#9. The Effective Aperture Number
True or False?
The effective aperture number A(1 + m), where A is the aperture number and m is the magnification, is an unwarranted and ineffective attempt to replace the quantity A/(1 – f/u) in the exposure equation with the quantity A(1 + m).
True.
Many authors attempt to correct an approximate distance dependence in photography equations by replacing the aperture number A with an effective aperture number A(1 + m), where m is the magnification m = v/u = (image distance)/(subject distance). The Manual of Photography by Jacobson, Ray and Attridge, as an in-depth text, offers a variety of examples of this replacement, including: the depth of field (field of focus) equations (p42), the close up depth of field (p44), the depth of focus (p42), the image illuminance or Illumination Equation (p50), the exposure compensation for close-up photography (p53).
The Manual of Photography offers a definition for the effective aperture number as the aperture number corrected for lenses not focused at infinity. Each of these replacement opportunities arise because the derivations of the equations involve the assumption that the lens is focused at infinity so that the image distance v can be replaced by the focal length f. This is appropriate based on the Focusing Equation
1/v = 1/f – 1/u
when the subject or focusing distance u gets very large and its reciprocal 1/u gets very small. When 1/u approaches zero:
1/v = 1/f or v = f
The assumption can in most places be avoided by leaving the expression for the image distance in the derivations of the equations in place and unchanged.
v = f / (1 – f/u)
This expression for v can certainly be approximated as v = f when u is very large, but it can also be carried through a derivation in full producing a result that is exact for all subject (focusing) distances u.
The typical treatment of the Illumination Equation (or the Exposure Equation that depends on it) approximates the quantity A/(1 – f/u) with its value for large u, that is, simply A. Going back to address smaller values of u (and larger values of the magnification m), the aperture number A is replaced with the effective aperture number A(1 + m).
The use of the effective aperture number is unwarranted because the quantity A/(1 – f/u) can be written exactly as A(1 + m)/m. These expressions are equal to A(1 + m) only in the trivial case of m = 1. The effective aperture number is not an effective replacement because it is incorrect (not just inexact) and because an exact expression for all distances can be easily carried through the derivation.
The derivation of the Illumination Equation for all subject distances is demonstrated and discussed in the book Photographic Exposure Calculations and Camera Operation.
Copyright 2008 Michael G. Prais, Ph.D.
For a readable but in-depth analysis of this concept along with many other concepts associated with photographic exposure, take a look at the book Photographic Exposure Calculations and Camera Operation. This book provides insight into the equations that govern exposure, exposure meters, photosensitive arrays (both solid-state and emulsion) and the Zone System as well as concepts associated with resolution, dynamic range, and depth of field.
The book is available through Amazon.com (ISBN 978-1-4392-0641-6) where you can Search Inside!™.
Check https://michaelprais.me under Photography for the table of contents, an extensive list of the topics and subtopics covered, the preface describing the purpose of the book, and a diagram central to the concepts in the book.
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