Abstract

Registration and reconstruction of high-quality digital holograms with a large view angle are intensive computer tasks since they require the space-bandwidth product (SBP) of the order of tens of gigapixels or more. This massive use of SBP severely affects the storing and manipulation of digital holograms. In order to reduce the computer burden, this work focuses on the generation and reconstruction of very large horizontal parallax only digital holograms (HPO-DHs). It is shown that these types of holograms can preserve high quality and large view angle in x direction while keeping a low use of SBP. This work first proposes a numerical technique that allows calculating very large HPO-DHs with large pixel size by merging the Fourier holography and phase added stereogram algorithm. The generated Fourier HPO-DHs enable accurate storing of holographic data from 3D objects. To decode the information contained in these Fourier HPO-DHs (FHPO-DHs), a novel angular spectrum (AS) technique that provides an efficient use of the SBP for reconstruction is proposed. Our reconstruction technique, which is called compact space bandwidth AS (CSW-AS), makes use of cylindrical parabolic waves that solve sampling issues of FHPO-DHs and AS. Moreover, the CSW-AS allows for implementing zero-padding for accurate wavefield reconstructions. Hence, suppression of aliased components and high spatial resolution is possible. Notably, the imaging chain of Fourier HPO-DH enables efficient calculation, reconstruction and storing of HPO holograms of large size. Finally, the accuracy and utility of the developed technique is proved by both numerical and optical reconstructions.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The aim of 3D display technology is to create the illusion of depth from a 2D plane. Among the several solutions that have been proposed in the last three decades, digital holography and holographic displays constitute the best-known framework for truly 3D imaging and video. Holography has shown the potential to encode accurately the required information from the optical field of recorded scene - digital holography (DH) or computer - generated holography (CGH). Full parallax holograms can provide high quality reconstruction of 3D objects with wide angle view in vertical and horizontal directions (up to 47°) [1]. Such holograms are spatially large (around 10 cm) with pixel pitch Δ = 0.8 µm [1,2]. These requirements generate an optical and numerical signal with space bandwidth product (SBP) of the order of tens of gigapixels [13]. Alternative is the employment of horizontal parallax only digital holograms (HPO-DHs). HPO-DH ignores wide angle view in vertical direction and preserves parallax in the horizontal direction. In consequence, HPO-DHs are small in vertical direction, and thus the amount of SBP for capturing is substantially reduced [46]. Moreover, HPO-DHs are supported by most of the current light field displays [4,7,8]. Nevertheless, HPO-DHs must be still captured with small Δ, which limits the view angle. The means of reducing the SBP without sacrificing the viewing angle and quality are the primary importance in this work. Therefore, it is required that hologram capturing is carried out with large pixel pitch. This can be achieved by means of Fourier DH, where spherical reference illumination is employed. When combining Fourier DH and synthetic aperture concept, wide angle registration of HPO-DHs, which we call as Fourier HPO-DH (FHPO-DH), can be carried out [9,10]. Notably, unlike classical HPO-DH where diffraction in vertical direction is neglected [5], FHPO holography is based on the accurate propagation scheme. Thus, objects larger than the vertical size of the hologram can be captured and consequently SBP can be further reduced. Experimental recording of FHPO holograms is limited by physical properties of the optical set-up (up to 23 cm [9]). This work expands the concept of FHPO showing that holograms of size in meters can be numerically generated and reconstructed. Provided phase-space analysis investigates 3D limits of viewing zone and related change of resolution. To compute holograms of such size, this work proposes a novel computer holographic method that employs concept of Fourier holography to the phase added stereogram (PAS) algorithm [11,12], which divides the hologram area in square segments. It is shown that our modified version of PAS, called SA-PAS, enables calculation of Fourier Computer Generated Holograms (FCGH) with much larger segments. Hence, FCGH has key properties as: (i) accurate and fast calculation of FCGH from clouds with millions of point sources; (ii) calculation of FHPO holograms of very large size, up to 1500 mm; (iii) angle view beyond 80°; (iv) convenient observation distance; and (v) very small depth of focus. When reconstructing these large hologram trough z axis, sharp and thin layers of the object can be observed.

Reconstruction of FHPO-DHs, either experimental or synthetic, requires non-paraxial propagation algorithms like the Angular Spectrum (AS) [9]. However, accurate reconstruction with the AS needs large amount of zero padding in x and y direction, demanding a huge SBP. This makes impossible to process numerically the object wave field from FHPO-DHs. Recently, the multi-FFT AS method has been proposed for reconstruction of FHPO-DHs [10]. The proposed method can implement the required zero-padding strategies for accurate reconstruction of FHPO-DHs while using a reduced SBP. Nevertheless, the larger size of the FHPO-DH is, the larger the amount of required zero-padding, which is reflected in a growth of SBP and execution time. Thus, the multi-FFT AS cannot retrieve the wavefield from FHPO-DHs larger than 250 mm. In this work, we develop a novel numerical method for reconstructing the wavefield from very large FHPO holograms, as the ones calculated with the SA-PAS. The algorithm applies the compact space bandwidth (CSW) representation [13,14] to the AS approach. Previous implementation of CSW-AS uses paraxial spherical wavefronts for signal compression. But this approach cannot be applied in FHPO-DH due to large resolution difference in the image plane. In this work, CSW schemes are applied only for the critical dimension providing bandwidth reduction for the corresponding direction, which solves sampling issues of FHPO-DHs and AS. It is shown that our reconstruction algorithm allows reducing significantly the SBP, which improves the efficiency of storing, calculation, and reconstruction of FHPO holograms of large size. The substantial SBP reduction enables to surpass reconstruction limits of any known propagation technique. Moreover, we propose a phase-space representation (PSR) analysis that calculates the necessary amount of zero-padding in frequency and space domain for accurate wavefield reconstructions, which allows suppression of aliased components and high spatial resolution. Nevertheless, the longer is the horizontal size of the FHPO-DH, the larger amount of SBP required by the CSW-AS. We numerically found that holograms of sizes above 800 mm need a larger amount of zero-padding, which means that CSW-AS is not able to retrieve the corresponding wavefields. To solve this problem, a variant of the CSW-AS is proposed, which is called split CSW-AS (CSWSAS). The CSW-SAS divides the FHPO hologram in x direction into P sub-holograms, and thus, each sub-hologram is processed separately. The processing of sub-holograms is based on frequency interpolation and tile summation for vertical direction [10]. In this way, FHPO-DH larger than 800 mm can be reconstructed accurately.

Theoretical capabilities of SA-PAS and CSW-AS and CSW-SAS algorithms are tested. These tests prove high accuracy of computations. Finally, CSW-AS and CSW-SAS algorithms reconstruct holographic data obtained with the Fourier synthetic aperture DH system [9] and the SA-PAS, respectively. Reconstruction size for experimental FHPO-DH is 256 mm x 7 mm (the largest that can be captured) and for the FCGH the maximum size is 1500 mm x 7 mm. This paper is structured as follows: Section 2 studies properties of large FHPO hologram. Section 3 describes the principle of working of the CSW-AS. Section 4 introduces the SA-PAS. Section 5 shows the reconstruction results of the CSW-AS using FHPO-DH and FCGHs. Discussion and conclusions of this work are finally presented in Section 6.

2. Fourier HPO hologram

2.1 HPO hologram recording

A digital hologram H is a discrete real-valued signal that comes from measuring the intensity from linear superposition between object O and reference R wave fields with a digital camera. The detected intensity consists in a set of dark and bright fringes that encode amplitude and phase information of the object. The mathematical expression for the captured digital hologram is given by

$$H = {|O |^2} + {|R |^2} + OR^{\ast}{+} O^{\ast} R,$$

Selection of the reference wavefield is relevant when considering the extent of object wavefield that can be captured by devices with predetermined SBP. Here, we follow the concept of the Fourier holography where a spherical reference wave is used [15,16]. This type of hologram enables us to capture large wavefields with larger pixel pitch, thus smaller SBP is required. The simplified scheme of recording Fourier holograms is presented in Fig. 1(a), where (x1, y1) and (x2, y2) are hologram and object planes, respectively. Object plane contains the reference source point, which is placed at the origin of this plane. Separation between the planes is given by zR. Considered geometry with spherical reference wave is optimal solution for registering wide-angle digital holograms by means of synthetic aperture method [9], where large hologram is synthetized via sequential recording of small holograms that are shifted in horizontal direction. In Ref. [10], it is shown that FHPO-DHs up to 26 cm length in horizontal direction and 7 mm in vertical direction can be captured when using spherical reference wave.

 figure: Fig. 1.

Fig. 1. a) Geometry of the experiment for recording the FHPO-DH. b) Viewing zone for FHPO-DH of size 1000 mm x 7 mm and zR = 800 mm.

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Fourier holography is an efficient codding approach but for spatially limited object that are localized around the reference point source. This important limitation can be analyzed by using viewing zone analysis approach, as shown in Fig. 1(b). The evaluated image zone is a volumetric limit for the object placement. The viewing zone can be found by considering the boundaries points $x_a^{\prime}$= −500 mm, xa = 500 mm and the center point x = 0 at the hologram plane. The points $x_a^{\prime}$ and xa generate two light cones that intersects over the longitudinal axis. The illumination angles of these cones can be calculated from the frequency limit of the object wavefield in x direction. Points xc and xd show the maximal spatial extension that the projected image can take from the observation point xa. When projecting the limiting rays from the boundary points, a diamond shape area is formed. Points A and B represent axial limit of the viewing zone. To find transverse size of the viewing zone, the observation point x = 0 and the corresponding light cone, which is showed with red lines, is analyzed. The corresponding light cone limits the transversal size of the viewing zone. Final shape of the viewing zone is shown in Fig. 1(b) by the blue area. Since Fourier holography shares the property of Fresnel holograms for the object size, the spatial extension of the object in the image plane is given by Bx2 = Bx2 = λzR/Δ. It is worth mentioning that for small FHPO-DH the maximum transversal extension of the viewing zone is equal to Bx2 as shown by Ref. [17]. The longitudinal size of the viewing zone is 329.85 mm, which limits the depth of the object that can be used for recording the hologram, and the transversal size is given by Bx2 ≈ 123 mm when employing zR = 800 mm, Δ = 3.45 µm and λ = 0.532 µm.

With the development of computing technology, computer generated holograms (CGHs) can be captured without the necessity of a physical set-up. CGHs does not have limitations of physical systems [10], such as extent of object/reference wave or accurate camera translation. Thus, a Fourier CGH (FCGH) can be much larger. When synthetizing FCGHs, only the complex-valued signal of OR* from Eq. (1) is calculated. FCGH can be calculated by different approaches [1820]. The most popular approach for calculating this term considers that the 3D object is a composite of light point sources distributed in a volume. Each point source will propagate a spherical wave field that reaches the hologram plane. The FCGH will be result of summing all these propagated wave fields at the that plane. Hence, the term OR*, denoted as OR, can be written as

$${O_R} = \frac{1}{r}\textrm{exp} (ikr)\sum\limits_{p = 1}^N {\frac{{{a_p}}}{{{r_p}}}\textrm{exp} (ik{r_p})} ,$$
where ${r_p} = \sqrt {{{({x_2} - {x_p})}^2} + {{({y_2} - {y_p})}^2} + z_p^2}$, $r = \sqrt {x_2^2 + y_2^2 + z_R^2}$ and k = 2π/λ. The position of an arbitrary point source within the cloud and its amplitude are given by (xp, yp, zp) and ap, respectively. Equation (2) represents the synthetic object wave compressed with non-paraxial spherical wave [13], which is the most straightforward way of generating the term OR. However, calculation time can range from hours to days depending on the number of points and length of the CGH [18,19]. This makes Eq. (2) an impractical approach for large non-paraxial FCGHs of detailed objects. Calculation time of OR can be significantly reduced by the application of accelerating algorithms like the wavefront recoding plane approach [18], the layer-based method [21], or the phase added stereogram (PAS) [22].

2.2 Resolution and image zone of the holographic image

Previous works [9,10,17,23,24] have shown that frequency components of a FHPO-DH can be stored by using a reduced SBP, this is true for very large FHPO-DH, as well. However, the large size of the FHPO-DH introduces untypical phase-space representation (PSR). Figure 2(a) depicts the PSR of recorded and reconstructed hologram of 1 m size, only for x direction. The red, and green areas show PSRs of the term OR and O for hologram plane, respectively. Blue region represents the PSR for object plane. The red area in Fig. 2(a) shows that the local bandwidth of the object is reduced for larger spatial coordinates. This is observed clearer in the zoomed box. This means that for large holograms some PS support determined by the sampling frequency Δ−1 is not fully utilized. This shrinking effect is not observed for FHPO-DHs smaller than 26 cm [9,10,23]. On the other hand, the parameter Bfx2, which is the height of the area that represents the PSR of O, exceeds the sampling frequency Δ−1, as shown in Fig. 2(a). The exceeding frequencies will be rewrapped as aliasing errors, which can be removed by increasing the sampling rate for x direction. Finally, the PSR at object plane has constant and large local bandwidth. This constant extension is possible due to the progressive frequency reduction of the signal OR. Note that the frequency support for the image plane is the same as for the object wave O, which is much larger than hologram sampling frequency Δ−1. Classical AS method cannot accommodate this frequency growth since sampling frequency at the hologram and at the image must be the same. This property presents the limit for reconstruction of large holograms when using classical diffraction method, which makes reconstructions computationally expensive or impossible [10]. For y coordinate, we have that By2 >> By1, and thus, at the object plane the image bandwidth is reduced Bfy2 = By1z << Δ−1. Since AS must keep the same bandwidth in both planes, zero-padding is required. However, the amount of zero padding is large and consequently this solution is also ineffective for this direction [10]. Finally, the resolution on the image plane can be calculated by taking the inverse of Bfx2 and Bfy2.

 figure: Fig. 2.

Fig. 2. PSR of a FHPO-DH of size 1000 mm x 7 mm and zR = 800 mm, Δ = 3.45 µm, λ = 0.532 µm. a) x direction. b) z direction.

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Above discussion considered point sources placed at the object plane. This shows relation between PSRs at hologram and object planes. Here, the PSR of on-axis holographic signal OR along the z-axis is analyzed. The area with inclined green lines in Fig. 2(a) represents the region where the signal OR can evolve without exceeding the frequency sampling Δ−1. The limiting distances of this area are represented with the dot-dash green line and the double-dot-dashed green line for zA and zB, respectively. Any propagation distance smaller than zA or larger than zB will result in a PSR that exceeds the frequency sampling, and in consequence, aliasing errors will be obtained. Notably, the sweeping depth allows utilizing the full frequency support of the hologram. Moreover, the limiting distances zA and zB are the same as obtained from the viewing analysis of Fig. 1(b). PSR of the deep object is presented in Fig. 2(a) by the area with inclined red lines. Limiting distances are shown with red-dotted and red-double-dotted lines for zA and zB, respectively. Figure 2(b) shows the progressive bandwidth reduction given by the sweeping of the longitudinal distances. The large arrow in this figure depicts the frequency support for the smallest distance zA. Progressive increasing of sweeping distance results in a decreasing of the frequency support. When using the distance zR, the frequency support is equal to Bfx2. Finally, the smallest arrow in Fig. 2(b) represents the frequency support when the largest distance zB is reached. The frequency support area for the sweeping distances zA and zB is depicted by the zone with inclined red lines.

3. Hologram reconstruction with CSW-AS

3.1 CSW-AS propagation method

In this section, the compact space bandwidth-based angular spectrum (CSW-AS) method is introduced, which enables accurate and efficient reconstruction of FHPO holograms. According to the PS analysis of section 2, the object wavefronts of hologram (O1) and the image (O2) cannot be sampled within the frequency support Δ−1. For O1, for x direction extreme frequency up-sampling is required [10] to enable supporting high frequencies of object wavefront in the hologram. On the other hand, for O2, there is a need of large spatial padding in y direction for reconstructing the entire size of the object. To solve these issues, we propose to apply the compact space bandwidth (CSW) sampling of O1 and O2 [14]. CSW schemes are applied only for the critical dimension providing bandwidth reduction for the corresponding direction. It has to be noted that in the literature, paraxial spherical waves are applied for both directions [6]. Here, paraxial cylindrical waves are used. Thus, for hologram plane the CSW representation is applied for x direction while for reconstruction plane for y as follows

$${O_{c1}}({{x_1},{y_1}} )= {O_1}({{x_1},{y_1}} )\textrm{exp} \left\{ {\frac{{i\pi x_1^2}}{{\lambda {z_{cx}}}}} \right\},$$
$${O_2}({{x_2},{y_2}} )= {O_{c2}}({{x_2},{y_2}} )\textrm{exp} \left\{ { - \frac{{i\pi x_2^2}}{{\lambda {z_{cy}}}}} \right\},$$
where subscript cx,y denotes CSW representation and parameter zcx,y is the curvature of the cylindrical waves employed for bandwidth reduction. Using Oc1 larger holograms can be captured, stored, and reconstructed with x-sampling rate slightly larger than Δ−1. On the other hand, Oc2 enables reduction of y sampling rate of O2 to the frequency support Bfy2, which is close to the physical resolution of image for y direction. The relation between the wavefields Oc1 and Oc2 can be related by employing the AS representation in the frequency coordinates (see Refs. [13,14] for more clarity) as follows
$$\begin{array}{l} {{\tilde{O}}_{c2}}({{f_{x2}},{f_{y2}}} )\otimes \textrm{exp} \{{ - i\pi \lambda {z_{cy}}f_{y2}^2} \}= [{{{\tilde{O}}_{c1}}({{f_{x2}},{f_{y2}}} )\otimes \textrm{exp} \{{ - i\pi \lambda {z_{cx}}f_{x2}^2} \}} ]\\ \begin{array}{{cccc}} {}&{}&{}&{\begin{array}{{cccc}} {}&{}&{}&{\begin{array}{{cccc}} {}&{}&{}&{\begin{array}{{cc}} {}& \times \end{array}} \end{array}} \end{array}} \end{array}\textrm{exp} \{{ - ikzf_z^{}({{f_{x2}},{f_{y2}}} )} \}, \end{array}$$
where
$$f_z^{}({{f_x},{f_y}} )= \sqrt {1 - {\lambda ^2}f_x^2 - {\lambda ^2}f_y^2} ,$$
is the longitudinal frequency of phase accommodation.

By expanding the convolutions in Eq. (5), the algorithm of CSW-AS method is obtained. Calculation frame of the CSW-AS can be presented in four steps. In the first step, the 2D FT is applied to the wavefield Oc1 and multiplied with the parabolic x-cylindrical phase factor as follows

$${g_1}({{f_{x1}},{f_{y1}}} )= \textrm{exp} \{{ - i\pi \lambda {z_{cx}}f_{x1}^2} \}\int {{O_{c1}}({{x_1},{y_1}} )} \textrm{exp} \{{ - 2\pi i({{x_1}{f_{x1}} + {y_1}{f_{y1}}} )} \}d{x_1}d{y_1}.$$

The resulting signal g1 is sampled with [Δfx1 = 1/NxΔ, Δfy1 = 1/NyΔ].

In the step two, a scaled x-directional 1D FT is applied to the function g1. The result of this transformation is multiplied by three phase exponential functions, which yields

$$\begin{array}{l} {g_2}({{f_{x2}},{f_{y1}}} )= \textrm{exp} \{{ - ikz{f_z}({{f_{x2}},{f_{y1}}} )} \}\textrm{exp} \{{i\pi \lambda ({{z_{cx}}f_{x2}^2 + {z_{cy}}f_{y1}^2} )} \}\\ \begin{array}{{cccc}} {}&{}&{}&{\begin{array}{{cc}} {}& \times \end{array}} \end{array}\int {{g_1}({{f_{x1}},{f_{y1}}} )} \textrm{exp} \{{ - 2\pi i\lambda {z_{cx}}{f_{x1}}{f_{x2}}} \}d{f_{x1}}, \end{array}$$

The signal g2 is sampled with [Δfx2= Δ/λzcx, Δfy1 = 1/NyΔ]. In the third step, 1D scaled FT in respect to y direction is carried out on g2 and the result is multiplied with a parabolic y-cylindrical phase factor as

$${\tilde{O}_{c2}}({{f_{x2}},{f_{y2}}} )= \textrm{exp} \{{i\pi \lambda {z_{cy}}f_{y2}^2} \}\int {{g_2}({{f_{x2}},{f_{y1}}} )} \textrm{exp} \{{2\pi i\lambda {z_{cy}}{f_{y1}}{f_{y2}}} \}d{f_{y1}}.$$

The obtained distribution has the sampling [Δfx2 = Δ/λzcx, Δfy2 = Δ/λzcy]. Finally, the result is calculated in final step 4 via application of the 2D FT as

$${O_{c2}}({{x_2},{y_2}} )= \int {{{\tilde{O}}_{c2}}({{f_{x2}},{f_{y2}}} )} \textrm{exp} \{{ - 2\pi i({{x_2}{f_{x2}} + {y_2}{f_{y2}}} )} \}d{f_{x2}}d{f_{y2}}.$$

Note that the reconstructed object wave field is sampled with [Δx2= λzcx/NxΔ, Δy2= λzcy/NyΔ].

3.2 PS analysis of the CSW-AS algorithm

Accurate reconstruction of large holograms with CSW-AS algorithm requires calculation with extended SBP [13]. The increase of SBP is needed to correctly represent all frequencies of the object wavefront captured in the hologram and to avoid aliasing errors of the algorithm. The necessary SBP extension is carried out by applying spatial and frequency zero-padding for both directions. Proper amount of zero-padding is found when analyzing the space-frequency response of all steps of the CSW-AS algorithm for the point source with coordinate (xp = Bx2/2, yp = By2/2, zR). This point source represents the boundary point of the object that can be used within the Fourier DH configuration. This inspection gives the maximum SBP needed.

The following analysis is visually supported with PS plots for hologram size of 1 m and for the object point at coordinate [xp = 61.68, yp = 61.68, zR = 800] mm. CSW-AS algorithm processes hologram data for x, and y direction in different ways, and thus, separate analyses are provided. Let first focus on x dimension. For this direction, the outgoing spherical wave at the detector plane in CSW representation generates local spatial frequencies as follows

$$f_{x1}^l ={-} \frac{{{x_1} - {x_p}}}{{\lambda {r_p}}} + \frac{{{x_1}}}{{\lambda {z_{cx}}}}.$$

This produces the PS representation:

$$PSR({{O_{c1}},{x_1},f_{x1}^{}} )= \delta \left( {f_{x1}^{} + \frac{{{x_1} - {x_p}}}{{\lambda {r_p}}} - \frac{{{x_1}}}{{\lambda {z_{cx}}}}} \right).$$

Figure 3(a) presents Oc1 in the PSR by using black continous line. In the first step of the AS CSW algorithm, the signal Oc1 is transformed into the signal g1 using FT and addition of parabolical phase factor. This phase introduces a spatial shear in the PS space represenation of Oc1 as

$$PSR({{g_1},{x_1},f_{x1}^{}} )= PSR({{O_{c1}},{x_1} - f_{x1}^{}\lambda {z_{cx}},f_{x1}^{}} ).$$

The transformed signal is presented using a black dotted line in Fig. 3(a). For the illustrated case and for right corner point the spatial shift is 76 mm. Next step includes 1D scaled FT and multiplication with three frequency phase factors. The scaled FT introduces modification of space-frequency axes, the spatial axis is changed to the frequency axis, while frequency into space. The phase factors give frequency phase shift, which changes the signal to a Dirac delta at the location of the object point. The PSR of this signal is contained within the PS limits of g1. Consequently, for evaluation of the maximum SBP needed, this step does not have to be considered. Thus, required spatial zero-padding in x direction can be found from Eq. (13). For this direction, y-dimension can be disregarded since the hologram has small extent.

 figure: Fig. 3.

Fig. 3. a) PSRs of Oc1 and g1 for x direction; b) PSRs of Oc1 and g2 for y direction and for two hologram cross sections.

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For y direction, the spherical wave for the same object point at the hologram plane in CSW representation has local spatial frequencies

$$f_{y1}^l ={-} \frac{{{y_1} - {y_p}}}{{\lambda {r_p}}}.$$

Thus PSR of this input signal is given as

$$PSR({{O_{c1}},{y_1},f_{y1}^{}} )= \delta \left( {f_{y1}^{} + \frac{{{y_1} - {y_p}}}{{\lambda {r_p}}}} \right).$$

In this case, CSW-AS algorithm starts modifying PSR of the input signal in steps two and three. The PS transformation includes a FT and two frequency phase factors, which introduce frequency dependent spatial shift to the input signal as

$$PSR({{g_2},{y_1},f_{y1}^{}} )= PSR({{O_{c1}},{y_1} + f_{y1}^{}\lambda {z_{cy}} - zf_{y1}^{}f_z^{ - 1}(f_{x1}^{},f_{y1}^{}),f_{y1}^{}} ).$$

Similarly, like for x dimension, the remaining part of the algorithm converts the signal into a Dirac delta having smaller PS coverage than g2. Consequently, it can be disregarded for evaluation of SBP extension. Nevertheless, for y analysis the x dependence cannot be disregarded. To analyze this relationship, we examine two cross-sections of the hologram for x1 = 0 and x1 = max(x1). Figure 3(b) presents PSRs calculated via Eq. (15) and (16) for both cross sections of the hologram. Both PSRs are different and they must be examined to find the required space frequency zero padding. Also, the relative shift for y direction is much larger than x. Furthermore, the shearing effect is not so evident since smaller frequency coverage. Finally, it must be noted that they are depended on the values zcx and zcy.

3.3 Space-frequency zero-padding for the CSW-AS

Above analysis found that estimation of the amount of zero-padding requires evaluation of PS support of g1 and g2 for x and y directions via Eq. (13) and Eq. (16), respectively. Figure 4 illustrates the required extension of space-frequency support for the example case investigated in Fig. 3. Evaluation of PSRs is carried out for two boundary object points: [xp= max(x2), yp= max(y2)] and [xp = min(x2), yp= min(y2)]. The obtained distributions are symmetrical, and thus, evaluation of PSRs for one point is enough. Figures 4(a) and 4b) are evaluated for fx1-x1 and fy1-y1 coordinates, respectively. The smaller dashed boxes in those figures depict PS dimensions of the input hologram: SBPx1 = Bx1 × Bfx1, SBPy1 = By1 × Bfy1. Aliasing free calculations with CSW-AS needs space-frequency extension to SBPx’ = Bx×Bfx and SBPy’ = By×Bfy. This extension is illustrated in Fig. 4 by the larger dashed line boxes. Notably, these four quantities Bx, Bfx, By, and Bfy, which defines the amount of space-frequency zero-padding, are depended on zcx, and zcy. Evaluation of minimal PS support needs determination of the optimal values of these two coefficients and SBPx’, SBPy’ denotes minimum size. The PS support for given zcx, and zcy can be evaluated from limits of the distributions of PSR(g1,x1,fx1) and PSR(g1,y1,fy1). For the example case shown in Fig. 4, accurate computations of CSW-AS requires small padding SBPx’ = 1.28×SBPx1 for x while the padding needed for y is larger SBPy’ = 4.7×SBPy1.

 figure: Fig. 4.

Fig. 4. Illustration of PS coverage and space-frequency zero padding of CSW-AS algorithm for a) x, and b) y coordinates for hologram 1000 mm × 7 mm and reconstruction distance z = 800 mm.

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The dependence of both SBPs on zcx and zcy is shown in Fig. 5(a). For illustrated case, the optimal SBPx’ = 373500 and SBPy’ = 9356 are obtained for zcx = 883.4 mm and zcy= 932.6 mm, respectively. Calculations of SBPx’, SBPy’ were repeated for different sizes of hologram and results are presented in Fib 5b). The presented plot shows that for all hologram sizes the most of extension goes for y direction. Moreover, the y component of spatial padding is illustrated in Fig. 5(b), which is main factor of SBP enlargement. Notably, the larger holograms are, the larger the zero padding; for example, for investigated hologram of size 1000 mm × 7 mm the SBP enlargement factors are marked by dashed vertical line in Fig. 5(b) and are equal to 1.28 and 4.54 for x and y directions, respectively. For a half meter longer hologram (1500 mm x 7 mm) the SBP must be enlarged much more: 1.8 and 14.7 times for x and y directions, respectively. Especially SBP growth in y direction is considerable, where its largest component, that is spatial zero-padding for y, equals 10.7.

 figure: Fig. 5.

Fig. 5. a) SBPx,y’ as a function of zcx,y, for hologram 1000 mm × 7 mm and reconstruction distance z = 800 mm. b) SBP enlargement ratio in x and y directions in relation to hologram size.

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3.4 Split CSW-AS algorithm

Section 3.1 develops a four step CSW-AS algorithm involving two 2D FFT and two 1D FFT. It has been shown in former sections that computations of this algorithm involve multiplications with corresponding phase exponential functions that expands PS dimensions of the signal. Thus, accurate reconstruction with the CSW-AS algorithm requires space-frequency zero-padding before the first step of CSW-AS. Nevertheless, application of zero-padding generates an input signal that consumes a significant amount of memory. For large holograms, this problem is worsened in such a way that the input signal might not fit in the computer memory. To solve this problem, we have developed the split CSW-AS (CSW- SAS) algorithm which in additions features faster computations for larger holograms. In this new algorithm the spatial zero-padding for y, taking most of memory resources, is realized in step 3. Also, the signal of size SBPx’× SBPy’ is never allocated. These elements improve calculation efficiency enabling reconstruction of very large FHPO holograms.

CSW-SAS algorithm improves calculations for large FHPO holograms by dividing the FT of the signal g1 into P sub-signals on x-direction. The sub-matrix has size SBPx’/P. Then each sub-signal is processed separately. The first element of the processing is frequency padding for y to SBPy’, which is performed using frequency interpolation. The frequency dimension of this sub-matrix corresponds to spatial size of the input signal in y direction. Thus, this operation is equivalent to the spatial zero-padding at input. Interpolated matrix is processed by step 2 and 3 of the CSW-AS algorithm. The important part of the CSW-SAS algorithm is a tiling process that reduces the signal SBP back to the size By. This tiling process enables reduction of the sampling rate of the calculated diffracted field and it is based on the periodic property of the FFT that was proposed in Refs. [25,26]. This property enables reduction of required bandwidth, which avoids computing a FFT of full-sized vector. Thus, the CSW-SAS algorithm does not require allocation and processing of full-size signal. It has to be noted that reconstructed hologram using CSW-SAS has By’/By larger pixel pitch for y than the CSW-AS. Finally, Fig. 6 present the flux diagram of the CSW-SAS.

 figure: Fig. 6.

Fig. 6. Split CSW-AS scheme.

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3.5 Accuracy and speed test of the CSW-AS reconstruction method

Accuracy of the proposed algorithms is tested by reconstructing the hologram of eleven-point sources Pn distributed linearly from the center of image point P0(0, 0, zR) up to corner point P11=(−0.48Bx2, −0.48By2, zR), where zR = 800 mm. Propagated fields of point sources are calculated analytically and superimposed at the hologram plane. The size of the calculated hologram is 1500 mm. Calculation of the FHPO-DH and reconstruction of the CSW-AS and CSW-SAS are carried out in a PC equipped with Intel i9-9940X processor, 128 GB of RAM, without support of GPU. Maximum hologram size that can be processed with the CSW-AS implementation is 800 mm while the CSW-SAS is able to reconstruct the complex field from the 1500 mm hologram.

Cross-sections of reconstructed point P0 in x and y directions for CSW-AS and CSW-SAS are presented in Fig. 7(a). Notably, the larger hologram is, the higher the resolution in x direction. Full width at half maximum (FWHM) measures theoretical resolution in x direction for holograms of size of 800 mm and 1500 mm, which is given as0.716 µm and 0.469 µm, respectively. Average values of FWHMs of reconstructed points sources in x direction are 0.71 µm and 0.45 µm for hologram of size of 800 mm and 1500 mm, respectively. Notably, the obtained values for the FWHMs are very close to the corresponding theoretical values, which is an expected result. Cross-sections for direction y require comment. At first glance, reconstruction of smaller hologram seems to have better resolution, but it has larger sampling rate, not resolution, due to spatial applied y zero-padding of basic implementation of CSW-AS. Thus, physical resolution for y direction of both holograms is the same. Accuracy of calculations across the field is presented in Fig. 7(b), where normalized maximum amplitude of 11-point sources is presented. Amplitude drops for the most off-axis point P11 is 0.9907 and 0.9229 for Bx1 = 800 mm and Bx1 = 1500 mm, respectively. Amplitude for other off axis simulated points are close to 1 (loss of amplitude is <1%) what means the proposed reconstruction process is aliasing free in the image space. Moreover, it is clearly visible that tiling strategy, which requires interpolation, does not introduce significant errors. Reconstruction time for CSW-AS and CSW-SAS algorithms was measured, and results are presented in Fig. 7(c). Up to hologram of size Bx1 = 500 mm, the more efficient method is the CSW-AS, where reconstruction took 113 s against the 130 s for the CSW-SAS. For holograms larger than 500 mm, performing multiple small FFTs accelerates the reconstruction process, i.e., for hologram of size of 800 mm the CSW-AS took 294.1 s and CSW-SAS 234.5 s respectively, which means ∼21% faster computation. For holograms of size over 800 mm wide, use of CSW- SAS is the only option due to PC memory limitations.

 figure: Fig. 7.

Fig. 7. a) Cross-sections of points Pc in x and y direction. b) Normalized amplitude of reconstructed points in relation to their position along image space diagonal. c) Speed test.

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According to [27], the root mean square of a wavefront aberration of a spherical wave is proportional to the peak intensity. Thus, by analyzing variations of the peak intensity at the focus plane as a function of the transverse position of the point source, it is possible to compare the value of wavefront aberrations encoded in the calculated focus intensity. This allows us to investigate the spatial dependence of computational accuracy [13].

4. Fast and accurate calculation of FCGHs

One class of CGH, including WRP and layer method, is based on multiple propagations in the object space [18,28]. However, propagation-based methods are not optimal solution for synthetizing Fourier CGH (FCGH) since they do not offer the best solution to calculate the field of objects larger than the size of the hologram [29]. Moreover, optical field propagation through object volume requires dynamics parameters in sampling and size. These two effects are challenging for rigorous propagation algorithms. PAS algorithms do not have these restrictions since they are based solely on summation of point source responses at the hologram plane.

CGH calculation with the PAS consists in dividing the hologram area in segments of size Ns × Ns. At each segment, PAS approximates the spherical wavefront that comes from the i-th point source of the object with a local tilted plane wave [11]. The tilt is estimated according to the value of the local instantaneous frequency at the center of the segment $(x_{mn}^c,y_{mn}^c)$, where n = 1, …, N, m = 1, …, M indicate the position of the segment over the horizontal and vertical directions, respectively. Typically sizes of the segment are taken among 8 and 32 for minimizing the wavefront approximation error [11,22]. Thus, PAS takes relatively large computational time when calculating large CGHs.

Since frequency information into each segment is coded according to the local approximation of the tilted plane, the PAS can calculate the diffracted field of objects larger than the hologram size. This property fits perfectly to the FHPO approach. However, calculation of CGHs with classical PAS-based algorithms require having absolute value of local instantaneous frequencies smaller than 0.5Δ−1, otherwise aliasing will be introduced. Thus, the concept of Fourier holography is needed for calculating efficiently the FHPO hologram. With this consideration, the term OR for a given segment mn can be calculated as

$$O_R^{mn} = \sum\limits_{p = 1}^P {\frac{{{a_p}}}{{{r_{pmn}}}}} \textrm{exp} [{ik({{r_{pmn}} - {r_{mn}}} )} ]\textrm{exp} ({ik[{{{\hat{f}}_{pxmn}}({{x_1} - x_{mn}^c} )+ {{\hat{f}}_{pymn}}({{y_1} - y_{mn}^c} )} ]} ),$$
where rpmn is the distance from p-th point source to the center of the segment mn, and
$${\hat{f}_{pxmn}} = \left\lfloor {\frac{1}{{\lambda \Delta f}}\left( {\frac{{{x_1} - x_{mn}^c}}{{{r_{pmn}}}} - \frac{{x_{mn}^c}}{{{r_{mn}}}}} \right)} \right\rfloor \Delta f,$$
$${\hat{f}_{pymn}} = \left\lfloor {\frac{1}{{\lambda \Delta f}}\left( {\frac{{{y_1} - y_{mn}^c}}{{{r_{pmn}}}} - \frac{{y_{mn}^c}}{{{r_{mn}}}}} \right)} \right\rfloor \Delta f,$$
are the corresponding discretized local spatial frequencies, Δf = 1/NsΔ is the frequency sampling and $\lfloor. \rfloor $ is the rounding operation. Note that Eq. (17) includes, unlike the classical PAS, the subtraction between the reference and point source distances to the center of the segment. This is done for matching the phase difference. Moreover, the second term into the parenthesis of Eq. (18) and (19) is a constant in classical PAS [11,22].

Equation (17) can be efficiently computed with the help of the FT [11]. When representing Eq. (17) in the frequency domain, the contained information consists of sets of Dirac deltas. The position of the corresponding Dirac deltas is given by Eq. (18) and (19). After placing all the Dirac deltas for the 3D point distribution, the inverse FT is applied. The employment of a spherical reference into the PAS brings two important advantages: (i) the absolute value of local frequencies is much smaller than 0.5Δ−1 even for segments that are far from the center of the hologram plane, (ii) it allows choosing larger segment sizes since the synthetized OR term has a much smaller curvature. Therefore, this Spherical-PAS (S-PAS) allows synthetizing large FCGH in one direction.

Discretization of the instantaneous local frequencies given by Eq. (18) and (19) introduces a truncation error [12], which depends on the segment size. Thus, the larger is the segment, the smaller the error in the approximation. However, arbitrarily large segment cannot be chosen since the wavefront at the hologram plane may not be well represented. Thus, proper size of the segment is a kind of compromise for PAS. Accurate PAS (A-PAS) overcomes the discretization problem while fulfilling the sampling theory. The A-PAS considers that the segment Ns × Ns is enlarged in the discrete frequency domain by a factor of q, where q is an integer. In this way, the sampling is increased q-times, which allows better approximation of the discrete values of the frequencies to their respective continuous values. After applying the inverse FT to the enlarged segment, only a segment of size Ns × Ns is taken. Naturally, the benefits of the A-PAS can be introduced into the S-PAS for obtaining the improved SA-PAS. In this way, the SA-PAS reduces the discretization error and consequently improve the accuracy of the hologram.

4.1 Accuracy and speed test of the SA-PAS

Speed of calculation with SA-PAS algorithm is improved when using large segments sizes and smaller q parameter, but the wavefront, despite high accuracy of discrete local frequency calculations, will be ill-represented. In contrast, good representation of the wavefront requires small size of segment and large q factor to maintain the accuracy of the discrete local instantaneous frequency calculations, but the price to pay is computation time. Because of that trade-off behavior, speed and accuracy tests of the SA-PAS algorithm are carried out to find the optimal Ns and q parameters. In order to test the accuracy of the SA-PAS, one meter FCGH of a point source at the position P11 = (−0.48Bx2, −0.48By2, zR), where zR = 800 mm, is calculated. Moreover, SA-PAS has been implemented with the parallel computing toolbox from MATLAB. Reconstruction of the FCGH requires multiplication by the reference field R and then the obtained field is numerically reconstructed with the CSW-SAS. Several values for Ns and q were employed and results are shown in Fig. 8(a). The normalized value of maximum amplitude of reconstructed point P11 respect to P0 is a measure of accuracy.

 figure: Fig. 8.

Fig. 8. a) Normalized reconstructed amplitude of the point source P11 for different Ns and q of an FCGH. b) Execution time of FCGH of 100k points for different segment Ns and increase factors q.

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As shown in Fig. 8(a), the maximum amplitude of the recovered point source is below 0.8 when using the original segment size (q = 1), which means that efficiency of the SA-PAS is not high. This situation can be corrected by increasing the value of q. When using large values of q (q>2), the value of the maximum amplitude is near to one. However, this behavior is not shared for all Ns. When using Ns= 1024 the maximum amplitude of the reconstructed point drops below 0.8 independently of the employed q. Hence, the employment of too large segment produces a wavefront representation that is not accurate. From the accuracy test, it is found that segment sizes up to 512 allow accurate calculation of FCGHs.

Computation efficiency is a primary concern, and hence, speed test of the algorithm is performed. In the speed test instead of the single point, 100k points randomly distributed in image space are taken. The geometry of calculated FCGHs is the same as in the accuracy test. The results illustrating calculation times for segment sizes from 32 to 1024 and several q factors are presented in Fig. 8(b). The total computation time of hologram with SA-PAS is composed of two parts: points processing and FFT. The points processing is made of scalar calculations, thus its computation time per segment is almost constant regardless the segment size. However, the computation time of FFT grows with the segment size, for Ns below 256, the FFT is very fast and in its contribution to the total time is negligible. Thus, the q has no significant effect on computation speed. The q parameter starts to be significant for larger segments, when the execution time of FFTs grows becoming comparable to the points processing time. From the accuracy and speed tests, we find that the optimal set of parameters is Ns = 512 with q = 2 or 3.

5. Registration and reconstruction of FHPO holograms with the CSW-AS

This section illustrates the visual aspects of the introduced FHPO computer and digital holography, combining hologram generation with the SA-PAS, numerical (using CSW-AS) and optical reconstruction. In the numerical reconstruction experiment, two aspects are investigated. First, it compares quality of reconstructed digital and computer holograms, of size 0.26 m. The second part of numerical experiment shows visual effects obtained from reconstruction of very large holograms, up to 1.5 m. The optical experiment shows the possibilities of wide-angle observation of object in the Fourier holographic display configuration.

5.1 Registration of FHPO holograms

Experimental FHPO-DH is registered by the Fourier synthetic aperture digital holographic system described in Ref. [23]. Employed camera has a resolution of 2448 × 2050 with pixel size Δ = 3.45 µm. The employed object for experimental capturing of the FHPO-DH is a reindeer figurine of dimensions 110 mm (height) × 110 mm (depth) × 60 mm (width). The reference source is zR = 770 mm distant from the CCD camera, which gives an image space of size Bx2 = 112 mm. Dimension of registered FHPO-DH are 258 × 7 mm2, which is equivalent to 74872 × 2048 pixels2.

FCGH does not have the limitations of physical capturing setup, it can be much larger. SA-PAS is used for calculating the FCGH with the parameters Ns = 512, q = 2, Δ = 3.45 µm and zR = 800 mm, which results in Bx2 = 123 mm. Here, the maximum size of the calculated FCGH is 1500 × 7 mm2 (434176 × 2048 pixels2), two smaller holograms of size 254 × 7 mm2 (73728 × 2048 pixels2) and 500 × 7 mm2 (145408 × 2048 pixels2), which are center part of large hologram, are also investigated. Note that the smallest FCGH is comparable in size to the experimental FHPO-DH. The employed virtual object is a point cloud of a gargoyle figurine of size 117 mm (height) × 54 mm (depth) × 84 mm (width). Used point cloud represents only the front view of the gargoyle, which consists in 4.4M points and has density of 253 points/mm2. Required time for calculation of the corresponding FCGH with the SA-PAS algorithm is 4132 s. Notably, computation time for FCGHs for the employed cloud of points varies linearly with the size of the hologram. Hence, calculation of smaller FCGHs takes 734 s and 1450 s for hologram 254 mm and 500 mm long, respectively.

5.2 Numerical reconstruction of FHPO holograms

Reconstructions of the smaller FCGH and FHPO-DH are carried out with the CSW-AS, described in section 3.1, which means that no space-frequency zero-padding was applied. Reconstructions without padding operations can be applied for holograms of width smaller than 500mm. Reconstruction distances for the FHPO-DH are selected as z = 714.8, 718.8, 724 mm. The results are presented in Fig. 9(a)-(d). Figure 9(a) shows the reconstructed object when using the first distance z = 714.8 mm. This distance allows obtaining a sharp image of the left horn, which is shown by the area enclosed with red box. This area is zoom in Fig. 9(b) for better observation. Second distance brings to focus the region of the left eye and parts of the forehead, as seen in Fig. 9(c). Finally, Fig. 9(d) presents a sharp image of the ears and a shadow on the right horn that are obtained when applying the third reconstruction distance. These reconstructions were completed in 22 s each.

 figure: Fig. 9.

Fig. 9. CSW-AS reconstruction results. a) FHPO-DH of a Reindeer figurine and zooms of b) horn tip for z = 714.8 mm, c) eye for z = 818.8 mm, d) ear and casting shadow for z = 724 mm. e) FCGH of a Gargoyle with zooms of f) eye lid region for z = 808.3 mm and g) left ear for z = 819.5 mm. For comparing the through focus reconstruction of the object see Visualization 1.

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For reconstruction of the FCGH selected distances are z = 808.3 mm and 819.5 mm. Figure 9(e) presents whole reconstructed object when using the first distance. This distance allows obtaining a sharp image of the gargoyle eye lid and neighboring face regions. This area is marked with red box in Fig. 9(e) and zoomed in Fig. 9(f). When applying the second distance, the right ear of the gargoyle is focused (Fig. 9(g). Notably, reconstructions of DH and FCGH are aliasing-free, despite not using the space-frequency padding scheme. This is a result of applying the described CSW-AS compression schemes (see Fig. 11 in Ref. [10] for comparison). Notably, calculation of a FCGH hologram with a dense point clouds allows obtaining visual effects of the virtual object similar to the real object encoded by the DH. Finally, reconstruction of both object along the z axis is presented in Visualization 1. Notably, the reconstruction process through focus does not shows evidence of aliasing.

Second part of numerical experiment investigates reconstruction of very large FCGHs of width 500 mm and 1500 mm. Reconstruction of both holograms requires space-frequency zero-padding for suppression of aliasing artifacts, and thus, CSW-SAS is employed. It is worth noting that the maximum hologram size that the multi FFT algorithm can process is 445 mm x 7mm. Thus, the multi-FFT is unable to process the holograms studied in this section. Reconstruction distance is set to z = 808.3 mm, which allows obtaining a sharp view of the gargoyle eye lid. Figure 10(a)-(b) present the reconstruction of the whole gargoyle for FCGHs of 500 mm and 1500 mm, respectively. Notably, these reconstructions show that the larger is the horizontal size of the CGHs, the smaller the depth of focus. When comparing these figures, the increasing of the blur effect, which occurs on object parts placed out of focus plane, becomes stronger due to the small depth of focus. Decrease of depth of focus can be easily observed when comparing zooms of sharply reconstructed parts presented in Fig. 10(c)-(d). For the 1500 mm long hologram, blur becomes very strong overlapping the sharply reconstructed parts, which causes loss of contrast. Nevertheless, it can be observed that the sharp slice of gargoyle is much smaller for 1.5 m hologram than for the 0.5 m hologram. Investigated FCGHs of width 254 mm, 500 mm, and 1500 mm have the depths of focus δz = 21 µm, 5.4 µm and 0.6 µm, respectively. The dependence of the depth of focus regarding the size of the reconstructed hologram is presented in Visualization 2, where size of the reconstructed hologram changes in the range of 7-1500 mm in length. Notably, the larger the hologram is, the thinner the slice where the object can be sharply reconstructed. However, the blurring of regions out of focus increases as well, which results in a loss of contrast. Reconstruction time with the CSW-SAS algorithm is 58s, 122 s and 1190 s for 254 mm, 500 mm and 1500 mm long hologram, respectively.

 figure: Fig. 10.

Fig. 10. Reconstruction results of large FCGH a) 500 mm and b) 1500 mm. c-d) Respectively zooms of sharp regions lying in focus plane.

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5.3 Optical reconstruction of the FCGH

In this section, optical reconstructions of the FCGH with the Fourier holographic display [6] with laser source (λ = 0.64 µm) and 4K SLM (HoloEye GAEA-2, 2160×3840 pixels, pixel pitch Δ = 3.74 µm) giving a viewing aperture of physical size 8×14.3 mm2 are shown. The scheme of the system is presented in Fig. 11. Detailed description of the display can be found in Ref. [6]. In the display, the 1.5 m hologram discussed in section 5.1 is optically reconstructed. The gargoyle object is reconstructed at the distance R = 800 mm from the viewing window (VW).

 figure: Fig. 11.

Fig. 11. Fourier holographic display scheme. HW – halfwave plate, MO – micro-objective, PH – pinhole, Lc - collimating lens (Fc = 300 mm), BS – beamsplitter, L1 (F1 = 100 mm), L2 (F2 = 600 mm), Lf (Ff = 600 mm) – field lens.

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Shifting the aperture along FCGH allows reconstructing the full spectrum of object perspectives. For the optical experiment, three positions of the aperture were chosen: in the center (0°), on the left and right ends of hologram (±42°). Figure 12(a) presents numerical reconstruction of center view. For coherent light source and small aperture, a strong speckle noise is inherent part of the reconstruction, as shown in Fig. 12(a). To suppress speckle noise in optical experiment a time-multiplexing technique is applied [30]. Results of optical reconstructions captured with zoom lens camera are presented in Fig. 12(b)-(d). For all views, reconstructed object has high quality, and its details are well visible. Although, for the left and right views of gargoyle slight loss of sharpness can be observed. The reason for this is that geometrically the distance between object and hologram plane is much larger for boundary views than for the middle view. As result, object occupies smaller area of the available hologram SBP, and thus, loss of higher frequencies is inevitable. Visualization 3 presents numerical reconstructions of all views of the gargoyle from −42° to +42°.

 figure: Fig. 12.

Fig. 12. Experimental angle view (a) numerical and (b) optical reconstructions of center view (0°) and optical reconstructions of (c) left end (-42°) and (d) right end (+42°) view.

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Notably, there are differences in wavelength λ and pixel pitch Δ used for hologram registration, and optical reconstruction. The discrepancies are handled by the holographic image processing using the display geometry, which leads to image magnification [6].

6. Conclusions

In this work, the imaging chain for registration and reconstruction of very large FHPO holograms has been provided. For the registration part, the paper proposed generation of very large FHPO-DH by using the concept of Fourier holography. The first advantage is the possibility of registration of objects of large vertical size. The second is large viewing angle that can be obtained with large pixel pitch. The next important advantage is given by the SA-PAS that enables use of segments of large size for generation of FHPO holograms without compromising accuracy. It is shown that the size of the segment used by the SA-PAS is sixteen times larger than the typical size employed by the standard PAS. This provides hologram generation 260 times faster than the standard PAS, as shown in Table 1. Consequently, fast calculation of very large FHPO holograms from very dense cloud of points can be carried out. This speed of calculation has not been shown by any know hologram generator algorithm. Finally, the SA-PAS enables calculating FHPO holograms up to 1500 mm with pixel pitch Δ = 3.45 µm providing 84° angle view.

Tables Icon

Table 1. Processing time for generation and reconstruction of FCGHs.

For reconstruction of the holographic data, this work has proposed the CSW-AS method. The CSW-AS enables reconstruction of holographic data from the synthetic aperture DH system and the SA-PAS. In previous publication [10], the multi-FFT AS was proposed for retrieving information from FHPO holograms. However, the maximum size that can be processed by the multi-FFT AS is 440 mm x 7 mm. The CSW AS easily overcomes this limit while keeping a low usage of the SBP, as shown by the right side of Table 1. It is interesting to note that when reconstructing the largest FCGH, the obtained depth of focus is around 0.6 µm. Another advantage of developed algorithm is that accurate reconstruction can be performed. It was found that accurate reconstruction with the CSW-AS is limited to FCGH of size up to 800 mm x 7 mm. To break this limit, the CSW-SAS algorithm was developed. The strategy of breaking down the hologram into small pieces and tiling summation in y direction allows accurate wavefield reconstructions of very large FCGH (up to 1500 mm x 7 mm). Finally, the FCGH of size 1500 mm x 7 mm allows object reconstruction that can be viewed from many directions.

Funding

Politechnika Warszawska (Statutory funds); Narodowe Centrum Nauki (UMO-2018/31/B/ST7/02980).

Acknowledgements

The authors of this work are thankful to Weronika Finke for providing the reindeer hologram. We would like to acknowledge the support of the statutory funds of Warsaw University of Technology.

Disclosures

The authors declare no conflicts of interest.

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28. Y. Zhao, L. Cao, H. Zhang, D. Kong, and G. Jin, “Accurate calculation of computer-generated holograms using angular-spectrum layer-oriented method,” Opt. Express 23(20), 25440 (2015). [CrossRef]  

29. J. Weng, T. Shimobaba, N. Okada, H. Nakayama, M. Oikawa, N. Masuda, and T. Ito, “Generation of real-time large computer generated hologram using wavefront recording method,” Opt. Express 20(4), 4018 (2012). [CrossRef]  

30. B. Lee, D. Yoo, J. Jeong, S. Lee, D. Lee, and B. Lee, “Wide-angle speckleless DMD holographic display using structured illumination with temporal multiplexing,” Opt. Lett. 45(8), 2148 (2020). [CrossRef]  

References

  • View by:

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  26. M. Hillenbrand, A. Hoffmann, D. P. Kelly, and S. Sinzinger, “Fast nonparaxial scalar focal field calculations,” J. Opt. Soc. Am. A 31(6), 1206–1214 (2014).
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  27. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 518–522, Chap. 9.
  28. Y. Zhao, L. Cao, H. Zhang, D. Kong, and G. Jin, “Accurate calculation of computer-generated holograms using angular-spectrum layer-oriented method,” Opt. Express 23(20), 25440 (2015).
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  30. B. Lee, D. Yoo, J. Jeong, S. Lee, D. Lee, and B. Lee, “Wide-angle speckleless DMD holographic display using structured illumination with temporal multiplexing,” Opt. Lett. 45(8), 2148 (2020).
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2020 (2)

2019 (1)

2018 (3)

2017 (1)

M. Yamaguchi, “Full-Parallax Holographic Light-Field 3-D Displays and Interactive 3-D Touch,” Proc. IEEE 105(5), 947–959 (2017).
[Crossref]

2016 (4)

T. Kozacki and K. Falaggis, “Angular spectrum method with compact space–bandwidth: generalization and full-field accuracy,” Appl. Opt. 55(19), 5014 (2016).
[Crossref]

T. Shimobaba, T. Kakue, and T. Ito, “Review of Fast Algorithms and Hardware Implementations on Computer Holography,” IEEE Trans. Ind. Inf. 12(4), 1611–1622 (2016).
[Crossref]

H. Kang, E. Stoykova, and H. Yoshikawa, “Fast phase-added stereogram algorithm for generation of photorealistic 3D content,” Appl. Opt. 55(3), A135 (2016).
[Crossref]

A. Gołoś, W. Zaperty, G. Finke, P. Makowski, and T. Kozacki, “Fourier RGB synthetic aperture color holographic capture for wide angle holographic display,” Opt. Photonics Inf. Process. X 9970, 99701E (2016).
[Crossref]

2015 (3)

2014 (1)

2012 (2)

J. Weng, T. Shimobaba, N. Okada, H. Nakayama, M. Oikawa, N. Masuda, and T. Ito, “Generation of real-time large computer generated hologram using wavefront recording method,” Opt. Express 20(4), 4018 (2012).
[Crossref]

C. Tang and B. Yao, “Fast computation for generating CGH of a 3D object by employing connections between layers,” J. Mod. Opt. 59(16), 1406–1409 (2012).
[Crossref]

2010 (1)

2009 (2)

2008 (1)

2007 (1)

H. Kang, “Compensated phase-added stereogram for real-time holographic display,” Opt. Eng. 46(9), 095802 (2007).
[Crossref]

2002 (1)

S. Paśko and R. Jóźwicki, “Novel Fourier approach to digital holography,” Opto-electronics Rev. 10, 80822E (2002).
[Crossref]

2001 (1)

D. Dirksen, H. Droste, B. Kemper, H. Deleré, M. Deiwick, H. H. Scheld, and G. Von Bally, “Lensless Fourier holography for digital holographic interferometry on biological samples,” Opt. Lasers Eng. 36(3), 241–249 (2001).
[Crossref]

1993 (1)

M. E. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 2(1), 28 (1993).
[Crossref]

Baba, T.

Blinder, D.

D. Blinder and T. Shimobaba, “Efficient algorithms for the accurate propagation of extreme-resolution holograms,” Opt. Express 27(21), 29905 (2019).
[Crossref]

A. Symeonidou, D. Blinder, B. Ceulemans, A. Munteanu, and P. Schelkens, “Three-dimensional rendering of computer-generated holograms acquired from point-clouds on light field displays,” in Proceeding of SPIE (2016), Vol. 9971, p. 99710S.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 518–522, Chap. 9.

Cao, L.

Ceulemans, B.

A. Symeonidou, D. Blinder, B. Ceulemans, A. Munteanu, and P. Schelkens, “Three-dimensional rendering of computer-generated holograms acquired from point-clouds on light field displays,” in Proceeding of SPIE (2016), Vol. 9971, p. 99710S.

Chlipala, M.

Deiwick, M.

D. Dirksen, H. Droste, B. Kemper, H. Deleré, M. Deiwick, H. H. Scheld, and G. Von Bally, “Lensless Fourier holography for digital holographic interferometry on biological samples,” Opt. Lasers Eng. 36(3), 241–249 (2001).
[Crossref]

Deleré, H.

D. Dirksen, H. Droste, B. Kemper, H. Deleré, M. Deiwick, H. H. Scheld, and G. Von Bally, “Lensless Fourier holography for digital holographic interferometry on biological samples,” Opt. Lasers Eng. 36(3), 241–249 (2001).
[Crossref]

Dirksen, D.

D. Dirksen, H. Droste, B. Kemper, H. Deleré, M. Deiwick, H. H. Scheld, and G. Von Bally, “Lensless Fourier holography for digital holographic interferometry on biological samples,” Opt. Lasers Eng. 36(3), 241–249 (2001).
[Crossref]

Droste, H.

D. Dirksen, H. Droste, B. Kemper, H. Deleré, M. Deiwick, H. H. Scheld, and G. Von Bally, “Lensless Fourier holography for digital holographic interferometry on biological samples,” Opt. Lasers Eng. 36(3), 241–249 (2001).
[Crossref]

Falaggis, K.

Finke, G.

A. Gołoś, W. Zaperty, G. Finke, P. Makowski, and T. Kozacki, “Fourier RGB synthetic aperture color holographic capture for wide angle holographic display,” Opt. Photonics Inf. Process. X 9970, 99701E (2016).
[Crossref]

Finke, W.

Futterer, G.

S. Reichelt, R. Haussler, N. Leister, G. Futterer, H. Stolle, and A. Schwerdtner, “Holographic 3-D Displays - Electro-holography within the Grasp of Commercialization,” Adv. Lasers Electro Opt. (2010).

Golos, A.

A. Gołoś, W. Zaperty, G. Finke, P. Makowski, and T. Kozacki, “Fourier RGB synthetic aperture color holographic capture for wide angle holographic display,” Opt. Photonics Inf. Process. X 9970, 99701E (2016).
[Crossref]

Graulig, C.

Haussler, R.

S. Reichelt, R. Haussler, N. Leister, G. Futterer, H. Stolle, and A. Schwerdtner, “Holographic 3-D Displays - Electro-holography within the Grasp of Commercialization,” Adv. Lasers Electro Opt. (2010).

Hayasaki, Y.

Hillenbrand, M.

Hoffmann, A.

Huang, F. C.

L. Shi, F. C. Huang, W. Lopes, W. Matusik, and D. Luebke, “Near-eye light field holographic rendering with spherical waves for wide field of view interactive 3D computer graphics,” ACM Trans. Graph. 36 (2017).

Igarashi, S.

Ito, T.

Jackin, B. J.

Jeong, J.

Jin, G.

Józwicki, R.

S. Paśko and R. Jóźwicki, “Novel Fourier approach to digital holography,” Opto-electronics Rev. 10, 80822E (2002).
[Crossref]

Kakue, T.

T. Shimobaba, T. Kakue, and T. Ito, “Review of Fast Algorithms and Hardware Implementations on Computer Holography,” IEEE Trans. Ind. Inf. 12(4), 1611–1622 (2016).
[Crossref]

Kang, H.

Kanka, M.

Kelly, D. P.

Kemper, B.

D. Dirksen, H. Droste, B. Kemper, H. Deleré, M. Deiwick, H. H. Scheld, and G. Von Bally, “Lensless Fourier holography for digital holographic interferometry on biological samples,” Opt. Lasers Eng. 36(3), 241–249 (2001).
[Crossref]

Kong, D.

Kozacki, T.

Kukolowicz, R.

T. Kozacki, J. Martinez-Carranza, R. Kukolowicz, and W. Finke, “Accurate reconstruction of horizontal parallax-only holograms by angular spectrum and efficient zero-padding,” Appl. Opt. 59(27), 8450 (2020).
[Crossref]

T. Kozacki, J. Martinez-Carranza, and R. Kukołowicz, “Numerical reconstruction of large HPO Fourier holograms,” in Proc. of SPIE (2020), pp. 1–10.

Lee, B.

Lee, D.

Lee, S.

Leister, N.

S. Reichelt, R. Haussler, N. Leister, G. Futterer, H. Stolle, and A. Schwerdtner, “Holographic 3-D Displays - Electro-holography within the Grasp of Commercialization,” Adv. Lasers Electro Opt. (2010).

Lopes, W.

L. Shi, F. C. Huang, W. Lopes, W. Matusik, and D. Luebke, “Near-eye light field holographic rendering with spherical waves for wide field of view interactive 3D computer graphics,” ACM Trans. Graph. 36 (2017).

Lucente, M. E.

M. E. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 2(1), 28 (1993).
[Crossref]

Luebke, D.

L. Shi, F. C. Huang, W. Lopes, W. Matusik, and D. Luebke, “Near-eye light field holographic rendering with spherical waves for wide field of view interactive 3D computer graphics,” ACM Trans. Graph. 36 (2017).

Makowski, P.

A. Gołoś, W. Zaperty, G. Finke, P. Makowski, and T. Kozacki, “Fourier RGB synthetic aperture color holographic capture for wide angle holographic display,” Opt. Photonics Inf. Process. X 9970, 99701E (2016).
[Crossref]

Makowski, P. L.

Martinez-Carranza, J.

T. Kozacki, J. Martinez-Carranza, R. Kukolowicz, and W. Finke, “Accurate reconstruction of horizontal parallax-only holograms by angular spectrum and efficient zero-padding,” Appl. Opt. 59(27), 8450 (2020).
[Crossref]

T. Kozacki, J. Martinez-Carranza, and R. Kukołowicz, “Numerical reconstruction of large HPO Fourier holograms,” in Proc. of SPIE (2020), pp. 1–10.

Masuda, N.

Matsushima, K.

Matusik, W.

L. Shi, F. C. Huang, W. Lopes, W. Matusik, and D. Luebke, “Near-eye light field holographic rendering with spherical waves for wide field of view interactive 3D computer graphics,” ACM Trans. Graph. 36 (2017).

Munteanu, A.

A. Symeonidou, D. Blinder, B. Ceulemans, A. Munteanu, and P. Schelkens, “Three-dimensional rendering of computer-generated holograms acquired from point-clouds on light field displays,” in Proceeding of SPIE (2016), Vol. 9971, p. 99710S.

Nakahara, S.

Nakamura, T.

Nakayama, H.

Ohkawa, T.

Oikawa, M.

Okada, N.

Ootsu, K.

Pasko, S.

S. Paśko and R. Jóźwicki, “Novel Fourier approach to digital holography,” Opto-electronics Rev. 10, 80822E (2002).
[Crossref]

Reichelt, S.

S. Reichelt, R. Haussler, N. Leister, G. Futterer, H. Stolle, and A. Schwerdtner, “Holographic 3-D Displays - Electro-holography within the Grasp of Commercialization,” Adv. Lasers Electro Opt. (2010).

Riesenberg, R.

Scheld, H. H.

D. Dirksen, H. Droste, B. Kemper, H. Deleré, M. Deiwick, H. H. Scheld, and G. Von Bally, “Lensless Fourier holography for digital holographic interferometry on biological samples,” Opt. Lasers Eng. 36(3), 241–249 (2001).
[Crossref]

Schelkens, P.

A. Symeonidou, D. Blinder, B. Ceulemans, A. Munteanu, and P. Schelkens, “Three-dimensional rendering of computer-generated holograms acquired from point-clouds on light field displays,” in Proceeding of SPIE (2016), Vol. 9971, p. 99710S.

Schwerdtner, A.

S. Reichelt, R. Haussler, N. Leister, G. Futterer, H. Stolle, and A. Schwerdtner, “Holographic 3-D Displays - Electro-holography within the Grasp of Commercialization,” Adv. Lasers Electro Opt. (2010).

Shi, L.

L. Shi, F. C. Huang, W. Lopes, W. Matusik, and D. Luebke, “Near-eye light field holographic rendering with spherical waves for wide field of view interactive 3D computer graphics,” ACM Trans. Graph. 36 (2017).

Shimobaba, T.

Sinzinger, S.

Stolle, H.

S. Reichelt, R. Haussler, N. Leister, G. Futterer, H. Stolle, and A. Schwerdtner, “Holographic 3-D Displays - Electro-holography within the Grasp of Commercialization,” Adv. Lasers Electro Opt. (2010).

Stoykova, E.

Symeonidou, A.

A. Symeonidou, D. Blinder, B. Ceulemans, A. Munteanu, and P. Schelkens, “Three-dimensional rendering of computer-generated holograms acquired from point-clouds on light field displays,” in Proceeding of SPIE (2016), Vol. 9971, p. 99710S.

Tang, C.

C. Tang and B. Yao, “Fast computation for generating CGH of a 3D object by employing connections between layers,” J. Mod. Opt. 59(16), 1406–1409 (2012).
[Crossref]

Von Bally, G.

D. Dirksen, H. Droste, B. Kemper, H. Deleré, M. Deiwick, H. H. Scheld, and G. Von Bally, “Lensless Fourier holography for digital holographic interferometry on biological samples,” Opt. Lasers Eng. 36(3), 241–249 (2001).
[Crossref]

Watanabe, S.

Weng, J.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 518–522, Chap. 9.

Wuttig, A.

Yamaguchi, M.

Yamaguchi, T.

Yao, B.

C. Tang and B. Yao, “Fast computation for generating CGH of a 3D object by employing connections between layers,” J. Mod. Opt. 59(16), 1406–1409 (2012).
[Crossref]

Yatagai, T.

Yokota, T.

Yoo, D.

Yoshikawa, H.

Zaperty, W.

A. Gołoś, W. Zaperty, G. Finke, P. Makowski, and T. Kozacki, “Fourier RGB synthetic aperture color holographic capture for wide angle holographic display,” Opt. Photonics Inf. Process. X 9970, 99701E (2016).
[Crossref]

P. L. Makowski, T. Kozacki, P. Zdankowski, and W. Zaperty, “Synthetic aperture Fourier holography for wide- angle holographic display of real scenes,” Appl. Opt. 54(12), 3658–3665 (2015).
[Crossref]

Zdankowski, P.

Zhang, H.

Zhao, Y.

Appl. Opt. (7)

IEEE Trans. Ind. Inf. (1)

T. Shimobaba, T. Kakue, and T. Ito, “Review of Fast Algorithms and Hardware Implementations on Computer Holography,” IEEE Trans. Ind. Inf. 12(4), 1611–1622 (2016).
[Crossref]

J. Electron. Imaging (1)

M. E. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 2(1), 28 (1993).
[Crossref]

J. Mod. Opt. (1)

C. Tang and B. Yao, “Fast computation for generating CGH of a 3D object by employing connections between layers,” J. Mod. Opt. 59(16), 1406–1409 (2012).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

H. Kang, “Compensated phase-added stereogram for real-time holographic display,” Opt. Eng. 46(9), 095802 (2007).
[Crossref]

Opt. Express (5)

Opt. Lasers Eng. (1)

D. Dirksen, H. Droste, B. Kemper, H. Deleré, M. Deiwick, H. H. Scheld, and G. Von Bally, “Lensless Fourier holography for digital holographic interferometry on biological samples,” Opt. Lasers Eng. 36(3), 241–249 (2001).
[Crossref]

Opt. Lett. (4)

Opt. Photonics Inf. Process. X (1)

A. Gołoś, W. Zaperty, G. Finke, P. Makowski, and T. Kozacki, “Fourier RGB synthetic aperture color holographic capture for wide angle holographic display,” Opt. Photonics Inf. Process. X 9970, 99701E (2016).
[Crossref]

Opto-electronics Rev. (1)

S. Paśko and R. Jóźwicki, “Novel Fourier approach to digital holography,” Opto-electronics Rev. 10, 80822E (2002).
[Crossref]

Proc. IEEE (1)

M. Yamaguchi, “Full-Parallax Holographic Light-Field 3-D Displays and Interactive 3-D Touch,” Proc. IEEE 105(5), 947–959 (2017).
[Crossref]

Other (5)

S. Reichelt, R. Haussler, N. Leister, G. Futterer, H. Stolle, and A. Schwerdtner, “Holographic 3-D Displays - Electro-holography within the Grasp of Commercialization,” Adv. Lasers Electro Opt. (2010).

A. Symeonidou, D. Blinder, B. Ceulemans, A. Munteanu, and P. Schelkens, “Three-dimensional rendering of computer-generated holograms acquired from point-clouds on light field displays,” in Proceeding of SPIE (2016), Vol. 9971, p. 99710S.

L. Shi, F. C. Huang, W. Lopes, W. Matusik, and D. Luebke, “Near-eye light field holographic rendering with spherical waves for wide field of view interactive 3D computer graphics,” ACM Trans. Graph. 36 (2017).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 518–522, Chap. 9.

T. Kozacki, J. Martinez-Carranza, and R. Kukołowicz, “Numerical reconstruction of large HPO Fourier holograms,” in Proc. of SPIE (2020), pp. 1–10.

Supplementary Material (3)

NameDescription
Visualization 1       This visualization presents reconstruction through-focus of small FHPO-DH and FCGH with the CSW-AS algorithm, respectively. The FHPO-DH is is registered by the Fourier synthetic aperture digital holographic system described in reference [23] while th
Visualization 2       Presented video shows the dependence of the depth of focus regarding the size of the reconstructed FHPO hologram. The video was generated by employing a FCGH with maximum size of 1500 × 7 mm. The size of the hologram to be reconstructed changes in
Visualization 3       This video presents the horizontal perspectives of the gargoyle object when placing an aperture of size of 7 x 7 mm over the FCGH. Horizontal displacement of the aperture along the FGCH allows to reconstruct the different perspectives of the gargoyle

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Figures (12)

Fig. 1.
Fig. 1. a) Geometry of the experiment for recording the FHPO-DH. b) Viewing zone for FHPO-DH of size 1000 mm x 7 mm and zR = 800 mm.
Fig. 2.
Fig. 2. PSR of a FHPO-DH of size 1000 mm x 7 mm and zR = 800 mm, Δ = 3.45 µm, λ = 0.532 µm. a) x direction. b) z direction.
Fig. 3.
Fig. 3. a) PSRs of Oc1 and g1 for x direction; b) PSRs of Oc1 and g2 for y direction and for two hologram cross sections.
Fig. 4.
Fig. 4. Illustration of PS coverage and space-frequency zero padding of CSW-AS algorithm for a) x, and b) y coordinates for hologram 1000 mm × 7 mm and reconstruction distance z = 800 mm.
Fig. 5.
Fig. 5. a) SBPx,y’ as a function of zcx,y, for hologram 1000 mm × 7 mm and reconstruction distance z = 800 mm. b) SBP enlargement ratio in x and y directions in relation to hologram size.
Fig. 6.
Fig. 6. Split CSW-AS scheme.
Fig. 7.
Fig. 7. a) Cross-sections of points Pc in x and y direction. b) Normalized amplitude of reconstructed points in relation to their position along image space diagonal. c) Speed test.
Fig. 8.
Fig. 8. a) Normalized reconstructed amplitude of the point source P11 for different Ns and q of an FCGH. b) Execution time of FCGH of 100k points for different segment Ns and increase factors q.
Fig. 9.
Fig. 9. CSW-AS reconstruction results. a) FHPO-DH of a Reindeer figurine and zooms of b) horn tip for z = 714.8 mm, c) eye for z = 818.8 mm, d) ear and casting shadow for z = 724 mm. e) FCGH of a Gargoyle with zooms of f) eye lid region for z = 808.3 mm and g) left ear for z = 819.5 mm. For comparing the through focus reconstruction of the object see Visualization 1.
Fig. 10.
Fig. 10. Reconstruction results of large FCGH a) 500 mm and b) 1500 mm. c-d) Respectively zooms of sharp regions lying in focus plane.
Fig. 11.
Fig. 11. Fourier holographic display scheme. HW – halfwave plate, MO – micro-objective, PH – pinhole, Lc - collimating lens (Fc = 300 mm), BS – beamsplitter, L1 (F1 = 100 mm), L2 (F2 = 600 mm), Lf (Ff = 600 mm) – field lens.
Fig. 12.
Fig. 12. Experimental angle view (a) numerical and (b) optical reconstructions of center view (0°) and optical reconstructions of (c) left end (-42°) and (d) right end (+42°) view.

Tables (1)

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Table 1. Processing time for generation and reconstruction of FCGHs.

Equations (19)

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H = | O | 2 + | R | 2 + O R + O R ,
O R = 1 r exp ( i k r ) p = 1 N a p r p exp ( i k r p ) ,
O c 1 ( x 1 , y 1 ) = O 1 ( x 1 , y 1 ) exp { i π x 1 2 λ z c x } ,
O 2 ( x 2 , y 2 ) = O c 2 ( x 2 , y 2 ) exp { i π x 2 2 λ z c y } ,
O ~ c 2 ( f x 2 , f y 2 ) exp { i π λ z c y f y 2 2 } = [ O ~ c 1 ( f x 2 , f y 2 ) exp { i π λ z c x f x 2 2 } ] × exp { i k z f z ( f x 2 , f y 2 ) } ,
f z ( f x , f y ) = 1 λ 2 f x 2 λ 2 f y 2 ,
g 1 ( f x 1 , f y 1 ) = exp { i π λ z c x f x 1 2 } O c 1 ( x 1 , y 1 ) exp { 2 π i ( x 1 f x 1 + y 1 f y 1 ) } d x 1 d y 1 .
g 2 ( f x 2 , f y 1 ) = exp { i k z f z ( f x 2 , f y 1 ) } exp { i π λ ( z c x f x 2 2 + z c y f y 1 2 ) } × g 1 ( f x 1 , f y 1 ) exp { 2 π i λ z c x f x 1 f x 2 } d f x 1 ,
O ~ c 2 ( f x 2 , f y 2 ) = exp { i π λ z c y f y 2 2 } g 2 ( f x 2 , f y 1 ) exp { 2 π i λ z c y f y 1 f y 2 } d f y 1 .
O c 2 ( x 2 , y 2 ) = O ~ c 2 ( f x 2 , f y 2 ) exp { 2 π i ( x 2 f x 2 + y 2 f y 2 ) } d f x 2 d f y 2 .
f x 1 l = x 1 x p λ r p + x 1 λ z c x .
P S R ( O c 1 , x 1 , f x 1 ) = δ ( f x 1 + x 1 x p λ r p x 1 λ z c x ) .
P S R ( g 1 , x 1 , f x 1 ) = P S R ( O c 1 , x 1 f x 1 λ z c x , f x 1 ) .
f y 1 l = y 1 y p λ r p .
P S R ( O c 1 , y 1 , f y 1 ) = δ ( f y 1 + y 1 y p λ r p ) .
P S R ( g 2 , y 1 , f y 1 ) = P S R ( O c 1 , y 1 + f y 1 λ z c y z f y 1 f z 1 ( f x 1 , f y 1 ) , f y 1 ) .
O R m n = p = 1 P a p r p m n exp [ i k ( r p m n r m n ) ] exp ( i k [ f ^ p x m n ( x 1 x m n c ) + f ^ p y m n ( y 1 y m n c ) ] ) ,
f ^ p x m n = 1 λ Δ f ( x 1 x m n c r p m n x m n c r m n ) Δ f ,
f ^ p y m n = 1 λ Δ f ( y 1 y m n c r p m n y m n c r m n ) Δ f ,

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