Some Principles of Photographic Optics
FOCAL LENGTH AND FOCUS
Focal length of a lens
An important optical parameter of a lens is its focal length. Consider a basic,
“thin”, single-element lens. Imagine an object at a very great distance (actually, an
“infinite” distance). All the light rays from a given patch ( very small area on an object (that is, part of a scene) or
image, which we will refer to as a “patch”) on that object that are able
to pass through the lens are brought to convergence (“focus”) at a point behind the
lens at a distance f from the lens. f is said to be the focal length of the lens.
To be precise, this is only exactly true:
• For an object patch that is on the lens axis
• For those rays from the patch that pass through a very small circle in the
center of the lens.
With respect to the latter restriction, if we in fact consider all the rays from the
patch that enter the lens, the ones entering the outermost portion of the lens will
be converged at a point closer to the lens than the rays entering the center of the
lens. This is the manifestation of spherical aberration, one of the classical lens
aberrations (imperfections in behavior).
The image
In a camera, the collection of patches created behind the lens, each resulting from
the convergence of the rays of light emanating from a patch in the scene, is called
the image. We allow it to fall on the film in order to be recorded.
The focus equation
If we have an object not at an “infinite” distance, but rather at a distance P in front
of the lens, the rays from a patch on the object will converge into an image patch
lying a distance Q behind the lens. P and Q are related by:
 Equation 1
where f is the focal length of the lens.
(As before, this relationship is only precise for patches lying on the lens axis and
for rays entering at the center of the lens.)
We can rewrite this as:
 Equation 2
Note that if P, the object distance, becomes infinite, this degenerates into:
Q = f Equation 3
the result we in fact stated earlier for an infinite object distance.
Beyond the thin lens
If the lens of interest is not a “thin” lens, either in that it is a single element of
substantial thickness or (as in the case of most photographic lenses) that it is made
of a number of separate elements, the equations above still hold. However, the
distances P and Q cannot now just be said to be measured “from the lens”. Rather P is measured to a point known as the first principal point of the lens, and Q is
measured to a point within the lens known as the second principal point of the
lens.
Note that while both of the principal points are generally within the lens assembly
itself, there are some special, widely-used lens designs in which one of the points
or the other is outside it.
MAGNIFICATION OF A CAMERA LENS
The magnification of a camera lens is defined as the ratio of the size of some
feature on the image to the size of the corresponding feature on the object itself. It
is solely a function of the object and image distances, as follows:
where m is the magnification and Q and P are the image and object distances,
measured to the appropriate principal point. (Again, the familiar condition of an
object on the lens axis applies.)
Substituting from Equation 1, we then get:
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Equation 5 |
| Equation 6 |
two forms that can be useful in various further work.
Since magnification is a ratio, we can express it in various ways, including the
following (the examples are for a magnification of one-fifth):
• 1/5
• 1/5X
• 1:5
• 0.2
• 0.2X
• 0.2:1
We are often especially interested in magnification in connection with closeup
photography (including macrophotography, which is defined as the photography of
very small, but not microscopic, objects). There, it is typically the maximum
magnification of the lens which is of interest. This occurs with the closest available
focusing distance (For any particular focal length, in the case of a zoom lens. We cannot conclude that the greatest
maximum magnification occurs with the longest focal length or the shortest—it could even occur for
some focal length in between, depending on the lens design ).
Our interest there in magnification is due to our desire to have the image of the
small object fill a substantial portion of the film frame. If we know the sizes of the
object and the film frame, we can determine the magnification required to meet
that objective.
Note that it is not usually possible to calculate the maximum magnification of a
lens just from knowledge of the focal length and the specified closest focusing
distance (perhaps using Equation 5), since the specified closest focusing distance is
usually defined from the film plane and not the first principal point of the lens (that
is, we do not actually know P),
The receptor for the image developed by the lens in a camera may be photographic
film, an analog electronic “target” (as in an analog television camera), or a digital
sensor array. Most of the principles described here apply equally to any of these.
For conciseness, we will just refer to the receptor consistently as the film.
Other uses of the term magnification
The term magnification with respect to a lens is sometimes used with other
meanings, some of which are ambiguous, some questionable, and some downright
invalid. We will not discuss those meanings here.
EXPOSURE AND APERTURE
Exposure
The quantitative phenomenon that characterizes the impact on the film that causes
the film to record the image is known as exposure. It is defined as the product of the illuminance upon the film (luminous flux per unit area) and the length of time
that illuminance persists (the “exposure time”). ( In fact, for an exposure involving extreme values of exposure time, the impact on the film may not
be the same as for the same exposure involving a shorter exposure time (and correspondingly
greater illuminance). This phenomenon is known as “reciprocity failure”.)
If we set aside a couple of small wrinkles, which we will look into later, the value
of exposure on a patch of the image results from the interaction of these three
factors:
1. The luminance (brightness) of the corresponding patch of the object
(”scene”)
2. The exposure time (“shutter speed”)
3. The relative aperture of the lens (expressed as an “f/number”).
Just to confuse things, we must note that a second, equally-legitimate use of the
term exposure is to represent the combined effect of only factors 2 and 3 (This is the factor that, in a logarithmic form, as designated “exposure value” (Ev).). Thus,
when we encounter the term, we must carefully consider the context to be certain
that we appreciate it with the proper meaning.
Relative aperture
The relative aperture of a lens (usually just called aperture) is described by the
f/number, defined as the ratio of the focal length of the lens to the diameter of the
entrance pupil.
What is the entrance pupil? In most lenses, we have an aperture stop, an opening
of adjustable diameter in an opaque plate (the diaphragm) someplace in the path of
the light through the lens. The diameter of the opening is changed in order to
control the amount of light passing through the lens and thus to affect exposure.
The entrance pupil is the aperture stop as it appears (in both location and diameter)
from in front of the entire lens.
The relative aperture is commonly stated as an “f/number”, this way: “f/3.5”. This
in fact means that the diameter of the entrance pupil is the focal length of the lens
(f) divided by 3.5. (The numerical value is a ratio and can also be stated as such. Under the provisions of an
international standard, the maximum relative aperture of a lens is marked on the lens this way: “1:3.5” (for an f/3.5 maximum aperture).)
People are often mystified as to why the factor involves the focal length of the
lens. After all, isn’t it merely the diameter of the entrance pupil (and thus its area)
that governs how much of the light emitted by a patch on the object is collected by
the camera?
Indeed it is, but it is not only the amount of light gathered in which we are
interested.( It is sometimes said that the f/number describes “the light-gathering power” of the lens, but that
concept is misleading, as we will see shortly.) Rather, what we are concerned with is how the lens transforms the
luminance (brightness) of a patch on the object into the illuminance delivered to the
film to form the corresponding patch of the image. This involves both the amount
of light gathered through the entrance pupil from the object patch and the size of
the resulting image patch. Because of the latter, the distance from the lens to the
film is also involved.
That transformation ( The equation is given for SI (metric) units.) is described by :
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Equation 7 |
where E is the illuminance given to a patch on the image (Note that the symbol E is also used for exposure (in the first sense mentioned above).) , L is the luminance of the
source object patch, D is the diameter of the entrance pupil, and Q is the distance
from the second principal point of the lens to the film.
This can be rewritten as:
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Equation 8 |
From this form we can clearly see that the ratio Q/D describes the performance of
the lens in turning object luminance into image illuminance.
Now recall that, for a large value of P (that is, for an object whose distance is large
compared to the focal length):
Q = f [Equation 3]
Thus, in that situation, the factor of importance becomes f/D, which is of course
the f/number.
Since most (but certainly not all) of our photographic work is done with objects at
substantial distances, we can in most cases conveniently use the f/number of the
lens to characterize its role in affecting exposure. We will in other page talk about the case
where the assumption of great object distance does not hold (in “closeup”
photography).
The “stop”
In some early cameras, adjustment of the aperture was achieved by having a metal
strip, carrying holes of different diameter, which was slid through the lens. The
photographer positioned the strip so the hole of the desired diameter was in line
with the lens axis. The different holes were said to be different “stops”.
Most commonly, the sizes of the holes were such that the area of adjacent holes
differered by the factor 2 (the diameter by the square root of 2), as this provided a
sufficiently-fine adjustment for the technique of the time.
As a result, a change in aperture giving twice (or half) the area is said to be a “onestop”
change. This notation is also extended to other factors affecting exposure,
including shutter speed, where a change in shutter speed by a factor of two is said
to be a “one-stop” change.
Today, apertures and shutter speeds can often be set in the camera with an
increment of one-half or one-third stop. For example, in the case of aperture area, a
change of one-third stop represents a ratio of the third root of two; for aperture
diameter, the sixth root of two. The ratios involved for these different increments
are given in this table:
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