Coherent backscattering of Raman light

Transcription

1 In the format provided by the authors and unedited. SUPPLEMENTARY INFORMATION DOI: /NPHOTON Coherent backscattering of Raman light Barbara Fazio 1 *, Alessia Irrera 1, Stefano Pirotta 2, Cristiano D Andrea 3,4, Salvatore Del Sorbo 2, Maria Josè Lo Faro 1,3,5, Pietro Giuseppe Gucciardi 1, Maria Antonia Iatì 1, Rosalba Saija 6, Maddalena Patrini 2, Paolo Musumeci 3,5, Cirino Salvatore Vasi 1, Diederik S. Wiersma 7,8,9, Matteo Galli 2 * and Francesco Priolo 3,4,5,10* 1 CNR-IPCF, viale F. Stagno d Alcontres 37, Faro Superiore, Messina, Italy. 2 Dipartimento di Fisica, Università degli Studi di Pavia, via Bassi 6, Pavia, Italy. 3 MATIS IMM-CNR, via S. Sofia, 64, Catania, Italy. 4 CSFNSM, Viale A. Doria, 6, Catania. Italy 5 Dipartimento di Fisica e Astronomia, Università di Catania, via S. Sofia, 64, Catania, Italy. 6 Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Università di Messina, I Messina, Italy 7 LENS, Università di Firenze, via Nello Carrara, 1, Sesto Fiorentino (Firenze), Italy. 8 Dipartimento di Fisica e Astronomia, Largo Enrico Fermi, 2, Firenze, Italy. 9 INRIM - Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce, 91, Torino, Italy 10 Scuola Superiore di Catania,Università di Catania, via Valdisavoia, 9, Catania, Italy. * Corresponding authors: address: fazio@ipcf.cnr.it, Tel.: ; fax: ; address: matteo.galli@unipv.it, Tel , fax: , address: francesco.priolo@ct.infn.it, Tel.: ; fax: ; These authors contributed equally Now at CNR-IFAC, via Madonna del Piano 10, I Sesto Fiorentino (Firenze), Italy. NATURE PHOTONICS Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

2 Section S1. Raman Coherent Backscattering model for finite slab We consider the backscattered intensity in terms of the bistatic coefficient γ, defined as the measured scattered flux per solid angle and per unit of the probed area, normalized to the incident flux. In particular the coherent intensity γ c for a finite slab of thickness L can be modelled by the following expression, as obtained by the integral over all scattering paths 1-3 : γ c ( ψ ) = 3e ul 2αl 3 t sinh α L+2z 0 u 2 +η 2 α 2 2( α 2 +u 2 +η ) 2 cosh( 2αz 0 )cos( Lη )+ 4αηsinh( 2αz 0 )sin( Lη ) ( 2αη ) 2 +2( α 2 u 2 η 2 )cos( Lη ) 2 α u α 2 u 2 +η 2 +2( α 2 u 2 η 2 )cosh α ( L+2z 0 ) sinh α ( L+2z 0 ) sinh( ul) cosh( ul) 4αusinh( αl)sinh( ul) (S1.1) +2( α 2 +u 2 +η 2 )cosh( αl)cosh ul where ψ is the angle formed between the incoming and the outgoing wavevectors. The γ c ( ψ ) expression is a function of the following parameters: η k( 1 µ s ), u 1 2 κ 1+ µ ext _C s 1 and α. Here k = 2π λ is the light wavevector, µ s = cosψ, z 0 = al t is the extrapolation length accounting for internal reflections at the boundaries ( z = 0 and z = L ) 4,5,! κ ext _C is an effective extinction coefficient accounting for the attenuation of scattered intensity and α is the diffusive extinction coefficient. We introduce two linked characteristic lengths l d1 and = 2! d2 d1 2 t, which account for the Raman dephasing effect, as detailed in the main text and in the following Section S3. These dephasing lengths are set to very large value (infinite) in the fitting function of the Rayleigh cones. Thus, the 2

3 extinction coefficient becomes κ ext _C = l 1 t + l 1 i il 1 d1 + l 1 d2. The introduction of these dephasing lengths implies the addition of two diffusion lengths in the coefficientα : L d1 =! t i d1 3 and L d2 =! t d2 3. As a consequence α L 2 + L 2 abs d1 + L 2 d2 + q 2 with L abs = l l t i 3 and q = ksinψ. For the expression describing the diffuse background we also introduce the parameter v 1 2 κ ext _D 1 µ 1 ( s ). In this case the extinction coefficient does not include the effect of dephasing, thus it becomes κ ext _D = l 1 t + l 1 i. Since γ is evaluated at q l = 0, as a direct consequence α 2 L abs and the bistatic coefficient representing the diffuse background contribution, after integration over all scattering paths becomes: ( ψ ) = 3 Z 1 ( 1+ e 2L )+ Z 2 1 e 2L 2αl 3 t sinh α ( L+2z 0 ) u u 2 α 2 L u+v + Z 3 e 2 + v ( 2 v 2 2α 2 2u ) 2 (S1.2) where, Z 1 = u( v 2 u 2 +α 2 )cosh α ( L+2z 0 ) +u ( u2 v 2 +α 2 )cosh αl 2uvα sinh α L+2z 0 uvα v2 α 2 3u 2 u 2 α 2 sinh( αl) Z 2 = v( u 2 v 2 +α 2 )cosh α ( L+2z 0 ) 2u2 α cosh αl +α ( u 2 + v 2 α 2 )sinh α ( L+2z 0 ) +u2 v v2 u 2 3α 2 u 2 α 2 cosh( αl) Z 3 = 2u( v 2 u 2 α 2 )+2u( u 2 v 2 α 2 )cosh( 2z 0 α )+ 4uvα sinh( 2z 0 α ) (S1.3) (S1.4) 3

4 Finally, the fitting function of the coherent backscattering cone is defined as follows: = 1+ E 1 exp ( E 1) γ c I ψ (S1.5) where E exp is the experimental enhancement factor, which deviates from its theoretical value 2 due to residual of single scattering and the stray light, and E = 1+ γ c is the theoretical enhancement factor. Notice that E=2 in the Rayleigh scattering case since γ c = 1, while E=E Raman < 2 since γ c <1 in the Raman case. This expression for the fitting function is particularly useful for RCBS because it allows us to extract the value of the enhancement factor E=E Raman as a fitting parameter for the backscattering cone excluding any spurious effects, which may reduce the measured intensity. The inelastic (absorption) mean free path l i is maintained fixed in all fitting procedures; in particular, l i values are estimated by scaling the absorption length of bulk silicon to the percentage of silicon present in nanowires materials 6. In the case of Rayleigh cones (ECBS), the fitting parameters are the transport mean free path l t and L, that we named effective thickness L eff. In the case of Raman cones (RCBS) the free fitting parameters are l d1 and E Raman, while l t and L eff are fixed to the values found in the Rayleigh case (note that = 2! d2 d1 2 t is not a free parameter). S1.1 Role of the anisotropy and definition of effective thickness Due to the strong vertical orientation of the Si NWs, as shown in Fig. 1 of the main text, light transport in these systems is expected to be highly anisotropic, with a transport mean free path l z in 4

5 the vertical direction (parallel to the NWs) which is much larger than l x and l y in the xy plane (orthogonal to the NWs). Nevertheless, coherent backscattering can take place, and be observed, in systems with strong anisotropies in either direction 7,8. The mean free paths in the x and y direction (in the sample plane) determine the width of the backscattering cone - the cone being the Fourier transform of the intensity distribution on the output (back-reflection) surface. The z direction comes into play with respect to the (optical) thickness of the sample L/ l z. Some scattering in the z direction is of course required, otherwise the sample will appear transparent, although the mean free path in z can be much larger (or smaller) than that in x and y. However, the precise value of l z is difficult to be determined and the situation is further being complicated in our case by some roughness of the sample substrate, which will provide an efficient coupling mechanism of the scattering from the z direction into the xy plane. This helps to sustain multiple scattering and will make the sample more opaque (and the coherent backscattering cone more easily observable). A simple and efficacious way to quantify this is to define an effective sample thickness, which will capture most of the effects that are relevant for the coherent backscattering cone, and which will allow to compare theory with experiments with reasonable accuracy. Following this reasoning, we fitted the data using a finite-slab model for the ECBS intensity, with a transport mean free path l t l x =l y and an effective optical thickness L eff /l t = L/l z, where L eff is an effective thickness that is left as a free fitting parameter. This way, the effect of the larger (or smaller) mean free path along the z direction will be accounted for by the value of L eff. We remark that the two free parameters l t and L eff entering the fit function have very different effects on the shape of the backscattering cone, which leads to a rapid convergence of the fit towards reliable values of these parameters. A characteristic inelastic (absorption) mean free path l i has also been included in the fit function as a fixed parameter, which was independently evaluated for the two samples from the effective Si fraction in the NW layer, as determined by combined SEM analyses and energy dispersive x-ray (EDX) spectroscopy (see Table 1 in the main text). Model fits to the ECBS data yielding very small 5

6 values of the transport mean free path l t = µm and l t = µm combined with an effective optical thickness L eff /l t = 17 and L eff /l t = 5 for sample 1 and sample 2, respectively. These relatively large values of the effective optical thickness point towards a strong effective scattering in the z direction, despite the strong anisotropy of the NWs material. We ascribe this observation to the strong light diffusion by the sample substrate (as we assessed by measuring the diffused reflectance from a stripped NWs sample), which determines an effective transport mean free path along the z direction that may be substantially lower than what expected from an ideal free-standing layer of vertically aligned NWs (with no/flat substrate). We also notice that the values of the effective optical thickness derived from the fit to the ECBS data are consistent with those inferred from the extremely low values (below one per thousand) of the normal incidence reflectance on the same samples 6. Thus, the roughness of the sample substrate provides a very efficient coupling of the scattering from the z direction into x and y plane, and makes the ECBS cone easily observable despite the strong anisotropy of the system. Section S2. Experimental data for Raman Coherent Backscattering The best fits to the experimental data are shown in Fig.S2.1 for samples 1 and 2, and the corresponding relevant parameters are resumed in the table 1 of the main text. In particular in Fig.S2.1a and c we show the angular dependence of the Raman shifted light by exciting at normal incidence to the sample surface, and the backscattered Stokes Raman peak was collected as a function of the scattering angle in the helicity-conserving channel (blue dots), i.e. by selecting the same circular polarization of the pump beam. Similarly, the Raman shifted light was also measured for the two polarization non-conserving channels as a function of the detection angle and averaged (dark yellow dots). Here a clear enhancement around the backscattering direction, i.e. for ψ=0, is observed for the helicity-conserving channel on both samples, in contrast to a smooth Lambertian-like diffusing behavior for the polarization non-conserving channel. The normalization 6

7 of the two scattering channels for the Raman signal provides clear backscattering cones, as shown in Fig.S2.1b and d (see Methods section in the main text for details). a Intensity (arb. units) c Intensity (arb. units) Raman ( λ exc = 532 nm) error bars Raman ( λ exc = 532 nm) error bars Si Nanowires - sample 1 linear polarization non conserving channel Helicity conserving channel Si Nanowires - sample 2 linear polarization non conserving channel Helicity conserving channel b Normalized coherent intensity d Normalized coherent intensity Raman ( λ exc = 532 nm) error bar Raman ( λ exc = 532 nm) error bar Scattering angle Ψ(deg) Si Nanowires - sample 1 Si Nanowires - sample Figure S2.1 Coherent backscattering of Raman light. Raman backscattered intensities as a function of detection angle for sample 1 (a,b) and sample 2 (c,d). In a and c the helicity conserving channel and the linear polarization non conserving channel configurations are represented by blue and dark yellow dots respectively, while in b and d the Raman interference cones are plotted, as obtained by the normalization procedure to the Raman background signals. The best fitting curves (red continuous lines) are obtained by the finite slab model for RCBS theory, as discussed in the text. In a and c the intensities are scaled such that the incoherent contributions at the exact backscattering direction are 1. The error bars are indicated as a legend in the graph. Aiming to exclude geometrical effects, the experiments for the determination of RCBS shape have been repeated with an incident laser beam forming an angle of 20 with the normal to the sample surface. The experimental results are shown in Fig.S2.2. Blue and dark yellow dots in Fig.S2.2a are the helicity conserving channel and the linear polarization not conserving channel, respectively. After the normalization of the two curves, an enhanced backscattering cone is obtained (blue dots in Fig.S2.2b). Similar values of parameters ( l d1 = 3.6 ± 0.4 and E Raman =1.63 ± 0.08 µm with l t fixed at µm) are found from the best fit (red continuous line), as compared to the values obtained for normal incidence. Zero is the exact backscattering angle. 7

8 a λ exc θ Ψ b λ exc θ Ψ Figure S2.2 Coherent Backscattering of Raman light with excitation deviating from the normal to the sample surface. In a the Raman backscattered intensities as a function of the scattering angle are shown: the helicity conserving channel (blue dots) and the linear polarization non conserving channel configurations (dark yellow dots). The intensities are scaled such that the incoherent contribution at the exact backscattering direction is 1. Notice that the excitation angle θ is defined with respect to the perpendicular to the sample surface, while the backscattering angle Ψ is defined with respect to the excitation beam direction. In b the Raman interference cones is plotted, as obtained by the normalization procedure to the Raman background signal. The red continuous line is the best fitting curve as obtained by the finite slab model for RCBS theory. The error bars are indicated as a legend in the graph. Section S3. Effect of dephasing on the enhancement factor The effect of dephasing on the RCBS cone can be evaluated using the expressions for the time integrated diffusion probability (albedo) γ = γ c + and including the coherence function, a damped cosine as described in the main text, in the term γ c only. If we include also the effect of the inelastic mean free path we have the following expressions: 8

9 1 dt 1 e t/τ t 0 t 3/2 e t/τ i (S3.1) 1 γ c dt e 1 ( 3 kl t ψ ) 2 t/τ t cos ( 3DΔkt 2lt )e 3DΔk2 t 2 1 e t/τ t e t/τ i 0 t 3/2 Where the term 1 e t/τ t has been added to cut-off the integrals at small times. (S3.2) For convenience, we can express the integrals as a function of the scattering path length, which for ψ=0 (i.e. in the exact backscattering direction) reads: 1 dl 1 e l/l t 0 l 3/2 1 γ c dl 1 e l/l t 0 l 3/2 e l/l i e l/l i e l/l d 2 cos l/l d1 (S3.3) (S3.4) The calculation of these integrals allows us to estimate the enhancement factor γ c + as a function of the relevant parameters l, l t and l d1, as shown in Fig. 5a and 5b of the main text. i Fig. S3.1 shows the enhancement factor E Raman, calculated using the above defined relations, as a function of both l i and l d1, normalized to the scattering mean free path. As expected, for a given value of l i, E Raman increases by increasing the dephasing length l d1. On the contrary, for a fixed value of l d1, the dependence of E Raman on l i shows the opposite trend, leading to a lower enhancement for a longer l i. This unusual behaviour is determined by the oscillating coherence function entering the integral expression of the bistatic coefficient, which must be multiplied by the exponentially decaying function describing the absorption. Indeed, for large values of l i > l d1 this 9

10 results in a strongly damped cosine function, whose average value may be significantly different from zero. l = l d1 l = l i Figure S3.1. Theoretical Raman Enhancement plotted versus the dephasing length l d1 (blue) fixing l i =8.1 µm and versus the inelastic mean free path l i (red) at l d1 =3.38 µm. 10

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