Abstract

When the high coherence laser beam is homogenized by microlens array (MLA), interference fringes will be generated reducing the uniformity of homogenized spot. A novel close packed random rectangular microlens array (rRMLA) is proposed to solve this problem. By designing the MLA with random apertures and random focal lengths of sub-lenses, the phase regulation can be realized, so as to disturb the coherent superposition fringes for improving the uniformity. To realize the dense arrangement of a MLA with random rectangular aperture, an iterative segmentation method is proposed to design the structure of rRMLA with controllable divergence angle and high filling factor. Theoretical simulations and experimental results both demonstrate the improvement of uniformity of the homogenized spot based on the proposed rRMLA.

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

1. Introduction

With the continuous development of laser technology, laser is widely used in industry, medical treatment, military, scientific research and many other fields. When it is applied to laser projection [1], laser background lighting [2,3] and other fields, the laser beam is always need to be shaped and homogenized. Diffractive optical element (DOE) and microlens array (MLA), as highly integrated beam control elements, have high practical value in laser beam homogenization and have been widely studied. The DOE can accurately control the laser beam with Gaussian distribution and realize the output of homogenized spot with arbitrary shape [4,5]. When the DOEs are used to homogenize the laser beam, the more the number of steps in the structure, the better the homogenization effect will, while the processing difficulty will increase as well. Meanwhile, the energy utilization rate, that is, diffraction efficiency will depend on the number of steps. For example, when the number of steps is two, four, eight, or sixteen, the energy utilization rate will be $40.5\%$, $81\%$, $94.9\%$, and $98.6\%$, respectively [6]. In addition, the DOEs operate in a very narrow wavelength band and are sensitive to the change of the wavelength, which limits the applicability of lasers with different wavelengths. The binary DOEs is always designed for a single wavelength. When the other wavelength is used to irradiate the DOEs, the central zero order strong intensity will be produced, which greatly reduces the uniformity of the homogenized spot. The method of MLA has the advantages of high energy utilization rate, small volume and high integration. In addition, it is not sensitive to the intensity distribution of incident light. The incident laser beam is divided into a series of sub-beams, which are superimposed on each other in the far field to eliminate the inhomogeneity between different sub-beams and form a homogenized spot [7,8].

Because the MLA is a refractive continuous surface structure with less stray light and it is suitable for beam homogenization of different lasers with a high energy utilization rate, researchers have carried out a lot of researches on beam homogenization by using MLA. M. Zimmermann et al. used diffraction theory to explain the diffraction effect of MLA with small aperture and found that periodic MLA is only suitable for laser beam homogenization with poor coherence [9]. For high coherence laser beams, due to the periodic structure of MLA, the divided sub-beams will interfere, resulting in interference fringes and periodic strong dots in the homogenized spot, which greatly reduces the uniformity. In order to disturb the interference fringes of periodic MLA during beam homogenization, researchers propose to change the polarization state or modulate the phase of the laser to disturb the spatial coherence. A. S. Victoria et al. proposed to use the random phase plates (RPP) of an optical birefringent material to smooth the homogenized spot [10]. It is proposed that the polarization direction of the incident beam is modulated by randomly etching structure at a specific depth on the surface, so as to disrupt the coherence condition of the beam. For the beam homogenization method based on MLA, the uniformity is usually improved by phase modulation. F. Wippermann et al. proposed that chirped MLA can eliminate the periodic lattice effect of homogenized spot [11]. However, due to the wedge-shaped distribution of sub-lenses of chirped MLA, it is difficult to process and prepare in practice, which is far from practical application.

In recent years, free-form surfaces MLA have also been widely studied in beam shaping [1214]. In 2017, Y. Jin et al. proposed a free-form surface random MLA for beam homogenization of excimer lasers [12]. Taking advantage of the high degree of freedom in the design of free-form surface microlens, each free-form surface in the array is designed separately. By introducing an appropriate aberration to redistribute the irradiance of the beam, a higher beam homogenization effect can be obtained. This method can effectively reduce the diffraction effect caused by the small aperture of sub-lens. In 2019, Z. Liu et al. proposed random free-form surface MLA (rfMLA) for beam homogenization of digital micromirror device (DMD) lithography system [13]. The rfMLA with different structural parameters are designed to break the periodicity of MLA and achieve the goal of weakening the interference pattern in the homogenized spot to improving the uniformity of the spot projected onto the DMD. In 2020, W. Zhang et al. proposed a randomly distributed free-form surface cylindrical microlens for laser beam shaping and homogenization [14]. A linear uniform spot is generated by controlling the light field through a free-form surface cylindrical microlens. Although the scheme of free-form surface can improve the uniformity of coherent light homogenization, it is worth noting that the design of free-form surface lens depends on the irradiance distribution and beam profile of the incident light beam. Therefore, the designed free-form surface lens belongs to static optical system. If the energy distribution of the input beam changes or the beam shape is distorted, the homogenization effect will be adversely affected. Therefore, for different input laser beams, a new free-form surface lens needs to be redesigned to match it, which limits the application flexibility of this method.

In order to make the random MLA not only homogenize the laser beam with one type of energy distribution, it is considered that the free-form surface is not introduced to regulate the energy distribution of the input light field when designing the random MLA, under the condition of ensuring the homogenization effect. Our research group proposed to design and fabricate random MLA with small apertures to realize laser beam homogenization [15,16]. The designed MLA is fine enough to divide the input beam, so that the uniformity of the spot will be greatly improved. Meanwhile, the randomness of the MLA is used to reduce the interference fringes in the homogenized spot. However, the specific divergence angle needs to be controlled by adjusting the process parameters in the preparation process. The preparation results are easily affected by the process parameters and environmental factors, which reduces the robustness of the preparation results during the overall process, leading to the phenomenon that the intermediate energy is strong and the edge energy decreases gradually.

Based on the previous researches, this paper proposes to design a close packed random rectangular microlens array (rRMLA) for laser beam homogenization. The apertures and sag heights of the sub-lenses are manipulated and designed separately to make it a random distribution. While breaking the periodicity of the MLA, a high uniformity homogenized spot is obtained. Theoretical simulations and experiments are carried out to demonstrate the improvement of uniformity of the homogenized spot based on the proposed rRMLA. The main arrangement of this paper is as follows: Section 2 describes the beam homogenization principle of the proposed rRMLA. Section 3 introduces the design of rRMLA and simulations. Section 4 describes the experimental verification. Section 5 is the discussion, and Section 6 is the summary of the whole paper.

2. Principle

The principle of laser beam homogenization based on rRMLA is shown in Fig. 1. The collimated laser beam is incident on the surface of rRMLA1 and divided into several small sub-beams by the sub-lenses. These modulated sub-beams pass through the objective lens system composed of rRMLA2 and Fourier lens (FL), and finally get a uniform spot on the focal plane (FP). Due to the random aperture, radius of curvature and arrangement mode of each sub-lens in rRMLA, the beam is divided into several independent sub-beams after passing through the rRMLA, disrupting the interference conditions between the sub-beams and improving the uniformity of the output spot.

 figure: Fig. 1.

Fig. 1. Principle of laser beam homogenization.

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Assuming that rRMLA1 and rRMLA2 are composed of $2N + 1$ sub-lenses, the transmittance function of MLA in y direction can be expressed by Eq. (1).

$$t({y_1}) = [rect(\frac{{{y_1}}}{{{p_n}}})\cdot \exp ( - ik\frac{{y_1^2}}{{2{f_n}}})] \otimes \sum\limits_{n ={-} N}^N {\delta ({{y_1} - {p_n}n} )} .$$

where, ${p_n}$ is the aperture size of the $nth$ sub-lens in rRMLA, and the value range of n is $[{ - N,N} ]$. ${f_n}$ is the focal length of the $nth$ sub-lens. The apertures and focal lengths of different sub-lenses are different, which are randomly distributed. k is the wave number and its value is $2\pi /\lambda$. $rect$ is a rectangular function, which controls the aperture of sub-lens to be rectangular $\delta$ is an impulse function, which is used to produce an arrayed structure.

Fourier optics [17] is used to analyze the homogenized light field obtained by the beam passing through the whole homogenized structure. Assuming that the input laser is a collimated parallel light, so that the light field distribution close to the front surface of the rRMLA1 is constant 1, that is $E({y_1},{z_1}^ - ) = 1$, where z is the propagation direction of the light field. After the incident light passes through rRMLA1, the light field distribution function $E({y_1},{z_1}^ + )$ is obtained on the rear surface of rRMLA1, as shown in Eq. (2).

$$E({y_1},{z_1}^ + ) = E({y_1},{z_1}^ - )\cdot t({y_1}).$$

After transmission through distance d, the field distribution $E({y_2},{z_2}^ - )$ on the front surface of rRMLA2 can be calculated by Fresnel diffraction formula, as shown in Eq. (3).

$$\begin{aligned} E({y_2},{z_2}^ - ) &= \frac{1}{{i\lambda d}}\exp (ikd)\exp (ik\frac{{y_2^2}}{{2d}})\cdot \int {\left\{ {E({y_1},{z_1}^ + )\cdot \exp [i\frac{{ky_1^2}}{{2d}}]\exp ( - i2\pi \frac{{{y_2}}}{{\lambda d}}{y_1})} \right\}} d{y_1}\\ &= \frac{1}{{i\lambda d}}\exp (ikd)\exp (ik\frac{{y_2^2}}{{2d}})\cdot { {{{\cal F}}\{{E({y_1},{z_1}^ + )} \}} |_{f^{\prime} = \frac{{{y_2}}}{{\lambda d}}}} \otimes {\left. {{{\cal F}}\left\{ {\exp [i\frac{{ky_1^2}}{{2d}}]} \right\}} \right|_{f^{\prime} = \frac{{{y_2}}}{{\lambda d}}}}, \end{aligned}$$
where, ${{\cal F}}$ is Fourier transform function. The Fourier transform of the light field $E({y_1},{z_1}^ + )$ is shown in Eq. (4).
$$\begin{aligned} {{\cal F}}\{{E({y_1},{z_1}^ + )} \}&= {{\cal F}}\{{t({y_1})} \}= {\left. {{{\cal F}}\left\{ {[rect(\frac{{{y_1}}}{{{p_n}}})\cdot \exp ( - ik\frac{{y_1^2}}{{2{f_n}}})] \otimes \sum\limits_{n ={-} N}^N {\delta ({{y_1} - {p_n}n} )} } \right\}} \right|_{f^{\prime} = \frac{{{y_2}}}{{\lambda d}}}}\\ &= [{p_n}\sin c({p_n}f^{\prime}) \otimes \sqrt { - i\lambda {f_n}} \exp ({i\pi \lambda {f_n} \cdot f{^{\prime}2}} )]\cdot \sum\limits_{n ={-} N}^N {\exp ({ - i2\pi {p_n}nf^{\prime}} )} \\ &= Diff({p_n},{f_n},f^{\prime})\cdot Interf(N,{p_n},f^{\prime}). \end{aligned}$$
where, the function $Diff({p_n},{f_n},f^{\prime}) = [{p_n}\sin c({p_n}f^{\prime}) \otimes \sqrt { - i\lambda {f_n}} \exp ({i\pi \lambda {f_n} \cdot f{^{\prime 2}}} )]$ describes the diffraction effect of the $nth$ sub-lens, and the function $Interf(N,{p_n},f^{\prime}) = \sum\limits_{n ={-} N}^N {\exp ({ - i2\pi {p_n}nf^{\prime}} )}$ describes the light field distribution of multi-slit interference with $2N + 1$ slits. Due to the random value of ${p_n}$ and ${f_n}$ in design, the phase and amplitude of the light field $E({y_2},{z_2}^ - )$ incident on the surface of rRMLA2 change randomly and are no longer periodic distribution. After passing through rRMLA2, the light field distribution is $E({y_2},{z_2}^ + )$, as shown in Eq. (5).
$$\begin{aligned} E({y_2},{z_2}^ + ) &= E({y_2},{z_2}^ - )\cdot t({y_2}),\\ t({y_2}) &= [rect(\frac{{{y_2}}}{{{p_n}}})\cdot \exp ( - ik\frac{{y_2^2}}{{2{f_n}}})] \otimes \sum\limits_{n ={-} N}^N {\delta ({{y_1} - {p_n}n} )} . \end{aligned}$$

After passing through a Fourier lens with focal length ${f_F}$, the light field distribution $E({y_{{f_F}}},{z_{{f_F}}})$ on the FP is Fourier transform of $E({y_2},{z_2}^ + )$, as shown in Eq. (6).

$$\begin{aligned} &E({y_{{f_F},}}{z_{{f_F}}}) = {{\cal F}}\{{E({{y_2},{z_2}^ + } )} \}\\ &= {{{\cal F}}\{{E({{y_2},{z_2}^ - } )} \}} |{\textrm{ }_{{f^{^{\prime\prime}}} = \frac{{{y_f}}}{{\lambda {f_F}}}}} \otimes {{{\cal F}}\{{t({{y_2}} )} \}} |{\textrm{ }_{{f^{^{\prime\prime}}} = \frac{{{y_f}}}{{\lambda {f_F}}}}}. \end{aligned}$$

According to Eqs. (3)–(5), the random ${p_n}$ and ${f_n}$ are introduced into the amplitude distribution in the final light field distribution $E({y_{{f_F}}},{z_{{f_F}}})$, so that the homogenized spot is no longer a periodic lattice distribution.

3. Design and simulations

When designing rRMLA, if the aperture and focal lengths of the sub-lenses change randomly and are not constrained, the divergence angle of the beam passing through rRMLA will not be controllable, so that the dimension and shape of the obtained homogenized spot will not be controllable. This section will introduce the design method of rRMLA and simulate its beam homogenization effect.

3.1 Control of the dimension of homogenized spot

In order to obtain a uniform spot with a target size of S, it is necessary to control the divergence angle (${\theta _n}$) of the beam after passing through rRMLA. Therefore, it is necessary to analyze the structural parameters of the sub-lens of the rRMLA in the optical path.

The overall optical path is shown in Fig. 2(a), and the value of the sub-lens aperture ${p_n}$ in rRMLA is shown in Eq. (7).

$${p_n} = {p_0} \pm {C_{rand}}\cdot \Delta .$$

 figure: Fig. 2.

Fig. 2. (a) Optical path of beam homogenization. (b) Structural parameters of the sub-lens.

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where, ${p_0}$ is the intermediate aperture value of the sub-lens. ${C_{rand}}$ is the random coefficient and varies randomly from $0$ to $1$. $\Delta$ is the maximum aperture variation of the sub-lens. That is, the aperture value of the sub-lens fluctuates up and down around the central value ${p_0}$, and the fluctuation value is ${\pm} {C_{rand}}\cdot \Delta$.

In the paraxial approximation, the divergence angle of the light emitted from the position ${x_i}$ of the sub-lens satisfies the equation $\tan {\theta _i} ={-} \frac{{{x_i}}}{{{f_n}}}$ (Fig. 2(b)). When ${x_i} = \frac{{{p_n}}}{2}$, it is the maximum divergence angle of the sub-lens, and its calculation formula is Eq. (8).

$$\tan {\theta _n} = \frac{{{p_n}}}{{2{f_n}}}.$$

When using rRMLA1 and rRMLA2 for beam homogenization, the sub-lenses correspond one to one, so the size S of the homogenized spot is calculated by Eq. (9), which can be derived from matrix method in optics [18]. It should be noted that the distance z between rRMLA2 and FL (Fig. 2(a)) has no effect on the size S of the final homogenized spot, so it can be adjusted freely according to the needs of practical application.

$$S = {p_n}\frac{{{f_F}({f_n} + {f_n} - d)}}{{{f_n}\cdot {f_n}}},(d \le 2{f_n}).$$

Equation (10) can be obtained by calculating and simplifying in combination with Eqs. (8) and (9).

$$S = 4\tan {\theta _n}{f_F}(1 - \frac{{\tan {\theta _n}d}}{{{p_n}}}).$$

According to Eqs. (7) and (10), in order to obtain a uniform spot of a certain size, the fluctuation range ${\pm} {C_{rand}}\cdot \Delta$ in the aperture need to be controlled as a relatively small value. Besides, $\tan {\theta _n}$ needs to be controlled as a fixed value, that is ${\theta _n}$ is a fixed value such as ${\theta _0}$. Under this constraint, the surface profile of the sub-lens in rRMLA can be further designed.

From the knowledge of geometric optics [19], the focal length of the plano convex lens is shown in Eq. (11).

$${f_n} = \frac{{{R_n}}}{{{n_\lambda } - 1}}.$$
where, ${R_n}$ is the curvature radius of the lens and ${n_\lambda }$ is the refractive index of the optical material at the wavelength of $\lambda$. Combining Eqs. (8) and (11), the curvature radius expression of the sub-lens can be obtained, as shown in Eq. (12).
$${R_n} = \frac{{{p_n}({n_\lambda } - 1)}}{{2\tan {\theta _0}}}.$$

In the practical design of rRMLA, we control the diagonal divergence angle of each sub-lens to a unified fixed value. The curvature radius of the sub-lens can be calculated according to Eq. (12). At this time, the ${p_n}$ value is $\sqrt {p_{n - x}^2 + p_{n - y}^2}$, where ${p_{n - x}}$ is the aperture size in the x direction and ${p_{n - y}}$ is the aperture size in y the direction.

The surface profile function of the sub-lens can be calculated by Eq. (13).

$$sa{g_n}({x,y} )= \frac{{\frac{1}{{{R_n}}}({x^2} + {y^2})}}{{1 + \sqrt {1 - \frac{1}{{{R_n}^2}}({x^2} + {y^2})} }},\textrm{ }\left\{ \begin{array}{l} - \frac{{{p_{n - x}}}}{2} \le x \le \frac{{{p_{n - x}}}}{2},\\ - \frac{{{p_{n - y}}}}{2} \le y \le \frac{{{p_{n - y}}}}{2}. \end{array} \right.$$

When the sub-lens surface function is known, the phase of the sub-lens can be calculated, as shown in Eq. (14).

$${\Phi _n}(x,y) = \frac{{2\pi }}{\lambda }({n_\lambda } - 1)\cdot sa{g_n}(x,y).$$

It can be seen from Eq. (14) that the random change of the surface profile of the sub-lens leads to the random change of phase ${\Phi _n}$, which leads to the random phase difference between the divided sub-beams. It is further proved that the design method disrupts the interference conditions between the sub-beams divided by the sub-lenses, so that the homogenized spot pattern is no longer a periodic lattice.

3.2 Dense arrangement of rRMLA

Since the shape of the homogenization spot is determined by the aperture shape of the sub-lens in rRMLA. In order to obtain a rectangular homogenized spot, all of the apertures of the sub-lenses must be rectangular. Therefore, it is necessary to study how to realize the dense arrangement of rectangular sub-lenses with different aperture sizes to achieve high filling factor. In this paper, the iterative segmentation method is proposed to realize the design of rRMLA.

Assuming that the intermediate value of the aperture of each sub-lens in the direction x and direction y are both ${p_0}$. During iterative segmentation, the rectangular window with area $M{p_0} \times M{p_0}$ ($M$ is a positive integer) is cut into the array with the array number of $M \times M$. The aperture value of the sub-lens fluctuates around the intermediate value ${p_0}$ in the direction x and direction y, and the fluctuation range is ${\pm} {C_{rand}}\cdot \Delta$. The overall segmentation principle is shown in Fig. 3. Since the MLA with the number of $2 \times 2$ can be generated by each segmentation, the total number of iterations satisfy $num = {\log _2}M$. During segmentation, the median line of the rectangular window is taken as the benchmark and the random offset value for cutting meets $Offse{t_i} ={\pm} \frac{{{C_{rand}}\Delta }}{{{2^{num - k + 1}}}}$, where i is the number of offset corresponding to the number of cutting line, k is the iterative order, and the maximum value of k is $num$.

 figure: Fig. 3.

Fig. 3. Iterative segmentation principle: (a) First iteration; (b) Second iteration; (c) Third iteration; (d) $kth$ iteration.

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In the first iteration, the rectangular window with area $M{p_0} \times M{p_0}$ is divided into rRMLA with array number of $2 \times 2$, as shown in Fig. 3(a). In this process, the rectangular window is divided for three times, and the cutting lines are cut1, cut2 and Cut3 respectively to generate sub-lenses with aperture shapes of A1, A2, A3 and A4. The values of the offset between the cutting line and the median lines of Mid1 and Mid1 equal to ${\pm} \frac{{{C_{rand}}\Delta }}{{{2^{num - 1 + 1}}}}$, so the apertures of the sub-lenses after the first iteration satisfy $p{x_1} = \frac{{M{p_0}}}{2} \pm \frac{{{C_{rand}}\Delta }}{{{2^{num - 1 + 1}}}}$.

The second iteration is based on the completion of the first iteration, and the rectangular window with area $M{p_0} \times M{p_0}$ is further divided into rRMLA with array number of $4 \times 4$, as shown in Fig. 3(b). The apertures of the sub-lenses after the second iteration satisfy $p{x_2} = \frac{{{p_1}}}{2} \pm \frac{{{C_{rand}}\Delta }}{{{2^{num - 2 + 1}}}}$.

By analogy, the third iteration is to further divide the rectangular window with area $M{p_0} \times M{p_0}$ into rRMLA with array number of $8 \times 8$ based on the completion of the second iteration, as shown in Fig. 3(c). The apertures of the sub-lenses satisfy $p{x_3} = \frac{{{p_2}}}{2} \pm \frac{{{C_{rand}}\Delta }}{{{2^{num - 3 + 1}}}}$.

The $kth$ iteration is based on the completion of the $(k - 1)th$ iteration, and the rectangular window with area $M{p_0} \times M{p_0}$ is further divided into rRMLA with array number of ${2^k} \times {2^k}$, as shown in Fig. 3(d). The apertures of the sub-lenses satisfy $p{x_k} = \frac{{{p_{k - 1}}}}{2} \pm \frac{{{C_{rand}}\Delta }}{{{2^{num - k + 1}}}}$.

When $k = num$, the segmentation is completed, and the apertures of the sub-lenses in direction x and direction y are a series of random values around ${p_0}$. It can be seen that the iterative segmentation method can complete the dense arrangement between sub-lenses with different apertures on the basis of controlling the random variation range, so that the filling factor of the structure can reach $100\%$.

3.3 Simulations

Based on the above design method, the MLAs with a fixed divergence angle ${\theta _0}$ in the diagonal direction of $4^\circ$ and the number of arrays of $8 \times 8$ were designed. The sub-lens apertures of the periodic MLA are $500\mu m \times 500\mu m$, and the sub-lens apertures of the rRMLA are random values that fluctuate around ${p_0} = 500\mu m$ in the direction x and direction $y$ with fluctuation range of ${\pm} {C_{rand}}\cdot 50\mu m$. The design results are shown in Fig. 4. The surface profile and phase of the periodic MLA are periodically distributed (Fig. 4(a)-(c)), while the sub-lenses in rRMLA are randomly distributed (Fig. 4(d)-(f)). Due to the different apertures of the sub-lenses in rRMLA, the curvature radii of the sub-lenses are different under the condition of unchanged ${\theta _0}$ according to Eq. (12), resulting in different sag heights (Fig. 4(e)) and phase distributions (Fig. 4(f)).

 figure: Fig. 4.

Fig. 4. Design results: (a), (b) and (c) are the top view, three-dimensional view and phase view of the periodic MLA, respectively; (d), (e) and (f) are top view, three-dimensional view and phase view of rRMLA, respectively.

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Furthermore, the numerical analysis software is used to simulate and compare the homogenization effect. During simulation, the focal length of Fourier lens was set as ${f_F} = 15cm$, and the distance between the two rRMLAs was selected as $d = 5.5mm$. After the beam was modulated by the two types of periodic MLA and rRMLA respectively, the homogenized spots on the focal plane of the Fourier lens were obtained with a size of $13.6mm \times 13.6mm$, as shown in Fig. 5. It can be seen that the light field distribution after periodic MLA homogenization is periodic strong dot distribution (Fig. 5(a) and 5(b)), while the light field distribution after rRMLA modulation is random distribution pattern (Fig. 5(c) and 5(d)), indicating that rRMLA can eliminate the phenomenon of periodic strong dot distribution to improve the uniformity of homogenized spot.

 figure: Fig. 5.

Fig. 5. Simulation results: (a) and (b) are the homogenized spot and energy profile generated by the periodic MLA, respectively; (c) and (d) are the homogenized spot and energy profile generated by the designed rRMLA, respectively.

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4. Experiments

In order to verify the simulation results, the preparation of periodic MLA and rRMLA were carried out. The specific preparation principle can refer to the previous researches of our group [20]. The preparation results are shown in Fig. 6(a) and 6(b) respectively, and the beam homogenization experiments have been carried out with the prepared structure. The MLA in Fig. 6(a) is a periodic structure, the apertures and focal lengths of different sub-lenses are the same, as shown in Fig. 6(c). The apertures are the same as $500\mu m$ and the measured sag heights of sub-lenses are the same as $7.52\mu m$. According to Eqs. (11) and (13), the focal length can be calculated as $4.94mm$. The divergence angle in the diagonal direction can be calculated as $4.\textrm{09}^\circ$ according to Eq. (8), which is basically consistent with the design parameters. The apertures and focal lengths of different sub-lenses are the same, and the probability is $1$. The rRMLA in Fig. 6(b) is a random structure, the apertures and focal lengths of different sub-lenses are different. So, the sag heights of different sub-lenses are different as shown in Fig. 6(d). According to Eqs. (8), (11) and (13), the focal length can be calculated as $4.66mm$, $5.00mm$ and $4.64mm$, and the divergence angle in the diagonal direction can be calculated as $4.16^\circ$, $4.15^\circ$ and $4.29^\circ$. The divergence angle is approximate to the design value $4^\circ$ with a slight deviation, which may have a certain impact on the final size of the homogenized spot. The probability distribution of the apertures is determined by the mask. Therefore, when the mask is designed, the probability distribution of the apertures in the prepared rRMLA will be determined. Since the designed mask was converted according to the design parameters, the aperture distribution of the prepared rRMLA must be consistent with the design parameters. The probability distribution of focal lengths can be obtained by indirectly measuring the sag height of sub-lenses. It is calculated that the focal lengths of sub-lenses with different apertures are different. However, according to Eq. (8), it is found that the relationship between aperture and focal length follows the rule that the divergence angle in the diagonal direction is a fixed value close to $4^\circ$. Therefore, it can be considered that the probability distribution of focal length is consistent with that of aperture.

 figure: Fig. 6.

Fig. 6. Preparation results of periodic MLA and rRMLA: (a) and (c) are microscopic enlarged view and cross-section profile of periodic MLA, respectively; (b) and (d) are the microscopic enlarged view and cross-section profile of rRMLA, respectively.

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In the experiment, a laser beam with wavelength of $650nm$ was used as the incident beam. The focal length of the Fourier lens was consistent with the simulation parameters, which was $15cm$. After the beam was phase modulated by the MLA, the homogenized spot was obtained on the focal plane of the Fourier lens and collected by CCD, as shown in Fig. 7(a) and 7(c), respectively. In the experiment, the misalignment was avoided by observing the homogenized spot in the CCD. Further, the collected spots were analyzed by using the numerical analysis software. When the periodic MLA is used for laser beam homogenization, the spot distribution is consistent with the simulation results, which is a typical periodic lattice distribution (Fig. 7(b)), and the spot size is $14.3mm$, which is calculated by multiplying the pixel number occupied by the homogenized spot by the CCD pixel size. The pixel number is $1934$ and the pixel size is $7.4\mu m$. When the designed rRMLA is used for laser beam homogenization, the lattice of homogenized spot is effectively eliminated (Fig. 7(d)), and the spot size is $14.1mm$ calculated by $1907 \times 7.4\mu m$. The results of homogenization experiment are consistent with the simulation results, which proves that the homogenized spot of rRMLA has better uniformity. However, there is a deviation in the size of the homogenized spot, which is mainly due to the fact that the distance between the two MLA cannot be accurately controlled and measured in the optical test experiments. Although there is a certain deviation in the size of the homogenized spot, it does not have a fundamental impact on the verification of the design principle of the whole article.

 figure: Fig. 7.

Fig. 7. Homogenization effect of periodic MLA and rRMLA: (a) and (b) are the homogenization spot and energy profile of periodic MLA, respectively; (c) and (d) are the homogenized spot and energy profile of rRMLA, respectively.

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Next, the homogenized spot formed by this method is evaluated quantitatively. If the uniformity is evaluated by using the root mean square error (RMSE) of the traditional evaluation function, the uniformity of the homogenized spot of the periodic MLA and rRMLA captured in the experiment is $82.9\%$ and $80.6\%$, respectively. The calculation results are not very different, so the homogenized spots with these two different characteristics cannot be well identified and distinguished. Therefore, this paper proposes to use the effective speckle density $\rho$ to evaluate the uniformity of the homogenized spot. That is, the proportion of the area with normalized energy I greater than ${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {{e^2}}}} \right.}\!\lower0.7ex\hbox{${{e^2}}$}}$ in the overall homogenized spot is counted. The evaluation method is shown in Eq. (15). After beam homogenization by periodic MLA and rRMLA, the effective speckle density is $17.1\%$ and $96.9\%$, respectively. It can be seen that when the laser beam is homogenized by rRMLA, the effective speckle density and uniformity in the overall homogenized spot are higher.

$$\rho = \frac{{{S_{(I > {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {{e^2}}}} \right.}\!\lower0.7ex\hbox{${{e^2}}$}})}}}}{S} \times 100\%.$$

5. Discussion

In Section 3.1, it has beam analyzed that, the divergence angle in the diagonal direction of each sub-lens is controlled to be a unified fixed value ${\theta _0}$ in the actual design of rRMLA. The influence of divergence angle constraints on the structural parameters of the overall rRMLA will be discussed below.

If the divergence angle in the diagonal direction of each sub-lens is controlled to be a fixed value, the random probability distribution of the focal length of the sub-lens will be consistent with the aperture in the diagonal direction, as shown in Fig. 8(a) and 8(b). The larger the aperture, the longer the focal length will, while the ratio of aperture to focal length is fixed. If the divergence angle is taken randomly, the focal length and aperture of the sub lens will obey different probability distribution functions, as shown in Fig. 8(c) and 8(d). The homogenized spot will be as shown in Fig. 8(e). In this case, the energy distribution of the homogenized spot is uneven. The middle energy of the spot is strong while the edge energy is weak, and the size of the homogenized spot is not controllable.

 figure: Fig. 8.

Fig. 8. (a) and (b) show the probability distribution of aperture and focal length when the divergence angle of the sub-lens in rRMLA is fixed. (c), (d) and (e) are the probability distribution of aperture and focal length and homogenized spot respectively when the divergence angle is a random value.

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It can be seen from the above analysis that the divergence angle is a very important parameter in the design process of rRMLA. Next, the influence of the change of divergence angle on the homogenization effect will be analyzed. The divergence angle was changed and the homogenization effect was evaluated by using the effective speckle density. The results are shown in Fig. 9. From the overall trend, with the increase of divergence angle, the effective speckle density increases and the uniformity of spot improves. When the divergence angle satisfies $\tan {\theta _0} < 0.05$, the speckle density increases linearly with the divergence angle. When the divergence angle satisfies $\tan {\theta _0} \ge 0.05$, the speckle density tends to be stable. It can be seen that when using the design method in this paper to improve the beam homogenization effect, the divergence angle of rRMLA is designed to be preferably greater than $0.05$.

 figure: Fig. 9.

Fig. 9. Effect of variation of divergence angle on homogenization effect.

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Meanwhile, different array numbers illuminated to yield sufficient smoothing have been discussed, as shown in Fig. 10. When the number of arrays is greater than $7 \times 7$, the periodicity of homogenization spot is completely broken, and a better homogenization effect can be obtained.

 figure: Fig. 10.

Fig. 10. Homogenized spot with different array numbers of rRMA illuminated: (a) $2 \times 2$ illuminated; (b) $3 \times 3$ illuminated; (c) $4 \times 4$ illuminated; (d) $5 \times 5$ illuminated; (e) $6 \times 6$ illuminated; (f) $7 \times 7$ illuminated; (g) $8 \times 8$ illuminated.

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6. Conclusion

In this paper, a novel rRMLA is designed to homogenize the laser beam. By designing the MLA with random apertures and random focal lengths, the phase regulation of the laser beam is realized, so as to disturb the coherent superposition fringes between the sub-beams divided by MLA, improving the uniformity of the homogenized spot. By studying the dense arrangement method of random rectangular aperture MLA, the structure with controllable divergence angle is designed and verified by simulations. Finally, the prepared rRMLA has been used to test the homogenization effect. The effective speckle density of the homogenized spot is increased from $17.1\%$ to $96.9\%$.

Funding

National Natural Science Foundation of China (61905251).

Disclosures

The authors declare no conflicts of interest.

Data availability

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

References

1. A. Bewsher, I. Powell, and W. Boland, “Design of single-element laser-beam shape projectors,” Appl. Opt. 35(10), 1654–1658 (1996). [CrossRef]  

2. B. Kondász, B. Hopp, and T. Smausz, “Homogenization with coherent light illuminated beam shaping diffusers for vision applications: spatial resolution limited by speckle pattern,” J. Eur. Opt. Soc.-Rapid Publ. 14(1), 27 (2018). [CrossRef]  

3. J. S. Liu and M. R. Taghizadeh, “Iterative algorithm for the design of diffractive phase elements for laser beam shaping,” Opt. Lett. 27(16), 1463–1465 (2002). [CrossRef]  

4. S. Tao and W. Yu, “Beam shaping of complex amplitude with separate constraints on the output beam,” Opt. Express 23(2), 1052–1062 (2015). [CrossRef]  

5. Y. Li, C. Qiu, P. Li, T. Xing, W. Lin, and C. Zhou, “Shape the unstable laser beam using diffractive optical element array,” Proc. SPIE 7848, 78481X (2010). [CrossRef]  

6. W. Veldkamp and T McHugh, “Binary optics,” Sci. Am. 266(5), 92–97 (1992). [CrossRef]  

7. R. Voelkel and K. J. Weible, “Laser Beam Homogenizing: Limitations and Constraints,” Proc. SPIE 7102, 71020 (2008). [CrossRef]  

8. F. M. Dickey, Laser Beam Shaping: Theory and Techniques, 2nd ed. (CRC, 2014, Chap. 8).

9. M. Zimmermann, N. R. V. Lindlein, and K. J. Weible, “Microlens laser beam homogenizer - from theory to application,” Proc. SPIE 6663, 666302 (2007). [CrossRef]  

10. A. S. Victoria, K. K. Galina, M. S. Maxim, A. Z Roman, and B. Y Evgeniy, “Speckle-free smoothing of coherence laser beams by a homogenizer on uniaxial high birefringent crystal,” Opt. Mater. Express 9(5), 2392–2399 (2019). [CrossRef]  

11. F. Wippermann, UD. Zeitner, P. Dannberg, A. Bräuer, and S. Sinzinger, “Beam homogenizers based on chirped microlens arrays,” Opt. Express 15(10), 6218–6231 (2007). [CrossRef]  

12. Y. Jin, A. Hassan, and Y. Jiang, “Freeform microlens array homogenizer for excimer laser beam shaping,” Opt. Express 24(22), 24846–24858 (2016). [CrossRef]  

13. Z. Liu, H. Liu, L. Z. Lu, Q. Li, and J. Lu, “A beam homogenizer for digital micromirror device lithography system based on random freeform microlenses,” Optics Communications 443, 211–215 (2019). [CrossRef]  

14. W. Zhang, L. Xia, M. Gao, and C. Du, “Laser beam homogenization with randomly distributed freeform cylindrical microlens,” Opt. Eng. 59(6), 065103 (2020). [CrossRef]  

15. L. Xue, Y. Pang, W. Liu, L. Liu, H. Pang, A. Cao, L. Shi, Y. Fu, and Q. Deng, “Fabrication of Random Microlens Array for Laser Beam Homogenization with High Efficiency,” Micromachines 11(3), 338–342 (2020). [CrossRef]  

16. W. Yuan, C. Xu, L. Xue, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Integrated Double-Sided Random Microlens Array Used for Laser Beam Homogenization,” Micromachines 12(6), 673–678 (2021). [CrossRef]  

17. J. W. Goodman, Introduction To Fourier Optics (Roberts & Company Publishers, 2004, Chap. 3).

18. A. Gerrard and J. M. Burch, “Introduction to Matrix Methods in Optics,” Am. J. Phys. 44(8), 338–341 (1976).

19. K Iizuka, Engineering Optics (Springer, 1985, pp. 105–144).

20. W. Yuan, Y. Cai, C. Xu, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Fabrication of Multifocal Microlens Array by One Step Exposure Process,” Micromachines 12(9), 1097–1102 (2021). [CrossRef]  

References

  • View by:

  1. A. Bewsher, I. Powell, and W. Boland, “Design of single-element laser-beam shape projectors,” Appl. Opt. 35(10), 1654–1658 (1996).
    [Crossref]
  2. B. Kondász, B. Hopp, and T. Smausz, “Homogenization with coherent light illuminated beam shaping diffusers for vision applications: spatial resolution limited by speckle pattern,” J. Eur. Opt. Soc.-Rapid Publ. 14(1), 27 (2018).
    [Crossref]
  3. J. S. Liu and M. R. Taghizadeh, “Iterative algorithm for the design of diffractive phase elements for laser beam shaping,” Opt. Lett. 27(16), 1463–1465 (2002).
    [Crossref]
  4. S. Tao and W. Yu, “Beam shaping of complex amplitude with separate constraints on the output beam,” Opt. Express 23(2), 1052–1062 (2015).
    [Crossref]
  5. Y. Li, C. Qiu, P. Li, T. Xing, W. Lin, and C. Zhou, “Shape the unstable laser beam using diffractive optical element array,” Proc. SPIE 7848, 78481X (2010).
    [Crossref]
  6. W. Veldkamp and T McHugh, “Binary optics,” Sci. Am. 266(5), 92–97 (1992).
    [Crossref]
  7. R. Voelkel and K. J. Weible, “Laser Beam Homogenizing: Limitations and Constraints,” Proc. SPIE 7102, 71020 (2008).
    [Crossref]
  8. F. M. Dickey, Laser Beam Shaping: Theory and Techniques, 2nd ed. (CRC, 2014, Chap. 8).
  9. M. Zimmermann, N. R. V. Lindlein, and K. J. Weible, “Microlens laser beam homogenizer - from theory to application,” Proc. SPIE 6663, 666302 (2007).
    [Crossref]
  10. A. S. Victoria, K. K. Galina, M. S. Maxim, A. Z Roman, and B. Y Evgeniy, “Speckle-free smoothing of coherence laser beams by a homogenizer on uniaxial high birefringent crystal,” Opt. Mater. Express 9(5), 2392–2399 (2019).
    [Crossref]
  11. F. Wippermann, UD. Zeitner, P. Dannberg, A. Bräuer, and S. Sinzinger, “Beam homogenizers based on chirped microlens arrays,” Opt. Express 15(10), 6218–6231 (2007).
    [Crossref]
  12. Y. Jin, A. Hassan, and Y. Jiang, “Freeform microlens array homogenizer for excimer laser beam shaping,” Opt. Express 24(22), 24846–24858 (2016).
    [Crossref]
  13. Z. Liu, H. Liu, L. Z. Lu, Q. Li, and J. Lu, “A beam homogenizer for digital micromirror device lithography system based on random freeform microlenses,” Optics Communications 443, 211–215 (2019).
    [Crossref]
  14. W. Zhang, L. Xia, M. Gao, and C. Du, “Laser beam homogenization with randomly distributed freeform cylindrical microlens,” Opt. Eng. 59(6), 065103 (2020).
    [Crossref]
  15. L. Xue, Y. Pang, W. Liu, L. Liu, H. Pang, A. Cao, L. Shi, Y. Fu, and Q. Deng, “Fabrication of Random Microlens Array for Laser Beam Homogenization with High Efficiency,” Micromachines 11(3), 338–342 (2020).
    [Crossref]
  16. W. Yuan, C. Xu, L. Xue, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Integrated Double-Sided Random Microlens Array Used for Laser Beam Homogenization,” Micromachines 12(6), 673–678 (2021).
    [Crossref]
  17. J. W. Goodman, Introduction To Fourier Optics (Roberts & Company Publishers, 2004, Chap. 3).
  18. A. Gerrard and J. M. Burch, “Introduction to Matrix Methods in Optics,” Am. J. Phys. 44(8), 338–341 (1976).
  19. K Iizuka, Engineering Optics (Springer, 1985, pp. 105–144).
  20. W. Yuan, Y. Cai, C. Xu, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Fabrication of Multifocal Microlens Array by One Step Exposure Process,” Micromachines 12(9), 1097–1102 (2021).
    [Crossref]

2021 (2)

W. Yuan, C. Xu, L. Xue, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Integrated Double-Sided Random Microlens Array Used for Laser Beam Homogenization,” Micromachines 12(6), 673–678 (2021).
[Crossref]

W. Yuan, Y. Cai, C. Xu, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Fabrication of Multifocal Microlens Array by One Step Exposure Process,” Micromachines 12(9), 1097–1102 (2021).
[Crossref]

2020 (2)

W. Zhang, L. Xia, M. Gao, and C. Du, “Laser beam homogenization with randomly distributed freeform cylindrical microlens,” Opt. Eng. 59(6), 065103 (2020).
[Crossref]

L. Xue, Y. Pang, W. Liu, L. Liu, H. Pang, A. Cao, L. Shi, Y. Fu, and Q. Deng, “Fabrication of Random Microlens Array for Laser Beam Homogenization with High Efficiency,” Micromachines 11(3), 338–342 (2020).
[Crossref]

2019 (2)

Z. Liu, H. Liu, L. Z. Lu, Q. Li, and J. Lu, “A beam homogenizer for digital micromirror device lithography system based on random freeform microlenses,” Optics Communications 443, 211–215 (2019).
[Crossref]

A. S. Victoria, K. K. Galina, M. S. Maxim, A. Z Roman, and B. Y Evgeniy, “Speckle-free smoothing of coherence laser beams by a homogenizer on uniaxial high birefringent crystal,” Opt. Mater. Express 9(5), 2392–2399 (2019).
[Crossref]

2018 (1)

B. Kondász, B. Hopp, and T. Smausz, “Homogenization with coherent light illuminated beam shaping diffusers for vision applications: spatial resolution limited by speckle pattern,” J. Eur. Opt. Soc.-Rapid Publ. 14(1), 27 (2018).
[Crossref]

2016 (1)

2015 (1)

2010 (1)

Y. Li, C. Qiu, P. Li, T. Xing, W. Lin, and C. Zhou, “Shape the unstable laser beam using diffractive optical element array,” Proc. SPIE 7848, 78481X (2010).
[Crossref]

2008 (1)

R. Voelkel and K. J. Weible, “Laser Beam Homogenizing: Limitations and Constraints,” Proc. SPIE 7102, 71020 (2008).
[Crossref]

2007 (2)

M. Zimmermann, N. R. V. Lindlein, and K. J. Weible, “Microlens laser beam homogenizer - from theory to application,” Proc. SPIE 6663, 666302 (2007).
[Crossref]

F. Wippermann, UD. Zeitner, P. Dannberg, A. Bräuer, and S. Sinzinger, “Beam homogenizers based on chirped microlens arrays,” Opt. Express 15(10), 6218–6231 (2007).
[Crossref]

2002 (1)

1996 (1)

1992 (1)

W. Veldkamp and T McHugh, “Binary optics,” Sci. Am. 266(5), 92–97 (1992).
[Crossref]

1976 (1)

A. Gerrard and J. M. Burch, “Introduction to Matrix Methods in Optics,” Am. J. Phys. 44(8), 338–341 (1976).

Bewsher, A.

Boland, W.

Bräuer, A.

Burch, J. M.

A. Gerrard and J. M. Burch, “Introduction to Matrix Methods in Optics,” Am. J. Phys. 44(8), 338–341 (1976).

Cai, Y.

W. Yuan, Y. Cai, C. Xu, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Fabrication of Multifocal Microlens Array by One Step Exposure Process,” Micromachines 12(9), 1097–1102 (2021).
[Crossref]

Cao, A.

W. Yuan, C. Xu, L. Xue, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Integrated Double-Sided Random Microlens Array Used for Laser Beam Homogenization,” Micromachines 12(6), 673–678 (2021).
[Crossref]

W. Yuan, Y. Cai, C. Xu, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Fabrication of Multifocal Microlens Array by One Step Exposure Process,” Micromachines 12(9), 1097–1102 (2021).
[Crossref]

L. Xue, Y. Pang, W. Liu, L. Liu, H. Pang, A. Cao, L. Shi, Y. Fu, and Q. Deng, “Fabrication of Random Microlens Array for Laser Beam Homogenization with High Efficiency,” Micromachines 11(3), 338–342 (2020).
[Crossref]

Dannberg, P.

Deng, Q.

W. Yuan, C. Xu, L. Xue, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Integrated Double-Sided Random Microlens Array Used for Laser Beam Homogenization,” Micromachines 12(6), 673–678 (2021).
[Crossref]

W. Yuan, Y. Cai, C. Xu, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Fabrication of Multifocal Microlens Array by One Step Exposure Process,” Micromachines 12(9), 1097–1102 (2021).
[Crossref]

L. Xue, Y. Pang, W. Liu, L. Liu, H. Pang, A. Cao, L. Shi, Y. Fu, and Q. Deng, “Fabrication of Random Microlens Array for Laser Beam Homogenization with High Efficiency,” Micromachines 11(3), 338–342 (2020).
[Crossref]

Dickey, F. M.

F. M. Dickey, Laser Beam Shaping: Theory and Techniques, 2nd ed. (CRC, 2014, Chap. 8).

Du, C.

W. Zhang, L. Xia, M. Gao, and C. Du, “Laser beam homogenization with randomly distributed freeform cylindrical microlens,” Opt. Eng. 59(6), 065103 (2020).
[Crossref]

Evgeniy, B. Y

Fu, Y.

W. Yuan, C. Xu, L. Xue, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Integrated Double-Sided Random Microlens Array Used for Laser Beam Homogenization,” Micromachines 12(6), 673–678 (2021).
[Crossref]

W. Yuan, Y. Cai, C. Xu, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Fabrication of Multifocal Microlens Array by One Step Exposure Process,” Micromachines 12(9), 1097–1102 (2021).
[Crossref]

L. Xue, Y. Pang, W. Liu, L. Liu, H. Pang, A. Cao, L. Shi, Y. Fu, and Q. Deng, “Fabrication of Random Microlens Array for Laser Beam Homogenization with High Efficiency,” Micromachines 11(3), 338–342 (2020).
[Crossref]

Galina, K. K.

Gao, M.

W. Zhang, L. Xia, M. Gao, and C. Du, “Laser beam homogenization with randomly distributed freeform cylindrical microlens,” Opt. Eng. 59(6), 065103 (2020).
[Crossref]

Gerrard, A.

A. Gerrard and J. M. Burch, “Introduction to Matrix Methods in Optics,” Am. J. Phys. 44(8), 338–341 (1976).

Goodman, J. W.

J. W. Goodman, Introduction To Fourier Optics (Roberts & Company Publishers, 2004, Chap. 3).

Hassan, A.

Hopp, B.

B. Kondász, B. Hopp, and T. Smausz, “Homogenization with coherent light illuminated beam shaping diffusers for vision applications: spatial resolution limited by speckle pattern,” J. Eur. Opt. Soc.-Rapid Publ. 14(1), 27 (2018).
[Crossref]

Iizuka, K

K Iizuka, Engineering Optics (Springer, 1985, pp. 105–144).

Jiang, Y.

Jin, Y.

Kondász, B.

B. Kondász, B. Hopp, and T. Smausz, “Homogenization with coherent light illuminated beam shaping diffusers for vision applications: spatial resolution limited by speckle pattern,” J. Eur. Opt. Soc.-Rapid Publ. 14(1), 27 (2018).
[Crossref]

Li, P.

Y. Li, C. Qiu, P. Li, T. Xing, W. Lin, and C. Zhou, “Shape the unstable laser beam using diffractive optical element array,” Proc. SPIE 7848, 78481X (2010).
[Crossref]

Li, Q.

Z. Liu, H. Liu, L. Z. Lu, Q. Li, and J. Lu, “A beam homogenizer for digital micromirror device lithography system based on random freeform microlenses,” Optics Communications 443, 211–215 (2019).
[Crossref]

Li, Y.

Y. Li, C. Qiu, P. Li, T. Xing, W. Lin, and C. Zhou, “Shape the unstable laser beam using diffractive optical element array,” Proc. SPIE 7848, 78481X (2010).
[Crossref]

Lin, W.

Y. Li, C. Qiu, P. Li, T. Xing, W. Lin, and C. Zhou, “Shape the unstable laser beam using diffractive optical element array,” Proc. SPIE 7848, 78481X (2010).
[Crossref]

Lindlein, N. R. V.

M. Zimmermann, N. R. V. Lindlein, and K. J. Weible, “Microlens laser beam homogenizer - from theory to application,” Proc. SPIE 6663, 666302 (2007).
[Crossref]

Liu, H.

Z. Liu, H. Liu, L. Z. Lu, Q. Li, and J. Lu, “A beam homogenizer for digital micromirror device lithography system based on random freeform microlenses,” Optics Communications 443, 211–215 (2019).
[Crossref]

Liu, J. S.

Liu, L.

L. Xue, Y. Pang, W. Liu, L. Liu, H. Pang, A. Cao, L. Shi, Y. Fu, and Q. Deng, “Fabrication of Random Microlens Array for Laser Beam Homogenization with High Efficiency,” Micromachines 11(3), 338–342 (2020).
[Crossref]

Liu, W.

L. Xue, Y. Pang, W. Liu, L. Liu, H. Pang, A. Cao, L. Shi, Y. Fu, and Q. Deng, “Fabrication of Random Microlens Array for Laser Beam Homogenization with High Efficiency,” Micromachines 11(3), 338–342 (2020).
[Crossref]

Liu, Z.

Z. Liu, H. Liu, L. Z. Lu, Q. Li, and J. Lu, “A beam homogenizer for digital micromirror device lithography system based on random freeform microlenses,” Optics Communications 443, 211–215 (2019).
[Crossref]

Lu, J.

Z. Liu, H. Liu, L. Z. Lu, Q. Li, and J. Lu, “A beam homogenizer for digital micromirror device lithography system based on random freeform microlenses,” Optics Communications 443, 211–215 (2019).
[Crossref]

Lu, L. Z.

Z. Liu, H. Liu, L. Z. Lu, Q. Li, and J. Lu, “A beam homogenizer for digital micromirror device lithography system based on random freeform microlenses,” Optics Communications 443, 211–215 (2019).
[Crossref]

Maxim, M. S.

McHugh, T

W. Veldkamp and T McHugh, “Binary optics,” Sci. Am. 266(5), 92–97 (1992).
[Crossref]

Pang, H.

W. Yuan, Y. Cai, C. Xu, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Fabrication of Multifocal Microlens Array by One Step Exposure Process,” Micromachines 12(9), 1097–1102 (2021).
[Crossref]

W. Yuan, C. Xu, L. Xue, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Integrated Double-Sided Random Microlens Array Used for Laser Beam Homogenization,” Micromachines 12(6), 673–678 (2021).
[Crossref]

L. Xue, Y. Pang, W. Liu, L. Liu, H. Pang, A. Cao, L. Shi, Y. Fu, and Q. Deng, “Fabrication of Random Microlens Array for Laser Beam Homogenization with High Efficiency,” Micromachines 11(3), 338–342 (2020).
[Crossref]

Pang, Y.

L. Xue, Y. Pang, W. Liu, L. Liu, H. Pang, A. Cao, L. Shi, Y. Fu, and Q. Deng, “Fabrication of Random Microlens Array for Laser Beam Homogenization with High Efficiency,” Micromachines 11(3), 338–342 (2020).
[Crossref]

Powell, I.

Qiu, C.

Y. Li, C. Qiu, P. Li, T. Xing, W. Lin, and C. Zhou, “Shape the unstable laser beam using diffractive optical element array,” Proc. SPIE 7848, 78481X (2010).
[Crossref]

Roman, A. Z

Shi, L.

L. Xue, Y. Pang, W. Liu, L. Liu, H. Pang, A. Cao, L. Shi, Y. Fu, and Q. Deng, “Fabrication of Random Microlens Array for Laser Beam Homogenization with High Efficiency,” Micromachines 11(3), 338–342 (2020).
[Crossref]

Sinzinger, S.

Smausz, T.

B. Kondász, B. Hopp, and T. Smausz, “Homogenization with coherent light illuminated beam shaping diffusers for vision applications: spatial resolution limited by speckle pattern,” J. Eur. Opt. Soc.-Rapid Publ. 14(1), 27 (2018).
[Crossref]

Taghizadeh, M. R.

Tao, S.

Veldkamp, W.

W. Veldkamp and T McHugh, “Binary optics,” Sci. Am. 266(5), 92–97 (1992).
[Crossref]

Victoria, A. S.

Voelkel, R.

R. Voelkel and K. J. Weible, “Laser Beam Homogenizing: Limitations and Constraints,” Proc. SPIE 7102, 71020 (2008).
[Crossref]

Weible, K. J.

R. Voelkel and K. J. Weible, “Laser Beam Homogenizing: Limitations and Constraints,” Proc. SPIE 7102, 71020 (2008).
[Crossref]

M. Zimmermann, N. R. V. Lindlein, and K. J. Weible, “Microlens laser beam homogenizer - from theory to application,” Proc. SPIE 6663, 666302 (2007).
[Crossref]

Wippermann, F.

Xia, L.

W. Zhang, L. Xia, M. Gao, and C. Du, “Laser beam homogenization with randomly distributed freeform cylindrical microlens,” Opt. Eng. 59(6), 065103 (2020).
[Crossref]

Xing, T.

Y. Li, C. Qiu, P. Li, T. Xing, W. Lin, and C. Zhou, “Shape the unstable laser beam using diffractive optical element array,” Proc. SPIE 7848, 78481X (2010).
[Crossref]

Xu, C.

W. Yuan, C. Xu, L. Xue, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Integrated Double-Sided Random Microlens Array Used for Laser Beam Homogenization,” Micromachines 12(6), 673–678 (2021).
[Crossref]

W. Yuan, Y. Cai, C. Xu, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Fabrication of Multifocal Microlens Array by One Step Exposure Process,” Micromachines 12(9), 1097–1102 (2021).
[Crossref]

Xue, L.

W. Yuan, C. Xu, L. Xue, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Integrated Double-Sided Random Microlens Array Used for Laser Beam Homogenization,” Micromachines 12(6), 673–678 (2021).
[Crossref]

L. Xue, Y. Pang, W. Liu, L. Liu, H. Pang, A. Cao, L. Shi, Y. Fu, and Q. Deng, “Fabrication of Random Microlens Array for Laser Beam Homogenization with High Efficiency,” Micromachines 11(3), 338–342 (2020).
[Crossref]

Yu, W.

Yuan, W.

W. Yuan, C. Xu, L. Xue, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Integrated Double-Sided Random Microlens Array Used for Laser Beam Homogenization,” Micromachines 12(6), 673–678 (2021).
[Crossref]

W. Yuan, Y. Cai, C. Xu, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Fabrication of Multifocal Microlens Array by One Step Exposure Process,” Micromachines 12(9), 1097–1102 (2021).
[Crossref]

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W. Zhang, L. Xia, M. Gao, and C. Du, “Laser beam homogenization with randomly distributed freeform cylindrical microlens,” Opt. Eng. 59(6), 065103 (2020).
[Crossref]

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Y. Li, C. Qiu, P. Li, T. Xing, W. Lin, and C. Zhou, “Shape the unstable laser beam using diffractive optical element array,” Proc. SPIE 7848, 78481X (2010).
[Crossref]

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M. Zimmermann, N. R. V. Lindlein, and K. J. Weible, “Microlens laser beam homogenizer - from theory to application,” Proc. SPIE 6663, 666302 (2007).
[Crossref]

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[Crossref]

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W. Yuan, Y. Cai, C. Xu, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Fabrication of Multifocal Microlens Array by One Step Exposure Process,” Micromachines 12(9), 1097–1102 (2021).
[Crossref]

L. Xue, Y. Pang, W. Liu, L. Liu, H. Pang, A. Cao, L. Shi, Y. Fu, and Q. Deng, “Fabrication of Random Microlens Array for Laser Beam Homogenization with High Efficiency,” Micromachines 11(3), 338–342 (2020).
[Crossref]

W. Yuan, C. Xu, L. Xue, H. Pang, A. Cao, Y. Fu, and Q. Deng, “Integrated Double-Sided Random Microlens Array Used for Laser Beam Homogenization,” Micromachines 12(6), 673–678 (2021).
[Crossref]

Opt. Eng. (1)

W. Zhang, L. Xia, M. Gao, and C. Du, “Laser beam homogenization with randomly distributed freeform cylindrical microlens,” Opt. Eng. 59(6), 065103 (2020).
[Crossref]

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Opt. Mater. Express (1)

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Z. Liu, H. Liu, L. Z. Lu, Q. Li, and J. Lu, “A beam homogenizer for digital micromirror device lithography system based on random freeform microlenses,” Optics Communications 443, 211–215 (2019).
[Crossref]

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M. Zimmermann, N. R. V. Lindlein, and K. J. Weible, “Microlens laser beam homogenizer - from theory to application,” Proc. SPIE 6663, 666302 (2007).
[Crossref]

R. Voelkel and K. J. Weible, “Laser Beam Homogenizing: Limitations and Constraints,” Proc. SPIE 7102, 71020 (2008).
[Crossref]

Y. Li, C. Qiu, P. Li, T. Xing, W. Lin, and C. Zhou, “Shape the unstable laser beam using diffractive optical element array,” Proc. SPIE 7848, 78481X (2010).
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Data availability

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

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Figures (10)

Fig. 1.
Fig. 1. Principle of laser beam homogenization.
Fig. 2.
Fig. 2. (a) Optical path of beam homogenization. (b) Structural parameters of the sub-lens.
Fig. 3.
Fig. 3. Iterative segmentation principle: (a) First iteration; (b) Second iteration; (c) Third iteration; (d) $kth$ iteration.
Fig. 4.
Fig. 4. Design results: (a), (b) and (c) are the top view, three-dimensional view and phase view of the periodic MLA, respectively; (d), (e) and (f) are top view, three-dimensional view and phase view of rRMLA, respectively.
Fig. 5.
Fig. 5. Simulation results: (a) and (b) are the homogenized spot and energy profile generated by the periodic MLA, respectively; (c) and (d) are the homogenized spot and energy profile generated by the designed rRMLA, respectively.
Fig. 6.
Fig. 6. Preparation results of periodic MLA and rRMLA: (a) and (c) are microscopic enlarged view and cross-section profile of periodic MLA, respectively; (b) and (d) are the microscopic enlarged view and cross-section profile of rRMLA, respectively.
Fig. 7.
Fig. 7. Homogenization effect of periodic MLA and rRMLA: (a) and (b) are the homogenization spot and energy profile of periodic MLA, respectively; (c) and (d) are the homogenized spot and energy profile of rRMLA, respectively.
Fig. 8.
Fig. 8. (a) and (b) show the probability distribution of aperture and focal length when the divergence angle of the sub-lens in rRMLA is fixed. (c), (d) and (e) are the probability distribution of aperture and focal length and homogenized spot respectively when the divergence angle is a random value.
Fig. 9.
Fig. 9. Effect of variation of divergence angle on homogenization effect.
Fig. 10.
Fig. 10. Homogenized spot with different array numbers of rRMA illuminated: (a) $2 \times 2$ illuminated; (b) $3 \times 3$ illuminated; (c) $4 \times 4$ illuminated; (d) $5 \times 5$ illuminated; (e) $6 \times 6$ illuminated; (f) $7 \times 7$ illuminated; (g) $8 \times 8$ illuminated.

Equations (15)

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t ( y 1 ) = [ r e c t ( y 1 p n ) exp ( i k y 1 2 2 f n ) ] n = N N δ ( y 1 p n n ) .
E ( y 1 , z 1 + ) = E ( y 1 , z 1 ) t ( y 1 ) .
E ( y 2 , z 2 ) = 1 i λ d exp ( i k d ) exp ( i k y 2 2 2 d ) { E ( y 1 , z 1 + ) exp [ i k y 1 2 2 d ] exp ( i 2 π y 2 λ d y 1 ) } d y 1 = 1 i λ d exp ( i k d ) exp ( i k y 2 2 2 d ) F { E ( y 1 , z 1 + ) } | f = y 2 λ d F { exp [ i k y 1 2 2 d ] } | f = y 2 λ d ,
F { E ( y 1 , z 1 + ) } = F { t ( y 1 ) } = F { [ r e c t ( y 1 p n ) exp ( i k y 1 2 2 f n ) ] n = N N δ ( y 1 p n n ) } | f = y 2 λ d = [ p n sin c ( p n f ) i λ f n exp ( i π λ f n f 2 ) ] n = N N exp ( i 2 π p n n f ) = D i f f ( p n , f n , f ) I n t e r f ( N , p n , f ) .
E ( y 2 , z 2 + ) = E ( y 2 , z 2 ) t ( y 2 ) , t ( y 2 ) = [ r e c t ( y 2 p n ) exp ( i k y 2 2 2 f n ) ] n = N N δ ( y 1 p n n ) .
E ( y f F , z f F ) = F { E ( y 2 , z 2 + ) } = F { E ( y 2 , z 2 ) } |   f = y f λ f F F { t ( y 2 ) } |   f = y f λ f F .
p n = p 0 ± C r a n d Δ .
tan θ n = p n 2 f n .
S = p n f F ( f n + f n d ) f n f n , ( d 2 f n ) .
S = 4 tan θ n f F ( 1 tan θ n d p n ) .
f n = R n n λ 1 .
R n = p n ( n λ 1 ) 2 tan θ 0 .
s a g n ( x , y ) = 1 R n ( x 2 + y 2 ) 1 + 1 1 R n 2 ( x 2 + y 2 ) ,   { p n x 2 x p n x 2 , p n y 2 y p n y 2 .
Φ n ( x , y ) = 2 π λ ( n λ 1 ) s a g n ( x , y ) .
ρ = S ( I > 1 / 1 e 2 e 2 ) S × 100 % .

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