Fundamentals of Nonlinear Optics ECED 6400 Lecture Notes C 2016 Sergey A

Fundamentals of Nonlinear Optics ECED 6400 Lecture Notes c 2016 Sergey A. Ponomarenko January 11, 2018 Contents 1 Introduction 3 2 Plane electromagnetic waves in linear media 8 2.1 Plane waves in free space . 8 2.2 Plane waves in homogeneous dielectrics . 12 2.2.1 Plane waves in homogeneous isotropic media with no spatial dispersion . 16 2.2.2 Plane waves in uniaxial crystals . 18 2.2.3 Faraday effect and polarization rotation . 20 2.3 Refraction and reflection of plane waves at the interface of homogeneous media . 22 2.3.1 Reflection of plane waves at oblique incidence: Generalized Snell’s law . 22 2.3.2 Reflection of plane waves at oblique incidence: Fresnel Formulae 24 2.3.3 Brewster angle and surface plasmon polaritons . 28 2.3.4 Total internal reflection . 30 2.4 Refraction and reflection from dielectric slab: Multi-wave interference 32 2.5 Classical theory of optical dispersion and absorption . 38 2.5.1 Lorentz-Kramers expression for dielectric permittivity . 38 2.5.2 Classical theory of Faraday effect . 43 3 Pulses and beams in linear optics 45 3.1 Pulse propagation in dispersive media: non-resonant case . 45 3.2 Resonant pulse propagation in linear absorbers . 48 3.2.1 Resonant interaction of short pulses with linear media: Homo- geneous line broadening . 48 3.2.2 Inhomogeneous broadening . 50 3.2.3 Maxwell-Lorentz pulse evolution equations and classical area theorem . 52 3.3 Paraxial wave equation and Gaussian beam optics . 55 3.4 Plane wave decomposition of beams: Angular spectrum . 57 1 4 Nonlinear optics 60 4.1 Introduction. Qualitative description of nonlinear optical processes . 60 4.2 Nonlinear processes generated by arbitrary fields: Spatial and temporal dispersion . 65 4.3 Formal properties of nonlinear optical susceptibilities . 67 4.4 Nonlinear wave equation approach: Classical coupled-wave equations 71 4.5 Second-harmonic generation . 74 4.5.1 Coupled wave equations and phase matching considerations . 74 4.5.2 Second-harmonic generation: Beyond the undepleted pump approximation . 79 4.6 Sum-frequency generation . 82 4.6.1 Coupled wave equations and their solution in the undepleted pump approximation . 82 4.6.2 Manley-Rowe relations . 84 4.7 Difference-frequency generation (parametric down-conversion) . 86 4.8 Four-wave mixing: General considerations . 90 4.9 Third harmonic generation . 91 4.10 Self-focusing, nonlinear absorption, and spatial solitons . 93 4.11 Z-scan measurement of nonlinear refractive index . 101 4.12 Polarization dynamics of third-order processes . 106 4.13 Electro-optical Kerr effect . 111 4.14 Spontaneous and stimulated Raman scattering: CW case . 114 4.15 Transient stimulated Raman scattering . 123 4.16 Spontaneous Brillouin scattering . 129 4.17 Brillouin phonon propagation . 131 4.18 Stimulated Brillouin scattering . 133 2 Chapter 1 Introduction In this course, we will be describing all optical phenomena classically within the framework of macroscopic Maxwell’s equations written in terms of macroscopic electromagnetic fields. The latter are obtained by averaging rapidly varying microscopic fields over spatial scales much larger than characteristic material microstructure scales (atomic size, lattice scale, etc). The averaging procedure is examined in detail in stan- dard electrodynamics textbooks1. Within the framework of such a phenomenological approach, which circumvents a detailed microscopic light-matter interaction description, external or driving volume charge and current densities, ρex and Jex, give rise to the electromagnetic fields obeying the Maxwell equations in the form r · D = ρex; (1.1) r · B = 0; (1.2) r × E = −@tB (1.3) and r × H = Jex + @tD: (1.4) The set of equations (1.1) through (1.4) is not closed, however, until we provide any information about the material media. Such information is furnished by supplying phenomenological constitutive relations among the four fields, E, D, B, and H. Without much loss of generality we will assume hereafter that all material media are nonmag- netic, which holds true for virtually all natural media at optical frequencies 2. We can then represent the magnetic constitutive relation in its simplest form as B = µ0H; (1.5) with µ0 being the free space permeability in the SI units we will be employing hereafter. A general electric constitutive relation states D = 0E + P; P = P(E); (1.6) 1J. D. Jackson, Classical Electrodynamics (Wiley, New York, NY, 1999) 3rd edition. 2This criterion, however, breaks down for some artificial materials, the so-called metamaterials, which we will not consider in this course. 3 where 0 is the free space permittivity and P is a macroscopic polarization field. The latter is in turn a function of the applied electric field. For sufficiently weak applied fields, P is a linear function of E; this is a regime of linear optics. However, even in the linear optics regime, the dependence of the polarization on the applied electric field can be rather complicated to account for possible medium anisotropy and–temporal and sometimes even spatial–dispersion. While the former implies that the medium response in a particular direction can be affected by the electric field components or- thogonal to this direction, the latter acknowledges the fact that the medium response at a given space-time point can depend on the applied electric field in the past (temporal dispersion) and/or on the fields in the neighborhood of the spatial point (spatial dispersion). We will study all these cases in detail in the subsequent chapters. As the magnitude of the applied electric field increases, the linear relationship be- tween P and E breaks down and we enter the realm of nonlinear optics. If the electric 9 field intensity is far below a critical value, Ecr ∼ 10 V/cm needed to ionize a material atom, the resulting polarization can be expressed as a series in increasing powers of the electric field. Schematically, such a series can be expressed as (1) (2) 2 (3) 3 P = 0(χ E + χ E + χ E + :::); (1.7) where we ignored the vector nature of the fields as well as dispersion, for simplicity. The expansion coefficients, χ(1) and χ(2), etc., are identified as linear and nonlinear susceptibilities, respectively. The linear and nonlinear susceptibilities should be treated as phenomenological constants in our classical description. The condition E Ecr is typically met with a vast majority of laser sources which rarely generate fields in excess 6 of 10 V/cm. However, even if the applied field does not exceed Ecr, the power series expansion can fail, provided the carrier frequency of the field lies close to any internal resonance of the medium. In the latter case, the material response tends to saturate at high enough field intensities. The proper quantitative description of such nonlinear saturation phenomena calls for a quantum mechanical treatment of the medium. When- ever, the power expansion of P is valid, though, we shall refer to the lowest-order term in the expansion as a linear contribution and designate the rest to be nonlinear polarization such that P = PL + PNL: (1.8) Next, the external charge and current densities are not independent from each other. Rather they are related by another fundamental law, the charge conservation law, which takes the form of a well-known continuity equation viz., @tρex + r · Jex = 0: (1.9) The external ρex and Jex drive the electromagnetic fields which, in turn, induce internal charge and current densities, ρ and J, inside a medium. The induced charges and currents can be of either free (conduction) or bound (polarized) type and they also obey the continuity equation, @tρ + r · J = 0: (1.10) We stress that charge conservation (1.9) amounts to a fundamental law which does not follow from Maxwell’s equations. The electromagnetic energy conservation law, 4 however, does follow from the Maxwell equations by the same token as the mechanical energy conservation follows from Newton’s laws. To derive the electromagnetic energy conservation law or the Poynting theorem, we take dot products of the both sides of Eqs. (1.3) and (1.4) with H and E, respectively, and use the constitutive relations (1.5) and (1.6), yielding H · (r × E) = −µ0H · @tH (1.11) and E · (r × H) = Jex · E + 0E · @tE + E · @tP: (1.12) On subtracting Eq. (1.11) from Eq. (1.12) term by term, we obtain 0 2 µ0 2 2 @tE + 2 @tH + Jex · E = E · (r × H) − H · (r × E) − E · @tP: (1.13) Further, using the vector identity r · (E × H) = H · (r × E) − E · (r × H) (1.14) we arrive, after minor algebra, at a differential form of the electromagnetic energy conservation equation @twem + r · S = −Jex · E − E · @tP: (1.15) Here the electromagnetic energy density wem is defined in the same way as in free space, 1 2 1 2 wem = 2 0E + 2 µ0H ; (1.16) and we introduced the electromagnetic energy flux density, the so-called Poynting vector, by the expression S = E × H: (1.17) Equation (1.15) is often referred to as Poynting’s theorem. In essence, it implies that the time rate of change of the electromagnetic energy density is determined by the energy flux density minus losses associated with external as well as internal currents. The second term on the r.h.s of Eq. (1.15) describes Ohmic losses associated with external currents and the third one is identified with the energy loss caused by induced polarization currents, including the ones associated with the generation of nonlinear polarizations. To reexpress the right-hand side of Eq. (1.15) in a more symmetric from, we can explicitly define the induced polarization currents as J = @tP; (1.18) and introduce the corresponding induced charge densities as ρ = −∇ · P; (1.19) such that the continuity equation (1.10) is respected.

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