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 Hong Kong Polytechnic University 1

One-way propagation in photonic circuit  Topological Photonics ◦ Term often used: Time-reversal symmetry (TRS) Edge mode breaks spectral reciprocity: 𝜕 ( 𝑙 ) ≠ 𝜕 ( −𝑙 ) 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

This talk focuses on breaking spectral reciprocity 𝜕 ( 𝑙 ) ≠ 𝜕 ( −𝑙 ) There is a difference between spectral reciprocity and Lorentz reciprocity.  Our recent works related to spatial-temporal symmetries such as PT symmetry are also provided as examples: ◦ Asymmetric bands in a “diatomic” plasmon waveguide ◦ Phys. Rev. B 92, 165430 (2015) ◦ Non-reciprocal 𝜈 -near-zero surface modes ◦ Phys. Rev. B 91, 235410 (2015) 3

Spectral Reciprocity & PT Symmetries 𝜕 Spectral reciprocity - band structure is symmetric 𝜕 𝑙 = 𝜕 ( −𝑙 ) 𝑙 0 PT symmetry - system is invariant by P and T operations together. T: time reversal ( 𝑦 , 𝑧 , 𝑨 , 𝑢 ) → ( 𝑦 , 𝑧 , 𝑨 , −𝑢 ) P: spatial inversion ( 𝑦 , 𝑧 , 𝑨 , 𝑢 ) → ( −𝑦 , −𝑧 , −𝑨 , 𝑢 ) P x : spatial inversion ( 𝑦 , 𝑧 , 𝑨 , 𝑢 ) → ( −𝑦 , 𝑧 , 𝑨 , 𝑢 ) 4

Quiz 1 Static magnetic field  Can this periodic system support asymmetric band 𝜕 ( 𝑙 ) ≠ 𝜕 ( −𝑙 ) ? B x One-way? Unit cell 5

Quiz 2 Static magnetic field  Can this 1D periodic system support asymmetric band 𝜕 ( 𝑙 ) ≠ 𝜕 ( −𝑙 ) ? B x One-way? Unit cell A waveguide 6

Quiz 3  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

Symmetries and spectral reciprocity  We need to break enough symmetries to achieve spectral non-reciprocity 𝜕 ( 𝑙 ) ≠ 𝜕 ( −𝑙 )  Well-known examples of symmetries to break ◦ P: spatial inversion symmetry ◦ T: time reversal symmetry (TRS) 8

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 𝛎 are still satisfied by time- reversing the oscillating fields, To have TRS, we want the following Original: after time-reversal of fields: 𝛼 × 𝐅 = 𝑗𝜕𝛎 ∙ 𝐈 𝛼 × ( 𝐅 ∗ ) = 𝑗𝜕𝛎 ∙ ( −𝐈 ∗ ) 𝛼 × 𝐈 = −𝑗𝜕𝛇 ∙ 𝐅 𝛼 × ( −𝐈 ∗ ) = −𝑗𝜕𝛇 ∙ ( 𝐅 ∗ ) 𝛇 ∗ = 𝛇 These new equations may NOT be satisfied. 𝛎 ∗ = 𝛎 If they are satisfied, then we have these conditions on 𝛇 and 𝛎 . 9

Consequence of TRS on Band Structures Ref: Optical Properties of Photonic Crystals If 𝐹 ∗ ( 𝑦 ) 𝑓 𝑗 ( 𝑙𝑙−𝜕𝜕 ) is a solution, by K. Sakoda 𝐹 ( 𝑦 ) 𝑓 𝑗 ( −𝑙 ∗ 𝑙−𝜕𝜕 ) is also a solution. Symmetry in band structure 𝜕 𝜕 ( 𝑙 ∗ ) = 𝜕 ( −𝑙 ) For pass band with real 𝑙 , we have 𝜕 ( 𝑙 ) = 𝜕 ( −𝑙 ) even when there is no spatial 𝑙 symmetry other than periodicity 10

Symmetries and spectral reciprocity  We need to break enough symmetries to achieve spectral non-reciprocity 𝜕 ( 𝑙 ) ≠ 𝜕 ( −𝑙 )  Well-known examples of symmetries to break ◦ P: spatial inversion symmetry ◦ T: time reversal symmetry (TRS) Enough? 11

Symmetries and spectral reciprocity  We need to break enough symmetries to achieve spectral non-reciprocity 𝜕 ( 𝑙 ) ≠ 𝜕 ( −𝑙 )  Well-known examples of symmetries to break ◦ P: spatial inversion symmetry ◦ T: time reversal symmetry ◦ Symmetric permittivity and permeability tensor Lorentz Reciprocity 12

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 𝛎 T = 𝛎 13

A static magnetic field breaks both 1) T reversal symmetry (TRS) & 2) Symmetry in 𝛇 and 𝛎 (Lorentz reciprocity) Broken TRS Observe the difference 𝛇 ∗ ≠ 𝛇 e.g., gyromagnetic materials 𝛎 ∗ ≠ 𝛎 𝜈 𝑗𝜆 𝑛 0 −𝑗𝜆 𝑛 𝜈 0 𝛎 = Broken Lorentz 0 0 𝜈 3 reciprocity 𝜁 𝑗𝜆 𝑓 0 𝛇 T ≠ 𝛇 −𝑗𝜆 𝑓 𝜁 0 𝛇 = 𝛎 T ≠ 𝛎 0 0 𝜁 3 (representation in frequency domain) 14

Quiz 3 (simple)  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

Quiz 2 (not very simple) Static magnetic field  Can this 1D periodic system support asymmetric band 𝜕 ( 𝑙 ) ≠ 𝜕 ( −𝑙 ) ? B x One-way? Unit cell A waveguide 16

Symmetries and spectral reciprocity  We need to break enough symmetries to achieve spectral non-reciprocity 𝜕 ( 𝑙 ) ≠ 𝜕 ( −𝑙 )  Well-known examples of symmetries to break ◦ P: spatial inversion symmetry ◦ T: time reversal symmetry ◦ Symmetric permittivity and permeability tensor Enough now? The answer is still NO! Lorentz Reciprocity 17

Quiz 2 Static magnetic field  Can this 1D periodic system support asymmetric band 𝜕 ( 𝑙 ) ≠ 𝜕 ( −𝑙 ) ? B x NO One-way? Unit cell A waveguide 18

Analysis for Quiz 2 Static  To break magnetic field ◦ P: spatial inversion symmetry (already broken) ◦ T: time reversal symmetry B (already broken) x ◦ Symmetric 𝛇 and 𝛎 (already broken) Unit cell A waveguide 19

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

Example: Plasmonic nanoparticle chain “Diatomic” chain of nanoparticles 2 𝜕 𝑞 𝜁 𝑗𝜆 𝑓 0 𝜕 𝑞 : Plasma frequency 𝜁 = 1 − 2 𝜕 2 − 𝜕 𝑑 −𝑗𝜆 𝑓 𝜁 0 𝛇 = 𝜕 𝑑 : Cyclotron frequency 0 0 𝜁 3 2 𝜕 𝑞 𝜆 𝑓 = − 𝜕 𝑑 2 𝜕 2 − 𝜕 𝑑 𝜕 21

Example: Plasmonic nanoparticle chain “Diatomic” chain of nanoparticles 2 𝜕 𝑞 𝜁 𝑗𝜆 𝑓 0 𝜕 𝑞 : Plasma frequency 𝜁 = 1 − 2 𝜕 2 − 𝜕 𝑑 −𝑗𝜆 𝑓 𝜁 0 𝛇 = 𝜕 𝑑 : Cyclotron frequency 0 0 𝜁 3 2 𝜕 𝑞 𝜆 𝑓 = − 𝜕 𝑑 2 𝜕 2 − 𝜕 𝑑 𝜕 Light line Notes: 4 bands due to 4 degrees of freedom in unit cell. 22

We want this asymmetric angry face! 23

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

Why are the bands still symmetric in k? “Diatomic” chain of nanoparticles Protected by RT symmetry T = time reversal R = 180° 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

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

Break inversion Non-reciprocal bands! ( 𝐬 → −𝐬 ) 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

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

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

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

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

This one has PT symmetry T: time reversal ( 𝑦 , 𝑧 , 𝑨 , 𝑢 ) → ( 𝑦 , 𝑧 , 𝑨 , −𝑢 ) y Domain 1 P: spatial inversion ( 𝑦 , 𝑧 , 𝑨 , 𝑢 ) → ( −𝑦 , −𝑧 , −𝑨 , 𝑢 ) Domain 2 x P y : spatial inversion in y ( 𝑦 , 𝑧 , 𝑨 , 𝑢 ) → ( 𝑦 , −𝑧 , 𝑨 , 𝑢 ) 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