Phys 402: Nonlinear Spectroscopy: SHG and Raman Scattering

Requirements: Polarization of Electromagnetic Waves

General consideration of polarization How Polarizers work Phys 402: Representation of Polarization: Jones Formalism Nonlinear Spectroscopy: Polarization of light and materi SHG and Raman Scattering understand nonlinear effects ! als properties are important to Spring 2008

Andrei Sirenko, NJIT Lecture 2 1 2

Broad field of nonlinear effects Linear vs. Nonlinear Spectroscopy G ||P GG P = ε 0 χE Linear spectroscopy:

GG GGG DE==+=+ε εε(1 χ ) EEP ε ε G 00 00 ||E GG ⎡⎤DE⎡⎤ ⎡⎤ xxxx xy xz PE= ε 0 χ ⎢⎥⎢⎥εεε ⎢⎥ DEyyxyyyzy=⋅0 ⎢⎥ ⋅ ⎢⎥εεεε ⎢⎥ ⎢⎥⎢⎥ ⎢⎥ ⎣⎦DEzzxzyzzz⎣⎦ ⎣⎦ G ||P εεε GGG GGG P =+εχEP PEP=+εχ 0 NL Nonlinear effects: 0 NL PE= (2) 2 NL χ G G ||E ()2PNL i= dEE ijk j k We will consider in details only SHG and Raman Scattering

Induced polarization vs. electric field in linear dielectric and in a 3 crystal without center of inversion, where electrons move in 4 asymmetric potential. Electric field and Polarization Microscopic understanding of nonlinearity GGGG Ert(,)=− E0 exp[( ikrω t )] Electronic contribution to susceptibility (linear response)

In vacuum In a materials media G ||P GG For simplicity consider one-dimensional case (∆r parallel to x) Electric field P = ε 0 χE

Polarization Displaced electronic cloud feels a restoring force, which is linear (for small displacements) G - time ||E e ∆r p G + E

G Electric field ||P GGG Polarization P =+εχ0 EPNL

time G ||E

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Microscopic understanding of nonlinearity -iωt Now, have electromagnetic wave with field E(t)=E0e -iωt Electronic contribution to susceptibility (nonlinear response) Force F(t)=eE0e d 2 x Electron is moving in an asymmetric potential with damping eE e−iωt − mω 2 x = m Equation of motion becomes o 0 2 dt ∂∂2 xt() xt () e (forced oscillator) +++=γω22x()tDxt () Ee−itω ∂∂tt2 0 2 mo ω -iωt Look for a solution x(t)=x0e ω and get e m e m x – deviation from potential minimum x(t) = E e−iωt = E(t) 2 2 o ω 2 2 2 0 − 0 −ω mDx() t Anharmonic restoring force Linear response: ∂xt() Expect strong response (large x), ⇒ large susceptibility χ⇒ γ Damping ∂t large refractive index n at ω ≈ ω0 −−itωω i2 t Solution: xt(,)ω =+( qeqe12) Dipole moment p = qx, so polarization P = eNZx eE0 1 (N atoms per unit volume, Z electrons per atom) ⇒ q1 =⋅22 miω0 −+⋅ωγω NonLinear 2 2 e ZN m NZe 1 −De22 E response: P = ω E Recall P = ε0χE and get χω()= 0 2 2 22 q2 = ε m ()ωω− 22 22 2 2 (second harmonic 0 −ω 007 2(2)2mi⎡⎤⎡⎤ω −+⋅ωγωω − ω +⋅ i γω 8 Linear response: ⎣⎦⎣⎦00 generation) Why nonlinear effects are usually weaker than linear ones? For NL polarization at the second harmonic frequency:

22(2)22ω −−itωω it PNeqe==⋅20χNL Ee Vx() For correct power consideration we need to take the complex r0 x conjugate part of the electromagnetic wave 11 P222222ωωω=+=⋅+ NeqeedEee()()−+it it −+ it ωω it 2220NL

For nonlinear susceptibility we have: xr<< 0 χ 2 3 2 (2) mD()χLL()ωχωε⋅⋅ (2) 0 −3e 223 = D ≈≈; rnm 0.5 mm23 e−5.83 x x NL 23 4 0 Vx( )=+ x Dx = ( + 24.1 − 13.3 + ...) 2||Ne ε00mr 23 23 4πε 00rrr 0 0

Why nonlinear effects are weaker than linear effects?

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Symmetry of nonlinear susceptibility tensor Two-wave mixing GGG (2) ||PEE=⋅⋅χˆ | | General case of two-wave mixing: ω12+=ωω 3 GNL 12 ()2PdEE= ()ω312=+ωω NL i ijk j k GG G 2 kk12+= k 3 ⎡Ex ⎤ GG ⎢ ⎥ G ∂∂2 EE E 2 Wave equation for linear process ∇=2 E εµ σµ + ⎢⎥y ∂∂tt2 ⎡⎤Pddddddx ⎡⎤11 12 13 14 15 16 ⎢⎥ Using Laplace operator: GGG ⎢⎥ E 2 G 2 2 ⎢⎥⎢⎥z 2 ∂∂EE∂ PNL ⎢⎥Pddddddy =⋅21 22 23 24 25 26 Wave equation for nonlinear ∇=E εµ σµ µ + + ⎢⎥⎢⎥2EE 22 ⎢⎥⎢⎥dd ddd d zy process: ∂∂∂ttt ⎣⎦Pz ⎣⎦31 32 33 34 35 36 ⎢⎥ ⎢⎥2EEzx Wave propagation along z; ⎢⎥ i, j, k index are permutations of x and y coordinates ⎣2EEx y ⎦ dE σ 1i =−1 µµEi − dEEe' * −−−ik()321 k k z For cubic, tetragonal, and orthorhombic crystals: dz 2 11iijkjk 32 11εε * ω ⎡⎤0 0 0 d14 0 0 ⎡⎤0 0 0 d ' 0 0 dE 2k 2 **−−ikkkz()321 + + ⎢⎥⎢⎥ =−σ Ei22kijkij − dEEe' 13 0 0 0 0 dd 0= 0 0 0 0 ' 0 dz 2 µµ ⎢⎥25 ⎢⎥ 22 εε ⎢⎥0 0 0 0 0 d ⎢⎥0 0 0 0 0 d " dE3 j ⎣⎦36 ⎣⎦ 3 ω −−ikkkz()321 + + 11 =−σ Ei33j − dEEe' ijk 12 i k 12 dz 2 33µµ εεω Second-harmonic generation Second-harmonic generation GG dE1i Small loss of power in the primary beam SHG: ω +=ωωω2 (31 = 2 ω ) ≈ 0 If in birefringent crystal nn(2ωω )=⇒= ( ) 2 kk GG dz eO ω 2ω 2kk= ωω2 dE µ ω 2 3 j =−idEEe2'ω −∆ikz ε NZe ⎛⎞1 ijk11 i k n ()==+ Re Re1 dz ε R ⎜⎟22 ε 000εωωωγmi⎝⎠−+ e−∆ikz−1 n()ω ωωEz()=− i 2ω dµ ' EE 311j ijk i k ik∆ ε Transparent crystal Strong absorption 2 *2222µ sin (∆kL / 2) Power(2 )== E33j ( L ) E j ( L ) 4 ( d ' ijk ) E 11 i E k ε (/2)∆kL 2 3 nO ()ω 2π Coherence length: l = ωω ∆k G ω 2ω ne (2ω ) Phase matching requirement: 2⋅=||knω 2( ) = n ( ) ω ccω G ωω2 ||(2)kn= ⎡⎤⎡⎤2 2 ω ω 2 ω ω εεεxx xy xz n e 0 0 c ω 0 εεεε⎢⎥⎢⎥2 ==⎢⎥yx yy yz⎢⎥ 0 n O 0 nn()≠ (2) ! ⎢⎥⎢⎥2 Blue waves propagate with the same velocity in the crystal ⎣⎦zxεεε zy zz ⎣⎦0 0 n O GG 13 14 If in birefringent crystal nneO(2ωω )=⇒= ( ) 2 kkω 2ω

Experimental setup for Second-harmonic generation Applications of Second-harmonic generation ω +=ωω2 KHPO24 GG Lasers (Nd:YAG, second harmonic) 2kkωω= 2 KDP crystal Coherent anti-Stokes Raman scattering nneO(2ω )= (ω ) Bio-imaging Materials Physics Fine tuning of refractive index for Phase matching in uniaxial Solar Physics Nonlinear crystals: Quantum cryptography (two-wave mixing) θ 1cossinθ θ 222=+ nnn() Oe

2 ⎡⎤n e 0 0 ⎢⎥2 ε = ⎢⎥0 n O 0 ⎢⎥2 ⎣⎦0 0 n O

15 16 Brillouin and Raman spectroscopy Raman scattering in crystalline solids Inelastic light scattering mediated by the electronic polarizability of the medium Not every crystal lattice vibration can be probed by Raman • a material or a molecule scatters irradiant light from a source scattering. There are certain Selection rules: • Most of the scattered light is at the same wavelength as the laser source (elastic, or Raileigh scattering) 1. Energy conservation:

• but a small amount of light is scattered at different wavelengths (inelastic, =ωi = =ωs ± =Ω; or Raman scattering) 2. Momentum conservation: I Elastic 4πn ks k ћΩ 0 ≤ q ≤ 2k q ≈ 0 s q ћΩ β (Raileigh) ki = k s ± q ⇒ ⇒ 0 ≤ q ≤ λ ki k α Scattering i i α q ≈ 2k β ћωs k k λ ~ 5000 Å, a ~ 4-5 Å ⇒ λ >> a s i ћω Anti- i 0 phonon 0 ћωi s ћω Stokes i Stokes ω Stokes ωi Anti-Stokes 0 0 Raman Raman ⇒ only small wavevector (cloze to BZ center) phonons are seen in Scattering Scattering the 1st order (single phonon) Raman spectra of bulk crystals Raileigh ωi- Ω(q) ωi+ Ω(q) Analysis of scattered light energy, polarization, relative intensity 3. Selection rules determined by crystal symmetry provides information on lattice vibrations or other excitations17 18

Raman scattering in crystalline solids Example of Raman scattering in crystalline solids

Raman scattering

G GGG qkkk=±∆ =±||is −

2×+TO LO 3S = 15 modes 11 2×+TO22 LO Phonon Energy 3 acoustic modes 2×+TO33 LO 12 optical modes; 3× 4 2×+TO44 LO

Mandelstam-Brillouin Phonon wavevector 19 20 scattering