782 Nucl ear I nst rument s and Methods mPhysi cs Resear ch A304 ( 1991) 782- 785 Nort h-Hol l and Numer i cal st udi es of r esonat or s with on- axi s hol es in m i r r or s f or FEL appl i cat i on M . Kesel br ener , S. Ruschi n, B. Li ssak and A . Gover Facul t y of Engi neer i ng, Tel - Avi v Uni ver si t y, Ramai -Avi v 69978, I sr ael An opt i cal r esonat or with hol es on- axi s f or appl i cat i ons in a FEL osci l l at or was r ecent l y pr oposed by Pant el l et al . [ Nucl . I nst r and Meth. A296 ( 1990) 6381 . These aut hor s pr esent ed an anal yt i cal appr oach based on approxi mat i ng t he f i el d i nsi de t he r esonat or by a t r uncat ed expansi on mGauss- Laguer r e modes . We per f or m f ur t her i nvest i gat i on of t hi s conf i gur at i on by usi ng a Fox and Li t ype code whi ch was r ecent l y appl i ed to t he anal ysi s of a l aser r esonat or with i nt er nal ci r cul ar aper t ur e Thi s numeri cal appr oach al l ows us to check t he r ange of val i di t y of Pant el l ' s sol ut i ons, t he par amet er s f or whi ch t hey const i t ut e t he l owest - l oss mode and to map t he t r ansver se pr of i l e of t he mode at di f f er ent pl anes i nsi de t he r esonat or 1 . I nt r oduct i on A r esonat or with on- axi s hol es to t he m i r r or s, is an ef f ect i ve met hod f or el i m i nat i ng space, wei ght and cost of bendi ng magnet s f or di r ect i ng t he el ect r on beam i nt o t he r esonat or of a FEL . Accor di ng to t he anal ysi s of Pant el l et al . [ 1] , a mode can be bui l t ma r esonat or whi ch shows on one hand, a zer o val ue f or t he f i el d at t he cent er of t he hol es and on t he ot her hand a f ocusi ng pr oper t y al ong t he nud- par t of t he r esonat or , a desi r ed pr oper t y f or enhanced el ect r on beam over l ap and cor r e- spondi ngl y enhanced gai n . The use of hol es in nur r or s in or der to coupl e out power f r om l aser r esonat or s was i mpl ement ed a l ong time ago [ 3] . Al t hough t he conf i gur at i ons t her e wer e basi cal l y si m i l ar to t hose anal ysed her e, t he r ol e and t he r equi r ement s of t he hol e ar e basi cal l y di f f er ent . In a hol e coupl ed r esonat or , t he hol e ser ves usual l y as t he out put coupl i ng por t and its l osses ar e t her ef or e consi d- er ed as usef ul , so t hat t he desi gn is ai med at opt i m i zed out put coupl i ng vi a t he aper t ur e. In t he devi ce di s- cussed her e, t he hol e ser ves as an i nser t i on means, t he r adi at i on escapi ng t hr ough it bei ng essent i al l y usel ess . The out put coupl i ng is in t hi s case i mpl ement ed by means of an addi t i onal el ement in t he cavi t y, and t he desi gn r equi r ement s on t he aper t ur e ar e m i ni mal l osses compat i bl e with t he m i ni mal si ze of t he hol e, whi ch is det er m i ned by t he di amet er of t he el ect r on beam . In r ef . [ 1] it was shown t hat by a sui t abl e combi na- t i on of l ow or der Gauss- Laguer r e modes, a f i el d di st r i - but i on on t he r esonat or can be const r uct ed with t he r equi r ement of cancel l at i on of t he f i el d at t he cent er of t he m i r r or ( " hol e avoi di ng" ef f ect ) . The s ame mode combi nat i on f i l l ed t he vol ume at t he cent er of t he r esonat or as r equi r ed f or FEL gai n opt i m i zat i on . Thi s 0168- 9002/ 91/ $03 . 50 © 1991 - El sevi er Sci ence Publ i sher s B.V. ( Nor t h- Hol l and) anal yt i cal appr oach, however , di d not t ake i nt o account di f f r act i on ef f ect s of t he hol es and f i ni t e si zes of t he m i r r or s . Fur t her mor e, t he exi st ence of t hese di st r i bu- t i ons as l owest - l oss ei genmode sol ut i ons of t he ent i r e cavi t y was not pr oved . In t hi s wor k we under t ake a numeri cal anal ysi s of t he pr oposed conf i gur at i on . We show t hat t he hol e- avoi di ng di st r i but i ons ar e val i d f or a lim i t ed r ange of par amet er s . We check t he mode shapes and l osses f or a r ange of val ues of bot h t he m i r r or di amet er and hol e si zes . The shape of t he mode is i nvest i gat ed al so at pl anes i nsi de t he cavi t y . In t he next sect i on we br i ef l y descr i be t he numeri cal pr ocedur e and f ol l owi ng we pr e- sent s ome sel ect ed r esul t s and di scussi on on t he r ol e of t he di f f er ent r esonat or var i abl es. 2 . Comput at i onal pr ocedur e Our i nvest i gat i on is based on a Fox and Li t ype code whi ch was - r ecent l y appl i ed to t he anal ysi s of a l aser r esonat or with an i nt er nal aper t ur e [ 2] . The wor k was lim i t ed to cyl i ndr i cal l y symmet ri cal pat t er ns, si nce t he l owest r esonat or mode we ar e i nt er est ed in is expect ed to be such . The r esonat or consi st s of t wo f i ni t e m i r r or s , of di amet er 2a t and 2a 2 , and cur vat ur e r adi us equal to R, and R 2 , r espect i vel y . The m i r r or s ' separ at i on al ong t he r esonat or axi s is L . Ci rcul ar hol es of r adi us h t and h 2 exi st at t he cent er of m i r r or 1 and m i r r or 2, r espec- t i vel y . For t he anal ysi s, it is assumed t hat t he m i r r or sur f ace be per f ect l y r ef l ect i ng, t he di mensi ons of t he r esonat or be l ar ger compar ed to t he wavel engt h X of t he l i ght , and t he r esonat or l engt h L be muc h l ar ger t han t he m i r r or ' s di amet er . In our anal ysi s we do not consi der t he edge