Fundamentals of Photonics Chapter 8 NONLINEAR OPTICS Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Question: Is it possible to change the color of a monochromatic light? output NLO sample input Answer: Not without a laser light Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Nicolaas Bloembergen (born 1920) has carried out pioneering studies in nonlinear optics since the early 1960s. He shared the 1981 Nobel Prize with Arthur Schawlow. Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Part 0：Comparison Linear optics: ★Optical properties, such as the refractive index and the absorption coefficient independent of light intensity. ★ The principle of superposition, a fundamental tenet of classical, holds. ★ The frequency of light cannot be altered by its passage through the medium. ★ Light cannot interact with light; two beams of light in the same region of a linear optical medium can have no effect on each other. Thus light cannot control light. Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Part 0： Comparison Nonlinear optics: ★The refractive index, and consequently the speed of light in an optical medium, does change with the light intensity. ★ The principle of superposition is violated. ★ Light can alter its frequency as it passes through a nonlinear optical material (e.g., from red to blue!). ★ Light can control light; photons do interact Light interacts with light via the medium. The presence of an optical field modifies the properties of the medium which, in turn, modify another optical field or even the original field itself. Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Part 1：phenomena involved frequency conversion Second-harmonic generation (SHG) Parametric amplification Parametric oscillation third-harmonic generation self-phase modulation self-focusing four-wave mixing Stimulated Brillouin Scatteirng Stimulated Raman Scatteirng Optical solitons Optical bistability Two-order Three-order Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics 19.1 Nonlinear optical media Origin of Nonlinear the dependence of the number density N on the optical field the number of atoms occupying the energy levels involved in the absorption and emission if Hooke’s law is satisfied Linear! if Hooke’s law is not satisfied Noninear! Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Figure 19.1-1 The P-E relation for (a) a linear dielectric medium, and (b) a nonlinear medium. Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics The nonlinearity is usually weak. The relation between P and E is approximately linear for small E, deviating only slightly from linearity as E increases. In centrosymmetric media, d vanish, and the lowest order nonlinearity is of third order basic description for a nonlinear optical medium Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics In centrosymmetric media: d=0 the lowest order nonlinearity is of third order Typical values Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics The Nonlinear Wave Equation nonlinear wave equation Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics There are two approximate approaches to solving the nonlinear wave equation: ★The first is an iterative approach known as the Born approximation. ★ The second approach is a coupled-wave theory in which the nonlinear wave equation is used to derive linear coupled partial differential equations that govern the interacting waves. This is the basis of the more advanced study of wave interactions in nonlinear media. Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics 19.2 Second-order Nonlinear Optics Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics A. Second-Harmonic Generation and Rectification complex amplitude Substitute it into (9.2-l) Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics This process is illustrated graphically in Fig. 9.2-1. P PNL(t) + E t t t dc second-harmonic E(t) t Figure 9.2-1 A sinusoidal electric field of angular frequency w in a second-order nonlinear optical medium creates a component at 2w (second-harmonic) and a steady (dc) component. Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Second-Harmonic Generation SHG SFG DHG Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Component of frequency 2w SHG complex amplitude intensity The interaction region should also be as long as possible. Guided wave structures that confine light for relatively long distances offer a clear advantage. Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Figure 9.2-2 Optical second-harmonic generation in (a) a bulk crystal; (b) a glass fiber; (c) within the cavity of a semiconductor laser. Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Optical Rectification The component PNL(0) corresponds to a steady (non-time-varying) polarization density that creates a dc potential difference across the plates of a capacitor within which the nonlinear material is placed. An optical pulse of several MW peak power, may generate a voltage of several hundred uV. Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics B. The Electra-Optic Effect Substitute it into (9.2-l) 9.2-8 Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics If the optical field is substantially smaller in magnitude than the electric field Can be negleted Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics 9.2-8 a linear relation between PNL(w) and E(w) incremental change of the refractive index 9.2-9 Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics the nonlinear medium exhibits the linear electro-optic effect Pockels effect Pockels coefficient Comparing this formula with (9.2-9) Fundamentals of Photonics 2017/4/11

Frequency up-conversion Frequency down-conversion C. Three-Wave Mixing Frequency Conversion E(t) comprising two harmonic components at frequencies w1 and w2 Frequency up-conversion Frequency down-conversion Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Figure 9.2-5 An example of frequency conversion in a nonlinear crystal 点击查看flash动画 Although the incident pair of waves at frequencies w1 and w2 produce polarization densities at frequencies 0, 2wl, 2w2, wl+w2, and w1-w2, all of these waves are not necessarily generated, since certain additional conditions (phase matching) must be satisfied, as explained presently. Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Phase Matching where Frequency-Matching Condition Phase-Matching Condition Figure 9.2-6 The phase-matching condition Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics ★same direction: nw3/c0=nw1/c0+ nw2/c0, w3=w1+w2 frequency matching ensures phase matching. ★different refractive indices, nl, n2, and n3: n3w3/c0=n1w1/c0+n2w2/c0 n3w3=n1w1+n2w2 The phase-matching condition is then independent of the frequency-matching condition w3=w1+w2; both conditions must be simultaneously satisfied. Precise control of the refractive indices at the three frequencies is often achieved by appropriate selection of the polarization and in some cases by control of the temperature. Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Three- Wave Mixing We assume that only the component at the sum frequency w3=w1+w2 satisfies the phase-matching condition. Other frequencies cannot be sustained by the medium since they are assumed not to satisfy the phase-matching condition. Once wave 3 is generated, it interacts with wave 1 and generates a wave at the difference frequency w2=w3-w1. Waves 3 and 2 similarly combine and radiate at w1. The three waves therefore undergo mutual coupling in which each pair of waves interacts and contributes to the third wave. three-wave mixing parametric interaction Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics parametric interaction ◆Waves 1 and 2 are mixed in an up-converter, generating a wave at a higher frequency w3=w1+w2. A down-converter is realized by an interaction between waves 3 and 1 to generate wave 2, at the difference frequency w2=w3-w1. ◆ Waves 1, 2, and 3 interact so that wave 1 grows. The device operates as an amplifier and is known as a parametric amplifier. Wave 3, called the pump, provides the required energy, whereas wave 2 is an auxiliary wave known as the idler wave. The amplified wave is called the signal. ◆ With proper feedback, the parametric amplifier can operate as a parametric oscillator, in which only a pump wave is supplied. Fundamentals of Photonics 2017/4/11

Up-converted signal w3=w1+w2 Crystal w1, w2 Pump w2 Filter w3 w3 w1 Pump signal Amplified signal (b) Crystal w1 w2 Filter w2 Pump w3 (c) Crystal w1 w1 Figure 9.2-7 Optical parametric devices: (a) frequency up-converter; (b) parametric amplifier; (c) parametric oscillator. Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Two-wave mixing can occur only in the degenerate case, w2=2w1, in which the second-harmonic of wave 1 contributes to wave 2; and the subharmonic w2/2 of wave 2, which is at the frequency difference w2-w1, contributes to wave 1. Parametric devices are used for coherent light amplification, for the generation of coherent light at frequencies where no lasers are available (e.g., in the UV band), and for the detection of weak light at wavelengths for which sensitive detectors do not exist. Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Wave Mixing as a Photon Interaction Process conservation of energy and momentum require Figure 9.2-8 Mixing of three photons in a second-order nonlinear medium: (a) photon combining; (b) photon splitting. Fundamentals of Photonics 2017/4/11

Photon-Number Conservation Manley-Rowe Relation Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics 19.3 Coupled-wave theory of three-wave mixing Coupled- Wave Equations Rewrite in the compact form Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Frequency-Matching Condition Three-wave Mixing Coupled Equations Fundamentals of Photonics 2017/4/11

slowly varying envelope approximation Mixing of Three Collinear Uniform Plane Waves slowly varying envelope approximation Three-wave Mixing Coupled Equations Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics A. Second-Harmonic Generation a degenerate case of three-wave mixing w1=w2=w and w3=2w Two forms of interaction occur: ☆ Two photons of frequency o combine to form a photon of frequency 2w (second harmonic). ☆ One photon of frequency 2w splits into two photons, each of frequency w. ☆ The interaction of the two waves is described by the Helmholtz with equations sources. k3=2k1 Fundamentals of Photonics 2017/4/11

(Second-Harmonic Generation) Coupled- Wave Equations for Second-Harmonic Generation. where perfect phase matching Coupled Equations (Second-Harmonic Generation) Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics the solution Consequently, the photon flux densities Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Figure 9.4-1 Second-harmonic generation. (a) A wave of frequency w incident on a nonlinear crystal generates a wave of frequency 2w. (b) Two photons of frequency w combine to make one photon of frequency 2w. (c) As the photon flux density f1(z) of the fundamental wave decreases, the photon flux density f3(z) of the second-harmonic wave increases. Since photon numbers are conserved, the sum f1(z)+2f3(z)= f1(0) is a constant. Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics The efficiency of second-harmonic generation for an interaction region of length L is For large gL (long cell, large input intensity, or large nonlinear parameter), the efficiency approaches one. This signifies that all the input power (at frequency w) has been transformed into power at frequency 2w; all input photons of frequency w are converted into half as many photons of frequency 2w. For small gL (small device length L, small nonlinear parameter d, or small input photon flux density f1(0)), the argument of the tanh function is small and therefore the approximation tanhx=x may be used. The efficiency of second-harmonic generation is then Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Effect of Phase Mismatch Solution Efficiency Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Figure 9.4-2 The factor by which the efficiency of second-harmonic generation is reduced as a result of a phase mismatch △kL between waves interacting within a distance L. Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics B. Frequency Conversion A frequency up-converter converts a wave of frequency w1 into a wave of higher frequency w3 by use of an auxiliary wave at frequency w2, called the “pump.” A photon from the pump is added to a photon from the input signal to form a photon of the output signal at an up-converted frequency w3=w1+w2. The conversion process is governed by the three coupled equations. For simplicity, assume that the three waves are phase matched (△k = 0) and that the pump is sufficiently strong so that its amplitude does not change appreciably within the interaction distance of interest. Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Efficiency Figure 9.4-3 The frequency up-converter: (a) wave mixing; (b) photon interactions; (c) evolution of the photon flux densities of the input w1-wave and the up-converted w3-wave. The pump w2-wave is assumed constant Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics C. Parametric Amplification and Oscillation Parametric Amplifiers The parametric amplifier uses three-wave mixing in a nonlinear crystal to provide optical gain. The process is governed by the same three coupled equations with the waves identified as follows: ★ Wave 1 is the “signal” to be amplified. It is incident on the crystal with a small intensity I(0). ★ Wave 3, called the “pump,” is an intense wave that provides power to the amplifier. ★ Wave 2, called the “idler,” is an auxiliary wave created by the interaction process Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Parametric Amplifier Gain Coefficient Figure 9.4-4 The parametric amplifier: (a) wave mixing; (b) photon mixing; (c) photon flux densities of the signal and the idler; the pump photon flux density is assumed constant. Fundamentals of Photonics 2017/4/11

Fundamentals of Photonics Parametric Oscillators A parametric oscillator is constructed by providing feedback at both the signal and the idler frequencies of a parametric amplifier. Energy is supplied by the pump. Figure 9.4-5 The parametric oscillator generates light at frequencies w1 and w2. A pump of frequency w3=w1+w2 serves as the source of energy. Fundamentals of Photonics 2017/4/11

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