Monochromaticity of coherent Smith-Purcell radiation from finite size grating A. Aryshev,1, A. Potylitsyn,2 G. Naumenko,2, M. Shevelev,1 K. Lekomtsev,1,3 ∗ † L. Sukhikh,2 P. Karataev,3 Y. Honda,1,4 N. Terunuma,1,4 and J. Urakawa1 1KEK: High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan 2Tomsk Polytechnic University, Institute of Physics and Technology, Lenin Avenue 30, Tomsk 634050, Russian Federation 3John Adams Institute at Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom 4SOKENDAI: The Graduate University for Advanced Studies, 1-1, Oho, Tsukuba, Ibaraki 305-0801, Japan Investigation of coherent Smith-Purcell Radiation (SPR) spectral characteristics was performed both experimentally and by numerical simulation. The measurement of SPR spectral line shapes of different diffraction orders was carried out at KEK LUCX facility. A pair of room-temperature 7 Schottky barrier diode (SBD) detectors with sensitivity bands of 60−90 GHz and 320−460 GHz 1 was used in the measurements. Reasonable agreement of experimental results and simulations per- 0 formedwithCSTStudioSuitejustifiestheuseofdifferentnarrow-bandSBDdetectorstoinvestigate 2 different SPR diffraction orders. It was shown that monochromaticity of the SPR spectral lines in- n creases with diffraction order. The comparison of coherent transition radiation and coherent SPR a intensities in sub-THz frequency range showed that the brightnesses of both radiation mechanisms J were comparable. A fine tuning feasibility of the SPR spectral lines is discussed. 1 1 PACSnumbers: 23.23.+x,56.65.Dy,41.60.-m,41.75.Ht Keywords: CoherentSmith-PurcellRadiation;TransitionRadiation;MonochromaticTHzRadiationBeams ] h p I. INTRODUCTION appearingwhenachargedparticlemovesaboveandpar- - c alleltoagratingisaresonantprocesswithspectrallines c IntenseTHzradiationiswidelyusedfordifferentappli- defined by the well-known dispersion relation: a . cationsincludingTHzdiffractionandspectroscopy[1,2]. s The interest appears due to the fact that THz radiation d (cid:18)1 (cid:19) c λ = −cosθ . (1) i is non-ionizing, which prevents destruction of a sample k k β s y and enables investigation of living cells without radia- h tion damage. Furthermore, most of the molecules oscil- Here λk is the wavelength of the resonance order k, d is p lateatTHzfrequenciesprovidingdistinctspectralsigna- the grating period, β is the particle velocity in the speed [ turesofdifferentmaterialswhentheradiationpropagates of light units, and θ is the observation angle. 1 through the sample [3]. The use of coherent SPR generated by short electron v These days THz radiation can be produced by table- bunches(orbyatrainofbunches)asthebasisofTHzra- 5 top type thermal or laser mixer based generators [4–6]. diationsourceswasproposedbyauthorsoftheRefs.[13– 7 However, the power of these sources is rather low. The 16]. Coherent radiation emission occurs when the bunch 8 state-of-the-art methodology for high power THz gener- length is comparable to or shorter than the radiation 2 ation is based on particle accelerators, e.g. ultra-short wavelength resulting in the SPR intensity being deter- 0 pulse compact accelerators [7] or free-electron lasers [8]. minedbythesquarenumberofelectronsperbunch. The . 1 Nevertheless, an optimal mechanism for THz production spectral-angular distribution of coherent SPR produced 0 is still under consideration. The problem of designing by a strip grating with a finite number of periods N can 7 of a cost-effective, compact, adjustable THz source with be written as: 1 short pulse duration has to be resolved. : v There are several approaches based on electron beam d2W d2W sin2[Nφ] Xi technologies(see,forinstance,[9–12])proposedtodesign dνdΩ = dνdΩ0 sin2[φ] (Ne+Ne(Ne−1)F). (2) suchasource. Inmostcasesanadjustablemonochroma- r a tor needs to be foreseen to achieve a narrow-band THz Here φ=dπν (cid:0)β 1−cosθ(cid:1), d2W /dνdΩ is the spectral- output, but tunable in a broad spectral range. However, c − 0 angular distribution of the radiation from a single elec- theusageofanykindofdiffractometerorbandpassfilter tron, ν is the radiation frequency (ν =c/λ), N is the reducesthetransmittedpowerandmayintroduceanun- e bunch population, and F is the bunch form-factor [17]. desirablespectradistortioninvirtueofdiffractioneffects. According to Eq. (2) extension of the number of peri- In this respect, a THz source based on Smith-Purcell ods to infinity N → ∞ results in Dirac’s delta function radiation (SPR) mechanism is promising, because SPR that defines dispersion relation in Eq.(1). Also intensity d2W /dνdΩ decreases if the relation γλ ≤ b is not ful- 0 k filled [18], where γ is the the electron Lorenz-factor, b is ∗ Correspondingauthor: [email protected] the grating width, which is the size along the direction † Correspondingauthor: [email protected] perpendicular to electron momentum. 2 II. PRINCIPLES They have measured the SPR spectrum generated by the 42 MeV electrons from a 4 mm period grating with One of the most important characteristics of any com- N =20periodsusingthegrating-typespectrometerwith pactacceleratorbasedTHzsourceisitsmonochromatic- a focusing mirror. The observed SPR line at θ = 70◦ ity. FromEq.(2)themonochromaticityoftheSPRspec- for k = 1 had the wavelength of λ1 = 2.68 mm with trallineis(∆λ /λ ∝1/kN)forthefinitelengthgrating FWHM ∆λ = 0.21 mm. Taking into account the band- k k Nd. Using full width at half maximum (FWHM) as an width of the spectrometer in use, ∆λint = 0.06 mm, it absolutespectrallinewidth,themonochromaticityisde- is possible to conclude that the observed line broaden- fined as: ing was caused by both above-mentioned reasons (see Eqs. (5) and (3)). In [22] the SPR spectrum generated ∆λ ∆ν 0.89 k = k = , (3) by the 2.3 MeV electrons was measured with a Fabry- λk νk kN Perot interferometer. The grating with 2.5 mm period andeffectivelengthof20mmwasused. Fromtheexper- stating the fact that higher SPR orders are more imental results obtained for the angle θ = 90 and the monochromatic than the fundamental one. ◦ first diffraction order (k =1, λ =2.5 mm) presented in As it was shown in [17] the value of the SPR line 1 Fig.1[22],onecanestimatethemonochromaticityofthe widthmeasuredbyadetectorplacedintheso-calledpre- measured spectral line as ∆λ /λ = 15%. Such a large wavezonebecomesbroaderthantheestimationgivenby 1 1 valueforinterferometercanbeexplainedbybothfactors Eq. (3). If the grating-to-detector distance is L, then (5), (3)and, probably, bytheprewavezoneeffect(inad- the far-field zone (or wave zone) condition is determined dition, an extra uncertainty is coming from the fact that by [18]: thegeometryoftheexperimentwasnotpresentedbythe L(cid:29)L =kN2d(1+cosθ). (4) authors). In [23] the coherent SPR characteristics from ff the finite length and width grating were considered. In Contrariwise when L≤L the condition complies with ordertoavoidlossesoftheradiatedpowerincomparison ff the prewave zone and the simplest way to avoid spectral with the conventional case (i.e. when transverse grating line broadening is to use focusing optical elements (e.g. size tends to infinity), the requirements for the grating lensesorfocusingmirrors)infrontofthedetector[19,20]. width were formulated and the power spectrum as well However,suchaneffectdoesnotinfluencethemonochro- astheradiatedenergyusingdifferentmodelswerecalcu- maticity ∆λ/λ. The expression for monochromaticity in lated. However, the simulation of the SPR line widths this case can be derived directly from the Eq. (1) as: was not considered. ∆λ ∆ν sinθ = = ∆θ. (5) λ ν 1/β−cosθ III. EXPERIMENT Equation (5) determines the monochromaticity of ra- A. Methods and techniques diation generated from an infinite grating (N →∞) and measured with a finite detector aperture ∆θ. Neverthe- less, Eq. (5) allows to estimate the broadening of SPR The experiment was carried out at the KEK LUCX spectral line due to finite aperture of the detector used facility. Schematic diagram and the photograph of the during the experiment. In addition, any spectrometer experimental setup are shown in Fig. 1. possesses its own intrinsic resolution; hence, the spectral Short electron bunches were generated in the RF gun line shape measurements always include systematic dis- viaCs Tephotocathodeilluminationbythefemtosecond 2 tortion. Assumingthattheinitiallineshape∆λSPR and laser pulses with wavelength of 266 nm. Then electron spectrometerresolution∆λint canbeapproximatedbya bunches were accelerated to the energy of 8 MeV in the Gaussiandistributions, onecanusethefollowingexpres- 3.6 cell RF gun. The experiment was conducted with sion for the FWHM value of the measured line width: electron beam parameters given in Table I. The trans- verse shape of the electron beam in experimental area (cid:113) ∆λ = (cid:0)∆λSPR(cid:1)2+(cid:0)∆λint(cid:1)2. (6) was measured using a scintillating screen, which was lo- k k k cated ∼400 mm downstream the vacuum chamber with According to Eq. (3) transition to the higher diffrac- installed gratings. The longitudinal electron bunch pro- tion orders (k > 1) allows to narrow the coherent SPR file measurements were well described in [24]. spectral line ∆λSPR significantly if the angular accep- The vacuum chamber for experimental investigation tanceformingtheradiationbeamischosentobeassmall was installed after the RF gun. The chamber vacuum as practically possible. The main objective of this pa- window was made of the 12 mm thick 2 wedged sap- ◦ per is to show a possibility to generate SPR beams with phire and mounted into an ICF-203 flange, which pro- monochromaticitybetterthan1−2%choosingthehigher vided an effective aperture of 145 mm. A 5-axis manip- diffraction order k >1 even for N of about 10. ulator system was integrated on the top of the chamber. To the best of our knowledge, the authors of the It was used for fine adjustment of grating’s position in 3 Ref. [21] have observed coherent SPR for the first time. orthogonal directions and also for the control of the two 3 Grating SPRgeometry,thedistancebetweengratingandtheelec- Linac tronbeamwas0.6mm. Theradiationspectralcharacter- Screen isticsweremeasuredbytheMichelsoninterferometer(de- RF Gun Sapphire scribed in [26]) installed directly in front of the chamber Beam direction Soleniod window vacuum window (see Fig. 1). The main interferometer Laser pulse opticalaxiscoincidedwiththedirectionperpendicularto BS the electron beam propagation and corresponded to the M PM observation angle θ = π/2. Two Schottky barrier diode 1 detectors (SBD) with different regions of spectral sensi- tivity 60−90 GHz and 320−460 GHz were used in the Detector M experiment. Detailed detectors’ parameter list is shown 2 inTableII.Theelectronbeamparameters(bunchlength ≈0.5ps)ensuredcoherentradiationemissionwithinthe Grating Beam direction measurement spectral regions. In this case the radia- tion intensity was scaled by the bunch population factor Sapphire N ≈ 1.56·108 and the detectors described above could e window be used for radiation intensity measurements. BS PM TABLE II. Detector parameters. Parameter SBD 60-90 GHz SBD 320-460 GHz Frequency range 60−90 GHz 320−460 GHz M SBD 320-460 Wavelength range 3.3−5.0 mm 0.94−0.65 mm 1 Response time ∼250 ps sub-ns M 2 Antenna gain 24 dB 25 dB SBD 60-90 Input aperture 30×23 mm 4×4 mm Video sensitivity 20 V/W 1250 V/W FIG. 1. Top: experimental schematics. Bottom: photograph of the experimental station. Abbreviations: M - fixed inter- 1 ferometer mirror, M - movable interferometer mirror, BS - 300 2 splitter, PM - off-axis parabolic mirror. 300 TABLE I. KEK: LUCX, beam parameters at the RF gun 4 mm section. 60 mm Parameter Value Beam energy 8 MeV FIG. 2. SPR grating geometry. Charge per bunch 25 pC Bunch rms length 0.5 ps Transverse rms size 230×230 µm The 60 × 30 mm2 echellete profile grating shown in Repetition rate 3.13 bunch/s Fig. 2 was placed in the vacuum chamber to generate Normalized emittance, typ. 1.5×1.5 mm mrad Smith-Purcell radiation. The opposite side of the grat- ing was flat that allowed to use this surface as coherent transition radiation (TR) source. The grating could be rotated around its vertical axis for TR orientation de- rotationanglesofthegratingwithrespecttotheelectron pendencemeasurement(co-calledΘ-scan,whereΘisthe beam propagation direction. The mechanics of the ma- angle between TR target surface and electron beam di- nipulator were based on the linear and rotation stages rection). During such a scan the interferometer movable driven by stepping motors with resistive encoders. All mirror was set to the position of zero path difference. motors were remotely controlled by an industrial-grade Oriental Motor CRK-series controllers in the open-loop mode [25, 26]. The positioning accuracies were better than 5 µm and 0.02 for the linear and rotation stages, B. Transition radiation characteristics ◦ respectively. Thisallowedtocontrolthegratingposition with respect to the electron beam trajectory. Prior sub- From the theory [17] the TR spectrum emitted by a THz radiation properties measurements, the grating was single electron is supposed to be constant within detec- aligned with respect to electron beam using the forward torsensitivitybands. Thereforethemeasurednormalized bremsstrahlungappearingduetodirectinteractionofthe TRspectrumcanbeusedastheentiremeasurementsys- electron beam with the grating material. In the case of tem spectral efficiency, including spectral transmission 4 efficiencyofthevacuumwindow,detectorwavelengthef- lines correspond to the SPR diffraction orders k calcu- ficiency,splitterefficiency,reflectioncharacteristicsofthe latedusingthedispersionrelationEq.(1)forexperimen- mirrors and air absorption. The typical TR orientation tal grating parameters and observation angle θ = π/2. dependenceobtainedbyrotationoftheTRtargetatthe The resonances k = 1 and k = 5 were within the detec- angleΘandmeasuredwith320−460GHzSBDisshown tor sensitivity bands. in Fig. 3. C. SPR characteristics 1.2 Measured SPR interferograms are shown in Fig. 5. 1.0 Since there are different definitions of Fourier spectrom- s) t ni 0.8 Interferometermirrordisplacement(mm) u 6 4 2 0 2 4 6 2 1 0 1 2 b. − − − − − r a 0.6 1.0 1.0 ( y sit 0.5 0.5 Inten 0.4 nits)0.0 a b 0.0 u 0.2 b.0.8 k=1 k=5 0.3 r a ( 0.0 y0.6 t 30 35 40 45 50 55 60 si 0.2 n OrientationangleΘ(deg) nte0.4 6.4GHz 15.7GHz I 0.1 FIG. 3. TR orientation dependence measured in the fre- 0.2 quency range 320−460 GHz. 0.0 0.0 20 40 60 80 100 120 200 300 400 500 600 InterferogramsmeasuredusingMichelsoninterferome- Frequency(GHz) teratthemaximaoforientationdependencieswithboth detectors are shown in Fig. 4a and Fig. 4b. FIG. 5. SPR spectral measurements results. a and b are interferogramsmeasuredintherangeof60−90GHzand320− Interferometermirrordisplacement(mm) 460 GHz respectively. Bottom plots: reconstructed spectra. 6 4 2 0 2 4 6 2 1 0 1 2 − − − − − 1.0 1.0 eters resolution, it is important to mention that the cri- 0.5 0.5 terion, which defines ∆λiknt as FWHM of the apparatus ts) 0.0 a b 0.0 spectralpeakfrommonochromaticsourcewiththewave- uni length λk was chosen: k=1 k=3k=4k=5k=6 rb. 1.0 1.0 ∆λint λ (a k =1.21 k , (7) y 0.8 0.8 λ 2L t k int si n 0.6 0.6 e where L is the interferometer maximal optical paths nt int I 0.4 0.4 difference from zero position. For Michelson interferom- eter in case of symmetrical interferogram this value co- 0.2 0.2 incides with the full length of the interferogram. Apply- 0.0 0.0 ing this criterion to the interferograms shown in Fig. 5 20 40 60 80 100 120 200 300 400 500 600 (top part) it is possible to find ∆λint/λ = 4.2% for Frequency(GHz) 1 1 60−90GHzand∆λint/λ =2.1%for320−460GHz. The 5 5 FIG. 4. TR spectral measurements results. a and b are spectra recovered from these interferograms are shown interferograms measured in the range of 60−90 GHz and in Fig. 5 (bottom part). As one can see the spectral 320 − 460 GHz respectively. Bottom plots: reconstructed peak measured in the range 60−90 GHz corresponds to spectra. k =1 and the spectral peak measured in 320−460 GHz range corresponds to the 5th SPR resonance k = 5. Fourier transform algorithm was used for spectral re- The peaks’ relative line widths are ∆λ /λ = 8.8% and 1 1 construction [27]. Two normalized TR spectra measured ∆λ /λ = 4.3%, which are close to the estimated spec- 5 5 inarangeof60−90GHzand320−460GHzareshownin tralresolution∆λint/λ obtainedthroughanalysisofthe k k Fig. 4 (bottom part). Subsequently, these spectra were interferometercharacteristics. Itisobviousthatthemea- used as the entire measuring system spectral efficiency sured spectra peaks’ widths are defined by the interfer- for the SPR spectra renormalization. Vertical dashed ometer spectral resolution and the real peak widths are 5 narrower. To compare the radiation intensities at k = 1 andk =5theSPRspectrawerenormalizedbyTRspec- tra. It is also important to notice that the background contribution (both coherent from the accelerator beam- lineandassociatedwithenvironmentalnoise)totheSBD signal was constantly low during the experimental run. IV. SIMULATIONS ThespectrumofSPRfromthegrating,identicaltothe one used in the experiment (see section III), was simu- latedusingComputerSimulationTechnology(CST)Par- t=0.05 ns t=0.10 ns t=0.15 ns t=0.20 ns ticle In Cell solver [28]. According to Eq. 4 even for the first diffraction order the far-field condition is not ful- FIG. 6. Electric field representation of beam propagation filled. However, taking into account focusing parabolic through small calculation domain near grating. optics used in experiment it was assumed that prewave zone effect was minimized [19, 20]. The distance from thegratingtotheelectricfieldprobe, locatedperpendic- Frequency(GHz) 0 20 40 60 80 100 ularly to the grating surface, was equal to L=500 mm. 2.0 1.0 L was also the distance from the grating to the movable SSmmaallllccaallccuullaattiioonnddoommaaiinn 0.8 mirror (M2) in the interferometer (see Fig. 1). The sim- LLaarrggeeccaallccuullaattiioonnddoommaaiinn 0.6 ulations were carried out using two calculation domains 0.4 in order to show the influence of prewave zone effect for nits) 1.5 00..02 thefirstdiffractionorderofradiation. Themaincalcula- u b. tionwasdoneforthefrequencyrange0−400GHzusing r a 1.0 ( so-called “small calculation domain” that assumed cal- y t culation of the electric fields in the calculation domain nsi withthesize16×32×70mm3,toencloseentire3Dgrat- te n 0.5 I ing, and the following field propagation based on CST far-field monitor. It was confirmed that the spectrum calculated by the monitor was not sensitive to increase 0.0 of the domain size (60% increase was considered), and, 0 50 100 150 200 250 300 350 400 therefore, the smaller domain was chosen to reduce cal- Frequency(GHz) culation time [29]. In the case of the small domain the prewave zone effect was not taken into account. A so - FIG. 7. Calculated SPR spectrum in small calculation do- called“largecalculationdomain”thatcoveredthewhole main. Inset: comparison with SPR spectrum calculated in area of the radiation propagation was used for the fre- large calculation domain. quencies up to 100 GHz only because of simulation time limitations. In this domain the prewave zone effect was taken into account due to the fact that L = 900 mm. ing both calculation domains at frequencies ν = 0 − ff 1 Beam propagation and electric field distribution in the 100 GHz. The inset of Fig. 7 shows the spectra calcu- 2D cross-section of the small calculation domain at four lated using the two methods: first - when the electric consecutive time steps are shown in Fig. 6. The electron field values at the border of the small domain were ex- beam parameters used in the simulation are shown in trapolated using the far field monitor (blue curve), and Table I. The spectrum of SPR was obtained by record- the second - when the electric field values were recorded ing the dominant component of the electric field at the at the probe in the large domain without extrapolation probe location as a function of time and, then, by per- (red curve). For the blue curve ∆λ /λ = 8.7%, for the 1 1 formingFouriertransform. Thisprocedureremainedthe red curve ∆λ /λ =9.3%, and the theoretical value cal- 1 1 same for both calculation domains, the only difference culatedusingEq.(3)is∆λ /λ =5.9%. Increasedwidth 1 1 wasthatinthecaseofthesmalldomaintheelectricfield of the spectrum for the second method agrees well with valuesattheborderwereextrapolatedtotheprobeloca- the fact that the prewave zone affects the spectrum. In tion, and for the large domain they were recorded at the thecaseof1st orderthebroadeningeffectshouldbemore probe without extrapolation. Five diffraction orders in pronounced and it should be always taken into account. the SPR spectrum calculated for the far field are shown Noise in the red curve is most likely caused by a low in Fig. 7. signal to numerical noise ratio for the large domain due The influence of the prewave zone effect was investi- to the probe being located at the large distance L (cid:29) λ gated by comparison of radiation spectra obtained us- from the grating. Table III summarizes the comparison 6 oftheSPRlinewidthssimulatedusingCSTstudiosuite, detector with horn antenna is defined as: Eq. (3) and measured SPR relative line widths. The 5th λ2 diffraction order in Fig. 7 corresponds to the far field. A = G, eff 4π Monochromaticity of this peak is ∆λ /λ = 1.6% and 5 5 the theoretical value is ∆λ /λ =1.2%. The monochro- where G is the antenna gain. For the SBD detectors’ 5 5 maticities of the 1st and 5th diffraction orders in the far parameters shown in Table II and constant wavelengths field spectrum are 1.5 and 1.3 times larger than the cor- λ1 = 4mm and λ5 = 0.8mm the angular acceptance of responding theoretical values. Nevertheless, the spectral 320−460GHzSBDismorethan5timeslargerthanthat line of the 5th order is narrower than the line of the 1st of 60−90 GHz SBD. order, which agrees well with the theoretical predictions. It was shown that the monochromaticity of the SPR spectral lines increases with diffraction order k. If angu- lar aperture is defined by: TABLEIII.ComparisonoftheSPRspectrallinewidths: CST tan(θ/2) simulation, theoretical and measured values. ∆θ < , kN k Theory Simulation Measurements the monochromaticity will be determined by the diffrac- 1 5.9% 8.7% 8.8% tion order k and the number of grating periods N. The 2 3.0% 3.3% − relative line width smaller than 1% can be achieved. 3 2.0% 2.6% − 4 1.5% 2.0% − We have compared intensities of coherent TR and co- 5 1.2% 1.6% 4.3% herent SPR in GHz frequency range measured in identi- calconditionsandshowedthatthebrightness(energyper unit solid angle and per unit frequency range) is practi- cally comparable for both mechanisms. In order to com- pare coherent TR and SPR intensities at lower frequen- V. RESULTS AND CONCLUSION cies(aspresentedin[32])furtherexperimentalworkisre- quired. The simulation results showed that the intensity We have investigated coherent SPR spectral charac- of the fifth order spectral line is about two times smaller teristics both experimentally and by numerical simula- incomparisonwiththelineintensitiesofthefirstandthe tion. Our experimental apparatus included the Michel- second orders at θ = π/2. Different energy distribution soninterferometerwithSBDdetectorsplacedinthefocal between diffraction orders can be achieved by selecting planeoftheparabolicmirrorlocatedbehindtheinterfer- either different grating parameters or observation angle ometer. To measure SPR spectral line width of differ- θ, which is usually limited by the experimental geome- entorderstherequireddetectorspectralrangeshouldbe try. Also some adjustment of the SPR spectral line can rather broad. Room-temperature detectors such as SBD be achieved by the grating tilt angle θ (cid:28) 1 [33] with gr cover half an octave bandwidths, i.e. 60−90 GHz and the line shift described by the following relation: 320−460 GHz in our case. The spectral range and effi- λsinθ ciency of each detector were investigated experimentally ∆λ=− ∆θ . 1−cosθ gr by measuring coherent TR spectrum. The measured re- sults in comparison with theoretical simulations showed Another important approach was demonstrated by au- apossibilitytousedifferentSBDdetectorstoinvestigate thors of the Ref. [34] where a train of short bunches two different SPR spectral lines (k =1 and k =5). with the fixed spacing was used to generate quasi- For experimental parameters (E = 8 MeV, N = 15, monochromatic radiation using the emission mechanism e d=4mm) the natural (FWHM) spectral line widths ac- characterised by a continuous spectrum (TR, for in- cording to Eq. (7) are equal to ∆λint/λ = 4.2% and stance). One can expect that a train consisting of N 1 1 b ∆λint/λ = 2.1%. Thus it is possible to estimate ab- bunches may be used for SPR monochromaticity im- 5 5 solute (FWHM) SPR line widths as ∆λ /λ = 7.7% at provement if condition N >N is fulfilled. 1 1 b k = 1 and ∆λ /λ = 3.7% at k = 5, using Eq. (6) and 5 5 taking into account the interferometer resolution. It is inareasonableagreementwiththesimulatedvalues(see ACKNOWLEDGMENTS Tab. III). Small discrepancy at high frequencies related to the fact that the limited detectors apertures result in The work was performed by international collabora- increased angular acceptance as: tion AGTaX. Authors would like to thank S. Araki and M. Fukuda for valuable help, useful discussion (cid:112) A and support of the LUCX accelerator operation and eff ∆θ = , 2f maintenance. This work was supported by the Pho- par ton and Quantum Basic Research Coordinated Devel- whereA istheeffectiveareaofthedetectorandf = opment Program from the Ministry of Education, Cul- eff par 152mm is the parabolic mirror focal distance (PM at ture, Sport, ScienceandTechnology, Japan, JSPSKAK- the Fig. 1). 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