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 ( 29 ( 29 2 1 2 2 1 1 2 1 2 2 2 1 2 2 1 1 cos ) sin( ) sin( 2 ) cos( ) cos( d d d k d k k k k k d k d k + = - - β (2) d 1 d 2 d 1 +d 2 ≡Lattice Constant λ i =wavelength in region “i” Wave, β=2π/λ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 1 0 1 2 R.H.S of Eq. 1 Frequency (GHz) Left Hand Side of Equation 1 speed wave 2 i i = = = v v k i i λ π ϖ + = - 2 1 1 L.H.S. cos d d β For b idden Ban d Objective Experiment Outline Dispersion Engineering: Impurity States d 1 =d 2 =7 mm Periodic microwave transmission lines are used to create dispersive effects which mimic those associated 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 w 1 w 2 d 2 d 1 ε s ε 2,eff ε 1,eff ) / ( 12 1 2 1 2 1 , i S S eff i w h + - + ≈ ε ε ε Periodic variation in ε eff produces a periodic impedance mismatch of the wave. Signal in S 11 ≡ Measured reflection coefficient magnitude and phase S 21 ≡ Measured transmission coefficient magnitude and phase 21 2 11 2 2 21 2 11 2 21 2 11 ) ( 2 ) 2 ( ) 1 ( 1 S S S S S S e L j j - - + + + - = + β α Propagation constant, solved by inverting this equation Attenuation coefficient (ref. 1) d 1 +d 2 ≡Lattice Constant Band Gap ! Dispersion Constant ⇒ ≠ β ϖ d d β⋅ ( d 1 +d 2 )/2 π 0 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 ϖ⋅ ( d 1 +d 2 )/2 π c Brillouin Zone Edge β 2 1 d d + π 0 (1) Periodic Transmission Lines 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 Dispersion Curves 4 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 Sim Current Distr: Using λ observed in the current distribution to find β . d 2 d 1 Results Experiment This transmission line was fabricated using photolithography. A corporate sponsor of R&D at Hope College Dean 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. 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. 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 ϖ β dispersionless case: |LHS|<1 Simulated & Measured Transmission, 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 doping Increased interstitial spacing simulates n-type doping Reference 3 Copper tape Paper design layout L