Wide-viewing integral imaging using fiber-coupled monocentric lens array

JianLei Zhang; XiaoRui Wang; XiongXiong Wu; Chen Yang; YuJiao Chen

doi:10.1364/OE.23.023339

1. Introduction

Various methods have been studied to implement three dimensional (3D) image display, which is regarded as the final form of display techniques [1–3 ]. Among them, integral imaging is a promising 3D display technique with advantages such as quasi-continuous, full-color viewpoints within a viewing zone, full parallax and working with incoherent light [4]. In an integral imaging system, a lens array spatially samples and captures the light field from the 3D scene, and then using another lens array with the captured elemental image array (EIA) can display 3D image.

Despite its many advantages, integral imaging suffers from the inherent drawbacks such as limited viewing angle, small image depth, and low viewing resolution. Especially, the small viewing angle is regarded as a primary disadvantage of integral imaging display to reach a commercial level. To overcome the limitation of viewing angle, various methods have been proposed using time multiplexing schemes, viewer tracking, etc [5–12 ]. However, the time multiplexing schemes require either mechanical dynamic movement or high speed polarization switching, and the viewer tracking has a complex control system. In addition, modifying the configuration of the lens array can also extend the viewing angle, such as the fresnel lens array and negative index planoconcave lens array [13, 14 ]. However, in the previous integral imaging system, the lens array is coupled with the flat-panel display device directly. The off-axis aberrations of the lens array limit the viewing angle. Conventional fisheye lenses can map flat image onto wide field of view, but suppressing field curvature and other aberrations needs multi-layer components, and thus imposes harsh design tradeoffs.

Recent developments in monocentric optical systems have opened up the possibility for commercially viable wide-field gigapixel imaging [15–17 ]. Monocentric lenses consist entirely of hemispherical optical surfaces that share a common center of curvature. The symmetry architecture makes it focus light identically coming from any direction, thus eliminates most of the off-axis aberrations. The monocentric lenses are capable of generating a high-resolution wide-angle on a spherical image surface, which can be coupled to the conventional planar focal plane by relay optics [18, 19 ]. These features are useful for wide viewing display. However, to our knowledge, no investigation has been done on the wide viewing property of monocentric lens with regard to integral imaging display. In this paper, the monocentric lens array (MoLA) coupled with fiber bundle has been proposed to improve the viewing angle of integral imaging display.

To present the new approach, we have organized this paper as follows. In Section 2, conceptual prototype of integral imaging using fiber-coupled MoLA is present, and the viewing angle characteristic is analyzed. Section 3 is devoted to the imaging analysis of the MoLA and verifications of the proposed method. Finally in Section 4, we summarize the main achievements of this paper.

2. Integral imaging using fiber-coupled MoLA

In conventional integral imaging display, the pitch of the elemental lens is equal to the size of the corresponding elemental image, which is displayed by a flat-panel monitor directly. According to paraxial optics theorem, the viewing angle is limited by the elemental lens pitch and the gap between the lens array and the display device. The viewing angle of the conventional integral imaging can be expressed as:

θ = 2 arc \tan (Δ p / 2 f)

where Δp is the pitch of the elemental lens, and f is the focal length of the elemental lens. According to Eq. (1), one easy way to improve the viewing angle is to enlarge the pitch of the elemental lens or shorten the focal length. However, enlarging lens pitch degrades the viewing resolution. Thus, the pitch of the elemental lens should be no larger than several millimeters to avoid observing the grid structure effect. In addition, because the conventional lens array consists of one-shell elemental lenses, small focal length leads to unbearable off-axis aberrations. Formulas for the calculation of aberration coefficients are relatively complex and thus an approximation with thin lenses is used during the initial optical design. The results form a very useful tool for preliminary analytical optical system design. Equation (2) is the primary aberration function of the thin lens with the aperture stop located at the lens [20]:

W_{l} (r, θ; h') = a_{s} r^{4} + a_{c} h' r^{3} \cos θ + a_{a} h'^{2} r^{2} \cos^{2} θ + a_{d} h'^{2} r^{2}

where

\begin{matrix} a_{s} = - \frac{1}{32 n (n - 1) f^{3}} [\frac{n^{3}}{n - 1} + (3 n + 2) (n - 1) p^{2} + \frac{n + 2}{n - 1} q^{2} + 4 (n + 1) p q] \\ a_{c} = \frac{1}{4 n f^{2} S'} [(2 n + 1) p + \frac{n + 1}{n - 1} q] \\ a_{a} = - 1 / 2 f S'^{2} \\ a_{d} = - (n + 1) / 4 n f S'^{2} \end{matrix}

The notation a_s, a_c, a_a and a_d represent the coefficients of spherical aberration, coma, astigmatism and field curvature, respectively. And f is the focal length, n is the refractive index, $S'$ is the imaging distance, $h'$ is the image height, $(r, θ)$ is the polar coordinate in the plane of exit pupil, p and q are position factor and shape factor, respectively [20]. According to Eq. (2) and Eq. (3), it can be seen that the off-axis aberrations increase with the decline of focal length. Small-focal-length lens array with tolerable aberration always requires a number of optical elements, leading to excessive levels of complexity, weight and size.

In the proposed method, the conventional lens array is replaced by the MoLA, and a fiber bundle is used as relay optics for the EIA. Figure 1 shows the concept of the proposed method using one-glass MoLA. The MoLA is an array configuration of multiple elemental monocentric lenses, whose surfaces share a common center of curvature. The symmetrical design of MoLA makes it focus light identically coming from any direction. Thus most of the aberrations can be eliminated except for spherical aberration and longitudinal chromatic aberration. In addition, an aperture stop is placed inside the MoLA with its center at the center of the spherical surface of the MoLA in order to control the spherical aberration, according to Eq. (2). With this configuration, the back focal plane of each elemental monocentric lens is a spherical surface rather than a plane. The backlight unit which provides homogeneous backlight and the liquid crystal which modulate the light intensity compose the conventional flat-panel display device. The EIA for display is loaded onto the liquid crystal faceplate. A fiber bundle array can couple the planar EIA displayed by conventional flat-panel display device to the spherical back focal plane of the MoLA. Since the fiber bundle has its numerical aperture, which limits the cone angle of output light, a transmission-type isotropic scattering membrane is placed on the sphere surface of the fiber bundle to couple the emerging beam into the corresponding elemental monocentric lens. Note that, the diameter of each fiber must be smaller than the pitch of the pixel on the display device, according to the Nyquist sampling theorem.

Fig. 1 Integral imaging display with one-glass fiber-coupled MoLA.

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For simplicity, we first analyze the imaging property of one-glass monocentric lens array without considering the chromatic aberration. By paraxial optical approximation, the effective focal length of the MoLA is:

f = \frac{n r}{2 (n - 1)}

where n is the refractive index and r is the radius of curvature of the MoLA. For depth priority integral imaging case [21], the curve surface of the fiber bundle should be placed at the back focal plane of the MoLA. From Fig. 1, the viewing angle of the proposed method is:

θ = 2 arc \sin (r / f)

From Eq. (4) and Eq. (5), it can be seen that the viewing angle of the MoLA-based integral imaging is only limited by the refractive index of the glass. To show the advantage of the proposed method over the conventional case, Fig. 2 plots the relationship between the viewing angle and the refractive index for the MoLA-based integral imaging and that for the conventional case, according to Eq. (1) and Eq. (5). Here we assume that the fill factor of the conventional lens array is 1 and the lens array is equiconvex with the curvature radius equal to the pitch. It can be seen that with the increase of the refractive index, the advantage of the proposed method over the conventional case becomes obvious. In conventional case, the viewing angle is difficult to reach 50 degree without using other improvement method. In contrast, the viewing angle of integral imaging using fiber-coupled MoLA is larger than 100 degree with high refractive index material, which is enough in the practical application.

Fig. 2 The viewing angle of the proposed method and that of the conventional lens array.

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Although the one-glass fiber-coupled MoLA can enlarge the viewing angle, using one-glass MoLA has no ability to correct chromatic aberration. Hence, a two-or-more-glass MoLA structure should be adapted to decrease the chromatic aberration. As a matter of fact, a two-glass monocentric lens has excellent imaging performance [22]. So in the next, we just analyze the two-glass MoLA case, as shown in Fig. 3 , which consists of two shell glass. According to the geometric optics, the focal length of the two-glass MoLA is:

f = \frac{R_{1} n_{1} R_{2} n_{2}}{2 (n_{1} R_{2} n_{2} - R_{2} n_{2} + R_{1} n_{2} - R_{1} n_{1})}

where R ₁ and n ₁ are the radius of curvature and refractive index of the inner sphere ball lens, R ₂ and n ₂ are the radius of curvature and refractive index of the outer sphere shell. If we substitute Eq. (6) into Eq. (5), we can obtain the viewing angle expression for integral imaging with two-glass fiber-coupled MoLA.

Fig. 3 Integral imaging display with the two-glass fiber-coupled MoLA.

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In order to find the achromatic condition, the focal length is set equal at the red C line wavelength 656.27 nm and the blue F line wavelength 486.13 nm, i.e. f _C = f _F. The central wavelength is usually chosen as the D line, which is 587.56 nm. If the refractive indices at the C and F spectral lines for the inner sphere ball lens are denoted by the symbols n _1C and n _1F, and the corresponding refractive indices for the outer shell are given by n _2C and n _2F, the achromatic condition of MoLA becomes:

(\frac{1}{R_{2}} - \frac{1}{R_{1}}) (\frac{1}{n_{1 C}} - \frac{1}{n_{1 F}}) = \frac{1}{R_{2}} (\frac{1}{n_{2 C}} - \frac{1}{n_{2 F}})

According to Eq. (7), a combination with a high-index flint glass for the outer shell and low-index crown for the internal ball lens with moderate difference in Abbe number can solve the chromatic aberration. However, because the nonlinear dispersion characteristics of the inner and outer shell do not match up, so that the focal points of other wavelengths do not coincide exactly with the common focal point of the two selected colors. Hence, more glasses is need to eliminate the secondary spectrum phenomenon.

3. Experiment and analysis

To confirm the validity of the proposed method, the imaging qualities of the MoLA are analyzed and integral imaging simulated reconstruction experiment is performed. We use optical design software ZEMAX to model all the components in the system. For a comparative study, we also analyze the imaging characteristic of the conventional lens array coupled with flat-panel display device. With one-glass MoLA architecture, the fused silica is a suitable material, because it has low chromatic dispersion. The parameters used in the experiment are listed in Table1 .

Table 1. Parameters of integral imaging display

View Table

To assess the image qualities, a comparison of the spot diagrams for conventional lens array coupled with planar monitor and that for MoLA coupled with fiber bundle are shown in Fig. 4 . In each diagram, the dimensions of the plotted spot diagram are 0.20 mm on a side, and blue, green, and red symbols are the positions in the image plane for the traced rays of 486, 588 and 656 nm wavelengths respectively. Four field positions for the conventional lens array are $0 °$ , $6 °$ , $12 °$ , $18 °$ , and four field positions for the MoLA are $0 °$ , $10 °$ , $20 °$ and $30 °$ . From Fig. 4(a), it can be seen that the spot size for the conventional lens array increases with the field angle (RMS spot size: 18.1 μm at $0 °$ , 17.3 μm at $6 °$ , 24.7 μm at $12 °$ , 53.1 μm at $18 °$ ), which is due to the off-axis aberrations increasing with the field angle. This phenomenon limits the viewing angle and leads to the degradation of the viewing quality of the display 3D image when the observer views the edge viewpoints. In contrast, with the proposed method, fiber bundle can relay the planar EIA displayed by flat-panel monitor onto the spherical focal plane of the MoLA. Note that, using the MoLA can eliminate the off-axis aberrations except for the field curvature. The fiber bundle can compensate the field curvature produced by the MoLA. Hence, the spot size for the MoLA remains almost the same at different field angle (RMS spot size: 9.2 μm at $0 °$ , 8.9 μm at $10 °$ , 8.3 μm at $20 °$ , 7.7 μm at $30 °$ ), which is due to the symmetry architecture of MoLA. Hence, integral imaging using fiber-coupled MoLA has the ability of wide viewing display.

Fig. 4 Spot diagram for the conventional lens array (a) and the MoLA (b).

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Due to the limitation of the experimental condition, we do not have fiber bundle and MoLA in our laboratory. Although the fiber bundle is commercial available [18, 19 ], the manufacture of MoLA remains a challenge, so a new development is needed. Hence, computational integral imaging (CII) reconstruction [23] is used to verify the validity of the proposed method. The 3D target object is a parrot positioned at depth of 33 mm. First, the planar EIA of a parrot is captured with our self-developed integral imaging pickup software. A lens array, 60 × 60, with 2 mm × 2 mm rectangular aperture is used in the pickup process. Each EI has 101 × 101 pixels. The refractive index is assumed to be 1.5 with no chromatic dispersion. Hence the focal length of the MoLA is 1.118 mm according to Eq. (4).

The camera array captures the EIA by perspective projection onto a plane, but the fiber-coupled MoLA works with perspective projection onto a spherical surface. To avoid distortion occurring in the display 3D image, a transformation for the EIA is needed from plane perspective projection to spherical perspective projection before display [24]. Figure 5(a) shows the EIA captured by the pickup system based on plane perspective projection and Fig. 5(b) is the EIA for display using the MoLA coupled with fiber bundle after the perspective projection transformation. Compare to Fig. 5(a), distortion can be observed in Fig. 5(b) from the enlargement of one elemental image located in the bottom right.

Fig. 5 (a) EIA captured by CII pickup system based on plane perspective projection; (b) EIA for display using the MoLA after the perspective projection transformation.

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In order to compare with the proposed method, simulated reconstruction using the conventional integral imaging display is performed with the same refractive index i.e. 1.5. The lens array is assumed to be equiconvex with the curvature radius equal to the pitch. Hence the viewing angle of the conventional method is $36.8 °$ according to Eq. (1). Figure 6 shows the displayed 3D images on different viewpoints with the conventional method. A minus and a plus sign represent the left and right directions of the observing position from the center of the lens array, respectively. As shown in Fig. 6, the perspectives can be observed continuous within an angle of $36 °$ in accordance with the theoretical result. In contrast, the displayed 3D images on different viewpoints using the proposed method are shown in Fig. 7 . The different perspectives of the images can be seen continuously up to $80 °$ , which is in accordance with the theoretical result according to Eq. (5).

Fig. 6 The displayed 3D images on different viewpoints with the conventional method.

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Fig. 7 The displayed 3D images on different viewpoints with the proposed method.

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Note that, we do not consider the MTF of the fiber bundle in the experiment. In fact, the resolution of the transmitted image by fiber bundle is limited by the pixilated nature of the fiber bundle as well as by the degree of coupling between cores. The index contrast between cores and cladding should be increased from that of standard single mode fiber in order to more tightly confine the light and reduce crosstalk between cores. In our future work, we will focus on the effect of fiber bundle on the quality of displayed 3D image.

4. Conclusion

In this paper, we propose an integral imaging system using fiber-coupled MoLA to improve the viewing angle. In conventional integral imaging display, the planar EIA is mapped to the viewing zone by a lens array, thus the off-axis aberrations of the conventional lens array limit the viewing angle. The MoLA consists entirely of hemispherical optical surfaces that share a common center of curvature, which eliminates most of the off-axis aberrations. A fiber bundle is used to transfer the planar EIA onto the spherical focal plane of the MoLA to compensate the field curvature introduced by the MoLA. Viewing angle more than 80 degree can be achieved by the proposed method.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (61377007, 61575152 and 61007014) and the Fundamental Research Funds for the Central Universities (NSIZ011401 and NSIY151410).

References and links

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	Conventional case	Proposed method
Radius of curvature	2 mm	1 mm
Pitch	2 mm	2 mm
Material	Fused silica
Aperture stop	1 mm
Number of array	60 (horizontal) by 60 (vertical)

Wide-viewing integral imaging using fiber-coupled monocentric lens array

Abstract

1. Introduction

2. Integral imaging using fiber-coupled MoLA

3. Experiment and analysis

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Tables (1)

Equations (7)

Optics Express