1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Self consistency of adjoint analysis for topology optimization in
frequency domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Dielectric material based topology optimization for wave optics . . . . 4
1.3 Metal material based topology optimization for wave optics . . . . . . . 5
1.4 Topology optimization on two dimensional manifolds for wave
optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Self-consistent adjoint analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 Topology optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Split of wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Adjoint analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.1 Optical cloak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5.2 Nanostructures for localized surface plasmon resonances . . . 36
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.7.1 Adjoint analysis of topology optimization problem for two
dimensional optical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.7.2 Adjoint analysis of topology optimization problem for
three dimensional optical waves . . . . . . . . . . . . . . . . . . . . . . . . 49
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Dielectric material based topology optimization for wave optics . . . . . 55
3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.1.1 Magnetic field formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.1.2 Adjoint analysis for magnetic field based topology
optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.1.3 Electric field formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1.4 Adjoint analysis for electric field based topology
optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1.5 Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2.1 Cloak for perfect conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2.2 Dielectric resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.3 Beam splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2.4 Cloak for dielectric resonator . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2.5 Metalens with optical vortices . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4 Metal material based topology optimization for waves optics . . . . . . . . 103
4.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.1.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.1.2 Analyzing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.1.3 Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.2.1 Nanostructures for localized surface plasmonic resonances . 109
4.2.2 Nanoslits for extraordinary optical transmission . . . . . . . . . . . 111
4.2.3 Nanoantennas for coupling free space and metal-insulatormetal
waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.2.4 Cloak for surface plasmonic polaritons . . . . . . . . . . . . . . . . . . 133
4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5 Topology optimization on two dimensional manifolds for wave optics . 157
5.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.1.1 Two dimensional manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.1.2 PDEs for physical fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.1.3 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.1.4 Topology optimization problem . . . . . . . . . . . . . . . . . . . . . . . . 160
5.1.5 Adjoint analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.1.6 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Yongbo Deng is a professor at Changchun Institute of Optics Fine Mechanics and Physics (CIOMP), Chinese Academy of Sciences, China. In 2012, he received his Ph.D. from Changchun Institute of Optics Fine Mechanics and Physics (CIOMP), Chinese Academy of Sciences (CAS). In 2018, he derived a Humboldt Research Fellowship for Experienced Researchers and was elected as the member of Youth Innovation Promotion Association of Chinese Academy of Sciences in the same year. During the period from September of 2018 to February of 2020, he worked in the Institute of Microstructure Technology, Karlsruhe Institute of Technology as a Humboldt Fellow. During the periods from May to July of 2016 and from September to November of 2017, he worked in the Institute of Microstructure Technology, Karlsruhe Institute of Technology, based on support of a Guest Professor Fellowship. During the period from December of 2014 to March of 2015, he worked in IMTEK, University of Freiburg, for his research collaboration on electromagnetic metamaterial. His research mainly focuses on topology optimization, microfluidics, and nano-optics.
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