Redrawing an 1888 lens


Dr. Hugo Schröder's concentric lens
British Patent 5,194 (1888)
US Patent 404,056 (1889)

This was the first modern photographic lens, designed in 1888. It was made possible by new optical glasses developed by Ernst Abbe and Otto Schott. This ‘aplanatic’ lens couldn't be manufactured until 1892, though, because of Schott's initial quality control problems with the exotic glass.

Meanwhile in 1890, Dr. Paul Rudolph designed an ‘anastigmatic’ lens using a combination of old glasses and new Jena glasses, but of an asymmetric form (later called the Protar), which was first to see mass production.

Some history
Ernst Abbe: Microscopy by Michael W. Davidson PhD, Laboratory Medicine, August 1, 2009
Schott's Glass by Andrea Sella, Chemistry World, April 29, 2015
The Zeiss Works and the Carl-Zeiss Stiftung in Jena by Felix Auerbach, 1904
Jena Glass and its Scientific and Industrial Applications by Dr. H. Hovestadt, 1902
The Development of the Photographic Objective by Rudolph Kingslake, 1934
Dioptrique Photographique by Eric Beltrando

The original lens

A cross section of the concentric lens:

Schröder's patent doesn't describe a specific lens, but sets out design guidelines. The listed dimensions are from Eric Beltrando's recreation of the concentric lens, included among hundreds of other historical lenses at Dioptrique Photographique. Beltrando calculated some data and two pages of spot diagrams for it (you can click each thumbnail to pop up a full size spot diagram).

A variant using modern glasses

Can we make make an improved lens of this type, with currently available optical glasses?

One of the historical Jena glasses, first listed in the 1888 catalog supplement, No. 56 O.381 Dispersive Crown, is within the range of refractive index and dispersion specified in Schröder's patent. The closest modern equivalent is Schott N-KF9.

For the other glass in the new lens, several types from Schott, Ohara and others fall within the range that Schröder gave. The glass used here, N-SK16, is a bit outside that range. Modern equivalents of early Jena glass are now inexpensive and consistent, but a few still have poor resistance to moisture and chemicals and should have a protective anti-reflection coating, and that's true of N-SK16.

My unconcentric variant with RoHS glasses, with the same 60 degree field of view:

Old and new lens formulas

Schröder's formula is more compact than mine, for the same focal length and f/20 aperture. Why is this so much larger? Spherochromatism is significant here, the milder radii mitigate it, and the element thicknesses follow from that. Moving the stop increases the diameter of the front elements, but gives better correction.

It's still relatively small and light. People usually use f/22 or f/32 with view camera lenses, and this one doesn't need to be fast enough to show a focusing image on a ground glass. The original lens would be 18mm (0.71 inch) in diameter for a 180mm lens for a simple 5x7 box camera. The new version would be 27.7mm (1.09 inches) in diameter.

The original is completely symmetrical. My variant is symmetrical except that the stop isn't exactly in the center. It also departs somewhat from concentricity.

Beltrando's historical example appears to reach its highest resolution at about f/32, while the new lens' best resolution is at about f/19. One reason is that N-SK16 pairs better with N-KF9's dispersion curve in this design than did the glass types that fell within Schröder's design parameters in 1888.

Image quality

These spot diagrams show the same six wavelengths that Schneider uses in their resolution charts. For reference, the Airy disks drawn around the spots are for 546nm at f/20. But the spots themselves are purely geometric and do not show the effects of diffraction.

The on-axis spot, in the center of the image, is at upper left. 30 degrees off axis is at the end of the bottom row:

  405nm 436nm 480nm 546nm 588nm 644nm
    7.1% 15.2% 12.4% 24.6% 22.1% 18.6%

The next plot shows spot diagrams out to 350nm, because some orthochromatic or panchromatic black and white sheet films have lots of violet and ultraviolet sensitivity.

Schott N-KF9 glass has good transmittance at 365nm, but lower at 350nm — roughly 90% total through the thin centers of the two negative elements and 75% at the thicker edges.

On axis, at the center of the image, is the top row. 30 degrees off axis is the bottom row:

These spot diagrams show 350nm to 665 nm. The Airy disks are for 539nm at f/20:

Point spread function at f/20 — for 0°, 6°, 12°, 18°, 24°, 30° off axis — also before considering diffraction:

405 - 644nm (0° at upper left, 30° at lower right)

350 - 665nm (0° at upper left, 30° at lower right)

Astigmatism isn't corrected quite as fully as it could be here, as a compromise with overall image quality:

Lateral color:

Secondary color:

Distortion is greater than in the original (which Beltrando gives as 0.08%), but it wouldn't be conspicuous for pictorial use. It's 0.06% at 50% out from the center of the field, 0.13% at 70%, and 0.31% in the corners.

These modulation transfer function (MTF) graphs show how much contrast is preserved in the image, compared to the original being photographed, at any given fineness of detail. Here, “0.6 modulation” means 60% contrast, “0.4” means 40% contrast, etc.

Coarse detail at only a few line pairs per millimeter can be imaged at nearly the original contrast. Detail at 6 line pairs/mm is reproduced with about 90% contrast (in the upper left corner of the diagram), but detail at 30 lp/mm is reproduced with about 50% contrast over much of the field. Beyond that, finer and finer detail is reproduced at lower and lower contrast, because of the effects of lens abberations and diffraction. Finally, past the diffraction limit, everything becomes a fuzzy detail-less blur.

The diffraction limit (Rayleigh limit at 9% contrast) for any f/20 lens is about 75 line pairs/mm in green light. The average of the wavelengths shown here is a bit shorter than that. In the diagram, 9% contrast is shown by the red line at .09 modulation.

At f/20, resolution is not quite diffraction limited on axis, decreases to about 65 lp/mm at 22° off axis, and falls to about 50 lp/mm at 30° in the corners. An actual lens as built would not reach these numbers:

Close focusing
The ‘unconcentric’ lens is designed to focus at infinity. It retains good sharpness at distances down to about 1 meter, but becomes less sharp at 500mm and closer focusing distances.

The first chart, at left, is for focus at one meter. The second chart, at right, is for focus at infinity.

MTF at 10, 20 and 40 lp/mm — lens image in air, neglecting manufacturing tolerances, etc.:

  405nm 436nm 480nm 546nm 588nm 644nm
    7.1% 15.2% 12.4% 24.6% 22.1% 18.6%


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