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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 16 — Jul. 30, 2012
  • pp: 18303–18312
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Digital Fresnel holography beyond the Shannon limits

Mathieu Leclercq and Pascal Picart  »View Author Affiliations


Optics Express, Vol. 20, Issue 16, pp. 18303-18312 (2012)
http://dx.doi.org/10.1364/OE.20.018303


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Abstract

This paper presents a detailed analysis of the influence of the pixel dimension in digitally-recorded holograms. The investigation is based on both theoretical and experimental viewpoints for recordings beyond the Shannon limits. After discussing the pixel paradox, the sinc amplitude modulation is experimentally demonstration. The experimental analysis is well correlated to the theoretical basics; in addition, the filling factor of the sensor can be estimated. The analysis of the phase changes of the object show that they can be obtained with a very good contrast and that they are only limited by the decorrelation noise, as when the Shannon conditions are fulfilled.

© 2012 OSA

1. Introduction

2. Theoretical basics

2.1 Digital hologram recording and reconstruction

Let us consider the imaging of digital holograms in a smooth-plane reference wave scheme. In the recording plane, where an object wave interferes with a reference wave, the hologram is expressed as:

H=|R|2+|O|2+R*O+RO*,
(1)

where R(x,y) = aRexp[2iπ(u0x + v0y)] is the reference wave with spatial frequencies {u0,v0}, and O is the wave diffracted in the recording plane by the object located at distance d0 from this plane. In the Fresnel approximations, we have [23

23. J. W. Goodman, Introduction to Fourier Optics (Second Edition, McGraw-Hill Editions, 1996).

]:

O(x,y,d0)=iλd0exp(2iπd0λ)exp(iπλd0(x2+y2))×A(X,Y)exp(iπλd0(X2+Y2))exp(2iπλd0(xX+yY))dXdY,
(2)

and A(X,Y) = A0(X,Y)exp[0(X,Y)] is the object wave front. Due to the spatial extension of the pixel, the detected holograms, which will be transferred to the digital acquisition board, are composed of elementary spatially-integrated zones of the real ones. At any pixel coordinate (lpx,kpy) at which the digital hologram is recorded (l, k: integers; px, py: pixel pitches), we have:

HD(lpx,kpy,d0)=lpxΔx/2lpx+Δx/2kpyΔy/2kpy+Δy/2H(x,y,d0)dxdy,
(3)

Introducing the pixel function defined as an even “rect” function,

rect(xΔx,yΔy)={1if|x|Δx/2and|y|Δy/20ifnot,
(4)

Equation (3) is rewritten according to the following convolution formula [12

12. P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A 25(7), 1744–1761 (2008). [CrossRef] [PubMed]

]:

HD(x,y,d0)x=lpx,y=kpy=H(x,y,d0)rect(xΔx,yΔy).
(5)

In Eq. (5), * means two-dimensional convolution. Note that the above-mentioned papers describe the pixel effect by almost the same parameter: a narrow rectangular function as given in Eq. (4).

Considering the discrete recording, the reconstruction of the object field at distance −d0 from the recording plane is given by the discrete Fresnel transform [3

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

,6

6. Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997). [CrossRef]

]:

Ar(X,Y,d0)=iexp(2iπd0/λ)λd0exp[iπλd0(X2+Y2)]×k=0k=K1l=0l=L1HD(lpx,kpy,d0)exp[iπλd0(l2px2+k2py2)]exp[2iπλd0(lXpx+kYpy)].
(6)

2.2 The pixel paradox

Several authors discussed the influence of the pixel function on the reconstructed object. According to [12

12. P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A 25(7), 1744–1761 (2008). [CrossRef] [PubMed]

], the pixel induces a spatial low-pass filtering of the reconstructed object. This means that, mathematically, the reconstructed object is a convolution between the real object and the pixel function. This point was thoroughly investigated in references [13

13. D. P. Kelly, B. M. Hennelly, C. McElhinney, and T. J. Naughton, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008). [CrossRef]

15

15. D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5D imaging performance of digital holographic systems,” JEOS rapid publication 6, 11034 (2011). [CrossRef]

,20

20. A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43(1), 239–250 (2004). [CrossRef]

]. This result is quite compatible with optical imaging properties. Indeed, in classical imaging, where the object is projected onto the sensor plane by means of an optical lens, the image is the real one convoluted by the point spread function of the lens and convoluted by the pixel function (see Eq. (5)). So, in classical imaging, the filtering of the real object is due to the lens and the pixel dimensions. Digital holography is an interferometric method to image objects. In such non-conventional imaging, filtering is not processed by a lens since there is no lens in the set-up. However, it is performed by the double free-space propagation, due, first, to the physical diffraction from object to sensor, and, second, to the numerical diffraction from sensor plane to the reconstructed plane. In addition, the results obtained with digital image-plane holography are quite similar to those obtained with digital Fresnel holography [24

24. M. Karray, P. Slangen, and P. Picart, “Comparison between digital Fresnel holography and digital image-plane holography: the role of the imaging aperture,” Exp. Mech. (2012), doi:. [CrossRef]

]. Finally, the fundamental results given in references [12

12. P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A 25(7), 1744–1761 (2008). [CrossRef] [PubMed]

15

15. D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5D imaging performance of digital holographic systems,” JEOS rapid publication 6, 11034 (2011). [CrossRef]

,20

20. A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43(1), 239–250 (2004). [CrossRef]

] are common sense and can be intuitively understood.

Although it may seem strange regarding this analysis, previously published papers [9

9. C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. 38(22), 4812–4820 (1999). [CrossRef] [PubMed]

,21

21. 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]

] argued that the pixel function induces a multiplicative sinc function to the reconstructed object amplitude. In order to explain this point of view as straightforwardly as possible, let us consider that the object amplitude in the recorded plane is slowly varying compared to the pixel size. As a result, Eq. (3) can be re-written as:

HD(lpx,kpy,d0)|O(lpx,kpy,d0)|2+|R(lpx,kpy)|2+O(lpx,kpy,d0)R*(lpx,kpy)sinc(πu0Δx)sinc(πv0Δy)+O*(lpx,kpy,d0)R(lpx,kpy)sinc(πu0Δx)sinc(πv0Δy).
(7)

In Eq. (7), sinc(x) = sin(x)/x. According to Eq. (7), the useful + 1 order is amplitude-modulated by the sinc functions; therefore, the reconstructed object will also be amplitude-modulated by the sinc functions [21

21. 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]

].

These two approaches stand for what can be called the pixel paradox. Indeed, the first approach states that the pixel function is a convolution function, whereas the latter states that it is a multiplicative sinc function. One may wonder what magical reason explains why the same element provides different influences. Furthermore, note that the Fourier transform of the recorded hologram provides the spatial frequency spectrum of the object. Thus, it is multiplied by the sinc function according to Eq. (5). Nevertheless, the Fourier transform of the hologram is also the discrete Fresnel transform for a reconstruction distance tending toward infinity, leading to an out-of-focus object. There is no logical reason for the out-of-focus object to be not convoluted by the pixel function in the same manner as the focused object will be. But it is quite common sense that the Fourier transform of the hologram is multiplied by the sinc function.

2.3 The Shannon limits

The Shannon limits correspond to the recording conditions compatible with the Shannon theorem. There are several ways to optimize the recording condition, so that the three diffraction orders may not overlap [12

12. P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A 25(7), 1744–1761 (2008). [CrossRef] [PubMed]

,16

16. N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt. 48(34), H186–H195 (2009). [CrossRef] [PubMed]

,25

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

]. This paper deals with an off-axis configuration; the optimization proposed in [12

12. P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A 25(7), 1744–1761 (2008). [CrossRef] [PubMed]

] is considered. The Shannon theorem applied to off-axis digital holography, resulting in the spatial separation of the three diffraction orders, leads to the optimal recording distance. Practically, the spatial frequencies can be adjusted following this method: the reference beam is perpendicular to the recording plane but the object is laterally shifted [22

22. N. Demoli, H. Halaq, K. Sariri, M. Torzynski, and D. Vukicevic, “Undersampled digital holography,” Opt. Express 17(18), 15842–15852 (2009). [CrossRef] [PubMed]

,24

24. M. Karray, P. Slangen, and P. Picart, “Comparison between digital Fresnel holography and digital image-plane holography: the role of the imaging aperture,” Exp. Mech. (2012), doi:. [CrossRef]

]. We consider here that the spatial Shannon limits are given by the maximum lateral shift, as long as that the center of the object fulfills the Shannon theorem. When translating the object along the horizontal axis of a quantity ΔX, the mean spatial frequency of the hologram becomes u0 = ΔX/λd0. When laterally moving the object in the field of view, the reconstructed object will be localized at the corresponding spatial shift until spatial aliasing occurs. Indeed, the field of view reconstructed by the discrete Fresnel transform is sized λd0/px × λd0/py. Because of the sampling of recording and reconstruction spaces, the field of view is virtually periodic. This means that the reconstructed field is repeated infinitely, with a period equal to its width. However, the numerical computation can only give the data of the main window corresponding to the field of view. Figure 1
Fig. 1 Virtual periodicity of the reconstructed field and lateral shift of the object.
illustrates the virtual periodicity of the reconstructed field. The central window (not translucent) is that reconstructed by the FFT algorithms, whereas the 8 other ones (translucent) are not reachable by FFT calculation. However, these 8 neighborhood zones indicate that, when the object is shifted from the central window to one of the adjacent ones, its reconstructed image comes from the symmetric zone. Figure 1(left) shows the aliasing that occurs when exceeding Shannon limits: the under-sampled part of the object is shifted out of the main window and re-appears in the opposite corner. Thus, for a physical object shifted along the x-axis with ΔX∈{(2k−1)λd0/2px,(2k + 1)λd0/2px}, k integer, the position of the numerically-reconstructed object is given by ΔXr = ΔXkλd0/px. Beyond the Shannon limits, the reconstructed object appears in the field of view at cyclic positions and with an amplitude attenuated by the sinc(πΔXΔx/λd0) function.

As an illustration, Fig. 2
Fig. 2 Shannon limits and amplitude modulation vs lateral shift of the object.
shows the Shannon limits given by the numerical sampling and the amplitude modulation of the reconstructed object according to both the mean spatial frequency of the digital hologram and the lateral shift of the object. In this example, the filling factor of the sensor is taken to equal to 1 (i.e. Δx = px).

3. Experimental set-up and methodology

The experimental set-up is described in Fig. 3
Fig. 3 Optical setup for investigating the pixel influence.
. It is based on an off-axis Mach Zehnder configuration in which the sensor has 8 bits digitization with L × K = 1024 × 1360 pixels sized px = py = 4.65μm. The laser is a continuous frequency doubled NdYAG laser (λ = 532nm, P0 = 150mW) and the object is a 50 cents euro coin, 24mm in diameter. It is illuminated with a circular spot. The object is mounted on a translation stage (Fig. 3); the mean horizontal spatial frequencies of the digital hologram, {u0}, may thus vary continuously. The amplitude modulation due to the pixel is then formulated with sinc(πΔxΔX/λd0). In order to measure the amplitude modulation of the object, a power meter estimates the injected laser power into the interferometer. A rotation stage (mirror M2) ensures that the object is correctly illuminated when translating in the x-axis direction. Note that change in the illuminating angle, due to the object translation, requires the measurement of the bidirectional reflectance distribution function (BRDF) of the object for angles similar to those provided by the illumination.

The BRDF was taken into account for the amplitude modulation of the reconstructed object so that it may only be due to the pixel effect and not to photometric considerations (an experiment not described in this paper). The object is localized at a constant distance d0 = 900mm during the experiment. The object can be slightly tilted on its mount. This facility provides the possibility to study the effect of the pixel dimension on phase differences extracted between two object states, especially the decorrelation noise and the coherence degree of the two speckle phases [24

24. M. Karray, P. Slangen, and P. Picart, “Comparison between digital Fresnel holography and digital image-plane holography: the role of the imaging aperture,” Exp. Mech. (2012), doi:. [CrossRef]

].

p(ε)=1|μ|2π(1β2)3/2(βsin1β+πβ2+1β2).
(8)

Equation (8) can be used as a relevant indicator so as to compare the noise sensitivity at different lateral shifts, the correlation factor |μ| being a quality marker extracted from experimental data.

4 Experimental results

The object is shifted along the X-axis direction from 0 to 280mm. The Shannon limits for the set-up is 51.48mm for the lateral position. The BRDF for the experimental conditions was measured and was transposed as a lateral shift variation. It was evidenced that its contribution to the amplitude modulation is a multiplication by exp(−ΔX/b). This means that the amplitude modulation on the reconstructed object is a fX) = |sinc(πΔX/a)exp(−ΔX/b)| function rather than |sinc(πΔX/a)|. Figure 4
Fig. 4 Amplitude modulation vs lateral object shift.
shows the measured amplitude modulation from 0 to 280mm. The experimental curve is compared to the analytical model fX), giving a = 119.15mm and b = 168.35mm (99.824% correlation). The |sinc(πΔX/a)| curve is also plotted. Fit fX) is in excellent agreement with the experimental curve. This confirms a part of the pixel paradox discussed in this paper: the pixel dimension has a sinc influence on the reconstructed object amplitude. Furthermore, this validates the modeling for the BRDF of the object, that must be taken into account to measure pixel influence.

For each object shift, the phase changes are extracted from the two recordings without and with tilt. Then, the noise and the probability density are estimated; Eq. (8) is fitted to the experimental data; the noise rms and the correlation factor are calculated. Figure 6
Fig. 6 Phase changes extracted from the tilt measurement, (a) raw, (b) filtered
shows the raw and filtered phases (5 × 5 moving average according to [29

29. H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4-6), 205–210 (1999). [CrossRef]

]) extracted from measurements. Tilt was applied to the object with an almost good reproducibility. This acceptable reproducibility can be appreciated on experimental results in Fig. 6, although the tilt angles are not exactly identical. However, this reproducibility is quite sufficient to maintain the speckle decorrelation as constant, so that the variation of noise may be attributed to out-of-Shannon conditions. Figure 6(a) and 6(b) show the 6 phase changes corresponding to 6 red points (1 to 6 in Fig. 4).

Figure 6 shows that the fringe contrast is relatively good for recordings beyond the Shannon limits. The fringes obtained for points 1 and 2 do not seem to be altered by the sinc. The fringe maps obtained for point 3 (near zero-crossing) and point 5 (second lobe) reveal good quality. This qualitative analysis is confirmed by a quantitative measurement of the noise rms and speckle correlation factor [24

24. M. Karray, P. Slangen, and P. Picart, “Comparison between digital Fresnel holography and digital image-plane holography: the role of the imaging aperture,” Exp. Mech. (2012), doi:. [CrossRef]

,26

26. P. Picart, R. Mercier, M. Lamare, and J.-M. Breteau, “A simple method for measuring the random variation of an interferometer,” Meas. Sci. Technol. 12(8), 1311–1317 (2001). [CrossRef]

]. Figure 7
Fig. 7 Noise rms and correlation factor vs the lateral shift of the object
shows the noise rms and the correlation factor |μ| of Eq. (8) according to the object lateral shift. The Shannon limit and the red points of Fig. 4 are also shown.

The two curves are well correlated to the sinc variation. The zero-crossings of sinc induce an increase in noise (two visible peaks) and a decrease of the correlation between the two phases (two visible craters). Figure 7 shows that the noise rms and the correlation factor remain constant beyond the Shannon limit, until a shift of about ΔX≈90mm, corresponding to 1.74 times the Shannon limit.

The filling factor of the CCD sensor can be estimated thanks to approximating the amplitude modulation of Fig. 1. The first zero-crossing of the sinc function is obtained for a = 119.15mm. Thus, the pixel width is estimated to Δx = λd0/a = 4.01μm, leading to a filling factor of Δx/px = 86.4%.

5. Conclusion

This paper has experimentally demonstrated the influence of the pixel dimension in digitally-reconstructed holograms. Extended pixels induce a sinc-type amplitude modulation which depends on the mean spatial frequency of the hologram. As a practical consequence of the experimental results, the filling factor of the sensor can be estimated. The demonstration of the sinc modulation contributes to solving a part of the pixel paradox. The paper has demonstrated that recording beyond the Shannon limits is possible since both amplitude and phase information can be retrieved. Furthermore, the phase changes of the object can be obtained with a very good contrast and may be only limited by the decorrelation noise, as when the Shannon conditions are fulfilled. As a result, classical Shannon constraints can be relaxed without deteriorating the quality of the image reconstruction.

Acknowledgments

This research is funded from the French National Agency for Research (ANR) under grant agreement n°ANR 2010 BLAN 0302.

References and Links

1.

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

2.

M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

3.

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

4.

L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18(11), 846–848 (1993). [CrossRef] [PubMed]

5.

Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Image reconstruction for in-line holography with the Yang-Gu algorithm,” Appl. Opt. 42(32), 6452–6457 (2003). [CrossRef] [PubMed]

6.

Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997). [CrossRef]

7.

Th. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41(4), 771–778 (2002). [CrossRef]

8.

Th. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41(8), 1829–1839 (2002). [CrossRef]

9.

C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. 38(22), 4812–4820 (1999). [CrossRef] [PubMed]

10.

M. Liebling, T. Blu, and M. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A 21(3), 367–377 (2004). [CrossRef] [PubMed]

11.

I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt. 40(34), 6177–6186 (2001). [CrossRef] [PubMed]

12.

P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A 25(7), 1744–1761 (2008). [CrossRef] [PubMed]

13.

D. P. Kelly, B. M. Hennelly, C. McElhinney, and T. J. Naughton, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008). [CrossRef]

14.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48(9), 095801 (2009). [CrossRef]

15.

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5D imaging performance of digital holographic systems,” JEOS rapid publication 6, 11034 (2011). [CrossRef]

16.

N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt. 48(34), H186–H195 (2009). [CrossRef] [PubMed]

17.

P. Picart, P. Tankam, and Q. Song, “Experimental and theoretical investigation of the pixel saturation effect in digital holography,” J. Opt. Soc. Am. A 28(6), 1262–1275 (2011). [CrossRef] [PubMed]

18.

N. Pandey and B. M. Hennelly, “Quantization noise and its reduction in lensless Fourier digital holography,” Appl. Opt. 50(7), B58–B70 (2011). [CrossRef] [PubMed]

19.

C. S. Guo, L. Zhang, Z. Y. Rong, and H. T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the paper,” Opt. Eng. 42(9), 2768–2772 (2003). [CrossRef]

20.

A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43(1), 239–250 (2004). [CrossRef]

21.

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]

22.

N. Demoli, H. Halaq, K. Sariri, M. Torzynski, and D. Vukicevic, “Undersampled digital holography,” Opt. Express 17(18), 15842–15852 (2009). [CrossRef] [PubMed]

23.

J. W. Goodman, Introduction to Fourier Optics (Second Edition, McGraw-Hill Editions, 1996).

24.

M. Karray, P. Slangen, and P. Picart, “Comparison between digital Fresnel holography and digital image-plane holography: the role of the imaging aperture,” Exp. Mech. (2012), doi:. [CrossRef]

25.

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

26.

P. Picart, R. Mercier, M. Lamare, and J.-M. Breteau, “A simple method for measuring the random variation of an interferometer,” Meas. Sci. Technol. 12(8), 1311–1317 (2001). [CrossRef]

27.

D. Middleton, Introduction to Statistical Communication Theory (McGraw Hill, 1960).

28.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw Hill, 1958).

29.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4-6), 205–210 (1999). [CrossRef]

OCIS Codes
(090.0090) Holography : Holography
(090.1760) Holography : Computer holography
(100.3010) Image processing : Image reconstruction techniques
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
(090.1995) Holography : Digital holography

ToC Category:
Holography

History
Original Manuscript: June 1, 2012
Revised Manuscript: July 5, 2012
Manuscript Accepted: July 16, 2012
Published: July 25, 2012

Citation
Mathieu Leclercq and Pascal Picart, "Digital Fresnel holography beyond the Shannon limits," Opt. Express 20, 18303-18312 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-18303


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References

  1. J. W. Goodman and R. W. Laurence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett.11(3), 77–79 (1967). [CrossRef]
  2. M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys.17, 333–334 (1972).
  3. U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt.33(2), 179–181 (1994). [CrossRef] [PubMed]
  4. L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett.18(11), 846–848 (1993). [CrossRef] [PubMed]
  5. Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Image reconstruction for in-line holography with the Yang-Gu algorithm,” Appl. Opt.42(32), 6452–6457 (2003). [CrossRef] [PubMed]
  6. Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE3098, 224–233 (1997). [CrossRef]
  7. Th. Kreis, “Frequency analysis of digital holography,” Opt. Eng.41(4), 771–778 (2002). [CrossRef]
  8. Th. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng.41(8), 1829–1839 (2002). [CrossRef]
  9. C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt.38(22), 4812–4820 (1999). [CrossRef] [PubMed]
  10. M. Liebling, T. Blu, and M. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A21(3), 367–377 (2004). [CrossRef] [PubMed]
  11. I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt.40(34), 6177–6186 (2001). [CrossRef] [PubMed]
  12. P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A25(7), 1744–1761 (2008). [CrossRef] [PubMed]
  13. D. P. Kelly, B. M. Hennelly, C. McElhinney, and T. J. Naughton, “A practical guide to digital holography and generalized sampling,” Proc. SPIE7072, 707215 (2008). [CrossRef]
  14. D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng.48(9), 095801 (2009). [CrossRef]
  15. D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5D imaging performance of digital holographic systems,” JEOS rapid publication6, 11034 (2011). [CrossRef]
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