Diffuser screen with flat-top angular distribution of scattered light realized by a dual-beam holographic speckle

Chaoxiong Yang; Chaoxiong Yang; Chaoxiong Yang; Haibo Jiang; Haibo Jiang; Jinyu Wang; Xiuhui Sun; Zheng Yang; Jianjun Chen; Yongmo Lv; Shaoyun Yin; Shaoyun Yin

doi:10.1364/OE.420910

1. Introduction

Diffuser screens are used in many fields, such as the head up display (HUD) which is a driving assistance system applied to automobiles and airplanes, enabling drivers to view information without lowering their heads [1,2,3]. In recent years, the functions of the HUD system have been enriched by combining with augmented reality technology, and a rapid growth of HUD is expected in the application of automobiles. In the HUD system, the images from liquid crystal on silicon (LCOS), digital micromirror device (DMD), or liquid crystal display (LCD) are projected to the HUD’s eye-box through an optical system. Diffuser screens are usually added at the intermediate image plane to increase the étendue so that the image observed in the eye box has the appropriate size and field angle [4].

According to the characteristics of microstructure, the diffuser screens can be divided into three categories. The first is the ground glass or plastic plate with sandblasted particles. These diffuser screens have a low production cost, but light transmittance is low (< 70%), and the angular distribution of scattered light cannot be accurately controlled [5]. These disadvantages greatly limit their application in HUD. The second type is the diffuser with a relief structure fabricated by micro lithography, such as laser direct writing lithography. The typical microstructure includes pyramids, microlens arrays, etc. [6–9]. The size of the scattering unit is usually more than 50µm and the distribution of scattered light can be precisely controlled. However, due to the large feature size of the microstructure, used as a screen in the intermediate image plane of HUD, this diffuser screen will have a strong graininess when viewed and the periodic structure renders the defects very visible. The third type is the holographic diffuser formed by speckle exposure. The size of the speckle is about 1/10 of the pixel size of the projection image, making the display effect more detailed and comfortable. Its light transmittance can reach more than 90% [10–14]. The angle of the scattered light can be controlled by adjusting the exposure parameters. Due to the advantages listed above, it is often used as the diffuser screen of the intermediate image.

In order to enhance display brightness, it is necessary to control the angular distribution of scattered light of holographic speckle screens. Lu et al. researched the relation between the scattering angle of fabricated the holographic speckle screens and the shape of illuminating area of the ground glass [10]. By controlling the slit width, the distance between the ground glass and photoresist, and exposure time, diffuser screens with different scattering angles can be fabricated. Sakai et al. used azobenzene functionalized polymers films as the holographic recording medium to fabricate surface-relief holographic diffusers, and the scattering angle can be expanded by controlling the polarization state, aperture size, and ground glass material [15]. Ganzherli et al. used silver halide emulsions as the holographic recording medium to fabricate holographic speckle screens [16]. The theoretical and experimental results show that the maximum scattering angle is proportional to the ratio of aperture size and diffraction distance, and the full width at half maximum (FWHM) of scattering angle can be controlled in the range of 0 ° to 8 °. Kevin Murphy et al. reported a method for fabricating diffractive holographic optical diffusers by using acrylamide-based volume photopolymer. By controlling the size and shape of the speckle, the shape and angular size of scattered light are controlled [17]. However, the probability density of the microstructure height of the holographic diffuser screens proposed above is in the form of Gaussian distribution, which cause a Gaussian or approximately Gaussian angular luminance distribution [13–17]. This type of energy distribution makes the brightness of the HUD system change obviously when viewed at different positions of the eye box. In order to overcome this problem, the scattering angle has to be designed much larger than the viewing angle, which leads to a waste of energy and reduction of image brightness. Diffuser screens with flat-top angular distribution can solve this problem.

In this paper, the spatial frequency spectrum (SFS) of the speckle field generated by the ground glass illuminated with two laser beams at different angles is analyzed theoretically. A fabrication method for holographic speckle screens with flat-top angular distribution by using a dual-beam speckle field is proposed. The optical path for speckle exposure is built, and the speckle screen is fabricated. Compared with the Gaussian holographic speckle screen, it has high energy efficiency under the requirement of highly uniform display. The proposed speckle screen is easy to be fabricated and can improve the brightness effectively used in the HUD system.

2. SFS of dual-beam holographic speckles

The surface-relief structure on the holographic speckle screen is formed by recording the far-field speckles produced by laser illuminated on the photoresist through ground glass [17,18,19], as shown in Fig. 1(a). When illuminated by a collimating laser beam, the angular distribution of the scattered light is the reconstruction of the SFS of speckles intensity received on the photoresist. By changing size of the optical aperture or the distance between the photoresist and ground glass, the intensity as a function of the scattering angle of the holographic speckle screen can be accurately controlled. The SFS of the scattered light from the ground glass illuminated by collimated laser is Gaussian-like, leading to the Gaussian-like angular distribution of the scattering light. The main idea to form scattered light with flat-top angular spectrum distribution is to use the SFS superimposition of multiple beams irradiating ground glass at different angles. Next, we will concentrate on the analysis of SFS of the dual-beam speckle field.

Fig. 1. (a) the schematic diagram of fabricating holographic speckle screens, (b) the schematic diagram of dual-beam speckle field superposition.

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As shown in Fig. 1(b), two light beams from the same laser are incident on the ground glass at ${\theta _1}$ and ${\theta _2}$, respectively, and the cross sections of the two beams with the ground glass are ${S_1}$ and ${S_2}$, respectively. When the surface correlation length of ground glass is much larger than the wavelength of the laser and the scattering field is in the far field, the scattering electric fields $E_{sc}^1({\xi ,\eta } )$ and $E_{sc}^2({\xi ,\eta } )\; $ produced by the two incident beams can be expressed by using the scalar Kirchhoff diffraction theory as [20]:

(1)$$E_{sc}^m({\xi ,\eta } )\; = \frac{{ikexp ({i{n_1}kd} )}}{{4\pi d}}\mathop \int\!\!\!\int\nolimits_{S_m} \left( {{a_m}\frac{{\partial h}}{{\partial x}} + {b_m}\frac{{\partial h}}{{\partial y}} - {c_m}} \right) \times exp ({ik[{{A_m}x} } + { {{B_m}y + {C_m}h({x,y} )} ]} )dxdy$$

where m=1,2, represents the index of incident laser beams; $h({x,y} )$ is the height of the rough surface at the position $({x,y} )$, and 〈〉 represents the average value. Other parameters are defined as follows:

(2)$$\begin{array}{*{20}{c}} {{A_m} = {n_1}sin {\theta _m} - {n_2}\xi \lambda ,{B_m} = - {n_2}\eta \lambda ,{C_m} = - {n_1}cos {\theta _m} + {n_2}\gamma \lambda }\\ {{a_m} = - {T_0}\left( {{n_2}\xi \lambda + \frac{{n_1^2sin {\theta _m}cos {\theta _m}}}{{{n_2}cos {\theta _{\textrm{spec }}}}}}\; \right),{b_m} = - {T_0}{n_2}\eta \lambda }\\ {{c_m} = {T_0}\left( {{n_2}\gamma \lambda + {n_2}cos {\theta _{\textrm{spec }}}}\; \right),{n_1}sin {\theta _m} = {n_2}sin {\theta _{\textrm{spec}}}} \end{array}$$

where $({\xi ,\eta } )$ is the spatial frequency of the scattered light, which is defined as $\xi = sin {\theta _s}cos {\varphi _s}/\lambda $, $\eta = sin {\theta _s}sin {\varphi _s}/\lambda $, $\gamma = cos {\theta _s}/\lambda = \sqrt {{\lambda ^{ - 2}} - {\xi ^2} - {\eta ^2}} $ . ${\theta _s}$ is the angle between the scattered light and the z axis; ${\varphi _s}$ is the azimuth of the direction of scattered light, $\lambda $ is the wavelength of the laser; ${n_1}$ and ${n_2}$ are the refractive indices of ground glass and its surrounding medium respectively; $k = 2\pi /\lambda $, ${T_0}$ is the transmittance coefficient of ground glass. Thus, the superposition of the scattering electric fields from ground glass can be expressed as follows:

(3)$${E_{sc}}({\xi ,\eta } )= E_{sc}^1({\xi ,\eta } )+ E_{sc}^2({\xi ,\eta } )$$

The average intensity of the scattering field is as follows:

(4)$$\begin{aligned} \left\langle {I_t}({\xi ,\eta } ) \right\rangle &= \left\langle {E_{sc}}({\xi ,\eta } )E_{sc}^\ast ({\xi ,\eta } ) \right\rangle = \left\langle E_{sc}^1E_{sc}^{1\ast } \right\rangle + \left\langle E_{sc}^2E_{sc}^{2\ast } \right\rangle + \left\langle E_{sc}^1E_{sc}^{2\ast } \right\rangle + \left\langle E_{sc}^2E_{sc}^{1\ast } \right\rangle\\ &= \left\langle {I_{t1}}({\xi ,\eta } ) \right\rangle + \left\langle {I_{t2}}({\xi ,\eta } ) \right\rangle + \left\langle {I_{t12}}({\xi ,\eta } ) \right\rangle + \left\langle {I_{t21}}({\xi ,\eta } ) \right\rangle \end{aligned}$$

The first and second terms of Eq. (4) corresponding to the average intensity of scattering light from the first and second beams, and the third and fourth terms are the interference of the diffused field produced by the two beams. When the angle between the two incident beams satisfies $|{{\theta_1} - {\theta_2}} |\ge {5^o}$, the latter two terms can be ignored [21,22]. Therefore, the average intensity of the superimposed scattering field is the sum of the average scattering field intensity of each single beam irradiating ground glass, as follows:

(5)$$\left\langle {I_t}({\xi ,\eta } ) \right\rangle \approx \left\langle {I_{t1}}({\xi ,\eta } ) \right\rangle + \left\langle {I_{t2}}({\xi ,\eta } ) \right\rangle $$

The rough surface of ground glass usually meets the following requirements: (I) the surface roughness is large enough; (II) the height distribution function ${P_h}(h )$ of random rough surface is Gaussian distribution, as follows:

(6)$$({{n_2} - {n_1}} )\sqrt {\left\langle{h^2}\right\rangle_{{s_1}}} = ({{n_2} - {n_1}} )\delta \ge \frac{\lambda }{{2\sqrt {2Ln2} }}$$

(7)$${P_h}(h )= \frac{1}{{\delta \sqrt {2\pi } }}exp ( - \frac{{{h^2}}}{{2{\delta ^2}}})$$

So, the average intensity of the scattered field after the ground glass irradiated by each single beam can be simplified as follows:

(8)$$\begin{aligned} \left\langle {I_{tm}}({\xi ,\eta } ) \right\rangle &= \frac{{{k^2}F_m^2{\beta ^2}}}{{4\pi {d^2}}}\frac{1}{{{g_m}}}{S_m}exp \left( { - \frac{{{k^2}({A_m^2 + B_m^2} ){\beta^2}}}{{4{g_m}}}} \right) \\ &= \frac{{{k^2}F_m^2{\beta ^2}}}{{4\pi {d^2}}}\frac{1}{{{g_m}}}{S_m}exp \left( { - \frac{{n_2^2{\lambda^2}{\beta^2}\left[ {{{(\xi - \frac{{{n_1}sin {\theta_m}}}{{{n_2}\lambda }})}^2} + {\eta^2}} \right]}}{{4{\delta^2}C_m^2}}} \right) \end{aligned}$$

where $\delta $ and $\beta $ are the variance and correlation length of rough surface height function respectively, ${S_m}$ is the aperture area of the rough surface, ${g_m} = {k^2}{\delta ^2}C_m^2$, ${F_m} = 0.5 \times [{({{A_m}{a_m}/{C_m}} )+ ({{B_m}{b_m}/{C_m}} )+ {c_m}} ]$, and $A_m^2 + B_m^2 = n_2^2{\lambda ^2}\{{{{[\xi - {n_1}sin {\theta_m}/({{n_2}\lambda } )]}^2} + {\eta^2}} \}$.

According to Eqs. (5) and (8), the SFS of the scattered light will be shifted by $sin {\theta _m}/({{n_2}\lambda } )$ from the normal incident light. And only when ${\theta _2} ={-} {\theta _1}$, $\left\langle {I_{t2}}({\xi ,\eta } ) \right\rangle = \left\langle {I_{t1}}({ - \xi ,\eta } ) \right\rangle$ is obtained, the intensity of scattered light is symmetrically distributed.

Figure 2(a) shows the normalized angular distribution of scattered light of ground glass. The SFS is Gaussian-like, and its width decreases gradually with the increase of polish grade of ground glass. With the increase of incident angle ${\theta _m}$, the peak of SFS deviates from the center.

Fig. 2. (a) Normalized spatial frequency of scattered light as a function of scattering angle for different roughness of ground glass and different incident beam angles, (b) The normalized SFS of the scattered light as a function of scattering angle for two beams with different included angles, (c) $\sigma /{\sigma _s}$ changes with glass roughness $\delta $ and angle $\theta $ between two incident beams.

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Figure 2(b) shows the SFS of the scattered light obtained by illuminating the ground glass with two beams of light at the included angles of ±10°, ±20°, ± 25° and ±30° when the ground glass roughness is $\delta = 1.8$µm, and the wavelength is $\lambda = 325$ nm. It can be seen that with the increase of the included angle, the profile of SFS changes from Gaussian to the flat-top, then to that with central depression.

In order to quantitatively evaluate the flat-top degree of scattered light, combined with the requirement of 70% uniformity commonly required in display, we define the nonuniformity sigma is:

(9)$$\sigma = \sqrt {{{\left\langle (I - {I_{ave}})\right\rangle} ^2}_{I \ge 0.7}} $$

where I is the normalized angular spectrum intensity, ${I_{ave}} = {\left\langle I \right\rangle_{I \ge 0.7}}$. The closer the $\sigma $ value is to zero, the closer the angular distribution of the scattered light is flat-top. As a comparison, ${\sigma _s}$ represents the nonuniformity of the scattered light of the speckle screen formed when a single beam irradiates the same ground glass.

According to Eqs. (5) and (9), we calculate the value of $\sigma /{\sigma _s}$ as a function of $\delta $ and $\theta $, as shown in Fig. 2(c). The corresponding the grit of ground glass in the figure is 120–600, and the wavelength is 325 nm. It can be seen that the angle $\theta $ between two incident beams for producing flat-top angular distribution is different for different ground glass. The larger $\delta $ is, the larger $\theta $ is, and there is a suitable $\theta $ value to determine the perfect flat top effect. When the angle $\theta $ deviates from the appropriate value, the $\sigma /{\sigma _s}$ approaches to 1, and the homogenization effect will be eliminated. Flat-top angular distribution with different angles can be achieved by reasonably selecting the included angle $\theta $ and the specification of ground glass, corresponding to the blue area in Fig. 2(c).

3. Experimental results

In order to verify the above analysis, we set up the optical path as shown in Fig. 3(a). A beam from He-Cd laser ($\lambda = 325$ nm) is focused by an aspheric lens and filtered by a pin hole with size 15um, then it is collimated by two cylindrical lenses with effective focal length (EFFL) of 20 mm and 200 mm respectively. The collimating beam passes through the beam splitting prism (BS) and is reflected to the ground glass (240 grit polish) at an angle ${\theta _m}$ of by two mirrors M₁ and M₂. The roughness $\delta $ of the ground glass is about 1.8µm which is measured using the white light interferometer (Zygo NewView 9000). The speckles formed by scattered light of ground glass are recorded on the photoresist.

Fig. 3. (a). Experimental diagram of speckle recording; (b). Soft lithography.

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The manufacturing process of the speckle screen is shown in Fig. 3(b). The photoresist is developed after recording the speckle. The mold is made by transfering the pattern on the photoresist to the Polydimethylsiloxane (PDMS), and then the photosensitive adhesive is coated on the quartz substrate to replicate the microstructure of the PDMS. In the experiment, AZ4562 photoresist was applied at 1500r/min for 45s, and the thickness was about 5µm. The exposure dose is about 40mJ/cm². The average thickness of the PDMS mold is 5 mm, and it is formed after baking at 60°C for 12 hours. Finally, a layer of photosensitive adhesive (Norland Optical adhesive, NOA61) was coated on the surface of PDMS mold, and cured by ultraviolet light.

Figure 4(a) shows the photograph of the holographic speckle screen and the scanning electron microscopy (SEM) image of the microstructure. The feature size of the microstructure is about 20 µm, and by measuring the optical power before and after passing through the holographic speckle screen with a power meter (PM100D, Thorlabs), it is obtained that the transmittance is about 92%. Figure 4(b) shows the scattering light distribution when the laser ($\lambda = 650nm$) irradiates on the speckle screen. The scattered light is elliptical due to cylindrical lenses with different focal lengths are used in the two directions.

Fig. 4. (a) the photograph of holographic speckle screen and the SEM image of its microstructure. (b) the scattered light distribution of holographic speckle screen irradiated by a laser with $\lambda = 650nm$.

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The angular distribution of scattered light is measured by a photometer (LED620, EVERFINE). The principle of the testing optical path is shown in Fig. 5(a). The test system consists of a rotating platform with an accuracy of 0.1° and a photodetector with an aperture of 1cm² installed 31.6 cm away from the center of the rotating axis; a blue semiconductor laser ($\lambda = 450nm$) and the holographic speckle screens are fixed on the platform. In the experiment, we used diffuser screen samples obtained in the condition of light beam included angles of ±10°, ± 25°, and ±30°. The measurement results of the scattered light in the direction that the SFS superposition of the speckle field occurs are as shown in Fig. 5(b) (the solid line). For comparison, the theoretical results are also shown in Fig. 5(b) as the dotted line. It can be seen that the theoretical values are in good agreement with the experimental results.

Fig. 5. (a) Optical schematics for testing the angular distribution of the speckle screen. (b) Theoretical and experimental results of angular distribution of the scattered light.

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The holographic speckle screens placed at the plane of real image scatter the light into the designed direction, and the angular intensity determines the brightness of the image observed at different positions of the eye box [5]. In the display system with a specific field of view, we prefer to see the same brightness in each position. For Gaussian type scattering screen, it cannot meet this requirement. In practice, the usual way is to increase the scattering angle of the screen, which inevitably leads to a waste of energy and a reduction of the overall brightness. However, the diffuser screen with flat-top angle distribution is expected to alleviate this problem and achieve the effect of brightness enhancement.

The energy efficiency of different brightness uniformity requirements can be obtained by integrating the normalized energy of the scattered light within the viewing angle. Table 1 shows the calculated results of the holographic speckle screens fabricated by our proposed method and the one beam exposure method. It shows that the higher the requirement for uniformity of the field of view, the more obvious the advantages of our holographic speckle screens. The energy efficiency is defined as ${\eta _{diffuser}} = ({{\eta_s} - {\eta_f}} )/{\eta _f}$, where ${\eta _s}$ and ${\eta _f}$ are the energy efficiency of the holographic speckle screen proposed in this paper and that fabricated by classical method respectively.

Table 1. Energy efficiency comparison

View Table

4. Conclusion

Based on the speckle field generated by dual-beam illuminated rough surface, a method for fabricating diffuser screens with flat-top angular spectrum distribution of scattered light is proposed. The experimental results and the theoretical calculations are in good agreement with each other. The diffuser screens realized by this method can be applied to HUDs for vehicle and airborne, LCD backlight, and other fields to enhance brightness uniformity or increase display brightness.

Funding

National Key Research and Development Program of China (2017YFB1002902); National Natural Science Foundation of China (61475199); Chongqing Science and Technology Commission (cstc2019jscx-mbdxX0019).

Acknowledgments

The author thanks the National Science Foundation of China for helping to identify the collaborators of this work.

Disclosures

The authors declare no conflicts of interest.

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Brightness uniformity requirements	Classical speckle screen	Holographic speckle screens	$η_{d i f f u s e r}$
0.5	78.46%	91.18%	18.00%
0.6	71.28%	86.32%	21.10%
0.7	62.80%	80.60%	28.34%
0.8	49.53%	72.39%	46.16%
0.9	38.25%	59.25%	55.19%

Diffuser screen with flat-top angular distribution of scattered light realized by a dual-beam holographic speckle

Abstract

1. Introduction

2. SFS of dual-beam holographic speckles

3. Experimental results

4. Conclusion

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (5)

Tables (1)

Equations (9)

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