Fresnel equations and energy ratios

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Discussion Overview

The discussion revolves around the Fresnel equations, specifically focusing on the concepts of reflectivity and transmissivity, and the treatment of energy ratios associated with incident and reflected waves at a boundary between different media. Participants explore the implications of superposition of fields and the calculation of energy density in this context.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the conventional approach of separating incident and reflected energies to calculate total energy entering a boundary, suggesting that the superposition of fields should be considered instead.
  • Another participant argues that because the incident and reflected waves travel in opposite directions, they do not interfere, making it appropriate to take the difference of their energies.
  • A different viewpoint is presented, stating that reflected and incident waves do interfere, particularly at normal incidence, where the reflected wave can nearly annihilate the incident wave if the second medium's index of refraction is sufficiently high.
  • One participant seeks clarification on the logic behind neglecting interference when deriving reflectivity and transmissivity coefficients, advocating for the use of the Poynting formula on the sum of the two waves.
  • Another participant confirms that the cross terms cancel when applying the Poynting vector to the combined fields, supporting the argument about the treatment of energy calculations.

Areas of Agreement / Disagreement

Participants express differing views on whether the incident and reflected waves interfere and how this should affect the calculation of energy ratios. There is no consensus on the correct approach to take regarding the treatment of these waves in the context of reflectivity and transmissivity.

Contextual Notes

The discussion highlights potential limitations in the assumptions made about wave interference and energy calculations, particularly in the context of deriving the Fresnel equations. The dependence on specific conditions, such as the polarization of light and the indices of refraction, is also noted but remains unresolved.

becko
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I'm having trouble understanding the reflectivity and transmissivity (the ratios of energy reflected and transmitted through a boundary separating different media).

Since energy is proportional to the square of the field, if you have a superposition of two fields at a point, to obtain the energy density at that place you cannot just sum the energies of the separate fields. You have to sum the fields and then square the total field.

So my problem is that to calculate the total energy that enters a patch of boundary, on the media of the incident wave, there are two fields, the incident and the reflected. Yet somehow everywhere I look they separate the energies, as the incident energy and the reflected energies, and then assume that the total energy entering the patch is the sum (incident energy - reflected energy) (with a minus sign because the reflected energy goes away from the patch). I don't understand this. Can someone help me here?
 
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Because the waves are going in opposite directions, they don't interfere, so taking the difference of the energies is appropriate.
 
As far as I know, the reflected and incident waves do interfere. In fact, at normal incidence, with light polarized perpendicular to the plane of incidence, if the index of refraction of the second media is large enough, the reflected wave will almost annihilate the incident wave.
 
So back to my question, what is the logic of neglecting this interference in deriving reflectivity and transmissivity coefficients?

Shouldn't one take into account that there are two fields in the first media, the incident and the reflected?

I think one should use Poynting formula on the sum of the two waves, instead of calculating the Poynting vector of each field separately, which is what I see is done to calculate the reflectivity.
 
If you take (E+E')X(B+B').n (where n is the normal to the plane), the cross terms cancel.
 
Last edited:
You're right ! . The cross terms do cancel in the component normal to the boundary plane. Thanks !
 

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