## Optical coherence correlation spectroscopy (OCCS) |

Optics Express, Vol. 22, Issue 1, pp. 782-802 (2014)

http://dx.doi.org/10.1364/OE.22.000782

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### Abstract

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

## 1. Introduction

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]

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## 2. Theory

### 2.1. Principle of OCCS

*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

*l*

_{c}. This allows to differentiate axial sampling volumes [25]. Supposing a Gaussian-shaped source spectrum,

*l*

_{c}is given as where

*n*is the refractive index of the medium,

*λ*

_{0}is the central source wavelength in vacuum and Δ

*λ*its spectral bandwidth at full width at half-maximum (FWHM). The 1/

*e*

^{2}half width of the axial coherence gate is then

*z*

_{0}axial length and spaced by the center to center distance [25] where Δ

*k*

_{s}is the spectral bandwidth of the spectrometer. If the available source spectrum is not fully imaged on the spectrometer, the axial resolution will be inferior to the value

*l*

_{c}in Eq. (1). Choosing Δ

*k*

_{s}such that Δ

*z*≲

*l*

_{c}/2 enables two consecutive sampling volumes to be resolved. In our case, Δ

*z*= 1.66μm and

*l*

_{c}= 3.07μm, and Δ

*z*≃

*l*/2.

_{c}*t*at wavenumber

*k*after mapping from

*λ*to

*k*= 2

*π/λ*can be written as [25]

*α*stands for the electric field reflectivity of each scatterer

_{p}*p*and

*z*their positions.

_{p}*α*

_{r}is the reflectivity and

*z*

_{r}the optical path length of the reference arm.

*S*(

*k*) represents the power spectrum of the light source. The first term (i) is the back-reflected average power and is path-length independent. As the reference intensity is much stronger than the reflected sample intensity, subtracting the reference power spectrum eliminates the path-length independent contribution. The second term (ii) corresponds to the interference between the reference field and each scatterer’s fields. This is the term of interest in OCCS because it depends on the path-length difference between the reference arm and the scatterers. The third term (iii) corresponds to the interference occurring between the fields of the scatterers. Due to the strong reference intensity, this term can be neglected. Therefore, Eq. (3) can be reduced to the second term.

*α*=

_{p}*α*

_{s}∀

*p*) and assuming only one single moving particle, the inverse Fourier transform yields the complex signal in a sampling volume

*V*where

_{m}*z*is the position of the center of the sampling volume

_{m}*V*and

_{m}*k*

_{0}is the central wave number of the source spectrum.

*W*(

_{m}*ρ*,

*z*) of the sampling volume

*V*as defined in Eq. (16). Neglecting the small phase contribution due to the radial displacement

_{m}*ρ*, the signal

*I*

_{d}

*(*

_{,m}*t*) becomes Taking the modulus of this signal, we obtain the OCCS signal which is a similar expression as in incoherent methods like FCS. For OCCS, this expression is limited to the single particle regime (

*N*≪ 1). This equation shows that

*I*scales with the scattering amplitude in volume

_{m}*V*because the signal is proportional to the particle electric field reflectivity

_{m}*α*

_{s}.

#### Auto- and cross-correlation in the single particle regime

*n*≠

*m*, the cross-correlations

*G*and

_{mn}*G*are different. The directed flow from sampling volume

_{nm}*V*to

_{m}*V*results in a shift of the maximum cross-correlation amplitude towards a lag time

_{n}*τ*equal to the mean transit time

*τ*a NP requires to move from

_{mn}*V*to

_{m}*V*. Hence, a flow from

_{n}*V*to

_{m}*V*can be evidenced by taking the difference

_{n}*G*−

_{nm}*G*. With a net flow, this difference shows a non-zero amplitude with a maximum at

_{mn}*τ*=

*τ*.

_{mn}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]

*C*〉 is the average concentration of the particles in the sampling volume,

*D*is the diffusion coefficient and

*W*(

**r**) is the brightness profile. For a three-dimensional Gaussian volume with 1/

*e*

^{2}radii of

*r*

_{0}laterally and

*z*

_{0}axially, we would obtain where

*N*is the average number of particles in the volume and

*γ*is the volume contrast [28

28. T. Wohland, R. Rigler, and H. Vogel, “The standard deviation in fluorescence correlation spectroscopy,” Biophys. J. **80**, 2987–2999 (2001). [CrossRef] [PubMed]

*N*in a sampling volume is calculated using the formula where

*V*

_{eff}is the effective sampling volume.

*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 as with

*A*

_{b}and the characteristic length

*r*account for the radial Bessel profile. The characteristic length

_{b}*r*depends on the distance between the lobes and is thus constant for a given setup.

_{b}*A*

_{b}is linked to the visibility of the Bessel side lobes. The visibility increases with the particle brightness because it depends on the signal to noise ratio (SNR). Eq. (11) closely matches the experimental correlation curve as shown in Fig. 2.

#### Auto-correlation in the few particles regime

*N*particles within a sampling volume as In the image plane, the different field contributions from all particles in one sampling volume create a speckle pattern that fluctuates in time due to the particles’ mutual movements (stochastic phase contributions). In a heterodyne detection, the signal of scattered light for diffusing particles decorrelates at a rate of

*q*= 2

_{c}*nk*

_{0}[16]. The signal analyzed in OCCS is a product of two intensities formed in a heterodyne light scattering geometry for which Kalkman

*et al.*[29

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]

*A*

_{c}has to be proportional to

*N*because the speckle fluctuations contrast is independent of the number of particles.

#### Auto-correlation in the many particles regime

*C*〉 increases, all terms in Eq. (13) decrease with 1/

*N*,

*i.e.*1/〈

*C*〉 as in incoherent methods like FCS, but the

*A*coefficient is proportionally growing with

_{c}*N*. In consequence, the auto-correlation is dominated by the coherent particle interaction. In the many particles regime (

*N*≫ 1) and because

*τ*

_{c}≪

*τ*

_{b}, Eq. (13) simplifies to which corresponds to the theory for dynamic light scattering (DLS).

## 3. Methods

### 3.1. OCCS setup

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]

*F*

_{ill}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

*F*

_{det}conjugated to the back focal plane of the objective. Overall, this interferometer implements a so-called Bessel-Gauss configuration [33

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]

*F*

_{ill}and

*F*

_{det}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.

*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

*μ*-Slide 8 well, uncoated, sterile, Ibidi GmbH) with a single well-volume of 300 μl.

### 3.3. Characterization of the sampling volumes

*r*

_{0},

*z*

_{0},

*A*

_{b}and

*r*is crucial for an appropriate fit model, which requires an accurate characterization of the OCCS sampling volumes. The spatial light field distribution

_{b}*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.

*I*〉 for the sampling volume

_{m}*V*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 Δ

_{m}*z*= 1.66μm (Eq. (2)).

### 3.4. Data analysis

*z*

_{0}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

*r*

_{0}and

*r*

_{b}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

*r*

_{0},

*r*

_{b}and

*z*

_{0}were then kept fixed and Eq. (11) was used to fit the auto-correlations shown in Fig. 6(a), where only

*A*

_{b},

*N*and

*D*were free fit parameters.

*A*

_{b}is obviously linked to the visibility of the side lobes as explained in section 2.3 (single particle regime). We calibrated

*A*

_{b,PS}by using the ∅109nm polystyrene microspheres. Taking the maximum signal

*h*

_{s}of the first side lobe with respect to the profile maximum signal in Fig. 4(c), we estimated the value

*A*

_{b}=

*A*

_{b,PS}

*h*

_{s,PS}/

*h*

_{s}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

*A*

_{b}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.

*r*

_{0},

*r*

_{b},

*z*

_{0}and

*A*

_{b,PS}are used in the fit model. The free parameters are

*D*,

*N*and

*A*

_{c}. 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

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]

36. A. Tcherniak, J. Ha, S. Dominguez-Medina, L. Slaughter, and S. Link, “Probing a century old prediction one plasmonic particle at a time,” Nano Lett. **10**, 1398–1404 (2010). [CrossRef] [PubMed]

*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

*et al.*[29

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]

*B*〉/〈

*I*〉)

_{m}^{−2}, where

*B*is the measured background intensity [38

38. S. Hess and W. Webb, “Focal volume optics and experimental artifacts in confocal fluorescence correlation spectroscopy,” Biophys. J. **83**, 2300–2317 (2002). [CrossRef] [PubMed]

39. D. Koppel, “Statistical accuracy in fluorescence correlation spectroscopy,” Phys. Rev. A **10**, 1938–1945 (1974). [CrossRef]

*A*

_{c}and

*N*with the number of particles estimated with Eq. (10). Concentrations below 70pM correspond to the single particle regime, for which no meaningful

*A*

_{c}can be extracted. According to our measurements,

*A*is indeed equal to

_{c}*N*, as has been derived by Berne and Pecora [16].

*C*〉 increases, all contributions in Eq. (13) decrease with 1/

*N*,

*i.e.*1/〈

*C*〉, but the

*A*coefficient is growing proportionally with

_{c}*N*. In consequence, the auto-correlation is dominated by the coherent particle interaction. In this regime, Eq. (13) simplifies to Eq. (15).

*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

*V*to

_{m}*V*results in a characteristic shift of the maximum correlation amplitude towards a lag time

_{n}*τ*equal to the typical transit time

*τ*between these volumes. This peak shift is evidenced by the difference of the cross-correlation curves Δ

_{mn}*G*=

_{n,m}*G*−

_{nm}*G*. Obviously, no axial net flow results in Δ

_{mn}*G*= 0.

_{n,m}*et al.*[11

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]

*G*

_{m}_{+2,}

*. The axial distance 2Δ*

_{m}*z*is slightly greater than the coherence length

*l*

_{c}, 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.*Δ

*G*

_{m}_{+2}

*≃ 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 ≳ 5*

_{,m}*mW*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

40. W. Wright, G. Sonek, and M. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. **33**, 1735–1748 (1994). [CrossRef] [PubMed]

41. M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: A review,” J. Nanophotonics **2**, 021875 (2008). [CrossRef]

*V*being the focal volume

_{m}*V*

_{0}(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(Δ

*G*

_{m}_{+2}

*) (blue crosses in Fig. 9(a)). Taking into account the distance 2Δ*

_{,m}*z*= 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 (Δ

*G*

_{m}_{+2}

*≃ 0).*

_{,m}*G*

_{m}_{+2}

*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 [41*

_{,m}41. M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: A review,” J. Nanophotonics **2**, 021875 (2008). [CrossRef]

*G*

_{m}_{+2}

*in different sampling volumes for the same three cases.*

_{,m}## 5. Conclusion

^{2}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

*occsTrajectory.m*## Nanoparticles trajectories

*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 Δ

*t*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

*D*are the diffusion constants of the particles.

*i.e.*its

*x*

^{2}+

*y*

^{2}>

*R*

^{2}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.

*occsBrightness.m*## Brightness profile

*E*

_{ill}(

*ρ,z,k*) and the field amplitude

*E*

_{det}(

*ρ,z,k*) detected by the single-mode fiber.

*k*= 2

*πn/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

*W*(

*ρ,z,k*) =

*E*

_{ill}(

*ρ,z,k*)

*E*

_{det}(

*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*.

*J*

_{0}(2.404

*kρ/k*

_{c}

*ρ*

_{0}) of the first kind, where

*ρ*

_{0}= 410nm is the radius of the first Bessel zero at the central wave vector

*k*

_{c}=

*nk*

_{0}= 2

*πn/λ*

_{0}. The axial illumination profile is given by the axial spread

*z*

_{0}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 where

*θ*, which is given by sin

*θ*= 2.404/

*k*

_{c}

*ρ*

_{0}.

*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 −0.481

*z*

_{0}≲

*z*≲ 0.652

*z*

_{0}.

*R*(

*z,k*) is the radius of wavefront curvature,

*w*(

*z,k*) is the beam waist and

*Z*(

*k*) is the Rayleigh length. With good approximation, the single-mode fibers serving as illumination source and detection pinhole have non-dispersive numerical apertures (NA). Also, the axicon and lenses show only a small dispersion of the refraction index. Neglecting this material dispersion, the Gaussian beam waist

*w*(0,

*k*) and the Bessel beam waist

*ρ*

_{0}scale with the wavelength (factor

*k*

_{c}/

*k*) and need only be given for the central wave vector

*k*

_{c}.

*k*=

*k*

_{c}multiplied by the axial phase factor:

`occsInterferogram`function. This requires only little memory such that multi-threaded calculations on graphics cards can be used.

*occsInterferogram.m*

*occsDetect.m*## Detected interferogram

*E*

_{s}(

*k*) back-scattered from the particles and the reference field

*E*

_{r}= exp(2i

*z*

_{r}

*k*). The sample field is given by where we assume that the particles reflectivity

*α*is approximately constant across the source spectrum. The signal on the spectrometer is then where

_{p}*S*(

*k*) is the intensity spectrum of the source. The source’s spectral power is expressed in photons per detection interval per pixel (spectral channel). Hence,

*I*(

*k*) yields the number of photons received during the exposure time by each detector pixel. The detector converts these photons in digital values. Firstly, shot noise is applied to the signal by drawing random numbers from a Poisson distribution with average values

*I*(

*k*). Next, these detected photon numbers are scaled by the detector sensitivity

*s*and an eventual bias (dark current, offset

*o*) is added. Finally, Gaussian read noise is added (standard deviation

*σ*) and the result is rounded and limited to the numeric range of the detector.

*occsTomogram.m**A*(

*z*)| from interference spectrum

*I*

_{D}(

*k*).

## Tomogram amplitude

*Ī*

_{D}(

*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}(

*I*

_{D}(

*k*) −

*Ī*

_{D}(

*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

*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

*V*

_{0}are shown in Fig. 11 and were fitted using Eq. (11) with

*r*

_{0},

*A*

_{b}and

*N*as free parameters.

*r*

_{0}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

*r*

_{0}values of the two simulated curves in Fig. 11 to fit the auto-correlation curves of the measurements of the ∅30nm gold NPs.

## Supporting information

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.

## Acknowledgments

## 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. |

2. | R. Rigler and E. Elson, |

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 |

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. |

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. |

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 |

7. | P. Dittrich and P. Schwille, “Spatial two-photon fluorescence cross-correlation spectroscopy for controlling molecular transport in microfluidic structures,” Anal. Chem. |

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. |

9. | T. Hellerer, A. Schiller, G. Jung, and A. Zumbusch, “Coherent anti-stokes raman scattering (cars) correlation spectroscopy,” Chem. Phys. Chem. |

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 |

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. | 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 |

13. | V. Octeau, L. Cognet, L. Duchesne, D. Lasne, N. Schaeffer, D. Fernig, and B. Lounis, “Photothermal absorption correlation spectroscopy,” ACS Nano |

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 |

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. |

16. | B. Berne and R. Pecora, |

17. | D. Boas, K. Bizheva, and A. Siegel, “Using dynamic low-coherence interferometry to image brownian motion within highly scattering media,” Opt. Lett. |

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 |

19. | S. Wennmalm and J. Widengren, “Interferometry and fluorescence detection for simultaneous analysis of labeled and unlabeled nanoparticles in solution,” J. Am. Chem. Soc. |

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 |

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. |

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. |

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. |

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. |

25. | J. Izatt and M. Choma, |

26. | P. Schwille, “Fluorescence correlation spectroscopy and its potential for intracellular applications,” Cell Biochem. Biophys. |

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 |

28. | T. Wohland, R. Rigler, and H. Vogel, “The standard deviation in fluorescence correlation spectroscopy,” Biophys. J. |

29. | J. Kalkman, R. Sprik, and T. Van Leeuwen, “Path-length-resolved diffusive particle dynamics in spectral-domain optical coherence tomography,” Phys. Rev. Lett. |

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. | M. Villiger, C. Pache, and T. Lasser, “Dark-field optical coherence microscopy,” Opt. Lett. |

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 |

33. | M. Villiger and T. Lasser, “Image formation and tomogram reconstruction in optical coherence microscopy,” J. Opt. Soc. Am. A |

34. | M. Leutenegger, R. Rao, R. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express |

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. |

36. | A. Tcherniak, J. Ha, S. Dominguez-Medina, L. Slaughter, and S. Link, “Probing a century old prediction one plasmonic particle at a time,” Nano Lett. |

37. | N. Cheng, “Formula for the viscosity of a glycerol-water mixture,” Ind. Eng. Chem. Res. |

38. | S. Hess and W. Webb, “Focal volume optics and experimental artifacts in confocal fluorescence correlation spectroscopy,” Biophys. J. |

39. | D. Koppel, “Statistical accuracy in fluorescence correlation spectroscopy,” Phys. Rev. A |

40. | W. Wright, G. Sonek, and M. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. |

41. | M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: A review,” J. Nanophotonics |

42. | W. Singer, M. Totzeck, and H. Gross, |

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. |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(170.4500) Medical optics and biotechnology : Optical coherence tomography

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: November 6, 2013

Revised Manuscript: December 19, 2013

Manuscript Accepted: December 19, 2013

Published: January 7, 2014

**Virtual Issues**

Vol. 9, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Stephane Broillet, Akihiro Sato, Stefan Geissbuehler, Christophe Pache, Arno Bouwens, Theo Lasser, and Marcel Leutenegger, "Optical coherence correlation spectroscopy (OCCS)," Opt. Express **22**, 782-802 (2014)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-22-1-782

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### References

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