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Applying an Abbe Criterion to Photomacrography
by 
Ted Clarke, Scientific Photographer and Instrument Maker 
Microscopists are generally familiar with Abbe’s
criterion for useful magnification being between 500 and 1000 times the
numerical aperture. This subject and its relation to digital imaging are
covered in my previous Modern Microscopy article “Pixel
Array Size Requirement to Replace Photomicrographs on Film”. Since
the readers of this article are most likely to be microscopists, the intent
of this article is to show how the Abbe criterion can be applied to photomacrography
over a magnification range between 1X and 50X. The macro lenses used for
photomacrography have iris diaphragms calibrated in f/numbers or exposure
factors (in the case of Zeiss Luminar lenses) and the problem for the
scientific photographer is what aperture setting to use when the subject
is three dimensional and the usual goal is to have the entire depth of
field in focus, which is more difficult as the magnification increases.
These macro lenses can be used over ranges of image magnification for
which they are aberration corrected, making their numerical aperture a
function of both the diaphragm setting and the camera magnification. The
term numerical aperture, N.A., is not common in the literature on photomacrography. The
purpose of this article is to show by example that an Abbe criterion chosen
within a three f/stop range provides a good balance between image resolution
and depth of field. The examples cover a magnification range of 1X through
50X, including contact printing of 1X negatives through 50X digital images. The
three f/stop range corresponds to an Abbe criterion of final magnification
between 320 and 660 times the N.A. The equations for calculating the f/stop
values are given in Table 1. The mathematical basis for Table 1 is given
in Appendix
I: Resolution Considerations for Photomacrography, a shortened version
of my article on the subject in Microscopy Today ^{1}.
TABLE
1 
Max
Resolution of Final Image 
Abbe
Criterion 
f/number 
Depth
of Field
mm for
8 lines/mm 
Depth
of Field
mm for
6 lines/mm 
Depth
of Field
mm for
3 lines/mm 
9 lines/mm
0.11 mm
(0.22 mm Airy Disk) 
320
N.A. 
160/(M_{Tot}+M_{enl}) 
38/M_{Tot}^{2} 
85/M_{Tot}^{2} 
200/M_{Tot}^{2} 
7 lines/mm
0.15 mm
(0.29 mm Airy Disk) 
440
N.A 
220/(M_{Tot}+M_{enl}) 
 
70/M_{Tot}^{2} 
260/M_{Tot}^{2} 
4.5 lines/mm
(0.44 mm Airy Disk) 
660
N.A 
330/(M_{Tot}+M_{enl}) 
 
 
330/M_{Tot}^{2} 
The first experiments were with a finely detailed
graphic arts pattern mounted on a 45 degree incline shown in Figure 1. The
pattern was photographed with a 35mm camera at magnifications of 1/8X,
1/4X, 1/2X, and 1X. An Olympus 50mm focal length macro lens was used for
apertures between f/8 and f/22. A 50 mm focal length enlarging lens modified
to use inserted Waterford stops in place of the iris diaphragm, shown
in Figure 2, was used for apertures between f/25 and f/100.
click image to enlarge (65K)
Figure 1 
click image to enlarge (125K)
Figure 2 
The half of the field below the focal plane of the
camera lens was recorded at 1X using two exposures to cover the field
width, followed by contact printing and splicing the two images together
to make the montage shown in Figure 3. Figures 46 show the other montages
for the patterns recorded at lower magnification and subsequently enlarged
to 1X during printing of the negatives. These montage images are from
scanned slides taken of the original montages published in my 1984 article
in The Microscope ^{2}.
click image to enlarge (132K)
Figure
3 
click image to enlarge (153K)
Figure
4 
click image to enlarge (164K)
Figure
5 
click image to enlarge (190K)
Figure
6 
