SECOND ORDER NONLINEAR PROCESSES AT SURFACES AND INTERFACES

Transcription

1 SECOND ORDER NONLINEAR PROCESSES AT SURFACES AND INTERFACES C.Stancu, R.Ehlch /A1 Boundary condtons of a polarzed sheet Radaton from a polarzed sheet Surface nonlnear response Bulk nonlnear response The equvalence of surface and bulk contrbutons The thn flm geometry - example of an sotropc flm Rotatonal ansotropy; examples

2 Bblography: 1. T.F.Henz, Second-order nonlnear optcal effects at surfaces and nterfaces, n Nonlnear Surface Electromagnetc Phenomena, Eds. H.- E.Ponath and G.I.Stegeman, B.Koopmans, Interface and bulk contrbutons n optcal secondharmonc generaton, Ph.D.thess, N.Bloembergen and P.S.Pershan, Lght waves at the boundary of nonlnear meda, Phys.Rev. 18, 606 (196) 4. P.Guyot-Sonnest, W.Chen and Y.R.Shen, General consderatons on optcal second-harmonc generaton from surfaces and nterfaces, Phys.Rev.B 33, 854 (1986) 5. P.Guyot-Sonnest and Y.R.Shen, Bulk contrbuton n surface secondharmonc generaton, Phys.Rev.B 38, 7985 (1988) 6. J.E.Spe, D.J.Moss and H.M. van Drel, Phenomenologcal theory of optcal second- and thrd- harmonc generaton from cubc centrosymmetrc crystals, Phys.Rev.B 35, 119 (1986) 7. T.F.Henz, M.M.T.Loy and W.A.Thompson, Phys.Rev.Lett. 54, 63 (1985) 8. D.Wlk, D.Johannsmann, C.Stanners and Y.R.Shen, Phys.Rev.B 51, (1994)

3 Applcatons of nonlnear optcal phenomena: - development of optcal-devce technology and laser systems - the use of these phenomena as a tool for materal characterzaton - partcular case: surface specfcty of the second order nonlnear processes for centrosymmetrc materals Wthn the electrc dpole approxmaton the polarzaton can be wrtten as an expanson of the electrc feld: P = τ χ () 1 E + τ χ () :EE + τ χ () 3 :EEE +... The nverson symmetry s reflected n the susceptblty tensor τ χ ( n) whch s nvarant under the transformaton r -r; ths gves τ( n) τ( ) χ ( ) n n = χ any even-order response s forbdden wthn the electrc dpole approxmaton for a centro-symmetrcal materal At the nterface between two centro-symmetrc meda, the nverson symmetry s broken by defnton, gvng rse to an ED allowed second order susceptblty second order processes are useful nondestructve technques for the study of surfaces and bured nterfaces, wth a resoluton better than the nherent penetraton depth of the probe

4 Boundary condtons of a polarzed sheet ε ε 1 ε Ω k 1 (Ω) k (Ω) Ω Medum 1 Medum k 1 (ω 1 ) k 1 (ω ) ω 1 ω τ ( ) χ q τ ( ) χ s τ ( ) χ q The nonlnearty of the nterface s treated as a sheet of generalzed nonlnear source polarzaton. From Maxwell s equatons wth a nonlnear source polarzaton P nls occupyng a fnte volume: D = -4π P nls c E + B t = 0 B = 0 c H - D = 4 π nls P t t the boundary condtons for the electrc and magnetc felds on ether nls sde of ths polarzed sheet P s (x,y) (wth a volume polarzaton nls P s (x,y)δ (z)) are: nls Dz = -4π Ps Et = - 4π nls t P s, z ε ' B z = 0 Ht = 4π nls Ps z ) c t

5 Radaton from a polarzed sheet nls P s (x,y,t) = P s e px- Ω t + c.c. The reflected beam s gven by the wave vector k 1 = p x ) ) - q 1 z, wth q 1 = [ε1k - p ] 1/ ; smlarly, the transmtted beam s gven by k = p x ) ) + q 1 z. From the boundary condtons, the radated felds are (wth ε1 = ε = ε = 1): E (x,y,z,t) = E (Ω )e k r- Ω t + c.c. where: E = πk [P q s - k ) ( k ) Ps )] The Fresnel correctons are made for the waves propagatng towards the nterface by replacng e ) ) wth e = F e j, where the Fresnel transformaton has the dagonal elements (for sotropc meda): q xx F = ε j j q + ; ε εq F yy q = j ; q + q F zz = εε q 1 j ε' ε q + εq j j j j j The radated feld can be wrtten lke ) e πk E = sec θ ( e P) 1 / s ε / ε ( ) where sec θ ( Ω) Ω 1 = Ω defnes, for a lossless medum, the angle cq ( Ω) between the wave at Ω and the surface normal.

6 Surface nonlnear response When consderng the nonlnear polarzed sheet as arsng from the surface nonlnear susceptblty tensor τ ( ) χ s : P s (Ω) = τ ( ) χ s (Ω=ω 1 +ω ):E(ω 1 )E(ω ) or (Ω=ω 1 +ω ):e 1 (ω 1 )e (ω )E 1 (ω 1 )E (ω ) P s (Ω) = τ ( ) χ s The radated felds can be now wrtten as ) πωsec θ ( Ω) ( ) e( Ω) E ( Ω) = [( e Ω) χ :( e ω )( e ω )] E ( ω ) E ( ω ) 1 / s cε ( Ω) When absorpton s present n the bulk meda, the electrc feld ampltudes correspond to the ncomng or outgong felds near the nterface. For an arbtrary geometry of the pump beams, the drecton of the radated beam s gven by the n-plane wave vector components p(ω) = p(ω1) + p(ω) - the nonlnear Snell s law In terms of rradances (I = cε 1/ E /π for a plane wave n a medum ε), the felds are gven by I ( Ω) 8 3 π Ω sec θ ( Ω) ( e ) ( ) = Ω χ :( e ω )( e ω ) I ( ω ) I ( ω ) 3 s c [ ε ( Ω) ε ( ω ) ε ( ω ) 1 1

7 Bulk nonlnear response The nonlnear source polarzaton n the bulk of a materal, as a multpole expresson n succesve degrees of nonlocalty, s: P nls (Ω ) = χ () (Ω = ω 1 +ω ):E(ω 1 )E(ω ) + 1 τ ( χ q )(Ω = ω 1 +ω ):E(ω 1 ) E(ω ) + 1 τ ( χ q )(Ω = ω 1 +ω ):E(ω ) E(ω 1 ) +... The relatve contrbutons of the succesve terms vary lke (ka), wth a the typcal atomc dmenson. For centrosymmetrc meda, the leadngorder response conssts of the electrc-quadrupole and magnetc-dpole terms comprsed of products of E and the spatal dervatves of E. Up to the frst order spatal dervatves of the electrc feld, we can express the generalzed polarzaton lke: P () (Ω ) = () Pd (Ω ) - Q τ c ( ) (Ω ) + Ω M () (Ω ) where, n the case of SHG, we can wrte: P () d = τ τ χ D :E(ω )E( ω ) + χ P :E(ω ) E( ω τ ) Q ( ) = τ χ Q :E(ω )E( ω ) M () = τ χ M :E(ω )E( ω ) τ wth χ D descrbng a local response, whle τ χ P, nonlocal. τ χ Q and τ χ M are For nonmagnetc materals (nverson symmetry n both space and tme), M () (Ω ) = 0.

8 The effectve SH polarzaton can be wrtten as: P (Ω ) = γ (E E) + ( δ - β - γ )(E )E + β ( E)E + ζ E E β, γ and δ descrbe the sotropc response of the medum. * for a homogeneous medum, E = 0 * (δ-β-γ)(e )E = 0 when only a sngle plane wave s present n the medum (reflectons from deeper lyng nterfaces are gnored) * n an sotropc materal, ζ = 0 * the only bulk term contrbutng to the sgnal n a smple measurement of an sotropc medum remans γ (E E) whch s a longtudnal feld for any pump feld E, gvng rse to radaton felds ndstngushable from those of an equvalent surface nonlnear response

9 The equvalence of surface and bulk contrbutons For a bulk nonlnear source polarzaton gven by a sngle wave vector k b : P(x,y,z,t) = P b (Ω,kb )e k b x-ω t ϑ (z) + c.c. one can defne an equvalent nonlnear polarzaton sheet at the nterface between meda 1 and : P s eq (x,y,z,t) = P s eq (Ω,kb )e k b x-ω t δ (z) + c.c. Takng the bulk polarzaton as a successon of polarzed sheets and consderng the propagaton of the nonlnear wave n medum, the equvalent polarzaton sheet s gven by P s eq (Ω,kb ) = q q P k x ) P k y ) ε'( Ω) P k z ) [ ( Ω, ) + ( Ω, ) + ( Ω, ) ] bx, b by, b bz, b + ε ( Ω) b where q b s the z-component of k b and q s the z-component of the Ω- wave n medum. The bulk nonlnear polarzaton P b (Ω ) = γ (E E) excted by a sngle plane wave can be wrtten as P b (Ω ) = γ ks [E(ω ) E( ω )]. Defnng an effectve surface polarzaton P s eq (Ω ) correspondng to ths term and lookng at the p-polarzed radaton whch t produces, one fnds: e p P eq s (Ω) = ) C[q x+ ε ε ' pz) ] {[- γ (E E)(qs +q ) -1 ][p x+ ) ε ' ε = -Cγ p(e E) = e p [- γ ε ' (E E) z ) ] ε γ = e p P s ( Ω) where we have defned a surface polarzaton γ P s ( Ω) = - γ ε ' (E(ω) E(ω)) z ) ε qz ) ]}

10 By specfyng the radated feld n terms of a surface polarzaton γ ) P s z one can reproduce the angular and polarzaton dependence of the bulk longtudnal polarzaton, wthout ntroducng the vector k s. It s therefore mpossble to separate the bulk longtudnal polarzaton from the z-component of the surface polarzaton. Ths equvalence has also been demonstrated for an arbtrary pump feld, n sotropc as well as n cubc materals. The SH response from an nterface between two bulk centrosymmetrc meda can be descrbed by an effectve surface nonlnear tensor ( ) ( ) ( ) χ seff, = χ s + χ s, γ wth the radated feld expressed lke ) e πωsec θ ( Ω) ( ) ( Ω) E ( Ω) = [( e Ω) χ :( e ω )( e ω )] E ( ω ) E ( ω ) 1 / seff, cε ( Ω)

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12 Thn flm geometry * offers addtonal possbltes for the problem of the SHG vs. bulk separaton. * mportant for the study of bured nterfaces n case of overlayers havng thcknesses of the order of the wavelength of lght The total SH feld generated by a nonlnear flm s gven by: E(Ω ) = E B (Ω ) + E S (Ω ) + E I (Ω ) wth a total SH output: 3πω 3 ( ) S(Ω ) = θ χ I ω AT 3 1/ eff ηc ε ( Ω) ε ( ω) sec ( ) Ω q where τ ( ) τ τ τ ( ) χ = e ( Ω) L( Ω, z = 0): χ : L( ω, z = 0) ( e ω) L( ω, z = 0) ( e ω) eff S τ τ τ τ ( ) + e ( Ω) L( Ω, z = d): χ : L( ω, z = d) ( e ω) L( ω, z = d) ( e ω) I d τ τ τ τ ( ) + e ( Ω) L( Ω, z'): χ : L( ω, z') ( e ω) L( ω, z') ( e ω) dz' B 0 q θ Ω s the ext angle, A the beam cross secton, τ T the pulse wdth, e ( Ω) the unt vector of the feld at Ω and L( ω, z) s a dagonal tensor descrbng the feld nsde the flm, wth the elements: p k fzz p k fz d z f L (, z) t ( e r e ) cos θ ( ) ω = xx 1 I cos s k fzz s k fz ( d z ) L ( ω, z) = t ( e + r e ) yy 1 p k fzz p k fz ( d z ) ε1 L ( ω, z) = t ( e + r e ) zz 1 I ε t h ts = h 1 rre h 1 S I h k d fz I 1 / 1 / f θ ω

13 τ L, the bulk term n χ eff Because of the phase factors n wll exhbt a composte nterference pattern as the thckness d vares; the nterface, surface and bulk term can have relatve phase dfferences and ther nterference also changes wth d. By fttng S(ω ) versus d one can get to the dfferent surface and bulk susceptbltes. The effectve bulk SH polarzaton s wrtten as: P (Ω ) = γ (E E) + ( δ - β - γ )(E )E + β ( E)E + ζ E E For an sotropc medum, one can express the parameters β, γ, δ and ζ as a functon of the non-vanshng bulk susceptbltes: β = ( τ P χ jj γ = τ P χ jj - τ Q χ jj - τ Q χ jj ) δ = ( τ P χ - τ Q χ ) ζ = 0 δ - β - γ = ( τ P χ jj where τ Q χ jj = τ Q χ jj - τ Q χ jj and ) τ PQ, χ = τ PQ, χ jj + τ PQ, χ jj + τ PQ, χ jj Snce γ s ndstngushable from surface contrbuton, t wll be accounted for by takng effectve surface susceptbltes for the z drecton. There wll be 7 ndependent susceptblty elements, probed for varous polarzatons combnatons: s/p m/s p/p m/p χ S,zyy - γ χ I,zyy + γ χ S,zzz - γ χ I,zzz + γ χ S,yzy χ S,yzy δ - β - γ ( )

14 D.Wlk, D.Johannsmann, C.Stanners and Y.R.Shen, Phys.Rev.B 51, (1994)

15 Rotatonal ansotropy P (Ω ) = γ (E E) + ( δ - β - γ )(E )E + ζ E E ζ 0 The cartesan coordnate axes concde, n ths expresson, wth the prncpal crystallographc axes. When estmatng the values of the generated felds n dfferent polarzaton combnatons, one needs to transform χq() nto the laboratory axes. As a general rule, rotatonal measurements of the L-th order multpole contrbuton to the N-th order NLO process of a materal wth q-fold rotatonal symmetry can only show ansotropc response f N + L q. For cubc centrosymmetrc materals (m3m- or 43-symmetry), the varaton of the SH felds wth an arbtrary azmuthal angle can be wrtten as E p (Ω) = a + b (m) cos(mϕ) E s (Ω) = c (m) sn(mϕ) where m = 3 for (111) crystal faces and m = 4 for (001) crystal faces. For a (110) face, E p (Ω) = a + b () cos(ϕ) + b (4) cos(4ϕ) E s (Ω) = c () sn(ϕ) + c (4) sn(4ϕ)

16 Photopolymerzaton of a crystallne C 60 flm unpolymerzed polymerzed p-p sgnal I SHG (arb.u.) 0,35 0,30 0,5 0,0 0,15 0,10 0,05 0,00 0,35 0,30 0,5 0,0 0,15 0,10 0,05 0, Azmuthal angle (deg) ps sgnal unpolymerzed polymerzed

17 Reconstructon of S(111) surfaces The polarzaton dependence for a SH sgnal assocated only wth the surface nonlnear susceptblty, for normal ncdence: * S(111)-7x7: ( ) I x (Ω) = A χ s, xxx cos ϕ I y (Ω) = A χ ( ) s, xxx sn ϕ * S(111)-x1 I x (Ω) = A χ ( ) s, xxx I y (Ω) = A χ ( ) s, xxx sn ϕ cos ϕ + χ ( ) s, xyysn ϕ T.F.Henz, M.M.T.Loy and W.A.Thompson, Phys.Rev.Lett. 54, 63 (1985)