Davidson, R.C.

Hartemann, F.

Mourier, G.

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PFC/JA-89-51ANOMALOUS (STIMULATED) REFRACTION INDUCED BY THEFREE-ELECTRON LASER INTERACTIONF. Hartemann*G. Mourier*R.C. DavidsonOctober, 1989Plasma Fusion CenterMassachusetts Institute of TechnologyCambridge, MA 02139*Permanent Address: Thomson-CSF/TTE, 78141 Velizy, France.Submitted to Physics Letters A.Anomalous (Stimulated) Refraction Induced by theFree-Electron Laser InteractionF. Hartemann*, G. Mourier* and R.C. DavidsonPlasma Fusion Center, MIT, Cambmdge MA. 02139, USA.AbstractThe stimulated refraction effect (optical guiding) in a free-electron laser (FEL) is studiedtheoretically. The analysis of the 3-dimensional FEL wave equation is performed accord-ing to two different formal methods. First, the microwave field is expanded into vacuumwaveguide modes, in which case the optical guiding appears as an active mode couplingeffect. In the second method, the eigenmodes of the FEL radiation field are found, whichyields both the gain of the system (complex eigenvalues) and the radial intensity distribu-tion of the interacting waves, describing the guiding effect. Computer calculations showthat optical guiding may have a strong influence on both the gain and mode content ofmillimeter-wave FELs.* Permanent address : Thomson-CSF/TTE, 78141 Ve1izy, France.1Free-electron lasers (FELs) are tunable sources of coherent electromagnetic radiation [1].One of their most remarkable properties is the large phase shift [2-4] that the resonantbeam-wave interaction induces on the amplified electromagnetic wave. Under proper cir-cumstances, the refraction associated to this phase shift can be such that the electro-magnetic wave is confined within the electron beam, in a manner somewhat akin to theguiding properties in an optical fiber. This stimulated refraction effect, referred to as "opti-cal guiding" [5-131, has many important implications. For short wavelength FELs, it wouldmitigate the effects of diffraction, therefore allowing the length of FEL wigglers to exceedthe Rayleigh range, and the obtention of higher single-pass gains. Such long wigglers areneeded if FELs are to operate either in the vacuum-ultraviolet or at high efficiencies inthe infrared wavelength regime. For millimeter-wave FELs, optical guiding can also beassociated to higher gains due to a better coupling (overlap) between the electron beamand the FEL radiation mode (higher electromagnetic wave filling factor) ; in addition, thestimulated refraction effect can strongly affect the mode purity of waveguide FELs.The purpose of this Letter is to present a simple theoretical analysis of optical guidingin a waveguide FEL. In contrast with the conventional vacuum waveguide mode expan-sion, we perform an eigenanalysis of the 3-D FEL wave equation, yielding the eigenmodes(guided modes) and eigenvalues of the problem. In particular the gain is calculated in aself-consistant manner, without using the so-called electromagnetic wave filling factor, byevaluating the imaginary part of the complex eigenwavenumber.The FEL interaction can be described, within the framework of a linearized fluidmodel, by a set of 4 coupled PDEs describing the evolution of the 4-potential vectorperturbation 6A, (60/c,6A)[&1 -iso - + i . x x g :)# bAIr- + '6A - '--. c c-- &i- 6+,60 X t +014 X1 -t bf =0, 1ti &+-iV 6A = 0. (2)whereC2 r eis the d'Alembertian operator (electromagnetic wave propagator). Here, the quantities2indexed "0" refer to the unperturoed fluid equilibrium. Note that we can easily identify thedifferent terms in equation (1) as a beam-mode type operator coupled to an electromagneticwave propagator, and a beam coupling term proportional to the beam density (w2) andcontaining the ponderomotive force. Here, we have assumed that the equilibrium field ispurely magnetic (E0  0), and we have defined the following parameterse L0 4 noe2  _-o 0m I 70- C o=y OM 2 0 'YOm 0  cAt this point, different beam-wave interactions are characterized by different fluid equilibriaand different boundary conditions for 6A, and its derivatives.We now consider a FEL with an axial guide field, pumped by a helically polarizedwiggler. The waves propagate in a cylindrical waveguide of radius r = a. The externallyapplied fields are purely magnetic, and we neglect the self-fields generated by the beam.We haveBo(9, z) = B + Bw(i cos + O sin &), (3)where B is the strength of the axial guide magnetic field, B, is the amplitude of thewiggler field and = ksz - 0, where f, = 27r/kw is the wiggler period. Note that here,for the sake of simplicity, we consider a radially uniform wiggler field and that we neglectthe space-charge effects of the electron beam on the fluid equilibrium. The correspondingequilibrium fluid velocity field is3 (0, z) = i3 + 3 (r cos V + sinv), (4)where the normalized fluid velocity components 31 and Oi are constrained by energyconservation12 - -(5)20and related by the following equation(6)Here, we have defined the relativistic cyclotron frequencies Q1 and 9., in the guide andwiggler fields, respectively. Because the divergence of the equilibrium fluid velocity field(4) is zero, the beam density profile can be described by a step functionnn 0. for r < rb: (7)0 , for rb < r < a;3where rb is the beam radius, which satisfies the equilibrium continuity equation everywhere,except at the edge of the beam. The surface effects can then be modelled by non-uniformsurface charge and current distributions. Making use of equations (3) to (7) into the initialsystem (1-2) yields the 3-D FEL wave equation describing the evolution of the microwavelaser field in the linear regime.At this point, the 3-dimensional analysis of the FEL radiation field can be performedaccording to two different formal methods. One can either find the eigenmodes of thesystem, which yields both the gain (eigenvalue) of the system and the radial intensitydistribution of the interacting waves., or expand the microwave field into vacuum waveguidemodes and make use of the orthonormality of these modes to study their coupling. In thelatter case, the guiding appears as an active mode conversion effect. We first briefly reviewthe coupled-mode analysis. The general form of the 3-D wave equation is[ w(r)1E c2 C(r, w, k) 6'A= 0, (8)where the coupling operator C contains the space-charge waves; the beam dynamics effects,etc. The solution is expressed as a superposition of vacuum waveguide modes6= 6Am0m, (9)msatisfying the following eigenvalue equationLC C - - k2 _ 7, -2 (10)Here, xm corresponds to the cutoff frequency of the m-th vacuum mode, determined fromthe boundary conditions at the waveguide walls. For TEI, modes in a cylindrical pipeof radius a, j, = x',n/a. The wave equation is then reduced to a set of coupled linearequations by integrating over the waveguide cross-section and using the orthonormality ofthe Um's/a -2 _ 2 - )LdS U*>Z - -k C(r, w, k) 6Amm = 0, (11)to obtain-2  - X 2 + L5 4 C 2C- k2 - x2 + Can(, k) 2A Cmn(w,k)A, = 0. (12)27n C 2 ] m n4Here, the guiding effect appears as active mode coupling. In the conventional theory, oneneglects the coupling terms C,,, and introduces the electromagnetic wave filling factor,defined asP2 = dS U[-) U,, (13)to obtain the usual FEL dispersion relation for cylindrical TEI, modes22C2 a c (11 (kc)]"[ + 'P(kV-01 I c 0 1(ky +ck )-P2 wPo kkW. (14)Here, 4 is a correction to the space-charge dispersion equation due to the combined presenceof the axial and wiggler magnetic fields [14'. From this relation, it is straightforward toevaluate the Raman growth rate6 2 1-P2 = )1/2 p2(iI'o)2. (15)We now study the eigenmodes of the system, for TE (bA = 60 = 0) geometry. Theinteraction region is divided into two areas. Region 1 (rb < r < a) corresponds to thevacuum surrounding the electron beam. The plasma frequency W,(r) = 0, and the generalsolution to the 3-D wave equation for the azimuthal component of the 4-potential vectoris of the form6A 0 (r, 0, z,t) =[.4J, (xir) -- BY, (xir)] exp[i(wit - k i z + 11)], (16)where k, and x 1 are constrained by the vacuum dispersion relationD (ki, xi) = C2 - ki - X = 0. (17)Note that in this region of space, the Bessel function of the second kind can be included inthe general solution because r > Tb T 0. Region 2 (0 < r < rb) corresponds to the electronbeam. Here the plasma frequency is constant w,(r) = wpo, and the general solution for6A 8 is of the form6 (r, 9, z t) CJ 2 (X2r) expi(w21 - k2 z + 129)], (18)5where k2 and x2 are constrained by the beam dispersion relation [D2 (k2 , X2 ) = 01,2-2 - 2 [ - 3(k 2 4- k .. ) + "']" -U;2 (k2 + k.) - _V/0]c I [c Y c c 'yic1 W c2 k2k. (19)The boundary conditions are the following. At the waveguide wall (r = a), the tangentialelectric field component (bEE = -iwi5A) must be zero, which yieldsbAo(r = a) = 0. (20)At the beam edge (r = rb), we require that the 4-vector potential be continuousAbA(r = rb) = 0, (21)which is equivalent to having 6E and 6B, continuous. Finally, we use the dielectricboundary conditiona6rAo(r = rb) = 0. (22)This last condition implies that we have neglected the surface current distributions at thebeam edge. The continuity of 6.4 0 at the beam edge (21) yields the following relationsk,= k2 = k, (23)11 12 = 1, (24)W1 =U2 = W, (25)which correspond to the fact that continuity must hold at any z, 0 and t, respectively. Inaddition, we haveAJ;(xlrb) -- BY'(xirb) = CJj(x2 rb). (26)The continuity of a6Ao at r = rb (22) yieldsX[AIJ'(x 1rb) + BI"'(xIrb)] = x 2CJI'(x2 rb). (27)Finally, the boundary condition at the waveguide wall (20) results inAJI(x ia) + B1Y'(xia) = 0. (28)6Combining equations (26), (27) and (28), and upon elimination of the constants A, B andC, we obtain a third relation [Do(xi, x 2 ) = 01,YI" (xirbl x2 Jj(x2rb ) '(la Y '(x IrbJi(xrb) -J(x 1a) ,a Jijxrb) - Jxixa) , =0. (29)Jj ~ I" (xi1a) X1J,(X2rb) I (Xia) . (9Equation (29) and the two dispersion relations in vacuum (17) and inside the beam (19)form a system of 3 nonlinear equations in the complex eigenvalues k, x, and x 2Do(xl,x 2 ) = 0 , D 1(k,xi) = 0 , D2 (k, x2 ) = 0.We can solve these equations as functions of the frequency w to obtain the complex eigen-wavenumber k(w), where Im(k) is the 3-D growth rate of the FEL instability, and x1 (w)and x 2 (w) which determine the radial profile of the electromagnetic waves (eigenmodes)in the interaction region, and therefore describe the optical guiding effect in a waveguideFEL.To illustrate the derivations presented above, we have done some computer calcula-tions. comparing the FEL gain and radiation field profile with and without guiding. Theexample we study corresponds to the design parameters of a FEL experiment plannedat MIT in collaboration with TTE. The beam voltage is V = 500 kV, the beam currentdensity is kept constant at j 0.337 kA/cm2 . The nominal wiggler field amplitude isB, =500 G, with a period 1, 3.0 cm. The guide field strength is Bg = 2.5 kG, and thewaveguide radius is a = 10 mm. The microwave are launched in the fundamental TE11cylindrical mode. For these parameters, the operation frequency is w+/27r ~ 42 GHz. Onthe drawings, the TE1 1 mode and the eigenmode are normalized so that they have thesame power flux through the waveguide. In Figure 1, we show the radial profile of thenormalized azimuthal electric field component for a 5-mm-radius electron beam, and 3different amplitudes of the pump : B, = 0, 100 and 500 G. As the optical activity (gainand phase shift of the FEL interaction) of the beam increases, the guiding effect appearsclearly and the eigenmode deviates substantially from the vacuum TE11 mode. In Fig. 2,the wiggler field is held constant at B, = 500 G, and we plot the normalized microwaveintensity profile as the beam radius is varied from 4 mm to 9 mm. The guiding effectis strong at small beam radii (strong transverse "gradient" of the anomalous refractiveindex) and, as expected, the eigenmode relaxes towards the vacuum TE11 mode as the7beam radius increases. Note that if the beam fills the waveguide, no guiding is expected.Finally, in Fig. 3 we plot the ratio of the imaginary part of the eigen-wavenumber Im(k) tothe TE 1 gain Fo,/j-, calculated from Eq. (15). For large beam radii, the guiding is smalland both gains are comparable and close to the 1-D gain, Fo. However, as the beam radiusdecreases, the electromagnetic wave filling factor for the TE1 1 mode P2 is strongly reduced,while Im(k) remains close to Po. At still smaller radii, the radiation appears to decouplefrom the beam, which may be due to strong diffraction effects. It should be noted, how-ever, that at the present time our model does not include the effects of the surface chargeand current distributions, which clearly become predominant at small beam radii.In conclusion, we have described a theoretical model of the FEL interaction in the lin-ear regime that includes the stimulated refraction effect which is the physical phenomenonat the origin of optical guiding. Indeed, we find that the eigenmodes of the problem cansubstantially deviate from the vacuum waveguide modes used in other models. In addition,the imaginary part of the complex eigen-wavenumber directly yields the gain, making thecalculation of the so-called electromagnetic wave filling factor an unnecessary step. We findthat this gain can be higher than that calculated from the vacuum mode theory. Finally,it should be pointed out that our model can be improved by using more complete bound-ary conditions at the beam/vacuum interface, in particular by including the non-uniformcharge and current distributions at the beam edge resulting from the wiggling of the beam.8References[1] C.W. Roberson and P. Sprangle, Phys. Fluids B1, 3 (1989).[21 J. Fajans and G. Bekefi. Phys. Fluids 29, 3461 (1986).[3] J. Fajans. J.S. Wurtele, G. Bekefi, D.S. Knowles and K. Xu, Phys. Rev. Lett. 57,579 (1986)F41 T.J. Orzechowski, E.T. Scharlemann and B.D. Hopkins, Phys. Rev. A35, 2184(1987).[5] P. Sprangle and C.M. Tang, Appl. Phys. Lett. 39, 677 (1981).[6] W. M. Kroll. P.L. Morton and M.W. Rosenbluth, IEEE J. Quantum Electron. 17,1436 (1981).[7] J.1M. Slater and D.D. Lowenthal, J. Appl. Phys. 52, 44 (1981).[8: D. Prosnitz, A. Szoke and V.K. Neil, Phys. Rev. A24, 1436 (1981).L9] E.T. Scharlemann, A.M. Sessler and J.S. Wurtele, Phys. Rev. Lett. 54, 1925 (1985).110 G.T. Moore, Opt. Commun. 52, 46 (1984).[11] G.T. Moore, Nucl. Instrum. Methods Phys. Res. Sect. A239, 19 (1985).[12, F. Hartemann, K. Xu, G. Bekefi, J.S. Wurtele and J. Fajans, Phys. Rev. Lett. 59,1177(1987).'13' A. Fruchtman, Phys. Rev. A37, 2989 (1988).14] H.P Freund and P. Sprangle. Phys. Rev. A28, 1835 (1985).Figure CaptionsFig. 1 Normalized azimuthal E-field component of the eigenmode as a function of the nor-malized radius for B, = 0, 100 and 500 G. The beam radius is rb = 5 mm.Fig. 2 Normalized microwave intensity profile for the eigenmode (solid line) and the TE11mode (dashed line) for different beam radii. The wiggler field B, = 500 G. The beamedges are marked by the arrows.Fig. 3 Ratio of the imaginary part of the eigenvalue k to the TE11 gain as a function of thenormalized beam radius, for different values of B,2.91.00---- Bw =0--- Bw =100Gu-.8 - B =500G0.6~0.4LUN%S0.2-BEAM EDGE0o 0.2 0.4 0.6 0.8 1.0NORMALIZED RADIUS (r/a)Fig. 1101.00.20.80.6---- TEII MODEGUIDED MODEBw 500G -rb/a =.9- - 0.210.4 0.6 - 0.8 1.0 0.00.2 0.4 0.6 0.8 1NORMALIZ1.0-0.4V)zwz O.2w: 0.002 1.00-0.80.60z.0 0.00.2 0.4 0.6 0.8 .CTEI MODEGUIDED MODE- -% Bw= 500Grb/a=.40.2 0.4 0.6 0.8 1.0ED RADIUS (r/a)Fig. 2110.80.61.00.8.0.60.40.20.40.210.0TE1 I MODE--- GUIDED MODE-Bw 500Grb/o =.5- -6- \ -0.4---- TE11  MODEGUIDED MODE- Bw 500Grb/a .71.5.0 -0.5 -0--+--0Bw= IO0G-E-Bw= 500G0 0.2 0.4 0.6 0.8 1.0NORMALIZED BEAM RADIUS (rb/a)Fig. 312t

Anomalous (stimulated) refraction induced by the free-electron laser interaction

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Anomalous (stimulated) refraction induced by the free-electron laser interaction

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