THE EFFECT OF UNDULATOR HARMONICS FIELD ON FREE-ELECTRON LASER HARMONIC GENERATION *

The harmonics field effect of planar undulator on FreeElectron Laser (FEL) harmonic generation has been analyzed. For both the linear and the nonlinear harmonic generation, the harmonic generation fraction can be charactered by the coupling coefficients. The modification of coupling coefficients is given when third harmonics field component exist, thus the enhancement of the harmonic radiation can be predicted. With the third harmonics magnet field being 30 percent of the fundament, for both the small signal gain and the nonlinear harmonic generation, the intensity of thirdharmonic radiation can maximally be doubled. INTRODUCTION Using the higher harmonic is a way of Free Electron Laser (FEL) developing towards the shorter wavelength ranges [1-4]. For a planar undulator with a sinusoidal periodic magnetic field, the electrons also radiate at odd harmonics on-axis due to their non-uniform axial motion. In actual planar undulators, the magnetic field is nonsinusoidal and with harmonics field components. The harmonic radiation can be enhanced by aptly increase the harmonic field component [5]. Therefore some methods for this purpose were proposed, such as putting high permeability shims inside the undualtor [6], optimizing magnetic blocks size in a standard Harbch undulator [7]. In this paper, we analysis the effect of undulator harmonics field on FEL harmonic generation, the case of third harmonics magnetic field is considered specially. ANALYSIS In a planar undulator with an ideal sinusoidal periodic magnetic field, the electrons oscillate at odd harmonics frequency in the transverse direction, thus lead to the odd harmonics radiations in the direction forward [8]. For a FEL utilizing such undulator, the nth harmonic optical field equation and the phase equation in onedimensional mode are [9] [ , ] = in e e u n sn sn d r n a J J a e dz φ λ γ − % (1)


INTRODUCTION
Using the higher harmonic is a way of Free Electron Laser (FEL) developing towards the shorter wavelength ranges [1][2][3][4].For a planar undulator with a sinusoidal periodic magnetic field, the electrons also radiate at odd harmonics on-axis due to their non-uniform axial motion.In actual planar undulators, the magnetic field is nonsinusoidal and with harmonics field components.The harmonic radiation can be enhanced by aptly increase the harmonic field component [5].Therefore some methods for this purpose were proposed, such as putting high permeability shims inside the undualtor [6], optimizing magnetic blocks size in a standard Harbch undulator [7].In this paper, we analysis the effect of undulator harmonics field on FEL harmonic generation, the case of third harmonics magnetic field is considered specially.

ANALYSIS
In a planar undulator with an ideal sinusoidal periodic magnetic field, the electrons oscillate at odd harmonics frequency in the transverse direction, thus lead to the odd harmonics radiations in the direction forward [8].For a FEL utilizing such undulator, the nth harmonic optical field equation and the phase equation in onedimensional mode are [9] [ , ] = J is integer order Bessel function.From Eq.(1-2) for nth harmonic optical field, the small signal gain in low gain FEL and the nonlinear harmonic generation in high gain FEL can be given by [9] , ] 2 where, ρ is the Pierce parameter, x=nφ 0 'L, L=Nλ u is the length of undulator,.P e is the power of the electron beam 1 exp[ / ] / 9 ef g P P z L = is the fundamental power; ef P is the effective start-up shot noise power, it is equal to the fraction of the spontaneous undulator radiation in one power gain length [9].Thus the harmonic generation are charactered by the coupling coefficients for both linear case and non-linear case.The coupling coefficients therefore the harmonic generation increase with undulator deflection parameter but very slowly after a u >2.One can expect enhancement of the laser harmonics by adding a harmonic field to the fundamental sinusoidal undulator field.
In actual planar undulators, the magnetic field is nonsinusoidal, when expanded in Fourier series the field includes odd spatial harmonics due to the symmetry of the magnetic structure.Therefore the magnetic fields and corresponding dimensionless vector potential can be expressed by sin( ) where k u =2π/λ u is the wave number of fundamental magnetic fields; B um and a um is the peak of mth harmonics magnetic fields and corresponding dimensionless vector potential,.m is all or part odd numbers depending on the ____________________________________________ *Work supported by the the National Nature Science Foundation of China under Grant No. 10975137 # jiaqk@ustc.edu.cn

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Proceedings of FEL2010, Malmö, Sweden magnetic structure.Generally all harmonics components are much smaller than the fundamental component: % , the electrons velocity with the harmonics undulator fields is [cos(( where and rms value of the field is used.Accordingly the resonance condition is The longitudinal motion of electron is where = II z ct ) ) um ul m l ml ui a a r r m l a m l ξ ξ For the case that all magnetic harmonic components are much smaller than the fundamental component, all m ξ , ml ξ ± terms that don't contain the fundament are much smaller then one and can be neglected.Then Eq.( 7) became as ) Including the harmonic magnetic fields (Eq.( 6)) and with the optical field sin[ ( ) ] the phase equation (Eq.( 2)) now is   16) the condition k s >>k u is used.
Similarly when magnetic harmonic field existed the nth harmonic optical equation becomes Compare Eq.(15,17) with Eq.(1-2), it can be seen that when magnetic harmonic fields are included the coupling coefficient is modified as In the exponential of the modified coupling coefficient (Eq.( 16),) many terms are small and oscillate fast, a average over undulator period will eliminate these small contribution terms.Among all the harmonics, the third harmonic is the most important.Next we consider the case that only third harmonic field exist, and all other harmonics are neglected.In this case, Eq.(12 ) becomes 1 2 sin(2 ) sin( 4) where and have Then the modified coupling coefficient (Eq.( 16)) is here n=1,3.After a average over undulator period, the dominant product terms in the sum of above equation are those with For the small arguments, only Bessel functions of zero order will contribute.Because 2 can be further simplified by taking h 2 =0.Then at last we give the modified coupling coefficient as According above formulas, the modified harmonic coupling coefficients as function of undulator parameter are numerical calculated for different harmonic magnetic field fraction.The results reveal that the harmonic coupling coefficients so the harmonic emission are enhanced when B u3 have an opposite sign to B u1 , and are suppressed when the magnetic fields has the same sign (Fig. 1).The results also show that the fundament has been less affected by harmonic magnetic field (Fig. 2).For both the small signal gain and the nonlinear harmonic generation in high gain, the harmonic FEL radiation is proportional to the square of the coupling coefficient (Eq.( 4) and ( 5)).Therefore comparing with the case without harmonic magnetic field presents, the enhancement of the 3rd harmonic radiation is The dependence of third harmonic generation on the ratio of B u3 to B u1 is shown in Fig. 3.The enhancement of third harmonic generation is shown in Fig. 4.While in the calculation of Eq.( 24), the fundamental magnetic field in the coupling coefficients and in the modified coupling coefficients are taken with a little difference to keep the resonant wavelengths same, it is a u in the former, and a u1 in the later: We can see that harmonics radiation enhancement increase with the magnetic field ratio of the harmonics to the fundamental, and for a given harmonic magnetic field

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Proceedings of FEL2010, Malmö, Sweden fraction, the enhancement is larger when the magnetic field is weaker.With the third harmonics magnetic field 30 percent of the fundament, this field ratio translates into a vector potential ratio of 10 percent, the intensity increase of third-harmonic radiation is about 40 percent, and is more larger in smaller undulator deflecting parameter case, maximally doubled for undulator deflecting parameter K(= 2 u a ) =1.

SUMMARY
In summary, we have analysed the effects of undulator harmonics field on the harmonic coupling coefficients and FEL harmonic generation.For the case where third harmonic field present, analytical and easy to calculate expressions of the modified fundamental and harmonic coupling coefficients are given, and used to predict the effects of undulator harmonics field on the small signal gain in low gain FEL and the non-linear harmonic generation in high gain FEL.The numerical results demonstrate that the third harmonic emission can be distinctly enhanced by the undulator third harmonic field that has an opposite sign to the fundament field, while the fundamental emission has been less affected.With a third harmonic field 30 percent of the fundamental field, the third-harmonic emission can be enhanced about 40 percent, and can maximally be doubled.
Owing to that the FEL simulation codes we had can't include the harmonic magnetic fields, here only analytic results are given.For the next work, we'll cast about for the numerical simulation study.

Figure1:Figure 2 :
Figure1: Modified third harmonic coupling coefficient due to the harmonic magnetic field

Figure 3 :Figure 4 :
Figure 3: the effect of B 3 on FEL harmonic generation (mc 2 k u ) are dimensionless vector potential of the rms nth harmonic radiation field E sn and undulator field B u , respectively; k sn =2π/λ sn , k u =2π/λ u are the corresponding wave number; ϕ sn is the phase of the radiation field; r e is the classical electron radius; n e and γ is the density and energy of electrons; φ is the ponder motive phase of electron φ=(k s +k u )z-ω s t. the angular bracket represents the average over the electron's initial phases and initial phase velocitiesφ 0 '.[J,J] n is the coupling coefficient