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Geometric Mechanics Part I May 30, 2011 1:17pm Holm Vol 1 WSPC/Book Trim Size for 9in by 6in 1 FERMAT’S RAY OPTICS Contents 1.1 Fermat’s principle 3 1.1.1 Three-dimensional eikonal equation 4 1.1.2 Three-dimensional Huygens wave fronts 9 1.1.3 Eikonal equation for axial ray optics 14 1.1.4 The eikonal equation for mirages 19 1.1.5 Paraxial optics and classical mechanics 21 1.2 Hamiltonian formulation of axial ray optics 22 1.2.1 Geometry, phase space and the ray path 23 1.2.2 Legendre transformation 25 1.3 Hamiltonian form of optical transmission 27 1.3.1 Translation-invariant media 30 1.3.2 Axisymmetric, translation-invariant materials 31 1.3.3 Hamiltonian optics in polar coordinates 34 1.3.4 Geometric phase for Fermat’s principle 35 1.3.5 Skewness 36 1.3.6 Lagrange invariant: Poisson bracket relations 39 1.4 Axisymmetric invariant coordinates 43 1.5 Geometry of invariant coordinates 45 1.5.1 Flows of Hamiltonian vector fields 47 1.6 Symplectic matrices 51 GEOMETRIC MECHANICS - Part I: Dynamics and Symmetry (2nd Edition) © Imperial College Press http://www.worldscibooks.com/mathematics/p801.html May 30, 2011 1:17pm Holm Vol 1 WSPC/Book Trim Size for 9in by 6in 2 1 : FERMAT’S RAY OPTICS 1.7 Lie algebras 57 1.7.1 Definition 57 1.7.2 Structure constants 57 1.7.3 Commutator tables 58 1.7.4 Poisson brackets among axisymmetric variables 59 3 1.7.5 Noncanonical R Poisson bracket for ray optics 60 1.8 Equilibrium solutions 63 1.8.1 Energy-Casimir stability 63 1.9 Momentum maps 66 ∗ 2 2 2 1.9.1 The action of Sp(2; R) on T R ' R × R 66 1.9.2 Summary: Properties of momentum maps 70 1.10 Lie–Poisson brackets 72 3 1.10.1 The R bracket for ray optics is Lie–Poisson 72 1.10.2 Lie–Poisson brackets with quadratic Casimirs 73 1.11 Divergenceless vector fields 78 1.11.1 Jacobi identity 78 1.11.2 Geometric forms of Poisson brackets 82 1.11.3 Nambu brackets 84 1.12 Geometry of solution behaviour 85 1.12.1 Restricting axisymmetric ray optics to level sets 85 2 2 1.12.2 Geometric phase on level sets of S = pφ 88 1.13 Geometric ray optics in anisotropic media 90 1.13.1 Fermat’s and Huygens’ principles for anisotropic media 90 1.13.2 Ibn Sahl–Snell law for anisotropic media 94 1.14 Ten geometrical features of ray optics 95 GEOMETRIC MECHANICS - Part I: Dynamics and Symmetry (2nd Edition) © Imperial College Press http://www.worldscibooks.com/mathematics/p801.html May 30, 2011 1:17pm Holm Vol 1 WSPC/Book Trim Size for 9in by 6in 1.1 FERMAT’S PRINCIPLE 3 1.1 Fermat’s principle Fermat’s principle (1662) states that the path between two points taken by a ray of light leaves the optical length stationary un- der variations in a family of nearby paths. This principle accurately describes the properties of light that is reflected by mirrors, refracted at a boundary between different media or transmit- ted through a medium with a continu- ously varying index of refraction. Fer- mat’s principle defines a light ray and Pierre de Fermat provides an example that will guide us in recognising the principles of geome- tric mechanics. The optical length of a path r(s) taken by a ray of light in passing from point A to point B in three-dimensional space is defined by Z B A := n(r(s)) ds ; (1.1.1) A 3 where n(r) is the index of refraction at the spatial point r 2 R and ds2 = dr(s) · dr(s) (1.1.2) is the element of arc length ds along the ray path r(s) through that point. Definition 1.1.1 (Fermat’s principle for ray paths) The path r(s) tak- en by a ray of light passing from A to B in three-dimensional space leaves the optical length stationary under variations in a family of nearby paths r(s; ") depending smoothly on a parameter ". That is, the path r(s) satis- fies Z B d δA = 0 with δA := n(r(s; ")) ds ; (1.1.3) d" "=0 A GEOMETRIC MECHANICS - Part I: Dynamics and Symmetry (2nd Edition) © Imperial College Press http://www.worldscibooks.com/mathematics/p801.html May 30, 2011 1:17pm Holm Vol 1 WSPC/Book Trim Size for 9in by 6in 4 1 : FERMAT’S RAY OPTICS where the deviations of the ray path r(s; ") from r(s) are assumed to vanish when " = 0, and to leave its endpoints fixed. Fermat’s principle of stationary ray paths is dual to Huygens’ prin- ciple (1678) of constructive interference of waves. According to Huygens, among all possible paths from an object to an image, the waves corresponding to the stationary path contribute most to the image because of their constructive interference. Both principles are models that approximate more fundamental physical results de- rived from Maxwell’s equations. The Fermat and Huygens principles for geometric optics are also foundational ideas in mechanics. Indeed, the founders of mechan- ics Newton, Lagrange and Hamilton all worked seriously in optics as well. We start with Fermat and Huygens, whose works preceded Newton’s Principia Mathematica by 25 years, Lagrange’s Mécanique Analytique by more than a century and Hamilton’s On a General Method in Dynamics by more than 150 years. After briefly discussing the geometric ideas underlying the principles of Fermat and Huygens and using them to derive and interpret the eikonal equation for ray optics, this chapter shows that Fermat’s principle naturally introduces Hamilton’s principle for the Euler–Lagrange equations, as well as the con- cepts of phase space, Hamiltonian formulation, Poisson brack- ets, Hamiltonian vector fields, symplectic transformations and momentum maps arising from reduction by symmetry. In his time, Fermat discovered the geometric foundations of ray optics. This is our focus in the chapter. 1.1.1 Three-dimensional eikonal equation Stationary paths Consider the possible ray paths in Fermat’s prin- ciple leading from point A to point B in three-dimensional space as 2 3 belonging to a family of C curves r(s; ") 2 R depending smoothly GEOMETRIC MECHANICS - Part I: Dynamics and Symmetry (2nd Edition) © Imperial College Press http://www.worldscibooks.com/mathematics/p801.html May 30, 2011 1:17pm Holm Vol 1 WSPC/Book Trim Size for 9in by 6in 1.1 FERMAT’S PRINCIPLE 5 on a real parameter " in an interval that includes " = 0. This "- family of paths r(s; ") defines a set of smooth transformations of the ray path r(s). These transformations are taken to satisfy r(s; 0) = r(s) ; r(sA;") = r(sA) ; r(sB;") = r(sB) : (1.1.4) That is, " = 0 is the identity transformation of the ray path and its endpoints are left fixed. An infinitesimal variation of the path r(s) is denoted by δr(s), defined by the variational derivative, d δr(s) := r(s; ") ; (1.1.5) d" "=0 where the fixed endpoint conditions in (1.1.4) imply δr(sA) = 0 = δr(sB). With these definitions, Fermat’s principle in Definition 1.1.1 im- plies the fundamental equation for the ray paths, as follows. Theorem 1.1.1 (Fermat’s principle implies the eikonal equation) Stationarity of the optical length, or action A, under variations of the ray paths r Z B dr dr 0 = δA = δ n(r(s)) · ds ; (1.1.6) A ds ds defined using arc-length parameter s, satisfying ds2 = dr(s) · dr(s) and 3 j_rj = 1, implies the equation for the ray path r 2 R is d dr @n n(r) = : (1.1.7) ds ds @r In ray optics, this is called the eikonal equation.1 Remark 1.1.1 (Invariance under reparameterisation) The integral in (1.1.6) is invariant under reparameterisation of the ray path. In 1The term eikonal (from the Greek ικoνα meaning image) was introduced into optics in [Br1895]. GEOMETRIC MECHANICS - Part I: Dynamics and Symmetry (2nd Edition) © Imperial College Press http://www.worldscibooks.com/mathematics/p801.html May 30, 2011 1:17pm Holm Vol 1 WSPC/Book Trim Size for 9in by 6in 6 1 : FERMAT’S RAY OPTICS particular, it is invariant under transforming r(s) ! r(τ) from arc length ds to optical length dτ = n(r(s))ds. That is, r r Z B dr dr Z B dr dr n(r(s)) · ds = n(r(τ)) · dτ ; (1.1.8) A ds ds A dτ dτ in which jdr/dτj2 = n−2(r(τ)). 2 We now prove Theorem 1.1.1. Proof. The equation for determining the ray path that satisfies the stationary condition (1.1.6) may be computed by introducing the "- family of paths into the action A, then differentiating it with respect to " under the integral sign, setting " = 0 and integrating by parts with respect to s, as follows: Z B p 0 = δ n(r(s)) _r · _r ds A Z B @n _r = j_rj · δr + n(r(s)) · δ_r ds A @r j_rj Z B @n d _r = j_rj − n(r(s)) · δr ds ; A @r ds j_rj where we have exchanged the order of the derivatives in s and ", and used the homogeneous endpoint conditions in (1.1.4). We choose the arc-length variable ds2 = dr · dr, so that j_rj = 1 for _r := dr=ds.
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