Abstract

We present a detailed analysis of image formation in digital Fresnel holography. The mathematical modeling is developed on the basis of Fourier optics, making possible the understanding of the different influences of each of the physical effects invoked in digital holography. Particularly, it is demonstrated that spatial resolution in the reconstructed plane can be written as a convolution product of functions that describe these influences. The analysis leads to a thorough investigation of the effect of the width of the sensor, the surface of pixels, the numerical focusing, and the aberrations of the reference wave, as well as to an explicit formulation of the Shannon theorem for digital holography. Experimental illustrations confirm the proposed theoretical analysis.

© 2008 Optical Society of America

1. INTRODUCTION

Digital holography found its origin in the early 1970s [1], but it became properly available only once its confirmation was established in 1994 [2]. Since then, many spectacular applications have been demonstrated such as microscopic imaging and phase-contrast digital holographic microscopy [3, 4, 5, 6, 7, 8, 9], three-dimensional object recognition and securing information [10, 11, 12, 13], polarization imaging [14, 15], surface shape measurement and contouring and material property investigations [16, 17, 18, 19, 20, 21, 22, 23], vibrations analysis with pulsed lasers and time averaging [24, 25, 26, 27, 28, 29, 30, 31], and multidimensional dynamic investigations [32, 33]. Furthermore, the possibility of correcting aberrations in the digital holographic process has been demonstrated [34, 35]. The particular properties of in-line holography for imaging have also been investigated for particle field extraction [36, 37, 38, 39, 40, 41]. Theoretical support of all these applications have been presented in previous works, which exposed theory and reconstruction algorithms for digital holography, according to the different possible schemes for the recording, i.e., in-line holography [42, 43, 44], off-axis Fresnel holography [45, 46, 47], digital Fourier holography [48], and reconstruction with Fresnelets in off-axis holography [49].

In the past, Yamaguchi et al. proposed an analysis of the image formation process in phase-shifting digital holography [50]. The modeling was based on Fourier optics, and it was shown that magnified images can be reconstructed by use of a divergent reference beam or by addition of a quadratic phase function in the numerical reconstruction. Furthermore, the effect of finite resolution of a CCD camera on the quality of the reconstructed image by using computer simulations was investigated. In their paper, Wagner et al. also used a mathematical model based on Fourier optics to describe discretization effects and to determine lateral resolution [48]. Sampling was modeled by multiplication of the original function with a comb function in digital signal processing. For Fourier holograms, it was found that the spatial extension of pixels affects the intensity of the reconstructed image by multiplication with a sinc function. It was concluded that the intensity at the border of the reconstructed picture decreases to 41% if a fill factor of 100% is used. More recently, several authors focused on some influences in the full recording/reconstruction process such as speckle effects [51, 52] or quantization effects. Baumbach et al. proposed to reduce the speckle noise by using digital reconstructions from laterally shifted holograms and pixelwise averaging [51]. The technique is based on averaging several phase or intensity images with different speckle patterns of the same object. The different speckle patterns are generated experimentally by different lateral positions of the CCD camera when recording the hologram. This can be interpreted as a generation of a large synthetic aperture consisting of the many small apertures given by the single CCD. Note that this approach is quite similar to that proposed in synthetic aperture digital holography [53, 54]. Finally, Mills et al. discussed theoretically and experimentally the influence of bit-depth limitation in quantization [55]. They found that an adequate visual recognition of the reconstructed image can be obtained with the use of at least 4bits. Xu et al. also discussed image analysis of digital holography by considering the space–bandwidth product of the setup [56]. This analysis leads to conclusions similar to those of [48].

As a general rule, it appears that a detailed mathematical model of the recording and the reconstruction processes on the basis of Fourier optics makes it possible to understand the different influences of each of the invoked physical effects. These physical effects concern the technology used for the recording, especially their active surface; they also concern the interferometric process associating the diffracted wave and the reference wave, which as its name indicates must be a reference. The recording of a digital hologram invokes the Shannon theorem. However, there is no explicit formulation for digital Fresnel holography, since no simple relation indicates the “good distance” or “good size” of the considered object. Furthermore, the reconstruction process implies a digital focusing whose influence, although intuitive, is not explicit. The reader can easily apprehend the role of such contributions: All of them influence the resolution of the full process. Thus, to get an explicit analytic formulation of digital holography, this paper focuses on the object–image relation in the full recording–reconstruction holographic process in the context of the Fresnel approximations.

The paper is organized as follows: In Sections 2, 3, an in-depth description of the reconstruction principle leads to a general modeling that will state the notion of intrinsic resolution. The focus will be on the mathematical development of the resolution function in the image plane. Section 4 deals with the influence of the active surface of pixels. Sections 5, 6 discuss, respectively, the influence of focus and aberrations of the reference wave. Section 7 presents a simple formulation of the Shannon theorem for digital Fresnel holography. Some experimental results confirming the theoretical analysis are presented in Sections 3, 7. Section 8 presents the conclusions of the study.

2. GENERAL MODELING OF THE OBJECT–IMAGE RELATION

As a general rule, phenomena involved in a digital holographic process are linear processes. Thus, it seems to be a pertinent way to search for a general relation between object and image that includes convolution products. The main processes that must be taken into account are the following: diffraction, interferences, spatial integration and sampling by pixels, and digital reconstruction. The relation between the object and its digital holographic image can be decomposed in several steps, which are summarized in Fig. 1 . The aim of this paper is to propose a general formulation describing the reconstructed field written in the form of a convolution product between the real object and the impulse response of the full digital holographic process according to

AR(x,y)=κA(x,y)*Rxy(x,y),
where * means convolution, A(x,y) is the real object, AR(x,y) is the reconstructed field, function Rxy(x,y) is related to the image quality in terms of the spatial resolution, and κ is a constant. This paper focuses on each of these steps.

2A. Fundamental Basics

Consider a reference set of coordinates {x,y} attached to the surface of the object of interest; the z axis is perpendicular to the surface and is considered to be the propagation direction of a diffracted light beam. The object surface illuminated by a coherent beam of wavelength λ produces an object wavefront noted A(x,y)=A0(x,y)exp[jψ0(x,y)]. Phase ψ0 is random because of the roughness of the surface, and it will be considered to be uniform over [π,+π]. The object may not be perfectly centered in the reference set of the coordinates, but it can be slightly laterally shifted at coordinates (x0,y0,z). In what follows, it is considered that the object wave propagates through a distance d0, at which it interferes with a plane and a smooth reference wave having spatial frequencies noted as {uR,vR}. At distance d0, the reference set of coordinates is chosen to be {x,y,z}, and in the case where (x0,y0)(0,0) the diffracted field produced by the object is given in the Fresnel approximations by [57]

O(x,y,d0)=jexp(2jπd0λ)λd0exp[jπλd0(x2+y2)]×++A(x,y)exp[jπλd0((xx0)2+(yy0)2)]exp[2jπλd0((xx0)x+(yy0)y)]dxdy.
The general relation for the reference wave is
r(x,y)=aRexp[2jπ(uRx+vRy)+jΔΨab(x,y)].
The term ΔΨab(x,y) added to the reference phase corresponds to aberrations of the reference wavefront. The choice for a plane reference wave is motivated by some considerations: In the case where the reference wave is spherical, its curvature can be inserted in the computation of the diffracted field [45], but if the curvature is false, this results in a focusing error, which will be discussed further in the paper. Furthermore, in off-axis Fresnel holography, the main parameter is the spatial frequencies of the reference wave, even if it is plane or spherical. Note also that in the case of Fourier holograms, focusing is naturally done by the geometry of the setup [48], and a priori focusing error cannot occur. Now, in the interference plane, the hologram H is written as
H(x,y,d0)=O(x,y,d0)2+r(x,y)2+r*(x,y)O(x,y,d0)+r(x,y)O*(x,y,d0).
Note that in the case where (x0,y0)(0,0), the +1 order of the hologram can be rewritten as
r*(x,y)O(x,y,d0)=jexp(2jπd0λ)λd0exp[jπλd0(x02+y02)]exp[jπλd0(x2+y2)]aRexp[2jπ(uRx0λd0)x2jπ(vRy0λd0)yjΔΨab(x,y)]×++A(x,y)exp[jπλd0(x2+y2)]exp[2jπλd0(x(x+x0)+y(y+y0))]dxdy.
The influence of lateral shift of the object comes apparent in Eq. (5). Indeed, there appears an irrelevant phase term exp[jπ(x02+y02)λd0], a modification of the spatial frequencies of the reference wave that become, respectively, uRx0λd0 and vRy0λd0, and the Fresnel transform of the object field is evaluated at coordinates (x+x0,y+y0) instead of at (x,y) as is classically the case. Spatial frequencies of the reference wave are then increased/decreased by quantities (x0λd0,y0λd0). The influence of the evaluation of the Fresnel transform at coordinates (x+x0,y+y0) instead of (x,y) can be understood by simple means. The Fresnel transform is proportional to a Fourier transform [Eq. (1)]. Shifting the coordinate of the Fresnel transform will lead to multiplication by a biased phase when computing the reconstructed field of the +1 order by using another Fresnel transform. Thus, the mathematical expression of the biased phase will be exp[+2jπ(Xx0+Yy0)], where (X,Y,z) is a set of coordinates attached to the reconstructed plane. So the main influence of the lateral shift of the object is a contribution to the effective spatial frequencies of the reference wave. This point will be discussed further in the paper.

In digital Fresnel holography, the interferogram is recorded with a matrix of pixels. Figure 2 illustrates the general geometry for the recording. Each pixel induces a sampling of the hologram and also a spatial integration due to its extended surface. Generally, the detector includes M×N pixels of pitches px and py, each of them sized Δx×Δy. In a previous paper, the recorded hologram at point (kpx,lpy) was given to be written as [58]

HPIX(kpx,lpy,d0)=[H(x,y,d0)*ΠΔx,Δy(x,y)](kpx,lpy),
with the even pixel function
ΠΔx,Δy(x,y)={1Δx×1ΔyifxΔx2andyΔy20ifnot.
According to diffraction theory, the diffracted field in the +1 order, noted AR+1, at any arbitrary distance dR from the recording plane can be computed with (K,L)(N,M) data points by evaluating the following equation [2, 3, 45]:
AR+1(X,Y,dR)=jexp(2jπdRλ)λdRexp[jπλdR(X2+Y2)]×k=0k=K1l=0l=L1HPIX+1(kpx,lpy)exp[jπλdR(k2px2+l2py2)]×exp[2jπλdR(kXpx+lYpy)].
with
HPIX+1(kpx,lpy)={[(r*(x,y)O(x,y,d0)]*ΠΔx,Δy(x,y)}(kpx,lpy)).
Note that with the properties of Fourier transform, the conjugated order (1 order) is simply given by
AR1(X,Y,dR)=AR+1(X,Y,dR)*.
Thus, the structure of the diffracted field in the 1 order can be easily deduced from that of the +1 order. As a consequence, this paper will focus only on the +1 order.

The general modeling of the image formation process can be globally established by taking simultaneously into account all of the parameters contributing to the full digital holographic process; nevertheless, the diffracted field can also be evaluated by considering only one parameter at each evaluation. This last approach is easier to develop and will be applied in this paper.

2B. Different Contributions to Image Formation

As a first step, consider that (x0,y0)=(0,0); that the reference wave is perfectly plane, i.e., ΔΨab(x,y)=0; that the image is in focus in the +1 order, i.e., dR=d0; and that the pixel has an extended surface, i.e., ΠΔx,Δy(x,y)δ(x,y). In [58], it was demonstrated that the diffracted field in the +1 order is given by

AR+1(X,Y,d0)=λ2d02exp[jπλd0(uR2+vR2)]r*(X,Y)×A(X,Y)*ΠΔx,Δy(X,Y)*W̃NM(X,Y,d0)*δ(XλuRd0,YλvRd0),
where W̃NM(X,Y,d0) is the filtering function of the bidimensional discrete Fourier transform, given by
W̃NM(x,y,d0)=exp[jπ(N1)xpxλd0jπ(M1)ypyλd0]×sin(πNxpxλd0)sin(πxpxλd0)sin(πMypyλd0)sin(πypyλd0).
The first fundamental relation in Eq. (11) needs some comment. It indicates that the reconstructed object is related to the real one by a convolution relation with different contributions. The first is the pixel function, the second is due to the finite size of the recording, and the last is a localization function in the reconstructed field. The convolution with the sinc-type function of Eq. (12) is a result quite coherent with that presented by Kreis [46, 47] and discussed by Guo [59]. As was discussed in references [27, 30, 58], the last convoluting function indicates that the +1 order is localized at coordinates (X0,Y0)=(uRλd0,vRλd0). These coordinates constitute the paraxial position of the reconstructed object when there is no aberration of the reference wavefront. The object localization depends on the spatial frequencies of the reference wave, the reconstruction distance, and the wavelength of the illuminating beam. In the case where (x0,y0)(0,0), the +1 order becomes localized at coordinates (X0,Y0)=(uRλd0x0,vRλd0y0). So the lateral shift of the object contributes to the paraxial localization of the reconstructed image. Note also that in Eq. (11), the reconstructed field is multiplied by the complex conjugate of the reference wave, which is due to the fact that the reconstruction modeling is performed by considering a plane wave with normal incidence; in the case of a spherical reference wave, the complex term, including its curvature, can be inserted into the double summation of Eq. (8). Equation (11) states that the reconstructed object is the real one convoluted by the pixel function. Although it seems to be surprising given previous published papers [48, 56], this result is quite compatible with optical imaging properties. Indeed, in classical imaging where the object is projected on the sensor 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. In classical imaging, the filtering of the real object is due to the lens and the pixel surface. Digital holography is an interferometric method for imaging objects. In such an imaging, the filtering is not performed by a lens, since there is no lens in the setup. However, it is performed by the double free-space propagation due first to the physical diffraction from object to sensor and last to the numerical diffraction from sensor plane to reconstructed plane. So Eq. (11) indicates that the object is convoluted by the point-spread function of the filtering, which is given in Eq. (12), and by the pixel function. Note that Eq. (12) is the diffraction pattern of a numerical rectangular aperture. In classical imaging, one of the convoluting terms is the diffraction pattern of the pupil of the lens. Finally, the fundamental result given in Eq. (11) follows common sense and can be intuitively understood. From this it follows that the influence of the pixel function is a convoluting argument and not a multiplicative sinc function as was proposed in [48, 56].

As a second step, consider that (x0,y0)=(0,0); that the reference is not perfectly plane, i.e., ΔΨab(x,y)0; that the focus is right, i.e., dR=d0; and that the pixel is not extended, i.e., ΠΔx,Δy(x,y)=δ(x,y). The diffracted field in the +1 order is now written as

AR+1(X,Y,d0)=aRλ2d02exp[jπλd0(X2+Y2)]×k=0k=K1l=0l=L1exp[jΔΨab(kpx,lpy)]exp[2jπ(kpxuR+lpyvR)]×F(kpxλd0,lpyλd0)exp[2jπλd0(Xkpx+Ylpy)],
with
F(xλd0,yλd0)=++F̃(x,y)exp[2jπλd0(xx+yy)]dxdy=FT[F̃(x,y)](xλd0,yλd0),
where F̃(x,y)=A(x,y)exp[jπ(x2+y2)λd0] and FT means Fourier transform. Invoking the same properties as in [58], we get
AR+1(X,Y,d0)=λ2d02exp[jπλd0(uR2+vR2)]r*(X,Y)×A(X,Y)*W̃ab(X,Y)*W̃NM(X,Y,d0)*δ(XλuRd0,YλvRd0).
As can be expected, the filtering function and the localization function are included in Eq. (15). Also appearing is a new convoluting function noted W̃ab(x,y), which is due to aberrations of the reference wavefront. Its analytical formulation is related to the Fourier transform
W̃ab(x,y)=exp[jΔΨab(x,y)]exp[2jπλd0(xx+yy)]dxdy.
As a last step, consider that (x0,y0)=(0,0); that the reference wave is perfectly plane, i.e., ΔΨab(x,y)=0; that the pixel is not extended, i.e., ΠΔx,Δy(x,y)=δ(x,y); and that reconstruction is not in focus in the +1 order, i.e., dRd0. The diffracted field in the +1 order is given by
AR+1(X,Y,dR)=exp[2jπ(dR+d0)λ]λ2d0dRexp[jπλdR(X2+Y2)]×k=0k=K1l=0l=L1r*(kpx,lpy)F(kpxλd0,lpyλd0)×exp[jπλ(1dR+1d0)(k2px2+l2py2)]exp[2jπλdR(Xkpx+Ylpy)].
We have
FT1[FT[F̃(x,y)](xλd0,yλd0)](XλdR,YλdR)=λ2d02[F̃(λd0u,λd0v)](XλdR,YλdR)=λ2d02F̃(d0dRX,d0dRY),
where FT1 means inverse Fourier transform. For the reference wave, we have
FT1[aRexp[2jπ(uRx+vRy)]](XλdR,YλdR)=λ2dR2aRδ(X+uRλdR,Y+vRλdR).
If (x0,y0)(0,0), the spatial frequencies of the reference wave become, respectively, (uRx0λd0,vRy0λd0), including the irrelevant phase terms due to (x0,y0), i.e., exp[jπ(x02+y02)λd0] and exp[+2jπ(Xx0+Yy0)], and we get
AR+1(X,Y,dR)=λ2d0dRexp[jπλd0(uR2+vR2)]exp[jπλd0(x02+y02)]r(Xd0dR,Yd0dR)exp[2jπ(d0+dR)λ]exp[+2jπ(Xx0+Yy0)]A(Xd0dR,Yd0dR)*W̃dR(X,Y)*W̃NM*(X,Y,dR)*δ(X+λuRdRx0dRd0,Y+λvRdRy0dRd0).
The function due to the focusing error appears in Eq. (20), and it is noted by W̃dR(x,y). This function is also related to a Fourier transform:
W̃dR(x,y)=exp[jπλ(1d0+1dR)(x2+y2)]exp[2jπλdR(xx+yy)]dxdy.
Note that putting dR=d0 in Eq. (20), and with (x0,y0)=(0,0), leads to the ideal situation and we get the diffracted field
AR+1(X,Y,d0)=λ2d02exp[jπλd0(uR2+vR2)]r*(X,Y)×A(X,Y)*W̃NM(X,Y,d0)*δ(XλuRd0,YλvRd0),
which includes the filtering function and the localization function.

2C. General Linear Relation

As a conclusion of Subsections 2A and 2B, the general theoretical formulation of image formation in digital Fresnel holography is

AR+1(X,Y,dR)=κ×A(Xd0dR,Yd0dR)*W̃ab(X,Y)*W̃dR(X,Y)*ΠΔx,Δy(X,Y)*W̃NM*(X,Y,dR)*δ(X+λuRdRx0dRd0,Y+λvRdRy0dRd0),
where κ includes irrelevant constants and phase terms of Eq. (20). This relation can also be written in the general form of Eq. (1) by introducing the impulse response of the process
Rxy(x,y)=W̃ab(x,y)*W̃dR(x,y)*ΠΔx,Δy(x,y)*W̃NM*(x,y,dR)*δ(x+λuRdRx0dRd0,y+λvRdRy0dRd0).
In what follows, function Rxy(x,y) is called the resolution function of digital Fresnel holography.

3. INTRINSIC RESOLUTION

In the ideal case and with (x0,y0)=(0,0), Eqs. (22, 24) indicate that the resolution function of digital Fresnel holography is

Rxy(x,y)=W̃NM(x,y,d0)*δ(xλuRd0,yλvRd0).
The localization function does not really contribute to resolution, since it includes only information on spatial localization of the paraxial image and not on image quality. Furthermore, it vanishes in the case of in-line holography because of the three-order overlap (uR,vR)=(0,0). The main function that imposes the spatial resolution is the function W̃NM(x,y,d0). Thus, this is the one that gives the intrinsic resolution in digital Fresnel holography, i.e., the ultimate achievable resolution. Its interpretation is simple: It is a digital diffraction pattern of a rectangular digital aperture with size (Npx×Mpy) and uniform transmittance. Note that domain (Npx×Mpy) corresponds to the size of the recording area and thus of the observation zone. In what follows, this observation zone will be called the “observation horizon.” The mathematical expression of W̃NM(x,y,d0) appears to be equivalent to that of the classical analogical diffraction pattern of a rectangular pupil, i.e., two-dimensional sinc function [46, 47, 48]. Both functions have similar profiles, except for their maximum value, since W̃NM(0,0,d0)=NM. Considering the Rayleigh criterion for determining the width of the function leads to
ρx=λd0Npxandρy=λd0Mpy,
where ρx and ρy are the widths of W̃NM(x,y,d0) at its first zeros. The larger the dimensions of the image sensor (Npx×Mpy), the smaller ρx and ρy, the narrower W̃NM, and the better the resolution. Note that the image produced by digital holography has an intrinsic resolution that is degraded compared to that of classical holography. An experimental comparison between digital and classical holography is proposed in [58].

In order to enhance spatial resolution, several authors proposed to increase the observation horizon with synthetic aperture strategies. Jacquot et al. used a matrix composed of submicrometer-sized dots coupled with scanning, which permitted the increase of the observation horizon and thus the resolution [60]. Other authors [53, 54] used shifting of the sensor in the recording plane to reconstruct a synthetic hologram with dimensions greater than that given by the sensor. Another strategy can also be found in the selection of the wave vector in the Fourier plane using a pinhole [61]. However, these strategies are not suitable for studying dynamic phenomena, such as vibrations or acoustics.

Figure 3 represents the modulus of the intrinsic resolution for N=M=1024, px=py=4.65μm, λ=632.8nm, and d0=500mm. It is found ρx=ρy=66.4μm, giving a bandwidth of 15mm1; this will be the ultimate achievable resolution of such a sensor.

In digital holography, the diffracted field must be computed over a finite number of sampling points. Remember also that a digital hologram is a two-dimensional signal sampled over N×M points. Computation of the discrete Fresnel transform can be performed with K×L points such that (K,L)(N,M). If (K,L)=(N,M) then the raw hologram is used for computation. If (K,L)(N,M), this case is called “zero padding,” and it consists in adding (KN,LM) zeros to the hologram matrix. Fundamentally, these zeros do not add pertinent information; however, they modify the sampling of the diffracted field. In the case where the reconstruction is performed with fast Fourier transform algorithms, the pixel pitches of the reconstructed plane are given by

Δξ=λd0KpxandΔη=λd0Lpy.
Image plane sampling is simply ξ=nΔξ and η=mΔη, where n and m vary from K2 to K21 and from L2 to L21. Consequently, the paraxial +1 order is localized at pixel coordinates (uRKpx,vRLpy). In the case where (K,L)=(N,M), then ρx=Δξ and ρy=Δη; in the case where (K,L)>(N,M), then ρx>Δξ and ρy>Δη. This means that the intrinsic resolution is not modified because it is imposed by the number of useful pixels (M,N) of the detector area and not by the number of data points of the reconstructed field. However, there is a decrease of the sampling step, inducing an increase of the “definition” of the image plane. Concretely, this means that one will see more texture in the image: The resolution function W̃NM will be finely sampled, and the granular structure of the object will appear to the observer. Zero padding of the hologram will make fine speckles of the image appear but without decreasing their size. Indeed, the presence of speckles in the image is intuitive considering Eq. (22) because the initial rough object is convoluted with enlarging functions. This aspect is illustrated in Fig. 4 in the case of a 2 euro coin 25mm in diameter illuminated with a HeNe laser that was placed at distance d0=661mm from the sensor (1024×1360 pixels, px=py=4.65μm). The number of reconstructed points is chosen to be (512,1024,2048,4096). When K=L=512, there is a strong reduction of the observation horizon of the hologram, since the number of data points used for the computation is smaller than the initial matrix. In this case, the intrinsic resolution decreases and the reconstructed image appears very bad. When K=L=1024, the number of data points is approximately equal to that given by the sensor (1024 against 1360 in the horizontal direction). The image sampling corresponds also approximately to the intrinsic resolution. Thus the image appears “pixelized.” For K=L=2048, zero padding is effective and image sampling is two times smaller than the intrinsic resolution. So the resolution function is sampled with a better definition, and this allows the observation of the fine texture of the image, particularly its speckle. For K=L=4096, image sampling is now four times smaller than the intrinsic resolution. The definition of the image plane is again increased, but the speckle does not change its size, since it is imposed by the intrinsic resolution.

The following section is devoted to the characterization of the influence of the different contributions to image formation.

4. INFLUENCE OF THE ACTIVE SURFACE OF PIXELS

4A. Resolution Function

Consider that the focus is perfect and that the reference wave is aberration free. From the previous section, the resolution function is

Rxy(x,y)=ΠΔx,Δy(x,y)*W̃NM(x,y,d0).
The pixel function ΠΔx,Δy(x,y) is convoluted with the intrinsic resolution. If we consider that the pixel size is “large,” then this leads to a low-pass filtering of the intrinsic resolution function. It follows that the active surface of pixels may have a significant effect on the spatial resolution of reconstructed images. Figure 5 illustrates this effect and shows the modulus of the intrinsic resolution function compared to that of the effective one in the x direction. Numerical values are N=M=1024, px=py=10μm, Δx=Δy=9.5μm, λ=632.8nm, and d0=150mm. Figure 5 shows that the effective shape of Rxy undergoes a significant decrease of its maximum value and attenuation of its undulations; thus the contrast between peak and foot of the function decreases. This decrease induces a blurring in the image. Figure 6 shows the normalized degradation of the resolution function versus pixels size along the x profile. The numerical values are λ=632.8nm and d0=500mm; the observation horizon is imposed equal to Npx=6.75mm, corresponding to the usual size of the CCD sensor commercially available; and the pixel width Δx varies from 3μm to 10μm.

4B. Criterion for Influence

The degradation of the resolution function due to the active surface of pixel depends on the width of the intrinsic resolution and then on the reconstruction distance. If the distance is “large” and the active surface “small,” there is no degradation of the intrinsic resolution. In the opposite case, where the distance is “small” and the active surface “large,” degradation will be significant, as illustrated in Fig. 6. As can be seen, it is difficult to summarize these notions of “small,” “large,” “moderated,” etc. One can chose a criterion on the resolution function. It can be retained as

C=++Rxy(x,y)2dxdy++RxyΔx=Δy=0(x,y)2dxdy,
where RxyΔx=Δy=0(x,y)=W̃NM(x,y,d0). This criterion corresponds to the ratio of the energies of impulse responses with extended and nonextended pixels; the criterion tends to 1 if the pixel is pointwise and tends to 0 if the pixel is largely extended compared to W̃NM. In order to simplify this, one can consider the one-dimensional problem. So consider a one-dimensional problem along the x coordinate, for which we have
Cx=+Rx(x)2dx+RxΔx=0(x)2dx
and RxΔx=0(x)=W̃N(x). Analytical development with the mathematical expression of W̃NM is not straightforward, but one can advantageously replace it by a sinc function. Indeed, note that
W̃N(x)=RxΔx=0(x)Nej(N1)xpxλd0sinc(πNpxλd0x),
with sinc(x)=sin(x)x. Thus energy of RxΔx=0(x) can be evaluated and is found to be N2λd0Npx. Note that since λd0Npx is also equal to the width of W̃N(x), then the Rayleigh criterion is equivalent to stating that (1N2)+RxΔx=0(x)2dx is equal to the width of the diffraction pattern and then equal to the intrinsic resolution.

The evaluation of energy (1N2)+Rx(x)2dx leads to (Appendix A)

1N2+Rx(x)2dx=λd0Npxk=0k=(2πNpxΔxλd0)2k2×(1)k(2k+1)(2k+2)!;
Cx is then written as
Cx=2k=0k=(2πNpxΔxλd0)2k(1)k(2k+1)(2k+2)!,
giving
Cx=14π2N2px2Δx236λ2d02+16π4N4px4Δx41800λ4d0464π6N6px6Δx6141120λ6d06+.
The criterion is equal to 1 if Δx=0, and it decreases if Δx increases. Figure 7 shows the criterion versus distance d0 and the parameters of data acquisition Δx and N. The numerical values are λ=632.8nm, Δx={10,8,6,4}μm, and N={512,1024,2048} for d0 varying from 100mm to 1000mm.

The curves represent Eq. (30), and the circles represent Eq. (34). Note that some curves overlap others, thus masking them. Figure 7 shows that the greater the number of pixels is, the greater the influence of the active surface is. The criterion allows us to answer a question that the reader can legitimately ask: What is the distance for which the pixel width has significant influence on the impulse response? To answer this question, one can impose that Cx does not vary by more than 10%, so Cx0.9, corresponding to the dashed horizontal line in Fig. 7. It can be seen than for 2048 pixels of size 10μm, the minimal distance for which Cx0.9 is at least 1000mm. The distance will be only 200mm for 512 pixels.

5. INFLUENCE OF FOCUS

5A. Analytical Formulation

In this section, consider that the pixel is not extended and that the focus is not perfect. So we have Rxy=W̃dR*W̃NM*. In the resolution function there appears an enlarging function given by a Fourier transform, and it is written as

W̃dR(x,y)=Npx2+Npx2Mpy2+Mpy2exp[jπλ(1d0+1dR)(x2+y2)]exp[2jπ(xλdRx+yλdRy)]dxdy.
It disappears if dR=d0, since the Fourier transform of 1 is the Dirac distribution δ(x,y). Because this function is related to the focus in the digital reconstruction, its spectral expansion is limited by the size of the image sensor; this means that limiting its observation horizon imposes a limitation of its energy and also of its spatial bandwidth. The extension of coordinates x and y is limited to (xmax,+xmax) and (ymax,+ymax), which correspond to the spatial extension of recording (M×N pixels of pitches px×py; sensor of size Npx×Mpy); thus xmax=Npx2, and ymax=Mpy2. The function W̃dR is related to the Fourier transform of a quadratic phase function for which the local spatial frequencies are [57]
ui=xλ(1d0+1dR)andvi=yλ(1d0+1dR).
Then the maximal spatial frequencies considered on the observation horizon are
uimax=Npx2λ(1d0+1dR)andvimax=Mpy2λ(1d0+1dR).
The analytical evaluation of W̃dR must take this property into account; thus this is why the Fourier transform is evaluated on the horizon of the sensor such as written in Eq. (35). There is no simple analytical formulation for W̃dR. However, from previous discussion, the spatial frequency bandwidth Δu generates a spatial width ΔX in the reconstructed plane via the relation ΔX=λdRΔu. So the width of the defocusing function is related to the spatial frequency bandwidth, given by
Δu=2uimax=Npxλ(1d0+1dR).
Considering the Rayleigh criterion and the profile of W̃dR, the resolution in the presence of numerical defocus in the reconstructed plane can be considered to be
ρxdR=Npx(1+dRd0).
A similar relation holds for the y dimension. This expression gives the contribution of the focus error to the blurring of the image. It depends only on the observation horizon in the considered direction and on the ratio dRd0. It vanishes in the case of perfect focus for the +1 order, since dR=d0. Figure 8 shows Rxy(x,0) for λ=632.8nm, px=4.65μm, N=1024, d0=250mm, and dR varying from 350mm to 152mm. Profiles along the x direction are represented in Fig. 9 . Enlarging due to W̃dR when dR+d00 is visible, and this induces a blur in the reconstructed image because of the enlarging of each point constituting the image. Applying Eq. (39) with d0=250mm and dR+d0=100mm gives ρxdR=1.9mm, which corresponds to the width of the corresponding curve in Fig. 9; in this example, resolution is not better than 1.9mm.

5B. Comparison with a Digital Holographic Simulation

Simulation of the full digital holographic process (recording and reconstruction) was performed using a digital pinhole as an object. Reconstruction is computed with the convolution method [45] such that sampling remains invariant. The parameters are λ=632.8nm, px=4.65μm, N=1024, d0=250mm, and dR varying from 350mm to 152mm. The reference wave has spatial frequencies uR=vR=54mm1. Figure 10 shows the comparison between the resolution functions Rxy obtained with the full process simulation and those with the computation of Eq. (35). The two pictures are quite similar: They have the same width around perfect focusing (dR=d0). However, their amplitudes are slightly different, and this can be seen with the slight variation of the color map representation (grayscale inprint). The similitude of the curves can be appreciated in Figs. 11, 12 , which show the x profile for perfect focusing and the z profile along the z axis. It can be seen that the x profiles overlap, whereas the z profiles are slightly different but have equivalent width according to the Rayleigh criterion. No explications were found for understanding this slight difference. Globally, it can be admitted that the modeling presented in this section is a good approximation of the spatial resolution in digital Fresnel holography in the presence of focusing error.

6. INFLUENCE OF REFERENCE WAVE ABERRATION

6A. Modeling for Aberration

The presence of aberration in the reference wavefront depends on how the reference wave is generated. Note that in this paper, focus and tilt are not considered to be aberrations. It is reasonable to consider the first primary aberrations, which are spherical and coma. Figure 13 shows some cases of setups that produce aberrations in the off-axis reference wave. In cases (a)–(c), the aberration is produced by the lens used for the collimation of the light, whereas in case (d), the aberration is due to the cube and the reference wave is spherical. Whatever the manner used for producing the reference wave, the aberration, if it exists, can be described by the supplementary phase term of Eq. (3). This phase term can be written with a polynomial expansion for spherical aberration and coma (third-order aberrations):

ΔΨab(ξ,η)=2πλ[CS(ξ2+η2)2+CCξ(ξ2+η2)],
where the set of coordinates (ξ,η) is related to the reference wavefront. The aberration “seen” by the sensor depends on how the reference wave is incident on the recording plane.

6B. Aberration in the Recording Plane

Now focus on the aberration in the recording plane at pixel coordinates (x,y). Figure 14 illustrates the two main situations for the incidence on the detector area. The first case (a) deals with the propagation axis of the reference wave incident at the center of the pixel matrix. The last case (b) is when its axis is not centered and is shifted from quantities {δx,δy} along the x and y directions. For both cases, the aberration in the recording plane is obtained by changing ξ into x+δx and η into y+δy in Eq. (40). So it follows that

ΔΨab(x,y)=2πλ[ax4x4+ay4y4+ax3x3+ay3y3+ax2x2+ay2y2+ax1x+ay1y+ax2y2x2y2+ax2y1x2y+ax1y2xy2+ax1y1xy+c].
The reader can refer to Appendix B for more details on the coefficients of the polynomial expansion.

6C. Resolution in the Presence of Aberrations

The Fourier transform expressed in Eq. (16) is not analytically calculable. However, as was described for W̃dR, it is relevant to consider the spatial frequency bandwidth of such a function. The local spatial frequencies are given by [57]

{ui=1λ(4ax4x3+3ax3x2+2ax2x+ax1+2ax2y2xy2+2ax2y1xy+ax1y2y2+ax1y1y)vi=1λ(4ay4y3+3ay3y2+2ay2y+ay1+2ax2y2x2y+ax2y1x2+2ax1y2xy+ax1y1x).
The estimation of the bandwidth allows the determination of the enlarging due to aberration in the reconstructed plane. Since the spatial expansion in the recording area is limited by the detector size, this gives the resolution in the presence of aberration:
{ρxab=λd0Δu2=d0(ax4N3px32+ax2Npx+ax2y2NpxM2py24+ax1y1Mpy2)ρyab=λd0Δv2=d0(ay4M3py32+ay2Mpy+ax2y2N2px2Mpy4+ax1y1Npx2).
This expression is obtained by considering the Rayleigh criterion. Note that constant terms ax1 and ay1 in Eq. (42) contribute to the mean spatial frequencies of the reference wave. Since frequencies (uR,vR) localize the paraxial image, it follows that if ax1 and ay1 are not zero (i.e., CS0, CC0 and δx0 or δy0), the reconstructed object under aberration will be slightly translated by an amount given by
{δX=d0ax1=4d0CS(δx3+δxδy2)+d0CC(3δx2+δy2)δY=d0ay1=4d0CS(δx2δy+δy3)+2d0CCδxδy.
This shift does not alter the image quality but moves the paraxial image. In the case where there is aberration, the spatial resolution is given by
{ρxab=d0(CSN3px32+CSNpx(6δx2+2δy2)+3CCδyNpx+CSNpxM2py22+4CSδxδyMpy+CCδyMpy)ρyab=d0(CSM3py32+CSMpy(2δx2+6δy2)+CCδxMpy+CSN2px2Mpy2+4CSδxδyNpx+CCδyNpx).
In the case of pure spherical aberration (i.e., CC=0), the spatial resolution is now given by
{ρxab=d0(CSN3px32+CSNpx(6δx2+2δy2)+CSNpxM2py22+4CSδxδyMpy)ρyab=d0(CSM3py32+CSMpy(2δx2+6δy2)+CSN2px2Mpy2+4CSδxδyNpx).
In case (b) of Fig. 14, where {δx,δy}{0,0}, the resolution is degraded by the presence of {δx,δy} in Eqs. (45, 46). In case (a), with {δx,δy}={0,0} and with pure spherical aberration, the resolution is simply ρxab=d0NpxCS(N2px2+M2py2)2 and ρyab=d0MpyCS(N2px2+M2py2)2. In the case of pure coma (i.e., CS=0), the spatial resolution is now given by ρxab=d0(3CCδxNpx+CCMpy) and ρyab=d0(CCδxMpy+CCNpx). In case (a), where δx=0, the resolution is simply ρxab=d0CCMpy and ρyab=d0CCNpx.

6D. Numerical Simulation

Simulation of the digital holographic process, according to Subsection 5B, was used for simulating the influence of aberrations. The object is a digital pinhole, and the parameters are λ=632.8nm, px=py=4.65μm, N=M=1024, d0=dR=250mm, and (x0,y0)=(0,0). The reference wave has spatial frequencies uR=vR=54mm1. The setup producing the reference wave is considered to be case (a) of Fig. 13 [21, 22, 27, 30, 31, 33, 58]. The lens is considered to be plano-convex and 50.8mm in diameter, with a focal length of 200mm and a glass index n=1.5168. Note that the lens can be an aberration-free lens such as a Clairaut or a Van Heel objective. However, the choice for the plano-convex lens is useful to illustrate the influence of aberrations on the digital holographic process when one uses low-cost optical components. Spherical aberration of such a lens is about 21 fringes at the border of the pupil. Taking into account the angle producing spatial frequencies uR and vR, coma is about 4 fringes of coma at the border of the pupil. Thus this gives CS=3.4262×108mm3 and CC=1.6581×107mm2. With the results of previous sections, the localization of the paraxial image is (uRλd0,vRλd0)=(8.50mm,8.50mm). Figures 15, 16, 17 show the resolution function obtained with the full process simulation for the digital pinhole and three shifts corresponding to, respectively, no shift (δx,δy)=(0,0), the median shift (δx,δy)=(7.64mm,7.64mm), and the maximum available shift (δx,δy)=(15.3mm,15.3mm). The shift values are computed by taking into account the size of the sensor and the diameter of the lens (see Fig. 14). Figures 15, 16, 17 show a rectangular zone of ±0.5mm around the coordinates of the paraxial image (8.50mm, 8.50mm) in the reconstructed plane. Table 1 summarizes the values obtained for the different contributions to the spatial resolution. The total width of the resolution function is estimated by computing ρx,y_totalab=ρx,y_sphericalab+ρx,y_comaab. Figure 15 shows that in the case of a geometry such as that of Fig. 14a, the influence of aberration is quite minimized, since there is no enlarging of the intrinsic resolution function. Figures 16, 17 show that aberration has a significant influence on the resolution in the case of Fig. 14b. Indeed, the greater is the shift in the pupil, the greater is the enlarging of the intrinsic resolution. The two white circles are localized at coordinates (δX+ρx_totalab,δY+ρy_totalab) and (δXρx_totalab,δYρy_totalab), and they show that the theoretical estimations for the resolution are quite valid. Furthermore, these results show that even if the collimating lens has an aberration, it is possible to optimize the setup such that its influence is irrelevant. For this, the wavefront “seen” by the sensor must be the portion corresponding to the paraxial region, since in this region the reference wavefront will be only slightly distorted.

7. SHANNON THEOREM FOR DIGITAL HOLOGRAPHY

Subsection 2A states that the hologram is a coherent two-wave mixing recorded with discrete spatial sampling. Thus, the signal must be recorded while taking into account the Shannon theorem. Classically, it is admitted that the maximum angle between the two waves must satisfy the relation [2]

θmax2arcsin(λ4max(px,py)).
This equation depends only on the wavelength and on the maximum pixel pitch of the recording area. Note that it does not include explicit information about the object size and the distance at which it is localized from the detector. Equation (47) is a necessary condition for recording but not a sufficient one, because it does not indicate whether the three diffraction orders overlap or not when reconstructing the object field, or whether the spatial frequencies of the reference wave have “good” values. So, with only this assumption on the maximum angle, it is quite difficult to optimize the setup. However, the structure of the diffracted field gives information on how to adjust the experimental setup. In previous sections, interest was focused on the +1 order, with the 1 order being directly related to the latter. In the diffracted field, the last order is the zero order, which is classically seen as a foe in digital Fresnel holography. Some authors have proposed methods to remove its contribution to image formation. For example, the mean value of the hologram can be subtracted [62], or the filtering can be performed in the spectral domain [63]. The only efficient method for suppressing the zero order is that proposed by Yamaguchi; phase shifting digital holography [64], which needs the recording of at least three holograms. The other methods require an optimization of the recording such that there is no overlapping of the three orders. In this section, the zero order is considered as a friend because it allows a new formulation of the Shannon theorem for digital Fresnel holography.

7A. Structure of the Diffracted Field

The zero order is related to the Fresnel transform of O2+r2. Since the discrete Fresnel transform of the constant r2 is an approximately uniform spatial area of size Npx×Mpy, the main contribution to the zero-order width is given by O2. Under an advantageous hypothesis (no aberration, nonextended pixel, no focusing error, and no lateral shift), it is found that

O(x,y,d0)2=1λ2d02++A(x,y)exp[jπλd0(x2+y2)]exp[2jπλd0(xx+yy)]dxdy×++A*(x,y)exp[jπλd0(x2+y2)]exp[+2jπλd0(xx+yy)]dxdy.
The power spectrum of the zero order is related to the convolution of the Fourier transform of the diffracted object field,
Õ0(u,v)=FT[O](u,v)*FT[O*](u,v).
Taking into account the properties of the Fourier transform and dilatation theorem, the convolution product of Eq. (49) is written as
Õ0(u,v)=λ4d04exp[jπλd0(u2+v2)]++A(λd0u1,λd0v1)A*(λd0uλd0u1,λd0vλd0u1)exp[2jπλd0(u12+v12)]du1dv1.
Let us consider the spectral expansion of the zero order by considering the modulus of the spectrum. We have
Õ0(u,v)=λ4d04++A(λd0u1,λd0v1)A*(λd0uλd0u1,λd0vλd0u1)exp[2jπλd0(u12+v12)]du1dv1,
from which comes the inequality
Õ0(u,v)λ4d04++A0(λd0u1,λd0v1)A0(λd0uλd0u1,λd0vλd0u1)du1dv1,
since A0(x,y)=A(x,y) is the modulus of the amplitude of the object. Equation (52) shows that the spectral expansion of the zero order is limited by the spatial expansion of a scaled version of the autocorrelation function of the object. If CA0A0(x,y) is the spatial autocorrelation of A0(x,y), then
Õ0(u,v)λ4d04CA0A0(λd0u,λd0v).
The spectral expansion of the zero order induces its spatial expansion since X=λd0u and Y=λd0v. If AR0(X,Y) is the reconstructed field in the zero order, one gets
AR0(X,Y)λ4d04CA0A0(X,Y).

7B. Optimization of Recording

The spatial expansion of AR0 is given by the autocorrelation function of the object and is limited by twice the lateral dimensions of the object. The optimal experimental recording is dependent on the spatial extension of the object as illustrated in Figs. 18a, 18b . In Fig. 18 it is considered that the optimal recording is obtained if the three orders do not overlap and if the object is fully included in the diffracted field. This means that for a square object of width a, the full width of the reconstructed field (λd0px and λd0py, respectively, in the x and y directions) must be equal to four times the width of the object (i.e., λd0px=4a or λd0py=4a). Note that in the case where pxpy, the smallest width given by the largest pixel pitch [max(px,py)] must be taken into account so that the condition becomes λd0max(px,py)=4a. However, for a circular object of diameter 2a, the smallest diagonal of the reconstructed field must be taken into account [i.e., 2×λd0max(px,py)]. So, due to geometrical considerations, the condition now becomes 2λd0max(px,py)2a(21)=8a. These considerations are graphically explained in Figs. 18a, 18b. The optimal couple of the spatial frequencies of the reference wave can be determined by considering that the reconstructed object must be localized at the paraxial coordinates (X0,Y0)=(±3λd0px(6+22),±3λd0px(6+22)) for a circular object and at the paraxial coordinates (X0,Y0)=(±3λd08px,±3λd08py) for a square object [Figs. 18a, 18b].

Thus for a circular object of diameter 2a [Fig. 18a], the optimal recording distance is given by

d0=(2+32)max(px,py)λa,
and the optimal spatial frequencies are given by
{uR,vR}{±1px36+22+x0λd0,±1py36+22+y0λd0}.
Considering a square object of size a, one gets [Fig. 18b]
d0=4max(px,py)λa,
and the optimal spatial frequencies are given by
(uR,vR)=(±38px+x0λd0,±38py+y0λd0).
Note that for (x0,y0)=(0,0), frequencies {uR,vR} can take independently two values with opposite sign and that in the case of the square object, uR or vR can be chosen to be zero. In the case of the phase-shifting method [64], the zero order and the 1 order are naturally removed by the demodulation algorithm. Thus, the spatial frequencies of the reference wave can be null (in-line holography), and the +1 order can overlap the entire reconstructed field. For the circular object, the optimal recording distance becomes
d0=2max(px,py)λa.
It is about three times smaller than in the previous case. For the square object, it now gives
d0=max(px,py)λa.
It is about four times smaller than in off-axis holography. As a conclusion to this section, Eqs. (55, 56, 57, 58, 59, 60) constitute the Shannon theorem for digital Fresnel holography.

In order to illustrate optimization of the recording by considering the Shannon theorem, consider a circular object of diameter 2a=48mm illuminated by a 2ω NdYAG laser at λ=532nm. Holographic recording is performed with a pixel matrix with (M×N)=(1024×1360) pixels with pitches px=py=4.65μm, and the object is centered on the optical axis [i.e., (x0,y0)=(0,0)]. Digital reconstruction is computed with zero padding with 2048×2048 pixels in the Fresnel transform. With Eq. (55), the recording distance must be set to d0=1186mm and the spatial frequencies are adjusted to uR67mm1 and vR68.2mm1, which are a little bit less than the value of 73mm1 obtained with Eq. (56). Considering the results of subsection 4B, it can be seen that the active surface of pixels (Δx=Δy4.2μm) has no influence on the intrinsic resolution. The reconstructed field is presented in Fig. 19 when the focus is set in the +1 order. The intrinsic resolution is ρx=99.7μm and ρy=132.5μm for a sampling step of Δξ=Δη=66.2μm. The reconstructed field covers an area of 135.7mm×135.7mm, and the paraxial object is localized at coordinates (X0,Y0)=(42.25,43.06)mm, corresponding to pixel coordinates (uRKpx,vRLpy)(775,790). It can be seen that spatial optimization is fulfilled since there is no overlapping and the maximum spatial width is occupied. Note that there is no distortion of the amplitude in the corner of the field by the sinc function as proposed in [48, 56]. When the reconstructing distance is set to dR=+1186mm, the focus is on the 1 order, the reconstructed field having the same dimensions as before. With the hermitic properties, the 1 order is localized at (X0,Y0)=(42.25,43.06)mm as can be seen in Fig. 20 . If the reconstruction distance is different from d0, the object will be blurred as in Fig. 21 . For example, with dR=1000mm, each point of the real object is represented in the diffracted field by a pattern if about a 992μm width [Eq. (39)]; it is also the approximate value of the spatial resolution in the image plane.

8. CONCLUSION

This paper has presented a detailed analysis of the image-to-object relation in digital Fresnel holography. It is demonstrated that the spatial resolution in the reconstructed plane can be written as a convolution product of functions that describe the influence of the different parameters of the full process. These functions are explicated and studied with Fourier analysis and the concept of local spatial frequencies. It is found that these functions are, first, the intrinsic resolution, depending only on the geometry of the detector, on the wavelength, and on the reconstruction distance; second, the active surface of pixels, for which the influence is described by a criterion corresponding to the ratio between energy included in the impulse responses with the extended pixel and with the nonextended pixel; third, the focusing function, depending on the ratio between the reconstruction distance and the recording distance and depending on the observation horizon; and last, aberrations of the reference wavefront, with their influence depending on how the wave is incident on the recording area and wether the setup can be optimized to avoid their effect even if the lens has strong aberrations. The theoretical analysis has taken into account the influence of the lateral shift of the object. It was demonstrated that the shift contributes to the spatial frequencies of the reference wave. The analysis of the full reconstructed field by considering also the zero order leads to a new formulation of the Shannon theorem for digital holography. Its expression allows users to efficiently design the experimental setup by considering the object size, the wavelength, and the pixel pitches. Experimental illustrations confirm the analysis proposed in this paper. Further works will focus on some other sources of influence in digital holography, such as, for example, the influence of pixel saturation or nonlinearity in the full process.

Appendix A

We have

+RxΔx=0(x)2dx=N2+sinc2(πNpxλd0x)dx=Nλd0px=N2(λd0Npx).
According to Parseval’s theorem,
+Rx(x)2dx=+R̃x(k)2dk,
with
R̃x(u)=FT[W̃N(x)](u)×FT[ΠΔx(x)](u),
and
FT[W̃N(x)](u)=λd0pxΠNpxλd0(u(N1)px2λd0)λd0pxΠNpxλd0(uNpx2λd0),
FT[ΠΔx(x)](u)=sinc(πΔx,u).
The Fourier transform of Rx(x) is
R̃x(u)=sinc(πΔxu)λd0pxΠNpxλd0(u),
thus
+Rx(x)2dx=λ2d02px2+sinc(πΔxu)ΠNpxλd0(uNpx2λd0)2du,
giving
+Rx(x)2dx=λ2d02px20+Npxλd0sinc2(πΔxu)du.
With
sin2(πΔxu)(πΔxu)=k=0k=(1)k22k+1(πΔxu)2k(2k+2)!,
we have
sin2(πΔxu)(πΔxu)2du=k=0k=(1)k(πΔx)2k22k+1u2k+1(2k+1)(2k+2)!.
So
+Rx(x)2dx=2N2λd0Npxk=0k=(2πNpxΔxλd0)2k(1)k(2k+1)(2k+2)!.

Appendix B

In Table 2 , the coefficients of the polynomial expansion are prescribed in detail.

Tables Icon

Table 1. Numerical Values for the Resolution Function Computed from Theory

Tables Icon

Table 2. Coefficients of the Polynomial Expansion of Aberration in the Recording Plane

 

Fig. 1 Schematic diagram of image formation in digital holography.

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Fig. 2 Basics of recording a digital hologram.

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Fig. 3 Intrinsic resolution function.

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Fig. 4 Illustration of zero padding.

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Fig. 5 Profiles of the resolution function with nonextended and extended pixels.

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Fig. 6 Influence of the active surface of pixels.

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Fig. 7 Plot of criterion quantifying the influence of the active surface of pixels [circles, Eq. (34); curves, Eq. (30)].

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Fig. 8 Evolution of the resolution function versus the reconstruction distances.

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Fig. 9 Profiles of the resolution function for different reconstruction distances.

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Fig. 10 Comparison between the analytical formulation and the numerical simulation versus the reconstruction distance.

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Fig. 11 Profiles along x for the analytical formulation and numerical simulation at the best focus.

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Fig. 12 Profiles along z for the analytical formulation and full numerical for x=y=0.

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Fig. 13 Different schemes for generating a reference wave.

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Fig. 14 Zone of the wavefront “seen” by the sensor.

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Fig. 15 Resolution for (δx,δy)=(0,0).

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Fig. 16 Resolution for (δx,δy)=(7.64mm,7.64mm).

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Fig. 17 Resolution for (δx,δy)=(15.29mm,15.29mm).

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Fig. 18 Schematic structure of the reconstructed field.

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Fig. 19 Focus on the +1 order.

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Fig. 20 Focus on the 1 order.

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Fig. 21 Defocused image.

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

7. C. Mann, L. Yu, L. Chun-Min, and M. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express 13, 8693–8698 (2005). [CrossRef]   [PubMed]  

8. K. Chalut, W. Brown, and A. Wax, “Quantitative phase microscopy with asynchronous digital holography,” Opt. Express 15, 3047–3052 (2007). [CrossRef]   [PubMed]  

9. J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15, 7231–7242 (2007). [CrossRef]   [PubMed]  

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

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

12. Y. Frauel and B. Javidi, “Neural network for three-dimensional object recognition based on digital holography,” Opt. Lett. 26, 1478–1480 (2001). [CrossRef]  

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

14. T. Nomura and B. Javidi, “Object recognition by use of polarimetric phase-shifting digital holography,” Opt. Lett. 32, 2146–2148 (2007). [CrossRef]   [PubMed]  

15. T. Nomura, B. Javidi, S. Murata, E. Nitanai, and T. Numata, “Polarization imaging of a 3D object by use of on-axis phase-shifting digital holography,” Opt. Lett. 32, 481–483 (2007). [CrossRef]   [PubMed]  

16. I. Yamaguchi, J. Kato, and S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85–89 (2001). [CrossRef]  

17. I. Yamaguchi, T. Ida, M. Yokota, and K. Yamashita, “Surface shape measurement by phase shifting digital holography with a wavelength shift,” Appl. Opt. 45, 7610–7616 (2006). [CrossRef]   [PubMed]  

18. M. Mosarraf, G. Sheoran, D. Singh, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684–689 (2007). [CrossRef]  

19. S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103–126 (2001). [CrossRef]  

20. Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, “Deformation measurement by phase shifting digital holography,” Exp. Mech. 45, 65–70 (2005). [CrossRef]  

21. P. Picart, E. Moisson, and D. Mounier, “Twin sensitivity measurement by spatial multiplexing of digitally recorded holograms,” Appl. Opt. 42, 1947–1957 (2003). [CrossRef]   [PubMed]  

22. P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. (Bellingham) 43, 1169–1176 (2004). [CrossRef]  

23. T. Baumbach, W. Osten, C. von Kopylow, and W. Juptner, “Remote metrology by comparative digital holography,” Appl. Opt. 45, 925–934 (2006). [CrossRef]   [PubMed]  

24. G. Pedrini and H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 18, 251–260 (1995). [CrossRef]  

25. Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal Fourier analyses of a digital hologram sequence,” Appl. Opt. 46, 5719–5727 (2007). [CrossRef]   [PubMed]  

26. G. Pedrini, S. Schedin, and H. J. Tiziani, “Pulsed digital holography combined with laser vibrometry for 3D measurements of vibrating objects,” Opt. Lasers Eng. 38, 117–129 (2002).

27. P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Time averaged digital holography,” Opt. Lett. 28, 1900–1902 (2003). [CrossRef]   [PubMed]  

28. N. Demoli and I. Demoli, “Dynamic modal characterization of musical instruments using digital holography,” Opt. Express 13, 4812–4817 (2005). [CrossRef]   [PubMed]  

29. A. Asundi and V. R. Singh, “Time-averaged in-line digital holographic interferometry for vibration analysis,” Appl. Opt. 45, 2391–2395 (2006). [CrossRef]   [PubMed]  

30. J. Leval, P. Picart, J.-P. Boileau, and J.-C. Pascal, “Full field vibrometry with digital Fresnel holography,” Appl. Opt. 44, 5763–5772 (2005). [CrossRef]   [PubMed]  

31. P. Picart, J. Leval, F. Piquet, J.-P. Boileau, Th. Guimezanes, and J.-P. Dalmont, “Tracking high amplitude auto-oscillations with digital Fresnel holograms,” Opt. Express 15, 8263–8274 (2007). [CrossRef]   [PubMed]  

32. T. Saucedo, F. M. Santoyo, M. De la Torre Ibarra, G. Pedrini, and W. Osten, “Simultaneous two-dimensional endoscopic pulsed digital holography for evaluation of dynamic displacements,” Appl. Opt. 45, 4534–4539 (2006). [CrossRef]  

33. P. Picart, J. Leval, M. Grill, J.-P. Boileau, J. C. Pascal, J.-M. Breteau, B. Gautier, and S. Gillet, “2D full field vibration analysis with multiplexed digital holograms,” Opt. Express 13, 8882–8892 (2005). [CrossRef]   [PubMed]  

34. A. Stadelmaier and J. H. Massig, “Compensation of lens aberration in digital holography,” Opt. Lett. 25, 1630–1632 (2000). [CrossRef]  

35. S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattin, “Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,” Opt. Lett. 26, 974–976 (2001). [CrossRef]  

36. L. Onural and M. T. Ozgen, “Extraction of three-dimensional object-location information directly from in-line holograms using Wigner analysis,” J. Opt. Soc. Am. A 9, 252–260 (1992). [CrossRef]  

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

38. F. Nicolas, S. Coetmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” J. Opt. Soc. Am. A 268, 27–33 (2006).

39. 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). [CrossRef]   [PubMed]  

40. M. Malek, D. Allano, S. Coëtmellec, and D. Lebrun, “Digital in-line holography: influence of the shadow density on particle field extraction,” Opt. Express 12, 2270–2279 (2004). [CrossRef]   [PubMed]  

41. L. Denis, C. Fournier, T. Fournel, C. Ducottet, and D. Jeulin, “Direct extraction of the mean particle size from a digital hologram,” Appl. Opt. 45, 944–952 (2006). [CrossRef]   [PubMed]  

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

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

44. Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Reconstruction of in-line digital holograms from two intensity measurements,” Opt. Lett. 29, 1787–1789 (2004). [CrossRef]   [PubMed]  

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

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

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

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

49. M. Liebling, “On Fresnelets, interferences fringes, and digital holography Ph.D. thesis (Ecole Polytechnique Fédérale de Lausanne, 2004).

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

51. T. Baumbach, E. Kolenovic, V. Kebbel, and W. Jüptner, “Improvement of accuracy in digital holography by use of multiple holograms,” Appl. Opt. 45, 6077–6085 (2006). [CrossRef]   [PubMed]  

52. X. Cai and H. Wand, “The influence of hologram aperture on speckle noise in the reconstructed image of digital holography and its reduction,” Opt. Commun. 281, 232–237 (2008). [CrossRef]  

53. R. Binet, J. Colineau, and J. C. Lehureau, “‘Short-range synthetic aperture imaging at 633nm by digital holography,” Appl. Opt. 41, 4775–4782 (2002). [CrossRef]   [PubMed]  

54. J. H. Massig, “Digital off-axis holography with a synthetic aperture,” Opt. Lett. 27, 2179–2181 (2002). [CrossRef]  

55. G. A. Mills and I. Yamaguchi, “Effects of quantization in phase-shifting digital holography,” Appl. Opt. 44, 1216–1225 (2005). [CrossRef]   [PubMed]  

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

57. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

58. P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Some opportunities for vibration analysis with time-averaging in digital Fresnel holography,” Appl. Opt. 44, 337–343 (2005). [CrossRef]   [PubMed]  

59. 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. (Bellingham) 42, 2768–2772 (2003). [CrossRef]  

60. M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87–94 (2001). [CrossRef]  

61. F. Le Clerc, L. Collot, and M. Gross, “Numerical heterodyne holography with two-dimensional photo detector arrays,” Opt. Lett. 25, 716–718 (2000). [CrossRef]  

62. Th. Kreis and W. Juptner, “Suppression of the DC term in digital holography,” Opt. Eng. (Bellingham) 36, 2357–2360 (1997). [CrossRef]  

63. E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000). [CrossRef]  

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

References

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  5. P. Ferraro, D. Alferi, S. De Nicola, L. De Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett. 31, 1405-1407 (2006).
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  6. G. Pedrini and H. J. Tiziani, “Short-coherence digital microscopy by use of a lensless holographic imaging system,” Appl. Opt. 41, 4489-4496 (2002).
    [CrossRef] [PubMed]
  7. C. Mann, L. Yu, L. Chun-Min, and M. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express 13, 8693-8698 (2005).
    [CrossRef] [PubMed]
  8. K. Chalut, W. Brown, and A. Wax, “Quantitative phase microscopy with asynchronous digital holography,” Opt. Express 15, 3047-3052 (2007).
    [CrossRef] [PubMed]
  9. J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15, 7231-7242 (2007).
    [CrossRef] [PubMed]
  10. B. Javidi and E. Tajahuerce, “Three-dimensional object recognition by use of digital holography,” Opt. Lett. 25, 610-612 (2000).
    [CrossRef]
  11. B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. 25, 28-30 (2000).
    [CrossRef]
  12. Y. Frauel and B. Javidi, “Neural network for three-dimensional object recognition based on digital holography,” Opt. Lett. 26, 1478-1480 (2001).
    [CrossRef]
  13. B. Javidi and D. Kim, “Three-dimensional-object recognition by use of single-exposure on-axis digital holography,” Opt. Lett. 30, 236-238 (2005).
    [CrossRef] [PubMed]
  14. T. Nomura and B. Javidi, “Object recognition by use of polarimetric phase-shifting digital holography,” Opt. Lett. 32, 2146-2148 (2007).
    [CrossRef] [PubMed]
  15. T. Nomura, B. Javidi, S. Murata, E. Nitanai, and T. Numata, “Polarization imaging of a 3D object by use of on-axis phase-shifting digital holography,” Opt. Lett. 32, 481-483 (2007).
    [CrossRef] [PubMed]
  16. I. Yamaguchi, J. Kato, and S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85-89 (2001).
    [CrossRef]
  17. I. Yamaguchi, T. Ida, M. Yokota, and K. Yamashita, “Surface shape measurement by phase shifting digital holography with a wavelength shift,” Appl. Opt. 45, 7610-7616 (2006).
    [CrossRef] [PubMed]
  18. M. Mosarraf, G. Sheoran, D. Singh, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
    [CrossRef]
  19. S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103-126 (2001).
    [CrossRef]
  20. Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, “Deformation measurement by phase shifting digital holography,” Exp. Mech. 45, 65-70 (2005).
    [CrossRef]
  21. P. Picart, E. Moisson, and D. Mounier, “Twin sensitivity measurement by spatial multiplexing of digitally recorded holograms,” Appl. Opt. 42, 1947-1957 (2003).
    [CrossRef] [PubMed]
  22. P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. (Bellingham) 43, 1169-1176 (2004).
    [CrossRef]
  23. T. Baumbach, W. Osten, C. von Kopylow, and W. Juptner, “Remote metrology by comparative digital holography,” Appl. Opt. 45, 925-934 (2006).
    [CrossRef] [PubMed]
  24. G. Pedrini and H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 18, 251-260 (1995).
    [CrossRef]
  25. Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal Fourier analyses of a digital hologram sequence,” Appl. Opt. 46, 5719-5727 (2007).
    [CrossRef] [PubMed]
  26. G. Pedrini, S. Schedin, and H. J. Tiziani, “Pulsed digital holography combined with laser vibrometry for 3D measurements of vibrating objects,” Opt. Lasers Eng. 38, 117-129 (2002).
  27. P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Time averaged digital holography,” Opt. Lett. 28, 1900-1902 (2003).
    [CrossRef] [PubMed]
  28. N. Demoli and I. Demoli, “Dynamic modal characterization of musical instruments using digital holography,” Opt. Express 13, 4812-4817 (2005).
    [CrossRef] [PubMed]
  29. A. Asundi and V. R. Singh, “Time-averaged in-line digital holographic interferometry for vibration analysis,” Appl. Opt. 45, 2391-2395 (2006).
    [CrossRef] [PubMed]
  30. J. Leval, P. Picart, J.-P. Boileau, and J.-C. Pascal, “Full field vibrometry with digital Fresnel holography,” Appl. Opt. 44, 5763-5772 (2005).
    [CrossRef] [PubMed]
  31. P. Picart, J. Leval, F. Piquet, J.-P. Boileau, Th. Guimezanes, and J.-P. Dalmont, “Tracking high amplitude auto-oscillations with digital Fresnel holograms,” Opt. Express 15, 8263-8274 (2007).
    [CrossRef] [PubMed]
  32. T. Saucedo, F. M. Santoyo, M. De la Torre Ibarra, G. Pedrini, and W. Osten, “Simultaneous two-dimensional endoscopic pulsed digital holography for evaluation of dynamic displacements,” Appl. Opt. 45, 4534-4539 (2006).
    [CrossRef]
  33. P. Picart, J. Leval, M. Grill, J.-P. Boileau, J. C. Pascal, J.-M. Breteau, B. Gautier, and S. Gillet, “2D full field vibration analysis with multiplexed digital holograms,” Opt. Express 13, 8882-8892 (2005).
    [CrossRef] [PubMed]
  34. A. Stadelmaier and J. H. Massig, “Compensation of lens aberration in digital holography,” Opt. Lett. 25, 1630-1632 (2000).
    [CrossRef]
  35. S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattin, “Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,” Opt. Lett. 26, 974-976 (2001).
    [CrossRef]
  36. L. Onural and M. T. Ozgen, “Extraction of three-dimensional object-location information directly from in-line holograms using Wigner analysis,” J. Opt. Soc. Am. A 9, 252-260 (1992).
    [CrossRef]
  37. S. Coetmellec, D. Lebrun, and C. Oskul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A 19, 1537-1546 (2002).
    [CrossRef]
  38. F. Nicolas, S. Coetmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” J. Opt. Soc. Am. A 268, 27-33 (2006).
  39. 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).
    [CrossRef] [PubMed]
  40. M. Malek, D. Allano, S. Coëtmellec, and D. Lebrun, “Digital in-line holography: influence of the shadow density on particle field extraction,” Opt. Express 12, 2270-2279 (2004).
    [CrossRef] [PubMed]
  41. L. Denis, C. Fournier, T. Fournel, C. Ducottet, and D. Jeulin, “Direct extraction of the mean particle size from a digital hologram,” Appl. Opt. 45, 944-952 (2006).
    [CrossRef] [PubMed]
  42. L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846-848 (1993).
    [CrossRef] [PubMed]
  43. Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Image reconstruction for in-line holography with the Yang-Gu algorithm,” Appl. Opt. 42, 6452-6457 (2003).
    [CrossRef] [PubMed]
  44. Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Reconstruction of in-line digital holograms from two intensity measurements,” Opt. Lett. 29, 1787-1789 (2004).
    [CrossRef] [PubMed]
  45. Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224-233 (1997).
    [CrossRef]
  46. Th. Kreis, “Frequency analysis of digital holography,” Opt. Eng. (Bellingham) 41, 771-778 (2002).
    [CrossRef]
  47. Th. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. (Bellingham) 41, 1829-1839 (2002).
    [CrossRef]
  48. C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. 38, 4812-4820 (1999).
    [CrossRef]
  49. M. Liebling, “On Fresnelets, interferences fringes, and digital holographyPh.D. thesis (Ecole Polytechnique Fédérale de Lausanne, 2004).
  50. I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase shifting digital holography and application to microscopy,” Appl. Opt. 40, 6177-6186 (2001).
    [CrossRef]
  51. T. Baumbach, E. Kolenovic, V. Kebbel, and W. Jüptner, “Improvement of accuracy in digital holography by use of multiple holograms,” Appl. Opt. 45, 6077-6085 (2006).
    [CrossRef] [PubMed]
  52. X. Cai and H. Wand, “The influence of hologram aperture on speckle noise in the reconstructed image of digital holography and its reduction,” Opt. Commun. 281, 232-237 (2008).
    [CrossRef]
  53. R. Binet, J. Colineau, and J. C. Lehureau, “'Short-range synthetic aperture imaging at 633nm by digital holography,” Appl. Opt. 41, 4775-4782 (2002).
    [CrossRef] [PubMed]
  54. J. H. Massig, “Digital off-axis holography with a synthetic aperture,” Opt. Lett. 27, 2179-2181 (2002).
    [CrossRef]
  55. G. A. Mills and I. Yamaguchi, “Effects of quantization in phase-shifting digital holography,” Appl. Opt. 44, 1216-1225 (2005).
    [CrossRef] [PubMed]
  56. L. Xu, X. Peng, Z. Guo, J. Mia, and A. Asundi, “Imaging analysis of digital holography,” Opt. Express 13, 2444-2452 (2005).
    [CrossRef] [PubMed]
  57. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  58. P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Some opportunities for vibration analysis with time-averaging in digital Fresnel holography,” Appl. Opt. 44, 337-343 (2005).
    [CrossRef] [PubMed]
  59. 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. (Bellingham) 42, 2768-2772 (2003).
    [CrossRef]
  60. M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87-94 (2001).
    [CrossRef]
  61. F. Le Clerc, L. Collot, and M. Gross, “Numerical heterodyne holography with two-dimensional photo detector arrays,” Opt. Lett. 25, 716-718 (2000).
    [CrossRef]
  62. Th. Kreis and W. Juptner, “Suppression of the DC term in digital holography,” Opt. Eng. (Bellingham) 36, 2357-2360 (1997).
    [CrossRef]
  63. E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070-4075 (2000).
    [CrossRef]
  64. I. Yamaguchi and T. Zhang, “Phase shifting digital holography,” Opt. Lett. 22, 1268-1270 (1997).
    [CrossRef] [PubMed]

2008 (1)

X. Cai and H. Wand, “The influence of hologram aperture on speckle noise in the reconstructed image of digital holography and its reduction,” Opt. Commun. 281, 232-237 (2008).
[CrossRef]

2007 (8)

M. Mosarraf, G. Sheoran, D. Singh, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
[CrossRef]

T. Nomura, B. Javidi, S. Murata, E. Nitanai, and T. Numata, “Polarization imaging of a 3D object by use of on-axis phase-shifting digital holography,” Opt. Lett. 32, 481-483 (2007).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

K. Chalut, W. Brown, and A. Wax, “Quantitative phase microscopy with asynchronous digital holography,” Opt. Express 15, 3047-3052 (2007).
[CrossRef] [PubMed]

J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15, 7231-7242 (2007).
[CrossRef] [PubMed]

P. Picart, J. Leval, F. Piquet, J.-P. Boileau, Th. Guimezanes, and J.-P. Dalmont, “Tracking high amplitude auto-oscillations with digital Fresnel holograms,” Opt. Express 15, 8263-8274 (2007).
[CrossRef] [PubMed]

T. Nomura and B. Javidi, “Object recognition by use of polarimetric phase-shifting digital holography,” Opt. Lett. 32, 2146-2148 (2007).
[CrossRef] [PubMed]

Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal Fourier analyses of a digital hologram sequence,” Appl. Opt. 46, 5719-5727 (2007).
[CrossRef] [PubMed]

2006 (8)

T. Baumbach, W. Osten, C. von Kopylow, and W. Juptner, “Remote metrology by comparative digital holography,” Appl. Opt. 45, 925-934 (2006).
[CrossRef] [PubMed]

L. Denis, C. Fournier, T. Fournel, C. Ducottet, and D. Jeulin, “Direct extraction of the mean particle size from a digital hologram,” Appl. Opt. 45, 944-952 (2006).
[CrossRef] [PubMed]

A. Asundi and V. R. Singh, “Time-averaged in-line digital holographic interferometry for vibration analysis,” Appl. Opt. 45, 2391-2395 (2006).
[CrossRef] [PubMed]

P. Ferraro, D. Alferi, S. De Nicola, L. De Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett. 31, 1405-1407 (2006).
[CrossRef] [PubMed]

T. Saucedo, F. M. Santoyo, M. De la Torre Ibarra, G. Pedrini, and W. Osten, “Simultaneous two-dimensional endoscopic pulsed digital holography for evaluation of dynamic displacements,” Appl. Opt. 45, 4534-4539 (2006).
[CrossRef]

T. Baumbach, E. Kolenovic, V. Kebbel, and W. Jüptner, “Improvement of accuracy in digital holography by use of multiple holograms,” Appl. Opt. 45, 6077-6085 (2006).
[CrossRef] [PubMed]

I. Yamaguchi, T. Ida, M. Yokota, and K. Yamashita, “Surface shape measurement by phase shifting digital holography with a wavelength shift,” Appl. Opt. 45, 7610-7616 (2006).
[CrossRef] [PubMed]

F. Nicolas, S. Coetmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” J. Opt. Soc. Am. A 268, 27-33 (2006).

2005 (9)

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, “Deformation measurement by phase shifting digital holography,” Exp. Mech. 45, 65-70 (2005).
[CrossRef]

P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Some opportunities for vibration analysis with time-averaging in digital Fresnel holography,” Appl. Opt. 44, 337-343 (2005).
[CrossRef] [PubMed]

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

G. A. Mills and I. Yamaguchi, “Effects of quantization in phase-shifting digital holography,” Appl. Opt. 44, 1216-1225 (2005).
[CrossRef] [PubMed]

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

N. Demoli and I. Demoli, “Dynamic modal characterization of musical instruments using digital holography,” Opt. Express 13, 4812-4817 (2005).
[CrossRef] [PubMed]

J. Leval, P. Picart, J.-P. Boileau, and J.-C. Pascal, “Full field vibrometry with digital Fresnel holography,” Appl. Opt. 44, 5763-5772 (2005).
[CrossRef] [PubMed]

C. Mann, L. Yu, L. Chun-Min, and M. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express 13, 8693-8698 (2005).
[CrossRef] [PubMed]

P. Picart, J. Leval, M. Grill, J.-P. Boileau, J. C. Pascal, J.-M. Breteau, B. Gautier, and S. Gillet, “2D full field vibration analysis with multiplexed digital holograms,” Opt. Express 13, 8882-8892 (2005).
[CrossRef] [PubMed]

2004 (4)

M. Malek, D. Allano, S. Coëtmellec, and D. Lebrun, “Digital in-line holography: influence of the shadow density on particle field extraction,” Opt. Express 12, 2270-2279 (2004).
[CrossRef] [PubMed]

Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Reconstruction of in-line digital holograms from two intensity measurements,” Opt. Lett. 29, 1787-1789 (2004).
[CrossRef] [PubMed]

P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. (Bellingham) 43, 1169-1176 (2004).
[CrossRef]

M. Liebling, “On Fresnelets, interferences fringes, and digital holographyPh.D. thesis (Ecole Polytechnique Fédérale de Lausanne, 2004).

2003 (4)

2002 (7)

2001 (6)

M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87-94 (2001).
[CrossRef]

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103-126 (2001).
[CrossRef]

I. Yamaguchi, J. Kato, and S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85-89 (2001).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattin, “Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,” Opt. Lett. 26, 974-976 (2001).
[CrossRef]

Y. Frauel and B. Javidi, “Neural network for three-dimensional object recognition based on digital holography,” Opt. Lett. 26, 1478-1480 (2001).
[CrossRef]

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

2000 (5)

1999 (2)

1998 (1)

1997 (3)

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

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

Th. Kreis and W. Juptner, “Suppression of the DC term in digital holography,” Opt. Eng. (Bellingham) 36, 2357-2360 (1997).
[CrossRef]

1996 (1)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

1995 (1)

G. Pedrini and H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 18, 251-260 (1995).
[CrossRef]

1994 (1)

1993 (1)

1992 (1)

1972 (1)

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).

Adams, M.

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

Alferi, D.

Allano, D.

Asundi, A.

Baumbach, T.

Berthelot, J.-M.

P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. (Bellingham) 43, 1169-1176 (2004).
[CrossRef]

Bevilacqua, F.

Binet, R.

Boileau, J.-P.

Breteau, J.-M.

Brown, W.

Brunel, M.

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).
[CrossRef] [PubMed]

F. Nicolas, S. Coetmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” J. Opt. Soc. Am. A 268, 27-33 (2006).

Cai, X.

X. Cai and H. Wand, “The influence of hologram aperture on speckle noise in the reconstructed image of digital holography and its reduction,” Opt. Commun. 281, 232-237 (2008).
[CrossRef]

Chalut, K.

Charrière, F.

Chun-Min, L.

Coetmellec, S.

F. Nicolas, S. Coetmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” J. Opt. Soc. Am. A 268, 27-33 (2006).

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

Coëtmellec, S.

Colineau, J.

Collot, L.

Colomb, T.

Cuche, E.

Dalmont, J.-P.

De la Torre Ibarra, M.

De Nicola, S.

De Petrocellis, L.

Demoli, I.

Demoli, N.

Denis, L.

Depeursinge, C.

Diouf, B.

P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. (Bellingham) 43, 1169-1176 (2004).
[CrossRef]

Ducottet, C.

Emery, Y.

Ferraro, P.

Finizio, A.

Fjigaki, M.

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, “Deformation measurement by phase shifting digital holography,” Exp. Mech. 45, 65-70 (2005).
[CrossRef]

Fournel, T.

Fournier, C.

Frauel, Y.

Fu, Y.

Gautier, B.

Gillet, S.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gougeon, S.

Grill, M.

Gross, M.

Guimezanes, Th.

Guo, C. S.

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. (Bellingham) 42, 2768-2772 (2003).
[CrossRef]

Guo, Z.

Ida, T.

Jacquot, M.

M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87-94 (2001).
[CrossRef]

Javidi, B.

Jeulin, D.

Juptner, W.

T. Baumbach, W. Osten, C. von Kopylow, and W. Juptner, “Remote metrology by comparative digital holography,” Appl. Opt. 45, 925-934 (2006).
[CrossRef] [PubMed]

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103-126 (2001).
[CrossRef]

Th. Kreis and W. Juptner, “Suppression of the DC term in digital holography,” Opt. Eng. (Bellingham) 36, 2357-2360 (1997).
[CrossRef]

Jüptner, W.

Kato, J.

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

I. Yamaguchi, J. Kato, and S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85-89 (2001).
[CrossRef]

Kebbel, V.

Kim, D.

Kim, M.

Kolenovic, E.

Kreis, Th.

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

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

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

Th. Kreis and W. Juptner, “Suppression of the DC term in digital holography,” Opt. Eng. (Bellingham) 36, 2357-2360 (1997).
[CrossRef]

Kronrod, M. A.

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).

Kühn, J.

Le Clerc, F.

Lebrun, D.

Lehureau, J. C.

Leval, J.

Liebling, M.

M. Liebling, “On Fresnelets, interferences fringes, and digital holographyPh.D. thesis (Ecole Polytechnique Fédérale de Lausanne, 2004).

Lolive, E.

P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. (Bellingham) 43, 1169-1176 (2004).
[CrossRef]

Malek, M.

Mann, C.

Marquet, P.

Massig, J. H.

Merzlyakov, N. S.

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).

Mia, J.

Mills, G. A.

Mizuno, J.

Moisson, E.

Montfort, F.

Morimoto, Y.

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, “Deformation measurement by phase shifting digital holography,” Exp. Mech. 45, 65-70 (2005).
[CrossRef]

Mosarraf, M.

M. Mosarraf, G. Sheoran, D. Singh, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
[CrossRef]

Mounier, D.

Murata, S.

Nicolas, F.

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).
[CrossRef] [PubMed]

F. Nicolas, S. Coetmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” J. Opt. Soc. Am. A 268, 27-33 (2006).

Nitanai, E.

Nomura, T.

Numata, T.

Ohta, S.

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

I. Yamaguchi, J. Kato, and S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85-89 (2001).
[CrossRef]

Onural, L.

Oskul, C.

Osten, W.

Ozgen, M. T.

Pascal, J. C.

Pascal, J.-C.

Pedrini, G.

Peng, X.

Picart, P.

Pierattin, G.

Pierattini, G.

Piquet, F.

Rong, Z. Y.

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. (Bellingham) 42, 2768-2772 (2003).
[CrossRef]

Sandoz, P.

M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87-94 (2001).
[CrossRef]

Santoyo, F. M.

Saucedo, T.

Schedin, S.

G. Pedrini, S. Schedin, and H. J. Tiziani, “Pulsed digital holography combined with laser vibrometry for 3D measurements of vibrating objects,” Opt. Lasers Eng. 38, 117-129 (2002).

Schnars, U.

Seebacher, S.

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103-126 (2001).
[CrossRef]

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

Shakher, C.

M. Mosarraf, G. Sheoran, D. Singh, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
[CrossRef]

Sheoran, G.

M. Mosarraf, G. Sheoran, D. Singh, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
[CrossRef]

Singh, D.

M. Mosarraf, G. Sheoran, D. Singh, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
[CrossRef]

Singh, V. R.

Stadelmaier, A.

Tajahuerce, E.

Takahashi, I.

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, “Deformation measurement by phase shifting digital holography,” Exp. Mech. 45, 65-70 (2005).
[CrossRef]

Tiziani, H. J.

Tribillon, G.

M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87-94 (2001).
[CrossRef]

von Kopylow, C.

Wagner, C.

Wand, H.

X. Cai and H. Wand, “The influence of hologram aperture on speckle noise in the reconstructed image of digital holography and its reduction,” Opt. Commun. 281, 232-237 (2008).
[CrossRef]

Wang, H. T.

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. (Bellingham) 42, 2768-2772 (2003).
[CrossRef]

Wax, A.

Xu, L.

Yamaguchi, I.

Yamashita, K.

Yaroslavskii, L. P.

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).

Yokota, M.

Yoneyama, S.

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, “Deformation measurement by phase shifting digital holography,” Exp. Mech. 45, 65-70 (2005).
[CrossRef]

Yu, L.

Zhang, L.

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. (Bellingham) 42, 2768-2772 (2003).
[CrossRef]

Zhang, T.

Zhang, Y.

Appl. Opt. (18)

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

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

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070-4075 (2000).
[CrossRef]

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

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

R. Binet, J. Colineau, and J. C. Lehureau, “'Short-range synthetic aperture imaging at 633nm by digital holography,” Appl. Opt. 41, 4775-4782 (2002).
[CrossRef] [PubMed]

P. Picart, E. Moisson, and D. Mounier, “Twin sensitivity measurement by spatial multiplexing of digitally recorded holograms,” Appl. Opt. 42, 1947-1957 (2003).
[CrossRef] [PubMed]

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

P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Some opportunities for vibration analysis with time-averaging in digital Fresnel holography,” Appl. Opt. 44, 337-343 (2005).
[CrossRef] [PubMed]

G. A. Mills and I. Yamaguchi, “Effects of quantization in phase-shifting digital holography,” Appl. Opt. 44, 1216-1225 (2005).
[CrossRef] [PubMed]

T. Baumbach, W. Osten, C. von Kopylow, and W. Juptner, “Remote metrology by comparative digital holography,” Appl. Opt. 45, 925-934 (2006).
[CrossRef] [PubMed]

L. Denis, C. Fournier, T. Fournel, C. Ducottet, and D. Jeulin, “Direct extraction of the mean particle size from a digital hologram,” Appl. Opt. 45, 944-952 (2006).
[CrossRef] [PubMed]

A. Asundi and V. R. Singh, “Time-averaged in-line digital holographic interferometry for vibration analysis,” Appl. Opt. 45, 2391-2395 (2006).
[CrossRef] [PubMed]

T. Saucedo, F. M. Santoyo, M. De la Torre Ibarra, G. Pedrini, and W. Osten, “Simultaneous two-dimensional endoscopic pulsed digital holography for evaluation of dynamic displacements,” Appl. Opt. 45, 4534-4539 (2006).
[CrossRef]

T. Baumbach, E. Kolenovic, V. Kebbel, and W. Jüptner, “Improvement of accuracy in digital holography by use of multiple holograms,” Appl. Opt. 45, 6077-6085 (2006).
[CrossRef] [PubMed]

I. Yamaguchi, T. Ida, M. Yokota, and K. Yamashita, “Surface shape measurement by phase shifting digital holography with a wavelength shift,” Appl. Opt. 45, 7610-7616 (2006).
[CrossRef] [PubMed]

J. Leval, P. Picart, J.-P. Boileau, and J.-C. Pascal, “Full field vibrometry with digital Fresnel holography,” Appl. Opt. 44, 5763-5772 (2005).
[CrossRef] [PubMed]

Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal Fourier analyses of a digital hologram sequence,” Appl. Opt. 46, 5719-5727 (2007).
[CrossRef] [PubMed]

Exp. Mech. (1)

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, “Deformation measurement by phase shifting digital holography,” Exp. Mech. 45, 65-70 (2005).
[CrossRef]

J. Opt. Soc. Am. A (3)

Measurement (1)

G. Pedrini and H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 18, 251-260 (1995).
[CrossRef]

Opt. Commun. (2)

M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87-94 (2001).
[CrossRef]

X. Cai and H. Wand, “The influence of hologram aperture on speckle noise in the reconstructed image of digital holography and its reduction,” Opt. Commun. 281, 232-237 (2008).
[CrossRef]

Opt. Eng. (Bellingham) (5)

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

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

Th. Kreis and W. Juptner, “Suppression of the DC term in digital holography,” Opt. Eng. (Bellingham) 36, 2357-2360 (1997).
[CrossRef]

P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. (Bellingham) 43, 1169-1176 (2004).
[CrossRef]

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. (Bellingham) 42, 2768-2772 (2003).
[CrossRef]

Opt. Express (9)

C. Mann, L. Yu, L. Chun-Min, and M. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express 13, 8693-8698 (2005).
[CrossRef] [PubMed]

P. Picart, J. Leval, M. Grill, J.-P. Boileau, J. C. Pascal, J.-M. Breteau, B. Gautier, and S. Gillet, “2D full field vibration analysis with multiplexed digital holograms,” Opt. Express 13, 8882-8892 (2005).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

K. Chalut, W. Brown, and A. Wax, “Quantitative phase microscopy with asynchronous digital holography,” Opt. Express 15, 3047-3052 (2007).
[CrossRef] [PubMed]

J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15, 7231-7242 (2007).
[CrossRef] [PubMed]

P. Picart, J. Leval, F. Piquet, J.-P. Boileau, Th. Guimezanes, and J.-P. Dalmont, “Tracking high amplitude auto-oscillations with digital Fresnel holograms,” Opt. Express 15, 8263-8274 (2007).
[CrossRef] [PubMed]

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

N. Demoli and I. Demoli, “Dynamic modal characterization of musical instruments using digital holography,” Opt. Express 13, 4812-4817 (2005).
[CrossRef] [PubMed]

M. Malek, D. Allano, S. Coëtmellec, and D. Lebrun, “Digital in-line holography: influence of the shadow density on particle field extraction,” Opt. Express 12, 2270-2279 (2004).
[CrossRef] [PubMed]

Opt. Lasers Eng. (3)

G. Pedrini, S. Schedin, and H. J. Tiziani, “Pulsed digital holography combined with laser vibrometry for 3D measurements of vibrating objects,” Opt. Lasers Eng. 38, 117-129 (2002).

M. Mosarraf, G. Sheoran, D. Singh, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
[CrossRef]

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103-126 (2001).
[CrossRef]

Opt. Lett. (17)

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

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

F. Le Clerc, L. Collot, and M. Gross, “Numerical heterodyne holography with two-dimensional photo detector arrays,” Opt. Lett. 25, 716-718 (2000).
[CrossRef]

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

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

T. Zhang and I. Yamaguchi, “Three-dimensional microscopy with phase shifting digital holography,” Opt. Lett. 23, 1221-1223 (1998).
[CrossRef]

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

A. Stadelmaier and J. H. Massig, “Compensation of lens aberration in digital holography,” Opt. Lett. 25, 1630-1632 (2000).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattin, “Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,” Opt. Lett. 26, 974-976 (2001).
[CrossRef]

Y. Frauel and B. Javidi, “Neural network for three-dimensional object recognition based on digital holography,” Opt. Lett. 26, 1478-1480 (2001).
[CrossRef]

J. H. Massig, “Digital off-axis holography with a synthetic aperture,” Opt. Lett. 27, 2179-2181 (2002).
[CrossRef]

Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Reconstruction of in-line digital holograms from two intensity measurements,” Opt. Lett. 29, 1787-1789 (2004).
[CrossRef] [PubMed]

P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Time averaged digital holography,” Opt. Lett. 28, 1900-1902 (2003).
[CrossRef] [PubMed]

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

T. Nomura, B. Javidi, S. Murata, E. Nitanai, and T. Numata, “Polarization imaging of a 3D object by use of on-axis phase-shifting digital holography,” Opt. Lett. 32, 481-483 (2007).
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P. Ferraro, D. Alferi, S. De Nicola, L. De Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett. 31, 1405-1407 (2006).
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T. Nomura and B. Javidi, “Object recognition by use of polarimetric phase-shifting digital holography,” Opt. Lett. 32, 2146-2148 (2007).
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Opt. Rev. (1)

I. Yamaguchi, J. Kato, and S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85-89 (2001).
[CrossRef]

Proc. SPIE (1)

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[CrossRef]

Sov. Phys. Tech. Phys. (1)

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Other (2)

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Figures (21)

Fig. 1
Fig. 1

Schematic diagram of image formation in digital holography.

Fig. 2
Fig. 2

Basics of recording a digital hologram.

Fig. 3
Fig. 3

Intrinsic resolution function.

Fig. 4
Fig. 4

Illustration of zero padding.

Fig. 5
Fig. 5

Profiles of the resolution function with nonextended and extended pixels.

Fig. 6
Fig. 6

Influence of the active surface of pixels.

Fig. 7
Fig. 7

Plot of criterion quantifying the influence of the active surface of pixels [circles, Eq. (34); curves, Eq. (30)].

Fig. 8
Fig. 8

Evolution of the resolution function versus the reconstruction distances.

Fig. 9
Fig. 9

Profiles of the resolution function for different reconstruction distances.

Fig. 10
Fig. 10

Comparison between the analytical formulation and the numerical simulation versus the reconstruction distance.

Fig. 11
Fig. 11

Profiles along x for the analytical formulation and numerical simulation at the best focus.

Fig. 12
Fig. 12

Profiles along z for the analytical formulation and full numerical for x = y = 0 .

Fig. 13
Fig. 13

Different schemes for generating a reference wave.

Fig. 14
Fig. 14

Zone of the wavefront “seen” by the sensor.

Fig. 15
Fig. 15

Resolution for ( δ x , δ y ) = ( 0 , 0 ) .

Fig. 16
Fig. 16

Resolution for ( δ x , δ y ) = ( 7.64 mm , 7.64 mm ) .

Fig. 17
Fig. 17

Resolution for ( δ x , δ y ) = ( 15.29 mm , 15.29 mm ) .

Fig. 18
Fig. 18

Schematic structure of the reconstructed field.

Fig. 19
Fig. 19

Focus on the + 1 order.

Fig. 20
Fig. 20

Focus on the 1 order.

Fig. 21
Fig. 21

Defocused image.

Tables (2)

Tables Icon

Table 1 Numerical Values for the Resolution Function Computed from Theory

Tables Icon

Table 2 Coefficients of the Polynomial Expansion of Aberration in the Recording Plane

Equations (71)

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A R ( x , y ) = κ A ( x , y ) * R x y ( x , y ) ,
O ( x , y , d 0 ) = j exp ( 2 j π d 0 λ ) λ d 0 exp [ j π λ d 0 ( x 2 + y 2 ) ] × + + A ( x , y ) exp [ j π λ d 0 ( ( x x 0 ) 2 + ( y y 0 ) 2 ) ] exp [ 2 j π λ d 0 ( ( x x 0 ) x + ( y y 0 ) y ) ] d x d y .
r ( x , y ) = a R exp [ 2 j π ( u R x + v R y ) + j Δ Ψ a b ( x , y ) ] .
H ( x , y , d 0 ) = O ( x , y , d 0 ) 2 + r ( x , y ) 2 + r * ( x , y ) O ( x , y , d 0 ) + r ( x , y ) O * ( x , y , d 0 ) .
r * ( x , y ) O ( x , y , d 0 ) = j exp ( 2 j π d 0 λ ) λ d 0 exp [ j π λ d 0 ( x 0 2 + y 0 2 ) ] exp [ j π λ d 0 ( x 2 + y 2 ) ] a R exp [ 2 j π ( u R x 0 λ d 0 ) x 2 j π ( v R y 0 λ d 0 ) y j Δ Ψ a b ( x , y ) ] × + + A ( x , y ) exp [ j π λ d 0 ( x 2 + y 2 ) ] exp [ 2 j π λ d 0 ( x ( x + x 0 ) + y ( y + y 0 ) ) ] d x d y .
H P I X ( k p x , l p y , d 0 ) = [ H ( x , y , d 0 ) * Π Δ x , Δ y ( x , y ) ] ( k p x , l p y ) ,
Π Δ x , Δ y ( x , y ) = { 1 Δ x × 1 Δ y if x Δ x 2 and y Δ y 2 0 if not .
A R + 1 ( X , Y , d R ) = j exp ( 2 j π d R λ ) λ d R exp [ j π λ d R ( X 2 + Y 2 ) ] × k = 0 k = K 1 l = 0 l = L 1 H P I X + 1 ( k p x , l p y ) exp [ j π λ d R ( k 2 p x 2 + l 2 p y 2 ) ] × exp [ 2 j π λ d R ( k X p x + l Y p y ) ] .
H P I X + 1 ( k p x , l p y ) = { [ ( r * ( x , y ) O ( x , y , d 0 ) ] * Π Δ x , Δ y ( x , y ) } ( k p x , l p y ) ) .
A R 1 ( X , Y , d R ) = A R + 1 ( X , Y , d R ) * .
A R + 1 ( X , Y , d 0 ) = λ 2 d 0 2 exp [ j π λ d 0 ( u R 2 + v R 2 ) ] r * ( X , Y ) × A ( X , Y ) * Π Δ x , Δ y ( X , Y ) * W ̃ N M ( X , Y , d 0 ) * δ ( X λ u R d 0 , Y λ v R d 0 ) ,
W ̃ N M ( x , y , d 0 ) = exp [ j π ( N 1 ) x p x λ d 0 j π ( M 1 ) y p y λ d 0 ] × sin ( π N x p x λ d 0 ) sin ( π x p x λ d 0 ) sin ( π M y p y λ d 0 ) sin ( π y p y λ d 0 ) .
A R + 1 ( X , Y , d 0 ) = a R λ 2 d 0 2 exp [ j π λ d 0 ( X 2 + Y 2 ) ] × k = 0 k = K 1 l = 0 l = L 1 exp [ j Δ Ψ a b ( k p x , l p y ) ] exp [ 2 j π ( k p x u R + l p y v R ) ] × F ( k p x λ d 0 , l p y λ d 0 ) exp [ 2 j π λ d 0 ( X k p x + Y l p y ) ] ,
F ( x λ d 0 , y λ d 0 ) = + + F ̃ ( x , y ) exp [ 2 j π λ d 0 ( x x + y y ) ] d x d y = F T [ F ̃ ( x , y ) ] ( x λ d 0 , y λ d 0 ) ,
A R + 1 ( X , Y , d 0 ) = λ 2 d 0 2 exp [ j π λ d 0 ( u R 2 + v R 2 ) ] r * ( X , Y ) × A ( X , Y ) * W ̃ a b ( X , Y ) * W ̃ N M ( X , Y , d 0 ) * δ ( X λ u R d 0 , Y λ v R d 0 ) .
W ̃ a b ( x , y ) = exp [ j Δ Ψ a b ( x , y ) ] exp [ 2 j π λ d 0 ( x x + y y ) ] d x d y .
A R + 1 ( X , Y , d R ) = exp [ 2 j π ( d R + d 0 ) λ ] λ 2 d 0 d R exp [ j π λ d R ( X 2 + Y 2 ) ] × k = 0 k = K 1 l = 0 l = L 1 r * ( k p x , l p y ) F ( k p x λ d 0 , l p y λ d 0 ) × exp [ j π λ ( 1 d R + 1 d 0 ) ( k 2 p x 2 + l 2 p y 2 ) ] exp [ 2 j π λ d R ( X k p x + Y l p y ) ] .
F T 1 [ F T [ F ̃ ( x , y ) ] ( x λ d 0 , y λ d 0 ) ] ( X λ d R , Y λ d R ) = λ 2 d 0 2 [ F ̃ ( λ d 0 u , λ d 0 v ) ] ( X λ d R , Y λ d R ) = λ 2 d 0 2 F ̃ ( d 0 d R X , d 0 d R Y ) ,
F T 1 [ a R exp [ 2 j π ( u R x + v R y ) ] ] ( X λ d R , Y λ d R ) = λ 2 d R 2 a R δ ( X + u R λ d R , Y + v R λ d R ) .
A R + 1 ( X , Y , d R ) = λ 2 d 0 d R exp [ j π λ d 0 ( u R 2 + v R 2 ) ] exp [ j π λ d 0 ( x 0 2 + y 0 2 ) ] r ( X d 0 d R , Y d 0 d R ) exp [ 2 j π ( d 0 + d R ) λ ] exp [ + 2 j π ( X x 0 + Y y 0 ) ] A ( X d 0 d R , Y d 0 d R ) * W ̃ d R ( X , Y ) * W ̃ N M * ( X , Y , d R ) * δ ( X + λ u R d R x 0 d R d 0 , Y + λ v R d R y 0 d R d 0 ) .
W ̃ d R ( x , y ) = exp [ j π λ ( 1 d 0 + 1 d R ) ( x 2 + y 2 ) ] exp [ 2 j π λ d R ( x x + y y ) ] d x d y .
A R + 1 ( X , Y , d 0 ) = λ 2 d 0 2 exp [ j π λ d 0 ( u R 2 + v R 2 ) ] r * ( X , Y ) × A ( X , Y ) * W ̃ N M ( X , Y , d 0 ) * δ ( X λ u R d 0 , Y λ v R d 0 ) ,
A R + 1 ( X , Y , d R ) = κ × A ( X d 0 d R , Y d 0 d R ) * W ̃ a b ( X , Y ) * W ̃ d R ( X , Y ) * Π Δ x , Δ y ( X , Y ) * W ̃ N M * ( X , Y , d R ) * δ ( X + λ u R d R x 0 d R d 0 , Y + λ v R d R y 0 d R d 0 ) ,
R x y ( x , y ) = W ̃ a b ( x , y ) * W ̃ d R ( x , y ) * Π Δ x , Δ y ( x , y ) * W ̃ N M * ( x , y , d R ) * δ ( x + λ u R d R x 0 d R d 0 , y + λ v R d R y 0 d R d 0 ) .
R x y ( x , y ) = W ̃ N M ( x , y , d 0 ) * δ ( x λ u R d 0 , y λ v R d 0 ) .
ρ x = λ d 0 N p x and ρ y = λ d 0 M p y ,
Δ ξ = λ d 0 K p x and Δ η = λ d 0 L p y .
R x y ( x , y ) = Π Δ x , Δ y ( x , y ) * W ̃ N M ( x , y , d 0 ) .
C = + + R x y ( x , y ) 2 d x d y + + R x y Δ x = Δ y = 0 ( x , y ) 2 d x d y ,
C x = + R x ( x ) 2 d x + R x Δ x = 0 ( x ) 2 d x
W ̃ N ( x ) = R x Δ x = 0 ( x ) N e j ( N 1 ) x p x λ d 0 sinc ( π N p x λ d 0 x ) ,
1 N 2 + R x ( x ) 2 d x = λ d 0 N p x k = 0 k = ( 2 π N p x Δ x λ d 0 ) 2 k 2 × ( 1 ) k ( 2 k + 1 ) ( 2 k + 2 ) ! ;
C x = 2 k = 0 k = ( 2 π N p x Δ x λ d 0 ) 2 k ( 1 ) k ( 2 k + 1 ) ( 2 k + 2 ) ! ,
C x = 1 4 π 2 N 2 p x 2 Δ x 2 36 λ 2 d 0 2 + 16 π 4 N 4 p x 4 Δ x 4 1800 λ 4 d 0 4 64 π 6 N 6 p x 6 Δ x 6 141120 λ 6 d 0 6 + .
W ̃ d R ( x , y ) = N p x 2 + N p x 2 M p y 2 + M p y 2 exp [ j π λ ( 1 d 0 + 1 d R ) ( x 2 + y 2 ) ] exp [ 2 j π ( x λ d R x + y λ d R y ) ] d x d y .
u i = x λ ( 1 d 0 + 1 d R ) and v i = y λ ( 1 d 0 + 1 d R ) .
u i max = N p x 2 λ ( 1 d 0 + 1 d R ) and v i max = M p y 2 λ ( 1 d 0 + 1 d R ) .
Δ u = 2 u i max = N p x λ ( 1 d 0 + 1 d R ) .
ρ x d R = N p x ( 1 + d R d 0 ) .
Δ Ψ a b ( ξ , η ) = 2 π λ [ C S ( ξ 2 + η 2 ) 2 + C C ξ ( ξ 2 + η 2 ) ] ,
Δ Ψ a b ( x , y ) = 2 π λ [ a x 4 x 4 + a y 4 y 4 + a x 3 x 3 + a y 3 y 3 + a x 2 x 2 + a y 2 y 2 + a x 1 x + a y 1 y + a x 2 y 2 x 2 y 2 + a x 2 y 1 x 2 y + a x 1 y 2 x y 2 + a x 1 y 1 x y + c ] .
{ u i = 1 λ ( 4 a x 4 x 3 + 3 a x 3 x 2 + 2 a x 2 x + a x 1 + 2 a x 2 y 2 x y 2 + 2 a x 2 y 1 x y + a x 1 y 2 y 2 + a x 1 y 1 y ) v i = 1 λ ( 4 a y 4 y 3 + 3 a y 3 y 2 + 2 a y 2 y + a y 1 + 2 a x 2 y 2 x 2 y + a x 2 y 1 x 2 + 2 a x 1 y 2 x y + a x 1 y 1 x ) .
{ ρ x a b = λ d 0 Δ u 2 = d 0 ( a x 4 N 3 p x 3 2 + a x 2 N p x + a x 2 y 2 N p x M 2 p y 2 4 + a x 1 y 1 M p y 2 ) ρ y a b = λ d 0 Δ v 2 = d 0 ( a y 4 M 3 p y 3 2 + a y 2 M p y + a x 2 y 2 N 2 p x 2 M p y 4 + a x 1 y 1 N p x 2 ) .
{ δ X = d 0 a x 1 = 4 d 0 C S ( δ x 3 + δ x δ y 2 ) + d 0 C C ( 3 δ x 2 + δ y 2 ) δ Y = d 0 a y 1 = 4 d 0 C S ( δ x 2 δ y + δ y 3 ) + 2 d 0 C C δ x δ y .
{ ρ x a b = d 0 ( C S N 3 p x 3 2 + C S N p x ( 6 δ x 2 + 2 δ y 2 ) + 3 C C δ y N p x + C S N p x M 2 p y 2 2 + 4 C S δ x δ y M p y + C C δ y M p y ) ρ y a b = d 0 ( C S M 3 p y 3 2 + C S M p y ( 2 δ x 2 + 6 δ y 2 ) + C C δ x M p y + C S N 2 p x 2 M p y 2 + 4 C S δ x δ y N p x + C C δ y N p x ) .
{ ρ x a b = d 0 ( C S N 3 p x 3 2 + C S N p x ( 6 δ x 2 + 2 δ y 2 ) + C S N p x M 2 p y 2 2 + 4 C S δ x δ y M p y ) ρ y a b = d 0 ( C S M 3 p y 3 2 + C S M p y ( 2 δ x 2 + 6 δ y 2 ) + C S N 2 p x 2 M p y 2 + 4 C S δ x δ y N p x ) .
θ max 2 arcsin ( λ 4 max ( p x , p y ) ) .
O ( x , y , d 0 ) 2 = 1 λ 2 d 0 2 + + A ( x , y ) exp [ j π λ d 0 ( x 2 + y 2 ) ] exp [ 2 j π λ d 0 ( x x + y y ) ] d x d y × + + A * ( x , y ) exp [ j π λ d 0 ( x 2 + y 2 ) ] exp [ + 2 j π λ d 0 ( x x + y y ) ] d x d y .
O ̃ 0 ( u , v ) = F T [ O ] ( u , v ) * F T [ O * ] ( u , v ) .
O ̃ 0 ( u , v ) = λ 4 d 0 4 exp [ j π λ d 0 ( u 2 + v 2 ) ] + + A ( λ d 0 u 1 , λ d 0 v 1 ) A * ( λ d 0 u λ d 0 u 1 , λ d 0 v λ d 0 u 1 ) exp [ 2 j π λ d 0 ( u 1 2 + v 1 2 ) ] d u 1 d v 1 .
O ̃ 0 ( u , v ) = λ 4 d 0 4 + + A ( λ d 0 u 1 , λ d 0 v 1 ) A * ( λ d 0 u λ d 0 u 1 , λ d 0 v λ d 0 u 1 ) exp [ 2 j π λ d 0 ( u 1 2 + v 1 2 ) ] d u 1 d v 1 ,
O ̃ 0 ( u , v ) λ 4 d 0 4 + + A 0 ( λ d 0 u 1 , λ d 0 v 1 ) A 0 ( λ d 0 u λ d 0 u 1 , λ d 0 v λ d 0 u 1 ) d u 1 d v 1 ,
O ̃ 0 ( u , v ) λ 4 d 0 4 C A 0 A 0 ( λ d 0 u , λ d 0 v ) .
A R 0 ( X , Y ) λ 4 d 0 4 C A 0 A 0 ( X , Y ) .
d 0 = ( 2 + 3 2 ) max ( p x , p y ) λ a ,
{ u R , v R } { ± 1 p x 3 6 + 2 2 + x 0 λ d 0 , ± 1 p y 3 6 + 2 2 + y 0 λ d 0 } .
d 0 = 4 max ( p x , p y ) λ a ,
( u R , v R ) = ( ± 3 8 p x + x 0 λ d 0 , ± 3 8 p y + y 0 λ d 0 ) .
d 0 = 2 max ( p x , p y ) λ a .
d 0 = max ( p x , p y ) λ a .
+ R x Δ x = 0 ( x ) 2 d x = N 2 + sinc 2 ( π N p x λ d 0 x ) d x = N λ d 0 p x = N 2 ( λ d 0 N p x ) .
+ R x ( x ) 2 d x = + R ̃ x ( k ) 2 d k ,
R ̃ x ( u ) = F T [ W ̃ N ( x ) ] ( u ) × F T [ Π Δ x ( x ) ] ( u ) ,
F T [ W ̃ N ( x ) ] ( u ) = λ d 0 p x Π N p x λ d 0 ( u ( N 1 ) p x 2 λ d 0 ) λ d 0 p x Π N p x λ d 0 ( u N p x 2 λ d 0 ) ,
F T [ Π Δ x ( x ) ] ( u ) = sinc ( π Δ x , u ) .
R ̃ x ( u ) = sinc ( π Δ x u ) λ d 0 p x Π N p x λ d 0 ( u ) ,
+ R x ( x ) 2 d x = λ 2 d 0 2 p x 2 + sinc ( π Δ x u ) Π N p x λ d 0 ( u N p x 2 λ d 0 ) 2 d u ,
+ R x ( x ) 2 d x = λ 2 d 0 2 p x 2 0 + N p x λ d 0 sinc 2 ( π Δ x u ) d u .
sin 2 ( π Δ x u ) ( π Δ x u ) = k = 0 k = ( 1 ) k 2 2 k + 1 ( π Δ x u ) 2 k ( 2 k + 2 ) ! ,
sin 2 ( π Δ x u ) ( π Δ x u ) 2 d u = k = 0 k = ( 1 ) k ( π Δ x ) 2 k 2 2 k + 1 u 2 k + 1 ( 2 k + 1 ) ( 2 k + 2 ) ! .
+ R x ( x ) 2 d x = 2 N 2 λ d 0 N p x k = 0 k = ( 2 π N p x Δ x λ d 0 ) 2 k ( 1 ) k ( 2 k + 1 ) ( 2 k + 2 ) ! .

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