Benjamin D. Braaten* North Dakota State University, Fargo ND, USA David A. Rogers

1. Current Distribution of a Printed Dipole with Arbitrary Length Embedded in Layered Uniaxial Anisotropic Dielectrics Benjamin D. Braaten* North Dakota State University, Fargo ND, USA David A. Rogers North Dakota State University, Fargo ND, USA Robert M. Nelson University of Wisconsin – Stout, Menomonie WI, USA North Dakota State University

2. Topics • Problem Definition • Spectral domain immittance functions • Spectral domain moment method • Results/Discussion • Conclusion North Dakota State University

3. Problem Definition Consider: North Dakota State University

4. The spectral domain immittance functions Start with the following Hertz vector potentials: Electric Hertz potential and Magnetic Hertz potential North Dakota State University

5. The spectral domain immittance functions • Next, only the y-direction of the Hertz vector potential is needed. and • This is because the optical axis is in the y-direction and • this component satisfies the higher order TE and TM tangential boundary conditions. North Dakota State University

6. The spectral domain immittance functions Now define the following expression for the magnetic and electric field: where the Hertzian vector potentials are solutions to the following equations: North Dakota State University

7. The spectral domain immittance functions and North Dakota State University

8. The spectral domain immittance functions To simplify evaluating the previous expressions, we define the following Fourier transform: This results in the following relations: North Dakota State University

9. The spectral domain immittance functions This results in the following simplified expressions: where and North Dakota State University

10. The spectral domain immittance functions Similarly for and North Dakota State University

11. The spectral domain immittance functions Next, consider: Enforcing the B.C. with the previous expressions results in the following simple relation: North Dakota State University

12. The Spectral Domain Moment Method Next, the x-component of the electric field in each region can be written in the spatial domain as: Since is in the spatial domain, the two-dimensional inverse Fourier transform will need to be applied to North Dakota State University

13. The Spectral Domain Moment Method Proceeding in this manner results in the following expression: Next, the current in terms of the basis functions: North Dakota State University

14. The Spectral Domain Moment Method Defining weighting functions and rearranging gives: where and North Dakota State University

15. The Spectral Domain Moment Method Notice that the following expression is in the previous rearrangement: This is the 2D FT of the basis function. This can be useful if a basis function with an analytical FT is chosen. North Dakota State University

16. The Spectral Domain Moment Method Using this simplification results in the following expression: Integration only on a single plane Notice North Dakota State University

17. The Spectral Domain Moment Method For this work, PWS basis functions were used: North Dakota State University

18. The Spectral Domain Moment Method The alpha-beta plane of integration: North Dakota State University

19. Numerical Results Single Layer L = .5λ0 W = .0004λ0 d1 = .1016λ0 1V source North Dakota State University

20. Numerical Results Single Layer Notice: Notice that the imaginary part can be individually modified (compare the solid lines with the dashed lines) North Dakota State University

21. Numerical Results Triple Layer L = .25λ0 W = .00083λ0 d1 = .0026λ0 d2 = .0026λ0 d3 = .0026λ0 ε1 = 3.25 (iso.) 1V source North Dakota State University

22. Numerical Results Triple Layer Notice: Notice that both the real and imaginary parts of the current change from the isotropic case when each permittivity component is modified. North Dakota State University

23. Conclusion • A summary on the spectral domain moment method has been presented. • A single printed dipole on a single anisotropic substrate has been investigated. • It was shown that with certain permittivity components, the imaginary part of the current could be modified while the real part of the current remained unchanged. • A single printed dipole in three anisotropic layers has been investigated. • It was shown that each component of the permittivity in the superstrate and substrate had an effect on both the real and imaginary part of the current. North Dakota State University

24. Questions Thank you for listening North Dakota State University