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Appnote --> Para. 4.3
4. Resolution and Related Topics (continued)
4.3 Depth of Focus
With the (idialized) "camera obscura" you get well-defined
images at any distance.
Object distance as well as image distance may be varied without drawback.
But with a lens, things change: departing from the
plane of best definition (that is, "defocussing") will render the image
No matter whether you defocus at the object- or image side.
Looking at geometrical optics, we see that only infinitely small errors
in object- or image distance will produce infinitely small blur.
But the image is very often examined or used with finite
resolution. For example with a CCD- or CMOS image sensor.
In this case, we can easily establish a limit of image detail size, below
which we need no further resolution. And you would certainly like to calculate
the allowable image- (or object-)distance error that results from this
finite resolution requirement.
So let us examine quantitatively, how blur increases with defocussing.
Geometrical optics and figure 4.3 will lead the way.
Important are these two rays coming from
the aperture rim. Here they are separated by the diameter of the exit pupil
DAP . Then they intersect in the
image point. As they propagate beyond the image plane, their distance
increases again. Between these rays is everything that geometrical optics
can predict for the diameter of blur circle DBC :
(b' - b) / b = DBC / DAP ---> DBC = DAP * (b' - b) / b
This equation clearly shows that DBC increases linearly
with DAP (lens speed) and with the relative error of image distance.
Or, if an upper DBC limit is defined, simply divide it by DAP and
result is the allowable relative image distance error.
We have looked at effects of image distances that are
too long. But naturally, the same applies at image distances that are too
short. For remaining within a DBC limit, you may move the image plane
to and fro by the same allowable error. So depth of focus, DOF , is:
DOF = +/- b * DBC / DAP
Having the image sensor focussed exactly, you may want to allow some defocussing
in the object plane. How much?
Like all sizes and distances DOF , too, in the object space is 1/m
times as big as in the image space ( m is scale or magnification as described
in paragraph 3):
DOF = +/- b * DBC / (m * DAP) (object side)
But remember: you cannot use up both! Full DOF is only available either at the image side,
or at the object side.
Doing some more arithmetics, you'll be able to show that you cannot achieve
a different DOF by choosing different focal lengths f .
(Geometrically) DOF depends on scale m , and on lens speed, and on
the allowable blur circle diameter ... and on nothing else.
In paragraph 4.2 the modulation transfer function MTF
has been introduced. It's the application engineer's most powerful tool for
assessing a lens. And it can give you a very good DOF criterion if it is
For a certain spatial frequency and in a certain location and direction
within the image field, modulation transfer factors are measured for various
displacement values of the image plane, of the object plane, or of the
If evaluating a through-focus MTF, be sure that spatial frequency, location
and direction do have the values you are interested in.
But if they do so, you can readily see the influence of the minimum MTF
value (that you pose) on the DOF (that you get).
Including not only effects of geometrical optics, but also diffractive
effects and all sorts of corrections that your lens may contain.
continue: Diffraction Limit
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Last modified Nov.28th, 2002