The Schoenberg Lens
Background
Lens design has much in common with composing music. A successful optical system is more than a collection of individual lens elements – it is a carefully orchestrated arrangement in which each component plays a distinct role while contributing to the performance of the whole. A small change to one element can influence the entire system, much like changing a single note or instrument can alter the character of a musical composition. Although lens design is grounded in mathematics and physics, creating an elegant optical system also requires creativity, intuition, and a sense of balance and harmony. The most memorable designs, like the most memorable music, achieve far more than their individual parts.
In 1923, the musical composer Arnold Schoenberg developed a musical technique he called the “law of the twelve tones.” This required all twelve notes of the chromatic scale to be sounded equally often while preventing the emphasis of any one note. There were many rules associated with it, but the main idea was that for every successive group of twelve notes, each note in the chromatic scale could be played only once. Thus, all twelve notes were given equal importance.
Problem Description
The Schoenberg Lens applies this technique to lens design for the ten digits 0-9, where each of the 10 digits are given equal importance. For each lens in a lens system, there are four parameters (not counting the glass type): the front and back radii, the center thickness, and the spacing to the next element or image plane. For each lens in a Schoenberg Lens, the values of these four parameters use the 10 digits 0 to 9 only once. All 10 digits must be used and can be in any order (radii can be positive or negative). No digit can be used again until the next lens. The digit 0 must have significance, either as a leading digit (e.g., 0.5) or embedded in the other digits (e.g., 10.2 or 1.05). A trailing zero, such as 2.40, does not count as a use of the digit 0. A plano surface, which is conventionally represented in lens design programs as a radius of 0, counts as a use of the digit 0.
For example:
| Radius | Thickness | Glass |
|---|---|---|
| 15.3 | 2.8 | Index = 1.49032 @587.6 nm |
| -40.6 | 7.9 |
You can have more digits after the decimal (e.g., a center thickness of 1.023), but this takes digits away from the other parameters. Similarly, you can have several digits before the decimal (e.g., a radius of 1483), but this removes all those digits from being used by the other parameters. Note that for a given lens, the lens thickness and the spacing to the next lens cannot both be less than 1, as this would require two uses of the leading digit zero.
Problem Statement
Design a lens with maximum etendue where for each lens element, the values for its two radii, center thickness, and spacing to the next surface are formed using all the digits 0 to 9 only once – the “Schoenberg Rule.”
Specifications
| Focal length: | 98.7 ± 0.1 mm |
| Overall length: | ≤ 523.46 mm from the first optical surface vertex to the image. |
| Entrance pupil diameter: | Not specified. |
| Semi-field of view: | Not specified. |
| Wavelength: | Monochromatic at 587.6 nm. |
| Glass: | Index = 1.49032. |
| Number of lenses: | Not specified. |
| Maximum diameter: | Not specified. |
| Lens type: | All refractive; no reflectors or TIR. |
| Lens configuration: | Rotationally symmetric. All lenses must have non-negative thicknesses, non-negative edge thicknesses at the clear apertures, and non-negative spacings at the axis and at the clear apertures. |
| Cemented doublets: | Not allowed. |
| Intermediate images: | Not allowed. |
| Surface shape: | Spherical or plano only (no aspheres or diffractives). |
| Object: | Flat, at infinity. |
| Image: | Flat image plane in air. |
| Image clearance: | Non-negative; this includes axial distance and edge thickness. |
| Stop location: | Must be on a plano dummy surface between two lenses. |
| Vignetting: | The stop surface must be fully filled over the field of view. |
| Dummy surfaces: | Only two dummy surfaces are allowed in the system. The first is a plano stop surface whose radius of infinity counts as a “0” and the airspace after the stop needs to use the rest of the 1-9 digits. The second is optional. A dummy surface may be inserted before the final image plane, again with a radius of infinity (counts as a “0”), and the airspace after the dummy surface to the final image plane must use the rest of the 1-9 digits. Note the airspaces before each of these dummy surfaces are still part of the Schoenberg Rule for the lens preceding the dummy surface. |
| Distortion: | ≤ 9.865432% |
| Image quality: | RMS wavefront error ≤ 70.1 milliwaves over the field of view (piston and tilt removed, focus not removed). |
See the Frequently Asked Questions (FAQ) on the IODC web site for more details on the specifications.
Merit Function
The merit function for comparing entries is the etendue of the lens (entrance pupil diameter in mm times the semi-field of view in degrees).
Submissions
The coveted Shafer Cup will be awarded to the entry with the highest merit function and a separate Shafer Cup will be awarded to the entry with the highest merit function from a student (undergraduate or graduate).
Send your entry to bentley@optics.rochester.edu. Entrants may submit more than one entry, but only the one with the highest merit function will be considered. Please include the following information with your submission:
- Name,
- Affiliation (if an educational institution, indicate if you are a student),
- Country or nationality,
- Approximate number of years of lens design experience you have,
- Lens design program(s) used,
- Lens files (text format) or lens prescriptions,
- Lens layouts (to help the evaluators verify the prescription is correct),
- Your value for the etendue (although it will be verified by the evaluators),
- Approximate number of hours you spent on the problem (not counting the time any global optimizers were grinding away on their own), and
- Indicate whether you used a global optimizer on the problem or not.
- (Optional) Describe your design methodology (warning – may be used in the talk and/or in the written paper if it is unique enough or interesting enough).
Lens files for CODE V and Zemax can be read directly by the evaluators. For all other programs, include lens prescriptions in a readily understandable text format. All entries will be converted to CODE V format for common verification of compliance to the specifications and evaluation of the merit function.
All entries must be received by midnight, Eastern Daylight Time, April 1, 2027 (midnight Universal Time plus five hours). If you have any questions about the problem, refer to the frequently asked questions (FAQ) page on the IODC web site, or contact Julie Bentley at bentley@optics.rochester.edu.