Experimental Observation and
Control of Wave Dispersion
Dispersive Effects
Analysis and Modeling
Kyle McLellan
• Electrons in a Lattice
• EM wave in a solid
• Sound in elastic media
Dr. Stephen Remillard, faculty advisor
Hope College Department of Physics
(A/m)
Method-of-Moments simulation using IE3D (Ref 5) yields
both scattering parameters and surface current distribution.
Condensed Abstract
• Dispersion: Deviations from the simple model of constant phase velocity
Kronig-Penny potential in the Schrödinger equation
• Goal 1: Find a hands-on way to investigate the dispersion of electron waves in a crystal lattice
has a transcendental solution:
Left Hand Side of Equation 1
cos( k1d1 ) cos( k 2 d 2 ) −
k12 − k 22
sin(k1d1 ) sin(k 2 d 2 ) = cos (β (d1 + d 2 ))
2k1k 2
2
R.H.S of
E q. 1
1
For
Ba bidde
n
nd
0
ki =
ω
vi
=
2π
λi
• Analysis: Convert the transmission and reflection coefficients into band structure
vi = wave speed
• Purpose: Results in an experimental examination of the band theory of solids
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
d1 d2 d1+d2≡Lattice Constant
λi=wavelength in region “i”
Frequency (GHz)
dω
≠ Constant ⇒ Dispersion!
dβ
 L.H.S. 

 d1 + d 2 
β = cos −1 
dispersionless case:
ω
0.7
ω⋅(d 1+d 2)/2πc
• Findings: Defects produce engineered states in the band gap
K.-P. potential
d1=d2=7 mm
2
• Process: Simulate crystals using hand-made transmission lines with periodic variations
Wave, β=2π/λ
1
Matlab is used to invert
Equation 2 and to calculate
the wave number, β.
• Goal 2: Introduce defects into the crystal which result in controllable states in the band gap
(1)
0.6
Experiment
β
0.4
Band Gap
0.3
Transmission line equations
from Ref 4.
Code Written in MATLAB.
(Ref 6)
The author uses IE3D to
layout the structure and
to perform method-ofmoments EM simulation.
|LHS|<1
0.5
Transmission & reflection parameters of the
dispersive structure are measured with a vector
network analyzer.
The 400+ lines of Matlab
code are used to process raw
transmission & reflection
data.
0.2
Paper design layout
0.1
π
d1 + d 2
β⋅(d1+d 2)/2π
Copper tape
The dispersive
structure is handfabricated using an
Exacto knife.
Periodic Transmission Lines
d1+d2≡Lattice Constant
d1
w2
w1
d2
h
ε1,eff
εs
L
ε i ,eff ≈
ε S +1
Signal in
2
+
ε S −1
2 1 + 12(h / wi )
S11 ≡ Measured reflection coefficient magnitude and phase
S21 ≡ Measured transmission coefficient magnitude and phase
e j ( α + jβ ) L =
Results
Periodic variation in εeff produces a
periodic impedance mismatch of
the wave.
ε2,eff
2 2
1 − S112 + S 212 + (1 + S112 − S 21
) − (2 S11 ) 2
2 S 21
Matlab graphs
b.
d1
d2
As the size of the gap
‘G’ increases, the state
in the gap appears at a
continually lower
frequency until it
reaches the critical
point, where the defect
jumps back to the far
right of the band gap.
Distance of defect from
the center of the band gap
(Delta F)
∆F
This transmission line was fabricated using photolithography.(Ref 2)
(2)
Propagation constant, solved by inverting this equation
Attenuation coefficient (Ref 1)
Simulated & Measured
Transmission, α and β
Dispersion Engineering: Impurity States
p-Silicon (IV)
doped with
Al (III)
G
n-Silicon (IV)
doped with
As (V)
Defect Location From Center
Reduced interstitial spacing simulates p-type doping
Increased interstitial spacing simulates n-type doping
2.5
Experiment Outline
∆ F > 0 N-type
defect state
Delta F (GHz))
1.5
)
1. Write C-based code to evaluate Equation 1 and to invert Equation 2
2. Design a periodic transmission line using an EM field simulator
3. Fabricate the periodic transmission line
4. Measure the transmission and reflection coefficients vs. frequency
5. Use the computer program to compute β vs. frequency with Eq. 2
6. Plot the dispersion relation in the extended or reduced zone scheme
7. Attempt some “dispersion engineering” with an impurity
2
The dispersion curve generated using the above transmission
line is generated three ways:
Analytic: Using Equation 1
Simulated S-Parameters: Using Equation 2
and T&R coefficients from EM sim.
Measured: Using Equation 2 and T&R
coefficients from measurement
1
0.5
0
-0.5
0
1
2
3
4
5
6
7
8
9
10
P type gap state
N type gap state
Ref 3
-1
∆ F < 0 P-type
defect state
-1.5
-2
Gap Size (mm)
Acknowledgements
References
1. W.R. Eisenstadt and Y. Eo, “S-Parameter Based IC Interconnect Transmission Line Characterization,” IEEE Trans. Components, Hybrids and Manufacturing
Technol., 15, no. 4, 483-490 (1992).
2. C. Isaac Angert and S.K. Remillard, "Dispersion in One-Dimensional Photonic Band Gap Periodic Transmission Lines," Microwave and Optical Technology
Letters, 51, no. 4, 1010-1013 (2009).
3. E. Yablonovitch, et. al, “Donor and Acceptor Modes in Photonic Band Structure,” Phys. Rev. Lett., 67, no. 24, 3380-3383 (1991).
4. Brian C Wadell, Transmission Line Design Handbook, Artech House, Inc, Norwood, MA, 1991, Page 94.
5. IE3D EM Design System, Zeland Software Inc, Fremont, CA.
6. MATLAB, The MathWorks, Natick, MA.
A corporate sponsor of R&D at Hope College
Dean of Natural and Applied Science
This work was supported by an R&D contract from Mesaplexx, pty ltd., by the National Science Foundation under
NSF-REU Grant No. PHY-0452206, and by the Hope College Division of Natural and Applied Science.