An Application of Negative Index Metamaterials: by eL3VEnx

VIEWS: 18 PAGES: 32

									“Metamaterial Loaded Compact Cavity Resonators”


                                 E
                             k       H




                     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
                          Overview of Presentation
Part I:   Metamaterials Overview
 History and Theory of Operation

Part II: Applying Metamaterials to Create Thin Subwavelength Cavity Resonators
 Engheta’s Idea
 Hrabar’s Study
 Kong’s 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
               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.
                        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
εeff 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 λo as
to prevent diffraction effects that would ruin this effective medium picture.

                Size Restrictions:

                                                 (Pendry, et. al. “Magnetism from Conductors and
           a << λo = 2πcoω-1                     Enhanced Nonlinear Phenomena):



 In practice, we like to keep the unit cell dimensions around λo/10, although cell
sizes on the order of λo/6 have proved to keep the effective medium picture intact.
           (Smith, et. al., “Electromagnetic parameter retrieval from inhomogeneous metamaterials):
              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     Pendry          Pendry proposes         Smith is the first in the
       the effect a negative      proposes wire   Split Ring              world to physically
       permittivity and           structures to   Resonators (SRR’s)      realize a medium with
       permeability has on        realize a       to realize a negative   an effective negative
       wave propagation           negative        permeability            index of refraction
                                  permittivity
           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 ε and μ allow for a broader electromagnetic palette

Example: No Cut-Off Waveguide:
Dispersion Relation in Rectangular WG loaded with anisotropic NIM:

                                      y 2
                                        2               As can be seen, by choosing εx <0, μy < 0
                     kz   x  y 2 
                      2
                                         ky              and μz > 0 , kz will always be positive and
                                 c    z                 there will be no lower cutoff frequency.
                             …Negative Refraction Continued
Snell’s Law at the interface between a
negative index material and a positive index
material:

      n>0                   n<0               n>0




  http://sagar.physics.neu.edu/wavepacket_refraction.htm   http://www.utexas.edu/research/cemd/nim/Intro.html



                               Light Bending the Wrong Way?
n1 sin i  n2 sin t ,                       and for n1 > 0 and n2 < 0,

           n1      
 t  sin  sin  i 
          n
            1
                    
                                              Refracted beam will be opposite to the normal
                                              as shown in the animation above.
           2       
                                   Realizing a Negative Permittivity
        1968             1996          1999          2000
                                                               Time

Veselago first      Pendry         Pendry           Smith is the first
studies the         proposes       proposes Split   in the world to
effect a negative   wire           Ring             realize a medium
permittivity and    structures     Resonators       with an effective
permeability has    to realize a   (SRR’s) to       negative index of
on wave             negative       realize a        refraction
propagation         permittivity   negative
                                   permeability




 The Drude Model of Permittivity:

                                      2                                   2
                                                                                  2co   2

              r  1 
                                                              2         Ne
                                                        p                     2
                                       p
                                                                      
                                       2                                me o  a Ln ( a / r )

  We want εr to be small and negative since a large and negative εr could shrink λeff
 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.)
                                        Realizing a Negative Permeability
        1968             1996           1999          2000
                                                                Time
                                                                             The Lorentz Model of Permeability:
Veselago first      Pendry          Pendry           Smith is the first



                                                                                       r  1                F 2
studies the         proposes        proposes Split   in the world to
effect a negative   wire            Ring             realize a medium


                                                                                                                          o
permittivity and    structures      Resonators       with an effective
permeability has    to realize a    (SRR’s) to       negative index of
on wave
propagation
                    negative
                    permittivity
                                    realize a
                                    negative
                                                     refraction
                                                                                                         o  2 
                                                                                                          2
                                                                                                                     j
                                    permeability
                                                                                                                          Q

                                    Pendry proposed split ring resonators (SRR’s) 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.

                                                                                       o H inc A / L      r Is related to this
                                                                            I ind             o
                                                                                                2              current
                                                                                          1   2
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:                                                                 Kong:
Short Paper that summarized Engheta’s                                          More in depth experimental
theory. They measured the spatial phase                                      investigation behind Engheta’s
variation in the cavity.                                                            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
                               Engheta’s 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

      k2                  k1
         tan(k1d1 )         tan(k2 d 2 )  0
      2                  1

When the metamaterial            d1  2
slab has an effective                
negative permeability as         d2      1
well as being electrically
thin, we are left with the ( 1  1 , 2   2 )
constraint:
The electric fields in both slabs can be expressed as:

                    Ex1  Eo sin(k2 d 2 )sin(k1 z )
                    Ex 2  Eo sin(k1d1 )sin(k2 (d1  d 2  z ))
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!
 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!

 Thus, if d1<  2k , Then to satisfy the dispersion relation, d2 must be >  2k
                   1                                                             2



 This constrains      And makes the resonant cavity dependent on d1+d2 !




So clearly using the metamaterial slab allows us to build a
more compact resonator than if we had used a RH bilayer.
                       Hrabar’s Experiment
 First experimental validation of Engheta’s 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
                       Kong’s Experiment
 Expands upon Hrabar’s experiment

 Shows that the resonant frequency is invariant for various slab
  thicknesses, as long as Engheta’s 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.
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.
                Field Analysis Inside the Cavity
                                                    Using a 1-D Cavity Topology, we assume a
                                                   metamaterial slab of thickness d and effective
                                                   material parameters ε2 and μ2.

                                                    The goal is to show that the metamaterial
                                                   permittivity ε2 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:
                                                                                      
                                                                                      
            E x1  Eo  e  jko ( z  d )  1                         e jko ( z  d ) 
                                                          2
                                                    k                                
                                               1  j o 2 tan (k 2 d )                 
                                                    k2 o                                Electrically
                                                                                             Thin Layers
                                        1  jko ( z  d )  
                          E x1  2 Eo 1  
                                           1  jk  d   
                                                              
                                                 o r2     
For there to exist a null in Ex1, we require


            1  jko ( z  d )  1  jko r 2 d
And since in our domain z < -d, equality of imaginary parts tells us that μr2
must be < 0 as well!


   zo  d ( r 2  1)             (Null Location, where μr2 < 0)

 From this requirement, the null location will be dependent on the
metamaterial’s permeability and thickness

 Thus, we can vary the null position zo by controlling the properties of the
metamaterial slab!
Varying the Material Parameters of the Metamaterial Slab




        Holding εr2 constant at +1                             Holding μr2 constant at -1

   Showing how variations in the metamaterial’s permeability will affect the electric
   field. In this plot, f = 2.5 GHz, d2 = 5 mm (k2d < 0.3)



 It is evident that this field structure is a strong function of the metamaterial’s
  effective permeability.
 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.
              Designing the Metamaterial Slab
 Used resonant rings to realize the effective negative permeability

 No need for wire structures (negative ε) 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.”
                 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.”
                                    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      Above: Fabricated rings on FR4. In total, there were 10
achieve a negative permeability at about 2.5   rings in the transverse section of the waveguide, and two
GHz.                                           rings in the vertical direction.
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
                                        mc
                                   fc                       vacuum, h is the height of the
                                                             waveguide (hmax in the bottom
                                        2h                   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.
                              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       Above: Side view of experimental setup. The white
center slot. Markings are spaced 1mm    outlined box shows where the other copper slab can be
apart from each other allowing for      placed to realize the thin cavity resonator, and the green
highly resolved electric field          dashed line shows the theoretical electric field amplitude
magnitude measurements.                 distribution inside the cavity
                                      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 λo/2 = 5.4 cm)
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 δ = 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.
Effective Medium Field Behavior as Permeability approaches zero

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


                                                                 
Ex 2  E e
          2x       jk2 z
                             e   jk2 z
                                             0   Since   k2 
                                                                  c
                                                                         r 2 r 2          0

                                                      Fields Plotted for various
                                                       permeabilities
                                  r 2  1

                                                      Permittivity of the slab is held
                   r 2  0.5                         constant at +1.


               r 2  0.1                            Notice how effective medium
                                                       fields tend to vanish as μr2 tends
                                                       to zero ?
                                 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        We have successfully made the
to a resonant mode at f = 2.776 GHz for a 3 cm          cavity 55% its unloaded size!
cavity
                          Project Summary
 Verified theoretically that electrically thin slabs can realize the thin resonator,
where the resonance depends on the metamaterial’s 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!
   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.
                                                                     References
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517- 526, 1967), Soviet Phys. Uspekhi, vol. 10, no. 4, pp. 509- 514, 1968.              efficients," Phys. Rev. B, vol. 65, No. 195104, pp. 1-5, 2002.

[2] N. Engheta, “An Idea for Thin Subwavelength Cavity Resonators Using Meta-           [12] Koschny, et. al., “Resonant and Anti-resonant frequency dependence of the ef-
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[3] A.E. Centeno and P.S. Excell, “High-Q Dielectrically Loaded Electrically Small      [13] Bogdan-Ioan Popa, Steven A. Cummer, “Determining the effective electromag-
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[4] Z.M. Hejazi, P.S. Excell, “Miniature HTS spiral cavity resonator for mobile tele-   [14] B. J. Justice, J. J. Mock, L. Guo, A. Degiron, D. Schurig, and D. R. Smith,
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