Chapter 1 Foundations of geometrical optics

1.1 Electromagnetic waves in a homogeneous medium: the refractive index

The refractive index n of the medium, in which light propagates, is of crucial importance to the vast variety of optical phenomena. Further, we calculate the refractive index using a general approach based on Maxwell’s equations:

Here ρf is the density of free (noncompensated) charges, which for nonintentionally charged materials equals zero:

Differential operators in eqs. (1.1.) and (1.2) are called divergence (∇⋅), whereas those in eqs. (1.3) and (1.4) are rotor or curl (∇×). The displacement field D and magnetic induction B are connected, correspondingly, to the electric field E and magnetic field H as follows:

where ε0  and μ0 are, respectively, the dielectric permittivity and magnetic permeability of a vacuum, while εm and μm are, respectively, the dielectric and magnetic constants of a medium. Note that in anisotropic media, parameters εm and μm are, in fact, tensors of second rank. We will discuss this issue in more detail in Chapter 4, in relation to light polarization in anisotropic crystals. Here we consider isotropic materials such as glass, which is often used for fabricating conventional optical components. In this case, parameters εm and μm are scalars, though frequency dependent.

To proceed, we recall the linear relationship between the vectors of current density J and electric field E (Ohm’s law), which are interconnected via the specific electrical conductivity σ:

Together with eqs. (1.5)–(1.8), this yields:

Applying a rot (curl) vector operator (∇×) to both sides of eq. (1.11) and using eq. (1.12), one obtains:

Since rotrotE=graddivE−grad⋅grad E, then with the aid of eq. (1.9) we find:

where ∇2=∂2∂x2+∂2∂y2+∂2∂z2 is the Laplace operator. Substituting eq. (1.14) into eq. (1.13) yields

In a vacuum, μm= εm= 1, whereas the conductivity σ = 0. Therefore, eq. (1.15) converts to the standard wave equation

with the light velocity in a vacuum equal to

In fact, substituting the plane wave

into eq. (1.16) yields the dispersion law, that is, the relationship between the light frequency ω and wave vector k

where k0 is the wave vector in vacuum. One can rewrite the dispersion law (1.19) in its more familiar linear form

justifying definition (1.17).

In the case of a homogeneous medium, differing from a vacuum (i.e., with μm≠1, εm≠ 1, and generally nonzero conductivity σ > 0), a similar procedure leads to the modified wave equation:

Substituting the plane wave (1.18) into eq. (1.21) yields

and

Expression (1.23) is, in fact, a new dispersion law for an electromagnetic wave in a medium. This dispersion law (1.23) is very different from that of an electromagnetic wave in a vacuum (1.20):

Using eqs. (1.23) and (1.24), one can calculate the refractive index n=k/k0, which determines the key optical phenomena for electromagnetic waves interacting with a material

Note that the refractive index comprises the real and imaginary parts, being responsible, respectively, for light refraction and attenuation. We elaborate further on this issue in Chapter 6.

In dielectrics (e.g., in a glass), the conductivity σ = 0; correspondingly, the wave equation (1.21) transforms into

with the phase velocity of light in a medium equal to

In other words, the refractive index

shows in what proportion the phase velocity of light propagation in a medium (Vp) is decreased with respect to that in a vacuum c.

For nonmagnetic materials (μm = 1), eqs. (1.25) and (1.28) are transformed into

and further to the well-known expression for nonmagnetic dielectrics (σ = 0):

In most materials used for visible light optics, nd is around 1.3−1.8. In contrast, the refractive index for X-rays is a bit less than 1, that is, n=1−δ, with δ≈10−5 – 10−6. Despite the smallness of δ, the fact that n<1 leads to some interesting optical phenomena that are absent in the optics of visible light. We discuss the related consequences in more detail in the following chapters (e.g., in Chapters 2 and 3).

1.2 Inhomogeneous medium: eikonal equation and mirage formation

As we will learn later in this book, the wave aspects in light propagation through material objects are not important (or play a little role) when the light wavelength λ=2π/k is much smaller than the characteristic sizes of these objects. In this limit (λ → 0), which is called geometrical optics, the basic phenomena in light propagation, such as reflection and refraction, are completely determined by the refractive index n only.

In this context, let us consider the more general case of light propagation through an inhomogeneous medium in which the refractive index is the coordinate-dependent function, nx,y,z. In this case, the solution of the wave equation (1.26) is not more the plane wave. For simplicity, we use the wave equation for the scalar function fx,y,z:

and considering the...