arXiv:1306.1401v1 [cond-mat.soft] 6 Jun 2013 ohi ieadsae n hudide distinguish indeed should one space, and between time fluctuates in field co- the both Janus, because transitions, two-faced: however and is beginnings herence of god Roman the ffc,i novsa motn ocpuldistinction: matter conceptual a a is important coherence As an temporal involves whereas approach. it practical fact, useful is of a it can feasible, than they is more since this far when possible, but always intermixed, intrinsically not be is aspects two these in r ieyebde ntecnetof concept the in correla- in embodied and nicely fluctuations described are optics, tions physically In are correlations. fields of which optical terms turning signals, detection in random the concur into effects by these influenced with All further field process. are probing the and of sample, interaction the system, the investigated by the from modified probe to are stem used already source optical which the fluctuations, statistical volve ne odsigihi rmcreain fthe of coherence correlations temporal from coher- it optical the distinguish order” the to “first to or of ence, refer fields, concepts optical first basic of shall properties some We recalling by latter. is shall start it we to Nevertheless, problems useful coherence. physical temporal is the than coherence for investigate, a spatial important, always) this, more of ex- not far spite source (but in the usually Curiously, with is problem. do it to radiation, so mostly detected tension, the has is of coherence which “source” spatial sample, actual the with in- the the field therefore by incoming generated the signal of optical teraction the of spectrum the to h rqec oan hr ti aial eae to related time–varying basically generically field is real a it For where non-monochromaticity. domain, frequency the icse later. discussed AI OCPSI TTSIA OPTICS STATISTICAL IN CONCEPTS BASIC cteigo irsoyeprmnsncsaiyin- necessarily experiments microscopy or Scattering eprlflcutoscnb qiaetydsusdin discussed equivalently be can fluctuations Temporal iatmnod hmc,MtraieIggei hmc “G Chimica Ingegneria e Materiali Chimica, di Dipartimento temporal u R ( rtflt h ulse o aiggatdm h permissi the me granted having for publisher the to grateful htcmieterpwr.A xeddvrino hspaper this of version extended An powers. their combine that d nclodlsine npriua,Ipa oso htf that show n to designing plan allows I microscopy particular, and In scattering light science. between colloidal in ods fNanoscience of t ihFuirtransform Fourier with ) hsrve ist rvd ipeitouto oteappl the to introduction simple a provide to aims review This eprlcoherence Temporal and spatial dtdb .BriadG aaz,Esve IB 978-0-444 (ISBN Elsevier Palazzo, G. and Berti D. by edited , oeec.Stigapart Setting coherence. o h netgto fclodlsystems colloidal of investigation the for physical pia orlto techniques correlation Optical F ocp,related concept, [ coherence u R geometrical ˜ = ] intensity u oet Piazza Roberto R As . ( ω ), , function”) coherence order ln h mltd ftepstv ns[]Fra For ones.[2] positive band the of amplitude the bling a ikbtentetm n rqec ecito is description frequency and the by time provided the then between link tal ut4 just rmtedfiiin()i a eesl hw htthe that shown easily be can spectrum it power (1) definition the From hc ttsta Γ( that states which ∆ ntm clsmc hre hna than shorter much scales time on ∆ bandwidth with easm h rcs ob ttoay ota does Γ that = so Γ(0) stationary, value on be explicitly to depend not process the time assume initial the we over performed is average the where hence hc ste ope uniyotie ysuppress- of by components obtained frequency quantity negative the complex ing a then is which rvddta inlhsafiieaeaepwrw can we power average finite a its has define signal a that Provided utain:seicly h envelope the specifically, fluctuations: ℓ ucino h nltcsga,or signal, analytic the of function 2 the n inlwt nt bandwidth finite with signal any ti sflt nrdc h associated the introduce to useful is it c π/ u ω h rca on htw r on odsusi that is discuss to going are we that point crucial The = ( ui at” oienc iMln,Mln,Italy Milano, Milano, di Politecnico Natta”, iulio ∆ t ope envelope complex ≪ = ) cτ ω inl aigasetu etrdon centered spectrum a having signal, P u c ω owihw a soit oeec length coherence a associate can we which to , u R ( nt otti rpito arXiv. on preprint this post to on oseti,ltu nrdc h iecorrelation time the introduce us let this, see To . vlpwru netgto techniques investigation powerful ovel 0 t π ( 1 l pria fteitmt relation intimate the of appraisal ull = ) ecnwrite can we , oe pcrldensity spectral power ω ilapa in appear will Z for ) P cto fotclcreainmeth- correlation optical of ication 0 u R U ∞

| ( ( u ω d t g uulysml ubd“edcorrelation “field dubbed simply (usually Γ( ( ω )e P ω lim = ) 1 t ( ) u ω τ ≥ − u ˜ τ | ( 2 = ) i R = ) ω τ osntapeibycag ntime in change appreciably not does ω

. T ,ad0ohrie h fundamen- The otherwise. 0 and 0, ( and ) 0 t ftecmlxaayi inlis signal analytic complex the of ) Wiener-Kintchine →∞ ω t = where , olia Foundations Colloidal h )e 5516.Ia very am I -59541-6). h u t u I omlzn Γ( Normalizing . − ∗ u T ∗ 1 ( eoti the obtain we , i R ( t P ωt ) t ( Z u ) u t − u ( 2 = = ) ( U T ω T ( I t t are ) ( + d efchrnefunction self-coherence must t + Z = ) u t τ 0 A oeec time coherence τ ) ∞ R ) i ( nltclsignal analytical i ore transform Fourier t t A ( d t cos[ ) ipa temporal display , t U ν ( )e (WK) t u ( u )e ˜ i τ t R ereo first of degree ωt R fasignal a of ) oi initial is to ) i ( ω φ ω ( t ( ν 0 n dou- and ) 0 t t )e ) theorem fwidth of scalled is − narrow- − i2 t φ and , πν τ ( c t (4) (2) (1) (3) [1] , )], = , 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. Even when operating on a single lon- Fringe visibility is actually a manifestation of the spa- gitudinal mode, like the diode–pumped solid–state lasers tial coherence of the fields at the pinholes. Consider in- (DPSS) now extensively used in light scattering measure- deed two points U and V on Σ, which we assume to be a ments, a laser is not an ideal monochromatic source, for is thermal source made of many independent and spatially displays phase fluctuations due to the intrinsic nature of uncorrelated emitters, and call ui and vi the amplitude of the lasing process but also, in practice, to coupling with the fields reaching pinhole Pi from U and V respectively. mechanical vibrations of the cavity mirrors. Well above If P and P are very close, so that u u , v v ), the 1 2 1 ≃ 2 1 ≃ 2 lasing threshold and at steady–state, the field amplitude fields U(P1)= u1 + v1, U(P2)= u2 + v2, will be strongly can be written as[4] correlated (they are almost the same field!), even if the fields u and v are fully uncorrelated. Namely, propa- R u (t)= A cos[ω0ϑ(t)] + un(t). gation from Σ to the screen induces spatial correlations even if different points of the source are uncorrelated. where un(t) is a narrowband noise due to spontaneous However, if P2 is moved apart from P1, the phases of emission, while phase fluctuations are embodied in ϑ(t). the fields coming from U and V change differently. If Neglecting the additive noise contribution, which is usu- ru(Pi) and rv(Pi) are the distances of U and V from pin- ally very small, neither the amplitude nor the intensity hole P , putting ∆r = r (P ) r (P ), ∆r = r (P ) i u u 1 − u 2 v v 1 − probability densities differ however from those of an ideal rv(P2) we have at first order ∆ru = ∆rv dD/z, monochromatic source. Mechanical stability usually sets where D is the distance UV . Spatial− field correlation≃ a lower limit of the order of tens of MHz to the laser is retained only provided that ∆ru ∆rv λ, namely, bandwidth, which is far wider than the extremely nar- d λz/D. In the Young setup, the− fields≪ coming from row line of an ideal single–mode laser: yet, this is mostly U ≪and V form two displaced sets of fringes. However, due to phase fluctuations, hence intensity fluctuations are if the pinhole are sufficiently close, fringe oscillations are usually negligible. However, scattering measurements are coarse, the shift of the two patterns is a small fraction of often still made using common lab sources, such as sim- their period, and the sum of the two interference patterns ple He-Ne lasers, which oscillates on many longitudinal still shows fringes. Conversely, if the pinholes are moved modes separated by c/2L, where L is the cavity length. apart, fringe oscillation becomes more rapid and the two By increasing the number of oscillating modes, and pro- sets of fringes soon gets strongly out of phase, canceling vided that coupling between different modes is weak, the out. intensity fluctuations approach those of a thermal source When U and V are taken as far as possible, so that D is with a bandwidth equal to that of the atomic gain line the maximal lateral extension of the source, the pinholes of the laser. 2 must therefore lie within a coherence area Ac (zλ/D) . To the source is then associated a “coherence≃ cone” with solid angle at vertex ∆Ω (λ/D)2, which corresponds Spatial coherence to an angular aperture 2α≃ λ/D. Conversely, the solid angle under which the source≃ is seen from the pinhole Suppose we illuminate with a laser beam a light dif- plane is ∆Ω′ = D/z2, so the coherence area can also be 2 ′ fuser, for instance a window made of ground glass: then, conveniently expressed as Ac λ /∆Ω . For example, a complex figure made of many irregular spots forms on the coherence area at a distance≃ of 1m of a thermal source a screen placed beyond the diffuser, which is what we of diameter D = 1 mm emitting at λ = 0.5 µm is Ac call a speckle pattern. If we insert a lens and enlarge 0.25 mm2, whereas at the same wavelength the coherence≃ the beam spot on the diffuser, the speckle size reduces. area for the sun, which has an apparent angular diameter ′ ′ −5 −3 2 Conversely, if we move the diffuser towards the lens focus 2α 32 (∆Ω 7 10 sr), is Ac 4 10 mm . plane, the speckle pattern becomes much coarser. Hence, Note≃ that for a≃ star× like Betelgeuse (≃α Orionis),× with ′′ 2 the speckle size depends on the extension of the illumi- 2α 0.047 , Ac is conversely as large as about 6 m . nated region on the diffuser. This≃ last example, showing that the coherence area of Again, reflecting upon an interferometric experiment, the light emitted by a star is fully coherent over the size in this case made with a classical two-pinhole Young’s of our eye pupil, actually explains why stars “twinkle”, setup, sheds light on the origin of this effect. When an while a planet with a sizeable angular size does not. Of absorbing screen pierced by two pinholes P1 and P2 sepa- course, air turbulence, which is the physical mechanism rated by a distance d is illuminated by a monochromatic generating intensity fluctuations, affects the light coming point-like source, fringes with a spatial period ∆x = lλ/d from a planet too, but these fluctuations gets averaged form on a plane placed at distance l from the screen. out if the number of coherence areas on our eye pupil is However, if we illuminate the pinholes with an extended large. source Σ of size D made of independent emitters and The former considerations can be made quantitative placed at distance z from the screen, the fringe pattern by introducing the key concept of mutual intensity. Still forms only provided that Dd/z λ. considering a quasi-monochromatic source, so that all de- ≪ 4 lays in propagation are much shorter than τc, we call mu- Intensity correlation tual intensity the spatial correlation of the field at two different points In section we have investigated the temporal coher- ∗ ∗ ence properties of optical fields. Scattering techniques, J12 = J(r1, r2)= u (r1,t)u(r2,t) = U (r1,t)U(r2,t) , h i h (12)i however, usually probe intensity correlations, which are which, when r1 = r2 = r, becomes just the intensity I(r) described by means of the normalized time–correlation in r. The normalized mutual intensity is called degree of function spatial coherence I(t)I(t + τ) u∗(t)u∗(t + τ)u(t + τ)u(t) g (τ)= h it = h it . 2 2 ∗ 2 J(r1, r2) I(t) u (t)u(t) µ = . (13) h it h it √I1I2 (16) Note that, for τ , g (τ) 1, whereas g (τ) 0. An extremely interesting result about spatial coherence → ∞ 2 → 1 → While for an ideal monochromatic source g2(τ) = 1 for comes from considering how J12 propagates from a given surface, where it is known, to another surface. The gen- all values of τ, for a random source, we should evaluate eral problem is rather complicated, but it considerably the rather complicated double sum simplifies if the first surface is actually a planar source N Σ that can be considered as fully spatially incoherent, by I(t)I(t + τ) = u∗(t)u∗(t + τ)u (t + τ)u (t) . h i i j i j which we mean that, over Σ, i,j=1 X (17) J(ρ , ρ )= I(ρ )δ(ρ ρ ). 1 2 1 2 − 1 Due to the independence of the emitters, however, a given Denoting by ρ the coordinates on the source plane, term averages to zero unless it contains only products of and r those on an observation plane further down the a field times its complex conjugate relative to the same propagation axis, one indeed obtains in the paraxial emitter. For a very large number N of emitters, splitting approximation[5] the averages and taking into account that all emitters are −iψ identical, the dominant contribution to the sum, which e 2 2π J(r1, r2)= d ρ I0(ρ)exp i ρ r (14) is of order N 2, is found to be (λz)2 λz · ZΣ   2 2 2 ∗ 2 ∗ 2 where ∆r = r1 r2 and ψ = π[r r )]/λz. Hence, apart I(t)I(t + τ) N u (t)u (t) + u (t)u (t + τ) , − 1 − 2 h i≃ h i i i |h i i i| from a scaling and phase factor, the mutual intensity is h i 2 ∗ 2 2 the Fourier transform of the intensity distribution across which, noticing that N ui (t)ui(t) = I(t) , yields the source. Eq. (14) is the Van Cittert-Zernike (VCZ) the important Siegert relationh : i h i theorem, arguably the most important result in statistical g (τ)=1+ g (τ) 2. (18) optics.[6] By means of the VCZ theorem, it can be shown 2 | 1 | that the coherence area is quantitatively given by Hence, for a random source, g2(τ) does not yield any ad- 2 2 2 2 I(x, y) dxdy (λz) I ditional information, and can be directly obtained from Ac = (λz) | | = , (15) 2 A 2 g1(τ); in particular, for a collision-broadened thermal R I(x, y)dxdy s I h i source g2(τ) = 1+exp( 2 τ /τc). Nevertheless, the dis- where As is the R area of the source. For a incoherent tinctive difference in the− long-time| | asymptotic behavior source with uniform intensity (which may be an in- between g2(τ) and g1(τ) yields, as we shall see, a crucial coherently and uniformly illuminated sample), so that advantage for intensity correlation techniques. 2 2 2 I = I , Ac = (λz) /As, consistently with our quali- tative approach.h i

The coherence area basically yields the size of the DYNAMIC LIGHT SCATTERING (INTENSITY speckles produced by a source or a diffuser around each CORRELATION SPECTROSCOPY) point P on the screen. Since the field in P is a ran- dom sum of the contributions coming from all points The most popular optical correlation technique in col- on the source, which are independent emitters, the to- loid science is Dynamic Light Scattering, which I shall tal amplitude has a Gaussian statistics. The distribution also call “Intensity Correlation Spectroscopy”, a denom- of the speckle intensity (namely, the distribution of the ination that captures much better, as we shall see, the intensity at different points on the screen) is hence ex- essence of the method. This short presentation is mostly ponential, so there are many more “dark” speckles than meant to stress those fundamentals of the technique that “bright” speckles. What is more important, according to are essential to grasp the more recent advancement we the VCZ theorem the “granularity” of the speckle pat- shall later discuss. For the same reason, we shall just tern should depend only on the geometry of the source, discuss DLS from a system of non-interacting particles, and not on its physical nature. We shall later see that referring to excellent books and reviews[7–11] for a more this is not always necessarily true. comprehensive treatment. 5

To spot the key feature of an intensity correlation mea- tude. However, this is not true for many systems of in- surement, let us make a comparison with a simple spec- terest in colloid science, such as glasses and gels: we shall troscopic or interferometric experiment, where the sig- comment on these “nonergodic” systems shortly. Second, nal is related to the spectrum E(ω), and therefore to the Siegert relation connecting field and intensity corre- the field time correlation function of the source, which lations is violated when the number N of particles in V in our case is the scattering volume. To select a given is very small, which may be the case when performing frequency, we have to insert a filter (such as a monochro- measurements on very diluted suspensions under a mi- mator) on the optical path, and then detect the signal croscope, if the coherence area of the illuminating source at the selected frequency. The basic strategy of DLS is small. In this case, by retaining the terms of order is simply moving the filter after the detector, so that 1/N in Eq. (17), one can show that Eq. (18) contains an the photocurrent output i(t) of the detector, instead of additional a number fluctuation term: the optical signal, is filtered. Any optical detectors is 2 δN(0)δN(τ) necessarily quadratic, namely, it detects a signal propor- g2(τ)=1+ g1(τ) + h 2 i , (19) ∗ | | N tional to the time–averaged intensity I(t) = E (t)E(t): h i hence, by using a filter whose central frequency can be where δN(τ)= N(τ) N decays on a time scale com- swept through a given range, the power spectrum of the parable to the time it− takes h i for a particle to move across signal can be obtained. Because of Wiener–Kintchine the scattering volume. theorem, an equivalent procedure is measuring the time The field scattered by a particle suspension can be correlation function of i(t), which is directly related to written as I(t)I(t + τ) . Whatever the choice, we shall see that E (q,t)= E b (q,t)eiq·ri(t). (20) hoperating oni the photocurrent is a winning strategy for s 0 i i a basic reason: at variance with field correlation spec- X troscopy or interferometry the spectral bandwidth ∆ωs If particles are all identical, and provided that the scat- (or the correlation time τs) of the source illuminating the tering amplitudes do not depend on time (which holds scattering volume poses no limitation to the measure- true for optically isotropic particles), the normalized field ments, even when the spectral bandwidth of the scat- correlation function is then given by ∗ tered field ∆ω ∆ωs (corresponding to a correlation E (q, 0)E (q, τ) ≪ s s −iωτ time τ τ ). The first approach, based on using a spec- g1(q, τ)= h 2 i = F (q, τ)e s Es(0) trum analyzer,≫ was mostly used at the dawn of DLS. The | | invention of the digital correlator (once a complex dedi- where we have defined the intermediate scattering func- cated instrument, now just a PC data acquisition board), tion (ISF) which allows to work in the time domain, has however − q· r −r been crucial to make DLS the spectroscopic method with F (q, τ)= e i [ i(0) j (τ)] , (21) * i,j + the highest resolving power ever devised. X which is nothing but the FT (in frequency) of the dy- namic structure factor S(q,ω) measured in quasi-elastic Time-dynamics of the scattered field neutron scattering experiments.[12] Neglecting interac- tions amounts of course to assume that the position of In a scattering experiment, the linear dimension of different particles are uncorrelated, so g1(q, τ) is propor- the scattering volume V is usually much larger than the tional to the self ISF range ξ of the structural and hydrodynamic correlations F (q, τ)= exp[iq ∆r(τ)] (22) of the systems, even when the latter extend over large s h · i spatial scales compared to the particle size. Hence, V where ∆r(τ) = r(τ) r(0). Therefore, Fs(q, τ) is the can ideally be split into volume elements δV satisfying average value of exp[iq− ∆r(τ)] over the probability dis- ξ3 δV V . Consequently, V can be regarded as tribution p(∆r, τ) of the· particle displacement in a time ≪ ≪ a random source, where these uncorrelated volumes δV τ. Note that, as a matter of fact, q ∆r is just the com- · play the role of “elementary emitters”. We may then ponent ∆rq of the particle displacement in the direction expect the scattered field and intensity to display, re- of the wave-vector q. Hence, Fs(q, τ) can be seen as the spectively, a gaussian and an exponential statistics, and Fourier transform F [p(∆rq , τ)], which is the characteris- the time correlation functions of Es and Is to be dic- tic function of p(∆rq , τ). Given the characteristic func- tated by the temporal correlation of the field emitted by tion, all the moments of a probability distribution are a single elementary emitter, which will be related to the easily calculated. For instance, the mean square particle particle dynamics in δV . There are however a couple of displacement along q is given by warnings. First, the total scattered field has a gaussian ∂g (q, τ) statistics only provided that the field scattered by each ∆2r (τ) = 1 (23) q − ∂q2 single emitter is fully fluctuating in phase and/or ampli-  q=0

6

Time-correlation of the field scattered by Brownian slow: for instance, expressing its radius R in nanome- particles ters, a spherical particle in water at 20◦C has D (2.15/R) 10−6 cm2/s. Since the largest accessible q≃- × − The simplest model of a freely–diffusing Brownian par- values in light scattering are about 3 105 cm 1, even × ticle is that of a mathematical random walk. In one di- for a small surfactant micelle with a radius R = 2 nm the mension, the particle motion is seen as a sequences of spectral broadening is of the order of 0.1 MHz, which is random “steps” xi along the positive or negative direc- negligible compared to the bandwidth of a spectral lamp, tion, so that xi = 0 and, if we assume the steps to be or of a common laser with no longitudinal mode selection. h i 2 2 uncorrelated xixj = xi δij = ∆ δij . Then, because For “usual” colloids with a size in the tenths of a micron of the Centralh Limiti Theorem, the total displacement range, the situation is obviously far worse. Measuring a N x = i=1 xi for a large number N of steps is a gaussian spectral broadening that is much smaller than the source 2 2 2 random variable with x = 0 and σx = x = N∆ . intrinsic bandwidth is of course extremely challenging: ThisP corresponds, in ah continuumi description, to a dif- as a matter of fact, it is totally out of question for any fusion process with a diffusion coefficient D = ∆2/2∆t, spectroscopic method relying on field correlations. where ∆t is the time it takes for a step. Generalizing to Yet, things change dramatically if we consider inten- 3D, the particle mean square displacement is then given sity correlations. This is probability easier to see in by r2(t) =6Dt, where, because of the celebrated Ein- the time domain. Assume that a source has a band- stein’s relation, the diffusion coefficient is related to the width ∆ωs, hence a coherence length ℓc 2πc/∆ω. ≃ hydrodynamic friction coefficient[13] ζ by D = kBT/ζ . If the scattering volume has linear dimensions ℓ = 1/3 For t 0, the random walk model yields however a (Vs) ℓc, which is usually the case,[15] each point → ≪ rather unphysical result, because the particle velocity di- in Vs basically “sees” the same incident field. Hence, −1/2 verges as t . A more consistent description is ob- we can write Es(q,t) = B(q,t)E0(t), where E0(t) tained from the Langevin equation,[14] whose solution is the incident field and B(q,t) = bi(q) exp[iq i · shows that the particle motion becomes diffusive only af- ri(t)] the total scattering amplitude. However, E0(t) P ter the hydrodynamic relaxation time τB = m/ζ, where and B(q,t) are clearly independent random vari- ∗ ∗ m is the particle mass, which is the decay time of the ve- ables, so we have: B (q, 0)E0 (0)B(q, τ)E0(τ) = ∗ ∗ h i locity time-correlation function. It is also useful to note B (q, 0)B(q, τ) E (0)E0(τ) . Hence, the field correla- h i h 0 i that the diffusion coefficient is just the time integral of tion function factorizes as the latter S B g1(q, τ)= g1 (τ)g1 (q, τ) 1 ∞ D = v(0) v(t) dt (24) S 3 0 h · i where g1 (τ) is the time correlation function of the source Z B and g1 (q, τ) is the sample correlation function due to For t τB, the probability for a particle to be in r if it S ≫ particle Brownian motion. Since g1 (τ) decays to zero on was in the origin at t = 0 is then a gaussian. Note how- the correlation time τc of the source, which is far shorter ever that we need only the component of the displace- than the Brownian correlation time, there is no way to ment in direction of q (which can in fact be taken as the B follow the decay of g1 . Consider however the intensity x axis), hence p(∆rq, τ) is a gaussian with ∆rq = 0 correlation function. Again, we can write and variance σ2 = 2Dτ. Being the characteristich i func- S B tion of a gaussian centered on the origin, Fs(q,t) is it- g2(q, τ)= g2 (τ)g2 (q, τ) self a gaussian in q with variance 1/σ2 = (2Dτ)−1, 2 Yet, in this case, for τ τ , gS(τ) decays to one, and Fs(q, τ) = exp( Dtq ). Then as a function of τ, the ISF ≫ c 2 decays exponentially− with a rate Γ = Dq2. The field and we have:

(because of the Siegert relation) the intensity correlation B g2(q, τ) g2 (q, τ) (26) functions are given by t−→≫τc g (τ) = exp( iωt) exp( Γτ) which is exactly what we want to measure. In other 1 (25) g (τ) = 1+exp(− 2Γτ)−. words, we actually want to avoid using a source with  2 − a very long coherence time, for we need τc to be much shorter than the physical fluctuation time of the DLS, the ultimate spectroscopy sample.[16] S Of course, using single longitudinal mode lasers g2 1, Brownian motion gives then rise to a spectral broad- even if the effective laser bandwidth is not negligible,≡ be- ening Γ = Dq2 that, because D is related to the par- cause the spectral broadening is due to pure phase fluctu- ticle radius, should allow for particle sizing. The prob- ations. The latter, however, still affect g1(τ), thus ham- lem, however, is that these spectral broadenings are ex- pering spectroscopic and interferometric measurements. tremely small, because colloidal diffusion is extremely Quantitatively,[10] one finds that the scattered field is not 7

with a low-pass filter. This strategy, which is called ho- modyne detection (the signal is “mixed with itself”), is again the result of using a quadratic detector. In DLS, the photodetector plays a role quite similar to the galena crystal, with B(q,t) as modulating signal, although in the form f(t)vc(t) instead of [1 + mf(t)]vc(t).[18] The net effect of the “self–beating” of the scattered field on the quadratic detector is reconstructing a copy of the spectrum of B(q,t) in baseband, but with all frequencies doubled. FIG. 1. Behavior of the field (left) and intensity (right) corre- lation functions, using a temporally partial coherent thermal S B source with τc = 0.05τc . Spatial coherence requirements in DLS

Intensity correlation measurements have several re- gaussian, so that, in terms of the full correlation func- quirements in terms of spatial coherence for what con- tions g (q, τ) =1+ g (q,t) 2; yet, g (τ)=1+ gB(q,t) 2, 2 6 | 1 | 2 | 1 | cerns both the illuminating source and the detection thus intensity correlation measurements still yield what is scheme. Maximizing the DLS signal requires indeed to needed. Even if useful, using single–mode lasers in DLS is illuminate the scattering volume with a spatially coher- not at all compulsory, so much that the first attempts to ent beam. Yet, we have seen that a source of area A study Brownian motion by analyzing the intensity fluc- emits a spatially coherent field only within a solid angle tuations of speckle patterns were performed by Raman ∆Ω λ2/A: the useful emitted power is then just the using a conventional mercury-arc lamp.[17] Hence, lasers amount≃ contained in ∆Ω, namely, P = SL∆Ω, where L, are not used in DLS setups because of they are particu- the power emitted per unit area and solid angle, is the larly monochromatic but, as we shall shortly see, just for radiance of the source (sometimes also called “bright- practical reasons related to their unique spatial coherence ness”, or “brilliance”). The crucial difference between properties. a laser and a spectral lamp is actually its enormously In the frequency domain, we can see that the “magic” higher spatial coherence, which is strictly related to its of intensity correlation comes from the fact that doing directionality. In fact, a gaussian beam emitted by a laser DLS is like playing a kind of “optical radio”. To broad- is perfectly coherent over its whole section, and diverges cast an audio signal vs = f(t) we can for instance modu- 2 2 with the diffraction angle ∆Ω = λ /w0, where w0 is the late the amplitude of a carrier wave at a radio frequency minimum beam–spot size. The section of the emitted ωc much larger than the frequency components of f(t): beam can therefore be regarded as a “speckle” emitted by a source of size w ; a source, however, that emits all v(t)= A[1 + mf(t)]cos ω t. 0 c its power on a single speckle. It is actually their high brilliance that make lasers practically indispensable in Then, to “decode” the signal, we use again a quadratic DLS. detector, which basically consist of a rectifier (a simple galena crystal in the first radios, a diode later). Suppose Let us now consider detection. The scattering volume for simplicity that we wish to transmit a simple sinusoidal behaves as a random source, with a size that is just the projection perpendicular to q of the illuminated volume. signal cos ωmt, with ωm ωc. Before the rectifier, the broadcast field is: ≪ As a consequence, there is no advantage in using a de- tector with an area A larger than a coherence area of mA this source. Namely, increasing the detector area beyond v(t)= A cos ω t + [cos(ω + ω )t + cos(ω ω )t]. c 2 c m c − m the size of the speckles made by the scattered field in- creases the detected power, but this additional power is This contains, besides the original carrier frequency, of no use, for different speckles are uncorrelated. If the two symmetric sidebands with ∆ω = ωm but, because ± number N = A/Ac of collected speckles is large, intensity ∆ω ω , no resonant filter can resolve them. After 1/2 ≪ c fluctuations will grow just as σ(I) N (it is a Poisson the rectifier, supposing that the modulation depth m is ∼ 2 statistics). Hence g (0) g ( ) = σ2(I)/ I N −1, small, we have: 2 2 so we just loose contrast.− For∞ a generic valueh i of∼N, one A2 can actually write a “corrected” Siegert relation of the 2 2 2 v (t) [1+cos 2ωct+m cos(2ωc ωm)t]+mA cos ωmt, kind g (τ) = 1+ f(N) g (τ) , where the spatial co- ≃ 2 ± 2 1 herence factor f(N) can| be approximately| written as namely, besides a zero–frequency component and three f(N) (1 + N)−1. To get a high contrast (a “good components at radio-frequency (RF), we have obtained a intercept”,≃ in the jargon of DLS) , the detector aperture signal at the modulation frequency that can be extracted should be considerably smaller then a coherence area. 8

In the earliest schemes of a DLS apparatus, the angu- tional to the (generally weak) amplitude of the carrier lar extent of the scattered light reaching the photodetec- wave detected by an aerial. Radios became much more tor was limited by means of two pinholes aligned along efficient with the development of the “heterodyne” re- the selected scattering direction. However, a much more ceiver, where the signal power is “pumped up” by mixing efficient detection scheme, which consists in forming by it with the signal vL(t) = AL cos ωct from a local oscil- a lens an image of the scattering volume on a slit that lator (LO) at the frequency of the carrier wave. Indeed, can be closed or opened by micrometers to select a sin- using a mixer that multiplies the incoming and LO sig- gle speckle, was soon adopted. The real novelty is that nals, we get again the audio signal, but amplified by vL: the effective size of speckle on the slits can be tuned by V (t)= v(t)v (t)= AA (1+cos ω t)cos2 ω (t)= RFsignals +AA stopping–down the lens with an iris diaphragm, because L L m p { } L the image of a speckle gets convoluted with the lens pupil, so that by reducing the lens aperture the size of a coher- A very similar trick is used in heterodyne DLS, where ence area on the image plane increases.[19] We shall re- the LO is simply a fraction of the incident beam (even turn to this idea of performing a “spatial coarse-graining” simply a reflection from the cell windows) which “beats” on the image plane in section . With these “traditional” with the scattered field on the photodetector. We have detection schemes it is however very hard to reach a con- then dition close to the “ideal” contrast g2(0) g2( ) = 1, HD 2 2 I(0)I(τ) = Es(0) + EL(0) Es(τ)+ EL(τ) . which is conversely ensured by novel detection− ∞ schemes h i | | | | using single-mode fibers that have become widespread Neglecting fluctuations in the incident field (hence in in the last two decades. Understanding fiber detection EL), observing that EL and Es are uncorrelated, and as- requires however to forget all about “geometrical” ar- suming that EL Es (which is almost unavoidable), guments: neither the size of the fiber to be used, nor one obtains after| | some ≫ | calculation| the distance of its opening from the sample, have indeed HD anything to do with the speckle size. Rather, an optical g2 (τ)=1+ kRe[g1(τ)] (27) fiber has to be regarded as an “antenna”, which can res- where k = I /I . The important difference with re- onate only on well-defined proper “modes”. A monomode h si L fiber, in particular, allows for a single propagating mode, spect to homodyne DLS is that, by heterodyning, we whose spatial structure is very similar to the fundamental also detect the real part of oscillating terms of the form transversal mode of a laser and display therefore full spa- exp(iωτ). Consider for instance a colloidal suspension tial coherence. The field detected by such a fiber is noth- in flow with a uniform velocity v. The field correla- ing but the projection (in the full mathematical sense) tion function can be evaluated by adding to the diffu- sion equation an advective term v ∇c. Using the same of the scattered field on the single fiber mode. The am- · plitude of the field collected by the fiber can vary by method we have described earlier, one finds g1(q, τ) = exp(iq vτ)exp( Dq2τ). The first phase term is totally changing the size of the scattering volume or of the fiber · − core but, because of the full spatial coherence of the fiber “invisible” in homodyne detection, whereas: mode, the field and intensity correlation functions always HD 2 g2 (τ)=1+ k exp( Dq τ)cos(q vτ) show full contrast, with values g1(0) = 1 and g2(0) = 2 − · at zero delay. One can show that the amplitude of the Heterodyne detection is therefore at the roots of Laser projected component can be maximized by matching the Doppler Velocimetry, which allows to study hydrody- angular aperture of a speckle with the acceptance angle namic motion using particles as tracers, or the drift par- of the fiber. Besides being much simpler both conceptu- ticle motion induced by an external field, such as in elec- ally and practically, fibers receivers present another very trophoresis. interested feature: if a laser beam is fed into the fiber from the opposite terminal (the one usually bringing the collected light to the photodetector) and launched to- NOVEL INVESTIGATION METHODS BASED wards the scattering cell from the receiver input, its spa- ON INTENSITY CORRELATION tial intersection with the incident beam allows to pre- cisely define the scattering volume. By this trick, optical Multi–speckle DLS and Time-Resolved Correlation alignment, which is time–consuming in traditional DLS (TRC) setups, becomes much simpler.[20] Colloidal gels and glasses are a class of materials of prominent interest characterized by an extremely low, Heterodyne detection and Doppler velocimetry quasi–arrested dynamics where each single particle per- forms a restricted motion around a fixed position. Be- In radio engineering, homodyne detection has the dis- cause of the limited particle displacement, the scattered advantage of generating a signal at ωm which is propor- field can be written as the sum Es(q,t)= Ef (q,t)+Ec(q) 9 of a fully fluctuating component Ef (q,t) plus a time– Multi–speckle methods are also ideal for investigating independent contribution Ec(q). As a main consequence, systems displaying heterogeneous temporal dynamics in Es(q,t) is not anymore a fully–fluctuating gaussian ran- glasses, foams, and a variety of jammed systems that of- dom variable, and its statistical properties of are very ten evolve in time through intermittent rearrangements. different from those of the light scattered by free Brow- This is the principle of the Time–Resolved Correlation nian particles. The value of Ec(q) depends indeed on (TRC) technique,[23, 24] where the change of the sample the specific configuration of the scatterers as seen from configuration is obtained by calculating the degree of in- a given detection point, hence it is different from speckle tensity correlation between pairs of images taken at time to speckle because each coherence area comes from a t and t + τ, which explicitly depends on t unique combination of the phases of the individual fields Ip(t)Ip(t + τ) scattered by each particle. Therefore, while evaluat- c (t, τ)= h ip 1. I I (t) I (t + τ) − ing the ensemble average of the scattered field over h p ip h p ip many speckles we get E (q,t) = 0, the time aver- h s ie The amplitude of the fluctuations in the temporal dy- age of Es(q,t) does not vanish, but is rather given by namics can then be quantified by the variance χ(τ) = Es(q,t) = Ec(q). Retrieving sound structural infor- 2 2 h it cI (t, τ) cI (t, τ) , which is directly related to the mation by DLS on gels and glasses requires then to mea- so-called dynamical− h i susceptibility χ used to character- 4 sure ensemble–averaged correlation functions. The latter ize dynamic heterogeneity in computer simulations of the can be of course obtained with a “brute force” method glassy state. In the last section we will see that an ex- by very slowly displacing or rotating the cell between tension of TRC, allowing to resolve g2(τ) both in time distinct acquisitions of g2(t), so that the detector is se- and space, provides a basic link between scattering and quentially illuminated by many independent speckles. A imaging. different and far less time–consuming strategy was how- ever proposed by Pusey and van Megen, who showed that the correct, ensemble-averaged correlation function may Near Field Scattering (NFS) be reconstructed from the intensity correlation function measured in a single run on a fixed speckle, provided that Because the scattered intensity has (for ergodic media) the ensemble–average of just the static intensity I 2 h iE an exponential distribution with I = I , Eq. (15) ba- is carefully measured. The correct intensity correlation sically states that the size of a speckle ish justi fixed by the function is obtained from the single-run g2(τ) and the ra- geometry of the scattering volume, and does not contain tio I / I with a well–defined, although non trivial, h it h iE any information about the physical mechanisms that pro- correction scheme.[21] duce scattering (see section ). This is a consequence of Investigating “non-ergodic” media by traditional DLS the VCZ theorem, which is however strictly valid only is anyway laborious. Luckily, we can actually take ad- when the source is not spatially correlated. In fact, it is vantage from the very slow dynamics of colloidal gels definitely not true for by a “structured source”, by which and glasses. In fact, neither a fast detectors as a photo- we mean a sample scattering light because of the pres- multiplier, nor a real–time digital correlator are needed: ence of correlated regions of size ξ & λ, due for instance to a digital camera with a moderately fast data acquisition an inhomogeneous refractive index distribution.[25] For and transfer rate fully suffices, and the calculation of g2 example consider, as in Fig. 2a, the scattering pattern can still be made in real time via software. CCD and generated on a close-by plane at distance z from the cell CMOS cameras are moreover multi-pixel devices, where by a suspension of colloidal particles contained in a thin each pixel acts as a detector, hence, in principle, we have cell, and illuminated with a beam spot of diameter D. a way to perform DLS measurements simultaneously on Particles with a size ξ & λ scatter light mostly within a vary large number of speckles. The outcome of such a cone of angular aperture ϕ λ/ξ (which, for very a multi-speckle experiment is a series of speckle images, large particles, coincides with≃ the angular aperture of where the intensity for each pixel p and time t is recorded. their diffraction pattern). By reciprocity, light can reach The intensity correlation function is then obtained as a given point P on the observation plane only from a re- gion of size d zϕ. Hence, P sees an “effective” source Ip(t)Ip(t + τ) ≃ p with a size that, provided that z

NA of the objective, and can be enlarged at will by re- ducing the latter. This trick of magnifying speckles by just stopping-down the imaging optics, is in fact sim- ilar to what is done in DLS detections by closing the diaphragm of the lens that images the sample volume on the slits. A second important advantage is that, be- cause the scattered and transmitted beams are perfectly superimposed, NFS is an ideal heterodyne method that provides an absolute measure of the scattering cross sec- tions, since the strength of the local oscillator is exactly known. An example of NFS experiment, made in our lab to obtain the form factor of very diluted polystyrene particles, in shown in Fig. 2b.

FIG. 2. Speckles in near field (a) and sketch of a NFS exper- iment (b). NFS velocimetry

Besides providing a simple and efficient tool to obtain More quantitatively, it turns out that, for a structured the structure factor of a suspension at very small angles, source with a generic mass distribution, the intensity cor- heterodyne NFS can be used as a very accurate technique relation function of the scattered light in the near–field to measure the local motion in a fluid, using colloidal is proportional to the radial distribution function g(r), particles as “tracers” like in Particle Imaging Velocime- which yields, for non-interacting scatterers with a finite try (PIV[29]). In a PIV measurement, a fluid containing size, the average value for the speckle size we found with tracer particles is illuminated by a thin sheet of light the former qualitative argument.[26] The intensity dis- and imaged in the perpendicular direction. By measur- tribution I(q) measured in the usual far–field scatter- ing the tracer displacement between two closely spaced ing experiments, which is conversely proportional to the times, the two-dimensional in-plane velocity of the fluid structure factor of the sample, can then also be obtained is recovered, whereas a full 3-D reconstruction of the field by evaluating the power spectrum of the intensity on a profile can be obtained by holographic methods.[30] Of near field plane. These conclusions are fully confirmed course, tracking individual particles requires the latter by a reassessment of the VCZ theorem for a source with to be large enough to be imaged, namely, the particle finite spatial correlation, which leads to conclude that, size must be larger than the resolution limit of the imag- within the so-called “deep Fresnel region” (DFR) zholography, yet it has do not cross any≃ more, and≃ each speckle preserves its been the subject of relatively few theoretical investiga- own coherence in propagation (no “crosstalk” between tions (for a review till 2007, see.[33]) the speckles). As a consequence, the 3-D geometrical 12 shape of a speckle changes dramatically from a jelly bean a faithful copy of a planar section of an object onto an- to a pencil. Hence, in the Fraunhofer region, δz . other plane, apart from a change of scale (magnification). → ∞ “Full” Fresnel region zc

−2πi(fxx+fy y) A(fx,fy,z)= dxdyU(x,y,z)e . Z Spatial frequencies be given a simple geometric interpre- tation by expressing the amplitude of a simple plane wave in terms of the director cosines (α,β,γ) it makes with the axes (x,y,z) as

P (x,y,z) = exp[i(2π/λ)(αx + βy)] exp[i(2π/λ)γz].

Thus, across the plane z = 0, exp[2πi(fxx + fyy)] may be seen as a plane wave traveling with director cosines α = λfx, β = λfy. However, the director cosines are not independent, because γ = 1 α2 γ2. The physical meaning of this relation can be− grasped− by ob- p serving that U(x,y,z) satisfies the Helmholtz equation FIG. 3. Sketch of the longitudinal coherence profile of speck- ( 2 + k2)U(r) = 0, with k = 2π/λ. Hence, writing les from an ideal monochromatic sources in the deep Fresnel, A∇(α,β,z)= dxdyU(x,y,z)e−ik(αx+βy), we have Fresnel, and Fraunhofer regions. ∂2A(α,β,z) R +k2(1 α2 β2)A(α,β,z)=0= A(α,β,z)= A(α, β, 0)e ∂z2 − − ⇒ SPATIAL COHERENCE AND IMAGING 2 2 2 2 −2 For α +β 1 ( fx +fy λ ) γ is real, hence propaga- tion just amounts≤ to a change≤ of the relative phases of the In its simplest acceptation, imaging consist in produc- components of the angular spectrum, because each wave ing, by means of optical elements like lenses or mirrors, travels a different distance between constant-z planes, 13 which brings in phase delays. Conversely, for α2 +β2 > 1 and find what is the stop that limits its angular aperture: γ is imaginary, and α, β cannot be regarded anymore this is the field stop FS. For example, Fig. 4a, shows the as true direction cosines. Rather, we have an evanes- aperture and field stops for a simple propagation between cent wave, whose amplitude decays as exp( 2π γ z) and two diaphragms, whereas in the so–called 2f1 2f2 lens becomes negligible as soon as z is a few times− |λ.| Wave system shown in Fig. 4b (a very convenient combination− propagation in free space can then be regarded as a “low– for spatial filtering) AS is the diaphragm placed in the pass dispersive filter”, since only those spatial frequencies common focus of the two lenses, while FS is the pupil of 2 2 −2 such as fx + fy λ can propagate, with a phase shift the lens that limits more the angular aperture. that depends however≤ on frequency. b) Fourier–Transform properties of a lens Suppose we illuminate with uniform amplitude A a flat object, for instance a transparency transmitting an amplitude U(x, y) = At(x, y), placed against a thin lens of focal length f. Then, if the object is much smaller than the lens aperture, so that we can neglect the effect of the fi- nite size of the latter, the amplitude distribution Uf (x, y) in the focal plane of the lens is the Fraunhofer diffraction pattern of the object transmittance t(x, y), aside from a pure phase factor that does not change the intensity.[38] The former phase factor exactly cancels out when the object is placed at a distance f before the lens. In other words, the front and back focal planes of a lens are re- lated by a FT or, as we shall say are reciprocal Fourier planes. Finally, is an object is placed before a thin lens at a distance z1 then (except again for phase factors) an image of the object, inverted and magnified by the ratio M = z2/z1, forms at a distance z2 such that −1 −1 − −1 z1 + z2 = f , which is of course the simple lens law from geometrical optics. Moreover, the back focus FIG. 4. Aperture and field stops for free propagation between is exactly a Fourier plane for the object, so we can “ma- two apertures (a) and for a 2f1 2f2 lens system (b). − nipulate” the image, for instance by “cutting out” some spatial frequencies or by selectively changing their rela- Partially–coherent sources According to footnote , tive phases. This “spatial filtering” technique, besides the fundamental gaussian mode emitted by a laser has a being at the roots of the whole field of optical commu- far–field angular divergence θ λ/(πw0), where w0 is the nication, is fully exploited in phase–contrast microscopy. beam waist, which is the spread≃ expected for a spatially The effect of the lens pupil is very similar, since also the coherent wavefront because of diffraction. For a partially lens plane is (aside from a phase factor) a Fourier plane 2 coherent circular source of area σ0 = πw0, which can be for the object. Hence, reducing the lens diameter D (or pictured as “speckle mosaic” made of N σ /ξ2 un- c ∼ 0 0 better, its numerical aperture NA = D/f) corresponds correlated coherence regions of size ξ0 (see figure 5), the to cut out the high-frequency Fourier components, in fact divergence is found to be Nc times larger.[40] It is how- reducing the image resolution. ever interesting to investigate how the correlation length c) Aperture and field stops In free space, all spa- changes upon propagation. We have seen that, in the − tial frequencies with f 2 + f 2 λ 2 propagate, whereas deep Fresnel region, the propagation of the spatial co- x y ≤ evanescent waves die out. When an optical signal is fed herence is very different from what predicted by the VCZ through a generic imaging system, however, there are fur- theorem for a fully uncorrelated source. Here, however, ther limitations to the spatial frequencies that can reach we wish to find how a similar source behaves in far field, the image plane, because the finite size of the optical namely, in the Fraunhofer diffraction regime. Without components limits the angular extent of the radiation entering into details, which involve rather tedious calcu- emitted by the object that can propagate through the lations, we just state the main result. In far field, the system. Crucial to the analysis of spatial coherence in area σ of the source and the correlation length ξ grow an optical system are the concepts of aperture and field upon propagation by a distance z as stops, which are defined as follows. Let first look at the 2 optical system from the image plane, and find what is the (λz) σ = 2 aperture which most limits the incoming light: this is the ξ0 σ σ0 2  = 2 = 2 . (30) aperture stop AS, or simply the “pupil” of the system.[39] (λz)  ⇒ ξ ξ0 ξ2 =  Now project of cone from the center of the aperture stop, σ0   14

power fed into the fiber is W = W0σf Ωf /G. It is then easy to show that, for a monomode fiber collecting scat- tered radiation, W coincides with the power scattered by the sample within one speckle.

Microscope structure: coherence of illumination and resolution limit

FIG. 5. Propagation of the spatial coherence for a partially– coherent source and ´etendue.

Hence, the “expansion rate” of the source area is deter- mined by the area of a coherence region and vice versa. It is therefore useful to define a quantity with the dimen- sions of an area called the ´etendue σ G = λ2 . (31) ξ2 which, because of Eq. (30), has the very important prop- erty of being conserved upon free–space propagation.[41] Moreover, introducing as in Fig. 5 the solid angles Ω0 = 2 2 σ/z and Ω = σ0/z , we can also write G = σ0Ω0 = σΩ. Physically, the ´etendue is a “combined extension” of the FIG. 6. Structure of a microscope with K¨ohler illumination. source, given by the the product of its area in the real The illumination path consists of the collector lenses L1 and space times its far–field diverging angle, which is related L2 that generate an image of the illumination source on the to the region in the Fourier space of the spatial frequen- plane of the condenser, which focuses the light on the object cies that propagate from σ. For a uniform source, we plane. In the imaging path, the transmitted light is collected can the write the total emitted power as the product by an infinity–corrected objective and made parallel by the W = GL of the ´etendue times the radiance: since W is tube lens. The two conjugate sets of planes where the illu- of course fixed, the invariance of the ´etendue upon free– mination source and the sample are in focus are shown by corresponding symbols. space propagation is equivalent to the conservation of the source brightness. The ´etendue is however not conserved in the presence Fig. 5 shows the basic structure of an optical micro- of limiting apertures. Suppose for instance that a fully scope using K¨ohler illumination. This setup provides a coherent planer wavefront of infinite lateral extent im- uniform illumination of the sample by placing the latter pinges on the simple system in figure 4a, where the aper- on the focal plane of the condenser, which is a conjugate ture stop AS limits the source size, while the field stop FS Fourier plane for the illuminating lamp. As a matter of its angular divergence: it is easy to show that the effec- fact, in the configuration shown in Fig. 6 there are actu- ′ 2 tive ´etendue is limited to Gt = AsΩ= Af Ω = AsAf /z , ally two sets of planes where the source S and the object where As and Af are the areas of the aperture and field (sample) plane are, respectively, imaged. Set 1 is com- stop respectively. This is called the throughput of an posed of the lamp filament, the source aperture stop ASs optical system. Using the double–diaphragm setup in at the front focal plane of the condenser, and the image figure 4, we can actually increase the effective spatial aperture stop ASi at the back focal plane of the objec- coherence of a source: this happens whenever the solid tive. All these planes are Fourier planes for set 2, which angle subtended by FS is smaller than than Ω0 (we loose comprises the field stop FSs at the back focal plane of of course some power). This also helps to understand the collector lens L1, the object plane where the sample fiber–optic detection in DLS. A monomode optical fiber is placed, and the image plane (in visual observation, the 2 has by definition an ´etendue G = σf Ωf = λ where σf latter is further imaged by the eyepiece). and Ωf are the area of the fiber core and its solid accep- Understanding the reciprocal nature of these two sets tance angle. For a source of ´etendue G, the maximum of planes is crucial to describe the way a microscope 15 works. In particular, it is important to stress that the tive index of the medium the objective is immersed in size of the illumination source and its spatial coherence (which may not be air). It is then not hard to deduce properties can be controlled independently. The former is that the Rayleigh limit is generalized by the celebrated simply tuned by opening or closing the diaphragm ASs. Abbe criterion, stating that the minimal separation dis- The field stop FSs conversely controls the angular aper- tance is: ture of the light reaching the sample from a given point λ λ on the source plane. Since the latter lies on the focal δ 1.22 =1.22 , ≃ n sin ϑ NAobj plane of L1, where we have the FT of the source, closing down FSs corresponds to filtering the spatial frequencies For fully coherent illumination, however, things are quite of S and therefore to tuning the spatial coherence proper- different, even in the paraxial approximation, and the re- ties of the source. By increasing the condenser aperture, sult depends on phase difference ϕ of the illumination at the illuminating optics becomes more and more similar to the two point sources. Indeed, one finds that the situ- a fully incoherent source, whereas by progressively stop- ation is identical to the incoherent case only when the ping it down we approach the coherent illumination limit. phases are in quadrature (ϕ = π/2), whereas, when the From what we have seen in the previous section, the il- two sources are fully in phase (ϕ = 0), the two Airy disks lumination on the object plane has then in general the conversely merge into a single peak centered at x = 0: form of a “speckle mosaic” similar to the one sketched in thus, at the Rayleigh limit, they are not resolved at all. figure 5, where the speckle size ξ is fixed by the condensed If the sources are in counter-phase (ϕ = π), however, at numerical aperture. In section , we shall see how novel the Rayleigh limit they are fully separated, hence resolu- correlation methods in microscopy exploit this peculiar tion actually doubles. Stating that coherent illumination tunability of the spatial coherence of illumination. is “worse” than incoherent illumination, as often made The degree of spatial coherence of the illumination at in elementary textbooks, is therefore incorrect. With co- the sample plane has noticeable effects on the resolving herent illumination, the resolution actually depends on power of the microscope. For of a telescope with an ob- the specific way we illuminate the object: whereas in jective of radius w, the determination of the resolving a standard geometry two close-by points are usually il- power is particularly simple, because two close-by stars luminated with the same phase, with a suitable oblique we may wish to resolve behave as mutually incoherent illumination (a technique which has often been used in point sources. Moreover, since the telescope is focused microscopy) one can obtain a counter–phase condition. ai infinity, each one of them is imaged on the focal plane Even with a standard illumination geometry, the best of the objective as an “Airy disk” (namely, the Fraun- resolution is not obtained by increasing as much as possi- hofer diffraction pattern of a circular aperture) of diam- ble the condenser aperture. A detailed calculation shows eter d 0.6λf/w. A reasonable criterion for separation, indeed that it is not worth increasing the condenser nu- suggested≃ by Rayleigh, is that they are “barely resolved” merical aperture NAcon to more than about 1.5NAobj, if the center of the Airy disk of one star coincides with and that in these conditions the resolving power is[42] the first minimum of the second one, namely, if their λ angular separation is larger than ϑ 0.6λ/w. For δ 1.22 . (32) min ≃ NAobj + NAcon microscope, however, the problem is more≃ complicated, first because this simple result from Fraunhofer diffrac- tion holds only provided that ray propagation is parax- SCATTERING AND IMAGING: TOWARDS A ial, which is the case of a telescope but surely not of JOINT VENTURE a microscope; second, because we are considering non self-luminous objects, hence the spatial coherence of the We have seen how statistical optics concepts can de- light generated at the object plane depend on the coher- scribe both DLS and imaging by a microscope. Yet, com- ence of the illuminating source. Consider first the sit- munication between these two worlds has been rather uation where the illumination is fully incoherent, which limited till a few years ago. The main reason is that the can be obtained for instance by opening up completely description of particle scattering necessarily requires a the field stop FSs of the condenser. If we take a look to full 3-D treatment of the electromagnetic problem lead- the “imaging path” to the right of figure 6, we can see ing, even in the case of spherical particles, to the com- that the spatial frequencies of the light produced at the plicated Lorenz-Mie solution. On the other hand, most object plane that can reach the image plane are basically traditional microscopy problems can be discussed using limited by the aperture stop ASi. Since the object plane the simpler language of diffraction, which is basically 2- lies very close to the front focal plane of the objective, D. Recent advancements in imaging, such as the devel- the maximum spatial frequency that enters the imaging opment of confocal microscopy and of accurate particle– path is determined by the numerical aperture of the lat- tracking methods, have led to investigate many aspects ter NAobj = n sin ϑ, where ϑ is the angle subtended by of imaging of 3-D objects, and to reconsider the relation ASi when viewed from the image plane, and n is refrac- between scattering and microscopy. 16

The latter is far from being trivial. It is not easy even that, according to Eq. (33), the same qq vector may actu- to state when we can actually see under a microscope ally correspond, for two distinct wavelengths, to different a particle made of a non-absorbing material and with q vectors. Nevertheless, it is not difficult to show that a size much larger than the wavelength. If we regard this effect is small as long as the difference in wavelength −1 them as two dimensional sources and just apply the ba- ∆λ qq , which is of order λ/θ: hence at small col- sics of Fourier optics, the answer is simple: never. A non lection≪ angles, the speckle patterns formed by different absorbing particle just modulates the phase of the illu- wavelengths superimpose. minating radiation, and does not change its amplitude: in other words, they are phase diffractive elements, and the image of a phase element is again a phase element, Photon Correlation Imaging (PCI) with no intensity contrast.[19] Cells and other optically transparent biological samples object are indeed practi- TRC is a very powerful method to investigate the het- cally invisible, except at their contour boundaries, but, as erogeneous and intermittent time–dynamics of restruc- a matter of fact, large polystyrene particles can be seen turing processes in gels and glasses. However, glassy dy- under a microscope, even when they are right on focus. namics is also very heterogeneous in space, behaving very This must have therefore to do both with the 3D nature differently in different regions of the sample at equal time. of the particles, and with the difference np ns between Photon Correlation Imaging,[48] a simple extension of − the refractive indexes of the particle and of the solvent. TRC, allows to detect these spatial heterogeneities by In fact, particles of a size a such that np ns a/λ 1 means of measurements of space and time resolved cor- | − | ≪ (namely, Rayleigh-Gans scatterers) cannot be visualized relation functions. With respect to the TRC scheme, at all, and the same is true for particles scattering in the the major change concerns the collection optics. Instead so-called “anomalous diffraction” regime,[43] where re- of collecting the light scattered in far–field, one forms flections and refractions at the particle/solvent interface an low-magnified image of the scattering volume onto a can be neglected.[44] A detailed analysis of the visibil- multi-pixel detector, using only the light scattered in a ity problem for a generic scatterer is however still lack- narrow cone centered around a well defined scattering ing. Scattering from non-absorbing objects is in any case angle. Of course, since the magnification M is low, the rather weak, whatever their refractive index with the sur- scatterers themselves are not resolved, but a speckle pat- rounding medium, hence a common way to increase their tern is visible, because we are actually collecting the light visibility is “de-focusing”, namely, focusing the objective within all the depth of field of the imaging lens, hence also on a plane outside the particle. However, it is worth the near–field scattering from the sample. In fact, we can noticing that, with this methods, evaluating particle size tune the size of the speckles by adjusting the NA of the or interparticle distances is not trivial, and may lead to imaging lens with an iris diaphragm, exactly as when, in serious errors.[45] In fact, quantifying how the imaging heterodyne NFS, the near–field speckle pattern is mag- optics collects the intensity distribution generated on a nified using an objective. In contrast to far field speckles generic plane from particles situated at various distances that are formed by the light coming from the whole scat- z from it requires a full 3D treatment of the imaging pro- tering volume, however, each speckle in a PCI experiment cess. In the simplest case of a Rayleigh–Gans scatterer, receives only the contribution of scatterers located in a one finds that the intensity pattern consists of a cen- small volume, centered about the corresponding object tral disk surrounded by a set of concentric fringes that point in the sample. The linear size of this volume will get the coarser the farther is the particle from the plane be of order (λ/Md)z, where d is the diameter of the lens z = 0, and that a particle displacement at constant z0 pupil, and z the lens-detector distance. As a result of amounts to a rigid translation of this fringe pattern, sim- the imaging geometry, the fluctuations of the intensity ilarly to what is observed in out-of-focus microscopy ob- of a given speckle are thus related to the dynamics of a servations. A full discussion of 3D imaging can be found well localized, small portion of the illuminated sample. in Ref.[46, 47]. Hence, the local dynamics can be probed by dividing the For what follows it is also useful relating the scatter- image in “Regions of Interest” (RoI) which contains a ing wave-vector q to its projection qq on the observa- sufficient number of speckles and measuring their time- tion plane z = 0. Since in the paraxial approximation fluctuations. q =2k sin(θ/2) kθ, where θ is the scattering angle, we This method was developed to study slow or quasi– ≃ have arrested systems, but it works also for free particles in 2 Brownian motion too, provided that the speckle size is 2 2 qq q qq 1+ , (33) ≃ 2k sufficiently enlarged by stopping down the imaging lens     and that a fast detector is used. For instance, in our lab 2 so that the perpendicular component of q is qz qq /2k. we were able to obtain very good measurements for dilute The second term in square brackets is of order≃θ, so it suspensions of particles with a size of about 50 nm using is negligible for small scattering angles. Notice however a fast CMOS camera. Of course, because one measures 17

−1 many speckles simultaneously, the averaging process is that ∆λ qq is uniquely associated to a single scat- ≪ very fast, and very good correlation functions can be ob- tering wave-vector q = 2πf0. Second, at variance with tained in a few seconds, but there is much more than this. a standard NFS experiment with a laser source, using Indeed, when the particles, besides performing Brownian a microscope we can vary the spatial coherence of the motion, are also moving as a whole, the overall motion of illuminating source. This means that the deep Fresnel the speckle pattern is then a faithful reproduction of the region where NFS is observed depends on the numerical local hydrodynamic motion within the sample. Hence, if aperture of the condenser: in fact, if the condenser is the speckle correlation time is sufficiently long, the local fully opened, no appreciable speckle pattern is observed. flow velocity can be obtained by monitoring the motion What is more important, this also amounts to change the of the speckle pattern: for instance, for particles settling thickness of the sample region which is coherently illu- under gravity, the local sedimentation velocity can be minated: we have indeed seen that the speckles have a obtained. This strategy has allowed to investigate the “jelly bean” structure, with a longitudinal size δz ξ2/λ, relation between microscopic dynamics and large-scale where ξ is the transversal coherence of the source∼ on the restructuring in depletion[49] and biopolymer[50] gels. object plane. By micrometrically translating the objec- tive, a “z-scan” through the sample can be made. The typical longitudinal resolution is is of the order of tens of Differential Dynamic Microscopy (DDM) microns, which is much larger than the resolution achiev- able with a confocal microscope, but still sufficient for The powers of microscopy and DLS are perfectly com- many purposes. bined in Differential Dynamic Microscopy (DDM), a sim- ple but very powerful technique that can be set up on a standard microscope and does not even require a coherent laser source.[51, 52] Let us see how it works by retracing the original steps made by R. Cerbino and V. Trappe.[51] The image under a conventional microscope of a sus- pension of particles having a size much smaller than the wavelength is just an uniform white field with spurious disturbances due to dust or defects in the optics, like in the image to the left of figure 7a. However, taking a sec- ond images after a time delay, and subtracting from it the first one, a well-defined speckle pattern appears, and gets the sharper the longer the delay time t (see figure 7a, right). In fact, calling ∆I(x, y; t) = I(x, y; t) I(x, t; 0) the difference in intensity at a given point on− the image plane, one finds that the total variance

σ2(t)= ∆I(x, y; t) 2dxdy | | Z FIG. 7. Panel A: “Extraction” of the speckle pattern by image grows with time, progressively reaching a plateau. subtraction in DDM. The images refer to a suspension of PS Why the speckles? Collecting just the field originating particles with a diameter of about 0.1 µm at a concentration from the object plane, we would not see any intensity of about 0.2%, imaged with a 0.5NA objective and a stopped- difference between the two frames,[53] but, as in PCI, we down condenser. Panel B: Time-evolution of the correlation are also collecting the scattering in the near field. Actu- peak (top) and of the structure factor (bottom) in a GPV experiment. ally, DDM has many points in common with near–field scattering, but with two crucial advantages. First, we do not need at all a monochromatic source because, as With DDM, one can in fact obtain fast measurements discussed in the last section, the speckle patterns gener- of the intensity correlation function at very low angles. ated by different wavelengths fully superimpose at small Recalling that there is a one-to-one correspondence be- angles. To make it clearer, it is sufficient to observe tween the spatial frequencies of the image and the scat- tering wave-vectors, and using the Parseval’s theorem, that each spatial frequency fo of the object behaves as a grating, diffracting in paraxial approximation at an an- which states that the integral of the square of a func- gle θ = sin−1(λf). This diffraction pattern generates on tion is equal to the integral of the square of its Fourier the image plane a set of fringes with spatial frequency transform,[19] the total variance can indeed be written also as fi = sin(θ)/λ = fo that does not depend on λ. Hence, each different wavelength generates an identical interfer- 2 2 σ (t)= ∆I(fx,fy; t) dfxdfy, ence pattern which depends only on fo, which, provided | | Z f 18 where ∆I(fx,fy; t)= F [∆I(x, y; t)]. Hence, by Fourier- What is really interesting, however, is that the size of the transforming the image differences, one can extract the particles used as tracers does not matter, as long as they Brownianf dynamics of the particles.[28] scatters sufficiently strong (remember indeed that, even for scatterers with a size a λ, the near–field speckle size cannot be smaller than about≪ λ, whereas their size on Ghost Particle Velocimetry (GPV) the image plane is just fixed by the NA of the objective). In fact, using GPV one can perform a detailed analysis of Particle Imaging Velocimetry is extensively used to hydrodynamic flow using as tracers nanometric “ghost” monitor fluid flow in microfluidics systems, which are be- particles that are far smaller than the microscope resolu- coming widespread in academic and company research tion limit.[55] labs. Individual tracking, however, requires particles PCI, DDM, and GPV are just some examples of how a large enough to be optically resolved, which therefore careful application of statistical optics concepts can help perturb the flow over spatial scales that, in microfluidics, in devising novel powerful optical methods that bring may be comparable to those of the investigated struc- together scattering and imaging. In fact, these tech- tures. This limitation can be overcome by resorting to niques, and DDM in particular, are deeply related to more sophisticated methods such as micro-scale Particle other methods that fully exploit coherence effects, such Imaging Velocimetry (µPIV), which exploits small fluo- as Digital Holography and Optical Tomography. It is rescent tracers that do not need to be individually re- therefore highly probable that in the next future these solved. In this alternative approach, the fluid average new fascinating approaches will gain more importance in velocity within a small region is rather found by detect- the investigation of colloidal systems. ing fluorescence intensity fluctuations and evaluating the spatial cross-correlation of two images taken at different times with a suitable frame rate.[54] However, µPIV in- strumentation requires a rather expensive optical setup, [1] J. W. Goodman, Statistical Optics (Wiley, New York, usually including a pulsed laser source synchronized with 1985). a high resolution fast CCD camera. [2] This is nothing but an extension of what is done in rep- R As we mentioned in section , NFS techniques provides resenting a monochromatic signal u (t)= A cos(ω0t φ) − a simple, efficient, and much cheeper method for track- as u(t) = A exp[ i(ω0t φ)], as can be appreciated by − − ing fluid motion that overcomes the main limitation of looking at the Fourier transform (FT) in time of these standard PIV, since particles that are smaller than the two functions: R iφ −iφ iφ optical resolution limit can be used. Microfluidic appli- F [u ]=(A/2)[e δ(ω ω0)+e δ(ω+ω0)] ; F [u]= Ae δ(ω ω0), − − cations, however, require velocimetry to be made under . a microscope on microfluidic chips that have generally a [3] R. Loudon, The Quantum Theory of Light (Oxford Uni- poor optical quality: feeding in an additional laser source versity Press, 2000). and setting the configuration required to measure near- [4] J. A. Armstrong and A. W. Smith, in Progress in Optics, field scattering is surely inconvenient, if not unfeasible. Vol. VI, edited by E. Wolf (North–Holland, Amsterdam, An alternative approach to quantitatively map fluid flow 1967) p. 211. in microfluidic devices is what we call “Ghost Particle [5] Namely, for small propagation angles with respect to the optical axis, which is the condition required for the Fres- Velocimetry” (GPV), which uses the same procedures of nel approximation in diffraction to hold. NFS velocimetry, but within a DDM optical scheme.[55] [6] As a matter of fact, no real source can truly be δ- Figure 7b, for instance, which refer to an experiment correlated in space. The minimum “physical size” of a made using a standard microscope and white light, shows source is indeed of the order of the wavelength λ, for that two basic strategies for extracting the local fluid ve- smaller sources would emit only evanescent waves, ex- locity discussed in Section can be used with no relevant ponentially decaying with the distance from the source: change in a DDM configuration. hence, spatial correlations must extend over a distance comparable to λ. Nevertheless, in terms of propagating At variance with a standard NFS experiment, however, waves, a source of size λ is equivalent to a point source. the depth of the region probed in GPV is extremely lim- [7] B. J. Berne and R. Pecora, Dynamic Light Scattering: ited, because of the very small size of coherence area of With Applications in Chemistry, Biology and Physics the illumination source: in fact, it is much smaller than (Wiley, New York, 1976). the depth of focus of the objective, so that mapping of [8] B. Chu, Laser Light Scattering: Basic Principles and the velocity field can be done by focusing the objective on Practice (II edition) (Academic Press, New York, 1991). the object plane itself. In a microfluidic geometry, this [9] C. C. Han and A. Z. Akcasu, Scattering and Dynamics of Polymers: Seeking Order in Disordered Systems (Wiley- allows to simultaneously obtain, for instance, a detailed Blackwell, Singapore, 2011). image of the channel. GPV also allows for an appreciable [10] P. N. Pusey, in Photon Correlation Spectroscopy and Ve- resolution along the optical axis, yielding 2D sections of locimetry, edited by H. Z. Cummins and E. R. Pike the flow pattern separated by a few tens of micrometers. (Plenum, New York, 1977) p. 45. 19

[11] P. N. Pusey, in Neutron, X-rays and Light: Scattering [34] M. D. Alaimo, Ph.D. thesis, University of Milan (2006). Methods Applied to Soft Condensed Matter, edited by [35] A. Gatti, D. Magatti, and F. Ferri, Phys. Rev. A, 88, P. Lindner and T. Zemb (North–Holland, Amsterdam, 191101 (2008). 2002) Chap. 9. [36] We recall that the spot size w(z) and radius of cur- [12] If the system is spatially isotropic, F (q,τ) does not de- vature of a gaussian laser beam focused in z = 0 to pend on the direction of q, but only on its modulus a minimum spot size (beam waist) w0 are given by 2 2 q = q . In Eq. (21) the average is of course made over w(z) = w0p1+(z/zR) , R(z) = z 1+(zR/z) , where | | 2 the statistical distribution of the particle positions. zR = πw0/λ is called the Rayleigh range. Hence, the [13] ζ = 6πηa for a spherical particle of radius a in a solvent wavefront at z = 0 is flat, whereas both R(z) and w(z) of viscosity η. grow linearly with z for z zR, corresponding to an ≫ [14] R. Kubo, M. Toda, and N. Hashitsume, Statistical angular divergence of the beam θ λ/(πw0). Note that nd ≃ Physics II (2 ed.) (Springer-Verlag, Heidelberg, 1993) w( zR)= w0√2, so that within the Rayleigh range, the Chap. 1. cross-section± of the beam changes only of a factor √2, [15] Even for a bandwidth of the order of the GHz, ℓc is of whereas the curvature radius is maximal at the Rayleigh the order of a few centimeters. range, R( zR) = 2zR. [16] Note that g2(τ) decreases from four to one because the [37] J. Mertz, ±Introduction to Optical Microscopy (Roberts & scattered field is, at least in the case of a pure thermal Co., Greenwood Village, Colorado, 2010). source, the product of two gaussian processes. [38] If the finite size of the lens cannot be neglected, Uf (x,y) [17] C. V. Raman, Lectures in Physical Optics, Part 1 (Ban- is actually proportional to the FT of the product of t(x,y) galore: Indiamn Academy of Sciences, 1959) p. 160. times the pupil of the lens (see the next paragraph). [18] The exact analogous in radio engineering is dubbed [39] Actually, in optics it is more customary to define an “en- “carrier–suppressed AM”. trance” and an “exit” pupil as the images of the aperture [19] J. W. Goodman, Introduction to Fourier Optics, III Ed. stop seen through all the optics before or, respectively, (Roberts & Co. Publ., Greenwood Village, CO, 2005). after the aperture stop. These can be real or virtual im- [20] J. Riˇcka, Appl. Opt., 32, 2860 (1993). ages, depending on the location of the aperture stop. [21] P. N. Pusey and W. van Megen, Physica A, 157, 705 [40] The same applies to the higher transversal modes of a (1989). laser. [22] The order in which these two averages is taken is cru- [41] Note that for a fully coherent source the ´etendue attains cial to obtain a correct ensemble–averaged g2(τ). This its minimum value G = λ2. is evident for fully–arrested sample, where one expects [42] M. Born and E. Wolf, Principles of Optics, VI Ed. (Cam- g2(τ) 1 for all τ, whereas I is constant in time but ≡ bridge Univ. Press, Cambridge, 1997). varies from pixel to pixel, so that reversing the order of [43] H. C.van de Hulst, Light Scattering by Small Particles averaging we would obtain g2(τ) 1. ≡ (Dover, New York, 1957). [23] L. Cipelletti, H. Bissig, V. Trappe, P. Ballesta, and [44] It is indeed because of the latter that pure phase fluctu- S. Mazoyer, J.Phys: Cond. Matt., 15, S257 (2003). ations on the object plane yield amplitude fluctuations [24] A. Duri, H. Bissig, V. Trappe, and L. Cipelletti, Phys. when propagated to a following plane, because of an ef- Rev. E, 72, 051401 (2005). fect similar to shadowgraphy in geometrical optics. [25] M. Giglio, M. Carpineti, and A. Vailati, Phys. Rev. Lett., [45] J. Baumgartl and C. Bechinger, Europhys. Lett., 71, 487 85, 1416 (2000). (2005). [26] M. Giglio, M. Carpineti, A. Vailati, and D. Brogioli, [46] N. Streibl, J. Opt. Soc. Am. A, 2, 121 (1985). Appl. Opt., 40, 4036 (2001). [47] I. Nemoto, J. Opt. Soc. Am. A, 5, 1848 (1988). [27] We recall that, in diffraction optics, the far-field Fraun- [48] A. Duri, D. A. Sessoms, V. Trappe, and L. Cipelletti, hofer diffraction pattern from a source of size D is ob- 2 Phys. Rev. Lett., 102, 085702 (2009). served only for z zF = D /λ, whereas the more com- ≫ [49] G. Brambilla, S. Buzzaccaro, R. Piazza, L. Berthier, and plex Fresnel diffraction regime corresponds to z