Dome Theory

by Julian Scheunemann Sunday, October 3rd, 2004

dome theory - dioptre & focus distance

A frequently asked question is if a dioptre lens is required with a certain dome/lens combination. The following is a theoretical approach which just could give an idea when choosing a lens/dome combination. This could be important as there are lenses which do not allow mounting a dioptre.

Before determining the minimum required focusing distance some dome design aspects are to be mentioned.

dome design

The radial light rays go thru the dome and meet in a virtual point, called lens entrance pupil or centre of perspective (not one of the nodal points!). Virtual, because the rays are refracted already upon entering. Distance between dome glass and the entrance pupil is equal to the dome radius ( r ) if the dome (and port extension) is correctly designed for a certain lens.
Unfortunately I found only Sigma and Zeiss providing the position of the entrance pupil (Zeiss officially and Sigma Japan on email request.). You also could take a laser pointer to find the position. You can rotate the lens around this point and the red ray would have to go thru the lens (like the pano shooters do, sometimes it is erroneously called nodal point).

Sigma 12-24mm @12mm, not to scale:

dioptre required?

If a certain lens works without a dioptre it has to be capable of focusing at least to infinity underwater. Therefore it has to be determined how far the virtual image is in front of the dome.
This is influenced by the dome radius. Smaller radius gives a nearer virtual image. Considering the following:
refraction index water = 1.33, refraction index air = 1.0, dome radius = average between outer and inner radius - the simplified formula can be used including these values:

L(inf) = 3.03 * r

L(inf) = distance between dome and virtual image (at underwater infinity).
To be strictly correct the result would have to be a negative value due to the negative refraction index of the dome: L(inf) = -3.03 * r
If the dome thickness (inner and outer radius) would be included into the calculations the distance would even become a little shorter.

Example: dome radius 10cm gives L(inf) = 30.3cm

This is the distance from the dome glass. A lens would have to focus at least to this distance in order to

Four approximations of simulation models were developed to predict the behaviour of a real cone speaker. Most of the modern tweeters used in multi-way loudspeakers are rigid or soft domes. To describe the geometry of a spherical segment, the approximation with truncated cones can be used. Each cone can be presented as a number of ring elements. -------------------- williamgeorge widecircles
Posted by williamgeorge on 2008-09-03 23:36:15 Of course you can "see" the entrance pupil when you look at a lens. If you shine a light through the viewfinder, hence running the optical path in reverse, you can see it emerge from the entrance pupil, looking at the front of the lens. If you then hold up an empty toilet paper/smarties tube to the surface of the port (round near the edge of the glass), you can evaluate how close the pupil is to the centre of curvature. Most Ports don't let you get a fisheye in the right place due to the port flange placement. That's what my O-level Physics is telling me, so if an optics guru can show me what I've got wrong I'd love to know :)
Posted by Rattus on 2006-09-29 09:26:44 Joe Wisniewski gives a nice table, including entrance pupil data at:
http://www.swissarmyfork.com/lens_table_1.htm
Interestingly, about 4mm's of change over the focal range of the Nikon 12-24 (avg about 110 mm) -- all relative to lens flange.

Also, "plattro" at http://www.kekus.com/forum/showthread.php?t=329&goto=nextoldest says
the Nikon 10.5 has En of 87mm and the 14/2.8 is 97 mm (both +/- 5mm).

I'm still shopping for WA DSLR stuff, but it'd be interesting to know if the ports we see make sense with respect to these measurements.
Posted by Chris White on 2005-01-31 12:27:55 There is a nice procedure to obtain those data (without lens manufactures providing them). According to your equipment signature you might already enjoy the results :-) I'll send you an email.
Posted by Julian on 2004-11-02 11:00:59 You might be interested to take a look at http://www.canon.com/camera-museum/ where some Canon lenses are shown in diagramatic form. Assuming that they are dimensionally correct (ie to scale) then it should be possible to get some idea of the position of the entrance pupil. Unfortunately the zooms are only shown at one setting and its focal length is not stated, but it is not too much of a problem to figure out with the lens in front of you. I started to sort out similar data to that you produced above for Canon's 17~40mm lens and will post it when I've got sufficient details calculated. There is undoubtedly a similar shift to that of the Sigma and the lens tends to exhibit poor corners behind a small dome.
Posted by Paul Kay on 2004-10-28 08:41:12
Posted by Paul Kay on 2004-10-28 08:35:34 Paul, thank you for the comment. The issue of focusing on a curved image is seldom taken into consideration. The image centre has the largest focus distance. The corners are always closer in terms of focus distance. Huge radius is the best way to minimize this problem and finally of course resulting in diameter increase depending on the Dome
Posted by Julian on 2004-10-16 10:33:22 Just a couple of points I'd like to add to this very useful piece. Firstly, the entrance pupil can shift with wide-angle zooms as their focal length changes, so the dome/lens/dioptre set-up may only be optimised for a specific focal length (I've found that in practice this is usually the widest setting as this is probably where the lens will be used most!). Secondly, using a dome produces a c'urved image field' (that is to say that the image on the flat sensor isn't flat) and roughly speaking, the smaller the dome diameter, the greater the curvature. So corners may simply never be sharp except at extremely small apertures. There is (apparently) empirical evidence which suggests that a simple dioptre lens (curved on the outside surface, flat on the inside or lens side) tends to curve the image field in an opposing direction so can provide better overall correction as well as adjusting for focus. I've never looked into the optics of why this is so but must do one day. In general terms its best to have as big a dome as possible (correctly set up with appropriate extenders) and a lens that has as good close-focus capabilities as possible. For an optical explanation, Sidney Ray has a chapter in his massive volume on photography (which unfortunately costs around
Posted by Paul Kay on 2004-10-08 07:08:48