Transmitted Field (with greater than critical-angle Incidence)

[jeff1evesque]

Definition:
The phase velocities of the incident, reflected, and transmitted fields must be equal on the boundary. Another way to represent this relationship for the incident and transmitted fields is:
$$\beta_{1}sin(\theta_{i}) = \beta_{2}sin(\theta_{t} \Rightarrow sin(\theta_{t}) = \frac{\beta_{1}}{\beta_{2}}sin(\theta_{i}$$

Question:
Could someone elaborate on the definition above-

...must be equal on the boundary?

Thank you.

 

[cepheid]

The boundary refers to the boundary between the two media between which the light is propagating. Equal "on the boundary" means that right at that interface between the two media, the velocity of the wave is continuous. There is no abrupt discontinuity. The waves that are incident upon, reflected from, and transmitted through the boundary all have the same speed.

 

[Mene]

The definition is referring to the principle of equal phase velocities at the boundary between two media. When an electromagnetic wave travels from one medium to another, its velocity changes according to the properties of the new medium. The phase velocity is the speed at which the wave's phase (or position of its crests and troughs) travels. In order for the wave to continue propagating smoothly without any disruptions at the boundary, the phase velocities of the incident, reflected, and transmitted fields must be equal. This means that the wave's crests and troughs must line up at the boundary, otherwise there would be a mismatch and the wave would break or scatter. Mathematically, this is represented by the equation where the phase velocity of the incident field (β1) multiplied by the sine of the incident angle (θi) is equal to the phase velocity of the transmitted field (β2) multiplied by the sine of the transmitted angle (θt). This ensures that the wave's phase is conserved at the boundary and it can continue to propagate smoothly into the new medium.