Method of color holographic display with speckle noise suppression

Zhao-Song Li; Yi-Wei Zheng; Yi-Long Li; Di Wang; Di Wang; Qiong-Hua Wang; Qiong-Hua Wang

doi:10.1364/OE.461294

1. Introduction

The holographic display technology can provide complete encoding and accurate control of the wavefront information of the 3D object, so it can provide the viewer with the full depth information of the 3D objects. However, holographic display technology is still accompanied by problems such as noise interference, small field of view, and slow calculation speed, which severely limit the further development of this technology. At present, these problems are gradually being resolved [1–10]. Generally, holographic display requires the laser to illuminate the computer-generated holograms (CGHs), but the high coherence of the laser will introduce speckle noise into the reconstructed image. Besides, the iterative optimization algorithms for the CGHs such as the Gechberg-Saxton algorithm [11] often require the addition of completely random phases, which makes the speckle noise of the reconstructed images uncontrollable [12]. The speckle noise is manifested as the change of bright and dark spots randomly appearing on the holographic reconstructed image, which reduces the quality of the reconstructed image [13,14].

To suppress the speckle noise of the reconstructed image, there are two commonly used methods. The first method is to suppress the speckle noise by reducing the coherence of the light source [15]. For example, by using the light emitted from red, green and blue (RGB) light-emitting diodes as the reference light for the CGHs [16,17] or using the specially designed low-spatial-coherence laser to illuminate the CGHs, low speckle noise image display can be achieved [18]. Although improving the light source can greatly suppress the speckle noise of the CGHs, special light sources are not only difficult to obtain, but also increase the manufacturing cost of the system.

The second method is to optimize the calculation process of the CGHs [15]. For example, the reduction of speckle noise has been achieved by encoding multiple incoherent reconstructed images for a single complex amplitude wavefront [19]. Some researchers have used the double-phase encoding method to convert the complex amplitude CGHs into the phase-only CGHs and combine it with the 4-f optical filtering system to suppress the speckle noise [20]. In addition, the use of spatial-multiplexing technology [21–23], time-multiplexing technology [24–26] and pixel separation methods [27,28] to suppress speckle noise are also very effective. These methods can help to improve the quality of reconstructed images, but the calculation of the multi-depth CGHs for 3D objects and the loading of the multi-frame CGHs require stronger computational power and the spatial light modulators (SLMs) with higher refresh rate [29].

Currently, the SLM is widely used in the holographic display, and can be divided into the phase-only SLM and amplitude-only SLM. Although the phase-only SLM has high diffraction efficiency, the refresh rate severely limits its ability to modulate the CGHs. In contrast, the digital micromirror device (DMD), a kind of amplitude-only SLM, has a high refresh frame rate and can load the multi-frame CGHs, thus, it gains much attentions [30–32]. However, in addition to speckle noise, there are also zero-order light and conjugate diffraction light in the reproduced image when the DMD is used for holographic reconstruction. In order to improve the display effect, it requires a set of 4-f optical filtering system to strip the requisite signals from zero-order and conjugate diffraction light [33]. Therefore, the system is complex and its structure is not compact enough. Since the color holographic reproduction usually requires the RGB channels light multiplexing [34–36], the color holographic display with speckle noise suppression is still difficult to realize.

In this paper, a method to suppress the speckle noise in color holographic display is proposed. The proposed method consists of the following three main processes. Firstly, the intensity information of the RGB channels of the object is extracted and the iterative process is employed to generate a constrained random initial phase to reduce the incidental speckle noise of the completely random phase. Secondly, the filter plane position of the filter used to remove the stray light, zero-order and conjugate diffraction light is preset in the double-step Fresnel diffraction (DSFD) algorithm. The DSFD algorithm is used to calculate the multi-frame CGHs of the RGB channels of the object. The discrete spherical phases are added to the CGHs to discretize the speckle noise. Thirdly, the CGHs are sequentially loaded on the DMD with a high-speed refresh rate. Based on the time-multiplexing technology, the color reconstructed images with speckle noise suppression can be captured. Compared with the previous work [22,29], the proposed method has the following advantages: 1) The speckle noise of the proposed method is reduced obviously and the quality of monochrome reconstructed images is greatly improved. 2) The stray light, zero-order and conjugate diffraction light are eliminated without using the conventional 4-f optical filtering system, and the system complexity by using the proposed method is optimized. 3) Color holographic display with speckle noise suppression is realized without chromatic aberration. When the refresh rate of the CGHs increases, the display effect will be better. The proposed method is expected to be applied in fields such as holographic AR display to promote the development of holographic display technology.

2. Principle of the method

The flowchart of the method is shown in Fig. 1. It consists of three steps. The first step is to extract the intensity information of the RGB channels of the object. Through an m-times iterative algorithm, a band-limited phase is obtained as the initial phase to avoid speckle noise caused by the completely random phase. In the second step, the DSFD algorithm is used to calculate the complex amplitude distribution on the display plane. The divergent spherical phase factor is added to the CGH to further suppress the speckle noise, and then the required CGH is obtained by amplitude coding. The third step is to load the obtained CGHs on the DMD. Under illumination of the RGB lights, a color holographic display with speckle noise suppression is obtained.

Fig. 1. Schematic diagram of the proposed method.

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In the first step, to eliminate magnification chromatic aberration, the intensity information of the RGB channels of the object is extracted after scaling the object for each color channel, and then the CGHs of each color channel are calculated independently. The phase information and intensity information are superimposed to form the complex amplitude distribution of the object. To simulate the scattering of the recording light on the surface of the object, a completely random phase is often superimposed on the initial phase, but it will bring serious speckle noise interference to the reconstructed image. In order to weaken the randomness, a bandwidth constraint is imposed on the initial complex amplitude distribution. The initial complex amplitude distribution is the multiplication result of the intensity information of the object and the completely random phase. It is used as the input for the iterative loop. The initial complex amplitude distribution after the forward Fresnel diffraction (FFD) is multiplied by a band-limited mask to preserve the complex amplitude information within a fixed window. The rectangular functions are used to express the band-limited masks. In this way, the completely random phase information can be constrained so that the speckle noise does not occupy the whole plane during filtering process. The phase part of the constrained complex amplitude distribution after the inverse Fresnel diffraction (IFD) is used as the initial phase for the next iteration. The diffraction distances of the FFD and IFD are the sum of the recording distance of the object and the distance from the filter plane to the hologram plane. The iterative process is cycled for m-times, and finally obtained is the band-limited phase. Since the complete randomness of the input phase at the beginning of the iteration has not changed, the band-limited phase is still random, but its contribution to the speckle noise has been greatly reduced.

In the second step, the amplitude of the object is multiplied by the band-limited phase to obtain the band-limited complex amplitude distribution, which is used as the input of the DSFD algorithm. For the amplitude-only CGHs, the diffraction efficiency of zero-order diffraction light is more than 60%, so the zero-order diffraction light is a strong noise for reconstructed images. In addition, the existence of conjugate light and stray light also introduce additional interference. In order to eliminate their influence on the reconstructed image, they should be filtered out as completely as possible. To achieve this purpose, a filter plane is designed in the DSFD algorithm which is set to a specific position in the optical path. Firstly, the FFD is used to transfer the object intensity information I(x₀, y₀) to the filter plane, and the complex amplitude distribution is expressed as U(x, y). The calculation formula is expressed as follows:

(1)$$ \begin{aligned} U(x, y) &=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} I\left(x_{0}, y_{0}\right) \cdot \exp \left[j \varphi_{\mathrm{n} \text { band-limited }}\left(x_{0}, y_{0}\right)\right] \\ & \cdot \exp \left\{\frac{j \pi}{\lambda_{\mathrm{n}}(s+d)}\left[\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}\right]\right\} \mathrm{d} x_{0} \mathrm{~d} y_{0}(\mathrm{n}=\mathrm{r}, \mathrm{g}, \mathrm{b}) \end{aligned} $$

where φ_{nband-limited}(x₀, y₀) represents the band-limited phase related to the wavelength superimposed on the object, λ_n represents the wavelength of the coherence light, s represents the recording distance from the object to the hologram plane, d represents the distance from the hologram plane to the filter plane, and j represents the imaginary number.

To eliminate the influence of the zero-order and conjugate diffraction light on the reconstructed images, the recording distance s of the object is designed based on digital holographic theory, and the formula is expressed as follows:

(2)$$ s=\max \left\{\rho \cdot \frac{2 D_{0} p}{\lambda_{\mathrm{n}}}\right\}, $$

where max{·} means taking the maximum value of the expression in the symbol, ρ represents a real number slightly greater than ‘1’, D₀ represents the width of the recorded object, and p represents the pixel pitch. Due to the constraint of the random phase in the first step, the speckle noise does not cover the whole filtering plane. The zero-order and conjugate diffraction light is completely separated from the reconstructed image. As long as the size of the filtering window is reasonably set on the filtering plane, the required reconstructed image can be separated from the stray light, zero-order and conjugate diffraction light. Afterwards, U(x, y) is transferred to the hologram plane by using the IFD, and the calculation formula is expressed as follows:

(3)$$ U\left(x_{\mathrm{h}}, y_{\mathrm{h}}\right)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} U(x, y) \cdot \exp \left\{\frac{-j \pi}{\lambda_{\mathrm{n}} d}\left[\left(x_{\mathrm{h}}-x\right)^{2}+\left(y_{\mathrm{h}}-y\right)^{2}\right]\right\} \mathrm{d} x \mathrm{~d} y $$

The divergent spherical phase factor is superimposed on the generated complex amplitude distribution U(x_h, y_h). In holographic display, due to the converging effect of the imaging lens, the energy of the laser is concentrated on the reconstructed image plane, which aggravates the interference of speckle noise. The divergent spherical phase factor can further separate the reconstructed image from other interference information on the filter plane, thereby suppressing the speckle noise generated by zero-order and conjugate diffraction light on the display plane. The updated complex amplitude distribution formula is expressed as follows:

(4)$$ U_{\mathrm{c}}\left(x_{\mathrm{h}}, y_{\mathrm{h}}\right)=U\left(x_{\mathrm{h}}, y_{\mathrm{h}}\right) \cdot \exp \left[\frac{j \pi}{\lambda_{\mathrm{n}} f}\left(x_{\mathrm{h}}^{2}+y_{\mathrm{h}}^{2}\right)\right] $$

where f represents the focal radius of the divergent spherical phase factor, and its value is also set to d, so as to maximize the dispersion of speckle noise at the position of the filtering plane. Finally, the amplitude coding for the complex amplitude distribution U_c(x_h, y_h) is conducted, and the amplitude-only CGH H(x_h, y_h) obtained after coding is shown as follows:

(5)$$ H\left(x_{\mathrm{h}}, y_{\mathrm{h}}\right)=\left\|\left[U_{\mathrm{c}}\left(x_{\mathrm{h}}, y_{\mathrm{h}}\right)+R\left(x_{\mathrm{h}}, y_{\mathrm{h}}\right)\right]^{2}\right\|, $$

where R(x_h, y_h) is the set reference parallel light, and ||·|| means taking the modulus of the expression in the symbol. The reference light R(x_h, y_h) is a parallel light with (cosα, cosβ, cosγ) as the direction cosine.

(6)$$ R\left(x_{\mathrm{h}}, y_{\mathrm{h}}\right)=\exp \left[\frac{j 2 \pi}{\lambda_{\mathrm{n}}}\left(x_{\mathrm{h}} \cos \alpha+y_{\mathrm{h}} \cos \beta+s \cos \gamma\right)\right] . $$

In order to better realize the separation of the reconstructed image and the interference field, the reference light R(x_h, y_h) is set to the parallel light propagating in the following directions:

(7)$$ \left\{\begin{array}{l} \cos \alpha=\cos \beta=\frac{\rho D_{0} / 2}{\sqrt{\rho^{2} D_{0}^{2} / 4+s^{2}}} \approx \frac{\lambda_{\mathrm{n}}}{4 p} \\ \cos \gamma=\frac{s}{\sqrt{D_{0}^{2} / 4+s^{2}}} \approx 1 \end{array}\right. $$

It can be found that when the wavelength of the reference light is changed, the direction cosine of the parallel reference light will change, which leads to the inconsistent position of the reconstructed images of the CGHs calculated according to the different color channels on the reconstructed plane. In order to solve this problem, the reference light of each color channel is compensated by using grating phase P_n(x_h, y_h) (n = r, g, b) before the amplitude encoding, so that the CGHs of each color channel are reconstructed at the same position in the reconstruction plane. The grating phase equation is expressed as follows:

(8)$$ P_{\mathrm{n}}\left(x_{\mathrm{h}}, y_{\mathrm{h}}\right)=\exp \left(j\left(\frac{2 \pi}{\lambda_{\mathrm{n}}}\right) \cdot \sin \left(\arctan \left(\frac{\frac{\sqrt{2}}{4 s p}\left(\lambda_{\mathrm{n}}-\lambda_{\mathrm{g}}\right)}{1+\left(\frac{\sqrt{2}}{4 s p}\right)^{2} \lambda_{\mathrm{n}} \lambda_{\mathrm{g}}}\right) \frac{180}{\pi}\right) \cdot p \sqrt{x_{\mathrm{h}}^{2}+y_{\mathrm{h}}^{2}}\right) . $$

Therefore, the amplitude-only CGH H_n(x_h, y_h) with the compensated grating phase is generated as follows:

(9)$$ H_{\mathrm{n}}\left(x_{\mathrm{h}}, y_{\mathrm{h}}\right)=\left\|\left[U_{\mathrm{c}}\left(x_{\mathrm{h}}, y_{\mathrm{h}}\right)+R\left(x_{\mathrm{h}}, y_{\mathrm{h}}\right) \cdot P_{\mathrm{n}}\left(x_{\mathrm{h}}, y_{\mathrm{h}}\right)\right]^{2}\right\| $$

After normalizing H_n(x_h, y_h), the amplitude-only CGHs corresponding to the RGB channels can be obtained respectively.

In the third step, the above steps are repeated in the program until the frames of the CGHs of each color channel reach the required number. The same frames for the CGHs of the RGB channels are calculated respectively. In the reconstructed process of the holographic display, due to the high coherence of the laser and the diffraction effect of the display device, the pixel of the reconstructed image can be regarded as the Airy disk. If the distance between two adjacent pixels in the reconstructed image is too close, there will be a superposition area between the Airy disks. The interference occurs in the superposition areas, and it is embodied in the unevenly distributed bright and dark spots on the reconstructed image, which are also called speckle noise, as shown in Fig. 2. Essentially, the interference occurs due to the randomness of the phase of the diffraction light. When the multiple diffraction lights are superimposed on each other, the randomness of the phase is diminished. Therefore, if multi-frame of the CGHs are displayed simultaneously at a frequency higher than the human vision persistence effect, the speckle noise on the reconstructed image can be homogenized due to the time-averaging effect, and the reconstructed image will tend to be more similar to the recorded object.

Fig. 2. Phenomenon of uneven distributions of the bright and dark spots in the superposition areas of the Airy disks.

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When loading the multi-frame CGHs on the DMD for holographic display, the converging spherical light is used to illuminate the CGHs. There are two reasons for using the converging spherical light. One is to compensate for the discrete spherical phase set in the program, and the other is to converge the zero-order diffraction light into a single point on the filter plane. In this way, the interference of zero-order diffraction light to the reconstructed image is reduced. For the CGH of each color channel, different random band-limited phases are added, and different random band-limited phases bring different signal randomness. When the CGHs with different random band-limited phases of each color channel are loaded on the DMD, different random noise distribution is generated on the reconstructed image and that approximately obeys the negative exponential probability distribution. The speckle noise between the reconstructed images is not correlated with each other.

The peak signal-to-noise ratio (PSNR) is used to quantify the quality of the reconstructed images, which can be expressed as follows:

(10)$$PSNR = 20 \cdot \lg \left( {\frac{{255}}{{\sqrt {\frac{1}{{mn}}\sum\limits_{i = 0}^{m - 1} {\sum\limits_{j = 0}^{n - 1} {{{||{X(i,j) - Y(i,j)} ||}^2}} } } }}} \right),$$

where X(i,j) and Y(i,j) represent the object image and the reconstructed image with a size of m×n, respectively. The 24-bit depth bitmaps is used in the proposed method.

Besides, the speckle noise contrast (SNC) is used to measure the speckle noise of the reconstructed image. According to the statistical law of the speckle noise, the SNC of the M frames CGHs can be expressed as follows:

(11)$$SNC = \frac{\sigma }{{I\sqrt M }},$$

where σ is the standard deviation of the reconstructed image, and I is the average intensity of the reconstructed image. It can be found from the formula that when the average intensity I of the reconstructed image remains unchanged, the more frames CGHs are loaded, the smaller SNC will be.

3. Experiment and result

Based on the proposed method to suppress the speckle noise, a color holographic display optical path is built for the experiments, as shown in Fig. 3. The main components in the optical path are: RGB color lasers, shutters, reflectors, beam splitters (BSs), spatial filters, lenses, DMD and filter. The display device used in the optical path is a DMD with a resolution of 1920×1080 and a pixel pitch of 7.56µm produced by Xi'an CAS Microstar Science and Technology LTD. The refresh frame rate can reach 247 Hz in 8-bit grayscale display mode. The shutters are used to make the lasers with wavelengths of 671 nm, 532 nm and 473 nm emit in time sequence. The emitted light passes through the spatial filter and lens to form the uniform parallel light. The parallel RGB lights pass through a reflector and two BSs and then overlap. The overlapping RGB lights are reflected by the reflector. Then, the RGB lights pass through lens 1 with a focal length of 300 mm to form the convergent spherical lights. The DMD with an inclination angle of 24° (In the “on/off state”, the flip angle of the micro-mirror of the DMD is ±12°) is illuminated by the RGB spherical lights. The distance between lens 1 and the DMD is 60 mm. Lens 2 with a focal length of 200 mm is added at 140 mm behind the DMD to shorten the receiving distance of the reconstructed image. The filter is located at 60 mm behind lens 2. The camera without the eyepiece (Canon EOS 77D) located at 250 mm behind the filter is used to capture the reconstructed image separated from the stray light, zero-order and conjugate diffraction light.

Fig. 3. Optical path structure of the color holographic display system.

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In the traditional method, the CGHs for the RGB channels of the object are calculated by using the completely random phase. During the encoding of the CGHs, no discrete spherical phases are superimposed and the incident angles of the reference lights are not compensated. The letter ‘G’ of size 1000×1000 is used for the experiments in the traditional method. The single frame CGH of the letter ‘G’ for each color channel is loaded on the DMD separately and illuminated by the monochromatic light corresponding to the color channel. The experimental results are shown in Figs. 4(a)–4(c). It can be observed that the reconstructed images of the RGB channels of the letter ‘G’ are masked by the zero-order light. The high-order diffraction images exist simultaneously with the reconstructed images, and the speckle noise of the reconstructed images is also very serious.

Fig. 4. Experimental results of the letter ‘G’ by using the traditional method under the illumination of (a) red light, (b) green light and (c) blue light.

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In the proposed method, the single frame CGH for each color channel of the object is calculated by using the band-limited phase. During the encoding of the CGH, the discrete spherical phase is superimposed and the incident angle of the reference light is compensated. The letter ‘G’ is also used for the experiments in the proposed method. In the calculation of the band-limited phase, the number of iterations m is set to 16. According to Eq. (2), the recording distance of the letter ‘G’ is calculated as 250 mm by taking ρ=1.03. The focal radius of the divergent spherical phase factor is set to 300 mm. According to Eq. (8), the compensated incident angle of the red reference light is calculated as 1.47° and the blue reference light is calculated as -0.63°. After the stray light, zero-order and conjugate diffraction light are filtered out on the filter plane, the reconstructed images of the CGHs loaded on the DMD are shown in Fig. 5. The reconstructed results of the 10 frames CGHs of each color channel of the letter ‘G’ are shown in Figs. 5(a)–5(c). They are loaded on the DMD separately and illuminated by the monochromatic light of the corresponding color channel. Figure 5(d) is the reconstructed result when the DMD is refreshed to display 10 frames CGHs for each color channel under the illumination of the time-sequential emitted RGB lights. Compared with Fig. 4, it can be found that after filtering and grating phase compensation of the reference light incidence angle, not only the reconstructed image quality is improved, but also the positional chromatic aberration is eliminated, and the speckle noise is suppressed.

Fig. 5. Experimental reconstruction results of the letter ‘G’ by using the proposed method under the illumination of the (a) red light, (b) green light, (c) blue light and (d) time-sequential emitted RGB lights.

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Predictably, the more frames of the CGHs are refreshed on the DMD, the more speckle noise suppressed significantly. To validate this idea, the quality of the reconstructed images is compared and analyzed when different frames of the CGHs are loaded on the DMD. The recorded object is a ‘dice’ of size 1024×1024. Taking the green channel of the ‘dice’ as an example, the wavelength of the recording light is set to 532 nm, the recording distance is 250 mm (ρ=1.01), and the divergent spherical phase factor focal radius is 300 mm. The reconstructed results of the 1 frame, 10 frames and 30 frames CGHs of the ‘dice’ are shown in Figs. 6(a)–6(c) respectively. They are loaded on the DMD separately and illuminated by the green light. In this way, the reconstructed results of the red channel and the blue channel of the ‘dice’ can also be obtained, as shown in Figs. 6(d)–6(f) and Figs. 6(g)–6(i).

Fig. 6. Experimental reconstruction results. (a)-(c) Reconstructed images of the 1 frame, 10 frames and 30 frames CGHs of the green channel of the ‘dice’; (d)-(f) reconstructed images of the 1 frame, 10 frames and 30 frames CGHs of the red channel of the ‘dice’; (g)-(i) reconstructed images of the 1 frame, 10 frames and 30 frames CGHs of the blue channel of the ‘dice’; (j)-(l) reconstructed images of the 1 frame, 10 frames and 30 frames CGHs of the RGB channels of the ‘dice’.

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It can be observed that as the frames of the loaded CGHs increase, the speckle noise of the reconstructed image is gradually reduced. For example, Fig. 6(a) is the reconstructed image of the 1 frame CGH, and it can be observed that the details of the reconstructed image of the ‘dice’ have been completely covered by the speckle noise. Figure 6(b) is the reconstructed image of the 10 frames CGHs and the details of the reconstructed images are recovered, but the speckle noise can still be seen. As shown in Fig. 6(c), the speckle noise is further suppressed when the DMD is loaded with the 30 frames CGHs.

For the other color channels, the improvement of the quality of the reconstructed images follows a similar pattern. The color reconstructed images of the ‘dice’ are shown in Figs. 6(j)–6(l). The reconstructed results of the 1 frame and 10 frames CGHs of the RGB channels of the ‘dice’ are shown in Fig. 6(j) and Fig. 6(k). The CGHs are loaded on the DMD sequentially according to the emission temporal order of the RGB lights. The reconstructed results of the 30 frames CGHs of the RGB channels of the ‘dice’ are shown in Fig. 6(l). It is found that the speckle noise in the color reconstructed image is homogenized and the image details are restored as the frames of the CGHs loaded on the DMD increase.

Table 1 shows the PSNR of the reconstructed images in Figs. 6(a)–6(l). It can be found that the PSNR increases with the frames of the CGHs. It proves that increasing the number of refreshing frames of the CGHs helps to improve the quality of the reconstructed images.

Table 1. Relationship between PSNR and the frames of the CGHs.

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To quantify and analyze the speckle noise, the reconstructed images of 300×300 in the red-box region of Figs. 6(a)–6(l) are analyzed and their normalized intensity 3D distributions are shown in Figs. 7(a)–7(l). To facilitate the visualization of the data, the image intensity values are inverted. It can be found that as the frames of the CGHs increase, the speckle noise in the intensity distribution of the images becomes gradually sparse and the image details become gradually clearer.

Fig. 7. Normalized intensity 3D distribution. (a)-(c) ‘1 point’ intensity of the 1 frame, 10 frames and 30 frames green ‘dice’; (d)-(f) ‘1 point’ intensity of the 1 frame, 10 frames and 30 frames red ‘dice’; (g)-(i) ‘1 point’ intensity of the 1 frame, 10 frames and 30 frames blue ‘dice’; (j)-(l) ‘1 point’ intensity of the 1 frame, 10 frames and 30 frames color ‘dice’.

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Table 2 shows the SNC of the 300×300 red-boxed regions in Figs. 6(a)–6(l). It can be found that the more frames of the CGHs are loaded, the more effective the speckle noise suppression is. Especially for the green channel, loading 30 frames CGHs can reduce the SNC by 88% compared with loading 1 frame CGH. For the RGB channels, loading 30 frames CGHs per color channel can reduce the SNC by 62% compared with loading 1 frame CGH per color channel.

Table 2. Relationship between the SNC and frames of the CGHs.

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In addition, the advantages of the proposed method are verified through comparison experiments. The letter ‘BH’ of size 320×180 is used for the experiments. The simulation result of the previous method based on two SLMs [22] is shown in Fig. 8(a). The GS algorithm based on pixel separation with 20 iterations and pixel separation interval of 2 is used in the previous method. It can be noticed that around the letter ‘BH’ there is a significant background noise. The simulation result of the traditional amplitude-only holography method [29] is shown in Fig. 8(b). The reconstruction result of the single frame CGH for the letter ‘BH’ calculated by using the completely random phase is shown in Fig. 8(c). It can be found that the reconstructed image is completely covered by the speckle noise. The reconstruction result of the single frame CGH for the letter ‘BH’ calculated by using the band-limited phase is shown in Fig. 8(d). It is clear that the quality of the reconstructed image by using the band-limited phase is better than Fig. 8(c). In the proposed method, the refreshed frames of the CGHs are 30. It can be seen from Fig. 8(e) that the background noise is eliminated and the speckle noise is suppressed. The PSNR of Figs. 8(a)–8(e) is 14.4610, 18.7636, 12.3519, 20.2746 and 26.1118 respectively.

Fig. 8. Simulation reconstruction results. (a) Reconstructed image based on two SLMs; (b) reconstructed image of the traditional amplitude-only holography method; (c) reconstructed image without using the band-limited phase; (d) reconstructed image by using the band-limited phase; (e) reconstructed image by using the proposed method.

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The previous method based on interference fringes superposition [37] is also compared with the proposed method. The ‘duck’ with resolution of 576×324 is used for the simulation comparison experiments. To ensure the same comparison conditions, the number of interference fringes [37] is set to 10 and the number of CGHs frames of the proposed method to 10. Figure 9(a) is the original image of the duck. Figure 9(b) is the reconstructed image of the green channel CGHs under 10 interference fringes calculated by using the method of Ref. [37]. Figure 9(c) is the reconstructed image of 10 frames CGHs of the green channel calculated by using the proposed method. Figure 9(d) is the reconstructed image of the duck calculated by using the method of Ref. [37]. Figure 9(e) shows the color reconstruction of the duck calculated by using the proposed method. The PSNR of Figs. 9(b)–9(e) is 16.0225, 24.0166, 15.3385 and 26.6599 respectively. By enlarging the image details, it can be found that there is still some speckle noise in the reconstructed images by using the method of Ref. [37]. The speckle noise of the reconstructed images obtained by the proposed method is significantly less than that of the method used in Ref. [37].

Fig. 9. Simulation reconstruction results. (a) Recorded object ‘duck’; (b) green reconstructed image of the 10 fringes CGHs based on Ref. [37]; (c) green reconstructed image of the 10 frames CGHs based on the proposed method; (d) color reconstructed image based on Ref. [37]; (e) color reconstructed image based on the proposed method.

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Compared with the previous methods, the proposed method demonstrates its advantages. The speckle noise of the reconstructed images of the CGHs calculated with by the proposed method is significantly reduced, and the quality of the reconstructed images is greatly improved compared to the previous work. Besides, by using the proposed method, the conventional 4-f optical filtering system can be avoided. The structure of the system is optimized. The stray light, zero-order and conjugate diffraction light can also be effectively filtered out. Moreover, the proposed method enables color holographic display with elimination of chromatic aberration.

In the experiments, the computation of all the CGHs is done by a workstation on Intel Core i9-10980XE, 3.00 GHz, and 128GB RAM. Taking ‘dice’ as an example, the calculation time for a single frame CGH of each color channel takes about 2.5s, which includes the iteration time of the band-limited phase and the time of diffraction calculation. If each color channel requires 30 frames CGHs, it takes about 4 minutes to complete the computation of all the CGHs of the RGB channels, which is obviously not enough to meet the demand for real-time display. The maximum memory of the DMD in the experiments is 100MB, and the single frame CGH calculated is about 2MB, so the DMD can only store up to 50 frames CGHs. Because of the memory size limitation of the DMD, Fig. 6(l) is obtained by merging the color channels by computer. Next work will be focused on reducing the memory size of a single frame CGH and continuing to reduce the calculation time of a single frame CGH. In addition, the power mismatch of the red, green and blue lasers causes the white in Fig. 5(d) and Figs. 6(j)-(l) to be impure and reddish. It is worthwhile to continue exploring how to generate the truly white light.

4. Conclusion

In this paper, a method of color holographic display with speckle noise suppression is proposed. Firstly, the band-limited initial phase is used to reduce the randomness of the phase, and the contribution of the completely random phase to the speckle noise is effectively reduced. Secondly, the filter plane is designed in the optical path to place a filter to filter out zero-order and conjugate images to eliminate their effects on the display quality. Then, the laser energy is also diffused by using the divergent spherical phase factor, which effectively suppresses the speckle noise. Finally, the DMD instead of a phase-only SLM is used to achieve high frame rate refreshment for the color amplitude-only CGHs and different recorded objects are reconstructed by using a color holographic display system. The related experimental results show that the proposed method can effectively suppress the speckle noise, the SNC can be reduced by about 60% compared with that when loading a single frame CGH, and the color holographic display quality is effectively improved. The proposed method is expected to be applied in the fields such as holographic VR/AR display and so on.

Funding

National Key Research and Development Program of China (2021YFB2802100); National Natural Science Foundation of China (62011540406, 62020106010).

Acknowledgment

The authors would like to thank Dr. Nan-Nan Li, Dr. Wei Duan and Dr. Chen-Liang Chang for their contribution to this paper.

Disclosures

The authors declare that there are no conflicts of interest related to this paper.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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PSNR	Frames of the CGHs
PSNR	1	10	30
Green channel	17.1606	19.5352	19.5443
Red channel	16.8854	18.2834	18.9607
Blue channel	16.6649	17.2719	17.5659
RGB channels	16.9036	18.3635	18.6903

SNC	Frames of the CGHs
SNC	1	10	30
Green channel	0.0979	0.0170	0.0116
Red channel	0.0376	0.0249	0.0135
Blue channel	0.2762	0.0996	0.0631
RGB channels	0.0176	0.0109	0.0067

PSNR	Frames of the CGHs
PSNR	1	10	30
Green channel	17.1606	19.5352	19.5443
Red channel	16.8854	18.2834	18.9607
Blue channel	16.6649	17.2719	17.5659
RGB channels	16.9036	18.3635	18.6903

SNC	Frames of the CGHs
SNC	1	10	30
Green channel	0.0979	0.0170	0.0116
Red channel	0.0376	0.0249	0.0135
Blue channel	0.2762	0.0996	0.0631
RGB channels	0.0176	0.0109	0.0067

Method of color holographic display with speckle noise suppression

Abstract

1. Introduction

2. Principle of the method

3. Experiment and result

4. Conclusion

Funding

Acknowledgment

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Tables (2)

Equations (11)

Optics Express