We present a new method called optical coherence correlation spectroscopy (OCCS) using nanoparticles as reporters of kinetic processes at the single particle level. OCCS is a spectral interferometry based method, thus giving simultaneous access to several sampling volumes along the optical axis. Based on an auto-correlation analysis, we extract the diffusion coefficients and concentrations of nanoparticles over a large concentration range. The cross-correlation analysis between adjacent sampling volumes allows to measure flow parameters. This shows the potential of OCCS for spatially resolved diffusion and flow measurements.
© 2014 Optical Society of America
The observation of dynamic processes at the nanoscale allows the study of kinetics at the single particle level. Fluorescence Correlation Spectroscopy (FCS) [1, 2] is a widely used technique to study enzyme reaction kinetics  and molecular dynamics in cells , but also allows investigating coalescence and aggregation , the interaction between nanoparticles and membranes  or flow in micro-channels . Using fluorescence as a reporter signal has a number of advantages, in particular molecular specificity, high contrast and single molecule sensitivity. However, fluorophores tend to photo-bleach, especially when observed at high illumination intensities and/or over long time periods. This typically limits the total observation time for classical FCS to a few seconds, which complicates characterizing slow processes. On the other hand, the optical emission from fluorophores saturates at high excitation intensities, which imposes constraints for the study of fast molecular processes.
Novel techniques such as Raman correlation spectroscopy (RCS) , coherent anti-Stokes Raman scattering correlation spectroscopy (CARSCS) [9, 10], correlation spectroscopy based on second and third harmonic non-linear light-matter interactions (NLCS)  and FCS on quantum dots  have been conceived to overcome these drawbacks. Based on the absorption properties of gold nanoparticles (NPs) and an auto-correlation analysis, photothermal correlation spectroscopy (PCS) has been demonstrated recently [13, 14]. These gold NPs are photo-stable and available with biocompatible surface coatings and have been proven to be alternative markers for time-lapse observations. NPs can also be detected using their light scattering properties . For instance, their diffusion properties can be measured by dynamic light scattering (DLS) , low-coherence interferometry (LCI) , scattering correlation spectroscopy (SCS)  or scattering interference correlation spectroscopy (SICS) . The luminescence properties of gold NPs can also be used to perform FCS experiments , overcoming the photobleaching but not the signal limitation. Whereas DLS and SICS provide good temporal resolution, they have limited spatial resolution because the long coherence length of the monochromatic illumination integrates the signal due to particle dynamics over a wide depth range. In contrast, FCS, PCS, SCS and LCI measure particle dynamics locally but need longer time spans for measuring the spatial distribution of particle dynamics in larger volumes (using for instance raster image correlation spectroscopy (RICS)  in the case of FCS). Dual and multiple focal FCS systems are known to measure lateral flow and diffusion [22–24] but entail a higher complexity of the optical system.
Here we introduce Optical Coherence Correlation Spectroscopy (OCCS), which exploits the backscattered light of NPs illuminated by a broadband light source. Over a large concentration range, this technique allows the extraction of the diffusion coefficient and concentration of NPs using an auto-correlation analysis. OCCS gives simultaneous access to several sampling volumes along the optical axis, which allows assessing flow parameters through cross-correlation analysis. Thus, OCCS enables spatially resolved diffusion and flow measurements.
2.1. Principle of OCCS
A typical OCCS experiment is shown in Fig. 1. The sample is composed of NPs diffusing in a liquid (Fig. 1(a)). The OCCS system is a low coherence interferometer that illuminates the sample with an elongated Bessel beam (Fig. 1(b)). The LCI principle allows a multitude of sampling volumes along this Bessel beam to be interrogated simultaneously, i.e. the total illuminated volume can be subdivided into several sampling volumes along the optical axis. The lateral extent of a single sampling volume is determined by the numerical aperture whereas the axial extent is governed by the temporal coherence of the broadband light source. The superposition of the backscattered sample field and the reference field results in a spectral interference signal which is recorded via a spectrometer (Fig. 1(c)). The acquired spectrum is then resampled at equidistant wavenumbers (λ to k mapping, Fig. 1(d)). By taking the Fourier transform of the resampled spectrum, we obtain the time-dependent signal traces extending over several sampling volumes (Fig. 1(e)). Concentration and diffusion coefficients of identical NPs are then extracted by fitting the auto-correlations of these time-dependent signals by corresponding correlation model functions (Fig. 1(f) top). Additionally, cross-correlating time traces between adjacent sampling volumes yields access to the mean transit times of NPs moving across the axially aligned sampling volumes (Fig. 1(f) bottom). OCCS correlations therefore allow assessing diffusion, concentration and directed flow along the optical axis.
Signal acquisition in OCCS
The spectrum of the interfering fields resulting from the superposition of the reference light and the light back-scattered from the sample is measured in OCCS. Due to the used broadband light source, OCCS has a limited temporal coherence characterized by a coherence length lc. This allows to differentiate axial sampling volumes . Supposing a Gaussian-shaped source spectrum, lc is given as25] Eq. (1). Choosing Δks such that Δz ≲ lc/2 enables two consecutive sampling volumes to be resolved. In our case, Δz = 1.66μm and lc = 3.07μm, and Δz ≃ lc/2.
In the plane-wave approximation, the detected signal spectrum as a function of time t at wavenumber k after mapping from λ to k = 2π/λ can be written as Eq. (3) can be reduced to the second term.
Considering the case where the sample contains only scatterers of the same kind (αp = αs ∀p) and assuming only one single moving particle, the inverse Fourier transform yields the complex signal in a sampling volume Vm
We now take into account the brightness profile Wm(ρ, z) of the sampling volume Vm as defined in Eq. (16). Neglecting the small phase contribution due to the radial displacement ρ, the signal Id,m (t) becomes
Auto- and cross-correlation in the single particle regime
In general, the cross-correlation function Gmn(τ) of the fluctuating signals Im(t) and In(t) in the sampling volumes Vm and Vn is
In case of a directed net flow and n ≠ m, the cross-correlations Gmn and Gnm are different. The directed flow from sampling volume Vm to Vn results in a shift of the maximum cross-correlation amplitude towards a lag time τ equal to the mean transit time τmn a NP requires to move from Vm to Vn. Hence, a flow from Vm to Vn can be evidenced by taking the difference Gnm − Gmn. With a net flow, this difference shows a non-zero amplitude with a maximum at τ = τmn.28]. and are the lateral and axial diffusion times. The average number of particles N in a sampling volume is calculated using the formula
This 3D Gaussian auto-correlation model has initially been used for interpreting our OCCS measurements (Fig. 2). It is obvious that this simple model is inadequate, because the shape of the measured correlation curve is significantly different from the model. As shown by the brightness profile in Fig. 4, the Gaussian approximation is only valid in the axial direction. Therefore, we applied Eq. (8) on the numerically calculated brightness profile for better estimating the shape of the auto-correlation curve. Figure 2 clearly shows the much improved match of the experimental curve and the numerical model. We attribute the residual difference to the coherent nature of OCCS that we ignored in this calculation. For a further improvement we designed a Monte Carlo based model (appendix A) taking into account the brightness W (r) and all coherent light interactions. The simulated correlations closely approach the experimental results, which confirms that the lateral Bessel illumination profile has a significant impact on the shape of the correlation curve. The Bessel profile leads to an autocorrelation that does not monotonically decrease as the particle moves away from the center of the sampling volume. Instead, the particle transiently disappears when moving through the minima between the ”Bessel” lobes. Therefore, we introduce an exponential decay term and write the auto-correlation term asEq. (11) closely matches the experimental correlation curve as shown in Fig. 2.
Auto-correlation in the few particles regime
We need an extended fit model when measuring at higher particle concentrations. Similarly to Eq. (5), we write the total signal of N particles within a sampling volume as16]. The signal analyzed in OCCS is a product of two intensities formed in a heterodyne light scattering geometry for which Kalkman et al.  showed that it is a Lorentzian with half linewidth . Integrating this coherent contribution into Eq. (11) yields
Auto-correlation in the many particles regime
As the concentration 〈C〉 increases, all terms in Eq. (13) decrease with 1/N, i.e. 1/〈C〉 as in incoherent methods like FCS, but the Ac coefficient is proportionally growing with N. In consequence, the auto-correlation is dominated by the coherent particle interaction. In the many particles regime (N ≫ 1) and because τc ≪ τb, Eq. (13) simplifies to
3.1. OCCS setup
The broadband light source (Ti-Sa laser, Femtolasers Inc.) delivers output powers up to 400mW with a central wavelength of 790 nm and FWHM bandwidth of 130 nm. The linearly polarized output of the laser is attenuated (down to 45mW), coupled into the illumination fiber (mode field diameter of 4.2 μm, single mode polarization maintaining fiber, Fibercore Ltd.) in order to adjust the power at the sample between 2–20mW (depending on the experiment). The collimated illumination is split into a reference and an illumination arm by an unpolarized beam-splitters 50/50. BK7 glass prisms located in the reference arm compensate for the dispersion mismatch due to the different optical components (objective, lenses, etc.) inside the illumination and detection arms. An axicon (176° apex angle, Del Mar Photonics Inc.) generates a radial zero-order Bessel beam, which is further imaged in the objective’s focal plane by telescopes. In order to suppress residual stray light from the tip of the axicon, an illumination mask Fill is placed in the intermediate focal plane of the following telescope. The epi-illuminated OCCS setup contains a 164 mm tube lens (Carl Zeiss) and a Zeiss plan apochromat water immersion objective (25×, NA 0.8) for illumination and detection. As illustrated in Fig. 3, the illumination and detection fields do not overlap in the back-focal plane of the objective. This corresponds to a dark-field configuration, which is generated by the detection aperture Fdet conjugated to the back focal plane of the objective. Overall, this interferometer implements a so-called Bessel-Gauss configuration . The complementary apertures Fill and Fdet ensure the dark-field effect. This is essential for a high SNR while measuring the weak backscattered light from NPs. The illumination field corresponds to a radial zero-order Bessel distribution in the focal plane with the first minimum located at 0.41 μm lateral radius, whereas the detection mode is Gaussian with a smaller numerical aperture (NA) of about 0.18.
The backscattered light (Fig. 1(b)) is superimposed with the reference light and focused into a single mode fiber (mode field diameter of 4.6 μm, Corning Inc.) guiding the collected light to the spectrometer (Fig. 1(c)). The customized spectrometer decomposes the input field with a transmission grating (1200 lines/mm, Wasatch Inc.). The spectral interferogram was registered with a linear array (Atmel Aviiva M2 - 2048 pixels). The spectrum is recorded from 720nm to 860nm wavelength. The detector was set to a line rate of 10kHz with an integration time of 43 μs. The reference arm intensity is adjusted to fill about 75% of the available dynamic range of the camera. The sample spectrum is obtained by subtracting the reference spectrum from the measured spectrum. The sample spectrum is then re-sampled at equidistant wavenumbers (λ to k mapping, Fig. 1(d)) and the residual dispersion is compensated by multiplying with calibrated phase factors. The depth profile containing a sequence of sampling volumes (center to center distance of 1.66 μm in water) is then extracted by computing the fast Fourier transform (FFT) of the corrected spectrum (Fig. 1(e)). Auto-correlations of these time-dependent signals and cross-correlations of time traces between different sampling volumes are then calculated (Fig. 1(f)).
3.2. Sample preparation
Sample solutions of monodisperse polystyrene microspheres (PS MSs) with a diameter of 109nm (POLYBEAD Microspheres 0.109 μm, Polysciences, Inc.) and gold colloids (gold NPs) with diameters of 30, 50, 80 and 100nm (EM.GC30, EM.GC50, EM.GC80, EM.GC100, British Biocell International) were used. For all measurements in the single particle regime, the concentration of particles was 9.3pM, which is the supplier’s given concentration of 100nm gold NPs stock solution. The solutions containing other particles were prepared by diluting an adequate volume of stock solution in ultrapure water or in glycerol. To obtain solutions of the desired concentration for the measurements in the few particles and many particles regimes, the samples were prepared using the same procedure. All sample measurements were performed in plastic wells (μ-Slide 8 well, uncoated, sterile, Ibidi GmbH) with a single well-volume of 300 μl.
3.3. Characterization of the sampling volumes
The determination of r0, z0, Ab and rb is crucial for an appropriate fit model, which requires an accurate characterization of the OCCS sampling volumes. The spatial light field distribution W (r) = W (x,y,z) (brightness profile) was characterized by imaging individual gold NPs and polystyrene microspheres. The scattering particles were dispersed in an agarose gel with a 0.3% weight/volume ratio. We imaged individual NPs using a two axis piezoscanning stage (x-y, resolution 0.12 μm) for displacing the NPs and an illumination power of 9mW.
We compared the brightness profile measurements with ab initio calculations using the focus field calculation framework by M. Leutenegger et al. . Based on the detection fiber specification (core diameter and NA), we calculated the conjugated Gaussian field Edet(x,y,z) in the object space. As OCCS is an interferometric technique, it measures the field amplitude and not the intensity. Therefore, the spatial distribution Wm(x,y,z) of the detected signal in the sampling volume Vm was related to the product of the illumination field amplitude Eill(x,y,z), the detection field amplitude Edet(x,y,z) and the coherence gate g(2n(z − zm)) where zm is the center position of sampling volume Vm.Figure 4(a) shows the measured brightness profile W (r) in the x-y cross-section with an individual ∅100nm gold NP, whereas the axial x-z cross-section is shown in Fig. 4(b). In Fig. 4(c) we compare the radial illumination profile (averaged on 10 particle observations) W (r) of individual ∅100nm, ∅109nm PSs MSs with the calculation (red line). As the particle signal decreases, so does the observed strength of the side lobes because the signal in these lobes approaches the noise level. Figure 4(d) shows the measured axial brightness profiles W (r) compared with the coherence gate determined by the measured source spectrum S(k) of our OCCS system and calculated using the Wiener-Khinchin relation. The calculated coherence gate is 3.07 μm long. We obtain a good match except for the ∅30nm gold NPs, for which the axial brightness profile seems to be larger due to a lower signal-to-noise ratio (SNR).
The depth of field (DOF) assessment is important to evaluate the number of useful sampling volumes. We used stock solutions of ∅100nm gold NPs (concentration of 9.3pM) freely diffusing in water, an illumination power of 2mW and measured the time-averaged signal amplitude 〈Im〉 for the sampling volume Vm along the optical axis. Figure 4(e) compares the average DOF from 10 measurements of 100 seconds with an ab initio calculation. Within the DOF of about 40 μm FWHM, 20 sampling volumes can be observed simultaneously in axial separation steps of Δz = 1.66μm (Eq. (2)).
3.4. Data analysis
In the single particle regime, the extent z0 was extracted from the coherence length given by the spectrum of the light source. Figure 4(d) confirms the good agreement between the calculated and the measured axial brightness profile. The lengths r0 and rb were calibrated with an autocorrelation measurement of ∅109nm polystyrene microspheres matched to Eq. (11) by using the theoretical value for the diffusion coefficient D. These characteristic lengths r0, rb and z0 were then kept fixed and Eq. (11) was used to fit the auto-correlations shown in Fig. 6(a), where only Ab, N and D were free fit parameters.
Ab is obviously linked to the visibility of the side lobes as explained in section 2.3 (single particle regime). We calibrated Ab,PS by using the ∅109nm polystyrene microspheres. Taking the maximum signal hs of the first side lobe with respect to the profile maximum signal in Fig. 4(c), we estimated the value Ab = Ab,PShs,PS/hs for other particles. We then fitted the autocorrelation curves using Eq. (11) with only N and D being free parameters. These results differ by less than 6% with respect to the previous fit with Ab as a free parameter. Considering this small difference, the fitting can be done without having to measure first the brightness profile of each particle type. However, for the smallest NPs of ∅30nm, a correction factor has been used, as is explained in details in appendix B.
In the few particles regime, the same calibration values r0, rb, z0 and Ab,PS are used in the fit model. The free parameters are D, N and Ac. The theorical number of particles in a sampling volume is calculated from the brightness profile using Eq. (10).
4. Experimental results
4.1. Proof-of-principle experiments
It has been shown that the scattered field amplitude is proportional to the NPs volume up to diameters of 100nm [35, 36]. The expected third power dependency of the scattering signal versus NP diameter is well confirmed by our measurement as shown by the fitted trend line in Fig. 5.
In proof-of-principle experiments, we performed OCCS on freely diffusing gold NPs by varying the NP diameter and the viscosity of the solvent. Figure 6 summarizes the results of these experiments. The normalized auto-correlations of differently sized gold NPs (concentration: 9.3pM; ∅100nm, ∅80nm, ∅50nm illuminated with 2mW and ∅30nm illuminated with 8mW) are shown in Fig. 6(a). These NPs are freely diffusing in water inside the focal sampling volume V0 (Fig. 4(e)). Extracting the diffusion coefficients is based on a straightforward fitting procedure as described in the Data Analysis section. The fit residuals are on the order of a few percent, thus confirming the quality of our method. The extracted diffusion coefficients (Fig. 6(b)) are related by the Stokes-Einstein relation, D = kBT/3πηd, where kB is the Boltzmann constant, T the absolute temperature, η the fluid viscosity and d the particle diameter. Diffusion constants of up to 15 μm2 s−1 were measured. In a following experiment, we varied the viscosity by measuring ∅80nm gold NPs (illuminated with 2mW) diffusing in different glycerol/water solutions, over a range of 0% w/w up to 80% w/w of glycerol. The extracted diffusion data (Fig. 6(c)) matches well the viscosity versus glycerol concentration model according to Cheng .
So far, we introduced OCCS for the single particle regime. As OCCS is based on a coherent scattering process, the coherent interaction among signals from several particles cannot be neglected if more than one particle is contained in a sampling volume. The scattered field originating from several particles results in a speckle field, i.e. an additional coherent contribution that is absent in intensity based methods like FCS. This speckle field fluctuates in time due to the particles’ mutual movements and decorrelates at a rate of as shown by Kalkman et al. . Integrating this coherent contribution into Eq. (11) yields Eq. (13).
The auto-correlations (Fig. 7(a)) were obtained by measuring ∅109nm polystyrene microspheres (PS MSs) at different concentrations in 80% glycerol/water solutions with an illumination power of 10mW. These auto-correlations have been background corrected by multiplying with (1 − 〈B〉/〈Im〉)−2, where B is the measured background intensity [38, 39]. The measured auto-correlations fit our phenomenological model well (Eq. (13)) as supported by the small residuals in Fig. 7(a). Figure 7(b) relates the extracted Ac and N with the number of particles estimated with Eq. (10). Concentrations below 70pM correspond to the single particle regime, for which no meaningful Ac can be extracted. According to our measurements, Ac is indeed equal to N, as has been derived by Berne and Pecora .
In the limit of high concentration we enter the many particles regime. As the concentration 〈C〉 increases, all contributions in Eq. (13) decrease with 1/N, i.e. 1/〈C〉, but the Ac coefficient is growing proportionally with N. In consequence, the auto-correlation is dominated by the coherent particle interaction. In this regime, Eq. (13) simplifies to Eq. (15).
Figure 7(a) shows auto-correlations as a function of concentrations extending over two orders of magnitude. This demonstrates the transition from the single particle regime to the many particles regime and underlines the differences due to the coherent interactions appearing in the few particles regime. In the few particles regime, the diffusion coefficient D is also related to the decorrelation time τc. Figure 7(d) shows τc extracted from the auto-correlations (Fig. 7(c)), where the small residuals show the good agreement with our fit model. For these experiments in different glycerol/water solutions we used ∅109nm polystyrene MSs at a concentration of 332pM (illumination power of 10mW). The background-corrected amplitudes of the auto-correlations (Fig. 7(c)) have been used to determine τc. The measured τc matches well the relation due to Cheng  at high viscosities (trendline), whereas deviations at low viscosity (c.f. the residuals) are mainly due to our limited sampling rate (10 kHz).
In summary, the OCCS based auto-correlation analysis applies to a wide range of particle concentrations subdivided into three regimes illustrated by Fig. 8. (i) In the single particle regime (N ≪ 1), Eq. (11) allows assessing the mean concentration and diffusion coefficient of NPs. (ii) With increasing NP concentration (few particles regime: N ∼ 1), the coherent interaction becomes significant and Eq. (13) adequately fits the measured auto-correlations. (iii) For high concentrations, i.e. the many particles regime (N ≫ 1), the fit model simplifies to Eq. (15) and only the diffusion coefficient can be extracted.
4.2. Multiplex advantage and cross-correlations
OCCS has an intrinsic multiplex advantage and allows cross-correlating signals from several sampling volumes. A directed flow from sampling volume Vm to Vn results in a characteristic shift of the maximum correlation amplitude towards a lag time τ equal to the typical transit time τmn between these volumes. This peak shift is evidenced by the difference of the cross-correlation curves ΔGn,m = Gnm − Gmn. Obviously, no axial net flow results in ΔGn,m = 0.
As already observed by M. Geissbühler et al.  light scattering on NPs induces directed NPs movements. This induced net flow depends on the impinging illumination power. At low illumination power, the diffusion will dominate and the additional induced flow is insignificant (case (i)). By increasing the illumination power a notable directed flow appears (case (ii)). At low illumination power but with an additional external pump laser, the induced directed flow due to the optical forces can be easily assessed via OCCS (case (iii)).
Figure 9(a) shows the experimental result when measuring the OCCS signal of a ∅100nm gold NP sample illuminated with increasing powers (2–20mW) and evaluating the cross-correlations ΔGm+2,m. The axial distance 2Δz is slightly greater than the coherence length lc, minimizing the spatial overlap while increasing the sensitivity for slow axial flow. For the lowest power of 2mW negligible directed flow is perceived (case (i)), i.e. ΔGm+2,m ≃ 0. Therefore, we conclude that diffusion is the dominant process, which can be measured by OCCS. For higher illumination powers a directed flow with decreasing transit time appears (case (ii)). At powers of ≳ 5mW the notable directed flow is due to the force equilibrium between the optical forces induced by the illumination beam and the counteracting drag force (Stokes’ law) [40,41]. All measurements have been performed with Vm being the focal volume V0 (Fig. 4(e)). Due to the Bessel illumination, the optical force varies strongly with the lateral NP position within the sampling volume, which causes a spread in transit times that is difficult to model. Therefore, we estimated the mean transit time by the lag time of max(ΔGm+2,m) (blue crosses in Fig. 9(a)). Taking into account the distance 2Δz = 3.32μm between the sampling volumes, a mean transit speed of 20μm/s at 5mW illumination power and of 80μm/s at 20mW has been measured. For all the previous concentration and diffusion measurements based on the auto-correlation analysis, we kept the illumination power small and checked for a negligible net flow (ΔGm+2,m ≃ 0).
To study case (iii), we used an additional laser beam at 532nm with 20mW and 40mW power to push differently sized gold NPs through the sampling volumes (2mW illumination power). This extra laser beam with a Gaussian beam shape had been focused 25 μm further away from the objective than the observation focus with an effective NA of 0.19 (source: diode pumped solid state laser, Roithner Lasertechnik, Austria). The cross-correlations differences ΔGm+2,m are shown in Fig. 9(b) and the observed mean transit speeds are shown in Fig. 9(c). OCCS clearly resolves the induced flow speed exerted on identical NPs by different power levels of the extra laser source at 532 nm (see Fig. 9(c), mean transit speed between 2.0 and 2.2 times greater at 40mW than at 20mW) and, as expected, the optical forces are lower for smaller particles .
Figure 10 presents information on the axial flow by calculating the cross-correlations ΔGm+2,m in different sampling volumes for the same three cases.
For case (i), Fig. 10(a) shows the results when measuring a ∅100nm gold NP sample with an illumination power of 2mW. In that case, the cross-correlation curves do not show a clear flow signature because the diffusion is the dominant process for the NP mobility. The volumes are numbered according to Fig. 4(e). Figure 10(b) shows the resulting cross-correlations measured with ∅100nm gold NPs in case(ii) with the illumination laser beam at a power of 20mW. Finally, in case (iii), the second laser beam was focused 25 μm further away from the objective than the observation focus. Therefore, in the sampling volumes close to the objective (negative indices), the pushing laser intensity was almost constant within a sampling volume. This led to a well defined transition time showing up a sharp peak in the cross-correlations with small lag time spread. On the other hand, closer to the focus of the green laser beam (positive indices), the green laser beam illuminated a region smaller than the sampling volume, which led to a notable transit time spread.
In conclusion, we presented a new spectroscopic technique named Optical Coherence Correlation Spectroscopy. We used numerical simulations to estimate the performance of our method and to build an analytical model for the auto-correlation from the single particle regime up to the many particles regime. The mean NP concentration can be measured over a large range well above the single particle limit. Based on OCCS, the diffusion coefficient of gold NPs down to ∅30nm in water (approximately 15 μm2 s−1) and in glycerol/water solutions with varying viscosities has been investigated. A key feature of OCCS is the simultaneous probing of several sampling volumes in the axial direction. The cross-correlation between signal traces originating from different volumes enabled the measurement of the mean transit speed along the axial direction (up to 700 μm s−1 was shown). In summary, OCCS opens the door to fast 3D flow and diffusion measurements.
Appendix A. Monte-Carlo simulation
This Monte-Carlo simulation models the essential parts of OCCS in order to investigate numerically the main concepts and to propose better fit functions for the auto- and cross-correlations. According to Fig. 1, the simulation starts by calculating random trajectories of particles due to Brownian motion within a cylindrical volume. At each time step, the back-scattered light is calculated for each particle and the summed field is superimposed with the reference field. The resulting interference spectra are then ”detected” by modeling shot noise, read-out noise and truncation to discrete values. This yields the time-dependent interferograms which are then processed as real data from the spectrometer.
Single particle trajectory (free Brownian motion).
Within a cylindrical volume of lateral radius R and axial length L, we simulate the free Brownian motion of N NPs. The initial position (x,y,z) of each NP is randomly chosen in the volume by assuming a uniform distribution. The simulated measurement time T is divided in small time intervals Δt less than or equal to the sampling time of the spectrometer. The trajectory is then obtained step by step by adding normally distributed random numbers of standard deviation to each coordinate of the NPs, where D are the diffusion constants of the particles.
If a particle exits the simulation volume, i.e. its x2 + y2 > R2 and/or |z| > L/2, its position is reset randomly with a uniform distribution on the surface of the cylindrical volume. Therefore, the particle concentration in the entire volume is kept constant, whereas the local concentration varies due to Brownian motion of the particles. The random reseting of the position avoids long-term correlations between particles leaving and entering the simulation volume.
Approximative OCCS brightness profile modeling a Bessel illumination profile and a Gaussian detection profile.
The brightness profile of the back-scattered light by a small particle can be approximated by the scalar product of the illumination field amplitude Eill(ρ,z,k) and the field amplitude Edet(ρ,z,k) detected by the single-mode fiber. k = 2πn/λ is the wave vector inside the sample medium with refractive index n. We assume rotational symmetry around the optical axis z, and use cylindrical coordinates (ρ,z), where . Within the illumination and detection volume, the light keeps to a large extent its linear polarization, such that the polarization of the field can be neglected. The amplitude W (ρ,z,k) = Eill(ρ,z,k)Edet(ρ,z,k) approximates then the detected amplitude spectrum of a point-like scatterer at the position (ρ,z), that is the brightness profile for detecting a small particle at the wavenumber k.
The lateral illumination profile is modeled as a zero-order Bessel profile J0(2.404kρ/kcρ0) of the first kind, where ρ0 = 410nm is the radius of the first Bessel zero at the central wave vector kc = nk0 = 2πn/λ0. The axial illumination profile is given by the axial spread z0 as determined by the waist of the Gaussian amplitude profile on the conical wavefront. The maximum field shall be reached at z = 0, which yields the illumination profile
For negative v(z), the axial field is null because the illumination cone does not reach the optical axis. The present approximation does not model the non-overlap zone for v(z) < 0 in which the beam profile is a converging half-Gaussian ring. For 0 < v(z) ≲ 0.2, the calculated beam profile approaches the real beam profile and becomes sufficiently accurate for v(z) ≳ 0.2. We observe this range by limiting the calculations to the FWHM of the axial profile, that is to the range −0.481z0 ≲ z ≲ 0.652z0.
The Gaussian detection profile is accurately calculated by the well-known formula for the propagation of the amplitude in a Gaussian beam .
In order to speed up the simulation, we further approximated the brightness profile by splitting off the wavelength dependency and by neglecting small phase contributions in the Gaussian detection mode. Using these simplified Eq. (17) and Eq. (18), we approximated the brightness profile by the amplitude for k = kc multiplied by the axial phase factor:
Interference spectrum intensity for scattered field(s).
Detected intensity converted to digital values.
The signal intensity on the spectrometer is given by the interference between all field contributions Es(k) back-scattered from the particles and the reference field Er = exp(2izrk). The sample field is given by
Tomogram amplitude |A(z)| from interference spectrum ID(k).
From here on, the simulated interference spectra are processed in the same way as the measurements. First, the average spectrum ĪD(k) (background and reference spectrum ) is subtracted and the spectrum is interpolated to an equidistant wavenumber sampling. Then, the fast Fourier transform A(z) = ℱ−1 (ID(k) − ĪD(k)) is calculated. The resulting tomogram is cropped to the region of interest (ROI) and only the tomogram amplitude |A(z ∈ ROI)| is retained.
B. Data analysis
For the smallest NPs of ∅30nm, the low SNR led to a low visibility of the lateral side lobes. In this case, the value of the diffusion coefficient D seemed to increase. We investigated if the Monte-Carlo simulation could reproduce this effect. For this purpose, we simulated two kinds of particles: both had the diffusion coefficient of the ∅100nm gold NPs. One kind was given the brightness of ∅100nm gold NPs with an illumination power of 2mW, while the other was given the brightness of the ∅30nm gold NPs with an illumination power of 8mW as used in the experiment. This way, we eliminated the influence of the diffusion coefficient when comparing the effect of the SNR, that is the NP brightness. The Monte-Carlo simulated auto-correlation curves for the sampling volume V0 are shown in Fig. 11 and were fitted using Eq. (11) with r0, Ab and N as free parameters.
It turned out that dim NPs seem to move faster. We interpret this finding as an apparent decrease of the parameter r0 for low SNR. Because we set the brightness of the low SNR particles at the same value as the ∅30nm gold NPs in our measurements, we could use the ratio (0.84) between the r0 values of the two simulated curves in Fig. 11 to fit the auto-correlation curves of the measurements of the ∅30nm gold NPs.
A Matlab implementation of the algorithm developed here can be downloaded from our website .
We acknowledge funding by the Swiss National Foundation (grant 205321L_135353).
References and links
1. D. Magde, E. Elson, and W. Webb, “Thermodynamic fluctuations in a reacting system measurement by fluorescence correlation spectroscopy,” Phys. Rev. Lett. 29, 705–708 (1972). [CrossRef]
2. R. Rigler and E. Elson, Fluorescence Correlation Spectroscopy: Theory and Applications (Springer, 2001). [CrossRef]
3. K. Hassler, P. Rigler, H. Blom, R. Rigler, J. Widengren, and T. Lasser, “Dynamic disorder in horseradish peroxidase observed with total internal reflection fluorescence correlation spectroscopy,” Opt. Express 15, 5366–5375 (2007). [CrossRef] [PubMed]
4. P. Schwille, U. Haupts, S. Maiti, and W. Webb, “Molecular dynamics in living cells observed by fluorescence correlation spectroscopy with one- and two-photon excitation,” Biophys. J. 77, 2251–2265 (1999). [CrossRef] [PubMed]
5. D. Schaeffel, R. Staff, H.-J. Butt, K. Landfester, D. Crespy, and K. Koynov, “Fluorescence correlation spectroscopy directly monitors coalescence during nanoparticle preparation,” Nano Lett. 12, 6012–6017 (2012). [CrossRef] [PubMed]
6. K. Jaskiewicz, A. Larsen, D. Schaeffel, K. Koynov, I. Lieberwirth, G. Fytas, K. Landfester, and A. Kroeger, “Incorporation of nanoparticles into polymersomes: Size and concentration effects,” ACS Nano 6, 7254–7262 (2012). [CrossRef] [PubMed]
7. P. Dittrich and P. Schwille, “Spatial two-photon fluorescence cross-correlation spectroscopy for controlling molecular transport in microfluidic structures,” Anal. Chem. 74, 4472–4479 (2002). [CrossRef] [PubMed]
8. W. Schrof, J. Klingler, S. Rozouvan, and D. Horn, “Raman correlation spectroscopy: A method for studying chemical composition and dynamics of disperse systems,” Phys. Rev. E. 57, R2523–R2526 (1998). [CrossRef]
10. J. Cheng, E. Potma, and S. Xie, “Coherent anti-stokes raman scattering correlation spectroscopy: Probing dynamical processes with chemical selectivity,” J. Phys. Chem. A 106, 8561–8568 (2002). [CrossRef]
11. M. Geissbuehler, L. Bonacina, V. Shcheslavskiy, N. Bocchio, S. Geissbuehler, M. Leutenegger, I. Maerki, J. Wolf, and T. Lasser, “Nonlinear correlation spectroscopy (nlcs),” Nano Lett. 12, 1668–1672 (2012). [CrossRef] [PubMed]
12. T. Liedl, S. Keller, F. Simmel, J. Radler, and W. Parak, “Fluorescent nanocrystals as colloidal probes in complex fluids measured by fluorescence correlation spectroscopy,” Small 1, 997–1003 (2005). [CrossRef]
14. P. Paulo, A. Gaiduk, F. Kulzer, S. Gabby Krens, H. Spaink, T. Schmidt, and M. Orrit, “Photothermal correlation spectroscopy of gold nanoparticles in solution,” J. Phys. Chem. C 113, 11451–11457 (2009). [CrossRef]
15. J. Yguerabide and E. Yguerabide, “Light-scattering submicroscopic particles as highly fluorescent analogs and their use as tracer labels in clinical and biological applications i. theory,” Anal. Biochem. 262, 137–156 (1998). [CrossRef] [PubMed]
16. B. Berne and R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology and Physics (John Wiley and Sons, New-York, 1976).
17. D. Boas, K. Bizheva, and A. Siegel, “Using dynamic low-coherence interferometry to image brownian motion within highly scattering media,” Opt. Lett. 23, 319–321 (1998). [CrossRef]
18. S. Dominguez-Medina, S. McDonough, P. Swanglap, C. Landes, and S. Link, “In situ measurement of bovine serum albumin interaction with gold nanospheres,” Langmuir 28, 9131–9139 (2012). [CrossRef] [PubMed]
19. S. Wennmalm and J. Widengren, “Interferometry and fluorescence detection for simultaneous analysis of labeled and unlabeled nanoparticles in solution,” J. Am. Chem. Soc. 134, 19516–19519 (2012). [CrossRef] [PubMed]
20. J. Chen and J. Irudayaraj, “Quantitative investigation of compartmentalized dynamics of erbb2 targeting gold nanorods in live cells by single molecule spectroscopy,” ACS Nano 3, 4071–4079 (2009). [CrossRef] [PubMed]
21. M. Digman, C. Brown, P. Sengupta, P. Wiseman, A. Horwitz, and E. Gratton, “Measuring fast dynamics in solutions and cells with a laser scanning microscope,” Biophys. J. 89, 1317–1327 (2005). [CrossRef] [PubMed]
22. M. Brinkmeier, K. Doerre, J. Stephan, and M. Eigen, “Two-beam cross-correlation: A method to characterize transport phenomena in micrometer-sized structures,” Anal. Chem. 71, 609–616 (1999). [CrossRef] [PubMed]
23. M. Gosch, H. Blom, S. Anderegg, K. Korn, P. Thyberg, M. Wells, T. Lasser, R. Rigler, A. Magnusson, and S. Hard, “Parallel dual-color fluorescence cross-correlation spectroscopy using diffractive optical elements,” J. Biomed. Opt. 10, 054008 (2005). [CrossRef] [PubMed]
24. T. Dertinger, V. Pacheco, I. Von Der Hocht, R. Hartmann, I. Gregor, and J. Enderlein, “Two-focus fluorescence correlation spectroscopy: A new tool for accurate and absolute diffusion measurements,” Chem. Phys. Chem. 8, 433–443 (2007). [CrossRef] [PubMed]
25. J. Izatt and M. Choma, Optical Coherence Tomography: Technology and Applications (Springer Verlag, Berlin, 2008).
26. P. Schwille, “Fluorescence correlation spectroscopy and its potential for intracellular applications,” Cell Biochem. Biophys. 34, 383–408 (2001). [CrossRef]
27. M. Leutenegger, C. Ringemann, T. Lasser, S. Hell, and C. Eggeling, “Fluorescence correlation spectroscopy with a total internal reflection fluorescence sted microscope (tirf-sted-fcs),” Opt. Express 20, 5243–5263 (2012). [CrossRef] [PubMed]
29. J. Kalkman, R. Sprik, and T. Van Leeuwen, “Path-length-resolved diffusive particle dynamics in spectral-domain optical coherence tomography,” Phys. Rev. Lett. 105, 198302 (2010). [CrossRef]
30. R. A. Leitgeb, M. Villiger, A. H. Bachmann, L. Steinmann, and T. Lasser, “Extended focus depth for fourier domain optical coherence microscopy,” Opt. Lett. 31, 2450–2452 (2006). [CrossRef] [PubMed]
32. C. Pache, N. Bocchio, A. Bouwens, M. Villiger, C. Berclaz, J. Goulley, M. Gibson, C. Santschi, and T. Lasser, “Fast three-dimensional imaging of gold nanoparticles in living cells with photothermal optical lock-in optical coherence microscopy,” Opt. Express 20, 21385–21399 (2012). [CrossRef] [PubMed]
33. M. Villiger and T. Lasser, “Image formation and tomogram reconstruction in optical coherence microscopy,” J. Opt. Soc. Am. A 27, 2216–2228 (2010). [CrossRef]
35. M. A. van Dijk, A. L. Tchebotareva, M. Orrit, M. Lippitz, S. Berciaud, D. Lasne, L. Cognet, and B. Lounis, “Absorption and scattering microscopy of single metal nanoparticles,” Phys. Chem. Chem. Phys. 8, 3486–3495 (2006). [CrossRef] [PubMed]
37. N. Cheng, “Formula for the viscosity of a glycerol-water mixture,” Ind. Eng. Chem. Res. 47, 3285–3288 (2008). [CrossRef]
39. D. Koppel, “Statistical accuracy in fluorescence correlation spectroscopy,” Phys. Rev. A 10, 1938–1945 (1974). [CrossRef]
41. M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: A review,” J. Nanophotonics 2, 021875 (2008). [CrossRef]
42. W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems: Vol. 2 Physical Image Formation (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2005).
43. S. Broillet, A. Sato, S. Geissbuehler, C. Pache, A. Bouwens, T. Lasser, and M. Leutenegger, “Matlab OCCS Experiment,” http://lob.epfl.ch/page-103066.html.