Details of total internal reflection

[ShayanJ]

Consider snell's law $n_1 \sin{\theta_1}=n_2 \sin{\theta_2}$($n_1$ and $n_2$ are real).
We know that if $n_2<n_1$, there exists an incident angle called critical angle that gives a refraction angle of ninety degrees i.e. $\sin{\theta_c}=\frac{n_2}{n_1}$.

But if the incident angle is greater than the critical angle(i.e. $\sin{\theta_1}>\frac{n_2}{n_1}$),Then:$\sin{\theta_2}=\frac{n_1}{n_2}\sin{\theta_1}>1$

But we know that $\sin{\theta}>1$ can happen for no real $\theta$,so we say that $\theta_2$ should be complex:
$\theta_2=\alpha+i \beta$ and $\sin{\theta_2}=\sin{(\alpha+i \beta)}=\sin{(\alpha)}\cos{(i \beta)}+\sin{(i \beta)}\cos{(\alpha)}=\sin{(\alpha)}\cosh{(\beta)}+i\cos{(\alpha)}\sinh{(\beta)}$

But from snell'w law,we know that $\sin{\theta_2}$ should be real and so we should always have $cos{\alpha}=0 \Rightarrow \alpha=\frac{\pi}{2}$ and so $\sin{\theta_2}=\cosh{\beta}$.

This means that the only variable which is capable of giving information about the Total reflected ray,is $\beta$. But how?

Thanks

 

[mfb]

But from snell'w law,we know that $\sin{\theta_2}$ should be real

You cannot use a law in a parameter range where it does not apply.

This means that the only variable which is capable of giving information about the Total reflected ray,is $\beta$. But how?

What else do you need? The angle is just the same as the incident angle.

 

[dauto]

No reason to use complex angles here. You simply defined $\beta$ such that $$\cosh \beta = \frac{n_1}{n_2}\sin \theta_1$$

 

[dauto]

Now, what you do get from the treatment of this problem with complex angles is an expression for the evanescent waves $F = A e^{i\vec k_2\cdot \vec x},$ where $\vec k_2 = cos\theta_2\hat i + sin\theta_2\hat j$, and $\vec x = x\hat i + y\hat j$. Now if you plug in your complex parametrization $\theta_2 = \frac{\pi}{2} + \beta$, than you get $$F = A exp [iy\,cosh\beta - x sinh\beta]$$

 

[sardar]

for providing the details of total internal reflection and considering Snell's law. Total internal reflection occurs when a light ray travels from a denser medium to a less dense medium, and the incident angle is greater than the critical angle. In this case, the refracted angle becomes imaginary, as you have explained. This phenomenon is possible because of the complex nature of the refracted angle, which is dependent on the imaginary component, \beta . The value of \beta determines the intensity of the reflected ray, with higher values resulting in stronger reflection. This is why \beta is important in understanding the behavior of total internal reflection. Additionally, the angle of incidence, \theta_1 , also plays a crucial role in determining the critical angle and the resulting refracted angle. As a scientist, it is important to consider both the real and imaginary components of total internal reflection in order to fully understand and predict its behavior.