## Abstract

We propose a modified method of acquisition and analysis of Spectral Optical Coherence Tomography (SOCT) data to provide information about flow velocities. The idea behind this method is to acquire a set of SOCT spectral fringes dependent on time followed by a numerical analysis using two independent Fourier transformations performed in time and optical frequency domains. Therefore, we propose calling this method as joint Spectral and Time domain Optical Coherence Tomography (joint STdOCT). The flow velocities obtained by joint STdOCT are compared with the ones obtained by known, phase-resolved SOCT. We observe that STdOCT estimation is more robust for measurements with low signal to noise ratio (SNR) as well as in conditions of close-to-limit velocity measurements. We also demonstrate that velocity measurement performed with STdOCT method is more sensitive than the one obtained by the phase-resolved SOCT. The method is applied to biomedical imaging, in particular to *in vivo* measurements of retinal blood circulation. The applicability of STdOCT different measurement modes for *in vivo* examinations, including 1, 5 and 40 µs of CCD exposure time, is discussed.

© 2008 Optical Society of America

## 1. Introduction

Optical Coherence Tomography (OCT) provides cross-sectional images of internal structure of objects with micrometer resolution, and can be either performed in time [1] or frequency domain [2]. The latter is performed by analysis of interferometric modulations of light intensity versus optical wavelengths (spectral fringe signal). In Spectral OCT known also as Spectral domain OCT (SOCT/SdOCT) spectral intereferometric fringes are registered by a spectrometer. This modality is particularly useful for ophthalmic examinations since it offers high speed of more than 20 000 A-scans per second and detection sensitivity of more than 95dB [3–6]. The spectral interferometric fringe signals are collected for each lateral position of the scanning beam and numerically processed to obtain two-dimensional cross-sectional images representing the amount of back-reflected light versus depth and lateral positions of the elements of internal structure of an object.

In addition to morphological imaging, SOCT can provide visualization of physiological parameters [7–11]. At present the retinal blood flow attracts attention as a potentially important physical parameter in the functional OCT studies. The measurements of bidirectional flow by Spectral OCT have been demonstrated by many groups [8, 12, 13]. In all of these publications, authors used the phase-resolved technique based on linear relationship between the phase difference of consecutive spectral fringe signals and the velocity of the moving sample. This technique is analogous to phase-resolved approach applied in the OCT methods with the time domain detection [14].

Two main factors deteriorate and even preclude velocity recovery in the phase-resolved OCT techniques: its vulnerability to low signal-to-noise ratio (SNR) [15] and motion artifacts causing severe phase instabilities [12, 16]. Both problems frequently occur in OCT measurements of biomedical objects *in vivo*. Recently, a novel spectral method has been proposed in order to minimize the influence of phase instabilities so called resonant Doppler imaging [17], which determines flow velocities on the intensity basis without the need of extracting the signal phase. Moreover, this method overcomes a phenomenon of blurring of interference fringes caused by any sample movement during CCD camera integration time. Another phase independent method was proposed by Wang, *et al.*, [18]. Since this technique separates the moving and static components within a sample, only blood perfusion imaging is possible without any flow velocity estimations. This optical angiography relies on introducing a constant Doppler frequency to modulate the spatial OCT spectral interferograms what was initially introduced to Spectral OCT by Yasuno, *et al.*, in 2006 [19].

In this paper we present an alternative method of measuring and processing OCT signals providing information on the spatial distribution of flow velocities. We propose joint time and frequency domain detection of interferometric OCT signals. The information about the velocity is obtained directly – similarly to first velocity estimation techniques in TdOCT [20, 21] – from the time dependent beating frequency due to the Doppler shift between the reference and the sample light beams. Since the velocity estimation is not based on explicit phase information extracted from interferometric fringes, the proposed variant of Doppler SOCT is significantly less sensitive to undesired phase instabilities present in low SNR conditions. Since it does not require any phase wrapping and averaging procedure, it is accurate for flows close to the upper limit of measurable velocities. This approach does not require any modifications in hardware of a standard SOCT instrument. High sensitivity of this method facilitates flow velocity estimation within the time frame required by the regular OCT imaging. In many biomedical applications, especially in ophthalmology, there are severe limitations in optical power, which can be delivered to the sample. In such cases a preservation of high sensitivity requires fixed value of exposure time of the CCD camera collecting the spectral fringe data. In this case the multi-shot measurements required in STdOCT can be balanced by reduction of CCD exposure time. And the same it is possible to keep sensitivity and the optical power delivered to the object at the same levels like in regular SOCT imaging. In such case the spectral fringe signals can be first processed and then superimposed giving the structural reconstruction, while time dependent Fourier transformation will yield information about flow velocities. Comparing to phase-resolved techniques our method can operate in conditions of much lower SNR still preserving high accuracy in the whole velocity range, what is crucial in any quantitative measurements of biomedical samples *in vivo* in OCT functional studies.

## 2. Theory

In Spectral OCT Fourier transformation (FT) of the spectral fringe signal measured by a single exposure of a CCD camera creates one line of a structural cross-sectional image (A-scan). In order to asses a velocity of a moving interface most of known methods require at least two spectral fringes acquired in the same lateral position of a sampling light beam [8] or almost the same lateral position [15]. The acquired set of spectral interferometric fringes can be described as a function of wavenumber *k* and time *t* according to the following equation:

where *I*(*k*,*t*) is the spectral fringe signal, *I*
_{0}(*k*) is spectral density of the light source, *R _{l}* and

*R*denote the reflectivity of the sample and reference mirror, respectively;

_{r}*z*(

_{l}*t*) denotes the optical path difference between the reference mirror and the

*l*-th interface in the sample, which is time dependent due to the movement of the reference mirror and/or due to the displacement of the

*l*-th interface in the sample. The displacement of the interfaces within the sample is usually caused either by a movement of the entire sample itself or by a motion of the specific interface

*z*within the sample. If we assume that both, reference mirror velocity and velocities in the sample are constant during the acquisition of the spectral fringes, Eq. (1), then the time-dependent position of the

_{l}*l*-th interface

*z*(

_{l}*t*) can be expressed as:

In this relation, *z _{l}* is the depth position of the

*l*-th interface at the beginning of data acquisition and

*v*is the difference between the velocity of the reference mirror and an axial component of velocity (parallel to the direction of the probing beam propagation) of the

_{l}*l*-th interface. If the

*l*-th interface moves with velocity

*V*at an angle

_{l}*α*to the probing beam and the velocity of the reference mirror is equal to

*v*this can be expressed as:

_{r}Here velocities directed towards a beam-splitter are regarded positive.

One can rewrite Eq. (1) making use of Eqs. (2) and (3).

Although the above equations represent the same interference pattern, they emphasize its different properties. Phase of the oscillatory component visible in Eq. (4) is a function of wavenumber and its modulation frequency depends on static position *z _{l}* of

*l*-th interface and small additional change of

*δz*, that occurs if the

*l*-th interface is moving. Equation (5) highlights the time-dependence of the interferometric fringes and shows that signal is modulated in time with frequency

*ω*. This beat frequency is caused by a Doppler effect, that arises for each

_{l}*l*-th interface along the time axis. This frequency depends on the velocity

*v*and is different for each wavenumber

_{l}*k*:

The phase-resolved methods of velocity estimation enables extracting and using the phase of the signal Eq. (4) while the joint Spectral and Time domain OCT uses the Fourier transformation to analyze the time-dependent frequency of the signal Eq. (5).

#### 2.1 Velocity measurement using phase-resolved SOCT

The idea behind the phase-resolved OCT is to determine the phase difference between points at the same depth in consecutive A-scans. Knowing the change of position on *δz* of *l* -th interface, that arises during the time Δ*t* between two consecutive measurements, the velocity *v _{l}* of the

*l*-th interface can be calculated. Since

*δz*is much smaller than

*z*, the difference between two consecutive measurements appears as a phase change ΔΦ of interferometric fringes:

_{l}Here ΔΦ is the phase difference between successively recorded depth profiles at the same location of the probing beam. The time between successive profiles acquisition Δ*t* is approximately equal to the exposure time of the detector, therefore 1/Δ*t* is the frame rate of an array detector (or equivalently ‘A-scan’ rate). It is important to ensure that ΔΦ is less than 2*π*. Since the phase can be unambiguously determined in the range of 2*π*, and the phase difference is within the range of 4π a procedure of phase wrapping has to be performed to transform the phase differences to the range (-*π*,*π*) [12, 13]. In the procedure adapted to recovery of bidirectional flows the following algorithm is used: if |ΔΦ|<*π*, the phase shift it is left as it is, and when *π*<|ΔΦ|<2*π*, the phase shift is replaced by ΔΦ-sign(ΔΦ)2*π*. The phase differences ΔΦ from the range (0,*π*) are considered positive, while those form (-*π*,0) negative. Usually more than two measurements are used to estimate the phase differences. Wrapped phase differences ΔΦ are averaged out to increase the sensitivity and accuracy of the velocity estimation [8, 12, 15].

The upper and lower velocity measurement ranges are limited by maximum and minimum detectable phase difference, respectively. Hence the maximal flow velocity possible to detect without ambiguity 2*π*(ΔΦ_{±max}=±*π*) is:

and the minimal one is:

The value of minimal velocity defines velocity sensitivity. The minimum phase difference ΔΦ_{min} is equal to the standard deviation of the estimator of the phase differences *σ*
_{ΔΦ} and is limited by signal-to-noise ratio (SNR) [15, 22, 23]. Assuming signal amplitude much higher than noise amplitude SNR≫1 the minimum phase difference can be determined as ΔΦ_{min}=(*SNR*)^{-1/2} [15]. Vakoc, *et al.*, suggest that retinal blood flow velocities <0.2 mm/s are out of the scope of the standard phase-resolved OCT method in the case of *in vivo* imaging [23].

#### 2.2 Velocity measurement using joint Spectral and Time domain OCT

We propose a direct detection of Doppler frequency from a set of *M* spectra collected in time increments Δ*t* at the same transverse location of the probing beam, Eq. (5). As the data are registered and analyzed in wavenumber and time space simultaneously, we called this method joint Spectral and Time domain OCT (STdOCT).

In order to explain the idea of the method it is convenient to discuss a simple experiment. We assume that an objective mirror in standard Michelson interferometer is driven with a constant speed. *M* measurements of spectral interferometric fringes are performed at the same lateral location (Eq. 5). Collected spectral fringe signals undergo standard SOCT preprocessing consisting of background removal and rescaling to wavenumber domain [24]. Then spectral fringes are plotted as rows, so the abscissa corresponds to wavenumber and the ordinate to time (*k* – *t* plane, Fig. 1(a)). A Doppler frequency arising from a movement of *l* -th interface is visible as a frequency of the signal along the *t* -axis while the modulation frequency along *k* -axis provides information on location of *l* -th interface. The two-dimensional set of spectral fringes is analyzed by Fourier transformations, that can be applied in two separate ways. First FT can be performed “horizontally” thus it converts the STdOCT data from wavenumber domain to the depth (*z* – *t* plane, Fig. 1(b)). The second FT acts “vertically” and converts data from time domain to Doppler frequency, that corresponds to velocity (*k* – *ω* plane, Fig. 1(c)).

Using both Fourier transforms, one after another, 2D spectral fringes are converted to velocity distribution in depth h (*z* – *ω* plane, Fig. 1(d)). Note that panels (b), (c), (d) display only amplitudes of complex valued functions. The top-right panel (Fig. 1(b)) corresponds to data processed in standard SOCT, where the structure of the object is reconstructed. Standard SOCT uses the modules of Fourier transforms of data to create structural A-scans and the phases to calculate the velocities with phase-resolved method. Here, *M* registered spectra result in *M* structural A-scans and only structural information is presented with no velocity information. Maximal optical path difference between the mirror in the reference arm and the reflecting interface in the sample arm define imaging range in depth *z*
_{±max}. It is connected with the sampling interval in wavenumber domain Δ*k* of recorded spectra:

The variable *z*, that encodes the position of the sample is chosen to be positive if the sampling arm is longer then the reference arm, and negative in the opposite case. The image of the objective mirror is visible as a single interface which apparently is fixed in time. This image is doubled due to the fact that registered interferogram is a real-valued function [25]. In STdOCT as well as in standard SOCT complex conjugation of the image is considered unwanted, thus not displayed in resulting cross-sectional images. Therefore, in the practical applications only positive depths are displayed.

The bottom left panel (Fig. 1(c)) corresponds to the one-dimensional distribution of velocity of moving object with no information about structure of the sample. To increase the sampling density, zero-padding in time domain is applied. The velocity is recovered from the Doppler frequency *ω _{l}*, according to Eq. (6). For each known

*k*the velocity can be calculated separately. Therefore, this representation of data can be also used to find exact relationship between wavenumbers and pixels in an array detector, and the same to calibrate the spectrometer very accurately. In this particular case the velocity of moving mirror is measured to be 14.25 kHz (0.95 mm/s) for k=7.5·10

^{6}m

^{-1}, Fig. 1(f). The question of velocity distribution within the object is trivial in the case of a mirror. If the object is more complex, magnitude and direction of the movement will be known but there would be no information about the position of the moving interface. Similar to the phase-resolved OCT, the maximal value of bidirectional flow velocity

*v*

_{±max}is given by the time interval Δ

*t*between consecutive measurements of the spectral fringes:

and for Δ*t*=40us it becomes *v*
_{±max}=±5.2mm/s.

The bottom right panel, Fig. 1(d) shows the result of two-dimensional Fourier transformation of the set of *M* spectral fringes. Coordinates of displayed signals link positions of all measured interfaces with corresponding velocities. Each interface *z _{l}* is represented by two symmetric points appearing with respect to the zero-path-delay and zerovelocity. The interpretation of resulting points, shown in Fig. 1(d), is following: the mirror surface localized at

*z*=140um (Fig. 1(e)) moves with the velocity of 14.2 kHz (Fig. 1(f)). The sign of velocity value indicates forward or backward direction. The point localized at (

*z*,

*ω*)=(-0.14 mm, -14.2 kHz) is its complex conjugate.

The points are in fact fuzzy structures. This is due to the fact that their width along the z axis depends on the axial resolution and the spread along the *ω* – axis is caused by the dependence of the Doppler frequency on wavenumber, *ω*=2*vk*, Eq. (6). A velocity value for each *z* position is calculated from Doppler frequency indicated by the point with maximal amplitude.

#### 2.3 Conditions of reliable velocity measurement for phase-resolved SOCT and joint Spectral and Time domain OCT – SNR analysis

In order to compare both methods we determine conditions under which each of them fails in velocity estimation. As phase-resolved SOCT operates on phases and STdOCT on signal amplitudes, we have to analyze how decreasing SNR affects distributions of measured phases and measured amplitudes.

Since every registered signal can be considered as a deterministic signal and a random noise, we assume that our interferometric fringe signal *I*(*k*,*t*) [Eq. (1)] is a sum of a harmonic component *S* and a noise component *X*.

The harmonic component can be expressed by a real-valued function with a given amplitude *s _{kt}*, frequency

*ω*and the initial phase set to be zero (random variable and its specific value are denoted by capital and lowercase letter, respectively):

_{kt}The noise component in turn, can be expressed as a sum of harmonic components with random phases Φ* _{n}* (with uniform distribution) and random amplitudes

*α*:

_{n}and its statistical properties can be described by a Gaussian function with a mean value *x̄*=0 and a variance *σ*
^{2}
* _{kt}*:

The distribution of amplitudes *α _{n}* is identical for all frequencies and has a mean value equal to zero and a variance equal to

*σ*

^{2}

*. The relation between*

_{α}*σ*and

_{α}*σ*is following:

_{kt}As SOCT measurements are performed in *k* – *t* space (Fig. 1(a)), the phase-resolved method operates in *z* – *t* space (Fig. 1(b)) and STdOCT in *z* – *ω* space (Fig. 1(d)), the amplitudes *s _{kt}*,

*s*,

_{zt}*s*

_{zω}are coupled via Fourier transformations. If the Fourier transformation is defined to conserve power of the signal (

*E*[

*I*

^{2}]=

*E*[

*Z*

^{2}],

*Z*=FT(

*I*)), the amplitudes are amplified with respect to a number of points in Fourier transforms

*N*,

*M*:

while the energy of the noise is preserved and equally distributed among the real and imaginary part of the transform:

In order to describe the relation between the signal amplitude and the distribution of the noise with the same frequency we introduce a parameter κ:

and we define SNR as a quantity that describes parameters of standard structural tomograms (*z* – *t* space):

and any further comparisons between joint STdOCT and phase-resolved SOCT are based on above definition of SNR.

Table 1 presents relations between the amplitudes and the variances under condition that Fourier transformation is scaled to preserve signal energy. These results are similar to analysis performed by Leitgeb, *et al.* [26]. Comparing the values of signal-to-noise parameter after one- and two-dimensional Fourier transformation one can see that *κ _{ωz}* (STdOCT) is

*M*

^{1/2}times higher comparing to

*κ*(phase-resolved SOCT).

_{zt}The distributions of phase and amplitude of signal G is based on the formalism presented by Goodman to describe the phase and amplitude distribution of a sum of a constant known phasor *S* and a random phasor *X* [27].

After Goodman, the probability density function for the phase is given by the following expression:

where function $\Omega \left(b\right)=\frac{1}{\sqrt{2\pi}}\underset{-\infty}{\overset{b}{\int}}{e}^{\frac{-{y}^{2}}{2}}dy$ .

With increasing *κ _{zt}* the density function becomes more narrow, converging towards a Dirac delta function centered at Φ=0, whereas when the length of the known phasor

*s*decreases to zero (

*κ*→0), this distribution converges to uniform distribution, Fig. 2. As the phase-resolved SOCT requires phase subtraction, the width of the final distribution broadens two times. The broader distribution is, the more random wrapped phase differences are detected, and in turn the averaged value of phase differences is closer to zero.

_{zt}Quoting from Goodman, the probability density function of the amplitude *A* of the sum of a constant phasor and a random phasor is given by a Rician density function:

Where *I*
_{0}(·) is a modified Bessel function of the first kind, zero order. As the length of the known phasor *s* increases, the shape of density function *p _{A}*(

*a*) changes from that of a Rayleigh density to approximately a Gaussian density with mean equal to

*s*.

Joint STdOCT uses the time dependent modulation of the signal, therefore it is successful, when the amplitude of signal is higher than the maximal amplitude of noise component. This occurs when the distributions of signal amplitude (*κ _{zω}*≠0, Eq. (21)) and noise (

*κ*=0) are separated. The minimal value of κ

_{zω}_{zω}, which almost always meets this requirement is

*κ*=7, Fig. 3.

_{zω}When *κ _{zω}* converges to zero, the probability of detecting the correct position of signal amplitude decrease. Every detection of noise causes indication of random velocity, therefore, the distribution of recovered velocity broaden and its mean value converges to the center of the available velocity range (usually to zero).

To determine critical values of SNR below which STdOCT and the phase-resolved method give false readings, we performed computer simulations based on provided theoretical model. In order to reconstruct the process of velocity estimation, multiple signals (*M*=30) were generated with respect to the shape of spectrum and the probability density function of amplitude (Eq. (22)) and phase (Eq. (21)). The magnitude of change in harmonic component between consecutive signals was set to correspond to 0.35 *ν*
_{max} and 0.75 *ν*
_{max}. Both methods operates on exactly the same amount of generated signals. The velocity was recovered for different SNR and the results are shown in Fig. 4.

Both methods fail for certain SNR, however STdOCT is more robust under low SNR. The failure appears as an underestimate of retrieved velocity values. In joint STdOCT the critical SNR above which recovered velocity is reliable does not depend on the magnitude of the set velocity, whereas phase-resolved SOCT fails earlier for higher velocity.

To explain this effect we analyzed the velocity recovery process in the phase-resolved method within the entire theoretical range. Four different velocities: 0.05, 0.35, 0.75 and 0.95 of v_{max} were chosen and the velocity estimation for each of them was performed. The dependence of velocity reading on proximity to the theoretical limits of velocity are shown in Fig. 5(a). One can see that there is no significant difference in critical values of SNR for velocities <0.5 v_{max}, whereas for higher velocities the critical SNR shifts substantially towards higher values and for 0.95 v_{max} it is 25dB higher than for 0.05 v_{max}. In other words, for a given value of SNR the phase-resolved method can give correct velocity readings for lower velocities and underestimated values for higher velocities, for example for 20dB 0.95 v_{max} is estimated to be 0.75 v_{max}, whereas lower velocities are correctly recovered. Figure 5(b) shows simulated values of velocity retrieved by the phase-resolved method versus the velocities set in the simulation as real ones. In both cases normalization was applied to display the velocities in respect to the maximum measurable velocity V_{max}. It is visible that for higher SNRs it is possible to measure correctly higher values of velocity. SNR defines the shape of phase difference distribution. When the width of distribution becomes broad comparing to phase range, the wrapping procedure causes that some of the phase differences are found as positive and the other as negative (Fig. 5(b) right panel). Since the final velocity estimation is based on calculation using several phase differences, its averaged value decreases to zero for the phase difference close to ±*π*. Hence, the decreased value of averaged phase difference underestimates the value of velocity in the phase resolved OCT.

Basing on performed analysis we can determine the conditions under which joint STdOCT and phase-resolved SOCT fail in velocity measurement. For the phase-resolved method the critical SNR, that guarantees reliable velocity detection in whole velocity range is estimated to be >30 dB (Fig. 5), whereas the corresponding value for STdOCT is >6 dB (Fig. 4). Additionally, since signal amplitude ude *A _{zω}* used in joint STdOCT to retrieve velocity value depends on number of spectra registered in time (

*M*), the critical SNR can be improved

*M*

^{1/2}times with increasing number of measurements (Table 1.). In phase-resolved SOCT increasing number of spectra does not improve measurement sensitivity, however it facilitates detection of mean value of phase difference distribution.

Performed simulations do not take into consideration the washout of interference fringes [17]. This phenomenon deteriorate SNR, hence in this way it affects velocity recovery. Because both methods suffer from the blurring of interference fringes in the same degree, the conducted comparison is still valid.

## 3. Experiment

We use laboratory high resolution Spectral OCT system comprising a broadband light source (*Broadlighter*, Superlum, Δ*λ*=90nm, central wavelength 840 nm), a fiber Michelson interferometer with fixed reference mirror and custom designed spectrometer with a volume phase holographic grating DG (1200 grooves/mm) and an achromatic lens focusing spectrum on 12-bit CCD line-scan camera (*Aviiva M2*, Atmel), Fig. 6. The experiments were performed for three different objects: moving mirror, capillary flow and blood flow in human retina. In measurements of the velocity of moving mirror, a silver mirror was attached to a piezo-actuator (Physik Instrumente) and it was driven by a triangular voltage signal. The exact velocity was calculated at the moment of a linear slope of the driving signal from trajectory registered by the position sensor mounted inside the actuator. Measurements were performed with A-scan rate of 40.4 µs.

To investigate flows in scattering media, we used a water solution of Intralipid flowing through capillaries. Two 700 µm thick glass capillaries with flow in opposite directions were mounted at the angle of 88 deg to the direction of the probing beam (*z* -axis) and stable, laminar flow was ensured by a medical drip system. The sets of 40 spectra were collected at the same transversal position of the light beam. The acquisition time was set to 52 µs including 10 µs dead time needed for stabilization of the position of galvo scanner driven by the stepwise signal. The optical power of the light illuminating the sample was 3.3 mW.

For all retinal blood flow examination the optical power of light illuminating the cornea was set to 750 µW. In the case of regular exposure time (40 µs) the velocity recovery is based on 20 spectra, each recorded with 43 µs of repetition time. In measurements with short exposure time (5 µs and 1 µs) 40 spectra were collected for each lateral position of the scanning beam. Despite such a short exposure times, the repetition time was >41 µs due to the dead time of 40 µs between consecutive measurements.

The velocity estimations in phase-resolved SOCT and joint STdOCT are always based on exactly the same registered data in all comparative experiments. This guarantees that the differences in velocity recovery are solely caused by the methods themselves, not by experimental environment or different amount of processed data.

## 4. Results and discussion

#### 4.1 Moving mirror

In order to validate provided theoretical analyses we performed an experiment with a moving mirror as an object. To investigate the relation between the velocity estimation and SNR, the light intensity in objective arm had been reduced by a neutral density filter from 20 dB to -6 dB (from κ=10 to κ=0.5) for two velocities 1.9 mm/s and 3.9 mm/s corresponding to 0.35 and 0.95 of v_{max}. The sets of 30 spectra were collected and then processed to obtain phase-resolved and STdOCT velocity estimations. Figures 7(a), 7(b) presents achieved velocity values, which are displayed together with theoretical results demonstrated in Fig. 4.

In the next step we verified the capability of measuring velocities close to the upper limit. The objective mirror was driven with different velocities within the whole theoretical range. The intensity of the light in objective arm was constant and resulted in SNR=17.5 dB in structural tomogram. A single velocity value was calculated from 18 spectra. Retrieved values of velocities were marked in Fig. 7(c) together with theoretical prediction.

The experimental results are in good agreement with the theoretical model. Although theoretical ranges for both methods are identical, the phase-resolved SOCT fails earlier than STdOCT and its correctness depends on the magnitude of measured velocity. STdOCT is able to detect a velocity for SNR ~30 dB lower than the phase-resolved method. This leads to conclusion that useful velocity range in phase-resolved SOCT is significantly narrower than in STdOCT.

#### 4.2 Capillary flow

Two experiments were designed, one to verify the method of STdOCT in case of bidirectional flow in scattering media and the other to compare with phase-resolved SOCT. Figure 8 presents STdOCT images achieved as individual steps during velocity recovery (section 2.2).

The structural tomogram of the capillaries and the velocity map that indicates bidirectional flow are shown in Figs. 8(a), 8(b). Single lines in structural and velocity image are obtained from a set of 40 A-scans. As a first step in velocity recovery, the procedure of zero-padding to 128 points in time space was applied. The signal underwent 2D Fourier transformation and formed the Doppler shift distribution in depth as shown in Fig. 8(c). The positions of maximal intensities for each depth *z* were detected and points that most likely correspond to noise (*κ _{zω}*≤2) were removed by thresholding procedure, Fig. 8(d). Images (c) and (d) correspond to a single line in 2D velocity map (Fig. 8(b)), on which the values of velocity are encoded using false colors. Figure 8(e) presents a single 1D velocity distribution along the transversal direction indicated by green horizontal line on the velocity map. All presented velocity distributions have parabolic shapes, what implies that measured flow is laminar.

To compare the STdOCT and phase-resolved flow velocity estimation in scattering media, the flow measurements for different concentrations of Intralipid solution and different flow rates were performed. Concentration of scattering medium affects signal intensity and its change yields different characteristics of SNR decrease in depth. The acquisition parameters of OCT data were unchanged. To take full advantage from recording multiple spectra, we averaged all single A-scans and they were displayed as a single line in structural tomograms. The results of both methods of flow estimation are presented in Fig. 9. The experiment was performed under three different flow conditions. In the first case, Fig. 9(a), the concentration of the Intralipid solution was chosen in such a way that the SNR changes significantly between front and back side of the capillary. The flow velocity was set to approximately 0.75 of maximal velocity. Both methods return a parabolic distribution of flow velocity, however phase-resolved method exhibits a slight asymmetry, which increases with depth. Then the Intralipid concentration was changed to maximize SNR at the back side of capillary and the flow velocity was increased to exceed v_{max}. (Fig. 9(b)) We can observe that both methods give similar readings to approximately half of the velocity range. For higher velocities phase-resolved method dramatically underestimates the velocity values, and for v_{max} returns zero. In STdOCT velocities beyond the range are wrapped and found as negative values. The distortions of velocity distributions in phase-resolved method appear when SNR decreases or when velocities are too high (however still in the theoretically achievable range).

The result of velocity estimation in conditions when both effects occur is shown in Fig. 9(c). It is evident that for the illuminated side of the capillary, the SNR is sufficient to return accurate velocity values for both methods. With decreasing SNR and increasing velocity phase-resolved method starts failing, while STdOCT remains unaffected. These results are in a good agreement with theoretical predictions and with experiments performed with moving mirror as an object.

#### 4.3 Retinal blood flow, in vivo

As a final test of joint STdOCT and phase-resolved SOCT capabilities in velocity estimations, the measurement of blood flow in human retina *in vivo* was performed. Figure 10(a) demonstrates cross-sectional image of human retina scanned through the region of optic disc. Figures 10(b) and 10(c) show two-dimensional maps of the flow velocity distribution obtained with SOCT and STdOCT, respectively.

The velocity distributions inside the vessel indicated by green lines in Fig. 10 are presented in Fig. 11. Although blood flow in large vessels is evident in both methods, the quantitative velocity estimations differ significantly. The magnitude of the blood velocity in the center of the vessel is 1mm/s for the phase-resolved method and 4mm/s for STdOCT. The recovered shapes of velocity distribution also differ greatly. In the phase-resolved method we observe distortion in the center of the vessel, what may cause a misinterpretation of nature of flow.

The underestimate in phase-resolved SOCT arises from getting out of the useful velocity range. Beyond the useful range, velocity estimators became progressively underestimated with decreasing signal and finally decayed to zero. The useful range in STdOCT is wider, but of course also limited. Another difference is that in STdOCT decreasing signal does not influence the accuracy of velocity estimation but only the probability of its detection. It gives confidence that if signal is distinguished from noise and measured velocity does not exceed the upper limit, measured velocity is correct. This feature is especially valuable in biomedical imaging, where diagnoses are based on measured functional parameters.

#### 4.4 Retinal blood flow imaging with ultra-short CCD exposure time

Joint STdOCT comparing to phase-resolved SOCT requires more data to be collected to estimate velocity value. This experiment is performed to give the proof of concept of time effective STdOCT. To compensate longer scanning protocols shorter CCD exposure time is proposed. We assume that higher sensitivity offered by this technique enables measuring the velocity without any time extension comparing to regular imaging. The capability of STdOCT to estimate the flow velocity in human retina with extremely short CCD exposure time of 5 and 1 µs is presented in Fig. 12.

The maps in the top line (5 µs) present spatial distribution of the blood flow velocities only in the larger vessels. The velocity underestimation in the phase-resolved method results in vanishing of the middle-size vessels in the flow image. STdOCT results obtained for 1 µs exposure time enable reconstructing one large and three smaller vessels. The map based on the phase-resolved method shows only faded velocity image of the large vessel.

Unfortunately, the present state-of-the-art of CCD technique does not allow taking full advantage of extremely short exposure time because the relatively long dead time of the CCD camera limits the duty ratio to 0.025. In general in OCT studies, there is a pressing need to collect more data in examination time acceptable for patients. However, in many cases optical power delivered to the object has to be limited either by the safety regulations or by the power limitations of the light sources and/or optical components. Recent developments in CCD and CMOS technologies probably will be soon completed with ultra-fast line scan cameras. This experiment shows that, in contrast to the phase-resolved method STdOCT is able to benefit from these improvements. Phase-resolved SOCT requires higher SNR than it is possible to achieve with ultra-short exposure time in CCD technique, whereas STdOCT can still operate reliably in these conditions.

## 5. Conclusions

We demonstrate the potential of Joint Spectral and Time domain OCT to estimate flow velocities accurately. In this approach, SOCT measurements are repeated in one position of scanning beam to register interferogram, that simultaneously depends on optical frequencies and time. Intensity modulation along axis of optical frequencies encodes information about structure and the modulation along time axis contains information on velocity.

We analyze known phase-resolved SOCT method under low SNR conditions. It appears that retrieved velocities are dramatically underestimated and have tendency to decay to zero. This strong dependence on SNR is especially adverse in the case of measurements of highly scattering media, where the contribution of noise increases with depth. This may lead to considerable corruption of the velocity profile for points located deeper. Another cause of distortion may occur if the velocity is close to the maximal limit of velocity measurable by the phase-resolved method. Since velocity estimation in STdOCT is based on Doppler shifts, it is significantly less vulnerable to both effects and is more reliable for any qualitative and quantitative analysis as it is demonstrated using the same sets of OCT data. The possibility to unequivocal assessment of blood circulation in human retina renders STdOCT especially valuable.

Additionally, STdOCT is more sensitive and it is able to detect a correct value of velocity for SNR lower at ~30 dB than the phase-resolved method. This unique feature can be used to compensate longer scanning procedure by shortening CCD exposure time. The proof of concept of time effective STdOCT is ascertained by presented measurements of blood flow in human retina *in vivo* for 5 µs and 1 µs exposure time.

## Acknowledgments

This work was supported by EURYI grant/award funded by the European Heads of Research Councils (EuroHORCs) and the European Science Foundation (ESF). Maciej Wojtkowski acknowledges additional support of Foundation for Polish Science (Homing project and EURYI) and Rector of NCU for the scientific grant 504-F. Maciej Szkulmowski acknowledges support of Polish Ministry of Science, grants for years 2005/2008. Anna Szkulmowska acknowledges support of Polish Science Foundation FNP’2008 scholarship for young researchers.

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