31st October, HKUST, Hong-Kong Electromagnetic wave propagation through ultra-narrow channels filled with an ENZ material Mário G. Silveirinha
How to have ε-nearε zero (ENZ) Media? 2 ω p ε r ~1 ω ω ( + iγ) Metal near ~ 0 ε r ε r 2 2 p ω p ω ε m < 0 ω Γ ε > 0 d αε 1 1+ αε 2 2 0 0 α1+ α2 εt 0 0 αε 1 1+ αε 2 2 ε = ε 0 0 εt 0 = ε0 0 0 α1 α2 0 0 ε + z ( α 1+ α2) ε1ε 2 0 0 αε 2 1+ αε 1 2 2
How to have ε-nearε zero (ENZ) Media? (contd.) 3
What if ε is near zero? 2-D Scenario with TE z polarization H int z (, ) = H x y uˆ z 1 E= H iωε ε o r 1 1 ˆ iωε ε u int = H z z o r The magnetic field must be constant inside id a connected ENZ region 4
Magnetic field inside the ENZ material Applying Faraday s law to the ENZ contour: E.dl A =+ iωμ μ H A int 0 r, p z p The magnetic field inside the ENZ material can be written in simple form in terms of the electric field evaluated at the outer side of the boundary 5
Formal solution of a scattering problem involving ENZ objects (Object is replaced by PMC) int Hz = Hz = const. + (Dirichlet problem) 6
Formal solution of a scattering problem involving ENZ objects (contd.) The total magnetic field outside the object can thus be written as: PMC int?s H z =ψ + H z ψ 1 The electric field is given by:? 1 1 ˆ PMC int s E = ψ H ψ1 iωε u + z z z 0 iωε u 0 ˆ (Object is replaced by PMC) + (Dirichlet problem) H z = ψ PMC H z = ψ s 1 7
Formal solution of a scattering problem involving ENZ objects (contd.) Calculation of the magnetic field inside the ENZ material: E.dl =+ iωμ μ H A A int 0 r, p z p E 1 u 1 ˆ u ˆ = PMC int s ψ z H z ψ1 z iωε + 0 iωε 0 H int z = A A s 1 ψ n PMC dl ψ 2 dl + k0 μr, p A n p 8
Formal solution of a scattering problem involving ENZ objects (conclusion) It is possible to completely determine the field scattered by an ENZ object by solving two external problems! PMC int s H z =ψ + H z ψ 1 H int z = A A s 1 ψ n PMC dl ψ 2 dl + k0 μr, pa n p 9
Application to the theory of parallel plate waveguides 10
General waveguide transition a 1 H inc E inc PEC PEC PEC PEC PEC E inc H inc PEC a 2 11
General waveguide transition a 1 H inc E inc PEC PEC PEC PEC PEC ε 0 E inc H inc a 2 PEC 12
Calculation of the scattering parameters (Object is replaced by PMC) + (Dirichlet problem) These problems have a trivial solution!!! 13
The scattering parameters for an ENZ channel with arbitrary geometry ρ = ( ) ( ) a a + ik μ A 1 2 0 r, p a + a ik μ A 1 2 0 r, p p p τ = 1+ ρ 14
Tunneling EM energy through tight ENZ channels If a=a 1 =a 2 the reflection coefficient is: ρ = k0 μr, pap i a k0 μr, pa 2 i a p The reflectivity can be made very small provided: k μ 0 r, p a A p << 1 15
Tunneling EM energy through tight ENZ channels (contd.) k μ Possibilities: 0 r, p a A p << 1 Very low frequency (static-like behavior). Permeability of the ENZ material is near zero. The channel is very tight so that t A p /a is small 16
Tunneling EM energy through tight ENZ channels (contd.) Despite the huge wave impedance contrast, in the ε=0 lossless limit it is possible to squeeze more and more energy through an ENZ channel by making the channel tighter and tighter! 17
U-shaped Waveguide Transition & EM Squeezing ρ = ( ) ( ) a a jk μ A 1 2 0 a + a + jk μ A 1 2 0 r r D D where A = al + al + a L D 1 1 2 2 ch 18
U-shaped Waveguide Transition & EM Squeezing (contd.) a1 = a2 = L a L1 = L2 = ach = 0.1a = 1 μ r 2 ω p ε = 1 ω ω ( jγ) ω a p / c = π /2 19
Field concentration in the ENZ channel a = a = L a 1 2 L1 = L2 = 0.1a = 1 μ r E E inc a a ch 20
Poynting vector a1 = a2 = L a L1 = L2 = 0.1a μ r = 1 Γ / ω p = 0.001 21
Field concentration and confinement in a small air cavity a1 = a2 = L a L1 = L2 = ach = 0.1a = 1 μ r
Waveguide with 90-deg Bend μ r = 1 aimp = 0.9a ω a / c = p 3 π /4
Experimental Demonstration of Energy Squeezing at Microwaves
Artificial plasma emulated by a waveguide below cut-off 2 π = ωεμε b TE10 mode: 2 γ 10 0 0 d ε eff γ = ω μ 2 10 2 0 b π b ω / c = ε0 εd 2 a ε d W. Rotman, IRE Trans. Antennas Propag. AP-10, 82 1962. R. Marqués, J. Martel, F. Mesa, and F. Medina, Phys. Rev. Lett. 89, 183901 2002.
Artificial plasma emulated by a waveguide below cut-off (contd.) The plasma frequency is controlled by the H-plane width, b. ε = 1 d b L 1 L L 2 How to emulate the free-space regions in the waveguide scenario?
Emulation of the free-space regions in the waveguide scenario ε = 1 d b L 1 L L 2 ε eff 2 π = ε 0 1 = 0 bω / c (at the plasma freq.) ε d = 2.0 2 π ε eff = ε 0 2 = ε0 b ω / c (at the plasma freq.) The waveguide sections that emulate the free-space regions are filled with a dielectric i with ε diel =2.0 20
Emulation of the 2D-problem using a waveguide setup Incident wave is TEM Incident wave is the TE10 mode ε = 1 d ε = d 2.0 ε d = 2.0 b L 1 L L 2
Simulation of an ENZ filled channel a1 = a2 a L1 = L2 = ach = 0.1a ω a / c p = π /2 Artificial plasma Artificial plasma L L = = 3.0a 10 1.0 a a 2D setup 2D setup
Simulation of an unfilled channel a1 = a2 = L a L1 = L2 = ach = 0.1a 3D waveguide uniformly filled with ε diel =2.0. 2D setup with unfilled narrow channel.
Experimental Demonstration
Experimental Demonstration (contd.)
Experimental Demonstration (contd.)
Tunneling oblique waves through an ENZ material
Oblique Incidence Total reflection ENZ n=0 How Since can the we magnetic make a field wave is that constant t impinges i inside idon the ENZ interface medium off the phase variation normal at the tunnel air side through cannot the be reproduced ENZ slab? inside the slab
Idea: Convert the incident EM wave into TEM modes Provided d a is much smaller than the wavelength, the incident id field can be sampled and propagated through different channels
Transmission as a function of the extension of the metallic plates LENZ = 025 0.25 λ 0 ε = 0.001 a = 0.03λ L = L 0 1ef 2ef When the total thickness L 1 +L 2 is a odd multiple of 0.5λ 0, the wave can tunnel through the ENZ medium, even for wide incident angles
Compression of the modal fields of a dielectric waveguide
Geometry ε s = 22 2.2 Standard dielectric waveguide Array of 10 waveguides filled with ENZ material Standard dielectric waveguide
Poynting vector ε i0.01 0 01 LENZ = 0.75λd L1 = L2 = 0.25λd ach = 0.1a hs = 0.87λd
Electric Field (movie)
Transmission of modal fields in a dielectric waveguide 42
Transmission of an image through sub-λ aperture D h = W s /2.5 W = λ 0 /2 s M. Silveirinha, N. Engheta, Phys. Rev. Lett., 102, 103902, 2009 43
Transmission of an image through sub-λ aperture (contd.) 44
Summary ENZ media have interesting wave guiding properties; EM waves may be squeezed through very tight and narrow channels, with great electric field enhancement. These structures may enable bending and compressing waves in the subwavelength scale.
Thank you! 46