Depth of Field and Bokehby

H. H. Nasse

Carl Zeiss

Camera Lens Division

March 2010

Preface"Nineroundeddiaphragm

bladesguarantee

images with exceptional bokeh"
Wherever there are reports about a new camera lens, thissentence is often found. What characteristic of the image isactually 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 thatyou will not be bored. I will use a rather unusual method toshow how to use a little geometry to very clearly understandthe 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 tounderstand the connections and learn a few practical rulesof thumb. You will find useful figures worth knowing in a few
graphics and tables.
Then it only takes another small step to understand what isbehind the rather secretive sounding term "bokeh". Bothparts 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 oflenses 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 besaid here.
There are also plenty of pictures for illustrating this topic for
those who do not want to get deeply involved in the theoryof their camera, so we really wish everyone a lot of fun withthe reading.

Carl Zeiss

Camera Lens Division

March 2010

"Schärfentiefe" or "Tiefenschärfe" for depth of field?
When searching the net, there is aseemingly endless amount of entries
about our topic and much of what is there
for reading is of course incorrect orincomplete. It is therefore not surprisingthat photography forums like to spend somuch time discussing it.
There was a particular increase in theinterest 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 thelens still applies or not.

In the German forums we even find someheavy debate about the proper term for thedepth 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 avery precise feature anyway. Both terms
have been in common use for a while now.
And both refer to the same characteristic ofphotographic imaging - namely that a clear
two-dimensional photographic image can bemade of objects in a three-dimensional
space under certain conditions, eventhough 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. Thisdebate 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 behindthe 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 bringthe 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 theacceptableblurrinessandistherefore essentially based on
arbitrary specifications. But it is not
the case that the sharpness of theimage 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 withthe distance of the object.

Camera Lens Division3

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 tocontradict that. After all, practicalexperience has shown us that wide-angle
lenses make images with a large depth
and telephoto lenses have a rather
selective sharpness. Despite this, theperson making the original claim may beright, 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 depthof 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 spaceinside 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 tounderstand:
Every picture element is generated by alarge number of beams of light that shine
through the aperture and combine in the
picture element. In doing so, they form alight cone whose area is the image of theaperture 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 toa 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, theresulting intersection is the circle ofconfusion marked red in above drawing.
The total image-side depth of focus (theblue section of the image space in the
diagram above) is twice the product of thediameter of the circle of confusion (z) andthe f-number (k):depth ⋅ of ⋅ focus ≈ 2 ⋅ z ⋅ kThis simple equation can be seen in theengraved depth of field scales:
The rotary angle on the focusing ring is
proportional to the image-side focus
adjustment
andthedepth-of-field
markings on the lens barrel are therefore
proportional to the f-number.
The f-number is the ratio of the distancefrom the image plane to the exit pupil andthe diameter of the exit pupil. The angularaperture of the light cone therefore only
depends on the f-number.

Carl Zeiss(Strictly speaking, the image-side depth offocus behind the image plane is justslightly larger, but this can be ignored.)

Camera Lens Division

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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 areproportional 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 theleft 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 withthe depth of the three-dimensional image.
Short focal lengths only have a very shortfocus 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 requirea 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 forvery short focal lengths and then the entire
motif will appear to be slightly blurry. With along 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 tothe misconception that short focal lengths havea short image-side depth of focus.

Carl Zeiss

It is true, however, that the depth of fieldin the object space is also (almost)independent of the focal length if wecompare the respective imaging of theobject at the same imaging scale. For
photographs with different focal lengths
and the same image format, of course,
this means that the photographs are takenfrom 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 atcloser taking distance, two photographs ofan object will not be identical if they aretaken with two different focal lengths, even
if the depth of focus is practically identical.
Besides the perspectives, the maximum
blurriness of the distant backgrounddiffers. 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 actuallyarise and take a look at the space in frontof the lens in order to understand why thatis the case.

Camera Lens Division5

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 thesensor 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 thespace 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 offocus 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 ofconfusion 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 basearea is the entrance pupil. That is the imageof the aperture that we see when we look at abright surface through the front of a lens froma certain distance:

Carl Zeiss

The entrance pupil can also be located far
in the back of the lens, so we should not befooled by its name. In the case of the long
Tele-Tessar lenses for the Hasselblad, theentrance pupil is in the film magazine.
A virtual plane in front of the lens within thefocus 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 ofsomething that is not even physicallypresent is the trick to simplifying theconcept.

Camera Lens Division6

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 theimage space on the left, closer to the lens. The
blue line in the object space is the image of thesensor 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 acorresponding 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, thenthe 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 lightcone?
Conditions 2 and 3 determine how narrow an
object-side light cone is. And condition 1 thendetermines the relative size of the intersectionof 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 along 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 hf − number

With a little geometry, we can now easily
see how the depth of field depends on thetaking distance, the focal length and theaperture:
1. Distance
If we double the focusing distance, the sizeof the object field in the focal plane also
doubles - not its area, but rather the width,height and diagonal lengths. At the sametime a light cone from a point behind the
focal plane will be twice as narrow, because
the base area remains the same and weinfer the length of the cone. As a result, theratio of the field diagonal and circle ofconfusion 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 thesame time, half the focal length means half
the diameter of the entrance pupil, whichthen makes the light cone twice as narrowfrom a point behind the focal plane. As aresult, the ratio of the field diagonal andcircle of confusion becomes four times aslarge as before or, in other words: thedepth of field with equal focusing
distance is inversely proportional to thesquare of the focal length.

3. Aperture
If we stop down the aperture of the lens, wereduce the area of the entrance pupil. Its
diameter decreases by a factor of 0.71 witheach 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 thesame, 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 Division7

Makro-Planar 2/100 ZF Format 24 x 36 z= 0.029 mm D/1500
0.1

1Distance [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

1Distance [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 oneach axis is divided logarithmically so that distance always changes by the same factor for eachequally 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 andthe 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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Now we can also just as easily explain what
happens during a change in the film format:

4. Smaller film format
with the same lensIf we remove a lens from an old analogue
camera and attach it to a digital camera of thesame 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, thelens does not know how much of its image
circle we are capturing with our sensor.
The size of the object field is reduced by thecrop factor while the object-side light cones
remain the same, as long as we use the samelens 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 wewant to maintain the same ratio of diagonal tocircle of confusion. Reducing the size of thefilm format therefore reduces the depth offield by the crop factor.

5. Different film formats
with the same object fieldIf we select the suitable focal length to ensurethat we always display the same field withdifferent film formats, then things go just theother way round: reducing the size of thesensor 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 asmaller 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, eventhough 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 haveto calculate with the 0.03 mm of the smaller
format.
The acceptable diameter of the circle ofconfusion 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 aparadoxical characteristic whereby large
formats have a smaller object-side depthof field and simultaneously a larger imageside depth of focus with the same aperturesand object fields. This is also reflected in
the mechanical tolerances of cameras:
Large-format cameras can be built withcarpenter precision, and the camera modulein a mobile phone requires µm (micrometer)
precision. Those are the extremes, but in
SLR photography we can already see thedifference between APS-C and full-frame
format with regard to the requirements forfocusing accuracy.
It appears to be a confusing paradox at firstglance, but of course it has a very simple
explanation. We just photographed object
fields of the same size with different sizes ofimage formats. If the acceptable blurrinessis supposed to be the same with thesedifferent cameras, it means that the ratio of
the object field diagonal and the "object-sidecircle of confusion" should be the same.
The object-side light cones travelling from apoint behind the focal plane, for example,
should therefore be the same for allcompared cameras. If the images havedifferent 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 thesame if all entrance pupils are of the same
size, however. But because object fields ofthe same size mean longer focal lengths forlarger image formats, the f-numbers must
be different.

Camera Lens Division

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The big format comparison
We now know that the depth is only dependent
on the size of the entrance pupil if we have thesame distance and the same angular field. The
pupil diameter is the quotient of the focal lengthand the f-number.

An aperture of 2.8 in 2/3" format therefore
approximately corresponds to an apertureof 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, aslong as we have the same angular field.

If the focal length then changes by a factor
determined by the image format, we only haveto 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 havethe same depth of field relationships.

The widely spread practice of describing theangular field of lenses by calculating theequivalent 35 mm focal length is therefore
inconsistent if it does not convert theaperture 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 novariation possibilities of the depth of field
and hence the look of images.

Diagonal [mm]Formatk/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 aspectratio, 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 Division10

Depth of field with the same imaging scale
The imaging scales are different in eachcolumn in the format comparison on theprevious page because we are looking atdifferent cameras. In our photographic practice
it is more common that we have one single
camera and different lenses for it. For thatreason, we are sometimes faced with thequestion of which focal length to use. The
decisive criteria are the room conditions,
intended perspective, and background.
Are there also differences with regard to thedepth of field if we want to display a motif in thesame size? Would the 2/50 or 2/100 macrolens be better, for instance?

The depth of field (almost) does notdepend on the focal length at all but
rather on the imaging scale, and we canunderstand that as follows:
A focal length that is twice as long creates
an image of the same size from anapproximately doubled distance, and withthe same f-number its entrance pupil
diameter is twice as large. Because of theincreased focusing distance the object sidecone of light is nevertheless the same. As aresult, the “object side circles of confusion”
are also the same.

However:theinfinitely
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, andtherefore 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 samefor 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 offield 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 intwo.
But the same rules apply in the format
comparison for the hyperfocal distance, theshortest focus distance where the depth of fieldreaches infinity. We can easily understand thiswith 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 sameas the diameter of the entrance pupil. The
hyperfocal distance is therefore the distance
where the acceptable "object-side circle ofconfusion diameter" is as large as theentrance pupil.
And once again the rule applies that thesmaller sensor format has the smaller entrance
pupil if it has the same angular field and thesame aperture. The acceptable object-sidecircle of confusion is therefore already insmaller object fields, meaning it is reached at ashorter distance.
Looking at the cones of light we can easily seethat the front end of the depth of field is
located at half of the hyperfocal distance. Thatis because the beam cone, whose rear
extension is as large as the entrance pupil in
the hyperfocal object plane, has its point rightin the middle between the entrance pupil andthe 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 cancalculate them in our head:

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 theimage-side circle of confusion z’ results from it throughthe 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-sidecircle of confusion, Dist = distancefield at hyperfocal distancefront edge of depth

Carl Zeiss

Camera Lens Division12

It is especially easy to do calculations with these sizes if we relate everything to the diagonal of thesensor 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 inour heads:

D D ⎛ f '⎞Dist hyperfocal ≈ ⋅ ⋅ ⎜ ⎟z' k ⎝ D ⎠2

A 35 mm format lens with the focal length f’ = 85 mm and f-number k=2, a sensor diagonalof 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 longas the sensor diagonal; the f-number 2 is about 1/20 (=0.05) of the diagonal: so we can find thehyperfocal distance by starting from 2 on the horizontal scale and moving upward until we reach thethin yellow line for k/D=0.05.
The hyperfocal distance is often underestimated; in order to check whether the infinity alignment ofa 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

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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 ourconception 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 theacceptable 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 thehyperfocal 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

100depth=0.2m

depth=1m
depth=2m
depth=5m

10depth=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 thehyperfocal distance, then the depth of fieldis 1/5th of the focus distance.""If the focus distance is 0.4 times thehyperfocal distance, then the total depth offield is of the same amount as the focus
distance.""If the focus distance is one third of thehyperfocal 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
distances and gradually becomes lesssymmetrical at longer distances, which is very
obvious when we approach the hyperfocal
distance.
There is a relationship between the distancefrom the camera to the near and far limits ofthe depth of field and the focus distance thatapplies to all apertures and distances:

Dist =

Carl Zeiss

2 ⋅ Near ⋅ Far
Near + Far

To put that into words, the focus distance isthe product of the near limit and the far limit
divided by the average of the near limit andfar limit. (Also called ‘harmonic mean’, forexample: 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 thefront:back relationship is only 1:2 if the
distance to the far limit is twice as far as tothe near limit. In other words, the total depthof field is as large as the distance between
the camera and the near limit.
For those who enjoy the beauty ofmathematical relationships, it should be
noted that this is the precisely the case forthe distance where the size of the "objectside circle of confusion" is 1/3 of theentrance pupil, therefore 1/3 of therespective hyperfocal distance.
For a 50 mm lens with 35 mm film format
having a circle of confusion of 0.03 mm andaperture 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 andmacro range in particular is very
symmetrical. Reversing the lens does notchange anything about this either, but rather
only influences the correction condition.
If we use relatively long focal lengths with avery large hyperfocal distance, then wemust 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 macrolenses or close-up accessories like extension
rings ore bellows, depth of field becomes verysmall. The scales engraved on the lens mountthen 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 fieldskip over macro photography because theusual formulas and tables for long distances do
not apply in those cases. At close range, thelens no longer actually has the f-number that isengraved on the ring; we have to calculate
using the effective aperture - some cameras
display it, others do not. The amount that thiseffective aperture value deviates from thenominal value depends not only on the scale
but also on the construction of the lens.

Telephoto lenses show a heavier loss of theeffective f-number at close range thansymmetrically constructed lenses do.
Modern macro lenses have lens groups thatmove relative to each other in order to keep
the correction stable at all distances. As aresult, their focal length also changes withthe focusing. So there are plenty ofcomplications.
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 readerswith the most important figures for our two
2/50 and 2/100 macro lenses for 35 mm
format, first as a graphical overview andthen 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 tothe one on page 8, although here the depth offield is not displayed over the focusing distancebut rather the imaging scale or magnification of
two lenses at the same time. The fact that thesame imaging scale is achieved from different
distances can be seen with the two black lines.
The yellow, green, and red lines show thedepth of field for full aperture, f/8 and f/22 forthe 2/50. The values for the same apertures ofthe 2/100 are drawn as dotted black lines.
These lines are congruent almost everywhere another nice proof that the depth of field ismostly only dependent on the imaging scale.
There are only deviations at the ends: on theright side at magnification 0.01 and at f/22, therear 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 1/3of an aperture stop higher, as we can see withthe 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 apertureis only reduced to f/3.2. This difference
between our two macro lenses is an indicationthat depth of field does not come free and wehave to pay for it with exposure time. In fact, avery fundamental general physical law isbehind it: the law of conservation of energy.
That is because the angular aperture of theobject-side light cone also determines how
much optical radiant energy enters into thelens. And only this energy can be distributed
onto the image surface. If we compare two
images of the same sensor size, then the onethat needs a longer exposure time with thesame sensitivity has the larger depth of field
because it has collected less energy on theimage-side with a narrower light cone (we mustof course rule out absorption by filters etc. - weare only dealing with geometric efficiency). The
specific optical construction of a lens istherefore in the end meaningless for the depthof field.

Carl Zeiss

Telephoto lens constructions lose more lightat close range; that is because theirentrance 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 aswith 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 offield 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 thesame emulsions in analogue photography,
this meant that small formats were always
advantageous if a large depth of field had tobe achieved at fast shutter speeds. If thesignal-to-noise ratio of the sensor increases
with the format size, which is to someamount the case with digital cameras, thenwe can compensate for the increased needfor 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 canbe any length, then there is no difference atall between different film formats withregard to the maximum achievable depth offield.
Because light travels in waves, thediffraction determines how far we can close
the aperture without losing picture quality inthe 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 fieldat the diffraction limit.

Camera Lens Division17

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

Total Depth-of-Field [cm]
5.6
8
11

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

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 tolimitations by diffraction. This means that even with 24MP full frame cameras there is only a verysmall loss of sharpness that can still be balanced out with digital edge enhancement. Combinationsof scale and f-number that no longer meet this requirement are listed in gray in the table. The depthof 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 havebeen calculated assuming a circle of confusion
diameter that fits 1,500 times into the diagonalof the image. But we must explain why this size
is so often chosen and why we should
sometimes choose another one. Depth of fieldis 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 handthe eye will perceive an image as blurry if theeye is capable of seeing significantly more thanis shown. The resolution that the eye canrecognize 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 theeye. At longer distances, of course the eye isless capable of recognizing as much; at two
meters away it is barely possible to distinguisha 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 theeye independently of its distance from theobject, then we use the angular resolution. It
thereby matches the numbers above, that theeye can distinguish the smallest details fromone another if they appear at a visual angle of
one arc minute. This is the physiologicalcritical 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, thenwe 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 withjust a few mouse clicks. Then we canalready view 1/3000 of the picture diagonal
at a visual angle of four arc minutes if weare still viewing the image from 25 cm
away; the eye can then see much smaller
details.
However, the entire image width thenappears to us at an angle of 110°; wecannot overlook that entirely and still see
the smallest details everywhere in theimage at the same time. If we look at it inthis 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 thestrictest 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 canno longer be met with this usual circle ofconfusion 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 Division19

In the 50s, the depth of field for 35 mm lenses
was often calculated with a circle of confusionof 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 thesomewhat 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 findreasons for to use different circle of confusion
sizes for different focal lengths of a camera:

If we view images "from the rightperspective", meaning closer to the sameangle at which they were really viewed by
the camera when they were taken, then wemust view the wide-angle images from acloser 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 morediscerningly for the DISTAGON 4/40 fromthe old C series for the HASSELBLAD thanfor all other lenses in the series. Becauseeven without viewing them from the rightperspective, the details that interest us inwide-angle images are usually smaller andtherefore place stricter requirements on the
sharpness of the image.

How precise are tables and depth of field calculators?
Usually most tables pretend to have aprecision that is neither available nor sensible
in reality. That is partly because the values
calculated in the tables are based on thearbitrary 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 basedon the image content in addition to the viewing
conditions. There is therefore no clear limit!
The from-to tables with millimetre precision atmeter distances, on the other hand, easily givethe impression that there are two precisely
positioned flat limit surfaces in front of thecamera 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 modelof 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 idealizationof the real optical processes in a lens. That is
because this model does not recognize
aberrations, colours, or diffraction. In thegeometric model, the circle of confusion is adisc with even brightness.

Carl Zeiss

In reality, however, the distribution ofbrightness in the focused and lightly
unfocused point image is uneven. We willdeal with that in more detail below. All typesof aberrations from real lenses cause aseries of deviations from the geometricmodel:
•

•

•

•
•

In ideal lenses, the image-side depth offocus grows nearer and farther away at thesame rate when the aperture is madenarrower. In real lenses, however, there canbe a displacement to one side called thefocus shift. When it is very large, the nearlimit of the depth of field range might remainunchanged when the aperture is narrowed.
Away from the optical axis, this shift often
has opposite direction compared to thecentre of the image, the depth of field spaceis then curved.
If lenses suffer form vignetting by parts of thelens barrel, then the depth of field at theedge of the frame is larger than in the centre
because the size of the pupil decreases dueto the vignetting.
Depending on the lens aberration, the typeof blurriness can be different in front of andbehind 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, butthey should not be taken too seriously.

Camera Lens Division20

Depth of field and modulation transfer MTF
The meaning of depth of field can also be
understood if one measures how the contrastof the image in terms of the modulation transfer
MTF (see CLN30 and CLN31) changes withdeviation from the best focus. Curves that arealmost bell shaped are found in this measuring
process; these curves show very well that the
sharpness of the picture is in no way constantwithin the depth of field but is continuously
changing. The curves also show that not somuch remains of the best performance of alens 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 fromthe maximum to about 20-30 %.

Biogon 2/35 ZM

Such measurements of MTF related tofocus also indicate the limitations of thesimple geometric model as an explanationof 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 thesame f-number, the depth of field varies insize, not because we are looking at different
spatial frequencies because of different
formats, but because the curves of thesame 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 Zeisstan 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 distancefilm or sensor
depth of field

Camera Lens Division

21

The black curve shows the relationshipbetween the distances in the object space andthe associated positions in the image spacewhen 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 theimage-side depth of focus between the black
triangle marks goes with the object-side depthof field between the blue marks; these can also
be seen in the picture below showing the depthof field scale on the lens barrel.
If you read off the MTF values at the limits ofthe depth of field area near the triangularmarks, you will still find 10 to 20% at 40
Lp/mm. If one then takes account of theadditional 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 somethingfrom a distance of 25 cm; the image is still
perceived as being sharp. However, in thecase 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 theimage and the lens distance applies forfocusing 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 distancefilm or sensor
depth of field

Compared with the geometric depth of focus
marks the curves have become a shadenarrower. At the same time the maximum
MTF values at the best focus are somewhat
higher. It is due to the incipient influence ofdiffraction 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 theexample of the Sonnar 2/85 ZM. This istherefore proof that the image-side depth offield 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 spaceposition, now looks quite different. It is now
much flatter because the image of the same
deep object has greater depth than with ashorter focal length. Therefore a smaller
object-side depth of field now goes with thesame 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 distancefilm 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'= 0100
MTF [%] 20, 40 Lp/mm

The position of each measuring point in theimage space is indicated by the focus
deviation on the horizontal axis; negativevalues are closer to the lens. At zero we haveno deviation and hence the best contrast; thefilm or sensor should be there - represented bya 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 theychange 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 thecase; 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]60tan 20 LP/mm f= 120 f/ 32 u'= 0

tan 40 LP/mm f= 120 f/ 32 u'= 040

depth of focus for z'=0.033
object distancefilm or sensor

A macro lens very sharply stopped down to f/32; theMTF is reduced by diffraction and more spreading in
depth.

20
0-0.1

f= 120 f/ 32 u'= 0100
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 distancefilm 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 aperturesand high performance level one can see how high thedemands 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 distancefilm 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 thegeometric 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 distancefilm 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 distancefilm 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 andtherefore a slightly increased depth of field; and within thedepth 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 pixeldata) varies with the focus, the curves look different:
they appear more rounded and flat. This is no surprisesince the low-pass filter is also a kind of image
degradation, similar to the aberrations or the diffractionin the previous cases. These data are from a good lens
at f/11, so you may compare with page 21. Highsharpening 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 interms of MTF measurement the question istranslated into: ”At what spatial frequency does
the contrast transfer (MTF) drop below acertain threshold (e.g. 10 %)?”
Typical values which answer this question canbe seen in the following chart which shows
how contrast transfer gradually decreases withincreasing spatial frequency, in other words
with of the structures getting finer and finer. In
order to make the figures independent of theformat size, the spatial frequency is not given
in absolute terms in line pairs per mm but inline pairs per image height. The blue curveshows 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 bydiffraction 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 thecontrast transfer at the edge of the depth offield area at a circle of confusion diameterz’. As the f-number the circle diameter isrelated 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 inmany fields of practical photography
because they can only be achieved withextremely tight focus tolerances and withvery flat objects.

Resolving power of the lens at different permitted degrees of blurriness described by thediameter 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 popularvalue of 0.03mm, the resolution drops to about 1200 line pairs per image height.

Carl Zeiss

Camera Lens Division24

Bokeh – properties of blurriness
Large depths of field can be desirable; inmacro-photography it would be great to havemore of it than is possible. However, it is as
well often undesirable, as a good image isusually characterised by the absence ofsuperfluous and distracting items.
A composition parameter which can help us toachieve this objective is the adjustment of theblurring 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 thecomposition of the picture.
We therefore want to deal with blurring in thefollowing 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 forcontrast, 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 istherefore perfectly right that the Japanese word
“bokeh“ is used around the world as acollective term for all attributes of blurring.

The root of the Japanese word boke or bokeh*
actual means nothing good; its meaning issimilar to “confused” or “dizzy” and is used toname mental states in exactly the same way.
In photography the term ”confused“ relates
naturally to light beams which no longer cometogether 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 themeanings.

Carl Zeiss

In spite of the subjective nature of thematter we nevertheless want to attempt toremain faithful to the style and character of
our technical articles by describing bokehwith 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 ofparameters. But figures can help us to
improve our understanding of connections.

All the parameters listed here influence thephenomena outside the focal plane:
•
•
•
•
•
•
•
•
•
•
•

Picture format
Focal length
f-number
The camera-to-subject distance
Distance to the background or theforeground
Shapes and patterns of the subject
Aperture iris shape
Aberrations of the lens
Speed of the lensForeground/background brightness
Colour

It is therefore not surprising that one often
hears different and sometimes contradictoryjudgements about the bokeh of many lenses.
Undue generalisations are all too often drawn
from single observations.
Many effects are attributed to the lens eventhough they are mainly caused by the subject
in front of the camera. Differences between
lenses are often very marginal but are thengrossly exaggerated.
In principle one should not turn the ranking inthe 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 thedeciding 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 theirown yard-sticks.

Camera Lens Division

25

The quantity of blurriness
The most important and clearest attribute ofblurring is simply the amount of it. Whenconsidering the depth of field we have been
concerned with permissible blurring; this blurring
is allowed if it is quite unnoticeable in theconditions in which the picture is viewed. We
have learnt that limits are fluid in this process.

But outside these limits where one clearlysees 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 thediameter of the circle of confusion.
This means that we now will deal with verylarge circles of confusion, and to understandthe 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; thevertical 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 thefirst 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 ofabout 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 withour camera , we learn that we need circles of confusion larger than 1/100 of the diagonal in order toseparate the main subject from the background.

Carl Zeiss

Camera Lens Division

26

Field width 70 cm

100

1

Ratio Diagonal / Circle ofConfusion

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 thecamera is situated on the left. At a distance of 1m from the foreground and 1.8m focus distance thehorizontal 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 thezoo, 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), thehighest achievable blurriness decreases. In order to reach the same degree of backgroundblurriness as at closer distance, one has to use wider apertures or take care that the distance to thebackground is larger.

Carl Zeiss

Camera Lens Division27

Focus Distance 2mRatio 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 ofconfusion 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, thesharpness 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 ofConfusion

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 fromthe 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 thesame distance in different formats, in other words with different focal lengths but in each case withthe same angular field. If equivalent aperture numbers are chosen (see the table on page 10), theattributes of the depth representation of different formats are identical and the curves are thencongruent. 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 ofblurriness.
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 andtherefore 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 verysmall levels of blurriness on the left and determines the order of the curves, in the far distantbackground the influence of focal length predominates. If the subject is to be truly separated fromthe background, one ideally needs both – a longer focal length and a high speed lens.

All these curves of the large circles ofconfusion can be easily understood if you lookback once again to page 11 and examine thesketch you see there. In your imagination or ona piece of paper let the point of the light conemove 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 whichforms on the sensor.
The decisive parameter for the quantity of theblurriness is therefore the physical size of theentrance pupil. If by “bokeh’ you mean
principally the ability to be able to represent thebackground 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 lightcone entering the lens from the subject is so
important that this information is written on thelens barrel:

Carl Zeiss

The number 0.75 on this microscope lens fora 20x magnification is the sine of half the light
acceptance angle of 48.5° and tells us thatthe lens has a resolution limited by diffraction
at about 2300 Lp/mm with a minute depth offield of 0.001mm.
As a comparator: a photographic lens with animaging scale of 1:10 and a nominal f-numberof 8 has a light acceptance angle of 0.6°; theresolving power limited by diffraction is then
16 Lp/mm measured in the subject and thedepth in which this resolution is achieved is
20mm.

Camera Lens Division30

Aperture iris images
It often happens that many circles of confusionof similar brightness overlay each other andintermingle in a picture in such a way that theindividual circle can no longer be recognised.
This causes the flat, smooth character of avery 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 ofwater. In such a case the associated circle ofconfusion is always accentuated beyond its
surroundings in the picture such that it ispossible to see its geometric shape. In thiscase 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 brightthey attract the view of the observer. A‘beautiful’ geometry of the iris is therefore
desirable. But an iris image reminiscent of asaw blade as in the example below on theleft 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 thecone at a position where is cross-section area
is still very large.

At the edge of the photo iris images arealso altered by vignetting if the light cones
going to the edge of the photo at wide
apertures are intersected by front or rearparts of the lens barrel or by the limited
diameter of rear and front elements:

Four examples of iris structures with 5, 6, 8
and 9 blades which are made visible by a veryout 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 curvesof the circular full aperture between the fivestraight 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 ofsegments 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 ispossible to come close to the ideal of acircular 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 isnaturally a matter of taste.

Camera Lens Division31However,alongsidethedesign-based
properties of the lens, quite natural,
unavoidable effects give reason that images ofhighlight areas not only display perfect circles.
To be specific, if many highlight areas arelocated in close proximity to each other – areflective water surface would be an example –the bright areas created by each individual
highlight area overlap and the points ofbrightness 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 arespecks of dust floating in the air close to thecamera 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 muchout of focus. Their inner structure andtransparency therefore create in manyobservers the perception of transparent
spheres floating in the room. If you search
the Internet for ‘orbs’ you will find dozens ofarticles in which this phenomenon isinterpreted as mysterious ghosts. But inreality the reason is a diffraction pattern of
the light waves travelling through the lenssurfaces.

If the iris images of out of focus highlight areas
overlap, these bright areas cumulate andcreate new geometrical shapes in this way.
If you look at this picture very closely, you can
see another interesting effect – all thedefocused spot images contain a circular
structure. It is possible to recognise from thisthat the lens has an aspherical surface asthese surfaces are often not as smooth as aconventionally polished lens. Particularly in thecase of lenses which are manufactured bypressing hot liquid glass it is possible torecognise the traces of the turning process withwhich the mould was manufactured.
It is possible to combine rotational and pivotal
movements when polishing spherical surfaces
because the curvature of the surface is thesame everywhere; in this case no traces areleft. In the case of aspherical surfaces thecurvature is variable and therefore demands
otherprocessing
techniques.
Residual
unevenness of these surfaces becomes visible
if a very small light source is reproduced verymuch out of focus.

‘Light orbs’ from a compact camera withintegrated flash.
Such disruption to light waves is particularly
significant if “soft filters” are used on thelens. In the case of the Zeiss ‘Softar’ filter
the effect is caused by small lens-shaped
bumps distributed across the surface of thefilter. These, too, are visible in the iris
images:Iris images with the Zeiss “Softar’ andMinolta “Portrayer’ soft filters

Carl Zeiss

Camera Lens Division32Sometimes the phenomenon of the individual
iris images is equated with “bokeh’; under thisheading one finds collections of pictures inwhich iris images are mixed with photos ofsoap 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 whatsignificance has all this for the reproduction ofimage areas in which there are no highlightareas?
In the following examples of photos we will seethat one should not over-estimate thesignificance 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 tosee the geometrical shape of the iris only
indirectly in the alternating wide and narrow
beams of light which radiate out from the lightsource. 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 betraythe exact shape of the iris. This is because in
the slight out of focus which has been set
here, the circular illuminated surface of thelight source is still relatively large incomparison to the tiny pentagonal image ofeach individual point on the illuminatedsurface. Therefore the image appears to be
fairly round. But this changes if the out offocus 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 grassand, 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 offocus. Above, the focus is closer to theforeground; below, the camera is set to“infinity”.

Camera Lens Division

33

Thus whether we see the shape of a brightobject or the shape of the iris depends on asize 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 offocus. In between there is a transition zone inwhich both shapes are mixed.
It is apparent in the pentagonal diaphragm
images of the last two photos that they arereversed to each other. The reason is because
in the upper photo the sensor plane is behindthe 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 otherelement of the image. Lines and long edges
particularly generate an image of manyhighlight areas blurred in one direction – theshape of the iris is unimportant in this.
Only in the slightly out of focus iris image atthe bottom of the previous page can we see a
couple of gentle highlight areas at the edge ofthe flowers showing the pentagonal shape ofthe blades. They disappear as the image goes
further out of focus because the amount oflight in the point is then distributed across
such a large area that it is no longer noticed.

Lens with 8 iris blades

Lens with 9 iris blades

In summary we can say that the shape of
the iris can become visible in the picture
either obviously as a decorative feature oras a disturbing artefact and that it can betrayinteresting facts about the lens to us.
However, the iris can remain totally invisible
in many pictures. Yes, and if we use a lenswith the aperture fully open, it can of course
play no role at all.
Lens with 6 iris blades

Carl Zeiss

Nevertheless or perhaps in just such a casethere can be major differences in the bokeh.

Camera Lens Division

34

The Nature of BlurrinessWe have come to understand the basic
characteristics of depth of field with the aid ofa little geometry: we have taken a look at light
cones that are intersected in different placesby the sensor of the camera (see pages 4 and
6). The intersections are the circles ofconfusion, 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 geometricalfactors of the lens that can be entered into a
depth of field calculator, for instance. All
lenses would then have to be the same whenusing the same f-number and same focus
deviation.

We know, however, that lenses are not at allalike at their best focus, especially not with awide aperture. Differences in contrast andsharpness 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 havealready shown us how different lenses canbe, not only at best focus but also at thecalculated limit of the geometric depth offield (see page 23). Let us find out why thatis the case.
The geometric theory of depth of field is an
idealized model that does not takeaberrations into account; it simply assumes
that all light cones intersect at one point:

25

Half Exit Pupil [mm]

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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 raymarked blue that is more on the inside is travelling from a point of the pupil plane that is 14mm 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 Division35

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 asingle common point, and then leave the focal point in just as orderly of a way on thebackside. 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 havethe same point of intersection, but rather each zone of the pupil has its own point ofintersection. 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 themarginal rays of light is closer to the lens for simple collective lenses and this natural defect issimilar 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 thebest focus is at full aperture. If the aperture is narrowed, the focus moves to the black dot - thelens 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 lightcone is no longer ideally arranged, and we
could say that the rays of light are "confused."This is the original meaning of theJapanese 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 issomewhat less than in the ideal geometriclight cone. The circle of confusion istherefore larger behind the focal point thanin 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 ringaround the outside.

There are so many rays that overlap in thiszone 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 athin green border because on the outside we
see the rays of light whose focal point isclosest to the lens. Since green light has theclosest focal point in the normal chromaticaberration it dominates the cover surface ofthe light cone behind the focus.

Background blurriness with Sonnar 1.5/50
ZM

Camera Lens Division37

• If we want to generate a noticeably beautiful
bokeh in the background, then we must
make the under-correction so noticeable thatthe focus shift is also very large and makes
focusing difficult.
• In addition, the contrast rendition of the lensis overall poor by necessity. Because theoutside 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 generaluse and have to limit the spherical undercorrection. In any case, we should avoid
spherical over-correction. This is not to saythat the lens is now better than good overcorrection just means that the sphericalaberrations 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
foregroundcharacteristics
withundercorrection are found in the background in caseof overcorrection. And because background isalmost always more important, it would be theless desired balancing of the lens.
But even spherical aberration which remains
completely within the range of the mild undercorrection, yet shows clear signs of themeasures that are intended to limit the growthof the spherical aberration. It can alreadycause 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 amore modest maximum aperture where thespherical correction is much simpler either.

Carl Zeiss

20

15

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h [mm]

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h [mm]-0.20• The more appealing the blurriness is in the
background, the less appealing it is in theforeground. There it often seems harsh anddisturbing. It generates swirls of small
highlights and transforms lines into doublelines.

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10

-0.25

There are disadvantages to this imaging
property, however:25

ds' [mm]Common diagram of the longitudinal sphericalaberration in optics: the vertical axis shows thestarting point of a ray in the pupil plane,
expressed by the distance from the opticalaxis, and the horizontal axis shows thedeviation from the focus position. The
directions correspond to the graphics on theprevious pages. The diagram on the left
shows a strongly under-corrected lens.

But even with the well-corrected lens on theright, the brightness profile of the circles ofconfusion already shows a "thin ring"emphasizing the edge:
0.6

Circle of confusion, radius [mm]The nature of the background blurriness of aspherically 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 willfind examples illustrating this.

0.5

0.4

0.3

0.2

0.1

0.0

-1

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Brightness rel. ideal [EV]Brightness variation for circles of confusion ofvarious sizes in the background with a lensshowing mild spherical under-correction. The
more it is defocused, the smaller are thedeviations from the ideal disc withhomogenous brightness throughout.

Camera Lens Division

38

25

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00

In reality the brightness gradient at the edgeof the blur circle is not as high as shown inabove charts. They had been calculated fora single wavelength, but in reality different
colours have different circles, which makes
things a bit more smooth. Real lenses haveas well a longitudinal chromatic aberration;
the focus of the rays depends on the
wavelength. And these deviations are ofsimilar amounts as the spherical ones:

0.00

-0.15

ds' [mm]-0.05

5ds' [mm]ds' (Lambda) [mm]0.6

Circle of confusion, radius [mm]But if the lens is stopped down one stop
(charts on the bottom and above on theright-hand side), the reversal point of thelongitudinal spherical aberration is excluded
and the brightness profiles look pleasantagain.
In this example we can also see that thebrightness profiles become flatter when theimage is blurred more.

10

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Somewhat stronger counteractive measures
towards spherical overcorrection strongly
increasethebrightness
aroundthecircumference of the circles of confusion:

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Thus the focus deviations of the colours aredifferent and the according circles ofconfusion have slightly different size:

Brightness rel. ideal [EV]Circle of confusion, Radius [mm]0.50

Circle of confusion, radius [mm]

0.6

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Brightness rel. ideal [EV]Brightness rel. ideal [EV]Carl Zeiss

Camera Lens Division39This combined effect of different aberrations
causes that in the background of the image
the inner area of the blur circles is dominatedby the colours which have the longer focal
distance. These are usually the colours fromthe ends of the visible spectrum which mix to a
purple shade. The edge of the blur circle isdominated by the colours from the middle ofthe spectrum. This explains the green fringes
which we see in the blurred image of a whitespot.

This leads to the following rules about
bokeh:•

Some caution is advised when making
judgments about the bokeh dependingon the lens correction, because bokeh isextremely variable.

•

The correction balancing has anespecially strong influence on theblurriness of the rendition at small
deviations from the focal point. If there isa 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 alsochanges 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 coreis due to the too high brightness of thehighlights.
As with the brightness profiles of blur circles
the handwriting of the lens with respect tocolour-bokeh disappears more and more,when it is strongly out-of-focus, or when it isstopped down.

Carl Zeiss

Camera Lens Division40

After working through many difficult diagrams, we will now relax and use a few example pictures toillustrate the influence of spherical aberration, which should also invite you to have a look to the image
files available for download:

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 ispresent at the best edge sharpness (top picture), many artefacts can be seen in the blurry picture.

Carl Zeiss

Camera Lens Division41

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. Itscharacteristics: a veil is present at the best edge sharpness (second picture from the top), the contourof the triangle remains for a long time. Because of the veil of the spherical aberration, the very brighthighlight appears much larger.

Carl Zeiss

Camera Lens Division42

A 700x820 pixel crop from a 24MP-image, above with strong spherical aberration, below withgood correction. The hatching in the hair is at about 40 Lp/mm on the sensor.
Scale ratio was 1:10. These images are available as download files.
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 files for download
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 windowof 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 4Circle 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 blurcircle due to spherical and chromatic aberration•

Image 5Circle 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 circleof the highlight has nothing to do with aberrations of the lens, it is caused bydiffraction.

••

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 softversion as an image background. The other two don’t appear calm, somehow noisy. But they had beentaken 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 aprime lens Makro-Planar 2/100 on the right. Both shots taken at f/5.6

Image 13This 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 theimage at around 40 Lp/mm. Each horizontal line of nine images is a focus sequence: on the left side thecamera is most close to the subject, to the right the subject moves through the focus into thebackground. 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 apermitted circle of confusion of 1/500 of the field results in a depth of filed of 1.9cm. Two images on theleft 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
show some loss already.
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 overa larger range than compared to images taken at smaller aperture f/2.8. This demonstrates the limits ofall simple calculations about depth of field.

Image 14 to 17

Carl Zeisssome illustrations of the text as jpg-file.

Camera Lens Division45