Hamiltonian optics

A different approach to solving this problem consists in defining a Hamiltonian (taking a Legendre transform of the Lagrangian) as

This derivation is the same as in Hamiltonian mechanics, only with time t now replaced by a general parameter σ.

The general results presented above for Hamilton's principle can be applied to optics.

In general, as light travels, it moves in a medium of variable refractive index which is a scalar field of position in space, that is,

The Euler-Lagrange equations with parameter σ =x3 and N=2 applied to Fermat's principle result in

It is assumed that light travels along the x3 axis, in Hamilton's principle above, coordinates

Also shown is optical momentum p, tangent to a light ray and perpendicular to the wavefront.

The gradient theorem can then be applied to the optical path length (as given above) resulting in

For segment BC the optical momentum p has the same direction as displacement ds and

For segment DA the optical momentum p has the opposite direction to displacement ds and

If p1 is given, p3 may be calculated (given the value of the refractive index n) and therefore p1 suffices to determine the direction of the light ray.

This ray may then be defined by a point rC=(xB,p1C) in space x1p1 as shown at the bottom of the figure.

As such, ray rD shown at the top is represented by a point rD in phase space at the bottom.

Accordingly, all rays crossing axis x1 at coordinate xA contained between rays rA and rB are represented by a vertical line connecting points rA and rB in phase space.

In general, all rays crossing axis x1 between xL and xR are represented by a volume R in phase space.

If p1 and p2 are given, p3 may be calculated (given the value of the refractive index n) and therefore p1 and p2 suffice to determine the direction of the light ray.

In particular, an infinitesimal area dA with outward pointing unit normal n moves with a velocity v. This leads to a volume variation

which means that the phase space volume is conserved as light travels along an optical system.

Liouville’s theorem is essentially statistical in nature, and it refers to the evolution in time of an ensemble of mechanical systems of identical properties but with different initial conditions.

Each point in phase space, which in this example has 2N dimensions, where N is the number of molecules, represents one of an ensemble of identical containers, an ensemble large enough to permit taking a statistical average of the density of representative points.

[3]Figure "conservation of etendue" shows on the left a diagrammatic two-dimensional optical system in which x2=0 and p2=0 so light travels on the plane x1x3 in directions of increasing x3 values.

Light rays crossing the input aperture of the optic at point x1=xI are contained between edge rays rA and rB represented by a vertical line between points rA and rB at the phase space of the input aperture (right, bottom corner of the figure).

All rays crossing the input aperture are represented in phase space by a region RI.

Also, light rays crossing the output aperture of the optic at point x1=xO are contained between edge rays rA and rB represented by a vertical line between points rA and rB at the phase space of the output aperture (right, top corner of the figure).

All rays crossing the output aperture are represented in phase space by a region RO.

Conservation of etendue in the optical system means that the volume (or area in this two-dimensional case) in phase space occupied by RI at the input aperture must be the same as the volume in phase space occupied by RO at the output aperture.

This ensures that an image of the input is formed at the output with a magnification m. In phase space, this means that vertical lines in the phase space at the input are transformed into vertical lines at the output.

A more general situation can be considered in which the path of a light ray is parametrized as

For that reason not all solutions of the Euler-Lagrange equations will be possible light rays, since their derivation assumed an explicit dependence of L on σ which does not happen in optics.

A particular case is obtained when these vectors form an orthonormal basis, that is, they are all perpendicular to each other.