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Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Stephen A. Burns
  • Vol. 25, Iss. 7 — Jul. 1, 2008
  • pp: 1744–1761
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General theoretical formulation of image formation in digital Fresnel holography

Pascal Picart and Julien Leval  »View Author Affiliations


JOSA A, Vol. 25, Issue 7, pp. 1744-1761 (2008)
http://dx.doi.org/10.1364/JOSAA.25.001744


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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

In the past, Yamaguchi et al. proposed an analysis of the image formation process in phase-shifting digital holography [50

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]

]. 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

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]

]. 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

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

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]

] or quantization effects. Baumbach et al. proposed to reduce the speckle noise by using digital reconstructions from laterally shifted holograms and pixelwise averaging [51

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]

]. 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

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

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

]. Finally, Mills et al. discussed theoretically and experimentally the influence of bit-depth limitation in quantization [55

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

]. 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

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]

]. This analysis leads to conclusions similar to those of [48

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]

].

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

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

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

]
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.
(2)
The general relation for the reference wave is
r(x,y)=aRexp[2jπ(uRx+vRy)+jΔΨab(x,y)].
(3)
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

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

], 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

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]

], 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).
(4)
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.
(5)
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.

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

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),
(23)
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).
(24)
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).
(25)
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

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

, 47

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

, 48

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]

]. 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,
(26)
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

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]

].

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

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

]. Other authors [53

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

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

] 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

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]

]. 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.
(27)
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

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,
(29)
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
(30)
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),
(31)
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)!;
(32)
Cx is then written as
Cx=2k=0k=(2πNpxΔxλd0)2k(1)k(2k+1)(2k+2)!,
(33)
giving
Cx=14π2N2px2Δx236λ2d02+16π4N4px4Δx41800λ4d0464π6N6px6Δx6141120λ6d06+.
(34)
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

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

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

] 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

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].
(41)
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

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

]
{ui=1λ(4ax4x3+3ax3x2+2ax2x+ax1+2ax2y2xy2+2ax2y1xy+ax1y2y2+ax1y1y)vi=1λ(4ay4y3+3ay3y2+2ay2y+ay1+2ax2y2x2y+ax2y1x2+2ax1y2xy+ax1y1x).
(42)
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).
(43)
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.
(44)
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).
(45)
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).
(46)
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

7. SHANNON THEOREM FOR DIGITAL HOLOGRAPHY

7A. Structure of the Diffracted Field

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].

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

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]

, 56

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]

]. 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).
(A1)
According to Parseval’s theorem,
+Rx(x)2dx=+R̃x(k)2dk,
(A2)
with
R̃x(u)=FT[W̃N(x)](u)×FT[ΠΔx(x)](u),
(A3)
and
FT[W̃N(x)](u)=λd0pxΠNpxλd0(u(N1)px2λd0)λd0pxΠNpxλd0(uNpx2λd0),
(A4)
FT[ΠΔx(x)](u)=sinc(πΔx,u).
(A5)
The Fourier transform of Rx(x) is
R̃x(u)=sinc(πΔxu)λd0pxΠNpxλd0(u),
(A6)
thus
+Rx(x)2dx=λ2d02px2+sinc(πΔxu)ΠNpxλd0(uNpx2λd0)2du,
(A7)
giving
+Rx(x)2dx=λ2d02px20+Npxλd0sinc2(πΔxu)du.
(A8)
With
sin2(πΔxu)(πΔxu)=k=0k=(1)k22k+1(πΔxu)2k(2k+2)!,
(A9)
we have
sin2(πΔxu)(πΔxu)2du=k=0k=(1)k(πΔx)2k22k+1u2k+1(2k+1)(2k+2)!.
(A10)
So
+Rx(x)2dx=2N2λd0Npxk=0k=(2πNpxΔxλd0)2k(1)k(2k+1)(2k+2)!.
(A11)

Appendix B

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

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

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

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Fig. 1 Schematic diagram of image formation in digital holography.
Fig. 2 Basics of recording a digital hologram.
Fig. 3 Intrinsic resolution function.
Fig. 4 Illustration of zero padding.
Fig. 5 Profiles of the resolution function with nonextended and extended pixels.
Fig. 6 Influence of the active surface of pixels.
Fig. 7 Plot of criterion quantifying the influence of the active surface of pixels [circles, Eq. (34); curves, Eq. (30)].
Fig. 8 Evolution of the resolution function versus the reconstruction distances.
Fig. 9 Profiles of the resolution function for different reconstruction distances.
Fig. 10 Comparison between the analytical formulation and the numerical simulation versus the reconstruction distance.
Fig. 11 Profiles along x for the analytical formulation and numerical simulation at the best focus.
Fig. 12 Profiles along z for the analytical formulation and full numerical for x=y=0.
Fig. 13 Different schemes for generating a reference wave.
Fig. 14 Zone of the wavefront “seen” by the sensor.
Fig. 15 Resolution for (δx,δy)=(0,0).
Fig. 16 Resolution for (δx,δy)=(7.64mm,7.64mm).
Fig. 17 Resolution for (δx,δy)=(15.29mm,15.29mm).
Fig. 18 Schematic structure of the reconstructed field.
Fig. 19 Focus on the +1 order.
Fig. 20 Focus on the 1 order.
Fig. 21 Defocused image.
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).

2.

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

3.

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

4.

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

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). [CrossRef] [PubMed]

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.

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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]

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

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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]

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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]

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OCIS Codes
(090.0090) Holography : Holography
(090.2870) Holography : Holographic display
(090.2880) Holography : Holographic interferometry
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.4630) Instrumentation, measurement, and metrology : Optical inspection
(090.1995) Holography : Digital holography
(110.3010) Imaging systems : Image reconstruction techniques

ToC Category:
Holography

History
Original Manuscript: December 19, 2007
Revised Manuscript: April 8, 2008
Manuscript Accepted: May 4, 2008
Published: June 25, 2008

Citation
Pascal Picart and Julien Leval, "General theoretical formulation of image formation in digital Fresnel holography," J. Opt. Soc. Am. A 25, 1744-1761 (2008)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-25-7-1744


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References

  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).
  2. U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179-181 (1994). [CrossRef] [PubMed]
  3. T. Zhang and I. Yamaguchi, “Three-dimensional microscopy with phase shifting digital holography,” Opt. Lett. 23, 1221-1223 (1998). [CrossRef]
  4. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase contrast imaging,” Opt. Lett. 24, 291-293 (1999). [CrossRef]
  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). [CrossRef] [PubMed]
  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]
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  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]
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