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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 21 — Oct. 8, 2012
  • pp: 23398–23413
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Dispersion compensation in Fourier domain optical coherence tomography using the fractional Fourier transform

Norman Lippok, Stéphane Coen, Poul Nielsen, and Frédérique Vanholsbeeck  »View Author Affiliations


Optics Express, Vol. 20, Issue 21, pp. 23398-23413 (2012)
http://dx.doi.org/10.1364/OE.20.023398


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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

Developed in the early 1990’s, optical coherence tomography (OCT) is an established and successful imaging technology that enables high resolution, cross-sectional imaging of biological tissues and materials [1

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

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]

]. Nowadays, it is commonly implemented in the frequency domain with so-called Fourier-domain OCT (FD OCT), which has significantly improved sensitivity and imaging speed [3

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

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]

]. Frequency domain systems can be implemented either by using a low coherent light source and a spectrometer (SD OCT) [6

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

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]

], or by using a coherent swept-source laser (SS OCT) that is sampled by a photo diode [9

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

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]

].

OCT imaging has proven to be a powerful diagnostic tool in many medical fields. In some systems, OCT tomograms have image quality comparable to histology with resolutions down to the 1 μm level [13

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]

]. Obtaining high isotropic resolution over a large imaging depth range is however hindered by two factors. First, the high NA objectives traditionally used to obtain high lateral resolution limit the depth of field and hence the imaging depth. This can be circumvented by using axicons [14

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

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]

], binary-phase filters [17

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]

], or multi-beam OCT [18

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 684768470O (2008). [CrossRef]

, 19

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]

]. The second problem arises from the broadband nature of the light source. Increasing the axial resolution requires larger optical bandwidths, which make OCT systems more sensitive to chromatic dispersion, especially for a long sensing range. For example, using a center wavelength around 800nm, imaging the human macula with an axial resolution below 15 μm makes dispersion compensation necessary in order to account for the dispersion produced by the vitreous body of the human eye [20

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

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]

]. In ophthalmology, it is therefore advantageous to operate close to the zero dispersion wavelength of water, λ0 = 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]

].

With this in mind, it is no surprise that dispersion compensation methods have become increasingly important for OCT. Traditional methods rely on placing the right amount of dispersion balancing material in one interferometer arm of the OCT setup [23

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:Al2O3 laser source,” Opt. Lett. 20, 1486–1488 (1995). [CrossRef] [PubMed]

25

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]

], but this is usually only practical for 2nd-order dispersion. Grating-based phase delay scanners [26

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]

] and dual optical fiber stretchers [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]

] can also be used for 2nd order dispersion compensation and present some degree of tunability. However, these approaches require bulky components and the latter is difficult to adapt for depth-dependent compensation of sample dispersion. We note that a fiber-stretching-based dispersion compensator has been recently combined with a grating-based, scanning free, time domain OCT system to compensate for both 2nd and 3rd-order dispersion [28

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]

], but to the detriment of added complexity.

Because of these drawbacks, OCT systems increasingly rely on numerical dispersion compensation techniques, which offer continuous adjustment capabilities and, in theory, can be optimized for any amount of dispersion. This was first demonstrated in time-domain systems [21

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

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

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]

]. It is however more readily implemented with FD OCT systems which have direct access to the phase information of the signal as was originally demonstrated almost simultaneously by two groups, Cense et al. [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]

] and Wojtkowski et al. [33

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]

]. Their method is based on the complex conjugate of the dispersive spectral phase. It compensates for 2nd and higher-order dispersion by adding an optimized phase term to the analytical expression of the measured spectral fringe signal. The phase term is found using a sharpness function that searches for maximum signal magnitude in a given depth range. Note that FD OCT systems still require nearly dispersion-matched interferometer arms: too much dispersion spreads the point-spread-function (PSF) over a wide range of depths, leading to sensitivity decay and information loss, even after numerical dispersion compensation.

In our work, we revisit the problem of numerical dispersion compensation based on the use of the fractional Fourier transform (FrFT). In doing so, our aim is not to provide a replacement to the dispersive spectral phase compensation technique discussed above and which is rapidly becoming the de-facto standard of numerical dispersion compensation. Rather, we wish to illustrate how the FrFT provides a new fundamental perspective on the nature and role of group-velocity dispersion in Fourier domain OCT and how it highlights in a visual manner the physics behind dispersion compensation. Our work also provides new general insights into advanced problems associated with dispersion compensation such as defining a sharpness function that is not dependent on the presence of an isolated single back-scatterer, the issue of depth-dependent dispersion, or the prospect of differentiating materials by dispersion mapping [34

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

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]

]. In the following, we first briefly present the theory of the FrFT and then demonstrate numerical dispersion compensation using FrFT experimentally. Furthermore, we show theoretically how we can extend our method, and use the FrFT for depth-dependent sample dispersion compensation. Finally, we show how we can extract the 2nd-order group-velocity dispersion coefficient β2 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

The 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]

]
fa(ua)=Fa[f(u)]=Aϕexp[iπ[ua2cot(ϕ)2uuacsc(ϕ)+u2cot(ϕ)]]f(u)du,
(1)
where
Aϕ=exp{iπsign[sin(ϕ)]4+iϕ2}|sin(ϕ)|1/2andϕ=aπ2.
(2)
Here u is the independent variable of the transform input function while ua is that of the transform output, with corresponding fractional transform order a. These independent variables are assumed to be dimensionless (see below).

When a = 1, Aϕ 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 u1 and u axes represent the (normalized) time τ and optical frequency ν axes, respectively (the normalization is such that uu1 = 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.

For a → 0 and a → ±2 the integral kernel approaches δ(uau) and δ(ua + 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]

]. Only for 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.

To illustrate these concepts, we consider a simple linear chirp signal
S(ω)=Re{I(ω)exp[j(ωτ+β2l(ωωc)2)]}.
(5)
In an FD OCT setup, such a spectral signal would be observed in the presence of residual chromatic dispersion from a single reflection occurring at a delay τ from the point of zero-path difference of the interferometer, with β2 the group-velocity dispersion coefficient of the dispersive element of optical thickness l. Here ω = 2πν is the angular optical frequency, and ωc the center frequency of the source spectrum I(ω). After normalization, u = ν/η, u1 = ητ, Eq. (5) reads
S(u)=Re{I(u)exp[j(2πuu1+(2πη)2β2l(uuc)2)]}=Re{I(u)exp[jφ]}.
(6)
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]

],
η=ΔνΔτ=dνN,
(7)
where Δν and Δτ are the width of the spectral and temporal intervals over which our signal is represented, respectively, = Δν/N = 1/Δτ 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 ΔνΔτ=N, the Wigner distribution is confined to a circle, and the samples in both domains are spaced 1/N apart.

Fig. 1 Wigner distribution of a chirped signal in the time-frequency plane (u1, u). It illustrates how 2nd-order dispersion affects a spectral interferogram and how the continuum of domains of the FrFT can be interpreted geometrically by the rotated frame (ua, ua+1). The order parameter a chosen for the plot is optimal for the signal considered here, i.e., it leads to a dispersion-compensated OCT depth signal after FrFT.

In essence, the FrFT order 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.

The optimized FrFT order parameter 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

To experimentally demonstrate numerical dispersion compensation using the FrFT, we have used an SD OCT configuration as shown in Fig. 2. It is based on a Michelson interferometer built around a 50/50 fiber coupler made up of SM800 single-mode optical fiber (Thorlabs). The sample arm incorporates a galvanometric mirror for transverse scanning so that we can obtain 2D images. On the detector side, we use a custom-built spectrometer with a 30 mm focal length achromatic collimating lens at the input. The light is then spectrally dispersed with a transmission volume phase holographic grating (1200 lines/mm, Wasatch Photonics Inc.) before being imaged onto a CMOS line scan camera (BASLER Sprint spl2048–70km) using another achromatic doublet lens with 75 mm focal length. The camera offers 2048 pixels, 12 bit resolution, and a maximum acquisition rate of 70 kHz. The spectrometer provides a spectral bandwidth of 220 nm with a spectral resolution of 0.1 nm. It has been carefully calibrated to avoid any coupling with dispersion compensation [41

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

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]

].

Fig. 2 SD OCT configuration used to demonstrate FrFT-based dispersion compensation. A 26 mm water cell induces a dispersion imbalance between the interferometer arms. M: mirror, L: lens, PC: polarization controller, FC: fiber collimator, GM: galvo mirror, S: sample, SLD: superluminescent diode, Disp: dispersion matching material, G: grating, LSC: line scan camera.

The light source of our OCT system is a superluminescent diode with a center wavelength of 843 nm and an optical bandwidth (full width at half maximum, FWHM) of ΔλFWHM = 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.

Fig. 3 PSF of our OCT system after coarse physical dispersion compensation (dotted red) compared with the theoretical one (solid blue). Inset: corresponding source spectrum.

For all our measurements, the signal coming from the reference arm, measured by blocking the sample arm, and averaged over 100 spectra, is subtracted to obtain the spectrally modulated signal only. The modulated signal is then re-sampled to account for the hyperbolic dependence between wavelength and frequency. Finally, zero padding is performed in order to improve the digital sampling resolution in the transformed domain. The sensitivity of our OCT system was measured as 98 dB at 50 μs exposure time and 100 μm depth. A sensitivity fall off of 11.6 dB was measured at a depth of 1 mm.

4. Results

4.1. Point-spread function measurements

To test the efficiency of the FrFT for numerical dispersion compensation, the interferometer was first purposely dispersion mismatched by placing a 26 mm long water cell in the sample arm. Applying the traditional FT to the acquired spectrum resulted in a dispersed PSF 49 μm wide (FWHM). This is shown as the solid blue curve in Fig. 4(a) and compared with the PSF obtained without the dispersive water cell (dotted red curve), which is the same as that plotted in Fig. 3. Using the FrFT with an optimized order parameter aopt = 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.

Fig. 4 (a) The dotted red curve is the PSF of our OCT system (3.8 μm FWHM) using only BK7 coarse physical dispersion compensation and the traditional FT (same as in Fig.3). The two other curves are obtained with an additional 26 mm long dispersive water cell. Using the traditional FT (solid blue curve), the PSF broadens to 49 μm FWHM, while the FrFT with optimized order aopt = 1.0555 numerically compensates dispersion, leading to a 3.65 μm FWHM PSF (dashed black curve). (b) Sharpness function as a function of fractional Fourier transform order, a.

To provide additional physical insights into FrFT-based dispersion compensation, Fig. 5(a) compares the measured unwrapped spectral phase and Fig. 5(b) the instantaneous spectral fringe frequency [i.e., 1st order derivative of the spectral phase, see Eq. (8)] without the water cell (dotted red), with the water cell (solid blue), and with the water cell plus FrFT dispersion compensation (dashed black). As can be seen, the presence of the water cell induces a strong spectral frequency modulation visible in the inclination of the solid blue curve in Fig. 5(b) compared with the dotted red curve, as was already discussed schematically in Fig. 1. After FrFT dispersion compensation, the curve becomes horizontal. It is even flatter than before the water cell is introduced, showing that the FrFT numerically compensates for both the dispersion of the water cell and the residual dispersion of the setup, i.e., what was left uncompensated by the physical introduction of the BK7 glass. Figure 5(c) and Fig. 5(d) show the Wigner distribution of the signal, respectively before and after FrFT dispersion compensation. It is easily seen that the signal energy is entirely preserved and only rotated about the domains after FrFT.

Fig. 5 (a) Spectral phase and (b) instantaneous spectral fringe frequency without (dotted red) and with the water cell (solid blue), and with the water cell plus FrFT dispersion compensation (dashed black). (c) and (d) shows the Wigner distribution of the spectrum acquired with the water cell before and after FrFT dispersion compensation.

For completeness, let us point out that the spectral phase of the FrFT dispersion compensated signal [dashed black curve in Fig. 5(a) and Fig. 5(b)] was obtained from the inverse Fourier transform of the FrFT-compensated complex spectrum (i.e. analytic signal),
φFrFT=arg{F1{Faopt{S(u)+iS^(u)}}}.
(9)
Here 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, WS(u, u1) = WS(u, − u1). Upon projection on the optimal rotated ua 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 ≠ 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.

Finally, the width of the PSF obtained with the water cell after FrFT numerical dispersion compensation was measured for different axial delays. These measurements are plotted in Fig. 6 and compared with that obtained without water cell (and using the traditional FT). The same FrFT order was used for all depths. The resolution observed in the two cases are in very good agreement. Higher order dispersion terms did not disturb our measurements. However, we need to point out a slight increase (7 %) of asymmetric side lobes of the PSF after numerical dispersion compensation, which we attribute to the third order dispersion of the water cell. A similar effect may also arise if higher bandwidths were used. In our current state of knowledge, we cannot readily compensate higher order dispersion using the FrFT but we do not preclude that this may be possible using transformations of the time-frequency plane more complex than rotations (i.e. linear projections).

Fig. 6 Measured PSF for different delays for matched dispersion (dots) and FrFT compensated dispersion (cross).

4.2. Imaging

To further validate our method, we have performed FrFT numerical dispersion compensation on two particular samples. The first consists of two stacked 100 μm thick microscope cover slides. The poor surface flatness of the slides produced an air gap of approximately 20 μm between them. Figure 7(a) was obtained using the traditional FT. The gap is blurred due to the poor axial resolution caused by the presence of the dispersive water cell. Using the optimized FrFT, the air gap between the two microscope slides can be resolved as seen in Fig. 7(b).

Fig. 7 OCT depth scan and intensity image of two cover slides (a) using the traditional FT and (b) using FrFT numerical dispersion compensation with the optimized order parameter. The 26 mm-long dispersive water cell is present in the setup in both cases.

Fig. 8 (a) Radon transform of the spectrogram of one A-scan of a grape. (b) B-scan of the grape using the traditional FT (orange line). (c) B-scan of the grape using the optimized FrFT (aopt = 1.04, green line). The same FrFT order was used for all A-scans. The white bars correspond to 500 μm. Note that the x-axis of (a) only corresponds strictly to depth for the optimized FrFT so is labeled in terms of the pixels of the projections (or Radon bins).

To complete this Section, we must note that the algorithm used for numerical computation of the FrFT, and which was introduced by Ozaktas et al. [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]

], is currently one order of magnitude slower than the Fast Fourier Transform (FFT) algorithm used to compute traditional FTs. With current computing technology, this would however not preclude the use of the FrFT for real time processing.

5. Depth-dependent sample dispersion compensation

An axial resolution below 3 μm is generally sensitive to sample dispersion, so that even thin sample layers cause a broadening of the PSF during imaging. In such situations, the depth-dependent dispersion of the sample is not negligible and must be taken into account in order to get the sharpest image at all depths. For a Gaussian source spectrum, the factor of axial resolution broadening σPSF due to a dispersive element of thickness l and group velocity dispersion coefficient β2 is given by
σPSF=1+(π2c22ln(2)lβ2ΔλFWHM2λc4)2.
(10)
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 (β2 = 41 ps2/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 ΔλFWHM = 210 nm at λ0 = 710 nm center wavelength.

When the sample dispersion is not negligible, the optimal FrFT order required for dispersion compensation becomes a function of imaging depth. Figure 9 highlights this issue for a simulated sample made up of five identical 100 μm-thick microscope cover slides, stacked against each other, and for which we assume an average sample group velocity dispersion of β2 = 54 ps2/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.

Fig. 9 (a) Sharpness function at each interface of a simulated sample made up five identical 100 μm-thick cover slides as a function of FrFT order. (b) Corresponding optimal FrFT order as a function of imaging depth.

Fig. 10 (a) Color plot of OCT depth scans obtained for a range of FrFT orders for a simulated sample made up five identical 100 μm-thick cover slides. Three cross-sections are highlighted: (b) traditional FT (a = 1, solid green line), (c) average sample dispersion compensation (dotted red line), and (d) depth-dependent dispersion compensation (dashed yellow line).

Performing depth-dependent dispersion compensation by finding an optimized cross-section of the Radon plot of the signal can be generalized to more complex samples. In Fig. 11, we show similar plots to Fig. 10 but for a simulated sample which presents normal dispersion in a first 150 μm-thick layer, with a group-velocity dispersion coefficient β2 linearly increasing with depth from 54 ps2/km to 97 ps2/km, followed by a second layer with uniform anomalous dispersion, β2 = −38 ps2/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.

Fig. 11 Same as in Fig. 10 but for a simulated sample which presents linearly increasing normal dispersion in a first 150 μm-thick layer, followed by a second layer with uniform anomalous dispersion. (b) is obtained with the traditional FT (solid green line), while (c) comes from the optimized cross-section of the Radon plot (dashed yellow line).

OCT signals from real samples may be more challenging to process, but the problem of finding an optimal cross-section passing through maxima of the Radon plot as defined above seems amenable to an appropriate algorithm and may even be applicable to samples with no isolated scatterers. Indeed, a quick look at Fig. 8(a) obtained for our grape image shows that this procedure would have led to the horizontal dashed green line that corresponds to the optimal FrFT order, 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 (λ0 < 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]

]. If water is the dominant medium in the sample, only if a better resolution is needed would the approach described here break down, as compensation of higher order terms may become necessary.

6. Group velocity dispersion measurement using FrFT

While dispersion causes broadening of the PSF and blurs images, it has the potential to provide additional functional information. For example, in tissues which exhibit regions with different group-velocity dispersion coefficients, dispersion information can be used for tissue differentiation. Liu et al. proposed to use material dispersion in order to differentiate water and lipid as a diagnosis of vulnerable plaques in the coronary arteries [34

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]

]. Obviously, this requires a mean to extract absolute or relative information about the group-velocity dispersion coefficient β2. We show below that our FrFT-based dispersion compensation routine makes such information readily available. Note that the experimental estimation of the dispersion coefficient β2 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.

From the analysis of a chirp signal, Eq. (5), we have shown in Fig. 1 how the time-frequency representation provides a simple geometric relationship between the group-velocity dispersion coefficient β2 and the FrFT order required for optimal dispersion compensation, aopt. 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:
tan(πϕ)=πlβ2(2πη)2,
(11)
where ϕ = aoptπ/2 defines the orientation of the optimal fractional Fourier domain. Solving for the group-velocity dispersion coefficient, and with η=dν/N the normalization parameter used in our digital implementation [Eq. (7)], yields
β2=πlNdω2tan(aoptπ/2),
(12)
where = 2πdν is the angular frequency spacing between the samples of the OCT spectral signal. The above equation clearly shows that the absolute value of β2, including its sign, can be extracted once the optimal FrFT order for dispersion compensation has been obtained.

To demonstrate this capability, we have measured with this technique the group velocity dispersion of a short length of single mode fiber and of distilled water, both at a center wavelength of 843 nm. The fiber sample was a 244 mm long SM800 (Thorlabs) fiber, which has a mode field diameter of 5.6 μm. The water sample was the same as used for the dispersion compensation experiments, i.e., 26 mm thick. In both cases, we first determined the residual dispersion of the setup (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 (2)(total) and normalized to the sample length to yield the dispersion coefficient β2 of the sample itself,
β2(sample)=1lsample[(lβ2)(total)(lβ2)(setup)].
(13)
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]

] while that of water is in good agreement with Van Engen et al., who measured a group velocity dispersion of water of approximately 22 ps2/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, aopt(setup)<1, for the fiber sample measurement but normal, aopt(setup)>1, for the water measurement. This is due to the empty water cell being included in the setup dispersion in the latter case.

Table 1. FrFT-based measurements of the group velocity dispersion coefficient β2 of a single mode fiber and of distilled water. All the measurements have been done at 843 nm but the reference value for water is for a 840 nm wavelength. N = 1001, = 34.5×1010 rad/s.

table-icon
View This Table

In OCT one generally has no information about the absolute thickness of depth layers in a cross sectional image as the refractive index of the material is unknown. However, using Eq. (12), one can readily obtain relative dispersion information by using the optimized FrFT order and the relative thickness of a layer. The relative thickness corresponds to l = l̄/n, where n may be assumed as 1.33 and l̄ is the layer thickness obtained from the OCT depth scan.

7. Conclusion

The fractional Fourier transform (FrFT) is a generalization of the traditional Fourier transform. The FrFT provides a new fundamental perspective on the nature and role of group-velocity dispersion in Fourier domain OCT and can be seen as a visual tool to highlight the physics behind dispersion compensation. Using the FrFT one obtains depth information in FD OCT while simultaneously compensating for group velocity dispersion. Our theoretical axial resolution of 3.6 μm was recovered by optimizing the order parameter of the FrFT to compensate the group velocity dispersion induced by a 26 mm long water cell for a source spectral bandwidth of 110 nm. The technique was successfully demonstrated on a biological sample. Furthermore we provided new insights on the issue of depth-dependent dispersion. Simulations have shown that both normal and anomalous sample dispersion can be addressed by dynamically adapting the order parameter as a function of depth. This method can be seen as analogous to a “short time” FrFT but is more efficient and intuitive and may even be applicable to samples without isolated scatterers. From the optimized FrFT order parameter one also readily obtains some quantitative information about the amount of dispersion in an OCT configuration and sample. We have derived the relationship between the FrFT order parameter and group velocity dispersion and used it to successfully measure the group velocity dispersion coefficient of distilled water and some single mode fiber (at 840 nm).

Acknowledgments

This work was supported by a NERF grant from the Foundation for Research Science and Technology from the New Zealand government.

References and links

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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]

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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]

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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 684768470O (2008). [CrossRef]

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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]

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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]

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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:Al2O3 laser source,” Opt. Lett. 20, 1486–1488 (1995). [CrossRef] [PubMed]

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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. 24, 1221–1223 (1999). [CrossRef]

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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]

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]

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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]

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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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References

  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,” Science254, 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. Express11, 889–894 (2003). [CrossRef] [PubMed]
  4. 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]
  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. Express11, 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]
  7. M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography”, Journal of Biomedical Optics7, 457–463 (2002). [CrossRef] [PubMed]
  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. Express12, 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]
  10. B. Golubovic, B. E. Bouma, G. J. Tearney, and J. G. Fujimoto, “Optical frequency-domain reflectometry using rapid wavelength tuning of a Cr4+: forsterite laser,” Opt. Lett.22, 1704–1706 (1997). [CrossRef]
  11. S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftimia, and B. E. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express11, 2953–2963 (2003). [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. Express14, 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]
  15. K.-S. Lee and J. P. Rolland, “Bessel beam spectral-domain high-resolution optical coherence tomography with micro-optic axicon providing extended focusing range,” Opt. Lett.33, 1696–1698 (2008). [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. Express19, 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. A25, 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 SPIE684768470O (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]
  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:Al2O3 laser source,” Opt. Lett.20, 1486–1488 (1995). [CrossRef] [PubMed]
  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.24, 1221–1223 (1999). [CrossRef]
  25. W. Drexler, U. Morgner, R. K. Ghanta, F. X. Krtner, J. S. Schuman, and J. G. Fujimoto, “Ultrahighresolution ophthalmic optical coherence tomography,” Nature Medicine7, 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]
  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. Express9, 610–615 (2001). [CrossRef] [PubMed]
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