## Transit-time analysis based on delay-encoded beam shape for velocity vector quantification by spectral-domain Doppler optical coherence tomography

Optics Express, Vol. 18, Issue 2, pp. 1261-1270 (2010)

http://dx.doi.org/10.1364/OE.18.001261

Acrobat PDF (309 KB)

### Abstract

We propose a transit-time based method to ascertain the azimuth angle of a velocity vector by spectral-domain Doppler optical coherence tomography (DOCT), so that three-dimensional (3-D) velocity vector can be quantified. A custom-designed slit plate with predetermined slit orientation is placed into the sample beam to create three delay-encoded sub-beams of different beam shape for sample probing. Based on the transit-time analysis for Doppler bandwidth, the azimuth angle within 90° range is evaluated by exploitation of the complex signals corresponding to three path length delays. 3-D velocity vector is quantified through further estimating of Doppler angle and flow velocity by combined Doppler shift and Doppler bandwidth measurements. The feasibility of the method is demonstrated by good agreement between the determined azimuth angles and the preset ones, and further confirmed by velocity vector measurement of flowing solution inside a capillary tube.

© 2010 OSA

## 1. Introduction

1. Z. Chen, Y. Zhao, S. M. Srinivas, J. S. Nelson, N. Prakash, and R. D. Frostig, “Optical Doppler Tomography,” IEEE J. Sel. Top. Quantum Electron. **5**(4), 1134–1142 (1999). [CrossRef]

2. D. P. Davé and T. E. Milner, “Doppler-angle measurement in highly scattering media,” Opt. Lett. **25**(20), 1523–1525 (2000). [CrossRef]

3. C. J. Pedersen, D. Huang, M. A. Shure, and A. M. Rollins, “Measurement of absolute flow velocity vector using dual-angle, delay-encoded Doppler optical coherence tomography,” Opt. Lett. **32**(5), 506–508 (2007). [CrossRef] [PubMed]

4. 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]

5. A. Royset, T. Storen, F. Stabo-Eeg, and T. Lindmo, “Quantitative measurements of flow velocity and direction using Transversal Doppler Optical Coherence Tomography,” Proc. SPIE **6079**, 607925 (2006). [CrossRef]

6. Y. C. Ahn, W. Jung, and Z. Chen, “Quantification of a three-dimensional velocity vector using spectral-domain Doppler optical coherence tomography,” Opt. Lett. **32**(11), 1587–1589 (2007). [CrossRef] [PubMed]

7. R. Michaely, A. H. Bachmann, M. L. Villiger, C. Blatter, T. Lasser, and R. A. Leitgeb, “Vectorial reconstruction of retinal blood flow in three dimensions measured with high resolution resonant Doppler Fourier domain optical coherence tomography,” J. Biomed. Opt. **12**(4), 041213 (2007). [CrossRef] [PubMed]

9. S. Makita, T. Fabritius, and Y. Yasuno, “Quantitative retinal-blood flow measurement with three-dimensional vessel geometry determination using ultrahigh-resolution Doppler optical coherence angiography,” Opt. Lett. **33**(8), 836–838 (2008). [CrossRef] [PubMed]

10. D. Piao, L. L. Otis, and Q. Zhu, “Doppler angle and flow velocity mapping by combined Doppler shift and Doppler bandwidth measurements in optical Doppler tomography,” Opt. Lett. **28**(13), 1120–1122 (2003). [CrossRef] [PubMed]

6. Y. C. Ahn, W. Jung, and Z. Chen, “Quantification of a three-dimensional velocity vector using spectral-domain Doppler optical coherence tomography,” Opt. Lett. **32**(11), 1587–1589 (2007). [CrossRef] [PubMed]

## 2. Theory

4. 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]

*B*in evaluation of the Doppler spectrum broadening is the inverse of the transit time, i.e.,As illustrated in Fig. 1 , V is the flow velocity,

*α*is the Doppler angle between the flowing direction and the optical axis of the probe beam,

*w*denotes the effective waist diameter of the probe beam in the focal area, which is the transit length of moving particles.

*B*

_{0}accounts for the contributions from other sources that are independent of the macroscopic flow velocity.

*V*, Doppler angle

*α*and azimuth angle

*φ*. As shown in Fig. 1(a), the slit plate is made of glass with air slit in the middle. The glass part of the plate has a thickness of

*t*with refractive index of

*n*. The slit with a width of

_{g}*d*is oriented along x axis. The plate is introduced in the sample arm of a spectral-domain DOCT system between the collimator and galvo mirror shown in Fig. 1(b). Collimated beam with a diameter of

*D*illuminates the plate and focused to the scattering particles by a focusing lens with the focal length of

*f*. Backscattered light from the moving particles is collected by the focusing lens and passes through the plate again. In double passing of the plate, the light could go through the plate via air slit first and then return via air slit or glass. Alternatively, the light could go through the plate via glass first and then return via air slit or glass. Therefore, the slit plate makes four different beam paths: air to air (AA), air to glass (AG), glass to air (GA) and glass to glass (GG). Since AG and GA have the same optical path delay, there are three independent path length delays, which can be denoted by 0,

*t*(

*n*-1) and 2

_{g}*t*(

*n*-1). In the following text, the double-subscript of the transit length

_{g}*w*and the Doppler bandwidth

*B*indicates the corresponding beam path.

*Mλ*

^{2}/4Δ

*λ*, where

*M*is the number of pixels of the CCD,

*λ*is the central wavelength of the light source, and Δ

*λ*is the full spectral range of the spectrometer. In the previously reported method [6

6. Y. C. Ahn, W. Jung, and Z. Chen, “Quantification of a three-dimensional velocity vector using spectral-domain Doppler optical coherence tomography,” Opt. Lett. **32**(11), 1587–1589 (2007). [CrossRef] [PubMed]

*Mλ*

^{2}/20Δ

*λ*will overlap, while in the proposed method this threshold is improved to be

*Mλ*

^{2}/12Δ

*λ*. Therefore, the demand on imaging depth for the spectral-domain DOCT system is relaxed.

*d*=1 mm) and glass, respectively.

*x*and

*y*are horizontal and vertical coordinates of the focal plane. To analyze the transit time, the beam size is estimated from the contour of the central maximum of the intensity pattern. Under illumination via air slit demonstrated in Fig. 2(a), the width of the central maximum along

*x*direction and

*y*direction are 2

*λf*/

*D*and 2

*λf*/

*d*, respectively. The fitted contour of the central maximum is almost a rectangle. Therefore, the transit length

*w*

_{AA}that a particle passes through the central pattern with an azimuth angle

*φ*can be expressed asWhen

*φ*approaches 90°,

*w*

_{AA}becomes very large, a small Doppler bandwidth comparable to the background

*B*

_{0}is resulted and the information on azimuth angle

*φ*is impossible to recover accurately.

*d*are plotted in Fig. 2(c). Obviously, the central maximum under illumination via glass approaches an Airy pattern as the air slit

*d*decreases. The transit length

*w*

_{GG}that a particle passes through this central pattern with an azimuth angle

*φ*is given by 2(

*x*

_{0}

^{2}cos

^{2}

*φ*+

*y*

_{0}

^{2}sin

^{2}

*φ*)

^{1/2}, where

*x*

_{0}and

*y*

_{0}represent the semimajor axis and semiminor axis of the fitted ellipse, respectively. For instance, with system parameters of

*λ*=0.84 μm,

*f*=75 mm,

*D*=5.2 mm and

*d*=1 mm, the fitted elliptic contour of the central maximum is (x/15.77)

^{2}+(y/11.50)

^{2}=1. Thus we have

*x*

_{0}=15.77 μm and

*y*

_{0}=11.50 μm.

*z*is the axial coordinate,

*A*(

*z*) and

*Φ*(

*z*) denote the amplitude and phase terms, respectively.

*A*(

*z*) is used for structural information reconstruction, and phase

*Φ*(

*z*) is used for Doppler information construction. Then

_{0}-

*δ*/2, z

_{0}+

*δ*/2), where

*δ*=

*t*(

*n*-1)/2 and z

_{g}_{0}represents the axial position corresponding to the center of sample depth.

*δ*in axial direction compared with

*δ*in axial direction compared with

12. N. Iftimia, B. E. Bouma, and G. J. Tearney, “Speckle reduction in optical coherence tomography by path length encoded angular compounding,” J. Biomed. Opt. **8**(2), 260–263 (2003). [CrossRef] [PubMed]

*B*

_{sum}, is independent on the azimuth angle

*φ*because of circular symmetry of the equivalent clear aperture. The transit length corresponding to the summed complex signal,

*w*

_{sum}, is the diameter of well-known Airy pattern given by 8

*λf*/π

*D*irrespective of the azimuth angle.

*B*

_{0}is subtracted, the ratio of the Doppler bandwidth obtained from

*B*and standard deviation

*σ*, i.e.

4. 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]

*T*is the time interval between two A-scans,

*N*is the number of A-lines used for averaging,

*j*th A-scan and its next A-scan,

*B*

_{AA},

*B*

_{GG}and

*B*

_{sum}, azimuth angle

*φ*within 90° range can be determined using Eq. (5) or (6). However, the determination of an unknown

*φ*through Eq. (5) is under a prior knowledge of its range, which can be obtained by Eq. (6). The other two quantities required for the quantification of the velocity vector, Doppler angle

*α*and flow velocity

*V*, are estimated by combined Doppler shift and Doppler bandwidth measurements from the coherently summed complex signal

10. D. Piao, L. L. Otis, and Q. Zhu, “Doppler angle and flow velocity mapping by combined Doppler shift and Doppler bandwidth measurements in optical Doppler tomography,” Opt. Lett. **28**(13), 1120–1122 (2003). [CrossRef] [PubMed]

*f*and flow velocity

_{d}*V*iswhere

*λ*is the central wavelength in the media.

*f*is calculated by phase difference of successive A-scans:

_{d}*d*. Therefore, both Eq. (5) and (6) are tried for the evaluation of azimuth angle in the following experiments.

## 3. Experiments

13. K. Wang, Z. Ding, T. Wu, C. Wang, J. Meng, M. Chen, and L. Xu, “Development of a non-uniform discrete Fourier transform based high speed spectral domain optical coherence tomography system,” Opt. Express **17**(14), 12121–12131 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-14-12121. [CrossRef] [PubMed]

14. K. Wang, Z. Ding, Y. Zeng, J. Meng, and M. Chen, “Sinusoidal B-M method based spectral domain optical coherence tomography for the elimination of complex-conjugate artifact,” Opt. Express **17**(19), 16820–16833 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-19-16820. [CrossRef] [PubMed]

*φ*starting from

*x*axis and Doppler angle

*α*starting from

*z*axis as depicted in Fig. 1. The air silt width

*d*is set to be 1 mm or 0.8 mm for comparable signal intensities of three sub-images corresponding to the three path length delays. The glass part of the plate is made of BK7, which has a thickness of 1.7 mm with a refractive index of 1.51 at 830 nm wavelength. In our experiment, group velocity mismatch among the different paths introduced by the slit plate is not considered since the thickness of the glass part is relatively small.

*d*of 1 mm and 0.8 mm. Under the condition that the orientation of capillary tube at a fixed Doppler angle of 76°, the volume velocity of flowing solution is varied from 1 μl/min to 5 μl/min at increment step of 1 μl/min, equivalent to an average velocity changing from 0.94 mm/s to 4.72 mm/s.

**27**(6), 409–411 (2002). [CrossRef]

*N*A-scans are performed for averaging (

*N*=1000). With A-scan rate at 20 KHz, the time needed for one M-mode data acquisition is 50 ms. The reason why we use M-mode in the experiment is to exclude other potential affecting factors which include: 1) deflexion of incident ray caused by geometric shape of capillary tube and the refraction index mismatch; 2) beam shape variation with the position of the scanning beam due to optical aberrations; 3) additional phase noise induced by the transversal scanning.

*B*

_{0}can be deduced through linear fitting and extending to zero velocity. The estimated background

*B*

_{0}for cases of

*d*=1 mm are 128.8, 140.4 and 136.0 Hz, respectively, and with

*d*=0.8 mm, those are 121.6, 139.2 and 133.2 Hz, respectively.

*φ*smaller than 80°. The accuracy for two values of

*d*is comparable. Invalid estimation of azimuth angle occurs when it approaches 90° due to the big influence of Doppler bandwidth background, thus the calculated data at 80° and 90° are discarded. Compared with the results shown in Fig. 5(a), the estimated azimuth angles in Fig. 5(b) are in better agreement with the preset ones especially for

*φ*near 90°. The data for

*φ*around 45° is in perfect match with the preset ones, since

*w*

_{GG}as a function of

*φ*is approximately linear in this situation. However, the data for

*φ*near 0° or 90° is not accurate for the reason that

*w*

_{GG}is insensitive to the variation of

*φ*in these two situations. As can be seen, the result for

*d*=1 mm is in better agreement with the preset ones compared with

*d*=0.8 mm. As

*d*becomes larger, the fitted elliptic contour of the central maximum under illumination via glass becomes more elongated (see Fig. 2(c)). Therefore the transit length

*w*

_{GG}is more sensitive to the variation of

*φ*. The data of 0° is discarded since the calculated cos

^{2}

*φ*(see Eq. (6)) is invalid owing to added noise. The RMS errors of the estimated angles corresponding to Fig. 5(a) and (b) for two values of

*d*are summarized in Table 1 . The difference between estimated azimuth angles based on Eq. (5) and preset ones is within the range of 0.29° to 9.37°, while based on Eq. (6) the difference is within the range of 0.19° to 12.90°. Higher accuracy of estimation on azimuth angle is achieved using Eq. (5) than using Eq. (6) if invalid estimations at 80° and 90° are excluded. On the other hand, method based on Eq. (6) can still obtain effective estimation of azimuth angle even it approaches 90° due to relatively high SNR.

*φ*is determined, the Doppler shift and Doppler bandwidth obtained from the coherently summed complex signal

*V*and Doppler angle

*α*. The estimated results are shown in Fig. 6(a) and (b) , where the capillary tube is positioned at azimuth angle of 45°. It can be seen that the estimated data for

*d*=1 mm and 0.8 mm are both in good agreement with the preset ones under different flow velocities. The estimated flow velocity vector is thus quantified and plotted in Fig. 7 with the preset flow velocity of 4.72 mm/s.

*w*is enlarged due to bigger lateral spot size. Therefore, according to the relationship of Doppler bandwidth and transit length in Eq. (2), the Doppler bandwidth

*B*decreases. In our method, the azimuth angle

*φ*is obtained by calculating the ratio of

*B*

_{AA}and

*B*

_{sum}or

*B*

_{GG}and

*B*

_{sum}according to Eq. (5) and (6). Although the Doppler bandwidth decreases, the variation of

*B*

_{AA}/

*B*

_{sum}or

*B*

_{GG}/

*B*

_{sum}is usually negligible since the decreasing of

*B*

_{AA},

*B*

_{GG}and

*B*

_{sum}is almost proportional, especially when the defocusing distance is small.

*w*is the theoretical transit length,

*f*is the Doppler shift which is not affected by defocusing. Therefore, the Doppler angle is underestimated.

_{d}## 4. Conclusion

## Acknowledgement

## References and links

1. | Z. Chen, Y. Zhao, S. M. Srinivas, J. S. Nelson, N. Prakash, and R. D. Frostig, “Optical Doppler Tomography,” IEEE J. Sel. Top. Quantum Electron. |

2. | D. P. Davé and T. E. Milner, “Doppler-angle measurement in highly scattering media,” Opt. Lett. |

3. | C. J. Pedersen, D. Huang, M. A. Shure, and A. M. Rollins, “Measurement of absolute flow velocity vector using dual-angle, delay-encoded Doppler optical coherence tomography,” Opt. Lett. |

4. | 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. |

5. | A. Royset, T. Storen, F. Stabo-Eeg, and T. Lindmo, “Quantitative measurements of flow velocity and direction using Transversal Doppler Optical Coherence Tomography,” Proc. SPIE |

6. | Y. C. Ahn, W. Jung, and Z. Chen, “Quantification of a three-dimensional velocity vector using spectral-domain Doppler optical coherence tomography,” Opt. Lett. |

7. | R. Michaely, A. H. Bachmann, M. L. Villiger, C. Blatter, T. Lasser, and R. A. Leitgeb, “Vectorial reconstruction of retinal blood flow in three dimensions measured with high resolution resonant Doppler Fourier domain optical coherence tomography,” J. Biomed. Opt. |

8. | 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. |

9. | S. Makita, T. Fabritius, and Y. Yasuno, “Quantitative retinal-blood flow measurement with three-dimensional vessel geometry determination using ultrahigh-resolution Doppler optical coherence angiography,” Opt. Lett. |

10. | D. Piao, L. L. Otis, and Q. Zhu, “Doppler angle and flow velocity mapping by combined Doppler shift and Doppler bandwidth measurements in optical Doppler tomography,” Opt. Lett. |

11. | V. L. Newhouse, E. S. Furgason, G. F. Johnson, and D. A. Wolf, “The dependence of ultrasound Doppler bandwidth on beam geometry,” IEEE Trans. Sonics Ultrason. |

12. | N. Iftimia, B. E. Bouma, and G. J. Tearney, “Speckle reduction in optical coherence tomography by path length encoded angular compounding,” J. Biomed. Opt. |

13. | K. Wang, Z. Ding, T. Wu, C. Wang, J. Meng, M. Chen, and L. Xu, “Development of a non-uniform discrete Fourier transform based high speed spectral domain optical coherence tomography system,” Opt. Express |

14. | K. Wang, Z. Ding, Y. Zeng, J. Meng, and M. Chen, “Sinusoidal B-M method based spectral domain optical coherence tomography for the elimination of complex-conjugate artifact,” Opt. Express |

**OCIS Codes**

(170.3340) Medical optics and biotechnology : Laser Doppler velocimetry

(170.3890) Medical optics and biotechnology : Medical optics instrumentation

(170.4500) Medical optics and biotechnology : Optical coherence tomography

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: November 10, 2009

Revised Manuscript: December 19, 2009

Manuscript Accepted: December 24, 2009

Published: January 11, 2010

**Virtual Issues**

Vol. 5, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Jie Meng, Zhihua Ding, Jiawen Li, Kai Wang, and Tong Wu, "Transit-time analysis based on delay-encoded beam shape for velocity vector quantification by spectral-domain Doppler optical coherence tomography," Opt. Express **18**, 1261-1270 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-2-1261

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### References

- Z. Chen, Y. Zhao, S. M. Srinivas, J. S. Nelson, N. Prakash, and R. D. Frostig, “Optical Doppler Tomography,” IEEE J. Sel. Top. Quantum Electron. 5(4), 1134–1142 (1999). [CrossRef]
- D. P. Davé and T. E. Milner, “Doppler-angle measurement in highly scattering media,” Opt. Lett. 25(20), 1523–1525 (2000). [CrossRef]
- C. J. Pedersen, D. Huang, M. A. Shure, and A. M. Rollins, “Measurement of absolute flow velocity vector using dual-angle, delay-encoded Doppler optical coherence tomography,” Opt. Lett. 32(5), 506–508 (2007). [CrossRef] [PubMed]
- 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]
- A. Royset, T. Storen, F. Stabo-Eeg, and T. Lindmo, “Quantitative measurements of flow velocity and direction using Transversal Doppler Optical Coherence Tomography,” Proc. SPIE 6079, 607925 (2006). [CrossRef]
- Y. C. Ahn, W. Jung, and Z. Chen, “Quantification of a three-dimensional velocity vector using spectral-domain Doppler optical coherence tomography,” Opt. Lett. 32(11), 1587–1589 (2007). [CrossRef] [PubMed]
- R. Michaely, A. H. Bachmann, M. L. Villiger, C. Blatter, T. Lasser, and R. A. Leitgeb, “Vectorial reconstruction of retinal blood flow in three dimensions measured with high resolution resonant Doppler Fourier domain optical coherence tomography,” J. Biomed. Opt. 12(4), 041213 (2007). [CrossRef] [PubMed]
- 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. 12(4), 041215 (2007). [CrossRef] [PubMed]
- S. Makita, T. Fabritius, and Y. Yasuno, “Quantitative retinal-blood flow measurement with three-dimensional vessel geometry determination using ultrahigh-resolution Doppler optical coherence angiography,” Opt. Lett. 33(8), 836–838 (2008). [CrossRef] [PubMed]
- D. Piao, L. L. Otis, and Q. Zhu, “Doppler angle and flow velocity mapping by combined Doppler shift and Doppler bandwidth measurements in optical Doppler tomography,” Opt. Lett. 28(13), 1120–1122 (2003). [CrossRef] [PubMed]
- V. L. Newhouse, E. S. Furgason, G. F. Johnson, and D. A. Wolf, “The dependence of ultrasound Doppler bandwidth on beam geometry,” IEEE Trans. Sonics Ultrason. SU-27, 50–59 (1980).
- N. Iftimia, B. E. Bouma, and G. J. Tearney, “Speckle reduction in optical coherence tomography by path length encoded angular compounding,” J. Biomed. Opt. 8(2), 260–263 (2003). [CrossRef] [PubMed]
- K. Wang, Z. Ding, T. Wu, C. Wang, J. Meng, M. Chen, and L. Xu, “Development of a non-uniform discrete Fourier transform based high speed spectral domain optical coherence tomography system,” Opt. Express 17(14), 12121–12131 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-14-12121 . [CrossRef] [PubMed]
- K. Wang, Z. Ding, Y. Zeng, J. Meng, and M. Chen, “Sinusoidal B-M method based spectral domain optical coherence tomography for the elimination of complex-conjugate artifact,” Opt. Express 17(19), 16820–16833 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-19-16820 . [CrossRef] [PubMed]

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