Time-resolved diffusing wave spectroscopy with a CCD camera

Katarzyna Zarychta; Eric Tinet; Leila Azizi; Sigrid Avrillier; Dominique Ettori; Jean-Michel Tualle

doi:10.1364/OE.18.016289

1. Introduction

The study of the propagation of diffuse light in biological tissue is a promising way towards new medical diagnostic tools, but this involves solving challenging inverse problems. Time-resolved measurements provide helpful additional information for that purpose: in the reflectance geometry, this information can be related to the in-depth position of an inhomogeneity [1–3] or of a layer [4–9], and the long-time behavior can be directly linked to the optical properties of the deepest layer [10]; in the transmittance geometry, the discrimination of short transit times allows an improvement of the transverse spatial resolution [11–14]. Numerous clinical setups now use such measurements [15–24], but even with recent technological improvements like picosecond laser diodes, they remain high-tech, delicate and expensive experiments.

We already proposed [25–27] a simple interferometric setup able to perform such time-resolved measurements, based on the use of a wavelength modulated continuous-wave source: if the optical pulsation ω was linearly varied, the time delay τ between the two waves in the interferometer would correspond to a proportional frequency shift, and to a beating at the same frequency shift like in heterodyne measurements. The discrimination of a beating frequency is therefore equivalent to the discrimination of a time delay. We proved that such a protocol can be extended to diffuse light: in our setup, interferences between the speckle pattern from the turbid medium and a reference beam are recorded by a lock-in detection. More precisely, the recorded photocurrent i(t) ≈i₀ + i_int(t), where i₀ is the photocurrent from the reference beam and where i_int(t) corresponds to interferences, is multiplied by a demodulation function Ref(t,τ); it is then integrated over half a period T/2 = 1/(2f) of the wavelength modulation, in order to obtain a demodulated signal S_DC(τ). A possible choice for the function Ref(t,τ) with a sine modulation of ω is:

Ref (t, τ) = \sin^{4} (2 π ​ f t) \cos [Δ Ω τ \cos (2 π ​ f t)]

where

Δ Ω

is the depth of the wavelength modulation. This function mainly contains a demodulation term cos[ΔΩτ cos(2π f t)], which follows the frequency shift due to the sine modulation of ω at a fixed transit time τ ; the sin⁴(2π f t) term is an anti-windowing filter. Such a demodulation procedure discriminates light traveling through the turbid medium with a transit time τ, and we proved [26] that the ensemble averaging <S_DC(τ)²> is proportional to the average time-resolved flux ϕ(τ). Furthermore, we also proved [28,29] that correlations like <S_DC,i(τ)S_{DC,i + p}(τ)>, where integers i and i + p denote two different integration periods separated by a time pT, constitute a unique way to measure the time-resolved normalized correlation function of the field, g ₁(t,τ), for a correlation time t and a photon transit time τ. We have (see Appendix):

〈 S_{D C, i} (τ) S_{D C, i + p} (τ) 〉 \approx g^{2} (0) \frac{η e i_{0}}{4 f} \int Π (τ - τ') φ (τ') g_{1} (p T / 2, τ') d τ'

where

φ (τ)

is the time-resolved diffuse light flux that we would measure in a real time-resolved experiment with the same incident energy, expressed as a photon flux; η is the detector quantum efficiency, e the electron charge and

Π (τ) = g^{2} (τ) / g^{2} (0)

is the normalized temporal response of the system, with:

g (τ) = 2 f \int_{0}^{T / 2} Ref(t, τ)dt

The measurement of the correlation function of diffuse light, also called Diffuse Correlation Spectroscopy (DCS) or Diffusing Wave Spectroscopy (DWS) [30–33], is a great challenge in biomedical optics, due to the difficulty to reach a reasonable signal to noise ratio (SNR). Such measurements however seem to have a real potential, with very promising results already obtained for tumors detection [34] or blood flow monitoring [35–37]. The present interferometric method enables to take advantages of both DCS and time-resolved measurements.

In previous experiments [26–29] we obtained results with a simple photodiode for the photodetection, and we computed the demodulated signal S_DC(τ) from the sampled photocurrent. In order to improve the SNR, or equivalently to reduce the acquisition time, it is possible to increase the detectors number. Interferometric measurements are indeed limited to a coherence area, and a detector much larger than the speckle grain size is useless. The use of N detectors however ideally allows an improvement of the SNR by a factor N. Lock-in detection schemes using CCD cameras exist [38–40], and we already proposed [25,41] an adaptation of such a scheme to our setup: the multiplication by the function Ref(t,τ) can be performed before photodetection, through the modulation of the intensity of the light source by the positive function

ξ (t) = Ref(t, τ) + Ref(t, 0)

Let us insist on the fact that this modulation of the intensity of the source is completely disconnected from the wavelength modulation, and that the frequencies implies in ξ(t) or in Ref(t,τ) can be up to one hundred times higher than the modulation frequency f (see equ. 1). The half-period integration is performed on each pixel. In such a setup, the DC component due to the reference beam is no more cancelled, and a difference between two frames is needed for this cancellation. Each pixel then corresponds to a value S_DC,i, and correlations between two different frames should then be computed pixel by pixel in order to get the average value <S_DC,i S_{DC,i + p} >.

We need quite a high modulation frequency f in order to ‘freeze’ speckle pattern fluctuations, and to subsequently measure the correlations in such fluctuations. A high speed camera is therefore expected in order to obtain a measurement for each modulation period, and to compute frames correlations with short correlation times. High speed cameras however usually have low quantum efficiency, and a high spatially correlated electronic noise that prevents to reach the SNR expected from a multi-pixel detection [41]. In this paper we show the possibility to compute correlations using a standard CCD camera, with great SNR improvement despite the low duty cycle of such an acquisition process.

2. Time-resolved measurements

2.1 The experimental setup

Our experimental setup is depicted on Fig. 1 . The source is a continuous wave Distributed feedback (DFB) laser diode cheetah from Sacher Lasertechnik, emitting at about 780 nm. One can obtain a modehop free wavelength modulation up to relatively high frequency through a weak modulation of the power current (the resulting modulation of the laser intensity, lower than 5%, will have a negligible impact in the following): the experiments reported in the present paper are performed with a frequency modulation f = 500Hz, but we can obtain frequencies higher than 10kHz with such a system. The frequency modulation depth is $Δ Ω = 2 π \cdot 2.3 G H z$ . This corresponds to a time-resolution of 270ps. Of course we could obtain a better time-resolution with a greater modulation depth, but the counterpart is a decrease of the SNR. We chose a compromise corresponding to the expected nanosecond scale of the time-resolved signal.

Fig. 1 Experimental setup: two weak reflectivity (15%) beamsplitters (BS) constitute a two-arms interferometer; a lens (L₁, f₁ = −12mm) allows uniform illumination of the CCD camera; a lens (L₂, f₂ = + 50mm) project the transmitted scattered light on the camera, which registers the interferometric signal with a typical grain size of 3μm, that is of about half the pixel size; the value of the BS reflectivity was chosen in order to have both a correct detection of the reference beam and a maximal illumination of the scattering sample; a (3:1) anamorphic prisms pair allows to correct the ellipticity of the laser beam; an acousto-optic modulator (AOM) allows lock-in detection through the multiplication by the positive function ξ(t).

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We use the diffraction order + 1 of an acousto-optic modulator (AOM) which allows a high-frequency modulation of the laser beam intensity, in order to perform the multiplication by the positive function $ξ (t)$ defined in Eq. (4). If i(t) ≈i₀ + i_int(t) is the photocurrent that a pixel should record without this modulation, then the charge stored in this pixel is:

Q = \int ξ (t) i (t) d t

At first we will consider that for each acquisition frame the integral runs over only one modulation half-period. Let us note here that the integration interval is not controlled by the frame acquisition shutter (which has a 20ms duration in the present experiment), but by the function $ξ (t)$ which is set to zero outside this half-period integration interval.

The photocurrent i₀ from the reference beam leads in (5) to a contribution Q₀ (with Q₀ = [g(0) + g(τ)]i₀T/2 ) while, from the expression (4) of $ξ (t)$ , the photocurrent i_int leads to the terms S_DC(τ) and S_DC(0). A time delay δτ = 2060 ps is introduced between the two arms of the interferometer (Fig. 1): transit times are therefore always higher than this value, and the term S_DC(0) can be neglected (like the term g(τ) in the expression of Q₀).

As the reference beam illuminates the CCD sensor quite evenly, we should be able to cancel the term Q₀ through a simple high-pass spatial digital filter. The signal is however really low, and even small heterogeneities on the acquired frame could lead to non-negligible components with high spatial frequencies. So, in addition to digital filtering (with a cutoff of ~0.17 pixel⁻¹; see ref [42]. for details), we perform the difference between two images (a) and (b) in order to complete this cancellation and to leave only the S_DC terms, which appear as a uniform white noise:

Δ ​ Q = Q^{(a)} - Q^{(b)} \approx S_{D C}^{(a)} (τ) - S_{D C}^{(b)} (τ)

Note that digital filtering also has an incidence on the signal $Δ Q$ : the averaged squared value of this quantity over all pixels is mathematically identical to the averaged squared modulus of the Fourier components; a spatial filter reduces the effective number of Fourier components, which is statistically equivalent to a reduction of the effective pixels number. We estimate that our digital filtering implies a 58% reduction of this effective pixels number. The CCD camera is a Hamamatsu C8484-05G, with 1.37 million pixels (i.e. an effective pixel number equals to N≈0.8million) for a frame rate ~9 images/s, and with quantum efficiency η ~25% @800nm. The time correlations we are going to consider in this paper are in the millisecond range, so that there is no correlation between two successive frames and the variance of $Δ Q$ can be written:

〈 Δ^{2} Q 〉 \approx 2 〈 S_{D C}^{2} (τ) 〉 = g^{2} (0) \frac{η e i_{0}}{2 f} \int Π (τ - τ') φ (τ') d τ' = g^{2} (0) \frac{η e i_{0}}{2 f} Π * φ (τ)

Note that the factor 2 in Eq. (7) does not correspond to a real signal enhancement: the use of two frames to compute this quantity will also induce the same enhancement for the noise, as shown is next section.

The phantom used for the experiments presented in this paper is a scattering slab with a 4 cm thickness, made with a suspension of calibrated microspheres (Estapor K050, 520 ± 37nm diameter) in glycerol. The microspheres concentration was calculated in order to have a reduced scattering coefficient μ’_s = 10cm⁻¹. The absorption coefficient of glycerol at 800nm is μ_a = 0.03cm⁻¹ [43]. Such optical coefficients are representative of bulk optical properties of breast tissue [44,45], so this phantom is a reasonable model for a compressed breast. We observe the diffuse light transmitted through this phantom. A lens (L₂) images a small region facing the illumination beam.

The temporal response of the setup can be measured using a sheet of tracing paper instead of the breast mimicking phantom. The recorded response is plotted on Fig. 2 : we observe a peak centered on the time delay δτ with the expected full width at half maximum (FWHM) equal to 270 ps. A time origin t₀ can be defined in the following way: in a real time-resolved experiment, the pulse would arrive on the input face of the sample at time t = t₀, and would then reach the detector 4cm farther; if there is no scattering medium but only air between these positions, light will only experience a 4cm/c₀ ≈130ps free-space travel, so that one has to set:

Fig. 2 Temporal response of the setup, which has the expected position (δτ = 2060 ps) and width (270 ps).

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t_{0} = d τ - 130 p s = 1930 p s .

2.2 Influence of the quantum noise

Quantum noise is the main source of noise in this experiment: a Gaussian white noise δi has to be added to the photocurrent, with <δi(t)δi(t’)> = ei₀(t)δ(t-t’), resulting in an additional random term δS = ∫δi(t)dt in the definition of the stored charge Q. This term is of course not correlated with the signal S_DC, and its variance is [41]:

〈 δ^{2} S 〉 = \int 〈 δ i (t) δ i (t') 〉 d t d t' = e i_{0} \int_{0}^{T / 2} ξ (t) d t \approx e i_{0} g (0) T / 2

This therefore leads to an additional constant $S N L = 2 〈 δ^{2} S 〉$ in the variance (7) of $Δ Q$ :

〈 Δ^{2} Q 〉 \approx 2 〈 S_{D C}^{2} (τ) 〉 + 2 〈 δ^{2} S 〉 = 2 e Q_{0} {1 + η \frac{g (0)}{2} Π * φ (τ)}

with

g (0) = 3 / 8

.

This shot noise level (SNL) enables a calibration of the received photon number Π *ϕ(τ). Figure 3 shows raw data recorded with the breast phantom illuminated by a 5mW laser beam. The evolution of the transmittance with the transit time appears as a small variation above the SNL, indicated by the dotted line: the transmittance at its maximum is 15 times lower than the SNL, which corresponds to 1.5 photons per pixel and per acquisition. The relevant signal is in fact quite lower than noise, and only an average over a high number of measurements – here a high pixel number – enables us to extract it from noise.

Fig. 3 Raw data recorded with the breast phantom illuminated by a 5 mW laser beam. The shot noise level (SNL) is indicated by a dotted line. The transmittance at its maximum is 15 times lower than the SNL. A spurious peak is surrounded by a dotted line.

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Due to the low level of the signal, quantum noise gives the main contribution to the overall noise. From the Gaussian nature of this noise, it can be easily shown [41] that the variance of $δ^{2} S$ is $2 < δ^{2} S >^{2}$ , and the standard deviation σ of the noise on the value of $Δ^{2} Q$ averaged over N pixels is simply:

σ = 2 〈 δ^{2} S 〉 \sqrt{\frac{2}{N}} = 2 e Q_{0} \sqrt{\frac{2}{N}}

The standard deviation σ is therefore the SNL times 2, and divided by a factor N called “improvement factor” in the following, which should be 890 in the present experiment. This highlights the challenge of this project: to perform a measurement of the SNL with an accuracy of about 10⁻³. This implies the cancellation of pixel-correlated classical noises that do not decrease as $1 / \sqrt{N}$ with pixel averaging. This cancellation is the role of the image difference and of the high-pass real-time digital filter. But small fluctuations of the variance of the quantum noise could significantly reduce the performance of this setup. For instance, fluctuations of the average power of the reference beam, ie of Q₀, could preclude the improvement factor N. This problem can be circumvented as Q₀ is measured by the camera: $< Δ^{2} Q >$ in Eq. (9) is proportional to Q₀, and the multiplicative factor is the only relevant quantity. For each pixel i we have a value x_i ≈αQ₀/e for each acquired image (where α is the gray level/ charge conversion factor), and a value $Δ^{2} x_{i}^{} = α^{2} Δ^{2} Q_{i}^{} / e^{2}$ for each squared image difference. The signal plotted in Fig. 3 is actually the optimal slope of $Δ^{2} x_{i}^{}$ as a function of $x_{i}$ from the least squares method:

\frac{\sum_{i} x_{i} Δ^{2} x_{i}^{}}{\sum_{i} x_{i}^{2}} \approx 2 α {1 + η \frac{g (0)}{2} Π * φ (τ)}

The quantity defined in Eq. (11) is not very different than the average of $Δ^{2} Q$ , but it is free of any fluctuations of Q₀ . Of course, fluctuations of the conversion factor α can be a problem too. Some peaks appear for instance on the raw data in Fig. 3 (one of them is surrounded by a dotted line): the origin of these peaks is not clear, but they can be easily removed as outliers (cancellation of values farther than 3σ from the local mean). With all these precautions, we reached the expected performance, with an improvement factor of about 910 ± 30.

2.3 Analysis of the experimental results

We can obtain the time-resolved transmittance past the time origin t₀ from the raw data of Fig. 3 after subtraction of the shot noise level. In fact, a careful examination of Fig. 3 shows a small difference between the raw data and the SNL at short times: we haven’t yet mentioned another signal linked to interferences between different paths in the scattering medium, leading to another term [26] proportional to the autocorrelation ∫ϕ(τ + τ’)ϕ(τ’)dτ’ of $φ (τ)$ . This term is small, but not completely negligible and we use an iterative algorithm in order to remove it from the knowledge of $φ (τ)$ . We present the result of this processing on Fig. 4 , together with a theoretical fit (red line) based on diffusion approximation [46] with extrapolated boundary conditions [47], with $μ_{s}' = 10 c m^{- 1}$ and $μ_{a} = 0.03 c m^{- 1}$ (an arbitrary amplitude scaling factor was included in this fit, and the temporal response of Fig. 2 was included in the fitting). The experimental results therefore correspond quite well with the theoretically expected ones. Much more work is needed to characterize the metrological performance of this setup for optical parameters measurement [48], but with a signal to noise ratio of about 45 with only 2 frames, we have here a great improvement compared to our previous results, and compared to those given by other related experiments [49,50]. The results presented in Fig. 3 and 4 correspond to one measurement (ie 2 frames) at each picosecond, what represents several thousand measurements and about half an hour overall acquisition time. But so many points are not needed, and many applications only involve measurements at one transit time, such as transillumination measurements for projection imaging [29]. We are now going to deal with such an application.

Fig. 4 (a)- Recorded transmittance as a function of the transit time, together with a theoretical fit (red line) based on diffusion approximation with extrapolated boundary conditions (the setup temporal response was included in the fitting; we have used an isotropic source at a depth z₀ = 1/μ’_s, and an extrapolated boundary condition at a distance z_s = 2/μ’_s [47]). (b)-same as (a) in a logarithmic scale.

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3. Time-resolved correlations

We mentioned in the introduction the possibility to perform correlation measurements through the measurement of <S_DC,i(τ)S_{DC,i + p}(τ)>, where S_DC,i(τ) and S_{DC,i + p}(τ) correspond to two different acquisitions. We also mentioned the fact that, with a standard CCD camera, the images on two successive frames are completely decorrelated.

The possibility to record correlations of the scattered light with a CCD camera using different exposure times was already reported [51]. In the present paper we will not propose to vary the exposure time but, for each acquired image, to integrate over two modulation half-periods separated by a time interval pT, as depicted on Fig. 5 .

Fig. 5 Acquisition protocol for correlation measurements: the red curve symbolizes the wavelength modulation, and the blue curve represents the modulation function ξ(t), which is zero excepted on two modulation half-periods separated by a time interval pT.

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Two successive images are acquired using the same protocol, and their pixel to pixel difference $Δ Q$ is now:

Δ ​ Q \approx S_{D C, i}^{(a)} (τ) + δ ​ S_{i}^{(a)} + S_{D C, i + p}^{(a)} (τ) + δ ​ S_{i + p}^{(a)} - S_{D C, j}^{(b)} (τ) - δ ​ S_{j}^{(a)} - S_{D C, j + p}^{(b)} (τ) - δ ​ S_{j + p}^{(a)}

where we recall that

δ ​ S

accounts for quantum noise. This leads for the variance of

Δ ​ Q

to:

〈 Δ^{2} ​ Q 〉 = 4 (〈 δ^{2} S 〉 + 〈 S_{D C}^{2} (τ) 〉 + 〈 S_{D C, i} (τ) S_{D C, p} (τ) 〉)

or, in term of the measured quantity defined in Eq. (11):

\frac{\sum_{i} x_{i} Δ^{2} x_{i}^{}}{\sum_{i} x_{i}^{2}} \approx 4 α {1 + η \frac{g (0)}{2} Π * φ (τ) [1 + g_{1} (p T, τ)]}

Experiments were performed at a fixed transit time τ = 1.5ns, and experimental results for each value of the correlation time pT was averaged over 25 measurements, with cancellation of the outliers. The shot noise level was estimated from measurements performed without modulation (ie without the term $Ref(t, τ)$ in Eq. (3). With the term $4 α η g (0) φ (τ) / T$ obtained from measurements at large correlation times (t = 7T = 14ms), we can derive values of the correlation function g ₁ for p running from 1 to 5. The logarithm of these values are plotted on Fig. 6 , exhibiting the linear dependence on the correlation time t expected from the Brownian motion of the scatterers:

\ln g_{1} (t, τ) = - 2 μ'_{s} c τ \frac{t}{t_{0}}

where n = 1.45 is the glycerol refractive index [52], k_B the Boltzmann constant, T_a the temperature and a the microspheres radius.

Fig. 6 Experimental values of ln[g₁(t,τ)] for a transit time τ = 1,5ns and a correlation time t = pT (T = 2ms) with p running from 1 to 5. The red line is a fit, weighted according to the statistical error, by the function α t with α = −0.35 ± −0.015 ms ⁻¹.

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The fit in Fig. 6 leads to $t_{0} = 1.8 \pm 0.1 s$ , which is consistent with the viscosity of glycerol with 12% water at 19°C (η_g≈200 mPa.s [53] leads to t₀≈1.8s). Each point in Fig. 6, which corresponds to correlation measurements performed at only one fixed photon transit time through a thick turbid medium, is taken within a reasonably low acquisition time of a few seconds.

4. Conclusion

We have shown the possibility to perform time-resolved measurements of diffuse light from an interferometric setup using a CCD camera. Using a camera with one million pixels, this setup allows us to obtain the theoretical improvement of the signal to noise ratio by almost three orders of magnitude compared to a single pixel. This represents a great advantage despite the low duty cycle implied by an acquisition rate lower than 10 frames per second: we estimate that we earned about a factor 40 for the acquisition time compared to our previous setups using only one photodiode. This enables measurements through thick tissue with a reasonably low acquisition time. We have furthermore shown that time-resolved correlations at fixed photon transit times can be measured by this method in spite of the low acquisition rate. These results are all the more impressive than the incident illumination is still quite low, of about 5mW, and can be increased by at least one order of magnitude. This last point opens the possibility to increase the modulation frequency f: another advantage of the system presented in this paper is indeed that there is no difficulty to use high frequency modulations (in the MHz range) with an acousto-optic modulator, when our previous photodiode systems were limited by their cutoff frequencies.

Appendix

Let us briefly recall the theoretical basis of the experiment [26,28]. Using a scalar model, the monochromatic (CW) optical fields can be written $s_{0}_{ω} (t)$ and $s_{ω} (t)$ for the reference beam and the scattered light, respectively. These quantities are proportional to the Fourier components ${\tilde{s}}_{0}$ and $\tilde{s}$ of a time-resolved experiment [26], with a proportionality factor K(ω) that can be determined from energy considerations in the following way: let us write the spectral power of the pulsed source ${| {\tilde{s}}_{0} (ω) |}^{2} = F f (ω)$ , where F is the pulse fluence and $f (ω)$ the normalized spectral profile ( $\int f (ω) d ω = 2 π$ ); we will set an equivalence between pulsed and CW experiments by assuming that the fluence ${| s_{0}_{ω} |}^{2} T / 2$ in the CW case during the acquisition time T/2 is equal to the pulse fluence F, leading to:

s_{0}_{ω} (t) = K (ω) {\tilde{s}}_{0} [ω (t)]

We consider that the spectral profile is large compared to the wavelength modulation, so that $f (ω) \approx f (ω_{0})$ . The Fourier component $\tilde{s}$ slowly depends on time t because of the movements of the scatterers: this leads to time-resolved decorrelation. One can introduce the correlation function:

〈 \tilde{s} [ω_{1}, 0] {\tilde{s}}^{*} [ω_{2}, t] 〉 \equiv \tilde{ϕ} (ω, Ω, t)

where $ω \equiv (ω_{1} + ω_{2}) / 2$ refers to light pulsation, and where $Ω \equiv ω_{1} - ω_{2}$ is conjugate to the time of flight $τ \equiv (τ_{1} + τ_{2}) / 2$ . The inverse Fourier transform $ϕ (ω, τ, t)$ of $\tilde{ϕ}$ can be interpreted as a time-resolved averaged spectral intensity, the decorrelation being accounted through an effective absorption coefficient $μ (t)$ . This function essentially follows the source spectrum [28], and $ϕ (ω, τ, t) \equiv f (ω) ϕ (τ) g_{1} (τ, t)$ , where $ϕ (τ)$ is the time-resolved averaged intensity and where $g_{1} (τ, t)$ is the time-resolved normalized correlation function of the field. With a phase definition so that $s_{0}_{ω} \equiv s_{0}$ is a constant, and for a quasi point-like detector with an area A (see ref [41]. for a complete computation taking into account the spatial extension of the detector), the interferometric signal i_int(t) is:

i_{int} (t) = S A K (ω) {s_{0} {\tilde{s}}^{*} [ω (t), t] + s_{0}^{*} \tilde{s} [ω (t), t]}

where $S = η e / (h ν)$ is the sensitivity of the detector. At this stage, we did not take the intensity modulation ξ(t) into account: as explained just before Eq. (5), one only has for that purpose to multiply the photocurrent that should be recorded without modulation by this function ξ(t), leading to the stored charge $Q = \int ξ (t) i (t) d t$ and subsequent terms like S_DC. From $S_{D C} = \int Ref (t, τ) i_{int} (t) d t$ and assuming [28] $〈 \tilde{s} \tilde{s} 〉 = 〈 {\tilde{s}}^{*} {\tilde{s}}^{*} 〉 = 0$ one has quite directly (with $φ (τ) \equiv A ϕ (τ)$ ):

\begin{array}{l} 〈 S_{D C, i} S_{D C, i + p} 〉 = \frac{S i_{0}}{T} \int d t_{1} d t_{2} d τ' φ (τ') g_{1} (t_{2} - t_{1} + p T / 2, τ') \times \\ [Ref (t_{2}, τ - τ') + Ref (t_{2}, τ + τ')] [Ref (t_{2}, τ - τ') + Ref (t_{2}, τ + τ')] \end{array}

When one assumes [28] that the wavelength modulation is fast enough to freeze the speckle fluctuations linked to the movement of the scatterers, i.e. $g_{1} (t_{2} - t_{1} + p T / 2, τ') \approx g_{1} (p T / 2, τ')$ and with Eq. (3), one has:

〈 S_{D C, i} S_{D C, i + p} 〉 = \frac{S i_{0}}{4 f} \int φ (τ') g_{1} (p T / 2, τ') [g (τ - τ') + g (τ + τ')] ​^{2} d τ'

As $φ (τ) = 0$ when $τ < 0$ , the time gate $g (τ + τ')$ can be neglected, leading to Eq. (2).

Acknowledgements

The authors acknowledge Thierry Billeton for technical support. This work has been supported by the Agence Nationale de la Recherche through the project ANR-08-PCVI-0038-01.

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Time-resolved diffusing wave spectroscopy with a CCD camera

Abstract

1. Introduction

2. Time-resolved measurements

2.1 The experimental setup

2.2 Influence of the quantum noise

2.3 Analysis of the experimental results

3. Time-resolved correlations

4. Conclusion

Appendix

Acknowledgements

References and links

Cited By

Figures (6)

Equations (25)

Optics Express