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, 2]. The complex wavefront in the object plane is reconstructed by applying the Fresnel integral to the recorded diffraction pattern [3–5]. 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, 6]. 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]. As a consequence of this undersampling, Onural demonstrated that we get superposed, shifted replicas of the original object function [8]. 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] 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]. 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]. 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]. Consider a vertical fiber of diameter d illuminated by a monochromatic plane wave and located at a distance z_e from a camera (see Fig. 1).

 

Fig. 1. Fiber hologram recorded in the Gabor configuration

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

where

and

The fringe spacing of the chirp function [h_ze (x, λ) + h_ze¯ (x,λ)]depends on x and is given by $i\left(x\right)=\frac{\lambda z_e}{x}$. 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], the center of one of the moiré pattern is located at the abscissa x_M such that the fringe spacing i exactly corresponds to the sampling period:

Figure 2 shows a simulation of the intensity distribution of a fiber (d= 40 μm) recorded at a distance z_e=30mm by a CCD camera. I_ze 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 3^rd lateral side-lobes. As described in Ref. [8], 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:

 

Fig. 2. Intensity distribution of the diffraction pattern sampled by the imaging system (p=11 μm, d = 40 μm, z_e = 30 mm and λ=0,6328 μm).

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

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 $\omega =\frac{2\mathrm{\pi c}}{\lambda }$:

with:

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]:

where C is a normalized coefficient.

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

where

and

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

where

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 [h_ze (x,ω ₀) + h_ze¯ (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 $\frac{1}{e^2}$ attenuation of the chirp function is necessary at x=x_M and by introducing (5) in (17), it leads to:

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 I_ze(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 z_e=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, z_e = 30 mm and λ₀=800 μm.

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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 z_e=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] 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, z_e = 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.

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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.

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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]. 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]. 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:

I₁ can be calculated:

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

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

By taking into account the order of magnitude of the variables δ and γ, we can notice that the term $\sqrt{\frac{{\omega }_0\chi }{2\mathrm{\pi c}{z}_e}}\mathit{sin}c\left(\mathrm{\delta \chi }\right)$ 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]:

By using again the equality (A1) we obtain:

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