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 4x5 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 (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.

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 (M_camera) 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:

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".

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.