Mathematical Insights into Surface Plasmon Polaritons

This mathematical explanation delves into the intricacies of surface plasmon polaritons, covering Maxwell's equations, wavenumbers, bulk and surface plasmon-polariton formulations, medium considerations, excitation conditions, losses, and matching examples with air, gold, and sapphire prisms. Detailed equations and explanations elucidate the behavior and characteristics of surface plasmon polaritons in various mediums.

• Surface Plasmon Polaritons
• Mathematical Explanation
• Maxwell Equations
• Medium Considerations
• Excitation Conditions

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1. Mathematical explanation for Surface Plasmon Polaritons Benji B rner 1

2. Revision of the problem photon SPP 2

3. Mathematical explanation bulk plasmon-polariton Maxwell equation: ? ? = ? + ? ?? 1 2 2 ??2 ?0? ? ? = 1 ?? ?2 ??? ??= Wavenumber: ?2= ?2?? 3

4. Mathematical explanation surface plasmon-polariton ??1= ?? ?(??1?+??1?)+ ???(??1? ??1?) ???1 ?? 1 ??1 ??1?? ? ??1?+??1?+ ??? ??1? ??1? ??1= = ???1 4

5. Mathematical explanation surface plasmon-polariton ??2= ?? ?(??2?+??2?) ??2=??2 ?2? ?(??2?+??2?) 5

6. Mathematical explanation surface plasmon-polariton Medium 1 Medium 2 ??1= ???1?? ???? ??2= ?? ?2?? ???? ?1 ?2 ???1???1?? ???? ?? ?2?? ???? ??1= ??2= ???2 ??1= ?? ??2= ?? ??1???1?? ???? ??2?? ?1?? ???? 6

7. Mathematical explanation surface plasmon-polariton ? = ? ?1 =?2 ?1 ?2 1 2 ??=? ?1?2 ?1+ ?2 ? 7

8. Mathematical explanation SPP losses ??? ??= ?? ??2= ? ?? 2 2?? ?? ?? ? = 1 ? = ?2+ ??2 ?(?2+ ??2) 8

9. Mathematical explanation excitation conditions genral condition: ?1?2< 0 and ?1+ ?2< 0 matching condition for prisms ? =? ??????sin? ? matching condition for gratings ? =? ?sin? ?? with G =2? ? 9

10. SPP matching Example: air, gold and sapphire prism 1 ?1= 1 ? = 1.1?? ? = 5.9 1022 ??3 ??= 9.24 1012 ?? ???= 6.53 1012 ?? ??????= 9.3 ????? ? = 36 10

11. SPP matching 11 J.R. Sambles et.al. Optical excitation of surface plasmons: an introduction

12. SPP matching 12 J.R. Sambles et.al. Optical excitation of surface plasmons: an introduction

13. SPP matching 13