Two-Plate Waveguide PEC plate Subject to b.c. Subject to b.c. 𝐷 𝑥 𝑧 𝐸 𝑥 = 𝐴 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 Subject to b.c. 𝐸 𝑦 = 𝐶 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 Subject to b.c. 𝐷 𝐸 𝑧 = −𝐵 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝑥 𝑧 PEC plate 𝛽 𝑧 = 𝑚𝜋 𝐷 , 𝑚=0,1,… Boundary conditions: sin 𝛽 𝑧 𝑧 =0 𝛽 2 = 𝛽 𝑥 2 + 𝛽 𝑧 2 𝛽 𝑥 = 𝛽 2 − 𝑚𝜋 𝐷 2 𝑧=0, 𝐷

Two-Plate Waveguide Boundary conditions: PEC plate 𝐷 𝑥 PEC plate 𝑧 𝐸 𝑥 = 𝐴 𝑚 sin 𝑚𝜋 𝐷 𝑧 𝑒 −𝑗 𝛽 2 − 𝑚𝜋 𝐷 2 𝑥+ 𝜋 2 𝐸 𝑦 = 𝐶 𝑚 sin 𝑚𝜋 𝐷 𝑧 𝑒 −𝑗 𝛽 2 − 𝑚𝜋 𝐷 2 𝑥+ 𝜋 2 𝐷 𝐸 𝑧 = − 𝐵 𝑚 cos 𝑚𝜋 𝐷 𝑧 𝑒 −𝑗 𝛽 2 − 𝑚𝜋 𝐷 2 𝑥 𝑥 PEC plate 𝑧 𝛽 𝑧 = 𝑚𝜋 𝐷 , 𝑚=0,1,… Boundary conditions: 𝛽 𝑥 = 𝛽 2 − 𝑚𝜋 𝐷 2 𝐸 𝑥 and 𝐸 𝑦 vanish on the PEC surface sin 𝛽 𝑧 𝑧 =0, 𝑧=0, 𝐷 𝛽 2 = 𝛽 𝑥 2 + 𝛽 𝑧 2 𝛽=𝜔 𝜇𝜀

Two-Plate Waveguide PEC plate 𝐷 𝑥 PEC plate 𝑧 𝐻 𝑥 =− 𝐶 𝑚 1 𝜂 𝛽 𝑧 𝛽 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝐸 𝑥 = 𝐴 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝐻 𝑦 = 𝐴 𝑚 2 + 𝐵 𝑚 2 1 𝜂 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝐸 𝑦 = 𝐶 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝐷 𝐸 𝑧 = − 𝐵 𝑚 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝐻 𝑧 = 𝐶 𝑚 1 𝜂 𝛽 𝑥 𝛽 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝑥 PEC plate 𝑧 𝛻× 𝐸 =− 𝜕 𝐵 𝜕𝑡 𝐻 = 𝛻× 𝐸 −𝑗𝜔𝜇 𝛻× 𝐸 =−𝑗𝜔𝜇 𝐻

Two-Plate Waveguide 𝐷 𝑥 𝑧 Field modes: standing wave patterns 𝐻 𝑥 =− 𝐶 𝑚 1 𝜂 𝛽 𝑧 𝛽 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝐸 𝑥 = 𝐴 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝐻 𝑦 = 𝐴 𝑚 2 + 𝐵 𝑚 2 1 𝜂 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝐸 𝑦 = 𝐶 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝐷 𝐸 𝑧 = − 𝐵 𝑚 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝐻 𝑧 = 𝐶 𝑚 1 𝜂 𝛽 𝑥 𝛽 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝑥 𝑧 TM: 𝐻 𝑥 = 𝐻 𝑧 =0 TE: 𝐸 𝑥 = 𝐸 𝑧 =0 Field modes: standing wave patterns One m value corresponds to one mode TM: 𝐶 𝑚 =0, 𝐸 𝑦 =0 TE: 𝐴 𝑚 = 𝐵 𝑚 =0, 𝐻 𝑦 =0 Field modes: TM mode and TE mode TM: Transverse magnetic field → 𝐻 𝑥 = 𝐻 𝑧 =0 TE: Transverse electric field → 𝐸 𝑥 = 𝐸 𝑧 =0

Two-Plate Waveguide 𝐷 𝑥 𝑧 𝐸 𝑥 = 𝐴 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 TM modes 𝐻 𝑦 = 𝐴 𝑚 2 + 𝐵 𝑚 2 1 𝜂 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝐷 TM modes 𝐸 𝑧 = − 𝐵 𝑚 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝑥 𝑧

Two-Plate Waveguide 𝑥 𝑡=0 Plane wave: TEM mode 𝑚=0 TM 0 mode TM modes 𝐸 𝑥 = 𝐴 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝐸 𝑥 =0 𝐸 𝑧 = − 𝐵 𝑚 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝐸 𝑧 =− 𝐵 𝑚 𝑒 −𝑗 𝛽 𝑥 𝑥 =− 𝐵 𝑚 cos 𝜔𝑡− 𝛽 𝑥 𝑥 𝐻 𝑦 = 𝐴 𝑚 2 + 𝐵 𝑚 2 1 𝜂 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥

Two-Plate Waveguide 𝑥 𝑡=0 𝑚=1 TM 1 mode TM modes 𝐸 𝑥 = 𝐴 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝐸 𝑥 = 𝐴 1 sin 𝜋 𝐷 𝑧 cos 𝜔𝑡− 𝛽 𝑥 𝑥− 𝜋 2 𝐸 𝑧 = − 𝐵 𝑚 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝐸 𝑧 =− 𝐵 1 cos 𝜋 𝐷 𝑧 cos 𝜔𝑡− 𝛽 𝑥 𝑥 𝐻 𝑦 = 𝐴 𝑚 2 + 𝐵 𝑚 2 1 𝜂 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥

Two-Plate Waveguide 𝑥 𝑡=0 𝑚=2 TM 2 mode TM modes 𝐸 𝑥 = 𝐴 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝐸 𝑥 = 𝐴 1 sin 2𝜋 𝐷 𝑧 cos 𝜔𝑡− 𝛽 𝑥 𝑥− 𝜋 2 𝐸 𝑧 = − 𝐵 𝑚 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝐸 𝑧 =− 𝐵 1 cos 2𝜋 𝐷 𝑧 cos 𝜔𝑡− 𝛽 𝑥 𝑥 𝐻 𝑦 = 𝐴 𝑚 2 + 𝐵 𝑚 2 1 𝜂 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥