Modeling third harmonic generation from layered materials using nonlinear optical matrices: erratum

Cristina Rodríguez; Wolfgang Rudolph

doi:10.1364/OE.23.026670

Abstract

We present corrected versions of Equations (11), (13), and (14) where typos were made.

After manuscript [1] was published we realized there was a typo in Equation (11). The factor

\frac{H_{0, 1} L_{1}^{- 1}}{M_{11}^{- 1}} (\begin{matrix} 1 & 0 \\ M_{21}^{- 1} & - M_{11}^{- 1} \end{matrix}), should read \frac{1}{M_{11}^{- 1}} (\begin{matrix} 1 & 0 \\ M_{21}^{- 1} & - M_{11}^{- 1} \end{matrix}) H_{0, 1} L_{1}^{- 1} .

Thus, Equation (11) should read as follows:

\begin{array}{l} (\begin{matrix} E_{r} \\ E_{0, ℓ}^{'} \end{matrix}) & = \frac{1}{M_{11}^{- 1}} (\begin{matrix} 1 & 0 \\ M_{21}^{- 1} & - M_{11}^{- 1} \end{matrix}) H_{0, 1} L_{1}^{- 1} [{\vec{Δ}}_{1} + H_{1, 2} L_{2}^{- 1} {\vec{Δ}}_{2} + \dots + H_{1, 2} L_{2}^{- 1} \dots H_{n - 1, n} L_{n}^{- 1} {\vec{Δ}}_{n}] \\ + \frac{1}{M_{11}^{- 1}} (\begin{matrix} 1 & - M_{12}^{- 1} \\ M_{21}^{- 1} & M_{11}^{- 1} M_{22}^{- 1} - M_{12}^{- 1} M_{21}^{- 1} \end{matrix}) (\begin{matrix} E_{0, r}^{'} \\ E_{ℓ} \end{matrix}) . \end{array}

Equations (13) and (14) had a sign typo in the exponent of the last factor. They should read

\begin{array}{l} Δ E_{i, r} & = & - \frac{9 i ω^{2} χ_{i}^{(3)} d_{i} e^{- i κ_{i} d_{i}}}{2 κ_{i} c^{2}} \times {F_{i, r}^{3} e^{- i Δ k_{i}^{-} d_{i} / 2} sinc (Δ k_{i}^{-} d_{i} / 2) + F_{i, ℓ}^{3} e^{+ i Δ k_{i}^{+} d_{i} / 2} sinc (Δ k_{i}^{+} d_{i} / 2) \\ + & 3 F_{i, r}^{2} F_{i, ℓ} e^{- i Δ ϕ_{i}^{-} d_{i} / 2} sinc (Δ ϕ_{i}^{-} d_{i} / 2) + 3 F_{i, r} F_{i, ℓ}^{2} e^{+ i Δ ϕ_{i}^{+} d_{i} / 2} sinc (Δ ϕ_{i}^{+} d_{i} / 2)}, \end{array}

and

\begin{array}{l} Δ E_{i, ℓ} & = & - \frac{9 i ω^{2} χ_{i}^{(3)} d_{i}}{2 κ_{i} c^{2}} \times {F_{i, r}^{3} e^{- i Δ k_{i}^{+} d_{i} / 2} sinc (Δ k_{i}^{+} d_{i} / 2) + F_{i, ℓ}^{3} e^{+ i Δ k_{i}^{-} d_{i} / 2} sinc (Δ k_{i}^{-} d_{i} / 2) \\ + & 3 F_{i, r}^{2} F_{i, ℓ} e^{- i Δ ϕ_{i}^{+} d_{i} / 2} sinc (Δ ϕ_{i}^{+} d_{i} / 2) + 3 F_{i, r} F_{i, ℓ}^{2} e^{+ i Δ ϕ_{i}^{-} d_{i} / 2} sinc (Δ ϕ_{i}^{-} d_{i} / 2)}, \end{array}

respectively.

Because the mistakes were made when typing the equations for the manuscript, all results and conclusions discussed remain unchanged, since all calculations were performed with the correct relations. The authors regret these accidental typos and would like to acknowledge Zev Montz and Professor Amiel Ishaaya, from the Department of Electrical and Computer Engineering at Ben-Gurion University of the Negev, for bringing our attention to these typos.

References and links

1. C. Rodriguez and W. Rudolph, “Modeling third harmonic generation from layered materials using nonlinear optical matrices,” Opt. Express 22, 25984–25992 (2014). [CrossRef]

Previous Article Next Article

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

\frac{H_{0, 1} L_{1}^{- 1}}{M_{11}^{- 1}} (\begin{matrix} 1 & 0 \\ M_{21}^{- 1} & - M_{11}^{- 1} \end{matrix}), should read \frac{1}{M_{11}^{- 1}} (\begin{matrix} 1 & 0 \\ M_{21}^{- 1} & - M_{11}^{- 1} \end{matrix}) H_{0, 1} L_{1}^{- 1} .

\begin{array}{l} (\begin{matrix} E_{r} \\ E_{0, ℓ}^{'} \end{matrix}) & = \frac{1}{M_{11}^{- 1}} (\begin{matrix} 1 & 0 \\ M_{21}^{- 1} & - M_{11}^{- 1} \end{matrix}) H_{0, 1} L_{1}^{- 1} [{\vec{Δ}}_{1} + H_{1, 2} L_{2}^{- 1} {\vec{Δ}}_{2} + \dots + H_{1, 2} L_{2}^{- 1} \dots H_{n - 1, n} L_{n}^{- 1} {\vec{Δ}}_{n}] \\ + \frac{1}{M_{11}^{- 1}} (\begin{matrix} 1 & - M_{12}^{- 1} \\ M_{21}^{- 1} & M_{11}^{- 1} M_{22}^{- 1} - M_{12}^{- 1} M_{21}^{- 1} \end{matrix}) (\begin{matrix} E_{0, r}^{'} \\ E_{ℓ} \end{matrix}) . \end{array}

\begin{array}{l} Δ E_{i, r} & = & - \frac{9 i ω^{2} χ_{i}^{(3)} d_{i} e^{- i κ_{i} d_{i}}}{2 κ_{i} c^{2}} \times {F_{i, r}^{3} e^{- i Δ k_{i}^{-} d_{i} / 2} sinc (Δ k_{i}^{-} d_{i} / 2) + F_{i, ℓ}^{3} e^{+ i Δ k_{i}^{+} d_{i} / 2} sinc (Δ k_{i}^{+} d_{i} / 2) \\ + & 3 F_{i, r}^{2} F_{i, ℓ} e^{- i Δ ϕ_{i}^{-} d_{i} / 2} sinc (Δ ϕ_{i}^{-} d_{i} / 2) + 3 F_{i, r} F_{i, ℓ}^{2} e^{+ i Δ ϕ_{i}^{+} d_{i} / 2} sinc (Δ ϕ_{i}^{+} d_{i} / 2)}, \end{array}

\begin{array}{l} Δ E_{i, ℓ} & = & - \frac{9 i ω^{2} χ_{i}^{(3)} d_{i}}{2 κ_{i} c^{2}} \times {F_{i, r}^{3} e^{- i Δ k_{i}^{+} d_{i} / 2} sinc (Δ k_{i}^{+} d_{i} / 2) + F_{i, ℓ}^{3} e^{+ i Δ k_{i}^{-} d_{i} / 2} sinc (Δ k_{i}^{-} d_{i} / 2) \\ + & 3 F_{i, r}^{2} F_{i, ℓ} e^{- i Δ ϕ_{i}^{+} d_{i} / 2} sinc (Δ ϕ_{i}^{+} d_{i} / 2) + 3 F_{i, r} F_{i, ℓ}^{2} e^{+ i Δ ϕ_{i}^{-} d_{i} / 2} sinc (Δ ϕ_{i}^{-} d_{i} / 2)}, \end{array}

Abstract

References and links

Cited By

Equations (4)

Optics Express