Wave Mode Couplings in a Free-Electron Laser with Axial Magnetic Field in the Presence of Self-Fields

Transcription

1 J Plama Fuion SEIES Vol 8 (9) Wa Mod Coupling in a Fr-Eltron ar ith Axial Magnti Fild in th Prn of Slf-Fild hrou Maraghhi Halh Mahdai Taghi Mohnpour Dpartmnt of Phyi Amirair Unirity of Thnology PO ox Thran Iran (id: 3 Augut 8 / Aptd: 5 Nomr 8) Th on-dimnional analyi of th ollti intration in a fr ltron lar (FE) ith omind hlial igglr and axial guid magnti fild in th prn of lf-fild i prntd Contrary to th priou intigation rlatiiti trm to all ordr of th igglr amplitud i rtaind in th linarid quation A diprion rlation for th untal oupling of a i drid Thi diprion rlation i old numrially to intigat th uual FE intailiti ith rlatiiti trm inludd It a found that lf-fild lor th maximum groth rat and narro th idth of th untal ptrum Kyord: Fr ltron lar hlial igglr lf-fild diprion rlation intaility Introdution Th fr-ltron lar (FE) thory in th ollti or aman rgim rli on th untal oupling tn th radiation and th ngati-nrgy pa-harg a [-3] Mhdian t al did a rlatiiti thory for an FE and drid a diprion rlation (D) [4] ntly Mohnpour t al old th D numrially and found additional oupling tn a [5] In thi high-gain rgim du to th high dnity and lo nrgy of th ltron am an axial magnti fild i uually mployd to fou on th am In uh a onfiguration quilirium lf-ltri and lf-magnti fild du to th harg and urrnt dniti of th am ha onidral fft on quilirium orit It ha n hon that lf-fild an indu hao in th ingl-partil trajtori in th iinity of th gyroronan Frund t al ha alulatd th firt-ordr lf-magnti fild gnratd y th igglr-indud tranr loity [6] ntly th fft of lf-fild on th taility of quilirium trajtori ha n tudid in a FE ith a on-dimnional hlial igglr and an axial magnti fild [7-8] Th purpo of th prnt intigation i to u a rlatiiti thory to dri a D for th intration of all th a in a rlatiiti ltron am that pa through a on-dimnional hlial igglr magnti fild and an axial magnti fild in th prn of lf fild Thi D i old numrially for group I orit to tudy th rlatiiti fft on th FE intaility i th untal oupling tn th ngati nrgy pa-harg a and th ltromagnti radiation Thi D may alo tudid furthr to intigat th oupling tn othr a mod in th ytm [5] author -mail: hrou@autair Slf-Fild Calulation Conidr a rlatiiti ltron am moing along th axi of an idalid hlial igglr magnti fild drid y xˆ o yˆ in () and in th prn of an axial tati magnti fild ẑ Hr i th igglr a numr and i th igglr alngth Th tranr part of th tady-tat hlial trajtori of ltron nglting th lf-fild of th am an found a xˆ o yˆ in () ( m (3) hr ) m ) ( m i th ltron rt ma i th magnitud of th harg of an ltron and i th pd of light in auum Th lf-ltri and lf-magnti fild ar indud y th tady-tat harg dnity and urrnt of th non-nutral ltron am In ordr to modl th lf-fild ma th aumption of a homognou ltron dnity profil n ont r r n r (4) r r hr n i th numr dnity of th ltron and r i th am radiu Soling Poion quation yild th lf-ltri fild in th form E n r rˆ n x xˆ y yˆ (5) Th lf magnti fild i indud y th tady-tat urrnt dnity of th ltron am and may otaind y Ampr la 4 J (6) 55 9 y Th Japan Soity of Plama Sin and Nular Fuion arh

2 Maraghhi t al Wa Mod Coupling in a Fr-Eltron ar ith Axial Magnti Fild in th Prn of Slf-Fild hr J n x o yˆ in ˆ (7) ˆ i th am urrnt dnity y th mthod of f [8] may found a (8) n ry xˆ x yˆ hr 4 n m y oling th quation of motion of an ltron in th prn of lf-fild E and ill find th tady-tat orit xo yˆ in ˆ (9) hr ˆ () Figur ho th ariation of axial loity ith normalid ylotron frquny for group I and group II orit 3 Th paramtr ar m n m 3 and 5 G Fig Axial loity a a funtion of th normalid axial magnti fild 3 Diprion lation An analyi of th propagation of ltromagnti/ltrotati a in th ltron am may ad on th ontinuity quation n n () t th rlatiiti momntum quation E E () t m and th a quation E 4 E n (3) t t Hr n i th ltron dnity i th ltron loity i th ornt fator orrponding to E i th ltri fild and i th magnti fild With th unprturd ltron dnity n tan to indpndnt of poition and tim th ltron and fild arial may xprd in th form n n n (4) (5) E E E (6) (7) (8) hr i th radial ditan from th axi Th linarid quation for th ontinuity quation th rlatiiti momntum quation and th a quation may drid a n n n (9) t E E t m () E E E 4 E n n t t () y introduing a n t of ai tor ˆ xˆ i yˆ ˆ xˆ i yˆ and ˆ ˆ th unprturd paramtr an rittn a E i ˆ ˆ xp( i) xp( i) () i n ˆ ˆ xp( i) xp( i) xp( i )ˆ xp( i )ˆ ˆ n xp( i )ˆ xp( i )ˆ ˆ (3) (4) (5) hr i th radiu of th quilirium orit of ltron and (6) Th prturd tat i aumd to onit of a longitudinal pa-harg a and right and lft irularly polarid ltromagnti a rfrrd hr a radiation ith all prturd a propagating in th poiti dirtion Aordingly olution of th ytm of quation (9)-() may aumd a ˆ ˆ Zˆ (7) E n ˆ E n (8) E ˆ E ˆ Z 55

3 Maraghhi t al Wa Mod Coupling in a Fr-Eltron ar ith Axial Magnti Fild in th Prn of Slf-Fild ˆ (9) ˆ ˆ ˆ (3) n n xp[ i( t)] (3) xp[ i( t)] (3) xp[ i( t)] (33) and E ar analogou to n ; E and ar analogou to ; E and ar analogou to ; th a numr ar rlatd to y (34a) (34) Sutituting Eq (7)-(33) in Eq (9)-() gi th xprion for prturd fild a follo: D E M E E D 3 3 M E hr D 3 E E D ( ) p E E E (35) (36) (37) (38) (39) (4) and M M 3 3 ar dfind in th Appndix Hr D D and ar th unoupld diprion rlation i in th an of th igglr for th right and lft irularly polarid ltromagnti a and th pa-harg a rptily Equation (35) and (36) ho that th D for th right and lft a alon in th an of th othr to a ar D D (4) D D (4) hih indiat that th igglr ha dirt fft on th right and lft a and th igglr fft on thir D ar of th ond ordr in th igglr amplitud On th othr hand Eq (4) ho that th D for th pa-harg a in th an of th right and lft a i hih indiat that th igglr ha no dirt fft on th pa-harg a Th raon i that th tranr hlial motion of ltron du to th igglr ha no fft on th longitudinal oillation of th pa-harg a Th nary and uffiint ondition for a nontriial olution onit of th dtrminant of offiint in Eq (35)-(37) quatd to ro Impoing thi ondition yild th diprion rlation D D D 3 M D 3 M 3 M Equation (43) i th D for oupld ltrotati and ltromagnti a propagating along a rlatiiti ltron am in th prn of a igglr magnti fild and an axial guid magnti fild Thi D (43) ill old numrially in th nxt tion to intigat th rlatiiti and lf-fild fft on th FE intaility 3 M (43) 4 Coupling tn Wa for Group I Orit In th tal group I orit th igglr indud loity i not o larg thrfor hould xpt to Fig Coupling of right ap and ngati pa-harg a for group I orit or th at oupling that an indud y a rlatily mall In group I orit ith largr axial magnti fild and in group II orit nar th ronan rgion ith 7 i largr and hould ha additional oupling that ar drin y th rlatiiti fft of Th root of th D (43) ar found numrially for group I orit ith Th rt of th paramtr ar th am a in Fig Th poiti and ngati-nrgy pa-harg a S and th ap ranh of right irular a ar hon in Fig Thr ar to oupling tn th mod and th S mod and ar hon y dottd lin Th id ptrum oupling at larg alu i th ll non FE ronan Solid lin ho on th lft rtial axi Cirl ho th normalid imaginary part of a numr Im for th to oupling in Fig In th group I orit indud loity 553

4 i lo and thrfor only th ll non FE oupling tn th right irularly polarid ltromagnti a and th ngati-nrgy pa-harg a i poil In ordr to orrt lf-fild analyi it i onnint to introdu an ffti igglr magnti fild ff hr i gin y quation (6) In th an of lf-fild i unity For group I orit i onidraly lo unity thrfor th oupling agnt om a and maximum groth rat i drad from to 8 in omparion ith th an of lf-fild Th idth of th untal ptrum in Fig i 5 hih it a 35 7 in th an of lf-fild [5] Thrfor lf-fild ha lord th groth rat ha mad th untal ptrum narror and ha mod it toard th longr alngth Du to th larg dnity and lo nrgy of th ltron am in thi analyi an axial magnti fild i mployd to fou th am againt it lf-fild Moror in th am i intn th ltrotati potntial of th pa-harg a i not ngligil ompard to th pondrmoti potntial Thrfor th FE oprat in th high-gain aman rgim and in th on-pa amplifir mod Th prolm undr onidration i in th linar tag of th FE intaility hih an drid y th paramtri intaility i aman aattring of th pump a (igglr) in th am fram into a forard attrd pa-harg a and a aattrd ltromagnti radiation In ordr to h th alidity of our rult a on dimnional nonlinar omputr imulation i prformd ith th am paramtr a in Fig Th imulation od i th am a in f 9 Th rult ho that th mall ignal domain oupi th ntir injtion lngth from to m 34 along th undulator Th linar domain tart from m 38 and nd at m 75 and th radiation aturat at m 86 Th groth rat in th linar domain i 7 Im hih i in a ry good agrmnt ith our thory ithout th lf-fild ith Im 5 Appndix: Dfinition of Quantiti Th folloing quantiti ar ud in quation (34)- (36) 3 M 554 Maraghhi t al Wa Mod Coupling in a Fr-Eltron ar ith Axial Magnti Fild in th Prn of Slf-Fild

5 3 M 6 Anoldgmnt M ould li to than th Cntr of Exlln in Computational Aropa Enginring for finanial upport 7 frn [] T Kan and JM Daon Phy Fluid 89 (979) [] HP Frund P Sprangl D Dillnurg EH da Jornada S Shnidr and irman Phy A 6 4 (98) [3] JE Willtt olon U-H Hang and Y Ata J Plama Phy 66 3 () [4] H Mhdian JE Willtt and Y Ata Phy plama (988) [5] T Mohnpour Maraghhi and S Miranjhad Phy Plama (7) [6] HP Frund H Jaon and D E Prhing Phy Fluid 5 38 (993) [7] S Miranjhad Maraghhi and T Mohnpour Phy Plama 4776 (4) [8] M adh J E Willtt and J Willtt J Plama Phyi (5) [9] H P Frund Phy A (983) 555 Maraghhi t al Wa Mod Coupling in a Fr-Eltron ar ith Axial Magnti Fild in th Prn of Slf-Fild