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Princeton University Press eBooks

Scattering

Sönke Johnsen

Published Dec 25, 2011• 0 citations
Publisher

Abstract

This chapter discusses scattering. As mentioned already, optics is primarily photons interacting with electrons, atoms, and molecules. If the energy of the photon matches the difference in energy between two levels of excitation, then the photon vanishes, its energy converted into other forms. However, if the photon's energy does not find a match among the differences in possible energy states, a new photon will quickly be emitted. The new photon is usually the same energy—and thus the same wavelength—as the old one, making it appear as if the original photon bounced, which is why this nonabsorptive interaction is known as “scattering.” When the energy of the new photon matches that of the old one, the scattering is known as elastic. When the energy of the new photon is lower, the scattering is known as inelastic. All the observed phenomena of scattering—refraction, reflection, structural colors, turbidity, and more—are based on these few principles.

DOI
10.23943/princeton/9780691139906.003.0005

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This chapter discusses scattering. As mentioned already, optics is primarily photons interacting with electrons, atoms, and molecules. If the energy of the photon matches the difference in energy between two levels of excitation, then the photon vanishes, its energy converted into other forms. However, if the photon's energy does not find a match among the differences in possible energy states, a new photon will quickly be emitted. The new photon is usually the same energy—and thus the same wavelength—as the old one, making it appear as if the original photon bounced, which is why this nonabsorptive interaction is known as “scattering.” When the energy of the new photon matches that of the old one, the scattering is known as elastic. When the energy of the new photon is lower, the scattering is known as inelastic. All the observed phenomena of scattering—refraction, reflection, structural colors, turbidity, and more—are based on these few principles.
Light gets scattered when it encounters an obstacle or inhomogeneity even on a microscopic scale. A well-known example is the blue color of the sky, resulting from Rayleigh scattering of light by molecules in the air. Such redirection of energy can be used to amplify signals by taking power from a “pump” wave co-propagating with the signal in an appropriate optical medium. An example of this is provided by Raman scattering. Having said that, it is important to realize that scattering does not always occur when light interacts with a material [1]. In some cases, photons get absorbed in the medium, and their energy is eventually dissipated as heat. In other cases, the absorbed light may be re-emitted after a relatively short time delay in the form of a less energetic photon [2], a process known as fluorescence. If fluorescence takes place after a considerable delay, the same process is called phosphorescence [3]. For a photon to get absorbed by a material, its energy must correspond to the energy required by the atoms or molecules of that material to make a transition from one energy level to a higher energy level. In contrast, the scattering of photons from a material can take place without such a requirement. However, if the energy of the incident photon is close to an allowed energy transition, significant enhancement of scattering can occur.
Most of our knowledge about the structure of matter and interaction between particles is derived from scattering experiments. From a theoretical point of view, scattering problems are concerned with the continuous (and positive) energy eigenvalues and unbound eigenfunctions of the Schro¨dinger equation. We have already encountered one-dimensional examples of scattering in Chapter 4, in the discussion of reflection and transmission of an incident particle with definite momentum from a potential step or barrier. In this chapter we consider scattering from a more formal point of view. We will confine our discussion to elastic scattering (scattering without loss or gain of energy by the projectile) from central potentials although many of the concepts and results are applicable to inelastic scattering.
Raman scattering is a two-photon process [1]. As illustrated in Fig. la, an incident photon is annihilated and a new photon of a different frequency is created. This implies that the scattering is inelastic, and the medium is left in a different energy state. When the scattered photon is lower in energy, or frequency, it is said to be Stokes shifted. The shift in frequency of the Stokes photon is related to a characteristic frequency of the medium. This is illustrated in Fig. 2a. Calling this characteristic material excitation frequency ω)v, the Stokes frequency is then ωs = ΩL — ωv, where ωL is the incident laser frequency. Thus, Raman (Stokes) scattering results in an excitation from the ground state of the medium to an excited state mediated by a two-photon inelastic scattering. The internal material excitation can involve the creation of an excited electronic state, an excited vibrational-rotational state, a lattice vibration, a spin flip in semiconductors, or electron waves in plasmas [2]. The phenomenon occurs in gases, liquids, solids, and plasmas.
This chapter studies wave interference. Light does not bend in a lens, it does not bounce off the surface of glass, and it does not spread out after passing through a small hole. All of these are illusions. It only appears that light bends, bounces, and spreads out. The effects that people see—reflection, refraction, colors in soap bubbles—come from the fact that photons interact with one another in an unusual way. The interaction is often referred to as wave interference because it can be described mathematically much like the interference of water waves, but the reality of it is considerably stranger. The strangeness is discussed in more detail in the last chapter, but a hint of it can be seen from the fact that photons appear to interfere with one another even if they are emitted one at a time, which water waves obviously do not do.
The interaction of light with nanometer-sized structures is at the core of nano-optics. It is obvious that as the particles become smaller and smaller the laws of quantum mechanics will become apparent in their interaction with light. In this limit, continuous scattering and absorption of light will be supplemented or replaced by resonant interactions if the photon energy hits the energy difference of discrete internal (electronic) energy levels. In atoms, molecules and nanoparticles, like semiconductor nanocrystals and other “quantum confined” systems, these resonances are found at optical frequencies. Due to the resonant character, the light–matter interaction can often be approximated by treating the quantum system as an effective two-level system, i.e. by considering only those two (electronic) levels whose difference in energy is close to the interacting photon energy ħ ω 0 . In this chapter we consider single-quantum systems that are fixed in space, either by deposition to a surface or by being embedded into a solid matrix. The material to be covered should familiarize the reader with single-photon emitters and with concepts developed in the field of quantum optics. While various theoretical aspects related to the fields emitted by a quantum system have been discussed in Chapter 8, the current chapter focuses more on the nature of the quantum system itself. We adopt a rather practical perspective since more rigorous accounts can be found elsewhere (see e.g. [1–4]).
Introduction The emission and absorption of radiation, as well as the conversion of radiation into other modes of energy such as heat or electricity, all involve interaction between electromagnetic waves and atoms, molecules, or free electrons. Such daily phenomena as the radiative emission by the sun, the shielding of earth from harmful UV radiation by the ozone layer, the blue color of the sky, and red sunsets are all – despite their celestial magnitude – generated by microscopic particles. Most lasers depend on emission by excited atoms (e.g. the He-Ne laser), ionized atoms (the Ar + laser), molecules (CO or CO 2 lasers), impurities trapped in crystal structures (Nd: YAG or Ti:sapphire lasers), or semiconductors (GaAs diode lasers). Similarly, many scattering processes of interest (e.g., Rayleigh or Mie scattering) result from the exchange of energy and momentum between incident radiation and atomic or molecular species. In the previous chapter we saw that the energy of microscopic particles is quantized: their energy can be acquired, stored, or released only in fixed lumps called quanta . The example of the “particle in the box” (eqn. 8.19) illustrated that these energy quanta are specific not only to the particle itself but to the system to which it belongs. Thus, in the box, the energy of the particle is specified by its own mass and by the dimension of the box; in a different box, the same particle will have an entirely different system of energy levels and the quanta will have different magnitudes.