【文献精读】Memetic算法与混合整数规划优化多层光学薄膜

一、文献核心概览 (Literature Core Overview)

1.1 基本信息 (Basic Information)

1.2 核心结论 (Core Conclusions)

1. 算法创新: 提出了一种基于混合整数规划(MIP)的Memetic算法实现，将材料选择（离散变量）与厚度优化（连续变量）相结合，特别适合宽带薄膜的实际优化问题。

2. 材料色散处理: 与传统方法将折射率作为连续变量不同，该方法直接从预选定材料列表中进行选择，能够自然处理材料在宽带范围内的色散特性。

3. 性能提升: 在辐射制冷器优化中，使用6层结构实现了比文献报道的7层结构更好的性能（制冷功率45.7 vs 38.0 W/m²，温降23.4K vs 19.5K）。

4. 层数减少: 在白炽灯滤光片优化中，使用41层结构超越了90层文献结构的性能（发射增强因子14.8 vs 12.8），层数减少超过50%。

5. 材料多样性价值: 通过使用更多种类的材料（7种 vs 2种），算法能够在减少层数的同时显著提升器件性能。

1.3 核心价值 (Core Value)

1.4 研究方法 (Research Methods)

Memetic算法核心流程（5步迭代）:

1. 种群生成: 随机生成N个K层结构作为初始种群
2. 交叉操作: 模拟染色体交换，将父代结构的上/下部分组合产生子代
3. 变异操作: 以概率μ随机改变某层的材料或厚度，保持种群多样性
4. 种群重选: 按适应度排序，以比例r保留优秀个体，(1-r)保留较差个体以维持探索性
5. 精英优化: 每G代对前P个最优个体使用拟牛顿法进行局部厚度优化

收敛判据: 当种群平均merit function与最小值的差异小于2%时判定收敛。

评价函数(Merit Function): $$F(\tilde{n}, d) = \sum_{\lambda,\theta} W(\lambda)\left(R(\theta, \lambda; \tilde{n}, d) - R^*(\theta, \lambda)\right)^2$$

二、全文双语对照 (Full Bilingual Text)

Abstract 摘要

English: Multilayer optical films have been extensively used in optical technology, but the design of multilayer structures for broadband applications is often challenging due to the need to incorporate material dispersion. Here, we present an implementation of a memetic algorithm based on mixed integer programming, which is especially suited for practical broadband optimization of layered thin-film materials. In our implementation, the optimization variables consist of a list of discrete variables that represents different dielectric materials, along with a list of continuous variables that represents the thicknesses of each material. As a set of concrete demonstrations, we optimize the spectra of a radiative cooling device and an incandescent light bulb filter. The resulting structures from the optimization can, by using more materials, achieve better performance than their counterparts in the literature while using fewer numbers of material layers.

中文: 多层光学薄膜在光学技术中得到了广泛应用，但用于宽带应用的多层结构设计往往具有挑战性，因为需要考虑材料色散。本文提出了一种基于混合整数规划的Memetic算法实现，特别适合分层薄膜材料的实际宽带优化。在我们的实现中，优化变量包括代表不同介电材料的离散变量列表，以及代表每种材料厚度的连续变量列表。作为一系列具体演示，我们优化了辐射制冷器件和白炽灯滤光片的光谱。优化得到的结构通过使用更多种类的材料，能够在使用更少材料层数的同时实现比文献中同类结构更好的性能。

Introduction 引言

Paragraph 1:

Multilayer optical thin films have been widely used to achieve specific optical transmission and reflection spectra for applications in many important areas of optical technology in the last century.

多层光学薄膜在上个世纪已被广泛用于实现特定的光学透射和反射光谱，应用于光学技术的许多重要领域。

In recent years, there has also been significant interest in designing multilayer optical structures for energy and thermal applications.

近年来，设计用于能源和热应用的多层光学结构也引起了极大关注。

Notable examples include efforts to improve the efficiency of solar and thermophotovoltaic cells, to achieve passive radiative cooling, to attain broadband spectral filtering, and to enhance the efficiency of thermal and optical emitters.

著名的例子包括提高太阳能电池和热光伏电池效率的研究、实现被动辐射制冷、实现宽带光谱滤波以及增强热发射体和光发射体效率的工作。

A common characteristic for energy and thermal applications is the need to control optical properties over a broad range of wavelengths.

能源和热应用的一个共同特征是需要在宽波长范围内控制光学特性。

For example, in the passive radiative cooling application, one needs to simultaneously achieve particular spectral properties across both solar and thermal wavelength ranges.

例如，在被动辐射制冷应用中，需要同时在太阳和热波长范围内实现特定的光谱特性。

For such solar and energy applications, a systematic and robust multilayer design technique for broadband optical spectrum engineering is thus highly desirable.

因此，对于这种太阳能和能源应用，一种系统且稳健的多层宽带光谱工程设计技术是非常需要的。

Paragraph 2:

There exist many methods of designing multilayer structures.

存在许多设计多层结构的方法。

One can certainly rely on physical intuition, but doing so tends to explore only a tiny subspace of the entire allowed design phase space.

人们当然可以依靠物理直觉，但这样做往往只能探索整个允许设计相空间的极小一部分。

Therefore, a wide range of systematic approaches in designing optical multilayer structures have been explored.

因此，人们已经探索了多种设计光学多层结构的系统方法。

A few prominent examples include the jump method, spiral search method, particle swarm optimization, needle optimization, and evolutionary algorithms.

一些突出的例子包括跳跃方法、螺旋搜索方法、粒子群优化、针优化和进化算法。

When implementing these methods, the optimization parameters are usually chosen to be the thicknesses and refractive indices of each layer.

在实现这些方法时，优化参数通常选择为每层的厚度和折射率。

As it turns out, the choice of refractive indices as optimization parameters strongly limits the practical usefulness of these algorithms in applications that require optimization over broad bandwidths.

事实证明，将折射率作为优化参数的选择，强烈限制了这些算法在需要宽带优化的应用中的实用价值。

Paragraph 3:

In practice, one typically fabricates multilayer structures with a discrete set of materials.

在实践中，人们通常使用离散的材料集合来制造多层结构。

Over a broad bandwidth, the refractive indices of all materials vary as a function of wavelength.

在宽带范围内，所有材料的折射率都随波长变化。

In the usual implementations, incorporating such material dispersion is difficult since with material dispersion, a given material does not have a single corresponding refractive index.

在通常的实现中，纳入这种材料色散是困难的，因为由于材料色散的存在，给定材料没有单一的对应折射率。

As an indication of this difficulty, many previous works on optical multilayer design in fact assume nondispersive materials.

作为这种困难的一个表现，许多先前关于光学多层设计的工作实际上假设材料无色散。

In order to design a multilayer structure for broadband energy and thermal applications, it is essential to develop a practical algorithm that starts from a given list of materials with their realistic dispersions.

为了设计用于宽带能源和热应用的多层结构，开发一种从给定材料列表及其真实色散出发的实用算法是至关重要的。

Paragraph 4:

In this work, we propose and test an implementation of a memetic algorithm, the combination of an evolutionary algorithm and local optimization method, to design multilayer devices starting from a list of preselected materials.

在这项工作中，我们提出并测试了一种Memetic算法的实现，即进化算法与局部优化方法的结合，用于从预选材料列表出发设计多层器件。

In contrast to previous work in this space, we implement the optimization algorithm as a mixed-integer programming (MIP) problem, where the optimization variables consist of a list of discrete variables that represent different optical materials, along with a list of continuous variables that represent the thicknesses of each material.

与这一领域的先前工作不同，我们将优化算法实现为一个混合整数规划(MIP)问题，其中优化变量包括代表不同光学材料的离散变量列表，以及代表每种材料厚度的连续变量列表。

In the following sections, we present the details of this algorithm’s formulation, and as a demonstration, we apply the memetic algorithm to the optimization of a passive radiative cooling device and incandescent light bulb filter.

在接下来的章节中，我们介绍该算法公式的详细内容，并作为演示，将Memetic算法应用于被动辐射制冷器件和白炽灯滤光片的优化。

We show that the structures produced from this algorithm have better performance than those reported in literature while using fewer numbers of layers, demonstrating that our memetic algorithm is especially suitable for broadband thin-film optimization.

我们展示了该算法产生的结构在使用更少层数的同时比文献报道的结构具有更好的性能，证明了我们的Memetic算法特别适合宽带薄膜优化。

Memetic Algorithm Implementation Memetic算法实现

Paragraph 5:

To start, we outline how we implement the memetic algorithm for thin-film optimization.

首先，我们概述如何实现用于薄膜优化的Memetic算法。

First, we suppose that the incident and substrate media have refractive indices $n_{in}(\lambda)$ and $n_{out}(\lambda)$, respectively, where $\lambda$ is the wavelength.

首先，我们假设入射介质和基底介质的折射率分别为 $n_{in}(\lambda)$ 和 $n_{out}(\lambda)$，其中 $\lambda$ 是波长。

To design a multilayer structure, we specify a target reflection spectrum denoted as $R^*(\lambda, \theta)$, where $\theta$ is the incident angle.

为了设计多层结构，我们指定目标反射光谱为 $R^*(\lambda, \theta)$，其中 $\theta$ 是入射角。

Given a structure that consists of K-layers of materials with refractive indices $n(\lambda) = [n_1(\lambda), …, n_K(\lambda)]^T$ and thicknesses $d = [d_1, …, d_K]^T$, the reflection viewed from the incident medium, $R(\lambda, \theta; n, d)$, can be efficiently calculated using the impedance method as described in ref 28.

给定一个由K层材料组成的结构，其折射率为 $n(\lambda) = [n_1(\lambda), …, n_K(\lambda)]^T$，厚度为 $d = [d_1, …, d_K]^T$，从入射介质观察到的反射 $R(\lambda, \theta; n, d)$ 可以使用文献28中描述的阻抗方法高效计算。

To find an optimum thin film structure, we wish to minimize the residual between the reflection spectrum of a given structure and the target spectrum as described by the merit function $F(n, d)$.

为了找到最优薄膜结构，我们希望最小化给定结构的反射光谱与目标光谱之间的残差，如评价函数 $F(n, d)$ 所描述的。

Paragraph 6:

For broadband optimizations, where material refractive indices can vary greatly among wavelengths of interest, it is necessary to accurately account for material dispersion.

对于宽带优化，材料的折射率可能在感兴趣的波长范围内变化很大，因此有必要准确考虑材料色散。

In many practical situations, one chooses to fabricate a thin film structure using only a discrete set of materials.

在许多实际情况下，人们选择仅使用离散的材料集合来制造薄膜结构。

To account for this, instead of treating the list of refractive indices, $n(\lambda)$, as continuous variables, we treat the material composition as a list of integers that varies from 1 to M denoted as $\tilde{n}$.

为了考虑这一点，我们不将折射率列表 $n(\lambda)$ 视为连续变量，而是将材料组成视为从1到M变化的整数列表，记为 $\tilde{n}$。

Each of the entries in $\tilde{n}$ represents a material chosen from a list of M realistic materials.

$\tilde{n}$ 中的每个条目代表从M种真实材料列表中选择的一种材料。

Given a desired spectrum $R^*(\lambda, \theta)$, the optimization problem can be formulated as the following mixed-integer problem.

给定期望光谱 $R^*(\lambda, \theta)$，优化问题可以表述为以下混合整数问题。

Paragraph 7:

Having defined the optimization problem in eq 2, we now outline the use of the memetic algorithm in finding the optimal multilayer structure.

在定义了方程2中的优化问题后，我们现在概述如何使用Memetic算法来找到最优多层结构。

The memetic algorithm belongs to the class of evolutionary algorithms, which adapt the biological rule of survival-of-the-fittest to conduct optimization searches that are theoretically and empirically robust in complex spaces.

Memetic算法属于进化算法的一类，它适者生存的生物学规则来进行优化搜索，这在复杂空间中在理论和经验上都是稳健的。

In particular, evolutionary algorithms are not constrained by the domain of the input parameters, and therefore they are especially suited for mixed-integer optimization problems such as the one described in eq 2.

特别是，进化算法不受输入参数域的约束，因此特别适合于方程2所描述的混合整数优化问题。

Our implementation of the memetic evolutionary algorithm consists of the following operations.

我们的Memetic进化算法实现包括以下操作。

Paragraph 8 - Step 1: Generating a Population:

We first randomly generate N number of K-layered structures as a set that we call “population”: $P^{(0)} = {[\tilde{n}^{(1)},d^{(1)}], [\tilde{n}^{(2)},d^{(2)}], …, [\tilde{n}^{(N)},d^{(N)}]}$.

我们首先随机生成N个K层结构作为一个集合，我们称之为"种群"：$P^{(0)} = {[\tilde{n}^{(1)},d^{(1)}], [\tilde{n}^{(2)},d^{(2)}], …, [\tilde{n}^{(N)},d^{(N)}]}$。

Each structure $[\tilde{n}^{(i)},d^{(i)}]$ is called an “individual”.

每个结构 $[\tilde{n}^{(i)},d^{(i)}]$ 被称为一个"个体"。

The superscript in the population set $P$ denotes the “generation”, which is a counter variable that captures how many times the population has gone through a cycle of steps 2 through 5.

种群集合 $P$ 中的上标表示"代"，这是一个计数器变量，记录种群经历步骤2到5的循环次数。

Paragraph 9 - Step 2: Crossover:

We mimic the crossover operation to the way that chromosomes exchange genetic information.

我们模拟染色体交换遗传信息的方式来执行交叉操作。

We group individuals into random pairs ${[\tilde{n}^{(i)},d^{(i)}], [\tilde{n}^{(j)},d^{(j)}]}$, which are called “parents”.

我们将个体分组为随机对 ${[\tilde{n}^{(i)},d^{(i)}], [\tilde{n}^{(j)},d^{(j)}]}$，称为"父代"。

For each set of parents, we randomly select an integer k between 2 and K − 1, which denotes the delimiter that separates the parents into their upper sections and lower sections.

对于每组父代，我们随机选择2到K-1之间的整数k，它表示将父代分离为上部分和下部分的分隔符。

The upper section of one parent is combined with the lower section of the other, and vice versa, which produces two new individuals that are called “children”.

一个父代的上部分与另一个父代的下部分组合，反之亦然，产生两个称为"子代"的新个体。

Along with the parents, the population size grows to 2N.

连同父代一起，种群规模增长到2N。

Paragraph 10 - Step 3: Mutation:

In every generation, each individual has a small probability $\mu$ of experiencing mutation, where we randomly vary the material and thickness of a layer in an individual structure that is randomly chosen.

在每一代中，每个个体有很小的概率 $\mu$ 经历变异，我们在随机选择的个体结构中随机改变某一层的材料和厚度。

This is to mimic the mutation during biological evolution and allows for the exploration of other search regions.

这是为了模拟生物进化过程中的变异，并允许探索其他搜索区域。

Paragraph 11 - Step 4: Reselection of Population:

We first evaluate eq 1 for each individual and sort the individuals according to their merit functions.

我们首先为每个个体评估方程1，并根据其评价函数对个体进行排序。

We then reselect the next generation of population at a heuristic reselection rate r between 0.8 and 0.99, such that rN individuals are randomly drawn from the top 50th percentile of the current population, and (1 − r)N individuals are randomly drawn from the bottom 50th percentile of the current population.

然后我们以0.8到0.99之间的启发式重选率r重选下一代种群，其中rN个个体从当前种群的前50%中随机抽取，(1-r)N个个体从当前种群的后50%中随机抽取。

In doing so, the population size is kept at N at the end of each generation.

通过这样做，每一代结束时种群规模保持在N。

By selecting from both the fit and the unfit portions of the population, we ensure that the population remains sufficiently diverse, which helps in comprehensively exploring the search space.

通过从种群中的适应和不适部分都进行选择，我们确保种群保持足够的多样性，这有助于全面探索搜索空间。

Paragraph 12 - Step 5: Refinement of Elite Individuals:

This step is carried out for every G generations.

这一步每G代执行一次。

In this step, we use the quasi-Newton method to perform a local optimization on the thicknesses of the layers in the top P best individuals while keeping the material choice of each layer fixed.

在这一步中，我们使用拟牛顿法对前P个最优个体的层厚度进行局部优化，同时保持每层材料选择固定。

Such local optimization in combination with evolutionary algorithm is known to speed up convergence.

这种与进化算法结合的局部优化已知可以加速收敛。

On the other hand, the process of local optimization is much more computationally demanding than other steps, and therefore, we only perform this refinement to a subset of the population periodically.

另一方面，局部优化过程比其他步骤计算量大得多，因此我们仅定期对种群的一个子集执行此优化。

Paragraph 13 - Convergence:

Steps 2 through 5 are performed until convergence is reached, which we define as the case when the population’s average merit function is within 2% difference compared to the population’s minimum merit function.

重复执行步骤2到5直到达到收敛，我们将收敛定义为种群的平均评价函数与最小评价函数的差异在2%以内的情况。

When this occurs, the majority of the individuals will have identical properties, and their merit functions will have arrived at a local minimum that is robust to small fluctuations.

当这种情况发生时，大多数个体将具有相同的特性，它们的评价函数将到达一个对微小波动稳健的局部最小值。

Therefore, when the population becomes sufficiently uniform, there is negligible probability for further improvement of the population.

因此，当种群变得足够均匀时，进一步改进种群的概率可以忽略不计。

Paragraph 14 - Layer Removal:

It is important to note that when setting up the optimizations for a K-layer structure, it is possible for the algorithm to remove layers during the “crossover”, “mutation”, and “refinement of elite individuals” steps.

重要的是要注意，当为K层结构设置优化时，算法可能在"交叉"、“变异"和"精英个体优化"步骤中移除层。

This can happen when two identical materials are placed in adjacent layers, or when a particular layer’s thickness is set to zero.

当两个相同的材料放置在相邻层中，或当某一层的厚度被设为零时，这种情况可能发生。

Thus, the optimized multilayer structure will always have fewer than or equal to K layers, which complies with realistic fabrication constraints.

因此，优化的多层结构总是具有少于或等于K层，这符合实际的制造约束。

Optimization of a Radiative Cooling Device 辐射制冷器件优化

Paragraph 15:

We first demonstrate the application of the memetic algorithm to optimize a radiative cooling device that can operate under direct sunlight.

我们首先展示Memetic算法在优化可在直射阳光下运行的辐射制冷器件方面的应用。

Cooling is a major end-use of energy globally that drives the demand for electricity.

制冷是全球能源的主要终端用途，驱动着电力需求。

For instance, in the United States, powering air conditioning and refrigeration systems accounts for approximately 15 percent of energy used by buildings.

例如，在美国，为空调和制冷系统供电占建筑物能源使用的约15%。

Therefore, a passive cooling device without any external energy input can have a significant role in reducing global energy usage.

因此，无需任何外部能量输入的被动制冷器件可以在减少全球能源使用方面发挥重要作用。

Paragraph 16:

Recently, passive radiative cooling under direct sunlight was demonstrated in ref 7, where a multilayer structure consisting of seven alternating layers of HfO₂ and SiO₂ was placed above a silver reflector, as schematically shown in Figure 1a.

最近，文献7展示了在直射阳光下的被动辐射制冷，其中将七层交替的HfO₂和SiO₂多层结构放置在银反射器上方，如图1a示意所示。

The multilayer structure was designed to both be highly reflective over the solar spectrum and highly emissive in the wavelength range between 8 and 13 μm, where the atmosphere has a high transparency window.

该多层结构被设计为在太阳光谱上具有高反射率，同时在8至13μm的波长范围内具有高发射率，这是大气具有高透明度的窗口。

In the absence of parasitic heating from the environment, and using the atmospheric condition assumed in the theoretical calculations of ref 7, we computed that the structure reported in ref 7 can attain a temperature that is 19.5 K below an ambient temperature of 293 K, even under direct exposure to sunlight.

在没有来自环境的寄生加热的情况下，使用文献7理论计算中假设的大气条件，我们计算出文献7报道的结构可以达到比293K环境温度低19.5K的温度，即使在直射阳光下也是如此。

Paragraph 17:

There are two inter-related metrics by which one can evaluate the performance of radiative cooling device: the net cooling power and the minimum temperature it can attain (its steady-state temperature).

有两个相互关联的指标可以评估辐射制冷器件的性能：净制冷功率和它能够达到的最低温度（其稳态温度）。

On both metrics, the performance of the multilayer structure in ref 7 can be improved.

在这两个指标上，文献7中的多层结构的性能都可以得到改善。

First, the emissivity of the structure in the atmosphere transparency window between 8 and 13 μm can be further increased for more cooling power.

首先，结构在8至13μm大气透明度窗口内的发射率可以进一步提高以获得更多的制冷功率。

Second, within the atmospheric transparency window, there is a significant dip in the atmospheric transmittance between 9.3 and 10 μm due to ozone absorption.

其次，在大气透明度窗口内，由于臭氧吸收，9.3至10μm之间的大气透射率有显著下降。

Thus, within this wavelength range, there is significant downward radiation from the atmosphere.

因此，在这个波长范围内，有大量来自大气的向下辐射。

An ideal radiative cooling device should have low absorptivity/emissivity within this ozone absorption region in order to achieve a lower steady-state temperature.

理想的辐射制冷器件应该在这个臭氧吸收区域内具有低吸收率/发射率，以实现更低的稳态温度。

Paragraph 18:

Here, we employ the memetic algorithm to further optimize the performance of the passive radiative cooling device according to the considerations described above.

在这里，我们采用Memetic算法根据上述考虑进一步优化被动辐射制冷器件的性能。

Compared with ref 7, we choose to have a more diverse set of materials to design a spectrum more favorable for cooling performance.

与文献7相比，我们选择使用更多样化的材料集合来设计更有利于制冷性能的光谱。

We aim to achieve an objective spectrum that has minimal absorptivity over solar wavelengths, high emissivity in the atmosphere transparency window, and suppressed emissivity between 9.3 and 10 μm.

我们的目标是实现一个目标光谱：在太阳波长上具有最小吸收率，在大气透明度窗口具有高发射率，在9.3至10μm之间具有抑制的发射率。

As was done in ref 7, the superstrate and substrate of the cooling device are chosen as air and silver, respectively.

按照文献7的做法，制冷器件的覆盖层和基底分别选择为空气和银。

We select from the following seven common dielectric materials: Al₂O₃, HfO₂, MgF₂, SiC, SiN, SiO₂, and TiO₂.

我们从以下七种常见介电材料中进行选择：Al₂O₃、HfO₂、MgF₂、SiC、SiN、SiO₂和TiO₂。

Paragraph 19:

Since the substrate is made of silver, which light cannot transmit through, the emissivity/absorptivity $\epsilon(\lambda, \theta)$ is simply $\epsilon(\lambda, \theta) = 1 - R(\lambda, \theta)$, where $R(\lambda, \theta) = \frac{1}{2}(R_{TE}(\lambda, \theta) + R_{TM}(\lambda, \theta))$ is the average reflectivity over the transverse electric (TE) and transverse magnetic (TM) polarizations.

由于基底由银制成，光无法透过，因此发射率/吸收率 $\epsilon(\lambda, \theta)$ 简单地等于 $\epsilon(\lambda, \theta) = 1 - R(\lambda, \theta)$，其中 $R(\lambda, \theta) = \frac{1}{2}(R_{TE}(\lambda, \theta) + R_{TM}(\lambda, \theta))$ 是横电(TE)和横磁(TM)偏振的平均反射率。

To obtain the angle-averaged emissivity $\epsilon_{avg}(\lambda)$, we perform the solid angle integral over a hemisphere.

为了获得角度平均发射率 $\epsilon_{avg}(\lambda)$，我们对半球进行立体角积分。

Paragraph 20:

To find the optimum structure for radiative cooling given the objective spectrum specified above, we perform a broadband optimization on the emissivity spectrum.

为了找到满足上述目标光谱的辐射制冷最优结构，我们对发射率光谱进行宽带优化。

We choose to optimize in the normal direction for simplicity.

为了简化，我们选择在正入射方向进行优化。

For such a broadband device, we anticipate that while certain sharp spectral features generally exhibit a blue shift at oblique angles of incidence, overall its spectrum should not exhibit a strong angular dependence as it is inherently nonresonant.

对于这种宽带器件，我们预期虽然某些尖锐的光谱特征通常在斜入射角处表现出蓝移，但总体上其光谱不应该表现出强烈的角度依赖性，因为它本质上是非共振的。

Paragraph 21:

We set up the optimization for a maximum of K = 9 layers of materials, and we first randomly generate a population size of N = 1000 with different materials that have thicknesses randomly and uniformly distributed between 0 and 2 μm.

我们为最多K=9层材料设置优化，并首先随机生成种群大小N=1000，包含不同材料，其厚度在0到2μm之间随机均匀分布。

We heuristically set the reselection rate r = 0.90 and the mutation rate μ = 0.01, and we perform local optimization for refinement every G = 15 generations for the top P = 60 individuals.

我们启发式地设置重选率r=0.90和变异率μ=0.01，并每G=15代对前P=60个个体进行局部优化精炼。

Over the course of optimizing for 75 generations, in Figure 1d, we plot the best and average merit function in each generation while indicating the generations where refinement occurred.

在75代的优化过程中，在图1d中，我们绘制了每一代的最优和平均评价函数，同时标明了发生精炼的代数。

As expected, the evolutionary algorithm systematically produced structures that generally decreased the merit function from one generation to the next, and the optimization converged after 60 generations to a six-layer structure.

正如预期的那样，进化算法系统地产生了从一代到下一代普遍降低评价函数的结构，优化在60代后收敛到一个六层结构。

Paragraph 22:

We now analyze the emissivity properties of the structure in Table 1.

我们现在分析表1中结构的发射率特性。

In Figure 1b, we plot the normal-direction emissivity with red dotted lines.

在图1b中，我们用红色虚线绘制了正入射方向的发射率。

The emissivity spectrum has the same feature as the target emissivity, including the suppression of absorption in both the solar spectrum and the ozone absorption line, as well as the enhancement of emissivity in the wavelengths in the range of 8 to 13 μm that is outside the ozone absorption line.

发射率光谱具有与目标发射率相同的特征，包括抑制太阳光谱和臭氧吸收线的吸收，以及增强8至13μm范围内（臭氧吸收线之外）波长的发射率。

Paragraph 23:

We plot the emissivity spectra at different angles in Figure 1c.

我们在图1c中绘制了不同角度的发射率光谱。

The emissivity in general shows only weak angular dependency as expected.

发射率总体上如预期的那样仅表现出微弱的角度依赖性。

In particular, the prominent dip in the emissivity spectrum around the wavelength of 9.5 μm, which is desirable, persists over a wide range of angles.

特别是，在9.5μm波长附近发射率光谱中的显著下降（这是理想的）在宽角度范围内持续存在。

The overall angle-averaged emissivity, as calculated according to eq 4, is shown as the blue solid line in Figure 1b.

根据方程4计算的整体角度平均发射率如图1b中的蓝色实线所示。

While we do not directly optimize for such angle-averaged emissivity, the resulting angle-averaged emissivity reproduces the main features of the target emissivity spectrum.

虽然我们不直接优化这种角度平均发射率，但得到的角度平均发射率再现了目标发射率光谱的主要特征。

Paragraph 24:

To assess the cooling performance of the device above, we compare the lowest temperature achievable between our device and the device in ref 7 under the same thermal insulation and solar illumination.

为了评估上述器件的制冷性能，我们在相同的热绝缘和太阳照射下比较我们的器件和文献7中器件可达到的最低温度。

From the emissivity spectra in Figure 1b, we find that this structure absorbs 3.5% of the incident sunlight (compared to 3% from ref 7), and it has a cooling power of 45.7 W/m² (compared to 38.0 W/m² from ref 7) at the ambient temperature.

从图1b的发射率光谱中，我们发现该结构吸收了3.5%的入射阳光（相比文献7的3%），在环境温度下具有45.7 W/m²的制冷功率（相比文献7的38.0 W/m²）。

Assuming the absence of parasitic heat load, we find that the six-layer structure can achieve a steady-state temperature that is 23.4 K below ambient temperature, as compared with 19.5 K below ambient for the device in ref 7.

假设没有寄生热负荷，我们发现该六层结构可以达到比环境温度低23.4K的稳态温度，而文献7的器件为19.5K。

This temperature reduction directly results from the reduction of emissivity between the 9.3 to 10 μm wavelength range.

这种温度降低直接源于9.3至10μm波长范围发射率的降低。

Furthermore, in Figure 1e, we compare the cooling power between our six-layer device and the one in ref 7 at a given temperature, and we find that our optimized structure has a higher cooling power than the device in ref 7 at all temperatures, which enables our device to more efficiently achieve lower temperatures.

此外，在图1e中，我们在给定温度下比较了我们的六层器件与文献7器件的制冷功率，我们发现我们的优化结构在所有温度下都具有比文献7器件更高的制冷功率，这使得我们的器件能够更有效地实现更低的温度。

Optimization of Incandescent Light Bulb Filter 白炽灯滤光片优化

Paragraph 25:

We now demonstrate the use of the memetic algorithm in the optimization of thin-film filter for an incandescent light bulb.

我们现在展示Memetic算法在白炽灯薄膜滤光片优化中的应用。

While incandescent light bulbs have attractive properties such as having a high color rendering index (CRI), they suffer from poor efficiencies of around 3% because most of their emission is lost as near-infrared radiation (which does not contribute to illumination in the visible wavelength range).

虽然白炽灯具有高显色指数(CRI)等吸引人的特性，但它们效率低下，约为3%，因为大部分发射都作为近红外辐射损失（这不贡献于可见光波长范围的照明）。

The efficiency of incandescent light bulbs, however, can be improved by enclosing them inside a spectral filter, which has high transmittance in the visible wavelength range and high reflection in the near-infrared wavelength range.

然而，白炽灯的效率可以通过将它们封装在光谱滤光片中来提高，该滤光片在可见光波长范围内具有高透射率，在近红外波长范围内具有高反射率。

Paragraph 26:

By limiting infrared transmission, the reflected infrared light can be recycled to heat up the light bulb’s emitter.

通过限制红外透射，反射的红外光可以被回收以加热灯泡的发射体。

The emitter can thus achieve a higher temperature at a lower input power, which is conducive to enhanced emission in the visible spectrum and therefore improved efficiency.

因此，发射体可以在较低的输入功率下达到更高的温度，这有利于增强可见光谱的发射，从而提高效率。

This concept was demonstrated by Ilic et al., who fabricated a filter structure using 90 layers of alternating Ta₂O₅ and SiO₂ to surround a high-temperature tungsten emitter.

Ilic等人证明了这一概念，他们使用90层交替的Ta₂O₅和SiO₂制造滤光片结构来包围高温钨发射体。

Ilic et al. experimentally demonstrated that such a 90-layer structure could improve the efficiency of incandescent bulbs from around 3% to 6.6%.

Ilic等人通过实验证明了这种90层结构可以将白炽灯的效率从约3%提高到6.6%。

However, from economic, fabrication, and structure stability considerations, it would be preferable to use fewer layers of materials.

然而，从经济、制造和结构稳定性的考虑，使用更少的材料层是更优选的。

Paragraph 27:

Here, we implement the memetic algorithm to design an improved spectral filter for incandescent light bulbs.

在这里，我们实现Memetic算法来设计改进的白炽灯光谱滤光片。

In particular, we show that the memetic algorithm can design a structure that achieves more efficient visible light emission as compared to the device in ref 14 while using fewer than half the number of material layers.

特别是，我们展示了Memetic算法可以设计一种结构，与文献14的器件相比，使用少于一半的材料层数实现更高效的可见光发射。

Similar to the radiative cooler design in the previous section, we design for spectrum in the normal direction and then consider the angular properties of the resulting structure after optimizations.

与上一节的辐射制冷器设计类似，我们在正入射方向设计光谱，然后在优化后考虑所得结构的角度特性。

Paragraph 28:

The target reflection spectrum in the normal direction is shown in Figure 2b, and we expect the broadband transmission window in the visible wavelengths to persist over a wide range of angle as was the case in ref 14.

正入射方向的目标反射光谱如图2b所示，我们预计可见光波长的宽带透射窗口将在宽角度范围内持续存在，正如文献14的情况一样。

We optimize with the maximum number of layers being K = 45 and a population size of N = 3000, and the materials are selected from the same seven materials as is used in the previous section.

我们以最大层数K=45和种群大小N=3000进行优化，材料从与上一节相同的七种材料中选择。

The algorithm converged in 132 generations to a 41-layer structure, whose parameters are listed in Table 2.

算法在132代后收敛到一个41层结构，其参数列于表2中。

Paragraph 29:

We now analyze the optical properties of the structure described in Table 2.

我们现在分析表2所述结构的光学特性。

In Figure 2b, we plot the structure’s normal-direction reflectivity, which reproduces excellently the main requirements of the target spectrum.

在图2b中，我们绘制了结构的正入射反射率，它极好地再现了目标光谱的主要要求。

We further calculate the structure’s angular dependence and plot it in Figure 2c.

我们进一步计算了结构的角度依赖性并在图2c中绘制。

With this, we can compute the average reflection across all angles, which is plotted as the blue solid line in Figure 2b.

据此，我们可以计算所有角度的平均反射率，如图2b中的蓝色实线所示。

As desired, the angle-averaged reflection is low in the visible range to allow high visible light transmission between 400 and 700 nm, but the reflection is high in the infrared part so that infrared radiation can be recycled to further heat up the emitter.

如所期望的，角度平均反射率在可见光范围内较低，以允许400至700nm之间的高可见光透射，但红外部分反射率高，以便红外辐射可以被回收以进一步加热发射体。

Paragraph 30:

To directly evaluate the structure in Table 2 against the structure in ref 14, we surround incandescent emitters with the respective structures and compare the enhancement factor of visible light emission at a fixed input power.

为了直接评估表2中的结构与文献14中的结构，我们用各自的结构包围白炽发射体，并比较在固定输入功率下可见光发射的增强因子。

For simplicity, we assume that the emitter is a perfect blackbody, and the multilayer filter encloses the emitter with a view factor of f = 0.95, as was the case in ref 14.

为简单起见，我们假设发射体是完美的黑体，多层滤光片以f=0.95的视角因子包围发射体，正如文献14的情况。

Paragraph 31:

To measure the enhancement of visible light emission due to the spectral filter, we compute the emissivity enhancement factor χ.

为了测量由于光谱滤光片导致的可见光发射增强，我们计算发射率增强因子χ。

Note that χ = 1 when there is no spectral filtering, and χ > 1 for filters that favor emission in visible wavelengths over infrared and ultraviolet wavelengths.

注意，当没有光谱滤波时χ=1，对于在可见光波长上优于红外和紫外波长的发射的滤光片，χ>1。

Paragraph 32:

We can now numerically compare the performance of our stack structure with the structure in ref 14.

我们现在可以数值比较我们的堆叠结构与文献14中结构的性能。

First, we fix the input power to the emitter to be P_emitter = 100 W, and we assume that the surface area of the emitter is A = 20 mm², as is reasonable for an incandescent light bulb.

首先，我们将发射体的输入功率固定为P_emitter=100W，并假设发射体的表面积为A=20mm²，这对于白炽灯泡来说是合理的。

We assume that the heat in the emitter is entirely dissipated through radiation, i.e. there is no conductive or convective heat loss.

我们假设发射体中的热量完全通过辐射耗散，即没有传导或对流热损失。

With the angle-averaged emissivity spectrum of our structure and the structure in ref 14, we can then solve for the temperature to find that our structure reaches a temperature of T₁ = 3743 K, whereas the structure from ref 14 reaches a temperature of T₂ = 3599 K.

使用我们结构和文献14结构的角度平均发射率光谱，我们可以求解温度，发现我们的结构达到T₁=3743K的温度，而文献14的结构达到T₂=3599K。

For comparison, at the same input power of 100 W, a blackbody would reach a temperature of T_BB = 2577 K.

作为比较，在相同的100W输入功率下，黑体将达到T_BB=2577K的温度。

Paragraph 33:

The overall thermal emission intensity spectrum of our structure and Ilic et al.’s structure are plotted along with the eye’s sensitivity spectrum in Figure 2d, where we see that the stack achieved χ₁ = 14.8, compared to Ilic et al.’s 90-layer structure that achieves χ₂ = 12.8.

我们结构和Ilic等人结构的整体热发射强度光谱与眼睛敏感度光谱一起绘制在图2d中，我们可以看到该堆叠达到χ₁=14.8，相比之下Ilic等人的90层结构达到χ₂=12.8。

Therefore, our 41-layer stack achieves a larger enhancement factor for visible-light emission than the 90-layer structure in ref 14.

因此，我们的41层堆叠比文献14中的90层结构实现了更大的可见光发射增强因子。

Moreover, the overall thickness of the 41-layer stack is 7.4 μm as compared to 15.8 μm for the structure in ref 14, and such thickness reduction is beneficial from an economic and fabrication perspective.

此外，41层堆叠的总厚度为7.4μm，而文献14中的结构为15.8μm，从经济和制造角度来看，这种厚度减少是有益的。

Through this example, we have demonstrated that the memetic algorithm allows us to design an incandescent light bulb filter with improved performance, while using fewer than half the number of layers compared to the previous design.

通过这个例子，我们证明了Memetic算法使我们能够设计性能改进的白炽灯滤光片，同时使用比先前设计少一半的层数。

Discussion and Summary 讨论与总结

Paragraph 34:

In both of the memetic algorithm demonstrations above, we have heuristically chosen the reselection rate to be r = 0.90 and mutation rate to be μ = 0.01.

在上述两个Memetic算法演示中，我们启发式地选择重选率为r=0.90，变异率为μ=0.01。

On the other hand, it is well-known that in evolutionary algorithms, the choices of r and μ is important and can influence the performance of the algorithm depending on the properties of the search landscape.

另一方面，众所周知，在进化算法中，r和μ的选择很重要，可以根据搜索景观的特性影响算法的性能。

In part 1 of the Supporting Information, we provide a study that examines the effects of using r = [0.80,0.90,0.99] and μ = [0.01,0.05,0.10], as well as fixing r = 0.90 and having very high mutation rates of μ = [0.50,0.80].

在支持信息的第1部分，我们提供了一项研究，检验使用r=[0.80,0.90,0.99]和μ=[0.01,0.05,0.10]的效果，以及固定r=0.90并使用非常高的变异率μ=[0.50,0.80]。

Paragraph 35:

These examples both show that a high reselection rate generally accelerates the decrease in the merit function during early generations.

这些例子都表明，高重选率通常加速早期代数中评价函数的下降。

As for the end-result of these optimizations, while a moderate reselection (r = 0.90) and high mutation rate (μ = 0.10) are beneficial for the radiative cooling device optimization, a low reselection rate (r = 0.80) and high mutation rate (μ = 0.10) more consistently generate higher-performing structures in the light bulb filter optimization.

至于这些优化的最终结果，虽然适度的重选(r=0.90)和高变异率(μ=0.10)有利于辐射制冷器件优化，但低重选率(r=0.80)和高变异率(μ=0.10)在灯泡滤光片优化中更一致地产生更高性能的结构。

In addition, having very high mutation rates of μ = 0.50 and 0.80 could help the memetic algorithm converge to better-performing structures for both optimizations.

此外，具有非常高的变异率μ=0.50和0.80可以帮助Memetic算法在两种优化中收敛到性能更好的结构。

This study illustrates that an optimal choice of the reselection and mutation rates is problem-specific, and it is important to diversify the population in order to explore more regions of a search landscape.

这项研究说明，重选率和变异率的最优选择是问题特定的，为了探索搜索景观的更多区域，保持种群多样性很重要。

Paragraph 36:

From both the radiative cooling device and incandescent light bulb filter examples above, the resulting structures as obtained from our memetic algorithm have higher performance than those reported in the literature while using fewer number of layers.

从上述辐射制冷器件和白炽灯滤光片两个例子来看，从我们的Memetic算法得到的结构在使用更少层数的同时比文献报道的结构具有更高的性能。

In these examples, we use a larger set of materials as compared to refs 7 and 14.

在这些例子中，与文献7和14相比，我们使用了更大范围的材料集合。

As an additional comparison with the multilayer structures used in refs 7 and 14, we apply this memetic algorithm to optimize the layer thicknesses of seven layers of SiO₂ and HfO₂ for the radiative cooling device and 90 layers of SiO₂ and Ta₂O₅ for the incandescent light bulb filter.

作为与文献7和14中使用的多层结构的额外比较，我们将这种Memetic算法应用于优化辐射制冷器件的SiO₂和HfO₂七层层厚，以及白炽灯滤光片的SiO₂和Ta₂O₅ 90层层厚。

The details of these optimization results are provided in part 2 of the Supporting Information, where we show that the memetic algorithm produced a seven-layer radiative cooling device that could reach an equilibrium temperature of 20.8 K below an ambient temperature of 293 K, and an 80-layer incandescent light bulb filter with an emissivity enhancement factor of χ = 14.1.

这些优化结果的细节在支持信息的第2部分提供，其中我们展示了Memetic算法产生的七层辐射制冷器件可以达到比293K环境温度低20.8K的平衡温度，以及发射率增强因子χ=14.1的80层白炽灯滤光片。

Both of these devices could out-perform those reported in the literature, but they have lower performance compared to the structures in Tables 1 and 2, which were optimized using a more diverse set of materials.

这两种器件都可以胜过文献报道的器件，但与使用更多样化材料集合优化的表1和表2中的结构相比，它们的性能较低。

These two examples demonstrate that the memetic algorithm could more effectively perform optimizations with the same materials that were used in refs 7 and 14.

这两个例子证明，Memetic算法可以更有效地使用与文献7和14中相同的材料进行优化。

Furthermore, the results suggest that when designing spectral properties of a structure, it is advantageous to build the structure from a diverse set of materials.

此外，结果表明，在设计结构的光谱特性时，使用多样化的材料集合来构建结构是有利的。

Paragraph 37:

In summary, we have presented an implementation of a memetic algorithm for the design of multilayer optical thin-films that is especially suited for broadband thermal and energy applications.

总之，我们提出了一种用于多层光学薄膜设计的Memetic算法实现，特别适合宽带热和能源应用。

Given a target optical spectrum, we construct the optimization as a mixed integer problem, where the optimization variables consist both a list of discrete variables that represent different materials and a list of continuous variables that represent each material’s thickness.

给定目标光学光谱，我们将优化构建为一个混合整数问题，其中优化变量包括代表不同材料的离散变量列表和代表每种材料厚度的连续变量列表。

As a set of demonstrations, we have employed the memetic algorithm to optimize the performances of both a passive radiative cooling device and an incandescent light bulb filter.

作为一系列演示，我们采用Memetic算法优化了被动辐射制冷器件和白炽灯滤光片的性能。

In both examples, by using a diverse set of materials, the multilayer structures obtained from the optimization have fewer layers but better performances as compared to those presented in the literature.

在两个例子中，通过使用多样化的材料集合，从优化得到的多层结构具有更少的层数但更好的性能，与文献中报道的相比。

Because of its versatility and effectiveness, our algorithm should prove to be a powerful tool in the optimization of multilayer optical materials, not only for thermal and energy applications, but also for imaging, display, and information processing applications.

由于其多功能性和有效性，我们的算法应该被证明是多层光学材料优化的强大工具，不仅适用于热和能源应用，也适用于成像、显示和信息处理应用。

三、语言学习 (Language Learning)

3.1 雅思词汇 (IELTS Vocabulary)

3.2 科研术语 (Technical Terms)

3.3 学术表达 (Academic Expressions)

3.3.1 研究背景与动机

3.3.2 方法描述

3.3.3 结果与讨论

3.3.4 结论与展望

3.3.5 逻辑连接

四、数学公式汇总 (Key Equations)

评价函数 (Merit Function)

$$F(\tilde{n}, d) = \sum_{\lambda,\theta} W(\lambda)\left(R(\theta, \lambda; \tilde{n}, d) - R^*(\theta, \lambda)\right)^2$$

混合整数规划问题 (Mixed-Integer Problem)

$$[\tilde{n}^, d^] = \arg\min_{\tilde{n} \in \mathbb{Z}^K, d \in \mathbb{R}^K} F(\tilde{n}, d)$$

发射率与反射率关系

$$\epsilon(\lambda, \theta) = 1 - R(\lambda, \theta) = 1 - \frac{1}{2}\left(R_{TE}(\lambda, \theta) + R_{TM}(\lambda, \theta)\right)$$

角度平均发射率

$$\epsilon_{avg}(\lambda) = \int d\Omega \cos\theta , \epsilon(\lambda, \theta) = 2\pi \int_0^{\pi/2} d\theta \sin\theta \cos\theta , \epsilon(\lambda, \theta)$$

黑体辐射 (Planck’s Law)

$$I_{emitter}(\lambda, T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^{hc/(\lambda k_B T)} - 1}$$

发射率增强因子

$$\chi = \frac{\int d\lambda , \epsilon_{eff}(\lambda) I_{emitter}(\lambda) V(\lambda)}{\int d\lambda , I_{emitter}(\lambda) V(\lambda)}$$

五、关键图表说明 (Key Figures)

Figure 1 - 辐射制冷器件

• (a) 辐射制冷器原理示意图
• (b) 设计的发射率/吸收率光谱与太阳辐射光谱、大气透射窗口的关系
• (c) 发射率光谱的角度依赖性
• (d) 每代最优和平均评价函数的收敛曲线
• (e) 优化器件与文献器件的温度-制冷功率对比

Figure 2 - 白炽灯滤光片

• (a) 白炽灯滤光片结构示意图
• (b) 反射率光谱（正入射与角度平均）
• (c) 反射率光谱的角度依赖性
• (d) 相同输入功率下的发射光谱对比
• (e) 优化过程的收敛曲线

六、延伸阅读 (Further Reading)

算法基础

1. Goldberg, D. E. (1989). Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley.
2. Moscato, P. (1989). On Evolution, Search, Optimization, Genetic Algorithms and Martial Arts - Towards Memetic Algorithms. Caltech.

相关应用

3. Raman, A. P., et al. (2014). Passive Radiative Cooling below Ambient Air Temperature under Direct Sunlight. Nature, 515, 540-544.
4. Ilic, O., et al. (2016). Tailoring High-Temperature Radiation and the Resurrection of the Incandescent Source. Nature Nanotechnology, 11, 320-324.

计算方法

5. Haus, H. A. (1983). Waves and Fields in Optoelectronics. Prentice Hall.
6. Palik, E. D. (1997). Handbook of Optical Constants of Solids. Academic Press.

Published: 2018 | Journal: ACS Photonics | DOI: 10.1021/acsphotonics.7b01136