Abstract
We show that the Green’s function proposed in a recent paper by M. Fu and Y. Zhao [Opt. Express 29(4), 6257 (2021) [CrossRef] ] does not satisfy Helmholtz equation and thus cannot be used to describe monochromatic light propagation from a spherical surface.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
The authors of the recent paper [1] proposed the Green’s function, Eq. (14) of [1], that has zero value at the surface of the sphere of radius b. We present it here in a slightly more detailed form as
(1)$${G_2}({\textbf{r},{\textbf{r}_0}} )= \frac{1}{{|{\textbf{r} - {\textbf{r}_0}} |}}\exp ({ik|{\textbf{r} - {\textbf{r}_0}} |} )- \frac{b}{{|{{\textbf{r}_0}} ||{\textbf{r} - \textbf{r}{^{\prime}_0}} |}}\exp \left( {ik\frac{{|{{\textbf{r}_0}} |}}{b}|{\textbf{r} - \textbf{r}{^{\prime}_0}} |} \right)$$
Here $\textbf{r} = ({x,y,z} )$ is observation point, ${\textbf{r}_0} = ({0,0,c} )$ is the source point, and $\textbf{r}{^{\prime}_0} = ({0,0,{{{b^2}} / c}} )$ is the auxiliary (mirror) source point. These points are denoted as ${P_x}$, ${P_1}$ and ${P_2}$ accordingly in [1].
It is straightforward to check that ${G_2}({\textbf{r},{\textbf{r}_0}} )= 0$ at ${\vert\mathbf{r}\vert=b}$. It is also well-known that the first term
(2)$$G({\textbf{r},{\textbf{r}_0}} )= \frac{1}{{|{\textbf{r} - {\textbf{r}_0}} |}}\exp ({ik|{\textbf{r} - {\textbf{r}_0}} |} )$$
satisfies inhomogeneous Helmholtz equation (3)$${\nabla ^2}G({\textbf{r},{\textbf{r}_0}} )+ {k^2}G({\textbf{r},{\textbf{r}_0}} )={-} 4\pi \delta ({\textbf{r} - {\textbf{r}_0}} ). $$
Consequently, the second term,
(4)$$G^{\prime}({\textbf{r},{\textbf{r}_0}} )= \frac{b}{{|{{\textbf{r}_0}} ||{\textbf{r} - \textbf{r}{^{\prime}_0}} |}}\exp \left( {ik\frac{{|{{\textbf{r}_0}} |}}{b}|{\textbf{r} - \textbf{r}{^{\prime}_0}} |} \right)$$
satisfies inhomogeneous Helmholtz equation with respect to the $\textbf{r}$ variable (5)$${\nabla ^2}G^{\prime}({\textbf{r},{\textbf{r}_0}} )+ {k^2}\frac{{{{|{{\textbf{r}_0}} |}^2}}}{{{b^2}}}G({\textbf{r},{\textbf{r}_0}} )={-} 4\pi \frac{b}{{|{{\textbf{r}_0}} |}}\delta \left( {\textbf{r} - {\textbf{r}_0}\frac{{{b^2}}}{{{{|{{\textbf{r}_0}} |}^2}}}} \right). $$
Equation (5) describes propagation with wave number $k^{\prime} = k{{|{{\textbf{r}_0}} |} / b}$, and demonstrates that the sum of two terms does not satisfy Helmholtz equation, Eq. (7) of [1], in the source-free domain. Same conclusion is valid for the field inside the spherical shell, Eq. (22) of [1].
The only instance when the development of [1] is valid is the electrostatic case, $k = 0$, where it is well-known as method of images, e.g., Eq. (2.16) of [2].
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
References
1. M. Fu and Y. Zhao, “Rigorous expression of Huygens’ principle in scalar theory,” Opt. Express 29(4), 6257–6270 (2021). [CrossRef]
2. J. D. Jackson, Classical Electodynamics, 3rd ed. (Wiley, 1998).
Data availability
No data were generated or analyzed in the presented research.
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