Optical correlation techniques for the investigation of colloidal systems Roberto Piazza Dipartimento di Chimica, Materiali e Ingegneria Chimica “Giulio Natta”, Politecnico di Milano, Milano, Italy This review aims to provide a simple introduction to the application of optical correlation meth- ods in colloidal science. In particular, I plan to show that full appraisal of the intimate relation between light scattering and microscopy allows designing novel powerful investigation techniques that combine their powers. An extended version of this paper will appear in Colloidal Foundations of Nanoscience, edited by D. Berti and G. Palazzo, Elsevier (ISBN 978-0-444-59541-6). I am very grateful to the publisher for having granted me the permission to post this preprint on arXiv. Scattering or microscopy experiments necessarily in- it is useful to introduce the associated analytical signal[1] volve statistical fluctuations, which already stem from 1 ∞ ∞ the optical source used to probe the investigated system, u(t)= dω u˜R(ω)e−iωt =2 dν u˜R(ν)e−i2πν , are modified by the interaction of the probing field with π 0 0 Z Z the sample, and are further influenced by the detection (1) process. All these effects concur in turning optical fields which is then a complex quantity obtained by suppress- R into random signals, which are physically described in ing the negative frequency components of u (t) and dou- terms of correlations. In optics, fluctuations and correla- bling the amplitude of the positive ones.[2] For a narrow- tions are nicely embodied in the concept of coherence. As band signal, having a spectrum centered on ω0 of width R the Roman god of beginnings and transitions, Janus, co- ∆ω ω0, we can write u (t) = A(t) cos[ω0t φ(t)], ≪ −iω0t iφ(t) − herence is however two-faced: because the field fluctuates hence u(t)= U(t)e , where U(t)= A(t)e is called both in time and space, one should indeed distinguish the complex envelope. between temporal and spatial coherence. Setting apart The crucial point that we are going to discuss is that these two aspects is not always possible, since they can any signal with finite bandwidth must display temporal be intrinsically intermixed, but when this is feasible, it is fluctuations: specifically, the envelope U(t) of a signal far more than a useful practical approach. As a matter with bandwidth ∆ω does not appreciably change in time of fact, it involves an important conceptual distinction: on time scales much shorter than a coherence time τc = whereas temporal coherence is a physical concept, related 2π/∆ω, to which we can associate a coherence length to the spectrum of the optical signal generated by the in- ℓc = cτc. To see this, let us introduce the time correlation teraction of the incoming field with the sample, which is function of the analytic signal, or self-coherence function therefore the actual “source” of the detected radiation, Γ(τ)= u∗(t)u(t + τ) , (2) spatial coherence has mostly to do with the source ex- h it tension, so it is usually (but not always) a geometrical where the average is performed over the initial time t, and problem. Curiously, in spite of this, spatial coherence is we assume the process to be stationary, so that Γ does far more important, for the physical problems we shall not depend explicitly on t. Normalizing Γ(τ) to is initial investigate, than temporal coherence. Nevertheless, it is 2 value Γ(0) = u(t) t = I, we obtain the degree of first useful to start by recalling some basic concepts of the order coherence| (usually| simply dubbed “field correlation latter. We shall first refer to the temporal coherence function”) properties of optical fields, or “first order” optical coher- ence, to distinguish it from correlations of the intensity, u∗(t)u(t + τ) arXiv:1306.1401v1 [cond-mat.soft] 6 Jun 2013 g (τ)= h it (3) discussed later. 1 I Provided that a signal has a finite average power we can define its power spectral density T BASIC CONCEPTS IN STATISTICAL OPTICS R 1 R iωt Pu (ω) = lim dtu (t)e (4) T →∞ T Z−T Temporal coherence From the definition (1) it can be easily shown that the power spectrum Pu(ω) of the complex analytic signal is R Temporal fluctuations can be equivalently discussed in just 4Pu (ω) for ω 0, and 0 otherwise. The fundamen- the frequency domain, where it is basically related to tal link between the≥ time and frequency description is non-monochromaticity. For a generically time–varying then provided by the Wiener-Kintchine (WK) theorem, R R R real field u (t) with Fourier transform F [u ]=˜u (ω), which states that Γ(τ) and Pu(ω) are Fourier transform 2 pairs. If we define the normalized power spectrum of the its value averaged over many optical cycles that, for a 2 real signal as narrowband signal, is Irad = (ǫ0c/2)A , where ǫ0 is the vacuum permittivity and c the speed of light. Follow- R Pu (ω) ∞ for ω 0 ing a common convention, rather than the “radiometric” P (ω)= dω P R(ω) ≥ (5) 0 u intensity Irad, we shall simply call “intensity” the quan- 0 for ω< 0 tity I = A2 (actually an irradiance). Changing again R variable, we get the WK theorem can be restated in the form ∞ iωτ 1 I 1 I P (ω)= F [g1(τ)] = −∞ dτg1(τ)e PI (I)= exp = exp . (9) (6) 2σ2 −2σ2 I − I F −1 ∞ −iωτ h i h i ( g1(τ)= [P (ω)]R = −∞ dωP (ω)e , The intensity has therefore an exponential probability which will be particularly usefulR for our purposes. The density, with a decay constant given by its average value degree of temporal coherence is strongly related to I . h i the signals detected in classical interferometric mea- These probability distributions for the field and inten- surements, such as those obtained with a Michelson sity apply for instance to a spectral lamp, but also, as we interferometer.[3] Qualitatively, the beams propagating shall see, to a medium containing scatterers. As a matter in the two arms of the interferometer can interfere only of fact, a gaussian distribution for the field characterizes if the difference ∆l between the optical paths is smaller any “random” optical source. However, the spectrum than the coherence length of the source ℓc. Quantita- and the time–correlation function depend on the physi- tively, one finds that the time dependence of the detected cal origin of the frequency broadening. Indeed, for in- intensity is given by dependent emitters, we have ui(0)uj (t) = 0 for i = j. Hence: h i 6 I = I0 1 + Re[g1(∆t)] , (7) { } N Γ(τ)= U ∗(0)U(τ) = u (0)u (τ) = N u(0)u(τ) . with ∆t = ∆l/c, which is then proportional to the real h i h i i i h i i=1 part of the time correlation function, evaluated at the X delay ∆t. The field correlation function of the system coincides As an important example for what follows, we briefly therefore with the correlation function for a single emit- describe the temporal properties of a narrowband thermal (i) ter, g1(τ) g1 (τ), which is determined by a spe- source, defined as a collection of many microscopic inde- cific physical≡ mechanism. Let us for instance con- pendent emitters, such as a collection of thermally ex- sider the model we formerly introduced, correspond- cited atoms, all radiating at the same frequency ω , but 0 ing to a “collision-broadened” source, where g1(τ) = undergoing collisions that induce abrupt phase jumps. e−iω0τ ei[φ(τ)−φ(0)] . The phases φ(0) and φ(t) are cor- With N identical emitters, the total signal amplitude related only if the atom does not undergo collisions in τ, (the complex envelope) can be written so the phase correlation function is proportional to the N N probability of colliding at any t > τ, which is easily found iφ(t) iφi(t) to be exp( τ/τ ), where τ is the average time between U(t)= A(t)e = ui(t)= a e , − c c i=1 i=1 collisions. Hence X X iφi(t) g (τ) = exp( iω τ τ/τ ), (10) where ui(t)= ae is the complex envelope for a single 1 − 0 − c emitter. This is nothing but a N-step random walk in the with τ playing therefore the role of coherence time (for complex plane. For large N, u = Re(U)= A cos(φ) and c r a gas at 300K, 105 Pa, τ 30ps and ℓ 1cm). It is for u = Im(U) = A sin(φ) have therefore a joint Gaussian c c i instance easy to show that,≃ in a Michelson≃ interferometer, statistics the fringe visibility is related to τc by 2 2 1 ur + ui p(r, i)= exp , (8) Imax Imin − 2πσ2 − 2σ2 − =e ∆t/τc , Imax + Imin with σ = a√N. By a standard transformation of vari- where ∆t is the difference in propagation time between ables, it is easy to show the probability density for the the two arms. Fourier–transforming g1(t), we obtain a amplitude is a Rayleigh distribution Lorenzian lineshape for the power spectrum A A2 1 1 pA(A)= exp (A 0) P (ω)= (11) σ2 −2σ2 ≥ πτ (ω ω )2 + (1/τ)2 − 0 A photodetector does not respond to the instantaneous In view of our application to light scattering, it is also optical intensity associated to the signal, but rather to useful to have a brief look to the temporal coherence of 3 a laser source.