## Dispersion compensation in Fourier domain optical coherence tomography using the fractional Fourier transform |

Optics Express, Vol. 20, Issue 21, pp. 23398-23413 (2012)

http://dx.doi.org/10.1364/OE.20.023398

Acrobat PDF (3152 KB)

### Abstract

We address numerical dispersion compensation based on the use of the fractional Fourier transform (FrFT). The FrFT provides a new fundamental perspective on the nature and role of group-velocity dispersion in Fourier domain OCT. The dispersion induced by a 26 mm long water cell was compensated for a spectral bandwidth of 110 nm, allowing the theoretical axial resolution in air of 3.6 μm to be recovered from the dispersion degraded point spread function. Additionally, we present a new approach for depth dependent dispersion compensation based on numerical simulations. Finally, we show how the optimized fractional Fourier transform order parameter can be used to extract the group velocity dispersion coefficient of a material.

© 2012 OSA

## 1. Introduction

1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science **254**, 1178–1181 (1991). [CrossRef] [PubMed]

2. A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. **117**, 43–48 (1995). [CrossRef]

3. 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]

5. 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]

6. G. Häusler and M. W. Lindner, “Coherence radar and spectral radar — New tools for dermatological diagnosis,” J. Biomed. Opt. **3**, 21–31 (1998). [CrossRef]

8. N. A. Nassif, B. Cense, B. H. Park, M. C. Pierce, S. H. Yun, B. E. Bouma, G. J. Tearney, T. C. Chen, and J. F. de Boer, “*In vivo* high-resolution video-rate spectral-domain optical coherence tomography of the human retina and optic nerve,” Opt. Express **12**, 367–376 (2004). [CrossRef] [PubMed]

9. S. R. Chinn, E. A. Swanson, and J. G. Fujimoto, “Optical coherence tomography using a frequency-tunable optical source,” Opt. Lett. **22**, 340–342 (1997). [CrossRef] [PubMed]

12. R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography,” Opt. Express **14**, 3225–3237 (2006). [CrossRef] [PubMed]

13. L. Liu, J. A. Gardecki, S. K. Nadkarni, J. D. Toussaint, Y. Yagi, B. E. Bouma, and G. J. Tearney, “Imaging the subcellular structure of human coronary atherosclerosis using microoptical coherence tomography,” Nature Med. **17**, 1010–1014 (2011). [CrossRef] [PubMed]

14. R. A. Leitgeb, M. Villiger, A. H. Bachmann, L. Steinmann, and T. Lasser, “Extended focus depth for Fourier domain optical coherence microscopy” Opt. Lett. **31**, 2450–2452 (2006). [CrossRef] [PubMed]

16. C. Blatter, B. Grajciar, C. M. Eigenwillig, W. Wieser, B. R. Biedermann, R. Huber, and R. A. Leitgeb, “Extended focus high-speed swept source OCT with self-reconstructive illumination,” Opt. Express **19**, 12141–12155 (2011). [CrossRef] [PubMed]

17. L. Liu, F. Diaz, L. Wang, B. Loiseaux, J.-P. Huignard, C. J. R. Sheppard, and N. Chen, “Superresolution along extended depth of focus with binary-phase filters for the Gaussian beam,” J. Opt. Soc. Am. A **25**, 2095–2101 (2008). [CrossRef]

18. J. Holmes, S. Hattersley, N. Stone, F. Bazant-Hegemark, and H. Barr, “Multi-channel Fourier domain OCT system with superior lateral resolution for biomedical applications,” Proc. of SPIE **6847**68470O (2008). [CrossRef]

19. B. A. Standish, K. K. C. Lee, A. Mariampillai, N. R. Munce, M. K. K. Leung, V. X. D. Yang, and I. A. Vitkin, “*In vivo* endoscopic multi-beam optical coherence tomography,” Phys. Med. Biol. **55**, 615–622 (2010). [CrossRef] [PubMed]

20. C. K. Hitzenberger, A. Baumgartner, W. Drexler, and A. F. Fercher, “Dispersion effects in partial coherence interferometry: implications for intraocular ranging,” J. of Biomed. Opt. , 144–151 (1999). [CrossRef]

21. A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Comm. **204**, 67–74 (2002). [CrossRef]

*λ*

_{0}= 1 μm [22

22. A. G. Van Engen, S. A. Diddams, and T. S. Clement, “Dispersion measurements of water with white-light interferometry,” Appl. Opt. **37**, 5679–5686 (1998). [CrossRef]

23. B. Bouma, G. J. Tearney, S. A. Boppart, M. R. Hee, M. E. Brezinski, and J. G. Fujimoto, “High-resolution optical coherence tomographic imaging using a mode-locked Ti:Al_{2}O_{3} laser source,” Opt. Lett. **20**, 1486–1488 (1995). [CrossRef] [PubMed]

25. W. Drexler, U. Morgner, R. K. Ghanta, F. X. Krtner, J. S. Schuman, and J. G. Fujimoto, “Ultrahighresolution ophthalmic optical coherence tomography,” Nature Medicine **7**, 502–507 (2001). [CrossRef] [PubMed]

26. G. J. Tearney, B. E. Bouma, and J. G. Fujimoto, “High-speed phase- and group-delay scanning with a grating-based phase control delay line,” Opt. Lett. **22**, 1811–1813 (1997). [CrossRef]

27. S. Iyer, S. Coen, and F. Vanholsbeeck, “Dual-fiber stretcher as a tunable dispersion compensator for an all-fiber optical coherence tomography system,” Opt. Lett. **34**, 2903–2905 (2009). [CrossRef] [PubMed]

28. L. Froehly, S. Iyer, and F. Vanholsbeeck, “Dual-fibre stretcher and coma as tools for independent 2nd and 3rd order tunable dispersion compensation in a fibre-based ‘scan-free’ time domain optical coherence tomography system,” Opt. Commun. **284**, 4099–4106 (2011). [CrossRef]

21. A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Comm. **204**, 67–74 (2002). [CrossRef]

29. A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Numerical dispersion compensation for Partial Coherence Interferometry and Optical Coherence Tomography,” Opt. Express **9**, 610–615 (2001). [CrossRef] [PubMed]

31. D. L. Marks, A. L. Oldenburg, J. J. Reynolds, and S. A. Boppart, “Digital algorithm for dispersion correction in optical coherence tomography for homogeneous and stratified media,” Appl. Opt. **42**, 204–217 (2003). [CrossRef] [PubMed]

32. B. Cense, N. A. Nassif, T. C. Chen, M. C. Pierce, S.-H. Yun, B. H. Park, B. E. Bouma, G. J. Tearney, and J. F. de Boer, “Ultrahigh-resolution high-speed retinal imaging using spectral-domain optical coherence tomography,” Opt. Express **12**, 2435–2447 (2004). [CrossRef] [PubMed]

33. M. Wojtkowski, V. J. Srinivasan, T. H. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express **12**, 2404–2422 (2004). [CrossRef] [PubMed]

34. B. Liu, E. A. Macdonald, D. L. Stamper, and M. E. Brezinski, “Group velocity dispersion effects with water and lipid in 1.3 μm optical coherence tomography system,” Phys. Med. Biol. **49**, 923–930 (2004). [CrossRef] [PubMed]

35. J. Liebermann, C. Brckner, B. Grajciar, J. Haueisen, and A. F. Fercher, “Dual-band refractive Low Coherence Interferometry in the spectral domain for dispersion measurements,” Proc. of SPIE **7889**, 788922 (2011). [CrossRef]

*β*

_{2}of a sample from the order parameter of the fractional Fourier transform. All of our techniques are general and can be applied to SD as well as SS OCT, but also more generally to interferometry or optical coherence domain reflectometry.

## 2. Theory

*a*-th order FrFT is a linear transform. For the range 0 < |

*a*| < 2 it is defined as [40

40. H. M. Ozaktas, O. Arıkan, M. A. Kutay, and G. Bozdağı, “Digital computation of the fractional Fourier transform,” IEEE Trans. Sig. Proc. **44**, 2141–2150 (1996). [CrossRef]

*u*is the independent variable of the transform input function while

*u*is that of the transform output, with corresponding fractional transform order

_{a}*a*. These independent variables are assumed to be dimensionless (see below).

*a*= 1,

*A*becomes unity, the first and third terms in the argument of the exponential in Eq. (1) vanish, and the FrFT reduces to the traditional FT (i.e., the FrFT to the power of 1). Therefore, the

_{ϕ}*u*

_{1}and

*u*axes represent the (normalized) time

*τ*and optical frequency

*ν*axes, respectively (the normalization is such that

*uu*

_{1}=

*ντ*). Note that, in our work, these domains are swapped compared to traditional terminology because we deal here with FD OCT, i.e., the measurements

*f*(

*u*) are performed in the frequency domain.

*a*→ 0 and

*a*→ ±2 the integral kernel approaches

*δ*(

*u*−

_{a}*u*) and

*δ*(

*u*+

_{a}*u*), sampling

*f*(

*u*) as the identity and parity operators, respectively [40

40. H. M. Ozaktas, O. Arıkan, M. A. Kutay, and G. Bozdağı, “Digital computation of the fractional Fourier transform,” IEEE Trans. Sig. Proc. **44**, 2141–2150 (1996). [CrossRef]

*a*= 1 does the transform of a real function yield mirrored counterparts in the two halves of the Fourier space. Otherwise the transform yields different energy distributions in the two (fractional) Fourier half spaces. Except for the special cases

*a*= 0 and

*a*= 1, the transform output lies in neither the traditional frequency nor time domain.

*τ*from the point of zero-path difference of the interferometer, with

*β*

_{2}the group-velocity dispersion coefficient of the dispersive element of optical thickness

*l*. Here

*ω*= 2

*πν*is the angular optical frequency, and

*ω*

_{c}the center frequency of the source spectrum

*I*(

*ω*). After normalization,

*u*=

*ν*/

*η*,

*u*

_{1}=

*ητ*, Eq. (5) reads For digital computation, the normalization factor

*η*is set as [40

40. H. M. Ozaktas, O. Arıkan, M. A. Kutay, and G. Bozdağı, “Digital computation of the fractional Fourier transform,” IEEE Trans. Sig. Proc. **44**, 2141–2150 (1996). [CrossRef]

*ν*and Δ

*τ*are the width of the spectral and temporal intervals over which our signal is represented, respectively,

*dν*= Δ

*ν*/

*N*= 1/Δ

*τ*is the spectral resolution, and

*N*is the number of sampling points. With this scaling, the normalized length of both intervals are equal to the dimensionless quantity

*a*can be chosen to adjust the chirp rate of a dispersed OCT signal. This amounts to a rotation of the time-frequency plane. In the optimally rotated frame, the instantaneous frequency of the spectral fringes remains constant. The fractional Fourier transformed OCT signal only results from a different projection angle in the time-frequency distribution compared to the standard FT, while entirely preserving the energy of the original spectrum. With this in mind, one could state that group-velocity dispersion does not degrade the axial resolution in FD OCT but only causes one to observe an OCT depth signal from a perspective in the time-frequency plane that is not suitable for imaging. The FrFT allows one to correct for that perspective.

*a*that leads to a dispersion-compensated OCT depth signal can be interpreted as an intuitive measure of the amount of chromatic dispersion present in an FD OCT system. If the optimized order is

*a*= 1, the OCT A-scan is dispersion free. Normal and anomalous dispersion can easily be distinguished by

*a*> 1 and

*a*< 1, respectively.

## 3. Experimental setup

41. C. Dorrer, N. Belabas, J. P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B **17**, 1795–1802 (2000). [CrossRef]

42. M. Mujat, B. H. Park, B. Cense, T. C. Chen, and J. F. de Boer, “Autocalibration of spectral-domain optical coherence tomography spectrometers for in vivo quantitative retinal nerve fiber layer birefringence determination,” J. Biomed. Opt. **12**(4), 041205 (2007). [CrossRef] [PubMed]

*λ*

_{FWHM}= 110 nm. The theoretical axial resolution, calculated by inverse Fourier transforming the source spectrum (Wiener-Khinchin theorem), is 3.6 μm FWHM. Although care was taken to equalize the fiber lengths between the two arms of the interferometer, dispersion was not completely balanced in the initial setup. Coarse dispersion compensation was done physically by inserting two BK7 microscope slides into the reference arm, which led to an actual axial resolution of 3.8 μm that was close to the theoretical minimum. Dispersion may be matched more precisely e.g. by dispersion compensating prisms of the correct material. The corresponding point-spread-function (PSF), obtained by using a mirror in place of the sample, is plotted as the dashed red curve in Fig. 3. Note that here the measured spectrum has been processed with the traditional FT to obtain the PSF. Figure 3 also shows the theoretical PSF (solid blue curve) for comparison. The difference can be explained by the fact that the two microscope slides do not exactly compensate the residual setup dispersion.

## 4. Results

### 4.1. Point-spread function measurements

*a*

_{opt}= 1.0555, one obtains the OCT depth scan while simultaneously fully compensating for group velocity dispersion, as revealed by the dashed black curve in Fig. 4(a). Using the FrFT leads to an axial resolution of 3.65 μm, closer to the theoretical minimum than that observed with coarse physical dispersion compensation before introducing the water cell (3.8 μm). This clearly highlights that the FrFT can efficiently compensate dispersion with continuous adjustment capabilities.

*a*

_{opt}was found by looking for the FrFT order

*a*for which the peak intensity of the PSF is maximized. Figure 4(b) presents a graph of the PSF peak intensity versus

*a*, which can be interpreted as a sharpness function. The peak intensity obtained with the optimized FrFT order is approximately 3.3 times higher than that obtained with the traditional Fourier transform (i.e., without dispersion compensation). This optimization only needs to be done occasionally and subsequent images can be analyzed using the same order value. Note that during in-vivo imaging it can be challenging to find a reference-interface that is suitable for optimization. For retinal imaging, it has been suggested to use the center of the fovea (foveal umbo) for this purpose [32

32. B. Cense, N. A. Nassif, T. C. Chen, M. C. Pierce, S.-H. Yun, B. H. Park, B. E. Bouma, G. J. Tearney, and J. F. de Boer, “Ultrahigh-resolution high-speed retinal imaging using spectral-domain optical coherence tomography,” Opt. Express **12**, 2435–2447 (2004). [CrossRef] [PubMed]

*Ŝ*(

*u*) is the Hilbert transform of the acquired spectrum

*S*(

*u*), and

*F*

^{−1}represents the traditional inverse Fourier transform. To understand this expression, we need to recall that the Wigner transform of a real signal

*S*(

*u*) is symmetric with respect to the transformed axis,

*W*(

_{S}*u*,

*u*

_{1}) =

*W*(

_{S}*u*, −

*u*

_{1}). Upon projection on the optimal rotated

*u*axis [which amounts to perform the FrFT, see Eq. (4)], out of the two mirror parts of the Wigner transform only one leads to a dispersion compensated response while the other leads to a broadened response in the other half of the transformed domain (not shown in Fig. 1 for simplicity). The broadened part of the response is in essence affected by twice the amount of dispersion present. It is the equivalent of the complex conjugate term encountered while using the traditional FT on a real signal. We will refer to it as such, although strictly speaking it is not, in the general case (

_{a}*a*≠ 1), the mirror image of the other half of the fractional Fourier domain. This “complex conjuguate” term needs to be eliminated or it will distort the retrieved spectral phase after inverse Fourier transform. We solve this problem by starting from the analytic signal.

### 4.2. Imaging

*a*

_{opt}. Instead, we have looked for the maximum of the Radon transform of the spectrogram of the full complex depth signal of one random A-scan. This Radon transform plot is shown in Fig. 8(a). Recall that the Radon transform of a two-dimensional distribution consists of a set of integral projections for various projection angles

*δ*[43]. In essence, this procedure is therefore similar to taking the FrFT of the signal for a range of order parameter

*a*[which would correspond to the Radon transform of the Wigner distribution, see Eq. (4)] where we can relate the projection angle

*δ*and the order parameter

*a*through

*δ*=

*aπ*/2. The important difference is that here we used the

*spectrogram*of the signal rather than the

*Wigner distribution*. The spectrogram is another time-frequency distribution that employs a windowed FT of the complex signal, i.e., the short-time FT of the complex signal. It has the advantage of not exhibiting cross-terms, in contrast to the Wigner distribution, because it is phase insensitive. Consequently, the point of the Radon transform where the energy converges determines the optimized order parameter. For our grape image, a quick examination of Fig. 8(a) suggests an average optimized FrFT order

*a*

_{opt}= 1.04 highlighted by the green line. It was used for all A-scans of the tomographic B-scan shown in Fig. 8(c). In contrast, the orange line in Fig. 8(a) represents the traditional FT which leads to Fig. 8(b). Note that the grape was imaged from the bottom through a microscope cover slide and that the front interface of that slide was positioned ahead of the point of zero path difference of the interferometer in order to minimize the depth dependent sensitivity fall-off. The intensity maximum at approximately 2000 Radon bins and FrFT order

*a*= 0.96 therefore corresponds to the complex conjugate term of the front interface of the glass slide. Its intensity maximum is at FrFT order 2 −

*a*

_{opt}as it has opposite dispersion and therefore experiences twice the amount of dispersion after compensation compared to the traditional FT.

*et al.*[40

**44**, 2141–2150 (1996). [CrossRef]

## 5. Depth-dependent sample dispersion compensation

*σ*

_{PSF}due to a dispersive element of thickness

*l*and group velocity dispersion coefficient

*β*

_{2}is given by Applying this relation to the parameters of the source used in our experiments (see Section 3) reveals that sample dispersion is negligible in our case: a 1 mm thick slide of BK7 glass (

*β*

_{2}= 41 ps

^{2}/km) would only broaden the PSF by a factor of 1.2. Given the limitations of our equipment, we have therefore been unable to investigate experimentally depth-dependent dispersion compensation at this stage. However, in order to demonstrate that the FrFT can be used to handle this problem, and for the completeness of this article, we present below a proof-of-principle demonstration of such capability based on numerical simulations. In all these simulations, we consider a source bandwidth Δ

*λ*

_{FWHM}= 210 nm at

*λ*

_{0}= 710 nm center wavelength.

*β*

_{2}= 54 ps

^{2}/km. In Fig. 9(a), we have plotted the sharpness function [refer to Fig. 4(b)] at each interface, i.e., the intensity of the OCT signal at those depths, versus the FrFT order. Because dispersion gradually increases across the sample, choosing the FrFT order to compensate dispersion and maximize the PSF at a certain depth means the signals simultaneously obtained at the other depths are not optimal. For completeness, Fig. 9(b) shows the optimum FrFT order required for dispersion compensation at each depth.

*β*

_{2}linearly increasing with depth from 54 ps

^{2}/km to 97 ps

^{2}/km, followed by a second layer with uniform anomalous dispersion,

*β*

_{2}= −38 ps

^{2}/km. Here the optimized cross-section is made up of a quadratic and a linear section, which leads to the optimized A-scan plotted in Fig. 10(c). For comparison, Fig. 10(b) is the less satisfactory result obtained with the traditional FT.

*a*= 1.04. Only one maximum, on the left, is excluded, but we have identified this as an artefact resulting from a spurious reflection in our OCT setup. It explains the thin horizontal line visible in both Fig. 8(b) and Fig. 8(c), which always appear at the same depth. We must also point out that third-order dispersion is too small to have an impact on depth dependent dispersion compensation and can therefore be neglected up to an axial resolution of 2 μm, for a center wavelength in the normal dispersion regime of water (

*λ*

_{0}< 1 μm) [44

44. T. R. Hillman and D. D. Sampson, “The effect of water dispersion and absorption on axial resolution in ultrahigh-resolution optical coherence tomography,” Opt. Express **13**, 1860–1874 (2005). [CrossRef] [PubMed]

## 6. Group velocity dispersion measurement using FrFT

34. B. Liu, E. A. Macdonald, D. L. Stamper, and M. E. Brezinski, “Group velocity dispersion effects with water and lipid in 1.3 μm optical coherence tomography system,” Phys. Med. Biol. **49**, 923–930 (2004). [CrossRef] [PubMed]

*β*

_{2}. We show below that our FrFT-based dispersion compensation routine makes such information readily available. Note that the experimental estimation of the dispersion coefficient

*β*

_{2}of a material is not only of interest for imaging, but has also applications in many other fields such as in fiber optics or in non-linear optics.

*β*

_{2}and the FrFT order required for optimal dispersion compensation,

*a*

_{opt}. In particular, the modulation rate of the spectral fringes of the OCT signal [see Eq. (8)] can be readily extracted from Fig. 1 without the need of additional phase analysis: where

*ϕ*=

*a*

_{opt}

*π*/2 defines the orientation of the optimal fractional Fourier domain. Solving for the group-velocity dispersion coefficient, and with

*dω*= 2

*πdν*is the angular frequency spacing between the samples of the OCT spectral signal. The above equation clearly shows that the absolute value of

*β*

_{2}, including its sign, can be extracted once the optimal FrFT order for dispersion compensation has been obtained.

*lβ*

_{2})

^{(setup)}, using Eq. (12) without the sample-under-test, by optimizing the point spread function with only a mirror in the sample arm. This was then subtracted from the corresponding measurement with the sample present (

*lβ*

_{2})

^{(total)}and normalized to the sample length to yield the dispersion coefficient

*β*

_{2}of the sample itself, The results are shown in Table 1, and are in excellent agreement with reference values. The measured fiber dispersion agrees with a measurement that was obtained by traditional white light interferometry [27

27. S. Iyer, S. Coen, and F. Vanholsbeeck, “Dual-fiber stretcher as a tunable dispersion compensator for an all-fiber optical coherence tomography system,” Opt. Lett. **34**, 2903–2905 (2009). [CrossRef] [PubMed]

^{2}/km at 840 nm. The table also shows the optimal FrFT orders that were obtained first without then with the sample, the former value being reminiscent of the residual uncompensated dispersion of the setup. Note how the residual setup dispersion is anomalous,

*l*=

*l*̄/

*n*, where

*n*may be assumed as 1.33 and

*l*̄ is the layer thickness obtained from the OCT depth scan.

## 7. Conclusion

## Acknowledgments

## References and links

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24. | W. Drexler, U. Morgner, F. X. Kartner, C. Pitris, S. A. Boppart, X. D. Li, E. P. Ippen, and J. G. Fujimoto, “In vivo ultrahigh-resolution optical coherence tomography,” Opt. Lett. |

25. | W. Drexler, U. Morgner, R. K. Ghanta, F. X. Krtner, J. S. Schuman, and J. G. Fujimoto, “Ultrahighresolution ophthalmic optical coherence tomography,” Nature Medicine |

26. | G. J. Tearney, B. E. Bouma, and J. G. Fujimoto, “High-speed phase- and group-delay scanning with a grating-based phase control delay line,” Opt. Lett. |

27. | S. Iyer, S. Coen, and F. Vanholsbeeck, “Dual-fiber stretcher as a tunable dispersion compensator for an all-fiber optical coherence tomography system,” Opt. Lett. |

28. | L. Froehly, S. Iyer, and F. Vanholsbeeck, “Dual-fibre stretcher and coma as tools for independent 2nd and 3rd order tunable dispersion compensation in a fibre-based ‘scan-free’ time domain optical coherence tomography system,” Opt. Commun. |

29. | A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Numerical dispersion compensation for Partial Coherence Interferometry and Optical Coherence Tomography,” Opt. Express |

30. | J. F. de Boer, C. E. Saxer, and J. S. Nelson, “Stable carrier generation and phase-resolved digital data processing in optical coherence tomography,” Appl. Opt. |

31. | D. L. Marks, A. L. Oldenburg, J. J. Reynolds, and S. A. Boppart, “Digital algorithm for dispersion correction in optical coherence tomography for homogeneous and stratified media,” Appl. Opt. |

32. | B. Cense, N. A. Nassif, T. C. Chen, M. C. Pierce, S.-H. Yun, B. H. Park, B. E. Bouma, G. J. Tearney, and J. F. de Boer, “Ultrahigh-resolution high-speed retinal imaging using spectral-domain optical coherence tomography,” Opt. Express |

33. | M. Wojtkowski, V. J. Srinivasan, T. H. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express |

34. | B. Liu, E. A. Macdonald, D. L. Stamper, and M. E. Brezinski, “Group velocity dispersion effects with water and lipid in 1.3 μm optical coherence tomography system,” Phys. Med. Biol. |

35. | J. Liebermann, C. Brckner, B. Grajciar, J. Haueisen, and A. F. Fercher, “Dual-band refractive Low Coherence Interferometry in the spectral domain for dispersion measurements,” Proc. of SPIE |

36. | L. Cohen, “Time-frequency distributions — A review,” Proc. of the IEEE |

37. | V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” IMA J. Appl. Math. |

38. | D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. |

39. | L. Durak and S. Aldirmaz, “Adaptive fractional Fourier domain filtering,” Sig. Proc. |

40. | H. M. Ozaktas, O. Arıkan, M. A. Kutay, and G. Bozdağı, “Digital computation of the fractional Fourier transform,” IEEE Trans. Sig. Proc. |

41. | C. Dorrer, N. Belabas, J. P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B |

42. | M. Mujat, B. H. Park, B. Cense, T. C. Chen, and J. F. de Boer, “Autocalibration of spectral-domain optical coherence tomography spectrometers for in vivo quantitative retinal nerve fiber layer birefringence determination,” J. Biomed. Opt. |

43. | A. C. Kak and M. Slaney, |

44. | T. R. Hillman and D. D. Sampson, “The effect of water dispersion and absorption on axial resolution in ultrahigh-resolution optical coherence tomography,” Opt. Express |

**OCIS Codes**

(110.4500) Imaging systems : Optical coherence tomography

(170.3890) Medical optics and biotechnology : Medical optics instrumentation

(070.2575) Fourier optics and signal processing : Fractional Fourier transforms

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: July 3, 2012

Revised Manuscript: September 6, 2012

Manuscript Accepted: September 14, 2012

Published: September 26, 2012

**Citation**

Norman Lippok, Stéphane Coen, Poul Nielsen, and Frédérique Vanholsbeeck, "Dispersion compensation in Fourier domain optical coherence tomography using the fractional Fourier transform," Opt. Express **20**, 23398-23413 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-21-23398

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