Abstract
We report and correct errors in our recently published paper [J. Opt. Soc. Am. A 32, 1403 (2015) [CrossRef] ].
© 2016 Optical Society of America
In Eq. (17) of our recently published paper [1], vector had been omitted after . The correct equation reads
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
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 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: and .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
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.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 of Fig. 20 according to
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]