Lens Scaling: What it is and why it is important

Gerald · Joined: 25 Mar 2014 Posts: 1196 Location: Brazil

Scaling can be defined as a geometrical operation in which all the dimensions of an object are multiplied by the same factor. What happens when a lens is scaled? From a conceptual standpoint, the question is interesting because it allows a better understanding of the factors that affect the realizability of a lens. For example, scaling can explains why a 500mm F2 wouldn't be practical with the current optical technology. From a practical point of view, lens scaling can be used, for example, as a starting point to redesign a lens for a format different from the original.

When a lens is scaled, but the optical and mechanical materials used in its construction are kept the same, some optical and physical parameters don't change, some simply change in proportion to the scaling factor, while others change dramatically.The list below shows what happens with several parameters when a lens is scaled up by a factor of 2, that is, when all linear dimensions are multiplied by 2.

Diameter, length ....................................... 2x
Area ........................................................ 4x
Volume ….................................................. 8x
Mass, weight ............................................. 8x
FL (Focal Length) ...................................... 2x
Aperture .................................................. same
IC (Image Circle) diameter.......................... 2x
Angular coverage ...................................... same
Light collecting capability............................ 4x
Aberration (blur) absolute .......................... 2x
Aberration (blur) relative to IC diameter ...... same
Distortion ................................................ same
Resolution absolute (lpmm) ....................... 1/2x
Resolution relative (lwph) .......................... same

Suppose, for example, that a Pentacon 50mm F1.8 lens is scaled by a factor 2. The focal length changes to 100mm, but the maximum aperture remains the same F1.8. The optical parts of the Pentacon 50mm weighs 63 g, so the weight of the optics of the scaled "Pentacon 100mm" would increase to 504 g! By the way, if you had used a scaling factor of 4 instead of 2, the weight would increase by a factor 4x4x4 = 64. This way, just the optics of a "Pentacon 200mm F1.8" would weigh 4032 g! However, this is not too far from 3010 g, the weight of a Canon 200mm F1.8. You could argue that 3010 g is the total weight of the Canon lens, not the weight of the optical parts alone. It is true, but it incidentally shows why Double Gauss is not the most appropriate scheme for telephoto lenses. Indeed, most real telephoto lenses use asymmetrical structures which are lighter than a Double Gauss.

Of course, scaling also works in the opposite direction (scaling down). When you go from FF format to M43, for example, focal lengths are divided by 2, but the weights of the optical parts as predicted by scaling are reduced by 8. This remarkable weight reduction is one of the motives for Olympus and Panasonic having embarked on developing the M43 system. An Olympus 40-150mm F2.8 weighs very reasonably 880g. Can you imagine the weight of an 80-300 F2.8 for FF?

Another example: superzoom cameras use sensor with dimension of 1/2.3", whose factor is 5.6x compared to FF. The "scaling theory" predicts a weight reduction of 5.6^3, which is about 175 times. Consider now a 15,700 g Sigma 200-500 F2.8 lens which is scaled down from FF to the format 1/2.3". The scaled down lens would have a weight of only 90 g, which I believe is not too far from the real weight of the zoom lens of a Panasonic FZ200/300 (24-600mm eq).

Please feel free to make any comments, raise questions, or add some interesting information about lens scaling.
Later I intend to present some real lenses that were designed using lens scaling.

Himself · Expire: 2013-05-30

Interesting read!
Where did you get the data?

AE Conrady · Joined: 16 Jan 2016 Posts: 16 Location: Hamburg Germany

Lens can be scaled, that is asolutely right and you can find the proof if you study the lens design history. There were times where computing a lens needed month making simply scaling a practical solution. Looking into patent literature you will find that a lot of lens designs given for f=100mm or even f=1. That is because designs can be described by geometrical data.

However, scaling was practical when glass plates and sheet film dominated. For 35mm or 60mm films this is neither good solution, nor is it for digital sensors. Above all, lens abberations can be scaled, but MTF can't. Hence resolution can not be derived by a simple scaling factor although as a thumb rule it might be acceptable.

For the mechanical design rather then the optical design, your assumptions are quite good, but not perfect.

visualopsins · Expire: 2025-04-11

Welcome AE Conrady!

Thank you Gerald. I too like simple summaries of the complicated which retain the technicalities.

Gerald · Joined: 25 Mar 2014 Posts: 1196 Location: Brazil

Himself, visualopsins,
Thank you!

Galileo probably was the first who used the concept of scaling, a concept widely used in modern engineering and physics. It serves to explain many things in nature, for example, "why large mammals like elephants have a harder time cooling themselves than small ones like mice, and why building taller and taller skyscrapers is increasingly difficult." This article about the so-called "square-cube law" is very interesting:
https://en.wikipedia.org/wiki/Square-cube_law

Gerald · Joined: 25 Mar 2014 Posts: 1196 Location: Brazil

Now, a case in which Zeiss explicitly acknowledged that scaling was used in the actual design of a lens. The Distagon 60mm F4 from 1956 for the Hasselblad 1000F was scaled down to the Distagon 35mm F4 from 1958 for the Contarex 35mm. Note that the optical diagrams of the two lenses are identical, but of course, the optical elements of Distagon 60mm F4 are larger by a factor of 60/35 = 1.71.

Zeiss Distagon 60mm F4 for Hasselblad 1000F:

(extracted from: http://lenspire.zeiss.com/en/wp-content/uploads/sites/2/2015/09/en_CLB41_Nasse_LensNames_Distagon.pdf)

Zeiss Distagon 35mm F4 for Contarex 35mm:

(extracted from: https://sites.google.com/site/renatogucciardi/carl-zeiss/carl-zeiss-objektive/distagon)

The design of enlarger lenses is another important case where lens scaling seems to have often been used by optical designers. See for example, the table below with the technical data of the professional Fujinon EP enlarger lenses:

Fujinon EP enlarger lenses:

(extracted from: http://forum.mflenses.com/viewtopic.php?t=46651&view=previous)

The lenses of greater focal length, namely 70mm, 90mm, 105mm and 135mm have all the same aperture F5.6. Their distortions are very similar, and back focal lengths are almost in proportion to their focal lengths. The small discrepancies are due, probably, to the different image scales for which the parameters were measured. It seems pretty clear to me that with the exception of Fujinon 38mm and 50mm, all the other lenses were designed using some form of lens scaling.

AE Conrady · Joined: 16 Jan 2016 Posts: 16 Location: Hamburg Germany

I think it comes down to the point what is meant by lens scaling. If you think of a fully developped design and just try to produce another one by geometric scaling and put it into production without further elaboration, it would not work (at least it did work after late 1890ies). On the other hand lots of lens have been developed by means of scaling, but only as a part of the design process. This process would include:

- taking an existing design (a split Dagor type, if we take your Fuji example)
- scale it to a new focal length
- check and optimize all lens radii for the new format
- check all lens thickness and optimize
- include all constrains provided by an existing lens barrel you want to use and or lens shutter.
- Fix all thickness and radii to production standard.
- Check the solution

At the end, you will get a split Dagor again, if your design boundaries will allow it and it will look like a scaled lens. But this will never be 100% true. If you accept the whole process as scaling then it is fine. But you have to understand that optical companies will provide only rough drawings, no production figures and there shall be in fact a deviation between similar looking lens (Your Distagon example). A representative of Zeiss told me that they constantly change designs of production lens, even for lens which where in production for 50 years (e.g. 5.6/250 Sonnar) and still look the same.

Regards Roland

Gerald · Joined: 25 Mar 2014 Posts: 1196 Location: Brazil

danfromm · Joined: 04 Sep 2011 Posts: 576

Hmm. I'd always thought that when a prescription is scaled by, say, a factor of n everything about it is scaled by n. Radii; distances, including elements' thicknesses; diameters. Number of elements doesn't change.

Gerald, y'r Distagon fisheye example clearly involves two different prescriptions.

Gerald · Joined: 25 Mar 2014 Posts: 1196 Location: Brazil

paulhofseth · Expire: 2018-06-28

When it comes to dating Zeiss versions i find Hartmut Thieles list "versuchsobjektive" most useful.

The F-Distagon 3,5\30 is listed as case number 104813 for Hasselblad C, 104877 for Hasselbled CF , both dated 1972, as well as case number 104905 in 1974 for the Hasselblad and a "korbfassung", number 104825 in 1972 together with number 104828 for the Rollei66.

The 2.8\16 is listed for Contarex as number 104822 in 1970 and as number 104839 for Contax YB in the same year.There is also an unspecified M42 version , case number 104812 in 1971

From the list of prototype numbers the experiment numbers seem to be similar for many Distagons (104195), but for the 16mm the test 15.12.1970 is labelled Rollei SL 35 , and the 30mm labelled Hasselblad was tested on the 27.01.1972.

My conclusion is that the smaller format version preceeded the larger format version.

p.

Gerald · Joined: 25 Mar 2014 Posts: 1196 Location: Brazil

Abbazz · Joined: 23 Jun 2007 Posts: 1098 Location: Jakarta

If you look at original patents for lenses, you will find out that they usually describe a theorical 100mm lens. The patent drawing is just a scaled version of the actual lens.

For example, if you look at the original 1961 patent (US 2 975 673 I) by W. Mandler for the Leitz Summilux 35mm F/1.4 lens, you will see this:

Credit : E. Beltrando - Dioptrique.info (a wonderful web resource about optical design of vintage lenses)

Of course, there is no 100/1.4 Summilux, but the description in the patent is for a "normalized" 100mm version of the lens. The angle of view (64°) tells us that this is a 35mm lens on 24x36 format. The actual lens is just a scaled down version of this diagram.

Cheers!

Abbazz

danfromm · Joined: 04 Sep 2011 Posts: 576

Interesting. But there were 50/1.4 Summiluxes too.

There's no mention of 35 mm in the patent. It does say "In presently available Gauss-type objectives of a focal length of 50 mm"

The patent gives two examples of prescriptions that embody the patent's ideas, both scaled to "Focal length f = 1.0" One's claimed coverage is 64 degrees, the other's is 45 degrees. The two prescriptions are quite different, rescaling won't transform one into the other.

Gerald · Joined: 25 Mar 2014 Posts: 1196 Location: Brazil

An interesting application of lens scaling is the design of perspective-control lenses (also called shift-lenses). This type of lens emulates the movements of view camera optics. In general, a shift-lens is a wide angle lens with a mechanical construction that allows a displacement of the optical axis relative to the center of the sensor.

Of course, a possible way to design a shift-lens is designing it from scratch. However, lens scaling provides an alternative design path that has effectivelly been used by several lens manufacturers. The idea is very simple: Take an ultra wide-angle lens and scale it to a larger focal length, such as 24, 28 or 35mm. The image circle will increase proportionally, so if the ultra wide-angle lens barely covered the sensor format, the image circle of the scaled lens will have room for a useful shift.

Pentax seems to have just followed that route when designing the SMC PENTAX SHIFT 28mm F3.5. Note the amazing similarity to the ultra-wide angle SMC PENTAX 18mm F3.5:

We can use the relations for lens scaling (see table in the original post) to estimate the diameter of image circle of the PENTAX SMC SHIFT 28mm F3.5. If we guess that the image circle of the SMC PENTAX 18mm F3.5 has a diameter of 45mm, the image circle of the SMC PENTAX SHIFT 28mm F3.5 will have a diameter equal to 45 x 28 / 18 = 70mm. This means a room for a shift of (70 - 45) / 2 = 12.5mm in any direction.

Olympus is another lens manufacturer that seems to have used lens scaling to design its Zuiko 24mm and Zuiko 35mm shift lenses from the ultra wide-angle lenses Zuiko 18mm and Zuiko 24mm, respectively, as the diagrams below strongly suggest.

(picture from http://www.marcocavina.com/articoli_fotografici/zuiko_shift/00_pag.htm)

In a certain sense, a shift lens for 35mm can be seen as an ultra wide angle lens for MF that was lent for use with a "cropped" sensor (the 35mm sensor). From this point of view, it is easy to understand why a shift lens is never a cheap lens as this type of lens brings together three features which increase the costs of any lens: 1) ultra-wide angle, 2) medium format, and 3) complex and precise mechanics.

CuriousOne · Joined: 31 Dec 2013 Posts: 669 Location: Home

This scaling is great, but aperture scaling would be much more interesting.

Say there's 100mm F4 lens. Can it be "scaled" to 100mm F2, without changing it's optical design?

Gerald · Joined: 25 Mar 2014 Posts: 1196 Location: Brazil

Gerald · Joined: 25 Mar 2014 Posts: 1196 Location: Brazil

We will see now a case where the relations of lens scaling help to better understand a real lens. In 1954, the State Optical Institute (GOI) in Leningrad designed an unusually small lens, the Helios-57 9mm F3.5, which was a Double-Gauss lens with 6 elements and just 1.5 grams weight, mounted on a panoramic micro-spy camera. In Marco Cavina's words, "a jewel of miniaturization, very rare, and perhaps for the KGB". The full article can be read here:

http://www.marcocavina.com/articoli_fotografici/Helios_57_micropanoramic_camera/00_pag.htm

Marco Cavina marveled at the small weight of the Helios-57 lens, just 1.5 grams, and the high resolution of more than 80 lines per mm in the center of the field. Do those numbers represent indeed a great feat of optical engineering that deserve our admiration? To answer this question, let's assume the Helios-57 could be designed from a down-scaling of a 6-element Double Gauss lens like the Fujinon EP 50mm F3.5 enlarger lens shown below:

Fujinon EP 50mm F3.5:

The downscaling factor is 50 / 9 = 5.56. If the Fujinon is down scaled by a factor 5.56, the aperture remains the same, but the weight is reduced by a factor 5.56 ^ 3 = 171.9. Since the Fujinon weighs 124 grams, the down scaled lens would have a weight equal to 0.72 grams, what is less than half the weight of the Helios-57. One possible explanation for the Helios-57 being so "heavy" is the use of a flat glass elemento in front for protection, and/or the possible use of brass for the barrels.

As for the resolution, we can say with some certainty that the Fujinon EP 50mm F3.5 resolves at least 40 lines/mm in the center of the field. The scaling would increase the resolution to 40 x 5.56 = 222 lines/mm, which is considerably higher than 80+ lines /mm of the Helios-57. To be fair, the resolution of the Helios-57 as measured by GOI was the combined resolution of lens plus film. In any case, a resolução of 80+ lines/mm does not impress much after you compare this value with the estimated resolution of the Fujinon scaled down to a focal length of 9 mm. By the way, modern lenses for compact cameras with 1/2.3" sensors routinely reach resolution of hundreds of lines per millimeter.

CONCLUSION
The low weight and high resolution of the Helios-57 may seem impressive at first glance, but they are not spectacular. In a sense, they are even disappointing. Remember that in 1954, the manufacturing technology of lenses of small size, such as those used in microscopes, and the art of designing Double Gauss lenses were quite developed.