image sharpness part 6:
updated Nov. 5, 2004
of field and diffraction
this pagr we discuss depth of field (DOF) and deal
with such questions as, How sharp is the image at the DOF limits? Can
trust DOF scales? (I wouldn't.) What aperture should you choose for
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
member of a maligned and misunderstood but highly elite fellowship.
The text in green is for you. If you're normal or mathematically
you may skip these sections. You'll never know what you missed.
of field: introduction
far we've only considered images in exact focus. That's all you need
if you only photograph distant landscapes or two-dimensional objects
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
known: The more you stop down a lens (the larger the f-stop number),
larger the DOF. Wide angle lenses appear to have much larger DOF than
Telephotos are often used to intentionally limit DOF, for example in
where you want the subject to be in focus, but you want a distracting
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
field (DOF) scales. Your camera's instruction manual states that if you
stop down your lens, for example to f/8, everything at distances
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
is its MTF?) at the DOF limit?" To answer these questions we begin with
the diagram below, representing a lens with aperture a
s on the film plane at d.
object at a distance s
in front of the lens is focused
at a distance d behind it,
according to the lens equation:
= 1/f - 1/s,
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.
at sf ,
in front of s,
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
Likewise, an object at sr,
dr, in front
Its circle of confusion at d has
The depth of field (DOF) is the range of distances
and sr, (Dr
where the circles of confusion, Cf
are small enough so the image appears to be "in focus." The standard
for choosing C (the largest
allowable value of Cf
is that on an 8x10
inch print viewed at a distance of 10 inches, the smallest
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
close to the standard 0.03 mm used by 35mm lens manufacturers to
their DOF scales. If you've ever had a close look at a fine contact
from 4x5 or 8x10
film, you'll doubt that 0.01 inch feature size is a good criterion.
on human visual
that the smallest feature an eye with 20:20 vision can distinguish is
one minute of an arc: 0.003 inches at a distance of 10 inches. But
prevails: 0.01 inch is universally used to specify DOF.
Lens and Depth of Field equations.
+ 1/d = 1/f
= focal length— the lens's most important
lens-to-film plane distance. If object
is located an infinite distance from the lens (s
>> f), the
image is focused at a distance f from the lens,
i.e., d =
|N = f-stop = f /a
= aperture diameter.
= a|(df -d)/df
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.
confusion at the film plane (d) for
object located at Sr(behind s), which focuses on
Derived from simple geometry using 1/sr
|M = d/s
f / (s-f ) (5)
|Focus = F = s+d(6)
||The focus scale
of most lenses is the distance
from the object to the film plane. >= 4*f.
= s - 1/(1/f-(1-C/a)/d)
(front depth of field limit relative to s) derived
from (1) and
= s - sf ;
N = f-stop = f
= s - Df ;
Lens to front DOF limit.
There equations are in
agreement with Sushkin.
(rear depth of field limit relative to s) derived
from (1) and (4)
using Dr = sr
= infinity when demoninator fa-c(s-f
<= 0 ( f 2-CN(s-f
= s + Dr ;
Lens to rear DOF limit.
at DOF limits
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
box at the bottom (for
math geeks only). For a circle of confusion C
(no diffraction), the spatial frequencies for MTF =
50%, 20% and 10% ( f50 ,
= 0.72/C ; f20
= 1/C ; f10
is familiar from
film, lenses and scanners. f20
because it is the inverse of C.
For C = 0.03mm,
= 24 lp/mm. An excellent lens
for the 35mm
format has f50 ~=
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
for the depth of field scale on your lens with a simple procedure,
an equation derived from the box above.
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.
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
- Note the distance s+d
where the lens is focused.
- The circle of confusion is
C = f2/(Nmax(s-f))
- 2 f)).
At this setting, s
>> f and s
>> d, so the
equation can be simplified to C
Focus) with little loss
old (medium format) chrome-barrel Hasselblad (Zeiss)
= 0.055 mm, corresponding to the same 0.01 inches on an 8x10
print, enlarged 5x
for this format
cm, but actually 5.6x5.6
Zeiss has a good
deal to say
about DOF in Camera Lens News No. 1. Inadequate depth of field turns
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
for the 35mm format), when producing their depth of
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,
the overall sharpness at the DOF limits will be inevitably worse than
simple circle of confusion would indicate. For this reason alone, the
for setting the circle of confusion is a bit loose.
- "The international depth of field
standard, the basis for all
manufacturers to calculate their depth of field scales and tables,
back to a time when image quality was severely limited by the films
- "Those who use depth of field scales, tables, and formulas
(e. g. for hyperfocal
settings), restrict themselves – most probably without
knowing why – to
the image quality potential of an average pre-World-War-II emulsion."
myth of hyperfocal distance
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
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.
Neither does Harold M. Merklinger
in his page,
of Field Revisited. Nor does Zeiss.
If the part of the scene at infinity is at all important in
it's often visually dominant— you'll be disappointed with the
which is only 40% that of a high quality lens in focus; about one
third what the eye can distinguish. Merklinger recommends
at infinity— you lose very little forward depth of field. I
setting infinity focus opposite the far DOF mark corresponding to 2
larger than the actual f-stop setting (half the number). For example,
you are using f/8, it's safe to put the far f/4 DOF mark opposite
It's a judgment call. When you make it, think about what parts of the
will be dominant. There is no rule to blindly follow.
doesn't lie perfectly flat— especially roll film (35mm and
format). Sheet film is better. Film flatness is probably the least
of the factors that degrade image sharpness. According to Robert
Monaghan, "film often buckles in 60% of 35mm SLRs tested, and
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
number depends on the f-stop. The equation for the circle of confusion
due to film bulge is (for focus near infinity: s
>> f ),
For a 0.08 mm bulge at f/5.6, Cbulge
For a 0.2mm bulge at f/5.6, Cbulge
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
the film. And it's different for 35mm and medium
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
format film: it's small at 5 minutes, significant at 15 minutes and
after 2 hours. One solid piece of useful information from Zeiss: the
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
The Zeiss rule of thumb is, " For best sharpness in medium format,
220 type roll film and run it through the camera rather quickly."
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
For much more detail on film flatness, I recommend Robert
Monaghan's exhaustive discussion (with reader comments).
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
of daylight situations. This bending, called diffraction,
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
diffraction limit, adapted from R.
N. Clark's scanner detail page, is,
(line pairs per mm) = 1/(1.22 N ω)
The MTF at the Rayleigh limit is about 9%. Most lenses for 35mm and
cameras are aberration-limited— relatively unaffected by
N = f/8 and below. The spatial
frequencies for 10% and
50% MTF for diffraction-limited lenses are,
= 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
= 2.44 N ω
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
results in the best sharpness over a range of distances. But how to
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.
To learn more about diffraction, see Sean McHugh's superb Tutorial: Diffraction & Photography.
your depth of field scale to find the optimum aperture
The following procedure lets you determine
where you can obtain optimum sharpness over a range of
It is derived in the green box, below.
- Determine the closest and farthest distances you
want to be in sharp
- Focus the lens so identical DOF marks on either side of the
are aligned with these distances. For example, if you are using the
lens in the above illustration and want sharp focus between 14 and 25
you would place identical DOF marks, in this case, f/4, at these
Focus would be around 17 feet.
- Make the adjustments indicated in the table below, based on
of the DOF marks at the desired focus limits. These adjustments are
for the 35mm format (C = 0.03 mm)
adjustment for optimum sharpness
focusing over a range of distances
for DOF marks
desired focus limits
example, if marks indicate
down to f/8.
by 1 f-stop (!)
camera territory, where
a big bite out of
of aperture for optimum sharpness over a range of distances
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
because they are dependent on lens design and manufacturing quality.
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
for exact focus at f/4 through f/8 (upper left in both tables) are
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
omitted. For reference, Fuji
which is regarded as one of the sharpest and finest grained slide
has f50 = 40
= 70 lp/mm and f10
= 110 lp/mm (the latter two
estimated by extrapolating the manufacturer's MTF plot). This is
to diffraction at f/16.
The column on the left
indicates the actual f-stop setting. The row on the top indicates the
of field marks on the lens. The cells contain
table) and f20 (
lower table) at the distances
corresponding to the DOF marks on the top when the lens is set to the
on the left.
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
diffraction is insignificant for this case. The f/16 DOF marks are
6.5 and 28 feet; f50
= 12.1 lp/mm.
Now, suppose I wanted maximum
at the 8 and 14 feet limits, corresponding to the f/8 DOF marks. I
set the aperture (actual f-stop) to f/16 (the pale yellow cell in the
at the limits would be 37.8 lp/mm; the in-focus
be a maximum of 45.2 lp/mm (not bad). f/11 might give a better
result— sharper in the center but not quite as sharp ( f50
= 32.6 lp/mm) at the DOF limits— it's an aesthetic judgment
at f/16 diffraction takes a big bite out of sharpness where the image
in focus; there is no advantage in stopping down further.
(lp/mm) for 35mm format (C = 0.03
mm) at DOF mark
of field mark
(lp/mm spatial frequency for 50% MTF)
misfocus and diffraction,
(lp/mm) for 35mm format (C = 0.03
mm) at DOF mark
of field mark
(lp/mm spatial frequency for 20% MTF; ~=1/circle diameter)
misfocus and diffraction,
and focal length
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
DOF expressed in distance is nearly independent of focal length. It
smaller with telephoto lenses because it is smaller when expressed as a
fraction of the lens-to-subject distance, s.
for Total Depth of Field
the equations for Df
and Dr from the first box of
equations, we can obtain
the total depth of field.
DOF = Df
+ Dr = sC(s-f
+ sC(s- f )/(
Now, substitute magnification M
into the equation
using M = d/s = f /
(s-f ); s-f
= f / M.
= 2 fasC(s- f
DOF = 2 fasC(
/M )/(( fa)2-C2
/ M2) = 2asCM /
using s = f
+ f / M = f (1
+ 1/M) = Na (1
+ 1/M), where N = f-stop = f
DOF = 2Na2C
- C2) = 2NC
(M2 - (CN / f )2)
No approximations yet, but we
haven't entirely eliminated
the focal length f. Fortunately, the (CN
term is usually much smaller than
M 2, except for very
distant images (with very small magnification). As we point out
is a constant, independent of format, equal to about 1/1600 for a
lens. For example, for the 35mm format with a standard 50 mm lens at
0.03*8/50 = 0.0048 ~= 1/200. So the (cN /
term can be eliminated from the equation (the error will be less than
for magnifications M larger than 1/20 (a 20x30
inch or smaller field for 35mm format), which covers most portraits and
DOF ~= 2NC (M+1)/
This approximation holds for large
portraits, still lives, etc. (M > 1/20 in the
Now let's look at Depth of Field for M
~= 1/20 at f/8 for several focal lengths, using Jonathan
of Field Calculator set for 30 lp/mm resolution (the
f/8, M ~= 1/20, 35mm format
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
(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.
limits, diffraction, and format
can draw an interesting conclusion about depth of field for
film formats (35mm up to 8x10
rearranging the equation for for total depth of field.
|Total DOF = Df
+ Dr = 2 fasC(s-
f )/(( fa)2-C2(s-f
is difficult to interpret,
but we can arrive at
an interesting result if we assume that the subject is relatively
from the lens, i.e., s >> f
. We can then simplify the equation,
i.e., it becomes an approximation.
2 faCs2/(( fa)2-(sC)2)
= 2 as2(
circle of confusion C
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
to the format size (cropped for an 8x10
inch image) and inversely proportional to the magnification.
is therefore a constant, independent of the format, about 1600 for a
lens. The following table shows approximate values of key parameters
is a constant, independent
of format, depth of field is constant for constant aperture opening a.
And since f-stop N = f
of field is constant when the f-stop is proportional to the format
i.e., DOF is the same for a 35mm image taken at f/11, a 6x7 image at
a 4x5 image at f/45 or an 8x10 image at f/90.
This has important consequences when the lens sharpness becomes
limited— beyond around f/11 for 35mm; slightly larger for
(High quality lenses become diffraction-limited at larger
The f-stop at which diffraction becomes dominant increases rather
with format size.)
A lens is likely
to be diffraction-limited
when a large depth of field is required; the larger the
more it must be stopped down; hence the more likely it is to be
Once a lens is diffraction-limited its resolution is inversely
proportional to its f-stop. This leads to a rather surprising
a lens is stopped down so to achieve a large depth of field, and is
increasing the format size does not increase image sharpness, i.e.,
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
by film resolution. Fuji
one of the finest grained slide films, has resolution roughly
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
the individual components, you can see some improvement in overall
for lens apertures as wide as f/8. You must choose
with care for optimum sharpness in the 35mm format. Film resolution
limits the sharpness of medium format images, but this is only
on images larger than 13x19
the maximum for inexpensive consumer printers.
When large depth of field is needed, lenses usually have to be
down beyond their optimum aperture, especially for large formats, where
very small apertures are required. Diffraction in digital
is discussed here.
spot and format
large format images you've seen that were thrillingly sharp—
that tempted or inspired you to schlep a view
camera— were taken
at f-stops near the lens's optimum
between large apertures where it is aberration-limited and small
(with large depth of field) where it is diffraction-limited. Optimum
is typically around 2 to 4 f-stops below maximum aperture; in the
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
particularly the tilt, which allows the plane of focus to be altered
effect). Virtually all large format cameras have these movements; they
are a major advantage. (Another, lesser, advantage is that sheet film
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
(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
large apertures, where lenses are aberration-limited, and small
where they are diffraction-limited. Let's take a closer (but rough,
look. Good 35mm lenses tend to be sharpest around f/8,
starting around f/5.6, and diffraction-limited starting around f/11.
total detail a lens can resolve at large apertures, where performance
aberration-limited, is relatively independent of format. It is a
of lens quality and design. A good lens can resolve about the same
at f/5.6 for 35mm as for 4x5,
the image is much larger, but 4x5
will have more detail because 35mm images are limited by film
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
(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.
larger the format, the larger the sweet spot.
Lenses have their optimum
their highest total resolution— near the center of the sweet
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
is more practical for 8x10
depth of field is severely limited for large formats. Testing and
will teach you which apertures are sharpest for your individual lens,
these numbers are good estimates. Optimum aperture is not sharply
for example, a good 4x5
lens with an
optimum aperture around f/16 should produce excellent image quality
f/11 and f/32. Since large format
lenses tend to
be diffraction-limited at optimum aperture,
resolution at optimum aperture scales roughly with the square root of
format size for large formats.
This rough but useful approximation applies to lenses only. When film
dominate image quality, as it does for 35mm and medium format (recall,
Provia 100F has MTF comparable to diffraction at f/16), total
scales linearly with format size. There is a greater advantage to
I need to stress that the advantage of large formats is
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
f/8.) But even though lens resolution is less than for 35mm film
tiny digital cameras still produce very sharp images at f/4 and f/5.6
their tiny pixels— 4 micron spacing or less with no
have far better lp/mm resolution than 35mm film. Image resolution is
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
prints, although traditional 8x10
prints have a unique tonal beauty, particularly when made on special
papers such as Azo.
I've recently seen some incredibly sharp huge prints (over 40x50
inches) made from 8x10—
you could achieve with 4x5.
is the largest practical format for carrying on hikes, and it has its
"sweet spot" for inexpensive flatbed film scanners such as the Epson
2450 and 3200. Even though these scanners have somewhat
than dedicated film scanners, their resolution is sufficient to make
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!
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
Also visit his index
and list of PDF
Jonathan Sachs' Depth of Field
Calculator— Yet another DOF calculator
(this one a stand-alone
program) from the creator of my favorite
Andrzej Wrotniak's Photo Tidbits: Depth
of Field and your Digital Camera— A nice
includes DOF tables for digital cameras.
equations for circle of confusion and diffraction.
equations below are from David
Jacobson's Lens Tutorial. MTF is the
absolute value of the optical
transfer function OTF (MTF = |OTF|),
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
languages, but which can be adequately approximated using|
2 J1(x)/x ~= sinc(0.84x) where sinc(x)
below the first null of sin(x); x < π.
= λ N fsp ; a
= π C fsp
where λ = wavelength of light
(typically 0.0005 or 0.000555 mm for green or yellow-green, near the
of the visible spectrum; N =
frequency; π = 3.14159; C
Circle of confusion. s
and a are dimensionless.
For pure diffraction (no focus
= 2/π (arccos(s) -
for s < 1
For focus error only with circle of
confusion C (no
diffraction; s = 0),
= 0 for s
= 2 J1(a)/a
J1 is a first order Bessel function.
For combined diffraction and focus
up to the first
null (0.84a < π)
Because of phase effects implicit in OTF,
the combined diffraction and focus error is not
of the OTF's for diffraction and misfocus. (You can
for separate components, e.g., lens and flim, because phase is lost
you go from one to another.) The combined equation is relatively easy
solve numerically. It appears to work well in the limits of a
(diffraction dominant) and a << c
Here is a plot of the individual and combined terms for N
= 22, C = 0.03mm and lambda (wavelength of light) =
same data as
Jacobson's plot). The pale green dotted line is the sin(0.84x)/(0.84x)
to the Bessel function for C, which works up to the
Using the OTF/MTF
equations we can
find the spatial frequencies where MTF from
misfocus and diffraction
is 50%, 20% and 10% ( f50 ,
These frequencies are shown in the graph below.
The solid curves are f10
(middle) and f50
(lower), derived from the OTF equation for combined
and focus error. The peaks in f10
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,
and f10 as
functions of f-stop
N and circle
of confusion C are,
= c10 / sqrt(d102
- .5 d10 + 1)
; c10 =
1.10/C ; d10
= 1.27 λN c10
If we neglect diffraction (let λN
approach zero), we can use the simple approximations,
= c20 / sqrt(d202
- .7 d20 + 1)
; c20 =
0.99/C ; d20
= 1.49 λN c20
= c50 / sqrt(d502
- .7 d50 + 1)
; c50 =
0.71/C ; d50
= 2.49 λN c50
= 1/C ;
f10 = 1.11/C
and text copyright © 2000-2014 by Norman
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.