Transfer matrix for treating stratified media including birefringent crystals: erratum

Thomas Essinger-Hileman

doi:10.1364/AO.53.004264

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 $Λ = (E_{x}, H_{y}, E_{y}, - H_{x})$ . The corrected equations are

(\begin{matrix} (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} \end{matrix}) = T (\begin{matrix} E_{t}^{p} \cos θ_{3} \\ n_{3} E_{t}^{p} \\ E_{t}^{s} \\ E_{t}^{s} n_{3} \cos θ_{3} \end{matrix})

and

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} .

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 = (\begin{matrix} {(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 \end{matrix}),

in terms of components of

{(ϵ^{- 1})}^{'}

, the inverse of the dielectric tensor for the rotated layer. This makes

Λ_{I} = Ψ Φ_{I} X

and

Λ_{II} = Ψ Φ_{II} X

with

Φ_{II} = Φ_{I} P

. The transfer matrix for the slab of material is then

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} .

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

D_{r^{'} II}^{'} = | D_{r^{'} II}^{'} | ({\hat{D}}_{t I}^{' (x)}, {\hat{D}}_{t I}^{' (y)}, - {\hat{D}}_{t I}^{' (z)}) D_{r^{'} II}^{''} = | D_{r^{'} II}^{''} | ({\hat{D}}_{t I}^{'' (x)}, {\hat{D}}_{t I}^{'' (y)}, - {\hat{D}}_{t I}^{'' (z)}) H_{r^{'} II}^{'} = \frac{1}{n^{'}} | D_{r^{'} II}^{'} | (- {\hat{H}}_{t I}^{' (x)}, - {\hat{H}}_{t I}^{' (y)}, {\hat{H}}_{t I}^{' (z)}) H_{r^{'} II}^{''} = \frac{1}{n^{''}} | D_{r^{'} II}^{''} | (- {\hat{H}}_{t I}^{'' (x)}, - {\hat{H}}_{t I}^{'' (y)}, {\hat{H}}_{t I}^{'' (z)})

and

Φ_{I} = (\begin{matrix} {\hat{D}}_{t I}^{' (x)} & {\hat{D}}_{t I}^{'' (x)} & {\hat{D}}_{t I}^{' (x)} & {\hat{D}}_{t I}^{'' (x)} \\ \frac{1}{n^{'}} {\hat{H}}_{t I}^{' (y)} & \frac{1}{n^{''}} {\hat{H}}_{t I}^{'' (y)} & - \frac{1}{n^{'}} {\hat{H}}_{t I}^{' (y)} & - \frac{1}{n^{''}} {\hat{H}}_{t I}^{'' (y)} \\ {\hat{D}}_{t I}^{' (y)} & {\hat{D}}_{t I}^{'' (y)} & {\hat{D}}_{t I}^{' (y)} & {\hat{D}}_{t I}^{'' (y)} \\ - \frac{1}{n^{'}} {\hat{H}}_{t I}^{' (x)} & - \frac{1}{n^{''}} {\hat{H}}_{t I}^{'' (x)} & \frac{1}{n^{'}} {\hat{H}}_{t I}^{' (x)} & \frac{1}{n^{''}} {\hat{H}}_{t I}^{'' (x)} \end{matrix}) .

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

{(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

and

n^{''} = n_{e} \sqrt{1 + (n_{e}^{- 2} - n_{o}^{- 2}) n_{1}^{2} \sin^{2} θ_{1} \cos^{2} χ},

with

\sin θ^{''} = (n_{1} / n^{''}) \sin θ_{1}

.

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]

Transfer matrix for treating stratified media including birefringent crystals: erratum

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Applied Optics