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CHAPTER 8----NONLINEAT OPTICS Chapter 8 NONLINEAR OPTICS Fundamentals of Photonics 2017/5/24 1 CHAPTER 8----NONLINEAT OPTICS Question: Is it possible to change the color of a monochromatic light? NLO sample input output Answer: Not without a laser light Fundamentals of Photonics 2017/5/24 2 CHAPTER 8----NONLINEAT OPTICS 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 2017/5/24 3 CHAPTER 8----NONLINEAT OPTICS 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 2017/5/24 4 CHAPTER 8----NONLINEAT OPTICS 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 2017/5/24 5 CHAPTER 8----NONLINEAT OPTICS 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 Two-order Three-order Optical solitons Optical bistability Fundamentals of Photonics 2017/5/24 6 CHAPTER 8----NONLINEAT OPTICS 19.1 Nonlinear optical media Origin of Nonlinear PNp p ex, F eE the dependence of the number density N on the optical field if Hooke’s law is satisfied pxE the number of atoms occupying the energy levels involved in the absorption and emission Fundamentals of Photonics Linear! if Hooke’s law is not satisfied pxE Noninear! 2017/5/24 7 CHAPTER 8----NONLINEAT OPTICS P P E E Figure 19.1-1 The P-E relation for (a) a linear dielectric medium, and (b) a nonlinear medium. Fundamentals of Photonics 2017/5/24 8 CHAPTER 8----NONLINEAT OPTICS 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. 1 1 P a1 E a2 E 2 a3 E 3 2 6 P 0 E 2dE 4 E 2 1 a2 4 (3) 3 1 a3 24 In centrosymmetric media, d vanish, and the lowest order nonlinearity is of third order basic description for a nonlinear optical medium Fundamentals of Photonics 2017/5/24 9 CHAPTER 8----NONLINEAT OPTICS In centrosymmetric media: d=0 the lowest order nonlinearity is of third order Typical values d 1024 1021 (3) 1034 1029 Fundamentals of Photonics 2017/5/24 10 CHAPTER 8----NONLINEAT OPTICS The Nonlinear Wave Equation 1 2 E 2 P E 2 2 0 2 c0 t t 2 n2 1 P 0 E PNL c0 1/( 0 0 )1/ 2 PNL 2dE 2 4 (3) E 3 c c0 / n 2 1 E 2 E 2 2 J c0 t 2 PNL J 0 t 2 nonlinear wave equation Fundamentals of Photonics 2017/5/24 11 CHAPTER 8----NONLINEAT OPTICS 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 2017/5/24 12 CHAPTER 8----NONLINEAT OPTICS 19.2 Second-order Nonlinear Optics P(2) (t ) 2 (2) EE* ( (2) E 2e2it C.C.) 2 ( 2) PNL 2dE 2 Fundamentals of Photonics 2017/5/24 13 CHAPTER 8----NONLINEAT OPTICS A. Second-Harmonic Generation and Rectification complex amplitude E (t ) Re{E ( ) exp( jt )} Substitute it into (9.2-l) PNL (t ) PNL (0) Re{PNL (2 ) exp( j 2t )} PNL (0) dE ( ) E ( ) * PNL (2 ) dE ( ) E ( ) Fundamentals of Photonics 2017/5/24 14 CHAPTER 8----NONLINEAT OPTICS This process is illustrated graphically in Fig. 9.2-1. P PNL(t) 0 E t t dc + t 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 2017/5/24 15 CHAPTER 8----NONLINEAT OPTICS Second-Harmonic Generation P(2 ) 2dE 2 E 2 1 1 i1t * i1t E ( E1 E2 ) [ ( A1e A1 e ) ( A2ei2t A2*e i2t )]2 2 SHG 2 SFG 1 2 i 21t ( A1 e A12 A1 A2ei (1 2 )t A1 A2*ei (1 2 )t 4 DHG i (1 2 ) t 2 2 i 21t * * * i (1 2 ) t A1 A1 e A1 A2e A1 A2 e 2 2 A1 A2 ei (1 2 )t A1* A2 ei (1 2 )t A22ei 22t A22 A1 A2*ei (1 2 )t A1* A2*e i (1 2 )t A22 A22e i 22t ) Fundamentals of Photonics 2017/5/24 16 CHAPTER 8----NONLINEAT OPTICS SHG Component of frequency 2w complex amplitude intensity S (2 ) 4 0 2 dE ( ) E ( ) S (2 ) d I d 2 4 2 4 2 E ( ) 2 2 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 2017/5/24 17 CHAPTER 8----NONLINEAT OPTICS 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 2017/5/24 18 CHAPTER 8----NONLINEAT OPTICS 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 2017/5/24 19 CHAPTER 8----NONLINEAT OPTICS B. The Electra-Optic Effect E (t ) E (0) Re{E ( ) exp( jt )} Substitute it into (9.2-l) PNL (t ) PNL (0) Re{PNL ( ) exp( jt )} Re{PNL (2 ) exp( j 2t )} PNL (0) d [2 E (0) E ( ) ] 2 2 PNL ( ) 4dE (0) E ( ) 9.2-8 PNL (2 ) dE ( ) E ( ) Fundamentals of Photonics 2017/5/24 20 CHAPTER 8----NONLINEAT OPTICS If the optical field is substantially smaller in magnitude than the electric field E ( ) PNL (2 ) 2 E (0) PNL (0) 2 PNL ( ) Can be negleted Fundamentals of Photonics 2017/5/24 21 CHAPTER 8----NONLINEAT OPTICS PNL (0) d [2 E (0) E ( ) ] 2 2 PNL ( ) 4dE (0) E ( ) 9.2-8 PNL (2 ) dE ( ) E ( ) a linear relation between PNL(w) and E(w) PNL ( ) 0 E ( ) (4d / 0 ) E (0) incremental change of the refractive index n Fundamentals of Photonics 2d E (0) 9.2-9 n 0 2017/5/24 22 CHAPTER 8----NONLINEAT OPTICS the nonlinear medium exhibits the linear electro-optic effect Pockels effect 1 3 n n rE (0) 2 Pockels coefficient Comparing this formula with (9.2-9) r Fundamentals of Photonics 4 d 4 0n 2017/5/24 23 CHAPTER 8----NONLINEAT OPTICS C. Three-Wave Mixing Frequency Conversion E(t) comprising two harmonic components at frequencies w1 and w2 E (t ) Re{E (1 ) exp( j1t ) E (2 ) exp( j2t )} PNL 2dE 2 PNL (0) d [ E (1 ) E (2 ) ] 2 2 PNL (21 ) dE (1 ) E(1 ) PNL (22 ) dE (2 ) E (2 ) Frequency up-conversion PNL ( ) 2dE (1 ) E (2 ) PNL ( ) 2dE (1 ) E * (2 ) Fundamentals of Photonics 2017/5/24 Frequency down-conversion 24 CHAPTER 8----NONLINEAT OPTICS 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 2017/5/24 25 CHAPTER 8----NONLINEAT OPTICS Phase Matching E(1 ) A1 exp( jk1 r ) E(2 ) A2 exp( jk2 r ) PNL (3 ) 2dE(1 ) E(2 ) 2dA1 A2 exp( jk3 r ) where 3 1 2 Frequency-Matching Condition k 3 k1 k 2 Phase-Matching Condition Figure 9.2-6 The phase-matching condition Fundamentals of Photonics 2017/5/24 26 CHAPTER 8----NONLINEAT OPTICS ★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 2017/5/24 27 CHAPTER 8----NONLINEAT OPTICS 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 w 2=w3-w1. Waves 3 and 2 similarly combine and radiate at w 1. 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 2017/5/24 28 CHAPTER 8----NONLINEAT OPTICS parametric interaction ◆Waves 1 and 2 are mixed in an up-converter, generating a wave at a higher frequency w 3=w1+w2. A down-converter is realized by an interaction between waves 3 and 1 to generate wave 2, at the difference frequency w 2=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 2017/5/24 29 CHAPTER 8----NONLINEAT OPTICS signal w1 (a) Up-converted signal w3=w1+w2 Crystal w1, w2 Pump w2 Filter (b) Pump signal w3 w3 w1 Crystal Amplified signal w2 w1 Filter (c) Pump w2 w3 w1 Crystal w1 Figure 9.2-7 Optical parametric devices: (a) frequency upconverter; (b) parametric amplifier; (c) parametric oscillator. Fundamentals of Photonics 2017/5/24 30 CHAPTER 8----NONLINEAT OPTICS Two-wave mixing can occur only in the degenerate case, w 2=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 2017/5/24 31 CHAPTER 8----NONLINEAT OPTICS Fundamentals of Photonics 2017/5/24 32 CHAPTER 8----NONLINEAT OPTICS Wave Mixing as a Photon Interaction Process conservation of energy and momentum require 3 1 2 k 3 k1 k 2 Figure 9.2-8 Mixing of three photons in a second-order nonlinear medium: (a) photon combining; (b) photon splitting. Fundamentals of Photonics 2017/5/24 33 CHAPTER 8----NONLINEAT OPTICS d3 d1 d2 dz dz dz Photon-Number Conservation d I1 d I2 d I3 ( ) ( ) ( ) dz 1 dz 2 dz 3 Fundamentals of Photonics 2017/5/24 Manley-Rowe Relation 34 CHAPTER 8----NONLINEAT OPTICS 19.3 Coupled-wave theory of three-wave mixing Coupled- Wave Equations E (t ) q 1,2,3 Re[ Eq exp( jq t )] 1 [ Eq exp( jq t ) Eq* exp( jqt )] q 1,2,3 2 Rewrite in the compact form E (t ) 1 2 E E 2 2 J c t 2 q q E q Eq* 1 Eq exp( jq t ) q 1, 2, 3 2 2 PNL J 0 t 2 PNL (t ) J PNL 2dE 2 1 d Eq Er exp[ j (q r )t ] 2 q ,r 1,2,3 1 0 d (q r ) 2 Eq Er exp[ j (q r )t ] 2 q , r 1, 2, 3 Fundamentals of Photonics 2017/5/24 35 CHAPTER 8----NONLINEAT OPTICS E (t ) 1 Eq exp( jq t ) 2 q 1, 2, 3 J 1 0 d (q r ) 2 Eq Er exp[ j (q r )t ] 2 q , r 1, 2, 3 2 1 E 2 E 2 2 J c t Frequency-Matching Condition 3 1 2 ( 2 k12 ) E1 S1 S1 2012 dE3 E2* ( 2 k22 ) E2 S 2 S 2 2 022 dE3 E1* ( 2 k32 ) E3 S3 S3 2032 dE1 E2* ( 2 k12 ) E1 2 012 dE3 E2* Three-wave Mixing Coupled Equations ( 2 k22 ) E2 2 022 dE3 E1* ( 2 k32 ) E3 2032 dE1 E2 Fundamentals of Photonics 2017/5/24 36 CHAPTER 8----NONLINEAT OPTICS Mixing of Three Collinear Uniform Plane Waves Eq Aq exp( jkq z ) kq q / c aq Aq /(2 q )1/ 2 , 0 / n,0 (0 / 0 )1/ 2 Eq (2 q ) aq exp( jkq z), q 1, 2,3 q 1/ 2 slowly varying envelope approximation (2 kq2 )[aq exp( jkq z )] j 2kq Three-wave Mixing Coupled Equations ( 2 k12 ) E1 2 012 dE3 E2* ( 2 k22 ) E2 2 022 dE3 E1* ( 2 k32 ) E3 2032 dE1 E2 Fundamentals of Photonics daq dz q aq 2 exp( jkq z ) da1 jga3 a2* exp( j kz ) dz da2 jga3a1* exp( j kz ) dz da3 jga1a2 exp( j kz ) dz g 2 2 123 3 d 2 2017/5/24 Iq k k3 k2 k1 37 CHAPTER 8----NONLINEAT OPTICS 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 2017/5/24 38 CHAPTER 8----NONLINEAT OPTICS Coupled- Wave Equations for Second-Harmonic Generation. da1 jga3 a1* exp( jkz ) dz da3 g j a1a1 exp( j kz ) dz 2 where k k3 2k1 g 2 4 perfect phase matching k 0 da1 jga3a1* dz da3 g j a1a1 dz 2 Fundamentals of Photonics 3 3d 2 2017/5/24 Coupled Equations (Second-Harmonic Generation) 39 CHAPTER 8----NONLINEAT OPTICS the solution ga1 (0) z a1 ( z ) a1 (0)sec h 2 a3 ( z ) ga (0) z j a1 (0) tan h 1 2 2 Consequently, the photon flux densities 1 ( z ) 1 (0) sec h 2 1 2 z 2 3 ( z ) 1 (0) tan h 2 z 2 2 2 g 2 a12 (0) 2 g 21 (0) 8d 2 3 31 (0) 8d 2 3 2 I1 (0) Fundamentals of Photonics 2017/5/24 40 CHAPTER 8----NONLINEAT OPTICS 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 1(z) of the fundamental wave decreases, the photon flux density 3(z) of the second-harmonic wave increases. Since photon numbers are conserved, the sum 1(z)+23(z)= 1(0) is a constant. Fundamentals of Photonics 2017/5/24 41 CHAPTER 8----NONLINEAT OPTICS The efficiency of second-harmonic generation for an interaction region of length L is I 3 ( L) 33 ( L) 23 ( L) L tanh 2 I1 (0) 11 ( L) 1 (0) 2 For large L (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 L (small device length L, small nonlinear parameter d, or small input photon flux density 1(0)), the argument of the tanh function is small and therefore the approximation tanhx=x may be used. The efficiency of secondharmonic generation is then 2 2 I 3 ( L) 3 2 d L 20 3 P I1 (0) n A Fundamentals of Photonics 2017/5/24 42 CHAPTER 8----NONLINEAT OPTICS Effect of Phase Mismatch da1 jga3 a1* exp( jkz ) dz da3 g j a1a1 exp( j kz ) dz 2 k 0 Solution a3 ( L) j L g 2 g a1 (0) exp( jkz ')dz ' ( )a12 (0)[exp( jkL) 1] 0 2 2k Efficiency I 3 ( L) 23 ( L) 1 2 2 kL g L 1 (0) sin c 2 I1 (0) 1 (0) 2 2 Fundamentals of Photonics 2017/5/24 43 CHAPTER 8----NONLINEAT OPTICS sin c 2 ( kL ) 2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -2/L -1/L 0 1/L 2/L k 2 Figure 9.4-2 The factor by which the efficiency of secondharmonic generation is reduced as a result of a phase mismatch △kL between waves interacting within a distance L. Fundamentals of Photonics 2017/5/24 44 CHAPTER 8----NONLINEAT OPTICS 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 2 is added to a photon 1 from the input signal to form a photon 3 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. da1 jga3 a2* exp( j kz ) dz da2 jga3a1* exp( j kz ) dz da3 jga1a2 exp( j kz ) dz Fundamentals of Photonics a1 ( z ) a1 (0) cosh da1 j a2* dz 2 da2 j a1* dz 2 2017/5/24 z a2 ( z ) ja1 (0) sinh 1 ( z ) 1 (0) cos 2 2 z 2 z 2 z 3 ( z ) 1 (0) sin 2 2 45 CHAPTER 8----NONLINEAT OPTICS Efficiency I 3 ( L) 3 z sin 2 I1 (0) 1 2 2 2 I 3 ( L) 3 2 d L 20 3 3 P2 I1 (0) n A 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 w3wave. The pump w2-wave is assumed constant Fundamentals of Photonics 2017/5/24 46 CHAPTER 8----NONLINEAT OPTICS 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 da1 j a2* dz 2 da2 j a1* dz 2 a1 ( z ) a1 (0) cosh z a2 ( z ) ja1 (0) sinh Fundamentals of Photonics 2 z 2017/5/24 2 1 ( z ) 1 (0) cosh 2 3 ( z ) 1 (0) sinh 2 z 2 z 2 47 CHAPTER 8----NONLINEAT OPTICS Parametric Amplifier Gain Coefficient d 2 P3 1/ 2 [8 12 3 ] n A 3 0 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 2017/5/24 48 CHAPTER 8----NONLINEAT OPTICS 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 2017/5/24 49 CHAPTER 8----NONLINEAT OPTICS Frequency Upconversion 返回