In 1969, Arnaud described the use of Gaussian beamlets as a basis set from which an arbitrary complex field could be synthesized. The optics community soon came to embrace this approach as a practical method of ray tracing coherent fields, and is widely known as Gaussian Beam Decomposition (GBD).
The basic tenet of GBD is that the individual Gaussian beamlets which make up a given field distribution remain Gaussian, to within some reasonable tolerance, as they propagate through an optical system. By definition Gaussian beams carry a quadratic wavefront, such that any surface they intersect must be locally quadratic to avoid inducing wavefront error. This condition requires careful sampling of each surface in an optical system. However, the relationship
tan(q) = l/[ pwo]
relating the divergence angle, q, of a Gaussian beam to its beam waist, wo, is in direct conflict with the sampling requirement. As a result, the functionality of GBD breaks down as geometric spatial variations begin to approach the wavelength of light.
As long as its limitations are respected, GBD provides a reliable method of evaluating diffraction and coherence for a wide range of optical applications. By employing GBD, FRED can calculate the Point Spread Function for an imaging lens, the interference pattern from an interferometer, or edge diffraction effects.
It is recommended that, when possible, Source Primitives be used to describe coherent sources. This will ensure that all necessary settings are configured in order to generate a valid coherent field.
If your coherent source is not described by one of the Source Primitive constructs, then it will be necessary to configure the source representation using an appropriate combination of Detailed Source settings.
Coherent sources in FRED are constructed from ray grids and the coherent properties of the source are invoked by assigning a Gaussian beamlet to each ray in the grid. This application note examines how the individual Gaussian beamlets are constructed and traced through an optical system.
Each ray in the grid serves as a "base ray" that is traced along the axis of each beamlet. By default, FRED assigns eight secondary rays to each base ray (see the Coherence Tab on Detailed Source dialog) and it is the relationship of the secondary and base rays which allow computation of the beam waist and beam divergence at any plane as the beam is propagated.
For illustration purposes, the Figure below shows a base ray and two in-plane pairs of secondary rays. An additional set of four secondary rays lie in the orthogonal plane. The waist rays initially trace parallel to the base ray at a distance determined internally by the grid spacing and the overlap factor and corresponding to Overlap*wo. The divergence rays begin coincident with the base rays and trace a trajectory asymptotic to the far-field divergence angle, q.
NOTE: The inverse relationship between the beamwaist, wo, and the far field divergence, q, (as defined at the beginning of this help article) carries an important implication with regard to sampling of surfaces. The spatial extent over which the secondary rays intersect any given surface increases as the initial waist wo of the individual Gaussian beamlets decrease. A common mistake in setting up coherent sources is to create a grid of rays so finely spaced that the individual beamlets so divergent as to prevent adequate sampling of subsequent optical elements. The user is urged to carefully consider their choice of grid density based upon the discussion herein.
The following two rules pertaining to Gaussian beamlets must be respected:
Consider Case 1 below in which a coherent beamlet intersects a lens surface near the edge of its clear aperture. The red ray is the base ray, the blue rays are the waist rays and the green rays are the divergence rays. Based upon the diagram, one might be lead to conclude the the outermost divergence ray is clipped by the lens aperture; this is not the case! FRED actually extends the surface profile based on its mathematical definition, and in the particular case shown FRED has no problem intersecting the outermost divergence ray with the surface.
Case 1
Consider now a sphere, cylinder or ellipse as shown in Case 2 below. There exists no mathematical extension of the surface profile such that the upper divergence ray have an intersection. As a result, this beamlet fails the intersection criterion and is halted at the sphere.
Case 2
Finally, consider Case 3 below in which the surface can be extended mathematically but the functional variation is higher order than quadratic. Such a situation is characteristic of aspherics, general polynomials and NURBs. This beamlet can intersect the surface but will have some non-quadratic error induced on its wavefront. As a result, the beamlet is no longer Gaussian and will not be included in the analysis calculations.
Case 3
The individual beamlets comprising these sources can overlap one another. In all cases except the Point Source, FRED determines the width of each Gaussian beamlet based upon the user-defined grid spacing and an overlap factor. The default value for this Adjacent Beams Overlap Factor is 1.5, though the user has control of the overlap value through the Coherence Tab on the Detailed Source dialog by specifying the "Adjacent Beams Overlap Factor".
When the overlap factor is equal to the grid spacing (overlap=1.0), the source exhibits amplitude and phase variation indicative of the individual beamlet profiles as shown below. As the overlap factor is increased, amplitude becomes smoother while the roll off becomes more shallow.
Coherent Field Calculations on Analysis Surfaces The sampling behavior for analysis surfaces is different when using coherent and incoherent sources. For incoherent rays, the power of each ray is simply accumulated into the pixel that the ray intercepts ("bucket counting"). For coherent sources, however, the local coherent field represented by that ray may subtend many pixels. Consequently, the optical field of each coherent ray will be evaluated at the center of each pixel of the analysis surface and that value will be treated as a constant over the area of the pixel. Then, the complex field for each ray in each pixel (at each unique wavelength) is coherently summed together to retrieve the total complex field distribution across the analysis surface area.
As noted in the Individual Beamlets section above, the phase of an individual beamlet must remain quadratic to within a l/4 tolerance in order to be considered a "valid" Gaussian beamlet. This determination is made at the time a coherent field calculation is performed and the number of beamlets included, or excluded, in the calculation will be reported in the output window at the time of the analysis. If rays are excluded from an analysis, the user should test the sensitivity of the excluded ray counts by returning to the source definition, decreasing (or increasing) the ray sampling of the source, and performing the raytrace and analysis again. The point at which there are no coherent ray errors reported in the output window, for either the raytrace or the analysis calculation, represents the best sampling of the source that can be achieved for the system without further user intervention. Advanced methods for performing spatial or angular resampling along the beam propagation path are available, if the source sampling is determined to be insufficient for the application.
Coherent Beam Propagation References
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