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Design and collaborative optimization method of multilayer coatings to correct polarization aberration of optical systems

Wentao Jia, Wenjun He, Yahong Li, Yuegang Fu, and Lei Zhang

Open Access Open Access

Abstract

Multilayer coatings induce a significant polarization aberration in optical systems with high numerical aperture (NA) and wide field of view, which will cause wavefront distortion and imaging degradation. Studies have used low-polarization coatings (LPC) design to reduce the coating-induced polarization aberration. However, the polarization aberration caused by LPC remain evident in systems with large incident angles and many coated surfaces. In this paper, a hybrid optimization algorithm (HOA) is proposed to enhance the design accuracy of LPC. Based on the HOA, a collaborative optimization method is developed to simultaneously design coatings with different polarization properties for multiple surfaces, which can correct polarization aberration by mutual compensation between the coated surfaces and other optical elements in a single system. Finally, a high NA lithographic lens is simulated as an example to verify the collaborative optimization method. The simulation demonstrates that this method is superior to conventional methods. This research provides a new way to correct the polarization aberration and is applicable to any systems coated with multilayer coatings.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

As the demands for optical imaging quality have continued to increase, many optical systems have become increasingly sensitive to polarization, especially for those with high numerical aperture (NA), large incident angle and non-paraxial application [1–3]. Polarization aberration, defined as variations of amplitude, phase and polarization associated with ray paths through optical systems, is produced by the intrinsic polarization effects of optical elements [4,5]. For the most imaging systems, polarization aberration will cause wavefront distortion and image placement error, such as high NA lithography lens and microscopy objectives [6–8]. In addition, measurement accuracy for polarization states is affected by the polarization aberration in applications such as polarimetry, polarization sensing and coherent laser communications [9–11].

A multilayer coating, as a key optical element, is usually used to control the light transmission in optical systems [12]. Whereas, optical admittance of s- and p-polarized light are not generally equal at non-normal incident in coatings systems, which introduces phase difference and amplitude separation between them [13]. This polarization variation directly affects wavefront aberration and point-spread function for the majority of optical systems [13–15]. Especially in the case of large incident angle, the polarization aberration introduced by coatings has become the dominant factor affecting the system performance [15,16]. In recent years, some researchers have reduced the coating-induced polarization aberration by designing coatings with low-polarization effect [17–19]. The polarization aberration of Terrestrial Planet Finder Coronagraph is compensated by changing the thickness of a dielectric layer on the metal surface [20]. Low diattenuation cancellation coatings is designed to reduce the polarization aberration in a three-mirror off-axis camera [21]. However, the low-polarization coatings (LPC) still exhibit residual polarization, and this polarization increases as the incident angle increases. In the context of the system with large number of coated surfaces, a significant polarization aberration will be accumulated by the LPC. Therefore, the LPC has limitations in regards to correcting the polarization aberration.

Interesting studies have been conducted on correcting polarization aberrations through mutual compensation of polarization effects between certain elements. For example, the polarization aberration of crossed fold mirror was balanced by orienting the p-polarized light of the second mirror with the s-polarized light of the first mirror [22]. The polarization aberration caused by the birefringence effect in optical lithography lens was corrected by rotating crystal orientation of calcium fluoride lens [23,24]. Inspired by these studies, we attempted to correct polarization aberrations by polarization compensation between the coated interfaces. To achieve a better correction effect, we designed coatings with different polarization properties for different interfaces of an optical system potentially consisting of multiple optical elements and materials. However, it is a very complicated problem to design and evaluate multiple coatings systems. Therefore, we firstly study the optimization strategy of multilayer coatings and combine two intelligent algorithms to enhance LPC design accuracy. Then, based on this hybrid optimization mechanism, we developed a collaborative optimization method of multiple coatings systems to automatically correct the polarization aberration of system.

The remainder of this paper is organized as follows. In Section 2, a hybrid optimization algorithm is provided to design the LPC. In Section 3, the collaborative optimization method of multilayer coatings is proposed and the optimization process of polarization aberration is introduced in detail. In Section 4, the LPC is designed for simulation system, and the polarization aberration of the system is optimized twice using collaborative optimization method. Finally, our conclusions are summarized in Section 5.

2. Hybrid optimization algorithm of low-polarization coatings

Genetic algorithm (GA) [25] and particle swarm optimization (PSO) [26] are two common intelligent optimization algorithms, which are stochastic global search method to imitate natural evolution and biological behaviors. Recently, they have been successfully applied to design various types of multilayer coatings, such as anti-reflection (AR) coatings, high-reflection (HR) coatings and beam splitter [27–29]. These coatings design only consider the transmittance and spectral characteristics. In the case of LPC, the amplitude and phase difference between s- and p-polarized light should also be considered in the design process. The LPC design is more complex than other types of coatings design. In general, hybrid intelligent algorithms outperform single algorithms in terms of solving complex problems. Therefore, we propose a hybrid optimization algorithm (HOA) to improve the optimization accuracy of LPC. The HOA combines global optimization performances of GA and PSO, and introduce K-means algorithm [30] to enhance local search ability. The following section introduces the HOA infrastructure for optimizing the LPC.

In the beginning of optimization, initial structural parameters of multilayer coatings should be determined. The number of layers, wavelength range, incident angle interval, refractive index of materials and sequence of materials are set to appropriate values, and they are fixed throughout the optimization program. The thickness parameter of each layer is defined as design variable during the optimization process and is optimized using the HOA.

Figure 1 depicts the optimization process of HOA. Firstly, the 2N individuals are generated in the initial population. Suppose there are M layers in the coating stack, each individual is encoded into M random floating-point numbers, which indicate the potential solution to thickness variables. Based on these structural parameters, the fitness value is calculated to assess the optical performance of each individual. The best N individuals are selected by fitness as optimized population. Then, we use parallel optimization strategy to evolve population based on the GA and PSO. As shown in Fig. 1, the PSO updates N individuals according to the best individual in the previous generations and the best individual of the whole population. Meanwhile, GA evolve the same population through crossover and mutation operations, which produces N offspring. To improve the local search ability, the newly generated 2N individuals are reassigned using the K-means method. The N individuals whose Euclidean distances are closest to the global optimal solution in the M dimensional space are selected as the elite population, and proceed to the next-generation. Until the end of the genetic algebra, the best individual of each generation is compared to select the appropriate coating structure. The optimization process of HOA is described in Table 1 in the form of pseudo-code.

 figure: Fig. 1.

Fig. 1. The framework of the HOA to optimize LPC.

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Tables Icon

Table 1. Pseudo-code of HOA flow.

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3. Collaborative optimization method to correct polarization aberration

To avoid the polarization accumulation caused by the LPC, we propose correcting the coating-induced polarization aberration by using polarization compensation between the coated interfaces. Designing reasonable coating structures for different surfaces of an optical system potentially consisting of multiple optical elements and materials is essential for achieving this polarization compensation. In this section, we firstly develop a collaborative optimization method to simultaneously design multilayer coatings for multiple surfaces. Then, based on the polarization aberration function of the system, a polarization-dependent merit function (PDMF) is established to evaluate the design effect of collaborative optimization method. Finally, we introduce in detail the workflow of the polarization aberration correction procedure.

3.1 Collaborative optimization method of multilayer coatings

To obtain a better polarization compensation effect between different coated surfaces, it is necessary to optimize the polarization characteristics of coatings on multiple surfaces. As the HOA is the optimization process for a single multilayer coating (as described in Section 2), we first optimize the coatings on different surfaces separately. As shown in Fig. 2(a), two coatings stacks (P1 and P2) are optimized as examples. The color bar of P1 and P2 represents two evolved population, where the colored box represents an individual. Based on the HOA, they generate new population ($P_1^{\prime}$ and $P_2^{\prime}$) in each generation separately. In order to evaluate the effect of coating optimization on the polarization characteristics of the entire system, we use the polarization aberration function of the system as the evaluation criterion. We select an individual from each population to calculate the PDMF of system. After matching individuals in two populations one by one, each generation needs to calculate N×N times for PDMF. Assuming that m films stacks are optimized, m populations are evolved and evaluated Nm times for each generation. Therefore, the data calculation of this optimization strategy increases exponentially as the population increase, and the direction of optimization between populations is no correlation, which leads to a slow convergence rate.

 figure: Fig. 2.

Fig. 2. Collaborative optimization strategy for multiple coatings stacks, taking two coatings stacks as example: (a) Two populations generate N new individuals separately, and evaluate N×N fitness value after one-to-one matching; (b) Two populations are combined into a new population, which establishes an evolutionary link between them. N fitness value are evaluated after evolution. Then, the best individuals are selected.

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To accelerate the search efficiency, a collaborative optimization strategy is constructed to realize the co-evolution between the different coating stacks, as shown in Fig. 2(b). Firstly, the two evolved populations are merged, that is, the N pairs of short-stranded individuals are combined into the N long-stranded individuals of population P. At this point, the number of layers of the long-stranded individual is the sum of those of the two stacks. Then, a new individual is obtained after the evolution of the long-stranded individuals using the HOA, thereby realizing the co-evolution between the different populations. Finally, the individual of P’ is split into two coatings stacks to calculate the PDMF of system, and the best individual combination is selected in each generation. In the evolution of each generation, this method only optimized once and calculates N values of the PDMF, and these data are independent of the number of coatings stacks. To accelerate the convergence of the optimal solution, individuals in each population are initially designed prior to population merging operations. Based on the HOA, some individuals with high transmittance and low polarization properties are generated in each population. Individuals are sorted by fitness value to form populations P1 and P2 in Fig. 2(b). This provides better initial conditions for satisfying the basic performance requirements of the system, and reduces the search range for the collaborative optimization algorithm. Contrast with the strategy in Fig. 2(a), the collaborative optimization method establishes co-evolution mechanism for multiple population, which performs the same operation on individuals in different populations. Meanwhile, this method simplifies the amount of data and improve search efficiency. Therefore, the collaborative optimization method is used to design multilayer coatings for polarization aberration correction in this study.

3.2 Polarization-dependent merit function model

The Jones pupil represents the polarization properties of a system, and can be used to construct the PDMF for the collaborative optimization method. Jones pupil is precisely calculated by the three-dimensional (3D) polarization ray-tracing matrix method [31–33], which is equal to the cascade of the polarization ray-tracing matrix Jq of each optical surface.

$${J_{pupil}} = \prod\limits_{q = n, - 1}^1 {{J_q}} = {J_n} \cdot {J_{n - 1}} \cdot{\cdot} \cdot {J_2} \cdot {J_1}, $$

For the dielectric, metal and coated interfaces, the 3D polarization ray-tracing matrix can be expressed as

$${J_q} = {O_{out,q}} \cdot {W_q} \cdot O_{in,q}^{ - 1} = \left( {\begin{array}{ccc} {{{\vec{s}}_{x,q}}}&{\vec{p}{^{\prime}_{x,q}}}&{{{\vec{k}}_{x,q}}}\\ {{{\vec{s}}_{y,q}}}&{\vec{p}{^{\prime}_{y,q}}}&{{{\vec{k}}_{y,q}}}\\ {{{\vec{s}}_{z,q}}}&{\vec{p}{^{\prime}_{z,q}}}&{{{\vec{k}}_{z,q}}} \end{array}} \right) \cdot \left( {\begin{array}{ccc} {{\alpha_{s,q}}}&0&0\\ 0&{{\alpha_{p,q}}}&0\\ 0&0&1 \end{array}} \right) \cdot \left( {\begin{array}{ccc} {{{\vec{s}}_{x,q}}}&{{{\vec{s}}_{y,q}}}&{{{\vec{s}}_{z,q}}}\\ {{{\vec{p}}_{x,q}}}&{{{\vec{p}}_{y,q}}}&{{{\vec{p}}_{z,q}}}\\ {{{\vec{k}}_{x,q - 1}}}&{{{\vec{k}}_{y,q - 1}}}&{{{\vec{k}}_{z,q - 1}}} \end{array}} \right), $$
where ${O_{out,q}}$ and ${O_{in,q}}$ denote the orthonormal basis matrix of the emergent and incident light on the optical surface, which consists of s-, p-light vector and wave vector. Wq represent the Jones matrix in local coordinates, ${\alpha _{s,q}}$ and ${\alpha _{p,q}}$ are the transmittance or reflection coefficients of s- and p-polarized light at qth optical surface. The singular value decomposition of the Jones pupil can decompose a series of basic physical meanings,
$${J_{pupil}} = t{e^{i\phi }}{J_{dia}} \cdot {J_{ret}}, $$
where t is transmittance and $\phi$ is scalar phase. Jdia and Jret represent the diattenuation and retardance, respectively. These two parameters describe the ability of the system to modulate the polarization state, and are important indicators to assess the impact of the polarization aberration on image formation. Thus, the PDMF of the collaborative optimization method is defined as
$$MF = \left[ {\frac{{{y_d}}}{{mn}}\sum\limits_{i = 1}^m {{{\sum\limits_{j = 1}^n {\left( {\frac{{D({i,j} )- \tilde{D}({i,j} )}}{{\Delta D({i,j} )}}} \right)} }^2}} } \right. + {\left. {\frac{{{y_r}}}{{mn}}\sum\limits_{i = 1}^m {{{\sum\limits_{j = 1}^n {\left( {\frac{{R({i,j} )- \tilde{R}({i,j} )}}{{\Delta R({i,j} )}}} \right)} }^2}} } \right]^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}} \right.}\!\lower0.7ex\hbox{$2$}}}}, $$
where D and R are the value of diattenuation and retardance, respectively. $\tilde{D}$ and $\tilde{R}$ are target values, ΔD and ΔR are specified tolerance, yd and yr denote weight. Index n and m mark the ray grid points in the x- and y-axis directions on the exit pupil plane. In addition, the transmission characteristics of the coatings must be satisfied during the optimization process. Thus, we add the constraint to the PDMF,
$$\sqrt {\frac{4}{{\pi mn}}\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {t({i,j} )} } } \ge \widetilde t, $$
where $\tilde{t}$ is the target transmittance on the exit pupil. When the optimized individual satisfies the constraints, the PDMF is calculated.

3.3 Polarization aberration correction workflow

In this section, we introduce the polarization aberration correction process based on the collaborative optimization method. As multilayer coatings are very thin, the thickness of the coatings is usually ignored in the ray-tracing procedure. Therefore, the coatings only affect the light transmission of the interface, and does not change the ray-tracing path. When the coatings structure on the surface change, the polarization tracing matrix in Eq. (2) only update the parameters ${\alpha _{s,q}}$ and ${\alpha _{p,q}}$ to quickly calculate the Jones pupil. Based on the collaborative optimization method, an automatic optimization program for polarization aberration correction is implemented in this study using MATLAB code.

The optimization flowchart is shown in Fig. 3. First, polarization ray-tracing data are exported from Code V, including the incident angle and s- and p-light vectors of each ray path. These data are used to construct the polarization ray-tracing matrix in Eq. (2). Next step, the collaborative optimization method optimizes the coatings structures for multiple surfaces based on the MATLAB code. After updating the coatings on the interfaces, the transmission coefficient of each ray is calculated according to the incident angle. The optimized individuals satisfying the transmittance requirements in Eq. (5) are selected to calculate the PDMF, and the other individuals continue to be optimized in the next generation. The minimum PDMF value and the best individual are recorded in each generation. The polarization aberration is calculated until the end of the genetic algebra, and the best coating structure is selected based on the global minimum PDMF.

 figure: Fig. 3.

Fig. 3. The optimization flowchart for the polarization aberration correction of optical system.

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4. Example

As described this section, an immersion lithographic lens with a NA of 1.2 was used for a simulation of correcting a polarization aberration. Figure 4 shows the layout of the optical system. The height of object surface is 66 mm and the illumination wavelength is 193.37 nm. The material of the last lens is calcium fluoride crystal and other lens material is fused silica. As described below, we design an LPC for the simulation system. Then, the polarization aberration of the entire system is corrected using the collaborative optimization method.

 figure: Fig. 4.

Fig. 4. The layout of a NA 1.2 lithographic lens in the simulation.

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4.1 LPC design

The LPC design of the lens in Fig. 4 needs to meet the requirements of high transmittance and low-polarization effects in the incident angle range. The edge of the field of view has a large range of incident angle in the system. As shown in Fig. 5, the incidence interval for all interfaces is concentrated in the range of 0–60°. Due to the small incident angle of two reflecting surfaces, the AR coatings with low-polarization effects is designed in this interval.

 figure: Fig. 5.

Fig. 5. The incident angle ranges of each interface of the lithographic lens in the case of edge of the field of view.

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Owing to the small number of materials in the deep-ultraviolet band, the refractive index difference is small. Two dielectric materials with relatively large refractive index differences are chosen: HfO2 and SiO2. Six-layer are set as optimization variables. After determining the coatings parameters, the optimization parameters of HOA in Table 1 are set as Table 2. Then, the AR coatings is designed to use GA, PSO and HOA respectively. And their design results are compared under the same genetic algebra.

Tables Icon

Table 2. Initial parameters of HOA.

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Figure 6(a) shows the polarization characteristic curves of AR coatings designed by three algorithms. These coating structures all have high transmittance in the incident interval, where the curves of the s and p light as designed by the HOA change smoothly, and exhibit the smallest separation. The phase difference of s and p light increases with the incidence angle and this polarization effect cannot be eliminated by the coatings design. Figure 6(b) shows the convergence curve of fitness value in Table 1 during the optimization process. The convergence speed of the HOA is better than those of the other two algorithms. Both the GA and PSO quickly fall into local optimal solutions and stop evolving, while the HOA continues to evolve the data as the genetic algebra increases. Therefore, the HOA has high accuracy for the design of LPCs. The simulation system also has two reflective surfaces and the design results of HR coatings based on the HOA are shown in Fig. 7. From the above design results of LPC, it can be shown that the HOA is more suitable for optimizing the polarization effects and transmission characteristics of the coatings. Therefore, the HOA is applied to the collaborative optimization method to improve the optimization efficiency.

 figure: Fig. 6.

Fig. 6. The AR coatings is designed by GA, PSO and HOA method: (a) Transmittance and phase difference of s- and p-polarized light of AR coatings; (b) The converged curves for fitness value.

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 figure: Fig. 7.

Fig. 7. The reflectivity and phase difference of s- and p-polarized light of HR coatings based on HOA design.

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4.2 Polarization aberration correction

Based on the polarization aberration caused by the LPC, a collaborative optimization method is used to further correct the polarization aberration at the edge of the field of view. Because of the large number of surfaces in the system, we only select coating structures on a portion of the surfaces for optimization to reduce the design variables. As shown in Fig. 6(a), the polarization effect of coatings designed by the HOA increases significantly above 20°. Therefore, we choose the surfaces where the root mean square (RMS) values of incident angle are greater than 20° in Fig. 5. Surfaces 3, 5, 8, 9, 11, 13, 15, 17, 25, 31, 32, 33, 39, 46 and 47 are selected to redesign coatings, and other surfaces are coated with LPC. In general, the coating design must consider the design requirements over the entire range of incident angles. Smaller incident angle intervals make it easier to design coatings with high transmittance. As shown in Fig. 5, the incident angle intervals of most surfaces is smaller than the design range of LPC. If we design AR coatings for a single surface, it may have a higher transmittance than the LPC. In the following, we use the collaborative optimization method to redesign the coatings on the selected optical surfaces. And the transmittance and polarization aberration of the system coated with LPC are considered as initial reference values. It is possible to improve the transmittance of the system when the coatings on a part of the optical surface are redesigned. Therefore, we can divide the constraints in Eq. (5) into two cases, i.e., above or below the initial transmittance value. We simulate the effects of two different constraints on the polarization aberration optimization, and the results are discussed below.

When the simulated system is uncoated, Figs. 8(a1)-8(d1) show the distribution map of four physical properties in Eq. (3) in the exit pupil. The retardance map in Fig. 8(c1) is caused by birefringence effect of calcium fluoride lens in the system. Whereas, the system has significant diattenuation aberration, and its RMS is as high as 50.676%. As shown in Fig. 8(b1), the RMS of diattenuation drops to 12.454% when the system coated with LPC. This shows that the design of LPC can effectively to control the diattenuation aberration. However, a retardance aberration is accumulated by the LPC as shown in Fig. 8(c1), indicating that the LPC has a weak ability to adjust the retardance aberration. Thus, the retardance aberration is taken as the main optimization target in the PDMF. We set the parameters yd and yr to 0.3 and 0.7 in Eq. (4), respectively.

 figure: Fig. 8.

Fig. 8. The transmission map, diattenuation map, retardance map and scalar phase map of the lithographic lens: (a1) - (d1) No coatings; (a2) - (d2) Coated with LPC; (a3) - (d3) First optimization results for improving transmittance; (a4) - (d4) Second optimization results for reducing transmittance.

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A 75.187% transmittance of the system coated with LPC is used as a reference value for constraints. For the first optimization, the target transmittance is higher than 1% of reference value. The optimization results are provided in Fig. 8(a3)-8(d3). The RMA values of these physical pupil maps are shown in Table 3. Compared with the data of the LPC, the transmittance of the first optimization is increased by 1.08%. The RMS values of the diattenuation map and retardance map of the first optimization are 4.35% and 31.99% less than those of the system coated with the LPC, respectively. This simulation result shows that the collaborative optimization method can effectively correct the polarization aberration and also improve the transmittance of system. For the second optimization, the transmittance target not less than 3% of the reference value. As shown in Fig. 8(a4)-8(d4), the polarization aberration is further corrected relative to the first optimization results. At this time, the retardance aberration in Fig. 8(c4) is smaller than that produced by the birefringent crystal in Fig. 8(c1). This indicates the collaborative optimization method can not only reduce the coating-induced polarization aberration, but also compensate for the polarization effects of other components. Meanwhile, it is shown that the loss of transmittance is more conducive to the polarization aberration correction. The two simulation results from the collaborative optimization method are clearly better than those of the system coated with the LPC, and the collaborative optimization method can realize a simultaneous design for the polarization and apodization aberrations of the system.

Tables Icon

Table 3. The RMS values of transmission, diattenuation, retardance and scalar phase in Fig. 8.

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4.3 Tolerance analysis

Tolerance analysis is an important step in coatings design. The thickness error of each layer in coatings process will change the transmittance and polarization effect of the system. Currently, magnetron sputtering technology can control the layer thickness error to within 1 nm. Within this variation interval, we tested 1000 random samples for the simulation system. As shown in Fig. 9, the differences between the sample and simulation values in Table 3 conform to a normal distribution law. An interesting phenomenon in Fig. 9(a)–9(b) is the negative value of the difference of diattenuation and retardance for some samples, which indicate that the error of layer thickness could potentially compensate for the polarization aberration. However, the thickness error of layers reduces the transmittance of the system in Fig. 9(c). Therefore, the influence of the tolerance value on the transmittance should be considered in the coating design.

 figure: Fig. 9.

Fig. 9. The variation in diattenuation, retardance and transmittance.

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5. Conclusion

In this work, a hybrid optimization algorithm is first applied to enhance the design accuracy of an LPC. Based on this optimization strategy, a collaborative optimization method is proposed to simultaneously design coatings with different polarization properties for different surfaces. This method utilizes the polarization difference between coated surfaces to balance coating-induced polarization aberration, and can also compensate for polarization effects of other components. Subsequently, a high-NA lithographic lens is simulated to verify the collaborative optimization method. The simulation shows that the collaborative optimization method can further degrade the polarization aberration caused by the LPC and has better correction effect when reducing transmittance requirement. In particular, the collaborative optimization method has a great ability to modulate the retardance aberration of the system. In conclusion, this research can realize the automatic correction of the polarization aberration in optical systems with multilayer coatings, and is also applicable to the optimization of apodization aberration. Additionally, this paper provides a theoretical foundation for the integration of lenses and coating designs.

Funding

National Natural Science Foundation of China (61805025).

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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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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Figures (9)
Fig. 1.
Fig. 1. The framework of the HOA to optimize LPC.

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Fig. 2.
Fig. 2. Collaborative optimization strategy for multiple coatings stacks, taking two coatings stacks as example: (a) Two populations generate N new individuals separately, and evaluate N×N fitness value after one-to-one matching; (b) Two populations are combined into a new population, which establishes an evolutionary link between them. N fitness value are evaluated after evolution. Then, the best individuals are selected.

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Fig. 3.
Fig. 3. The optimization flowchart for the polarization aberration correction of optical system.

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Fig. 4.
Fig. 4. The layout of a NA 1.2 lithographic lens in the simulation.

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Fig. 5.
Fig. 5. The incident angle ranges of each interface of the lithographic lens in the case of edge of the field of view.

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Fig. 6.
Fig. 6. The AR coatings is designed by GA, PSO and HOA method: (a) Transmittance and phase difference of s- and p-polarized light of AR coatings; (b) The converged curves for fitness value.

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Fig. 7.
Fig. 7. The reflectivity and phase difference of s- and p-polarized light of HR coatings based on HOA design.

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Fig. 8.
Fig. 8. The transmission map, diattenuation map, retardance map and scalar phase map of the lithographic lens: (a1) - (d1) No coatings; (a2) - (d2) Coated with LPC; (a3) - (d3) First optimization results for improving transmittance; (a4) - (d4) Second optimization results for reducing transmittance.

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Fig. 9.
Fig. 9. The variation in diattenuation, retardance and transmittance.

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Tables (3)

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Table 1. Pseudo-code of HOA flow.

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Table 2. Initial parameters of HOA.

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Table 3. The RMS values of transmission, diattenuation, retardance and scalar phase in Fig. 8.

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Equations (5)

Equations on this page are rendered with MathJax. Learn more.

J p u p i l = ∏ q = n , − 1 1 J q = J n ⋅ J n − 1 ⋅ ⋅ ⋅ J 2 ⋅ J 1 ,
J q = O o u t , q ⋅ W q ⋅ O i n , q − 1 = ( s → x , q p → x , q ′ k → x , q s → y , q p → y , q ′ k → y , q s → z , q p → z , q ′ k → z , q ) ⋅ ( α s , q 0 0 0 α p , q 0 0 0 1 ) ⋅ ( s → x , q s → y , q s → z , q p → x , q p → y , q p → z , q k → x , q − 1 k → y , q − 1 k → z , q − 1 ) ,
J p u p i l = t e i ϕ J d i a ⋅ J r e t ,
M F = [ y d m n ∑ i = 1 m ∑ j = 1 n ( D ( i , j ) − D ~ ( i , j ) Δ D ( i , j ) ) 2 + y r m n ∑ i = 1 m ∑ j = 1 n ( R ( i , j ) − R ~ ( i , j ) Δ R ( i , j ) ) 2 ] 1 / 1 2 2 ,
4 π m n ∑ i = 1 m ∑ j = 1 n t ( i , j ) ≥ t ~ ,
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