19. Nonlinear Optics19. Nonlinear Optics

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19.19. NonlinearNonlinear OpticsOptics

NonlinearNonlinear opticsopticsPolarization : P ε 0 χ ESusceptibility : χ χ1 χ 2 E χ 3 E 2 "D εE ε 0 E ε 0 χE ε ε 0 (1 χ ) n vε 1 χcε0P P1 P2 P3 " ε 0 χ1 E ε 0 χ 2 E 2 ε 0 χ 3 E 3 "

2P εχESecond-orderNonlinearopticsSecond-order Nonlinear optics 20 2Second-harmonic generation (SHG) and rectificationE E (ω ) optical P2 E 2 (ω ) P2 (ω ω ) P2 (2ω ),P2 (0)ÆFrequency doublingÆ RectificationElectro-optic (EO) effect (Pockell’s effect)E E (0) electrical , DC E (ω ) optical P2 E 2{{but, E (0) E (ω ) }} P2 (0) E 2 (0) , P2 (ω ){ E (0) E (ω )}, P2 (2ω ){ E (ω ) E (ω )} P2 (0), P2 (ω ){ E (0) E (ω )} Δn E (0) electric , DC Æ Index modulation by DC E-fieldThree-wave mixingE E (ω1 ) optical E (ω2 ) optical P2 E 2{}{} P2 (2ω1 ) E 2 (ω1 ) , P2 (2ω2 ) E 2 (ω2 ) ,P2 (ω1 ω2 ){ E (ω1 ) E (ω2 )},P2 (ω1 ω2 ){ E (ω1 ) E (ω2 )}Æ SHGÆ Frequency up-converterÆ Parametric amplifier, parametric oscillator

Third-orderThird-order NonlinearNonlinear opticsopticsThird-harmonic generation (THG){P3 ε 0 χ 3 E 3}E E (ω ) optical P3 E 3 (ω ) P3 (ω ) E (ω ) E (ω ) , P3 (3ω ){ E 3 (ω )}2Æ Frequency triplingElectro-optic (EO) Kerr effectE E (0) electrical , DC E (ω ) optical{but, E (0) E (ω ) } P3 (ω ) E (0) electric , DC E (ω ) Δn E (0) electric , DC22Æ Index modulation by DC E2Optical Kerr effectP3 (ω ) E (ω ) E (ω ) I (ω ) E (ω ) Δn I (ω )2n n0 Δn( I ) ϕ ϕ 0 Δϕ ( k 0 ΔnL )Æ Index modulation by optical IntensityÆ Self-phase modulationn n0 Δn{I ( x)} Δn{I ( x )} n0Æ Self-focusing, Self-guiding (Spatial solitons)n n0 Δn{I ( x )} Δn{I ( x )} n0Æ Self-defocusing

P3 ε 0 χ 3 E 3Third-orderThird-order NonlinearNonlinear opticsopticsFour-wave mixingE E (ω1 ) optical E (ω2 ) optical E (ω3 ) optical P3 E 3 ( ω1 , ω2 , ω3 ) 63 216 terms3 One example : P3 (ω1 ω2 ω3 ω4 ) E (ω1 ) E (ω2 ) E (ω3 ) If ω1 ω2 ω3 ω4 3ωÆ Frequency up-converterÆ THG Another example : P3 (ω1 ω2 - ω3 ω4 ) E (ω1 ) E (ω2 ) E * (ω3 ) ω1 ω2 ω3 ω4 If ω1 ω2 ω3 ω4Æ Degenerate four-wave mixing Assume two waves among them areplane waves traveling in opposite directions P3 (ω4 ω ) E (ω ) E (ω ) E * (ω )Æ Optical phase conjugation

24-2.24-2. SecondSecond harmonicharmonic generationgeneration (SHG)(SHG)P2 ε 0 χ 2 E2E Eo cos ω t: Only for non-centro-symmetry crystals[GaAs. CdTe, InAs, KDP, ADP, LiNbO3, LiTaO3, ]P P1 P2 ε 0 χ1 Eo cos ω t ε 0 χ 2 Eo2 cos 2 ω t1 2 cos θ ( 1 cos 2θ ) 2 112 ε 0 χ1 Eo cos ω t ε 0 χ 2 Eo ε 0 χ 2 Eo2 cos 2ω t22 1 2 1P2 ( t ) ε 0 χ 2 E0 ε 0 χ 2 E02 cos 2ω t P2 (0) P2 (2ω ) 2 2 Constant (DC) termÆ Optical rectificationSecond harmonic termÆ 2ω

SHG does not occur in isotropic, centrosymmetry crystalsP2 ε 0 χ 2 E 2If χ 2 is isotropic or centrosymmetric , both E and - E give the same P2 polarization that means the molecules are not polarized by the sencond χ effect .

SecondSecond harmonicharmonic generationgenerationP2 1 1 P2 ( t ) ε 0 χ 2 E02 ε 0 χ 2 E02 cos 2ω t P2 (0) P2 (2ω ) 2 2 P2(t)EE(t)From Fundamentals of Photonics (Bahaa E. A. Saleh)

SecondSecond harmonicharmonic generationgenerationE Eo cos ω tP P1 P2P1 ε 0 χ1 Eo cos ω t1P2 ε0χ2 Eo2 cos ( 2ωt )21 ε0χ2 Eo22

SecondSecond harmonicharmonic generationgeneration ω 2ω (λ λ / 2)From Fundamentals of Photonics (Bahaa E. A. Saleh)

SecondSecond harmonicharmonic generationgeneration

PhasePhase matchingmatching (index(index matching)matching) inin SHGSHGOutput intensity after second harmonic generation LΔk , Δk k2ω 2kωI sin c 2 2Phase matching : Δk 0Δk k2ω 2kω 2ω ω n2ω 2nω c c ω ( n2ω nω ) 2 0 c Optic axisDirection ofMatching(Δk 0)E-ray surface(n2ω)O-ray surface(nω)

FrequencyFrequency mixingmixing byby three-wavethree-wave mixingmixingfrequency up-converter (ω1 ω2 ω3 )parametric amplifier(ω3 ω1 ω2 ω3 ω2 ω1 )(ω2 idler, or parameter, 중개자 )parametric oscillator (ω3 ω1 ω2 ω3 ω2 ω1 )

ParametricParametric interactioninteractionE E (ω1 ) E (ω 2 ) Eo1 cos ω1t Eo 2 cos ω 2 t 11Eo1 {exp( iω1t ) exp( iω1t )} Eo 2 {exp( iω 2 t ) exp( iω 2 t )}22P2 ε 0 χ 2 E 2 ( ω1 ω1 2ω1 ) , ( ω2 ω2 2ω2 ) , ( ω1 ω2 ω3 ) , ( ω1 ω2 ω3 )Æ Frequency conservationÆ Momentum (phase) matching

Third-orderThird-order nonlinearnonlinear effecteffectIn media possessing centrosymmetry, the second-order nonlinearterm is absent since the polarization must reverse exactly when theelectric field is reversed.The dominant nonlinearity is then of third order,P3 ε 0 χ 3 E3The third-order nonlinear material is called a Kerr medium.P3E

Optical Kerr effectP3 (ω ) E (ω ) E (ω ) I (ω ) E (ω ) Δn I (ω )2n n0 Δn( I ) ϕ ϕ 0 Δϕ ( k 0 ΔnL )Æ Index modulation by optical IntensityÆ Self-phase modulationn n0 Δn{I ( x)} Δn{I ( x)} n0Æ Self-focusing, Self-guiding (Spatial solitons)n n0 Δn{I ( x)} Δn{I ( x)} n0Æ Self-defocusingn( I ) n n2 I

n( I ) n n2 ISelf-phase modulationThe phase shift incurred by an optical beam of power P andcross-sectional area A, traveling a distance L in the medium,Self-focusing (Optical Kerr lens)

Spatial Solitons Self-guided beamIn a linear medium,wave is spreading.In an optical Kerr medium,wave can be self-guided.(Also, see 19.8): Nonlinear Schrodinger equation

Raman Gain: Raman GainCoefficientCoupling of light to the high-frequency vibrational modes of the medium,which act as an energy source providing the gain.For low-loss media, the Raman gain may exceed the loss at reasonable levelsof power, so that the medium can act as an optical amplifier.Æ Fiber Raman lasers

Four-waveFour-wave mixingmixing (third-order(third-order nonlinearity)nonlinearity)Superposition of three waves of angular frequencies ω1, ω2, and ω3P3 ε 0 χ 3 E 3(as sum of 63 216 terms)If ω4 ω1 ω 2 ω 3 {ω3 ω4 ω1 ω2 }{G GG Gk3 k4 k1 k2}

OpticalOptical phasephase conjugationconjugation

Phase conjugate mirror (PCM)Conventional mirrorPCM

OpticalOptical phasephase conjugationconjugationWhen all four waves are of the same frequency Î degenerated four-wave mixing (DFWM)Assuming further that two waves (3,4) are uniform plane waves traveling in oppositedirections,A2 *A3 A4 A1A3: conjugated version of wave 1NonlinearA1Probe beamMediumA2(DFWM)PC beamA4Pump beamPump beam

Note: Phase Conjugation and Time ReversalIncidentPhase conjugationTime reversal[E1 (r, t ) Re ψ (r )ei (ωt kz )[]E2 (r, t ) Re ψ (r )ei (ωt kz )]i ω ( t ) kz E2 ( r, t ) Re ψ ( r ) e

Degenerate Four-Wave Mixing as a Form of Real-Time Holography

ImageImage restorationrestoration byby phasephase conjugationconjugationOptical reciprocity.

19.4.19.4. Coupled-waveCoupled-wave theorytheory ofof three-wavethree-wave mixingmixing

by equating terms on both sides at each of the frequencies w1, w2, and w3, separately,Coupled-wave Equationsin three-wave mixingHomework :

Mixing of Three Collinear Uniform Plane WavesHomework :

Second-harmonic generation (SHG)Z 0

Second-harmonic generation (SHG)Photon flux densitiesÆ photons of wave 1 are converted to half as many photons of wave 3.Æ photon numbers are conserved.

Second-harmonic generation (SHG)Efficiency of second-harmonic generationTo maximize the efficiency, we must confine the wave to the smallest possible area Aand the largest possible interaction length L.This is best accomplished with waveguides (planar or channel waveguides or fibers).

Second-harmonic generation (SHG)Effect of Phase MismatchFor weak-coupling case

19.6.19.6. AnisotropicAnisotropic nonlinearnonlinear mediamediaThree-Wave Mixing in Anisotropic Second-Order Nonlinear Media

Phase Matching in Three-Wave MixingThus the fundamental wave is an extraordinary waveand the second-harmonic wave is an ordinary wave.

Third-order nonlinear effectThird-order nonlinear effect In media possessing centrosymmetry, the second-order nonlinear term is absent since the polarization must reverse exactly when the electric field is reversed. The dominant nonlinearity is then of third order, 3 PE 303 εχ The third-order nonlinear material is called a Kerr medium. P 3 E

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Recommended reading -lasers and nonlinear optics: Lasers, by A. Siegman (University Science Books, 1986) Fundamentals of Photonics, by Saleh and Teich (Wiley, 1991) The Principles of Nonlinear Optics, by Y. R. Shen (Wiley, 1984) Nonlinear Optics, by R. Boyd (Academic Press, 1992) Optics, by Eugene Hecht (Addison-Wesley, 1987)

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