Metamaterial Loaded Compact Cavity Resonators

1
Metamaterial Loaded Compact Cavity Resonators
E
H
k
Thomas Henry Hand
Duke University Department of Electrical and
Computer Engineering Ph.D. Qualifying Exam
Presentation
Friday, October 27th, 2006
Dr. Steven Cummer
Dr. William Joines
Dr. David Smith
Dr. Qing Liu
2
Overview of Presentation
Part I Metamaterials Overview

• History and Theory of Operation

Part II Applying Metamaterials to Create Thin
Subwavelength Cavity Resonators

• Enghetas Idea

• Hrabars Study

• Kongs Study

Part III My Work on the Metamaterial Loaded
Cavity Resonator

• Motivation

• Design Approach

• Simulation Results

• Experimental Results

Part VI Conclusions

• Project Summary

• Upcoming Publications

3
Part I Metamaterials Overview
Metamaterial Meta Material
Meta Greek prefix meaning Beyond

• Metamaterials are synthetic structures that
  possess electromagnetic properties beyond
  conventional materials

• They gain electromagnetic properties from their
  structure as opposed to their intrinsic material
  property

• The goal is to give a structure an effective
  permittivity and permeability by providing
  electric and magnetic responses using artificial
  metallic inclusions. These effective parameters
  are the result of averaging the spatial fields
  across the material.

4
The Effective Medium Picture

• Metamaterials are useful because they are
  designed to function as continuous effective
  media to electromagnetic radiation

• We want to be able to characterize a sample with
  effective material parameters eeff and µeff ,
  which result from spatial averaging of the
  electric and magnetic fields

• Since the metamaterial is composed of tiny
  metallic inclusions, we want to be sure these
  elements are significantly smaller than the free
  space wavelength ?o as to prevent diffraction
  effects that would ruin this effective medium
  picture.

Size Restrictions
(Pendry, et. al. Magnetism from Conductors and
Enhanced Nonlinear Phenomena)
a

• In practice, we like to keep the unit cell
  dimensions around ?o/10, although cell sizes on
  the order of ?o/6 have proved to keep the
  effective medium picture intact.

(Smith, et. al., Electromagnetic parameter
retrieval from inhomogeneous metamaterials)
5
What are Negative Index Metamaterials (NIMs)?

• Metamaterials that provide a structure with an
  effective negative index of refraction.

• First conceptualized by V.G Veselago in 1968

• Pendry proposed physical structures in 1996 and
  1999 that lead to the their physical
  realization

• First physically realized by Smith, et. al. in
  2000.

• Since metamaterials were first physically
  realized in 2000, many research groups have
  exploited these synthetic structures to create
  novel devices and components.

Timeline
1968
1996
1999
2000
Time
Veselago first studies the effect a negative
permittivity and permeability has on wave
propagation
Pendry proposes wire structures to realize a
negative permittivity
Pendry proposes Split Ring Resonators (SRRs) to
realize a negative permeability
Smith is the first in the world to physically
realize a medium with an effective negative index
of refraction
6
Negative Index Metamaterial Features

• Negative Permittivity and Permeability will
  cause the phase velocity and power flow to be
  anti-parallel

NIM Slab
Phase velocity
Power Flow
(Borrowed from physics.ucsd.edu/drs/left_home.htm
)

• Negative e and µ allow for a broader
  electromagnetic palette

Example No Cut-Off Waveguide
Dispersion Relation in Rectangular WG loaded with
anisotropic NIM
As can be seen, by choosing ex 0 , kz will always be positive and there will
be no lower cutoff frequency.
7
Negative Refraction Continued
Snells Law at the interface between a negative
index material and a positive index material
n 0
n 0
n
http//sagar.physics.neu.edu/wavepacket_refraction
.htm
http//www.utexas.edu/research/cemd/nim/Intro.html
Light Bending the Wrong Way?
and for n1 0 and n2
Refracted beam will be opposite to the normal as
shown in the animation above.
8
Realizing a Negative Permittivity
1968
1996
1999
2000
Time
Veselago first studies the effect a negative
permittivity and permeability has on wave
propagation
Pendry proposes wire structures to realize a
negative permittivity
Pendry proposes Split Ring Resonators (SRRs) to
realize a negative permeability
Smith is the first in the world to realize a
medium with an effective negative index of
refraction
The Drude Model of Permittivity

• We want er to be small and negative since a
  large and negative er could shrink ?eff to the
  point where the effective medium picture
  disappears.

• In 1996, Pendry proposed a way to reduce the
  plasma frequency using a periodic wire lattice
  structure (Pendry, et. al. Extremely Low Freq.
  Plasmons in Metallic Mesostructures.)

9
Realizing a Negative Permeability
The Lorentz Model of Permeability
1968
1996
1999
2000
Time
Veselago first studies the effect a negative
permittivity and permeability has on wave
propagation
Pendry proposes wire structures to realize a
negative permittivity
Pendry proposes Split Ring Resonators (SRRs) to
realize a negative permeability
Smith is the first in the world to realize a
medium with an effective negative index of
refraction

• Pendry proposed split ring resonators (SRRs) to
  achieve the necessary resonant magnetic response
  (Pendry, et. al. Magnetism from Conductors and
  Enhanced Nonlinear Phenomena)

• Any LC resonant particle will realize the
  negative permeability, such as the single ring
  particle I employ in practice.

Is related to this current
10
Part II Applying Metamaterials to Create Thin
Subwavelength Cavity Resonators

• My Work is based primarily on Three Research
  Papers Devoted to the Thin Cavity Resonator
  Concept

Engheta Originally proposed metamaterial loaded
cavity resonator
Hrabar Short Paper that summarized Enghetas
theory. They measured the spatial phase variation
in the cavity.
Kong More in depth experimental investigation
behind Enghetas cavity resonator.
Hand, Cummer, Engheta Expanded upon these ideas
in addition to measuring the spatial electric
field distribution for further clarification of
the physics inside the cavity
11
Enghetas Study

• Engheta proposed theoretically that negative
  index metamaterials could be used to create thin
  subwavelength cavity resonators

• In his paper, he analyzed a 1D cavity loaded
  with a bilayer composed of dielectric and
  negative index slabs

• His aim was to show that by loading a cavity
  with dielectric and negative index slabs, the
  resonance depends on the ratio of slab
  thicknesses, and not their sum

When the metamaterial slab has an effective
negative permeability as well as being
electrically thin, we are left with the
constraint
12
The electric fields in both slabs can be
expressed as
So for slab thicknesses d1 d2 d 1cm, and f
2.5 GHz, the electric field distribution inside
the cavity is
Note the change in electric field slope at the
interface between slabs. This discontinuity in
dE/dz arises due to the discontinuity in
effective permeability
And it is this triangular field distribution that
I seek to measure experimentally!
13
But what if we had used two conventional RH slabs
instead?

• Would have remained unchanged, and since
  µ1 and µ2 would be positive, if one tangent term
  is positive, then the other would have to be
  negative!

Then to satisfy the dispersion relation, d2 must
be

• Thus, if d1

• This constrains

And makes the resonant cavity dependent on d1d2 !
So clearly using the metamaterial slab allows us
to build a more compact resonator than if we had
used a RH bilayer.
14
Hrabars Experiment

• First experimental validation of Enghetas
  metamaterial loaded
• resonator

• Made a resonant ring structure to function as
  the LH layer

• This resonant ring structure was then placed
  inside an evanescent
• waveguide to realized a LH wave!

• Coupling loops were used to excite the loop and
  measure the
• phase of S21

S21 phase was then measured, showing that the
metamaterial behaves as a phase compensator
15
Kongs Experiment

• Expands upon Hrabars experiment

• Shows that the resonant frequency is invariant
  for various slab
• thicknesses, as long as Enghetas dispersion
  relation is satisfied

Experiment

• Metamaterial was fabricated and transmission
  properties were measured to verify LH behavior

• The phase difference across the bilayer cavity
  was measured to see whether or not it approached
  zero as predicted by Engheta

• The slab thicknesses (d1 and d2) were
  simultaneously varied to see if the resonant
  frequency in the cavity remained unchanged.

16
Part III My Work on the Metamaterial Cavity
Resonator

• To help make more lucid the relationship between
  the properties of the metamaterial slab and the
  field structure inside the cavity, I decided to
  measure the spatial electric field magnitude
  distribution inside the cavity.

• This study provides a more in depth look at the
  behavior of the fields inside the cavity.

• It allows us to see some interesting physical
  effects, such as how well defined the boundary is
  between the air and metamaterial layers, and how
  the effective permeability changes as frequency
  is shifted.

17
Field Analysis Inside the Cavity

• Using a 1-D Cavity Topology, we assume a
  metamaterial slab of thickness d and effective
  material parameters e2 and µ2.

• The goal is to show that the metamaterial
  permittivity e2 has little effect on the electric
  field in the air region.

• After applying boundary conditions at d and 0,
  the electric field in the air region is expressed
  as

Electrically Thin Layers
18
For there to exist a null in Ex1, we require
And since in our domain z imaginary parts tells us that µr2 must be well!
(Null Location, where µr2

• From this requirement, the null location will be
  dependent on the metamaterials permeability and
  thickness

• Thus, we can vary the null position zo by
  controlling the properties of the metamaterial
  slab!

19
Varying the Material Parameters of the
Metamaterial Slab
Holding µr2 constant at -1
Holding er2 constant at 1
Showing how variations in the metamaterials
permeability will affect the electric field. In
this plot, f 2.5 GHz, d2 5 mm (k2d

• It is evident that this field structure is a
  strong function of the metamaterials
• effective permeability.

20
How do Metamaterial Losses Affect the Field
Structure?

• As we add losses to the metamaterial slab, the
  electric field becomes unable to reach a true
  null.

• This is understood from the analytical electric
  field magnitude in the air adding an imaginary
  component to µr2 makes it essentially impossible
  for Ex1 to reach a null.

21
Designing the Metamaterial Slab

• Used resonant rings to realize the effective
  negative permeability

• No need for wire structures (negative e) if the
  phase variation across the cavity is small.

Extracted effective permeability using method
discussed in Smith, et. al. Determination of
effective permittivity and permeability of
metamaterials from reflection and transmission
coefficients.
22
Where should we probe the fields?

• Because the metamaterial is composed of discrete
  metallic inclusions, we must be careful where we
  place our thin wire probe when making field
  measurements.

• We do not want to probe along (line 1) where the
  quasi-static fields associated with it dominate
  over the effective medium fields.

• In experiment, we will probe somewhere between
  lines 2 and 3, where the effective medium fields
  dominate the response.

• This idea was explored in the paper Cummer,
  Popa, Wave Fields measured inside a negative
  refractive index metamaterial.

23
Fabricating the Rings

• The optimal ring design from HFSS was then
  realized on FR4. I utilized standard
  photolithography methods to etch the ring design.

• Because the gap g 0.3 mm was at the limit of
  the resolution of this process, the actual
  fabricated rings were resonant at a slightly
  higher frequency (2.7 GHz).

Above Dimensions of the ring designed to achieve
a negative permeability at about 2.5 GHz.
Above Fabricated rings on FR4. In total, there
were 10 rings in the transverse section of the
waveguide, and two rings in the vertical
direction.
24
Creating the Parallel Plate Waveguide for Field
Measurements
Because all of the simulations and analysis thus
far assumed the unit cell(s) to be illuminated by
a uniform plane wave, it would be inappropriate
to make measurements in a closed waveguide. The
parallel plate waveguide allows TEM propagation
at 0 Hz, and the higher order modes propagate
according to
Where c is the speed of light in a vacuum, h is
the height of the waveguide (hmax in the bottom
diagram), and m 1, 2,3,
Slit for probing the Waveguide
Unit Cells (10 in transverse direction)
Copper Wall
Because only the TEM mode is propagating the
waveguide, the slit running down the propagation
axis will not disturb the field patterns, because
the current induced in the top and bottom plate
of the waveguide to support this mode will run
parallel to this slit.
25
Experimental Setup
Left The experimental setup showing the parallel
plate waveguide with slot for field measurements.
Slot used for Field Measurements
Copper plate to act as PEC
Unit Cell Structures
1.5cm
3 cm
Above Top view of waveguide with center slot.
Markings are spaced 1mm apart from each other
allowing for highly resolved electric field
magnitude measurements.
Above Side view of experimental setup. The white
outlined box shows where the other copper slab
can be placed to realize the thin cavity
resonator, and the green dashed line shows the
theoretical electric field amplitude distribution
inside the cavity
26
Experimental Results
Above Overlay plot showing Analytical,
Simulated, and Measured Spatial Electric Field
Magnitude distributions inside the cavity
resonator.

• It is clear from the measurements that the
  electric field forms a triangular shape, a
  characteristic predicted by Engheta.

• It is clear that Ex1 reaches a null close to the
  interface, thus allowing us to form a cavity that
  is 3 cm in thickness (At this frequency, an
  unloaded cavity would require a thickness of ?o/2
  5.4 cm)

27
The effect of frequency variation on the field
structure

• It is seen that as frequency is increased to
  where µr2 approaches zero, the field null pushes
  closer and closer to the interface as predicted
  in my analysis.

• Loss tangent tan d µr2/µr2 becomes more
  sensitive to subtle changes in µr2 as µr2
  approaches zero. This explains why the loss seems
  to increase as frequency is increased towards the
  µr2 0 frequency.

• Notice as frequency increases, quasi-static
  fields inside the metamaterial slab tend to
  dominate over the effective medium fields.

28
Effective Medium Field Behavior as Permeability
approaches zero

• As µr2 approaches zero, the effective medium
  fields in the slab must vanish

Since

• Fields Plotted for various
• permeabilities

• Permittivity of the slab is held
• constant at 1.

• Notice how effective medium
• fields tend to vanish as µr2 tends
• to zero ?

29
Coupling to S11
No slab present
Resonance due to the shrunken 1.5 cm cavity (no
air layer present)
Resonance due to 3 cm cavity (d1 d2 1.5 cm)
Unloaded 3 cm cavity
Loaded 3 cm cavity

• So the S-Parameter data proves we have coupled
  to a resonant mode at f 2.776 GHz for a 3 cm
  cavity

We have successfully made the cavity 55 its
unloaded size!
30
Project Summary

• Verified theoretically that electrically thin
  slabs can realize the thin resonator, where the
  resonance depends on the metamaterials effective
  permeability, as well as the ratio of the two
  slab thicknesses.

• Field magnitude measurements inside the cavity
  suggest that we can reduce the size of the
  resonator significantly

• Showed that the thin cavity functions as desired
  by measuring both spatial electric field
  magnitude, as well as the coupling to S11 to
  verify resonant characteristics.

• Can better understand the effect that the slab
  parameters has on the electric field structure
  inside the cavity.

• Reduced the cavity to 55 its unloaded size!

31
Upcoming Publications and Projects
Hand, Cummer, Engheta, The Measured Electric
Field Spatial Distribution Within A Metamaterial
Subwavelength Cavity Resonator (Submitted to
IEEE Transactions on Antennas and
Propagation)
Cummer, Popa, Hand Accurate Q-Based Design
Constraints for Resonant Metamaterials and
Experimental Validation
Hand, et. al., Accurate Method for Determining
Q, F and effective Material Parameters of
Magnetically Resonant Materials (Journal and
Submission Date TBD)
Laboratory Manual for EE 53 Steady State and
Transients on Transmission Lines, Crosstalk, and
Antenna Experiments.
32
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