Method of chromatic aberration elimination in holographic display based on zoomable liquid lens

Di Wang; Chao Liu; Qiong-Hua Wang

doi:10.1364/OE.27.010058

1. Introduction

As an ideal three-dimensional (3D) display technology, holographic display technology can bring a real visual experience to the viewer and has become one of the research hotspots in 3D display technology [1–6]. For a recorded color scene, it is necessary to use a coherent monochromatic light to ensure the quality of the reconstructed image. In order to reproduce the color scene more realistically, the research of true color holographic display is more important. With the development of optoelectronic device, holographic display based on the spatial light modulator (SLM) has become more and more popular [7,8]. The traditional methods to realize color holographic display are time-division method and spatial multiplexing method. Time-division method uses red, green and blue lights to illuminate the SLM in turns based on one SLM [9,10]. The SLM displays three color holograms respectively and outputs synchronized signals. Once the switching frequency of three color holograms is fast enough, color reconstructed image could be seen due to the afterimage effect. Spatial multiplexing method uses three SLMs to record three color holograms respectively [11,12]. When three color light beams are used to eliminate the SLMs, color reconstructed image can be seen due to the space synthetic registration. However, due to the different wavelengths in the holographic reconstruction, the position and size of the reconstructed images are different. So the magnification chromatic aberration and axial chromatic aberration exist and the quality of the reconstructed images is affected. In the previous researches, a method to eliminate the axial chromatic aberration is proposed by adding a phase factor on the hologram [13,14], and digital lens is also used to replace the conventional solid lens to eliminate the chromatic aberration [15,16]. In recent years, zoomable lenses (such as liquid crystal lens and liquid lens) have been widely used in imaging system, zoom system and holographic display [17–20]. In 2014, a method to eliminate the high-order diffraction light of the holographic display is proposed by using a mechanically controlled liquid lens [21]. High-order diffraction light and images can be separated from the reconstructed image by using the zoomable lens. Usually these zoomable component devices are smaller and more flexible than the conventional solid devices. At present, the proposed researches use commercially available liquid lenses in holographic display driven electrically or mechanically. Mechanically controlled liquid lens has slow response time due to the manual operation. The electrically controlled liquid lens has a fast response, but its types of structures are fewer.

In this paper, we propose a method to eliminate chromatic aberration in color holographic display. A liquid lens is developed based on electrowetting. Different from the structure of the conventional electrically driven liquid lens, the proposed device consists of a circular hole in the center of the middle substrate and several small holes around the center together to achieve focal length control. By controlling the focal length of the liquid lens, three color holographic reconstruction can display in the same position clearly. More importantly, color holographic reproduction can be achieved at any specific position without chromatic aberration.

2. Principle of the proposed method

In the holographic reproduction, when a parallel laser is used to illuminate the SLM loaded with hologram, holographic reproduction can display on the receiving screen after the modulation of the SLM, as shown in Fig. 1(a). The diffraction distribution at the distance of s from the SLM can be calculated according to Fresnel diffraction principle:

U_{s} (x, y) = \frac{e^{i k s}}{i s λ} \exp [\frac{i π}{s λ} (x^{2} + y^{2})] \int \int_{- \infty}^{\infty} {U (u, v) \exp [\frac{i k}{2 s} (u^{2} + v^{2})} \exp [\frac{- 2 i π}{s λ} (x u + y v)] d u d v,

where λ is the wavelength of the incident light,

k = 2 π / λ

, U_s(x, y) is the distribution of the reconstructed image, and U(u, v) is the distribution of the hologram. Then a lens (with focal length f) is introduced in the system, as shown in Fig. 1(b). When the SLM is located on the front focal plane of the lens, and the imaging plane is located on the back focal plane of the lens, the relationship between U_s(x, y) and U(u, v) meets the exact Fourier transform. Then the diffraction distribution on the receiving screen U_f(x, y) can be expressed as follows according to the Fourier transform:

Fig. 1 Principle of the holographic reproduction. (a) Fresnel diffraction; (b) Fourier transform.

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U_{f} (x, y) = \frac{e^{i k f}}{i f λ} \exp [\frac{i π}{f λ} (x^{2} + y^{2})] \int \int_{- \infty}^{\infty} U (u, v) \exp [\frac{- 2 i π}{f λ} (x u + y v)] d u d v .

For a lens, the focal length f can be calculated as follows [16,22]:

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

where r is the curvature radius of the lens and n is the refractive index. Then f_r, f_g and f_b can be calculated according to Eq. (3). We know that the refractive index n decreases with the increasing of the wavelength. So in color holographic reconstruction, when red, green and blue lights are used to illuminate the lens, the focal lengths of red, green and blue lights are f_r, f_g, f_b, respectively,

f_{r} \neq f_{g} \neq f_{b}

. So the position of the reconstructed images are different correspondingly. For a simple holographic system, color chromatic aberration can be shown in Fig. 2. According to the principle of the Fourier holographic diffraction, the size of the reconstructed image is h = fλ/p.

Fig. 2 Color reconstruction with chromatic aberration.

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So, in order to get a better viewing effect, we propose a method to eliminate the chromatic aberration by using a zoomable liquid lens to replace the traditional solid lens. In the proposed method, the zoomable liquid lens is produced based on electrowetting effect. The structure of the liquid lens is shown in Fig. 3. The device consists of an upper substrate, a bottom substrate, the dielectric layer, transparent conductive liquid, transparent non-conductive liquid and ITO film, and the ITO film is coated with electrode layer. A middle substrate with several holes is placed between the upper and bottom substrates. The middle substrate is coated with dielectric layer and ITO film sequentially. In initial state, the liquid-liquid interface forms a convex surface. The light beam converges after passing through the device and the device can realize the function of the positive lens, as shown in Fig. 3(a). When a voltage is applied on the liquid lens, the device can function as a negative lens due to the electrowetting effect, as shown in Fig. 3(b). By changing the number of the holes, the power of the liquid lens can be adjusted. By adjusting the voltage applied on the device, the liquid lens can realize the function of zoomable lens.

Fig. 3 Structure of the liquid lens. (a) State without voltage; (b) state when the voltage is applied on the device.

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In order to keep the position of three color reconstructed images in the same position, the liquid lens is used to adjust the reconstruction position. When the red hologram is loaded on the SLM, we use the red laser to illuminate the SLM and adjust the focal length of the liquid lens to $f_{r} = d$ . When the green hologram is loaded on the SLM, we use the green laser to illuminate the SLM and adjust the focal length of the liquid lens to $f_{g} = d$ . Similarly, When the blue hologram is loaded on the SLM, we use the blue laser to illuminate the SLM and adjust the focal length of the liquid lens to $f_{b} = d$ . By changing the voltage of the liquid lens, we can control the focal length $f_{r} = f_{g} = f_{b} = d$ . In this way, three color reconstructed images can coincide in the same position without axial chromatic aberration, as shown in Fig. 4.

Fig. 4 Color reconstruction with the proposed method.

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3. Experiments and results

In order to verify the feasibility of the proposed method, three different wavelength solid-state lasers are used in the experiment. The wavelengths of three color lasers are 633nm, 532nm and 471nm, respectively. A reflective phase SLM is used. The pixel size of the SLM is 8μm and the pixel number is 1920 × 1080. NaCl solution is used as the conductive liquid and silicon oil is used as the non-conductive liquid to produce the liquid lens. In order to eliminate the gravity effect of the device itself, the density of the two solutions is adjusted to be similar (the density 1.09 g/cm³). Firstly, the red, green and blue lasers are used to test the liquid lens, respectively. Figure 5 is the relationship between the focal length and the voltage of the liquid lens. From Fig. 5 we can see that when different voltages are applied on the liquid lens, the focal length changes accordingly. For the same voltage, the focal length of the red color is larger than that of the green color, and the focal length of the blue color is the smallest.

Fig. 5 Relationship between the focal length and the voltage of the liquid lens.

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A color lotus is used as the object to verify the proposed method. We process color separation of the lotus for green, blue and red components by using MATLAB software. Here the iterative Fourier transform algorithm is used to generate the holograms, as shown in Fig. 6. Firstly, the amplitude and phase of the object wave is initialized, and the frequency domain amplitude is set to B. The amplitude a and random phase φ constitute an original complex amplitude f. Then the complex amplitude f is Fourier transformed, yielding the complex amplitude F. Finally, the error value e is calculated. When e is less than the threshold ε, φ is the desired phase, or the calculations will be continued. In this way, three color holograms are generated by calculating the corresponding color scenes.

Fig. 6 Flow chart of the iterative Fourier transform algorithm.

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Then the liquid lens is used to reconstruct the image, as shown in Fig. 7. The laser, the filter and the solid lens are used to generate the collimated light. The liquid lens locates between the SLM and the receiving screen. When green laser is used to illuminate the SLM loaded with the hologram, green image can be reconstructed at the focal plane of the liquid lens. We record the voltage applied on the liquid lens when the green image on the receiving screen is the clearest. When the voltage is adjusted, the position of the reconstruction can be changed accordingly. Figure 8 is the green reconstructed image when the voltage of the liquid lens changes, where d_g is the distance between the liquid lens and the receiving screen. From the result we can see that the position of the green reconstructed image changes with the voltage of the liquid lens.

Fig. 7 Experimental setup of the holographic reproduction.

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Fig. 8 Green reconstructed image when the voltage of the liquid lens changes. (a) d_g = 40cm; (b) d_g = 42.5cm; (c) d_g = 65cm.

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Then, the liquid lens is used in the holographic system to eliminate color chromatic aberration. The red laser is used to illuminate the SLM loaded with the red hologram. By adjusting the voltage of the liquid lens, we record the position of the red reconstruction when the image on the receiving screen is the clearest, as shown in Fig. 9(a). The distance between the liquid lens and the receiving screen is recorded as d. We keep d unchanged and use the green laser to illuminate the SLM loaded with the green hologram. By adjusting the voltage of the liquid lens, the green reconstruction can focus on the receiving screen clearly. Similarly, when blue and green lasers are used to illuminate the SLM loaded with the corresponding holograms, blue and green reconstructed images can focus on the receiving screen clearly by adjusting the voltage of the liquid lens respectively. Figures 9(b) and 9(c) show the results before adjusting the voltage. The results show that the blue and green images are blurred due to the chromatic aberration. Figures 9(d) and 9(e) are the results after adjusting the voltage. The voltage is set to be U = 42.8V for the blue color and the voltage is set to be U = 43.6V for the green color. From the results we can see that by controlling the voltage of the liquid lens, three color reconstructed image can coincide in the same position without axial chromatic aberration.

Fig. 9 Reconstructed image when the voltage of the liquid lens changes. (a)-(c) are the images when U = 44.3V; (d) blue image when U = 42.8V; (e) green image when U = 43.6V.

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According to the principle of the Fourier holographic diffraction, the size of three color reconstructed images are h_r = f_rλ_r/p, h_g = f_gλ_g/p, h_b = f_bλ_b/p, respectively. So in order to keep the same size of three color images, the resolutions of three monochrome objects R_r, R_g, R_b are adjusted, where R_r: R_g: R_b = (λ_b/λ_r):(λ_b/λ_g):1, as shown in Fig. 10. In this way, the three reconstructed image can be adjusted to the same size, as shown in Fig. 11.

Fig. 10 Three color object after adjusting the resolutions. (a) Blue object; (b) green object; (c) red object.

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Fig. 11 Color holographic reconstruction.

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In the experiment, we use spatial multiplexing method with three SLMs and three liquid lenses to reconstruct the color image, as shown in Fig. 11, which is the color reconstruction of lotus without chromatic aberration. The experimental device of a single color is shown in Fig. 7. When using three SLMs, three beam splitting prisms are used to achieve color reproduction. The response time of the produced liquid lens is ~110ms, which is related to the viscosity of the liquid and the structure of the device. In the subsequent work, the response time can be improved by reducing the thickness of the dielectric layer. By optimizing the device, the response time can be faster. Then the device may be used to realize color holographic reconstruction with one SLM based on the time-division method. When the switching time is fast enough, color reconstructed image can be seen due to the afterimage effect. The effective aperture of the liquid lens can be fabricated about ~15mm. However, when the aperture becomes larger, the focal lengths changes becomes smaller since the electrowetting effect cannot provide enough driving force. Moreover, when the operating voltage is enhanced further, the lifetime of the liquid lens will be shortened. These are the bottlenecks of the liquid lens. The produced liquid lens has some advantages compared with the commercially liquid lens. The commercial liquid lens are usually driven electrically or mechanically. The electrically controlled liquid lens has a fast response, but it is always at great cost for about 1000 dollars. The focal length changes of the mechanical controlled liquid lens are usually limited. So in this paper, we design a new type of liquid lens actuated by electrowetting. The main advantages of our design are the low cost and simple fabrication. Besides, a middle substrate with several holes is placed between the upper and bottom substrates. At the same driving voltage, the curvature of the interface of the liquid lens can be improved greatly, thereby the curvature of the liquid-liquid interface and the optical power can be increased accordingly. In Fig. 3 we show the schematic of the lens magnification to demonstrate the mechanism of proposed liquid lens clearly. The liquid lens with high optical power can be used to increase the viewing angle of the holographic display in the future.

Besides, the liquid lens may have a certain influence on the imaging quality of the system, so the light intensity of the reconstruction is weak. Due to the small aperture of the liquid lens, the size of the reproduced image is limited. However, from Fig. 8 we can see that the liquid lens can be used to adjust the reconstruction size. At present, an important problem of the holographic display is that the size and viewing angle of the reconstruction are relatively small. Maybe the liquid lens can be used in the holographic system to improve the size and viewing angle. We believe that through further in-depth research, a holographic reproduction system with better viewing effect can be built, which makes the proposed method more practical.

In the proposed method, the liquid lens is used to replace the traditional solid lens and color chromatic aberration can be eliminated by controlling the focal length of the liquid lens. Compared with the method by correcting the hologram itself, the proposed method cannot only eliminate the chromatic aberration easily, but also change the size and position of the reconstructed image according to the requirement. From Fig. 8 we can see that the size and the position of the reconstructed image change with the voltage. At present, the size of the holographic reproduction cannot meet the viewing requirements. With the proposed liquid lens, holographic zoom system can be realized without changing the position of the system components. In order to verify the feasibility of the holographic zoom display, experiments are done by using the system of Fig. 7. A digital lens is loaded on the SLM and the green laser is used for reconstruction. By controlling the focal lengths of the digital lens and the liquid lens, zoom function can be realized without moving any components, as shown in Fig. 12. However, for the zoom system with solid lens, we need to use several solid lenses with different focal lengths and change the relative positions of the lenses to realize the zoom function. Moreover, the method by correcting the hologram is not easy to realize the zoom function. So the proposed liquid lens has unique advantages. Besides, since the liquid lens has a relatively small aperture, it also functions as a filter. When diffraction light illuminates the liquid lens, high-order reconstructed images cannot pass through the liquid lens. As shown in Fig. 13, the blue results by using the traditional method of correcting the hologram are given. Figure 13(a) is the result when using the traditional solid lens to reconstruct the image, while Fig. 13(b) is the result when correcting the hologram to change the position of the reconstructed image. The results show that the reconstructed image has many high-order reconstructed images.

Fig. 12 Result of the holographic zoom display. (a) - (c) are the reconstructed images of different magnifications.

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Fig. 13 Results when using the traditional method. (a) Result when using the traditional solid lens to reconstruct the image; (b) result when correcting the hologram to change the position of the reconstructed image.

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In order to verify the color coincidence effect, three SLMs are used in the experiment. The position and size of the reconstructed image can be adjusted easily. However, it is not easy to change the position and size of the reconstructed image by using three fixed lenses instead of tunable focus. Since the liquid lens has a relatively small aperture, it can also function as a filter. While the system without high-order images by using three fixed focus lenses is usually complicated. For the 3-SLMs system with three fixed lenses, the position of the reconstructed image cannot be changed easily. For the zoom system with solid lens, we need to use several solid lenses with different focal lengths and change the relative positions of the lenses to realize the zoom function. In this paper, the chromatic aberration is discussed in the Fourier holography. As shown in Fig. 1(b), only the front and back focal planes of the lens satisfy the exact Fourier transform relationship. So, 2D images are used in the experiment. In order to show the experiment more clearly, the video result of a moving object is given (see in Visualization 1). Moreover, 3D object is calculated based on the novel-look-up-table algorithm and the reconstructed image can be displayed on the receiving screen without the extra lens. Here we use the letters “BH” as the 3D object, where “B” and “H” are located at two different depths. When the liquid lens is placed behind the SLM, the reconstructed image can be displayed. When the focal length of the liquid lens changes, the object has a clear image at different depths. Zoom function can also be realized, and the reconstructed image can achieve a good scaling effect when passing through the liquid lens, as shown in Fig. 14.

Fig. 14 Result of the 3D object. (a) Result when focusing on the depth of “B”; (b) result when focusing on the depth of “H”.

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4. Conclusion

In this paper, a method to eliminate chromatic aberration in color holographic display is proposed by using a zoomable structure liquid lens. Experimental results verify that by changing the voltage of the liquid lens, three color holographic reconstructions can coincide in the same position without chromatic aberration. In addition, the liquid lens is easy to be combined with other methods to improve the holographic display performance, so it has practical significance.

Funding

National Natural Science Foundation of China (61805130, 61805169, 61535007).

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Method of chromatic aberration elimination in holographic display based on zoomable liquid lens

Abstract

1. Introduction

2. Principle of the proposed method

3. Experiments and results

4. Conclusion

Funding

References

Supplementary Material (1)

Cited By

Figures (14)

Equations (3)

Optics Express