Engineering of Radiation of Optically Active Molecules with Chiral Nano-Meta Particles - Singular and Chiral Nanoplasmonics - page 139

wherethevectorcylindricalharmonics n ( + ) and m ( + ) aredefinedin

nq σ nq σ

Appendix 4.A.
Tofindtheunknowncoe cientsofexpansions(4.18)and(4.19),
one can use the continuity of the tangential components of the
electric and magnetic fields at the interface [17]. As the molecule
is placed out of the chiral half-space, then to find the spontaneous
emission decay rate, we need to know only the coe cients C nqe and
D nqe for which one can obtain the following expressions ( n = 0, 1;
σ = e , o ):
C nq σ =− b q C ( − )

nq σ − c q D ( − )

, D nq σ =− a q D ( − )

nq σ − c q C ( − )

, (4.20)

nq σ nq σ

where the functions were introduced:
a q = A q ( L ) W q ( R ) + A q ( R ) W q ( L ) / ,
b q = V q ( L ) B q ( R ) + V q ( R ) B q ( L ) / ,
c q = B q ( L ) W q ( R ) − B q ( R ) W q ( L ) / ,
= V q ( L ) W q ( R ) + V q ( R ) W q ( L ),

(4.21)

in which ( J = L , R )
A q ( J ) = S ( k J ) / Z − S ( k 0 ), B q ( J ) = S ( k J ) − S ( k 0 ) / Z ,
V q ( J ) = S ( k J ) / Z + S ( k 0 ), W q ( J ) = S ( k J ) + S ( k 0 ) / Z ,
(4.22)

k J − q 2

where S ( k J ) =

/ k J .
In the present wo rk, we assume that k 0 −

i q 2 k 0

q 2

= −

k 0 ,and k J − q 2

√ k J − q √ k J + q . For dielectric

> =

for q

( k J

>

0), th is definition provides fulfillment of the re lations

Re k J − q 2 q , and also Im k J − q 2

> > >

0,
when k J < q . For DNG-metamaterial, for whi ch the r efractive index
is negative ( k J

0, when k J

=−| k J | ), we obtain Re k J − q 2

< 0, when

| k J | > q , and also Im k J − q 2

> 0, when | k J | < q . For metal
and MNG-metamaterial, we have Im k J − q 2

> 0. In addition,

in the case of ma terials with losses (i.e., Im( k J )

>

0), this definition

provides Im k J − q 2

> 0.