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Abstract
We propose a full color computer generated holographic near-eye display (NED) based on white light illumination. The method inspired from color rainbow holography is used for calculation of 2D and 3D color holograms. The parameters of the color hologram calculation are designed based on the parameters of the spatial light modulator (SLM) with 4K resolution. A slit type spatial filter is designed in frequency domain to extract red, green and blue frequency components for full color display. A NED system including a white light source, an achromatic collimating lens, a 4K SLM, a 4f optical filtering system, and an achromatic lens as eyepiece is designed and developed. The main contribution of this paper is the first time to apply the rainbow holography concept to the dynamic full color NED with a compact display system. The optical experiments prove the feasibility of the proposed method.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Head-mounted near-eye display (NED) brings unprecedented and exciting experiences of human-computer interaction for human and is promising for augmented reality (AR) and virtual reality (VR) applications [1,2]. However, most of the current NEDs are stereoscopic systems that are illuminated by incoherent lights [3–6]. The stereoscopic 3D displays always lead to the vergence accommodation conflict (VAC) [7,8], which is caused by a mismatch between the binocular disparity of a stereoscopic image and the optical focus cues provided by the display. This conflict can be reduced using integral imaging or pinhole based display techniques at the cost of reduced resolution [9,10]. Additive light field imaging and varifocal displays have also been proposed to alleviate the VAC [11,12]. Alternatively, holographic 3D display is able to resolve the VAC because it directly presents viewers a portion of light field that emanates from the displayed 3D objects rather than only relying on the view differences of left-right eyes. Progress has been made for high resolution holograms such as Fresnel holograms mostly for static 3D display [13]. Apart from the static display, recent works were also conducted for dynamic holographic display [14,15]. Nevertheless, the following challenges still need to be resolved for better application of holographic display technique: computational burden for hologram calculation, the limited bandwidth of SLM and the speckle noise of the reconstructed 3D images due to the use of coherent sources. Therefore, the currently available SLM based holographic 3D displays are mostly restricted to small objects and narrow viewing angles.
Multiple approaches have been proposed for holographic color display including time multiplexing [16,17], spatial multiplexing [18], frequency domain multiplexing [19–23] and three SLMs based methods [24,25]. The time multiplexing holographic color display involves the use of red, green and blue laser light sources as well as a synchronized high speed SLM. For spatial multiplexing holographic color display, the SLM area is typically divided into three regions for loading and modulating red, green, and blue holograms. For spatial domain multiplexing holographic color display, however, the optical system must be accurately manipulated so that each color of light only illuminates the corresponding area. In contrast, the frequency domain multiplexing holographic color display depends on designing the frequency distribution of each color components. Advantages of those methods are that they are applicable for 3D displays with large depth but at the cost of speckle noise in reconstructed images and relative bulk optical system [19–22]. A white light illumination based frequency domain multiplexing holographic color display has been proposed in Ref. [23] by filtering in frequency domain with a designed color filter. This concept is excellent for compact color NED free of speckle noise. However, the spatially separated desired frequencies of red, green and blue components mean that the propagation angles of each color components are different toward to the human eye. In addition, the three SLMs based color displaying methods, however, are not suitable for portable head-mounted NED for the expensive system costs and bulk optical system.
Another holographic display technology with white light illumination is rainbow holography, which was firstly proposed by Benton [26]. The computer generated color rainbow holograms are always used for high resolution color static 3D display [27–29]. Recently, a white light illumination-based holographic color display method, which is also known as the Fourier rainbow holography was proposed for dynamic color 3D display [30]. The advantage of this technique is that it is free of speckle noise during optical reconstruction. In this method, a grating is used to diffract different wavelengths into different directions to illuminate the hologram. The spectral of different wavelengths are then formed at different positions in the frequency domain. One disadvantage is that the visual perception of the different colors corresponds to different spectral bands at different positions, which means that the color display is not a full color display, but rather a rainbow color display. For addressing this, an improved display technique has been proposed for full color 3D display for real 3D object with time multiplexing of red, green and blue LED as illumination [31].
The combination of holography and NED to achieve near-eye holographic color display is quite intuitive and thus desirable. With this research, a rainbow hologram alike method based on frequency domain multiplexing is proposed. Within our proposed method, a simple optical setup consisting of a white light illumination, an achromatic collimating lens, a 4K resolution LCoS-SLM, a 4f optical filtering system and an achromatic lens as eyepiece for NED is demonstrated. The experimental results show that the proposed method is flexible to realize full color 2D and 3D NED without speckle noise.
In contrast to the previous methods [30,31], advantages of the proposed method are that a grating for spectroscopic illumination or time multiplexing of red, green and blue LED illumination for color display are not needed. This work is organized as follows: Section 2 discusses the concept of full color rainbow holographic alike method as well as the frequency design parameters for hologram calculation based on the parameters of the 4K LCoS. The calculation of 2D color hologram and 3D color hologram are also demonstrated; In section 3, the holographic color display system and experimental results for 2D and 3D color displays are presented; Section 4 discusses the mechanisms that affect the color of display; Section 5 is the conclusion of our research.
2. Principle of the proposed method
2.1. Overview
Inspired by color rainbow holography that uses a slit in the spatial domain for multiplexing colors, the overall of our idea is to use a slit in the frequency domain to prevent color blur [27–29]. Figure 1 illustrates the concept of the proposed method. The hologram is set on the front focal plane of the lens and a white light plane wave illuminates the hologram with an incline angle${\theta _{refy}}$ between the y direction and z axis. Due to the diffraction of the hologram H, a dispersive 3D “image” is formed. On the back focus of the lens, the “+1” order light corresponding to the frequency of the color image and the zero order corresponding to the frequency of illumination light denoted as “0” in Fig. 1 exist. The “−1” order light and higher orders light are not presented in Fig. 1 for simplicity. Within the first order frequency, the spectral distributions of the three longitudinal dispersion bands are overlapped. The first order frequency distribution is enlarged for simplicity illustration. And a slit is used to extract the desired red, green and blue frequency components for correct full color display. For calculation of the hologram, the diffraction angle in the y direction is limited, which corresponds to the bandwidth of $\Delta {f_y}$ in the enlarged part of the Fig. 1. Because the$\Delta {f_y}$ is small and the diffraction angle corresponding to red, green and blue components of color image can be approximately expressed as:
where${\lambda _r}$, ${\lambda _g}$ and ${\lambda _b}$ are three primary wavelengths used for hologram calculation, respectively. $\Delta {\theta _{yr}}$, $\Delta {\theta _{yg}}$ and $\Delta {\theta _{y\textrm{b}}}$ denote the diffraction angles of red, green and blue light in the y direction, respectively.
The conjugate waves of three primary color illumination lights are used as reference wave, which can be expressed as: $\exp [\textrm{ - }i\frac{{2\pi }}{{{\lambda _{r,g,b}}}}\sin ({\theta _{refy}})y]$. The off-axis amplitude type hologram is used for hologram calculation, which can be expressed as [29]:
The frequency shifts corresponding to the phase of the reference waves in the frequency domain can be expressed as follows:
where ${f_{sr}}$, ${f_{sg}}$ and ${f_{sb}}$ denote the frequency shifts of red, green and blue color frequency components of the color object, respectively.We analyze the diffraction angles both in the x and y directions for hologram calculation in the frequency domain based on the existing LCoS with a resolution of 4096×2160 pixels. Assume the wavelength ${\lambda _r}$ is 632nm, ${\lambda _g}$ is 547nm and ${\lambda _b}$ is 473nm and the pixel pitch of the LCoS ${d_h}$ is 3.74µm, then the maximum frequency of the LCoS is as follows:
The relationship between the angle of the reference light and the amount of frequency shift is shown in Fig. 2 according to Eqs. (5)–(7). The horizontal axis is the angle of the reference light, and the vertical axis is the frequency shift amount. The black dotted line indicates the maximum frequency of the LCoS. If the sampling theorem is satisfied, the maximum spatial frequency on the hologram needs to be lower than the maximum spatial frequency of the LCoS. The red, green and blue lines correspond to ${f_{sr}}$, ${f_{sg}}$ and ${f_{sb}}$, respectively. It is evident that through Fig. 2, the frequency shift of different color wave increases as the angle of the reference light increases. Under a certain condition, the frequency distributions of the three primary colors of light can be designed centering on the corresponding frequency shift. For example, if ${f_{sr}}$ is 82.8line/mm, ${f_{sg}}$ is 95.6line/mm and ${f_{sb}}$ is 112.7line/mm, then $\Delta {f_y}$ can be set as 6line/mm, which means that red, green, blue color information of the object light corresponding to three non-overlapping frequency band distributions exist in the y direction. In this case, the reference light angle is ${\theta _{refy}} = 3^\circ$. The corresponding angles $\Delta {\theta _{yr}}$, $\Delta {\theta _{yg}}$ and $\Delta {\theta _{yb}}$ can be calculated by Eqs. (1)–(3), which are written as follows:In the x direction of the hologram, the diffraction angle is not limited, and the corresponding diffraction angles determined by the maximum frequency of LCoS are as follows:
The 2D frequency domain distribution of object light that takes consideration of the frequency shift caused by the reference light is as shown in Fig. 3. The red, green, and blue bars represent the red, green and blue frequency components of the color object light, respectively. The positions of the three strips indicate the range of the frequency. The construction of such a spectral distributed frequency distribution of color object is a key point in our proposed method. The computational methods for 2D and 3D color holograms are described later.
Fig. 3. 2D frequency domain distribution of object wave light considering of the frequency shift caused by reference wave.
2.2. Hologram calculation method for 2D color image
The 2D color display refers to the color display of a 2D color image on the holographic plane. The calculation of complex amplitudes of red, green and blue components of 2D color image is demonstrated in Fig. 4. A color image with the same resolution as LCoS is decomposed into three monochrome images $i{m_r}$, $i{m_g}$ and $i{m_b}$. The monochrome images are converted to frequency domain with 2D Fourier transformation. The frequency of ${F_r}$, ${F_g}$ and ${F_b}$ corresponding to three color components are obtained and only the frequency information within frequency limit discussed in Section 2.1 is kept and other frequency part are all set as zeros. The useful frequency information is demonstrated in Fig. 4 inside the red, green and blue rectangles. The resolution of useful information in the y direction identical among red, green, and blue components, which is represented as follows:
where $N = 2160$ is the resolution of LCoS in the y direction, and ${d_{fy}} = \frac{1}{{N{d_h}}}$ is the frequency interval in the y direction.Fig. 4. Calculation the complex amplitudes of the red, green and blue components of the 2D color image.
The calculation of complex amplitudes can be expressed as follows:
where $fft2[{\ast} ]$ and $ifft2\{{\ast} \}$ mean 2D Fourier transformation and inverse Fourier transformation of *, respectively.${U_r}(x,y)$, ${U_g}(x,y)$ and ${U_b}(x,y)$ denote the complex amplitudes of red, green and blue components, respectively.$i{m_r}(x,y)$, $i{m_g}(x,y)$ and $i{m_b}(x,y)$ are red, green and blue channel of the color image, respectively.$mask$ is a 4096×2160 binary mask with 1 in a rectangular center at the resolution of 4096×48 and 0 in the other part. After ${U_r}$, ${U_g}$ and ${U_b}$ are obtained, the amplitudes and phases of the red, green and blue wave-front components on holographic plane can be calculated, and finally the hologram can be calculated through the coding method illustrated in Eq. (4).2.3. Hologram calculation method for 3D color image
Hologram calculation method with multi-view projected images for monochrome holographic 3D display has been proposed in our earlier work [32]. In this study, this method is further extended for color 3D hologram calculation. A 3D object located near the holographic plane is projected onto a projection plane to obtain a plurality of projected images at different angles. The hologram is calculated by superposition of projected images and convolutions with the corresponding point spread function (PSF). For simplicity, two projection planes are shown in Fig. 5 for explanation. As shown in Fig. 5(a), A and B are two points in the space, and layer1 and layer2 are two projection planes. Two projected images can be obtained by projecting 3D object on the two layers. In the hologram calculating process, the projected image emits small beam of light whose direction corresponds to the direction of projection, and the angular extent of the beam determines its area on the holographic plane H. The complex amplitude light field from the projected image is recorded on the holographic plane H and encoded as a hologram, as shown in Fig. 5(b). In the reconstruction of hologram, the overlapped areas of the small beam of light from hologram forms the 3D image. This method is similar to the integral imaging 3D display [33], where the propagation direction of the beamlets determines the outcome of 3D display. However, the computational time can be reduced compared with the object point based calculation method [32].
Fig. 5. Calculation method of the hologram for 3D color object: (a) Projecting of object points to different layers, (b) Hologram calculation from projected images.
The color hologram calculation only requires the projected image in the x direction in this study. LCoS has different diffraction capabilities for different wavelengths. The diffraction angles in the x direction are set as the same angle for color 3D display. The color 3D object is projected at angle range of $[ - {3.5^\circ },{3.5^\circ }]$ in the x direction with angular interval of 1°. And a total of seven perspective images are used for hologram calculation for the three primary wavelengths. Assuming that there is only one projection plane, the 3D object is projected at different directions to obtain a plurality of projection images. Figure 6 shows the calculation of the PSF corresponding to different projected images. P is the projection plane. Assuming that O is an on-axis point on the projection plane and the projection angles are ${\theta _{ix}}$ and ${\theta _{jy}}$ in the x and y directions, respectively. The emitting angle intervals are $\Delta {\theta _x}$ and $\Delta {\theta _y}$ in the x and y directions, respectively. The area of the PSF corresponding to the projected image on the hologram is determined by the projection angles and the angular intervals. The PSF for this projection image can be expressed as follows:
The red, green, and blue images of each view are convolved with the corresponding PSF and superimposed to obtain the complex amplitude of each color components on the holographic plane, which can be expressed as follows:
3. Holographic display system setup and experimental results
The experimental setup for holographic color NED is shown in Figs. 8(a) and 8(b). In this setup, halogen lamp is used and the white light is coupled into an optical fiber. The fiber head has a certain translation in the y direction with respect to the optical axis to ensure that the direction of illumination to the LCoS corresponds to the direction of the reference light. The LCoS we used for the holographic display is a phase type LCoS, the amplitude type hologram directly loaded into the phase LCoS for display is approximately the same as the traditional optical recorded hologram with silver salt holographic plate after bleaching. The divergent illumination light from fiber head is collimated by lens 1, and the LCoS is illuminated by the reflected light from beam splitter (BS). The computer-generated hologram is loaded on the LCoS, and the illumination light is modulated. The modulated light is filtered by the 4f optical filtering system with a slit type spatial filter and a real 2D or 3D color image free of zero order and high order noises is reconstructed near the focal plane of the 4f optical filtering system. Lens 4 is used as eyepiece for near eye display. In the optical display system, a camera is used at the exit pupil of lens 4 to capture the reconstructed color image. In order to avoid color dispersion problems, the lenses used for collimating, the 4f optical filtering system and eyepiece are all cemented doublet achromatic lenses. Lenses 1 and 4 are with focal length of 50mm and diameter of 30mm. And the focal length of lenses 2 and 3 in the 4f optical filtering system is 75mm and the apertures of those lenses are all 30mm. In this case, the distance from the LCoS to the human eye is less than 35cm, which means this setup is a relatively compact display system. The color of the reproduced image can be changed by changing the position of the slit as well as the width of the slit.
Fig. 7. Calculation of complex amplitudes of the three primary color light on the holographic plane.
Fig. 8. Holographic display system: (a) A schematic diagram of display system, (b) A photograph of display system.
The proposed full color NED technique is evaluated by three experiments. The first experiment presents the reconstruction of 2D color images. The second experiment shows the 3D parallax effect with a simple 3D color model with four-layer color rings while the third experiment illustrates a complex 3D color animation. The camera used here is a USB camera (F# 1.4, Focal length = 12mm) to simulate the conditions of perception of human eye. It should be noted that the quality of the reconstructed images observed with naked eye is higher than those recorded with the use of camera.
Figure 9(a) is the original 2D color image. Figures 9(b)–9(d) are three reconstructed images while the slit type spatial filter is set to three different positions. The original 2D color image has the resolution of 1774×2160, which is zeros padded to the resolution of 4096×2160 for hologram calculation. The calculation parameters and method have been discussed in Sections 2.1 and 2.2. The size and position of the slit type spatial filter have an important influence on the color selection of the reproduced image. When the slit moves up and down, the reproduced color image appears continuously changing rainbow colors, which is similar to the effect of traditional color rainbow holography [27–29]. In the experiment, the width of slit is about 1mm in the y direction and 20mm in the x direction, respectively. Approximate accurate color display can be achieved at a certain slit position, as shown in Fig. 9(b). In order to present the continuous change of color, the slit in the y direction is opened about 10mm and at the exit pupil of the eyepiece and the camera is placed on a lifting platform. Visualization 1 is captured by manually adjusting the position of the lifting platform from the bottom to the top of the exit pupil of the eyepiece.
Fig. 9. The reconstructions of 2D color image: (a) Original image, (b) Reconstructed image when the slit is on the first position, (c) Reconstructed image when the slit is on the second position, (d) Reconstructed image when the slit is on the third position. Visualization 1 shows the continuous change of color when the camera moves from the bottom to the top of exit pupil of the eyepiece.
Figure 10 shows the geometric relationship in the second experiment and reconstructed results. Figure 10(a) is a side view. P is the projection plane and $r{i_1}$, $r{i_2}$, $r{i_3}$ and $r{i_4}$ are four-layer rings with 2mm between each other. The distance between $r{i_1}$ and $r{i_2}$ is set to 2mm. The distance between projection plane P and holographic plane H is set to 15mm. The four rings are all symmetrical rings with radii of 0.97mm, 1.34mm, 1.72mm and 2.09mm. Figure 10(b) is a top view of the four-layer rings 3D color model. The 3D model is firstly projected to projected planes with seven projection angles as discussed in Section 2.3 and the projected images are interpolated and zero-padded to the images with the same resolution as LCoS. Figures 10(c)–10(e) are three reconstructed images when camera laterally moved into three positions. Six areas of interest are framed by green rectangles and enlarged. From the results we can see that the 3D parallax effect is perceivable but small because of the small diffraction angle limited by LCoS.
Fig. 10. The reconstruction of four-layer rings 3D color model: (a) Side view of geometric relationship of holographic plane, color 3D model and projection plane, (b) Top view of the color 3D model, (c) Left view of reconstructed image, (d) Middle view of reconstructed image, (d) Right view of reconstructed image.
The final experiment demonstrates a 3D color animation, which contains a fixed background color image and a horizontally rotated 3D color model. The side view of the relationship of holographic plane H, background color image bk, the 3D color model and the projected plane P is demonstrated in Fig. 11(a). The background color image has a size of 6.7 mm×6.7mm and the 3D color model has a size of 6.7mm ×4.2mm ×3.5mm (H×W×D). Figure 11(b) is the top view of the color 3D color model. The distance between projection plane P and holographic plane H is 15mm, which is the same as in the second experiment. The distances between the center of 3D color model to background color image and projection plane are both 4mm. In the animation, the 3D color model rotates horizontally from −75° to 75° with an angular separation of 1° while the background color image keeps still. For each view, seven projected images are calculated and used for hologram calculation. Visualization 2 demonstrates the reconstructed 3D color animation and Figs. 11(c)–11(f) are the 50th, 75th, 100th and 125th reconstructed images.
Fig. 11. The reconstruction of 3D color animation: (a) Side view of geometric relationship of holographic plane, color 3D model and projection plane, (b) Top view of the color 3D model, (c) 50th reconstructed image, (d) 75th reconstructed image, (e) 100th reconstructed image, (f) 125th reconstructed image. Visualization 2 shows the reconstructed 3D animation.
The results of the three experiments show that the proposed method can achieve high-quality color 2D and 3D display without speckle noises, but chromatic aberration problem due to the use of white light as illumination is existence. In the fourth part, we will discuss the factors affecting color quality.
4. Factors affecting color quality
Assuming that the illumination light on hologram is a white light plane wave, the grating equation can be used to analyze the spectrum expansion. Figure 12(a) shows the spectrum expansion in the y direction. Assume ${\theta _{cy}}$ is the illumination angle of white light plane wave, the diffraction angle ${\theta _\lambda }$ of $\lambda$ from the hologram H can be expressed as:
Fig. 12. Spectrum expansion in the y direction: (a) Diffraction by grating, (b) Relationship between diffraction angle and wavelength.
As we discussed in the first part, the diffraction angles of red, green and blue components on the holographic plane is $\Delta {\theta _{yr}} = {0.2173^\circ }$, $\Delta {\theta _{yg}} = {0.1880^\circ }$ and $\Delta {\theta _{yb}} = {0.1595^\circ }$. The spectral range of the three color components corresponding to the wavelength are
where ${\lambda _{rc}}$, ${\lambda _{gc}}$ and ${\lambda _{bc}}$ are wavelength within the designed bandwidth for each color components. The calculations show that when the color hologram is displayed, the spectrum is expanded within the designed bandwidth, and the extended spectrum is an important factor affecting the color display. In addition, the spectral distribution of the white light source still affects the color of reconstructed images.The fiber head we used here has a diameter of 4mm, which means the plane wave after passing through the collimating lens itself has a certain divergence angle, so that the direction of the light illuminating the hologram is slightly different from the direction of the reference light. This situation causes other wavelengths of light to be imaged through the designed slit, which also has a certain effect on the color of the reproduced image. In the experiments, the width of the slit used is about 1 mm, and when the width of the slit is reduced, the spectral component for imaging was reduced, but the brightness of the reproduced image is also lowered. Therefore, there is a trade-off between brightness and spectral composition.
Further improvement of the color quality in our proposed method is still possible. For example, a white light source composed of three-color LEDs has narrow spectral bandwidths for each color, which can be used as illumination to avoid imaging of more wavelength components compared with using continues white light source as illumination. The color system of the holographic color display is different from the standard color system of the electronic display, so the hologram can be calculated by an accurate color conversion relationship instead of directly using the red, green, and blue components of the color data as the amplitudes of the 2D or 3D color object [34,35]. The use of a smaller size optic fiber head will improve the divergence of the illumination light and lead to increase the color quality of the reproduced image. These works will be conducted in the near future.
In the calculation of 3D color hologram, the multi-view image based hologram calculation method is used in this study. However, the point cloud calculation method [14] or polygon-based calculation method [36] by limiting the diffraction angle in the y direction for each color components can also be used to calculate the 3D color hologram.
5. Conclusion
This paper proposes a full color holographic NED based on white light direct illumination, including 2D and 3D color hologram calculation methods. The experimental results show that the proposed technique can realize 2D and 3D dynamic color display without speckle noise. Several reasons for affecting color quality are discussed as well as the guidelines for color quality improvement. The white light illumination based technique with compact display system and convenient illumination compared to traditional holographic display allows us to push the holography based AR display further.
Funding
National Key R&D Program of China (2017YFB1002900); National Natural Science Foundation of China (61535007).
Acknowledgments
Thanks to the supports provided by Lochn Optoelectronics Co., Ltd.
Disclosures
The authors declare no conflicts of interest.
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