Abstract

In a previous paper, there were three errors or unclear statements. This erratum corrects them.

© 2014 Optical Society of America

1. In a previous paper [1], there were three errors or unclear statements. To properly match boundary conditions, Eqs. (3) and (4) needed to be changed to represent relations between E instead of D, with the corresponding change that Λ=(Ex,Hy,Ey,Hx). The corrected equations are

((Eip+Erp)cosθ1n1(EipErp)(Eis+Ers)(EisErs)n1cosθ1)=T(Etpcosθ3n3EtpEtsEtsn3cosθ3)
and
α=(t11cosθ3+t12n3)/cosθ1β=(t13+t14n3cosθ3)/cosθ1γ=(t21cosθ3+t22n3)/n1δ=(t23+t24n3cosθ3)/n1η=(t31cosθ3+t32n3)κ=(t33+t34n3cosθ3)ρ=(t41cosθ3+t42n3)/(n1cosθ1)σ=(t43+t44n3cosθ3)/(n1cosθ1)Γ=[(α+γ)(κ+σ)(β+δ)(η+ρ)]1.

To accommodate the change from D to E, a new matrix needed to be introduced to Eq. (28) to transform the components of D to components of E. This matrix is

Ψ=((ϵ1)xx0(ϵ1)xy00100(ϵ1)yx0(ϵ1)yy00001),
in terms of components of (ϵ1), the inverse of the dielectric tensor for the rotated layer. This makes ΛI=ΨΦIX and ΛII=ΨΦIIX with ΦII=ΦIP. The transfer matrix for the slab of material is then
ΛI=ΨΦI(ΨΦIP)1ΛII=TΛIIT=ΨΦIP1ΦI1Ψ1.

2. A sign flip was introduced in Eq. (3′) that carries through to Eqs. (25) and (26). These equations are now

DrII=|DrII|(D^tI(x),D^tI(y),D^tI(z))DrII=|DrII|(D^tI(x),D^tI(y),D^tI(z))HrII=1n|DrII|(H^tI(x),H^tI(y),H^tI(z))HrII=1n|DrII|(H^tI(x),H^tI(y),H^tI(z))
and
ΦI=(D^tI(x)D^tI(x)D^tI(x)D^tI(x)1nH^tI(y)1nH^tI(y)1nH^tI(y)1nH^tI(y)D^tI(y)D^tI(y)D^tI(y)D^tI(y)1nH^tI(x)1nH^tI(x)1nH^tI(x)1nH^tI(x)).

3. While not in error, Eqs. (16) and (17) are more clearly written as

(1/n)2=cosψ/no2+sinψ/ne2=(1/ne)2+(no2ne2)sin2θcos2χ
and
n=ne1+(ne2no2)n12sin2θ1cos2χ,
with sinθ=(n1/n)sinθ1.

The author thanks Monika Pietrzyk of the University of St. Andrews, U.K., for bringing item 1 to our attention.

Reference

1. T. Essinger-Hileman, “Transfer matrix for treating stratified media including birefringent crystals,” Appl. Opt. 52, 212–218 (2013). [CrossRef]  

References

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  1. T. Essinger-Hileman, “Transfer matrix for treating stratified media including birefringent crystals,” Appl. Opt. 52, 212–218 (2013).
    [Crossref]

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Equations (8)

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( ( E i p + E r p ) cos θ 1 n 1 ( E i p E r p ) ( E i s + E r s ) ( E i s E r s ) n 1 cos θ 1 ) = T ( E t p cos θ 3 n 3 E t p E t s E t s n 3 cos θ 3 )
α = ( t 11 cos θ 3 + t 12 n 3 ) / cos θ 1 β = ( t 13 + t 14 n 3 cos θ 3 ) / cos θ 1 γ = ( t 21 cos θ 3 + t 22 n 3 ) / n 1 δ = ( t 23 + t 24 n 3 cos θ 3 ) / n 1 η = ( t 31 cos θ 3 + t 32 n 3 ) κ = ( t 33 + t 34 n 3 cos θ 3 ) ρ = ( t 41 cos θ 3 + t 42 n 3 ) / ( n 1 cos θ 1 ) σ = ( t 43 + t 44 n 3 cos θ 3 ) / ( n 1 cos θ 1 ) Γ = [ ( α + γ ) ( κ + σ ) ( β + δ ) ( η + ρ ) ] 1 .
Ψ = ( ( ϵ 1 ) x x 0 ( ϵ 1 ) x y 0 0 1 0 0 ( ϵ 1 ) y x 0 ( ϵ 1 ) y y 0 0 0 0 1 ) ,
Λ I = Ψ Φ I ( Ψ Φ I P ) 1 Λ II = T Λ II T = Ψ Φ I P 1 Φ I 1 Ψ 1 .
D r II = | D r II | ( D ^ t I ( x ) , D ^ t I ( y ) , D ^ t I ( z ) ) D r II = | D r II | ( D ^ t I ( x ) , D ^ t I ( y ) , D ^ t I ( z ) ) H r II = 1 n | D r II | ( H ^ t I ( x ) , H ^ t I ( y ) , H ^ t I ( z ) ) H r II = 1 n | D r II | ( H ^ t I ( x ) , H ^ t I ( y ) , H ^ t I ( z ) )
Φ I = ( D ^ t I ( x ) D ^ t I ( x ) D ^ t I ( x ) D ^ t I ( x ) 1 n H ^ t I ( y ) 1 n H ^ t I ( y ) 1 n H ^ t I ( y ) 1 n H ^ t I ( y ) D ^ t I ( y ) D ^ t I ( y ) D ^ t I ( y ) D ^ t I ( y ) 1 n H ^ t I ( x ) 1 n H ^ t I ( x ) 1 n H ^ t I ( x ) 1 n H ^ t I ( x ) ) .
( 1 / n ) 2 = cos ψ / n o 2 + sin ψ / n e 2 = ( 1 / n e ) 2 + ( n o 2 n e 2 ) sin 2 θ cos 2 χ
n = n e 1 + ( n e 2 n o 2 ) n 1 2 sin 2 θ 1 cos 2 χ ,

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