高精密成像系统膜层引入的波像差

王庆国; 赵尚男; 张纪鹏; 吴庆; 史广维

doi:10.37188/CO.2025-0136

高精密成像系统膜层引入的波像差

doi: 10.37188/CO.2025-0136

cstr: 32171.14.CO.2025-0136

王庆国^{1, 2, 3,},
赵尚男^{1, 3, ,},
张纪鹏^{1, 2, 3},
吴庆^{1, 3},
史广维^{1, 3}

1.
中国科学院长春光学精密机械与物理研究所, 吉林长春 130033
2.
中国科学院大学, 北京100049
3.
光学系统先进制造全国重点实验室, 吉林长春 130033

基金项目: 中国科学院长春光学精密机械与物理研究所青年科学基金项目（No. 62005271）；自平衡多模内力/力矩纵弯复合变形驱动式大口径空间主动反射镜设计方法研究（No. 12473084）；国家自然科学基金（No. 62475122）

详细信息

作者简介:
王庆国（2001—），男，山东淄博人，硕士研究生，2023年于东北石油大学获得学士学位，主要从事光学系统设计研究。E-mail:1733641607@qq.com

赵尚男（1993—），女，吉林长春人，博士，副研究员，主要从事光学设计仿真、超构表面设计的研究。E-mail：18810575846@163.com

中图分类号: TP394.1;TH691.9
计量
- 文章访问数: 10
- HTML全文浏览量: 6
- PDF下载量: 1
- 被引次数: 0
出版历程
- 收稿日期: 2025-10-27
- 录用日期: 2026-02-03
- 网络出版日期: 2026-04-24

Wavefront aberrations induced by coatings in high-precision imaging systems

WANG Qing-guo^{1, 2, 3
,},
ZHAO Shang-nan^{1, 3
, ,},
ZHANG Ji-peng^{1, 2, 3},
WU Qing^{1, 3},
SHI Guang-wei^{1, 3}

1.
Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences
2.
University of Chinese Academy of Sciences
3.
State Key Laboratory of Advanced Manufacturing for Optical Systems

Funds: Youth Science Fund Project of Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences (No. 62005271); Research on the Design Methodology of a Self-Balanced, Multimodal Internal-Force/Moment–Driven, Axial–Bending Compound-Deformation Actuation Scheme for Large-Aperture Space Active Reflectors (No. 12473084); National Natural Science Foundation of China (NSFC) (No. 62475122)

More Information

Corresponding author: 18810575846@163.com

摘要

摘要:
为提高系统透过率，高精密成像光学系统表面常需镀制多层膜。然而在短波段光学系统中，多层膜不仅改变表面透过率，还会引入显著的相位效应和横向位移，从而导致系统产生额外的波像差。本文针对短波段小入射角系统，系统分析了多层膜对全视场成像质量的影响。首先，利用膜层断点追迹算法，将膜层效应与光线追迹过程相结合，比较了可见光、红外及极紫外（EUV）波段系统的膜层引起的波像差。而后，以数值孔径为0.25的六反EUV投影系统为例，分析了均匀的40层Mo/Si多层膜引入的波前变化。在此基础上，提出一种基于Gram–Schmidt正交化（GSO）的EUV系统波前全视场分析方法，对弧形视场下的Zernike像差分布进行分析。结果表明，膜层引起的波像差在长波段系统中确实可忽略，而在短波段系统中则十分显著；膜层对EUV引入明显的倾斜和离焦，使得波前RMS由0.016λ增至0.842λ；全视场分析表明，膜层引入了0.727λ全视场倾斜和0.034λ视场无关的离焦，倾斜主要引起平移、倍率和低阶畸变等视场相关波前变化。研究表明，膜层引起EUV系统剧烈的像面变形，应在设计阶段将膜层影响纳入考虑范围。
- 多层膜 /
- 波像差 /
- 横向位移 /
- 极紫外(EUV)光学系统 /
- Gram–Schmidt正交化(GSO) /
- 光学设计
Abstract:
Multilayer coatings are widely applied to high-precision imaging optics to improve throughput. In short-wavelength systems, however, coatings not only alter transmittance/reflectance but also introduce pronounced phase effects and coating-induced lateral shifts, which collectively manifest as additional wavefront aberrations at the system level. This work systematically investigates coating-induced full-field degradation in short-wavelength imaging systems operated at small angles of incidence. A multilayer-coating break-point ray-tracing algorithm is used to incorporate coating-induced phase and lateral-shift effects into the geometrical ray-tracing workflow, enabling a comparative evaluation of coating-induced wavefront aberrations in the visible, infrared, and extreme ultraviolet (EUV) bands. A six-mirror EUV projection system (NA = 0.25) is then analyzed to quantify the wavefront changes introduced by a uniform 40-bilayer Mo/Si multilayer coating. Furthermore, a full-field wavefront analysis method based on Gram–Schmidt orthogonalization (GSO) is developed to characterize the field dependence of Fringe-Zernike aberration coefficients over a curved image field. The results indicate that coating-induced wavefront aberrations are negligible for long-wavelength systems but become significant in the short-wavelength regime. In the EUV example, the coating introduces strong tilt and defocus, increasing the RMS wavefront error from 0.016λ to 0.842λ. Full-field analysis shows a 0.727λ field-dependent tilt component and a 0.034λ field-independent defocus component; the tilt terms primarily correspond to image translation, magnification variation, and low-order distortion. These results demonstrate that multilayer coatings can induce severe image-plane deformation in EUV systems and therefore must be accounted for during the optical design stage.
- multilayer coating /
- wavefront aberration /
- lateral displacement /
- extreme ultraviolet (EUV) projection optics /
- Gram–Schmidt orthogonalization (GSO) /
- optical system design

HTML全文

图 1 EUV投影系统结构，物方离轴视场高度134 mm−142 mm

Figure 1. Optical layout of the EUV projection system; the object-side off-axis field height ranges from 134 mm to 142 mm.

下载: 全尺寸图片幻灯片

图 2 修正均匀膜层的反射相位和反射率随入射角的变化，实线、虚线和点划线分别表示表面1、3、5膜层反射特性

Figure 2. Reflection phase and reflectivity of the corrected uniform multilayer coating versus angle of incidence; the solid, dashed, and dash-dotted curves correspond to surfaces 1, 3, and 5, respectively.

下载: 全尺寸图片幻灯片

图 3 横向位移和额外相移随入射角的变化，入射角范围0~28°

Figure 3. Coating-induced lateral shift and the associated additional phase versus angle of incidence (AOI = 0–28°).

下载: 全尺寸图片幻灯片

图 4 膜层断点追迹算法与有效反射深度算法计算的横向位移比较，二者在小角度下具有良好的一致性

Figure 4. Comparison of the lateral shift computed by the multilayer break-point ray-tracing method and the effective reflection-depth method; the two approaches agree well at small angles of incidence.

下载: 全尺寸图片幻灯片

图 5 FFTM与HFTM计算的反射场振幅与相位随入射角分布，振幅、相位曲线对应左y轴，差值曲线对应右y轴

Figure 5. Reflected-field amplitude and phase versus angle of incidence calculated using FFTM and the (HFTM): the amplitude/phase curves correspond to the left y-axis, and the difference curves correspond to the right y-axis.

下载: 全尺寸图片幻灯片

图 6 比较不同波段膜层影响的反射系统

Figure 6. Test reflective system used to compare coating-induced effects at different wavelength bands.

下载: 全尺寸图片幻灯片

图 7 弧形视场形状及视场采样数

Figure 7. Curved scanning field and the corresponding field-sampling grid.

下载: 全尺寸图片幻灯片

图 8 裸系统、镀膜系统及膜层引入波像差的前16项Zernike系数全视场分布，（a）、（b）、（c）中相同Zernike项绘制范围一致

Figure 8. Full-field maps of the first 16 Fringe-Zernike coefficients for (a) the uncoated system, (b) the coated system, and (c) the coating-induced aberration (coated minus uncoated); for each Zernike term, the color scale limits are kept identical in (a)–(c).

下载: 全尺寸图片幻灯片

图 9 Z2～Z4拟合系数及拟合误差

Figure 9. Fitting coefficients and fitting errors of Z2–Z4

下载: 全尺寸图片幻灯片

表 1 EUV系统各表面平均入射角和膜层厚度

Table 1. Average incidence angle and coating thickness of each surface in the EUV system

表面序号	平均入射角/°	Si/nm	Mo/nm
1	6.5034	4.2272	2.8181
2	6.5397	4.2275	2.8183
3	17.6011	4.4063	2.9375
4	8.3387	4.2449	2.8299
5	10.6555	4.2737	2.8491
6	3.7275	4.2089	2.8059

下载: 导出CSV

表 2 倾斜平面镜镀膜前后波前RMS，对600 nm和1100 nm采取2种膜层形式，括号中数据是镀有膜系一的波前RMS

Table 2. RMS wavefront error of a tilted plane mirror before and after coating. Two alternative multilayer designs are used for the 600-nm and 1100-nm cases; values in parentheses correspond to the alternative design.

波长/nm	镀膜前RMS(λ)	镀膜后RMS(λ)
13.4	0.0000	0.1691
600	0.0000	0.0440(0.0069)
1100	0.0000	0.0203(0.0038)

下载: 导出CSV

表 3 裸系统和镀膜系统的各Zernike项沿视场变化的RMS（单位：λ@13.4 nm）

Table 3. Field-dependent RMS of each Fringe-Zernike term for the uncoated and coated systems (unit: λ@13.4 nm).

Zernike项	未镀膜	镀膜
Z2	0.00121	0.31954
Z3	0.00537	1.41882
Z4	0.01855	0.05474
Z5	0.01623	0.01565
Z6	0.00752	0.00727
Z7	0.00242	0.00223
Z8	0.01071	0.00995
Z9	0.00323	0.00387
Z10	0.00165	0.00213
Z11	0.00225	0.00296
Z12	0.00605	0.00660
Z13	0.00278	0.00304
Z14	0.00184	0.00181
Z15	0.00806	0.00790
Z16	0.00143	0.00111

下载: 导出CSV

表 4 对x^2myⁿ正交化的前8项系数矩阵

Table 4. Coefficient matrix of the first eight terms in the orthogonalization of x^2myⁿ

1	y	y²	y³	y⁴	x²	x²y	x²y²
1.000	0	0	0	0	0	0	0
−34.910	36.832	0	0	0	0	0	0
1.015E3	−2.153E3	1.142E3	0	0	0	0	0
−2.993E4	9.555E5	−1.016E5	3.600E4	0	0	0	0
4.701E3	−1.531E4	1.654E4	−5.934E3	5.5765	0	0	0
6.466E3	−1.799E4	1.646E4	−4.941E3	−301.235	319.7468	0	0
−8.479E4	2.652E5	−2.763E5	9.594E4	1.101E4	−2.341E4	1.244E4	0
4.543E4	−1.434E5	1.509E5	−5.291E4	−1.167E5	3.743E5	−4.001E5	1.425E5

下载: 导出CSV

表 5 正交多项式拟合的Zernike系数的视场分布（单位:λ@13.4 nm）

Table 5. Field dependence of Zernike coefficients described by the orthogonal polynomial fitting (unit: λ@13.4 nm).

m	n	Z2	Z3	Z4	Z5	Z6	Z7	Z8	Z9
0	0	−0.278	−1.419	0.059	0.012	−0.005	−0.001	−0.004	0.001
0	1	0.118	−0.053	−0.001	0.002	0.002	0.000	0.000	0.000
0	2	0.000	−0.005	0.000	0.000	0.000	0.000	0.000	0.000
0	3	−0.014	−0.003	0.000	0.000	0.000	0.000	0.000	0.000
1	0	−0.099	−0.016	0.000	0.000	−0.002	0.000	0.001	0.000
1	1	−0.027	−0.004	0.000	0.000	0.000	0.000	0.000	0.000
1	2	−0.005	0.001	0.000	0.000	0.000	0.000	0.000	0.000
1	3	−0.005	0.001	0.000	0.000	0.000	0.000	0.000	0.000
2	0	0.014	−0.004	0.000	0.000	0.000	0.000	0.000	0.000
2	1	0.011	0.003	0.000	0.000	0.000	0.000	0.000	0.000
2	2	0.005	0.000	0.000	0.000	0.000	0.000	0.000	0.000
2	3	0.002	0.000	0.000	0.000	0.000	0.000	0.000	0.000
3	0	−0.005	0.003	0.000	0.000	0.000	0.000	0.000	0.000
3	1	−0.006	0.000	0.000	0.000	0.000	0.000	0.000	0.000
3	2	−0.003	0.000	0.000	0.000	0.000	0.000	0.000	0.000
3	3	−0.001	0.000	0.000	0.000	0.000	0.000	0.000	0.000
RMS		0.320	1.420	0.059	0.013	0.006	0.001	0.004	0.001
RMSE		0.0086	0.0015	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000

下载: 导出CSV

表 6 EUV裸系统和镀膜后的波前RMS（单位：λ@13.4 nm）

Table 6. Full-field RMS wavefront error of the EUV system before and after coating (unit: λ at 13.4 nm).

	RMS	RMS（去除倾斜）
裸系统	0.01619	0.01588
镀膜	0.8422	0.04486
镀膜后离焦	0.6576	0.01707

下载: 导出CSV

参考文献(24)

[1]	CIESIELSKI R, SAADEH Q, PHILIPSEN V, et al. Determination of optical constants of thin films in the EUV[J]. Applied Optics, 2022, 61(8): 2060-2078. doi: 10.1364/AO.447152
[2]	MACLEOD H A. Thin-Film Optical Filters[M]. 4th ed. Boca Raton: CRC Press, 2010.
[3]	MA J Y, REN J L, ZHANG J H, et al. Quantum imaging using spatially entangled photon pairs from a nonlinear metasurface[J]. eLight, 2025, 5(1): 2. doi: 10.1186/s43593-024-00080-8
[4]	MA SH J, YAN L, YE H K, et al. Polarization aberration in catadioptric anamorphic optical system[J]. Proceedings of SPIE, 2023, 12765: 127651S.
[5]	HE CH, ANTONELLO J, BOOTH M J. Vectorial adaptive optics[J]. eLight, 2023, 3(1): 23. doi: 10.1186/s43593-023-00056-0
[6]	吕金锦, 祝青霞, 路易, 等. 正交线偏振光成像制导镜头的设计与仿真[J]. 飞控与探测, 2023, 6(2): 23-28. LYU J J, ZHU Q X, LU Y, et al. Design and simulation of orthogonal linearly polarized imaging guidance lens[J]. Flight Control & Detection, 2023, 6(2): 23-28. (in Chinese).
[7]	罗敬, 陈兴达, 吕凝睿, 等. 光学系统偏振特性影响抑制方法综述[J]. 中国光学(中英文), 2025, 18(5): 979-1015. doi: 10.37188/CO.2025-0066 LUO J, CHEN X D, LV N R, et al. A review of methods for suppressing the influence of polarization characteristics in optical systems[J]. Chinese Optics, 2025, 18(5): 979-1015. (in Chinese). doi: 10.37188/CO.2025-0066
[8]	LIU X L, HUANG Y, YAN X, et al. The correction method for wavefront aberration caused by spectrum-splitting filters in multi-modal optical imaging system[J]. Photonics, 2024, 11(9): 876. doi: 10.3390/photonics11090876
[9]	BAL M F, SINGH M, BRAAT J J M. Optimization of multilayer reflectors for extreme ultraviolet lithography[J]. Journal of Micro/Nanolithography, MEMS, and MOEMS, 2004, 3(4): 537-544. doi: 10.1117/1.1793171
[10]	REILEY D J, CHIPMAN R A. Coating-induced wavefront aberrations[J]. Proceedings of SPIE, 1992, 1746: 139-146.
[11]	LIANG CH, DESCOUR M R, SASIAN J M, et al. Multilayer-coating-induced aberrations in extreme-ultraviolet lithography optics[J]. Applied Optics, 2001, 40(1): 129-135. doi: 10.1364/AO.40.000129
[12]	SMITH B W. Optical considerations of EUVL wavelength, NA, and multilayers at large angles[J]. Proceedings of SPIE, 2025, 13424: 1342402.
[13]	BROVELLI L R, KELLER U. Simple analytical expressions for the reflectivity and the penetration depth of a Bragg mirror between arbitrary media[J]. Optics Communications, 1995, 116(4-6): 343-350. doi: 10.1016/0030-4018(95)00084-L
[14]	王君, 金春水, 王丽萍, 等. 极紫外光刻投影物镜中多层膜分析模型的建立及应用[J]. 光学学报, 2014, 34(8): 0811002. doi: 10.3788/AOS201434.0811002 WANG J, JIN CH SH, WANG L P, et al. Foundation and application of model for multilayers analysis in extreme ultra-violet lithography projection[J]. Acta Optica Sinica, 2014, 34(8): 0811002. (in Chinese). doi: 10.3788/AOS201434.0811002
[15]	来搏, 蒋励, 齐润泽, 等. 40~90 nm波段极紫外多层膜研究进展[J]. 光学精密工程, 2024, 32(9): 1293-1306. doi: 10.37188/OPE.20243209.1293 LAI B, JIANG L, QI R Z, et al. Research developments of extreme ultra-violet multilayers for 40-90 nm[J]. Optics and Precision Engineering, 2024, 32(9): 1293-1306. (in Chinese). doi: 10.37188/OPE.20243209.1293
[16]	HENKE B L, GULLIKSON E M, DAVIS J C. X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30, 000 eV, Z = 1-92[J]. Atomic Data and Nuclear Data Tables, 1993, 54(2): 181-342. doi: 10.1006/adnd.1993.1013
[17]	ZHANG Y, ZHANG S T, ZHENG Y H, et al. Algorithm for accurate and efficient calculation of coating-induced effects[J]. Optics Express, 2024, 32(17): 29279-29290. doi: 10.1364/OE.527842
[18]	GOODMAN J W. Introduction to Fourier Optics[M]. 3rd ed. Englewood: Roberts & Co. Publishers, 2005.
[19]	MANSURIPUR M. Classical Optics and its Applications[M]. 2nd ed. Cambridge: Cambridge University Press, 2009.
[20]	WANG Z ZH, BALADRON-ZORITA O, HELLMANN C, et al. Theory and algorithm of the homeomorphic Fourier transform for optical simulations[J]. Optics Express, 2020, 28(7): 10552-10571. doi: 10.1364/OE.388022
[21]	SASIÁN J. Introduction to Aberrations in Optical Imaging Systems[M]. Cambridge: Cambridge University Press, 2012.
[22]	CHAI X Y, ZHANG H B, LIN X Y, et al. Method for orthogonal fitting of arbitrary shaped aperture wavefront and aberration removal[J]. Optical Engineering, 2024, 63(5): 054112. doi: 10.1117/1.oe.63.5.054112
[23]	BAUER A, TAKAKI N, ROLLAND J P. Design methods for imaging with freeform optics[J]. Optica, 2025, 12(11): 1775-1793. doi: 10.1364/OPTICA.575611
[24]	CHAI X Y, LIN X Y, CHEN H T, et al. Zernike polynomials fitting of arbitrary shape wavefront[J]. Proceedings of SPIE, 2024, 13069: 1306912. doi: 10.1117/12.3023145