Vectorial ray-based diffraction integral: erratum

Birk Andreas; Giovanni Mana; Carlo Palmisano

doi:10.1364/JOSAA.33.000559

In Eq. (17) of our recently published paper [1], vector $\hat{N}$ had been omitted after $(\hat{r} • \hat{N})$ . The correct equation reads

{\hat{r}}_{t} = \frac{n_{1}}{n_{2}} [\hat{r} - (\hat{r} \cdot {\hat{N}}_{1}) {\hat{N}}_{1}] + σ (\hat{r} \cdot {\hat{N}}_{1}) \sqrt{1 - {(\frac{n_{1}}{n_{2}})}^{2} [1 - {(\hat{r} \cdot {\hat{N}}_{1})}^{2}]} {\hat{N}}_{1} .

Equation (22) in [1] is only correct for use by the stepwise method and must be corrected when it is used at interfaces during the tracing of ray tubes as is mentioned in Section 3.C. Then, it reads

d E_{1, t} = [t_{TM} (d E_{1} \cdot \hat{ξ}) {\hat{ξ}}_{t} + t_{TE} (d E_{1} \cdot \hat{η}) \hat{η}] \sqrt{\frac{\cos θ_{t}}{\cos θ}} .

In Eq. (2) the intensity law of geometrical optics has been taken into account as pointed out by Wyrowski and Kuhn [2]. In the appendix of their paper the factor

{(\frac{{\hat{k}}_{q} \cdot {\hat{a}}_{q}}{{\hat{k}}_{q - 1} \cdot {\hat{a}}_{q}})}^{1 / 2}

is established, which is equivalent to the square root factor which corrects Eq. (22) for use with ray tubes. In Eqs. (24) and (25) of [1] the Fresnel coefficients for TE and TM polarizations must be interchanged:

r_{T M} \leftrightarrow r_{T E}

and

t_{T M} \leftrightarrow r_{T E}

.

All vectorial ray-based diffraction integral (VRBDI) results in our paper had been obtained by use of the uncorrected formula. A recalculation of the results of Section 4 yields a significantly improved agreement between stepwise and VRBDI methods. The test of power conservation now gives

• δ_{SW, 2} = \frac{P_{SW} - P_{2}}{P_{2}} = - 2.1 \cdot 10^{- 10},

• δ_{PW, 2} = \frac{P_{PW} - P_{2}}{P_{2}} = - 7.8 \cdot 10^{- 9} .

The recalculated irradiance comparison between VRBDI with spherical wave decomposition of the input field (VRBDI-SW) and stepwise is shown as an example in Fig. 1. The ring structure is no longer visible, and the residual difference is three orders of magnitude lower and dominated by sampling artifacts. Therefore, the discussion about momentum uncertainty and Eq. (40) are irrelevant at the obtained accuracy level.

Fig. 1. Recalculated irradiance comparison between VRBDI-SW and stepwise from Fig. 11 in [1].

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The statement that the VRBDI is a good approximation when the apertures are much larger than the wavelength is not correct. Apertures can be considered large if no significant part of the beam, as it would be calculated by the diffraction integral at the aperture plane, is cut by the aperture rim. In such a case, the VRBDI result is as accurate as that obtained by applying the stepwise method.

In Eqs. (B22) and (B23) and consequently Eqs. (37) and (38) the optical path length OPL has to be modified by the same $Δ z$ of Fig. 20 according to

\tilde{O} P L = O P L + n_{m - 1} Δ z .

The acknowledgment reads incorrectly. The correct acknowledgment is presented here.

Acknowledgment

This work is a study for the EMRP Joint Research Project SIB08: Traceability of sub-nm length measurements (subnano). The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. The authors gratefully acknowledge funding by the EMRP.

REFERENCES

1. B. Andreas, G. Mana, and C. Palmisano, “Vectorial ray-based diffraction integral,” J. Opt. Soc. Am. A 32, 1403–1424 (2015). [CrossRef]

2. F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58, 449–466 (2011). [CrossRef]

Vectorial ray-based diffraction integral: erratum

Abstract

Acknowledgment

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Journal of the Optical Society of America A