Objectves: 1. Develop a smple conceptual 1D model for NLO effects. Extend to 3D and relate to computatonal chemcal calculatons of adabatc NLO polarzabltes. 2. Introduce Sum-Over-States (SOS) approaches derved from tme-dependent perturbaton theory for resonant NLO polarzabltes. 3. Reduce the general SOS forms down to smple, ntutve expressons. 4. Introduce three dfferent coordnatendependent vsualzaton approaches for representng the molecular tensor (sagttary and vector-sphere), and relate them back to the tabulated tensor elements. 5. Introducton to smplfcatons from molecular symmetry. Department of Chemstry Purdue Unversty 1
I. Classcal 1D Model for Nonlnear Optcs The lnear polarzablty s gven by the curvature of the potental energy surface descrbng the nduced dpole vs. appled feld 2
I. Classcal 1D Model for Nonlnear Optcs In lnear nteractons descrbng reflecton, refracton, and lght-scatterng, the drvng feld nduces a polarzaton at the same freuency. Tme 2 H α = 2 E The lnear polarzablty s gven by the curvature of the potental energy surface descrbng the nduced dpole vs. appled feld. 3
I. Classcal 1D Model for Nonlnear Optcs When the feld strength s ncreased, addtonal anharmonc contrbutons become sgnfcant, resultng n dstortons n the nduced polarzaton. Tme 3 H β = 3 E The adabatc hyperpolarzablty s gven by the asymmetrc anharmoncty n the PES. 4
I. Classcal 1D Model for Nonlnear Optcs These dstortons n the tme-doman are recovered n the freuency-doman by contrbutons at the hgher harmoncs. Tme 5
I. Classcal 1D Model for Nonlnear Optcs The net polarzaton at the second harmonc freuency arses from the coherent nterference of all oscllators. Tme 6
I. Classcal 1D Model for Nonlnear Optcs When the feld strength s ncreased, addtonal anharmonc contrbutons become sgnfcant, resultng n dstortons n the nduced polarzaton. Tme 4 H γ = 4 E The adabatc second hyperpolarzablty s gven by the symmetrc dstorton away from a uadratc PES. 7
Extenson of the adabatc polarzablty to 3D α j 2 H = E E j β 3 H = E E E j k γ l 4 H = E E E E j k l 8
The adabatc approxmaton mplctly assumes that all resonances are suffcently hgh n energy that the lneshape functons S are constant, such that the energy weghtng s smlar for generatng each vrtual state. As such, the adabatc polarzablty clearly cannot be used to descrbe spectroscopc measurements and resonanceenhanced measurements. Even n nonresonant SHG, the energy gap between the two vrtual states s 1%. n 9
Although potentally useful for ualtatvely descrbng nonresonant SHG (e.g., n collagen) and the background n CARS, the adabatc nonlnear polarzablty cannot be relably used to descrbe the most nterestng measurements, performed near a vbratonal or electronc resonance. Alternatve formalsm: Tme-dependent perturbaton theory. Fnal result: ( 1) β ( 2 ω; ωω, ) = Sm( 2ω ) µ m αm Electronc SHG ( ;, ) S ( ) m ( 1) β ω ω ω = ω α µ 3 1 2 n 2 n n n Two-photon absorpton matrx Transton moment Raman polarzablty matrx Complex-valued lneshape functon Vbratonal SFG 1
From the top: The near-resonant polarzablty descrbes a two-wave process ncludng both exctaton and coherent emsson. n (Near resonance) α j j j n n n n ( ωω ; ) = + j µ nµ n + ω ω Γ n n µ µ µ µ ω ω Γ ω + ω+ Γ n n n n n NR 11
For systems ntally n the ground state, only the frst (red) term contrbutes sgnfcantly. α j j j j n n n n ( ωω ; ) = + n µ µ µ µ ω ω Γ ω + ω+ Γ n n n n n n In blue, the ω term comes frst n the tmeorderng, correspondng to enhancement wth a lower lyng state (stmulated emsson). * *Note the sgn conventon on the dampng term +Γ, whch must be postve for stmulated emsson to arse. 12
β j k jk j k j k µ α α µ NR ω 2 ω Γ = ω ω Γ + ( n n 2 ωωω ;, ) = + NR ( ) n n β ω ;, sum ωvs ωr n n n In SHG, the hyperpolarzablty s domnated by the one-photon transton moment µ n and the two-photon absorpton matrx α n. End results: n n r n In SFG, the hyperpolarzablty s domnated by the Raman polarzablty matrx α n and the one-photon nfrared transton moment µ n. How do we get ths outcome from tme-dependent perturbaton theory? We start wth 3! = 6 dfferent pathways to consder. 13
Next, we pck the two pathways relevant to the resonance of nterest. For vbratonal SFG (one-photon resonance wth ω 2 ): The two terms from the sx possble that descrbe ths nteracton: j k j k 1 µ rµ rpµ p µ rµ rpµ p β ( ωsum; ωvs, ωr ) = NR 2 + + 2 jk p ( ω ω Γ )( ω ω Γ ) ( ω ω Γ )( ω + ω1+γ ) p r p r p r sum r p r p r r j j k 1 1 µ µ µ µ µ = + + NR 2 p r ( ωr ωsum Γ r ) ( ωr + ω1+γ r ) ( ωp ωr Γ p) 1 = 2 α k µ p p ( ωp ωr Γ p) r rp r rp p + NR 14
The answer: Contracton of the SOS expressons! Group the pathways nto pars sharng the desred resonance enhancement wth ω 2. ω 1 -ω sum ω 2 + ω 2 -ω sum ω 1 β ( ω ; ω, ω ) 1 µ µ µ µ µ µ j k j k m mn n m mn n 3 1 2 = 2 + 2 n m n 2 n m 3 m n 2 n m 2 m ( ω ω Γ )( ω ω Γ ) ( ω ω Γ )( ω + ω +Γ ) Close to vb. resonance, the molecular tensor s gven the by the drect product of the ant-stokes Raman tensor and the onephoton transton moment. β ( ω ; ω, ω ) k j j n m mn m mn 3 1 2 = + 2 n n 2 n m 3 m m 2 m β 1 µ 1 µ µ µ µ ( ω ω Γ ) m ( ω ω Γ ) ( ω + ω +Γ ) = + NR j k 1 α µ ( ω ; ω, ω ) n n 3 1 2 2 n ωn ω2 But ths s just the SOS expresson for ant-stokes Raman! 15
γ l ( ω ; ω, ω, ω ) 4 1 2 3 kl jl ( ) µ p ( βp) ( βp) ( βp) lkj j k l 1 β p µ 3 p µ p µ PA AHR AHR AHR p = + + + * 4 p ωpi + ω4 +Γ p ωpi ω1 Γ p ωpi ω2 Γ p ωpi ω3 Γ p 1 = 4 The CATS sgnal ω -ω responsble for the 3 2 nonresonant bkg n CARS s domnated by two-photon ω 1 resonances. l γ ( ω ; ω, ω, ω ) l jk k jl j ( ) ( ) ( ) ( ) ( ) ( ) kl α α α α α α AR 2PA AR 2PA AR + + 2 PA ωi ω1 ω2 Γ ωi ω1 ω3 Γ ωi ω2 ω3 Γ kj l lj k lk j ( α) ( α ) ( ) ( ) ( ) ( ) 2 α PA AR α 2PA α AR α 2PA AR + + + * * * ω + ω1+ ω2 +Γ ω + ω1+ ω3+γ ω + ω2 + ω3+γ I I I jkl lk lj ( βr ) ( β ) ( ) 3 r βr * * 4 1 2 ββ 2 ( ) j k l kj 1 µ r µ PA r µ SHR r µ SHR r β r SHR = + + + * 4 r ωri ω Γ r ωri + ω + Γ r ωri + ω +Γ r ωri + ω3 +Γ r sum -ω ω 2 3 -ω sum ω1 -ω sum 1 2 3 l jk ( α) ( α) 2PE 2PA + k jl j kl ( α) ( α) + ( α) ( α) 1 4 ω ( ω + ω ) Γ ω ω + ω Γ AR SR AR SR I 1 2 I 1 3 The CARS sgnal s domnated by the drect product of two coherent Raman transtons. ETC. 16
γ l ( ω ; ω, ω, ω ) 4 1 2 3 kl jl ( ) µ p ( βp) ( βp) ( βp) lkj j k l 1 β p µ 3 p µ p µ PA AHR AHR AHR p = + + + * 4 p ωp + ω4 +Γ p ωp ω1 Γ p ωp ω2 Γ p ωp ω3 Γ p 1 = 4 The CATS sgnal ω -ω responsble for the 3 1 nonresonant bkg n CARS s domnated by two-photon ω 1 resonances. l γ ( ω ; ω, ω, ω ) l jk k jl j kl ( α) ( α) ( α) ( α) ( α) ( α) AR 2PA AR 2PA AR 2PA + + ω ω1 ω2 Γ ω ω1 ω3 Γ ω ω2 ω3 Γ kj l lj k lk j α α 2 α α PA AR 2PA α AR α 2PA AR + + + * * * ω + ω1+ ω2 +Γ ω + ω1+ ω3+γ ω + ω2 + ω3 +Γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 µ µ µ µ = + + + 4 ( ) jkl j lk k lj l kj r βr 3 r βr r βr r β PA SHR SHR r SHR * * * r ωr ω4 Γ r ωr + ω1+γ r ωr + ω2 +Γ r ωr + ω3 +Γ r sum -ω ω 1 3 -ω sum ω1 -ω sum 1 1 3 l jk k jl j kl ( α) ( α) ( α) ( α) + ( α) ( α) 1 1 4 ω 2ω Γ 2! ω ω + ω Γ 2PE 2PA + AR SR AR SR I 1 I 1 3 The CARS sgnal s domnated by the drect product of two coherent Raman transtons. ETC. 17
1. Tme-dependent perturbaton theory provdes a framework for recoverng both the resonant and nonresonant molecular response. 2. Contracton allows re-expresson wthout addtonal assumpton or approxmaton. 3. Smple nspecton of the energy level dagram allows dentfcaton of the contracted form that s consstent wth the partcular resonant NLO process probed, wth the remanng contrbutons contaned wthn a nonresonant background term. References: -Moad, A. J.; Smpson, G. J. A Unfed Treatment of Selecton Rules and Symmetry Relatons for Sum-Freuency and Second Harmonc Spectroscopes J. Phys. Chem. B. 24, 18, 3548-3562. -Moad, A. J.; Smpson, G. J. Self-Consstent Approach for Smplfyng the Molecular Interpretaton of Nonlnear Optcal and Mult-Photon Phenomena J. Phys. Chem. A 25, 19, 1316-1323. -Davs, R. P.; Moad, A. J.; Goeken, G. S.; Wampler, R. D.; Smpson, G. J. Selecton Rules and Symmetry Relatons for Four-Wave Mxng measurements of Unaxal Assembles J. Phys. Chem. B 28, 112, 5834-5848. 18