Applying an Abbe Criterion to Photomacrography

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 below  in Appendix I: Resolution Considerations for Photomacrography, a shortened version of my article on the subject in Microscopy Today ¹.

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.

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 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.

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.

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 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.

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. 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 (M_cameraM_enlarging + M_enlarging)
      6.7 x 10^-4 (in mm). Where the maximum print resolution
      is equal to one-half the final Airy disk diameter after enlarging.

1. 1. 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:

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

1. 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:

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 field¹⁰. 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

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

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

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

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

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

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

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

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

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

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