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- Homework Statement
- This is more so a conceptual question than a homework question, from my instructors notes;

(I'm assuming that the first medium is air, even though my instructor does not explicitly say so, I figure I can address this case first and then generalize)

(The subscript

"If for example the medium on which the radiation is incident is a conductor then the refractive index ##\eta_t## will be complex.

In the case that the angle of incidence is greater than the critical angle (##\theta_c##), then

$$\frac{ \sin \theta_i}{\sin \theta_t} = \frac{\eta_t}{\eta_i} = \sin \theta_{c} \Rightarrow \sin \theta_t = \frac{\sin \theta_i}{\sin \theta_c} \gt 1$$

and then the sine and cosine of the transmission angle must be complex. This serves to indicate that the radiation is rapidly attenuated inside the medium.

- Relevant Equations
- Snell's Law

$$\eta_i \sin \theta_i = \eta_t \sin \theta_t$$

Incident Angle

$$\theta_i = \sin^{-1} \left( \frac{\eta_t}{\eta_i} \sin \theta_t \right)$$

Transmitted Angle

$$\theta_t = \sin^{-1} \left( \frac{\eta_i}{\eta_t} \sin \theta_i \right)$$

Critical Angle (corresponds to ##\theta_t = \frac{\pi}{2}##)

$$\theta_{ic} = \sin^{-1} \left( \frac{\eta_t}{\eta_i}\right)$$

I for one don't see how ##\sin \theta_t \gt 1## is possible, even when you extend into the complex numbers. Is there even a way to order the complex numbers? Does he mean to say that the

Anyway here's my attempt at interpreting what my instructor is trying to say. Forgive the egregious handwaving and heuristic arguments, I'm not well versed in complex analysis.

If ##\theta_i \gt \theta_{ic}## then

$$\sin^{-1} \left( \frac{\eta_t}{\eta_i} \sin \theta_t \right) \gt \sin^{-1} \left( \frac{\eta_t}{\eta_i}\right)$$

##\sin^{-1}## is strictly increasing over one period so it follows that if ##\sin^{-1} x \gt \sin^{-1} y## then ## x \gt y##

So we have

$$\left( \frac{\eta_t}{\eta_i} \sin \theta_t \right) \gt \left(\frac{\eta_t}{\eta_i} \right)$$

Multiply through by ##\frac{\eta_i}{\eta_t}## and we get

$$\sin \theta_t \gt 1$$

But how can ## \sin \theta_t## be greater than ##1## (even if we extend into the complex plane), and how does that imply ##\theta_t## must be complex and how does that in turn imply "rapid attenuation inside the medium"?

I'm at an impasse because I don't know enough complex analysis to resolve this in my opinion, nor do I know enough about attenuation to see the connection.

As always any helpful hints and insight is greatly appreciated.

__magnitude__is greater than 1?Anyway here's my attempt at interpreting what my instructor is trying to say. Forgive the egregious handwaving and heuristic arguments, I'm not well versed in complex analysis.

If ##\theta_i \gt \theta_{ic}## then

$$\sin^{-1} \left( \frac{\eta_t}{\eta_i} \sin \theta_t \right) \gt \sin^{-1} \left( \frac{\eta_t}{\eta_i}\right)$$

##\sin^{-1}## is strictly increasing over one period so it follows that if ##\sin^{-1} x \gt \sin^{-1} y## then ## x \gt y##

So we have

$$\left( \frac{\eta_t}{\eta_i} \sin \theta_t \right) \gt \left(\frac{\eta_t}{\eta_i} \right)$$

Multiply through by ##\frac{\eta_i}{\eta_t}## and we get

$$\sin \theta_t \gt 1$$

But how can ## \sin \theta_t## be greater than ##1## (even if we extend into the complex plane), and how does that imply ##\theta_t## must be complex and how does that in turn imply "rapid attenuation inside the medium"?

I'm at an impasse because I don't know enough complex analysis to resolve this in my opinion, nor do I know enough about attenuation to see the connection.

As always any helpful hints and insight is greatly appreciated.