Phys 774: Nonlinear Spectroscopy: SHG and Raman Scattering

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1 Last Lcturs: Polaraton of Elctromagntc Wavs Phys 774: Nonlnar Spctroscopy: SHG and Scattrng Gnral consdraton of polaraton Jons Formalsm How Polarrs work Mullr matrcs Stoks paramtrs Poncar sphr Fall 7 Polaraton s mportant to undrstand nonlnar ffcts! Broad fld of nonlnar ffcts W wll consdr n dtals only SHG and Scattrng Lnar spctroscopy: Nonlnar ffcts: Lnar vs. Nonlnar Spctroscopy D E ( + χ) E E+ P D y E D E y y yy y y D y E P χe P χ E+ P () P χ E ( P ) d E E jk j k P P P χe E P χe + P E 3 Inducd polaraton vs. lctrc fld n lnar dlctrc and n a crystal wthout cntr of nvrson, whr lctrons mov n 4 asymmtrc potntal.

2 In vacuum In a matrals mda Elctrc fld and Polaraton Ert (,) Ep[( kr t)] tm Elctrc fld Polaraton P P χe E y y Nonlnar polaraton n crystals wth th cntr of nvrson Iˆ ( P ) ( P ) Iˆ ( P ) djk ( Ej ) ( Ek ) P () P or ˆ χ d jk In gnral stuaton: Elctrc fld Polaraton tm P P χe + P E P χ E + d E E + 4χ E E E + (3) j j jk j k jkl j k l In cntr-symmtrc matrals: P χ E + d E E + 4χ E E E + (3) j j jk j k jkl j k l 5 6 Mcroscopc undrstandng of nonlnarty Now, hav lctromagntc wav wth fld E(t)E -t Elctronc contrbuton to suscptblty (lnar rspons) For smplcty consdr on-dmnsonal cas ( r paralll to ) Dsplacd lctronc cloud fls a rstorng forc, whch s lnar (for small dsplacmnts) r E - + p Forc F(t)E -t Equaton of moton bcoms (forcd oscllator) Look for a soluton (t) -t ( t) m E t o m E t o and gt m E( t) d m dt Epct strong rspons (larg ), larg suscptblty χ larg rfractv nd n at Dpol momnt p q, so polaraton P NZ (N atoms pr unt volum, Z lctrons pr atom) 7 ZN m P E Rcall P χe and gt Lnar rspons: χ ( ) NZ m ( ) 8

3 Mcroscopc undrstandng of nonlnarty Elctronc contrbuton to suscptblty (nonlnar rspons) Elctron s movng n an asymmtrc potntal wth dampng t () t () t D t E + γ + () + () t t m dvaton from potntal mnmum md () t t () γ t q Anharmonc rstorng forc Dampng E q + γ m D E m + γ ( ) + γ t o Lnar rspons: t t Soluton: (, t) ( q + q ) NonLnar rspons: (scond harmonc 9 gnraton) For polaraton at th scond harmonc frquncy: P Nq χ E t () t For corrct powr consdraton w nd to tak th compl conjugat part of th lctromagntc wav t t t t P Nq( + ) d E( ) For nonlnar suscptblty w hav: χ () md ( ) 3 χ ( ) χ ( ) L L 3 N 3 D ; r.5 nm mr 4 Why nonlnar ffcts ar wakr than lnar ffcts? V ( ) Why nonlnar ffcts ar usually wakr than lnar ons? r << r m m 5.83 V D 3 3 ( ) + ( ) 3 3 4π r r r Symmtry of nonlnar suscptblty tnsor () P ˆ χ E E ( P ) d E E jk j k E Ey P d d d3 d4 d5 d6 E Py d d d3 d4 d5 d 6 EE y P d 3 d3 d33 d34 d35 d 36 EE EE y For cubc, ttragonal, and orthorhombc crystals: d4 d ' d d' d 36 d " 5

4 Two-wav mng Gnral cas of two-wav mng: + 3 ( 3 + ) k+ k k3 E E Wav quaton for lnar procss: E µ + σµ t t E E Wav quaton for nonlnar E µ σµ µ + + procss: t t t Wav propagaton along ;, j, k nd ar prmutatons of and y coordnats σ µ µ E d' jk E3jEk d k σ µ µ E k d' jk E E3j d σ µ µ E d' E E d ( k3 k k) ( k3+ k+ k) 3 j 3 ( k3+ k+ k) 3j 3 jk k 3 3 P 3 Scond-harmonc gnraton SHG: + ( 3 ) k k Phas matchng rqurmnt: y n y yy y n y n If n brfrngnt crystal n ( ) n ( ) k k d 3 j d µ d' jk E Ek µ E ( ) d' E E 3j jk k µ sn ( kl / ) 3j 3j jk k 3 ( kl /) Powr( ) E ( L) E ( L) 4 ( d ' ) E E Cohrnc lngth: π l k k n( ) n( ) c c k n ( ) c n( ) n( )! Small loss of powr n th prmary bam k k k 4 Scond-harmonc gnraton If n brfrngnt crystal n ( ) n ( ) k k n( ) Transparnt crystal n ( ) n ( ) NZ nr ( ) R R + m + γ Strong absorpton Blu wavs propagat wth th sam vlocty n th crystal 5 Eprmntal stup for Scond-harmonc gnraton + KHP4 k k KDP crystal n ( ) n ( ) Fn tunng of rfractv nd for Phas matchng n unaal Nonlnar crystals: cosθ snθ + n( θ ) n n n n n 6

5 Applcatons of Scond-harmonc gnraton Lasrs (Nd:YAG, scond harmonc) Cohrnt ant-stoks scattrng Bo-magng Matrals Physcs Solar Physcs Quantum cryptography (two-wav mng) 7 Brlloun and spctroscopy Inlastc lght scattrng mdatd by th lctronc polarablty of th mdum a matral or a molcul scattrs rradant lght from a sourc Most of th scattrd lght s at th sam wavlngth as th lasr sourc (lastc, or Ralgh scattrng) but a small amount of lght s scattrd at dffrnt wavlngths (nlastc, or scattrng) α β ћ ћω ћ s Stoks Ralgh β α ћ ћω ћ s Ant- Stoks I Stoks Scattrng - Ω(q) Elastc (Ralgh) Scattrng Ant-Stoks Scattrng + Ω(q) Analyss of scattrd lght nrgy, polaraton, rlatv ntnsty 8 provds nformaton on lattc vbratons or othr ctatons scattrng n crystalln solds Not vry crystal lattc vbraton can b probd by scattrng. Thr ar crtan Slcton ruls:. Enrgy consrvaton: ± Ω; s scattrng n crystalln solds scattrng. Momntum consrvaton: 4πn k ± q q k q λ λ ~ 5 Å, a ~ 4-5 Å λ phonon >> a only small wavvctor (clo to BZ cntr) phonons ar sn n th st ordr (sngl phonon) spctra of bulk crystals q k q k k k q Phonon Enrgy q ± k ± k k s 3. Slcton ruls dtrmnd by crystal symmtry 9 Mandlstam-Brlloun scattrng Phonon wavvctor