by Q.-Tuan Luong for the Large Format Page
f16 f22 f32 f45 f64 5% 30% 50% 20% 5%Would the project have been Smell the tidepools, the breakdown would have been probably quite different, but this still gives you an idea of what f-stops are commonly used.
If you don't want to read the explanations and wade in the math which follows, here are the important points explaned in more details in this article:
D(mm) F 1 16.6 2 22.6 3 32.2 4 32.6 5 32.9 6 45.2 7 45.4 8 45.6 9 45.8 10 64
Now time to get technical...
If the circle of confusion is large, the image will look fuzzy.
If the circle of confusion is small, then it
will be indistinguishable to the eye from a point, and the image will
look sharp. There is not a universal boundary between fuzzy and sharp.
The blurring of an out of focus point occurs gradually. The
calculations done in this document should be taken just as general guidelines.
When you use a depth of field scale, you rely on
the manufacturer's choice of a value for the acceptable circle of
confusion. This value might not be the best for your use.
The brightness within the circle of confusion is constant.
If we use the criterion that two points are resolved when their
circle of confusion do not overlap (if the circles overlap, the two
points are imaged in a continuous bright spot) then
the resolution
R is related to
the diameter of the circle of confusion c by
R = 1/c. This is the number of non-overlapping circles of
confusion that can be fit in a unit length. If c is
expressed in millimeters, then R represents the number of
lines pairs per millimiters (lp/mm). One precision is necessary here. We have
considered two bright points, separated by a dark space. This is the
same situation as two bright lines separated by a dark line. Each
element consists of a dark line and and bright line. The separation
between each element is c.
A source of confusion is that some
authors consider a line pair (a dark line and a white line) to be two
lines, so they would use R = 2/c. This corresponds to
the number of pixels.
The smallest resolvable distance c0
is such that two dark lines separated
by more than this distance (by a white line) are seen as two lines,
while two dark lines separated by less than this distance would be
seen as one line.
It is generally agreed that at normal minimal viewing distance
d0 (reading distance, around 25cm), the eye can resolve
elements spaced at 0.2mm. A good choice for the diameter of the acceptable
circle of confusion c on a print viewed at that distance
is therefore 0.2mm. The resolution is R0 = 1/c0 = 5 lp/mm
Some people feel that the eye can sense finer details.
The resolving power of the eye is approximatively
inversely proportional to the
viewing distance d.
A common practice is that a print should be viewed from a distance
approximately equal to the diagonal measurement of the print. The
rationale is that the eye would then be at the center of
projection if the image was taken with a normal lens (whose focal is
equal to the diagonal measurement of the negative), resulting in a
correct perspective. Personally I am not fond of this rule. After
viewing the image as a whole, I always like to come closer to see the
details. It is in order to preserve those details, even in extreme
enlargements, that you use a large format camera in the first place !
The enlargement factor is the ratio of the linear
dimension of a feature on the print by its corresponding feature on
the negative. If the print and the negative don't have the same aspect
ratio, it is not the ratio of the dimensions of the print and the negative.
Lets illustrate the formula c = c0/m * d/d0 by a few
examples. We assume c0 = 0.2 (average eye resolution), and d/d0 = 1
(print viewed at the minimal distance). The formula becomes
c = 0.2/m
To make a full-format 6x8 (150mm x 200mm) from a 35mm
negative (24mm x 36mm), you match the two smaller dimensions, so
the magnification is m = 150/24 = 6.25, and the
acceptable circle of confusion on the negative is c= 0.2 * 1/6.25 =
0.032mm. This value appears to be used by most of the 35mm lens
manufacturers on their depth of field scale. The diagonal of a
150mm x 200mm (6x8inch) print is 250mm (10inch), which corresponds to
a normal minimal viewing distance.
To make a full-frame 8x10 (200mm x 250mm) from a 35mm
negative (24mm x 36mm), you match the two larger dimensions, so
the magnification is m = 250/36 = 7, and the
acceptable circle of confusion on the negative is c = 0.2 * 1/7 = 0.028mm.
This indicates the limitation of the depth of field scales commonly
used: even a 8x10 looked closely won't appear critically sharp.
To make the same print from a 4x5 (100mm x 125mm) negative, the
magnification is 250/125 = 2, and the
acceptable circle of confusion on the negative is 0.2 * 1/2 = 0.1mm.
This value is often used for LF depth of field tables, but personally
I don't find it adequate.
In summary, the acceptable circle of confusion on the negative
depends on a number of factors, and it is better to determine the one
which is best for you, rather than rely on the manufacturer's which is
often not critical enough.
The defocus circle of confusion
Let consider a bright point in space surrounded by a dark background, or
equivalently
a point light source. It is converged by the lens into a cone. The image
of the point is the intersection of the film plane with the cone. If
the point in space lies in the plane of focus, the tip of the cone coincides
with the film plane, so that the image is a point.
If the point in space
lies outside the plane of focus, the cone is not intercepted by the
film plane at the right place, therefore
the image is a circle, called the
circle of confusion. The more the point is out of focus, the larger is
the circle.
If tilts are used, the image is
actually an ellipse, however for reasonable amounts of tilts, the
circle is a good approximation.
What is an acceptable circle of confusion ?
The diameter of the largest circle of confusion which is perceived by
the eye as a point is called the acceptable (or permissible) circle of
confusion. The acceptable circle of confusion c
on the negative depends on:
The equation is c = c0/m * d/d0 .
The relation between focus spread and the circle of confusion
If you have followed the method
II.3, you have chosen a point
of nearest focus and a point of furthest focus, measured the distance
D (called here the focus spread
between the two focussing rail positions.
If the distance D was zero, then everything
is exactly in focus because your subject is planar or at infinity.
Calculations (see references) show that for small magnifications, the relation between the focus spread D and the diameter of the circle of confusion c is extremely simple, and gives you the admissible f-number:
N = D/(2c)where N is the f-stop number, regardless of the focal length or format used. Once you have determined what is an acceptable circle of confusion for you, this formula gives you the f-stop, proportional to the focus spread. As an example, here are the tabulated values that I would use, based on c=0.1 for 5x7. This gives a critically sharp 11x14 at minimum viewing distance, and has the advantage of being easy to remember. I've also included in the table the equivalent for 4x5 based on c=0.066, and 8x10 based on c=0.133 Distances are in mm.
focus spread admissible 4x5 5x7 8x10 fstop D (mm) N 1.4 2.2 2.8 11 2.1 3.2 4.2 16 3.0 4.5 6.0 22 4.2 6.4 8.4 32 6.0 9.0 12.0 45 8.5 12.8 17.0 64
The focus spread is sometimes refered to as depth of focus,
the acceptable distance between imaging planes for a given circle of
confusion and f-stop D = 2cN.
In the previous formula N is the effective f-number.
For close-ups, you would replace the f-stop by the effective f-stop,
and use instead the relation:
N = f/v * D/(2c) = 1/(1+M) * D/(2c)
where f is the focal and v your total
belows extension. M is the magnification.
Since the ratio f/v is always smaller
than 1, by using the previous formula, you
always end up stopping down more than necessary. Note that at infinity,
the ratio is exactly 1.
I think this approach is most useful for judging whether within the constraints dictated by the shutter speeds that you can use, you can get your subjects sufficiently in focus. For instance, you are trying to get a field of flowers in focus at the same time as a tree behind. You measure a focus spread of 10mm between the closest flower and the tree. In 5x7, you'd need to stop down at 45. Is the resulting shutter speed acceptable, given the small breeze ? If yes, you just go ahead and take the picture. If no, you might want to change the composition so that the closest flower would be a bit further.
As a general approach (ie independently from the constraints), the
drawback of this approach is that it relies on the choice of an acceptable
circle of confusion, which in turn depends on a number of factors,
including the dimensions and display of the final print. Since you
don't necessarily know what you'd like to do with your negative in the
future, why don't you just use a very tight acceptable circle of
confusion and stop down more ? The next two sections answer this from an
optical point of view.
Strictly speaking, diffraction is a function of aperture size or the
physical size of the hole and that is how it would be defined in a
physics textbook. Which means that the larger area aperture in a
300mm lens at f/16 (as compared to a 50mm lens at f/16) should
provide lower diffraction. However, diffraction patterns are
angular patterns and as such are dependent on how far from the
aperture you place the screen used to view it also. In
photography, the aperture is at the optical center of the lens and
the screen is (for infinity focus) one focal length away. The
physical size of the diffraction blur is then the focal length
divided by the apparent size of the aperture i.e., the definition
of the f stop. Thus, in photography, diffraction is only a
function of f stop and not a function of the focal length. In
simpler terms, the larger aperture of the 300mm lens does offer
lesser diffraction at the diaphragm (i.e., less bending around the
diaphragm) but since the light now has a longer distance to travel
(as compared to the 50mm lens), the smaller bending still results
in a fair bit of blur at the viewing screen. N Dhananjay
The image
of a point light source (such as a star)
after diffraction appears not as a point, but as a central circular
spot
surrounded by a series of dark and bright rings. See
references for details and formulas. Because
a point source is imaged as a disk rather than a point, it will appear
blured, and this limits the maximum resolution of the lens. The formula
R = 1500/N explaned below, gives the maximum resolution of the lens
as a function of the f-number N.
The central spot size due to diffraction, called the Airy disk,
has a diameter of d =
2.44 x lambda x N. Note that this formula depends on the f-stop
and not on the physical aperture of the lens.
For
a wavelength near the middle of the visible,
lambda = 550 x 10 exp(-9)
meters
,
d = 1.34 x 10 exp(-6) x N meters, or d = N /
750 if d is in mm. No lens can make a spot
smaller than that value.
86% of the power is in the Airy disk, so it can be viewed as the image
of the point. However unlike in the case of defocus, the brightness
within the central diffraction spot is not constant, but it decreases
from the center to its border, where it reaches 0, resulting in a dark
ring. If we use the Rayleigh's criterion that two points are resolved when the
dark ring of one coincides with the center of the Airy disk of the
other, the corresponding resolution is
R = 2/d.
As it can be seen from
these values, stopping down the lens beyond a certain point degrades
considerably the resolution. However, this is more of a concern to
small camera users than to large format users who don't need a large
enlargement ratio. At f22, the diffraction spot equals the acceptable
circle of confusion for 35mm. At f64, it barely equals the acceptable circle
of confusion for 4x5. There is still plenty of sharpness left, and
these values indicate that you shouldn't be too worried about stopping
down a lot if necessary. Some folks on the East Coast thought that
that Ansel Adams f64 Group was lame and started a f180 Group.
An aggravation to note is that the fstop N is the effective
aperture. Therefore, if you are working at 1:1, your f-number would be
doubled (two f-stops), and the resulting resolution would be
divided by two. Add to that the fact that you need to stop down a lot
a close distances to get enough depth of field, and you see why there
are not many folks doing macro work with LF
Although the more rigorous way to combine both effects is the use the
MTF (see references), a good approximation is to
use the root mean square error of the two diameters. The total circle of
confusion is:
This choice of f-stop is optimal, in the
sense that it will yield the sharpess possible image at the depth of
field limits. However, the larger your focus spread, the less sharp it
will be. Sometimes you want to know whether it results in
enough resolution for the enlargements you want to make.
The resulting total circle of confusion
has a diameter:
The defocus and the diffraction result in two different
light patterns, which in turn lead to two different definitions
of resolution. Remember that in the defocus case, brightness in the
circle is uniform, and
two points are considered resolved when their circle of confusion do
not overlap, resulting in a resolution R = 1/c, while
in the diffraction case, brightness is more concentrated at the
center. If one accepts the technical definition that
two points are resolved using the Rayleigh's
criterion, this would result in the resolution R = 2/d.
For this reason, Bob Wheeler advocated combining the circle of
confusion with the central half of the diffraction spot, ie using
N/1500 instead of N/750, resulting in the optimum f-number:
N = sqrt(750 D).
In practice, this would mean using the next
f-stop in the table below (ie for D=1.3mm, you'd use f32 instead of
f22), which would imply stopping down even more.
However, Paul Hansma suggests that
from a photographic point of view, the
Rayleigh criterion is not really enough. The brightness between the two
"resolved" objects is very close to the peak brightness, so
we would not call them resolved if that is what we saw in a photo. To
have a level of resolution comparable to that obtained with the
circle of confusion, where there is no overlap, would call for having
no overlap of the two diffraction spots. This would explain why N/750
fits his experimental results very well.
To summarize, if D is the focus spread expressed in
millimeters, then the optimal f-stop
which yields the sharpest possible image
at the depth of field limits is N = sqrt(375 D). This
works regardless of focal lengths, formats, and movements. The
resulting resolution at the limits of depth of field (ie for your far
and near points) cannot be improved in anyway and determine the
maximum possible enlargment. Here are some tabulated values by whole fstops:
The previous calculations were done at zero magnification. As before,
or close-ups, you would replace the f-stop by the effective f-stop.
Diffraction
A beam of light passing through a circular aperture spreads out a
little, a phenomenon known as diffraction. Diffraction is a physical
phenomena which is unescapable. The smaller the aperture,
the more the spreading. For photographic
lenses, diffraction depends only on the f-number N.
fstop resolution limit
N R (lp/mm) d (mm)
11 136 .014
16 93 .021
22 68 .029
32 46 .042
45 33 .059
64 23 .085
These values correspond to the "diffraction limit" of resolution, the
maximum resolution which could be achieved on perfect film by a
perfect lens. Don't expect a real lens to deliver 136 lp/mm at f11.
Finding the optimal f-stop to balance defocus and diffraction
The more you stop down, the more you gain sharpness for the out of
focus points at the
depth of field limit,
but at one point, diffraction effects outweight whatever
gain is obtained. Intuitively, the optimal point is when the two effects are equal.
ct = sqrt( (D/(2N))^2 + (N/750)^2 )
This quantity is mimimal when N = sqrt(375 D).
For such a f-number, the diameter of the defocus circle of confusion
is exactly equal to the diameter of the diffraction spot:
D/(2N) = N/750.
ct = sqrt(2)/750 N = N/530 = .0365 sqrt(D)
If you have predetermined a value for the acceptable circle of
confusion ct, this formula gives you the maximum focus
spread and f-stop that you can use:
D = 750 ct^2 N = 530 ct
Let suppose that you are assuming average eye resolution and minimal
viewing distance, then ct = 2 * 0.2/m, where m
is the enlargement factor. This gives you the following values:
m D(mm) N
2 30 106
3 13 70
4 7.5 53
5 4.8 42
6 3.3 35
Table of optimal fstops (1)
focus optimal resolution resolution maximum enlargment
spread fstop at DOF limits for in-focus based on DOF limits
D (mm) N ct(mm) R(lp/mm) d (mm) R(lp/mm) m
.3 11 .021 95 .014 136 19
.7 16 .030 65 .021 93 13
1.3 22 .042 47 .029 68 9
2.7 32 .061 32 .042 46 6.4
5.4 45 .085 23 .059 33 4.6
11.0 64 .122 16 .085 23 3.2
Here are tabulated values by whole millimeters focus spread. I give
the f-stop is given in decimal values ("F"), followed by actual
f-number ("N").
Now what does
"decimal value" mean ? It is what you would read on a meter
graduated in 1/10th of fstops, like many hand-held meters. For
instance, "decimal"
22.33 means 22 and
one third of a stop (0.33),
"decimal" 22.5 means 22 and a half stop (0.5),
whereas in f-numbers this would be respectively
25.4 and 26.9 which certainly speaks less to me. On my camera, I
have stuck only the decimal values.
Table of optimal fstops (2)
This shows you the best value of the f-stop to use.
D(mm) F N
1 16.6 19.4
2 22.6 27.4
3 32.2 33.5
4 32.6 38.7
5 32.9 43.3
6 45.2 47.4
7 45.4 51.2
8 45.6 54.8
9 45.8 58.1
10 64 61.2
The fstop given on these tables is optimal in the sense that it is
the choice which yields the sharpest results at the DOF limits among
all the other possible f-stops.
Remark that this table implies that you have to stop down a lot !
This assumes that other considerations (wind, reciprocity...) do not
prevent you from doing so, in which case you'd have to compromise.
By compromising, I mean you would use an f-stop other than the
optimal value. Then by applying the admissible
f-stop above, you would see if that non optimal f-stop gives you
a result which is acceptable, although you know now that it would not
be the best possible. In general, calculations show that results
pretty close
to the optimal one are obtained within 1 fstop of the optimal value.
Satisfying sharpness requirements for the horizon
If you focus on the point which is 2/3 from the near,
1/3 from the far, as suggested in the
focussing method II.3, and the focus spread between near
and far was D then the actual focus spread is
4/3 * D. This is because you have "wasted" 1/3 behind the far point to
ensure that the horizon is critically sharp.
Because of various optical aberrations, lenses are not perfect. However, stopping down enough corrects those aberrations pretty well in modern lenses, well enough that they can be considered diffraction limited at f22 and smaller apertures.
The final resolution r_final that you get on film depends not only on the aerial resolution of the lens r_lens but also on the resolution of the film r_film. However, when r_lens becomes critical, the effect of r_film becomes quite neglectible. In LF, since you have to stop down so much, this is almost always the case.
Several sources use the formula 1/r_final = 1/r_lens + 1/r_film to compute the final resolution. This formula is an approximation to the exact calculation consisting of the convolution of the response of the film and the response of the lens. This approximation is most valid when both the film and lens are being used near their resolution limits (spatial frequencies with very low contrast). This corresponds roughly to f-stops up to f16. Thus the formula is pretty good for 35mm and MF work. However, when the frequencies involved are nowhere near the film limits, the formula is a poor approximation which predits a worse 1/r_final than what you actual get. For f-stops of f32 and higher, what you get on film is in fact practically equal to the aerial resolution, and the formula shouldn't be used in that case.
Some people think that instead of using all that math, you should just rely on inspecting the ground glass. You would watch it through a lupe as you stop down, and see when the out of focus areas become sharp enough. For several years I have relied on this approach. I eventually found, when I switched to a higher power lupe for evaluating the transparency on the light box, that this yielded results less sharp than I wanted. The problem is that it is not that easy to make a good judgement of sharpness on the ground glass. The image is grainy, and when you stop down, it gets pretty dim, especially through a lupe in the corners of a wide-angle view.
Material on diffraction includes the Melles Griot catalog, and Jacobson's lens tutorial (original, improved math typeset (HTML), improved math typeset (PDF)). In the latter reference, defocus and diffraction are combined using the MTF. The resulting on-film resolution values were calculated by Bob Atkins. Using the formulas in this article, for c=0.035, at f32, D= 2 * 0.035 * 32 = 2.24, ct = .0365 * sqrt(2.24) = .054, which gives the resolution R = 2/.054 = 37 lp/mm, which is in the ballpark of Bob Atkins calculation of 43.5 lp/mm. Note that his data at f32 supports the observation that at high f-numbers, on-film resolution is pretty close to aerial resolution.
The idea of combining the effects of diffraction and defocus to find the optimal f-stop was introduced in the article View camera focussing in practice, page 1, page 2, page 3, page 4 (gif files, reproduced with the author's permission) by Paul Hansma in Phototechniques March/April 96. Most of the ideas developed in this page are from this article. Later, Bob Wheeler, in his Notes on view camera geometry, section 9.3, advocated combining the radius of the diffraction spot N/1500 with the circle of confusion, rather than the diameter N/750. His own data support this argument, which is theoretically sound. However Paul Hansma's experiments seem to have been conducted more carefully.
