Phys 402: Nonlinear Spectroscopy: SHG and Raman Scattering

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1 Rquirmnts: Polariation of Elctromagntic Wavs Phys : Nonlinar Spctroscopy: SHG and Scattring Gnral considration of polariation How Polarirs work Rprsntation of Polariation: Jons Formalism Polariation of light and matrials proprtis ar important to undrstand nonlinar ffcts! Spring 8 Andri Sirnko, NJIT Lctur 1 Broad fild of nonlinar ffcts W will considr in dtails only SHG and Scattring Linar spctroscopy: Nonlinar ffcts: Linar vs. Nonlinar Spctroscopy D E (1 + χ) E E+ P D y E D E y y yy y y D y E P χe P χ E+ PNL () PNL χ E ( P ) d E E NL i j k P P P χe E P χe + P NL E 3 Inducd polariation vs. lctric fild in linar dilctric and in a crystal without cntr of invrsion, whr lctrons mov in asymmtric potntial.

2 In vacuum In a matrials mdia Elctric fild and Polariation Ert (,) Ep[( ikr t)] tim Elctric fild Polariation P P χe E Microscopic undrstanding of nonlinarity Elctronic contribution to suscptibility (linar rspons) For simplicity considr on-dimnsional cas ( r paralll to ) Displacd lctronic cloud fls a rstoring forc, which is linar (for small displacmnts) r E - + p Elctric fild Polariation P P χe + P NL tim E 5 6 Now, hav lctromagntic wav with fild E(t)E -it Forc F(t)E -it Equation of motion bcoms (forcd oscillator) Look for a solution (t) -it ( t) m E it o m E it o and gt m E( t) d m dt Epct strong rspons (larg ), larg suscptibility χ larg rfractiv ind n at Dipol momnt p q, so polariation P NZ (N atoms pr unit volum, Z lctrons pr atom) ZN m P E Rcall P χe and gt Linar rspons: NZ 1 χ ( ) m ( ) 7 Microscopic undrstanding of nonlinarity Elctronic contribution to suscptibility (nonlinar rspons) Elctron is moving in an asymmtric potntial with damping t () t () t D t E + γ + () + () t t m dviation from potntial minimum md () t t () γ t q Anharmonic rstoring forc Damping E 1 q + iγ 1 m D E m + iγ ( ) + iγ it o Linar rspons: it it Solution: (, t) ( q 1 + q ) NonLinar rspons: (scond harmonic 8 gnration)

3 For NL polariation at th scond harmonic frquncy: P Nq χ E i t () it NL For corrct powr considration w nd to tak th compl conjugat part of th lctromagntic wav 1 i t i t 1 i t i t P Nq( + ) dnl E( ) For nonlinar suscptibility w hav: χ () NL md ( ) 3 χ ( ) χ ( ) L L 3 N 3 D ; r.5 nm mr V ( ) Why nonlinar ffcts ar usually wakr than linar ons? r << r m m 5.83 V D 3 3 ( ) + ( ) 3 3 π r r r Why nonlinar ffcts ar wakr than linar ffcts? 9 1 Symmtry of nonlinar suscptibility tnsor () P ˆ NL χ E1 E ( P ) d E E NL i j k E Ey P d11 d1 d13 d1 d15 d16 E Py d1 d d3 d d5 d 6 EE y P d 31 d3 d33 d3 d35 d 36 EE EE y For cubic, ttragonal, and orthorhombic crystals: d1 d ' d5 d' d 36 d " 11 Two-wav miing Gnral cas of two-wav miing: 1+ 3 ( 3 1+ ) k1+ k k3 E E Wav quation for linar procss E µ + σµ t t Using Laplac oprator: E E Wav quation for nonlinar E µ σµ µ + + procss: t t t Wav propagation along ; i, j, nd ar prmutations of and y coordinats 1 i σ 1 µ µ E1 i i1 d' E3jEk d 1 1 k σ µ µ E k i d' E1 ie3j d σ µ µ E i d' E E d i( k3 k k1) i( k3+ k+ k1) 3 j 3 i( k3+ k+ k1) 3j 3 1i k 3 3 P NL 1

4 Scond-harmonic gnration SHG: + ( 3 1) k k Phas matching rquirmnt: y n y yy y n y n If in birfringnt crystal n ( ) n ( ) k k d 1i 3 j d µ i d' E1 ie1k µ E ( ) i d' E E 3j 1i 1k µ sin ( kl / ) 3j 3j 1i 1k 3 ( kl /) Powr( ) E ( L) E ( L) ( d ' ) E E Cohrnc lngth: π l k k n( ) n( ) c c k n ( ) c n( ) n( )! Small loss of powr in th primary bam i k i k 1 i k 13 Scond-harmonic gnration If in birfringnt crystal n ( ) n ( ) k k n( ) Transparnt crystal n ( ) n ( ) NZ 1 nr ( ) R R 1+ m + iγ Strong absorption Blu wavs propagat with th sam vlocity in th crystal 1 Eprimntal stup for Scond-harmonic gnration + KHP k k KDP crystal n ( ) n ( ) Fin tuning of rfractiv ind for Phas matching in uniaial Nonlinar crystals: 1 cosθ sinθ + n( θ ) n n Applications of Scond-harmonic gnration Lasrs (Nd:YAG, scond harmonic) Cohrnt anti-stoks scattring Bio-imaging Matrials Physics Solar Physics Quantum cryptography (two-wav miing) n n n 15 16

5 Brillouin and spctroscopy Inlastic light scattring mdiatd by th lctronic polariability of th mdium a matrial or a molcul scattrs irradiant light from a sourc Most of th scattrd light is at th sam wavlngth as th lasr sourc (lastic, or Railigh scattring) but a small amount of light is scattrd at diffrnt wavlngths (inlastic, or scattring) α β ћ i ћω ћ s Stoks Railigh β α ћ i ћω ћ s Anti- Stoks I Stoks Scattring i - Ω(q) i Elastic (Railigh) Scattring Anti-Stoks Scattring i + Ω(q) Analysis of scattrd light nrgy, polariation, rlativ intnsity 17 provids information on lattic vibrations or othr citations scattring in crystallin solids Not vry crystal lattic vibration can b probd by scattring. Thr ar crtain Slction ruls: 1. Enrgy consrvation: i ± Ω; s. Momntum consrvation: πn ± q q k q λ λ i ~ 5 Å, a ~ -5 Å λ phonon >> a only small wavvctor (clo to BZ cntr) phonons ar sn in th 1 st ordr (singl phonon) spctra of bulk crystals 3. Slction ruls dtrmind by crystal symmtry i q q k ki 18 q scattring in crystallin solids Eampl of scattring in crystallin solids scattring Phonon Enrgy q ± k ± k k i s 3S 15 mods 3 acoustic mods 1 optical mods; 3 T + L 1 1 T + L T + L 3 3 T + L Mandlstam-Brillouin scattring Phonon wavvctor 19