Introduction One of the most impressive accomplishments of wave optics is its success in providing a coherent and succinct description of the interactions between electromagnetic waves and matter (gases, solids, etc.). Maxwell's equations, which describe the propagation of light, and the Laplace–Lorentz equations, which describe the source terms of light, allow one to take into account the phenomena of refraction, diffusion, and diffraction of light by dense media. It is amazing – to say the least – that such a theory can account for the complex interactions of an electromagnetic wave with an immense ensemble of atoms (each approximated in terms of individual harmonic oscillators), by means of a simple optical index n op . Such an achievement is reminiscent and, indeed, on par with the level of concision achieved by the concept of effective mass in representing the interaction of a conduction electron with a crystalline lattice. In the description of all these effects, an electromagnetic wave with (angular) frequency ω forces free carriers into oscillatory motion at the same frequency, leading to radiative re-emission at this same frequency. This behaviour is a natural by-product of the linear equations we have employed up until now. In this chapter, we will see that non-linear media (i.e. materials whose response to external excitations contains non-linear terms), may by used to perform frequency conversion as evidenced, for example, by second harmonic light generation or optical parametric oscillations.
Introduction One of the most impressive accomplishments of wave optics is its success in providing a coherent and succinct description of the interactions between electromagnetic waves and matter (gases, solids, etc.). Maxwell's equations, which describe the propagation of light, and the Laplace–Lorentz equations, which describe the source terms of light, allow one to take into account the phenomena of refraction, diffusion, and diffraction of light by dense media. It is amazing – to say the least – that such a theory can account for the complex interactions of an electromagnetic wave with an immense ensemble of atoms (each approximated in terms of individual harmonic oscillators), by means of a simple optical index n op . Such an achievement is reminiscent and, indeed, on par with the level of concision achieved by the concept of effective mass in representing the interaction of a conduction electron with a crystalline lattice. In the description of all these effects, an electromagnetic wave with (angular) frequency ω forces free carriers into oscillatory motion at the same frequency, leading to radiative re-emission at this same frequency. This behaviour is a natural by-product of the linear equations we have employed up until now. In this chapter, we will see that non-linear media (i.e. materials whose response to external excitations contains non-linear terms), may by used to perform frequency conversion as evidenced, for example, by second harmonic light generation or optical parametric oscillations.
Nonlinear optics is the study of the response of dielectric media to strong optical fields. The fields are sufficiently strong that the response of the medium is, as its name implies, nonlinear. That is, the polarization, which is the dipole moment per unit volume in the medium, is not a linear function of the applied electric field. In the equation for the polarization, there is a linear term, but, in addition, there are terms containing higher powers of the electric field. This leads to significant new types of behavior, one of the most notable being that frequencies different from that of the incident electromagnetic wave, such as harmonics or subharmonics, can be generated. Linear media do not change the frequency of light incident upon them. The first observation of a nonlinear optical effect was, in fact, second-harmonic generation – a laser beam entering a nonlinear medium produced a second beam at twice the frequency of the original. Another type of behavior that becomes possible in nonlinear media is that the index of refraction, rather than being a constant, is a function of the intensity of the light. For a light beam with a nonuniform intensity profile, this can lead to self-focusing of the beam. Most nonlinear optical effects can be described using classical electromagnetic fields, and, in fact, the initial theory of nonlinear optics was formulated assuming the fields were classical. When the fields are quantized, however, a number of new effects emerge. Quantized fields are necessary if we want to describe fields that originate from spontaneous emission. For example, in a process known as spontaneous parametric down-conversion, a beam of light at one frequency, the pump, produces a beam at half the original frequency, the signal. This second beam is a result of spontaneous emission.
Thus far, we have developed the properties of the electromagnetic field at optical frequencies, based on Maxwell’s equations. These equations further give a classical macroscopic perspective on the coupling of the propagation media to the field, as presented in Chapter 2. The macroscopic properties of a medium are based on averaged microscopic properties. The microscopic energy structure of matter was presented in Chapter 3, covering gases, solids, and liquids by employing mostly quantum models. We now proceed to the next level of development, the dynamic description of the interaction between the optical field and the propagation medium as a function of the field frequency and propagation media variables (e.g., energy structure, temperature, and pressure). In this chapter, the classical electromagnetic field is coupled to discrete frequency oscillators via Newton’s equation of motion. This approach leads to the popular classical oscillator model, often presented in introductory books on lasers. The classical oscillator model is an incomplete theory and can be only a semiempirical model. In the next chapter, a more detailed and comprehensive approach, which also includes statistical and quantum mechanics, is used leading to robust semiclassical and quantum oscillator models. This chapter and the next are the basis for the applied models presented in Part II of this book. Classical electrodynamics is based on Maxwell’s equations, as given in Chapter 2, and the Lorentz force relation, as given below: . . . F = q[E + (v × B)]. (4.1) . . . These equations cover the classical description of the interaction of light and matter. The first term in Eq. 4.1 represents coupling of the electric field to the medium. As discussed in Chapter 2 (Section 2.2), the leading mechanism for this is the electric dipole moment. To see that this is the coupling mechanism in the first term, consider the potential function driving this force, F = −∇V(r) = −∇(−qr · E). The above expression contains the dipole moment, as defined in Chapter 2. The second term in Eq. 4.1 represents coupling of the magnetic field to the medium.
Photons and ElectronsHistorical SketchThe Wave Nature of LightPolarizationThe Electromagnetic SpectrumEmission and Absorption ProcessesPhoton Statistics The Behaviour of Electrons LasersSummaryWave Properties of LightThe Electromagnetic SpectrumWave RepresentationElectromagnetic WavesReflection and RefractionTotal Internal ReflectionInterference of LightLight WaveguidingInterferometersDiffractionGaussian Beams and Stable Optical ResonatorsPolarization OpticsThe Polarization EllipseCrystal OpticsRetarding WaveplatesA Variable Waveplate: The Soleil-Babinet Compensator Polarizing PrismsLinear BirefringenceCircular BirefringenceElliptical BirefringencePractical Polarization EffectsPolarization AnalysisThe Form of the Jones MatricesLight and Matter Emission, Propagation, and Absorption ProcessesClassical Theory of Light Propagation in Uniform Dielectric Media Optical Dispersion Emission and Absorption of LightOptical Coherence and CorrelationIntroductionMeasure of Coherence Wiener-Khinchin TheoremDual-Beam Interference Practical Examples Some Essential Physics of Radiation and Solids RadiationElectrons in Solids Optical Sources, Modulators, and DetectorsOptical SourcesOptical ModulatorsPhotodetectorsOptical WaveguidesThe Planar Waveguide Integrated OpticsCylindrical WaveguidesOptical FibresOptical Fibres for CommunicationsPolarization-Holding Waveguides Photonic Crystal FibresNonlinear OpticsNonlinear Optics and Optical FibresThe Formalism of Nonlinear OpticsSecond-Harmonic Generation and Phase MatchingOptical MixingIntensity-Dependent Refractive IndexFour-Photon Mixing (FPM)Parametric and Inelastic ProcessesSolitonsPhotosensitivityNon-Linear Effects in Photonic Crystal Fibres (PCF)Photonics in ActionAntireflective Coatings Optical-Fibre Current MeasurementThe Integrated Optical Spectrum AnalyserThe Audio Compact Disc (CD) The Optical-Fibre GyroscopeHolographyOptical-Time-Domain Reflectometry (OTDR) and Its Use in Distributed Optical-Fibre Raman Temperature Sensing (DTS)Measurement of Very Short Optical Pulses: The AutocorrelatorTopics in Optical-Fibre CommunicationsRegimes-Correlation LengthEpilogue And What of the Future?AppendicesAppendix I: Maxwell's EquationsAppendix II: The Fourier Inversion TheoremAppendix III: Symmetry of the Permittivity TensorAppendix IV: The Polarization EllipseAppendix V: Radiation from an Oscillating DipoleAppendix VI: The d-FunctionAppendix VII: The Fermi-Dirac Function Appendix VIII: Second Harmonic Generation Appendix IX: The Sampling TheoremAppendix X: The Semiconductor EquationAppendix XI: The Formal Analysis of Polarization-Mode Dispersion (PMD)ReferencesEach chapter includes an introduction, conclusion, problems, references, and suggestions for further reading.
This first complement is devoted to a completely classical approach of lightmatter interaction which was proposed by Lorentz at the end of the nineteenth century, before the advent of quantum mechanics, but after the discovery of the electron. Lorentz' phenomenological model is based on the experimental fact that atoms have well-defined and sharp absorption lines: he assumed that atoms behaved like harmonic oscillators, in which the electrons are bound to the atomic nucleus by a restoring force which varies linearly with its displacement (from its equilibrium point close to a nucleus), and makes them oscillate at a given frequency ω 0 equal to the experimentally determined absorption frequency. Within the frame of this model, we first calculate the electromagnetic field radiated by an oscillating electron. We show that in the absence of an externally applied force the free oscillations of the electron are damped, because electromagnetic energy is radiated at the expense of mechanical energy. We then study the characteristics of the radiation that is emitted when the oscillations are forced by the application of an external oscillatory electromagnetic field of angular frequency ω. We characterize the different regimes of this scattering of the incident electromagnetic wave and finally determine the polarization induced in the atomic medium by the incident electromagnetic wave. The Lorentz model can be considered as a lowest order approximation to a description of the light–matter interaction, a better approximation being the semi-classical treatment presented in Chapter 2, and the rigorous treatment being the completely quantum mechanical model presented in Chapter 6.
In the previous chapter we examined nonlinear propagation in a two-level medium, which leads to amplification or absorption of the radiation field. As we will see further on in this volume, when the medium is placed in an optical cavity one realizes a laser in the case of amplification or an optically bistable system in the absorptive configuration. On the other hand, the nonlinear response of the atomic polarization to the electric field leads to further fascinating scenarios [29, 51–60]. For example, a field with central ω 0 can generate harmonic components with central frequencies 2 ω 0 , 3 ω 0 etc., processes that are called second-harmonic generation, third-harmonic generation etc. Furthermore, the interaction of two fields with central frequencies ω 1 and ω 2 , with ω 1 > ω 2 , can generate components with central frequency ω 1 + ω 2 (sum-frequency generation) and ω 1 − ω 2 (difference-frequency generation). Such phenomena have not yet appeared in this volume because we have assumed that the electric field corresponds to a narrow frequency band around a central frequency ω 0 only, whereas to describe them it is necessary to assume that the electric field is the superposition of a number of frequency bands. In this chapter we will provide a description of the basic elements which are necessary to describe these phenomena and we will discuss a small number of them for the case of materials with a quadratic nonlinearity, referring to textbooks in nonlinear optics for the description of other phenomena. The following chapter is devoted, instead, to the case of materials with cubic nonlinearities. The starting point of nonlinear optics is the expansion of the atomic polarization in powers of the electric field, when the electric field is not too intense. In our treatment in Chapter 5 we saw that the expansion of the nonlinear susceptibility contains only even powers of the electric field (see Eq. (5.27)), and therefore the expansion of the polarization includes only odd powers.
In this book we treat the interaction of radiation with matter with special application to gain (amplifying) media typically existing in lasers. Tradition ally the interaction of radiation with matter is treated in terms of the index of refraction and the absorption of light, utilizing the model of electrons on springs. In Chap. 3 we show that this model is reasonable for some linear absorbing systems but is very awkward, if not outright unworkable, for a microscopic description of gain media. The essential failure of the electron-spring model is its difficulty in yielding net stimulated emission (and therefore gain) for an ensemble of systems.t In contrast, a more physically sound, quantum treatment of matter provides for the concept of population inversion and explains stimulated emission in a natural way. Hence from the very start we are compelled to base our understanding of the interaction of radiation with matter on a quantum-mechanical description of atoms. In Chaps. 1-13 a classical field interacts with quantum-mechanical atoms; in the remainder of the book, both field and atoms are treated quantum mechani cally.