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Abstract
The main issue in applying a microlens array as a beam homogenizer is the high intensity contrast resulting from the periodic structure. We present a practical solution to eliminate speckle patterns by using a diffuser that can convert a coherent source into a Gaussian Schell source with an adjustable coherence length. Based on numerical simulation and experimental results, the microlens should have the Fresnel number above 15, a large pitch when the target plane is fixed, and an appropriate D/σf ratio to obtain a spot with high uniformity. Moreover, a slight defocus of the observation surface within the range of ±1mm has a negligible effect on the uniformity of shaped spots the homogenized spot. Finally, the applications of the findings are presented.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In various applications, such as optical projection, imaging, and lithography [1–5], the conversion of an output beam into a uniform distribution is commonly required. A few studies [6–8] have demonstrated that the beam integration method is reliable for the shaping of arbitrary spectral and radiation sources. Over the past decades, several optical components have been utilized in beam integration systems, such as prisms [9], digital micro-mirror devices (DMDs) [10], spatial light modulators (SLMs) [11,12], and microlens arrays (MLAs) [13,14]. Both array components have the same mechanism—they have multiple elements that split the incident beam into modulated beamlets and the beamlets superimpose on the target surface after an integral component such as lenses. DMD and SLM, in particular, can regulate their initial sources in real-time by controlling their single lens or pixel units; these methods have high response speeds and reliability. However, they have strict requirements for the field distribution of the incident source, and their damage-tolerance threshold is low. The compatibility of MLA with various sources makes it an effective and widely used element in integration methods. However, while suppressing speckle noise, energy loss also occurs. In recent years, various techniques, such as gray-tone, e-beam lithography, and proton beam writing [15–17] have been introduced to the MLA fabrication procedures, and high accuracy and low surface roughness have been realized. Studies on beam intensity distribution, including geometrical optics, ABCD-matrix, finite time difference domain methods, and physical optics, have attracted extensive attention [18–22]. Thus, the diffraction and interference of the wavefront can be included by considering the changes in both the amplitude and phase.
However, speckle patterns are generated because of the roughness of the irradiated surface, particularly when highly coherent sources are used. Existing works on speckle suppression, can be divided into two categories:
- (1) Multiple speckle patterns are generated over a certain period of time with the use of dynamic diffractive optical elements [23–25], rotating diffusers [4], and ultrasonic waves [26]. However, the conclusions obtained in these literatures — such as those relating to oscillator frequency, amplitude, and vibration mode — are applicable only to the specific experimental setup in the papers and cannot be generalized.
- (2) Multiple speckle patterns are instantly generated. The focus here is on directly changing the incident light source by using options such as broadband light sources [27], laser arrays [28], or random lasers [29]. This presents a major limitation for the homogenization of various light sources. For field homogenization systems using MLAs, the speckle patterns are interference fringes arranged periodically due to the periodicity of the array elements. Previous studies have proposed several schemes such as double-sided MLA and irregular MLA [4,13, 30–32] to improve the uniformity of the target plane. Irregular MLAs are the most effective; however, such devices that have very specific individual sources of light are more complicated to design and manufacture. Therefore, researchers are still interested in developing simple and economical methods for reducing speckle patterns. A few studies have focused on temporal smoothing techniques [33], but they are more applicable to the smoothing of wide-bandwidth sources. In this study, we use a combination of MLA and a spatially partially coherent source with a narrow spectral width to reduce speckles. Although a few effects of source coherence using MLA for smoothing have been verified through experimentation [31,34], the relationship among the spatial coherence, parameters of MLA, position of lens, and uniformity of intensity distribution in the target plane are not investigated.
Partially coherent beams (PCBs) have received considerable attention in recent years [35] owing to their controllable spatial coherence. The cross-spectral density (CSD) function can characterize PCBs in the space-frequency domain; thus, we intend to obtain a transverse distribution of PCBs passing through free space or a linear optical system. The coherence characteristics of PCBs are intuitively described by the degree of coherence (DOC) in the CSD function. The Gaussian-Schell (GS) beam is a particular class of PCBs since its CSD function only depends on the difference of two arbitrary points and characterized by the property that the intensity and DOC distribution are both Gaussian. The researchers have applied them widely in the field of free-space optical communications [36], optical imaging [37], and generating other types of PCBs [38] etc. In the current research, we use a GS beam that is convenient to generate from experiment [39] with different DOC as the initial source.
The aim of our study is to investigate the feasibility of the reduction of speckle and distribution nonuniformity of focal spot on the DMD in the projection system. In the present research, we first derive the formulas for the cross-spectral density function of the GS beam in the transverse plane after allowing it to pass through a microlens array and Fourier lens, based on the expression in the source plane. Based on the formals, we can correctly capture the relevant spot homogenization parameters, particularly to quantify the value of spatial coherence length. Thereafter, we conclude the relationship between the uniformity of the target spot and the parameters via numerical simulation. In Section 4, we discuss the approach to controlling and determining the DOC of the source and present an experiment to record the intensity distribution. The characteristics of parameter cv describing the distribution uniformity of the focal spot are analyzed. The results show the viability of improving the intensity distribution uniformity using a partially coherent beam smoothing system in a single/double MLA homogenizer. Furthermore, we also test the energy utilization rate of sources with varying coherence lengths and investigate the influence of the defocus distance of the detection sensor on the uniformity of the target spot. Finally, our main conclusions are summarized in Section 5. Compared with other methods, this method can investigate and quantify the relationship between the spatial coherence length of the light source and the uniformity of the target surface spot, moreover it does not require additional design and manufacturing costs and can be integrated and installed separately in imaging or projection systems.
2. Theory
Figure 1 shows the experimental system including the generation of a partially coherent beam and the beam integrator optical configuration based on MLAs. The PCB generation setup incorporates convex lenses L1 and L2 and a diffuser disk S. After the laser beam passes through L1, it is focused on S. We can alter the spot size on the diffuser and control the value of DOC of the beam propagation towards the homogenization system by changing the distance between L1 and S. S is placed in the front focal plane of lens L2 to ensure that PCBs are collimated. A conventional MLA integrator system includes MLAs (a single MLA or double MLAs), integrating lens and a sensor such as a CCD camera, to measure the intensity distribution of the shaped beam.
The fundamental principle of homogenizers is that the beam is divided into different radiation patterns by an MLA that converge and overlap in the back focal plane of the integrating lens L3, thereby developing speckle patterns. The distance between the MLA and L3 is arbitrary; generally, we set it equal to the focal length of L3. To obtain a clear view of the optical schematic in Fig. 1, consistent orientation of the two-lens array is ensured. We set the arrays in inverse orientations to offset the spherical aberration. In our discussion, the two arrays have the same parameters. The distance between the double microlens arrays is controlled to be within two times the focal length. To avoid crosstalk when beamlets pass through MLAs, we set the distance between MLA1 and MLA2 to equal the focal length of the MLA.
We define the coordinate symbols (xs, ys), (x0, y0), (xM, yM), (x0’, y0’), and (x, y) refer to the source plane, the input plane of MLA1, the input plane of MLA2, the back focal plane of MLA2, and the output plane (the plane of CCD) as shown in Fig. 1; the focal lengths L1 and L2 are f1, f2, respectively.
In Fig. 2, we present the schematic for the MLAs smoothing methods, the focal lengths of Fourier lens L3 is f; the focal length of the MLA is fp and p is its diameter. To present the influence of the degree of coherence on the characteristics of shaped spots, we consider a partially coherent beam propagation in a beam-smoothing system.
2.1 Homogenizer with single MLA
Assuming a scalar, quasi-monochromatic, and statistically stationary optical field in the output plane, the field can be described by
The expression h(x0, y0; x, y) denotes the transmission function of the beam homogenizer system in Fig. 1. Considering the MLAs to have a low aspect ratio (i.e., h/p <<1, where h is the height of the lens), we performed a two-step approximation. First, the distance R between the points (x0, y0) and (x, y) in Green’s function $G\textrm{(}{x_0},{y_0};x,y\textrm{) = }\frac{{\textrm{exp(ik}R\textrm{)}}}{{\textrm{i}\lambda R}}$ can be approximated as:
Second, the phase modulation function of a single sub-microlens can be approximated as $\exp\left[ {\frac{{ - \textrm{i}k({x_0^2 + y_0^2} )}}{{2{f_\textrm{p}}}}} \right]$. Therefore, after considering the phase modulation of the Fourier lens L3, the transfer function can be written as:
Investigating a partially coherent beam that satisfies the Gaussian-Schell model, whose intensity profiles and DOC function correspond to a Gaussian distribution, the cross-spectral density W is expressed as
Equation (5) can be further simplified as
In particular, the beam approximates a totally coherent source while σ→∞; thus, the intensity can be expressed as
Furthermore, if the beam coherence length is much smaller than the diameter of the microlens unit, I(x,y) has a finite value as Δx, Δy≤σ, and tp (x0, y0) ≈ tp (x0-Δx0, y0-Δy0), and F in Eq. (7) is consistent, then the intensity I(x, y) becomes
When h/p ∼1 and h/p >1, as is the case with cylindrical MLAs, the incident beams would undergo a large spreading. Thus, we must reconsider the non-paraxial approximation in the previous derivation. The distance R in Green’s function should now be written as:
In Eq. (13), the term $\textrm{exp}[{\textrm{ik}({x_0^2 + y_0^2} )({1/2r - 1/2f} )} ]$ can be regarded as a part of the phase modulation, and we note that ${t^{\prime}_\textrm{p}}\textrm{(}{x_0},{y_0}\textrm{)} = {t_\textrm{p}}\textrm{(}{x_0},{y_0})\textrm{exp}[{\textrm{ik}({x_0^2 + y_0^2} )({1/2r - 1/2f} )} ]$. However, when the divergence angle of the microlens is large, the focal length satisfies the relation fp << (1/2r-1/2f) in the case of non-paraxial approximation; therefore, this part of the modulation can be ignored in practice. The term $\textrm{exp}[{\textrm{ik}({x{x_0} + y{y_0}} )/r} ]$ can be regarded as the continuous variation of the focal length of the Fourier lens, which causes a point spread (i.e., aberrations) of the transfer function. However, if the homogenized area generated is used in a illumination system, we only need to concern ourselves with the energy distribution of the target spot, so smaller diffuse spots can be tolerated. However, if the diffuse spots are distributed on the edges of the homogenized spots, edge sharpness decreases; this would reduce the uniformity of the target spots.
Bringing Eq. (13) into Eq. (5) and assuming an off-axis angle α, the intensity distribution on the target plane can be written as:
2.2 Homogenizer with double MLAs
We consider that the parameters of the two arrays in the double MLAs system are exactly the same; thus, the transmittance function can be simplified as
The intensity can be written as
The conditions described in Eq. (9) and (10) are also applicable to double MLAs. From Eq. (8) and Eq. (16), the formulas suggest that the profiles are related to three parameters: (1) the Fresnel coefficient FN that determines the diffraction effect produced by the sub-refractive lens in the array, (2) the interference period Δx that determines the interval between interference fringes, and (3) the parameter σf related to the degree of coherence and the focal length of the Fourier lens that determines the speckle contrast. In section 3, we will discuss how these parameters affect the intensity distribution specifically based on the numerical results.
If an MLA with h/d∼1 or h/d>1 is used in a double MLAs homogenization system, it will cause serious crosstalk of the sub-beams, which will not achieve the shaping effect. Therefore, the non-paraxial condition is not considered in double MLAs.
3. Numerical calculation
Initially, we studied the intensity distribution properties in the target plane with the coherence length varying from ∞ to 0.01mm. In the simulation, the wavelength was 532nm, the width of the Gaussian-Schell beam ω was 10mm, the pitch of the MLA was set as 110µm, the focal length of the MLA was set as 0.4mm, and the focal length of the Fourier lens was 25mm. The shaped beam profile was rendered as a rectangle with D = 6.875mm. Figure 3 illustrates the numerical results that show the evolution of I(x) as a function of the coherence length.
Fig. 3. Intensity distribution along x axis for different coherence length of initial beam using a microlens array
As shown in Fig. 3, as σ decreases, the speckle patterns are gradually smoothed. However, the enhancement at the edge of the shaped spots caused by diffraction is challenging to eliminate. In addition, we have also observed that when the coherence length is reduced to 0.01mm, the shaped intensity profiles become Gaussian.
The irradiance distribution produced by the diffraction pattern of a sub-microlens is shown in Fig. 4. All these patterns have a similar intensity distribution: the central intensity has an obvious fluctuation that is lower than the edge of the spots that exhibit a sharp increase. The Fresnel number FN = p2/λfp can characterize the diffraction effect. Figure 4(a) and (b) are obtained by setting p = 0.11, fp = 0.4, f = 25mm and p = 0.22, fp = 1.6, f = 50mm, respectively; the Fresnel number in both the figures are the same. Moreover, Fig. 3(c) and (d) are obtained by setting p = 0.11, fp = 0.2, f = 50mm and p = 0.11, fp = 2, f = 25mm, that is, FN ≈ 110 and 11, respectively. The higher the FN, the smaller the variation in the shaped beam profile. After many simulations and practical experiments, when FN>15, the influence of single-lens diffraction on the uniformity is gradually indistinguishable.
Fig. 4. Intensity distribution of diffraction patterns using a microlens array: (a) p = 0.11, fp = 0.4, f = 25 mm; (b) p = 0.22, fp = 1.6, f = 50 mm; (c) p = 0.11, fp = 0.2, f = 50 mm; (d) p = 0.11, fp=2, f = 25 mm.
The interference fringes are due to the periodic structure of the MLA and the high coherence of the initial beam. The parameter Δx = λf/p is defined as the interference period. Intuitively, the smaller the value of Δx, more concentrated the fringes become, and the easier it becomes to obtain an appropriate uniformity. Figure 5(a), (b), and (d) correspond to Δx = 0.12mm and (c) correspond to Δx = 0.24mm; thus, it is obvious that the interference period is smaller in (a), (b), and (d).
Fig. 5. Intensity distribution of interference patterns: (a) p = 0.11, fp = 0.4, f = 25 mm; (b) p = 0.22, fp = 1.6, f = 50 mm; (c) p = 0.11, fp = 0.2, f = 50 mm; (d) p = 0.11, fp = 2, f = 25 mm.
From Eq. (8) and (16), the third term is not only related to the coherence length, but is also affected by the focal length of the Fourier lens, implying that it is determined by the value ${\sigma _\textrm{f}} = \sqrt 2 \lambda f/\mathrm{\pi }\sigma$ that represents the width after the Fourier transform of the DOC function. In Fig. 6, the blue lines represent the Fourier transform of the DOC function with σ = 0.03mm, and the red lines represent the cross-sectional distribution of the final intensity profiles I(x) in Eq. (8) with consistent parameters, as shown in Fig. 4 and Fig. 5. For the same size of the target spot, as shown in Fig. 6(a) and (b), the intensity profiles become smoother and approximate Gaussian distribution as σf increases.
Fig. 6. Intensity distribution of Fourier transform of DOC (blue curves) and shaped spot (red curves): (a) p = 0.11, fp = 0.4, f = 25 mm; (b) p = 0.22, fp = 1.6, f = 50 mm; (c) p = 0.11, fp = 0.2, f = 50 mm; (d) p = 0.11, fp = 2, f = 25 mm.
To evaluate the uniformity of the intensity distribution after homogenizers accurately, the coefficient of variation (cv) was used to calculate the vibration intensity.
where ${\sigma _{\textrm{rms}}} = \sqrt {\frac{1}{{{N_0}}}\sum\limits_{i = 1}^{{N_0}} {{{({I_i} - \bar{I})}^2}} }$ denotes the standard deviation of intensity over the shaped spots, N0 is the total sampling number, Ii is the intensity at the corresponding point, and Ī represents the average intensity in the observation plane.After multiple simulations (as shown in Fig. 7), we conclude that the shaped spots after a single MLA homogenizer assume a Gaussian distribution when the ratio of D/σf is less than 7, as shown in Fig. 6(d). When D/σf is approximately 35, a spot with uniform intensity distribution can be obtained; a more accurate experimental value is mentioned in Section 4.
Figure 8 shows the cross-sectional intensity distribution versus coherence length after beam shaping by the double MLA homogenizer. We set the distance between the two arrays as fp, such that the size of homogenized area is the same as that of the single MLA homogenizer. Compared with Fig. 3, the only difference is that there is no obvious enhancement at the sides of the shaped spot. The change is mainly caused by the diffraction of the double lens array as shown in Fig. 9, and the overall diffraction pattern after the double array has a more uniform distribution. In addition, comparing Fig. 10 and Fig. 11, if D/σf is gradually decreased, the intensity shaped by the double MLA homogenizer can reach a homogeneous distribution with a larger value than that obtained using a single MLA.
Fig. 8. Intensity distribution along x axis for different coherence lengths of initial beam using double microlens array
Fig. 9. Intensity distribution of diffraction patterns using double microlens array: (a) p = 0.11, fp = 0.4, f = 25 mm; (b) p = 0.22, fp = 1.6, f = 50 mm; (c) p = 0.11, fp = 0.2, f = 50 mm; (d) p = 0.11, fp = 2, f = 25 mm.
Fig. 10. Intensity distribution of Fourier transform of DOC (blue curves) and shaped spot (red curves) using double microlens array: (a) p = 0.11, fp = 0.4, f = 25 mm; (b) p = 0.22, fp = 1.6, f = 50 mm; (c) p = 0.11, fp = 0.2, f = 50 mm; (d) p = 0.11, fp = 2, f = 25 mm.
4. Experimental testing and results
4.1 Experimental results
The method and device for generating the GS beams are shown in Fig. 1. To avoid the influence of temporal coherence, we chose a single-frequency laser (≤300 kHz) with a wavelength of 532 nm. The M2 factor can roughly related to the coherence of a partially coherent beam, and the relationship between M2 and σ is [41]:
Therefore, we can determine the coherence length of the beam by measuring the M2 factor and waist radius of the beam incident on the microlens; this was performed by using the experimental optical path shown in Fig. 12(b) to measure the beam radius at different positions, where l < f2. By fitting the curves of the beam radius with the transmission distance, we obtained the divergence angle and waist radius of the GS beam, from which the value of M2 factor could be determined.
Fig. 12. Principle of M2 factor measurement: (a) collimated optical path of PCBs; (b) the measuring method of M2.
The parameters in the experiment are the same as those in the simulation, and the elements used in the factual experiment are given by Fig. 13.
Figure 14 shows four experimental results of the shaped spots obtained using a single MLA homogenizer with different coherence lengths, including diagrams from the sensor, 3D diagrams, and cross-sectional distribution. The intensity distribution has a high contrast when using the homogenizer without a diffuser, as shown in Fig. 14(a). By altering the coherence length of the initial beam to 0.3350 mm and 0.0675 mm, respectively, we obtained an optimized distribution, as shown in Fig. 14(b) and (c). Figure 14(d) shows the shaped spot when the coherence length is equal to 0.01 mm, and the cross-sectional distribution is consistent with the simulation results.
Fig. 14. Normalized 3D distribution of shaped intensity using a single MLA homogenizer with partially coherent beam: (a) σ = 1.046 mm; (b) σ = 0.335 mm; (c) σ = 0.0675 mm; (d) σ = 0.0127 mm.
Figure 15 shows the variation of the value of cv versus the coherence length, the dash indicates cv = 0. The diffusers are used in the experiment offered in 1500, 600, 200 grit polishes and Gaussian transmittance, we numbered the three diffusers in sequence as diffuser 1, 2 and 3. Both the simulation and experimental results confirm that a partially coherent beam facilitates reducing the fluctuation of intensity. These curves clearly indicate that the coefficient of variation drops significantly with the decrease in the coherence length from 1 mm to 0.3mm. When σ is less than 0.1mm, the values of cv do no change significantly. The intensity distribution has a most steady profile as σ approximately equals 0.036mm in the experiments; thus, σf ≈ 0.17mm and D/σf ≈ 40. It is applicable to homogenizers using MLA with different parameters.
Fig. 16. Normalized 3D distribution of shaped intensity using double MLA homogenizer with partially coherent beam: (a) σ = 1.046 mm; (b) σ = 0.335 mm; (c) σ = 0.0675 mm; (d) σ = 0.0127 mm.
Zone 1 in Fig. 15 shows that when the coherence length is large, the values of cv in the experiment are smaller than the simulation results. This phenomenon can be explained as follows: in the practical experiment, the beam intensity is attenuated to avoid over-saturation of the CCD that causes the minimum value of the shaped spot to be unaffected. Thus, we obtain a homogenization spot with a lower speckle contrast. All the curves exhibit a modest rise in Zone 2 because the intensity profile becomes Gaussian. However, the increased width of the Fourier transform of the DOC function still maintains cv at a low value.
The effectiveness of vibration reduction in a shaped beam with double MLAs is further examined, as shown in Fig. 16. Fig. 17 indicates that cv shows a trend similar to that shown in Fig. 15, corresponding to the change in the coherence length. However, the effectiveness of homogenizing the beam is slightly better than that of a single MLA system. Meanwhile, the coherence length corresponding to the most homogenized spot obtained in the experiment was 0.0832 mm, and D/σf ≈ 94. Both in Fig. 15 and Fig. 17, we obtain the same trend as that in the existing theory, demonstrating the feasibility of improving the uniformity of intensity distribution using a partially coherent beam smoothing system in a single or double MLA homogenizer.
Finally, we discuss the effect of inserting a diffuser on the energy efficiency and the variation in the uniformity of the shaped spots when the observation sensor is defocused. Figure 18 shows the variation of the value of cv when the output plane in the single or double MLA system is defocused. The blue lines and red lines represent the single and double MLA systems, respectively, and (a), (b), and (c) correspond to the cases with different coherence lengths. All the six curves have the same change trend, proving that a slight defocus has a negligible effect on the uniformity of shaped spots; even when the defocus distance is ±1mm, the uniformity is slightly improved, owing to the spread of the spot.
Fig. 18. Coefficient of variation cv of spot distribution in target plane as a function of the defocus distance for single MLA homogenizer (blue line) and double MLA homogenizer (red line) with coherence length (a) σ = 1.072 mm; (b) σ = 0.3310 mm; (c) σ = 0.2520 mm.
Both simulation and experimental results show that the uniformity of the spot depends not only on adjusting one parameter, such as p, fp or f, but also on the result of several factors. However, f and D are often fixed in practical applications. Therefore, we need to adjust the coherence length based on the results obtained before.
4.2 Optical efficiency
The energy efficiency of the entire homogenization system used in the experiment depends on three factors: the transmittance of the optical elements in the system, the Fresnel loss of the microlens, and the transmittance of the diffuser. In the system, lenses L1 and L2 were coated with an antireflective film that was at 532 nm and their reflectivity was about 3%; the surface of lens L3 was uncoated, and its reflectivity was about 10%.
The transmission coefficient R and reflection coefficient T of light on the surface of the microlens are determined by the refractive indices n1, n2 on both sides of the microlens interface and the angle of incidence θ. For the microlens surface, the incident angle changes with the coordinates. Therefore, the reflectance and transmittance can be defined as [22]:
The reflection and transmission coefficients for the s and p components are given by the expressions:
On calculation, the reflectance and transmittance were respectively found to be 4% and 96%. This result means that the surface of MLA used in the experiment has almost no effect on Fresnel losses. In addition, the surface of MLA has been coated with anti-reflection film (@400∼870nm), so the reflectance can be reduced to 1.2%.
In the system, the low transmittance of the diffuser is the main reason for the energy loss from the system. We used polishes of three different specifications on the diffusers: the 200 grit polish belongs to coarse sand, with a 70% transmittance at 532nm; this is followed by the 600 and 1500 grit polishes, whose transmittances were 83% and 85%, respectively. Due to variations in the roughness of the diffuser surface, the scattering angle of the light beam is also different. The beam divergence angles of diffusers 1 (200 grit polishes), 2 (600 grit polishes), and 3 (1200 grit polishes) are 4°, 7°, and 10° respectively.
Figure 19 presents the energy efficiency after MLA homogenizers when using the 3 diffusers separately. Once the diffuser is utilized in the system, the energy efficiency drops sharply. And we found in the experiments that the coherence length can be controlled more precisely when using a diffuser with fine sand. In particular, the excess energy is emitted from the edge of the MLA when using diffuse 3, since the divergence angle of the diffuser is large and the size of spot (≈17.63mm) at the back focal plane is larger than the size of our MLA (15mm×15mm). The curves exhibit a significant decline when the coherence length is approximately 0.01 mm that can be explained by: the homogenized area with a Gaussian distribution no longer has a sharp boundary.
5. Conclusion
In this study, we modeled the intensity distribution characteristics shaped by the MLA homogenizer with the parameters FN, Δx, and σf to realize more uniform spots. The model clearly indicates that a low coefficient of variation can be obtained by setting the parameters of the microlens, the focal length of the Fourier lens, and the coherence length of the initial source. To obtain a spot with high uniformity, the microlens should have FN>15, a larger pitch when the target plane is fixed, and an appropriate ratio between D and σf. This conclusion is applicable to any homogenizer using MLAs.
In the simulation, we utilized a Gaussian-Schell beam with different coherence lengths in the MLA. Comparing the uniformity measurements in the experiment with the simulation results indicates that our insight into the uniformity-increase factors, particularly the determination of coherence length, is appropriate. The experimental results show that the value of the coefficient of variation decreases to 0.0253 when a double MLAs homogenizer is utilized and the coherence length is adjusted to 0.0832mm and the value of D/σf is roughly equal to 94.
In addition, we considered the defocus of the observation surface. A slight defocus has a negligible effect on the homogenized spot, and the uniformity of the distribution is optimized within the range of ±1mm.
The proposed model is flexible and can be embedded in an optical structure as per requirement; thus, it can be independently applied to various devices and different application conditions, such as the illumination path of microscope systems or the uniform illumination required by a lithography machine, etc., that require uniform illumination.
Funding
Strategic Priority Research Program of China Academy of Sciences (XDA17040501); Instrument Developing Project of the Chinese Academy of Sciences (YJKYYQ20200047).
Acknowledgments
The authors would grant their appreciation to the support from the Strategic Priority Research Program of China Academy of Sciences, and the Instrument Developing Project of the Chinese Academy of Sciences.
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 may be obtained from the authors upon reasonable request.
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