## Abstract

How to balance the optical efficiency, illumination uniformity and the size of the illumination system is a challenging task in projector design. In this paper, we present a mathematical model describing the relationship between optical energy of the illumination system and the optical parameters and an optimization design method considering the light intensity distribution of the light source. By using the proposed method, two illumination systems are designed with different types of the digital micromirror device chips. In addition, we also propose a non-coaxial system to solve the deformation problem caused by the large flip angle of the DMD chip and further improve the illumination uniformity based on Scheimpflug principle. The optical efficiency and illumination uniformity of the illumination systems were verified and analyzed. The results indicate that the systems designed by the proposed method can provide a good design scheme and obtain a satisfactory utilization rate of the optical energy and higher uniformity.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Digital light processing (DLP) is a technology used in an all-digital projection system that is based on Texas Instruments Digital micromirror device (DMD) [1–3]. Compared with traditional projectors, the projectors using DLP technology has better optical efficiency, higher image quality and a smaller system size [4]. The DLP projectors have been extensively used in many fields from entertainment, medicine to industrial applications [5–7].

The illumination system is the key to the design of the projector optical engine. An illumination system with high uniformity and high optical efficiency can make the projection system have excellent projection effects and high energy utilization rate, thereby reducing the pressure on thermal dissipation [8]. Therefore, the high optical efficient and uniform illumination system is a critical issue in projector design, especially for the pico-projector.

Light emitting diode (LED) is the most suitable light source for the DLP projector, because LED light source has the advantages of small volume, long lifetime, affordable price, fast on-off speed and high efficiency [9–11]. However, the optical power per unit of étendue (luminance) of LED is significantly lower than that of other light sources, such as ultra-high performance (UHP) mercury halide arc-lamp [12]. Moreover, the light output from LEDs that can leave the illumination system as usable light is limited. To increase the acceptance étendue, it is necessary to effectively collimate the LED light using non-imaging optical devices [13,14], which will also lead to the increase of the total size of the projector. The system design always requires a trade-off between system complexity and achievable luminous output. Therefore, how to balance the optical efficiency, illumination uniformity and the size of the illumination system is a challenging task in projector design.

The compound-eye has received much attention in recent years because of its remarkable properties, such as its large field of view (FOV), low aberration and distortion, and compact structure [15]. Therefore, it is commonly used in the illumination system of projectors [16–19]. The total internal reflection prism (TIR) [20] and the reverse total internal reflection prism (RTIR) [21] have been proposed to connect the non-imaging system with the imaging system, which can be used to reduce the size of the projector. However, in the optical system design of the projector, the selection of optical materials and structural parameters generally depends on manual selection and simulation iterative optimization. To the best of our knowledge, a systematic, accurate and effective design method of optical parameters has not been discussed in the previous work about DLP projection system.

In this paper, a mathematical model is proposed to describe the relationship between optical energy and optical parameters of each part in the illumination system. Meanwhile, the optimal design function and method has been proposed considering the intensity distribution of the light source. The proposed design method is verified and analyzed by two illumination systems with different types of the DMD chips. In addition, the projector optical system is improved based on Scheimpflug principle, so as to further enhance the illumination uniformity.

## 2. Design methodology

#### 2.1 Overview of the illumination system

The illumination system in this paper composes of the LED light source, the collimation lens, fly’s eye lens array (FELA), and the DMD chip, as shown in Fig. 1. The collimation lenses are used to collect the energy from light source as the concentrator. The illumination system requires a uniform light distribution in order to achieve the best possible results. The characteristics of FELA are efficient homogeneous illumination in working plane, compact design, beam shaping and convenient to use. Therefore, the FELA is utilized as uniform illumination device.

Referring to Fig. 2, the beam irradiated from the LED light source passes through the collimation lens, and incidences to the first surface S_{1}. Then each microlens of the FELA on the surface S_{1} focuses on their vertexes on the second surface S_{2}. The FELA and the DMD chip should be arranged in the front and back focal plane of relay lens, respectively, so that the focal point of microlenses illuminates the DMD chip as an individual point light source, and every pixel of the DMD chip receive a portion energy from every focal point of the FELA. The relay lens can make the sub-apertures of first surface S_{1} of FELA relay to the aperture of DMD chip. Therefore, the illumination distribution of DMD chip can be approximated to the sum of illumination distribution of sub-apertures. In order to ensure that the output beam passing through the relay lens could uniformly distributed on the DMD chip, and to minimize the energy loss, the rectangular FELA is used in the illumination system to redistribute the circular beam into rectangular (a flat-topped profile). Therefore, based on the working mechanism, the relationship between parameters of the FELA, including the focal length of relay lens *f _{r}*, the size of illumination area

*S*, the aperture angle of the FELA

_{x}*U’*and the aperture angle of DMD pixel

_{x}*V*can be described as the followed equations.

_{x}Meanwhile, *V _{x}* should be also considered based on the projection system, which is shown in Eq. (3).

*F*is F-number of the projection system. To guarantee that all the rays can enter into the projection system,

_{p}*V*should be equal to the aperture angle of projection system.

_{x}#### 2.2 Mathematic model of optical energy

As the Fig. 2 shown, the x-axis is used to describe the position coordinates in the meridian plane, and the z-axis is the direction of the light propagation. S_{c} is the principal plane of the collimation lens, and the energy on the surface S_{c} can be described as:

*x*is the position coordinate along the x-axis on the surface S

_{c}_{c},

*θ*is the corresponding aperture angle at

_{c}*x*, ${I_1}({{x_c}} )$ is the energy in the circle of the radius

_{c}*r*=

*x*on the surface S

_{c}_{c}, $\theta $ is the radiation angle of light source. $L(\theta )$ is the light intensity distribution of LED light source fitting with the original intensity data. It can be expressed by the polynomial fitting:

Meanwhile, *x _{c}* and

*θ*have to satisfy Eq. (6) according to the imaging principle:

_{c}*fc*is the focal length of the collimation lens.

The light beam passes through the collimating lenses and enters the FELA. The energy distribution is divided into many parts with microlenses on the surface S_{1}. The energy distribution of each microlens can be defined by:

*x*is the coordinate of the center of the

_{n}*n*th microlens, Δ

*x*is the diameter of the microlens, ${x_n}\textrm{ - }\frac{{\Delta x}}{2}$, ${x_n} + \frac{{\Delta x}}{2}$ can be transformed into

*θ*based on Eq. (6) for simple calculation by substituting

*x*. The beam passing through the FELA is divided into separate beams with each lens element. Therefore, the energy of separate beams can be written as:

_{c}*n*is an integer.

*N*is the number of microlenses in the

_{x}*x*direction, which is always odd. Equation (8) is the energy of the

*n*th microlens. The energy of each microlens is further normalized by the Eq. (9), which facilitates the subsequent calculation.

Based on the working principle of the relay lens, the relation between the position of microlens and the incidence angle can be shown as:

where ${V_{xn}}$ is the incidence angle caused by the*n*th microlens on the DMD surface.

The RTIR prism is used to connect the illumination system with the projection system, which can reduce the size of projector hardware and ensure that the incidence ray can enter into projection system when the DMD chip is in the ON state. The RTIR consists of a wedge prism and an equilateral triangular prism. There is a 5µm air gap between these two prisms [22]. It is can be seen from Fig. 3 that the light enters into the RTIR and reaches the DMD chip, and then reflects to the interface between the equilateral triangular prism and the air gap, resulting in total internal reflection, so as to change the direction of light for entering the projection system. In the process of system design, to maximize the light energy on the DMD chip, three conditions should be considered:

- 1). Total reflection cannot occur at the interface between the wedge prism and the air gap.
- 2). Total reflection must occur at the interface between the rectangular prism and the air gap when the rays is reflected from DMD chip.
- 3). The light cone angle of each DMD pixel must not be greater than the aperture angle of the projection system.

However, not all the DMD chips can meet the above conditions in practical application. In projector design, higher illumination uniformity and optical efficiency at the DMD’s active area is a critical issue. Therefore, energy design can be transformed into a mathematical optimization problem with nonlinear constraints. The design variables are refractive index of the wedge prism and the triangular prism ${n_w}$, ${n_t}$, the installation angle of the wedge prism ${\theta _{in}}$, the incidence angle of the DMD chip ${\theta _{DMD}}$. The energy on the DMD chip can be written as:

*n*th microlens,

By means of the sequential unconstrained minimization technique method (SUMT), the optimal problem with the constraint is converted to a minimum value problem of merit function:

${\theta _{DMD}}$ in the above equation has to satisfy the constraints: $|{\theta _{DMD}} - 2{\theta _{ON}}|\le 3^\circ$, where ${\theta _{ON}}$ is the flip angle of the DMD chip in the ON state.According to Eq. (5)-Eq. (12), the energy distribution of each component of the illumination system can be described by the four parameters in Eq. (13). Then the parameters of the RTIR prism can be easily optimized by Eq. (13). The method described above can also be applied to the sagittal plane.

#### 2.3 Design of the FELA

In this subsection, the numbers of microlenses are discussed based on optical efficiency. Generally speaking, a part of light energy is wasted on the DMD chip, because the spot size of the light beam from FELA and relay lens is slightly larger than the size of the DMD chip. Considering that the machining accuracy can reach 0.2mm, the edge of the illumination area on the DMD chip is designed to be 0.2mm larger than the chip, which also depends on the alignment error of the system. The layout of the illumination spot on the DMD chip is shown in Fig. 4. The aspect ratio of illumination spot is described as:

where*a*and

*b*are the length and width of the DMD chip, respectively. The yellow area represents the illumination spot, and the array of blue blocks represents the pixels of the DMD chip. The RTIR prism has an anamorphic effect [23], and the lateral magnification is calculated by Eq. (15), which has been discussed in Ref. [21].

The layout of the FELA and the change of the illumination spot shape before and after the light beam passes through the FELA are shown in Fig. 5(a) and Fig. 5(b), respectively. The illumination spot shape on the DMD surface should match the shape of the DMD chip to improve energy utilization rate. Therefore, the aspect ratio of the illumination spot after the beam passes through the FELA is equal to the ratio of the numbers of microlenses in x- and y-direction. Considering the lateral magnification caused by the RTIR prism, the width of microlenses in x- and y-direction *W _{x}* and

*W*can be calculated by the follwing equation:

_{y}## 3. Optical simulation and results

#### 3.1 Systems layout

In this section, the proposed method is verified in terms of optical efficiency and illumination uniformity by two illumination systems. The illumination systems were designed based on DMD4500 and DMD3010 chips, respectively. The parameters of these DMD chips are listed in Table 1. The systems were simulated by Zemax optics software. The optical efficiency and illumination uniformity was analyzed by LightTools illumination design software.

As Fig. 6(a) shown, the illumination system based on DMD4500 uses RGB LEDs as the light source, which is usually used for entertainment such as home theatre projector. In this system, an optical lens L1 is added behind the first mirror to correct the difference caused by different sources, and a convex lens L2 is added behind the FELA to collect more light from the large incident angle. The illumination system based on DMD3010 adopts blue LED as shown in Fig. 6(b), which is usually used in industrial measurement, such as 3D camera. The size of both systems is marked in the Fig. 6, which is very suitable for compact projectors. The parameters of both optical systems are listed in Table 2. The prototypes of DMD4500 system and DMD3010 system are shown in Fig. 7, respectively.

#### 3.2 Optical efficiency analysis

The optimal parameters of the RTIR prism are obtained by Eq. (13). According to ${n_w}$ and ${n_t}$, the material of RTIR prism can be determined. Material of the prism, ${\theta _{in}}$ and ${\theta _{DMD}}$ are listed in Table 3. *F _{p}* is determined by the tradeoff between design capability of projection systems and ${\theta _{DMD}}$. For DMD4500,

*F*is chosen based on the maximum value calculated by ${\theta _{ON}}$. For DMD3010, the maximum value calculated by ${\theta _{ON}}$ is difficult for projection design, and 2 is a reasonable value for optical design.

_{p}The simulation result of the optical efficiency is calculated by LightTools and the energy loss caused by the FELA is considered. The optical efficiency of DMD4500 system and DMD3010 system is 55.2% and 67%, respectively. 55.2% is the average value of RGB LEDs, which is 54%, 55% and 56.6%. The measured result of the optical efficiency is defined as the ratio of the energy incident on the screen to the energy emitted from the LEDs. The optical energy on the projection screen is measured by using the illuminometer, and the energy of LEDs is measured by the lumenmeter under the same electric current with energy measurement of projection. The measured values of the DMD4500 system and the DMD3010 system are 52% and 62%, respectively. The results show that the simulated optical efficiency is in good agreement with the measured optical efficiency, indicating that the systems designed by the proposed method can obtain a satisfactory utilization rate of the optical energy.

Obviously, the utilization rate of the optical energy of DMD3010 system is higher than DMD4500 system. The most important reason is ${\theta _{ON}}$, which has an effect of total reflection at the interface of wedge prism and the rectangular prism. The larger angle ${\theta _{ON}}$ has a better comprise between them. Therefore, the angle is a key factor in the selection of the DMD chips with high requirements for optical energy.

#### 3.3 Illumination uniformity analysis

The illumination uniformity of on the DMD surface is calculated by using 500×500 points, and one million rays have been traced in LightTools. There are three indicators to evaluate the illumination uniformity. The illumination area is divided into nine zones as shown in Fig. 8. The *Uniformity* values are defined as follows:

In the above equations, *Uniformity1* represents the illumination difference between four corners and the central region. *Uniformity2* and *Uniformity3* not only represents the difference in illumination between the maximum and minimum value, but also can eliminate the evaluation error of higher *Uniformity1* caused by image plane tilt. Then, a complete evaluation system of the illumination uniformity can be established by using three *Uniformity* values. By measuring the irradiance of each zone on the DMD surface, the evaluation result of illumination uniformity can be calculated, as shown in Table 4. The simulated irradiance distribution on the DMD surface of both systems is shown in Fig. 9. The 2D-plots of Fig. 9 have been smoothed, and the smoothing parameters are set by default in LightTools, including the integration width of 3×3 specifies, without edge preservation. The total flux difference is 0.02%, far less than 10%. Therefore, the data after smoothing in LightTools is credible.

As shown in Table 4, the uniformity evaluation results of DMD4500 system and DMD3010 system are both greater than 90%, which indicates that the system designed by the proposed method can obtain a satisfactory illumination uniformity.

The experiment was arranged to examine the illumination uniformity for the performance of the prototypes by the industrial monochrome camera, which has better measurement accuracy than the color camera. The illumination patterns and normalized illumination distribution are shown in Fig. 10. In Fig. 10, the blue and yellow curves represent the normalized illuminance distribution of the horizontal and vertical center axes of the illumination spot, respectively. The normalized illuminance distribution is calculated by dividing the gray value by the maximum gray value. The *Uniformity* values of the experimental results are calculated and shown in Table 5.

As shown in Table 5, the experimental results of illumination uniformity of DMD4500 system and DMD3010 system are both greater than 89%, indicating that the illumination uniformity of the prototypes is basically consistent with the simulation results in Table 4. The experimental results show that the normalized illuminance distribution of DMD4500 system and DMD3010 system are both greater than 0.87, which are relatively consistent with the simulation results.

It should be noted that the uniformity of DMD3010 system is slightly low, according to the above analysis results. It mainly caused by the spot deformation on DMD chip, which can be explained by Eq. (15). Obviously, *M _{RTIR}* is proportional to ${\theta _{DMD}}$ and DLP3010 has a larger ${\theta _{DMD}}$. To solve this problem, a non-coaxial system is proposed in the next section.

## 4. Uniformity improvement with a non-coaxial system

As mentioned above, illumination uniformity is a critical issue of the illumination system in projector design. However, in the design process of the illumination system, it was found that the RTIR prism will cause the spot deformation with the larger ${\theta _{in}}$, which has an adverse impact on illumination uniformity. Therefore, we propose a non-coaxial system based on the Scheimpflug principle to solve this problem.

Firstly, an equivalent optical path of relay system and RTIR prism is proposed, which is shown in Fig. 11. The system can be regarded as focusing from infinity. The second surface of the FELA is equivalent to a stop. The size and the exit angle of the FELA is equal to the stop size and field angle of the system. The DMD plane is equal to the image plane. Obviously, the location of the DMD chip is not on the real image plane due to the insertion of the RTIR prism. In other words, the spot deformation caused by RTIR prism can be equivalent to projection from the real image plane to the DMD chip plane.

According to the above analysis, it is necessary to make the DMD chip coincide with the real image plane in order to eliminate the deformation. Hence, a non-coaxial system based on the Scheimpflug principle is proposed. The Scheimpflug principle is the basis for a number of devices and imaging systems, such as high-precision measurement [24], Optical coherence tomography (OCT) [25], and ophthalmology [26] and so on. The Scheimpflug principle refers to a concept in geometric optics whereby the object plane that is not parallel to the image plane can be rendered maximally focused given certain angular relations among the object plane, the lens plane, and the image plane [27].

In the illumination system, the relation of the non-coaxial system and the Scheimpflug principle can be described as shown in Fig. 12. The RTIR prism and the DMD plane form an oblique image plane. Then the angle of relay lens is adjusted reasonably, which can make the DMD plane become real image plane. With this condition, the light output from the relay lens can be maximally in focus. We modified the DMD 3010 illumination system by rotating the relay lens with an angle of 2.4°(Orange dotted line in Fig. 11). The corresponding irradiance distribution is shown in Fig. 13. The *uniformity* values of simulation enhanced from 92%, 93%, 92% to 98%, 97%, 98%, respectively. The *uniformity* values of the experimental measurement result enhance to 97%, 98%, 97%, which was achieved to a satisfactory result.

## 5. Conclusions

In this paper, we propose a design method of the illumination system, which provides a mathematical model and optimization algorithm to achieve high optical efficiency and illumination uniformity in the design of projector. The precise structural and optical parameters of the RTIR prism are obtained by nonlinear constrained optimization, and the parameters of the FELA are determined based on machining and aligning error in the proposed method. This method is verified and analyzed by optical simulation of two illumination systems with different DMD chips. The simulation results of optical efficiency are 55.2% and 67%, respectively, which is in good agreement with the measured optical efficiency. In addition, we also propose a non-coaxial system to solve the deformation problem caused by the large flip angle of the DMD chip, and further improve the illumination uniformity based on the Scheimpflug principle. The illumination uniformity of two systems reaches to 97%. The results indicate that the systems designed by the proposed method can obtain a satisfactory utilization rate of the optical energy and higher uniformity. The proposed method plays an important role in projector design, provide the optimal design scheme, and obtain the optimal design result with high optical efficiency and high illumination uniformity. This design method is applicable for all types of LEDs and DMD chips and has great potential in projector design.

## Funding

National Natural Science Foundation of China (61975012).

## Disclosures

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

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