How Accurate is NURBS?
http://www.zemax.com/kb/articles/225/1/How-Accurate-is-NURBS-/Page1.html
By Mark Nicholson
Published on 7 April 2008
This article explains the range of ZEMAX surfaces and objects than can be exactly defined using NURBS, and gives an example of exporting high-order aspheres.
How Accurate is NURBS?
When sequential surfaces, or non-sequential objects, are exported to CAD programs, ZEMAX describes them as NURBS (Non-Uniform Rational B-Splines). A useful overview is given here.
A very useful property of the rational b-spline is that conic aspheres are represented exactly by the NURBS formalism. Therefore spheres, ellipses, parabolas and hyperbolas can be represented exactly by the NURBS surface form. This is great news for optical engineers, of course, because these shapes are so commonly used in optical design.
Higher-order aspheres, like polynomial aspheres, are approximated by using multiple segmented splines, which is in general a piece-wise fit over the whole optical surface using multiple lower-order polynomials. Typically, multiple third or fourth order polynomials are used to approximate the surface. This is adequate for mechanical design, but should be carefully tested for optical precision ray tracing, where surfaces may be needed to be known to fractions of the wavelength of light.
This problem often arises when a high optical precision surface is modeled in ZEMAX, then exported as a CAD file to have mechanical sections added, and then imported as a CAD file for subsequent ray tracing. The optical precision of non-conic-asphere parts is reduced upon exporting the native ZEMAX asphere as a NURBS surface form. For non-imaging optics, the precision of the CAD representation is usually adequate, but for imaging systems, care must be taken to verify that the imported CAD part is a suitably accurate description of the desired shape. Note ZEMAX uses a relative internal optical precision of about 10-12 for ray tracing. Most CAD representations of objects are many orders of magnitude more coarse.
A very useful property of the rational b-spline is that conic aspheres are represented exactly by the NURBS formalism. Therefore spheres, ellipses, parabolas and hyperbolas can be represented exactly by the NURBS surface form. This is great news for optical engineers, of course, because these shapes are so commonly used in optical design.
Higher-order aspheres, like polynomial aspheres, are approximated by using multiple segmented splines, which is in general a piece-wise fit over the whole optical surface using multiple lower-order polynomials. Typically, multiple third or fourth order polynomials are used to approximate the surface. This is adequate for mechanical design, but should be carefully tested for optical precision ray tracing, where surfaces may be needed to be known to fractions of the wavelength of light.
This problem often arises when a high optical precision surface is modeled in ZEMAX, then exported as a CAD file to have mechanical sections added, and then imported as a CAD file for subsequent ray tracing. The optical precision of non-conic-asphere parts is reduced upon exporting the native ZEMAX asphere as a NURBS surface form. For non-imaging optics, the precision of the CAD representation is usually adequate, but for imaging systems, care must be taken to verify that the imported CAD part is a suitably accurate description of the desired shape. Note ZEMAX uses a relative internal optical precision of about 10-12 for ray tracing. Most CAD representations of objects are many orders of magnitude more coarse.
An Example
For example, consider an even-aspheric lens, consisting of a surface with a conic asphere surface with added r4 and r6 terms. This is probably the most common aspheric element:
The ray-fan diagram shows balancing focus, spherical, and higher-order spherical, and unbalanced r8 higher-order-higher-order spherical:
The spot diagram shows that the performance is diffraction limited:
This lens is then exported using the default settings:
The default settings are that 8 spline segments are used to represented non-conic-asphere surfaces, and the approximate accuracy of the export is 10-4 lens units. Since lens units in this file were millimeters, the approximate accuracy is 0.1 micron. This number should be compared to the manufacturing method used to make the component, and is normally reduced only if the manufacturing method employed justifies it. These settings yield the following optical performance:
It can be seen that eight low-order splines do not describe the r6 order asphere particularly well. However, the basic shape of the ray-fan is reasonable: there is a lower amplitude region of balancing (oscillating?) performance and a flaring tail at the edge of the pupil. The spot diagrams are very similar, and both are well within the diffraction-limited region.
We must not fall into the trap of thinking that just because we can see a difference in performance, that the difference must be important and must be eliminated! In reality, only the engineer responsible for meeting specification can decide whether this difference is significant or not. In this example, the author has chosen a very sensitive way of showing differences. Other analysis features may not be so sensitive. The MTF performance, for example, is unaffected by the use of the CAD object, because the design is still well within the diffraction-limited performance range:
So always assess the quality of the CAD export using the real-world, physically-significant performance criterion for your optical system, and not just some test that shows a difference!
Nevertheless, if we export the lens again, using 32 spline segments and an approximate accuracy of 10-5 (0.01 microns, 10 nanometers!) we obtain the original ray-fan plot, with only some very small ripple on it caused by the failure of 32 low-order polynomials to perfectly describe a single higher order polynomial:
Further tightening of the export settings can reduce even this ripple until a ray-fan indistinguishable from the original native ZEMAX asphere results. However export settings tighter than the manufacturing method can provide should not be used.
The ray-fan diagram shows balancing focus, spherical, and higher-order spherical, and unbalanced r8 higher-order-higher-order spherical:
The spot diagram shows that the performance is diffraction limited:
This lens is then exported using the default settings:
The default settings are that 8 spline segments are used to represented non-conic-asphere surfaces, and the approximate accuracy of the export is 10-4 lens units. Since lens units in this file were millimeters, the approximate accuracy is 0.1 micron. This number should be compared to the manufacturing method used to make the component, and is normally reduced only if the manufacturing method employed justifies it. These settings yield the following optical performance:
It can be seen that eight low-order splines do not describe the r6 order asphere particularly well. However, the basic shape of the ray-fan is reasonable: there is a lower amplitude region of balancing (oscillating?) performance and a flaring tail at the edge of the pupil. The spot diagrams are very similar, and both are well within the diffraction-limited region.
We must not fall into the trap of thinking that just because we can see a difference in performance, that the difference must be important and must be eliminated! In reality, only the engineer responsible for meeting specification can decide whether this difference is significant or not. In this example, the author has chosen a very sensitive way of showing differences. Other analysis features may not be so sensitive. The MTF performance, for example, is unaffected by the use of the CAD object, because the design is still well within the diffraction-limited performance range:
So always assess the quality of the CAD export using the real-world, physically-significant performance criterion for your optical system, and not just some test that shows a difference!
Nevertheless, if we export the lens again, using 32 spline segments and an approximate accuracy of 10-5 (0.01 microns, 10 nanometers!) we obtain the original ray-fan plot, with only some very small ripple on it caused by the failure of 32 low-order polynomials to perfectly describe a single higher order polynomial:
Further tightening of the export settings can reduce even this ripple until a ray-fan indistinguishable from the original native ZEMAX asphere results. However export settings tighter than the manufacturing method can provide should not be used.
Summary
NURBS is a flexible method of describing surfaces. It is exact when modeling conic aspheres, which is great news for optical engineers! This means that spheres, ellipses, paraboloids and hyperboloids can be exported and imported exactly.
Higher order aspheres are modeled using multiple lower-order polynomial segments, and arbitrary accuracy can be achieved by simply changing the number of spline segments and export tolerances. Before simply setting extreme values however, careful thought should be given to the performance of the component in its intended application, and to the manufacturing tolerances that can be achieved in practice.
Higher order aspheres are modeled using multiple lower-order polynomial segments, and arbitrary accuracy can be achieved by simply changing the number of spline segments and export tolerances. Before simply setting extreme values however, careful thought should be given to the performance of the component in its intended application, and to the manufacturing tolerances that can be achieved in practice.