Unusual Optical Properties of Helical Metallic Photonic Crystals and Chiral Channels in Dielectric Photonic Crystals - Singular and Chiral Nanoplasmonics - page 192

coordinateswhere k i isthetransversecomponentofwavevector k in
the vacuum. I n ( x ) = dI n ( x ) / dx and K n ( x ) = dK n ( x ) / dx . A n , B n , C n ,
and D n are the coe cients to be determined. We follow Sensiper to
assume uniformly distributed surface current flow along the metal
wires [42], and the metal wires are treated as thin conducting
tapes in the so-called tape-helix model. Under that assumption, the
boundary conditions require that the local electric field on metal
wires must be perpendicular to the line of metal wire, so that we
canassignthesurfacecurrentcomponent J ⊥ oralternatively E ⊥ ( a )
tobezerowhere J ⊥ =

ψ − ψ

, E // =

φ φ/ π =

J z cos J φ sin E t ( a , , p 2 )

. The coe cients A n , B n , C n ,and D n can be
determined by considering the boundary conditions that link the
wave fields at interface of regions I and II. At the interface (the
surface ofthe conducting tapes), wehave

ψ + ψ

E φ cos E z sin

E t 1 ( a ) = E t 2 ( a ) H t 2 ( a ) − H t 1 ( a ) = α f
E z 1 ( a )

= E z 2 ( a ) H z 2 ( a ) − H z 1 ( a ) =− J s φ ( a )

(5.5)

E φ 1 ( a )

= E φ 2 ( a ) H φ 2 ( a ) − H φ 1 ( a ) = J sz ( a ),

where J s φ ( a )and J sz ( a )referstotheangularandradialcomponents
ofsurfacecurrentsontheinterface,respectively.Thesecomponents,
flowingontheconductingtapes(ormetallicwires),mustpropagate
along the z direction for the sake of boundary requirements so that
they shall bedecomposed in form of harmonic waves in space

= n J φ n e − in φ e i β n z J sz ( a ) = n J zn e − in φ e i β n z ,

J s φ ( a )

(5.6)

where β n = β 0 + 2 π n / p is the Bloch wavevector in the nth order.
J s φ ( a ) and J sz ( a ) can also be expressed as the superposition of
the components J // and J ⊥ that are along and perpendicular to the
metallic wire, as

J s φ = J // cos ψ − J ⊥ sin ψ J sz = J // sin ψ + J ⊥ cos ψ . (5.7)

Since J ⊥ = 0, for the Fourier components of J s φ ( a ) and J sz ( a ),we

always have

J φ n = J // n cos ψ J zn = J // n sin ψ .

(5.8)

As the surface currents oscillate along the line of metallic wires in
uniform distribution for the incidence waves along the direction of