Metal-dielectric-metal multilayer structure with tunable Fabry-Perot resonance for highly sensitive refractive index sensing

Zhang Xiang-Yu; Liu Hui-Gang; Kang Ming; Liu Bo; Liu Hai-Tao

doi:10.7498/aps.70.20202058

Abstract

Authors and contacts

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1. 引　言
2. 方法与实验
2.1 测量原理
2.2 贫化铀装置
2.3 活化探测器
2.4 中子源及产额监测
2.5 γ射线探测系统
3. 结果与分析
3.1 实验结果及分析
3.2 蒙特卡罗模拟与实验比较
4. 结　论

References

Abstract

The metal-dielectric-metal multilayer structure sensor with a transparent top layer and an opaque bottom layer is proposed, which can provide a perfect narrow-band absorption resonance and is suitable for sensing refractive index change of the liquid to be measured in dielectric layer. The Fabry-Perot resonance analytical model that can accurately reproduce response spectrum and theoretically analyze the mechanism of the dielectric layer thickness to tune resonance wavelength and linewidth of response spectrum is constructed. Theoretical analysis shows that the resonance wavelength is directly proportional to the thickness of dielectric layer, and the full width at half maximum is inversely proportional to the thickness of dielectric layer. The analytical expressions for its resonance wavelength, quality factor, full width at half maximum and sensitivity are also given. When used for the refractive index sensing, the quality factor and figure of merit of the proposed multilayer structure based on the 8th order Fabry-Perot resonance are 2162.8 and 1648.1 RIU^–1, respectively. However, due to the influence of the minimum resolution of the spectrometer, the conventional method of measuring resonance wavelength shift to achieve refractive index sensing has a high measurement limit. For the sensing of weaker refractive index perturbation, with the help of superposition of exceptional point degenerate state and tuning mechanism of Fabry-Perot resonance, in this paper proposed is a method of tunably sensing the liquid refractive index by measuring the increase of reflection coefficient or splitting of eigenvalue at a specific wavelength. Here, we take for example the metal-dielectric-metal multilayer structure sensor based on the 8th order Fabry-Perot resonance. According to the calculation results of Fabry-Perot model, when the change in refractive index of liquid to be measured is 10^–4 RIU, the increase of forward reflection coefficient and the splitting of two eigenvalues of the scattering matrix are 0.319 and 1.1279, respectively.

Keywords:

PACS:

25.85.Ec	(Neutron-induced fission)
28.20.Gd	(Neutron transport: diffusion and moderation)
28.41.Ak	(Theory, design, and computerized simulation)

Authors and contacts

1.
Engineering Research Center of Thin Film Optoelectronics Technology, Ministry of Education, Nankai University, Tianjin 300350, China
2.
Tianjin Key Laboratory of Optoelectronic Sensor and Sensing Network Technology, Tianjin 300350, China
3.
Department of Microelectronic Engineering, College of Electronic Information and Optical Engineering, Nankai University, Tianjin 300350, China
4.
College of Physics and Materials Science, Tianjin Normal University, Tianjin 300387, China
5.
Institute of Modern Optics, College of Electronic Information and Optical Engineering, Nankai University, Tianjin 300350, China
6.
Tianjin Key Laboratory of Micro-scale Optical Information Science and Technology, Tianjin 300350, China

Corresponding author: Liu Hui-Gang, liuhg@nankai.edu.cn ; Liu Hai-Tao, liuht@nankai.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 62075104, 61775105, 11674244)

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1. 引　言

大量基础研究认为, 基于聚变中子源^[1]和次临界堆包层的聚变-裂变混合型次临界能源堆^[2]是解决能源问题比较值得研究的途径. 在这些概念研究中, 我国的研究团队开发了基于MCNP^[3]程序的耦合程序, 如MCORGS, MCBURN, MOCOUPLES等^[4-7], 并配套开发了多群截面库^[8,9], 为校验程序和截面库的可靠性, 有必要在一系列模拟装置上开展基准实验.

次临界能源堆能源供应主要依靠裂变包层中贫化铀裂变释放裂变能来实现, 裂变反应率的准确性在设计能量放大指标中异常重要. 英国的Weale等^[10]用直径29.2 mm的天然铀棒搭建了一个高1066 mm, 直径990 mm的准圆柱装置, 用圆柱形裂变电离室测量了²³⁸U(n, f)及²³⁵U(n, f)裂变反应率分布. 日本东京大学的Akiyama等^[11]用铀块堆砌了等效半径为457.2 mm的准球形装置, 利用柱形裂变室测量了14 MeV中子源条件下装置内侧²³⁸U(n, f)及²³⁵U(n, f)反应率, 不确定度为5.5%—6.0%. Afanas’ev等^[12]在聚变堆包层模拟装置上开展14 MeV中子学积分实验, 用固体径迹探测器测量²³⁸U(n, f)及²³⁵U(n, f)反应率, 相对不确定度在8%. 国内朱通华等^[13]在贫化铀聚乙烯交替球壳上用长杆平板型贫化铀裂变室测量了裂变反应率径向分布, 不确定度为3.4%.

Weale实验与Akiyama实验所用的装置均采用现有材料堆砌而成, 基准性不足. 另外, 上述实验测量无论采用裂变室^[14]还是固体径迹探测器^[15,16], 均在被测位置引入了其他材料, 破坏了测量对象的原始状态, 使得测量点的物理参数发生改变, 要对测量值进行修正才能得到所需的物理量. 为了解决以上探测手段对测量的影响, 本文采用与宏观贫化铀球壳丰度完全相同的贫化铀片状样品作为活化探测器^[17,18], 活化探测器可以看作被测模型的一部分, 最大限度降低了对被测对象的影响. 利用中国工程物理研究院现有系列贫化铀球壳, 构建五种不同厚度的实验模型, 选择与D⁺粒子(D⁺粒子指氘气电离后形成的带正电的原子核, 经引出、加速、然后轰击氚靶产生中子)水平呈45°角的测量孔道, 采用活化法基于14 MeV D-T中子开展实验研究. 选择裂变碎片中的¹⁴³Ce^[19]作为测量对象, 通过测量其发射的γ射线, 实现¹⁴³Ce核子数的测量, 再根据裂变产额数据和中子产额监测数据, 获得裂变反应率沿45°方向分布规律, 为裂变放能与球壳厚度关系研究提供数据支持.

2. 方法与实验

2.1 测量原理

绝对裂变反应率是指一个源中子诱发一个核材料原子核产生裂变的概率^[20]. 通过测量装置内被测位置的核裂变计数, 可得到该处的裂变反应率. 裂变率计算公式为

$f = \frac{{{N_{\rm{f}}}}}{{\phi \times {t_1} \times m}}.$

(1)

式中, ${N_{\rm{f}}}$ 为裂变数, $\phi$ 为中子产额, ${t_1}$ 为辐照时间, m为活化探测器核子数.

本实验选取裂变产额比较准确的裂变碎片¹⁴³Ce作为测量对象, 核裂变数 ${N_{\rm{f}}}$ 与¹⁴³Ce核子数 ${N_{{\rm{Ce}} \text{-} 143}}$ 有如下关系:

${N_{\rm{f}}} = {N_{{\rm{Ce}} \text{-} 143}}/{Y_{{\rm{Ce}} \text{-} 143}}.$

(2)

式中 ${Y_{{\rm{Ce}} \text{-} 143}}$ 为 ¹⁴³Ce总裂变产额.

贫化铀主要成分为²³⁸U和²³⁵U, 将中子能量从0 keV到15 MeV分成8个能段(²³⁸U只计算后7个能段), ¹⁴³Ce总裂变产额 ${Y_{{\rm{Ce}} \text{-} 143}}$ 可用(3)式表示:

${Y_{{\rm{Ce}} \text{-} 143}} = {R_5}\sum\limits_1^8 {{Y_{5i}}{R_{5i}}} + {R_8}\sum\limits_2^8 {{Y_{8i}}{R_{8i}}},$

(3)

式中, ${R_5}$ 为 ²³⁵U裂变占总裂变的百分比; ${Y_{5i}}$ 为第i个能量段²³⁵U裂变¹⁴³Ce的产额; ${R_{5i}}$ 为第i个能量段裂变占总²³⁵U裂变的百分比; ${R_8}$ 为²³⁸U裂变占总裂变的百分比; ${Y_{8i}}$ 为第i个能量段²³⁸U裂变¹⁴³Ce的产额; ${R_{8i}}$ 为第i个能量段裂变占总²³⁸U裂变的百分比.

¹⁴³Ce的半衰期为33 h, 能量为293 keV的γ射线的分支比是42%. 通过测量这条γ射线可以得到¹⁴³Ce核子数. ¹⁴³Ce核子数 ${N_{{\rm{Ce}} \text{-} 143}}$ 与293 keV γ射线计数有如下关系:

$\begin{split} & {N_{{\rm{Ce}} \text{-}143}} \\ = \;& \frac{{{C_{293.3}}}}{{A{{(d)}_{293.3}}{b_{293.3}}{\eta _{293.3}}k{{\rm{e}}^{ - {\lambda _{{\rm{Ce}} \text{-} 143}}\tau }}(1 - {{\rm{e}}^{ - {\lambda _{{\rm{Ce}} \text{-} 143}}t}})}}.\end{split}$

(4)

式中, ${C_{293.3}}$ 为293.3 keV γ射线计数, $A{(d)_{293.3}}$ 为293.3 keV γ射线自吸收修正因子, ${b_{293.3}}$ 为293.3 keV γ射线分支比, ${\eta _{293.3}}$ 为293.3 keV γ射线探测效率, k为辐照过程修正因子, $\lambda {}_{{\rm{Ce}} \text{-} 143}$ 为 ${C_{{\rm{Ce}} \text{-} 143}}$ 衰变常数, $\tau$ 为冷却时间, ${t_2}$ 为测量时间.

由(1), (2)和(4)式可得裂变反应率f, 如(5)式所示:

$f = \frac{{{C_{293.3}}}}{{A{{(d)}_{293.3}} \cdot {b_{293.3}} \cdot {\eta _{293.3}} \cdot k \cdot {{\rm{e}}^{ - {\lambda _{{\rm{Ce}} \text{-} 143}}\tau }} \cdot (1 - {{\rm{e}}^{ - {\lambda _{{\rm{Ce}} \text{-} 143}}{t_2}}}) \cdot {Y_{{\rm{Ce}} \text{-} 143}} \cdot \phi \cdot {t_1} \cdot m}}.$

(5)

(5)式中各参数的意义同上.

2.2 贫化铀装置

贫化铀装置^[21]共分五种模型, 内半径R_in均为13.1 cm, 外径及组合厚度见表1. 贫化铀的密度为(18.8 ± 0.1) g/cm³, 其中²³⁸U和²³⁵U同位素丰度分别为99.58%和0.416%^[22]. 在装置的水平径向留有放置D⁺离子束流漂移管的靶室孔道, 水平方向与D⁺粒子漂移方向呈45°和90°方向留有两条测量孔道, 竖直方向有一条测量(吊装)孔道, 靶室孔道及测量孔道直径均为44 mm. 本次实验在45°孔道开展实验.

表 1 五种贫化铀球壳的外径及厚度

Table 1. Radius and thickness of depleted uranium shells.

模型编号	外半径R_out/cm	厚度L/cm
1	18.10	5.00
2	19.40	6.30
3	23.35	10.25
4	25.40	12.30
5	28.45	15.35

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实验测量时需在测量孔道内放置一个贫化铀套筒, 与套筒匹配有不同厚度(5, 10, 20和30 mm)的圆柱形贫化铀塞块, 用于填充活化探测器之外的空间, 以尽可能保证贫化铀球壳的完整性, 最大限度避免空腔效应及其他材料对贫化铀球壳中子场的扰动. 贫化铀装置及其蒙特卡罗模型分别见图1(a)和图1(b), 贫化铀装置置于一铁支架上, 其中心与中子源中心重合(偏差 < 3 mm). 活化探测器与塞块交替放置在套筒内, 它们的材料成分均与贫化铀装置一样的, 5片活化探测器在装置中的分布情况见图1(b)中“1, 2, 3, 4, 5”. 为降低实验大厅散射中子本底影响, 实验装置距离实验大厅周围墙壁、地面及屋顶的距离均在3.5 m以上.

$图 1 (a)贫化铀装置实物图; (b)蒙特卡罗模型5片活化探测器分布情况(45°方向中的1, 2, 3, 4, 5)\r\nFig. 1. (a) Physical map of depleted uranium device; (b) distribution of five activation detectors in Monte Carlo Model 5 (1, 2, 3, 4, 5 in the direction of 45°).$

图 1 (a)贫化铀装置实物图; (b)蒙特卡罗模型5片活化探测器分布情况(45°方向中的1, 2, 3, 4, 5)

Fig. 1. (a) Physical map of depleted uranium device; (b) distribution of five activation detectors in Monte Carlo Model 5 (1, 2, 3, 4, 5 in the direction of 45°).

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2.3 活化探测器

实验时, 每种模型均布放5片活化探测器, 活化探测器位置p_i见(距中子源距离)表2. 测量孔道与D⁺粒子入射方向呈45°, 测量孔道内放置贫化铀套筒, 活化探测器置于贫化铀套筒内, 活化探测器之间用贫化铀塞块填充. 套筒内径Φ3.2 cm, 外径Φ4.2 cm, 活化探测器直径Φ2.4 cm, 标称厚度0.2 mm, 塞块直径Φ3.15 cm.

表 2 五种模型中活化探测器的布放位置

Table 2. Position of activation detector in various models

模型编号	L/cm
模型编号	p₁	p₂	p₃	p₄	p₅
1	13.60	14.62	15.64	16.16	17.18
2	14.60	15.62	16.64	17.66	18.68
3	14.60	16.62	18.64	20.66	21.68
4	15.60	18.62	20.64	22.66	24.68
5	15.60	18.62	20.64	24.16	27.18

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2.4 中子源及产额监测

中子源由中国工程物理研究院核物理与化学研究所的中子发生器提供, 中子源靶室置于装置中心. 直管式铝制靶室外径26 mm, 靶管外为铝制水套, 内外直径分别为37 mm和39 mm, 靶管与水套之间为循环冷却水. TiT靶活性区直径为12 mm, TiT靶为厚靶(D⁺粒子全部阻止在T-Ti层内), 平均入射D⁺粒子能量为135 keV, 对应最大中子能量为14.9 MeV.

绝对中子产额通过伴随α粒子法监测, 探测器置于漂移管中与D⁺束流方向夹角178.2°, 束流强度为300 μA时, 中子产额约为3 × 10¹⁰—4 × 10¹⁰ s^–1. 中子产额监测采用分时记录系统, 该记录系统时间步长为10 s, 能够精确反映实验期间中子产额波动, 实验期间中子产额波动修正由(4)式中的“辐照过程修正因子”k完成.

2.5 γ射线探测系统

测量γ射线的探测系统为ORTEC公司的TRANS-SPEC-DX100电制冷HPGe探测器和GammaVision谱分析软件. Ge晶体直径67.0 mm, 长度51.7 mm, 死层厚度0.7 mm. HPGe探测器置于屏蔽体内, 工作高压为–4500 V, 对⁶⁰Co的1.33 MeV γ射线能量分辨率为1.87 keV, 相对探测效率40%.

利用到⁶⁰Co, ¹³³Ba, ¹⁵²Eu等γ点源对HPGe探测器效率刻度, 得到探测器表面中心位置点源的探测效率曲线. 面源的探测效率采用积分的方法获得. 在探测器表面径向每隔3 mm测量一个点源探测效率, 拟合出径向的探测效率变化曲线, 在半径为12 mm的圆面内积分得到面源探测效率. 经标定, 位于探测器端面直径为24 mm的面源发射的293 keV γ射线的探测效率为7.44%. 293 keV的γ射线在贫化铀片中的自吸收因子由理论模拟得到(已由实验验证), 每片贫化铀片的自吸收因子根据实际厚度进行计算, 实验所用25片贫化铀片自吸收因子介于85%—92%之间.

3. 结果与分析

3.1 实验结果及分析

辐照实验完成后, 为降低短半衰期的γ本底, 将活化探测器冷却2 h, 然后在HPGe探测器表面对活化探测器进行了实验测量, 得到裂变碎片¹⁴³Ce发射的293.3 keV γ射线能谱, 如图2所示. 与γ射线测量相关的自吸收修正因子、探测效率、辐照过程修正因子等实验前都进行了测量及标定.

$图 2 HPGe探测器测量的贫化铀活化探测器发射的γ谱\r\nFig. 2. γ spectrum of depleted uranium activation detector, detected by using HPGe detector.$

图 2 HPGe探测器测量的贫化铀活化探测器发射的γ谱

Fig. 2. γ spectrum of depleted uranium activation detector, detected by using HPGe detector.

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为了获得¹⁴³Ce的总裂变产额, 分能区理论模拟了不同位置²³⁸U和²³⁵U的裂变反应率, 得到了 ${R_5}$ , ${R_{5i}}$ , ${R_8}$ 和 ${R_{8i}}$ , ${Y_{5i}}$ 和 ${Y_{8i}}$ 取文献值^[23]. 根据(3)式得到了五种模型不同测量位置的¹⁴³Ce总裂变产额Y_Ce-143, 结果见表3.

将293.3 keV γ射线总计数、自吸收修正因子、射线分支比、探测效率、辐照过程修正因子、 ${C_{{\rm{Ce}} \text{-} 143}}$ 衰变常数、冷却时间、测量时间、裂变产额、活化探测器核子数及源中子数代入(5)式, 可得到各模型相应位置处的裂变反应率, 归一到一个源中子一个铀原子核, 结果见图3.

表 3 Y_Ce-143值

Table 3. Values of Y_Ce-143.

Model No.	p₁/%	p₂/%	p₃/%	p₄/%	p₅/%
1	4.29	4.32	4.33	4.34	4.34
2	4.33	4.35	4.37	4.38	4.37
3	4.36	4.41	4.45	4.45	4.46
4	4.40	4.47	4.49	4.50	4.49
5	4.41	4.48	4.51	4.55	4.55

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$图 3 五种模型中的裂变反应率分布情况\r\nFig. 3. Fission reaction rate distribution for five models.$

图 3 五种模型中的裂变反应率分布情况

Fig. 3. Fission reaction rate distribution for five models.

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从图3可以看出: 1)每种模型, 随着距中子源距离L的增加裂变反应率逐渐变小, 厚模型变化幅度比薄模型变化幅度更大, 主要原因是距中子源越远中子通量密度越小, 裂变反应率自然降低; 2)相同测量位置, 模型越厚, 裂变反应率越大, 原因是外层贫化铀球壳的屏蔽与反射作用使得测点处中子通量密度增大, 为此用蒙特卡罗模拟计算了各模型测量位置处的中子通量密度, 结果见图4.

实验结果的不确定度主要来自中子产额、裂变产额和γ射线测量三个方面. 实验前对α探测器和靶片在靶管中的几何位置进行了准确测量, 对准直光栏孔径采用显微镜进行了测量, 各向异性修正因子通过查表得到, 保证了中子产额的不确定度小于2.5%^[24]. 裂变产额不确定度小于5%. γ射线测量的不确定度主要来自HPGE探测器探测效率、γ射线自吸收因子^[25]及γ谱解谱. 探测效率用系列标准源进行了标定, 不确定度小于2%^[26], 自吸收因子进行了理论模拟及实验验证, 不确定度小于0.5%, 解谱不确定度为2%—10% (不同位置处活化探测器不确定度不同). 五种模型各位置裂变反应率的总不确定度见表4.

$图 4 五种模型不同测量点处的中子通量密度(蒙特卡罗模拟计算)\r\nFig. 4. Neutron flux density at various measuring positions of five models (Monte Carlo simulation).$

图 4 五种模型不同测量点处的中子通量密度(蒙特卡罗模拟计算)

Fig. 4. Neutron flux density at various measuring positions of five models (Monte Carlo simulation).

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表 4 裂变反应率总不确定度

Table 4. Synthesize uncertainty of fission reaction rate.

Position	Model 1	Model 2	Model 3	Model 4	Model 5
p₁/%	6.5	6.5	6.5	7.4	6.1
p₂/%	6.2	6.3	5.7	7.2	7.0
p₃/%	6.5	5.8	6.7	8.6	10.0
p₄/%	6.5	6.3	6.5	9.5	9.5
p₅/%	6.5	6.1	7.0	10.9	10.9

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3.2 蒙特卡罗模拟与实验比较

利用MCNP5程序和ENDF/VI.8数据库对五种模型进行理论模拟, 用F4栅元卡配合计数乘子卡F4得到了不同测量位置的裂变反应率, 不确定度小于3.4%. 为更清晰地分析计算值与实验值的差异, 用计算值与实验值的比值(C/E)来进行表征, 结果见图5. 可以看到, 五种实验模型C/E值介于0.9至1.1之间, 表明计算与实验在10%以内符合, 且对于大部分测量位置其比值都落在测量标准不确定度范围之内.

$图 5 贫化铀装置中不同位置裂变率C/E值\r\nFig. 5. C/E ratio of fission reaction rate for various measuring position in depleted uranium assembly.$

图 5 贫化铀装置中不同位置裂变率C/E值

Fig. 5. C/E ratio of fission reaction rate for various measuring position in depleted uranium assembly.

下载: 全尺寸图片幻灯片

4. 结　论

采用与宏观贫化铀球壳丰度完全相同的贫化铀活化片作为探测器, 采用活化法测量方法得到了不同模型裂变反应率随径向距离的变化. 同种模型随径向距离增大裂变率逐渐降低, 主要是通量密度逐渐变小及能谱逐渐变软造成. 不同模型相同测量位置裂变反应率随模型厚度的增大逐渐变大, 主要原因是外层贫化铀球壳的屏蔽与反射作用使得测点处中子通量密度增大. 利用MCNP5程序和ENDF/VI.8数据库对上述实验模型进行了模拟, 理论与实验结果在不确定度范围内一致, 验证了蒙特卡罗输运程序及ENDF/VI.8数据库的可靠性.

本方法克服了裂变室及固体径迹探测器的不足, 最大限度降低了探测器对测量对象的影响, 使得测量结果更加真实反映宏观模型内部裂变放能特性; 细分了测量位置的中子能谱, 采用最新的¹⁴³Ce裂变产额数据, 提高了总裂变产额 ${Y_{{\rm{Ce}} \text{-} 143}}$ 的精度. 该研究成果对校验贫化铀材料核参数及中子输运特性具有重要意义, 为贫化铀裂变放能包层设计提供实验支撑.

References

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图 1 (a) MDM多层结构示意图, 箭头表示均匀平面波的传播方向, 红色、蓝色箭头对应前向、后向入射均匀平面波; (b) 前向反射率R、透射率T和吸收率A的光谱; (c) 不同玻璃层厚度d_Glass对应的反射率谱, 蓝色实线表示玻璃厚度无限大时的反射率谱(即图(b)中的红色实线), 插图为上下玻璃层均为有限厚度时的结构示意图; (d) 双层增透膜结构的反射率谱, 插图表示位于空气、玻璃之间的双层增透膜结构; (e) 不同玻璃层厚度d_Glass对应的反射率谱, 蓝色实线表示玻璃厚度无限大时的反射率谱, 插图为上下玻璃层均为有限厚度且玻璃表面镀双层增透膜的结构示意图

Figure 1. (a) Schematic of MDM multilayer structure. The solid arrows indicate the propagation directions of plane waves. The red and blue arrows correspond to forward and backward travelling incident plane waves, respectively. (b) Spectra of forward reflectance R, transmittance T and absorptance A of the proposed structure. (c) Reflectance spectra for different thicknesses d_Glass of the glass layer, where the blue solid curve corresponds to infinite glass thickness [i.e. the red solid curve in Fig. (b)]. The inset shows the structure diagram with upper and lower glass layers set to have a finite thickness. (d) Reflectance spectra of the double-layer antireflection coating. The inset shows the structure of the double-layer antireflection coating between the air and glass regions. (e) Reflectance spectra for different thicknesses d_Glass of the glass layer, where the blue solid curve corresponds to infinite glass thickness. The inset shows the structure diagram where the upper and lower glass layers are both set to have a finite thickness and both glass surfaces are coated with the double-layer antireflection coating.

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图 2 (a) 不同折射率n对应的反射率谱线; (b) 谐振波长随被测液体折射率n变化的曲线

Figure 2. (a) Reflectance spectra for different refractive indices n; (b) resonance wavelength plotted as a function of the refractive index n of the measured liquid.

DownLoad: Full-Size Img PowerPoint

图 3 Fabry-Perot模型　(a1) MDM结构中待求解的反射系数r_FP, 透射系数t_FP, 介质层中模式系数a, b的定义; (a2)—(a4) Fabry-Perot模型中双金属界面散射系数r₁, t₁, r₂, t₂, r₃, t₃的定义; (b1)—(b4) 对于r₁, t₁的Fabry-Perot模型, 中间金属层模式系数a', b'的定义, 以及单界面散射系数定义; (c1)—(c4) 对于r₂, t₂的Fabry-Perot模型, 相应的金属层模式系数和单界面散射系数定义; (d1)—(d4) 对于r₃, t₃的Fabry-Perot模型, 相应的金属层模式系数和单界面散射系数定义

Figure 3. Fabry-Perot model: (a1) Definitions of reflection coefficient r_FP, transmission coefficient t_FP to be solved in MDM structure, and mode coefficients a, b in dielectric layer; (a2)–(a4) definitions of bimetal interface scattering coefficients r₁, t₁, r₂, t₂, r₃, t₃ in Fabry-Perot model; (b1)–(b4) Fabry-Perot model of r₁ and t₁, definitions of mode coefficients a', b' in intermediate metal layer, and single interface scattering coefficients; (c1)–(c4) Fabry-Perot model of r₂ and t₂, the corresponding metal layer mode coefficients and single interface scattering coefficients are defined; (d1)–(d4) Fabry-Perot model of r₃ and t₃, the corresponding metal layer mode coefficients and single interface scattering coefficients are defined.

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图 4 (a) 不同介质层厚度d对应的反射率谱曲线, 点表示RCWA数值仿真结果, 实线表示Fabry-Perot模型计算结果; (b) 谐振波长随介质层厚度d的变化, 红色和蓝色实线表示Fabry-Perot模型和方程(4)的计算结果; (c) 不同谐振阶次m对应同一谐振波长1035.967 nm时(m越大则d越大), RCWA计算(点)、Fabry-Perot模型(实线)给出的反射率谱曲线; (d) Q, FOM, S随d的变化, 红色、蓝色和灰色实线代表由Fabry-Perot模型计算的反射率谱得到的Q, S, FOM, 粉色和青色虚线代表方程(5)和(6)计算的Q, S

Figure 4. (a) Reflectance as a function of wavelength for variable dielectric layer thicknesses d, dots represent RCWA numerical simulation results, solid lines represent Fabry-Perot analytical model calculation results; (b) resonance wavelength as a function of dielectric layer thickness d, red and blue solid lines represent the calculation results of Fabry-Perot model and Eq. (4); (c) reflectance spectrums given by RCWA calculation (dots) and Fabry-Perot model (solid lines), and different resonance orders m correspond to the same resonance wavelength 1035.967 nm (larger m is, larger d is); (d) Q, FOM, S as a function of d, red, blue and gray solid lines represent Q, S, FOM obtained from reflectance spectrum calculated by Fabry-Perot model, pink and cyan dotted lines represent Q, S calculated by Eqs. (5) and (6).

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图 5 奇异点简并　(a) 前向(红色实线)和后向(蓝色虚线)反射系数光谱; (b) 前向(红色实线)和后向(蓝色虚线)透射系数光谱; (c), (d) 散射矩阵本征值ν₁(红色实线)、ν₂(蓝色虚线)的实部和虚部随波长变化的曲线

Figure 5. Exceptional point degeneracy: (a) Spectra of forward (red solid line) and backward (blue dashed line) reflection; (b) spectra of forward (red solid line) and backward (blue dashed line) transmission; (c), (d) real and imaginary parts of eigenvalue ν₁ (red solid line), ν₂ (blue dashed line) of the scattering matrix vary with wavelength.

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图 6 (a) 基于8阶Fabry-Perot共振的EP简并系统; (b) EP简并系统的前向反射系数光谱随折射率n的变化, 竖直黑色实线表示观测波长; (c) DP简并系统; (d) 对称银层系统; (e) 前向反射系数随折射率微扰的变化曲线, 圆圈实线对应基于Fabry-Perot共振(取不同阶次m)的EP简并系统, 三角和方形实线分别对应DP简并系统和对称银层系统; (f) m = 0和m = 8时本征值分裂量随折射率微扰的变化曲线

Figure 6. (a) EP degenerate system based on 8 th order Fabry-Perot resonance; (b) forward reflection coefficient spectra of the EP degenerate system versus refractive index n, where the vertical black solid line represents the observation wavelength; (c) DP degenerate system; (d) symmetrical silver layer system; (e) forward reflection coefficient as a function of refractive index perturbation, where circle solid lines correspond to the EP systems based on Fabry-Perot resonance (taking different orders m), and triangular and square solid lines correspond to the DP degenerate system and the symmetrical silver layer system, respectively; (f) eigenvalue splitting amount as a function of the refractive index perturbation for m = 0 and m = 8.

DownLoad: Full-Size Img PowerPoint

[1]	Kocer H, Butun S, Palacios E, Liu Z, Tongay S, Fu D, Wang K, Wu J, Aydin K 2015 Sci. Rep. 5 13384 Google Scholar
[2]	Kocer H, Butun S, Li Z, Aydin K 2015 Sci. Rep. 5 8157 Google Scholar
[3]	Gomes de Souza I L, Rodriguez-Esquerre V F 2019 Sci. Rep. 9 7045 Google Scholar
[4]	Meng Y L, Tan J, Xu K, Chen J, Jin G J, Sun Y, Wang L L, Zuo Z, Qin H Y, Zhao Y, Guo J 2019 Appl. Opt. 58 6700 Google Scholar
[5]	Williams C, Rughoobur G, Flewitt A J, Wilkinson T D 2016 Appl. Opt. 55 9237 Google Scholar
[6]	Gomes de Souza I L, Rodriguez-Esquerre V F, Rêgo D F 2018 Appl. Opt. 57 6755 Google Scholar
[7]	Zhao H, Chen Z, Zhao R, Feng L 2018 Nat. Commun. 9 1764 Google Scholar
[8]	Yim J, Zhao H, Midya B, Feng L 2019 Opt. Lett. 44 1626 Google Scholar
[9]	Qin L, Wu S, Zhang C, Li X 2019 IEEE Sens. J. 19 2924 Google Scholar
[10]	Katyal J, Soni R K 2014 Plasmonics 9 1171 Google Scholar
[11]	王文慧, 张孬 2018 物理学报 67 247302 Google Scholar Wang W H, Zhang N 2018 Acta Phys. Sin. 67 247302 Google Scholar
[12]	Lu X, Zhang T, Wan R, Xu Y, Zhao C, Guo S 2018 Opt. Express 26 10179 Google Scholar
[13]	Liao Y L, Zhao Y 2020 Sci. Rep. 10 1480 Google Scholar
[14]	Chen L, Cao Z, Shen Q, Deng X, Ou F, Feng Y 2007 J. Lightwave Technol. 25 539 Google Scholar
[15]	Chen L, Cao Z, Ou F, Li H, Shen Q, Qiao H 2007 Opt. Lett. 32 1432 Google Scholar
[16]	Liao Y L, Zhao Y 2015 Plasmonics 10 1219 Google Scholar
[17]	Li Q, Li Z, Xiang X, Wang T, Yang H, Wang X, Gong Y, Gao J 2019 Coatings 9 393 Google Scholar
[18]	Midya B, Zhao H, Feng L 2018 Nat. Commun. 9 2674 Google Scholar
[19]	Miri M A, Alù A 2019 Science 363 eaar7709 Google Scholar
[20]	张翔宇, 康明, 刘会刚, 刘海涛 2020 中国激光 47 0300001 Google Scholar Zhang X Y, Kang M, Liu H G, Liu H T 2020 Chin. J. Lasers 47 0300001 Google Scholar
[21]	Wiersig J 2020 Photonics Res. 8 1457 Google Scholar
[22]	Huang Y, Shen Y, Min C, Fan S, Veronis G 2017 Nanophotonics 6 977 Google Scholar
[23]	Bai R, Zhang C, Gu X, Jin X R, Zhang Y Q, Lee Y 2017 Sci. Rep. 7 10742 Google Scholar
[24]	Pommet D A, Grann E B, Moharam M G, Gaylord T K 1995 J. Opt. Soc. Am. A 12 1068 Google Scholar
[25]	Wang N F, Kuo T W, Tsai Y Z, Lin S X, Hung P K, Lin C L, Houng M P 2012 Opt. Express 20 7445 Google Scholar
[26]	Do T T, Gubanova L A, Putilin E S, Khoa P V 2014 J. Opt. Technol. 81 612 Google Scholar
[27]	Lalanne P, Yan W, Vynck K, Sauvan C, Hugonin J P 2018 Laser Photonics Rev. 12 1700113 Google Scholar
[28]	Wan J, Zhu J, Zhong Y, Liu H 2018 J. Opt. Soc. Am. A 35 880 Google Scholar
[29]	Meng Z, Cao H, Liu R, Wu X 2020 Sensors 20 2301 Google Scholar
[30]	Ren X, Ren K, Ming C 2018 Sensors 18 1376 Google Scholar
[31]	祁云平, 张婷, 郭嘉, 张宝和, 王向贤 2020 物理学报 69 167301 Google Scholar Qi Y P, Zhang T, Guo J, Zhang B H, Wang X X 2020 Acta Phys. Sin. 69 167301 Google Scholar
[32]	Lan G, Jin Z, Nong J, Luo P, Guo C, Sang Z, Dong L, Wei W 2020 Appl. Sci. 10 2295 Google Scholar
[33]	Liu H, Zheng L, Ma P, Zhong Y, Liu B, Chen X, Liu H 2019 Opt. Express 27 13252 Google Scholar
[34]	Chen J, Nie H, Tang C, Cui Y, Yan B, Zhang Z, Kong Y, Xu Z, Cai P 2019 Appl. Phys. Express 12 052015 Google Scholar
[35]	Vassallo C 1991 Optical Waveguide Concepts (Netherlands: Elsevier) pp18−24
[36]	Demange G, Graefe E M 2011 J. Phys. A: Math. Theor. 45 25303 Google Scholar
[37]	Wiersig J 2016 Phys. Rev. A 93 033809 Google Scholar

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