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/(MTot+Menl)
38/MTot2
85/MTot2
200/MTot2
7 lines/mm
0.15 mm
(0.29 mm Airy Disk)
440
N.A
220/(MTot+Menl)
70/MTot2
260/MTot2
4.5 lines/mm
(0.44 mm Airy Disk)
660
N.A
330/(MTot+Menl)
330/MTot2

 

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.

 



Figure 1


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 4-6 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.

 


Figure
3

Figure
4

Figure
5

Figure
6

 

Figure 7 is a higher resolution close- up showing
the finest detail at the object focal plane in the transition region where
the loss in detail with higher f/numbers (smaller apertures) becomes evident. My
own preference is to use an Abbe criterion value of final magnification
equal to about 440X N.A. when depth of field is a key issue. The corresponding
calculated f/numbers from this Abbe criterion are f/110, f/73, f/44, and
f/24 for the 1X test patterns shown in Figure 7.  

 


Figure
7

Figure
8

 

The next experiment consisted of photographing a
dried butterfly at 1/4X magnification using the Olympus 135 mm f/4 macro
lens with a tripod mounted OM-4 camera and dual flash system shown in
Figure 8. The tripod mounted camera was kept focused on the body of the
insect while exposures were made at whole f-stop increments between f/16
and f/45 while using a cable release. The resulting test images in Figures
9-12 were first shown at Inter/Micro-84. Note that the depth of field
increases with smaller apertures. The loss in fine detail for the f/32
and f/45 images is evident in the close-up montage of Figure 13.  My preference
for the optimum aperture from study of these images is f/22 and in agreement
with an Abbe criterion of 440 times N.A. This is the aperture I would
choose for an 8X enlargement. I would use an aperture of f/45 for a 4X
enlargement to obtain an optimum balance of depth of field and high resolution.

 

click image to enlarge (614)

Figure
9
click image to enlarge (617)

Figure
10
click image to enlarge (610)

Figure
11
click image to enlarge (603)

Figure
12

Figure
13

 

The final part of this article demonstrates application
of the optimum aperture concept to digital imaging with a sequence of
increasing magnification images obtained with a scientific grade digital
camera, the Kodak (now Redlake Imaging) MegaPlus 1.6i/AB. This camera
is equipped with a Nikon “F” lens mount so it can be used with Nikon 35
mm camera lenses. My previous article “Pixel
Array Size Requirement to Replace Photomicrographs on Film
” demonstrated
that the 1024 x 1534 pixel image recorded by 9X13mm grayscale sensor of
this camera can meet an Abbe criterion of 500 times the N.A. in small
format print equivalent to a 10X enlargement of a negative or slide. The
camera is shown mounted to an Olympus 35mm camera bellows in Figure 14.

 


Figure
14

Figure
15

 

Figure 15 is a close-up view of the Olympus Auto
Bellows and the adapters made to mount the MegaPlus camera and Zeiss macro
lenses to the bellows. My specialty later in my working career was failure
analysis of gears. A test gear with a tooth broken out by cyclic fatigue
loading near the tooth tip is shown on a copy-stand in Figure 14. The
fracture surface left when a tooth was broken out of this gear is shown
at increasing magnifications in Figures 16-19. The scientific interpretation
of the topographical features of the gear tooth fracture surface is called
fractography. This test gear of medium carbon steel has been heat treated
using a new process called contour induction hardening. The arrow in Figure
16 indicates a smoother, circular region of slow crack growth from each
cycle of loading with radial markings indicating that the tooth had a
primary fatigue crack origin in the center of the circular area in the
interface between the hardened surface zone and the much softer core. The
origin region is at a nonmetallic inclusion stringer evident in Figure
19. Figures 16 and 17 were recorded using a 60 mm f/2.8 Nikon Micro Nikkor
macro lens instead of a bellows mounted lens. The recording conditions
were with camera magnifications of 0.5X and 1X at f/11. Figure 18 was
recorded with the Zeiss 100 mm f/6.3 Luminar lens with 2.5X camera magnification
using an aperture setting of f/6.3. Figure 19 was recorded with the Zeiss
63 mm f/4.5 Luminar lens at 5X camera magnification using an aperture
setting of f/4.5. Small format photographic prints of these digital images
for Figures 16-19 would have, respectively, the following Abbe criterion
values for total magnification: 330N.A., 440N.A., 440N.A., and 540N.A. The
plastic dovetail inserts of the Olympus Auto Bellows were found to be
cracked a few years after the gear fracture was photographed.    

 


Figure
16

Figure
17

Figure
18

Figure
19

 

All-metal heavy duty sliders and brackets for the
lens and camera were made in my home shop to correct this problem. The
modified bellows is shown in my Microscopy Today article “Heavy Duty Camera
Bellows for Digital Imaging” along with resolution test results for the
Zeiss 63mm f/4.5 Luminar lens and the Olympus 38mm f/2.8 macro lens 3.
The transition from photomacrography with a view camera and 4X5 Polaroid
film to digital imaging with the MegaPlus camera occurred very rapidly
in early 1995 in our metallography laboratory because of savings in time
and elimination of film cost. Polaroid film is still used in the laboratory
with the Zeiss Ultraphot II for low magnification photomacrographs obtained
with the Luminar lenses and their vertical illuminators. Polaroid film
is also used with a view camera system dedicated to brightfield imaging
of complete metallographic samples (see "Brightfield Illumination
of Complete Metallographic Specimens
")4. The Polaroid
film images are subsequently scanned to digital files.

 

 


Appendix I: Resolution Considerations for Photomacrography

My Inter/Micro-84 presentation and subsequent article in
The Microscope dealt with optimizing diffraction limited depth
of field in photomacrography and is referenced by Brian Bracegirdle in
his “Scientific Photomacrography5.  My analysis was based upon
the classical solution for the diffraction pattern image of a point source
of light, using the Rayleigh criterion of resolution. H. Lou Gibson’s
method of treating combined diffraction and geometric blurring away from
the object focal plane was used6. The calculations and experimental
results indicated that a final print resolution, for the object focal
plane, of 7 lines/mm (0.29 mm Airy disk) gives an optimum balance of depth
of field and resolution. The 7 lines/mm criterion is equivalent to an
Abbe criterion of total magnification equal to 440 times the numerical
aperture. The 500 times numerical aperture Abbe criterion for microscopists
with very acute vision corresponds to a maximum resolution of 6 lines/mm
in the final image. This resolution is not as good as our Oce 3045 office
copier used to duplicate our metallurgical reports containing 4×5 Polaroid
images.  The Oce, when properly adjusted, resolves 8 lines/mm on standard
copier paper in photo mode.

 

 

Basic Equations

  1. The relationship between the maximum print resolution, lens f/number setting, and magnification is as follows (assuming λ = 5.5 x 10-4 mm):

    Maximum print resolution
    = f/number (McameraMenlarging + Menlarging)
    6.7 x 10-4 (in mm). Where the maximum print resolution
    is equal to one-half the final Airy disk diameter after enlarging.

  2. The relationsip between the enlarged circle of confusion diameter in the final print, C, and the geometric depth of field is given by the following equation:

     

  3. The relationship between the numerical aperature (N.A.), f/number, and camera magnification (Mcamera) is given by the following equation:7

     

  4. Lou Gibson’s method is uaed to calculate the final image resolution at the depth of field/limits:

    (2 Print Resolution)2 = (2 Maximum Print Resolution)2 +
    C2 at depth of field limits.

Using the basic equations 1-4 (above), the following table (Table 1) of depth of field and resolution is derived for full f/stop increments:

 

TABLE
1
Max
Resolution of Final Image
Abbe
Criterion
f/number
Depth
of Field
mm for
8lines/mm
Depth
of Field
mm for
8lines/mm
Depth
of Field
mm for
8lines/mm

9 lines/mm
0.11 mm
(0.22 mm Airy Disk)

320
N.A.
160/(MTot+Menl)
38/MTot2
85/MTot2
200/MTot2
7 lines/mm
0.15 mm
(0.29 mm Airy Disk)
440
N.A
220/(MTot+Menl)
70/MTot2
260/MTot2
4.5 lines/mm
(0.44 mm Airy Disk)
660
N.A
330/(MTot+Menl)
330/MTot2

 

 

Discussion of Optimum Aperture Concept

H. Lou Gibson, at first, strongly objected to the results
given in Table 1. His previous analysis included all sources of image
blur, including recording and enlarging losses. His main objection was
my finding that, for a given maximum final image resolution for the object
focal plane, the depth of field was inversely proportional to the square
of the final magnification. He had concluded that the greatest depth of
field could only be obtained with a large format camera and no subsequent
enlarging. Our results were in agreement when an enlarging magnification
of 1X was used with my f/number and depth of field equations for a maximum
final image resolution of 7 lines/mm. Gibson’s original conclusion that
depth of field for 6 lines/mm resolution decreased when a significant
part of the final magnification was obtained by enlarging was the result
of a mathematical reasoning error in Gibson’s ray optics based depth of
field, which was correct only for 1X enlarging magnification (contact
printing). Subsequent correspondence, facilitated by Dr. Walter C. McCrone,
led Gibson to acknowledge my work was valid.  Gibson’s error was subsequently
corrected in his 1986 BPA article8, as noted in my letter to the editor
published in The Microscope9. I have copies of the
Gibson correspondence if anyone is interested in doing a historical study
of photomacrography. Gibson was the pioneer and great contributor to this
field.  Gibson agreed with me that the image did not “fall apart” until
the resolution was less than 3 lines/mm.

 

A more stringent criterion than 6 lines/mm is used
for the depth of field in conventional photography, where the circle of
confusion should not exceed 0.25mm (8 lines/mm) within the depth of field
10. A one stop decrease in f/number from that
giving 7 lines/mm in Table 1 gives a maximum resolution of 9 lines/mm
(0.22mm Airy disk diameter), a zone with 8 lines/mm resolution and a maximum
depth of field for 6 lines/mm resolution, but at the expense of a significant
loss of depth of field for 3 lines/mm print resolution. John Gustav Delly
preferred a 4.5 lines/mm resolution requirement, which gives the greatest
depth of field, when required, reaching a minimum of 3 lines/mm resolution. The
test images did not appear to be blurred until their resolution was less
than 3 lines/mm. These results are consistent with Abbe’s criterion that
useful magnification ranges between 6 lines/mm and 3 lines/mm. These values
of image resolution measured on the final print can be readily calculated
from the common definition of light microscope resolution:

 


 

Three dimensional objects requiring greater depth
of field than can be obtained with apertures calculated from Table 1 of
this article can be recorded without the depth of field limitations by
using scanning light photomacrography, as demonstrated in my previous
article "Constructing a Scanning Light Photomacrography System".

 

References

  1. Clarke, T. M. (1996) Resolution Considerations for
    Photomacrography and Photomicroscopy. Microscopy Today, May,
    10-11.

     

  2. Clarke, T. M. (1984) Method for Calculating Relative
    Apertures for Optimizing Diffraction-Limited Depth of Field in Photomacrography.
    Microscope, 32, 219-258.

     

  3. Clarke, T. M. (1998) Heavy Duty Camera Bellows for
    Digital Imaging. Microscopy Today, April, 12-13.

     

  4. Clarke, T. M. (2003). Brightfield Illumination of
    Large Field Sizes. Microscopy Today, July/August, 22-25.

     

  5. Bracegirdle, B. (1995) Scientific Photomacrography.
    RMS Microscopy Handbook, No. 3. Oxford, U.K.: BIOS Scientific
    Publishers Ltd.

     

  6. Gibson, H. Lou. (1969). Photomacrography: Mathematical
    Analysis of Magnification and Depth of Detail. Kodak Publication,
    No. N-15.

     

  7. Shillaber, C. P. (1944). Photomicrography in
    Theory and Practice
    . John Wiley & Sons Inc.

     

  8. Gibson, H. Lou (1986). Depth and Enlarging Factors
    in Ultra-Close-up and Photomacrographic Prints from Slides. BPA, 54, 127-142.

     

  9. Clarke, T. M. (1987). Letter to the Editor. The
    Microscope
    , 35, 332-336 (1987).

     

  10. Neblette, C. B. & Murray, A. E. (1973). Photographic
    Lenses.
    Morgan & Morgan, Inc.