04. Circle of confusion, depth of field and hyperfocal distance.
Description
This article is from the Photographic Lenses Tutorial, by David Jacobson with contributions by others.
04. Circle of confusion, depth of field and hyperfocal distance.
The light from a single object point passing through the aperture is converged by the lens into a cone with its tip at the film (if the point is perfectly in focus) or slightly in front of or behind the film (if the object point is somewhat out of focus). In the out of focus case the point is rendered as a circle where the film cuts the converging cone or the diverging cone on the other side of the image point. This circle is called the circle of confusion. The farther the tip of the cone, ie the image point, is away from the film, the larger the circle of confusion.
Consider the situation of a "main object" that is perfectly in focus, and an "alternate object point" this is in front of or behind the main object.
Soa alternate object point to front principal point distance Sia rear principal point to alternate image point distance h hyperfocal distance C diameter of circle of confusion c diameter of largest acceptable circle of confusion N f-stop (focal length divided by diameter of entrance pupil) Ne effective f-stop Ne = N * (1+M/p) D the aperture (entrance pupil) diameter (D=f/N) M magnification (M=f/(So-f))
The diameter of the circle of confusion can be computed by similar triangles, and then solved in terms of the lens parameters and object distances. For a while let us assume unity pupil magnification, i.e. p=1.
When So is finite
C = D*(Sia-Si)/Sia = f^2*(So/Soa-1)/(N*(So-f))
When So = Infinity,
C = f^2/(N Soa)
Note that in this formula C is positive when the alternate image point is behind the film (i.e. the alternate object point is in front of the main object) and negative in the opposite case. In reality, the circle of confusion is always positive and has a diameter equal to Abs(C).
If the circle of confusion is small enough, given the magnification in printing or projection, the optical quality throughout the system, etc., the image will appear to be sharp. Although there is no one diameter that marks the boundary between fuzzy and clear, .03 mm is generally used in 35mm work as the diameter of the acceptable circle of confusion. (I arrived at this by observing the depth of field scales or charts on/with a number of lenses from Nikon, Pentax, Sigma, and Zeiss. All but the Zeiss lens came out around .03mm. The Zeiss lens appeared to be based on .025 mm.) Call this diameter c.
If the lens is focused at infinity (so the rear principal point to film distance equals the focal length), the distance to closest point that will be acceptably rendered is called the hyperfocal distance.
h = f^2/(N*c)
If the main object is at a finite distance, the closest alternative point that is acceptably rendered is at distance
Sclose = h So/(h + (So-F))
and the farthest alternative point that is acceptably rendered is at distance
Sfar = h So/(h - (So - F))
except that if the denominator is zero or negative, Sfar = infinity.
We call Sfar-So the rear depth of field and So-Sclose the front depth field.
A form that is exact, even when P != 1, is
depth of field = c Ne / (M^2 * (1 +or- (So-f)/h1))
= c N (1+M/p) / (M^2 * (1 +or- (N c)/(f M))
where h1 = f^2/(N c), ie the hyperfocal distance given c, N, and f and assuming P=1. Use + for front depth of field and - for rear depth of field. If the denominator goes zero or negative, the rear depth of field is infinity. ("!=" means "is not equal to".)
This is a very nice equation. It shows that for distances short with respect to the hyperfocal distance, the depth of field is very close to just c*Ne/M^2. As the distance increases, the rear depth of field gets larger than the front depth of field. The rear depth of field is twice the front depth of field when So-f is one third the hyperfocal distance. And when So-f = h1, the rear depth of field extends to infinity.
If we frame an object the same way with two different lenses, i.e. M is the same both situations, the shorter focal length lens will have less front depth of field and more rear depth of field at the same effective f-stop. (To a first approximation, the depth of field is the same in both cases.)
Another important consideration when choosing a lens focal length is how a distant background point will be rendered. Points at infinity are rendered as circles of size
C = f M / N
So at constant object magnification a distant background point will be blurred in direct proportion to the focal length.
This is illustrated by the following example, in which lenses of 50mm and 100 mm focal lengths are both set up to get a magnification of 1/10. Both lenses are set to f/8. The graph shows the circles of confusions as a function of the distance behind the object.
circle of confusion (mm)
#
# *** 100mm f/8
# ... 50mm f/8
0.8 # *******
# *********
# *********
# ****
# *****
# ****
0.6 # ****
# ***** .......
# *** ..................
# ** .............
0.4 # **** .........
# *** ....
# ** .....
# * ....
# **..
0.2 # **.
# .*.
# **
#*
*######################################################################
0 #
250 500 750 1000 1250 1500 1750 2000
distance behind object (mm)
The standard .03mm circle of confusion criterion is clear down in the ascii fuzz. The slope of both graphs is the same near the origin, showing that to a first approximation both lenses have the same depth of field. However, the limiting size of the circle of confusion as the distance behind the object goes to infinity is twice as large for the 100mm lens as for the 50mm lens.
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