Refraction

Light changes speed when it enters a different transparent medium. At an oblique boundary the direction of its transmitted ray generally changes too; this is refraction. The refractive index is $n=c/v$, where c is the vacuum light speed and v is the light speed in the material. A higher n means slower light. Use $c\approx3.00\times10^8\;\mathrm{m/s}$ in these calculations.

At a stationary boundary the frequency stays fixed by the source. In the wave-speed relationship, $v=f\lambda$, f denotes frequency (cycles per second, in Hz) and λ denotes wavelength (distance between successive equal-phase fronts, in metres). This frequency f is different from the focal-length f used in mirror and lens equations. With frequency fixed, a lower speed means a shorter wavelength, not a lower frequency. At normal incidence the direction stays straight even though speed and wavelength change. A refracted-ray diagram should show both media, their indices, the boundary, a normal, and arrows crossing it.

Snell’s law is $n_1\sin\theta_1=n_2\sin\theta_2$. Medium 1 is where the incident ray travels; medium 2 is where the transmitted ray would travel. Both angles are measured from the same interface normal. For lower-to-higher index, the ray bends toward the normal (smaller θ). For higher-to-lower index, a transmitted ray bends away from the normal (larger θ). At normal incidence θ₁ = 0 gives θ₂ = 0: no directional bending.

For a parallel-sided slab surrounded by the same medium, Snell’s law at the second surface restores the original direction, but the exiting ray can be shifted sideways. A partly reflected ray can coexist with the refracted ray at a boundary; refraction does not require all light energy to be transmitted.

Total internal reflection is possible only for light approaching a lower-index medium from a higher-index medium, $n_1>n_2$. Increase the incident angle and the transmitted angle grows until it reaches 90°. At this critical angle the transmitted ray runs along the interface. Setting sin θ₂ = 1 in Snell’s law gives $\theta_c=\sin^{-1}(n_2/n_1)$.

For θ₁ > θ_c there is no real transmitted-ray angle: Snell would require sin θ₂ > 1. In the ideal lossless geometric model, all incident light reflects back into medium 1 and no light is transmitted into medium 2. Do not draw an impossible bent ray. At θ₁ = θ_c the boundary case is a grazing transmitted ray; total internal reflection is the beyond-critical case. Lower-to-higher index has no critical angle for this purpose, no matter how large its incident angle is.

For n₁ = 1.50 and n₂ = 1.00, θ_c = 41.8°. An internal 30° ray transmits and bends away; a 41.8° ray is the grazing limit; a 50° ray reflects entirely in the ideal model. A real device can have absorption or surface imperfections; “total” here refers to the ideal reflection-versus-transmission boundary condition.