Selective
focus (depth of field control) is just one of many creative
techniques used in practical photography. To a standard observer,
the depth of field is that zone of a photograph (from foreground to
background) in acceptably sharp focus. However, the boundary from sharp to unsharp
focus is not a discontinuity but a uniform gradation.
To select the optimal camera settings for a particular photo 
opportunity, some appreciation of the concept of depth of field is
necessary. Though most cameras feature a depth of field preview,
because the viewfinder darkens as the lens is closed, to gauge the
sharpness of an image is easier said than done. Furthermore, most
modern lenses have no depth of field scale. On a
digital camera the LCD screen allows images to be reviewed, (select
an area of the image and
zoom  in), even so, to estimate the depth of field is
not straightforward.
Hyperfocal
distance (HD) is a photographic measure that
is closely related to depth of
field (DOF).
Focus at infinity and the nearest distance in acceptably
sharp focus is
the hyperfocal
distance.
To maximise the available depth of field: refocus at the hyperfocal
distance and the zone of acceptably sharp focus extends from half
the hyperfocal distance to infinity.
Hyperfocal
focusing is an effective technique that is relevant to many
photographic disciplines (particularly landscape photography).
However, to apply the principles to real scenes (without
consideration) may not guarantee the desired visual effect,
you may
prefer to trade the
sharpness in
featureless regions
for more sharpness elsewhere.
Hyperfocal distances may vary from a few metres for a wide angle
lens (typical is the 17  35 mm zoom) to many metres for a telephoto
lens (typical is the 70  300 mm zoom). In practice you can
estimate the hyperfocal distance, precise focusing is
unnecessary,
focus further than the hyperfocal distance and close the lens (one
stop). Most scene features
are rendered in acceptably
sharp focus,
but how is
acceptably sharp defined.
A camera lens can focus in
only one plane. For a three dimensional scene (a typical scene has
spatial depth), objects in front and behind the point of focus are
imaged
at the focal plane as a blur circle. The blur circle is the
surface generated by a converging (object in front of the point of
optimum focus) or diverging
(object
behind the point of optimum focus) cone of light that intersects the focal plane
(see
Depth of Field Calculation). The so  called
circle
of confusion
is the maximum circle of light (blur circle) that is unresolved
(perceived as a point) by the unaided eye, the resolving power
of the human eye is limited. The criterion for 'acceptably
sharp' is based on the visual examination of a photograph,
traditionally this is an
8 x 10 inch print, viewed at a distance of
about 10 inches (distance of most distinct vision). That the human
eye can just resolve spatial detail of 0.01 inches at
this distance is rooted in the photographic (film) technology of
former times.
In fact,
the normal eye can resolve about one minute of arc,
20/20 vision, that corresponds to spatial detail of 0.003
inches, nonetheless, the accepted standard used by optics
manufacturers is 0.01 inches (0.25 mm).
Spatial detail of 0.25 mm in the print corresponds to spatial detail of 0.03 mm
in the image
(0.25/8), since a magnification of about 8x (aspect ratios may differ)
is necessary to produce an 8 x 10 inch photo enlargement from a
35 mm format image.
Hence, the diameter of the circle of confusion is 0.03 mm.
For the advanced photographic system (APS) format, the diameter of the circle of confusion is
less
than 0.03 mm, since a higher magnification is necessary to produce
an 8 x 10 inch photo enlargement, for the medium and large formats, the
diameter
of the circle of confusion is more
than
0.03 mm, since a lower magnification is necessary to produce an
8 x 10 inch photo enlargement.
Since the resolving power of the normal eye varies with luminance and
retinal location (under ideal conditions the human eye can
resolve about thirty seconds of arc, 20/10 vision), at the depth of field limits the focus is not
exceptionally sharp. However,
some blurring of the image is not
necessarily ruinous and may even be desirable for
aesthetic effects. Even though
0.03 mm (or
thereabouts) is the accepted standard that is
used by optics manufacturers for depth of field tables and lens
markings,
depth of field is a subjective property (related to the
physiology and psychology of
visual perception), the circle of
confusion is
an
arbitrary limit that may be changed to suit the
desired sharpness (within the limitations of diffraction effects) or particular style of photography.
To recap,
depth of field is not
a fundamental
(lens  film/image sensor)
parameter
that is measurable.
The circle of confusion, our criterion for acceptably sharp focus
determines the near and far focus limits and the extent of the depth of field.
The circle of confusion is related to the
viewing geometry
(the print size and the viewing
distance) and the resolving power (the reciprocal of the
visual acuity) of the human eye. Change the
viewing geometry, the circle of confusion changes and the depth of
field changes.
For
a given sharpness criterion that is characterised by the diameter of the
circle of confusion,
Depth of
Field is dependent on
f  number of the lens (relative aperture)
focal length of the lens
focus distance (camera to subject distance)
The highlighted
quantities are lens parameters, closing or
opening the lens aperture
(f  number increasing or decreasing) changes the hyperfocal
distance and the depth of field.
The effect of aperture (f/#) on the depth of field
(the focal length and focus distance are constant) is illustrated (the focus point is the nearest golf ball).
Since the hyperfocal
distance and depth of field are influenced by the same lens
parameters, depth of field may
be recast in terms of the hyperfocal distance and
focus distance.
Show the way, build a
Hyperfocal Distance/Depth of Field Calculator
for your general photography.
The
simple
Hyperfocal Distance/Depth of Field
Calculator
operates thus, determine the hyperfocal distance from the focal
length and f  number of the lens, then determine the depth of field
from the hyperfocal distance
and focus
distance.
The
Hyperfocal Distance (H) and Depth of Field (DOF) are given by
where
f,
f/#
are the focal length and f  number of the lens, c is the diameter of the
circle of confusion
(0.03 mm for the 35 mm format) and
u
is the focus distance.
To understand the detailed derivation of hyperfocal distance and depth of
field is
unnecessary (some algebraic manipulation is used), for the
mathematically minded,
(see
Depth of Field Calculation).
Procedures based on
this simplified formulation are ideal for programmable handheld devices,
use
separate memory areas to store
the hyperfocal distance and depth of field procedures.
Please read your
calculator/palmtop manual to code the procedures.
Units must be consistent throughout (feet or metres or ...), enter
the focal length of the lens in millimetres.
In the field, to calculate
the
Hyperfocal
Distance
and
Depth of Field
takes less than one minute.
So that unforeseen photo  opportunities are not missed, you may
prefer to create a
Depth of Field/Hyperfocal
Distance
chart for the most useful camera settings.
For Digital Cameras that use image sensors of different size
and type (aspect ratios may differ), calculate the diagonal
(width^{2 }+ height^{2
}= diagonal^{2})
of the
image sensor active area
(the effective
pixels that contribute to each image). Consult your digital camera handbook for the active area
of the image sensor.
The Focal Length Multiplier (FLM) or crop factor is the ratio of the 35 mm format
diagonal to the image format
(film gate/image sensor) diagonal.
The
focal length
multiplier for most digital compacts is typically between 2x and 8x
and for most DSLRs is
typically between
1.3x and 2x, (see
Digital Cameras).
Because there are many digital camera types (compact and SLR), image formats differ,
hence for the same field of view the focal length must change
proportionally.
In order to
avoid confusion, most manufacturers of digital cameras quote the
35 mm equivalent focal
length, the combination of 35
mm (film) format and
focal length that gives the same field of view.
A lens of focal length f
mounted on a digital body (focal length multiplier, FLM) gives the
same field of view as a lens of focal length (f x FLM) mounted
on a 35 mm body, where (f x FLM) is the
35 mm equivalent focal length.
For
digital cameras, the diameter of the
circle of confusion is (0.03/FLM).
You can obtain FLM from the image format diagonal or
from the ratio of the
35 mm equivalent focal length to the real focal length.
Use the calculator for general photography
(not close  up photography)
to determine Hyperfocal Distance and Depth of Field, note the approximation used (H
>> u >> f). Enter the REAL
FOCAL LENGTH of the lens and for the diameter of the circle of confusion,
enter (0.03/FLM). For 35 mm film
(full frame digital), FLM = 1. You can compare the depths
of field of film cameras,
digital cameras, any camera, however, image quality
may not be the same (see
Digital Cameras). 
Notice
the depth
of field is directly proportional to the focal length multiplier.
The smaller the image format, the larger the focal length multiplier
and the greater
the depth of field. My Nikon D80 has 1.5x more depth
of field than a 35 mm format camera (for the same field of view).
For digital cameras that use even smaller image sensors (1/3", 2/3",
...) and short focal length lenses, the hyperfocal distance
is
very small and the depth of field is very large. Users of digital
compacts may
find that selective focusing to isolate subject matter
from the background can be
especially challenging.
Provided that focus distance and focal
length are changed proportionally
(to maintain the magnification), the
total
depth of field (for a given format)
is
effectively
constant.
Though the total depth of field is effectively constant, the front
and rear depths of field are not evenly divided. For long focal length
lenses, the front and rear depths of field are similar, for short
focal length lenses, there is less front depth of
field and more rear
depth of field. Zooming (in or out) from a set location changes the
focal length (the magnification) and the depth of field (see Depth of Field Calculation).
Diffraction and Depth of Field
DIFFRACTION
places a fundamental limit on image quality, the lens aperture
cannot be progressively closed to extend the depth of field (zone of acceptably
sharp focus).
A
perfect imaging system maps object points to corresponding image
points, consistent with the rectilinear propagation of light and the laws of geometrical optics. In the real world a point in
object space is mapped to a blur circle (a diffuse patch of light)
in image space due to DIFFRACTION a fundamental property of
the wave nature of light. Fraunhofer (far  field) diffraction
phenomena relate to
collimated (parallel) light, where the effects are observed in the
image plane. The image is just the superposition of the blur circles
that correspond to all object points. Diffraction describes the
spreading of waves (into a region of geometrical shadow) as they pass through a
finite aperture or the edge of an obstacle.
This is illustrated for light of wavelength λ.
Diffraction phenomena are not restricted to light, all waves
diffract, matter (electron, neutron, ...), radio, sound, water, .... . Familiar examples
of light diffraction are the haloes that encircle street lights and the
stripes of colour on the surface of
a CD.
According
to the
Huygens  Fresnel principle, points on a wavefront are a
source of diverging secondary waves (wavelets) that can interfere
constructively or destructively, dependent on their phase
relationship. Since the lateral extent of the wavefront is bounded, no
optical system (camera) can be diffraction free.
Though aberrations, (spherical aberration, coma,
astigmatism, field curvature, distortion and chromatic aberration),
diffraction and defocus all contribute to the image blur,
diffraction sets the upper limit on spatial resolution. For an ideal lens
at best focus, the diffraction
pattern of a point source has the form of a circular
core of light, the Airy disc (after G Airy, 1801  1892) surrounded by
fainter concentric
rings of light that gradually fade. For a
non ideal lens that has residual aberrations,
light is
redistributed to the rings (decreasing the contrast) and can form irregular
shaped patterns
(aberrated Airy pattern).
The
onset of noticeable diffraction blur can be estimated from the
diameter of the Airy disc, the bright circular core of the
diffraction pattern that contains about 84% of the light energy. For a uniform intensity distribution, the diameter of the Airy disc
in the focal
plane (Φ) is given by
where, λ is the wavelength of the light and f/#
is the f  number of the lens,
the ratio of the focal length
to the entrance pupil diameter (lens aperture).
Notice that
Φ
is
wavelength dependent, different for each of the primary colours,
Φ_{R}
> Φ_{G}
> Φ_{B}.
Longer wavelengths
red light are diffracted more than
shorter wavelengths
blue light,
and hence the diffraction blur (measured by the Full  Width at Half
 Maximum (FWHM) of the Airy
disc) decreases from red to blue.
Call to mind the circle of confusion, our criterion for acceptably sharp
focus or tolerable blur. It is instructive to compare the
diameters of the Airy disc and
circle of confusion.
For light at the centre wavelength
(λ = 0.555 μm) of the visible spectrum, there is
increasing image degradation due to diffraction
at f  numbers (f/#)
≥
(circle
of confusion (mm)/0.0014).
The corresponding f  numbers for
red light and blue light are lower and
higher.
For any image format, the diameter of the circle of confusion is
0.03/FLM (mm), where FLM
is the focal length multiplier. Of course, the pixel size must not
limit the resolution, provided that several pixels are used to sample the circle
of confusion,
the conclusions should be plausible.

For
the
(full frame) 35 mm
format, the simple analysis demonstrates that blurring due to
diffraction becomes the dominant factor at about f/22.
Beyond f/22 the diameter of the Airy disc exceeds the diameter
of the circle of confusion and the camera performance is
degraded more by diffraction.

For the
(sub full frame)
digital formats, the simple analysis demonstrates that blurring due to
diffraction becomes the dominant factor at
about
f/#
(see table, FLM and f/# are rounded). Beyond f/# the diameter of the Airy disc exceeds
the diameter of the circle of confusion and
the camera performance is degraded more by diffraction. For
(sub full frame)
digital compacts a limiting aperture
of f/8 is typical.
Format 
FLM 
f/#

1/3" 
7.2 
3.0

1/2.5" 
6.0 
3.6 
1/2" 
5.4 
4.0 
1/1.7" 
4.6 
4.7 
2/3" 
3.9 
5.4

1" 
2.7 
7.9 
4/3" 
2.0 
10.7 
Cn APS  C 
1.6 
13.2

Nikon DX 
1.5 
14.1 
Cn APS  H 
1.3 
17.1 
The effect of diffraction on image quality is gradual, there is no
step change of sharpness as the lens aperture is
closed, even so, there may be 'smoothing'
of the image. A lens functions as a low  pass
filter (the cutoff frequency
is inversely proportional to
the f  number)
of the scene detail, passing lower frequencies (coarse detail) and attenuating higher
frequencies (fine detail). Needless to say, there are trade  offs,
lower f 
numbers to reduce diffraction, higher f  numbers to reduce
aberrations. The lens may be aberration limited or diffraction
limited. Close the lens, typically 2  4 stops to balance the
aberration and diffraction blur, thus maximising
the performance. Close the lens further and the performance
is degraded more by diffraction. The depth of field (zone of acceptably sharp focus) does not continue to increase as the lens
aperture is progressively closed, from the near to far focus, the
image quality deteriorates. Though diffraction blur permeates the
recorded scene,
the loss of sharpness (low contrast fine detail may no longer be
distinguished) is most noticeable at the plane of focus.
A
simple pinhole camera has a vast depth of field, however, the
image is blurred, everywhere. As you reduce the hole diameter, the
geometric
blur decreases but the diffraction blur
increases. There are many formulae for the optimal pinhole diameter,
the classical formula is 1.9 √f λ, where f is the
focal length and λ is the centre wavelength of the visible spectrum.
For any image forming system, the resolution limit
(the ability to distinguish
small angular separations) increases
with decreasing aperture and with increasing wavelength. To minimise
diffraction effects (reduce the diameter of the Airy disc),
so that
fine detail is resolvable, telescopes use large apertures, microscopes
use
short wavelength UV light.
Given that image
formation is the end result of
diffraction and interference, these processes must decide the
achievable
resolution. According to the Rayleigh criterion, two points
of light A and B
(for example, distant car headlights or a double star) are just
resolved when the central maximum of the diffraction pattern of A
falls on the first minimum of the diffraction pattern of B (linear
separation ≥ 1.22 λ f/#).
At the Rayleigh limit, the mid  point intensity is
about 0.81 of the maximum intensity and the corresponding MTF is
about 0.09 (see Modulation Transfer
Function).
Be aware,
this is an optical resolution limit, the diffraction
resolution limit is quite apart from the film/image sensor resolution
limit, set by the dye  silver grain/pixel size.
The Rayleigh limit (close to the empirical Dawes' limit) is intended for visual
instruments (micro, spectro, ..., scopes), nonetheless, the
criterion is applicable (with
some adaptation) to modern cameras. In practice, the spatial
resolution of the lens, characterised by the diameter of the Airy
disc must match the spatial resolution of
the image sensor, characterised by the pixel pitch. This is a
simplification, numerous factors contribute. For monochrome cameras,
the necessary condition to avoid undersampling (aliasing) is
(2 x
pixel pitch = diameter of the Airy disc).
For colour cameras that use a standard Bayer colour filter (the
sampling rate is that
of the RGBG
2 x 2 array), the
necessary condition to avoid undersampling (aliasing) is
(4 x
pixel pitch = diameter of the Airy disc).
Bear in mind, increasing the MegaPixel count
(smaller pixels for a fixed image sensor size) is not synonymous
with enhanced resolution and image quality, the resolution may be
imposed by the optics, and for smaller pixels, photon noise and the
effects of decreasing signal to noise ratio may become performance issues.
Diffraction is a fundamental property of wave propagation that
limits the resolution of an optical system. I have said that
aberrations, diffraction and defocus all contribute to the image
blur and that diffraction sets the upper limit on spatial
resolution
(for an aberration
 free optical system at best focus, the image of a point source is
the Airy pattern). However, for practical photography, residual
aberrations (modern lenses
are not aberration  free), camera and
subject movement, focus error and the atmospheric conditions (haze
and turbulence reduce the
image contrast) are often
more significant than quality limitations resulting from diffraction
phenomena.
All
images and text © imajtrek

