B. Faatz and G. P. Gallerano Vol. 12, No. 12 /December 1995 /J. Opt. Soc. Am. B 2475

Stacking of free-electron-laser output pulses

B. Faatz* and G. P. Gallerano

Ente per le Nuovo Tecnologie, l’Energia e l’Ambiente, Dipartimento Innovazione,Settore Fisica, CRE Frascati, P.O. Box 65, 00044 Frascati, Rome, Italy

Received June 30, 1995

We investigate the possibility of stacking light pulses produced by a free-electron laser with wavelengths in themillimeter regime in an external cavity. An aperture in one of the mirrors is used to couple the radiation intothe cavity. The field is extracted by means of an optical switch, such as a YAG-illuminated semiconductorslab. Some general design considerations of the cavity are discussed, and results of numerical simulationsare shown as verification of the applicability of the principle. 1995 Optical Society of America

1. INTRODUCTIONIn IR and millimeter free-electron lasers (FEL’s) drivenby radio-frequency accelerators, radiation is created bya sequence of electron pulses with durations of severalpicoseconds separated by the period Trf of the radio fre-quency, which is in the range of a few hundred picosec-onds to several nanoseconds. For some applications(nonlinear optics, time-resolved spectroscopy), intensepulses with a larger separation and possibly an evenlarger peak power are required. One of the options forachieving these specifications, pulse stacking, is investi-gated in this paper for the millimeter-wavelength regime.The feasibility of this method was demonstrated at theSanta Barbara FEL facility and is under investigationat the FELIX facility at Rijnhuizen in The Netherlandsfor micrometer wavelengths.1,2 This technique involvesthe use of a separate cavity in which the individual mi-cropulses created by the FEL are stacked one on top of oneanother. An aperture in one of the mirrors is used to cou-ple the successive micropulses into the cavity (see Fig. 1).The cavity length is chosen to be equal to the repetitionfrequency of the FEL’s electron gun, resulting in a cav-ity round-trip time equal to the separation between themicropulses. Consequently one single, high-intensitypulse builds up inside the cavity. After each macropulsethe radiation is extracted by means of an optical switch(similar to the silicon slab illuminated by a YAG laserused as discussed in Ref. 1). One could consider a cav-ity length equal to twice the FEL’s repetition frequency,resulting in two pulses that can be extracted, using thesame slab.

This paper is organized as follows. In Section 2 somegeneral criteria and requirements for the cavity are in-vestigated. In Section 3 a brief discussion of the numeri-cal simulation code used is given. Section 4 is dedicatedto results of numerical simulations in various possibleconfigurations. Finally, in Section 5 the results are dis-cussed and conclusions drawn.

2. GENERAL CONSIDERATIONCONCERNING PULSE STACKINGIn the pulse-stacking system one can distinguish three ra-diation fields, namely, the field that is coupled in through

0740-3224/95/122475-07$06.00

the aperture, the intracavity field, and the extracted field.All three are given by a Rayleigh length l and waist po-sition z. Boundary conditions of the three fields are de-termined by the FEL and the external cavity. The mainrequirements for the total system are

(1) Need for the maximum fraction of the FEL outputpower to be coupled in through the hole,

(2) Minimum loss of the intracavity field through thehole and at mirror edges,

(3) Periodic solution of the intracavity field.

In addition, the quality of the extracted field is animportant feature. The transverse mode preferred is aGaussian, as this can be focused to the smallest possiblespot size. At the moment, however, the main concern isthe maximum achievable power inside the stacker cavity.The mode structure of the extracted field is considered tobe of less importance and is briefly discussed in Section 5.

It is assumed that the optical switch is transparentuntil illuminated. Therefore all calculations performedin this paper can be assumed to have axial symmetry,consistent with the FEL output radiation pulse and theaxial symmetry of the stacker cavity. A convenient setof orthonormal functions in which such a field can beexpressed is the set of Gauss–Laguerre (GL) functions.3 – 5

The transverse shape of the FEL output field is de-termined by the extraction mechanism and by the opticsbetween the FEL and the stacker cavity. At this pointthe field is assumed to consist of a single Gaussian modewith a waist position z0 and a Rayleigh length l0 thatcan be chosen freely. In Section 4 the influence of devia-tions from this field is investigated in more detail. Thusa large fraction of the field is coupled in when z0 ­ 0 andl0 # 2Ra

2yls, where Ra is the aperture radius and ls isthe wavelength of the radiation extracted from the FEL.The largest possible Rayleigh length is therefore given bythe diffraction limit l0 ­ 2Ra

2yls and results in an incou-pled fraction of 95%.

Before looking at the other two requirements, we firstexplore the easiest case to investigate, the one in whichthe intracavity field is dominated by a Gaussian compo-nent, in greater detail. The fractional loss of such a fieldthrough the aperture is given by

1995 Optical Society of America

2476 J. Opt. Soc. Am. B/Vol. 12, No. 12 /December 1995 B. Faatz and G. P. Gallerano

Fig. 1. Layout of the pulse-stacking scheme.

Z ja

0e2jdj ø ja ­ 2pRa

2l1ylssl12 1 z1

2d , (1)

where j ­ 2pl1r2ylsfl12 1 sz 2 z1d2g, where l1 and z1 are

the Rayleigh length and the waist position of the intra-cavity mode, respectively, with respect to the upstreammirror. Both field parameters can be expressed in cav-ity radii of curvature of the upstream, R1, and the down-stream, R2, mirrors and the cavity length Lc by6

z1 ­sR2 2 LcdLc

R1 1 R2 2 2Lc

, l1 ­ fz1sR1 2 z1dg1/2. (2)

Minimizing loss through the hole means a large value ofz1 or a small value of either Ra or l1.

Because the modes inside the cavity have z1 and l1 asgiven by Eqs. (2), the obvious choice is to express the ex-ternal field in these parameters as well. The conversionfrom the external Gaussian mode to the intracavity GLmodes is given by7,8

cpsz1, l1d ­

√2l0

l0 1 l1 1 iz1

!√l1 2 l0 2 iz1

l0 1 l1 1 iz1

!p

, (3)

where cp are the coefficients of the Gauss–Laguerremodes of the external field expressed in cavity eigen-modes normalized to unity, i.e., normalized to the ex-ternal Gaussian mode c0sl0, 0d ­ 1. When the Rayleighlength and the waist position of both fields are equal, weget cp ­ dp,0, the Gaussian cavity eigenmode of interest.This mode, however, by definition loses its power afterone round trip through the aperture. Thus the externalfield cannot be a system eigenmode, and only a fractionof the field is converted into one. The power fraction ofthe Gaussian mode is given by

jc0j2Pjcpj2

­4l0l1

sl1 1 l0d2 1 z12

­4l1

l0 1 s2 1 pyjadl1

, (4)

where we have taken the maximum value for l0. Becauseja, the loss factor, has to be small, this term is alwayssmall. Therefore small loss always means that a smallfraction of the total power of the external field is allowedin the Gaussian mode.

We thus have to expect many modes in the cavity ifthe loss is to be small or, possibly in some cases, a modedifferent from the Gaussian mode to be dominant. Be-cause higher-order GL modes have to be taken into ac-count, loss at the mirror edge becomes important as wellas loss through the aperture. Taking into account allmodes cp inside the cavity, there are two contributions

to both loss and gain of each mode. The obvious loss andgain are due to the fraction of the intracavity field that islost through the hole or at the edge of the mirror and thefraction of the input field that is in this mode, expressedin cavity modes. Additional loss and gain terms are dueto conversion of one mode to another because of the fi-nite dimensions of the mirrors. Because the transfer ofenergy between modes, given by the mirror transfer ma-trix (see Section 3), is proportional to the amplitude of themode, a mode with high power loses more than it gainsfrom other modes. Therefore the direct power-loss term(diagonal term) has to be as close to unity as possible fora mode to reach a high power level.

For higher-order GL modes the energy loss through thehole is, similar to the case for the Gaussian mode, inleading order (after expansion in ja) equal to ja. Theedge loss is harder to estimate, but it will be assumedto be small when jm

2p exps2jmd is small, where p is theGL mode number and jm ­ jasRmyRad2 is the normalizedmirror dimension (Rm is the mirror radius). This relationgives an overestimate of the edge loss, and the actual lossis determined by numerical simulation.

The third point mentioned at the beginning of thissection is periodicity of the solution. One can demandthat a mode have a periodic solution after each roundtrip, which is expressed as

ksLc 2 s2p 1 1darctan

√Lc 2 z1

l1

!2 s2p 1 1darctan

√z1

l1

!­ np , (5)

where p is the mode number, ks is the wave number, andn is an arbitrary integer. Because z1 and l1 are cavity-related quantities, constants after they have been set bythe mirrors’ radii of curvature, this relation holds for spe-cific values of ks only. Especially in case of millimeterwaves, in which the FEL output pulse is only a few wave-lengths long, the cavity cannot be detuned much. Thusin these cases this condition has to be fulfilled withoutan extra shift of one of the mirrors. Because the spac-ing of different longitudinal modes under the broad FELgain profile is close in k space compared with the modespacing in the stacker cavity, there is always one modethat satisfies Eq. (5). It is also obvious that when thisrelation is fulfilled for one value of p it will in general notbe fulfilled for another. Under the assumption that ksLc

is a multiple of p, it is interesting to investigate the tworemaining terms, which can be cast in the form

s2p 1 1darctan

"Lcl1

sR1 2 Lcdz1

#­ np . (6a)

In the case of a symmetric resonator, with z1 ­ Lcy2,Eq. (6a) reduces to

2s2p 1 1darctan

√Lc

2l1

!­ np . (6b)

Equation (6b) is of special interest at larger wavelengths,when a nonsymmetrical resonator gives more loss at theedge of one of the mirrors. In this case it follows that

R1 ; R2 ­Lc

s1 2 cos fd,

ja ­2pRa

2

lsLcsin f ­

pl0

Lcsin f , (7)

B. Faatz and G. P. Gallerano Vol. 12, No. 12 /December 1995 /J. Opt. Soc. Am. B 2477

where f ­ npys2p 1 1d. When p ­ 0, the Gaussianmode, this condition can be fulfilled only when R1 ­ `,i.e., for plane mirrors, or R1 ­ Lcy2, for a concentric reso-nator. Thus the Gaussian mode has no stable solutionwhen ksLc is a multiple of 2p. For higher values of p, aperiodic solution can always be found.

If the transverse dimensions of the mirrors are largeenough, one can choose a nonsymmetric resonator. Weknow that choosing z1 large results in small loss throughthe aperture. For given z1 the choice of R1 resulting ina periodic solution is given by

R1 ­ Lc 1Lc

2

2k2z1

6Lc

2

2k2z1

√1 1 4k2 z1

Lc2 4k2 z1

2

Lc2

!1/2

, (8)

where k ­ tanfnps2p 1 1dg. For given k, and the maxi-mum possible value for z1, which is equal to Lcs1 1p

1 1 1yk2 dy2, we obtain

R1 ­ 2R2 ­Lc

sin f, z1 ­

Lcssin f 1 1d2 sin f

,

l1 ­Lc cos f

2 sin f, ja ­

pl0 cos f sin f

Lcssin f 1 1d(9)

in terms of the angle f ­ npys2p 1 1d. Thus, for thehole loss to be minimized, f has to be as close to py2 aspossible. This condition can be achieved only for largevalues of p and n ­ p, i.e., for high mode numbers. Forthe minimum value of z1, equal to Lcssin f 2 1dy2 sin f,the values for R1 and R2 are interchanged, and

ja ­pl0 cos f sin f

Lcs1 2 sin fd.

In this case f has to be as close to zero as possible, i.e.,also high mode numbers. In principle, to obtain maxi-mum power enhancement in the stacker cavity one has totry all the different periodic cavity configurations. By nomeans do the two given solutions in Eqs. (7) and (9) en-sure maximum intracavity power. An important featureof these solutions is that usually only one mode is periodic.Thus only one mode will grow inside the cavity.

We would like to mention a different method of ob-taining high intracavity power, namely, having differentmodes inside the cavity with a phase difference such thatthey give zero amplitude on axis at the aperture position,thus minimizing the loss.9 – 10 The influence of mirror tiltand misalignment on the loss is believed to be very largein this case.11 – 12 This method is not persued in this pa-per. Some results can be found in Ref. 2 for micrometer-radiation wavelengths.

3. SIMULATION CODE FORPULSE STACKINGThe simulation code to describe pulse stacking (SCOPUS)consists of two parts. The first part converts the externalfield, the output field of the FEL, into eigenmodes of thestacker cavity. The second part calculates the (complex)reflection matrix of the cavity, i.e., the matrix that is theproduct of the two mirror reflections, and determines the

field after one complete round trip. All that the programdoes then is to perform an arbitrary number of multiplica-tions of matrix and field vector, adding the external fieldafter each round trip, to determine the stationary fieldinside the cavity. The entire program is based on an ex-pansion of the field into GL modes, the set of orthonormalmodes that form a complete solution of the paraxial, axi-ally symmetric, homogeneous wave equation. Thus onlythe coefficients of the GL modes at a given Rayleigh lengthand waist position have to be given.

To convert the external field into intracavity eigen-modes, one has to truncate the field at the apertureboundaries and transfer the coefficients, given with theirRayleigh lengths and waist positions, into another set ofcoefficients, with Rayleigh lengths and waist positions de-termined by the radii of curvature and the length of thestacker cavity. Truncation of the field means solving13

ap exps2izpd ­Xq

aq exps2izqdZ ja

0LqsjdLpsjde2jdj ,

(10)

where j is given in relation (1) and Lp is the pth-orderLaguerre polynomial. The choice of the number of modesto be taken into account is critical, and one determines itin the program by keeping track of the error in energy con-servation. A calculation is accepted only if the error isof the order of 10% of the micropulse power after the sta-tionary field is reached. The problems that occur whena field is expanded in a finite set of orthonormal modes isusually not evident, because one does not add coefficientsof an external field to those of a field already present in-side the cavity. When two fields are added, the power isnot conserved in the calculation because the input fieldis not zero beyond the edge of the aperture and thus con-tributes to the power at positions where the intracavityfield or the input field should be zero. In case of holecoupling, in which an aperture is used to extract the ra-diation energy from the cavity, the error is small. Theoutput power is calculated by the law of energy conserva-tion. In this technique the modes present in the outputfield are unknown, however. Because, for pulse stacking,the modes are of crucial importance, this method will notbe sufficient in this case. Another method is renormal-izing the input field to ensure energy conservation, butinasmuch as the overlap between the fields in question de-pends on the intracavity field, which changes after eachround trip, renormalization is different after each pass.Therefore this method is not particularly attractive. Inview of the straightforward interpretation with GL modesused in terms of Rayleigh length and waist position aswell as the periodicity condition, this method gives moreinsight into important parameters than would solving aFresnel integral numerically, which does not suffer fromthe mentioned problems.

Transferring the (truncated) input field to cavity modesis described in Ref. 8. In the case of a Gaussian field,the conversion is given by Eq. (3). The description of thereflection by the two mirrors when the field is decomposedin a set of GL modes results in the evaluation of theintegral equation (10), with boundaries ja and jm for eachof the two mirrors. For the intracavity field the extraphase shift given by Eq. (5) has to be included. Using

2478 J. Opt. Soc. Am. B/Vol. 12, No. 12 /December 1995 B. Faatz and G. P. Gallerano

pXk­0

√pk

!s21dkfk,q ­

Z jp

p!Lqsjde2jdj

and integrating by parts results in

fp,q ­Z Rm

Ra

LqsjdLpsjde2jdj

­p21Xk­0

√p 2 1

k

!s21dk1p

"√qp

2k

p 2 k

!fk,q 2

qp

fk,q21

#

1 s21dp11 xp

p!Lqsjde2xj

jmja ,

for p . 0 and in

f0,q ­Z jm

ja

Lqsjde2jdj ­qX

k­1

√q 2 1q 2 k

!s21dk11 jk

k!e2jj

jmja ,

f0,0 ­Z jm

ja

e2jdj ­ 2e2jjjmja

for p ­ 0. These equations turn out to give very accuratenumerical results (confirmed to p ­ 25), and, because onecan calculate each matrix element by using all elementswith a lower mode number, it is fast.

Both truncation and conversion of the external field,and the total complex matrix, have to be calculated onlyonce at the beginning of each multipass calculation. Inmost cases some 10 to 15 GL modes are needed to describethe field including up to 95% of its power.

4. RESULTS OF NUMERICALSIMULATIONSIn the remainder of this paper we use the parametersof the millimeter-wave Ente per le Nuovo Tecnologie,l’Energia e l’Ambiente (ENEA) compact FEL to calcu-late the performance of the stacker cavity (see Table 1).Only the first 10 GL-modes have been taken into account.All modes with a higher mode number lose 50% or moreof their power at the mirror edge. The total absorptionmentioned in Table 1 includes absorption by mirrors andoptical switch.

As is stated in Section 2, the relations derived there byno means guarantee that the intracavity power reachesits maximum for the values calculated with the equations.They merely give a guideline along which solutions can beexpected. Another issue that has not yet been addressedis the value of the Rayleigh length of the external field.It has merely been given as a relation between l0, theaperture radius, and the wavelength, but whether sucha length, which will in most cases be very short, can bereached in an experimental setup is sometimes doubtful.Therefore this parameter is varied first.

In Fig. 2 the saturated intracavity power is shown asa function of the Rayleigh length of the external field(solid curve). The radius of curvature of both mirrors is27.6 mm. Thus f ­ 4py5 [see Eqs. (7)] and ,1 ­ 8 mm.With the value for f, the only modes with a periodic so-lution according to Eqs. (6) are L2 and L7, of which thelatter has a large loss at the mirror edge. The expo-nential decay of intracavity power is caused by the de-creased fraction of the power that is coupled in throughthe aperture, with its maximum at ,0 ­ 2Ra

2yls. A fur-

ther decrease in Rayleigh length does not increase theintracavity power, because the power loss at the down-stream mirror increases and the fraction of the power inthe L2 cavity mode becomes smaller [see Eq. (3)]. Forcomparison, the intracavity power, normalized to the mi-cropulse power in the first 10 GL modes, that actuallyenters the cavity is shown (dashed curve). At larger val-ues of the Rayleigh length the normalized power remainsconstant, which indicates that indeed the exponential de-crease in power is due only to a reduced incoupled frac-tion. The increase in normalized power for large valuesof ,0 is due to round-off errors in the programs and is notbelieved to be a real effect. One might argue that whenpulse stacking is used to create one super pulse the realpower rather than the normalized power is the relevantquantity. However, one of the aims is to create one singlepulse, and high power is not always a demand. Further-more, normalized power is easy to measure and can beused to optimize the stacker cavity.

Thus far the shape of the FEL output pulse has beenassumed to be Gaussian. The longitudinal shape is notaddressed in this paper. However, it is reasonable to as-sume that the transverse and longitudinal shape are in-dependent, in which case the longitudinal shape can bethought of consisting of a number of slices that propagateindependently. The transverse shape is strongly influ-enced by the extraction scheme used in the FEL. Oneusually assumes a Gaussian distribution. When holecoupling is used to extract radiation from the FEL (asis the case for FELIX), the output field will not be purelyGaussian. The influences of the transverse shape on theintracavity and normalized intracavity powers are shown

Table 1. ENEA Millimeter-WaveCompact FEL Parameters

Parameter Value Definition

ls 2.5 mm Radiation wavelengthf 3 GHz Micropulse reference frequency

60 ps Micropulse durationLc 50 mm Stacker cavity length

S1 ­ S2 25 mm Mirror radiusRa 5 mm Aperture radius

3% Total absorption

Fig. 2. Power (solid curve) and power normalized to the in-put power in the first 10 GL modes (dashed curve) versus theRayleigh length of the input field. The radius of curvature ofboth mirrors is 27.6 mm.

B. Faatz and G. P. Gallerano Vol. 12, No. 12 /December 1995 /J. Opt. Soc. Am. B 2479

Table 2. Influence of the Input Field Profilein Terms of GL Modes on the PowerEnhancement and Normalized PowerEnhancement in the Stacker Cavitya

No. Input Field Intracavity Power Normalized Power

1 L0 6.355 6.8482 L1 0.642 1.7303 L2 0.243 1.0434 L0 1 L1 4.803 5.8505 L0 2 L1 2.052 4.5306 L0 1 iL1 3.011 4.4677 L0 2 iL1 3.844 6.4078 L0 1 L2 3.250 5.9749 L0 2 L2 3.174 5.14410 L0 1 iL2 3.315 5.43411 L0 2 iL2 3.110 5.644

aThe Rayleigh length of the input field is 20 mm. The sign betweenthe GL modes in column 2 gives their phase difference when enteringsi ­

p21 d.

in Table 2 for different input fields. In the second col-umn only that part of the input field that depends on themode number is given. When more than one GL modeis present in the input field, the phase between them isgiven by 61 or 6i si ­

p21 d. For example, for L0 –L1,

the two GL components have opposite phases at the aper-ture position and thus give zero amplitude on axis. Inthis case one expects the input power to be low, becausea large fraction is off axis and does not enter the cavity.When no Gaussian component L0 is in the input field, as isthe case in rows 2 and 3, both intracavity and normalizedpowers are lower by a factor of 5 or more compared withall other cases in the table, even compared with rows 5and 9, where the input field has a zero on axis. Calcu-lations of a combination of the first and the second GLmodes (not shown in the table) show similar low powerlevels. One of the reasons for the low power is that partof the radiation of higher-order modes is off axis. There-fore the difference in normalized power is not so large.The difference in normalized power shows, however, thatpart is also due to mirror edge loss and less efficient cou-pling into cavity modes [see Eq. (3) for the case of a Gauss-ian beam].

The latter problem can be partially solved by a differentchoice of the mirror’s radius of curvature. Calculationscomparable with those of Fig. 2 but with a radius of cur-vature of 33.3 mm have been performed. Owing to theangle f ­ 2py3, both L1 and L4 have periodic solutionsaccording to Eqs. (6). Because the loss in L4 at the mir-ror edge is not as large as in L7 in the case of Fig. 2,energy converted into both cavity modes contributes tothe intracavity power in this case. The results are sim-ilar, but the power saturates at a lower level because ofhigher loss through the aperture. If the input field nowdeviates from a Gaussian, both contributions to L1 andL4 add coherently on successive passes through the cav-ity. Because two transverse modes can exist inside thecavity, the actual intracavity mode present can changebetween the two, thus ensuring a higher power over theentire range of input fields. The intracavity power variesby a factor of approximately 2 for the input fields givenin Table 2, i.e., also when the field is non-Gaussian. The

result in Fig. 2 thus shows a high power level at satura-tion in one single mode, L2; the power level is, however,sensitive to the input field profile. When a different reso-nator geometry is used, the power variation as functionof input profile can be reduced. This is done, however,at the cost of having two different transverse modes in-side the resonator. A similar set of simulations has beendone in case of a nonsymmetrical resonator [with a radiusof curvature, given by Eqs. (9), of 78 mm], showing simi-lar results. In this case, the transverse dimension of theupstream mirror had to be enlarged from 25 to 45 mm toreduce side loss significantly.

Thus far the influence of the input field on the in-tracavity field has been studied. The results show that,when the input field is close to a Gaussian with Rayleighlength ,0 ­ 2Ra

2yls, the intracavity field can reach apower level as high as seven times the micropulse power.These calculations were done, however, only for certainresonant GL modes. Because the loss of a single mode,either through the aperture or at the mirror edge, de-pends on mode number and geometry, calculations in theremainder of this section have been done to determinewhat mode gives the highest intracavity field. Thereforeall resonators given by Eqs. (7) and (9) are invesigatedup to mode numbers p ­ 4. Results for a symmetric[Eqs. (7)] resonator are shown in Fig. 3, for a nonsym-metric resonator [Eqs. (9)] in Fig. 4. In both cases the

Fig. 3. Total power (filled circles) and power in the dominantmode (open circles) versus the radius of curvature for the case ofa symmetrical resonator.

Fig. 4. Total power (filled circles) and power in the dominantmode (open circles) versus the radius of curvature for the case ofa nonsymmetrical resonator.

2480 J. Opt. Soc. Am. B/Vol. 12, No. 12 /December 1995 B. Faatz and G. P. Gallerano

total amplification is given as function of the radius ofcurvature of the mirrors. Only when the power of theexpected dominant mode, as calculated from Eqs. (7) and(9), differs by more than 5% from the total power, is thislevel also shown. Furthermore, when the power of theexpected dominant mode was less than 50% of the totalpower the entire point was omitted from the figure. Inmost of these cases, of which there are four for p ­ 4, thetotal power was also low owing to high loss at the mirroredge. In Fig. 3 many resonant modes can be found atvalues for the radius of curvature that are close together(note the logarithmic horizontal scale). This has the ad-vantage that many different modes have points close tothe chosen radius where they have a stable solution with-out much loss. Thus, by shifting one of the mirrors, onecan always find power enhancement. The disadvantageis that the system can easily have a mode structure dif-ferent from the one expected. If the radius of curvatureis not as specified, many modes may be present insidethe cavity, usually not a desirable situation. This doesnot occur for the nonsymmetric case in Fig. 4, where thespacing between the modes is large and the power en-hancement is larger (where the radial extension of theupstream mirror is 45 mm). A remark on these resultsis in order. When the mirror size is increased, loss at themirror edge for higher-order modes becomes smaller. InFig. 4 this was the only way to enhance the power. Com-pared with (proposed) experiments, where the wavelengthwould be 1–2 orders of magnitude shorter, the mirror sizecuts away part of the power at the tails. An increasedmirror size would prevent this. This is the main reasonthat high powers cannot be obtained. In this paper wehave chosen not to increase the mirror size further to getpower levels as high as to 20 times the input power be-cause this would require large mirrors (of the order of20 cm). However, one can easily calculate what mirrorsize is needed for a certain transverse mode by calculatingthe edge loss for this specified mode.

Calculations, both analytical and numerical, have as-sumed that ksLc must be a multiple of p to be detuned.If this is not the case, the cavity has to be detuned. Theinfluence of a small scan of the cavity length is shownin Fig. 5. The radii of curvature of the two mirrors areR1 ­ 2R2 ­ 77.8 mm. The peaks in the total power(solid curve) are due to creation of one or more differ-ent resonant modes in the cavity. The power level ofthe expected dominant mode (L4) is also shown (dashedcurve). From this figure it can be seen that if the cavityis not properly aligned then high power can be achieved,but the mode profile will be different from the oneexpected.

Additional simulations have been done with a cavitylength of 10 cm. For the solution to be stable, the ra-dius of curvature has to be doubled in these cases [ascan be seen from Eqs. (7) and (9)]. Most modes did notshow any enhancement as the result of high loss at theedge of the mirror. In the symmetric case a power en-hancement of 5.76 was found for a radius of curvatureof 12.86 cm (two times 6.43 cm, where L3 is expected tobe dominant), almost completely in L3. This could beachieved only for a doubling of the transverse dimensionof the mirror. In the nonsymmetric case, only the case inwhich L1 is dominant gave a significant increase in intra-

cavity power (from 3.43 to 5.75 for R1 ­ 2R2 ­ 11.5 cm).Again, the situation changes if one allows a large mirrorsize, because in all cases the main loss occurred at themirror edge.

In one symmetric and one nonsymmetric case we inves-tigated the influence of phase jitter between micropulsesby adding a random phase to the input field within ksLc ­6py10. In the case of a normal random variation, hardlyany change in enhancement appeared. However, whenthe random phase was cumulative, i.e., when the randomphase was added to the input field, which was then con-sidered to be the input field for the next pass (adding anew random phase), the results were confusing. In thesymmetric case the intracavity power increased by ,15%,whereas in the nonsymmetric case the power was reducedby a factor of 2. The reason for the reduction is that thefields are no longer added coherently. No reasonable ex-planation could be found for the enhancement.

5. DISCUSSION AND CONCLUSIONSAs can be seen from the results given in Section 4, pulsescan be stacked with this setup. The amplification fac-tor depends on both the geometry of the stacker cavityand the input field. The latter depends on the extrac-tion mechanism of the FEL, which is a crucial parameterto consider. It seems that, assuming a Gaussian inputfield, a nonsymmetric resonator gives the highest ampli-fication (up to a factor of 10 for the parameters used).Such a cavity is, however, highly sensitive to misalign-ment of the mirrors and for the radiation wavelengthstudied here needs at least one mirror with a large ra-dial extension. If the output field of the FEL is non-Gaussian, at least a large fraction of the field has to beGaussian for the stacker cavity to show amplification.The power will in most cases be reduced by as muchas a factor of 2. In comparing the micropulse power inthe FEL and in the stacker cavity, it is good to keep inmind that the number of longitudinal modes supportedby the FEL cavity, which is usually much longer thanthe stacker cavity, is larger by a factor equal to the ra-tio of the two lengths. Therefore, for there to be ap-proximately the same power in one micropulse the powerhas to be at least this fraction larger in the stacker,where the bandwidth will be reduced. Increasing the

Fig. 5. Total power (solid curve) and the power in thedominant L4 mode for the case of dL ­ 0 (dashed curve)versus the cavity detuning for a nonsymmetrical resonator withR1 ­ R2 ­ 77.8 mm.

B. Faatz and G. P. Gallerano Vol. 12, No. 12 /December 1995 /J. Opt. Soc. Am. B 2481

size of the stacker by a factor of 2, thus creating twopulses instead of one, thus has the additional advantagethat the number of longitudinal modes supported is twiceas large. Even though the power enhancement is gen-erally lower according to the calculations in Section 4,in fact the real power might be larger. Additional en-hancement might be possible by construction of a stackercavity that supports several transverse modes such thatthe field amplitude would have a minimum on axis atthe aperture position to minimize loss. This method, ofwhich a preliminary study was performed in the microm-eter region in Ref. 2, has shown promising results so far.Also, nonsymmetric modes, which can be created by aslight tilt of one of the mirrors, can enhance intracav-ity power (see, for example, Ref. 12). This also seems tobe confirmed by experiments, such as the one described inRef. 1. To simulate this, the present simulation code hasto be extended to include either generalized GL modes orHermite–Gaussian modes.

Finally a remark is in order concerning the radialprofile extracted from the stacker. As was shown inSection 2, transforming the field from one set of modesto another is not efficient. Thus the only way to obtainhigh power in one single mode is not to change the modestructure. In practical cases, however, this is impossi-ble, because the modes created in the cavity have a smallRayleigh length, which cannot be transported over largedistances, for example, to a user facility. The shape ofthe output field has not been investigated here, but it isbelieved that the only way to get a single mode is to trans-form the mode into many modes for transport and at theuser facility transform them back into the original field,which should be possible on the basis of linearity.

ACKNOWLEDGMENTB. Faatz was financially supported by the EuropeanInfrared Free Electron Laser network under contractCHRX-CT93-0109.

*Permanent address, Fundamenteel Onderzoek deMaterie-Instituut voor Plasmafysica Rijnhuizen, Edis-onbaan 14, 3439 MN Nieuwegein, The Netherlands.

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