How does the Beer Lambert law work at non-normal incidences?

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Discussion Overview

The discussion centers on the application of the Beer-Lambert law in calculating the intensity of transmitted light through a medium at non-normal incidence, specifically at an angle of 45 degrees. Participants explore the interaction of light with a thin film of aluminum on a glass substrate, considering both reflection and absorption.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant seeks to determine the proportion of intensity lost due to reflection when light hits a thin aluminum film at 45 degrees.
  • Another participant suggests using the Fresnel equations to calculate the reflected light, emphasizing the need for the complex index of refraction of aluminum.
  • A participant inquires whether to use the real index of refraction for reflectivity calculations and the complex part for attenuation calculations.
  • It is noted that reflectivity is influenced by both the real and imaginary parts of the index of refraction.
  • A participant calculates a reflectivity of 93% but questions the applicability of this value considering the film thickness.
  • Another participant acknowledges the calculated reflectivity and suggests using the Beer-Lambert law to determine the intensity reaching the lower surface of the film, factoring in reflectivity again for the final intensity in the glass.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate use of the Beer-Lambert law and the Fresnel equations in this context. There is no consensus on the implications of film thickness on the calculations or the feasibility of light transmission through the aluminum film.

Contextual Notes

Participants highlight the complexity of the problem, noting the dependence on the thickness of the film and the angle of incidence, as well as the need for accurate values of the complex index of refraction.

warfreak131
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I want to calculate the intensity of a transmitted wave in a medium, but not at normal incidence. I want to consider the case of 45 degrees. What proportion of the intensity of the wave is lost by reflection?

Specifically I am looking at a the reflection off of a thin film of aluminum deposited on a glass substrate.
 
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That's not so much a question about Lambert Beer. Rather you have to use the Fresnel equation
http://en.wikipedia.org/wiki/Fresnel_equations
with some expression for the complex index of refraction of aluminium, to first determine the amount of reflected light from the aluminium surface. The latter one can be found e.g. in the Handbook of physics.
 
Thanks Dr Du. Since my light source is going to be unpolarized, the total reflectance is going to be (Rs+Rp)/2. Since both Rs and Rp are dependent on n2, would I just use the real index of refraction to calculate the amount reflected, and then when calculating what is attenuated, use the complex part as well?
 
No, reflectivity also depends on the imaginary part,too.
 
Okay, so I calculated Rp+Rs / 2, and I got 93% reflectivity, but that doesn't take into account the thickness of the film. I think it's more appropriate as a way to model a thick film where practically nothing makes it through the aluminum film.

EDIT: Or is it just impossible, regardless of the film thickness?
 
Last edited:
93% reflectivity sounds at least reasonable. The remaining 7% will get either absorbed or transmitted. Given the thickness of the film and the angle of refraction you calculated, you can calculate the effective path length in the aluminium film and use Lambert Beer to calculate what reaches the lower surface. There you should take reflectivity into account again, to calculate the final intensity in the glas.
 

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