In tale regime propagativo, considereremo sempre valide le condizioni cosθ ≃ 1 e 1/r ≃ 1/z, per l’esponenziale possiamo ulteriormente semplificare l’e- spressione (A.11). Abbiamo visto, infatti, che nell’approssimazione di Fre- snel: r = [ z + x 2+ y2+ ξ2+ η2− 2xξ − 2yη 2z ]
ora quando z tende ad infinito i termini ξ2ed η2tendono a dare un contributo molto trascurabile alla fase percui possiamo eliminarli ed ottenere quella che va sotto il nome di “approssimazione di Fraunhofer” del campo:
V (x, y, z) = −ie i[kz+2zk(x2+y2)] λz ∫∫ ∞ V0(ξ, η)e− 2πi λz(xξ+yη)dξdη (A.13)
L’integrale che compare nella (A.13) coincide esattamente con la trasformata di Fourier di V0 calcolata in x/(λz) e y/(λz).
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Indice analitico
bagno termico, 6, 17 temperatura, 18 biosensore
schema di principio, 39 cut-off, frequenza di, 7, 11 Damping, 6
Disturbanza, 7
effetto tunnel ottico, 9, 36 equazione d’onda, 7, 12, 16, 20, 45, 46 di Helmholtz, 12 di Liouville, 18 di Schr¨odinger, 36 equazioni di Maxwell covarianti, 14 metrica, 14 lagrangiana, 14 fotone, 5 distribuzione spettrale, 6 Energia, 5 polarizzazione, 5 quantit`a di moto, 5 spazio di Fock, 6 gauge, 15 di Coulomb, 30, 32, 49, 50 di coulomb, 51 di Feynman, 15 Gauss legge di, 50, 55 Goos-H¨anchen, 43
guida d’onda rettangolare, 11 automodi, 12, 13 frequenza di cut-off, 13 relazione di dispersione, 13 onda evanescente, 41 onde p o onde TM, 41 s o onde TE, 41 onde evanescenti applicazioni, 38 biosensori, 38 bidimensionali, 46 monodimensionali, 45 70
onde piane, 8
non omogenee, 8, 46 omogenee, 8
Open Quantum Systems, 6, 17 bagno termico, 6, 17
hamiltoniana d’interazione, 17 operatori
di creazione e distruzione, 15 parassialit`a, 46, 48
confronto con gauge, 49 longitudinalit`a, 49 soluzioni, 47 spettro angolare, 47 SVEA, 47, 48 polarizzazione ellittica, 40 lineare, 40 longitudinale, 49, 50 trasversa, 50–52 potenziale scalare, 14 potenziale vettore, 14 propagazione guidata, 7 quadri-potenziale vettore, 14 quantizzazione canonica, 14 momento coniugato, 14 policromaticit`a, 16 Quasi Normal Modes, 7, 19
discontinuity condition, 20 incoming waves conditions, 22 non hermiticit`a, 22
norma complessa, 21
outgoing waves conditions, 20 quasi-frequenze, 21 rappresentazione multi particella, 27 singola particella, 25 regione di Fraunhofer, 9 regole di commutazione, 15 riflessione totale, 7, 9, 40, 41 angolo critico, 10
campi spazialmente limitati, 43 frustrata, 11
piano d’incidenza, 10
superficie di separazione, 11 Shift di Goos-H¨anchen, 43
longiudinale, 44 trasversale, 44 Snell, legge di, 10 stato coerente, 25, 26
distribuzione Poisson, 26 multimodo, 28
tensore metrico, 15 trasversalit`a
condizione di, 4, 50 tunneling, 36