Besides the wavefront aberration theory described in Chapter 7, geometrical optics can be formulated in a manner in direct analogy with the Hamiltonian theory of classical dynamics.¹⁷⁹ The basis for this analogy is the recognition that Fermat's principle, which requires that a physical light path be an extremum, is equivalent to the requirement that this physical light path follow a trajectory governed by a Hamiltonian. The coordinates and momenta of a particle in classical mechanics therefore correspond to the coordinates and direction cosines of a light ray in optics, and the tools of classical mechanics can be applied directly to geometrical optics.

The characterization of optical systems using Lie transformations, hitherto applied to dynamical systems, was first developed by Dragt, who considered axially symmetric systems.¹⁸⁰ Later this formulation was extended by Goto and Kurosaki to optical systems without axial symmetry (but with a plane of symmetry).¹⁸¹

Dragt considered the coordinates x and y of a point in the object plane, as well as their direction cosines p and q, as the object phase space variables and primed quantities (x', y', p' and q') as their corresponding image phase space variables (see Figure B-1), and expressed the transformation of the ray in object space to the ray in image space (due to the optical system) as

where M is a mapping (or simply map), an operator that transforms coordinates in object space to corresponding coordinates in image space.

A transformation that maps coordinates in object space into coordinates in image space according to Fermat's principle is called symplectic. Dragt and Finn showed in 1976 that a symplectic map can be expressed in terms of Lie transformations.¹⁸² More specifically, this map can be expressed as the product of Lie transformations, each of which is homogeneous in the object space coordinates (that is, all terms in each Lie transformation are of the same power in the independent variables). Furthermore, truncating the product at any power leaves a symplectic transformation, so lower-order imaging properties can be examined without considering the higher-order Lie transformations in the map. Goto and Kurosaki used Dragt's formalism to derive, using Lie algebraic theory rather than wavefront aberration theory, equations that are formally identical to the aberration coefficients F₂₀, F₀₂, F₃₀, etc. seen in Chapter 7.

Figure B-1.   Definition of the object space variables. The optical ray in object space has coordinate (x, y) in the object plane and direction defined by angles q and f (shown), whose direction cosines are p and q.

The transformation from object phase space to image phase space may be represented as a sequence of operations; for example, for diffraction by a grating, these operations are (in sequence)

1. transit from the object plane to the grating, through the distance r,
2. rotation, through the angle a, to a coordinate frame centered at the grating center and oriented with an axis along the grating normal,
3. diffraction in this coordinate frame,
4. rotation through the angle b, and
5. transit from the grating to the image plane, through the distance r'.

An advantage of the Lie transformation approach over the wavefront aberration technique is that general points (x, y) in the object plane are naturally considered; which this is also true of wavefront aberration theory, the algebra is cumbersome and, as a result, most authors consider only a point source in the dispersion plane.

Another immediate advantage of the Lie transformation approach is that systems with more than one optical element can be addressed in a computationally straightforward manner, simply by appending transformations (in the sequence in which the optical ray encounters the optical elements).¹⁸³ Using wavefront aberration theory this is not straightforward, in large part because the grating surface coordinates appear explicitly in the optical path difference (see Eq. (7-5)). Many researchers overcame this complication by imposing intermediate foci between successive elements,¹⁸⁴ though Chrisp introduced the use of toroidal reference surfaces and provided an important advance in the development of wavefront aberration theory for multielement systems.¹⁸⁵

As an example of the power of Lie algebraic techniques in the analysis of multielement optical system, below is the equation for defocus for a system of two aberration-reduced concave holographic gratings:¹⁸⁶

(B-2)

Here the subscripts on the angles and distances refer to each grating, and we have defined

as the distance between the two gratings. The quantity depends on the substrate curvature and groove pattern for the i^th grating.