Influence of an imperfect energy profile on a seeded free electron laser performance

A single-pass high-gain x-ray free electron laser (FEL) calls for a high quality electron bunch. In particular, for a seeded FEL amplifier and for a harmonic generation FEL, the electron bunch initial energy profile uniformity is crucial for generating an FEL with a narrow bandwidth. After the acceleration, compression, and transportation, the electron bunch energy profile entering the undulator can acquire temporal nonuniformity. We study the influence of the electron bunch initial energy profile nonuniformity on the FEL performance. Intrinsically, for a harmonic generation FEL, the harmonic generation FEL in the final radiator starts with an electron bunch having energy modulation acquired in the previous stages, due to the FEL interaction at those FEL wavelengths and their harmonics. The influence of this electron bunch energy nonuniformity on the harmonic generation FEL in the final radiator is then studied.


I. INTRODUCTION
The free electron laser (FEL) is perceived as one of the candidates for the fourth generation light source.Success in commissioning the world's first x-ray (1:5-15 # A) FELthe LINAC Coherent Light Source (LCLS)-at SLAC National Accelerator Laboratory opens the gate for new science [1].Further improving the FEL spectrum bandwidth is urged by various potential users.One of the possibilities to generate narrow bandwidth FEL is to invoke a coherent seed laser to start the FEL process, which is generally referred to as a seeded FEL.With a coherent seed laser, the radiator can set to have the resonant wavelength the same as the seed laser to simply form a FEL amplifier or an optical klystron (OK) [2].An OK has two undulators with a magnetic buncher in between.For an OK, indeed the radiator can have the resonant frequency as one of the harmonics of the seed laser.In such an operation mode, a harmonic generation free electron laser (HGFEL) can be configured [3,4].Because of the fact that the buncher between the two undulators will rotate the phase space on the seed wavelength scale, the electron bunch entering the radiator will have multifrequency components in its energy spectrum.We investigate its impact on the radiator FEL performance, in particular the FEL bandwidth from this multifrequency energy spectrum.In general, the electron bunch generated from the photoinjector has a very small energy spread and small emittance.During the acceleration, bunch compression, and transportation, the electron bunch will experience the rf curvature, the second-order effect in the chicane, and collective effects, which will all lead to a nonuniform energy profile [5].In addition, the electron bunch is subject to microbunching instability [6][7][8][9][10][11][12][13].Thus, the electron bunch entering the undulator can have an energy modulation with multiple frequencies.Such energy modulation will impact the FEL performance and affect the FEL bandwidth.As mentioned above, there is interest in this subject because of the general attention to the problem of increasing the temporal coherence properties of single-pass short wavelength FEL sources.The seeded FEL can start the FEL process with a coherent laser seed; yet, the nonuniform beam energy profile along the electron bunch, which changes the resonance condition and the local gain factor driving part of the beam out of resonance, is one of the causes of degradation of the coherence length during the amplification process.Hence, the flatness requirement of the beam parameters for the operation of a seeded FEL was addressed [14], with the idea of reverse tracking the particles dynamics in order to provide a ''flat'' beam in the sense of uniform parameters at the entrance of the undulator.Along this line of thinking, the dynamical problem of the interaction of a seeded FEL with an energy chirped (linear and second-order) electron beam was also addressed [15][16][17][18].
A study conducted for single frequency energy modulation was recently reported [19].In this paper, we consider multifrequency initial energy modulation which is closer to the reality as we described above.We study the influence of the energy profile nonuniformity on the free electron laser (FEL) performance for a FEL amplifier as well as for a harmonic generation (HG) FEL.The theory is compared to three-dimensional simulation with GENESIS [20].
The paper is organized as follows.In Sec.II, the theory frame is laid out.We formulate it as an initial value problem within the Vlasov-Maxwell coupled equation framework.The seeded FEL evolution with an electron bunch having an initial multifrequency energy spectrum is derived.The FEL bandwidth with such a nonuniform energy profile electron bunch is computed and compared to that of an ideal monoenergetic electron bunch.The expression is cross-checked with GENESIS numerical simulation in Sec.III.In Sec.IV, we then discuss the impact of the electron bunch multifrequency energy spectrum on the harmonic generation FEL performance in the radiator.As mentioned above, intrinsically, a HGFEL in the radiator starts from an electron bunch with multifrequency energy spectrum.Some discussion is drawn in Sec.V.

II. VLASOV-MAXWELL ANALYSIS FOR AN INITIAL VALUE PROBLEM
For a FEL amplifier, the FEL process starts from a coherent seed; while for an optical klystron [2] and (high-gain) harmonic generation FEL [4,21,22], the FEL radiation in the radiator starts from coherent emission from a microbunched electron bunch.Nevertheless, the coherent emission once generated will be decomposed into the FEL guided modes and will be amplified due to the same FEL process.The FEL amplification process by an electron bunch with multifrequency energy spectrum is the same and applicable to all these different FEL configurations.Hence, in the following let us formulate the FEL start-up and evolution process when the electron bunch has energy nonuniformity.We will postpone the discussion of the coherent emission for an optical klystron or a harmonic generation FEL in Sec.IV.
To analyze the start-up of a seeded FEL amplifier, we use the coupled set of Vlasov and Maxwell equations which describes the evolution of the electrons and the radiation fields [15].This approach is used as well for the self-amplified spontaneous emission (SASE) FEL [23].We will work with a one-dimensional system in this section.

A. Vlasov-Maxwell equations
We follow the analysis and notation of Refs.[15,23].Dimensionless variables are introduced as , and k w ¼ 2= w with 0 being the radiation wavelength, w being the undulator period, and c being the speed of light in vacuum.We also introduce p ¼ 2ð À 0 Þ= 0 as the measure of energy deviation, with the Lorentz factor of an electron in the electron bunch, and 0 the resonant energy defined by 0 ¼ w ð1 þ K 2 =2Þ=ð2 2 0 Þ, for a planar undulator, where the undulator deflecting parameter K % 93:4B w w with B w the peak magnetic field in Tesla and w the undulator period in meter.The electron distribution function c ð; p; ZÞ is normalized, i.e., R c ð; p; ZÞddp ¼ 1, with c 0 ð; p; ZÞ describing the slow-varying unperturbed component.The FEL electric field is written as Eðt; zÞ ¼ Að; ZÞe iðÀZÞ with Að; ZÞ being the slow-varying envelope function.
The one-dimensional linearized Vlasov-Maxwell equations are and where in SI units , with e and m being the charge and mass of the electron; " 0 % 8:85 Â 10 À12 F=m being the vacuum permittivity; n 0 being the electron bunch density in units of 1=m 3 ; and , where the dimensionless rms undulator parameter a w K= ffiffiffi 2 p and J 0 ðxÞ and J 1 ðxÞ are the zeroth and first order Bessel functions.Equation (1) gives a general solution as Plugging Eq. (3) into Eq.(2), we have with ð2Þ 3 ð2D 1 D 2 Þ= 3 0 defining the FEL parameter [24,25].

B. Initial energy imperfectness-General solution
To model an energy imperfectness in the electron bunch coming into the undulator, we assume that the initial energy distribution function is where the initial discrete radiators (electrons) are modeled as P j ð À j Þ½p þ gð j Þ for the longitudinal coordinates following Eq.( 5).This term is for modeling the shot noise which is responsible for the SASE FEL.In addition, we also introduce a component Bð; ZÞ, which is related to the bunching factor, b, as b 1 2 This second term (given in the curly brackets) is written separately for a premicrobunched electron bunch.As written explicitly, after integration over , the second term shows the evolution of the initial bunching factor following the pendulum equation, i.e., the bunch factor bðZÞ is a function of Z.
To further work on Eq. ( 6), we now introduce the Laplace transform, fð; sÞ ¼ Z 1 0 dZe ÀsZ Að; ZÞ: Likewise, we introduce for the premicrobunched component.With this, Eq. ( 6) is now cast in the frequency domain as which yields the general solution as Notice that, in the square brackets in Eq. ( 11), the first term Að; 0Þ characterizes the initial seed for a seeded FEL, the second term models a premicrobunched electron bunch, while the third term models the SASE FEL.In the following, let us focus on a seeded FEL, so that the second term and third term in the square brackets will be ignored in the derivation.

C. Initial energy modulation-An example
In this section, the general function gðÞ as in Eq. ( 5) characterizing the nonuniform energy profile is represented as a Fourier series as in the following.Indeed, for electron bunch experienced microbunching instability, or in the harmonic generation FEL as explained above and detailed in the following Sec.IV B, there can be an energy modulation along the electron bunch as where !m characterizes the mth component of the energy modulation.The initial energy distribution function is then where m 2" m = 0 and !m !m =! 0 .For such a sinusoidal modulation, we have to the leading order in m .

The FEL solution
For a seeded FEL, let us throw away the initial value term, the premicrobunched term, as well as the SASE term, and keep only the seed in Eq. (11): Obviously, once we know the initial seed field envelope Að; 0Þ, we can obtain the seeded FEL field envelope Að; ZÞ along the undulator.The double integral in Eq. ( 16) can be evaluated by first performing the contour integral to get with the Green function Gð; ; Z; s; Þ and the corresponding phasor F ð; ; Z; s; Þ defined as where ¼ f 1 ; 2 ; . . .; 1 g.In Eq. ( 17), we implicitly introduce À 0 .The Green function can be estimated by saddle point approximation.The saddle point s s is found from and the Green function is approximated as For an initial Gaussian seed, we model it as where 0 ¼ 1=ð4 2 t0 Þ with t0 being the initial seed rms pulse duration.According to Eq. ( 17), the FEL pulse is where It is interesting to find that to the first order in , in the exponential function, the microbunching energy modulation only leads to a pure phase modulation, but does not affect the power.To see this more explicitly, one can exponentiate the small correction term in front of the exponential function as we show above.

Bandwidth
As we find above, the first order correction is a pure phase modulation; we would like to investigate this phase modulation on the FEL coherence.Recall that one of the most important purposes of a seeded FEL is to generate transform limited light; let us now find the FEL spectrum: Notice that Eðt; zÞ $ e Ài! 0 t , hence the Fourier transform is defined as in Eq. ( 24).First, we rewrite Eðt; zÞ to have t dependence explicit, i.e., JIA, WU, BISOGNANO, CHAO, AND WU Phys.Rev. ST Accel.Beams 13, 060701 (2010) where v g !0 =ðk 0 þ 2k w =3Þ is the FEL group velocity for a coasting beam without energy nonuniformity [15,23,26,27], and B ¼ 9! 2 0 =ð2ZÞ.Completing the integral in Eq. ( 24), we have For the following calculation, the FEL energy density is then introduced as where ẼÃ ð!; zÞ is the complex conjugate of Ẽð!; zÞ.
With this FEL energy density function, we can compute the FEL average frequency as So, we see clearly how the energy nonuniformity can drive part of the beam out of the resonant to the seed laser as mentioned in Sec.I. To characterize the influence more quantitatively, we now compute the standard deviation, which is Notice that, for ¼ 0, is the well-known 1D rms bandwidth of the FEL Green function for a coasting electron beam without energy nonuniformity [23,26,27].
To be explicit, we have Recall that !m ¼ !m =! 0 is the ratio of the m th component microbunching frequency to the FEL frequency.Before we give a detailed study in the following, let us look at an example.For LCLS 1:5 # A FEL [1], the undulator period w ¼ 3 cm and the FEL parameter $ 5:0 Â 10 À4 for nominal operation.The microbunching instability has peak gain modulation with period around 0:5 m entering the undulator.Assuming m ¼ 0:2, Eq. ( 31) predicts about 10% bandwidth increment due to this energy modulation along the undulator.
For a cascaded high-gain harmonic generation FEL [28], the most serious degradation will be at the first stage, where the FEL frequency is the lowest.This is due to the n 2 amplification of any degradation from the first stage into the final FEL radiation, assuming that we are doing n th harmonic generation.So, the first stage degradation is always the most severe one regarding this imperfectness and also shot noise [29,30].
We make some general comments here.First, we rewrite the oscillation phase in Eq. (31) where the 1D power gain length is defined as So, if ! m is a few times larger than , then during the exponential growth region, i.e., z can be up to about 20L 1D G , this cosð!m zk w =3Þ term can lead to a few oscillations.In this case, the exp½À! 2 m k w z=ð18 ffiffiffi 3 p Þ term will exponentially decrease quickly along the undulator z making the overall degradation small along the undulator.So, it is worthwhile to study when the phase is not too large.In that case, let us simply look at the oscillation amplitude only, i.e., we can study C m =ð2BÞ, which is maximum at At this location defined in Eq. ( 34), we have One can also find that the bandwidth degradation is maximized for That said, we expect the bandwidth degradation to be small for both !m small and large limits.We will explore this more in Sec.III.For this particular energy modulation frequency defined in Eq. (36), we have Of course, with the oscillation term as in Eq. ( 31), more detailed analysis is needed for discussion of the maximum degradation.We prefer not to analytically elaborate it more here, but rather leave the discussion to Sec.III.

III. SIMULATION
In this section, we would like to compare the analytical calculations with numerical simulations.The simulations are carried out by GENESIS working in time dependent mode [20].The bandwidth degradation shown by our theory is quantitatively measured by the bandwidth degradation which is defined as Á !ðzÞ !ðzÞ À !;0 ðzÞ !;0 ðzÞ ; where !;0 ðzÞ is the FEL bandwidth with perfect monoenergetic initial electron beam energy " m ¼ 0, !ðzÞ is bandwidth with modulated initial beam energy, and the degradation Á !ðzÞ is the absolute value of the relative ratio as defined in Eq. ( 38).
In order to compare with the theory, proper simulation configurations have to be carefully set.The theory is in accord with the high-gain FEL situation so the FEL characteristic parameter could not be small compared to electron bunch intrinsic energy spread.On the other hand, as shown in Eq. ( 29), the degradation is characterized by the oscillations with period of 3 w =! m , so in order to have a few oscillations in the exponential growth region, could neither be so large that saturation happens too soon.In addition, the modulation m of the initial beam energy should be large enough to overwhelm the internal numerical simulation noise.Yet, m should also be smaller than , otherwise the FEL exponential growth is undermined.The parameters for the simulation are listed in Table I.
In Table I, we introduce a parameter ã, whose square is defined as where r 0 is the transverse radius of the electron bunch assuming a transverse uniform hard edge distribution [31].The parameter ã characterizes the 3D effects.Following Ref. [31], we will set ã 2 ð2; 6Þ in the following simulation.This is the interesting region where current designing or operating x-ray FEL projects sit.For example, for LCLS [1], ã is about 4. For ã 2 ð2; 6Þ, the FEL power growth rate is close to 1D results and single mode dominates.Gain guiding is very effective and transverse coherence of the FEL mode is well achieved.So, this is the range where high-gain FELs are designed.Therefore, we will focus in this range to compare the theory with full 3D simulation.
We first show the bandwidth degradation of a single frequency electron bunch energy modulation.The theoretical calculation of the bandwidth curves !;0 ðzÞ, !ðzÞ and the degradation Á !ðzÞ curves are shown in Fig. 1.The simulation bandwidth curves and the degradation curve are shown in Fig. 2 as a comparison to the theoretical curves in Fig. 1.
As can be seen from Fig. 1, with imperfect initial beam energy, the bandwidth is oscillating around the nonmodulated one.This oscillation contributes to the degradation Á !ðzÞ which is shown as a few consecutive humps.Figure 2 shows the simulation and compares the simulated degradation Á !ðzÞ with the theory calculation.As can be seen, from about 1.9 m to about 3.6 m, Á !ðzÞ of the simulation and the theory are in accord with each other reasonably well in terms of both pattern and value.The discrepancy outside of the (1.9, 3.6) m region is explained in Fig. 3.
As shown in Fig. 3, the simulation gain length are flat and close to the theoretical number within a certain region.
Before 2 m and after 3.5 m, the gain length difference is huge so the theoretical Á !ðzÞ does not follow the simulation curve well within this region as shown in Fig. 2. The region where the theory and simulation have discrepancy is either the start-up region or the saturation region.This is expected, since the theory is developed for the exponential growth region only.A similar situation is found for the case of multifrequency modulation as in Fig. 4. In Fig. 4, a  The theoretical gain length is a fixed number.The number is plotted as a straight line (dot-dashed) and extended horizontally.The solid curve and the dashed curve are the simulation gain length for perfect initial beam energy and modulated initial beam energy, respectively.The simulation parameters are the same as in Fig. 2 and listed in Table I.Bandwidth curves are with perfect beam energy and modulated beam energy, respectively.The beam energy modulation for the degraded bandwidth curve is " m ¼ 0:6, i.e., m ¼ 0:0024.The theoretical calculation is based on the parameters listed in Table I. number of simulations are carried out to eliminate the impact of numeric noise.The simulation sample statistics such as the first-and the second-order moments are calculated; and the average simulated bandwidth degradation with confidence intervals is constructed accordingly.As can be seen, the theory and simulation agree with each other in the exponential growth region from 2.0 to 3.7 m within the confidence interval.
Our theory is a 1D theory which is valid when ã is large enough.In Fig. 5, we show series simulations with a few different values of ã.It can be seen, at smaller ã (ã ! 2) where the 1D limit is not achieved, the theory does not match the simulation.In addition, at a higher peak current, the FEL enters saturation faster so the top plot of Fig. 5 shows the bandwidth degradation follows the theory up to a shorter range before saturation starts.
In Sec.II C 2, we state that the degradation is small for both small and large !m .In the single modulation frequency case, where m ¼ 1 and Eq.(38) becomes Eq. ( 31), we plot Eq. ( 31) at a fix z ¼ 3 m as a function of !m as shown in Fig. 6.Our theory predicts that the maximum bandwidth degradation is around ! m ¼ 0:01 (following the parameters listed in Table I).Figure 7 shows the bandwidth degradation within the shaded area in Fig. 6.As can be seen in Fig. 7, around the peak of Fig. 6, degradations follow the theoretical prediction and at the boundaries of the shaded area, degradations start to vanish.Figure 8 shows the extreme cases where !m is either too large or too small.The simulation bandwidth degradation is dominated by noise which is expected.
It can be noticed that there is a noise beating pattern for the simulations we have shown.The internal numerical noise of the simulation contributes to the noise beating.Figure 9 shows the bandwidth comparison of two simulations with perfect initial bunch energy.The calculation is done in the manner similar to Eq. ( 38).The curve shows the absolute value of the two bandwidth curve difference divided by one of the bandwidth curves.As can be seen,  I. Bandwidth degradation at the shaded area is shown in Fig. 7. different random number generator seeds could introduce the noise beating and the two bandwidth curves are not identical.This is also a measure of the statistical fluctuation of the numerical simulation, which is quite challenging itself.

IV. IMPACT ON A SEEDED FEL
The work developed in the previous sections is sufficient to study the bandwidth degradation of a seeded FEL amplifier when the electron bunch has a nonuniform energy profile.Yet, for a harmonic generation FEL or an optical klystron configuration, there is no initial radiation seed, but rather the FEL will start from a premicrobunched electron bunch.In fact, this can be done by keeping the prebunched term (given in the curly brackets) and throwing away the seed term and the SASE term in Eq. (11).Therefore, Eq. ( 15) should be modified accordingly.The follow-up analysis will have to be modified as well.This is being reported in another publication.In the following, we will get the coherent radiation power and study the bandwidth degradation in the radiator treating the coherent radiation power as a seed.

A. Coherent emission power
In the radiator of a HG FEL [3,4], the coherent radiation power of the premicrobunched electron bunch at the fundamental frequency is [22,32] where we assume that the radiator is resonant at the l th harmonic of the modulator in a HG FEL with the corresponding bunching factor being b l .This is the start-up power which is treated as the coherent seed.In Eq. ( 40), Z 0 is the vacuum impedance, N e is the number of electrons in the bunch, N u is the number of undulator periods, k r ¼ 2= r with r being the radiator resonant wavelength, and ? is the electron bunch transverse rms size.

B. Electron energy profile into the radiator
Since we are working with a cold electron bunch without intrinsic energy spread, the phase space distribution function at the exit of the modulator in a HG FEL will be ð À Á sinÞ; where ð À 0 Þ= 0 with 0 as the electron centroid energy, and ðxÞ is the Dirac delta function.After the buncher, the phase space distribution is then where d=d characterizes the buncher strength and 0 for an overall phase shift.
Based on the reversion of series method [33], Eq. (42) yields a formal series expansion as With this formal expression, is ready to be further expressed as a Fourier series, where the Fourier coefficient is calculated as  where c k n is the binomial coefficient.As mentioned above, assuming that the radiator is resonant at frequency !0 which is the l th harmonic of the first undulator-the modulator-fundamental frequency, then we can rewrite Eq. (45) according to Eqs. ( 12) and ( 13).We have Combining Eqs. ( 31) and ( 46), one can estimate the impact on the FEL bandwidth from the energy modulation generated in the modulator.As an example, assuming a seed laser with a wavelength of 240 nm, via HG FEL, the final FEL wavelength is 0 ¼ 4 nm.Further assuming that the final 4 nm FEL has FEL parameter ¼ 3:0 Â 10 À3 , undulator period of w ¼ 5 cm, and the generic energy modulation at 240 nm leads to m ¼ 0:3, then over the 10 m long undulator the bandwidth increment is about 5%.

V. DISCUSSION
As a conclusion, in this paper, we study the effect on a seeded FEL amplifier performance due to an initial energy nonuniformity when the electron bunch enters the undulator.Such nonuniformity can come from the rf curvature, the collective effect induced microbunching instability, and also generic energy modulation in a HG FEL.We derived the bandwidth degradation to quantitatively measure this effect.The simulation and theory match reasonably well when FEL is close to the 1D limit and in the exponential growth region.We then discuss the FEL bandwidth degradation due to the generic energy modulation in a HG FEL treating the initial coherent emission as the seed into the radiator.
INFLUENCE OF AN IMPERFECT ENERGY . . .Phys.Rev. ST Accel.Beams 13, 060701 (2010) 060701-3The inverse Laplace transform then gives us the FEL electric field slow-varying envelope function as FIG. 2. (Color) Simulations of bandwidth curves and bandwidth degradation.The theoretical degradation curve is copied from Fig.1.The beam energy modulation for the degraded bandwidth curve is the same as in Fig.1.The simulation parameters are listed in TableI.

FIG. 3 .
FIG. 3. (Color) Theoretical and simulation gain length curves.The theoretical gain length is a fixed number.The number is plotted as a straight line (dot-dashed) and extended horizontally.The solid curve and the dashed curve are the simulation gain length for perfect initial beam energy and modulated initial beam energy, respectively.The simulation parameters are the same as in Fig.2and listed in TableI.
FIG.1.(Color) Theoretical bandwidth curves and bandwidth degradation.Bandwidth curves are with perfect beam energy and modulated beam energy, respectively.The beam energy modulation for the degraded bandwidth curve is " m ¼ 0:6, i.e., m ¼ 0:0024.The theoretical calculation is based on the parameters listed in TableI.
FIG. 7. (Color) Bandwidth degradation for different !m .The simulation parameters are listed in TableI.The initial bunch energy modulation frequency !m is a single value for each plot.

TABLE I .
Parameters for simulation.