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I'm studying reflection of EM Waves at dielectric boundary from Optics by Eugene Hecht. But despite many attempts I am unable to understand

What is meant by statement "Here ϵr and ϵt are phase constant relative to EI and are introduced because position is not unique."?

Picture from book:

Page 112 Optics by Eugene Hecht

Thank You.

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2 Answers 2

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Actually one can shift the expression along the interface. For instance, consider the transformation $$ \mathbf{r} \rightarrow \mathbf{r} + \mathbf{r}_0 $$ where $\mathbf{r}_0$ is a vector lying in the plane of the interface. Then the term in the argument would become $$ \mathbf{k}_x\cdot\mathbf{r} \rightarrow \mathbf{k}_x\cdot\mathbf{r} + \mathbf{k}_x\cdot\mathbf{r}_0 , $$ where the subsript $x$ is either $r$ or $t$. Here $\mathbf{k}_x\cdot\mathbf{r}_0$ is a constant. Since one can do any such shift this constant represents an unspecified phase. Therefore , one should include these arbitrary phases.

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$\epsilon_r$ and $\epsilon_t$ appear to be phase terms to shift the reference plane of the reflection. If the reference plane or origin was chosen to be the interface then $\epsilon_r$ , $\epsilon_t$ $=0$. However, if you chose a reference plane that was not at the interface then you would add a phase term $\epsilon_r$ , $\epsilon_t$ $\neq$ $0$. These values should be proportional to the distance from the interface and the $\vec{k}$. Hope this helps.

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  • $\begingroup$ @flippiefanus is correct. If you are considering cases besides normal incidence. (Which you should.) Then there will be non-zero phase differences at different points along the boundary. $\endgroup$
    – ljs
    Commented Aug 20, 2016 at 15:03

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