## Improved tomographic imaging of wavelength scanning digital holographic microscopy by use of digital spectral shaping

Optics Express, Vol. 15, Issue 3, pp. 878-886 (2007)

http://dx.doi.org/10.1364/OE.15.000878

Acrobat PDF (863 KB)

### Abstract

The technique of wavelength scanning digital holographic microscopy (WSDHM) is improved by use of a digital spectral shaping method which is used to suppress the sidelobes of the amplitude modulation function in WSDHM for non-Gaussian-shaped source spectra. Spurious structures caused by sidelobes can be eliminated in tomographic imaging and the performance of the tomographic system greatly improved. Detailed theoretical analysis is given. Both simulation and experimental results are presented to verify the idea.

© 2007 Optical Society of America

## 1. Introduction

1. U. Schnars and W. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. **13**,85–101 (2002). [CrossRef]

2. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. **24**,291-3 (1999). [CrossRef]

3. G. Indebetouw, “Properties of a scanning holographic microscope: improved resolution, extended depth-of-focus, and/or optical sectioning,” J. Mod. Opt. ,**49**,1479–1500 (2002). [CrossRef]

4. E. Wolf, “Three-dimensional structure determination of semitransparent object from holographic data,” Opt. Commun. **1**,153–156 (1969). [CrossRef]

8. F. Charrière, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett. **31**,178–180 (2006) [CrossRef] [PubMed]

9. M. K. Kim, “Tomographic three-dimensional imaging of a biological specimen using wavelength-scanning digital interference holography,” Opt. Express **7**,305–310 (2000). [CrossRef] [PubMed]

12. L. Yu and M. K. Kim, “Wavelength scanning digital interference holography for variable tomographic scanning,” Opt. Express. **13**,5621–5627 (2005). [CrossRef] [PubMed]

13. F. Montfort, T. Colomb, F. Charrière, J. Kühn, P. Marquet, E. Cuche, S. Herminjard, and C. Depeursinge, “Submicrometer optical tomography by multiple-wavelength digital holographic microscopy,” Appl. Opt. **45**,8209–8217 (2006) [CrossRef] [PubMed]

9. M. K. Kim, “Tomographic three-dimensional imaging of a biological specimen using wavelength-scanning digital interference holography,” Opt. Express **7**,305–310 (2000). [CrossRef] [PubMed]

## 2. Principle

9. M. K. Kim, “Tomographic three-dimensional imaging of a biological specimen using wavelength-scanning digital interference holography,” Opt. Express **7**,305–310 (2000). [CrossRef] [PubMed]

*λ*, any point

*P*on the object at

**r**scatters the incident beam into a Huygens wavelet

_{p}*A*(

**r**), so that the resultant field

_{p}*E*(

**r**) at

**r**is

*k*is the wave number and the integral is over the whole object volume. Now let us repeat the above holographic process using different wave numbers, and all the other conditions of the object or illumination are kept the same. If the reconstructed fields are all superposed together with infinite wave numbers, then the resultant field is

*k*

_{min},

*k*

_{max}], with a bandwidth of Δ

*k*=

*k*

_{max}-

*k*

_{min}. Practically, if one uses a finite number

*N*of wave numbers at regular intervals of

*k*

_{min},

*k*

_{max}], the above equation can be written as

*π*[

*dk*]

^{-1}and the axial resolution

*δ*=

*Λ*/

*N*= 2

*π*/Δ

*k*. Fig. 1 shows an example of an absolute AMF. Obviously, since the wavelength bandwidth of the light source is fixed,

*δ*is always the same for different

*N*. But the more wave numbers used, the bigger interval Λ between the two peaks of the modulation function. Thus for tomographic imaging, other than the diffraction or defocusing effect of propagation, the reconstructed object image

*A*(

**r**) will also repeat itself at a beat wavelength Λ with axial resolution of δ. By using appropriate values of

*dk*and

*N*, the beat wavelength Λ can be matched to the axial extent of the object and

*δ*to the desired level of axial resolution.

*k*

_{min},

*k*

_{max}] are continuously scanned for illumination and reconstruction, it can be easily shown that the normalized amplitude modulation function finally become a sinc function as

14. 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]

**r**and k space form a Fourier transform pair, equally k-spaced wavelengths are always preferred for scanning. A tunable laser is normally used as a light source which is sequentially scanned to obtain the equally k-spaced wavelengths. Each wavelength corresponds to a quasi-Dirac spectrum (very narrow compared to the tuning range) of the source. However, all previously reported WSDHM systems gave these different wavelengths the same weight, or the reconstructed wavefields from each wavelength were numerically normalized. This resulted in a synthetic rectangular or limited-sampled-rectangular shape spectra of the light source; thus either the AMF from Eq.(4) or Eq.(5) will cause big sidelobes which are not well suppressed. These sidelobes will generate severe spurious structures in tomographic imaging and increase the average noise level of the reconstruction.

15. W. Drexler, U. Morgner, F. X. Krtner, C. Pitris, S. A. Boppart, X. D. Li, E. P. Ippen, and J. G. Fujimoto, “Invivo ultrahigh-resolution optical coherence tomography,” Opt. Lett. **24**,1221–1223 (1999) [CrossRef]

16. I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, “Ultrahigh-resolution optical coherence tomography using continuum generation in an air silica microstructure optical fiber,” Opt. Lett. **26**,608–610 (2001) [CrossRef]

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

19. A F Fercher, W Drexler, C K Hitzenberger, and T Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. **66**,239–303 (2003) [CrossRef]

*k*̄ is the center wave number and 2

*σ*is the standard deviation power spectral bandwidth.

_{k}*M*(|

**r**-

**r**|) now becomes the Fourier transform of the Gaussian spectra

_{p}*S*(

*k*-

*k*̄). Thus, the amplitude modulation function contains a Gaussian envelope as well with a characteristic standard deviation spatial width 2

*σ*that is inversely proportional to the power spectral bandwidth, which means that

_{x}*σ*= 1. This is the limiting case of a general inequality on the product of variances of Fourier transform pairs. In general, if

_{x}σ_{k}*S*is an arbitrary distribution and

*M*is its Fourier transform, then the product of the variations is greater than one. This confirms the Fourier uncertainty relation that the product of variances of a Fourier transform pair reaches its minimum for Gaussian functions. If the above product is not minimized, then the AMF must not be a Gaussian. In this paper, a final Gaussian shape is always aimed by spectral shaping. In numerical implementation, an

*N*-point Gaussian spectra covering the spectrum range of [

*k*

_{1},

*k*

_{N}] is obtained by digitizing Eq. (6) as

*n*≤

*N*-1, and

*α*is a parameter introduced to adjust the width of the Gaussian spectra. We have ignored the constant before the exponential term in Eq.(6). The influence of the parameter

*α*on spectral shaping will be studied below.

## 3. Simulations and Experiments

*μm*which can now be considered as the axis resolution of the WSDHM system.

*α*on spectral shaping is also studied. Fig. 3 shows the amplitude modulation function in linear and decibel scales when different values of a are used in Eq.(9) for Gaussian spectral shaping. According to the property of Gaussian function, the width of the Gaussian envelope is inversely proportional to the value of

*α*: a bigger value of

*α*(

*α*≥ 2) induces a more narrow spectra but a broader AMF. A smaller

*α*(

*α*< 2), however, will cut a portion of a broader Gaussian envelope; thus, sidelobes will appear in AMF, as shown in Fig. 3. Therefore,

*α*≥ 2 is preferred in our application.

*α*= 2.5 will normally guarantee both a good axial resolution and well-suppressed sidelobes in AMF. We will use this value for spectral shaping in the following section.

*μm*pitch size and 12-bit gray scale output. A camera link cable connected the camera to the desktop computer which processed the acquired images and calculated the holographic diffraction using a number of programs based on LabVIEW® and MatLab®.

^{2}, 256×256 pixels, was observed by the WSDHM system. There are two ways to perform reconstruction. If we consider the reconstruction in the “image domain,” then the CCD camera records the hologram, and we use the distance between the image plane (image of the object through L2) and the CCD plane for reconstruction. Alternatively, we could also consider the reconstruction in the “object domain.” In this case, the image of the CCD plane (S plane) records a hologram of the object, then the distance between the S plane and the object plane is directly used for reconstruction. We are using the latter way for reconstruction. The reconstruction distance z, representing the distance from the object to S plane in Fig. 4 was about 3mm. A series of holograms were recorded using 50 equally separated wave numbers. As has been discussed above, this gave a beat wavelength of 1034.1 μm. The image volume was calculated from each of the holograms by use of numerical algorithms in the computer and all such image volumes were numerically superposed to create the 3D tomographic image [11

11. 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]

12. L. Yu and M. K. Kim, “Wavelength scanning digital interference holography for variable tomographic scanning,” Opt. Express. **13**,5621–5627 (2005). [CrossRef] [PubMed]

*α*= 2.5 in Eq. (9) to guarantee both a good axial resolution and well suppressed sidelobes in AMF. The experimental AMF from a cross line is also shown in Fig. 6(b), and it fits well with its theoretical prediction. Clearly, the sidelobes have now been greatly suppressed in AMF. If we consider the spurious ripples as background noise and analyze the experimental AMFs of both the above cases in decibel scale, it shows that there is about 15 dB improvement in the average noise level, which results in a signal-to-noise ratio gain by using Gaussian spectral shaping.

11. 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]

## 4. Conclusion

**7**,305–310 (2000). [CrossRef] [PubMed]

## References and links

1. | U. Schnars and W. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. |

2. | E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. |

3. | G. Indebetouw, “Properties of a scanning holographic microscope: improved resolution, extended depth-of-focus, and/or optical sectioning,” J. Mod. Opt. , |

4. | E. Wolf, “Three-dimensional structure determination of semitransparent object from holographic data,” Opt. Commun. |

5. | W. H. Carter, “Computational reconstruction of scattering objects from holograms,” J. Opt. Soc. Am. |

6. | R. Dändliker and D. Weiss, “Reconstruction of three dimensional refractive index from scattered waves,” Opt. Commun. |

7. | A. F. Fercher, H. Bartelt, H. Becker, and E. Wiltschko, “Image formation by inversion of scattered field data: experiments and computational simulation,” Appl. Opt. |

8. | F. Charrière, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett. |

9. | M. K. Kim, “Tomographic three-dimensional imaging of a biological specimen using wavelength-scanning digital interference holography,” Opt. Express |

10. | M. K. Kim, L. Yu, and C. J. Mann, “Interference techniques in digital holography,” J. Opt. A, Pure Appl. Opt. |

11. | 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. |

12. | L. Yu and M. K. Kim, “Wavelength scanning digital interference holography for variable tomographic scanning,” Opt. Express. |

13. | F. Montfort, T. Colomb, F. Charrière, J. Kühn, P. Marquet, E. Cuche, S. Herminjard, and C. Depeursinge, “Submicrometer optical tomography by multiple-wavelength digital holographic microscopy,” Appl. Opt. |

14. | 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 |

15. | W. Drexler, U. Morgner, F. X. Krtner, C. Pitris, S. A. Boppart, X. D. Li, E. P. Ippen, and J. G. Fujimoto, “Invivo ultrahigh-resolution optical coherence tomography,” Opt. Lett. |

16. | I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, “Ultrahigh-resolution optical coherence tomography using continuum generation in an air silica microstructure optical fiber,” Opt. Lett. |

17. | Y. Zhang, M. Sato, and N. Tanno, “Resolution improvement in optical coherence tomography by optimal synthesis of light-emitting diodes,” Opt. Lett. |

18. | R. Tripathi, N. Nassif, J. S. Nelson, B. H. Park, and J. F. de Boer, “Spectral shaping for non-Gaussian source spectra in optical coherence tomography,” Opt. Lett. |

19. | A F Fercher, W Drexler, C K Hitzenberger, and T Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. |

20. | M. Alonso and G. W. Forbes, “Measures of spread for periodic distributions and the associated uncertainty relations,” Am. J. Phys. |

**OCIS Codes**

(090.1760) Holography : Computer holography

(110.0180) Imaging systems : Microscopy

(110.6880) Imaging systems : Three-dimensional image acquisition

**ToC Category:**

Holography

**History**

Original Manuscript: December 4, 2006

Revised Manuscript: January 11, 2007

Manuscript Accepted: January 11, 2007

Published: February 5, 2007

**Virtual Issues**

Vol. 2, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Lingfeng Yu and Zhongping Chen, "Improved tomographic imaging of wavelength scanning digital holographic microscopy by use of digital spectral shaping," Opt. Express **15**, 878-886 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-3-878

Sort: Year | Journal | Reset

### References

- U. Schnars and W. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, 85-101 (2002). [CrossRef]
- E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase-contrast imaging," Opt. Lett. 24, 291-3 (1999). [CrossRef]
- G. Indebetouw, "Properties of a scanning holographic microscope: improved resolution, extended depth-of-focus, and/or optical sectioning," J. Mod. Opt., 49, 1479-1500 (2002). [CrossRef]
- E. Wolf, "Three-dimensional structure determination of semitransparent object from holographic data," Opt. Commun. 1, 153-156 (1969). [CrossRef]
- W. H. Carter, "Computational reconstruction of scattering objects from holograms," J. Opt. Soc. Am. 60, 306-314 (1970). [CrossRef]
- R. Dändliker and D. Weiss, "Reconstruction of three dimensional refractive index from scattered waves," Opt. Commun. 1, 323-328 (1970). [CrossRef]
- A. F. Fercher, H. Bartelt, H. Becker, and E. Wiltschko, "Image formation by inversion of scattered field data: experiments and computational simulation," Appl. Opt. 18, 2427-2439 (1979). [CrossRef] [PubMed]
- F. Charrière, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, "Cell refractive index tomography by digital holographic microscopy," Opt. Lett. 31, 178-180 (2006) [CrossRef] [PubMed]
- M. K. Kim, "Tomographic three-dimensional imaging of a biological specimen using wavelength-scanning digital interference holography," Opt. Express 7, 305-310 (2000). [CrossRef] [PubMed]
- M. K. Kim, L. Yu, and C. J. Mann, "Interference techniques in digital holography," J. Opt. A, Pure Appl. Opt. 8, S518-S523 (2006). [CrossRef]
- 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]
- L. Yu and M. K. Kim, "Wavelength scanning digital interference holography for variable tomographic scanning," Opt. Express. 13, 5621-5627 (2005). [CrossRef] [PubMed]
- F. Montfort, T. Colomb, F. Charrière, J. Kühn, P. Marquet, E. Cuche, S. Herminjard, and C. Depeursinge, "Submicrometer optical tomography by multiple-wavelength digital holographic microscopy," Appl. Opt. 45, 8209-8217 (2006) [CrossRef] [PubMed]
- 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]
- W. Drexler, U. Morgner, F. X. Krtner, C. Pitris, S. A. Boppart, X. D. Li, E. P. Ippen, and J. G. Fujimoto, "Invivo ultrahigh-resolution optical coherence tomography," Opt. Lett. 24, 1221-1223 (1999) [CrossRef]
- I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, "Ultrahigh-resolution optical coherence tomography using continuum generation in an air silica microstructure optical fiber, " Opt. Lett. 26, 608-610 (2001) [CrossRef]
- Y. Zhang, M. Sato, and N. Tanno, "Resolution improvement in optical coherence tomography by optimal synthesis of light-emitting diodes, " Opt. Lett. 26, 205-207 (2001) [CrossRef]
- R. Tripathi, N. Nassif, J. S. Nelson, B. H. Park, and J. F. de Boer, "Spectral shaping for non-Gaussian source spectra in optical coherence tomography," Opt. Lett. 27, 406-408 (2002) [CrossRef]
- A F Fercher, W Drexler, C K Hitzenberger and T Lasser, "Optical coherence tomography - principles and applications," Rep. Prog. Phys. 66, 239-303 (2003) [CrossRef]
- M. Alonso, and G. W. Forbes, "Measures of spread for periodic distributions and the associated uncertainty relations," Am. J. Phys. 69, 340-347 (2001). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.