The discrete energies En of bound states are not the only indicators of quantum-mechanical properties and quantization. Rotational motion of atoms or molecules, associated with angular momentum, is also quantized, both in magnitude and in direction; only discrete orientations are allowed with respect to any selected (but arbitrary) axis of quantization [Sho90, §18.1]. In the absence of external fields, the energy of a free atom or molecule does not depend on this orientation, and the energy states are degenerate. This chapter discusses that degeneracy, and the theory of coherent excitation of such degenerate quantum states.

Angular momentum degeneracy. The theoretical building blocks for describing rotational motion are angular momentum states |J,M,〉 discussed in App.A, associated with a dimensionless vector operator Ĵ and its component Ĵz along a quantization axis, taken as defining the z axis. The label J, the angular momentum quantum number, derives from the eigenvalue J(J + 1) of the operator Ĵ2 and therefore quantifies the magnitude of the angular momentum. It may be an integer or half integer. The label M, the magnetic quantum number, is the eigenvalue of Ĵz; it quantifies the projection of angular momentum along a reference axis. The values of M differ by integers and range from −J to +J in integer steps. The total number of such values, 2J +1, is an integer, the degeneracy of the quantum state |J,M〉.

Angular momentum of an isolated quantum system, such as an atom or molecule, refers always to the center of mass (which may be moving).

From prehistoric times has come recognition that sunlight and firelight provide warmth, and that such illumination casts shadows. Expressed in more contemporary terms one would say that light travels in rays, and that this radiation has the potential to provide heat energy to absorbing material. From the time of Newton it has been known that light sources emit radiation comprising a distribution of colors. During the nineteenth century it was recognized that radiation had characteristics of transverse waves (with wavelength associated to color) but until the late twentieth century, when lasers became laboratory tools, it was hardly necessary to delve into the equations of electromagnetic theory to treat such experiments as were then possible; interest lay primarily with thermodynamic considerations of energy flow or with measurements of the dark or bright lines seen in the distribution of light that, after passing through a slit, was dispersed by a prism or grating into constituent colors. Although laser light sources are essential for the types of atomic excitation considered in this monograph, the legacy from thermal radiation still influences many interpretations of the interaction between radiation and matter, and it is therefore useful to summarize some of those concepts.

The mathematics needed for describing laser radiation, or polarized light in general, has much in common with the mathematics of quantum theory discussed starting in Sec. 3.5 and specialized to two-level atoms in Sec. 5.6. In both cases one deals with two complex-valued functions – independent electric field amplitudes or probability amplitudes – whose absolute squares are measurable.

The quantum world within an atom or molecule that once attracted explorations only by academic physicists now provides fertile sustenance for chemists seeking control of chemical reactions and for engineers developing ever smaller electronic devices or tools for processing information with greater security. Whereas the first pioneers could only discover the most elementary properties – the discrete energy levels that characterize the internal structures of atoms and the radiative transitions that link these structures to our external world – it is now possible to alter that structure at will, albeit briefly.

Objective

This monograph presents the physical principles that describe such deliberately crafted changes, namely how single atoms or molecules (or other simple quantum systems) are affected by coherent interactions, primarily laser light – a subject that has been regarded first as a part of quantum electronics and then quantum optics but is most generally described as coherent atomic excitation [Sho90].

This physics has relevance to such basic concerns as the detection and quantitative analysis of trace amounts of chemicals, the catalysis or control of chemical reactions, the alignment of molecules, and the processing of quantum information. The physics necessarily involves elementary quantum mechanics, but it has many associations to the classical dynamics that governs macroscopic objects – waves and particles. The mathematics that quantifies the changes is that of differential equations, specifically coupled ordinary differential equations (ODEs), whose parameters incorporate the controls of experimenters and whose solutions, appropriately interpreted, quantify the resulting changes.

The philosophers who first hypothesized the existence of “atoms” had in mind the smallest particles of matter that could preserve identifiable chemical properties – building blocks that could be assembled into familiar substances. Little more than this definition – tiny masses that carry kinetic energy and undergo collisions – led to the fruitful quantitative explanation of vapor properties in the kinetic theory of gases, and to such devices as mass spectrometers and ion accelerators.

Atoms and molecules. As became clear during the early twentieth century, these “chemical atoms’ from which materials are constructed have internal structure that endows them with their chemical attributes: one or more positively charged nuclei, each a few femtometers (1 fm = 10−13 cm = 10−15 m) in diameter, surrounded by one or more much lighter negatively charged electrons whose motion fills a volume of at least a few cubic angstroms (1 Å= 10−8 cm = 10 nm) in diameter. Nowadays we distinguish between particles having multiple nuclei (molecules) and those with a single nucleus (atoms); when the positive and negative charges are unbalanced these are ions (positive or negative). The simplest atom, hydrogen, has a single electron; the most complex atoms have more than a hundred electrons. Although I will often refer to “atoms”, usually the discussion applies equally well to molecules or to any other structure whose constituents exhibit distinct quantum properties, as manifested by discrete energies.

Nuclei. The nuclei of atoms are, in turn, composed of protons and neutrons.

Throughout this monograph the effect of radiation upon a quantum system has been presented as an interaction between an atomic moment (usually the electric dipole moment) and a pulse of nearly periodic electromagnetic radiation, typically the electric field of that radiation. The strength of the interaction of such a field with an individual atom or molecule is typically parametrized by the field intensity (or radiation irradiance) and a transition moment (or oscillator strength). When the atoms form macroscopic aggregates, through which the laser radiation must pass, the laser-induced alteration of atomic states produces new fields that subtract from or add to the original field [Sho90, Chap. 12]. As a result, pulses propagating through matter become altered. The pulse amplitude will, at first, decrease as energy is absorbed by the atoms, but more dramatic effects can occur that drastically alter the shape of the pulse as it travels through greater thicknesses of matter [Sho90, Chap. 9]. Furthermore, new frequencies may be generated – the phenomena of nonlinear optics [Rei84; Gae06; Boy08].

Prior to the advent of laser radiation there was little interest in short pulses, and the equations describing radiation dealt with the flow of energy through matter that could absorb or divert the radiation, embodied in the theory of radiative transfer [Cha60; Tuc75; Ryb85; Car07]. Such descriptions characterized the radiation–matter interaction by a static complex-valued index of refraction, whose imaginary part produced absorption while the real (dispersive) part altered the propagation velocity [Bre32; Mea60; Bor99].

The fields within cavities differ in an important respect from those of a laser beam in free space: The bounding surfaces that define the cavity enclosure impose constraints that limit the fields to discrete modes, characterized in part by discrete frequencies. The surfaces enclosing a cavity are boundaries where the dielectric properties change abruptly; they are idealized as discontinuities of the susceptibility ∊ and permeability μ. Across any such surface the normal component of the B field is continuous, as is the transverse component of the E field. These conditions imply, for example, that along a perfectly conducting surface (an idealized mirror) the electric field has a node. The allowed fields (the discrete mode fields) are then particular solutions to the Maxwell equations, or their conversion into Helmholtz equations, that vanish along bounding surfaces.

Figure 4.2 of Sec. 4.1.3 depicts two classes of cavities. Frame (a) shows a cylindrical cavity used for microwave radiation. The cavity completely encloses the field, apart from a small aperture through which the atoms pass. Frame (b) shows a prototype optical cavity, in which the cavity field is that of a beam confined along one axis. Idealized as perfect conductors, the confining mirrors permit only integral half waves between them, and the frequencies of such plane waves are correspondingly discrete. Because the enclosing endmirrors are not perfectly reflecting there will occur some loss through them, along the cavity axis, and the field is not strictly monochromatic.

Introduction

A reader of this book will quickly see that structured surfaces, whether deterministic or random, can reflect, transmit, refract, and amplify volume or surface electromagnetic waves in ways that naturally occurring surfaces cannot. They can also change the nature of an electromagnetic field incident on them. For example, they can change a beam with one intensity distribution into a beam with a different intensity distribution, or they can convert a plane wave into a beam. The use of structured surfaces, specifically randomly rough surfaces, to effect such transformations of optical fields is the subject of this chapter, where two examples of this use are presented, namely beam shaping and the formation of pseudo-nondiffracting beams.

The creation of optical elements that transform an electromagnetic beam with a specified transverse intensity distribution into a beam with a different specified transverse intensity distribution, especially those that transform a laser beam with a Gaussian intensity profile into a beam with a constant intensity profile – a flat top beam, has been studied theoretically and experimentally for many years [1–38]. The interest in beam shaping is due to a wide range of applications for beams with a variety of non-Gaussian intensity distributions. These applications include laser surgery [39], laser radar [40], laser microstructuring of materials [41], metal hardening [42], optical communication [43], and optical scanning [44], among others. Some of them and other applications of beam shaping are discussed in the recent book by Dickey et al. [45].