Introduction

When considering how to design coma correcting eyepieces, I started from the expectation that the number of elements in the modern widefield eyepieces – typically 7 to 10 – would allow the coma correction to be built-in without adding new elements, and without adding significantly to the cost of the eyepieces.

To test this hypothesis, I created an eyepiece designer application. I had already written some ray-tracing and optical system visualization code back in 1999, using Delphi, and I recently started revamping this code – literally 24 years after last touching it. I added an eyepiece quality scoring system and automated optimizer so that optical designs could automatically be improved. The result is an application called the “Cruxis Eyepiece Designer”.

The goal of the Cruxis Eyepiece Designer is to optimize eyepieces by adapting the positions and powers of the optical surfaces, and selecting the best glass types, so that an overall “eyepiece quality score” is optimized.

For each configuration the Eyepiece Designer traces many thousands of rays at various wavelengths. Usually red, green, and blue rays are used, but any combination of colors is possible. Rays are traced from the objective (a parabolic mirror or a perfect flat-field objective) towards the eyepiece.

Behind the eyepiece, at the exit pupil, the direction and distribution of the rays coming out of the eyepiece are converted to a spot diagram as the human eye would see it, allowing for some limited focus accommodation of the human eye.

The spot diagram, magnification, eye relief, curvature and vignetting for a series of field angles (typically 10 field angles to cover the whole field) are combined and weighted into a single score that expresses the “quality” of the eyepiece design.

Eyepiece Quality Score

• Spot diagram. The size of the diagram showing all the rays as they are seen by the human eye at the exit pupil is the most important quality factor of an eyepiece. The theoretical resolution of the eye is around 1’ (arc minute), but in practice up to about 5’ is great performance for an eyepiece, and at the edge of the field up to 10’ is considered very good. The Eyepiece Designer uses the RMS of the rays for minimizing the spot diagrams but also shows the spot diameter that contains 80% of the rays.
• Eye relief. Around 18 to 20 mm eye relief makes an eyepiece usable with glasses, and this can be achieved by 80° eyepieces. For 100° eyepieces the practical limit for the eye relief is around 15 mm; beyond the eye lens would become huge. Eyepiece manufacturers usually specify the eye relief for the central part, when in reality the eye relief at the edge of the field is most relevant. The designer takes into account the maximum and minimum eye relief across the field.
• Spherical aberration of the exit pupil (SAEP). The eye relief should be as constants as possible across the whole field, to facilitate the eye positioning and avoid “kidney bean” effect. The Designer limits the difference between the highest and lowest eye relief to less than 15%.
• Angular magnification distortion (AMD). In wide field eyepieces for astronomical observing the best solution is to have minimal AMD, so that the radial magnification at the edge of the field is identical to the center. In other words, a planet at the edge of the field would have the same size as in the center.
  Note that lack of AMD produces pincushion distortion which is acceptable for astronomical observations.
• Field curvature. Field curvature is accommodated by the human eye. The eye has focus capabilities that diminish with age from over 10 diopters at age 15 to around 1 diopter above age 60. For an eyepiece up to 1 diopter of accommodation is deemed acceptable.
• Vignetting. Some vignetting at the edge of the field is acceptable. At the edge of large fields the vignetting by the secondary mirror will dominate, usually the eyepiece will not really add to this.
• Size. Smaller is better. To push the designs towards “small” a weight is given to the total distance between the first and last glass element. This prevents the optimization from needlessly growing the eyepiece to achieve tiny gains.
• Transmission. Light passing through glass is partially absorbed. A good eyepiece should have very high transmission in the visual spectrum. This boils down to picking the correct glasses (some glass types are OK in red and green but have poor transmission in blue), and making the glass path lengths as short as possible.
  Note that this does not take into account reflection losses, for which multi-coating is needed on all the surfaces.
• Weight. Lighter is better for an eyepiece! Like the transmission, it boils down to picking the correct glasses (glass density varies from 2.4 to 5.5) and keeping the eyepiece as small as possible.
• Cost. The total glass cost of the eyepiece depends on the mass and the relative price of each lens. It would be interesting to include grinding and polishing costs for each glass element depending on their dimensions, but currently I have no accurate information about this.

The Eyepiece Designer takes into account all these parameters with configurable weights so that one can easily nudge the design towards any target. The final solution will be the best compromise of all the factors above – by adapting the various scoring weights the application can force a design to go towards being smaller or cheaper, have more eye relief, less AMD, less curvature or tighter spot diagrams.

The optimization uses a Monte Carlo approach with random changes applied to every optical surface to move the optical design towards a better quality score.

The Eyepiece Designer uses the two main glass catalogs (Schott and Ohara) and can improve the eyepiece by selecting different glasses. You can filter the glasses on their relative price and their transmission in green and blue. Typically about 40 glass types from are considered in the optimization process.

Compared to generic optical system design solutions like Zemax or Oslo the Cruxis Eyepiece Designer adds the specific know-how about eyepiece quality directly into the optical design evaluation, so that the automated optimization takes into account all factors that matter for an astronomical eyepiece and steers towards a design that will work.

The Eyepiece Designer can scale eyepieces to different power or field targets. For example, starting from a good 12 mm eyepiece it's easy to create an eyepiece line from, say, 8 mm to 18 mm. Or starting from a 100° design it's easy to scale down to 85° field of view.

User Interface

The application looks like the following:

The screen shot shows a design for a 16 mm 100° eyepiece in a 300 mm f/4 Newtonian telescope.

The Eyepiece Designer uses a simple text file to describe the optical system and the optimization control parameters. There are different sections for the optical layout (lens position, thickness, curvature, glass type) and for the optimization settings.

The settings contain various parameters including the weights for the Eyepiece Quality Score, for example:

Settings
    TargetPower=75 MultiCPU=1
    TraceAngle=0.667 TraceSteps=9
    PencilType=HexaPolar Wavelength=RGB
    RaysGreen=1800 RaysRed=600 RaysBlue=600
    SpotScore=1 PowerScore=4 PowerVarScore=0.4
    EyeReliefScore=0.0002 EyeReliefMin=17 EyeReliefMax=20 EyeReliefUsable=0
    SaepScore=0.05 SaepAllowed=1.17
    CurvatureScore=0.003 CurvatureAllowed=0.6
    DistortionScore=0.01 VignettingScore=0.01
    LengthScore=0.00001 MassScore=0.001 CostScore=0.001
    OptDeltaCurvature=0.0002 OptDeltaPosition=0.1 OptDeltaRadius=0.2
    OptFixedPct=70 OptShiftAllPct=2
    MinimumEdge=2.5 MinimumAir=0.1
    GlassStart=5 GlassBrand=Schott GlassMaxPrice=3.6 GlassSkipDuplicate=1
    GlassMinTransm546=0.99 GlassMinTransm460=0.975

The first output for a single configuration is a table of spot circle sizes, eye relief, curvature, vignetting, and radial and tangential powers at the various field angles. Usually 10 field angles are analyzed to cover the whole field.

Below the results for the 16 mm 100° eyepiece at f/4:

   Angle    Field  Spot Circle     RMS    Relief   Curvature   Vignet   Power rad/tan
   0.00°    0.07°   1.0'x 1.0'    0.42'    19.49    0.242 /m    0.00%    74.86/ 74.86
   0.07°    5.55°   1.1'x 1.2'    0.45'    19.43    0.288 /m    0.00%    74.88/ 74.75
   0.15°   11.10°   1.1'x 1.6'    0.52'    19.28    0.415 /m    0.00%    74.94/ 74.42
   0.22°   16.66°   1.2'x 2.0'    0.64'    19.02    0.592 /m    0.00%    75.04/ 73.87
   0.30°   22.22°   1.2'x 2.5'    0.77'    18.65    0.771 /m    0.00%    75.21/ 73.10
   0.37°   27.80°   1.4'x 2.9'    0.93'    18.17    0.898 /m    0.00%    75.39/ 72.13
   0.44°   33.40°   1.7'x 3.5'    1.14'    17.58    0.930 /m    0.00%    75.45/ 70.92
   0.52°   38.97°   2.1'x 4.0'    1.31'    16.92    0.869 /m    0.00%    74.96/ 69.47
   0.59°   44.48°   2.5'x 4.7'    1.51'    16.30    0.774 /m    0.00%    73.27/ 67.71
   0.67°   49.83°   2.7'x 4.8'    1.60'    16.46    0.291 /m   13.11%    71.89/ 65.64

  Spot Diagram     0.002702
  Power Target     0.000014
  Eye Relief       0.000097
  SA Exit Pupil    0.000032
  Field Curvature  0.000023
  Distortion-AMD   0.000024
  Vignetting       0.000066
  Size             0.001125
  Glass mass       0.000476
  Cost             0.000867
  =========================
  Total Score      0.005426

Performance

A typical ray trace of a system will consist of 3000 rays (1800 green, 600 red, 600 blue) for 10 different field angles, in other words 30,000 rays are traced to compute a single Eyepiece Quality Score.

On a modern CPU the application can perform about 10 to 15 million ray traces per second and evaluate the Eyepiece Quality Score for about 400 layouts per second.

Optimizing an eyepiece layout without changing its glass types is very fast, it will typically take less than a minute. On the other hand, optimizing the glass types is very lengthy and can take several hours, processing millions of configurations.