Derivation of Mathematical Formulas for Exposure Compensation in Close-up Photography

January 01, 2016

Derivation of the Net Exposure Factor

 

                                                                                                                                                           (Eq 1)

 

 

 

Thin Lens Equation

 

 

The Inverse Square Law: the intensity of light (energy per unit area) emanating from a point light source is inversely proportional to the square of the distance from the light source at that point in space and time.


First, let’s derive the light intensity lost due to magnification (i.e., extending the lens from the film plane). From the Inverse Square Law, in terms of the light traveling from the lens plane a distance Di to reach the film plane, the intensity of light striking the film plane before (I1 ) and after (I2 ) the lens extension is given by the ratio:

              (Eq 2)

Where Di1 is the original distance between the image and lens plane and Di2 is the new distance between the image and lens plane. In other words, Di2 is the sum of the original image distance and the Extension of the lens from the film plane. From the basic lens diagram and the geometric law of similar triangles, the Magnification is given by:

                                                           

Thus, magnification is the ratio of the image height, Hi, on the film plane to the subject height, HO. Intuitively, it should also make sense that the greater the distance between the image and lens plane (i.e., the more the lens is extended from the film plane), the greater the magnification.

Rearranging the equation for magnification (in terms of the image and subject distances) followed by substitution into Equation 2 gives:

 

                                                                                                   (Eq 3)

 

Equation 3 represents the light intensity lost at the film plane due to magnification. Next, let’s derive the light intensity gained at the film plane due to close focusing (i.e., moving the camera/lens toward the subject). From the Inverse Square Law, in terms of the light traveling from the subject a distance DO to reach the lens plane, the intensity of light striking the film plane before (I1 ) and after (I2 ) the camera/lens is moved closer to the subject to achieve focus is given by the ratio:

                                              (Eq 4)

 

Starting with the classic thin lens equation and applying algebraic manipulations, we eventually have:

 

Since we know from above that magnification is the ratio of the image distance to subject distance, we have:

 

Next, substituting this new relationship for DO into Equation 4 gives:

                                                                     (Eq 5)

Equation 5 represents the light intensity gained at the film plane due to close focusing (i.e., decreasing the lens-subject distance). Next, substituting both Equation 3 and Equation 5 into Equation 1, we have:

 

Since M2 >M1, the net Exposure Factor is less than unity. Hence, there is net light loss at the film plane. In the case of starting off from zero magnification (infinity focus), M1 is zero and we have

 

In order to neutralize this inevitable light loss at the film plane and achieve a nominal exposure density on the film, this same factor (converted to a reciprocal factor) must be applied to the exposure indicated by the light meter. Hence, the Exposure Factor correction becomes:

 


 

To derive relationships for the Exposure Factor and Magnification expressed in terms of the extension (or bellows), we start with the thin lens equation:

The expression for magnification then becomes:

 

 

By definition, the image distance, Di, equals the focal length, f, when the lens is focused at infinity. Magnification occurs when the lens is extended beyond infinity focus. The difference between Di and the focal length equals the extension, EXT, of the lens past infinity focus. Therefore, the magnification can be expressed as:

 

The salient feature of this expression is that magnification is directly proportional to the lens extension in use:

Intuitively, what this expression tell us is that by allowing the light to travel farther between the lens and the image plane, the projected image, and thus the projected subject, will enlarge.

In the case of a camera using solely a bellows attachment directly between the film plane and the lens, we know that when the lens is focused at infinity, by definition the bellows length equals focal length. Subsequently, when the lens is extended from infinity to magnify, the bellows length is the sum of the focal length and the extension of the lens past infinity focus B = EXT + f . With a little more algebra, the magnification becomes:

 

Expressed In terms of the bellows length, with more algebraic manipulations the Exposure Factor correction becomes:

 

Intuitively, what this expression tell us is that by allowing the light to travel farther between the lens and the image plane to magnify, the exposure loss to be compensated varies with the square of that distance (i.e., the extension):

 

Thus, even a small increase in the lens extension results in a substantial loss of exposure at the image plane.

 


                                                                                                             

 

Derivation of the Subject-Lens Distance Threshold for Exposure Compensation


 

Rule of thumb: the threshold lens-subject distance at and within which exposure compensation must be applied for light loss due to magnification corresponds to a minimum distance of nine focal lengths. If the lens-subject distance is exactly nine focal lengths, an exposure compensation of + stops of light must be applied.

 

A subject distance of 9 focal lengths corresponds to a magnification of:

 

 

Of course, at a subject distance of twice the focal length, the magnification will be 1:1 (100%), and the exposure compensation will be +2 stops of light.

 


 

 

Derivation of the “Effective f-stop"

 

By definition, the physical size of the aperture equals the ratio of the focal length to the f-stop. Let f1 be the focal length at infinity and f2 the "effective  focal length" -- the sum of the focal length at infinity and the extension of the lens from infinity focus. Simmilarly, let F stop 1 be the actual F-stop at infinity and F stop 2 be the "effective F stop" at the point of lens extension: 

 

 

 
 

 

 

 

 

 

 

 

 

 

In the special case of a 1:1 magnification where by definition two stops of exposure are lost at the film plane, the “effective f-stop” is equal to twice the working F stop:

 

 

Thus, a working f-stop of f/16 used to make an exposure at 1:1 behaves “as if” the real f-stop were f/32 to make the exposure, since two stops of exposure are lost at the film plane, yet the actual physical size of the aperture is unchanged.

If the photographer chooses to make use of an “effective f-stop” to calculate Exposure Factor, then this “effective f-stop” can be used on the light meter to determine the corrected shutter speed to make the exposure.

For example, in traditional fashion, if the spot light meter gives an EV of 9.0 at f/16 with ISO 100 film, the shutter speed would be ½ second. Multiplying this shutter speed by 4x, or two stops, gives a corrected shutter speed of 2 seconds. Alternatively, if the photographer sets an “effective aperture” of f/32 on the light meter, then the corrected shutter speed is read off directly from the meter as 2 seconds.

In terms of the “effective focal length”, the focal length at very close range behaves “as if” it were increased by an amount equivalent to the lens extension. That is,

For example, at 1:1 magnification a lens with a focal length of 150 mm behaves “as if” it has an actual focal length of 300 mm, since the lens must be extended by 150 mm to create a 100% magnification. Thus, since light must travel twice the distance to reach the film plane compared to infinity focus at 150 mm, the Inverse Square Law dictates that the light intensity exposing the film is reduced by ¼, or two stops of light.

 


 

 
Derivation of the Image Circle Diameter at 1:1
 
 
 

 

 

 

By the Inverse Square Law,

By similar triangles,

The Inverse Square Law becomes:

 

By definition, at 1:1 magnification the exposure at the film plane is reduced by ¼. The relationship becomes:

Thus, at 1:1 the diameter of the image circle doubles.


 

If the photographer prefers to work at a set “working distance” (closely approximated by the subject-lens distance) and does not wish to compromise this distance for technical or aesthetic reasons, then more magnification can be attained by simply using a lens of a longer focal length, provided of course, the photographer has the means to increase the lens extension. Mathematically, this is shown by the following relation:

 

At a fixed subject distance (working distance), the magnification can be increased with a longer focal length. But to obey the thin lens equation, the image distance, DI (lens extension), must also be increased.

Alternatively, if the photographer has a limited amount of lens extension with which to work but does not care about the working distance, then increasing magnification can be achieved by using a lens of a shorter focal length, but this of course would involve moving the lens even closer to the subject. Mathematically, this is shown by the following relation:

At a fixed lens extension, the magnification can be increased with a shorter focal length. But to obey the thin lens equation, the subject distance, DO (working distance), must also be decreased.



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