Some Principles of Photographic Optics
Closeup photography
When we work in a regime where the assumption of a great object distance is no
longer valid (such as closeup photography), the f/number is no longer a good
approximation to Q/D, which as we recall is the actual lens parameter affecting exposure. We usually overcome this not by actually using the ratio Q/D instead of
the f/number, but rather by using a correction factor by which we multiply the f/number to get the “effective f/number” for the situation (which is in fact Q/D).
This correction factor is often called the “bellows factor” ( On cameras in which the lens was connected to the body with a bellows in order to accommodate
movement of the lens for focusing, values of this correction factor were described as depending on “bellows extension”. Values of the factor were sometimes presented on a scale on the rail on which
the lens board traveled.) . Thus:
N' = BN
where N’ is the “effective” f/number, B is the correction factor, and N is the actual
f/number of the lens.
B can be calculated as:
 |
Equation 9 |
Of course, we rarely know Q. But often in closeup photography, we may know the
magnification involved. Then we can determine the correction factor, B, as:
B = m +1 Equation 10
where m is the magnification.
Thus, at a magnification of 1.0 (1:1), B becomes 2, and the effective f/number, N’,
is twice the f/number itself. This represents a “two-stop” decrease in the
photometric performance of the lens compared to what the f/number would imply.
Lens transmission
The demonstration above that the f/number is the indicator of lens behavior in “connecting” object luminance to image illuminance—for the case of an object
distance that is great compared to the focal length—assumes that all the light
gathered by the lens ends up on the image.
In reality, a portion of the light gathered by the lens is redirected by reflections at
the various glass-to-air surfaces of the lens, and is lost to the image. The ratio of
the light delivered by the lens to the image to the light collected by the lens is
described by the factor T, the lens transmission. We ignore this when we treat the
f/number as the indicator of lens photometric performance.
However, in fields such as professional motion picture photography, where it is
perhaps more critical to calculate the exposure factors precisely, there is a system
called the “T-stop” system that does reflect the effect of lens transmission.
The T-stop rating of a lens is essentially the “effective f/number” of a lens, taking
transmission into account. It can be used in place of the f/number in precise exposure calculation. It is often expressed this way: T/3.5 (by parallel to the
f/number system), or sometimes as T3.5 or T-3.5.
The T-stop value, NT, is defined as:
where N is the f/number of the lens and T is the transmission.
Note that the T-stop doesn’t wholly replace the f/number in the cinematographer’s
concern. It is still the f/number that controls such matters as depth of field.
Field of view
Field of view refers to the amount of “space” taken in by the camera in forming the
image. It is properly described in terms of the angle(s) subtended by the view. If
the image is rectangular (as in most cameras), we may choose to describe the size
of the field of view in terms of its width, height, and/or diagonal size (angle).
If the camera is focused at infinity, the angular field of view is closely given by:
 |
Equation 12 |
where θ (Greek letter theta) is the angular size of the dimension of interest of the
field of view, x is the size of that dimension of the camera film frame, F is the focal
length of the lens, and arctan represents the trigonometric function arc tangent
(inverse tangent). (The arc tangent of x is the angle whose tangent is x. The field of view angle can also be expressed in terms of the size of the field at a
stated distance, as “353 feet wide at a distance of 1000 feet” (corresponding to a
horizontal field of view angle of 20º). This was formerly the practice for stating the
field of view of binoculars, but has today been replaced by the angle in degrees.
Most photographers are not accustomed to thinking of field of view in terms of
angle (An exception is the case of lenses having a very large angular field of view, such as “fisheye”
lenses, for which the field of view is commonly in fact expressed in degrees). Rather, they learn what the photographic effect is of the field of view
afforded by lenses of various focal length. Of course this relationship varies with
the frame size of the camera. Since for many decades the most common type of
still film camera used by advanced amateurs (and by many professional
photographers) was the full-frame 35 mm camera, it is widely considered today
that a useful way to describe a field of view is in terms of the focal length lens that
would produce that field of view on a 35 mm camera.
Thus, when dealing with a camera having a frame size different than that of the 35
mm camera (usually smaller, as for many digital cameras), and considering a lens of
a certain focal length, we often (in effect) ask the question, “what focal length lens
used on a 35 mm camera would give the same field of view as this lens will give
on this camera?” That focal length is often called the “35 mm equivalent focal
length” of the lens of interest when used on the camera of interest. It may be
calculated thus:
 |
Equation 13 |
where f35 is the “35 mm equivalent focal length”, f is the (actual) focal length of
the lens of interest, and K is the ratio of some dimension of the image frame of the
camera of interest to the corresponding dimension of the film frame of a 35 mm
camera (Note that if the frame of the camera of interest does not have the same aspect ratio (ratio of
horizontal to vertical size) as the frame of a 35 mm camera (3:2), a unique value of this ratio does
not exist. In such case, we often nevertheless still use the concept, based on the ratio of the
diagonal dimensions of the respective frames.).
More commonly, we define a factor J as the reciprocal of K, so that:
 |
Equation 14 |
Thus, for a camera whose frame is 62.5% the size of a 35 mm camera frame (in
linear dimensions), the 35 mm equivalent focal length of any lens used on that
camera is 1.6 times the (actual) focal length of the lens. ( The factor we call here J is called by some the “field of view crop factor”. The rationale is that
the difference between the fields of view exhibited by any given focal length lens on a 35 mm
camera and a smaller-frame camera is a result of the fact that the image that would have beencaptured by the 35 mm camera is “cropped” by the smaller frame of the camera of interest. We do
not find that term attractive, and discourage its use).
Note that this does not mean that the focal length of a lens is dependent on the
frame size or any other parameter of the camera on which it is used. The focal
length is a property of the lens itself. The “35 mm equivalent focal length” is not a
focal length of the lens of interest. It is merely a number that can be used to allow
appreciation of the field of view given by the lens on a particular camera in terms
of familiar 35 mm camera experience (which of course many users of smaller-frame
cameras don’t have!).
Depth of field
Strictly speaking, when the lens is set at a certain focus position, only an object
patch at precisely the corresponding distance will be truly focused on the film
plane.
Of course, in almost all real-life photography, we are interested in capturing scene
elements lying at varying distances from the lens. We are able to do so only by
accepting the fact that the degree of imperfect focus afforded objects at other
distances than the ideal one is “acceptable”.
The range of object distances over which misfocus is considered acceptable is
known as the depth of field of the camera.
To be able to objectively predict the depth of field we will obtain under any given
situation, we must establish some objective criterion for how much misfocus we
will consider acceptable.
We define our choice of this criterion on the concept of the circle of confusion.
When focus is imperfect, the image of an infinitesimal patch of the object is not an
infinitesimal patch on the image, but rather a roughly-circular pattern of finite
diameter. This pattern is known as the circle of confusion. We express our adopted criterion of acceptable misfocus by stating a maximum acceptable diameter of the
circle of confusion.
The actual diameter of the circle of confusion (not our criterion for its maximum
acceptable diameter) depends on four parameters of the optical system:
• The distance to the object patch of interest
• The distance to the plane of perfect focus (the “focus distance”)
• The focal length of the lens
• The actual diameter of the aperture, or, if we prefer, the aperture as an
f/number ( Since focal length is a parameter anyway, we can recast the defining equation to accept aperture
as an f/number)
The selection of a maximum acceptable diameter of the circle of confusion is not a
simple one, and does not flow automatically from any simple combination of
technical properties. The choice, for one thing, must be based upon some
assumptions about how the image is to be viewed, and against what norms are we
to judge “acceptable” misfocus.
Under one set of such guidelines, a maximum acceptable diameter of the circle of
confusion is selected based on a fixed fraction of the diagonal size of the camera
format (film frame or digital sensor size). Often a fraction of 1/1400 is used.
With the various factors in hand, the depth of field can be calculated approximately
as:
 |
Equation 15 |
where Dd is the depth of field, S is the distance to the plane of perfect focus, f is
the focal length of the lens, N is the lens aperture as an f/number, and c is the adopted maximum circle of confusion diameter, Dd, S, f, and c in the same unit.
The approximation is closely valid for values of S which are many times the focal
length, f.
Although it is difficult to see from this equation the effect of changes in the various
parameters, perhaps most important is the fact that, for any given values of S, f,
and c, the depth of field increases as the f/number (N) increases; that is, the smaller relative aperture gives greater depth of field.
The hyperfocal distance
For given values of f, N, and c, there is a focus distance S such that the far limit of
the depth of field reaches just to infinity. That value of S is called the hyperfocal
distance for that camera setup, Sh. With the camera focused at distance Sh, the
near limit of the depth of field is at Sh/2.
Thus, in situations in which it is not possible to focus the camera (perhaps even in
a “fixed-focus” camera), setting the focus distance to the hyperfocal distance
yields the greatest possible field of view, which hopefully will accommodate most
of the photographic needs of the user.
The hyperfocal distance is given approximately by:
 |
Equation 16 |
Depth of focus
A related property, depth of focus, is often confused with depth of field.
If we have an object lying in only one plane, and move the film forward or backward from its position that gives perfect focus, we find that the image
becomes misfocused. In effect, moving the film changes the object distance for
perfect focus so it no longer corresponds to the actual distance to our object.
Depth of focus is the range of positions of the film plane over which acceptable
focus is maintained, for an object at a given distance.
This is reckoned in a way parallel to the concept of reckoning depth of field, and
involves the now familiar concept of an adopted criterion for the maximum
acceptable diameter of the circle of confusion resulting from imperfect focus.
Depth of focus is of greatest interest in considering such things as the impact of
accidental shift in the position of the film plane owing to imperfect film guidance.
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