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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 blurred.
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.

fig.4.3: geometrical blur (10 kByte)
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           (image side)

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 measured "through-focus":
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 lens.

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.

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Last modified Nov.28th, 2002 20:18