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 ο Topological Photonics β¦ Term often used: Time-reversal symmetry (TRS) Edge mode breaks spectral reciprocity: π ( π ) β π ( βπ ) 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 π ( π ) β π ( βπ ) 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 βdiatomicβ plasmon waveguide β¦ Phys. Rev. B 92, 165430 (2015) β¦ Non-reciprocal π -near-zero surface modes β¦ Phys. Rev. B 91, 235410 (2015) 3
Spectral Reciprocity & PT Symmetries π Spectral reciprocity - band structure is symmetric π π = π ( βπ ) π 0 PT symmetry - system is invariant by P and T operations together. T: time reversal ( π¦ , π§ , π¨ , π’ ) β ( π¦ , π§ , π¨ , βπ’ ) P: spatial inversion ( π¦ , π§ , π¨ , π’ ) β ( βπ¦ , βπ§ , βπ¨ , π’ ) P x : spatial inversion ( π¦ , π§ , π¨ , π’ ) β ( βπ¦ , π§ , π¨ , π’ ) 4
Quiz 1 Static magnetic field ο Can this periodic system support asymmetric band π ( π ) β π ( βπ ) ? B x One-way? Unit cell 5
Quiz 2 Static magnetic field ο Can this 1D periodic system support asymmetric band π ( π ) β π ( βπ ) ? B x One-way? Unit cell A waveguide 6
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
Symmetries and spectral reciprocity ο We need to break enough symmetries to achieve spectral non-reciprocity π ( π ) β π ( βπ ) ο Well-known examples of symmetries to break β¦ P: spatial inversion symmetry β¦ T: time reversal symmetry (TRS) 8
What is TRS for EM waves? We say that a system of given π ( π² ) and π ( π² ) has TRS if The macroscopic Maxwellβs equations and the constitutive relations for the same π and π are still satisfied by time- reversing the oscillating fields, To have TRS, we want the following Original: after time-reversal of fields: πΌ Γ π = πππ β π πΌ Γ ( π β ) = πππ β ( βπ β ) πΌ Γ π = βπππ β π πΌ Γ ( βπ β ) = βπππ β ( π β ) π β = π These new equations may NOT be satisfied. π β = π If they are satisfied, then we have these conditions on π and π . 9
Consequence of TRS on Band Structures Ref: Optical Properties of Photonic Crystals If πΉ β ( π¦ ) π π ( ππβππ ) is a solution, by K. Sakoda πΉ ( π¦ ) π π ( βπ β πβππ ) is also a solution. Symmetry in band structure π π ( π β ) = π ( βπ ) For pass band with real π , we have π ( π ) = π ( βπ ) even when there is no spatial π symmetry other than periodicity 10
Symmetries and spectral reciprocity ο We need to break enough symmetries to achieve spectral non-reciprocity π ( π ) β π ( βπ ) ο Well-known examples of symmetries to break β¦ P: spatial inversion symmetry β¦ T: time reversal symmetry (TRS) Enough? 11
Symmetries and spectral reciprocity ο We need to break enough symmetries to achieve spectral non-reciprocity π ( π ) β π ( βπ ) 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
Lorentz Reciprocity Lorentz reciprocity can be written as Case A: or symmetry in Greenβs Function β‘ T π² 2 , π² 1 S πππππ β ππππππππ β‘ π² 1 , π² 2 = π π Conditions of reciprocal medium: Case B: π T = π π T = π 13
A static magnetic field breaks both 1) T reversal symmetry (TRS) & 2) Symmetry in π and π (Lorentz reciprocity) Broken TRS Observe the difference π β β π e.g., gyromagnetic materials π β β π π ππ π 0 βππ π π 0 π = Broken Lorentz 0 0 π 3 reciprocity π ππ π 0 π T β π βππ π π 0 π = π T β π 0 0 π 3 (representation in frequency domain) 14
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
Quiz 2 (not very simple) Static magnetic field ο Can this 1D periodic system support asymmetric band π ( π ) β π ( βπ ) ? B x One-way? Unit cell A waveguide 16
Symmetries and spectral reciprocity ο We need to break enough symmetries to achieve spectral non-reciprocity π ( π ) β π ( βπ ) 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
Quiz 2 Static magnetic field ο Can this 1D periodic system support asymmetric band π ( π ) β π ( βπ ) ? B x NO One-way? Unit cell A waveguide 18
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 π and π (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 βDiatomicβ chain of nanoparticles 2 π π π ππ π 0 π π : Plasma frequency π = 1 β 2 π 2 β π π βππ π π 0 π = π π : Cyclotron frequency 0 0 π 3 2 π π π π = β π π 2 π 2 β π π π 21
Example: Plasmonic nanoparticle chain βDiatomicβ chain of nanoparticles 2 π π π ππ π 0 π π : Plasma frequency π = 1 β 2 π 2 β π π βππ π π 0 π = π π : Cyclotron frequency 0 0 π 3 2 π π π π = β π π 2 π 2 β π π π 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? βDiatomicβ 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
Why are the bands still symmetric in k? βDiatomicβ 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
What about this one? (Case B) βDiatomicβ 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! ( π¬ β βπ¬ ) Break reflection & T Result: Non-reciprocal bands Plasmon frequency Did we break enough symmetries? Yes, It breaks P , T, RT, β¦ 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 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 π -near-zero surface modes β¦ 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 ( π¦ , π§ , π¨ , π’ ) β ( π¦ , π§ , π¨ , βπ’ ) y Domain 1 P: spatial inversion ( π¦ , π§ , π¨ , π’ ) β ( βπ¦ , βπ§ , βπ¨ , π’ ) Domain 2 x P y : spatial inversion in y ( π¦ , π§ , π¨ , π’ ) β ( π¦ , βπ§ , π¨ , π’ ) 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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