Question

Consider a TEM plane wave with electric field Ei) = Â E)e-ikız normally-incident on the finite-thickness slab shown below. E 1 €2 E 1 ZEL Z=L Write the reflected electric field as E(") = EY") eikız, the field inside the slab as E(2) = î (Ae-ik3z + Beikaz), and the transmitted field as E(t) = Â Ce-iki(z-L). (a) Determine the magnetic fields H and enforce the boundary conditions at z = EL to obtain 4 equations in the 4 unknowns E", A, B, and C. (b) Define a reflection coefficient Eteiki(-) EO T = (2.18) E"e-iki(-) and numerically solve for the unknowns in MATLAB assuming a unity incident amplitude, f = 1 GHz, 2L = 1/6, €1 = €0, and €2 = 4€0. Compare the reflection coefficient with the well-known result T= I 12 + I23e-21k22L 1+T 121 23e-2ik 2L (2.19) where I 12 = (12 – mi)/ (12+ni) and I 23 = (n3 - 12)/(13 +12) are interfacial reflection coefficients (and here, region 3 is the same as region 1, E3 = E1, so that I 23 = -I 12). Note that failing to include your MATLAB code as a part of your solution will result in zero credit. Submission of your code by any other means, e.g., a separate .m file, a separate PDF file, etc., will result in a reduction in points.

Answer 1

Problem : € С. X E å E' .KI. Z which on the ZEL Incident electric field E! e incident nom normally binite thickness. ík, Z The reflected electric field ² =ñere and the field inside the slab ikzz ) and the trans- mitted field as a) ÊXĒi ixãei eik, z Likiz as 2 (A è ikzz + Ве E i i eik, (2-2) # ܕܐܦ e H etikz ikzz) e t H of . H² - Ý (A/C alkz. ŷ § ¢ ik, (7-2) Boundary conditions are the normal component the magnetic intensity vector is continuous across the b the boundary H' (Zz :-) H² (Z --4) Eelte elet Like H² (z =2) Ht (Z=L) or, Eero 1 Again, E" (Z --2) = E(Z 2-2) E e-ikyl - Apr A ekz2 + Belkzz ET (Z =) EL (Z22) IKIL Like Er e A ei ku ikat را and. LA де + Betina

b) Reflection coefficient is defined as En eik, (-2) r i-iki (-4) e well-known result for the reflection Again, the coefficient is given by > T2 + 123 ezik₂ 22 Г é zî kz 2L 2 Y 50, Eo و درد را e 2 Esie ikyr 1-Tz e i ikit Tz or It Tz 123 Tz %-n, and 23 3월 2+1, 3+12 LikL ézik, 2L é zik, 21 E -PKL (-12 Ta ezik, 22 J = Elé (1-é 21 kq 22 ) or E (1-T2 e Lik₂2) El Te eziki (1-re virkey Tiz (1-12 étikal) - zik, L (1-é 41142) Eor (12 e-zikl ora ye 50, 53 -1) e (112h etike (1-é dikely