Math & Engineering

Design of a 30 GHz Bragg reflector for a Raman FEL

Description
Design of a 30 GHz Bragg reflector for a Raman FEL
Published
of 5
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Transcript
  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
Search
Similar documents
View more...
Tags
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks