## Transverse motion as a source of noise and reduced correlation of the Doppler phase shift in spectral domain OCT

Optics Express, Vol. 17, Issue 22, pp. 19698-19713 (2009)

http://dx.doi.org/10.1364/OE.17.019698

Acrobat PDF (2922 KB)

### Abstract

Recently, a new phase-resolved Doppler model was presented for spectral domain optical coherence tomography (SD OCT) showing that the linear relation between the axial velocity component of the obliquely moving sample and the phase difference of consecutive A-Scans does not hold true in the presence of a transverse velocity component which is neglected in the widely-used classic Doppler analysis. Besides taking note of the new non-proportional relationship of phase shift and oblique sample motion, it is essential to consider the correlation of the phase shift and its specific characteristic at certain Doppler angles for designing Doppler experiments with SD OCT. A correlation quotient is introduced to quantify the correlation of the backscattering signal in consecutive A-Scans as a function of the oblique sample motion. It was found that at certain velocities and Doppler angles no correlation of the phases of sequential A-Scans exists, even though the signal does not vanish. To indicate how the noise of the Doppler phase shift behaves for oblique movement, the standard deviation is determined as a function of the correlation quotient and the number of complex Doppler data averaged. The detailed theoretical model is validated by using a flow phantom model consisting of a 1% Intralipid flow through a 310 µm capillary. Finally, a short discussion of the presented results and the consequence for performing Doppler experiments is given.

© 2009 OSA

## 1. Introduction

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*in vivo*investigation of the birefringence properties of human tissue [4

4. J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. **22**(12), 934–936 (1997). [CrossRef] [PubMed]

5. M. Pircher, E. Goetzinger, R. Leitgeb, and C. Hitzenberger, “Three dimensional polarization sensitive OCT of human skin in vivo,” Opt. Express **12**(14), 3236–3244 (2004). [CrossRef] [PubMed]

*in vivo*imaging of internal organs [6

6. Z. G. Wang, C. S. D. Lee, W. C. Waltzer, J. X. Liu, H. K. Xie, Z. J. Yuan, and Y. T. Pan, “In vivo bladder imaging with microelectromechanical-systems-based endoscopic spectral domain optical coherence tomography,” J. Biomed. Opt. **12**(3), 034009 (2007). [CrossRef] [PubMed]

7. G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science **276**(5321), 2037–2039 (1997). [CrossRef] [PubMed]

8. U. Morgner, W. Drexler, F. X. Kärtner, X. D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. **25**(2), 111–113 (2000). [CrossRef]

9. R. Leitgeb, M. Wojtkowski, A. Kowalczyk, C. K. Hitzenberger, M. Sticker, and A. F. Fercher, “Spectral measurement of absorption by spectroscopic frequency-domain optical coherence tomography,” Opt. Lett. **25**(11), 820–822 (2000). [CrossRef]

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12. X. J. Wang, T. E. Milner, and J. S. Nelson, “Characterization of fluid flow velocity by optical Doppler tomography,” Opt. Lett. **20**(11), 1337–1339 (1995). [CrossRef] [PubMed]

15. J. A. Izatt, M. D. Kulkarni, S. Yazdanfar, J. K. Barton, and A. J. Welch, “In vivo bidirectional color Doppler flow imaging of picoliter blood volumes using optical coherence tomography,” Opt. Lett. **22**(18), 1439–1441 (1997). [CrossRef]

12. X. J. Wang, T. E. Milner, and J. S. Nelson, “Characterization of fluid flow velocity by optical Doppler tomography,” Opt. Lett. **20**(11), 1337–1339 (1995). [CrossRef] [PubMed]

15. J. A. Izatt, M. D. Kulkarni, S. Yazdanfar, J. K. Barton, and A. J. Welch, “In vivo bidirectional color Doppler flow imaging of picoliter blood volumes using optical coherence tomography,” Opt. Lett. **22**(18), 1439–1441 (1997). [CrossRef]

16. Z. Xu, L. Carrion, and R. Maciejko, “A zero-crossing detection method applied to Doppler OCT,” Opt. Express **16**(7), 4394–4412 (2008). [CrossRef] [PubMed]

17. H. Ren, K. M. Brecke, Z. Ding, Y. Zhao, J. S. Nelson, and Z. Chen, “Imaging and quantifying transverse flow velocity with the Doppler bandwidth in a phase-resolved functional optical coherence tomography,” Opt. Lett. **27**(6), 409–411 (2002). [CrossRef]

18. B. White, M. Pierce, N. Nassif, B. Cense, B. Park, G. Tearney, B. Bouma, T. Chen, and J. de Boer, “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical coherence tomography,” Opt. Express **11**(25), 3490–3497 (2003). [CrossRef] [PubMed]

27. B. Vakoc, S. Yun, J. de Boer, G. Tearney, and B. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express **13**(14), 5483–5493 (2005). [CrossRef] [PubMed]

28. S. H. Yun, G. Tearney, J. de Boer, and B. Bouma, “Motion artifacts in optical coherence tomography with frequency-domain ranging,” Opt. Express **12**(13), 2977–2998 (2004). [CrossRef] [PubMed]

29. R. Leitgeb, C. Hitzenberger, and A. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express **11**(8), 889–894 (2003). [CrossRef] [PubMed]

30. M. Choma, M. Sarunic, C. Yang, and J. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express **11**(18), 2183–2189 (2003). [CrossRef] [PubMed]

18. B. White, M. Pierce, N. Nassif, B. Cense, B. Park, G. Tearney, B. Bouma, T. Chen, and J. de Boer, “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical coherence tomography,” Opt. Express **11**(25), 3490–3497 (2003). [CrossRef] [PubMed]

25. T. Schmoll, C. Kolbitsch, and R. A. Leitgeb, “Ultra-high-speed volumetric tomography of human retinal blood flow,” Opt. Express **17**(5), 4166–4176 (2009). [CrossRef] [PubMed]

31. M. Szkulmowski, A. Szkulmowska, T. Bajraszewski, A. Kowalczyk, and M. Wojtkowski, “Flow velocity estimation using joint Spectral and Time domain Optical Coherence Tomography,” Opt. Express **16**(9), 6008–6025 (2008). [CrossRef] [PubMed]

32. A. Szkulmowska, M. Szkulmowski, D. Szlag, A. Kowalczyk, and M. Wojtkowski, “Three-dimensional quantitative imaging of retinal and choroidal blood flow velocity using joint Spectral and Time domain Optical Coherence Tomography,” Opt. Express **17**(13), 10584–10598 (2009). [CrossRef] [PubMed]

33. R. K. Wang, S. L. Jacques, Z. Ma, S. Hurst, S. R. Hanson, and A. Gruber, “Three dimensional optical angiography,” Opt. Express **15**(7), 4083–4097 (2007). [CrossRef] [PubMed]

*et al.*using a modified Hilbert transform algorithm for 3D flow detection without the use of spatial frequency modulation. This technique called single-pass volumetric bidirectional blood flow imaging (SPFI) [34

34. Y. K. Tao, A. M. Davis, and J. A. Izatt, “Single-pass volumetric bidirectional blood flow imaging spectral domain optical coherence tomography using a modified Hilbert transform,” Opt. Express **16**(16), 12350–12361 (2008). [CrossRef] [PubMed]

35. Y. K. Tao, K. M. Kennedy, and J. A. Izatt, “Velocity-resolved 3D retinal microvessel imaging using single-pass flow imaging spectral domain optical coherence tomography,” Opt. Express **17**(5), 4177–4188 (2009). [CrossRef] [PubMed]

36. A. H. Bachmann, M. L. Villiger, C. Blatter, T. Lasser, and R. A. Leitgeb, “Resonant Doppler flow imaging and optical vivisection of retinal blood vessels,” Opt. Express **15**(2), 408–422 (2007). [CrossRef] [PubMed]

28. S. H. Yun, G. Tearney, J. de Boer, and B. Bouma, “Motion artifacts in optical coherence tomography with frequency-domain ranging,” Opt. Express **12**(13), 2977–2998 (2004). [CrossRef] [PubMed]

_{z}by using its proportionality to the calculated phase shift Δφ between adjacent A-Scans [18

18. B. White, M. Pierce, N. Nassif, B. Cense, B. Park, G. Tearney, B. Bouma, T. Chen, and J. de Boer, “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical coherence tomography,” Opt. Express **11**(25), 3490–3497 (2003). [CrossRef] [PubMed]

25. T. Schmoll, C. Kolbitsch, and R. A. Leitgeb, “Ultra-high-speed volumetric tomography of human retinal blood flow,” Opt. Express **17**(5), 4166–4176 (2009). [CrossRef] [PubMed]

_{0}is the center wavelength of the OCT system, f

_{A-Scan}the A-Scan frequency and n the refractive index of the medium. With known Doppler angle, the absolute sample velocity v can be calculated. In a laminar flow which can be assumed for flowing blood in most cases, phase shifts of more than π can be identified uniquely by phase unwrapping techniques [37

37. V. Yang, M. Gordon, B. Qi, J. Pekar, S. Lo, E. Seng-Yue, A. Mok, B. Wilson, and I. Vitkin, “High speed, wide velocity dynamic range Doppler optical coherence tomography (Part I): System design, signal processing, and performance,” Opt. Express **11**(7), 794–809 (2003). [CrossRef] [PubMed]

_{Int}larger than the sample beam diameter, one would expect deviations from the classic Doppler model. The effect of the sample beam movement on the standard deviation of the phase shift was investigated by B. H. Park

*et al.*[38

38. B. H. Park, M. C. Pierce, B. Cense, S. H. Yun, M. Mujat, G. J. Tearney, B. E. Bouma, and J. F. de Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 microm,” Opt. Express **13**(11), 3931–3944 (2005). [CrossRef] [PubMed]

*et al.*[39

39. B. J. Vakoc, G. J. Tearney, and B. E. Bouma, “Statistical properties of phase-decorrelation in phase-resolved Doppler optical coherence tomography,” IEEE Trans. Med. Imaging **28**(6), 814–821 (2009). [CrossRef] [PubMed]

*in vitro*experiments were carried out using a capillary model with a 1% Intralipid emulsion.

## 2. Theory of the New Phase-dependent Doppler Model

_{z}⋅T

_{Int}and a transverse one Δx = v

_{x}⋅T

_{Int}where v

_{z}and v

_{x}are the axial and transverse velocity components and T

_{Int}is the integration time of the CCD line detector. For the theoretical consideration, the blurring of the Gaussian beam outside the focal plane as well as the Gouy phase shift are neglected [41

41. G. Lamouche, M. L. Dufour, B. Gauthier, and J. Monchalin, “Gouy phase anomaly in optical coherence tomography,” Opt. Commun. **239**(4-6), 297–301 (2004). [CrossRef]

_{0}corresponds to the beam width (FWHM) of the Gaussian sample beam, λ

_{0}is the center wavelength of the OCT system and n represents the refractive index of the investigated medium.

*et al.*[28

28. S. H. Yun, G. Tearney, J. de Boer, and B. Bouma, “Motion artifacts in optical coherence tomography with frequency-domain ranging,” Opt. Express **12**(13), 2977–2998 (2004). [CrossRef] [PubMed]

_{1}und T

_{2}are the borders of the time interval T

_{Int}, x

_{m}is the normalized coordinate in x-direction at t’ = 0 and a

_{m}is the complex amplitude of a scatterer m combining the amplitude and the phase of the backscattered light. The sum in front of the integral corresponds to the summation of all individual backscattered signals a

_{m}of multiple continuously moving scatterers with identical velocity but different positions x

_{m}.

^{*}of the second A-Scan [23

23. H. Wehbe, M. Ruggeri, S. Jiao, G. Gregori, C. A. Puliafito, and W. Zhao, “Automatic retinal blood flow calculation using spectral domain optical coherence tomography,” Opt. Express **15**(23), 15193–15206 (2007). [CrossRef] [PubMed]

_{x}of zero (δx = 0), the integral resulting from Eq. (6) can be solved analytically, leading to the linear relation between Δφ and v

_{z}in accordance to Eq. (1) [40]. Unfortunately, no analytical solution of the integral can be given for finite δx [42

42. J. Walther, A. Krüger, M. Cuevas, and E. Koch, “Effects of axial, transverse and oblique sample motion in FD OCT in systems with global or rolling shutter line detector,” J. Opt. Soc. Am. A **25**(11), 2791–2802 (2008). [CrossRef]

_{1},T

_{2},δx,δz) = N(T

_{1},T

_{2},0,0) does not depend on time and corresponds to the complex result of the Fourier transform of the integrated photocurrent. The signal from different scatterers adds up to the total signal N. The product N(T

_{1},T

_{2,}δx,δz)⋅N

^{*}(T

_{1},T

_{2,}δx,δz) corresponds to the intensity caused by the interference from the different scatterers which consists of the sum of squares of each signal, the first part in Eq. (8), and the interference term between all the pairs (m and k) of scatterers, which relates to the second part. While the mean value of the second part is zero over all possible combinations of scatterers, the rms-value of both sums is similar leading to the huge span of intensities (speckle noise). For finite sample velocities, as in the case of zero velocity, the average of the second term will disappear due to the arbitrary phases. As a result, the averaged value of C

_{cor}(Eq. (8)) can be described by Eq. (9).

_{m}. As we are only interested in relative values, we normalize the integral to the average value at zero velocity.

_{m}|

^{2}cancels down. Consequently, the following integral in Eq. (11) has to be solved:where the value

_{m}for zero velocity.

^{®}6.0 (Wolfram Research, Inc.). The double integral in Eq. (11) can be evaluated for any δx and δz without any further transformation. The resulting phase shift as a function of δx and δz is shown by a contour plot in Fig. 1a . The data points for the contour plot were selected on an adaptive grid with smaller step sizes in the regions of large changes of the function. The computation of the plot takes several hours on a normal PC.

_{0}and wavelength λ

_{0}in our experiment, the value ϑ of 3.75° corresponds to an angle ϑ’ of 54°. All phase differences Δφ arising from an obliquely moving sample at a constant Doppler angle ϑ are found on a line through the origin as exemplarily shown by the black solid line in Fig. 1a for ϑ being 3.75°.

## 3. Correlation and Noise of the Doppler Phase Shift

### 3.1 The Correlation Quotient

### 3.2 The Effect of Finite Spectral Bandwidth

36. A. H. Bachmann, M. L. Villiger, C. Blatter, T. Lasser, and R. A. Leitgeb, “Resonant Doppler flow imaging and optical vivisection of retinal blood vessels,” Opt. Express **15**(2), 408–422 (2007). [CrossRef] [PubMed]

_{0}used in the OCT system. Therefore, the effect of a finite spectral bandwidth can be considered by looking at the results of the phase shift Δφ and the correlation quotient CQ for different center wavelengths λ

_{0}. In most cases the beam diameter w

_{0}will change only slightly with λ

_{0}, while the axial displacement δz is inversely proportional to λ

_{0}(cp. Equation (3)). Consequently, the phase shift Δφ and the correlation quotient CQ as functions of the wavelength can be determined from small, almost vertical lines (δx ≈const.) starting at the smallest quotient of δz

_{L}= 2n⋅Δz/λ

_{L}and ending at δz

_{S}= 2n⋅Δz/λ

_{S}, where λ

_{L}and λ

_{S}are the long and short wavelength limits of the spectral range. As the curves of I

_{cor,mean}, I

_{mean}and Δφ are quite smooth in most areas, the influence of a large spectral bandwidth will be small in these areas leading only to a minor broadening of the axial peak shape of the SD OCT signal. In the areas where I

_{cor,mean}disappears for monochromatic light, the effect of finite spectral bandwidth will be similar to the effect shown by A. H. Bachmann

*et al.*[36

36. A. H. Bachmann, M. L. Villiger, C. Blatter, T. Lasser, and R. A. Leitgeb, “Resonant Doppler flow imaging and optical vivisection of retinal blood vessels,” Opt. Express **15**(2), 408–422 (2007). [CrossRef] [PubMed]

### 3.3 Tradeoff between Beam Size and Integration Time

_{0}and integration time T

_{Int}have to be chosen carefully to achieve optimal imaging results. The universal contour plots as shown in Figs. 1 and 2 are applicable for any center wavelength λ

_{0}and beam size w

_{0}, independent of T

_{Int}. By considering Eq. (13), it can be noted that the larger the spot size w

_{0}is chosen the larger becomes the angle ϑ’ at constant center wavelength λ

_{0}and Doppler angle ϑ. As a result, the influence of the transverse velocity can be reduced by choosing a larger w

_{0}. The disadvantage of this approach is that the transverse resolution of the OCT system decreases which is unfavorable for high resolution morphological and flow imaging of especially small blood vessels. A small quotient of vessel diameter to beam size will certainly reduce the accuracy of blood flow measurement. Additionally, an increase of the spot size results in a decrease of the numerical aperture of the objective of the sample beam and with this in a decrease of the backscattered light collected which in turn causes a reduced intensity signal. The second important parameter for the Doppler measurement is the integration time T

_{Int}which represents the limiting factor for the maximum axial velocity component v

_{z}measurable without using phase unwrapping techniques (cp. Equation (1)). In order to avoid the observed effects, it is helpful to choose a smaller T

_{Int}. But it has to be noted that the smaller T

_{Int}is used the more the SNR of the signal decreases and with this the minimum detectable axial velocity v

_{z}increases [38

38. B. H. Park, M. C. Pierce, B. Cense, S. H. Yun, M. Mujat, G. J. Tearney, B. E. Bouma, and J. F. de Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 microm,” Opt. Express **13**(11), 3931–3944 (2005). [CrossRef] [PubMed]

_{Int}has to be adapted to the region of interest of the phase shift which could possibly be difficult for the quantification of pulsatile flow velocities under

*in vivo*conditions.

## 4. Experimental Setup

### 4.1 The SD OCT System and Doppler Analysis

43. S. Meißner, G. Muller, J. Walther, A. Krüger, M. Cuevas, B. Eichhorn, U. Ravens, H. Morawietz, and E. Koch, “Investigation of murine Vasodynamics by Fourier Domain Optical Coherence Tomography,” Proc. SPIE **6627**, 66270D (2007). [CrossRef]

_{0}= 840 nm and a full width half maximum of FWHM = 50 nm. Furthermore, the system consists of a directional coupler, a fiber coupled 3D-scanner head with integrated reference arm and a self-developed spectrometer. The sample beam has a measured FWHM of the intensity profile of w

_{0}= 6.7 µm. The line scan detector (Dalsa IL C6, DALSA) in the spectrometer operates at a read-out rate of 11.88 kHz for one interference spectrum with a duty cycle of almost 100%. As shown in Eq. (1), the maximum of the linearly measured axial velocity component v

_{z}before phase unwrapping is limited by f

_{A-Scan}and amounts to 2.5 mm/s at f

_{A-Scan}= 11.88 kHz and λ

_{0}= 840 nm. The minimum value of v

_{z}that can be measured is limited by the SNR of the signal and the transverse scanning [38

38. B. H. Park, M. C. Pierce, B. Cense, S. H. Yun, M. Mujat, G. J. Tearney, B. E. Bouma, and J. F. de Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 microm,” Opt. Express **13**(11), 3931–3944 (2005). [CrossRef] [PubMed]

_{J + 1}(z) with the complex conjugate coefficient of the subsequent A-Scan Γ

_{J}

^{*}(z) in each depth z where J is the A-Scan number [44]. The result is in turn a complex value Γ

_{res}with the phase shift Δφ(z) as the argument.

_{res}has a random part. Accordingly, it is essential to average sequent complex data Γ

_{res}. The effective Δφ is computed from the argument of the averaged Γ

_{res}as shown in Eq. (18). As presented by B. J. Vakoc

*et al.*[39

39. B. J. Vakoc, G. J. Tearney, and B. E. Bouma, “Statistical properties of phase-decorrelation in phase-resolved Doppler optical coherence tomography,” IEEE Trans. Med. Imaging **28**(6), 814–821 (2009). [CrossRef] [PubMed]

45. A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, and M. Wojtkowski, “Phase-resolved Doppler optical coherence tomography--limitations and improvements,” Opt. Lett. **33**(13), 1425–1427 (2008). [CrossRef] [PubMed]

### 4.2 The Capillary Model for the Doppler Measurement

46. J. Walther and E. Koch, “Flow measurement by using the signal decrease of moving scatterers in spatially encoded Fourier domain optical coherence tomography,” Proc. SPIE **7168**, 71681S (2009). [CrossRef]

_{cor,mean}, I

_{mean}and CQ as functions of the flow velocity, a parabolic flow profile was assumed because the capillary had a length of 80 mm, the inlet path was considered and the Reynolds number was less than 10 in all cases. The set Doppler angles ϑ between the transverse direction and the flow velocity were measured by a volume scan and result in 3.75° and 12.1°, respectively. The refractive index n of the diluted Intralipid is determined by the ratio of the optical path length of the inner diameter of the Intralipid-filled capillary and the one with air-filling. To quantify the flow velocities, cross-sectional images of the capillary with the flowing Intralipid were acquired with a transverse displacement per A-Scan of the scanning sample beam of 0.5 µm. After this, the complex data according to Eq. (17) from nine consecutive A-Scans were averaged to remove the strong oversampling effect. Further reduction of the noise was achieved by averaging the complex signals of fifteen B-Scans.

## 5. Results and Discussion

### 5.1 Validation of the Correlation Quotient CQ

_{res}are calculated for some flow measurements at ϑ = 3.75°. In Fig. 5 , the complex values of Γ

_{res}of the 135 single measurements are presented for five different correlation quotients CQ = 0.99, 0.95, 0.88, 0.81 and 0.02. For a better comparison, all data sets were normalized to a mean absolute value of 1.

47. B. Karamata, K. Hassler, M. Laubscher, and T. Lasser, “Speckle statistics in optical coherence tomography,” J. Opt. Soc. Am. A **22**(4), 593–596 (2005). [CrossRef]

_{res}| obeys an exponential distribution at least for small velocities. Therefore, the 135 individual signals span a large range. As the individual signals are not on a line through the origin, the phase differences calculated from adjacent A-Scans are not constant. Note that Δφ calculated from small values will have a larger spread than the one from large values (cp. Figure 5a) which again emphasizes the kind of averaging used (Section 4.1). The next points were selected with phase shifts of π/4, π/2 and π. Due to the higher velocity, CQ becomes smaller and results in 0.95, 0.88 and 0.81, respectively. The last point was chosen with nearly zero correlation (CQ = 0.02). In this case, the noise is too high to calculate a mean phase shift from the 135 data points. Note that the correlation between the phases of subsequent A-Scans does not only disappear for high transverse velocity components (ϑ' < 45°) but also in the areas around δx ≈0.64 and δz ≈j (j being an integer).

### 5.2 Mean Error of the Averaged Phase Shift

_{res}for ten different flow velocities and Doppler angles ϑ of 3.75° and 12.1°, respectively. The black theoretical curve shows σ for K = 1 as a function of CQ and is based on the following model. If all the phasors in the numerator of CQ would have the same length, one would expect that the standard deviation σ follows an arccos function with a value of a = 1, since the denominator is virtually the sum of the absolute values of the numerator.

_{res}| are not identical for the 135 single values and small values of |Γ

_{res}| contribute to a large spread of phases. The fit of a function to the experimental data for high correlation quotients results in a function according to

^{b}+ y

^{b}]

^{1/b}approaches the larger one for any positive b. For negative values of b, the function converges to the smaller one. In order to have the specific value for CQ = 0, the third term was added in the brackets. In Eq. (22), the fitting parameters a and b are presented as mean values with their standard deviation. Note that the reason for the large standard error is the high correlation of the parameters a and b. Increasing one can be compensated to a high extend by decreasing the other.

_{mean}also decreases with rising sample velocity and with this other noise factors increase additionally until the phase measurement is not realizable any longer. Even at zero flow conditions, the experimentally measured CQ will always be smaller than one because of Brownian motion in the Intralipid, residual scanner movement and other sources of noise.

^{h}with h = −0.62 as determined experimentally (cp. Figure 6). Unfortunately, such a general statement of the reduction of σ with K cannot be given for larger σ. The orange colored areas in the diagram correspond to CQ ≥ 5 dB with a maximum σ of up to π/√3 relating to an isotropic distribution of the phase shifts. At those regions, an averaging of the measured phase shift reduces the noise only very slowly and should be avoided for Doppler measurements.

## 6. Summary and Conclusion

_{0}and beam size w

_{0}. Besides the non-proportional relationship between phase shift and axial velocity component, it is essential to consider the correlation and the noise of the phase shift for designing Doppler SD OCT setups and measurements with optimal imaging results.

*et al.*[48

48. J. W. You, T. C. Chen, M. Mujat, B. H. Park, and J. F. de Boer, “Pulsed illumination spectral-domain optical coherence tomography for human retinal imaging,” Opt. Express **14**(15), 6739–6748 (2006). [CrossRef] [PubMed]

## Acknowledgements

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7. | G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science |

8. | U. Morgner, W. Drexler, F. X. Kärtner, X. D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. |

9. | R. Leitgeb, M. Wojtkowski, A. Kowalczyk, C. K. Hitzenberger, M. Sticker, and A. F. Fercher, “Spectral measurement of absorption by spectroscopic frequency-domain optical coherence tomography,” Opt. Lett. |

10. | R. K. Wang, Z. Ma, and S. J. Kirkpatrick, “Tissue Doppler optical coherence elastography for real time strain rate and strain mapping of soft tissue,” Appl. Phys. Lett. |

11. | S. J. Kirkpatrick, R. K. Wang, and D. D. Duncan, “OCT-based elastography for large and small deformations,” Opt. Express |

12. | X. J. Wang, T. E. Milner, and J. S. Nelson, “Characterization of fluid flow velocity by optical Doppler tomography,” Opt. Lett. |

13. | Z. Chen, T. E. Milner, S. Srinivas, X. Wang, A. Malekafzali, M. J. C. van Gemert, and J. S. Nelson, “Noninvasive imaging of in vivo blood flow velocity using optical Doppler tomography,” Opt. Lett. |

14. | Z. Chen, T. E. Milner, D. Dave, and J. S. Nelson, “Optical Doppler tomographic imaging of fluid flow velocity in highly scattering media,” Opt. Lett. |

15. | J. A. Izatt, M. D. Kulkarni, S. Yazdanfar, J. K. Barton, and A. J. Welch, “In vivo bidirectional color Doppler flow imaging of picoliter blood volumes using optical coherence tomography,” Opt. Lett. |

16. | Z. Xu, L. Carrion, and R. Maciejko, “A zero-crossing detection method applied to Doppler OCT,” Opt. Express |

17. | H. Ren, K. M. Brecke, Z. Ding, Y. Zhao, J. S. Nelson, and Z. Chen, “Imaging and quantifying transverse flow velocity with the Doppler bandwidth in a phase-resolved functional optical coherence tomography,” Opt. Lett. |

18. | B. White, M. Pierce, N. Nassif, B. Cense, B. Park, G. Tearney, B. Bouma, T. Chen, and J. de Boer, “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical coherence tomography,” Opt. Express |

19. | R. Leitgeb, L. Schmetterer, W. Drexler, A. Fercher, R. Zawadzki, and T. Bajraszewski, “Real-time assessment of retinal blood flow with ultrafast acquisition by color Doppler Fourier domain optical coherence tomography,” Opt. Express |

20. | L. Wang, Y. Wang, S. Guo, J. Zhang, M. Bachman, G. P. Li, and Z. Chen, “Frequency domain phase-resolved optical Doppler and Doppler variance tomography,” Opt. Commun. |

21. | Y. Wang, B. A. Bower, J. A. Izatt, O. Tan, and D. Huang, “In vivo total retinal blood flow measurement by Fourier domain Doppler optical coherence tomography,” J. Biomed. Opt. |

22. | B. A. Bower, M. Zhao, R. J. Zawadzki, and J. A. Izatt, “Real-time spectral domain Doppler optical coherence tomography and investigation of human retinal vessel autoregulation,” J. Biomed. Opt. |

23. | H. Wehbe, M. Ruggeri, S. Jiao, G. Gregori, C. A. Puliafito, and W. Zhao, “Automatic retinal blood flow calculation using spectral domain optical coherence tomography,” Opt. Express |

24. | Y. Wang, A. Fawzi, O. Tan, J. Gil-Flamer, and D. Huang, “Retinal blood flow detection in diabetic patients by Doppler Fourier domain optical coherence tomography,” Opt. Express |

25. | T. Schmoll, C. Kolbitsch, and R. A. Leitgeb, “Ultra-high-speed volumetric tomography of human retinal blood flow,” Opt. Express |

26. | A. Mariampillai, B. A. Standish, N. R. Munce, C. Randall, G. Liu, J. Y. Jiang, A. E. Cable, I. A. Vitkin, and V. X. D. Yang, “Doppler optical cardiogram gated 2D color flow imaging at 1000 fps and 4D in vivo visualization of embryonic heart at 45 fps on a swept source OCT system,” Opt. Express |

27. | B. Vakoc, S. Yun, J. de Boer, G. Tearney, and B. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express |

28. | S. H. Yun, G. Tearney, J. de Boer, and B. Bouma, “Motion artifacts in optical coherence tomography with frequency-domain ranging,” Opt. Express |

29. | R. Leitgeb, C. Hitzenberger, and A. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express |

30. | M. Choma, M. Sarunic, C. Yang, and J. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express |

31. | M. Szkulmowski, A. Szkulmowska, T. Bajraszewski, A. Kowalczyk, and M. Wojtkowski, “Flow velocity estimation using joint Spectral and Time domain Optical Coherence Tomography,” Opt. Express |

32. | A. Szkulmowska, M. Szkulmowski, D. Szlag, A. Kowalczyk, and M. Wojtkowski, “Three-dimensional quantitative imaging of retinal and choroidal blood flow velocity using joint Spectral and Time domain Optical Coherence Tomography,” Opt. Express |

33. | R. K. Wang, S. L. Jacques, Z. Ma, S. Hurst, S. R. Hanson, and A. Gruber, “Three dimensional optical angiography,” Opt. Express |

34. | Y. K. Tao, A. M. Davis, and J. A. Izatt, “Single-pass volumetric bidirectional blood flow imaging spectral domain optical coherence tomography using a modified Hilbert transform,” Opt. Express |

35. | Y. K. Tao, K. M. Kennedy, and J. A. Izatt, “Velocity-resolved 3D retinal microvessel imaging using single-pass flow imaging spectral domain optical coherence tomography,” Opt. Express |

36. | A. H. Bachmann, M. L. Villiger, C. Blatter, T. Lasser, and R. A. Leitgeb, “Resonant Doppler flow imaging and optical vivisection of retinal blood vessels,” Opt. Express |

37. | V. Yang, M. Gordon, B. Qi, J. Pekar, S. Lo, E. Seng-Yue, A. Mok, B. Wilson, and I. Vitkin, “High speed, wide velocity dynamic range Doppler optical coherence tomography (Part I): System design, signal processing, and performance,” Opt. Express |

38. | B. H. Park, M. C. Pierce, B. Cense, S. H. Yun, M. Mujat, G. J. Tearney, B. E. Bouma, and J. F. de Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 microm,” Opt. Express |

39. | B. J. Vakoc, G. J. Tearney, and B. E. Bouma, “Statistical properties of phase-decorrelation in phase-resolved Doppler optical coherence tomography,” IEEE Trans. Med. Imaging |

40. | E. Koch, J. Walther, and M. Cuevas, “Limits of Fourier domain Doppler-OCT at high velocities,” Sensors and Actuators A , |

41. | G. Lamouche, M. L. Dufour, B. Gauthier, and J. Monchalin, “Gouy phase anomaly in optical coherence tomography,” Opt. Commun. |

42. | J. Walther, A. Krüger, M. Cuevas, and E. Koch, “Effects of axial, transverse and oblique sample motion in FD OCT in systems with global or rolling shutter line detector,” J. Opt. Soc. Am. A |

43. | S. Meißner, G. Muller, J. Walther, A. Krüger, M. Cuevas, B. Eichhorn, U. Ravens, H. Morawietz, and E. Koch, “Investigation of murine Vasodynamics by Fourier Domain Optical Coherence Tomography,” Proc. SPIE |

44. | J. Walther, G. Muller, H. Morawietz, and E. Koch, “Analysis of in vitro and in vivo bidirectional flow velocities by phase-resolved Doppler Fourier-domain OCT,” Sens. Actuators A . |

45. | A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, and M. Wojtkowski, “Phase-resolved Doppler optical coherence tomography--limitations and improvements,” Opt. Lett. |

46. | J. Walther and E. Koch, “Flow measurement by using the signal decrease of moving scatterers in spatially encoded Fourier domain optical coherence tomography,” Proc. SPIE |

47. | B. Karamata, K. Hassler, M. Laubscher, and T. Lasser, “Speckle statistics in optical coherence tomography,” J. Opt. Soc. Am. A |

48. | J. W. You, T. C. Chen, M. Mujat, B. H. Park, and J. F. de Boer, “Pulsed illumination spectral-domain optical coherence tomography for human retinal imaging,” Opt. Express |

**OCIS Codes**

(110.4500) Imaging systems : Optical coherence tomography

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(280.2490) Remote sensing and sensors : Flow diagnostics

(110.4153) Imaging systems : Motion estimation and optical flow

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: September 3, 2009

Revised Manuscript: October 12, 2009

Manuscript Accepted: October 14, 2009

Published: October 15, 2009

**Virtual Issues**

Vol. 4, Iss. 12 *Virtual Journal for Biomedical Optics*

**Citation**

Julia Walther and Edmund Koch, "Transverse motion as a source of noise and reduced correlation of the Doppler phase shift
in spectral domain OCT," Opt. Express **17**, 19698-19713 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-19698

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- M. Szkulmowski, A. Szkulmowska, T. Bajraszewski, A. Kowalczyk, and M. Wojtkowski, “Flow velocity estimation using joint Spectral and Time domain Optical Coherence Tomography,” Opt. Express 16(9), 6008–6025 (2008). [CrossRef] [PubMed]
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- S. Meißner, G. Muller, J. Walther, A. Krüger, M. Cuevas, B. Eichhorn, U. Ravens, H. Morawietz, and E. Koch, “Investigation of murine Vasodynamics by Fourier Domain Optical Coherence Tomography,” Proc. SPIE 6627, 66270D (2007). [CrossRef]
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- A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, and M. Wojtkowski, “Phase-resolved Doppler optical coherence tomography--limitations and improvements,” Opt. Lett. 33(13), 1425–1427 (2008). [CrossRef] [PubMed]
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- J. W. You, T. C. Chen, M. Mujat, B. H. Park, and J. F. de Boer, “Pulsed illumination spectral-domain optical coherence tomography for human retinal imaging,” Opt. Express 14(15), 6739–6748 (2006). [CrossRef] [PubMed]

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