## Improved lateral resolution in optical coherence tomography by digital focusing using two-dimensional numerical diffraction method

Optics Express, Vol. 15, Issue 12, pp. 7634-7641 (2007)

http://dx.doi.org/10.1364/OE.15.007634

Acrobat PDF (684 KB)

### Abstract

This paper proposes a non-iterative, two-dimensional numerical method to alleviate the compromise between the lateral resolution and wide depth measurement range in optical coherence tomography (OCT). A two-dimensional scalar diffraction model was developed to simulate the wave propagation process from out-of-focus scatterers within the short coherence gate of the OCT system. High-resolution details can be recovered from outside the depth-of-field region with minimum loss of lateral resolution. Experiments were performed to demonstrate the effectiveness of the proposed method.

© 2007 Optical Society of America

## 1. Introduction

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

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

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

7. S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftimia, and B. E. Bouma, “High-speed optical frequency domain imaging,” Opt. Express **11**, 2953–2963 (2003). [CrossRef] [PubMed]

12. R. A. 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]

^{2}) of the optical system. Only a very small range around the DOF will exhibit the desired lateral resolution of the system, and the OCT image in the out-of-focus range is blurred laterally. A typical DOF for a small NA system will be several hundred microns, which is about 10 orders smaller than the scanning depth range of an OCT system. Adaptive optics [15

15. B. Hermann, E. J. Fernandez, A. Unterhuber, H. Sattmann, A. F. Fercher, W. Drexler, P. M. Prieto, and P. Artal, “Adaptive-optics ultrahigh-resolution optical coherence tomography,” Opt. Lett. **29**, 2142–2144 (2004). [CrossRef] [PubMed]

16. Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett. **27**, 243–245 (2002). [CrossRef]

17. Y. Wang, Y. Zhao, J. S. Nelson, and Z. Chen, “Ultrahighresolution optical coherence tomography by broadband continuum generation from a photonic crystal fiber,” Opt. Lett. **28**, 182–184 (2003). [CrossRef] [PubMed]

19. T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Inverse scattering for optical coherence tomography,” J. Opt. Soc. Am. A **23**, 1027–1037 (2006). [CrossRef]

20. M. D. Kulkarni, C. W. Thomas, and J. A. Izatt, “Image enhancement in optical coherence tomography using deconvolution,” Electron. Lett. **33**1365–1367 (1997). [CrossRef]

19. T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Inverse scattering for optical coherence tomography,” J. Opt. Soc. Am. A **23**, 1027–1037 (2006). [CrossRef]

24. Y. Yasuno, J. -i. Sugisaka, Y. Sando, Y. Nakamura, S. Makita, M. Itoh, and T. Yatagai, “Non-iterative numerical method for laterally superresolving Fourier domain optical coherence tomography,” Opt. Express **14**, 1006–1020 (2006). [CrossRef] [PubMed]

25. T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, “Inverse scattering for high-resolution interferometric microscopy,” Opt. Lett. **31**, 3585–3587 (2006). [CrossRef] [PubMed]

26. T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart. “Interferometric Synthetic Aperture Microscopy,” Nature Physics **3**,129–134, (2007). [CrossRef]

## 2. Principle

27. R. M. Lewis, “Physical optics inverse diffraction,” IEEE Trans. Antennas Propag. **AP-17**, 308–314 (1969). [CrossRef]

29. L. Yu and M. K. Kim, “Wavelength-scanning digital interference holography for tomographic three-dimensional imaging by use of the angular spectrum method,” Opt. Lett. **30**, 2092–2094 (2005). [CrossRef] [PubMed]

*E*(

*x,y*;0) is the en-face wave field distribution at plane

*z*= 0, the corresponding angular spectrum of the field at this plane can be obtained by taking the Fourier transform:

*k*and

_{x}*k*are corresponding spatial frequencies of

_{y}*x*and

*y*. The field

*E*(

*x,y*;0) can be rewritten as the inverse Fourier transform of its angular spectrum,

*i*(

*k*+

_{x}x*k*)] may be regarded as a projection, onto the plane

_{y}y*z*= 0, of a wave propagating with a wave vector (

*k*,

_{x}*k*,

_{y}*k*), where

_{z}*k*=[

_{z}*k*

^{2}-

*k*

_{x}^{2}-

*k*

_{y}^{2}]

^{1/2}and

*k*= 2

*π/λ*. Thus, the field

*E*(

*x,y*;0) can be viewed as a superposition of many wave components propagating in different directions in space and with complex amplitude of each component equal to

*S*(

*k*,

_{x}*k*;0). After propagating along the

_{y}*z*axis to a new plane, the new angular spectrum,

*S*(

*k*,

_{x}*k*;

_{y}*z*), at plane

*z*can be calculated from

*S*(

*k*,

_{x}*k*;0) as

_{y}*S*(

*k*,

_{x}*k*;

_{y}*z*) =

*S*(

*k*,

_{x}*k*;0)exp[

_{y}*ik*]. Thus, the complex field distribution of any plane perpendicular to the propagating

_{z}z*z*axis can be calculated from Fourier theory as

*I*(

*x, y*;

*z*). Then the angular spectrum

_{i}*S*(

*k*,

_{x}*k*;

_{y}*z*) of each en-face image was calculated by Eq. (1), and the new angular spectrum after propagating a distance

_{i}*z*was calculated by multiplying

*S*(

*k*,

_{x}*k*;

_{y}*z*) with a z-dependent exponential term as exp[

_{i}*ik*] with

_{z}z*k*= [

_{z}*k*

^{2}-

*k*

_{x}^{2}-

*k*

_{y}^{2}]

^{1/2}. Finally, the en-face field distribution at plane (

*z*+

_{i}*z*) was calculated from Eq. (3). Thus, by selecting a correct reconstruction distance z, the proposed method can be used to numerically cancel the lateral defocus and improve the lateral resolution.

*z*represents the double-pass delay considering the reflection geometry in the OCT system and is thus given as

*z*= 2

*n*Δ

*d*, where

*n*is the refractive index of the tissue and Δ

*d*is the actual deviation from the focal plane. The propagation distance

*z*is determined either by prior knowledge or by automatic maximum-sharpness searching algorithms.

## 3. Experiments

11. R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: unidirectional swept laser sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett. **31**, 2975–2977 (2006). [CrossRef] [PubMed]

## 4. Conclusion

*x*or

*y*direction can be improved by use of a non-iterative numerical diffraction algorithm, and high-resolution details can be reconstructed from outside the depth-of-field region without any special hardware in system design. Although a spectrometer-based system is considered in the paper, the proposed method is also applicable to swept-source or full-field OCT systems.

## References and links

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 |

2. | A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. Elzaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. |

3. | G. Hausler and M. W. Linduer, “Coherence radar and spectral radar-new tools for dermatological diagnosis,” J. Biomed. Opt. |

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

5. | B. Cense, N. A. Nassif, T. C. Chen, M. C. Pierce, S. 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 |

6. | S. Jiao, R. Knighton, X. Huang, G. Gregori, and C. A. Puliafito, “Simultaneous acquisition of sectional and fundus ophthalmic images with spectral-domain optical coherence tomography,” Opt. Express |

7. | S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftimia, and B. E. Bouma, “High-speed optical frequency domain imaging,” Opt. Express |

8. | S. R. Chinn, E. A. Swanson, and J. G. Fujimoto, “Optical coherence tomography using a frequency-tunable optical source,” Opt. Lett. |

9. | S. H. Yun, C. Boudoux, G. J. Tearney, and B. E. Bouma, “High-speed wavelength-swept semiconductor laser with a polygon-scanner-based wavelength filter,” Opt. Lett. |

10. | R. Huber, M. Wojtkowski, K. Taira, J. G. Fujimoto, and K. Hsu, “Amplified, frequency swept lasers for frequency domain reflectometry and OCT imaging: design and scaling principles,” Opt. Express |

11. | R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: unidirectional swept laser sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett. |

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

13. | 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. |

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

15. | B. Hermann, E. J. Fernandez, A. Unterhuber, H. Sattmann, A. F. Fercher, W. Drexler, P. M. Prieto, and P. Artal, “Adaptive-optics ultrahigh-resolution optical coherence tomography,” Opt. Lett. |

16. | Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett. |

17. | Y. Wang, Y. Zhao, J. S. Nelson, and Z. Chen, “Ultrahighresolution optical coherence tomography by broadband continuum generation from a photonic crystal fiber,” Opt. Lett. |

18. | M. J. Cobb, X. Liu, and X. Li, “Continuous focus tracking for real-time optical coherence tomography,” Opt. Lett. |

19. | T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Inverse scattering for optical coherence tomography,” J. Opt. Soc. Am. A |

20. | M. D. Kulkarni, C. W. Thomas, and J. A. Izatt, “Image enhancement in optical coherence tomography using deconvolution,” Electron. Lett. |

21. | J. M. Schmitt, “Restoration of optical coherence images of living tissue using the clean algorithm,” J. Biomed. Opt. |

22. | D. Piao, Q. Zhu, N. Dutta, S. Yan, and L. Otis, “Cancellation of coherent artifacts in optical coherence tomography imaging,” Appl. Opt. |

23. | J. Hsu, C.W. Sun, C.W. Lu, C. C. Yang, C. P. Chiang, and C.W. Lin, “Resolution improvement with dispersion manipulation and a retrieval algorithm in optical coherence tomography,” Appl. Opt. |

24. | Y. Yasuno, J. -i. Sugisaka, Y. Sando, Y. Nakamura, S. Makita, M. Itoh, and T. Yatagai, “Non-iterative numerical method for laterally superresolving Fourier domain optical coherence tomography,” Opt. Express |

25. | T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, “Inverse scattering for high-resolution interferometric microscopy,” Opt. Lett. |

26. | T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart. “Interferometric Synthetic Aperture Microscopy,” Nature Physics |

27. | R. M. Lewis, “Physical optics inverse diffraction,” IEEE Trans. Antennas Propag. |

28. | J. W. Goodman. |

29. | L. Yu and M. K. Kim, “Wavelength-scanning digital interference holography for tomographic three-dimensional imaging by use of the angular spectrum method,” Opt. Lett. |

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(100.6950) Image processing : Tomographic image processing

(170.4500) Medical optics and biotechnology : Optical coherence tomography

**ToC Category:**

Image Processing

**History**

Original Manuscript: April 20, 2007

Revised Manuscript: May 25, 2007

Manuscript Accepted: May 31, 2007

Published: June 6, 2007

**Virtual Issues**

Vol. 2, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Lingfeng Yu, Bin Rao, Jun Zhang, Jianping Su, Qiang Wang, Shuguang Guo, and Zhongping Chen, "Improved lateral resolution in optical coherence tomography by digital focusing using two-dimensional numerical diffraction method," Opt. Express **15**, 7634-7641 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-12-7634

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### References

- 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]
- A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. Elzaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117, 43-48 (1995). [CrossRef]
- G. Hausler and M. W. Linduer, "Coherence radar and spectral radar-new tools for dermatological diagnosis," J. Biomed. Opt. 3, 21-31 (1998). [CrossRef]
- 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]
- B. Cense, N. A. Nassif, T. C. Chen, M. C. Pierce, S. 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]
- S. Jiao, R. Knighton, X. Huang, G. Gregori, and C. A. Puliafito, "Simultaneous acquisition of sectional and fundus ophthalmic images with spectral-domain optical coherence tomography," Opt. Express 13, 444- 452 (2005). [CrossRef] [PubMed]
- S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftimia, and B. E. Bouma, "High-speed optical frequency domain imaging," Opt. Express 11, 2953-2963 (2003). [CrossRef] [PubMed]
- 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]
- S. H. Yun, C. Boudoux, G. J. Tearney, and B. E. Bouma, "High-speed wavelength-swept semiconductor laser with a polygon-scanner-based wavelength filter," Opt. Lett. 28, 1981-1983 (2003). [CrossRef] [PubMed]
- R. Huber, M. Wojtkowski, K. Taira, J. G. Fujimoto, and K. Hsu, "Amplified, frequency swept lasers for frequency domain reflectometry and OCT imaging: design and scaling principles," Opt. Express 13, 3513- 3528 (2005). [CrossRef] [PubMed]
- R. Huber, D. C. Adler, and J. G. Fujimoto, "Buffered Fourier domain mode locking: unidirectional swept laser sources for optical coherence tomography imaging at 370,000 lines/s," Opt. Lett. 31, 2975-2977 (2006). [CrossRef] [PubMed]
- R. A. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, "Performance of fourier domain vs. time domain optical coherence tomography," Opt. Express 11, 889-894 (2003). [CrossRef] [PubMed]
- J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, "Improved signal-to noise ratio in spectral-domain compared with time-domain optical coherence tomography," Opt. Lett. 28, 2067-2069 (2003). [CrossRef] [PubMed]
- M. A. Choma, M. V. Sarunic, C. H. Yang, and J. A. Izatt, "Sensitivity advantage of swept source and Fourier domain optical coherence tomography," Opt. Express 11, 2183-2189 (2003). [CrossRef] [PubMed]
- B. Hermann, E. J. Fernandez, A. Unterhuber, H. Sattmann, A. F. Fercher, W. Drexler, P. M. Prieto, and P. Artal, "Adaptive-optics ultrahigh-resolution optical coherence tomography," Opt. Lett. 29, 2142-2144 (2004). [CrossRef] [PubMed]
- Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, "High-resolution optical coherence tomography over a large depth range with an axicon lens," Opt. Lett. 27, 243-245 (2002). [CrossRef]
- Y. Wang, Y. Zhao, J. S. Nelson, and Z. Chen, "Ultrahighresolution optical coherence tomography by broadband continuum generation from a photonic crystal fiber," Opt. Lett. 28, 182-184 (2003). [CrossRef] [PubMed]
- M. J. Cobb, X. Liu, and X. Li, "Continuous focus tracking for real-time optical coherence tomography," Opt. Lett. 30, 1680-1682 (2005). [CrossRef] [PubMed]
- T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, "Inverse scattering for optical coherence tomography," J. Opt. Soc. Am. A 23, 1027-1037 (2006). [CrossRef]
- M. D. Kulkarni, C. W. Thomas, and J. A. Izatt, "Image enhancement in optical coherence tomography using deconvolution," Electron. Lett. 331365-1367 (1997). [CrossRef]
- J. M. Schmitt, "Restoration of optical coherence images of living tissue using the clean algorithm," J. Biomed. Opt. 3, 66-75 (1998). [CrossRef]
- D. Piao, Q. Zhu, N. Dutta, S. Yan, and L. Otis, "Cancellation of coherent artifacts in optical coherence tomography imaging," Appl. Opt. 40, 5124-5131 (2001). [CrossRef]
- J. Hsu, C.W. Sun, C.W. Lu, C. C. Yang, C. P. Chiang, and C.W. Lin, "Resolution improvement with dispersion manipulation and a retrieval algorithm in optical coherence tomography," Appl. Opt. 42, 227-234 (2003). [CrossRef] [PubMed]
- Y. Yasuno, J. -i. Sugisaka, Y. Sando, Y. Nakamura, S. Makita, M. Itoh, and T. Yatagai, "Non-iterative numerical method for laterally super resolving Fourier domain optical coherence tomography," Opt. Express 14, 1006-1020 (2006). [CrossRef] [PubMed]
- T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, "Inverse scattering for high-resolution interferometric microscopy," Opt. Lett. 31, 3585-3587 (2006). [CrossRef] [PubMed]
- T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart. "Interferometric Synthetic Aperture Microscopy," Nat. Phys. 3, 129-134, (2007). [CrossRef]
- R. M. Lewis, "Physical optics inverse diffraction," IEEE Trans. Antennas Propag. AP-17, 308-314 (1969). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
- L. Yu and M. K. Kim, "Wavelength-scanning digital interference holography for tomographic three-dimensional imaging by use of the angular spectrum method," Opt. Lett. 30, 2092-2094 (2005). [CrossRef] [PubMed]

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