Thermal Radiative Properties of Uniaxial Anisotropic Materials and Their Manipulations [1st ed.] 9789811578229, 9789811578236

Table of contents :
Front Matter ....Pages i-xix
Introduction (Xiaohu Wu)....Pages 1-14
Calculation Method for Slab and Grating Structure Made of Anisotropic Materials (Xiaohu Wu)....Pages 15-28
Unidirectional Transmission of Light (Xiaohu Wu)....Pages 29-45
Broadband Perfect Absorption (Xiaohu Wu)....Pages 47-73
Near-Field Radiative Heat Transfer Between Anisotropic Materials (Xiaohu Wu)....Pages 75-86
Conclusions and Recommendations (Xiaohu Wu)....Pages 87-89
Back Matter ....Pages 91-91

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Springer Theses Recognizing Outstanding Ph.D. Research

Xiaohu Wu

Thermal Radiative Properties of Uniaxial Anisotropic Materials and Their Manipulations

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

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Xiaohu Wu

Thermal Radiative Properties of Uniaxial Anisotropic Materials and Their Manipulations Doctoral Thesis accepted by Peking University, Beijing, China

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Author Dr. Xiaohu Wu Shandong Institute of Advanced Technology Jinan, Shandong, China

Supervisor Associate Professor Ceji Fu LTCS and Department of Mechanics and Engineering Science College of Engineering Peking University Beijing, China

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-15-7822-9 ISBN 978-981-15-7823-6 (eBook) https://doi.org/10.1007/978-981-15-7823-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Supervisor’s Foreword

It is a great pleasure to introduce the work of Dr. Xiaohu Wu’s doctoral thesis, which is accepted for publication in Springer Theses after being awarded excellent Ph.D. thesis by Peking University in 2019. Dr. Wu received his Ph.D. degree from Peking University in 2019. Before that, he received his B.S. degree from China University of Mining and Technology (Beijing) in 2014. Supported by China Scholarship Council (CSC), he, as a visiting student, did research at Georgia Institute of Technology under supervision of Prof. Zhuomin Zhang from September 2017 to September 2018. Dr. Wu’s doctoral thesis work deals with theoretical and numerical investigations on the influence of orientation of the optical axis of uniaxial anisotropic materials on the thermal radiative properties and related potential applications of such materials. The motivation of his work comes from the fact that unlike isotropic materials, the electromagnetic response of uniaxial anisotropic materials and metamaterials shows some particular characteristics that are closely related to the orientation of the optical axis, suggesting the possibility for flexible manipulation of thermal radiation with such materials. Through his thesis work, Dr. Wu revealed the mechanisms for realizing unidirectional transmission and ultrabroadband perfect absorption with uniaxial anisotropic materials combined with gratings or photonic crystals, and near-field radiative heat transfer controlled with uniaxial anisotropic materials. Numerical algorithms for conveniently calculating the thermal radiative properties of uniaxial anisotropic materials that are applicable for grating structures were also developed. The research results presented in this thesis will not only deepen our understanding of the thermal radiative properties of anisotropic materials, but also provide theoretical guidance in design of novel devices for energy conversion and energy harvesting.

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Supervisor’s Foreword

A significant part of Dr. Wu’s thesis has been published in high-quality journals, some of which are fruit of joint collaborations with Prof. Zhuomin Zhang’s group at Georgia Institute of Technology in USA. Beijing, China September 2020

Ceji Fu, Ph.D. Associate Professor

Acknowledgements

My experience at Peking University has been a wonderful journey of learning, sharing, and growing. I would like to give my sincerest appreciation to my advisor, Dr. Ceji Fu, for his tutorship, and guidance in my research career. He gave me much freedom to do what I was interested and gave me fully support. Besides, to gain more research experience, he encouraged me to go abroad for study. I would like to extend my gratitude to my abroad advisor, Dr. Zhuomin Zhang. During the study at Georgia Institute of Technology, he gave me so much care not only in research but also in daily life. Besides, his inspiration in work also encouraged me to work harder and harder. Gratitude is extended to friends in Peking University. In particularly, I would like to thank Dr. Yi Zhao, Yang Yang, Jianshu Fan, Chengshuai Su, and Yao Hong for help, discussion, and collaborations. Gratitude is also extended to friends in Georgia Institute of Technology. Especially, I would like to thank Dr. Peiyan Yang, Dr. Eric Tervo, Dudong Feng, and Chuyang Chen, for their help in my daily life and research. I would also like to thank CSC for financial support. Finally, I want to express my sincere gratefulness to my family.

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Summary

The thermal radiative properties of materials are significantly relevant to the dielectric function and magnetic permeability. Non-magnetic materials are classified into two types according to the form of the materials’ dielectric function, namely isotropic and anisotropic materials. The thermal radiative properties of the former have been extensively investigated by researchers. However, not too much attention was paid to the latter one. With the advancement of chemical processing technology and micro/nanoscale fabrication technology, dramatically increasing number of anisotropic materials have been manufactured and applied, examples being hexagonal boron nitride (hBN), metamaterials and metasurfaces that consisted of periodically arrayed metals and dielectrics. Compared with isotropic materials, several unique characteristics of the thermal radiative properties are presented by anisotropic materials because of the orientation of the materials’ optic axes. Accordingly, it results in more diverse means of manipulations for anisotropic materials. Studying the thermal radiative properties of anisotropic materials is not only helpful to deepen our understanding of their influencing factors and involved manipulation methods, but also has important theoretical guiding significance in energy conversion, energy-saving technologies and design of novel devices. However, due to the existence of polarization coupling and conversion in anisotropic materials, the analysis and calculation methods of thermal radiative properties of such materials are much more complex than those of isotropic materials. Therefore, studies on the thermal radiative properties are far less than fruitful for anisotropic materials. With this in mind, the author mainly presents a systematic of studies for uniaxial anisotropic materials in this dissertation, involving the calculation methods of thermal radiative properties of uniaxial anisotropic materials, unidirectional transmission and ultrabroadband perfect absorption achieved with uniaxial anisotropic materials, and near-field radiative heat transfer controlled with uniaxial anisotropic materials. Some innovative results have been obtained through these studies. The main content of this dissertation includes the following four parts:

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Firstly, we have improved and expanded the 4  4 transfer matrix method and the rigorous coupled-wave analysis method so that they can be used to accurately and efficiently obtain the electromagnetic waves in anisotropic multi-layered structures and grating structures, respectively. By introducing the method of enhanced transmittance matrix, we solve the numerical overflow problem in these two methods. Secondly, based on the uniaxial anisotropic material hexagonal boron nitride (hBN), three different structures are proposed to realize unidirectional transmission of light in different spectral bands: the combination of two anisotropic slabs, the combination of an anisotropic slab and a grating, and the combination of an anisotropic slab and a photonic crystal. Each structure consists of two parts, one of which can fulfill conversion between two types of linearly polarized waves, and the other can realize selective transmission of linearly polarized waves. We find that unidirectional transmission of light can be achieved by the combination of polarization conversion and selective transmission, which provides a new idea for the design of novel unidirectional transmission devices. This design scheme can not only obtain unidirectional transmission with high transmission contrast, but also control the transmission by manipulating the optic axis of the uniaxial material. Thirdly, we study the influence of the orientation of the optic axis of hBN on its direction and hemispheric emissivity. In the calculations, we propose to use the 4  4 transfer matrix method combined with coordinate rotational transformations to circumvent projection operations so as to simplify the calculation process. It is found that the orientation of the optic axis has a great influence on the emissivity of hBN in its two hyperbolic bands. In addition, we propose a structure based on multi-layered hBN slabs to achieve perfect absorption. The absorptance of a structure consisting of 91 hBN slabs can reach 0.94 in the spectral band of 1367– 1580 cm−1. The physical mechanism of the broadband perfect absorption is attributed to the matching of impedance at each interface and the enlarged wave vector in each layer of the structure. At the same time, we also studied the influence of the number of layers and the thickness of each layer on the absorption of the whole structure. Finally, we propose to realize ultrabroadband perfect absorption by using a hyperbolic metamaterial consisting of doped silicon nanowires embedded in the TPX matrix. We find that at an incident angle of 60o, the absorptance can reach 0.99 in the spectral band from 2 to 50 lm, and its physical mechanism is similar to that in the structure made of multi-layered hBN. Finally, we study the near-field radiative heat transfer between hBN slabs as well as between graphene/hBN heterostructures, and analyze the effect of optic axis orientation on the near-field radiative heat transfer. The numerical results show that the near-field radiative heat flux between two semi-infinite hBN slabs reaches its maximum when the tilting angle of the optic axis is 0o, and it decreases with the increase of the tilting angle. However, the near-field radiative heat flux between two graphene-covered semi-infinite hBN plates is maximized when the tilting angle of the optic axis is 90o, and it decreases gradually with the decrease of the tilting angle. It is found that the enhanced near-field radiative heat flux between two semi-infinite hBN slabs originates from the excitation of hyperbolic phonon

Summary

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polaritons and hyperbolic surface phonon polaritons, while the enhanced near-field radiative heat flux between two graphene-covered semi-infinite hBN slabs is mainly due to the excitation of surface plasma polaritons at the air/graphene and the graphene/hBN interfaces.

Contents

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2 Calculation Method for Slab and Grating Structure Made of Anisotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 4  4 Transfer Matrix Method . . . . . . . . . . . . . . . . . . . 2.2 Rigorous Coupled-Wave Analysis . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction . . . . . . . . . . . . . . . . . . . . . . 1.1 Unidirectional Transmission of Light . 1.2 Broadband Perfect Absorption . . . . . . 1.3 Near-Field Radiative Heat Transfer . . References . . . . . . . . . . . . . . . . . . . . . . . .

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3 Unidirectional Transmission of Light . . . . . . . . . . . . . . . . . . . . . . 3.1 Realization of Unidirectional Transmission with Two Twisted Slabs Made of Anisotropic Material . . . . . . . . . . . . . . . . . . . . . 3.2 Realization of Unidirectional Transmission Based on Excitation of Magnetic Polaritons or Surface Plasmon Polaritons . . . . . . . 3.3 Realization of Unidirectional Transmission Based on Photonic Bandgap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Broadband Perfect Absorption . . . . . . . . . . . . . . . . . 4.1 Directional and Hemispherical Emissivity of hBN 4.2 Broadband Perfect Absorption of hBN . . . . . . . . . 4.3 Broadband Perfect Absorption of Metamaterial . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Near-Field Radiative Heat Transfer Between Anisotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Conclusions and Recommendations . . . . . . . . . . . . . . . . . . . . . . . . .

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Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Figures

Fig. 1.1

Fig. 1.2 Fig. 2.1 Fig. 2.2 Fig. 3.1 Fig. 3.2 Fig. 3.3

Fig. 3.4

Fig. 3.5

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Fig. 3.7

Schematic of unidirectional transmission devices. The transmission of forward illumination approaches unity, while that of backward illumination is close to zero . . . . . . . . The near-field radiative heat transfer between two objects with different temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . The single slab made of anisotropic material with thickness of d, and the plane of incidence is x-z plane . . . . . . . . . . . . . . The single grating made of anisotropic material with thickness of d, and the plane of incidence is x-z plane . . . . . . . . . . . . . . The unidirectional transmission device consists of two slabs with a relative rotation angle. . . . . . . . . . . . . . . . . . . . . . . . . . The real parts of the permittivity components of hBN variate with wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Reflectivity as a function of wavelength for TM wave and TE wave. b Transmissivity as a function of wavelength for TM wave and TE wave . . . . . . . . . . . . . . . . . . . . . . . . . . . a Transmissivity as a function of wavelength for normal incidence of a TM wave in two opposite directions. b Transmissivity of the unconverted TM wave component and the converted TE wave component for normal incidence of a TM wave from two opposite directions . . . . . . . . . . . . . . Transmissivity contrast ratio for normal incidence of a TM wave from two opposite directions as a function of wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Transmissivity and b transmissivity contrast ratio for normal incidence of a TE wave from two opposite directions as a function of wavelength . . . . . . . . . . . . . . . . . . The unidirectional transmission structure consists one hBN slab and one 1D grating. The optical axis of the top hBN slab is in the x–y plane and is tilted off the x-axis by an angle of /, and the thickness of the slab is d1 . . . . . . . . . . . . . . . . . . . . .

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Fig. 4.1

List of Figures

The transmittance and absorptance of the device varying with wavenumber for normal incidence of a a TE plane wave and b a TM plane wave from opposite sides of the device . . . Distribution patterns of a and b in the SiC grating for incidence of a TE and a TM plane waves, respectively, on the grating side at wavenumber 900 cm−1 . . . . . . . . . . . . . . . . . . . . . . . . The transmittance and the absorptance of the device varying with wavenumber for normal incidence of a a TE plane wave and b a TM plane wave from opposite sides of the device . . . The transmittance and absorptance of the device varying with wavenumber for normal incidence of a a TE plane wave and b a TM plane wave from opposite sides of the device . . . The unidirectional transmission structure consists of one hBN slab and one photonic crystal. The optical axis of the top hBN slab is in the x–y plane and is tilted off the x-axis by an angle of /, and the thickness of the slab is d1 . . . . . . . . . . . . . . . . . a The transmissivity of the PC structure as a function of wavelength for TE and TM waves for various number of unit cells. b The transmissivity ratio of the PC structure as a function of wavelength for various number of unit cells . . . . . a The transmissivity as a function of wavelength for incidence in the forward and backward directions. b The transmissivity contrast ratio as a function of wavelength. The number of unit cells N is equal to 14, 21 and 28 . . . . . . . . . . . . . . . . . . . . . . The transmissivity components of a TE wave under a forward incidence and b backward incidence . . . . . . . . . . . . . . . . . . . . Variation of Tf and Tb of the unidirectional transmission device with the rotation angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Variation of Tf and Tb with wavelength for normal incidence of a TM wave. b Variation of the transmissivity contrast ratio with wavelength for normal incidence of a TM wave . . . . . . . a Variation of Tf and Tb with wavelength for normal incidence of a TM wave. b Variation of the transmissivity contrast ratio with wavelength for normal incidence of a TM wave . . . . . . . Schematic of the coordinate systems used for calculating the reflection coefficients. Here, the upper half air and the lower half is a uniaxial crystal whose optical axis is tilted in the x–z plane by an angle b with respect to the z-axis. a The plane of incidence is the x–z plane with an incidence angle h; b the plane of incidence is rotated around z-axis by an azimuthal angle /. In the coordinate system, the plane of incidence is the x–z plane, where the zʹ-axis and z-axis are the same . . . . . . . .

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List of Figures

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Calculated emissivity spectra of hBN for different tilting angles: a hemispherical emissivity; b normal emissivity, i.e., h = 0° . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contour plots for the spectral directional emissivity of hBN with respect to the azimuthal angle and tilting angle at wavenumber of 1400 cm−1: a TM wave, h = 0°; b TE wave, h = 0°; c TM wave, h = 60°; d TE wave, h = 60° . . . . . . . . . The emissivity averaged over TM and TE waves as a function of azimuthal angle and tilting angle at the wavenumber of 1400 cm−1: a h = 0°; b h = 60° . . . . . . . . . . . . . . . . . . . . . Reflectivity components for a plane wave at normal incidence and at 1400 cm−1: a |rpp|2; b |rps|2; c |rss|2; d |rsp|2 . . . . . . . . . The spectral directional emissivity of hBN of the medium at the wavenumber of 800 cm−1 as a function of azimuthal angle and tilting angle: a TM wave, h = 0°; b TE wave, h = 0°; c TM wave, h = 60°; d TE wave, h = 60° . . . . . . . . . . . . . . . The normalized a impedance for TM wave incidence and b admittance for TE wave incidence at 1400 cm−1, and the normalized c impedance for TM wave incidence and d admittance for TE wave incidence at 800 cm−1 . . . . . . The spectral directional emissivity at 1400 cm−1: a TM wave, b = 0°; b TE wave, b = 0°; c TM wave, b = 45°; d TE wave, b = 45°; e TM wave, b = 90°; f TE wave, b = 90° . . . . . . . . The directional emissivity at 800 cm−1: a TM wave, b = 0°; b TE wave, b = 0°; c TM wave, b = 45°; d TE wave, b = 45°; e TM wave, b = 90°; f TE wave, b = 90° . . . . . . . . . . . . . . . a Schematic of the coordinates, the optical axis of hBN is tilted off the z-axis by an angle in the x–z plane. b The asymmetric isofrequency curves for the tilted hBN . . . . . . . . . . . . . . . . . . The apparent permittivity of the tilted hBN. a The real part; b the imaginary part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a The absorptivity of a 20 nm thick hBN slab as a function of wavenumber and the tilted angle for normal incidence of a TM wave. b The absorptivity of the hBN slabs as a function of slab thickness at wavenumber 1527 cm−1 when tilted angle is 45° . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a The absorptivity of three single hBN slabs with tilted angle equal to 60°, 45°, and 30°, respectively, and that of a layered structure by stacking the three slabs together varying with wavenumber. b Similar to (a) except that the slab thickness is selected based on the FP resonance condition . . . . . . . . . . . . .

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Fig. 4.14

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Fig. 5.1

Fig. 5.2

Fig. 5.3

List of Figures

The absorptivity of the layered structure varying with wavenumber for different number of slabs: a the difference of tilted angles between adjacent slabs is 1°; b the difference of tilted angles between adjacent slabs is 10°, 3°, 2°, and 1°, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Real and b imaginary parts of the normalized impedance of a hBN medium as a function of the tilted angle for different values of the wavenumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a The absorptivity of a 31-layer structure varying with wavenumber. b The absorptivity of a 91-layer structure varying with wavenumber. The thickness of each slab is the same and is taken to be 20, 40, and 60 nm, respectively . . . . . . a Cross-section view and b top view of the composite made of silicon nanowire arrays embedded in the host medium of polymethylpentene (TPX). c Schematic view of a metamaterial slab. d Schematic view of the stacked metamaterial slabs. . . . . . Absorptance of the structure as a function of wavelength for different number of slabs in the structure . . . . . . . . . . . . . . . a The real and imaginary parts of the normalized impedance of the first slab as a function of wavelength. b The real part of the normalized impedance in each slab of the structure for L = 41 and for different wavelengths. c The imaginary part of the normalized impedance in each slab of the structure for L = 41 and for different wavelengths . . . . . . . . . . . . . . . . . . a The attenuation coefficient in each slab of the structure (L = 41) for different wavelengths. b The absorptance of the structure when the tilted angle of each slab is set the same and is equal to 0° and 90° . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a The absorptance of the structure (L = 41) as a function of wavelength and angle of incidence. b The real part of the normalized impedance of the first slab as a function of wavelength at different angles of incidence. c The imaginary part of the normalized impedance of the first slab as a function of wavelength at different angles of incidence . . . . . . . . . . . . . . Schematic of near-field radiative heat transfer between two graphene-hBN heterostrucutures. The optical axis (OA) of hBN is in the x-z plane and tilted off the z-axis by an angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NFRHF between a two bare hBN slabs and b two graphenecovered hBN slabs as a function of the tilting angle of hBN optical axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral NFRHF between a bulk hBN slabs, b hBN slabs with thickness of 50 nm, c graphene-covered bulk hBN slabs, and d graphene-covered hBN slabs with thickness of 50 nm . . . .

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List of Figures

Fig. 5.4

Fig. 5.5

Fig. 5.6

Fig. 5.7

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Fig. 5.10

Energy transmission coefficient between two bulk hBN slabs varying with wavevector components kx and ky at 1.55  1014 rad/s: a a = 0°, b a = 90°, and c a = 45° . . . . Energy transmission coefficient between two bulk hBN slabs varying with wavevector components kx and ky at 2.85  1014 rad/s: a a = 0°, b a = 90°, and c a = 45° . . . . Energy transmission coefficient between two bulk hBN slabs varying with wavevector components kx and ky at 2.96  1014 rad/s: a a = 0°, b a = 90°, and c a = 45° . . . . Energy transmission coefficient between two bulk hBN slabs varying with wavevector components kx and ky at 1.0  1014 rad/s: a a = 0°, b a = 90° . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy transmission coefficient between two bulk hBN slabs varying with wavevector components kx and ky at 1.47  1014 rad/s: a a = 0°, b a = 90° . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy transmission coefficient between two bulk hBN slabs varying with wavevector components kx and ky at 1.58  1014 rad/s: a a = 0°, b a = 90° . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of different oriented optical axes of hBN on the NFRHF between a two bulk hBN slabs and b two graphene-covered bulk hBN slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Introduction

Thermal radiation is a fundamental way of heat transfer, and its mechanism is essentially different from that of heat conduction and convection [1]. Thermal radiation occurs in everything whose temperature is greater than absolute zero, so it is of universal significance to study the properties of thermal radiation of materials. There are direct and indirect methods to calculate the radiative emissivity of an object [2]: the direct method is based on Function-dissipation theorem (FDT) and Green’s function to calculate the emissivity while the indirect method is based on Kirchhoff’s law to calculate this value. As Kirchhoff’s law states: the emissivity and the absorptivity of one surface at a given temperature and wavelength are equal and one could obtain this value by making one minus the sum of transmissivity and reflectivity. Therefore, to study the radiative property of an object, it is indispensable to study its reflectivity, transmissivity, as well as absorptivity. Wang et al. theoretically deduced that these two methods mentioned above are equivalent when calculating the emissivity of the multilayer planar structures [2]. It is worth noting that Miller [3] and his coworkers have recently shown that in reciprocal systems, the directional emissivity of a material is equal to the directional absorptivity. However, in non-reciprocal systems, there is no such relationship between the directional emissivity and the directional absorptivity. Here, only reciprocal systems are under consideration. The reflectivity, transmissivity, and absorptivity of a material are essentially determined by the permittivity of this material. In this dissertation, the magnetic materials are out of scope, so the permeability of a material is equal to the permeability of vacuum. The permittivity of a material can be represented by a diagonal matrix of 3 by 3 [4]. For isotropic materials, the three main diagonal elements of the permittivity tensor are equal, and the permittivity can thus be considered as a scalar while with respect to the anisotropic materials, these three main diagonal elements are not simultaneously equal. In comparison with the isotropic materials, the thermal

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 X. Wu, Thermal Radiative Properties of Uniaxial Anisotropic Materials and Their Manipulations, Springer Theses, https://doi.org/10.1007/978-981-15-7823-6_1

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1 Introduction

radiation property of anisotropic materials has a closer relationship with the direction, so the anisotropic materials have abundant thermal radiation properties. With the improvement of chemical fabrication technology, the preparation of two-dimensional (2D) materials have become easier and easier, and many 2D materials have been highlighted, such as graphene [5], hexagonal boron nitride (hBN) [6], black phosphorus(BP) [7], and α-MoO3 [8]. These 2D materials are anisotropic because their optical properties are not identical in different directions. The optical properties of graphene and BP can be regulated by applying voltage, so they have good performance in manipulating the thermal radiation characteristics [9, 10]. α-MoO3 and hBN both are hyperbolically natural materials, whose photon excitation can be utilized to achieve perfect absorption [11–14]. In addition, with the promotion of microfabrication, the preparation of metamaterials (MMs) has been established [15]. MMs can be obtained by periodically arranging dielectric and metal components, and the permittivity is derived by the effective dielectric theory. Most MMs reported are anisotropic and represent hyperbolic property at specific wavelength, which means two of the permittivity components are opposite [16, 17]. Compared with the common materials, hyperbolic metamaterials have tunable optical properties, making themselves useful in many important applications, such as negative refraction [18, 19], sub-diffraction imaging [20, 21] and thermal radiation regulation and control [22, 23], etc. No matter 2D materials or metamaterials, the advancement of their preparation technology has greatly promoted the research of anisotropic materials. From this point of view, it is of great significance to study and control the thermal radiation properties of anisotropic materials. This dissertation concentrated on the uniaxial anisotropic materials, studying the unidirectional transmission of light, broadband perfect absorption and near-field radiative heat transfer. The following part will introduce the recent development of unidirectional transmission of light, broadband perfect absorption, near-field radiative heat transfer and the related work done by the author.

1.1 Unidirectional Transmission of Light In recent years, the devices of unidirectional transmission of light (UDT) have shown significant effect on information processing and optical computing systems, since these devices can eliminate the impact of reflected light, which has been making UDT an attractive research direction. The devices of UDT have selectivity to the transmission direction of light, i.e., the light can propagate in one specific direction, but cannot in the opposite direction, an analogy to the principle of electronic diodes that regulate the transmission of electrons, as shown in Fig. 1.1. The important parameter to measure the performance of UDT devices is the transmission contrast between forward and backward directions, which is defined as the ratio of the transmissivity in two directions. Up to now, there have been three main approaches to realize UDT. One method is to use the nonlinear optics [24–29]. For example, Xie et al. realized UDT in a system composed of a nonlinear layer and two photonic crystals

1.1 Unidirectional Transmission of Light

3

Fig. 1.1 Schematic of unidirectional transmission devices. The transmission of forward illumination approaches unity, while that of backward illumination is close to zero

[29]. The transmission contrast between forward and backward directions can reach 15 dB when power of incident light is 1.5 kW cm−2 . At communication band, Fan [28] et al. realized UDT within two circular rings made of nonlinear materials, in which the transmission contrast can reach 28 dB when the power of incident light is 0.85 kW cm−2 . However, a drawback of this method is the need of high-power input to induce the nonlinearity of materials, which means that the higher intensity of incident light is needed. Therefore, the intensity threshold used to induce the nonlinear effect will limit the practical utilization. Another method is to use the magneto-optic materials that can break Lorentz symmetry [30–32]. For example, Wang et al. [32] realized UDT in a system composed of a metal grating and a magneto-optic material. The transmission ratio of this configuration between forward and backward directions can reach 83% when external magnetic field of 1.1 T is applied. However, high magnetic field is generally required to generate the magneto-optic effect and the size of this system is too big to be compatible to the integrated system. Although these two approaches mentioned above can realize UDT, they have some intrinsic limitations. The third method is to use the reciprocal linear system to realize UDT. Because of no external magnetic field and low intensity of incident light, it is of great value to realize UDT in the reciprocal linear system. The most commonly used configurations are photonic crystal [33–37], gratings [38–40], mode converter [41, 42], meta-surface [43, 44] and chiral metamaterials [45, 46]. For instance, Wang [33] et al. realized UDT at wavelength 1550 μm in an asymmetric configuration composed of two photonic crystals, where the transmission difference between forward and backward directions can achieve up to two orders and the bandwidth is around 50 nm. Stolarek et al. designed double-layer metal gratings and realized UDT at terahertz band [39]. However, the transmission contrast reported in their work was not high due to the limited thickness of the metal grating. In addition, due to the current limitation of micro-fabrication technology, the size of metal grating in this configuration is difficult to reduce, so UDT is not easy to be realized at optical band. Although reciprocal linear system can realize UDT, the transmission contrast between forward and backward directions is not as high as expected. Besides, all

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1 Introduction

configurations mentioned above are complex, needing sophisticated equipment to fabricate such configurations. Therefore, it is of critical significance to design a simple configuration which can realize UDT with high transmission contrast. At present, most devices of UDT are based on isotropic materials. Compared with isotropic materials, anisotropic materials are more suitable to realize UDT due to the outstanding properties of guiding and controlling the light. For example, anisotropic materials can not only be used to realize the polarization conversion between two kinds of linearly polarized waves, but also be used to realize selective transmission of two linearly polarized waves. Therefore, it might be a good choice to choose anisotropic materials to realize UDT.

1.2 Broadband Perfect Absorption It is of great significance to realize broadband perfect absorption at different wavelength because of its wide applications, such as thermophotovoltaic (TPV), radiative cooling, infrared imaging, and detection [47]. Thermal energy can be converted to electric energy in a TPV system, which is clean, safe, and efficient, but its conversion efficiency needs to be improved. The most effective method is to realize perfect absorption for the radiator at specific wavelength. Radiative cooling is one of the most appealing concepts in the twenty-first century, which has drawn much attention. With radiative cooling technology, the temperature of an object can be cooled below the ambient temperature without applying electricity, which has a positive influence on reducing the global energy consumption and protecting the global environment [48]. The main task of radiative cooling at night is to enhance the emissivity of objects in the transparent window of the atmosphere at 8–13 μm. Many approaches can be used to realize the broadband perfect absorption, such as Fabry–Perot (FP) resonance [49, 50], critical coupling [51, 52], excitation of surface plasmon polaritons (SPPs) [53, 54] and magnetic polaritons (MPs) [55–59]. At present, some detailed works have reported that their configurations based on these methods can realize broadband perfect absorption. For example, Greffet et al. demonstrated that the excitation of surface phonon polaritons (SPhPs) in SiC gratings could realize the perfect absorption [53]. Yang et al. recently reported that the MPs on deep metal gratings could realize perfect absorption [59]. Fan et al. theoretically deduced that graphene could realize perfect absorption at the near infrared and visible band via critical coupling [51]. As for SPhPs and MPs, their excitations require greater transverse wavevector, which is generally induced by grating configurations. In addition, the excitation of SPhPs is sensitive to the incident angle, which means perfect absorption is limited in a rather narrow range of the incident angle. What is more, even though these methods can be used to realize perfect absorption. However, the perfect absorption can usually be realized at specific wavelength. To realize broadband perfect absorption, superimposing resonance might be an appropriate method. Bai et al. realized dual-band infrared perfect absorption via Lshaped metamaterials, in which there were two L-shaped metal blocks in each unit

1.2 Broadband Perfect Absorption

5

[60]. Cui et al. observed perfect absorption at four wavebands using metal gratings, where there were four nanoantennas of different widths in each unit [61]. Although broadband perfect absorption can be realized through superimposing resonances, the bandwidth is still limited. Besides, it is of demanding to design many absorption units in a single period using the present micro-fabrication technology. Xu et al. designed a flat multi-layer structure and realized perfect absorption at 0.1–20 terahertz [62]. The flat multi-layer structure composed of many planar layers, which are made of doped silicon with different doping concentration. The permittivity of this flat multi-layer structure is distributed in a gradient so that impedance matching between each interface can be satisfied and the reflectivity of this structure can be dramatically lowered. However, it is difficult to change the configuration parameter due to the limitation of micro-fabrication technology. As a result, the absorption waveband cannot be shifted from terahertz waveband to visible or infrared waveband. Recently, Zhou et al. reported an absorber whose absorptivity can exceed 0.99 [63]. Through assembling different metal nanoparticles on the nanoporous template, they obtained this absorber and found that the efficiency of perfect absorption is mainly determined by the impedance matching between the excitation of localized surface plasmon polaritons of each nanoparticle and the nano-porous template of low refractivity. As seen from these two examples, impedance matching is an effective method to realize broadband perfect absorption. As for isotropic materials, it is not easy to adjust their impedance while for anisotropic, one can adjust the orientation of optic axis to match their impedance with that of air at specific wavelength. Nefedov et al. realized perfect absorption in a tilted planar structure made of metamaterials [64–66]. There are two main physical mechanisms: impedance matching reducing the reflectivity and the rapid decay of light in materials. This method is especially effective in the ultrathin slab. As reported, it is easy to realize perfect absorption if one of the permittivity components is equal to one and the other component is approximately to minus one. However, it is difficult to design the metamaterials to satisfy such demanding condition. In addition, perfect absorption only occurs at certain wavelength instead of a broadband spectral range. It is also worth noting that these previous works are focused on the single flat structure and thus the absorptivity must be considered when the structure is multilayer. One might obtain broadband absorption by stacking multi-layer slabs.

1.3 Near-Field Radiative Heat Transfer When the distance between two objects of different temperatures is larger than the characteristic wavelength, the radiation heat transfer between them can be derived using Stefan-Boltzmann formula [1]. Supposing the temperature of emitter and receiver are T 1 and T 2 , respectively, and the far-field radiation heat transfer between them can be described as [1]

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1 Introduction

Fig. 1.2 The near-field radiative heat transfer between two objects with different temperature

=

  σ T14 − T24 1 + e12 e1

−1

(1.1)

where e1 and e2 are the emissivity of emitter and receiver, respectively and σ = 5.67 × 10–8 W/(m2 K4 ) is the Stefan-Boltzmann constant. As seen from Eq. (1.1), the far-field radiation heat transfer between the two objects is independent on the distance. However, this formula only works if the distance between these two objects is greater than the characteristic wavelength, where the characteristic wavelength can be deduced from Wien’s displacement law λT =

2897.6 μm K T1

(1.2)

As seen from Fig. 1.2, if the distance between these two objects is equal or less than the characteristic wavelength, the radiation heat transfer cannot be deduced from Eq. (1.1). Previous studies have shown that the near-field radiative heat transfer between two plates could be much greater than the black body radiation heat transfer at the same temperature, and there are two main reasons to explain this phenomenon [67]. On one hand, when the distance between two objects is equal or less than the characteristic wavelength, electromagnetic waves will be emitted from one medium to the other medium, propagating forth and backward within the vacuum gap between these two media, in which the interference might either be constructive or destructive, depending on the phase difference between these reflective wave. On the other hand, when the distance is less than the characteristic wavelength, photon tunneling will result in near-field energy transfer. These two reasons mentioned above can explain why the near-field radiation energy is greater than the black body radiation heat transfer. It is of great significance to introduce this mechanism into non-contact devices, such as near-field imaging, thermal regulation and radiant heat rectifier [68].

1.3 Near-Field Radiative Heat Transfer

7

Theoretically, the radiation heat transfer between two objects can be calculated according to Fluctuation dissipation theorem [69] and Green’s function [70]. The electromagnetic field induced by the random fluctuation current J(r, ω) of the emitter follows Maxwell’s equations [68]. ∇ × E(r, ω) = iωμ0 H(r, ω)

(1.3)

∇ × H(r, ω) = iωε0 εE(r, ω) + J(r, ω)

(1.4)

where ω is angular frequency, and ε0 and μ0 and ε are vacuum permittivity, permeability and the relative permittivity, respectively. One can obtain induced electric field intensity E and magnetic field intensity H respectively in the form of Green’s function and fluctuation current. In anisotropic materials, the correlation function of fluctuation current at different positions can be expressed as [68] 

   4ωε0 (ω, T )εmn    (r, ω)  δ r − r δ ω − ω Jm (r, ω)Jn∗ r , ω = π

(1.5)

where m and n are the subscripts that denote the components of a vector, and < > is the system average. (ω, T ) is the average energy of Planck harmonic oscillator,  which is a function of angular frequency ω and temperature T. εmn is imaginary part of the relative permittivity and δ is Dirac function. Thus, the near-field radiative heat transfer between two slabs of different temperatures T 1 and T 2 can be expressed as ∞ q=

 1  Re E(r, ω) × H∗ (r, ω) dω 2

(1.6)

0

No matter isotropic or anisotropic materials, the near-field radiative heat transfer between them has been widely studied, such as polar materials [71, 72], doped silicon [73–75], hyperbolic materials [76–80], metasurfaces [81, 82], magneto-optic materials [83, 84] and anisotropic magneto-dielectric uniaxial materials [85–87]. Compared with the calculation of near-field radiative heat transfer of isotropic materials, it is more complex to calculate the near-field radiative heat transfer of anisotropic materials because of polarization conversion. When calculating the nearfield radiative heat transfer, the transmissivity and reflectivity can be obtained via transfer matrix and scattering matrix. Experimentally, many research groups have independently demonstrated that the near-field radiative heat transfer is greater than blackbody radiative heat transfer. Watjen et al. measured the near-field radiative heat transfer of two doped-silicon plates and found it eleven times of that of black body [88]. They concluded that the enhanced radiative heat transfer was resulted from the excitation of SPhPs of doped silicon. Lim et al. measured the near-field radiative heat transfer between

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1 Introduction

metal-dielectric multilayer and found that the excitation of SPPs on the interface dramatically enhanced the radiative heat transfer [89]. Yang et al. measured the nearfield radiative heat transfer of graphene sheets and found that the near-filed radiative heat transfer was more than four times of that of black body when the interlamellar spacing was 430 nm [90]. Bernardi et al. measured the near-field radiative heat transfer of silicon plates of 150 nm thickness and found that it was over eight folds of that of black body [91]. Ghashami et al. measured that the near-filed radiative heat transfer between quartz crystals was around forty times of that of black body [92]. As for near-field radiative heat transfer, the radiative heat transfer can be dramatically enhanced via the excitation of different modes. The SPPs excited from metal and SPhPs excited from polar materials can enhance the near-filed radiative heat transfer to a great extent. With the development of anisotropic materials, the research materials of near-filed radiative heat transfer has been gradually shifted to anisotropic materials. Among anisotropic materials, the hyperbolic materials are rather unique because they can support hyperbolic modes, which dramatically enhance the nearfield radiative heat transfer [77]. Most of the hyperbolic materials studied at present are metamaterials with periodic arrangement of metal and dielectric. As for this type of materials, when the wave vector is greater than π /P (P is the periodicity of the metamaterials), the effective dielectric theory does not work anymore. The research on near-field radiative heat transfer of two-dimensional materials has become a hot topic, such as graphene [93–96], hBN [97, 98] and BP [99, 100]. Liu et al. studied the near-field radiative heat transfer between two sheets of hBN, and found that the heat transfer of these two infinite sheets is about one hundred and twenty times of that of black body at the same temperature when the distance between two sheets is 20 nm [97]. The enhanced radiative heat transfer is mainly attributed the excitation of hyperbolic surface phonon polaritons in the first hyperbolic band of hBN. Zhang et al. studied the near-field radiative heat transfer between multilayers of BP [100], and found that the enhanced radiative heat transfer is mainly attributed the excitation of anisotropic SPPs on the surface of BP. Besides, they also studied another three factors that might control the near-field radiative heat transfer, i.e., the orientation of BP, the density of electrons and the layers of BP. Zhao et al. studied the near-field radiative heat transfer between the heterojunction of graphene/hBN, and mainly analyzed the coupling effect between SPPs from the surface of graphene and hyperbolic phonon polaritons (HPPs) of hBN [94]. From their research results, the near-field radiative heat transfer can be effectively controlled via changing the chemical potential of graphene. hBN is a natural hyperbolic material, whose hyperbolic modes excited in two hyperbolic bands can significantly enhance the near-field radiative heat transfer [97]. However, the impact of the orientation of the optical axis on the regulation of the near-field radiative heat transfer has not been fully explored. There are few works on the thermal radiation properties of anisotropic materials, so it is of significance to further study this subject. We have designed different structures to realize unidirectional transmission and broadband perfect absorption. In addition, we have studied the effect of the optic axis of anisotropic materials on

1.3 Near-Field Radiative Heat Transfer

9

the near-field radiative heat transfer. The main work presented in this dissertation is briefly described as follows: In Chap. 2, the transfer matrix method and rigorous couple-wave analysis (RCWA) are introduced in detail. To avoid the numeric overflow problem, the enhanced transmission matrix is applied. The transmission matrix method can be used to calculate the optical properties of anisotropic planar multi-layer structures, and RCWA can be used to calculate the optical properties of gratings made of anisotropic media. All the calculations in this dissertation are based on these two methods. In Chap. 3, three different structures are proposed to realize unidirectional transmission, including the combination of two anisotropic slabs, a combination of an anisotropic slab and a grating, and a combination of an anisotropic slab and a photonic crystal. These three structures can realize the unidirectional transmission of light at different wavelength, and each has its own advantages and disadvantages. The first type of structure is the simplest but can only transmit light in a single direction at a specific wavelength. Although the second structure can realize the unidirectional transmission, the transmission contrast between the forward and backward directions is not as large as expected. The third structure can realize the unidirectional transmission of light with high transmission contrast, but the fabrication is relatively demanding. In Chap. 4, the influence of the orientation of the optic axis of hBN on its direction and hemispheric emissivity is firstly analyzed. The calculation results indicate that the orientation of the optic axis has a great influence on the emissivity of hBN in its two hyperbolic bands. In addition, a structure based on multi-layered hBN slabs to achieve perfect absorption in the spectral band of 1367 – 1580 cm−1 is proposed. In such multi-layer structure, the tilted angles of the optic axes of the slabs are distributed in a gradient, so that the adjacent slabs can meet impedance matching, achieving weak reflection but strong absorption at the interface. Finally, the similar designed structure is applied in the metamaterials composed of doped silicon nanowires and TPX matrix, and the absorptance of this system can reach 0.99 in the band from 2 to 50 μm. In Chap. 5, the influence of the optic axis orientation on the near-field radiative heat transfer between anisotropic materials is studied. The numerical results show that the orientation of optic axis has a great influence on the near-field radiative heat transfer. Besides, the near-field radiative heat flux between two semi-infinite hBN slabs reaches its maximum when the orientation of optic axis is along the direction of heat transfer. It is found that the enhanced near-field radiative heat flux between two semi-infinite hBN slabs originates from the excitation of hyperbolic phonon polaritons and hyperbolic surface phonon polaritons. In addition, the near-field radiative heat flux between two graphene-covered semi-infinite hBN slabs is maximized when the orientation of the optic axis is perpendicular to the direction of heat transfer. It is demonstrated that the enhanced near-field radiative heat flux between two graphenecovered semi-infinite hBN plates is mainly due to the excitation of surface plasmon polaritons at the air/graphene and the graphene/hBN interfaces. In Chap. 6, the summary and outlook and the innovation of this dissertation is given.

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48. Raman AP, Anoma MA, Zhu LX, Rephaeli E, Fan SH (2014) Passive radiative cooling below ambient air temperature under direct sunlight. Nature 515:540–550 49. Wang LP, Lee BJ, Wang XJ, Zhang ZM (2009) Spatial and temporal coherence of thermal radiation in asymmetric Fabry-Perot resonance cavities. Int J Heat Mass Transf 52(13–14):3024–3031 50. Wu JP, Guo J, Wang X, Jiang LY, Dai XY, Xiang YJ, Wen SC (2017) Dual-band infrared nearperfect absorption by Fabry-Perot resonances and surface phonons. Plasmonics 13(6876):1–7 51. Piper JR, Fan SH (2014) Total absorption in a graphene monolayer in the optical regime by critical coupling with a photonic crystal guided resonance. ACS Photonics 1(4):347–353 52. Jiang X, Wang T, Xiao S, Yan X, Cheng L, Zhong Q (2018) Approaching perfect absorption of monolayer molybdenum disulfide at visible wavelengths using critical coupling. Nanotechnology 29(33):335205 53. Greffet JJ, Carminati R, Joulain K, Mulet JP, Mainguy S, Chen Y (2002) Coherent emission of light by thermal sources. Nature 416(6876):61 54. Marquier F, Joulain K, Mulet JP, Carminati R, Greffet JJ, Chen Y (2004) Coherent spontaneous emission of light by thermal sources. Phys Rev B 69(15):155412 55. Lee BJ, Wang LP, Zhang ZM (2008) Coherent thermal emission by excitation of magnetic polaritons between periodic strips and a metallic film. Opt Exp 16(15):11328–11336 56. Zhao B, Zhao JM, Zhang ZM (2014) Enhancement of near-infrared absorption in graphene with metal gratings. Appl Phys Lett 105(3):031905 57. Wang LP, Haider A, Zhang ZM (2014) Effect of magnetic polaritons on the radiative properties of inclined plate arrays. J Quant Spectrosc Radiat Transfer 132:52–60 58. Bai Y, Zhao L, Ju DQ, Jiang YY, Liu LH (2015a) Wide-angle, polarization-independent and dual-band infrared perfect absorber based on L-shaped metamaterial. Opt Exp 23(7):8670– 8680 59. Yang PY, Ye H, Zhang ZM (2019) Experimental demonstration of the effect of magnetic polaritons on the radiative properties of deep aluminum gratings. J Heat Transfer 141(5):052702 60. Bai Y, Zhao L, Ju DQ, Jiang YY, Liu LH (2015b) Wide-angle, polarization-independent and dual-band infrared perfect absorber based on L-shaped metamaterial. Opt Exp 23(7):8670– 8680 61. Cui YX, Xu J, Fung KH, Jin Y, Kumar A, He SL, Fang NX (2011) A thin film broadband absorber based on multi-sized nanoantennas. Appl Phys Lett 99:253101 62. Xu GJ, Zhang J, Zang XF, Sugihara O, Zhao HW, Cai B (2016) 0.1–20 THz ultra-broadband perfect absorber vai a flat multi-layer structure. Opt Exp 24(20):23177–23185 63. Zhou L, Tan YL, Ji DX, Zhu B, Zhang P, Xu J, Gan QQ, Yu ZF, Zhu J (2016) Self-assembly of highly efficient, broadband plasmonic absorbers for solar steam generation. Sci Adv 2(4):e1501227 64. Hashemi SM, Nefedov IS (2012) Wideband perfect absorption in arrays of tilted carbon nanotubes. Phys Rev B 86:195411 65. Nefedov IS, Valagiannopoulos CA, Hashemi SM, Nefedov EI (2013) Total absorption in asymmetric hyperbolic media. Sci Rep 3:2662 66. Nefedov IS, Valagiannopoulos CA, Melnikov LA (2013) Perfect absorption in graphene multilayers. J Opt 15(11):114003 67. Basu S, Zhang ZM, Fu CJ (2009) Review of near-field thermal radiation and its applications to energy conversion. Int J Energ Res 33(13):1203–1232 68. Liu XL, Wang LP, Zhang ZM (2015) Near-field thermal radiation: recent progress and outlook. Nanoscale Microscale Thermophys Eng 19:98–126 69. Polder D, Hove MV (1971) Theory of radiative heat transfer between closely spaced bodies. Phys Rev B 4(10):3303–3314 70. Sipe JE (1987) New Green-function formalism for surface optics. J Opt Soc Am B B 4(4):481– 489 71. Liu XL, Zhang ZM (2014) Graphene-assisted near-field radiative heat transfer between corrugated polar materials. Appl Phys Lett 104:251911

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72. Messina R, Abdallah PB, Guizal B, Antezza M (2017) Graphene-based amplification and tuning of near-field radiative heat transfer between dissimilar polar materials. Phys Rev B 96:045402 73. Fu CJ, Zhang ZM (2006) Nanoscale radiation heat transfer for silicon at different doping levels. Int J Heat Mass Transf 49(9–10):1703–1718 74. Liu XL, Zhang RZ, Zhang ZM (2014) Near-field radiative heat transfer with doped-silicon nanostructured metamaterials. Int J Heat Mass Transf 73:389–398 75. Liu XL, Zhao B, Zhang ZM (2015) Enhanced near-field thermal radiation and reduced Casimir stiction between doped-Si gratings. Phys Rev A 91:062510 76. Biehs SA, Tschikin M, Abdallah PB (2012) Hyperbolic metamaterials as an analog of a blackbody in the near field. Phys Rev Lett 109:104301 77. Biehs SA, Tschikin M, Messina R, Abdallah PB (2013) Super-Planckian near-field thermal emission with phonon-polaritonic hyperbolic metamaterials. Appl Phys Lett 102:131106 78. Liu XL, Zhang RZ, Zhang ZM (2013) Near-field thermal radiation between hyperbolic metamaterials: graphite and carbon nanotubes. Appl Phys Lett 103:213102 79. Biehs SA, Abdallah PB (2017) Near-field heat transfer between multilayer hyperbolic metamaterials. Zeitschrift Fur Naturforchung A 72(2):115–127 80. Messina R, Abdallah PB, Guizal B, Antezza M, Biehs SA (2016) Hyperbolic waveguide for long-distance transport of near-field heat flux. Phys Rev B 94:104301 81. Liu XL, Zhang ZM (2015) Near-field thermal radiation between metasurfaces. ACS Photonics 2(9):1320–1326 82. Hurtado VF, Vidal FJG, Fan SH, Cuevas JC (2017) Enhancing near-field radiative heat transfer with Si-based metasurfaces. Phys Rev Lett 118:203901 83. Zhu LX, Fan SH (2016) Persistent directional current at equilibrium in nonreciprocal manybody near field electromagnetic heat transfer. Phys Rev Lett 117:134303 84. Villa EM, Hurtado VF, Vidal FJG, Martin AG, Cuevas JC (2015) Magnetic field control of near-field radiative heat transfer and the realization of highly tunable hyperbolic thermal emitters. Phys Rev B 92:125418 85. Wu HH, Huang Y, Zhu KY (2015) Near-field radiative transfer between magneto-dielectric uniaxial anisotropic media. Opt Lett 40(19):4532–4535 86. Song JL, Cheng Q (2016) Near-field radiative heat transfer between graphene and anisotropic magneto-dielectric hyperbolic metamaterials. Phys Rev B 94:125419 87. Song JL, Lu L, Cheng Q, Luo ZX (2018) Three-body heat transfer between anisotropic magneto-dielectric hyperbolic metamaterials. J Heat Transfer 140(8):082005 88. Watjen JI, Zhao B, Zhang ZM (2016) Near-field radiative heat transfer between doped-Si parallel plates separated by a spacing down to 200 nm. Appl Phys Lett 109:203112 89. Lim M, Song J, Lee SS, Lee BJ (2018) Tailoring near-field thermal radiation between metallicdielectric multilayers using coupled surface plasmon polaritons. Nat Commun 9(1):4302 90. Yang J, Du W, Su YS, Fu Y, Gong SX, He SL, Ma YG (2018) Observing of the super-Planckian near-field thermal radiation between graphene sheets. Nat Commun 9(1):4033 91. Bernardi MP, Milovich D, Francoeur M (2016) Radiative heat transfer exceeding the blackbody limit between macroscale planar surfaces separated by a nanosize vacuum gap. Nat Commun 7:12900 92. Ghashami M, Geng HY, Kim T, Iacopino N, Cho SK, Park K (2018) Precision measurement of phonon-polaritonic near-field energy transfer between macroscale planar structures under large thermal gradients. Phys Rev Lett 120:175901 93. Zhao B, Zhang ZM (2017c) Enhanced photon tunneling by surface plasmon-phonon polaritons in graphene/hBN heterostructures. J Heat Transfer 139(2):022701 94. Zhao B, Guizal B, Zhang ZM, Fan SH, Antezza M (2017) Near-field heat transfer between graphene/hBN multilayers. Phys Rev B 95:245437 95. Shi KZ, Liao R, Cao GJ, Bao FL, He SL (2018) Enhancing thermal radiation by grapheneassisted hBN/SiO2 hybrid structures at the nanoscale. Opt Exp 26(10):A591–A601 96. Shi KZ, Bao FL, He SL (2017) Enhanced near-field thermal radiation based on multilayer graphene-hBN heterostructures. ACS Photonics 4(4):971–978

14

1 Introduction

97. Liu XL, Xuan YM (2016) Super-Planckian thermal radiation enabled by hyperbolic surface phonon polaritons. Sci China Technol Sci 59(11):1680–1686 98. Liu XL, Shen JD, Xuan YM (2017) Pattern-free thermal modulator via thermal radiation between van der Waals materials. J Quant Spectrosc Radiat Transfer 200:100–107 99. Shen JD, Guo S, Liu XL, Liu BA, Wu WT, He H (2018) Super-planckian thermal radiation enabled by coupling quasi-elliptic 2D black phosphorus plasmons. Appl Therm Eng 144:403– 410 100. Zhang Y, Yi HL, Tan HP (2018) Near-field radiative heat transfer between black phosphorus sheets via anisotropic surface plasmon polaritons. ACS Photonics 5(9):3739–3747

Chapter 2

Calculation Method for Slab and Grating Structure Made of Anisotropic Materials

In isotropic slab, the transverse electric (TE) wave and transverse magnetic (TM) wave are decoupled, and thus they can be solved separately. Traditional 2 × 2 transfer matrix method can be employed to calculate the reflection, transmission, as well as absorption of multilayer slabs made of isotropic materials. However, the TM and TE waves are always coupled in anisotropic slab, which requires the 4 × 4 transfer matrix method (TMM) to calculate the radiative properties. When there is evanescent wave, the numerical overflow problem is possible to occur in the 4 × 4 TMM. It has been reported that the scattering matrix method could be used to suppress this problem, but it does not work when the thickness of the slab is very larger. Therefore, it is necessary to find a method to completely solve the numerical flow problem in the 4 × 4 TMM. Besides, RCWA is a relatively straightforward technique for obtaining the exact solution of Maxwell’s equations for the electromagnetic diffraction by grating structures. Compared with commercial software FDTD Solutions and COMSOL Multiphysics, the RCWA do not need to deal with mesh process, which is more convenient to solve nanoscale structures. Moharam et al. have done extensively research about the RCWA of isotropic materials. Moreover, they proposed that the enhanced transmittance matrix method could be used to eliminate the numerical flow problem. Although the RCWA for anisotropic materials has been studied, however, the numerical overflow problem has not attracted much attention. To study the optical properties of planar or grating structure made of anisotropic materials, it is very necessary to remove the numerical overflow problem in the 4 × 4 TMM and RCWA for anisotropic materials. In the following, we will present these two methods in detail. In addition, we further expand the enhanced transmittance matrix method presented in paper [1] to ensure that there is no numerical overflow problem in our methods.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 X. Wu, Thermal Radiative Properties of Uniaxial Anisotropic Materials and Their Manipulations, Springer Theses, https://doi.org/10.1007/978-981-15-7823-6_2

15

16

2 Calculation Method for Slab and Grating Structure Made …

2.1 4 × 4 Transfer Matrix Method As shown in Fig. 2.1, we first consider single slab made of anisotropic material. Supposing the permittivity tensor of the anisotropic material is ⎞ εx x εx y εx z ε = ⎝ ε yx ε yy ε yz ⎠ εzx εzy εzz ⎛

(2.1)

Here we consider x-z plane as the plane of incidence. When the incident light is TM wave, the electromagnetic fields in the anisotropic material are   H = U(z) exp( jωt − jβx), U = Ux , U y , Uz

(2.2)

  1 2   E = j μ0 ε0 / S(z) exp( jωt − jβx), S = Sx , S y , Sz

(2.3)

where β is the wavevector component along the x-axis, ω is the angular frequency.

The wavevector component in the z-axis in the vacuum is k z = k02 − β 2 . ε0 and μ0 are the permittivity and permeability in vacuum, respectively. Ux , U y and Uz are the magnetic fields in three directions, Sx , S y and Sz are the electric fields in three directions.  Submitting Eqs. (2.1–2.3) into Maxwell’s equation, letting K x = β k0 , one can get ⎛

⎞ ⎛ ⎞ Sx Sx ⎟ ⎟ ⎜ d ⎜ ⎜ S y ⎟ = k0 A⎜ S y ⎟ ⎝ ⎠ ⎝ Ux ⎠ dz Ux Uy Uy where the coefficient is Fig. 2.1 The single slab made of anisotropic material with thickness of d, and the plane of incidence is x-z plane

(2.4)

2.1 4 × 4 Transfer Matrix Method

17

⎡

j K x εzy /εzz j K x εzx /εzz ⎢ 0 0 A=⎢ ⎣ ε yz εzx /εzz − ε yx ε yz εzy /εzz + K 2 − ε yy x εx x − εx z εzx /εzz εx y − εx z εzy /εzz

⎤ 0 K x2 /εzz − 1 ⎥ 1 0 ⎥ 0 − j K x ε yz /εzz ⎦ 0 j K x εx z /εzz

(2.5)

The electromagnetic field inside the media can be expressed using the eigenvalue and eigenvectors of matrix coefficient A. Supposing the thickness of the slab is d, the solution of Eq. (2.4) can be expressed as Sx (z) =

2 

w1,m cm+ exp(k0 qm z) +

m=1

S y (z) =

2 

Ux (z) =

w2,m cm+ exp(k0 qm z) +

2 

2 

(2.6)

  w2,m+2 cm− exp k0 qm+2 (z − d)

(2.7)

  w3,m+2 cm− exp k0 qm+2 (z − d)

(2.8)

  w4,m+2 cm− exp k0 qm+2 (z − d)

(2.9)

m=1

w3,m cm+

exp(k0 qm z) +

m=1

U y (z) =

  w1,m+2 cm− exp k0 qm+2 (z − d)

m=1

m=1 2 

2 

2  m=1

w4,m cm+ exp(k0 qm z) +

m=1

2  m=1

where wi,m is the elements of eigenvectors W of A, qm is the eigenvalues of A. The real parts of q1 and q2 are negative, and the real parts of q1 and q2 are positive. cm+ and cm− are unknown constants, which can be solved by applying boundary conditions. The boundary conditions are that the tangential electromagnetic fields at the two sides of each interface of the slab should be the same. In the upper side of the slab, the boundary conditions are  ⎞ ⎛  ⎞ − jk z k0 jk z k0 0     ⎟ ⎜ ⎟ ⎜ 0  +  0 − ⎜ ⎟+⎜  j ⎟ r pp = W1 W2 X C− ⎝ ⎠ ⎝ 0 C 0 k z k0 ⎠ r ps 1 0 1 ⎛

(2.10)

In the lower side of the slab, the boundary conditions are  ⎞ 0 − jk z k0    + ⎜   C 0 −j ⎟ ⎟ t pp ⎜ = W1 Y W2 ⎝ C− 0 −k z k0 ⎠ t ps 1 0 ⎛

(2.11)

  where W = W1 W2 is the eigenvectors of A, C+ and C− are unknown constants, X is diagonal matrix with elements exp(−k0 qm d), m = 3, 4; Y is diagonal matrix

18

2 Calculation Method for Slab and Grating Structure Made …

with element exp(−k0 qm d), m = 3, 4. r pp and r ps are the TM and TE components in the reflected wave, t pp and t ps are the TM and TE components in the transmitted wave. It should be noting that when the permittivity is a diagonal matrix, Eq. (2.4) can be decoupled into      d Uy 0 ε Uy x x  = k0 2 K x εzz − 1 0 Sx dz Sx

(2.12)

     d Sy 0 1 Sy = k0 Ux K x2 − ε yy 0 dz Ux

(2.13)

and

It means that there is no polarization conversion between TM and TE waves, and they can be solved separately. Likewise, when the incident light is TE wave, the electromagnetic fields in the media can be expressed as   E = S(z) exp( jωt − jβx), S = Sx , S y , Sz   1 2   H = − j ε0 μ0 / U(z) exp( jωt − jβx), U = Ux , U y , Uz

(2.14) (2.15)

Submitting Eqs. (2.1), (2.14) and (2.15) into Maxwell’s equations, one can get the same differential equations as Eq. (2.4). The boundary conditions of in the upper and lower sides of the slab are respectively  ⎞ ⎛ ⎞ −k z k0 0 0     ⎜ ⎜  +  0 1 ⎟ 1 ⎟ ⎟+⎜ ⎟ rsp = W1 W2 X C ⎜ ⎝ − jk z k0 ⎠ ⎝ C− 0 jk z k0 ⎠ rss 0 j 0 ⎛

(2.16)

and ⎞ ⎛  0 k z k0    + ⎜ 0   C 1 ⎟ ⎟ tsp ⎜ = W1 Y W2 ⎝ 0 − jk z k0 ⎠ tss C− j 0

(2.17)

where r sp and r ss are the TM and TE components in the reflected wave, while t sp and t ss are the TM and TE components in the transmitted wave. We can apply the above analysis to arbitrary L-layer structure. Taking TM wave for an example, all the boundary conditions are

2.1 4 × 4 Transfer Matrix Method

19

 ⎞ ⎛  ⎞ − jk z k0 jk z k0 0     ⎟ r pp ⎜ ⎟ ⎜ 0   C+ 0 − j (1) ⎟ ⎜ ⎟+⎜  = W(1)1 W(1)2 X(1) (2.18) ⎝ ⎠ ⎝ 0 C− 0 k z k0 ⎠ r ps (1) 1 0 1      C+  C+   (l−1) (l) = W(l)1 W(l)2 X(l) (2.19) W(l−1)1 Y(l−1) W(l−1)2 C− C− (l−1) (l)  ⎞ ⎛ − jk z k0 0     + ⎜  C(L)  0 −j ⎟ ⎟ t pp ⎜ (2.20) = W(L)1 Y(L) W(L)2 ⎝ C− 0 −k z k0 ⎠ t ps (L) 1 0 ⎛

where l = 2, 3, …, L. W(l), X(l) and Y(l) have the same definition as W, X and Y. For each l, there is W(l) = W(l)1 W(l)2 . It is possible to have numerical overflow problem by straightly solving the Eqs. (2.18–2.20). To remove the numerical overflow problem, we adopt the enhanced transmittance approach [1]. From Eq. (2.20), we can get 

C+ (L) C− (L)



  −1 f L+1  t = W(L)1 Y(L) W(L)2 g L+1

(2.21)

where  t=

       t pp − jk z k0 0 0 −k z k0 , f L+1 = , g L+1 = t ps 1 0 0 −j

When Y(L) is very small, the matrix in the right side of Eq. (2.21) is an ill matrix, which may cause numerical flow under inverse operation. To ensure that there is no numerical flow problem in the inverse operation, we suggest 

W(L)1 Y(L) W(L)2

−1

 =

 −1 −1 0  Y(L) W(L)1 W(L)2 0 I

(2.22)

Therefore, Eq. (2.21) can be rewritten as 

C+ (L) C− (L)



 =

−1 0 Y(L) 0 I



 aL t bL

where I is the unit matrix.      −1 f L+1 aL = W(L)1 W(L)2 bL g L+1

(2.23)

(2.24)

20

2 Calculation Method for Slab and Grating Structure Made …

Letting t = a−1 L Y(L) t L , submitting it into Eq. (2.23), one can get 

C+ (L) C− (L)



 =

I b L a−1 L Y(L)

 tL

(2.25)

When l = L, submitting Eq. (2.23) into Eq. (2.19), one can get 

C+ (L−1) C− (L−1)





= W(L−1)1 Y(L−1)

  −1 f L tL W(L−1)2 gL

(2.26)

where 

fL gL



= W(L)1 + W(L)2 X(L) b L a−1 L Y(L)

(2.27)

For each slab, repeating the above process, one can finally get  ⎞ ⎛  ⎞ − jk z k0 jk z k0 0     ⎟ ⎜ ⎟ ⎜ 0 0 − ⎜ ⎟+⎜  j ⎟r = f1 t1 , r = r pp ⎝ ⎠ ⎝ 0 g1 r ps 0 k z k0 ⎠ 1 0 1 ⎛

(2.28)

Solving Eq. (2.28), one can get r pp , r ps and t1 . The two transmission coefficients are     1 t pp = am−1 Y(m) t1 (2.29) t ps m=L

When the incident light is TE wave, the process is the same, and we do not repeat here. Besides, the above method can obtain transmission and reflection coefficients at the same time. Under some particular cases, only transmission coefficients or reflection coefficients are important. To solver the transmission coefficients or reflection coefficients more effectively, we can adopt the following method. Equation (2.20) can be rewritten as 

−W(L)12 f L+1 −W(L)22 g L+1



C− (L) C+ (L+1)



 W(L)11 Y(L) + C = W(L)21 Y(L) (L) 

(2.30)

T  where C+ . From Eq. (2.30), we can see that the relation between C− (L) = t pp t ps (L) + and C(L) is + C− (L) = a L C(L)

(2.31)

2.1 4 × 4 Transfer Matrix Method

21

where 

aL bL





−W(L)12 f L+1 = −W(L)22 g L+1

−1 

W(L)11 Y(L) W(L)21 Y(L)

 (2.32)

The right side of Eq. (2.19) can be rewritten as 

W(l)11 W(l)12 X(l) W(l)21 W(l)22 X(l)



C+ (l) C− (l)



 =

 fL C+ (l) gL

(2.33)

where 

fL gL





W(l)11 + W(l)12 X(l) a L = W(l)21 + W(l)22 X(l) a L

 (2.34)

Equation (2.19) can be rewritten as 





 W(L−1)11 Y(L−1) + C W(L−1)21 Y(L−1) (L−1)

(2.35)

 ⎞ ⎛  ⎞ − jk z k0 jk z k0 0     ⎟ ⎜ ⎟ ⎜ 0 0 − ⎜ ⎟+⎜  j ⎟ r pp = f1 C+ ⎝ ⎠ ⎝ 0 g1 (1) 0 k z k0 ⎠ r ps 1 0 1

(2.36)

−W(L−1)12 f L −W(L−1)22 g L

C− (L−1) C+ (L)

 =

Repeating the above process, one can get ⎛

Solving Eq. (2.36), we can get reflection coefficients. The above analysis is applied for single slab with finite thickness and multilayer structures. If considered structure is semi-infinite, the above process can be further simplified. For semi-infinite slab, the electromagnetic fields in the media can be expressed as Sx (z) = w1,1 c1 exp(k0 q1 z) + w1,2 c2 exp(k0 q2 z)

(2.37)

S y (z) = w2,1 c1 exp(k0 q1 z) + w2,2 c2 exp(k0 q2 z)

(2.38)

Ux (z) = w3,1 c1 exp(k0 q1 z) + w3,2 c2 exp(k0 q2 z)

(2.39)

U y (z) = w4,1 c1 exp(k0 q1 z) + w4,2 c2 exp(k0 q2 z)

(2.40)

22

2 Calculation Method for Slab and Grating Structure Made …

where wi,m is the elements in the eigenvectors W of coefficient A. The imaginary parts of q1 and q2 are negative. When the incident light is TM wave, the boundary conditions of the surface are   ⎞ ⎞⎛ ⎞ ⎛ ⎛ jk z k0 r pp jk z k0 0 w11 w12 ⎟⎜ ⎟ ⎜ ⎜ 0 ⎟ − ⎜  j w21 w22 ⎟⎜ r ps ⎟ = ⎜ 0 ⎟ (2.41) ⎝ ⎝ ⎠ ⎠ ⎝ 0 k z k0 w31 w23 −c1 0 ⎠ 1 0 w41 w42 −c2 −1 When the incident light is TE wave, the boundary conditions of the surface are  −k z k0 0 ⎜ 0 1 ⎜ ⎝ 0 jk z k0 j 0 ⎛

w11 w21 w31 w41

⎞ ⎞⎛ ⎞ ⎛ rsp 0 w12 ⎜ ⎟ ⎟ ⎜ w22 ⎟ ⎟⎜ rss ⎟ = ⎜ −1  ⎟ ⎝ ⎝ ⎠ ⎠ jk z k0 ⎠ w23 −c1 w42 −c2 0

(2.42)

2.2 Rigorous Coupled-Wave Analysis The one-dimensional grating along the x-axis is shown in Fig. 2.2, the period, filling ratio, and height of the grating are Λ, f and d, respectively. We first consider single grating structure, later we consider multilayer grating structure. Here we consider conical incidence, and TM wave and TE wave are two special cases of the conical incidence. In the incident media, the incident electric fields can be expressed as E inc = u exp[− jk0 n 1 (sin θ cos φx + sin θ sin φy + cos θ z)] where Fig. 2.2 The single grating made of anisotropic material with thickness of d, and the plane of incidence is x-z plane

(2.43)

2.2 Rigorous Coupled-Wave Analysis

23

u = (cos ψ cos θ cos φ − sin ψ sin φ exp( jη))x + (cos ψ cos θ sin φ + sin ψ cos φ exp( jη)) y − cos ψ sin θ z

(2.44)

where ψ is the angle between the electric field vector and the plane of incidence. φ and θ are azimuthal angle and angle of incidence, respectively, η is the phase different between the electric field in the plane and out the plane. The electric fields in the incident and transmitted media (refractive index is n2 ) can be expressed as 

   Ri exp − j k xi x + k y y − k1,zi z

(2.45)

   Ti exp − j k xi x + k y y + k2,zi (z − d)

(2.46)

E1 = Einc +

i

and E2 =

 i

where

 kl,zi =

    k xi = k0 n 1 sin θ cos φ − i λ0 Λ

(2.47)

k y = k0 n 1 sin θ sin φ

(2.48)

 1 2 2 + (k0 n l )2 − k xi − k 2y /  2 1 2 − j k xi + k 2y − (k0 n l )2 /

 2  k xi + k 2y < (k0 n l )2 , l = 1, 2  2  k xi + k 2y > (k0 n l )2

(2.49)

Ri is the normalized vector electric field amplitude of the ith reflected wave in region I, Ti is the normalized electric field vector amplitude of the ith transmitted wave in region II. The magnetic field vectors in region I and II can be obtained from Maxwell’s equations. Note that the output plane of diffraction for the ith propagating diffraction order has an inclination angle given by    ϕi = tan−1 k y k xi

(2.50)

Here we consider that the permittivity of the anisotropic material is ⎛

⎞ εx x εx y 0 ε = ⎝ ε yx ε yy 0 ⎠ 0 0 εzz The electromagnetic fields in the anisotropic media can be expressed as

(2.51)

24

2 Calculation Method for Slab and Grating Structure Made …

Eg =



    Sxi (z)x + S yi (z)y + Szi (z)z exp − j k xi x + k y y

(2.52)

i



ε0 Hg = − j μ0

1/ 2 

     Uxi (z)x + U yi (z)y + Uzi (z)z exp − j k xi x + k y y

i

(2.53) Submitting Eqs. (2.51–2.53) into Maxwell’s equations, one can get ⎛

⎞ ⎛ ⎞ Ux Ux ⎟ ⎟ ⎜ d ⎜ ⎜ Sx ⎟ = k0 A⎜ Sx ⎟ ⎝ ⎠ ⎝ Sy ⎠ dz S y Uy Uy

(2.54)

where the coefficient A is     −1 ⎤ o o −Kx K y − E ε yx Kx2 − E 1 ε yy −1 ⎢ −Kx E(εzz )−1 K y o o Kx E(εzz ) Kx − I ⎥ ⎥ A=⎢ ⎣ I − K y E(εzz )−1 K y o o K y E(εzz )−1 Kx ⎦ −1     2 o E 1 εx x − K y E εx y + K y Kx o ⎡

where E(εzz ) is the matrix formed by the permittivity harmonic components, with  the i, p element being equal to εzz(i− p) . εzz(h) = (εzz − 1) sin(π h f ) π h is the hth Fourier component of the relative permittivity in the grating the 0th Fourier   region, component is εzz(0) = f + (1 − f ). The definition of E εx y is similar to that of E(εzz ). The inverse of the permittivity εx x is used for the sake of fast and guaranteed convergence according to the Fourier factorization rule [2, 3]. n is the number of space harmonics retained in the field expansion. The size of A is 4n × 4n, and thus the matrix has 4n eigenvalues. The solutions of the Eq. (2.54) can be expressed as 2n 

Uxi (z) =

m=1

Sxi (z) =

2n 

wi,m cm+ exp(k0 qm z) +

2n 

  wi,m+2n cm− exp k0 qm+2n (z − d)

(2.55)

m=1

wi+n,m cm+ exp(k0 qm z) +

m=1

2n 

  wi+n,m+2n cm− exp k0 qm+2n (z − d)

m=1

(2.56) S yi (z) =

2n  m=1

wi+2n,m cm+ exp(k0 qm z) +

2n 

  wi+2n,m+2n cm− exp k0 qm+2n (z − d)

m=1

(2.57)

2.2 Rigorous Coupled-Wave Analysis

U yi (z) =

2n 

25

wi+3n,m cm+ exp(k0 qm z) +

m=1

2n 

  wi+3n,m+2n cm− exp k0 qm+2n (z − d)

m=1

(2.58) where wi,m is the elements of eigenvectors W of A, qm is the eigenvalues of A. The real parts of qm (m = 1, 2, …, 2n) are negative, and the real parts of qm (m = 2n + 1, 2n + 2, …, 4n) are positive. cm+ and cm− are unknown constants, which can be solved by applying boundary conditions. In the upper side of the grating, the boundary conditions are ⎤ ⎞ ⎡ I 0 sin ψ exp( jη)δi0     +  ⎜ j sin ψn 1 cos θ exp( jη)δi0 ⎟ ⎢ − jY1 0 ⎥ Rs I 0 c ⎥ ⎟+⎢ ⎜ = BW ⎦ ⎠ ⎣ ⎝ cos ψ cos θ δi0 Rp c− 0 − jZ1 0X − j cos ψn 1 δi0 0 I (2.59) ⎛ ⎞ o −Fs Fc o ⎜ −Fc o o −Fs ⎟ ⎟ B=⎜ (2.60) ⎝ o Fc Fs o ⎠ Fs o o −Fc ⎛

where Fc and Fs are diagonal matrices with the diagonal elements cos ϕi and sin ϕi , respectively. When the plane of incidence is x-z plane, there is ϕi = 0o . As such, Fc is the identity matrix, and Fs is the zero matrix. The boundary conditions of in the lower side of the grating are  BW

Y0 0 I



 +

c c−

⎡

I ⎢ jY2 =⎢ ⎣ 0 0

⎤ 0   0 ⎥ ⎥ Ts jZ2 ⎦ T p I

(2.61)

where W is the eigenvectors of A, C+ and C− are unknown constants X is diagonal matrix with elements exp(−k0 qm d), m = 1, 2, …, 2n; Y is diagonal matrix with element exp(−k0 qm d), m = 2n + 1, 2n + 2, …, 4n. We can apply the above analysis to arbitrary L-layer grating. For L-layer grating, all the boundary conditions are ⎤ ⎞ ⎡ I 0 sin ψ exp( jη)δi0    +   ⎜ j sin ψn 1 cos θ exp( jη)δi0 ⎟ ⎢ − jY1 0 ⎥ Rs I 0 c1 ⎥ ⎟+⎢ ⎜ ⎠ ⎣ 0 − jZ1 ⎦ R p = BW1 0 X1 ⎝ cos ψ cos θ δi0 c1− − j cos ψn 1 δi0 0 I (2.62) ⎛

26

2 Calculation Method for Slab and Grating Structure Made …



 +  I 0 cl+1 = BWl+1 − cl+1 0 Xl+1 ⎡ ⎤ I 0     +  ⎢ jY2 0 ⎥ Ts cL YL 0 ⎢ ⎥ BW L = ⎣ 0 jZ2 ⎦ T p c− 0 I L 0 I

Yl 0 BWl 0 I



cl+ cl−





(2.63)

(2.64)

where l = 2, 3, …, L. W(l) , X(l) and Y(l) have the same definition as W, X and Y. To remove the numerical overflow problem, we adopt the enhanced transmittance approach [1]. From Eq. (2.64), we can get 

c+ L c− L





   0 Y−1 −1 −1 f L+1 L = T WL B g L+1 0 I

(2.65)

where  T=

     I 0 0 jZ2 Ts , f L+1 = , g L+1 = Tp 0 I jY2 0

Equation (2.65) can be rewritten as 

c+ L c− L





Y−1 L 0 = 0 I



 aL T bL

(2.66)

where 

aL bL



−1 = W−1 L B



f L+1 g L+1

 (2.67)

Letting t = a−1 L Y(L) t L , and submitting it into Eq. (2.66), one can get 

c+ L c− L



 =

I b L a−1 L YL

 TL

(2.68)

Submitting Eq. (2.68) into Eq. (2.63), one can get 

c+ L−1 c− L−1



   0 Y−1 −1 −1 f L L−1 TL W L−1 B = gL 0 I 

(2.69)

where 

fL gL



 = BW

I X L b L a−1 L YL

 (2.70)

2.2 Rigorous Coupled-Wave Analysis

27

Repeating the above process for each grating, one can finally get ⎤ ⎞ ⎡ I 0 sin ψ exp( jη)δi0     ⎜ j sin ψn 1 cos θ exp( jη)δi0 ⎟ ⎢ − jY1 0 ⎥ Rs f1 ⎥ ⎜ ⎟+⎢ ⎝ ⎠ ⎣ 0 − jZ1 ⎦ R p = g1 T1 cos ψ cos θ δi0 − j cos ψn 1 δi0 0 I ⎛

(2.71)

Solving Eq. (2.71), one can get Ts , Tp and T1 . The transmission coefficients are 

Ts Tp



 =

1 

 am−1 Ym

T1 .

(2.72)

m=L

The diffraction efficiencies are defined as 

  k1,zi n 21 k0 n 1 cos θ        2 2 k2,zi n 22 k 2,zi + T p,i  Re DEti = Ts,i  Re k0 n 1 cos θ k0 n 1 cos θ

 2 DEri =  Rs,i  Re



k1,zi k0 n 1 cos θ



 2 +  R p,i  Re

(2.73)

(2.74)

The above analysis is applied for arbitrarily polarized light. TE and TM waves are commonly considered incident light. When the incident light is TE wave, there is ψ = 90o and η = 0, Eq. (2.62) can be simplified into ⎤ ⎞ ⎡ δi0 I 0    +   ⎜ jn 1 cos θ δi0 ⎟ ⎢ − jY1 0 ⎥ Rs I 0 c1 ⎥ ⎜ ⎟+⎢ = BW 1 ⎝ ⎠ ⎣ 0 − jZ1 ⎦ R p 0 c1− 0 X1 0 I 0 ⎛

(2.75)

When the incident light is TM wave, there is ψ = 0o and η = 0, Eq. (2.62) can be simplified into ⎤ ⎞ ⎡ I 0 0     +  ⎟ ⎢ − jY1 0 ⎥ Rs ⎜ I 0 c1 0 ⎥ ⎟+⎢ ⎜ = BW 1 ⎝ cos θ δi0 ⎠ ⎣ 0 − jZ1 ⎦ R p c1− 0 X1 − jn 1 δi0 0 I ⎛

(2.76)

28

2 Calculation Method for Slab and Grating Structure Made …

References 1. Moharam MG, Pommet DA, Grann EB, Gaylord TK (1995) Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach. J Opt Soc Am A 12(5):1077–1086 2. Zhao B, Zhang ZM (2017) Perfect mid-infrared absorption by hybrid phonon-plasmon polaritons in hBN/metal-grating anisotropic structures. Int J Heat Mass Transfer 106:1025–1034 3. Li LF (1996) Use of Fourier series in the analysis of discontinuous periodic structures. J Opt Soc Am A 13(9):1870–1876

Chapter 3

Unidirectional Transmission of Light

Compared with methods which use nonlinear optical materials or magneto-optical materials, realizing unidirectional transmission in reciprocal system does not need applied magnetic field or have limitation for the intensity of incident light. At present, most of the unidirectional transmission devices have been realized in reciprocal systems made of isotropic materials, few studies have been carried with anisotropic materials. Based on anisotropic materials, we proposed three simple structures to realize unidirectional transmission, which will be presented in the following part.

3.1 Realization of Unidirectional Transmission with Two Twisted Slabs Made of Anisotropic Material As shown in Fig. 3.1, the unidirectional transmission device consists of two hBN slabs with different orientation. hBN is a natural anisotropic material, which has two hyperbolic bands. Supposing the thicknesses of the top and bottom slabs are d 1 and d 2 , respectively. The optical axis of the bottom slab is along the y-axis, and thus its permittivity can be expressed as ⎞ ε⊥ 0 0 ε = ⎝ 0 ε|| 0 ⎠ 0 0 ε⊥   ω2L O,m − ωT2 O,m εm = ε∞,m 1 + 2 ωT O,m − ω2 + jωm ⎛

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 X. Wu, Thermal Radiative Properties of Uniaxial Anisotropic Materials and Their Manipulations, Springer Theses, https://doi.org/10.1007/978-981-15-7823-6_3

(3.1)

(3.2)

29

30

3 Unidirectional Transmission of Light

Fig. 3.1 The unidirectional transmission device consists of two slabs with a relative rotation angle

where j is the imaginary unit, m = ⊥,  indicates the component perpendicular or parallel to the optical axis, ω is the wavenumber, the other parameters are: ωT O,⊥ = 1370 cm−1 , ωT O,|| = 780 cm−1 , ω L O,⊥ = 1610 cm−1 , ω L O,|| = 830 cm−1 , ε∞,⊥ = 4.87, ε∞,|| = 2.95, ⊥ = 5 cm−1 , || = 4 cm−1 . Figure 3.2 shows the real parts of ε⊥ and ε|| in the wavelength range from 6 to 14 µm. It can be seen that Re(ε⊥ ) has a sharp change from a large negative value  to a large positive value at wavelength 7.3 µm (corresponding to ωT O,⊥ ), while Re ε|| stays the value of ε∞,|| in the wavelength range around 7.3 µm. On the other hand,  at Re ε|| has a sharp change from a large negative value to a large positive value at wavelength 12.82 µm (corresponding to ωT O,|| ), while Re(ε⊥ ) stays at the value 12.82 µm. In the two bands of wavelength of ε∞,⊥ in the wavelength rangearound lower than 7.3 and 12.82 µm, Re ε|| and Re(ε⊥ ) have opposite signs, and the dispersion relation of a TM wave propagation in the medium is hyperbolic in these two Fig. 3.2 The real parts of the permittivity components of hBN variate with wavelength

3.1 Realization of Unidirectional Transmission with Two Twisted Slabs …

31

bands. The two hyperbolic bands of hBN make the slab exhibit different reflection and transmission phenomena for an electromagnetic wave of different polarizations incident on it, as will be discussed. Now rotating the top slab around the z-axis by an angle φ, we can get a new permittivity tensor for this slab in the xyz coordinate system as ⎞ εx x εx y 0 ε = ⎝ ε yx ε yy 0 ⎠ 0 0 εzz ⎛

(3.3)

  where εx x = ε|| + ε⊥ − ε|| cos2 φ, εx y = ε yx = 0.5 ε⊥ − ε|| sin 2φ, εx x = ε|| +  ε⊥ − ε|| sin2 φ, and εzz = ε⊥ . In this work, our analysis is based on the xyz coordinate system. We assume normal incidence of a plane wave on the structure and the x–z plane is the plane of incidence. In this case, the incident wave is a TE wave if its electric field is along the y-axis, while it is a TM wave if its magnetic field is along the y-axis. In the crystal, the electromagnetic fields can be expressed in the following form  E = S(z) exp( jωt), S(z) = Sx , S y , 0

(3.4)

 1 2  H = − j ε0 μ0 / U(z) exp( jωt), U(z) = Ux , U y , 0

(3.5)

where ε0 and μ0 are the permittivity and the permeability of vacuum, the vectors S and U denote the magnitude and direction of the electric and the magnetic fields, respectively. Submitting Eqs. (3.4) and (3.5) into Maxwell’s equations, we get the following differential equations ⎛

⎞ ⎛ ⎛ ⎞ Sx 0 0 Sx ⎟ ⎜ 0 ⎜ Sy ⎟ d ⎜ S 0 y ⎜ ⎟ = k0 A⎜ ⎟, A = ⎜ ⎝ −εx y −ε yy ⎝ Ux ⎠ dz ⎝ Ux ⎠ Uy Uy εx x εx y

0 1 0 0

⎞ −1 0 ⎟ ⎟ 0 ⎠ 0

(3.6)

From the coefficient matrix A, it can be seen that TE wave and TM wave couple together and conversion of polarization can occur if εx y = 0. but when εx y = 0, i.e., when the rotation angle φ = 0o , Eq. (3.6) can be decomposed into

   d Sx 0 −1 Sx = k0 Uy ε⊥ 0 dz U y and

(3.7)

32

3 Unidirectional Transmission of Light

Fig. 3.3 a Reflectivity as a function of wavelength for TM wave and TE wave. b Transmissivity as a function of wavelength for TM wave and TE wave

   d Sy 0 1 Sy = k0 Ux −ε|| 0 dz Ux

(3.8)

Equations (3.7) and (3.8) indicate that TM wave and TE wave are decoupled and they can be solved separately. In this case, the propagation of a TM wave in the slab is only related to ε⊥ while the propagation of a TE wave in the slab is only related to ε|| . Therefore, incidence of a TM wave on hBN slab will have different reflection and transmission from incidence of a TE wave, and there is no conversion of polarization between them. In the following, we focus our attention on the hyperbolic bands, in which the optical responsefor TE and TM waves can be significantly different due to the opposite signs of Re ε|| and Re(ε⊥ ). Figure 3.3a, b show respectively the calculated reflectivity and transmissivity of an hBN slab of thickness equals to 2 µm, whose permittivity tensor is in the form of Eq. (3.1), for normal incidence of TE and TM waves in the wavelength range from 6.16 to 7.4 µm. It can be seen from Fig. 3.3a that the reflectivity is close to zero for incidence of TE wave while the reflectivity is high for incidence of TM wave in the whole specified wavelength range. Especially, the reflectivity for the latter is close to unity in the wavelength range from 6.4 to 7.3 µm. Figure 3.3b shows that the transmissivity for incidence of TE wave is close to unity in the while specified wavelength range. However, the transmissivity for incidence of TM wave is almost zero in the wavelength range from 6.4 to 7.4 µm. These different reflection and transmission phenomena for incidence of TE and TM waves in the slab are important factors, though not sufficient, for realizing the optical diode. Furthermore, it can be found from Fig. 3.3 that absorption in the hBN slab is small, since hBN is a low-loss material. To calculate the transmission of the whole structure, the 4 × 4 transfer matrix method presented in Chap. 2 is employed. To realize the asymmetric transmissivity of an optical diode, conversion of polarization by using the top hBN slab is required in the structure. For the purpose of achieving maximum polarization conversion efficiency between TM wave and TE

3.1 Realization of Unidirectional Transmission with Two Twisted Slabs …

33

Fig. 3.4 a Transmissivity as a function of wavelength for normal incidence of a TM wave in two opposite directions. b Transmissivity of the unconverted TM wave component and the converted TE wave component for normal incidence of a TM wave from two opposite directions

wave, the rotation angle is set as φ = 45o . On the other hand, the conversion efficiency depends on the thickness of the top slab, which should be taken to ensure large transmission of the converted wave through the top slab. For the bottom slab, its thickness should be taken to ensure that the slab block a TM (or TE) wave while permitting large transmission of a TE (or TM) wave. Here, we firstly consider incidence of a TM wave, and the thickness of the top and the bottom slabs are taken by optimization to be 1 and 2 µm, respectively. Figure 3.4a shows the calculated transmissivity of the structure for the TM wave incident from the top and from the bottom, respectively. It can be seen the incidence cannot pass the structure from the bottom slab to the top slab in the wavelength range from 6.40 to 7.38 µm, while the incidence can pass the structure from the opposite direction. For better understanding the underlying mechanism, we define T pp and T ps as the transmissivity of the unconverted TM wave component and the TE wave component converted from TM wave, respectively. Figure 3.4b shows the calculated T pp and T ps for incidence from the opposite directions. As shown, the incident TM wave from the top can be partly converted to TE wave in the top slab. This TE wave can then pass the bottom slab while the unconverted TM wave is blocked by the bottom slab in the wavelength range from 6.40 to 7.38 µm. Therefore, the total transmissivity is not zero and is contributed from the converted TE wave. In contrast, when the TM wave is incident from the bottom, it cannot pass the bottom slab, thus leading to a nearly zero total transmission. As a result, an optical diode working in a broadband is realized in this structure. Another key characteristic for an optical diode is the transmissivity contrast ratio (TCR), which is defined as T 1 /T 2 , where T 1 is the transmissivity for incidence from the top and T 2 is the transmissivity for incidence from the opposite direction. Figure 3.5 shows the TCR for normal incidence of a TM wave from the top and the bottom varying with wavelength, a value beyond 1000 is revealed in the wavelength range from 6.63 to 7.33 µm. The maximum TCR can reach 1026 , which is much larger than the best available value in published literature [1].

34

3 Unidirectional Transmission of Light

Fig. 3.5 Transmissivity contrast ratio for normal incidence of a TM wave from two opposite directions as a function of wavelength

The high TCR of the optical diode shown Fig. 3.5 stems from the hyperbolicity of hBN. In the hyperbolic band of hBN from 6.4 to 7.3 µm, the real part of ε⊥ is negative while the real part of ε|| is positive. In the bottom slab, the propagation of TM wave is only related to ε⊥ while the propagation of TE wave is only related to ε|| . Therefore, the TM wave cannot pass the bottom slab while the TE wave can. This is the reason for the high TCR of the optical diode. We should emphasize that an optical diode can also be realized when the real part of ε|| and ε⊥ share the same sign. However, a high TRC is hard to obtain in that case. This explains why a hyperbolic material should be selected for making the optical diode with a high contrast ratio. Likewise, an optical diode can also be achieved for incidence of a TE wave by using the proposed structure, but in a different wavelength range. In this case, the thickness of the top slab is remained to be 1 µm but the thickness of the bottom slab is changed to 6 µm. Figure 3.6a shows that there is almost no transmission in the wavelength range from 12.2 to 12.9 µm when the TE wave is normally incident

Fig. 3.6 a Transmissivity and b transmissivity contrast ratio for normal incidence of a TE wave from two opposite directions as a function of wavelength

3.1 Realization of Unidirectional Transmission with Two Twisted Slabs …

35

from below, while the transmissivity is higher than 0.2 in the whole wavelength range of interest for incidence from the opposite direction. Figure 3.6b illustrates the TCR varying with wavelength for normal incidence of the TE wave from the two opposite directions. A waveband is found from 12.33 to 12.88 µm for the TCR higher than 1000. Furthermore, the maximum TCR at 12.8 µm is higher than 1017 . The underlying mechanism is similar to that for incidence of a TM wave. When the incident TE wave is from the top, it can be partly converted to TM wave in the top slab, which can pass the bottom slab, thus the total transmissivity is not zero and is contributed from the converted TM wave. But when the incident TE wave is from below, it is blocked by the bottom slab, thus leading to a nearly zero total transmissivity. It should be pointed out that although our above analysis is based on a natural hyperbolic crystal hBN, on which the obtained wavebands for an optical diode are not tunable, this setback can be overcome by replacing the hBN slabs with hyperbolic metamaterials slabs. Because a hyperbolic metamaterial is an artificial uniaxial crystal, whose hyperbolic band can be tailored by tuning the geometric parameters of the metamaterial. Therefore, it is possible to realize an optical diode in a targeted waveband by tuning the geometric parameters of the metamaterial.

3.2 Realization of Unidirectional Transmission Based on Excitation of Magnetic Polaritons or Surface Plasmon Polaritons To realize unidirectional transmission in other waveband, one can use the structure shown in Fig. 3.7. The device is constructed by attaching on one side of the uniaxial Fig. 3.7 The unidirectional transmission structure consists one hBN slab and one 1D grating. The optical axis of the top hBN slab is in the x–y plane and is tilted off the x-axis by an angle of φ, and the thickness of the slab is d 1

36

3 Unidirectional Transmission of Light

crystal slab with a one-directional (1D) grating, with the grooves of which perpendicular to the plane of incidence. Depending on the wavenumber range within which unidirectional transmission to be achieved, the material of the grating can be selected differently. We firstly investigated the case in which the grating is made of SiC. The period, thickness and filling ratio of the grating are taken as 1 µm, 4 µm, and 0.5, respectively. The dielectric function of SiC was represented by the Lorentz model as [2] εSiC = ε∞

ω2L − ω2 − jω ωT2 − ω2 − jω

(3.9)

The thickness of the uniaxial crystal slab is set as d 1 = 3.8 µm, and its orientation is 45°. We extended the rigorous coupled-wave analysis (RCWA) algorithm to permit conversion of polarization and used it to calculate the transmittance of the device. Assuming normal incidence of a TE plane wave on either side of the device, the calculated transmittance and absorptance of the device varying with the wavenumber are shown in Fig. 3.8a. Clearly, the results reveal that when the wave is incident on the grating side, the transmittance is closed to zero in the specified wavenumber range from 860 to 910 cm−1 . On the other hand, when the wave is incident on the slab side, the wave can transmit through the structure and the transmittance is higher than 0.8 for wavenumber around 900 cm−1 . Therefore, unidirectional transmission is realized with the proposed structure in the specified spectral band. On the other hand, if a TM plane wave is assumed normally incident on either side of the device, the calculated transmittance and absorptance curves are shown in Fig. 3.8b. It can be seen in this case, the UDT is for incidence on the grating side and the transmittance is almost zero for incident on the slab side in the same wavenumber range as in Fig. 3.8a. To reveal the physical mechanism for the unidirectional transmission with the device, we calculate the electromagnetic field distribution inside the grating at wavenumber 900 cm−1 . The patterns of the electric field |E| and magnetic field

Fig. 3.8 The transmittance and absorptance of the device varying with wavenumber for normal incidence of a a TE plane wave and b a TM plane wave from opposite sides of the device

3.2 Realization of Unidirectional Transmission Based on Excitation …

37

Fig. 3.9 Distribution patterns of a and b in the SiC grating for incidence of a TE and a TM plane waves, respectively, on the grating side at wavenumber 900 cm−1

|H| side the grating are plotted in Fig. 3.9a, b for incidence on the grating side of a TE and a TM plane waves, respectively. The outline of the grating is also illustrated in the figures for ease of reading. It can be seen in Fig. 3.9a that the electric field amplitude inside the grating decreases quickly from the grating surface. This is because the real part of the dielectric function of SiC is negative at wavenumber 900 cm−1 , which makes SiC highly reflective. In addition, the width of the grooves is much smaller than the wavelength of incidence. Therefore, the TE incident wave can hardly penetrate the grating. In contrast, when the incident is a TM wave, as shown in Fig. 3.9a, the magnetic field is enhanced in grooves of the grating, due to the excitation of magnetic polaritons (MPs). According to Lenz’s law, an antiparallel oscillating current will be produced in the two walls of the groove. These anti-parallel currents will introduce a diamagnetic response which in turn couple with the incident wave to result in the excitation of MPs. The enhanced fields in the grooves make the incident be able to penetrate the grating and thus be transmitted through the device. However, MPs cannot be excited by a TE wave in the 1D grating so that the incident can hardly penetrate the grating. The same explanation applies for the incidence on the slab side. In this case, incidence of a TE wave will be converted into TM wave when travelling through the uniaxial crystal slab, which then excites MPs in the grating. In contrast, incident of a TM wave will be converted into TE wave by the uniaxial crystal slab, which cannot excite MPs in the grating. It should be pointed out here that excitation of MPs in the grating also enhances the absorption of wave in the grating, which can reduce the transmittance. As can be seen in Fig. 3.8a, b that there is an absorption peak at 878 cm−1 , at which the corresponding transmittance is greatly reduced. We have checked that at this wavenumber the magnetic field is enhanced in the grating and the magnetic field amplitude patterns are similar to those in Fig. 3.8a, b, which demonstrate that the enhanced absorptance results from excitation of MPs. Realization of UDT with the device in the optical and the ultraviolet regions can also be achieved by using silver (Ag) grating and changing the geometric parameters of the device. In our numerical simulations we set the period, thickness, and filling

38

3 Unidirectional Transmission of Light

Fig. 3.10 The transmittance and the absorptance of the device varying with wavenumber for normal incidence of a a TE plane wave and b a TM plane wave from opposite sides of the device

ratio of the Ag grating as 0.15 µm, 0.25 µm, and 0.3, respectively. The dielectric function of Ag was described using the Drude model [3] εAg = ε∞ −

ω2p ω2 + jω

(3.10)

where ε∞ = 3.4, ω p = 1.39 × 1016 rad/s, and  = 2.7 × 1013 rad/s. The thickness of the uniaxial crystal slab is set as d = 0.5 µm, and its orientation is 45°. Again, assuming normal incidence of a TE plane wave on either side of the device, the calculated transmittance is shown in Fig. 3.10a in the wavenumber range from 12,500 cm−1 to 20,000 cm−1 . While the transmittance for incidence on the grating side is almost completely equal to zero, the transmittance for incidence on the slab side is higher than 0.4 in the while specified wavenumber ranger. Especially, the transmittance at wavenumber 16,900 cm−1 is higher than 0.9. The physical mechanism for this UDT is also attributed to excitation of MPs, the same as that for Fig. 3.8. It should be emphasized that oscillation of the transmittance with the wavenumber is not due to enhancement of absorption in the grating, but due to oscillation of the PCR, since the absorption is very low in almost the whole specified wavenumber range. The calculated transmittance for normal incidence of a TM plane wave is shown in Fig. 3.10a. In this case, high transmittance is found for incidence on the grating side, due to excitation of MPs in the grating. However, the transmittance for incidence on the slab side is not close to zero in the whole specified wavenumber ranger. Especially, the transmittance is higher than 0.4 for the wavenumber around 14,000 cm−1 . This is because the incident TM wave is not converted completely to TE wave when travelling through the uniaxial crystal; the unconverted TM wave can still excite MPs in the gratings. As a consequence, the structure does not exhibit UDT for wavenumber in the ranger from 12,500 cm−1 to 15,000 cm−1 . Here, it deserves to emphasize that the physical mechanism for the UDT achieved with the proposed device is not limited to excitation of MPs. We show below that

3.2 Realization of Unidirectional Transmission Based on Excitation …

39

Fig. 3.11 The transmittance and absorptance of the device varying with wavenumber for normal incidence of a a TE plane wave and b a TM plane wave from opposite sides of the device

the UDT may also be realized by excitation of surface plasma polaritons (SPPs) on the surface of the Ag grating. In so doing, the period, thickness and filling ratio of the grating are changed to be 0.3 µm, 0.1 µm, and 0.1, respectively. The thickness of the uniaxial crystal slab is set as d = 0.3 µm. The calculated transmittance and absorptance of the device varying with the wavenumber from 28,500 cm−1 to 31,000 cm−1 for normal incidence of a TE plane wave on opposite sides of the device are shown in Fig. 3.11a. Clearly, the results reveal that when the wave is incident on the slab slide, the absorptance is close to zero while the transmittance is above 0.8 in the whole specified range of wavenumber. But on the other hand, when the wave is incident on the grating side, the absorptance is greatly enhanced to nearly 0.5 while the transmittance is greatly reduced to below 0.15 at wavenumber 30,413.6 cm−1 . Therefore, unidirectional transmission is realized at this wavenumber. The calculated results for normal incidence of a TM plane wave are shown in Fig. 3.11b. In this case, the UDT realized at wavenumber 30,413.6 cm–1 is for the incidence on the slab side, in contrast to that shown in Fig. 3.11a. The reason for the UDT is that, when a TE wave is incident on the slab side, this TE wave is converted to TM wave when it travels through the slab. We carefully select the thickness of the slab such that the TE wave is almost completely converted into TM wave when it reaches the grating. The diffracted TM wave by the grating then excites SPPs on the surface of the Ag grating at wavenumber 30,413.6 cm−1 and thus causing large absorption in the grating. As a consequence, the transmittance is reduced. The same explanation applies for the results shown in Fig. 3.11b, in which the enhanced absorption is attributed to SPPs excited by the TM plane wave incident on the grating.

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3 Unidirectional Transmission of Light

3.3 Realization of Unidirectional Transmission Based on Photonic Bandgap The third structure to realize UDT is shown in Fig. 3.12, which consists of one uniaxial slab on the top and one photonic crystal (PC) made of anisotropic materials. The optical axis of the top uniaxial slab is in the x–y plane and is tilted off the x-axis with an angle of φ, and its thickness is d 1 . The unit cell of PC structure consists of two hBN slabs, but with differently oriented optical axis. For the first slab, the optical axis is along the y-axis, and thus its permittivity tensor is ε = diag ε⊥ , ε|| , ε⊥ . For the second slab, the optical axis is along the z-axis, thus its permittivity tensor is in the form of ε = diag ε⊥ , ε⊥ , ε|| . According to Eqs. (3.8) and (3.9), we can see that TM wave and TE wave are only related to ε⊥ and ε|| in the first slab, respectively. But the TM wave and TE wave are both related to ε⊥ in the second slab. Therefore, we can achieve the photonic bandgap (PBG) for TE wave by periodically arranging these two slabs. Let λ be the center of the forbidden band such that [4] k1 d2 = k 2 d3 = 0.5π

(3.11)

√ √ where k1 = k0 ε|| , k2 = k0 ε⊥ , d 2 and d 3 are the thicknesses of the first and the second slabs, respectively. Supposing this PC has N periodic unit cells, the total

Fig. 3.12 The unidirectional transmission structure consists of one hBN slab and one photonic crystal. The optical axis of the top hBN slab is in the x–y plane and is tilted off the x-axis by an angle of φ, and the thickness of the slab is d 1

3.3 Realization of Unidirectional Transmission Based on Photonic …

41

Fig. 3.13 a The transmissivity of the PC structure as a function of wavelength for TE and TM waves for various number of unit cells. b The transmissivity ratio of the PC structure as a function of wavelength for various number of unit cells

thickness of the PC is N(d 2 + d 3 ). To achieve as much transmission as possible for TM wave, we need N (d2 + d3 )k2 → mπ, m = 1, 2, 3 . . .

(3.12)

Taking the wavelength of incidence as 0.6 µm and according to Eq. (3.11), we have d 2 = 87 nm and d 3 = 68 nm. According to Eq. (3.12), we choose N = 14, 21 and 28 in our analysis. The transmissivity of the PC as a function of wavelength for TE and TM waves are shown in Fig. 3.13a. It can be seen transmission of TE wave is prohibited in the waveband from 0.56 to 0.64 µm while TM wave can pass through with the highest transmissivity close to unit for all the three values of N. The transmissivity ratio of the PC structure, defined as the ratio of the transmissivity for TM wave to that for TE wave, is shown in Fig. 3.13b, which reveals that the transmissivity ratio can be greatly enhanced in the waveband 0.6 µm by increasing the number of the unit cells in the PC structure. Based on the above analysis, the thickness of the top hBN slab is taken as d 1 = 615 nm and its optical axis is arranged to rotate around the z-axis off the y-axis by an angle φ = 45o . The PC is taken as that shown in Fig. 3.12 with d 2 = 87 nm and d 3 = 68 nm. Assuming normal incidence of a TE plane wave from the top and the bottom sides of the structure, i.e. in the forward and the backward directions, respectively, we calculate the transmissivity of the structure as a function of the wavelength. The results are shown in Fig. 3.14a for the number of unit cells N equals to 14, 21, and 28. It can be seen that the optical diode is realized in a broadband for all the three values of N, where T f denotes the transmissivity for incidence from the forward direction while T b for incidence from the backward direction. The transmission contrast ratio, defined as T f /T b , is shown varying with wavelength in Fig. 3.14b, which reveals that the contrast ratio is higher than 10,000 in the wavelength range from 0.583 to 0.617 µm when N = 21, and is in the wavelength range from 0.567 to 0.636 µm

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3 Unidirectional Transmission of Light

Fig. 3.14 a The transmissivity as a function of wavelength for incidence in the forward and backward directions. b The transmissivity contrast ratio as a function of wavelength. The number of unit cells N is equal to 14, 21 and 28

when N = 28. Therefore, we can control the transmission contrast ratio by changing the number of the unit cells. The highest contrast ratio obtained with the proposed structure is much larger than these obtained by coupling of nonlinear PC defects and based on resonator with pump-assisting [5,6]. To better understand the physical mechanisms of the optical diode, we plot the transmission components varying with wavelength for forward incidence of the TE wave in Fig. 3.15a. In this case, the incident TE wave is partly converted to TM wave in the top hBN slab, which can then pass the PC. But the unconverted TE wave cannot pass the PC because of the PBG. On the other hand, when incidence is from the backward direction, as shown in Fig. 3.15b, the PBG can forbid transmission of the TE wave through the PC, and thus the transmission is close to zero in this direction and a high-contrast ratio optical diode is realized. From the above analysis, we can draw that the transmission for the backward incidence is mainly determined by the

Fig. 3.15 The transmissivity components of a TE wave under a forward incidence and b backward incidence

3.3 Realization of Unidirectional Transmission Based on Photonic …

43

Fig. 3.16 Variation of Tf and Tb of the unidirectional transmission device with the rotation angle

PBG. However, the transmission for the forward incidence is mainly determined by the PRC of the top slab. Figure 3.16 shows T f and T b of the optical diode varying with the rotation angle φ for the incident TE wave in forward and backward directions with its wavelength at 0.6 µm. It can be seen that while T b is close to zero in the whole range of φ, T f increases monotonically with φ and reaches a maximum value of 0.67 when φ is close to 45°, from which T f decreases monotonically with φ. Only when φ is 0° or 90° does T f equal T b , due to the zero PCR at these two angles. In short, the contrast ratio of the optical diode can be effectively tunable by choosing different parameters. In principle, an optical diode can also be realized with the proposed structure under incidence of a TM wave. Figure 3.17a shows the calculated results of T f and T b varying with wavelength under normal incidence of a TM wave from the forward and backward directions, respectively. In the calculations, the rotation angle is set as 45°, the number of unit cells is 21 and the value of d 1 , d 2 , and d 3 remain unchanged.

Fig. 3.17 a Variation of Tf and Tb with wavelength for normal incidence of a TM wave. b Variation of the transmissivity contrast ratio with wavelength for normal incidence of a TM wave

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3 Unidirectional Transmission of Light

It can be seen in this case T f is small while T b is large in the specified wavelength range, which is in contrast to the case in Fig. 3.14a. In addition, the transmission contrast ratio, defined as T f /T b , is shown in Fig. 3.17b as a function of wavelength. It shows that the contrate ratio around wavelength 0.6 µm and the highest contrast ratio are both much smaller than those in Fig. 3.14b. The reason is that the incident TM wave cannot be totally converted into TE wave in the top slab under the forward incidence and the unconverted TM wave can pass the PC. Therefore, the value of T f is not close to zero in this case. However, we can adopt a different arrangement of the slabs for the PC to achieve high-contrast ratio for incidence of TM wave. If the first slab of the unit cell is set with the optical axis along the x-axis, its permittivity tensor becomes ε = diag ε|| , ε⊥ , ε⊥ , this change can be done by rotating the first slab, as shown in Fig. 3.13, around the z-axis by an angle of 90°. The second slab remains the same, then its permittivity tensor is still ε = diag ε⊥ , ε⊥ , ε|| . Therefore, by stacking periodically these two slabs, a PBG for TM wave can be achieved. By replacing the PC in the structure shown in Fig. 3.13 with this PC, we calculate the transmission of the structure under normal incidence of a TM wave. In the calculations, the number of unit cells N is set as 21 and the other parameters remain the same as in Fig. 3.14. The calculated T f and T b varying with wavelength are shown in Fig. 3.18a. We can see that the two curves of T f and T b are similar to those in Fig. 3.14a. In this case, T b is close to zero since the PBG formed by the PC cannot allow a TM wave to pass under the backward incidence. The transmission contrast ratio, defined as T f /T b , is also calculated and its variation with wavelength is shown in Fig. 3.18b. The contrast ratio higher than 10,000 is found in the wavelength range from 0.582 to 0.615 µm in this case, again is similar to the results in Fig. 3.14b. Apparently, a broadband and high-contrast-ratio optical diode can also be realized in other wavelength ranges by choosing proper parameters. These are five tunable parameters in the structure, namely, the rotation angle φ, the thicknesses d 1 , d 2 and d 3 , and the number of the unit cells N. We usually choose φ as 45° for getting a large PCR, the rest four variables can be determined according to Eqs. (3.8), (3.11) and (3.12). Therefore, it is a very straightforward way to realize an optical diode

Fig. 3.18 a Variation of Tf and Tb with wavelength for normal incidence of a TM wave. b Variation of the transmissivity contrast ratio with wavelength for normal incidence of a TM wave

3.3 Realization of Unidirectional Transmission Based on Photonic …

45

with anisotropic materials of appropriate permittivity tensor. The performance of the optical diode can be tuned by adjusting the number of unit cells and the rotation angles.

References 1. Gao H, Zhou YS, Zheng ZY (2016) Broadband unidirectional transmission realized by properties of the Dirac cone formed in photonic crystals. J Opt 18(10):105102 2. Wang WJ, Fu CJ, Tan WC (2014) Thermal radiative properties of a photonic crystal structure sandwiched by SiC gratings. J Quant Spectrosc Radiat Transfer 132:36–42 3. Zhao B, Zhang ZM (2014) Study of magnetic polaritons in deep gratings for thermal emission control. J Quant Spectrosc Radiat Transfer 135: 81–89 4. Yeh P (2005) Optical waves in layered media. Wiley 5. Lin XS, Wu WQ, Zhou H, Zhou KF, Lan S (2006) Enhancement of unidirectional transmission through the coupling of nonlinear photonic crystal defects. Opt Express 14: 2429–2439 6. Lin XS, Yan JH, Wu LJ, Lan S (2008) High transmission contrast for single resonator based all-optical diodes with pump-assisting. Opt Express 18: 20949–20954

Chapter 4

Broadband Perfect Absorption

As said in Chap. 1, impedance matching is an effective way to realize broadband perfect absorption. Nefedov et al. has proposed to realize perfect absorption using an asymmetric hyperbolic metamaterial (AHM) slab [1–3]. The mechanism for the perfect absorption were attributed to matching of impedance and amplified wavevector. They pointed out that these two conditions could be satisfied simultaneously if one of the permittivity components was approximately 1 and the other was approximately −1. However, it is not easy to design metamaterials to meet this strict requirement. Besides, the perfect absorption only occurs at certain wavelengths rather than in a broadband. It is also worth noting that these previous researches are focused on the single flat structure. It is possible to get broadband perfect absorption if multi slabs are stacked. Here we will present how to realize broadband perfect absorption in multilayer structure. Before introducing the broadband perfect absorption, we will study the impact of the orientation of optical axis on the emissivity of a uniaxial crystal. To investigate the emissivity of anisotropic materials, it is necessary to calculate the emissivity for different combination of polar angle and azimuthal angle. The traditional 4 × 4 transfer matrix method can be used to calculate the optical properties of anisotropic materials. However, this method is quite complex and tedious for use when there is an angle between the plane of incidence and the coordinate system. A modified 4 × 4 transfer matrix method is employed to calculate the spectral directional emissivity of the materials for each polarization, in which a rotational transformation technique is implanted to avoid the projection operations when solving for the reflection coefficient matrix.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 X. Wu, Thermal Radiative Properties of Uniaxial Anisotropic Materials and Their Manipulations, Springer Theses, https://doi.org/10.1007/978-981-15-7823-6_4

47

48

4 Broadband Perfect Absorption

4.1 Directional and Hemispherical Emissivity of hBN The emissivity of a semi-infinite medium can be obtained from the reflectivity based on Kirchhoff’s law. For an isotropic medium, the reflectivity for each polarization can be obtained based on Fresnel’s coefficients. For an anisotropic medium, however, cross-polarization can occur. Therefore, one needs to obtain the reflection coefficient matrix defined as follows:   r r (4.1) R = pp ps rsp rss Here, the first and the second subscripts of each reflection coefficient describe the polarization status of the incident and the reflected waves, respectively. Although there have been numerous studies on the optical properties of anisotropic materials, conventional approaches are usually cumbersome and complicated due to anisotropy. Here, we present a concise procedure for calculating the spectral hemispherical emissivity of a semi-infinite uniaxial crystal. The permittivity tensor of the uniaxial crystal  medium can be expressed as ε = diag ε⊥ , ε⊥ , ε|| when its optical axis is along the z-axis of the coordinate system. When the optical axis is tilted off the z-axis by an angle β in the x–z plane, as shown in Fig. 4.1a, the permittivity tensor in the xyz coordinate system can be expressed as ⎛

⎞ ⎛ ⎞ ε⊥ 0 0 cos β 0 − sin β ⎝ 0 1 ε = T y ⎝ 0 ε⊥ 0 ⎠T−1 0 ⎠ y , Ty = 0 0 ε|| sin β 0 cos β

(4.2)

When β is equal to 0° or 90°, the optical axis of the crystal is perpendicular or parallel to the surface of the material, respectively. These two cases have been

Fig. 4.1 Schematic of the coordinate systems used for calculating the reflection coefficients. Here, the upper half air and the lower half is a uniaxial crystal whose optical axis is tilted in the x–z plane by an angle β with respect to the z-axis. a The plane of incidence is the x–z plane with an incidence angle θ; b the plane of incidence is rotated around z-axis by an azimuthal angle φ. In the coordinate system, the plane of incidence is the x–z plane, where the z -axis and z-axis are the same

4.1 Directional and Hemispherical Emissivity of hBN

49

considered previously [4]. Here, we consider the general case when is between 0° and 90°. To calculate hemispherical emissivity of the uniaxial medium, it is necessary to calculate the emissivity for different combinations of polar angle and azimuthal angle. When the azimuthal angle φ is 0°, the plane of incidence is the x–z plane, as shown in Fig. 4.1a. When φ is not equal to 0°, the plane of incidence is rotated off the x–z plane as shown in Fig. 4.1b. It is cumbersome to calculate the emissivity in the xyz coordinate system when φ is not equal to 0°. Here, we choose the x  y z coordinate system to evaluate the reflection coefficients and use them to obtain the directional emissivity. The permittivity tensor in x  y z coordinate system can be expressed as ⎛

⎞ ⎡ ⎛ ⎞ ⎤ εx x εx y εx z ε⊥ 0 0 ⎦T−1 ε = ⎝ ε yx ε yy ε yz ⎠ = Tz ⎣T y ⎝ 0 ε⊥ 0 ⎠T−1 y z εzx εzy εzz 0 0 ε||

(4.3)

where Tz is the coordinate rotational transformation matrix, which is given by ⎛

⎞ cos φ sin φ 0 Tz = ⎝ − sin φ cos φ 0 ⎠ 0 0 1

(4.4)

It should be noted that when φ is 0°, Tz is a unit matrix, and the plane of incidence is the x–z plane. So, we can always calculate the emissivity in the x  y z coordinate system no matter whether the azimuthal angle is 0° or not. While the two approaches are very equivalent, carrying out the calculations in the x  y z coordinate system has two advantages. Firstly, in the xyz coordinate system, the wavevector has three components. However, the wavevector has only two components with respect to the x  y z coordinate system. Secondly, if the calculations are performed in the xyz coordinate system, all the electromagnetic field vectors need to be projected onto the coordinate axes when dealing with the boundary condition. This process can be avoided if the calculations are performed in the x  y z coordinate system. Similar technique has been used by Rose et al. in their analysis of the Casimir interactions for anisotropic magnetodielectric metamaterials, although their analysis is only for the case when the optical axis is parallel or perpendicular to the surface of the medium [5]. Here, we consider the general case when the optical axis can be arbitrarily oriented with respect to the surface. Once the reflection coefficients are obtained, the spectral directionalhemispherical reflectivity for TM and TE waves are given respectively as  2  2  2 ρTM,λ (θ, φ) = r pp  + r ps  ; ρTE,λ (θ, φ) = rsp  + |rss |2

(4.5)

The directional absorptivity is obtained by applying the energy balance at the interface for each polarization, viz.

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4 Broadband Perfect Absorption

αTM,λ (θ, φ) = 1 − ρTM,λ (θ, φ); αTE,λ (θ, φ) = 1 − ρTE,λ (θ, φ)

(4.6)

According to Kirchhoff’s law, the directional emissivity for each polarization reads [4] εTM,λ (θ, φ) = αTM,λ (θ, φ); εTE,λ (θ, φ) = αTE,λ (θ, φ)

(4.7)

Finally, the hemispherical emissivity can be calculated from the spectral directional emissivity as 1 ελ = 2π

2π π/2 0

εTM,λ (θ, φ) + εTE,λ (θ, φ) sin(2θ)dθ dφ 2

(4.8)

0

All the above-mentioned radiative properties are spectral quantities. If the total emissivity is needed, one can perform a weighted integration over Planck’s blackbody distribution function evaluated at a given surface temperature. This algorithm has been validated by comparing Fresnel’s coefficients obtained with it with those obtained with the traditional 4 × 4 matrix method, and by comparing the calculated spectral emissivity of a uniaxial medium when the optical axis is either perpendicular or parallel to the surface with published results. It should be pointed out the spectral hemispherical emissivity is the same as long as the tilting angle is the same, no matter whether the optical axis is tilted in the x–z plane or not. Although other methods have also been developed for calculation of Fresnel’s coefficients of uniaxial materials with arbitrarily oriented optical axis, it is very convenient to calculate the spectral hemispherical emissivity of such materials with the algorithm presented here. The calculated spectral hemispherical emissivity of the hBN varying with wavenumber for different tilting angles is shown in Fig. 4.2a. It can be seen the emissivity drops abruptly and is also affected significantly by the tilting angle in the

Fig. 4.2 Calculated emissivity spectra of hBN for different tilting angles: a hemispherical emissivity; b normal emissivity, i.e., θ = 0°

4.1 Directional and Hemispherical Emissivity of hBN

51

two hyperbolic band. The normal emissivity of the hBN medium is shown in Fig. 4.2b for comparison. It can be seen that the normal and hemispherical emissivity has very similar trends, expect at 830 cm−1 when the tilting angle is 0° and at 1610 cm−1 when the tilting angle is 90°. Similar feature to the drop of emissivity for hBN in the two hyperbolic bands has been demonstrated in the experimental study of pyrolytic boron nitride (pBN) at elevated temperatures [6]. However, since their study dealt with a randomly oriented polycrystalline material, the dielectric function of pBN was modeled based on the effective medium approaches using a weighted average of the two polarizations. Therefore, all the anisotropic features were obscure and only the average effect was observed in the experiment. It is impossible to give quantitative comparison. Our numerical model allows the calculation of the detailed contributions from different hBN orientation as well as emission angles. In the following, we pick one wavenumber in each of the hyperbolic band to illustrate the effect of orientation and polarization on the emissivity. Taking ω = 1400 cm−1 as an example, which is within the type II hyperbolic band, we calculate the spectral directional emissivity as a function of the tilting angle β and the azimuthal angle φ. The results for TM and TE waves at φ = 0o are shown in Fig. 4.3a and b, and those at φ = 60o are in Fig. 4.3c and d, respectively. It can be seen that when φ = 0o , the emissivity patterns for TM and TE waves are the

Fig. 4.3 Contour plots for the spectral directional emissivity of hBN with respect to the azimuthal angle and tilting angle at wavenumber of 1400 cm−1 : a TM wave, θ = 0°; b TE wave, θ = 0°; c TM wave, θ = 60°; d TE wave, θ = 60°

52

4 Broadband Perfect Absorption

same except that there is a shift in the azimuthal angle of φ = 90o between the two patterns. As shown in Fig. 4.3a and b, the emissivity of TM wave is high only in the regions when φ is around 0° and 180° and β > 20o , while the emissivity for TE wave is high only in the regions when φ is around 90° and 270° and β > 20o . When, the regions of high emissivity for TM wave becomes plumper but the regions of high emissivity for TE wave shrinks compared to the cases at θ = 0o . This suggests that the emissivity increases for TM wave but decreases for TE wave as θ increases from 0° to 60°. The spectral directional emissivity averages over TM and TE wave is shown in Fig. 4.4a and b for θ = 0o and θ = 60o , respectively. It can be seen the averaged emissivity is independent of the azimuthal angle when θ = 0o . However, when θ = 60o , the averaged emissivity becomes a periodic function of the azimuthal angle β with a period of 180°. It should be pointed out occurrence of polarization conversion between TM and TE waves gives rise to a small difference of the emissivity pattern in the azimuthal angle region [0°, 180°] from that in [180°, 360°]. consequently, the emissivity for a TM or TE wave is fact a periodic function of φ with a period of 360°. But the difference is so small that the emissivity patterns shown in Fig. 4.3 exhibit a period of 180°. Similar results have been found for the averaged reflectivity of an anisotropic metamaterials [7]. In order to weight the impact of polarization conversion on the reflectivity and emissivity of the medium, the values of |r pp |2 and |r ps |2 varying with the angles φ and β for normal incidence of a TM wave, and those of |r ss |2 and |r sp |2 for normal incidence of a TE wave are plotted in Fig. 4.5a–d, respectively. The reflectivity components resulting from polarization conversion can be clearly observed in Fig. 4.5b and d for φ around 45°, 135°, 225°, and 315°. In addition, the values of |r ps |2 and |r sp |2 increase with the tilting angle and their maxima appear at β = 90°. The occurrence of the maximum polarization conversion can also impact near-field radiative heat transfer as demonstrated by Liu et al. [8]. The directional emissivity of the medium at 800 cm−1 , which is within the type I hyperbolic band, varying with the azimuthal angle and the tilting angle is shown in

Fig. 4.4 The emissivity averaged over TM and TE waves as a function of azimuthal angle and tilting angle at the wavenumber of 1400 cm−1 : a θ = 0°; b θ = 60°

4.1 Directional and Hemispherical Emissivity of hBN

53

Fig. 4.5 Reflectivity components for a plane wave at normal incidence and at 1400 cm−1 : a |r pp |2 ; b |r ps |2 ; c |r ss |2 ; d |r sp |2

Fig. 4.6a and b for normal incidence of TM and TE waves, respectively. The corresponding results for incidence at θ = 60o are plotted in Fig. 4.6c and d. Interestingly, the emissivity patterns are found to be complementary to those at wavenumber 1400 cm−1 , i.e., the regions of high emissivity in Fig. 4.6 are almost those exhibiting low emissivity in Fig. 4.3. As shown in Fig. 4.6a, the emissivity for TE wave is small only in the regions when φ is around 0° and 180° and β > 40o , But reveals that the emissivity for TE wave is small only in the regions when φ is around 90° and 270° and β > 40◦ . This is anticipated since the lattice vibration for hBN is parallel to the optical axis in the type I hyperbolic band, while it is perpendicular to the optical axis in the type II hyperbolic band. To gain deeper understanding of these results, we analyze the impedance od the medium for incidence of a TM wave and the admittance of the medium for incidence of a TE wave on it. For convenience, let us focus on the four azimuthal angles: φ = 0◦ , 90◦ , 180◦ , and 270°. When φ = 0◦ or 180◦ , the optical axis is in the plane of incidence and there is no coupling between TM and TE waves in this case regardless of the angle of incidence. The normalized impedance is defined as the ratio of the impedance of hBN over that of vacuum. For TM wave incidence, the normalized impedance for φ = 0◦ or 180◦ is expressed as [9, 10]

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4 Broadband Perfect Absorption

Fig. 4.6 The spectral directional emissivity of hBN of the medium at the wavenumber of 800 cm−1 as a function of azimuthal angle and tilting angle: a TM wave, θ = 0°; b TE wave, θ = 0°; c TM wave, θ = 60°; d TE wave, θ = 60°

 Z=

ε|| cos2 β + ε⊥ sin2 β − sin2 θ ε⊥ ε|| cos2 θ

(4.9)

Similar definition applies for the admittance of the medium for TE wave incidence. Therefore, the normalized admittance for φ = 0◦ or 180◦ is written as  ε⊥ − sin2 θ Y = cos θ

(4.10)

The reflection coefficients r pp and r ss can be expressed respectively in terms of the normalized impedance and the normalized admittance as follows r pp =

1− Z 1−Y , rss = 1+ Z 1+Y

(4.11)

The normalized impedance of the hBN medium for φ = 0◦ or 180◦ is calculated using Eq. (4.9) for TM wave incidence at 1400 cm−1 . The real and imaginary parts of the normalized impedance versus the tilting angle β at θ = 0◦ or 60◦ are shown in Fig. 4.7a. It can be seen that when θ = 0◦ , the imaginary part of the normalized

4.1 Directional and Hemispherical Emissivity of hBN

55

Fig. 4.7 The normalized a impedance for TM wave incidence and b admittance for TE wave incidence at 1400 cm−1 , and the normalized c impedance for TM wave incidence and d admittance for TE wave incidence at 800 cm−1

impedance is very small when β > 20°, while the real part increases monotonically with β and exceeds 0.5 when β > 60°. This implies that matching of impedance between vacuum and hBN is getting better and better as β increases, resulting in a reducing of the reflectivity and enhancement of the absorptivity, according to Eq. (4.11). When θ = 60◦ , Im(Z) is also very small when β > 20°, but Re(Z) is closer to unity than the case for θ = 0◦ , as revealed in Fig. 4.3. Hence, applying the concept of impedance helps to elucidate the trend of the emissivity varying with the angle of incidence. When the incidence is a TE wave, the real and imaginary parts of the normalized admittance are plotted in Fig. 4.7b for θ = 0◦ and 60◦ , respectively, when φ = 0◦ or 180◦ . The admittance is independent of β in this case. While the real part is small, the magnitude of the imaginary part is much larger than unity, especially for the case when θ = 60◦ . This is due to the fact that the real part of ε⊥ is negative at 1400 cm−1 . A large admittance will result in a larger reflection from the surface and thus the emissivity is small, as already seen from Fig. 4.3b and 4.3d at φ = 0◦ or 180◦ . When the azimuthal angle φ is 90° or 270°, TE and TM waves inside the medium are decoupled only when the angle of incidence is 0°. In this case, the normalized impedance for a TM wave incidence is

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4 Broadband Perfect Absorption

1 Z=√ ε⊥

(4.12)

and the normalized admittance of the medium for a TE wave incidence is  ε⊥ ε|| (4.13) Y = 2 ε|| cos β + ε⊥ sin2 β Note that the impedance and the admittance in Eqs. (4.12) and (4.13) are the reciprocals of the admittance in Eq. (4.10) and the impedance in Eq. (4.9) when θ = 0◦ , respectively. Therefore, in can be inferred from Eq. (4.11) that the emissivity for TE wave at φ = 90◦ or 270◦ is the same as the emissivity for TE wave at φ = 0◦ or 180◦ and vice versa, as seen in Fig. 4.5a and c. Impedance and admittance cannot be defined when TM and TE waves inside the medium are coupled to each other. Therefore, they fail to be applicable for interpreting the reflectivity and emissivity when coupling and polarization conversion occur between TM and TE waves. Similar discussion can be made for the case at wavenumber equals to 800 cm−1 . The real and imaginary parts of the normalized impedance varying with the tilting angle β for a TM wave at θ = 0◦ and 60◦ are shown in Fig. 4.7c, when the azimuthal angle is 0° or 180°. It can be seen for either value of the angle of incidence, the real part of the normalized impedance is far away from unity when β > 40°. On the other hand, the imaginary part increases with β. Hence, the reflectivity is large and the emissivity is small, which agrees with the results shown in Fig. 4.6a and c. The corresponding results of the admittance for incidence of a TE wave are shown in Fig. 4.7d, when the flat lines indicate that the admittance is independent of β. However, in contrast to the case at 1400 cm−1 , the imaginary part here is close to zero while the real part is larger than unity. This is because ε⊥ has a positive real part at 800 cm−1 . From these values of the admittance and according to Eq. (4.11), the emissivity at 800 cm−1 is not small, especially for the case when θ = 0◦ . This explains the calculated results shown in Fig. 4.6b and d. When the azimuthal angle φ is 90° or 270°, and the angle of incidence is 0°, the emissivity for TM wave is the same as that for TE wave at φ = 0◦ or 180◦ and vice verse. The same reason could be used to explain result at wavenumber 1400 cm−1 . We now turn our attention to the directional emissivity of the medium when the tilting angle β is fixed. Figure 4.8a, c and e shows the emissivity as a function of the angles φ and θ for TM wave at the wavenumber 1400 cm−1 , when β is 0°, 45°, and 90°, respectively. It can be seen from figure that the emissivity is independent of φ This is because the optical axis is along the z-axis when β is 0°; thus, there is no coupling effect between TE and TM waves. However, the emissivity first increases with θ , reaching a maximum (though small) at angle greater than 80°, then drops rapidly to zero when θ = 90◦ . In contrast, due to existence of polarization conversion, the emissivity depends on both φ and θ when β is greater than 0°, as shown in Fig. 4.8c and e. In addition, the emissivity is high around φ = 0◦ and 180◦ . It first increases with θ and then drops rapidly when θ exceeds about 80°. The corresponding results

4.1 Directional and Hemispherical Emissivity of hBN

57

Fig. 4.8 The spectral directional emissivity at 1400 cm−1 : a TM wave, β = 0°; b TE wave, β = 0°; c TM wave, β = 45°; d TE wave, β = 45°; e TM wave, β = 90°; f TE wave, β = 90°

for TE wave at wavenumber 1400 cm−1 are shown in Fig. 4.8b, d and f, respectively. In this case, the dependence of the emissivity on φ is similar to that for TM wave, except that the regions for high values of the emissivity are around φ = 90◦ and 270◦ , as shown in Fig. 4.3b and d. Furthermore, the emissivity decreases as θ increases, which is different from the case form TM wave. The corresponding cases at wavenumber 800 cm−1 are shown in Fig. 4.9a–f for comparison. It can be seen the distribution patterns are quite similar to those in Fig. 4.8a–f. However, when β = 0°, the emissivity for both TM and TE waves are much higher than that at wavenumber 1400 cm−1 . In addition, for the cases of β = 45° and 90°, high emissivity values for TM wave are found around φ = 90◦ and 270◦ ,

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4 Broadband Perfect Absorption

Fig. 4.9 The directional emissivity at 800 cm−1 : a TM wave, β = 0°; b TE wave, β = 0°; c TM wave, β = 45°; d TE wave, β = 45°; e TM wave, β = 90°; f TE wave, β = 90°

while those for TE wave are found around φ = 0◦ and 180◦ , in contrast to those at wavenumber 1400 cm−1 . The above analysis can be applied to the spectral directional emissivity of hBN in other wavenumber regions. However, the influence of the optical axis orientation in other wavenumber regions is not as significant as in the hyperbolic bands, since the real parts of ε⊥ and ε|| have the same sign and the imaginary parts are relatively small.

4.2 Broadband Perfect Absorption of hBN

59

4.2 Broadband Perfect Absorption of hBN Here we study the absorption of hBN. When the optical axis of hBN is along the z-axis system xyz, its permittivity tensor can be expressed as ε =   of the coordinate diag ε⊥ , ε⊥ , ε|| . When the optical axis is tilted off the z-axis by an angle in the x–z plane, as shown in Fig. 4.10a, the permittivity tensor of hBN can be written as   ⎞ ⎞ ⎛ εx x 0 εx z ε⊥ cos2 φ + ε|| sin2 φ 0 ε|| − ε⊥ sin φ cos φ ⎠ (4.14) ε = ⎝ 0 ε yy 0 ⎠ = ⎝ 0 ε⊥  0 2 2 ε|| − ε⊥ sin φ cos φ 0 ε⊥ sin φ + ε|| cos φ εzx 0 εzz ⎛

Here we consider the incidence wave is a TM wave with the x–z plane as the plane of incidence. In this case, there is no polarization coupling in the slab and for a transverse electric (TE) wave, the slab behaves like an isotropic medium with a permittivity of ε⊥ , regardless of the tilting. In general, when the incidence angle is θ , the parallel wavevector component is k x = k0 sin θ , where k0 = ω/c and c is the speed of light in vacuum. The relation between the parallel and vertical components of the wavevector in the tilted hBN is εzz k z2 + 2εx z k x k z + εx x k x2 = ε|| ε⊥ k02

(4.15)

The diagram in Fig. 4.10b illustrates the isofrequency curves governed by Eq. (4.15), when ε⊥ and ε|| have opposite signs. We can see that the isofrequency curves are asymmetric about the two wavevector components k x and k z . When the tilted angle is zero, then εx z = 0, Eq. (4.15) reduces to ε|| k z2 + ε⊥ k x2 = ε|| ε⊥ k02

(4.16)

This relation has been studied in many papers. One conclusion about this relation is that if a large wavevector is to be achieved, the vertical and the parallel wavevector components must be large simultaneously. In contrast, if we consider the parallel wavevector component k x = 0 in Eq. (4.15), we obtain  kz = k =

Fig. 4.10 a Schematic of the coordinates, the optical axis of hBN is tilted off the z-axis by an angle in the x–z plane. b The asymmetric isofrequency curves for the tilted hBN

ε|| ε⊥ k0 ε|| cos2 φ + ε⊥ sin2 φ

(4.17)

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4 Broadband Perfect Absorption

Fig. 4.11 The apparent permittivity of the tilted hBN. a The real part; b the imaginary part

The above equation indicates that the wavevector can be large enough if εzz = ε|| cos2 φ + ε⊥ sin2 φ is small. This is possible when ε|| and ε⊥ have opposite signs and the value of φ is properly selected. The large wavevector in this case is illustrated schematically in Fig. 4.10b. For conciseness, we defined under normal incidence an apparent permittivity εa as εa =

ε||

cos2

ε|| ε⊥ φ + ε⊥ sin2 φ

(4.18)

√ Such that k z = εa k0 in the slab. In this paper we investigate the behavior of the tilted hBN in the wavenumber range from 1350 to 1600 cm−1 . hBN exhibits the property of hyperbolicity in the band from 1370 to 1600 cm−1 . The εa varying with wavenumber and the tilted angle is shown in Fig. 4.11b. It can be seen that when the tilted angle is zero, the imaginary part of the permittivity is rather small in the hyperbolic band except for wavenumber close to ωT O,⊥ . When the tilted angle is not zero, the wavenumber for maximum value of Im(εa ) changes with the tilted angle, and it moves towards higher wavenumber when the tilted angle increases. According to Eq. (4.18), a large amplitude of εa will ensure that the wavevector is also large in the slab. The property of the enhanced wavevector shifting with the tilted angle makes it potential to achieve strong absorption in hBN at tunable wavenumbers. We study the absorption property of layered structures made of hBN slabs with varied tilted angles of the optical axis for normal incidence of a TM plane wave on the surface of the structure. We first calculate the absorptivity of a hBN slab of thickness 20 nm as a function of wavenumber and the tilted angle for normal incidence of a TM plane wave from air. The result is shown in Fig. 4.12a. The bright color in the graph indicates large absorption, which is in accordance with the behavior of εa depicted in Fig. 4.11. Because the reflection at the interface between air and the hBN is large due to the impedance mismatching, the maximum absorptivity does not exceed 0.5. We also investigate the impact of the slab thickness on the absorption. Taking wavenumber

4.2 Broadband Perfect Absorption of hBN

61

Fig. 4.12 a The absorptivity of a 20 nm thick hBN slab as a function of wavenumber and the tilted angle for normal incidence of a TM wave. b The absorptivity of the hBN slabs as a function of slab thickness at wavenumber 1527 cm−1 when tilted angle is 45°

1527 cm−1 for an example, this point corresponds to the maximum amplitude of εa when the tilting angle is 45°. Figure 4.12b shows that absorptivity as a function of the slab thickness. It can be seen an absorptivity peak of 0.5 appears at 20 nm. This is due to the combination of the enlarged wavevector and the Fabry–Perot (FP) resonance slab. The FP quantization condition can be written as βa + Re(k z )d = mπ

(4.19)

where m is an integer, d is the thickness of the hBN slab, Re(k z ) is the real part of k z , βa represents the phase of the reflection coefficient ra at the air/hBN interface, which is expressed for normal incidence as ra =

√  √  εa − 1 / εa + 1

(4.20)

The slab thickness of 20 nm satisfies the FP resonance condition with m = 1. As the slab thickness increases further, the absorptivity curve does not show higher-ordered FP resonances, but saturates at a value around 0.21. This is because the imaginary part of k z is so large that the wave decays quite fast during propagation in the slab. In fact, when the slab is as thick as 441 nm, it also satisfies the FP resonance condition. But the backward propagating wave in the slab by reflection at the second interface is too weak to produce strong interference with the forward propagating wave. Therefore, the maximum absorptivity corresponds to the first-ordered FP resonance. Based on the above analysis, we can see that the enlarged wavevector can be shifted by tilting the optical axis, thus enhanced absorption can be achieved at different wavenumbers. The enhanced absorptivity depends on the thickness of the slab, which can be optimized based on the FP resonance condition. However, there two issues to be considered if hBN slabs were used for constructing efficient absorbers. On the one hand, the absorptivity of one hBN slab is below 0.5 due to the impedance

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4 Broadband Perfect Absorption

Fig. 4.13 a The absorptivity of three single hBN slabs with tilted angle equal to 60°, 45°, and 30°, respectively, and that of a layered structure by stacking the three slabs together varying with wavenumber. b Similar to (a) except that the slab thickness is selected based on the FP resonance condition

mismatching. On the other hand, realizing broadband absorption in hBN slabs by employing stacked hBN slabs of different tilted angles. Figure 4.13a shows the absorptivity curves varying with wavenumber for three single hBN slabs with tilted angle equal respectively to 60°, 45°, and 30°, and that of a layered structure by stacking the three slabs together with the one of tilted angle equal to 60° on the top. Each slab has the thickness of 20 nm. It can be seen that for the case of a single hBN slab there is a single absorptivity peak appearing at wavenumber 1574, 1527, and 1463 cm−1 , which corresponds respectively to the tilted angle of 60°, 45°, and 30°. But, interestingly, when these three slabs are stacked together, there exhibits three peaks in the absorptivity curve completely overlapping with those corresponding to the single slab case. The absorptivity peaks are caused by the enlarged wavevector in the slabs and because the tilted angles are separated from each other by at least 15° so that the enhanced absorption in each slab of the structure is independent to each other. This is the reason why the absorptivity peaks of the layered structure overlap completely with those of the single slabs. Note that the two peaks corresponding to the tilted angles 60° and 30° are lower than that for the tilted angle of 45°. This is because the slab thickness of 20 nm does not satisfy the FP resonance condition Eq. (4.19) for these two tilted angles. We select the slab thickness based on Eq. (4.19) for the tilted angles of 60° and 30°, and recalculate the absorptivity of the layered structure. The results are shown in Fig. 4.13b, in which the peaks are all equal to 0.5 and they overlap completely with those of the single slabs. Keep in mind that the peaks in Fig. 4.13b result from the combined effects of the enlarged wavevector and the FP resonance. The reason why the FP resonance condition in the layered structure can be accurately estimated by the simple relation in Eq. (4.19) is that the value of Re(k z ) at wavenumber corresponding to the absorptivity peak is much larger than that at other wavenumbers. It follows that the phase difference Re(k z )d in the slab with enlarged wavevector is much larger than that in the other two slabs where there is no enlarged wavevector. In other words, the former dominates

4.2 Broadband Perfect Absorption of hBN

63

Fig. 4.14 The absorptivity of the layered structure varying with wavenumber for different number of slabs: a the difference of tilted angles between adjacent slabs is 1°; b the difference of tilted angles between adjacent slabs is 10°, 3°, 2°, and 1°, respectively

the phase different Re(k z )d in the layered structure so that Eq. (4.19) can be used to accurately determine the slab thickness of the layered structure for FP resonance. We now investigate the absorptivity of a layered structure made of varied numbers of hBN slabs, with each slab having the thickness of 20 nm and the tilted angle of lth slab equals to (91-l)° (l = 1, …, L). The calculated results are shown in Fig. 4.14a for the total slab number L equals to 1, 31, 61, and 91, respectively. From Fig. 4.11, the wavenumber for maximum amplitude of εa can be found to increase with the tilted angle and this maximum amplitude is getting small and smaller as the tilted angle approaches 90°. It should be noted that when the tilted angle is equal to 90°, εa reduces to ε|| whose maximum amplitude is located at about 780 cm−1 . Therefore, for the case of L = 1, i.e., the structure consists of only one hBN slab with the tiled angle equal to 90°, there is no enlarged wavevector in the wavevector range from 1350 to 1650 cm−1 and the absorptivity is close to zero, as shown in Fig. 4.14a. However, when L is taken as 31, 61 and 91, a band for large enhancement of the absorptivity is found to form and the larger the value of L, the broader the band is. The mechanism for the enhanced absorptivity in the band is attributed to the enlarged wavevectors in the slabs of different tilted angles. Enlarged wavevectors in the structure occur at more and more wavenumbers when L is getting larger and larger, which makes the enhanced absorptivity band getting broader and broader. Most importantly, when the absorptivity peaks are getting close to each other, interplay between these peaks can cause the absorptivity to be further enhanced significantly. As a consequence, the absorptivity higher than 0.94 is achieved in a broadband from 1367 to 1580 cm−1 for L = 91, as can be seen from Fig. 4.14a. Figure 4.14b shows a very different absorptivity characteristic of the structure with the number of slabs L taken as 10, 31, 46 and 91, respectively. The corresponding tilted angle of the lth slab varies as 10(L-l)°, 3(L-l)°, 2(L-l)°, and (L-l)°, respectively. As such, the difference of the tilted angles between adjacent slabs is 10°, 3°, 2°, and 1°, respectively. Therefore, the tilted angle of the top slab of the structure is 90° while that of the bottom slab is 0° in this case. It can be seen for each value of L, the absorptivity is greatly enhanced in a

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4 Broadband Perfect Absorption

Fig. 4.15 a Real and b imaginary parts of the normalized impedance of a hBN medium as a function of the tilted angle for different values of the wavenumber

broadband except that separated absorptivity peaks can be clearly seen for the case L = 10. More and more absorptivity peaks appear and they get closer to each other as the value of L increases, such that they overlap with each other when L = 91, which enhances the absorptivity to higher than 0.94 in the whole band. To further elucidate the mechanisms for the absorptivity enhancement in the structure, we investigate the normalized impedance of a hBN medium with its optical axis at different tilted angles. The normalized impedance of the hBN medium is defined as the ratio of its impedance over the impedance of vacuum. For normal incidence of a TM plane wave, the normalized impedance of the medium can be expressed as 1 Z=√ εa

(4.21)

We calculate the normalized impedance of the hBN medium based on Eq. (4.12) and its real and imaginary parts are shown in Fig. 4.15a and b respectively as a function of the tilted angle for different values for the wavenumber. It can be seen from Fig. 4.15 that as the tilted angle increases from 0° to 90°, the real part of the normalized impedance first stays close to zero, then increases monotonically with the tilted angle. For all given values of the wavenumber, the real part of the normalized impedance approaches a value of 0.6 at the tiled angle of 90°. At the meanwhile, the imaginary part of the normalized impedance also increases monotonically with the tilted angle and it eventually converges to zero at the tilted angle of 90° for all given values of the wavenumber. A simple calculation can reveal that the normalized impedance at the tilted angle of 90° makes the reflectivity at the air/hBN interface smaller than 0.0625. In other words, matching of impedance at the air/hBN interface is the best, though not perfect, at the tilted angle of 90°. This is the reason why we set the tilted angle of the top slab of the structure to be 90°. Furthermore, since difference of the normalized impedances of two adjacent slabs in the structure is small, matching of impedance at the two slabs’ interface can also be satisfied, thus reflection at the interfaces between adjacent slabs can be neglected. Therefore, it is

4.2 Broadband Perfect Absorption of hBN

65

Fig. 4.16 a The absorptivity of a 31-layer structure varying with wavenumber. b The absorptivity of a 91-layer structure varying with wavenumber. The thickness of each slab is the same and is taken to be 20, 40, and 60 nm, respectively

clear to this end that the greatly enhanced absorption in a broadband is contributed from the enlarged wavevectors in different slabs of the structure and matching of impedance at interfaces of the slabs. It is worthy to investigate the effect of the slab thickness on the absorptivity of the layered structure as the number of slabs increases. In Fig. 4.14a and b, the absorptivity curves are calculated by fixing the thickness of each slab to be 20 nm, without considering slab thickness optimization. This is because as the number of slabs L increases and the tilted angle difference between adjacent slabs decreases, the absorptivity peaks due to the enlarged wavevectors in different slabs are no longer mutually independent, but instead, they get close and overlap with each other. In addition, although the phase difference Re(k z )d through a single slab is small if there is no enlarged wavevector, the phase difference through many slabs without enlarged wavevector may no longer be small and may not be neglected. Therefore, the simple relation for FP resonance in Eq. (4.21) cannot no longer be used for optimization of the slab thickness of the structure. In Fig. 4.16a, we show the absorptivity of the layered structure composed of 31 slabs varying with wavenumber for three cases of the slab thickness. The thickness of each slab is the same and is taken to be 20, 40, and 60 nm, respectively. It can be seen close interplay of the absorptivity peaks enhances the absorptivity to be much higher than the optimized value shown in Fig. 4.13. In addition, the absorptivity increases with the thickness of the slabs. On the other hand, the tilted angle difference is only 1° between adjacent slabs in the structure when the number of slabs is increased to 91. Thus, the intervals between wavenumber for enlarged wavevectors in the structure are small enough such that the corresponding absorptivity peaks can overlap well with each other. The close interplay between the absorptivity peaks not only enhances the absorptivity to be higher than 0.94, but also makes the absorptivity curve flattened in a broadband, as shown in Fig. 4.16b. Furthermore, the enhanced absorptivity is insensitive to the slab thickness in this case as the slab thickness increases from 20 to 60 nm.

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4.3 Broadband Perfect Absorption of Metamaterial Here we introduce how to realize broadband perfect absorption of metamaterials. Figure 4.17a and b schematically shows the metamaterial made of doped silicon nanowire arrays embedded in the host medium of polymethylpentene (TPX), we select TPX as the host medium because of its excellent transmittance and the fact that its relative permittivity can be stabilized at around εh = 2.12 for a broadband. For doped silicon, its relative permittivity in the infrared region (λ > 2 μm) can be described by the Drude model as εD − Si (ω) = ε∞ −

ω2p ω(ω + jγ )

(4.22)

where ω is the angular frequency and ε∞ is the limiting value of the relative permittivity at high frequencies and is equal to 11.7. ω p and γ denote the plasma frequency and scattering rate, respectively. In this work, we consider an n-type doped silicon with doping concentration equals to 5 × 1020 cm−3 . The corresponding plasma frequency ω p and scattering rate γ are equal to 2.4275 × 1015 rad/s and 1.9579

Fig. 4.17 a Cross-section view and b top view of the composite made of silicon nanowire arrays embedded in the host medium of polymethylpentene (TPX). c Schematic view of a metamaterial slab. d Schematic view of the stacked metamaterial slabs

4.3 Broadband Perfect Absorption of Metamaterial

67

× 1014 rad/s, respectively. Assuming that f is the volume filling ratio of the doped silicon nanowires, the effective relative permittivity tensor of the composite in the x*y*z* coordinate system shown in Fig. 4.17a and b can be expressed as ⎞ ε⊥ 0 0 ε = ⎝ 0 ε⊥ 0 ⎠ 0 0 ε|| ⎛

(4.23)

where the z* direction is along the orientation of the nanowires that represents also the direction of the optical axis of the metamaterials. ε⊥ are ε|| the permittivity components that are perpendicular and parallel to the optical axis, respectively, and can be expressed as ε⊥ = εh

εD − Si + εh + f (εD − Si − εh ) , ε|| = εh + f (εD − Si − εh ) εD − Si + εh − f (εD − Si − εh )

(4.24)

Note that if f is taken as 0.1 and with the selected parameters mentioned   earlier, the metamaterial exhibits the property hyperbolicity, i.e., Re(ε⊥ )Re ε|| < 0 for wavelength larger than 5 μm. This is critical for the perfect absorber to be discussed below. Figure 4.17c schematically illustrates a tilted AHM slab where the optical axis of the AHM is tilted by an angle of φ, i.e., the angle between the z-axis and the z-axis of the coordinate system is equal to φ. Figure 4.17d schematically illustrates the proposed structure, which is composed of L AHM slabs stacked one by one with the tilted angle and the thickness of the lth slab equal to φl and dl (l = 1,2,…, L). By performing a rotational transformation, the relative permittivity tensor of the lth AHM slab in the xyz coordinate system can be obtained as ⎛

⎛ ⎞ ⎞ ε⊥ 0 0 cos φl 0 − sin φl εl = T⎝ 0 ε⊥ 0 ⎠T−1 , T = ⎝ 0 1 0 ⎠ 0 0 ε|| sin φl 0 cos φl

(4.25)

This leads to the expression of εl in the following form ⎛

⎞ εx x 0 εx z εl = ⎝ 0 ε yy 0 ⎠ εzx 0 εzz

(4.26)

  where εx x = ε⊥ cos2 φl + ε|| sin2 φl , εx z = εzx = ε⊥ − ε|| sin φl cos φl , ε yy = ε⊥ , εzz = ε⊥ sin2 φl + ε|| cos2 φl . It can be seen from the above equation that by appropriately selecting the tilted angle, εzz can  be very small at certain frequencies within the hyperbolic band where Re(ε⊥ )Re ε|| < 0. We consider incidence of a TM plan wave from vacuum with the x–z plane as the plane of incidence, i.e., the magnetic field vector is parallel to the y-axis. Assuming

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4 Broadband Perfect Absorption

that the angle of incidence is θ, the wavevector component in the x-direction is k x = k 0 sinθ and the wavevector components in the z-direction in the slab can be calculated as    εx z ± εx2z − εx x εzz k x2 − εzz k02 k zl1,2 = (4.27) εzz Here, one of these two wavevector components has a positive imaginary part and the other has a negative imaginary part, which are respectively associated with propagation of forward and backward waves in the lth slab. Note that k zl1 and k zl2 are not opposite numbers if εx z is not equal to zero and they depend highly on the value of εzz , as can be seen in Eq. (4.27). To investigate the property of matching of impedance at the interface of the structure, we also define the normalized impedance of the slab by diving its impedance (assuming an infinite thickness) 

Zl =

2 2 η0 k x − εzz k0  k0 εx2z − εzz εx x

(4.28)

by the corresponding impedance of vacuum Z 0 = η0 cos θ as 

εzz k02 − k x2 Z l /Z 0 = √ k0 cos θ ε⊥ ε||

(4.29)

where η0 is the product of c and the absolute permeability of vacuum μ0 and is equal to 120π . We investigate the relationship between the absorption and the number of slabs L in the structure. The absorptance α of the structure can be calculated according to Kirchhoff’s law as α = 1 − R − T, where R and T denote the reflectance and the transmission of the structure, respectively. Since the composite is lossy and the structure is multilayered, we applied the enhanced transmittance matrix approach for calculating R and T of the structure to avoid the problem of numerical overflow. In our calculations, the filling ratio and the thickness of the slab were taken as f l = 0.1 and d l = 0.4 μm, respectively, while the tilted angle φl was arranged in the way as φl = (39 + l − 1)◦ (l = 1, 2, . . . , L). Figure 4.18 shows the calculated absorptance α of the structure as a function of wavelength and the number of slabs L for the angle of incidence θ equal to 60°. It can be seen the absorptance is enhanced in a broader and broader wavelength range as L increases such that the absorptance is higher than 0.99 in an ultra-broadband ranging from 2 to 50 μm when L = 41. In order words, perfect absorption in an ultra-broadband is realized with the structure. We discuss in the following the underlying mechanisms for the realization of broadband perfect absorption. One necessary condition for achieving perfect absorption is that the reflection at the surface of the structure is small. Here, we first consider

4.3 Broadband Perfect Absorption of Metamaterial

69

Fig. 4.18 Absorptance of the structure as a function of wavelength for different number of slabs in the structure

a simple situation. Supposing that the structure has only one slab and the thickness is infinite. At the interface between the structure and vacuum, the boundary condition for the electromagnetic field is 

1 1 Z 0 −Z 0

     1 1 1 t = r Z −Z 0

(4.30)

where r and t are the reflection and the transmission coefficients, respectively. From Eq. (4.30), we can obtain r=

1 − Z /Z 0 Z0 − Z = Z0 − Z 1 − Z /Z 0

(4.31)

The reflectance can be obtained as R = |r|2 . From Eq. (4.31), one can see that if there is matching of impedance the reflectance will be small. Figure 4.19a shows the calculated real and imaginary parts of the normalized impedance of the first slab of the structure for wavelength between 2 and 50 μm. It can be seen the real part is close to one, while the imaginary part is much less than one for the whole specific wavelength range, especially for wavelength larger than 5 μm. Therefore, matching of impedance is satisfied so that the reflectance at the surface is small. Figure 4.19b and c shows, respectively, the calculated real and imaginary parts of the normalized impedance of each slab (denoted by the slab index number l) of the structure for L = 41 and for different wavelength, according to Eq. (4.29). It can be seen the differences between neighboring slabs of both real and imaginary parts of the impedance are small. In other words, matching of impedance is satisfied well at all interfaces between neighboring slabs inside the structure. As a consequence, reflection from the structure is very small due to the matching of impedance at the interfaces, which is a necessary condition for perfect absorption. Here, the main

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4 Broadband Perfect Absorption

Fig. 4.19 a The real and imaginary parts of the normalized impedance of the first slab as a function of wavelength. b The real part of the normalized impedance in each slab of the structure for L = 41 and for different wavelengths. c The imaginary part of the normalized impedance in each slab of the structure for L = 41 and for different wavelengths

reason why the impedance at 5 μm deviates from and becomes much larger than that at the other wavelengths as the slab index number increases, as shown in Fig. 4.19b and c, is that the real part of ε|| is close to zero, while the imaginary part is also very small at wavelength of 5 μm. Recently, Xu et al. also proposed to realize perfect absorption in a broadband based on the mechanism of matching of impedance by using graded index slabs, where the graded refractive index of each layer can be tuned by changing the dopants and doping concentrations [11]. Other cascaded structures have also been proposed to achieve broadband absorption, but the matching of impedance condition was found to be difficult to satisfy at each interface, which deteriorates the absorption performance of the structure. In addition to the condition of matching of impedance, we now look at another important factor that contributes to broadband perfect   absorption. We define the attenuation coefficient of the lth slab as δl = Im k zl+ dl , where k zl+ is one of k zl1,2 the that has a positive imaginary part, i.e., the wavevector component representing wave propagation in the positive z-direction. Figure 4.20a presents the calculated attenuation coefficient of each slab (denoted by the slab index number l) of the structure for L = 41 and for different wavelengths. It can be seen the attenuation

4.3 Broadband Perfect Absorption of Metamaterial

71

Fig. 4.20 a The attenuation coefficient in each slab of the structure (L = 41) for different wavelengths. b The absorptance of the structure when the tilted angle of each slab is set the same and is equal to 0° and 90°

coefficient can be quite large in some slabs, depending on the wavelength and the tilted angle. Specifically, large attenuation takes place in the slabs with small tilted angles for incidence of shorter wavelengths. As the wavelength of the incidence increases, occurring of large attenuation will move to the slabs with increased tilted angles. As large attenuation coefficient means that absorption in the slab is large, and since reflection is small due to matching of impedance, the effect of the backward wave in the slab by reflection can be neglected. The variation of the attenuation coefficient with wavelength and the tilted angle clearly explains the results shown in Fig. 4.18 that the band for perfect absorption is getting broader and broader as the number of slabs increases. It can be seen from Eq. (4.27) that appropriately selecting the tilted angle can make the magnitude of k zl+ much larger than the case without tilting at certain frequencies within the band where Re(ε⊥ )Re ε|| < 0, since in this case εzz can be very small. As a consequence, a large attenuation coefficient can be obtained in a slab of small thickness. It has been demonstrated a small imaginary part of the permittivity can cause great absorption in a slab due to the asymmetric wave propagation in the slab with tiled optical axis. As a reference, Fig. 4.20b shows the absorptance of the structure (with L = 41) as a function of wavelength, when the tilted angle of each slab is set to be the same and is equal to 0° and 90°, respectively. In either case, the off-diagonal element of the permittivity tensor is zero. Comparing Fig. 4.18 with Fig. 4.20b, we can see that the waveband of perfect absorption is greatly broadband with the asymmetric structure. It should be pointed out the large absorption in Fig. 4.20b for wavelength between 2 and 5 μm comes from the inherent dielectric properties of doped silicon. Finally, the effect of the angle of incidence on the absorption is investigated. Shown in Fig. 4.21a is the calculated absorptance of the structure (L = 41) as a function of wavelength and the angle of incidence. It can be seen when the angle of incidence is in the range from 38° to 74°, the absorptance is above 0.9 in the whole wavelength range from 2 to 50 μm. Even in the much wider range of angle of incidence from 0° to 80°, the absorptance is still higher than 0.75 in the whole specified wavelength

72

4 Broadband Perfect Absorption

Fig. 4.21 a The absorptance of the structure (L = 41) as a function of wavelength and angle of incidence. b The real part of the normalized impedance of the first slab as a function of wavelength at different angles of incidence. c The imaginary part of the normalized impedance of the first slab as a function of wavelength at different angles of incidence

range. Therefore, large absorption is realized in a broad wavelength range as well as in a wide range of angle of incidence. Figure 4.21b and 4.21c shows, respectively, the real and the imaginary parts of the normalized impedance of the first slab as a function of wavelength for the angle of incidence equal to 0°, 38°, 60°, and 74°. It can be seen while the real part of the impedance changes significantly with the angle of incidence, the imaginary part changes little and is close to zero except at wavelength around 5 μm. Therefore, the reflectance R of the structure can be estimated to be lower than 0.1 when the real part of the normalized impedance of the first slab is within the range between 0.52 and 1.92, according to Eq. (4.31). From Fig. 4.21b, only the real part of the normalized impedance for θ equals to 0° falls beyond this range, agreeing well with the numerical result in Fig. 4.21a. It should be mentioned that the thickness of each slab has impact on the absorption. It is possible to optimize the results by tuning the thickness of the slabs. However, our calculated results reveal that the absorptance of the structure changes little if the slab thickness is larger than 0.4 μm. Therefore, we do not show the results for varied slab thickness.

References

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References 1. Nefedov IS, Valagiannopoulos CA, Hashemi SM, Nefedov EI (2013) Total absorption in asymmetric hyperbolic media. Sci Rep 3:2662 2. Nefedov IS, Valagiannopoulos CA, Melnikov LA (2013) Perfect absorption in graphene multilayers. J Opt 15(11):114003 3. Basu S, Zhang ZM, Fu CJ (2009) Review of near-field thermal radiation and its applications to energy conversion. Int J Energy Res 33(13):1203–1232 4. Rosa FSS, Dalvit DAR, Milonni PW (2008) Casimir interactions for anisotropic magnetodielectric metamaterials. Phys Rev 78:032117 5. Malone CG, Choi BI, Flik MI, Cravalho EG (1993) Spectral emissivity of optically anisotropic solid media. ASME J Heat Transfer 115(4):1021–1028 6. Arrieta IGD, Echaniz T, Fuente R, Campo LD, Meneses DDS, Lopez GA, Tello MJ (2017) Mid-infrared optical properties of pyrolytic boron nitride in the 390–1050 °C temperature range using spectral emissivity measurements. J Quant Spectrosc Radiat Transfer 194:1–6 7. Zhao B, Sakurai A, Zhang ZM (2016) Polarization dependence of the reflectance and transmittance of anisotropic metamaterials. J Thermophys Heat Transfer 30(1):240–246 8. Liu XL, Zhang RZ, Zhang ZM (2014) Near-field radiative heat transfer with doped-silicon nanostructured metamaterials. Int J Heat Mass Transf 73:389–398 9. Zhang RZ, Zhang ZM (2015) Negative refraction and self-collimation in the far infrared with aligned carbon nanotube films. J Quant Spectrosc Radiat Transfer 158:91–100 10. Wang XJ (2012) Study of the radiative properties of aligned carbon nanotubes and silver nanorods. Ph.D. Dissertation, Georgia Institute of Technology, Atlanta, GA 11. Bai Y, Zhao L, Ju DQ, Jiang YY, Liu LH (2015) Wide-angle, polarization-independent and dual-band infrared perfect absorber based on L-shaped metamaterial. Opt Express 23(7):8670– 8680

Chapter 5

Near-Field Radiative Heat Transfer Between Anisotropic Materials

As an anisotropic material, the optical response of hBN is related to the orientation of its optical axis. Here, we studied the near-field radiative heat transfer between a planar emitter and a planar receiver separated by a vacuum gap as shown in Fig. 5.1. The structures of the emitter and the receiver, which consist of bare hBN or graphenecovered hBN slabs, are essentially the same. For convenience, the optical axis of hBN is considered in the x-z plane of the coordinate system xyz and it is tiled off the z-axis by angle of α 1 and α 2 for the emitter and the receiver, respectively. In addition, the width of vacuum gap and the thickness of hBN are denoted by d and h, respectively. Graphene is modeled as a layer of thickness  = 0.3 nm with an effective dielectric function εeff = 1 −

jσs ε0 ω

(5.1)

where ε0 is the absolute permittivity of vacuum and σs is the sheet conductivity that includes the contributions from both the intraband and intranband transitions. In the mid- and far-infrared region, σs is dominated by the intraband transitions and can be approximately written as σs =

τ e2 μ 2 π  1 + jωτ

(5.2)

where e is the electron charge,  is the reduced Planck constant, τ is the relaxation time and μ is the chemical potential. Based on the fluctuation–dissipation theorem and the reciprocity of the dyadic green function, the NFRHF between anisotropic media can be expressed as

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 X. Wu, Thermal Radiative Properties of Uniaxial Anisotropic Materials and Their Manipulations, Springer Theses, https://doi.org/10.1007/978-981-15-7823-6_5

75

76

5 Near-Field Radiative Heat Transfer Between Anisotropic Materials

Fig. 5.1 Schematic of near-field radiative heat transfer between two graphene-hBN heterostrucutures. The optical axis (OA) of hBN is in the x-z plane and tilted off the z-axis by an angle

1 Q= 8π 3

∞

2π ∞ ξ (ω, β, φ)βdβdφ

[ (ω, T1 ) − (ω, T2 )] 0

0

(5.3)

0

  where (ω, T ) = ω eω/ k B T − 1 is the average energy of a Planck oscillator, φ is the azimuth angle, and ξ (ω, β, φ) is called the energy transmission coefficient or the phonon tunneling probability, which can be expressed as  ξ (ω, β, φ) =

     Tr I −  R2∗ R2 − T∗2T2 D I − R1∗ R1 − T∗1 T1 D∗ , β < k0 (5.4) β > k0 Tr R2∗ − R2 D R1 − R1∗ D∗ e−2|kz |d ,

where β is the wavevector component parallel to the x-y plane. k z = k02 − β 2 is the wavevector component along the z-axis in vacuum. Note that the asterisk denotes conjugate transpose, and Tr(•) takes the trace of a matrix. I is a 2 × 2 unit matrix and

R1,2 =

  (1,2) (1,2) (1,2) rss tss(1,2) tsp rsp , T1,2 = (1,2) (1,2) r (1,2) r (1,2) t ps t pp ps pp

(5.5)

and the matrices that include the reflection and transmission coefficients for incident s- and p-polarized plane waves from vacuum to the emitter or the receiver, respectively. The first and second letters of the subscript in each coefficient denote the polarization state of incident and reflected (transmitted) waves, respectively. These coefficients can be obtained using a modified 4 × 4 transfer matrix method. Note that T = 0 if the incidence is on a semi-infinite medium. The matrix D is given as −1  D = I − R1 R2 e−2 jkz d . In this work, we set the emitter temperature T 1 = 300 K, the receiver temperature T 2 = 0 K, d = 20 nm, μ = 0.37 eV, and τ = 10−13 s. In addition, the tilting angles α 1 and α 2 are first assumed as α 1 = α 2 = α. Four cases are investigated and compared with each other: bulk hBN, hBN slab with thickness of 50 nm, graphene-covered bulk hBN, and graphene-covered hBN slab with thickness of 50 nm. The NFRHT

5 Near-Field Radiative Heat Transfer Between Anisotropic Materials

77

Fig. 5.2 NFRHF between a two bare hBN slabs and b two graphene-covered hBN slabs as a function of the tilting angle of hBN optical axis

as a function of the tilting angle is shown in Fig. 5.2 for the four cases. One can see from Fig. 5.2(a) that the NFRHF between two bulk hBN slabs (i.e., h = +∞) is larger than that between two hBN slabs of h = 50 nm at α = 0°. The heat fluxes of both cases decrease with the tilting angle, but the decrease is faster for the former than for the latter such that the heat flux of the former becomes smaller than that of the latter when is larger than 65°. On the other hand, as shown in Fig. 5.2b, the NFRHF between two graphene/hBN heterostructures of h = +∞ is also larger than that between two graphene/hBN heterostructures of h = 50 nm. But the heat fluxes of these two cases both increase with the tilting angle, and the increase is faster for the former than for the latter. In addition, one can find by comparing Fig. 5.2b with Fig. 5.2a that the NFRHF between two graphene/hBN heterostructures is larger than that between two bare hBN slabs by around one order of magnitude. To elucidate the effect of the tilting angle of the hBN optical axis on the near-field radiative heat transfer between the two media, the corresponding spectral NFRHF is shown in Fig. 5.3 for α equals to 0°, 45°, and 90°, respectively. One can see that the NFRHF between two bare hBN slabs is mainly contributed from the two hyperbolic bands as shown in Fig. 5.3a, b, respectively. For ease of analysis, the dispersion relation for EM wave propagating in bulk hBN is given below as ω 2 2 k||2 k⊥ + = ε|| ε⊥ c

(5.6)

where k⊥ and k|| denote, respectively, the wavevector component perpendicular and parallel to the optical axis. Note that the expression in Eq. (5.6) is in a different form from that in Refs. [1, 2] because ε⊥ and ε|| are defined differently from these 2 = k x2 + k 2y when the optical axis is in in Refs. [1, 2]. Note also that k|| = k z and k⊥ 2 2 the z-direction, whereas k|| = k x and k⊥ = k y + k z2 when the optical axis is in the x-direction, corresponding to the tilting angle α equal respectively to 0° and 90° in

78

5 Near-Field Radiative Heat Transfer Between Anisotropic Materials

Fig. 5.3 Spectral NFRHF between a bulk hBN slabs, b hBN slabs with thickness of 50 nm, c graphene-covered bulk hBN slabs, and d graphene-covered hBN slabs with thickness of 50 nm

this work. In the two hyperbolic bands with opposite signs of ε⊥ and ε|| , the solution to Eq. (5.6) for a given frequency are three-dimensional (3-D) open hyperboloids. As a consequence, thermal emission waves in hBN are propagating waves for k⊥ and k|| far exceeding k 0 , which provides more channels for photon tunneling and thus greatly enhances the NFRHF compared with that in the other frequency bands. These channels for the enhanced NFRHF have previously been termed as hyperbolic phonon polaritons (HPPs) or hyperbolic modes. But the heat flux contributed from these two hyperbolic bands is sensitive to the tilting angle of optical axis. For two bulk hBN slabs, the heat flux contributed from the hyperbolic bands of types I and II is 43.18 and 17.27 kW/m2 , respectively, when the tilting angle is 0°. The contribution from these two hyperbolic bands are 22.08 and 24.59 kW/m2 when the tilting angle is 45° and are 14.25 and 21.92 kW/m2 when the tilting angle is 90°. Therefore, the heat transfer contributed from the hyperbolic band of type I decreases dramatically with the tilting angle while that contributed from the hyperbolic band of type II does not change much. There is a significant decrease in the heat flux around ω = 1.55 × 1014 rad/s as the tilting angle increases. In addition, there appears a peak at ω = 2.85 × 1014 rad/s for α = 45° and at ω = 2.96 × 1014 rad/s for α = 90°. However, for the graphene/hBN heterostructures, the spectral NFRHF is greatly

5 Near-Field Radiative Heat Transfer Between Anisotropic Materials

79

Fig. 5.4 Energy transmission coefficient between two bulk hBN slabs varying with wavevector components k x and k y at 1.55 × 1014 rad/s: a α = 0°, b α = 90°, and c α = 45°

enhanced in two much wider frequency bands below and between the two hyperbolic bands, as shown in Fig. 5.3c, d. This is due to excitation of surface plasmon polaritons (SPPs) at the vacuum/graphene and graphene/hBN interfaces. Therefore, the NFRHF between the graphene/hBN heterostructures is dominated by SPPs, instead of HPPs. In addition, the excited SPPs can couple with HPPs, resulting in the formation of a hybrid mode. To be discussed below. For comparison, the NFRHF between two free standing graphene sheets is also shown in Fig. 5.3c, which is much lower than that between two graphene/hBN heterostructures. Note that from Fig. 5.3, not much change is found for the results when the hBN slab thickness h is changed from 50 nm to ∞, which comes from the fact that for d 2 = 20 nm, a thickness of 50 nm is almost large enough for a hBN slab to be treated as bulk for near-field radiative heat transfer. The underlying mechanism of the effect of the tilted optical axis of hBN can be better understood by presenting the energy transmission coefficient ξ in the k x − k y plane. Figure 5.4a, b show the energy transmission coefficient ξ between two bulk hBN slabs varying with the wavevector components k x and k y at ω = 1.55 × 1014 rad/s when the tilting angle α is equal to 0° and 90°, respectively. When α is equal to 0°, the optical axis is along the z-axis and the permittivity tensor of hBN possesses rotational symmetry in the hBN plane, which results in the solution to

80

5 Near-Field Radiative Heat Transfer Between Anisotropic Materials

Eq. (5.6) possessing rotational symmetry in the k x − k y plane. Hence, the energy transmission coefficient also exhibits rotational symmetry in the k x − k y plane. As shown in Fig. 5.4a, the bright circular region centered at the origin and with radius far exceeding k 0 indicates excitation of HPPs in the hBN slabs, which greatly enhances the spectral NFRHF between the slabs. However, when α is equal to 90°, the optical axis is along the x-axis and the permittivity tensor of hBN has no rotational symmetry in the x–y plane in this case. In fact, the projected HPP isofrequency surface in the k x − k y plane is the region bounded by the hyperbolas corresponding to k z = 0 in Eq. (5.6). Those of the hyperbolas corresponding to k z = 0 in Eq. (5.6) are given as  ky ε|| =± − kx ε⊥

(5.7)

assuming ε⊥ and ε|| are both real numbers. Considering the values of ε⊥ and ε|| at ω = 1.55 × 1014 rad/s and noting that εd = 1, the square root on the right-hand side of Eq. (5.7) is obtained as 0.297. For comparison, the asymptotes based on Eq. (5.7) are also added in Fig. 5.4b, which are found in excellent agreement with the numerical results. The enhanced radiative heat transfer in this case is weaker than that due to HPPs along for α = 0°, as seen from Fig. 5.3a. Shown in Fig. 5.4c are the contours of the energy transmission coefficient for α = 45°, which are very similar to those in Fig. 5.4b except that the angle between the two lines bounding the bright color region is larger than that in Fig. 5.4b. The energy transmission coefficient distribution at ω = 2.85 × 1014 rad/s for two bulk hBN slabs is shown in Fig. 5.5a, b when the tilting angle is 0° and 90°, respectively. This frequency is within the hyperbolic band of type II, at which ε⊥ = −3.6397 − 0.1572 j and ε|| = 2.8085 − 0.0005 j. Hence, the bright circular region shown in Fig. 5.5a for α = 0° is attributed to HPPs. In addition, it has been shown that hyperbolic surface phonon polaritons (HSPhPs), which are resonant modes confined on the surface of hBN, can be excited in this case due to the strong anisotropy of the surface, i.e., opposite signs of ε⊥ and ε|| . The dispersion of HSPhPs, when the values of k x and k y are far exceeding the wavevector k 0 , can be written concisely as 2    2  2 ω − εd2 = ε|| ε⊥ k x2 ε|| ε⊥ − εd2 + k 2y ε⊥ 2 c

(5.8)

where εd denotes the dielectric function of the medium adjacent to hBN and εd = 1 for vacuum. At this angular frequency, the relations 1−ε|| ε⊥ > 0−ε|| ε⊥ > 0 and 2 < 0, can be satisfied when neglecting the imaginary parts of ε⊥ and ε|| . 1 − ε⊥ Therefore, the dispersion in Eq. (5.8) is hyperbolic. In this case, the asymptotes of the HSPhP dispersion curves in the k x -k y plane can be expressed as

5 Near-Field Radiative Heat Transfer Between Anisotropic Materials

81

Fig. 5.5 Energy transmission coefficient between two bulk hBN slabs varying with wavevector components k x and k y at 2.85 × 1014 rad/s: a α = 0°, b α = 90°, and c α = 45°

 ε2 − ε|| ε⊥ ky = ± d2 kx ε⊥ − εd2

(5.9)

When α = 90°, HSPhPs can be excited due to the strong anisotropy of the surface. 2 Keep in mind that ε⊥ < 0 and ε|| > 0,ε⊥ ε|| − 1 < 0 and ε⊥ − 1 > 0 when the imaginary parts of ε⊥ and ε|| are neglected in this case. These inequalities give rise to the HSPhP dispersion curves being hyperbolas with the two focuses on the k y -axis, which can be determined from Eq. (5.8). Excitation of HSPhPs is clearly shown in Fig. 5.5b by the hyperbolas with focuses on the k y -axis. The two dashed lines in this figure also represent the asymptotes of the HSPhPs dispersion curves drawn based on Eq. (5.9), which are clearly seen to be in excellent agreement with the numerical results. Furthermore, the projected HPP isofrequency surface in the k x − k y plane is the region bounded by the hyperbolas that extend in the direction of k x and whose asymptotes are still described by Eq. (5.7). Enhanced radiative heat transfer due to HPPs is also clearly manifested in Fig. 5.5b by the bright color in the region near the origin. Because the asymptotes of the HPP hyperbolas described by Eq. (5.7) are close to those of in this case, they are not shown in Fig. 5.5b. Plotted in Fig. 5.5c are the contours of the energy transmission coefficient for α = 45°. Similar results to those in Fig. 5.5b are also found except that the angle between the asymptotes

82

5 Near-Field Radiative Heat Transfer Between Anisotropic Materials

Fig. 5.6 Energy transmission coefficient between two bulk hBN slabs varying with wavevector components k x and k y at 2.96 × 1014 rad/s: a α = 0°, b α = 90°, and c α = 45°

of the dispersion curves becomes smaller while that between the asymptotes of the hyperbolas that bound the region for getting larger. We also attribute the enhanced energy transmission in this case to the interaction of excited HPPs and HSPhPs, which may be the cause that broadens the dispersion curves, as shown in Fig. 5.5c. Figure 5.6a, b show the energy transmission coefficient between two bulk hBN slabs varying with the wavevector components k x and k y at ω = 2.96 × 1014 rad/s, which is also within the hyperbolic band of type II, when the titling angle α is equal to 0° and 90°, respectively. As this frequency, ε⊥ = −1.0421 − 0.0788 j and ε|| = 2.8222 − 0.0004 j.The energy transmission coefficient ξ distribution shown in Fig. 5.6a is similar to that in Fig. 5.5a, owing to the contribution from HPPs whose relation in this case possesses rotational symmetry in the k x − k y plane. When α = 2 < 0 are satisfied when 90°, as the relation 1−ε⊥ ε|| > 0−ε⊥ ε|| > 0 and 1 − ε⊥ 2 is very neglecting the imaginary parts of ε⊥ and ε|| , can be excited. However, 1 − ε⊥ close to zero in this case. As such, the dispersion curves solved from Eq. (5.8) is in a transitional state from hyperbolas to an ellipse, corresponding to a transitional resonant mode from to elliptical surface phonon polaritons (ESPhPs). Figure 5.6b clearly shows this situation, which reveals that the dispersion curves at large |k y | values have changed from hyperbolas to be flattened and closed. Similar results have also been obtained by Liu and Xuan. In addition, similar to Fig. 5.5b, the bright

5 Near-Field Radiative Heat Transfer Between Anisotropic Materials

83

Fig. 5.7 Energy transmission coefficient between two bulk hBN slabs varying with wavevector components k x and k y at 1.0 × 1014 rad/s: a α = 0°, b α = 90°

color regions on the two sides of the origin shown in Fig. 5.6b manifest enhanced radiative heat transfer due to HPPs. Note that the two dashed lines in Fig. 5.6b represent the asymptotes of the hyperbolas and are drawn based on Eq. (5.7). Those asymptotes based on Eq. (5.9) are not shown as in this case, the resonant mode is in a transitional state. Shown in Fig. 5.6c is the energy transmission coefficient ξ distribution for α = 45°. Interestingly, the bright color region looks very different from that shown in Fig. 5.6b. In fact, the bright color region is in an elliptical shape, indicating that the excited surface phonon polaritons are ESPhPs, instead of HSPhPs in this case. Therefore, the enhanced radiative heat transfer is due to the combined effect of ESPhPs and HPPs. It should be pointed out the above contour plots are only for the energy transmission coefficient between two bulk hBN slabs. The results for the NFRHF between two hBN slabs of h = 50 nm are similar except that the HPP modes are discrete in this case due to wave interference effect in the slabs. More importantly, HSPhPs can be excited on both surfaces of the slab at large tilting angles, the coupling of which can strengthen the radiative heat transfer. As a consequence, the NFRHF between two hBN slabs of h = 50 nm can exceed that between two bulk hBN slabs as shown in Fig. 5.2a. The contours of the energy transmission coefficient ξ between two graphene-covered bulk hBN slabs in the k x -k y plane and at ω = 1.00 × 1014 rad/s are shown in Fig. 5.7a, b for α equal to 0° and 90°, respectively. It has been shown that SPPs can be excited at vacuum/graphene interface such that the NFRHF between two graphene sheets can be enhanced in a broadband. However, the enhancement is not as significant as shown in Fig. 5.3c, d because the thickness of graphene is too small. For the case of near-field radiative heat transfer between two graphene-covered bulk hBN slabs, SPPs can be excited at the graphene/hBN interface besides at the vacuum/graphene interface. Although ε⊥ and ε|| of hBN are positive at ω = 1.00 × 1014 rad/s, the real part of the dielectric function of graphene is negative at this frequency. Therefore, the dispersion of SPPs at the graphene/hBN interface can be satisfied, which in the

84

5 Near-Field Radiative Heat Transfer Between Anisotropic Materials

Fig. 5.8 Energy transmission coefficient between two bulk hBN slabs varying with wavevector components k x and k y at 1.47 × 1014 rad/s: a α = 0°, b α = 90°

case of α = 0° can be written as      2 2 2 ε|| k x2 + k 2y = εd2 ε⊥ ε|| − ε⊥ εd ε|| k02 εd ε⊥ − ε⊥

(5.10)

where εd represents the dielectric function of graphene as shown in Eq. (5.3). The dispersion relation shown in Eq. (5.10) has the property of rotational symmetry in the k x − k y plane, so does that of SPPs at the vacuum/graphene interface. The enhanced NFRHF shown in Fig. 5.7a, represented by the bright contours of the energy transmission coefficient, is due to the coupling of SPPs excited at the vacuum/graphene interface and at the graphene/hBN interface. Rotational symmetry of the SPP dispersions in the k x − k y plane can be clearly observed. In this case that α = 90°, though the dispersion of SPPs at the vacuum/graphene interface still retains the rotational symmetry in the k x − k y plane, the dispersion of SPPs at the graphene/hBN interface, the same as Eq. (5.7), represents an ellipse with its long axis along k y as ε⊥ = 7.0531 − 0.0036 j and ε|| = 3.6763 − 0.0047 j at ω = 1.00 × 1014 rad/s. It can be seen clearly in Fig. 5.7b that the inner ring of the bright contours of indeed exhibits an ellipse with its long axis along k y , which, due to coupling of SPPs, causes the outer ring to deviate from the circular shape. Change in the dispersion of SPPs at the graphene/hBN interface with the tilting angle α changes the NFRHF as shown in Fig. 5.3c, d, respectively. Similar mechanisms apply to the enhancement of the NFRHF with the graphene coatings at other frequencies shown in Fig. 5.3c, d except for the regions close to the hyperbolic bands, where SPPs may strongly interact with HPPs to induce the formation of a hybrid mode. For example, it can be seen from Fig. 5.3c, d and that the radiative heat flux between the graphene/hBN heterostructures at ω = 1.47 × 1014 rad/s is influenced significantly by the tilting angle of the optical axis of hBN. This frequency is at the edge of the hyperbolic band of type I so that SPPs may interact strongly with HPPs. To see this, we plotted in Fig. 5.8a, b the energy transmission coefficient distribution at ω = 1.47 × 1014 rad/s when the tilting angle α is 0° and 90°,

5 Near-Field Radiative Heat Transfer Between Anisotropic Materials

85

Fig. 5.9 Energy transmission coefficient between two bulk hBN slabs varying with wavevector components k x and k y at 1.58 × 1014 rad/s: a α = 0°, b α = 90°

respectively. At this frequency,ε⊥ = 7.6153−0.0084 j and ε|| = 8.2826−75.7521 j, both ε⊥ and ε|| are positive. It can be seen from Fig. 5.8a that for α = 0°, the bright contours of the energy transmission coefficient ξ are only concentrated around the origin of the k x -k y plane, which indicates that the NFRHF is very small in this case and is in accordance with the value shown in Fig. 5.3c. This small radiative flux is the result of mode repulsion between excited SPPs and HPPs supported by the structure. But when α = 90°, the radiative heat flux is significantly enhanced compared with the case of α = 0°, as shown in Fig. 5.3c. The corresponding contours of the energy transmission coefficient ξ are shown in Fig. 5.8b, which indicate a hybrid mode excitation that comes from the coupling of SPPSs and HPPs. In fact, HPPs are bulk modes whose impact is weak if the thickness of the hBN slab is small. We have checked that the distribution ξ is very similar to that shown in Fig. 5.7b if the thickness of the hBN slab is on the order of 10 nm. As the hBN slab is getting thicker and thicker, the pattern shrinks in the k x direction and gradually changes to the hybrid mode pattern as shown in Fig. 5.8b. The reason is that the HPPs become stronger and stronger as the slab thickness increases, and they eventually can couple with SPPs to form the hybrid mode. Note from Fig. 5.8b that the NFRHF due to HPPs is quite weak, which comes from the fact that the imaginary part of ε|| is very large in this case. The effect of the tilting angle α on the hybrid mode is further shown in Fig. 5.9a, b, where the energy transmission coefficient ξ between two graphene-covered bulk hBN slabs at ω = 1.58 × 1014 rad/s is plotted for α equal to 0° and 90°, respectively. This frequency is beyond but close to the hyperbolic band of type I, at which ε⊥ = 7.8360 − 0.0106 j and ε|| = 0.4324 − 0.0896 j. Note that ε|| is close to zero in this case. From Fig. 5.9a, strong mode repulsion is not seen for α = 0°, and the distribution pattern of ξ clearly indicates the characteristic of coupled SPPs in this case. When α = 90°, however, the distribution pattern of is tortured dramatically due to the interaction of SPPs and HPPs and the formation of a hybrid mode. Especially, the ellipse due to the excited SPPs at the graphene/hBN interface is shrunk significantly

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5 Near-Field Radiative Heat Transfer Between Anisotropic Materials

Fig. 5.10 Effect of different oriented optical axes of hBN on the NFRHF between a two bulk hBN slabs and b two graphene-covered bulk hBN slabs

in the k x direction and the hyperbolas corresponding to HPPs in the hyperbolic band are clearly seen at small |k y | values. Finally, the effect of different tilting angles α 1 and α 2 on the NFRHF is investigated. The results are shown in Fig. 5.10a, b for the near-field radiative heat transfer between two bulk hBN slabs and between two graphene-covered bulk hBN slabs, respectively. It can be seen in both cases maximum NFRHF is obtained when the values of α 1 and α 2 are equal. Furthermore, Fig. 5.10a shows that the largest NFRHF between two bulk hBN slabs is obtained when α 1 = α 2 = 0°, the NFRHF decreases with an increase in the tilting angle. In contrast, the NFRHF between two graphenecovered bulk hBN slabs is the largest when α 1 = α 2 = 90°, and it decreases with a decrease of the tilting angle. In other words, to enhance the NFRHF between two pure hBN slabs, the slabs should be arranged with in-plane isotropy of the surface. But for the graphene-covered hBN slabs, the slabs should be arranged with strong in-plane anisotropy of the surface.

References 1. Biehs SA, Tschikin M, Abdallah PB (2012) Hyperbolic metamaterials as an analog of a blackbody in the near field. Phys Rev Lett 109:104301 2. Biehs SA, Tschikin M, Messina R, Abdallah PB (2013) Super-Planckian near-field thermal emission with phonon-polaritonic hyperbolic metamaterials. Appl Phys Lett 102:131106

Chapter 6

Conclusions and Recommendations

Three research works have been explored based on anisotropic materials in this dissertation, including unidirectional transmission of light, broadband perfect absorption, and near-field radiative heat transfer. To calculate the radiative properties of multilayer anisotropic slabs and gratings, the 4 × 4 transfer matrix method and rigorous couple-wave analysis (RCWA) are further improved and developed, enabling one to stably and effectively obtain the electromagnetic wave propagating in slabs or gratings. The main research and conclusions are summarized below: (1) To realize the unidirectional transmission of light, three different structures have been proposed: the combination of two layers of anisotropic slabs, the combination of one anisotropic slab and one grating, and the combination of one anisotropic slab and one photonic crystal. Each of these three structures has its own advantages and disadvantages, and they can support unidirectional transmission of light at different bands. Though the combination of two layers of anisotropic slabs is the simplest, it realizes the unidirectional transmission of light at the hyperbolic bands of hBN, where the transmission contrast reaches as high as 1000. The second structure allows light to unidirectionally transmit at visible and infrared bands, but the transmission contrast is small between two direction. The combination of one anisotropic slab and one photonic crystal is relative complex but applicable for unidirectional transmission of light at visible bands. What is importance is that the transmissivity of opposite directions can be dynamically tuned via the orientation of optic axis of the anisotropic slab. (2) The influence of the orientation of optic axis on the directional and semispherical emissivity has been studied in this dissertation. By means of 4 × 4 transfer matrix method and rotational transform, the projection operation is circumvented and the calculation can be simplified as well. As the results shown, the orientation of optic axis has great impact on the emissivity of hBN at hyperbolic bands.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 X. Wu, Thermal Radiative Properties of Uniaxial Anisotropic Materials and Their Manipulations, Springer Theses, https://doi.org/10.1007/978-981-15-7823-6_6

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6 Conclusions and Recommendations

When optic axis deviates from z axis, the spectral direction emissivity of TM and TE wave periodically varies with the azimuth angle due to the polarization transformation. Furthermore, a structure consists of multi-layer slabs of hBN that can realize perfect absorption has been proposed, and when the structure consists of 91 layers, the absorptivity reaches up to 0.94 at 1367–1580 cm−1 bands. The physical mechanism is the impedance matching at each interface and the enhanced wave vector within each layer. Meanwhile, the number of layers and the thickness of each layer have also been explored to find out the influence on the absorption of the whole structure. Moreover, the perfect absorption at ultrabroadband has been realized in hyperbolic materials which is made of doped-silicon nanowires and TPX. When the incident angle is 60°, the absorptivity of TM wave can reach 0.99 in 2–50 μm, which can be explained by the similar theory of multi-layer flat structure of hBN. As is demonstrated here, the absorption characteristic can be regulated and controlled by manipulating the optic axis of anisotropic materials. (3) The optic axis, which has an influence on the near-field radiative heat transfer of hBN and graphene/hBN heterostructure, is thoroughly studied. As the numerical results shown, the near-field radiative heat transfer between two infinite hBN slabs originates from the excitation of HPPs and HSPhPs, which reaches the maximum when the tilted angle of optic axis is 0°, and increases as the tilted angle decreases. By comparison, the near-field radiative heat transfer between two infinite hBN slabs covered by graphene is mainly from the excitation of SPPs at the interfaces of air/graphene and graphene/hBN, which reaches its maximum when the tilted angle of optic axis is 90°, and decreases as the tilted angle decreases. Some fundamental work has been done based on theoretical calculation in this dissertation, which includes unidirectional transmission of light, broadband perfect absorption, and near-field radiative heat transfer. Though the results derived from theoretical calculation are encouraging, these designs, nevertheless, have been waiting for subtle experiment to demonstrate. Therefore, it is pointed out that some of the primary research work need to be further developed, which are listed as follows: Firstly, the size of unidirectional transmission devices is demanding in integrated light circuits. Though three structures that allow light unidirectionally transmit have been designed and theoretically proved, their sizes are still too large to meet the requirement. To realize unidirectional transmission of light at a smaller scale, one can use metasurface, in which the size is nanoscale and the energy loss can thus be reduced. In addition, the metasurface allows the linear polarized light to be able to convert to each other, and it has selective transmissivity to different linear polarized light. Therefore, two layers of metasurface can theoretically realize unidirectional transmission of light at small scale, although the fabrication of such structures are complex and challenging. Secondly, perfect absorption was realized in hBN and a hyperbolic material comprised of nanowire doped with silicon and TXP. It is rather difficult to fabricate an hBN slab with tilted optic axis due to the fact that hBN is a 2D material.

6 Conclusions and Recommendations

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With the development of micro-nano fabrication technology, it is foreseeable to prepare metamaterials with tilted optic axis, which would be an alternative material to demonstrate the perfect absorption theory mentioned in this dissertation. Thirdly, by comparing with uniaxial crystal hBN, biaxial crystal α-MoO3 has a wider hyperbolic region, and thus the near-field radiative heat transfer is relatively high. However, the near-field radiative heat transfer of this material has not yet been explored, waiting for further research. Lastly, the object of study in this dissertation is hBN, one of anisotropic materials, and thus the theories and calculating results might be extended to other anisotropic materials. According to effective dielectric theory, the permittivity tensor of some metamaterials can be expressed as those of optically uniaxial materials, so one could analogously study such materials. If appropriate metamaterials are chosen, the perfect absorption could be realized in visible and infrared bands, which is worthwhile studying.

Curriculum Vitae

Xiaohu Wu Xiaohu Wu was born in Jingzhou, Hubei province, China in 1992. He received his doctor’s degree from the Peking University in 2019 with a best thesis award. His research is concentrated on the thermal properties of anisotropic materials and nanophotonics. During his Ph.D., he coauthored over 20 journal publications in Optica, International Journal of Heat and Mass Transfer, Journal of Quantitative Spectroscopy and Radiative Transfer, and other peer-reviewed journals. He has given three conference presentations. He is a reviewer for Journal of Quantitative Spectroscopy and Radiative Transfer, Journal of Heat Transfer, and Materials Research Express, and so on.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 X. Wu, Thermal Radiative Properties of Uniaxial Anisotropic Materials and Their Manipulations, Springer Theses, https://doi.org/10.1007/978-981-15-7823-6

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