## Abstract

The luminance uniformity of the backlight module (BLM) importantly depends on the microstructure distribution on the bottom surface of the light guide plate (LGP). Based on the small-size integrated LGP (ILGP) proposed, we put forward a distribution expression of micro-prisms on the bottom surface of the ILGP, and present the relational expressions between the coefficients of the analytical expression and the structural parameters of the ILGP, such as the light guide length *L*, width of the ILGP *W*, thickness of the ILGP *H*, and space between light emitting diodes (LEDs) *d*. Then, the research results above are applied to the design of the small-size ILGPs. Not only can the micro-structure distributions on the bottom surface of the ILGPs be directly given, but also the simulation results show that the luminance uniformities of the integrated BLMs are higher than 85%. The research indicates that the expressions proposed in this paper are correct and effective, and have important guiding significances and referential value.

© 2015 Optical Society of America

## 1. Introduction

Liquid crystal displays (LCDs) are non-active light-emitting devices, and they are provided with uniform surface light sources by backlight modules (BLMs) [1]. Luminance uniformity is one of the most important performance parameters in the BLM, and the high luminance uniformity can be achieved by adjusting the distribution of dots or micro-structures fused on the bottom surface of the light guide plate (LGP) [2]. Since the structure of the typical BLM is too complex and the lights in the LGP are non-sequentially propagating, it will waste a lot of time to design the distribution of the dots or micro-structures on the bottom surface of LGPs by experience. In order to save the design time and improve the luminance uniformity of the BLM simultaneously, some research institutes introduced theoretical models of LGPs and optimization algorithms to design and optimize the distribution of dots or micro-structures on the bottom surface of the LGP, but the bottom surface of the LGP always needed to be partitioned beforehand and mass data needed to be simulated to establish the theoretical model [3–6]. Some researchers also explored the distribution expressions of micro-structures on the bottom surface of the LGP to make high luminance uniformity of the BLMs. In 2006, Luo *et al*. proposed a dot distribution formula for the typical BLM [7], but it was only suitable for the BLM, of which the light source is a high luminous flux line source cold cathode fluorescent lamp (CCFL). While at present, point light sources, light emitting diodes (LEDs), are commonly used as the light sources of the BLM. In 2008, based on the least squares method, Li *et al*. proposed an expression to optimize the radius of hemispheric micro-structure on the bottom surface of the LGP [2], but the study above was based on the wedge-shape LGP with CCFL as the light source, and the bottom surface of the LGP needed to be partitioned in advance. In 2011, according to the law of light propagation and the distribution of light energy inside the LGP, Kim *et al*. proposed a exponential function of the dot density distribution [8]. In 2012, Zhi *et al*. proposed a pattern distribution formula in accordance with the emission characteristics of LED [9]. Among the researches above [2,7–9], the expression coefficients were determined by simulation, and the relationship between the coefficients and the structural parameters of the BLM has not yet been built. In 2013, Kim *et al*. investigated the pattern density function coefficients for various LGP thickness further based on the study in 2011 [10], but the relational expression between the coefficients and the thickness of the LGP wasn’t yet given.

In this paper, based on the small-size integrated LGP (ILGP) presented in our lab [11,12], we propose an expression of the micro-prism distribution on the bottom surface of the ILGP and present the relational expressions between coefficients of the expression above and structural parameters of the ILGP, such as the light guide length *L*, width of the ILGP *W*, thickness of the ILGP *H*, and space between light emitting diodes (LEDs) *d*, by using the simulation of optical software Lighttools and the analysis of software OriginLab. They are very important in the design of the ILGP because when we know the structural parameters of the ILGP, the micro-prism distributions on the bottom surface of the regular small-size ILGPs can be directly given by use of the expressions above. Neither the micro-structure distributions on the bottom surface will need to be partitioned beforehand, nor the coefficients of distribution expression will need to be determined by simulation of optical software. And the luminance uniformities of the integrated BLMs are greater than 85%. The study will help save a lot of design time and has important guiding significances and referential value.

## 2. Micro-prism distribution expression on the bottom surface of the ILGP

Based on the small-size ILGP we proposed [11], we first explore a universal expression of the micro-prism distribution to make high luminance uniformity of the BLM.

Figure 1 shows the diagram of the integrated BLM composed of an ILGP and the light sources LEDs. The ILGP is fused with the designed aspheric semi-cylindrical micro-concentrator structure (ASCMCS) arrays on the top surface and the designed convex micro-prism arrays on the bottom surface coated with a high reflective film (RF) [11], which can achieve the functions of five complex-structure films in the typical BLM, including double crossed bright enhancement films, a diffusion film, a LGP and a flat RF. In Fig. 1, *L* represents the length of the ILGP in a direction perpendicular to LEDs arranged, called as the light guide length. *W*, *H* and *d* represent the width, thickness of the ILGP and space between two adjacent LEDs, respectively.

The structural parameters of the integrated BLM are listed in Table 1.

For exploring the law of the micro-prism distribution, the integrated BLM in Fig. 1 is set up and simulated in the optical software Lighttools. After fitting the simulation results by OriginLab, it is found that the micro-prism spacing decreases in a quadratic polynomial way with the increase of the distance far from light source. So, according to the fitting of mass design data, the micro-prism distribution expression for the small-size ILGP is proposed as following.

Where,*x*represents the distance between the current micro-prism and the light sources LEDs. Δ

*x*represents the space between the current micro-prism and the next one, as shown in Fig. 2.

*A*,

*B*and

*C*are the coefficients of the quadratic polynomial.

*A*mainly controls the maximum space of the adjacent micro-prisms,

*B*shows the changing slope of the micro-prism space with the distance from the light sources, and

*C*appropriately controls the bending degree of the quadratic polynomial curve so that the luminance uniformity can be improved.

Only if the relationship between the coefficients of the quadratic polynomial and the structural parameters of the ILGP is built, could the distribution expression in Eq. (1) have a universal significance.

## 3. Establishment of the relationship between the coefficients of the micro-prism distribution expression and the structural parameters of the ILGP

The structural parameters of the ILGP, such as the light guide length *L*, width of the ILGP *W*, thickness of the ILGP *H*, space between LEDs *d*, are shown in Fig. 1. In this section, we will study the relationship between the coefficients of the distribution expression Eq. (1) and the structural parameters of the ILGP.

#### 3.1 Relationship between the coefficients and the light guide length L

For exploring the relationship between the coefficients in Eq. (1) and *L*, some integrated BLMs in Fig. 1 are set up and simulated in Lighttools, where *W* = 68.7mm, *H* = 0.5mm and *d* = 1.73*d*_{0} (*d*_{0} represents the length of LED, and *d*_{0} = 3.8mm). *L* is 61.9, 93.6, 126.8 and 155.8mm, respectively, which equals to the length of the LGP in 2.8, 4.0, 5.0 and 7.0inch BLM, respectively. After adjusting the micro-prism distribution, the coefficients of the distribution expression Eq. (1) and the corresponding high luminance uniformities of BLMs with different *L* are shown in Table 2, where the luminance uniformity *U* equals to the ratio of the minimum and maximum of luminance measured by 9-points measurement method.

After fitting the data above by OriginLab, it’s found that the coefficients *A*, *B*, *C* in Table 2 and the light guide length *L* satisfy the following relationship.

According to Eqs. (2a)-(2c), the coefficient *A* is linearly increasing with the increase of light guide length *L*, *B* is exponentially decreasing, and *C* is a relatively fixed constant.

#### 3.2 Relationship between the coefficients and the width of the ILGP W

Then, the relationship between the coefficients in Eq. (1) and *W* is studied. Several integrated BLM models are set up in Lighttools, where *L* = 61.9mm, *H* = 0.5mm, *d* = 1.73*d*_{0} and the values of *W* are different as shown in Table 3. By use of Eqs. (1) and (2a)-(2c) in Section 3.1, the micro-prism distributions on the bottom surface of ILGPs in these BLMs are directly gotten, and the coefficients in Eq. (1) are listed in Table 3, which are calculated by using Eqs. (2a)-(2c) directly. The luminance uniformities of BLMs with different *W* are shown in Table 3.

Table 3 shows that when *L*, *H* and *d* are constants, the luminance uniformities of the integrated BLMs change within the range of ± 3% for various *W*. This indicates that the effect of *W* on the luminance uniformity of the BLM is small. So, the width of the ILGP *W* has little effect on the coefficients in Eq. (1).

#### 3.3 Relationship between the coefficients and the thickness of the ILGP H

During the study above, the thickness of the ILGP *H* is fixed at 0.5mm. While, in practice, the thickness of the LGP *H* will change with various sizes of BLMs. Especially when the ILGP prototype is made to validate the effectiveness of the ILGP, the thickness of the ILGP becomes more important. Because of the limitation on current manufacture technique, the thickness of the fabricated ILGP prototype is much thicker than that in the theoretical design [11].

The relationship between the coefficients in Eq. (1) and *H* is studied in this sub-section. Using the similar analysis method in Sub-section 3.1, some integrated BLMs are set up and simulated in Lighttools, where *W* = 68.7mm and *d* = 1.73*d*_{0}. *L* is 61.9, 93.6, 126.8 and 155.8mm, respectively. *H* is 0.5, 1.5 and 2.5mm, respectively. Adjusting the coefficients in Eq. (1) to make high luminance uniformities of the BLMs with different *L* and *H*, the relationship between the coefficients in Eq. (1) and *H* are shown in Fig. 3.

From Figs. 3, they are shown that the thickness of the ILGP *H* has a great impact on the coefficients *A*, *B* and *C* in Eq. (1). Therefore, the coefficients in Eq. (1) are not only related to *L*, but also associated with *H*, which can be expressed as *A*(*L*,*H*), *B*(*L*,*H*) and *C*(*L*,*H*).

Firstly, the relationship between *A* and *H* is analyzed. The correspondence relationship between the *A* variation Δ*A* and the *H* variation Δ*H* is gotten from the data in Fig. 3(a), as shown in Table 4.

Here,

*A*(

*L*,

*H*) presents the coefficient

*A*in Eq. (1),

*L*

_{0}= 61.9mm and

*H*

_{0}= 0.5mm. Since Eq. (2a) is proposed in that case that

*H*= 0.5mm,

*A*(

*L*,

*H*

_{0}) is the expression of

*A*in Eq. (2a) which is that

*A*(

*L*,

*H*

_{0}) = −0.04875 + 0.01514 ×

*L*. Obviously,

*A*(

*L*

_{0},

*H*

_{0}) is also directly calculated by using Eq. (2a), that is equal to 0.888416.

By fitting the data in Table 4, *α*(*L*) can be expressed as

Therefore, if the expression of *A*(*L*_{0},*H*) can be solved, *A*(*L*,*H*) could be obtained by Eq. (3). In Fig. 3(a), when *L* = 61.9mm the curve between *A* and *H* is expressed as *A*(*L*_{0},*H*). For fitting more accurately, more coefficients *A* are obtained by adjusting and simulating, which are corresponding to high-luminance-uniformity integrated BLMs with *L* = *L*_{0}, *H* is 1.0 and 2.0mm respectively. Then the simulation results above and the data in Fig. 3(a) are drawn to be *A*(*L*_{0},*H*), that are the triangles as shown in Fig. 4.

The solid-line curve in Fig. 4 is gotten by fitting the simulation results in Fig. 4. *A*(*L*_{0},*H*) is expressed as

The expression of *A*(*L*,*H*) is obtained with substituting Eq. (5) into Eq. (3).

Where, the constant coefficients *a*_{1}~*a*_{8} in Eq. (5) are shown in Table 5.

Next, the data in Fig. 3(b) are analyzed and it is found that *B* and *A* have the similar rule. When the light guide length *L* is a constant, the ratio of {*B*(*L*,*H*)-*B*(*L*,*H*_{0})}/{*B*(*L*_{0},*H*)-*B*(*L*_{0},*H*_{0})} is approximately associated with *L*. Using the method which is similar to the one to analysis the coefficient *A*, *B*(*L*,*H*) finally satisfies the following relationship.

*b*

_{1}~

*b*

_{9}in Eq. (7) are shown in Table 6.

Based on the fitting analysis of the data in Fig. 3(c), the relationship between *C* and the structural parameters *L* and *H* are satisfied as following.

*c*

_{1}~

*c*

_{3}and

*c*′

_{1}~

*c*′

_{3}in Eq. (8) are shown in Table 7.

The analysis above shows that the coefficients in Eq. (1) and the structural parameters of the ILGP, such as the light guide length *L* and the thickness of the ILGP *H*, have close relationship, and they satisfy Eqs. (6)-(8). The constant coefficients in Eqs. (6)-(8) are obtained by fitting, as listed in Table 5-7. It indicates that the distribution of the micro-prisms depends on the structural parameters of the ILGP, such as the light guide length *L* and the thickness of the ILGP *H*. So, the micro-prism distribution of the ILGP with high luminance uniformity can be obtained by solving Eqs. (6)-(8) and Eq. (1), when the light guide length *L* and the thickness of the ILGP *H* are known.

#### 3.4 Relationship between the coefficients and the space between two adjacent LEDs d

Finally, we explore the relationship between the coefficients in Eq. (1) and *d*.

Similarly, two integrated BLMs are set up and simulated, where *W* = 68.7mm, *H* = 0.5mm, *L* is 79.0 and 93.6mm respectively. Those micro-prism distributions on the bottom surface of the ILGPs are directly obtained from Eqs. (6)-(8) and Eq. (1). The curve of luminance uniformities with different *d*/*d*_{0} is shown in Fig. 5.

From Fig. 5, it’s found that when *d/d*_{0}≤2.5, the luminance uniformity of the integrated BLM is changed within ± 3% and the values of luminance uniformity are all higher than 85%, which indicates that the effect of *d* on the coefficients in Eq. (1) is small. When *d/d*_{0}>2.5, the luminance uniformity of the integrated BLM sharply declines with the increase of *d/d*_{0}. In this case, the space between LEDs *d* has a great effect on the coefficients in Eq. (1). The main reason is that with the increase of *d*, the output surface of the ILGP near the LEDs is serious dark, which significantly affects the luminance uniformity and further affects the coefficients in Eq. (1).

Above all, when *d/d*_{0}≤2.5, the luminance uniformity of the integrated BLM is less affected by *d*. That means Eqs. (6)-(8) are independent of *d*. While when *d/d*_{0}>2.5, Eqs. (6)-(8) will be no longer effective. According to the market research, the space between LEDs *d* in the typical small-size BLMs is not greater than 2.5*d*_{0}. Therefore, this paper will not further explore the relationship between the coefficients in Eq. (1) and *d* in the case of *d/d*_{0}>2.5.

## 4. Design and verification

Based on the analysis in Section 3, it’s known that when *d/d*_{0}≤2.5, the coefficients in Eq. (1) are only related to the light guide length *L* and the thickness of the ILGP *H*, while have nothing to do with the width of the ILGP *W* and the space between LEDs *d*. The micro-prism distribution of the ILGP with high luminance uniformity can be obtained directly by solving Eqs. (6)-(8) and Eq. (1).

For verifying the scope of the conclusions above, the micro-prism distributions of the integrated BLMs with different structural parameters are designed by the quadratic polynomial Eq. (1) and the coefficients expressions Eqs. (6)-(8), then the luminance uniformities of the integrated BLMs are simulated, where the structural parameters of the integrated BLMs and their luminance uniformities are shown in Fig. 6. In Fig. 6, the width of the ILGP *W* = 68.7mm, the space between LEDs *d* = 1.73*d*_{0}, the light guide length *L* is 61.9, 79.0, 93.6, 100.2, 116.3, 126.8, 137.8, 155.8mm respectively, and the thickness of the ILGP *H* is from 0.5mm to 2.5mm. Because the thickness of the LGP in small-size BLMs is generally from 0.5mm to 0.7mm, the step is 0.1mm when *H* is within the range of 0.5~0.7mm, while the step is 0.5mm when *H* is within the range of 1.0~2.5mm. The efficiencies of light energy utilization of the integrated BLMs corresponding to Fig. 6 are also calculated, as shown in Fig. 7. It can be seen from Fig. 7 that all values of light efficiencies are greater than 84%, which is much bigger than the one (about 50%) in the typical BLM [11].

Figure 6 shows that when the micro-prism distributions on the bottom surface of the ILGPs are directly obtained based on Eq. (1) and Eqs. (6)-(8), most values of the luminance uniformities of the integrated BLMs are higher than 85%, especially for those of which the ILGP thickness is in the range of 0.5~0.7mm, the general range in the small-size BLMs. For example, in the ILGP with *L* = 93.6mm and *H* = 0.5mm corresponding to the size of 4.0inch BLM, the coefficients of *A*, *B* and *C* are 1.36600mm, −0.01852 and 5.5 × 10^{−5}mm^{−1} respectively, which are calculated directly by using Eqs. (6)-(8). Then the micro-prism distribution is gotten by substituting the obtained *A*, *B* and *C* into Eq. (1), and the simulation result shows that the luminance uniformity of the corresponding integrated BLM is 91.21%, of which the luminance and angular luminance of output light is shown in Fig. 8. Of course, there are still several integrated BLMs of which luminance uniformities are less than 85%. While, after slightly adjusting the coefficients of *A*, *B* and *C* calculated by using Eqs. (6)-(8), the micro-prism distributions on the bottom surface of the ILGPs obtained by Eq. (1) could also make that luminance uniformities of the corresponding integrated BLMs are greater than 85%.

Therefore, for the small-size ILGP of which the size is 7.0 inch or less (*L* from 61.9~155.8mm, *H* from 0.5~2.5mm), the coefficients *A*, *B* and *C* in Eq. (1) can be directly solved by using Eqs. (6)-(8), while not need to be determined by simulation of optical software. By using Eqs. (6)-(8) and Eq. (1), the micro-prism distributions can be given directly for any small-size ILGP of which the size is 7.0 inch or less to make high luminance uniformity, which saves a lot of design time.

## 5. Conclusion

Based on the small-size ILGP presented, by using mass simulations of optical software Lighttools and the fitting of OriginLab, we propose a expression of the micro-prism distribution in regular ILGP for 7.0 inch or less in size, as shown in Eq. (1), and present the relational expressions between the coefficients in the distribution expression of Eq. (1) and the structural parameters of the ILGP, such as the light guide length *L*, the width of the ILGP *W*, the thickness of the ILGP *H*, and the space between LEDs *d*, as shown in Eqs. (6)-(8). Equations (6)-(8) indicate that the coefficients in Eq. (1) are only related to the light guide length *L* and the thickness of the ILGP *H*, while have no relationship with the width of the ILGP *W* and the space between LEDs *d* in the case of *d*≤2.5*d*_{0}. [Equations below in this order: Eq. (1), Eq. (6), Eq. (7), Eq. (8).]

*x*represents the distance between the current micro-prism and the light sources LEDs. Δ

*x*represents the space between the current micro-prism and the next one.

*A*,

*B*and

*C*are the coefficients of the quadratic polynomial. The constant coefficient values

*a*

_{1}~

*a*

_{8},

*b*

_{1}~

*b*

_{9},

*c*

_{1}~

*c*

_{3}and

*c*′

_{1}~

*c*′

_{3}are shown in Table 5-7.

The research results above are applied to design small-size integrated BLMs with different structural parameters, and the simulation results show that the expressions in this paper are correct and effective. Once the structural parameters of the ILGP are known, the micro-prism distribution of the ILGP to make high luminance uniformity can be directly given by applying the research results in this paper, which will save a lot of design time. The study verifies the expressions’ universality in the design of the small-size ILGP, of which size is 7.0 inch or less.

## Acknowledgments

This work was supported by National Natural Science Foundation of China (No. 61108053, and No. 61275167), by Science and Technology Development Funds of Shenzhen, China (No. JCYJ20130329103020637, No. JCYJ20140418095735591, No. JC201005280533A, No. JCYJ20120613174700014 and No. GJHS20120621160334587).

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