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D.D. Holm Y K.B. Wolf, Lie-Poisson Description of Hamiltonian Ray Optics, Physica D 51

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D.D. Holm Y K.B. Wolf, Lie-Poisson Description of Hamiltonian Ray Optics, Physica D 51 Physica D 51 (1991) 189-199 North-Holland Lie-Poisson description of Hamiltonian ray optics Darryl D. Holm a and K. Bernardo Wolf b aCenter for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, MS B284, Los Alamos, NM 87545, USA bInstituto de Investigaciones en Matemdticas Aplicadas yen Sistemas, Universidad Nacional Autonoma de Mexico-Cuernavaca, Apdo. Postal 20-726, 01000 Mexico D.F., Mexico We express classical Hamiltonian ray optics for light rays in axisymmetric fibers as a Lie-Poisson dynamical system defined in R 3, regarded as the dual of the Lie algebra sp(2, •). The ray-tracing dynamics is interpreted geometrically as motion in ~3 along the intersections of two-dimensional level surfaces of the conserved optical Hamiltonian and the skewness invariant (the analog of angular momentum, conserved because of the axisymmetry of the medium). In this geometrical picture, a Hamiltonian level surface is a vertically oriented cylinder whose cross section describes the radial profile of the refractive index, and a level surface of the skewness function is a hyperboloid of revolution around a horizontal axis. Points of tangency of these surfaces are equilibria, which are stable when the Gaussian curvature of the Hamiltonian level surface (constrained by the skewness function) is negative definite at the equilibrium point. Examples are discussed for various radial profiles of the refractive index. This discussion places optical ray tracing in fibers into the geometrical setting of Lie-Poisson Hamiltonian dynamics and provides an example of optical ray trapping within separatrices (homoclinic orbits). 1. Optical phase space qy The phase space of geometrical optics is four- //~'~q,z) dimensional. Referred to a standard planar screen, a ray is determined by two position coor- dinates q = (qx, qy) defining its intersection with the screen, and two momentum coordinates p = x (Px, Py) that cue the projection onto the screen of a three-vector ff tangent to the ray, whose length n(q) is the refractive index of the medium at that point. See fig. 1. Only rays perpendicular to the optical axis cannot be parametrized in this man- Fig. 1. Image screen phase space for geometrical optics. ner. The coordinate normal to the screen, z, ex- tends along the optical axis. The projection along and the evolution of the system is governed by the optical axis of the vector if(q, z)-which in Hamilton's equations, general is allowed to depend on z- is given by h = yn2[q,z)/ ~ x 2 = _nopt. 'q = {q, H °pt} OH °pt dz 0p ' The manifold (q, p) is symplectic, with Poisson bracket OF 0G OF 0G d__p_p = {p, HOpt } OH °pt {F,G} =-~" Op - O---/)- " Oq' dz Oq 190 D.D. Holm, K.B. Wolf/Lie-Poisson description of Hamiltonian ray optics 2. Generators of flow scribed by Hamiltonians, To any differentiable function F(q, p) we asso- .: -- ¢/~(~2)2 --p 2 , ciate its Lie operator if(q, p; O/Oq, O/Op) [6], OF O OF 0 which Poisson commute with the skewness invari- F= {F, o} = -~-.~ - O---p'Oq" ant (i.e. exhibit {H,S 2} = 0). We define the ax- isymmetric coordinates A Lie operator ff generates the flow of phase space (q, p) given by the vector field X=q2>0, y=p2~0, Z=q'p, (- OF/Op, OF/Oq), which is orthogonal to the gra- dient of F, (OF/Oq, OF/Op), and, hence, is con- for which {S 2, X} = {S2, y} = {S 2, Z} = 0. In fact, tained within the submanifolds F(q, p) = any translation-invariant Hamiltonian describing constant, i.e. within its "level surfaces". The flow a system that is symmetric under rotations around expressed by Hamilton's equations is the particu- the optical axis (and under reflections across lar case for F = -H °pt= h. The evolution of an planes containing this axis) may be expressed as a observable g(q,p) under the flow generated by function of only these axisymmetric variables, an ff will be governed by the analogous equation H=H(X,Y,Z). Actually, the general fiber dg/dz =fig, for flow parameter ~-, and will be Hamiltonian is a function of only two of these canonical, i.e. it will preserve Poisson brackets. If variables, H = H(X, Y), since no terms appear in F is independent of ~" we may integrate the flow H that depend on the relative angle q~ between q equation into a Lie transformation, and p through q • p = qp cos ~ = Z. The Hamiltonian flow will take place on level g(q, p; ~') = exp(rP) g(q, p). surfaces H = constant, or In particular, the skewness function y=nZ(X) -H 2. S = q ×p = qxPy - qyPx These are surfaces generated by a line parallel to the Z-axis whose intercept describes the refrac- generates rotations of phase space, of q and p tive index profile in the Z = 0 plane, and which jointly, each in its plane, around the optical axis. are restricted to the first quadrant X > 0, Y > 0. Its square, S 2, is called the Petzval invariant, and See fig. 2. is conserved for ray optics in axisymmetric media. 2 ~Y : n(X) \\ 3. lnvariants in axisymmetric systems ~\x'~xx H = c ons t. When two functions, F and G, Poisson com- mute, {F, G} = 0, the flow generated by F takes place on the G = constant submanifolds, and vice 2 versa. This leads to the natural introduction of symmetry-adapted coordinates in phase space. Fig. 2. Level surfaces of the Hamiltonian are cylinders whose Specifically, we take advantage of the fact that intersection with the X-Y plane describes the radial refrac- axisymmetric, translation-invariant fibers are de- tive-index profile. D.D. Holm, K.B. Wolf / Lie-Poisson description of Hamiltonian ray optics 191 and S 2 are conserved, and that the motion of the system takes place in •3 along the intersections of the level surfaces of these two conserved quan- : pa Sy tities. The level surfaces of H are cylinders extending along the Z-axis with cross-sections determined I~-. -':"/. - Sx by the refractive index profile, while level sur- faces of S 2 are hyperboloids of revolution around the line X = Y in the positive quadrant of the X-Y plane. At the points of tangency of these level surfaces, the gradients of H and S 2 become collinear. These tangency points are the equilib- ria, i.e., the fixed points of the flow in ~3. Fig. 3. Level surfaces of the skewness function are hyper- At an equilibrium point, the constrained sur- boloids of revolution around the X = Y axis at Z = 0. face composed of the sum Concurrently, the flow will lie on level surfaces ns(x ) =H+AS 2 S 2 = constant, determined by (q ×p)2 = q2pZ _ (q. p)2 = constant, or has a critical point, for a multiplier A that may depend upon the value of the equilibrium point, Xy- Z2 ~-.~ 82 > 0. x~. At such a point, we have The S = constant submanifolds are seen to be 8H s = 0 = DHs( x~) " ~x. hyperboloids of revolution around the X = Y axis, extending up through the interior of the S = 0 Written out, this becomes cone, and lying between the 3(- and Y-axes. See fig. 3. The intercepts between the hyperboloids 1 dn 2 ) and the axis of revolution of the cone occur at ~Hs= 2-H dX + AY ~X X = Y = S. Through the discussion and examples in the following sections, the reader should come +(_ I~+AX)SY_2aZSZ. to appreciate the geometry introduced by the (X, Y, Z) coordinates and the level surfaces of H At criticality each coefficient vanishes, so and S 2. 1 _ dn z Ze=0, A- 2H~ <0, X~Tv-+Y,=0. lazx e -- 4. R 3 vector formulation of ray optics: equilibria and stability conditions Hence, critical points occur in the Z = 0 plane only, for values of H and S 2 such that the Hamilton's equations for ray optics may be gradients VH= (1/2H)(dn2/dX,-1) and rewritten in terms of the ~3 coordinates x= VS 2 = (Y, X) are collinear. These are the equilib- (X, Y, Z) in vector form as ria of the ray optics system. The second variation, .~" = V2S 2 X VH. 82Hs = 8x t " D2Hs( xe) • 8x, This vector-cross-product form of the optics equations ensures that both of the quantities H is conserved by the linearized equations. (This is 192 D.D. Holm, K.B. Wolf/Lie-Poisson description of Hamiltonian ray optics because ~2H S is the Hamiltonian for the lin- equilibrium and earized dynamics around x e. See ref. [3], ap- pendix A, and ref. [7] for discussions of linearized -K= det(D2Hs) Hamiltonian dynamics.) When the equilibrium point x~ satisfies the conditions for DZHs(xe) to be definite in sign, the second variation ~2H s may 2A (2X n2 + =n2 + 2H 2 ) 8H4X 2 ~ 2 X -d-~ . be taken as a norm for measuring perturbations from equilibrium, and its conservation in that case implies stability of the equilibrium. (Phase The determinant is positive (and the Gaussian points that start near the equilibrium stay near it curvature of Hs(x ~) is negative), provided in the linearly conserved norm ~2H S when D2Hs(xe) is definite; so in that case the equilib- d2( X2n 2) x2d2n 2 + dn2 rium is stable.) Consequently, x¢ will be a stable dX 2 dX 2 4X~-R- + 2n 2 < O, equilibrium when the following symmetric matrix is definite: which means that the profile of XZnZ(X) is con- cave downward at a stable equilibrium.
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