Understanding image sharpness part 6:
Depth of field and diffraction
by Norman Koren

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Table of contents

for the
image sharpness
series

Part 1: Introduction
Part 1A: Film and lenses
Part 2: Scanners and sharpening
4000 vs. 8000 dpi scans
Part 3: Printers and prints
Part 4: Epson 1270 results
Part 5: Lens testing
Part 6: Depth of field and diffraction
Introduction | Sharpness at DOF limits
Hyperfocal distance | Film flatness
Diffraction
Using the DOF scale to find optimum aperture
DOF and focal length
DOF limits, diffraction and format
Sweet spot | Links | MTF equations
Digital cameras vs. film, part 1 | part 2
Part 8: Grain and sharpness: comparisons
In this pagr we discuss depth of field (DOF) and deal with such questions as, How sharp is the image at the DOF limits? Can you trust DOF scales? (I wouldn't.) What aperture should you choose for optimum sharpness when the subject spans a range of distances?
Green is for geeks. Do you get excited by an elegant equation? Were you passionate about your college math classes? Then you're probably a math geek a member of a maligned and misunderstood but highly elite fellowship. The text in green is for you. If you're normal or mathematically challenged, you may skip these sections. You'll never know what you missed.

Depth of field: introduction

So far we've only considered images in exact focus. That's all you need if you only photograph distant landscapes or two-dimensional objects like paintings. But most subjects are three-dimensional: you want to capture objects clearly over some range of distance from near to far; hence you need to be concerned with depth of field (DOF). The basics of DOF are well known: The more you stop down a lens (the larger the f-stop number), the larger the DOF. Wide angle lenses appear to have much larger DOF than telephotos. Telephotos are often used to intentionally limit DOF, for example in portraits where you want the subject to be in focus, but you want a distracting foreground or background to be out of focus. But if you read on you'll discover telephotos don't actually have less DOF.

Most 35mm and medium format prime lenses and some zooms have depth of field (DOF) scales. Your camera's instruction manual states that if you stop down your lens, for example to f/8, everything at distances between the two f/8 DOF marks will appear to be "in focus." Of course, not exactly in focus. You may therefore ask the question, "How sharp is the image (what is its MTF?) at the DOF limit?" To answer these questions we begin with the diagram below, representing a lens with aperture a imaging an object at s on the film plane at d.
.

An object at a distance s in front of the lens is focused at a distance d behind it, according to the lens equation: 1/d = 1/f - 1/s, where  f is the focal length of the lens. If the lens were perfect (no aberrations; no diffraction) a point at s would focus to an infinitesimally tiny point at d. An object at sf , in front of s, focuses at df , behind d. At the film plane d, the object would be out of focus; it would be imaged as a circle whose diameter Cf  is called its circle of confusion. Likewise, an object at sr, behind s, focuses at dr, in front of d. Its circle of confusion at d has diameter Cr.

The depth of field (DOF) is the range of distances between sf and sr, (Dr + Df ), where the circles of confusion, Cf and Cr, are small enough so the image appears to be "in focus." The standard criterion for choosing C (the largest allowable value of Cf and Cr) is that on an 8x10 inch print viewed at a distance of 10 inches, the smallest distinguishable feature is (allegedly) 0.01 inch. That was the assumption in the 1930's when film was much softer than it is today. At 8x magnification this corresponds to 0.00125 inches = 0.032 mm on 35mm film, close to the standard 0.03 mm used by 35mm lens manufacturers to calculate their DOF scales. If you've ever had a close look at a fine contact print from 4x5 or 8x10 film, you'll doubt that 0.01 inch feature size is a good criterion. Studies on human visual acuity indicate that the smallest feature an eye with 20:20 vision can distinguish is about one minute of an arc: 0.003 inches at a distance of 10 inches. But inertial prevails: 0.01 inch is universally used to specify DOF.
 

 Basic Lens and Depth of Field equations.
1/s + 1/d = 1/f
(1)
f = focal length— the lens's most important parameter.
s = lens-to-object distance.
d = lens-to-film plane distance. If object s is located an infinite distance from the lens (s >> f), the image is focused at a distance f from the lens, i.e., d = f.
N = f-stop = f /a
(2)
a = aperture diameter.
Cf = a|(df -d)/df |
(3)
Circle of confusion at the film plane (d) for object located at Sf (closer than s), which focuses on df . Derived from simple geometry using 1/sf + 1/df = 1/f.  |...| denotes absolute value.
Cr = a|(d-dr)/dr|
(4)
Circle of confusion at the film plane (d) for object located at Sr(behind s), which focuses on dr. Derived from simple geometry using 1/sr + 1/dr = 1/f.
M = d/s = f / (s-f ) (5)
Magnification
Focus = F = s+d(6)
The focus scale of most lenses is the distance from the object to the film plane. >= 4*f.
Df  = s - 1/(1/f-(1-C/a)/d)
= sC(s-f )/( fa+C(s-f ))
= sCN(s-f )/( f 2+CN(s-f ))
(7)
Df (front depth of field limit relative to s) derived from (1) and (3) using
Df = s - s;   N = f-stop = f /a
sf = s - D;   Lens to front DOF limit.
There equations are in agreement with Sushkin.
Dr = 1/(1/f-(1+C/a)/d) - s
= sC(s-f )/( fa-C(s-f ))
= sCN(s-f )/( f 2-CN(s-f ))
(8)
Dr (rear depth of field limit relative to s) derived from (1) and (4) using Dr = sr - s.
Dr = infinity when demoninator  fa-c(s-f ) <= 0  ( f 2-CN(s-f ) <= 0).
sr = s + D;   Lens to rear DOF limit.

Sharpness at DOF limits

How sharp is an image with a 0.03 mm circle of confusion? Not difficult to answer. The circle has an MTF,  whose equation is derived in the box at the bottom (for math geeks only). For a circle of confusion C only (no diffraction), the spatial frequencies for MTF = 50%, 20% and 10% (  f50 , f20 and  f10 ) are,
f50 = 0.72/C ;    f20 = 1/C ;    f10 = 1.11/C
 f50 is familiar from film, lenses and scanners.  f20 is interesting because it is the inverse of C. For C = 0.03mm, f50 = 24 lp/mm. An excellent lens for the 35mm format has  f50 ~= 60 lp/mm.
The sharpness at the DOF limit for C = 0.03mm (typical for the 35mm format) is 24 lp/mm, about 40% of an excellent lens in focus (60 lp/mm). Not great!
You can determine the precise circle of confusion C for the depth of field scale on your lens with a simple procedure, using an equation derived from the box above.
  1. Canon 50mm lens at hyperfocal distance for f/16.Set the lens's focus so the infinity mark is opposite the far DOF mark for the largest f-stop, Nmax , typically between f/16 and f/32. This is equivalent to setting sr to infinity.
  2. Note the distance s+d where the lens is focused.
  3. The circle of confusion is C = f2/(Nmax(s-f)) ~=  f2/(Nmax(Focus - 2 f)). At this setting, s >> f and s >> d, so the equation can be simplified to  C = f2/(Nmax Focus) with little loss of accuracy.
For example, the largest f-stop on my Canon FD 50mm f/1.4 lens is f/16. When the f/16 DOF mark is set at infinity the lens is focused at 5 meters = 5000 mm (be sure to use the same units for the lens focal length and the distance). C = 502/(16*(5000-100)) = 0.032 mm. I have found this to be typical of a large sample of Canon FD and older Leica M-series lenses. It also holds for the 45mm lens on my (35mm panoramic) Hasselblad XPan.

For my old (medium format) chrome-barrel Hasselblad (Zeiss) lenses, C = 0.055 mm, corresponding to the same 0.01 inches on an 8x10 print, enlarged 5x for this format (nominally 6x6 cm, but actually 5.6x5.6 cm).

Zeiss has a good deal to say about DOF in Camera Lens News No. 1. Inadequate depth of field turns out to be their number one customer complaint. They state, "All the camera lens manufacturers in the world including Carl Zeiss have to adhere to the same principle and the international standard that is based upon it (0.03mm for the 35mm format), when producing their depth of field scales and tables." They summarize,

Of course the actual image sharpness at the DOF limits is degraded by diffraction, lens aberrations, film properties and possible lack of film flatness, so the overall sharpness at the DOF limits will be inevitably worse than the simple circle of confusion would indicate. For this reason alone, the standard for setting the circle of confusion is a bit loose.
Imatest: affordable software for measuring sharpness and image quality

The myth of hyperfocal distance

The focus point in the above example is called the hyperfocal distance for f/16. When you focus at this distance, everything between the front DOF mark (about 2.7 meters in the example) and infinity is supposed to be "in focus." Well, sort of. Some authors, for example, photofocus.com, recommend focusing at the hyperfocal distance if you want a large range of focus out to infinity. I don't. Neither does Harold M. Merklinger in his page, Depth of Field Revisited. Nor does Zeiss.

If the part of the scene at infinity is at all important in the image— it's often visually dominant— you'll be disappointed with the sharpness, which is only 40% that of a high quality lens in focus; about one third what the eye can distinguish. Merklinger recommends focusing at infinity— you lose very little forward depth of field. I feel safe setting infinity focus opposite the far DOF mark corresponding to 2 stops larger than the actual f-stop setting (half the number). For example, if you are using f/8, it's safe to put the far f/4 DOF mark opposite infinity. It's a judgment call. When you make it, think about what parts of the image will be dominant. There is no rule to blindly follow.

Film flatness

Film doesn't lie perfectly flat— especially roll film (35mm and medium format). Sheet film is better. Film flatness is probably the least predictable of the factors that degrade image sharpness. According to Robert Monaghan, "film often buckles in 60% of 35mm SLRs tested, and virtually all medium format backs - by an average of 0.2mm (on 35mm). Yet even a 0.08 mm film bulge can reduce contrast by an astonishing 48%!" The latter number depends on the f-stop. The equation for the circle of confusion due to film bulge is (for focus near infinity: s >> f ),
Cbulge = bulge/f-stop
For a 0.08 mm bulge at f/5.6, Cbulge = 0.014mm. For a 0.2mm bulge at f/5.6, Cbulge = 0.036mm— worse that the circle of confusion at the DOF limit. Pretty bad. That's why we sometimes need to stop down a little more than optimum.

To further confound you, film flatness is a function of time after winding the film. And it's different for 35mm and medium format. According to Robert Monaghan, film gets flatter if you wait up to 30 minutes after winding 35mm film, but according to both Mohaghan and Zeiss (in Camera Lens News No. 10) the bulge increases with time after winding medium format film: it's small at 5 minutes, significant at 15 minutes and maximum after 2 hours. One solid piece of useful information from Zeiss: the bulge is only half as much for 220 film as it is for 120. (That means I have to buy a new back if I go back to using my old Hasselblad; a great temptation.) The Zeiss rule of thumb is, " For best sharpness in medium format, prefer 220 type roll film and run it through the camera rather quickly." Temperature and humidity probably also affect flatness.

Oh yes, digital cameras don't suffer from film flatness problems. That's one reason why their performance is expected to exceed 35mm with only 6 to 10 megapixel sensors (multiplied by 3 when converted to RGB file formats). For much more detail on film flatness, I recommend Robert Monaghan's exhaustive discussion (with reader comments).

Enter diffraction

Light bends when it passes near a boundary. "Near" is defined as a few wavelengths of light, where the wavelength ω at the middle of the visible spectrum— green to yellow-green— is 0.0005 to 0.000555 mm (500 to 555 nanometers). The eye is most sensitive at 0.00055 mm, but 0.0005 may be more characteristic of daylight situations. This bending, called diffraction, is an unavoidable physical effect that happens regardless of lens quality.

The smaller the aperture— the larger the f-stop ( N )— the more the image is degraded by diffraction. The equation for the Rayleigh diffraction limit, adapted from R. N. Clark's scanner detail page, is,

Rayleigh limit (line pairs per mm) = 1/(1.22 N ω)
The MTF at the Rayleigh limit is about 9%. Most lenses for 35mm and larger cameras are aberration-limited— relatively unaffected by diffraction— at N = f/8 and below. The spatial frequencies for 10% and 50% MTF for diffraction-limited lenses are,
f10 = 0.77/(N ω) ;     f50 = 0.38/(N ω)
These numbers are derived from the OTF equation and figure in David Jacobson's Lens Tutorial, Part V (where MTF = |OTF|); see MTF equations, below. The diameter of the corresponding circle, known as the Airy disk, is,
CAiry = 2.44 N ω     (The 1.22 term in the Rayleigh limit comes from the radius.)
Now here's the rub. If it weren't for diffraction, you could stop down a lens as much as you needed to get the depth of field you desired. But in the real world you reach a point where diffraction starts degrading the image more than misfocus. There is an optimum aperture that results in the best sharpness over a range of distances. But how to find that optimum isn't exactly common knowledge.

You can measure your lens's sharpness and learn how it varies with aperture using the Imatest program, which works with a simple target you can print yourself.
Imatest: affordable software for measuring sharpness and image quality
To learn more about diffraction, see Sean McHugh's superb Tutorial: Diffraction & Photography.

Use your depth of field scale to find the optimum aperture

The following procedure lets you determine the aperture where you can obtain optimum sharpness over a range of distances  It is derived in the green box, below.
  1. Determine the closest and farthest distances you want to be in sharp focus.
  2. Focus the lens so identical DOF marks on either side of the focus mark are aligned with these distances. For example, if you are using the 50mm lens in the above illustration and want sharp focus between 14 and 25 feet, you would place identical DOF marks, in this case, f/4, at these distances. Focus would be around 17 feet.
  3. Make the adjustments indicated in the table below, based on the f-stop of the DOF marks at the desired focus limits. These adjustments are valid for the 35mm format (C = 0.03 mm)
Aperture adjustment for optimum sharpness
when focusing over a range of distances
f-stop for DOF marks
at desired focus limits
 Decrease aperture
by
Comments
f/4-f/8 2 f-stops For example, if marks indicate f/4,
close down to f/8. 
f/11-f/16 1 f-stop  
f/22-f/32 No change  
f/45 and above Increase by 1 f-stop (!) View camera territory, where diffraction
takes a big bite out of sharpness.
 
Derivation of aperture for optimum sharpness over a range of distances
The tables below contain the sharpness (50% MTF spatial frequency f50 in the upper table; 20% MTF spatial frequency f20 in the lower table) for distances corresponding to a lens's DOF marks. These numbers are the result of misfocus and diffraction; they do not include lens aberrations or film bulge. There is no general way to include lens aberrations in this table because they are dependent on lens design and manufacturing quality. Aberration correction is a major factor in distinguishing mediocre from excellent lenses. The excellent lens used in this series has  f50 = 61 lp/mm at f/8, which is probably close to its best performance. The  f50 numbers for exact focus at f/4 through f/8 (upper left in both tables) are shown in gray because they are unrealistically high.  f50 = 61 lp/mm is nearly as good as a 35mm format lens gets.

Film sharpness, which is the critical factor in limiting the sharpness of 35mm images (assuming high quality lenses are used), and which affects medium format images to a lesser degree, is also omitted. For reference, Fuji Provia 100F, which is regarded as one of the sharpest and finest grained slide films, has  f50 = 40 lp/mm,  f20 = 70 lp/mm and  f10 = 110 lp/mm (the latter two estimated by extrapolating the manufacturer's MTF plot). This is comparable to diffraction at f/16.

The column on the left indicates the actual f-stop setting. The row on the top indicates the depth of field marks on the lens. The cells contain f50 (upper table) and  f20 ( lower table) at the distances corresponding to the DOF marks on the top when the lens is set to the f-stop on the left.
Example: Suppose the Canon FD 50mm f/1.4 lens is focused at 10 feet (3 meters) and set to f/8. The f/4 DOF marks are opposite 9 and 11.5 feet. At these distances  f50 = 47 lp/mm. The f/8 DOF marks are opposite 8 and 14 feet;  f50 = 24.4 lp/mm— due almost entirely to the 0.03 mm circle of confusion; diffraction is insignificant for this case. The f/16 DOF marks are opposite 6.5 and 28 feet;  f50 = 12.1 lp/mm.

Now, suppose I wanted maximum sharpness at the 8 and 14 feet limits, corresponding to the f/8 DOF marks. I would set the aperture (actual f-stop) to f/16 (the pale yellow cell in the table). f50 at the limits would be 37.8 lp/mm; the in-focus f50 would be a maximum of 45.2 lp/mm (not bad). f/11 might give a better all-around result— sharper in the center but not quite as sharp ( f50 = 32.6 lp/mm) at the DOF limits— it's an aesthetic judgment call. Starting at f/16 diffraction takes a big bite out of sharpness where the image is in focus; there is no advantage in stopping down further.

  f50 (lp/mm) for 35mm format  (C = 0.03 mm)  at DOF mark
Depth of field mark
Actual
f-stop 
Diffraction-
limited f50
in focus
4 5.6 8 11 16 22 32 45 64
4 181 lp/mm 24.3 17.2 12.0 8.7 6.0 4.3 3.0 2.1 1.5
5.6 129 34.2 24.4 17.0 12.3 8.4 6.1 4.2 3.0 2.1
8 90.5 47.0 34.6 24.4 17.7 12.1 8.8 6.0 4.3 3.0
11 65.8 53.4 43.4 32.6 24.3 16.8 12.2 8.3 5.9 4.1
16 45.2 45.5 42.9 37.8 31.4 23.5 17.6 12.2 8.7 6.1
22 32.9 34.0 33.8 30.9 26.7 22.0 22.0 16.3 11.9 8.4
32 22.6 23.1 23.3 23.4 23.3 22.7 12.5 18.9 15.5 11.8
45 16.1 16.3 16.4 16.5 16.5 16.6 16.5 16.1 15.2 13.4
64 11.3 11.4 11.4 11.5 11.5 11.6 11.6 11.7 11.6 11.4
f50 (lp/mm spatial frequency for 50% MTF)
Includes misfocus and diffraction, but not lens aberrations.
.
  f20 (lp/mm) for 35mm format  (C = 0.03 mm)  at DOF mark
Depth of field mark
Actual
f-stop 
Diffraction-
limited f20
in focus
4 5.6 8 11 16 22 32 45 64
4 302 lp/mm 33.7 24.0 16.7 12.1 8.3 6.0 4.1 2.9 2.1
5.6 216 47.7 33.9 23.6 17.1 11.7 8.5 5.8 4.1 2.9
8 151 66.9 48.6 34.1 24.7 16.9 12.2 8.4 5.9 4.2
11 110 80.6 63.2 46.2 34.0 23.4 17.0 11.6 8.2 5.7
16 75.6 74.1 67.8 57.3 46.1 33.5 24.7 17.0 12.1 8.4
22 55.0 56.7 55.9 53.3 48.6 40.3 32.0 23.1 16.6 11.7
32 37.8 38.8 39.0 39.0 38.6 37.0 34.1 28.7 22.6 16.7
45 26.9 27.3 27.4 27.6 27.7 27.7 27.4 26.3 24.0 20.2
64 18.9 19.1 19.1 19.2 19.3 19.4 19.5 19.5 19.3 18.5
f20 (lp/mm spatial frequency for 20% MTF; ~=1/circle diameter)
Includes misfocus and diffraction, but not lens aberrations.

.DOF and focal length

It is well known that short focal length lenses have large apparent depths of field and long telephoto lenses have small apparent depths of field. There are some very practical reasons for this conception, but it isn't quite true. DOF is much more closely related to magnification and f-stop; DOF expressed in distance is nearly independent of focal length. It appears smaller with telephoto lenses because it is smaller when expressed as a fraction of the lens-to-subject distance, s.
 
Equations for Total Depth of Field
Combining the equations for Df and Dr from the first box of equations, we can obtain the total depth of field.
Total DOF = Df + DrsC(s-f )/( fa-C(s- f )) + sC(s- f )/( fa+C(s- f ))
                 =  2 fasC(s- f )/(( fa)2-C2(s-f )2)
Now, substitute magnification M into the equation using M = d/s = f / (s-f );  s-f  = f / M.
Total DOF = 2 fasC( f /M )/(( fa)2-C2 f 2 / M2) = 2asCM / (M2a2 - C2)
Eliminate s using s  = f + f / M = f (1 + 1/M) =  Na (1 + 1/M), where N = f-stop = f / a.
Total DOF = 2Na2C (M+1)/ (M2a2 - C2) = 2NC (M+1)/ (M2 - (CN / f )2)
No approximations yet, but we haven't entirely eliminated the focal length  f. Fortunately, the (CN / f )2 term is usually much smaller than M 2, except for very distant images (with very small magnification). As we point out below, c/f  is a constant, independent of format, equal to about 1/1600 for a "normal" lens. For example, for the 35mm format with a standard 50 mm lens at f/8, cN / f = 0.03*8/50 = 0.0048 ~= 1/200. So the (cN / f )2 term can be eliminated from the equation (the error will be less than 1%) for magnifications M larger than 1/20 (a 20x30 inch or smaller field for 35mm format), which covers most portraits and still lives.
Total DOF ~= 2NC (M+1)/ M2
This approximation holds for large magnifications: portraits, still lives, etc. (M > 1/20 in the above example).

Now let's look at Depth of Field for M ~= 1/20 at f/8 for several focal lengths, using Jonathan Sachs' Depth of Field Calculator set for 30 lp/mm resolution (the default).

DOF for f/8, M ~= 1/20, 35mm format
Focal length
f mm
 Distance
S mm
Near DOF
limit mm
Far DOF
limit mm
Total
DOF mm
Total
DOF/S %
20 mm 400 319 536 217 54.2
50 mm 1000 908 1113 205 20.5
100 mm 2000 1904 2107 203 10.1
200 mm 4000 3901 4104 203 5.1
1000 mm 20000 19899 20102 203 1.0
For a specific format, depth of field, expressed as distance, is independent of focal length. But depth of field, expressed as a percentage of the distance to the subject (Total DOF/s %), is inversely proportional to focal length. It can be very small for long telephoto lenses.
Using a long telephoto lens is an effective way of isolating a subject from busy, uncontrolled backgrounds without sacrificing actual depth of field.

DOF limits, diffraction, and format

We can draw an  interesting conclusion about depth of field for varous film formats (35mm up to 8x10 in) by rearranging the equation for for total depth of field.
 
Total DOF = Df + Dr =  2 fasC(s- f )/(( fa)2-C2(s-f )2) ,

is difficult to interpret, but we can arrive at an interesting result if we assume that the subject is relatively distant from the lens, i.e., s >> f . We can then simplify the equation, i.e., it becomes an approximation.

Total DOF(s>>f ) ~=  2 faCs2/(( fa)2-(sC)2) =  2 as2( f/C)/(( f/C)2a2 - s2)

The circle of confusion C at the DOF limit is based on the 0.01 inch = 0.25 mm feature in an 8x10 inch print. On the film, C (mm) = 0.25/(magnification for an 8x10 print). For a constant angle of view, lens focal length  f  is proportional to the format size (cropped for an 8x10 inch image) and inversely proportional to the magnification. f/C  is therefore a constant, independent of the format, about 1600 for a "normal" lens. The following table shows approximate values of key parameters for various formats.
Format
 f  (mm)
"normal" lens
Magnification
8x10 print
C (mm)
35mm (24x36 mm) 50 8x 0.032
6x6 cm 80 5x 0.055
6x7 cm 100 4x 0.064
4x5 in 200 2x 0.128
5x7 in 250 1.6x 0.16
8x10 in 400 1x 0.25

Since f/C  is a constant, independent of format, depth of field is constant for constant aperture opening a. And since f-stop Nf /a,

Depth of field is constant when the f-stop is proportional to the format size, i.e., DOF is the same for a 35mm image taken at f/11, a 6x7 image at f/22, a 4x5 image at f/45 or an 8x10 image at f/90.
This has important consequences when the lens sharpness becomes diffraction limited— beyond around f/11 for 35mm; slightly larger for large formats. (High quality lenses become diffraction-limited at larger apertures. The f-stop at which diffraction becomes dominant increases rather slowly with format size.) A lens is likely to be diffraction-limited when a large depth of field is required;  the larger the format, the more it must be stopped down; hence the more likely it is to be diffraction-limited. Once a lens is diffraction-limited its resolution is inversely proportional to its f-stop. This leads to a rather surprising observation.
When a lens is stopped down so to achieve a large depth of field, and is diffraction-limited, increasing the format size does not increase image sharpness, i.e., total resolution. For example, an 8x10 image taken at f/64 will be no sharper than a 4x5 image taken at f/32.
This statement applies primarily to large formats (4x5 and above). For small formats, particularly 35mm, image sharpness is limited by film resolution. Fuji Provia 100F, one of the finest grained slide films, has resolution roughly equivalent to diffraction at f/16 ( f50 = 40 lp/mm; f20 = 70 lp/mm), but since the total system MTF is the product of the MTF of the individual components, you can see some improvement in overall sharpness for lens apertures as wide as f/8. You must choose film with care for optimum sharpness in the 35mm format. Film resolution also limits the sharpness of medium format images, but this is only noticeable on images larger than 13x19 inches— the maximum for inexpensive consumer printers.

When large depth of field is needed, lenses usually have to be stopped down beyond their optimum aperture, especially for large formats, where very small apertures are required.  Diffraction in digital cameras is discussed here.

Sweet spot and format

The large format images you've seen that were thrillingly sharp— the images that tempted or inspired you to schlep a view camera— were taken at f-stops near the lens's optimum aperture, between large apertures where it is aberration-limited and small apertures (with large depth of field) where it is diffraction-limited. Optimum aperture is typically around 2 to 4 f-stops below maximum aperture; in the neighborhood of f/11 for medium format, f/16 for 4x5, and f/32 for 8x10. Many of these ultra-sharp images are distant landscapes that don't require large DOF.

If large DOF was required, it was obtained by using the camera's movements, particularly the tilt, which allows the plane of focus to be altered (via the Scheimpflug effect). Virtually all large format cameras have these movements; they are a major advantage. (Another, lesser, advantage is that sheet film tends have better flatness than roll films.) Few medium format cameras have these movements. (The Rollei SL66 was a rare and wonderful exception.) A few 35mm camera systems (most notably Canon) offer specialized lenses with movements. I love my old Canon FD 35mm f/2.8 TS lens, despite its manual aperture.

There is a sweet spot between large apertures, where lenses are aberration-limited, and small apertures, where they are diffraction-limited. Let's take a closer (but rough, qualitative) look. Good 35mm lenses tend to be sharpest around f/8, aberration-limited starting around f/5.6, and diffraction-limited starting around f/11. The total detail a lens can resolve at large apertures, where performance is aberration-limited, is relatively independent of format. It is a function of lens quality and design. A good lens can resolve about the same detail at f/5.6 for 35mm as for 4x5, where the image is much larger, but 4x5 images will have more detail because 35mm images are limited by film resolution.

The total detail a lens can resolve at small apertures, where performance is diffraction-limited, is proportional to to the format size and inversely proportional to the f-stop. A 35mm lens at f/11 resolves about the same total detail as a medium format (6x7) lens at f/22, a 4x5 lens at f/45, or an 8x10 lens at f/90. Resolutions at these apertures are roughly comparable to resolution of a high quality lens at f/5.6. (Disclaimer: this estimate is very rough! Variations between lenses make a huge difference.)

The sweet spot— the range of apertures with excellent sharpness, tends to be between f/5.6 and the aperture corresponding f/11 for the 35mm format (f/22 for medium format, f/45 for 4x5, and f/90 for 8x10). It comprises about 3 f-stops for 35mm, 5 f-stops for medium format, 7 f-stops for 4x5, and 9 f-stops for 8x10.

The larger the format, the larger the sweet spot.
Lenses have their optimum aperture— their highest total resolution— near the center of the sweet spot, typically around f/8 for 35mm, f/11 for medium format, f/16 for 4x5, and f/22 for 8x10. In practice, f/16 may be more practical for 4x5 and f/32 is more practical for 8x10 because depth of field is severely limited for large formats. Testing and experience will teach you which apertures are sharpest for your individual lens, but these numbers are good estimates. Optimum aperture is not sharply defined: for example, a good 4x5 lens with an optimum aperture around f/16 should produce excellent image quality between f/11 and f/32. Since large format lenses tend to be diffraction-limited at optimum aperture,
Total resolution at optimum aperture scales roughly with the square root of the format size for large formats.
This rough but useful approximation applies to lenses only. When film losses dominate image quality, as it does for 35mm and medium format (recall, Fuji Provia 100F has MTF comparable to diffraction at f/16), total resolution scales linearly with format size. There is a greater advantage to larger formats.

I need to stress that the advantage of large formats is greatest when lenses are not stopped down to achieve extreme depth of field.

For very small formats— for compact digital cameras with 11mm diagonal or smaller sensors (1/4 the size of 35mm), the sweet spot is extremely small. Lenses are aberration and diffraction-limited at the same aperture, around f/4 to f/5.6. They are severely diffraction-limited at f/8, where DOF is equivalent to f/32 or more in 35mm. (They rarely go beyond f/8.) But even though lens resolution is less than for 35mm film cameras, tiny digital cameras still produce very sharp images at f/4 and f/5.6 because their tiny pixels— 4 micron spacing or less with no anti-aliasing filters— have far better lp/mm resolution than 35mm film. Image resolution is almost entirely dominated by the lens.

Detail can be quite stunning in well-made large format images, particularly in very large prints— beyond the 13x19 inch maximum size of most consumer digital printers. Large formats have little advantage for 8½x11 inch prints, although traditional 8x10 contact prints have a unique tonal beauty, particularly when made on special contact papers such as Azo. I've recently seen some incredibly sharp huge prints (over 40x50 inches) made from 8x10— sharper than you could achieve with 4x5. But 4x5 is the largest practical format for carrying on hikes, and it has its own "sweet spot" for inexpensive flatbed film scanners such as the Epson 2450 and 3200. Even though these scanners have somewhat poorer resolution than dedicated film scanners, their resolution is sufficient to make sharp 32x40 inch prints from 4x5 film (the same magnification as 8x10 prints from 35mm). Of course I'd need a wide body printer, like the 24 inch wide Epson 7600, which could make extremely sharp 24x30 inch prints from 2450/3200 scans. Tempting!

Links

David Jacobson's Lens Tutorial—  An excellent introduction to optics, with a moderate number of equations.
Bob Atkins' pages on Technical Optics—  The Depth of Field Calculator contains a nice applet.
Nicholas V. Sushkin's Online Depth of Field Plotter—  Contains some of the equations I use and another DOF calculation applet.
Harold M. Merklinger's Depth of Field Revisited—  An excellent explanation of why using hyperfocal distance results in underwhelming sharpness in distant parts of landscape. Also visit his index and list of PDF articles.
Jonathan Sachs' Depth of Field Calculator—  Yet another DOF calculator (this one a stand-alone program) from the creator of my favorite image editor.
Andrzej Wrotniak's Photo Tidbits: Depth of Field and your Digital Camera—  A nice introduction, which includes DOF tables for digital cameras.
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 MTF equations for circle of confusion and diffraction.
The equations below are from David Jacobson's Lens Tutorial. MTF is the absolute value of the optical transfer function OTF (MTF = |OTF|), which includes phase and hence can go negative. The OTF for the circle of confusion C is based on a mathematical function called a Bessel function of the first kind (J1), which you won't find in simple programming languages, but which can be adequately approximated using
2 J1(x)/x ~= sinc(0.84x) where sinc(x) = sin(x)/x;     below the first null of sin(x);  x < π.

Let  s = λ N fsp ;   a = π fsp

where λ = wavelength of light (typically 0.0005 or 0.000555 mm for green or yellow-green, near the middle of the visible spectrum; N = f-stop;  fsp = spatial frequency;  π = 3.14159;  C = Circle of confusion. s and a are dimensionless.
For pure diffraction (no focus error;  a= C = 0),
OTF(s) = 2/π (arccos(s) -sqrt(1-s2 ))      for s < 1
             = 0     for s >= 1
For focus error only with circle of confusion C (no diffraction; s = 0),
OTF(a) = 2 J1(a)/a ;     J1 is a first order Bessel function.
            ~= sin(0.84a)/(0.84a)     up to the first null (0.84aπ)
For combined diffraction and focus error,

Because of phase effects implicit in OTF, the combined diffraction and focus error is not the product of the OTF's for diffraction and misfocus. (You can multiply MTF's for separate components, e.g., lens and flim, because phase is lost when you go from one to another.) The combined equation is relatively easy to solve numerically. It appears to work well in the limits of a >> c (diffraction dominant) and a << c (misfocus dominant). Here is a plot of the individual and combined terms for N = f-stop = 22, C = 0.03mm and lambda (wavelength of light) = 0.000555mm (the same data as David Jacobson's plot). The pale green dotted line is the sin(0.84x)/(0.84x) approximation to the Bessel function for C, which works up to the first null.
Using the OTF/MTF equations we can find the spatial frequencies where MTF from misfocus and diffraction is 50%, 20% and 10% (  f50 f20 and f10 ). These frequencies are shown in the graph below.
The solid curves are  f10 (upper),  f20 (middle) and  f50 (lower), derived from the OTF equation for combined diffraction and focus error. The peaks in  f10 and  f20 are due to idiosyncracies of the combined OTF equation. The dotted lines are analytic approximations— much easier to work with, and I suspect more trustworthy. The approximations for  f50, f20 and  f10 as functions of f-stop N and circle of confusion C are, 
f10 = c10 / sqrt(d102 - .5 d10 + 1) ;   c10 = 1.10/C ;   d10 = 1.27 λN c10
f20 = c20 / sqrt(d202 - .7 d20 + 1) ;   c20 = 0.99/C ;   d20 = 1.49 λN c20
f50 = c50 / sqrt(d502 - .7 d50 + 1) ;   c50 = 0.71/C ;   d50 = 2.49 λN c50
If we neglect diffraction (let λN approach zero), we can use the simple approximations,
 f50 = 0.72/C ;    f20 = 1/C ;    f10 = 1.11/C


Images and text copyright © 2000-2014 by Norman Koren.
Norman Koren lives in Boulder, Colorado, founded Imatest LLC in 2004, previously worked on magnetic recording technology. He has been involved with photography since 1964.