## 2 Solution of the Paraxial Equation in Elliptic Coordinates
 \tag{9}\]  
Both Equation (9) reduce to the Ince equation, with its canonical form being given by [1; 2; 3]:  
\[\bigg{(}\frac{\partial^{2}}{\partial t^{2}}+\varepsilon\sin 2t\frac{ \partial}{\partial t}+(\lambda-p\varepsilon\cos 2t)\bigg{)}N(t)=0, \tag{10}\]  
where \(p\) is an integer nonnegative number; \(\varepsilon\) is a positive ellipticity parameter, as mentioned in the Introduction; and \(\lambda=\lambda(p,\varepsilon)\) is some constant. In works [1; 2; 3], it is supposed that \(\varepsilon>0\). This assumption can also be seen from comparison of Equations (9) and (10), as \(\varepsilon=2\sigma^{2}>0\). Below, in the current work, we show that \(\varepsilon\) in the solution (10) can be of any sign. The constant \(\lambda\) in Equation (9) appears on separating the variables \(\xi\) and \(\eta\) in Equation (2). For Equation (10), the constant is a real number, introduced for the existence of solution \(N(t)\) in the form of a trigonometric polynomial of degree \(p\) (called the Ince polynomial). Once the form of the solution is restricted in this way, we find that \(\lambda\) should be a root of a certain polynomial of degree \((p+1)\), whose coefficients depend on \(\varepsilon\) (i.e., root of a characteristic equation). Thus, in total, \((p+1)\) roots appear as \(\lambda_{0},\lambda_{1},\ldots,\lambda_{p}\), and they are all real-valued. In addition, if these roots are sorted in ascending order at \(\varepsilon=0\), their ascending order does not change with increasing values of \(\varepsilon\) (for all positive \(\varepsilon\), the characteristic equation has no multiple roots). Thus, by choosing a root \(\lambda_{q}(p,\varepsilon)\) and by constructing a trigonometric Ince polynomial \(N(t)=N_{p,q}(t,\varepsilon)\), solution (7) can be written via the solutions to Equation (10) as follows:  
\[\mathcal{I}_{p,q}(\mathbf{r},z)=G(\mathbf{r},z)\bigg{(}\frac{1-iz}{1+iz}\bigg{)} ^{p/2}N_{p,q}(i\xi,\varepsilon)N_{p,q}(\eta,\varepsilon), \tag{11}\]where \(\varepsilon=2\sigma^{2}\). It can be shown that the characteristic equation is factorized and splits into two equations. If \(p\) is even, then there are \(p/2\) odd Ince polynomials (they are written via the sines of multifold angles and usually indexed as \(q=2,4,\ldots,p\)) and \(p/2+1\) even Ince polynomials (they are written via the cosines of multifold angles and indexed as \(q=0,2,4,\ldots,p\)). If \(p\) is odd, there are equally \((p+1)/2\) even and odd Ince polynomials (indexed as \(q=1,3,\ldots,p\)). Thus, even and odd IG modes (11) are usually written as  
\[ \begin{split}\mathbf{IG}_{p,q}^{\varepsilon}(\mathbf{r},z)& =G(\mathbf{r},z)\left(\tfrac{1-iz}{1+iz}\right)^{p/2}\!\!\!\!C_{p}^{ \left(q\right)}(i\xi,\varepsilon)C_{p}^{\left(q\right)}(\eta,\varepsilon),\\ \mathbf{IG}_{p,q}^{o}(\mathbf{r},z)&=G(\mathbf{r},z )\left(\tfrac{1-iz}{1+iz}\right)^{p/2}\!\!\!S_{p}^{\left(q\right)}(i\xi, \varepsilon)S_{p}^{\left(q\right)}(\eta,\varepsilon),\end{split}  \tag{12}\]  
where \(C_{p}^{\left(q\right)}(t,\varepsilon)\) and \(S_{p}^{\left(q\right)}(t,\varepsilon)\) are even and odd Ince polynomials, respectively, and indices \(p\) and \(q\) are nonnegative integers (\(q\leq p\)). Unfortunately, a tradition of writing the even and odd IG modes differently and studying them separately leads to unjustified complications in the designations. The index \(q\) is chosen to be of the same parity as \(p\), i.e., \(q=\{p,p-2,p-4,\ldots\}\). If \(p\) is odd, then this list stops at \(q=1\), but if \(p\) is even., it stops at \(q=0\) for modes \(\mathrm{IG}^{o}\) and at \(q=2\) for modes \(\mathrm{IG}^{o}\). This separation is probably related to the initial form of the Ince polynomials \(N(t)\), as expressed via \(\sin kt\) or \(\cos kt\). If, instead of the sines and cosines, exponentials \(e^{ikt}\) were used and if the even and odd IG modes were not separated, then, for any fixed \(p\), the index \(q\) would range from \(0\) to \(p\), and respective numbers \(\lambda_{q}(p,\varepsilon)\) would be sorted in ascending order.