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Abstract
The viewing-angle enlargement of a holographic image is a crucial factor for realizing the holographic display. The numerical aperture (NA) of digital hologram other than a pixel specification has been known to determine the angular field extent of image. Here, we provide a valid foundation for the dependence of viewing angle on the hologram numerical aperture by investigating mathematically the internal structure of the sampled point spread function showing a self-similarity of its modulating curve. The enhanced-NA Fresnel hologram reconstructs the image at a viewing angle larger than a diffraction angle by a hologram pixel pitch where its angle value is expressed in terms of the NA of whole hologram aperture, which is observed systematically by optical experiments. Finally, we found that the aliased replica noises generated in the enhanced-NA Fresnel diffraction regime are effectively suppressed within the diffraction scope by a digitized pixel. This characteristic enables us to overcome the image reduction and to remove the interference of high-order images, which leads to the wide viewing-angle holographic display.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The holographic display is an optical imaging system that reconstructs the real or imaginary image in a free space from the digital hologram [1–4]. The spatial resolution of the numerically reconstructed image in digital holography is subject to the Abbe-Rayleigh diffraction limit related to the hologram numerical aperture [5–7]. We studied the angular field view of optically reconstructed image in holographic display by investigating the diffraction fringes propagating from the sampled hologram loaded on a pixelated spatial light modulator [8]. We found that the viewing angle of holographic image is fundamentally determined by the numerical aperture (NA) of digital hologram, whereas the pixelated structure invokes only the generation of high-order images. That is, the viewing angle changes with an image position even in the spatial light modulator with a finite pixel pitch. The viewing angle $\varOmega$ depends on only the geometrical structures of lateral size, $L = N {\mathit{\Delta}} x$ and distance $z$:
The angle value does not directly relate to the diffraction ability from the pixel size ${\mathit{\Delta}} x$ or a wavelength $\lambda $ of incident wave. This appears to be unintuitive in some ways because the lateral scope of diffractive wave from the pixel would define the viewing zone. However, we investigated in detail that the variation of viewing angle well obeys the relation of Eq. (1) [8].The extension of viewing angle is a crucial task for realizing the holographic display [9–11]. Based on Eq. (1), the hologram numerical aperture can be enhanced in the digital hologram synthesized at a closer distance. When the digital hologram is made at a distance lower than a critical distance, ${z_c} = N{\mathit{\Delta}} {x^2}/\lambda $, the viewing angle would be higher than the diffraction angle by a pixel pitch. We defined this type of hologram as the enhanced-NA Fresnel hologram [12]. In such circumstance, the hologram fringe undergoes the aliasing errors when being sampled. The sampling condition has been well understood in the Nyquist sampling theorem with respect to both planes or by the Wigner domain description [13–16]. The window extent of both planes is calculated by the pixel pitch of opposite plane as with a feature of simple Fourier transform, where only a limited size object could be recoverable. Onural and Stern et al. [17,18] researched that from the Fourier analysis of sampling theory, the object smaller than the diffraction range by a hologram pixel can be fully reconstructed even in severe aliased environment.
We also studied the aliased error of hologram fringe and the image recovery from this type of hologram [12]. The aliased fringes are due to the undersampling of high-frequency components, which appears in the form of replica patterns. Here, the object size decreases even though the viewing angle increases. When the object larger than the diffraction extent by a hologram pixel pitch is used, the interference of high-order images is inevitable. Particularly, we found that the replica fringes in the enhanced-NA hologram are nearly suppressed due to the near-field diffraction property, which gives an opportunity for accomplishing a viewing-angle enlargement without contracting the image size.
Generally, the resolution of the numerically reconstructed image has been interpreted by the window function covering the whole hologram aperture. Despite some related researches, there remains a question about defining the hologram numerical aperture in optical reconstruction. The deficiency of high-frequency components may affect the image formation. Therefore, a further study is required to clarify the definition of NA in above type of digital hologram.
In this study, we investigate mathematically the internal structure of the sampled point spread function (PSF) showing a self-similarity of its modulating curve. The PSF becomes a single Fresnel zone forming the object image. The diffraction-limited resolution relates to the resolving ability of two closest point images. From this, we characterize the dependence of viewing angle on the hologram numerical aperture, especially based on quantum mechanical framework. Subsequently, we carry out the optical experiments for systematically observing the viewing angle variation dependent on the hologram numerical aperture, and finally, suggest a scheme for realizing a wide viewing-angle holographic display.
2. Viewing angle of holographic image dependent on hologram numerical aperture
2.1. Fractal structure of modulating curves in sampled point spread function
The Fresnel diffraction field is well described by the convolutional kernel function $h({x,y} )$ of the PSF [1],
Figure 1 displays a quadratic sinusoid of sampled kernel function. The magnitude of curves seems to be very irregular unlike the analog signal, but the curves include their internal structure. If one divides the function of Eq. (4) into two parts, sampled by even numbers ${n_e}$ and odd numbers ${n_o}$, i.e., $\mathop \sum \limits_{{n_e}} h({{n_e}\varDelta x} )\delta ({x - {n_e}\varDelta x} )+ \mathop \sum \limits_{{n_o}} h({{n_o}\varDelta x} )\delta ({x - {n_o}\varDelta x} )$, each term also consists of replica functions with an interval $\lambda z/({2\varDelta x} )$:
Fig. 1. Quadratic phase curves of (a) real-valued and (b) angle-valued PSFs are drawn in one-dimensional digitized space. The specifications of the blue curve are as follows; wavelength λ = 532 nm, distance z = 30.8 mm, pixel number N = 256, and pixel pitch Δx = 8 µm. The red and green curves are sampled by pixels of 2Δx and 4Δx, respectively.
Furthermore, the sampled kernel function has the modulating curves of original function according to its geometrical placement. Let us incorporate two terms of Eq. (5). The incorporation of two terms results in a twofold increase in sampling rate. The primary Fresnelet centered on the axis simply becomes original form. Meanwhile, the secondary Fresnelet should be made using the specifications ($z/2$, $\varDelta x$) in a view of its placement [12], and thus, they would be modified in the form using a sampling rate $\varDelta {x^{ - 1}}$:
Above characteristics induces a self-similar structure of modulating curves of PSF. The similar subsidiary curves are recursively created in arbitrary sampled PSF, depending on two internal variables (z, $\varDelta x$) of distance and sampling period. Figure 2 shows the fractal structure of two-dimensional Fresnelet for angle-valued kernel function. The similar zones in a small scale are emerged when magnifying the pattern, where the number of smaller zones increases fourfold at half the scale parameter. This is a typical fractal characteristic with a Hausdorff fractal dimension of two [20]. However, we can observe that the structure in two-dimensional space is somewhat complicate. Since the description based on Eq. (5) relates to only the multiple of sampling period $2\varDelta x$, other sampling periods as like $3\varDelta x$ and $5\varDelta x$ should be considered to analyze the similar patterns completely. The subsidiary zones created by sampling period of prime number pixel $3\varDelta x$ appear in Fig. 2, which could also create their self-similar zones, where a Hausdorff dimension is still two. We find that in the Fresnelet fractal, the fractal structures with respect to various sampling periods are mixed.
Fig. 2. Fractal structure of the sampled two-dimensional PSF. Angle-valued two-dimensional PSF, known as Fresnel zone plate, are drawn at specifications; distance of 7.7 mm and 512${\times} $512 pixels with a pixel pitch of 4 $\mu $m. Four aliased fringes are generated in this specification, and $\delta X$ indicates the width of core shell of the primary Fresnelet.
2.2. Viewing-angle analysis of reconstructed image from enhanced-NA Fresnel hologram based on quantum mechanical approach
We investigate the property of viewing angle of the holographic image reconstructed from the enhanced-NA Fresnel hologram. The object field $o({x^{\prime}, y^{\prime}} )$ reconstructed from the digital hologram $g({x,y} )$ with a finite lateral size L is represented by,
In an on-axis point object of delta function, the real or imaginary value of the kernel function $h({x,y} )$ becomes a hologram in the form of a Fresnel zone. Here, the replica patterns of Fresnel zone can be generated at below the ${z_c}$-distance. In digital hologram synthesized at a half of ${z_c}$, the modulating curve of Eq. (6) becomes the form of Fresnel zones. The fractal property of modulating curve makes adaptively the well-behaved replications of Fresnel zones in accordance with a decrease of distance. The digital hologram for the object as a collection of points is synthesized by summing up the Fresnel zones [21].
Let us unwrap the phase hologram of point object. As illustrated in Fig. 3(a), the unwrapped phase synthesized at a critical distance ${z_c}$ reveals the single Fresnel lens with smooth curvature. The single Fresnelet acts like Fourier lens to focus the incident wave to a point spot. In case of the enhanced-NA hologram synthesized at a distance lower than ${z_c}$, the multiple Fresnel lens are formed in Fig. 3(b). They can generate the multiple point images from the incident plane wave.
Fig. 3. Unwrapped phase of Fresnel zones synthesized at (a) a critical distance zc and (b) a half of distance zc. The specifications are as follows; wavelength λ = 532 nm, distance zc = 30.8 mm, pixel number N = 256, and pixel pitch Δx = 8 µm.
The minimum resolvable distance between two point objects is interpreted within the Abbe-Rayleigh diffraction theory. The point image is reconstructed from only the Fourier transform of aperture window because the hologram having an optical kernel function is cancelled out by its inverse term. We described the spatial resolution R by hologram numerical aperture [8]:
Numerical aperture is expressed as the geometric structure of aperture size, $L = N\varDelta x$ and distance z, $\textrm{NA} = \textrm{sin}{\varOmega _{\textrm{NA}}} = N\varDelta x/({2z} )$. The total spatial extent L of digital hologram becomes the aperture window in a single Fresnel lens, whereas in case of multiple lenses, it is not straightforward for defining the aperture window. We know that even in the digital hologram properly sampled at critical distance zc, the symmetrically distributed point sources can generate the similar Fresnel patterns. Here, it is natural that the total area plays a role in an aperture, because the individual Fresnel patterns are spread over the whole area and overlapped each other. However, as depicted in Fig. 1, the formation of multiple zones is due to the undersampling of high-frequency components in the Fresnelet. This process is rather similar to that of high-order diffraction patterns by a relatively large pixel pitch. Meanwhile, the components of each Fresnelet should be still the high-frequency components of adjacent Fresnelets. The higher angular spectrum components of digital hologram survive.We try a quantum mechanical approach to clarify the range of aperture window in the enhanced-NA hologram. The photon field incident on the hologram aperture forms the multiple point images in the image plane. Let us represent the spatial image mode by a density matrix $\rho $,
where ${e_q}$ is probability. This coefficient has the same quantity, because it has been studied that the point images have an equally distributed intensity [22,23]. The spatial mode state $|\psi \rangle $ is given by a sinc or jinc function in accordance with an aperture shape. This state is calculated by the inverse transform of Eq. (9):Considering the rectangular aperture, the state mode has the form of a sinc function distributed with $({\lambda z/\varDelta x,\lambda z/\varDelta y} )$ intervals:
The wave field incident on the hologram acts as individual photons with a momentum, $p = \hbar k$, where $\hbar $ is the reduced Planck constant. The particular spatial mode will be selected by the interaction of photon field with respective Fresnel zone, which is a measurement process quantum mechanically. The spatial resolution of image could be also interpreted from the Heisenberg uncertainty relation [24]. Putting the photon momentum in the lateral direction as ${p_{x^{\prime}}} = p\sin ({\varOmega /2} )$, the general uncertainty relation is written by
where ${p_{x^{\prime}}}$ could be regarded as the maximum uncertainty of photon momentum in the lateral coordinate. From this, we find that the spatial resolution of the corresponding images to the replica Fresnel lenses has the same value generated from the hologram numerical aperture with a whole aperture window:Considering a real object, we obtain the following convolution relation for the restored image with the sinc function related to aperture size [8,18]:
The self-similar property of kernel function appearing in the form of modulating curve makes it possible to form well-behaved replica fringes. The replicated subfunctions are continuous over the entire area of digital hologram. The incident photon fields on the digital hologram experience a whole area as one aperture. The optically reconstructed images from the enhanced-NA hologram would show the increase of viewing angle, but there appear the image reduction and the interference of adjacent images when viewing one with a large depth.
3. Optical hologram imaging for enhanced-NA Fresnel hologram
Figure 4 illustrates a schematic diagram of the in-line holographic system used to synthesize the enhanced-NA Fresnel hologram at a distance ${z_{en}}$ lower than a critical distance ${z_c}$. In order to avoid the aliased error in the process of hologram synthesis, the object space should be confined to the diffraction region by a hologram pixel, and thus, it decreases with decreasing the synthesis distance.
Fig. 4. Schematic of enhanced-NA Fresnel hologram synthesis via the in-line holographic system. The physical sizes of the hologram and object are the same at a distance zc. The amplitude hologram synthesized by Gerchberg-Saxton iterative algorithm is displayed. θ is the diffraction angle by a pixel pitch Δx of digital hologram.
The synthesized digital holograms are shown in Fig. 5. For convenience, in order to discriminate apparently the high-order fringes, the digital hologram with 256×256 pixels of a pixel pitch 8 µm is displayed. We used two letter objects separated from each other in the axial direction and vertically stacked on the coaxial x-axis. This configuration is very useful to measure a smaller viewing angle. The viewing angle is measured by observing the maximum parallax. The Gerchberg-Saxton iterative algorithm is applied to extract the phase hologram [28]. The four replica fringes in the digital hologram made at half the distance of ${z_c}$, 15.4 mm are spatially created in Fig. 5(a). We note that the replica fringe of the former letter is placed at a shifted position vertically or horizontally. In the reconstruction process, the replica fringes will restore the corresponding images at their respective locations, which coincide with the high-order reconstructed images by the pixel pitch of the digital hologram.
Fig. 5. Enhanced-NA Fresnel hologram synthesized through the in-line holographic system. The critical distance ${z_c}$ is 30.8 mm for this specification. (a) Phase hologram of two letter objects synthesized at half the distance of zc by the Gerchberg-Saxton iterative algorithm with 30 steps. Two letters are separated by 5 mm. The red arrow indicates the shift of the first letter fringes. (b) Phase hologram synthesized at a quarter of the distance of zc. Two letters are separated by 2 mm.
Figure 6 shows the optically reconstructed images for the enhanced-NA digital holograms. We used a phase spatial light modulator (Holoeye PLUTO) with 1920${\times} $1080 pixels and a pixel pitch of 8 $\mu $m. The blue laser of a 473-nm wavelength is utilized as the source of incident plane wave. The ${z_c}$-distance for x-direction is calculated to be 259.8 mm. The restored image from the hologram made using the objects located at half the distance of ${z_c}$, 129.9 mm, is displayed in Fig. 6(a). Here, two letter objects are located at a distance of 20 mm. Several high-order images are captured within the lens aperture of the camera. The high-order images of the reconstructed image are placed at the positions specified in the synthesized hologram. We also observe that the first-order image of the former letter is placed at a horizontally shifted position owing to a different perspective view. Since the viewing direction of captured image is set to be the zeroth-order image, the images adjacent to the central image show their perspective views. We can confirm that the perspective view of the central image is changed by moving the viewing direction to the adjacent image, in Fig. 6(b). The viewing angle is estimated to be ${7.1^ \circ }$ from the maximum perspective view of the reconstructed image, which is similar to the value ${6.8^ \circ }$ calculated from the whole aperture of the hologram. This means that all images reconstructed optically from the hologram fringes have the same viewing angle with respect to the whole aperture of the digital hologram. That is, the corresponding aperture size is not limited to the area within the boundary of each fringe.
Fig. 6. Optically reconstructed images and their viewing angle variation for the enhanced-NA Fresnel hologram. The images are captured at a slightly inclined vertical-direction to avoid the directed beam. (a) Reconstructed image of the digital hologram made at a distance of 129.9 mm. (b) Reconstructed image captured in the viewing direction of the first-order image. (c) Reconstructed image for the digital hologram made at a distance of 64.9 mm. (d) Viewing-angle variations of the holographic image as a function of the reconstruction distance.
Figure 6(c) is the restored image for the objects located at a quarter of the distance of ${z_c}$, 64.9 mm. For convenience, two letter objects are separated by a distance of 10 mm. We can see up to the second-order restored image within the diffraction zone with respect to an object pixel pitch of 2 $\mu $m. The former letter image is shifted twice in comparison to the first-order image, and the viewing angle of the second images is estimated to be ${14.8^ \circ }$. In the restored image from the holograms synthesized using the objects placed at a distance of 32.5 mm, we can see up to the fourth-order images. Here, the estimated maximum viewing angle is appeared to be ${28.1^ \circ }$ from two object images separated by 5 mm. The perspective views of all orders of images are clearly confirmed by changing the viewing direction to the respective images. Figure 6(d) is a plot of the variations in the viewing angle of holographic image as a function of the reconstruction distance, where the simulated curve is based on Eq. (1). It is noted that the holographic image with a high viewing angle can be obtained at a shorter distance.
4. Wide viewing-angle holographic display
For realizing the wide viewing-angle holographic display, the increase of viewing angle should be feasible without sacrificing the image size. Furthermore, the interference of high-order images should be prevented. The sampling condition of a digitized space in the Fresnel hologram synthesis is defined by the Nyquist sampling theorem. The pixel specifications of both hologram and object planes are as follows,
In the enhanced-NA Fresnel hologram, although the sampling criterion with respect to the object plane is well satisfied, the aliased error in the hologram plane occurs. The point object generates the replica Fresnel zone patterns from the undersampling of digital hologram, as described in Section 2. However, as studied in our previous work [12], the replica fringes are suppressed as the point objects are aggregated, and thus, the object with a finite size does not form the replica fringes. This unusual behavior is resulted from the intrinsic property of the near-field diffraction in the enhanced-NA Fresnel hologram:In this case, the digital Fresnel hologram with no aliased fringes could be synthesized even using the extended object out of diffraction region by a hologram pixel, as illustrated in Fig. 7. Likewise, this feature is inversely applicable to the image reconstruction process, because the reconstruction process is carried out by the near-field diffraction of the enhanced-NA Fresnel hologram. The original extended object could be well reconstructed only from the upsampled digital hologram without the interference of high-order images [8].
Fig. 7. Configuration of the enhanced-NA Fresnel hologram synthesis by using an extended object out of a diffraction scope. The specifications of hologram synthesis are as follows; wavelength λ = 532 nm, pixel number N = 256, and hologram pixel 8 µm. The distances of ${z_1}$, ${z_2}$, and ${z_3}$ are a half, a quarter, and one eighth of ${z_\textrm{c}}$ of 30.8 mm, respectively. Real-valued digital hologram computed at a distance of a half of zc, 15.4 mm is displayed. The diffraction angle θ by a hologram pixel is calculated to be ${3.8^ \circ }$, where all the blue boxes indicate the diffraction range by a hologram pixel pitch at respective object spaces. The red line relates to the diffraction angle ${\theta _{\textrm{u}1}}$ of ${7.6^ \circ }$ by upsampling pixel to be 4 µm. The viewing angles of ${\varOmega _1}$, ${\varOmega _2}$, and ${\varOmega _3}$ are calculated to be ${7.6^ \circ }$, ${15.3^ \circ }$, and ${30.9^ \circ }$, respectively.
Figure 7 presents the configuration of digital hologram synthesis by using the extended object in the enhanced-NA Fresnel diffraction regime. The distances of ${z_1}$, ${z_2}$, and ${z_3}$ are a half, a quarter, and one eighth of ${z_\textrm{c}}$, respectively. The critical distance ${z_\textrm{c}}$ is 30.8 mm for this specification. All the blue boxes indicate the diffraction range by a hologram pixel pitch, 8 μm at respective object spaces. Real-valued digital hologram computed at a distance, 15.4 mm of a half of ${z_\textrm{c}}$ is displayed, where the diffraction scope in the object plane is two times smaller than the hologram extent, based on Eq. (17). The Lena image with 512×512 pixels of a 4-μm resolution has a double size of 1024-μm diffraction scope. Nevertheless, the computed hologram does not show an aliased fringe due to the near-field diffraction property. Similarly, no aliased fringes are formed in the digital hologram made at distances of ${z_2}$ and ${z_3}$. Here, the whole area of digital hologram would act as one aperture despite the suppression of replica fringes. The hologram fringe suppression is unlike a spatial filtering occurring during the hologram synthesis by a convolutional or angular spectrum method [8]. The method using a double Fourier transform filters out the diffraction fringe by selecting the zeroth-order spectrum in the intermediate Fourier space.
The digitized Fresnel hologram reconstructs the high-order images at a shifted position of $\lambda z/\varDelta x$ in the object plane, which could appear in the form of modulating a high frequency wave [18]:
Fig. 8. Reconstructed high-order images from the enhanced-NA Fresnel hologram made by using a circular object. (a) The first-order images appear in the form of modulating a high frequency wave. (b) Red and blue graphs are the intensity and imaginary value profiles in the horizontal direction at the center of image. The frequency of a modulation carrier is $1/({2\varDelta x^{\prime}} )$.
Figure 9(a) is the restored image from the digital hologram made at a distance of a half of ${z_\textrm{c}}$, in Fig. 7. The restored image is mixed with the modulated high-order noises. That is because the diffraction scope by a hologram pixel is a half of the image size of 2048 μm. Here, we can expect that if the hologram pixel is upsampled to be 4 µm, the high-order noises are effectively removed. As depicted in Fig. 7, the upsampling process enlarges two times the diffraction scope, indicated by a diffraction angle ${\theta _{\textrm{u}1}}$. Since the diffracted field from the upsampled hologram pixel occupies a full extent of image plane, the first-order image signals become a kind of the replica noises within a diffraction scope, which would be suppressed due to the near-field diffraction property. Figure 9(b) shows that the twofold upsampled hologram well removes the first-order noises to obtain the original Lena image.
Fig. 9. Characteristics of reconstructed images from the enhanced-NA Fresnel hologram synthesized by using an extended object. All blue boxes indicate the diffraction range by a hologram pixel, $N\varDelta x^{\prime} = \lambda z/\varDelta x$. Restored images from the digital hologram made at a distance ${z_1}$ (a) without an additional upsampling process and (b) with two-fold upsampling process. (c) Restored image from the fourfold upsampled hologram made at a distance ${z_2}$. (d) Restored image from the eightfold upsampled hologram made at a distance ${z_3}$. Red and blue lines of graphs are the intensity and real value profiles in the horizontal direction at the center of image. (e) The upper and (f) lower graphs correspond to (a) and (b) images, respectively.
The intensity and real value profiles in the horizontal direction at the center of image are drawn in Fig. 9(e) and 9(f), respectively. It looks like that the sinusoidal wave is clearly filtered out by an upsampling process. But, this process is fundamentally different from the simple spatial filtering. As stated previously, although the high frequency term is deleted by upsampling a hologram pixel, the increase of object resolution could prohibit this removal. Optically, the object space is continuous. Furthermore, the high-order terms survive despite a removal of high frequency wave, in Eq. (19). We confirmed that the suppression of high-order images arises irrespective of the object resolution even in a numerical simulation. For this reason, we conclude that the property of the near-field diffraction of the enhanced-NA Fresnel hologram allows it to implement this extraordinary phenomenon.
Figure 9(c) shows the well restored images from the digital hologram computed at a distance ${z_2}$, 7.7 mm, where the object size is four times larger than the diffraction scope. The digital hologram is upsampled fourfold to diffract the incident wave to object size. Restored image from the digital hologram made at a distance ${z_3}$ is displayed in Fig. 9(d). The eightfold upsampled hologram clearly reconstructs the Lena image, where its viewing-angle gets to ${30.9^ \circ }$, from Eq. (1). We find that it is possible to restore an original image even at a finer upsampling process for a high-NA hologram without higher-order noises. The upsampling process in optical experiment can be performed by using a simple grid. We surely know that the high-order image reappears outside the diffraction range by an upsampled hologram pixel. The finer upsampling would lead to a larger diffraction zone preventing high-order noises. Figure 10 shows the restored images from the synthesized hologram at a distance ${z_1}$ in the extended image space. The replica images are distributed at intervals of the diffraction range by a twofold upsampled hologram pixel, which is effectively suppressed by a fourfold upsampling process. As studied previously [8], above result supports that a wide viewing-angle holographic display could be realized even by using a commercial spatial light modulator.
Fig. 10. Reconstructed images in the extended image space. (a) Replica Lena images are distributed at intervals of the diffraction range by a twofold-upsampled hologram pixel. (b) The fourfold upsampling secures a larger diffraction zone preventing high-order noises.
5. Discussions
In practice, the optical reconstruction strongly depends on the performance of spatial light modulator. The present spatial light modulators are capable of only the phase or amplitude modulation. An amplitude or complex modulation could be applicable to this type of wide viewing-angle holographic display system. Complex hologram was used in this numerical simulation, and thus, it would be desirable to execute a complex modulation [29,30].
The practical phase hologram is obtained in terms of the iterative optimization algorithms. A convenient approach is to add a random phase to the object field. In such cases, since the replica fringes are formed during synthesizing an enhanced-NA hologram, the generation of high-order images is unavoidable, as described in Section 3. The replica fringe is formed due to the sampling rate lower than the proper sampling rate. We know that this phenomenon arises from the sub-sampling by a multiple pixel pitch, as displayed in Fig. 1. If the sub-sampling is nonuniformly performed, there appear no replica fringes. If there is also an irregularly pixelated spatial light modulator which has nonuniformly distributed pixels of irregular pixel size, it would effectively hinder the generation of high-order images. This also makes it possible to realize the wide viewing-angle holographic display, but the optimization procedure for the hologram synthesis is needed because of an irregularly pixelated hologram. Recently, various studies have been performed to synthesize the phase hologram without the iterative algorithm or random-phase addition [31].
From a different point of view, we find that on the basis of our interpretation for viewing angle of holographic image, a finer resolution of image would lead to a wider viewing-angle. In a classical wave optics, the typical methods to increase the image resolution are to reduce the wavelength or to build higher numerical-aperture optics. However, in a quantum imaging system, the Abbe-Rayleigh diffraction limit is secondary to the problem at hand, that is to say, it can achieve the ultimate optical resolution [25–27]. Quantum imaging by using N-correlated photons leads to in a $1/\sqrt N $ resolution enhancement called standard quantum limit. Furthermore, for N entangled photons one can achieve a $1/N$ scaling known as the Heisenberg limit.
If the image with higher resolution beyond the Rayleigh limit can be floated in a free space, the viewing angle would be larger than the angle estimated from the system NA. The image spot can be regarded as only a point source generating secondary wavelets, and thus, the diffraction angle would follow from the diffraction formula, $\varOmega = {\sin ^{ - 1}}({\lambda /\varDelta x^{\prime}} )$. The smaller image spot generates a wave with a higher angular spectrum. We find that the image resolution becomes a primary measure in defining the viewing angle of holographic image. The Fourier hologram generates the image with a constant spatial resolution dependent on only the focal length of lens, irrespective of image depth [32]. The digital hologram synthesized by a convolutional or angular spectrum method reveals a constant image resolution at a distance longer than a critical distance [8]. These properties of spatial resolution are not directly explained by a classical system NA. Here, we could infer the effective NA from the image resolution to interpret consistently this phenomenon. For a practical application based on quantum imaging, the effective imaging techniques without complicate processes such as a coincidence or multi-photon measurement should be developed.
6. Conclusions
The viewing angle of holographic image in the enhanced-NA Fresnel hologram is well understood from the analysis of the internal structure of the sampled PSF. The angle value is determined by the NA of whole aperture of digital hologram. Optical experiment shows the consistent result with the interpretation of viewing angle dependent on the hologram numerical aperture. The high-NA hologram reconstructs the image at a higher viewing-angle, while the image reduction is inevitable due to the Nyquist sampling criterion. The enhanced-NA Fresnel diffraction has an interesting characteristic that the aliased replica noises are effectively suppressed within the diffraction scope by a digitized pixel. The digital Fresnel hologram synthesized even using the extended object out of diffraction region shows no aliased fringes. Only the upsampling process of digital hologram results in the image reconstruction without the interference of high-order images. This approach gives the method for enlarging the viewing angle of holographic image without sacrificing image size.
Funding
Institute for Information and Communications Technology Promotion (2017-0-00049 and 2021-0-00745).
Disclosures
The author declares no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
2. T. Kozacki, M. Kujawińska, G. Finke, B. Hennelly, and N. Pandey, “Extended viewing angle holographic display system with tilted SLMs in a circular configuration,” Appl. Opt. 51(11), 1771–1780 (2012). [CrossRef]
3. J. Hahn, H. Kim, Y. Lim, G. Park, and B. Lee, “Wide viewing angle dynamic holographic stereogram with a curved array of spatial light modulators,” Opt. Express 16(16), 12372–12386 (2008). [CrossRef]
4. M. Park, B. G. Chae, H. Kim, J. Hahn, H. Kim, C. H. Park, K. Moon, and J. Kim, “Digital holographic display system with large screen based on viewing window movement for 3D video service,” ETRI J 36(2), 232–241 (2014). [CrossRef]
5. T. Latychevskaia and H.-W. Fink, “Inverted Gabor holography principle for tailoring arbitrary shaped three-dimensional beams,” Sci. Rep. 6(1), 26312 (2016). [CrossRef]
6. D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48(9), 095801 (2009). [CrossRef]
7. P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A 25(7), 1744–1761 (2008). [CrossRef]
8. B. G. Chae, “Analysis on angular field of view of holographic image dependent on hologram numerical aperture in holographic display,” Opt. Eng. 59(3), 035103 (2020). [CrossRef]
9. Y. Takaki and S. Uchida, “Table screen 360-degree three-dimensional display using a small array of high-speed projectors,” Opt. Express 20(8), 8848–8861 (2012). [CrossRef]
10. J. Park, K. Lee, and Y. Park, “Ultrathin wide-angle large-area digital 3D holographic display using a non-periodic photon sieve,” Nat. Commun. 10(1), 1304 (2019). [CrossRef]
11. L. Huang, S. Zhang, and T. Zentgraf, “Metasurface holography: from fundamentals to applications,” Nanophotonics 7(6), 1169–1190 (2018). [CrossRef]
12. B. G. Chae, “Analysis on image recovery for on-axis digital Fresnel hologram with aliased fringe generated from self-similarity of point spread function,” Opt. Commun. 466, 125609 (2020). [CrossRef]
13. D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculations,” Opt. Commun. 164(4-6), 233–245 (1999). [CrossRef]
14. A. Stern and B. Javidi, “Improved-resolution digital holography using the generalized sampling theorem for locally band-limited fields,” J. Opt. Soc. Am. A 23(5), 1227–1235 (2006). [CrossRef]
15. D. G. Voelz and M. C. Roggemann, “Digital simulation of scalar optical diffraction: revisiting chirp function sampling criteria and consequences,” Appl. Opt. 48(32), 6132–6142 (2009). [CrossRef]
16. J.-P. Liu, “Controlling the aliasing by zero-padding in the digital calculation of the scalar diffraction,” J. Opt. Soc. Am. A 29(9), 1956–1964 (2012). [CrossRef]
17. L. Onural, “Sampling of the diffraction field,” Appl. Opt. 39(32), 5929–5935 (2000). [CrossRef]
18. A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43(1), 239–250 (2004). [CrossRef]
19. L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in Fresnel diffraction simulations,” Opt. Eng. 43(11), 2557–2563 (2004). [CrossRef]
20. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, 1982).
21. T.-C. Poon and J.-P. Liu, Introduction to Modern Digital Holography with MATLAB (Cambridge, 2014).
22. D. M. Cottrell, J. A. Davis, T. Hedman, and R. A. Lilly, “Multiple imaging phase-encoded optical elements written as programmable spatial light modulators,” Appl. Opt. 29(17), 2505–2509 (1990). [CrossRef]
23. E. Carcole, J. Campos, I. Juvells, and S. Bosch, “Diffraction efficiency of low-resolution Fresnel encoded lenses,” Appl. Opt. 33(29), 6741–6746 (1994). [CrossRef]
24. R. Shanker, Principles of Quantum Mechanics (Springer, 2008).
25. P. Hong and G. Zhang, “A review of super-resolution imaging through optical high-order interference,” Appl. Sci. 9(6), 1166 (2019). [CrossRef]
26. M. Tsang, “Quantum imaging beyond the diffraction limit by optical centroid measurements,” Phys. Rev. Lett. 102(25), 253601 (2009). [CrossRef]
27. M. Unternährer, B. Bessire, L. Gasparini, M. Perenzoni, and A. Stefanov, “Super-resolution quantum imaging at the Heisenberg limit,” Optica 5(9), 1150–1154 (2018). [CrossRef]
28. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982). [CrossRef]
29. V. Arrizon, “Complex modulation with a twisted-nematic liquid-crystal spatial light modulator: double-pixel approach,” Opt. Lett. 28(15), 1359–1361 (2003). [CrossRef]
30. S. Reichelt, R. Häussler, G. Fütterer, N. Leister, H. Kato, N. Usukura, and Y. Kanbayashi, “Full-range, complex spatial light modulator for real-time holography,” Opt. Lett. 37(11), 1955–1957 (2012). [CrossRef]
31. P. W. M. Tsang and T.-C. Poon, “Novel method for converting digital Fresnel hologram to phase-only hologram based on bidirectional error diffusion,” Opt. Express 21(20), 23680–23686 (2013). [CrossRef]
32. B. G. Chae, “Analysis on viewing angle of holographic image reconstructed from a digital Fourier hologram in a holographic display,” OSA Continuum 4(2), 477–490 (2021). [CrossRef]
