2 pages, 1 figure.In the main text of the paper corresponding to the present document, WPP--SPP conversion
devices are considered. Reflection and radiation losses in such structures are evaluated by means of
overlap integrals. In this Auxiliary Material section details of such procedure are provided.Peer reviewe

Bozhevolnyi, Sergey I.

García-Vidal, F. J.

Martín-Moreno, Luis

Moreno, Esteban

Rodrigo, Sergio G.

Digital.CSIC

Evaluation of non-ohmic losses with overlap integralsEsteban Moreno,1 Sergio G. Rodrigo,2 Sergey I. Bozhevolnyi,3 L. Mart´ın-Moreno,2 and F. J. Garc´ıa-Vidal11Departamento de F´ısica Teo´rica de la Materia Condensada,Universidad Auto´noma de Madrid, E-28049 Madrid, Spain2Departamento de F´ısica de la Materia Condensada,Universidad de Zaragoza-CSIC, E-50009 Zaragoza, Spain3Department of Physics and Nanotechnology, Aalborg University, DK-9220 Aalborg Ost, DenmarkIn the main text of the paper corresponding to the present document, WPP↔SPP conversiondevices are considered. Reflection and radiation losses in such structures are evaluated by means ofoverlap integrals. In this Auxiliary Material section details of such procedure are provided.It can be shown that the electromagnetic eigenmodes supported by a cylindrical structure (e.g., a wedge of constantheight) are mutually orthogonal [1]. Let us denote such eigenmodes as|n〉 = |n(rT)〉 = {En(rT),Hn(rT)}, (1)where n = ±1,±2,±3, . . .. The fundamental mode is n = ±1, and negative indices correspond to modes propagatingin the negative z direction. {En,Hn} stands for the electric and magnetic field, and rT = (x, y) are coordinates inthe transverse plane. Eigenmode orthogonality reads〈n|m〉 = 〈n(rT)|m(rT)〉 =∫∫XY planedxdy ez · {En(rT)×H∗m(rT)} = sgn(m)δ|n||m|, (2)where ez is a unit vector along the longitudinal Z axis, the star denotes complex conjugate, and sgn(·) stands forthe sign function. Let us remark that: (i) The dependence on the z coordinate, exp(iknz), has been omitted (kn isthe modal wave vector). (ii) Orthogonality applies both for guided and radiation modes (continuous indices shouldbe used to label radiation modes, but we will avoid this to simplify notation). (iii) Counterpropagating modes withthe same index (e.g., |1〉 and | − 1〉) are not orthogonal. (iv) The scalar product of a mode with itself is proportionalto the power carried in the longitudinal Z direction. (v) In general, the integral should be carried out in the infinitetransverse XY plane. Nevertheless, when one of the modes is guided the integrand is non-negligible only in a finitepart of the XY plane, due to transverse localization of the guided mode. Thus, in our computations of scalar productsshown later, the integration area will be the transverse FDTD simulation window. (vi) In fiber and guided optics,orthogonality conditions are routinely used even when small losses are present.For a general non-cylindrical structure (e.g., a wedge with height varying along the z coordinate), a generic solution|f(x, y, z)〉 can be expanded in eigenmodes. For each z the eigenmodes corresponding to that particular transversecross section, |n(x, y, z)〉, should be used:|f(x, y, z)〉 =∑nan(z)|n(x, y, z)〉, (3)where the coefficients an(z) in the linear expansion are related to the projections (also termed overlaps) of the solution|f〉 on the various eigenmodes |n〉. For instance, the overlap with the fundamental WPP mode (n = +1) is〈f|1〉(z) = 〈f(x, y, z)|1(x, y, z)〉 =∫∫XY planedxdy ez · {Ef(x, y, z)×H∗1(x, y, z)}. (4)When absorption is present, it is convenient to normalize both |f〉 and |1〉 in a particular way that simplifiesthe bookkeeping of radiation leakage. Namely, at every transverse cross section, z = const, the functions |f〉 and|1〉 are normalized to unity in the chosen finite integration area. In the following we will plot the square of theoverlap integral, |〈f|1〉(z)|2, for the structures considered in the paper. Notice that, since |1〉 and | − 1〉 are notorthogonal, this function may include an oscillating term whenever reflection occurs, due to the interference of botheigenmodes and the subsequent formation of a standing wave. On the other hand, the function should be constantfor single mode propagation, no reflection, and negligible radiation losses (the mentioned constant is unity with thechosen normalization). This function is also smaller than unity when the linear expansion of |f〉 includes other modesdifferent from | ± 1〉. For our radiation evaluation purposes, the most important case of this possibility occurs when2radiation is present in the chosen finite normalization area. In this situation |〈f|1〉(z)|2 is smaller than unity. As themodes propagate and radiation leaves the normalization window, |〈f|1〉(z)|2 tends to a unit value.We consider here the structures related to the WPP↔SPP conversion device. We keep the same notation asin the paper, namely, structures I, II, and III correspond to wedges of constant height, linearly decreasing height,and an abrupt decreasing of height, respectively. Figure 1 renders the function |〈f|1〉(z)|2 for the three mentionedstructures. For the function associated to structure I (black line) we distinctly observe three phenomena: (i) smallripples, (ii) a value lower than unity for z < zt = 2µm, and (iii) a value about unity for z > zt. The ripples are dueto the interference of the incoming WPP and a reflected (counterpropagating) WPP. The period of the oscillationis consistent with the WPP wave vector. From the amplitude of the ripples it can be computed that the reflectioncoefficient is 0.1%. This tiny reflection is not physical and it is due to spurious reflection of the WPP mode atthe simulation boundary. The function being smaller than unity for z < zt is due to the fact that, in our FDTDsimulations, the source excites WPP modes and radiation modes. The displayed behavior of |〈f|1〉(z)|2 shows that thecontribution of radiation modes to the total field |f(x, y, z)〉 is negligible (in the transverse simulation window) afterthe excitation transient (i.e., for z > zt). Finally, a value of the function about unity for z > zt demonstrates that,after the excitation transient, radiation does not leak anymore (ohmic absorption is the only source of losses after theexcitation transient in structure I).The analysis of structure II (red line) is analogous: the reflection is still very small (0.2%) and not importantfor our purposes. This tiny reflection is most likely caused by the discontinuity (at zd = 3.9µm) in our conversiondevice. The radiation losses in the excitation transient are similar to those discussed for structure I. Finally, thefunction |〈f|1〉(z)|2 is plotted as long as the WPP mode exists (i.e., for z < zc = 5.8µm). The graph demonstratesthat radiation leakage induced by our conversion device is very small: less than 5% for zt < z < zc. These are theimportant results of the present document: our WPP↔SPP conversion device produces very little amountof reflection and radiation up to the coordinate zc where the WPP mode reaches the cutoff.The data corresponding to structure III (green dotted line) show large oscillations due to the reflection of the WPPmode at the abrupt height discontinuity (zIII = 4.6µm). Reflection is estimated to be about 20% in this case.In summary, reflection and radiation can be estimated with the help of overlap integrals. The performed tests showthat the WPP↔SPP conversion device with varying height produces very small reflection and radiation losses.FIG. 1: Squared overlap integral |〈f|1〉(z)|2 as a function of the longitudinal z coordinate for various structures. Black solidline: structure I, red line: structure II, green dotted line: structure III. The schematics on top of the graph shows the heightprofile for the three considered structures and the physical processes involved.[1] A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

Evaluation of non-ohmic losses with overlap integrals

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