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Abstract
We present an improved two-in-one polarization-interlaced liquid-crystal-on-silicon (LCoS) stereoscopic projection prototype employing a novel prism-array configuration and a specially designed illumination freeform lens group. The parallel prism configuration is designed based on the balance analysis between stereoscopic channels. For further system simplification, the illumination lens group, which consists of three prepositive aspherical surfaces and a single postpositive freeform one, is synthetically obtained from the Monge–Ampère method and feedback optimization. Design results show that the proposed prototype can well solve the problem of stereo-channel separation and integration, and provide both better performance and lower volume. It is proven to have potentiality replacing existing stereoscopic projectors.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The external world is a three-dimensional (3D) one captured through human eyes and processed by the human brain. The 3D display technologies, reproducing real-word information, provide the ultimate direction for information displays [1,2]. Stereoscopic vision can be achieved by some visually aided optical devices and some glass-free methods, namely, binocular stereoscopic display technologies and autostereoscopic 3D display technologies, respectively [3]. Two slightly different images (stereoscopic pairs) are captured and projected to the right and left eyes to yield 3D sensation. Compared with autostereoscopic 3D displays, binocular stereoscopic displays can adapt to larger amount of data processing and cause lower degree of visual fatigue, dizziness, or vomiting. The big challenge facing binocular stereoscopic display techniques, which can be referred to as the stereo-channel separation problem, is the proper separation of stereoscopic image pairs [4,5].
In current binocular stereoscopic display techniques, the stereoscopic projection displays, as the most promising competitor, are more suitable for high-quality 3D video output that can be watched simultaneously with different fields of view (FOVs). The optical performance and structure of a stereoscopic projector strongly depend on the built-in microdisplay panel, which usually has a two-dimensional pixel array. Based on the operational principles of various panels, different information sources can be used to separate the stereoscopic projection channels, such as the temporal characteristic, polarization state, and color spectrum. Stereoscopic projectors using a digital micromirror device (DMD) requires a strict constraint relationship of temporal synchronization between the image switching frequency and the optical shutter glasses [6]. Besides, the natural polarizing property of the light valves, such as the transmissive-type LCD and the reflective-type LCoS, provides another reliable stereo-channel separation solution [7]. The LCoS microdisplay technology, which has a liquid crystal screen injected and encapsulated in a reflected silicon backplane with a layer of indium tin oxide film, can provide extendible display resolution, high frame rate, and foldable optical architecture for stereoscopic projectors [8,9].
For polarization-interlaced stereoscopic displays, two separate projection engines are usually used to twist the original polarization direction orthogonally to project left- and right-eye images. The polarization-preserving screens, such as aluminized screens or translucent opaque acrylic screens, are needed to avoid binocular rivalry. Such a system allows multiple viewers to watch simultaneously without the spatial discontinuity caused by visual field switching. One of the typical applications is integral imaging 3D displays [10,11]. The polarization-interlaced projectors with two independent projection engines inevitably have inherent limitations of larger space occupation ratio and higher energy consumption. Moreover, the image and polarization direction need to be accurately aligned in real time [12]. For reducing the volume, Takizawa presented a polarization-interlaced stereoscopic projection system, which integrates two engines into a single one [13]. This type of system was further developed by using an LED illumination system, which is composed of a compound parabolic concentrator and a microlens array [14]. The critical challenge of designing a two-in-one polarization-interlaced stereoscopic projector lies in the problem of stereo-channel separation and integration while achieving a balance between performance improvement and structural miniaturization.
The illumination optical elements greatly affect both the performance and structure of the projection engine. Lens arrays or light pipes are usually employed as integrating elements of conventional LCoS projection engines [15]. Their limitations are obvious in that they need a complicated structure consisting of at least five optical elements, and the long back-working distance is unfavorable for structural miniaturization. These traditional illumination elements should not be the first choice in a two-in-one polarization-interlaced stereoscopic engine. Beside these, freeform optical elements appear more promising in that they can be described by any non-rotationally symmetric surfaces or microarray surfaces, i.e., NURBS, XY polynomial, and radial basis function representation, etc. Because freeform surfaces offer more design freedom than traditional ones, there has been no unified design method until now; thus, the methods are determined from specific applications. It can be concluded that two main categories are commonly used to design freeform optics: zero-étendue algorithms based on ideal source assumption, or optimization algorithms for extended light sources [16]. The rotationally symmetric freeform lens suffers significant energy loss due to the mismatch between the lens aperture and the rectangular active aperture of the LCoS panel [17,18]. Other non-rotationally symmetric lenses containing the functionality of circular-to-elliptical or circular-to-rectangular beam shaping can significantly improve the system efficiency [19–21]. Therefore, freeform illumination optics should be the most promising approaches for achieving the balance between structure and performance of stereoscopic projectors. However, the design principles of freeform optics need to be reconsidered under the specific application. To the best of our knowledge, freeform-surface design methods have not been reported for designing the two-in-one polarization-interlaced stereoscopic projection engines.
Based on this, this paper presents an improved two-in-one polarization-interlaced LCoS stereoscopic projection prototype employing a novel prism-array configuration and a specially designed freeform lens group. The design principles of the proposed illumination optics are presented in detail based on a thorough analysis of stereo-channel separation and integration. Design examples are given to explore the optical characteristics and performance of the designed two-in-one polarization-interlaced stereoscopic projection prototype. Existing prism configurations are also compared to show the effectiveness of our design.
2. Prism configuration for two-in-one polarization-interlaced stereoscopic projection
The optical path length and the polarization state are two essential factors for the balance between spectroscopic channels in two-in-one polarization-interlaced stereoscopic projection. The MacNeille-type polarizing beam splitter prisms (PBSP) are usually used for light separation and integration of two orthogonal polarization states. Two identical LCoS panels embedded in a single projection engine are employed for left- and right-eye image outputs, respectively, so that the functionality of two independent optical paths is incorporated into a single one. Before striking the left- and right-eye LCoS panels, the light beam should be separated into the parallel-polarized light (p-polarized light) and senkrecht-polarized light (s-polarized light) by a prism polarizing spectroscopic film (PPSF) coated on a certain interface of the PBSP. Then, polarization twisting takes place when the light beam passes through the LCoS modulator, leading to the interconversion between p- and s-polarized light.
2.1 Single prism configuration and the limitation
Figure 1 shows the light propagation in two spectroscopic channels where only one PBSP is used. Such an optical configuration denoted by the 1-PBSP configuration allows both the p- and s-polarized light to have the same optical path during the light propagation. In the left-eye channel shown in Fig. 1(a), the incident s-polarized light beam is firstly reflected by the PPSF and propagates toward the left-eye LCoS panel (L-LCoS). After modulation by the L-LCoS panel, the reflected s-polarized light beam is converted into p-polarized light, and then passes through the PPSF with its propagation direction unchanged. Figure 1(b) shows the light propagation process in the right-eye channel. From this figure, we can see that the incident p-polarized light beam passes through the PPSF with its propagation direction unchanged and then is reflected by the right-eye LCoS panel (R-LCoS) to become s-polarized light. After that, the s-polarized light beam is reflected by the PPSF and exits from the prism system. It is obvious that the optical path lengths of the p- and s-polarized light beams are the same in two spectroscopic channels.
Fig. 1 Light propagation process of the 1-PBSP configuration. (a) Left-eye channel modulated by the L-LCoS panel; (b) right-eye channel modulated by the R-LCoS panel. In this figure, p stands for p-polarized light and s for s-polarized light.
We assume that σ and τ denote the visible transmittance and reflectance of the PBSP, the corresponding subscript representing p- or s-polarized light, respectively. In the bright state of the LCoS panel, the total energy of the outgoing beam after propagating through the left- and right-eye channels can be given by
where ФLED is the energy emitted from the LED light sources and τLCoS denotes the average visible reflectance of the LCoS panel. Obviously, ФL1 = ФL2. Furthermore, we let Ф’ represent the total output energy of the dark state where the light beam will be reflected by the LCoS without twisting the original polarization state. From the left-eye channel, we know that the luminous flux of the dark state in the L-LCoS channel can be written asDividing ФL1 by Ф’L1 yields ΛL1 which is defined as the ratio between the energy of the bright and dark state in the left-eye channelSimilarly, the ratio ΛR1 of the right-eye channel is given by
Although the total output energies of the spectroscopic channels are the same in the bright state, as shown in Eq. (1), from Eqs. (3) and (4) we can clearly see that ΛL1 is not equal to ΛR1. This means that the balance between the stereoscopic channels cannot be achieved by using the 1-PBSP configuration. The unbalance is mainly caused by the transmittance/reflectance difference of p- and s-polarized light beams in the dark state of the LCoS. For a commonly used PBSP, ΛL1>1000 and ΛR1<100. Consequently, the images captured from the left- and right-eye channels should be relatively different. Such a significant imbalance between the two channels can dramatically deteriorate the stereoscopic visual performance. That is the main reason why the 1-PBSP configuration cannot be used in a two-in-one polarization-interlaced stereoscopic projector.2.2 Improved prism array configuration
To achieve a balance between stereoscopic channels, a new optical configuration consisting of two parallel PBSPs is presented in Fig. 2, namely, the 2-PBSP configuration. Although the optical path length is greater than that of the 1-PBSP configuration, the refraction/reflection times of propagation through PPSF still stay the same between the stereoscopic channels of the 2-PBSP configuration. In Fig. 2, a quarter wavelength retardation plate (QWP), together with a reflecting plate, is placed at one side of the PBSP, which is opposite to the LCoS panel, and a half-wave retardation plate (HWP) is placed between two PBSPs. These devices are used to realize the mutual conversion between p- and s-polarized light. The broadband QWPs have been developed by using a subwavelength grating structure [22], two twisted nematic liquid-crystal cells [23], and multiple twisted birefringent layers on a single substrate layer [24]. Excellent achromaticity QWPs are now commercially available during the visible light range. Figures 2(a) and 2(b) show the light propagation process in the left- and the right-eye channels, respectively. From these two figures, we can see clearly that the s-polarized light beam experiences reflection twice and the p-polarized light beam experiences refraction twice in both the right- and the left-eye channels. Moreover, the optical path lengths of the two stereoscopic channels are still the same.
Fig. 2 Light propagation process of the improved 2-PBSP configuration. (a) Left-eye channel modulated by the L-LCoS panel; (b) right-eye channel modulated by the R-LCoS panel.
In Fig. 2(a), the total energy of the outgoing beam, ΦL2, after passing through the left-eye channel can be written as
Similarly, the incident beam also experiences both reflection and refraction twice by the PPSF shown in Fig. 2(b), and the total energy of the outgoing beam, ΦR2, is given byFrom Eqs. (5) and (6), we can see the output energies from the two channels are equal to each other in the bright state of the LCoS panel. We assume that Ф’L2 and Ф’R2, respectively, represent the total energy of the dark state in the left- and right-eye channels. The light propagation from the stereoscopic channels tells uswhere, in the first equation, τP reveals that there still exists a small amount of p-polarized light reflected by PPSF without the modulation of the L-LCoS panel. σS indicates that this part is converted into s-polarized light while passing through the HWP, and transmitted through PBSP directly into the projection lens. Similarly, the second equation of Eq. (7) also reveals that a small amount of s-polarized light should directly enter the projection lens.We assume that ΛL2 and ΛR2 are the ratios of the total energy of the outgoing beam in the left- or right-channel to the energy of the dark state in this channel. Therefore, the contrasts from the light paths of L-LCoS and R-LCoS — ΛL2 and ΛR2 — can be expressed as
Obviously, ΛL2 = ΛR2. This means that a theoretical balance between the left- and right-eye channels can be achieved in the proposed 2-PBSP configuration.Crosstalk, also known as ghosting or leakage, is another primary factor affecting the image quality of stereoscopic displays. High levels of crosstalk can make stereoscopic images difficult to fuse and lack fidelity. Thus, it is important to achieve low levels of crosstalk in developing high-quality stereoscopic displays [25]. In the 2-PBSP configuration, the levels of crosstalk in the left- and right-eye channels are the same and can be written as [26]
where KL and KR are the levels of crosstalk in the two channels. As mentioned above, σP/σS>1000 and τS/τP<100. We know that the theoretical values of KL and KR are approximately equal to zero. The above analysis reveals that the performance balance could be achieved between stereoscopic channels within the proposed 2-PBSP configuration.3. Advanced illumination engine for stereoscopic channels
Besides the improved prism array presented above, the illumination engine is another important part of a stereoscopic projection display system, because the performance of the display system is also strongly determined by the performance of the illumination engine inside. For making the whole display system more compact and lightweight, freeform surfaces are used here to regulate the light propagation flexibly. The proposed illumination engine includes one aspherical lens and a freeform lens. The light rays emitted from the LED light source are firstly collected and redirected by the first aspherical lens, and then refracted by the second freeform lens to produce a predefined uniform rectangular illumination on the observation plane.
3.1 Design principle of the prepositive aspherical surfaces for beam collimation
Light refraction process is a key problem for a rotationally symmetric collimation system. We may use one or two aspherical surfaces instead of three; however, one or two aspherical surfaces may lead to more Fresnel loss on the optical surface of each lens when the spread angle of the incoming beam is large. Therefore, the deflection angles after each surface can be respectively predefined [17]. This means that each surface holds a definite relation between the incident and emergent light. According to these angles, the one-to-one relation among LED emitting distribution – aspherical surface I – aspherical surface II – aspherical surface III – freeform surface – prescribed collimated illumination on LCoS area, can be established based on the zero-étendue algorithm.
Figure 3 shows the lens group with three prepositive aspherical surfaces for light collection and collimation. θ and r represent the radiation angle of the LED chip and the illuminating radius on the LCoS area, respectively, which establishes an angle-to-area mapping relation. Therefore, the emitted light energy within the aperture angle will be directed toward the circular illumination spot with a maximum radius.
The LED’s Lambertian beam is narrowed by the collection lens into a smaller diffusion angle. The coordinates of any point P1 on the aspherical surface I are defined as (z1, y1); the relation between (z1, y1) and θ can be expressed as
where z1 and y1 are unknown parameters to be solved. Assuming that α1 is the angle between the normal direction of point P1 and the light axis, the Snell’s law for this surface can be written aswhere β1 is the deflection angle between the emergent light and the optical axis, and can be pre-defined as θ/2, which determines the slope of each point on the surface. According to the definition of slope [17], the slope of P1 can be extended as follows:where dy1/dz1 has a definite solution based on the predefined deflection angle. The relation between z1 and θ can be obtained by combining Eq. (12) and the derivation of Eq. (10).Equation (13) provides an angle-to-area relation between z1 and θ. Initial conditions for solving Eq. (13) are θ = 0 and z1 = d0.
A similar algorithm is applied to calculate aspherical surfaces II and III. We define the deflection angle of aspherical surface II as β2 = θ/3, The relation between a certain point P2(z2, y2) and θ can be expressed as
where z1 and dz1/dθ can be calculated from Eq. (13). The initial conditions of Eq. (14) are θ = 0 and z2 = d0 + d1.For aspherical surface III, the deflection angle should be theoretically considered zero due to the collimating characteristic. Similarly, a certain point P3(z3, y3) on the aspherical surface III can be obtained as
The initial conditions of Eq. (15) are θ = 0 and z3 = d0 + d1 + d2. Based on the predefined β1 = θ/2 and β2 = θ/3, the coordinates of aspherical surface I, II, and III can be uniquely determined by solving these differential equations iteratively. Finally, the design of the first three aspherical surfaces can be converted into three ordinary differential equations (ODEs) due to the rotational symmetry of each aspherical surface. By predefining the amount of deflection of light occurring on each aspherical surface for each ray, the three ODEs can be numerically solved.
3.2 Design principle of the postpositive freeform surface for beam shaping and homogenization
The emergent rays should be perpendicular to the planar surface of the second collimating lens, and the corresponding propagation direction will not be changed while passing through the planar surface. We could certainly design a smooth freeform surface with non-rotational symmetry to replace this planar surface to redirect those parallel light rays, as shown in Fig. 4(a). Compared with the design of the three aspherical surfaces, the design of the freeform surface is more challenging. We need to design a freeform surface by which the incident collimated rays are redirected to produce the prescribed uniform rectangular illumination on the target plane. The current methods that can be used to solve this design problem include the ray mapping method [27,28], supporting quadric method [29], and Monge–Ampère equation (MA) design method [30,31]. Among these methods, the MA method may be the most advanced zero-étendue algorithm that satisfies the integrability condition automatically and can be implemented efficiently [32]. Thus, the MA method is used here to design the exit freeform surface of the collimating lens. The MA method [33] tells us that, by Snell’s law and the local conservation law of energy, the design of the freeform surface can be converted into an MA equation with a nonlinear boundary condition, which can be written as
where I(x, y) is the intensity distribution of the collimated beam and E(tx, ty) is the prescribed illuminance distribution; S1 and S2 denote the cross-sections of the collimated beam and the target illumination area, respectively; ∂S1 and ∂S2 are the boundaries of S1 and S2, respectively, as shown in Fig. 4(b). The virtual receiver is a plane placed between the entrance and exit surfaces of the collimating lens. Thus, I(x, y) is the illuminance distribution produced by the collimated beam on the virtual plane. The freeform surface is numerically calculated by solving Eq. (16) using the Newton-Raphson method.Fig. 4 (a) Beam collimation by the first three aspherical surfaces and shaping by the freeform exit surface; (b) design principle of the freeform exit surface.
To make the beam-shaping system compact, the LED source is placed close to the collection lens so that the “five times rule” is not obeyed [34]. This means that the light source cannot be considered a point source and the influence of the spread angle of the incident beam on the design of the freeform surface cannot be ignored. Since the MA method is a zero-étendue algorithm and the collimated rays are assumed parallel to the optical axis, the actual illumination produced by the freeform surface will deviate significantly from the prescribed one due to the spread angle of the incident beam. However, this does not mean that the MA method becomes invalid when the “five times rule” is not obeyed. The feedback iteration is further employed here to improve the design [35].
We let E0(x, y) and E’i(x, y) denote the prescribed illuminance distribution and the actual illuminance distribution, respectively, obtained from the i-th iteration. Assuming that the stopping criterion is not met after the i-th iteration, the design still needs to be improved in the (i + 1)-th iteration. According to the feedback method [35], the updated target illuminance Ei+1(x, y), used to re-design the freeform surface in the (i + 1)-th iteration, can be written as
where Fi(x, y) is the feedback function for the i-th iteration, which can be defined aswhere γ is a parameter that can be changed from 0 to 1, and the total power of E’i(x, y) within the target area is normalized to be equal to that of E0(x, y). Equations (17) and (18) clearly reveal the differences between the actual design and the target accumulated and included in the feedback function. After Ei+1(x, y) is obtained, E(x, y) in Eq. (16) is replaced by Ei+1(x, y) and the freeform surface is redesigned. Because of the nature of the feedback design, the design can converge very fast and stably.4. Design results
The design parameters of the LCoS panel are given in Table 1. The illumination system and the imaging system are two basic parts of a complete two-in-one polarization-interlaced stereoscopic projection prototype. The two parts are designed separately and then assembled together to form the proposed projection system.
Table 1. Parameters of the LCoS Panel
4.1 Freeform illumination lens group
The lens parameters of the advanced illumination engine are listed in Table 2. The three aspherical surfaces and the planar surface are numerically calculated, and the surface profiles are defined by a series of coordinate point data, as shown in Fig. 5(a). One million rays are traced to evaluate the performance of the illumination optics. As can be observed in Fig. 5(b), the maximum divergence angle of the incident rays is less than ± 12°, and over 85% of the light energy can be concentrated within ± 8°. The collimation results can provide a good starting point for further freeform surface design.
Table 2. Lens Parameters of the Advanced Illumination Engine
Fig. 5 (a) Surface profile curves and (b) the spatial intensity distribution of the designed aspherical lens group with the planar exit surface. 3D model and the corresponding illuminance distribution of (c) the initial and (d) the optimized freeform surface.
Figures 5(c) and 5(d) show the initial and optimized freeform surfaces, and the corresponding illuminance distribution maps. The illumination spot can be converted into either a rectangular one by the initial or the optimized lens. Compared with the rotationally symmetric illumination optics, a larger part of the illumination energy can be collected into the active aperture of the LCoS panel, facilitating higher transmission efficiency. In Fig. 5(c), an obvious center illuminance defect can be observed while the initial lens is traced. However, after the feedback iteration method is applied, the uniformity of the illuminance distribution can be significantly improved, as can be seen in Fig. 5(d).
4.2 Improved prism array
Figure 6(a) gives the main characteristics of the PPSF within the 2-PBSP configuration, including the reflectivity varying with the divergence angle of incidence (DAOI) and the wavelength. It can be found that the reflectivity of both p- and s-polarized light is greater than 93% while the incident angle is over 40°. Reflectivity change is more stable against the incident angle than the wavelength. As shown in the sub-picture of Fig. 6(a), the reflectivity has an obvious fluctuation around 420 – 500 nm, and then tends to be stable with the wavelength range between 520 nm and 680 nm. Therefore, the central wavelength of the LED sources can be determined according to these curves. In this paper, the chosen central wavelengths of three primary colors are 458 nm, 525 nm, and 648 nm, corresponding to reflectivities of 86.6%, 97.2%, and 87.8%, respectively. Based on the above parameters, the 2-PBSP system is established and evaluated. Each prism is a cube structure with a side length of 16 mm and ZF6 (CDGM glass CO., LTD, China) as the material. Figures 6(b) and 6(c) show the ray tracing results of light separation and integration of stereoscopic channels among the 2-PBSP configurations, respectively.
Fig. 6 (a) Reflectivity of the PPSF varying with the DAOI and the wavelength. Ray-tracing results of the light separation and integration of (b) R-LCoS and (c) L-LCoS channels of the 2-PBSP system.
4.3 Projection imaging lenses
The characteristics of the illumination light, including the spread angle and beam aperture, are used as a reference for designing the projection imaging system. In this paper, the imaging lens group in Ref [36]. is chosen as the initial starting point. The comparisons of the main parameters between the original and optimized stereoscopic projection lens groups are listed in Table 3.
Table 3. Comparisons of the Parameters between the Original and Optimized Projection Lens Groups
Figures 7(a) and 7(b) show the light propagation of the projection lenses within the prisms for the left- and the right-eye channels, respectively. After the LCoS modulation, the s-polarized light beam experiences reflection once and the p-polarized light beam experiences refraction once in both the right- and the left-eye channels. The optical distances of the stereo-channel within the prims are the same, which is double the side length of a single prism, therefore, the 2-PBSP configuration can be equivalently expanded as a planar glass plate with the thickness of 32 mm and the material of ZF6. Moreover, the back-working distance of the projection lenses should be set at a proper length to accommodate the PBSP system, which is 32.83 mm. An image-space telecentric light path is employed to improve the uniformity between the imaging center and edges, which also helps to maintain the compact structure for stereoscopic projection. A larger relative aperture and lower F-number ensures lower luminous flux loss. The F-number of this lens group is 1.9, where the output light rays of the illumination system within the spread angle of 15.3° can be collected and projected onto the target-imaging plane. Figure 7(c) shows the optimized projection lens group and the corresponding modulation transfer function (MTF) curves. Totally 19 functional surfaces are included, i.e., 16 optical surfaces with certain radius, 2 planar surfaces for the prism configuration, and 1 aperture stop surface. The minimum MTF value is greater than 0.4 at a resolution of 33 cycles/mm. The longitudinal spherical aberration, astigmatic field curves, and distortion curves are shown in Fig. 7(d). The maximum chromatic aberration value is lower than 0.3 mm within the full aperture, while the field curvature and distortion on the full FOV are 0.19 and 2.71%, respectively. It is demonstrated that high imaging quality is obtained for the stereoscopic projection.
Fig. 7 The optical paths of projection imaging lenses within the prisms for (a) the left-eye and (b)the right-eye stereoscopic channels; (c) MTF curves and 2D cross-sectional plot; (d) optical aberration curves of the designed stereoscopic projection lens group.
The detailed lens data of the optimized projection imaging lens are listed in Table 4, including the surface number, the radius, the thickness, the glass, and the semi-aperture of each surface of the optimized projection imaging lens group. The lens surfaces are spherical and, therefore, easy to fabricate. The minimum thickness of the lens is 2 mm. The materials of the lenses, selected from CDGM glass CO., LTD, are commercially available. The listed semi-aperture reveals the actual aperture during simulated ray tracing process.
Table 4. The Lens Data of the Optimized Projection Imaging Lens Group
4.4 Complete stereoscopic projection prototype
The complete two-in-one polarization-interlaced stereoscopic projection prototype, shown in Fig. 8(a), is assembled with the designed illumination lenses, the 2-PBSP system, two separate LCoS panels, and the projection imaging lenses. The total volume is 182.98 cc. Without consideration of the reflectivity of LCoS panels, the transmission efficiencies are 34.59% and 34.60% for the left- and right-eye channels of the proposed prototype, respectively. The relative standard deviation (RSD) is applied to calculate the illuminance uniformity of the projection prototype, which is expressed as
where M is the sample number, Ej is the irradiance value of the j’-th sample point, and is the mean irradiance value of all the sample points. By choosing the common nine sample points on the projection screen, the RSD value of the proposed prototype is calculated to be 0.1456.Fig. 8 Ray tracing of (a) the complete projection prototype, (b) the left-eye and the right-eye stereoscopic channels. Results of the designed prototype, including (c) degree of polarization, (d) actual color mixing, and (e) 3D illuminance distribution.
Figure 8(b) shows the ray tracing of the stereoscopic channels referring to the L-LCoS and R-LCoS modulation, respectively. The results include the degree of polarization, actual color mixing, and illuminance distribution. As shown in Fig. 8(c), orthogonal polarization states are obtained on the screen, and the degree of polarization reaches higher than 0.9. By tracing three LEDs with primary colors simultaneously, the actual color mixing can be observed on the target projection screen, as seen in Fig. 8(d). The central color coordinates are (0.3105, 0.3234) and (0.3103, 0.3212), and the corresponding color temperatures are 6672 K and 6702 K. Although the vignetting effect occurs from the corners, the color difference is unobservable within the main projection area. The illuminance distribution is also evaluated and drawn in a 3D coordinate system. As seen in Fig. 8(e), the green peaks mean there exhibits a relatively uniform illuminance distribution within the main projection area, which ensures good stereoscopic visual effect. The above results also prove that the stereo-channel separation and integration can be well achieved.
In actual processing, both the physical and algorithm calibration can be applied for achieving sub-pixel alignment between the left- and right-eye LCoS panels [37–40]. The physical calibration method is similar to the common method for three-LCD-chip projectors. A grid pattern is first projected on the calibration screen by different panels sequentially. For further fine adjustment, the camera feedback and the corresponding algorithm compensation can be applied. By using these methods, the pixels on the stereo-channel LCoS panels can be precisely aligned.
5. Discussions
Optimal prism configuration is not only required for the stereo-channel separation and integration, but also determines the performance and structure of the stereoscopic projection prototype. Therefore, several potential prism configurations are compared and analyzed in this section. Apart from the previous configurations, Fig. 9 gives another existing configuration consisting of four PBSPs, namely, the 4-PBSP configuration [14,41]. From Fig. 9 we can see that the optical path lengths of the left- and right-eye channels are identical to those of the proposed 2-PBSP configuration, and the times of refraction/reflection that the light experiences through the PPSFs are also equal to each other. Besides, the equations in Sec. 2.2 can be used for the 4-PBSP configuration. It means that the theoretical performance of these two configurations cannot be differentiated. Thus, further analysis is needed.
Fig. 9 Light propagation process of the 4-PBSP configuration. (a) Left-eye channel modulated by the L-LCoS panel; (b) right-eye channel by from the R-LCoS panel.
Figure 10 shows the comparison results of the stereoscopic channels among the 1-PBSP, 2-PBSP, and 4-PBSP configurations. It is found that the transmission efficiency is largely affected by the DAOI. The transmission efficiencies of stereoscopic channels can reach 47% only when the incident light is collimated to less than 4°. For the 1-PBSP configuration shown in Fig. 10(a), different downtrends of transmission efficiency can be observed between stereoscopic channels. While the DAOI increases from 4° to 14°, the efficiency difference can increase from 1.99% to 4.74%, which inevitably causes luminance imbalance for the actual 3D visual effect. However, the 2-PBSP/4-PBSP configurations exhibit similar downtrends between the stereoscopic channels, as shown in Figs. 10(b) and 10(c), which agree well with previous theoretical analysis. It is also found that the efficiencies decease to lower than 30% at the DAOI of 14°, indicating that the beam collimation is an important issue for the system performance.
Fig. 10 Normalized transmission efficiencies varying with the DAOI among 1-PBSP, 2-PBSP, and 4-PBSP configurations.
The overall transmission efficiencies varying with the DAOI among the three configurations, by consideration of both stereoscopic channels, are shown in Fig. 11. The 1-PBSP configuration always provides a higher efficiency than other configurations with the increase in the DAOI. For the DAOI around ± 8°, the transmission efficiencies reach 92.30% for the 1-PBSP configuration, 78.64% for the 2-PBSP configuration, and 79.42% for the 4-PBSP configuration, respectively. It reveals that the optical path length affects the system efficiency significantly.
Fig. 11 Comparison of the overall transmission efficiencies among the 1-PBSP, 2-PBSP, and 4-PBSP configurations.
The complete prototypes are assembled for 2- and 4-PBSP configurations, respectively. Table 5 lists the volume and the optical performance of these two-in-one polarization-interlaced stereoscopic projection prototypes. The total volume of the proposed eqprototype with the 2-PBSP configuration is 37% lower than that with the 4-PBSP configuration due to the simplification of the prism system. Performance difference between the 2-PBSP and 4-PBSP configurations is relatively unobvious, especially for transmission efficiency. The stereo-channel differences of the L-LCoS and R-LCoS channels are 0.01%, and 0.09% within the 2-PBSP and 4-PBSP configurations, respectively. In an integrated view, the proposed prototype can achieve satisfactory performance and obvious volume reduction at the same time, which is superior to the existing designs.
Table 5. Comparisons of the Projection Prototypes Equipped with Different PBSP Configurations
6. Conclusions
This paper presents an improved prism array configuration and a novel illumination freeform lens group for two-in-one polarization-interlaced LCoS stereoscopic projection displays. The proposed design enables stereoscopic projection incorporating the functionality of two independent optical paths into a single optical engine, with both structural simplification and performance improvement. Several existing prism configurations are analyzed and compared in detail to show the superiority of the novel 2-PBSP configuration. Compared with the 4-PBSP configuration, the proposed prototype equipped with the 2-PBSP configuration can reduce the system volume by 37% to 182.98 cc. The design of the freeform illumination lens group, obtained by Monge–Ampère method and feedback optimization, also opens up new prospects in stereoscopic projection displays. Based on the effective beam collimation and shaping of the advanced illumination engine, the proposed prototype can provide high performance with a transmission efficiency of 69.19%, stereo-channel difference of 0.01%, and RSD of 0.1456, respectively. The proposed design is expected to be promising for replacing existing stereoscopic projectors. Future work will focus on processing and testing based on this prototype.
Funding
National Natural Science Foundation of China (61405037); National Key Research and Development Plan (2017YFB0404604); Fujian Science and Technology Key Project (2018H6011); Training Program of Fujian Excellent Talents in University (FETU).
Acknowledgments
The authors would like to express their sincere gratitude to the colleagues of Zhejiang University and Malata Technology Co., Ltd., for their assistance in this work.
References
1. I. Sexton and P. Surman, “Stereoscopic and autostereoscopic display systems,” IEEE Signal Process. Mag. 16(3), 85–99 (1999). [CrossRef]
2. J. Hong, Y. Kim, H. J. Choi, J. Hahn, J. H. Park, H. Kim, S. W. Min, N. Chen, and B. Lee, “Three-dimensional display technologies of recent interest: principles, status, and issues [Invited],” Appl. Opt. 50(34), H87–H115 (2011). [CrossRef] [PubMed]
3. J. Geng, “Three-dimensional display technologies,” Adv. Opt. Photonics 5(4), 456–535 (2013). [CrossRef] [PubMed]
4. J. Y. Hong, S. G. Park, C. K. Lee, S. Moon, S. J. Kim, J. Hong, Y. Kim, and B. Lee, “See-through multi-projection three-dimensional display using transparent anisotropic diffuser,” Opt. Express 24(13), 14138–14151 (2016). [CrossRef] [PubMed]
5. S. R. Soomro and H. Urey, “Integrated 3D display and imaging using dual purpose passive screen and head-mounted projectors and camera,” Opt. Express 26(2), 1161–1173 (2018). [CrossRef] [PubMed]
6. B. Park, J. Cho, S. Kim, H. Kim, J. Kwon, K. Sung, H. Lee, J. Lee, and D. Jang, “Field sequential type 3D pico-projector phone based on 1GHz mobile application processor and nHD DMD chipset,” SID Symp. Dig. Tech. Pap. 42(1), 1808–1809 (2011). [CrossRef]
7. S. C. Kim and E. S. Kim, “A novel configuration of LCD projectors for efficient orthogonal polarization of two projected views,” Opt. Commun. 266(1), 55–66 (2006). [CrossRef]
8. S.-T. Wu and C.-S. Wu, “Mixed-mode twisted nematic liquid crystal cells for reflective displays,” Appl. Phys. Lett. 68(11), 1455–1457 (1996). [CrossRef]
9. Y. Huang, E. Liao, R. Chen, and S.-T. Wu, “Liquid-crystal-on-silicon for augmented reality displays,” Appl. Sci. (Basel) 8(12), 2366 (2018). [CrossRef]
10. K. Takizawa, “Three-dimensional large screen display using polymer-dispersed liquid-crystal light valves and a schlieren optical system: proposal and basic experiments,” Opt. Rev. 13(1), 1–7 (2006). [CrossRef]
11. Q. H. Wang, C. C. Ji, L. Li, and H. Deng, “Dual-view integral imaging 3D display by using orthogonal polarizer array and polarization switcher,” Opt. Express 24(1), 9–16 (2016). [CrossRef] [PubMed]
12. S. Kim and E. Kim, “A novel configuration of LCD projectors for efficient orthogonal polarization of two projected views,” Opt. Commun. 266(1), 55–66 (2006). [CrossRef]
13. K. Takizawa, T. Fujii, T. Sunaga, and K. Kishi, “Three-dimensional large-screen display with reflection-mode spatial light modulators and a single-projection optical system: analysis of a retardation-modulation method,” Appl. Opt. 37(26), 6182–6195 (1998). [CrossRef] [PubMed]
14. L. Bogaert, Y. Meuret, J. Vanderheijden, H. D. Smet, and H. Thienpont, “Stereoscopic projector for polarized viewing with extended color gamut,” Displays 31(2), 73–81 (2010). [CrossRef]
15. B. Jacobson, R. Gengelbach, and J. Ferri, “Beam shape transforming devices in high-efficiency projection systems,” Proc. SPIE 3139, 141–150 (1997). [CrossRef]
16. R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Miñano, and F. Duerr, “Design of freeform illumination optics,” Laser Photonics Rev. 12(7), 1700310 (2018). [CrossRef]
17. E. Chen, P. Liu, and F. Yu, “Synchronized parameter optimization of the double freeform lenses illumination system used for the CF-LCoS pico-projectors,” Opt. Laser Technol. 44(7), 2080–2087 (2012). [CrossRef]
18. E. Chen, P. Liu, and F. Yu, “Optical design of single plano-convex freeform lens-based illumination system for color filter liquid crystal on silicon pico projectors,” Opt. Eng. 51(3), 033002 (2012). [CrossRef]
19. Y. Gao, D. Cheng, C. Xu, and Y. Wang, “Design of an ultra-short throw catadioptric projection lens with a freeform mirror,” Proc. SPIE 10154, 101540 (2016). [CrossRef]
20. Z. L. Zhenjie Liu, P. L. Peng Liu, and F. Y. Feihong Yu, “Parametric optimization method for the design of high-efficiency free-form illumination system with a LED source,” Chin. Opt. Lett. 10(11), 112201 (2012). [CrossRef]
21. E. Chen and F. Yu, “Design of an elliptic spot illumination system in LED-based color filter-liquid-crystal-on-silicon pico projectors for mobile embedded projection,” Appl. Opt. 51(16), 3162–3170 (2012). [CrossRef] [PubMed]
22. H. Kikuta, Y. Ohira, and K. Iwata, “Achromatic quarter-wave plates using the dispersion of form birefringence,” Appl. Opt. 36(7), 1566–1572 (1997). [CrossRef] [PubMed]
23. S. Shen, J. She, and T. Tao, “Optimal design of achromatic true zero-order waveplates using twisted nematic liquid crystal,” J. Opt. Soc. Am. A 22(5), 961–965 (2005). [CrossRef] [PubMed]
24. R. K. Komanduri, K. F. Lawler, and M. J. Escuti, “Multi-twist retarders: broadband retardation control using self-aligning reactive liquid crystal layers,” Opt. Express 21(1), 404–420 (2013). [CrossRef] [PubMed]
25. X. Zeng, X. Zhou, T. Guo, L. Yang, E. Chen, and Y. Zhang, “Crosstalk reduction in large-scale autostereoscopic 3D-LED display based on black-stripe occupation ratio,” Opt. Commun. 389, 159–164 (2017). [CrossRef]
26. A. J. Woods, “How are crosstalk and ghosting defined in the stereoscopic literature,” Proc. SPIE 7863, 78630Z (2011). [CrossRef]
27. A. Bruneton, A. Bäuerle, R. Wester, J. Stollenwerk, and P. Loosen, “High resolution irradiance tailoring using multiple freeform surfaces,” Opt. Express 21(9), 10563–10571 (2013). [CrossRef] [PubMed]
28. Z. Feng, B. D. Froese, and R. Liang, “Freeform illumination optics construction following an optimal transport map,” Appl. Opt. 55(16), 4301–4306 (2016). [CrossRef] [PubMed]
29. V. Oliker, “Controlling light with freeform multifocal lens designed with supporting quadric method(SQM),” Opt. Express 25(4), A58–A72 (2017). [CrossRef] [PubMed]
30. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002). [CrossRef] [PubMed]
31. R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38(2), 229–231 (2013). [CrossRef] [PubMed]
32. R. Wester, G. Müller, A. Völl, M. Berens, J. Stollenwerk, and P. Loosen, “Designing optical free-form surfaces for extended sources,” Opt. Express 22(S2), A552–A560 (2014). [CrossRef] [PubMed]
33. R. Wu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “A mathematical model of the single freeform surface design for collimated beam shaping,” Opt. Express 21(18), 20974–20989 (2013). [CrossRef] [PubMed]
34. I. Moreno and C. C. Sun, “LED array: where does far-field begin?” Proc. SPIE 7058, 70580R (2008). [CrossRef]
35. Y. Luo, Z. Feng, Y. Han, and H. Li, “Design of compact and smooth free-form optical system with uniform illuminance for LED source,” Opt. Express 18(9), 9055–9063 (2010). [CrossRef] [PubMed]
36. T. Ito and Y. Arai, “Wide-angle camera lens,” Japan patent 61–48128 (1986).
37. Y. Chen, D. W. Clark, A. Finkelstein, T. C. Housel, and K. Li, “Automatic alignment of high-resolution multi-projector display using an un-calibrated camera,” In Proc. of the IEEE conference on Visualization, 125–130 (2000).
38. J. C. Lee, P. H. Dietz, D. Maynes-Aminzade, R. Raskar, and S. E. Hudson, “Automatic projector calibration with embedded light sensors,” In Proc. of the 17th annual ACM symposium on User interface software and technology, 123–126, (2004). [CrossRef]
39. M. Brown, A. Majumder, and R. Yang, “Camera-based calibration techniques for seamless multiprojector displays,” IEEE Trans. Vis. Comput. Graph. 11(2), 193–206 (2005). [CrossRef] [PubMed]
40. T. Amano, “Projection center calibration for a co-located projector camera system,” In Proc. of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, 449 (2014). [CrossRef]
41. L. Bogaert, Y. Meuret, B. V. Giel, H. Murat, H. D. Smet, and H. Thienpont, “Projection display for the generation of two orthogonal polarized images using liquid crystal on silicon panels and light emitting diodes,” Appl. Opt. 47(10), 1535–1542 (2008). [CrossRef] [PubMed]
