Chapter 12 – Wave Optics

Daniel Malacara

Byron B. Taylor

This dissertation covers the field of atom optics and is divided into four main chapters: In Chapter 2 we investigate the effects of light forces on the center-of-mass motion of two-level atoms. This will lead to the discussion of two regimes: the "ray optic" and the "wave optic" regime. In the first case, an atom is well localized in the field which allows a comparison to be made with classical ray optics. In the second case, the atom is strongly delocalized which leads to a wave treatment and allows a comparison with diffractive optics. We finish this chapter with an example in each regime: Doppler cooling for ray optics and an atomic Fabry-Perot for wave optics. In Chapter 3 we extend the results of the previous chapter to the diffraction of atoms by a standing light field. We cover three regimes in the near resonant Kapitza-Dirac effect: the Raman-Nath, the Bragg and the optical Stern-Gerlach regime. In the Raman-Nath and Bragg regimes, the wave-packet is strongly delocalized compared to the period of the standing wave. In contrast, the Stern-Gerlach regime has a small spatial extent. The Raman-Nath and Bragg regimes are differentiated in their treatment of the kinetic energy. Initially we only discuss coherent interactions. In the later half of this chapter we introduce spontaneous emission and show how its presence affects the diffraction pattern in each of these regimes. In Chapter 4 we cover various atomic cooling schemes: strong field Sisyphus cooling, adiabatic cooling, evaporative cooling, polarization gradient cooling and velocity selective coherent population trapping. We begin with a brief discussion of atomic temperature. We then cover two cooling schemes for two-level atoms. We eventually move to multi-level atoms and end this chapter with a two-atom multi-level system. In Chapter 5 we conclude with a brief discussion of practical uses and devices that may arise from atom optics such as lenses, mirrors, gravitational interferometry, lithography and atomic clocks.

T. R. Groves

The central problem in this chapter is to solve for the wave function for a particle of charge q and rest mass m propagating in a general electromagnetic potential. It explores a few of the important implications for wave optics. The chapter begins with a review of basic quantum mechanics governing particle motion. It confines the discussion to only those topics which are relevant to the motion of a fast (unbound) charged particle in a general electromagnetic potential. The chapter seeks a dynamical equation to describe the motion of a single particle of charge q and rest mass m in the presence of a general electromagnetic potential. Physically, a close analogy exists with the wave optics of light propagation. In quantum mechanical wave optics, one starts with a surface of constant phase called a wave front. In wave optics, the aberrations are manifest as displacements in the surfaces of constant phase.

Marcel Wellner

Roland L. Omnes

Sergey V. Gaponenko

Let us return from optics to mechanics and explore the analogy to its fullest extent. In optics the old system of mechanics corresponds to intellectually operating with isolated mutually independent light rays. The new undulatory mechanics corresponds to the wave theory of light . Erwin Schrödinger, Nobel lecture , 1933 In this chapter we shall see that electromagnetic waves and electrons feature a number of common properties under conditions of spatial confinement. Simple and familiar problems from introductory quantum mechanics and textbook wave optics are recalled in this chapter to emphasize the basic features of waves in spatially inhomogeneous media. Herewith we make a first step towards understanding the properties of electrons and electromagnetic waves in nanostructures and notice that these properties in many instances are counterparts. Different formulas and statements of this chapter can be found in handbooks on quantum mechanics and wave optics. A few textbooks on quantum mechanics do consider analogies of propagation and reflection phenomena in wave optics with those in wave mechanics. Isomorphism of the Schrödinger and Helmholtz equations In Chapter 2 we discussed that an electron in quantum mechanics is described by the wave function, the square of its absolute value giving the probability of finding an electron at a specific point in space. This function satisfies the Schrödinger equation (2.56) which is the second-order differential equation with respect to space and the first-order differential equation with respect to time.