#### Doug Kerr

##### Well-known member

*luminance*of a scene "patch" into

*illuminance*on the focal plane (which, in combination with exposure time, constitutes the

*photometric exposure*to which the sensor responds).

If these conditions obtain:

a. The camera is focused at infinity (the distance to the object doesn't matter).

b. The object patch of interest is on the lens axis.

c. The lens transmission is 1.00 (that is, no light is lost through absorption,reflection, scattering, or the like).

then the ratio of illuminance on the focal plane to luminance on the object is dictated by the

*relative aperture*of the lens (to use the formal term), which we quantify by its f-number. The f-number is the ratio of the focal length of the lens to the diameter of its

*entrance pupil*. The entrance pupil is what the aperture stop appears to be from in front of the lens. It is the apparent "port" through which the lens gathers light from the object to deposit on the focal plane.

If condition "a" is not met (that is, focus is at some finite distance, and especially if it is at a short distance), then the f-number no longer indicates the pivotal photometric property of the lens. The ratio of the illuminance on the focal plane to the luminance on the object patch is now less than would be indicated by the f-number of the lens. We can calculate by how much (assuming we have the right information) - we'll do that a little later.

It is often useful, so that we can continue to apply familiar exposure calculations, to use a quantity known as the

*effective f-number*to describe the diminution of this ratio. The effective f-number is the f-number of a lens that, in a camera focused at infinity, would produce the same ratio of focal plane illuminance to object luminance as we have for this real situation.

Suppose that the f-number of our lens is f/2.0, and we are focused at a distance such that the ratio of focal plane illuminance to object luminance is half what it would be with the same lens in place and the camera focused at infinity. Then the diminution of the pivotal ratio is "one stop", equivalent to an f-number "one-stop higher", or f/2.8. Then f/2.8 is the

*effective f-number*for our actual setup - the f-number we would need to use to make exposure calculations.

The ratio of the effective f-number to the actual f-number is sometimes called the "bellows factor", a term that goes back to cameras that were focused by ,moving the lens on a track, there being a bellows between the lens and the camera body proper. The factor by which the effective f-number was greater than the actual f number depended on the position of the lens board - that is, the degree to which the bellows was "extended". Thus, we can say:

*N*'=

*BN*

where

*N*is the f-number,

*N*' is the effective f-number, and

*B*is the bellows factor.

If our lens design has what is called "unity pupil magnification" (which means that the entrance and exit pupils are located at the first and second principal planes), then the bellows factor can be calculated this way:

*B*=1+(1/

*m*)

where

*B*is the bellows factor and

*m*is the magnification of the lens at the focus setting of interest. (For any given lens, the magnification is determined by the distance at which the camera is focused.)

For example, for the magnification that is sometimes stated as "1:3",

*m*would be 0.333.Thus:

*N*'=

*N*(1+(1/

*m*))

If the lens does not have unity pupil magnification, the equation is a bit more complicated. For most of our uses, the basic equation will give sufficiently-accurate results.

For reference, for focus at a distance such that the magnification is unity ("1:1"), the bellows factor is 2. Thus the effective f-number is twice the f-number, representing a decline in the photometric relationship governing exposure of two stops.

A decline of one stop (a bellows factor of 1.414) occurs for focus at a distance at which the magnification is about 0.4 ("1:2.5").

Now, the effective f-number, even for object points on the lens axis, still does not precisely tell us the pivotal photometric ratio, since it ignores the matter of the lens transmission being less than 100%. (The same is true of the actual f-number for operation at infinity focus.) Thus, sometimes we will find effective f-numbers stated that take into account not only the bellows factor but also the actual transmission of the lens. I will not belabor that here - it is sufficient that we be alert to this possibility.

Note that the bellows factor, and the fact that the effective f-number is greater than the actual f-number, depends only on the magnification, which in turn depends on the distance at which the camera is focused. It does not matter whether this happens within the normal focusing range of the lens, or with the assistance of extension tubes.

It is sometimes erroneously thought that the decease in exposure when using an extension tube is a property of the extension tube itself. Rather, it just results from the "bellows factor" effect of the closer focus that we use, made possible of course by the extension tube.

Another small wrinkle of which we should be aware is that the actual f-number of a lens may not be exactly constant over its range of focus settings. Thus, as we focus at a closer distance, the effective f-number may change not just because of the increase in the bellows factor, but as well because of change in the f-number itself.

Best regards,

Doug