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Boyd University of Ottawa University of Rochester University of Glasgow Introduction to Nonlinear Optics Robert W. Boyd University of Ottawa University of Rochester University of Glasgow P resented at the International School of Physics “Enrico Fermi,” Course 190 - Frontiers in Modern Optics, 30 June - 5 July, 2014. Outline of Presentation Part I. Tutorial Introduction to Nonlinear Optics Part II. Recent Research in Quantum Nonlinear Optics Why Study Nonlinear Optics? It is good fundamental physics. It leads to important applications. It is a lot of fun. Demonstrate these features with examples in remainder of talk. 1. What is Nonlinear Optics? Nonlinear Optics and Light-by-Light Scattering The elementary process of light-by-light scattering has never been observed in vacuum, but is readily observed using the nonlinear response of material systems. Nonlinear material is fluorescein-doped boric acid glass (FBAG) 14 n2 n2(silica) [But very slow response!] M. A. Kramer, W. R. Tompkin, and R. W. Boyd, Phys. Rev. A, 34, 2026, 1986. W. R. Tompkin, M. S. Malcuit, and R. W. Boyd, Applied Optics 29, 3921, 1990. Opomjofbs!Pqujdt UIJSE!FEJUJPO Spcfsu!X/!Cpze Some Fundamental Nonlinear Optical Processes: I Second-Harmonic Generation 2 (2) 2 Dolgaleva, Lepeshkin, and Boyd Some Fundamental Nonlinear Optical Processes: II Sum-Frequency Generation 1 = + 2 (2) 3 1 2 2 3 1 Some Fundamental Nonlinear Optical Processes: III Difference-Frequency Generation amplified! 1 = 3 1 2 2 (2) 1 2 3 Optical Parametric Oscillation Parametric Downconversion: A Source of Entangled Photons The signal and idler photons are entangled in: (a) polarization (b) time and energy (c) position and transverse momentum (d) angular position and orbital angular momentum Entanglement is important for: (a) Fundamental tests of QM (e.g., nonlocality) (a) Quantum technologies (e.g., secure communications) Some Fundamental Nonlinear Optical Processes: III Third-Harmonic Generation (3) 3 3 Some Fundamental Nonlinear Optical Processes: IV Intensity-Dependent Index of Refraction i E( ) E( ) e NL self-phase modulation (3) cross-phase modulation E( ) (strong wave) i (3) NL E( ') E( ')e (probe wave) where Role of Material Symmetry in Nonlinear Optics (2) vanishes identically for a material possessing a center of inversion sym- metry (a centrosymmetric medium). non-centrosymmetric medium U(x) parabola parabola actual potential x centrosymmetric medium U(x) parabola parabola actual potential x 2. Coupled Wave Equations and Harmonic Generation Treatment of Second-Harmonic Generation – I z E ,ω E , 2ω 1 χ (2) 2 L Let i!t i(k z !t) E˜1(z,t)=E1(z)e− +c.c.=A1e 1 − +c.c. (1) i2!t i(k z 2!t) E˜2(z,t)=E2(z)e− +c.c.=A2(z)e 2 − +c.c. (2) where k1 = n1!/c and k2 = n22!/c. We have assumed that the pump wave E1 at frequency ! is unde- pleted by the nonlinear interaction. We take A2 to be a function of z to allow the second harmonic wave to grow with z.Wealsoset ˜ i2!t (2) 2 (2) 2 i2k1z P2(t)=P2 e− where P2 = ✏0χ E1 = ✏0χ A1 e (3) The generation of the wave at 2! is governed by the wave equation 2 2 ˜ 2 ˜ 2 ˜ n @ E2 1 @ P2 E2 2 2 = 2 2 . (4) r − c @t ✏0c @t Treatment of Second-Harmonic Generation – II Note that the first term in the wave equation is given by @2A˜ @A˜ 2 ˜ 2 2 2 2 3 i(k2z 2!t) E = 6 +2ik k A 7 e − (5) 2 6 2 2 2 27 6 7 r 6 @z @z − 7 6 7 4 5 @A˜ 2 2 2 3 i(k2z 2!t) 62ik k A 7 e (6) 6 2 2 27 − 6 7 ⇡ 6 @z − 7 6 7 The second form is the slowly4 varying amplitude5 approximation. Note also that @2A˜ @2P˜ 2 = (2!)2A ei(k2z 2!t) 2 = (2!)2P ei(2k1z 2!t) (7) @t2 − 2 − @t2 − 2 − By combining the above equations, we obtain dA 4!2 2ik 2 = − χ(2)A2ei∆kz where ∆k =2k k . (8) 2 dz c2 1 1 − 2 The quantity ∆k is known as the phase (or wavevector) mismatch factor, and it is crucially important in determining the efficiency of nonlinear optical interactions Treatment of Second-Harmonic Generation – III For the case ∆k =0,Eq.(8)becomes dA 4!2 2ik 2 = − χ(2)A2 (9) 2 dz c2 1 with solution evaluated at z = L of 2 2i! (2) 2 2 4! (2) 2 4 2 A2(L)= χ A1L or A2(L) = 2 2[χ ] A1 L . (10) n2c | | n2c | | Note that the SHG intensity scales as the square of the input intensity and also as the square of the length L of the crystal. 2ω 2ω dipole emitter phased array of dipoles Treatment of Second-Harmonic Generation – IV For the general case of ∆k =0,Eq.(8)canstillbesolvedtoyield 6 2 2 4! (2) 2 4 2 2 A2(L) = 2 2 [χ ] A1 L sinc (∆kL/2) (11) | | n2c | | Note that ∆kL must be kept much smaller than ⇡ radians in order for efficient SHG to occur. Second Harmonic Generation and Nonlinear Microscopy Nonlinear Optical Microscopy An important application of harmonic generation is nonlinear microscopy. Microscopy based on second-harmonic generation in the configuration of a confocal microscope and excited by femtosecond laser pulses was introduced by Curley et al. (1992). Also, harmonic-generation microscopy can be used to form images of transpar- ent (phase) objects, because the phase matching condition of nonlinear optics depends sensitively on the refractive index variation within the sample being imaged (Muller et al., 1998). Boyd, NLO, Subsection 2.7.1 Caution! Curley et al., not Curly et al. How to Achieve Phase Matching: Birefringence Phase Matching The phase matching condition ∆k =0requiresthat n ! n ! n ! 1 1 + 2 2 = 3 1 where ! + ! = ! c c c 1 2 3 These conditions are incompatible in an isotropic dispersive material. However, for a birefringent material phase matching can be achieved. cˆ ω θ 2ω k ordinary extraordinary Midwinter and Warner showed that there are two ways to achieve phase matching: Positive uniaxial Negative uniaxial (ne >n0)(ne <n0) 0 e e e o o Type I n3!3 = n1!1 + n2!2 n3!3 = n1!1 + n2!2 0 o e e e o Type II n3!3 = n1!1 + n2!2 n3!3 = n1!1 + n2!2 How to Achieve Phase Matching: Quasi Phase Matching single domain crystal periodically poled crystal Λ Sign of χ(2) is periodically inverted to prevent reverse power flow. (a) with perfect phase-matching (b) with quasi-phase-matching (c) with a wavevector mismatch field amplitude field 0 0 2 4 6 8 z / Lcoh Additional Studies of Wave Propagation Efects 3. Mechanisms of Optical Nonlinearity Typical Values of the Nonlinear Refractive Index a (3) n2 χ1111 Response time Mechanism (cm2/W) (m2/V2)(sec) 16 22 15 Electronic polarization 10− 10− 10− 14 20 12 Molecularorientation 10− 10− 10− 14 20 9 Electrostriction 10− 10− 10− 10 16 8 Saturated atomic absorption 10− 10− 10− 6 12 3 Thermal e↵ects 10− 10− 10− Photorefractive e↵ectb (large) (large) (intensity-dependent) a For linearly polarized light. b The photorefractive e↵ect often leads to a very strong nonlinear response. This response usually (3) cannot be described in terms of a χ (or an n2)nonlinearsusceptibility,becausethenonlinearpo- larization does not depend on the applied field strength in the same manner as the other mechanisms listed. Quantum Mechanical Origin of the Nonlinear Optical Suesceptibility ll l l (2) (! + ! ! ! ) ijk p q , q , p n n (a ) (a ) (a1) 1 2 i j k m m N (0) ω ln nm ml q ω p = 2 ll 2 0h [(!nl !p !q) i nl][(!ml !p) i ml] ω p lmn l l ω q ll i k j lnnmml + (a ) [(!nl !p !q) i nl][(!ml !q) i ml] 2 n n nn k i j lnnmml m m (a' ) + (a' ) 2 (a ) 1 [(!mn !p !q) i mn][(!nl + !p) + i nl] 1 ω q ω p j i k ω nm p ω q + ln ml ll ll [(! ! ! ) i ][(! + ! ) + i ] (a2) mn p q mn nl q nl j i k nn + ln nm ml nn (b1) [(!nm + !p + !q) + i nm][(!ml !p) i ml] (b ) 1 (b2) m m k i j ω q ω p lnnmml + ω ω [(! + ! + ! ) + i ][(! ! ) i ] (b2) p q nm p q nm ml q ml ll ll k j i lnnmml + (b ) [(!ml + !p + !q) + i ml][(!nl + !p) + i nl] 1 j k i mm mm lnnmml + b' ) (b2) 2 [(!ml + !p + !q) + i ml][(!nl + !q) + i nl] b' 1 n ω q n ω p ω p ω q l l ll Some Actual Z-Scan Data Closed aperture Open aperture 1.2 I0 = 0.176 GW/cm2 1.2 gold-silica composite CS 2 1.1 1.0 transmission 1.0 transmission n2 = Chang, Shin 0.0373 cm2/GW 0.8 Piredda 0.9 –20 0 20 z (mm) –125 0 125 z (mm) For closed aperture z-scan ΔTpv = 0.406 ΦNL where M. Sheik-Bahae et al., IEEE J. ΦNL = n2 (ω/c) Ι0 L Quantum Electron. 26 760 (1990). 6. Self-Action Effects in Nonlinear Optics Self-Action Efects in Nonlinear Optics Self-action efects: light beam modifes its own propagation tTFMGGPDVTJOH tTFMGUSBQQJOH d tTNBMMTDBMFöMBNFOUBUJPO Prediction of Self Trapping n = n0 d n = n0 +n n = n0 radial profle of self -trapped beam Optical Solitons Field distributions that propagate without change of form Temporal solitons (nonlinearity balances gvd) 1973: Hasegawa & Tappert 1980: Mollenauer, Stolen, Gordon Spatial solitons (nonlinearity balances difraction) 1964: Garmire, Chiao, Townes 1974: Ashkin and Bjorkholm (Na) 1985: Barthelemy, Froehly (CS2) 1991: Aitchison et al.
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