EJL R Sβ. sum. General objective: make sense of this cryptic expression. Garth J. Simpson. Department of Chemistry Purdue University. Garth J.

Transcription

1 e = EJL R Sβ General objective: make ene of thi cryptic expreion. Department of Chemitry Purdue Univerity Specific Objective: 1. Review of Jone vector, matrice, and tenor for decribing polarization phenomenologically. 2. Review rotation matrice and vector coordinate tranformation (1D, then 3D). 3. Demontrate two equivalent approache for rotating a matrix. 4. Tenor rotation: α, β, γ, etc. 5. Molecular frame to enemble frame and orientational averaging in uniaxial aemblie: 3WM with e = EJLRβ 6. Molecular frame to enemble frame and orientational averaging in uniaxial aemblie: 4WM with e = EJLRγ 1

2 A more general and robut approach i to go back a tep to the -frequency field, repreented a a Jone vector, a a function of the incident field. ω1 ω2 ω1 ω2 ω1 ω2 ω1 ω2 p ppp p p + pp p + pp p + p ω ω ω ω ω ω ω ω e e e e e e e e e e = e e + e e + e e + e e pp p p p p p p p p e e e e e e e e e ω ω ω ω ω ω ω ω p p p p p p = ω1 ω2 ω1 ω2 ω1 ω2 ω1 ω 2 e ep ep ep e e ep e e pp p p E ( ) T ( ) 1 2 E= e e e J = ω ω T ppp pp pp 2

3 The broader utility of thi Jone tenor repreentation may not be a obviou without a working familiarity with Jone vector and matrice. e ( ω ) i kz t + iδ x e e e = = y e y ( ω ) x x i kz t + iδx i( kz t) i i e ω + δ e e y e The normalized unit Jone vector conciely and completely decribe the pure polarization tate of plane-wave light. 2 2 x y Horiz: 1 eˆ = 0 1 ex eˆ = i e e ey e + 0 Vert: eˆ = 1 RCP: LCP: 1 1 eˆ = 2 i 1 1 eˆ = 2 i 3

4 All non-depolarizing optical element can be decribed by polarization tranfer matrice. eˆ out = M eˆ in Example: Half and quarter waveplate, mirror, window. HWP (0 o ) Perfect mirror QWP (0 o ) Window (ingle interface) i rp 0 0 r 4

5 y' y φ x' x vv V x' coφ inφ x R( φ ) v y' = = inφ coφ y Paive rotation. vv i fixed; coordinate ytem i rotated relative to it. HWP HWP (φ) ( φ ) QWP (φ) QWP ( φ ) 1 0 HWP = R R 0 1 ( ) ( ) ( ) ( ) co φ in φ 1 0 coφ inφ co 2φ in 2φ = in φ co φ = 0 1 inφ coφ in 2φ co 2φ ( ) ( ) ( ) ( ) ( φ) ( φ) ( φ) co φ in φ 1 0 coφ inφ co 2φ i in 2φ = in φ co φ 0 i = inφ coφ in 2φ co 2φ i BUT thi approach i not eaily generalized for rotation of a tenor! 5

6 y' y φ x' x vv V However, the ame net reult i alo obtained by rotating the vectorized matrix: Active rotation. Object i rotated in M ' = R( φ) R( φ) M a fixed reference frame. Mxx Mxx ' M xy co in co in Mxy ' φ φ φ φ = M yx inφ coφ inφ coφ M yx ' M yy M yy ' Mxx coφcoφ coφinφ inφcoφ inφinφ Mxx ' M xy coφinφ coφcoφ inφinφ inφcoφ Mxy ' = M yx inφcoφ inφinφ coφcoφ coφinφ M yx ' M yy inφinφ inφcoφ coφinφ coφcoφ M yy ' THIS approach i directly generalizable for rotation of a tenor! 7

7 V Worked example for a rotated HWP: M = R R M ( φ) ( φ) ' M xx coφcoφ coφinφ inφcoφ inφinφ 1 M xy coφinφ coφcoφ inφinφ inφcoφ 0 = M yx inφcoφ inφinφ coφcoφ coφinφ 0 M yy inφinφ inφcoφ coφinφ coφcoφ M xx co φ in φ co 2φ M xy 2in co in 2φ φ φ = = M yx 2inφcoφ in 2φ 2 2 M yy in φ co φ co 2φ M co 2φ in 2φ = in 2φ co 2φ Same reult obtained by reference frame rotation! THIS approach i directly generalizable for rotation of a tenor! 8

8 By analogy, we can decribe the NLO propertie of the ample in term of a Jone tenor, which i a direct higher-order extenion of the baic framework of Jone matrice. new HHH XXX HHV XXY HVH XYX HVV XYY = R( φ) R( φ) R( φ) VHH YXX VHV YXY VVH YYX VVV YYY old If the ample i rotated about Z (direction of beam propagation), the correponding Jone tenor i eaily recovered from the rotation matrix product. Thi ituation arie routinely in microcopy meaurement, when the local-frame (X,Y) coordinate are rotated relative to the horizontal H and vertical V axe of the image. = E R ( φ ) e J coφ inφ = inφ coφ 9

9 The preceding Jone framework enable model-independent meaurement, but i limited only to rotation in 1D about the (,p), (x,y), (h,v) frame. To decribe NLO microcopy meaurement of tilted and/or twited ytem a well a urface SFG and CARS meaurement, a more general rotation matrix i required. RXx' RXy ' R Xz ' v= RYx' RYy ' RYz ' v' RZx ' RZy ' R Zz ' R R R R R R R R R Xx' Xy ' Xz ' Yx' Yy ' Yz ' Zx' Zy' Zz ' = inψ inφ+ coθcoψ coφ = coψ inφ coθinψ coφ = inθcoφ = inψ coφ+ coθcoψ inφ = coψ coφ coθinψ inφ = inθinφ = inθcoψ = inθinψ = coθ 11

10 -There are twelve unique definition of the Euler angle, depending on the election and ordering of axe of rotation (boo!). -Two common convention are found in nonlinear optic: ZXZ and ZYZ (yay, ort of!). -Each of the two i interpreted differently for intrinic (paive) rotation or extrinic (active) rotation (boo!). Extrinic ZYZ Rotation coφ inφ 0 coθ 0 inθ coψ inψ 0 R θψφ = inφ coφ inψ coψ inθ 0 coθ Rotation about Z Rotation about Y Rotation about Z Intrinic ZY'Z' Rotation coψ inψ 0 coθ 0 inθ coφ inφ 0 R θψφ = inψ coψ inφ coφ in θ 0 coθ Rotation about Z Rotation about Y' Rotation about Z' Unle explicitly tated otherwie, ALL Euler angle preented herein will refer excluively to external rotation in the ZYZ convention! 12

11 Intrinic and extrinic rotation are invere operation. ex in R θψ,, φ R θψ,, φ = I ( ) ( ) Reviiting an example we ve ued but not explained in detail: Sign invert for the two rotation matrice in the andwich approach. coφ inφ 1 0 coφ inφ co 2φ in 2φ inφ coφ = 0 1 inφ coφ in 2φ co 2φ Combination of intrinic rotation and invere (extrinic) rotation Identical extrinic rotation matrice are ued in the vectorized approach. 1 co 2φ coφ inφ coφ inφ 0 in 2φ inφ coφ inφ coφ = 0 in 2φ 1 co 2φ Extrinic projection only 13

12 There i no limit to the order of tenor that can be rotated by vectorization and projection. v = R,, v' M = R,, R,, M ' ( θψ φ) ( θψ φ) ( θψ φ) tenor of rank 1 (vector) tenor of rank 2 (matrix) tenor of rank 3 β= R( θψ,, φ) R( θψ,, φ) R( θψ,, φ) β' Decribing econd order NLO (SFG, SHG, DFG) tenor of rank 4 γ= R( θψ,, φ) R( θψ,, φ) R( θψ,, φ) R( θψ,, φ) γ' Decribing third order NLO (CARS, THG, tranient aborption, etc.) etc. 14

13 v h v h If the focal plane cut right acro the center of a cell, each bilayer interface i perpendicular to the FoV, and related through rotation about φ. V H We could ue the known value of φ at each location at the cell urface to recover a ingle repreentative value of the Jone tenor from the collective et of bilayer meaurement. Converely, we could ue an independently determined Jone tenor to ae the local orientation of the cell membrane. 15

14 We will look more cloely at collagen when we cover the NLO propertie of protein. For now, we will ae a thin collagen fiber ha C ymmetry and that the tilt into and out of the image plane i negligible, uch that only azimuthal rotation i important. If the long collagen fiber axi i defined a the local x-axi, the collagen tenor for SFG or SHG i decribed by the unique nonzero element β xxx, β yyx, β xyy, and β zxy in φcoφ in φcoφ co φ (2) J, HHH (2) 0 in φ inφco φ inφco φ inφco φ (2) J, HHV βzxy (2) in co co in co J, HVV N φ φ φ φ φ (2) β b (2) = yyx (2) J, VHH ε inφco φ in φ inφco φ βxyy (2) (2) J, VVH βxxx 0 in φ in φcoφ in φcoφ in φcoφ (2) J, VVV L inφco φ inφco φ in φ l 16

15 ω ir ω vi p p p ω SFG Z Y X P,X V. S,Y Z ω SFG The rotation ha no component in φ or ψ for a uniaxial urface, but doe have rotation from θ. Since we will alo later be uing θ to decribe molecular tilt, we will include an ω upercript on all of thee to help differentiate. Pleae note the convention for the definition of θ relative to the urface normal! ω e vi X ωvi ωvi ep coθ 0 inθ e Y e = ez ω vi ω e ir X ωir ωir ep coθ 0 inθ e Y e = ez ω ir ω e X ω ω ep ± coθ 0 inθ e Y e = ez ω Note: the ± ymbol on the SFG output refer to the ign change in the X- polarized component for reflection detection. Sign will be (+) for tran, (-) for reflection. 17

16 ω ir ω vi p p p ω SFG Z Y X v. P,X S,Y Z ω SFG ω ωvi ωir J = R R R C, eff = J C, eff ω e = E = E J J C, eff ω e vi X ωvi ωvi ep coθ 0 inθ e Y e = ez e = R e ωvi ωvi ωvi J C, eff ω vi ω e ir X ωir ωir ep coθ 0 inθ e Y e = ez e = R e ωir ωir ωir J C, eff ω ir ω e X ω ω ep ± coθ 0 inθ e Y e = ez e = R e ω ω ω J C, eff ω The collective Kronecker product of R defined a J will be an 8 by 27 for 3WM (e.g., SFG/SHG), and a 16 by 81 for 4WM (e.g., CARS, CATS). 18

17 If each molecular tenor i expreed in the urface coordinate ytem, the Carteian tenor i imply given by mation over the all the molecule per unit area (with a number denity of N ). C C N = i If we can alo ae that each molecule i identical and uncoupled, changing only in it relative orientation, we can re-expre thi mation a an orientational average, connecting the molecular-frame tenor to the urface Carteian frame through rotation matrice. β Ci, = N R R R Mot commonly, urface are aed to be uniaxial with the urface normal paralleling a C ymmetry axi. In thi cae, a uniform ditribution in the azimuthal rotation angle φ can be aed, implifying the relationhip bridging the molecular and lab frame. β 19

18 For C urface ymmetry, it contain a a ubet: C 2 (z) X X Y Y Z Z C 4 (z) X Y Y X Z Z For C v urface ymmetry: σ(xz) σ(yz) X X Y Y Z Z X X Y Y Z Z 20

19 We can generalize thi approach for decribing the nonlinear optical propertie of nonlinear optical interaction in uniaxial aemblie by conidering the rotation operation explicitly. C = N R R R θψφ θψφ θψφ Uing the ZYZ extrinic definition for the rotation matrice: C = N Rφ Rθ R ψ Rφ Rθ R ψ Rφ Rθ R ψ From the propertie of Kronecker product (AB) (CD) = (A C)(B D), the above equation can be rewritten: C β ( ) ( ) ( ) = N Rφ Rφ Rφ Rθ Rψ Rθ Rψ Rθ R ψ Becaue the ditribution in φ i known to be uniform for a uniaxial aembly and therefore eparable (i.e., the ditribution in φ i independent of θ and ψ), it average can be independently evaluated: C β ( ) ( ) ( ) = N R R R R R R R R R φ φ φ θ ψ θ ψ θ ψ β β known a priori

20 Thi orientationally averaged matrix over the uniaxial ymmetry can be incorporated a an additional ymmetry matrix Q, which i a 27 by 27 matrix that remove and/or combine tenor element according to the enemble uniaxial ymmetry. C C = N R R R θψφ θψφ θψφ β ( ) ( ) ( ) = N Q R R R R R R θ ψ θ ψ θ ψ Q= Rφ Rφ Rφ Of the 27 poible nonzero tenor element, only even urvive in the cae of uniaxial ymmetry, uch that Q i in general a pare matrix. Note: Q i not a fundamental matrix in the EJL<R>Sβ product chain, a it i implicitly a part of the rotational averaging connecting the molecular to urface frame within <R>. However, it ue can implify the ubequent orientational average. β 22

21 Since only four nonzero (2) tenor element remain in a uniaxial aembly, we can write out analytical expreion for the product of Rφ Rφ Rφ ( Rθ Rψ ) ( Rθ Rψ ) ( Rθ Rψ ) for all 27 poible molecular tenor element for uniaxial achiral aemblie. 23

22 For chiral film, three additional nonzero tenor element are allowed in uniaxial C enemble 24

23 e = EJL R Sβ 1. The polarization-dependence of SFG and CARS are enitively coupled to molecular orientation. 2. Quantitative aement of molecular orientation require care in the mathematic, particularly for the Frenel factor due to uncertaintie in the optical contant. 3. The preence of molecular ymmetry can greatly reduce the number of unique, nonzero element within the molecular tenor. 4. In ytem of low ymmetry, analytical expreion for orientation analyi can be challenging unle the complexity of the molecular tenor can be reduced through ymmetry or an approximation thereof. Reference: -Begue, N. J.; Moad, A. J.; Simpon, G. J. Nonlinear optical Stoke ellipometry (NOSE) Part I: Theoretical Framework. J. Phy. Chem. C. 2009, 113, Moad, A. J., Moad, C. W.; Perry, J. M.; Wampler, R. D.; Begue, N, J.; Shen, T.; Goeken, G. S.; Heiland, R.; Simpon, G. J. NLOPredict: Viualization and Data Analyi Software for Nonlinear Optic J. Computational Chem. 2007, 28,