## Synthetic wavelength based phase unwrapping in spectral domain optical coherence tomography

Optics Express, Vol. 17, Issue 7, pp. 5039-5051 (2009)

http://dx.doi.org/10.1364/OE.17.005039

Acrobat PDF (1148 KB)

### Abstract

Phase sensing implementations of spectral domain optical coherence tomography (SDOCT) have demonstrated the ability to measure nanometer-scale temporal and spatial profiles of samples. However, the phase information suffers from a 2π ambiguity that limits observations of larger sample displacements to lengths less than half the source center wavelength. We introduce a synthetic wavelength phase unwrapping technique in SDOCT that uses spectral windowing and corrects the 2π ambiguity, providing accurate measurements of sample motion with information gained from standard SDOCT processing. We demonstrate this technique by using a common path implementation of SDOCT and correctly measure phase profiles from a phantom phase object and human epithelial cheek cells which produce multiple wrapping artifacts. Using a synthetic wavelength for phase unwrapping could prove useful in Doppler or other phase based implementations of OCT.

© 2009 Optical Society of America

## 1. Introduction

1. O. W. Richards, “Phase Difference Microscopy,” Nature **154**, 672 (1944). [CrossRef]

2. H. Gundlach, “Phase contrast and differential interference contrast instrumentation and applications in cell, developmental, and marine biology,” Opt. Eng. **32**, 3223–3228, (1993). [CrossRef]

3. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. **24**, 3053–3058, (1985). [CrossRef] [PubMed]

4. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital Holography for quantitative phase-contrast imaging,” Opt. Lett. **24**, 291–293, (1999). [CrossRef]

5. C. J. Mann, L. Yu, C. Lo, and M. K. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express **13**, 8693–8698, (2005). [CrossRef] [PubMed]

6. G. Popescu, L. P. Deflores, J.C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. **29**, 2503–2505, (2004). [CrossRef] [PubMed]

7. N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Quantitative phase imaging of live cells using fast Fourier phase microscopy,” Appl. Opt. **46**, 1836–1842, (2007). [CrossRef] [PubMed]

8. T. Ikeda, G. Popescu, R. R. Dasari, and M.S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. **30**, 1165–1167, (2005). [CrossRef] [PubMed]

9. G. Popescu, T. Ikeda, C. A. Best, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Erythrocyte structure and dynamics quantified by Hilbert phase microscopy,” J. Biomed. Opt. **10**, 060503 (2005). [CrossRef]

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

11. J.F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. **28**, 2067–2069, (2003). [CrossRef] [PubMed]

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

13. M. A. Choma, A. K. Ellerbee, C. Yang, T. L. Creazzo, and J. A. Izatt, “Spectral-domain phase microscopy,” Opt. Lett. **30**, 1162–1164 (2005). [CrossRef] [PubMed]

14. C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, “Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging,” Opt. Lett. **30**, 2131–2133, (2005). [CrossRef] [PubMed]

13. M. A. Choma, A. K. Ellerbee, C. Yang, T. L. Creazzo, and J. A. Izatt, “Spectral-domain phase microscopy,” Opt. Lett. **30**, 1162–1164 (2005). [CrossRef] [PubMed]

14. C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, “Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging,” Opt. Lett. **30**, 2131–2133, (2005). [CrossRef] [PubMed]

15. M. A. Choma, A. K. Ellerbee, S. Yazdanfar, and J. A. Izatt, “Doppler flow imaging of cytoplasm streaming using spectral domain phase microscopy,” J. Biomed. Opt. **11**, 024014 (2006). [CrossRef] [PubMed]

17. A. K. Ellerbee, T. L. Creazzo, and J. A. Izatt, “Investigating nanoscale cellular dynamics with cross-sectional spectral domain phase microscopy,” Opt. Express **15**, 8115 – 8124, (2007). [CrossRef] [PubMed]

20. C. R. Tilford, “Analytical procedure for determining lengths from fractional fringes,” Appl. Opt. **16**, 1857–1860 (1977). [CrossRef] [PubMed]

21. Y. Cheng and J. C. Wyant, “Two-wavelength phase shifting interferometry,” Appl. Opt. **23**, 4539–4543 (1984). [CrossRef] [PubMed]

22. J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. **28**, 1141–1143, (2003) [CrossRef] [PubMed]

23. J. Kuhn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Realtime dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express **15**, 7231–7242, (2007). [CrossRef] [PubMed]

24. S. Tamano, M. Otaka, and Y. Hayasaki, “Two-wavelength phase-shifting low-coherence digital holography,” Jpn. J. Appl. Phys. **47**, 8844–8847 (2008). [CrossRef]

25. N. Warnasooriya and M.K. Kim, “LED-based multi-wavelength phase imaging interference microscopy,” Opt. Express **15**, 9239–9247, (2007). [CrossRef] [PubMed]

26. C. Yang, A. Wax, R.R. Dasari, and M.S. Feld, “2π ambiguity-free optical distance measurement with subnanometer precision with a novel phase-crossing low-coherence interferometer,” Opt. Lett. **27**, 77–79, (2002). [CrossRef]

27. D. L. Marks, S.C. Schlachter, A.M. Zysk, and S.A. Boppart, “Group refractive index reconstruction with broadband interferometric confocal microscopy,” J. Opt. Soc. Am. A **25**, 1156–1164 (2008). [CrossRef]

_{o}/2. Image processing in OCT uses the Fourier transform of a broadband spectrum. By windowing the signal spectrum before applying the Fourier transform, phase information at multiple center wavelengths may be obtained. A similar procedure to the multi-wavelength method previously used in other phase imaging modalities may then be applied for correct phase unwrapping, except that in OCT only a single source is needed due to the large spectral bandwidth used. Such a procedure could be applied to phase based implementations of OCT in both the Fourier or time domain including Doppler, common path, or other phase based implementations.

## 2. SD-OCT dual-wavelength phase unwrapping theory

### 2.1 SD-OCT interferometric signal and phase wrapping

*S(k)*is the source spectral density,

*R*and

_{R}*R*are the reference and sample reflectivities respectively,

_{s}*n*is the index of refraction, and

*x*+

*Δx*is the distance between the reference and sample reflectors.

*x*accounts for the discrete sampling of the detector in the

*x*-domain while

*Δx*represents subresolution changes in the sample position around

*x*[15

15. M. A. Choma, A. K. Ellerbee, S. Yazdanfar, and J. A. Izatt, “Doppler flow imaging of cytoplasm streaming using spectral domain phase microscopy,” J. Biomed. Opt. **11**, 024014 (2006). [CrossRef] [PubMed]

*Δx*is the subresolution motion of the sample reflector,

*λ*is the center wavelength of the source,

_{o}*Δφ*is the phase difference between times

*t*and

_{j}*t*, and

_{o}*m*is an integer number of half wavelengths. The factor of 2 in the denominator accounts for the double pass optical path length due to the reflection geometry of the optical setup used in SDOCT.

*Δx*, but can potentially represent

*Δx*plus any integer number of half wavelengths. Without

*a priori*knowledge of the sample motion or structure, there is no way to know the exact value of

*m*in Eq. (3), as any displacement that is a multiple of

*λ*/2 will induce phase wrapping. If

_{o}*m*= ±1, simply adding or subtracting 2π to the phase can correctly unwrap the artifact. However, if |

*m*| ≤ 2, it will be impossible to unwrap the phase using this simple method. A larger

*Δx*may be correctly measured without phase wrapping if a larger

*λ*is used. This is the basis for multi-wavelength unwrapping in other phase imaging modalities and will be applied here to OCT.

_{o}### 2.2 Synthetic wavelength phase unwrapping

*λ*and

_{1}*λ*, a longer synthetic wavelength Λ may be defined as

_{2}*Δφ*, that can be calculated by the difference of the phase measurements made at each of the two single wavelengths then adding 2π to the result whenever

_{syn}*Δφ*is less than zero [22

_{syn}22. J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. **28**, 1141–1143, (2003) [CrossRef] [PubMed]

*Δφ*is the phase one would obtain had Λ been the actual illumination wavelength used and thus allows for wrap-free phase measurements of a large

_{syn}*Δx*.

*Δφ*may still suffer from phase wrapping if Λ is not chosen to be sufficiently large. It should also be noted that the noise level in the synthetic wavelength phase map suffers from noise magnification due to amplification by Λ, as is discussed in the next section.

_{syn}*λ*and

_{1}*λ*can correspond to the center wavelengths of two different subsets of the broadband source spectrum. The broadband spectrum acquired during SD-OCT imaging can be divided by applying Gaussian windows to different portions of the spectrum as shown in Fig. 1. The spectrum is first interpolated to be evenly spaced in wavenumber. The DC components of the signal are isolated by applying a Fourier transform low pass filter. The raw spectrum is then divided by the DC spectrum which leaves the interferometric term from Eq. (1). The Gaussian windows are applied to the remaining interference signal. The resulting signals have the form

_{2}*k*<

_{1}*k*being the centers of the two Gaussian windows and

_{2}*Δk*and

_{1}*Δk*as their bandwidths. Though in principle the windows may be of any desirable shape, Gaussian windows offer a convenient spectral shaping tool due to their Fourier transform properties and for their ability to suppress sidelobe artifacts, allowing for separation of closely spaced reflectors[28

_{2}28. R. Tripathi, N Nassif, J. S. Nelson, B. H. Park, and J. F. de Boer, “Spectral shaping for non-Gaussian source spectra in optical coherence tomography,” Opt. Lett. **27**, 406–408, (2002). [CrossRef]

29. D.L. Marks, P.S. Carney, and S.A. Boppart, “Adaptive spectral apodization for sidelobe reduction in optical coherence tomography images,” J. Biomed. Opt. **9**, 1281–1287 (2004). [CrossRef] [PubMed]

28. R. Tripathi, N Nassif, J. S. Nelson, B. H. Park, and J. F. de Boer, “Spectral shaping for non-Gaussian source spectra in optical coherence tomography,” Opt. Lett. **27**, 406–408, (2002). [CrossRef]

29. D.L. Marks, P.S. Carney, and S.A. Boppart, “Adaptive spectral apodization for sidelobe reduction in optical coherence tomography images,” J. Biomed. Opt. **9**, 1281–1287 (2004). [CrossRef] [PubMed]

### 2.3 Synthetic wavelength windowing and noise analysis

*δφ*, of a reflector in an acquired OCT interferogram is given by [15

15. M. A. Choma, A. K. Ellerbee, S. Yazdanfar, and J. A. Izatt, “Doppler flow imaging of cytoplasm streaming using spectral domain phase microscopy,” J. Biomed. Opt. **11**, 024014 (2006). [CrossRef] [PubMed]

*ρ(k)*is the detector responsivity,

*S(k)*is the spectral density function detected,

*Δt*is the integration time, and

*e*is the electronic charge [12

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

13. M. A. Choma, A. K. Ellerbee, C. Yang, T. L. Creazzo, and J. A. Izatt, “Spectral-domain phase microscopy,” Opt. Lett. **30**, 1162–1164 (2005). [CrossRef] [PubMed]

*m*= 0 giving

*cov(δφ*is the covariance of the phase noise between the two spectra. As a first simple model for the covariance, we assume

_{1},δφ_{2})_{1}Δk

_{2}+k

_{2}Δk

_{1})/(Δk

_{1}+Δk

_{2}) being the point at which the two windows intersect. If it is assumed that Δk

_{1}= Δk

_{2}, then as

*δφ*= 0 as

_{syn}*cov(φ*=φ

_{1}^{2},φ_{2}^{2})_{1}

^{2}+φ

_{2}

^{2}. If the two windows are greatly separated,

*a(k)*= 0, indicating that

*δφ*is equal to

_{syn}*δφ*and

_{1}*δφ*added in quadrature.

_{2}_{1}and has a bandwidth of

*Δk*. The SNR of the windowed spectrum is lower than that of the original spectrum. This effectively degrades

*δφ*, and the reduced bandwidth results in a loss of axial resolution in each of the images generated from the windowing procedure. It is desirable to use broad windows to preserve the axial resolution. If the synthetic wavelength Λ were to replace

_{syn}*λ*in Eq. (8) and

_{o}*δφ*replaced

_{syn}*δφ*, the synthetic wavelength displacement sensitivity,

*δφ*, is given as

_{syn}*δφ*also scales with Λ.

_{syn}*δφ*and

_{syn}*δx*in this analysis.

_{syn}*δφ*approaches 0 as the separation between the windows decreases because the synthetic phase is based upon the difference between the measured phases of each window. When the windows are completely overlapped, they possess complete correlation and thus their covariance is equal to the sum of their variances, which from Eq. (9) yields

_{syn}*δφ*= 0. However,

_{syn}*δx*is not zero because Λ approaches ∞ in this case. For situations where window separation is large, Λ becomes small while

_{syn}*δφ*becomes large because the SNR of the windowed spectra degrades with increasing distance from the source center. Thus Λ and

_{syn}*δφ*act to balance each other, keeping

_{syn}*δx*non-zero and finite.

_{syn}*δx*low. Figure 2 illustrates the results of this analysis. Experimental data is obtained from the surface of a glass coverslip as described in the Methods section. Equation (13) is plotted theoretically using Eq. (9) and the SNR dependent expressions to determine

_{syn}*δφ*. The measured SNR for the lower wavelength window decreased from 46.3 dB to 39.4 dB while that of the higher wavelength window decreased from 46.6 dB to 43 dB as the wavelength separation increased. Because the power of the source was less concentrated for wavelengths far from the source center, the strength of the interference fringes was also reduced at these wavelengths. Experimentally,

_{syn}*δφ*is determined by the standard deviation of the

_{syn}*δφ*measurements.

_{syn}*δx*increases (resolution degrades) as the two windows are moved very close together or very far apart. It is noted that both the independently measured displacement sensitivity and the SNR-calculated sensitivity follow the same general trend for windows that are separated by more than 20 nm. However, the theoretical model does not correctly predict the experimental trend of

_{syn}*δx*for highly overlapped windows (|λ

_{syn}_{1}- λ

_{2}| < 20 nm) as

*δφ*does not decrease fast enough to balance the rapid increase in Λ. The covariance model in Eqs. (10) and (11

_{syn}11. J.F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. **28**, 2067–2069, (2003). [CrossRef] [PubMed]

### 2.4 Noise reduction

*δφ*can be reduced to that of

_{syn}*δφ*by using the synthetic wavelength image as a reference for correctly unwrapping the single wavelength image [22

_{o}22. J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. **28**, 1141–1143, (2003) [CrossRef] [PubMed]

25. N. Warnasooriya and M.K. Kim, “LED-based multi-wavelength phase imaging interference microscopy,” Opt. Express **15**, 9239–9247, (2007). [CrossRef] [PubMed]

30. A. Khmaladze, A. Restrepo-Martínez, M.K. Kim, R. Castañeda, and A. Blandón, “Simultaneous Dual-Wavelength Reflection Digital Holography Applied to the Study of the Porous Coal Samples,” Appl. Opt. **47**, 3203–3210 (2008). [CrossRef] [PubMed]

*Δx*in Eq. (3) is determined but contains a high level of noise. Simply dividing the synthetic wavelength result by an integer number of

*λ*/2 allows for calculation of the appropriate value of

_{o}*m*. This allows the data to be recast in terms of the source center wavelength. The correctly unwrapped image has the same level of noise as is expected in the single wavelength case. However, areas of the image that possess noise levels of

*δx*>

*λ*would cause a miscalculation of

_{o}/4*m*resulting in spikes at these locations throughout the single wavelength corrected image. These spikes are within +/- 1 wrap and can be removed through a more simple unwrapping technique or filtering.

### 2.5 Algorithm summary

- Acquire raw OCT spectrum
- Interpolate data to be evenly spaced in k.
- Perform FFT on data, low pass filter, then perform iFFT to obtain DC spectrum.
- Divide interpolated spectrum by DC spectrum.
- Apply two different Gaussian windows to resulting interference fringes.
- Perform FFT separately to each of the two newly formed spectra.
- Extract phase information from each FFT.
- Subtract one phase dataset from the other and add 2π if the difference is less than zero.

*Λ*and

*Δφ*=

_{syn}*φ*into Eq. (3). Reducing the level of noise in the image can then be accomplished by dividing the resulting displacement profile by an integer number of

_{1}-φ_{2}*λ*and then adding the corresponding amount of wraps to the single wavelength data.

_{o}/2## 3. Methods

_{o}~ 790 nm Δλ ~ 70 nm) was used as the source with a measured axial resolution of 5.7 μm in air. The custom spectrometer contained a linescan CCD (Atmel, A VIVA, 2048 pixel, 19 kHz). The sample arm used two galvo mirrors to allow raster scanning of the sample. A microscope objective (Zeiss, 40x, 0.6 NA) focused light onto the sample giving a lateral resolution with a calculated diffraction limit of 1.6 μm. This objective was capable of resolving the smallest bars on a USAF test chart with 4.4 μm spacing per line pair (~ 2.2 μm per bar) in good agreement with the expected resolution.

*δφ*at the peak pixel location of the top coverslip surface in the A-scan profile.

_{syn}*δx*was calculated using the measured phase stability in Eq. (13). Theoretically, the phase stability depends on the SNR, which was calculated by taking the intensity at the A-scan peak squared divided by the variance of the noise floor over a region near the coverslip surface peak. The theoretical

_{syn}*δx*was calculated by combining Eqs. (6), (9) and (13). Figure 2 shows the results of this analysis.

_{syn}## 4. Results and discussion

31. A. K. Ellerbee and J.A. Izatt, “Phase retrieval in low-coherence interferometric microscopy,” Opt. Lett. **32**, 388–390, (2007). [CrossRef] [PubMed]

32. J.A. Izatt and M.A. Choma, “Theory of Optical Coherence Tomography,” in *Optical Coherence Tomography: Technology and Applications*,
W. Drexler and J.G. Fujimoto, eds. (Springer, 2008). [CrossRef]

_{2}steps coated with Si

_{3}N

_{4}on a silicon wafer, to obtain a dataset consisting of 100×50 A-scans covering a 20 × 6 μm area with an integration time of 150 μs. The 1.5 μm step height is roughly twice the center wavelength of the source and yet is less than its coherence length. Thus, the grating is expected to produce multiple wrapping artifacts in the phase data.

_{3}N

_{4}has a trivial complex component at 790 nm[33], implying normally incident light undergoes a π phase shift upon reflection. However, the index of refraction of silicon at 790 nm (n = 3.673 and k = 5×10

^{-3})[33] could potentially cause a non-π phase shift in reflected light. The phase shift deviation from π for light incident upon a silicon surface from air may be describe as [34

34. T. Doi, K. Toyoda, and Y. Tanimura, “Effects of phase changes on reflection and their wavelength dependence in optical profilometry,” Appl. Opt. **36**, 7157–7161, (1997). [CrossRef]

_{o}=1 is the index of air, n

_{1}=3.673 is the real component of the index of silicon, and k

_{1}= 5×10

^{-3}is its imaginary index. The value of p was calculated to be less than 1 mrad, which is much less than the phase difference caused by the calibration grating step height (approximately 400 mrad at the synthetic wavelength). Thus, the difference in materials between the peaks and valleys on the grating was not expected to affect the phase measurement significantly.

*Λ*=41 μm. The phase of the peak pixel location in an A-scan corresponding to the peak surface of the grating for each spectrum was used to calculate

*Δφ*and yielded an SNR of 32.3 dB and 33.0 dB for the lower and higher wavelength windows respectively. The step height was measured by taking the difference of the average peaks and valley of the grating as outlined in Fig. 4(b). The standard deviation over the valley indicated was 116 nm. The dual-wavelength method correctly measures an average step height of 1.51 +/- 0.14 μm. Figure 4(c) shows a cross-sectional profile of the grating comparing the synthetic wavelength result with unwrapping performed by a simple one dimensional 2π addition/subtraction algorithm from Matlab. The Matlab algorithm only measures an average step height of 70 nm due to the method’s inability to correctly detect multiple wrapping. AFM profiling of the grating (Digital Instruments 3100, 0.5Å height resolution) measured an average step height of 1.52 μm. This confirms that using a synthetic wavelength can correctly unwrap a phase profile, even in the presence of multiply wrapped phase. Using values for

_{syn}*Λ*from 16.5 μm to 94.7 μm (|λ

_{1}-λ

_{2}| = 40 nm to 8 nm, respectively) yielded similar results for the measured step height. Using a smaller

*Λ*resulted in underestimating the calculated step height, potentially due to errors in the phase measurement resulting from windowing the original spectral data far from the source center.

*Λ*to ensure correct unwrapping. This is possible because of the broadband spectrum of the source and the ability to access its phase information by the Fourier transform. The phase at a single pixel in a given phase image can be measured unambiguously regardless of the phase in the surrounding pixels.

_{o}/2 of the source to obtain an integer amount for

*m*in Eq. (3). This result was then added to the single wavelength phase map to produce an intermediate image. The result in Fig. 5(a) shows regions throughout the dataset containing spikes or regions where the phase was incorrectly determined. The image was corrected by subtracting the intermediate image from the synthetic wavelength image and then adding +/-2π to areas of the difference map in excess of |π|. However, some spikes still remained in regions of the image containing high amounts of noise[22

**28**, 1141–1143, (2003) [CrossRef] [PubMed]

_{o}/2) from the correct value and can be removed through a simpler software unwrapping method that searches for these sharp steps[30

30. A. Khmaladze, A. Restrepo-Martínez, M.K. Kim, R. Castañeda, and A. Blandón, “Simultaneous Dual-Wavelength Reflection Digital Holography Applied to the Study of the Porous Coal Samples,” Appl. Opt. **47**, 3203–3210 (2008). [CrossRef] [PubMed]

31. A. K. Ellerbee and J.A. Izatt, “Phase retrieval in low-coherence interferometric microscopy,” Opt. Lett. **32**, 388–390, (2007). [CrossRef] [PubMed]

*Λ*=20.4μm. As can be seen from the bright field microscope image in Fig. 6(c), single cells as well as a cluster of cells stacked together are present. The cell cluster should be expected to introduce multiple wrapping artifacts due to its thickness whereas the single cells may produce only a single wrap. After applying the synthetic wavelength algorithm, noise reduction to the single wavelength image was performed. Additionally, a 3×3 median filter was used to smooth the image. A clear picture of the cell height above the coverslip surface is obtained in Fig. 6(b). The heights of both single cells and the cell cluster are resolved with the regions of the original image containing either single or multiple phase wraps corrected. The standard deviation over the region indicated by the box in Fig. 6(b) was 31 nm for the synthetic wavelength image and 7 nm for the single wavelength corrected image demonstrating the effects of single wavelength noise reduction.

## 5. Conclusion

## Acknowledgments

## References and links

1. | O. W. Richards, “Phase Difference Microscopy,” Nature |

2. | H. Gundlach, “Phase contrast and differential interference contrast instrumentation and applications in cell, developmental, and marine biology,” Opt. Eng. |

3. | K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. |

4. | E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital Holography for quantitative phase-contrast imaging,” Opt. Lett. |

5. | C. J. Mann, L. Yu, C. Lo, and M. K. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express |

6. | G. Popescu, L. P. Deflores, J.C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. |

7. | N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Quantitative phase imaging of live cells using fast Fourier phase microscopy,” Appl. Opt. |

8. | T. Ikeda, G. Popescu, R. R. Dasari, and M.S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. |

9. | G. Popescu, T. Ikeda, C. A. Best, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Erythrocyte structure and dynamics quantified by Hilbert phase microscopy,” J. Biomed. Opt. |

10. | R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, “Performance of Fourier domain vs. time domain optical coherence tomography,” Opt. Express |

11. | J.F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. |

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

13. | M. A. Choma, A. K. Ellerbee, C. Yang, T. L. Creazzo, and J. A. Izatt, “Spectral-domain phase microscopy,” Opt. Lett. |

14. | C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, “Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging,” Opt. Lett. |

15. | M. A. Choma, A. K. Ellerbee, S. Yazdanfar, and J. A. Izatt, “Doppler flow imaging of cytoplasm streaming using spectral domain phase microscopy,” J. Biomed. Opt. |

16. | E. J. McDowell, A. K. Ellerbee, M. A. Choma, B. E. Applegate, and J. A. Izatt, “Spectral domain phase microscopy for local measurements of cytoskeletal rheology in single cells,” J. Biomed. Opt. |

17. | A. K. Ellerbee, T. L. Creazzo, and J. A. Izatt, “Investigating nanoscale cellular dynamics with cross-sectional spectral domain phase microscopy,” Opt. Express |

18. | D. C. Ghilglia and M. D. Pritt, |

19. | C.K. Hitzenberger, M. Sticker, R. Leitgeb, and A.F. Fercher, “Differential phase measurements in low-coherence interferometry without 2π ambiguity,” Opt. Lett. |

20. | C. R. Tilford, “Analytical procedure for determining lengths from fractional fringes,” Appl. Opt. |

21. | Y. Cheng and J. C. Wyant, “Two-wavelength phase shifting interferometry,” Appl. Opt. |

22. | J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. |

23. | J. Kuhn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Realtime dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express |

24. | S. Tamano, M. Otaka, and Y. Hayasaki, “Two-wavelength phase-shifting low-coherence digital holography,” Jpn. J. Appl. Phys. |

25. | N. Warnasooriya and M.K. Kim, “LED-based multi-wavelength phase imaging interference microscopy,” Opt. Express |

26. | C. Yang, A. Wax, R.R. Dasari, and M.S. Feld, “2π ambiguity-free optical distance measurement with subnanometer precision with a novel phase-crossing low-coherence interferometer,” Opt. Lett. |

27. | D. L. Marks, S.C. Schlachter, A.M. Zysk, and S.A. Boppart, “Group refractive index reconstruction with broadband interferometric confocal microscopy,” J. Opt. Soc. Am. A |

28. | R. Tripathi, N Nassif, J. S. Nelson, B. H. Park, and J. F. de Boer, “Spectral shaping for non-Gaussian source spectra in optical coherence tomography,” Opt. Lett. |

29. | D.L. Marks, P.S. Carney, and S.A. Boppart, “Adaptive spectral apodization for sidelobe reduction in optical coherence tomography images,” J. Biomed. Opt. |

30. | A. Khmaladze, A. Restrepo-Martínez, M.K. Kim, R. Castañeda, and A. Blandón, “Simultaneous Dual-Wavelength Reflection Digital Holography Applied to the Study of the Porous Coal Samples,” Appl. Opt. |

31. | A. K. Ellerbee and J.A. Izatt, “Phase retrieval in low-coherence interferometric microscopy,” Opt. Lett. |

32. | J.A. Izatt and M.A. Choma, “Theory of Optical Coherence Tomography,” in |

33. | D.R. Lide, ed. |

34. | T. Doi, K. Toyoda, and Y. Tanimura, “Effects of phase changes on reflection and their wavelength dependence in optical profilometry,” Appl. Opt. |

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

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

(170.4500) Medical optics and biotechnology : Optical coherence tomography

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: January 27, 2009

Revised Manuscript: March 7, 2009

Manuscript Accepted: March 10, 2009

Published: March 16, 2009

**Virtual Issues**

Vol. 4, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Hansford C. Hendargo, Mingtao Zhao, Neal Shepherd, and Joseph A. Izatt, "Synthetic wavelength based phase unwrapping in spectral domain optical coherence tomography," Opt. Express **17**, 5039-5051 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-5039

Sort: Year | Journal | Reset

### References

- O. W. Richards, "Phase Difference Microscopy," Nature 154, 672 (1944). [CrossRef]
- H. Gundlach, "Phase contrast and differential interference contrast instrumentation and applications in cell, developmental, and marine biology," Opt. Eng. 32, 3223-3228, (1993). [CrossRef]
- K. Creath, "Phase-shifting speckle interferometry," Appl. Opt. 24, 3053-3058, (1985). [CrossRef] [PubMed]
- E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital Holography for quantitative phase-contrast imaging," Opt. Lett. 24, 291-293, (1999). [CrossRef]
- C. J. Mann, L. Yu, C. Lo, and M. K. Kim, "High-resolution quantitative phase-contrast microscopy by digital holography," Opt. Express 13, 8693-8698, (2005). [CrossRef] [PubMed]
- G. Popescu, L. P. Deflores, J.C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, Fourier phase microscopy for investigation of biological structures and dynamics," Opt. Lett. 29, 2503-2505, (2004). [CrossRef] [PubMed]
- N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Quantitative phase imaging of live cells using fast Fourier phase microscopy," Appl. Opt. 46, 1836-1842, (2007). [CrossRef] [PubMed]
- T. Ikeda, G. Popescu, R. R. Dasari, and M.S. Feld, "Hilbert phase microscopy for investigating fast dynamics in transparent systems," Opt. Lett. 30, 1165-1167, (2005). [CrossRef] [PubMed]
- G. Popescu, T. Ikeda, C. A. Best, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Erythrocyte structure and dynamics quantified by Hilbert phase microscopy," J. Biomed. Opt. 10, 060503 (2005). [CrossRef]
- R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, "Performance of Fourier domain vs. time domain optical coherence tomography," Opt. Express 11, 889-894, (2003). [CrossRef] [PubMed]
- J.F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, "Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography," Opt. Lett. 28, 2067-2069, (2003). [CrossRef] [PubMed]
- M. A. Choma, M. V. Sarunic, C. Yang, and J. A. Izatt, "Sensitivity advantage of swept source and Fourier domain optical coherence tomography," Opt. Express 11, 2183-2189 (2003). [CrossRef] [PubMed]
- M. A. Choma, A. K. Ellerbee, C. Yang, T. L. Creazzo, and J. A. Izatt, "Spectral-domain phase microscopy," Opt. Lett. 30,1162-1164 (2005). [CrossRef] [PubMed]
- C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, "Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging," Opt. Lett. 30, 2131-2133, (2005). [CrossRef] [PubMed]
- M. A. Choma, A. K. Ellerbee, S. Yazdanfar, and J. A. Izatt, "Doppler flow imaging of cytoplasm streaming using spectral domain phase microscopy," J. Biomed. Opt. 11,024014 (2006). [CrossRef] [PubMed]
- Q1. E. J. McDowell, A. K. Ellerbee, M. A. Choma, B. E. Applegate, and J. A. Izatt, "Spectral domain phase microscopy for local measurements of cytoskeletal rheology in single cells," J. Biomed. Opt.04400, (2007).
- A. K. Ellerbee, T. L. Creazzo, and J. A. Izatt, "Investigating nanoscale cellular dynamics with cross-sectional spectral domain phase microscopy," Opt. Express 15, 8115 - 8124, (2007). [CrossRef] [PubMed]
- D. C. Ghilglia and M. D. Pritt, Two Dimensional Phase Unwrapping: Theory, Algorithms, and Software. (Wiley, 1998)
- C.K. Hitzenberger, M. Sticker, R. Leitgeb, and A.F. Fercher, "Differential phase measurements in low-coherence interferometry without 2π ambiguity," Opt. Lett. 26, 1864-1866, (2001). [CrossRef]
- C. R. Tilford, "Analytical procedure for determining lengths from fractional fringes," Appl. Opt. 16, 1857-1860 (1977). [CrossRef] [PubMed]
- Y. Cheng and J. C. Wyant, "Two-wavelength phase shifting interferometry," Appl. Opt. 23,4539-4543 (1984). [CrossRef] [PubMed]
- J. Gass, A. Dakoff, and M. K. Kim, "Phase imaging without 2π ambiguity by multiwavelength digital holography," Opt. Lett. 28, 1141-1143, (2003) [CrossRef] [PubMed]
- J. Kuhn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, "Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition," Opt. Express 15, 7231-7242, (2007). [CrossRef] [PubMed]
- S. Tamano, M. Otaka, Y. Hayasaki, "Two-wavelength phase-shifting low-coherence digital holography," Jpn. J. Appl. Phys. 47, 8844-8847 (2008). [CrossRef]
- N. Warnasooriya and M.K. Kim, "LED-based multi-wavelength phase imaging interference microscopy," Opt. Express 15, 9239-9247, (2007). [CrossRef] [PubMed]
- C. Yang, A. Wax, R.R. Dasari, and M.S. Feld, "2π ambiguity-free optical distance measurement with subnanometer precision with a novel phase-crossing low-coherence interferometer," Opt. Lett. 27, 77-79, (2002). [CrossRef]
- D. L. Marks, S.C. Schlachter, A.M. Zysk, and S.A. Boppart, "Group refractive index reconstruction with broadband interferometric confocal microscopy," J. Opt. Soc. Am. A 25, 1156-1164 (2008). [CrossRef]
- R. Tripathi, N Nassif, J. S. Nelson, B. H. Park, and J. F. de Boer, "Spectral shaping for non-Gaussian source spectra in optical coherence tomography," Opt. Lett. 27, 406-408, (2002). [CrossRef]
- D.L. Marks, P.S. Carney, and S.A. Boppart, "Adaptive spectral apodization for sidelobe reduction in optical coherence tomography images," J. Biomed. Opt. 9, 1281-1287 (2004). [CrossRef] [PubMed]
- A. Khmaladze, A. Restrepo-Martínez, M.K. Kim, R. Castañeda, and A. Blandón, "Simultaneous Dual-Wavelength Reflection Digital Holography Applied to the Study of the Porous Coal Samples," Appl. Opt. 47, 3203-3210 (2008). [CrossRef] [PubMed]
- A. K. Ellerbee and J.A. Izatt, "Phase retrieval in low-coherence interferometric microscopy," Opt. Lett. 32, 388-390, (2007). [CrossRef] [PubMed]
- J.A. Izatt and M.A. Choma, "Theory of Optical Coherence Tomography," in Optical Coherence Tomography: Technology and Applications, W. Drexler and J.G. Fujimoto, eds. (Springer, 2008). [CrossRef]
- D.R. Lide, ed. CRC Handbook of Chemistry and Physics, (CRC Press, 2001-2002).
- T. Doi, K. Toyoda, and Y. Tanimura, "Effects of phase changes on reflection and their wavelength dependence in optical profilometry," Appl. Opt. 36, 7157-7161, (1997). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.