Coherent X-rays with Tunable Time-Dependent Polarization
CCoherent X-rays with Tunable Time-Dependent Polarization
N. Sudar, R. Coffee, E. Hemsing
SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA (Dated: September 24, 2020)We describe a method for producing high power, coherent x-ray pulses from a free electron laserwith femtosecond scale periodic temporal modulation of the polarization vector. This approachrelies on the generation of a temporal intensity modulation after self seeding either by modulat-ing the seed intensity or the beam current. After generating a coherent temporally modulated s -polarization pulse, the electron beam is delayed by half a modulation period and sent into a shortorthogonally oriented undulator, serving as a p -polarization afterburner. We provide simulations ofthree configurations for realizing this polarization switching, namely, enhanced self seeding with anintensity modulation generated by 2 color self seeding, enhanced self seeding of a current modulatedbunch, and regular self seeding of a current modulated bunch. Start to end simulations for the LinacCoherent Light Source-II are provided for the latter. I. INTRODUCTION
The x-ray Free Electron Laser (FEL) has provided thescientific community with a tunable source of coherent,high peak power x-rays capable of probing phenomena atthe atomic scale [1]. Over the past decade, the scientificcapabilities of FEL facilities have been expanded by in-creases in the peak brightness and repetition rate [2, 3],further control of the spectral and temporal properties[4, 5], and improvements in the stability of FEL pulses[6]. In anticipation of the emerging needs of the usercommunity, investigation of novel configurations aimedat tailoring the spectral, spatial, temporal, and polar-ization properties of FEL photons is an active area ofresearch.Recently, several methods have been proposed to gen-erate x-ray pulses with structured spatial and temporalpolarization topology. These include the generation ofx-ray beams with few- µ m scale spatially varying polar-ization and 100 fs scale temporally varying polarization[7], the generation of Poincar´e beams [8], and the gener-ation of attosecond scale temporally varying polarization[9]. Each of these approaches share a common theme, us-ing an FEL afterburner consisting of crossed undulatorsto generate and overlap radiation with orthogonal polar-ization or opposite helicity. This purely electron beam-based approach removes the need for polarizing opticsnot readily available at x-ray wavelengths.One of the reasons for the growing interest in polariza-tion control [10] stems from the desire to probe electronicsymmetry as it relates to induced circular dichroism inatomic systems [11–13] and chirality in molecular sys-tems [14, 15]. The high degree of symmetry in atomicsystems enforces the familiarly strict transition selectionrules that dominate the photo-excitation process in theoptical regime. In the x-ray regime, however, transitionenergies lie far above valence ionization thresholds suchthat the electronic symmetries involved also couple tothe outgoing free electron partial wave. Induced chiral-ity can occur when these selection rules become modifiedby dressing the atomic ensemble with an optical field. Insuch cases, the various symmetries of resonant core-to- valence transitions produce lab-frame measurable mod-ulations to photo-electron spectra. Access to the poten-tially time-evolving electronic symmetries in molecularsystems as they explore internal degrees of freedom moti-vates targeted femtosecond (fs) scale polarization controlat x-ray FELs.Natural linewidths in the soft x-ray regime are in theorder of a few hundred meV. One therefore requires afew fs for a coherence to develop between the x-ray fieldand the molecular system. Resonant excitation is theresult of this developing coherence. Now, if the driving x-ray field switches polarization state, one could effectivelymodulate the core-to-valence resonant transition at anoptical frequency while allowing for many cycles of x-ray–molecule coherence to develop. Following a few opticalcycles of such a dithered excitation, one would expectthe preparation of an optically active coherent ensembleof mixed electronic nature.Given the rising interest in chirality in molecular sys-tems, here we propose a path toward tunable fs-scalepolarization switching in xFELs that may allow exami-nation of these systems. First, a longitudinally coherent,temporally modulated intensity profile is generated in the s -polarization. This intensity modulation will impart aperiodic modulation on the electron beam bunching pro-file. Introducing a magnetic chicane delay, the electronsare delayed by approximately half the intensity modu-lation period. Provided this delay is not too large, theelectron bunching is preserved, or potentially even en-hanced by the optical-klystron effect [16]. Sending theelectrons through an orthogonally oriented undulator sec-tion, an FEL pulse with nearly identical intensity profileis generated in the p -polarization, shifted in time relativeto the s -polarization pulse by half a modulation period.The preservation of the initial bunching between stagesgives a stable phase relationship between the two pulses.This allows for fine adjustment of the phase shift be-tween pulses, potentially producing a smooth temporalrotation of the polarization vector across the pulse. Thisscheme can be applied to the generation of time-varyingcircular polarization using a series of helical undulatorsof opposite helicity as well. a r X i v : . [ phy s i c s . acc - ph ] S e p The main challenge presented by this method is thegeneration of the required fs-level intensity modulation.Herein we describe three possible configurations, eachwith unique benefits. The first uses the enhanced selfseeding (ESS) method described in [17], using the two-color self seeding technique to generate a beatwave in-tensity modulation which is subsequently amplified bya fresh electron bunch with a flat current profile. Thesecond again uses the ESS method, this time generatinga monochromatic seed that is amplified by a fresh elec-tron bunch with a modulated current profile. The thirduses regular self seeding (SS) with a current modulatedelectron bunch as described in [18].In this paper, we first provide a simplified model ofthe generation of time-varying polarization with a briefoverview of the useful Stokes vector formalism. In sectionIII, we discuss the three methods for generating the requi-site intensity modulation in greater detail. This includessimulations with idealized conditions and uses undula-tor and electron beam parameters relevant to the LinacCoherent Light Source-II (LCLS-II), using the 3-D, timedependent FEL code, genesis [19]. Section IV describesa potential method for the generation of a fs-level cur-rent modulation at LCLS-II, including simulations of theLCLS-II linac. Included are full start to end simulationresults using the method described in section III.
II. TIME VARYING POLARIZATION: SIMPLEMODEL
Our method to produce time-dependent polarizationrelies on the combination of two orthogonally polarizedpulses with identical temporal structure, shifted in time.Consider an electric field produced by the emission of ra-diation from an electron beam propagating through twoconsecutive undulator sections. In the first undulator,the polarization of the emitted radiation is ˆe . The elec-tron beam then passes through a delay of ∆ t , and thenthrough another undulator where it radiates an identicalpulse, but with polarization ˆe . The total electric fieldvector can be written as E ( t ) = E ( t + ∆ t/ ˆe + e iψ E ( t − ∆ t/ ˆe = (cid:18) E x E y (cid:19) . (1)Here ψ is a small phase shift between the two pulses.To describe the polarization properties of the radia-tion, it is useful to examine the individual components ofthe Stokes vector S = ( S , S , S , S ); S = | E x | + | E y | S = | E x | − | E y | S = E x E ∗ y + E ∗ x E y S = i ( E x E ∗ y − E ∗ x E y ) (2)The first Stokes parameter S is the temporal intensityprofile. The S parameter describes linear polarization in FIG. 1. Example pulse intensity profiles from consecutivecrossed undulators. Each pulse carries two colors separatedby (cid:126) ∆ ω = 1 eV at (cid:126) ω = 1 .
24 keV. The pulses from eachundulator are interleaved with a time delay of ∆ t = π/ ∆ ω =2 fs. either the horizontal or vertical direction, depending onthe sign. S describes diagonal linear polarization in the ± ◦ directions, and S describes circular polarization.To produce time-dependent polarization in Eq. (1), weconsider each polarization component to be described bya field E ( t ) with carrier frequency ω and with a periodictemporal intensity modulation, E ( t ) = e iω t cos(∆ ω t/
2) = e iω t + e iω t . (3)This is mathematically equivalent to a pulse comprised oftwo frequencies separated by ∆ ω = ω − ω as would beradiated by an electron beam coherently bunched at thesetwo frequencies. Time-dependent polarization in Eq. (1)comes from shifting the differently polarized pulses intime by ∆ t to interleave the intensity modulations, andby finely adjusting their relative phase with ψ .Let us consider the case of two orthogonally polarizedplanar undulators. The field polarization unit vectors are ˆe = ˆe x = (cid:18) (cid:19) , ˆe = ˆe y = (cid:18) (cid:19) (4)The Stokes parameters for the crossed planar undula-tors are S = 1 + cos (∆ ω t ) cos (cid:18) ∆ ω ∆ t (cid:19) S = − sin (∆ ω t ) sin (cid:18) ∆ ω ∆ t (cid:19) S = cos ( φ ) (cid:20) cos (∆ ω t ) + cos (cid:18) ∆ ω ∆ t (cid:19)(cid:21) S = − sin ( φ ) (cid:20) cos (∆ ω t ) + cos (cid:18) ∆ ω ∆ t (cid:19)(cid:21) . (5)We have defined a total phase shift between the twopulses as φ = ω ∆ t − ψ .The intensity S shows the temporal beat modulationwith period 2 π/ ∆ ω . The amplitude of the modulationcan be controlled by adjustment of the delay ∆ t between FIG. 2. Temporal Stokes parameters in Eq. (6) for crossedplanar undulators, same parameters as FIG. 1. The relativephase is set to φ = 0 (top), φ = π/ φ = π/ ˆe R polarization green, and ˆe L polarization blue. the overlapping pulses. If the pulses are interleaved bysetting the delay to half of the beat period, ∆ t = π/ ∆ ω ,then the intensity modulation of the combined pulse van-ishes (see FIG. 1). In this case the Stokes vector is, S ( t ) = − sin (∆ ω t )cos ( φ ) cos (∆ ω t ) − sin ( φ ) cos (∆ ω t ) . (6)Each of the polarization parameters S , S , S has a timedependence with periodicity 2 π/ ∆ ω . The linear polar-ization parameter S has a fixed amplitude, while thediagonal S and circular polarization S can be tuned inquadrature by the phase φ . Figure 2 shows the time-dependent Stokes parameters and polarization ellipsesfor different phase φ values. For φ = 0, the circularcomponent vanishes S = 0 and the polarization is lin-ear at all times along the pulse, but evolves steadily be- FIG. 3. Distribution of Stokes parameters on the Poincar´esphere for two-color emission with crossed linear undulators(left) and crossed helical undulators (right). The time delayis ∆ t = π/ ∆ ω . tween horizontal, diagonal, and vertical. In terms of thePoincar´e sphere in FIG. 3, the polarization state evolvesstrictly along the equator defined by the ( S , S ) plane.For φ = π/
2, the diagonal component vanishes S = 0,and the polarization evolves from linear to circular po-larization purely in the ( S , S ) plane. For φ = π/
4, thepolarization involves all three parameters along a tiltedplane, but is never purely circularly or diagonally polar-ized.The situation is similar if we instead use two orthogo-nally polarized helical undulators. In this case, the fieldpolarization vectors are ˆe = ˆe R = 1 √ (cid:18) − i (cid:19) , ˆe = ˆe L = 1 √ (cid:18) i (cid:19) (7)The Stokes parameters are then modified to be S = 1 + cos (∆ ω t ) cos (cid:18) ∆ ω ∆ t (cid:19) S = cos ( φ ) (cid:20) cos (∆ ω t ) + cos (cid:18) ∆ ω ∆ t (cid:19)(cid:21) S = − sin ( φ ) (cid:20) cos (∆ ω t ) + cos (cid:18) ∆ ω ∆ t (cid:19)(cid:21) S = sin (∆ ω t ) sin (cid:18) ∆ ω ∆ t (cid:19) . (8)Again interleaving the two temporal modulations withthe delay ∆ t = π/ ∆ ω , for crossed helical undulators weobtain the Stokes vector S ( t ) = φ ) cos (∆ ω t ) − sin ( φ ) cos (∆ ω t )sin (∆ ω t ) . (9)Here, for φ = 0, the diagonal component S vanishes,and the polarization evolves between circular and linear.This is similar to the φ = π/ φ = π/ S component vanishes, andthe polarization state evolves through pure diagonal andcircular orientations in the ( S , S ) plane.Overall, we see that the Stokes parameters all evolvewith a period of 2 π/ ∆ ω . Therefore, at the shift ∆ t = π/ ∆ ω , the time it takes to change from one pure polar-ization state to another, (i.e., from ˆe x → ˆe R ) is τ ˆe i → ˆe j = π ω . (10)For example, consider the case of a (cid:126) ∆ ω = 1 eV en-ergy separation (∆ λ = 1 . µ m) between the two colors.This gives a temporal shift of ∆ t = π/ ∆ ω = 2 fs. Withcrossed planar undulators, the polarization changes frompure linear polarization to pure circular polarization in τ ˆe x → ˆe R = 1 fs. III. TIME VARYING POLARIZATION:METHODS
Here we consider three configurations for generating x-ray pulses with time varying polarization at the fs timescale as described in the previous section. Each configu-ration has trade-offs in the realms of available hardware,practical realization, stability and peak power. For eachcase we consider intensity modulations of 700 nm (2.33fs) and 1050 nm (3.5 fs), demonstrating the ability toachieve temporal structures in the visible and infrared. (I) Two Color Enhanced Self seeding
Enhanced self seeding is a recently proposed methodto improve intensity stability in the production of nar- row band, highly coherent x-ray pulses. Here, an ul-trashort electron bunch with pulse length comparableto the FEL cooperation length generates a single SASEspike in the initial undulator section, reaching satura-tion. The single spike nature of this pulse is character-ized by a wide, coherent bandwidth. Lasing in a nearbylong, lower current electron bunch is suppressed in theSASE stage by its reduced gain length and undulatortapering, thus these electrons remain “fresh”. For self-seeding, a narrow portion of the short spike spectrum isselected by a monochromater. This seed radiation is thusstretched, and then overlapped and amplified by the longfresh bunch in a subsequent undulator section.It was shown in [17] that the filtered seed can exhibit aperiodic intensity modulation by employing a monochro-mater capable of selecting two narrow regions of the sin-gle spike spectrum. Selecting two phase-stable frequen-cies with separation ∆ ω will generate a beat wave mod-ulation of the intensity with period λ m = 2 πc/ ∆ ω . Themodulation is then preserved and amplified by the longbunch. This two color enhanced self seeding provides asource of intensity stable, high power pulses with periodicintensity modulation at the femto-second time scale.We consider this as a candidate for arriving at theneeded temporal intensity modulation for the time vary-ing polarization scheme, as detailed in Figure 4, row I.In the seeded section, the intensity modulated two-colorseed generates radiation in the s -polarization. A smallchicane then delays the electron beam without destroy-ing the bunching in the regions of peak intensity, slippingthe s -polarization pulse ahead by a half modulation pe-riod, λ m /
2. A final crossed undulator section generatesa similar pulse in the p -polarization.Figure 5 shows results from 3-D time dependent gen-esis simulations considering idealized current profiles.Here we show simulations of the two-color ESS methodwith frequency separation (cid:126) ∆ ω = 1 .
77 eV and (cid:126) ∆ ω =1 .
33 eV corresponding to an intensity modulation of λ m = 700 nm and λ m = 1050 nm respectively. Simu-lation parameters are provided in Table I, following theelectron beam and undulator parameters available at theLCLS-II.The s -polarization pulse diffracts freely in the delayand p -polarization sections. In order to approximatelymatch the magnitude of the intensity from both polar-izations, we propagate the simulated radiation profilethrough a lens such that the s and p polarizations areupstream and downstream of a waist respectively. Withthis in mind, the normalized Stokes parameters are plot-ted for the field with an approximate π/ s and p states of ∼
90% purity.A gentle undulator taper is introduced in the s -polarization undulator to suppress lasing in the troughsof the intensity modulation, however some bunching is FIG. 4. Illustration of the methods considered for generation of time dependent polarization, showing the initial currentprofile (a), the SASE radiation generated before self seeding (b), the seed pulse after the self seeding monochromater (c), the s -polarization pulse (d), and the combined s and p -polarization (blue, yellow) after the delay and crossed undulator (e), fortwo-color ESS (I), ESS with current modulation (II) and regular SS with current modulation (III). generated in these regions. This bunching can be signif-icantly amplified by the delay between crossed undula-tors, spoiling the temporal structure of the electron beambunching entering the p -polarization undulator. Thisplaces a limitation on the achievable peak power, mod-ulation period, and resonant wavelength. Furthermore,the radiation generated in the troughs will degrade thepurity of the polarization states, since the troughs of onepolarization overlap the peaks of the other. The achiev-able peak power and polarization purity may be improvedby introducing a reverse taper in the undulator sectionfollowing self-seeding to generate bunching only in the de-sired temporal regions and with significantly less growthof the energy spread [20].The practical realization of this method relies on thedesign of a two-color soft x-ray monochromator. Whereasa comparable technique has been realized in hard x-rayself seeding [21], a soft x-ray counterpart would requirethe fabrication of gratings with superimposed or alter-nating line densities. This is currently being investigated[22]. The ability to produce the short bunch-long bunchcurrent profile in the LCLS-II copper linac was shown insimulations which are referenced in the supplementarymaterial of [17], and is a topic of active study. (II) Enhanced Self Seeding with Current Modulation Again taking advantage of the intensity stability of-fered by the ESS technique, in this method we can pro-duce a temporal intensity modulation after the self seed-ing monochromator by modulating the current profileof the long bunch, as illustrated in Figure 4, row II.Again a single SASE spike is generated by the short
FIG. 5. Temporal modulation produced with two-color ESS.Left: 700 nm modulation Right: 1050 nm modulation. (a)Power in s -polarization (blue) and p -polarization (yellow).(b) Normalized Stokes parameters, S /S (black, line), S /S (blue, dotted), S /S (red, dashed). (c) Spectral intensity in s -polarization (blue) and p -polarization (yellow). bunch in an initial undulator section, with lasing sup-pressed in the long bunch with undulator tapering. Thisshort pulse is then frequency filtered in a monochroma-tor, this time selecting a single frequency. A current-modulated beam seeded by a single color coherent sourcewill generate a series of coupled sidebands in the spec- FIG. 6. Temporal modulation produced with single color en-hanced self-seeding plus current modulation. Left: 700 nmmodulation Right: 1050 nm modulation. (Top) Power in s -polarization (blue) and p -polarization (yellow). (middle) Nor-malized Stokes parameters, S /S (black, line), S /S (blue,dotted), S /S (red, dashed). (Bottom) On-axis spectral in-tensity in s -polarization (blue) and p -polarization (yellow). trum, spaced at the current modulation frequency. Thisresults in a series of mode-locked pulses in the time do-main with s -polarization, as described in [23]. As before,a small chicane then delays the electron beam, preserv-ing the bunching in the current spikes and slipping the s -polarization pulse ahead by a half modulation period.A final crossed undulator section generates a similar pulsein the p -polarization.Figure 6 shows results from 3-D time dependent gene-sis simulations considering flattop electron beam currentprofiles superimposed with modulations of λ m = 700 nmand λ m = 1050 nm. Figure 6 also shows the normalizedStokes parameters for the field with a π/ s and p states of nearly 100% purity. The achiev-able peak power is increased compared to the two colorESS method due to the increased peak current. To op-timize this setup, both the s and p undulator sectionsare longer than in the first method (see Table 1), whichreduces the required delay. In this example we take theradiation closer to saturation, which lead to the growthof additional spectral modes and produces sharp inten-sity spikes [24, 25]. This has the effect of reducing theoverlap between polarizations and making the transitionbetween polarization states more abrupt. We note that,while the use of ESS may provide shot-to-shot stability in these temporal structures, the realization of the req-uisite short beam plus time-modulated current profile inthe long beam may prove difficult. TABLE I. Genesis simulation parametersParameter Valuee-beam energy (GeV) 4e-beam emittance (cid:15) x,y ( µ m) 0.4e-beam beta function β x,y (m) 12undulator period λ w (cm) 3.9undulator section length (m) 3.4break section length (m) 0.975fundamental photon energy (cid:126) ω (eV) 620Self seeding (S.S.) chicane R ( µ m) 750S.S. filter RMS (meV) 85Method I 700 nm mod. 1050 nm mod.short e-beam I (kA) 6 6short e-beam FWHM (fs) 0.67 0.67short e-beam σ E (MeV) 3 3long e-beam I (kA) 1 1long e-beam FWHM (fs) 30 30long e-beam σ E (MeV) 0.5 0.5pre S.S. N sec N sec R ( µ m) 0.36 0.6p-polarization N w
40 32p-polarization K 2.2937 2.2937Method II 700 nm mod. 1050 nm mod.short e-beam I (kA) 6 6short e-beam FWHM (fs) 0.67 0.67short e-beam σ E (MeV) 3 3long e-beam peak I (kA) 2 2long e-beam FWHM (fs) 30 30long e-beam σ E (MeV) 1 1pre S.S. N sec N sec R ( µ m) 0.2 0.5p-polarization N w
55 50p-polarization K 2.2912 2.2912Method III 700 nm mod. 1050 nm mod.long e-beam peak I (kA) 2 2long e-beam FWHM (fs) 30 30long e-beam σ E (MeV) 1 1pre S.S. N sec N sec R ( µ m) 0.3 0.7p-polarization N w
45 25p-polarization K 2.2967 2.2937 (III) Regular Self Seeding with Current Modulation
The previous method can be simplified in practice byusing the current modulated long bunch to generate itsown seed in a standard self seeding configuration, asshown in Figure 4, row III. In this case, a SASE pulseis generated by the long current-modulated bunch in aninitial undulator section. Due to the large number ofcooperation lengths in the beam this pulse is stochasticin nature, exhibiting significant fluctuations in temporaland spectral intensity. This pulse is then frequency fil-tered in a monochromator, and the seed is amplified inthe downstream section by the same current-modulatedbeam to produce a temporally-modulated pulse.Figure 7 shows results from simulations with a π/ s and p states of nearly 100%purity, however with a reduction of the interleaving he-lical states due to the even more pronounced sharpnessof the modulation. This method of conventional SS willlikely exhibit greater shot-to-shot fluctuations than theprevious schemes that use ESS, but it is simpler to setupand implement. Start to end simulations of this method,including the production of the modulated current pro-file, are presented in the next section. IV. START TO END SIMULATIONS
In order to generate a fs-scale current modulation weconsider seeding the micro-bunching instability at thelaser heater with a THz beatwave-modulated laser pulse,as was demonstrated in [26]. Using the chirped-pulsebeating technique, a THz scale intensity-modulated laserpulse is generated by the 1 µ m wavelength laser heaterlaser [27]. Overlapping this pulse with the electron beamin the laser heater undulator will generate a correspond-ing energy modulation. Bunch compression in the down-stream electron beam transport will convert this energymodulation to density modulation, with further ampli-fication of the energy modulation provided by longitu-dinal space charge impedance [28]. Bunch compressionalso serves to reduce the modulation period by the com-pression factor of the accelerator, reaching the desiredfemtosecond level.Figure 8 shows the laser heater laser intensity profileand energy modulation at the laser heater chicane exitfrom elegant simulations [29], with an initial beamdistribution generated from impact simulations of theLCLS-II gun [30, 31]. Simulation parameters are givenin Table II.The λ m = 920 nm current modulation at the undula-tor entrance is shown in Figure 9. To further increasethe current spikes, we pass this current modulated beamthrough a high impedance LCLS-II X-Leap wiggler mag-netic wiggler, which will produce an additional sinusoidal FIG. 7. Temporal modulation produced with single colorregular self-seeding plus current modulation. Left: 700 nmmodulation Right: 1050 nm modulation. (Top) Power in s -polarization (blue) and p -polarization (yellow). (middle) Nor-malized Stokes parameters, S /S (black, line), S /S (blue,dotted), S /S (red, dashed). (Bottom) On-axis spectral in-tensity in s -polarization (blue) and p -polarization (yellow).TABLE II. Parameters for start-to-end simulationsElegant parameters Valuelaser heater laser peak power (MW) 50laser heater laser waist ( µ m) 100laser heater laser wavelength ( µ m) 1.03laser heater modulation period ( µ m) 75e-beam charge (pC) 100e-beam energy at laser heater (MeV) 100e-beam energy at undulator (GeV) 4e-beam emittance (cid:15) x,y (mm-mrad) 0.39,0.37e-beam beta function β x,y (m) 12,12Compressed current modulation period ( µ m) 0.92X-leap undulator period (cm) 35X-leap undulator length (m) 2.1X-leap undulator strength (rms) K 17.92X-leap chicane R (mm) 1.2 genesis parameters No X-Leap w/ X-Leape-beam peak I (kA) 1.75 3e-beam modulation amplitude (kA) 1 2.75e-beam σ E (MeV) 0.4 1pre S.S. N sec s -polarization N sec s -polarization K (rms) 2.2937 2.2937chicane R ( µ m) 0.4 0.5p-polarization N w
40 45p-polarization K (rms) 2.2937 2.2937
FIG. 8. (left) Laser heater modulation. (right) longitudinalphase space at laser heater exit energy modulation driven by the wiggler’s short rangeCSR wake, as was demonstrated in [32]. Including anadditional R from a magnetic chicane will then pro-duce a sharper current modulation. This self modulationprocess can be simulated approximately in 1-D as de-scribed in [33]. Parameters are given in Table II, withthe final longitudinal phase space shown in Figure 9.Figure 9 also shows results from genesis simulationswith the final start-to-end beam. All simulations includethe effects of the undulator chamber resisitive wall wake-fields. From the normalized Stokes parameters we ob-serve rotation of the polarization vector between linearand helical polarization states with linear polarizationswitching between s and p states of nearly 100% purity.Non-linearities in the longitudinal phase space cause alengthening of the modulation period at the head andtail of the beam degrading the polarization switching.Lasing in the head and tail could perhaps be suppressedby additional shaping of the laser heater laser. FIG. 9. Left: No X-Leap undulator modulation Right:With X-Leap undulator modulation and R . (a) Longitu-dinal phase space at undulator entrance with current pro-file included (blue). (b) Power in s -polarization (blue) and p -polarization (yellow). (c) Normalized Stokes parameters, S /S (black, line), S /S (blue, dotted), S /S (red, dashed).(d) On-axis spectral intensity in s -polarization (blue) and p -polarization (yellow). V. CONCLUSION
The ability to generate x-ray pulses with femtosecondlevel polarization switching would provide a useful toolto the scientific community. Simulations of the threeaforementioned methods demonstrate tunability of thistemporal polarization modulation from the visible to in-frared. Each method exhibits benefits of stability, peakpower, or realizability with existing hardware at LCLS-II. Further investigation of the underlying scheme couldlead to increases in the achievable peak power, especiallyat longer modulation period. This could include imple-menting a reverse tapering scheme to better control themicro-bunching and energy spread in the s and p polar-ization stages. ACKNOWLEDGMENTS
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