User manual | Depth of Field and Bokeh

```Depth of Field and Bokeh
by
H. H. Nasse
Carl Zeiss
Camera Lens Division
March 2010
Preface
"Nine
rounded
diaphragm
guarantee
images with exceptional bokeh"
Wherever there are reports about a new camera lens, this
sentence is often found. What characteristic of the image is
actually meant by it? And what does the diaphragm have to
do with it?
We would like to address these questions today. But
because "bokeh" is closely related to "depth of field," I
would like to first begin with those topics on the following
pages. It is true that a great deal has already been written
about them elsewhere, and many may think that the topics
have already been exhausted. Nevertheless I am sure that
you will not be bored. I will use a rather unusual method to
show how to use a little geometry to very clearly understand
the most important issues of ‘depth of field’.
Don't worry, though, we will not be dealing with formulas at
all apart from a few exceptions. Instead, we will try to
understand the connections and learn a few practical rules
of thumb. You will find useful figures worth knowing in a few
graphics and tables.
Then it only takes another small step to understand what is
behind the rather secretive sounding term "bokeh". Both
parts of today's article actually deal with the same
phenomenon but just look at it from different viewpoints.
While the geometric theory of depth of field works with an
idealized simplification of the lens, the real characteristics of
lenses including their aberrations must be taken into
account in order to properly understand bokeh. The
diaphragm is not enough, and that is all that needs to be
said here.
There are also plenty of pictures for illustrating this topic for
those who do not want to get deeply involved in the theory
of their camera, so we really wish everyone a lot of fun with
Carl Zeiss
Camera Lens Division
March 2010
"Schärfentiefe" or "Tiefenschärfe" for depth of field?
When searching the net, there is a
seemingly endless amount of entries
about our topic and much of what is there
for reading is of course incorrect or
incomplete. It is therefore not surprising
that photography forums like to spend so
much time discussing it.
There was a particular increase in the
interest to understand depth of field when
the first digital SLR cameras were put on
the market in the smaller APS-C format,
which were compatible with "old" lenses
for the 24x36 mm format. But the question
was whether the engraved scale on the
lens still applies or not.
In the German forums we even find some
heavy debate about the proper term for the
depth of field - should it be "Schärfentiefe"
or "Tiefenschärfe", saying “depth of
sharpness” or “sharpness of the depth”?
We shouldn't split hairs over it, particularly
when we see that this depth itself is not a
very precise feature anyway. Both terms
have been in common use for a while now.
And both refer to the same characteristic of
photographic imaging - namely that a clear
two-dimensional photographic image can be
made of objects in a three-dimensional
space under certain conditions, even
though the camera can only be focused on
one specific distance.
Equipment details of a camera from 1934: a “Tiefenschärfe” table instead of a “Schärfentiefe”
one! Language is not always so strict, so we have to allow both terms to be used. This
debate about terms is of course useless for those who read the translated English version!
The fact that we can capture a
considerable portion of the threedimensional space in front of and behind
the optimally focused distance on the film
or chip is because we can obviously
tolerate or not even notice a certain
amount of blurriness.
It is really a blessing that this is the case,
because there is hardly any camera so
precise that it can be 100% sure to bring
the optimum performance of the lens onto
the film or sensor. That is because limited
film flatness in analogue times, focusing
errors, and other mechanical tolerances
make it more difficult.
Carl Zeiss
But as long as the errors are not too
great, we usually do not notice them.
Depth of field is based on the
acceptable
blurriness
and
is
therefore essentially based on
arbitrary specifications. But it is not
the case that the sharpness of the
image is actually constant at a certain
depth of space and then stops being so
in front of and behind it. The sharpness
is always continuously changing with
the distance of the object.
Camera Lens Division
3
When is the depth of field not dependent upon the focal length?
What is behind the scale on the lens?
When someone says that the depth of
filed is not at all dependent on the focal
length, of course we would like to
experience has shown us that wide-angle
lenses make images with a large depth
and telephoto lenses have a rather
selective sharpness. Despite this, the
person making the original claim may be
right, but must clarify which type of depth
is meant. Those speaking English have it
better because they use two clearly
different terms: depth of field and depth
of focus.
The former stands for what we generally
consider to be "Schärfentiefe" in German,
namely the depth in the object space. But
there is also a depth in the image space
inside the camera. This image-side depth,
called depth of focus in English, is not
actually dependent on the focal length but
rather on the f-number, which is easy to
understand:
Every picture element is generated by a
large number of beams of light that shine
through the aperture and combine in the
picture element. In doing so, they form a
light cone whose area is the image of the
aperture seen from the sensor. This
picture of the aperture is called the exit
pupil. You can easily see it when you look
into a lens from behind while you point it to
a light surface:
A large aperture (meaning a low f-number)
means a truncated light cone, and a small
aperture (meaning a higher f-number)
means a pointed light cone.
f − number =
DistEP
DiameterEP
If the sensor surface (yellow line)
intersects with the light cone at a certain
distance from the point of the cone, the
resulting intersection is the circle of
confusion marked red in above drawing.
The total image-side depth of focus (the
blue section of the image space in the
diagram above) is twice the product of the
diameter of the circle of confusion (z) and
the f-number (k):
depth ⋅ of ⋅ focus ≈ 2 ⋅ z ⋅ k
This simple equation can be seen in the
engraved depth of field scales:
The rotary angle on the focusing ring is
proportional to the image-side focus
and
the
depth-of-field
markings on the lens barrel are therefore
proportional to the f-number.
The f-number is the ratio of the distance
from the image plane to the exit pupil and
the diameter of the exit pupil. The angular
aperture of the light cone therefore only
depends on the f-number.
Carl Zeiss
(Strictly speaking, the image-side depth of
focus behind the image plane is just
slightly larger, but this can be ignored.)
Camera Lens Division
4
Examples of depth-of-field scales on lenses: engraved on the left, and a complex solution on
the right where the two red indicators are moved by a gear system when the aperture is set. In
both cases, the distances from the index in the middle to the depth-of-field markings are
proportional to the f-number. The intervals between the individual scale markings are of course
also dependent upon the specification of the acceptable circle of confusion and the thread
pitch of the focusing ring. That is why such scales are no longer useful on many modern AF
lenses if they have extremely steep focusing. The depth-of-field scales are symmetrical on the
left and right.
You may sometimes come across those who
hold the viewpoint that a longer focal length
has a larger image-side depth of focus. That is
not true, however, because the image-side
depth of focus is only dependent on the fnumber. This misconception comes from
confusing the image-side depth of focus with
the depth of the three-dimensional image.
Short focal lengths only have a very short
focus movement because they display
everything from the near foreground to the
distant background in a very short image space
- their image is flat. Long focal lengths require
a significantly larger focus movement because
the image of the same object space is much
deeper.
If cameras are poorly calibrated, the sensor
may be completely next to the flat image for
very short focal lengths and then the entire
motif will appear to be slightly blurry. With a
long focal length, on the other hand, despite
poor calibration it will still be perfectly clear
somewhere, even if it is not where it is
supposed to be. This experience also leads to
the misconception that short focal lengths have
a short image-side depth of focus.
Carl Zeiss
It is true, however, that the depth of field
in the object space is also (almost)
independent of the focal length if we
compare the respective imaging of the
object at the same imaging scale. For
photographs with different focal lengths
and the same image format, of course,
this means that the photographs are taken
from correspondingly different distances.
The fact that the depth of field is only
dependent on the imaging scale
regardless of the focal length no longer
applies with very large distances. Even at
closer taking distance, two photographs of
an object will not be identical if they are
taken with two different focal lengths, even
if the depth of focus is practically identical.
Besides the perspectives, the maximum
blurriness of the distant background
differs. It is lower for shorter focal lengths
than for longer ones.
In the following pages we will move on
from the image space inside of the camera
where the circles of confusion actually
arise and take a look at the space in front
of the lens in order to understand why that
is the case.
Camera Lens Division
5
Depth of field and the entrance pupil
On the last two pages we have taken a look at
the light cone on the image side and learned
that circles of confusion arise when these light
cones are truncated by the sensor surface. The
beams of light travelling from an object point
into the lens do not have an intersection on the
sensor surface in that case, but rather
somewhere in the space in front of it or behind
it. In either case, their energy is distributed
across an expanded spot on the sensor
surface that we may no longer perceive as a
sharp picture element.
The acceptable deviations of the best focus
point from the sensor surface in the camera
may be interesting for the camera
manufacturer, but when we are taking
photographs we are more concerned with the
space in front of the lens. All distance scales
on lenses refer to the object side. That is why
we have to convert the image-side depth of
focus into the object-side depth of field.
And at that point we usually face the trouble
with the formulas, which we try to avoid today.
The light cones that cause the circles of
confusion do not originate in the lens but rather
come from the corresponding object points.
This means that there are also light cones on
the object side in front of the lens. Their base
area is the entrance pupil. That is the image
of the aperture that we see when we look at a
bright surface through the front of a lens from
a certain distance:
Carl Zeiss
The entrance pupil can also be located far
in the back of the lens, so we should not be
fooled by its name. In the case of the long
Tele-Tessar lenses for the Hasselblad, the
entrance pupil is in the film magazine.
A virtual plane in front of the lens within the
focus distance is intersected by the light
cones travelling nearer from points further
away; it is intersected by the rear
extensions of light cones from closer object
points.
The intersections with this object-side plane
are the images of the circles of confusion in
the sensor plane - we call them "objectside circles of confusion" for simplicity.
Even if they are not physically present, we
can still say that because every beam path
can also be inverted. Making use of
something that is not even physically
present is the trick to simplifying the
concept.
Camera Lens Division
6
In the diagram on the previous page, the blue
image space is on the right, behind the lens. It
is the image of the object space marked yellow
on the left in front of the lens. The furthest
points on the left are also displayed in the
image space on the left, closer to the lens. The
blue line in the object space is the image of the
sensor surface marked yellow on the right in
the image space – it is the focal plane. The
circles of confusion that appear on the sensor
surface are marked red. They have a
corresponding mark in the object-side focal
plane.
If an image is made with an imaging scale of
1:100 in 35 mm format 24x36 mm allowing for
the usual 0.03 mm circles of confusion, then
the images of the circles of confusion in the
focal plane in the object space can be as large
as 3 mm maximum. The field of the focal plane
displayed on the sensor is 2.4 x 3.6 m. The
ratio of the diameter of the circle of confusion
and the field size is identical on both sides.
We will consider later how small this ratio
between the diameter of the circle of confusion
and the image size should be. At any rate, it is
the parameter of the acceptable blurriness.
And in the object space this ratio depends on
three things:
1. How big is the object field?
2. Where is the point of the light cone?
3. How big is the base area of the light
cone?
Conditions 2 and 3 determine how narrow an
object-side light cone is. And condition 1 then
determines the relative size of the intersection
of the cone with the focal plane.
The base area of the light cone is the entrance
pupil, and its diameter is the quotient of the
focal length and the f-number. Lenses with a
long focal length and wide-aperture lenses
(small f-numbers) have large entrance pupils,
and lenses with a short focal length and smallaperture lenses have small entrance pupils.
DiameterEP =
Carl Zeiss
FocalLengt h
f − number
With a little geometry, we can now easily
see how the depth of field depends on the
taking distance, the focal length and the
aperture:
1. Distance
If we double the focusing distance, the size
of the object field in the focal plane also
doubles - not its area, but rather the width,
height and diagonal lengths. At the same
time a light cone from a point behind the
focal plane will be twice as narrow, because
the base area remains the same and we
infer the length of the cone. As a result, the
ratio of the field diagonal and circle of
confusion becomes four times as large as
before or, in other words: the depth of field
grows with the square of the focusing
distance.
2. Focal length
The focal length behaves similarly: if we
halve it, for example, the size of the object
field in the focal plane also doubles. At the
same time, half the focal length means half
the diameter of the entrance pupil, which
then makes the light cone twice as narrow
from a point behind the focal plane. As a
result, the ratio of the field diagonal and
circle of confusion becomes four times as
large as before or, in other words: the
depth of field with equal focusing
distance is inversely proportional to the
square of the focal length.
3. Aperture
If we stop down the aperture of the lens, we
reduce the area of the entrance pupil. Its
diameter decreases by a factor of 0.71 with
each single f-stop, by a factor of 0.5 after
two stops. This also narrows the light cone.
If the size of the object field remains the
same, the depth of field increases
linearly with the f-number. Stopping down
the aperture two stops, for example from a
5.6 aperture to an 11 aperture, usually
doubles the depth of field.
Camera Lens Division
7
Makro-Planar 2/100 ZF Format 24 x 36 z= 0.029 mm D/1500
0.1
1
Distance [m]
10
100
10
1
0.1
0.01
Total Depth, Field Size [m]
100
0.001
Total depth k= 2
k= 5.6
k= 16
Object height
Object width
Distagon 2/35 ZF Format 24 x 36 z= 0.029 mm D/1500
0.1
1
Distance [m]
10
100
10
1
0.1
0.01
Total Depth, Field Size [m]
100
0.001
Total depth k= 2
k= 5.6
k= 16
Object height
Object width
Graphical representation of the relationships described on the previous page. The meter scale on
each axis is divided logarithmically so that distance always changes by the same factor for each
equally long increment. These types of scales are useful for displaying wide ranges of size
variations in one image and give us very simple curves. They are only a bit warped on the edges if
we come close to the lens or the infinity focus The focus distance runs along the horizontal axis and
the total depth of field runs along the vertical axis.
Logarithmic scales have ten intervals of varying length for the same steps of numbers, step size is 1 between
1 and 10, 10 between 10 and 100, 100 between 100 and 1000, 0.01 between 0.01 and 0.1 … and so on.
1
Carl Zeiss
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Camera Lens Division
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8
Now we can also just as easily explain what
happens during a change in the film format:
4. Smaller film format
with the same lens
If we remove a lens from an old analogue
camera and attach it to a digital camera of the
same system that has a somewhat smaller
APS-C sensor, then there is a "crop factor". We
do not talk about an extension of the focal
length, it doesn’t exist in this case. After all, the
lens does not know how much of its image
circle we are capturing with our sensor.
The size of the object field is reduced by the
crop factor while the object-side light cones
remain the same, as long as we use the same
lens and do not change the aperture setting.
That is why the points of the light cones may
not be located so far from the focal plane if we
want to maintain the same ratio of diagonal to
circle of confusion. Reducing the size of the
film format therefore reduces the depth of
field by the crop factor.
5. Different film formats
with the same object field
If we select the suitable focal length to ensure
that we always display the same field with
different film formats, then things go just the
other way round: reducing the size of the
sensor format increases the depth of field,
and enlarging the sensor format reduces
the depth of field, as long as we always use
the same aperture setting. That is because a
smaller sensor format displays the same object
field with an accordingly shorter focal length. If
the same f-number is used, then the entrance
pupil is reduced by the crop factor and the light
cones are narrower.
Carl Zeiss
For the same reason, medium format
photographs show a significantly smaller
depth of field with the usual apertures, even
though the absolute diameter of the imageside circles of confusion is larger, usually
0.05 mm as opposed to 0.03 mm in 35 mm
format. If the medium format lens is adapted
to a 35 mm camera, then of course we have
to calculate with the 0.03 mm of the smaller
format.
The acceptable diameter of the circle of
confusion is therefore not a characteristic of
the lens but rather the sensor format. A
feature of the lenses is only the smallest
possible circle of confusion, and this arises
from the correction of the lens aberrations.
At first glance we therefore observe a
formats have a smaller object-side depth
of field and simultaneously a larger imageside depth of focus with the same apertures
and object fields. This is also reflected in
the mechanical tolerances of cameras:
Large-format cameras can be built with
carpenter precision, and the camera module
in a mobile phone requires µm (micrometer)
precision. Those are the extremes, but in
SLR photography we can already see the
difference between APS-C and full-frame
format with regard to the requirements for
focusing accuracy.
It appears to be a confusing paradox at first
glance, but of course it has a very simple
explanation. We just photographed object
fields of the same size with different sizes of
image formats. If the acceptable blurriness
is supposed to be the same with these
different cameras, it means that the ratio of
the object field diagonal and the "object-side
circle of confusion" should be the same.
The object-side light cones travelling from a
point behind the focal plane, for example,
should therefore be the same for all
compared cameras. If the images have
different format sizes, however, the imaging
scale is different. Under these conditions,
the image-side circles of confusion must
therefore increase along with the scale
factor.
The object-side light cones can only be the
same if all entrance pupils are of the same
size, however. But because object fields of
the same size mean longer focal lengths for
larger image formats, the f-numbers must
be different.
Camera Lens Division
9
The big format comparison
We now know that the depth is only dependent
on the size of the entrance pupil if we have the
same distance and the same angular field. The
pupil diameter is the quotient of the focal length
and the f-number.
An aperture of 2.8 in 2/3" format therefore
approximately corresponds to an aperture
of 8-11 in 35 mm format and an aperture of
22 in a 6x7 medium format. With the APS
format we have to open the aperture one
stop in order to have the same depth of field
relationships as in the 35 mm format, as
long as we have the same angular field.
If the focal length then changes by a factor
determined by the image format, we only have
to multiply the f-number by the same factor.
Then the quotient, that is to say the entrance
pupil, has the same value again and we have
the same depth of field relationships.
The widely spread practice of describing the
angular field of lenses by calculating the
equivalent 35 mm focal length is therefore
inconsistent if it does not convert the
aperture as well. But on the other hand
there would be a conflict: a converted fnumber would be incorrect as an exposure
parameter.
There are therefore equivalent f-numbers for
all formats, corresponding to the linear format
size.
The table shows us that the small formats
have fewer or in some case nearly no
variation possibilities of the depth of field
and hence the look of images.
Diagonal [mm]
Format
k/D
6.6
8
3.96x5.28 4.8x6.4
1/2.5"
1/1.8"
11
6.6x8.8
2/3"
21.6
26
13x17.3 15.6x20.8
4/3"
APS
0.025
0.035
70
90
150
24x32
42x56
54x72
90x120
35mm
4.5x6
6x7
9x12
1
1.7
2.4
4
3.4
5.6
1.4
2.4
1.4
2
3.4
4.8
8
1.4
2
2.8
4.8
6.7
11
1.2
2
2.8
4
6.7
9.5
16
1.2
1.7
2.8
4
5.6
9.5
13
22
0.05
0.07
0.10
0.14
40
0.20
1.4
1.7
2.4
4
5.6
8
13
19
32
0.28
2
2.4
3.4
5.6
8
11
19
27
45
0.40
2.8
3.4
4.8
8
11
16
27
38
64
0.55
4
4.8
6.7
11
16
22
38
54
90
0.80
5.6
6.7
9.5
16
22
32
54
76
128
Each line of this table contains the equivalent f-numbers that have the same depth of
filed figures with the same angular fields. Formats are each cropped to the 3:4 aspect
ratio, aperture values are rounded to half-stops, and the left-hand column in blue shows
the f-number as a fraction of the format diagonals. The lower lines represent the maximum
reasonable f-numbers with respect to image degradation by diffraction..
Carl Zeiss
Camera Lens Division
10
Depth of field with the same imaging scale
The imaging scales are different in each
column in the format comparison on the
previous page because we are looking at
different cameras. In our photographic practice
it is more common that we have one single
camera and different lenses for it. For that
reason, we are sometimes faced with the
question of which focal length to use. The
decisive criteria are the room conditions,
intended perspective, and background.
Are there also differences with regard to the
depth of field if we want to display a motif in the
same size? Would the 2/50 or 2/100 macro
lens be better, for instance?
The depth of field (almost) does not
depend on the focal length at all but
rather on the imaging scale, and we can
understand that as follows:
A focal length that is twice as long creates
an image of the same size from an
approximately doubled distance, and with
the same f-number its entrance pupil
diameter is twice as large. Because of the
increased focusing distance the object side
cone of light is nevertheless the same. As a
result, the “object side circles of confusion”
are also the same.
However:
the
infinitely
distant
background is displayed with a different
amount of blurriness because the entrance
pupils are different.
from infinity
The geometric explanation for the rule that the depth of field is not dependent on the focal
length for a given size of the object field: with the same f-number, the size of the entrance
pupils is proportional to the focal length and focusing distance. The light cones, and
therefore also the circles of confusion, are always the same.
But the bundles of light entering from the infinite distance into the entrance pupils intersect
the object plain in different areas. That is why the blurriness in the image is not the same
for very distant objects. This tells us that the nice and simple rule explained on this page
does not accurately apply to all photographic cases. We will come back to the deviations
later.
Carl Zeiss
Camera Lens Division
11
The hyperfocal distance
If we think of conditions where the depth of
field stretches from the focus distance into
infinity, then it becomes clear that we may
have been a bit too naive when talking about
doubling or halving the depth of field. Infinite
distances can neither be doubled nor divided in
two.
But the same rules apply in the format
comparison for the hyperfocal distance, the
shortest focus distance where the depth of field
reaches infinity. We can easily understand this
with the help of our object-side light cones
again:
A light cone coming from infinity and entering
the lens is a bundle of parallel beams and its
angular aperture is 0°. Its diameter is the same
as the diameter of the entrance pupil. The
hyperfocal distance is therefore the distance
where the acceptable "object-side circle of
confusion diameter" is as large as the
entrance pupil.
And once again the rule applies that the
smaller sensor format has the smaller entrance
pupil if it has the same angular field and the
same aperture. The acceptable object-side
circle of confusion is therefore already in
smaller object fields, meaning it is reached at a
shorter distance.
Looking at the cones of light we can easily see
that the front end of the depth of field is
located at half of the hyperfocal distance. That
is because the beam cone, whose rear
extension is as large as the entrance pupil in
the hyperfocal object plane, has its point right
in the middle between the entrance pupil and
the object plane.
At this point we should make an exception
and use a few formulas, because they are
the most important ones of the whole topic
and are also so simple that we can
EP =
f'
k
The diameter of the entrance pupil is the focal length
divided by the f-number k
Z hyperfocal = EP = M ⋅ z '
The object-side circle of confusion Z at hyperfocal
distance is as large as the entrance pupil, and the
image-side circle of confusion z’ results from it through
the magnification M
M≈
Dist
f'
The magnification is approximately the ratio of the
distance and the focal length; from that follows:
Dist hyperfocal
2
(
f ')
≈
z '⋅k
EP = diameter of the entrance pupil, f’ = focal length,
k = f-number, M = magnification,
Z = object-side circle of confusion, z’ = image-side
circle of confusion, Dist = distance
field at hyperfocal distance
front edge of depth
Carl Zeiss
Camera Lens Division
12
It is especially easy to do calculations with these sizes if we relate everything to the diagonal of the
sensor format (D); then the formula of the hyperfocal distance looks more complicated at first but
finally results in very easy numbers that can actually be used to calculate the hyperfocal distance in
Dist hyperfocal
D D ⎛ f '⎞
≈ ⋅ ⋅⎜ ⎟
z' k ⎝ D ⎠
2
A 35 mm format lens with the focal length f’ = 85 mm and f-number k=2, a sensor diagonal
of 43 mm, and a requested circle of confusion diameter of D/1500 results in:
Dist hyperfocal ≈ 1.500 ⋅ 21.5 ⋅ (2 ) = 129m
2
(The factor of 1.5 must actually be doubled for the highest sharpness requirements!)
Hyperfocal Distance [m]
z'=Diagonal/1500
for
120
115
110
105
100
95
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
100
10
1
0.1
1
Ratio:
k/D = 0.025
k/D = 0.035
k/D = 0.05
k/D = 0.07
Diagonal Field Angle [°]
1000
k/D = 0.1
k/D = 0.14
k/D = 0.2
k/D = 0.28
k/D = 0.4
k/D = 0.56
k/D = 0.8
k/D = 1.13
Field Angle
10
Focal length / Image diagonal f'/D
Those who want to avoid the calculations can also use this chart that universally applies to all
formats because the aperture and focal length are not absolute but rather related to the diagonal of
the sensor format. The short telephoto lens in the example above has a focal length twice as long
as the sensor diagonal; the f-number 2 is about 1/20 (=0.05) of the diagonal: so we can find the
hyperfocal distance by starting from 2 on the horizontal scale and moving upward until we reach the
thin yellow line for k/D=0.05.
The hyperfocal distance is often underestimated; in order to check whether the infinity alignment of
a lens and a camera is correct, one has to look for very distant objects in case of longer focal
lengths.
Carl Zeiss
Camera Lens Division
13
The hyperfocal distance is a type of key variable for calculating the depth of field - if we know it,
then we can calculate the depth of field for any distance from that alone. That is because it is the
product of three ratios (see previous page) so it includes everything that we need for our
conception of "object-side circles of confusion":

The ratio of the focal length and the sensor diagonal determines how fast the object field
becomes larger with increasing distance from the camera.

The ratio of the focal length and the f-number determines the diameter of the entrance
pupil, and therefore how narrow the light cones are from points outside the focal plane.

The ratio of the sensor diagonal and diameter of the circle of confusion determines the
acceptable blurriness.
The following chart provides a very simple overview of the magnitudes of depths of field for normal
taking conditions. Each coloured line represents a certain constant depth of field beginning at 1cm
in the upper left-hand corner and ending at 100 meters at the black line. The axes of the chart are
only distances measured in meters, with the focusing distance on the horizontal axis and the
hyperfocal distance on the vertical axis. F-numbers, format sizes, and focal lengths are not listed
because they are already included in the hyperfocal distance. This chart is therefore universal for
all camera formats.
1000
depth=0.01m
depth=0.02m
depth=0.05m
depth=0.1m
100
depth=0.2m
depth=1m
depth=2m
depth=5m
10
depth=10m
depth=20m
depth=50m
depth=100m
1
Hyperfocal Distance [m]
depth=0.5m
1
100
10
Taking Distance [m]
Carl Zeiss
Camera Lens Division
14
Rules of thumb
The hyperfocal distance can be used to give a
few rules of thumb for the depth of field:
"If the focus distance is 1/10 of the
hyperfocal distance, then the depth of field
is 1/5th of the focus distance."
"If the focus distance is 0.4 times the
hyperfocal distance, then the total depth of
field is of the same amount as the focus
distance."
"If the focus distance is one third of the
hyperfocal distance, then the depth of field
behind the focal plane is twice as large as
the depth forward in front of the focal
plane."
A part of the last rule ("1/3 in front, 2/3 behind")
is often found in photography textbooks. But it
is not generally true. It only applies to a certain
focusing distance for each aperture. The
distribution is more symmetrical at shorter
symmetrical at longer distances, which is very
obvious when we approach the hyperfocal
distance.
There is a relationship between the distance
from the camera to the near and far limits of
the depth of field and the focus distance that
applies to all apertures and distances:
Dist =
Carl Zeiss
2 ⋅ Near ⋅ Far
Near + Far
To put that into words, the focus distance is
the product of the near limit and the far limit
divided by the average of the near limit and
far limit. (Also called ‘harmonic mean’, for
example: near limit 3 m, far limit 6 m, focus
distance 4 m, 18 divided by 4.5). From that
we can calculate that the distribution of the
front:back relationship is only 1:2 if the
distance to the far limit is twice as far as to
the near limit. In other words, the total depth
of field is as large as the distance between
the camera and the near limit.
For those who enjoy the beauty of
mathematical relationships, it should be
noted that this is the precisely the case for
the distance where the size of the "objectside circle of confusion" is 1/3 of the
entrance pupil, therefore 1/3 of the
respective hyperfocal distance.
For a 50 mm lens with 35 mm film format
having a circle of confusion of 0.03 mm and
aperture of 8, the focus distance to fulfil
above condition is 3.5 meters, a standard
picture taking situation. That is why this rule
continues to haunt through the literature.
But it does not generally apply in any way.
The distribution in the close range and
macro range in particular is very
symmetrical. Reversing the lens does not
only influences the correction condition.
If we use relatively long focal lengths with a
very large hyperfocal distance, then we
must assume a symmetrical distribution of
the depth of field in front of and behind the
focal plane.
Camera Lens Division
15
Close-up
With the usual maximum close-up lens setting
(scale 1:8 to 1:10), and even more with macro
lenses or close-up accessories like extension
rings ore bellows, depth of field becomes very
small. The scales engraved on the lens mount
then only provide little help. For many modern
lenses with their steep distance scales they are
not much more than a useless decoration.
Many explanations of the topic of depth of field
skip over macro photography because the
usual formulas and tables for long distances do
not apply in those cases. At close range, the
lens no longer actually has the f-number that is
engraved on the ring; we have to calculate
using the effective aperture - some cameras
display it, others do not. The amount that this
effective aperture value deviates from the
nominal value depends not only on the scale
but also on the construction of the lens.
Telephoto lenses show a heavier loss of the
effective f-number at close range than
symmetrically constructed lenses do.
Modern macro lenses have lens groups that
move relative to each other in order to keep
the correction stable at all distances. As a
result, their focal length also changes with
the focusing. So there are plenty of
complications.
An extensive and detailed explanation of
the optical rules in macro photography
including the field of magnified imaging
would therefore be too much to cover within
the framework of today's topic.
I would like to at least provide our readers
with the most important figures for our two
2/50 and 2/100 macro lenses for 35 mm
format, first as a graphical overview and
then as a table at the end of the chapter:
2.0
1000
Exposure Compensation [EV]
1.8
1.6
100
1.4
1.2
10
1.0
0.8
1
0.6
Depth of Field [cm],
Free
Working Distance [m]
Makro-Planar 2/50 and Makro-Planar 2/100 ZF.2 / ZE
depth at f/2
depth at f/8
depth at f/22
0.4
0.1
MP100
0.01
Working Distance [m]
0.2
0.0
1
0.1
0.01
Scale Ratio
MP100
Exposure Compensation
MP100
Depth of field, free working distance (without lens hood), and required compensation of exposure of
the two macro planar 2/50 and 2/100 lenses, calculated for 35 mm format and circle of confusion
diameter diagonal/1500.
Carl Zeiss
Camera Lens Division
16
The graph on the previous page is similar to
the one on page 8, although here the depth of
field is not displayed over the focusing distance
but rather the imaging scale or magnification of
two lenses at the same time. The fact that the
same imaging scale is achieved from different
distances can be seen with the two black lines.
The yellow, green, and red lines show the
depth of field for full aperture, f/8 and f/22 for
the 2/50. The values for the same apertures of
the 2/100 are drawn as dotted black lines.
These lines are congruent almost everywhere another nice proof that the depth of field is
mostly only dependent on the imaging scale.
There are only deviations at the ends: on the
right side at magnification 0.01 and at f/22, the
rear limit of the depth of field comes close to
the "infinity" value for the 2/50.
On the left side at imaging scale 1:2, the 2/100
has a bit more depth of field with the same
nominal value of the f-number, the dotted lines
are just a bit higher than the coloured ones.
Is that a benefit of the optical construction of
the 2/100 in comparison to the 2/50? No,
because the slightly larger depth of field is a
result of the 2/100's loss of speed, which is 1/3
of an aperture stop higher, as we can see with
the blue curves. At image scale 1:2 its
maximum aperture is no longer f/2 but rather
f/3.6, and with the 2/50 the maximum aperture
is only reduced to f/3.2. This difference
between our two macro lenses is an indication
that depth of field does not come free and we
have to pay for it with exposure time. In fact, a
very fundamental general physical law is
behind it: the law of conservation of energy.
That is because the angular aperture of the
object-side light cone also determines how
much optical radiant energy enters into the
lens. And only this energy can be distributed
onto the image surface. If we compare two
images of the same sensor size, then the one
that needs a longer exposure time with the
same sensitivity has the larger depth of field
because it has collected less energy on the
image-side with a narrower light cone (we must
of course rule out absorption by filters etc. - we
are only dealing with geometric efficiency). The
specific optical construction of a lens is
therefore in the end meaningless for the depth
of field.
Carl Zeiss
Telephoto lens constructions lose more light
at close range; that is because their
entrance pupil is located relatively far back,
so the object-side light cone becomes a bit
narrower if the distances to the object are
similar to the dimensions of the lens. But all
we have to do is simply use a wider
aperture to have the same depth of field as
with a symmetrical lens.
Different sensor formats with the same
sensitivity have the same object-side
depth of field if their exposure times
have the same ratio as their sensor
areas. That is because the same depth of
field means that the same amount of energy
is collected from the object for both pictures;
if this energy is distributed across a sensor
area twice as large, the light intensity is
divided in half and an exposure time twice
as long is required.
When practically all formats worked with the
same emulsions in analogue photography,
this meant that small formats were always
be achieved at fast shutter speeds. If the
signal-to-noise ratio of the sensor increases
with the format size, which is to some
amount the case with digital cameras, then
we can compensate for the increased need
for light of the larger format for the same
depth of field by increasing the sensitivity.
If we put aside the requirements of offhand
photography and photograph static objects
using a tripod so that the exposure time can
be any length, then there is no difference at
all between different film formats with
regard to the maximum achievable depth of
field.
Because light travels in waves, the
diffraction determines how far we can close
the aperture without losing picture quality in
the end. It ensures that a picture element
creates an Airy disk whose diameter in
micrometers is about the same as the fnumber. The relative size of the Airy disk
with regard to the format therefore allows
for smaller apertures with a larger format.
All formats have the same depth of field
at the diffraction limit.
Camera Lens Division
17
Makro-Planar 2/50
Scale
1: 100
1: 50
1: 40
1: 30
1: 25
1: 20
1: 15
1: 12
1: 10
1: 8
1: 6
1: 5
1: 4
1: 3
1: 2.5
1: 2
EC
[EV]
0.1
0.1
0.2
0.2
0.2
0.2
0.3
0.3
0.4
0.5
0.6
0.7
0.8
1.0
1.2
1.4
WD
[m]
5.15
2.57
2.06
1.54
1.28
1.02
0.77
0.61
0.51
0.41
0.30
0.25
0.20
0.15
0.12
0.10
k=2
2.8
4
Total Depth-of-Field [cm]
5.6
8
11
UF
16
22
118
167
245
361
581
1026
4592
2427
15.4
29.5
41.4
59.6
84
124
179
294
521
15.3
19.0
26.6
38.2
54
78
111
174
275
15.2
10.8
15.1
21.6
30.4
44
61
93
137
15.0
7.5
10.5
15.1
21.2
30.4
42
63
91
14.9
4.9
6.8
9.7
13.7
19.6
27.1
40
57
14.8
2.8
3.9
5.6
7.8
11.2
15.4
22.7
31.7
14.5
1.8
2.5
3.6
5.1
7.3
10.0
14.6
20.3
14.3
1.3
1.8
2.5
3.6
5.1
7.0
10.3
14.2
14.0
0.83
1.17
1.67
2.33
3.33
4.59
6.70
9.25
13.7
0.48
0.72
0.97
1.35
1.93
2.66
3.87
5.34
13.1
0.34
0.48
0.68
0.96
1.37
1.88
2.74
3.78
12.7
0.22
0.31
0.45
0.63
0.90
1.23
1.79
2.47
12.2
0.13
0.18
0.26
0.36
0.51
0.71
1.03
1.42
11.3
0.09
0.12
0.18
0.25
0.36
0.49
0.71
0.98
10.7
0.06
0.08
0.11
0.15
0.22
0.30
0.44
0.61
9.9
k=2
2.8
4
16
22
Makro-Planar 2/100
Scale
1: 100
1: 50
1: 40
1: 30
1: 25
1: 20
1: 15
1: 12
1: 10
1: 8
1: 6
1: 5
1: 4
1: 3
1: 2.5
1: 2
EC
[EV]
0.1
0.2
0.2
0.2
0.2
0.3
0.4
0.4
0.5
0.6
0.7
0.9
1.0
1.3
1.5
1.7
WD
[m]
9.81
4.93
3.96
2.98
2.50
2.01
1.52
1.23
1.03
0.84
0.64
0.54
0.45
0.35
0.30
0.25
Total Depth-of-Field [cm]
5.6
8
11
UF
117
164
236
335
493
715
1198
2209
15.4
29.6
41.5
59.3
83
120
167
251
364
15.1
19.1
26.7
38.2
54
77
107
158
225
15.0
10.9
15.2
21.8
30.5
44
60
89
124
14.8
7.6
10.7
15.3
21.4
30.6
42
62
86
14.7
4.9
6.9
9.9
13.9
19.8
27.3
40
55
14.5
2.8
4.0
5.7
8.0
11.4
15.7
22.9
31.6
14.1
1.9
2.6
3.7
5.2
7.5
10.3
15.0
20.6
13.8
1.3
1.9
2.6
3.7
5.3
7.3
10.6
14.6
13.5
0.87
1.22
1.74
2.44
3.48
4.79
6.98
9.61
13.1
0.51
0.72
1.02
1.43
2.05
2.81
4.09
5.63
12.4
0.37
0.51
0.73
1.02
1.46
2.01
2.93
4.03
11.9
0.24
0.34
0.49
0.68
0.97
1.33
1.94
2.67
11.2
0.14
0.20
0.28
0.40
0.57
0.78
1.14
1.56
10.3
0.10
0.14
0.20
0.28
0.40
0.55
0.80
1.10
9.6
0.06
0.09
0.12
0.17
0.25
0.34
0.50
0.69
8.8
Depth-of-field tables for the Makro-Planar 2/50 and 2/100 lenses. The f-numbers are the engraved
numbers. EC is the required exposure compensation in aperture stops [EV], WD is the free working
distance measured from the focal plane to the filter thread of the lens.
UF is the useful f-stop, where an MTF-figure of 10% for 90 linepairs/mm is achieved due to
limitations by diffraction. This means that even with 24MP full frame cameras there is only a very
small loss of sharpness that can still be balanced out with digital edge enhancement. Combinations
of scale and f-number that no longer meet this requirement are listed in gray in the table. The depth
of field is calculated for the standard 0.03 mm circle of confusion in 35 mm format. The best
performance with the useful f-stop is not achieved in the total depth, of course.
Carl Zeiss
Camera Lens Division
18
The diameter of the circle of confusion
All of the curves and tables shown so far have
been calculated assuming a circle of confusion
diameter that fits 1,500 times into the diagonal
of the image. But we must explain why this size
is so often chosen and why we should
sometimes choose another one. Depth of field
is the result of an arbitrary specification, or
rather it depends on the viewing conditions.
Whether we tolerate a small or large amount of
blurriness has no influence on the fundamental
characteristics of the depth of field.
The human eye will not perceive any loss of
sharpness in an image if the power of the eye
is the only thing determining which smallest
details can be recognized. On the other hand
the eye will perceive an image as blurry if the
eye is capable of seeing significantly more than
is shown. The resolution that the eye can
recognize must be the benchmark.
If we test the ability of the eye to recognize
resolution with periodic black & white patterns,
then we see that normally performing test
subjects have a limit of approximately 8 line
pairs per mm that they can recognize if the test
pattern is within a distance of 250 mm from the
eye. At longer distances, of course the eye is
less capable of recognizing as much; at two
meters away it is barely possible to distinguish
a pattern with one pair of lines per mm from a
simple gray surface of the same shape. This
experiment can be done easily using the lines
on a ruler.
If we want to describe the performance of the
eye independently of its distance from the
object, then we use the angular resolution. It
thereby matches the numbers above, that the
eye can distinguish the smallest details from
one another if they appear at a visual angle of
one arc minute. This is the physiological
critical angle of the human eye.
If we look at a 12x18 cm picture, for example a
5x magnification from the completely used 35
mm format viewed from a 25 cm distance, then
we see 1/3000 of the diagonal of this picture at
a visual angle of one arc minute.
That means that our eye would not even notice
if the picture had a higher sharpness. This
circle of confusion is therefore the strictest
sensible requirement for the given viewing
conditions.
Carl Zeiss
We could of course magnify the negative or
sensor image even more, for instance 20
times to the poster size of 48x72 cm. In
digital photography that can be done with
just a few mouse clicks. Then we can
already view 1/3000 of the picture diagonal
at a visual angle of four arc minutes if we
are still viewing the image from 25 cm
away; the eye can then see much smaller
details.
However, the entire image width then
appears to us at an angle of 110°; we
cannot overlook that entirely and still see
the smallest details everywhere in the
image at the same time. If we look at it in
this way then our eyes must wander about
in the image, and they see details but not
the entire image.
If we look at the poster from 1 meter away,
however, then we are looking at the image
width at an angle of 40° - such as with a
12x18 cm image from 25 cm away - which
we can comfortably view in its entirety.
Whenever we observe images in this way,
then 1/3000 of the picture diagonal is the
strictest sensible requirement for the circle
of confusion diameter. A circle of confusion
twice as large, 1/1500 of the diagonal,
viewed at a visual angle of 2 arc minutes,
still provides a satisfying sharpness even
then;
this
requirement
corresponds
approximately to the often used 0.03 mm
circle of confusion for the 35 mm format.
But we must not forget that our
expectations for the image sharpness can
no longer be met with this usual circle of
confusion if we make cropped enlargements
or view the details in large prints. After a
20x magnification we see the 0.03 mm
criterion of the 35 mm format from a
distance of 25 cm at a visual angle of over 8
arc minutes – it appears to be blurred.
Camera Lens Division
19
In the 50s, the depth of field for 35 mm lenses
was often calculated with a circle of confusion
of 0.05 mm, meaning 1/865 of the picture
diagonal. This can be viewed at 2 arc minutes
if we look at a 10x15 cm postcard image from a
distance of 35 cm. In those days of amateur
photography, that corresponded to the
somewhat more discerning viewing habits
when it was still most common to paste contact
prints from roll-film cameras into photo albums.
The depth of field is therefore a rather fuzzy
dimension that depends heavily on the viewing
conditions. Strictly speaking we can even find
reasons for to use different circle of confusion
sizes for different focal lengths of a camera:
If we view images "from the right
perspective", meaning closer to the same
angle at which they were really viewed by
the camera when they were taken, then we
must view the wide-angle images from a
closer distance than images from normal or
telephoto lenses. As a result, we must
calculate the depth of field in wide-angle
images using smaller circles of confusion.
The depth of field was calculated more
discerningly for the DISTAGON 4/40 from
the old C series for the HASSELBLAD than
for all other lenses in the series. Because
even without viewing them from the right
perspective, the details that interest us in
wide-angle images are usually smaller and
therefore place stricter requirements on the
sharpness of the image.
How precise are tables and depth of field calculators?
Usually most tables pretend to have a
precision that is neither available nor sensible
in reality. That is partly because the values
calculated in the tables are based on the
arbitrary specification of a limit value
(acceptable circle of confusion diameter).
In reality, however, the sharpness is
continuously changing in the depth and its
subjective perception is also somewhat based
on the image content in addition to the viewing
conditions. There is therefore no clear limit!
The from-to tables with millimetre precision at
meter distances, on the other hand, easily give
the impression that there are two precisely
positioned flat limit surfaces in front of the
camera between which everything is displayed
at constant sharpness. But there are a few
things wrong with this notion.
Most tables and calculation programs found on
the internet are based on the geometric model
of the light cones and circles of confusion that
we have also been using for illustration. Yet
despite how nice it is, it is only an idealization
of the real optical processes in a lens. That is
because this model does not recognize
aberrations, colours, or diffraction. In the
geometric model, the circle of confusion is a
disc with even brightness.
Carl Zeiss
In reality, however, the distribution of
brightness in the focused and lightly
unfocused point image is uneven. We will
deal with that in more detail below. All types
of aberrations from real lenses cause a
series of deviations from the geometric
model:
•
•
•
•
•
In ideal lenses, the image-side depth of
focus grows nearer and farther away at the
same rate when the aperture is made
narrower. In real lenses, however, there can
be a displacement to one side called the
focus shift. When it is very large, the near
limit of the depth of field range might remain
unchanged when the aperture is narrowed.
Away from the optical axis, this shift often
has opposite direction compared to the
centre of the image, the depth of field space
is then curved.
If lenses suffer form vignetting by parts of the
lens barrel, then the depth of field at the
edge of the frame is larger than in the centre
because the size of the pupil decreases due
to the vignetting.
Depending on the lens aberration, the type
of blurriness can be different in front of and
behind the focal plane.
The location of the depth of field range also
depends somewhat on the colour of the light.
The usual tables and calculators therefore
provide some useful clues for practice, but
they should not be taken too seriously.
Camera Lens Division
20
Depth of field and modulation transfer MTF
The meaning of depth of field can also be
understood if one measures how the contrast
of the image in terms of the modulation transfer
MTF (see CLN30 and CLN31) changes with
deviation from the best focus. Curves that are
almost bell shaped are found in this measuring
process; these curves show very well that the
sharpness of the picture is in no way constant
within the depth of field but is continuously
changing. The curves also show that not so
much remains of the best performance of a
lens at the edges of the usual depth of field
range.
As a rule of thumb we can say that when
defocusing the image by the length of k / R
(k = f-number, R = spatial frequency in line
pairs per millimetre), the MTF value falls from
the maximum to about 20-30 %.
Biogon 2/35 ZM
Such measurements of MTF related to
focus also indicate the limitations of the
simple geometric model as an explanation
of depth of field. We find many examples
where the image-side depth of focus is not
symmetrical to the best focus for various
reasons but is extended more to the front or
to the rear side.
We can also find examples where, at the
same f-number, the depth of field varies in
size, not because we are looking at different
spatial frequencies because of different
formats, but because the curves of the
same spatial frequency have different
widths.
f= 35 f/ 11 u'= 0
MTF [%] 20, 40 Lp/mm
80
4.5
60
40
20
Object Distance [m]
10
100
1.3
0
-1
-0.5
1
0
0.5
1
Focus deviation on image side [mm]
Carl Zeiss
tan 20 LP/mm f= 35 f/ 11 u'= 0
tan 40 LP/mm f= 35 f/ 11 u'= 0
depth of focus for z'=0.033
object distance
film or sensor
depth of field
Camera Lens Division
21
The black curve shows the relationship
between the distances in the object space and
the associated positions in the image space
when focussing on a distance of 2 metres. The
object distances can be read on the scale on
the right-hand edge of the chart. Thus the
image-side depth of focus between the black
triangle marks goes with the object-side depth
of field between the blue marks; these can also
be seen in the picture below showing the depth
of field scale on the lens barrel.
If you read off the MTF values at the limits of
the depth of field area near the triangular
marks, you will still find 10 to 20% at 40
Lp/mm. If one then takes account of the
additional losses by the sensor, the resolution
there should be a maximum of 40 Lp/mm, or if
in a five-fold enlargement, 8 Lp/mm. The eye
can do no more when examining something
from a distance of 25 cm; the image is still
perceived as being sharp. However, in the
case of greater enlargements it is obviously
necessary to further restrict the permissible
deviation from the ideal focus.
If the diaphragm is opened further, the curves
become narrower (please note the different
scale). The following curves were measured on
the same lens of the Biogon 2/35 ZM, again in
the middle of the image at aperture 4. The
black curve for the relationship between the
image and the lens distance applies for
focusing a distance of 4 metres:
Carl Zeiss
80
60
2.8
40
20
0
-0.4
-0.2
Object Distance [m]
10
6.9
1
0
0.2
0.4
Focus deviation on image side [mm]
tan 20 LP/mm f= 35 f/ 4 u'= 0
tan 40 LP/mm f= 35 f/ 4 u'= 0
depth of focus for z'=0.033
object distance
film or sensor
depth of field
Compared with the geometric depth of focus
marks the curves have become a shade
narrower. At the same time the maximum
MTF values at the best focus are somewhat
higher. It is due to the incipient influence of
diffraction that at f/11 the maximum
contrasts are reduced and that the curves
are wider than expected from the ratio of the
f-numbers 11 and 4.
If one measures a lens with a longer focal
length and similar performance at the same
f-number, curves of almost the identical
width are obtained as can be seen in the
example of the Sonnar 2/85 ZM. This is
therefore proof that the image-side depth of
field does not depend on the focal length but
only on the f-number. However, the black
curve, which shows the relationship between
the object distance and the image space
position, now looks quite different. It is now
much flatter because the image of the same
deep object has greater depth than with a
shorter focal length. Therefore a smaller
object-side depth of field now goes with the
same image side depth of focus.
Sonnar 2/85 ZM
f= 85 f/ 4 u'= 0
100
10
80
4.3
3.7
60
40
20
0
-0.4
Camera Lens Division
-0.2
1
0
0.2
0.4
Focus deviation on image side [mm]
tan 20 LP/mm f= 85 f/ 4 u'= 0
tan 40 LP/mm f= 85 f/ 4 u'= 0
depth of focus for z'=0.033
object distance
film or sensor
depth of field
22
Object Distance [m]
The black triangles on the image side focus
scale indicate the depth of field according to
the geometric model of the circle of confusion in the above example 11 x 0.033 mm in both
directions.
f= 35 f/ 4 u'= 0
100
MTF [%] 20, 40 Lp/mm
The position of each measuring point in the
image space is indicated by the focus
deviation on the horizontal axis; negative
values are closer to the lens. At zero we have
no deviation and hence the best contrast; the
film or sensor should be there - represented by
a yellow line.
Biogon 2/35 ZM
MTF [%] 20, 40 Lp/mm
The red curves in the above chart show the
MTF values for the spatial frequencies 20 and
40 line pairs per millimetre and also how they
change in longitudinal direction with focus
deviation if the lens is set to f/11.
The MTF curves of the previous page are very
similar to each other. This is not always the
case; at the edges of the depth range which is
calculated according to the geometrical theory
MTF figures might be quite different. This tells
us that this theory is a simplification of reality:
f= 40 f/ 1.5 u'= 0
80
60
40
20
100
0
-2.4
MTF [%] 20, 40 Lp/mm
80
-1.4
-0.4
0.6
1.6
Focus deviation on image side [mm]
60
tan 20 LP/mm f= 120 f/ 32 u'= 0
tan 40 LP/mm f= 120 f/ 32 u'= 0
40
depth of focus for z'=0.033
object distance
film or sensor
A macro lens very sharply stopped down to f/32; the
MTF is reduced by diffraction and more spreading in
depth.
20
0
-0.1
f= 120 f/ 32 u'= 0
100
MTF [%] 20, 40 Lp/mm
MasterPrime Distagon 1.2/40
Makro-Planar 4/120
-0.05
0
0.05
0.1
Focus deviation on image side [mm]
Planar 2/50 ZM
tan 20 LP/mm f= 40 f/ 1.5 u'= 0
tan 40 LP/mm f= 40 f/ 1.5 u'= 0
depth of focus for z'=0.033
object distance
film or sensor
f= 50 f/ 2.8 u'= 0
MTF [%] 20, 40 Lp/mm
100
High performance lens Zeiss MasterPrime for a 35mm
ARRIFLEX film camera at aperture 1.5. At such apertures
and high performance level one can see how high the
demands placed on the precision of the camera are; 1/100
mm changes the MTF at 40 Lp/mm by 20%!
80
60
40
20
0
Modern Lens 50mm f/1.4
-0.2
-0.1
0
0.1
0.2
Focus deviation on image side [mm]
MTF [%] 20, 40 Lp/mm
100
80
60
40
20
0
-0.1
-0.05
0
0.05
0.1
tan 20 LP/mm f= 50 f/ 2.8 u'= 0
tan 40 LP/mm f= 50 f/ 2.8 u'= 0
depth of focus for z'=0.033
object distance
film or sensor
A fast 50mm lens measured at f/2.8. The focus
reference (yellow line) is defined by the best MTF at f/2
and 20 Lp/mm. It shows a very slight displacement
due to a focus shift and slightly skewed curves. The
position of the depth of field does not agree with the
geometric theory of the circle of confusion.
Focus deviation on image side [mm]
tan 20 LP/mm f= 52 f/ 1.4 u'= 0
tan 40 LP/mm f= 52 f/ 1.4 u'= 0
depth of focus for z'=0.033
object distance
film or sensor
Depth variation of contrast and digital sharpening
CTF at 20 and 40 Lp/mm
120
Modern 1.4/50 first class 35 mm lens fully open
Vintage Lens 50mm f/1.5
MTF [%] 20, 40 Lp/mm
100
80
100
80
60
40
20
60
0
-0.8
40
low 20 Lp/mm
0
-0.05
0
0.05
0.1
Focus deviation on image side [mm]
tan 20 LP/mm f= 50 f/ 1.5 u'= 0
tan 40 LP/mm f= 50 f/ 1.5 u'= 0
depth of focus for z'=0.033
object distance
film or sensor
A vintage fast lens which has larger aberrations when fully
open at f/1.5; it has a much flatter and wider curve and
therefore a slightly increased depth of field; and within the
depth of field a smaller difference between the best values
and those that can still be tolerated.
Carl Zeiss
-0.4
-0.2
0
0.2
0.4
0.6
Focus deviation in image space [mm]
20
-0.1
-0.6
low 40 Lp/mm
high 20 Lp/mm
high 40 Lp/mm
When one measures how contrast transfer in the digital
image, (including the lens and the processing of pixel
data) varies with the focus, the curves look different:
they appear more rounded and flat. This is no surprise
since the low-pass filter is also a kind of image
degradation, similar to the aberrations or the diffraction
in the previous cases. These data are from a good lens
at f/11, so you may compare with page 21. High
sharpening increases the depth just a little bit, but it
may as well cause a more harsh transition.
Camera Lens Division
23
0.8
Resolution
Which resolution can be least achieved within
the depth of field range? If this is expressed in
terms of MTF measurement the question is
translated into: ”At what spatial frequency does
the contrast transfer (MTF) drop below a
certain threshold (e.g. 10 %)?”
Typical values which answer this question can
be seen in the following chart which shows
how contrast transfer gradually decreases with
increasing spatial frequency, in other words
with of the structures getting finer and finer. In
order to make the figures independent of the
format size, the spatial frequency is not given
in absolute terms in line pairs per mm but in
line pairs per image height. The blue curve
shows the relationship with the very familiar
35mm format: the corresponding absolute
spatial frequencies can be seen from the blue
scale on the right-hand side.
The data apply for a lens which is limited by
diffraction at aperture k = 0.2 x image
diagonal. In the 35 mm format this is about
f/8; in the 2/3“ format the corresponding
aperture is f/2 – at this high speed
diffraction limited performance is only
achieved by very elaborate and expensive
lenses such as the Zeiss DigiPrime.
The black curve applies to the best
focusing; the other curves indicate the
contrast transfer at the edge of the depth of
field area at a circle of confusion diameter
z’. As the f-number the circle diameter is
related to the format size, so that the chart
is valid for different sensor formats.
One can also see from these curves that
resolutions greater than 2000 line pairs per
image height cannot play a major role in
many fields of practical photography
because they can only be achieved with
extremely tight focus tolerances and with
very flat objects.
Resolving power of the lens at different permitted degrees of blurriness described by the
diameter of the circle of confusion as a fraction of the image diagonals. A lens which is
restricted by diffraction at aperture 8 achieves a resolution of about 4000 line pairs per image
height in 35mm format. At a circle of confusion of z’=D/1500, in other words at the popular
value of 0.03mm, the resolution drops to about 1200 line pairs per image height.
Carl Zeiss
Camera Lens Division
24
Bokeh – properties of blurriness
Large depths of field can be desirable; in
macro-photography it would be great to have
more of it than is possible. However, it is as
well often undesirable, as a good image is
usually characterised by the absence of
superfluous and distracting items.
A composition parameter which can help us to
achieve this objective is the adjustment of the
blurring in front of and behind the main subject
by a suitable combination of aperture, focal
length and taking distance. A blurred
background frees the main subject from
distracting unimportant details and increases
the three-dimensional illusion of the picture.
Blurred parts of the picture can also be
decorative and play a very important part in the
composition of the picture.
We therefore want to deal with blurring in the
following pages. This image attribute is indeed
more of an aesthetic and therefore subjective
nature and cannot be described as simply with
figures as it is the case with a well focused,
sharp image. Thus its subtleties in lens tests
play no important part sometimes. This is quite
different in Japan: as well as figures for
contrast, resolution etc., every test always
includes examples of images with blurred
flowers, leaves and other items which often act
as the background to photographs. It is
therefore perfectly right that the Japanese word
“bokeh“ is used around the world as a
collective term for all attributes of blurring.
The root of the Japanese word boke or bokeh*
actual means nothing good; its meaning is
similar to “confused” or “dizzy” and is used to
name mental states in exactly the same way.
In photography the term ”confused“ relates
naturally to light beams which no longer come
together at a single point in an orderly manner.
* I like to thank my colleague Hiromi Mori for the Japanese
Hiragana characters and for her explanations of the
meanings.
Carl Zeiss
In spite of the subjective nature of the
matter we nevertheless want to attempt to
remain faithful to the style and character of
our technical articles by describing bokeh
with some numbers. Of course, this cannot
be done on very simple scales, for example,
“a grade 5.5 bokeh“, because blurring
always depends on a large number of
parameters. But figures can help us to
improve our understanding of connections.
All the parameters listed here influence the
phenomena outside the focal plane:
•
•
•
•
•
•
•
•
•
•
•
Picture format
Focal length
f-number
The camera-to-subject distance
Distance to the background or the
foreground
Shapes and patterns of the subject
Aperture iris shape
Aberrations of the lens
Speed of the lens
Foreground/background brightness
Colour
It is therefore not surprising that one often
judgements about the bokeh of many lenses.
Undue generalisations are all too often drawn
from single observations.
Many effects are attributed to the lens even
though they are mainly caused by the subject
in front of the camera. Differences between
lenses are often very marginal but are then
grossly exaggerated.
In principle one should not turn the ranking in
the significance of the elements in a picture
on its head and raise small technical artefacts
to the rank of the most important part. In
many pictures the main subject is the
deciding moment – and all bokeh then literally
retreats into the background.
But in the beauty of calmly composed
pictures it can already mean the step towards
perfection. And here everyone can have their
own yard-sticks.
Camera Lens Division
25
The quantity of blurriness
The most important and clearest attribute of
blurring is simply the amount of it. When
considering the depth of field we have been
concerned with permissible blurring; this blurring
is allowed if it is quite unnoticeable in the
conditions in which the picture is viewed. We
have learnt that limits are fluid in this process.
But outside these limits where one clearly
sees the blurring, we can describe the extent
of the blurring in exactly the same way as
within the limits of the depth of field: by the
diameter of the circle of confusion.
This means that we now will deal with very
large circles of confusion, and to understand
the meaning of these numbers we should
connect them to our experience about a wellknown photographic situation:
Field width 70 cm
Ratio Diagonal / Circle of confusion
1000
100
10
0.01
0.1
1
10
100
Distance subject-to-background [m]
Format 24x36mm f=85mm f/1.4
Format 24x36mm f=85mm f/8
Format 24x36mm f=85mm f/2.8
Format 24x36mm f=85mm f/11
Format 24x36mm f=85mm f/5.6
Format 24x36mm f=85mm f/16
The chart describes a typical photographic situation e.g. in portrait photography: the object field is
70cm wide and photographed in 35 mm format with an 85mm lens. The focus distance to the main
subject set on the lens is therefore 1.8 metres.
The distance of the background from the main subject is indicated on the horizontal axis; the
vertical axis shows the size of the circle of confusion with reference to the image diagonal.
Therefore in this chart the region of the depth of field with which we have been concerned in the
first part is up at the top on the left, just outside the scale; at this point the circles of confusion are
diagonal/1500 or less; we are there still close to the focus; as we move to the right we move away
up to a distance of 100 metres in the background.
Each curve in the chart represents one of the aperture values specified in the legend and all curves
have the same character. Initially they fall uniformly (in this process the circles of confusion
gradually become larger) and then reach a kind of saturation beyond a background distance of
about 10 m. Thus, the blurring does not become any greater at larger distances. This limit depends,
of course, on the aperture and when we compare the figures with our experience or just try it with
our camera , we learn that we need circles of confusion larger than 1/100 of the diagonal in order to
separate the main subject from the background.
Carl Zeiss
Camera Lens Division
26
Field width 70 cm
100
1
Ratio Diagonal / Circle of
Confusion
1000
10
0.01
0.1
Distance subject-to-foreground[m]
Format 24x36mm f=85mm f/1.4
Format 24x36mm f=85mm f/2.8
Format 24x36mm f=85mm f/5.6
Format 24x36mm f=85mm f/8
Format 24x36mm f=85mm f/11
Format 24x36mm f=85mm f/16
This is how the corresponding curves for the foreground appear if we imagine once again that the
camera is situated on the left. At a distance of 1m from the foreground and 1.8m focus distance the
horizontal scale therefore commences 0.8m in front of the camera or, to put it more precisely,
before the sensor plane. There is no saturation to a limit in the close foreground; instead the curves
become increasingly steep; the blurriness becomes increasingly greater. It is thanks to this property
that it is possible to make filigree obstructions in the foreground, e.g. the wire mesh of a cage at the
zoo, disappear from view with lenses that are wide open.
Field width 200 cm
Ratio Diagonal / Circle of confusion
1000
100
10
0.01
0.1
1
10
100
Distance subject-to-background [m]
Format 24x36mm f=85mm f/1.4
Format 24x36mm f=85mm f/8
Format 24x36mm f=85mm f/2.8
Format 24x36mm f=85mm f/11
Format 24x36mm f=85mm f/5.6
Format 24x36mm f=85mm f/16
If the taking distance increases (in this case 4.8m is reached at an object field width of 2m), the
highest achievable blurriness decreases. In order to reach the same degree of background
blurriness as at closer distance, one has to use wider apertures or take care that the distance to the
background is larger.
Carl Zeiss
Camera Lens Division
27
Focus Distance 2m
Ratio Diagonal / Circle of confusion
10000
1000
100
10
0.01
0.1
1
10
100
Distance subject-to-background [m]
Format 24x36mm f=18mm f/5.6
Format 24x36mm f=50mm f/5.6
Format 24x36mm f=25mm f/5.6
Format 24x36mm f=85mm f/5.6
Format 24x36mm f=35mm f/5.6
Format 24x36mm f=180mm f/5.6
We are here comparing six different focal lengths in 35 mm format; each case is at the same
aperture 5.6 and with the same distance of the camera from the subject. The reproduction scales
for the pictures are therefore different. The two red diamonds on the vertical axis mark the circle of
confusion diameters “diagonal/1500“ and “diagonal/3000“ which have been assumed for the
calculation of the depth of field. The blue curve indicates that at a focal length of 18mm at least the
weaker of the two depth conditions is still maintained irrespective of the distance behind the subject
– the depth of field stretches into infinity. In the case of the other wide angle focal lengths, the
sharpness is no longer perfect in the distant background but neither has it fully disappeared.
Taking distance 2m
1000
100
1
0.1
Ratio Diagonal / Circle of
Confusion
10000
10
0.01
Distance subject-to-foreground[m]
Format 24x36mm f=18mm f/5.6
Format 24x36mm f=25mm f/5.6
Format 24x36mm f=35mm f/5.6
Format 24x36mm f=50mm f/5.6
Format 24x36mm f=85mm f/5.6
Format 24x36mm f=180mm f/5.6
The same lenses are featured here with respect to the foreground. If we compare the distance from
the main subject at which certain blurriness is reached in the upper and lower charts e.g. the value
1000, we then see that the spread before and after the focus is symmetrical with the longer focal
lengths, but is increasingly asymmetrical in the case of shorter focal lengths. The red curves for
35mm approximate very closely to the “1/3 in front – 2/3 behind“ rule.
Carl Zeiss
Camera Lens Division
28
Field width 70 cm
Ratio Diagonal / Circle of confusion
1000
100
10
0.1
1
10
100
Distance subject-to-background [m]
Format 36x48mm f=120mm f/4
Format 24x36mm f=85mm f/2.8
Format 15.6x23.7mm f=55mm f/5.6
Format 24x36mm f=85mm f/1.4
Format 15.6x23.7mm f=55mm f/2
Format 6.6x8.8mm f=21.6mm f/2
We have now returned to the first subject example but are now taking the photographs from the
same distance in different formats, in other words with different focal lengths but in each case with
the same angular field. If equivalent aperture numbers are chosen (see the table on page 10), the
attributes of the depth representation of different formats are identical and the curves are then
congruent. However, in the case of the very small format (2/3“) it is necessary to work with very wide
apertures and maintain sufficient distance from the background in order to achieve a good level of
blurriness.
Field width 100 cm
Ratio Diagonal / Circle of confusion
1000
100
10
0.1
1
10
100
Distance subject-to-background [m]
Format 24x36mm f=85mm f/2
Format 24x36mm f=180mm f/2.8
Format 24x36mm f=100mm f/2
Format 24x36mm f=180mm f/5.6
Format 24x36mm f=135mm f/2
Format 24x36mm f=300mm f/2.8
We are now comparing six different focal lengths and aperture values on 35 mm format. We are
therefore photographing from distances of between 2.5 and 8.5m. The first three curves (blue, green
and red) are all for aperture f/2; these curves all initially leave the zone of focus congruently and
therefore confirm that depth of field depends only on scale and the aperture figure. But at greater
distances behind the focal plane the longer focal distance creates increasing blurriness.
We also see the same if we compare 300mm and 180mm (black and yellow).
Carl Zeiss
Camera Lens Division
29
Field width 100 cm
Ratio Diagonal / Circle of confusion
1000
100
10
0.1
1
10
100
Distance subject-to-background [m]
Format 24x36mm f=25mm f/2
Format 24x36mm f=180mm f/2.8
Format 24x36mm f=35mm f/1.4
Format 24x36mm f=100mm f/2
Another comparison in 35 mm format with a larger range of focal lengths but each with the same
reproduction scale of the main subject: while the influence of aperture clearly dominates at very
small levels of blurriness on the left and determines the order of the curves, in the far distant
background the influence of focal length predominates. If the subject is to be truly separated from
the background, one ideally needs both – a longer focal length and a high speed lens.
All these curves of the large circles of
confusion can be easily understood if you look
back once again to page 11 and examine the
sketch you see there. In your imagination or on
a piece of paper let the point of the light cone
move behind the blue focal plane and see how
the cross section of the cone changes with the
focal plane. The cross section of the cone is
the image of the circle of confusion which
forms on the sensor.
The decisive parameter for the quantity of the
blurriness is therefore the physical size of the
entrance pupil. If by “bokeh’ you mean
principally the ability to be able to represent the
background as very blurred, soft and lacking
detail, it is necessary to have an entrance pupil
which is sufficiently large. A large photo format,
a high aperture lens and longer focal lengths
have the best potential in this direction.
There are lenses where the angle of the light
cone entering the lens from the subject is so
important that this information is written on the
lens barrel:
Carl Zeiss
The number 0.75 on this microscope lens for
a 20x magnification is the sine of half the light
acceptance angle of 48.5° and tells us that
the lens has a resolution limited by diffraction
at about 2300 Lp/mm with a minute depth of
field of 0.001mm.
As a comparator: a photographic lens with an
imaging scale of 1:10 and a nominal f-number
of 8 has a light acceptance angle of 0.6°; the
resolving power limited by diffraction is then
16 Lp/mm measured in the subject and the
depth in which this resolution is achieved is
20mm.
Camera Lens Division
30
Aperture iris images
It often happens that many circles of confusion
of similar brightness overlay each other and
intermingle in a picture in such a way that the
individual circle can no longer be recognised.
This causes the flat, smooth character of a
very blurred background. But sometimes one
point of the subject is much brighter than its
surrounding area - for instance, light sources
are reflected in glossy surfaces or drops of
water. In such a case the associated circle of
confusion is always accentuated beyond its
surroundings in the picture such that it is
possible to see its geometric shape. In this
case we can see that we are not always
dealing with circles because the entrance pupil
is an image of the mechanical iris blades.
Such iris images can be very decorative
items in a picture. It they are strikingly bright
they attract the view of the observer. A
‘beautiful’ geometry of the iris is therefore
desirable. But an iris image reminiscent of a
saw blade as in the example below on the
left is often perceived as disruptive.
The aperture of the lens determines the basic
area of the light cones which do not appear
exactly as the cones in our school books. We
therefore see the number and shape of the iris
blades if the sensor plane intersects with the
cone at a position where is cross-section area
is still very large.
At the edge of the photo iris images are
also altered by vignetting if the light cones
going to the edge of the photo at wide
apertures are intersected by front or rear
parts of the lens barrel or by the limited
diameter of rear and front elements:
Four examples of iris structures with 5, 6, 8
out of focus bright light source which is
depicted. The lens at the top on the left is only
stopped down a half stop from the full aperture,
which is why it is possible to see short curves
of the circular full aperture between the five
straight edges of the iris.
How vignetting becomes visible in iris
images: a circle in the middle of the picture
becomes a two-sided figure composed of
segments of circles at the edge; a pentagon
turns into a strange composite shape. Thus
it is only possible to see regular iris forms in
the overall surface of the photo if the lens is
stopped down so far that artificial vignetting
is no longer present.
Carl Zeiss
With a sufficiently large number of iris
blades and a suitable curvature it is
possible to come close to the ideal of a
circular aperture. Regular pentagons or
hexagons which were frequently seen in
earlier days are now felt to be too
‘technical’. But at the end of the day it is
naturally a matter of taste.
Camera Lens Division
31
However,
alongside
the
design-based
properties of the lens, quite natural,
unavoidable effects give reason that images of
highlight areas not only display perfect circles.
To be specific, if many highlight areas are
located in close proximity to each other – a
reflective water surface would be an example –
the bright areas created by each individual
highlight area overlap and the points of
brightness cumulate in these areas of overlap:
It is often possible to observe similar effects
in photos taken using flash photography
with digital compact cameras if there are
specks of dust floating in the air close to the
camera and these specks are illuminated by
the flash. They are very brightly illuminated
due to their close proximity to the flash but
at the same time are reproduced very much
out of focus. Their inner structure and
transparency therefore create in many
observers the perception of transparent
spheres floating in the room. If you search
the Internet for ‘orbs’ you will find dozens of
articles in which this phenomenon is
interpreted as mysterious ghosts. But in
reality the reason is a diffraction pattern of
the light waves travelling through the lens
surfaces.
If the iris images of out of focus highlight areas
overlap, these bright areas cumulate and
create new geometrical shapes in this way.
If you look at this picture very closely, you can
see another interesting effect – all the
defocused spot images contain a circular
structure. It is possible to recognise from this
that the lens has an aspherical surface as
these surfaces are often not as smooth as a
conventionally polished lens. Particularly in the
case of lenses which are manufactured by
pressing hot liquid glass it is possible to
recognise the traces of the turning process with
which the mould was manufactured.
It is possible to combine rotational and pivotal
movements when polishing spherical surfaces
because the curvature of the surface is the
same everywhere; in this case no traces are
left. In the case of aspherical surfaces the
curvature is variable and therefore demands
other
processing
techniques.
Residual
unevenness of these surfaces becomes visible
if a very small light source is reproduced very
much out of focus.
‘Light orbs’ from a compact camera with
integrated flash.
Such disruption to light waves is particularly
significant if “soft filters” are used on the
lens. In the case of the Zeiss ‘Softar’ filter
the effect is caused by small lens-shaped
bumps distributed across the surface of the
filter. These, too, are visible in the iris
images:
Iris images with the Zeiss “Softar’ and
Minolta “Portrayer’ soft filters
Carl Zeiss
Camera Lens Division
32
Sometimes the phenomenon of the individual
iris images is equated with “bokeh’; under this
heading one finds collections of pictures in
which iris images are mixed with photos of
soap bubbles. But this is not what is meant by
“bokeh”. In the iris image the lens is reading
the cards to a certain extent but what
significance has all this for the reproduction of
image areas in which there are no highlight
areas?
In the following examples of photos we will see
that one should not over-estimate the
significance of the shape of the iris:
The f-numbers chosen for these pictures
were exactly the same as for the iris images
on page 31. However here it is possible to
see the geometrical shape of the iris only
indirectly in the alternating wide and narrow
beams of light which radiate out from the light
source. These are caused by the diffraction of
the light at the edges of the iris blades.
The geometrical shape of the bright disc of
the image of the light source does not betray
the exact shape of the iris. This is because in
the slight out of focus which has been set
here, the circular illuminated surface of the
light source is still relatively large in
comparison to the tiny pentagonal image of
each individual point on the illuminated
surface. Therefore the image appears to be
fairly round. But this changes if the out of
focus is increased
Test subject: two flowers which are not wilting
under a spot-light, slivers of wood and metal
knitting needles as a model for blades of grass
and, in the background, a small, bright, circular
light source as an iris indicator.
Test subject photographed slightly out of focus
with the lens with 5 iris blades (see page 31)
Carl Zeiss
Test subject photographed very much out of
focus. Above, the focus is closer to the
foreground; below, the camera is set to
“infinity”.
Camera Lens Division
33
Thus whether we see the shape of a bright
object or the shape of the iris depends on a
size ratio. An object which is practically dotshaped always shows us the shape of the iris
if it is out of focus. Conversely the outline
shape of a rather larger object always
dominates if the image is only slightly out of
focus. In between there is a transition zone in
which both shapes are mixed.
It is apparent in the pentagonal diaphragm
images of the last two photos that they are
reversed to each other. The reason is because
in the upper photo the sensor plane is behind
the focus and in the lower image it is in front.
Behind their point of intersection with the focus
all the light beams exchange their position in
the light cone.
Except in the picture of the highlight areas we
do not find the shape of the iris in any other
element of the image. Lines and long edges
particularly generate an image of many
highlight areas blurred in one direction – the
shape of the iris is unimportant in this.
Only in the slightly out of focus iris image at
the bottom of the previous page can we see a
couple of gentle highlight areas at the edge of
the flowers showing the pentagonal shape of
the blades. They disappear as the image goes
further out of focus because the amount of
light in the point is then distributed across
such a large area that it is no longer noticed.
In summary we can say that the shape of
the iris can become visible in the picture
either obviously as a decorative feature or
as a disturbing artefact and that it can betray
interesting facts about the lens to us.
However, the iris can remain totally invisible
in many pictures. Yes, and if we use a lens
with the aperture fully open, it can of course
play no role at all.
Carl Zeiss
Nevertheless or perhaps in just such a case
there can be major differences in the bokeh.
Camera Lens Division
34
The Nature of Blurriness
We have come to understand the basic
characteristics of depth of field with the aid of
a little geometry: we have taken a look at light
cones that are intersected in different places
by the sensor of the camera (see pages 4 and
6). The intersections are the circles of
confusion, and so far we have always
assumed that they appear as homogenous
light disks.
If that were true, then the circles of confusion
would only depend on the purely geometrical
factors of the lens that can be entered into a
depth of field calculator, for instance. All
lenses would then have to be the same when
using the same f-number and same focus
deviation.
We know, however, that lenses are not at all
alike at their best focus, especially not with a
wide aperture. Differences in contrast and
sharpness naturally occur in such cases. But
why should these differences disappear
completely if we compare them at a
deviation from the best focus? The
measurements of the contrast transfer
depending on the depth in the image have
already shown us how different lenses can
be, not only at best focus but also at the
calculated limit of the geometric depth of
field (see page 23). Let us find out why that
is the case.
The geometric theory of depth of field is an
idealized model that does not take
aberrations into account; it simply assumes
that all light cones intersect at one point:
25
Half Exit Pupil [mm]
20
15
10
5
0
This graphic is simply a somewhat more abstract version of the sketch on page 4. Because
the lens is symmetrical, we are just looking at one half of the light cone to save space. We
have drawn 20 rays of light that are travelling from one half of the exit pupil and all intersect at
one point. The dimensions of the exit pupil are typical for a 1.4/50 mm lens.
We have highlighted two rays of light in particular: the marginal ray is marked red: the ray
marked blue that is more on the inside is travelling from a point of the pupil plane that is 14
mm away from the optical axis. If the aperture is narrowed from 1.4 to 2.4, the blue ray will
become a marginal ray and the rays on the outside will be blocked by the iris blades.
Carl Zeiss
Camera Lens Division
35
If we magnify the surroundings of the intersection, we see the ideal conditions: all rays
converge on the focal point in orderly lines travelling alongside each other, intersect there at a
single common point, and then leave the focal point in just as orderly of a way on the
backside. That is how we have always imagined it in all of our depth of field calculations - but
it is too perfect to be true.
A real lens can also look like this. The rays from different heights of the pupil no longer have
the same point of intersection, but rather each zone of the pupil has its own point of
intersection. They are all on the optical axis, but are at different distances from the lens. The
focus of the marginal rays is not as far away; the rays travelling with a flat slope that are close
to the optical axis intersect at the black point further away.
This image defect is called "spherical aberration." Because the point of intersection of the
marginal rays of light is closer to the lens for simple collective lenses and this natural defect is
similar to above example, the type described above is called "spherically under-corrected."
The greatest constriction of the double cone is in front of the black dot, and that is where the
best focus is at full aperture. If the aperture is narrowed, the focus moves to the black dot - the
lens has a positive focus shift.
Carl Zeiss
Camera Lens Division
36
There is a particularly interesting point further
to the left in the graphic above, about 0.4 mm
in front of the focal point of the paraxial rays:
there, the marginal rays seem to overtake
those travelling more on the inside. The light
cone is no longer ideally arranged, and we
could say that the rays of light are "confused."
This is the original meaning of the
Japanese word "bokeh."
There are no rays that intersect or overlap
behind the focal point. Quite the opposite;
the density of the rays on the outside is
somewhat less than in the ideal geometric
light cone. The circle of confusion is
therefore larger behind the focal point than
in geometric theory, and the brightness
decreases moving outward from the inside,
while the circle is smaller in front of the focal
point and is clearly bordered by a bright ring
around the outside.
There are so many rays that overlap in this
zone of intersection that a ring with increased
brightness results. This means that the circle
of confusion is not a disk with homogenous
brightness.
In practical photos, that can look like this:
Foreground blurriness with Sonnar 1.5/50 ZM,
a spherically under-corrected lens.
Carl Zeiss
On the outside, the circle of confusion has a
thin green border because on the outside we
see the rays of light whose focal point is
closest to the lens. Since green light has the
closest focal point in the normal chromatic
aberration it dominates the cover surface of
the light cone behind the focus.
Background blurriness with Sonnar 1.5/50
ZM
Camera Lens Division
37
• If we want to generate a noticeably beautiful
bokeh in the background, then we must
make the under-correction so noticeable that
the focus shift is also very large and makes
focusing difficult.
• In addition, the contrast rendition of the lens
is overall poor by necessity. Because the
outside rays form a halo surrounding the
spot where the inner rays form a small
image point, the contrast is reduced.
We must make use of this characteristic
moderately with lenses intended for general
use and have to limit the spherical undercorrection. In any case, we should avoid
spherical over-correction. This is not to say
that the lens is now better than good overcorrection just means that the spherical
aberrations now have a different signature.
The marginal rays then intersect far behind the
focal point of the paraxial rays. The bokeh
characteristics are then simply reversed. The
foreground
characteristics
with
undercorrection are found in the background in case
of overcorrection. And because background is
almost always more important, it would be the
less desired balancing of the lens.
But even spherical aberration which remains
completely within the range of the mild undercorrection, yet shows clear signs of the
measures that are intended to limit the growth
of the spherical aberration. It can already
cause a slight increase in the outward
brightness. That is why lenses with larger
apertures are usually not completely free of it.
They should not be compared to lenses with a
more modest maximum aperture where the
spherical correction is much simpler either.
Carl Zeiss
20
15
15
h [mm]
20
0.00
0.00
ds' [mm]
-0.05
0
-0.10
0
-0.15
5
-0.20
5
-0.05
-0.10
-0.15
10
-0.25
h [mm]
-0.20
• The more appealing the blurriness is in the
background, the less appealing it is in the
foreground. There it often seems harsh and
disturbing. It generates swirls of small
highlights and transforms lines into double
lines.
25
10
-0.25
There are disadvantages to this imaging
property, however:
25
ds' [mm]
Common diagram of the longitudinal spherical
aberration in optics: the vertical axis shows the
starting point of a ray in the pupil plane,
expressed by the distance from the optical
axis, and the horizontal axis shows the
deviation from the focus position. The
directions correspond to the graphics on the
previous pages. The diagram on the left
shows a strongly under-corrected lens.
But even with the well-corrected lens on the
right, the brightness profile of the circles of
confusion already shows a "thin ring"
emphasizing the edge:
0.6
The nature of the background blurriness of a
spherically under-corrected lens is appealing
to the human eye. The background is calming
and the contours of the object are retained
longer even in the blur. Further below you will
find examples illustrating this.
0.5
0.4
0.3
0.2
0.1
0.0
-1
-0.5
0
Brightness rel. ideal [EV]
Brightness variation for circles of confusion of
various sizes in the background with a lens
showing mild spherical under-correction. The
more it is defocused, the smaller are the
deviations from the ideal disc with
homogenous brightness throughout.
Camera Lens Division
38
25
25
20
20
15
15
h [mm]
0
0
In reality the brightness gradient at the edge
of the blur circle is not as high as shown in
above charts. They had been calculated for
a single wavelength, but in reality different
colours have different circles, which makes
things a bit more smooth. Real lenses have
as well a longitudinal chromatic aberration;
the focus of the rays depends on the
wavelength. And these deviations are of
similar amounts as the spherical ones:
0.00
-0.15
ds' [mm]
-0.05
5
ds' [mm]
ds' (Lambda) [mm]
0.6
But if the lens is stopped down one stop
(charts on the bottom and above on the
right-hand side), the reversal point of the
longitudinal spherical aberration is excluded
and the brightness profiles look pleasant
again.
In this example we can also see that the
brightness profiles become flatter when the
image is blurred more.
10
5
0.00
-0.05
-0.10
-0.15
10
-0.10
h [mm]
Somewhat stronger counteractive measures
towards spherical overcorrection strongly
increase
the
brightness
around
the
circumference of the circles of confusion:
0.5
0.4
0.3
0.1
0.08
0.06
0.04
0.02
0
-0.02400
-0.04
500
0.2
600
700
nm
0.1
0.0
-1
0
1
2
3
4
Thus the focus deviations of the colours are
different and the according circles of
confusion have slightly different size:
Brightness rel. ideal [EV]
0.50
0.6
0.5
0.4
0.3
0.2
0.40
0.30
0.20
0.10
0.1
0.00
-1
0.0
-1
-0.5
0
0
1
Brightness rel. ideal [EV]
Brightness rel. ideal [EV]
Carl Zeiss
Camera Lens Division
39
This combined effect of different aberrations
causes that in the background of the image
the inner area of the blur circles is dominated
by the colours which have the longer focal
distance. These are usually the colours from
the ends of the visible spectrum which mix to a
purple shade. The edge of the blur circle is
dominated by the colours from the middle of
the spectrum. This explains the green fringes
which we see in the blurred image of a white
spot.
bokeh:
•
Some caution is advised when making
on the lens correction, because bokeh is
extremely variable.
•
The correction balancing has an
especially strong influence on the
blurriness of the rendition at small
deviations from the focal point. If there is
a lot of blurriness, it usually becomes
more and more negligible.
•
The aperture has a strong influence;
even closing the aperture a small
amount can cause very visible changes
to the nature of the blurriness. Slower
prime lenses generally have smaller
spherical aberration by nature. So it is
no wonder that their bokeh is praised for
its appeal.
•
The spherical aberration of a lens also
changes depending on the imaging
scale. Bokeh characteristics therefore
depend on the focusing distance as
well.
Two examples of colour-bokeh near the focus.
Above all glossy details are in the background,
below they go through the focus. There you
can observe the reversal of the colour effects
in front of and behind the focus. That one sees
only the green fringes but not the purple core
is due to the too high brightness of the
highlights.
As with the brightness profiles of blur circles
the handwriting of the lens with respect to
colour-bokeh disappears more and more,
when it is strongly out-of-focus, or when it is
stopped down.
Carl Zeiss
Camera Lens Division
40
After working through many difficult diagrams, we will now relax and use a few example pictures to
illustrate the influence of spherical aberration, which should also invite you to have a look to the image
On the left is a focus series of images showing blurriness in the background using a lens with normal
spherical correction while the right side is using an overcorrected lens. Its characteristics: a veil is
present at the best edge sharpness (top picture), many artefacts can be seen in the blurry picture.
Carl Zeiss
Camera Lens Division
41
On the left is a focus series of images showing blurriness in the background using a lens with normal
spherical correction and five iris blades, while the right side is using an under-corrected lens. Its
characteristics: a veil is present at the best edge sharpness (second picture from the top), the contour
of the triangle remains for a long time. Because of the veil of the spherical aberration, the very bright
highlight appears much larger.
Carl Zeiss
Camera Lens Division
42
A 700x820 pixel crop from a 24MP-image, above with strong spherical aberration, below with
good correction. The hatching in the hair is at about 40 Lp/mm on the sensor.
Carl Zeiss
Camera Lens Division
43
The same lenses as the previous page, now defocused by 1.5x depth of field,
the subject is in the background. With a lens that is poorly corrected, the change in sharpness
and contrast is significantly less - but with compromises at maximum picture quality.
Carl Zeiss
Camera Lens Division
44
Image 1 to 7
These images show the Zeiss factory in Oberkochen with a different amount of blur, achieved by
combining different focus settings and f-numbers. The lens was a Planar 1.4/50 on an APS-C camera. In
some images a red arrow marks a prominent highlight where the sun was reflected in the screen window
of a car. Two blue arrows mark the width of a bright structure.
•
•
•
Image 1
Image 2
Image 3
Best focus shot
Circle of confusion is about 1/1000 of the image diagonal
Circle of confusion is about 1/200 of the image diagonal
•
Image 4
Circle of confusion is about 1/90 of the image diagonal;
The exposure with f/1.4 shows the bright green fringe at the edge of the blur
circle due to spherical and chromatic aberration
•
Image 5
Circle of confusion is again 1/90 of the image diagonal;
The exposure was now with f/11. The bright fringe at the edge of the blur circle
of the highlight has nothing to do with aberrations of the lens, it is caused by
diffraction.
•
•
Image 6
Image 7
Circle of confusion is about 1/45 of the image diagonal
Circle of confusion is about 1/10 of the image diagonal
Image 8 to 11
Dead leaves are shown with different blur. We would certainly prefer the very sharp or the very soft
version as an image background. The other two don’t appear calm, somehow noisy. But they had been
taken with f/2.8 and f/11, where unfavourable features of the bokeh are of very low importance. This tells
us, that unpleasant backgrounds have sometimes nothing to do with the lens.
Image 12
Two images with different background bokeh show a zoom lens at long focal length on the left and a
prime lens Makro-Planar 2/100 on the right. Both shots taken at f/5.6
Image 13
This is an arrangement of small crops from 24MP images of a full frame camera. The original detail is
5cm high and was imaged at scale 1:10. Thus the hatched structure in the hair of the lady is in the
image at around 40 Lp/mm. Each horizontal line of nine images is a focus sequence: on the left side the
camera is most close to the subject, to the right the subject moves through the focus into the
background. The distance step between neighbouring images is 4mm, hence 0.04 mm on the image
side.
The images of the top row have been taken with the Makro-Planar 2/100 at f/2.8. Calculating with a
permitted circle of confusion of 1/500 of the field results in a depth of filed of 1.9cm. Two images on the
left and the right of the centre image should be within this range. If you look carefully you can see that
only the two nearest neighbours fulfil the most demanding sharpness expectations, the two outer ones
In the second row a lens was used which was assembled in such a way that it had considerable
spherical aberration. It shows much less sharpness and brilliant contrast, but at the same time it shows
less variation over the depth. Highly corrected lenses have a more sudden transition from sharpness to
blur.
The tree lower rows compare three lenses at f/1.4, where the nature of the blur behind and in front of the
focus is very different. Especially in the background the resolution of some detail can be maintained over
a larger range than compared to images taken at smaller aperture f/2.8. This demonstrates the limits of
all simple calculations about depth of field.
Image 14 to 17
Carl Zeiss
some illustrations of the text as jpg-file.
Camera Lens Division
45
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