# Gaussian Beamlet Propagation¶

Gaussian beams are a solution to the paraxial wave equation. It turns out that the propagation of such modes can be represented by a chief ray and a number of skew marginal rays which define the 1/e^2 edge (and divergence) of the beam. It has been shown that such the propagation of such modes obeys the rules of geometric ray tracing (i.e. Snell’s law).

An arbirary wavefront can be decomposed into a set of paraxial Gaussian beamlets (called “Gausslets” henceforth) and propagated through an optical system using raytracing. This method provides a convenient route to extend a geometric ray-tracing algorithm to obtain a physical-optics modelling solution. The method has been successfully employed in a number of commercial optical modelling packages, such as CODE-V, FRED and ASAP. Given a set of Gausslets, the vector (E) field can be evaluated at any position by summing the fields from each Gausslet.

The method is accurate provided some restrictions are observed:

• The Gausslets must remain paraxial (i.e. more-or-less collimated).

• Individual Gausslets should interact with refracting or reflecting surfaces over a sufficiently small area such that the surface may be considered locally parabolic.

Typically, if either of these restrictions are violated, the solution is to interrupt the ray-tracing operation to evaluate the field, then decompose the field into a new set of Gausslets. In cases where the incident Gausslet is sampling too large an area of the surface for it to be considered parabolic, the field is decomposed into a grid of smaller Gausslets. At the other extreme, Gausslets incident on a small aperture may be decomposed into a set of spatially overlapped Gausslets with range of directions. Such an angle decomposition is sometimes known as a “Gabor Decomposition”.

## Raypier Gausslet Implementation¶

Raypier support Gausslet tracing through a GaussletCollection data-structure. Gausslets are an extension of the Ray data structure. In fact, the dtype for a Gausslet object looks like:

```para_dtype = [('origin', '<f8', (3,)), ('direction', '<f8', (3,)), ('normal', '<f8', (3,)), ('length', '<f8')]

gausslet_dtype = [('base_ray', ray_dtype), ('para_rays', para_dtype, (6,))]
```

I.e. we define a dtype for the parabasal rays which contains only the geometric information (origin, direction, length etc.) and omits any E-field polarisation or amplitude information. The gausslet is then composed of one base_ray (with regular ray_dtype) and 6 parabasal rays (with para_dtype).

Gausslets have their own set of predefined source-objects, found in `raypier.gausslet_sources`.

## Evaluating the E-field¶

Algorithms for evaluation of E-fields are provided in `raypier.core.fields`

The nice thing about Gausslet ray-tracing is that you can evaluate the E-field at any point in your model. For script-based analysis, you can give any GaussletCollection object (obtained from a source-object after a tracing operation) to the `raypier.core.fields.eval_Efield_from_gausslets()` function.

## Beam Decomposition¶

When a Beam-decomposition object intercepts a ray during the tracing operation, instead of immediately generating child rays as most other Traceable objects do, the decomposition objects simply store the intercepted ray. At the completion of tracing of the current ray generation (i.e. GaussletCollection), any decomposition-objects which have received one or more rays then perform their decomposition-algorithm to generate a new set of rays to be added to the other rays created in the last generation. The new rays created by the decomposition process will, in general, not join originate from the end-point of the input-rays.

High level Optics objects for beam decomposition are provided here.

class raypier.gausslets.PositionDecompositionPlane(BaseDecompositionPlane)

Defines a plane at which position-decomposition will be beformed.

Sets the radius used for capturing incoming rays. Rays outside of this will “miss”

curvature

An approximate radius-of-curvature to the beam focus. This is used to improve the phase-unwrapping of the wavefront. The default is zero, which means a plane-wave is assumed. Negative values imply a focus behind the decomposition plane (i.e. on the opposite side to the plane direction vector).

resolution

Sets the resampling density of the decomposition, in terms of the number of new rays per radius extent.

blending

Sets the blending values for the new rays. The new rays will have Gaussian 1/e**2 intensity widths equal to spacing/blending, where the spacing value is radius/resolution.

class raypier.gausslets.AngleDecomposition(BaseDecompositionPlane)

Defines a plane at which Gabor (angle)-decomposition is to be performed.

sample_spacing

Sets the sample-spacing at the decomposition plane, in microns.

width

A value in the range 1->512 to set the number of samples along the width of the sample-plane.

height

A value in the range 1->512 to set the number of samples along the height of the sample-plane.