Research Article Current Issue Versions 1 Vol 4 (4) : 21040401 2021
Download
Influence of Parameters in the Design of a Faceted Structure for Incoherent Beam Shaping
: 2021 - 12 - 01
: 2021 - 12 - 15
: 2021 - 12 - 31
1036 26 0
Abstract & Keywords
Abstract: A reflective faceted structure is proposed to reshaping an incoherent light beam into two focalized spots. To obtain the desired irradiance distribution on a detector, custom optimization function is written, and the two dimensional tilt angles of each facet are optimized automatically in a pure non-sequential mode in Zemax OpticStudio 16. The result is also confirmed inside LightTools 8.2 from Synopsys. For measuring the quality of the optimization result in the case of two spots focalization, four factors including efficiency on the detector, uniformity, the root mean square error and the correlation coefficient are calculated. These four factors are used to evaluate the influence of several parameters on the irradiance distribution. These parameters include the incidence angle, the divergence angle, the facet size, the source type and the resolution of the facet angular positions. Finally, an analysis of those parameters is made and the performance of this type of component is demonstrated.
Keywords: Incoherent beam shaping; micro lens array; custom optimization
1.   Introduction
Incoherent beam shaping plays a very important role in the areas of non-imaging optics and is applied today in different fields: light illumination, solar concentrators, multimode laser marking system, etc. Different types of optical component are employed for reshaping incoherent light beam: freeform system [1-5], digital mirror devices [6-8] or calculated roughness structure [9]. For example, one single free-shape mirror can be designed with a freeform shape, but the calculations are sometimes huge with complex surfaces [1,3]. Digital micro-mirror devices in connection with a telescope system are also used to reshape the incoherent light [6-8], but arbitrary targets without symmetry have not been reported in a static approach. Weyrich et al. obtain the custom reflectance by series of algorithms, in order to transform the highlight shape to produce the height fields [9]. Whereas the performance seems adapted to specific fields, the previous solutions seem difficult to implement for custom illumination system. We can note also the use of Fresnel lenses or equivalent structure in the field of solar energy where the idea of beam shaping is currently employed and a notion of symmetry appeared [10]. Another close case is the beam shaping of LED with lightpipe, but here, the goal is to obtain uniform illumination [11]. The method of design is really placed in the general case where the optical component and the result have no symmetry and the source is totally incoherent. The choice of the irradiance distribution is here two focalized spots to demonstrate the concepts and see the limitations.
We develop an optimization method to reshape incoherent light using Zemax OpticStudio 16 (Zemax) with a non-sequential approach [12]. At each loop inside the optimization process, the sum of the irradiance data on the non-target area of the detector is calculated automatically with the internal ZPL macro language. The two dimensional tilt angles of facets are modified iteratively by minimizing the sum of this irradiance value, while maximizing the target illumination. When the sum of the irradiance value is smaller to a preset standard minimum, the desired irradiance distribution is achieved. The verification of the optimization result is also realized with LightTools Illumination System Design 18.2 from Synopsys.
Using beam quality criteria in diffractive optics [13-16], the quality of the irradiance distribution is assessed in this paper by four factors including efficiency, uniformity, the root mean square and the correlation coefficient. Results on irradiance distribution are compared between Zemax and LightTools. The comparison is done by transferring the optimized model from Zemax into LightTools and setting up the same optical system. Identical non-sequential has been performed in both software tools leading to a more convincing comparison of the performances. Thus it is more convincing for providing the optimized model and to compare the performance.
As we will see, several parameters play a very important role in the quality of the irradiance. Among these are the wavelength, the incidence angle, the divergent angle and the facet size. The influence of these parameters is evaluated through the quality factors to see the real performance of our new component. Discussions about the acceptable tolerances will also be made with some recommendations.
In Section 2, details are given about the optical model. In Section 3, the optimization method and the optimization results are described. In Section 4, the numerical measurement theory is stated. Section 5 evaluates the influence of the parameters based on the beam quality assessment both with Zemax and LightTools.
2.   Optical Configuration
The optical configuration is composed of an incoherent light source (ILS), the faceted surface (FS) and the detector (D). To avoid the influence of the dispersion, the material of the faceted surface is defined to be a perfect mirror. The reflective sample is made of a number of square facets with an equal size. As the tilt angles of each facet are zero before the optimization, the reflective surface in the initial configuration is flat. As we aim at redistributing the incoherent light beam, the size of the facet needs to be large enough in respect to the wavelength in the case of the non-sequential optics. For the feasibility of the following optimization process, the distance between the detector and the faceted surface is chosen to be 10 times larger than the size of the complete faceted surface.


Figure 1.   Description of the optical configuration. A1 is the incidence angle between the incidence beam and the normal of the faceted surface. A2 is the divergence angle of the incidence beam. W is the width of the faceted surface. The width of the incidence beam is L.
As shown in Figure 1, the width of the incidence beam L is defined by two parameters, the width in X direction Lx and the width in Y direction Ly .Thecenter of the coordinate system is placed at the center of the faceted structure (FS). The divergence angle A2 is also determined by the angle in X direction A2x and theangle in Y direction A2y .In the initial configuration, A2x and A2y are defined to be zero. In our study, the incidence angle A1 is fixed to be 45 degrees, but this angle could be changed. In our choice of design, the reflective surface is made of 6×6 facets, and the size of each facet is 2 mm: the total size is then 12 mm by 12 mm. For totally fitting the overall reflective surface, the width of the source in direction X and Y needs to comply with the mathematical relation as, Lx² =W²/2, Ly =W. The size of the incidence beam is thus 8.5 mm by 12 mm. The incidence beam illuminates the faceted surface uniformly. The number of rays for the non-sequential ray tracing is 1 million. The number of pixels on the detector is 200 by 200 and the size of detector is 100 mm by 100 mm. The total power of the source is 1 Watt. The data type is incoherent irradiance, whose unit is watts/cm2. The choice of all these parameters is explained by a good convergence of the optimization algorithm explained below. Before optimization, the initial irradiance distribution on the detector is rectangular with the same size of the incidence beam. After iterative optimization of the tilt angles of the facets, different kinds of irradiance distribution could be achieved. The optimization process will be depicted in the following section.
3.   Design and Optimization Procedure
3.1.   Optimization Algorithm
Based upon the setup of the optical configuration, the reflective surface is composed with a matrix of facets. The reshaping of the incoherent beam is realized by optimizing the tilt angles of each facet along X and Y. The optimization is processed in non-sequential mode of Zemax. The 2D tilt angles of each facet are variables. The Orthogonal Descent algorithm is applied to optimize the variables. The merit function is written with the Equation (1):
(1)
In Equation (1), k1 represents the pixel with the non-zero irradiance value in the target area. The value k2 represents the pixel with the non-zero irradiance value in the non-target area. The value Rk is the actual irradiance value of the pixel k and Tk is the objective irradiance value for the pixel k. The value m is the weight of each item. For pixels in the target area, the target irradiance value Tk1 is defined to be 1. For pixels in the non-target area, the target irradiance Tk2 is defined to be 0. When the irradiance value of each pixel approaches the target, the value of the merit function decreases. One optimization loop is ended, when the Orthogonal Descent algorithm gets to the local minimum.
3.2.   Iteration of the Optimization Algorithm
The process is accomplished by optimizing the angular positions iteratively. During one optimization loop, the value of the merit function decreases to a local minimum. Every time after one optimization loop is ended, the non-sequential ray tracing is performed to obtain the current irradiance distribution on the detector. And the sum of all the irradiance data in the non-target area is calculated.
In the Equation (2), SF is sum of all the non-zero irradiance value in the non-target area.
(2)
If SF is below to a preset minimum, the optimization process will be finished. Otherwise, the merit function (1) will be refreshed by the pixels in the non-target area whose irradiance value are non-zero. Then, another optimization loop is processed. So the reshaping of the incoherent beam is achieved by several loops.
The automatic optimization process is controlled by the macro program written in ZPL Language, which contains the correspondent calculations and operations. Through directly executing the macro program, the optimization is started to run until the value of SF is below to the preset minimum. A detailed analysis of our optimization procedure is given in the reference [11].
3.3.   Design Result for Two Focalized Spots
In the case discussed here, the target is made of two horizontal spaced spots symmetric for focalization. The achieved irradiance distribution is shown in Figure 2. The coordinate of the center of the left spot is (-25.25 mm, 0.25 mm). The coordinate of the center of the right spot is (25.25 mm, 0.25 mm). The dimension of each spot is 4 mm × 4 mm. The source is monochromatic with a wavelength of 550 nm. The size of the facet is 2 mm. An obtained irradiance map is visible in the Figure 2 with the corresponding component.


Figure 2.   The optimization result for the target of two horizontal spaced spots. (a) The achieved irradiance distribution on the detector. (b) The optimized structure made of 6×6 facets, the size of each facet is 2 mm.
This result has been obtained in 3 loops of optimization, taking about 1.44 hours. The processor of the computer is an Intel Xeon E3-1270 3.50 GHz. The type of the operating system is with 64 bits, 4 cores and 32Go RAM. The number of rays for the non-sequential ray tracing is 1 million.
To verify the performance of our algorithm, we changed the size of the facet to be 1 mm and 3 mm in the flat initial configuration. By using the same optimization procedure, the tilt angles of the facets were re-optimized.
We achieved the same irradiance distribution with the new values of facet size demonstrating the reliability of our optimization flowchart. The influence of the size of the facet on the quality of the optimization result will also be assessed in the fifth section.
4.   Definition of Quality Criteria for Analysis
For measuring the quality of the achieved irradiance distribution on the detector, numerical calculation is done with these four factors: the efficiency, the root mean square error, the uniformity and the correlation coefficient.
The efficiency η is given by Equation (3):
(3)
In Equation (3), In,m is the actual irradiance distribution on the detector. R is the target area. N and M are the one-dimensional number of pixels on detector. In our case, N = M = 200. The efficiency η is the ratio of optical power in the target area to the total incidence power on the detector: it is a measurement of the useful energy in the irradiance map. In our case, we consider in the calculation any other energy losses as the reflective coefficient or roughness.
The root mean square error εi is given in Equation (4) for measuring a mathematical distance between the target and the achieved irradiance distribution [13, 14]. A is the number of pixels in the target area R. On,m is the desired irradiance distribution. The term ai is a constant that normalizes the calculated irradiance map in order to get consistent error values. The root mean square error in intensity is the most straightforward criterion to evaluate the quality result, and thus the convergence rate of the optimization process. The equation is given by:
(4)
where
(5)
The uniformity U is given by Equation (6), which is the standard deviation of the difference between the actual irradiance and the mean irradiance em[14] :
(6)
with the value em is the average of the irradiance in the target area:
(7)
The illumination uniformity U is small if the difference between irradiance at each point and the mean value em is small. Bokor et al. provide an equation for measuring the fidelity of an achieved irradiance distribution [13-15]. They introduce a mathematical quantity which measures the correlation between the calculated irradiance map and the desired irradiance. This correlation coefficient is always in the interval [0,1] The expression of the correlation coefficient C is given in the reference [13].
The quality factors will be used in the following sections to measure the quality of the irradiance distribution on the detector and evaluate the influence of several important parameters on the reshaping result of the incoherent beam. These four factors contain the required information to analyze the irradiance map properties. It is important to note that the analysis in the next section should always be done by observing the 4 criteria together. Indeed, one can easily notice that the behaviour of efficiency and correlation coefficient are linked together, and so are also the uniformity and the error.
Of course, in some applications, factors are more relevant than others, for example uniformity in high quality illumination. The most important in studying the quality factor is to observe the evolution of these values in a large interval, knowing that only efficiency is possible to link to the experiment and easily measurable.
5   Numerical Analysis of the Influence of Different Parameters and Discussion
Several optical parameters play a very important role in the quality of the irradiance distribution for our new faceted component. The optical parameters include the incidence angle A1, the divergence angle A2, the size of the facet, the material of the facet and the wavelength of the source. The quality of the optimization result is measured by the previous defined four factors. By calculating these quality factors, the influence of the optical parameters on the optimization result is assessed in this section.
To ensure the credibility of the results, the simulation is performed both in Zemax and LightTools software. After changing the value of the parameter, raytracing is performed. The number of rays for the raytracing is set to be 1 million, as before. As the number of pixels is 200 × 200 on the detector. 200 × 200 irradiance data are exported from Zemax or LightTools to Matlab, where the calculation of the quality criteria are made. The matrix of the objective irradiance On,m and the matrix of achieved irradianceIn,m must be normalized in preference before calculating the quality factors [13 ]. As the optimization process is made to obtain the best irradiance map, it is normal that the efficiency or the correlation coefficient will be more suitable for our analysis.
5.1.   Influence of the incidence angle
In this section, we study the influence of the incidence angle on an optimized structure with square facets of 1 mm, 2 mm and 3 mm, only with Zemax. The varying range of the incidence angle is between 30° and 50° with a step of 3 degrees. Outside this angular range, the quality factors visible in the Figure 3 cannot be calculated by the equations presented in the section 4 due to decrease of the irradiance information. When the incidence angle approaches 45 degrees, efficiency and correlation coefficient are on the top while the root mean square error and the uniformity are at the lowest point. This is normal because it is corresponding to the incident angle used for the optimization process: this point is the optimal point for the use of the component. The effect of the incidence angle is very sensible: a difference of 3 degrees of the incidence angle decreases the efficiency by 20%. We observe the same decrease with the correlation coefficient. A tolerance smaller than 1 degree on the incidence angle seems a reasonable choice to keep a usable efficiency and irradiance map.


Figure 3.   Four quality factors for different incidence angles. The optimized structures with the facet of 1 mm, 2 mm and 3 mm are analyzed separately with Zemax. (a) Efficiency (b) Root mean square error (c) Uniformity (d) Correlation coefficient.
The reasons why the case of 1 mm is more sensible are probably due to the shading effects of the facet and their corresponding losses. The quality factors for the structures with the facet of 2 mm and 3 mm are close comparing to the case of the facet of 1 mm, unless for angles smaller than 36°.
5.2.   Influence of the incidence angle in both software
In the Section 5.1, we analyze the influence of different incidence angles on the irradiance distribution only in Zemax. To further prove the credibility of the optimization results in Zemax, LightTools is employed to measure the quality factors for different incidence angles. The calculation results with Zemax and LightTools are compared in the same Figure 4.
In this section, we only present the optimized structure with the facet of 2 mm, but the results are comparable with the other size of facets. As shown in Figure 4, there is no significant difference between Zemax and LightTools. The coincidence of the data between the two software tools proves the validity of the calculation results for different incidence angles and different sizes of facets. For the case of uniformity, the small difference for the incidence angle in the range of [30°, 36°] can be explained by the non-sequential ray tracing algorithm used in the two software tools. This study confirms us to choose of 1 degree of tolerance for the incidence angle.
5.3.   Influence of the light source divergence
In our design, we state that the light source is collimated, i.e., the divergence angle is zero. We calculate the quality factors for different divergence angles both with Zemax and LightTools software. The varying range of the divergent angle is between 0° and 5°.


Figure 4.   Quality factors for different incidence angles calculated with Zemax and LightTools. The optimized structure with facet of 2 mm is analyzed. The irradiance distribution on the detector is two horizontal spaced spots. (a) Efficiency (b) Root mean square error (c) Uniformity (d) Correlation coefficient.


Figure 5.   Four quality factors for different divergent angle which are calculated with both Zemax and LightTools software. (a) Efficiency (b) Root mean square error (c) Uniformity (d) Correlation coefficient.
As we can see in the Figure 5, there is no difference between the simulation results of Zemax and LightTools for the four quality factors: that confirms the quality of the 2 software tools in non-sequential mode.
If the divergence increases with a small angle, the efficiency decreases drastically: the sample needs to be illuminated with a good collimated beam. As a matter of fact, this design requires us to take the divergence angle into account in the optimization process. In order to limit the decrease in either the efficiency or the correlation coefficient, namely 10%, the divergence angle should be lower than 0.5°.
We must note that the efficiency is defined here in this article as the ratio of the concentration of the energy inside the area of interest to the total energy in the irradiance map and not the incident energy: it is then an evaluation of the similarity between the target and the result. The root mean square error and the uniformity are more difficult to analyze, probably due to the fact that they are not considered as constraints inside our optimization algorithm: the behaviours are very close and the evolution is small inside the chosen interval.
5.4.   Influence of the size of facet
In this section, the optimized structure in the Figure 2(b) is employed with a 2 mm size of facet. We study here the four quality factors in function of the size of the facet between 0.5 mm and 5 mm with a step of 0.5 mm.
The angular positions of each facet are the same for each case. As before, the simulations are performed both in Zemax and LightTools, and we can see in Figure 6 that the results are very similar.
For the efficiency and the correlation coefficient, the performance is of course at the highest point when approaching to 2 mm facet size: the behaviour of these two quality factors is very similar.
For the root mean square error and the uniformity, the value is overall decreasing in the interval when the size of the facet is increased, but same remarks can be made as before as the optimization process does not use these constraints. The evolution of the error and the uniformity is however not standard. Then, we can only say that a tolerance of 0.5 mm of the flat facet size is compatible to keep good performance of our component and an acceptable relative decrease of 10 % in the total efficiency (taking account only perfect mirror).


Figure 6.   The quality factors for different size of facet of the optimized structure in the Figure 2(b). (a) Efficiency (b) Root mean square error (c) Uniformity (d) Correlation coefficient.
5.5.   Influence of the reflectivity of the material
Former analysis and calculation are based on the case of the perfect mirror whatever the wavelength. So, we change the material of our facets to be aluminium or gold without roughness, and compare the irradiance distribution with the perfect mirror. We study the case of a blue or green or red light source and also the combination of these three light sources called “white source” in the following. The wavelengths of the three unique sources are 611nm for red, 549nm for green, and 464nm for blue. The “white source” is the combination of the three wavelengths with the same weight. The refractive index and extinction coefficient of aluminium and gold for the three wavelengths are given in Table 1 and used in the two software tools.
After performing the non-sequential raytracing, the true color distributions for the blue, green, red and “white light” are studied in Zemax and LightTools. As expected, we have a convergence between the two software tools, which proves that the measurements of the quality factors for the material of aluminium and gold are correct.
Table 1.   The refractive index and the extinction coefficient of aluminium and gold for three wavelengths [17]. The surface is considered without roughness.
WavelengthAluminiumGold
611 nm1.1816+i7.04330.22273+i3.0325
549 nm0.96175+i6.3890.42791+i2.3345
464 nm0.66351+i5.4641.2597+i1.7531
We observe no change neither in the position of the two spots, nor in the irradiance map for the three cases: perfect mirror, aluminium and gold. Also, there is no color artifact in the irradiance map. It is quite normal that there is no influence of the wavelength on the irradiance distribution in this type of configuration. In the case of the perfect mirror [11], the quality factor does not change much in the studied interval of wavelength. The efficiency is about 90 %, the root mean square error is less than 0.2, the uniformity is about 0.5 (obviously due to the fact that we do not optimize this parameter) and the correlation coefficient is close to 0.9 for the case described in the section 3.3.
However, the most important change is the efficiency inside the irradiance map and the total reflectivity: we will focus on these two criteria using the optical component described in section 3.3. With LightTools, we study four cases summarized in the Table 2 and Table 3: these values are confirmed by Zemax at 0.1 % precision.
Table 2.   Different value of reflectance obtained in the case of two spots and aluminium material with LightTools. The definition of each value is given in the text.
Wavelength (nm)464549611
Efficiency η191.7 %91.7 %91.7 %
Reflectance R138.1 %37.9 %37.9 %
Reflectance R287.7 %87.3 %87.2 %
Reflectance R391.5 %91.0 %91.0 %
Table 3.   Different value of reflectance obtained in the case of two spots and gold material with LightTools. The definition of each value is given in the text.
Wavelength (nm)464549611
Efficiency η191.9 %91.7 %91.7 %
Reflectance R116.2 %32.0 %38.1 %
Reflectance R237.8 %74.0 %87.8 %
Reflectance R339.5 %77.1 %91.4 %
The efficiency η1 is the ratio described in Equation (3): this calculation shows us the distribution of the light in the area of interest in relation of the total irradiance impinging the detector. Reflectance R1 is the ratio between the irradiance in the detector and the incident irradiance of the source. Reflective R2 is the case of the reflectivity of a flat aluminium mirror or a flat gold mirror designed inside LightTools with the same optical configuration. Reflectance R3 is the mean value of the Fresnel reflectance coefficient in TE polarization and TM polarization calculated with a bulk material of aluminium or gold, which is the theoretical limit.
The first comment is that the efficiency 1 does not depend of the wavelength: the irradiance map is not perturbed by the choice of material. The reflectance 1 is stable for the aluminium and increases with the wavelength for the gold material: this point is compatible with the reflectance 2 and the theoretical calculation of reflectance 3. The choice of aluminium seems more interesting to apply with a “white source” and confirmed by the practical use of this material in the field of micro-optics. The total reflectivity coefficient R1 of the component is small due to different factors: the shadow of the facet, the choice of the optimization algorithm, the number of facet, etc.
5.6.   Influence of different light sources
In this part, we calculate efficiency and reflectance for different light source as black body with LightTools. The material of facet structure is aluminium and the perfect mirror is used as a reference. We use 4 different black bodies with a spectrum between 380 nm and 760 nm (visible spectrum). The parameters of the different used black bodies are listed in Table 4.
Table 4.   The parameters of different black bodies used with LightTools. The spectrum is calculated only between 380 nm and 760 nm. The number of samples in the range of spectrum is 36.
SourceTemperature of the black body
Lamp with filament in tungsten + argon gas2200 K
Lamp with filament in tungsten + krypton gas2500 K
Lamp with filament in tungsten + halogen gas3000 K
Sun5800 K
The number of samples in the spectral range is set to be 36, as a compromise of calculation time and sufficient definition of the spectral distribution. It means that for different black bodies, the weight for each wavelength is different to sample sufficiently the real distribution. An example of the used visible spectral distribution for the simulation is given in Figure 7. After setting the parameters of the black body (temperature, number of samples), the refractive index and extinction coefficient of aluminium for each sampling wavelength are given in LightTools.
For the case of the perfect mirror, the quality factors have no big difference between all studied cases: the values are close with the case given in the previous section. The calculation results prove that the choice of the polychromatic source has small influence on the irradiance distribution. Again, this is not surprising so far. So, the efficiency, the reflectance R1 and R2 defined in the previous section of the incoherent irradiance distribution on the detector for aluminium are calculated, which are given in the next Table 5.


Figure 7.   Normalized spectral distribution for a black body at 5800K: it is the theoretical spectrum used to simulate the sun as a light source. Table 5. Numerical simulation values with different light sources.
Table 5.   Numerical simulation values with different light sources.
SourceEfficiency η1Reflectance R1Reflectance R2
Lamp with filament in tungsten + argon gas91.7 %37.6 %86.6 %
Lamp with filament in tungsten + krypton gas91.7 %37.6 %86.7 %
Lamp with filament in tungsten + halogen gas91.7 %37.7 %86.8 %
Sun91.7 %37.9 %87.2 %
For different black bodies, the same irradiance distribution is achieved, as the efficiency η1 is stable between each case. The reflectance R1 and R2 are also constant in relation with the type of lamp or source. This simulation confirms that the irradiance distribution is not depending of the type of the incoherent light source. Moreover, the analysis confirms the choice of aluminium as a good material for the experiment.
5.7.   Tolerance on the angular position of the facets
In consideration of the future fabrication, the tolerance of the angular position of the facets is a main restriction. In the optimization process, the precision of the angular position is 1/1000 degree. In this section, the angular position is rounded at the precision of 1 degree, 0.1 degree and 0.01 degree in the two directions. The quality factors for the angular position with different tolerances are given in Table 6. We use the case described in the section 3.3 and a perfect mirror as the surface of each facet.
Table 6.   Quality factors for the angular position with different precision.
Angular position in degrees rounding at110-110-210-3
Efficiency34.1%87.3%89.6%89.8%
RMS Error0.370.150.170.17
Uniformity0.770.420.450.46
Correlation0.460.900.900.90
As we can see in Table 6, the value of the efficiency increases slowly when the angular position of the facets is given with a tolerance more than 0.1 degrees. When the tolerance is 0.001, 0.01 and 0.1 degree, the correlation coefficient between the actual irradiance distribution and the target is about 0.9. The difference of the quality factors between the tolerance of 0.01 and the tolerance of 0.001 is not large. This table demonstrates that the fabrication needs a tolerance better than 0.1 degree to obtain optimal results. Close to 0.1 degree is perfect: this is probably a limitation for the manufacture of such an optical component. The consequence of this angle tolerance is the required resolution for the height of the facet. This resolution is given by Equation (8):
(8)
where h(x) is the height of one facet in one direction, x is the size of the facet in one direction, △θ is the error angle of each facet, θ is the angle of each facet in one direction and △h is the error of the height of one facet in one direction. If the angle θ of the facet is at the maximum 45°, the acceptable error of the height of each facet is about 7 microns in one direction for 2 mm facet size. The minimal value of the tolerance is 3.5 microns. The Figure 8 presents the tolerances of the height in function of the angles in degrees in the case of tolerance of 0.1 degree (see Table 6 and Figure 8).


Figure. 8.   Tolerance of the height in function of the angle for △θ= 0.1 degree.
6.   Conclusion
In conclusion, we demonstrate in this paper the design of a new type of optical component containing only 6 x 6 facets to do beam shaping of incoherent light source to obtain two focalized spots. An automatic optimization process using internal macro language is explained and result is presented with a simple but non-standard irradiance map. Numerical measurements are made to analyze the influence of different parameters on the optimized result. We show also that the numerical simulations under Zemax OpticStudio 16 are comparable to those obtained with LightTools 8.2. This study shows us the limitation in using our optical component. First, the incidence angle of the incoherent light beam needs to be 45° ± 1° range. The divergence angle of the light beam should be smaller than 0.5°. The acceptable deviation from the size of the facet must be more accurate than 0.5 mm. The choice of aluminium as material for the facet is optimal for visible spectrum, but the total reflectivity is smaller than 40 %. This last value can be probably improved by changing the optimization process employing as constraints the total reflectivity for a future goal and not the number of illuminated pixels. The tolerance of the angular position of the facet is about 0.1 degree. However, this optical component has advantages comparing diffractive optics or other type of solution for beam shaping where tolerances of fabrication are sometimes harder and difficult to obtain. The influence of the wavelength is low and only related to the Fresnel losses if the quality of the surface is good. The flexibility of the design and the optimization process prove that this component can replace other type of solution in this field of incoherent light technology or nonimaging optics.
Acknowledgments
This article is supported by Natural Science Foundation of Shandong Province, China (No. ZR2019BF033).
[1] M.A. Moiseev, S.V. Kravchenko, and L.L. Doskolovich, “Design of efficient LED optics with two free-form surfaces,” Opt. Express 22, A1926 (2014).
[2] R. Wu, H. Li, Z. Zheng, and X. Liu, “Freeform lens arrays for off-axis illumination in an optical lithography system,” Appl. Opt. 50, 725 (2011).
[3] R. Wu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “A mathematical model of the single freeform surface design for collimated beam shaping,” Opt. Express 21, 20974 (2013).
[4] L. Liu, T. Engel, M. Flury, et al. “Faceted structure:a design for desired illumination and manufacture using 3d printing,”. OSA Continuum, 1(1), 26-39 (2018).
[5] C.Y. Tsai, “Optimized solar thermal concentrator system based on free-form trough reflector,” Solar Energy 125, 146 (2016).
[6] J. Liang, R.N. Kohn Jr, M.F. Becker, and D.J. Heinzen, “High-precision beam shaper for coherent and incoherent light using a DLP spatial light modulator,” Proc. SPIE 7932, 793208 (2011).
[7] J. Liang, R. N. Kohn Jr, M. F. Becker, and D. J. Heinzen, “Evaluation of DMD-based high-precision beam shaper using sinusoidal-flattop beam profile generation,” Proc. SPIE 8130, 81300C (2011).
[8] J. Liang, R.N. Kohn Jr, M.F. Becker, and D. J. Heinzen, “Homogeneous one-dimensional optical lattice generation using a digital micromirror device-based high-precision beam shaper,” J. Micro/Nanolith. MEMS MOEMS. 11, 023002 (2012).
[9] T. Weyrich, P. Peers, W. Matusik, and S. Rusinkiewicz, “Fabricating microgeometry for custom surface reflectance,” ACM Transactions on Graphics (Proc. SIGGRAPH) 28, 321 (2009).
[10] R. Leutz, and A. Suzuki, in Nonimaging Fresnel Lenses (Springer-Verlag Berlin Heidelberg, 2001), p. 77.
[11] F. Fournier, W.J. Cassarly, and J.P. Rolland, “Method to improve spatial uniformity with lightpipes,” Opt. Lett. 33, 1165 (2008).
[12] L. Liu, T. Engel, and M. Flury, “Simulation and optimization of faceted structure for illumination,” Proc. SPIE 9889, 98891A (2016).
[13] B.K. Jennison, J.P. Allebach, and D.W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 286629 (1989).
[14] M. Flury, P. Gérard, Y. Takakura, P. Twardworski, and J. Fontaine, “Investigation of M2 factor influence for paraxial computer generated hologram reconstruction using a statistical method,” Opt. Commun. 248, 347 (2005).
[15] N. Bokor, and Z. Papp, “Monte Carlo method in computer holography,” Opt. Eng. 36, 1014 (1997).
[16] N. Bokor, and Z. Papp, “Computational method for testing computer generated holograms,” Opt. Eng. 35, 2810 (1996).
[17] A.D. Rakić, A.B. Djurišić, J.M. Elazar, and M.L. Majewski, “Optical properties of metallic films for vertical cavity optoelectronic devices,” Appl. Opt. 37, 5271 (1998).
Photography & Biography
Lihong Liu is an assistant professor at Institute of Microelectronics, Chinese Academy of Sciences. Her research is majored in optical system design and modelling for nano-optics in Lithography.
Thierry Engel is an researcher in Team IPP - Photonics Instrumentation and Processes,ICube laboratory, University of Strasbourg, France. Thierry is working on laser micro-texturing, 3D metallic additive construction.
Huwen Ding is a PhD student at Institute of Microelectronics, University of Chinese Academy of Sciences. He is working on the design and modeling of nanolithography.
Yuanzhi Cui is an undergraduate student at North China University of Technology. He is working on the design and modeling of nanolithography based on surface plasma.
Jing Zhang is a professor at North China University of Technology. She is working on the design and fabrication of microelectronics.
Yayi Wei is a professor at Institute of Microelectronics, Chinese Academy of Sciences, and University of Chinese Academy of Sciences. Professor Wei has long been engaged in the research and development of semiconductor devices, materials and processes in the semiconductor lithography field.
Manuel Flury is an associate professor and is the co-leader of Team IPP - Photonics Instrumentation and Processes ICube laboratory, University of Strasbourg, France. Manuel is working on Micro and nano-optique, Microscopic and nanoscopic multimode, Laser micro-processing.
Article and author information
Lihong Liu
Thierry Engel
Huwen Ding
Yuanzhi Cui
Jing Zhang
Yayi Wei
Manuel Flury
Publication records
Published: Dec. 31, 2021 (Versions1
References
Journal of Microelectronic Manufacturing