## Single-shot digital holography by use of the fractional Talbot effect

Optics Express, Vol. 17, Issue 15, pp. 12900-12909 (2009)

http://dx.doi.org/10.1364/OE.17.012900

Acrobat PDF (626 KB)

### Abstract

We present a method for recording in-line single-shot digital holograms based on the fractional Talbot effect. In our system, an image sensor records the interference between the light field scattered by the object and a properly codified parallel reference beam. A simple binary two-dimensional periodic grating is used to codify the reference beam generating a periodic three-step phase distribution over the sensor plane by fractional Talbot effect. This provides a method to perform single-shot phase-shifting interferometry at frame rates only limited by the sensor capabilities. Our technique is well adapted for dynamic wavefront sensing applications. Images of the object are digitally reconstructed from the digital hologram. Both computer simulations and experimental results are presented.

© 2009 OSA

## 1. Introduction

1. U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. **13**(9), 85–101 (2002). [CrossRef]

2. J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. **11**(3), 77–79 (1967). [CrossRef]

6. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. **22**(16), 1268–1270 (1997). [CrossRef] [PubMed]

12. B. Javidi, S. Yeom, and I. Moon, “Real-time 3D sensing, visualization and recognition of biological microorganisms,” Proc. IEEE **94**(3), 550–567 (2006). [CrossRef]

13. B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. **25**(1), 28–30 (2000). [CrossRef]

15. O. Matoba and B. Javidi, “Optical retrieval of encrypted digital holograms for secure real-time display,” Opt. Lett. **27**(5), 321–323 (2002). [CrossRef]

16. B. Javidi and E. Tajahuerce, “Three-dimensional object recognition by use of digital holography,” Opt. Lett. **25**(9), 610–612 (2000). [CrossRef]

18. Y. Frauel, E. Tajahuerce, M.-A. Castro, and B. Javidi, “Distortion-tolerant 3D object recognition using digital holography,” Appl. Opt. **40**, 3887–3893 (2001). [CrossRef]

5. U. Schnars and W. P. O. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. **33**(2), 179–181 (1994). [CrossRef] [PubMed]

7. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. **24**(5), 291–293 (1999). [CrossRef]

13. B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. **25**(1), 28–30 (2000). [CrossRef]

19. L. Xu, X. Peng, Z. Guo, J. Miao, and A. Asundi, “Imaging analysis of digital holography,” Opt. Express **13**(7), 2444–2452 (2005). [CrossRef] [PubMed]

6. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. **22**(16), 1268–1270 (1997). [CrossRef] [PubMed]

9. G. Pedrini and H. J. Tiziani, “Short-coherence digital microscopy by use of a lensless holographic imaging system,” Appl. Opt. **41**(22), 4489–4496 (2002). [CrossRef] [PubMed]

14. E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. **39**(35), 6595–6601 (2000). [CrossRef]

20. X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. **31**(10), 1414–1416 (2006). [CrossRef] [PubMed]

21. A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. **39**(4), 960–966 (2000). [CrossRef]

27. T. Nomura, S. Murata, E. Nitanai, and T. Numata, “Phase-shifting digital holography with a phase difference between orthogonal polarizations,” Appl. Opt. **45**(20), 4873–4877 (2006). [CrossRef] [PubMed]

25. Y. Awatsuji, A. Fujii, T. Kubota, and O. Matoba, “Parallel three-step phase-shifting digital holography,” Appl. Opt. **45**(13), 2995–3002 (2006). [CrossRef] [PubMed]

26. Y. Awatsuji, T. Tahara, A. Kaneko, T. Koyama, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Parallel two-step phase-shifting digital holography,” Appl. Opt. **47**(19), 183–189 (2008). [CrossRef]

31. J. R. Leger and G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional Talbot planes,” Opt. Lett. **15**(5), 288–290 (1990). [CrossRef] [PubMed]

24. Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase shifting digital holography,” Appl. Phys. Lett. **85**(6), 1069–1071 (2004). [CrossRef]

27. T. Nomura, S. Murata, E. Nitanai, and T. Numata, “Phase-shifting digital holography with a phase difference between orthogonal polarizations,” Appl. Opt. **45**(20), 4873–4877 (2006). [CrossRef] [PubMed]

28. T. M. Kreis and W. P. O. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. **37**(8), 2357–2360 (1997). [CrossRef]

29. B. Javidi and D. Kim, “Three-dimensional-object recognition by use of single-exposure on-axis digital holography,” Opt. Lett. **30**(3), 236–238 (2005). [CrossRef] [PubMed]

30. J. T. Winthrop and C. R. Worthington, “Theory of Fresnel images: I,” J. Opt. Soc. Am. **55**(4), 373–381 (1965). [CrossRef]

30. J. T. Winthrop and C. R. Worthington, “Theory of Fresnel images: I,” J. Opt. Soc. Am. **55**(4), 373–381 (1965). [CrossRef]

36. A. Kolodziejczyk, Z. Jaroszewicz, A. Kowalik, and O. Quintero, “Kinoform sampling filter,” Opt. Commun. **200**(1-6), 35–42 (2001). [CrossRef]

31. J. R. Leger and G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional Talbot planes,” Opt. Lett. **15**(5), 288–290 (1990). [CrossRef] [PubMed]

36. A. Kolodziejczyk, Z. Jaroszewicz, A. Kowalik, and O. Quintero, “Kinoform sampling filter,” Opt. Commun. **200**(1-6), 35–42 (2001). [CrossRef]

37. A. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. **2**(9), 413–415 (1971). [CrossRef]

38. N. H. Salama, D. Patrignani, L. De Pasquale, and E. E. Sicre, “Wavefront sensor using the Talbot effect,” Opt. Laser Technol. **31**(4), 269–272 (1999). [CrossRef]

39. S. De Nicola, P. Ferraro, G. Coppola, A. Finizio, G. Pierattini, and S. Grilli, “Talbot self-image effect in digital holography and its application to spectrometry,” Opt. Lett. **29**(1), 104–106 (2004). [CrossRef] [PubMed]

## 2. Basic layout of the system for Talbot digital holography

*O*(

*x*,

*y*,

*z*), of the light field diffracted by the object at a location (

*x*,

*y*,

*z*) is the superposition of the spherical waves emitted by different points of the input object. Let us write the complex amplitude in the plane of the CCD, located at

*z*= 0, in the following form:where

*A*(

_{O}*x*,

*y*) and

*ϕ*(

_{O}*x*,

*y*) denote the amplitude and phase of the diffracted light field. Our objective is to measure both parameters to be able to reconstruct the complex amplitude distribution

*O*(

*x*,

*y*,

*z*) at different distances

*z*.

*t*(

*x*,

*y*) of the grating can be written as follows:where

*t*(

_{c}*x*,

*y*) is the amplitude transmittance of the unit cell, ⊗ denotes convolution,

*d*is the period of the array,

*P*is the number of periods,

*δ*(

*x*) represents the Dirac delta function, and

*j*and

*k*are integer numbers.

*t*(

*x*,

*y*) is a periodic function, the amplitude distribution associated with the grating produces self-images by free-space propagation, i.e., diffraction patterns that are a copy of the input distribution, but also Fresnel images, as is shown in Fig. 2 . In each transversal dimension, the former diffraction patterns consist of the superposition of

*r*phase-weighted copies of the input grating shifted by integer multiples of

*d*/

*r*. Under monochromatic illumination, the Fresnel images are obtained at distances [30

30. J. T. Winthrop and C. R. Worthington, “Theory of Fresnel images: I,” J. Opt. Soc. Am. **55**(4), 373–381 (1965). [CrossRef]

*z*

_{t}is the so-called Talbot distance,

*q*is an integer, and

*n*and

*m*are natural numbers with no common factor (with

*n*<

*m*). The ratio

*n*/

*m*determines the amplitude distribution at the unit cell of a particular Fresnel image. Different values of

*q*lead simply to different positions of the Fresnel image. The number of replicas,

*r*, associated with the Fresnel image of index

*n*/

*m*is given by

*m*/2 when

*m*is even and by

*m*when

*m*is odd. By choosing the opening ratio of the grating in accordance with the index

*n*/

*m*of the Fresnel image it is possible to get a uniform irradiance distribution with a periodic multilevel phase at the output plane.

**55**(4), 373–381 (1965). [CrossRef]

36. A. Kolodziejczyk, Z. Jaroszewicz, A. Kowalik, and O. Quintero, “Kinoform sampling filter,” Opt. Commun. **200**(1-6), 35–42 (2001). [CrossRef]

*n*/

*m*= 1/4 or 3/4. However other orders of the Fresnel image can be interesting to obtain more complex periodic phase distributions. For our case, the amplitude distribution generated by the reference beam at the output plane, at a distance given by Eq. (3), with

*n*/

*m*= 1/4 and

*q*arbitrary, is:where A is the constant amplitude of the reference beam just before the grating, assumed equal to unity in the analysis that follows. To get a Fresnel image with uniform irradiance, the amplitude transmittance,

*t*(

_{c}*x*,

*y*), of the unit cell of the grating in Eq. (2) should be given by

*P*of the grating and the number of periods

*P’*of the Fresnel image must fulfill two restrictions, the first one to guarantee the proper profile of the Fresnel image and the second imposed by the paraxial approximation [41

41. V. Arrizón and G. Rojo-VelázquezV. ArrizónG. Rojo-VelázquezVictor Arrizón and Gustavo Rojo-Velázquez, “Fractional Talbot field of finite gratings: compact analytical formulation,” J. Opt. Soc. Am. A **18**(6), 1252–1255 (2001). [CrossRef]

*O*(

*x*,

*y*,0), in Eq. (1) interferes with that of the Talbot-codified reference beam

*R*(

*x*,

*y*,0) in Eq. (4) we obtain a pixelated interferogram with different periodic phase shifts. Two approaches are now possible to reconstruct the light field diffracted by the object. In the first approach, three interferograms with the same size of the original one,

*I*(

*x*,

*y*,0),

*I*(

*x*,

*y*,π/2), and

*I*(

*x*,

*y*,π) are generated by extracting the values of the original interferogram periodically at locations with the same phase shift, and using linear interpolation to allocate the empty pixels. Interpolation is performed by averaging the values of adjacent pixels in a similar way as is done in Refs [24

24. Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase shifting digital holography,” Appl. Phys. Lett. **85**(6), 1069–1071 (2004). [CrossRef]

26. Y. Awatsuji, T. Tahara, A. Kaneko, T. Koyama, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Parallel two-step phase-shifting digital holography,” Appl. Opt. **47**(19), 183–189 (2008). [CrossRef]

*O*for each unit cell of the original interferogram. In this case, the final hologram is half the size of the previous one in each transversal dimension. We choose to use the first approach since it has been shown in other interferometry applications that it gives slightly better results than the second one [23

23. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. **44**(32), 6861–6868 (2005). [CrossRef] [PubMed]

*O*(

*x*,

*y,*0) allows us to reconstruct numerically the complex amplitude distribution,

*O*(

*x*,

*y*,

*z*), generated by the 3D object at plane located at a distance

*z*from the sensor. The reconstruction can be obtained by computing a discrete Fresnel integral or, alternatively, by using the propagation transfer function method, i.e.,where

*F*denotes the fast Fourier transform, (

*u*,

*v*) are discrete spatial frequency variables, (

*m*,

*n*) are discrete transversal spatial coordinates in both the CCD plane and the output plane, and

*N*

_{x}and

*N*

_{y}are the number of samples in the

*x*and

*y*directions. Note that negative values of

*z*are to be considered to simulate backward propagation in Eq. (8). In this approach, the resolution at the output plane is the same for any propagation distance

*z*, and is given by the resolution at the input plane, i.e., the size of the pixel (

*Δx*,

*Δy*) in the CCD sensor.

## 3. Numerical simulations

*λ*= 514.5 nm illuminating two 2D objects located at different distances from the CCD,

*z*

_{1}= 400 mm for the case of the object shown in Fig. 3(a) and

*z*

_{2}= 300 mm for the one in Fig. 3(b). A random phase mask was attached to the binary images to spread the diffraction patterns at the output plane. The CCD sensor is supposed to have N

_{x}× N

_{y}= 1024 × 1024 pixels with size

*Δx*=

*Δy*= 9 µm. Following our reasoning in the previous section we simulate a binary amplitude grating (see Fig. 2) with an open ratio equal to 0.5 and a period

*d*= 18 µm, double of the pixel size, in both dimensions located in the reference beam of the interferometer. By application of Eq. (3) we calculate the Talbot distance z

_{t}= 1.26 mm and the distance from the grating to the CCD to obtain the Fresnel image with

*q*= 1 and

*n*/

*m*= 3/4, i.e.,

*z*´ = 2.20 mm.

*z*

_{1}and

*z*

_{2}.

## 4. Experimental results

^{2}.

^{2}, period

*d*= 144 µm and opening ratio 0.5, manufactured in our laboratory by laser photolithography on a chrome photomask. The photomask was a quartz substrate (size 2 × 2 × 0.09 inch) coated with a low reflectivity chromium layer (thickness 120 nm) and S1805 photoresist film (

*Shipley*). The blank was irradiated by using a laser writing machine (

*Microtech, srl*). The final mask was obtained by developing the photoresist with MF319 developer (

*Shipley*), etching the unprotected chromium, and cleaning the remaining photoresist. The Talbot distance for this grating is

*z*

_{t}= 80.6 mm. Figure 5(a) shows a central region of the irradiance distribution of the first self-image of the grating recorded by the CCD. The distance from the grating to the CCD was then adjusted to obtain a Fresnel image with

*q*= 1 and

*n*/

*m*= 3/4. Figure 5(b) shows the irradiance distribution (approximately uniform) generated by the Fresnel image on the CCD while Fig. 5(c) is a gray level picture of the interference pattern generated when a parallel light beam is used as object beam in the interferometer. Note the periodic three-step phase distribution (with values 0, π/2 and π) associated to the Fresnel image. The requirements established in Eq. (6) to get high quality Fresnel images are fulfilled by using a parallel light beam with 15 mm diameter as. In this way, we illuminate around 100 periods of the grating.

^{2}located at different distances. The first transparency codifies the binary object shown in Fig. 3(a) and was located at a distance of 32 cm from the CCD. The second corresponds to the object in Fig. 3(b), at a distance of 37 cm, while the third object was a USAF resolution target located at 42 cm from the sensor.

^{2}, corresponding to 8 × 8 pixels of the camera, and the resolution of our final hologram (with only 256 × 256 complex values) is 8 times lower than the maximum achievable.

## 5 Conclusions

## Acknowledgments

## References and links

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

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

3. | M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavsky, “Reconstruction of holograms with a computer,” Sov. Phys. Tech. Phys. |

4. | L. Onural and P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. |

5. | U. Schnars and W. P. O. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. |

6. | I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. |

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

8. | 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. |

9. | G. Pedrini and H. J. Tiziani, “Short-coherence digital microscopy by use of a lensless holographic imaging system,” Appl. Opt. |

10. | L. Martínez-León, G. Pedrini, and W. Osten, “Applications of short-coherence digital holography in microscopy,” Appl. Opt. |

11. | Y. Frauel, T. Naughton, O. Matoba, E. Tajahuerce, and B. Javidi, “Three Dimensional Imaging and Display Using Computational Holographic Imaging,” Proc. IEEE |

12. | B. Javidi, S. Yeom, and I. Moon, “Real-time 3D sensing, visualization and recognition of biological microorganisms,” Proc. IEEE |

13. | B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. |

14. | E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. |

15. | O. Matoba and B. Javidi, “Optical retrieval of encrypted digital holograms for secure real-time display,” Opt. Lett. |

16. | B. Javidi and E. Tajahuerce, “Three-dimensional object recognition by use of digital holography,” Opt. Lett. |

17. | E. Tajahuerce, O. Matoba, and B. Javidi, “Shift-invariant three-dimensional object recognition by means of digital holography,” Appl. Opt. |

18. | Y. Frauel, E. Tajahuerce, M.-A. Castro, and B. Javidi, “Distortion-tolerant 3D object recognition using digital holography,” Appl. Opt. |

19. | L. Xu, X. Peng, Z. Guo, J. Miao, and A. Asundi, “Imaging analysis of digital holography,” Opt. Express |

20. | X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. |

21. | A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. |

22. | J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE |

23. | M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. |

24. | Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase shifting digital holography,” Appl. Phys. Lett. |

25. | Y. Awatsuji, A. Fujii, T. Kubota, and O. Matoba, “Parallel three-step phase-shifting digital holography,” Appl. Opt. |

26. | Y. Awatsuji, T. Tahara, A. Kaneko, T. Koyama, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Parallel two-step phase-shifting digital holography,” Appl. Opt. |

27. | T. Nomura, S. Murata, E. Nitanai, and T. Numata, “Phase-shifting digital holography with a phase difference between orthogonal polarizations,” Appl. Opt. |

28. | T. M. Kreis and W. P. O. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. |

29. | B. Javidi and D. Kim, “Three-dimensional-object recognition by use of single-exposure on-axis digital holography,” Opt. Lett. |

30. | J. T. Winthrop and C. R. Worthington, “Theory of Fresnel images: I,” J. Opt. Soc. Am. |

31. | J. R. Leger and G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional Talbot planes,” Opt. Lett. |

32. | A. W. Lohmann and J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. |

33. | V. Arrizón and J. Ojeda-Castañeda, “Multilevel phase gratings for array illuminators,” Appl. Opt. |

34. | J. Werterholm, J. Turunen, and J. Huttunen, “Fresnel diffraction in fractional Talbot planes: a new formulation,” J. Opt. Soc. Am. A |

35. | C. Zhou and L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illumination,” Opt. Commun. |

36. | A. Kolodziejczyk, Z. Jaroszewicz, A. Kowalik, and O. Quintero, “Kinoform sampling filter,” Opt. Commun. |

37. | A. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. |

38. | N. H. Salama, D. Patrignani, L. De Pasquale, and E. E. Sicre, “Wavefront sensor using the Talbot effect,” Opt. Laser Technol. |

39. | S. De Nicola, P. Ferraro, G. Coppola, A. Finizio, G. Pierattini, and S. Grilli, “Talbot self-image effect in digital holography and its application to spectrometry,” Opt. Lett. |

40. | A. Fajst, M. Sypek, M. Makowski, J. Suszek, and A. Kolodziejczyk, “Self-imaging phase mask used in digital holography with phase-shifting,” Proc. SPIE |

41. | V. Arrizón and G. Rojo-VelázquezV. ArrizónG. Rojo-VelázquezVictor Arrizón and Gustavo Rojo-Velázquez, “Fractional Talbot field of finite gratings: compact analytical formulation,” J. Opt. Soc. Am. A |

**OCIS Codes**

(050.5080) Diffraction and gratings : Phase shift

(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects

(090.0090) Holography : Holography

(090.1760) Holography : Computer holography

(100.3010) Image processing : Image reconstruction techniques

(100.6890) Image processing : Three-dimensional image processing

(120.3180) Instrumentation, measurement, and metrology : Interferometry

**ToC Category:**

Holography

**History**

Original Manuscript: May 22, 2009

Revised Manuscript: June 26, 2009

Manuscript Accepted: June 28, 2009

Published: July 13, 2009

**Citation**

Lluís Martínez-León, María Araiza-E, Bahram Javidi, Pedro Andrés, Vicent Climent, Jesús Lancis, and Enrique Tajahuerce, "Single-shot digital holography
by use of the fractional Talbot effect," Opt. Express **17**, 12900-12909 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12900

Sort: Year | Journal | Reset

### References

- U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13(9), 85–101 (2002). [CrossRef]
- J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967). [CrossRef]
- M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavsky, “Reconstruction of holograms with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).
- L. Onural and P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
- U. Schnars and W. P. O. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33(2), 179–181 (1994). [CrossRef] [PubMed]
- I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). [CrossRef] [PubMed]
- E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24(5), 291–293 (1999). [CrossRef]
- 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. 38(34), 7085–7094 (1999). [CrossRef]
- G. Pedrini and H. J. Tiziani, “Short-coherence digital microscopy by use of a lensless holographic imaging system,” Appl. Opt. 41(22), 4489–4496 (2002). [CrossRef] [PubMed]
- L. Martínez-León, G. Pedrini, and W. Osten, “Applications of short-coherence digital holography in microscopy,” Appl. Opt. 44(19), 3977–3984 (2005). [CrossRef] [PubMed]
- Y. Frauel, T. Naughton, O. Matoba, E. Tajahuerce, and B. Javidi, “Three Dimensional Imaging and Display Using Computational Holographic Imaging,” Proc. IEEE 94(3), 636–654 (2006). [CrossRef]
- B. Javidi, S. Yeom, and I. Moon, “Real-time 3D sensing, visualization and recognition of biological microorganisms,” Proc. IEEE 94(3), 550–567 (2006). [CrossRef]
- B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. 25(1), 28–30 (2000). [CrossRef]
- E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. 39(35), 6595–6601 (2000). [CrossRef]
- O. Matoba and B. Javidi, “Optical retrieval of encrypted digital holograms for secure real-time display,” Opt. Lett. 27(5), 321–323 (2002). [CrossRef]
- B. Javidi and E. Tajahuerce, “Three-dimensional object recognition by use of digital holography,” Opt. Lett. 25(9), 610–612 (2000). [CrossRef]
- E. Tajahuerce, O. Matoba, and B. Javidi, “Shift-invariant three-dimensional object recognition by means of digital holography,” Appl. Opt. 40(23), 3877–3886 (2001). [CrossRef]
- Y. Frauel, E. Tajahuerce, M.-A. Castro, and B. Javidi, “Distortion-tolerant 3D object recognition using digital holography,” Appl. Opt. 40, 3887–3893 (2001). [CrossRef]
- L. Xu, X. Peng, Z. Guo, J. Miao, and A. Asundi, “Imaging analysis of digital holography,” Opt. Express 13(7), 2444–2452 (2005). [CrossRef] [PubMed]
- X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31(10), 1414–1416 (2006). [CrossRef] [PubMed]
- A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39(4), 960–966 (2000). [CrossRef]
- J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004). [CrossRef]
- M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005). [CrossRef] [PubMed]
- Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004). [CrossRef]
- Y. Awatsuji, A. Fujii, T. Kubota, and O. Matoba, “Parallel three-step phase-shifting digital holography,” Appl. Opt. 45(13), 2995–3002 (2006). [CrossRef] [PubMed]
- Y. Awatsuji, T. Tahara, A. Kaneko, T. Koyama, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Parallel two-step phase-shifting digital holography,” Appl. Opt. 47(19), 183–189 (2008). [CrossRef]
- T. Nomura, S. Murata, E. Nitanai, and T. Numata, “Phase-shifting digital holography with a phase difference between orthogonal polarizations,” Appl. Opt. 45(20), 4873–4877 (2006). [CrossRef] [PubMed]
- T. M. Kreis and W. P. O. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. 37(8), 2357–2360 (1997). [CrossRef]
- B. Javidi and D. Kim, “Three-dimensional-object recognition by use of single-exposure on-axis digital holography,” Opt. Lett. 30(3), 236–238 (2005). [CrossRef] [PubMed]
- J. T. Winthrop and C. R. Worthington, “Theory of Fresnel images: I,” J. Opt. Soc. Am. 55(4), 373–381 (1965). [CrossRef]
- J. R. Leger and G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional Talbot planes,” Opt. Lett. 15(5), 288–290 (1990). [CrossRef] [PubMed]
- A. W. Lohmann and J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29(29), 4337–4340 (1990). [CrossRef] [PubMed]
- V. Arrizón and J. Ojeda-Castañeda, “Multilevel phase gratings for array illuminators,” Appl. Opt. 33(25), 5925–5931 (1994). [CrossRef] [PubMed]
- J. Werterholm, J. Turunen, and J. Huttunen, “Fresnel diffraction in fractional Talbot planes: a new formulation,” J. Opt. Soc. Am. A 11(4), 1283–1290 (1994). [CrossRef]
- C. Zhou and L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illumination,” Opt. Commun. 115(1-2), 40–44 (1995). [CrossRef]
- A. Kolodziejczyk, Z. Jaroszewicz, A. Kowalik, and O. Quintero, “Kinoform sampling filter,” Opt. Commun. 200(1-6), 35–42 (2001). [CrossRef]
- A. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2(9), 413–415 (1971). [CrossRef]
- N. H. Salama, D. Patrignani, L. De Pasquale, and E. E. Sicre, “Wavefront sensor using the Talbot effect,” Opt. Laser Technol. 31(4), 269–272 (1999). [CrossRef]
- S. De Nicola, P. Ferraro, G. Coppola, A. Finizio, G. Pierattini, and S. Grilli, “Talbot self-image effect in digital holography and its application to spectrometry,” Opt. Lett. 29(1), 104–106 (2004). [CrossRef] [PubMed]
- A. Fajst, M. Sypek, M. Makowski, J. Suszek, and A. Kolodziejczyk, “Self-imaging phase mask used in digital holography with phase-shifting,” Proc. SPIE 7141, 1–7 (2008).
- V. Arrizón, G. Rojo-Velázquez, and Victor Arrizón and Gustavo Rojo-Velázquez, “Fractional Talbot field of finite gratings: compact analytical formulation,” J. Opt. Soc. Am. A 18(6), 1252–1255 (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.