mathieu beams

Laguerre-Gauss and Bessel Beams are solutions to the wave equation in cylindrical coordinates. But solutions in other coordinates are possible, for example in elliptical coordinates one finds Mathieu Beams.

The literature in optical vortices (twisted light) focuses almost exclusively on beams with cylindrical symmetry, such as Laguerre-Gauss beams. And so we may think that the properties of optical vortices are exactly those of cylindrical beams. But interesting departures occur for optical vortices with different symmetry; this is for example the case of Mathieu beams.

Let us first remind that elliptical and Cartesian coordinates are related by the transformation $\{x = h \cosh(\xi) \cos(\eta), y = h \sinh(\xi) \sin(\eta), z = z \}$ which can be combined into a canonical ellipse equation
\begin{eqnarray}
\frac{x^2}{h^2 \cosh^2(\xi)} + \frac{y^2}{h^2 \sinh^2(\xi)} = 1
\end{eqnarray}
with $\xi \,\epsilon\, [0, \infty)$ and $\eta \,\epsilon\, [0,2\pi)$; note that curves of constant $\xi$ are ellipses.

The Helmholtz equation expressed in elliptical coordinates [1] is separable in longitudinal and transverse parts. The longitudinal part gives the common propagation factor $e^{i k_z z}$. The transverse part can be further separated into radial and angular parts, which are the Mathieu equations:
\begin{eqnarray}
\frac{d^2 R(\xi)}{d\xi^2} - [a-2q \cosh (2\xi)] R(\xi)&=&0 \\
\frac{d^2 \Theta(\eta)}{d\eta^2} + [a-2q \cosh (2\eta)] \Theta(\eta)&=&0 \\
\end{eqnarray}
where $a$ is the separation constant and $q$ is proportional to the ellipticity $h$ square. Combinations of solutions to these equations plus the longitudinal part yield travelling waves called Mathieu beams [2]. Solutions with different topological charge $\ell$ can be found.

A plot of the transverse profile of Mathieu beams show interesting features:

Figure 1: Transverse amplitude, phase and phase zoom-in of a 
Mathieu beam with $\ell=2$ showing elliptical amplitude and
two phase singularities on a transverse axis [3].

(see also Ref [4]), the amplitude does not have circular symmetry as expected and the phase shows the existence of displaced phase singularities, in contrast to the case of cylindrical vortices:

Figure 2: Cylindrical optical-vortex phase diagrama for different
topological charge $\ell$, showing a single phase singularity.

Other interesting features, such as fractional topological charge, are described in the references below.


REFERENCES

[1] E. Mathieu, Mémoire sur le mouvement vibratoire d’une membrane de forme elliptique. Journal de mathématiques pures et appliquées 2e série, tome 13 (1868), p. 137-203.

[2] Julio Cesar Gutierrez-Vega, Ramon M. Rodriguez-Dagnino, Marcelo David Iturbe Castillo, Sabino Chavez-Cerda, New class of nondiffracting beams: Mathieu beams, Proc. SPIE 4271, Optical Pulse and Beam Propagation III, (12 April 2001); doi: 10.1117/12.424720.

[3] D. O. Pabon, S. A. Ledesma, G. F. Quinteiro, and M. G. Capeluto, Design of a compact device to generate and test beams with orbital angular momentum in the EUV, Applied Optics 8048, Vol. 56, No. 29 (2017).

[4] R. J. Hernández-Hernández, R. A. Terborg, I. Ricardez-Vargas, and K. Volke-Sepúlveda, Experimental generation of Mathieu–Gauss beams with a phase-only spatial light modulator, Appl. Opt. 49, 6903-6909 (2010)