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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 3 — Feb. 5, 2007
  • pp: 887–895
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Suppression of the Moiré effect in sub-picosecond digital in-line holography

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun  »View Author Affiliations


Optics Express, Vol. 15, Issue 3, pp. 887-895 (2007)
http://dx.doi.org/10.1364/OE.15.000887


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Abstract

Digital in-line holography is widely used in flow studies where the 3D position, size and velocity of particles or fibers has to be known at a given time. When holograms are recorded by using a sub-picosecond laser, the enlargement of the spectral distribution acts as a spatial low-pass filter over the intensity distribution of the diffraction pattern. This is not a disadvantage as regards to the spatial sampling. Indeed, the Moiré effect, due to the sub-sampling of the fringe pattern is naturally reduced. Experimental results are provided.

© 2007 Optical Society of America

1. Introduction

In digital in-line holography, the diffraction pattern, produced by small particles illuminated by a coherent light source, is recorded by a CCD camera [1

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

, 2

2. S. Belaïd, D. Lebrun, and C. Özkul, “Application of two dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. 36,1947–1951, (1997). [CrossRef]

]. The complex wavefront in the object plane is reconstructed by applying the Fresnel integral to the recorded diffraction pattern [3–5

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

]. In the case of in-line holography, this integral can be seen as a wavelet transformation or as a Fractional Fourier transformation of the intensity distribution [2

2. S. Belaïd, D. Lebrun, and C. Özkul, “Application of two dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. 36,1947–1951, (1997). [CrossRef]

, 6

6. S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A 19,1537–1546 (2002). [CrossRef]

]. The in-line configuration is convenient for metrology in fluids where optical access is not easy. Furthermore, the hologram is produced by a small angle between the unscattered reference wave and the wave scattered by the object. As a result, the bandwidth of such digital hologram is generally well suited to the poor sampling rate of solid-state detectors (CCD, CMOS). Then the condition required for a digital recording system of hologram can be easily satisfied. However, under certain conditions (object near the CCD sensor, lower sampling rate), an under-sampling of the hologram may occur and gives rise to a moiré pattern which harms to a correct reconstruction of such holograms [7

7. D Lebrun, C Ozkul, D Allano, and A Leduc,“Use of the moiré effect to improve diameter measurements with charge coupled imagers,” J. Opt. 22175–184 (1991). [CrossRef]

]. As a consequence of this undersampling, Onural demonstrated that we get superposed, shifted replicas of the original object function [8

8. L. Onural, “Sampling of the diffraction field,” App. Opt. 39,5929–5935 (2000). [CrossRef]

]. The author concludes that this effect is not a severe problem as long as the object size is small. However, it may be more penalizing in the case of a particle field because a given replica may be superposed with another object function. In Ref. [9

9. M. Kato, Y. Nakayama, and T. Suzuki, “Speckle reduction in holography with a spatially incoherent source,” App. Opt. 14,1093–1099 (1975). [CrossRef]

] Kato et al. proposed to limit the transfer function of the hologram by using a spatially incoherent source. The authors show that the moiré pattern can be suppressed, provided that the cut-off frequency of the transfer function of the hologram check the Nyquist sampling criteria. The same idea has been used in Ref. [10

10. F. Dubois, L. Joannes, and J.C. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” App. Opt. 34,7085–7094 (1999) [CrossRef]

]. Here the coherence state of the laser source was intentionally decreased in order to reduce the speckle noise in digital holography microscopy. Recently, it has been shown that by illuminating particles by an ultrashort pulse laser, the same advantages can be expected [11

11. F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” Opt. Commun ,268,27–33 (2006). [CrossRef]

]. In this situation, the intensity distribution of the diffraction pattern is windowed by a super-Gaussian function. This smoothing effect, due to the spectral enlargement of the laser source, plays a convenient pre-filtering role of the optical signal. We show that the moiré effect is removed. Here, the example of small fibers is presented because the filtering effect is easily detectable.

2. Moiré effect of fiber holograms

Let us consider the theoretical formalism developed in Ref. [2

2. S. Belaïd, D. Lebrun, and C. Özkul, “Application of two dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. 36,1947–1951, (1997). [CrossRef]

]. Consider a vertical fiber of diameter d illuminated by a monochromatic plane wave and located at a distance ze from a camera (see Fig. 1).

Fig. 1. Fiber hologram recorded in the Gabor configuration

Under far-field approximation (i.e. πd 2/(2λze)≪1), the intensity distribution recorded by the camera is described by:

Izexλ=1[hzexλ+hzeˉxλ]Fxλ+1λzeF2xλ
(1)

where

rect(xd)={1ford2xd20elsewhere
(2)
hzexλ=1λzeexpj(πx2λzeπ4)
(3)

and

Fxλ=dsin(πdxλze)(πdxλze)=dsinc(dxλze)
(4)

The fringe spacing of the chirp function [hze (x, λ) + hze¯ (x,λ)]depends on x and is given by i(x)=λzex. It is well-known that when the pixel size p of the CCD camera is higher than the half period of the diffraction pattern (i<2p), the image is under-sampled and two unexpected moiré patterns appear. As described in Ref. [7

7. D Lebrun, C Ozkul, D Allano, and A Leduc,“Use of the moiré effect to improve diameter measurements with charge coupled imagers,” J. Opt. 22175–184 (1991). [CrossRef]

], the center of one of the moiré pattern is located at the abscissa xM such that the fringe spacing i exactly corresponds to the sampling period:

x=xM=±λzep.
(5)

Figure 2 shows a simulation of the intensity distribution of a fiber (d= 40 μm) recorded at a distance ze=30mm by a CCD camera. Ize is represented versus the spatial coordinate x. This distribution predicts the grey-level distribution delivered by the imaging system. The moiré effect (see black arrows) appears as a non-expected low spatial frequency region that can be observed on each side. In this example, the moiré patterns are centered on the 3rd lateral side-lobes. As described in Ref. [8

8. L. Onural, “Sampling of the diffraction field,” App. Opt. 39,5929–5935 (2000). [CrossRef]

], this pattern leads to unwanted reconstructed images (shift replicas). Consequently, an analogical spatial pre-filtering should be applied before sampling the intensity distribution by a discrete sensor. The following section shows that this operation is optically obtained by a short-time illumination.

3. Sub-picosecond recording of digital in-line holograms

In this article we consider the case where the fiber is illuminated by an ultra-short pulse u(t) described by the following temporal function:

u(t)=exp[jω0tt2T2]
(6)
Fig. 2. Intensity distribution of the diffraction pattern sampled by the imaging system (p=11 μm, d = 40 μm, ze = 30 mm and λ=0,6328 μm).

By a simple Fourier transformation, it leads to the spectrum expression:

U(ω)=Tπexp[T24(ωω0)2]
(7)

T represents the time when the amplitude drops to 1/e of the peak amplitude (e≈2.718) and ω 0 is the central pulsation of the illuminating beam.

For the analytical development it is more convenient to rewrite Eq. (1) as a function of the pulsation ω by using the relation ω=2πcλ:

Izexω=1[hzexω+hzeˉxω]Fxω+ω2πczeF2xω
(8)

with:

hzexω=ω2πczeexpj(ωx22czeπ4)andFxω=dsinc(ωdx2πcze)
(9)

Now if we consider the contribution of each frequency, it can be demonstrated that the intensity distribution recorded by the image sensor is the sum of each spectral component, weighted by |U(ω)|2 [11

11. F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” Opt. Commun ,268,27–33 (2006). [CrossRef]

]:

Ize(x)=C+IzexωU(ω)2
(10)

where C is a normalized coefficient.

By introducing Eqs. (7) and (8) in Eq. (10) and omitting the multiplicative constant, we obtain:

Ize(x)=I1I2I3+I4
(11)

where

I1=+exp[T22(ωω0)2],
(12)
I2=+hze(x,ω)F(x,ω)exp[T22(ωω0)2],
(13)
I3=+hzeˉxωFxωexp[T22(ωω0)2]
(14)

and

I4=+ω2πczeF2xωexp[T22(ωω0)2].
(15)

The calculation of each integral (see Appendix) leads to the following result:

Ize(x)=1[hzexω0+hzeˉxω0]WxTFxω0+ω02πczeF2xω0
(16)

where

WxT=exp[x48c2ze2T2]
(17)

Note that by comparing Eqs. (8) and (16), the time limitation of the laser source acts as a spatial windowing of the interference fringes. Consequently, the high spatial frequencies of the chirp function [hze (x,ω 0) + hze¯ (x,ω 0)] (high values of x) are naturally smoothed by a super-Gaussian function W(x,T). If we assume that, for removing the moiré effect, a 1e2 attenuation of the chirp function is necessary at x=xM and by introducing (5) in (17), it leads to:

T<λ02ze4cp2
(18)

Eq. (18) gives the maximum pulse duration that is necessary for avoiding the spatial sampling errors of the hologram. Figure 3 presents the variations of Ize(x) versus x. The representation of W(x,T) on the same graph shows the effect of a 85 fs pulse duration on the diffraction pattern. Here, the fiber is located at ze=30mm from the CCD camera, the pixel size p=11 μm and the central wavelength of the laser source is λ0= 800 nm

Fig. 3. Intensity distribution of the diffraction pattern : low-pass filtering effect produced by a short laser pulse (T=85 fs). p=11 μm, d = 40 μm, ze = 30 mm and λ0=800 μm.

Compared to Fig 2, Fig. 3 shows clearly that the undersampling effect is avoided and the moirés are removed. The pulse duration T could also be seen as an adjustable parameter that enables to adapt the bandwidth of the hologram to the sampling rate of the CCD sensor. As we will see on the following section, the central part of the diffraction pattern contains enough information for an optimal reconstruction of the digital hologram.

4. Experimental results and discussion

Figure 4 presents two experimental digital holograms of a 34 μm fiber located at ze=30 mm from the CCD camera. The first one is recorded by a He-Ne laser [Fig. 4(a)] and the second one by a commercial femtosecond Ti:Sapphire laser system [Fig. 4(b)] operating at λ 0 = 800nm with a repetition rate of 105 MHz. Autocorrelation measurements indicate that the pulse duration T=85 fs and the spectral bandwidth is about 14 nm. The smoothing effect and the suppression of the moiré effect is directly observable on the diffraction patterns.

Let us now consider the reconstructed images. Due to the finite size of the image sensor, the image boundaries are spoiled by square fringe patterns. The authors of Ref. [12

12. E. Cuche, P. Marquet, and C. Depeursinge, “Aperture apodization using cubic spline interpolation : application in digital holographic microscopy,” Opt. Commun. 182,59–69 (2000). [CrossRef]

] used a numerical apodization for removing this effect. In the present study, a pre-processing step consisting in dividing the intensity of the diffraction patterns by the background level intensity has been used. This operation enables to suppress the background variations and reduces the contrast of these unwanted fringes. In addition, the images of Fig. 5 have been cropped. This avoid to exhibit some residual fringes lying in the original images.

Fig. 4. Experimental intensity distribution in the diffraction pattern of a fiber. p=11 μm, d = 40 μm, ze = 30 mm. (a) with an He-Ne laser, λ0=632.8 nm, (b) with a femtosecond Ti:Sa laser, T=85 fs and λ0=800 nm.
Fig. 5. Digital reconstruction of the holograms of Fig. 4. (a). with an He-Ne laser, λ0=632.8 nm, (b). with a femtosecond Ti:Sa laser, T=85 fs and λ0=800 nm.

Here, the wavelet transformation has been used. The well-known diffraction pattern due to the defocused twin image is superimposed on the reconstructed focused image [2

2. S. Belaïd, D. Lebrun, and C. Özkul, “Application of two dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. 36,1947–1951, (1997). [CrossRef]

]. As we can see on Fig. 5(a), the moiré effect produces unexpected ghost images. This effect is removed by using an ultra-short pulse illumination [Fig. 5(b)]. Obviously, it can be also observed that the reduction of the bandwidth leads to a smoothing effect on the reconstructed images that may have an influence on the spatial resolution. The role played by a windowing function for the determination of 3D particle location by digital holography is detailed in Ref. [4

4. C. Fournier, C. Ducottet, and T. Fournel, “Digital in-line holography : influence of the reconstruction function on the axial profile of a reconstructed particle image,” Meas. Sci. Technol 15,686–693 (2004). [CrossRef]

]. In the present study, a numerical window function with a shape close to W(x,T) could also have been directly applied on the intensity distribution of Fig. 4(a). However, it leads one to suppose that the intensity distribution to be perfectly centered at the origin (x=0). Consequently, if more than one object were recorded, the windowing function should be recalculated in order to adapt it in each object. In our case, the filter is automatically and instantaneously applied on the whole field.

5. Conclusion

When fiber holograms are recorded by a digital image recording system, the spatial sampling error gives rise to a harmful moiré effect due to the aliasing. The reconstructed images are spoiled by unexpected ghost patterns. By recording the hologram with a sub-picosecond laser, the fringe pattern is optically smoothed and the moiré effect is avoided. Theoretical and experimental results are provided. These results may lead to the possibility of realizing a tunable laser source where the pulse duration is adjusted for filtering the diffraction pattern of an in-line hologram in order to adapt it to a given spatial sampling rate of a CCD camera.

6. Appendix

The objective of this section is to present the calculation of calculating I1, I2, I3 and I4. By using the following equality:

+exp(p2u2±qu)du=πpexp(q24p2)
(A1)

I1 can be calculated:

I1=2πT
(A2)

For calculating I2, I3 and I4, the following variables are introduced:

χ=ωω0,β=T2ω022,γ=ω0x22czeandδ=ω0dx2πcze
(A3)

The spectral bandwidth being narrow, χ is close to unity. By using these parameters the integral I2 becomes:

I2=ω0dexp(jπ4)+exp[β(χ1)2]ω0χ2πczeexp(jγχ)sinc(δχ)dχ
(A4)

By taking into account the order of magnitude of the variables δ and γ, we can notice that the term ω0χ2πczesinc(δχ) is much less oscillating than the function exp(jγχ). If β ≫ 1, the integral I2 can be approximated by applying the steepest descent method [11

11. F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” Opt. Commun ,268,27–33 (2006). [CrossRef]

]:

I2ω0dexp(jπ4)sinc(δ)ω02πcze+exp[β(χ1)2]exp(jγχ)
(A5)

By using again the equality (A1) we obtain:

I2ω0πβω02πczedsinc(δ)exp[j(γπ4)]exp[γ24β]
(A6)

I3 and I4 are calculated by using the same method. Finally, omitting a multiplicative constant, the intensity distribution recorded by the CCD camera is:

Ize(x)=1[hzexω0+hzeˉxω0]WxTFxω0+ω02πczeF2xω0
(A7)

References and links

1.

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

2.

S. Belaïd, D. Lebrun, and C. Özkul, “Application of two dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. 36,1947–1951, (1997). [CrossRef]

3.

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

4.

C. Fournier, C. Ducottet, and T. Fournel, “Digital in-line holography : influence of the reconstruction function on the axial profile of a reconstructed particle image,” Meas. Sci. Technol 15,686–693 (2004). [CrossRef]

5.

W. Xu, M. H. Jericho, and H. J. Kreuzer, “Tracking particles if four dimensions with in-line holographic microscopy,” Opt. Lett. 28,164–166 (2003). [CrossRef] [PubMed]

6.

S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A 19,1537–1546 (2002). [CrossRef]

7.

D Lebrun, C Ozkul, D Allano, and A Leduc,“Use of the moiré effect to improve diameter measurements with charge coupled imagers,” J. Opt. 22175–184 (1991). [CrossRef]

8.

L. Onural, “Sampling of the diffraction field,” App. Opt. 39,5929–5935 (2000). [CrossRef]

9.

M. Kato, Y. Nakayama, and T. Suzuki, “Speckle reduction in holography with a spatially incoherent source,” App. Opt. 14,1093–1099 (1975). [CrossRef]

10.

F. Dubois, L. Joannes, and J.C. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” App. Opt. 34,7085–7094 (1999) [CrossRef]

11.

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” Opt. Commun ,268,27–33 (2006). [CrossRef]

12.

E. Cuche, P. Marquet, and C. Depeursinge, “Aperture apodization using cubic spline interpolation : application in digital holographic microscopy,” Opt. Commun. 182,59–69 (2000). [CrossRef]

OCIS Codes
(090.0090) Holography : Holography
(090.1760) Holography : Computer holography
(100.3010) Image processing : Image reconstruction techniques
(100.6890) Image processing : Three-dimensional image processing

ToC Category:
Holography

History
Original Manuscript: September 27, 2006
Revised Manuscript: November 13, 2006
Manuscript Accepted: November 13, 2006
Published: February 5, 2007

Citation
F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, "Suppression of the Moiré effect in sub-picosecond digital in-line holography," Opt. Express 15, 887-895 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-3-887


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References

  1. L. Onural, "Diffraction from a wavelet point of view," Opt. Lett. 18, 846-848 (1993). [CrossRef] [PubMed]
  2. S. Belaïd, D. Lebrun and C. Özkul, "Application of two dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame," Opt. Eng. 36, 1947-1951, (1997). [CrossRef]
  3. U. Schnars and W. Jüptner, "Direct recording of holograms by a CCD target and numerical reconstruction," App. Opt. 33, 179-181 (1994). [CrossRef]
  4. C. Fournier, C. Ducottet and T. Fournel, "Digital in-line holography : influence of the reconstruction function on the axial profile of a reconstructed particle image," Meas. Sci. Technol 15, 686-693 (2004). [CrossRef]
  5. W. Xu, M. H. Jericho, and H. J. Kreuzer, "Tracking particles if four dimensions with in-line holographic microscopy," Opt. Lett. 28,164-166 (2003). [CrossRef] [PubMed]
  6. S. Coëtmellec, D. Lebrun and C. Özkul, "Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography," J. Opt. Soc. Am. A 19, 1537-1546 (2002). [CrossRef]
  7. D Lebrun, C Ozkul, D Allano, and A Leduc,"Use of the moiré effect to improve diameter measurements with charge coupled imagers," J. Opt. 22175-184 (1991). [CrossRef]
  8. L. Onural, "Sampling of the diffraction field," App. Opt. 39, 5929-5935 (2000). [CrossRef]
  9. M. Kato, Y. Nakayama, and T. Suzuki, "Speckle reduction in holography with a spatially incoherent source," App. Opt. 14, 1093-1099 (1975). [CrossRef]
  10. F. Dubois, L. Joannes and J.C. Legros, "Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence," App. Opt. 34, 7085-7094 (1999) [CrossRef]
  11. F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, "Digital in-line holography with a sub-picosecond laser beam," Opt. Commun,  268, 27-33 (2006). [CrossRef]
  12. E. Cuche, P. Marquet and C. Depeursinge, "Aperture apodization using cubic spline interpolation : application in digital holographic microscopy," Opt. Commun. 182, 59-69 (2000). [CrossRef]

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