Production and properties of two-color radiation generated by using a Free-Electron Laser with two orthogonal undulators

Nuclear Instruments and Methods in Physics Research A 767 (2014) 227–234

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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima

Production and properties of two-color radiation generated by using a Free-Electron Laser with two orthogonal undulators N.S. Mirian b,c,d, G. Dattoli a, E. DiPalma a, V. Petrillo c,d,n a

ENEA Centro Ricerche Frascati, via E. Fermi, 45, IT 00044 Frascati, Rome, Italy School of Particle and Accelerator Physics, Institute for Research in Fundamental Sciences (IPM), Post code 19395-5531, Tehran, Iran c Università degli Studi di Milano, via Celoria 16, IT 20133 Milano, Italy d INFN-Mi, via Celoria 16, IT 20133 Milano, Italy b

art ic l e i nf o

a b s t r a c t

Article history: Received 4 June 2014 Received in revised form 19 August 2014 Accepted 23 August 2014 Available online 1 September 2014

We present the analysis of the two-color Self Amplified Spontaneous Emission generated by a FreeElectron Laser amplifier constituted by two orthogonally polarized undulators with different periods and field intensities. Equations deduced in a non-averaged and in an averaged model have been integrated and compared. The two pulses have different frequencies, ruled by proper resonance conditions, and different polarizations, while the total length of the device does not change noticeably with respect to usual single color FELs. The wavelengths of two colors can be changed by choosing different periods, while variation in the magnetic strengths can be used to modify the gain lengths in view of various applications. & 2014 Elsevier B.V. All rights reserved.

Keywords: Time reversal T-violation Time Quantum theory

1. Introduction Free Electron Lasers (FELs) sources [1–4] play a key role in several applications belonging to the most various scientific and technical fields. Among the different formats in which radiation can be proposed to users, one of the most required is where the pulse is composed of two distinct spectral lines with a variable time delay between them. Matter can be indeed probed on the atomic scale in space and time [5] by means of two color X-rays, extending in this way the knowledge about the fundamental properties of materials and living systems with respect to the Nobel Prize work on femtochemistry of Zewail [12]. Pairs of colored X-ray pulses are particularly suitable to perform pump and probe experiments of structural dynamics, which are designed to monitor the ultrafast evolution of atomic, electronic and magnetic structures [6–8]. In pump-probe experiments, the process under study – e.g. a chemical reaction, an excitation or a structural change on the surface of a solid – is activated by means of a first pulse with a fixed frequency and then, after a delay, a second one of another frequency, or a sequence of pulses, records the event. In this way, following its time evolution, information on pathways, barriers and transition states of the phenomenon can be accessed. As regards the time scales of the dynamics of these atomic events, they can range from 10 fs in ultrafast processes as

n

Corresponding author. E-mail address: [email protected] (V. Petrillo).

http://dx.doi.org/10.1016/j.nima.2014.08.043 0168-9002/& 2014 Elsevier B.V. All rights reserved.

the dissociative ionization [9], to hundred femtoseconds for less energetic chemical mechanisms [10,11]. In another important field, the future color X-ray technology, the color component contains extra information and allows us to distinguish the chemical composition of the absorbing tissues [13,14], permitting the development of the diagnostic clinical imaging. Experiments on dual frequency production and use have been recently carried on with FELs [15–21] in many different ways. At the same time, several promising theoretical proposals aimed to generate two-color FEL emission in the X-ray wavelength regime [22–25] have been so far investigated. The first schemes dated back to more than 20 years ago [26,27], in a period when the tunability at infrared and visible wavelengths was a unique feature of the FELs, since optical parametric amplifiers were not yet commercially available. The fact of reusing the same hardware for generating two frequencies constitutes the main advantages of a two-color FEL, with respect to two independent FEL beamlines, because minimizes the time jitter between the two pulses. Some of the initially proposed designs were based on staggered undulator magnets having different strength, to achieve lasing at two distinct wavelengths [28–31]. This idea has been recently reconsidered at LCLS and implemented in the X-ray range. The emittance-spoiler technique with a magnetic chicane in the undulator section was used to control the pulse duration and relative delay of a two-color intense X-ray pulse generated by using two separate canted pole undulators tuned at different resonances [15]. However, the undulator length is essentially doubled and saturation is reached at power levels comparable

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with the single color configuration by using a complex scheme. Another option, recently demonstrated at the FERMI soft X-ray FEL, involves the use of either a chirped or a two-color seed laser which initiates the FEL instability at two different wavelengths within the modulator gain bandwidth [18,17]. The different approach of injecting in the FEL undulator a multi-energy electron beam [32] resonating at two different wavelengths permits the control of the frequency and time separation ranges of the FEL radiation, while maintaining similar saturated power levels and minimal undulator length [16,19–21]. In this configuration, the SASE lasing occurs from separated and nearly independent electron distributions [33]. In the present paper, we analyze the operation with a further different scheme: the FEL emission is obtained by assembling in a unique structure two orthogonally polarized undulators with different periods and field intensities. In this case, the two radiations have not only different frequencies, but also different polarizations, while the total length of the device does not change substantially with respect to usual single color FELs [34]. Producing two waves with orthogonal polarizations with comparable intensities is very important because permits the selective excitation of the molecular fluorescence, opening various possibilities to control the internal organization and space orientation of molecules. In this way, the techniques based on fluorescence anisotropy [35] and dichroism [36] could be significantly improved. An undulator with crossed polarizations in a delta like magnetic structure has been already constructed and measured [37], and the extension to the configuration discussed in this paper is in progress. The wavelengths of two different colors can be changed by setting different periods, while variations in the magnetic strengths have the effect of modifying the gain lengths. In the first two sections, non-averaged FEL equations, similar to those described in [38] which are at the basis of the code MEDUSA, and averaged ones [39] have been extended in the case of presence of detuning and commented upon. Analytical and numerical data are presented in Section 4. Comments and conclusions close the paper.

The momentum equations for each electron of the beam turn then out to be dpxi ¼ eβ zi Bw1 sin k01 z  ek1 ð1  βzi Þ½A1 eiα1i þ cc dt

ð4Þ

dpyi ¼ eβzi Bw2 sin k02 z  ek2 ð1  βz Þ½A2 eiα2i þ cc dt

ð5Þ

dpzi ¼  eβyi Bw2 sin k02 z ek2 β yi ½A2 eiα2i þcc dt eβxi Bw1 sin k01 z  ek1 β xi ½A1 eiα1i þ cc

ð6Þ

where

α1;2;i ¼ k1;2 zi  ω1;2 t:

ð7Þ

and βx;y;zj ¼ vx;y;zj =c are the normalized velocity components, and cc represents the complex conjugate. From Eq. (6) we obtain the following resonance conditions:

2. Model equations in a non-averaged SVEA treatment The FEL undulator shown in Fig. 1 is composed by two linear arrays of magnets orthogonally polarized and with periods given respectively by λ01 and λ02. The undulator magnetic field, in the paraxial approximation, is described by the following expression: B w ¼  Bw2 sin k02 ze x þ Bw1 sin k01 ze y

Fig. 1. Geometry of the undulator.

λ1;2 ¼

λ01;02 ð1 þ K 21 =2 þK 22 =2Þ 2γ 20

ð8Þ

with γ 0 ¼ 〈γ i 〉, the average value of the Lorentz factor of the electrons.

ð1Þ

where k01;02 ¼ 2π =λ01;02 and K 1;2 ¼ jeBw1;2 λ1;2 =mc2 j are the deflecting parameters of the undulators.

2.2. Field equation The wave equation can be written as

2.1. Momentum equation

∂ 1 ∂2 4π A  2 2A ¼  J ? : 2 c c ∂t ∂z

The complex radiation fields, in terms of the two orthogonal components, assume the form:

By expressing the transverse currents in terms of the particle density n:

2

J x;y ¼ ∑ecβ x;yj n δðz  zj Þ

E ¼ ½E1 eiðk1 z  ω1 tÞ e x þE2 eiðk2 z  ω2 tÞ e y  B ¼  B2 eiðk2 z  ω2 tÞ e x þ B1 eiðk1 z  ω1 tÞ e y ;

ð2Þ

while the vector potential may be written as iðk1 z  ω1 tÞ

A ¼  i½A1 e

iðk2 z  ω2 tÞ

e x þ A2 e

ð9Þ

e y ;

ð3Þ

where E1;2 , B1;2 and A1;2 are slow complex amplitudes and the two radiation wavelengths are λ1;2 ¼ 2π =k1;2 . The eventual presence of a frequency detuning δ1;2 for one of the two frequencies or both leads to the fact that the frequencies ω1;2 can be expressed as ω1;2 ¼ ωR1;2 þ δ1;2 , with ωR1;2 the nominal resonance frequency.

¼ ∑ec

K 1;2

γj

cos ðω01;02 tÞn δðz  zj Þ

ð10Þ

and by using the Slowly Varying Envelope Approximation (SVEA), we obtain for the two polarizations the following independent differential equations: ∂ 1∂ δ1 2π en A1 þ A1  i A1 ¼ ∑β xj δðz  zj Þe  iα1j : ∂z c ∂t k1 c

ð11Þ

∂ 1∂ δ2 2π en A2 þ A2  i A2 ¼ ∑β yj δðz  zj Þe  iα2j : ∂z c ∂t k2 c

ð12Þ

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Differently from the case where two waves have different wavelengths, but equal polarizations, the equations for the potential result to be coupled only through the longitudinal dynamics of the electrons appearing in α1;2j , while the transverse motion remains decoupled.

3. Model equations in an averaged SVEA treatment The zero order dimensionless transverse velocities can be written following the Colson's analysis [39] as K1

βx ¼ 

cos ðk01 zÞ  

γ0

K2

βy ¼ 

cos ðk02 zÞ  

γ0

K1

γ0

K2

γ0

cos ðω01 tÞ

3.2. Field equation Operating the same average on the potential equations leads to: ∂A1 1 ∂A1 δ1 en π K 1 þ  i A1 ¼ F 1 ∑e  iθ1;j ∂z c ∂t c k1 γ 0

ð21Þ

∂A2 1 ∂A2 δ2 en π K 2 þ  i A2 ¼ F 2 ∑e  iθ2;j : ∂z c ∂t c k2 γ 0

ð22Þ

In the limit of one undulator only, K 2 ¼ 0 (J 0 ð0Þ ¼ 1Þ and the usual Colson's equations are retrieved.

ð13Þ 3.3. Scaled equations and FEL parameter

cos ðω02 tÞ

ð14Þ

and the trajectories of the electrons become

λ01 K 1 λ02 K 2 r j ¼ β0 cte z  sin ðω01 tÞe x  sin ðω02 tÞe y 2π γ 0 2π γ 0
2
2 λ01 K 1 λ02 K 2  sin ð2ω01 tÞe z  sin ð2ω02 tÞe z 16π γ 0 16π γ 0 with β 0 ¼ 1  1=γ

229

In the universal scaling notation [40], the normalized fields take the expression: a1;2 ¼

ω1 eA1;2 pffiffiffiffiffiffiffiffiffiffi ωp ρ1 γ 0 mc2

ð23Þ

where ð15Þ

2 0

ρ1;2 ¼

  1 ωp K 1;2 F 1;2 2=3 γ 0 8ω01;02

ð24Þ

is the FEL parameter and ωp is the plasma frequency. In terms of the quantity Γ i ¼ γ i  γ 0 =ρ1 γ 0 and of scaled space ζ ¼ 2k01 ρ1 z and time τ ¼ 2k01 cρ1 t, the equations are therefore

3.1. Energy equation

dΓ i F 2 K 2 k02 iθ2i ¼ eiθ1i a1 þ e a2 þ cc: dτ F 1 K 1 k01

ð25Þ

∂a1 ∂a1 þ  iδ 1 a1 ¼ ∑e  iθ1;i : ∂τ ∂ζ

ð26Þ

where ξ1;2 ¼ K 21;2 =4ð1 þ K 21 =2 þ K 22 =2Þ, and, by defining the pondermotive phases

∂a2 ∂a2 K 2 k01 F 2 þ  iδ 2 a2 ¼ ∑e  iθ2;i ∂τ K 1 k02 F 1 ∂ζ

ð27Þ

θ1;2j ¼ ω01;02 t þ k1;2 βzj ct  ω1;2 t

where δ 1;2 ¼ δ1;2 =2k0;1 cρ1 , while the phases evolve according to

From Eq. (15), the longitudinal motion at zero order can be described by z ¼ β0 ct 

ξ1 k1

sin ð2ω01 tÞ 

the equation for

ξ2 k2

sin ð2ω02 tÞ

ð16Þ

ð17Þ

γj

dγ j e ¼  β  Re E mc j dt

ð18Þ

can be written as dγ j e K 1 iθ1j ¼ ðe k1 A1 ½e  iðξ1 sin ð2ω01 tÞ þ k1 ξ2 =k2 sin ð2ω02 tÞÞ dt 4mc γ 0

ð19Þ

We then introduce a double average along both directions x and y, over a length L multiple of the undulator periods L ¼ nλ01 ¼ mλ02 . By using the integral expression of the Bessel's functions: 1 2π

Zπ

dτeiðx sin τ  nτÞ

π

the equation for the energy becomes dγ j e K1 ¼ F 1 ðeiθ1j k1 A1 þ ccÞ dt 4mc γ 0 þ

e K2 F 2 ðeiθ2j k2 A2 þ ccÞ 4mc γ 0

dθ2i k02 ¼ Γ: dτ k01 i

ð29Þ

The gain lengths of both waves can be found by means of the standard linear analysis of Eqs. (25)–(29), leading to

e K 2 iθ2j þ ðe k2 A2 ½e  iðk2 ξ1 =k1 sin ð2ω01 tÞ þ ξ2 sin ð2ω02 tÞÞ 4mc γ 0

J n ðxÞ ¼

ð28Þ

3.4. Gain length

þ e  iðξ1 sin ð2ω01 tÞ þ k1 ξ2 =k2 sin ð2ω02 tÞÞ  i2ω01 t  þ ccÞ

þ e  iðk2 ξ1 =k1 sin ð2ω01 tÞ þ ξ2 sin ð2ω02 tÞÞÞ  2iω02 t  þ ccÞ:

dθ1i ¼ Γi dτ

ð20Þ

where F 1;2 ¼ J 0 ðk1;2 ξ2;1 =k2;1 Þ½J 0 ðξ1;2 Þ  J 1 ðξ1;2 Þ are the coupling Bessel factors, modified for the case of two undulators.

F 2 K2 δnm χ 2 F 1 K1
 F 2 K2 F K k g 32 χ 2 ¼ χ 1 δnm þ 2 2 2 χ 2 F 1 K1 F 1 K 1 k1 g 31 χ 1 ¼ χ 1 þ

ð30Þ

where, if the wave field is expressed as aj ¼ a0;j e  igj ζ , the positive imaginary part of the complex factor g j ¼ ηj þ iχ j is the field gain. From the above equations it turns out that χ 2j ¼ 3η2j and then the power gain length is Lgj ¼ λ01 =8πρχ j . When the two undulators have different periods m an (δnm ¼ 0), the above equations lead to
1=3
 λ01 λ01 F 2 K 2 2=3 Lg1 ¼ pffiffiffi ¼ Lg2 : ð31Þ F 1 K1 λ02 4 π 3 ρ1 When, instead, m¼ n (δnm ¼ 1), and therefore g 1 ¼ g 2 , then Lg1 ¼ Lg2 ¼

λ01


 2 1=3 : pffiffiffi 2 K2 4π 3 1 þ F F 1 K1

ð32Þ

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It can be noted that if the two undulators have equal periods and Bw1 ¼  Bw2 , and if the initial condition for the potentials is the same A1 ð0Þ ¼ A2 ð0Þ, the right hand of Eqs. (6) and (25) are null, and the evaluation of terms of second order is required. 3.5. Pendulum equation By differentiating Eqs. (28) and (29) with respect to τ and by using Eq. (25), the following pendulum equation can be obtained:
 F K k θ€ 1i ¼ eiθ1i a1 þ 2 2 02 eiðk02 =k01 Þθ1i a2 þ cc ð33Þ F 1 K 1 k01 showing the bi-harmonic electron oscillations along ζ with the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi two frequency f 1 ¼ a1 and f 2 ¼ ðF 2 =F 1 ÞðK 2 =K 1 Þa2 . We can then calculate analytically the separatrix by cross multiplying Eqs. (25) and (28), and integrating over τ:

F 2 K2 ðΔγ Þ2 ¼  2iðργ 0 Þ2 a1 eiθ1 þ a2 eiðk02 =k01 Þθ1 þ cc þH ð34Þ F 1 K1 where H, the constant of integration, is the Hamiltonian of the system.

and given by P 01;2 A1;2 eð0:223t=Z 1;2 Þ : P 1;2 ¼
P 01;2 1þ ðA1;2  1Þ P s1;2 pffiffiffi !

! 1 2 t 4 t 3t þ cos h A1;2 ¼ þ cos d cos h 3 9 Lg1;2 9 2Lg1;2 2Lg1;2

is also reported, with Z 1;2 ¼ 1:066Lg log ð9P s1;2 =P 01;2 Þ and P s1;2 ¼ 1:42ρ1;2 P b being the saturation power as a function of the beam power Pb. Formula (35) reproduces very accurately lethargy, growth and gain lengths of both waves, while the saturation value results of the same order only for one of the two polarizations. This is due to the interaction between the two waves occurring when the power in the two polarizations is large, an effect which is not taken into account in Eq. (35). Fig. 2 presents also the induced relative increase in the energy spread σ E =E occurring during the emission, as computed by the phase space (green stars), and evaluated by the analytical formula: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hσ i2 hσ i2 σE 1 2 ¼ ð36Þ þ E E E where

4. Two pulses SASE FEL emission In the approximation of the time independent scheme, the set of non-averaged (4)–(12) and averaged (25)–(29) equations have been numerically integrated with independent codes. The results demonstrate an overall satisfactory agreement between the two models, as always shown in similar situations [41]. The parameters chosen for the simulations, similar to the SPARC's values [42–44], are: λ01 ¼ 2:8 cm, K 1 ¼ 2:1. The electron current I has been fixed at 100 A, for a value of ρ1 ¼ 5:47  10  3 . 4.1. Different periods, same wiggler parameter (K 1 ¼ K 2 ) The comparison between the solutions of non-averaged (red curves) and averaged (blue curves) equations is shown in Fig. 2 for (a) λ02 ¼ 1:5λ01 , (b) λ02 ¼ 2λ01 and (c) λ01 ¼ 10λ02 , with the two magnetic strengths fixed at the same value K 1 ¼ K 2 and without detuning. The agreement is highly satisfactory up to the onset of saturation. Differences occur once that the saturation is reached, particularly when the two waves have similar intensity, and are probably due to the differences in the sets of equations and to the different method of integration. In the figure, the logistic map proposed in [45] (black solid and dotted curve, labeled with (T)),

ð35Þ

σ 1;2

3 ¼ 2 E

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ1;2 P 01;2 u u A t P 01;2 Pb 1 þ 1:24 ðA1;2 1Þ

ð37Þ

P s1;2

represented by the dark green dashed line. The growth of the energy spread continues up to the value of ρ2 (light green straight line) and then saturates, producing, as a consequence, the anticipated saturation of the radiation with the longest gain length. The numerical integration shows that the final value of power Ps is reached after a series of oscillations, due to the phenomenon of the mutual bunching described in Ref. [33]. Fig. 3 shows the values of ρ1 and ρ2 for different values of n/m. The power vs z, obtained with the non-averaged code, is presented in Fig. 4 for different values of m at fixed n ¼1. In this case the wavelength along the x direction is λ1 ¼ 800 nm, while λ2 (along the y axis) is equal to λ1 =m with m ¼ 1, 2, 4, 8 and 10, respectively. Since ρ1 remains unaltered, the gain length of the x-polarization does not change, while the gain length of the y-polarization wave first decreases as a function of m, and then increases, according to relation (31). For the singular case m ¼ n ¼ 1, the gain length of the x-polarization is different with respect to the other cases, according to (30). The trend of Lg1 =Lg2 vs m/n (for this figure is easier the use of this variable instead than n/m) as given by Eq. (31) is shown in Fig. 5 and confirms the fact that from m ¼1 to about m ¼9 the largest frequency wave has a

Fig. 2. Power P(W) in the x (solid curves) and y (dashed curves) polarizations vs z(m). Comparison between non-averaged (red curves) and averaged (blue curves) model for (a) n/m ¼ 0.66 and (b) n/m ¼ 0.5 and (c) n/m ¼ 0.1. The black line is the logistic map, Eq. (35). In green σ E =E; as given by (36). Green stars: energy spread computed by the phase spaces. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

N.S. Mirian et al. / Nuclear Instruments and Methods in Physics Research A 767 (2014) 227–234

Fig. 3. FEL parameter for x-polarization (black line) and y-polarization (red dashed line) versus of n/m. λ01 ¼ 800, K 1 ¼ K 2 ¼ 2:1. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

231

Fig. 6. (a) Saturation power and (b) saturation length of x-polarization (black crosses and lines) and y-polarization (red pluses and lines) versus n/m. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 4. Radiation growth of x-polarization (solid line) and y-polarization (dashed line) versus the coordinate inside the undulator for n¼ 1 and m ¼1, 2, 4, 8 and 10.

Fig. 7. Bunching factor of x-polarization (line curve) and y-polarization (dashed curve) versus length of the undulator in case of n/m ¼ 2 and n/m ¼ 1.5.

Fig. 5. Ratio of gain length of the pulses vs m/n, K 1 ¼ K 2 ¼ 2:1. The stars represent the values obtained by Fig. 4.

shorter gain length, while for m larger than 9 the opposite occurs. The stars are the values of Lg1 =Lg2 evaluated by the numerical calculations presented in Fig. 4. The saturation powers and the saturation lengths of both polarizations are compared in Fig. 6 for different value of n/m. Black line and crosses represent the x-polarization, red line and pluses the y-polarization. We identify the saturation with the point where the slope of the growth changes, and we indicate in the figure this condition with symbols connected by lines. Single black crosses and red pluses appearing in the graphs show, instead, the points of maximum power reached after saturation. These points have been evaluated for integer and low values of n/m or m/n. For larger integer values of n/m and m/n a significant reduction in the power growth takes place. In order to analyze the electron dynamics as the beam is advancing along the wiggler, the bunching factor for n/m ¼ 2 and n/m ¼1.5 is shown in Fig. 7. The microbunching process takes place on the scale of the wavelength of the corresponding

pondermotive potential and this process is completed at the onset saturation. So, first the beam is modulated and microbunched on the shorter wavelength, because, as shown in Fig. 3, the FEL parameter for the shortest wavelength is lower than that of the longest one. The first beam going in saturation influences, with its strong bunching, the growth of the other polarization. If one of the wavelengths is an harmonics of the other one, the slope of the longer wavelength growth changes due to their interaction, as it can be seen in Fig. 7, but the microbunching process in the long wavelength goes on. When the ratio n/m is not integer, the growth of the long wavelength radiation increases while the bunching of the other polarization remains low. The average energy deviation 〈Δγ 〉 versus the pondermotive phase of the y-polarization 〈θ2 〉 is presented in Fig. 8 for n/m ¼ 2 and n/m¼1.5. Both waves absorb first energy from the electrons. Referring to case (a), with n/m ¼2, the electrons lose energy from z¼ 0.5 m and z¼7.2 m, with a change in the y phase Δθ2  π =4: Using Eqs. (28) and (29), this value corresponds to a rotation of π =2, a quarter of wavelength, in the bucket of the x polarization. In this phase the x polarization is fed and grows up to saturation. Then, after its saturation at z ¼7.2 m, the short wavelength wave gives back energy to the electrons, its power decreases, while the power of the electrons and of the long wavelength increase. In fact, Fig. 8 shows that the electrons lose energy again between z¼ 8.33 m and z ¼10 m, during a phase change of Δθ2  π =2, a quarter of wavelength of the y polarization. The comparison of the energy deviation after the first saturation shows that the electrons compensate their lost energy and amplify the long wavelength wave.

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For case (b), instead, both phases where the electrons lose energy, first between z¼0.38 m and z¼7.194 m, and then after z¼8.14 m, correspond to a phase change of Δθ2  1:15, corresponding to a value of π =2 for the θ1 : In both these phases, in fact, the x polarization growths. The other polarization has a lower increase. The evolution of the dynamics of the electrons is presented in Fig. 9, where the longitudinal phase space is shown for n/m¼2 at different values of the coordinate z along the undulators. Black lines represent the separatrixes as given by Eq. (34), and dots show the position of macroparticles in the phase space (variation of relative energy Δγ =γ versus pondermotive phase along y, θ2). At the beginning of the process (Fig. 9(a) and (b)), since the potential amplitudes are small, the electrons perform stable oscillations. The microbunching process, due to the longitudinal motion, induces the exponential growth. Then, the FEL approaches the saturation (Fig. 9 (c)). The microbunching process is complete and the potential amplitude is still increasing. Further, the threshold between stable particle oscillations and the unstable regime is reached, above which the particles are no longer trapped within the radiation potential buckets. Afterward, the phase space begins to develop

Fig. 8. Average gamma variation 〈Δγ〉 vs average electron phase 〈θ2 〉 along the trajectory in case of (a) n/m¼ 2 and (b) n/m ¼ 1.5.

chaotic patterns, corresponding to deep saturation (Fig. 9(d–f)). Case (e) corresponds to a situation of relative maximum average energy of the electrons, while in (d) and (f) the electron energy is minimum. Since θ2 ¼ θ1 =2, the bunching on λ2 grows when the electron are grouped in areas with extension Δθ1  π =2; as occurs in Fig. 9(f). The frequencies of the two polarizations can be approached by varying the factor n/m. In the range around n/m¼1, the system produces two waves of close frequencies, with the largest one characterized by a shorter gain length. When n=m  1 the two saturation lengths and powers are very similar. Fig. 6 reports the radiation power and the position of saturation when n/m is very close to unity. 4.2. Different periods, different magnetic strength For balancing the level of the power in the two different polarizations at the end of the undulator, the magnetic strengths of the undulators can be varied. We fixed the vertical undulator properties (K 1 ¼ 2:1 and λ01 ¼ 2:8 cm), and varied K2 for different n/m. Fig. 10 shows the gain length ratio of both polarizations as a function of varying K2 for different value of n/m. For low values of K2, the slope of the curves increases with decreasing n/m. Since both saturation power and length depend on ρ, the ratio of the FEL parameters is shown in Fig. 11. The numerical simulations show that the interaction between the two waves can change the level of the saturation power. When one wave reaches saturation, the electrons are strongly influenced by its electric field, so the growth of the other one is affected. The ratio of the power of the pulses at the first saturation point is reported vs K2 in Fig. 12 for various n/m between 0.5 and 2. In the case n/m close to one, the waves saturate in similar positions. The saturation length does not depend strongly on K2, while, instead, the power ratio depends on it. For n/m¼ 1, the gain length follows Eq. (32) and the ratio between the powers has a different trend respect to the other cases, as shown in Fig. 12. For the cases n=m o 1, since the ratio between the FEL parameters is less than one (ρ1 =ρ2 o 1), the first wave going to saturation is the y-polarization. If m/n is integer (as,

Fig. 9. Electron orbits in phase space (red dots) and separatrix (black line) at (a) z ¼ 3.52 m, (b) z ¼4.7 m, (c) z¼ 5.9 m, (d) z ¼7 m, (e) z¼ 8.33 m and (f) z ¼10 m. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

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agreement as regards lethargy, growth and gain length of the radiation, while discrepancies appear in saturation. A theoretical logistic map fits well the simulations. The use of a device of this kind permits to produce two color radiations with an easy control of the frequencies and with different polarizations, while the total length of the device does not change respect to usual single color FELs. The possibility of changing independently the strength of the two magnetic fields allows us to rule the final power and the saturation length.

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Fig. 10. The gain length ratio of x-polarization to y-polarization for different values of n/m, while K 1 ¼ 2:1 and λ01 ¼ 2:8 cm.

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[6] [7] [8]

[9] [10] [11] [12] [13]

[14] [15] Fig. 11. The FEL parameter ratio between the x and y polarizations for different values of n/m, while K 1 ¼ 2:1 and λ01 ¼ 2:8 cm.

[16] [17] [18] [19]

[20] [21] [22] [23] [24] [25] [26] [27] [28] [29] Fig. 12. The power ratio of x-polarization to y-polarization for different values of n/m. K 1 ¼ 2:1 and λ01 ¼ 2:8 cm.

[30]

for instance, the case n/m¼ 0.5) the waves saturate in different points. If otherwise, as for instance, n/m ¼0.75, both waves saturate in the same position but at different power level.

[31]

5. Conclusion

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The case of two orthogonal undulators with different polarizations and periods have been studied. The proper resonance relation has been introduced. Non-averaged and averaged equations have been written and integrated. These two models present a significant

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