Преобразование полей точечных излучателей и плоских волн при помощи электромагнитных метаповерхностей тема диссертации и автореферата по ВАК РФ 00.00.00, доктор наук Глыбовский Станислав Борисович

• Глыбовский Станислав Борисович
• доктор наук
• 2023

Оглавление диссертации доктор наук Глыбовский Станислав Борисович

ВВЕДЕНИЕ

1. Используемые физические модели и предложенная классификация метаповерхностей по электромагнитным свойствам мета-атомов и по функциональному назначению

2. Преобразование полей плоских волн при дифракции на метаповерхностях

2.1. Преобразование поляризации при дифракции плоских волн на метаповерхностях с само-дополняющими элементами

2.2. Преобразование амплитуды и фазы прошедшей плоской волны с низким коэффициентом отражения в одномодовом режиме дифракции на метаповерхностях из сдвоенных разомкнутых кольцевых резонаторов

2.3. Метаповерхность для полного поглощения мощности при двух углах падения падающей плоской волны

3. Преобразование формы волнового фронта при дифракции плоских волн на неоднородных метаповерхностях

3.1. Пространственное разделение волновых пучков при дифракции плоской волны с круговой поляризацией на неоднородной метаповерхности из само-дополняющих элементов

3.2. Динамическое преобразование пространственного распределения поля при отражении от неоднородной отражающей поверхности, управляемой инфракрасным светом

4. Исследование и применение резонаторов на основе метаповерхностей

4.1. Возбуждение собственных мод плоских и объемных резонаторов с границами в виде метаповерхностей

4.2. Обзор возможностей применения резонаторов на основе метаповерхностей из параллельных проводов для преобразования радиочастотного магнитного поля в системах МРТ

4.3. Применение метаповерхности с высоким импедансом в качестве экрана радиочастотной катушки магнитно-резонансного томографа

4.4. Применение резонаторов на основе метаповерхностей для развязки излучателей магнитно-резонансного томографа

5. Исследование и применение волноведущих структур на основе метаповерхностей

5.1. Исследование возбуждения поверхностных волн на анизотропных метаповерхностях

5.2. Применение замедляющих структур на основе метаповерхностей для контроля пространственного распределения радиочастотного поля магнитно-резонансного томографа

ЗАКЛЮЧЕНИЕ

СПИСОК ЛИТЕРАТУРЫ

99

INTRODUCTION

1. The employed physical models and the proposed classification of metasurfaces according to the electromagnetic properties of meta-atoms and functionality

2. Transformation of plane-wave fields in the diffraction on metasurfaces

2.1. Polarization transformation of plane waves in the diffraction on metasurfaces with self-complementary meta-atoms

2.2. Amplitude and phase transformation of a transmitted plane wave with low reflection in the single-mode diffraction regime with metasurfaces of paired split-ring resonators

2.3. Metasurface for complete power absorption at two angles of plane-wave incidence

3. Wave-front shape transformation in the diffraction of plane waves on inhomogeneous metasurfaces

3.1. Splitting wave beams in the diffraction of a circularly polarized plane wave on an inhomogeneous metasurface with self-complementary meta-atoms

3.2. Dynamic transformation of the reflected field distribution by means of an inhomogeneous reflective surface controlled by infrared light

4. Investigation and application of resonators based on metasurfaces

4.1. Excitation of volumetric and planar metasurface-based resonators

4.2. Possibilities of using metasurface-based resonators of parallel wires for transforming RF magnetic fields in MRI systems

4.3. A high impedance metasurface as an artificial shield of a radio-frequency coil for magnetic resonance imaging

4.4. Decoupling of radiofrequency radiators for magnetic resonance imaging by means of metasurface-based resonators

5. Investigation and application of waveguides based on metasurfaces

5.1. Study of surface-wave excitation on anisotropic metasurfaces

5.2. Application of slow-wave structures based on metasurfaces to control the RF-field spatial distribution for magnetic resonance imaging

CONCLUSION

Приложение А. Список публикаций по теме диссертации

Приложение Б. Список используемых сокращений

Приложение В. Тексты основных публикаций по теме диссертации

Введение диссертации (часть автореферата) на тему «Преобразование полей точечных излучателей и плоских волн при помощи электромагнитных метаповерхностей»

ВВЕДЕНИЕ

Актуальность темы. Развитие современных радиочастотных систем передачи данных, позиционирования и медицинской диагностики ежегодно ставит множество новых задач, как в области электроники, так и в области прикладной электродинамики. Зачастую требуется повышение компактности применяемых в данных системах сложных многоэлементных излучателей при улучшении их полевых и радиотехнических характеристик, особенно при расположении в пределах ограниченного пространства корпуса прибора, а также - вблизи тела человека. Однако решение подобных задач с использованием излучателей стандартных типов чаще всего затруднено ввиду необходимости одновременного обеспечения заданного пространственного распределения поля как в зоне излучения (синтез требуемой диаграммы направленности), так и - в непосредственной близости от излучателя.

Устойчивым направлением развития методов синтеза пространственного распределения полей является применение решеток из пассивных элементов, расположенных в поле излучателя (в общем случае, на произвольном расстоянии от него) с некоторым субволновым пространственным шагом (периодом). Первые применения периодических структур для контроля пространственного распределения поля относятся к антенным рефлекторам из нерезонансных тонкопроволочных сеток [1-5], обеспечивающих близкий к единичному коэффициент отражения при малом весе и низкой ветровой нагрузке. Дальнейшее развитие получили резонансные субволновые решетки (частотно-селективные поверхности), способные обеспечивать различные отражающие свойства в зависимости от частоты [6]. Наконец, для управления формой пространственного распределения поля излучателя в режиме отражения или пропускания были предложены располагаемые в дальней зоне излучателя отражательные [7] и проходные [8] антенные решетки. Указанные структуры являются электрически тонкими аналогами фокусирующих зеркал и линз соответственно.

Вышеупомянутые периодические структуры объединяет режим дифракции на них поля излучателя, определяемый периодом, не превышающим половину длины волны. Также для их применения характерна медленная пространственная зависимость вдоль структуры как параметров элементов, так и самого поля излучателя. В указанном приближении можно описать локальное взаимодействие структуры с полем излучателя рассеянием на соответствующей бесконечной двумерно-периодической структуре из одинаковых (таких же, как в рассматриваемой точке) элементов. При этом, для

расчета локальных характеристик отражения и пропускания ввиду малости периода структуры достаточно ограничиваться основным дифракционным порядком. Связано это с тем, что в пределах одномодового режима дифракции высшие дифракционные порядки представляют собой нераспространяющиеся моды Флоке и не проявляются на расстояниях более одного периода от структуры [9]. Иными словами, не происходит возбуждения распространяющихся высших дифракционных порядков, присущих, к примеру, дифракционным решеткам. Благодаря данному свойству реальной дискретной структуре в каждом случае можно сопоставить эквивалентную сплошную поверхность с пассивно наведенными усредненными в пределах элементарных ячеек поверхностными электрическими и магнитными токами. Дифракция поля излучателя на такой сплошной поверхности будет неотличима на расстоянии более одного периода от дифракции на структуре из дискретных элементов, однако, при этом, расчет поля в математическом и вычислительном планах существенно упрощается. В результате можно говорить об «искусственной границе раздела» между различными областями пространства, граничные условия в каждой точке которой могут существенным образом отличаться от классических граничных условий на границе раздела двух естественных материальных сред, и могут быть заданы инженером исходя из желаемого распределения поля дифракции при фиксированном излучателе. Таким образом, в рассматриваемом подходе вместо синтеза распределения тока самого излучателя производится синтез макроскопических параметров окружающих его искусственных границ за счет подбора микроструктуры составляющих их дискретных элементов. Последние же, в свою очередь, обычно состоят из обычных металлов и диэлектриков. Переход от микроскопических свойств элементов к макроскопическому описанию граничного условия производится методами усреднения полей [1,9]. Усредненные граничные условия в их современной форме связывают усредненные по площади элементарной ячейки касательные составляющие электрического и магнитного полей (без учета вклада высших мод Флоке) с усредненными электрическим и магнитным поверхностными токами, наведенными излучателем в каждой точке эквивалентной сплошной поверхности [9].

По мере повышения сложности рассматриваемых элементов и решения различных задач дифракции на соответствующих субволновых решетках методами усредненных граничных условий и эквивалентных электрических цепей, к началу 2010-х годов в литературе устоялся термин «метаповерхности», объединяющий сетчатые экраны и поляризаторы,

частотно-селективные структуры и искусственные поглотители, экраны с высоким поверхностным импедансом, а также - отражательные и проходные решетки. Под метаповерхностями обычно понимаются двумерные структуры из периодически расположенных элементов (называемых мета-атомами) с субволновым периодом, взаимодействующие с полем заданного источника как сплошные границы и специально оптимизированные для обеспечения заданных характеристик поля дифракции. Часто в литературе метаповерхности называют двумерными аналогами метаматериалов. В отличие от классических частотно-селективных поверхностей и антенных решеток период метаповерхностей, в общем случае, существенно меньше половины длины волны, что позволяет считать их эффективно сплошными поверхностями даже по отношению к близкорасположенным точечным излучателям с широким пространственным спектром. Метаповерхности для различных назначений (частотная и пространственная селекция волн, поглощение мощности электромагнитной волны, преобразование поляризации и формы волнового фронта и др.) широко исследуются в литературе последних двух десятилетий.

Несмотря на неугасающий интерес к исследованию метаповерхностей, который подтверждается растущим ежегодным числом журнальных публикаций по теме (обзору существующих принципов построения и применений метаповерхностей посвящены, например, обзорные статьи [1012] и монографии [13-14]), многие уникальные и полезные функции метаповерхностей являются по сей день малоизученными, несистематизированными или вовсе могут оставаться неизвестными. При этом актуальными следует считать такие задачи, как: теоретическое и экспериментальное исследование реализуемых в радиодиапазоне усредненных граничных условий в зависимости от симметрии и состава мета-атомов, развитие методов синтеза метаповерхностей для достижения заданного распределения поля дифракции, анализ возбуждения метаповерхностей плоскими волновыми фронтами и полями точечных излучателей, расположенных на электрически малом расстоянии.

Успешное решение вышеуказанных задач применительно к радиочастотному диапазону обеспечит мощный инструмент для создания современных антенных систем с одновременным контролем ближнего поля и поля излучения, что, безусловно, приведет к новому уровню развития технологий связи, позиционирования и медицинской диагностики.

Целью диссертационной работы являются исследование и систематизация физических свойств метаповерхностей, а также -возможностей преобразования с их помощью характеристик волновых полей в радиочастотном диапазоне для создания задела для развития новых поколений систем связи, позиционирования и медицинской диагностики.

Для достижения данной цели в рамках диссертации были поставлены и решены следующие задачи:

1. Систематизировать возможности использования метаповерхностей для преобразования полей заданных излучателей и падающих плоских волн в радиочастотном диапазоне в зависимости от электромагнитного отклика (поляризации) отдельных элементов (мета-атомов) и симметрии их микроструктуры;

2. Развить подходы к достижению частотно-независимых фазовых и поляризационных характеристик плоской волны при ее отражении от метаповерхности и прохождении сквозь метаповерхность;

3. Развить систематические подходы к построению метаповерхностей, выполняющих функцию поглотителя по отношению к плоским волнам, падающим под различными углами;

4. Изучить эффекты возбуждения бесконечных метаповерхностей точечными излучателями, а также - возбуждения плоских и объемных резонаторов, стенки которых образованы конечными фрагментами метаповерхностей;

5. Развить подходы к применению конечных фрагментов метаповерхностей в радиочастотных системах магнитно-резонансных томографов для обеспечения требуемого преобразования пространственного распределения поля излучателей, а также - для электромагнитной развязки излучателей магнитно-резонансного томографа.

Научная новизна диссертационной работы заключается в следующем.

Во-первых, была расширена область применения метода усреднения полей для анализа метаповерхностей с новыми физическими свойствами. Так методы усредненных граничных условий и эквивалентных электрических цепей были впервые применены к анализу частотно-независимых свойств полей дифракции и собственных волн метаповерхностей с самодополняющей структурой мета-атомов, к анализу возбуждения резонаторов Фабри-Перо и цилиндрических объемных резонаторов со стенками в виде метаповерхностей, а также - к анализу и синтезу угловых свойств

поглощающих метаповерхностей. В результате работы удалось теоретически предсказать и экспериментально продемонстрировать ряд новых волновых эффектов, наблюдающихся при возбуждении метаповерхностей.

Во-вторых, были предложены новые применения метаповерхностей в радиочастотных системах магнитно-резонансных томографов, включая:

(а) улучшение однородности пространственного распределения радиочастотного магнитного поля томографа при помощи метаповерхности;

(б) искусственные экраны радиочастотных излучателей магнитно-резонансных томографов в виде поверхностей с высоким импедансом для улучшения соотношения сигнал-шум;

(в) электромагнитная развязка элементов многоэлементных излучателей томографа в рабочих диапазонах частот.

Практическая значимость работы определяется возможностью непосредственного применения разработанных методов для расчета и изготовления таких устройств на основе метаповерхностей, как экраны и резонаторы антенных систем СВЧ диапазона, пассивные устройства развязки элементов антенных решеток, антенные элементы радиочастотных катушек магнитно-резонансных томографов с уровнем постоянного поля магнита от 1.5 до 7 Тл (рабочие частоты от 64 до 300 МГц), а также - квазиоптические пассивные устройства (поляризаторы, частотные фильтры, фокусирующие и отклоняющие делители волновых пучков) субмиллиметрового диапазона. Отдельные метаповерхности и резонаторы на их основе были запатентованы и непосредственно используются для задач доклинических и медицинских исследований средствами магнитно-резонансной томографии в таких центрах, как ФГБУ «НМИЦ им. В. А. Алмазова» Минздрава России (Санкт-Петербург), NeuroSpin (Париж, Франция) и The Center for Magnetic Resonance in Biology and Medicine (Марсель, Франция).

Результаты работы в части методов построения управляемых отражающих поверхностей и методов расчета цилиндрических резонаторов с границами в виде метаповерхностей были использованы при выполнении научно-исследовательских работ, проводимых на Физическом факультете Университета ИТМО совместно с такими компаниями как ООО «Техкомпания Хуавэй» и ООО «Топкон Позишионинг Системс» в 2019-2022 гг., где автор выступал в качестве научного руководителя либо ответственного исполнителя проектов.

Методология и методы исследования. Для решения поставленных задач в работе использовались следующие методы:

Теоретические методы. Для решения задач дифракции плоских волн на бесконечных метаповерхностях из идентичных мета-атомов используется метод усредненных граничных условий, в рамках которого метаповерхность описывается плоским однородным листом с электрическим и (в отдельных случаях) магнитным поверхностными токами, создающими скачки усредненных тангенциальных составляющих магнитного и электрического поля соответственно. Далее, усредненные граничные условия используются при постановки граничных задач, решение которых ведет к получению комплексных коэффициентов отражения и пропускания. С целью определения макроскопических параметров усредненных граничных условий (к примеру, электрического импеданса решетки и ее магнитного адмиттанса - в случае магнитоэлектрической метаповерхности) используется метод эквивалентных цепей, позволяющий предсказать отклик структуры из мета-атомов рассматриваемой геометрической структуры в полосе частот. При этом номиналы элементов эквивалентных цепей, описывающих метаповерхности, определяются как аналитически (за счет решения вспомогательных квазистатических задач), так и численно (путем сопоставления с численно рассчитанными данными). При рассмотрении пространственно-неоднородных метаповерхностей используется подход локальных коэффициентов отражения и пропускания. Формы профилей амплитуды и фазы данных коэффициентов в плоскости метаповерхности, требуемые для обеспечения желаемой формы пространственного распределения поля в режиме отражения либо пропускания, рассчитывается отдельно методами геометрической оптики. При расчете дифракции полей точечных излучателей, расположенных вблизи метаповерхностей, применяется теория мнимых изображений (Exact Image Theory).

Численные методы. Для верификации полученных аналитически решений задач дифракции, для оптимизации микроструктуры мета-атомов, а также для непосредственного расчета распределения поля заданного излучателя в присутствии фрагмента метаповерхности конечных размеров, используются численные методы электродинамики (метод конечных элементов - FEM, метод конечных интегралов - FIT, а также - метод интегральных уравнений в тонкопроволочном приближении - IE), реализованные в коммерческих программных пакетах численного моделирования (CST Microwave Studio и Ansys HFSS). Данные пакеты также используются для анализа распределений поля и расчета радиотехнических

характеристик излучателей в присутствии метаповерхностей, а также - в присутствии детальных воксельных моделей тела человека (для задач магнитно-резонансной томографии).

Экспериментальные методы. Для экспериментального изучения физических свойств рассматриваемых метаповерхностей в работе развиты методы измерения пространственных распределений компонент электрического и магнитного полей. Для этого использован комплект оборудования лаборатории «Прикладная радиофизика» Университета ИТМО (г. Санкт-Петербург), состоящий из экранированной безэховой камеры, векторных анализаторов цепей с наборами калибровочных мер, сканера ближнего поля, поворотного стола, измерительных антенн, зондов ближнего поля, а также - измерительных волноводных секций. Измерения в дальней зоне проводились в пределах диапазона от 1 до 10 ГГц. Измерения поля в ближней зоне (в том числе - вблизи отдельных мета-атомов), а также -измерения коэффициентов матрицы рассеяния излучателей проводятся в диапазоне от 64 МГц до 10 ГГц. Для устранения систематических ошибок применяются специализированные калибровочные меры и известные методы калибровки векторных анализаторов цепей, а также - метод оконных функций во временной области. Дополнительно применяется метод пересчета поля из ближней зоны в зону излучения по дискретным измеренным данным и другие общепринятые экспериментальные методы СВЧ. Измерения характеристик метаповерхностей в субмиллиметровом диапазоне (от 100 до 400 ГГц) организованы на базе Центра коллективного пользования научным оборудованием «Высокие технологии и аналитика наносистем» (ЦКП ВТАН) НГУ (г. Новосибирск). Для измерения амплитудных и фазовых спектров отражения и пропускания метаповерхностей применяются квазиоптические интерферометры с различными оптическими схемами с источником в виде лампы обратной волны (спектроскопия в частотной области). Для изучения метаповерхностей при измерениях внутри действующих магнитно-резонансных томографов использованы методы исследования распределения радиочастотного магнитного поля путем обработки получаемых томографом магнитно-резонансных изображений фантомов - поглощающих объектов, имитирующих электрические свойства тканей тела человека. Наконец, эксперименты по изучению метаповерхностей в задачах МРТ, выполняемые в присутствии здоровых волонтеров, проводятся на основании разрешений местных этических комитетов с письменного согласия волонтеров при предварительном изучении условий радиочастотной безопасности (методами численного моделирования).

Научные положения, выносимые на защиту:

1. Анизотропные метаповерхности из само-дополняющих элементов имеют частотно-независимые коэффициенты отражения и пропускания плоских волн с круговой поляризацией в одномодовом режиме дифракции, а поддерживаемые ими поверхностные волны различной поляризации имеют в пределах того же диапазона одинаковые фазовые скорости;

2. Метаповерхности с элементами в форме парных разомкнутых кольцевых резонаторов со взаимным пространственным перекрытием обладают незначительным (менее 16% по мощности) уровнем коэффициента отражения в полосе частот одномодового режима дифракции, обеспечивая на рабочей частоте произвольную фазу коэффициента пропускания в пределах от 0 до 360 градусов;

3. Метод усредненных граничных условий позволяет получить аналитическое решение в замкнутой форме задачи дифракции поля горизонтального электрического диполя на резонаторе Фабри-Перо с зеркалами в форме тонкопроволочных сеток из параллельных проводов;

4. Метаповерхность с высоким поверхностным импедансом, образуя цилиндрический экран вокруг радиочастотной катушки магнитно-резонансного томографа, позволяет повысить соотношение сигнал-шум на величину до 30%;

5. Эффект возбуждения высших типов собственных колебаний фрагмента метаповерхности, расположенного вблизи излучателей магнитно-резонансного томографа, позволяет развязать их при снижении коэффициента вариации пространственного распределения поля внутри поглощающего объекта, имитирующего тело человека, в 1.9 раза по сравнению с известными методами;

6. Возбуждение замедленной волны, распространяющейся вдоль фрагмента метаповерхности, расположенной вблизи тела человека в магнитно-резонансном томографе, позволяет до двух раз снизить коэффициент вариации пространственного распределения радиочастотного поля внутри тела человека.

Степень достоверности и апробация результатов работы.

Достоверность полученных в работе результатов подтверждается следующим: (а) путем соответствующих предельных переходов подтверждается совпадение полученных аналитических выражений и результатов численных расчетов с известными в литературе результатами в более простых, частных случаях; (б) достигнуто точное совпадение результатов аналитических и численных расчетов между собой, а также - подтверждение численных результатов в экспериментах; (в) используются многократно апробированные методы теоретического и экспериментального исследования; (г) полученные результаты не противоречат известным в литературе результатам; (д) предложенные методы и результаты после их публикации в литературе были воспроизведены и развиты другими научными группами, что подтверждается соответствующими цитированиями работ.

Апробация результатов работы.

Результаты работы были представлены лично автором в виде устных и приглашенных докладов на следующих международных конференциях: RADIO, Маврикий, 2015; Days on Diffraction, Санкт-Петербург, 2015; Days on Diffraction, Санкт-Петербург, 2016; PIERS, Санкт-Петербург, 2017; METANANO, Владивосток, 2017; EuCAP, Париж, 2017; Metamaterials, Марсель, 2017; METANANO, Сочи, 2018; Metamaterials, Эспоо, 2018; EuCAP, Краков, 2019; ICEAA, Гранада, 2019; Metamaterials, Рим, 2019; METANANO, Санкт-Петербург, 2019; Metamaterials, ONLINE, 2020; METANANO, ONLINE, 2020; METANANO, ONLINE, 2021; Metamaterials, Сиена, 2022.

Результаты работ были представлены автором в виде докладов на научных семинарах в следующих организациях:

• Нижегородский Государственный Технический Университет им. Р.Е. Алексеева, г. Нижний Новгород;

• Московский физико-технический институт, г. Москва;

• Институт общей физики им. А.М. Прохорова РАН, г. Москва;

• Акционерное общество «Научно-производственное предприятие «Радиосвязь», г. Красноярск;

• Гомельский государственный университет имени Франциска Скорины, Гомель, Белоруссия;

• Институт Френеля, Марсель, Франция;

• Университет Аалто, Эспоо, Финляндия.

Личный вклад автора. Представленная работа содержит результаты теоретических и экспериментальных исследований, выполненных лично автором, либо при его непосредственном участии. Личный вклад включает постановку научных задач, проведение теоретических и экспериментальных исследований, планирование, организацию и координацию работы научного коллектива, анализ, обработку и обобщение полученных результатов, а также - подготовку и публикацию научных статей и патентов, и представление результатов на научных конференциях и семинарах. Личный вклад автора уточняется в Приложении А «Список публикаций по теме диссертации».

Автором была сформирована научная группа в Университете ИТМО, занимающаяся проблемами прикладной электродинамики и теории антенн. За время работы группы по направлениям исследований, представленным в данной диссертационной работе, под руководством автора были защищены: одна диссертация на соискание степени кандидата физико-математических наук по специальности 01.04.03 (радиофизика) и две диссертации на соискание степени кандидата технических наук по специальности 05.12.07 (антенны, СВЧ-устройства и их технологии), а также - 10 магистерских диссертаций. Отдельные результаты, представленные в настоящей работе, были получены в рамках следующих научно-исследовательских проектов, в которых автор выступал в качестве руководителя: РНФ 19-79-10260, грант Президента Российской Федерации МК-3620.2019.8, а также в качестве ответственного исполнителя: РНФ 15-19-20054, РНФ 18-19-00482, РНФ 21-7930038, ФЦП RFMEFI58717X0041.

Публикации. В период с 2014 по 2022 год результаты исследований, выполненных в рамках представленной работы, опубликованы в 31 статье (в 30 оригинальных работах и 1 обзоре) в журналах, индексируемых Scopus и Web of Science (27 статей в журналах Q1, 4 статьи в журналах Q2).

1. Используемые физические модели и предложенная классификация метаповерхностей по электромагнитным свойствам мета-атомов и по функциональному назначению

Все пассивные, взаимные и пространственно-однородные метаповерхности могут быть классифицированы в зависимости от поляризационного отклика их элементов на внешнее поле (а именно - от типа наводимых полем дипольных моментов мета-атомов при их расположении в составе бесконечной периодической структуры). Данная классификация, представленная в настоящем разделе, определяет макроскопические свойства метаповерхностей, которые, будучи составленными из дискретных мета-атомов, ведут себя при возбуждении падающими плоскими волнами как сплошные листы с наведенными усредненными поверхностными токами. Так, при малом по сравнению с длиной волны периоде метаповерхности реализуется одномодовый режим дифракции, при котором одна падающая плоская волна порождает лишь одну отраженную и лишь одну прошедшую плоские волны. При этом моды Флоке высших порядков, будучи нераспространяющимися волнами, вносят вклад только в распределение поля на расстоянии менее одного периода от плоскости метаповерхности (образуют ближнее поле периодической структуры), которое зачастую не представляет интереса. В одномодовом приближении решение задачи дифракции предполагает нахождение усредненных по площади элементарной ячейки поверхностных плотностей наведенных электрического и магнитного токов. Наличие токов данных двух типов, а также - наличие или отсутствие связи электрического тока с магнитным - при их одновременном возбуждении полем падающей волны, определяют основные классы метаповерхностей.

Рассмотрим задачу о возбуждении произвольной однородной метаповерхности плоскими волнами, падающими со стороны каждого из окружающих однородных полупространств (сред) с волновыми числами к12 и волновыми сопротивлениями (рисунок 1а). В одномодовом режиме дифракции, рассеянные плоские волны могут быть найдены с использованием усредненных граничных условий, описанных ниже, а также, в отсутствии анизотропии - с использованием эквивалентной электрической цепи, включенной между двумя полубесконечными линиями передачи (рисунок 1б). Эквивалентная цепь описывает метаповерхность как четырехполюсник с заданными частотно-зависимыми Z-, Y-, Б- или Т-матрицами. Как известно, на одной частоте произвольный пассивный

Список литературы диссертационного исследования доктор наук Глыбовский Станислав Борисович, 2023 год

СПИСОК ЛИТЕРАТУРЫ

1. Конторович М.И. Об экранирующем действии замкнутых сеток //ЖТФ. 1939. Т.9. Вып. 24. С. 2195-2210.

2. MacFarlane G. G. Surface impedance of an infinite parallel-wire grid at oblique angles of incidence //Journal of the Institution of Electrical Engineers-Part IIIA: Radiolocation. - 1946. - Т. 93. - №. 10. - С. 1523-1527.

3. Wait J. R. Reflection at arbitrary incidence from a parallel wire grid //Applied Scientific Research, Section B. - 1955. - Т. 4. - №. 1. - С. 393-400.

4. Kontorovich M. I. et al. Reflection factor of a plane electromagnetic wave reflecting from a plane wire grid //Radio Eng. Electron Phys. - 1962. - Т. 7. -С. 222-231.

5. Lindell I. V., Akimov V. P., Alanen E. Image theory for dipole excitation of fields above and below a wire grid with square cells //IEEE transactions on electromagnetic compatibility. - 1986. - Т. 28. - №. 2. - С. 107-110.

6. Munk B. A. Frequency selective surfaces: theory and design. - John Wiley & Sons, 2005.

7. Nayeri P., Yang F., Elsherbeni A. Z. Reflectarray antennas: theory, designs, and applications. - 2018.

8. Abdelrahman A. H. et al. Analysis and design of transmitarray antennas. -San Francisco, CA, USA : Morgan & Claypool, 2017. - С. 1288-1300.

9. Tretyakov S. Analytical modeling in applied electromagnetics. - Artech House, 2003.

10.Holloway C. L. et al. An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials //IEEE antennas and propagation magazine. - 2012. - Т. 54. - №. 2. - С. 10-35.

11.Yu N., Capasso F. Flat optics with designer metasurfaces //Nature materials.

- 2014. - Т. 13. - №. 2. - С. 139-150.

12.Glybovski S. B. et al. Metasurfaces: From microwaves to visible //Physics reports. - 2016. - Т. 634. - С. 1-72.

13.Simovski C., Tretyakov S. An introduction to metamaterials and nanophotonics. - Cambridge University Press, 2020.

14.Caloz C., Achouri K. Electromagnetic metasurfaces: Theory and applications.

- John Wiley & Sons, 2021.

INTRODUCTION

Relevance of the topic. The development of modern radiofrequency (RF) systems for data transmission, positioning and medical diagnostics continuously sets many new problems, both in the field of electronics and applied electromagnetics. Often, it is required to improve the compactness of complex multielement radiators used in these systems while improving their field and technical performance, especially when located within the limited space of the device case, and near the human body. However, the solution to such problems using standard types of radiators is most often difficult due to the need to simultaneously provide a given spatial field distribution, both in the radiation zone (synthesis of the required radiation pattern) and in the proximity of each radiator.

A clear trend in the development of synthesis methods for the spatial field distributions is the use of arrays of passive elements (scatterers) placed in the radiator field (in the general case, at an arbitrary distance from it) with a certain subwavelength spatial step (period). The first applications of periodic structures to control spatial field distributions was related to antenna reflectors made of non-resonant thin-wire meshes [1-5], which provide a reflection coefficient close to unity having low weight and low wind loads. Resonant subwavelength grids (frequency-selective surfaces) capable of providing different reflective properties depending on the frequency were further developed [6]. Finally, to control the wave-front shape of the radiator in the reflection or transmission regimes, reflect-[7] and transmit-arrays [8] located in the far-field region of the radiator were proposed. These structures are electrically thin analogues of focusing reflectors and lenses, respectively.

All above-mentioned periodic structures provide a common regime of the radiator field diffraction, determined by a period not exceeding half the wavelength. Also, their application is characterized by a slow spatial dependence along the structure of both the parameters of the elements and the radiator's field. In this approximation, it is possible to describe the local interaction of the structure with the radiator field by scattering on the corresponding infinite two-dimensionally periodic structure of identical elements (same as at the considered point). In this case, to calculate the local characteristics of reflection and transmission, due to the small period, it is sufficient to consider only the main diffraction order. This is because within the single-mode diffraction regime, higher diffraction orders are non-propagating Floquet modes having negligible contributions at distances of more than one period from the structure [9]. In other words, there is no excitation of the propagating higher diffraction orders inherent

in, for example, diffraction gratings. Due to this property, in each case it is possible to put in correspondence to a real discrete structure an equivalent continuous surface with passively induced surface electric and magnetic currents averaged within the unit cells. The diffraction of the radiator field on such a continuous surface will be indistinguishable at more than one period from the diffraction on the initial discrete structure. However, the field calculation will be greatly simplified in mathematical and computational terms. As a result, one can speak of an "artificial interface" between different regions of space, the boundary conditions at each point of which can differ significantly from the classical boundary conditions at the interface between two uniform materials. Moreover, the boundary conditions can be set by an engineer based on the desired distribution of the diffraction field for the given source. Thus, in the approach under consideration, instead of synthesizing the current distribution of the radiator itself, the macroscopic parameters of the artificial boundaries surrounding it are synthesized by selecting the microstructure of their constituent discrete elements. The latter, in turn, usually consist of ordinary metals and dielectrics. The transition from microscopic properties of elements to a macroscopic description of the boundary condition is carried out by methods of field averaging [1,9]. The averaged boundary conditions in their modern form relate the tangential components of the electric and magnetic fields averaged over the unit cell area (without considering the contribution of higher-order Floquet modes) to the averaged electric and magnetic surface currents induced by the radiator at each point of the equivalent continuous surface [9].

As the complexity of the elements under consideration increases and various diffraction problems for the corresponding subwavelength gratings are solved by the methods of averaged boundary conditions and equivalent electrical circuits, by the beginning of the 2010s, the term "metasurfaces" was established in the literature. This term combined wire-mesh screens and polarizers, frequency-selective structures and artificial absorbers, screens with high surface impedance, as well as reflect- and transmit-arrays. Metasurfaces are usually understood as two-dimensional structures of periodically arranged elements (called meta-atoms) with a subwavelength period, interacting with the field of a given source as uniform boundaries and specially optimized to provide the given diffraction field. Metasurfaces are often referred to in the literature as two-dimensional analogues of metamaterials. In contrast to classical frequency-selective surfaces and antenna arrays, the period of metasurfaces is, in general, much less than half the wavelength, which makes it possible to consider them effectively continuous surfaces even with respect to closely spaced point sources with a wide spatial

spectrum. Metasurfaces for various purposes (frequency and spatial selectivity of waves, absorption of electromagnetic power, polarization, and wavefront shape transformation, etc.) have been widely studied in the literature of the last two decades.

Despite the high interest in the study of metasurfaces, which is confirmed by the growing annual number of journal publications on the topic (for example, review articles [10-12] and monographs [13-14] are devoted to a review of the existing principles for constructing and applying metasurfaces), many their unique and useful functions have been still barely studied, unsystematized, or may even remain unknown. At the same time, the following tasks should be considered relevant for investigation: properties of averaged boundary conditions implemented in the RF depending on the symmetry and composition of metaatoms, development of methods for synthesizing metasurfaces to achieve a given distribution of the diffraction field, excitation of metasurfaces by plane wave fronts and point-source fields located at an electrically short distance.

The successful solution of the above problems in relation to the RF range will provide a powerful tool for creating modern antenna systems with simultaneous control of the near field and radiation field, which will certainly lead to a new level of development in communication, positioning, and medical diagnostics technologies.

The goal of the dissertation work is to study and systematize the physical properties of metasurfaces, as well as the possibilities of transforming the characteristics of wave fields in the radio frequency range using metasurfaces, to create a background for the development of new communication, positioning, and medical diagnostics systems.

To achieve this goal, the following tasks were set and solved within the framework of the dissertation:

1. To systematize the possibilities of using metasurfaces in transforming the fields of given sources and incident plane waves in the RF range, depending on the electromagnetic response (polarization) of individual elements (meta-atoms) and the symmetry of their microstructure.

2. To develop approaches to achieving frequency-independent phase and polarization characteristics of a plane wave when it is reflected from a metasurface or passes through a metasurface.

3. To develop approaches to the systematic construction of metasurfaces that act as absorbers with respect to plane waves incident at different angles.

4. To study the excitation of infinite metasurfaces by point sources, as well as the excitation of flat and volumetric resonators, the walls of which are formed by finite fragments of metasurfaces.

5. To develop approaches to the use of finite fragments of metasurfaces in RF systems of magnetic resonance scanners to ensure the required spatial distribution transformation of the source field, as well as for electromagnetic decoupling of radiators in a magnetic resonance scanner.

The scientific novelty of the dissertation work is as follows.

First, the application of the field averaging method was expanded to the analysis of metasurfaces with new physical properties. Thus, the methods of averaged boundary conditions and equivalent circuits were first applied to the analysis of: (a) the frequency-independent properties of diffraction fields and guided waves of metasurfaces with self-complementary meta-atoms; (b) the excitation of Fabry-Perot resonators and cylindrical cavity resonators with metasurface walls; (c) the angular properties of absorbing metasurfaces. As a result, it was possible to theoretically predict and experimentally demonstrate several new wave effects related to the excitation of metasurfaces.

Second, new applications of metasurfaces in RF systems of magnetic resonance scanners have been proposed, including: (a) improving the homogeneity of the spatial distribution of the radio frequency magnetic field; (b) artificial high-impedance shields of RF coils of magnetic resonance scanners to improve the signal-to-noise ratio; (c) electromagnetic decoupling of the elements of multielement radiators of the scanner at the operating frequency.

The practical significance of the work is determined by the possibility of direct application of the developed methods for the calculation and manufacture of such devices based on metasurfaces as screens and resonators of microwave antenna systems, passive devices for decoupling elements of antenna arrays, antenna elements of RF coils for magnetic resonance imaging with a constant magnet field levels from 1.5 to 7 T (operating frequencies from 64 to 300 MHz), as well as quasi-optical passive devices (polarizers, frequency filters, focusing and deflecting wave beam splitters) for the submillimeter range. Several metasurfaces and metasurface-based resonators have been patented and are directly used for tasks of preclinical and medical magnetic resonance imaging research in such centers as the Almazov National Medical Research Centre (St. Petersburg, Russia), NeuroSpin (Paris, France) and The Center for Magnetic Resonance in Biology and Medicine (Marseille, France).

The results of the work in terms of methods for constructing controlled reflective surfaces and methods for calculating metasurface-based cylindrical resonators were used in the research work carried out at the Faculty of Physics of ITMO University in cooperation with companies such as Huawei and Topcon Positioning Systems in 2019-2022, where the author acted as a supervisor or leading researcher of the project.

Methodology and research methods. The following methods were used to solve the tasks set in the work:

Theoretical methods. To solve the problems of plane-wave diffraction on infinite metasurfaces of identical meta-atoms, the method of averaged boundary conditions is used, in which the metasurface is described by a flat homogeneous sheet with electric and (in some cases) magnetic surface currents, which create steps in the averaged tangential components of the magnetic and electric fields, respectively. Further, the averaged boundary conditions are used in the formulation of boundary problems, the solution of which leads to obtaining complex reflection and transmission coefficients. To determine the macroscopic parameters of the averaged boundary conditions (for example, the electrical grid impedance and its magnetic admittance in the case of a magnetoelectric metasurface), the method of equivalent circuits is used, which makes it possible to predict the response of the structure of meta-atoms of the considered geometric structure in the frequency band. In this case, the values of the equivalent-circuit elements describing metasurfaces are determined either analytically (by solving auxiliary quasi-static problems) or numerically (by comparison with numerically calculated data). When considering spatially inhomogeneous metasurfaces, the approach of local reflection and transmission coefficients is used. The shapes of the amplitude and phase profiles in the metasurface plane, required to provide the desired shape of the spatial field distribution in the reflection or transmission mode, are calculated separately by methods of geometric optics. When calculating the diffraction of the fields of point sources located near metasurfaces, the Exact Image Theory is used.

Numerical methods. To verify the analytically obtained solutions of diffraction problems, to optimize the microstructure of meta-atoms, and also to directly calculate the field distribution of a given radiator in the presence of a metasurface fragment of finite size, numerical methods of electromagnetics are used (Finite Element Method - FEM, Finite Integral Technique - FIT, the method of Integral Equations in the thin-wire approximation - IE) implemented in commercial software for numerical simulations (CST Microwave Studio and Ansys HFSS). The software tools are also used to analyze field distributions and calculate the

characteristics of radiators in the presence of metasurfaces, as well as in the presence of detailed voxel models of the human body (for magnetic resonance imaging tasks).

Experimental methods. For studying the physical properties of the considered metasurfaces, methods for measuring the spatial distributions of the electric and magnetic field components are developed in this work. For this purpose, a set of equipment from the Applied Radiophysics Laboratory of ITMO University (St. Petersburg) was used, consisting of a shielded anechoic chamber, vector network analyzers with sets of calibration standards, a near-field scanner, a rotating table, measuring antennas, near-field probes, and - measuring waveguide sections. Measurements in the far-field region were carried out within the range from 1 to 10 GHz. Field measurements in the near-field region (including those near individual meta-atoms), as well as measurements of the scattering matrix coefficients of radiators, were carried out in the range from 64 MHz to 10 GHz. To eliminate systematic errors, specialized calibration standards and well-known methods for calibrating vector network analyzers are used, as well as the time-gating method. In addition, the method of converting the field from the near-field to the far-field region using discrete measured data and other generally accepted experimental microwave methods are used. Measurements of the characteristics of metasurfaces in the submillimeter range (from 100 to 400 GHz) are organized based on the Center for the Collective Use of Scientific Equipment "High Technologies and Analytics of Nanosystems" (CCU VTAN) of Novosibirsk State University. To measure the amplitude and phase spectra of reflection and transmission of metasurfaces, quasi-optical interferometers with various optical schemes with a source in the form of a backward-wave oscillator (spectroscopy in the frequency domain) are used. To study metasurfaces during measurements inside operating magnetic resonance scanners, methods were used to study the distribution of an RF magnetic field by processing magnetic resonance images of phantoms (absorbing objects imitating the electrical properties of human body tissues) obtained by a scanner. Finally, experiments with the metasurfaces for MRI applications, performed in the presence of healthy volunteers, are carried out on the basis of permissions from local ethical committees with the written consent of the volunteers with RF safety evaluated by numerical simulations.

The scientific statements presented for the defense:

1. Anisotropic metasurfaces with self-complementary meta-atoms have frequency-independent reflection and transmission coefficients for plane waves with circular polarization in the single-mode diffraction regime, and the surface waves of orthogonal polarizations supported by them have equal phase velocities within the same diffraction regime.

2. Metasurfaces with meta-atoms in the form of paired split-ring resonators with a spatial overlap have a small reflection coefficient (less than 16% of power) in the frequency band of the single-mode diffraction regime, providing an arbitrary transmission phase at the operating frequency in the range from 0 to 360 degrees.

3. The averaged boundary conditions method makes it possible to obtain an analytical closed-form solution to the diffraction problem of the horizontal electric dipole in the presence of a Fabry-Perot resonator of two parallel thin-wire meshes.

4. A metasurface with a high surface impedance, forming a cylindrical shield around the radio frequency coil of a magnetic resonance scanner, allows increasing the signal-to-noise ratio by up to 30%.

5. Due to higher-order mode excitation of a metasurface fragment located near two radiators of a magnetic resonance scanner, it possible to decouple them with the RF-field spatial variation coefficient in an absorbing object imitating the human body decreased by 1.9 times compared to known methods.

6. Excitation of a slow wave propagating along a metasurface fragment located near the human body in a magnetic resonance scanner makes it possible to reduce the RF-field spatial variation coefficient inside the human body up to two times.

The degree of reliability and approbation of the results of the work.

The reliability of the results obtained in the work is confirmed by the following: (a) the obtained analytical expressions and the results of numerical calculations coincide with the results known in the literature in the simpler, particular cases; (b) good agreement between the results of analytical and numerical calculations was achieved, as well as - between the numerical and experimental results; (c) commonly accepted methods of theoretical and experimental research are used; (d) the results obtained do not contradict any results known in the literature; (e) the proposed methods and results after their publication were reproduced and further developed by other scientific groups, which is confirmed by the corresponding citations.

Approbation of the results of the work.

The results of the work were presented personally by the author in the form of oral and invited talks at the following international conferences: RADIO, Mauritius, 2015; Days on Diffraction, St. Petersburg, 2015; Days on Diffraction, St. Petersburg, 2016; PIERS, St. Petersburg, 2017; METANANO, Vladivostok, 2017; EuCAP, Paris, 2017; Metamaterials, Marseille, 2017; METANANO, Sochi, 2018; Metamaterials, Espoo, 2018; EuCAP, Krakow, 2019; ICEAA, Granada, 2019; Metamaterials, Rome, 2019; METANANO, St. Petersburg, 2019; Metamaterials, ONLINE, 2020; METANANO, ONLINE, 2020; METANANO, ONLINE, 2021; Metamaterials, Siena, 2022.

The results of the work were reported by the author at scientific seminars in the following organizations:

• Nizhny Novgorod State Technical University n.a. R.E. Alekseev, Nizhny Novgorod, Russia

• Moscow Institute of Physics and Technology, Moscow, Russia

• Prokhorov General Physics Institute of the Russian Academy of Sciences, Moscow, Russia

• Joint-Stock Company "Scientific and Production Enterprise "Radiosvyaz", Krasnoyarsk, Russia

• Francisk Skorina Gomel State University, Gomel, Belarus

• Institute Fresnel, Marseille, France

• Aalto University, Espoo, Finland.

Personal contribution. The presented work contains the results of theoretical and experimental studies carried out personally by the author, or with his direct participation. Personal contribution includes setting scientific problems, conducting theoretical and experimental research, planning, organizing and coordinating the work of the research team, analyzing, processing and summarizing the results obtained, as well as preparing and publishing scientific articles and patents, and presenting the results at scientific conferences and seminars. The personal contribution of the author is specified in Appendix A "List of publications on the topic of the dissertation".

The author has established a research group at ITMO University working in the field of applied electromagnetics and antenna theory. During the work of the group in the areas of research presented in this dissertation, under the guidance of the

author, the following were defended: one dissertation for the degree of candidate of physical and mathematical sciences in the specialty 01.04.03 (Radio Science) and two dissertations for the degree of candidate of technical sciences in the specialty 05.12.07 (Antennas, Microwave Devices, and their Technologies), as well as 10 master's theses. Some of the results presented in this work were obtained within the framework of the following research projects, in which the author acted as a leader: RSF 19-79-10260, grant of the President of the Russian Federation MK-3620.2019.8, and also as a leading researcher: RSF 15-19-20054, RSF 18-19-00482, RSF 21-79-30038, RFMEFI58717X0041.

Publications. In the period from 2014 to 2022, the results of research carried out as part of the presented work were published in 31 articles (in 30 original papers and 1 review) in journals indexed by Scopus and Web of Science (27 articles in Q1 journals, 4 articles in Q2 journals).

1. The employed physical models and the proposed classification of metasurfaces according to the electromagnetic properties of meta-atoms and functionality

All passive, reciprocal, and spatially uniform metasurfaces can be classified depending on the polarization response of their elements to an external field (namely, on the type of induced dipole moments of meta-atoms located in an infinite periodic structure). This classification, discussed in this section, defines the macroscopic properties of metasurfaces, which, being composed of discrete metaatoms, behave as continuous sheets with induced average surface currents when excited by incident plane waves. So, for a small metasurface period compared to the wavelength, a single-mode diffraction regime is realized, in which one incident plane wave generates only one reflected and only one transmitted plane waves. In this case, higher-order Floquet modes, being evanescent waves, contribute only to the field distribution at less than one period from the metasurface plane (they form the near field of a periodic structure), which is often of no interest. In the singlemode approximation, the solution to the diffraction problem involves finding the surface densities of the induced electric and magnetic currents averaged over the unit cell area. The presence of currents of these two types, as well as the presence or absence of a coupling between the electric current and the magnetic one when they are simultaneously excited by the field of the incident wave, determine the main classes of metasurfaces.

Consider the excitation problem of an arbitrary homogeneous metasurface by plane waves incident from each of the surrounding homogeneous half-spaces (media) with wave numbers and wave impedances (Figure 1a). In the single-mode diffraction regime, scattered plane waves can be found using the averaged boundary conditions described below, and, in the absence of anisotropy, using an equivalent circuit connected between two semi-infinite transmission lines (Figure 1b). The equivalent circuit describes the metasurface as a two-port with given frequency-dependent Z-, Y-, 5-, or T-matrices. As it is known, at one frequency, an arbitrary passive two-port can be replaced, in general, with a T-circuit composed of three elements with different complex values. In the presence of anisotropy, it is necessary to consider the existence of waves of two orthogonal polarizations in each half-space, as well as their possible relations upon diffraction on a metasurface (cross-polarization effect). In this case, the scheme of Figure 1b can be generalized by considering vector transmission lines, and the coefficients of the above-mentioned matrices become tensor.

(a) (b)

(a) the original boundary problem of diffraction on a metasurface; (b) an equivalent problem of a two-port connected between two semi-infinite transmission lines

Figure 1 - To the problem of the diffraction of plane waves on a uniform metasurface

Metasurfaces of the electric type.

The simplest properties are possessed by electric metasurfaces, which are described only using the average electric surface current /e flowing along an infinitely thin sheet. Under the assumption of a small period compared to the wavelength, such structures include the following: inductive meshes of thin wires, capacitive meshes of metal patches and strips, self-resonant frequency-selective surfaces, as well as two-dimensionally periodic structures of subwavelength scatterers, and other structures in which the electric field of the incident wave predominantly induces electric dipole moments.

When describing such a metasurface, with elements located in the z = 0 plane, the following averaged boundary condition is valid:

/e = fc X [#t(z = +0) - tft(z = -0)]; fee/e = ^ = 0),

(1)

where Ft, //t are the tangential components of the electric and magnetic fields averaged over the unit cell area, zc is the unit vector of the normal, Zee is a dyadic macroscopic parameter of the metasurface, which depends on the frequency and a plane-wave incidence angle, called the grid impedance (in the absence of anisotropy, this value is scalar). The average tangential component of the electric field is continuous (£t(z = 0) = Ft(z = -0) = Ft(z = +0)) and proportional to the average surface current. The latter, in turn, determines the step in the average tangential component of the magnetic field when passing through the metasurface plane. Zee is determined by the microstructure of the meta-atoms. The equivalent circuit of a scalar metasurface of the electrical type is a special case of the circuit shown in Figure 1b, with Z12 = 0, Y = 1/Zee. Therefore, electrical surfaces are sometimes referred to as surfaces with shunt impedance. Due to the continuity of

£t for z = 0 we have 7t + fl = f, where fl, T are the dyadic reflection and transmission coefficients, and 7t is the unit planar dyadic. In other words, a change in the amplitude or phase of the transmittance of an electric metasurface always causes a change in the reflection coefficient. It is not possible to obtain an arbitrary transmission coefficient phase over the full range of 0 to 360 degrees (required to transform the wavefront shape of the transmitted wave) without causing significant reflection. Due to the above physical properties, the functions of electrical metasurfaces are limited to the following:

• mesh antenna reflectors replacing solid shields,

• spectral filters and diplexers for space-propagating plane waves and quasi-optical wave beams,

• plane wave polarization converters with reflection losses,

• cavity resonator walls,

• waveguides of slow surface waves.

Metasurfaces of the magnetic type.

Metasurface elements can occupy a layer that is small in thickness h in comparison with the wavelength. Then, in the depth of the layer (within each metaatom), circulating electric currents can exist, the excitation of which by the magnetic field of the incident wave generates a magnetic dipole moment in each meta-atom. In the absence of induced electric dipole moments, such metasurfaces are called magnetic type metasurfaces, modeled as sheets of average surface magnetic current density. The averaged boundary condition for magnetic

metasurfaces turns out to be dual with respect to (1) (Ht is continuous and

-» —>

proportional to the induced average surface magnetic current/m, while Ft suffers a step determined by the magnetic current). The corresponding equivalent circuit contains only a series element Z = Z1+Z2=Zmm, while 7 = 0. Possible functions and basic limitations for magnetic metasurfaces are the same as for electric ones.

Metasurfaces of magnetoelectric type.

With the simultaneous presence of electric dipole moments in meta-atoms induced by the electric field of the incident wave and magnetic dipole moments induced by the magnetic field of the incident wave, more degrees of freedom appear for independent transformation of the reflected and transmitted wave fronts. Metasurfaces consisting of such meta-atoms are called magnetoelectric and are described by a combination of sheets of electric and magnetic surface currents located in the same plane z = 0. In this case, when passing through the plane, both

Ft and Ht experience a step, and the following averaged boundary conditions are valid:

Ze = fc X pt(z = +0) - Ht(z = -0)]; fe Je = = 0); /m = -¿0 X [i?t(z = +0) - i?t(z = -0)]; fmmTm = = 0),

where, in the absence of anisotropy and a strong contrast between the parameters of the adjacent media 1 and 2, we can assume that £t(z = 0) = 0.5 [£t(z = +0) + Et(z = -0)] and tft(z = 0) = 0.5 [tft(z = +0) + tft(z = -0)]. Thus, a metasurface of this type has two independent macroscopic parameters: Zee and Zmm, which are determined by the microstructure of meta-atoms and depend on the frequency and angle of incidence.

With known induced average currents in the metasurface plane, and a given complex amplitude of the electric field of the incident wave £inc, the amplitude of the reflected £ref and transmitted £tr waves are defined as

f - v i

£ref = 9 7e X Jm

2 1 (3)

^tr = ^inc ^"/e + 2 Z0 X /m,

where the normal vector zc is assumed to be directed towards the wave source and the metasurface is in a homogeneous medium with wave impedance 7?c = = . It can be seen from (3) that in the presence of a magnetic current, it becomes possible to transform the fields of the reflected and transmitted waves in different ways. If the condition ^c/e = zc X /m is satisfied, the desired phase of the transmitted wave in the absence of reflection can be obtained. In this case, each meta-atom of the metasurface, under the influence of the field of the incident wave, simultaneously obtains electric and magnetic dipole moments, which are in the same ratio as the electric and magnetic fields in the incident plane wave (it is polarized as the Huygens' source). Magnetoelectric metasurfaces of such metaatoms are called Huygens' surfaces.

The T-shaped equivalent circuit of the magnetoelectric surface contains all three elements, however, the series elements are symmetrical (Z1 = Z2 = Z). In the absence of reflection and dissipative losses in the metasurface, the equivalent circuit is a matched reactive phase-shifting circuit, where the phase shift of the transmitted wave is determined by the ratio of Z and 7.

Magnetoelectric metasurfaces perform the following functions:

• phase-shifting non-reflective surfaces,

• thin focusing surfaces (meta-lenses), beam deflectors and phase holograms with a low reflection coefficient,

• symmetric resonant absorbers,

• non-reflective polarization converters.

The above functions are limited by the mirror symmetry of the microstructure of meta-atoms relative to the z = 0 plane: magnetoelectric metasurfaces generate the same scattering when a plane wave is incident from both sides.

Bianisotropic (biisotropic) metasurfaces.

While the transmission coefficient is the same when a plane wave is incident on both sides of the metasurface (due to reciprocity), the corresponding reflection coefficients may differ. To do this, the structure of the metasurface elements should have no mirror symmetry with respect to the z = 0 plane. Then, in each metaatom, the effect of bianisotropy (or biisotropy) is observed, that is, an electric dipole moment and a magnetic dipole moment are simultaneously induced by both electric and magnetic fields of the incident wave. The boundary conditions for the averaged electric and magnetic currents induced on bianisotropic metasurfaces turn out to be coupled and can be written in the generalized matrix form:

7 ^ee 7 em 7e"

_» — — — ->

A. 7 me 7 mm Jm.

In the case of biisotropic properties (in the z = 0 plane), instead of the dyadic macroscopic parameters Zee,Zmm, and Zem = Zme scalars should be used. The corresponding equivalent circuit corresponds to the general case of a T-shaped circuit with three different elements present, shown in Figure 1b. Due to their asymmetric properties, metasurfaces of this type can provide a different phase of the reflection coefficient when a plane wave is incident from two sides. Depending on the angle between the electric and magnetic dipole moments induced in metaatoms, chiral (moments are parallel) and omega-bianisotropic (moments are orthogonal) metasurfaces are distinguished. A combination of the above properties can also be implemented.

In practice, the case of an impenetrable metasurface is important (in the case of biisotropy, it corresponds to the condition Y ^ ro). In this case, there is no transmission through the metasurface, and the reflection coefficients when incident from both sides onto the metasurface are determined by two independent

impedance values: Z1 ^ Z2, which correspond to the loads for two decoupled semiinfinite transmission lines. On each side of the metasurface, there are, generally speaking, nonzero averaged tangential components of both the electric and magnetic fields (and hence the equivalent averaged surface magnetic and electric currents, respectively), related by the expression: Ft(z = ±0) = Z12 [zo X tft(z = ±0)]. In this case, the values Z12 are called surface impedances, which can be denoted as ZS12. In the case of anisotropy, the surface impedance is dyadic. Impenetrable metasurfaces include surfaces with high impedance (|Z1| » and/or |Z2|»^12), as well as asymmetric absorbers (Z1 = or Z2 = ^2) providing impedance matching when the wave impinges from the corresponding side.

The unique functions of bianisotropic and biisotropic surfaces are as follows:

• Asymmetric (or single-sided) absorbers,

• High-impedance surfaces,

• Non-reflecting deflectors of transmitted wave beams for deflection at an arbitrary angle from the normal,

• Surfaces selective with respect to the state of circular polarization, isotropic sheets for polarization conversion without reflection,

• Mirrors reflecting waves with different wavefront shapes when a plane wave is incident from opposite sides.

The above classification, which relates the physical properties of the metaatom polarizability and the corresponding achievable functions of metasurfaces for the transformation of electromagnetic fields, is given in the review article [A1].

2. Transformation of plane-wave fields in the diffraction on metasurfaces

This section is devoted to studying the characteristics of plane waves arising from diffraction on infinite uniform metasurfaces with and without anisotropy. Electric, magnetoelectric and impenetrable (absorbing) metasurfaces with the proposed structures of meta-atoms are considered.

2.1. Polarization transformation of plane waves in the diffraction on metasurfaces with self-complementary meta-atoms

Self-complementary patterns contain two different colors, which, when the colors are mutually replaced, turn into the same patterns, which differ only in a shift or rotation in the plane of the pattern. In this section, self-complementary metasurfaces are considered, formed by a periodically etched infinitesimally thin metal sheets, made in such a way that the pattern formed by metal areas and cuts is self-complementary. In other words, when cuts are mutually replaced with metal areas and vice versa, the metasurface structure transforms into itself, except for a small shift compared to the wavelength and (or) rotation. Self-complementary surfaces are electrical metasurfaces. Previously, the properties of isotropic self-complementary surfaces with a checkerboard structure and frequency-independent behavior of the scalar grid impedance Zee have been investigated. In this paper, we study the properties of anisotropic self-complementary surfaces, for which the dyadic properties Zee determine different frequency behaviors of the reflection coefficients rx and ry with respect to orthogonal linear polarizations, as well as the corresponding transmission coefficients tx and ty.

Based on the energy conservation law for an anisotropic surface without

i 12 | 12

dissipative losses and cross-polarization: |rxy| + |txy| = 1, as well as - averaged boundary conditions written for the reflection and transmission coefficients: 1 + rx,y = £x,y, as well as the Babinet principle (tyc + tx = 1), where the symbol "C" means the transmission coefficient for the complementary structure (where cuts and metal sections are replaced), the law was obtained: ty + tx = 1. In other words, an arbitrary self-complementary surface is characterized by a certain relationship between tx and ty, which is not inherent in an arbitrary anisotropic metasurface.

It was analytically shown that the phase difference between tx and ty does not depend neither on the shape of the meta-atoms of the self-complementary metasurface, nor the frequency, and is always equal to 90°. This property, which has been shown to be stable under oblique incidence of a plane wave, provides a

relatively simple construction of plane-wave linear-to-circular polarization converters (with reflection losses) compared to previously known polarizers based on electrical metasurfaces. So, to build such a converter, it is enough to provide the amplitude condition |tx| = |ty| at the operating frequency, while the phase condition arg(tx) = arg(ty) ±n/2 is certainly satisfied at any frequency. This makes it relatively easy to obtain transmission of circular polarization when a wave with linear polarization is incident at an angle n/4 (3n/4) in the XOYplane.

In the work, the above properties were numerically confirmed for several types of self-complementary meta-atoms, the structures of which are shown in Figure 2, and experimentally confirmed for one of them (shown in Figure 2c).

(a, b, c, d) four types of meta-atoms considered.

Figure 2 - Meta-atoms of self-complementary metasurfaces (top row) and numerically calculated spectra of amplitude (middle row) and phase (bottom row) of tx,ty,rx,ry [A2]

The white background of the spectrum plots corresponds to the single-mode diffraction regime, while the yellow and red backgrounds correspond to the appearance of higher-order diffraction at oblique and normal incidence, respectively. As can be seen, for all structures in the single-mode regime, the phase difference is invariably ±90° at any frequency.

In the work, the reflection and transmission spectra of the above metasurfaces were numerically calculated and analyzed, also under the condition of the finite conductivity of the metal and in the presence of a thin dielectric substrate. These factors violate the conditions of the Babinet's principle. As a result, the frequency-independent phase difference between tx and ty is violated. It was found that this distortion is minimal for the structures depicted in Figure 2c (resonant structure)

and Figure 2d (nonresonant structure). For further numerical and experimental studies, a metasurface from resonant patches and complementary rectangular cuts (Figure 2c) was chosen.

The experimental setup and metasurface sample, as well as the results of measuring the transmission coefficient tx and ty, are shown in Figures 3 and 4, respectively. Figure 3 also shows the dimensions of the meta-atoms of the metasurface, made in the form of a single-layer printed circuit board on the 0.5-mm-thick Arlon 25N substrate (sr = 3.38) with copper metallization. The measured normal incidence transmission coefficients were compared with the results of numerical simulations in Figure 4, which shows a good quantitative agreement in the frequency band from 3.5 to 7.0 GHz. As can be seen, in the obtained spectra of transmission coefficients, there is a frequency shift of approximately 0.3 GHz between the resonances for the x-polarization and y-polarization, which can be explained by the effect of the dielectric substrate, the presence of which affects the diagonal components of Zee to different degrees. In addition, the frequency shift affects the phase curves, and the corresponding phase difference is not exactly ±90°, but about 122° at the intersection of the amplitude curves below the resonance and -110° above the resonance.

Figure 3 - Experimental study (transmission coefficient measurement scheme and the sample) of a self-complementary metasurface made of rectangular patches and cuts [A2].

Figure 4 - Measured (solid lines) and numerically calculated (dashed lines) amplitudes (top) and phases (bottom) of tx and ty at normal incidence [A2].

Thus, it was shown that anisotropic metasurfaces of self-complementary elements can have a frequency-independent phase difference between tx and ty, but this effect is subject to the negative influence of the dielectric substrate. To best achieve a frequency-independent phase difference, it is necessary to reduce the thickness and/or permittivity of the substrate. Also, by measurements and numerical calculations, the stability of the phase curves was demonstrated at angles of incidence in two orthogonal planes up to 30° relative to the normal. These results are published in [A2].

At the next stage, a self-complementary structure of zigzag metal strips, shown in Figures 2d and Figure 5, was studied in detail. For the incident electric field component polarized along the y axis, the metasurface exhibits an inductive grid impedance (an equivalent circuit with a single shunt element, Y = 1/jwL), while for the x-polarized component, the structure exhibits a capacitive impedance (Y = jvC).

Figure 5 - Proposed self-complementary metasurface based on zigzag metal strips, designed to convert a linear polarization to circular one [A3].

In the problem with an incident plane wave with linear polarization at 45° to the x axis, both the incident, reflected and transmitted waves can be decomposed into orthogonal linearly polarized components. Circular polarization assumes the same amplitudes and 90° phase difference of the transmission coefficients tx and ty (and similarly for the reflection coefficients). In this work, an equivalent circuit was constructed for each of the two polarizations, and the frequency behavior of the transmission coefficients was predicted. In particular, it is shown that there is a frequency f0 at which |tx| = |ty| = |rx| = |ry| = 1/V2 in the nonresonant mode, while arg(tx) = arg(ty) ± n/2 at all frequencies of the single-mode diffraction regime. The frequency dependences of tx and ty were then numerically studied for

three variants of the shape of zigzag meta-atoms with the same period a = 6 mm but differing in the angle ft of the zigzag opening.

The results of the numerical comparison are summarized in Table 1. In all cases, the frequency independence of the phase difference (equal to 90°) was confirmed, and the frequency f0 decreases with decreasing angle ft, which also reduces the ratio of the period to the wavelength A0 at the frequency f0. The table also lists the values of the equivalent circuit parameters of the metasurface L and C, obtained by comparing the analytically and numerically calculated spectra of tx and ty. Relative bandwidth values are also given for the level of axial ratio smaller than 3 dB (3dB-AR-BW). As can be seen, the structure shown in Figure 6c (3dB-AR-BW=69.7%) has the best broadband properties when converting polarization from linear to circular, which is explained by the highest values of inductance and capacitance in the equivalent circuit. In this case, the theoretical limit is 70.5%, obtained from the analysis of the equivalent circuit.

(a) ft = 53.1°; (b) ft = 28.1°; (c) ft = 18.9°. Figure 6 - Meta-atoms of the studied self-complementary metasurfaces based on zigzag ideally

conductive strips depending on the angle ft [A3].

Table 1 - Numerical comparison of the characteristics of self-complementary metasurfaces made of meta-atoms shown in Figure 6.

ß, ° fo, GHz a/A0 L, nH C, pF 3dB-AR-BW, %

53.1 12.6 0.252 2.15 0.061 55.2

28.1 3.70 0.074 8.18 0.232 67.6

18.9 1.65 0.033 18.5 0.524 69.7

To study the effect of the dielectric substrate on the characteristics of polarization conversion, a numerical simulation of the structure with ft =53.1°, but with strips made of copper and placed on a 0.5-mm-thick substrate with relative permittivity £r = 3.38 — j0.0025 (corresponds to PCB substrate material Arlon 25N) was made. It was found that the presence of the substrate increases the capacitance C while keeping the inductance L unchanged, which leads to the |rx| and |tx| intersect at a lower frequency than the other two curves |rv| and |tv|. As

a result, the Babinet's principle, written for the parameters of the equivalent circuits related to mutually orthogonal polarizations, in the form L/C = iil/4 is no

longer valid. The constancy of the phase difference between tx and ty is also not fulfilled, which leads to the impossibility of obtaining circular polarization with an AR of 0 dB. Instead, the axial ratio remains above 2.73 dB for the reflection regime and 2.56 dB for the transmission regime.

A method was proposed to compensate for the effect of the substrate by restoring the condition L/C = iil/4. To do this, it was proposed to narrow the width of the zigzag metal strips to increase the inductance and, at the same time, reduce the capacitance due to the expansion of the gaps between the strips. As a result of the numerical simulation, it was shown that the expected intersection of

the spectra ltxl « ltyl « lrxl

lryl

1/^2 at a frequency f0 = 8.5 GHz with a frequency-independent phase difference of 90° is restored, however, compared to the case without a substrate, it is possible to obtain a lower value of 3dB-AR-BW=51.4% for reflection and 52.8% for transmission. At f0, as expected, AR reaches 0 dB.

For the experimental comparison of the self-complementing metasurface on the substrate and its capacitance-corrected version, both samples were fabricated as single-layer printed circuit boards on a 0.5-mm-thick Arlon 25N substrate. Both samples were 192 mm x 192 mm in size and contained 32 x 32 meta-atoms each. The experimental setup and samples are shown in Figure 7, and the results of the study are shown in Figure 8.

(a,b) scheme and photograph of the experimental setup; (c) a sample with self-complementary meta-atoms; (d) sample with capacitance correction

Figure 7 - To experimental study of the metasurface of zigzag strips [A3].

The transmittances were measured for three different angles of incidence (6 = 0°, 15°, 30°) in two orthogonal incidence planes (^ = 0° and 90°). The behavior of the measured spectra shows that the proposed capacitance correction to compensate for the substrate effect makes it possible to obtain an almost constant

phase difference of 90° between tx and ty, and amplitudes crossing at the « V2 level at 8.5 GHz, with an axial ratio close to 0 dB at the center frequency. Thus, a method of broadband linear-to- circular polarization conversion was proposed and experimentally demonstrated, in which the necessary phase relation is guaranteed by the self-complementary structure of meta-atoms, and the capacitance correction makes it possible to compensate for the substrate effect. The above results are published in [A3].

(a) a sample with self-complementary meta-atoms; (b) sample with capacitance correction Figure 8 - Measured transmittance spectra [A3].

Next, we studied the frequency-independent properties that arise when a plane wave with circular polarization is incident on a resonant self-complementary metasurface (considering the submillimeter range: 230-540 GHz). To do this, we studied a structure similar to that shown in Figure 2c, but due to the size of the meta-atoms, it has a resonance near 400 GHz. Numerical modeling of an idealized self-supplementary structure without a substrate showed that when a plane wave with circular polarization is incident, a wave with linear polarization is reflected and transmitted. The polarization plane of the transmitted wave smoothly rotates as frequency changes. In this case, the transmittance into the co-polarized circular component does not depend on the frequency and is equal to ]4. The transmittance into the cross-polarized circular component has a frequency-independent amplitude equal to and a frequency-dependent phase determined by the metaatom microstructure.

Numerical modeling of a real structure consisting of rectangular 0.4-^m-thick aluminum patches and complementary rectangular cuts on a 10-^m-thick polypropylene substrate (s = 2.25, tan5 = 0.001) showed the negative effects of resistive losses and the presence of the substrate, leading to ellipticity of both scattered waves. It was found that the finite conductivity of aluminum distorts the frequency-independent behavior of the transmission coefficient into a cross-polarized circular component, while the finite thickness of the metal and the presence of a dielectric substrate causes the resonance to shift down in frequency and distort the frequency-independent behavior of the transmission coefficient into a co-polarized circular component. To mitigate the above effects, a correction method was proposed based on the analysis of resonant RLC equivalent circuits, separately constructed for two orthogonal linear components of the incident wave. The structure of a self-complementary meta-atom and its version after correction, which is distinguished by a narrower rectangular patch (the gap width to the left and right of the patch is increased from 15 to 30 ^m with a period of meta-atoms along the horizontal direction being 170 ^m) and a reduced period along the vertical direction (245 ^m instead of 280 ^m) are shown in Figure 9a and 9b, respectively. The constructed equivalent RLC circuits are shown in Figures 9c and 9d.

(a) the structure of a self-complementary meta-atom; (b) the structure of the meta-atom with correction for the effects of the substrate and non-ideal metallization; (c,d) equivalent circuits for horizontal and vertical linear polarizations, respectively.

Figure 9 - Structure of meta-atoms and equivalent circuits of a self-complementary metasurface

of rectangular patches and rectangular cuts [A4].

As predicted by simulations, the amplitude deviation of the co- and cross-polarization transmission coefficients in the range of 230-540 GHz for the structure after correction does not exceed 12%. The phase of the cross-polarization transmittance covers the range from -159° to 155°, while the phase of the co-polarization one deviates from 0° by no more than 4°.

To experimentally confirm the expected properties, a metasurface sample with a corrected meta-atom structure was fabricated by contact photolithography on a polypropylene film (Figures 10a and 10b). The transmittance and axial ratio spectra were measured on a quasi-optical setup with a Mach-Zehnder interferometer with a source in the form of a backward wave oscillator (BWO), shown in Fig. 10c.

(a) a fabricated sample of the metasurface; (b) magnified image of meta-atom microstructure;

(c) scheme of the quasi-optical setup.

Figure 10 - Toward an experimental study of a self-complementary metasurface in the

submillimeter range [A4].

The results of the experimental study (dashed lines) in comparison with the results of numerical simulation (solid lines) are shown in Figure 11.

(a,b) amplitude and (c) phase of co- and cross-polarization transmission coefficients for circular polarizations; (d) inverse of the axial ratio; (e) amplitudes of the main axes of the polarization ellipse; (e) phase and polarization angle of the linearly-polarized transmitted wave.

Figure 11 - The measured transmitted wave polarization characteristics in comparison with the

simulation results [A4].

The experiment confirms the frequency-independent behavior of the transmission coefficients for the co- and cross-polarization components of circular polarization. When a wave with circular polarization is incident, the metasurface transmits a wave whose polarization is close to linear with an ellipticity coefficient no worse than 8, the amplitude is close to a constant level 1/V2 with an accuracy of 10%, and the polarization angle smoothly changes with frequency from -80° to 78° relative to the y direction. These results are published in [A4].

2.2. Amplitude and phase transformation of a transmitted plane wave with low reflection in the single-mode diffraction regime with metasurfaces of paired split-ring resonators

Huygens' metasurfaces, which are nonreflective in the entire frequency range of the single-mode diffraction regime, were previously proposed to be formed of resonant spirals with the opposite helicity contained in each meta-atom. In this work, an alternative structure is proposed, the meta-atoms of which include dual split-ring resonators, which has the advantage of a full range of phase coverage in the transmission regime being compatible with printed-circuit-board realization. The shape of a meta-atom with dimensions for the numerical study performed in the range from 0 to 10 GHz is shown in Figure 12.

Figure 12 - Meta-atom of the Huygens' metasurface with low reflection in the single-mode diffraction regime [A5]. The parameters shown in the figure have the following values (in mm): a = 4, c = 100, R = 1.7, w = 0.05, g = 0.05, 1 = 3 and d = 0.5. The mutual shift s of the two resonators varied in the range from 0 to 4 mm. The optimal value is 2.4 mm.

The meta-atom contains two split-ring resonators with a radius R and a ring width w, located in two close parallel planes, with a distance d in between (it can be filled with a dielectric substrate when printed). The period along the x and y axes is equal to a, and the centers of the rings are shifted relative to each other by a distance s along the z axis. The results of the numerical calculation of the coefficients tx and ty are shown in Figure 13.

<PO

0.8

0.6

0.4

0.2

0.0 1.0

0.8

0.6

0.4

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270 180 ■90

(a)

> 1.5 mm

■ a- - 2.4 mm

■ s = 3.5 mm

(b)

1 • '' I

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r

/7

0 123456789 10

Frequency (GHz) (a) amplitude ty; (b) amplitude ry; (c) phase ty.

Figure 13 - Numerically calculated transmission coefficients at s=1.5, 2.4, 3.5 mm. The optimal

value (s=2.4 mm) is shown with a solid line [A5].

As can be seen from the graphs in Figure 13, with the optimal shift of the centers of the two resonators (s = 2.4 mm), the amplitude of the ty coefficient remains greater than 0.9 over the entire frequency range, while the phase acquires different values in the range from 0 to 360 degrees. At the same time, the reflection coefficient remains below 0.35 in amplitude. These properties are explained by the excitation of the symmetric and antisymmetric modes of two split-ring resonators by the field of the incident wave. With an optimal shift, their inductive coupling is compensated and the above modes degenerate, both resonating at a frequency of 6.6 GHz. As a result, the magnetic and electric dipole moments of the meta-atom, for which the symmetric mode (induced by a magnetic field) and the antisymmetric mode (induced by an electric field) are separately responsible, are excited in the same ratio of complex amplitudes as the corresponding fields in the incident wave.

For the experimental demonstration, a metasurface sample was made and its transmission and reflection coefficients were measured in an anechoic chamber. The sample shown in Figure 14a consisted of 71 identical rows made in the form of printed circuit boards on a 0.5-mm-thick Arlon AD250 substrate. Each row contained 71 pairs of split-ring resonators. The measurements were carried out by the near-field method using a near-field scanner and an electric field probe. The measurement scheme is shown in Figure 14b. The results are shown for normal incidence and for four polarizations (TMy and TEx, TMx and TEy) at two angles of incidence (0 = 15° and 6 = 30°) in Fig. 15. The coefficient spectra measured in

the range from 4 to 10 GHz for all variants of the incident wave showed high values of the transmission coefficient amplitude (in the range from 0.79 to 0.96 in amplitude). For TMy and TEx polarizations, a range of variation in the phase of the transmitted wave was shown close to 360°. In this case, the obtained values of the power reflection coefficient during normal incidence did not exceed 0.16.

(a) the fabricated sample; (b) scheme of the measurement setup Figure 14 - To the study of the metasurface of paired split-ring resonators [A5]

Thus, it has been theoretically and experimentally shown that the proposed metasurfaces with elements in the form of paired split-ring resonators with mutual spatial overlap have a low level of the reflection coefficient in a wide frequency band of the single-mode diffraction regime, while providing a full-range phase coverage. Therefore, the use of the proposed metasurface is beneficial for constructing meta-lenses with low reflection in the microwave range. The results of this section were published in [A5].

(a) transmission coefficient amplitude; (b) reflection coefficient amplitude; (c) transmission coefficient phase

Figure 15 - Measured reflection and transmission coefficients at different incidence angles

and for different polarizations [A5].

Huygens' metasurfaces also include the so-called perfect (or resonant) absorbers (thin structures that provide complete power absorption of an incident plane wave at one particular frequency for a given angle of incidence and polarization). It has previously been noted that although the bandwidth of an absorption is always limited by the thickness of the absorber, the bandwidth in which it can remain non-reflective is not fundamentally limited.

In this work, based on the shape of meta-atoms considered above (Figure 12), including paired split-ring resonators, a design of a symmetric resonant absorber, with low reflection within the entire single-mode diffraction regime, was proposed. By increasing the loss tangent of the substrate by an order of magnitude, it is possible to achieve complete absorption at the frequency of the degenerate resonance of the symmetric and antisymmetric modes. Also, in comparison with the meta-atom shown in Figure 12, in this part of the work, an isotropic metasurface is studied, where in each meta-atom two orthogonally oriented pairs of split-ring resonators are contained, as shown in Figure 16a. By numerical simulation, the following geometric parameters of the resonators (in mm) were selected to obtain the resonance at 3.15 GHz: a = 15.2, b = 14.6, I = 5.13, s = 4.75, rR = 3.23, g = 035, w = 0.42, h = 0.20 (substrate thickness) and t = 0.018 (copper thickness). The substrate is made of FR4 material with £r = 4.32(1 — j0.015). This value of the substrate loss tangent, which is inherent in the FR4 material, ensures complete absorption at the normal incidence, as well as under the condition of choosing the optimal shift value s = 4.75, which ensures the degeneracy of the symmetric and antisymmetric modes.

As can be seen from the numerically calculated power coefficients spectra of the absorption A and reflection R in the 0-10 GHz range (Figure 16b) and in the vicinity of the resonance at 3.15 GHz (Figures 16c and 16d), almost complete resonant absorption is provided up to the incidence angle 6 = 45° from the normal for both polarizations of the incident wave. At the same time, the power reflection coefficient remains below 10% over the entire range from 0 to 10 GHz.

The angular characteristics of the absorber were also studied analytically depending on the existence of various nonzero components of the meta-atom polarizability tensor. It is shown that the theoretical maximum of the angular width of the absorption coefficient at a level of -3 dB with respect to the normal maximum of 1.0 is achieved for a metasurface with a zero normal polarizability of meta-atoms. It was numerically shown that the proposed absorber is close in its angular properties to this theoretical limit for both polarizations.

(a) the meta-atom shape; (b) numerically calculated power absorption A and reflection R coefficients; (c,d) resonance shape in absorption and reflection power spectra at different angles of incidence for TE and TM polarizations, respectively [A6].

Figure 16 - Geometric structure and results of numerical calculations for the proposed resonant absorber based on paired split-ring resonators on a lossy substrate.

For the experimental study, a metasurface sample was fabricated (Figure 17), composed of ~2500 meta-atoms, which were located on 35 narrow printed circuit boards in the form of 547-mm-long rows, each one containing 36 pairs of split-ring resonators.

Figure 17 - Photograph of the fabricated sample of the metasurface perfect absorber.

A close-up shows the structure of meta-atoms [A6].

The far-field method was used to measure the complex transmission and reflection coefficients in the frequency range from 2.5 to 4 GHz under the conditions of an anechoic chamber. The power coefficients A and R were then calculated. The resulting spectra are shown in Figure 18 for the normal incidence and for an incidence angle of 30°.

(a)

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0.8

0.6

A, R

0.4

0.2

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(b) 1.0

0.8

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0.6

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(a) 0 = 0°; (b) 0 = 30°

Figure 18 - Measured (dashed lines) and numerically calculated (solid lines) power reflection

coefficients R and absorption A [A6].

As can be seen from the obtained results, the resonant properties of absorption with the level A = 0.96 for 0 = 0° and A = 0.94 for 0 = 30° along with a low (less than 0.1) power reflection coefficient were confirmed experimentally. As a result, based on the proposed meta-atoms, the possibility of constructing an isotropic symmetric absorber with a resonant absorption peak at the given frequency and low reflection in the entire single-mode diffraction regime was demonstrated. These results were published in [A6].

2.3. Metasurface for complete power absorption at two angles of plane-wave incidence

In this section, we consider the problem of constructing a metasurface to completely absorb the power of an incident TM-polarized plane wave at two given angles of incidence at one frequency. Previously, a perfect absorber for the normal incidence was proposed based on the mushroom-type meta-atoms with increased loss tangent of the substrate dielectric and/or reduced conductivity of metallic parts. However, as the wave incidence angle 6 increases, the level of the absorption coefficient A decreases monotonically, reaching zero at 0 = 90°. In this work, it was proposed to modify the structure of mushroom-shaped meta-atoms by connecting the pins of square metal patches (thin metal vias passing through a layer of a dielectric substrate) to the ground plane through specially selected lumped loads, as shown in Figure 19a.

Lumped load ''-Ground plane

(a)

(b)

(a) idealized structure; (b) practical structure compatible with printed circuit boards

Figure 19 - Proposed absorber structure for two angles of incidence of a TM-polarized wave based on modified mushroom-shaped meta-atoms [A7].

The loads contain resistive and reactive parts: ZLoad = RLoad + y^Load, and the structure has period a « A and gap width between patches w « a. For the analytical calculation of the surface impedance ZTM for an incident TM-polarized wave at an arbitrary angle of incidence 0, a model of a substrate penetrated by periodic vertical conductors (vias), in the form of a homogenized wire medium layer was used, considering spatial dispersion. Inside the wire-medium layer of thickness h, the propagation of both TM waves and TEM waves was considered, and to find their complex amplitudes when excited by an incident TM-polarized wave, the method of Generalized Additional Boundary Conditions (GABC) in the planes z = 0 and z = —h was applied. As a result, for the absorber taken as a basis, optimized for the normal incidence (0t = 0°) due to dissipative losses in the substrate, it was possible to analytically calculate the dependence of the real and imaginary parts of the impedance of the lumped loads necessary to obtain complete absorption also for another desired angle of incidence (02 ^ 0°). Calculation results for frequency f0 = 4.68 GHz, geometry parameters: a = 10 mm, w = 0.5 mm, radius of vertical vias r0 = 0.3 mm, and for different values of substrate thickness h: 1, 2, 4 and 4.5 mm are shown in Figure 20. To ensure absorption at the normal incidence, the complex permittivity of the substrate was chosen to be £h = 11.4 — j 1.2, 5 — j 1.16, 1.9 — j 1.1, u 1.6 — j 1.0, respectively.

Examples of the analytically calculated dependence of the reflection coefficient R on the angle of incidence for a 2-mm-thick metasurface, optimized for 62 =45°,60°, are shown in Figure 21 with solid lines. For comparison, the dashed lines on the same graph show the results of the numerical calculation.

Although a slight frequency shift is observed in the numerical simulation (dashed lines correspond to the frequency f0 = 4.48 GHz instead of 4.68 GHz) due to the absence of higher-order Floquet modes in the analytical description, the proposed theory, in the first approximation, correctly predicts the numerically obtained behavior of the perfect absorber for two angles of incidence. It is worth

noting that the choice of 02 =60° makes it possible to achieve a reflection coefficient below -20 dB at 6 ranging from 0° to 65°.

(c) (d)

(a) h = 1 mm; (b) h = 2 mm; (c) h = 4 mm; (d) h = 4.5 mm.

Figure 20 - Analytically calculated complex impedance of lumped loads added to mushroom-shaped meta-atoms depending on the required angle 02 of total absorption [A7].

(a) perfect absorber optimized for = 0° and 02 = 45°; (b) perfect absorber optimized

for = 0° and 02 = 60°.

Figure 21 - Analytically (solid curves) and numerically (dashed curves) calculated dependences of the reflection coefficient on the incidence angle of a TM-polarized wave for

h = 2 mm [A7].

In this work, a practical structure of a dual-angle absorber (Figure 19b) that can be realized as a printed-circuit board was also proposed. Its properties were verified numerically for 02 =45°.

The above results were published in [A7].

3. Wave-front shape transformation in the diffraction of plane waves on inhomogeneous metasurfaces

This section is devoted to the study of wave fields resulting from the plane-wave excitation of inhomogeneous metasurfaces with spatial modulation of macroscopic parameters. Electric type metasurfaces are considered, for which Zee(x, y) is a function of coordinates in the metasurface plane, as well as impenetrable (reflective) metasurfaces, for which the surface impedance ZS(x,y) is modulated.

3.1. Splitting wave beams in the diffraction of a circularly polarized plane wave on an inhomogeneous metasurface with self-complementary meta-atoms

Previously, various designs of electric, magnetoelectric, and bianisotropic metasurfaces were proposed to transform the shape of a transmitted (or reflected) wavefront. In this work, a new, simpler method for the spatial separation of waves with right-hand (RHCP) and left-hand (LHCP) circular polarizations in the transmission and reflection regimes was proposed based on the diffraction on an anisotropic electric metasurface of self-complementary meta-atoms. The method is based on the effect described in Section 2.1 and in [A4]. Thus, in Section 2.1 it was shown that the amplitude |tco-pol| and phase arg(tc0_p0j) of the co-polarization transmittance (hereinafter, under "co-pol" and "cross-pol" the RHCP/LHCP basis is assumed) are frequency-independent within the single-mode diffraction regime and are equal to 0.5 and 0°, respectively, for any form of a self-complementing meta-atom. At the same time, the amplitude of the cross-polarization transmission coefficient has a constant amplitude |tcross-pol|, but its phase depends on the frequency and shape of the meta-atom. Let us now assume the presence of spatial modulation and excitation by an incident wave with circular polarization; we apply the approach of local reflection and transmission coefficients. Then, for an inhomogeneous metasurface of self-complementary meta-atoms, it is possible to calculate the required dependence of the phase of the cross-polarized transmitted wave on the coordinates (x,y) in the metasurface plane at a constant amplitude. This modulation is achieved by introducing local variation of the meta-atom's geometric parameters. In this case, despite the modulation, the phase and amplitude of the co-polarized transmitted wave will remain constant across the aperture. In other words, inhomogeneous metasurfaces of self-complementary elements offer an elegant and simple approach to spatial

phase coding and splitting the co- and cross-polarized waves. Next, the principle was demonstrated.

First, a quasi-optical splitter of wave beams (collimated waves with flat fronts) was proposed, designed, and experimentally studied, capable of deflecting a cross-polarized wave by a given angle, but passing a co-polarized wave in the forward direction. The study was carried out in the submillimeter range (frequencies from 0.3 to 0.37 THz). The form of a self-complementary meta-atom for the anisotropic metasurface was proposed, containing an I-shaped dipole and a slot of a similar shape. The metallization pattern is made of aluminum with a thickness of 0.4 |m deposited by contact photolithography on a polypropylene film with a thickness of t = 10 |m. The periods of meta-atoms along x and y are Px = 170 |m and Py = 210|m. The shape shown in Fig. 22, together with the principle of operation of the investigated splitter, is characterized by the following parameters: dipole and slot widths w = 15 |m, bend length H = 30 |m, and horizontal gap width gx = 15 |m, providing resonance at 350 GHz at half-width of the vertical gap gy = 20 |m. At the resonance frequency arg(tcross-pol) = 180°.

Figure 22 - The operation principle of the proposed polarization deflecting splitter of wave beams based on an inhomogeneous electrical metasurface of self-complementary meta-atoms;

the meta-atom shape with dimensions [A8].

To encode the spatial modulation of the value arg(tcross-pol), a numerical calculation of transmission coefficients at a given frequency depending on the parameter was done. The results shown in Figure 23a confirm the expected behavior of ico-pol and tcross-pol with variations in the shape of meta-atoms, i.e., constancy of amplitudes with a deviation of no more than 0.1 and constancy of arg(tco-pol) with a deviation of no more than 10°. As one can see, arg(tcross-pol) varies in the range from 40° to 360° while varying in the range of 7.5 to 52.5 |m. To form a linear phase modulation (with a drop by 360°) along x, 6 phase values

and 6 corresponding meta-atoms were chosen within the macro-period, differing in gy values: 40, 23.5, 20, 17.5, 14, and 7.5 |m. The macro period structure is shown in Figure 23b.

(a) values |tro-poi| and |tCToss-poi| (top), as well as arg(tCo_poi) and arg(tcross-poi) (bottom), depending on the parameter; (b) synthesized structure of the macro-period; (c) numerically

calculated scattering patterns for two polarizations

Figure 23 - Synthesis of the macro-period of the metasurface beam splitter [A8].

A metasurface fragment consisting of regularly arranged macro-periods (7 macro-periods along x and 17 along y, total dimensions 8A x 4.3A) was studied by numerical methods. The calculated scattering patterns for co- and cross-polarizations in the deflection plane (XOZ) for a frequency of 0.35 THz are shown in Figure 23c. As can be seen, the separation of polarizations occurs both in the transmission and reflection regimes. This is due to symmetric scattering, a common property of all electrical metasurfaces. In the transmission regime, the cross-polarized beam is deflected by 43°, while the co-polarized beam is passed in the normal direction. The opposite applies to the reflection regime.

To experimentally demonstrate the separation of circularly polarized beams in the transmission regime, a sample of the metasurface was made, shown in the inset of Figure 24. To measure the complex amplitudes of the beams transmitted in the normal and the deflected directions (separately two linear components in both cases) the Mach-Zehnder interferometer setup with additional adjustment mirrors was built, shown in Figure 24.

Figure 24 - Experimental setup [A8].

The measured transmission coefficients for linear and circular polarizations are shown in Figures 25a and 25b, respectively.

(a) Linear polarization (b) Circular polarization

(a) amplitude and phase for x and y polarizations; (b) amplitude for co- and cross-polarization

(RHCP/LHCP basis)

Figure 25 - Measured complex amplitudes in the straight and deflected transmitted beams [A8].

The measured spectra show the best circular co-polarization of the beam transmitted in the normal at 0.35 THz. The deflected beam has a maximum level of the dominant cross-polarization component at 0.33 THz, while the co-polarized component is 3.5 times lower in amplitude. Ellipticity is caused by the difference in wave impedances of TE- and TM-polarized waves when propagating at a nonzero angle from the normal. The above results were published in [A8].

Next, based on a similar physical principle, a focusing splitter (Figure 26a) with a focal length of 120 mm was proposed, designed, and experimentally studied. Since the encoding of the parabolic phase profile of the transmission coefficient requires its local values to vary in the full range from 0 to 360 degrees, a suitable meta-atom shape was proposed, containing a meandered dipole and a complementary slot cut in the metal sheet, as shown in Figure 26b. The parameters of the metal and the dielectric substrate layers are the same as in the deflecting splitter. The parameter responsible for arg(tcross-poi) in this case is the halfperimeter L = + L2 + L3 + L4 + L5 of the dipole/slot, while the other

parameters were fixed: Px = Py = 330 |m, i.e. 0.375A at 0.345 THz, w = 10 |m, = 20 |m. A fragment of the synthesized metallization pattern of the inhomogeneous metasurface is shown in Figure 26c.

(a) operational principle of the focusing beam splitter; (b) meta-atom shape; (c) a fragment of the synthesized metasurface

Figure 26 - Operational principle and practical realization of the focusing beam splitter based on an inhomogeneous electric metasurface of self-supplementary meta-atoms [A9].

The correspondence of the amplitude and phase of the co-polarization tRR and cross-polarization tRL transmission coefficients, as well as the radial coordinate p to the local parameter L, is shown in Figure 27. To study the focusing capability of the synthesized metasurface, a semi-analytical calculation of the field distributions of the transmitted wave in the Fresnel diffraction zone (including the focal plane) was done using aperture method. The results are shown in Figure 28.

(a) complex amplitudes of tRR, tRL; (b,d) amplitudes and phases; (c) correspondence of the phase of the parabolic profile and the calculated value of L to the radial coordinate p in the

metasurface plane

Figure 27 - Results of parametric calculation of transmission coefficients depending on L and synthesis of an inhomogeneous metasurface [A9].

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and vertical components.

Figure 28 - Calculated distributions of the electric field amplitude in the transmission regime

at 0.345 THz [A9].

As can be seen from the obtained results, the co-polarized beam passes through the inhomogeneous metasurface with practically no change in the wave front. In contrast, the cross-polarized beam is focused to a point at 120 mm.

For the experimental study, a metasurface sample with an optical aperture diameter of 50 mm was fabricated by contact photolithography. Its microstructure is shown in the photograph in Figure 29a. The shapes of the focal spots were measured separately for horizontal and vertical polarization components (shown in Figure 29b). It was found that the focal spots have the same width as the focal spot of a standard lens with the same focal length. The circular polarization of both beams was also confirmed (by investigating fields of a plane wave with co-polarization and in the center of the focal spot of a focused wave with cross-polarization).

The above results were published in [A9].

(a) a photograph of the manufactured sample microstructure; (b) measured intensity profiles of focal spots for x and y polarizations compared to the operation of a standard polyethylene lens

Figure 29 - To an experimental study of a focusing metasurface beam splitter of self-

supplementary meta-atoms [A9].

3.2. Dynamic transformation of the reflected field distribution by means of an inhomogeneous reflective surface controlled by infrared light

In this section, we study the proposed design of an inhomogeneous impenetrable metasurface, the local reflective properties of which (surface impedance and phase of the reflection coefficient) can be dynamically controlled due to varactor diodes included into the meta-atoms. Such structures can be used to expand the coverage area and improve the data transfer rate in wireless communication systems and are referred to in the literature as Reconfigurable Intelligent Surfaces (RISs).

The proposed RIS structure consists of identical and structurally independent blocks containing 4 independently phase-switchable meta-atoms (having two states of the reflection coefficient phase: the so-called 1-bit reflect-array). Each block has its own microcontroller that switches the phase value of the reflection coefficient for each meta-atom by applying a control voltage through the driver circuit in accordance with the control commands to the varactor diodes. The control commands are received by the controller through a photodiode that registers an infrared digital signal from a remote control device. The principle of controlling the position of the reflected beam when a wave is incident from the access point to direct it to the user terminal, as well as the scheme for constructing meta-atoms of an autonomous block, are shown in Figure 30. Meta-atoms are miniaturized

rectangular microstrip (patch) resonators with dimensions L = 30.5 mm and W = 19.6 mm made on a printed-circuit board, in which the metal strip is connected to the ground plane through two varactor diodes. The connection points of the diodes are selected in order to provide two specified values of the local reflection coefficient phase when applying two fixed levels of control voltage (states "0" with a phase of the reflection coefficient -90° and "1" with a phase of the reflection coefficient +90°, implemented at U = 0 and U = 3.2 V, respectively, based on the properties of the SMV2019-040LF diodes used at a frequency of 5.2 GHz).

The experimental RIS sample was made in the form of four joint panels, made in a form of multilayer printed circuit boards, where each panel contained 10 x 10 meta-atoms (25 blocks). On the front layer of the microwave substrate (Rogers 4003C material), patches were printed, and varactor diodes were mounted. On the back side of the structure, in several layers on FR4 substrates, control and power circuits for autonomous units were placed and mounted.

Figure 30 - The principle of operation of RIS, (top) and the sketch of the electronic components of the building block (bottom). A wireless access point combined with an infrared control device

is shown, as well as two user receivers [A10].

Numerically calculated and measured values of the reflection coefficient from the RIS with the same state of all meta-atoms (constant-phase aperture), as well as the phase difference between the states "0" and "1" are shown in Figure 31. As can be seen, in two states of the varactors, the phase difference is close to 180° at a

frequency of 5.2 GHz and is maintained in the range of 150°-210° from 5.12 to 5.28 GHz.

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