Depth of field is defined as the range of distances from a camera that
will be acceptably sharp in the finished picture.

For the photographer whose creative energies are often in competition with
the need to think analytically, this parameter of image quality is
probably second only to exposure determination in the attention it
demands.  This text will describe the variables that affect depth of
field, offering in summation, a simple formula and some techniques that
can be used to achieve acceptable sharpness from foreground to infinity.
This discussion will be limited to cameras that lack perspective control,
and will only provide the formulae associated with obtaining a depth of
field that includes infinity.  We are going to reverse the chronology of
image making, beginning with a discussion of the final event -- viewing
the print.  

Prints are often viewed at a distance equal to their diagonal.  More on
viewing distances later, but for the moment, imagine you are looking
straight down on a 6 inch by 8 inch print, from above its center.  This
print has a diagonal of exactly 10 inches.  Your eye is placed at the apex
of an equilateral triangle that is formed with the print diagonal as the
base leg.  Each side of this triangle is 10 inches long.  We will call
this a viewing distance of 10 inches (even though the distance from your
eye to the center of the print is a little less than that).

Let's have a better look at this print.  It's a classic lake and mountain
scenic with an out of focus foreground.  The foreground objects fell short
of the near-distance sharp.  The depth of field is bounded by the
near-distance sharp and the far-distance sharp.  Objects at distances
between these two sharps are perceived to be acceptably sharp in our
example print, when viewed at a distance of 10 inches.

The foreground appears unfocused because thousands of circles of confusion
are overlapping each other and they are of an unacceptable diameter.  A
circle of confusion is formed when the image for a given point on the
subject is focused in front of or behind the film plane instead of
precisely at the film plane.  The circles of confusion for those points in
the image that correspond to objects that are precisely focused are
extremely small.   Points imaged from objects nearer or farther from the
camera than those objects residing in the plane of sharpest focus have
increasingly larger circles of confusion at the film plane.   At the near-
and far-distance sharps, the circles of confusion have increased to a size
that is unacceptable.

The maximum acceptable diameter of these circles of confusion, as viewed
in the final print after possible cropping and enlargement, must be
subjectively selected before we can proceed to discussions of how to
control depth of field.  You can decide to what diameter these circles of
confusion should be limited and then implement controls to adhere to that
decision.

A survey of lens manufacturers revealed that their depth of field scales
and tables were based on values ranging from 1/70th to 1/200th of an inch
for 10 inch diagonal enlargements viewed at 10 inches.  The depth of field
scales on lenses you own are expanded or compressed by the manufacturer to
produce circles of confusion at a diameter of their choosing.

A conservative choice for the maximum acceptable circles of confusion,
after enlarging the format diagonal to a print diagonal of 10 inches, to
be viewed at a distance of 10 inches, is 1/175 inch.  Most would agree
that at a viewing distance of 10 inches, a print with circles of confusion
under 1/175 inch will be found acceptably sharp.  This would correspond to
a 35mm frame having circles of confusion no greater than 1/1021 inch
before enlarging the negative 5.87 times to reach the 10 inch diagonal of
the print.  1/1021 inch is about 0.025mm. 

Since our 6x8 print has a diagonal of 10 inches, we know that exactly 1750
acceptable circles of confusion of 1/175 inch diameter could fit
end-to-end along the print's diagonal.  Depending on the subject, of
course, there could actually be many more than this overlapping each other
along the diagonal.  If they were end to end, though, no more than 1750
would span the 10 inch length.  If this were a contact print, made without
enlarging the original negative, we would always have acceptable sharpness
as long as the diameters of the circles of confusion in the original
negative do not exceed 1/175th of  an inch.  If we enlarge the negative to
dimensions greater than 6x8 and increase our viewing distance
proportionately, there would be no perceived loss of sharpness.  What
happens if we make the same 6x8 print from a smaller negative?  Our first
formula will provide the answer.

For a specified format diagonal, what diameter circles of confusion are
acceptable at the film plane, prior to enlargement for viewing at a
distance equal to the print's diagonal, such that the perceived sharpness
will be equivalent to viewing 1/175 inch circles of confusion at a
distance of 10 inches?

-1-
Max. Circle of Con. at Film Plane = Format Diag./1750
(All lengths are expressed in inches. 1 inch=25.4 mm)
---

Where did the constant 1750 come from?  1750 circles of confusion, of
1/175 inch diameter, could fit end-to-end across a 10 inch print diagonal.
This formula shows that as the format diagonal decreases, the acceptable
diameter for circles of confusion at the film plane decreases, but for any
format, these circles of confusion will not be enlarged beyond the goal
diameter of 1/175 inch in a 10 inch diagonal print.  Using this formula to
determine the acceptable circle of confusion at the film plane, any
enlargement made from any format size, when viewed at a distance equal to
the print diagonal, will be found acceptably sharp.  (It will be perceived
to have the same sharpness as a 10 inch diagonal print viewed from 10
inches with circles of confusion not exceeding 1/175 inch.)

This calculation can be done for each format you work with, taking into
account any cropping you expect to do.  For example, the 35mm format has
an exposed area of roughly 24x36mm, but if this is regularly cropped to
extract 4:5 aspect ratio enlargements, only a 24x30 mm area will be used.
Thus the net format diagonal is the square root of the sum of the squares
of 24 and 30, which is 38.42mm, not the gross diagonal for 24x36 mm
(43.27mm).  Using the net diagonal, 38.42mm, converted to inches, 1.51
inches, let's calculate the maximum acceptable circle of confusion at the
film plane for a 35mm negative that will be used to make a 4:5 aspect
ratio enlargement.

Using formula -1-:

1.51 inches / 1750  =  0.000864 inch or 1/1157 inch

This maximum acceptable circle of confusion at the film plane is
considerably smaller than the 1/175 inch circles we will see after
enlargement.  The ratio of 1157 to 175 represents how much you have to
enlarge the 24x30mm portion of  a 35mm negative to expand its 1.51 inch
diagonal to the 10 inch print diagonal.

-2-
When a lens is focused on the hyperfocal distance, the depth of field
extends from infinity to one half the hyperfocal distance.
---

Armed with the figure 0.000864 inch and the focal length of our 35mm
camera's lens, we are prepared to calculate the hyperfocal distance.  We
can calculate a value for any selected aperture, to produce 4:5 aspect
ratio enlargements of any size that when viewed from a distance equal to
the print diagonals, will be found acceptably sharp.  To calculate the
hyperfocal distance, use the following formula:

-3-
Hyperfocal Distance = Focal Length^2 / (f-num. * Circle of Confusion @
Film)
(Remember to express all lengths in inches.)
---

For every lens you use for a given format, you can calculate hyperfocal
distances for each aperture. Let's calculate the hyperfocal distance for a
50mm (1.97 inch) lens, using the previously calculated maximum acceptable
circle of confusion for a 24x30mm negative, selecting an aperture of f/16:

Using formula -3-:
Hyperfocal Distance  = 1.97^2  /  (16 * 0.000864) =
281 inches  (23.4 feet)

The near-distance sharp is at half the hyperfocal distance.  This would be
140 inches (11.7 feet).  By focusing at the hyperfocal distance of 23.4
feet, the near-distance sharp will be at 11.7 feet and the far-distance
sharp will be infinity.  Points on every object at distances from 11.7
feet to infinity will be imaged with the circles of confusion not
exceeding 1/175 inch in a 10 inch diagonal print.  (If you focus on
infinity, the near sharp will be the hyperfocal distance of 23.4 feet.)

Once the hyperfocal distance is known, there are several ways to adjust
the depth of field.

-4-
Doubling the object distance makes the depth of field four times as large.
---

Similarly, four times the object distance yields sixteen times the depth
of field.  If you can not achieve the depth of field you need by other
methods, consider increasing the object distances to the camera.  Later,
when you crop the negative to achieve the perspective had before
increasing the object distances, the loss in sharpness due to cropping
will be linear, not exponential like the gain in depth of field. Grain
might  become an issue due to the increased magnification of the negative,
but the desired depth of field will have been achieved.  You could for
example, get a net gain of three times the depth of field by increasing
the object distance by a factor of three (a 9x increase in depth of field)
and later compensate for the change in perspective by cropping by a factor
of three (a 3x decrease in depth of field).  To implement this concept at
the time of exposure, simply move backward until the nearest object that
must be sharp has fallen within the calculated near-distance sharp
(one-half the calculated hyperfocal distance for the focal length and
aperture being used).

-5- 
Halving the focal length quadruples the depth of field, for a given
format.
---

For a given format, going from a 100mm lens to a 50mm lens yields four
times the depth of field, but if you have to close the distance to achieve
the same perspective with the 50mm lens that you would have had with the
100mm lens, you will have exactly undone the gain had by switching lenses.
By choosing the 50mm lens and later cropping to achieve the desired
perspective, instead of halving the object distance by moving closer,
there will still be a net gain in depth of field after enlargement,
similar to item -4- above.

-6-
Depth of field is directly proportional to the f-number.
---

If you double the f-number, you double the depth of field.  The depth of
field at f/16 is twice that at f/8.  If the hyperfocal distance for f/8 is
24 feet, it will be 12 feet at f/16.  Focusing at the hyperfocal distance
in both cases, would give far-distance sharps of infinity and
near-distance sharps of 12 feet for f/8 and 6 feet for f/16 (the
near-distance sharps being half the hyperfocal distances).

-7-
Best Perspective Viewing Distance = Focal Length * Negative-to-Print
Magnification
---

Thus far, we have assumed that all viewing distances would be equal to the
print diagonal.  This is not an unrealistic assumption, but it is said
that the best perspective is achieved at a viewing distance determined
using the formula above.  A 50mm (1.97 inch) lens on a 35mm camera,
generating a 10 inch diagonal print with a 4:5 aspect ratio, having a
negative-to-print magnification ratio of 6.62 (10 inch diagonal / 1.51
inch  diagonal), should be viewed at a distance of:

1.97 inch * 6.62 = 13.0 inches

For the 50mm focal length, a 13 inch viewing distance is greater than the
more critical 10 inch viewing distance for which we subjectively decided
to use 1/175 inch circles of confusion.  For any degree of magnification
from the 24x30mm useful area of the 35mm format (for prints with a 4:5
aspect ratio),  it is not until focal lengths under 1.5 inches (38mm) are
used that the so-called best viewing distance would bring the viewer
inside the 10 inch distance for a 10 inch diagonal print.  This would
jeopardize the apparent sharpness of our 1/175 inch circles of confusion
in our 10 inch print and is why the final print size and viewing distances
might be considered when selecting the constant used in formula -1-.  It
should be apparent by now that the constant 1750, is really a variable.
Fortunately, for any value selected as the maximum acceptable diameter of
circles of confusion, increasing the negative-to-print magnification
simultaneously moves the best viewing distance farther away, enhancing the
apparent sharpness at the same rate at which it is being lost due to
enlargement.  Only the focal length component of this equation presents a
threat.

-8-
To halve the diameter of circles of confusion in the final print, double
the calculated hyperfocal distance.
---

You do not have to generate several depth of field tables for each lens to
deal with various anticipated viewing distances.  Starting with the value
1750 in formula -1-, you can generate a list of the hyperfocal distances
for the apertures of a given lens that will give you 1/175 inch circles of
confusion in a 10 inch diagonal print.  From there, you can simply
increase or decrease the calculated hyperfocal distances to handle viewing
distances that are not equal to the print diagonal.

If you know at the time of the exposure that you intend the final
enlargement or projection to have a diagonal of 40 inches and expect it to
be viewed from a distance of 40 inches, your calculated hyperfocal
distances are fine.  If you anticipate 20 inch viewing distances for the
40 inch diagonal, you must double your calculated hyperfocal distance.  A
near-distance sharp would be doubled, too.  With this new information you
can alter your f-number, object distance or focal length to achieve the
necessary depth of field.

The limited scope of this article does not cover other factors that affect
sharpness.  Diffraction, for example, worsens as apertures are reduced to
improve depth of field.  So too, camera and subject movement become a
greater issue when slow shutter speeds are used with narrow apertures.
Other formulae are available that allow you to calculate near- and
far-distance sharps when focused at distances other than the hyperfocal
distance, including those typical in macro-photography.  The books I
reference below are highly recommended for additional information.



References / Suggested Reading:

*View Camera Technique*  Leslie Stroebel
6th Edition   c1993  Focal Press

*Basic Photographic Materials and Processes*
Leslie Stroebel, John Compton, Ira Current
and Richard Zakia   c1990  Focal Press

*The Camera*  Ansel Adams
c1980  Little, Brown and Company


-- 
/---------------------\
   Michael K. Davis              
  [email protected]                   
  MIME Attachments OK
\---------------------/
[Original posting:]
From: "Michael K. Davis" [email protected]

Newsgroups: rec.photo.misc

Subject: A Limited Look at the Math of DoF

Date: 15 Aug 1998

From: "Michael K. Davis" [email protected]
Newsgroups: rec.photo.misc
Subject: Re: A Limited Look at the Math of DoF
Date: 16 Aug 1998

I have an Excel-based Depth of Field Calculator that offers these features:

1) You can specify the desired maximum permissible diameter for Circles of Confusion at the print.

2) You are encouraged to use a sub-calculator to determine the diagonal of that portion of the image that will actually be used to make the print. This figure is used for Depth of Field calculations instead of making the assumption that the full format diagonal should be used.

3) The diffraction limited aperture is calculated for the format you are using -- taking cropping into account!

You can find it at Jon Grepstad's web page:

Excel Spreadsheet
(http://home.sol.no/~gjon/mdofcalc.xls)

Thanks Jon!

Mike

--
/---------------------\
Michael K. Davis
[email protected]
MIME Attachments OK
\---------------------/

From: "Michael K. Davis" [email protected]
Newsgroups: rec.photo.misc
Subject: Dof is Squelched by Diffraction
Date: 15 Aug 1998

Diffraction is the only aberration suffered by pinhole cameras.  It is an
image degrading phenomenon for which there is no means of correction.  The
smaller the aperture through which light must pass, the greater the effect
and the more an image at the film plane must be magnified to create a
print of a given size, the more visible the degradation in that print.
Small formats are proportionately more vulnerable to the effects of
diffraction than larger formats, so much so, that the smallest formats,
APS and to a lesser degree, 35mm, can not exploit the Depth of Field
advantage they have over large formats at their smallest available
apertures.  They are diffraction-limited to using wider apertures than
those which can be used by the larger formats.

Depth of Field is squelched by diffraction and the point at which this
happens moves to smaller f-number values (wider apertures) as the format
diagonal decreases. The good news is that at the diffraction-limited
apertures for each of the formats, the exact same depth of field can be
achieved, with the larger formats having a disadvantage of longer
exposures.  Let's look at why this is true.                 

John B. Williams authored a book called *Image Clarity, High-Resolution
Photography* that has a discussion of diffraction.  He gives the following
formula that calculates the radius (r) of an Airy disk
[#diameter?].  (G.B. Airy is the
astronmer who discovered diffraction in 1890 and diffraction's disks are
named after him, Airy, not airy.)  I can't type a lamda, so I have
substitued a "w" in the formula below, for wavelength.  "f" is for focal
length and "a" is for diameter of the aperture.

r = 1.22w(f/a)  [# see posting - diameter not radius?]

OK,  f/a can be renamed N where N is the familiar f-number that describes
the ratio of focal length to aperture diameter.  That gives us this:

r = 1.22wN

In his book, Williams selects 0.0005 mm as an average wavelength of light,
but I prefer 0.000555 mm, or 555 nanometers as being the wavelength that
is dead center in the spectrum of sensitivity.  It happens to be a nice
yellow-green, not far from William's choice anyway. 

OK, moving on, that gives us this formula, using 555 nanometers for all
future calculations:

r = 1.22 * 0.000555 * N

r =  0.0006771 N

To convert this formula to diameter (d) instead of radius (r):

2 * r = 2 * 0.0006771 N

d =  0.0013542 N  

This is the diameter in millimeters.

It was at about this point that I decided I didn't like the fact that
William's formula had so few significant digits in the constant 1.22, so I
went searching and found a longer, more accurate version of it and here it
is -- infinitely accurate:
      __
 1.21966   (66 repeating)

Not much different from 1.22, but it does change my formula for diameter
of an Airy disk to this:

d = 0.00135383 N

Previously, I introduced a bit of myself, so to speak, when choosing 555
nanometers as the average wavelength for calculating the diameter of Airy
disks and now I would like to state that I believe 1/175-inch is a good,
aggressive choice for maximum permissible circles of confusion when doing
depth of field calculations and thus, it is also my choice for the maximum
permissible diameter of Airy disks.  The reason this is expressed in
fractions of an inch instead of millimeters is because that is the
convention for discussions of circles of confusion and that convention
also adheres to stating such diameters not at the film plane, but rather
at the print, after magnification from a negative, and that print size is
a print with a 10-inch diagonal that is expected to be viewed at a
distance of ten inches. 

More specifically, the viewing distance is measured from the eye to any
equidistant corner, while centered over the print, not from the eye
straight down a line perpendicular to the plane of the print.  If the two
of us are discussing circles of confusion and are both adhering to this
convention, we will be comparing apples to apples even if you use 8x10 and
I use 35mm and better still, if you decide to make a 16x20 print and I
decide to make a 32x40 print, as long as viewing distances are equal to
the print diagonals in each case, they will both have the same perceived
sharpness (in so far as depth of field can effect sharpness) if we have
both chosen the same value (i.e. 1/175-inch) as the maximum permissible
diameter for circles of confusion for a 10-inch diagonal print to be
viewed at 10 inches.  Our mutual decision to limit circles of confusion to
1/175-inch in a 10-inch diagonal print would limit CoC's at the film plane
to 0.024724 mm for my 35mm, but your 8x10 could permit CoC's at the film
plane that are much larger, 0.185874 mm.  Both formats would however,
deliver the same illusion of depth of field after magnification to a given
print size, at a given viewing distance.

Quoting page 131 of "Basic Photographic Materials and Processes" by
Stroebel, Compton, Current, and Zakia (c1990 Focal Press):
"Permissible circles of confusion are generally specified for a viewing
distance of 10 inches, and 1/100 inch is commonly sited as an appropriate
value for the diameter. A study involving a small sample of cameras
designed for advanced amateurs and professional photographers revealed   
that values ranging from 1/70 to 1/200 inch were used -- approximately a
3:1 ratio."

It's somewhat subjective, but I like 1/175 inch, toward the more critical
end of the range used by manufacturers.  The larger the value you specify
for the denominator, the more conservative your calculated depths of field
will be.  The rotating-disk Depth of Field calculators published by Kodak
in their Photoguides use a generous, less critical value of 1/100 inch.

Using 1/175 inch, the maximum tolerable diameter of circles of confusion
for a given format can be calculated as the format diagonal divided by
1750.  (There would be 1750 circles set end to end along a print diagonal
that is 10 inches in length.)  As discussed above, the diameter of Airy
disks is calculated as 0.00135383 * f-number.

If we can calculate the aperture at which Airy disks become 1/175 inch
diameter when the format is enlarged or reduced to a 10-inch diagonal 
print, we will know the aperture at which it is pointless to make circles
of confusion any smaller than 1/175 inch.  This will be the aperture at
which a quest for more Depth of Field should be conducted using techniques
other than going to a smaller aperture (increasing the subject distance,
using tilts and swings, etc.)

Here we go: To set the size of the Airy disks equal to (and no larger
than) the tolerable diameter for circles of confusion for any format after
magnification or reduction to a 10-inch diagonal print to be viewed at 10
inches, I just have to equate to 1 the quotient had when circles of
confusion diameter is divided by Airy disk diameter, then reduce.

1 = (Format Diagonal mm / 1750) / (0.00135383 * f-number)

or

f-number = Format Diagonal mm / 2.36920501777           

Tah-dah!  My formula makes two assumptions.  This constant is specific
for yellow-green light at 555 nanometers and we don't want our Airy disks
to exceed 1/175-inch on a 10-inch print viewed at 10 inches.  You may
modify the constant proportionately if you want to change the values
0.000555 for wavelength or 1750 for the number of Airy disks set
end-to-end along a 10-inch diagonal print.

So, this calculated f-number is the aperture at which diffraction's Airy
disks would have a diameter of 1/175th inch in a 10-inch diagonal print.
As long as the viewing distance is equal to or greater than the print
diagonal, there would be no visible evidence of diffraction, no matter
how large the print is.  If however, the viewing distance were to be cut
to one half the print diagonal -- say a viewing distance of 12.5 inches
for a 16x20 print, then the aperture number would have to be cut in
half.


For example, using the formula above, the f-number at which the 8x10
format would begin to show evidence of diffraction in a print viewed at a
distance equal to its diagonal is:

  312.51 mm / 2.36920501777 = 132

So, according to the math, we can use f/90, but not f/128 (very near
f/132).

If we know in advance that we'll be viewing the final print at half the
print diagonal, we have to cut the f/number in half -- in this case,
from f/128 to f/64 -- a two-stop difference.  In this case, f/45 would be
acceptable, but not f/64.

Another easily overlooked point is that since this the formula uses format
diagonal, if you know in advance that you will be cropping to use only a
portion of the full image area, you should use the resulting cropped 
diagonal to calculate the f-number at which diffraction effects become
visible!  The smaller the diagonal, the greater the effects of diffraction
because of the increase in magnification necessary to yield a given print
size.


Format diagonal and maximum permissible diameter for the Airy disks are
the only variables for determining the f-number at which the effects of
diffraction become visible.  Using the formula given above, where Airy
disks will be limited to a diameter of 1/175 inch in a 10-inch diagonal
print, and which will be found equally acceptable in any size print as
long as the viewing distance is equal to or geater than the print diagonal
and where the full format diagonal is used, without cropping, I get the
following values for these formats:
                                                                        
Format                Full Diagonal   Diffraction     No Diffraction
                                      Visible at      Visible at

APS                    34.51 mm       f/14.56         f/11
35 mm                  43.27 mm       f/18.26         f/16
4.5x6 cm               69.70 mm       f/29.42         f/22
6x6 cm                 77.78 mm       f/32.83         f/22 + 1/2 stop
6x7 cm                 87.46 mm       f/36.92         f/32
6x9 cm                102.08 mm       f/43.09         f/32 + 1/2 stop
4x5 in                153.67 mm       f/64.86         f/45 + 1/2 stop
5x7 in                208.66 mm       f/88.07         f/64 + 1/2 stop
8x10 in               312.51 mm       f/131.9         f/90 + 1/2 stop
10x12 in              383.47 mm       f/161.9         f/128
11x14 in              447.78 mm       f/189.0         f/128 + 1/2 stop


Now let's generate the same table, but this time using the cropped image
diagonals that would be used to produce 4:5 aspect ratio prints, (8x10,
11x14, 16x20, etc.) instead of the full format diagonals.  Since the      
diagonals are smaller in some cases, where the full format diagonal is not
already a 4:5 aspect ratio, the resulting diffraction limits occur sooner,
at wider apertures!


Format                4:5 Cropped     Diffraction     No Diffraction
                      Diagonal        Visible at      Visible at

APS                    26.73 mm       f/11.28         f/8 + 1/2 stop
35 mm                  38.42 mm       f/16.22         f/11 + 1/2 stop
4.5x6 cm               66.43 mm       f/28.04         f/22
6x6 cm                 70.43 mm       f/29.73         f/22 + 1/2 stop
6x7 cm                 87.08 mm       f/36.75         f/32
6x9 cm                 88.04 mm       f/37.16         f/32 + 1/2 stop
4x5 in                153.67 mm       f/64.86         f/45 + 1/2 stop
5x7 in                193.69 mm       f/81.75         f/64
8x10 in               310.55 mm       f/131.1         f/90 + 1/2 stop  
10x12 in              377.78 mm       f/159.5         f/128
11x14 in              442.18 mm       f/186.6         f/128 + 1/2 stop


If viewing distance is one half of print diagonal, open up two more stops.
An acceptable aperture of f/16 for uncropped 35mm must be opened to f/8 if
the print will be viewed at a distance equal to half its diagonal!

At the top of this article I stated that at the diffraction-limited
apertures for each of the formats, the exact same depth of field can be
achieved, with the larger formats having a disadvantage of longer
exposures.  We've got the foundation to look at that, now.

Everybody laments that an 8x10 has less Depth of Field than a 4x5, than a
6x7, etc., assuming they are using equivalent focal lengths and people
argue that tiny formats like APS offer more depth of field, but thanks to
diffraction, the small format DoF advantage is squelched.    

The achievable diffraction-limited Near Sharp distances are IDENTICAL for
all formats, given that the ratio of focal length to image diagonal is the
same from one format to the next.  In other words, if several formats are
using focal lengths that are equivalent in their ratio to the format
diagonals, the effects of diffraction will limit each format to a unique
minimum aperture at which diffraction becomes visible AND it turns out
that if you calculate the Depth of Field for each focal length/image
diagonal pair AT THOSE UNIQUE APERTURES, you'll find that ALL the formats
can achieve the SAME Near Sharp (without movements and at different
f/stops, course).  The only disadvantage had by the larger formats is the
longer exposure times necessary to reach their diffraction-limited
apertures.  The smallest formats, can not use their smallest apertures,
but they too can achieve the same Near sharps had by the larger formats
using the apertures that aren't diffraction limited for them.  They have
no depth of field advantage, only the advantage of shorter exposures to
get the same depth of field larger formats have with longer exposures. 
(I'm compelled to mention here, that aside from the issues of depth of
field and diffraction, the larger formats benefit by all that comes with
having less magnification to get to a given print size and until someone
makes a 35mm with full movements, the larger formats also benefit by using
movements to control the position of the focus plane and perspective.)

In the table below, the third column (Near Sharp at f/22) was calculated
using a maximum permissible diameter for Circles of Confusion of 1/175th
of an inch.  The fourth column (Largest Aperture with Visible Diffraction)
was calculated with aerial disk diameters of 1/175th of an inch, also.
This reduces to the equation:

Format Diagonal in mm / 2.36920501777 = Aperture where diffraction becomes
visible.  Focal length is not a variable for this calculation.

Format,         Focal   Near Sharp      Largest         Near Sharp at
Using 4:5       Length  Distance        Aperture        Largest Aperture
Aspect Ratio    (mm)    at f/22         With Visible    With Visible
Diagonal                (feet)          Diffraction     Diffraction
                                        (f/stop)        (feet)

APS             24.6    3.0             11.3            5.7
35mm            35.3    4.2             16.2            5.7
6x7cm           80.0    9.6             36.8            5.7
4x5in          141.2    16.9            64.9            5.7
8x10in         285.3    34.2           131.1            5.7
11x14in        406.2    48.7           186.6            5.7

First, notice that at f/22, the Near Sharps are much closer for the
smaller formats.  (The Far Sharps are all nominally at infinity.)  So you
can see that 8x10 has half the Depth of Field enjoyed by 4x5, with a
resulting Near Sharp that's twice as far from the camera.  


But, also notice that the last column calculates the Near Sharp distances
for each format at each format's largest aperture with visible
diffraction.  The Near Sharps at THESE apertures all work out to be
exactly the same!  For these focal lengths, they are all 5.7 feet.  And
notice that thanks to diminished diffraction, the larger formats can bring
their Near Sharps to that had by APS and 35mm, just by stopping down to
apertures where small formats should not follow.

Diffraction limits them all to the SAME Near Sharp.  Large Format can get
just as much Depth of Field as small format because diffraction is getting
out of the way exactly in proportion to the loss of Depth of Field!
Guess what?  At the diffraction-limited apertures for each format, the
size of the hole the light passes through is identical in proportion to
the format diagonals!  That's why this whole discussion is true.

The above chart illustrates the fact that the impact of diffraction is
diminished linearly just as the Depth of Field diminishes with increase 
in format diagonal.

So, to avoid diffraction, the astute 35mm photographer stops down no
further than f/16, the APS photographer no further than f/11, etc. and the
depth of field promised at apertures smaller than f/16 can never actually
be enjoyed.  With a focal length of 35.3mm, the 35mm format can not enjoy
a Near Sharp closer than 5.7 feet.  BUT, this figure holds true for EVERY
format using equivalent focal lengths, at the diffraction-limited minimum
apertures for each format.  They can all achieve a Near Sharp of 5.7 feet
when stopped down to their respective diffraction-limited minimum
apertures.  Obviously, the larger formats will need longer exposures to
achive this Depth of Field (independent of movements) and the 11x14 format
would be hard pressed to find a lens with f/180.

So, in summary, I contend that even without their movements, large-format
cameras CAN achieve the exact SAME Depth of Field had by the smaller
formats (with longer exposures.)                

Mike

--
/---------------------\
   Michael K. Davis
  [email protected]
  MIME Attachments OK
\---------------------/ 

From: "Michael K. Davis" [email protected]
Newsgroups: rec.photo.misc
Subject: Re: Dof is Squelched by Diffraction
Date: 16 Aug 1998

I have an Excel-based DoF and Diffraction calculator that puts all this to practice.

1) You can specify the desired maximum permissible diameter for Circles of Confusion at the print.

2) You are encouraged to use a sub-calculator to determine the diagonal of that portion of the image that will actually be used to make the print. This figure is used for Depth of Field calculations instead of making the assumption that the full format diagonal should be used.

3) The diffraction limited aperture is calculated for the format you are using -- taking cropping into account!

You can find it at Jon Grepstad's web page:

http://home.sol.no/~gjon/mdofcalc.xls

Thanks Jon!

Mike

--
/---------------------\
Michael K. Davis
[email protected]
MIME Attachments OK
\---------------------/

From: Mark Herring [email protected]
Newsgroups: rec.photo.misc
Subject: Re: Diffraction
Date: Tue, 18 Aug 1998

To say that diffraction is not important in photography is like saying that tires are not important to cars. One thing I learned consistently (before ever learning the theory) is that almost all lenses are best in the middle of their aperture range----typically 2-3 stops down from wide open. Unless the lens is exceptional, the geometric abberations dominate when wide open. At the smallest aperature (highest f/no), diffraction dominates.

If you don't believe that this is important, try B and W printing. Looking thru a good grain focusser you can see the image degrade as you stop down past the optimum for the enlarging lens.

If you can't see the difference in your work--then by all means ignore the realities of physics.

From: [email protected] (Martin Tai)
Subject: DEPTH OF FIELD CALCULATIONS FOR MACRO
Date: Sun, 23 Aug 1998

   Most DOF equations for macro do not include diffraction into
consideration.

   When a lens  is stopped down, diffraction effect makes depth of
field shallower.

   I deduced the following equation which takes into account of
diffraction  effect


   DOF for macro = 2 * fstop * coc * R * (R+1 )  without diffraction
consideration      

   DOF for macro with diffraction effect included

       =  2* fstop * coc * R *(R+1 )/(1+0.02*fstop )


       R  = size of object/ size of image
       R=1  is life size,  R = 2 half life size

 Example

        R=1    f/16   coc  =0.03

        DOF  = 1.92   is reduced to  1.45mm

example 2

         R=2, f/2 


         DOF =  0.72  vs  0.69;  ie, at wide open , diffraction
effect is  minimal.

martin tai

From: [email protected] (David Rozen)
Newsgroups: rec.photo.equipment.35mm,rec.photo.misc
Subject: Re: DEPTH OF FIELD CALCULATIONS FOR MACRO
Date: 24 Aug 1998

Martin Tai ([email protected]) wrote:

:    Most DOF equations for macro do not include diffraction into
: consideration.

:    When a lens  is stopped down, diffraction effect makes depth of
: field shallower.

Since diffraction has a negative impact on sharpness, it will actully *increase* DoF, if DoF is defined as the range in which sharpness is not too different from the maximum sharpness at the focus point in the subject.

I have no argument with Martin's equations [although I haven't proofed them, or even read them.....] but Martin is too hung up on paperwork, and numerical predictions of results, and not enough concerned with actually getting results and seeing what really happens. In the real world, even camera shake has one positive attribute: it increases apparent DoF. Diffraction is not the monster that some fear it to be. When max DoF is barely enough to do the job, stop all the way down with no worry that ultimate theoretical sharpness is not achieved at any point in your shot. Or... you can worry about that ultimate lp/mm rating, shoot at "sharpest" aperture, and blow the whole shot to prove a theoretical point. There's good reason why even 50mm Micro-Nikkors close to f:32.

Regards, - dr

Date: 04 Feb 2000
From: [email protected]
To: [email protected]
Subject: AIRY DISK FORMULA

You mention in your article at: http://www.smu.edu/~rmonagha/mf/dofmath.html that the formula for the radius of the airy disk is:

r = 1.22w(f/a)

and I believe that formula is not for the radius but for the diameter of the airy disk, I found several web sites that confirm what I believe, for instance:

http://www.netacc.net/~poulsen/atmsect6.htm

Regards,

Guillermo

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