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Abstract
In this paper, a holographic display system with adjustable viewing angle is proposed. The system consists of a collimated beam, a spatial light modulator (SLM), a multi-focus optofluidic (MFO) lens and an aperture. The MFO lens with high focal power is produced and it consists of two substrates, one multilayer substrate and two chambers. When the liquids are pulled in/out from the channels, the curvature of the liquid-liquid interface changes due to the surface tension and adsorption between the liquids and the multilayer substrate. The relationship between the parameters of the MFO lens and the holographic display viewing angle is revealed for the first time. Based on the theoretical analysis, the mechanisms of the high focal power and mechanical stability of the proposed MFO lens are also clarified. The experiments show that the focal power of the proposed MFO lens can be varied from −20 D (m−1) to 4 D (m−1), respectively. By using the MFO lens, the viewing angle of the holographic display system can be adjusted without moving any components mechanically. Meanwhile the setup of the system is greatly simplified. The experimental results verify the feasibility of the system, and it is expected to bring new ideas to the holographic display with large viewing angle.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the rapid development of computer and optoelectronic devices, the research of computer-generated holography (CGH) has attracted extensive attention due to the advantages of high efficiency, low cost, convenient information storage and transmission [1–3]. However, some inherent problems like tigers on the way, which limit the further development of the holographic display. One of the most important problems is the small viewing angle [4–6]. The light wave diffraction is limited by the sampling rate of the hologram. Currently, the pixel size of device that can be dynamically displayed is generally just a few microns such that the viewing angle of the CGH is only a few degrees, which seriously affects the viewing effect. Researchers have made many innovations in order to enlarge the viewing angle. For instance, the method by arranging multiple spatial light modulators (SLMs) in parallel was proposed to expand the perspective of holographic reproduction [7]. Although the system can obtain a large field of view, there are certain gaps between the SLMs. In 2011, a circular holographic system by splicing twelve SLMs together was proposed with each tilted SLM loaded a hologram which contained the information of a certain viewing angle of the three-dimensional object [8]. The SLMs were used as different windows and the reconstructed images can be seen from the corresponding windows. In order to use less SLMs, in 2017, two SLMs and space-time multiplexing method were utilized to expand the perspective of color holographic display system [9]. Considering that multi-SLMs splicing will increase the cost and complexity of the system, the Japanese researchers expanded the viewing angle of holographic display by using convex parabolic mirrors in 2018 [10]. Although this method enlarged the viewing angle to a certain degree, yet it not suitable for different types of SLMs in the systems. Moreover, time multiplexing method based on a high frequency SLM was also proposed [11,12]. In this method, three-dimensional images can be observed in a certain range of viewing angles according to the persistence feature of the human eyes, but it has somewhat stringent requirements to SLM. Thereupon, a method of large viewing angle, together with low cost and high integration is demanded extremely.
In recent years, the optofluidic lenses have found widespread applications in optical communication and imaging. Liquid crystal (LC) lens and liquid lens are the two promising designs to replace the solid lens [13–15]. Scholars employ liquid crystal lens instead of the traditional solid lens to adjust the size of the reconstructed image [16], realizing the continuous zoom function of 3.7 × . However, the proposed LC lens is polarization-independent, and the light energy loss is nearly 50%. Different types of liquid lenses also have been studied by various driving methods including the electrowetting method [17–19], the dielectrophoresis method [20–23], the hydraulic pressure method [24–27], the electric field force method [28–30], and many others [31–33]. As the liquid lens has the advantages of easy fabrication, variable-focus, easy to integration and polarization-independent, it is expected to inject new vitality into the holographic display. Liquid lens and liquid iris have been introduced in holographic systems to improve the quality of the reconstructed image including eliminating the undesirable light, the higher-order diffraction images and the chromatic aberration [34,35]. However, to the best of our knowledge, the research about the influence of the liquid lens on the holographic viewing angle has not been reported up to now. The main reason should be attributed to the insufficient utilization of the liquid lenses’ advantages in holographic display, thus the liquid lenses which precisely match the holographic system have not yet been developed and the stability of the liquid lens is still needed to be improved. The commercial liquid lens has a relative reliable mechanical stability, whereas they are always at great cost.
In this paper, a multi-focus optofluidic (MFO) lens with high focal power is proposed, to be used as an important component in the holographic display system. The key novelty of the proposed MFO lens is that a multi-focus function can be achieved. Therefore, the MFO lens can be adapted to different holographic display systems and realize a high mechanical stability due to the multiple surface tensions in the multilayer substrate. In this work, the constraint relationship between the parameters of the MFO lens and the holographic display viewing angle is revealed for its debut. The mathematical relationship model is constructed in theory to clarify the mechanisms of the high focal power and mechanical stability of the MFO lens. Based on the established mathematical models, the viewing angle of the holographic display system can be adjusted expediently. We believe that, perhaps in the future, the holographic display based on the optofluidic lens can really go to application.
2. Mechanism and operating principle
2.1 Principle of the holographic display system
The proposed system consists of a collimated beam, an SLM, an MFO lens and an aperture. Figure 1(a) is the principle of the holographic display system, where the SLM and the MFO lens are the two core components of the system. When the collimated light illuminates the SLM, the diffraction light passes through the MFO, then the reconstructed image can be captured by the CCD camera. The viewing angle of the reconstructed image is affected by the maximum diffraction angle θm, where θm = acrsin(λ/2p). However, in the viewing area of the reconstructed image, the entire image can be seen only in a specific area. According to the principle of holographic display, the viewing angle, the size of the SLM array and the focal length of the MFO lens satisfy the following equations [36,37]:
where d is the size of the reconstructed image, F is the focal length of the MFO lens, λ is the wavelength, and p is the pixel size of the SLM. According to the geometric analysis of Fig. 1(a), the viewing angle can be expressed as follows:where θ is the viewing angle of the CGH display, w represents the size of the SLM array. In the traditional holographic display system, solid lens is used to reconstruct the image. For a solid lens with the focal length of 500 mm, the viewing angle is less than 1°. In order to get an adjustable large viewing angle, the MFO lens with high focal power is proposed. From Eq. (1) we know that when the focal length of the MFO lens is decreased from F1 to F2, as shown in Fig. 1, the viewing angle of the CGH system can be enlarged from θ1 to θ2.
Fig. 1 Principle of the holographic display system. (a) Original state of the CGH system. (b) State when the focal length of MFO lens is varied.
2.2 Structure and mechanism of the MFO lens
Figure 2(a) shows the structure of the MFO lens. Two substrates, one multilayer substrate and two chambers are stacked to form the whole structure of the MFO lens. The two chambers are filled with two immiscible transparent liquids, respectively. The two chambers are connected with pumping syringes through the channels for pulling in/out the liquids. When liquid-1 is pulled in from chamber-1 and liquid-2 is pulled out from chamber-2, the curvature of the liquid-liquid (L-L) interface changes accordingly on account of the surface tension and adsorption between the liquids and multilayer substrate, as shown in Fig. 2(b). The L-L interface is always concave during the driving process. Hence, the focal length of the MFO lens changes continuously as the yellow dotted line shows. Due to the multilayer substrate, the surface tension and adsorption are enlarged which can enhance the focal power of the MFO lens. While, when liquid-2 is pulled in from chamber-2 and liquid-1 is pulled out from cchamber-1, L-L interface can form lens types in different layers of the multilayer substrate. Each layer can make the L-L interface change from concavity to convexity, as the yellow dotted line shows in Fig. 2(c). Therefore, the MFO lens can achieve the multi-focus function in a relatively simple way.
Fig. 2 Mechanism of the proposed MFO lens. (a) Structure of the MFO lens. (b) State of injecting the liquid. (c) State of extracting the liquid.
2.3 Theory of the high focal power of the MFO lens and the fabrication
Most traditional mechanical-wetting liquid lens usually interact with one solid substrate, so the curvature change is limited. The proposed MFO lens has multilayer substrate, and it has multiple actions between the liquids and substrates. The schematic of the theoretical analysis is shown in Fig. 3.
We take first sublayer and second sublayer as a subject for study. According to the Laplace Equation, the total additional pressure ΔP can be calculated by the following equation:
where ΔP1 is the additional pressure derived from surface tension f1 without multilayer substrate, and ΔP2 is the additional pressure derived from surface tension f2 with the multilayer substrate. R is the radius of the MFO lens. α2 is the included angle between f2 and the vertical component f2v. Since ΔP > ΔP1, we can draw a conclusion that the proposed MFO lens has a higher focal power than that without multilayer substrate. ΔP1 and ΔP2 can be calculated by the fowling equations:
where γ is the surface tension coefficient, R1 is radius of the one liquid lens without multilayer substrate, a is the height of the second layer, b is the space between the two layers, β is the included angle between a and b, and r1, r2 is the radius of the two layers, respectively.For a lens, the focal length can be expressed as follows:
where F is the focal length of the lens, n1 and n2 are the refractive indexes of the two liquids in the MFO lens. Substituting Eqs. (3)-(8) into Eq. (2) we can obtain Eq. (9):Hence, by using the MFO lens, the viewing angle of the holographic system can be adjusted by controlling the corresponding parameters of the MFO lens.
To obtain a relationship between the volume change ΔV and the focal length of the MFO lens, we consider a general case wherein the interface curvature can be convex or concave during the actuating process. Assuming that, the L-L interface is spherical, the volume change is sum of the volume of the spherical cap of the interface and the half volume of the liquid on the sublayers, approximately. That is,
where Ri is the radius of the certain L-L interface, and ri is the radius of the certain sublayer. Based on Eqs. (9)-(11), the physical mechanism between the MFO lens and the viewing angle of the CGH system is established.The fabrication is described as follows. A polymethylmethacrylate (PMMA) substrate fabricated with a channel is designed as the chamber. The height and radius of the chamber are 5 mm and 9 mm, respectively. The diameters of the two channels are both 1 mm. The multilayer substrate is designed with four annuluses and their radiuses are 5.5 mm, 4.5 mm, 3.5 mm and 2.5 mm, respectively. The heights of the annuluses are both 1 mm, as shown in Fig. 4(a). Then the two cylindrical chambers, the multilayer substrate and two substrates are stacked together by using UV-331 glue, as shown in Fig. 4(b). The total height of the MFO lens is 15 mm. The normal saline is used as liquid-1 (a density of 1.03 g/cm3 @ 25°C, the viscosity is 1.1 mpa∙s, and the refractive index n1 = 1.33). Phenylmethyl silicone oil (a density of 1.03 g/cm3 @ 25°C, the viscosity is 150 mpa∙s, and the refractive index n2 = 1.48) is used as liquid-2. The densities of the liquids filled in the chambers are matched. Hence the MFO lens can have a reasonable mechanical stability.
3. Experiments, results and discussion
3.1 Experiments and optical properties of the MFO lens
To evaluate the performance of the MFO lens during actuation process, a CCD camera is used to record the image changes. The printed letter ‘a’ array is placed 3 mm away below the MFO lens. When liquid-2 is pulled out from chanmber-2 through channel-2, the images under different liquid volume are shown in Fig. 5. In the initial state, the L-L interface forms a convex shape, and the image is amplified, as shown in Fig. 5(a). When the volume of the injected liquid is increased from ~60 μl to ~260 μl, the shapes of the L-L interface changes from convex to concave and the size of the image is reduced accordingly, as shown in Figs. 5(b)-5(f). In our experiments, we inject the two liquids into the channels using a fluid pump (Longer Pump TS-1B, China). The speed of the pump is 50 μl/s. Thus, the measured response time is ~5.2 s. When the volume change ΔV >260 μl, the focal power cannot be enlarged anymore because liquid-1 has already filled with the multilayer substrate.
Fig. 5 Focal length changes of the MFO lens when liquid-2 is pulled out from channel-2. (a) ΔV = 0 μl. (b) ΔV = 20 μl. (c) ΔV = 40 μl. (d) ΔV = 120 μl. (e) ΔV = 200 μl. (f) ΔV = 260 μl.
The focal length changes of the MFO lens are also calculated, as shown in Fig. 6. The experiment demonstrates that the focal length can be varied from -∞ to −5 cm and 25 cm to + ∞ when the volume changes from 0 μl to 260 μl. That is to say, the focal power can be varied from −20 D (m−1) to 4 D (m−1).
When liquid-2 is pulled in from channel-2, the MFO lens switches to the function of the multi-focus. Each sublayer can form a single liquid lens. Hence, the focal length changes of the MFO lens in liquid-pulled-in model are quite different from the liquid-pulled-out model. The focal lengths of each single liquid lens are calculated in Table 1. We also measured the transmittance of the MFO lens using spectrograph, as shown in Fig. 7. The transmittance in visible light is above ~83%.
Table 1. Focal length changes of the MFO lens in liquid-pulled-in model.
3.2 Experiments of the holographic system
The structure of the holographic display system is shown in Fig. 8. A laser, a filter and a solid lens are used to generate a collimated light. The novel-look-up-table (NLUT) algorithm is used to generate the hologram. The principal fringe pattern (PFP) is pre-calculated and the hologram pattern can be generated by using shifting and adding operations of PFP based on the property of shift invariance. In the holographic system, the wavelength of the laser is 526.5 nm. The CCD camera is from Nikon (D810). The type of the SLM is reflective, and its pixel number and array size are 1920 × 1080 and 15.36 mm × 8.64 mm, respectively. Then a letter “W” is used as the object.
When a solid lens with the focal length of 500 mm is used to reconstruct the image based on the traditional holographic system, the viewing angle is less than 1° and completely reconstructed image cannot be seen out of the viewing angle. Then the experiments are conducted based on the proposed system. In order to record the viewing angle clearly, a scale is fixed behind the MFO lens. In the initial state, the camera is placed behind the scale. The focal length of the MFO lens is set to 200 mm. Figure 9(a) is the original calculated image of ‘W’. When the reconstructed image is displayed in the leftmost area, the position is recorded, as shown in Fig. 9(b). Next, the observing angle is gradually moved, as shown in Fig. 9(c). When the observing angle is moved to a specific position on the right side, we just cannot see the reconstructed image completely, as shown in Fig. 9(d). So, in this way, the viewing angle of the proposed system can be calculated experimentally.
Fig. 9 Reconstructed image using the proposed system. (a) Original calculated image of ‘W’; (b) Left viewing angle @ F = 200mm; (c) Middle viewing angle @ F = 200mm; (d) Right viewing angle @ F = 200mm.
Based on the mathematical model, the viewing angle of the reconstructed image can be controlled by adjusting the corresponding parameters of the SLM. When the focal length of the MFO lens changes, the reconstructed images are recorded from the left, middle and right sides respectively in order to observe the viewing angle. Figure 10 is the reconstructed image with different viewing angles. It is apparent that when the focal length of the MFO lens becomes smaller, the viewing angle of the reconstructed image becomes larger. Figure 10(a) is the original calculated image of ‘W’. When the focal length of the MFO lens is ~150 mm, the viewing angle is ~2.93°, as shown in Figs. 10(b)-10(d). While, when the focal length of the MFO lens is ~100 mm, the viewing angle of the proposed system is ~4.39°, as shown in Figs. 10(e)-10(g). Therefore, the reconstructed image can be seen with a larger viewing angle compared with the traditional holographic system. The proposed MFO lens has a higher power compared with the traditional liquid lens, so it can bring a large viewing angle to the holographic display. The experimental and theoretical results may be biased. There are two possible reasons: i) The boundary between left field of view and right field of view is determined by human vision. Thus, it may induce some errors; ii) The two liquids filled in the MFO lens are density-matched on the conditions of room temperature. The densities of the liquids may be varied due to the external environment such as temperature which would lead to the aberrations.
Fig. 10 Reconstructed image using the proposed system. (a) Original calculated image of ‘W’; (b) Left viewing angle @ F = 150mm; (c) Middle viewing angle @ F = 150mm; (d) Right viewing angle @ F = 150mm. (e) Left viewing angle @ F = 100mm; (f) Middle viewing angle @ F = 100mm; (g) Right viewing angle @ F = 100mm.
Besides, we also use a 3D object in the experiment. The letters ‘B’ and ‘H’ are locating at different depths. Figure 11(a) is the result of the reproduced images when ‘B’ is focused. When the focal length of the MFO lens changes, the result is shown in Fig. 11(b) and we can see that the letter ‘H’ is focused.
According to the established mathematical model, the viewing angle of the reconstructed image can be changed easily by adjusting the volume of injected liquid, where Fig. 12 is the relationship between the focal length of the MFO lens and the viewing angle. In the proposed system, only an SLM is used to reconstruct the image. The viewing angle can be further enlarged by using time multiplexing or spatial multiplexing based on the SLM.
3.3 Discussion
The unique advantages of using the MFO lens are its zoom function and the large range of focal power. The traditional zoom systems realize zoom function by changing the positions of several solid lenses mechanically, which are always complicated and cost much. With the proposed MFO lens, holographic zoom display can be realized easily by controlling the phase distribution of the hologram and the focal lengths of the MFO lens without moving the positions of system components. Holographic zoom system has been built and the results are shown in Fig. 13. From the result we can see that the reconstructed images with different magnifications M can be realized. As depicted in Fig. 13(a) and Fig. 13(c). The measured magnifications are ~0.9 and ~1.1, respectively.
In the experiment, the precise focal length control, repeatability and mechanical stability are the main technical problems for the MFO lens. For most of the electrowetting-actuated or electric control liquid lens, the repeatability can be severely affected due to the high driving voltage (usually above 60 V) for a long time. For the elastic membrane based liquid lens, driving for a long time, the elastic film will be also damaged. While the driving mechanism of the MFO lens are the so called mechanical-wetting. The MFO lens doesn’t need to employ the elastic membrane and external voltage. Thus, it has a reasonable mechanical stability. The liquids filled in the MFO lens are density-matched. Thus, it can avoid the effect of gravity. The MFO lens is controlled by a pumping syringe whose minimum flow is 2.3 nl/s. Thus, the focal length can be changed accurately. To sum up, the proposed MFO lens has a reasonable repeatability and precise focal length control. In the design of the MFO lens, we only used four sublayers as the middle substrate. The maximum focal power can be −20 D (m−1). If we add more sublayers, the focal power can be further enlarged. The response time of the MFO lens is ~5.2 s. So, it is not easy to realize the real-time switching. In the future, we will consider using another electric drive to increase the response speed of the MFO lens. When the properties of the MFO lens improve, holographic display systems with high performance will be explored further.
From Eqs. (1)-(2) we can see that the size and perspective of the reconstructed images are a pair of mutually restricted parameters. However, the space-bandwidth product is an important measurement for comparison in holographic display systems. When the viewing angle of the reconstructed image increases, the spatial resolution of the reconstructed images will be decreased. As can be seen from Eq. (2), another effective way to increase the viewing angle is to use a large SLM array. So, the viewing angle can be expanded by using multiple SLMs stitching methods.
Nowadays, many researchers hope to realize near-eye display system or AR display based on holographic method. The proposed system has a very simple structure and small size, so it has unique advantages in the near-eye display systems. In the next work, we will study holographic zoom system and near-eye display system further based on the proposed system. With the optimization of system performance, the proposed system will bring more new ideas to holographic display development.
4. Conclusion
A holographic system with adjustable viewing angle based on a new MFO lens is proposed in this paper. The relationship between the parameters of the MFO lens and the holographic display viewing angle in theory is exposed for the first time. Meanwhile, the mechanisms of the high focal power and mechanical stability of the proposed MFO lens are clarified as well. The proposed MFO lens can be used to enlarge the viewing angle of the CGH system. The experiments show that the focal power of the MFO lens can be varied ranging from −20 D (m−1) to 4 D (m−1). We hope that our research can make a modest contribution to the development of holography.
Funding
National Natural Science Foundation of China under Grant No. 61805169, 61805130, and 61535007; China Postdoctoral Science Foundation under Grant No. 2019M650421 and 2019M650422.
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