Pimp my FacebookParity, time-reversal and duality symmetry
Enrica Martini
Symmetry-protection against scattering
1)
2)
3)
T I= −2
flips the Poynting vector
Incident wave
waveguide 1
waveguide 2
No losses
PTD Symmetry
P – parity operator: flips one coordinate (perpendicular to the plane of propagation) (x,y,z)→(x,y,-z)
T – time reversal operator: reverses the time
D – duality operator: duality transformation (related to the symmetrical role of electric and magnetic fields in Maxwell's equations)
x
z
The PTD-operator is the combination of Parity, time-reversal and duality transformations
A structure is said to be PTD-symmetric if it is invariant under the combination of Parity, time-reversal and duality transformations
PTD Symmetry
The behavior of these symmetry-protected modes is similar to the one of topological modes, which allow energy to flow around large discontinuities without back-reflections, but it can also be obtained in reciprocal lossless and passive media
T = −S S
PTD symmetric systems
Passive, lossless reciprocal systems are PTD symmetric if they are invariant with respect to the application of parity and duality operations
V −
= ( ) ( )
( ) ( )
, , , ,
, , , ,
x y z V x y z V
ε µ
ξ ξ
PTD symmetric systems
Passive, lossless reciprocal systems are PTD symmetric if they are invariant with respect to the application of parity and duality operations
We can design PTD-symmetric lossless reciprocal WGs by only using the BCs
V −
= ( ) ( )
( ) ( )
, , , ,
, , , ,
x y z V x y z V
ε µ
ξ ξ
PTD symmetric structure based on MTS
• The field of the edge mode is concentrated close to the edge • The edge mode is robust against back-scattering from PTD-symmetric discontinuities
0 CZ j ζ
Inductive
Capacitive
• An edge mode is supported at the interface between two semi-infinite complementary impedance surfaces in free space
2 0C LZ Z ζ=
PTD symmetric structure based on MTS
• The field of the edge mode is concentrated close to the edge • The edge mode is robust against back-scattering from PTD-symmetric discontinuities
0 CZ j ζ
0LZ jαζ=
• An edge mode is supported at the interface between two semi-infinite complementary impedance surfaces in free space
2 0C LZ Z ζ=
PTD symmetric structure based on MTS
• The structure is open, and therefore characterized by a continuous spectrum of modes → it may radiate at discontinuities
• The two halves support SWs
To obtain robust propagation, the edge mode should be the only propagating one
• This structure was proposed and experimentally studied by Sievenpiper et al.*
PTD symmetric structure based on MTS
• This structure can be closed still preserving PTD symmetry
• The two halves of the structure have a bandgap starting fom 0 frequency, their combination supports an edge mode protected from back-scattering by PTD symmetry
0 CZ j ζ
PEC/PMC edge waveguide
A particularly interesting case is the one for α=0, for which the two impedance surfaces become PEC and PMC, respectively
PEC
z-axis inversion
This structure supports a TEM mode confined at the edge and is unimodal for d<λ/4
z PEC
PEC PMC
PEC/PMC edge waveguide
The edge mode supported by this structure is TEM, therefore it can be found by solving the electrostatic problem
The potential ψ(x,y) respects
( ) ( ) ( )2/4
21
F π
− = − ( )2 2
The plane wave is the solution in the z domain
( , ) ( ( ))x y z sψ = φ
( ) 0
F
ξ
PEC
( ) ( )
( ) ( ) ( )
( )
n
n n n
x y V c e y u x y u x
x y x y x V
d d n a nc n V n n
u x unit step function
∞ −ξ
( )( , ) sin xn xs n nx y y e−αψ = α
( )/ 2nd nα = π + π
PTD symmetry PTD symmetry
Field excited at the port respecting a certain symmetry is only compatible with propagation in one direction
E EH H
Port 1
Port 2
d L1
L1
L2
L3
L4
Implementation
In practice the PMC is substituted by a high impedance surface
Mushroom metasurface can provide high-impedance BC with a low profile
Unimodal band
Light line
Square PTD-symmetric waveguide
• supports a TEM mode without cutoff • the waveguide can be arbitrary small • unimodal for L<λ/2
Implementation through mushrooms
PTD-symmetric 90° bend
Propagation is robust wrt any discontinuity respecting the same PTD symmetry
L=11mm
2.1dB
Square PTD-symmetric waveguide
Matching can be spoiled for a generic arrangement of multiple WGs…
WG 1 WG 2
11mm
…but both reflections and coupling vanish if the arrangement respects PTD symmetry
Dual-polarized configuration
Dual-polarization can be obtained without breaking the PTD-symmetry
Can PTD-symmetry be exploited to obtain wide angle impedance matching (WAIM) in scanning arrays?
Port 1
Port 2
Port 3
Port 4
Scanning array
Very good matching is mantained when scanning along a symmetry plane
Scan plane
Scanning array
Very good matching is maintained when scanning along a symmetry plane
Scan plane
Scanning array
Performances are only slightly deteriorated when scanning along a different plane
Scan plane
Array efficiency
Part of the power flows in the hole between WGs and this decreases the structure efficiency
Array efficiency
Part of the power flows in the hole between WGs and this decreases the structure efficiency
Alternative configuration
Alternative configurations can be considered to solve the efficiency problem
Alternating PEC and PMC square patches on the aperture plane avoids holes while maintaining the PTD-symmetry
Alternative configuration
Alternating PEC and PMC square patches on the aperture plane avoids holes while maintaining the PTD-symmetry
Alternative configurations can be considered to solve the efficiency problem
References
1. M. G. Silveirinha, “PTD Symmetry protected scattering anomaly in optics,” Phys. Rev. B, vol. 95, 035153, 2017.
2. D. J. Bisharat and D. F. Sievenpiper, “Guiding Waves Along an Infinitesimal Line between Impedance Surfaces”, Phys. Rev. Lett., vol. 119, 106802, 2017.
3. *W.-J. Chen, Z.-Q. Zhang, J.-W. Dong, and C. T. Chan, “Symmetry- protected transport in a pseudospin-polarized waveguide,” Nat. Commun., vol. 6, 8183, 2015
4. E. Martini, M.G. Silveirinha, S. Maci, “Exact Solution for the Protected TEM edge mode in a PTD-Symmetric Parallel-Plate Waveguide,” IEEE Transactions on Antennas and Propagation, Jan. 2019.
Questions?