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A holographic display system combining the spatial- and time-multiplexing together in one system is proposed. The system is constructed by multiple planar aligned spatial-light-modulators (SLMs). A shiftable cylindrical lens is introduced in to build up an “equivalent SLM” by seamlessly linked horizontal images of the SLMs, which are tiled in a time-sequential manner. The proposed system can realize wide horizontal-viewing-angle holographic three-dimensional (3D) display through the “equivalent SLM”, but bear with low requirements on the number and frame rate of SLMs, and the numerical aperture of the optical system. In the proposed system, only one parallel incident beam is needed, leading to a simplified optical structure. Using two 60Hz phase SLMs, a 3D display with a horizontal viewing angle (VA) of 27.5° is implemented experimentally.
© 2014 Optical Society of America
1. Introduction
The computer-generated hologram (CGH) is thought as an ideal 3D display technique which provides a natural spatial effect. However, limited to the space-bandwidth characteristics of available SLMs, the display VA is very small [1]. For example, using a commercial SLM, which usually has a resolution of 100-200 lines/mm and 103 × 103 pixels, the displayed 3D object provides a VA of only about 4 degree when the displayed size is as large as 10 × 10 × 10mm3. Even for one viewer, this VA is too narrow for binocular observation.
In general, previous researches on enlarging VAs of those SLM-based holographic 3D displays can be classified into two categories. One is based on time-multiplexing, where a series of sub-images with different viewing ranges are projected from one SLM time-sequentially, and a wide VA is implemented through combining these viewing ranges via the “afterimage” effect. So, the frame rate of the used SLM determines the reachable VA and a SLM with a high frame rate is essential for this method. By using a digital-micromirror-device (DMD) of 15kHz, Y. Takaki et al implemented a holographic display with a VA of 15° in a horizontally scanning system [2]. D. Teng et al developed an all-around holographic 3D display system with a 60Hz SLM, but a viewer tracking system was necessary to address the problem of SLM’s low frame rate [3, 4]. The other research direction is spatial-multiplexing, where multiple SLMs project sub-images with different viewing ranges synchronously and an ensemble of viewing ranges together gives a wide VA whose value depends on the number of employed SLMs. Fukaya et al implemented the alignment of three transmission-type-SLMs in a planar configuration by a beam-splitter [5, 6]. In the work of Slinger et al [7–9], 25 de-magnified images generated by a high-speed electrically addressed SLM were tiled in a time-sequential manner onto an optically addressed SLM, which stored these images by planar layout, to produce a wide-viewing-angle 3D holographic object. Such planar configurations require a Fourier-transform optical system which should cover the whole SLMs. Therefore, the reachable VA is limited by the numerical aperture (NA) of the optical system. Multiple SLMs aligned in a circular configuration were proposed to break through the NA induced bottleneck [10, 11]. But the incident light for each SLM needs to be different. Complicated optical structures were necessary for the adjustment of the incident light. Y.Z Liu et al [12] obtained equivalent curved SLMs array through superimposing different linear phase facor to a phase SLM sequentially. In their paper, only one SLM was used. Employing a pair of 4f cylindrical lenses and two pairs of mirrors, they tiled two horizontal images of the SLM onto a plane by a sequential mannar. The two horizontal images of the SLM can be linked together seamlessly. However, compared with the images of the SLM, the sizes of the cylindrical lens and used mirrors both are larger. If two or more SLMs are used in one system, seamless linkage of images from multiple SLMs are difficult to be realized, because the cylindrical lenses or the used mirrors for diffirent SLMs could not overlap with each other. Therefore, the system proposed by Y. Z. Liu et al is more like a time-multiplexing technology. Recently, T. Kozacki et al [13] presented a spatiotemporal multiplexing method. They arranged six SLMs on a circle for spatial-multiplexing. An additional SLM was placed in the display space for time-multiplexing to extend the horizontal VA to a maximum value of 35°. As a common deficiency of circular configurations, an optical structure was needed for the incident light’s adjustment. Furthermore, as T.Kozacki et al had pointed out in their paper, the assumption of six SLMs on a circle being parallel to the additional SLM could not be met in practice, which produced an error at the order of angular resolutions of human eyes.
In this paper, a new spatiotemporal multiplexing technology is proposed for holographic 3D display, where multiple SLMs aligned in a planar configuration are employed. Part of the idea has been patented in ZL201010142650.1 [14]. By introducing a cylindrical lens at the Fourier spectrum plane of the SLMs, the horizontal images of the SLMs will appear on the focal plane of the cylindrical lens. With positions of the cylindrical lens and spacings between adjacent SLMs being well pre-designed, horizontal images of the SLMs, corresponding to the cylindrical lens at different positions, can link up seamlessly along the horizontal direction in a time-sequential manner. Via the “after image” effect, an “equivalent SLM”, which is built up by all the horizontal images of the SLMs, is obtained. The “equivalent SLM” thus has an extended resolution and can give a wide horizontal VA to the displayed 3D image. Using two phase SLMs of 60Hz, a 27.5° horizontal VA is implemented experimentally.
The holographic display system based on the proposed spatiotemporal multiplexing technology needs only one parallel incident beam. So the optical structure is simplified greatly compared with previous work [2–13]. Other significant merits include that the number of employed SLMs is reduced and the required NA gets smaller compared with conventional spatial-multiplexing technologies when a same VA is pursued. Compared with traditional time-multiplexing methods, the proposed spatiotemporal multiplexing technology decreases the required frame rates of the employed SLMs. In addition, the error resulting from circularly arranged SLMs in the work of T.Kozacki et al [13] does not exist in the proposed technology.
The rest of this paper is organized as follows. In section 2, a new time-multiplexing technology with a shiftable cylindrical lens is explained. Based on the developed time-multiplexing, section 3 describes the spatiotemporal multiplexing technology with multiple planar aligned SLMs. Experiments and results are shown in Section 4. Section 5 analyzes the image blurring phenomena induced by the continuous rotating of the cylindrical lens. The display resolution of the system is calculated. Section 6 provides conclusions.
2. A new time-multiplexing technology with a shiftable cylindrical lens
The HEO-1080 SLM from Holoeye Photonics AG with a frame rate of 60Hz is used in this paper. The SLM is a pure phase modulator with a resolution of 1920 × 1080 and a pixel period of Δp = 8μm, i.e. an effective working area of D1 × D2 = 15.36 × 8.64mm2.
For a normal incident light, the VA of a single SLM, θ, is defined by the maximum diffraction angle θmax = 1.9° which is derived from the diffraction formula (), as shown in Fig. 1(a).The λ (532nm) is the wavelength of the laser used in this paper. The corresponding field of view (FOV) is the shadow zone. Following Ref [11], FOV denotes an area where the entire object can be seen while the full bandwidth of the SLM is utilized. To display an identical object, multiple SLMs in a planar configuration give a same value of VA as a single SLM, as shown in Fig. 1(b). Usually a Fourier-transform optical system is used in a planar configuration, as shown in Fig. 1(c). But the numerical aperture (NA) of the optical system, also the NA of the Fourier-transform lens in Fig. 1(c), must be large enough when more SLMs are used in the system for a wider VA. In the case of a circular array of SLMs, problems related with the NA do not exist. But a complex optical structure is necessary for providing different incident lights to different SLMs, as presented in Fig. 1(d). Physically it is impossible to arrange the effective working areas of SLMs closely due to their mechanical frames. So, the arrangement shown in Fig. 1(c) and Fig. 1(d) cannot be obtained actually and some parts of the FOV are missing. To compensate the missing segment, virtual images of the SLMs are usually used, as done in Ref [5, 6, 10, 11]. But, additional optical structures are necessary and the system becomes more complex.
Fig. 1 The FOVs for (a) a single SLM, (b) multiple SLMs in a planar configuration, (c) multiple SLMs in a planar configuration with a Fourier-transform optical structure, (d) multiple SLMs in a circular configuration, (e) the proposed spatiotemporal multiplexing technology with an “equivalent SLM” which is built up by the SLM’ images in a time-sequential manner.
In this paper, a new time-multiplexing technology is developed by introducing a shiftable cylindrical lens (f2) into the back focal plane (called IP in the paper) of the Fourier-transform lens (f1), as shown in Fig. 2. The optical axis of the Fourier-transform lens is taken as the main optical axis of the display system. The phase SLM is placed at the front focal plane of the Fourier-transform lens. Let a set of parallel edges (D1) of the SLM be along the horizontal x-direction. The CGH fed to the SLM generates a 3D sub-image around the IP. Along the x-direction, the sub-image locates between points O1 and O2 of the IP. The cylindrical lens is called the direction lens (DL) and used to refract the sub-image projected from the SLM. The refracted sub-image is the object displayed by the system.
Fig. 2 Schematic optical diagram of the proposed time-multiplexing technology with a shiftable cylindrical lens (DL).
The DL can move along the x-axis and its geometric axis is perpendicular to the x-z plane. Through the Fourier-transform lens and DL, the horizontal (x-direction) message of the SLM is imaged onto the focal plane (FP) of the DL, with a magnification of f2/f1. In the following sections, we refer to the image of the SLM as the “horizontal image”. It shall be minded that messages of the SLM along the vertical direction (y-direction) cannot be imaged onto the FP because the DL has no function of phase modulation along this direction.
With the DL being shifted to different positions, the image of the SLM is directed to different regions of the FP. Figure 2 demonstrates the idea of the new time-multiplexing technology through a simplified case, i.e. one SLM and two available positions of the DL. When the DL locates at Position 0 at the time t, the image is directed to P'1Q', which is denoted as “SLM image 0”. But when the DL moves to Position 1 at the time t + Δt/2, the corresponding “SLM image 1” is directed to P'Q'2. Conversely, the distribution spaces of the refracted sub-images do not translate with the shifting of the DL. They overlap with each other around the IP and stay between the point O1 and the point O2 along the x-axis. The overlapped space is the display space.
For a displayed point A in the display space, its horizontal VA changes from Q'AP'1 to Q'2AP' with the DL being shifted from Position 0 to Position 1. According to geometric optics, when the spacing between the two positions is set as
the SLM’s images are able to link up end to end along the x-direction spatially. As shown in Fig. 2, where the Position 0 and Position 1 are set as 0 × Δd and 1 × Δd away from the optical axis, respectively, the point P'1 will coincide with the point Q'2. With the DL being shifted to the two positions alternatively and corresponding CGHs of the point A being encoded onto the SLM synchronously, the two horizontal VAs (i.e. Q'AP'1 and Q'2AP') will appear alternatively. If the frame rate of the SLM and the shifting speed of the DL are high enough, the point A will become observable in an enlarged horizontal VA via the “afterimage” effect:The region corresponding to Q'AP' is the extended horizontal FOV of the point A. As an aggregation of many points, the displayed object’s horizontal VA, θ, depends on the overlapping area of all object points’ FOVs.From the perspective of practical effects on enlarging horizontal VA, the displayed object can be looked as being projected from a virtual SLM, which is built up by the two images of the SLM. This virtual SLM is denoted as “equivalent SLM” in this paper. Its resolution gets two-fold enlarged and the θ of the displayed objct gets about two-fold enlarged, as shown in Fig. 2. Obviously, a wider horizontal VA is reachable when the DL has more available positions and the SLM keeps synchronously with the refresh rate of corresponding CGHs.
The left part of Fig. 2 is re-drawn in Fig. 1(e) for the convenience of comparison. Compared with methods shown in Fig. 1(a-d), the developed time-multiplexing technology employs the SLM’s horizontal images to build an “equivalent SLM”. Such a technique takes a planar configuration similar to Fig. 1(b), but can realize the function of a circular configuration system (Fig. 1(d)).
For each position of the DL, the corresponding CGH is calculated through iteration algorithms, which involves an iterative loop of optical field propagation between the 2D slices of the target object and the SLM plane [15]. As shown in Fig. 3, when the DL is at a position, e.g. Position L, a Fourier-transform hologram CGHL is encoded onto the SLM. The projected 3D sub-image is denoted as IML. In order to make the refracted IML have the same intensity distribution as the target object, an algorithm for obtaining the correct CGHL is developed, as presented in Fig. 3. Assuming a target object being placed in the display space virtually, along the optical axis, the virtual target object is sliced into 2K + 1 2D slices with an equal spacing, δ = 0.2mm. The intensity distribution of the kth slice (k is an integer in [–K, + K]) is denoted by A(Pk). The maximum intensity value of A(Pk) is 255 in an 8-bit bitmap file, but the minimum intensity value is set to be 3 following the Ref [15]. Totally, four kinds of light field transformations are used: FT+, FT-, T+ and T-. The operator FT+ in Fig. 3 represents a Fourier transformation with an additional quadratic phase factor from the DL:
where is the position coordinate of the DL at the “Positon L”. The operator T+ works as a light field propagation from the (k-1)th slice to the kth slice by a short-distance δ:Here F and F−1 denote Fourier and inverse Fourier transform operators, respectively. FT- and T- are the inverse process of FT+ and T+. At each slice, the deduced intensity distribution through FT+, T+ or T- is overwritten by the target value while the phase remains unchanged. At the SLM plane, the deduced intensity distribution through FT- is replaced by a constant value, i.e. 1. On the IP plane, the light field from the SLMs is pre-modulated for including the modulation effect from the DL, then the intensity overwritten of all the object slices is carried out. So, although the phase modulation from the DL is asymmetric (only along the horizontal direction), distortion does not take place in the displayed image. Repeating the loop shown in Fig. 3 for 500 times, the obtained USLM is taken as the final CGH for the SLM when the DL moves to the Position L.Fig. 3 The scheme of the algorithm processes of obtaining the correct CGHs to make the refracted IMs have the same intensity distributions as the target object.
3. Spatiotemporal multiplexing with multiple planar aligned SLMs
For the time-multiplexing technology proposed above, the available maximum positions “m” of the DL are determined by the ratio of the SLM’ frame rate vs. the displaying frequency of the system. In order to avoid obvious image flicker, the displaying frequency of the system is set as 15Hz in this paper. For a SLM of 60Hz, “m” takes the value of 4. So, only fourfold enlargement of the horizontal VA is reached by using one SLM.
In order to further enlarge the horizontal VA, multiple (n) SLMs, being aligned along the x-direction with an equal spacing of 2D1, can be introduced into the above time-multiplexing system. The proposed time-multiplexing system thus becomes a spatiotemporal multiplexing system. A system with two SLMs is taken as an example to demonstrate above idea, as schematically drawn in Fig. 4. Four positions are available for the DL: ± 1.5Δd (Positions ± 1) and ± 2.5Δd (Positions ± 2). The “Imlj” refers to the horizontal image of the SLMj when the DL locates at the “Position l”. An “equivalent SLM” built up by n × m = 8 horizontal images will be obtained and a wider horizontal VA becomes realizable.
Since planar aligned SLMs are employed, NAs of the optical elements in the system need to be addressed. Geometrically, the Fourier-transform lens in the system needs to cover all the real SLMs. The minimum transmission aperture of the Fourier-transform lens shall be:
The DL’s aperture shall always ensure to cover the region O1O2, even when the DL is shifted to the farthest positions. So, its minimum transmission aperture shall be:Therefore, the required NAs of the Fourier-transform lens and DL can be expressed as:andrespectively.For comparison, if SLMs are aligned in the conventional planar configuration (Fig. 1(c)), to reach a similar VA, a quantity of 8 real SLMs are needed. To cover all the 8 SLMs, the minimum NA of the Fourier-transform lens will be:
Comparing Eq. (9) with Eq. (8) and Eq. (7), it is clear that the required NA is reduced effectively in the proposed display system. In fact, NAPLANAR = NADL + NAFL-λ/Δp and the third item is very small here. The lower demand on NA values lies in that the DL shares a part of the task on collecting beams emitting from the “equivalent SLM”.The iteration algorithm described above is suitable for a gapless working area. In order to use the iteration algorithms, the interval between two real SLMs is assumed to be filled with a SLM in the proposed display system. So, a virtual SLM with a resolution of (3 × 1920) × 1080 locates at the front focal plane of the Fourier-transform lens. The holographic codes for the virtual SLM can be calculated through the algorithm processes in Fig. 3. The specific codes for each real SLM is obtained from the whole holographic codes according to its corresponding spatial position. As a result, the calculated resolution of the target object’s slices is (3 × 1920) × 1080. However, the real SLMs do have a space interval, which makes the display resolution be determined by the resolution of a single real SLM, i.e. 1920 × 1080. In practice, the display resolution of the display system is smaller than 1920 × 1080, which is denoted as Kx × Ky. Reasons for the decrease of display resolution will be given in the later section. To settle the confliction, an image preprocessing based on the interpolation method is developed. For each slice from the target object with a practical display resolution Kx × Ky, each pixel is divided into [(3 × 1920/Kx) × 1080/Ky] sub-pixels, as shown in Fig. 5.The central region of a pixel containing the minimum number of sub-pixels is taken as the effective region. The sub-pixels in the effective region take the intensity value of the pixel. The other sub-pixels are assigned the minimum value of 3. Thus, a series of slices with (3 × 1920) × 1080 sub-pixels are obtained and the iteration algorithms given above can be run for holographic encoding.
Fig. 5 Image pre-processing to generate slices with a larger resolution based on the interpolation method.
4. Experiments and results
A system is set up to implement the idea described above, as shown in Fig. 6. A laser (Samba532nm100mw) from Cobolt AB of Sweden is used as the light source. The incident beam from the laser is converted into an elliptically polarized light through a 1/4 wave plate. Through a polarizing beam splitter, the proper light intensity is directed to the SLM. A 1/2 wave plate is placed in front of the SLMs to adjust the polarization direction of SLM’s incident beams. The two horizontal arranged SLMs (SLM1 and SLM2) are aligned in a plane with a small inclination angle of α = 4° to the incident beam.
Since the shuttle frequency of a translation stage is hard to reach 15Hz for a large travelling span, a rotating platform is used for DL’s addressing. Four kinds of square parts are cut out from the mother DL, with one set of parallel edges of each square along the geometric axial direction of the mother DL. Their geometrical centers are ± 1.5Δd and ± 2.5Δd away from the geometric axis of the mother DL along x-direction, respectively. These four kinds of partial cylindrical lenses are named as 1.5PL, 2.5PL, −1.5PL and −2.5PL accordingly and their side lengths are 13mm, as shown in Fig. 7. Two for each kind of partial cylindrical lenses are fabricated. In total, eight partial lenses are attached to the rotating platform with a sequence of −2.5PL, −1.5PL, 1.5PL, 2.5PL, −2.5PL, −1.5PL, 1.5PL and 2.5PL, as shown in Fig. 6. Their geometrical centers locate on a circle with a diameter d = 50mm surrounding the rotating axis and are separated by an equal angular interval. The rotating axis is parallel with the optical axis in the x-z plane. The axial direction of each partial cylindrical lens keeps being along the tangential direction of the circle.
With this architecture, addressing of the DL is implemented through rotating the partial cylindrical lenses. For example, when the geometrical center of the partial lens 1.5PL rotates to the optical axis, it is equivalent to the situation that the DL reaches the position which is 1.5Δd away from the optical axis along the x-direction, as in Fig. 2 and Fig. 4.
Let the platform rotate at a speed of 7.5rps. When a partial cylindrical lens will soon arrive at the optical axis, corresponding CGHs are fed onto the SLM1 and SLM2. The shutter in the light path remains closed until the geometrical center of the partial lens will soon arrive at the optical axis and keeps open for 7ms. The displaying frequency of the proposed system is 15Hz.
Encoding CGHs on a pixelated phase-only SLM will generate a zero-order beam at the Fourier plane as noises. In our experiments, a phase of −2π(x2 + y2)/2λr is added in pre-calculated CGHs [16]. The 3D sub-images projected from the SLMs are then shifted away from the back focal plane of the Fourier-transform lens by a distance of Δz = rf1(r-f1), while the zero-order noise remains unchanged. Placing a high-pass filter at the back focal plane, the zero-order noise can be blocked. The shifting of the projected sub-image along the optical axis doesn’t change its beam propagation directions. So, the introduction of the extra phase don’t change any equations discussed above, except for the distance between the Fourier-transform Lens and DL, which changes from f1 in Fig. 4 to f1 + Δz in Fig. 6.
Figure 8 shows the photograph of the experimental display system. f1 = 250mm and f2 = 200mm are adopted. The apertures of the used Fourier-transform lens and the customized DL (DDLx in Fig. 7, manufactured by Shanghai Institute of Optics and Fine Mechanics, CAS) are both 75mm, which satisfy Eq. (5) and (6). The value of r in the added spherical phase is 600mm.
To show the 3D effect along the depth direction, three square frames on three parallel planes with an equal spacing of 4mm along the optical axis are displayed. Side lengths of the square frames are 2mm (the 1st layer), 4 mm (the 2st layer) and 8mm (the 3st layer), respectively. The line-width is 0.3mm and the resolution is set as 200 × 200. When a camera is focused on one square frame along the optical axis, the other two square frames become blurred due to out-of-focus, as show in Fig. 9.To obtain obvious out-of-focus, the photos are captured at the 2 × optical magnification.
Fig. 9 Captured image when the the 1st layer, the 2nd layer and the 3rd layer are on focus in a sequence.
Finally, a teapot of 12 × 12 × 12mm3 is displayed to demonstrate the developed technology and system. The resolution of the 2D slices is Kx × Ky = 200 × 200 and the pitch between two adjacent slices is 0.2mm. Experimentally, the horizontal VA of the displayed 3D teapot reaches 27.5°. Replacing the viewer by a camera, captured images with the camera being at different angular positions around the display system are shown in Fig. 10. As in the Ref [13], in order to capture images simulating the naked-eye viewing, the diaphragm of the used camera is set as 5mm. The model of the teapot shown in Fig. 10 consists of a group of slices, which also are the slices used in the iteration algorithms. The distance between the camera and the displayed object is 250mm.
Fig. 10 Captured images when the HHD system works: the model of the teapot consisting of a group of slices and sequential views of the object captured with angular step of 2.7° from −13.5° to 13.5°.
5. Discussions
5.1 The display resolution
Because the DL keeps rotating, the displayed object point will shift in the slicing plane during the exposure process. It means that the displayed object point could not stay at the correct spatial location accurately. This deviation deteriorates the display resolution in the slice plane. For a point on the projected sub-image with a distance of |u| away from the DL plane, its corresponding refracted point on the refracted sub-image (also the target object) is of |v| away from the DL, as shown in Fig. 11.When the DL deviates from the correct position by a distance of de along the x-direction, the distance of the refracted point will deviate by a distance of:
When u takes the maximum value of 6mm, i.e. half of the display size along the optical axis, the maximum value of u/(f2-u) is 1/53.5.Fig. 11 Geometrical diagram showing the deviation distance of the point on the refracted sub-image when the cylindrical lens deviating from the correct position.
According to the geometrical relations shown in Fig. 12, when a partial cylindrical lens deviates from its correct angular position by θe, its geometrical center will deviate from its correct position, i.e. the point c, to the point ce. So, along the x-direction, the maximum deviation distance of the DL can be expressed as:
Fig. 12 Geometrical diagram showing the deviation of the partial cylindrical lens from the correct spatial position when it rotates around the optical axis during the exposure process.
A rotating speed of 7.5rps is equal to 2.7°/ms. Based on above two equations, when the exposure time is 7ms, the maximum deviation of displayed points is estimated as 36μm, where the point is on the plane being farthest away from the DL.
When the DL stays at the correct position, the physical resolution of a point A in the x-direction depends on its distance to the DL. The minimum resolving power along the x-direction is 7μm. Because the size of the 3D object is much less than the value of f2, the graded distribution of resolutions along the optical axis can be ignored. Along the y-direction, the resolving power is a constant value which depends on the diffraction spot of the SLM through the Fourier-transform lens. In our embodied experimental system, its value is 13μm.
Combining with the deviation distance ΔH, the reachable resolving power of the points on the displayed object is about 43 × 13μm2. This value is good enough for humans. The resolution of human eyes with a pupil diameter 4mm is about 100μm at the distance of distinct vision. In the experiment, the displayed resolution of the slices is set as 200 × 200 pixels, corresponding to a resolving power of 60 × 60μm 2.
5.2 Effects of the non-paraxial approximation on the display system
Another problem that needs to be mentioned is the non-paraxial approximation in the system. Equation (3) of the iteration algorithms, which describes the light field propagation between the SLM and the P0 plane, is based on the paraxial approximation. The aligned real SLMs should be close to the optical axis enough. Thus, the number of SLMs is limited. In our experiment, two SLMs with a space interval of 2D1 don’t bring obvious degradation of display qualities.
In Eq. (4), the frequency domain transfer function is used to describe the light field transmission between adjacent slices. The transfer function is an accurate solution. Therefore, a perfect DL does bring no aberration to the displayed object. However, Eq. (1) is based on geometrical optics principles. This equation gives the available positions of the DL. When the DL is shifted away from the optical axis, the geometrical optics principles become inaccurate. The adjacent images of the SLMs could not be joined end to end rigorously, leading to missing or overlapping of FOVs. But compared with the size of the viewer’s pupil, the portion of missing or overlapping is very small. Then, the viewing comfort is not influenced.
The imperfect DL would introduce aberrations into the displayed target object also, especially when it is shifted away from the optical axis by a maximum value of ± 2.5Δd. In our experiment, due to blurring introduced by the continuous rotating of the DL, the display resolution decreases to 200 × 200 pixels. Under this condition, the aberration from the off-axis DL is not obvious. When the exposure time of the shutter gets shorter and the display resolution becomes higher, this problem will become serious. Phase compensation shall be taken into consideration during the holographic encoding, which will be addressed in our further work.
5.3 Trade-off between display flickers and VA
Limited by the frame rate and number of the used SLMs, the displaying frequency of our protype system is 15Hz. It is close to the displaying frequency of the early silent films which is from 16 to 24Hz [17]. This value is enough for the sense of motion, but slight flicker is observed in the experiment. In the general PAL video standard, displaying frequency of 24Hz is required for flicker-free motion.
The displaying frequency can be increased, but there exists trade-off between the displaying frequency and the viewing angle in our system. To realize flicker-free display, SLMs with higher frame rates or more SLMs shall be introduced into the proposed system. For example, with a Fourier-transform lens of 100mm aperture, the experimental system shown in Fig. 8 can adopt 5 SLMs when their shorter sides (D2) are arranged along the horizontal direction. The maximum number of the used SLMs is limited by the aperture of the Fourier lens because the Fourier lens must keep covering all the light fields from different SLMs. With 5 SLMs being adopted, a displaying frequency of 30Hz can be realized when there are 2 available positions for the DL. Then, only 2 kinds of partial cylindrical lenses are needed. 4 for each kind will be attached to the rotating platform shown in Fig. 8. The rotating frequency of the platform keeps to be 30/4 = 7.5rps. That is to say, the rotating speed of the cylindrical lens is independent on the number of used SLMs. With all other system parameters being identical to those in Fig. 8, the reachable horizontal VA of the flicker-free system will be 20° . If the displaying frequency stays at 15Hz, 5 SLMs can realize a horizontal VA of 38°.
When SLMs with higher frame rates are used, the displaying frequency or VA can be improved effectively. But consequent increase of the rotating frequency of the platform will result in more serious blurring. To settle this problem, a non-mechanical optical element for light redirecting, such as a prism encoded on a phase SLM as shown in Ref [13], needs to replace the mechanical rotating cylindrical lens in our system for blurring suppression and high display resolution. This is the focus in our future work.
6. Conclusions
In conclusion, a shiftable cylindrical lens is employed to build up an “equivalent SLM” with extended resolutions by seamlessly linked horizontal images of multiple planar aligned SLMs. The developed spatiotemporal multiplexing technology paves a way for the wide-horizontal-viewing-angle holographic 3D display with lower demands on the number and frame rate of the SLMs, and NA of the optical elements. With two 60Hz SLMs, a teapot with a horizontal VA of 27.5° is displayed in the present work.
Acknowledgments
The authors gratefully acknowledge supports by the Natural Science Foundation of China, Grant No.10802101, the National High Technology Research and Development Program of China (No.2013AA03A106), and the National High Technology Research and Development Program of China (No. 2011BAE01B14).
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