Spinor Formalism for Waveguides [Pierre Hillion, Solange Quinnez; Ann.telecomm. 1985.40.5-6]

8/13/2019 Spinor Formalism for Waveguides [Pierre Hillion, Solange Quinnez; Ann.telecomm. 1985.40.5-6]

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pp 43 5

Spinor formalism for

Pierre H LL N*Sola ge QU NNE **


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S ,t) is he e ergy o a dt) he e ergy de si y o he spi or eld.Moreover o e has

( ") ( ,t)

= S( ,t)lUsi g S ,t a d ( ,t) ma es possible o com he po er o cross g he s o he spi or beam a d he el per i le g h of he beam

(3)

AiS ,t d

herej j , 3) are he compo e o he s rfaceA a d (3 ) A( ,t) d ,so ha he veloci y o e ergy proi he

HlLL N -

i7 H k 2

Ei Hy= k _

E

ha e m( ) EiEy E iEy i hese For aM ave eq a io( ) e


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H LL ON - S NOR ORMAL SM OR WA E D S

(24b) l (X , y ) = A c osY l X c osY 2 , A

and :

2 X, )=k +(- Y nY1 coY 2 +Y 2 nY 2 coy x),

A 2 2 ' ) S X, )=( k )Y 2 coYI 2Y 2 ,_ A 2Si x, )( k )Yl n 2Y1 x co2Y 2 ,

A 2Sx ) =A 2 co2y 1 x cO2Y 2 - ( k+) 2 x

( 2 2 2 2 2 2 )Y YX coY 2 +Y 2 Y 2 coYIXU ng(26 , t not d cult to chec on e pre on that the con t on9), 19 ) are ful lledThe olut on4a), ( 4b) of equat on(6 that from now on we note ':n x, ), ' CX, ) are the fundamental mode for propagat on ofd h l

and m proof that normal baWe pa completn co ectu e uat on6 w9 ) and w

( 27 a) o r (2 7 b) w it h CMoreove gonal, theE andMWe clo e nten tx, ) fo4b) and(4' we

29a) ex, )=n A 2A 2 n 2 2( k +) 2 Y coY x 29b) ex, )n A 2

nA 2 ( 2 2 Y Y1x


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248 wh ch g v sA ab A abP (31b)Ph - 4 k2 2 Uh 4- k2_2Th compa son o (3 0) w th (3 a) an (31 b) l a s to( 3 2 ) PtM U tM ) A P M U M )

Ph U h ) = A P U ) A ( k )n any cas , th v loc ty o ng has t sam p ss on 33) vg k 2 ,

wh ch s t g oup v loc ty

P fLL O SP

37)1 , cA -i1

m )2 T k12

ut to sa 3 an s n 3 ) , ) A ihe , ) k


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P H ON SP NOR ORM SM OR W E DES

with the cala p od ct 4 ) , < 1 Q r (r, +oJo

F(r, (r ) ) r dr d .U ing the following el io [n > 1)( 4 2) I n( o ) I n (

d = o 2 2

X

(J o x) : In( J x) ( I (o X ),\ 42') Jx J o x) d= 2o 2(o 2 x 2 n 2 ) J o x)

2 x J o x)riti ea to p ove the o thogona:

Let n fo the eU inga) anE H E H :

( k

( co s2 n e IEl + H /2 = Al

;S b titinteg ati w46) _


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250denotes conductivity foriwein the previous loss foranalysis. For a rectangular waveguide this leads tochange in eq a ion(21)k ntok2l - c/we)sothat we g t:

2 2 k 2- lk - Y \ 1 (k2_ y2)w e , Y Y1+ Y2

assumingIP2; we ,this give= k y

(1 - 2(P Z Z w e) which def nes the attenuation kcoe cient=2we(k2 _ y2 )1 IZ this resu t is also va id for circu ar waveguid s)

Let us now consider attenuation due to imperfectlyconducting boundaries

We rst note that according to0 a) 0b) one

P. H LL ON -

and similarly5b)

R

[W

=Aa

which are theformalism Oformulae stfor circular w

As well knoina ref rentia


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H ON S NO O MA SM O WAV U D

n a c nser ati e f rm, and there e ists a huge littture n this subject mainly in u d m chanic se]f r nstance) Recently ne f us(*) iscussed thethermal bl ming f laser beams thr ugh a numeris luti n f spin r wa e quati ns

APPENDIX

spin r x) is a g metrical being with twc mple c mp nents 1( x) hex) and with thepr p rty t transf rm under a r tati n f paramete, q (Euler's angles) acc rding t the relat n ( - ) (x) '( x)= \ i


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2 5 2 P HILLION. -

[ ] H LL ON(P Spi r representat n f electr magnet celds J opt Soc Am 976 nO 8 pp 865-866

2]HI LION(P An extens n f ge metr cal pt cs t p la zed l ght J. of optics, Fr 979 10 nO pp 2 -26

3] MO (S ,W NN RY (JR.), V N D Z R (Th F eldsand waves n c mmun cat n electr n csWiley, NewY rk 965 p 42

[4] W K R (E T ,W N (G N A c urse f m dernanalys s Camb Univ Press,L nd n 959 p 380

5] J N S S The the y f electr magnet sm Pergam nPress, xf rd 969 p 536] N .E eld the ry f gu ded waves Pergam nPress, xf rd 964 p 4 37 P YR ),T Y R (T D C mputat nal meth dsf r u d w Spr nger er ag 983 chap 2

8] (PFr 929 12

9]P RR N( . eChim. Phys 9

0]V N HU S (Cha m nn and ]S K S (G GSoc 852 9,

[ 2] F M H w th neut n Re iew, D 97

3] N K (H ) w th nternat n4 nO pp 455

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Spinor Formalism for Waveguides [Pierre Hillion, Solange Quinnez; Ann.telecomm. 1985.40.5-6]