Reflection of Optical Waves on a mirror may be modeled using the Helmholtz equation, k1 • r = k2 • r, where r = (x,y,0) (following the example below on the z = 0 plane), k1 = (k0*sinθ1, k0*cosθ1) and k2 = (k0*sinθ2, k0*cosθ2). The wavenumbers k1 and k2 in this formula, is assumed to be equal as this is one property of reflected waves. This simplifies to the expression θ1 = θ2, which means that the angle of incidence is equal to the angle of reflection.
Refraction of Optical Waves at a planar boundary can be described using the Helmholtz equation and Snell’s Law. For the below scenario, where k1 is the incident wave, k2 is the refracted wave and k3 is the reflected wave, the Helmholtz equation is satisfied by the following vectors:
Helmholtz equation: k1 • r = k2 • r = k3 • r , for r = (x,y,0)
k1 = (n1*k0*sinθ1, 0, n1*k0*cosθ1)
k2 = (n2*k0*sinθ2, 0, n2*k0*cosθ2)
k3 = (n1*k0*sinθ3, 0, -n1*k0*cosθ3)
This relationship may be simplified to show that θ1 = θ3 (essentially proven in the previous example of reflected waves) and n1*sinθ1 = n2*sinθ2 (Snell’s Law). Note that it is not possible with Wave Optics to describe the magnitudes of the incident, reflrected and refracted waves and a more rigorous method, such as Electromagnetic Optics is required to explain such phenomenon.
(1) B. E. A. Saleh and M. C. Teich, Fundamentals of photonics. Hoboken: Wiley, 2019.