Application Of Hamiltonian Methods To Analogous Problems In Optics: Optical Fibers

Abstract The formal analogy between Hamiltonian classical mechanics and quantum mechanics on the one hand and geometrical optics physical on the other is systematically presented. When the time is replaced by the axial coordinate of a cylindrically uniform optical system and Hamilton's principle replaced by Fermat's principle, classical particle trajectories correspond to rays and quantum mechanical wave functions to physical optics fields. The operator formalism of quantum mechanics then provides elegant solutions for problems associated with propagation of beams in multimode optical fibers.1. IntroductionHamiltonian methods in geometrical optics and physical optics provide a consistent frame­ work within which ray and wave results and interpretations articulate in an understandable way. Thanks to the labor and insights of many keen minds it has long been established that the principal features of propagation within a gradient index fiber (or a periodic sequence of focusing elements) are related more or less immediately to families rays within the fiber. 1 ' 12 Each method has its advantages and its adherents. The approach of this paper draws on formulations of classical and quantum physics noted for their power elegance. When the time is replaced by the axial coordinate of the fiber and the potential function by variation in the refractive index, classical particle trajectories correspond to rays and the stationary wave functions to modes of fiber.13 ? 14 The principal develop-, ments center on elaboration of a further analogy between coherent states and beam- modes. 15" 20 Off-axis beam-modes are generated from the (lowest) mode of the fiber by a unitary transformation, the parameters of which are average position and optical direction cosine (momentum). The latter are (classical) ray coordinates. off- axis mode is an eigenfunction of the operator analogous to annihilation operator. The (complex) eigenvalue of the annihilation operator can be employed as usual to label beam-mode and as the variable of the representative function. The real and imaginary parts of this complex variable again evolve like ray coordinates. A Gaussian beam different from the modes of the fiber may readily be represented in terms of beam-modes belonging to the fiber profile, and its evolution is then easily calculated.2_.__ Hamiltonian mechanics and opticsIn this section we introduce the elements of Hamiltonian mechanics and special form of Hamiltonian optics required for the analysis of fibers. For simplicity, we confine ourselves to mechanical systems specified by one generalized coordinate plus the time analogous to a planar fiber in which a single transverse coordinate specifies the displace­ ment of a ray at each axial point. The two systems are illustrated in Figure 1. The refractive index of the gradient fiber is a function transverse coordinate x. This variation is analogous to that of the potential energy in mechanical system. The propagation of the ray along system axis z compares with evolution mechanical system in time, t. We find that results in geometrical optics parallel classical mechanics, while those in physical optics parallel quantum mechanics. This formal analogy is displayed in Table 1. We will review the development optical terms.* On leave from Chengdu Institute of Radio Engineering, Chengdu, Sichuan, People's Republic of China.