Asphere Tolerancing

Written by Scott Sparrold Still under construction. (last edited 9-21-19)

The use of asphere has been on the rise in the last couple of decades. The process from design / tolerancing to manufacturing and testing is still evolving. This page will quickly outline how to tolerance an asphere in your system. Editorial: machining aspheres can be very difficult. Aspheric design is easy though:) The interplay between lens designer / tolerancer and the fabricator is critical to achieve the best optical performance!

Tolerances to include

Reference Equation and table
It has become standard practice to include the aspheric equation on the lens print, with the prescriptions in terms of radius and aspheric terms. It is advisable to supply a reference table of sag deviation versus semi-aperture. This allows the fabricator to ensure he has programmed the correct aspheric equation. Errors can be caused if the asphere is on the back surface of a lens or if significant figures get truncated or rounded from unit conversion. For reference, when you flip the lens, the radius and aspheric polynomial coefficients will change sign, while the conic constant will NOT change sign. Be very careful how the aspheric terms get rounded off between your lens prescription (in your software model) and subsequently transferred to a print.

Radius tolerance
“Radius” measured on an asphere is sometimes referred to as base radius or vertex radius and is specified as a ± value. Do not use a test plate radius tolerance and “power” for controlling radius: this is common for spherical production but creates ambiguity and difficulties for asphere manufacturers. Power or radius alone is sufficient. This value will be used to “radius optimize” the aspheric surface form error during metrology. Some lens designers allow radius optimization during aspheric surface form measurements, while others do not allow it. Personally, I allow it, otherwise you are asking your asphere fabricator to make a “perfect” radius of curvature. A change in radius is a change in total system power and will shift your systems focus slightly and probably not shift your aberration balances drastically. For the sake of keeping our asphere fabricators sane, please provide a single radius tolerance.

Surface Roughness
This is the measure of “micro” roughness and is measured on a very small patch of your optical surface. It does NOT control the macro shape of the asphere. Rather it is used to control surface scatter that will cause stray light in your system. There are two common parameters for surface roughness (insert link to taylor hobson here):
    1. Ra is the “..arithmetic mean of the absolute departures of the roughness line from the mean line” (insert equation) 
    2. Rq is the RMS (root mean square) parameter corresponding to Ra (insert equation) 
Polished surfaces have better surface roughness than say a diamond turned part. Typical values one can expect:
    • Ground glass: Rq = 5 to 100 angstroms # 
    • Polished glass, speed / pitch polishing : Rq = 1 to 30 angstroms # 
    • Polished glass, computer controlled: Rq = 0.05 angstroms # 
    • MRF (magneto-rheological finishing): Rq = 0.03 angstroms # 
    • Fluid jet: Rq 0.05 angstroms # 
    • IBF (Ion beam finishing): Rq = 0.02 angstroms # 
    • Injection molded plastic: Rq = 20 to 40 angstroms * 
    • Diamond turned metal : Rq = 0.5 to 2 angstroms  # 
    • Molded glass: Rq = 20 angstrom commercial, 10 angstrom precision, 5 angstrom manufacturing limits * 
    • Molded Chalcogenide glass: Rq = 1 angstroms typical * 
    • Diamond turned metal, post polished: Rq = locate reference 
* Reference source is “Molded optics, design and manufacture
# Reference source is “Advance optics using aspherical elements", table 5.1”

This is usually measured with a surface contact profilometer or a Mirau interferometer. This is NOT used to control scratch / dig… that is another specification and is not discussed here.

Surface Form Error
This is the sag deviation from a perfect asphere usually plotted versus clear aperture (sometimes diameter). Again, please allow your fabricators to optimize the base radius of curvature on an individual measurement to minimize this error. A good fabricator will ensure this tuned radius will not go outside the tolerance radius of curvature. This can be measured by:

1. A surface contact profilometer single cross section
2. A co-ordinate measurement machine(pretty sure that is coordinate-measuring machine, not co-ordinate) CMM which will get a surface map
3. A spherical interferometer with a null lens (diffractive, aspheric or spherical) which provides a surface map
4. A sub aperture stitching interferometer. SSI or ASI from QED or Verifire from Zygo. Provides an interferrometric map of the surface without contacting the surface and without the hassle of qualifying a null lens

There are several methods for controlling the surface form error. They are presented here and how a fabricator will interpret them.

1. Irregularity
This nomenclature is a hold out from test plate fitting and interferrometric readings from spherical surfaces. It is not the recommended method for aspheres, although some designers may use this. The irregularity will be converted to a surface sag deviation. A passing asphere will have a surface deviation from nominal whose total peak to valley error over the entire clear aperture is less than this sag deviation. 
This specification is a function of wavelength and can lead to errors if this is not clearly specified on the print.

Irregularity has been used with great success on spherical surfaces for decades if not centuries. It is a rather loose term, because the error could take many forms such as spherical (aperture^4), astigmatism, coma, trefoil, a bump or a divet, etc… In the late 2000s, I worked in an optics fabrication factory and I was quite surprised to hear that the conventional fabricators (spherical parts) included an aspheric term to irregularity... They described it as a rationally symmetric error where the fringe spacing across the aperture was off. Zemax models irregularity as 50% saddle (astigmatism) and 50% spherical. Code V models this as two cylinders at 45° roll relative to one another. I would love to see a distribution of these various errors on spherical and aspheric parts. In other words is the 50/50 split of spherical to saddle a good representation of a spherical / aspheric surface?

2. PV, PTV or Peak to Valley
Similar to irregularity but specifies linear deviation directly. If you allow 1 µm, the fabricator will not pass parts with a aspheric deviation > ±0.5 µm. Again this error can take any type of surface deformation such as coma or astigmatism. It eliminates the need for reference wavelength and is preferred over irregularity.

Special case: Some designers will specify the allowed PV for types of surfaces, such as 1 µm for spherical and 0.25µm for astigmatism. Or 1 µm for rotational errors (4rd order and higher spherical) and 0.25µm for non rotationally symmetric errors (all orders of coma, astig, trefoil etc…)

3. PV and RMS
This specification captures peak to valley (previous note) and RMS (root mean square) deviation across the part. Suppose you specify a 1 µm PV across the part. Without an RMS the error could be a pure cosine wave from the center to the edge. If it has one period across the semi-aperture it might not hurt your optical performance. If you have 100 periods it could be catastrophic. The added RMS specification is intended to control the “waviness”. RMS values can differ depending on if it's a cross section or an area basis.

4. PV and slope
What the rays / wavefront really care about is the localized surface slope. Some designers will specify a slope and / or a PV. The slope is usually a deviation over a given sub-aperture length. The slope can vary widely over the length interval selected. Some practitioners may even specify a slope plotted as a power spectral density versus sample length.

5. PVr
Chris Evans at Zygo has suggested a new metric for aspheric departure, PVr See link here.

6. Structure Function
Robert Parks has a new way to simulate aspheres using the structure function. See link here.

7. Rate of change of slope, derivative of slope or second derivative of the surface

This controls the inflection or or how rapidly the slope changes over the aperture. I'm still processing the importance of this spec

Implementing asphere tolerances in ray trace codes
There are many ways to tolerance an asphere and I will quickly outline them with advantages and disadvantages:

a. Power and irregularity: Power is simple radius error. The simple irregularity is modeled as a difference in radius in the X direction versus another in the Y direction and is therefore tolerancing pure astigmatism. The pitfall is that an irregularity spec could be interpreted to also control spherical aberration as is the case on a spherical part. The value of irregularity will be applied to a PV measurement by the fabricators / quality engineers (usually not interpreted as a root mean square value)
i. Zemax use “TIRR” for irregularity. Use TRAD or TFRN for radius (but not both). Irregularity is split 50/50 to Spherical and astigmatism
ii. Code V use IRR which creates CYN and CDN terms. Use DLF or DLR for radius (but not both). Irregularity is purely astigmatism and contains no spherical term. I my humble opinion this is a harsh tolerance for both spheres and aspheres. I actually do tolerance runs with and without irregularity tolerances. Sometimes the irregularity tolerances are simply ignored.

b. Aspheric terms and conic constants. These terms do not easily transfer to a total sag deviation error. I have seen the max sag deviation at the edge of the aperture from individual terms be RSS together to provide a total PV specification.
i. Zemax use TEDV line for each aspheric term. (TEDV = tolerance extra data values)
ii. Code V use DAK for conic, DAA for 4th order, DAB for 6th order etc….

c. Zernike orthogonal circle polynomials. These can be used to tolerance sag deviations. One or multiple terms can be used.   and  This author uses the Zernike spherical term to simulate a PV error. It includes a power term so it models the typical “gull wing” or “W” shape that the pofilometers measure after radius optimization. High order terms could be used to simulate a particular RMS value. (Add: Table showing RMS to PV ratios for each Zernike term. And input a conversion factor from Zernike coefficient to total sag departure for each Zernike term. And capture max slope for those who like to tolerance slope) There are two types of Zernike polynomials (fringe and standard): I will not be discussing the pros and cons of each here.
i. Zemax use TEXI for fringe Zernike or TEZI for standard Zernike.
ii. Code V use ZFR for fringe Zernike or ZRN for standard Zernike.

d. Forbes Asphere Coefficients. Greg Forbes from QED has proposed a new surface for designing, fabricating and testing aspheres. Their aspheric terms provide more intuitive understanding of what the surface looks like and how easy it is to fabricate. There are two types and I will leave it to the expert to describe the differences: QCON and QBFS.
i. Zemax models this surface as a user defined surface and one cannot tolerance them (at least I have not figured out how to tolerance these terms)
ii. Code V use QCN or QBF to tolerance these.****
**** Check out Code V’s asphere expert for “radical control” for your asphere! It helps locate optimum surface(s) for the asphere as well as the ability to easily constrain total departure and / or maximum slope. The max slope is critical to fabrication.

e. Pure cosine ripple. A periodic function could be used to tolerance a cosine function across your optical surface. This is the simplest form to intuitively model and understand RMS
i. Zemax requires a user defined surface
ii. Code V use RPA for cosine amplitude and RPS for cosine slope

f. Tolerance RMS directly.
i. Zemax Knowledge base recommends using the "TEZI" command and be found in a discussion here.
ii. Code V use RSE. Synopsys has assured me this is an RMS value as measured over a surface. (Other metrology engineers have reported slight differences in measured RMS values between a profilometer and a stitching interferometer)( are you talking about the difference of a linear and area RMS? The math is definitely different between the two). The RSE term is simple to use in sensitivity analysis (TOR). If one wants to use it for a Monte Carlo evaluations, Prateek from Synopsis has create a RMS from an FFT macro and it outlined in the 2017 Code V user group conference on using inteferrograms. Contact support for more information.

e. Slope derivative and Radius

Comment on PV to RMS ratios
Edmund Optics aspheric manufacturing engineers have noticed that the ratio between PV and RMS seems to be relatively constant based on how the asphere is fabricated. Here is a PRELIMINARY list of these ratios (quoted as PV to RMS):
  • 5:1 for molded glass optics whose tooling is directly from precision grinding 
  • 7:1 for molded glass optics whose tooling has been post polished 
  • 4.6:1 for computer controlled polished glass after radius optimization 
  • 3.6:1 for computer controlled polished glass without radius optimization 
  • 10:1 or higher (TBR) for MRFed parts (data point was a glass sub aperture pad polished F/2 parabola, post MRFed)
  • 2:1 to 6:1 for diamond turned parts. Variation in the ratio dependent on part, departure, aperture, geometry, etc..
  • TBD for injection molded plastic 
The feedback loop between design / tolerancing, fabrication and testing for aspheres is in still an evolving process in the optics industry. This author has successfully toleranced, procured, tested aspheres, integrated them into a full system and tested optical system performance. This was done by using PV and RMS surface form errors on the asphere drawing. PV was modeled in Code V using a Zernike spherical term for total departure. RMS was modeled using Code V’s “RSE” term. Predicted optical system yields were in line with actual measured hardware. 

Further work: 
Would like data, opinions or commentary on PDF (probability distribution functions) of PTV, irreg, RMS, etc...

Further Guidance for Aspheric Design for Lens Designers
Please do not design with a radius of curvature, a conic and a rho^2 term (rho = semi-aperture height). These are all “competing”. Zemax allows for a rho^2 term and it is “fighting” against the radius and conic. In other words a rho^2 term can be modeled with a radius and a conic. Some people argue that a radius, a conic and a rho^4 term also fight with each other, but this author has seen cases where having all three are beneficial (F/# < 0.8 for catalog singlets).( I would also add that you simply can’t test radius when there is a rho sq. term so you can’t have a radius tolerance and a rho sq. term)

If you plan on making your asphere with computer controlled manufacturing (sub-aperture pad polishing) and /or MRF the surface, please do not put the asphere on a concave surface. Also do not allow the local radius of curvature to get < 20 mm. Do not allow inflection points (local radius hits an asymptote)… or convex at the vertex and concave at the edge. A future blog entry will discuss this in more detail. See Zemax macro attached to bottom of page for calculation of local radius of curvature (written by E. Herman). Beta Code V macro available: contact

Post Edit: OptiPro machines use a ribbon as a polishing pad - kind of like a band saw. These are able to do some concave aspheres and some inflection aspheres. Ed Fess cautions that you still want to maintain the |local radius| > 15 mm.

If you plan on using injection molded plastics, the PV, RMS and especially the radius of curvature need to be large to accommodate large fluctuations in the fabrication process. It is advisable to diamond prototypes to save tooling costs in case your design needs to be modified. Once you have a plastic asphere design completed, the injection molder will need time to iterate the tooling due to part shrinkage. This is dependent on many competing factors, such as mold tooling, processing, machine quirks, sun spots etc….. Do not expect a plastic part to come out consistently from production run to production run. If you need tight tolerances, glass mold, machine or diamond turn your parts.

If you plan on glass molding, please account for the index of refraction drop due to the molding process. Consult your supplier for how much the index will drop based on what glass type you are using.
Scott Sparrold,
Sep 14, 2011, 8:37 AM