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
Digital holography found its origin in the early 1970s , but it became properly available only once its confirmation was established in 1994 . 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 , and reconstruction with Fresnelets in off-axis holography .
In the past, Yamaguchi et al. proposed an analysis of the image formation process in phase-shifting digital holography . 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 . 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 . 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 . They found that an adequate visual recognition of the reconstructed image can be obtained with the use of at least . Xu et al. also discussed image analysis of digital holography by considering the space–bandwidth product of the setup . This analysis leads to conclusions similar to those of .
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
2A. Fundamental Basics
Consider a reference set of coordinates 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 . Phase is random because of the roughness of the surface, and it will be considered to be uniform over . The object may not be perfectly centered in the reference set of the coordinates, but it can be slightly laterally shifted at coordinates . In what follows, it is considered that the object wave propagates through a distance , at which it interferes with a plane and a smooth reference wave having spatial frequencies noted as . At distance , the reference set of coordinates is chosen to be , and in the case where the diffracted field produced by the object is given in the Fresnel approximations by 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 , and a priori focusing error cannot occur. Now, in the interference plane, the hologram H is written as5). Indeed, there appears an irrelevant phase term , a modification of the spatial frequencies of the reference wave that become, respectively, and , and the Fresnel transform of the object field is evaluated at coordinates instead of at as is classically the case. Spatial frequencies of the reference wave are then increased/decreased by quantities . The influence of the evaluation of the Fresnel transform at coordinates instead of 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 order by using another Fresnel transform. Thus, the mathematical expression of the biased phase will be , where 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 pixels of pitches and , each of them sized . In a previous paper, the recorded hologram at point was given to be written as 2, 3, 45]:
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 ; that the reference wave is perfectly plane, i.e., ; that the image is in focus in the order, i.e., ; and that the pixel has an extended surface, i.e., . In , it was demonstrated that the diffracted field in the order is given by11) 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 . As was discussed in references [27, 30, 58], the last convoluting function indicates that the order is localized at coordinates . 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 , the order becomes localized at coordinates . 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 ; that the reference is not perfectly plane, i.e., ; that the focus is right, i.e., ; and that the pixel is not extended, i.e., . The diffracted field in the order is now written as58], we get15). Also appearing is a new convoluting function noted , which is due to aberrations of the reference wavefront. Its analytical formulation is related to the Fourier transform20), and it is noted by . This function is also related to a Fourier transform:20), and with , leads to the ideal situation and we get the diffracted field
2C. General Linear Relation20). This relation can also be written in the general form of Eq. (1) by introducing the impulse response of the process
3. INTRINSIC RESOLUTION46, 47, 48]. Both functions have similar profiles, except for their maximum value, since . Considering the Rayleigh criterion for determining the width of the function leads to58].
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 . 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 . 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 , , , and . It is found , giving a bandwidth of ; 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 points. Computation of the discrete Fresnel transform can be performed with points such that . If then the raw hologram is used for computation. If , this case is called “zero padding,” and it consists in adding 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 by22) 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 in diameter illuminated with a laser that was placed at distance from the sensor ( pixels, ). The number of reconstructed points is chosen to be (512,1024,2048,4096). When , 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 , 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 , 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 , 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 is5 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 , , , , and . Figure 5 shows that the effective shape of 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 and ; the observation horizon is imposed equal to , corresponding to the usual size of the CCD sensor commercially available; and the pixel width varies from to .
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
The evaluation of energy leads to (Appendix A)7 shows the criterion versus distance and the parameters of data acquisition and N. The numerical values are , , and for varying from to .
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 does not vary by more than 10%, so , corresponding to the dashed horizontal line in Fig. 7. It can be seen than for 2048 pixels of size , the minimal distance for which is at least . The distance will be only 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 . In the resolution function there appears an enlarging function given by a Fourier transform, and it is written as57]35). There is no simple analytical formulation for . However, from previous discussion, the spatial frequency bandwidth generates a spatial width in the reconstructed plane via the relation . So the width of the defocusing function is related to the spatial frequency bandwidth, given by8 shows for , , , , and varying from to . Profiles along the x direction are represented in Fig. 9 . Enlarging due to when 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 and gives , which corresponds to the width of the corresponding curve in Fig. 9; in this example, resolution is not better than .
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  such that sampling remains invariant. The parameters are , , , , and varying from to . The reference wave has spatial frequencies . Figure 10 shows the comparison between the resolution functions 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 . 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):
6B. Aberration in the Recording Plane
Now focus on the aberration in the recording plane at pixel coordinates . 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 along the and directions. For both cases, the aberration in the recording plane is obtained by changing ξ into and η into in Eq. (40). So it follows thatB 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 , it is relevant to consider the spatial frequency bandwidth of such a function. The local spatial frequencies are given by 42) contribute to the mean spatial frequencies of the reference wave. Since frequencies localize the paraxial image, it follows that if and are not zero (i.e., , and or ), the reconstructed object under aberration will be slightly translated by an amount given by14, where , the resolution is degraded by the presence of in Eqs. (45, 46). In case (a), with and with pure spherical aberration, the resolution is simply and . In the case of pure coma (i.e., ), the spatial resolution is now given by and . In case (a), where , the resolution is simply and .
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 , , , , and . The reference wave has spatial frequencies . 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 in diameter, with a focal length of and a glass index . 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 and , coma is about 4 fringes of coma at the border of the pupil. Thus this gives and . With the results of previous sections, the localization of the paraxial image is . 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 , the median shift , and the maximum available shift . 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 around the coordinates of the paraxial image (, ) 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 . 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 and , 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 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 order, with the 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 , or the filtering can be performed in the spectral domain . The only efficient method for suppressing the zero order is that proposed by Yamaguchi; phase shifting digital holography , 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 . Since the discrete Fresnel transform of the constant is an approximately uniform spatial area of size , the main contribution to the zero-order width is given by . Under an advantageous hypothesis (no aberration, nonextended pixel, no focusing error, and no lateral shift), it is found that49) is written as52) 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 is the spatial autocorrelation of , then
7B. Optimization of Recording
The spatial expansion of 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 ( and , respectively, in the x and y directions) must be equal to four times the width of the object (i.e., or ). Note that in the case where , the smallest width given by the largest pixel pitch must be taken into account so that the condition becomes . However, for a circular object of diameter , the smallest diagonal of the reconstructed field must be taken into account [i.e., ]. So, due to geometrical considerations, the condition now becomes . 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 for a circular object and at the paraxial coordinates for a square object [Figs. 18a, 18b].
Thus for a circular object of diameter [Fig. 18a], the optimal recording distance is given by18b]64], the zero order and the order are naturally removed by the demodulation algorithm. Thus, the spatial frequencies of the reference wave can be null (in-line holography), and the order can overlap the entire reconstructed field. For the circular object, the optimal recording distance becomes55, 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 illuminated by a laser at . Holographic recording is performed with a pixel matrix with pixels with pitches , and the object is centered on the optical axis [i.e., ]. Digital reconstruction is computed with zero padding with pixels in the Fresnel transform. With Eq. (55), the recording distance must be set to and the spatial frequencies are adjusted to and , which are a little bit less than the value of obtained with Eq. (56). Considering the results of subsection 4B, it can be seen that the active surface of pixels has no influence on the intrinsic resolution. The reconstructed field is presented in Fig. 19 when the focus is set in the order. The intrinsic resolution is and for a sampling step of . The reconstructed field covers an area of , and the paraxial object is localized at coordinates , corresponding to pixel coordinates . 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 , the focus is on the order, the reconstructed field having the same dimensions as before. With the hermitic properties, the order is localized at as can be seen in Fig. 20 . If the reconstruction distance is different from , the object will be blurred as in Fig. 21 . For example, with , each point of the real object is represented in the diffracted field by a pattern if about a width [Eq. (39)]; it is also the approximate value of the spatial resolution in the image plane.
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.
In Table 2 , the coefficients of the polynomial expansion are prescribed in detail.
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).
3. T. Zhang and I. Yamaguchi, “Three-dimensional microscopy with phase shifting digital holography,” Opt. Lett. 23, 1221–1223 (1998). [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]
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]
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]
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]
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]
24. G. Pedrini and H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 18, 251–260 (1995). [CrossRef]
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).
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).
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]
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]
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]
54. J. H. Massig, “Digital off-axis holography with a synthetic aperture,” Opt. Lett. 27, 2179–2181 (2002). [CrossRef]
57. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
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]