Dome Theory

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