Requirements: Polarization of Electromagnetic Waves General consideration of polarization How Polarizers work Phys 402: Nonlinear Spectroscopy: SHG and Raman Scattering Representation of Polarization: Jones Formalism Polarization of light and materials properties are important to understand nonlinear effects ! Spring 2008 Andrei Sirenko, NJIT Lecture 2 1 Broad field of nonlinear effects 2 Linear vs. Nonlinear Spectroscopy G |P| G G P = ε0χ E Linear spectroscopy: G G G G G D = ε 0ε E = ε 0 (1 + χ ) E = ε 0 E + ε 0 P ⎡ε xx ε xy ε xz ⎤ ⎡ Ex ⎤ ⎡ Dx ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ Dy ⎥ = ε 0 ⋅ ⎢ε yx ε yy ε yz ⎥ ⋅ ⎢ E y ⎥ ⎢ε ε ε ⎥ ⎢ E ⎥ ⎢D ⎥ ⎣ z⎦ ⎣ zx zy zz ⎦ ⎣ z ⎦ Nonlinear effects: We will consider in details only SHG and Raman Scattering 3 G |E| G G P = ε0χ E G G G P = ε 0 χ E + PNL PNL = χ (2) E 2 G ( PNL )i = 2dijk E j Ek G |P| G G G P = ε 0 χ E + PNL G |E| Induced polarization vs. electric field in linear dielectric and in a crystal without center of inversion, where electrons move in 4 asymmetric potential. Microscopic understanding of nonlinearity Electric field and Polarization GG G G E (r , t ) = E0 exp[i (kr − ωt )] In vacuum In a materials media Electric field G |P| Electronic contribution to susceptibility (linear response) G G P = ε0χ E Displaced electronic cloud feels a restoring force, which is linear (for small displacements) Polarization G |E| time Electric field For simplicity consider one-dimensional case (∆r parallel to x) G |P| ∆r e G E + p G G G P = ε 0 χ E + PNL Polarization time G |E| 5 6 Microscopic understanding of nonlinearity Now, have electromagnetic wave with field E(t)=E0e-iωt Electronic contribution to susceptibility (nonlinear response) Force F(t)=eE0e-iωt Equation of motion becomes (forced oscillator) eEo e −iωt − mω02 x = m d 2x dt 2 and get Look for a solution x(t)=x0e-iωt em em x(t ) = 2 E e −iωt = 2 E (t ) 2 o ω0 − ω ω0 − ω 2 Expect strong response (large x), ⇒ large susceptibility χ ⇒ large refractive index n at ω ≈ ω0 Dipole moment p = qx, so polarization P = eNZx (N atoms per unit volume, Z electrons per atom) ⇒ P= e 2 ZN m ω02 − ω 2 E Electron is moving in an asymmetric potential with damping NZe 2 1 Recall P = ε0χE and get χ (ω ) = 2 ε 0 m (ω07 − ω 2 ) Linear response: ∂ 2 x(t ) ∂x(t ) e +γ + ω02 x(t ) + Dx 2 (t ) = Eo e − iωt 2 ∂t ∂t 2m x – deviation from potential minimum mDx 2 (t ) Anharmonic restoring force Linear response: ∂x(t ) Damping γ ∂t x(ω , t ) = q1e − iωt + q2 e − i 2ωt Solution: ( q1 = q2 = eE0 1 ⋅ m ω02 − ω 2 + iγ ⋅ ω − De E 2 2 2 0 2m ⎡⎣ω − ω + iγ ⋅ ω ⎤⎦ ⎡⎣ω02 − (2ω ) 2 + iγ ⋅ 2ω ⎤⎦ 2 2 0 2 ) NonLinear response: (second harmonic 8 generation) Why nonlinear effects are usually weaker than linear ones? For NL polarization at the second harmonic frequency: (2) P 2ω = Neq2 e− i 2ωt = χ NL ⋅ E02 e − i 2ωt V ( x) For correct power consideration we need to take the complex conjugate part of the electromagnetic wave P 2ω = r0 x 1 1 Neq2 (e− i 2ωt + e+ i 2ωt ) = d NL ⋅ E02 (e − i 2ωt + e+ i 2ωt ) 2 2 x << r0 For nonlinear susceptibility we have: mD ( χ L (ω ) ) ⋅ χ L (2ω ) ⋅ ε 03 = 2 N 2 | e3 | 2 χ (2) NL −3e2 ; r0 ≈ 0.5 nm D≈ ε 0 mr04 m 2 m 3 e 2 −5.83 x2 x3 V ( x) = x + Dx = ( + 24.1 2 − 13.3 3 + ...) 2 3 4πε 0 r0 r0 r0 Why nonlinear effects are weaker than linear effects? 9 Symmetry of nonlinear susceptibility tensor G G G | PNL |= χˆ (2) ⋅ | E1 ⋅ E2 | G ( PNL )i = 2dijk E j Ek Two-wave mixing General case of two-wave mixing: d13 d14 d15 d 23 d 24 d 25 d33 d34 d35 ⎡E ⎤ ⎢ 2 ⎥ ⎢ Ey ⎥ d16 ⎤ ⎢ 2 ⎥ Ez ⎥ ⎢ ⎥ d 26 ⎥ ⋅ ⎢ 2 Ez E y ⎥ d36 ⎥⎦ ⎢ ⎥ ⎢ 2 Ez Ex ⎥ ⎢ ⎥ ⎣ 2 Ex E y ⎦ Wave propagation along z; i, j, k index are permutations of x and y coordinates dE1i σ µ µ =− 1 E1i − iω1 d ' E E * e − i ( k3 − k2 − k1 ) z 2 ε1 ε1 ijk 3 j 2 k dz For cubic, tetragonal, and orthorhombic crystals: ⎡0 0 ⎢ ⎢0 0 ⎢⎣0 0 0 ⎤ ⎡0 0 0 d 25 0 ⎥⎥ = ⎢⎢0 0 0 0 d36 ⎥⎦ ⎢⎣0 0 0 ⎤ 0 d ' 0 ⎥⎥ 0 0 d "⎥⎦ 0 d14 0 0 d' 0 0 0 0 0 ω1 + ω2 = ω3 (ω3 = ω1 + ω2 ) G G G k1 + k2 = k3 G G G ∂2E ∂E 2 ∇ E = εµ 2 + σµ Wave equation for linear process ∂t ∂t G Using Laplace operator: G G 2 G ∂ 2 PNL ∂ E ∂E 2 Wave equation for nonlinear ∇ E = εµ 2 + σµ +µ t t ∂ ∂ ∂t 2 process: 2 x ⎡ Px ⎤ ⎡ d11 d12 ⎢ ⎥ ⎢ ⎢ Py ⎥ = ⎢ d 21 d 22 ⎢ P ⎥ ⎢⎣ d31 d32 ⎣ z⎦ 10 dE *2 k σ =− 2 2 dz dE3 j 11 dz =− σ3 2 µ * µ E − iω2 d ' E E * e−i ( − k + k ε 2 2k ε 2 ijk 1i 3 j 3 µ µ E − iω3 d ' E E e−i (− k +k ε3 3 j ε 3 ijk 1i 2 k 3 2 + k1 ) z 2 + k1 ) z 12 Second-harmonic generation SHG: ω + ω = 2ω (ω3 = 2ω1 ) G G 2kω = k2ω Second-harmonic generation dE1i Small loss of power in the primary beam ≈0 dz dE3 j µ = −i 2ω d ' E E e − i∆kz dz ε ijk 1i 1k e − i∆kz − 1 µ d 'ijk E1i E1k i ∆k ε E3 j ( z ) = −i 2ω Power (2ω ) = E3 j ( L ) E *3 j ( L) = 4 Coherence length: l= Phase matching requirement: ⎡ε xx ε xy ε xz ⎤ ⎡ n 2 e 0 0 ⎤ ⎥ ⎢ ⎥ ⎢ ε = ⎢ε yx ε yy ε yz ⎥ = ⎢0 n 2O 0 ⎥ ⎢ε ε ε ⎥ ⎢0 0 n 2 ⎥ O⎦ ⎣ zx zy zz ⎦ ⎣ G G 2kω = k2ω ⎞ ε NZe 2 ⎛ 1 = Re 1 + ⎜ ⎟ ε0 ε 0 m ⎝ ω02 − ω 2 + iωγ ⎠ Strong absorption nO (ω ) ne (2ω ) G ω 2ω 2⋅ | kω |= 2n (ω ) = n (ω ) c c G 2ω | k2ω |= n (2ω ) c n(ω ) ≠ n (2ω ) ! ⇒ G G 2kω = k2ω ω 2ω ω0 ω Blue waves propagate with the same velocity in the crystal 13 Experimental setup for Second-harmonic generation G G 2kω = k2ω ne (2ω ) = nO (ω ) n (ω ) ⇒ 2 If in birefringent crystal ne (2ω ) = nO (ω ) ω + ω = 2ω nR (ω ) = Re Transparent crystal µ 2 sin ( ∆kL / 2) ω ( d 'ijk ) 2 E 21i E12k ε3 (∆kL / 2) 2 2π ∆k If in birefringent crystal ne (2ω ) = nO (ω ) KH 2 PO4 KDP crystal 14 Applications of Second-harmonic generation Lasers (Nd:YAG, second harmonic) Coherent anti-Stokes Raman scattering Bio-imaging Materials Physics Fine tuning of refractive index for Phase matching in uniaxial Nonlinear crystals: Solar Physics Quantum cryptography (two-wave mixing) 1 cos θ sin θ = + 2 2 n(θ ) nO 2 ne ⎡n2e 0 0 ⎤ ⎥ ⎢ ε = ⎢0 n 2O 0 ⎥ ⎢0 0 n 2 ⎥ O⎦ ⎣ 15 16 Brillouin and Raman spectroscopy Raman scattering in crystalline solids Inelastic light scattering mediated by the electronic polarizability of the medium • a material or a molecule scatters irradiant light from a source • Most of the scattered light is at the same wavelength as the laser source (elastic, or Raileigh scattering) • but a small amount of light is scattered at different wavelengths (inelastic, or Raman scattering) α β ћωi β α ћΩ ћωs Stokes ћωi 0 0 Raileigh ћΩ I =ωi = =ω s ± =Ω; λi λi ~ 5000 Å, a0 ~ 4-5 Å ⇒ λphonon >> a0 ω Stokes Raman Scattering ωi- Ω(q) 1. Energy conservation: 2. Momentum conservation: 4πn ki = k s ± q ⇒ 0 ≤ q ≤ 2 k ⇒ 0 ≤ q ≤ Elastic (Raileigh) Scattering ћωs AntiStokes Not every crystal lattice vibration can be probed by Raman scattering. There are certain Selection rules: ωi Anti-Stokes ks q≈0 ki ks q ≈ 2k ki ks q ki ⇒ only small wavevector (cloze to BZ center) phonons are seen in the 1st order (single phonon) Raman spectra of bulk crystals Raman Scattering ωi+ Ω(q) Analysis of scattered light energy, polarization, relative intensity provides information on lattice vibrations or other excitations17 3. Selection rules determined by crystal symmetry 18 Example of Raman scattering in crystalline solids Raman scattering in crystalline solids Phonon Energy Raman scattering G G G G q = ±∆k = ± | ki − ks | 3S = 15 modes 3 acoustic modes 12 optical modes; 3 × 4 Mandelstam-Brillouin scattering Phonon wavevector 19 2 × TO1 + LO1 2 × TO2 + LO2 2 × TO3 + LO3 2 × TO4 + LO4 20