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Derivation of the Optical Autocorrelation Function from Raman Scattering of Diffusing Particles Maki Nishida a & Edward R

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Derivation of the Optical Autocorrelation Function from Raman Scattering of Diffusing Particles Maki Nishida a & Edward R This article was downloaded by: [Georgetown University], [Edward R. Van Keuren] On: 27 February 2012, At: 16:14 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK 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 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Journal of Modern Optics Vol. 59, No. 2, 20 January 2012, 102–105 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 à à 0 ÀiqÁrj if!tþ’jkðtÞg G1ðÞ¼hE ðtÞEðt þ Þi: ð1Þ G1ðÞ¼ 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 iqÁrl Àif!ðtþÞþ’lpðtþÞg k  Al ðt þ Þ Blp e e : position of the particle, can be approximated as l¼1 p¼1 iqfrþDrkg Àif!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 cmÀ1. 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 iqÁrj Àif!tþ’kðtÞg Ej ¼ Aj ðtÞBke e ð3Þ k¼1 G ðÞ¼NeÀi! AÃðtÞAðtþÞ eÀiqÁrðtÞeiqÁrðtþÞ 1 * + XM XM where the amplitude Aj depends on the orientation of 0 i’kðtÞ Ài’kðtþÞ 0 i’kðtÞ Ài’pðtþÞ 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.
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