Gaussian Quadrature
The GQ algorithm uses a carefully selected and weighted ray set to accurately compute the RMS or PTV error over a uniformly illuminated entrance pupil (strictly speaking, the PTV algorithm is not a GQ algorithm, but it is very similar). The weighting for all rays is applied according to the weights set on the wavelength and field in the System Explorer and by the GQ merit function algorithm. For RMS merit functions, the weighting and ray set selection used is based on a method described in a paper by G. W. Forbes, JOSA A 5, p1943, 1988. For the PTV merit functions, the ray set is based on solutions to the Chebyshev polynomials, described in Numerical Recipes, Cambridge University Press (1989). If you are interested in detailed information on the basis and accuracy of these methods, see these references. GQ is much more accurate than any other known method, and requires fewer rays. Therefore, you get the best of both worlds: greater speed and greater accuracy.
The only drawback to GQ is that the algorithm assumes the pupil is a circle, or more generally, an ellipse. For non-elliptical pupils, GQ does not work accurately. For example, if there are surface apertures in the optical system that vignette enough rays to alter the effective shape of the pupil significantly, GQ should not be used.
One notable exception is when using circular pupils with central obscurations, such as a Newtonian telescope. In this case, an input value may be given to indicate the fraction of the pupil that is obscured (the "Obscuration"). The method used to account for obscuration of the circular pupil within the GQ algorithm is described in B.J. Bauman and H. Xiao, "Gaussian quadrature for optical design with non-circular pupil and fields, and broad wavelengths", Proc. SPIE, Vol. 7652, p76522S-1 (2010). While, in general, modest central obscurations do not affect the RMS significantly because the aberrations tend to be smaller in the central zone of the pupil, the use of a finite "Obscuration" factor within the GQ algorithm allows for a fully accurate representation of the obscuration in the merit function calculation.
Note also that GQ works fine when used with vignetting factors, since the ray pattern is redistributed from a circle to an ellipse. More details on the use of vignetting factors in conjunction with GQ may be found in the Knowledge Base article entitled "How to use Vignetting Factors".
The GQ algorithm requires specification of the number of "Rings", the number of "Arms", and the "Obscuration" factor.
Rings The "Rings" setting determines how many rays are traced at each field and at each wavelength. For on-axis fields (zero degrees field angle in a rotationally symmetric system), the number of rays is equal to the number of rings. For all other fields in symmetric systems, the number of rays traced per ring is equal to half the number of "arms" (defined in the next paragraph). Only half the rays are traced because the left-right symmetry of the system is exploited. Each set of rays is traced for each defined wavelength. For example, if you have one on-axis field, two off-axis fields, three wavelengths, four rings, and six arms selected, the number of rays traced is 3 * (4 + 4*3 + 4*3) = 84. For systems without rotational symmetry, the number of rays per ring is the number of arms independent of field. In the prior example, this means 3 * 3 * 4 * 6 = 216 rays. OpticStudio automatically calculates these numbers for you; the only reason it is described here is so you will understand how the default merit function is defined. Optimization runs are longer if more rays are traced.
Arms The "Arms" setting determines how many radial arms of rays in the pupil are traced. By default six equally spaced (in angle) arms are traced (or three if the system is rotationally symmetric). This number may be changed to eight, ten, or twelve. For most common optical systems, six is sufficient.
Sampling considerations for Rings and Arms
You should select the number of rings and the number of arms according to the order of aberrations present in your system. The highest order of aberration the GQ algorithm can accurately integrate is (2*n - 1) where n is the number of rings. If an optical design will be limited by 5th order ray aberrations n should be at least 3. A simple way of determining the correct number of rings is to select the minimum number, one. Then go to the optimization dialog box and note the merit function. Now go back to the default merit function tool, and select two rings. If the merit function changes by more than a few percent, go back and select three, and so on until the merit function does not change significantly (perhaps 1%). Repeat the procedure for the number of arms (six arms is almost always plenty). Selecting more rings or arms than required will not improve the optimization performance; it will only slow the algorithm down needlessly. Tracing more rays than required will not help you find better solutions!
Selecting more rings or arms than required will not improve the optimization performance; it will only slow the algorithm down needlessly.
Obscuration The "Obscuration" setting determines the fraction of the circular pupil that is vignetted, i.e. the fraction of the pupil through which rays cannot be traced. The default value is 0, i.e. the default assumes no obscuration. The maximum possible value is 1, which would indicate a fully-obscured pupil through which no rays could be traced. To avoid a situation in which no rays are defined in the merit function, OpticStudio will reset the input value to 0 if the user sets the (absolute value of the) input value to be > = 1.
Note: the obscuration fraction of the circular pupil is done as a linear fraction, not as a fraction of the pupil area. This means the pupil area that is obscured is the square of the input value. The obscuration is centered on the circular pupil.
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