Using Gaussian Beam Decomposition A new method for optics calculations unifies geometrical and physical optics.

 

Geometrical optics and physical optics are two entirely different approaches to solving optical problems. Geometrical optics is about tracing rays, Snell's law, reflections, distances, and angles. In contrast, the world of physical optics is about complex mathematical manipulations: Fourier transforms, differential equations, or matrix propagators. It almost seems as if these two approaches are describing two different phenomena. In recent years, however, a new method of performing optics calculations has arisen that unifies geometrical and physical optics within a simple mathematical formalism. This is the method of Gaussian beam decomposition.

The idea behind Gaussian beam decomposition is to decompose an arbitrary optical field into a coherent superposition of Gaussian beams.

Figure 1 illustrates this idea. The flat field on the left is broken into a series of Gaussian beams. The advantage to doing this is that propagation of Gaussian beams is well understood. Such beams retain their Gaussian profile as they propagate through free space. In fact, if the local curvature of a lens or mirror is gentle, then Gaussians even retain their shape after reflection or refraction from a mirror or through a lens. As a result, if the location, size, divergence, and phase of each beam is tracked as it is propagated, then the field can be reconstructed across any plane between and up to the final image plane, as illustrated by the middle and right-hand fields shown in Figure 1.

Figure 1. Illustration of how an arbitrary input field can be represented as a superposition of Gaussian beams.

Base and Parabasal

This method only works if there is a practical way to propagate Gaussians through optical systems. It so happens that propagation of the Gaussian beams can be done with standard ray-tracing techniques. This is because there is a one-to-one relationship between propagating Gaussian beams through an optical system and tracing rays. Figure 2 illustrates this idea.

Figure 2. Illustration of how parabasal rays are used to represent the characteristics and propagation of Gaussian beams.

The location and direction of the beam is represented by a single ray, called the base ray, at the peak of the Gaussian. Two additional rays, known as parabasal rays, together define the size and divergence of the Gaussian. These rays are traced in parallel with the base ray, so that the effect of refractive and reflective optical components on the size and divergence of the associated Gaussian beam is properly accounted for. Two additional parabasal rays are defined in a plane that is orthogonal to the first two. This permits the Gaussian to have a different diameter and divergence in the two dimensions.

Of course, as the base and parabasal rays are traced, they become thoroughly tangled together. Nevertheless, mathematical relations between the locations and directions of the base and parabasal rays and the size and divergence of the Gaussian beams exist that allow the Gaussian beams to be reconstructed at any point.

There are many advantages to this approach. First, because geometrical and physical optics calculations are both based on tracing rays, geometrical and physical optics propagation is now done through the single geometrical model. Second, the practical use of this method is simple: no matrix propagators, no complex field arrays, no differential equations, no Fourier transforms. Third, beamsplitting and recombination are combined within a single formalism. Beams approaching a partially transmitting component are simply split into reflected and transmitted components. These are then brought together to produce interference effects.

Fourth, nonsequential propagation through complex three-dimensional optomechanical geometries is now much simpler. The base and parabasal rays are simply traced in the usual way without regard to concepts such as an axis of propagation. Finally, by attaching polarization to each Gaussian beam, polarization effects, such as birefringence, can be carried with each beam and included in the final superposition of the field.

An Approach that Works

Does this approach work? In a word, yes. Figures 3 and 4 show a model of an unstable laser resonator, and propagation of the output at three propagation distances. Initially, the central obscuration and spider shadows are clearly imprinted on the beam. After propagating 1000 cm, complex diffraction phenomena appear, including a spot of Arago in the center of the obscuration. After propagating 100,000 cm, a far-field pattern is produced that is reminiscent of a classic Airy-disk pattern. Propagation through a complex three-dimensional geometry is illustrated in Figure 5. Here light is propagated nonsequentially through a laser printer model.

The Gaussian decomposition method is most appropriate for workers who do diffraction calculations in systems that have no axis of symmetry, or systems that have three-dimensional apertures and obscuration, or for those who must propagate fields that are not Gaussian or flat-top. These types of systems do not have a single well-defined exit pupil, so traditional Fourier transform techniques are difficult to use. Also, matrix methods are often cumbersome when there is not a single optical axis.

Figure 4. Irradiance calculation of the output from an unstable resonator at three propagation distances: from left, 1 cm, 1000 cm, and 100,000 cm.

Gaussian decomposition is also useful to designers or users of interferometers. These workers must understand the interference patterns produced by ghosts or component misalignments and tolerances. In these systems, base and parabasal rays are split at partially transmitting surfaces during a nonsequential ray trace, and both parent and ghost rays are traced to the focal plane. Gaussian beamlets are then synthesized from these rays at the focal plane, and superposed to calculate the total field, including all interference effects. Tracing the base and parabasal rays is easy with modern nonsequential ray tracers, but complex three-dimensional folds and component tilts make it difficult to define a unique propagation axis or pupil location for standard Fourier transforms or matrix propagation methods.

Figure 5. Illustration of nonsequential beam propagation through a laser printer model.

Gaussian beam decomposition is an easier approach to performing physical optics calculations within a wide class of fully three-dimensional systems. Because the method is based on ray tracing, users are freed of concerns about propagation axes, differential equation stability, and aliasing. In fact, in most conventional optical systems, performing a wave optics calculation is not much more difficult than the geometrical calculations that are so familiar.

For more information, contact Gary L. Peterson, the author of this article, at Breault Research Organization Inc., 6400 East Grant Rd., Suite 350, Tucson, AZ 85715; (800) 882-5085; (520) 721-0500; fax: (520) 721-9630; E-mail: gpeterson@breault.com.