## Optimal spectral reshaping for resolution improvement in optical coherence tomography

Optics Express, Vol. 14, Issue 13, pp. 5909-5915 (2006)

http://dx.doi.org/10.1364/OE.14.005909

Acrobat PDF (172 KB)

### Abstract

We analyze the resolution limit that can be achieved by means of spectral reshaping in optical coherence tomography images and demonstrate that the resolution can be improved by means of modelessly reshaping the source spectrum in postprocessing. We show that the optimal spectrum has *a priory* surprising “crater-like” shape, providing 0.74 micron axial resolution in free-space. This represents ~50% improvement compared to resolution using the original spectrum of a white light lamp.

© 2006 Optical Society of America

## 1. Introduction

*in vivo*imaging technique, which has undergone rapid development since its invention in 1991 [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 (1991). [CrossRef] [PubMed]

*l*

_{c}of a light source. If the spectrum of the light source has a Gaussian shape,

*l*

_{c}is proportional to

*/Δλ*with

*λ*

_{0}the central wavelength and

*Δλ*the bandwidth of the emission spectrum, respectively [2]. Depending on the choice of a light source, the axial resolution of a typical OCT system varies from ~1 to 30 µm. Several techniques have been proposed to improve the resolution of OCT. One possible strategy is to develop new light sources with shorter coherence lengths. For example, a Kerr-lens mode-locked Ti:sapphire laser [3

3. B. Bouma, G. Tearney, S. Boppart, M. Hee, M. Brezinski, and J. Fujimoto, “High-resolution optical coherence tomographic imaging using a mode-locked Ti:Al2O3 laser source,” Opt. Lett. **20**1486–1488 (1995) [CrossRef] [PubMed]

4. B. Povazay, K. Bizheva, A. Unterhuber, B. Hermann, H. Sattmann, A. F. Fercher, W. Drexler, A. Apolonski, W. J. Wadsworth, J. C. Knight, P. St. J. Russell, M. Vetterlein, and E. Scherzer, “Submicrometer axial resolution optical coherence tomography,” Opt. Lett. **27**1800–1802 (2002) [CrossRef]

5. Wolfgang Drexler, “Ultrahigh-resolution optical coherence tomography,” J. Biomed. Opt. **9**, 47–74 (2004) [CrossRef] [PubMed]

6. A. Dubois, G. Moneron, K. Grieve, and A.C. Boccara, “Three-dimensional cellular-level imaging using full-field optical coherence tomography,” Phys. Med. Biol. **49**1227–1234 (2004) [CrossRef] [PubMed]

7. A. Wax, C.H. Yang, and J.A. Izatt, “Fourier-domain low-coherence interferometry for light-scattering spectroscopy,” Opt. Lett. **28**1230–1232 (2003) [CrossRef] [PubMed]

3. B. Bouma, G. Tearney, S. Boppart, M. Hee, M. Brezinski, and J. Fujimoto, “High-resolution optical coherence tomographic imaging using a mode-locked Ti:Al2O3 laser source,” Opt. Lett. **20**1486–1488 (1995) [CrossRef] [PubMed]

4. B. Povazay, K. Bizheva, A. Unterhuber, B. Hermann, H. Sattmann, A. F. Fercher, W. Drexler, A. Apolonski, W. J. Wadsworth, J. C. Knight, P. St. J. Russell, M. Vetterlein, and E. Scherzer, “Submicrometer axial resolution optical coherence tomography,” Opt. Lett. **27**1800–1802 (2002) [CrossRef]

5. Wolfgang Drexler, “Ultrahigh-resolution optical coherence tomography,” J. Biomed. Opt. **9**, 47–74 (2004) [CrossRef] [PubMed]

6. A. Dubois, G. Moneron, K. Grieve, and A.C. Boccara, “Three-dimensional cellular-level imaging using full-field optical coherence tomography,” Phys. Med. Biol. **49**1227–1234 (2004) [CrossRef] [PubMed]

7. A. Wax, C.H. Yang, and J.A. Izatt, “Fourier-domain low-coherence interferometry for light-scattering spectroscopy,” Opt. Lett. **28**1230–1232 (2003) [CrossRef] [PubMed]

8. Yan Zhang and Manabu Sato, “Resolution improvement in optical coherence tomography by optimal synthesis of light-emitting diodes,” Opt. Lett. **26**, 205–207 (2001) [CrossRef]

9. Renu Tripathi, Nader Nassif, J. Stuart Nelson, Boris Hyle Park, and Johannes F. de Boer, “Spectral shaping for non-Gaussian source spectra in optical coherence tomography,” Opt. Lett. **27**, 406–408 (2002) [CrossRef]

10. E. Smith, S. C. Moore, N. Wada, W. Chujo, and D. D. Sampson, “Spectral domain interferometry for OCDR using non-Gaussian broadband sources,” IEEE Photon. Technol. Lett. **13**, 64–66 (2001) [CrossRef]

11. A. Ceyhun Akcay, Jannick P. Rolland, and Jason M. Eichenholz, “Spectral shaping to improve the point spread function in optical coherence tomography,” Opt. Lett. **28**, 1921–1923 (2003) [CrossRef] [PubMed]

12. Daniel Marks, P. Scott Carney, and Stephen A. Boppart, “Adaptive spectral apodization for sidelobe reduction in optical coherence tomography images,” J. Biomed. Opt. **9**, 1281–1287 (2004) [CrossRef] [PubMed]

## 2. Theory and simulation

*I*(

*k,r*) is given by [2]

*k=2π/λ*is the wave number;

*n*is the refractive index of the sample;

*z*

_{0}is the path length from the beam splitter to the top surface of the sample,

*z*is the depth of the sample,

*r*is the path difference between the reference mirror and the top surface of the sample;

*a*(

*z*) is the depth-resolved backscattered amplitude of the sample;

*P*(

*k*),

*Q*(

*k*),

*R*(

*k*) and

*S*(

*k*) are the source spectrum, spectral response of the detector, reflection spectrum of the reference mirror, and the scattering spectrum in the sample (dispersion within the sample was neglected), respectively;

*B*≡

*a*(z)a(z′)exp[2

*ink*(

*z-z*

^{′})]

*dzdz*

^{′}describes the mutual interference of the light scattered from the sample;

*I*

_{d}(

*k, r*)=2

*a*(

*z*)cos2

*k*(

*r-nz*)

*dz/B*

^{1/2}decodes the depth-resolved OCT signal from the sample a(z), which we aim to obtain. In an experiment, by blocking the reference or sample arm, respectively,

*P*(

*k*)

*Q*(

*k*)

*S*(

*k*)

*B*and

*P*(

*k*)

*Q*(

*k*)

*R*(

*k*) can be recorded, and

*I*

_{d}(

*k,r*) can then be obtained as

*k*

_{1}and

*k*

_{2}, respectively, the sum of the interference signal weighted with a distribution

*V*(

*k*) within the range [

*k*

_{1},

*k*

_{2}] can be expressed as follows:

*V*(

*k*) can be reshaped in postprocessing. In order to improve the resolution, the width of PSF has to be minimized.

*rect*(

*k*

_{1},

*k*

_{2}) is the rectangle function which equals to 1 within the range [

*k*

_{1},

*k*

_{2}] and zero elsewhere, and

*U*(

*Z*)=

*π*(

*k*

_{2}-

*k*

_{1})cos[

*k*

_{1}+

*k*)2]sinc[

*k*

_{2}-

*k*

_{1})Z]/4 is a high frequency oscillation term whose envelope is a sinc function. If

*V*(

*k*) is a constant (i.e. white light source), the first term in Eq. (5) would be a delta function, and PSF would equal to

*U*(

*Z*). The peak of this PSF is narrow; however the sidelobes are large. If

*V*(

*k*) is Gaussian or Hamming windowed, the sidelobes can be significantly compressed, however the peak of the PSF is broadened. In this paper we optimized

*V*(

*k*) so that

*U*(

*Z*) functions with different shifts compensate each other to create a narrower PSF peak than that of white light, at the same time the sidelobes are suppressed to a lower level than that of white light also.

*V*(

*k*). We discretize the optical path difference (OPD) and wave number as

*Z*

_{i}(

*i*=1, 2,…

*M*) and

*k*

_{j}(

*j*=1, 2,…

*W*), respectively, and rewrite Eq. (5) in matrix notations:

*f*

_{i}is the PSF for

*Z*

_{i}

*, v*

_{j}

*=V*(

*k*

_{j}), and

*g*

_{ij}=cos2

*k*

_{j}

*Z*

_{i}. In order to find the optimal spectrum

*V*, we solved the nested optimization problem:

*FWHM*represents the full-width half-maximum. The optimization procedure is shown as Fig. 1. Since the shape of the PSF of a broad-band source is similar to the sinc function, we selected the sinc function with a variable width

*T*, sinc(

*Z/T*), as the target PSF. The initial width

*T*was set to

*1/k*

_{1}. In the

*m*-th iteration, the target function

*F*

_{m}was chosen as sinc(

*Z/T*

_{m}), the half-optimal spectrum

*V*

_{m}was found so that ‖

*GV*

_{m}-

*F*

_{m}

*D*

_{m}, was calculated by Hilbert transform. In order to optimize

*F*, we applied a 0.1% perturbation to

*T*

_{m}to obtained another FWHM

*T*

_{m}was set as

*T*

_{m-1}+

*γ*(

*D*

_{m-1}-

*D*

^{′m-1}to give a PSF with narrower peak. γ is a proportional coefficient. Normally a smaller value of

*γ*leads to more stable convergence but also increases the complexity of computations. In our computations, the convergence criterion was set as

*γ*(

*D-D*

^{′})<10

^{-5}

*T*

_{m}, and

*γ*was chosen as 10 to ensure the convergence after 100–1000 iterations. The reflective Newton iterative method [13

13. T.F. Coleman and Y. Li, “A Reflective Newton Method for Minimizing a Quadratic Function Subject to Bounds on Some of the Variables,” SIAM J. Optimiz. **6**, 1040–1058 (1996) [CrossRef]

*v*

_{j}, was chosen to solve the matrix-form least-squares problem, and set the lower limit of

*v*

_{j}to zero to satisfy the physics of real light sources. The system is spectrally sensitive, and the signal-to-noise ratio (SNR) values differ significantly at different wave numbers. In order to limit the use of noisy spectral data, we applied an additional condition

*v*

_{w}

*/SNR*

_{w}<

*α*for the optimization, where

*SNR*

_{w}is the signal-to-noise ratio at wave number

*v*

_{w}which was obtained by dividing the signal level at at wave number

*v*

_{w}by the amplitude of background white noise.

*α*is a threshold controlling the trade-off of the SNR and the FWHM. For a less

*α*value, the SNR has larger effect while FWHM has lower effect on the optimization procedure. After a few attempts we selected

*α*as

*W*, the number of the wave numbers.

*a priory*surprising asymmetric crater-like shape with two peaks close to the lower and upper bounds of the spectrum, and the peak at shorter wavelength is higher. It can be understood with this way: the resolution is primarily limited by the shortest and longest detectable wavelengths, especially the shortest detectable wavelength, so the side-spectrum enhancement, especially the enhancement at the shorter wavelength side, helps improve the resolution. In Fig. 2 we also show five other spectra: the modified optimal spectrum which is more symmetric with the amplitude of the two peaks being the same, Gaussian profile with FWHM =300 nm, white light, white light weighted with Hamming window, and the xenon lamp spectrum. To be consistent with our experimental system, a wavelength band from 421 nm to 895 nm was used. The envelopes of theoretical PSFs of the six spectra are shown in Fig. 4(a), and their FWHM values are given in the notation text. Evidently, the optimal spectrum provides the highest FWHM resolution, 0.44 µm in free space, with sidelobes lower than those of the PSF generated by the white light spectrum. The PSF peak of the modified optimal spectrum is slightly broader than that of the optimal spectrum, which proves that an asymmetric spectral profile may lead to a higher resolution than a symmetric spectral profile. To quantitatively compare the sidelobes effect of the PSFs, we calculated the root-mean square width (RMSW) [∫

*z*

^{2}|

*PSF*(

*z*)|

^{2}

*dz*/∫|

*PSF*(

*z*)|

^{2}

*dz*]

^{1/2}which represents the deviation of the PSF from the central position

*z*=0 [12

12. Daniel Marks, P. Scott Carney, and Stephen A. Boppart, “Adaptive spectral apodization for sidelobe reduction in optical coherence tomography images,” J. Biomed. Opt. **9**, 1281–1287 (2004) [CrossRef] [PubMed]

## 3. Experimental results

*I*(

*λ, r*) was recorded at each OPD step. For calibration,

*I*(

*λ, r*) reflected by the reference mirror and the sample were also recorded at both the beginning and the end of each imaging experiment. We then followed Eq. (2) to obtain

*I*

_{d}(

*k,r*) for different

*k*and

*r*values, and applied different spectra

*V*(

*k*) to get the corresponding

*I*

_{d}(

*r*) values according to Eq. (3).

*a*(

*z*) can be regarded as a delta function. Thus the resulted

*I*

_{d}(

*r*) gave the PSFs for different source spectra

*V*(

*k*). Fig. 3(b) shows the envelope of the experimental PSFs. There are multi-peaks in the theoretical PSF, with about 100nm between the adjacent peaks. In the experiments, the un-flatness of components surfaces and the dispersion induced optical path mismatch, which are in the order of 100nm, smooth the peaks. So there is no multiple peaks in the experimental PSFs. The corresponding FWHM and RMSW values are given in Table 1. As shown in Fig. 3(b) and Table 1, the modeless spectral reshaping indeed provides resolution superior to those obtained with other conventionally used spectra including the original xenon-lamp spectrum, white light, Gaussian spectrum, and Hamming windowed profile, with the FWHM reduced by 51%, 20%, 28%, and 30%, respectively. Although the spectrum was not purposely reshaped to optimize the sidelobes, we also notice that the optimal spectrum suppress sidelobes compared to the original xenon-lamp spectrum with RMSW reduced by 54%. The sidelobes are larger, however, than those of corresponding to the Gaussian and Hamming profiles.

## 4. Conclusions

*a priory*surprising crater-like spectral shape, which is different from the conventionally adopted bell-shape spectra. Although the resolution provided by the optimal spectrum is primarily determined by the spectral range, we further demonstrated, for the first time to our knowledge, the optimal spectral reshaping results in an extra improvement in resolution. In particular, in our experiments, the resolution of 0.74 µm in free space was obtained, which is 51% higher than the one provided by the original xenon-lamp spectrum. Therefore, the modeless spectral reshaping technique provides a flexible way to improve the resolution of OCT and may help achieve sub-micron resolution for subcellular OCT imaging.

## Acknowledgments

## 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. | Brett E. Bouma and Guillermo J. Tearney, |

3. | B. Bouma, G. Tearney, S. Boppart, M. Hee, M. Brezinski, and J. Fujimoto, “High-resolution optical coherence tomographic imaging using a mode-locked Ti:Al2O3 laser source,” Opt. Lett. |

4. | B. Povazay, K. Bizheva, A. Unterhuber, B. Hermann, H. Sattmann, A. F. Fercher, W. Drexler, A. Apolonski, W. J. Wadsworth, J. C. Knight, P. St. J. Russell, M. Vetterlein, and E. Scherzer, “Submicrometer axial resolution optical coherence tomography,” Opt. Lett. |

5. | Wolfgang Drexler, “Ultrahigh-resolution optical coherence tomography,” J. Biomed. Opt. |

6. | A. Dubois, G. Moneron, K. Grieve, and A.C. Boccara, “Three-dimensional cellular-level imaging using full-field optical coherence tomography,” Phys. Med. Biol. |

7. | A. Wax, C.H. Yang, and J.A. Izatt, “Fourier-domain low-coherence interferometry for light-scattering spectroscopy,” Opt. Lett. |

8. | Yan Zhang and Manabu Sato, “Resolution improvement in optical coherence tomography by optimal synthesis of light-emitting diodes,” Opt. Lett. |

9. | Renu Tripathi, Nader Nassif, J. Stuart Nelson, Boris Hyle Park, and Johannes F. de Boer, “Spectral shaping for non-Gaussian source spectra in optical coherence tomography,” Opt. Lett. |

10. | E. Smith, S. C. Moore, N. Wada, W. Chujo, and D. D. Sampson, “Spectral domain interferometry for OCDR using non-Gaussian broadband sources,” IEEE Photon. Technol. Lett. |

11. | A. Ceyhun Akcay, Jannick P. Rolland, and Jason M. Eichenholz, “Spectral shaping to improve the point spread function in optical coherence tomography,” Opt. Lett. |

12. | Daniel Marks, P. Scott Carney, and Stephen A. Boppart, “Adaptive spectral apodization for sidelobe reduction in optical coherence tomography images,” J. Biomed. Opt. |

13. | T.F. Coleman and Y. Li, “A Reflective Newton Method for Minimizing a Quadratic Function Subject to Bounds on Some of the Variables,” SIAM J. Optimiz. |

**OCIS Codes**

(100.2980) Image processing : Image enhancement

(170.4500) Medical optics and biotechnology : Optical coherence tomography

(350.5730) Other areas of optics : Resolution

**ToC Category:**

Image Processing

**History**

Original Manuscript: March 16, 2006

Revised Manuscript: May 30, 2006

Manuscript Accepted: June 4, 2006

Published: June 26, 2006

**Virtual Issues**

Vol. 1, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Jianmin Gong, Bo Liu, Young L. Kim, Yang Liu, Xu Li, and Vadim Backman, "Optimal spectral reshaping for resolution improvement in optical coherence tomography," Opt. Express **14**, 5909-5915 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-13-5909

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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 (1991). [CrossRef] [PubMed]
- Brett E. Bouma, Guillermo J. Tearney, Handbook of optical coherence tomograph, (Marcel Dekker, New York, 2002)
- B. Bouma, G. Tearney, S. Boppart, M. Hee, M. Brezinski, and J. Fujimoto, "High-resolution optical coherence tomographic imaging using a mode-locked Ti:Al2O3 laser source," Opt. Lett. 201486-1488 (1995) [CrossRef] [PubMed]
- B. Povazay, K. Bizheva, A. Unterhuber, B. Hermann, H. Sattmann, A. F. Fercher, W. Drexler, A. Apolonski, W. J. Wadsworth, J. C. Knight, P. St. J. Russell, M. Vetterlein and E. Scherzer, "Submicrometer axial resolution optical coherence tomography," Opt. Lett. 271800-1802 (2002) [CrossRef]
- Wolfgang Drexler, "Ultrahigh-resolution optical coherence tomography," J. Biomed. Opt. 9, 47-74 (2004) [CrossRef] [PubMed]
- A. Dubois, G. Moneron, K. Grieve and A.C. Boccara, "Three-dimensional cellular-level imaging using full-field optical coherence tomography," Phys. Med. Biol. 491227-1234 (2004) [CrossRef] [PubMed]
- A. Wax, C.H. Yang, J.A. Izatt, "Fourier-domain low-coherence interferometry for light-scattering spectroscopy," Opt. Lett. 281230-1232 (2003) [CrossRef] [PubMed]
- Yan Zhang, Manabu Sato, "Resolution improvement in optical coherence tomography by optimal synthesis of light-emitting diodes," Opt. Lett. 26, 205-207 (2001) [CrossRef]
- Renu Tripathi, Nader Nassif, J. Stuart Nelson, Boris Hyle Park and Johannes F. de Boer, "Spectral shaping for non-Gaussian source spectra in optical coherence tomography," Opt. Lett. 27, 406-408 (2002) [CrossRef]
- E. Smith, S. C. Moore, N. Wada, W. Chujo, and D. D. Sampson, "Spectral domain interferometry for OCDR using non-Gaussian broadband sources," IEEE Photon. Technol. Lett. 13, 64-66 (2001) [CrossRef]
- A. Ceyhun Akcay, JannickP. Rolland and Jason M. Eichenholz, "Spectral shaping to improve the point spread function in optical coherence tomography," Opt. Lett. 28, 1921-1923 (2003) [CrossRef] [PubMed]
- Daniel Marks, P. S. Carney, Stephen A. Boppart, "Adaptive spectral apodization for sidelobe reduction in optical coherence tomography images," J. Biomed. Opt. 9, 1281-1287 (2004) [CrossRef] [PubMed]
- T.F. Coleman, and Y. Li, "A Reflective Newton Method for Minimizing a Quadratic Function Subject to Bounds on Some of the Variables," SIAM J. Optimiz. 6, 1040-1058 (1996) [CrossRef]

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