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1 Suppoting Infoation fo Spin annihilations of and spin siftes fo tansvese electic and tansvese agnetic waves in co- and counte-otations Hyoung-In Lee,3 and Jinsik Mok Addess: Reseach Institute of Matheatics, Seoul National Univesity, Seoul, Koea, Dept. of Matheatics, Sunoon Univesity, Asan, Chungna Koea and 3 School of Coputational Sciences, Koea Institute fo Advanced Study, Seoul Koea Eail: Hyoung-In Lee - hileesa@nave.co Coesponding autho Matheatical deivations This Suppoting Infoation consists of 0 sections. Most of the deivations and discussions ae caied out twice: fist fo the co-otational case, and second fo the counte-otational case. The pupose of this Suppoting Infoation is to keep all the inteediate steps as detailed as possible, because one is liable to coit istakes in dealing with coplex paaetes and vaiables and thei coplex conugates. In paticula, the Sections S6-S8 ae tageted at finding the obital pat of angula oentu with i i and E E E E x e which we believe has neve been explicitly attepted. H H H H x e in the pola coodinate, i i S. Fundaental Foulas Conside the Maxwell's euations i E k H and i H k Hee, all the field vaiables follow the popotionality exp it E with k k. fo both otational cases. Fo siplicity, conside dielectic edia without loss. Both H, H ae thus elated to H S

2 though H i E and H i E fo the TE ode. Fo the TM ode, both E, E ae expessible likewise in tes of E though E i H and feuency doain by E i dh d. Let us goup these fou elations in the E E TE : H i, H i. (S.) H H TM : E i, E i Fo the co-otational case, the cobined electic and agnetic fields ae hence given below. E H f e, f e, f e i i i h e, h e, h e i i i. (S.) Fo this co-otational case, E. (S.) tanslate into df TE : h f, h i d. (S.3) dh TM : f h, f i d Fo the counte-otational case, the cobined electic and agnetic fields ae witten below. E H f e, f e, f e i i i h e, h e, h e i i i. (S.4) Fo this counte-otational case, E. (S.) tanslate into S

3 df TE : h f, h i ; d. (S.5) dh TM : f h, f i d Note the diffeence that h f fo the co-otational TE ode in E. (S.3), wheeas h f fo the counte-otational TE ode in E. (S.5). But, the othe thee elations eain the sae. Fo a geneic coplex vaiable A, we ake use of a set of the siple coplex-vaiable identities I A I A I ia Re A, AA Re A, ReiA I A, and aong othes. In addition, we need to take caution in taking the following steps h i df d i df d fo h idf d f i dh d. Meanwhile, f f f and. Siilaly, f i dh d h h h. fo We ecall that the axial pofiles f F and h NF have been constucted to satisfy the continuity elations E E and H H acoss the thin laye at R kr. F F J J H H R, R, R R. (S.6) Hee, J and H the fist kind, espectively. ae the Bessel functions of fist kind and Hankel functions of Thanks to these noaliation schees, it is specified that F R F R In addition, F F, although J J and. S3

4 H H. This fact F F will be epeatedly eployed in, dealing with the counte-otational case. We will also utilie the notation that F F. In addition, we define the fist-ode gadient function function K in the adial diection. G and the second-ode gadient d d G ln F F d, (S.7) F d K d F d d d d d d F d ln F. (S.8) To ephasie that these gadients take non-eo value only fo spatially inhoogeneous fields, we ay call these gadients "inhoogeneity gadients". In ode to copute G and K, we need to have evaluate deivatives of the Bessel functions of fist kind and Hankel functions of the fist kind. Fo the Bessel functions, the ecusion elations ae dj0 J d d J0 dj d d dj J J d d J dj dj d d d, (S.9) J, 0. (S.0) The Hankel functions follow a siila set of ules fo deivatives. In addition, we need to pay special attentions to the aiuthally stationay case with 0. Figue S. pesents the adial pofiles of G plotted against k c with S4

5 R kr. In paticula, the aiuthally stationay state with 0 is dawn in a solid black cuve. Panels (a) and (b) display ReG and IG the fact that, espectively. In paticula, I G 0 in the inteio will play a doinant ole in chaacteiing vaious wave popeties such as angula oentu and spins. Figue S.: The adial pofiles of the fist-ode gadient function G with the thin laye located at R kr and fo 0,,,3,4. (a) Real pats. (b) Iaginay pats. Figue S.: The adial pofiles of the second-ode gadient function K with the thin laye located at R kr and fo 0,,,3,4. (a) Real pats. (b) Iaginay pats. S5

6 Figue S. displays a siila pai fo K. In shot, all the fothcoing foulas end up with expessions involving the functions F, G, and K (S.6)-(S.8). In siilaity to IG 0, we notice povided in Es. I K 0 in the inteio, which will be useful fo the fothcoing E. (S8.4) in evaluating the adial coponent of the obital angula oentu. In addition, we will eploy the "helicity" and "chiality". I, I exp i. (S.) Of couse, fo expi. S. Enegy Density The enegy density defined by 4 4 w E H is fully witten to be w f e f e f e h e h e h e 4 i i i i i i,,,, f f f h h h 4. (S.) By eithe E. (S.3) o E. (S.5), E. (S.) is cast into the following foula in tes of f, h. dh df w h f f h 4 d d (S.) The final step is to call upon the pai of solutions f F and h NF find S6 to

7 N w G F 4. (S.3) Fo anothe viewpoint, w is sepaable into w TE and w TM fo the TE and TM odes. w w TE w TM. (S.4) TE We then obtain 4 w G F and eains the sae fo both otating cases. w TM N wte. It is obvious that w Figue S: (a) Enegy density w TE fo the TE waves. (b) The scaled enegy density 4 TE w fo the TE waves. Both ae plotted against k c with the data R kr 5. Cuves ae dawn with vaying aiuthal index 0,,,3,4. Figue S(a) displays the negy density w TE fo the TE waves plotted against k c. The data is R kr 5. Cuves ae dawn with vaying aiuthal index 0,,,3,4. It tuns out that the diffeent cuves ae difficult to esolve aong the. S7

8 Theefoe, Fig. S(b) shows the scaled enegy density 4 w instead of ust w TE. We find that on the w 4 -scale, the effects of vaying ae easily discenible. TE TE S3. Optical Chiality The optical chiality C n I E H is fully expessed as follows fo the cootational case. C I f e, f e, f e h e, h e, h e i i i i i i. (S3.) Upon scala ultiplication, the exponential factos e i ae cancelled so that C I f h f h f h I f h f h h f. (S3.) By E. (S.3), E. (S3.) is cast into the following foula only in tes of f, h. dh df C I h f i i h f d d. (S3.3) dh df I h f h f d d Recalling that I f, the final step is to call upon the pai of solutions F and h NF to aive at N C I F G, (S3.4) S8

9 C C, 4 C N F G. (S3.5) As expected, the chiality coefficient is positive fo all, naely, C 0. In addition, C does not explicitly depend on. Most ipotantly, C density w in E. (S.3) in its adial dependence. is popotional to the enegy sgn C C Figue S3. pesents contou plots of the unscaled C in (a), and the scaled on the kx, ky -plane fo the co-otational case. The data ae R kr and i 5. We clealy obseve only a adial inhoogeneity. The white egion in panel (a) signifies highe values of optical chiality. Theefoe, the scaled one in panel (b) offes a bette view ost of the tie. Figue S3.: (a) A contou plot of the optical chiality C. (b) A contou plot of the scaled 5 optical chiality sgn C C. Both ae plotted on the, case. The data ae R kr, 3., and i kx ky -plane fo the co-otational Fo the counte-otational case, the optical chiality C n I E H expessed as follows. is fully S9

10 C I f e, f e, f e h e, h e, h e i i i i i i. (S3.6) Even upon siplification, the aiuthal angle suvives explicitly such that C I e f h e f h e f h i i i I e f h f h h f i. (S3.7) By E. (S.5), E. (S3.7) is cast into the following foula in tes of f, h. i dh df C I e h f i i h f d d i dh df I e f h f h d d. (S3.8) The final step is to call upon the pai of solutions f F and h NF aive at to C C, 4 C N F G. (S3.9) Hee, is the chiality defined in E. (S.). As expected, the aiuthal te in C fo this counte-otational case changes its sign, in copaison to (S3.5) fo the co-otational case. in E. Figue S3. shows a contou plot of C, tanslated into, plane fo with the data R kr and i because of the facto cosi with. C kx ky on the kx, ky -. Hee, the patten epeats twice S0

11 Figue S3.: A contou plot fo the scaled optical chiality 0. sgn C C on the, plane fo the counte-otational case. The data ae R kr,, and kx ky - i. S4. Poynting Vecto Poyn Fo the co-otational case, the Poynting vecto P n ReE H be is fully witten to n P f e f e f e h e h e h e Poyn i i i i i i Re,,,,. (S4.) Poyn Poyn Poyn Poyn As a esult, we obtain the coponent-wise elations fo P P e P e P e as follows. Poyn n P Re f h f h Poyn n P Re f h f h. (S4.) Poyn n P Re f h fh S

12 Note that i e cancels out e i following elations in tes of in evey tes. By E. (S.3), these thee ae cast into the f, h. Poyn n dh df P Re i h i f d d Poyn n P Re f f h h. (S4.3) Poyn n df dh P Re h i i f d d With the help of vaious elations fo coplex vaiables, these thee ae futhe siplified. Poyn n df dh P I f h d d Poyn n h f P. (S4.4) Poyn n df dh P I h f d d The final step is to call upon the pai of solutions f F and h NF find to Poyn n df df P I F N F d d Poyn n N P F. (S4.5) Poyn n df df P I N F. F d d By way of the helicity I, two of E. (S4.5) get futhe down to S

13 P F G Poyn n P N F ReG Poyn n N I. (S4.6) The taectoies fo the Poynting-vecto flows ae evaluated fo the following diffeential fos. Poyn Poyn d P d k P,. Poyn Poyn (S4.7) d P d n P We then ewite these two elations into the following diffeential taectoies with as a paaete, based on Es. (S4.5) and (S4.6). d d IG d k N d k N Re d n N ReG I d n N IG G I. (S4.8) Poyn Fo the counte-otational case, the Poynting vecto P n ReE H witten to be is fully n P f e f e f e h e h e h e Poyn i i i i i i Re,,,,. (S4.9) Poyn Poyn Poyn Poyn As a esult, we obtain the coponent-wise elations fo P P e P e P e as in (S4.). Note that i e cancels out e i cast into the following elations in tes of f, h. in evey tes. By E. (S.5), these thee ae S3

14 Poyn n dh df P Re i h i f d d Poyn n P Re f f h h. (S4.0) Poyn n df dh P Re h i i f d d Notice the sign change in the expession fo Poyn P and Poyn P (in ed colos) in the above euation. With the help of vaious elations fo coplex vaiables, these thee ae futhe siplified. Poyn n df dh P I f h d d Poyn n h f P. (S4.) Poyn n df dh P I h f d d The final step is to call upon the pai of solutions f F and h NF find to Poyn n df df P I F N F d d Poyn n N P F. (S4.) Poyn n df df P I N F. F d d By way of the helicity I, two of E. (S4.) get futhe down to S4

15 P F G Poyn n P N F IG Poyn n N I. (S4.3) Poyn It should be noticed that P ReG Poyn in E. (S4.6) and I espectively fo the co- and counte-otational cases. P G in E. (S4.3) The taectoies fo the Poynting-vecto flows ae evaluated fo E. (S4.7). We then ewite these two elations into the following diffeential taectoies with as a paaete, based on Es. (S4.) and (S4.3). d N d N IG d k N I d n N d k N I d n N G I. (S4.4) S5. Enegy Flow Density and Poynting Vecto Via k c and the Maxwell's euations i E k H and ih k E with k k, we have the total enegy flow density (FD) tot P as follows. 4n k k tot P I E E H H I E i H H i E 4n I 4n 4n ReE H E H ReE H P 4n n ie H ih E I ie H ie H S5 Poyn. (S5.)

16 As a esult, the total enegy flow density (FD) is identical to the Poynting vecto (PV), naely, P tot Poyn P fo dielectic edia. This fact holds tue to both otational cases. Meanwhile, the decoposition of tot P into its obital FD O P and spin FD handle in the Catesian coodinates by use of the epeated indices. S P is easie to I, (S5.) O nkp I E E H H I S nkp I E E H H. (S5.3) tot O S In ode pove the identity P P0 P0, let us pove the following identity fo the electic field fist. I I I E E E E E E fist. To pove E. (S5.4), conside E E lk E E E E e l x k n lk kneen el kn kl el x x E E e E E E E e l n nl n l l l l x x EE. (S5.4). (S5.5) Note that i and lk ae the Konecke and Levi-Civita sybols, espectively. Besides, e i is the unit basis vecto of the Catesian coodinates. We ade use of the identity kn kl ln nl. Hence, taking the iaginay pats on both sides of E. (S5.5) leads to S6

17 I I E E E l E E El el x. (S5.6) E E l I E I I l E el E el El el x x x Fo dielectic edia without space chages, we have E x E 0. As a esult, E. (S5.6) is educed to E I I 0. (S5.7) l E E E e l x On the othe hand, conside next E E. E n E E E E lkel E kn lkel k x. (S5.8) E E E E E e E e E e E e n n l kn kl l l n nl l l l x x xl x Because of the defining elation E E E E x e, the cobination of Es. (S5.7) i i and (S5.8) ushes us to the desied identity in E. (S5.4). In siilaity to E. (5.4), we could pove its agnetic countepat. I I I H H H H H H. (S5.9) tot O S As a conseuence of Es. (S5.), (S5.3), (S5.4), (S5.9), we have poved that P P0 P0. tot O S It is woth ephasiing that the identity P P0 P0 has been poved not by the identity E E E E E E fo coplex fields, but by its espective iaginay pats. In addition, we should eak that the divegence-fee conditions E 0 and S7

18 H 0 have been incopoated in the pocedue. In othe wods, hoogeneous dielectic edia ae assued without any pesence of space chages. S6. Obital Flow Density in the Catesian Coodinates Now, let us futhe pocess the obital pat f F and h NF i E k H and i H k E with via the popotionality exp it O P fo ou paticula waves descibed by. We ely on the Maxwell's euations k k, expessed in the feuency doain. Fo diensional easons, let us intoduce x, y, nk x, y,. By this way, we obtain x, y cos,sin based on the peviously intoduced adial coodinate n k. Fo axial-coodinate-independent field vaiables, we educe i E H and k ih E to the following. k f f TE : hx i, hy i y x h h TM : fx i, f y i y x. (S6.) E E E E x e. Hee, one is Conside the electic field fist by noticing that pone to aking a gave istake when setting i i I E E xi ei I xi E E ei I xi E ei 0, which is absolutely incoect!. In othe wods, although E E E, we find that E i i xi xi EE E e e. (S6.) S8

19 Nonetheless, thee is a oe infoative intepetation fo E field as E E expi with as its eal-valued phase. As a esult, E if we wite the electic E E exp i E E E e E i e exp i i xi xi E exp i E exp exp exp i i ei E i ei xi xi. (S6.3) E E exp exp ei i E i i ei x x i E e i i E e i xi xi i This expansion ake a coection fo the above-entioned istake in E. (S6.). Let us explain the suation conventions eployed hee. Any double indices ean a suation ove that index as in the Einstein's notation. Fo the tiple (3 ties) o uatic (4 ties) indices, we eploy the suation sybol in an explicit way. Theefoe, E E E E E. (S6.4) E ei E E ei E E ei xi xi i xi By this way, we have sepaated E E into its eal and iaginay pats. The fist te is eal-valued, and it is exactly what we have entioned in the pevious paagaph fo waning fo E. (S6.). The second additional te is iaginay-valued, and it is what we should not iss out. This second te is the intensity of the electic field ultiplied by the phase gadient. We cannot ephasie too stongly the ipotance of the phase gadients in the business of angula oentu. Besides, we could teat H H in a siila way. Now, we poceed with the electic-field pat in tes of the noalied vaiables. S9

20 E E E I E ei nk xi Ex Ey E I E x Ey E ex x x x Ex Ey E I E x Ey E ey y y y f f x y f I f x f y f ex x x x. (S6.5) I f f y f x x y f y f f ey y y Hee, we utilied eithe E. (S.) o (S.4) and the axial-coodinate independence of the electic field. Siilaly, we teat the agnetic field to obtain H H H I H ei nk xi H H x y H I H x H y H ex x x x H H x y H I H x H y H ey y y y hx h y h I h x hy h ex x x x. (S6.6) I h x hx y h y h y h ey y h y We expect instantly that both E E in E. (S6.5) and H H in E. (S6.6) do not diffeentiate between the co- and counte-otational cases, because all coplex vaiables show up in coplex-conugate pais. Anothe obsevation is that thee is no intefeence te, since only shows up. Hence, the obital FD is sepaable into the TE and TM odes. Suing up Es. (S6.5) and (S6.6), S0

21 I 4nk 4nk O P I E E H H f f x y f I fx f y f ex 4 x x x f f x y f I f x f y f ey 4 y y y hx h y h I h x hy h ex 4 x x x. (S6.7) hx h y h I h x hy h ey 4 y y y Let us go futhe to sepaate O P into fou pats.,,,, O O TE O TM O TE O TM P P x Px ex P y Py ey. (S6.8) Hee, the idea is to collect tes espectively fo the TE and TM odes in the two in-plane Catesian diections x, y. O, TE h h I x y f Px hx hy f 4 x x x O, TM I f f x y h Px fx f y h 4 x x x. (S6.9) O, TE h I h x y f Py hx hy f 4 y y y O, TM f I f x y h Py fx f y h 4 y y y Hee, the additional supescipts "TE" and "TM" efe espectively to the TE and TM odes. We can check a patial validity of these ites by finding that the espective sets hx, hy, f and fx, f y, h appea in the ites epesenting the TE and TM odes, espectively. S

22 Via eithe E. (S.3) o E. (S.5), the above E. (S6.9) can be expessed solely in tes of the two axial coponents f, h. O, TE f I f f f f Px f 4 x x y xy x O, TM h I h h h h Px h 4 x x y xy x. (S6.0) O, TE f I f f f f Py f 4 x xy y y y O, TM h I h h h h Py h 4 x xy y y y We find that the iaginay unit do not explicitly appea in these fou defining euations due to self-cancellations. As a conseuence, Es. (S6.0) holds tue fo both otational cases. In the Catesian coodinates, E. (S6.0) lends itself easily to a vecto fo. To this goal, the tes in E. (S6.0) ae plugged back into E. (S6.8). O O TE O TE O TM O TM P P e P e P e P e 4,,,, x x y y x x y y f f f f f f ex x x y xy x I 4 f f f f f. (S6.) f e y x xy y y y h h h h h h ex x x I y xy x h h h h h h e y x xy y y y At this oent, it is helpful to intoduce the following in-plane gadient fo a geneic vecto A A, A in the Catesian coodinates. x y S

23 A A x y A e x ey. (S6.) x y Hence, E. (S6.) is eoganied as follows. O f f f f I P f f 4 x x y y. (S6.3) h h h h I h h 4 x x y y Let us intoduce anothe diffeential vecto opeato. A A A A A A. (S6.4) x x y y This fo A A is the convective-deivative vecto, which often occus in classical fluid dynaics. E. (S6.3) is then cast into O P I f f f f 4 I h h h h 4 I 4 f f f I 4 h h h. (S6.5) This vecto fo cannot be easily tanslated into its countepat in the pola coodinates. The E E E E x e and afoeentioned special fos i i i i H H H H x e fo the obital flow density (FD) ake the cuent investigation both uniue and tie-consuing. These fos ae unlike the usual gadient and Laplacian opeatos, fo which we find petinent tansfoation ules eadily fo handbooks and S3

24 textbooks. The difficulty with these fos aises essentially fo the fact that both E and H H efe to diffeential fos opeated not on scala but on vecto uantities. Fo this pupose, we have expessed the obital FD in anothe vecto fo in (S6.5). E S7. Vecto Laplacians As anothe exaple of the opeatos on vectos, conside the following vecto Laplacian fo a geneic vecto V., (S7.) V Vxex Vyey Ve Vx x y y Vx Vx Vx e x, etc. (S7.) Hee, V V e V e V e x x y y o V Vx, Vy, V in a shot-hand notation fo the Catesian coodinates. We can cast this vecto in the cylindical coodinates such that V V e V e V e o V V, V, V in a shot-hand notation fo the cylindical coodinates. The atheatical coplications associated with V is that it is not easy to expess V in the cylindical coodinates. Fo instance, we ay assue that V V e V e V e with. (S7.3) In this case, what is issing is the tes like the vecto gadients e, e, e vecto Laplacian e, e, e. It is because, the unit vectos e, e, e and the fo the cylindical coodinates ae spatially vaying o inhoogeneous. We will get these exta tes in fothcoing E. (S7.7). Instead, we would bette deal with Es. (S7.) and (S7.). In this appoach, let us genealie S4

25 V V x, y, so that it depends on all the thee space coodinates. In ode tanslate E. x x (S7.) in the Catesian coodinates into that in the cylindical coodinates, let us conside seveal tansfoation ules. Vx cv sv Vy sv cv e x ce se e y se ce, (S7.4), (S7.5) s c x. (S7.6) c s y Hee, we eployed a shot-hand notation fo a pai of angles c cos and s sin. All the thee tansfoation ules in Es. (S7.4)-(S7.6) have the sae tansfoation atix, naely, c s, which has a unit deteinant. s c Now, in E. (S7.) is expanded in full as follows. V Vxex Vyey Ve V V e V e V e x x y y x y y x y y x y y Vx Vx V Vy Vy V x y V V V e x ey e Vx Vx Vx V V x y Vy Vy V e x V V V V e y e y. (S7.7) This step woks fine, since V x and the likes ae the scala Laplacians, and theefoe they can be witten in pola coodinates without incuing any eos. Since the axial te Ve does not change even in the pola coodinates, it will be not S5

26 consideed any futhe. With the help of Es. (S7.4) and (S7.5), the fist two tes of E. (7.7) is now tansfoed as follows. Vxex Vyey Vxex Vyey cv sv cv sv cv sv cv sv ce se sv cv sv cv sv cv sv cv se ce. (S7.8) At this point, let us define V e V e D e D e. (S7.9) x x y y Collecting tes of the siila sots in E. (S7.8), we ascetain c V csv c V csv c cv sv c V csv D s V csv s V csv s sv cv s V csv D csv s V csv s V s cv sv csv s V csv c V csv c V c sv csv cv c V. (S7.0) Exploiting seveal cancellations, we obtain cv sv sv cv V V V c s D D cv sv sv cv V V V s c. (S7.) While poceeding, we need to pay a special attention to the factos like cos c sin. S6

27 cv sv sv cv V V V c s D V V V s c D cv sv sv cv. (S7.) We take deivatives of the above with espect to the aiuthal angle to obtain V e V e D e D e x x y y V V V c V V sv cv c s e s V V cv sv s c V V V s V V sv cv c s e c V V cv sv s c. (S7.3) Taking the aiuthal deivatives once oe, V e V e D e D e x x y y V V V c V V V V V V cv sv s c s c c s e s V V V V V V sv cv c s c s s c V V V s V V V V V V cv sv s c s c c s e. (S7.4) c V V V V V V sv cv c s c s s c S7

28 Via the tigonoetic euality c s, the final siplification leads to V e V e D e D e x x y y V V V V V V e V V V V V V e. (S7.5) Hee, thee ae two exta tes. Both V and addition, both V and V V ae the centipetal tes. In ae the Coioli's tes. The eual-signed featue of the centipetal tes in the adial diection ecus in ou photon poble. In a siila vein, the opposite-signed featue of the Coioli's tes in the aiuthal diection ecus in ou photon poble. E. (S7.5) is nothing but V V V V Vxex Vyey V e V e. (S7.6) Meanwhile, E. (S7.5) is cast soeties in the following fo. V e V e D e D e x x y y V V V V e. (S7.7) V V V V e E. (S7.7) is in a oe consevation-like fo than E. (S7.5), although E. (S7.7) is still in a non-consevation fo. This non-consevative natue of the vecto Laplacian (S7.) gave ise to all the exta tes, and it ende ou investigation into obital pat of S8 V in E. angula oentu both difficult and challenging. We eak that this kind of coplication aises fo V, the vecto Laplacian, in contay to the conventional scala Laplacian. In a siila context, we have encounteed the convective-deivative vecto f and h in E. (S6.5) fo the obital FD.

29 We encounte this vecto Laplacian while teating the inteaction of chaged paticles with the electoagnetic field, whee the vecto potential shows up though the kinetic oentu te in the Hailtonian fo electons. This Diac euation o the Schödinge euation in the pesence of an extenal axial agnetic field helps to establish the discete Landau levels. It is no supise that the esulting foulation obtained in the condensedatte physics help to constitute the basic odel fo what is ecently called topological photonics. In the case of the Navie-Stokes euations in classical fluid dynaics, the vecto Laplacian V appeas as the diffusion of the velocity vecto V. S8. Obital Flow Density fo the Rotating Cases We eak that both otating cases cannot be esolved in the Catesian coodinates in a pope way. In ode to diffeentiate between the co- and counte-otating cases, we should esot hence to the pola coodinates. To this goal, let us teat E. (S6.8) given in the Catesian coodinates into the pola coodinates via E. (S7.5).,, x x O O TE O TM P P P ce se O, TE O, TM P y Py se ce c P s P cp sp e O, TE O, TE O, TM O, TM x y x y O, TE O, TE O, TM O, TM s P x c Py spx cpy e cp sp cp sp e O, TE O, TE O, TM O, TM x y x y O, TE O, TE O, TM O, TM sp x cpy spx cpy e. (S8.) Ou goal is to euate the above elation to the following elation in the pola coodinates. S9

30 ,,,, O O TE O TM O TE O TM P P P e P P e. (S8.) Theefoe, the fou coefficient functions in the pola coodinates ae elated to the othe fou in the Catesian coodinates in the following way. P cp sp P cp sp P sp cp P sp cp O, TE O, TE O, TE x y O, TM O, TM O, TM x y O, TE O, TE O, TE x y O, TM O, TM O, TM x y. (S8.3) Obviously, the tansfoation between that between O, TM O, TM O, TM O, TM Px, P y and, O, TE O, TE O, TE O, TE Px, P y and, P P. P P is the sae as Conside the fou ebes sepaately. Fist, O, TE I O, TE O, TE O, TE P P 4 cpx spy, (S8.4) O, TE f f f f f f f f f f P c f s f x x y xy x x xy y y y. (S8.5) Via E. (S7.6), the following coon opeato is futhe teated. s c c s cc s s x y. (S8.6) cs cs c s c s Hence, we poceed with E. (S8.5) while keeping the ode of diffeentiations to obtain S30

31 f f f f f x x y y O, TE f P f s f f s f c c f c f f c f f s s f f s f f s f s f c c. (S8.7) f c f f c f c f f s s f It is siplified futhe. f f cs f f cs f f O, TE c P cs f f s f f s f f 3 f f cs f f cs f f s cs f f c f f c f f f f 3. (S8.8) It siplifies once again to the final elation, thus leading to no explicit appeaance of the aiuthal angle. O, TE f f f f f f f P f 3. (S8.9) O, TM h h h h h h h P h 3 Fo the second elation in E. (S8.9), we have taken an advantage of the syety between the TE and TM odes. Second, the peceding entie pocedue is epeated in the aiuthal diection. S3

32 O, TE I O, TE O, TE O, TE P P sp 4 x cpy, (S8.0) O, TE f f f f f f f f f f P s f c f x x y xy x x xy y y y. (S8.) Via E. (S7.6), the cobination opeato becoes now the aiuthal deivative. s c s c sc c s x y. (S8.) s c cs cs c s Hence, we poceed with E. (S8.) while keeping the ode of diffeentiations to obtain f f f f f x x y y O, TE f P f s f f s f c c f c f f c f f s s f f s f f s f c c. (S8.3) f c f f c f f s s f Futhe siplifying, O, TE c f f cs f f cs f f s f f P 3. (S8.4) s f f cs f f cs f f c f f f f 3 Finally, we get two elations without any explicit dependence on the aiuthal angle. S3

33 O, TE f f f f f P f 3. (S8.5) O, TM h h h h h P h 3 In the second elation of E. (S8.5), we have siilaly teated the TM ode. Hence, we O O O have obtained the two desied in-plane coponents P P e P e. Now, we ae to eplace the aiuthal deivatives with the aiuthal ode index accoding to e i. Let us wok on the co-otational case fist. On the basis of the field vaiables in E. (S.), we eploy f if, h h ih. Fo the TE ode, ih, f if O, TE, and P in E. (S8.9) becoes O, TE f 3 P ln f f 3 f f f f f f f f f f f f f 3 f f f f ln f f f ln f ln f 3 f ln f ln f ln 3 f f. (S8.6) We teat P in E. (S8.8) fo the TM ode in a siila way. O, TM S33

34 O, TM h 3 P ln h h 3 h h h h h h h h h h h h h 3 h h h h h ln h h h ln h ln h h 3 h ln h ln h ln 3 h. (S8.7) In both Es. (S8.6) and (S8.7), we notice that the aiuthal deivatives appea even nubes of ties. Theefoe, E. (T6.7) could follows diectly fo E. (T6.6) ust by substituting f with h. Meanwhile, conside the aiuthal coponents. Fo the TE ode, E.(S8.5) becoes f f f f f O, TE f 3 P 3 i f f i i f f f f 3 f f i f f i f ln f. (S8.8) Likewise, fo the TM ode, E. (S8.5) becoes h h h h h O, TM h 3 P 3 i h h i i h h h h 3 h h i h h i h ln h. (S8.9) S34

35 In both Es. (S8.8) and (S8.9), we notice that the aiuthal deivatives appea odd nubes of ties. Theefoe, E. (S8.9) could follow diectly fo E. (S8.8) ust by substituting f with h. Next, conside the counte-otational case. On the basis of E. (S.4) fo the counteotational case, we poceed by eplacing f if, and h f if, h ih, ih. Howeve, the end foulas eain the sae as those fo the co-otational cases when it coes to the adial coponents P and O, TE P, O, TM because the aiuthal deivatives ae applied to both a field vaiable and its conugate within the identical te. The eason is again that the adial coponent is akin to the centipetal o centifugal foce so that it is independent of the diection of the aiuthal popagations. That is why we consideed E. (S7.5) o E. (7.6) fo a vecto Laplacian to explicitly show the appeaance of both V and V of eual signs. In contast, let us ewok the aiuthal coponents O, TE P in E. (S8.8) and O, TM P in E. (S8.9) now fo the counte-otational case. f f f f f O, TE f 3 P 3 i f f i i f f f f 3 f f i f f i f ln f, (S8.0) h h h h h O, TM h 3 P 3 i h h i i h h h h 3 h h i h h i h ln h. (S8.) Quite obviously, O, TE P fo the counte-otational case has its sign opposite to that fo the co- S35

36 otational case, since we assued opposite otating diections. In copaison, O, TM P eains the sae, because we assued the sae otating diection. To suaie, the adial coponents of the obital FD is found as follows fo both otational cases. O, TE ln f f ln f P f I ln 4 3. (S8.) O, TM ln h h ln h P h I ln 4 3 The sign of the cobined adial coponent cannot be easily figued out fo these foulas alone, so that we would find it though nueical eans. To this goal, let us now eploy the solution pofiles f F and h NF the gadient functions G and K ecall the siple algebaic fact that ln f lnh ln f ln h.. In addition, we will ake use of in E. (S.7) and (S.8). It will be helpful to and Now, both P and O, TE O, P TM ae expessible in tes of F, G, and K., O TE P F I 4 G K G 3. (S8.3) O, TM O, TE P N P The adial coponent of the obital FD is thus given by O N I 3 P F G K G. (S8.4) 4 As expected, the aguent of I in the above elation takes eal values in the inteio. S36

37 O Conseuently, P 0 in the inteio, whee fo the obital FD ae puely cicula in the inteio. O P is non-eo. Conseuently, the taectoies Let us tun to the aiuthal coponents. Fo the co-otational case, the aiuthal coponents of the obital FD is found as follows, when taking the espective iaginay pats. O, TE ln f P 4 f O, TM ln h P h 4. (S8.5) Hee, the supescipts " " and " " efe espectively to the clockwise and counte-clockwise otations. In copaison, fo the counte-otational case, the aiuthal coponents of the obital FD is found as follows, when taking the espective iaginay pats. O, TE ln f P 4 f O, TM ln h P h 4. (S8.6) Now, the aiuthal functions ae expessed in tes of F, G, and K. O, TM O, TE P N P O, TE O, TE P P O, TM O, TE P N P O, TE P G 4 F. (S8.7) The cobined adial coponent of the obital FD is given fo the co-otating and counteotational cases by S37

38 O N P G F 4 O N P G F 4. (S8.8) O Fo the co-otational case, it is inteesting that P w. In wods, the aiuthal coponent of the obital FD is -ties the enegy density w, the latte being defined in E. (S.3). In addition, thee is a possibility of a vanishing aiuthal coponent fo the counte-otational case if N. As with the taectoies fo the Poynting vectos, we can find the taectoies fo the obital FD. Conside a taectoy foed by the obital FD. G F N O d P 4 O d P N F I G K G 3 4. (S8.9) G N N I G K G 3 As usual, the double signs efe to the co- and counte-otational cases, espectively. We find that both taectoies do not depend on F, the function itself. Instead, they depend stongly on the logaithic deivatives G and K. S9. Spin Vecto The spin vecto is given by S I 4 E E H H. Fo dielectic and non- S38

39 agnetic edia, S I E E H H 4 I E E H H I n E E H H n n. (S9.) Fo the appeaance of E. (S9.), we ay call E E and oute poduct" and "inta-agnetic-field oute poduct, espectively. H H "inta-electic-field Conside fist the co-otational case accoding to E. (S.). Fo convenience, let us intoduce S 4 I S 0. (S9.) It is thus convenient to exaine S 0 fist. i i i i i i S0 f e, fe, f e fe, fe, f e i i i i i i h e, he, he he, he, he f f f f e f f ff e f f f f e h h h h e hh h h e h h h h e. (S9.3) We note that both i e and (S.3) into (S9.3) to obtain e i cancel each othe in evey tes. We substitute E. dh dh S 0 i f i f e f h hf e d d dh dh df df h i i h e i h hi e. (S9.4) d d d d df df h f f h e f i i f e d d S39

40 Taking its iaginay pat, we find dh dh IS0 Ii f I i f e f h hf e d d dh dh I h i i h e d d df df I i h I hi e h f f h e d d df df I f i i f e d d. (S9.5) dh dh IS0 Ii f I I e h f e ih e d d. (S9.6) df df Ii h I I e h f e if e d d Hee, we have eployed the fact that I A A I A A. Besides, we have tied to keep athe than fo a genetic coplex vaiable in all the tes fo the convenience in the ensuing anipulations. We now eploy the fact that IiA Re A coplex vaiable A. E. (S9.6) then siplifies futhe to fo a genetic dh dh I S0 Re f I Re e h f e h e d d. (S9.7) df df Re h I Re e h f e f e d d Theefoe, fo S 4 I S 0 in E. (S9.), we obtain dh df S I S0 Re f h e 4 d d dh df d d I h Re f e h f e S40. (S9.8)

41 Expessed in tes of F in E. (S.6) and G in E. (S.7), the above becoes, N df df S Re F F e d d,, N, I F Re F e F e N df d N df i F e d, Re I N F I e F G e N N F I IG e Re N F I e F G e N Re. (S9.9) Hee, we utilied the following fact fo a genetic coplex vaiable A and. Re I Re I I i i Re Re Re I I A A A i A A i A i A. (S9.0) Finally, E. (S9.9) is educed to S N F IG S N F S F G N Re. (S9.) We notice that the helicity-dependent in-plane coponents S S ae not sepaable into, the TM and TE odes due to non-ectilinea otions, wheeas the axial spin coponent S S4

42 is sepaable. In the inteio, we have siply S 0. In the exteio, not only 0 o I 0 but also the spatial inhoogeneity of F addition to the non-eo facto I which efes to an aiuthal inhoogeneity. ae euied fo S 0 in G. By the sae token, S 0 fo 0 in E. (), Figue S9.: Spin coponents S, S, S fo the co-otational case plotted ove 0 k 4 fo 3, R kr, and i. We define the noalied spin Sw pe photon. Fo a bette viewing, we intoduce scaling, fo instance, such that sgn 4. Figue S9. pesents such S, S, S S S S w ove 0 k 4 with R kr and 3. In paticula, we povided a non-tivial coupling i S w and. Futheoe, we find though nueical coputations that S w as well. The aveage spin Savg 3 S S S w is plotted in geen cuve. All the cuves in Figue S9. display discontinuities in the adial pofiles acoss the thin laye. Fist, S 0 in the inteio, wheeas 0 S in the exteio. This spin flips take place due to the thin laye, whee excitations ae supplied. Second, S 0 ove all k. In addition, we find fo E. (S9.) the atio of the in-plane coponents is found to be S S I G. As a esult, photon spins fo cicula stealines in the inteio o S4

43 S S 0 because IG 0 accoding to Fig. S.. In copaison, S S 0, because I G 0 accoding again to Fig. S.. Hence, the spin taectoies in the exteio ae diected adially outwad and they point to the counte-clockwise diection. Let us tun to the counte-otational case via E. (S.4). i i i i i i 0,,,, i i i i i i h e, he, he he, he, he i i i i f f f f e i i i i h he hh e e hhe h he e h h hh e S f e f e f e f e f e f e f f e f f e e f f e f f e e. (S9.) We note that both i e and aiuthal facto exp i e i collaboate fo the in-plane coponents so that suaed continues to appea. Howeve, they cancel each othe in the axial tes. As a esult, E. (S9.) is uite diffeent fo its countepat in E. (S9.3) fo the co-otational case. To ou geat supise, the following pocedue leads us to an entiely adical esult. To see this fact, let us substitute E. (S.5) into (S9.) to obtain dh dh S 0 i f e i f e e d d i i dh dh f h e h f e e h i i h e d d i i df df i h e h i e e d d i i h f e f h e i i df df e f i i f e d d. (S9.3) Theefoe, taking the iaginay pat leads to S43

44 dh dh IS0 Ii f e i f e e d d i i dh dh Ih f e h f e e ih i h e d d i i I df df Ih i e i h e e d d i i i i Ih fe h f e e I df d df d if i f e. (S9.4) If futhe pocessed, it gets to dh i i I S0 I i f I e e h f e e d dh dh Iih i h e d d df i i I ih I e e h fe e d df df if i f e d d I Hee, we have eployed the fact that I A A I A. (S9.5) fo a genetic coplex vaiable A. Cuiously enough, we find that the two tes of E. (S9.5) in the aiuthal diection cancel each othe (as aked in ed colos). The inute step involving this cancellation in the aiuthal diection will lead to the biae esult as entioned befoe. Via IiA Re A futhe siplifies to and Re A Re A fo a genetic coplex vaiable A, E. (S9.5) S44

45 dh dh IS0 Re f e e Reh e d d i df df Re h e e f e d d i Re dh df dh df Re e f h e h f e d d d d i Re By S 4 I S 0. (S9.6) in E. (S9.), we go back to the spin vecto to have i dh df S I S0 Re e f h e 4 d d. (S9.7) dh df Re h f e d d Expessed in tes of siplified to F in E. (S.6) and G in E. (S.7), E. (S9.7) is futhe N df df S Re e F F e d d N, df Re F, i, e d. (S9.8) N i df, N Re ie I F Re e F G e d N F I e G e N i I F Re G e Hee, we utilied E. (S9.0) fo a genetic coplex vaiable A and. Theefoe, fo the counte-otational case, we obtain the following with the chiality defined in E. (S.). S45

46 S N F IG S 0 S F G N Re. (S9.9) Figue S9.: Spin coponents S, S, S fo the counte-otational case plotted ove 0 k 4 fo 3, R kr, and i. Figue S9. pesents such S, S, S ove 0 k 4 with R kr, 3, and i. Notice hence that. Futheoe, we set i exp in E. (S.) fo convenience so that. The efactive indices ae n (say, glass) and n (say, ai) fo panel (a), wheeas they ae evesed such that n (say, ai) and n (say, glass) fo panel (b). We find though nueical coputations that S, S, S w. The aveage spin avg 3 S S S S w is plotted in geen cuve. All the cuves in Figue S9. display discontinuities in the adial pofiles acoss the thin laye. Howeve, S shows an appoxiately siila behavio on both panels. S46

47 On panel (a) of Figue S9., it happen on panel (a) that S 0 in the inteio fo E. (S9.9) fo the counte-otational case, because N in inteio. In copaison, S 0 in the exteio. On panel (b), 0 still in inteio. Only diffeence is that N and S again in the inteio, because N N in the exteio fo panels (a) and (b), espectively. In evaluating S in E. (S9.9), N fo panel (a) and N fo panel (b), espectively. As a esult, S 0 on panel (a) and S 0 on panel (b), espectively. Let us copae E. (S9.) to E. (S9.9) fo the thee coponents of the co- and counteotational cases. Fo instance, S undegoes a little change in the ultiplying facto fo fo the co-otational case to fo the counte-otational case. Notice that depends stongly on the aiuthal angle as is the optical chialty C pesented in E. (S3.9). We can easily find that the usual sign changes ae encounteed fo S. Figue S9.3: (a) A contou plot of the adial coponent of spin S. (b) A contou plot of its 5 scaled value S S. Both ae plotted on the, sgn case. The coon data ae R kr, 3 kx ky -plane fo the counte-otational., and i S47

48 Figue S9.3(a) exhibits a contou plot of the adial coponent of spin S, wheeas panel (b) 5 shows a contou plot of its scaled value S S. Both ae plotted on the, sgn plane fo the counte-otational case. The coon data ae R kr, 3 kx ky -., and i We find that S on panel (a) exhibits a blued iage so that its spatial distibution is had to eveal itself. 5 Figue S9.4: Contou plots of S S on the, sgn case. The coon data ae R kr and 3 witten on each panel. S48 kx ky -plane fo the counte-otational. Besides,, i, i, i as

49 5 Figue S9.4 shows contou plots of the scaled value S S on the, sgn kx ky -plane fo the counte-otational case. The coon data ae R kr and 3. The TE-TM coupling coefficient is vaied such that, i, i, i as witten on each panel. Hence, panel (c) is what we have aleady pesented. Figue S9.4 is theefoe shows the i effect of the facto Ie on S. It is S 0 fo the counte-otational case, which is adically diffeent fo S N F in E. (S9.) fo the co-otational case. Note fo E. (S9.5) that the tivial identity S 0 stes fo the cancellation between the electic- and agneticfield contibutions duing counte-otations of the TE and TM waves. To take a close look at this fact, conside the aiuthal pat of E. (S9.) i i i i i i i i e f f e f f e hh e hh S f f e f f e h h e h h e 0,. (S9.0) Taking its iaginay pats, i i S 0, e f f e f f I I I I i i Ie h h I e h h i i e f f Ie h h. (S9.) Again fo E. (S.5), we obtain i i I S0, I e h f I e h f. (S9.) i i Ie h f Ie h f 0 S49

50 We find that we have neve eployed the paticula solutions F in E. (S.6) and G E. (S.7) in ode to obtain the conclusion that S 0 fo the counte-otational case. in Theefoe, S 0 holds tue fo any concentic cylindes as long as the efactive index can be expessed by any piecewise-continuous functions. S0. Spin Vecto fo Counte-Rotations with Diffeing Angula Speeds In the peceding section, one kind of waves popagate accoding to exp i t wheeas the othe kind of waves popagate accoding to exp i t,. Conseuently, ou paticula pai of the TE and TM waves leads to S 0, when they ae counteotational. We then aise a uestion: "Would any othogonal pai of two would lead to S 0?". In ode to answe this uestion, let us conside the two waves in counte-otations but with diffeent angula speeds. To this goal, let us odify both Es. (S.4) and (S.5) as follows fo the counte-otational case. E H f e, f e, f e i i il h e, h e, h e il il i. (S0.) l df TE : h f, h i ; d. (S0.) :, dh TM f h f i d Hee, we assue that l, 0 being integes. Theefoe, the TE waves follow exp i l t, while the TM waves follow exp i t. Naely, S50

51 TE : exp i l t. (S0.3) TM : exp i t In both euations, the ites in ed colo indicate what have been changed in copaison to Es. (S.4) and (S.5). Conside i i il i i il 0,,,, il il i i il i h e, he, he he, he, he S f e f e f e f e f e f e. (S0.4) Let us conside only the aiuthal coponent fo siplicity. il il il il il il il il e f f e f f e hh e hh S f f e f f e h h e h h e 0,. (S0.5) Taking its iaginay pats, il il 0, I S I e f f I e f f I il il I I il il e f f Ie h h e h h e h h. (S0.6) Again fo E. (S0.), we obtain il il 0, l I S I e h f I e h f. (S0.7) l il Ie h f By S 4 I S 0 in E. (S9.), we go back to the spin vecto to have S5

52 l il 0, S I S I e h f 4 4 l i l I N F e. (S0.8) As a conseuence, S 0 hold tue if and only if two waves popagate in counte-otations with the sae agnitude. This esult S 0 is theefoe a vey special. By the way, S 0 coesponds to the in-plane spin being adially polaied. In diensional tes, we note that il I S I e h f H E, which is nothing but the uantity popotional to the optical chiality in the axial diection. In this espect, we notice that C n I E H n I H E. S5