The Optical Lagrangian

Geometrical optics can be stated in the form

where ds is an element of the ray path from PI to P2. This can be written in the form where

and the dots denote differentiation with respect to z. Also, we define

The analogy is to regard L as the Lagrangian function of a mechanical system in which x and у are two generalized coordinates, p and q are the corresponding generalized momenta, and z corresponds to the time axis. On this basis the ordinary development of mechanics can be carried out, such as by Luneburg (1964, Article 18), by solving the variational problem of Equation (2.7). The Hamiltonian is found to have the value

In other words, it is -nN where N is the z-direction cosine, and, of course, p and q as just defined are respectively equal to nL and nM. The phase space for this system has the four coordinates (x, y,p, q), and Liouville’s theorem in statistical mechanics can be invoked immediately to state that phase space volume is conserved (Goldstein, classical mechanics).

FIGURE 2.14 Inside a medium of refractive index n the etendue becomes n²a²(P.

The Fermi Vector Method

“An important result deserves more than one proof.”

Consider the Legendre transformation of the Optical Lagrangian:

Construct p_xx + p_yy - C(x,y, x,y) = H

This results in — = ~p_x,— = -= x,-= y, which is similar to

8x 8y • 8p_x dp_y

Hamiltonian equations.

_XT . , 8p_x 8 (8H 8 (дН) dx

Notice that -2— =--— =--- =—, etc.

8p_x 8p_x l, dx ) 8x у 8p_x ) 8x

Now construct a vector W = [x,y,p_x, />,,), and notice that

„ ,_Г/ 8W 8W 8W 8W 8p_x дх 8p_y 8y _л _ . . . .. .

V • W =-н--+--h-= н--+ -f— + — = 0. This means the field of

ox 8y 8p_x 8p_v др_х dx dp_v dy

four-dimensional vector W has the important property of divergence being zero. In other words, the four-dimensional hyperspace of (x, y, p_x, /;_v) has the property of conservation of volume as all the light rays evolve in an optical system.

Here we will also offer an intuitive analogy using fluid (Figure 2.15).

Let V be the volume surrounded by a closed Surface in a field with small integral elements denoted as ds, where v is the speed of small particles of an imaginary fluid. As the fluid starts to flow according to the change of t, and the Surfaced) starts to evolve into shape as Surfaced+At), the enclosed volume of V(t + At) will also change as:

Where ds is the surface vector pointing along the normal direction outwards.

Using Gauss’s theorem:

Where dr is the volume integral.

If V • v = 0 everywhere, then obviously:

or.

Implementing this in the etendue conservation, we find that a four-dimensional volume (x, y, p_x, p) remains constant, as it evolves according to an optical axis (for exmple direction z). The vector field is W = (x,y,p_x, /)_v), instead of v. And V • IV = 0 just as V • v = 0. This not only implies that the etendue is conserved but also it results in there being no source or sink of field W as the light propagates.