## Numerical dispersion compensation for Partial Coherence Interferometry and Optical Coherence Tomography

Optics Express, Vol. 9, Issue 12, pp. 610-615 (2001)

http://dx.doi.org/10.1364/OE.9.000610

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### Abstract

Dispersive samples introduce a wavelength dependent phase distortion to the probe beam. This leads to a noticeable loss of depth resolution in high resolution OCT using broadband light sources. The standard technique to avoid this consequence is to balance the dispersion of the sample by arranging a dispersive material in the reference arm. However, the impact of dispersion is depth dependent. A corresponding depth dependent dispersion balancing technique is diffcult to implement. Here we present a numerical dispersion compensation technique for Partial Coherence Interferometry (PCI) and Optical Coherence Tomography (OCT) based on numerical correlation of the depth scan signal with a depth variant kernel. It can be used *a posteriori *and provides depth dependent dispersion compensation. Examples of dispersion compensated depth scan signals obtained from microscope cover glasses are presented.

© Optical Society of America

## 1. Introduction

1. A.F. Fercher and E. Roth, “Ophthalmic laser interferometry,” Proc. SPIE **658**, 48–51 (1986). [CrossRef]

2. 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. Dispersion in PCI and OCT

*λ*

_{0}is the center wavelength and Δ

*λ*is the FWHM spectral width of the emitted light. In PCI ranging and OCT imaging the optical distances are determined by the group index

*n*rather than the phase index

_{g}*n*, used in classical interferometry. Furthermore, back-reflected light is used, hence, depth resolution is Δ

*z/l*/2

_{C}*n*.

_{g}*n*includes the phase index

_{g}*n*and the phase index dispersion by

*ν*is the light frequency. Group dispersion increases coherence length. For example, a Fourier transform limited Gaussian pulse the temporal 1/

*e*-width

*π*

_{0}is extended by a factor

*f*[7]:

*d*is the distance travelled by the wave in the dispersive material. Low coherence light can be considered as a random temporal distribution of ultrashort Fourier transform limited Gaussian pulses [7], hence, the dispersed coherence length of low coherence light is of the order of

*f*.

*f*[7].

4. C.K. Hitzenberger, A. Baumgartner, and A.F. Fercher, “Dispersion induced multiple signal peak splitting in partial coherence interferometry,” Opt. Commun. **154**, 179–185 (1998). [CrossRef]

## 3. Dispersed Light Waves

*G*(

*ω*) and the real coherence function

*G*(

*τ*) form a Fourier transform pair [9]:

*E*(

*t*) is the electric field component of the light wave. A time delay

*τ*in the coherence function, as it occurs in the depth scan, leads to a phase Φ

_{0}_{0}=

*ωτ*

_{0}, linear in

*ω*, in the spectral density:

*(*

_{Disp}*ω*) depending on the properties of the dispersive sample:

*(*

_{Disp}*ω*) can be developed into a Taylor series [10

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

^{th}order dispersion;

*z*is the path length in the dispersive medium.

*k*

^{(1)}(

*ω*

_{0}) adds a phase term linear in

*ω*to the intensity spectrum. It determines the group velocity [9]:

*ω*and, therefore, increase the coherence length.

11. T. Fuji, M. Miyata, S. Kawato, T. Hattori, and H. Nakatsuka, “Linear propagation of light investigated with a white-light Michelson interferometer”, J. Opt. Soc. Am. B **14**1074–1078, 1997. [CrossRef]

*E*(

*t*) as a Fourier integral:

*(*

_{Disp}*ω*) to the spectral components of the light in the probe beam. Introducing Φ

*(*

_{Disp}*ω*) to the phases of the waves in equation (11) yields the cross-correlation:

*II*(

_{T}*τ*) at the interferometer exit with dispersion in the sample arm equals the Fourier transform of the spectral intensity

*G*(

*ω*) at the interferometer exit without dispersion multiplied by the dispersion phase coefficient

## 4. Numerical Dispersion Compensation

*i*Φ

*(*

_{local}*ω*)).

*I*(

_{local}*τ*) is the local interference term.

*K*(

_{local}*τ*) is the response of a light remitting site at depth

*z*, subject to the dispersion at position

*z*or the corresponding time delay

*τ*with respect to the reference beam. The (undispersed) kernel at the surface of the dispersing object is the auto-correlation of the light wave exiting the source. This can be obtained from the light source data using standard textbook formulas. In most cases the spectrum can be approximated by a Gaussian and, therefore, the undispersed kernel too can be approximated by a Gaussian temporal distribution:

*K*(

_{local}*τ*) is obtained from the undispersed kernel

*K*(

*t*) by adding a depth dependent phase nonlinear in

*ω*caused by the second and higher order dispersion terms to the Fourier components

*Ê*(

*ω*). This phase leads to a broadening of the light pulses and, therefore, generates chirpening of these pulses: their instantaneous frequency changes with time [7]. The local kernel

*K*(

_{local}*τ*) equals the cross-correlation signal of the interferometer and, therefore, its FWHM duration increases with increasing coherence length of the light remitted at the corresponding object depth.

*µm*thick) to demonstrate the power of this technique: Two non-dispersed front signals and two dispersed back signals were mathematically added with a mutual shift of 2,5

*µm*. In fact this is an indirect proof but based on experimental data. To provide a reasonable picture most of the data between the two interfaces have been omitted. The undispersed Gaussian kernel was based on a mean wavelength of

*λ*

_{0}=710

*nm*and

*τ*

_{0}=0,35.10

^{-14}

*s*. Only second order dispersion has been compensated using a dispersed kernel based on 144

*µm*path length in the dispersive medium.

12. SCHOTT’96 for Windows Catalog Optical Glass, Schott Glaswerke Mainz, Germany, 1996, http://us.schott.com/sgt/english/products/catalogs.html.

*B*

_{1}=1.03961212;

*B*

_{2}=0.231792344;

*B*

_{3}=1.01046945;

*C*

_{1}=0.00600069867;

*C*

_{2}=0.0200179144;

*C*

_{3}=103.560653. Using equation (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

*λ*

_{0}=710

*nm*.

*µm*distance of two back interfaces the numerical correlation shows a pronounced dip between the two correlation peaks. The shift of the numerical correlation peaks to the left is the width of the correlation kernel. It has not been removed to ease comparison. The amplitude of the dispersed depth scan signal (right) has been increased mathematically to ease the comparison with the non-dispersed depth scan signal.

## 5. Conclusions

*a posteriori*and can easily realize a dynamic dispersion scheme. Dispersion compensation is achieved by correlating the depth scan signal with a depth-dependent correlation kernel.

*a posteriori*. Biological tissue contains about 70 % water and 30 % mainly proteins. Hence dispersion properties of such tissue are mainly determined by these components. For a first approximation water data might be used which are easily available [10

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

13. J.M. Schmitt and G. Kumar, “Optical scattering properties of soft tissue: a discrete particle model,” Appl. Opt. **37**, 2788–2797, 1998. [CrossRef]

*Fonds zur Förderung der Wissenschaftlichen Forschung*(Project No. 10316) and from the Jubiläumsfonds (Project No. 7428) of the Austrian National Bank is acknowledged.

## References and links

1. | A.F. Fercher and E. Roth, “Ophthalmic laser interferometry,” Proc. SPIE |

2. | 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 |

3. | A.F. Fercher, C. K. Hitzenberger, M. Sticker, E. Moreno-Barriuso, R. Leitgeb, W. Drexler, and H. Sattmann, “A thermal light source technique for optical coherence tomography,” Opt. Commun. |

4. | C.K. Hitzenberger, A. Baumgartner, and A.F. Fercher, “Dispersion induced multiple signal peak splitting in partial coherence interferometry,” Opt. Commun. |

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

6. | A.F. Fercher, C.K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “A new dispersion compensation technique for Partial Coherence Interferometry (PCI) and Optical Coherence Tomography (OCT),” Proc SPIE4431 (to be published). |

7. | A. Ghatak and K. Thyagarajan, |

8. | C.K. Hitzenberger, A. Baumgartner, W. Drexler, and A.F. Fercher, “Dispersion effects in partial coherence interferometry: implications for intraocular ranging,” J. Biomed. Opt. |

9. | M. Born and E. Wolf, |

10. | A.-G. Van Engen, S. Diddams, and T.-S. Clement, “Dispersion measurements of water with white-light interferometry,” Appl. Opt. |

11. | T. Fuji, M. Miyata, S. Kawato, T. Hattori, and H. Nakatsuka, “Linear propagation of light investigated with a white-light Michelson interferometer”, J. Opt. Soc. Am. B |

12. | SCHOTT’96 for Windows Catalog Optical Glass, Schott Glaswerke Mainz, Germany, 1996, http://us.schott.com/sgt/english/products/catalogs.html. |

13. | J.M. Schmitt and G. Kumar, “Optical scattering properties of soft tissue: a discrete particle model,” Appl. Opt. |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(170.4500) Medical optics and biotechnology : Optical coherence tomography

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 6, 2001

Published: December 3, 2001

**Citation**

Adolf Fercher, Christoph Hitzenberger, Markus Sticker, Robert Zawadzki, Boris Karamata, and Theo Lasser, "Numerical dispersion compensation for Partial Coherence Interferometry and Optical Coherence Tomography," Opt. Express **9**, 610-615 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-12-610

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### References

- A. F. Fercher and E. Roth, "Ophthalmic laser interferometry," Proc. SPIE 658, 48-51 (1986). [CrossRef]
- 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, "Optical coherence tomography," Science 254, 1178-1181 (1991). [CrossRef] [PubMed]
- A. F. Fercher, C. K. Hitzenberger, M. Sticker, E. Moreno-Barriuso, R. Leitgeb, W. Drexler, H. Sattmann, "A thermal light source technique for optical coherence tomography," Opt. Commun. 185, 57-64 (2000). [CrossRef]
- C. K. Hitzenberger, A. Baumgartner, A. F. Fercher, "Dispersion induced multiple signal peak splitting in partial coherence interferometry," Opt. Commun. 154, 179-185 (1998). [CrossRef]
- W. Drexler, U. Morgner, F. X. K�rtner, C. Pitris, S. A. Boppart, X. D. Li, E. P. Ippen, J. G. Fujimoto, "In vivo ultra-high resolution optical coherence tomography," Opt. Lett. 24, 1221-1223 (1999). [CrossRef]
- A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, T. Lasser, "A new dispersion compensation technique for Partial Coherence Interferometry (PCI) and Optical Coherence Tomography (OCT)," Proc SPIE 4431 (to be published).
- A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge University Press, 1998).
- C.K. Hitzenberger, A. Baumgartner, W. Drexler, A.F. Fercher, "Dispersion effects in partial coherence interferometry: implications for intraocular ranging," J. Biomed. Opt. 4, 144-151, 1999. [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1998).
- A.-G.Van Engen, S. Diddams, T.-S.Clement, "Dispersion measurements of water with white-light interferometry," Appl. Opt. 37, 5679-5686, 1998. [CrossRef]
- T. Fuji, M. Miyata, S. Kawato, T. Hattori, H. Nakatsuka, "Linear propagation of light investigated with a white-light Michelson interferometer", J. Opt. Soc. Am. B 141074-1078 (1997). [CrossRef]
- SCHOTT'96 for Windows Catalog Optical Glass, Schott Glaswerke Mainz, Germany, 1996, http://us.schott.com/sgt/english/products/catalogs.html.
- J. M. Schmitt and G. Kumar, "Optical scattering properties of soft tissue: a discrete particle model," Appl. Opt. 37, 2788-2797, 1998. [CrossRef]

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