## Surpassing digital holography limits by lensless object scanning holography |

Optics Express, Vol. 20, Issue 9, pp. 9382-9395 (2012)

http://dx.doi.org/10.1364/OE.20.009382

Acrobat PDF (1587 KB)

### Abstract

We present lensless object scanning holography (LOSH) as a fully lensless method, capable of improving image quality in reflective digital Fourier holography, by means of an extremely simplified experimental setup. LOSH is based on the recording and digital post-processing of a set of digital lensless holograms and results in a synthetic image with improved resolution, field of view (FOV), signal-to-noise ratio (SNR), and depth of field (DOF). The superresolution (SR) effect arises from the generation of a synthetic aperture (SA) based on the linear movement of the inspected object. The same scanning principle enlarges the object FOV. SNR enhancement is achieved by speckle suppression and coherent artifacts averaging due to the coherent addition of the multiple partially overlapping bandpass images. And DOF extension is performed by digital refocusing to different object’s sections. Experimental results showing an impressive image quality improvement are reported for a one-dimensional reflective resolution test target.

© 2012 OSA

## 1. Introduction

## 2. Theoretical analysis

### 2.1 System description

### 2.2 Mathematical analysis

*O(x*This object is illuminated by a spherical divergent wave

_{1}, y_{1}).*Aexp[jk(x*coming from the point source that is initially located at an effective distance

_{1}^{2}+ y_{1}^{2})/(2D_{0})]*D*. This effective distance takes into account the passing of the spherical wave through the higher index medium of the BS. This is not troublesome since it affects equally to both interferometric branches.

_{0}*d*. In this particular case, the object is placed at half the distance between the illuminating point source and the CCD. Moreover the distance

_{1}= d_{2}*D*between the object and the CCD becomes equal to

*D*Thus, just after reflection on the input object, except for constant amplitude factors, the field distribution provided by the imaging arm is

_{0}.*(x*where the CCD is located can be written aswhere

_{2}, y_{2})_{2}/(λD), y

_{2}/(λD)) and C = 1/(jλD).

*D*in front of the CCD. If, moreover, the reference mirror is slightly tilted, the reference virtual point source becomes shifted a distance

*b*outside the optical axis. In our case, this shift is achieved by tilting both the BS and the reference mirror in the direction that is perpendicular to the object scanning direction. Assuming that the object is moved along the

*x*-direction,

*b*is perpendicular to Fig. 1 (

*y*-direction). For instance, when

*b<0*, the virtual reference point is located below the axis along the

*y*-direction. Thus, the reference beam can be written as

*I(x*provided by the addition of Eqs. (2) and (4). Assuming a linear relation between amplitude response and intensity in the reconstruction process, the two diffracted terms that give rise to the images at the Fourier domain can be written aswhere

_{2}, y_{2}) = |U(x_{2}, y_{2}) + R(x_{2}, y_{2})|^{2}*O(x*and

_{1}, y_{1})*O*(-x*. In any case and to get the final images, we need a spherical lens to carry out the FT of the hologram transmittance. If the focal length of the lens is

_{1}, -y_{1})*f*and we propagate to the back focal plane

*(x*of the lens, we get a complex amplitude proportional tobeing

_{3}, y_{3})*α = D/f*. Aside of phase factors, Eq. (7) represents a direct image of the input object centered at

*(0, b/α)*and an inverted image of the input object centered at

*(0, -b/α).*Obviously, all this reconstruction process is performed numerically.

*x*-direction). Then, each hologram provides a recovered image corresponding to a different area of the input object (that one illuminated during each specific recording). Finally, an image with improved performance is synthesized by properly assembling all those images.

*(0, b)*, the object point with coordinates

*(x*is encoded as a sinusoidal fringe pattern of spatial frequency given by

_{0}, y_{0})*L*centered on-axis, that is,

*O(x*, the intervals of frequencies where the object points are encoded are

_{1}, y_{1})rect(x_{0}/L, y_{0}/L)*P*the pixel size. Thus, the minimum fringe period is defined by

*2P*which corresponds with a maximum frequency in line pairs of

*ν = 1/(2P)*. And that frequency corresponds to the farthest point of the object with respect to the reference point. This condition can be written aswhere, for a given object and a given CCD detector, the maximum value of

*b*to assure that all the points of the object are recorded is given by

*y*-direction (as well as in the

*x*-direction) is limited by the size of the pixels in the detector. This limitation of the FOV is characteristic of the FT holograms. If we consider the experimental conditions of the proposed experimental setup, we are far from this limit when considering our fully recovered image.

*b/α = bf/D*along the

*y*-direction, and each one has to be contained in, at most, half of that matrix. In addition, the background terms (zero order term of the hologram) are located around the origin of the coordinate system and limit even more the available portion of the matrices for displaying the images. Note that, since both twin images are symmetrically placed from the center of the digital matrix, we focus only in one of them (the direct image).

*(0, b/α)*and it is inscribed in a square of side

*L/α*. Its farthest point is placed at a distance of “

*L/(2α)+b/α*”, while the closest one is at “

*-L/(2α)+b/α*”. Since, the background term has a half-height equal to

*L/α*, to avoid overlapping between the background and the image it is necessary that

*b*in order to optimize the experimental setup.

*x*-direction. Let us suppose that we move the object a distance

*ξ*from its initial position. The corresponding new intervals of frequencies are nowthat is, the same one in the

*y*-direction [

*f’*] and a new one in the multiplexed direction [

_{y1}, f’_{y2}*f’*]. Thus, when performing object scanning, the CCD records the information of different object regions in different spatial frequency bands. In other words, the new frequency interval [

_{x1}, f’_{x2}*f’*] contains higher spatial frequency information of the input object than the case where no scanning is considered. And this procedure is valid as far as the frequency of the interferometric pattern is lower than the Nyquist sampling frequency of the CCD. The farthest object point that can be recorded in the hologram is given by

_{x1}, f’_{x2}*u*, where NA is the numerical aperture of the detector and K = 0.82 for coherent imaging systems, approximately ([41], page 471). Then, the spatial frequency band transmitted by the system for the central object point along the direction of interest (

_{c}= NA/(kλ)*x*-direction) is

*ξ*, the detector will provide a spatial frequency band depending on the value of

*ξ*. For the central point, the transmitted band is

*ξ = H/2*, the transmitted band becomesand for

*ξ = H*, the band transmitted is

*x*-direction. Or in other words, a SA increasing by a factor of 3 the initial cutoff frequency can be numerically generated after properly managing all the information contained in the set of recorded holograms. Then, the resolution limit becomes also improved by a factor of 3. For those displacement values

*ξ≤H*, we will get partial overlapping of the different bands transmitted by the system. In particular, for

*ξ = H/2*, the aperture is expanded by a factor of 2.

### 2.3 Algorithmic

## 3. Experimental validation

### 3.1 FOV enlargement

*D*between the test target and the CCD (see subsection 3.2).

_{0}= 13.6 cm### 3.2 Resolution improvement

*Δs = e(1-1/n)*, being

*e*and

*n*the width and the refractive index of the plate, respectively. According to technical specifications provided by Schott (http://www.schott.com), the refractive index of BK7 optical glass at the VCSEL wavelength is 1.51, and then the shift introduced by the BS cube is around

*Δs = 8.6 mm*. Thus, the effective distance (

*D*) between the test target and the CCD is

_{0}*D*, approximately. We want to recall that, in our setup,

_{0}= 13.6 cm*D = D*.

_{0}*NA ≅ H/(2D*, being H the CCD width. This value (NA = 0.02) provides a theoretical resolution limit equal to

_{0})*R = kλ/NA = 35 μm*. Figure 5(a) depicts the low resolution image provided by the proposed layout and corresponding with the case when the last resolved element of the test is placed on-axis, that is, centered in the FOV. Otherwise, the element will be illuminated with a slightly tilted beam and will produce an image with improved resolution. We can see as the last resolved element is the 5th element (starting from the highest period and going down) of the 2nd group (identified by two black squares in the test) which is marked with a black arrow and corresponding with 25 lp/mm or, equivalently, 40 μm pitch. This experimental result matches with the theoretical one (35 μm) since the next element in the resolution test (6th element of 2nd group corresponding with 30 lp/mm or 33 μm) is below the theoretical resolution limit.

### 3.3 SNR enhancement

### 3.4 DOF extension

## 4. Conclusions

## Acknowledgment

## References and links

1. | L. P. Yaroslavsky, |

2. | U. Schnars and W. P. O. Jüpter, |

3. | J. W. Goodman, |

4. | J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected
holograms,” Appl. Phys. Lett. |

5. | T. Huang, “Digital holography,”
Proc. IEEE |

6. | F. Le Clerc, M. Gross, and L. Collot, “Synthetic-aperture experiment in the visible with
on-axis digital heterodyne holography,” Opt. Lett. |

7. | J. H. Massig, “Digital off-axis holography with a synthetic
aperture,” Opt. Lett. |

8. | P. Almoro, G. Pedrini, and W. Osten, “Aperture synthesis in phase retrieval using a
volume-speckle field,” Opt. Lett. |

9. | J. Di, J. Zhao, H. Jiang, P. Zhang, Q. Fan, and W. Sun, “High resolution digital holographic microscopy with a
wide field of view based on a synthetic aperture technique and use of linear CCD
scanning,” Appl. Opt. |

10. | V. Micó, L. Granero, Z. Zalevsky, and J. García, “Superresolved phase-shifting Gabor holography by CCD
shift,” J. Opt. A, Pure Appl. Opt. |

11. | B. Katz and J. Rosen, “Super-resolution in incoherent optical imaging using
synthetic aperture with Fresnel elements,” Opt.
Express |

12. | C. Yuan, H. Zhai, and H. Liu, “Angular multiplexing in pulsed digital holography for
aperture synthesis,” Opt. Lett. |

13. | P. Feng, X. Wen, and R. Lu, “Long-working-distance synthetic aperture Fresnel
off-axis digital holography,” Opt. Express |

14. | L. Granero, V. Micó, Z. Zalevsky, and J. García, “Synthetic aperture superresolved microscopy in digital
lensless Fourier holography by time and angular multiplexing of the object
information,” Appl. Opt. |

15. | V. Micó and Z. Zalevsky, “Superresolved digital in-line holographic microscopy
for high-resolution lensless biological imaging,” J. Biomed.
Opt. |

16. | L. Granero, Z. Zalevsky, and V. Micó, “Single-exposure two-dimensional superresolution in
digital holography using a vertical cavity surface-emitting laser source
array,” Opt. Lett. |

17. | Y. Kuznetsova, A. Neumann, and S. R. Brueck, “Imaging interferometric microscopy-approaching the
linear systems limits of optical resolution,” Opt.
Express |

18. | V. Micó, Z. Zalevsky, C. Ferreira, and J. García, “Superresolution digital holographic microscopy for
three-dimensional samples,” Opt. Express |

19. | T. R. Hillman, T. Gutzler, S. A. Alexandrov, and D. D. Sampson, “High-resolution, wide-field object reconstruction with
synthetic aperture Fourier holographic optical microscopy,”
Opt. Express |

20. | M. Kim, Y. Choi, C. Fang-Yen, Y. Sung, R. R. Dasari, M. S. Feld, and W. Choi, “High-speed synthetic aperture microscopy for live cell
imaging,” Opt. Lett. |

21. | A. Calabuig, V. Micó, J. Garcia, Z. Zalevsky, and C. Ferreira, “Single-exposure super-resolved interferometric
microscopy by red-green-blue multiplexing,” Opt. Lett. |

22. | C. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, “Super-resolution digital holographic imaging
method,” Appl. Phys. Lett. |

23. | M. S. Hezaveh, M. R. Riahi, R. Massudi, and H. Latifi, “Digital holographic scanning of large objects using a
rotating optical slab,” Int. J. Imaging Syst. Technol. |

24. | M. Paturzo, F. Merola, S. Grilli, S. De Nicola, A. Finizio, and P. Ferraro, “Super-resolution in digital holography by a
two-dimensional dynamic phase grating,” Opt. Express |

25. | L. Granero, V. Micó, Z. Zalevsky, and J. García, “Superresolution imaging method using phase-shifting
digital lensless Fourier holography,” Opt. Express |

26. | M. Paturzo and P. Ferraro, “Correct self-assembling of spatial frequencies in
super-resolution synthetic aperture digital holography,” Opt.
Lett. |

27. | R. Binet, J. Colineau, and J.-C. Lehureau, “Short-range synthetic aperture imaging at 633 nm by
digital holography,” Appl. Opt. |

28. | Y. Zhang, X. Lu, Y. Luo, L. Zhong, and C. She, “Synthetic aperture holography by movement of
object,” Proc. SPIE |

29. | C. Ventalon and J. Mertz, “Quasi-confocal fluorescence sectioning with dynamic
speckle illumination,” Opt. Lett. |

30. | J. García, Z. Zalevsky, and D. Fixler, “Synthetic aperture superresolution by speckle pattern
projection,” Opt. Express |

31. | P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential
intensity measurements of a volume speckle field,” Appl.
Opt. |

32. | A. Anand, V. K. Chhaniwal, P. Almoro, G. Pedrini, and W. Osten, “Shape and deformation measurements of 3D objects using
volume speckle field and phase retrieval,” Opt. Lett. |

33. | P. F. Almoro and S. G. Hanson, “Wavefront sensing using speckles with fringe
compensation,” Opt. Express |

34. | F. Dubois, L. Joannes, and J.-C. Legros, “Improved three-dimensional imaging with a digital
holography microscope with a source of partial spatial coherence,”
Appl. Opt. |

35. | J. Maycock, B. M. Hennelly, J. B. McDonald, Y. Frauel, A. Castro, B. Javidi, and T. J. Naughton, “Reduction of speckle in digital holography by discrete
Fourier filtering,” J. Opt. Soc. Am. A |

36. | T. Nomura, M. Okamura, E. Nitanai, and T. Numata, “Image quality improvement of digital holography by
superposition of reconstructed images obtained by multiple
wavelengths,” Appl. Opt. |

37. | L. Rong, W. Xiao, F. Pan, S. Liu, and R. Li, “Speckle noise reduction in digital holography by use of
multiple polarization holograms,” Chin. Opt. Lett. |

38. | F. Pan, W. Xiao, S. Liu, F. J. Wang, L. Rong, and R. Li, “Coherent noise reduction in digital holographic phase
contrast microscopy by slightly shifting object,” Opt.
Express |

39. | Y. K. Park, W. Choi, Z. Yaqoob, R. Dasari, K. Badizadegan, and M. S. Feld, “Speckle-field digital holographic
microscopy,” Opt. Express |

40. | J. Goodman, |

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

42. | T. Colomb, J. Kühn, F. Charrière, C. Depeursinge, P. Marquet, and N. Aspert, “Total aberrations compensation in digital holographic
microscopy with a reference conjugated hologram,” Opt.
Express |

**OCIS Codes**

(090.4220) Holography : Multiplex holography

(100.2000) Image processing : Digital image processing

(100.6640) Image processing : Superresolution

(090.1995) Holography : Digital holography

**ToC Category:**

Holography

**History**

Original Manuscript: February 3, 2012

Revised Manuscript: March 9, 2012

Manuscript Accepted: March 11, 2012

Published: April 9, 2012

**Citation**

Vicente Micó, Carlos Ferreira, and Javier García, "Surpassing digital holography limits by lensless object scanning holography," Opt. Express **20**, 9382-9395 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-9-9382

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