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Journal of Modern Optics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tmop20 Derivation of the optical autocorrelation function from Raman scattering of diffusing particles Maki Nishida a & Edward R. Van Keuren a a Department of Physics, Georgetown University, Washington, DC, 20057, USA Available online: 10 Nov 2011
To cite this article: Maki Nishida & Edward R. Van Keuren (2012): Derivation of the optical autocorrelation function from Raman scattering of diffusing particles, Journal of Modern Optics, 59:2, 102-105 To link to this article: http://dx.doi.org/10.1080/09500340.2011.631053
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Derivation of the optical autocorrelation function from Raman scattering of diffusing particles Maki Nishida and Edward R. Van Keuren* Department of Physics, Georgetown University, Washington, DC, 20057, USA (Received 19 August 2011; final version received 4 October 2011)
Raman scattering is an inelastic scattering process with chemical specificity to molecular bonds. Because it is a coherent process, the autocorrelation function of its intensity fluctuations can be treated similarly as in the case of the quasi-elastic scattering that is the basis of photon correlation spectroscopy (PCS). This article discuses the possibility of such a new optical characterization method, Raman correlation spectroscopy (RCS). If the phase behavior of Raman scattering is constant or slow enough compared to the diffusion of particles in a dispersion, RCS can work as a variation of PCS with similar instrumentation as PCS and could become a useful addition for nanoparticle characterization research. This article also discusses the effect of incoherent fluorescence in the data analysis of RCS. Keywords: autocorrelation function; Raman scattering; multicomponent nanoparticles; particle sizing; temporal coherence
1. Introduction That is, the Raman scattering process itself is coherent Photon correlation spectroscopy (PCS), also known as with the incident light [2,3]. However, even though the dynamic light scattering or quasi-elastic light scatter- Raman effect produces very narrow spectral line- ing, is a well-established characterization technique for widths, the scattering from the individual molecules determining dynamical information such as the diffu- could have a slightly different phase from one another, sion coefficients of nanoparticles in a dispersion. In depending on the phase of the final vibrational state of PCS, the intensity fluctuations of the coherent light the molecule. Therefore, while the individual scattering scattering due to the quasi-elastic collision of photons events may be coherent, the total scattered field of the from the diffusing particles (Rayleigh scattering) is particles would be incoherent [4]. As we show in this measured, and their field or intensity autocorrelation letter, the fluctuation of this phase shift in time will functions (ACFs) are analyzed [1]. Simultaneous with determine the temporal coherence of the Raman this type of scattering, however, Raman scattering that scattered light and thus explain whether the ACFs of is from inelastic collisions is also coherently emitted the Raman scattering will contain the dynamical from the particles. In this letter, we derive the information on the Brownian diffusion of the particles. expressions for the ACFs of Raman scattered light If the phase behavior of Raman scattering of and discuss the possibility of using this method for diffusing particles is constant or slowly changing particle characterization with chemical specificity. This compared to the sampling rate of the signal detector, method would be useful in the characterization of the ACFs of the Raman signals from a specific complex mixtures since the Raman line from a specific molecular band of diffusing particles could produce molecular species contained in the mixture could be similar information to that obtained in PCS. This Downloaded by [Georgetown University], [Edward R. Van Keuren] at 16:14 27 February 2012 used to determine the diffusing transport of only that Raman version of PCS, or Raman correlation spec- component. troscopy (RCS), could be used to obtain diffusion Raman scattering occurs with a change in the coefficients and particle sizes from a specific chemical vibrational or rotational energy of a molecule. The species in a multicomponent system. time-scale of the spontaneous, non-resonant Raman An earlier version of RCS was developed by Schrof scattering is instantaneous, as in the case of quasi- et al., using number fluctuation analysis with confocal elastic scattering, since the energy transition occurs via optics, as in fluorescence correlation spectroscopy [5]. a virtual energy state of the molecules in a particle. This implementation requires small sampling volume
*Corresponding author. Email: [email protected]
ISSN 0950–0340 print/ISSN 1362–3044 online ß 2012 Taylor & Francis http://dx.doi.org/10.1080/09500340.2011.631053 http://www.tandfonline.com Journal of Modern Optics 103
0 and low concentration. Combined with the weak The subscripts of the factor Bk and ’k(t) change to jk nature of Raman scattering, this limits the range of since each scatter has M molecules. materials for which this method will work. As men- Following the derivation of the ACF for PCS tioned above, however, an implementation with a by Cummins and Swinney, the average total Raman standard scattering apparatus, similar to PCS, should scattered intensity would be given by the time be possible due to the coherent nature of the Raman average [6]: scattering process. This could result in higher sensitiv- ¼ 2 ð Þ ity and the ability to characterize a wider range of IRS ERS 5 materials than with the confocal optics implementa- which is equivalent to the first-order, field ACF in tion, but most importantly it could also provide a Equation (1). detailed picture of the individual species in a multi- The positions of the scattering particles are not component system. correlated. Hence, all cross-terms average to zero as XN 2 2 IRS ¼ Aj ¼ N Aj : ð6Þ 2. Derivation of the autocorrelation function j PCS employs the ACF of the scattered wave. The field Consequently, the ACF of the Raman scattered ACF, a measure of temporal correlation of the fields is scattered electric field E, is defined as * XN XM 0 iq rj if!tþ’jkðtÞg G1ð Þ¼hE ðtÞEðt þ Þi: ð1Þ G1ð Þ¼ Aj ðtÞ Bjk e e j¼1 k¼1 + For Raman scattering, the scattered field from the kth XN XM molecule in a particle at position r þ Dr , where r is the 0 iq rl if!ðtþ Þþ’lpðtþ Þg k Al ðt þ Þ Blp e e : position of the particle, can be approximated as l¼1 p¼1
iq frþDrkg if!tþ’kðtÞg ð7Þ Ek ¼ Bke e ð2Þ Because the scatterers, or particles, are in random where the amplitude B is a constant that depends on k motion and their translations are not correlated to the Raman cross-section and ! is the particular Raman each other, the cross-term from the fields from frequency of the molecules that is observed. The different scatterers ( j 6¼ l) averages out to zero as in scattering vector q depends on the experimental the case of PCS. Hence those terms are dropped, and geometry since it is the difference between the only the case where j ¼ l is considered, which turns the wavevectors of the incident and scattered waves. ’ ’ Each of the molecules may have different phase shifts subscripts of the phase factor lk (or lp) back to the ’ (t). Raman linewidths are typically a few cm 1. This original subscripts k (or p). We assume that the time- k scale of the functions A(t), r(t), and ’(t) are not narrow linewidth allows us to assume that Bk depends only on the particular Raman band being observed, correlated to one another because one is a function of therefore, it is independent of the particular molecule. the molecular orientation time-scale, the second is a Since the scattered field of the jth particle at the function of the diffusional translation time, and the third is a function of the phase change. To simplify the position rj is the sum of all the molecules in the particle, if there are M molecules in the particle, the scattered derivation without loss of the fundamental physics, field from a single particle becomes the N scatterers are assumed to be identical, so that each particle has M molecules. Therefore, Equation (7) XM can be rewritten as Downloaded by [Georgetown University], [Edward R. Van Keuren] at 16:14 27 February 2012 0 iq rj if!tþ’kðtÞg Ej ¼ Aj ðtÞBke e ð3Þ k¼1 G ð Þ¼Ne i! A ðtÞAðtþ Þ e iq rðtÞeiq rðtþ Þ 1 * + XM XM where the amplitude Aj depends on the orientation of 0 i’kðtÞ i’kðtþ Þ 0 i’kðtÞ i’pðtþ Þ 0 Bkp e e þ Bkpe e : the scatter, and Bk incorporates the factor Bk and the k¼p k6¼p exponential term with Drk since both terms are constant in time. ð8Þ The total scattered Raman field from N scatters can The last average term has two components: The then be written as contributions from the interactions between (a) the XN XM same molecules in the same scatterer (k ¼ p) and 0 iq rj if!tþ’jkðtÞg ERS ¼ Aj ðtÞ Bjk e e : ð4Þ (b) different molecules in the same scatterer (k 6¼ p). j¼1 k¼1 Since the Raman scatterings from different molecules 104 M. Nishida and E.R. Van Keuren
are not in-phase with each other [4], then the correla- temporal intensity fluctuation method to characterize tions of the molecules for case (b) in Equation (7) particle diffusion. The above arguments suggest that average out to zero. This implies that only the terms the intensity fluctuation from Raman scattering would with case (a), or the correlation from the same yield the same data as that of elastic scattering if the molecule, provide meaningful data. Therefore, the phase difference from an individual molecule does not case (b) terms can be ignored, and Equation (8) is affect the autocorrelation. That is, a similar apparatus rewritten: as used in PCS can be applied for RCS. One potential issue with using the intensity fluctu- G ð Þ¼Ne i! ½G ð Þ ½G ð Þ ½G ð Þ ð9Þ 1 A r ’ ation analysis of Raman scattering for particle char- where acterization is that fluorescence could be present in the scattering as background noise, and the incoherence of ½ ¼G ð Þ A ðtÞAðt þ Þ A this fluorescence signal would affect the coherence iq rðtÞ iq rðtþ Þ ½ ¼Grð Þ e e factor of the ACF of Raman scattering. However, this ð10Þ XM incoherent signal can be taken into account in the data 0 i’kðtÞ i’kðtþ Þ analysis. Flammer and Ricˇka have shown the effect of ½G’ð Þ ¼ Bk e e : k¼1 such a background signal in PCS obtained by using a single mode fiber [7]. Their analysis holds for RCS as The amplitude autocorrelation function [G ( )] is A well. For the case of light collection with a single mode usually constant, so it does not affect the decay rate fiber, the predicted signal transmitted to the detector is of the ACF, but the autocorrelation function [G ( )] is r J(t) ¼E(t)E (t) where E(t) is the signal amplitude. The due to the position of the diffusing scatterer and so it electric field in RCS may be a sum of two independent depends on the particle translational diffusion. The contributions: E(t) ¼E (t) þE (t), where E (t) is from ACF in PCS is characterized by these two autocorre- R F R Raman scattering and E (t) is from fluorescence lation functions [6]. However, the contribution of the F emission. That is, in RCS, the signal J(t) can be phase autocorrelation function [G ( )] of the Raman ’ expressed as scattering is a key factor in the utility of RCS. If there is a phase shift in Raman scattering as in JðtÞ¼JRðtÞþJFðtÞþERðtÞEFðtÞþEFðtÞERðtÞð11Þ Equation (9), the simplest case would be that ’k(t) is not a function of time but rather constant. This where JR(t) and JF(t) are the signals from Raman or fluorescence emission, respectively, as functions of assumption makes the function [G’( )] constant, which make the derivation of the ACF of Raman fields very time. Both Raman and fluorescence contributions are similar to the case of the Rayleigh fields in PCS. At assumed to obey complex Gaussian statistics, but while ¼ 0, the time average of the phase also becomes the fluctuations in the Raman scattering are much constant. Therefore, it has no effect on the time decay longer than the sampling time of the detector, the but only contributes as a multiplier of the ACF. Thus, fluorescence is incoherent and thus produces fluctua- the ACF of RCS would become almost the same as tions faster than the sampling time [2,7,8]. Since its that of PCS, except for additional constant terms [6]. emission also occurs at random times, JF(t) is replaced Since the additional terms would not contribute to the with its average JF, and the cross-terms with EF(t) then diffusional decay rate, the PCS theory could be directly also average to zero. With these approximations, the normalized intensity ACF, which is defined as g2( ) ¼ applied to the RCS method. If this is the case, it can be 2 seen that the coherence of the Raman effect allows the hI(t) I(t þ ) i/h I(t) i, can be rewritten as in the Siegert ACF of its intensity fluctuation to yield meaningful relation; data as in PCS. 2 2 g2ð Þ¼1 þ jR½ g1ð Þ : ð12Þ Downloaded by [Georgetown University], [Edward R. Van Keuren] at 16:14 27 February 2012 On the other hand, if ’k(t) is not constant, the phase correlation will affect the decay of the ACF. For The factor jR is equivalent to a coherence factor and is a time T that is much longer than the time-scale of defined as the phase shift (T t’), the cross-term of the phase hJRi contribution would become zero. If it were shorter jR ¼ : ð13Þ J þhJ i than the typical time-scale of diffusional translation, F R the ACF would not be useful, since the correlation Therefore, although the ideal coherence factor is unity, would rapidly decay to zero and the data would rather the fluorescence reduces the signal-to-noise ratio in a appear as noise, with no dynamic information on the known way and can be accounted for in the data particle diffusion. analysis. If, however, jR is too small or the fluorescence Therefore, the phase autocorrelation function is too strong, the ACF would become nearly constant, [G’( )] determines how effective RCS would be as a so that the accuracy of the fitting of the decay rate of Journal of Modern Optics 105
the particle diffusion would be lost. Therefore, the References fluorescence must be minimized experimentally. [1] Berne, B.J.; Pecora, R. Dynamic Light Scattering: with Applications to Chemistry, Biology, and Physics; John Wiley: New York, 1976. [2] Mukamel, S. Principles of Nonlinear Optical 3. Conclusion Spectroscopy; Oxford University Press: New York, We have recently conducted an experimental study of 1995. RCS [9]. The study shows that challenges due to the very [3] Levenson, M.D.; Kano, S.S. Introduction to Nonlinear weak nature of Raman scattering, which can easily be Laser Spectroscopy. Revised; Academic Press: New York, masked by the fluorescence, and the necessity of 1988. maintaining the balance between the coherence area [4] Long, D.A. The Raman Effect: A Unified Treatment of the and the detected area of the sample for the intensity Theory of Raman Scattering by Molecules; John Wiley: Chichester, 2002. fluctuation analysis require precision instrumentation [5] Schrof, W.; Klingler, J.; Rozouvan, S.; Horn, D. Phys. with high sensitivity and resolution. However, we have Rev. E 1998, 57, R2523–R2526. shown here that if the phase fluctuations are sufficiently [6] Cummins, H.Z.; Swinney, H.L. Light beating slow, then RCS will work as a variation of PCS, and it spectroscopy. In Progress in Optics, Vol. III: could become an important addition to the collection of Wolf, E., Ed.; North-Holland Publishing: Amsterdam, nanoparticle characterization techniques for studying 1970; Chapter 3, pp. 133–200. colloidal dispersions due to its capability to distinguish [7] Flammer, I.; Ricˇka, J. Appl. Opt. 1997, 36, 7508–7517. the specific chemical bonds within a complex mixture. [8] Rousseau, D.; Williams, P. J. Chem. Phys. 1976, 64, 3519–3537. [9] Nishida, M. Raman Correlation Spectroscopy: A Feasibility Study of a New Optical Correlation Technique and Development of Multi-component Acknowledgement Nanoparticles Using the Reprecipitation Method. Ph.D. This material is based upon work supported by the National Dissertation, Georgetown University, Washington, D.C., Science Foundation under Grant No. DMR 0348955. 2011. Downloaded by [Georgetown University], [Edward R. Van Keuren] at 16:14 27 February 2012