Edwin Ellefsen’s (President, Opticote, Inc.) response to a readers question concerning a mini article
appearing in the April 98 issue of Eyecare Business:
A reader found a difference in the answers for in total light transmission when using Fresnel’s equations (1)
and the formula published in “Ask the Labs” section of the April 98 issue. For example, using the Fresnel
equations the transmission through a non-absorbing substrate of refractive index 1.523 is 91.6%. The
article’s formula results in 91.76% transmission. Which one is correct?
Both are correct. The difference is in the question. When light strikes a lens, a fraction of the light is
reflected off the front surface. The remaining light continues through the lens and exits the back surface.
Again, a fraction of the remaining light is reflected off the back surface. In this case, the answer of 91.6%
transmission is correct considering only the front and back surface reflections.
But what happens to the back surface reflection? It goes back through the lens. Some exits the front
surface and a fraction is reflected towards the back surface where again some will exit the back and a
fraction will be reflected yet again towards the front. This will continue as ever-smaller amounts of light
exit the back surface. If light exiting the back surface from all these reflections is summed up we get
91.76% or the answer from the article’s formula.
Both answers represent two special cases. One is the minimum transmission the other is the maximum
transmission. In actual practice the amount of light transmitted through a lens and reaching the eye will be
between the two extremes. Conditions will vary as the wearer moves and changes the type and direction of
light entering the spectacles. Changes in the iris and lens will also affect the amount reaching the eye’s
light sensitive detectors.
If the majority of light rays striking a lens are at glancing angles the multiple internal reflections will exit
over a wider geometric area. The total amount of transmitted light entering the eye would be closer to the
91.6% result from the first example. If however the light is entering normal (perpendicular to the lens) and
the lens surfaces are parallel (little or no power) all the multiple reflections would reach the eye and the
result would approach the 91.76%.
The article does assume, as noted in the small print, light striking perpendicular to the lens. However, no
explicit mention of the lenses power or the size and placement of the detector (eye) was printed. I used the
qualifier close approximation instead.
The reader also noted that the formula could be reduced to the following.
2ns
(1 + ns 2 )
The above equation is the mathematical equivalent of formula given in the article. I used a more verbose
form of the equation because it is easier to show how the equation is derived from the Fresnel equations. I
assumed those who wanted a quick reference would use the pre-calculated table. For the technically
inclined readers, the following is additional detail illustrating the origins of the Fresnel equation in the
equation published. I have omitted some algebraic manipulation for the sake of brevity.
Tfs ∗ Tbs
Tavg =
1 − Rfs ∗ Rbs
4ns ∗ na
T=
(ns + na )2
T = 1− R
ns − na
R=
ns + na
2
From the above formulas, you can see the origins in the Fresnel equation. Tavg is in the form 1/(1-x) which
greatly simplifies the calculation of the infinite polynomial series that accounts for all multiple reflections.
(1)
(2)
Fannin & Grosverner’s CLINICAL OPTICS 2nd ed. Page 167
Angus Macleod’s THIN FILM OPTICAL FILTERS 2nd ed. Pages 67 & 68