On Classical Electromagnetic Fields (cont.)

II. RAYS: The Eikonal Treatment of Geometric Optics (pdf copy)[1]
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 pseudo-simplewaves  -- viz.
     [ II-1a ]
     [ II-1b]
It is assumed that  and  are weak functions of position.  The scalar phase function  is the spatially varying phase of the pseudo-simplewave.  For the cases of pseudo-plane waves and pseudo-spherical waves the phase function is given, respectively, by
pseudo-plane waves
     [ II-2a]
pseudo-spherical waves     [ II-2b]
We now substitute these pseudo-simple wave expressions (i.e.Equations [ II-1] ) into Maxwell's equations to obtain
     [ II-3a]
     [ II-3b]
     [ II-3c]
     [ II-3d]
Rearranging, we obtain
     [ II-4a]
     [ II-4b]
     [ II-4c]
     [ II-4d]
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 [ II-4a ] ) as follows:
Equation [ II-4a ]
     [ II-5a]
     [ II-5b]
Applying the "abc = bac - cab" rule [2] we obtain
     [ II-5c]
which becomes upon substitution from the second Equation [ II-4b ]
     [ II-5d]
From Equation [ II-4c ], we see that the first term vanishes in the geometric approximation  --  i.e., if we neglect the term . Therefore, for non-vanishing  we obtain the following  reduction of Maxwell's equations:
     [ II-6]
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 
     [ II-7]
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.
     [ II-8a ]
However, from the definition of the grad operator we know that
so that
     [ II-8b ]
     [ II-8c ]
Thus we have obtain the  eikonal[3]  equation for the ray vector -- viz.
     [ II-9 ]


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
     [ II-10 ]
where xis the vertical height above the planar surface, ng  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  xo and at an angle qo with respect to the surface of the earth.


     [ II-11a ]
or from Equation [ II-10 ]
     [ II-11b ]
Thus, the ray trajectory is given by
     [ II-12 ]


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 two-dimensional paraxial approximation, we assume that   and write

         [ II-13 ]
where  .


We may write results of this sort in the form of the famous and highly useful ray transformor ABCD matrix -- viz.
     [ II-14 ]
In the case of a uniform dielectric
     [ II-15 ]
      so that A = 1, B = L, C = 0, and D = 1
    2. A dielectric discontinuity:
      Starting with Equation [ II-7 ] and noting, once again, that


      we see
         [ II-16 ]
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
     [ II-17 ] 
    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
         [ II-18 ]
    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]
         [ II-19 ]
Within such a GRIN rod, we write  for the ray coordinates and  for the index variation. Using the eikonal equation -- i.e. Equation [ II-9 ] -- in the paraxial approximation, we find
     [ II-20a ]
     [ II-20b ]
     [ II-20c ]
     [ II-20d ]
Doubtless, the simplest and most valuable instance is m = 2 -- i.e. what is usually called a parabolic or quadratic material -- wherein
     [ II-21 ]
so that
     [ II-22a ]
    [ II-22b ]
In terms of a ray transform matrix

     [ II-23 ]
where  .

Ray trajectories confined in a GRIN rod.



A Schematic SELFOC Lense

An Array of SELFOC Lenses

Alternative (Hamiltonian) Derivation of Eikonal Equation:


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.

     [ II-24 ]
where the line element, , is measured along a ray and the two end-points 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,
     [ II-25 ]
where  and , Fermat's variation principle is transformed into the more familiar Lagrangian form -- viz.
     [ II-26 ]
     [ II-27 ]
The minimization procedure is then well-known in the variational calculus and leads to the famous Euler-Lagrangian equations -- i.e.
     [ II-28a ]
     [ II-28b ]
When applied to the Fermat Lagrangian, as defined in Equation [ II-27 ], these equations yield
     [ II-29a ]
     [ II-29b ]
Using Equation [ II-25 ] we see that the Euler-Lagrangian equations may be expressed in the vector form as
     [ II-30 ]
which is precisely the content of Equation [ II-9 ] -- QED.


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

     [ II-31 ]
The Hamiltonian is then define in terms of the generalized momentum by the relation
     [ II-32 ]
With the assumed functional dependence of the Hamiltonian, we form the derivatives
     [ II-33a ]
     [ II-33b ]
Given the definitional relationships embodied in Equation [ II-31 ] we see that these expression reduce to one set of Hamilton's equation -- viz.
     [ II-34 ]
The other set of Hamilton's equation -- viz.
     [ II-35 ]
follow directly from the Euler-Lagrangian equations -- i.e. Equations [ II-28a ] and [ II-28b ] -- and the definitions embodied in Equation [ II-31 ]. Using the Fermat Lagrangian we see that
     [ II-36 ]
and consequently that we may solve for  in terms of  as
     [ II-37 ]
Substituting into Equation [ II-32 ], we find an expression for the Fermat or ray optics Hamiltonian -- viz.
     [ II-38 ]
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 non-relativistic particle -- viz.
     [ II-39 ]
when . [9]


[1] See, for example, Max Born and Emil Wolf, Principle of Optics, Pergamon Press (1986), Chapter 3.

[2] Again using

[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 [ II-7 ] we see that

.[8] The formal theory of optical systems was developed by Sir W. R. Hamilton in 1828-37.

[9] Applying the quantization rules of quantum mechanics to these Hamiltonians, we can go full circle and recover wave optics from ray optics. Equation [ II-38 ] leads directly to the equivalent of the relativistic Klein-Gordon equation while the equivalent of the nonrelativistic Schrödinger equation follows directly from Equation [ II-39 ].


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This page was prepared and is maintained by R. Victor Jones, jones@deas.harvard.edu
Last updated May 15, 2000