The author uses IE3D to layout the structure and to perform method-of-moments EM simulation.

Experimental Observation and Control of Wave Dispersion

Kyle McLellan, C. Isaac Angert and S.K. Remillard

Hope College Physics Department

Matlab graphs

• Electrons in a Lattice• EM wave in a solid• Sound in elastic media

Dispersive Effects

Acknowledgements

Analysis and Modeling

This material is based upon work supported by the National Science Foundation under NSF-REU Grant No. 0452206

Kronig-Penny potential in the Schrödinger equation

( )( )21221121

22

21

2211 cos)sin()sin(2

)cos()cos( dddkdkkk

kkdkdk +=−− β

(2)

d1 d2 d1+d2≡Lattice Constantλi=wavelength in region “i”

Wave, β=2π/λ

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 152

1

0

1

2R.H.S of Eq. 1

Frequency (GHz)

Left

Han

d S

ide

of E

quat

ion

1

speed wave 2

ii

=== vv

ki

i λπω

+= −

21

1 L.H.S.cos

ddβ

Forbidden

Band

Objective

Experiment Outline

Dispersion Engineering: Impurity States

d1=d2=7 mm

Periodic microwave transmission lines are used to create dispersive effects which mimic thoseassociated with the band theory of solids. Students are equipped with a simple method to construct a crystal lattice using a hobby knife. The measured frequency response is then analyzed with a student-generated code using MatLab and reduced to an experimental dispersion curve.

h

w1

w2

d2d1

εs

ε2,effε1,eff

)/(1212

1

2

1,

i

SSeffi

wh+−++≈ εεε

Periodic variation in εeff produces a periodic impedance mismatch of the wave.

Signal in

S11 ≡ Measured reflection coefficient magnitude and phaseS21 ≡ Measured transmission coefficient magnitude and phase

21

211

2221

211

221

211)(

2

)2()1(1

S

SSSSSe Ljj −−+++−

=+ βα

Propagation constant, solved by inverting this equationAttenuation coefficient (ref. 1)

d1+d2≡Lattice Constant

Band Gap

!Dispersion Constant ⇒≠βω

d

d

β⋅(d1+d2)/2π0 0.1 0.2 0.3 0.4 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ω⋅ (d

1+d 2)

/2π c

Brillouin Zone Edge

β 21 dd +π0

(1)

Periodic Transmission Lines

1. Write C-based code to evaluate Equation 1 and to invert Equation 22. Design a periodic transmission line using an EM field simulator3. Fabricate the periodic transmission line4. Measure the transmission and reflection coefficients vs. frequency5. Use the computer program to compute β vs. frequency with Eq. 26. Plot the dispersion relation in the extended or reduced zone scheme7. Attempt some “dispersion engineering” with an impurity

Dispersion Curves 4 Ways

Analytic: Using Equation 1Simulated S-Parameters: Using Equation 2

and T&R coefficients from EM sim.Measured: Using Equation 2 and T&R

coefficients from measurementSim Current Distr: Using λ observed in the

current distribution to find β

b.d2d1

Results

Experiment

This transmission line was fabricated using photolithography.

A corporate sponsor of R&D at Hope CollegeDean of Natural and Applied Science

The dispersive structure is hand-fabricated using an Exacto knife.

T & R parameters of the dispersive structure are measured with a vector network analyzer.

See Reference 2.

References1. 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.

P type gap state N type gap state

Transmission line equations from Reference 4.Code Written in MATLAB (ref 6).

Method-of-Moments simulation using IE3D (Ref. 5) yields both scattering parameters and surface current distribution.

(A/m)

Matlab is used to invert Equation 2 and to calculate the wave number, β.

has a transcendental solution:

K.-P. potential

ω

β

dispersionless case:

|LHS|<1

Simulated & MeasuredTransmission, α and β

The 400 lines of Matlab code are used to process raw T&R data. Parts of the code are left blank as a programming exercise.

p-Silicon (IV)doped with Al (III)

n-Silicon (IV)doped with As (V)

Reduced interstitial spacing simulates p-type dopingIncreased interstitial spacing simulates n-type doping

Reference 3

Copper tape

Paper design layout

L