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Handedness reversal of circular Bragg phenomenon due to negative real permittivity and permeability

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Abstract

When the real parts of the permittivity and the permeability dyadics of a structurally chiral, magnetic-dielectric material are reversed in sign, the circular Bragg phenomenon displayed by the material is proved here to suffer a change which indicates that the structural handedness has been, in effect, reversed. Additionally, reflection and transmission coefficients suffer phase reversal.

©2003 Optical Society of America

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Figures (1)

Fig. 1.
Fig. 1. Computed reflectances of a chiral ferrosmectic slab in free space, for Ω=140 nm, L=60Ω, and χ=30°. Case (i): h=1, ψ=35°, ∊a=2.7(1+iδ), ∊ b =3.3(1+iδ), ∊ c =3(1+iδ), µa=1.1(1+iδµ), µ b =1.4(1+iδµ), µ c =1.2(1+iδµ), δ=2δµ=2× 10-3. Case (ii): Same as (i) except h=-1 and ψ=-35°. Case (iii): Same as (i) except that ψ=215° and the real parts of ∊ a,b,c and μ a,b,c are negative.

Equations (27)

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¯ ¯ ( r ) = 0 S ¯ ¯ z S ¯ ¯ y [ a u z u z + b u x u x + c u y u y ] S ¯ ¯ y T S ¯ ¯ z T μ ¯ ¯ ( r ) = μ 0 S ¯ ¯ z S ¯ ¯ y [ μ a u z u z + μ b u x u x + μ c u y u y ] S ¯ ¯ y T S ¯ ¯ z T } , 0 z L ,
E ( r ) = E ˜ ( z ) exp [ ( x cos ψ + y sin ψ ) ] H ( r ) = H ˜ ( z ) exp [ ( x cos ψ + y sin ψ ) ] } , < z < ,
d dz [ f ¯ ( z ) ] = i [ P ¯ ¯ ( z ) ] [ f ¯ ( z ) ] , 0 < z < L .
[ P ¯ ¯ ( z ) ] = [ P ¯ ¯ 0 ( z ) ] + κ k 0 [ P ¯ ¯ 1 ( z ) ] + ( κ k 0 ) 2 [ P ¯ ¯ 2 ( z ) ] ,
[ P ¯ ¯ 0 ( z ) ] = ω { [ 0 0 0 μ 0 μ c + μ ˜ d 2 0 0 μ 0 μ c + μ ˜ d 2 0 0 0 c + ˜ d 2 0 0 0 c + ˜ d 2 0 0 0 ]
+ [ 0 0 μ 0 μ c μ ˜ d 2 sin 2 ζ μ 0 μ c μ ˜ d 2 cos 2 ζ 0 0 μ 0 μ c μ ˜ d 2 cos 2 ζ μ 0 μ c μ ˜ d 2 sin 2 ζ 0 c ˜ d 2 sin 2 ζ 0 c ˜ d 2 cos 2 ζ 0 0 0 c ˜ d 2 cos 2 ζ 0 c ˜ d 2 sin 2 ζ 0 0 ] } ,
[ P ¯ ¯ 1 ( z ) ] = ( k 0 sin χ cos χ ) ×
{ ˜ d ( a b ) a b [ cos ζ cos ψ 0 0 0 0 sin ζ sin ψ 0 0 0 0 sin ζ sin ψ 0 0 0 0 cos ζ cos ψ ]
+ μ ˜ d ( μ a μ b ) μ a μ b [ sin ζ sin ψ 0 0 0 0 cos ζ cos ψ 0 0 0 0 cos ζ cos ψ 0 0 0 0 sin ζ sin ψ ]
+ ˜ d μ ˜ d ( a μ b b μ a ) a b μ a μ b [ 0 sin ζ cos ψ 0 0 cos ζ sin ψ 0 0 0 0 0 0 sin ζ cos ψ 0 0 cos ζ sin ψ 0 ] } ,
[ P ¯ ¯ 2 ( z ) ] = ω ×
[ 0 0 μ 0 ˜ d a b cos ψ sin ψ μ 0 ˜ d a b cos 2 ψ 0 0 μ 0 ˜ d a b sin 2 ψ μ 0 ˜ d a b cos ψ sin ψ 0 μ ˜ d μ a μ b cos ψ sin ψ 0 μ ˜ d μ a μ b cos 2 ψ 0 0 0 μ ˜ d μ a μ b sin 2 ψ 0 μ ˜ d μ a μ b cos ψ sin ψ 0 0 ] .
[ f ¯ ( L ) ] = [ M ¯ ¯ ] [ f ¯ ( 0 ) ] ,
e inc ( r ) = [ ( i s p + ) 2 a L ( i s + p + ) 2 a R ] e i k 0 z cos θ , z 0 ,
e ref ( r ) = [ ( i s p ) 2 r L + ( i s + p ) 2 r R ] e i k 0 z cos θ , z 0 ,
e tr ( r ) = [ ( i s p + ) 2 t L ( i s + p + ) 2 t R ] e i k 0 ( z L ) cos θ , z L ,
[ r L r R ] = [ r LL r LR r RL r RR ] [ a L a R ] , [ t L t R ] = [ t LL t LR t RL t RR ] [ a L a R ] .
{ Re [ ¯ ¯ ( r ) ] Re [ ¯ ¯ ( r ) ] , Re [ μ ¯ ¯ ( r ) ] Re [ μ ¯ ¯ ( r ) ] , 0 z L } .
z [ 0 , L ] , [ P ¯ ¯ ( z ; ¯ ¯ ( r ) , μ ¯ ¯ ( r ) ; h , ψ ) ] = [ R ¯ ¯ ] [ P ¯ ¯ ( z ; ¯ ¯ ( r ) , μ ¯ ¯ ( r ) ; h , ψ ) ] [ R ¯ ¯ ]
= [ P ¯ ¯ ( z ; ¯ ¯ * ( r ) , μ ¯ ¯ * ( r ) ; h , π + ψ ) ] * ,
z 0 , L , [ f ¯ ( z ; ¯ ¯ ( r ) , μ ¯ ¯ ( r ) ; h , ψ ) ] = [ R ¯ ¯ ] [ f ¯ ( z ; ¯ ¯ ( r ) , μ ¯ ¯ ( r ) ; h , ψ ) ]
= [ f ¯ ( z ; ¯ ¯ * ( r ) , μ ¯ ¯ * ( r ) ; h , π + ψ ) ] * .
{ h h , ψ ψ } { a L a R , r L r R , t L t R } ;
{ Re [ a , b , c ] Re [ a , b , c ] Re [ μ a , b , c ] Re [ μ a , b , c ] ψ π + ψ } { a L a R * , a R a L * r L r R * , r R r L * t L t R * , t R t L * } .
{ h h , ψ ψ } { r LL r RR , r LR r RL t LL t RR , t LR t RL } ,
{ Re [ a , b , c ] Re [ a , b , c ] Re [ μ a , b , c ] Re [ μ a , b , c ] ψ π + ψ }
{ r LL r RR * , r RR r LL * , r LR r RL * , r RL r LR * t LL t RR * , t RR t LL * , t LR t RL * , t RL t LR * } .
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