Неэрмитовы интерференционные эффекты при рассеянии света высокоиндексными полупроводниковыми наночастицами тема диссертации и автореферата по ВАК РФ 01.04.05, кандидат наук Канос Валеро Адриа

• Канос Валеро Адриа
• кандидат наук
• 2022

Оглавление диссертации кандидат наук Канос Валеро Адриа

1.7 Size sorting of Au-NPs

1.8 Conclusion

2 Non-Hermitian description of the Hybrid Anapole Regime

2.1 Motivation and relevance

2.2 Design and observation of the HA regime

2.3 QNM description of the HA regime

2.4 Model describing the interaction with a dielectric substrate

2.5 Short pulse modulation in the transient regime

2.6 Conclusion and outlook

3 Superscattering of Interfering Resonances

3.1 Motivation and Relevance

3.2 Problem Formulation

3.3 Toy model of scattering by an arbitrary object

3.4 Dielectric Nanocavities

3.5 Conclusion and Outlook

4 Engineering Non-Hermitian Singularities in Single Dielectric Nanocavities 126 4.1 Motivation and Relevance

4.2 Two-mode model of a dielectric nanoparticle

4.3 Existence of EPs in a single dielectric nanoparticle

4.4 Magnetoelectric response in the strong coupling regime

4.5 Substrate-mediated EP control

4.6 Design of EP-based local refractive index sensors

4.7 Conclusion and Outlook

Conclusion

Acknowledgments

Bibliography

Appendix

A Scattered field formulation, Multipolar and Modal representations

A.1 Scattered field formulation

A.2 Multipole theory

A.3 QNM expansions

B QNM expansion of multipolar scattering

C Derivation of the TCMT equations

D Elements of QNM perturbation theory

D.1 General principles and formulae

D.2 Connection between PT and TCMT

E Condition for a source to be nonradiating: The case of the anapole

F Derivation of the viscosity tensor

G Main publications

Реферат

Общая характеристика работы

Введение диссертации (часть автореферата) на тему «Неэрмитовы интерференционные эффекты при рассеянии света высокоиндексными полупроводниковыми наночастицами»

Актуальность

Способность управлять взаимодействием света с веществом на нано-уровне обладает фундаментальным значением для современных фотонных технологий, начиная от современных методов сбора энергии или биосенсинга и заканчивая квантовыми коммуникациями. Уже более десяти лет на переднем крае исследований в области нанофотоники находится плазмоника. Наночастицы из благородных металлов поддерживают локализованные поверхностные плазмонные резонансы (ЛППР), которые способны усиливать и запасать электромагнитное поле на наноуровне. Это создало исключительные возможности для управления светом за пределами дифракционного предела [1]. Однако в настоящее время эта область находится в тупике; джоулевы потери в металлах неизбежны и сильно ухудшают работу оптических устройств. Как мы увидим в этой диссертации, потери (вызванные поглощением или излучением) могут быть плюсом в зависимости от применения. Тем не менее, в большинстве рассматриваемых случаев поглощение является вредным, к примеру, в плоской оптике или оптических соединителях [2]. Другой важный недостаток связан с тем, что благородные металлы несовместимы с производством комплементарных структур металл-оксид-полупроводник (КМОП). Интеграция электронных и фотонных устройств на одном чипе имеет решающее значение для следующего поколения оптических соединителей [3].

Несмотря на то, что научное сообщество предприняло серьезные попытки для преодоления этих узких мест, ни одна из предложенных стратегий до сих пор не оказалась достаточно универсальной. К счастью, несколько лет назад стало очевидно, что диэлектрики с высоким показателем преломления и полупроводниковые наночастицы, например, из кремния также могут поддерживать

сильные оптические резонансы. Помимо того, что диэлектрики предоставляют разнообразные резонансы, недоступные в обычных плазмонных наночастицах, они также преодолевают две фундаментальные проблемы своих металлических предшественников, а именно: в них почти нет джоулевых потерь, и они совместимы с КМОП. Однако вскоре исследователи поняли, что диэлектрические наноча-стицы — это больше, чем просто выигрышная платформа для реализации всё той же старой физики; оказалось, что новые удивительные эффекты в двух шагах, их оставалось обнаружить. В отличие от обычных плазмонных частиц, изотропные диэлектрические наночастицы поддерживают широкий класс мультипольных ре-зонансов как электрического, так и магнитного типа, которые могут взаимно интерферировать, порождая замечательные явления, такие как эффект Керкера [4], неизлучающие анаполи [5], [6], преобразование спинового углового момента в орбитальный угловой момент [7], или магнитный свет [8].

Большинство эффектов в полностью диэлектрической нанофотонике описываются в рамках упрощенных мультипольных моделей. Однако при усложнении геометрии необходимо учитывать громоздкие мультипольные вклады высокого порядка. Еще более важно то, что некоторые недавно открытые физические эффекты требуют геометрии, значительно отличающейся от сферы - среди них связанные состояния в континууме [9], спиновый эффект Холла [10] и усиленный круговой дихроизм [11]. В этом случае динамика мультипольных резонансов может стать очень сложной, и при этом обычно используются полуаналитические мультипольные разложения, которые могут привести к нелогичным выводам, таким как очевидные несоответствия между разработанными мультиполями и степенями свободы структуры [5], [12]. Таким образом, необходима более прозрачная, физически осмысленная теория.

Функциональный отклик как плазмонных, так и диэлектрических на-ноантенн в основном определяется резонансами. Они связаны с дискретными частотами, при которых свет эффективно локализуется внутри оптического 'резонатора' или в его ближней зоне, что максимизирует взаимодействие света с ве-

ществом. Однако в пассивных структурах действует правило взаимности: любое распределение токов, которое можно возбудить из дальней зоны, также должно излучать. Следовательно, любая резонансная мода должна затухать по прошествии некоторого конечного времени жизни. Кроме того, если есть поглощение, то количество фотонов при упругом рассеянии не сохраняется, что приводит к дополнительным каналам затухания для резонансных мод, особенно важным в плазмон-ных структурах. Более того, эти два общих процесса потери энергии приводят к уширению пиков физических измеряемых величин на резонансных частотах, таких как сечение рассеяния или фактор Парселла.

В математической постановке задачи резонансные моды соответствуют собственным модам оператора Максвелла, дополненного набором граничных условий. Для описания открытых систем, таких как диэлектрические нанорезона-торы, необходимы излучательные граничные условия. Это делает лежащий в основе оператор неэрмитовым в том смысле, что собственные моды обращенной во времени задачи отличаются от исходной задачи. То же самое происходит, если система замкнута, но тензоры диэлектрической и/или магнитной проницаемости неэрмитовы (т.е. е^ = е), что происходит при наличии поглощения или усиления. В этих рассматриваемых сценариях собственные частоты резонансных мод являются комплексными, в них действительная часть - это резонансная частота, а мнимая часть пропорциональна потерям, как на излучение, так и на поглощение. В настоящее время эти резонансные моды обычно называют квазинормальными модами (КНМ), поскольку они уже не подчиняются нормировке энергии, используемой в замкнутых эрмитовых системах. Важно, что в нескольких работах уже показано, что можно синтезировать отклик произвольных электромагнитных резонаторов как сумму возбужденных КНМ [13]. Приведенного базиса КНМ в интересующем спектральном диапазоне достаточно, чтобы дать интуитивное объяснение для ряда физических явлений [13], в дополнение к тому, что в существующие теории нормальных мод вносится необходимая поправка [14].

Цель диссертации

Из вышеизложенного становится ясно, что взаимодействие света с веществом при посредстве диэлектрических наночастиц можно естественным образом описать в рамках неэрмитовой физики. Последняя обеспечивает общую постановку задачи, которая применима ко всем волновым системам, в которых важную роль потери играют (на излучение или поглощение). Задействуя эти новые теоретические инструменты, эта работа ставит целью введение в полностью диэлектрическую нанофотонику новой физики на основе неэрмитовых эффектов интерференции мод. Такая интерференция может возникать в разнообразных пространствах параметров, например, при настройке частоты (главы 1,2) или геометрии (главы 3, 4).

С учетом этого обоснования были поставлены следующие цели:

Главные цели

• Исследовать новые подходы к управлению поведением света на наноуровне с помощью диэлектрических наночастиц.

• Разработать качественные и количественные физические модели рассеяния такими частицами и их взаимодействия с окружающей средой, в частности, связи возбужденных КНМ с мультипольным излучением, а также модели неконсервативных сил, возникающих в мультипольных полях.

• Спрогнозировать новые эффекты рассеяния с помощью упрощенных аналитических теорий с учетом неэрмитовости.

• Создать реалистичные устройства для реализации ряда задач технологического значения в области оптики, химии и биологии.

Научные положения

В результате исследования, проведенного в рамках данной диссертации, на защиту выносятся следующие положения:

Положения, выносимые на защиту

• Преобразование спинового углового момента лазерного пучка в орбитальный угловой момент в ближнем поле субволнового диэлектрического рассеивателя приводит к появлению неконсервативных сил, способных передавать момент силы поглощающим наночастицам. Из-за компоненты, пропорциональной ротору плотности спинового углового момента, сила не обязательно параллельна потоку электромагнитной энергии.

• Неконсервативные силы ближнего поля, возникающие вблизи диэлектрического рассеивателя, можно применить для создания устройства, способного смешивать водные растворы в субволновом объеме, т.е. 'наномиксера', с использованием плазмонных золотых наночастиц для перемешивания жидкости. Кроме того, если использовать смену знака их поляризуемости вблизи плазмонного резонанса, устройство также можно применять для реализации полностью оптической сортировки наночастиц золота по размеру.

• В полупроводниковой наночастице с высоким показателем преломления и с цилиндрической симметрией, освещенной плоской волной, падающей по нормали, могут перекрываться четыре анаполя всего с двумя степенями свободы из-за смешанного (электрического и магнитного) мультипольного характера лежащих в основе квазинормальных мод. Таким образом, мы можем наблюдать гибридный ана-поль при рассеянии в видимом диапазоне. Этот режим нарушается при переходном процессе, в котором сигналом рассеяния можно управлять с помощью расположенной под частицей диэлектрической подложки.

• Для рассеивателя без сферической симметрии можно превзойти предел канала рассеяния путем связывания двух резонансных мод Ми различной мультипольной природы, в соответствии с механизмом Фридриха-Винтгена. Этот новый режим сверхрассеяния возникает как аналог квази-ССК в изолированных рассеивателях.

• В изолированной полупроводниковой наночастице с высоким показателем преломления с перекрывающимися электрическими и магнитными дипольными модами введение бианизотропного возмущения приводит к образованию пары ис-

ключительных точек и 'дуги Ферми' в пространстве параметров.

Ключевые научные достижения

Следующие пункты представляют собой наиболее важные достижения, представленные в этой диссертации:

1. Впервые было обнаружено, что преобразование спинового углового момента лазерного пучка с круговой поляризацией в орбитальный угловой момент диэлектрической наночастицы (преобразование СУМ-ОУМ) может привести к появлению поля неконсервативных сил вблизи последней, не обязательно параллельное потоку энергии. Они вызывают спиральное движение поглощающих дипольных наночастиц.

2. Предложена и численно подтверждена концепция инновационного типа полностью оптической наномашины, способной смешивать растворы и сортировать плазмонные наночастицы на наноуровне.

3. Теория, разработка и первое наблюдение четырехкратного гибридного ана-поля в кремниевом наноцилиндре в видимом диапазоне.

4. Первый анализ взаимодействия гибридных анаполей с подложкой и исследование их разрушения при импульсном освещении.

5. Открытие, формальное объяснение и наблюдение нового механизма для достижения эффекта сверхрассеяния, основанного на интерферирующих ре-зонансах диэлектрического резонатора без сферической симметрии. Показано существование нового 'супердипольного' резонанса, образующегося за счет конструктивной интерференции двух резонансов, в противоположность тому, как деструктивная интерференция приводит к квази-связанным состояниям в континууме.

6. Формулировка строгой теории для создания неэрмитовых сингулярностей (так называемых исключительных точек) в одиночной диэлектрической на-ночастице с использованием только настройки геометрии, и первая численная модель реалистичного резонатора, демонстрирующая этот эффект, для апробации концепции.

Научная ценоностъ

Полученные результаты носят фундаментальный характер и открывают новые парадигмы в полностью диэлектрической нанофотонике, а также создают актуальные ответвления в химии, биологии и квантовой оптике. Наличие сил, пропорциональных ротору плотности спинового углового момента в ближнем поле диэлектрических рассеивателей, может положить начало для новых степеней свободы в оптическом управлении. Разработанный наномиксер можно применить в новом поколении установок "лаборатория на чипе" в качестве этапа в рабочем цикле химического синтеза или для обработки образцов. Гибридные анаполи являются перспективными для применения в плоской оптике и беспроводной передаче энергии, и они могут обеспечить реализацию мета-устройств с двойной функциональностью: полная прозрачность при непрерывном освещении и импульсная модуляция в переходном режиме. Гибридный анаполь обладает надежностью в присутствии диэлектрической подложки, что может упростить их реализацию в реальных установках, в отличие от обычного анапольного эффекта. Хотя исследования проводятся в контексте электромагнитного рассеяния, существование су-пердипольных резонансов - это общий физический эффект, который может существовать в любой открытой системе, п оддерживающей два интерферирующих резонанса, имеющих утечку по крайней мере в два открытых канала континуума. Возможность передачи энергии от нескольких входящих волн в один исходящий канал может открыть дорогу для наноразмерных сумматоров мощности и усиленного взаимодействия света с веществом. Наконец, неэрмитовы сингулярности

являются предметом активных исследований не только в фотонике, но и в области открытых квантовых систем и играют важную роль в развитии теорий неэрмитовой топологии. Представленные здесь правила для их разработки представляют собой первый шаг к их реализации и наблюдению на наномасштабе. Кроме того, численные результаты демонстрируют, что потенциально их можно применять в качестве сверхчувствительных объемных датчиков показателя преломления , которые широко используются для биосенсинга и отслеживания реакций.

Надежность и достоверность

Надежность и достоверность всех результатов и выводов в этой диссертации гарантируются воспроизводимостью численного моделирования и выполненных измерений. Достоверность аналитических выражений и численного моделирования дополнительно подтверждается тем, что теоретические и экспериментальные данные находятся в количественном соответствии.

Апробация результатов

Результаты были представлены на ряде международных конференций, где автор участвовал в качестве докладчика. Среди них METANANO 2019 (Санкт-Петербург, Россия), METANANO 2020 (Онлайн), SNAIA2020 (Париж, Франция), METANANO 2021 (Онлайн), SNAIA 2021 (Париж, Франция), CLEO 2022 (Сан-Хосе, США). Кроме того, результаты также были представлены и обсуждались в рамках серии теоретических семинаров физического факультета Университета ИТ-МО. Работа выполнена при поддержке программы «Приоритет 2030».

Публикации

Основные результаты диссертации представлены в 6 рецензируемых статьях, опубликованных в журналах, входящих в перечень ВАК РФ и индексируемых в базах Scopus и Web of Science, а также в 14 рецензируемых материалах

конференций, также индексируемых в Scopus и Web of Science. Вклад автора

Автор сыграл главную роль в разработке оригинальных идей, проработке целей исследования, математических методов и численного моделирования, которые привели к получению основных результатов (преобразование СУМ-ОУМ, динамика плазмонных наночастиц в оптическом вихре, теория и разработка гибридного анаполя, теория супердипольного резонанса и правила проектирования исключительных точек в диэлектрических наночастицах). Автор принимал активное участие в подготовке рукописей.

Структура и объем диссертации

Диссертация состоит из введения, 4 глав, заключения и списка литературы. Общий объем диссертации составляет 235 стр. Библиография содержит 129 ссылок. Работа содержит 58 рисунков.

Обзор результатов

Глава 1 исследует неконсервативные силы (НКС), которые генерируются при преобразовании спинового углового момента лазерного пучка в орбитальный угловой момент субволнового диэлектрического рассеивателя (преобразование СУМ-ОУМ). Обсуждается взаимодействие между СУМ и ОУМ, происходящее при посредстве субволновой диэлектрической наночастицы, и выясняется его связь с неконсервативными силами. В результате усиленной спин-орбитальной связи между наночастицей и плоской волной с круговой поляризацией в ближнем поле возникает оптический вихрь. Показано, что последний усиливает неконсервативные силы двух разных видов: более распространенное давление излучения и менее изученную силу, пропорциональную ротору плотности спинового углового

момента (далее будем называть ее спин-роторной силой). Они вызывают спиральное движение поглощающих наночастиц золота (Аи НЧ) в видимом диапазоне. Основываясь на результатах, предлагается и численно проверяется полностью оптическая платформа «лаборатория на чипе» (ЛНЧ) для реализации смешивания на-ножидкостей (показано на рисунке 1а) и сортировки частиц (показано на рисунке

Рисунок 1. - (а) Художественное представление наносмешивания, вызванного НКС в оптическом вих ре. Свет с круговой поляризацией падает на диэлектрический рассеиватель (нанокуб кремния), который генерирует сильные мультипольные поля и НКС на поглощающих нанорассеивателях, таких как Аи НЧ вблизи их плазмонного резонанса. Их вращение передает импульс окружающей жидкости, уменьшая длину диффузии растворенных в ней веществ. (Ь) Оптическая сортировка Аи НЧ по размеру. Градиентные силы притягивают мелкие частицы к нанокубу, а крупные отталкиваются от него за счет сочетания градиента и НКС. (с) Нормированное поле НКС, создаваемое точечным магнитным квадруполем с круговой поляризацией в ближнем поле при z=0.

Рассмотрим плоскую волну с круговой поляризацией с выражением Ещс (г) = Ё0(г) (ех + ¡аеу), и амплитудой £0(г) = Ё0е-гк°г. & можно интерпретировать как "спин" электромагнитной волны. Нам интересно понять, как УМ может передаваться от падающего пучка к рассеивателю, обозначенному А. Поверхностную плотность УМ рассеянного поля (усредненная по времени) можно выразить как векторное произведение радиус-вектора г и потока рассеянной энергии (Б5)

(Л> =

г х (Б5>

с

(1)

Оценивая ]г, мы видим, что (]г> гс г^5 ]. Таким образом, прямая передача УМ от падающего пучка к рассеивателю может произойти только при ненулевом азимутальном потоке рассеянной энергии . Во-вторых, (]г> является функцией начала системы координат, что делает ее так называемой внешней величиной. Поскольку СУМ по своей сути является внутренней величиной, то (]г > обязательно содержит ОУМ. Таким образом, преобразование СУМ-ОУМ может происходить при ] = 0, т.е. при наличии оптического вихря. В качестве частного примера этого процесса исследуется диэлектрический нанокуб с па ~ 4, эмулирующий Si в видимом спектре (Рисунок 2). Предполагается, что рассеиватель погружен в воду и оптимизирован таким образом, что он поддерживает магнитную квадрупольную резонансную моду на длине волны зеленого лазера (532 нм), что подтверждается мультипольным разложением на рисунке 2Ь. Во всех последующих результатах использовался свет с левой круговой поляризацией.

Рисунок 2. - (а) Оптимизация размера МК КНМ нанокуба. Сторона Ь нанокуба выбрана так, чтобы резонансная длина волны была близка к длине волны зеленого лазера (зеленая линия, 532 нм в вакууме). Как ЭД, так и МК КНМ демонстрируют линейную зависимость от Ь. В воде (пт = 1.33) ЭД испытывает значительное синее смещение, а МК - лишь небольшое красное смещение. Вставки: распределения полей ЭД и МК КНМ. (Ь) Мультипольное разложение сечения рассеяния для выбранного нанокуба (размеры указаны на вставке). Предполагается, что рассеиватель размещается на подложке с низким показателем преломления, такой как стекло. На резонансной длине волны МК-моды наблюдается хорошо выраженный МК-резонанс.

Исследованы НКС при поглощении дипольными наночастицами в ближнем поле А. Для сильно поглощающей наночастицы консервативные силы пренебрежимо малы, и НКС становятся основным членом в уравнении. Они задаются выражениями

<F> = ^ (^ + Vx<L, >), (2)

£0 * С !

где а" - мнимая часть их поляризуемости, определяемая потерями на поглощение и излучение. НКС включают вклады от давления излучения (первый член в правой части уравнения) и спин-роторной силы (второй член). Интересно, что азимутальная составляющая вектора Пойнтинга, возникающая из-за спин-орбитальной связи, обязательно вызывает тангенциальные силы, действующие на частицу и, следовательно, смещение в полярном направлении. Чтобы подтвердить это, можно вывести аналитические выражения для НКС, предполагая, что магнитный квад-руполь является основным вкладом в поле вблизи А:

, 2 (к4г4 + 3к 2г2 + 9)

(к, г, в) = 25л&е2~-í6kV-" sin (3)

такое идеализированное распределение полей показано на рисунке 1. Становится очевидно, что любая частица, попадающая под его действие, будет испытывать спиральное движение вокруг начала координат.

Рассчитанное распределение вектора Пойнтинга для оптимизированного нанокуба в ближней зоне действительно показывает оптический вихрь, ограниченный в субволновой области с центром в рассеивателе (Рисунок 3a). Вклады от давления излучения и спин-роторной силы в орбитальный момент силы , испытываемый маленькой поглощающей наночастицей, показаны на Рисунке 3b как функция координаты z, где начало отсчета расположено в центре нанокуба. Весьма неожиданно то, что существуют области, в которых момент спин-роторной силы значительно больше, чем давление излучения, что означает, что наночастицы не следуют вдоль силовых линий вектора Пойнтинга. Кроме того, в определенных

положениях один из двух членов может быть отрицательным. Это означало бы вращение, противоположное знаку падающего лазерного пучка. Однако полный момент силы, ощущаемый наночастицей, подчиняется спиновой синхронизации и почти не меняется с координатой ъ. Это полезно с практической точки зрения, поскольку позволяет сообщать одинаковые оптические моменты силы наночасти-цам, расположенным на разной высоте.

|E|(107V/m) I з

ч \ \ \ \ /;////

ШШ:

/ / I 1 \ \ \ \4

-50 0 50 100

z (nm)

Рисунок 3. - (а) Стрелки: Распределение поперечной части распределения вектора Пойнтинга S = (Sx,Sy, 0). Цветовая шкала: нормировка электрического поля. Срез соответствует поперечной плоскости x-y при z=70 нм. Падающая плоская волна с интенсивностью 70 мВт/мкм2. (b) Моменты силы давления излучения (P), спин-роторной силы (Spin) и суммарные моменты силы, испытываемые идеально поглощающими НЧ (а = 0, а = 1). Они были усреднены для каждого z по нескольким круговым кольцам в параллельных плоскостях x-y.

Необычные характеристики поля НКС вблизи диэлектрического на-нокуба можно использовать для реализации функций ЛНЧ по требованию. Здесь предлагается реалистичное устройство ЛНЧ, способное смешивать растворы, растворенные в воде, в субволновом объеме, т. е. 'наномиксер', использующее для смешивания жидкости плазмонные Аи НЧ. Идея состоит в том, чтобы передать оптический момент силы раствору инертных дипольных наночастиц, растворенных в воде, окружающей нанокуб. Тогда их динамика будет результатом взаимодействия между броуновской силой (Гв) и силой вязкого сопротивления (Го), возникающими в жидкости, и оптической силы. В результате жидкость будет получать импульс, и будут стимулироваться процессы диффузии растворенных в ней хими-

ческих примесей в субволновом объеме.

Вблизи своего плазмонного резонанса Аи НЧ подходящего размера удовлетворяют условию а' = 0 при больших а . Удобно, что в исследованном диапазоне размеров это условие выполняется между 500-540 нм, близко к длине волны выбранного зеленого лазера (Л = 532 нм). Далее рассматривается ансамбль Аи НЧ с Я = 40 нм. Затем изучается их динамика в оптическом нановихре во временной области, путем решения уравнений движения Ланжевена.

В момент времени ? = 0 Аи НЧ равномерно распределяются вокруг диэлектрического нанокуба. Примеры их траекторий показаны на Рисунке 4а. Включение пучка с ЛКП с интенсивностью около 50-80 мВт/мкм2 приводит к спиралевидному движению Аи НЧ в ближней зоне. В дальней зоне преобладает радиальная составляющая вектора Пойнтинга, поэтому передается только линейный импульс. Чтобы оценить эффект преобразования СУМ-ОУМ, можно определить эффективный радиус гт, чтобы ограничить область, в которой оптический вихрь заметно влияет на динамику. Из результатов на Рисунке 4а можно сделать вывод, что гт приблизительно равен Л/2 (где Л = Л0/пт), и, таким образом, момент силы, передаваемый вихрем, можно ощутить только в субволновой области. В настоящее время невозможно достичь такого уменьшенного масштаба с использованием луча, возбуждаемого из дальнего поля, т.е. сфокусированного пучка Гаусса или Бесселя. Это первое предложение, в котором обеспечиваются оптические нановихри, созданные в простой реализуемой установке, которая не требует плаз-монных наноантенн с потерями, волноводных мод [16] или сложных хиральных мета-элементов [17].

Затем было выполнено гидродинамическое моделирование, чтобы визуализировать эффект передачи импульса воде в вихре. Поля скорости и напряжения, возникающие в результате «нано-перемешивания», показаны на Рисунке 4Ь-с. Были сделаны два разных снимка экрана в течение полного времени моделирования 260 мкс, при этом предполагалось, что исходная жидкость находится в состоянии покоя. При начальной скорости гу = 0 Аи НЧ все больше ускоряются под

действием НКС оптического нановихря. В результате импульс передается жидкости, и устанавливается вихреобразное течение, как показано на линиях скорости на Рисунке 4Ь-с.

Список литературы диссертационного исследования кандидат наук Канос Валеро Адриа, 2022 год

Список публикаций

1. A. Canos Valero, D. Kislov, E. A. Gurvitz и др., "Nanovortex-Driven all-dielectric optical diffusion boosting and sorting concept for lab-on-a-chip platforms", Advanced Science, т. 7, № 11, с. 1 903 049, 2020

2. A. Canos Valero, E. A. Gurvitz, F. A. Benimetskiy и др., "Theory, observation, and ultrafast response of the hybrid anapole regime in light scattering", Laser & Photonics Reviews, т. 15, № 10, с. 2 100 114, 2021

3. A. C. Valero, H. Shamkhi, A. S. Kupriianov и др., "Superscattering Empowered by Bound States in the Continuum", arXivpreprint arXiv:2105.13119, 2021

4. A. V. Kuznetsov, A. C. Valero, M. Tarkhov и др., "Transparent hybrid anapole metasurfaces with negligible electromagnetic coupling for phase engineering", Nanophotonics, т. 10, № 17, с. 4385—4398, 2021

5. E. Zanganeh, M. Song, A. C. Valero и др., "Nonradiating sources for efficient wireless power transfer", Nanophotonics, т. 10, № 17, с. 4399—4408, 2021

6. H. K. Shamkhi, A. Sayanskiy, A. C. Valero и др., "Transparency and perfect absorption of all-dielectric resonant metasurfaces governed by the transverse Kerker effect", Physical Review Materials, т. 3, № 8, с. 085 201, 2019

7. A. C. Valero, H. K. Shamkhi, A. S. Kupriianov и др., "Reaching the superscattering regime withBIC physics", в Journal of Physics: Conference Series, IOP Publishing, т. 2172, 2022, с. 012 003

8. A. C. Valero, "Exceptional points of all-dielectric nanoresonators", в Journal of Physics: Conference Series, IOP Publishing, т. 2015, 2021, с. 012 028

9. A. C. Valero, E. A. Gurvitz, A. Miroshnichenko и др., "Hybrid anapoles: Near-zero scattering States driven by high order modal interference", в AIP Conference Proceedings, AIP Publishing LLC, т. 2300, 2020, с. 020 015

10. A. C. Valero, D. Kislov, E. A. Gurvitz h gp., "Spin-locked scattering forces in the near field of high index particles", b AIP Conference Proceedings, AIP Publishing LLC, t. 2300, 2020, c. 020 016

11. A.C. Valero h A. Shalin, "Optically-driven Rotation of Perfectly Absorbing Nanopartic

b Journal of Physics: Conference Series, IOP Publishing, t. 1461,2020, c. 012 021

12. A. C. Valero, E. Gurvitz, A. Miroshnichenko h gp., "Nontrivial invisibility induced by optical hybrid anapole", b Journal of Physics: Conference Series, IOP Publishing, t. 1461,2020, c. 012 020

13. A. Shalin, A. Kuznetsov, V. Bobrovs h gp., "Novel Hybrid anapole state and non-Huygens' transparent metasurfaces", b Journal of Physics: Conference Series, IOP Publishing, t. 2172, 2022, c. 012 001

14. H. Shamkhi, A. C. Valero h A. Shalin, "Effective electromagnetic fields of a particle situated near a substrate", b AIP Conference Proceedings, AIP Publishing LLC, t. 2300, 2020, c. 020115

15. D. Borovkov h A. C. Valero, "On the link between mean square-radii and highorder toroidal moments", b Journal of Physics: Conference Series, IOP Publishing, t. 2015, 2021, c. 012 021

16. A. Kuznetsov h A. C. Valero, "Non-Huygens transparent metasurfaces based on the novel Hybrid anapole state", b Journal of Physics: Conference Series, IOP Publishing, t. 2015, 2021, c. 012 079

17. A. Kuznetsov, A. C. Valero, P. Terekhov h gp., "Various multipole combinations for conical Si particles", b Journal of Physics: Conference Series, IOP Publishing, t. 2015, 2021, c. 012 080

18. A. V. Kuznetsov, A. C. Valero h A. S. Shalin, "Optical properties of a metasurface based on silicon nanocylinders in a hybrid Anapole state", b AIP Conference

Proceedings, AIP Publishing LLC, т. 2300, 2020, с. 020 075

19. H. K. Shamkhi, K. V. Baryshnikova, A. Sayanskiy и др., "Non-Huygens invisible metasurfaces", в Journal of Physics: Conference Series, IOP Publishing, т. 1461, 2020, с. 012156

20. H. Shamkhi, K. Baryshnikova, A. Sayanskiy и др., "Full Transmission through a metasurface beyond Kerker conditions", в XIII международные чтения по квантовой оптике (IWQO-2019), 2019, с. 116—118

Synopsis

General description of the thesis

Context of the thesis

The ability to control light-matter interactions at the nanoscale is of fundamental importance for modern photonic technologies, ranging from advanced energy harvesting or biosensing to quantum communications. For more than a decade, the forefront of research in nanophotonics has been centered around plasmonics. Nanoparticles constituted of noble metals support localized surface plasmon resonances (LSPR) capable of enhancing and storing the electromagnetic field at the nanoscale. This has led to extraordinary opportunities to control light beyond the diffraction limit [1]. However, the field is currently at an impasse; metals suffer from unavoidable Joule losses, which strongly hinder the performance of optical devices. As we shall see during this thesis, losses (whether induced by absorption or radiation) can be a plus depending on the application. Nevertheless, absorption is detrimental in most cases of interest, such as flat optics or optical interconnects [2]. Another important disadvantage corresponds to the incompatibility of noble metals with Complementary Metal-Oxide Semiconductor (CMOS) manufacturing. The integration of electronic and photonic devices on a single chip is critical for the next generation of optical interconnects [3].

Despite the significant attempts made by the community to overcome these bottlenecks, none of the proposed strategies has proven versatile enough so far. Fortunately, a few years back, it became evident that high-index dielectrics and semiconductor nanoparticles, such as Silicon (Si) could also support strong optical resonances. Besides offering a diversity of resonances inaccessible to conventional plasmonic nanopar-ticles, dielectrics overcome the two fundamental problems of their metallic predecessors, namely, they have almost no Joule losses and are CMOS-compatible. Soon, however, researchers realized that dielectric nanoparticles were more than just an advanta-

geous platform for the same old physics; new surprising effects were lying just around the corner to be discovered. Unlike conventional plasmonic particles, isotropic dielectric nanoparticles support a broad class of multipolar resonances of both electric and magnetic type, which can mutually interfere leading to remarkable phenomena such as the Kerker effect [4], nonradiating anapoles [5], [6], spin to orbital angular momentum conversion [7], or magnetic light [8].

Most effects in all-dielectric nanophotonics are described in terms of simplified multipolar models. However, as design complexity grows, cumbersome higher order multipolar contributions must be considered. More importantly, some recently unveiled physical effects require the design of shapes differing significantly from a sphere, e.g. bound states in the continuum [9], spin-Hall effect [10] or enhanced circular dichro-ism [11]. The dynamics of multipolar resonances can then become very complex, and semi-analytical multipolar decompositions are generally employed, which can lead to counterintuitive conclusions, such as apparent mismatches between engineered mul-tipoles and degrees of freedom of the structure [5], [12]. Thus, a more transparent, physically insightful theory is needed.

The functional response of both plasmonic and dielectric nanoantennas is primarily governed by resonances. They are associated to discrete frequencies at which light is efficiently confined within or in the near field of the optical 'resonator', maximizing light-matter interactions. However, in passive structures, reciprocity dictates that any current distribution that can be excited from the far field must also radiate away. Therefore, any resonant mode must decay after some finite lifetime. In addition, in the presence of absorption, the number of photons in elastic scattering is not conserved, leading to extra decay channels for the resonant modes which become particularly relevant in plasmonic structures. Moreover, these two general energy dissipation processes result in a broadening of the peaks in physical observables centered around the resonance frequencies, such as the scattering cross section or the Purcell factor.

In a mathematical setting, the resonant modes correspond to the eigen-modes of the Maxwell operator, supplemented with a set of boundary conditions. To

describe open systems, such as dielectric nanoresonators, radiation boundary conditions are needed. This renders the underlying operator non-Hermitian, in the sense that the eigenmodes of the time-reversed problem are different from the original problem. The same occurs if the system is closed, but the permittivity and/or permeability tensors are non-Hermitian (i.e. s^ = s), something that occurs in the presence of absorption or gain. In these scenarios of interest, the eigenfrequencies of the resonant modes are complex, where the real part indicates the resonance frequency, and the imaginary part is proportional to the losses, both radiative and absorptive. Nowadays, these resonant modes are commonly referred to as quasinormal modes (QNMs), since they no longer obey the energy normalization akin to closed, Hermitian systems. Importantly, several works have already shown that the response of arbitrary electromagnetic resonators can be synthesized as the sum of the excited QNMs [13]. A reduced basis of QNMs in the spectral range of interest is sufficient to deliver an intuitive explanation of a number of physical phenomena [13], in addition to introducing a necessary correction to existent normal mode theories [14].

Aim of the thesis

From the above, it becomes clear that light-matter interactions mediated by dielectric nanoparticles can be naturally described in the framework of non-Hermitian physics. The latter provides a general setting that deals with all wave systems where losses (radiative or absorptive), play an important role. Armed with these new theoretical tools, this work aims at bringing new physics to all-dielectric nanophotonics, based on non-Hermitian modal interference effects. Such interference can occur in diverse parameter spaces, for instance by tuning frequency (Chapters 1,2), or geometry (Chapters 3, 4). With this rationale in mind, the following main objectives were posed:

Main Objectives

• To explore new approaches to control the behavior of light at the nanoscale with the help of dielectric nanoparticles.

• To develop qualitative and quantitative physical models of scattering by such particles and their interaction with the environment, particularly a connection between the excited QNMs and multipolar radiation, as well as models of non-conservative forces arising in multipolar fields.

• To predict new scattering effects with simplified analytical theories, taking into account non-Hermiticity.

• To design realistic devices for the realization of a number of tasks of technological relevance in the field of optics, chemistry and biology.

Scientific statements

As a result of the research conducted within the framework of this thesis, the following statements are presented for the defense:

Scientific Statements

• Spin to orbital angular momentum conversion from a laser beam to the near field of a subwavelength dielectric scatterer results in a non conservative force field capable of imparting torques to absorbing nanoparticles. Due to the spin-curl component, the force is not necessarily parallel to the electromagnetic energy flow.

• The near field non conservative forces arising in the vicinity of a dielectric scatterer can be exploited to design a device capable of mixing solutions dissolved in water within a subwavelength volume, i.e. a 'nanomixer', using plasmonic gold nanoparticles as fluid stirrers. In addition, by exploiting a sign switch in their polarizability near the plasmon resonance, the device can also be used to realize all-optical size sorting of gold nanoparticles.

• In a high-index semiconductor nanoparticle with cylindrical symmetry, illuminated by a normally incident plane wave, four anapoles can be overlapped with just two degrees of freedom, due to the mixed (electric and magnetic) multipolar nature of the underlying quasinormal modes. We are thus able to observe a Hybrid Anapole in scattering in the visible range. The regime breaks down in the transient, where the scattering

signal can be manipulated with the help of an underlying dielectric substrate.

• For a scatterer without spherical symmetry, the single channel limit can be overcome by coupling two resonant Mie modes of different multipolar nature, following the Friedrich-Wintgen mechanism. This new superscattering regime arises as the counterpart of quasi-bound states in the continuum in isolated scatterers.

• In an isolated high-index semiconductor nanoparticle with overlapped electric and magnetic dipole modes, introducing a bianisotropic perturbation in the geometry induces the formation of a pair of exceptional points and a 'Fermi arc' at some given complex frequencies, under a continuous variation of at least two geometrical parameters.

Key scientific achievements

The following points constitute the most relevant achievements presented in this thesis:

1. For the first time, it has been found that SAM-OAM conversion from a circularly-polarized laser beam to a dielectric nanoparticle can induce a non-conservative force field in the vicinity of the latter, not necessarily parallel to the energy flow. They induce spiral motion of absorbing dipolar nanoparticles.

2. The concept of a novel type of an all-optical nanomachine capable of mixing solutions and sorting plasmonic nanoparticles at the nanoscale has been proposed and verified numerically.

3. The theory, design and first observation of a four-fold hybrid anapole in a Si nanocylinder in the visible range.

4. The first analysis of the interaction of hybrid anapoles with a substrate, and the study of their breakdown under pulsed illumination.

5. The discovery, formal explanation, and observation of a new mechanism to achieve the superscattering effect, based on interfering resonances of a dielectric cavity

without spherical symmetry. A new 'super-dipole' resonance is shown to exist, forming by constructive interference of two resonances, in contrast to destructive interference leading to quasi bound states in the continuum.

6. The formulation of a rigorous theory for the design of non-Hermitian singularities (the so-called exceptional points) in a single dielectric nanoparticle based solely on geometry tuning, and the first proof-of-concept numerical model of a realistic cavity displaying the effect.

Scientific novelty

The obtained results are fundamental in nature, and open new paradigms in all-dielectric nanophotonics, with also relevant ramifications in chemistry, biology and quantum optics. The presence of curl-spin forces in the near field of dielectric scatterers might pave the way to new degrees of freedom for optical manipulation. The designed nanomixer could be implemented in a new generation of lab-on-a-chip setups as part of a chain of operations for chemical synthesis or sample treatments. Hybrid anapoles show potential for applications in flat optics and wireless power transfer, and might enable the realization of metadevices with a double functionality: full transparency under continuous illumination and pulse modulation in the transient regime. The robustness of the hybrid anapole in the presence of a dielectric substrate might facilitate their implementation in realistic setups, in contrast with the conventional anapole effect. Although studied in the context of electromagnetic scattering, the existence of super-dipole resonances is a general physical effect akin to any open system supporting two interfering resonances leaking to at least two open channels of the continuum. The possibility to transfer power from several incoming waves to one outgoing channel can pave the way to nanoscale power combiners and enhanced light-matter interactions. Finally, non-Hermitian singularities are the subject of intense research not only in photonics, but also in the field of open quantum systems, and play an important role in developing theories of non-Hermitian topology. The design rules presented here constitute the first

step for their realization and observation at the nanoscale. Furthermore, the numerical results demonstrate their potential applicability as ultrasensitive bulk refractive index sensors, having wide use in biosensing and reaction tracking.

Reliability and validity

The reliability and validity of all the results and conclusions in this thesis are guaranteed by the reproducibility of the numerical simulations and the measurements performed. The validity of the analytical expressions and the numerical simulations is further ensured by a quantitative agreement between theoretical and experimental data.

Approbation of the results

The results were presented in a number of international conferences where the author participated as a speaker. They include METANANO 2019 (Saint Petersburg, Russia), METANANO 2020 (Online), SNAIA 2020 (Paris, France), METANANO 2021 (Online), SNAIA 2021 (Paris, France), CLEO 2022 (San Jose, USA). In addition, they were also presented and discussed in a series of theoretical seminars of the department of physics of ITMO University. This research was supported by Priority 2030 Federal Academic Leadership Program.

Publications

The main results of the thesis have been published in 6 peer-reviewed papers published in journals included in the list of the Higher Attestation Commission of the Russian Federation and indexed by Scopus and Web of Science, as well as 14 peer-reviewed conference proceedings, also indexed by Scopus and Web of Science.

Author contribution

The author played a major role in the conception of the original ideas, the elaboration of research objectives, the mathematical methods and the numerical simulations leading to

the main results (SAM-OAM conversion, dynamics of plasmonic nanoparticles in the optical vortex, theory and design of the hybrid anapole, theory of the super dipole resonance and design rules for exceptional points in dielectric nanoparticles). The author participated actively in the preparation of the manuscripts.

Structure and scope of the thesis

The thesis consists of an introduction, 4 chapters, the conclusion and a list of references. The total volume of the thesis is 235 pages. The bibliography contains 129 references. The work contains 58 figures.

Result overview

Chapter 1 investigates non-conservative forces (NCFs) generated by spin to orbital angular momentum conversion (SAM-OAM conversion), from a laser beam to a subwave-length dielectric scatterer. The interplay between SAM and OAM mediated by a sub-wavelength dielectric nanoparticle is discussed, and its connection with non-conservative forces is elucidated. As a result of an enhanced spin-orbit coupling between the nanoparticle and a circularly polarized plane wave, an optical vortex emerges in the near field. It is demonstrated that the latter boosts non-conservative forces of two different kinds: the more common radiation pressure and the less studied curl-spin force. They induce spiral motion of absorptive gold nanoparticles (Au-NPs) in the visible range. Based on the findings, an all-optical lab-on-a-chip (LOC) platform to realize nanofluid mixing (illustrated in Figure 1a), and particle sorting (illustrated in Figure 1b) is proposed and verified numerically.

Consider a circularly polarized plane wave with the expression Einc (r) = Eo(z) (ex + i<rey), and amplitude E0(z) = E0e-lk0z. < can be interpreted as the "spin" of the electromagnetic wave. We are interested in understanding how AM can be transferred from an incident beam to a scatterer, labeled A. The (time-averaged) AM surface density of the scattered field can be expressed as the cross product of the position vector

Figure 1 - (a) Artistic representation of nanomixing induced by NCFs in an optical vortex. Circularly-polarized light impinges on a dielectric scatterer (a Si nanocube), which generates strong multipolar fields and NCFs on absorbing nanoscatterers, such as Au-NPs near their plasmon resonance. Their rotation transfers momentum to the surrounding fluid, reducing the diffusion length of the species dissolved within it. (b) Optical size sorting of Au-NPs. Gradient forces attract small particles towards the nanocube, while large particles are repelled from it due to a combination of gradient and NCFs. (c) Normalized NCF field produced by a circularly-polarized point magnetic quadrupole in the near field at z=0

r and the scattered energy flow (Ss) [15]:

<J> =

r x<S5)

c

By evaluating Jz it can be seen that (Jz) rc rSs]. Thus, direct AM transfer from the beam to a scatterer can only occur with nonvanishing azimuthal scattered energy flow. Second, (Jz) is a function of the origin of the coordinate system, which renders it a so-called extrinsic quantity. Since SAM is an inherently intrinsic quantity, (Jz) necessarily contains OAM. Thus, SAM-OAM conversion can occur when SSi = 0, i.e. in the presence of an optical vortex.

As a particular example of this process, a dielectric nanocube with ha - 4, emulating Si in the visible spectrum, is investigated (Figure 2). The scatterer is assumed to be submerged in water, and is optimized so that it supports a magnetic quadrupole resonant mode at the green laser wavelength (532 nm), as confirmed by the multipole decomposition in Figure 2b. Left-circularly polarized light was used in all the following results.

Figure 2 - (a) Size optimization of the MQ QNM of the nanocube. The side L of the nanocube is chosen so that the resonant wavelength is in the vicinity of the green laser wavelength (green line, 532 nm in vacuum). Both the ED and MQ QNMs display a linear dependence with L. In water, (hm = 1.33) while the ED significantly blueshifts, the MQ only slightly redshifts. Insets: field distributions of the ED and MQ QNMs. (b) Multipolar decomposition of the scattering cross section for the chosen nanocube (dimensions given in the inset). The scatterer is assumed to be placed on top of a low-index substrate, such as glass. A well-defined MQ resonance is observed at the resonance wavelength of the MQ mode

The NCFs felt by absorbing dipolar nanoparticles in the near field of A are

investigated. For a strongly absorbing nanoparticle, conservative forces are negligible, and NCFs become the dominant term. They are given by

(F> = ^ (^ + Vx(L, >), (2)

where a" is the imaginary part of their polarizability, determined by absorption and radiation losses. NCFs include contributions of the radiation pressure (first term on the rhs), and curl-spin forces (second term). Interestingly, the azimuthal component of the Poynting vector originated from spin-orbit coupling would necessarily induce tangential forces on the particle, and therefore a displacement in the polar direction. To confirm this, analytical expressions for the NCFs can be derived assuming the magnetic quadrupole is the dominant contribution to the field in the vicinity of A:

2 (k4r4 + 3k2r2 + 9) (k,r,6) = 25™^-i6kV-" sin (3)

such an idealized field distribution is displayed in Figure 1. It becomes clear that any particle subject to it will experience spiral motion around the origin.

The calculated Poynting vector distribution of the optimized nanocube in the near-field indeed displays an optical vortex confined in a subwavelength region centered at the scatterer (Figure 3a). The radiation pressure and curl-spin contributions to the orbital torque experienced by a small absorbing nanoparticle are plotted in Figure 3b as a function of the z-coordinate, where the origin is taken at the center of the nanocube. Quite surprisingly, there exist regions where the curl-spin torque is significantly larger than radiation pressure, implying that nanoparticles do not follow along the Poynting vector field lines. In addition, at certain positions, one of the two terms can be negative. This would imply a rotation opposite to the sign of the incident laser. However, the total torque felt by the nanoparticle remains spin-locked and almost does not change with the z-coordinate. This is useful from a practical perspective, since it allows to impart similar optical torques to nanoparticles located at different heights.

The unusual characteristics of the NCF field in the vicinity of the dielec-

v \ \ \ \ \ t / / , ,

mm:

\ WS-

/ / I 1 \ \ \ \N

Figure 3 - (a) Arrows: Distribution of the transverse part of the Poynting vector distribution S = (Sx,Sy, 0). Colorplot: Electric field norm. The slice corresponds a transverse x-y plane at z=70 nm. The incident plane wave has an intensity of 70 mW/um2. (b) radiation pressure (P), spin-curl (Spin) and total torques experienced by perfectly absorbing NPs (a = 0, a = 1). For each z they have been averaged over several circular rings in parallel x-y planes

tric nanocube can be exploited to implement on-demand LOC functionalities. Here, a realistic LOC device is proposed, capable of mixing solutions dissolved in water within a subwavelength volume, i.e. a 'nanomixer', using plasmonic Au-NPs as fluid stirrers. The idea is to transfer an optical torque to a solution of inert, dipolar nanoparticles dissolved in the water surrounding the nanocube. Their dynamics will then be a result of the interplay between Brownian (Fb) and viscous drag (Fd) forces induced in the fluid, together with the optical force. As a result, the fluid will gain momentum, stimulating diffusion processes of chemical admixtures dissolved in it within a subwavelength volume.

Near their plasmon resonance, Au-NPs of a suitable size fulfill the condition a' = 0, with large a". Conveniently, in the range of sizes studied, this condition occurs between 500-540 nm, close to the chosen green laser (A = 532nm). In the following, an ensemble of Au-NPs with R = 40nm is considered. Their dynamics in the optical nanovortex are then studied in the time domain, by solving their Langevin equations of motion.

At time t = 0, the Au-NPs are uniformly spread around the dielectric nanocube. Examples of their trajectories are shown in Figure 4a. Switching on an LCP

beam with intensities about 50-80 mW /im-2 results in a spiral motion of the Au-NPs in the near-field. In the far-field zone, the radial component of the Poynting vector dominates, and thus only linear momentum is transferred. To estimate the effect of SAM-OAM conversion, an effective radius rm can be defined to delimit the area in which the optical vortex appreciably influences the dynamics. From the results in Figure 4a, it can be concluded that rm is approximately A/2 (where A = A0/Hm), and thus the torque imparted by the vortex can only be felt within a subwavelength region. Currently, such a reduced scale cannot be reached using a beam excited from the far-field, e.g. focused Gaussian or Bessel beams. This is the first proposal providing optical nanovortices created in a simple, realizable setup avoiding the need of lossy plasmonic nanoantennas, guided modes [16], or complex chiral meta-elements [17].

Next, fluid dynamics simulations are performed to visualize the effect of the momentum transfer to the water in the vortex. The induced velocity and stress fields arising from the "nanostirring" are displayed in Figure 4b-c. Two different screenshots are taken during a total simulation time of 260is, where the initial fluid is assumed to be at rest. With a starting velocity of rj = 0, the Au-NPs increasingly accelerate under the effect of the NCFs of the optical nanovortex. As a result, momentum is transferred to the fluid and a vortex-like current is established, as demonstrated by the velocity streamlines in Figure 4b-c.

In addition, by exploiting a sign switch of the polarizability near the plas-mon resonance, the suggested device could be utilized to sort by size an ensemble of gold nanoparticles. To see this, the polarizability is plotted as a function of the Au-NP radius. From Figure 5a, it can be seen that a' displays a sign change near the plas-mon resonance. The switch reverses the direction of the radial gradient force on the Au-NPs. At a given incident wavelength, the behavior of the Au nanoparticles can be separated into regions I and II. In region I, a' > 0 and the Au-NPs are brought towards the nanocube. Those with dimensions within region II, however, are repelled from it.

Chapter 2 presents a theoretical study on the physics of the novel Hybrid

Figure 4 - (a)Trajectories followed by Au-NPs in the vicinity of the optical vortex and in the far field. Within a radius of action rm the NCFs of the optical vortex impart strong torques, inducing spiral paths of the Au-NPs in a direction which is locked by the spin of the incident plane wave (here LCP). These can be well appreciated in the enlarged area within rm. Far from the vortex, conventional radiation pressure kicks in and the Au-NPs are simply pushed outwards. (b)-(c) Fluid nanovortex. Screenshots at 150 us and 260 us of a simulation of the fluid dynamics in the laminar regime. Streamlines show the fluid velocity, and the colorplot displays the stress fields. The fluid was initially static at t=0

Figure 5 - Optical size sorting of Au-NPs. (a) a and a for the vacuum wavelength of 532 nm as a function of Au-NP size. The red dashed line shows the border between repulsive (I) and attractive (II) regions. The blue shaded area indicates where Brownian motion dominates. Insets: schemes depicting the radial direction of the optical forces acting on Au-NPs in different regions. Gradient forces attract small particles towards the nanocube, while large particles are repelled from it due to a combination of gradient and NCFs

Anapole, previously unknown non-scattering regimes requiring the simultaneous destructive interference of electric and magnetic quasi-static multipoles with their toroidal counterparts. Surprisingly, only two design parameters are required to superpose four multipole anapoles at the same wavelength. To explain this effect beyond the ED approximation, a solid physical picture has been developed on the basis of quasinormal mode (QNM) theory.

A dielectric nanodisk of radius r, height H and refractive index n - 4 is illuminated by a normally incident, linearly polarized plane wave. In the initial design, the nanodisk is embedded in an homogeneous space with nm = 1 (air). For a limited range of values of kr, H, a strong scattering minimum appears in the spectra (Figure 6a). Remarkably, scattering is an order of magnitude more suppressed than in conventional anapole disks, which are usually considered nonradiating (for a comparison, refer to [5]). This suggests the appearance of a HA, and is confirmed by a multipole decomposition of the scattering cross section (Figure 6b). The results show pronounced scattering dips of the four dominant multipoles, both the electric and magnetic dipoles and quadrupoles.

Following the theoretical results, the HA was designed in a Si nanodisk on top of a glass substrate, as depicted in the artistic representation in Figure 7a. A series of Si nanodisks were fabricated and characterized in the optics range with the help of dark field spectroscopy measurements (Figure 7b). Nanodisks of different radius were measured and compared with the simulations, achieving very good agreement. It was confirmed that the most pronounced scattering dip coincided with the maximal overlap of the anapoles stemming from all the dominant multipoles.

Unlike the conventional electric dipole anapole (EDA), the interpretation of the HA in terms of quasistatic multipoles is quite complex [5]. Namely, each multipolar anapole must be described as the destructive interference of two partial contributions to the cross section, giving a total of 8 terms, which must be tuned in phase and amplitude to achieve the desired regime at a given wavelength. However, as already hinted in the previous section, the multipolar anapoles appear to be linked with each other. Oth-

Figure 6 - (a) Scattering cross section of Si nanodisks in {kr,H} space. The HA emerges as a strong scattering dip, reaching 10 times more efficient scattering suppression than the conventional electric dipole anapole. (b) Multipole decomposition of the scattering cross section of a nanodisk with H=367 nm. The HA occurs in the vicinity of ^=780 nm, where the four dominant multipoles display a dip in their contributions to the cross section. Inset: total Ex component at ^=780 nm. The incident wave is clearly seen to be unperturbed by the scatterer

650 700 750 800 850

A (nm)

Figure 7 - (a) Artistic representation of the fabricated HA nanodisks. The sample is made of amorphous Si (aSi) and deposited on top of a glass substrate. Under incident plane wave illumination, strong near fields are excited within the nanodisk and the wave traverses the sample without being altered by scattering. (b) Comparison between measurements (solid lines) and theory (dashed lines), for different nanocylinders with increasing diameters D. The most pronounced dip is identified for D=251 nm (R= 126 nm), corresponding to the HA nanodisk. (right: SEM micrographies of selected samples). Details on the fabrication, material dispersion and measurements can be found in the Supplementary Information of Ref.[5]

erwise, it would be impossible to design a HA with just two degrees of freedom. Here, with the help of a QNM expansion, a much simpler explanation of the HA regime is elucidated, beautifully describing all the physical aspects of it, and resolving the apparent mismatch between multipoles and degrees of freedom.

In the absence of absorption, the extinction cross section can be estimated with a finite set of QNMs as

^ext * ^'m{ 2 OuM J d3rE^Eu (r)}. (4)

Moreover, o-ext = ^sca. Due to the cylindrical symmetry of the scatterer, it is convenient to use the standard notation for the normal modes of dielectric cavities [18]. Specifically, the QNMs can be separated into Transverse Electric (TE,EZ * 0), and Transverse Magnetic (TM,#Z * 0). The modes are characterized by a set of quantum numbers u,v,w, which determine the number of standing wave maxima in the azimuthal (u), radial (v), and axial (w) directions. From here on, the QNMs will be labeled as TE (TM )uvw.

Three resonant QNMs, the TEm, TMn3 and TE\20, are excited in the visible range. Their field distributions are shown in Figure 8e. On the one hand, the fields of the TE12o QNM resemble that of a conventional EDA. Since w = 0, this QNM can be visualized as a standing wave formed from reflections in the lateral walls of the cavity. It can be interpreted as a Mie-like mode, since it resembles the first resonant dipole mode of a dielectric sphere or an infinitely long dielectric rod [9]. The TM\\3, on the other hand, has no analogue in a canonical geometry. It develops as a result of the finite size of the rod in the z-direction, due to standing waves bouncing from the top and bottom walls. Therefore, it can be thought of as a Fabry-Perot mode [12]. Altogether, changing the height of the cavity will shift strongly the resonant frequency of the Fabry-Perot mode, but will barely affect the Mie mode. To first order, the latter depends solely on the radius of the cavity. Thus, the resonant frequencies of the two mode types can be brought together.

The HA is produced by the overlap of the Fano dips stemming from the

Figure 8 - Modal explanation of the HA effect. (a) QNM reconstruction of the extinction cross section. (b) QNM expansion of MQ scattering at the HA, where 'Bckg' includes all contributions besides TE 120. The dip in scattering is clearly shown to originate from the interference of a resonant partition (TE 120), with the background partition formed by the Born term and the non-resonant QNMs. (c)-(d) Contributions to multipolar scattering of TM113 (red) and TE 120 (blue) for two cylinder heights: H=400 nm (multipolar anapoles not overlapped), and H=367 nm (HA regime). Each QNM radiates as a combination of electric and magnetic multipoles with the same parity. The HA regime appears when the resonant responses of the two QNMs are superposed. Inset of (d): x-z field distributions of TM113 (right) and TE 120 (left). Yellow arrows indicate their symmetry with respect to the x-y plane. The first is odd, while the second is even. (e) QNM reconstruction of the HA internal field

TM113 and TE120 modes. At the dip, both QNMs have a negative contribution to extinction, at first sight contradicting the optical theorem [19].This would be the case if a single QNM with negative extinction could be independently excited by the plane wave. However, the plane wave always excites a combination of QNMs with positive and negative contributions, so that the total sum remains positive and in agreement with the optical theorem. In fact, negative contributions are a signature of modal interference [20], which was found to be at the origin of EDAs.

More insight can be obtained by directly reconstructing the multipolar spectra with QNMs, using the novel formalism developed in Appendix B. Figure 8b displays the result of the expansion for MQ scattering (the remaining multipoles can be reconstructed in the same fashion). For a given multipole, for instance the MQ (denoted by M), its total scattering cross section is proportional to hij IZmMmm |2. Due to this, the QNM-multipole expansion does not produce negative contributions, unlike the decomposition of extinction. This expression includes direct terms of the form | Mm. |2, as well as cross-terms 2Re{Mm Mil}, in addition to a 'Born-like' scattering term (refer

iJ iJ

to Appendix B). The direct terms are Lorentzians centered at the resonant frequencies of each QNM, i.e. u = Re{¿^}, excepting static poles, which only contribute as a background. The sum of the direct and cross-terms gives rise to Fano resonances and multipolar anapoles, as shown in Figure 8b. In the case of the MQ, the contribution of TE120 is resonant and interferes with a background formed by the other QNMs and the Born scattering term. This results in a suppression of scattering in the vicinity of its resonance frequency.

Defying conventional intuition, TM113 and TE120 do not radiate as one multipole, but as a combination of dipoles and quadrupoles (Figure 8c-d). The direct contributions of TM113 result in sharp Lorentzians in the MD and EQ cross sections. Similarly, the ED and MQ multipoles are resonantly enhanced by TE 120. The remaining QNMs act as a background that interferes with the resonant QNMs, resulting in the anapoles. Note that shifting the resonant frequencies of TE120 or TM113 implies a spectral shift of the Lorentzian peaks in a pair of multipoles. This can be appreciated

when shifting them with a variation of the nanodisk height, as shown in Figure 8c-d. Indeed, the TMn3 and TE\20 are odd and even, respectively, with respect to a mirror plane (refer to insets in Figure 2.4d). Thus, the first can only radiate as a multipole(s) with odd symmetry, and viceversa for the second.

The HA is almost unaltered when placing the nanodisk on top of a dielectric substrate, as shown in Figure 9b, where ^sca is calculated for different substrates with lowering contrast with the nanodisk, ns =1.5 (glass), 2, 3, 3.87 (aSi). The HA is unchanged, with the exception of zero contrast (silicon particle over silicon substrate). An increase of ns can be viewed as a decrease in the effective reflectivity of the bottom wall supporting the cavity. In consequence, QNMs of Fabry-Perot type are strongly affected and radiate more energy to the substrate, with the subsequent drop in their Q-factors (Figure 2.5d).

For a simplified 1D cavity, it is found that the resonance condition is fulfilled when

rtop^bot b2 = 1, (5)

where b = exp(1ikwnH) and rtoprbot are the Fresnel reflection coefficients from the cavity-air (top) and the cavity-substrate (bottom) interfaces, respectively. Solving Eq. ??, the quality factor of a QNM with axial index w is calculated as

wn

Qw = -T. (6)

ln(r top^bot)

If both rtop, rbot are close to unity, (i.e. large index contrasts), the energy is well stored within the dielectric cavity, and the opposite occurs for small reflections. This results in a lower Qw. This explains the strong leakage to the substrate of TMn3 (Figure 9c).

In strong contrast with TMu3, TE\20 does not depend on the reflection coefficients in the axial direction, due to its 'Mie-like' nature. Thus, its Q-factor is barely affected by changes in ns. This QNM is responsible for preserving the low scattering regime. As a result, a direct calculation of the multipolar content at low contrasts reveals

(a) (b) (c)

n A (nm)

s

Figure 9 - Behavior of the HA regime for different dielectric substrates. (a) Artistic representation of the HAnanodisk on top of a dielectric substrate with refractive index ns. TM113 and TE120 are represented with different colored arrows. (b) Evolution of the scattering cross section at the HA for different ns. (c) Field distributions of TM113 and TE 120 for two different substrates. (d) Calculated Q-factors of the two resonant QNMs as a function of ns. (e) Multipoles for ns = 3 (log scale). The HA is now primarily dominated by the ED contribution

a drop in the contributions of the MD and EQ multipoles, which were mainly excited by TM113, while the ED remains dominant (Figure 9e).

Furthermore, detailed investigations of the the breakdown of the HA regime in a sharp temporal transient have been performed. Extending earlier works [21], a QNM expansion in the time domain is developed, with the objective of revealing the underlying physics of such transients, focusing in the HA regime. In the time domain under pulsed illumination, the HA regime is shown to display energy spikes and a nontrivial transient evolution for different ns (Figure 10). Their origin is shown to be the coherent interference between the resonant TE 120 , TM113 QNMs and the background (Figure 11). In parallel, the same mechanism produces signal beating of the transient scattered power, opening the possibility to realize ultrafast time modulation of the scattered signal in the transient regime.

Figure 10 - Numerical simulations of the transient response at the HA. (a) Scattered power as a function of time for a the HA nanodisk when excited by a square pulse. (b) Scattered power at the transient for different ns. (c)-(d) Spatiotemporal maps of Ex for ns = 1,3, respectively

Chapter 3 deals with bright scattering regimes. Enhancing scattering by subwavelength objects can be highly beneficial for a number of practical applications, such as energy harvesting [22], sensing or magnetic resonance imaging [23]. There exists a strict bound to the scattering cross section known as the single-channel limit, being defined for a dipole as o-0 = 3^2/2n [24]. The recently observed phenomena of superscattering [24], [25], allows to overcome it, enabling physically small objects to capture energy from an area much larger than their geometrical cross section [26].

(a) (b)

580 590 600 610 620 630 640 650

t(fs)

Figure 11 - QNM reconstruction of the scattered power in the time domain. (a) FDTD simulations of the scattered power. (b) QNM reconstruction with 50 QNMs. (c) Simplified model taking into account only three QNMs: TM113, TE 120, and the most relevant low Q mode

In essence, it is possible to exceed <0 by spectrally overlapping the resonant frequencies of QNMs matched to different multipoles [24]. This can be achieved by carefully engineering the geometrical degrees of freedom, for instance adding layers of different refractive index to a sphere [27].

Here, the superscattering effect is discussed from a completely different perspective; employing non-Hermitian physics. Being inspired by the physics of Bound States in the Continuum (BICs), a new mechanism to reach the superscattering regime is unveiled and demonstrated experimentally. In this process, the widely accepted single channel limit ceases to exist, and the scattering cross section of just a dipole can exceed <0, giving rise to a 'super' dipole resonance.

The approach is based on a non-Hermitian phenomenological model of two QNMs interacting with at least two multipole moments (two 'open' channels of the continuum), implemented with temporal coupled mode theory [28]. Despite its simplicity, this model and formalism allows to describe the Friedrich-Wintgen mechanism of interfering resonances leading to quasi-BICs in isolated cavities [29]. Similar to the other chapter, here scattering by a single resonator is considered, see Figure 12a. The goal is

to understand how the internal resonances of the resonator i.e. the QNMs, redistribute the power from an incident wave to the scattered fields, to determine the rules that maximize the cross section. Remarkably, it is found that a new regime of superscattering naturally arises as a counterpart of quasi-BICs in finite scatterers, (see Figure 12b). While destructive interference of the modes leads to the suppressed dipole scattering and a high Q-factor (quasi-BICs can only scatter through high-order multipoles), constructive interference can significantly enhance it, surprisingly surpassing the allowed limit for its cross section, as proven by the calculations in Figure 13 a. This regime is termed a 'super-dipole' resonance. In contrast with conventional superscattering, the limit is not exceeded by overlapping high order multipolar resonances. Instead, here the power is transferred by the resonant states from the high order multipoles to the low order ones. The effect is unique for scatterers without spherical symmetry.

Figure 12 - Illustration of BIC-inspired superscattering from a subwavelength resonator. (a) Incident plane wave excites the quasi-normal modes of a resonator which scatter outgoing waves. (b) A dielectric cylinder supporting mutually coupled quasi-normal modes is tuned to reach the novel superscattering regime. The mode mixing induced by the coupling can produce either a quasi-BIC (upper left) or a super-dipole mode (upper right)

Based on the new theoretical insight, subwavelength dielectric nanocylin-ders are designed to display both quasi-BICs and super dipoles as a function of the aspect ratio, as shown in Figure 13b. To observe experimentally a super dipole, the results are scaled to the microwave range (Figure 13c). At the super dipole resonant wavelength, the cross section exceeds the limit, while a significant distortion of the incident field

can be appreciated in both simulated and experimental results, both clear signatures of superscattering.

A h/r

Figure 13 - Theory, design, and experimental observation of BIC-inspired superscattering. (a) Analytical two-state model reproducing the main physics of the Friedrich-Wintgen mechanism. By varying two system parameters, the evolution of the scattering cross section of a multipole is calculated and normalized by the single channel limit for a dipole. Near the avoided crossing, the formation of a quasi-BIC is observed. Remarkably, the same process can enhance scattering from a multipole beyond the traditional limit, i.e., superscattering. (b) Q-factors of the QNMs in a dielectric cylinder and regions of enhanced scattering, as a function of a height perturbation, (r cylinder radius). Red indicates super dipole modes. Blue corresponds to pure magnetic quadrupole scattering near the quasi-BIC (point B). (c) Experimental confirmation of a super dipole mode. In the upper plot the numerical simulations are compared for the lossless ceramic material (black curve), and the lossy ceramic (red), as well as the experimental measurements (blue curve). Lower panels: The simulated and measured electric field norms for the two dielectric cylinders

Chapter 4 the existence of non-Hermitian singularities, EPs, have been predicted in a single dielectric nanoparticle, subject solely to geometrical perturbations. To do so, a modal theory evidencing the critical role of vertical symmetry breaking to obtain the EP has been proposed and validated. The latter is general, and tells us that any two resonant modes of a dielectric disk whose eigenfrequencies cross with the aspect ratio and have opposite sign with respect to a mirror reflection perpendicular to the z-axis can be brought to an EP by a continuous perturbation that breaks the mirror plane of the disk, as schematically shown in Figure 14a-c for the example of the electric dipole and magnetic dipole QNMs. For example, as shown in Figure 14c, the cylinder can

be transformed into a truncated cone. This asymmetric perturbation drastically alters the field distributions of the original modes. Its effect can be graphically visualized by decomposing the perturbed field into the sum of the original and the perturbation (Figure 14c). The new modes are of mixed electric and magnetic nature, enabling them to mutually couple.

Figure 14 - Coupling magnetic and electric dipole modes by symmetry breaking. (a) Symmetry operations defining a cylinder of revolution, where C< || z indicates the principal axis of rotation, and az a mirror plane perpendicular to the latter. (b) Symmetry of the ED and MD modes with respect to . (c) Effect of an out-of-plane perturbation on the MD mode, and equivalent decomposition into a sum of ED and MD contributions

For the example of the electric and magnetic dipole modes, the evolution of the eigenfrequencies as a function of two parameters is calculated analytically with

the aid of rigorous non-Hermitian coupled mode theory. The detuning can be set with the bottom radius R, while as confirmed in Figure 15f-g, coupling can be induced by modifying the top-to-bottom ratio r/R. The resonant frequencies and loss rates of the two modes are displayed in Figure 16 as functions of the two parameters (Riemann Surfaces). Starting with a Si nanocylinder with height 100 nm, increasing or decreasing r/R leads to the transition from a crossing of the resonant frequencies to an avoided crossing, a signature of strong coupling. The opposite occurs for the loss rates. Marking the transition from one regime to another, a pair of second order EPs can be found (green dots in Figure 16a-b). They are connected in parameter space by an open arc (highlighted with a dashed line in Figure 16), known as a bulk Fermi arc, along which the resonant frequencies of the two modes are degenerate. Their existence was previously pinpointed in photonic crystals when forcing the coupling between two Floquet modes in momentum space [30]. In contrast, here the latter is obtained via geometry tuning of a single dielectric nanoparticle.

Finally, magnetoelectric EPs are shown to be promising for the detection of small local changes in the refractive index of an aqueous environment (refer to scheme in Figure 17a). The idea is based on tracking the split between the resonant frequencies of the two modes (Figure 17a). The numerical results demonstrate that much larger sensitivities can be obtained for the EP in comparison with the electric and magnetic dipole modes of a nanodisk designed to be overlapped together, forming a diabolic point (DP). This is due to the anomalous square-root dispersion characteristic of EPs, in contrast to the conventional linear one. The results suggest a route towards novel biosensing schemes based on single dielectric nanoparticles tuned near an EP. To further confirm the power laws, Figure 17b displays a log-log plot of Af as a function of the change in the environment refractive index, Anenv. The dashed lines correspond to a linear fit, with slope m. The DP displays a slope of 1, whereas the EP is well fitted with m=0.5.

Figure 15 - (a) Crossing of the real parts of the two resonant modes in the weak coupling regime, before transitioning through an EP (r/R=1). Scheme depicts a cross section of the unperturbed geometry. (b) Avoided crossing of the imaginary parts for the same ratio as (a). (c)-(d) r/R=0.95. Avoided crossing of the real parts in the strong coupling regime, after transitioning through the EP, and crossing of the imaginary parts. The scheme in (c) depicts a cross section of the truncated cone (over-deformed for a better visualization). In all plots, lines of different colors indicate different QNMs. Only when r/R=1 we can associate to each of them a pure electric or magnetic character

Figure 16 - Calculated Riemann surfaces of the two eigenfrequencies in the vicinity of a pair of EPs. To obtain them, two parameters of the structure need to be tuned: (i) the radius of a Si cylinder in the vicinity of the Kerker condition (where the ED and MD modes overlap), (ii) the conicity of the particle (ratio between top and bottom radii). (a) Real part, (b) imaginary part (loss rate). White dashed lines indicate the region in parameter space displaying a Fermi arc. Green dots correspond to the EPs, where both real and imaginary parts coalesce

Figure 17 - Refractive index sensitivity near a magnetoelectric EP. The nanocone dimensions are R=191nm, r/R=0.575, H=100 nm. (a) Split between the resonance frequencies of the two modes when collapsed in an EP or simply brought together by tuning the radius of the nanocylinder, forming a Diabolic Point (DP). The local refractive index is varied from pure water to a solution of ethylene glycol in water (nenv = 1.34), emulating realistic experimental conditions 35. (b) Log-Log plot of (a) and linear fit (dashed lines). The slopes (m) show a linear dependence of the modes at the DP with Anenv, while a square-root dependence (m=0.5), is confirmed for the EP

Publication list

1. A. Canos Valero, D. Kislov, E. A. Gurvitz, et al., "Nanovortex-driven all-dielectric optical diffusion boosting and sorting concept for lab-on-a-chip platforms," Advanced Science, vol. 7, no. 11, p. 1 903 049, 2020

2. A. Canos Valero, E. A. Gurvitz, F. A. Benimetskiy, et al., "Theory, observation, and ultrafast response of the hybrid anapole regime in light scattering," Laser & Photonics Reviews, vol. 15, no. 10, p. 2 100 114, 2021

3. A. C. Valero, H. Shamkhi, A. S. Kupriianov, et al., "Superscattering empowered by bound states in the continuum," arXiv preprint arXiv:2105.13119, 2021

4. A. V. Kuznetsov, A. C. Valero, M. Tarkhov, et al., "Transparent hybrid anapole metasurfaces with negligible electromagnetic coupling for phase engineering," Nanophotonics, vol. 10, no. 17, pp. 4385-4398, 2021

5. E. Zanganeh, M. Song, A. C. Valero, et al., "Nonradiating sources for efficient wireless power transfer," Nanophotonics, vol. 10, no. 17, pp. 4399-4408, 2021

6. H. K. Shamkhi, A. Sayanskiy, A. C. Valero, et al., "Transparency and perfect absorption of all-dielectric resonant metasurfaces governed by the transverse kerker effect," Physical Review Materials, vol. 3, no. 8, p. 085 201, 2019

7. A. C. Valero, H. K. Shamkhi, A. S. Kupriianov, et al., "Reaching the superscattering regime with bic physics," in Journal of Physics: Conference Series, IOP Publishing, vol. 2172, 2022, p. 012 003

8. A. C. Valero, "Exceptional points of all-dielectric nanoresonators," in Journal of Physics: Conference Series, IOP Publishing, vol. 2015, 2021, p. 012 028

9. A. C. Valero, E. A. Gurvitz, A. Miroshnichenko, et al., "Hybrid anapoles: Near-zero scattering states driven by high order modal interference," in AIP Conference Proceedings, AIP Publishing LLC, vol. 2300, 2020, p. 020 015

10. A. C. Valero, D. Kislov, E. A. Gurvitz, et al., "Spin-locked scattering forces in the near field of high index particles," in AIP Conference Proceedings, AIP Publishing LLC, vol. 2300, 2020, p. 020 016

11. A. C. Valero and A. Shalin, "Optically-driven rotation of perfectly absorbing nanoparticles," in Journal of Physics: Conference Series, IOP Publishing, vol. 1461, 2020, p. 012 021

12. A. C. Valero, E. Gurvitz, A. Miroshnichenko, et al., "Nontrivial invisibility induced by optical hybrid anapole," in Journal of Physics: Conference Series, IOP Publishing, vol. 1461, 2020, p. 012 020

13. A. Shalin, A. Kuznetsov, V. Bobrovs, et al., "Novel hybrid anapole state and non-huygens' transparent metasurfaces," in Journal of Physics: Conference Series, IOP Publishing, vol. 2172, 2022, p. 012 001

14. H. Shamkhi, A. C. Valero, and A. Shalin, "Effective electromagnetic fields of a particle situated near a substrate," in AIP Conference Proceedings, AIP Publishing LLC, vol. 2300, 2020, p. 020 115

15. D. Borovkov and A. C. Valero, "On the link between mean square-radii and highorder toroidal moments," in Journal of Physics: Conference Series, IOP Publishing, vol. 2015, 2021, p. 012 021

16. A. Kuznetsov and A. C. Valero, "Non-huygens transparent metasurfaces based on the novel hybrid anapole state," in Journal of Physics: Conference Series, IOP Publishing, vol. 2015, 2021, p. 012 079

17. A. Kuznetsov, A. C. Valero, P. Terekhov, et al., "Various multipole combinations for conical si particles," in Journal of Physics: Conference Series, IOP Publishing, vol. 2015, 2021, p. 012 080

18. A. V. Kuznetsov, A. C. Valero, and A. S. Shalin, "Optical properties of a metasur-face based on silicon nanocylinders in a hybrid anapole state," in AIP Conference Proceedings, AIP Publishing LLC, vol. 2300, 2020, p. 020 075

19. H. K. Shamkhi, K. V. Baryshnikova, A. Sayanskiy, et al., "Non-huygens invisible metasurfaces," in Journal of Physics: Conference Series, IOP Publishing, vol. 1461,2020, p. 012156

20. H. Shamkhi, K. Baryshnikova, A. Sayanskiy, et al., "Full transmission through a metasurface beyond kerker conditions," in XIII международные чтения по квантовой оптике (IWQO-2019), 2019, pp. 116-118

Introduction

The ability to control light-matter interactions at the nanoscale is of fundamental importance for modern photonic technologies, ranging from advanced energy harvesting or biosensing to quantum communications. For more than a decade, the forefront of research in nanophotonics has been centered around plasmonics. Nanoparticles constituted of noble metals support localized surface plasmon resonances (LSPR) capable of enhancing and storing the electromagnetic field at the nanoscale. This has led to extraordinary opportunities to control light beyond the diffraction limit [1]. However, the field is currently at an impasse; metals suffer from unavoidable Joule losses, which strongly hinder the performance of optical devices. As we shall see during this thesis, losses (whether induced by absorption or radiation) can be a plus depending on the application. Nevertheless, absorption is detrimental in most cases of interest, such as flat optics or optical interconnects [2]. Another important disadvantage corresponds to the incompatibility of noble metals with Complementary Metal-Oxide Semiconductor (CMOS) manufacturing. The integration of electronic and photonic devices on a single chip is critical for the next generation of optical interconnects [3].

Despite the significant attempts made by the community to overcome these bottlenecks, none of the proposed strategies has proven versatile enough so far. Fortunately, a few years back, it became evident that high-index dielectrics and semiconductor nanoparticles, such as Silicon (Si) could also support strong optical resonances. Besides offering a diversity of resonances inaccessible to conventional plasmonic nanopar-ticles, dielectrics overcome the two fundamental problems of their metallic predecessors, namely, they have almost no Joule losses and are CMOS-compatible. Soon, however, researchers realized that dielectric nanoparticles were more than just an advantageous platform for the same old physics; new surprising effects were lying just around the corner to be discovered. Unlike conventional plasmonic particles, isotropic dielectric nanoparticles support a broad class of multipolar resonances of both electric and magnetic type, which can mutually interfere leading to remarkable phenomena such as

the Kerker effect [4], nonradiating anapoles [5], [6], spin to orbital angular momentum conversion [7], or magnetic light [8].

Most effects in all-dielectric nanophotonics are described in terms of simplified multipolar models. However, as design complexity grows, cumbersome higher order multipolar contributions must be considered. More importantly, some recently unveiled physical effects require the design of shapes differing significantly from a sphere, e.g. bound states in the continuum [9], spin-Hall effect [10] or enhanced circular dichro-ism [11]. The dynamics of multipolar resonances can then become very complex, and semi-analytical multipolar decompositions are generally employed, which can lead to counterintuitive conclusions, such as apparent mismatches between engineered mul-tipoles and degrees of freedom of the structure [5], [12]. Thus, a more transparent, physically insightful theory is needed.

The functional response of both plasmonic and dielectric nanoantennas is primarily governed by resonances. They are associated to discrete frequencies at which light is efficiently confined within or in the near field of the optical 'resonator', maximizing light-matter interactions. However, in passive structures, reciprocity dictates that any current distribution that can be excited from the far field must also radiate away. Therefore, any resonant mode must decay after some finite lifetime. In addition, in the presence of absorption, the number of photons in elastic scattering is not conserved, leading to extra decay channels for the resonant modes which become particularly relevant in plasmonic structures. Moreover, these two general energy dissipation processes result in a broadening of the peaks in physical observables centered around the resonance frequencies, such as the scattering cross section or the Purcell factor.

In a mathematical setting, the resonant modes correspond to the eigen-modes of the Maxwell operator, supplemented with a set of boundary conditions. To describe open systems, such as dielectric nanoresonators, radiation boundary conditions are needed. This renders the underlying operator non-Hermitian, in the sense that the eigenmodes of the time-reversed problem are different from the original problem. The same occurs if the system is closed, but the permittivity and/or permeability tensors

are non-Hermitian (i.e. s^ = s), something that occurs in the presence of absorption or gain. In these scenarios of interest, the eigenfrequencies of the resonant modes are complex, where the real part indicates the resonance frequency, and the imaginary part is proportional to the losses, both radiative and absorptive. Nowadays, these resonant modes are commonly referred to as quasinormal modes (QNMs), since they no longer obey the energy normalization akin to closed, Hermitian systems. Importantly, several works have already shown that the response of arbitrary electromagnetic resonators can be synthesized as the sum of the excited QNMs [13]. A reduced basis of QNMs in the spectral range of interest is sufficient to deliver an intuitive explanation of a number of physical phenomena [13], in addition to introducing a necessary correction to existent normal mode theories [14].

From the above, it becomes clear that light-matter interactions mediated by dielectric nanoparticles can be naturally described in the framework of non-Hermitian physics. The latter provides a general setting that deals with all wave systems where losses (radiative or absorptive), play an important role. Armed with these new theoretical tools, this work aims at bringing new physics to all-dielectric nanophotonics, based on non-Hermitian modal interference effects. Such interference can occur in diverse parameter spaces, for instance by tuning frequency (Chapters 1,2), or geometry (Chapters 3, 4). With this rationale in mind, the following objectives were posed:

Main Objectives

• To explore new approaches to control the behavior of light at the nanoscale with the help of dielectric nanoparticles.

• To develop qualitative and quantitative physical models of scattering by such particles and their interaction with the environment, particularly a connection between the excited QNMs and multipolar radiation, as well as models of non-conservative forces arising in multipolar fields.

• To predict new scattering effects with simplified analytical theories, taking into account non-Hermiticity.

• To design realistic devices for the realization of a number of tasks of technological relevance in the field of optics, chemistry and biology.

As a result of the research conducted within the framework of this thesis, the following statements are presented for the defense:

Statements for the defense

• Spin to orbital angular momentum conversion from a laser beam to the near field of a subwavelength dielectric scatterer results in a non conservative force field capable of imparting torques to absorbing nanoparticles. Due to the spin-curl component, the force is not necessarily parallel to the electromagnetic energy flow.

• The near field non conservative forces arising in the vicinity of a dielectric scatterer can be exploited to design a device capable of mixing solutions dissolved in water within a subwavelength volume, i.e. a 'nanomixer', using plasmonic gold nanoparticles as fluid stirrers. In addition, by exploiting a sign switch in their polarizability near the plasmon resonance, the device can also be used to realize all-optical size sorting of gold nanoparticles.

• In a high-index semiconductor nanoparticle with cylindrical symmetry, illuminated by a normally incident plane wave, four anapoles can be overlapped with just two degrees of freedom, due to the mixed (electric and magnetic) multipolar nature of the underlying quasinormal modes. We are thus able to observe a Hybrid Anapole in scattering in the visible range. The regime breaks down in the transient, where the scattering signal can be manipulated with the help of an underlying dielectric substrate.

• For a scatterer without spherical symmetry, the single channel limit can be overcome by coupling two resonant Mie modes of different multipolar nature, following the Friedrich-Wintgen mechanism. This new superscattering regime arises as the counterpart of quasi-BICs in isolated scatterers.

• In an isolated high-index semiconductor nanoparticle with overlapped electric and magnetic dipole modes, introducing a bianisotropic perturbation induces the formation of a pair of exceptional points and a 'Fermi arc' in parameter space.

1 | Non-conservative forces for sorting and mixing at

the nanoscale

1.1 Motivation and relevance

In this chapter, the interplay between spin and orbital angular momentum of a subwave-length dielectric nanoparticle is discussed, and its connection with non-conservative forces is elucidated. As a result of an enhanced spin-orbit coupling between the nanopar-ticle and a circularly polarized plane wave, an optical vortex emerges in the near field. I demonstrate that the latter boosts non-conservative forces of two different kinds: the more common radiation pressure and the less studied curl-spin force. They induce spiral motion of absorptive Au nanoparticles in the visible range. Based on the findings, an all-optical lab-on-a-chip (LOC) platform to realize nanofluid mixing and particle sorting is proposed and verified numerically.

Optofluidics, i.e. the manipulation of fluids by optical means, plays a paramount role in the operation of novel LOC devices [49]. Controlling fluid flow in microchannels is essential in a number of multidisciplinary applications, ranging from analytical chemistry to biotechnology [50], [51]. Within the scales of hundreds of nanometers, optical force fields can be designed with the help of plasmonic nanores-onators [52], [53]. Unfortunately, noble metals suffer from detrimental ohmic losses. In microfluidics, this is particularly problematic, since strong heating can start unde-sired reactions or rapidly degrade biological tissue. Thus, due to their low intrinsic absorption, all-dielectric nanostructures seem to be a natural alternative for implementing photonic concepts to control fluids at the micro and nanoscales. Surprisingly, such hybrid all-dielectric/microfluidic platforms remain largely unexplored. It is evident that the multipolar modes supported by a high-index nanoparticle can potentially enhance the electromagnetic field in the vicinity of the latter, inducing strong gradient forces [54]. Here, however, we are interested in a different aspect, namely, on

the role played by non-conservative forces (NCFs). Earlier works have mostly considered them as parasitic contributions to the total force, which can prevent trapping in optical tweezers. Due to their detrimental effect in optical trapping schemes, most of the research on NCFs is centered around minimizing their influence. Here, we depart from this trend, and investigate the force fields that arise from pure NCFs generated by a properly designed subwavelength dielectric scatterer. It is demonstrated that a pure magnetic multipole, when excited by an incident light with nonzero spin angular momentum (SAM), gives rise to spin-orbit coupling phenomena and the emergence of optical vortices. Within its radius of influence, unusual NCF fields arise, which are not necessarily parallel to the energy flow. The dynamics of absorbing nanoparticles (e.g. Au nanoparticles near their plasmon resonance), can be strongly affected by such NCFs. Specifically, NCFs impart them an extrinsic optical torque, making them rotate within the nanovortex. Based on the findings, a LOC-compatible device capable of performing fluid nanomixing and size sorting ofAu nanoparticles is designed.

1.2 Multipolar Spin-Orbit Conversion

Let us start by considering a circularly polarized plane wave with the expression Einc (r) = Eo(z) (ex + i<rey), and amplitude E0(z) = E0e-lk0z. < can be interpreted as the "spin" of the electromagnetic wave. This can be understood by evaluating the total angular momentum (AM), using expressions for a beam in the paraxial limit [55]. The projection of AM to the z-axis (the direction of propagation) is given by (in a host environment with refractive index nm):

Jz(r) = 20 (E^ • LEinc) [ + 2c <L,>|z. (1.1)

The rhs of Eq.1.1 is general for any paraxial wave, and consists of two terms. The first one is the orbital angular momentum (OAM) of the beam, generated by projecting the field to the OAM operator L = -ir xV. The second one quantifies the SAM flux density [55], as (L^ > = s0nm E x E*/4 io. A straightforward evaluation of Eq.1.1 shows

that Einc has no OAM, but nonzero SAM flux density, so that J™c = <rI0/o>, where I0 is the incident intensity. The analogy with spin in quantum mechanics now becomes clear: for a circularly polarized plane wave, < is proportional to the projection of AM to the z-axis.

We are interested in understanding how AM can be transferred from an input wave to a scatterer, labeled A. The (time-averaged) AM surface density of the scattered field can be expressed as the cross product of the position vector r and the scattered energy flow (Ss) [15]:

(J) = ^. (1.2)

c

Eq.1.2 deserves a few comments. First, by evaluating Jz we see that (Jz) <x rSs^]. Thus, direct AM transfer from the beam to a scatterer can only occur with nonvanishing az-imuthal scattered energy flow. Second, (Jz) is a function of the origin of the coordinate system, which renders it a so-called extrinsic quantity. Since SAM is an inherently intrinsic quantity, (Jz) necessarily contains OAM. Thus, SAM-OAM conversion can occur when S^ = 0. We model the scatterer as a current distribution j(r) localized in a volume Vs and characterized by source multipoles, as described in Appendix A. For reasons made clear in the following paragraphs, we particularize our treatment to the magnetic quadrupole (MQ). However, an analogous procedure can be used to describe AM transfer for more complex distributions. The incident field exerts work on the current jackson, according to

^ = ^1 E*nc • j(rM3r

(1.3)

recalling the results in Appendix A, the first order current distribution radiating as an MQ has the form (in tensor notation):

Ji(r) « -1 Siyady (MfiadfiS(r)) . (1.4)

Plugging Einc and Eq.1.4 in Eq.1.3, the delta function removes the integral, so that Pext

simplifies to

Pext =--Re {i(TMzx + MZy} . (1.5)

From Eq.1.5, it appears that only the Mzx and Mzy components of the MQ tensor can interact directly with the incident field. Next, we assume the scatterer belongs to a Dnh point group, with n > 4. This is typically the case in state-of-the-art all-dielectric nanostructures implemented in experiment [5], [56], [57]. Then, we can write Mzx = aei01 and Mzy = aei02, where a = |Mzx |. The phase difference between the two components is subject to the constraint 02 - 01 = ±^/2, as dictated by Einc. Inserting the previous assumptions, Eq.1.5 yields:

Re {i<Mzx + Mzy} = a (< cos (01 + n/2) + cos (02)). (1.6)

As a result, non-vanishing contributions to Pext have the form:

Mzy = i(Mzx. (1.7)

The Poynting vector radiated by this current distribution can be calculated in a number of ways. One simple approach is to use the Green's function formalism in Appendix A to recover the field at any point outside the smallest spherical volume enclosing the scatterer. A similar approach can be used to find the magnetic field. Combining the two results, (and taking the origin as the center of the scatterer) we find that the Poynting vector has a nonzero azimuthal component, with the following explicit form:

3 |Mzx|2 (9 + 3r2k2 + r4k4) 2

S0 = <F (0,r) = ^tJ -r°7-- cos (0)2 sin (0). (1.8)

v lo^cjuo k0r'

Eq.1.8is expressed in spherical coordinates with origin at the scatterer center of mass, so that 0 is the polar angle, and r is the radial distance. It confirms that the direct excitation of a magnetic multipole by a circularly polarized incident wave indeed causes SAM-OAM conversion. This conversion has an observable consequence: the azimuthal energy flow is associated with the formation of an optical nanovortex, which is confined

Figure 1.1 - (a) Artistic representation of nanomixing induced by NCFs in an optical vortex. Circularly-polarized light impinges on a dielectric scatterer (a Si nanocube), which generates strong multipolar fields and NCFs on absorbing nanoscatterers, such as Au-NPs near their plasmon resonance. Their rotation transfers momentum to the surrounding fluid, reducing the diffusion length of the species dissolved within it. (b) Optical size sorting of Au-NPs. Gradient forces attract small particles towards the nanocube, while large particles are repelled from it due to a combination of gradient and NCFs. (c) Normalized NCF field produced by a circularly-polarized point magnetic quadrupole in the near field at z=0

in the near field (see Figure 1.2b).

The origin of this confinement can be grasped by a direct inspection of S0 and Jz. The two quantities scale with |Mzx |2, while their values drop following an inverse power law with the radial distance from the scatterer. Taking the limit of the radial terms in F(0, r) for r << A, we find that

9

Fi(r ) = k^,

3

F2(r) = k^,

and

/ \ k F3(r) = 73.

gain importance with an increasing distance r from the scatterer. The first two are near-field quantities, while the third is intermediate-field. Most importantly, no contribution survives in the far-field. Therefore, no vorticity can be felt by an observer located in the far-field. Interestingly, higher order multipoles can feature even more strongly confined vortices, since their fields are more concentrated in the vicinity of the scatterer.

1.3 Non-conservative forces in the optical vortex

SAM-OAM conversion has important implications in the forces felt by nanoparticles in the near and intermediate-fields, as we show in the following. First, consider a spherical nanoparticle, labeled B, with radius R, fulfilling the condition R << A, so that its optical response can be well described with an effective electric dipole peff (dipole approximation, DA). The latter is induced by the background field E^ (r) felt by the nanoparticle as peff = aE^ (dependences on space, wavevector and geometry are omitted for brevity). a can be obtained analytically by comparing the fields generated by

peff with those from exact Mie theory [58]:

a(m,kR) = i a\(m,kR).

(1.9)

a\(m, kR) is the first electric (TM) Mie coefficient, and m = ^sb(m)/n, where sb(m) is the permittivity dispersion of B.

Within the DA, the optical force felt by the nanoparticle is [59]:

(Fo >

(1.10)

Conservative Forces

Non Conservative Forces

In Eq.1.10 we further neglected self-induced back-action [59] (that is, the effect of the dipole fields on the background field). Conservative forces simply follow the intensity gradient, and are not connected to AM. However, Eq.1.10 features two types of non-conservative forces related, respectively, to the Poynting vector and the curl of the local SAM density. The first one is well-known; it is often referred to as radiation pressure. The second one is less common, usually negligible, and was only recently brought to to the attention of the optics community [60]; from now on it will be called curl-spin force.

Assume now a hypothetical scenario where a" >> a', i.e. the case of a strongly absorbing nanoparticle. B would then be mostly affected by non-conservative forces. If B is located close to the scatterer A, the former will feel a combination of the incident field Einc and the scattered field from A, EA, so that the quantities entering Eq.1.10 are associated with the field E = Einc + E A. If the two objects are located in free space, the dominant contribution to the non-conservative force will be the z-component proportional to I0. Thus, B will simply be pushed in the -ez direction. However, in a realistic scenario both A and B would be located on top of a substrate, typically SiO2 (glass) or Al2O3 (sapphire). The z-component of the force would be canceled out by the reaction

forces from the substrate 1. Then, the in-plane components of the Poynting vector play an important role. If A is excited at a resonance, in the near-field, EA > Einc, and so in a first approach we neglect the incident field and study solely the effect of A. Via the Green's function approach, we calculate the field and the non-conservative force with the last two terms in Eq.1.10. This yields a non-zero azimuthal component:

, 2 (k4r4 + 3k2r2 + 9) F<(k, r, 0) = 25^<E2--^-sin 0 (1.11)

However, by comparing Eq.1.11 with the expression of S0 in Eq.1.8, there seems to be an apparent contradiction. Intuitively, one would expect F< to be proportional to S0. Strikingly, this is not the case, as can be seen by regarding the force at z = 0 (0 = n/2), displayed in Figure 1.1c. In this case, S0 = 0 according to Eq.1.8, but the azimuthal force is maximal. The explanation lies in the curl-spin force. It turns out that the latter has an important contribution to the optical force induced by a multipole field, and can even become the only non-conservative term acting on a dipolar particle. This is one of the main results of this chapter.

The curl-spin force is associated with a non-uniform spatial distribution of the SAM flux density in the near-field. Interestingly, a straightforward calculation of (L^) of the MQ in the x-y plane yields exactly zero. This is in agreement with earlier works quantifying the SAM flux density in the far field zone for an arbitrary scatterer [55]. However, Vx (L^) is not zero, since it involves spatial derivatives over the radial and polar components of (L^), which do not vanish.

In general, when B is placed at an arbitrary point in the near-field of A, the NCFs are a combination of radiation pressure (FP) and curl-spin (Fs) contributions. Remarkably, the azimuthal nature of the NCFs will make B rotate with respect to A. This is due to an orbital torque produced by the NCFs, directed towards <ez,

xThe effect of VanDer Waals interactions between B and the substrate can be safely neglected since the latter are proportional to the strength of the fields reflected from the substrate [61], which are generally small for small index contrasts between the substrate and the surrounding environment.

rz = r ±((FP )(p + (F^ Note that the sign of the torque can be immediately switched upon a change of the incident polarization. From this realization, it becomes clear that NCFs can potentially provide the means to control the rotation of absorbing nanoparti-cles in the near field of a multipolar scatterer. In particular, as we show in the following, NCFs arising from dielectric scatterers can be used to realize all-optical fluid mixing and sorting of nanoparticles at the nanoscale. In the next section, I propose a realistic design enabling these exciting functionalities.

1.4 Semianalytical design of a dielectric nanocube

To obtain a pure multipolar response emulating the theory from the previous sections, I take advantage of the rich variety of resonant modes (QNMs) supported by dielectric nanoparticles. Specifically, I investigate the response of a dielectric nanocube with nA ^ 4, emulating Si in the visible spectrum. The illumination scheme is the same as discussed earlier: a CP plane wave propagating towards -ez. As an example, I choose left-circularly polarized (LCP, < = 1).

First, the nanocube is embedded in vacuum, and the eigenspectra of the QNMs that are compatible with Einc is obtained numerically. The internal fields of the two most relevant ones are shown in Figure 1.2a, with their internal fields displayed in the inset. They can be straightforwardly identified as an ED and an MQ mode, respectively. It is convenient to select a QNM with high Q-factor in order to have a well-defined multipolar response. For that reason, the MQ QNM is well-suited for our purposes. Indeed, the multipolar spectrum at real wavelengths shows a strong MQ peak, with high signal-to-noise ratio with respect to other contributions to the scattering cross section (Figure 1.2b).

Emulating realistic experimental conditions, the nanocube is placed on top of a glass substrate and embedded in water with n = 1.33 (inset of Figure 1.2b). As a

(a) (b)

150 160 170 180 190 200 400 500 600 700 800

L(nm) A,(nm)

Figure 1.2 - (a) Size optimization of the MQ QNM of the nanocube. The side L of the nanocube is chosen so that the resonant wavelength is in the vicinity of the green laser wavelength (green line, 532 nm in vacuum). Both the ED and MQ QNMs display a linear dependence with L. In water, (nm = 1.33) while the ED significantly blueshifts, the MQ only slightly redshifts. Insets: field distributions of the ED and MQ QNMs. (b) Multipolar decomposition of the scattering cross section for the chosen nanocube (dimensions given in the inset). The scatterer is assumed to be placed on top of a low-index substrate, such as glass. A well-defined MQ resonance is observed at the resonance wavelength of the MQ mode

result, there is a drop in the Q-factor and a slight red-shift in the resonant wavelength of the MQ mode. The MQ mode can then be brought to a suitable laser wavelength, for instance a green laser, by size engineering of the nanocube (Figure 1.2a).

1.5 Spin-locked optical torques

In agreement with the theory, the Poynting vector distribution in the near-field displays an optical vortex confined in a subwavelength region centered at the scatterer (Figure 1.3a). The radiation pressure and curl-spin contributions to the orbital torque experienced by a small absorbing nanoparticle are plotted in Figure 1.3b as a function of the z-coordinate, where the origin is taken at the center of the nanocube. The calculation is performed by averaging the torques over several circular rings on parallel planes, (placed perpendicularly to the incident k-vector). As could be expected from the discussion in section 1.3, nanoparticles whose centroids are positioned at different heights "feel" torques with varying combinations of the curl-spin and radiation pressure contri-

butions 2. Surprisingly, at certain positions, one of the two terms can be negative. This would imply a rotation opposite to the sign of the incident laser. However, the total torque felt by the nanoparticle remains spin-locked and almost does not change with the z-coordinate. This is useful from a practical perspective, since it allows to impart similar optical torques to nanoparticles located at different heights.

(a) (b)

, \ \ \ \

|E| (107 V/m)

3

\ \ \ \

0 50

z (nm)

Figure 1.3 - (a) Arrows: Distribution of the transverse part of the Poynting vector distribution S = (Sx, Sy, 0). Colorplot: Electric field norm. The slice corresponds a transverse x-y plane at z=70 nm. The incident plane wave has an intensity of 70 mW/^m2. (b) radiation pressure (P), spin-curl (Spin) and total torques experienced by perfectly absorbing NPs (a = 0, a = 1). For each z they have been averaged over several circular rings in parallel x-y planes

Summarizing, in this section we have performed numerical simulations demonstrating the existence of strongly confined (subwavelength) optical torques with respect to the propagation direction of the incident laser. They are mediated by the MQ fields of a resonant dielectric nanocube, and feature both radiation pressure and curl-spin contributions. In particular, the results show how the optical torque is nonzero even in the absence of radiation pressure. Furthermore, we reveal that curl-spin and radiation pressure contributions can induce torques with opposite sign. Despite this unusual effect, the total torque remains locked by the spin of the incident wave.

2The attentive reader will notice that rf = 0 at z = 0, contrarily to what one might initially expect from Eq.1.8. To understand this, we recall that the Poynting vector entering in Eq.1.10 must also include the incident field, neglected in the analysis of section 1.4. In our case, the latter contribution displaces the zero to a negative z-coordinate.

1.6 Design of an all-optical nanomixer

We now evaluate the applicability of the optical nanovortex as a new nanomixing method for microfluidics. The idea is to transfer an optical torque to a solution of inert, dipolar nanoparticles dissolved in the water surrounding the nanocube. Their dynamics will then be a result of the interplay between Brownian (Fb) and viscous drag (F^) forces induced in the fluid, together with the optical force. As a result, the fluid will gain momentum, stimulating diffusion processes of chemical admixtures dissolved in it within a subwavelength volume.

Au nanoparticles (Au-NPs) are the natural choice to act as mixing mediators, since they are chemically inert, and rarely interact with biological media. Their use is generalized in microfluidics [62], [63].

In our case, gradient forces must be strongly limited to avoid Au-NPs to be attracted towards the nanocube. As shown earlier, within the EDA, the ratio (Fnc) /(F0) will be maximal when a' = 0. For high enough ratios, NCFs will drive their dynamics.

To evaluate the polarizability of Au-NPs we utilize the Mie formulae given in Eq.1.9, together with the permittivity of Au, calculated as

Eq.1.12 includes a size correction of the bulk dispersion sau (m) [64], by taking into account the Drude correction due to the limitation of the electron mean free path in small metallic particles [65]. wpl, yb,vp are the plasma frequency, the damping constant from the free electron path and the Fermi velocity.

The real and imaginary parts of the polarizability are shown in Figure 1.4 for different R. It can be clearly appreciated how, near the plasmon resonance, Au-NPs with sizes R > 35nm can fulfill the condition a' = 0, with large a". Conveniently, in the

(1.12)

(1.13)

Figure 1.4 - (a) Real (a) and (b) imaginary (a) parts of the polarizability of Au-NPs of different radius as a function of wavelength in vacuum. Near the plasmon resonance, a is enhanced while a can be fully suppressed. This effect conveniently takes place in the vicinity of the green laser wavelength

range of sizes studied, this condition occurs between 500-540 nm, close to the chosen green laser (A = 532nm). In the following, we consider Au-NPs with R = 40nm.

Having selected the ideal size of the 'nanostirrers', we now predict their dynamics in the optical vortex. To simplify the problem, we assume equilibrium has been established in z, and study solely a two dimensional problem in a cross section along the x - y plane. The Langevin equation of motion can be written as:

<Fnc > + FB + Fd = Mj r j,

where my is the mass of the jth Au-NP and f is the acceleration. While the analysis is restricted to 2D, the expression for the viscosity of water is modified as to include the additional drag introduced by the nanocube walls and the glass substrate. In consequence, the Brownian and drag forces assume the forms:

^ 12 nkBT-ri FB = $ -y —dj,zj)

FD = -6nRjl(dj,zj) - tj.

(1.14)

(1.15)

j(dj, Zj) is the viscosity tensor in the fluid, which varies with the distance between the center of the j th Au-NP and the nearest wall of the nanocube dj, and the distance to the

substrate Zj. An approach to numerically estimate ¡2(dj, Zj) can be found in Appendix F $ is a dimensionless vector function of randomly distributed numbers with zero mean, T is the temperature of the system (assumed to be at ambient temperature), ks is Boltz-mann's constant and rc « 3ns [66] is the momentum relaxation time for Au-NPs in water.

Figure 1.5 - (a)Trajectories followed by Au-NPs in the vicinity of the optical vortex and in the far field. Within a radius of action rm the NCFs of the optical vortex impart strong torques, inducing spiral paths of the Au-NPs in a direction which is locked by the spin of the incident plane wave (here LCP). These can be well appreciated in the enlarged area within rm. Far from the vortex, conventional radiation pressure kicks in and the Au-NPs are simply pushed outwards. (b)-(c) Fluid nanovortex. Screenshots at 150 ¡s and 260 ¡s of a simulation of the fluid dynamics in the laminar regime. Streamlines show the fluid velocity, and the colorplot displays the stress fields. The fluid was initially static at t=0

Eq.1.6 is solved numerically after retrieving the optical force distribution from FEM simulations. The equations are nonlinear ordinary differential equations (ODEs) which require a spatial distribution of the Au-NPs at time t = 0 as an initial

(a)

b)

condition. For this, the Au-NPs are uniformly spread around the dielectric nanocube. Examples of their trajectories are shown in Figure 1.5a. Switching on an LCP beam with intensities about 50-80 mW¡m-2 results in a spiral motion of the Au-NPs in the near-field. In the far-field zone, the radial component of the Poynting vector dominates, and thus only linear momentum is transferred. To estimate the effect of SAM-OAM conversion, an effective radius rm can be defined to delimit the area in which the optical vortex appreciably influences the dynamics. From the results in Figure 1.5a, it can be concluded that rm is approximately A/2 (where A = A0/nm), and thus the torque imparted by the vortex can only be felt within a subwavelength region. Currently, such a reduced scale cannot be reached using a beam excited from the far-field, e.g. focused Gaussian or Bessel beams. This is the first proposal providing optical nanovortices created in a simple, realizable setup avoiding the need of lossy plasmonic nanoanten-nas, guided modes [16], or complex chiral meta-elements [17]. They are a promising strategy for on-a-chip SAM-OAM conversion to mediate light-matter interactions at the nanoscale (e.g., controlled optical emission from quantum dots, superresolution, [67], [68] and nano-object manipulation [69], [70]. Quantitative estimations of the enhancement of linear and angular velocity of the Au-NPs with respect to pure Brownian motion can be found in Ref. [7], attached in proof to this thesis.

Finally, fluid dynamics simulations are performed to visualize the effect of the momentum transfer to the water in the vortex. The simulations are implemented in COMSOL Multiphysics. At a given time, the model couples the governing equations of the Au-NPs and the fluid:

• r j and r j for every Au-NP are obtained by coupling Eq.1.6 with the Navier-Stokes equation and the mass balance for the fluid [71].

• The pressure and velocity of water are obtained from the Navier-Stokes equation.

At such reduced scales, the flow is laminar (i.e. Reynolds numbers much smaller than 1000, so that shearing forces strongly dominate inertial ones). This con-

siderably simplifies the Navier-Stokes equation 3 [71]. Open boundary conditions are imposed to emulate a large fluid domain.

The induced velocity and stress fields arising from the "nanostirring" are displayed in Figure 1.5b-c. Two different screenshots during a total simulation time of 260js, where the initial fluid is assumed to be at rest. With a starting velocity of r j = 0, the Au-NPs increasingly accelerate under the effect of the NCFs of the optical nanovortex. As a result, momentum is transferred to the fluid and a vortex-like current is established, as demonstrated by the velocity streamlines in Figure 1.5b-c.

1.7 Size sorting of Au-NPs

Here, it is shown that the proposed setup can also be used to sort by size an ensemble of Au-NPs. To see this, the polarizability is plotted as a function of the Au-NP radius. From Figure 1.6a, it can be seen that a' displays a sign change near the plasmon resonance. The switch reverses the direction of the radial gradient force on the Au-NPs. At a given incident wavelength, we can split the behavior of the Au nanoparticles into regions I and II. In region I, a' > 0 and the Au-NPs are brought towards the nanocube. Those with dimensions within region II, however, are repelled from it. Note that in region I there is a competition between Brownian, gradient, and NCFs. A more detailed study of their role is necessary to determine the viability of the sorting method.

Devices based on the effect described here can pave the way towards all-optical size separation of Au-NPs on-chip. It might prove beneficial in a plethora of relevant processes, such as the regulation of biological cell uptake rates, [72], [73] toxicity control [74], and plasmonics [75].

3Refer also to Appendix F for more details.

(a)

x Id-32

(b)

x 10

-2

15 nm 50 nm

10 20 30 40 50 60 70 Rp(nm)

Figure 1.6 - Optical size sorting of Au-NPs. (a) a and a for the vacuum wavelength of 532 nm as a function of Au-NP size. The red dashed line shows the border between repulsive (I) and attractive (II) regions. The blue shaded area indicates where Brownian motion dominates. Insets: schemes depicting the radial direction of the optical forces acting on Au-NPs in different regions. Gradient forces attract small particles towards the nanocube, while large particles are repelled from it due to a combination of gradient and NCFs

In this chapter, NCFs arising from SAM-OAM conversion have been investigated. They are shown to be a natural consequence of the multipolar fields produced by a dielectric scatterer. As a result of SAM-OAM conversion, an optical vortex arises in the near-field. In stark contrast with conventional intuition, small dipolar nanoparticles experience a NCF that is not parallel to the direction of the energy flow. This behavior is due to strong spin-curl forces, attributed to spatial inhomogeneities in the SAM flux density of a multipole field. The NCFs impart orbital torques on the nanoparticles, which in turn can be used to transfer momentum to a surrounding fluid. Based on the theoretical findings, a realistic LOC device capable of mixing solutions dissolved in water within a subwavelength volume, i.e. a 'nanomixer', using plasmonic Au-NPs as fluid stirrers, is proposed and validated numerically. In addition, by exploiting a sign switch of the polarizability near the plasmon resonance, the suggested device could be utilized to sort by size an ensemble of Au-NPs.

1.8 Conclusion

2 | Non-Hermitian description of the Hybrid Anapole

Regime

2.1 Motivation and relevance

Non-radiating sources are localized current distributions that do not radiate to the far-field. They are a subject of great interest in electrodynamics, dating back to descriptions of electronic configurations of stable atoms [76]. Anapoles, ('without poles' in Greek) are one of the most elementary nonradiating current distributions observed in nanophotonics. Conventionally, the latter is understood as arising from the destructive interference between a quasistatic electric dipole moment pq and an electric toroidal dipole moment Tq. The two moments possess the same far-field pattern, and so, when out of phase, can cancel completely. This results in a significant drop of the scattering cross section, but nonzero internal fields [6]. Nowadays, anapoles have been observed in a number of plasmonic and all-dielectric structures from the microwave to the optics range [77], having found application in metamaterials, lasing or second and third harmonic generation [56].

Rigorously speaking, anapoles cannot be fully described by considering only pq and Tq. Higher order mean square radii T^n) also come into play. In fact, as shown in Appendix A, all pq, Tq, Tqn) are terms of increasing order of a Taylor series of the exact electric dipole moment p observed from the far-field. Formally, the anapole condition corresponds to p = 0. It is a special case of a general nonradiating current distribution as discussed by Devaney and Wolf in their seminal work [78], where they elaborate the sufficient condition for any charge-current to be nonradiating. However, as I show in Appendix E, it is worth mentioning that the explanation with pqand Tq becomes exact and coincides with that of Devaney and Wolf in the limit when the source is much smaller than the wavelength (i.e. kr < 1, where r is the radius of the smallest spherical shell surrounding the source).

In all-dielectric nanophotonics, anapoles are supported by resonators of relatively simple shape, such as spheres or disks [6] under normally incident plane wave illumination. In the first case, their appearance can be predicted with Mie theory as zeros of the electric dipole coefficient a\ [6]. This notion was later extended to other multipoles [79]. In a sphere of radius r and refractive index n embedded in an homogeneous medium and illuminated by a normally incident plane wave, a zero or anapole of a given electric or magnetic multipole of order I can be found, respectively, as a solution to the secular equations [79]:

n^ (kr (nkr) - (kr (nkr) = 0, n^(nkr(kr) - ^(nkr(kr) = 0,

(2.1) (2.2)

where ^(z) is a Ricatti-Bessel function of order I [79] and ^(z) is the first order derivative of (z) with respect to its argument.

Figure 2.1 - (a) Trajectories of the multipole anapoles of the ED, MD, MQ and EQ as a function of kr and n. The blue-shaded area indicates the region with kr < 2 (b) Partial contributions to the scattering cross section of the first four multipoles for a dielectric sphere with n = 3.87, shown by the black-dashed line in (a)

Refs.[79], [80] studied the possibility to overlap the ED and MD anapoles

to form hybridanapoles (HAs). These unconventional regimes should present a number of peculiarities previously unobserved, such as helicity singularities [80], and storage of both electric and magnetic energy in the absence of far-field radiation. In Fig.2.1(a), I go a step further and study the trajectories in the {kr, w}-space not only of the dipolar anapoles, but also the quadrupolar ones. I restrict the analysis to nanospheres with kr < 2, in order for them to be subwavelength, i.e. r/A << 1, but featuring significant contributions from the quadrupoles to scattering. Interestingly, it can be seen that HAs with mixed ED-MD character only appear for kr >> 2, at the crossings between two solid lines. However, quadrupole anapoles almost overlap with their dipolar counterparts in a broad region of the parameter space for kr < 2. To confirm this, the partial contributions of each multipole to the scattering cross section are calculated as [81] = 2/(kr)2(2£ + 1)|(a, h)£|2. They are plotted in Figure 2.1b for the nanosphere with the parameters of the dashed black line in Figure 2.1a.

The nanosphere studied indeed shows the presence of MD and EQ anapoles in close proximity to each other, manifesting as sharp dips in the multipolar scattering spectrum. Isolated ED and MD anapoles can also be appreciated. Unfortunately, the observation of the (quasi) HA is hindered by strong ED scattering. Further calculations (not shown) confirm this is the case for all the nanospheres studied. Doubly degenerate dipolar HAs in microspheres are also 'hidden away' by high order multipolar scattering, as pointed out in Refs. [79], [80].

From the above, it is evident that additional degrees of freedom are necessary to observe the scattering signature of HAs experimentally. In the next section, it is demonstrated that dielectric nanoresonators with broken spherical symmetry, such as disks, support not only doubly, but quadruply degenerate HAs, allowing for their direct experimental verification.

2.2 Design and observation of the HA regime

A dielectric nanodisk of radius r, height H and refractive index n ^ 4 is illuminated by a normally incident, linearly polarized plane wave. In the initial design, the nanodisk is embedded in an homogeneous space with nm = 1 (air). For a limited range of values of kr, H, a strong scattering minimum appears in the spectra (Figure 2.2a). Remarkably, scattering is an order of magnitude more suppressed than in conventional anapole disks, which are usually considered nonradiating (for a comparison, refer to [5]). This suggests the appearance of a HA. This first intuition is confirmed by a multipole decomposition of the scattering cross section, performed using Eqs.A.27-A.30 in Appendix A (Figure 2.2b). In particular, a nanodisk with r = 126 nm and H =367 nm is investigated, so that the effect can be observed in the near-IR part of the visible range. The results show pronounced scattering dips of the four dominant multipoles, both the electric and magnetic dipoles and quadrupoles.

In Ref.[5], the HA was confirmed experimentally. Following the theoretical results, the HA was designed in a Si nanodisk on top of a glass substrate, as depicted in the artistic representation in Figure 2.3a. A series of Si nanodisks were fabricated and characterized in the optics range with the help of dark field spectroscopy measurements (Figure 2.3b). Nanodisks of different radius were measured and compared with the simulations, achieving very good agreement. It was confirmed that the most pronounced scattering dip coincided with the maximal overlap of the anapoles stemming from all the dominant multipoles.

(a) (b)

log cr (a.u.)

H (nm) A (nm)

Figure 2.2 - (a) Scattering cross section of Si nanodisks in {kr, H} space. The HA emerges as a strong scattering dip, reaching 10 times more efficient scattering suppression than the conventional electric dipole anapole. (b) Multipole decomposition of the scattering cross section of a nanodisk with H=367 nm. The HA occurs in the vicinity of ^=780 nm, where the four dominant multipoles display a dip in their contributions to the cross section. Inset: total Ex component at ^=780 nm. The incident wave is clearly seen to be unperturbed by the scatterer

650 700 750 800 850

A (nm)

Figure 2.3 - (a) Artistic representation of the fabricated HA nanodisks. The sample is made of amorphous Si (aSi) and deposited on top of a glass substrate. Under incident plane wave illumination, strong near fields are excited within the nanodisk and the wave traverses the sample without being altered by scattering. (b) Comparison between measurements (solid lines) and theory (dashed lines), for different nanocylinders with increasing diameters D. The most pronounced dip is identified for D=251 nm (R= 126 nm), corresponding to the HA nanodisk. (right: SEM micrographies of selected samples). Details on the fabrication, material dispersion and measurements can be found in the Supplementary Information of Ref.[5]

2.3 QNM description of the HA regime

Unlike the conventional electric dipole anapole (EDA), the interpretation of the HA in terms of quasistatic multipoles is quite complex [5]. Namely, each multipolar anapole must be described as the destructive interference of two partial contributions to the cross section, giving a total of 8 terms, which must be tuned in phase and amplitude to achieve the desired regime at a given wavelength. However, as already hinted in the previous section, the multipolar anapoles appear to be linked with each other. Otherwise, it would be impossible to design a HA with just two degrees of freedom. Here, with the help of a QNM expansion, a much simpler explanation of the HA regime is elucidated, beautifully describing all the physical aspects of it, and resolving the apparent mismatch between multipoles and degrees of freedom.

To simplify the problem, the frequency dispersion of Si is neglected, setting n = 3.87, corresponding to amorphous Si (aSi) in the near-IR part of the visible range 1. As derived in Appendix A, the scattered field within the cavity can be expanded with a finite set of QNMs:

Es(w; r) * ^ avMEi(r), (2.3)

v

with the av (w) given by Eq.??. All fields are normalized with Eq.??. Using the eigen-solver incorporated in COMSOL Multiphysics, 50 QNMs of the nanocylinder are calculated around 780 nm. The extinction cross section can then be estimated with

^ext * WT^m {2 av (w)J d3rE\ EM (r)}. (2.4)

In the absence of absorption, o-ext = ^sca. Due to the cylindrical symmetry of the scatterer, it is convenient to use the standard notation for the normal modes of dielectric cavities [18]. Specifically, far from avoided crossings (discussed in the next chapter), the QNMs can be classified into Transverse Electric (TE,Ez * 0), and

xThe value chosen is not arbitrary: it corresponds to the refractive index of aSi of the real experimental samples at the HA wavelength, c.a. 780 nm.

Transverse Magnetic (TM,Hz * 0). The modes are characterized by a set of quantum numbers uvw, which determine the number of standing wave maxima in the azimuthal (u), radial (v), and axial (w) directions. From here on, the QNMs will be labeled as

TE (TM )u

luvw-

Figure 2.4 - Modal explanation of the HA effect. (a) QNM reconstruction of the extinction cross section. (b) QNM expansion of MQ scattering at the HA, where 'Bckg' includes all contributions besides TE120. The dip in scattering is clearly shown to originate from the interference of a resonant partition (TE 120), with the background partition formed by the Born term and the non-resonant QNMs. (c)-(d) Contributions to multipolar scattering of TM113 (red) and TE 120 (blue) for two cylinder heights: H=400 nm (multipolar anapoles not overlapped), and H=367 nm (HA regime). Each QNM radiates as a combination of electric and magnetic multipoles with the same parity. The HA regime appears when the resonant responses of the two QNMs are superposed. Inset of (d): x-z field distributions of TM113 (right) and TE 120 (left). Yellow arrows indicate their symmetry with respect to the x-y plane. The first is odd, while the second is even. (e) QNM reconstruction of the HA internal field

Each term entering the sum in the rhs of Eq.2.4 can be interpreted as the contributions of a QNM ¡1 to extinction. The QNM reconstruction of <ext is displayed

in the first panel of Figure 2.4. The results are validated by comparing the sum in Eq.2.4 with standard frequency domain calculations. The resonant frequencies of the remaining QNMs used for the reconstruction lie outside the visible range. For clarity in the results, their contributions are summed together in a "background" term, which will be shown later to be of great importance to understand the scattering suppression.

Three resonant QNMs, the TEm, TM113 and TE120, are excited in the visible range. Their field distributions are shown in Figure 2.4e. On the one hand, the fields of the TE120 QNM resemble that of a conventional EDA. Since w = 0, this QNM can be visualized as a standing wave formed from reflections in the lateral walls of the cavity. It can be interpreted as a Mie-like mode, since it resembles the first resonant dipole mode of a dielectric sphere or an infinitely long dielectric rod [9]. The TM113, on the other hand, has no analogue in a canonical geometry. It develops as a result of the finite size of the rod in the z-direction, due to standing waves bouncing from the top and bottom walls. Therefore, it can be thought of as a Fabry-Perot mode [12]. Altogether, changing the height of the cavity will shift strongly the resonant frequency of the Fabry-Perot mode, but will barely affect the Mie mode. To first order, the latter depends solely on the radius of the cavity. Thus, the resonant frequencies of the two mode types can be brought together. In the limit of infinitely large Q-factor (no radiation losses), analytical expressions for the resonant frequencies can be derived [18]. It is found that two generic modes of the form TEuv0 and TMu'v' w coincide when the aspect ratio fulfills the condition:

r \ 2 xlv - ^2'v'

2 ,2 ' (2.5)

n) n2 w'2

where xuv is the vth root of the uth Bessel function, so that Ju (xuv) = 0, and yuv is the vth root of the first derivative of the uth Bessel function, so that J'u (yuv) = 0. Eq.1.5 is a formal confirmation of the previous physical considerations. Note that, in order for both modes to be able to overlap, xuv > yu'v'. This is fulfilled in the particular case of the TM113 and TE12o modes, since xi2 = 3.83 and yu = 1.84.

Interestingly, all three resonant QNMs display a Fano-like response. The

HA is produced by the overlap of the Fano dips stemming from the TM113 and TE120 modes. At the dip, both QNMs have a negative contribution to extinction, at first sight contradicting the optical theorem [19].This would be the case if a single QNM with negative extinction could be independently excited by the plane wave. However, the plane wave always excites a combination of QNMs with positive and negative contributions, so that the total sum remains positive and in agreement with the optical theorem. In fact, negative contributions are a signature of modal interference [20], which was found to be at the origin of EDAs.

The above analysis has shown that the HA appears when the resonant frequencies of two particular QNMs with negative extinction and Fano-like response are positioned near each other. However, it does not answer why those QNMs in particular are interfering, with what are they interfering, nor what is their connection with the dips observed in the multipolar spectra in Figure 2.2b. More insight can be obtained by directly reconstructing the multipolar spectra with QNMs, using the formalism developed in Appendix B.

Figure 2.4b displays the result of the expansion for MQ scattering (the remaining multipoles can be reconstructed in the same fashion). For a given multipole, for instance the MQ (denoted by M), its total scattering cross section is proportional to Hz/|Zm Mm|2. Due to this, the QNM-multipole expansion does not produce negative

J lj

contributions, unlike Eq.2.4. This expression includes direct terms of the form | Mm |2,

l/

as well as cross-terms 2Re{M^M".}. In addition, the Born scattering term must also

l/ l/

be included in general (refer to Appendix B). The direct terms are Lorentzians centered at the resonant frequencies of each QNM, i.e. u = Re {¿3M }. The sum of the direct and cross-terms gives rise to Fano resonances and multipolar anapoles, as shown in Figure 2.4b. In the case of the MQ, the contribution of TE120 is resonant and interferes with a background formed by the other QNMs and the Born scattering term. This results in a suppression of scattering in the vicinity of its resonance frequency.

Defying conventional intuition, TM113 and TE120 do not radiate as one multipole, but as a combination of dipoles and quadrupoles (Figure 2.4c-d). The direct

contributions of TM113 result in sharp Lorentzians in the MD and EQ cross sections. Similarly, the ED and MQ multipoles are resonantly enhanced by TE120. The remaining QNMs act as a background that interferes with the resonant QNMs, resulting in the anapoles. Note that shifting the resonant frequencies of TE120 or TM113 implies a spectral shift of the Lorentzian peaks in a pair of multipoles. This can be appreciated when shifting them with a variation of the nanodisk height, as shown in Figure 2.4c-d. The scattering response of the background QNMs is not strongly modified by geometrical perturbations. Thus, the resonant QNMs represent the true physical entities that are being engineered to achieve the HA regime. This is consistent with the number of degrees of freedom of the structure: while height H (or H/r), must be modified to bring close to each other the resonant frequencies of the two QNMs, the remaining degree of freedom (r), is used to bring both resonant frequencies to the desired spectral range (in our case the visible).

The fact that each QNM radiates as a combination of multipoles is no coincidence, but a direct consequence of the cylindrical symmetry. Since the nanodisk has a mirror symmetry in the x-y plane, all the QNMs are either even or odd with respect to reflection. Indeed, the TM113 and TE120 are odd and even, respectively (refer to insets in Figure 2.4d). Thus, the first can only radiate as a multipole(s) with odd symmetry, and viceversa for the second. Under normally incident plane wave illumination, if their scatterer has no in-plane bianisotropy, only magnetic and electric multipoles with opposite mirror symmetry can be excited [82], (e.g. px and my). Furthermore, mirror symmetry is reversed with multipolar order [83]. Consequently, if the QNM in question radiates as a combination of multipoles, it can only radiate as a sum of multipoles of increasing order and alternate electric or magnetic nature (i.e. if even, ED, MQ, EO...; if odd, MD, EQ, MO... ).

The QNM expansion approach provides a simple, general explanation of multipolar anapoles in particles of arbitrary shape. Using this formalism, all the features of the newly found HA regime can be quantitatively explained with a minimal-istic model. Remarkably, tuning one QNM allows modifying two resonant multipole

moments simultaneously. This peculiar feature allows the overlap of four multipole anapoles within the same spectral range, forming a HA.

2.4 Model describing the interaction with a dielectric substrate

The HA regime is almost unaltered when placing the nanodisk on top of a dielectric substrate, as shown in Figure 2.5b, where <sca is calculated for different substrates with lowering contrast with the nanodisk, ns =1.5 (glass), 2, 3, 3.87 (aSi). This unusual behavior is unlike that of conventional EDAs. The HA regime is barely unchanged in amplitude or spectral position, with the exception of zero contrast (silicon particle over silicon substrate). Interestingly, however, the multipolar content varies significantly. This seems to suggest an underlying mechanism "protecting" the HA from the substrate.

A simplified physical picture can be drawn after inspecting the evolution of TMn3 and TE120 with ns (Figure 2.5c). An increase of ns can be viewed as a decrease in the effective reflectivity of the bottom wall supporting the cavity [34]. In consequence, QNMs of Fabry-Perot type are strongly affected and radiate more energy to the substrate, with the subsequent drop in their quality factors (Figure 2.5d). Their resonances are damped and visually vanish for small contrasts (refer to Figure 2.5b).

More insight can be obtained by regarding the exact solution of the dispersion of a 1D Fabry Perot QNM. This is equivalent to neglecting the standing waves in the azimuthal and radial directions. Following a similar set of steps as in Ref.[84], (see also Supplementary Material of Ref.[5]), it can be found that the resonance condition is fulfilled when