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Analytic solutions to closed and eclipsing aperture Z-scan power transmission: erratum

Esa Jaatinen, Dinithi Namarathne, and Robert Donaldson

Open Access Open Access

Abstract

In our paper J. Opt. Soc. Am. B 39, 1130 (2022) [CrossRef]   there are typographical mistakes and omissions in four mathematical expressions that appear in Section 2.A of the published paper. This erratum corrects those errors to allow the normalized power transmission to be calculated correctly.

© 2022 Optica Publishing Group

This erratum corrects typographical errors that appear in Section 2.A of the published paper [1].

The expression for $L_{\rm Eff}$, the effective medium length, that appears after Eq. (1) is missing a closing parenthesis. The correct expression is

$${L_{\text{Eff}}} = {\raise0.7ex\hbox{${\left({1 - {\rm Exp}[{- \sigma L} ]} \right)}$} \!\mathord{\left/ {\vphantom {{\left({1 - Exp[{- \sigma L} ]} \right)} \sigma}}\right.}\!\lower0.7ex\hbox{$\sigma$}}.$$

The unwanted parameters $\textit{ss}$ appear in the last exponential factor of Eq. (26). The correct expression for Eq. (26) is

$$\begin{split}{P^{(3 )}} = & \frac{{{I_0}\pi w_{00}^2\Delta {\phi _0}^3}}{{24{{({1 + {x^2}} )}^3}}}\\[-3pt]&\left(- {\rm Sin}\left[{\frac{{48{\alpha ^2}x}}{{49 + {x^2}}}} \right]e^{- \left({{\raise0.7ex\hbox{${8{\alpha ^2}({{x^2} + 7} )}$} \!\mathord{\left/ {\vphantom {{8{\alpha ^2}({{x^2} + 7} )} {({49 + {x^2}} )}}}\right.}\!\lower0.7ex\hbox{${({49 + {x^2}} )}$}}} \right)} \right.\\[-3pt]& + \left.3{\rm Sin}\left[{\frac{{16{\alpha ^2}x({{x^2} + 1} )}}{{({9 + {x^2}} )({{x^2} + 25} )}}} \right]{e^{- \left({8{\alpha ^2}\frac{{({{x^2} + 1} )({{x^2} + 15} )}}{{({{x^2} + 9} )({{x^2} + 25} )}}} \right)}} \right).\end{split}$$

Equation (27) is missing the term $({1 + {x^2}})$ in the numerator of the sinusoidal term. The correct expression for Eq. (27) is

$$\begin{split}&P_{\text{ODD}}^{(n )} = {I_0}\pi w_{00}^2\frac{{{{({- 1} )}^{\frac{{n + 1}}{2}}}}}{{n + 1}}{\left({\frac{{\Delta {\phi _0}}}{{1 + {x^2}}}} \right)^n}\mathop \sum \limits_{i = 0}^{\frac{{n - 1}}{2}} \frac{{{{({- 1} )}^{i + 1}}}}{{i!({n - i} )!}} \\[-3pt]& \times {\rm Exp}\left[\!{- 2({n + 1} ){\alpha ^2}\frac{{({({2i + 1} )({2({n - i} ) + 1} ) + {x^2}} )({1 + {x^2}} )}}{{({{{({2i + 1} )}^2} + {x^2}} )({{{({2({n - i} ) + 1} )}^2} + {x^2}} )}}}\! \right] \\[-3pt]& \quad\times{\rm Sin}\left[{{\alpha ^2}x\frac{{({{{({2({n - i} ) + 1} )}^2} - {{({2i + 1} )}^2}} )({1 + {x^2}} )}}{{({{{({2i + 1} )}^2} + {x^2}} )({{{({2({n - i} ) + 1} )}^2} + {x^2}} )}}} \right] \\[-3pt]& {\rm where}\, n \in 1,{}3,{}5,{}7,{} \ldots .\end{split}$$

The starting limit for the summation expressed in Eq. (28) is $i = 0$. Equation (28) is also missing the term $\;({1 + {x^2}})$ in the numerator of the cosinusoidal term. The correct expression for Eq. (28) is

$$\begin{split}&P_{\text{EVEN}}^{(m )} = {I_0}\pi w_{00}^2\frac{{{{({- 1} )}^{\frac{m}{2}}}}}{{m + 1}}{\left({\frac{{\Delta {\phi _0}}}{{1 + {x^2}}}} \right)^m}\mathop \sum \limits_{i = 0}^{m/2} \frac{{{{({- 1} )}^{i + 1}}}}{{i!({m - i} )!}}\frac{1}{{1 + \delta ({i,m - i} )}} \\ &\quad\times {\rm Exp}\left[{- 2({m + 1} ){\alpha ^2}\frac{{\left({({2i + 1} )({2({m - i} ) + 1} ) + {x^2}} \right)({1 + {x^2}} )}}{{\left({{{({2i + 1} )}^2} + {x^2}} \right)\left({{{({2({m - i} ) + 1} )}^2} + {x^2}} \right)}}} \right] \\& \quad\times {\rm Cos}\left[{{\alpha ^2}x\frac{{\left({{{({2({m - i} ) + 1} )}^2} - {{({2i + 1} )}^2}} \right)({1 + {x^2}} )}}{{\left({{{({2i + 1} )}^2} + {x^2}} \right)\left({{{({2({m - i} ) + 1} )}^2} + {x^2}} \right)}}} \right]\end{split}$$
where $\delta ({i,m - i})$ is the Kronecker delta function and
$$m \in 2,\;4,\;6,\;8,\; \ldots .$$

Acknowledgment

The authors are indebted to T. Vogel from Photonics and Ultrafast Laser Science, Ruhr University Bochum, for bringing these errors to our attention.

REFERENCE

1. E. Jaatinen, D. Namarathne, and R. Donaldson, “Analytic solutions to closed and eclipsing aperture Z-scan power transmission,” J. Opt. Soc. Am. B 39, 1130–1140 (2022). [CrossRef]  

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Equations (5)

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L Eff = ( 1 − E x p [ − σ L ] ) / ( 1 − E x p [ − σ L ] ) σ σ .
P ( 3 ) = I 0 π w 00 2 Δ ϕ 0 3 24 ( 1 + x 2 ) 3 ( − S i n [ 48 α 2 x 49 + x 2 ] e − ( 8 α 2 ( x 2 + 7 ) / 8 α 2 ( x 2 + 7 ) ( 49 + x 2 ) ( 49 + x 2 ) ) + 3 S i n [ 16 α 2 x ( x 2 + 1 ) ( 9 + x 2 ) ( x 2 + 25 ) ] e − ( 8 α 2 ( x 2 + 1 ) ( x 2 + 15 ) ( x 2 + 9 ) ( x 2 + 25 ) ) ) .
P ODD ( n ) = I 0 π w 00 2 ( − 1 ) n + 1 2 n + 1 ( Δ ϕ 0 1 + x 2 ) n ∑ i = 0 n − 1 2 ⁡ ( − 1 ) i + 1 i ! ( n − i ) ! × E x p [ − 2 ( n + 1 ) α 2 ( ( 2 i + 1 ) ( 2 ( n − i ) + 1 ) + x 2 ) ( 1 + x 2 ) ( ( 2 i + 1 ) 2 + x 2 ) ( ( 2 ( n − i ) + 1 ) 2 + x 2 ) ] × S i n [ α 2 x ( ( 2 ( n − i ) + 1 ) 2 − ( 2 i + 1 ) 2 ) ( 1 + x 2 ) ( ( 2 i + 1 ) 2 + x 2 ) ( ( 2 ( n − i ) + 1 ) 2 + x 2 ) ] w h e r e n ∈ 1 , 3 , 5 , 7 , … .
P EVEN ( m ) = I 0 π w 00 2 ( − 1 ) m 2 m + 1 ( Δ ϕ 0 1 + x 2 ) m ∑ i = 0 m / 2 ⁡ ( − 1 ) i + 1 i ! ( m − i ) ! 1 1 + δ ( i , m − i ) × E x p [ − 2 ( m + 1 ) α 2 ( ( 2 i + 1 ) ( 2 ( m − i ) + 1 ) + x 2 ) ( 1 + x 2 ) ( ( 2 i + 1 ) 2 + x 2 ) ( ( 2 ( m − i ) + 1 ) 2 + x 2 ) ] × C o s [ α 2 x ( ( 2 ( m − i ) + 1 ) 2 − ( 2 i + 1 ) 2 ) ( 1 + x 2 ) ( ( 2 i + 1 ) 2 + x 2 ) ( ( 2 ( m − i ) + 1 ) 2 + x 2 ) ]
m ∈ 2 , 4 , 6 , 8 , … .
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