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Design of a 30 GHz Bragg reflector for a Raman FEL

Design of a 30 GHz Bragg reflector for a Raman FEL
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  NUCLEARINSTRUMENTS Nuclear Instruments and Methods in Physics Research A 341 (1994) 484-488   8R METHODS IN PHYSICS RESEARCH Section A North-Holland Design of a 30 GHz Bragg reflector for a Raman FEL P . Zambon  , P .J .M vander Slot b,* University of Twente, Departmentof Applied Physics, P .O Box 217, 7500 AE Enschede, The Netherlands n Nederlands Centrum voor Laser Research BV   P .O Box 2662, 7500 CR Enschede, The Netherlands A design of a Bragg reflectorfora Raman FEL is described It is shown that mode conversion occurs whenever the axial wavenumbers ofthe two modes fulfil the Bragg condition . Wth aconstant ripple ofthe corrugation it is shown that the reflectedradiation also containshigher order modes, assumng that theincidentradiation consists onlyofa TE zyxwvutsrqponmlkjihgfed mode . Themode purity can be increased by increasingthelengthof the reflectoratthe expense of a smaller reflection bandwidth . A more flexible method is byapplying a Hammng window to the corrugation ofthe reflector . Contributions of other modes to thereflectedradiation can m that case be neglected . The reflector wll be installed in a Raman laser to be able to compare theamplifier with the oscillator configuration . Therefore some prelimnary results are also presented about the start-up ofthe Raman laser . 1 . Introduction The Raman type FEL situated at the Universityof Twente has so far been operated in an amplifier con- figuration with a maximum current I b of the electron beam of about 900 A maximum [1,2] . In order to compare the behaviour of the laser in an amplifier and oscillator configuration the current of the electron beam has been reduced to about250 A maximum [3 ] . Wth this current the laser can be operatedwith the two configurations . Several choices for the mrrors for the oscillator areavailable . As the beam line contains no bends, the mrror situated at the upstream side of theundulator must contain a hole . The electron beam is dumped in the waveguide wall before it reachesthe mrror at the downstream side of theundulator . Sincethe laser operates at wavelengths in the mm-region, one option forthe downstream mrror is a mesh . Another option is to use a platewith a hole in it through withthe electron beam passes . In the former case the electron beam is influenced by the mesh, but for the radiation field itis almost a perfect reflecting mrror, whereas in the latter case the radiation field is modified by the hole and the electron beam is hardlyinfluenced . The effects of this hole canbe reduced by increasing thearea of the mrror, keeping the hole diameter constant . However,one then needs to taper the waveguide to the * Corresponding author . 0168-9002/94/ 07 .00 C 1994 - Elsevier Science B .V All rights reserved SSDI 0168-9002(93)E0898-3 outer diameter of the mrror to avoid impedance ms- match . A drawback of these type of mrrors is that there is no mode selectivity and it is difficultto change the reflectivity . This canbe overcome by using a Bragg type reflectorconsisting ofa section of waveguide with a corrugated wall(see e .g   ref . [41 for anintroduction) . The electron beam will be undisturbed, and the prop- erties of the mrror, such as reflected power, spectral width of reflection band and centre frequency of re- flection band,can be controlled by changing parame- tersas lengthof the mrror, corrugation height and period of the corrugation . It will be shown that, by applying a spatial filter, the mode purity of the re- flected radiation canbeimproved . The space availableinside the axial guide magnet forcedus to choose a plate with a hole as the upstream mrror . For the downstream mrror a Bragg reflector is chosen because it gives the designer a flexibility not offered by the other typeof mrrors . 2 . Design of a Bragg reflector The theory of Bragg reflectors is well known [4-61 and only the coupled mode equations will be given here . Consider a sectionof waveguide with a corru- gated wall for which theradius is given by r=r,+10 cos(kbz+00),   (1) where r is the mean radius of thewaveguide, 1, is the amplitude of the corrugation, k b =27r/A b   A b being  the periodicity of thecorrugation, and 0   is an arbi- trary starting phase . The coupled mode equations for the cylindrical waveguide Bragg reflector are [5,6] c UNO U ô U m Q N3 0 a 1 00 0 80 0 60 040 0.20 000 P Zambon, P .J M uander Slot / Nucl lnstr k z beingthe axial wavenumber of the incident mode . Since the corrugation is azimuthallysymmetric, the Bragg reflector will notcouple modes with different azimuthal indices [4] . The indices in theaboveequa- tionsrefer to the radial indicesof the modes . f,+ is the amplitude of the (incident) waveguide mode propagat- ing in the forward direction and f, is the amplitude of the backward propagating (reflected) wave . G zyxwvutsrqponmlkjihgfedcbaZYXWVUT   G, P , and   P arethe coupling coefficient between the for- ward and backward components of the incident mode, the coupling coefficient between two TE or TM modesand the cross-coupling coefficient between a TE (H) and TM (E) mode respectively . Expressions for these coefficients can befound inrefs . [5,6] . Solving thesystem of equations given by (2) and (3) fromz = 0 to z = L, L beingthe length of the mrror, one finds the mode amplitudes fromwhich the reflection R and 25 30 354045 and Meth . m Phys . Res . A 341 (1994) 484-488   485 transmssion T coefficients for each mode can be cal- culated R=1f (z=0)I Z /If+(z=0)I Z , T=1f+(z=L)Iz/If+(z= (» Iz . The phase shift of the reflected (¢,) and transmtted (0,) components of awaveguide mode can also be obtained fromtheamplitudes f, +and f,   0, = -tan-'(f   n,(z-L)/fc(Z- L)), (hT=-tan-'(f,»(z-0)/fe(z-0)), (8) where the subscripts im and re stand for imaginary and real part of the complex amplitude . In the Raman laser the FEL interaction basically takes place with the TE   mode, though cyclotron instabilities can result in interaction with other, higher order, modes [2] . The Bragg reflector must therefore reflect the TE   mode and preferably none of the other modes   The central frequency, chosen to be30 GHz, is given by the Bragg condition, k Z =k h /2 . The periodof the corrugation for reflection of the TE   mode at 30 GHz thus becomes .t i , = 5 .37 mm The system of equations (2) and (3) canbe solved for any number of modes, but for practical reasons  i .e . for reasonable computing time) the number of modes simultaneously present is limted to two in the (mod- ified) code [5] used . The incident mode atz = 0 is assumed to beapure TE S mode whereas theother mode is chosen to be the TM t   mode . The reflector is matched at z = L, i .e . nowaves arereflected at the end of themrror . The power reflectioncoefficient is shown in Fig . l a forthecase of L = 120 mm and 1, = 0 .5 mm . The transmssion coefficient is shown in c   1 00 m UO U (D   0 80 ô   0 60 NNy 040 C c0 020 3 IL   0 00 Frequency (GHz)   Frequency (GHz) (5)(6) 25 30 354045 Fig 1 . Power reflection coefficient(a) and transmssion coefficient (b) forthe TE I mode (solid line) and TM I mode (dashed line) for the Bragg reflector versus frequency . The incident mode is assumed to be a pure TE  mode . The parameters of the Bragg reflector are d n = 5 .37 mm 1,   = 0 .5 mm and L = 120 mm IX   OPTICAL TECHNOLOGY 'f - _ id,,f +  iG f, z - iY_H,1,f,,P - - i Y_ G,fti, P , (2) P _dfr _- JA   f=   1G   f +  iY_H P ft' . P c3z P - J Y_ G,PfH,P > (3) P where =k z   kt,/2, (4)  486 P Zambon, P .J .M cander Slot I Nucl . Instr . and Meth . i n Phys   Res . A 341 (1994) 484-488 Fig . l b . It can clearly be seen that the incident TE zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON mode is primarilyreflected asa TE, 1 mode at 30 GHz (FWHM = 1 .7 GHz) and as a TM t, mode around 33 .4 GHz (FWHM = 2 .4 GHz) . Note that the Bragg reflec-toractsas a mode converter atthis frequency which is given by the Bragg condition kT E +kZ M =k B .   (10) At the designfrequency of 30 GHz not all the power is reflected in the TE   mode because of the side lobesof the TM   reflection . Almost all the trans- mtted power is in the TE   mode, while only a mnor fraction is transmtted in the TM, mode (see Fig . 1b) #' . The side lobes may be undesired in high gain FELs . One way to decrease the effects of side lobes is to make the reflection band around the centre fre- quency smaller, by increasing the length of the mrror, so that overlap between reflections will be less . A different way to decrease or even remove the effects of side lobes and to obtainhigh mode purity is to apply a Hammng window to thecorrugated partof the mrror . This is particularly interestingfor over- moded waveguides where higher order modes are not damped . This typeof waveguide is often used in FELs because propagation of the electron beam requires waveguide dimensions corresponding to overmoded operation for the radiation generated . A Hammng window can be applied to the Bragg reflector by modi- fying the corrugation height 1 according to Trz 1=100 .54-0 .46* cos 2 (   ~~,   (ll) where L is the length of thecorrugated section which starts atz = 0 . The Raman FEL for which this mrror is designedhas anelectron beam duration of 100 ns . The radiation field can thereforeonly haveabout 6 round trips during which the electron beam is present . As thesystem starts from noise, and as we want to investigate the difference between an amplifier and oscillator con- figuration it was decided to use a power reflectivity around the 40 forthe TE   mode at the design frequency of 30 GHz . The single passgain will be changed in the futureby varying the lengthof theundulator . Again thesystem of coupled equations is solved with the Hammng window applied to the reflec-tor with a corrugation amplitude of 0 .2 mm The power reflectioncoefficient is shown in Fig . 2 . The side lobes havedisappeared and the reflection bands forthe twomodes are completely separated . Even for the 0 .5 mm ripplereflector one finds that the two modes are still  t This is a general property of the reflector, i .e . the mode ofthe transmtted radiation is basically theincident mode . U m 0 U ô U m Q m 3 0- 1 000750 .50 0250 .00 25   30   35   40 Fig . 2 Power reflection coefficient for the TE   and TM   mode   The parameters ofthe Bragg reflector are Ab = 5 .37 mm t o = 0 .2 mm and L= 270 mm (solid line) and 1   = 0 .17 mm (dashed line) . The crosses indicate measured valuesofthe reflection coefficient . completelyseparated if the Hammng window is ap- plied . 3 . Measurement of reflection properties of the Bragg reflector A Bragg reflector with A,, = 5 .37 mm, 1, = 0 .2 mm and L= 270 mm has been constructed with the Ham- mng window (7) applied to the ripple . The reflection properties are measured in order to validate the calcu-lations made . The set-up is schematically shown in Fig . 3 . A HewlettPackard model HP8690B K .-band sweep generator(26 .5-40 .0 GHz) and a model HP8755C am- plitude analyser areused to generate the input power and analyse the forward and backward propagating power which are measured usingone-directionalcou- plers . The Raman FEL utilises a cylindrical waveguide . One-directional CouplerForward Power Amplitude Analyser frequency(GHz) Rectangular to CircularTransition One-directional Coupler BackwardPower Cylindrical TaperOutcouple Horn Fig . 3 . Schematic overview ofthe set-up used for measuring the power reflection of the Bragg reflector    P Zambon, P .J M van der Slot/ Nucl . Instr and Meth .i n Phys . Res   A 341 (1994) 484-488 Therefore a rectangular to cylindrical waveguide tran- sition is needed togetherwith a cylindrical taper to switch from standard K a -banddimensions to thecus- tom size of the Bragg reflector . The rectangular to cylindricaltransition converts the rectangular TE, waveguide mode to a cylindrical TE, mode . Higher order modes present in the cylindrical section of the measurements willin principle notcouple to the fun- damental TE, o rectangular waveguide mode asthey are reflected at the transition . However the Bragg reflector is reversible, i .e . if ata frequency fo an incomng TE, mode is reflectedinto a TM t   mode then at the same frequency an incomng TM t mode will be reflected in a TE zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB mode . Thus higherorder modes canbe measured with this setup, though the reflected power will depend on the double mode con-version that has takenplace . One side of the mrror is connected to the cylindri-cal taper while at the other side an outcouple horn (used in the Raman FEL) is mounted, i .e . the reflector is matched at thatside and there are no reflections . Spurious reflectionsinside the measurement system limted the dynamc range to about 20 dB . The results are also shown in Fig . 2 . The centre frequencies of the two reflection bands clearly coincidewith the calcu- lated ones . The total reflectivity measured is somewhat less than calculated . The amplitude ripple used in the calculation was 0 .2 mm The peak reflectivityat 30 GHz is reduced to 25 for a corrugation heightof 0 .15 mm whereas it is 30 for t o = 0 .17 mm The calculated reflection coefficientfor the latter value is also shown in Fig . 2 . For the TE, mode goodagreement is found between the measured values and the calculationfor t o = 0 .17 mm The double mode conversion, in the experiment, for the TM, mode should reduce the measured values to 87 of thetheoretical ones . One can thus state thatwithin themechanical limts of making the corrugation, good agreement is found be- tween thecalculations and the measurements for the TE, and the TM   modes . 4 . Start-upof the Raman FEL In this section some prelimnary results will be presentedobtainedwith a numericalcode . The code used is basically the one described in ref . [7] . The radius of the electron beam is such that the assumption ofperfect orbits and that of neglecting the radial dependence of the undulator field is not valid . Themodel has been modified to include the full three dimensional electron orbits . The current version  2  2 The model wll be described in more details in a future publication . m30 a 0 0m30 a 0 0 Z (m Fig . 4 . The logarithmcofthe power (solid line) and the radiation phase(dashed line) as afunctionofthedistance along the undulator . The initial power is assumed tobezero while the electronsare initialised with a uniform phase distri- bution with a small random disturbance (TDA-code) . The main simulation parameters areI = 200 A, B   = 0 .18 T, Au = 0 .03 m B z =1 .0 T, E = 10-S,R mrad and y=2assumesan idealhelical undulator field . For now par- ticular attention has been given to the start-up of the laser . Therefore the initial radiation amplitude is as- sumed to be zero in thecalculations and the laser starts from noise in theelectron beam Two methods ofelectron beam initialisation havebeenused . One is the initialisation used by the FRED codeand the other is used in the TDA code [8] . The TDA-initialisation has been modified tostart withzero radiation field and a uniformphase distribution modified bya small random number [9] . The code has beenrun for theparameters of the Twente Raman FEL [2] with an undulator field of 0 .18 T, a guide field of 1 .0 T, a current of 200 A and a normalisedemttance of10-5 -rr mrad . Growth is found for a radiation frequency of 30 GHz . The growth and the phase of the radiation field is given in Figs . 4 487 m L a - Z (m Fig . 5 . As Fig . 4 except that theelectrons are initialisedin the sameway as in the FRED-code . IX . OPTICAL TECHNOLOGY  488 5 . Conclusions P Zambon, P .J . AI van der Slot /Nucl   Instr   and Meth . t o Phys . Res . A 341 (1994) 484-488 and5 forthe TDA- and FRED-initialisation respec- tively . The lethargy  i.e . the distance before exponen- tial growth starts) is aboutthe same for both initialisa- tions . The FRED-initialisation gives however a lower field amplitude at the beginning of the exponential growth region resulting in a factor 5 loweroutput at the end of theundulator . The TDA-initialisation re-sultsin saturation atalevelof about 12 MW whereas this is notthe case for the FRED-initialisation . The phase of the radiation field changes more rapidly in the lethargyregion for the FRED-initialisation than for the TDA-initialisation whereas during exponential growth thefrequency shift is slightlyless . By varying the lengthof theundulator it becomes possible to investigate thebehaviour ofanamplifier and oscillator configuration for differentsingle passgains . A Bragg reflector has been designed with a power reflection coefficient of = 0 .4 at the design frequency of 30 GHz for an incomng and reflected wave in the TE tt mode . By applying a Hammng window to the corrugation itis possible toobtain high mode purity and still maintain arelative large FWHM of the reflec- tion peak . Itis found that the reflectivity is a sensitive functionof the ripple amplitude for the parameter region investigated . A Bragg reflector has been con-structedwith X1 6 = 5.37 mm, 1, = 0 .2 mm and L = 270 mm Allowing for mechanical tolerances in making the corrugation, good agreement is found between the measured and calculated reflectivity . Prelimnary calculations show that the laser may just orjust not saturate in a single pass dependingon the typeof initialisation used for the electron beam in the simulation . Varying the interactionlength makes it possible to investigate the different behaviour of an oscillator and amplifier configuration . Acknowledgements The authors would like to thank P .J .S . Teunisse from Hollandse Signaalapparaten B .V ., The Nether- lands and A F .M Bouman for their assistance with the rf diagnostics . References [1] P .J .M vander Slot, Ph .D . Thesis,Universityof Twente, The Netherlands (1992) . [2] P .J .M vander Slot and W J Wtteman, Nucl Instr andMeth   A 313 (1993) 140 . [3] P Zambon, W J . Wtteman and P .J .M vander Slot, these Proceedings (15th Int . Free Electron Laser Conf ., The Hague, The Netherlands,1993) Nucl . Instr and Meth . A 341 (1994) 88 . [4] V .L . Bratman, G .G . Denisov, N .S . Gnzburg and M I . Petelin . IEEE J . Quantum Electron . QE-19 (1983) 282 . [5] J.C . Cheng, B .S . thesis, MT, USA (1991) . [6] G .G . Denisov and M G . Reznikov . Izv . Vyssh . Uchebn . Zave d Radiofiz . 25 (5) (1982) 562,translated by AIP . [7] J .S . Wurtele, R Chu and J . Fauns, Phys Fluids B 2(1990)1626 . [8] Both typesof initialisation are present in TDA Version 0 .3 - T.M Tran and J .S . Wurtele, Comp . Phys . Commun . 54 (1989) 263 . [9] C Penman and B .WMcNed,Opt . Commun . 90(1992)82
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