( ) ( ) Incident light, p-polarization Z. We ll work through the Fresnel factor for the X-polarized incident field first. ω in.

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1 1 3 d Objecives: 1. Define he individual Fresnel correcion facors for inerfacial SFG and SHG.. Define Lorenz local field correcion facors for SHG and SFG microscopy. 3. Inroduce he problem of refracive index uncerainy. 4. Review differences in he lieraure and describe he evidence supporing he Fresnel facors used. 5. Inegrae he Fresnel facors ino he larger vecorized ensor archiecure for polarizaion analysis in linear and nonlinear opics. 1

2 Inciden ligh, p-polarizaion Z in 1 ambien 3 d Y X We ll work hrough he Fresnel facor for he X-polarized inciden field firs. The firs conribuion incorporaes he ampliude ransmission coefficien for raversing he 13 inerface, 13. p13 p13 r p3 p13 r p3 r p31 ec. The nex conribuion arises from reflecion a he 3 inerface, r 3. Noe he sign change in he X-polarized field upon reflecion and he phase shif e -iβ resuling from raversing he hin film. e e X = eˆ X X p13 X β β p13 p 13rp3e + p 13rp3rp31e i3 iβ ( p13rp3rp3rp31e β + p 13 rp3rp3 1e... ( ( 3 β β β 1 + rp3rp31e + rp3rp31e + rp3rp31e... = e ˆ β β β rp3e 1 + rp3rp31e + ( rp3rp31e +... ( 3 e e r e = p13 β ˆ X X 1 β p rp3rp31e d β = π n sinθ λ 3 3 p3 p13 L = β XX r e β rp3rp31e The infinie series can be arranged as a power series, wih he final correcion given by L XX.

3 The oher wo inciden correcion facors are generaed similarly. 1 3 d in s13 s13 r s3 s13 r s3 r s31 ec. p3 p13 β LXX = r e β rp3rp31e s3 s13 L = β YY r e β rs3rs31e + p3 p13 β LZZ = + r e β rp3rp31e In hen hin film limi, he phase erm e -iβ can be approximaed as 1. L L 1 L 0 0 XX = 0 LYY L ZZ 1 1 L 0 0 XX = 0 LYY L ZZ The wo ses of Fresnel facors for he inciden ligh are hen given by he following marices. The wo are idenical for SHG, bu can differ for SFG. 3

4 e sum X β sum ( sum sum sum sum i sum sum β p31 1 rp31 rp3 e rp31 rp3 e sum = PX sum sum sum sum sum β sum sum β sum sum β rp3 p31 e 1 + rp31 rp3 e + ( rp31 rp3 e +... NOTE: Wih his definiion of he Fresnel facor, he inernal angle θ wihin he NLO layer mus be used for he SFG/SHG oupu, and no he angle in he ambien or subsrae! = p31 xx β rp31rp3e p3 L r e β L sum sum L 0 0 XX sum = 0 LYY 0 sum 0 0 L ZZ L 1 L 0 0 XX = 0 LYY L ZZ 1 1 L L 0 0 XX = 0 LYY L ZZ = s31 yy β rs31rs3e + s3 L r e β p31 zz = + β rp31rp3e p3 L r e β sum 1 L=L L L χc, eff = L χc sum e = E χ = E J L χ J C 4

5 5

6 1. n ir, n vis, and n sfg wihin he inerfacial layer are ypically difficul o independenly deermine: (3 unknowns.. For measuremens a surfaces and inerfaces, he inerfaces will generally exhibi birefringence: (6 unknowns. n = n, n x y z ir ir ir x y z vis vis vis x y z sum sum sum n = n, n n = n, n 3. In specroscopic measuremens, he frequencies exhibiing resonance-enhancemen will be complexvalued opical consans: (ypically 8 and up o 1 unknowns. x y z x y z n = n' + ik n ' = n ', n ', k = k, k ir ir ir ir ir ir x y z x y z vis vis vis vis vis vis n ' = n ', n ', k = k, k x y z x y z nsum ' = nsum ', nsum ', ksum = ksum, ksum 6

7 A pah forward: Assume ha he effecive inerfacial refracive index is simply given by he numerical average of he subphase and superphase. Jusificaion: 1. This sraegy has worked splendidly for linear ellipsomery measuremens of rough surfaces for decades.. We are averaging over many molecules in many environmens, spanning subphase-like o superphaselike. 3. Should hold for dilue films wih minimal inerchromophore coupling. 4. I seems o work well experimenally, even for monolayer films of coupled chromophores! 7

8 For vib. SFG from a uniaxial achiral film wih C v symmery: χ zxx, χ zzz, χ xxz χ ppp, χ pss, χ sps, χ ssp => β (, f(θ,ψ For vib. SFG from a uniaxial chiral film wih C symmery: χ zxx, χ zzz, χ xxz, χ yzx χ ppp, χ pps, χ pss, χ spp, χ sps, χ ssp => β (, f(θ,ψ In general, each of hese ensor elemens will be complex-valued, doubling he oal number of observables possible. 8

9 In SHG, fewer parameers are available (C v symmery, achiral: χ zxx, χ zzz, χ xxz χ ppp, χ pss, χ ssp => β (, f(θ,ψ In a chiral film wih C symmery: χ ppp, χ pps, χ pss, χ spp, χ ssp => χ zxx, χ zzz, χ xxz, χ yzx β (, f(θ,ψ Two addiional measured parameers emerge, boh of which depend only on a single ensor elemen. 9

10 Nd:YAG Pol λ/ Nonlinear Film 1064 nm 53 nm Waveplae roaion angle: α H PMT # λ/4 λ/4 λ/ Pol α Q α -45 o H PMT #1 Insead of measuring inensiy, measure he complee polarizaion sae of he exiing beam. (13 Plocinik, R. M.; Simpson, G. J., Anal. Chim. Aca 003, 496, 133. (14 Plocinik, R. M.; Everly, R. M., Simpson, G. J., Phys. Rev. B. 005, 7,

11 The following relaionship is dicaed by symmery: χ XYZ = -χ YXZ χ χ XYZ YXZ = 1+ 0i From he measured values of χ PSP and χ PPS, can we find a se of opical consans ha recovers his required symmery condiion? 11

12 Oucomes: 1. Fresnel facors correc for differences beween he deeced fields and hose experienced and produced wihin he immediae environmen surrounding he molecules.. Ray-racing was used o generae Fresnel facors, which agree well for he inciden beam wih hose consruced based on he soluion o he wave equaion in he hin film limi. 3. Cauion he inernal angles from wihin he hin film should be used for he nonlinear field and exernal angles for he linear fields when using he ray-racing expressions. 4. The Fresnel facors can be sysemaically incorporaed ino he linear algebra framework hrough marix muliplicaion by L = L L L. 5. In principle, uncerainies in he local opical consans can complicae polarizaion analysis in hin films. In pracice, he effecive medium approximaion appears o allow he use of he average opical consans of he adjacen media. 1