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Abstract
In this paper, potential distortions corresponding to the hologram printed by a holographic wave-front printer are analyzed. Potential distortions are classified as the magnification(demagnification) distortion, barrel distortion, pincushion distortion, SLM mounting distortion, and translation distortion, respectively. These distortions are grouped as the optics distortion, SLM mounting distortion and the translation distortion depending on the process of recording the hologram in the holographic wave-front printer. In order to evaluate each distortion, a distortion analysis method based on a local spatial frequency is proposed. Through the proposed method, a diffracted wavefield reconstructed from a quantitatively distorted hologram is theoretically analyzed, and the validity of this analysis is verified by applying the numerical reconstruction method. In the numerical reconstruction, a propagation of a distorted wavefield reconstructed from the quantitatively distorted hologram is confirmed and contributed to generate the distorted reconstruction plane, such as a focal cloud plane and a convergence plane, depending on the types of distortion.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recently, a holographic printing technology is used to record a hologram of a real or virtual three-dimensional(3D) object in a holographic recording material [1–5]. This technology has been widely applied to various fields such as a military, architecture, commerce, automotive industry, and entertainment [6,7].
The holographic printing technology can be classified into two categories: the first is a holographic stereogram technology and the second is holographic wave-front printing technology. In the case of holographic stereogram technology, perspective images with a continuous parallax are acquired by a camera tracking method or a computational rendering method [8]. The interference pattern between each perspective image and a reference beam is recorded on the holographic recording material, and this recorded interference pattern is called a holographic element(hogel). when numerous recorded hogels are located in a tiled form, these hogels compose a printed hologram on the holographic recording material. The printed hologram illuminated by white light can reconstruct different perspective images, and the viewer to observe different parallax images can realize 3D perception by binocular disparity [6]. However, the massive calculation and the time-consuming recording are necessarily required for the hogel printing related to each perspective image. In addition, the hologram printed by the holographic stereogram printer cannot provide an actual 3D object because different images with binocular disparity are reconstructed.
The holographic wave-front printing method records an interference pattern of the wavefield propagated from the virtual 3D object [9–11]. The fringe pattern outgoing wavefield from the 3D object is numerically calculated, and the calculated fringe pattern is shown on a display device such as a spatial light modulator(SLM). The coherent light entering the SLM with the displayed fringe pattern is diffracted to the holographic recording material and this diffracted wavefield is interfered with the reference beam. This interference pattern is recorded on the holographic recording material and this recorded pattern is called an elemental hologram(EH).
In the same manner as the holographic stereogram technology, the holographic wave-front printing technology records the hologram consisted of multiple tiling EHs. Since the EH is recorded as the difference of refractive index in the holographic recording material, the printed hologram becomes a reflective volume-type hologram [12]. Also the large-sized and thin hologram with a high diffraction efficiency can be printed because the interval of the EH or the number of the tiled EHs can be controlled by the holographic printing technology. Moreover, this printed hologram can ideally reconstruct real 3D objects without any distortion. However, enormous calculations and time-consuming recording are unavoidably demanded printing hologram in the holographic printing system. Therefore, high-speed hologram generation algorithm and fast recording method are necessary [13]. While the hologram is being recorded by the traditional hologram recording system, unexpected distortions to the hologram can usually occur due to errors of optical elements in the hologram recording system or vibrations. So, various correcting methods for the distorted hologram have been proposed [14–21]. And, several pieces of research to correct a distorted hologram for a holographic display have been studied [20,21]. These pieces of research proposed distortion-correction methods to apply the holographic display consisting of a camera and an SLM. Consequently, some numerical distortion-correction methods from the research can effectively correct the distortion of the hologram. However, the recording structure of the holographic printing system is definitely different compared to a recording configuration of the traditional hologram recording system. So, a distortion analysis method particularized for the holographic printing system is required but the suitable analysis method has not been studied until now.
In this paper, potential distortions corresponding to the hologram printed by a holographic wave-front printer are analyzed. Potential distortions are classified as the magnification(demagnification) distortion, the barrel distortion, the pincushion distortion, the SLM mounting distortion, and the translation distortion, respectively. These distortions are grouped in the optics distortion, SLM mounting distortion and the translation distortion depending on the process of recording hologram in the holographic wave-front printer. In order to evaluate each distortion, we have built an optical model for the distorted propagating wavefield from the distorted hologram and a distortion analysis method using a local spatial frequency based on the paraxial approximation is proposed. Through the proposed method, a diffracted wavefield reconstructed from a quantitatively distorted hologram is theoretically analyzed. The validity of this analysis is verified by applying the numerical reconstruction method based on the angular spectrum method. In the numerical reconstruction, a propagation of distorted wavefield reconstructed from the quantitatively distorted hologram is confirmed and contributed to generate the distorted reconstruction plane such as a focal cloud plane and a convergence plane depending on the types of distortion. In further research, we will research an prediction and calibration method for distortions inherent in the printed hologram from optically distorted reconstruction of hologram.
2. Holographic wave-front printer and its distortions
2.1 Holographic wave-front printer
In the holographic wave-front printer, the interference pattern between the wavefield propagated from the SLM with the displayed fringe pattern and the reference beam is recorded on the holographic recording material [22]. This recorded interference pattern is named as a elemental hologram(EH) and a plurality of EHs recorded in the tiled form constitute the printed hologram, finally. For the explanation of the recording process, the structure of the holographic wave-front printer system is shown in Fig. 1.
Fig. 1. (a) System structure of holographic wavefront printer (b) distortions caused in a recording process
In Fig. 1(a), the holographic wave-front printer consists of two parts; The first part is for the controlling coherent light, and the second part, which can be called as an optical engine, is to print the hologram. In the first part, the controlled coherent light is illuminated to the SLM and the holographic recording material for the generating interference pattern between the wavefield and the reference beam. For this purpose, first part is composed of various optical elements such as a laser light source, an optical switch, a beam expander, a beam splitter, a lens and a mirror. The optical engine in the second part records the interference pattern between the wavefield from the SLM and the reference beam on the holographic recording material as described in Fig. 1(b). This optical engine consists simply of an SLM, a telecentric lens and an $\textit {x}$-$\textit {y}$ moving translation stage. Before EHs are recorded in the optical engine, each fringe pattern is numerically calculated and displayed on the SLM. The incoming coherent light to the SLM is diffracted by the displayed fringe pattern, and then the diffracted wavefield is propagated into the telecentric lens. After this wavefield passes through the telecentric lens, the size of the diffracted wavefield decreased by the ratio between each focal length of two lenses expressed as ${L_1}$ and ${L_2}$ [23]. The interference pattern between the size-reduced wavefield and the reference beam is recorded on the holographic recording material located on the $\textit {x}$-$\textit {y}$ moving translation stage. As this process is repeated in the optical engine, the hologram composed of sequentially tiled EHs is fabricated. However, various distortions depending on the optical and mechanical element of the optical engine may occur during the recording EH. In the next subsection, potential distortions in the holographic wave-front printer will be analyzed.
2.2 Distortions in holographic wave-front printer
Figure 2 shows a type of distortions,which can occur in the optical engine of the holographic wave-front printer.
First, when the SLM is rotated around the optical axis, the propagating wavefield diffracted from the SLM is rotated in the same manner. So, the interference pattern of the rotated wavefield is recorded on the holographic recording material. This distortion is defined as a SLM mounting distortion. Second, the longitudinal and transverse size of the diffracted wavefield can be unexpectedly distorted by several errors such as a performance error or the location error of each lens in the telecentric lens. When an interference pattern of the size-distorted wavefield is recorded on the holographic recording material as the EH, this phenomenon can be defined as an optics distortion. If the longitudinal, transverse or both size of the EH are constantly changed, this distortion applied in the EH means the magnification(demagnification) distortion [23]. However, when the size of the EH increases or decreases radially, this distortion is a radial distortion. Third, the interval between adjacent EHs on the holographic recording material may be distorted by the mechanical error of the $\textit {x}$-$\textit {y}$ moving translation stage. This is called a translation distortion because the interval of neighboring EHs increases or decreases regularly and irregularly.
These kind of distortions such as the SLM mounting distortion, optics distortion, and translation distortion should significantly contort the shape or the interval of the EH. When the hologram composed of distorted EHs is printed by the holographic wavefront printer, this printed hologram can reconstruct a distorted 3D object. So, an analysis method of the quantitative distortions, which may appear to the EH, is required for the holographic wave-front printer.
3. Numerical reconstruction of the distorted hologram
3.1 Desired hologram
In order to analyze the effect of the distorted hologram, the desired hologram is chosen to compare with the distorted hologram. The desired hologram is composed of tiled EHs without any distortions and designed to reconstruct a single point at a specific distance. Since the desired hologram is the simplest hologram, this is suitable for a distortion analysis in the holographic wave-front printer. In aid of the distortion analysis, an optical model of the diffracting wavefield from the desired hologram is built as shown in Fig. 3.
In Fig. 3(a), ${\Delta} {s}$ means the interval of adjacent EHs in the desired hologram to be printed, and $\textit {s}_1$ represents the half size of the desired hologram to be printed. $\textit {d}_1$ is the desired focal distance and $\theta _1$ is the diffraction angle of the wavefield propagated from the individual EH. As shown in Fig. 3(a), the propagating wavefield from the desired hologram is converged at the desired focal plane and then diverged within the diffraction angle along the $\textit {z}$-axis. Figure 3(b) shows the desired hologram, and this is composed of 4 $\times$ 4 EHs without any distortions here.
For a quantitative analysis of the wavefield, a local spatial frequency is very useful to analyze the relationship between the propagating wavefield and the hologram. In the case of the desired hologram, the original local spatial frequency ${\Delta} {\textit {f}}_1$ of the desired hologram is expressed as ${{\Delta} {f}_1}$ = sin($\theta _1$)/$\lambda$ where $\lambda$ is the wavelength [13]. Also, the diffraction angle $\theta _1$ of the desired hologram is represented by sin($\theta _1$) = $\lambda {{\Delta} {f}_1}$, tan($\theta _1$) = ${s_1/d_1}$,respectively. Since the relationship of sin($\theta _1$) $\approx$ tan($\theta _1$) is approximately derived by small-angle approximation, the $d_1$ of the desired hologram can be expressed in Eq. (1) as follows.
In Eq. (1), $d_1$ is proportional to $s_1$ but inversely proportional to $\lambda$ and ${\Delta} {\textit {f}}_1$. In other words, when each value of $s_1$ and $\lambda$ is constant, the $\theta _1$ also increases or decreases depending on the local spatial frequency of the printed hologram, inversely.
For the validity confirmation of the analysis method, the numerical reconstruction method based on an angular spectrum is adopted for the printed hologram. Also, this numerical reconstruction method can effectively confirm the complex-amplitude distribution of the diffracted wavefield at the specific plane since the angular spectrum is induced from the Rayleigh-Sommerfeld equation without any approximation [16,24]. The specification of the desired hologram used in the numerical reconstruction is summarized in Table 1.
Table 1. Specifications for the numerical reconstruction of the desired hologram
As described in Table 1, ${d_1}$ of the desired hologram is set up as 100 mm, and the range of the $\textit {z}$-axis for the numerical reconstruction is 0 mm $\sim$ 200 mm. In other words, the absolute complex-amplitude distribution of the diffracted wavefield at an arbitrary $\textit {z}$-plane limited from 0 mm to 200 mm can be simulated by means of the numerical reconstruction method. To confirm the absolute complex-amplitude distribution of the wavefield, the desired hologram is calculated based on specifications in Table 1 and assumed to be the printed hologram. And then, the numerical reconstruction of the printed hologram is carried out. Figure 4 shows the numerically calculated absolute complex-amplitude distribution of the wavefield, which is diffracted from the desired hologram.
Fig. 4. (a) Numerical reconstruction at desired focal plane (b) diffracted wavefield on longitudinal direction
Figure 4(a) reveals the result of numerical reconstruction at the desired focal plane(100 mm) According to Table 1. From the result of numerical reconstruction, we confirm that the diffracted wavefield from the desired hologram is converged to the single point at 100 mm from the printed hologram. Figure 4(b) shows the absolute complex-amplitude distribution of the diffracted wavefield from z = 0 mm to 200 mm. Since the desired focal plane is set up a 100 mm, the wavefield diffracted from the desired hologram is converged at z = 100 mm and diverged. In the next section, the distorted hologram depending on a type of distortion, and each result of the numerical reconstruction will be analyzed.
3.2 Optics distortion
An optics distortion in the EH may be caused by the telecentric lens in the optical engine of the holographic wave-front printer when the performance error and positional error of each lens in the telecentric lens occurs. In the below subsection, optics distortion will be discussed in detail.
3.2.1 Magnification(demagnification) distortion
Magnification(demagnification) distortion can be defined as when the vertical or horizontal size of the recorded EHs are changed at the same ratio compared with the size of the desired EH. When the size of the recorded EH is larger than the size of the desired EH, magnification distortion occurs. Whereas, the demagnification distortion is accompanied in the recorded EH when the size of the recorded EH becomes smaller than the size of the desired EH. Also, the analysis method of the magnification(demagnification) distortion can be applied to the demagnification distortion because demagnification distortion is reverse effect of magnification distortion. So, this section discusses only the optical model of the magnification distortion because this optical model can equally be applied to the demagnification distortion. Figure 5 shows the optical model for the magnification distortion and two holograms with magnification and demagnification distortion.
Fig. 5. (a) Optical model of magnification distortion (b) hologram with magnification distortion (c) hologram with demagnification distortion
As described in Fig. 5(a), $d_1$ signifies the desired focal distance, and $d_2$ is the distance of a focal cloud plane, which is a specific plane with multiple reconstructed points. $\textit {M}$ means the magnification factor at which can magnify or minify the size of the EH, and $\theta _2$ indicates the distorted diffraction angle according to the distorted EH. When a convergence plane is defined where the wavefield diffracted from the distorted EH regardless of the type of distortions is intersected on the $\textit {z}$-axis, the distance of the convergence plane from the printed hologram is expressed as $d_c$. As seen in Fig. 5(a), ${\Delta} s$ of the desired EH becomes $\textit {M}{\Delta} s$ because of $\textit {M}$. A region of the same size as ${\Delta} {s}$ from the center of each increased EH is selected, and then theoretically synthesized to form the magnification-distorted hologram to be printed. And, a overlapped area between the adjacent increased EHs is ignored in this optical model. In addition, this optical model describes that the distorted hologram can reconstruct multiple points at the focal cloud plane and form the convergence plane. So, we can confirm that the focal cloud planes and the convergence plane are important phenomenons to analyze the properties of the diffracted wavefield propagated from the distorted hologram.
In view of the local spatial frequency, ${\Delta} {f_2}$ is defined as a varied local spatial frequency of the distorted EH regardless of the kind of distortions. And, ${\Delta} {f_2}$ of magnification-distorted EH is expressed as ${\Delta} {f_2}$=${{\Delta} {f_1}/M}$. When a varied half size of the distorted hologram is $s_2$ which satisfies the relation of $s_2$ = $Ms_1$, $\theta _2$ of the magnification-distorted EH is expressed as sin($\theta _2$) = $\lambda {{\Delta} {f}_2}$, tan($\theta _2$) = ${s_2/d_2}$, respectively. Since the relationship of sin($\theta _2$) $\approx$ tan($\theta _2$) is satisfied by small-angle approximation, the focal cloud plane $d_2$ can be obtained from sin($\theta _2$) $\approx$ tan($\theta _2$) and Eq. (1). In the case of the magnification(demagnification) distortion, the focal cloud plane $d_2$ related to the distorted local spatial frequency ${\Delta} {f_2}$ is expressed in Eq. (2).
The focal cloud plane $d_2$ is changed depending on $\textit {M}^2$ under fixed $d_1$ and this equation can be adjusted in both cases of the magnification(demagnification) distortion. When ($\textit {M}$ > 1) or ($\textit {M}$ < 1) is satisfied except the case of ($\textit {M}$ = 1), the distortion to be applied in each EH becomes the magnification distortion($\textit {M}$ > 1) or the demagnification distortion($\textit {M}$ < 1). Since $\theta _2$ increases by the proportion of $1/M$ in the magnification distortion, the focal cloud plane is accordingly formed far from the desired focal plane. Whereas, the focal cloud plane in the demagnification distortion appears closer than the desired focal plane. Eq. (3) is a formula for the convergence plane caused by the magnification(demagnification) distortion.
When the magnification(demagnification) distortion is applied in the EH, a convergence plane appears where ${d_1}$ is proportionally shifted by the value of $\textit {M}$ by (3). To verify the validity of the analysis in the magnification distortion, the numerical reconstruction of the magnification-distorted hologram is performed. In order to change the interval of the desired EH, a bicubic interpolation method is used because cubic convolution (CC) interpolation method has become a standard in the image interpolation field [25,26]. Figure 6 shows the numerical reconstruction of the magnification-distorted hologram($\textit {M}$ = 1.2).
Fig. 6. (a) Numerical reconstruction of the magnification-distorted hologram at desired focal plane (b) numerical reconstruction of the magnification-distorted hologram at convergence plane (c) numerical reconstruction at focal cloud plane (d) diffracted wavefield from the magnification-distorted hologram on longitudinal direction
Figure 6(a) shows the absolute complex-amplitude distribution of the numerically reconstructed wavefield on the desired focal plane(100 mm). As seen here, this distribution of the diffracted wavefield from the magnification-distorted hologram to be printed is spread not focused because the diffracted wavefield will be converged at the focal cloud plane. The numerically reconstructed wavefield on the convergence plane is shown in Fig. 6(b). The convergence plane is formed at 120 mm far from the printed hologram according to Eq. (3). Moreover, this plane involves the narrowest wavefield distribution due to the intersection of the propagated wavefield but the narrowest wavefield distribution does not mean the reconstructed focal point. Figure 6(c) shows the absolute complex-amplitude distribution of the diffracted wavefield from the printed hologram at the focal cloud plane, and $d_2$ is calculated as 144 mm according to Equation (2) because the value of $\textit {M}$ is decided as 1.2. In the focal cloud plane, 4 $\times$ 4 points same as the number of tiled EHs are reconstructed because the diffraction angle of each magnification-distorted EH in the printed hologram is reduced by the decreased local spatial frequency as much as $\textit {M}$ times. The wavefield diffracted along the $\textit {z}$-axis is shown in Fig. 6(d). The range of the $\textit {z}$-axis is 0 $\sim$ 200 mm and the printed hologram is located at 0 mm. Since $\textit {M}$ is 1.2, the focal cloud plane and the convergence plane are formed farther than the desired focal plane according to Eqs. (2) and (3).
In order to confirm the complex-amplitude distribution of the wavefield in the demagnification distortion, the numerical reconstruction of the demagnification-distorted hologram($\textit {M}$ = 0.85) is performed. The following Figure 7 indicates the results of numerical reconstruction of demagnification-distorted hologram to be printed.
Fig. 7. (a) Numerical reconstruction of the demagnification-distorted hologram at focal cloud plane (b) numerical reconstruction of the demagnification-distorted hologram at convergence plane (c) numerical reconstruction from the demagnification-distorted hologram at desired focal plane (d) diffracted wavefield on longitudinal direction
Figure 7(a) shows the numerical reconstruction at the focal cloud plane. 4 $\times$ 4 points are reconstructed like the case of magnification distortion because wavefield diffracted from each demagnification-distorted EH is converged to its own reconstructed point at the focal cloud plane. Also, $d_2$ is calculated as 72.25 mm according to Eq. (2) because $\textit {M}$ is 0.85. Since the local spatial frequency of each demagnification-distorted EH increases the diffraction angle, the focal cloud plane is formed near the printed hologram. The numerically reconstructed wavefield at the convergence plane (85.00 mm) is shown in Fig. 7(b). Like the case of magnification distortion, the narrowest wavefield distribution in the convergence plane appears to be reconstructed. Figure 7(c) obtains the numerical reconstruction at the desired focal plane. Since the absolute complex-amplitude distribution of the propagating wavefield passed through the focal cloud plane is widely spread depending on the $\textit {z}$-axis, the result of the numerical reconstruction is similar to the case of the magnification distortion. Figure 7(d) indicates the wavefield from the demagnification-distorted hologram is diffracting along the $\textit {z}$-axis. Unlike the case of the magnification distortion, the focal cloud plane, the convergence plane, and the desired focal plane appear, sequentially.
3.2.2 Radial distortion
Radial distortion is caused by lenses constituting a telecentric lens, and typical radial distortions include barrel distortion and pincushion distortion [27,28]. The radial distortion that occurred in the holographic wave-front printer radially distorts the distance of the EH between the center $(x_c,y_c)$ in the element hologram and the arbitrary position $(x_o,y_o)$. If $\textit {k}$ is the distortion parameter and the distance from $(x_c,y_c)$ to $(x_o,y_o)$ in the EH is expressed as $r=\sqrt {(x_c-x_o)^2+(y_c-y_o)^2}$, the distorted position $(x_d,y_d)$ can be expressed as follows: $(x_d,y_d) = (x_o(1+kr^2),y_o(1+kr^2)$[29]. Thus, the distortion factor $M_k$ can be expressed as $M_k=1+{k}{r^2}$. Figure 8 shows an optical model of the radially distorted hologram in the holographic printing system.
Fig. 8. (a) Optical model of the magnification distortion (b) hologram with the barrel distortion (c) hologram with the pincushion distortion
In Fig. 8(a), the red line means the wavefield diffracted from the barrel-distorted hologram to be printed, and $k$ is typically negative. The diffracted wavefield near the center of the EH is relatively propagated far from the barrel-distorted hologram. On the other hand, the wavefield diffracted far from the center of barrel-distorted EH is propagated close to the distorted hologram. The phenomenon is caused by ${\Delta} {\textit {f}_2}$ of each EH because ${\Delta} {\textit {f}_2}$ is radially varied according to ${\Delta} {\textit {f}_2}$=${\Delta} {\textit {f}_1}/M_k$. When the value of $r$ is small, ${\Delta} {\textit {f}_2}$ slightly increases due to the small value of $M_k$. Therefore, the diffraction angle related to ${\Delta} {\textit {f}_2}$ increased less. On the other hand, when $r$ between the center and arbitrary point in each EH has a large value, the local spatial frequency depending on the point significantly increases compared to the case of small $r$. So, the diffraction angle of the wavefield largely increases by the increased local spatial frequency. In Fig. 8(a), the blue line means the diffracting wavefield from the pincushion-distorted hologram, and $k$ has a positive value. The diffracted wavefield from the printed hologram is radially propagated far from the desired focal plane depending on the ${r}$ because the pincushion distortion is the opposite effect of the barrel distortion. When $r$ is small value, ${\Delta} {\textit {f}_2}$ of the point related to $r$ in pincushion-distorted EH slightly decreases. Therefore, the diffraction angle related to the ${\Delta} {\textit {f}_2}$ slightly decreased due to the relation of ${\Delta} {\textit {f}_2}$=${\Delta} {\textit {f}_1}/M_k$. However, if $r$ from the center of the EH is longer, the ${\Delta} {\textit {f}_2}$ in the related point largely decreases depending on $M_k$. So, since the diffraction angle largely decreases by the significantly reduced ${\Delta} {\textit {f}_2}$, the wavefield from the related point with a large value of $r$ is propagated far from the desired focal plane. In other words, the diffraction angle of the pixel position in the distorted EH with barrel distortion or the pincushion distortion radially increased or decreased depending on the relative position. So, the focused point is not reconstructed in both cases.
Figures 8(b) and (c) shows the barrel- and pincushion-distorted hologram, respectively. The EH affected by the barrel distortion is mapped around a spherical shape, but the shape of the EH with the pincushion distortion is curved outward from the center. When each EHs is affected by the radial distortion, the wavefield propagated with the distorted diffraction angles is intersected within a certain $\textit {z}$-axis range. So, this phenomenon can be called a spread convergence plane and this spread convergence plane is expressed in Eq. (4).
When the value of ${k}$ is negative in the barrel distortion, the range of the spread convergence plane is decided by the relation of $min({M_k}{d_1})<{d_c}<{d_1}$. However, the range of the spread convergence plane in the pincushion distortion is determined like the relation of ${d_1}<{d_c}<max({M_k}{d_1})$ because of the positive ${k}$. In order to simulate the numerical reconstruction of the distorted hologram with the radial distortion, $k$ for the barrel distortion and the pincushion distortion is fixed as -0.3 and 0.3, respectively. The result of the numerical reconstruction for the barrel-distorted hologram with $k$ = -0.3 are shown in Fig. 9.
Fig. 9. (a) Numerical reconstruction of the barrel-distorted hologram at desired focal plane (b) diffracted wavefield on longitudinal direction
As shown in Fig. 9(a). the spread distribution of the wavefield is numerically reconstructed even though the reconstruction plane is the desired focal plane. In other words, the propagating wavefield from the barrel-distorted hologram cannot be converged because the diffraction angle related to the relative position from the center of barrel-distorted EH is radially distorted. The propagating wavefield within from the 0 mm to 200 mm of the $\textit {z}$-axis is described in Fig. 9(b). Since the diffraction angle on the barrel-distorted EH is radially varied, the spread convergence plane appears. The numerically calculated distribution of the wavefield from the pincushion-distorted hologram with $k$ = -0.3 is shown in Figure 10 as below.
Fig. 10. (a) Numerical reconstruction of the pincushion-distorted hologram at desired focal plane (b) diffracted wavefield on longitudinal direction
Figure 10(a) and (b) shows the spread distribution reconstructed from the pincushion-distorted hologram is reconstructed at the desired focal plane and the distribution of propagating wavefield, respectively. Also, the spread convergence plane is formed because of the radially distorted diffraction angle. In both cases of the barrel and pincushion distortion, the results are confirmed that the radial-distorted hologram cannot reconstruct a converged singe point.
3.3 SLM mounting distortion
The SLM mounting distortion is caused by the rotated SLM from the optical axis. When the SLM with the displayed fringe pattern is rotated, the diffracted wavefield from the SLM is also rotated and propagated into the free space. The optical model for a printed hologram with the SLM mounting distortion is presented in Figure 11.
Fig. 11. Optical model for the hologram with SLM mounted distortion (a) Optical model for the hologram with SLM mounted distortion (b) hologram with SLM mounting distortion
In Fig. 11(a), the distortion angle of SLM is referred to as $\theta _r$ and $\textit {s}_1$ is the half size of the desired hologram. When SLM mounting distortion is applied in each EH, the $d_1$ of the SLM-mounting-distorted EH is not changed. As shown in Fig. 11(b), each EH is rotated the same as the rotation angle of SLM. The formula of the convergence plane of the SLM-mounting-distorted hologram is expressed in Eq. (5).
When SLM mounting distortion occurred in each EH, the focal cloud plane is decided the same as the desired focal plane because the individually rotated wavefield from each SLM-mounting-distorted EH is converged at the desired focal plane. However, the same number of reconstructed points are also rotated in the same direction due to the angle of the SLM. In addition, the convergence plane appears because of the intersection of the propagating wavefield. In order to analyze the SLM mounting distortion, 5$^\circ$ of the distortion angle is applied to each EH and the numerical reconstruction is performed. This numerical result is shown in Fig. 12.
Fig. 12. (a) Numerical reconstruction at desired focal plane (b) numerical reconstruction at convergence plane (c) diffracted wavefield on longitudinal direction
In Fig. 12(a), the reconstructed point is shown in the focal cloud plane(100 mm). Since the rotated wavefield diffracted from SLM-mounting-distorted EH is converged at the focal cloud plane, the same number of points compared to the number of EHs are reconstructed. Also, each reconstructed point is rotated with 5$^\circ$ from the optical axis. Figure 12(b) presents the result of numerical reconstruction in the convergence plane. Since ${d_1}$ is 100 mm and the rotation angle is 5$^\circ$, the location of the convergence plane is 100.38 mm calculated by Eq. (5). Fig. 12(c) demonstrates a crosscut of the propagated wavefield along $\textit {z}$-axis. The range of the $\textit {z}$-axis is 0 $\sim$ 200 mm and the printed hologram is located at 0 mm. The propagating wavefield from SLM-mounting-distorted EH is rotatively converged in the focal cloud plane and diverged along the $\textit {z}$-axis.
3.4 Translation distortion
The translation distortion is caused by the mechanical error of the $\textit {x}$-$\textit {y}$ moving translation stage to distort the interval between adjacent EHs. when this distortion occurs to each EHs, the interval between adjacent EHs increases or decreases regularly or irregularly. Since the converging point of the wavefield diffracted from the distorted EH is shifted as much as the distorted interval, the same points as tiled EHs are reconstructed but shifted at the desired focal plane. Figure 13 shows the optical model of the translation distortion and the translation-distorted hologram.
Fig. 13. (a) Optical model of translation distortion (b) printed hologram with translation distortion
As shown in Fig. 13(a), ${\Delta} {s}$ and ${\Delta} {g}$ is the interval of the desired EH and the distortion gap between adjacent EHs, respectively. When translation distortion is involved in the EH, the interval between the current EH and the previous EH is distorted by distortion gap ${\Delta} {g}$ but the local spatial frequency of each EH is not changed. Accordingly, the diffraction angle of the wavefield spreading out from the translation-distorted EH is not changed, but only the position of the reconstructed point is shifted. So, points same as the number of EHs are reconstructed at the desired focal plane. In addition, a convergence plane is formed by the intersected wavefield from each translation-distorted EH. Figure 13(b) shows the translation-distorted hologram. The translation-distorted hologram involves empty space because of the additional distortion gap between the adjacent EHs. In analysis of the translation distortion, only the convergence plane can be considered because the focal cloud plane appears in the desired focal plane. Also, the convergence plane formed by the distorted wavefield from the translation-distorted hologram can be induced by the proportional expression among ${\Delta} {s}$, ${\Delta} {g}$ and $d_1$. $d_c$ as the distance of convergence plane is expressed by Eq. (6).
In Eq. (6), ${d_c}$ is proportional to ${d_1}$ and the absolute value of ${\Delta} {g}$. And, the location of the convergence plane is shifted by the sign of ${\Delta} {g}$. When the value of ${\Delta} {g}$ is positive, the convergence plane is located behind the focal cloud plane. Whereas, the negative value of ${\Delta} {g}$ contributes to forming the convergence plane in front of the focal cloud plane. So, the forming convergence plane closer to the focal cloud plane is affected by the small absolute value of ${\Delta} {g}$ but the larger absolute value of ${\Delta} {g}$ contribute the forming convergence plane far from the focal cloud plane. To simulate the numerical reconstruction of the translation-distorted hologram, the $\textit {x}$- and $\textit {y}$-axis distortion gap is equally set to 100 pixels, and other specifications in Table 1 are used. The numerical reconstruction result for the translation-distorted hologram is shown in Fig. 14.
Fig. 14. (a) Numerical reconstruction at desired focal plane (b) numerical reconstruction at convergence plane (c) diffracted wavefield on longitudinal direction
Figure 14(a) presents the numerical reconstruction result in the focal cloud plane. 4 $\times$ 4 points from the EH are reconstructed on the focal cloud plane. 4 $\times$ 4 reconstructed points from each EH are shifted as much as 100 pixels on the $\textit {x}$- and the $\textit {y}$-axis from the center of the focal cloud plane. Since the overlap between individually reconstructed points by the small value of ${\Delta} {g}$ is presented, a spreading distribution of the wavefield is numerically reconstructed. As shown in Fig. 14(b), the numerical reconstruction of the distorted hologram in the convergence plane. From the simulation result of the numerical reconstruction, the convergence plane is formed at 109.8 mm far from the distorted hologram according to Eq. (6) because ${\Delta} {s}$ and distortion gap ${\Delta} {g}$ is 10.24 mm and 0.1 mm, respectively. Since the diffracted wavefield is intersected at the convergence plane, the blurred single point is reconstructed. Fig. 14(c) shows the propagating wavefield along the $\textit {z}$-axis from the printed hologram with translation distortion. The range of the $\textit {z}$-axis is 0 $\sim$ 200 mm and the printed hologram is located at 0 mm. The propagating wavefield from each EH with translation distortion is converged in the focal cloud plane and intersected in the convergence plane.
4. Numerical verification
From the above sections, we confirm seven different distortions to each EH may occur individually or compositely while the EH is recording in the holographic wave-front printer. Also, different characteristics of the reconstructed image are presented depending on the type of distortion. Table 2 summarizes the characteristics by distortion.
Table 2. Summary of distortions
In the case of the optics distortion, the local spatial frequency and diffraction angle are changed due to the modified interval of the EH in the printed hologram. The reconstructed image at arbitrary plane is also distorted according to the change of diffraction angle. When magnification(demagnification) distortion occurred, the interval of the EH is changed by magnification of $\textit {M}$. The local spatial frequency and diffraction angle varied in an inverse proportion to $\textit {M}$. Therefore, the focal cloud plane in magnification-distorted hologram is calculated by the relation of ${\textit {M}^2}{\textit {d}_1}$ and moved far from the desired focal plane. But the focal cloud plane is located between the demagnification-distorted hologram and the desired focal plane. Furthermore, the convergence plane is formed according to the relationship with ${\textit {M}}{\textit {d}_1}$, and always located between $d_1$ and ${M_2}{d_1}$. In the radial distortion, however, the focal cloud plane and the convergence plane should not occur because the local spatial frequency and the diffraction angle of the EH are radially changed depending on the distance between the center and the arbitrary point in EH. The spread convergence plane is formed within a certain range along the $\textit {z}$-axis. In the case of SLM mounting distortion, each EH is rotated as much as $\theta _r$. Also, the wavefield is rotated and propagated along the $\textit {z}$-axis. The rotated propagating wavefield is converged at the desired focal plane because the local spatial frequency of the hologram with SLM mounting distortion is not changed. Thus, the desired focal plane and focal cloud plane are the same. And, the same points as the number of tiled EHs are rotatively reconstructed. In the case of the translation distortion, the focal cloud plane is located in the same as the desired focal plane because the diffraction angle of each EH with translation distortion is not changed. However, the reconstruction point converged by the diffracted wavefield from each EH is moved as much as the distortion gap ${\Delta} {g}$ applied in the individual EH. Since diffracted wavefield passes through the focal cloud plane and are intersected, the convergence plane is formed.
5. Conclusion
we analyzed potential distortions corresponding to the hologram printed by a holographic wave-front printer. Potential distortions are classified as the magnification(demagnification) distortion, the barrel distortion, the pincushion distortion, the SLM mounting distortion, and the translation distortion, respectively. These distortions are grouped in the optics distortion, SLM mounting distortion and the translation distortion depending on the process of recording hologram in the holographic wave-front printer. In order to evaluate each distortion, an optical model for the distorted propagating wavefield from the distorted hologram is built and a distortion analysis method based on a local spatial frequency is proposed. Through the proposed method, a diffracted wavefield reconstructed from a quantitatively distorted hologram is theoretically analyzed. The validity of this analysis is verified by applying the numerical reconstruction method based on the angular spectrum method. In the numerical reconstruction, a propagation of distorted wavefield reconstructed from the quantitatively distorted hologram is confirmed and contributed to generate the distorted reconstruction plane such as a focal cloud plane and a convergence plane depending on the types of distortion. In the case of the magnification(demagnification) distortion, as many points as the number of elemental holograms are reconstructed at the focal cloud plane. The focal cloud plane of the demagnification distortion is located closer than the desired focal plane. On the other hand, the focal cloud plane of the magnification distortion is far from the desired focal plane. Both distortions involve the convergence plane between the focal cloud plane and desired focal plane. In the case of the radial distortion, the focal cloud plane cannot be built; Points cannot be reconstructed because of radially varied diffraction angle. However, the convergence plane is spread within the range of the varied diffraction angle. In the case of SLM mount distortion, the same number of points as much as the number of elemental holograms is reconstructed at the focal cloud plane, which is the same as the desired focal plane. However, these reconstructed points are rotated as much as the rotation angle of SLM. Also, the convergence plane was presented. In the case of translation distortion, as many points as the number of elemental holograms is reconstructed at the focal cloud plane, which is the same as the desired focal plane, but shifted as much as the distortion gap. Also, the convergence plane appeared on both sides of the desired focal plane. In further research, we will establish a predictable model of distortions from the optically reconstructed image.
Funding
National Research Foundation of Korea (2020R1I1A3071771); Institute for Information and Communications Technology Promotion (2020-0-00924).
Disclosures
The authors declare no conflicts 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.
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