Dome port theory
by David Knight
Finding
the focal point of a dome port
The focal point of a lens is the point at which light from an object at
infinite distance is made to converge. Since an underwater
dome
port is a diverging lens, the focal length is strictly negative (the
image is virtual), but we will treat it as a positive quantity here
because the intention is to derive an expression that can be used to
estimate the distance from the camera to the virtual image.
In standard UW photographic practice
with lenses of
moderate to wide field of view (FOV), an attempt is made to place the
lens entrance
pupil at the centre of curvature of the dome. This choice
ensures
that rays normal to the dome surface are undeflected on their path to
the lens, thereby making the FOV underwater the same as it is in air.
Note however that pupil positioning is rarely accurate when
using an interchangeable lens camera, being dependent on generally
limited choices for port extension length, and being subject to the
fact that the camera is always mounted (inside the housing) in the same
place, regardless of which lens is in use. Thus, although we
must
assume that the lens is in the right place for the purposes of theory,
it is important to be aware that incorrect positioning will introduce
error in practice.
Thin dome
approximation
In the diagram below, the entrance pupil of a camera lens is
placed at the centre of curvarure C of an air-filled thin-walled dome
of radius r that is immersed in water. A ray travelling along
the lens axis is undeflected and goes straight to the camera, but an
off axis ray is deflected through an angle φ.
Thus rays
from infinity appear to come from a point F (the secondary focus).
If the camera lens is to be able to focus on the virtual
image at
F, it must have a minimum focusing distance of ≤ s + b (where b
is
the back-focal distance).
Using the Sine rule:
s / Sin(180-θ
a) = r / Sinφ
but Sin(180-θ) = Sinθ
Therefore
s / Sinθ
a = r / Sinφ
In the small angle limit, Sinθ
→ θ
, therefore
s = r θ
a / φ
but, by inspection, φ = θ
a
- θ
w , hence:
s = r θ
a /
( θ
a - θ
w
)
Factoring θ
a from the
denominator and cancelling gives:
s = r / ( 1 - θ
w
/ θ
a )
. . . . . . . . (
1)
By
Snell's law of
refraction:
n
w Sinθ
w = n
a
Sinθ
a
(where n
w and n
a are the
refractive indeces of water and air). In the small angle
limit this becomes:
θ
w / θ
a
= n
a / n
w
substituting this into (
1)
gives:
If we use the approximations n
a = 1 and n
w = 4/3,
then 1-n
a/n
w = ¼, and
Thus, for a thin dome immersed in water, the virtual image of an object
at infinity appears at a distance of about 4 radii from the centre of
curvature. On the somewhat unreliable assumption that the
lens
pupil is correctly placed at the centre of curvature; adding 4r to the
back focal distance b (which can be obtained from the lens data or
estimated by noting the apparent position of the iris) gives an
upper-limit estimate of the required minimum focusing distance for the
camera lens.
In practice, the use of very thin domes
is not
possible in pressure-resistant underwater housings. If the
dome
has finite thickness, and is made of material having a refractive index
greater than that of water (such as glass or acrylic), then the virtual
image will be moved slightly closer to the camera. Thus the
minimum focusing requirement is a little more severe than the simple 4r
rule would seem to imply.
Thick-walled
dome-port formula
The derivation of the focal distance for the general case is somewhat
more tricky than the thin-wall approximation, but we can start by
analysing the passage of a light ray striking the port at an arbitrary
angle.
In the diagram above, an incident ray (in water) meets the port surface
at an angle θ
w to the
perpendicular and emerges into the port material at an angle
θ
p. The
relationship between the two angles is given by Snell's law in the
small-angle limit:
n
w θ
w
= n
p θ
p
. . . . . . . . .
. . . . . . (
2)
As the ray traverses the port to strike the inner surface, the normal
to the surface undergoes an angular displacement δ .
Hence the angle of incidence at the inner surface
is θ
p + δ.
If the ray exits the port material into the air at an
angle θ
a , then Snell's law
gives:
n
p ( θ
p
+ δ ) = n
a θ
a
. . . . . . . . . . (
3)
By inspection of the diagram; using the Sine rule we get:
Sin(180 - θ
p
- δ) / ( r + δr) = Sinθ
p
/ r
Where r is the dome inner radius, and δr is the dome
thickness. Now, since
Sin(180-θ) = Sinθ ,
in the
small angle limit this becomes:
(θ
p + δ) / (r
+ δr) = θ
p
/ r
Rearrangement gives:
δ = θ
p δr
/ r . . . . . . .
. . . . . . . . . . (
4)
Substituting this into (
3)
gives:
n
p θ
p
(1 + δr / r ) = n
a θ
a
and using (
2)
to substitute for θ
p gives:
θ
w / θ
a
= (n
a / n
w) / (1
+ δr / r) . . . . (
5)
As shown below, an expression for the focal distance s is obtained in
the same way as for the thin-walled case:
Referring to the diagram above, using the Sine rule gives:
s / Sin(180 - θ
a) = r /
Sinφ
but Sin(180 - θ) = Sinθ , and so, in the
small angle limit:
s / θ
a = r
/ φ . . . . . . . . . . . . . (
6)
A problem that remains however, is that of
eliminating φ so
that s can be expressed in terms of fixed physical parameters.
The solution is obtained by drawing parallels, as shown in
the
diagram below:
by drawing a line a - a' parallel to the line c - c' against
which θ
w is defined, we can
see that:
θ
a = φ
+ θ
w + δ
i.e.,
φ = θ
a
- θ
w - δ
Using this in (
6)
gives:
s = r θ
a / (θ
a
- θ
w - δ)
Substituting for δ using (
4) gives:
s = r θ
a / (θ
a
- θ
w - θ
p δr
/ r)
and substituting for θ
p using
(
2)
gives:
s = r θ
a /
[ θ
a - θ
w
- θ
w (n
w
/ n
p) δr / r ]
i.e.,
s = r θ
a /
[ θ
a - θ
w
{ 1 + (n
w / n
p) δr
/ r } ]
Factoring out θ
a from the
denoninator and cancelling gives:
s = r / [ 1 - (θ
w
/ θ
a) { 1 + (n
w
/ n
p) δr / r } ]
and substituting for θ
w
/ θ
a using (
5) puts the
focal distance in terms of fixed system parameters and solves the
problem:
s = r / [1 - (na / nw)
{1 + (nw / np) δr
/r } / (1 + δr / r) ] |
Example
calculations
Some example calculations are shown below. Accurate data were
available for the ikelite 6" port, but the thicknesses of the boron
crown-glass (BK7) ports are guesses. Still, the calculations
give
the general idea, which is that the finite thickness of the port puts
the virtual image some 10 to 15mm closer to the camera than the simple
4r rule would suggest.
The infinity correction figure, in diopters, is simply the reciprocal
of s in metres, i.e., it is the magnifying power required to
make
rays from the virtual image become parallel if the correcting lens is
placed at the centre of curvature of the dome.
Don't
forget to add the back focal distance b when estimating the minimum
focusing distance requirement. The position of the focal
plane in
an interchangeable lens camera is often marked with the symbol
.
The position of the lens entrance pupil should be given in
the
data sheet, but it can also be estimated by stopping-down and noting
the apparent position of the iris.
You can download the open document spreadsheet used for the
calculations above and play with the numbers for yourself
→ Dome
port calculation spreadsheet dp_calcs.ods (right
click , 'save target as' ). You will need
Open
Office to open the file.
An alternative approach to the dome port formula is given here:
Secondary focal point of a dome port
© David Knight 2012. Updated Feb. 2018.
David W Knight asserts the right to be recognised as the author of this
work.