From Maxwell's Equations to Wave Optics and Geometrical Optics

Both geometric and wave optics stem from the same place. They both come from Maxwell's equations. Below are outlines of derivations of both views of optics.

A) Maxwell's Equations from Electromagnetic Theory (SI units)

Maxwell's Equations in Differential Form:

Reitz, Milford, Christy, Foundations of Electromagnetic Theory, 1972, Addison-Wesley, §16-2.

Faraday's Law of Electromagnetic Induction(1)

Generalized Ampere's Law(2)

There are no magnetic monopoles.(3)

Gauss's Law(4)

Constitutive Relations:

=(5)

=(6)

=(7)

Equation of Continuity:

Variables:

Variable / Name
/ Electric Field
/ Magnetic Field
/ Electric Displacement
/ Magnetic Induction
/ Current density
 / Charge density
 / Conductivity
c / Speed of Light:c = 2.9979x10 m/s
 / Permeability: 0 = 4x107 Ns/C
 / Permittivity:0 = 8.854x10 C/Nm
00c=1 / (8)
v=f / wave velocity=wavelength*frequency
f / angular frequency
k=2 / wave number
kc / angular frequency=wave number*speed of light (9)
n / index of refraction, (10)

B) Vector Identities

f is a scalar function of , , and perhaps time, t.

is a vector function of , , and perhaps time, t.

(11)

(12)

(13)

(14)

(Laplacian)(15)

(16)

C) Assumptions etc.

1)  and  are scalars, not tensors.

2)  is 0.

3) The wavelength, , is significantly smaller than any hardware sizes or propagation distances.

4)  is independent of time and piecewise slowly varying (variations are vanishingly small on the order of a wavelength except at interfaces).

5) and are zero.

7) A single frequency, , is chosen. Actual fields would be found by superposition of single frequency fields.

8) The time dependence is chosen to be~, where. This is a choice since Maxwell's Equations are real.

D) Maxwell's Equations Used in Derivations:

Replace equations (1) through (4) with

(17)

(18)

(19)

(20)

E) The Wave Equation

1) Take the curl of eqn. (17)

2) Apply eqn. (11) to the LHS and eqn. (18) to the RHS.

3) Use definitions (8), (9), and (10) on the RHS and eqns. (12) and (20) on the LHS. Rearrange the terms.

4) Using assumption (4), the RHS is assumed to vanish relative to the 2nd term on the LHS.

(21)

This is the wave equation. Although any of the four field vectors could have been chosen since they are interrelated by Maxwell's Equations, historically the electric field is chosen as the field of interest for the wave equation. Later, we will define the direction of the electric field to be the polarization vector.

F) Geometrical Optics

Marcuse, Dietrich, Light Transmission Optics, 2nd ed., 1982, 1972, Van Nostrand, §3.2 §3.4.

i) The Eikonal Equation

Besides the assumptions above, the main assumption used to derive geometrical optics is that the wave amplitude is slowly varying compared with the phase. Considering this assumption, we can write any component of the electric field as

E=ejkS()(22)

The two functions,  and S, are taken to be independent of wavelength,

to lowest order. The function S is called the eikonal and , the wave amplitude.

1) Plug eqn. (22) into the wave equation for one component.

2) Use (12), (14), (15) and (16) to get

3) Rearrange, dividing through by k.

4) Using assumption (3) and recalling that k=2, means that the RHS is vanishingly small. Therefore one has the eikonal equation:

(23)

This says that the magnitude of the gradient of the wavefront is the square of the index of refraction.

ii) The Ray Equation

How does this equation for the phase front relate to ray optics? Return to the wave equation. Assuming a region of constant index, a basis set of solutions is found to be plane waves:

i.e. the surfaces of constant phase are in the plane perpendicular to the direction, . The basis set consists of the set of such plane waves. "We are thus led to define light rays as the orthogonal trajectories to the phase fronts of a light wave. If we know the surface of constant phase, we can construct the light rays by drawing lines perpendicular to the phase fronts. As the phase fronts curve in space, owing to changes in the index of refraction, to do the light rays.

"It is desirable to be able to calculate the light rays directly without having to construct the phase fronts from the eikonal equation." Consider a small enough region that the light rays are locally linear. The following diagram identifies the parameters used below.

The equation for the ray can be written as a vector sum.

Therefore,

Since the gradient of the phase is in the direction, , with magnitude, n, we can write

(24)

Consider taking the gradient of the eikonal equation (why? because it will get us to the ray equation).

Using equation (24) and the chain rule, , we can rewrite this equation as

Dividing by n gives

(25)

This equation is the ray equation. Solving it for in index n, solves for the equation of the ray in that index starting from point P0. To cross the discontinuity at an interface, you would not use this equation but Snell's Law, which can be derived from Fermat's Principle. If you are tracking the amplitude of the wave lurking underneath all this, you would use the Fresnel equations for reflected and transmitted values of amplitude.

For systems in which the rays act like a pencil and all go primarily in the direction along some axis, say the z axis, the s in the derivatives can be safely replaced by z's. You would then solve for the (x,y) coordinate of a point on the ray as a function of z. This is one realization of the paraxial approximation.

iii) Optical Path Length and Fermat's Principle

Define a small amount of optical path length to mean dS. Now the question is, what is the total optical path length between two points in space? The total optical path length is given by

(26)

Again, we don't really want to calculate the eikonal but to use the index of refraction directly. From the definition of the gradient, (see the chain rule above). Both and are in the direction, so dS=n ds. The optical path length can therefore be written as

(27)

Equation (26) contains an exact differential, thus the OPL is independent of path. Now the question is, which path does the ray follow? Fermat said, the path with minimum OPL. If you divide equation (27) by c, the speed of light, the integral becomes the time of flight. Hence Fermat's principle can be stated as the observation that light travels from one point to another in the least amount of time. (The generalized Fermat principle says the integral is locally stationary, not necessarily a minimum).

As shown in your text, this leads to Snell's Law at an interface.

Prof. Townsend01/18/2019Page 1