Resonance

<p></p> <p>Preface xi</p> <p>Acknowledgments xiii</p> <p>1 State phase, rules, and theorems 1</p> <p>1.1 Introduction 1</p> <p>1.2 Finite, semi-infinite, and infinite systems 2</p> <p>1.3 State and resonance 2</p> <p>1.4 State phase 3</p> <p>1.4.1 State phase shift 4</p> <p>1.4.2 General state phase 5</p> <p>1.4.3 Discrete final states 6</p> <p>1.4.4 Bulk state phase shift 7</p> <p>1.4.5 Comments 7</p> <p>1.5 General rules and theorems 8</p> <p>1.5.1 Eigenfunction continuity rules 8</p> <p>1.5.2 General theorems 9</p> <p>1.6 Outlook 9</p> <p>References 10</p> <p>2 Photonic open loops 11</p> <p>2.1 Introduction 11</p> <p>2.2 Open loops 12</p> <p>2.2.1 Open-loop basic elements 12</p> <p>2.2.2 One finite open loop 17</p> <p>2.2.3 Two finite open loops 20</p> <p>2.2.4 N finite open loops 25</p> <p>2.3 Comb systems 30</p> <p>2.3.1 A finite comb system with two teeth and three teeth at its two ends 31</p> <p>2.3.2 A finite comb system with N teeth at M equidistant interface points 35</p> <p>2.3.3 Long-lived resonances: comb with two teeth per port 38</p> <p>2.4 Outlook 51</p> <p>References 52</p> <p>3 One photonic closed loop 53</p> <p>3.1 Introduction 53</p> <p>3.2 One closed loop and stubs 53</p> <p>3.2.1 Basic closed-loop elements 53</p> <p>3.2.2 One closed loop L and one stub L3 55</p> <p>3.2.3 One closed loop L and several stubs 70</p> <p>3.3 Simultaneous cross transmissions and disentanglement 79</p> <p>3.4 Outlook 79</p> <p>Acknowledgments 79</p> <p>References 79</p> <p>4 Two photonic closed loops 81</p> <p>4.1 Introduction 81</p> <p>4.2 Two tangent closed loops 81</p> <p>4.2.1 General results 81</p> <p>4.2.2 Two identical tangent loops L 83</p> <p>4.2.3 Two tangent closed loops L1 and L2 86</p> <p>4.3 Two tangent closed loops and stubs 89</p> <p>4.3.1 Two closed loops L1 and L2 and two stubs L1/4 and L2/4 89</p> <p>4.3.2 Two closed loops L +δ and L −δ and two stubs L/4 +δ/4 and L/4 −δ/4 93</p> <p>4.3.3 Two closed loops L +δ1 and L− δ1 and two stubs L/2+ δ2 and L/2 −δ2 96</p> <p>4.4 Outlook 98</p> <p>References 99</p> <p>5 Photonic two-port closed loop 101</p> <p>5.1 Introduction 101</p> <p>5.2 Closed-loop states 102</p> <p>5.3 Final system states 104</p> <p>5.3.1 BIC and SIBIC states 104</p> <p>5.3.2 Bulk state phase shift and state densities 105</p> <p>5.4 Transmission 108</p> <p>5.4.1 The transmission coefficient and the hybrid long-lived resonances 108</p> <p>5.4.2 Transmission phase and phase time 111</p> <p>5.5 States and transmission 113</p> <p>5.6 Stub hybrid resonances 115</p> <p>5.6.1 A general system 115</p> <p>5.6.2 Identical stubs L3 = L4 117</p> <p>5.7 Cross-transmission 125</p> <p>5.7.1 Cross-transmissions for any symmetric two-port system 125</p> <p>5.7.2 Cross-transmissions for the two-port closed loop 125</p> <p>5.7.3 Stub improved cross-transmissions 129</p> <p>5.8 Outlook 129</p> <p>Acknowledgments 130</p> <p>References 130</p> <p>6 Photonic spheres 133</p> <p>6.1 Introduction 133</p> <p>6.2 States of a two interface point sphere 134</p> <p>6.3 N closed loops: two tangent interface points and one port 135</p> <p>6.3.1 BIC and SIBIC states 136</p> <p>6.3.2 Long-lived transmission resonances in function of N 136</p> <p>6.4 Long-lived transmission resonances for N = 2: one port 137</p> <p>6.4.1 Two closed loops of length L 137</p> <p>6.4.2 Two closed loops L1 = L +δ, L2 = L −δ and stubs L1/4, L2/4 138</p> <p>6.4.3 Two closed loops (4 different parts): one L/4 stub 142</p> <p>6.4.4 Two closed loops (4 different parts): stubs L/4 and L/8 144</p> <p>6.5 N closed loops: two tangent interface points and two ports 145</p> <p>6.5.1 BIC and quasi-SIBIC states 147</p> <p>6.5.2 Long-lived transmission resonances as functions of N 148</p> <p>6.6 Long-lived transmission resonances for N = 2: two ports 149</p> <p>6.7 Outlook 152</p> <p>References 152</p> <p>7 Photonic triangular pyramid 153</p> <p>7.1 Introduction 153</p> <p>7.2 Triangular pyramid states 154</p> <p>7.2.1 Response function elements 154</p> <p>7.2.2 States of the pyramid 155</p> <p>7.3 The pyramid with two leads: one port 156</p> <p>7.3.1 BIC and SIBIC states 156</p> <p>7.3.2 Transmission, transmission phase, and state phase shift 157</p> <p>7.3.3 The one port long-lived resonances 160</p> <p>7.4 The pyramid with two leads: two ports 167</p> <p>7.4.1 BIC and SIBIC states 169</p> <p>7.4.2 Transmission, transmission phase, and state phase shift 169</p> <p>7.4.3 The two-port long-lived resonances 174</p> <p>7.5 Outlook 181</p> <p>References 182</p> <p>8 Square pyramid: one summit port 183</p> <p>8.1 Introduction 183</p> <p>8.2 Square pyramid states 184</p> <p>8.2.1 Response function: interface elements 185</p> <p>8.2.2 States of the square pyramid 186</p> <p>8.3 The pyramid with one summit port 187</p> <p>8.3.1 BIC and SIBIC states 188</p> <p>8.3.2 Transmission, state phase shift, and VADOS 188</p> <p>8.3.3 Transmission phase and phase time 191</p> <p>8.3.4 The long-lived resonances 193</p> <p>8.4 State and particle shifts, collapses, and sensing 199</p> <p>8.5 Outlook 202</p> <p>References 203</p> <p>9 Generalizations 205</p> <p>9.1 Introduction 205</p> <p>9.2 Other simple system geometries 206</p> <p>9.2.1 Open-loop chains 206</p> <p>9.2.2 Closed-loop chains 206</p> <p>9.2.3 Hexagons 206</p> <p>9.2.4 Squares and cubes 207</p> <p>9.2.5 Square pyramids 207</p> <p>9.2.6 Many-port systems 207</p> <p>9.3 Other generalizations 207</p> <p>9.3.1 Composite material systems 207</p> <p>9.3.2 Long-wavelength electronic waves 208</p> <p>9.3.3 Plasmonic waves 209</p> <p>9.3.4 Elastic waves 209</p> <p>9.3.5 Polaritonic waves 209</p> <p>9.3.6 Spin waves 209</p> <p>9.3.7 Atomic and continuous material edge states 209</p> <p>9.3.8 Simulations and state number conservations 209</p> <p>9.3.9 Exact models versus small deformation ones 210</p> <p>9.3.10 The attenuation effects 210</p> <p>9.4 Outlook 210</p> <p>References 210</p> <p>Index 213</p>