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Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique
1 April 2002
Optics Communications 204 (2002) 67–74 www.elsevier.com/locate/optcom
Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique A.F. Fercher a,*, C.K. Hitzenberger a, M. Sticker a, R. Zawadzki b, B. Karamata c, T. Lasser c a b
Institute of Medical Physics, University of Vienna, Waehringer Strasse 13, A-1090 Wien, Austria Physics Department, Nicholas Copernicus University, ul. Grudziadzka 5/7, PL-87-100 Torun, Poland c Institut d‘Optique Appliqu ee, EPFL, Ecublens, CH-1015 Lausanne, Switzerland
Received 26 October 2001; received in revised form 17 December 2001; accepted 10 January 2002
Abstract A new numerical a posteriori dispersion compensation technique for partial coherence interferometry and optical coherence tomography depth-scan signals is presented. This technique is based on numerical correlation of the depthscan interferometer signal with a depth-variant kernel. Examples of dispersion compensated depth-scan signals obtained from microscope cover glasses are presented. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.25 Bs; 42.25.Hz; 42.25.Kb; 42.30)d; 42.72.Bj Keywords: Partial coherence interferometry; Optical coherence tomography; Dispersion compensation
1. Introduction Optical coherence tomography (OCT) [1] has become one of the most promising new optical imaging technologies. Starting in the ophthalmologic field OCT has quickly found applications in other medical fields as well as in technical fields [2]. OCT images are synthesized from a series of transversely adjacent partial coherence interfer-
* Corresponding author. Tel.: +43-1-4277-60701; fax: +43-14277-9607. E-mail address: [email protected] (A.F. Fercher).
ometry (PCI) depth-scans. The basic optical schemes of PCI and OCT have already been described in many papers and some reviews [2–4], and therefore, will only be shortly mentioned here. Usually, a two-beam interferometer is used in PCI depth ranging. The output of a low coherence light source is split in a probe beam which is directed towards the sample and a reference beam which is directed towards the retro-reflecting reference mirror of the interferometer. Wave groups remitted and/or reflected from both, the sample and the reference mirror, are recombined at the beam splitter and propagated to a photodetector at the interferometer exit. Depth-scans are performed by shifting the reference mirror. Electronic signal
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processing is used to separate the interferometric depth-scan signal from the dc signal which carries no useful information. A cross-sectional image is produced by transversely scanning the probe beam across the sample and collecting the data of adjacent PCI depth-scans. The key to the internal structure of transparent and partially transparent objects by PCI is the correlation performance of white-light two-beam interferometry. Let us consider a Michelson interferometer configuration with a probe arm and a reference arm. During the depth-scan the reference mirror varies the optical length of the reference beam. An interferometric signal occurs at the photodetector if the optical distance from the beam splitter to the reference mirror matches the optical distance to light remitting sites in the sample within the coherence length. With a retro-reflecting mirror as a sample in the probe arm this signal equals the auto-correlation of the interferometer beams. With a transparent sample in the probe arm the interferometric signal equals the cross-correlation of the probe beam with the reference beam. In the linear regime the cross-correlation signal at the interferometer exit reveals, for example, the deformation of ultrashort light pulses caused by the sample in the interferometer probe arm. In fact, the cross-correlation is obtained from the autocorrelation by a convolution with the light pulse response function [5]. Low time coherence light is a sequence of random ultrashort light pulses [6]. Therefore, the response of a sample in the interferometer on low time coherence light too is reflected by the change of the auto-correlation signal at the interferometer exit to the cross-correlation signal. Hence, in the sense of linear system theory the interferometer cross-correlation signal depicts the internal structure of the sample with the autocorrelation signal as the impulse response. OCT imaging is based on PCI depth ranging with the probe beam penetrating the object material. Material is dispersive. One important consequence of light propagation through material is increased coherence length and, therefore, reduced depth resolution. For example, at a mean wavelength of k0 ¼ 890 nm and a wavelength bandwidth of Dk ¼ 320 nm a 1 mm thick microscope carrier glass degrades depth resolution from ini-
tially 1 lm at the front surface to 4 lm at the back surface [7]. The standard technique to avoid the dispersion related increase of coherence length is physical dispersion balancing. In this technique a dispersive plate which provides the same dispersion as the sample in the probe arm is introduced into the reference arm of the interferometer. However, in high-resolution PCI ranging and OCT imaging [8,9] dealing with variable object depths a depth-dependent dynamic dispersion balancing system is needed, which adapts its balancing power to the probing depth synchronously with the movement of the depth-scanning reference mirror. This requires, for example, mechanically moving optical components and, therefore, is slow and difficult to implement. An important field of PCI and OCT is depth ranging and tomographic imaging of the fundus of the human eye. Here one does not encounter dispersion problems if ‘‘conventional’’ depth resolution of down to 20 lm is used (at a mean wavelength of 800 nm) [10], see Fig. 1. However, ultrahigh resolution imaging of the fundus in the 1 lm range needs careful dispersion compensation [8]. The corresponding problem has already been encountered when testing integrated-optical waveguides. Brinkmeyer et al. [11,12] describe a numerical algorithm in the frequency domain. In the present paper we present a numerical technique based on correlation of time domain signals [13]. Both techniques can be used a posteriori to compensate dispersion related resolution loss and to perform depth-dependent dispersion compensation. In the next chapters we present a short analysis of dispersion in PCI depth-scans and describe the correlation technique together with examples of PCI depth-scans of glass plates.
2. Impact of dispersion on PCI and OCT A-scan signals We represent the optical probe and reference fields in the interferometer arms by the analytic signal V ðtÞ of the corresponding real electric field EðtÞ. The Fourier integral representation of EðtÞ is
A.F. Fercher et al. / Optics Communications 204 (2002) 67–74
69
Fig. 1. Resolution loss caused by second-order dispersion in ophthalmologic OCT. The coherence length lC at the fundus of a human eye (assumed to consist of 100% water; axial geometrical length 26.5 mm) increases dramatically if the coherence length l0 of the light illuminating the eye ðk 800 nmÞ is reduced below 10 lm. This graph has been obtained with Eq. (10) and does only include secondorder dispersion.
Z
where
1
E^ðxÞ expðixtÞdx 1 n o ¼ FT 1 E^ðxÞ ;
EðtÞ ¼
VP ðtÞ ¼ ð1Þ
and, therefore, the corresponding analytic signal is Z 1 V ðtÞ ¼ 2 E^ðxÞ expðixtÞdx: ð2Þ 0
The interference term at the exit of an empty interferometer with a mirror terminating the measurement arm is twice the real part of the autocorrelation of the corresponding analytic signals [14] J ðsÞ ¼ 2RefhV ðt þ sÞV ðtÞig n o ¼ 4FT 1 I^ðxÞ ;
ð3Þ
where J ðsÞ is the response of the interferometer caused by a d-like object represented by the mirror in the measurement arm; s is the time delay between the two interferometer arms. Hence, J ðsÞ can be considered as the impulse response function of the interferometer without sample. The interference term at the exit of an interferometer with a sample in the probe arm is twice the real part of the cross-correlation JS ðsÞ ¼ 2RefhVP ðt þ sÞ V ðtÞig ¼ J ðsÞ rðsÞ;
ð4Þ
Z
1
V ðt0 Þrðt t0 Þdt0 ¼ V ðtÞ rðtÞ
ð5Þ
1
is the analytic signal of the probe arm and rðtÞ is the response function of the sample. Hence, the response function of the interferometer with a sample in the probe arm equals the impulse response J ðsÞ of the empty interferometer convolved with the sample response function rðsÞ. A dispersive sample introduces a frequencyand length-dependent phase UDisp ðx; LÞ to the analytic signal of the probe arm beam ^rðx; LÞ ¼ exp½iUDisp ðx; LÞ;
ð6Þ
with UDisp ðx; LÞ ¼ ð2p=kÞL½nðxÞ 1 ¼ L½kðxÞ k0 ; L is the length of the dispersing path in the probe arm; nðxÞ is the sample refractive index. In other words, the transfer function of a homogeneous dispersive sample equals the dispersion phase coefficient of Eq. (6). If we insert the dispersion phase coefficient exp½iUDisp ðx; LÞ into Eq. (4) the interference term becomes JDisp ðs; LÞ ¼ J ðsÞ FT 1 fexp½iUDisp ðx; LÞg:
ð7Þ
JDisp ðs; LÞ is the impulse response function of an interferometer with a dispersing sample arm of length L. This response function equals the impulse response of the empty interferometer
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convolved with the inverse Fourier transform of the dispersion phase coefficient exp½iUDisp ðx; LÞ. Hence, dispersion adds a wavelength-dependent phase to the spectral components of the probe beam and thus increases the coherence length. The main consequence of dispersion in PCI and OCT is the corresponding loss of depth resolution. To analyse the impact of dispersion on low coherence depth-scan signals in more detail we expand the material dispersion relation k ¼ kðxÞ ¼ nðxÞx=c at k0 ¼ kðx0 Þ into various orders of dispersion [15] dk k k0 ¼ ðx x0 Þ dx x0 1 d2 k 2 ðx x0 Þ þ 2 dx2 x0 1 d3 k 3 þ ðx x0 Þ þ ð8Þ 6 dx3 x0 1. First-order dispersion dk=dx causes a change of the group velocity of a temporal light pulse with wavenumbers kðxÞ centered at k0 !1 dk c mG ¼ ¼ ; ð9Þ dx x0 nG where c is the speed of light in vacuum; nG is the group index. Since the group index nG is always larger than unity, first-order dispersion improves depth resolution. 2. Second-order dispersion degrades depth resolution in all substances. The corresponding coherence length is enlarged by a factor f [6,10,12] and depth resolution of PCI ranging and OCT imaging is degraded by the same factor, which in case of a Fourier transform limited Gaussian light pulse of initial pulse width s0 is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4L2 d2 k : ð10Þ f ¼ 1þ 4 s0 dx2 3. Group velocity dispersion furthermore leads to the phenomenon of chirping, i.e. the instantaneous frequency of light pulses changes. Chirped pulses are no more Fourier transform limited. For chirped pulses the temporal width can be much greater than the inverse of the
spectral width as predicted by the Fourier uncertainty relationship [6] whereas the auto-correlation width remains unchanged. Hence, chirping enables dispersion compensation by correlation as described in the next chapter. 4. Another consequence of second- and higher-order dispersion is a reduction of the signal to noise ratio: because of energy conservation of the light pulses any extension of their length causes a reduction of the pulse intensity even in non-absorbing media. Therefore, fringe visibilities of the PCI and OCT pffiffiffi signal amplitudes are reduced by a factor f and the signal to noise ratio is decreased by a factor f [6,10,12]. 5. Furthermore, dispersion gives rise to artifacts in PCI and OCT. For example, it has been shown, that a beat effect occurs if a thin object behind a dispersive medium is to be measured by PCI or imaged by OCT [16]. 6. Third-order dispersion does also contribute a wavelength-dependent phase to the spectrum. Using the data provided by Van Engen et al. [15] for water, however, the third-order term on the right-hand side of Eq. (8) is much less than 1% of the second-order term and, therefore, can be neglected.
3. Dispersion compensation The mean-square abscissa of a convolution of the two functions in Eq. (4) is equal to the sum of their mean-square abscissas [17]. This is another view at the loss of resolution caused by the dispersion phase term already mentioned above. Basically, deconvolution techniques could be used to compensate for that resolution loss. In a noise-free system, for example, the response of dispersion can be removed from the measured signals by dividing the signal by the dispersion response in the frequency domain. Severe problems are encountered if noise is present and if the response function has zeros. However, deconvolution has successfully been used to remove the dispersive broadening of experimental optical coherence domain reflectometry data of integrated-optical waveguides [11,12] and for OCT image sharpness enhancement [18,19].
A.F. Fercher et al. / Optics Communications 204 (2002) 67–74
Alternatively, a correlation technique can be used to compensate for dispersion induced resolution loss. The advantages of the correlation technique are that this technique is not sensitive to zeros in the response function and that correlation of the experimental signal with a mathematically defined function reduces noise. The correlation technique is based on the fact that the auto-correlation of a quadratic phase term QðsÞ yields a dfunction [20] QðsÞ QðsÞ ¼ dðsÞ:
ð11Þ
If a dispersive sample with second-order dispersion is positioned in the probe arm of the interferom^ ðx; LÞ is added to eter a quadratic phase factor Q the spectrum (we omit the linear phase-term which only changes the speed of light) of the probe beam h i 2 ^ ðx; LÞ ¼ exp iðx x0 Þ2 d kðxÞ L ; Q ð12Þ dx2
71
where L is the length of the dispersive sample. From Eq. (7) we have JDisp ðs; LÞ ¼ J ðsÞ Qðs; LÞ:
ð13Þ
Therefore, Qðs; LÞ broadens the impulse response function J ðsÞ of the interferometer and a corresponding loss of resolution results. Using the numerical dispersion compensation technique we correlate the response function of the interferometer with the corresponding dispersive impulse response function to obtain the dispersion-compensated impulse response function JDc ðs; LÞ JDc ðs; LÞ ¼ ½J ðsÞ Qðs; LÞ ½J ðsÞ Qðs; LÞ ¼ ½J ðsÞ J ðsÞ ½Qðs; LÞ Qðs; LÞ ¼ ½J ðsÞ J ðsÞ dðsÞ ¼ J ðsÞ J ðsÞ:
ð14Þ
Fig. 2. Dispersion compensation in a PCI system by depth-variant correlation. The double arrow s indicates the time delay introduced by the moving reference mirror. Dz is the depth extension of the correlation kernel. BS is the beam splitter; LS is the light source; PD is the photodetector; RM is the reference mirror; SP is the electronic signal processing unit. (a) Object space; (b) depth-scan signals occur at light remitting or reflecting sites at the object; (c) the correlation kernels are depth-variant. A is the non-dispersed coherence function, B and C are obtained from A by appending Ulocal ðxÞ; (d) corresponding correlation signals; ideally of same FWHM.
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Hence, the dispersion compensated impulse response function JDc ðs; LÞ of the interferometer equals the auto-correlation of the dispersion-free impulse response function J ðsÞ. The numerical implementation of that technique is explained with the help of Fig. 2. Since s and the depth coordinate have a one-to-one correspondence we shall replace s by z. Let JPCI ðzÞ be the interference term of the PCI system; its real part is the depth-scan signal IðzÞ, its imaginary part is obtained from IðzÞ by a Hilbert transform. IðzÞ is usually considered as the final PCI or OCT signal and used to synthesize the OCT image, for example. Basically, JPCI ðzÞ is the convolution of the scattering potential F ðzÞ of the object with the depth-dependent impulse response JDisp ðzÞ, where F ðzÞ ¼ k 2 ½1 m2 ðzÞ, and m is the complex refractive index [2]. It has already been shown, however, that back-scattering does only give access to a limited range of high spatial frequency Fourier components of the scattering potential. Low spatial frequency components of F ðzÞ do not contribute to the scattered light field [3]. Hence, a more or less constant refractive index distribution does not backscatter sufficient light for detection – but introduces dispersion and degrades resolution.
The interference term JPCI ðzÞ is obtained from the experimentally measured depth-scan signal IðsÞ, see Fig. 2, and numerically correlated with a depth-variant kernel KðzÞ ¼ JDisp ðzÞ a posteriori CðzÞ ¼ JPCI ðzÞ KðzÞ ¼ F ðzÞ JDisp ðzÞ JDisp ðzÞ ¼ F ðzÞ J ðzÞ J ðzÞ
ð15Þ
i.e. we obtain the object structure F ðzÞ convolved with the auto-correlation of the impulse response function J ðzÞ of the empty interferometer as point spread function. Fig. 3 presents an example. A microscope cover glass has been used as object in a Michelson interferometer PCI configuration. Light of a filtered (Schott OG 590) Hg high pressure lamp has been used. The FWHM width of the source was 210 nm at a central wavelength of 710 nm. The dispersion coefficient used in this work has been obtained from Sellmeier’s formula and BK7 refractive index data out of Schott Catalog Optical Glass [21] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 k2 k2 nðkÞ ¼ 1 þ B21 2 þ B2 2 þ B3 2 k C1 k C2 k C3 ð16Þ
µ
µ µ
µ
µ
Fig. 3. Depth variant dispersion compensation on depth-scan signals obtained from a microscope cover glass made of BK7 of 144 lm thickness. Shown are the depth-scan signals IðzÞ and the corresponding correlation signals CðzÞ. Structure-less data between front and back surface signals have been omitted. Light of a filtered Hg high pressure lamp is incident from left (thick arrow). Dispersion inreases the FWHM of the depth-scan signal from 1.03 to 3:23 lm. Numerical correlation results in FWHM’s of 1.2 and 1:32 lm, respectively.
A.F. Fercher et al. / Optics Communications 204 (2002) 67–74
µ
73
µ
Fig. 4. Numerical dispersion compensation on depth-scan signals obtained from a 2 0:5 lm thick nitrocellulose membrane behind a 144 lm thick microscope cover glass. Shown is the depth-scan signal IðzÞ and the corresponding correlation signal CðzÞ. The correlation signal clearly resolves the two surfaces in contrast to the dispersed depth-scan signal. There is a shift of 0:8 lm of the interferometer signal relative to the correlation peaks towards right due to the width of the correlation kernel.
with constants B1 ¼ 1:03961212, B2 ¼ 0:231792344, B3 ¼ 1:01046945, C1 ¼ 0:00600069867, C2 ¼ 0:0200179144, C3 ¼ 103:560653. Using Eq. (16) a significant variation of the second-order dispersion within the broad bandwidth of the light used in our experiment is found. We used a mean value of d2 n=dk2 ¼ 0:085 lm2 corresponding to the center wavelength of k0 ¼ 710 nm. The shift of the correlation peaks to left is the depth extension of the correlation kernels; it has not been removed to ease comparison. To provide a reasonable picture most of the data between the two interfaces have been omitted. The cross-correlation of the dispersed (right) signal demonstrates the power of the technique: the FWHM has been reduced from 3.23 to 1:32 lm, close to the FWHM of the undispersed case. The slight residual increase of the FWHM of the front signal might be due to noise and variable scanning speed of the reference mirror drive which lacked an independent position calibration. Variable scanning speed changes the chirp of the depth-scan signal and, therefore, has a large impact on the correlation signal. This might also be the cause for the even larger residual FWHM increase of the longer back surface signal. The amplitude of the dispersed depth-scan signal (right) has been increased
mathematically to ease the comparison with the non-dispersed depth-scan signal. Fig. 4 demonstrates another example of improved depth resolution attainable by the numerical dispersion compensation technique. This figure shows the PCI depth-scan signal of a pellicle beam splitter behind a microscope cover glass. Light of the same Hg high pressure lamp as described above is incident from left (thick arrow). The two pellicle interfaces are clearly resolved. Note that only the pellicle front surface is covered by a 50% reflective layer. Here too, we feel that the different FWHM widths might be caused by slight variations of the free running reference mirror drive.
4. Conclusion High resolution PCI and OCT in transparent media require dispersion compensation. Usually a dispersive plate is introduced into the reference arm of the interferometer to balance the dispersion of the sample in the probe arm. Variable object depths can be matched with a depth-dependent dynamic dispersion balancing system. A plate tilting synchronously with the movement of the depth-scanning reference mirror or synchronously shifted dispersive prisms can be used. This, however,
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requires mechanically moving components and, therefore, is difficult to implement. Furthermore, if the object and/or its dispersion changes, dispersion balancing components might have to be exchanged. As mentioned in the introduction, dispersion compensation is of decisive importance in highresolution PCI ranging and OCT imaging. In ophthalmology [8,22], for example, micrometer resolution is state of the art. Such high resolution low coherence interferometry is generally not available without dispersion compensation. Whereas a fixed probing depth in the eye can easily be balanced using solid state compensators, the depth-scan across the retina, for example, would need very fast dynamic dispersion compensation. Fast dynamic dispersion compensation is not easy to implement with physically moving components. Numerical compensation of dispersion in PCI and OCT depth-scan signals, however, does not need any mechanical movement, can be performed a posteriori, and can easily meet different dispersion requirements. Of course, object dispersion must be known in the numerical compensation technique as well as in the physical technique. In ophthalmology, dispersion data of different ocular components have been obtained just recently using PCI [23]. The numerical technique presented in this paper yields a depth resolution close to the undispersed coherence length. It can be combined with partial dispersion balancing of a fixed object depth and might provide a valuable tool for future clinical applications of PCI and OCT in ophthalmology.
Acknowledgements We gratefully acknowledge the hint of one of the referees who pointed to the work of E. Brinkmeyer et al. We acknowledge financial support from the Jubil€ aumsfonds (Project No. 7428) of the Austrian National Bank and from the Austrian Fonds zur F€ orderung der Wissenschaftlichen Forschung (FWF Project No. 10316).
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, J.G. Fujimoto, Science 254 (1991) 1178. [2] A.F. Fercher, J. Biomed. Opt. 1 (1996) 157. [3] A.F. Fercher, C.K. Hitzenberger, in: T. Asakura (Ed.), Optical Sciences, vol. 4, Springer, Berlin, 1999, p. 359. [4] J.M. Schmitt, IEEE J. Sel. Top. Quantum El. 5 (1999) 1205. [5] T. Fuji, M. Miyata, S. Kawato, T. Hattori, H. Nakatsuka, J. Opt. Soc. Am. B 14 (1997) 1074. [6] A. Ghatak, K. Thyagarajan, Introduction to Fiber Optics, Cambridge University Press, Cambridge, 1998 (Chapter 6). [7] A.F. Fercher, C.K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, T. Lasser, Proc. SPIE 4413 (2001) 12. [8] W. Drexler, U. Morgner, F.X. K€artner, C. Pitris, S.A. Boppart, X.D. Li, E.P. Ippen, J.G. Fujimoto, Opt. Lett. 24 (1999) 1221. [9] A.F. Fercher, C.K. Hitzenberger, M. Sticker, E. MorenoBarriuso, R. Leitgeb, W. Drexler, H. Sattmann, Opt. Commun. 185 (2000) 57. [10] C.K. Hitzenberger, A. Baumgartner, W. Drexler, A.F. Fercher, J. Biomed. Opt. 4 (1999) 144. [11] E. Brinkmeyer, R. Ulrich, El. Lett. 26 (1990) 413. [12] A. Kohlhaas, C. Froemchen, E. Brinkmeyer, J. Lightwave Tech. 9 (1991) 1493. [13] A.F. Fercher, C.K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, T. Lasser, Opt. Express 9 (2001) 610. [14] M. Born, E. Wolf, Cambridge University Press, Cambridge, 1998 (Chapter 10). [15] A.G. Van Engen, S. Diddams, T.S. Clement, Appl. Opt. 37 (1998) 5679. [16] C.K. Hitzenberger, A. Baumgartner, A.F. Fercher, Opt Commun. 154 (1998) 179. [17] R. Bracewell, The Fourier Transform and Its Applications, McGraw-Hill, New York, 2000 (Chapter 8). [18] M.D. Kulkarni, W.C. Thomas, J.A. Izatt, El. Lett. 33 (1997) 1365. [19] J.M. Schmitt, Z. Liang, Proc. SPIE 2981 (1997) 46. [20] J.D. Gaskill, Linear Systems, Fourier Transforms, and Optics, John Wiley & Sons, New York, 1978 (Chapter 8). [21] SCHOTT’96 for Windows Catalog Optical Glass, Schott Glaswerke Mainz, Germany, 1996, Available from http:// us.schott.com/sgt/english/products/catalogs.html. [22] A. Baumgartner, C.K. Hitzenberger, H. Sattmann, W. Drexler, A.F. Fercher, J. Biomed. Opt. 3 (1998) 45. [23] W. Drexler, C.K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, A.F. Fercher, Exp. Eye Res. 66 (1998) 25.
Optics Communications 204 (2002) 67–74 www.elsevier.com/locate/optcom
Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique A.F. Fercher a,*, C.K. Hitzenberger a, M. Sticker a, R. Zawadzki b, B. Karamata c, T. Lasser c a b
Institute of Medical Physics, University of Vienna, Waehringer Strasse 13, A-1090 Wien, Austria Physics Department, Nicholas Copernicus University, ul. Grudziadzka 5/7, PL-87-100 Torun, Poland c Institut d‘Optique Appliqu ee, EPFL, Ecublens, CH-1015 Lausanne, Switzerland
Received 26 October 2001; received in revised form 17 December 2001; accepted 10 January 2002
Abstract A new numerical a posteriori dispersion compensation technique for partial coherence interferometry and optical coherence tomography depth-scan signals is presented. This technique is based on numerical correlation of the depthscan interferometer signal with a depth-variant kernel. Examples of dispersion compensated depth-scan signals obtained from microscope cover glasses are presented. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.25 Bs; 42.25.Hz; 42.25.Kb; 42.30)d; 42.72.Bj Keywords: Partial coherence interferometry; Optical coherence tomography; Dispersion compensation
1. Introduction Optical coherence tomography (OCT) [1] has become one of the most promising new optical imaging technologies. Starting in the ophthalmologic field OCT has quickly found applications in other medical fields as well as in technical fields [2]. OCT images are synthesized from a series of transversely adjacent partial coherence interfer-
* Corresponding author. Tel.: +43-1-4277-60701; fax: +43-14277-9607. E-mail address: [email protected] (A.F. Fercher).
ometry (PCI) depth-scans. The basic optical schemes of PCI and OCT have already been described in many papers and some reviews [2–4], and therefore, will only be shortly mentioned here. Usually, a two-beam interferometer is used in PCI depth ranging. The output of a low coherence light source is split in a probe beam which is directed towards the sample and a reference beam which is directed towards the retro-reflecting reference mirror of the interferometer. Wave groups remitted and/or reflected from both, the sample and the reference mirror, are recombined at the beam splitter and propagated to a photodetector at the interferometer exit. Depth-scans are performed by shifting the reference mirror. Electronic signal
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 1 3 7 - 9
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processing is used to separate the interferometric depth-scan signal from the dc signal which carries no useful information. A cross-sectional image is produced by transversely scanning the probe beam across the sample and collecting the data of adjacent PCI depth-scans. The key to the internal structure of transparent and partially transparent objects by PCI is the correlation performance of white-light two-beam interferometry. Let us consider a Michelson interferometer configuration with a probe arm and a reference arm. During the depth-scan the reference mirror varies the optical length of the reference beam. An interferometric signal occurs at the photodetector if the optical distance from the beam splitter to the reference mirror matches the optical distance to light remitting sites in the sample within the coherence length. With a retro-reflecting mirror as a sample in the probe arm this signal equals the auto-correlation of the interferometer beams. With a transparent sample in the probe arm the interferometric signal equals the cross-correlation of the probe beam with the reference beam. In the linear regime the cross-correlation signal at the interferometer exit reveals, for example, the deformation of ultrashort light pulses caused by the sample in the interferometer probe arm. In fact, the cross-correlation is obtained from the autocorrelation by a convolution with the light pulse response function [5]. Low time coherence light is a sequence of random ultrashort light pulses [6]. Therefore, the response of a sample in the interferometer on low time coherence light too is reflected by the change of the auto-correlation signal at the interferometer exit to the cross-correlation signal. Hence, in the sense of linear system theory the interferometer cross-correlation signal depicts the internal structure of the sample with the autocorrelation signal as the impulse response. OCT imaging is based on PCI depth ranging with the probe beam penetrating the object material. Material is dispersive. One important consequence of light propagation through material is increased coherence length and, therefore, reduced depth resolution. For example, at a mean wavelength of k0 ¼ 890 nm and a wavelength bandwidth of Dk ¼ 320 nm a 1 mm thick microscope carrier glass degrades depth resolution from ini-
tially 1 lm at the front surface to 4 lm at the back surface [7]. The standard technique to avoid the dispersion related increase of coherence length is physical dispersion balancing. In this technique a dispersive plate which provides the same dispersion as the sample in the probe arm is introduced into the reference arm of the interferometer. However, in high-resolution PCI ranging and OCT imaging [8,9] dealing with variable object depths a depth-dependent dynamic dispersion balancing system is needed, which adapts its balancing power to the probing depth synchronously with the movement of the depth-scanning reference mirror. This requires, for example, mechanically moving optical components and, therefore, is slow and difficult to implement. An important field of PCI and OCT is depth ranging and tomographic imaging of the fundus of the human eye. Here one does not encounter dispersion problems if ‘‘conventional’’ depth resolution of down to 20 lm is used (at a mean wavelength of 800 nm) [10], see Fig. 1. However, ultrahigh resolution imaging of the fundus in the 1 lm range needs careful dispersion compensation [8]. The corresponding problem has already been encountered when testing integrated-optical waveguides. Brinkmeyer et al. [11,12] describe a numerical algorithm in the frequency domain. In the present paper we present a numerical technique based on correlation of time domain signals [13]. Both techniques can be used a posteriori to compensate dispersion related resolution loss and to perform depth-dependent dispersion compensation. In the next chapters we present a short analysis of dispersion in PCI depth-scans and describe the correlation technique together with examples of PCI depth-scans of glass plates.
2. Impact of dispersion on PCI and OCT A-scan signals We represent the optical probe and reference fields in the interferometer arms by the analytic signal V ðtÞ of the corresponding real electric field EðtÞ. The Fourier integral representation of EðtÞ is
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69
Fig. 1. Resolution loss caused by second-order dispersion in ophthalmologic OCT. The coherence length lC at the fundus of a human eye (assumed to consist of 100% water; axial geometrical length 26.5 mm) increases dramatically if the coherence length l0 of the light illuminating the eye ðk 800 nmÞ is reduced below 10 lm. This graph has been obtained with Eq. (10) and does only include secondorder dispersion.
Z
where
1
E^ðxÞ expðixtÞdx 1 n o ¼ FT 1 E^ðxÞ ;
EðtÞ ¼
VP ðtÞ ¼ ð1Þ
and, therefore, the corresponding analytic signal is Z 1 V ðtÞ ¼ 2 E^ðxÞ expðixtÞdx: ð2Þ 0
The interference term at the exit of an empty interferometer with a mirror terminating the measurement arm is twice the real part of the autocorrelation of the corresponding analytic signals [14] J ðsÞ ¼ 2RefhV ðt þ sÞV ðtÞig n o ¼ 4FT 1 I^ðxÞ ;
ð3Þ
where J ðsÞ is the response of the interferometer caused by a d-like object represented by the mirror in the measurement arm; s is the time delay between the two interferometer arms. Hence, J ðsÞ can be considered as the impulse response function of the interferometer without sample. The interference term at the exit of an interferometer with a sample in the probe arm is twice the real part of the cross-correlation JS ðsÞ ¼ 2RefhVP ðt þ sÞ V ðtÞig ¼ J ðsÞ rðsÞ;
ð4Þ
Z
1
V ðt0 Þrðt t0 Þdt0 ¼ V ðtÞ rðtÞ
ð5Þ
1
is the analytic signal of the probe arm and rðtÞ is the response function of the sample. Hence, the response function of the interferometer with a sample in the probe arm equals the impulse response J ðsÞ of the empty interferometer convolved with the sample response function rðsÞ. A dispersive sample introduces a frequencyand length-dependent phase UDisp ðx; LÞ to the analytic signal of the probe arm beam ^rðx; LÞ ¼ exp½iUDisp ðx; LÞ;
ð6Þ
with UDisp ðx; LÞ ¼ ð2p=kÞL½nðxÞ 1 ¼ L½kðxÞ k0 ; L is the length of the dispersing path in the probe arm; nðxÞ is the sample refractive index. In other words, the transfer function of a homogeneous dispersive sample equals the dispersion phase coefficient of Eq. (6). If we insert the dispersion phase coefficient exp½iUDisp ðx; LÞ into Eq. (4) the interference term becomes JDisp ðs; LÞ ¼ J ðsÞ FT 1 fexp½iUDisp ðx; LÞg:
ð7Þ
JDisp ðs; LÞ is the impulse response function of an interferometer with a dispersing sample arm of length L. This response function equals the impulse response of the empty interferometer
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convolved with the inverse Fourier transform of the dispersion phase coefficient exp½iUDisp ðx; LÞ. Hence, dispersion adds a wavelength-dependent phase to the spectral components of the probe beam and thus increases the coherence length. The main consequence of dispersion in PCI and OCT is the corresponding loss of depth resolution. To analyse the impact of dispersion on low coherence depth-scan signals in more detail we expand the material dispersion relation k ¼ kðxÞ ¼ nðxÞx=c at k0 ¼ kðx0 Þ into various orders of dispersion [15] dk k k0 ¼ ðx x0 Þ dx x0 1 d2 k 2 ðx x0 Þ þ 2 dx2 x0 1 d3 k 3 þ ðx x0 Þ þ ð8Þ 6 dx3 x0 1. First-order dispersion dk=dx causes a change of the group velocity of a temporal light pulse with wavenumbers kðxÞ centered at k0 !1 dk c mG ¼ ¼ ; ð9Þ dx x0 nG where c is the speed of light in vacuum; nG is the group index. Since the group index nG is always larger than unity, first-order dispersion improves depth resolution. 2. Second-order dispersion degrades depth resolution in all substances. The corresponding coherence length is enlarged by a factor f [6,10,12] and depth resolution of PCI ranging and OCT imaging is degraded by the same factor, which in case of a Fourier transform limited Gaussian light pulse of initial pulse width s0 is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4L2 d2 k : ð10Þ f ¼ 1þ 4 s0 dx2 3. Group velocity dispersion furthermore leads to the phenomenon of chirping, i.e. the instantaneous frequency of light pulses changes. Chirped pulses are no more Fourier transform limited. For chirped pulses the temporal width can be much greater than the inverse of the
spectral width as predicted by the Fourier uncertainty relationship [6] whereas the auto-correlation width remains unchanged. Hence, chirping enables dispersion compensation by correlation as described in the next chapter. 4. Another consequence of second- and higher-order dispersion is a reduction of the signal to noise ratio: because of energy conservation of the light pulses any extension of their length causes a reduction of the pulse intensity even in non-absorbing media. Therefore, fringe visibilities of the PCI and OCT pffiffiffi signal amplitudes are reduced by a factor f and the signal to noise ratio is decreased by a factor f [6,10,12]. 5. Furthermore, dispersion gives rise to artifacts in PCI and OCT. For example, it has been shown, that a beat effect occurs if a thin object behind a dispersive medium is to be measured by PCI or imaged by OCT [16]. 6. Third-order dispersion does also contribute a wavelength-dependent phase to the spectrum. Using the data provided by Van Engen et al. [15] for water, however, the third-order term on the right-hand side of Eq. (8) is much less than 1% of the second-order term and, therefore, can be neglected.
3. Dispersion compensation The mean-square abscissa of a convolution of the two functions in Eq. (4) is equal to the sum of their mean-square abscissas [17]. This is another view at the loss of resolution caused by the dispersion phase term already mentioned above. Basically, deconvolution techniques could be used to compensate for that resolution loss. In a noise-free system, for example, the response of dispersion can be removed from the measured signals by dividing the signal by the dispersion response in the frequency domain. Severe problems are encountered if noise is present and if the response function has zeros. However, deconvolution has successfully been used to remove the dispersive broadening of experimental optical coherence domain reflectometry data of integrated-optical waveguides [11,12] and for OCT image sharpness enhancement [18,19].
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Alternatively, a correlation technique can be used to compensate for dispersion induced resolution loss. The advantages of the correlation technique are that this technique is not sensitive to zeros in the response function and that correlation of the experimental signal with a mathematically defined function reduces noise. The correlation technique is based on the fact that the auto-correlation of a quadratic phase term QðsÞ yields a dfunction [20] QðsÞ QðsÞ ¼ dðsÞ:
ð11Þ
If a dispersive sample with second-order dispersion is positioned in the probe arm of the interferom^ ðx; LÞ is added to eter a quadratic phase factor Q the spectrum (we omit the linear phase-term which only changes the speed of light) of the probe beam h i 2 ^ ðx; LÞ ¼ exp iðx x0 Þ2 d kðxÞ L ; Q ð12Þ dx2
71
where L is the length of the dispersive sample. From Eq. (7) we have JDisp ðs; LÞ ¼ J ðsÞ Qðs; LÞ:
ð13Þ
Therefore, Qðs; LÞ broadens the impulse response function J ðsÞ of the interferometer and a corresponding loss of resolution results. Using the numerical dispersion compensation technique we correlate the response function of the interferometer with the corresponding dispersive impulse response function to obtain the dispersion-compensated impulse response function JDc ðs; LÞ JDc ðs; LÞ ¼ ½J ðsÞ Qðs; LÞ ½J ðsÞ Qðs; LÞ ¼ ½J ðsÞ J ðsÞ ½Qðs; LÞ Qðs; LÞ ¼ ½J ðsÞ J ðsÞ dðsÞ ¼ J ðsÞ J ðsÞ:
ð14Þ
Fig. 2. Dispersion compensation in a PCI system by depth-variant correlation. The double arrow s indicates the time delay introduced by the moving reference mirror. Dz is the depth extension of the correlation kernel. BS is the beam splitter; LS is the light source; PD is the photodetector; RM is the reference mirror; SP is the electronic signal processing unit. (a) Object space; (b) depth-scan signals occur at light remitting or reflecting sites at the object; (c) the correlation kernels are depth-variant. A is the non-dispersed coherence function, B and C are obtained from A by appending Ulocal ðxÞ; (d) corresponding correlation signals; ideally of same FWHM.
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Hence, the dispersion compensated impulse response function JDc ðs; LÞ of the interferometer equals the auto-correlation of the dispersion-free impulse response function J ðsÞ. The numerical implementation of that technique is explained with the help of Fig. 2. Since s and the depth coordinate have a one-to-one correspondence we shall replace s by z. Let JPCI ðzÞ be the interference term of the PCI system; its real part is the depth-scan signal IðzÞ, its imaginary part is obtained from IðzÞ by a Hilbert transform. IðzÞ is usually considered as the final PCI or OCT signal and used to synthesize the OCT image, for example. Basically, JPCI ðzÞ is the convolution of the scattering potential F ðzÞ of the object with the depth-dependent impulse response JDisp ðzÞ, where F ðzÞ ¼ k 2 ½1 m2 ðzÞ, and m is the complex refractive index [2]. It has already been shown, however, that back-scattering does only give access to a limited range of high spatial frequency Fourier components of the scattering potential. Low spatial frequency components of F ðzÞ do not contribute to the scattered light field [3]. Hence, a more or less constant refractive index distribution does not backscatter sufficient light for detection – but introduces dispersion and degrades resolution.
The interference term JPCI ðzÞ is obtained from the experimentally measured depth-scan signal IðsÞ, see Fig. 2, and numerically correlated with a depth-variant kernel KðzÞ ¼ JDisp ðzÞ a posteriori CðzÞ ¼ JPCI ðzÞ KðzÞ ¼ F ðzÞ JDisp ðzÞ JDisp ðzÞ ¼ F ðzÞ J ðzÞ J ðzÞ
ð15Þ
i.e. we obtain the object structure F ðzÞ convolved with the auto-correlation of the impulse response function J ðzÞ of the empty interferometer as point spread function. Fig. 3 presents an example. A microscope cover glass has been used as object in a Michelson interferometer PCI configuration. Light of a filtered (Schott OG 590) Hg high pressure lamp has been used. The FWHM width of the source was 210 nm at a central wavelength of 710 nm. The dispersion coefficient used in this work has been obtained from Sellmeier’s formula and BK7 refractive index data out of Schott Catalog Optical Glass [21] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 k2 k2 nðkÞ ¼ 1 þ B21 2 þ B2 2 þ B3 2 k C1 k C2 k C3 ð16Þ
µ
µ µ
µ
µ
Fig. 3. Depth variant dispersion compensation on depth-scan signals obtained from a microscope cover glass made of BK7 of 144 lm thickness. Shown are the depth-scan signals IðzÞ and the corresponding correlation signals CðzÞ. Structure-less data between front and back surface signals have been omitted. Light of a filtered Hg high pressure lamp is incident from left (thick arrow). Dispersion inreases the FWHM of the depth-scan signal from 1.03 to 3:23 lm. Numerical correlation results in FWHM’s of 1.2 and 1:32 lm, respectively.
A.F. Fercher et al. / Optics Communications 204 (2002) 67–74
µ
73
µ
Fig. 4. Numerical dispersion compensation on depth-scan signals obtained from a 2 0:5 lm thick nitrocellulose membrane behind a 144 lm thick microscope cover glass. Shown is the depth-scan signal IðzÞ and the corresponding correlation signal CðzÞ. The correlation signal clearly resolves the two surfaces in contrast to the dispersed depth-scan signal. There is a shift of 0:8 lm of the interferometer signal relative to the correlation peaks towards right due to the width of the correlation kernel.
with constants B1 ¼ 1:03961212, B2 ¼ 0:231792344, B3 ¼ 1:01046945, C1 ¼ 0:00600069867, C2 ¼ 0:0200179144, C3 ¼ 103:560653. Using Eq. (16) a significant variation of the second-order dispersion within the broad bandwidth of the light used in our experiment is found. We used a mean value of d2 n=dk2 ¼ 0:085 lm2 corresponding to the center wavelength of k0 ¼ 710 nm. The shift of the correlation peaks to left is the depth extension of the correlation kernels; it has not been removed to ease comparison. To provide a reasonable picture most of the data between the two interfaces have been omitted. The cross-correlation of the dispersed (right) signal demonstrates the power of the technique: the FWHM has been reduced from 3.23 to 1:32 lm, close to the FWHM of the undispersed case. The slight residual increase of the FWHM of the front signal might be due to noise and variable scanning speed of the reference mirror drive which lacked an independent position calibration. Variable scanning speed changes the chirp of the depth-scan signal and, therefore, has a large impact on the correlation signal. This might also be the cause for the even larger residual FWHM increase of the longer back surface signal. The amplitude of the dispersed depth-scan signal (right) has been increased
mathematically to ease the comparison with the non-dispersed depth-scan signal. Fig. 4 demonstrates another example of improved depth resolution attainable by the numerical dispersion compensation technique. This figure shows the PCI depth-scan signal of a pellicle beam splitter behind a microscope cover glass. Light of the same Hg high pressure lamp as described above is incident from left (thick arrow). The two pellicle interfaces are clearly resolved. Note that only the pellicle front surface is covered by a 50% reflective layer. Here too, we feel that the different FWHM widths might be caused by slight variations of the free running reference mirror drive.
4. Conclusion High resolution PCI and OCT in transparent media require dispersion compensation. Usually a dispersive plate is introduced into the reference arm of the interferometer to balance the dispersion of the sample in the probe arm. Variable object depths can be matched with a depth-dependent dynamic dispersion balancing system. A plate tilting synchronously with the movement of the depth-scanning reference mirror or synchronously shifted dispersive prisms can be used. This, however,
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requires mechanically moving components and, therefore, is difficult to implement. Furthermore, if the object and/or its dispersion changes, dispersion balancing components might have to be exchanged. As mentioned in the introduction, dispersion compensation is of decisive importance in highresolution PCI ranging and OCT imaging. In ophthalmology [8,22], for example, micrometer resolution is state of the art. Such high resolution low coherence interferometry is generally not available without dispersion compensation. Whereas a fixed probing depth in the eye can easily be balanced using solid state compensators, the depth-scan across the retina, for example, would need very fast dynamic dispersion compensation. Fast dynamic dispersion compensation is not easy to implement with physically moving components. Numerical compensation of dispersion in PCI and OCT depth-scan signals, however, does not need any mechanical movement, can be performed a posteriori, and can easily meet different dispersion requirements. Of course, object dispersion must be known in the numerical compensation technique as well as in the physical technique. In ophthalmology, dispersion data of different ocular components have been obtained just recently using PCI [23]. The numerical technique presented in this paper yields a depth resolution close to the undispersed coherence length. It can be combined with partial dispersion balancing of a fixed object depth and might provide a valuable tool for future clinical applications of PCI and OCT in ophthalmology.
Acknowledgements We gratefully acknowledge the hint of one of the referees who pointed to the work of E. Brinkmeyer et al. We acknowledge financial support from the Jubil€ aumsfonds (Project No. 7428) of the Austrian National Bank and from the Austrian Fonds zur F€ orderung der Wissenschaftlichen Forschung (FWF Project No. 10316).
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