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Real part and intensity of shifted-Fresnel diffraction at the propagation distance of 0.1 m, 0.05 m and 0.03 m.
Source publication
Numerical simulation of Fresnel diffraction with fast Fourier transform (FFT)
is widely used in optics, especially computer holography. Fresnel diffraction
with FFT cannot set different sampling rates between source and destination
planes, while shifted-Fresnel diffraction can set different rates. However, an
aliasing error may be incurred in shift...
Citations
... Diffraction calculations using the FFT are generally limited by the source and destination planes needing to be parallel, on-axis, and to have the same sampling pitch; however, there are methods that remove these limitations. Off-axis diffraction calculations include the shifted ASM [11], shifted band-extended ASM [12], shifted Fresnel diffraction [13], and aliasing-reduced shifted and scaled (ARSS) Fresnel diffraction [14]. Scaled diffraction calculations with different sampling pitches for the source and destination planes include the scaled ASM [15], double-step Fresnel diffraction [16], and band-limited doublestep Fresnel diffraction [17]. ...
Diffraction calculations in few-bit formats, such as single-precision floating-point and fixed-point numbers, are important because they yield faster calculations and lower memory usage. However, these methods suffer from low accuracy owing to the loss of trailing digits. Fresnel diffraction is widely known to prevent the loss of trailing digits. However, it can only be used when the paraxial approximation is valid. In this study, a few-bit diffraction calculation method that achieves high accuracy without using any approximation is proposed. The proposed method is derived only by rationalizing the numerator of conventional formulas. Even for scenarios requiring double-precision floating-point numbers using conventional methods, the proposed method exhibits higher accuracy and faster computation time using single-precision floating-point numbers.
... [24] Other methods constitute the Fresnel diffraction, the angular spectrum method which can be further accelerated to generate the CGH by Fast Fourier Transform (FFT) implemented convolution. [25] There are limitations to these diffraction calculation methods as the sampling rate of the source plane equates to the destination plane sampling rate due to the FFT method. Other studies proposed double-step Fresnel diffraction to extend the scale of the method of calculation for generating large CGHs with 8K × 4K px. ...
Identifying road obstacles hidden from the driver's field of view can ensure road safety in transportation. Current driver assistance systems such as 2D head‐up displays are limited to the projection area on the windshield of the car. An augmented reality holographic point cloud video projection system is developed to display objects aligned with real‐life objects in size and distance within the driver's field of view. Light Detection and Ranging (LiDAR) point cloud data collected with a 3D laser scanner is transformed into layered 3D replay field objects consisting of 400 k points. GPU‐accelerated computing generated real‐time holograms 16.6 times faster than the CPU processing time. The holographic projections are obtained with a Spatial Light Modulator (SLM) (3840×2160 px) and virtual Fresnel lenses, which enlarged the driver's eye box to 25 mm × 36 mm. Real‐time scanned road obstacles from different perspectives provide the driver a full view of risk factors such as generated depth in 3D mode and the ability to project any scanned object from different angles in 360°. The 3D holographic projection technology allows for maintaining the driver's focus on the road instead of the windshield and enables assistance by projecting road obstacles hidden from the driver's field of view.
... For instance, Muffoletto et al. introduced a fast shifted Fresnel transform enabling scaled diffraction calculation and hologram scaling [31]. Shimobaba et al. addressed aliasing conditions in short propagation distances and proposed the aliasing-reduced shift and scaled (ARSS) Fresnel diffraction algorithm, primarily applicable in lensless holographic projections [32,33]. Nascov et al. presented a rapid computational algorithm to evaluate Rayleigh-Sommerfeld diffraction with a scale parameter, allowing independent sampling intervals in the input and output computation windows [34]. ...
... Herein, we refer to the diffraction calculations using Eqs. (14) and (15) as polynomial-4 diffraction calculations. Equations (12)- (15) do not cause a loss of trailing digits from the addition of 2 to 1 or the subtraction of 2 from 1, and so high-accuracy calculations can be performed using single-precision floating-point numbers; however, they can only be used in paraxial approximations. ...
Diffraction calculations in few-bit formats, such as single-precision floating-point and fixed-point numbers, are important because they yield faster calculations and lower memory usage. However, these methods suffer from low accuracy owing to the loss of trailing digits. Fresnel diffraction is widely known to prevent the loss of trailing digits. However, it can only be used when the paraxial approximation is valid. In this study, a few-bit diffraction calculation method that achieves high accuracy without using any approximation is proposed. The proposed method is derived only by rationalizing the numerator of conventional formulas. Even for scenarios requiring double-precision floating-point numbers using conventional methods, the proposed method exhibits higher accuracy and faster computation time using single-precision floating-point numbers.
... The focal length can be determined by f = z/(1 − S h /S o ), where z is the distance between the image and hologram, and S h and S o are the sizes of the hologram and image, respectively. Subsequently, a scaled diffraction calculation [35], which can be set with different sampling pitches on the image and hologram planes, was utilized to record all the light information on the hologram. The final equation for the random phase-free method can be expressed as u(x , y ) = P s {o (x , y )s (x , y )}, where P s denotes a scaled diffraction calculation. ...
Utilizing computer-generated holograms is a promising technique because these holograms can theoretically generate arbitrary waves with high light efficiency. In phase-only spatial light modulators, encoding complex amplitudes into phase-only holograms is a significant issue, and double-phase holograms have been a popular encoding technique. However, they reduce the light efficiency. In this study, our complex amplitude encoding, called binary amplitude encoding (BAE), and conventional methods including double-phase hologram, iterative algorithm, and error diffusion methods were compared in terms of the fidelity of reproduced light waves and light efficiency, considering the applications of lensless zoomable holographic projection and vortex beam generation. This study also proposes a noise reduction method for BAE holograms that is effective when the holograms have different aspect ratios. BAE is a non-iterative method, which allows holograms to be obtained more than 2 orders of magnitude faster than iterative holograms; BAE has about 3 times higher light efficiency with comparable image quality compared to double-phase holograms.
... Therefore, the reconstructed 3D scene does not correctly reflect the original 3D scene in image size, and according to the perspective rule: large near and small far, the change in image size will make the observer misunderstand its true depth cues. The same problem exists in the Fresnel hologram, and the aliasing-reduced Fresnel diffraction with scale and shift operations (ARSS) method [42] and the nonuniform sampled wavefront recording method [36] can be used. However, the problem also needs to be addressed in the self-diffraction reconstruction method based on Fourier holograms. ...
The Fourier holographic projection method is compact and computationally fast. However, since the magnification of the displayed image increases with the diffraction distance, this method cannot be used directly to display multi-plane three-dimensional (3D) scenes. We propose a holographic 3D projection method of Fourier holograms by scaling compensation to offset the magnification during optical reconstruction. To achieve a compact system, the proposed method is also used to reconstruct 3D virtual images with Fourier holograms. Different from traditional Fourier holographic displays, images are reconstructed behind a spatial light modulator (SLM) so that the observation position can be placed close to the SLM. The effectiveness of the method and the flexibility of combining it with other methods are confirmed by simulations and experiments. Therefore, our method could have potential applications in the augmented reality (AR) and virtual reality (VR) fields.
... In addition, Tomoyoshi Shimobaba's group further proposed an improved algorithm ARSS (aliasing reduced Fresnel diffraction with scale and shift operations) to address the drawback of SFD that aliasing will occur with a short diffraction distance [18]. By introducing a rectangular window, the algorithm eliminates the aliasing phenomenon with a short diffraction distance. ...
Diffraction algorithms with adjustable magnification are dominant in holographic projection and imaging. However, the algorithms are limited by the Nyquist sampling conditions, and simulation results with inappropriate parameters sometimes appear with aliasing. At present, many diffraction algorithms have been proposed and improved, but there is a need for an overall analysis of their sampling conditions. In this paper, some classical diffraction algorithms with adjustable magnification are summarized, and their sampling conditions in the case of plane wave or spherical wave illumination are analyzed and compared, which helps to select the appropriate diffraction algorithm according to the specific parameter conditions of the simulation to avoid aliasing.
... Additionally, data size modification using zero padding is applied to conduct linear convolution with the fast Fourier transform, which is better in computational speed in methods based on convolutional calculations. Data size modification using zero padding is also applied when varying the sampling pitch before and after diffraction calculations [7][8][9] or when conducting diffraction calculations between nonparallel planes [10]. ...
... (2) holds for the diffraction calculations with changing sampling intervals [7][8][9] and diffraction calculations with an off-axis [21]. For the diffraction calculations based on convolution, zero padding is performed on u source (x 1 , y 1 ) to make the circular convolution a linear convolution. ...
Diffraction calculation techniques based on Fourier transform, such as Fresnel diffraction, are essential in computational optics. Notably, zero padding is applied in diffraction calculations to manipulate sampling pitch and convolution calculations. However, zero padding can generate ringing artifacts due to sudden changes in value, which affect hologram reconstructions, etc. Several existing methods reduce the ringing artifact by decreasing the sudden changes in values. Therefore, in this study, we propose and validate a “ringing artifact extraction method” that focuses on the pattern of ringing artifacts, which depends on the conditions of diffraction calculation.
... The optical wave fields scattered by the tire surface in two different physical states are reconstructed from the holograms using a Fresnel approximation based numerical reconstruction algorithm [27][28][29]: ...
Detection of tire defects is of prime importance for the on-road safety of vehicles. Hence, a quick non-invasive mechanism is required for frequent testing of tires in service as well as for quality checks of newly manufactured tires in the automotive industry. We propose a methodology based on the principle of double-exposure digital holographic interferometry, implemented with a portable digital holographic camera for successful detection and dimension measurement of tire defects. To implement the principle, mechanical load is applied to a tire to produce interferometric fringes by comparing the normal and stressed states of the tire surface. The defects in the tire sample are identified from discontinuities in the interferometric fringes. Quantitative analysis of the displacement of the fringes gives the dimensions of the defects. Some experimental results are presented, validated by a vernier caliper.
... When the hologram is reduced or enlarged with a magnification ratio M, the reconstructed scene is reduced or magnified by M time along the transverse direction but along the longitudinal direction, the reconstructed scene is reduced or magnificated by M 2 times because the depth information of an object is encoded by Fresnel zone plate that is a form of a chirp signal along the transverse direction that corresponds to lateral and longitudinal magnification mismatch of imaging using a lens [8,9]. In the field of computer-generated holograms for fictitious objects, equal scaling techniques in lateral and longitudinal direction have a long-standing history [10][11][12]. Recently, an equal scale hologram synthesizing technique using light field data of real objects was proposed [13]. ...
We propose an optical scanning holography (OSH) for optical reconstruction with equal scale magnification or demagnification ratio along the transverse and longitudinal directions when the magnification or demagnification ratio of the hologram is priori known. First, we review the principles of OSH. Second, we propose an equal scale OSH that encodes a 3D distribution of a 3D scene with a scanning beam pattern that is formed by the superposition of two spherical waves with the same curvature direction. The hologram recorded by the equal scale OSH reconstructs the 3D scene with equal scale magnification or demagnification along the transverse and longitudinal directions. Finally, we provide simulation and experiment results to illustrate and clarify the proposed idea.
