Course Nonlinear Optics 12 Lectures: 10 lectures and 2 tutorials (in Trinity term) Timetable of nonlinear optics lectures Nonlinear Optics NLO LB (8 lectures) and TK (2 lectures: slots to be determined) Timetable Dr. Louise Bradley Room 2:21 [email protected] Monday 4 pm Wednesday Friday 10 am 9 am Week 6 NLO NLO Week 7 NLO NLO NLO Week 8 NLO NLO NLO Week 9 NLO NLO Dr. L. Bradley Nonlinear optics isn’t something you see everyday. Suggested Reading Optical Electronics in Modern Communications – A. Yariv 1997 Nonlinear Optics – R.W. Royd Principles of Nonlinear Optics – Butcher and D. Cotter Applied Nonlinear Optics – Zernike and Midwinter Fundamentals of Photonics – Saleh and Teich 1991 and 2006 Handbook of Nonlinear optics – Sutherland 2003 Great web sites – used lots of figures/slides from these http://www.ph.surrey.ac.uk/intranet/undergraduate/3mol http://www.physics.gatech.edu/gcuo/UltrafastOptics/index.html Dr. L. Bradley What is NLO? What are nonlinear optical effects and why do they occur? Sending infrared light into a crystal yielded this display of green light (second-harmonic generation): Nonlinear optics allows us to change the colour of a light beam, to change its shape in space and time, and to create ultrashort laser pulses, the shortest events ever made by Man. NL is key element for optical data processing Why don't we see nonlinear optical effects in our daily life? 1. Intensities of daily life are too weak. 2. Normal light sources are incoherent. 3. The occasional crystal we see has the wrong symmetry (for SHG). 4. “Phase-matching” is required, and it doesn't usually happen on itsDr. own. L. Bradley Difference-frequency generation Examples SHG Medium ω 2ω ω1 SHG Medium ω1+ω2 ω1 ω3 ω2 ω2 = ω3 − ω1 ω1 ω3 ω2 Optical Parametric Amplification (OPA) THG Medium ω ω1 3ω And Frequency Sum Dr. L. Bradley ω2 = ω3 + ω1 Dr. L. Bradley 1 Self-diffraction Non co-linear – third order We can also allow two different input beams, whose frequencies can be different. So in addition to generating the third harmonic of each input beam, the medium will generate interesting sum frequencies, spatially separate. … but the frequencies don’t have to be different to generate new optical fields propagating in different directions… Signal #1 ω2 THG medium Signal #1 ω 2ω1 +ω2 2ω2 +ω1 ω1 Nonlinear medium ω ω Signal #2 Signal #2 ω Dr. L. Bradley Self-focusing Dr. L. Bradley Optical computing and optical data processing ¾ x ¾ ¾ n(x) ¾ Nonlinear absorption Two-photon absorption detectors Saturable Absorbers Optical limiting Nonlinearity key element for optical switches and optical bistability Optical logic gates, flip-flops Dr. L. Bradley Phase conjugation Dr. L. Bradley Questions Why do nonlinear-optical effects occur? How can we use them? Maxwell's equations in a medium Î Nonlinear-optical media Reflection of a plane wave from an ordinary and phase conjugate mirror Reflection of a spherical wave from an ordinary and phase conjugate mirror Second Order Effects: Second-harmonic generation Sum- and difference-frequency generation Autocorrelation Phase-matching and Conservation laws for photons Third Order Non-linear Effects: Mirror Distorting medium Phase conjugate mirror Dr. L. Bradley Frequency generation, Nonlinear refractive index, Phase conjugation… Consequences and Applications Dr. L. Bradley 2 Nonlinear Response Nonlinear Response Nonlinear response is not confined to optics R = ζ 1S + ζ 2 S 2 + ζ 3 S 3 + ζ 4 S 4 + ... Dr. L. Bradley Dr. L. Bradley The Fourier components Representation of Nonlinearity Linear R = ξ1S S R R R S s Non-Linear R R = ξ1S − ξ 2 S 2 The same frequency as the stimulus S R Double the frequency of the stimulus S A DC component Dr. L. Bradley Interaction of light and matter An applied electric (optical) field displaces the electrons from the nucleus in a medium Polarization = dipole moment per unit volume Separation of charges gives rise to a dipole moment P(t)=-Nex(t) + Dr. L. Bradley Polarization Optical polarization of dielectric crystals – mostly due to outer loosely bound valence electrons displaced by the optical electric field. Polarization is alternating with the same frequency as the applied E field. Electron oscillates about the equilibrium position – oscillating dipole is a source of EM radiation. - E Dr. L. Bradley Dr. L. Bradley 3 Linear Optics Linear optics Recall that, in normal linear optics, a light wave acts on a atom or atom, which vibrates and then molecule emits its own light wave that interferes with the original light wave. input emitted Consequence of induced charge photons photons polarization and re-radiation is a decrease in the speed f light in the medium i.e. increase in the refractive index relative to the vacuum Internal field due to nucleus ~1011 V/m Sunlight ~1 kV/m Consequently, for small E, P (t ) = ε 0 χ (1)E (t ) the linear approximation is very accurate P(t)=-Nex(t) x(t) is small, harmonic potential regime Dr. L. Bradley Nonlinear optics and anharmonic oscillators For field strength > 1 kV/m (i.e. lasers), X(t) is sufficiently large that the potential of the electron or nucleus (in an atom/molecule) is not a simple harmonic potential. Anharmonic motion occurs, and higher harmonics occur, both in the motion and the light emission Dr. L. Bradley Nonlinear Polarization In an anharmonic potential: Polarization expanded as a power series in E to give: ( r r r r P = ε 0 χ (1) E + χ ( 2 ) E 2 + χ ( 3) E 3 + ... ) χ(2) = 2nd order susceptibility χ(3) = 3rd order susceptibility Example: vibrational motion: atom or molecule input photons different colour! emitted photons In order for the series to converge: χ(3)E3<< χ(2)E2<< χ(1)E Accessed for optical intensities I ~1013 W/m2 Dr. L. Bradley Linear susceptibility Lorentz Model Lorentz model – analogous to a mass on a spring χ (1) = Electron of mass, m, and charge, e, is attached to the ion by a spring. External force applied by the E field drives the oscillation γ is the damping constant ω0 is the resonant frequency Anharmonic oscillator includes higher order terms Dr. L. Bradley Ne2 ε 0m ω − 2iγω1 − ω12 [ 2 0 r r P = ε 0 χ (1) E r r r r D = εE = ε 0 E + P ] ω0 Optical Frequency, ω F=-kx k=spring constant x=displacement The electric polarization P is defined as the difference between the electric fields D (induced) and E (imposed) in a dielectric due to bound and free charges, respectively. In MKS, r r r r r r D = ε 0 E + P = ε 0 (1 + χ (1) )E = εE = ε 0ε r E n= Dr. L. Bradley c = v μ0ε μ0ε 0 = ε = εr = ε0 (1 + χ ) (1) Dr. L. Bradley 4 Refraction and Absorption Second order susceptibility The refractive index is a complex quantity n = 1+ χ ε 02mξ ( 2 ) χ (2) = (1) N 2 e3 n = n0 − iκ ( ) κ = Im( 1 + χ ) n0 = Re 1 + χ (1) (1) Dispersion, frequency dependent speed of propagation 1 1 1 For a pure frequency applied field χ(2) Can be expressed in terms of the linear susceptibility χ(1) at two frequencies ω1 and 2ω1 The susceptibilty determines the magnitude of P and hence the strength of fields being reradiated Is proportional to the material absorption If we are far from the absorption, the imaginary parts are negligible. In the linear case the dipoles and the polarization oscillate at the same frequency as the incident field. χ ((ω1) ) χ ((ω1) ) χ ((21ω) ) Dr. L. Bradley Dr. L. Bradley Many interacting fields Third order susceptibility r r r P ( 2 ) = ∑∑ ε 0 χ ((ω2n) ,ωm ) E(ωn ) E(ωm ) e − i (ωn +ωm ) t n χ (2) (ω n ,ω m ) = Continuing our analysis of the anharmonic oscillator, for 1 driving field we will find m ε 02mξ ( 2 ) N 2 e3 χ (1) (ω n ) χ (1) (ω m ) χ χ ( 3) = (1) (ω n +ω m ) The 2nd order: interaction of two fields producing a third Result: all frequencies (ωn+ ωm) for all possible values of n,m For just two fields n,m = ±1,±2 Will get all the terms we had before: DC component Second harmonic generation Frequency sum and frequency difference terms ε 03mξ ( 3) N 3e 4 χ ((ω1) ) χ ((ω1) ) χ ((ω1) ) χ ((31ω) ) 1 1 1 1 Third order: 3 input fields producing a fourth Nonlinear polarization will find (ωn+ ωm+ ωp) for all possible values of n,m,p = ±1,±2, ±3 r r r r − i (ω +ω +ω ) t P ( 3) = ∑∑∑ ε 0 χ ((ω3)n ,ωm ,ω p ) E(ωn ) E(ωm ) E(ω p ) e n m p n χ ((ω3) ω n , m ,ω p ) m = p ε 03mξ ( 3) N 3e 4 χ ((ω1) ) χ ((ω1) ) χ ((ω1) ) χ ((ω1) +ω n m p n m +ω p ) Third harmonic generation and many frequency sum/frequency difference terms. Dr. L. Bradley Maxwell’s equation in a medium r r ∇⋅E = 0 r r ∇⋅B = 0 r r r ∂B ∇× E = − ∂t r r r r ∂D ∇ × B = μ0 J + μ 0 ∂t Dr. L. Bradley Solving the wave equation in the presence of linear induced polarization For low irradiances, the polarization is proportional to the incident field: P = ε 0 χE In this simple (and most common) case, the wave equation becomes: ∂ 2E 1 ∂ 2E − ∂z 2 c02 ∂t 2 These equations reduce to the wave equation: r r r 1 ∂2E ∂2 P ∇2 E − 2 2 = μ 2 v ∂t ∂t “Inhomogeneous Wave Equation” Simplifying: waves of all frequencies are solutions to the wave equation; it’s the polarization that tells which frequencies will occur. where The induced polarization, P, contains the effect of the medium or if you prefer The polarization is the driving term for the solutions to this equation. Dr. L. Bradley 1 ∂ 2E χ c02 ∂t 2 ∂ 2E 1 + χ ∂ 2E − 2 ∂z 2 c0 ∂t 2 This equation has the solution: Sine = ω=ck Using the fact that: ε 0 μ 0 = 1/ c02 = 0 E ( z, t ) ∝ E0 cos(ωt − k z) and c = c0 /n and n = (1+χ)1/2 The induced polarization, at the same frequency as the incident field and only changes the refractive index. Dull. If only the polarization contained other frequencies… Dr. L. Bradley 5 Maxwell's Equations in a Nonlinear Medium … and we now know this can occur Nonlinear optics is what happens when the polarization is the result of higher-order (nonlinear!) terms in the field: P = ε 0 ⎡⎣ χ (1)E + χ (2)E 2 + χ (3)E 3 + ...⎤⎦ And we saw the effects of such nonlinear terms: Light generated at many other frequencies, for example. Since E (t ) ∝ E exp(iωt ) + E * exp(−iωt ), E (t )2 ∝ E 2 exp(2iωt ) + 2 E + E *2 exp(−2iωt ) 2 2ω = 2nd harmonic! Harmonic generation is one of many exotic effects that can arise! Dr. L. Bradley 6