Angular Momentum Conservation of a Beam on the Interface between Two Homogeneous Media Liu Bao-Ying *

Transcription

1 Phoonic Ooelecronic (P&O Volume 4 6 wwwo-ournalorg Angular Momenum Conervaion of a Beam on he nerface beween Two Homogeneou Media Liu Bao-ing Elecronic nformaion Elecrical Engineering College Shangluo Univeri Shaani China lbl@6com Abrac The change of angular momenum of a ind of uniform olaried ligh beam refleced on he inerface beween wo homogeneou media ha been inveigaed according o claical elecromagneic heor i found ha he oal angular momenum conerve in he normal direcion of he inerface in he araial aroimaion he orbi angular er uni energ relaed o he orbi quanum number ha roaional invariance ranmiion invariance while he in angular change wih reflecion on he inerface Keword Vecor beam; eflecion; Angular momenum; Conervaion nroducion According o Mawell elecromagneic heor elecromagneic field have energ linear momenum angular momenum (AM The in angular momenum (SAM of quanum hoon of ligh field had been oberved earl in 96(Beh 9 The orbi angular momenum (OAM of Laguerre-Gauian (LG beam had been inveigaed in he cae of araial aroimaion b Allen hi co-reearcher in 99 i wa found ha he AM of LG beam can be earaed ino SAM OAM he SAM i onl relaed o he in quanum number he OAM i onl relaed o he angle quanum number reecivel (Allen e al 99 Laer he inveigaed he non-araial LG beam found ha he AM could no be naurall divided ino wo ar a he ame in he araial aroimaion (Barne e al 994; Allen e al 996; Allen e al 999; Allen e al O Neil hi co-reearcher found ha he SAM OAM of a non-araial beam la diinc role in he ineracion wih microcoic birefringen aricle raed off he beam ai (O Neil e al Furhermore he SAM had been found o be convered ino OAM b an inhomogeneou anioroic elemen or a high numerical aerure (Marrucci e al 6; Zhao e al 7; Heharia e al 4; ueca e al Thee eerimenal reul demonrae ha he SAM OAM of a nonaraial beam are differen in naure on one h are conneced omehow o each oher on he oher A he ame ime eole draw much aenion on he change of AM of he beam refleced on he inerface beween wo homogeneou media Bu he reul of he mber-fedorov (F effec relaed o SAM are conroverial u unil now (Onoda e al 4; Blioh e al 6; Hoen e al 8; Liu e al 8; Blioh 9; Li 9; Kong e al he focu of he conrover i he conervaion of he AM of he beam refleced on he inerface beween wo homogeneou media Hence he reearche abou he change of AM on he inerface beween wo media conribue o under he difference relaion of SAM OAM of beam n hi aer he AM of one ind of uniform olaried beam i change conervaion on he inerface beween wo homogeneou media are udied in deail i how in he cae of araial aroimael e( il he OAM er uni energ in he roagaion direcion decided he facor of wavefron hae ha roaional invariance ranmiion invariance while he SAM change in he roce of reflecion bu he oal AM of he beam conerve in he normal direcion of he inerface Angular Momenum of Monochromaic Beam in Free Sace Conider an arbirar elecromagneic beam in free ace elecric vecor in real ace can be ereed a an

2 wwwo-ournalorg Phoonic Ooelecronic (P&O Volume 4 6 inegral over he lane-wave ecrum where E( i he elecric vecor in recirocal ace E ( r e[ ( ] ( E( i r d c c ( e e e i he wave vecor i angular frequenc i ermiivi of vacuum i ermeabili of vacuum According o claical elecromagneic heor he ereion of energw in angular momenum S orbi angular L of he beam (Jacon 999; Li 9are given a follow So he angular momenum of he beam i W E ( E(d ( S E ( E( d i L E ( E( d i J L S E ( E( d E ( E( i d (5 i Conider a monochromaic wave roagaing in oiive direcion i elecric vecor aifing Helmhol equaion E(r E(r comling wih ranvere field condiion E(r can be ereed in erm of he angular ecrum a dd E(r E( e( i r (6 e( where he ime deendence E ( i imlied omied i angular ecrum ha i he amliude of elecric vecor in momenum ace(li 9 Subiuing Eq (6 ino Eq( ( (4 E ( i abbreviaed a E we ge he energ W he comonen of in angular momenum S orbi ( (4 angular momenum L of he beam a follow A we now W (7 E E d d e E E E E d d S i (8 E E L e E E d d i (9 E ( ha onl wo indeenden olariaion ae Denoing reecivel b he wo E E E E E E orhogonal linear olariaion ae we have E where amliude of he olariaion ae reecivel are heir reecive uni vecor Leing have e e e e e e ( are real number we E m ~ E( ( (

3 Phoonic Ooelecronic (P&O Volume 4 6 wwwo-ournalorg here m ( i a E ( ono he D Careian coordinae i( amliude eenion mari which eend he wo-form ~ i he normalied Jone vecor aociaed wih he angular ecrum can be regarded a olariaion elliici(li 8 The amliude ( calar E E( E( e( il i aumed here o include a hae facor in circular clindrical em E( where i quare inegrable i harl eaed a l i he angular quanum number relaed o he wavefron hae facor e( il Obvioul he olariaion ae can no be deermined b ranvere field condiion here ei a degree of freedom ha decide he vecor roerie of beam We can rereen uni vecor b inroducing a uni vecor a ( ( Leing lie in he lane o mari m ae he following form wih he roagaion ai e co e in he Auming m co ( in co ( co in in ( ( in co co in in Eq(he elecronic vecor ereed in Eq( can be rereened a E( [ ( e e ( e] E( (4 Thi ind of beam can be called uniform olaried beam ha he olariaion of he beam in cro ecion erendicular o he oical ai i no relaed o ranvere comonen of he wave vecor Linear olaried circular olaried fundamenal mode Gauian beam belong o hi ind of beam(li 8 Subiuing Eq (4 ino Eq (7 (8 (9 L l E ( i abbreviaed a E we have ( W E d d (5 S E dd (6 E dd ( E dd (7 Obvioul in he roagaion direcion he SAM i onl deenden on he olariaion elliici bu he OAM are boh deenden on he angular quanum number l he olariaion elliici he all relaed o beam rucure A he ame ime he AM er uni energ in he roagaion direcion i alo a conan a

4 wwwo-ournalorg Phoonic Ooelecronic (P&O Volume 4 6 J W l S L l W W wheher he beam i araial or no n he cae of araial aroimaion he OAM i onl relaed o l we have ha he AM can be naurall divided ino wo ar ha he SAM i onl relaed o Change Conervaion of he Angular Momenum of he Beam efleced on he nerface beween Two Homogeneou Media Conider he ligh beam ereed b Eq(4 roagaing from a homogeneou medium wih refracive inde n o anoher medium wih refracive inde n a deiced in Fig where Z i he inerface beween he wo media he laboraor reference frame i denoed b Z he reference frame aociaed wih he inciden beam i denoed b he reference frame aociaed wih he refleced beam i denoed b r r r he reference frame aociaed wih he ranmied beam i denoed b reecivel n he laboraor reference frame Z he wave elecronic vecor of he beam a ereed b Eq(4 in he reference frame can be rereened a follow r r n n O Z FG : EFEENCE FAMES Z AND Z ASSOCATED ESPECTVEL WTH THE NCDENT BEAM THE EFACTED BEAM AND THE LABOATO FO THE EFACTON AT AN NTEFACE BETWEEN TWO DFFEENT DELECTC MEDA N AND N Here D ( in co Z D( co in Z (8 Z in D ( m ~ Ee( il ~ E e( il co (9 EZ i roaion mari i he angle beween oical ai of he beam he normal of he wo media n he cae of araial aroimael we ge he energ he in angular momenum he orbi angular momenum a i W E d d ( 4

5 Phoonic Ooelecronic (P&O Volume 4 6 wwwo-ournalorg S ( E E E E d i d co E d d ( L i EZ EZ EZ EZ So he angular momenum of a ingle hoon i J W i d d l co l E d d ( ( When he beam i refleced a he inerface beween wo homogeneou media he angular ecrum of he E ( E( refleced beam can be ereed a r where i he reflecion coefficien e e olariaion mari are he reflecion coefficien for reecivel A he ame When he beam i refleced a he inerface beween wo homogeneou media he T T E ( TE( angular ecrum of he ranmied beam can be ereed a where T i he ranmiion coefficien mari reecivel T T are he ranmiion coefficien for olariaion To he reflecion beam he wave vecor vecor elecric field can be ereed a r r co in in co (4 co co ( co (5 E co ( ~ E e( il co ( in The energ he comonen of in angular momenum obi angular momenum of he beam are ereed a follow S n he cae of oal reflecion W r ( E d d (6 ( ( co ( E d d i co E d d (7 L l ( E d d (8 e( i e( i are he iniial hae of he reflecion coefficien for olariaion reecivel Obvioul he olariaion ae of he reflecive beam change relaivel o he inciden beam becaue he hae difference beween he wo reflecion coefficien which change wih he inciden angle f he hae difference hould be ignored in he condiion of aroimaion we 5

6 wwwo-ournalorg Phoonic Ooelecronic (P&O Volume 4 6 have S r E d d Wr W i J J n he cae of arial reflecion l co are all real number hence n he cae of oal reflecioni i ea o find ha J l co (9 W ( A he ame ime for he ranmied beam i wave vecor elecronic field vecor can be ereed a Baed on Eq( Eq( we have in co ( Z co in Z E Z ( T co T co co in T in ( T co T co in T co ( T co T co ~ E e( il in T in W ( T T E d d ( T T SZ co E d d ( LZ l ( T T E d d (4 From Eq(((4 i i ea o ge ( Z JZ l T T co (5 W ( T T n A we now he raio of inegral elemen beween medium (he region of inciden refleced beam medium n (he region of ranmied beam i inciden beam n co T n co T n n co co where i he incidence angle of he oical ai of i he correonding refracion angle According o Frenel formula here have T T n co T n co o he normal of he inerface are conervaion in he cae of ar reflecion Eq((9 (5 We have Obvioul he equaion n co T n co here Hence he energ he angular momenum in he direcion of W W W i r J J J Z From n co T Z (6 n co Z are imoible in he cae of ar reflecion for ingle hoon 6

7 Phoonic Ooelecronic (P&O Volume 4 6 wwwo-ournalorg Concluion The AM of he beam can be read ino SAM OAM he SAM i onl relaed o he olariaion of he beam wheher i i araial or no he OAM i no onl relaed o aial diribuion bu alo he olariaion i i onl relaed o aial diribuion in araial aroimaion The AM change wih beam refleced on he inerface beween wo media bu conervaion in normal direcion in he araial aroimaion he OAM i roaion invariance read invariance while he SAM i no For he ingle hoon he angular momenum of inciden reflecion ranmiion beam in he normal direcion of he inerface could no conerve reecivel onl aif Eq(6 he reaon abou he conrover of he mber-fedorov (F effec i baed on differen conervaion relaion of he AM er uni energ of he beam refleced on he inerface beween wo homogeneou media ACKNOWLEDGMENT Thi wor wa uored in ar b he Naional Naural Science Foundaion of China (Gran 845 Shangluo Science Technolog Bureau Diciline Program (SK4-- EFEENCES [] A Beh Mechanical Deecion Meauremen of he Angular Momenum of Ligh Ph ev 5 5(96 [] L Allen M W Beierbergen J C Sreeuw J P Woerdman Orbial angular momenum of ligh he ranformaion of Laguerre-Gauian laer mode Ph ev A (99 [] S M Barne L Allen Orbial angular momenum nonaraial ligh beam O Commun 67(994 [4] L Allen V E Lembei M Babier Sin-orbi couling in free-ace Laguerre-Gauian ligh beam Ph ev A 5 97(996 [5] L Allen M J Padge M Babier The orbial angular momenum of ligh Prog O 9 9(999 [6] L Allen M J Padge The Poning vecor in Laguerre-Gauian beam he inerreaion of heir angular momenum deni O Commun 84 67( [7] A T O Neil Mac Vicar L Allen M J Padge nrinic Erinic Naure of he Orbial Angular Momenum of a Ligh Beam Ph ev Le 88 56( [8] L Marrucci C Mano D Paaro Oical Sin-o-Orbial Angular Momenum Converion in nhomogeneou Anioroic Media Ph ev Le (6 [9] Zhao J S Edgar G D M Jeffrie D McGloin D TChiu Sin-o-Orbial Angular Momenum Converion in a Srongl Focued Oical Beam Ph ev Le 99 79(7 [] D Heharia M P van Eer W Löffler Saial coherence he orbial angular momenum of ligh in aronom Ph ev A 9 68 (4 [] Juliu ueca Viačelav Kudriašov e A u Gedimina Jueliūna Tranfer of orbial angular momenum of ligh uing wo-comonen low ligh Ph ev A ( [] M Onoda S Muraami N Nagaoa Hall Effec of Ligh Ph ev Le 9 89(4 [] K u Blioh u P Blioh Conervaion of Angular Momenum Tranvere Shif Sin Hall Effec in eflecion efracion of an Elecromagneic Wave Pace Ph ev Le 96 79(6 [4] Hoen P Kwia Obervaion of he Sin Hall Effec of Ligh via Wea Meauremen Science 9 787(8 [5] Bao-ing Liu Chun-Fang Li The mber Fedorov hif of araial ligh beam O Commun 847(8 [6] K Blioh Commen on The mber-fedorov hif of araial ligh beam O Commun 85(9 [7] Chun-Fang Li el o Commen on The mber Fedorov hif of araial ligh beam O Commun 85(9 [8] Ling-Jun Kong Sheng-ia Qian Zhi-Cheng en i-lin Wang Hui-Tian Wang Effec of orbial angular momenum on he geomeric in Hall effec of ligh Ph ev A ( [9] J D Jacon Claical Elecrodnamic rd ediion John Wile & Son nc

8 wwwo-ournalorg Phoonic Ooelecronic (P&O Volume 4 6 [] Chun-Fang Li Sin orbial angular momenum of a cla of nonaraial ligh beam having a globall defined olariaion Ph ev A 8 684(9 [] C-F Li ereenaion heor for vecor elecromagneic beam Ph ev A 78 68(8 8