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?