{a (1, nl- 1, vip) A(n, v, p) Tr~s,)(M) YJs,)(M) + a (1, nl- 1, vlp,p - 1) B(n, v,p) Tr~s,)(N) y:~s,)(N)} (6.3.16)

x {a(1,nl-1,vlp,p-1)B(n,v,P)Tr~S')(M)y:~s,)(M)

+ a (1, nl- 1, vlp)A(n, v, p) Tr~s,)(N) y:~s/)(N)}

(6.3.17)

A(n,v,p) = v(v+ 1) +n(n+ 1) -p(p+ 1) n (n + 1) 2v(v + 1) i n- v- p = (2v + 1) n(n + 1) a(n, v, p)

(6.3.18)

b(n, v,p) 2v(v + 1) ;n-v-p nn+1) (2v+1)

(3 ) 6.. 19

R2 Sp(k, KIRs,s/) = - K2 s:",s~2 . [kh~(kRsiS/)jp(I{Rsjs/) - Khp(kRs)s/)j~(KRsiS/)]

drr 2 [9s,s,(r) - 1] hp(kr)jp(Kr)

(6.3.20)

T(s/)(M) = _ [Ps,jn(PsJJ'jn((s/) - [(s/jn((s/)]'jn(Ps/) n [ps/hn(ps/)]'jn((s/) - [(s/jn((s/)]'hn(ps,) T(s,)(N) = _ [Ps,jn(Psl)]'(;,jn((sJ - [(s/jn((s/)]'p;/jn(Ps,) [ps,hn(ps,)]'(Un((.,/) - [(."jn((.,,)]'p;/hn(ps,) n

(6.3.22)

(6.3.23)

. r k d k = W vi JlE s / rp(sl)((M) an d T(sz)((N) are WIt h PSI = k 'as" '"s, = . ,as" an Sl .L n n the Mie scattering T-matrix coefficients for M and N vector spherical waves for species St. For each species St, we terminate at a multipole N s , depending on the radius as, of that species. Thus (6.3.16) and (6.3.17) form N e homogeneous .' equatIOns f It N e un k nowns v(sz)(M).and v(s/)(N), n - 1, 2, ... , N s, an d or le In In St = 1,2, ... ,L. Here N e = "Lf=l 2Ns ,. The generalized Lorentz-Lorenz law determines the effective propagation K. Setting the determinant of (6.3.16)-(6.3.17) equal to zero gives the nonlinear equation for the effective propagation constant K. After K is calculated, the complex transmitted angle Ot and K z can be determined from (6.3.12)-(6.3.15). The N e coefficients y;}sl)(M) and y;~sl)(N) are then reduced to only one arbitrary constant that is to be determined by the incident wave. That one equation to determine the arbitrary constant is provided by the generalized Ewald-Oseen extinction theorem, which gives a single inhomogeneous equation.

The generalized Ewald-Oseen extinction theorem is obtained by balancing the incident wave term of the second term of (6.3.5) and the term of the same phase dependence that is a result of the integral in (6.3.5). The result matrix equation can be reduced to one single equation for the set of coefficients v(s,)(M) d v(sl)(N) In an I n

In this case E vi i- 0 and E hi = O. The coherent transmitted wave is also vertically polarized. The generalized Ewald-Oseen extinction theorem gives rise to the following inhomogeneous equation

where the additional superscript (V) denotes the vertically polarized case. Note that the set of coefficients y~st}(M)(V) and y~st}(N)(V), n = 1, ... , N SI ' Sl = 1, ... , L, have been determined to within one arbitrary constant by the generalized Lorentz-Lorenz law of (6.3.16) and (6.3.17). Thus (6.3.24) provides the last equation that determines all those coefficients uniquely. The coherent reflected field is

(6.3.25a)

27fn sj i I)-l)n (2n + 1) kkiz(Kz + kiz) n n(n + 1)

P~ (cos( Oi + Ot))

sin(Oi+Ot)

+ T~N)(sj)y~sj)(N)(V) (n(n + l)Pn(cos(Oi + Ot))

+ cot(Oi + OdP~(cos(Oi + Od))]

(6.3.25b)

The Snell's law is K sin 0t = k sin Oi. The coherent transmitted field is the same as in Section 2 of (6.2.63) and (6.2.64).

For this case Evi = 0 and Ehi i' O. The coherent transmitted wave is horizontally polarized. The inhomogeneous equation for the generalized EwaldOseen extinction theorem is

+ 1) n(n + 1)

P~(COS(Oi - Ot))

~ n(n + l)Pn(coS(Oi - Ot))

+ T(SI)(N)y(st}(N)(H)

P~(COS(Oi -

Ot)))]

Isin(Oi-Ot)1

(6.3.26)

where the additional superscript (H) denotes horizontal polarization. Thus the homogeneous system of equations of (6.3.16) and (6.3.17) combined with the inhomogeneous equation (6.3.26) determines the coefficients y~sl)(M)(H)