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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) n2 14 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) 3 = + 1 2 2 3 2 1 Some Fundamental Nonlinear Optical Processes: III Difference-Frequency Generation amplified! 1 (2) 3 = 2 1 2 2 1 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 self-phase modulation cross-phase modulation E( ) E( ) (3) i NL (strong wave) E( ' ) (probe wave) where E( ) e (3) E( ' ) e i NL Role of Material Symmetry in Nonlinear Optics (2) vanishes identically for a material possessing a center of inversion symmetry (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 E1, ω χ (2) E2, 2 ω L Let Ẽ1(z, t) = E1(z)e i!t + c.c. = A1ei(k1z !t) + c.c. Ẽ2(z, t) = E2(z)e i2!t + c.c. = A2(z)ei(k2z 2!t) + c.c. where k1 = n1!/c and k2 = n22!/c. (1) (2) We have assumed that the pump wave E1 at frequency ! is undepleted by the nonlinear interaction. We take A2 to be a function of z to allow the second harmonic wave to grow with z. We also set P̃2(t) = P2 e i2!t where P2 = ✏0 (2)E12 = ✏0 (2)A21 ei2k1z (3) The generation of the wave at 2! is governed by the wave equation r2Ẽ2 n2 @ 2Ẽ2 1 @ 2P̃2 = . 2 2 2 2 c @t ✏0c @t (4) Treatment of Second-Harmonic Generation – II Note that the first 2term in the wave equation3 is given by 6 @ 2 Ã 7 @ Ã2 6 7 2 2 2 6 6 r Ẽ2 = 64 2 + 2ik2 k2 A27775 ei(k2z 2!t) (5) @z @z 2 3 6 7 @ Ã2 6 7 2 6 6 ⇡ 642ik2 k2 A27775 ei(k2z 2!t) (6) @z The second form is the slowly varying amplitude approximation. Note also that 2P̃ @ 2Ã2 @ 2 2 i(k z 2!t) 2P ei(2k1z 2!t) (7) 2 = (2!) A e = (2!) 2 2 @t2 @t2 By combining the above equations, we obtain dA2 4! 2 (2) 2 i k z 2ik2 = 2 A1 e where k = 2k1 k2. (8) dz c 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 dA2 4! 2 (2) 2 2ik2 = 2 A1 (9) dz c with solution evaluated at z = L of 2 2i! (2) 2 4! A2(L) = A1L or |A2(L)|2 = 2 2 [ (2)]2|A1|4L2. (10) n2 c n2 c 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 6= 0, Eq. (8) can still be solved to yield 2 4! |A2(L)|2 = 2 2 [ (2)]2 |A1|4L2 sinc2( k L/2) (11) n2 c Note that k L 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 transparent (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 = 0 requires that n1!1 n2!2 n3!1 + = where !1 + !2 = !3 c c c These conditions are incompatible in an isotropic dispersive material. However, for a birefringent material phase matching can be achieved. ω ordinary ĉ θ k 2ω extraordinary Midwinter and Warner showed that there are two ways to achieve phase matching: Positive uniaxial (ne > n0) Type I n03!3 = ne1!1 + ne2!2 Type II n03!3 = no1!1 + ne2!2 Negative uniaxial (ne < n0) ne3!3 = no1!1 + no2!2 ne3!3 = ne1!1 + no2!2 How to Achieve Phase Matching: Quasi Phase Matching single domain crystal periodically poled crystal Λ (2) is periodically inverted to prevent reverse power flow. (a) with perfect phase-matching (b) with quasi-phase-matching field amplitude Sign of (c) with a wavevector mismatch 0 0 2 4 z / Lcoh 6 8 Additional Studies of Wave Propagation Effects 3. Mechanisms of Optical Nonlinearity Typical Values of the Nonlinear Refractive Index Mechanism Electronic polarization Molecular orientation Electrostriction Saturated atomic absorption Thermal e↵ects Photorefractive e↵ectb a (3) 1111 2 2 n2 a (cm2/W) (m /V ) Response time (sec) 10 16 10 14 10 14 10 10 10 6 (large) 10 22 10 20 10 20 10 16 10 12 (large) 10 15 10 12 10 9 10 8 10 3 (intensity-dependent) For linearly polarized light. b The photorefractive e↵ect often leads to a very strong nonlinear response. This response usually cannot be described in terms of a (3) (or an n2) nonlinear susceptibility, because the nonlinear polarization 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 (2) ijk (!p l + !q , !q , !p ) N = 2 0 h2 (0) ll lmn (a1 ) iln jnm kml [(!nl !p !q ) inl ][(!ml !p ) iml ] iln knm jml + [(!nl !p !q ) inl ][(!ml !q ) iml ] kln inm jml + [(!mn !p !q ) imn ][(!nl + !p ) + inl ] jln inm kml + [(!mn !p !q ) imn ][(!nl + !q ) + inl ] (a1 ) (b2 ) jln knm iml + [(!ml + !p + !q ) + iml ][(!nl + !q ) + inl ] ωq ωp n m m ωp ωq l l n n (a ) 2 l l n n m m (a' ) 1 ωq (a' ) 2 ωp l (a2 ) + kln jnm iml + [(!ml + !p + !q ) + iml ][(!nl + !p ) + inl ] n 1 ωp (b1 ) [(!nm + !p + !q ) + inm ][(!ml !q ) iml ] l l (a2 ) jln inm kml + [(!nm + !p + !q ) + inm ][(!ml !p ) iml ] kln inm jml (a ) l l l n n ωq l n n (b ) 1 m (b 2 ) m ωq ωp ωp ωq l l l l (b1 ) m (b2 ) m m m b' ) 2 b' n 1 l l ωq ωp n ωp ωq l l Some Actual Z-Scan Data Closed aperture Open aperture 1.2 I0 = 0.176 GW/cm2 1.2 transmission gold-silica composite transmission CS2 1.0 Chang, Shin 0.8 –20 1.1 1.0 n2 = 0.0373 cm2/GW 0 z (mm) 20 0.9 –125 Piredda 0 z (mm) For closed aperture z-scan ΔTpv = 0.406 ΦNL where ΦNL = n2 (ω/c) Ι0 L M. Sheik-Bahae et al., IEEE J. Quantum Electron. 26 760 (1990). 125 6. Self-Action Effects in Nonlinear Optics Self-Action Effects in Nonlinear Optics Self-action effects: light beam modifies its own propagation tTFMGGPDVTJOH tTFMGUSBQQJOH tTNBMMTDBMFöMBNFOUBUJPO d Prediction of Self Trapping n = n0 d n = n0 + n n = n0 radial profile 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 diffraction) 1964: Garmire, Chiao, Townes 1974: Ashkin and Bjorkholm (Na) 1985: Barthelemy, Froehly (CS2) 1991: Aitchison et al. (planar glass waveguide 1992: Segev, (photorefractive) Solitons and self-focussing in Ti:Sapphire Diffraction-management controls the spatial selffocussing Dispersion-management controls the temporal self-focussing Beam Breakup by Small-Scale Filamentation Predicted by Bespalov and Talanov (1966) Exponential growth of wavefront imperfections by four-wave mixing processes exponential growth rate, / Sidemode amplitude grows as A1(z) = A1(0)ei z e z 2 q2 = 2 2k q2 , 2k = n2 ( / c)I0 , q = k transverse wavevector, q/qmax qmax = 2k qmax max = k Honeycomb Pattern Formation Output from cell with a single gaussian input beam At medium input power at cell exit in far field At high input power at cell exit in far field Quantum statistics? Input power 100 to150 mW Input beam diameter 0.22 mm Sodium vapor cell T = 220o C Wavelength = 588 nm Bennink et al., PRL 88, 113901 2002. Optical Radiance Limiter Based on Spatial Coherence Control Controlled small-scale filamentation used to modify spatial degree of coherence Alternative to standard appropches to optical power limiting spatially coherent light in spatially incoherent light out n2 medium 1.6 1.2 increasing power Schweinsberg et al., Phys. Rev. A 84, 053837 (2011). Far-field dif 0.8 0.4 0 0 10 20 Incident pulse ener 30 4. Local-Field Effects in Nonlinear Optics Local Field Effects in Nonlinear Optics – I Recall the Lorentz-Lorenz Law (linear optics) E E P loc (1) − 1 N α ϵ 4 (1) χ = or (1) = πN α. 4 1 − 3 πN α ϵ +2 3 This result follows from the assumption that the field that acts on a representative atom is not the macroscopic Maxwell field but rather the Lorentz local field given by Eloc = E + 43 πP where P = χ(1)E We can rewrite this result as ϵ(1) + 2 Eloc = LE where L = is the local field factor. 3 The Lorentz Red Shift See, for instance, H. A. Lorentz, Theory of Electrons, Dover, NY (1952). Observation of the Lorentz Red Shift Maki, Malcuit, Sipe, and Boyd, Phys. Rev. Lett. 68, 972 (1991). Local Field Effects in Nonlinear Optics – II For the case of nonlinear optics, Bloembergen (1962, 1965) showed that, for instance, χ(3)(ω = ω + ω − ω) = N γ (3)|L(ω)|2[L(ω)]2. where γ (3) is the second hyperpolarizability and where ϵ(ω) + 2 L(ω) = 3 For the typical value n = 2, L = 2, and L4 = 16. Local field effects can be very large in nonlinear optics! But can we tailor them for our benefit? We have been developing new photonic materials with enhanced NLO response by using composite structures that exploit local field effects. Nanocomposite Materials for Nonlinear Optics ) i 2a a h b b Fractal Str e Layer a b - In each case, scale size of inhomo eneity << optical avele - s all optical properties, ch as n an effective r ) s , can be escribe by Local Field Enhancement of the NLO Response - Under very general conditions, we can express the NL response as 2 (3) 2 (3) eff = fL L where f is the volume fraction of nonlinear material and L is the local-field factor, which is different for each material geometry. - Under appropriate conditions, the product f L2 L 2 can exceed unity. - For a homogeneous material +2 3 L= - For a spherical particle of dielectric constant of dielectic constant h 3h L= m +2 m embedded in a host h - For a layered geometry with the electric field perpendicular to the plane of the layers, the local field factor for component a is given by L= eff 1 a eff = fa a + fb b Enhanced NLO Response from Layered Composite Materials A composite material can display a larger NL response than its constituents! Quadratic EO effect Alternating layers of TiO2 and the conjugated polymer PBZT. gold electrode BaTiO3 signal generator 1 kHz AF-30 in polycarbonate ITO glass substrate photodiode polarizer Measure NL phase shift as a function of angle of incidence. 35% enhancement in (3) Fischer, Boyd, Gehr, Jenekhe, Osaheni, Sipe, and Weller-Brophy, Phys. Rev. Lett. 74, 1871 (1995). Diode Laser (1.37 um) dc voltmeter lockin amplifier 3.2 times enhancement! Nelson and Boyd, APL 74 2417 (1999) Accessing the Optical Nonlinearity of Metals with Metal-Dielectric Photonic Crystal Structures Metals have very large optical nonlinearities but low transmission Low transmission because metals are highly reflecting (not because they are absorbing!) Solution: construct metal-dielectric photonic crystal structure (linear properties studied earlier by Bloemer and Scalora) E Cu SiO2 E Cu PC structure 80 nm of copper (total) bulk metal 80 nm of copper T = 0.3% T = 10% z 0 L Bennink, Yoon, Boyd, and Sipe, Opt. Lett. 24, 1416 (1999). Lepeshkin, Schweinsverg, Piredda, Bennink, and Boyd, Phys. Rev. Lett. 93, 123902 (2004). bulk Cu M/D PC nm I = 500 MW/cm2 (response twelvetimes larger) z Gold-Doped Glass: A Maxwell-Garnett Composite Red Glass Caraffe Nurenberg, ca. 1700 Huelsmann Museum, Bielefeld Developmental Glass, Corning Inc. gold volume fraction approximately 10-6 gold particles approximately 10 nm diameter • Composite materials can possess properties very different from those of their constituents. • Red color is because the material absorbs very strong in the blue, at the surface plasmon frequency Counterintuitive Consequence of Local Field Effects Cancellation of two contributions that have the same sign Gold nanoparticles in a reverse saturable absorber dye solution (13 μM HITCI) normalized transmittance 1.3 SA β<0 9 1.2 8 1.1 Increasing Gold Content 7 1.0 < 0 6 0.9 0.8 2 0.6 1 -10 0 (3) 5 Im 4 β>0 3 0.7 0.5 Im (3) RSA (3) eff 2 = f L L 10 z (cm) 2 (3) i + 20 D.D. Smith, G. Fischer, R.W. Boyd, D.A.Gregory, JOSA B 14, 1625, 1997. (3) h 30 > 0 NLO in Plasmonics (Photonics Using Metals) Is there an intrinsic nonlinear reponse to surface plasmon polaritons (SPPs)? 2 is106 times that of silica) sp polarized Translation stage Au film z x Laser PD 6 y Aperture sp HWP PBS sp sp Rotation TM stage 1.0 0 Kretschmann dip 1 2 3 Ssp Reflectance, R 0.5 TIR onset increasing power Pi = 2.70 mW Pi = 8.70 mW Pi = 15.0 mW Pi = 22.8 mW Pi = 28.4 mW 0.1 0.05 41.5 42 42.5 43 Incidence angle, (deg.) 43.5 I. De Leon, Z. Shi, A. Liapis and R.W.Boyd, Optics Letters 39, 2274 (2014) Slow Light in a Fiber Bragg Grating (FBG) Structure (Can describe properties of FBGs by means of analytic expressions) forward and backward waves are strongly coupled dispersion relation slow light stored energy (same as group index) theory (Winful) KL = 4 k k Bhat and Sipe showed that the nonlinear coeficient is given by detuning from Bragg frequency where the slow-down factor S = ng/n Improved Slow-Light Fiber Bragg Grating (FBG) Structure Much larger slow-down factors possible with a Gaussian-profile grating Observation of (thermal) optical bistability at mW power levels group index approximately 140 H. Wen, M. Terrel, S. Fan and M. Digonnet, IEEE Sensors J. 12, 156-163 (2012). J. Upham, I. De Leon, D. Grobnic, E. Ma, M.-C. N. Dicaire, S.A. Schulz, S. Murugkar, and R.W. Boyd, Optics Letters 39, 849-852 (2014). The other Lake Como 5. Slow and Fast Light Controlling the Velocity of Light “Slow,” “Fast” and “Backwards” Light – Light can be made to go: slow: vg << c (as much as 106 times slower!) fast: vg > c backwards: vg negative Here vg is the group velocity: vg = c/ng ng = n + (dn/d ) – Velocity controlled by structural or material resonances absorption profile Review article: Boyd and Gaut ier, Science 326, 1074 (2009). Slow and Fast Light Using Isolated Gain or Absorption Resonances absorption resonance 0 0 n ng gain resonance g n slow light ng slow light fast light fast light ng = n + (dn/d ) Light speed reduction to 17 metres per second in an ultracold atomic gas |4〉 = |F =3, MF = –2 〉 |3〉 = |F =2, MF = –2 〉 60 MHz Lene Vestergaard Hau*², S. E. Harris³, Zachary Dutton*² & Cyrus H. Behroozi*§ ωc * Rowland Institute for Science, 100 Edwin H. Land Boulevard, Cambridge, Massachusetts 02142, USA ² Department of Physics, § Division of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA ³ Edward L. Ginzton Laboratory, Stanford University, Stanford, California 94305, USA ωp D2 line λ = 589 nm |2〉 = |F =2, MF = –2 〉 1.8 GHz |1〉 = |F =1, MF = –1〉 30 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 a 25 20 –30 1.006 Refractive index T = 450 nK τDelay = 7.05 ± 0.05 μs 1.004 –20 –10 0 10 20 30 b PMT signal (mV) Transmission NATURE | VOL 397 | 18 FEBRUARY 1999 |www.nature.com L = 229 ± 3 μm vg = 32.5 ± 0.5 m s–1 15 10 1.002 1.000 5 0.998 0.996 0 0.994 –30 –20 –10 0 10 20 30 –2 0 Probe detuning (MHz) Note also related work by Chu, Wong, Welch, Scully, Budker, Ketterle, and many others 2 4 Time ( μs) 6 8 10 12 Goal: Slow Light in a Room-Temperature Solid-State Material Crucial for many real-world applications We have identified two preferred methods for producing slow light (1) Slow light via coherent population oscillations (CPO) (2) Slow light via stimulated Brillouin scattering (SBS) Slow Light by Stimulated Brillouin Scattering (SBS) S = L - dI S = dz L g= S z gI LI S 2 2 e nv c3 0 B . refractive index SBS gain We often think of SBS as a pure gain process, but it also leads to a change in refractive index Stokes gain monochromatic pump laser anti-Stokes loss slow light The induced time delay is Td fast light G B where G= g Ip L and B is the Brillouin linewidth Okawachi, Bigelow, Sharping, Zhu, Schweinsberg, Gauthier, Boyd, and Gaeta Phys. Rev. Lett. 94, 153902 (2005). Related results reported by Song, González Herráez and Thévenaz, Optics Express 13, 83 (2005). Slow Light via Coherent Population Oscillations E3, + saturable medium E1, ng = n + dn/d b + measure absorption T2 << T1 2 ba 2 = T ba 2 = 1 T 1 a absorption profile 1/T1 1/T2 Want a narrow feature in absorption profile to give a large dn/d Ground state population oscillates at beat frequency (for < 1/T1). Population oscillations lead to decreased probe absorption (by explicit calculation), even though broadening is homogeneous. Ultra-slow light (ng > 106) observed in ruby and ultra-fast light (ng = –4 x 105) observed in alexandrite. Slow and fast light effects occur at room temperature! PRL 90,113903(2003); Science, 301, 200 (2003) Slow and Fast Light in an Erbium Doped Fiber Amplifier • Fiber geometry allows long propagation length • Saturable gain or loss possible depending on pump intensity Advance = 0.32 ms Output Input FWHM = 1.8 ms Time Fractional Advancement 0.15 6 ms in - 97.5 mW - 49.0 mW - 24.5 mW - 9.0 mW - 6.0 mW - 0 mW 0.1 0.05 0 -0.05 out -0.1 10 Schweinsberg, Lepeshkin, Bigelow, Boyd, and Jarabo, Europhysics Letters, 73, 218 (2006). 100 10 3 10 4 Modulation Frequency (Hz) 10 5 Observation of Backward Pulse Propagation in an Erbium-Doped-Fiber Optical Amplifier or ISO 80/20 coupler 980 nm laser Signal 1550 980 WDM EDF WDM We time-resolve the propagation of the pulse as a function of position along the erbiumdoped fiber. Procedure • cutback method • couplers embedded in fiber G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, R. W. Boyd, Science 312, 985 2006. 3 normalized intensity 1550 nm laser Ref pulse is placed on a cw background to minimize pulse distortion 2 in 1 out 0 0 2 4 time (ms) 6 Observation of Superluminal and “Backwards” Pulse Propagation s!STrONGLY counTerinTUITIvE PHENOMENON concEPTUALPrEDIcTION LABorATorYrESULTS ng negative sBUTEnTIrELY cONSISTenTWITH ESTABLISHEDPHySICS t=0 t=3 s0REDICTEDBY'ARRETT AND-C#UMBER AND#HIAO t=6 t=9 t = 12 s/BSERVEDBYGEHring, 3CHWEINSBERG"ARSI +OSTINSKIAND"OYD 3CIENcE, t = 15 propagation distance -1 -0.5 0 0.5 1x105 Normalized length |n g|Z (m) Causality? t4VQFSMVNJOBMWgD BOECBDLXBSETWgOFHBUJWF QSPQBHBUJPONBZTFFN DPVOUFSJOUVJUJWFCVUBSFGVMMZDPNQBUJCMFXJUIDBVTBMJUZ t5IFHSPVQWFMPDJUZJTUIFWFMPDJUZBUXIJDIQFBLPGQVMTFNPWFTJUJTOPU UIFiJOGPSNBUJPOWFMPDJUZw pulse amplitude t*UJTCFMJFWFEUIBUJOGPSNBUJPOJTDBSSJFECZQPJOUTPGOPOBMZUJDJUZPGBXBWFGPSN time sBroad spectral content at points of discontinuity sDISTURBANce moves at vacuum speed of light see, for instance, R. Y. Chiao 8. Spontaneous and Stimulated Light Scattering Experiment in Self Assembly Joe Davis, MIT Thank you for your attention!