Derivation of waveguide condition for two light rays

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Discussion Overview

The discussion revolves around the derivation of a specific equation related to the phase difference between two light rays in a waveguide context. Participants are exploring the mathematical relationships and geometric interpretations involved in this derivation, particularly focusing on the implications of total internal reflection and the distances traveled by the rays.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions the notation used in the derivation, specifically the use of -Φm instead of Φm, and discusses the implications of phase changes due to total internal reflection.
  • Another participant notes that both rays 1A and 2A' travel the same distance but incur different phase changes, leading to a phase difference that is central to the discussion.
  • There is a mention of the equation AC - A'C = 2*(a-y)*cos(θm), with some uncertainty about how this relationship is derived and its relevance to the overall problem.
  • One participant expresses frustration over the lack of clarity in the derivation and suggests that the reference to it being an exercise for the reader feels dismissive.
  • Another participant inquires about Equation (3), indicating a potential gap in understanding the context of the equations being discussed.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the derivation or the clarity of the equations presented. Multiple viewpoints and uncertainties remain regarding the geometric interpretations and the derivation process.

Contextual Notes

There are unresolved assumptions regarding the geometric relationships and the definitions of the angles involved, particularly the role of (a-y) and its interpretation as a hypotenuse. The relevance of certain equations to the overall chapter is also questioned.

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How did they derive equation (6)?

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I don't like how they say -Φm instead of Φm, but that aside, I get that both rays 1A and 2A' travel the same distance, but AC incurs a phase change due to total internal reflection (-Φm) and it travels AC from there. Meanwhile, A'C is the distance the other ray travels. So the difference between these two is what sets the phase difference between the two.

Somehow, they made AC - A'C = 2*(a-y)*cos(θm) it seems. I should note that the angle that is split into two by the perpendicular line (really an arrow) is 2θ, so the angle by A, C, and the interface is θ. Beyond that, I don't really know how to approach this. They conveniently said it is left as an exercise for the reader, but that always seems like laziness to me. It's not clear to me where it came from and I've spent several hours trying to "get it."
 
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Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 
I haven't figured this out since I posted, no. I'm still curious where this came from. It's not actually relevant to the rest of the chapter. I think somehow (a-y) might be a hypotenuse? But I don't see it.
 
What is Equation (3)?
 

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