1. Introduction

As sensors and communication equipment are getting smaller and at the same time the computation speed is increasing, a variety of immersive content playback systems are reported. A virtual or augmented display system is already paying attention from many fields, such as a medical or industrial purposes [1, 2]. Other next-generation display systems, such as an auto-stereoscopic display [3–7], aerial display [8–10], or holographic display [11–19], are widely developed from many research groups, and some of them achieved a significant goal with the advent of the high-resolution display panels, the advanced laser technology, and the graphic processor unit based powerful computation. Among the aforementioned realistic information playback methods, only the holographic video device is believed to be able to deliver a full three-dimensional depth cue toward the viewer’s eye [20, 21]. Therefore, the research and development for advanced holographic display technology are worth to continue.

Since the holographic display device mostly requires a wavefront information as an input source data, the ideal counterpart is the holographic video camera that faithfully records the wave field of the scene. However, the holographic recording systems announced so far have three disadvantages simultaneously or one at a time. First of all, the system size is inevitably bulky because they have a cumbersome interferometer structure or optoelectronic devices. Secondly, most systems use a laser to record interference patterns clearly, but this special light source is rarely used in our daily life. Lastly, the hologram video recording is hard to achieve because a time-division phase-shifting technique is performed to solve the bias and twin image problem that occurs in an on-axis holographic recording system. The three disadvantages of the existing holographic technique as described above greatly hinder various applications outside the laboratory.

Recently, our team proposed a relatively simplified digital holographic camera totally operated with geometric phase effect, under the low-coherent lighting condition [22, 23]. We call this system temporarily as GPSIDH, that is the geometric phase self-interference incoherent digital holography. This system can be categorized into the self-interference incoherent digital holography (SIDH) [24, 25], or Fresnel incoherent correlation holography (FINCH) [26–28], since the incoming wavefront is divided and modulated into two different curvatures of spherical wavefronts. Especially, in our system, the thin and light-weighted geometric phase (GP, or also known as Pancharatnam–Berry phase) lens is employed. The GP optical component is usually made from the polymerized liquid crystal (LC) that only the orientation state of the LC director induces the phase variation [29, 30]. Since this phase modulation is not affected by the physical path length or optical retardation, but only by the topological state of the polarization states, the phase called as a geometric phase. When the orientation map of LC directors corresponds to the quadratic phase profile, this component can serve as a lens. The very distinctive feature of this lens is that the lens serves either convex or concave lens according to the input handedness of the circular polarization (CP) states, due to the orthogonal ordinary and extraordinary axis of the LC. And if a linearly polarized light comes in, this light is considered to be a superposition of two orthogonal CP components, half converges and the other half diverges. In the GPSIDH system, the special property of the GP lens is utilized as a polarization selective wavefront modulator, so that opened a new possibility to make a digital holographic camera with portability and simplicity. This system solves two drawbacks of the digital holographic recording systems simultaneously, which are low-coherent light capability and relatively small and simple system size. But still, the system is not small enough to be implemented even inside the mobile devices, nor operated in real-time to be as a video camera, due to the time-division phase-shifting method.

Development of the compact incoherent video holographic camera is essential for the practical usage in various fields. Some of the reports related to the incoherent holographic camera, including our previous system [23], reached a significant goal toward such practical applications. For the case of single-exposure bias and twin image elimination, the off-axis system configuration [31–33], or parallel phase-shifting method is applied inside the SIDH or FINCH systems [34, 35]. Both methods sacrifice the spatial resolution instead of losing the temporal resolution, however, the off-axis method is less desirable since the limitation of the narrow bandwidth of the image sensor [36]. On the other hand, the parallel phase-shifting method, which the details are explained in the following section, requires four adjacent pixels to record a single complex-valued data, therefore the pixel numbers in a horizontal and vertical direction are reduced in half [37]. Even though the parallel phase-shifting method also sacrifices the spatial resolution, it is a good candidate to simplify the system structure, since this method employs the geometric phase-shifting, which is mostly performed only with the combination of the waveplates and polarizers, and their relative angle [38–40]. In the case of the wavefront division method, the implementation of a passive type common-path wavefront modulator is effective to build a compact and simple holographic system [23, 28], instead of using the bulky conventional interferometric structure [24, 25, 41, 42], or the phase-only spatial light modulator [26, 34, 35]. The birefringent crystal lens and the GP lens serve a similar role as a common-path wavefront modulator inside the incoherent holographic system. The FINCH system with the birefringent crystal lens has already proven its extreme capability as a holographic microscopy [28]. However, it requires a variable waveplate or the combination of the waveplates to obtain a phase-shifted digital hologram. On the other hand, the GP lens is originally designed as a half-waveplate, the geometric phase-shifting method with the rotation of the linear polarizer is effectively applied, which brings a great simplicity in the system design.

In this paper, we propose and analyze a compact holographic video camera, which is developed from the previous version of GPSIDH [23]. The system is only configured with the band-pass filter (BPF), convex lens, one polarizer, GP lens, and the polarized image sensor. The wavefront division is achieved by the GP lens, and the parallel phase-shifting is operated by the combination of the polarizer, GP lens, and the polarized image sensor. Unlike the previous study, the image sensor is located right behind the GP lens, that the system path length is significantly reduced. We believe the aforementioned studies about incoherent holographic systems are great candidates to function as a holographic video camera with compactness in size, with a simple modification. Though, we think that the proposed system in this study is highly compact and simple to operate than any other reported works above. In the following section, the working principle of the system is introduced, and the design rule of the compact system is analyzed. Then in the experimental section, the design-rule based recording results are presented. And the real-time hologram acquisition are demonstrated with the digitally reconstructed video. After the discussion, the summary of the paper follows.

 

Fig. 1 (a) the illustration of the GPSIDH system, (b) schematic diagram of the system with the parameter and field labels. Red and blue dashed lines after the GP lens indicate the diverging and converging rays, respectively, each of which modulated by the negative and positive focal length of the GP lens. BPF, band-pass filter (used if necessary); l_o, objective lens with focal length f_o; $P1,2$, first and last polarizer; l_gp, GP lens with focal length of $\pm f_{gp}$; z_o, object distance from l_o; d, l_o to l_gp distance; and z_h, l_gp to sensor distance.

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2. GPSIDH system structure and working principle

The basic structure of compact GPSIDH system is shown in Fig. 1(a). By referring to the expressions of the FINCH system proposed by Rosen, et al. and the related works [26, 43], let us describe the proposed GPSIDH system. If the object point source is located at z_o from the objective lens l_o, then the field before the lens is described with a diverging spherical wave, $C_1\left(\overline{r_o}\right)Q\left[z_o^{-1}\left]L\right[-\overline{r_o}/z_o\right]$. Here, $Q\left[z^{-1}\right]=\exp [j\pi z^{-1}{\lambda }^{-1}\left(x^2+y^2\right)]$ is the quadratic phase function. $L[\overline{r}\left]=\exp [j2\pi \left(r_{x}x+r_{y}y\right)/\lambda \right]$ is the linear phase function. λ is the central wavelength of the input light source. $\overline{r_o}=\left(x_o,y_o\right)$ is the object point coordinate, and $C_1\left(\overline{r_o}\right)$ is the complex constant of each object point source. The diverging spherical wave from the object point source experiences the objective lens with the transmission function of $Q\left[-f_o^{-1}\right]$, where f_o is the focal length of the lens [44]. Hence, the complex amplitude of the field after the objective lens becomes $C_1\left(\overline{r_o}\right)Q\left[z_o^{-1}\left]L\right[-\overline{r_o}/z_o\left]\times Q\right[-f_o^{-1}\right]$. The field propagates further to the GP lens with the distance d. And the field passes through the GP lens, where the transmission function is expressed as $\left(Q\left[-f_{gp}^{-1}\left]e^{j\delta /2}+Q\right[f_{gp}^{-1}\right]e^{-j\delta /2}\right)/2$. The GP lens transmission function has two terms, one for the lens with the focal length of f_gp and the positive phase shift of $\delta /2$. On the other hand, another term is described with the focal length of $-f_{gp}$ and the phase shift of $-\delta /2$. Here, δ is the total phase-shifting value and is discussed later in this section. After the GP lens, the field propagates to the image sensor with a distance z_h. Then the intensity of the Fresnel hologram from the single object point at $\left(x_o,y_o,z_o\right)$ is recorded as,

In Eq. (1), $C_2\left(\overline{r_o}\right)$ is the complex constant, and the propagation is denoted as a linear convolution with the operational symbol of $*$. As introduced in Eq. (9) of the work of Katz, et al. in [43], the spherical wave propagated from the original spherical wave $C\left(\overline{r_o}\right)Q\left[z_1^{-1}\left]L\right[\overline{A}\right]$ with the distance z₂ can be described as $C\left(\overline{r_o}\right)Q[\left(z_1+z_2{)}^{-1}\left]L\right[\overline{A}{z}_1/\left(z_1+z_2\right)\right]$. By following this rule, Eq. (1) is expressed as,

Here, z_i is the imaging distance deduced from the thin lens relation with the object distance z_o and the lens focal length f_o, that is $z_o^{-1}+z_i^{-1}=f_o^{-1}$. And, $z_p,z_n$ are the second imaging distance due to the positive and negative focal lengths of GP lens, respectively. These imaging distances are also derived by the thin lens relation, which are ${(d-z_i)}^{-1}+z_p^{-1}=f_{gp}^{-1}$ and ${(d-z_i)}^{-1}+z_n^{-1}=-f_{gp}^{-1}$. As the single object point source generates the Fresnel hologram $I_h\left(x_h,y_h;x_o,y_o,z_o\right)$ as shown in Eq. (2), the entire hologram $H\left(x_h,y_h\right)$ generated from the group of object points with $I_o\left(x_o,y_o,z_o\right)$ is expressed as Eq. (5).

According to Eq. (2), three terms remain in the equation, each of which is the bias, complex hologram, and the twin image components, respectively. To properly extract the complex hologram from the recorded intensity image, the phase-shifting method is utilized [45].

 

Fig. 2 Poincaré sphere representation of the proposed system. (a) the second polarizer is in 45° state, (b) the second polarizer is rotated to the vertically aligned state. The geometric phase difference δ between (a) and (b) is 90°. The coordinate labels, $S_{1,2,3}$ are the Stokes parameters.

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In this system, there is no conventional phase-shifting device such as a Piezo-adjuster, or variable retarder. In our previous research, we replaced such phase-shifting device into a geometric phase shifter, which is the combination of linear polarizer and GP lens [23]. More specifically, the polarization state of the incoming wave inside the GPSIDH system is defined firstly at point A on Fig. 2, and it is soon separated on two poles (point R and L) of the Poincaré sphere due to the GP lens, then they arerealigned with the second linear polarizer into the state of point B on the equator. The solid angle ${\mathrm{\Omega }}_{ABC}$ of the arbitrary spherical triangle surrounded by the points A, B, and C is given as Eq. (6), that is defined by three unit vectors $\widehat{a},\widehat{b},\widehat{c}$, from the sphere center O to each points A, B, and C, respectively [46].

The amount of geometric phase-shifting in the previous study and the proposed method can be deduced with Eq. (6). The arbitrary spherical triangular path $ABC$ is now specified by the path $ARB$ and $ALB$, as shown in Fig. 2. When the point A represents the horizontal and B is the 45° linearly polarized states, then their unit vector from the origin of the sphere O is $\widehat{a}=\left[1,0,0\right],\widehat{b}=\left[0,1,0\right]$, respectively. Since the right and left handedness of the circular polarization states are the north and south poles of the Poincaré sphere, the unit vector for the origin to the point R, and L is $\widehat{r}=\left[0,0,1\right],\widehat{l}=\left[0,0,-1\right]$. By using Eq. (6), the solid angle ${\mathrm{\Omega }}_{ARB}$ becomes $\pi /2$. And conversely, ${\mathrm{\Omega }}_{ALB}$ is $-\pi /2$. Owing to the geometric phase gain is the half of the solid angle, each spherical triangular path $ARB,ALB$ has $\pi /4$ and $-\pi /4$ geometric phase angle which is denoted as $\delta /2$ and $-\delta /2$ in Eq. (1) respectively [38, 47].

Therefore, their relative phase difference becomes $\pi /2$. By changing the relative angle between the two polarizers, let’s say from point B to B’ as in Figs. 2(a) to 2(b), the amount of geometric phase gain is controlled. The amount of geometric phase gain by changing the state between point A and B is also described by Jones matrix method with the same result in Eqs. (1) and (2) of the previous work [23]. If the rotational variation in four step with 45° per step, the four phase-shifted images with 90° per step can be obtained. Then the four recorded images are recombined to eliminate the bias and twin images.

However, even with this simple phase-shifting method, the real-time acquisition of a clear hologram is hard to achieve. If the object moves faster than the time period of the entire phase-shifting process, then the acquired and reconstructed hologram image would be blurred or nothing is observed. To simply solve this problem, the second polarizer and general image sensor of the previous system are replaced into the polarized image sensor, that the micro-polarizer is attached on every sensor pixels. By doing so, the motionless single-shot phase-shifting digital holography can be achieved. This spatial division phase-shifting method is referred as a parallel phase-shifting [37], and widely applied in various digital holography systems. The very recent successful implementation of the parallel phase-shifting method into the FINCH system is the work of Tahara et al. [34].

 

Fig. 3 Illustration of the structure of polarized image sensor, and parallel phase-shifting method

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The structure of the polarized image sensor is as shown in Fig. 3. The micro-polarizer array is attached on the pixel array. The micro-polarizers are arranged to be rotated at an interval of 45° with the adjacent micro-polarizer. Therefore, the intensity values of the polarization component of $0^{\circ },{45}^{\circ },{90}^{\circ }$, and 135° of light incident on the 2 x 2 array of pixel group are simultaneously recorded. Once the raw image file is obtained from the polarized image sensor, the four phase-shifted images are extracted with a proper sampling method. The sampled period of the phase-shifted images is considered as the twice of the original pixel size, as illustrated in Fig. 3. The four phase-shifted images, each of which has the hologram intensity by following Eqs. (1) to (5), are recombined into the complex-valued hologram (CH) as [45],

In Eq. (7), $H_{1,2,3,4}$ correspond the phase shifted images with $\delta =0^{\circ },{90}^{\circ },{180}^{\circ },{270}^{\circ }$, where the relative rotation angles between the first polarizer are $0^{\circ },{45}^{\circ },{90}^{\circ },{135}^{\circ }$, respectively. p and q is the pixel index of the two-dimensional array data. Then, the object field is retrieved by using the conventional Fresnel back propagation method, that is to correlate $CH\left[p,q\right]$ with the quadratic phase function $Q\left[-z_{rec}^{-1};p,q\right]=\exp \left[-\frac{j\pi }{\lambda z_{rec}}\left(p^2{\mathrm{\Delta }}_x^2+q^2{\mathrm{\Delta }}_y^2\right)\right]$ [44]. Here, $z_{rec}$ is a reconstruction distance, λ is the central wavelength of light, ${\mathrm{\Delta }}_{x,y}$ is the width and height of the sampling grid, respectively. The distance $z_{rec}$ is calculated as Eq. (3).

Analysis of required focal length of GP lens and GP lens to sensor distance for better hologram acquisition The interference due to the superposition of the two wavefronts can occur only when the optical path length difference($\mathrm{\Delta }OPL$) at a certain sensor location is shorter than the coherence length c_l, that $c_l={\lambda }^2/\mathrm{\Delta }\lambda$, where λ is the central wavelength and $\mathrm{\Delta }\lambda$ is the spectral bandwidth of the light source [48]. In the short focal length of GP lens, the curvature differences between the two converging and diverging waves after the GP lens is too large, that only a small overlapping area fulfills the condition of $\mathrm{\Delta }OPL\le c_l$. For this reason, in previous versions of the system, the relay lens is used to reduce the difference in curvature of the two wavefronts. Otherwise, due to the small interfering area, the obtained hologram looks faint, and its reconstructed image becomes also blurry and noisy. The overall hologram is obtained from the group of point source of the object that can be considered as a δ–function separately. Each δ–function creates their own fringe pattern by the transfer function of the system as expressed in Eq. (4). The fringe pattern is in the shape of Fresnel ring pattern as illustrated in Fig. 4, and the radius the pattern is denoted as r_h herein.

 

Fig. 4 (a) schematic optical diagram of the analyzed system, (b, c) conceptual illustration of the reconstructed image quality of hologram related to r_h, where (b) has a large r_h, and (c) has a small r_h.

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In this section, to set a basic design rule of the compact GPSIDH system without the relay lens, we analyze r_h of the recorded Fresnel hologram, which is closely related with the system parameters. Especially, r_h

determines the numerical aperture (NA) of the system input, as illustrated in Fig. 4 (blue solid line). Since the optical system with larger NA has improved imaging quality in terms of resolution either lateral or transverse, it is desirable to maintain the system NA higher to obtain a high quality hologram [44, 49–51].

As shown in Fig. 4(a), let us suppose that the δ–function like point light source at the front focal plane of lens l_o on the optic axis shines the system with a sufficient diverging angle. The rays are collected by l_o, then they simultaneously converge and diverge after passing through the GP lens l_gp, which is located behind the distance of d after l_o. Let us call each converging or diverging rays after passing through l_gp as a positive and negative ray respectively since they experience GP lens by its positive and negative focal lengths. Then these rays propagate with the distance z_h to reach the sensor surface with a radius of r_s. For the perfectly coherent light source, if r_s is smaller than the radius of the marginal positive ray r_p, the interference fringe pattern should be recorded all over the sensor area. However, due to the limitation of c_l of the incoherent light source, the radius r_h of the recorded fringe area is determined where the sensor position having the maximum $\mathrm{\Delta }OPL$ that is bounded by c_l. For every sensor pixels $r_s\left[n\right]$, where n is the pixel index, the $\mathrm{\Delta }OPL\left[n\right]$ is calculated by using Eq. (8), with simple geometries between the positive ($OPL{[n]}_{pos}$) and negative ($OPL{[n]}_{neg}$) rays drawn respectively as blueand red solid lines in Fig. 4(a).

The sensor position $r_s\left[n\right]$ where $\mathrm{\Delta }OPL\left[n\right]<c_l$ becomes the maximum fringe pattern radius r_h. This radius can be increased with the proper selection of the system parameters, f_gp and z_h. For example, when $f_o=$ 100 mm, $d=$ 18 mm, z_h = 20 mm, the $\mathrm{\Delta }OPL$ for every sensor pixels are presented as Fig. 5. The black-dashed horizontal line in Fig. 5 indicates the c_l boundary. The c_l in Fig. 5 is derived from the light source with the central wavelength of 525 nm, and the spectral bandwidth of 32 nm. This is the same light source used in the experimental results described in the following section. The colored arrows under the plot area indicate the bounded r_h of the system with different f_gp.

 

Fig. 5 Plot of $\mathrm{\Delta }OPL\left[n\right]$ to $r_s\left[n\right]$, when $f_o=$ 100 mm, $d=$ 18 mm, $z_h=$ 20 mm, $c_l=$ (525 nm)²/32 nm =9.76 μ m. The legend denotes f_gp. The solid horizontal arrows indicate the r_h defined by each f_gp.

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The entrance pupil r_o bounded by r_h is calculated as,

If the GP lens is large enough and the point object is located at the front focal point of l_o, then the NA of the system becomes,

According to above analysis, the relation of r_h to z_h and NA to z_h are plotted on Fig. 6. To obtain a wider Fresnel hologram as possible, it is desirable to select longer focal length of GP lens and locate the sensor closer to the GP lens. However, there are several trade-off relation with the long f_gp and z_h that should be taken account as well.

 

Fig. 6 (a) r_h to z_h relation, (b) system NA to z_h relation. The legend denotes f_gp.

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The Fresnel hologram can be considered as a digitized diffractive lens that has its own radius and focal length. In the case of hologram, its radius is r_h, and the focal length is the reconstruction distance, which can be derived as Eq. (3). The reconstruction distance equation is simply understood as a thin lens equation that the Fresnel hologram is a lens with a focal length of $z_{rec}$, when its object distance is $f_{gp}-z_h$ and the imaging distance is $f_{gp}+z_h$, where each object and image points correspond to the virtual and real imaging points due to the GP lens. Similar to the analysis about the NA of the hologram obtained in FINCH system [49], the NA of the hologram obtained in GPSIDH system can be represented as $r_h/z_{rec}$. For example, in Fig. 6(a), if f_gp increased from 100 mm to 500 mm, r_h increases about twice. However, $z_{rec}$ increases in five folds. Therefore, the NA of the hologram decreases faster when it comes to the longer f_gp.

The decreased z_h would demagnify the reconstructed image size severely due to the magnification factor of the system decreases as $z_h/f_o$. In increased r_h condition, but with the short $z_{rec}$, the fine fringe ring frequency can likely to generate the aliasing pattern. When the image sensor has a rectangular pixel where the size is ${\mathrm{\Delta }}_x={\mathrm{\Delta }}_y={\mathrm{\Delta }}_s$, the sampling frequency f_s is defined as $1/{\mathrm{\Delta }}_s$. The maximum local fringe frequency of the recorded Fresnel hologram have to be lower than f_s

to avoid the aliasing problem. This relation is described as Eq. (11) [36].

These kind of trade-off relations so far should be considered carefully to build the optimal GPSIDH system.

3. Experimental results

In this section, to demonstrate the compactness and the video recording capability of the proposed system, the recording experiments and their reconstruction results are presented. Firstly, to validate the system parameters with compactness, the GPSIDH system with three different focal lengths of the GP lens is evaluated. The first two GP lenses are the off-the-shelf model (ImagineOptix, USA), and their focal lengths are 50, and 100 mm at 550 nm. The lens diameter is about 1 inch. The other one is the custom-made model, where $f_{gp}\approx$ 164 mm at 525 nm, and its diameter is about 0.5 inch. The custom-made GP lens is fabricated by using a template convex lens with 200 mm focal length, and the recording beam wavelength of 457 nm. Even though the GP lens functions as a lens in a broadband spectral range, its focal length depends on the wavelength of input light, with the relation of $f_{gp}\left(\lambda \right)=f_t\left({\lambda }_t/\lambda \right)$ [52]. Ideally, the fabricated GP lens should have a focal length of 174 mm for the input wavelength $\lambda =$ 525 nm, which is calculated by the focal length of the template lens $f_t=$ 200 mm, writing beam wavelength ${\lambda }_t=$ 457 nm. The actual focal length is measured experimentally as 164 mm. The focal length is decided by repeatedly measuring the various object and imaging distances, using the resolution target shined by the 525 nm LED backward illumination. The disagreement between the ideal and experimental focal length is possibly due to the positioning issue of the optical components inside the fabrication interferometer, which is illustrated in Fig. 7.

 

Fig. 7 (a) illustration of the custom-made GP lens profile recording system with component labels, (b) polarization optical microscopic image of the fabricated GP lens sample. The axis with label P and A indicate polarizer and analyzer axis, respectively.

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To fabricate the custom-made GP lens, the similar methods are utilized such as the work of Kim et al. [29]. To begin with, a glass substrate is cleaned, then the photo-alignment layer of Brilliant Yellow (BY, Model No.: B0783, Tokyo Chemical Industry Co., Ltd., Japan) is formed above the glass to obtain the alignment properties of liquid crystalline polymer layer (RM257, Merck CO., Ltd. Germany). To implement the GP lens, the optical set-up with $\lambda =$ 457 nm (Cobolt Twist, Cobolt Co., Ltd., Sweden) is developed, whose two inferred beams of reference and writing beam have counterclockwise CP states, such as right handedness CP and left handedness CP, as shown in Fig. 7(a). A plano-convex lens with 200 mm focal length is utilized as a template lens inside the writing beam path to obtain the GP hologram interference pattern on the substrate. The writing beam and reference beam are illuminated on the BY coated substrate for 5 min with optical power of 50 mW/cm². After the photo-alignment process, a UV-curable liquid crystalline polymer, or the material referred as a reactive mesogen (RM), is spin-coated with 3000 rpm for 30 sec on the photo-aligned BY substrate to obtain the half wave plate optical property. The coated RM layer is baked on a hot plate to vaporize the residual solvent during 1 min with ${60}^{\circ }C$, then cured under the UV illumination with optical power of 10 mW/cm² for 1 min to implement the polymer film. The polarization optical microscopic (POM, Eclipse LV100POL, Nikon, Japan) image of thefabricated GP lens sample is presented in Fig. 7(b).

The photograph of the prototype system is presented in Fig. 8. Here, l_o is a general convex lens with $f_o=$ 100 mm. The monochromatic polarized image sensor (PHX050S-PC, LUCID Vision Labs, Canada) has the pixel resolution of 2448 × 2048, and only the central 2048 × 2048 pixels are utilized for the simplicity. Both horizontal and vertical pixel size of the image sensor is 3.45 μm. But to sample the complex hologram data, the sampling period can be regarded as the twice of the original pixel size, which is 6.9 μm, and the sampling number is 1024 × 1024. The bit-per-pixel, or the bit-depth of the sensor is set to 8-bit, which means that the obtained signal is sampled and stored with 256 levels. The sensor is connected to the personal computer with GigE Vision interface. The frame rate of the sensor is 25 frames per second, when the region of interest is set to 2048 × 2048 pixels. To hold all components as compact as possible, the 30 mm cage plates and SM1 lens tube from Thorlabs, Inc. are utilized, as shown in Fig. 8. The marginal area of GP lens is attached to the cage plate by using a sticky tape. The C-mount barrel of the image sensor is held by the custom-made adapter ring to fit into the cage plate.

 

Fig. 8 Photograph of compact GPSIDH video camera prototype

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Fig. 9 Reconstructed results from the various systems with $f_{gp}=$ 50 mm, 100 mm, 164 mm from the top to bottom rows, and $z_h=$ 18 mm, 23 mm, 28 mm from the left to right columns. The digits inside the figure indicate the PSNR values compared to the system results with $f_{gp}=$ 164 mm and the equivalent z_h.

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Figures 9 and 10 shows the recording results of the United States air force (USAF) 1951 negative target (R1DS1N, Thorlabs, USA) shined by the LED light (LED525E, Thorlabs, USA), where the lights are collected and converged by the 16 mm 1:1.4 lens. The target is located 100 mm in front of the entrance of the GPSIDH system. The central wavelength of the utilized LED is 525 nm, and the spectral bandwidth is 32 nm. The optical power measured at the entrance of the system is about 18.5 μW. The optical power measurement is operated using the off-the-shelf photodiode power sensor (S120C, Thorlabs, USA). On Fig. 9, f_gp of 50 mm, 100 mm, and 164 mm are employed inside the system subsequently from top to bottom rows, respectively. And z_h is varied with 18 mm, 23 mm, and 28 mm from left to right columns. The digits on the first and second rows indicate the peak signal-to-noise ratio (PSNR) values where the reference image is of $f_{gp}=$ 164 mm, with the equivalent z_h. The PSNR value represents the quality of the image compared to the referenced image, and the value gets lower if the compared image is degraded more [53]. The image size is almost same when z_h is equal. As z_h increases, the image size becomes larger. As confirmed in Fig. 5, the shorter focal length of the GP lens (f_gp) decreases the area of interference, where its radius is denoted as r_h. Accordingly, the NA of the system is decreased, as expected in Eqs. (9) and (10). The reconstruction results in Fig. 9 are also degraded as f_gp decreases when z_h is fixed. When z_h is increased, with the fixed f_gp, the reconstruction results are degraded by following Eqs. (9) and (10) as well, though the image is magnified more. The degradation of the reconstructed image with an inappropriate selection of f_gp and z_h is easily notified by examining the decreasing PSNR value as well.

 

Fig. 10 (a) reconstruction results of the system with $f_{gp}=$ 164 mm when z_h is changed from 18 mm to 63 mm, (b) chart of visibility values of group 2 element 2 of the USAF resolution target and the magnification factors of each sub-figures in (a).

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The target recording results for varying z_h are presented on Fig. 10(a), when $f_{gp}=$ 164 mm. And its visibility and magnification data of each reconstructed image is plotted on Fig. 10(b). Here, the visibility is calculated as $\left(I_{max}-I_{min}\right)/\left(I_{max}+I_{min}\right)$, where $I_{max}$ is the average intensity of target area, and $I_{min}$ is of the marginal area. In visibility calculation, the area of group 2 and element 2 is used. As expected, the magnification of the image increases with increasing z_h, whereas the visibility decreases due to the decreased NA of the system.

Lastly, to verify the video recording capability of the proposed system, the real-time holographic acquisition and its digital reconstruction are performed, as shown in Figs. 11 and 12. In this GPSIDH system, custom-made GP lens where $f_{gp}=$ 164 mm is utilized, and the GP lens to sensor distance z_h is 18 mm. The polarized beam splitter (CCM1-PBS251, Thorlabs, USA) is mounted between the l_o and l_gp, to serve as a horizontal polarizer to the transmitting beam, and also to be applied as a viewfinder by using the reflecting beam.

 

Fig. 11 (a) schematic illustration of the target configuration, (b) phase angle-only image of 91th frame of the recorded hologram video, (c,d) numerical reconstruction results of the hologram, where the best of focus is changed between the NBS and USAF targets. The images on (b-d) are cropped in 500 x 500 pixels. Also note Visualization 1 and Visualization 2, which is the hologram of two targets, while the digits on USAF target shifts vertically by the author’s manual control. On Visualization 1, the real-time captured hologram is represented as a phase-angle only video hologram, where the level of intensity from 0 to 255 represents the phase-angle from 0 to $2\pi$. On Visualization 2, the digitally reconstructed video hologram is presented.The first five seconds, the reconstruction plane is on the USAF 1951 target. The last five seconds, the plane is moved to focus on the NBS 1963A target. Visualization 2 is cropped to 500 x 500 pixels. BC: beam combiner.

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Two negative targets are used as the recording objects, one is the USAF 1951 target, and the other one is the National Bureau of Standards (NBS) 1963A resolution target (39-856, Edmund Optics, USA). Each target is arranged perpendicularly to one another and is aligned in one direction by a beam combiner to represent an object having the volume of two layers, as shown in Fig. 11(a). Both targets are shined by the green LED illuminating modules which are used in the previous result (on Fig. 9). The shined areas are the digits on the group 0 elements 2 to 6 of USAF 1951, and 4.0 cycles/mm of NBS 1963A targets. The NBS 1963A target is fixed and located at the front focal plane of l_o, and USAF 1951 target is located behind 12 mm from the NBS 1963A target. The real-time hologram of Fig. 11 and related movies Visualization 1 and Visualization 2 are taken while the USAF 1951 target is moving vertically, controlled manually by hand. The raw video data which contain four phase-shifted intensity profile for each frame is recorded and stored in the memory. Then the frames are digitally reconstructed sequentially and saved into a video file format. The digital reconstruction of each frame by using a conventional Fresnel back propagation method consumes about 0.5 seconds in MATLAB software without using a parallel computation (3.1 GHz Intel Core i5, 8GB memory). When it comes to the optical reconstruction, the processing of a raw image to an amplitude or phase-only hologram can be processed almost in real-time, with an optimized software design.

The recording of the reflective object is also demonstrated in Fig. 12, and Visualization 3 and Visualization 4. The white dice with 6 x 6 x 6 mm dimension is placed above the optical post and the post is rotated manually. The center of object to l_o distance is about 125 mm. The daily purpose LED spot light (JANSJÖ, Ikea, Sweden) is utilized to illuminate the dice. The band-pass filter (86-655, Edmund Optics, USA) is attached in front of the system to filter the 3000K broadband light into the narrow bandwidth of green light, where the central wavelength is 550 nm and the spectral bandwidth is 25 nm. In this case, the optical power measured at the entrance of the system is about 3.4 μW, after filtering with the band-pass filter. The numerical reconstruction results in Fig. 12 and the related video Visualization 3 contain higher noise level than the results from the transmissive object in Fig. 11, since the system is not fully optimized under such reflective lighting condition. If the wider selection of f_gp and z_h is possible, then the obtained hologram may have higher contrast in fringe pattern, and its reconstruction result is clearer than the current results. We think that the recording and reconstruction results present a potential of this system as a hologram video camera under a general illuminating condition. And we expect that the optical reconstruction of the real-time captured complex hologram is also available directly or with simple post-processing by using the advanced holographic display technologies [12–19].

 

Fig. 12 The recording and reconstruction result of the dice object fixed on the putty-like adhesive. (a) reference view, (b) the phase-only hologram data, (c) numerical reconstruction results of (b). Figures (b) and (c) are cropped in 500 x 500 pixels. The phase-angle only hologram and the digitally reconstructed videos of the rotating dice are presented on Visualization 3 and Visualization 4, respectively.

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4. Summary and conclusion

In summary, the compact and video recording capable GPSIDH system is developed and analyzed in this study based on our previous system. The system is configured with the band-pass filter, convex lens, polarizer, GP lens, and the polarized image sensor. The GP lens serves as a common-path polarization selective wavefront modulator. The polarized image sensor, collaborated with the polarizer and the GP lens, the parallel phase-shifting is achieved, which realizes the real-time hologram acquisition. The demonstration of the system performance under the compact form-factor within 4 cm is presented. The real-time hologram acquisition and its digitally reconstructed video is demonstrated in the speed of 25 frames per second, under the general incoherent light sources. Even with the compact optical path length of the system, the better hologram can be obtained by introducing an optimized selection of the focal length of the GP lens and the distance between the GP lens and sensor. We believe that the proposed system would be a great counterpart of the advanced future holographicdisplay.