Nonlinear Optics. Office: Tien s Photonic Research Hall 412. Tel : E

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1 Nonlnear Optcs Offce: Ten s hotonc Research Hall 1 Tel : mal:jyhuang@facultynctuedutw Ths course s amed to help students masterng the prncples and techncal materals of a graduate-level Nonlnear Optcs The followng topcs are covered: Macroscopc Theory of Optcal Susceptblty Tensors, Second-Order Nonlnear Optcal ffects, Thrd-Order Nonlnear Optcal ffects, and Tme-Resolved Nonlnear Optcal Spectroscopes, etc It s requred the audence of ths course to possess a background of Appled Mathematcs and lectromagnetsm Lecture notes wll be posted whenever possble Not all lectures are avalable for dstrbuton Textbook and References 1 Y R Shen, The rncples of Nonlnear Optcs, Wley-Interscence, 198 N Bu tcher & D Cotter, The lement of Nonlnear Optcs, Cambrdge Unversty ress, Robert Boyd, Nonlnear Optcs, Academc ress, Inc, 199

2 Course Content 1 Introducton 11 Hstorcal Background 1 Orgns of Optcal Nonlnearty 13 NLO rocesses Macroscopc Theory of Optcal Susceptbltes 1 Response Functon and Consttutve Relatons Optcal Susceptblty 3 Spatal Symmetry of hyscal Systems Symmetry ropertes of Optcal Susceptblty 5 Resonant Nonlnearty 3 General Descrpton of Wave ropagaton n Nonlnear Meda 31 Wave ropagaton n the Lnear Regme 3 Wave ropagaton n a Nonlnear Medum 33 Slowly Varyng Ampltude Approxmaton (SVA and Coupled Wave quatons 3 The Relatonshp between Macroscopc and Local-Feld Quanttes lectro-optcal and Magneto-Optcal ffects 1 Lnear Optcal ffect and ts Applcatons Quadratc O ffect 3 hyscal ropertes of O Coeffcents O ffect n Lqud Crystals 5 lectro-optc Devces 6 Magneto-Optcal ffect and Faraday Rotator 5 Second-Order NLO ffects 51 Crystal Optcs 5 hase Matchng Schemes 53 ffectve Nonlnearty 5 SFG wth Boundary Reflecton 55 SFG n Bulk wth Depleted ump Beams 56 Specal Cases 57 Summares of Nonlnear Frequency-Converson rocesses and Ther Lmtatons 6 Stmulated Lght Scatterng 61 Dagrammatc Densty Matrx Analyss Method 6 Stmulated Raman Scatterng (SRS 63 Coupled-Wave Descrpton of SRS 6 arametrc Couplng of Optcal and Materal xctatonal Waves

3 65 Transent SRS and Measurements of xctatonal Relaxaton Tmes 7 Four-Wave Mxng henomena 71 Thrd-Order Nonlnear Susceptbltes 7 Sngly Resonant Cases 73 Doubly Resonant Cases 7 Trply Resonant Cases 75 General Theory of WM 76 DFWM 8 Four-Wave Mxng Spectroscopy 81 CARS, CSRS 8 RIKS 83 Multply Resonant FWM 9 Nonlnear Tme-Resolved Spectroscopy 91 lementary Relaxaton rocesses 9 Tme-Resolved Spectroscopy 93 Local and Nonlocal Tme-Resolved Spectroscopc Technques

4 Chapter 1 11 Hstorcal Background Introducton Durng the pre-laser age, some NLO phenomena and research had been reported These nclude ockels and Kerr O effects, and some predctons by calculatons of two-photon absorpton publshed by Meyer et al 196 Inventon of Ruby laser by Maman Ths nventon provded a lght source wth hgh lumnosty whch had never been produced before 1961 SHG by Franken (see RL 7, 118 (1961 But the converson effcency s as low as 1-1 %! Nowadays typcal SHG has acheved as large as 9%! Ths s because phase matchng scheme was proposed by Gordman and Maker usng brefrngence of a crystal Currently research drecton of NLO can be separated nto: (1 NLO henomena: Investgatng varous NLO processes n meda excted by ultrashort, ultrastrong laser pulses ( Technques: Applyng varous NLO technques for characterzng materals to yeld spectroscopc or dynamcal nformaton of meda Common features n each NLO process (1 An ntense lght feld ( r, t nduced a response n a medum Ths part s usually governed by consttutve equatons local response = ( r r r t ( α α (, = α (, = + ( β (, + β β

5 ( Medum n reactng modfes the optcal felds n a nonlnear way Ths step s governed by Maxwell s quatons for nonmagnetc materals: D = ( + π = + π ( + B = 1 B 1 D = H = c t c t L NL By usng the last two curl equatons, we have 1 π ( ( c t c t L + NL = By combnng the lnear response nto second-order temporal dervatve of the feld, then the equaton becomes π c t c t ε π NL = Ths s vald for non-conductve meda wth = Orgn of Optcal Nonlnearty (Mcroscopc cture of Consttute quaton wth Classcal Approach Consder a delectrc optcal materal consstng of a collecton of charged partcles of electrons and on cores: In a conductor, the charged carrers are free to move n an electrc feld But for a delectrc materal, the charged partcles are bound together Note: ( the moton of the charged carrers s transtory when the feld s frst appled; ( That wll nduce a collecton of nduced electrc dpole moment The nduced polarzaton (r, t s defned to be the nduced dpole moment per unt volume ( Note lght wave oscllates at frequency of Hz (from IR to UV; (v Ion cores are massve compared wth electrons (Ths leads to the well-known Born-Oppenhemer (BO approxmaton Thus an anharmonc oscllator model can be constructed

6 damped harmonc oscllator + nonlnear response of the medum = m[ q ( t + γ q ( t + q( t ( ξ q + ξ q + ] = e( t ( (3 3 Here γ =dampng constant; = resonant frequency; 3eV; and q = dsplacement from the equlbrum poston Now neglect nonlnear response for a moment and consder harmonc response to an appled electrc feld t + t ( t = [ e + e ] t mployng the soluton an salz q( t = Ae + c c m[ A γ A + A] e = e e e q( t = m t t t e ( γ Now consder the polarzaton =polarzaton=dpole moment per unt volume= N e q χ e c c χ (1 t ( = + (1 = lnear susceptblty = Ne m 1 ( γ Note that the electrc dsplacement vector D D = + π = ε (1 ε = 1+ πχ = nr + κ We know that the lnear dependence of on s vald only when d s small, where oscllates at the same frequency as n the lnear approxmaton

7 When q s large, the restorng force of the medum could become sgnfcantly nonlnear When q s large, the response s dstorted and contans sgnfcant components oscllatng at the harmonc frequency, 3, etc, and a dc component ( e 1t 1t t t q ( t + γ q ( t + q( t + ξ q = [ 1 ( e + e + ( e + e ] m No analytc soluton for the above equaton But when technque can be used Let (1 ( q( t q ( t q ( t and note Neq = + + =, we frst lnearze the equaton to yeld q e ( = m (1 t e ( γ (1 (1 (1 1 q ( t = q ( + q ( + c c ( ξ q s small, perturbaton Then approxmate ( ξ q by ξ [ q ] Thus the solutons for ( (1 ( q become q = q ( + q ( + q ( + + q ( + q ( + c c, ( ( ( ( ( ( where q ξ ( e m e ( ( [ ( ( ], ( ξ ( e m 1 ( 1 ± t = e D( D( D( ± ( ( 1 ± t ( 1 ( 1 ± = γ1 1 γ γ 1 ± 1 ± 1 1 and D( = ( Γ

8 q ξ ( e m e ( ( ( ( ( ( = D D D t Therefore, we can express the nonlnear polarzaton as ( Neq ( ( ( χ ( ( χ ( ( ( = second order nonlnear susceptblty = Neξ e 1 ( m D( D( D( ( The resonances of χ ( ( can occur at = and = ( ( oscllates at and wll radate optcal feld wth frequency Several factors can be used to ncrease the response of medum (a hase matchng: The nduced dpoles are forced to radate n phase and therefore wll lead to coherent radaton D( 1 or (b Resonant enhancement: D( (c ( χ depends on ( ξ 1, e, depends on anharmoncty of medum When far below the absorpton edge of the medum, D(, Neξ ( e m D( eξ = D D en e m m ( ( ( 1 (1 ( 1 ( 1 ( Now note that m d e mξ d ( at m eat m d m = ξ ( ξ (

9 Therefore, ( (1 < at In general, ( n+ 1 ( n < at Typcally, for 8 at 3 1 V / cm and 1 3 V / cm for an optcal beam wth I W cm 5 /, ( 7 1 (1 at 13 Nonlnear Optcal rocesses As an example, let s consder NLO processes that modfy the ndex of refracton of medum Assumng ths medum s subject to an ntense electrc feld, and the polarzaton of the medum s weak compared to the bndng forces between the electrons and nucle, one can then express the polarzaton n a power seres of the feld strength = = χ + χ : + χ + (131 ( (1 ( (3 χ χ χ Here ( χ jk s a thrd-rank tensor, denotes the second-order nonlnear optcal susceptblty; (3 χ jkl s a fourth-rank tensor, denotes the thrd-order nonlnear optcal susceptblty, etc Now recast quaton (131 as ( (1 ( (3 = = χ + χ : + χ + = χ eff ( Here χ ( s an effectve optcal susceptblty and wll depend on the feld eff strength Now let = dc + optcal feld of frequency, then ( (1 ( = = χ [ ( + cos( t + kz] + χ [ ( + cos( t + kz] + χ [ ( + cos( t + kz] + (3 3

10 Consder cos( t kz + -term only: = = χ cos( t + kz + χ ( cos( t + kz + 3 χ ( cos( t + kz + ( (1 ( (3 3 (3 3 χ cos( t + kz = χ eff cos( t + kz We can now defne a nonlnear ndex of refracton to be (1 ( (3 3 (3 n = 1 + πχ eff = 1 + π [ χ + χ ( + 3 χ ( + χ The change of the ndex of refracton caused by the external feld becomes n n = ( n n ( n + n = 8 πχ ( + 1 πχ ( + 3 πχ ( (3 (3 πχ 6 πχ 3 πχ n = n ( (3 (3 ( ( n n n (13 We can also defne the lght ntensty n cgs unt as I c ε 8π = and rewrte quaton (13 as πχ 6πχ 1π χ n = n I ( (3 (3 ( ( n n cn = n + n ( + n ( ( + n ( I( 1 (a (b n = πχ n denotes a lnear electro-optcal effect, dscovered by ockels ( 1 n ( = 6 πχ n descrbes quadratc electro-optcal effect, whch s often also (3 called dc Kerr effect (c n ( = 1 π χ ( cn descrbes the optcal Kerr effect, whch s the bass of (3 all hgh-speed all-optcal swtchng 131 Second-Order Nonlnear Optcal ffects: χ ( ;, ( jk 3 1 Ths s a three-wave mxng process where the waves exchange energy wth one another through the nteracton of the nonlnear medum ( Second-Harmonc Generaton (SHG χ ( ( ;, : + jk ( Lnear lectro-optcal ffect (O χ ( ( ;, : + dc : Note the jk

11 change n refractve ndex due to the lnear electro-optcal effect s commonly defned as ( = [ ] [ ] r ( n j ε ( = j ε (, where j jk k r = 8 πχ n ( n cgs s also called O coeffcent jk ( ( Optcal Rectfcaton (OR χ ( ( ;, : : The nteracton of jk an optcal wave at frequency wth tself, generates a dc polarzaton n the medum Recall the Klenman symmetry (to be dscussed n the followng chapter, χ ( ;, = χ ( ;, ndcates the optcal ( ( jk jk rectfcaton and the dc electro-optcal effects are clearly related (v Sum-Frequency Generaton (SFG χ ( ;, : + ( jk (v ( Dfference-Frequency Generaton (DFG χ ( ;, : : jk The optcal parametrc amplfcaton (OA ( χ jk ( ;, : ( pump ( weak sgnal ( amplfed sgnal, ( dler, whch s smlar to DFG 13 Thrd-Order Nonlnear Optcal ffects: χ ( ;,, (3 jkl 1 3 ( Thrd-Harmonc Generaton (THG χ (3 ( 3 ;,, : jkl (3 1 (3 (3 = [ ] χ jkl ( 3 ;,, j( k ( l ( We have to nclude the number of dstnct permutaton possbltes of the ncdent felds, whch s equal to 1 n ths case Note that snce (, t and j j(3 (3, t are 1 ( t k z * ( t k z real, we must let j(, t = [ ][ j( e + j ( e ] and (3 1 (3 t k3 z * (3 t k3 z (3, [ ][ (3 (3 t = e + e ] Therefore, we fnd 1[# dstnct drvng feld permutaton] 1 [ from (3 ] = [ from each ncdent feld]

12 ( Self-Acton These phenomena are the thrd-order NLO processes wth a thrd-order NLO polarzaton 3 ( = [ ] Re[ χ ( ;,, ] ( ( (3 (3 kk k The nonlnear polarzaton s created n the medum at the same frequency as the ncdent beam t t e c c ( t k z (, = ( + = ( t e e φ ( t ( t k z A Self-Focusng/Defocusng These phenomena can be understood by combnng the effect of n ( and the spatal varaton of the laser ntensty: When χ >, n = 1 π χ ( cn > Then n( r = n + ni( r acts lke a (3 (3 focusng lens B Self-hase Modulaton (SM When consder the temporal varaton of the ncdent laser pulse, the temporal behavor of the nduced refractve ndex change can be generated n( t = n n = n I( t Ths can lead to a broadenng of the frequency profle of the laser pulse feld φ( t, z = phase front of the pulse n n + ni( t = t z = t z c c Defnng the nstantaneous frequency as nz di( t ( t d φ ( t dt =, whch shall depend on t c dt If n ( >, the nstantaneous frequency frst experences a frequency upchrpng at the leadng edge of the pulse and then down chrpng at the tralng edge of the pulse as depcted n the followng dagram

13 ( Two-hoton Absorpton (TA Ths NLO phenomenon can be explaned wth a NLO polarzaton nduced n a medum (3 3 (3 ( = [ ] Im[ χ kk ( ;,, ] ( k ( v Degenerate Four-Wave Mxng (DFWM (3 3 (3 ( = [ ] χ jkl ( ;,, j ( k ( l *( Note that: Although (, (, ( have the same frequency, they can have j k l dfferent propagaton drecton and polarzaton The generated beam can also propagate along a dfferent drecton wth dfferent polarzaton v Coherent Raman ffects = =Raman actve mode, we can observe resonant effect Ths s a If 1 = Q useful spectroscopc technque to determnng the structure of a molecular medum

14 Several techncal varatons can be developed from ths concept, whch nclude: v1 Raman Induced Kerr ffect (RIKS ( = [ 3 ] Re[ χ ( ;,, ] ( ( (3 (3 kk k v Coherent Stokes Raman Spectroscopy (CSRS (3 3 (3 * ( s = 1 = [ ] Im[ χ jkl ( s;,, 1] j( k ( l ( 1 where > 1 v3 Coherent Ant-Stokes Raman Spectroscopy (CARS (3 3 (3 * ( A = 1 = [ ] Im[ χ jkl ( A; 1, 1, ] j( 1 k ( 1 l ( where > 1 v lectrc Feld-Induced Second-Harmonc Generaton (FISH When an sotropc medum s based n a dc electrc feld, ts centro-symmetrc structure of electronc dstrbuton can be dstorted, renders t to become non centrosymmetrc structure The materal can therefore generate second harmonc output at when t s excted wth an optcal pulse wth a frequency of The phenomenon, whch s smlar to the optcal second-harmonc generaton from a noncentrosymmetrc crystal, s called lectrc Feld-Induced Second-Harmonc Generaton (FISH and n fact belongs to a thrd-order NLO process, descrbng by the followng NLO polarzaton nduced n the medum (3 3 (3 * ( 3 = 1 = [ ][ χ jkl ( 3; 1, 1, ] j( 1 k ( 1 l ( C Resonant Nonlnearty Under resonant condton (, one can also observe a large dynamc

15 nonlnearty derved from an optcal absorpton process, whch at suffcently hgh ntensty becomes lght-ntensty dependent α ( α ( ( I ncrease α (, I = 1 + ( I I From c d ' α ( ' n( = π ', ( s n s also lght-ntensty dependent In fact, any process whch can lead to an ntensty-dependent absorpton shall gve rse to a dynamc thrd-order optcal nonlnearty Summary of Chapter 1