Since ancient times, the notion of ray or beam propagation has been one of the most enduring and fundamental concepts in optical physics. As a zeroth order approximation we might consider a plane wave to be a model of a beam and its propagation vector to be a model of a ray. This is a reasonable start, but it is a much too restricted view and we can do much better. What we need is a solution to Maxwell's equations which is like a plane wave, but limited in spatial extent. One approach, the simplest, is called variously ray, Gaussian or geometric optics.
A Maxwellian Derivation of the Eikonal Equation of Ray Optics:
To fully understand geometric optics in the context of Maxwell's equations, we start by writing the electric and magnetic fields as pseudosimplewaves  viz.

[ II1a
]

[ II1b]

where
It is assumed that and are weak functions of position. The scalar phase function is the spatially varying phase of the pseudosimplewave. For the cases of pseudoplane waves and pseudospherical waves the phase function is given, respectively, by
pseudoplane waves 
[ II2a]


pseudospherical waves  [ II2b] 
We now substitute these pseudosimple wave expressions (i.e.Equations [ II1] ) into Maxwell's equations to obtain
[ II3a]

[ II3b]

[ II3c]

[ II3d]

Rearranging, we obtain
[ II4a]

[ II4b]

[ II4c]

[ II4d]

In the ray, Gaussianor geometric approximation we assume that we may neglect the RHS's of these equations. To get something useful we multiply through the first equation (i.e. Equation [ II4a ] ) as follows:
^{ Equation [ II4a ]} 
[ II5a]

[ II5b]

Applying the "abc = bac  cab" rule ^{[2]} we obtain
[ II5c]

which becomes upon substitution from the second Equation [ II4b ]
[ II5d]

From Equation [ II4c ], we see that the first term vanishes in the geometric approximation  i.e., if we neglect the term . Therefore, for nonvanishing we obtain the following reduction of Maxwell's equations:
[ II6]

where is the index of refraction. More explicitly, we may write an equation for a "ray vector"  i.e. the tangent to a space curve orthogonal to the surfaces of constant
[ II7]

We illustrate the geometric relationships below:
We may now derive the all important eikonal equation. To that end, we first take a derivative along the ray direction  viz.
[ II8a
]

However, from the definition of the grad operator we know that
so that
[ II8b
]

or
[ II8c
]

Thus we have obtain the eikonal[3] equation for the ray vector  viz.
[ II9
]

First Application of the Eikonal Equation: Mirages
Air adjacent to a hot surface rises in temperature and becomes less dense. Thus over a flat hot surface, such as a desert expanse or a sun drenched roadway, air density locally increases with height and the average refractive index may be approximated by a simple linear variation of the form
[ II10
]

where xis the vertical height above the planar surface, n_{g} is the refractive index at ground level, and k is a positive constant.
We may use the eikonal equation to find an equation for the approximate ray trajectory  i.e. an equation for ray height x as a function of ground distance z  of a light ray launched from a height x_{o }and at an angle q_{o }with respect to the surface of the earth.Therefore,
[ II11a
]

or from Equation [ II10 ]
[ II11b
]

Thus, the ray trajectory is given by
[ II12
]

Ray trajectories diverted by a hot surface
Second Application of the Eikonal Equation: The "ABCD" Ray Matrices  A Systems Approach to Optics
1. A uniform dielectric medium  i.e. is a constant
so that . Thus, the ray must be a straight line which may be written
In the twodimensional paraxial approximation, we assume that and write
[ II13 ]
where .
.We may write results of this sort in the form of the famous and highly useful ray transformor ABCD matrix  viz.
[ II14 ]
In the case of a uniform dielectric
[ II15 ]
so that A = 1, B = L, C = 0, and D = 1
2. A dielectric discontinuity:
s
Starting with Equation [ II7 ] and noting, once again, that
,we see
[ II16 ]
which is identical to the saltuscondition on the electric and magnetic fields at a dielectric interface! Hence is continuous across the dielectric boundary so that  i.e. Snell's law! This result in the paraxial approximation  i.e.,  may be written in ray matrix form as
[ II17 ]
3. A "Thin" lenses:
In passing we note that the ray matrix of a thin lens is given by or, perhaps more accurately, a thin lens is essentially defined by
[ II18 ]
4. An axially symmetric GRIN media:
Consider the use of GRaded INdex technology to obtain an axially symmetric variation in the index of refraction of the form [4]
[ II19 ]
Within such a GRIN rod, we write for the ray coordinates and for the index variation. Using the eikonal equation  i.e. Equation [ II9 ]  in the paraxial approximation, we find
[ II20a ]
or
[ II20b ]
Therefore
[ II20c ]
or
[ II20d ]
Doubtless, the simplest and most valuable instance is m = 2  i.e. what is usually called a parabolic or quadratic material  wherein
[ II21 ]
so that
[ II22a ][ II22b ]
In terms of a ray transform matrix
[ II23 ]
where .
Ray trajectories confined in a GRIN rod.
A Schematic SELFOC Lense
An Array of SELFOC Lenses
Alternative (Hamiltonian) Derivation of Eikonal Equation:
FERMAT'S PRINCIPLE [5]Like most laws of physics, the equations of geometric optics can be derived from a variation principle. In this context the variation principle is called the Fermat principle which states that a ray always chooses a trajectory that minimizes [6] the optical path length  viz.
[ II24
]

where the line element, , is measured along a ray and the two endpoints P1 and P2 are fixed in space.[7] Analysis of the variation problem is facilitated by choosing the projected coordinate z as the new variable of integration. Accordingly,
[ II25
]

where and , Fermat's variation principle is transformed into the more familiar Lagrangian form  viz.
[ II26
]

where
[ II27
]

The minimization procedure is then wellknown in the variational calculus and leads to the famous EulerLagrangian equations  i.e.
[ II28a
]

[ II28b
]

When applied to the Fermat Lagrangian, as defined in Equation [ II27 ], these equations yield
[ II29a
]

[ II29b
]

Using Equation [ II25 ] we see that the EulerLagrangian equations may be expressed in the vector form as
[ II30
]

which is precisely the content of Equation [ II9 ]  QED.
HAMILTONIAN FORMULATION OF RAY OPTICS
The analogy between ray optics and particle mechanics is most striking when the equations of ray optics are expressed in Hamiltonian form.[8] To that end, we define the generalized momentum which is canonically conjugate to x and y by the vector equation
[ II31
]

The Hamiltonian is then define in terms of the generalized momentum by the relation
[ II32
]

With the assumed functional dependence of the Hamiltonian, we form the derivatives
[ II33a
]

[ II33b
]

Given the definitional relationships embodied in Equation [ II31 ] we see that these expression reduce to one set of Hamilton's equation  viz.
[ II34
]

The other set of Hamilton's equation  viz.
[ II35
]

follow directly from the EulerLagrangian equations  i.e. Equations [ II28a ] and [ II28b ]  and the definitions embodied in Equation [ II31 ]. Using the Fermat Lagrangian we see that
[ II36
]

and consequently that we may solve for in terms of as
[ II37
]

Substituting into Equation [ II32 ], we find an expression for the Fermat or ray optics Hamiltonian  viz.
[ II38
]

which resembles the mechanical Hamilton of a relativistic particle  i.e., .
But the analogy is even stronger in the paraxial approximation where the Hamiltonian is approximated by an expression which is identical in form with the Hamiltonian of a nonrelativistic particle  viz.
[ II39
]

when . [9]
[3] The eikonal(from the Greek meaning image) was introduced in 1895 by H. Bruns..
[4] A note on GRIN technology: In GRIN technology one builds up a glass rod with a specific radial index of refraction distribution by fusing a sequence of coaxially arranged glass tubes with appropriate index and diameter as illustrated in the following:
See note on fiber optic manufacture
[5] See, for example,
Dietrich Marcuse, Light Transmission Optics, Van Nostrand Reinhold
(1972).
[6] More precisely, the path must be a local extremum and in rare cases may, in fact, be a maximum. See R. Y. Luneberg, Mathematical Theory of Optics, University of California Press, Berkeley and Los Angeles (1964).
[7] From Equation [ II7 ] we see that
.[8] The formal theory of optical systems was developed by Sir W. R. Hamilton in 182837.
[9] Applying the quantization
rules of quantum mechanics to these Hamiltonians, we can go full circle
and recover wave optics from ray optics. Equation [ II38 ] leads directly
to the equivalent of the relativistic KleinGordon equation while the equivalent
of the nonrelativistic Schrödinger equation follows directly from
Equation [ II39 ].
References
http://www.bostonoptical.com
http://www.nsgamerica.com