Single frontal projection autostereoscopic three-dimensional display using a liquid crystal lens array

Zi Wang; Zi Wang; Miao Xu; Guoqiang Lv; Qibin Feng; Anting Wang; Hai Ming

doi:10.1364/OE.381971

1. Introduction

Nowadays, the applications of three-dimensional (3D) display have been extended to many areas and developed a lot, especially in 3D cinema. In this multi-viewer occasion, a frontal projection is required for efficient space utilization. Generally, the 3D cinema uses two projectors with two polarizers to project left-eye and right-eye parallax images. The audiences must wear polarizing glasses to perceive the 3D images. However, wearing glasses influences the viewing experience.

To remove the glasses, rear projection can be used based on the parallax barrier method [1], the lenticular lens method [2], and the integral imaging (II) method [3,4]. In these methods, the projector is behind the screen, and the parallax barrier, lenticular sheet or lens array is in front of the screen. However, the rear projection has low space efficiency and is not suitable for 3D cinema. Several methods of reflection-type II 3D display were reported based on micro-convex or micro-concave mirror array [5–8]. However, the manufacturing cost of the mirror array is high, and the resolution is limited by the mirror number [9]. A frontal projection 3D display using a parallax barrier polarizer and a quarter-wave retarding film was reported [10], but the brightness was low due to the light block of the parallax barrier polarizer, and the dark spacings in the 3D images decreased the visibility of the 3D image. A frontal projection II 3D display using a pinhole array on a polarizer was also demonstrated with much lower brightness. Lenticular sheet could also be used for frontal projection 3D display [11–13], but in these methods, multiple projectors were required for multiple viewpoints, which increased the cost a lot and made the total system bulky.

In this paper, we introduce a frontal projection 3D display using a liquid crystal lens array (LCLA). The 3D display quality is the same as the lenticular lens method or II method with rear projection, but the space efficiency is improved much. The brightness is high compared with parallax barrier method [10] and only single projector is required. The demonstrated LCLA has the merits of no driving voltage, simple fabrication, and cost-effective. The proposed method could be a potential candidate for use in the future glasses-free 3D cinema.

2. Principle

Figure 1(a) shows the principle of the proposed scheme. It consists of a laser projector, a liquid crystal lens array, a quarter-wave retarding film and a polarization-preserving screen. The laser projector is to avoid using polarizer and to achieve high optical efficiency. The light projected by the laser projector is y-polarized. The LCLA generally consists of a molded plano-concave polymer lens array, an LC layer and an alignment layer. The rubbing direction of the alignment layer on the glass substrate is along the x-axis, so each LC molecule is oriented along the x-axis. The LC material has a birefraction of n_o=1.5216 and n_e= 1.7462. The solidified polymer has a refraction index of n_p∼1.524 which is very close to n_o. Now let us consider the light propagation in Fig. 1(b), the equivalent light path of Fig. 1(a). The y-polarized light projected by the laser projector first passes through the LCLA. Since the polarization direction is identical to that of the ordinary axes (n_o) of the LC, and n_o is very close to n_p, the LCLA presents the property of a transparency to the y-polarized light and allows it passing without bending. Then, the y-polarized light passes through the quarter-wave retarding film with its fast axis 45° oriented, and turns to be left-circularly polarized. The light is then reflected by the polarization-preserving screen which can preserve the polarization state and reverse the propagation of the light. The precalculated elemental image array is imaged on the screen. After reflection, the polarization state turns to be right-circularly polarized due to half wave loss. Since the propagation is reversed, the light will pass through the quarter-wave retarding film (with its fast axis 135° oriented) again and turns to a x-polarized light. When the x-polarized light passes through the LCLA, since the polarization direction is identical to that of the extraordinary axes (n_e) of the LC, and n_e is larger than n_p, the LCLA presents the property of a positive lens array to the x-polarized light. Finally, the light is refracted by the LCLA to reconstruct the 3D images based on the principle of integral imaging. Figure 1(b) shows an example of depth-priority integral imaging in which the light rays are collimated by the lens array. The elemental image array projected by the projector is calculated or captured with a lens array with the same parameters as the LCLA. In the display stage, each elemental image is imaged by the corresponding LC lens to integrate the 3D images. The proposed method can also realize a lenticular lens 3D display if the liquid crystal lenticular lens array is fabricated and used.

Fig. 1. (a) Principle of the proposed single frontal projection 3D display using a LCLA. (b) The equivalent light path of the proposed method.

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The proposed method operates depending on the different properties of the LCLA for different polarization direction. The above analysis assumes that the light passes through the LCLA perpendicularly. However, the incident angles of the light from the projector and the light reflected by the screen are not always 90°. When the light meets the LCLA at an oblique angle, the n_o keeps unchanged but the n_e changes as the incident angle changes. The incident angle will influence the refraction index of the light passing through the LCLA from behind, as indicated by green arrows in Fig. 1(a). When the light strikes an LC molecule with a tilted angle θ with respect to the extraordinary axes, the effective refraction index n_eff is expressed as

(1)$${n_{eff}} = \frac{{{n_o}{n_e}}}{{\sqrt {n_e^2{{\cos }^2}\theta + n_o^2{{\sin }^2}\theta } }}$$

Since n_eff decreases as θ increases, the LCLA has weaker refractive ability for the light with larger tilted angle. If the axial object point is considered only, this imaging aberration is just like the lens spherical aberration, and is a positive spherical aberration. Considering the convex spherical lens originally has a negative spherical aberration, these two aberrations will function together to decrease the total spherical aberration.

3. Crosstalk due to wavelength and incident angle

The wavelength and incident angle influence the phase retardation of the quarter-wave retarding film. Only the light with working wavelength and normal incidence will be transformed to a perfectly polarized light. The other light will cause crosstalk, that is, a part of light with y-polarization will be seen by the observer without the refraction of the LCLA. This light carries the direct information of the elemental image array without modulation of the lens array. This crosstalk can be estimated by the final degree of polarization of the light. The transmission of light through a quarter-wave retarding film is given by:[14]

(2)$$\begin{aligned} \left( {\begin{array}{@{}c@{}} {{{A^{\prime}}_s}}\\ {{{A^{\prime}}_p}} \end{array}} \right) &= \left( {\begin{array}{@{}cc@{}} {{{t^{\prime}}_s}}&0\\ 0&{{{t^{\prime}}_p}} \end{array}} \right)\left( {\begin{array}{@{}cc@{}} {\cos \psi }&{ - \sin \psi }\\ {\sin \psi }&{\cos \psi } \end{array}} \right)\left( {\begin{array}{@{}cc@{}} 1&0\\ 0&{\exp (i\pi /2)} \end{array}} \right)\left( {\begin{array}{@{}cc@{}} {\cos \psi }&{\sin \psi }\\ { - \sin \psi }&{\cos \psi } \end{array}} \right)\left( {\begin{array}{@{}cc@{}} {{t_s}}&0\\ 0&{{t_p}} \end{array}} \right)\left( {\begin{array}{@{}c@{}} {{A_s}}\\ {{A_p}} \end{array}} \right)\\ & = {T_o}R( - \psi )PR(\psi ){T_i}\left( {\begin{array}{@{}c@{}} {{A_s}}\\ {{A_p}} \end{array}} \right) \end{aligned}$$

and t_s, t_p, t_s′, t_p′ are the Fresnel coefficients given by

(3)$$\begin{array}{l} {t_s} = \frac{{2n\cos \theta }}{{n\cos \theta + {n_0}\cos {\theta _0}}},\\ {t_p} = \frac{{2n\cos \theta }}{{n\cos {\theta _0} + {n_0}\cos \theta }},\\ {{t^{\prime}}_s} = \frac{{2{n_0}\cos {\theta _0}}}{{{n_0}\cos {\theta _0} + n\cos \theta }},\\ {{t^{\prime}}_p} = \frac{{2{n_0}\cos {\theta _0}}}{{{n_0}\cos \theta + n\cos {\theta _0}}}, \end{array}$$

where we recall that n is the refractive index of air, n0 is the refractive index of the quarter-wave retarding film, θ is the incident angle, θ₀ is the refraction angle, and ψ is the angle for rotation of coordinate. The R(ψ) is simply the transformation matrix for the rotation of the coordinate by an angle ψ, which is expressed as:

(4)$$\cos \psi = \frac{{\cos {\theta _0}\sin \phi }}{{\sqrt {1 - {{\sin }^2}{\theta _0}{{\sin }^2}\phi } }}$$

where ϕ is the orientation angle of the optic axis of the quarter-wave retarding film. For different wavelength, the matrix representation of the quarter-wave retarding film P in Eq. (2) can be replaced by $\left( {\begin{smallmatrix} 1&0\\ 0&{\exp [i(\pi /2 + \Delta )]} \end{smallmatrix}} \right)$, where Δ is the extra phase retardation. To test the final degree of polarization (DOP) of the light, we consider a simple case in Fig. 2(a). The laser projector and the observer’s eye are placed at the same point. The projected light first passes through the quarter-wave retarding film with incident angle θ, and is reflected by the screen, and passes through the quarter-wave retarding film again with incident angle θ. It is noted that the returned light and the projected light have exactly opposite direction. Thus, on the screen plane, normal incidence happens so that the polarization state is not changed. We only need to calculate the transmission process 2 times in Eq. (2). Figure 2(b) shows that the DOP changes with wavelength and incident angle. A broadband quarter-wave retarding film (Mecan, MCR140N) was used for this simulation. The working wavelength is 550nm, and the retardations at 450nm and 650nm are 0.535π and 0.4825π, respectively. Generally, the DOP is not bad due to the dispersion compensation. The worst case happens at 450nm with incident angle of 30°, and about 4% of the light is responsible for the crosstalk. Of course, the simulated case here is simplified with the least crosstalk. With the observer’s eye moving away, the crosstalk will increase.

Fig. 2. (a) Simplified case for simulation in proposed method. (b) The DOP changes with wavelength and incident angle.

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4. Fabrication process

The fabrication procedures of the LCLA are shown in Fig. 3. A solid plano-convex lenticular lens array is chosen as a stamper. The pitch of lens unit is p = 1mm, and the surface curvature radius is r = 1.21mm. First, the stamper is slightly placed on the surface of a transparent Polyethylene terephthalate (PET) film, as shown in Fig. 3(a). Then the UV-curable monomer, here we choose NOA 65 (Norland Optical Adhesive) monomer, is filled in the empty space, as shown in Fig. 3(b). The monomer is exposed by the UV light (∼20mW / cm², λ∼365 nm) for 15 mins, as shown in Fig. 3(c). After UV exposure, the stamper is peeled off. Then a solidified polymer surface with concave pattern on the PET film as the top substrate is obtained, as shown in Fig. 3(d). The bottom substrate is a transparent glass substrate coated with an alignment material: polyimide (PI). After preheating at 120°C for 10mins and heating at 220°C for 2h, the PI layer is thoroughly polymerized. Then the PI layer is buffed in one direction using a rubbing roller to form uniform tiny grooves. After that the bottom substrate and the top substrate are assembled to form a cell, with the rubbing direction of the PI layer parallel to the direction of cylinder and the thickness of the cell is about 80μm, as shown in Fig. 3(e). Then the pure liquid crystal E7 (n_o=1.5216, n_e=1.7462, Merck) is injected into the cell and full fill the empty concave cavities, as Fig. 3(f) shows. The orientation of LC molecules is along the direction of cylinder.

Fig. 3. Schematic cross-section of the procedures for fabricating a LCLA.

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The focal length of the LCLA is calculated next. Assuming a parallel light comes from the side of the glass substrate, it will be refracted respectively at the surfaces between the LC layer, the polymer layer and the air. According to the sphere imaging equation:

(5)$$\frac{{n^{\prime}}}{{s^{\prime}}} + \frac{n}{s} = \frac{{n^{\prime} - n}}{r}$$

where n′ and n are the refractive indexes in the image space and object space, s′ and s are the image distance and object distance, and r is the curvature radius of the sphere, the focal length is calculated about f = 5.45mm. The viewing angle is then calculated as 2tan⁻¹(p/2f)≈10.5°. The viewing angle can be improved by decreasing the curvature radius of the lens surface or using a high birefringence LC. In addition, the glass substrate can be replaced by a flexible PET film and a curved LCLA is obtained for the viewing angle enhancement [15,16].

Figure 4 shows the principle of the polarization selectivity of the LCLA. The optic axis of each LC molecule is along the x-axis. When the y-polarized light (ordinary light) strikes the LCLA, the refractive index n_o is very close to the refractive index n_p of the solidified polymer. Thus, the LCLA presents the property of a transparency to the y-polarized light. When the x-polarized light (extraordinary light) strikes the LCLA, the refractive index n_e is larger than the refractive index n_p of the solidified polymer. Thus, the LCLA presents the property of a positive lens array to the x-polarized light.

Fig. 4. (a) The LCLA presents the property of a transparency to the y-polarized light. (b) The LCLA presents the property of a positive lens array to the x-polarized light.

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Two reasons will cause the aberration. The first reason is the simple spherical profile of the molded plano-concave polymer lens array. It can be improved by designing a special surface profile. The second reason is the non-perfect alignment of the LC molecules. The LC alignment layer (PI) only exists on the glass substrate, and no alignment layer is coated on the surface of the polymeric concave layer. Thus, the LC molecules near PI layer have a good alignment, while the LC molecules far away from the PI layer are relatively random. As a result, the actual refractive index is inconsistent with the theorical value, thus will cause aberration. To reduce the aberration, one method is to use photo alignment layer of LC on the surface of the polymeric concave layer, which do not need rubbing treatment.

5. Experimental results

Figure 5(a) shows the fabricated liquid crystal lenticular lens array (LCLLA) with size of 100mm × 100mm and lens pitch of 1mm. The LCLLA consists of 100 LC lenticular lens. The measured focal length is 5.1mm. The viewing angle is then calculated as about 11.2°. To check its utility, laser beams of different polarization directions are projected onto the LCLLA to form different light distributions, as shown in Figs. 5(b) and 5(c). The vertically polarized light is converged into a line array on the focal plane of the LCLLA, while the horizontally polarized light passes through without change. It confirms that the fabricated LCLLA has different optical properties for different polarized light. The demonstrated LCLA has the merits of no driving voltage, simple fabrication, and cost-effective.

Fig. 5. (a) The fabricated LCLLA. (b) The light distribution of the vertically polarized light after passing the LCLLA. (c) The light distribution of the horizontally polarized light after passing the LCLLA.

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Figure 6 shows the experimental setup. A laser projector (Green Orange MP1) based on MEMS scanning was used as the display device with horizontal polarization. The laser projector has a resolution of 1280 × 720 with three primary wavelengths of 638nm, 520nm and 450nm. It has a horizontal projection angle of 44° and vertical projection angle of 26°. A broadband quarter-wave retarding film (Mecan, MCR140N) was used for polarization modulation with its fast axis 45° oriented. The operating wavelength is at 550nm. The retardation errors caused by wavelength dispersion are 7% and 3.5% at 450nm and 650nm respectively. Thus, the retarding film has a generally good retardation over the entire wavelength range of the visible light from the laser projector. The rubbing direction of the alignment layer of the LCLLA is vertical. The polarization-preserving screen is coated with aluminum flakes, serving as the reflective polarization-preserving element. The distance between the polarization-preserving screen and the LCLLA was adjusted to be equal to the focal length of the LCLLA to implement a depth-priority II [17].

Fig. 6. Experimental setup.

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Figure 7(a) shows 2 elemental image arrays of the 3D models. Figure 7(b) shows the experimental results captured from different directions of the 3D image consisting of two letters “3” and “D”. The two letters “3” and “D” are designed 50 mm in front of and behind the LCLLA, respectively. The depth range is about 100mm. The change of the relative distance of the two letters confirms that they are reconstructed at different planes (Visualization 1). The frames in Visualization 1 are flickering since the laser projector is scanning the laser beam. Our eyes cannot perceive this flicker. The 3D image is captured at a viewing distance of L = 1500mm. The viewing angle of the 3D image is calculated as 2tan⁻¹[(L-fS/p)p/2fL]≈7.4°, where S is the size of the LCLLA. The viewing angle is small due to large focal length of the LCLLA. The viewing angle can be improved by decreasing the curvature radius of the lens surface or using a high birefringence LC. A flexible LCLLA can also be fabricated for viewing angle enhancement. Figure 7(c) shows the experimental results of a reconstructed bee model. The depth range of the bee model is also about 100mm, half behind the LCLLA and half in front of the LCLLA. It looks like the bee is facing different directions in the left and right figures (Visualization 2). The different perspectives confirm the successful reconstruction of the 3D image with continuous depth profile.

Fig. 7. (a) The elemental image arrays of the two 3D models. (b) Experimental results captured from different directions of the 3D image of two letters “3” and “D”. (c) Experimental results captured from different directions of the 3D image of a bee model.

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We compared the proposed method with the pinhole polarizer method [10]. There are 2 major problems in the pinhole polarizer method. One is that the optical efficiency is low, since the light can only pass through the pinhole and most light is blocked. The aperture ratio of the pinhole array in the pinhole polarizer method [10] is 3.1% so the optical efficiency is 3.1%. The other problem is that the visibility of the 3D image is low since the 3D image is seen through the pinholes. The dark spacings among the pinholes decrease the visibility of the 3D image. While the proposed method has no dark spacings in the 3D images, thus having high visibility. And the optical efficiency is high since it doesn’t block light. Compared with the micro-convex or micro-concave mirror array methods [5–8], the proposed method has improved resolution. Since the resolution-priority mode [16] can be applied in the proposed lens array method, the resolution is not limited by the lens number. Compared with the multi-projection method [11–13], the proposed method only needs a single projector. Thus, the proposed method has some merits superior to the previous methods.

6. Conclusion

In this paper, we introduce a single frontal projection 3D display method using a liquid crystal lens array. The LCLA has different optical properties, refraction and transparency, for different polarized light. By modulating the polarization state of the light directly passing through the LCLA using a quarter-wave retarding film, the returned light is refracted by the LCLA to reconstruct the 3D images. The fabrication of the LCLA is introduced in detail, and the fabricated LCLA needs no driving voltage and is simple and cost-effective. In addition, the LCLA can be made flexible to make a curved screen for the 3D cinema. Experimental results verify the proposed method. We believe this technique will contribute to the future glasses-free 3D cinema.

Funding

National Natural Science Foundation of China (61805065).

Disclosures

The authors declare no conflicts of interest.

References

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Single frontal projection autostereoscopic three-dimensional display using a liquid crystal lens array

Abstract

1. Introduction

2. Principle

3. Crosstalk due to wavelength and incident angle

4. Fabrication process

5. Experimental results

6. Conclusion

Funding

Disclosures

References

Supplementary Material (2)

Cited By

Figures (7)

Equations (5)

Optics Express

Name	Description
Visualization 1	Experimental results captured from different directions of the 3D image of two letters “3” and “D”
Visualization 2	Experimental results captured from different directions of the 3D image of a bee model.