## 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 , 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

$((Eip+Erp)cos θ1n1(Eip−Erp)(Eis+Ers)(Eis−Ers)n1cos θ1)=T(Etpcos θ3n3EtpEtsEtsn3cos θ3)$
and
$α=(t11 cos θ3+t12n3)/cos θ1β=(t13+t14n3 cos θ3)/cos θ1γ=(t21 cos θ3+t22n3)/n1δ=(t23+t24n3 cos θ3)/n1η=(t31 cos θ3+t32n3)κ=(t33+t34n3 cos θ3)ρ=(t41 cos θ3+t42n3)/(n1 cos θ1)σ=(t43+t44n3 cos θ3)/(n1 cos θ1)Γ=[(α+γ)(κ+σ)−(β+δ)(η+ρ)]−1.$

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

$Ψ=((ϵ−1)xx′0(ϵ−1)xy′00100(ϵ−1)yx′0(ϵ−1)yy′00001),$
in terms of components of $(ϵ−1)′$, 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
$ΛI=ΨΦI(ΨΦIP)−1ΛII=TΛIIT=ΨΦIP−1ΦI−1Ψ−1.$

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

$Dr′II′=|Dr′II′|(D^tI′(x),D^tI′(y),−D^tI′(z))Dr′II′′=|Dr′II′′|(D^tI′′(x),D^tI′′(y),−D^tI′′(z))Hr′II′=1n′|Dr′II′|(−H^tI′(x),−H^tI′(y),H^tI′(z))Hr′II′′=1n′′|Dr′II′′|(−H^tI′′(x),−H^tI′′(y),H^tI′′(z))$
and
$ΦI=(D^tI′(x)D^tI′′(x)D^tI′(x)D^tI′′(x)1n′H^tI′(y)1n′′H^tI′′(y)−1n′H^tI′(y)−1n′′H^tI′′(y)D^tI′(y)D^tI′′(y)D^tI′(y)D^tI′′(y)−1n′H^tI′(x)−1n′′H^tI′′(x)1n′H^tI′(x)1n′′H^tI′′(x)).$

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

$(1/n′′)2=cos ψ/no2+sin ψ/ne2=(1/ne)2+(no−2−ne−2)sin2 θ′′ cos2 χ$
and
$n′′=ne1+(ne−2−no−2)n12 sin2 θ1 cos2 χ,$
with $sin θ′′=(n1/n′′)sin θ1$.

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

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)

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