RMS spot radius vs field graph
  • Physics Optics Techniques

Gaussian Quadrature Wavelength Sampling in Optics

This article describes the use of Gaussian Quadrature (GQ) wavelength sampling to determine an optimal set of wavelengths for accurately optimizing and analyzing optical system performance over a broad input spectrum.

 

Introduction

Gaussian Quadrature (GQ) sampling has long been used in optical design to efficiently represent continuous variables with a limited number of discrete samples. In optical design software, GQ sampling was first applied to pupil sampling and has since been extended to wavelength spectrum sampling.

This article demonstrates how GQ wavelength sampling can be used to accurately model broadband optical performance while significantly reducing the number of wavelengths required compared to uniform sampling.

 

Gaussian Quadrature Wavelength Sampling in OpticStudio

OpticStudio incorporates a Gaussian Quadrature wavelength sampling algorithm originally developed by Dr. Bauman and Dr. Xiao. This algorithm automatically selects wavelength points and weighting factors that best represent the spectral behavior of the system.

To define a wavelength spectrum using GQ sampling, the user specifies:

  • Minimum wavelength
  • Maximum wavelength
  • Number of wavelength steps

All wavelength values are specified in microns.

The number of wavelength steps must be an even number between 2 and 12. Once the Gaussian Quadrature option is selected, OpticStudio calculates the optimal wavelength positions and corresponding weights to characterize the spectral range.

For example, defining a spectral range from 0.4 µm to 0.7 µm using four GQ wavelengths yields a set of non-uniformly spaced wavelengths with optimized weighting.

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Key Notes

  • The minimum and maximum wavelengths match the system-defined values (within round-off error).
  • OpticStudio automatically updates the GQ dialog box based on the system wavelength definition.
  • A suitable central wavelength is assigned as the Primary wavelength, which can be changed by the user at any time.

 

Example: RMS Spot Size Evaluation of an Apochromat

To illustrate the benefit of GQ wavelength sampling, consider the example lens file Apochromat3.zmx.

First, add a small off-axis field point to the system.

The performance of this system will be evaluated over the wavelength spectrum range of 0.4-0.7 microns. We’ll start by using four wavelengths uniformly spaced and weighted to define this range. Open the Wavelength Data dialog box and enter as show in figure below.

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A plot of the RMS spot size vs. field can be generated under Analyze…RMS…RMS vs. Field.

For the settings shown below, hit the “Save” button (for later use). The results are shown in the figure below:

To evaluate sensitivity to wavelength sampling, we can increase the number of wavelengths systematically. This process is automated using a ZPL macro, which performs the following steps:

  • Loads the lens file
  • Updates wavelength values (equal weights)
  • Extracts RMS spot size vs. field data using GETTEXTFILE
  • Plots results using the PLOT keyword

The macro is used to investigate the results for RMS spot size vs. field for 4, 6, 8, 10, and 12 wavelengths. In all cases, the spacing between wavelengths spectrum is constant making the sampling uniform. The results from uniform sampling are compared against four-wavelength Gaussian Quadrature sampling. The results are then plotted using the PLOT keyword:

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The plotted data represent six cases:

  • Blue: 4 uniform wavelengths
  • Green: 6 uniform wavelengths
  • Red: 8 uniform wavelengths
  • Gold: 10 uniform wavelengths
  • Purple: 12 uniform wavelengths
  • Black: 4 Gaussian Quadrature wavelengths

As the number of uniformly sampled wavelengths increases, the results gradually converge toward those obtained using GQ sampling. For this system, approximately three times more uniformly spaced wavelengths are required to achieve the same accuracy as four GQ wavelengths.

 

Important Considerations for GQ Wavelength Sampling

Gaussian Quadrature wavelength sampling is most effective when:

  • Glass dispersion follows the Sellmeier equation
    (e.g., Schott and Ohara glass catalogs)
  • Principal material absorption occurs near 0.1 µm, which is used as a reference in the algorithm

GQ sampling may be less suitable for:

  • Systems using dissimilar materials with significantly different dispersion behaviors
  • Materials with strong absorption at longer wavelengths
  • Systems operating beyond approximately 1.5 µm, where IR absorption becomes dominant

 

Chromatic Focal Shift as a Diagnostic Tool

The impact of wavelength sampling is most pronounced in chromatically dominated systems. One way to assess this sensitivity is by examining:

Analyze → Aberrations → Chromatic Focal Shift

Re-evaluating Apochromat3.zmx over the 0.4–0.7 µm range reveals a complex variation of back focal length with wavelength.

When the chromatic focal shift exhibits complex structure, it becomes difficult to capture system behavior using a small number of uniformly sampled wavelengths. Under these conditions, Gaussian Quadrature sampling provides a far more efficient and accurate characterization of wavelength-dependent performance.

 

Conclusion

Gaussian Quadrature wavelength sampling offers a powerful and efficient approach for modeling broadband optical systems. By intelligently selecting wavelength positions and weights, GQ sampling can achieve high accuracy with far fewer wavelengths than uniform sampling—making it especially valuable for systems dominated by chromatic effects.