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
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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 = ✏0 E1 = ✏0 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 = (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 e ciency of nonlinear optical interactions Treatment of Second-Harmonic Generation – III
For the case k =0,Eq.(8)becomes dA 4!2 2ik 2 = (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)= A1L or A2(L) = 2 2[ ] 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 [ ] A1 L sinc ( kL/2) (11) | | n2c | | Note that kL must be kept much smaller than ⇡ radians in order for e cient 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 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 (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 1111 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 (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. (planar glass waveguide 1992: Segev, (photorefractive) Solitons and self-focussing in Ti:Sapphire Diffraction-management controls the spatial self- focussing 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 Sidemode amplitude grows as i z z A1(z) = A1(0)e e / 2 2 2 q q e, = 2 , t a r 2k 2k wth n ( / c)I , q = k o = 2 0 r g tial q = 2k n max qmax transverse wavevector, q/qmax = xpone max e k Honeycomb Pattern Formation Output from cell with a single gaussian input beam At medium input power At high input power at cell exit in far field at cell exit in far field Quantum statistics? Input power 100 to150 mW Sodium vapor cell T = 220o C Input beam diameter 0.22 mm Wavelength = 588 nm Bennink et al., PRL 88, 113901 2002. Optical Radiance Limiter Based on Spatial Coherence Control Controlled small-scale flamentation used to modify spatial degree of coherence Alternative to standard appropches to optical power limiting spatially spatially coherent n2 medium incoherent light in light out 1.6 1.2 0.8 f increasing power 0.4 -field di r Fa 0 Schweinsberg et al., Phys. Rev. A 84, 053837 (2011). 0 10 20 30 Incident pulse ener 4. Local-Field Effects in Nonlinear Optics Local Field Effects in Nonlinear Optics – I Recall the Lorentz-Lorenz Law E loc (linear optics) EP Nα ϵ(1) − 1 4 χ(1) πNα. = 4 or (1) = 1 − 3πNα 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 4 (1) Eloc = E + 3πP where P = χ E We can rewrite this result as ϵ(1) +2 E = LE where L = is the local field factor. loc 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)=Nγ(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,andL4 = 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 , can be escribe by effective r ) s Local Field Enhancement of the NLO Response - Under very general conditions, we can express the NL response as (3) 2 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 fL 2 L 2 can exceed unity. + 2 - For a homogeneous material L = 3 - For a spherical particle of dielectric constant m embedded in a host of dielectic constant h 3 L = h m + 2 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 1 f f L = eff = a + b a eff a b Enhanced NLO Response from Layered Composite Materials A composite material can display a larger NL response than its constituents! Alternating layers of TiO2 and Quadratic EO effect the conjugated polymer PBZT. gold electrode BaTiO3 AF-30 in signal generator polycarbonate 1 kHz ITO glass substrate photodiode polarizer dc voltmeter Diode Laser Measure NL phase shift as a (1.37 um) lockin amplifier function of angle of incidence. 35% enhancement in (3) 3.2 times enhancement! Fischer, Boyd, Gehr, Jenekhe, Osaheni, Sipe, and Weller-Brophy, Phys. Rev. Lett. 74, 1871 (1995). 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) SiO2 Cu PC structure E Cu E 80 nm of bulk metal copper (total) 80 nm of copper T = 0.3% T = 10% z 0 L z Bennink, Yoon, Boyd, and Sipe, Opt. Lett. 24, 1416 (1999). bulk Cu M/D PC Lepeshkin, Schweinsverg, Piredda, Bennink, nm (response twelve- and Boyd, Phys. Rev. Lett. 93, 123902 (2004). times larger) I = 500 MW/cm2 Gold-Doped Glass: A Maxwell-Garnett Composite Red Glass Carafe 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 diferent 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) 1.3 9 SA 1.2 8 Increasing Gold β < 0 1.1 7 Content Im (3) < 0 1.0 6 (3) 0.9 5 Im > 0 0.8 4 β > 0 0.7 3 RSA normalized transmittance 2 2 0.6 (3) = fL L2 (3)+ (3) 1 eff ih 0.5 -10 0 10 20 30 z(cm) D.D. Smith, G. Fischer, R.W. Boyd, D.A.Gregory, JOSA B 14,1625,1997. NLO in Plasmonics (Photonics Using Metals) Is there an intrinsic nonlinear reponse to surface plasmon polaritons (SPPs)? 2 is106 times that of silica) sp Rotation Translation TM stage sp polarized Au film stage z Laser x PD y HWP PBS Aperture 6 sp sp 1.0 0 1 2 3 Kretschmann S dip sp 0.5 TIR onset R increasing power Pi = 2.70 mW P = 8.70 mW Reflectance, i 0.1 Pi = 15.0 mW Pi = 22.8 mW Pi = 28.4 mW 0.05 41.5 42 42.5 43 43.5 I. De Leon, Z. Shi, A. Liapis and R.W.Boyd, Optics Letters 39, 2274 (2014) Incidence angle, (deg.) Slow Light in a Fiber Bragg Grating (FBG) Structure (Can describe properties of FBGs by means of analytic expressions) forward and backward waves dispersion relation are strongly coupled slow light theory (Winful) k KL = 4 k Bhat and Sipe showed that the nonlinear coeficient is given by detuning from Bragg frequency where the slow-down factor S = ng/n stored energy (same as group index) stored energy Improved Slow-Light Fiber Bragg Grating (FBG) Structure Much larger slow-down factors possible Observation of (thermal) with a Gaussian-profile grating 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 g gain resonance resonance 0 0 n n slow light ng ng slow light fast light fast light ng = n + (dn/d ) Light speed reduction |4〉 = |F =3, M = –2〉 to 17 metres per second 60 MHz F |3〉 = |F =2, M = –2〉 in an ultracold atomic gas F Lene Vestergaard Hau*², S. E. Harris³, Zachary Dutton*² ωc ωp D2 line & Cyrus H. Behroozi*§ λ = 589 nm * Rowland Institute for Science, 100 Edwin H. Land Boulevard, Cambridge, 〉 〉 Massachusetts 02142, USA |2 = |F =2, MF = –2 ² Department of Physics, § Division of Engineering and Applied Sciences, 1.8 GHz Harvard University, Cambridge, Massachusetts 02138, USA |1〉 = |F =1, M = –1〉 ³ Edward L. Ginzton Laboratory, Stanford University, Stanford, California F 94305, USA NATURE | VOL 397 | 18 FEBRUARY 1999 |www.nature.com 30 0.8 0.7 a 0.6 25 T = 450 nK 0.5 0.4 τDelay = 7.05 ± 0.05 μs 0.3 20 L = 229 ± 3 μm Transmission 0.2 –1 0.1 vg = 32.5 ± 0.5 m s 15 –30 –20 –10 0 10 20 30 1.006 b PMT signal (mV) 1.004 10 1.002 1.000 5 0.998 Refractive index 0.996 0.994 0 –30 –20 –10 0 10 20 30 –2 0 2 4 6 8 10 12 Probe detuning (MHz) Time ( μs) Note also related work by Chu, Wong, Welch, Scully, Budker, Ketterle, and many others 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 = gILI S L dz 2 2 e S g = . nv c3 z 0 B 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 SBS gain anti-Stokes loss slow light fast light refractive index G The induced time delay is T d where G= g Ip L and B is the Brillouin linewidth B 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 b E3, + + saturable 2 1 2 = = medium measure ba T ba T 2 1 E1, absorption a absorption ng = n + dn/d T2 << T1 profile 1/T1 1/T Want a narrow feature in absorption 2 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 5 (ng = –4 x 10 ) 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 Advance = 0.32 ms • Saturable gain or loss possible depending on Output Input pump intensity FWHM = 1.8 ms Time 0.15 - 97.5 mW 0.1 - 49.0 mW - 24.5 mW - 9.0 mW 0.05 - 6.0 mW - 0 mW 0 6 ms Advancement Fractional -0.05 out in -0.1 10 100 10 3 10 4 10 5 Modulation Frequency (Hz) Schweinsberg, Lepeshkin, Bigelow, Boyd, and Jarabo, Europhysics Letters, 73, 218 (2006). Observation of Backward Pulse Propagation in an Erbium-Doped-Fiber Optical Amplifier or Ref 80/20 1550 nm laser ISO coupler WDM 980 nm laser EDF Signal 1550 980 WDM We time-resolve the propagation 3 of the pulse as a function of pulse is placed on a cw background to minimize 2 position along the erbium- pulse distortion doped fiber. Procedure 1 in • cutback method out • couplers embedded in fiber 0 normalized intensity normalized G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, 0 2 4 6 R. W. Boyd, Science 312, 985 2006. time (ms) Observation of Superluminal and “Backwards” Pulse Propagation LABorATorY rESULTS s ! STrONGLY concEPTUAL PrEDIcTION ng negative counTerinTUITIvE PHENOMENON s BUT EnTIrELY cONSISTenT WITH t = 0 ESTABLISHED PHySICS t = 3 s 0REDICTED BY 'ARRETT AND -C#UMBER t = 6 AND #HIAO t = 9 t = 12 s /BSERVED BY GEHring, 3CHWEINSBERG "ARSI t = 15 +OSTINSKI AND "OYD 3CIENcE , propagation distance -1 -0.5 0 0.5 1x105 Normalized length |n g|Z (m) Causality? t4VQFSMVNJOBM Wg D BOECBDLXBSET WgOFHBUJWF QSPQBHBUJPONBZTFFN DPVOUFSJOUVJUJWFCVUBSFGVMMZDPNQBUJCMFXJUIDBVTBMJUZ t5IFHSPVQWFMPDJUZJTUIFWFMPDJUZBUXIJDIQFBLPGQVMTFNPWFT JUJTOPU UIFiJOGPSNBUJPOWFMPDJUZw t*UJTCFMJFWFEUIBUJOGPSNBUJPOJTDBSSJFECZQPJOUTPGOPOBMZUJDJUZPGBXBWFGPSN pulse amplitude time s Broad spectral content at points of discontinuity s DISTURBANce 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!