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By §iikrii Ekin Kocabas

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Note that we have chosen to write the modal shape in terms of the surface mode formulation of [24, Sec. 3]. Surface modes are the main propagating modes for the MIM waveguide and have hyperbolic modal shapes. It is equivalently possible to describe the modes in terms of oscillatory shapes using trigonometric functions [24]—analogous to the modes of the dielectric CHAPTER 3 . 0 i~n n TM 6 TM 4 TM 0 TM, *TM! 497 and la = A/4 where A = 1550nm is the wavelength of operation. There are four real modes and an infinite number of complex modes, all denoted with the • symbol.

7 - Effects of loss on the spectrum. 517. 6 shows that the forward, proper, complex modes on the fourth quadrant have moved to the third quadrant and thus became leaky modes. 1 Birth of the Discretuum The presence of a continuous spectrum leads to the formation of integral equations when the mode-matching method is applied [55, Ch. 5]. The integral equation is then expanded using an orthogonal basis set—not necessarily that of the modes—to solve the scattering problem. Another way to approach the scattering problem is to limit the transverse coordinates by a PEC wall.

For the sake of clarity, we only drew one branch of modes after the bifurcation. 5, the quite small real part of kz for the complex modes after the bifurcation line is not visibly discernible, but is numerically there. 4. The two branches after the bifurcation form the forward and backward proper, complex modes. o,) are also shown. 7. 6 turn into leaky modes by migrating into the third quadrant of the complex Km plane. 4. The upper line in each row is the value for the em e R case, the lower line is for the em e C case.

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