## Abstract

In our previous article [J. Opt. Soc. Am. A **32**, 1236
(2015) [CrossRef] ] there is an
issue concerning the comparison of plane wave spectrum solutions of
paraxial and Helmholtz equations. We compared the angular plane wave
spectrum of Helmholtz solutions with the plane wave spectrum of the
paraxial solutions in terms of normalized projections of paraxial wave
vectors. We show that the proper comparison of plane wave spectra must
be done in terms of angles. The results presented in our previous work
are corrected accordingly. The most important change is that
Wünsche’s ${T}_{2}$ operator leads to a valid method.

© 2016 Optical Society of
America

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### Equations (17)

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(1)
$${\nabla}_{\perp}^{2}{\psi}_{P}+2ik{\partial}_{z}{\psi}_{P}+2\text{\hspace{0.17em}\hspace{0.17em}}{k}^{2}{\psi}_{P}=0,$$
(2)
$${\psi}_{P}(\mathbf{r})=\frac{ik}{8{\pi}^{2}}{\int}_{{C}_{q}}q\mathrm{d}q{\int}_{{C}_{\beta}}\mathrm{d}\beta F(q,\beta )\mathrm{exp}[i{\mathbf{k}}_{p}(q,\beta )\xb7\mathbf{r}],$$
(3)
$$q=\mathcal{N}(\alpha )\mathrm{sin}\text{\hspace{0.17em}}\alpha ,$$
(4)
$$\mathcal{N}(\alpha )=\frac{{[2-2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\alpha {(1+{\mathrm{sin}}^{2}\text{\hspace{0.17em}}\alpha )}^{1/2}]}^{1/2}}{{\mathrm{sin}}^{2}\text{\hspace{0.17em}}\alpha}.$$
(5)
$${\psi}_{P}(\mathbf{r})=\frac{ik}{8{\pi}^{2}}{\int}_{{C}_{\alpha}}\mathrm{sf}(\alpha )\mathrm{d}\alpha {\int}_{{C}_{\beta}}\mathrm{d}\beta F(\alpha ,\beta )\mathrm{exp}(i{\mathbf{k}}_{p}\xb7\mathbf{r}),$$
(6)
$$\mathrm{sf}(\alpha )=\frac{2-2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\alpha {(1+{\mathrm{sin}}^{2}\text{\hspace{0.17em}}\alpha )}^{1/2}}{{\mathrm{sin}}^{3}\text{\hspace{0.17em}}\alpha {(1+{\mathrm{sin}}^{2}\text{\hspace{0.17em}}\alpha )}^{1/2}},$$
(7)
$${\mathbf{k}}_{p}=k\mathcal{N}(\alpha )[{\mathbf{n}}_{\perp}(\beta )\mathrm{sin}\text{\hspace{0.17em}}\alpha +{\mathbf{e}}_{z}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\alpha ],$$
(8)
$$2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\frac{\gamma}{2}=q,$$
(9)
$$\gamma =2\text{\hspace{0.17em}}\mathrm{arcsin}\frac{\mathcal{N}(\alpha )\mathrm{sin}\text{\hspace{0.17em}}\alpha}{2}.$$
(10)
$${F}_{H}(\alpha ,\beta )=F(2\text{\hspace{0.17em}}\mathrm{arcsin}\frac{\mathcal{N}(\alpha )\mathrm{sin}\text{\hspace{0.17em}}\alpha}{2},\beta ).$$
(11)
$$F(\alpha ,\beta )-{F}_{H}(\alpha ,\beta )=\frac{1}{8}{\partial}_{\alpha}F(\alpha ,\beta ){|}_{\alpha =0}{\alpha}^{3}+\mathcal{O}({\alpha}^{4}).$$
(12)
$$\mathrm{sin}\text{\hspace{0.17em}}\gamma =q.$$
(13)
$$\gamma =\mathrm{arcsin}[\mathcal{N}(\alpha )\mathrm{sin}\text{\hspace{0.17em}}\alpha ].$$
(14)
$${F}_{H}(\alpha ,\beta )=\mathrm{cos}\text{\hspace{0.17em}}\alpha F\{\mathrm{arcsin}[\mathcal{N}(\alpha )\mathrm{sin}\text{\hspace{0.17em}}\alpha ],\beta \}.$$
(15)
$$F(\alpha ,\beta )-{F}_{H}(\alpha ,\beta )=\frac{1}{2}F(\alpha ,\beta ){|}_{\alpha =0}{\alpha}^{2}\phantom{\rule{0ex}{0ex}}+\frac{1}{2}{\partial}_{\alpha}F(\alpha ,\beta ){|}_{\alpha =0}{\alpha}^{3}+\mathcal{O}({\alpha}^{4}).$$
(16)
$${F}_{H}(\alpha ,\beta )=F\{\mathrm{arcsin}[\mathcal{N}(\alpha )\mathrm{sin}\text{\hspace{0.17em}}\alpha ],\beta \}.$$
(17)
$$F(\alpha ,\beta )-{F}_{H}(\alpha ,\beta )=-\frac{1}{8}{\partial}_{\alpha}F(\alpha ,\beta ){|}_{\alpha =0}{\alpha}^{5}+\mathcal{O}({\alpha}^{6}),$$