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2. Important Angles
2.1 Aperture Angle
Does the luminous incidance in the image plane depend on lens characteristics? -- Of course it does.

    fig.: Aperture Angle (10 kByte)

Consider figure 2.1.
Green rays: Object at infinity;     object rays parallel;  image distance   f .
Red rays:   Object distance finite; object rays divergent; image distance   b .
The image center is illuminated by the entire lens exit pupil.
So we call the angle of the rays   from the lens rim   to the image center   "aperture angle"   u .
This angle is clearly influenced by the image distance   b .
But we prefer a characteristic number that is independent of the special setup. And so we choose the aperture angle that occurs at infinite object distance to be the "nominal aperture angle".
From figure 2.1 you see that
    (nominal aperture angle) = u = arc tan ((DAP/2) / f)       (5)

Strictly speaking, it is a solid angle. It is defined approximately by the exit pupil area, divided by the square of the image distance.
But never mind: nearly all exit pupils are of circular shape. So we can simply take radius or diameter (DAP) of the pupil and divide it by the focal length, getting a characteristic number that describes lens speed.
But sad to say, in optics very often the inverse is established:

    nominal:    k = f# = f / DAP             (6)

The smaller the f#, the more light you will get.

And keep in mind: nominal f# is valid for object at infinity. Comparing different optical setups with finite object distances, you should use the effective f# (red rays in fig. 2.1):

  effective:    k = f# = b / DAP             (7)

At decreasing object distances, the effective f-number gets bigger by the factor   b/f   and hence, worse.

In some optical setups (especially at short and constant object distances) people prefer the "numerical aperture" against f#. What is "numerical aperture"?
Often it is designated NA or simply A. It is defined by

                A = n * sin(u)               (8)

Here,   n   is the refractive index of the medium between object and lens. And angle  u  is defined above in equation (5).

You may be annoyed looking at the "tan"-like definition of f# and the "sin"-like definition of A. No easy conversion. But if you use a good lens system and if you use it in the way it was intended (i.e. "aplanatically corrected" lens used at optimum magnification), then the "sine criterion" is fulfilled and for conversion,

                A = 1 / ( 2 * f#)            (9)
will hold.

For further details on image luminous incidance please refer to paragraph 5.

We have learnt that the aperture angle defines f# and hence, image luminous incidance --
but then, what is a field angle?

continue: "Field Angle"     back to index of Lens Appnote     back to index of

Last modified Nov.28th, 2002 11:25