periodic photonic systems
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periodic photonic systems by Kin Hung Fung Department of Applied - PowerPoint PPT Presentation

Pre se nta tio n in Ao E Wo rksho p (2016) Adva nc e d Co nc e pt in Wa ve Physic s T o po lo g y a nd PT Symme try PT symmetries & Non-reciprocity in periodic photonic systems by Kin Hung Fung Department of Applied Physics The


  1. Pre se nta tio n in Ao E Wo rksho p (2016) Adva nc e d Co nc e pt in Wa ve Physic s T o po lo g y a nd PT Symme try PT symmetries & Non-reciprocity in periodic photonic systems by Kin Hung Fung Department of Applied Physics The Hong Kong Polytechnic University 1

  2. One-way propagation in photonic circuit ο‚— Topological Photonics β—¦ Term often used: Time-reversal symmetry (TRS) Edge mode breaks spectral reciprocity: πœ• ( 𝑙 ) β‰  πœ• ( βˆ’π‘™ ) Static magnetic field B Z. Wang et al., PRL100, 013905 A recent review: F. D. M. Haldane and S. Raghu , PRL 100, 013904 Topological Photonics Nat. Photon. by Ling Lu et al. 2

  3. This talk focuses on breaking spectral reciprocity πœ• ( 𝑙 ) β‰  πœ• ( βˆ’π‘™ ) There is a difference between spectral reciprocity and Lorentz reciprocity. ο‚— Our recent works related to spatial-temporal symmetries such as PT symmetry are also provided as examples: β—¦ Asymmetric bands in a β€œdiatomic” plasmon waveguide β—¦ Phys. Rev. B 92, 165430 (2015) β—¦ Non-reciprocal 𝜈 -near-zero surface modes β—¦ Phys. Rev. B 91, 235410 (2015) 3

  4. Spectral Reciprocity & PT Symmetries πœ• Spectral reciprocity - band structure is symmetric πœ• 𝑙 = πœ• ( βˆ’π‘™ ) 𝑙 0 PT symmetry - system is invariant by P and T operations together. T: time reversal ( 𝑦 , 𝑧 , 𝑨 , 𝑒 ) β†’ ( 𝑦 , 𝑧 , 𝑨 , βˆ’π‘’ ) P: spatial inversion ( 𝑦 , 𝑧 , 𝑨 , 𝑒 ) β†’ ( βˆ’π‘¦ , βˆ’π‘§ , βˆ’π‘¨ , 𝑒 ) P x : spatial inversion ( 𝑦 , 𝑧 , 𝑨 , 𝑒 ) β†’ ( βˆ’π‘¦ , 𝑧 , 𝑨 , 𝑒 ) 4

  5. Quiz 1 Static magnetic field ο‚— Can this periodic system support asymmetric band πœ• ( 𝑙 ) β‰  πœ• ( βˆ’π‘™ ) ? B x One-way? Unit cell 5

  6. Quiz 2 Static magnetic field ο‚— Can this 1D periodic system support asymmetric band πœ• ( 𝑙 ) β‰  πœ• ( βˆ’π‘™ ) ? B x One-way? Unit cell A waveguide 6

  7. Quiz 3 ο‚— What if the materials have small gain/loss instead of magnetic field? The time reversal symmetry is still broken! One-way? Unit cell A waveguide 7

  8. Symmetries and spectral reciprocity ο‚— We need to break enough symmetries to achieve spectral non-reciprocity πœ• ( 𝑙 ) β‰  πœ• ( βˆ’π‘™ ) ο‚— Well-known examples of symmetries to break β—¦ P: spatial inversion symmetry β—¦ T: time reversal symmetry (TRS) 8

  9. What is TRS for EM waves? We say that a system of given 𝛇 ( 𝐲 ) and 𝝂 ( 𝐲 ) has TRS if The macroscopic Maxwell’s equations and the constitutive relations for the same 𝛇 and π›Ž are still satisfied by time- reversing the oscillating fields, To have TRS, we want the following Original: after time-reversal of fields: 𝛼 Γ— 𝐅 = π‘—πœ•π›Ž βˆ™ 𝐈 𝛼 Γ— ( 𝐅 βˆ— ) = π‘—πœ•π›Ž βˆ™ ( βˆ’πˆ βˆ— ) 𝛼 Γ— 𝐈 = βˆ’π‘—πœ•π›‡ βˆ™ 𝐅 𝛼 Γ— ( βˆ’πˆ βˆ— ) = βˆ’π‘—πœ•π›‡ βˆ™ ( 𝐅 βˆ— ) 𝛇 βˆ— = 𝛇 These new equations may NOT be satisfied. π›Ž βˆ— = π›Ž If they are satisfied, then we have these conditions on 𝛇 and π›Ž . 9

  10. Consequence of TRS on Band Structures Ref: Optical Properties of Photonic Crystals If 𝐹 βˆ— ( 𝑦 ) 𝑓 𝑗 ( π‘™π‘™βˆ’πœ•πœ• ) is a solution, by K. Sakoda 𝐹 ( 𝑦 ) 𝑓 𝑗 ( βˆ’π‘™ βˆ— π‘™βˆ’πœ•πœ• ) is also a solution. Symmetry in band structure πœ• πœ• ( 𝑙 βˆ— ) = πœ• ( βˆ’π‘™ ) For pass band with real 𝑙 , we have πœ• ( 𝑙 ) = πœ• ( βˆ’π‘™ ) even when there is no spatial 𝑙 symmetry other than periodicity 10

  11. Symmetries and spectral reciprocity ο‚— We need to break enough symmetries to achieve spectral non-reciprocity πœ• ( 𝑙 ) β‰  πœ• ( βˆ’π‘™ ) ο‚— Well-known examples of symmetries to break β—¦ P: spatial inversion symmetry β—¦ T: time reversal symmetry (TRS) Enough? 11

  12. Symmetries and spectral reciprocity ο‚— We need to break enough symmetries to achieve spectral non-reciprocity πœ• ( 𝑙 ) β‰  πœ• ( βˆ’π‘™ ) ο‚— Well-known examples of symmetries to break β—¦ P: spatial inversion symmetry β—¦ T: time reversal symmetry β—¦ Symmetric permittivity and permeability tensor Lorentz Reciprocity 12

  13. Lorentz Reciprocity Lorentz reciprocity can be written as Case A: or symmetry in Green’s Function ⃑ T 𝐲 2 , 𝐲 1 S 𝑝𝑝𝑝𝑝𝑓 ↔ 𝑆𝑓𝑝𝑓𝑗𝑆𝑓𝑝 ⃑ 𝐲 1 , 𝐲 2 = 𝐇 𝐇 Conditions of reciprocal medium: Case B: 𝛇 T = 𝛇 π›Ž T = π›Ž 13

  14. A static magnetic field breaks both 1) T reversal symmetry (TRS) & 2) Symmetry in 𝛇 and π›Ž (Lorentz reciprocity) Broken TRS Observe the difference 𝛇 βˆ— β‰  𝛇 e.g., gyromagnetic materials π›Ž βˆ— β‰  π›Ž 𝜈 π‘—πœ† 𝑛 0 βˆ’π‘—πœ† 𝑛 𝜈 0 π›Ž = Broken Lorentz 0 0 𝜈 3 reciprocity 𝜁 π‘—πœ† 𝑓 0 𝛇 T β‰  𝛇 βˆ’π‘—πœ† 𝑓 𝜁 0 𝛇 = π›Ž T β‰  π›Ž 0 0 𝜁 3 (representation in frequency domain) 14

  15. Quiz 3 (simple) ο‚— What if the materials have small gain/loss instead of magnetic field? The time reversal symmetry is still broken! One-way? Unit cell A waveguide NO because of Lorentz reciprocity. 15

  16. Quiz 2 (not very simple) Static magnetic field ο‚— Can this 1D periodic system support asymmetric band πœ• ( 𝑙 ) β‰  πœ• ( βˆ’π‘™ ) ? B x One-way? Unit cell A waveguide 16

  17. Symmetries and spectral reciprocity ο‚— We need to break enough symmetries to achieve spectral non-reciprocity πœ• ( 𝑙 ) β‰  πœ• ( βˆ’π‘™ ) ο‚— Well-known examples of symmetries to break β—¦ P: spatial inversion symmetry β—¦ T: time reversal symmetry β—¦ Symmetric permittivity and permeability tensor Enough now? The answer is still NO! Lorentz Reciprocity 17

  18. Quiz 2 Static magnetic field ο‚— Can this 1D periodic system support asymmetric band πœ• ( 𝑙 ) β‰  πœ• ( βˆ’π‘™ ) ? B x NO One-way? Unit cell A waveguide 18

  19. Analysis for Quiz 2 Static ο‚— To break magnetic field β—¦ P: spatial inversion symmetry (already broken) β—¦ T: time reversal symmetry B (already broken) x β—¦ Symmetric 𝛇 and π›Ž (already broken) Unit cell A waveguide 19

  20. Why is the answer still NO? My answer is: Nature is happy with symmetric bands. Let us consider the simplest example in plasmonics. 20

  21. Example: Plasmonic nanoparticle chain β€œDiatomic” chain of nanoparticles 2 πœ• π‘ž 𝜁 π‘—πœ† 𝑓 0 πœ• π‘ž : Plasma frequency 𝜁 = 1 βˆ’ 2 πœ• 2 βˆ’ πœ• 𝑑 βˆ’π‘—πœ† 𝑓 𝜁 0 𝛇 = πœ• 𝑑 : Cyclotron frequency 0 0 𝜁 3 2 πœ• π‘ž πœ† 𝑓 = βˆ’ πœ• 𝑑 2 πœ• 2 βˆ’ πœ• 𝑑 πœ• 21

  22. Example: Plasmonic nanoparticle chain β€œDiatomic” chain of nanoparticles 2 πœ• π‘ž 𝜁 π‘—πœ† 𝑓 0 πœ• π‘ž : Plasma frequency 𝜁 = 1 βˆ’ 2 πœ• 2 βˆ’ πœ• 𝑑 βˆ’π‘—πœ† 𝑓 𝜁 0 𝛇 = πœ• 𝑑 : Cyclotron frequency 0 0 𝜁 3 2 πœ• π‘ž πœ† 𝑓 = βˆ’ πœ• 𝑑 2 πœ• 2 βˆ’ πœ• 𝑑 πœ• Light line Notes: 4 bands due to 4 degrees of freedom in unit cell. 22

  23. We want this asymmetric angry face! 23

  24. Why are the bands still symmetric in k? β€œDiatomic” chain of nanoparticles Protected by RT symmetry T = time reversal R = 180Β° rotation about x-axis We need to break this RT symmetry too! CW Ling et al. Physical Review B 92 165430 (2015) 24

  25. Why are the bands still symmetric in k? β€œDiatomic” chain of nanoparticles Protected by RT symmetry T = time reversal R = 180Β° rotation about x-axis T R We need to break this RT symmetry too! CW Ling et al. Physical Review B 92 165430 (2015) 25

  26. What about this one? (Case B) β€œDiatomic” chain of nanoparticles Did we break enough symmetries? Yes. CW Ling et al. Physical Review B 92 165430 (2015) 26

  27. Break inversion Non-reciprocal bands! ( 𝐬 β†’ βˆ’π¬ ) Break reflection & T Result: Non-reciprocal bands Plasmon frequency Did we break enough symmetries? Yes, It breaks P , T, RT, … Wave vector k CW Ling et al. Physical Review B 92 165430 (2015) 27

  28. Why? Did we break enough symmetries? Yes. Case A Case B operation operation operation operation CW Ling et al. Physical Review B 92 165430 (2015) 28

  29. Unidirectional wave propagation Only forward propagation is allowed at some frequencies no matter what kind of excitation. CW Ling et al. Physical Review B 92 165430 (2015) 29

  30. This talk focuses on breaking spectral reciprocity & its relation to PT or RT symmetries ο‚— Conclusion of part A: β—¦ We need to break a lot of spatial temporal symmetries to achieve non-reciprocity β—¦ CW Ling et al. Physical Review B 92 165430 (2015) ο‚— Can we keep PT symmetry while having non-reciprocal bands? Yes. ο‚— Next part: β—¦ Non-reciprocal 𝜈 -near-zero surface modes β—¦ PT symmetric magnetic domains 30

  31. We now consider this We will focus on interface modes Domain 1 Domain 2 y x The surface mode dispersion has been considered before: H. Zhu and C. Jiang, Opt. Express 18 , 6914 (2010) but there is something missing… 31

  32. This one has PT symmetry T: time reversal ( 𝑦 , 𝑧 , 𝑨 , 𝑒 ) β†’ ( 𝑦 , 𝑧 , 𝑨 , βˆ’π‘’ ) y Domain 1 P: spatial inversion ( 𝑦 , 𝑧 , 𝑨 , 𝑒 ) β†’ ( βˆ’π‘¦ , βˆ’π‘§ , βˆ’π‘¨ , 𝑒 ) Domain 2 x P y : spatial inversion in y ( 𝑦 , 𝑧 , 𝑨 , 𝑒 ) β†’ ( 𝑦 , βˆ’π‘§ , 𝑨 , 𝑒 ) P operation The system has PT but not P y T symmetry. Domain 1 Domain 2 T operation We will focus on interface modes 32

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