a laser cavity to this precision, and the same should be possible in a storage ring with feedback stabilization through a path length correction insertion. For these conditions, the average output power delivered to the laser field is = 32 kW/amp

(stored current).

(67)

This should be contrasted to the synchrotron power which is at maximum = 100 W/amp,

(68)

for a damping time of 106 cycles. Not only is the power output enormous, but the efficiency is determined primarily by the RF losses, magnet losses, etc., rather than by an inherent competing process. These favorable characteristics justify a considerable expenditure of effort on the difficult aspects of the realization of an isochronous storage ring laser. While the numbers I have given describe an optimum configuration, it is interesting to examine the structure with the parameters I have assumed in the earlier sections (a = — l0~).This configuration is physically smaller, and less complex due to its lower tune. We found that for a tune of 11, an appropriate RF aperture was formed with the larger value for the momentum compaction. It is fairly easy to show that the implied value A = —28 still gives a system with very interesting characteristics. The maximum phase advance is not limited to the smaller value 4,max = 0.38. Ifwe reduce the intensity to 1.6 kW/cm2, and reduce the voltage by a factor of 25, the parameters can be brought within range: 4) = 0.24, and U/4,2 = 0.5. The biggest problem with such a non-optimal example is that the trap aperture shrinks, allowing particle diffusion across its boundary. The energy acceptance is now 0.6 o~.The power extracted from the storage charge is also reduced from the previous case, but remains at a relatively high level P 0~= 1.3 kW/amp

(A = —28).

(69)

The narrow energy acceptance of this example demonstrates the need for a solution to the electron distribution problem. Clearly, only a portion of the circulating charge will be stored in the traps, and the actual power output will depend on the magnitude of this fraction. Note that for this choice of parameters, the indicated gain per pass at equilibrium is about 80 percent. The pendulum equation, which rests on the low gainapproximation, is beginning to break down here, although the numerical deviations should not be too serious. Even if only 1% of the total current can be stored within the optical traps at equilibrium, the beam-to-beam efficiency remains above 10 percent. The output characteristics of the free electron laser operated in an isochronous storage ring are so good that with even a significant deviation from its optimum operation, it will produce interesting power and efficiency levels. 3. The general isochronism conditions in a storage ring We now return to the consideration of the isochronism of the storage ring. It is the purpose of this

David AG. Deacon, Basic theory of the isochronous storage ring laser

383

section to identify the important longitudinal effects of the electron motion around the arc of the storage ring and examine the implications for the design of an isochronous ring. In one transit of the arc, a given electron will acquire a certain displacement from the ideal particle. The expression for this displacement is derived and expanded order by order in the small coordinate deviations from the ideal particle. The isochronism conditions will be discussed for each order, generating a series of restrictions on various aspects of the design of the ring. The feasibility of satisfying the limit in each case is discussed with the aid of a numerical illustration, assuming a 1 p. wavelength, and typical magnitudes for the storage ring functions judged to be favorable, but at the same time physically realizable.

3.1. Derivation of the longitudinal coordinate shift To describe the motion of the electrons I use the usual coordinate system with origin attached to the ideal particle as it cycles through the system. The transverse coordinates are x and y (see fig. 19), the transverse angles are x’ dx/ds and y’, the longitudinal coordinate is the time deviation r from the bunch center, the energy deviation is 6y = (y — ,‘~)in units of mc2, and the independent variable is the azimuthal position s. The longitudinal deviation of a particle as it traverses the arc between two successive laser interaction regions can be found by comparing the path length swept out by an arbitrary particle with that of the central particle. For simplicity, I first consider the case of motion in the plane of the orbit. The ideal particle sweeps out a path length f ds = La, while an arbitrary particle covers f dl, taking the integral over the trajectory. In a curved section, the elements of length are related by the azimuthal angle swept out by the two trajectories (70)

dt1i=ds/p=dlCO~.

This relationship is also valid for the straight sections if the limit p —* is taken in the appropriate way. Here, p is the radius of curvature of the ideal orbit, taken to be positive as sketched in fig. 19. The path length difference in a finite arc is then

x

ds ti~

dl

~

iii ELECTRON BUN C H I

p(f)

IDEAL CLOSED-ORBIT PARTICLE

Fig. 19. The coordinate system used in the analysis of the electron motion through the storage ring.

David AG. Deacon, Basic theory of the isochronous storage ring laser

384

~L5=Jdl_fds=Jds[~o;~,—1].

(71)

In the time required for the ideal particle to travel the distance La, that is T = LJf3oc, the arbitrary 3o particle has of travelled the distance f3cT along its owninorbit (here,deviation /3 is the rnormalized velocity v/c, and f the velocity the synchronous particle). The change the time of the arbitrary particleis then proportional to the difference between the path actually travelled and the central path length around its arc La

c~r=/3c~——(La+ALa) poc =

‘La

+ La

(72)

(~)

(73)

La(~IdsIx,1l. \f3i

Lpcosx

.i

(74) j

The above argument can be generalized to include the case of motion out of the instantaneous orbit plane. The relation (70) becomes (75)

p +p.x

where x is the two component transverse deviation vector, 0 is the angular deviation in three dimensions (02 = x’2 + y’2), and p is the instantaneous radius vector. The general longitudinal shift is

ct~r_La~—Jcis

[~

2~t~5’—i}.

(76)

This expression can be simplified by breaking out the energy dependent transverse position x = r~+ ~ ôy/y, [23] and expanding 1/cos 0 and i~f3/f3 in terms of the small deviation vectors r, r’ and Sy. Arranging the result yields C

l~T=

—f ds ~

f

ds

2~

+(~)[~_Jth!i~+.. .]+(~z)

Jth+...]

(77)

where ~j is the dispersion function, the dot products imply a vector combination: r p x~p,+ y~py,and squares imply the vector dot products as in ~ ~ + i~ Once the exhibited terms have been made small enough to satisfy the isochronism conditions, the omitted terms, which contain higher powers of small parameters, become negligible. ~.

David AG. Deacon, Basic theory of the isochronous storage ring laser

385

3.2. The effects of transverse motion In the previous subsection, we found that the phase slip per pass D for a typical particle must be kept smaller than about D ~ 1— A (43) in order that all particles remain trapped. In what follows, we will simply require that the coordinate shift ~s satisfy the slightly more restrictive condition kAs
(78)

The first term in (77), linear in the transverse amplitude, is the most important term involving the transverse motion of a particle. Although completely negligible in normal storage rings where it averages to zero over many passes through the ring, it is very important to an isochronous ring in which it may produce unacceptably large phase excursions in a single pass. The design of an isochronous storage ring laser must take account of the necessity to make this term small. The transverse position can be expressed [23]as x~= aVf3(s) cos(4,(s) + ~)

(79)

where

(80)

The function /3(s) is the trajectory envelope function uniquely determined by the focussing characteristics of the ring 4+K(s)x~=0 /311

2

/312

4

2

(81)

l_ 2—K(s)

(8)

where K = (dB/dx)/pB is the focussing strength at any point on along the orbit. For almost planar motion [24], p~~ Px; the x component of the deviation f ds xe/p gives the most important contribution. In a ring with a constant and uniform magnetic guide field, its magnitude can be estimated by replacing the beta function by its average value

J ds xe/p = a’s/p J ds cos 4,(s) a”.Jfl~

(83)

If no special cancellation is induced, the phase integral in (83) is on the order of f~’.With an emittance at 100 MeV of about e 4 x iO~m rad, this term remains disastrously large. Since the contribution is oscillatory in the betatron phase, the simplest solution appears to be the introduction of a cancellation by

386

David AG. Deacon, Basic theory of the isochronous storage ring laser

choosing the appropriate phase advance. Taking the values /3 2 m and p 5 m, we can satisfy (78) if we require the betatron phase advance in the magnets to oscillate in such a way as to reduce the integral in (83) by the factor 40. Adjustability of the tune can be maintained through the use of matched focussing insertions in a straight section. For the general case of an arbitrary ring the betatron induced phase slip must satisfy the following two restrictions in the horizontal plane

Jds~cos4)i~J~

(84)

J ds

(85)

sin 4)

ô2 ~

5.j~,

where the integral is taken over the arc of the ring exterior to the laser. In other words the azimuthal phase slip must cancel for all values of the betatron phase. In practice, the betatron phase advance can be adjusted and measured to a high accuracy, so that it can be set to the correct value in a real ring, even if that value must in the last analysis be experimentally determined. For a particle with transverse amplitude a, the residual contribution to the optical phase slip on the nth pass oscillates according to the tune v c

=

—aVô~+c5~cos(n2irv+ 4)o)

(86)

with a mean square contribution of (c~r)2=a2~~ô2.

(87)

If the conditions (84) and (85) are satisfied, this term will produce no net phase slip on the particle, its effect being visible only through the resonance (47). The piece of this term contributed by the vertical motion, f ds(y~/p)(p5Ip),is much smaller than the horizontal term due to the presence of the small factor pr/p. The requirements (84, 85) need not be imposed on the vertical motion so long as p,,/p is sufficiently small

/~

Pz.<...L. p kf3~ e/3 ‘~

(88)

In planar rings, this ratio is determined predominantly by the alignment of the main bending magnets. For the numerical values quoted above, the alignment of the ring need be no better than pr/p 3 X i0~. Alignments as good as i0~are possible, leaving considerable latitude on the selection of a larger vertical /3 function. The ~r term in (77) which is second order in the horizontal coordinate gives a phase slip proportional to the emittance, and therefore establishes a limit on the magnitude of e. The integrand has oscillatory and constant parts of equal magnitude. Upon integration, the constant term becomes dominant, so let us

David AG. Deacon, Basic theory of the isochronous storage ring laser

387

consider it first. It produces a constant phase shift with magnitude dependent on the amplitude of the betatron motion. Again looking at an isomagnetic ring, approximating the derivative as x’mx/f3, and replacing the beta function by its average value this term becomes

J ds

~

(~ +

(89)

The contribution to (78) can be made small by reducing the emittances to the level ~4iT/3/kLa.

(90)

If the arc length La is about 100 m, this relation limits the emittances to e 4 x 108, a value which can be achieved in a ring of this energy. The oscillatory contribution of this term is reduced by the factor 4iw from (89). As discussed in the previous section, this component is only visible under resonant conditions, and even then, the amplitude of the induced longitudinal motion is stabilized by Landau damping. If the emittance has been reduced sufficiently to bring the constant component within the limit (78), the oscillatory part should have no detrimental effect. 3.3. The effects of energy deviations The collection of terms of various orders in the energy influence the behavior of the electrons on two different scales. On the microscopic level of the optical bunches, they produce a shear effect on the electron bunch as electrons are displaced according to their energies. As in the case of the emittance shifts, these terms must be restricted in magnitude to maintain the isochronism of the arc. On the macroscopic level, the higher order terms in the energy deviation produce the orbit distortions discussed in the previous section. The restrictions imposed on the storage ring design by these two effects will be considered here. The first order term in the energy is related to the momentum compaction factor of the storage ring. Defining the momentum compaction factor in an arc of length La as aa, the expression for the energy related coordinate shift in a complete circuit of the ring is ~

(91)

The isochronism condition must be applied to the arc of the storage ring between the end of one laser and the beginning of the next. The momentum compaction factor in the arc is

~

J~4ds_-~.

(92)

In an analogous fashion, the proportionality constant of the second order energy term in the arc has

David AG. Deacon, Basic theory of the isochronous storage ring laser

388

been defined as a2a L~ a2a

~

J

ds +

~-

4.

(93)

In typical low energy storage rings, the first order term lies beyond the bound set by (78). If no special steps are taken, the total momentum compaction factor a = 4irN(A

(94) 2 (ref. [16])so that aL is two orders of magnitude too large. would be in theonneighborhood of a design, 1/v from (66), is The condition the magnet lattice —

1)/kL

a,, ~ 4ITN/kLa.

(95)

This term is responsible for the phase focussing effect in the longitudinal phase space which maintains stability in the ring, and cannot be eliminated completely. Since aaLa is zero in the arc of the ring in the ideal case, only a small contribution remains in the laser itself. The magnitude of (92) changes with energy, so there will be an energy, not too far from the design energy, at which the net phase focussing goes to zero: the so-called transition energy. In other words, the ring must operate close to the transition energy at which the phase focussing goes to zero and the ring becomes unstable. The requirement that a must be small is therefore equivalent to the requirement that the ring be designed with the transition energy near the operating point at a relatively high energy. Several techniques for achieving a high transition energy have been considered in specific designs [25,26, 27] and the general design requirements are discussed in ref. [20]. In essence, the dispersion ~ must be made to oscillate in the bending magnets so that the negative portions of the integrand can cancel the positive portions. The situation is analogous to the lanes of a racetrack. Runners on the inside lane get the advantage; if all lanes are to cover the same distance, the lanes must be made to cross so that the runner on the inside track around one bend must get the outside track around the next one. However such a machine may be designed, the electron distribution would be sensitive to excitation and the ring would have only a narrow range of operating parameters. Since the ring must operate near transition, the suppression of noise in the ring elements, and the longer term stabilization of all components is a priority. The second order term in the energy makes a negligible contribution to the shape of the optical traps until they become very large. The most important effect of a 2, as discussed above, is to distort the large scale trajectories in the RF bucket. This term must therefore be limited by (64). The effect of the third order term is negligible compared to the second order (4aa3/a~~ 1), and presumably the effects of the higher order terms are also vanishingly small. Through the examination of the isochronism requirements on a storage ring it has now become clear that there are four independent conditions which must be satisfied. The focussing system must be designed so that the betatron phase advance satisfies the conditions (84, 85). The emittances during operating must be kept small, as specified by (90). The phase focussing (95) provided by the first order energy term must be kept small, and the second (93) and higher energy terms cannot exceed a maximum level (64).

David A.G. Deacon, Basic theory of the isochronous storage ring laser

389

4. Summary The isochronous storage ring laser is a possible solution to the problem of electron excitation which complicates the development of a high power, high efficiency storage ring free electron laser. The laser traps charge in optical potential wells which can be made deep enough to contain an electron distribution with the natural energy spread of the storage ring. If the ring does not perturb the traps too strongly, the trapped charge can be reinjected into the laser potential wells on the next transit. If the optical phase of each electron bunch is retained by this method, the trapped electrons can be made to radiate an amount proportional to the first power of the electric field. The first order interaction is strong, and not as sensitive to laser intensity fluctuations as the second order process. If the electrons are accelerated on each pass by an RF cavity, they will arrange themselves to radiate the same amount into the laser. In this action, they behave as the working fluid of a frequency transformer, coupling RF energy into coherent laser radiation. Since the optical phase of the electron on each pass is strongly correlated to its previous value, the trapped charge rotates along a smooth set of trajectories in its phase space (if the intensity is not so high as to produce the stochastic type of behavior shown in fig. 14). The energy spread is not excited beyond the natural value, and additional damping need not be introduced. The efficiency is therefore determined only by the losses in the accelerating cavity and the transport structure. The transformation produced by the storage ring perturbs the distribution and reduces the size of the traps. As shown in fig. 5, some traps exist at low intensity for any value of the momentum compaction. The optimization of their size requires that a be reduced as much as possible. The requirement of energy balance (38) limits the size of the applied RF voltage for a given intensity and therefore limits the laser power (52). As the voltage is increased towards its limit, the trap size shrinks as shown in fig. 10. The emittance must be kept small (42)so that the electron distribution will remain well contained within the transverse aperture illustrated in fig. 13. See table 1 for a summary of the key relations. For the parameters chosen in the example, the trap aperture is more than 10 times the natural energy spread, so that the equilibrium energy distribution is well contained deep inside the traps. Particles with transverse motion continue to be trapped out to seven standard deviations. The small damping always present in the ring cools the electron distribution into the traps, or if the traps are small compared to the natural spread, enhances the density there. Aligning the system so that the electrons see the same light on each pass results in the elimination of the absorbing traps. The remaining charge damps down into the emitting traps, where it contributes strongly to the output power of the laser. For 1 ampere of stored current, the extracted power is 32 kW (67). The sum of the requirements on the storage ring design and stabilization imply a unique structure very different from the presently existing designs which have been optimized either for colliding beams or for synchrotron radiation emission. Each of these requirements presents its own difficulty, but none is impossible to meet. Acknowledgments I would like to thank John Madey for a number of discussions on the implications of isochronism both on the storage ring and the laser, and Helmut Wiedemann and Phil Morton for their comments on the requirements which must be imposed on the storage ring. This work was supported in part by the National Research Council of Canada, and the United States Army ATC contract DASG 60-77-C-0083.

390

David A.G. Deacon, Basic theory of the isochronous storage ring laser

Appendix A. List of symbols Electron coordinates x, x’ y, y’ E ‘r s

transverse position and angle x’ = dxlds vertical position and angle electron energy time deviation from bunch center independent variable (arc length)

Storage ring parameters T0

orbit time total orbit length La length of arc outside laser p(s) radius of curvature V peak voltage of RF cavity w~ frequency of RF cavity E, synchronous energy (energy of central particle) L

Ring design parameters 1’V) dispersion function x(s) = xe(s) + fl(s)(&)~ transverse envelope function x~= a V~3(s)cos(4,(s) + &) 4)(s) betatron phase 4) = f ds/J3 p tune z’ = ~ d4)/2ir K(s) focussing constant (pB)”~1(dB/dx); x” + K(s)x = 0 0e natural energy spread of electron distribution s emittance (area of RMS transverse phase space ellipse) a momentum compaction coefficient aa momentum compaction in the arc of the storage ring exterior to the laser section a 2 second order momentum compaction coefficient a2,, second order momentum compaction in the arc ~ bucket aperture/separation of fixed point lines i’(s) /3(s)

Laser parameters A

emission wavelength

k B

2ir/A

N A0

L, K Er 0

magnetic field in wiggler magnet number of periods in wiggler wiggler wavelength (s~= 2irc/Ao) length of wiggler 2ko magnetic parameter eB/mc resonancefield energy optical phase k öz (z = z 0 + fl1~ct+ ôz(t))

David A.G. Deacon, Basic theory of the isochronoug storage ring laser

Y f~2 4, ~‘

U

~ D A 4, I

391

normalized (4irN/4))frequency ~y/y laser phase energy space rotation 2e2gB/ 2m2c2 7 optical phase advance in one laser transit IlL 1! c electric field strength of radiation field normalized voltage (4irNeV/Es) sin wpj~T~ phase shift/pass from cavity-ring length mismatch mean phase shift per pass normalized momentum compaction phase advance per pass in complete phase space radiation intensity

References [1[ The possible applications for lasers in chemistryhave been discussed in: The Laser Revolution in Energy-Related Chemistry, eds. J.I. Steinfeld and MS. Wrighton (1976); available from the Energy-Related General Research Program Office, National Science Foundation, Washington, D.C. 20550. A more recent discussion of laser chemistry appears in Physics Today, November 1980. [2) W.B. Colson, One Body Analysis of Free Electron Lasers, in: Physics of Quantum Electronics, Vol. 5 (1977). [31A. Renieri, The Free Electron Laser: The Storage Ring Operation, CNEN-Frascati Report 77-33 (1977), Edizioni Scientifiche, C.P. 65, 000044 Frascati, Rome, Italy. [4] J.M.J. Madey, D.A.G. Deacon and TI. Smith, 1. AppI. Phys. 50 (1979) 12. [51D.A.G. Deacon and J.M.J. Madey, Phys. Rev. Lett. 44 (1980) 449. [6] A discussion of the tolerances which must be held to realize the isochronous storage ring laser is contained in ref. [7]chapter 5. [7] D.A.G. Deacon, Ph.D. Dissertation, Stanford University, 1979. Available from University Microfilms International 300 N. Zeeb Rd., Ann Arbor, ML 48106. [8) W.B. Colson, Phys. Lett. MA (1977) 190. [9) J.M.J. Madey, D.A.G. Deacon, L.R. Elias and TI. Smith, II Nuovo Cimento 51B (1979) 53. [10)J.M.J. Madey and D.A.G. Deacon, Free Electron Lasers, in: Cooperative Effects in Matter and Radiation, eds. CM. Bowden, D.W. Howgate and HR. RobI (Plenum Press, 1979) p. 313. [11) W.H. L.ouisell, J.F. Lam and D.A. Copeland, Phys. Rev. A 18 (1978) 655. [121M. Sands, The Physics of Electron Storage Rings: An introduction, SLAC Report No. 121, Stanford Linear Accelerator Center, Stanford, CA 94305 (1965). [13] MV. Berry, Regular and Irregular Motion, in: Topics in Nonlinear Dynamics, AlP Conf. Proc., ed. S. Jorna (1978) p. 16. [14] 1. Ford, A Picture Book of Stochasticity, in: Topics in Nonlinear Dynamics, ALP Conf. Proc., ed. S. Jorna (1978) p. 121. [15]J. Le Duff, Beam-Beam interaction in e~—e Storage Rings, in: CERN 77-13, Theoretical Aspects of the Behavior of Beams in Accelerators and Storage Rings, Proc. Intern. School of Particle Accelerators, Erice (1976). [16) For a discussion of the perturbations produced by the Touschek effect on the distribution inside and around the optical traps, see ref. [71 chapter4. [17) L.R. Elias, J.M.J. Madey and TI. Smith, App. Phys. 23 (1980) 273. [181D.A.G. Deacon, Appl. Phys. 19 (1979) 97. [19] D.A.G. Deacon, AppI. Phys. 19 (1979) 295. [20] See appendix C of ref. [7]. [21] ~is calculated here for a = i0~,V = 5 kV, and h 10, values which correspond to the low tune machine example. [22] D. Einfeld and G. MOlhaupt, Choice ofthe Principal Parameters and Lattice ofBESSY, BESSY TB 13/1979, BESSY, Takustrasse 3, 1000 Berlin 33, West Germany. 2/B~)(B,JB [23) ED. Courant and H.S. Snyder, Ann. Phys. 3 (1958) 1. [24] In my notation, Py (ymc 5)is the projection of the radius vector on the y axis, and becomes small when the orbit becomes flat so that the radius points in the x direction. Other authors project the orbit on the y—z plane and use the radius of curvature of the projected orbit which is very different: it tends toward infinity as the horizontal field approaches zero and the orbit becomes flat. [25) RH. Helm, A Negative Momentum Compaction Lattice, PEP Note 60, 1973 PEP Summer Study, Stanford Linear Accelerator Center, Stanford, CA 94305. [26) L.C. Teng, Infinite Transition-Energy Lattice Using sr-Straight Sections, NAL-FN-232, Fermilab, Batavia, IL 60510 (1971). [27] A. Garren, A High Transition Energy Lattice, BNL 17279 CRISP 72-71, Brookhaven National Lab, Upton, NY 11973 (1972).