By Gang Bao; Lawrence Cowsar; Wen Masters
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Additional info for Mathematical modeling in optical science
Keywords Electromagnetic ﬁelds mathematical modeling Resonator Waveguide Homogeneous medium Boundary conditions Surface interface Á Á Á Á In this chapter, mathematical analysis of electromagnetic ﬁelds in rectangular dielectric resonator antenna (RDRA) has been introduced. The investigations are based on ﬁrst applying waveguide theory, then converting it to resonator by replacing Àc to d/dz. , exp(−γz). , the transverse coordinates). Waveguide models of four different rectangular DRAs with speciﬁed boundary conditions with homogeneous material having linear permittivity have been mathematically modeled.
Now, rewriting Hz and Ez Hz ¼ mpx npyo Xn C mn cos cos e Àjkz z a b m;n ð2:42Þ Ez ¼ mpx npyo Xn Dmn sin sin e Àjkz z a b m;n ð2:43Þ Here, Cmn and Dmn are the coefﬁcients of Fourier cosine and sine series. cm;n ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h2m;n À x2 le Hence, C mn and Dmn gives us relative amplitudes and phase. Hence, we get solution of possible amplitudes and phase of wave propagating through rectangular waveguide called as modes of propagation. 1. The MATLAB programs are being developed for modes graphical view.
1. Given top and bottom surfaces of RDRA as PMC at z ¼ 0; d; )nÂH ¼0 n Á E ¼ 0; Hy ¼ Hx ¼ 0; Ez ¼ 0; Rest of the other four walls are PEC. n Â E ¼ 0; n Á H ¼ 0; x ¼ 0; a; Ey ¼ Ez ¼ 0; Hx ¼ 0; Fig. 2 Rectangular DRA Mathematical Modeling 37 At, y ¼ 0; b; Ex ¼ Ez ¼ 0; Hy ¼ 0; From separation of variables (Refer Chap. 2), @Hz 1 @ 2 Ez Ex ¼ À 2 jxl @z@x @y jx 1 þ kc2 1 ! 1 @ 2 Ez @Hz À Ey ¼ À 2 jxl @z@y @x jx 1 þ kc2 1 À1 @Ez 1 @ 2 Hz À Hx ¼ 2 @y jx @z@x jxl 1 þ kc2 À1 1 @ 2 Hz @Ez À Hy ¼ 2 jxl 1 þ kc2 jx @z@y @x ð3:1Þ !