Ince-Gaussian Beams Find the Middle Ground
When optical engineers talk about laser modes, they're referring to the distinct spatial patterns of electric-field intensity within a resonator. These patterns are individual solutions to the paraxial wave equation subject to boundary conditions imposed by the resonator's mirrors, and they are well-known to anybody who has perused a laser textbook.
In rectangular coordinates, for example, the cross-sectional intensity distributions are referred to as Hermite-Gaussian beams. The lowest-order Hermite-Gaussian is the well-known TEM00 mode. A laser can be forced to oscillate in a single, higher-order mode by positioning appropriately spatial-dependent loss inside the resonator. The [4,0] Hermite-Gaussian beam, for example, may be obtained by placing a mask consisting of four vertical wires inside a laser resonator. Usually, however, a laser oscillates in a single, low-order mode or simultaneously in several higher-order modes.
The gain aperture in many lasers is circularly symmetric, and the paraxial wave equation can also be solved in polar coordinates, yielding the Laguerre-Gaussian beams.
These two examples are special cases -- rectangular and polar coordinates -- but recently researchers at the Tecnológico de Monterrey in Monterrey, Mexico, derived the general case for elliptical coordinates. Their "Ince-Gaussian" beams are the third complete family of orthogonal solutions to the paraxial wave equation (see figure). Laguerre-Gaussian beams and Hermite-Gaussian beams are the limiting cases of Ince-Gaussian beams in which the ellipticity parameter goes to zero or to infinity, respectively.
Well-known laser mode patterns in polar coordinates are shown in the left column, and corresponding patterns in rectangular coordinates are shown in the right. In the center are newly derived Ince-Gaussian patterns, of which the patterns on both the right and left are special cases.
Readers familiar with the concepts of optical polarization will find a useful analogy in elliptically polarized light. The two best-known states of polarization, linear and circular, occur when the tip of a propagating electric-field vector describes a straight line or a circle, respectively. Both polarization states are special cases of the general case, elliptical polarization. When the ellipse described by the tip of the electric-field vector collapses onto itself, the ellipse becomes a straight line and the polarization becomes linear. On the other hand, when the ellipse fills out to perfect roundness, the polarization is circular.
Similarly, Laguerre-Gaussian and Hermite-Gaussian beams are special cases of the general Ince-Gaussian beams. When the ellipticity of an Ince-Gaussian beam diminishes to zero, it is a Laguerre-Gaussian beam. When the ellipticity becomes infinite, the Ince-Gaussian beam becomes a Hermite-Gaussian beam.
The figure illustrates this transformation. A beam is Laguerre-Gaussian on the left when the ellipticity is zero, is Ince-Gaussian in the middle when the ellipticity is two and is Hermite-Gaussian on the right when the ellipticity is infinite.
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