In this paper, we have investigated on the potentialities of digital holography for whole reconstruction of wavefields. We show that this technique can be efficiently used for obtaining quantitative information from the intensity and the phase distributions of the reconstructed field at different locations along the propagation direction. The basic concept and procedure of wavefield reconstruction for digital in-line holography is discussed. Numerical reconstructions of the wavefield from digitally recorded in-line hologram patterns and from simulated test patterns are presented. The potential of the method for analysing aberrated wave front has been exploited by applying the reconstruction procedure to astigmatic hologram patterns.
© Optical Society of America
In recent years Digital Holography (DH) technique has been demonstrated to be a useful method in different fields of optics like microscopy [1,2,3], deformation analysis , object contouring , particles sizing and position measurement . In DH the recorded intensity distribution of the hologram is multiplied by the reference wavefield in the hologram plane and the diffracted field in the image plane is determined by the usual Fresnel-Kirchoff integral  to calculate the intensity and the phase distribution of the reconstructed real image. Compared to conventional holographic technique, digital recording and numerical reconstruction of holograms offer new possibilities in optical metrology. In fact, since the hologram is coded numerically as a digitized image, it is not necessary to process a photographic plate to reconstruct a real image. Moreover, the numerical reconstruction of the complex wavefield allows access to not only intensity, which is obtained by conventional photographic methods, but also to phase. Limitations of DH imposed by the low spatial resolution of the CCD detector have been widely discussed in literature .
Recently it has been shown that DH can be efficiently employed to compensate for aberrations [9,10] and for correcting image reconstruction in the presence of severe anamorphism . Further interesting applications of DH rely on the possibility of carrying out whole reconstruction of the recorded wave front, i.e. the determination of intensity and of the phase distribution of the wavefield at any arbitrary plane located between the object and the recording plane. To the best of our knowledge, the possibility of using DH for reconstructing whole optical wavefields has not been fully exploited in the framework of wave front sensing for optical testing. Quantitative determination of the complex amplitude of the field propagating away form the object allows investigation of the modifications suffered by the wavefield through phase-distorting media, e.g. lens with aberrations or ground-glass screen or atmospheric turbulence, to cite only some applications.
In this paper we discuss the principle of the method for numerical reconstruction of the wavefield complex amplitude, and we show that this technique can be used for simultaneously determining the intensity and phase distributions at locations along the propagation direction backward from the hologram plane. We present the numerical reconstruction of the wavefield from digitally recorded in-line hologram patterns and from simulated test patterns with the aim of examining the reliability of the method and its potential for analyzing wavefields.
2. Theoretical principle and experimental description
Holography is a method that allows reconstruction of whole optical wavefields. The hologram is recorded onto a high resolution CCD array and then multiplied by the reference wavefield in the hologram plane to calculate the diffraction pattern in the image plane. The reconstructed field b′(x′, y′) in the image plane is obtained by using the well known  Fresnel-approximation of the Rayleigh-Sommerfield diffraction formula, namely
where the quadratic phase function g(ξ,η)is the impulse response
d′ is the reconstruction distance, namely the distance backward measured from the hologram plane ξ-η to the image plane; the spatial frequencies are ν=x’/(d’λ), µ=y’/(d’λ); h(ξ,η) is the recorded hologram; r(ξ,η) represents the reference wavefield and λ is the wavelength of the laser source. The discrete finite form of equation (1) is obtained through the pixel size (Δξ, Δη) of the CCD array , which is different from that (Δx’,Δy’) in the image plane x′-y′ and related as follows
where N is the pixel number of the CCD array in each direction. We see that according to the equation (1), the wavefield b(x′, y′;d′)is determined essentially by the two-dimensional Fourier transform of the quantity h(ξ,η)r(ξ,η)g(ξ,η). Equation (1) is employed as the basic governing equation for determining both the light intensity distribution Id′(x′,y′)=b(x′,y′,d′)*b (x′, y′; d′) in the image plane at a distance d′ from the hologram plane and the phase distribution ψ(x′, y′;d′)=Arg[b(x′, y′;d′)]. It was pointed out that in the formulation based on equation (1) the reconstructed image is enlarged or contracted according to the reconstruction depth d’. An alternative approach is useful for keeping the size of the reconstructed image constant . In this formulation, the wavefield b(x′, y′;d′)can be computed by
where ℑ[g(ξ,η)]is the Fourier transform of the impulse response (cfr. Eq. (2)), namely
Taking into account the form of the impulse response in equation (2) we have that its Fourier transform is given by
With this method the size of the reconstructed image does not change, i.e., Δx′=Δξ,Δy′=Δη and one needs one Fourier transform and one inverse Fourier transform each to obtain one two-dimensional reconstructed image at a distance d′. Although the computational procedure is heavier in this case compared to the Fresnel approximation approach of equation (1), this method allows for easy comparison of the reconstructed images at different distances d′ since the size does not change with modifying the reconstruction distance. Furthermore, in this case we get an exact solution to the diffraction integral as far as the sampling Nyquist theorem is not violated.
2.1 Wavefield intensity reconstruction from digitized experimental holograms
In this section we present the numerical results obtained through the DH method for reconstructed intensity distribution of the object wavefield using two recorded holograms digitised with two different set-up conditions. The FFT digital reconstruction of the intensity was carried out at different locations z=d′ of the image plane along the z-axis propagation direction. A Mach-Zehnder interferometer (see Fig. 1) was used for the observation of in-line hologram patterns.
A collimated He-Ne laser beam (wavelength λ=632.8 nm) is divided by the beam splitter BS1 into two beams: one of these, the object beam, is a spherical wave produced by an achromatic doublet of focal length 300 mm (see Fig. 1); the other one is a reference plane wave, interfering with the object beam at the recombining beam splitter BS2. The hologram pattern was digitized by a CCD camera with pixel size Δξ=Λη=11 µm and recorded under two different conditions corresponding to two settings of the frame buffer. The hologram pattern shown in Fig. 2a was recorded with the right setting of the frame buffer corresponding to 736 columns ×572 row. The image shown is a digitized array of N×N=512×512 8-bit encoded numbers. In Fig. 2b the frame buffer setting was intentionally modified to 768 columns ×572 row in order to introduce a slight anamorphism, which changes the aspect ratio of the image  from the value 1. The effect of the anamorphism in the recorded hologram of Fig. 2b is to introduce a deformation along the x horizontal direction in the whole fringe pattern, thus obtaining elliptical interference fringes instead of the circular fringes shown in Fig. 2a.
In the case of the first recording condition, the sequence of digital reconstruction of the intensity distribution based on the discrete finite form of equation (4) with r(ξ,η)=1 was carried out for values of the reconstruction distance d’ ranging from 170 mm to 200 mm, with spatial discrete step of Δz=1mm. In the case of the aberrated hologram pattern in Fig. 2b the intensity distribution was determined for d’ ranging from 181 mm to 218 mm and with Δz=1mm. The sequence of intensity distributions were combined to obtain the two clip videos presented in Fig.3.
The reconstructed wavefield in the hologram plane contains three terms, which generate the zero order diffraction, the real and the unsharp virtual image of the object (here represented by the focal point of the object wavefield). The reconstructed intensities in Fig.3 show clearly the patterns of these three terms that are superposed because of the in-line geometry of the set-up (see Fig. 1). The clip video in Fig.3a shows that a point shaped intensity pattern is obtained at the reconstruction distance d’=D=180 mm from the hologram plane. According to simple geometrical considerations (see Fig. 4), this distance corresponds to the focusing distance of a converging spherical wavefield produced by the achromatic doublet.
The digital intensity reconstruction in the movie of Fig. 2b shows the focusing of a wavefield affected by anamorphism. It can be clearly seen that in this condition we have two line images: a line image at the horizontal focal line, occurring at d’≅183 mm, corresponding to the tangential focus, and a vertical line image at the sagittal focus reconstructed at a distance d’≅218 mm.
This simple example shows that numerical reconstruction of holograms provides an efficient method for visualizing qualitatively the influence of wavefield aberrations and makes it possible, in principle, to compensate for phase distortions suffered by the wave front along its propagation.
Quantitative analysis of optical aberrations of wavefields relies on the ability of DH to provide information not only about the intensity but also on the phase distribution of the optical field at different planes from the recorded hologram.
2.2 Reconstructing intensity and phase distributions from simulated holograms
Let us write the intensity distribution I(ξ,η)of the recorded hologram in the following form
Equation (6) describes elliptical interference fringes like those recorded in the experimental conditions of Fig. 2b. The two distances zx,zy correspond to the vertical and horizontal focal lines, respectively. Of course, in the case of circular fringes, as those recorded in Fig. 2a, we have simply that zx=zy=z. The floating-point numbers computed by equation (6), provide a reasonable approximation of the integer-number distribution that occurs from the frame store. Fig. 5a-5b shows respectively the density plot representation of the circular and elliptical fringe patterns computed for z=250 mm in the circular case, zx=300 mm and zy=250 mm in the elliptical one. The test hologram patterns were digitized as an array N×N=512×512 ; we have assumed λ=632.8 nm and step size 11 µm along the x and y directions. Equation (6) can be written in the following form
The first term in equation (6a) produces the zero order of diffraction in the reconstructed image; the other two terms generate the reconstruction of the object beam and that of the conjugate beam. This decomposition is more general than the specific example we are dealing with. In fact, it is well known that in classical holography these two terms correspond to the reconstruction of the virtual image and a real image of the object. In order to reconstruct the complex amplitude of the object beam, we have to isolate one of these two terms, say
where the phase distribution at the hologram plane is given by
After the object beam h(ξ,η) being extracted, a reconstruction procedure is employed to determine the complex amplitude of the wavefield. The extraction of the above terms can be carried out by applying for example the four-quadrature-phase shifting reconstruction algorithm as described in the case of the in-line digital holography  and in ref.  for the contrast enhancement of off-axis Fresnel holograms. In the following we will present numerical simulations to examine the reliability of digital holography for whole object wavefield reconstruction from the knowledge of its complex amplitude h(ξ,η) at the hologram recording plane.
The digital reconstruction of the intensity distributions for the two cases is shown in Fig. 6. Note that in Fig. 6b, the astigmatism of the wavefield results in a bright rectangular component, whereas in the case of the spherical wavefield (see Fig. 6a) we have a square component. The reconstructed image was obtained for a distance d’=180 mm from the hologram plane. In Fig. 6c the reconstruction distance is d’=250 mm. For this distance we have z=zy, the spherical wave front focuses at a single point (Fig. 6c) whereas the astigmatic wavefield focuses at a line image corresponding to the tangential focal line (Fig. 6d). These results reproduce quite well those obtained by the reconstruction procedure of the experimental hologram patterns (compare to the movies in Fig. 3a and 3b).
In Fig. 7a-7b we show the phase distribution of the phase values wrapped in the interval [-π,π]computed by the numerical reconstruction method at the reconstruction distance d’=180 mm. The density plot representations of the wrapped distributions in Fig. 7a and 7b correspond respectively to the simulated spherical and astigmatic wave fronts shown in Fig. 5a and 5b. Both phase distributions at the reconstructed image plane were computed in the restricted range of 140×140 pixels.
In order to compare the numerically reconstructed phase at different planes, we plotted in Fig. 9 the unwrapped phase distributions along the x-horizontal (straight line) and y-vertical (dashed line) phase distributions for the two considered cases and for different reconstruction distances. In Fig. 9a the two distributions are superposed owing to the spherical symmetry of the wave front, whereas in Fig. 9b they are clearly different due to the astigmatism. The vertical axis in Fig. 9a-9b is the z propagation axis along which the various phase distributions are evaluated for backward reconstruction distances ranging from d’=160 mm to d’=220 mm at step size of 10 mm. The scale of the horizontal axis of Fig. 9 is determined by the pixel size Δx′=Δξ of the reconstructed image, which does not change in the reconstruction method. The plots give a perspective of the wave front phase advance as one proceeds by reconstructing at distances closer to the focus in the case of the spherical wave front or to the tangential focal line in the case of the astigmatic wave front. Determination of the intensity, wrapped phase, unwrapped phase at different planes along the propagation direction of the wave front and wrapped phase show the potential of the DH for whole optical wavefield reconstruction and for qualitative and quantitative analysis of wavefield aberrations. We end this section by pointing out that once we have carried out the numerical procedure for computing sequence of the complex map of the field b(x′, y′; d′)for various reconstruction distances d’, the phase differences Δψ(x′, y′, Δz)at two planes separated by a distance Δz, can be easily evaluated in terms of the real and imaginary parts of the complex fields b(x′, y′;d′)and b(x′, y′; d′+Δz) by using the following relationship
Equation (7) determines the phase differences as wrapped values modulo 2π. Subsequent application of the unwrapping procedure allows calculation of the unwrapped map of the phase differences Δψ(x′, y′, Δz).
In this paper, we have investigated on the potential of digital holography for whole reconstruction of wavefields. We have shown that this technique can be efficiently used for simultaneously determining the intensity and phase distributions at different locations along the propagation direction backward from the hologram recording plane.
The advantage of the reconstruction method here used is that the size of the reconstructed image remains unchanged, this way allowing for easy comparison of the intensity and phase distributions along different reconstruction planes. We have presented numerical reconstructions of the wavefield from digitally recorded in-line hologram patterns and from simulated test patterns. Simulated test results have been found in good agreement with the experimental observations from recorded holograms. The potential of this method for analyzing aberrated wave front has been exploited by applying the reconstruction procedure to astigmatic hologram patterns.
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