Laws of Geometric Optics

(568) |

in the region , and

(569) |

in the region . Here, represents the magnetic component of the resultant light wave, the amplitude of the incident wave, the amplitude of the reflected wave, and the amplitude of the refracted wave. All of the component waves have the same angular frequency, , because this property is ultimately determined by the wave source. Furthermore, according to standard electromagnetic theory (Fitzpatrick 2008), if the magnetic component of an electromagnetic wave is specified then the electric component of the wave is fully determined, and vice versa.

In general, the wavefunction, , must be continuous at , because, according to standard electromagnetic theory (see Appendix C), there cannot be a discontinuity in either the normal or the tangential component of a magnetic field across an interface between two (non-magnetic) dielectric media. [The same is not true of an electric field, which can have a normal discontinuity across an interface between two dielectric media (ibid.). (See Section 8.8.) This explains why we have chosen to represent the magnetic, rather than the electric, component of the wave.] Thus, the matching condition at takes the form

(570) |

This condition must be satisfied at all values of , , and . This is only possible if

and

Suppose that the direction of propagation of the incident wave lies in the - plane, so that . It immediately follows, from Equation (572), that . In other words, the directions of propagation of the reflected and the refracted waves also lie in the - plane, which implies that , and are co-planar vectors. This constraint is implicit in the well-known laws of geometric optics (Hecht 1974).

Assuming that the previously mentioned constraint is satisfied, let the incident, reflected, and refracted wave normals subtend angles , , and with the -axis, respectively. (See Figure 44.) It follows that

(573) | ||

(574) | ||

(575) |

where is the vacuum wavenumber, and the velocity of light in vacuum. Here, we have made use of the fact that wavenumber (i.e., the magnitude of the wavevector) of a light wave propagating through a dielectric medium of refractive index is . (See Section 7.7.)

According to Equation (571), , which yields

(576) |

and , which reduces to

The first of these relations states that the angle of incidence, , is equal to the angle of reflection, . This is the familiar

Incidentally, the fact that a plane wave propagates through a uniform
dielectric medium with a constant wavevector, and, therefore, a constant direction of motion, is equivalent to the well-known *law of rectilinear propagation*, which
states that a *light ray* (i.e., the normal to a constant phase surface) propagates through a uniform medium in a straight-line (Hecht 1974).

It follows, from the previous discussion, that the laws of geometric optics (i.e., the law of rectilinear propagation, the law of reflection, and the law of refraction) are fully consistent with the wave properties of light, despite the fact that they do not seem to explicitly depend on these properties.