On Generation of Photons Carrying Orbital Angular Momentum in the Helical Undulator

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Edmund Ellis
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1 axv: [phss.a-ph] / 09/0 On naon of Phoons Cang Obal Angla Mon n h lal ndlao A.Afanasv * A.Mkhalhnko ** * Dpan of Phss og ashngon nvs ashngon DC 005 SA ** CLASS Ihaa NY 4853 SA Absa. anal pops of loagn adaon n hlal ndlaos wh a pala phass on h obal angla on of h adad phoons. donsa ha all haons hgh han h fs on adad n a hlal ndlao a an obal angla on. dsss so possbl applaons of hs phnonon and h was of ffv gnaon of hs phoons n a hlal ndlao. all fo vw of sls of pns pfod wh h hgh haons adad n a hlal ndlao gh b nvolvd. OVRVI Phoons a powfl pobs of h s of a. Dpndng on h wavlngh phoons gv s nsgh abo phnona a wdl dffn sals angng fo asophss o phss of lna pals. Movad b h nds of nla and pal phss h w wll fos on angla on pops of loagn adaon. Th fa ha lal polad plan-wav phoons a an angla on of ħ was donsad n a lassal pn []. Dspon of phoons n s of sphal wavs and lpol panson s also wll sablshd [] [3] fo adav posss a ao sals wh phoons haad b lpl ns of angla on. No howv ha spaaon of spn and obal angla on of a phoon anno b don n a gagnvaan wa [4]. Abo 0 as ago Alln and ollaboaos -dsovd ha a spal p of bas alld Lag-assan pdd as non-plan wav solons of Mawll qaons an a lag angla on assoad wh h hlal wav fons [5]. hn qand sh bas an b dsbd n s of wsd phoons [6]. Obal angla on OAM lgh bas fond nos applaons n ops onaons bophss hans of o-pals and pobs of Bos-nsn ondnsas [7]. I was donsad [8] ha n phoo onaon of aos b wsd phoons h nw slon ls appl ha nvolv o han on n of angla on. Oh possbl applaon gh b n h phoo-ahods fo gnaon of hghl polad bas of lon wh hgh bghnss wh anplaon wh h ng S f [] pag 40 Appnd.

2 ang of h phoons ang OAM gh hlp n losng ndsabl hannls n ng of phoolons fo h lvls pohbd b angla-on onsvaon laws. Rnl was pond o ha wsd phoons an b podd n MV and V ang va h hans of Copon baksang [9]. Bas of wsd phoons wh p-s angla on a g as a nw and podv ool n nla and pal phss wh loagn pobs ha wold hlp o onol angla on of pals o qan sas gnad n phoopodon posss. Applaons a nld: a podon of hgh-spn sas n h las-basd sahs fo dak a pal anddas [0] []; b son and baon sposop of hgh-spn sas; possbl foaon of nw hgh-spn pals a hgh ngs; d sds pa-volang ffs n ao ansons [] [3]; sahs fo aon-lk pals hogh analng polaaon pops of lgh passng hogh song agn flds [4]. Ths pap dals wh a hod of gnaon of OAM adaon basd on a s of a hlal ndlao. ndlao adaon R s a valabl so of adaon n h ng ag ndad. A lo of pblaons a ddad o h R pops [5]-[9]. On of h fas of R s ha adaon s d n haons [5] nω ω n β Cosϑ wh n3 nas h haons of fqn spaal pod of ndlao fld Ω β / D D s a β v / β v s h pal s avag longdnal vlo n h ndlao and ϑ s h ahal angl wh sp o h obsv. nfonal sng pblaons dd no laf h qson abo OAM ad b h phoons adad a haons hgh ha h fs on ospondng n. Ths s spall poan fo hlal ndlaos wh a hagd pal s lalpolad phoons. Alhogh fo h phoon h obal pa of angla on and h spn pa old no b spaad h plas of gnaon of adaon n a hlal ndlao allow o dsngsh h pa of adaon whh as awa h angla on of lon n an ndlao; hn s possbl o dsngsh wh OAM. As w wll donsa all haons wh n > a OAM. Th phss planaon of h fa wh h lon n a hlal ndlao gnas adaon wh OAM was no psnd lal n h la. In hs pap w anal ndlao adaon n a hlal ndlao fo s abl o gna h phoons wh OAM and o o a spl and la planaon wh all haons oh han h lows on n a OAM. Ths fa was no akn no onsdaon n h pas n a nb of pns n ao and nla phss oband wh h hgh haons fo hlal ndlaos; s Rf. [0] fo a fw apls. Sn spn polaaon ffs a b dffn f OAM phoons a nvolvd -analss of lvan pns a b n od. SORC OF ANLAR MOMNTA Angla on of adaon was dsbd fs n []. In [3] h onp of angla on of adaon was dvlopd fh. Mos gnal dspon of h

3 3 angla on was psnd n [4]; s also [5]-[7] [30]. Th angla on ad b M adaon s dfnd as dv dv M ] [ wh h ngaon s ov h n vol h ads vo s dd fo h oaon as o h pon n a vol wh h M flds a load s Fg. blow. hl h ads nass h flds dop down so h gon na h nsan poson of h so gvs h an onbon o h ngal. Sn w a nsd n a longdnal oponn of angla on dd along on an s ha h 0 M onl f 0 o 0 dv dv M ] [. Th ondons fo appaan of longdnal oponn of lgh a dsbd n Rfs. [4] [7]. To plan appaan of longdnal oponn n Rf.[4] h ahos onsdd h M wav wh sd ansvs dnsons donsang ha h longdnal oponn assoad wh dvav of h nvlop fnon of M fld n a ansvs don. Fo a hlall-polad M wav h s no nss fo hs howv as h longdnal oponn psss n hs p of wav nnsall whl h pon shfs fo h as. Th ass wa o pov hs s o onsd h dpndn pod M fld of h gnal na. ho an son o h oon as hs old b donsad wh TM fld. Ths p of fld old b dfnd b qaons 0 B 3 Th oponns of loagn fld old b psnd as followng [3] d ε d B B B B wh and s a longdnal oodna. 5 h h dfnon 5 Laplaan old b pssd as followng Th l and agn phoons ospondng o h psnaon of h fld as TM and T wavs.

4 Copl ponal sasfs h qaon B nodon of ponal / h longdnal oponn of l fld fo 4 an b psnd as 4 8 Fo h las psson on an s ha an plan loagn wav popagang n -don dos no hav a longdnal oponn and hn dos no ala pals ovng along h sagh ln -oodna and dos no a angla on. On an s fo q.8 ha h wav shold hav a ansvs s 0 / o b abl o a angla on and ala pals o o sasf h qaon 0. I old b sasfd f h ponal an b faod as ~ f wh ~ ~ ~ s a fnon of oodnas onl. Ths h so of h longdnal oponn s assoad wh a nono dvav / 0. Ths pop was sd n Rf.[4] o donsa appaan of longdnal oponn a h dg of a lndal M ba. Th oh wa o oban a nono sl fo psson 8 as w lad al s hlal s of h M wav. Indd h hlal s an b dsbd as [ ] / / / p Sn Cos 9 wh sands fo h spaal pod along h oodna wavlngh of adaon. Fola 9 dsbs h hlal fld wh lf hl.. whl h obsv ovs n a posv don along h ponal pan oas onlokws. Sn h spposon l s n ff h h fola 9 an b ad as psnaon of wo ohogonal polaaons. Th l fld an b psnd n h followng fo: d R 0

5 5 Mos poan s h as of a dpol haon whh dsbs a hlal ndlao o a wggl. In hs as h onl haons of ns a h ons havng dpol s. Fo a lpol haon h ponal solon of 8 looks as follows [3]: wh h fnons - dsb h val of lpol fld a h as 0. If hn all s n { } baks p h fs on n o o so ha -. Sbsng hs psson no 0 on an oban d R R. 3 In pola oodnas ϕ h la an b pssd as ϕ R. 4 On an s ha h longdnal oponn s qal o o on h as b s gowng whl h as off-s s nasng. Fo apl h dpol hlal longdnal oponn of h fld has a fo os ϕ D D. Th spls wa o ogn h so of angla ona s b onsdng a poss of adaon n a fn fa ovng wh avag vlo of h lon β β v v. In a ovng fa h so of adaon s an lon obng along h l wh a ads s Fg.. a nsd n h adaon dd angnall o h nsan ajo n hs ss wh v so h ngal an b valad as follows Id SdA d da d dv M v 5 wh da sands fo h ln of aa S s h Ponng vo SdA da I s h nns of adaon. On an s ha I d M d. 6

In a ovng fa h adaon has an lon as a pon so ovng along h l wh h ads D K / γ.

6 On h oh hand fo h lon losng s ng b snhoon adaon SR and ovng a a onsan ads h hang of ng n a ovng fa an b ansfod as h followng I d d dp d d p d p d M d d d 7.. h loss of on of lon s qal o h on ad awa b SR. Fg. In a ovng fa h adaon has an lon as a pon so ovng along h l wh h ads D K / γ. In h Lab fa h on of adaon s ld owad h don of oon -as b h angl /γ so h pojo-p adaon s d fo h off-as loaon. In fola 5 h ngal Id s h oal ng ad awa b h adaon o K 8 K L β γ Id 3 L β γ 3 L β γ 8 3 ε wh K D / s a so-alld ndlao paa o spl a K-fao. Toal on ad awa b h adaon os o 6

7 p o o L 3 ε v 8 K β γ 9 3 And h oal angla on ad awa b h adaon s o M [ ' p] [ ' p ] ' Kp 0 wold lk o phas h OAM ad awa b hgh haons as s psnd n Fg.. No ha h fs haon s no shown n Fg. s Fg.a. Sn h nns of adaon a h fs haon s nd a h as dos no dpnd on h nsananos poson of lon hn dos no a OAM. Th qan nb assoad wh h angla on an b psnd as follows M j d h h I an b sn n Fg. ha sn h so of adaon s load a h off-as poson a h l h adaon s ang awa a lna on and hn h angla on. Spl spakng h lon ob s dasng s ads. On an s ha b hangng h K fao s possbl o hang h angla on of adaon as o and o haons bo nvolvd n a poss of adaon. Th nns of adaon ad awa b h haons s dsbd b h fola [5] [6] [9] di n ω ϑ β nβ Snϑ Cos n 0 nβ Snϑ β J J 3 n n dο β osϑ ϑ β Cosϑ Sn β Cosϑ wh β β β β /. On an s f β 0 hn h fola 5 onds wh Sho s fola [33]. aphs of / ω di / 0 n dο fo h low haons n and n and n5 fo K0.7 psnd n Fg.. In h oodna ss ovng wh h avag vlo of lon h lon ovs b a la ajo wh an nsan ads β ε / Fg.. Th nns of adaon b h lon s gvn b saon of ov all haons and ngaon ov h pola angl so gvs agan 3 3 v γ I β γ 5 3 wh h ps dno h qans allad n h ovng ss of fn: v s h lon s vlo β v / γ ε / / β ε s h lon s nsan ng. / s h lassal lon ads. No also ha h ads s an nvaan nd Lon ansfoaon. 3 Sl spakng onl h pa of adaon wh n> as o h angla on owv fo K h nns of adaon a h fs haon s< 0% so fo valaon hs dffn gh b ngld. 7

8 q a aon n q q b aon n aon n5 Fg. Th angla dsbon of nns of adaon on haons n a ovng fa. In a ovng fa h lon s spnnng along h l wh a ads D K / γ adang SR whn h on whh angla opnng s dfnd b h loal gaa-fao n h ovng fa. So on an s ha h so of adaon n a ovng fa s load a hs ads and so h Ponng vo assoad wh lon has obal angla on wh sp o h as. 8

9 On pla assoad wh hlal oon n an ndlao s ha h obal on of lon nass whl h ng of h lon dops down. Indd as h ansvs on an b psnd as D K M p β γ K 4 γ wh was sd h dn Lon ansfoaons on an s ha K β γ and h fa ha p and a nvaans nd M ~ / γ. I s la ha h s of h so of ndlao adaon an no b lss han h ads of hl D K / γ wha fos sag of bas wh hghs γ and ndlao wh shos pod opang a low K. So fo h RL havng γ0 4 K and pod h nal ffv spo-s of h so wll b no lss han D K / γ / 3 µ whl fo h LCLS hs os o ~/6 µ. I s la also ha opaon wh low K-fao shold fo opaon wh hgh n o opnsa h loss of h phoon fl ~K. LICAL NDLATOR Paall an hlal ndlao whh s abl o gna h fld wh K ~ s sabl fo gnaon of hgh haons. B h pfn s gvn o h ons wh a possbl o hang polaaon fo apl h SC on dsbd n [8] allowng as hang of K and polaaon. Fg. SC hlal ndlao wh possbl o hang h K fao and polaaon. Th ols n an ndlao fo Fg. wond abov h Copp b wh s-w sands sk ogh n fla SC abl. A pod of wndng s 4.5. An o da s 0 and an nn da la fo h ba s Ø nn 8. Th don of n n h ospondng o-la ol s shown b h aows [8]. h sh ap h SC ndlao an gna K~.5 fo hs pod. Fo h lag pod h ahvabl vals of K fao a gowng ponnall fo h fd ap. 9

10 DISCSSION Th fa ha loagn adaon fo an hlal ndlao a hgh haons as h obal angla on was no ald fo an as. Th phss planaon of appaan of angla ona fo h phoons adad b lon n a hlal ndlao s la howv. I s assoad wh h fa ha h lon s ovng along a hlal ajo wsd aond h as and adas whl h pon of adaon s shfd awa fo h as. Sn n pns wh ndlao adaon pfod n h pas was no ald ha hgh haons a OAM s appopa o vw sls oband n sh pns spall h ons fo ansons bwn h sas wh dffn angla ona. Podon of polad posons wh hgh ndlao haons also nds vson. In h poson podon sh wh ndlao fo ILC h nal sggson was o opa a low K-fao K<0.4 [9] wh h onn of hgh haons s nal vn a K~0.7 onl 50% of adaon s ad b hgh haons. Manwhl h basln of ILC dals wh ndlao havng K~0.9 [3] wh onn of hgh haons s donang. As h lon-poson pa shold a awa h obal on h ngs shold b los o ah oh. I ans ha h ng of poson lon shold b abo ~half of h ng of qana whn naow agns. Thfo h polad poson podon b hgh ndlao haons shold b sppssd b h phas vol and b polaaon bhavo as h posons wh half ng a ~50% polaaon a os. SMMARY Th adaon n a hlal ndlao a hgh haons s donanl ang obal angla on whh shold b akn no aon. Phsal anng of hs phnonon s la: h adang lon ovs along h hl of nono ads so h so of nsananos adaon s shfd fo h as. Ths fa was gnod n all pns don wh hlal ndlaos n h pas. a sggsng vson of all sls oband wh sag of hgh ndlao haons gnad n a hlal ndlao. In addon h ho of OAM phoon naon n a onaon Copon sang and lon-poson pa podon also qs vson. Th dfnon of bghnss bllan fo h sos wh hlal ndlaos shold b odfd also. In onlson w no ha sag of phoons wh OAM gh opn nw was n obanng hghl polad lons fo phooahods. On of Ahos AM hanks vgn Bssonov fo sfl dsssons. RFRNCS [] R. A. Bh Phs. Rv [].l Th Qan Tho of Radaon Ofod 954. [3] R.Ro B.P.Nga Nla Phss John ll 967. [4] V.B. Bssk.M.Lfsh L.P.Pavsk Qan lodnas vol.4 of Cos of Thoal Phss Bwoh-nann 97. 0

11 [5] L.Alln M..Bjsbgn R.J.C.Spw J.P.odan Obal Angla Mon of Lgh and Tansfoaon of Lagg-assan Las Mods Phs.Rv. A Vol.45 No Jn 99. [6].sng A.Manll S.Rh J.Rosnwg Longdnal Dspson of Obal Angla Mon Mods n gh-an f-lon Lass Phs.Rv.Sp.Tops-Alaos and Bas M.Dnajsk Th Twsd Phoon Assoad o p-an Fo Manfolds Jonal of o and Phss Molna-Ta J.Tos LTon Twsd Phoons Na Phss vol 3 Ma 007. [7] A.ffn D..Snok S.Snga dos Bos-nsn ondnsaon Cabdg. Pss 995. [8] A.Pon al. Phooonaon wh Obal Angla Mon Bas Ops pss Fb. 00/Vol 8 No 4. [9].D.Jnsha V..Sbo naon of gh-ng Phoons wh Lag Obal Algla Mon b Baksang PRL [0] K. Van Bbb al. Poposd pn o Pod and D Lgh Psdosalas Phs.Rv.L.59: ; C. Robllad al. No Lgh Shnng Thogh a all Phs. Rv. L. 99: ; A. Cho al. Sah fo Aon-lk Pals sng a Vaabl Basln Phoon Rgnaon Thnq Phs.Rv.L.00: ; A.Afanasv.al. pnal L on Opal-Phoon Coplng o Lgh Nal Sala Boson Phs.Rv.L ; K.h al. Nw ALPS Rsls on ddn-so Lghwghs Phs.L.B689: [] L. Okn Ls of lodnas: Paaphoons? Sov.Phs.JTP 56:5098; A.Afanasv.al. Nw pnal L on Phoon ddn-so Paaphoon Mng Phs.L. B [] D. Bdk D.F. Kball S.M. Rohs V.V. Yashhk M. Zoloov Snsv Magno Basd on Nonlna Magno-Opal Roaon Phs.Rv. A [3] A.T.Ngn D.Bdk D.DMll M.Zoloov Sah fo Pa Nononsvaon n Ao Dspos Phs.Rv. A [4] P. Skv panal Tss of h Invsbl Aon Phs.Rv.L.5:45983 a-bd.5:695984; A.A. Ansl Aon <---> Phoon Osllaons In A Sad Magn Fld Yad. F ; L. Maan R. Pono and. Zavan ffs Of Nal Masslss Spn Zo Pals On Lgh Popagaon In A Magn Fld Phs. L. B ; R. Caon al. Sah fo nal asslss wakl opld pals b opal hnqs Phs.Rv.D47: ; C.Y.Sal al. An Anoalos Cva pn 840 BNL Ap [5] D.F.Alfov Y.A.Bashakov..Bssonov ndlao Radaon Sov. Phs. Th. Phs. Vol.8 No p [6] D.F.Alfov Y.A.Bashakov K.A.Blovnsv..Bssonov Th ndlao as a So of loagn Radaon Pa.Al

12 [7] B.M.Knad A Sho-Pod lal ggl as an Ipovd So of Snhoon Radaon Jonal of Appld Phss Vol.48 No 7 Jl 977. [8] A.A.Sokolov I.M.Tnov al. Ivsa Vov Fka N ; Zs. F. Phs [9] A.A.Sokolov I.M.Tnov Rlavs lon Mosow Naka DK [0] S.Sga. al. X-MCD wn-lal ndlao ba ln BL5S of Spng Jonal of Magns and Magn Maals vol. 33 sss - Jl 00 p.60. Y.Saon. al. Pfoan of a V gh Rsolon Sof X-a Baln BL5S wh a Twn-lal ndlao a SPng-8 Rv. S. Ins ; do:0.063/ S.Nas gh Pss pns wh Snhoon Radaon pfn Inaons Vol 3 Nbs DOI: 0.03/A: [] I.V.Sokolov Th Angla Mon of an loagn av Sadovsk ff and h naon of Magn Flds n a Plasa Phss-spkh Vol. 6 No0 O. 99 hp://fn./fn9/fn9_0/rssan/90g.pdf. [] M.Abaha D Dhpls ds Lhs Phskalsh Zshf XV 94 pp [3] C..Dawn Nos on h Tho of Radaon Po. R. So. Lond. A [4] J..Sons M.J.ann Sas avs and Phoons: A Modn nodon o Lgh Addson-sl 970. [5].K..Panofsk M.Phllps Classal l and Magns Addson- sl PC In956. [6] J.Jakson Classal lodnas J.l&Sons In [7].F.Fald On h longdnal Coponn n Lgh Phlosophal Magan and Jonal of Sn Vol. XLII No. CCLVI Jl-Db 896 pp [8] A.Mkhalhnko SC ndlao wh h Possbl o Chang s Sngh and Polaaon b Fdng Cn TP48 PAC NY 0. [9] A.Mkhalhnko ILC ndlao-basd Poson So Tss and Slaons ZAB0 PAC07 Albqq NM 007 Po. pp [30] A.M.Yao M.J.Padg Obal angla on: ogns bhavo and applaons Advans n Ops and Phoons [3] A.Mkhalhnko 3D loagn Fld. Rpsnaon and Masns CBN 95-6 Conll lson Lab [3] ILC Rfn Dsgn Rpo ILC RPORT vol III-Alao piii-48. [33].A.Sho On h lon Tho of Ma and on Radaon Phl. Magan Fb. 907; loagn Radaon Phl. Mag

9.4 Absorption and Dispersion

9.4 Absorption and Dispersion 9.4 Absoon and Dsson 9.4. loagn Wavs n Conduos un dnsy n a onduo ollowng Oh s law: J Th Maxwll s uaons n a onduo lna da should b: ρ B B B J To sly h suaon w agu ha h hag dsaas uly n a aoso od. Fo h onnuy