1 Nonlinear5. Nonlinear Opt Optics This chapter provides a brief introduction into the 5.4.6 Optical Phase Conjugation ............. 13 basic nonlinear-optical phenomena and discusses 5.4.7 Optical Bistability and Switching .... 14 some of the most significant recent advances and 5.4.8 Stimulated Raman Scattering......... 16 breakthroughs in nonlinear optics, as well as novel 5.4.9 Third-Harmonic Generation applications of nonlinear-optical processes and by Ultrashort Laser Pulses.............. 17 devices. 5.5 Ultrashort Light Pulses Nonlinear optics is the area of optics that in a Resonant Two-Level Medium: studies the interaction of light with matter in Self-Induced Transparency theregimewheretheresponseofthematerial and the Pulse Area Theorem.................. 22 system to the applied electromagnetic field is 5.5.1 Interaction of Light nonlinear in the amplitude of this field. At low with Two-Level Media .................. 22 light intensities, typical of non-laser sources, the 5.5.2 The Maxwell and Schrödinger properties of materials remain independent of Equations for a Two-Level Medium 22 the intensity of illumination. The superposition 5.5.3 Pulse Area Theorem ...................... 24 principle holds true in this regime, and light waves 5.5.4 Amplification can pass through materials or be reflected from of Ultrashort Light Pulses boundaries and interfaces without interacting with in a Two-Level Medium ................ 25 each other. Laser sources, on the other hand, can 5.5.5 Few-Cycle Light Pulses provide sufficiently high light intensities to modify in a Two-Level Medium ................ 27 the optical properties of materials. Light waves Part A can then interact with each other, exchanging 5.6 Let There be White Light: Supercontinuum Generation.................. 29 momentum and energy, and the superposition 5.6.1 Self-Phase Modulation, 5 principle is no longer valid. This interaction of light Four-Wave Mixing, waves can result in the generation of optical fields and Modulation Instabilities at new frequencies, including optical harmonics of in Supercontinuum-Generating incident radiation or sum- or difference-frequency Photonic-Crystal Fibers ................. 29 signals. 5.6.2 Cross-Phase-Modulation-Induced Instabilities ................................. 31 5.1 Nonlinear Polarization 5.6.3 Solitonic Phenomena in Media and Nonlinear Susceptibilities ............... 3 with Retarded Optical Nonlinearity. 33 5.2 Wave Aspects of Nonlinear Optics ........... 4 5.7 Nonlinear Raman Spectroscopy .............. 37 5.3 Second-Order Nonlinear Processes ......... 5 5.7.1 The Basic Principles ...................... 38 5.3.1 Second-Harmonic Generation........ 5 5.7.2 Methods of Nonlinear Raman 5.3.2 Sum- and Difference-Frequency Spectroscopy ............................... 40 Generation and Parametric 5.7.3 Polarization Nonlinear Raman Amplification............................... 7 Techniques.................................. 43 5.7.4 Time-Resolved Coherent 5.4 Third-Order Nonlinear Processes ............ 8 Anti-Stokes Raman Scattering........ 45 5.4.1 Self-Phase Modulation ................. 9 5.4.2 Temporal Solitons......................... 10 5.8 Waveguide Coherent Anti-Stokes 5.4.3 Cross-Phase Modulation ............... 10 Raman Scattering ................................. 46 5.4.4 Self-Focusing............................... 11 5.8.1 Enhancement of Waveguide CARS 5.4.5 Four-Wave Mixing........................ 13 in Hollow Photonic-Crystal Fibers... 46 2 Part A Basic Principles and Materials 5.8.2 Four-Wave Mixing and CARS 5.11 High-Order Harmonic generation........... 63 in Hollow-Core Photonic-Crystal 5.11.1 Attosecond Metrology – Fibers ......................................... 49 Historical Background................... 63 5.9 Nonlinear Spectroscopy 5.11.2 High-Order-Harmonic Generation with Photonic-Crystal-Fiber Sources....... 53 in Gases ...................................... 64 5.9.1 Wavelength-Tunable Sources 5.11.3 Microscopic Physics ...................... 66 and Progress 5.11.4 Macroscopic Physics...................... 69 in Nonlinear Spectroscopy ............. 53 5.12 Attosecond Pulses: 5.9.2 Photonic-Crystal Fiber Frequency Measurement and Application ............... 71 Shifters ....................................... 54 5.12.1 Attosecond Pulse Trains 5.9.3 Coherent Anti-Stokes Raman and Single Attosecond Pulses......... 71 Scattering Spectroscopy 5.12.3 Basic Concepts with PCF Sources .......................... 55 for XUV Pulse Measurement ........... 71 5.9.4 Pump-Probe Nonlinear 5.12.3 The Optical-Field-Driven XUV Streak Absorption Spectroscopy Camera Technique........................ 74 using Chirped Frequency-Shifted 5.12.4 Applications of Sub-femtosecond Light Pulses XUV Pulses: Time-Resolved from a Photonic-Crystal Fiber ........ 57 Spectroscopy of Atomic Processes ... 78 5.10 Surface Nonlinear Optics, Spectroscopy, and Imaging ........................................ 60 References .................................................. 80 Although the observation of most nonlinear-optical 1963 provided 20%–30% efficiency of frequency con- phenomena requires laser radiation, some classes of version [5.5, 6]. The early phases of the development nonlinear-optical effects were known long before the and the basic principles of nonlinear optics have been Part A invention of the laser. The most prominent examples of reviewed in the most illuminating way in the classi- such phenomena include Pockels and Kerr electrooptic cal books by Bloembergen [5.7] and Akhmanov and effects [5.1], as well as light-induced resonant absorp- Khokhlov [5.8], published in the mid 1960s. 5 tion saturation, described by Vavilov [5.2, 3]. It was, Over the following four decades, the field of nonlin- however, only with the advent of lasers that systematic ear optics has witnessed an enormous growth, leading studies of optical nonlinearities and the observation of to the observation of new physical phenomena and giv- a vast catalog of spectacular nonlinear-optical phenom- ing rise to novel concepts and applications. A systematic ena became possible. introduction into these effects along with a comprehen- In the first nonlinear-optical experiment of the laser sive overview of nonlinear-optical concepts and devices era, performed by Franken et al. in 1961 [5.4], a ruby- can be found in excellent textbooks by Shen [5.9], laser radiation with a wavelength of 694.2nm was Boyd [5.1], Butcher and Cotter [5.10], Reintjes [5.11] used to generate the second harmonic in a quartz crys- and others. One of the most recent up-to-date reviews of tal at the wavelength of 347.1 nm. This seminal work the field of nonlinear optics with an in-depth discussion was followed by the discovery of a rich diversity of the fundamental physics underlying nonlinear-optical of nonlinear-optical effects, including sum-frequency interactions was provided by Flytzanis [5.12]. This generation, stimulated Raman scattering, self-focusing, chapter provides a brief introduction into the main optical rectification, four-wave mixing, and many others. nonlinear-optical phenomena and discusses some of the While in the pioneering work by Franken the efficiency most significant recent advances in nonlinear optics, as of second-harmonic generation (SHG) was on the or- well as novel applications of nonlinear-optical processes der of 10−8, optical frequency doublers created by early and devices. Nonlinear Optics 5.1 Nonlinear Polarization and Nonlinear Susceptibilities 3 5.1 Nonlinear Polarization and Nonlinear Susceptibilities Nonlinear-optical effects belong to a broader class of We now represent the polarization P as a sum electromagnetic phenomena described within the gen- P = P + P , (5.8) eral framework of macroscopic Maxwell equations. The L NL Maxwell equations not only serve to identify and classify where PL is the part of the electric dipole polarization nonlinear phenomena in terms of the relevant nonlinear- linear in the field amplitude and PNL is the nonlinear optical susceptibilities or, more generally, nonlinear part of this polarization. terms in the induced polarization, but also govern the The linear polarization governs linear-optical phe- nonlinear-optical propagation effects. We assume the nomena, i. e., it corresponds to the regime where the absence of extraneous charges and currents and write optical properties of a medium are independent of the , the set of Maxwell equations for the electric, E(r t), field intensity. The relation between PL and the electric and magnetic, H(r, t), fields in the form field E is given by the standard formula of linear optics: ∂ 1 B ∇ × E =− , (5.1) P = χ(1)(t − t )E(t )dt , (5.9) c ∂t L 1 ∂D ∇ × B = , (5.2) where χ(1)(t) is the time-domain linear susceptibility c ∂t tensor. Representing the field E and polarization P in ∇·D = 0 , (5.3) L the form of elementary monochromatic plane waves, ∇·B = 0 . (5.4) E = E (ω) exp (ikr − ωt) + c.c. (5.10) Here, B = H + 4π M,whereM is the magnetic dipole polarization, c is the speed of light, and and t PL = PL(ω)exp ikr − ωt + c.c. , (5.11) D = E + 4π J(ζ)dζ, (5.5) we take the Fourier transform of (5.9) to find −∞ Part A P (ω) = χ(1)(ω)E(ω) , (5.12) where J is the induced current density. Generally, the L equation of motion for charges driven by the electromag- where 5.1 netic field has to be solved to define the relation between (1) (1) the induced current J and the electric and magnetic χ (ω) = χ (t)exp(iωt)dt . (5.13) fields. For quantum systems, this task can be fulfilled by solving the Schrödinger equation.