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--------------------------------------------------------- By Bill Claff

Many readers are aware that the
diameter of the entrance pupil relates to the f-number (f#) of the lens. The
formula is:

*P _{D} = f / f#*
or

where

This is an object space, light gathering property.

Mathematically the entrance and exit pupils are conjugates. In practical terms this simply means that all the light gathered by the entrance pupil is delivered to the exit pupil.

In image space the half-angle
the exit pupil makes with the image plane determines Numerical Aperture (NA).
The formula is:

*NA = sin **q*

where *NA* is Numerical
Aperture and *q*
is the half-angle.

Numerical Aperture and f-number
are related by this equation:

*f#
= 1 / (2 * NA)* or *NA =
1 / (2 * f#)*

Strictly speaking this the real
definition of f-number whereas *f
/ P _{D}* is a close approximation.

So, effective f-number and Numerical Aperture are closely related.

In any case, we can use the
entrance pupil diameter from our ray tracing to calculate f-number at infinity
and the exit pupil diameter at other than infinity to calculate NA and
effective f-number.

(Because the lens specifications don't state an aperture diameter I actually
tune the aperture diameter to meet the specification at infinity.)

Pupil magnification is the ratio
of exit pupil diameter to entrance pupil diameter:

*p = P' _{D} / P_{D}*

Note that pupil magnification can be less than one. Values tend to be less than 1 at longer focal lengths and greater than 1 at short focal lengths.

Because of geometry (similar triangles) effective f-number is normally stated in terms of pupil magnification rather than Numerical Aperture.

Pupil magnification and effective f-number are normally only important in close-up photography.

Here's another look at the 50mm
lens with cardinal points and the pupils:

For this lens the pupil magnification is close to 1.5

Note that P' is 1.5 times are far away from F' as H' and note that P is 1/1.5
times as far away from F as H.

Finally, for now, the entrance
pupil is the center of perspective.

To accomplish multi-image panoramic photographic, particularly at short
distances, you must rotate the lens around the entrance pupil to maintain perspective.

(Ignoring the nuance that entrance pupil, especially for wide angle lenses,
moves with viewing angle!)