Visualization of guided and leaky wave behaviors in an indium tin oxide metallic slab waveguide.

Visualization of guided and leaky wave behaviors in an indium tin oxide metallic slab waveguide Stephanie M. Teo,1 Christopher A. Werley,1,2 Congshun Wang,3 Kebin Fan,3,4 Benjamin K. Ofori-Okai,1 Xin Zhang,3 Richard D. Averitt,5,6 and Keith A. Nelson1,* 2

1 Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA Currently with Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA, 02139, USA 3 Department of Mechanical Engineering, Boston University, Boston, MA, 02215, USA 4 Currently with Department of Electrical and Computer Engineering, Duke University, Durham, NC, 27708, USA 5 Department of Physics, Boston University, Boston, MA, 02215, USA 6 Currently with Department of Physics, University of California San Diego, La Jolla, CA, 92023, USA * [email protected]

Abstract: We explored the use of the optically transparent semiconductor indium tin oxide (ITO) as an alternative to optically opaque metals for the fabrication of photonic structures in terahertz (THz) near-field studies. Using the polaritonics platform, we confirmed the ability to clearly image both bound and leaky electric fields underneath an ITO layer. We observed good agreement between measured waveguide dispersion and analytical theory of an asymmetric metal-clad planar waveguide with TE and TM polarizations. Further characterization of the ITO revealed that even moderately conductive samples provided sufficiently high quality factors for studying guided and leaky wave behaviors in individual transparent THz resonant structures such as antennas or split ring resonators. However, without higher conductive ITO, the limited reflection efficiency and high radiation damping measured here both diminish the applicability of ITO for high-reflecting, arrayed, or long path-length elements. ©2015 Optical Society of America OCIS codes: (070.7345) Wave propagation; (110.6795) Terahertz imaging; (130.2790) Guided waves; (230.7400) Waveguides, slab; (310.7005) Transparent conductive coatings.

References and links 1.

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz polaritonics,” Annu. Rev. Mater. Res. 37(1), 317–350 (2007). 2. C. A. Werley, K. A. Nelson, and C. Ryan Tait, “Direct visualization of terahertz electromagnetic waves in classic experimental geometries,” Am. J. Phys. 80(1), 72–81 (2012). 3. C. A. Werley, K. Fan, A. C. Strikwerda, S. M. Teo, X. Zhang, R. D. Averitt, and K. A. Nelson, “Time-resolved imaging of near-fields in THz antennas and direct quantitative measurement of field enhancements,” Opt. Express 20(8), 8551–8567 (2012). 4. C. A. Werley, S. M. Teo, B. K. Ofori-Okai, P. Sivarajah, and K. A. Nelson, “High-Resolution, Low-Noise Imaging in THz Polaritonics,” IEEE Trans. THz Sci. Tech. (Paris) 3(3), 239–247 (2013). 5. T. P. Dougherty, G. P. Wiederrecht, and K. A. Nelson, “Impulsive stimulated Raman scattering experiments in the polariton regime,” J. Opt. Soc. Am. B 9(12), 2179–2189 (1992). 6. C. Yang, Q. Wu, J. Xu, K. A. Nelson, and C. A. Werley, “Experimental and theoretical analysis of THzfrequency, direction-dependent, phonon polariton modes in a subwavelength, anisotropic slab waveguide,” Opt. Express 18(25), 26351–26364 (2010). 7. B. K. Ofori-Okai, P. Sivarajah, C. A. Werley, S. M. Teo, and K. A. Nelson, “Direct experimental visualization of waves and band structure in 2D photonic crystal slabs,” New J. Phys. 16(5), 053003 (2014). 8. F. Blanchard, K. Ooi, T. Tanaka, A. Doi, and K. Tanaka, “Terahertz spectroscopy of the reactive and radiative near-field zones of split ring resonator,” Opt. Express 20(17), 19395–19403 (2012). 9. M. Liu, H. Y. Hwang, H. Tao, A. C. Strikwerda, K. Fan, G. R. Keiser, A. J. Sternbach, K. G. West, S. Kittiwatanakul, J. Lu, S. A. Wolf, F. G. Omenetto, X. Zhang, K. A. Nelson, and R. D. Averitt, “Terahertz-fieldinduced insulator-to-metal transition in vanadium dioxide metamaterial,” Nature 487(7407), 345–348 (2012). 10. M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3(3), 152–156 (2009).

#233255 - $15.00 USD Received 27 Jan 2015; revised 11 Apr 2015; accepted 11 Apr 2015; published 29 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.014876 | OPTICS EXPRESS 14876

11. D. R. Ward, F. Hüser, F. Pauly, J. C. Cuevas, and D. Natelson, “Optical rectification and field enhancement in a plasmonic nanogap,” Nat. Nanotechnol. 5(10), 732–736 (2010). 12. D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B 23(3), 391–403 (2006). 13. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007). 14. T. Bauer, J. S. Kolb, T. Löffler, E. Mohler, H. G. Roskos, and U. C. Pernisz, “Indium-tin-oxide-coated glass as dichroic mirror for far-infrared electromagnetic radiation,” J. Appl. Phys. 92(4), 2210–2212 (2002). 15. C. G. Granqvist and A. Hultaker, “Transparent and conducting ITO films: new developments and applications,” Thin Solid Films 411(1), 1–5 (2002). 16. J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photonics 1(1), 58– 106 (2009). 17. E. Cassedy and M. Cohn, “On the existence of leaky waves due to a line source above a grounded dielectric slab,” IEEE Trans. Microw. Theory Tech. 9(3), 243–247 (1961). 18. H. A. Haus and D. A. B. Miller, “Attenuation of cutoff modes and leaky modes of dielectric slab structures,” IEEE J. Quantum Electron. 22(2), 310–318 (1986). 19. K.-H. Lin, C. A. Werley, and K. A. Nelson, “Generation of multicycle terahertz phonon-polariton waves in planar waveguide by tilted optical pulse fronts,” Appl. Phys. Lett. 95(10), 103304 (2009). 20. H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2(5), 295–298 (2008). 21. D. Marcuse and I. P. Kaminow, “Modes of a symmetric slab optical waveguide in birefringent media-Part II: Slab with coplanar optical axis,” IEEE J. Quantum Electron. 15(2), 92–101 (1979). 22. I. P. Kaminow, W. L. Mammel, and H. P. Weber, “Metal-clad optical waveguides: Analytical and experimental study,” Appl. Opt. 13(2), 396–405 (1974). 23. D. H. Auston and M. C. Nuss, “Electrooptic generation and detection of femtosecond electrical transients,” IEEE J. Quantum Electron. 24(2), 184–197 (1988). 24. D. B. Hall and C. Yeh, “Leaky waves in a heteroepitaxial film,” J. Appl. Phys. 44(5), 2271–2274 (1973).

1. Introduction Photonic elements such as metamaterials, bandgap crystals, and optical modulators are key components in next-generation optical signal processing and telecommunications systems. A powerful tool for studying and building intuition about such systems is the polaritonics platform [1], in which terahertz-frequency (THz) light is generated and detected in a thin slab of an electro-optic crystal, typically lithium niobate (LN). Metallic and dielectric photonic components can be fabricated directly onto the crystal slab [2,3], and THz waves can be visualized [4] as they propagate at the speed of light and interact with these components. These on-chip capabilities create an integrated environment for THz spectroscopy, signal processing, and the development of photonic devices. In this paper, we deposit indium tin oxide (ITO) on one surface of the LN slab to enable direct visualization of waveguide modes expected in an asymmetric metal-clad waveguide, and we assess the potential for using ITO in THz devices. The broad set of tools and capabilities in the polaritonics system results from its unique experimental geometry. THz waves are generated via impulsive stimulated Raman scattering [5] by focusing an intense, ultrafast near infrared (NIR) pump pulse through a thin LN crystal. Due to the large mismatch between refractive indices at THz and NIR frequencies, THz waves propagate laterally, perpendicular to the pump pulse if it is at normal incidence, and are waveguided within the slab [1,6] (see Fig. 1). As the THz waves propagate through the crystal, they modify the refractive index of LN via the electro-optic effect. With a phasesensitive imaging technique [4], an image of the spatially-dependent index change can be recorded using an expanded optical probe beam. By recording images with various pumpprobe delays, one can assemble a “movie” of the propagating THz wave. The method is capable of measuring THz electric fields (E-fields) quantitatively and non-invasively with deep-subwavelength resolution (~λ/100 at 0.5 THz). These capabilities have enabled the study of a diverse set of photonic components including diffractive elements, dipole antennas, waveguides, variable frequency filters or reflectors, and split ring resonators [2,3,6–8]. In the polaritonics platform, metals are an important class of materials because of their strong interactions with light and unique optical properties. Metal microstructures can be used, for example, to enhance E-fields and localize light well below the diffraction limit [3,9– 11], or to modify the effective dielectric properties of a substrate such as in metamaterials [12]. Solid sheets of metal modify boundary conditions and dispersion of waveguides [13]. A


drawback to using conventional metals on the polaritonics platform, however, is that they are opaque to optical light and thus the THz fields underneath the metal microstructures cannot be visualized using the method previously described [4]. Here we investigate the potential for the optically transparent electrical conductor ITO to substitute for ordinary metals in the polaritonics platform. As a highly doped, wide-bandgap semiconductor, ITO is transparent across much of the visible spectrum but has metal-like behavior at THz frequencies [14].

Fig. 1. A schematic illustration of the polaritonics platform consisting of a thin slab of LiNbO3 that allows for the generation, control, and detection of THz waves. Focusing an ultrafast nearinfrared pump pulse through the slab generates THz waves that are waveguided laterally down the slab.

2. Experimental methods The ITO-coated waveguides were fabricated by temporary attachment of a delicate, 54-μm thick x-cut LN slab (10 × 11 mm) to a silicon wafer with a thin layer of photoresist. Following this, a 2.7 µm-thick layer of ITO (10wt% of SnO2) was sputtered in a square pattern (5 × 5 mm) onto part of the slab as shown schematically in Fig. 1. Thermal annealing of the sample at 400 °C for one hour in a nitrogen atmosphere increased the DC conductivity of the ITO, as measured with a four-point probe, from σDC = 1.7 × 10−2 S/cm to 2 × 102 S/cm. Although this attained conductivity was lower than the characteristic optimized conductivity often claimed in literature for much thinner ITO films (σDC ~104 S/cm [15]), it is sufficiently high to waveguide THz fields with behavior expected for a metal-clad slab waveguide, as will be demonstrated in the next section. The NIR pump pulse (1.5 mJ, 80 fs, 800 nm center wavelength, 1 kHz repetition rate from a Ti:sapphire regenerative amplifier) was focused to a vertical line with a cylindrical lens to generate a pair of broadband counter-propagating, initially single-cycle THz waves in the slab that travel perpendicular to the line focus [see Fig. 2(b)]. TE modes (E-field polarized along the direction of the line focus) were generated when the pump light polarization and optic caxis [z-axis in Fig. 1] of the LN crystal were aligned along the direction of the line focus and perpendicular to the THz wave propagation direction (y-axis). The width of the “line” focus was ~20 μm, sufficiently narrow to provide access to a wide wave vector range. TM modes (E-field polarized out of the plane of the slab and along the THz wave propagation direction) were generated when the light polarization and c-axis were parallel to the propagation direction [6]. In this case, we used a ~100 μm cylindrically focused spot size, which allowed for close examination of the low-wave vector region. In both cases, the optical probe pulse (200 nJ, 100 fs, 532 nm center wavelength generated with a non-collinear optical parametric amplifier) was polarized at a 45° angle relative to the c-axis of LN. The sample was imaged onto a camera in the polarization gating configuration [4], where phase information in the sample was converted into amplitude information at the camera with the appropriate polarization optics. The imaging system used a magnification of 6.6 × and had an image resolution of 5 μm.


Fig. 2. (a) A raw optical image of an ITO-coated slab of LN (right side) that demonstrates the optical transparency of the ITO. The vertical dashed line represents the separation between uncoated LN (left) and the ITO-coated area (right). (b) A time series of selected images from Media 1 of a TE mode THz E-field that were collected using polarization gating imaging. A signal image, I(y,z,t), recorded at variable time t following THz field generation by the pump pulse was divided by a reference image, I0(y,z,t), recorded with the THz field absent and the signal modulation, [(I - I0)/I0](y,z,t), was determined to produce the images shown. The THz Efield is directly proportional to ΔI/I0. (c) THz E-field evolution as a function of space and time derived from averaging over the vertical dimension of a series of images, including those in (b).

Figure 2(b) shows a selection from the complete set of recorded THz E-field images at a series of pump-probe time delays (see Media 1). The leftward propagating wave travels unimpeded through the uncoated slab. Part of the rightward propagating wave reflects off the edge of the ITO-coated region of the slab [dashed white line in Fig. 2], and the rest propagates under the optically transparent ITO layer. THz E-fields underneath the ITO are directly and clearly observed. For each image in the series (i.e. each time delay) we averaged over the uniform, vertical dimension, conserving the information on lateral propagation, and placed the resulting vectors in different rows of a matrix. The compiled space-time plot, shown in Fig. 2(c), compactly shows the full evolution of the THz E-field, where dispersion, interference, and reflection of the waveguide modes are apparent. 3. Results and discussion An experimentally measured dispersion curve can be calculated by taking the 2D Fourier transform of a space-time plot like the one in Fig. 2(c). Figure 3 shows the dispersion curves determined for the TE and TM modes. Analytical waveguide dispersion curves for ideal metal-coated LN (red-dashed lines) are overlaid on the experimental results [see Appendix 1 for a theoretical derivation of bound TE and TM modes in this geometry]. For a LN waveguide with ITO coated on one side, the experimental TE and TM dispersion curves in Fig. 3 demonstrate good agreement with the theoretical solutions of their perfect metal analogs. In the TE-polarization, the cutoff frequency, evident at 270 GHz, is a feature that is non-existent in uncoated dielectric waveguides [see Appendix 1 for an analytical


description of the resulting cutoff frequency for the lowest order mode]. The calculated transverse field profiles for the two lowest order TE and TM modes at 1 THz are shown in the insets of Figs. 3(a) and 3(b), respectively; these field profiles assume a perfect metal cladding such that the field decays to zero at the metal-LN boundary. In reality, some of the field penetrates into the ITO layer due to its lower conductivity compared to an ideal metal. We expect a potential for further dispersion control in on-chip THz components, for example with the extension to a symmetric metal-clad waveguide geometry for both TE and TM mode excitations [see Appendix 1].

Fig. 3. Waveguide modes in a LN waveguide (thickness  = 54 µm) with ITO deposited on one surface. Dispersion of air and bulk LN extraordinary (eo) and ordinary (o) waves are shown in solid yellow. Analytical guided solutions in dashed red are overlaid on the experimentally observed results where we assume an ideal metal cladding. The insets show cross-section views of the slab. (a) TE guided modes. Inset: calculated Ez-field profiles of the two lowest order TE modes at 1 THz. (b) TM guided (red-dashed) and leaky (white-dot-dashed) modes. Inset: calculated Ex and Ey-field components of the two lowest order bound TM modes at 1 THz.

In Fig. 3(b), we clearly detected signal due to “leaky” modes, i.e. unbound modes that lie outside the light lines in the dispersion curves [16]; because these waves are not eigenmodes of the system, they are not guided within the slab. Leaky waves have complex propagation constants in both longitudinal and transverse directions and consequently attenuate as they propagate longitudinally and grow exponentially in the transverse direction. The TM leaky waves in Fig. 3(b) give rise to longitudinal phase velocities faster than the speed of light and radiate out of the waveguide. These characteristics are vastly different from those of surface waves, another set of proper modal solutions of the system, in addition to the guided modes shown in Fig. 3, which have slow longitudinal phase velocities and cannot radiate out of the system because they are bound to the surface [17]. The leaky TM waves were excited directly in the ITO-coated region of the slab by using a loosely focused optical pump to generate lowwave vector THz fields; leaky TE waves were not observed with the more tightly focused pump spot size that was used, but should be expected with larger spot sizes.


Because leaky modes are analogous to resonant states in a Fabry-Perot cavity [18], resulting from partial reflections off the substrate interfaces, we observed as expected that the real part of the propagation constants resembled the guided TM modes of a symmetric metalclad waveguide [see Appendix 1], with integral numbers m of half-wavelengths in the waveguide thickness. For an asymmetric metal-clad waveguide, the leaky waves only resemble the odd-m subset of these guided modes. This result arises because the Fabry-Perot solutions must be modified to account for reflections off the LN-air interface, where the waves, unlike at a metal-dielectric interface, do not undergo a π-phase shift. In this case, these leaky waves satisfy the Fabry-Perot boundary conditions when an odd-integral number of half-wavelengths constructively interfere in a cavity twice the length of that in the symmetric metal-clad geometry [see Appendix 2 for further discussion]. The odd-m order TM modes (with symmetric transverse Ey profiles) calculated for a symmetric metal-clad waveguide of twice the core thickness are overlaid as white-dot-dashed lines in Fig. 3(b) and show good agreement with the leaky TM dispersion curves.

Fig. 4. 2D space-time plot derived from Media 2 of a single-sided ITO-clad LN waveguide of thickness 30 μm, with the incident THz TE-polarized field tuned to 0.25 THz, well below the cutoff frequency at 0.50 THz. The contrast has been adjusted to saturate in the uncoated LN region such that unbound modes are more easily observed.

For the following analyses, we focused exclusively on the TE modes that have higher pumping efficiencies compared to that of TM modes since only E-field components of the modes along the c-axis (i.e. the pump polarization) inside the crystal can be generated [6]. Having demonstrated both guided and leaky wave behaviors predicted for a metal waveguide in the ITO-clad system, we further evaluated ITO’s potential for reflecting or scattering elements in the waveguide at frequencies well below the cutoff frequency. We measured the reflectivity of ITO in a thinner LN slab ( l = 30 µm) coated with ITO on one side, such that the cutoff frequency fell well within our THz pulse bandwidth and strong signals could be observed at lower frequencies. We generated a low frequency, TE-polarized, narrowband THz wave using a tilted optical pulse front [19] (see Media 2). This generation method, in which the optical pump profile moves laterally along with the THz wave for several mm at the speed of a selected frequency component, allows for high spectral brightness at this frequency, which was tuned to 0.25 THz, far below the cutoff frequency of 0.50 THz. The resulting multicycle THz wave images are compactly represented in the 2D space-time plot in Fig. 4. A majority of the incident field is reflected, but it is clear that a significant portion is also lost through scattering at the interface into free space. The transmitted component observed in the ITO-coated region was revealed to be a combination of the fast-decaying evanescent waves below the cutoff frequency and a weakly excited bound mode at 1.2 THz whose speed was also matched in the THz generation process. Returning to the broadband signals in the 54-μm thick LN slab, we calculated the frequency-dependent E-field reflection ratio for only the lowest TE waveguide mode using the incident and reflected waves near the ITO interface, as seen to the left of the dashed line in Fig. 2(c). The results are shown in Fig. 5(a). Even below the cutoff frequency (vertical dotdashed line), the reflection ratio is only in the 20-40% range. Approximately 60-80% of the #233255 - $15.00 USD Received 27 Jan 2015; revised 11 Apr 2015; accepted 11 Apr 2015; published 29 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.014876 | OPTICS EXPRESS 14881

field (or ~85-95% of the energy) is scattered into free space immediately at the interface or transmitted briefly into the metallic waveguide before radiating out. Below and above the cutoff frequency, the reflection efficiency decreases monotonically as the frequency increases, reaching nearly zero at 0.6 THz and above.

Fig. 5. Reflection efficiency, decay lengths, and Q factors measured for the lowest TE waveguide mode in an asymmetric ITO-clad waveguide. (a) The frequency-dependent amplitude reflection coefficient for a TE-polarized THz E-field incident from an uncoated region of a 54-μm thick LN slab into an ITO-coated region. (b) The frequency-dependent 1/e decay length (green) and quality factor (Q) (blue) measured for the corresponding transmission and propagation in the ITO-coated region of the THz E-field amplitude.

It is evident from Fig. 2 that the field that is transmitted into the ITO-coated region suffers far stronger losses at all frequencies than the field in uncoated LN. We show the frequencydependent 1/e decay length of the TE waves along the propagation direction upon transmission into the ITO-coated region in Fig. 5(b) (green). The guided modes above the cutoff frequency show longer decay lengths with increasing frequency, indicative of the loss being dominated by the Ohmic losses in ITO; higher-frequency waves are mostly confined in the core, with shallow penetration depths into the lossy ITO cladding. This is consistent with Fig. 3(a) which shows the linewidths for the two lowest TE waveguide modes narrowing as their frequencies increase; the decay lengths plotted in Fig. 5(b) are only calculated from the lowest mode since it contributes most strongly to the measured signal. Below the cutoff frequency, we measured the decay length of the leaky modes, which persisted in the waveguide for as long as some of the guided modes. Although the leaky modes characteristically experience high Fresnel losses, their propagation is facilitated by their shallow angles of incidence that are much less than the critical angle for total internal reflection. Far below the cutoff frequency, we observed the transmitted evanescent waves [also noted in Fig. 4], with purely imaginary propagation constants, which have decay lengths that are inversely proportional to the frequency. We also calculated the experimental cavity quality factor (Q) according to Q(ω ) = ω ⋅τ (ω ), where the envelope of the E-field at radial frequency ω decays exponentially as E (t ) = E0 exp ( −t / 2τ ) . The Q factor is plotted in Fig. 5(b) (blue); the results show that our ITO layer is a relatively poor metal in the THz frequency range, consistent with the low DC conductivity measurements. We note that some resonant structures of practical interest have #233255 - $15.00 USD Received 27 Jan 2015; revised 11 Apr 2015; accepted 11 Apr 2015; published 29 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.014876 | OPTICS EXPRESS 14882

extremely strong radiative damping, and in those cases it may be possible to use ITO for nearfield studies of individual THz resonant structures. For example, measurements have shown that Q ~1 for a gold dipole antenna at 255 GHz [3] and Q ~4 for a gold split-ring resonator at 1 THz [20]. For such functionalities, even quite high resistive loss could be tolerated. The radiation damping and scattering into free space will, however, still limit the use of ITO in devices based on metamaterial arrays or in other studies where long interaction lengths are required. 4. Conclusion In conclusion, direct visualization of THz wave propagation and dispersion in ITO-clad LN waveguides has been carried out, revealing bound and leaky waveguide modes and near-field evanescent waves. The results showed that the 2.7-μm thick ITO layer used in our structure behaves as a relatively poor metal at THz frequencies. It is possible that improved deposition methods may yield ITO layers with higher THz conductivity and lower loss, widening the range of waveguide applications that exploit the optical transparency and THz metallic properties of ITO. Extensive control over the waveguide behavior including dispersion and radiative coupling of leaky modes can be achieved by varying the slab thickness, THz polarization, and coating of the metal onto one or both slab surfaces. Despite the high losses observed, the ITO-coated LN exhibited sufficiently high Q for applications in which optically transparent, highly radiative resonant structures are desired in the THz frequency range. Direct observation of leaky modes and near-fields in the THz polaritonics platform may be exploited for characterization of various photonic devices. This may enable detailed testing of device designs even if other metals besides ITO are preferred for practical implementation. The study of THz fields in metamaterial components such as dipole antennas and split ring resonators, with direct visualization in the metal-coated regions as well as the near-fields between them, may also be possible. Appendix 1. Guided modes in metal-clad waveguides For the following derivation, we consider a slab waveguide consisting of an anisotropic core surrounded by an ideal metal cladding on one or both sides, referred to here as the asymmetric or the symmetric metal-clad waveguide system, respectively. The analysis closely follows the treatment of symmetric slab of an anisotropic medium by Marcuse and Kaminow [21] and Yang and coworkers [6]. We assume that the crystal is uniaxial and its optic axis is parallel to the slab surface (here as in x-cut LN). The y- and z- dimensions of the slab are assumed to extend infinitely, while the x-dimension, corresponding to the slab thickness, is confined. The waveguide modes propagate along the y-direction and extend infinitely along the z-direction. The metal claddings are assumed to be perfect conductors with lossless reflection, such that   E = 0 and H = 0 inside the metal itself. For simplicity we express the E-field as harmonic in space and time, having the general form:   E ( x, y, z , t ) = E ( x) exp[i ( β y − ωt )], (1) where β = k y is the propagation constant.


Fig. 6. (a) Asymmetric metal-clad waveguide experimental geometry with perfect conductor cladding (left), low-index, nl, cladding (right), and high-index, nh, anisotropic core. (b) Symmetric metal-clad waveguide experimental geometry with high-index core surrounded by perfect conductor cladding on both sides. ε and μ are the permittivity and permeability in each region.

In this derivation, we treat the TE and TM polarizations separately for each of the two experimental geometries shown in Fig. 6. Here we assume a nonmagnetic, lossless high-index core and a low-index cladding with purely real permittivities, ε h and ε l , respectively, and free-space permeability, μ0. Lossy waveguides have complex permittivities and wave vectors and have been treated by Kaminow and coworkers [22]. The general approach will be to firstly consider the bulk dispersion curves and polarizations that describe the bulk wave properties in the core and cladding. The waveguide modes are linear combinations of these bulk waves, which result from the geometric constraints of the system. The coefficients of the linear equation are determined by applying the boundary conditions across the interface described by Maxwell’s equations. The result is a set a homogeneous equations that may be recast into matrix notation, where solutions exist when the determinant is equal to zero. These solutions correspond to bound, propagating waveguide modes that only exist at specific pairs of propagation constant, β, and frequency, ω, i.e. the waveguide dispersion curves.

Fig. 7. (a) For an anisotropic system, the coordinate system for the derivations is defined as a function of θ, the angle between the z-axis and the optic c-axis of the crystal. (b) For TE waves, θ = 0°, such that the c-axis lies along the z-axis. (c) For TM waves, θ = 90°, such that the c-axis lies along the y-axis.

In this geometry, TE waves have E y = 0 while TM waves have H y = 0. TEM waves exist if both E y = 0 and H y = 0. TE modes have nonvanishing field components: Ez, Hx, and Hy. In contrast, the TM modes have nonvanishing field components: Hz, Ex, and Ey. Although the core is anisotropic, the TE-polarization modes simplify to the isotropic case since the E field, E ( y ), has only a single component, Ez, along the optic c-axis of the crystal, i.e. the extraordinary (eo) axis of LN [see Fig. 7(b)]. This does not hold for the TM-polarization since in our experiment we rotated the c-axis of the crystal by 90° relative to the z-axis; this kept the pump-polarization parallel to the eo-axis so that the largest electro-optic coefficient in LN, r33, was used for efficient THz generation [23]. The TM case requires consideration of the


anisotropy of LN since it has E-field components along the eo-axis, Ey, and the ordinary (o) axis, Ex [see Fig. 7(c)]. A. General waveguide information

The bulk dispersion curves are defined from the wave vector of the material: k =

2π

.

λ Straightforward substitution of λ = c / nf = 2π c / nω gives the relationship between wave 2

 ωn  vector and radial frequency, ω: k 2 = k x2 + k y2 + k z2 =   , where c is the speed of light and  c  n is the refractive index. As previously defined, the propagation constant is β = k y since the

propagation direction is along y. The wave vector orthogonal to the slab surface, k x , is defined differently for the core, k x = κ o or κ e , and the cladding, k x = iα . For bound modes, the field must evanescently decay in the cladding, which is indicated by the imaginary wave vector component along x. In contrast, propagating solutions exist for the bound modes in the lossless core, which have real wave vector components. Lastly, since the field extends infinitely along z, i.e. infinite wavelength, its wave vector component is kz = 0. Bulk dispersion relations of the anisotropic system describe both the ordinary and extraordinary waves of the core, LN, and of the cladding, air:  ωn  Core, ordinary : κ o 2 + β 2 =  o   c 

Core, extraordinary : κ e

2

2

   ωn  n2 + β  cos2 θ + e2 sin 2 θ  =  e  no    c  2

(2a) 2

(2b)

2

 ωn  Cladding : −α 2 + β 2 =  c  ,  c 

(2c)

where no and ne are the bulk ordinary and extraordinary index of LN, respectively, and nc is the index of the low-index cladding, air. θ is the angle between the z-axis and the c-axis of the crystal shown in Fig. 7(a). Accordingly, we see that θ = 0° for TE-polarization while θ = 90° for TM-polarization. It is important to note that in the following analyses, we are exclusively solving for the extraordinary TE and TM waves due to our experimental geometries, and study regarding the ordinary wave counterparts is limited. B. TE modes in asymmetric and symmetric metal-clad waveguides

The overarching goal of the following derivation is to arrive at the wave equation that relates β and ω; this requires the use of the bulk dispersion relations in Eq. (2) that express k x in terms of β and ω. For any pair of β and ω, the most general form has the sum of four possible plane waves in each region: two signs for k x and two polarizations (that map directly to TE or TM waves). We only measure the extraordinary TE waves where k x ,core = κ e = κ . For θ = 0°, Eq. (2b) simplifies and contains a single refractive index, ne. In an isotropic system, any two orthogonal polarizations may be chosen in the core and cladding. However here in this uniaxial crystal, there are two convenient polarizations that allow the TE modes to be treated isotropically. A unit vector along the z-axis, zˆ, represents one choice of polarization. The second polarization is orthogonal to both zˆ and k in each region. In the high-index core, the unit vector is:


 ± kh± × zˆ h = ± = kh × zˆ

−β   ±κ  = c 2 2  β + κ  0  ω nh  

−β   ±κ  ,    0 

(3a)

 −β   ±iα  = c  2 2  β − α  0  ω nl  

 −β   ±iα  .    0 

(3b)

1

similarly in the low-index cladding:  ± kl± × zˆ l = ± = kl × zˆ

1

The full form of the E-fields of the waveguide modes in each region is written as follows:  Ideal metal : E ( x) = 0 (4a) + −  Core : E ( x) = A1 zˆ exp[iκ x] + A2 zˆ exp[−iκ x] + A3 h exp[iκ x ] + A4 h exp[−iκ x] (4b)  Cladding : E ( x) = B1 zˆ exp[α ( x − )] + B2 zˆ exp[−α ( x − )] (4c)   + B3l − exp[α ( x − )] + B4 l + exp[−α ( x − )], where Ai and Bi are scalar constants. Upon inspection it is clear that the first polarization, zˆ, corresponds to TE modes that only have an Ez component (perpendicular to the propagation   direction along y) while the second polarization, h ± and l ± , corresponds to TM modes that have Ex and Ey components. Additionally, any E-field components that do not decay to zero as x → ∞ cannot describe the bound modes, so exponentially increasing terms are discarded. Therefore, the TE E-fields are simplified to:  Ideal metal : E ( x) = 0 (5a)  Core : E ( x) = A1 zˆ exp[iκ x] + A2 zˆ exp[−iκ x] (5b)  Cladding : E ( x) = Bzˆ exp[−α ( x − )] (5c) The boundary conditions are provided by Maxwell’s equations, which state that tangential E- and H-fields must be continuous across the boundary. Here, these are the y- and zcomponents of the E- and H-fields. Since the TE modes have been described in terms of Efield, we can use Faraday’s law to recast the H-field in terms of the E-field. Since we assume that the materials are nonmagnetic, μ = μ0 , and because the solutions are time-harmonic, we ∂ = −iω. Plugging in and rearranging Faraday’s law yields: ∂t      ∂H i   ∇ × E = −μ = iωμ0 H  H = − ∇× E (6) ∂t ωμ0

replace the time-derivative with

 ∂ ∂ = i β , = 0, which Additionally, the terms within the del operator, ∇, simplify to ∂y ∂z follows from the functional form of the E-field in Eq. (1). Consequently, the H-field may be re-written as follows:  ∂E y    ∂Ez −i  H= + zˆ  i β Ex −  −ixˆ β Ez + yˆ  . ωμ0  ∂x ∂x   

(7)


The complete set of boundary conditions after substituting for all the magnetic fields is: Ez ,clad = Ez ,core ∂Ez ,clad ∂x

=

(8a)

∂Ez ,core

(8b)

∂x

E y ,clad = E y ,core i β Ex ,clad −

∂E y ,clad ∂x

= i β Ex ,core −

(8c) ∂E y ,core ∂x

(8d)

For TE modes, only the first two conditions apply (the last two are for TM modes), and are evaluated at the boundaries of the geometries shown in Fig. 6. (i) TE modes in an asymmetric metal-clad waveguide For the asymmetric metal-clad waveguide in Fig. 6(a), the boundary conditions in Eqs. 8(a) and 8(b) are applied at the LN-metal interface ( x = 0 ) and LN-air interface ( x =  ) . At x = 0, the boundary conditions show that A1 = -A2 = A, which gives a sinusoidal E-field. At x =  , the equations yield a set of homogeneous equations that can be recast in the following matrix:  sin(κ ) −1  A κ cos(κ ) α   B  = 0    For a homogeneous system, a solution exists when the determinant is zero; the determinant can easily be calculated analytically from the matrix to yield: tan(κ  + mπ ) = −κ / α ,

(9)

where m = 0,1,2,… represents the mode number for the periodic solutions. The bulk dispersion curves in Eq. (2) are used to eliminate dependence on both κ and α such that we arrive at a relation between β and ω. The resulting transcendental equation may be solved using Newton’s method by bisection or other numerical solvers in Matlab. In both cases, accurate bounds must be given for each solution otherwise the method will not converge to the correct waveguide modes. In addition, we can determine the characteristic cutoff frequency, νc, in particular of the lowest order mode, which denotes the frequency below which the waveguide cannot operate or propagate signals and is dependent on the waveguide dimensions. In the case of a planar dielectric waveguide, the fundamental mode has no cutoff frequency and hence signals of all frequencies are allowed to propagate. However, there is a cutoff frequency in an asymmetric metal-clad waveguide because the transverse propagation constant of the lowest order mode has a finite lower bound: κ lowest ≥ π / 2 (i.e. it is nonzero) and so β and κ are never simultaneously zero. Using the bulk dispersion relation in Eq. 2(b), where θ = 0° for TE waves in our crystal geometry, we can substitute β lowest = κ lowest tan θ c (where the critical angle is θ c = sin −1 (nl / nh ) ) and rearrange to solve for the cutoff frequency since ωc = 2πν c : cκ lowest 1 + tan 2 θ c . (10) 2π ne Lastly, the waveguide mode profiles in Eq. (5) are solved for explicitly as follows for A = 1/2, B = sin(κ ) :

ν c , lowest =


 ˆ 0 E ( x) = zE

0

x 

where E0 is the amplitude of the E-field. The mode profiles and dispersion curves of the lowest three TE waveguide modes are shown in Fig. 8. The system has a waveguide cutoff frequency, as will also be observed in the other polarizations and geometries shown in the following sections.

Fig. 8. For the lowest three TE modes in an asymmetric metal-clad waveguide: (a) Ez-field profiles at 1.5 THz and (b) waveguide dispersion curves (solid lines) and bulk dispersion curves of LN core along the extraordinary axis and air cladding (dashed lines).

(ii) TE modes in a symmetric metal-clad waveguide The symmetric metal-clad waveguide in Fig. 6(b) is also evaluated at x = 0 or x = . Similarly here, at x = 0, the boundary condition shows that A1 = -A2 = A. With only a single unknown variable, A, we apply the boundary condition in Eq. 8(a) at x =  to yield the following homogeneous equation: A sin(κ ) = 0, when κ  = mπ

(12)

where m = 1,2,3,… represents the mode number. The propagation constant here is frequencyindependent for each mode. The waveguide mode profiles are solved in the core, given in Eq. 5(b), for A = 1/2: 0 x0

where E0 is the amplitude of the E-field. The cutoff frequency for the lowest order mode ( m = 1 ) in the symmetric metal waveguide is straightforward since κ lowest = π / . Just like in a Fabry Perot cavity, the modes are always confined in the slab regardless β (i.e. θc = 0° and so β lowest = 0 ). Substitution of κ lowest and β lowest in the bulk dispersion relation in Eq. 2(b) yields the cutoff frequency of the lowest order mode:

ν c , lowest =

c . 2ne 

(14)


The mode profiles and dispersion curves of the lowest three TE waveguide modes are shown in Fig. 9. In contrast to the TE modes in an asymmetric slab, in the limit of short wave vector, the modes have near zero group velocity, which is given by the slope of the curve, d ω dk .

Fig. 9. For the lowest three TE modes in a symmetric metal-clad waveguide: (a) Frequencyindependent Ez-field profiles and (b) waveguide dispersion curves (solid lines) and bulk dispersion curve of LN core along the extraordinary axis (dashed line).

C. TM modes in asymmetric and symmetric metal-clad waveguides

The TM-polarization case considers the anisotropy of the core. The polarizations are uniquely   defined by the extraordinary polarization, e , and the ordinary polarization, o. The dielectric cladding (air) is isotropic so we can choose any two orthogonal polarizations: vertical  polarization, v , and horizontal polarization, h. Therefore, the most general form of the Efields of the waveguide modes in each region can now be written as follows:  (15a) Ideal metal cladding: E ( x) = 0    Core : E ( x) = A1e + exp[iκ e x] + A2 e − exp[−iκ e x] (15b)   + A3 o + exp[iκ o x] + A4 o − exp[−iκ o x]    Cladding : E ( x) = B1v − exp[α ( x − )] + B2 v + exp[−α ( x − )] (15c) −  + B3 h exp[α ( x − )] + B4 h + exp[−α ( x − )], where Ai and Bi are scalar constants.  The extraordinary polarization is located in the plane containing the wave vector, k , and  the optic axis, c , while the ordinary polarization is located orthogonal to this plane:        e ∝ k × (k × c ) and o ∝ k × c . The relevant vectors are defined as follows:  0  ±kx    c =  sin θ  and k =  β  ,  cos θ   0 

(16)

where θ is the angle between the z-axis and the eo-axis of the crystal and the appropriate kx is   applied (κe for e and κo for o ). In an anisotropic waveguide, the angle must be accounted for   in both e and o.  In terms of defining e , which describes the polarization of extraordinary E-fields, the angle-dependence of the propagation constant as seen in Eq. (2), is taken into account by #233255 - $15.00 USD Received 27 Jan 2015; revised 11 Apr 2015; accepted 11 Apr 2015; published 29 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.014876 | OPTICS EXPRESS 14889

     expressing it in terms of the displacement field, D, since D ∝ k × (k × c ). The D-field is written as:  D=

 β  κ  e  2 2  β + κe  0   

D0

(17)

  where D0 is the amplitude of the D-field in C/m2. The relationship between D and E is   expressed by the constitutive relation: E = R(−θ )ε −1 R(θ ) D, where the dielectric tensor, ε , has been rotated around the x-axis by θ, as in Fig. 7(a). For a uniaxial crystal with the principal axes chosen in Fig. 7(a), the dielectric tensor has the form:

ε o ε =  0  0

0

εo 0

0 0  ε e 

(18)

where εo and εe are the ordinary and extraordinary permittivity, respectively, in C ⋅ V -1 ⋅ m -1 . The rotation around the x-axis is given by: ε ' = R(−θ )ε R(θ ), where: 0 1  R(θ ) = 0 cos θ  0 sin θ

0  − sin θ  cos θ 

(19)

We can see that if θ = 0°, as in for the TE cases, the rotation matrix would take on the identity form in which the dielectric tensor is unchanged. For θ = 90°, the y and z components of the dielectric tensor are switched, yielding: ε o ε ' =  0  0

0

εe 0

0 0  ε o 

(20)

Note: No off-diagonal terms exist since pure TE or TM modes do not couple to each other. For θ ≠ 0° or 90°, we have TE- or TM-like modes that are superpositions of the extraordinary and ordinary waves. The extraordinary and ordinary polarizations of the E-field for θ = 90° are then:  e± ∝

 β ε o   0  0   κ ε  and o ± ∝ 1  0  =  0  e e    κo  β 2 + κ e2  0   ±κ o  1   

1

(21)

We now see that in this geometry, the TM waves become the extraordinary waves while the TE waves become the ordinary waves, while the opposite pairing is true when θ = 0° [see Section B]. In contrast, for the isotropic cladding, any two orthogonal polarizations may be used,    where we substitute v and h in Eq. (15) by zˆ and l ± [from Eq. 3(b)], respectively:


0  −β  ±    zˆ =  0  and l ∝  ±iα  .  0  1 

(22)

 We can then simplify the E-field expressions in Eq. (15) since TM waves are described by e ± ± and l . Also, any E-field components that do not decay to zero as x → ∞ are discarded and consequently yield the following:  (23a) Ideal metal cladding: E ( x) = 0    Core: E ( x) = A1e + exp[iκ e x] + A2 e − exp[−iκ e x] (23b)   Cladding: E ( x) = Bl + exp[−α ( x − )] (23c)

(i) TM modes in an asymmetric metal-clad waveguide The TM modes in an asymmetric metal waveguide in Fig. 6(a) are determined by applying the boundary conditions in Eqs. 8(c) and 8(d) at x = 0 or x = . At x = 0, the boundary conditions show that A1 = -A2 = A. At x =  , the equations yield a set of homogeneous equations that can be recast in the following matrix: κe   ε sin(κ e ) α   A = 0  e  B  cos(κ e ) 1   

The resulting transcendental equation that describes the TM modes is: tan(κ e  + mπ ) =

αε e , κe

(24)

where m = 0,1,2,… represents the mode number for the periodic solutions. It can be shown that the TM waveguide modes derived here are simply the antisymmetric TM modes of a symmetric waveguide with dielectric claddings (i.e., a slab of LN surrounded by air cladding) for a slab of twice the thickness. In this case for the TM modes in the asymmetric metal-clad waveguide, the lowest order mode (m = 0) has no cutoff frequency since κ lowest = 0.


Fig. 10. In an asymmetric metal-clad waveguide, the TM E-field profiles (Ex and Ey) at 1.5 THz of (a) m = 0 mode, (b) m = 1 mode, and (c) m = 2 mode. (d) The dispersion curves of the lowest three TM modes (solid lines) and bulk dispersion curves of LN core along the ordinary axis and air cladding (dashed lines).

The waveguide mode profiles in Eq. (23) are solved for explicitly as follows with A = 1/ 2, B =

κe sin(κ e ) : ε eα

  0      E ( x) ∝ D0 iyˆ  (κ e / ε e ) sin(κ e x)   (κ / ε ) sin(κ ) exp[−α ( x − )] e e e   

(25) 0   x < 0   − xˆ  ( β / ε o ) cos(κ e x) 0 ≤ x ≤  ( βκ e / ε eα ) sin(κ e ) exp[−α ( x − )]  x >   The mode profiles and dispersion curves of the lowest three TM waveguide modes are shown in Fig. 10. The TM modes have components that travel at the speed of light in air at low wave vectors. (ii) TM modes in a symmetric metal-clad waveguide The TM modes in a symmetric metal-clad waveguide in Fig. 6(b) are similarly determined from the boundary conditions in Eq. 6(c) evaluated at x = 0 and x = . At x = 0, the boundary conditions show that A1 = -A2 = A. The resulting homogeneous equation is: A sin(κ e ) = 0, when κ e  = mπ

(26)

where m = 0,1,2,3,… represents the mode number. The solution here bears the same functional form as that for the extraordinary TE wave solution analog in the previous section.


The waveguide cutoff frequency of the lowest order TM mode in the symmetric metal-clad waveguide is the same as that described previously in Eq. (14). The TM waveguide mode profiles (for m > 0) are solved for explicitly in the core in Eq. 23(b) as follows, where A = 1/2:   κ e  β   (27) iyˆ  sin(κ e x)  − xˆ  cos(κ e x)   0 ≤ x ≤  β + κ   ε e   εo   The mode profiles and dispersion curves of the lowest three TM waveguide modes (excluding the m = 0 mode) are shown in Fig. 11. In this system, the lowest order TM mode, m = 0, is a TEM wave since it satisfies the previously described conditions: E y = 0 and  E ( x) =

D0 2

2 e

H y = 0, while Ex and Hz are non-zero. This is a special case since the TEM mode has no cutoff frequency. Additionally, TEM modes have perfectly linear dispersion, coinciding with the bulk dispersion curve in LN along the ordinary axis; since κe = 0, then in accordance with Eq. 2(b), β = ω no / c. It follows that both the group and phase velocities are equal to c / no . This means that the TEM wave travels through the waveguide at the speed of light in the medium, LN, without bouncing back and forth between the two metal boundaries. This differs from the higher order TM modes that have nonzero kx components. Another implication is that the TEM modes have no dependence on the transverse x-coordinate, such that the E-field [from Eq. (27)] and H-field [from Eq. (7)] profiles are constant across x in the core. The Efield is given as:  ˆ 0 E ( x) = xˆ D0 ε o = xE (28)

and consequently the H-field is:  β H ( x) = zˆ E0 .

ωμ0

(29)

This differs from the TE wave analog in the Appendix 1: Section B(ii) where m ≠ 0 and TEM mode solutions are not allowed since it cannot satisfy the required conditions without setting   all fields to zero ( E ( x) = 0 and H ( x) = 0).


Fig. 11. In a symmetric metal-clad waveguide, the TM E-field profiles (Ex and Ey) of (a) m = 1 mode, (b) m = 2 mode, (c) m = 3 mode. (d) The dispersion curves of the lowest four TM modes (solid lines) and bulk dispersion curves of LN core along the ordinary axis and air cladding (dashed lines).

Appendix 2. Physical interpretation of leaky waves in an asymmetric metal-clad waveguide

The physical interpretation of guided modes focused on in Appendix 1 is straightforward since the waves evanescently decay in the cladding (with purely imaginary wave vector) and oscillate in the core (with purely real wave vector for a lossless material); their behavior is illustrated in Fig. 12(a). Leaky modes, shown in Fig. 12(b), however, are more difficult to interpret because the wave vectors are complex and hence lead to unphysical behavior at the limits x → ±∞; they have been deemed ‘improper modal solutions’ to the waveguide. Here, we will elaborate on the leaky wave results in the paper by presenting an intuitive way to model their behavior in an asymmetric metal-clad waveguide in Fig. 6(a).

Fig. 12. The mode power, |E(x)|2, along the transverse direction of a 1D planar waveguide for (a) the guided modes and (b) the leaky modes.

Let us first consider the bound modes that are waveguided in the slab by total internal reflection (TIR) at every interface and consequently have the ability to propagate indefinitely (neglecting material damping). However, in general, waveguides also have waves that


propagate at angles less than the critical angle for TIR at the interfaces. Haus et al. refer to these waves as leaky modes, which exist from the partial reflections off the core-cladding interface [18]. In our experimental setup, because leaky modes are not bound (or guided by TIR), every successive reflection at the LN-air interface results in progressive leakage of field amplitude; we assume perfect reflection at the LN-ITO interface although in reality ITO is an imperfect metal and constitutes another interface with leakage. As previously described in the main text in Sec. 3, leaky modes resemble the resonant states of a Fabry-Perot cavity. If we only consider the real part of the wave vector, the wave behavior of leaky modes simply maps to a modified subset of the characteristic waveguide dispersion derived in Appendix 1: B(ii) and C(ii) for the guided TE and TM counterparts in a symmetric metal-clad waveguide (a geometry analogous to that of a Fabry-Perot cavity). The imaginary part of the propagation constant is the rate of leakage out of the waveguide, given by the decay constant or the inverse of the 1/e propagation distance, which has been calculated for a slab waveguide by Hall and Yeh [24]. In a Fabry-Perot cavity, the resonant states exist when the cavity length is an integral number of half-wavelengths: m(λ / 2) = L,

(30)

where m = 1, 2,3,... and L is the cavity length. These solutions are analogous to the modes in a symmetric metal-clad waveguide of thickness L = , discussed for TE and TM polarizations in Appendix 1. In the asymmetric metal-clad waveguide, leaky modes partially reflect off the LN-air interface and subsequently also resemble the resonant states in a Fabry-Perot cavity. Because a wave incident at the LN-air interface experiences no π-phase shift, since it is moving from higher to lower refractive index, the boundary condition at the LN-metal interface (where E = 0 ) is only satisfied in a round trip (i.e. LN-metal → LN-air → LN-metal). Consequently, Eq. (30) is modified to include twice the waveguide thickness, such that these leaky waves satisfy the following relationship: m(λ / 2) = 2,

(31)

where m = 1,3,5,... . This solution corresponds to only the odd-m states in a symmetric metalclad waveguide of twice the waveguide thickness.

Fig. 13. Leaky waves resemble Fabry-Perot cavity modes. (a) For an asymmetric metal-clad waveguide, the leaky waves resemble the odd order modes (m = 1,3,5,…) in a Fabry-Perot


cavity (of twice the thickness) that have symmetric transverse field profiles. This schematic illustration demonstrates constructive interference between incident (solid-blue line) and reflected (dashed-red line) waves at the LN-air interface for the m = 1 mode. (b) In an asymmetric metal-clad waveguide, none of the leaky modes resemble the even order modes (m = 2,4,6,…) of a Fabry-Perot cavity, which have antisymmetric transverse field profiles. This schematic illustration shows purely destructive interference between the incident and reflected waves at the LN-air interface for the m = 2 mode.

The waves of the odd-m states have symmetric transverse E-field profiles (Ez for TE waves and Ey for TM waves). In Fig. 13(a), the schematic illustration shows how the incident (solid-blue line) and reflected (dashed-red line) waves of the first odd mode (m = 1) constructively interfere upon completing a round trip of 2, while also satisfying the boundary condition at the LN-metal interface. In contrast, in Fig. 13(b), the incident and reflected waves of the first even mode (m = 2), have antisymmetric transverse E-field profiles and destructively interfere across the entire thickness of the waveguide. The geometry described here, which employs an ideal metal on one side, forms a system that is most easily reduced to the widely familiar Fabry-Perot cavity. The modifications to the Fabry-Perot solutions are indicative of the physical behavior of the leaky modes in the waveguide, which only exist for a set relationship between β and ω. Acknowledgments

This work was supported by NSF grant number 1128632. C.A.W. and B.K.O. were supported in part by NSF GRFP.


DNA adsorption by indium tin oxide nanoparticles.

Respirable indium exposures, plasma indium, and respiratory health among indium-tin oxide (ITO) workers.

Continuous leaky-wave scanning using periodically modulated spoof plasmonic waveguide.

Immune stimulation following dermal exposure to unsintered indium tin oxide.

Cytotoxicity and characterization of particles collected from an indium-tin oxide production facility.

Direct electroplated metallization on indium tin oxide plastic substrate.

Competitive Attachment of Gold Nanoparticles on an Indium Tin Oxide Electrode.

An indium tin oxide-free polymer solar cell on flexible glass.

Planar Indium Tin Oxide Heater for Improved Thermal Distribution for Metal Oxide Micromachined Gas Sensors.

Tissue distribution of indium after repeated intratracheal instillations of indium-tin oxide into the lungs of hamsters.

Pulmonary toxicity of indium-tin oxide production facility particles in rats.

Position- and orientation-controlled polarized light interaction of individual indium tin oxide nanorods.

Sintered indium-tin oxide particles induce pro-inflammatory responses in vitro, in part through inflammasome activation.

Properties of Transmission and Leaky Modes in a Plasmonic Waveguide Constructed by Periodic Subwavelength Metallic Hollow Blocks.

Correlative scanning electron and confocal microscopy imaging of labeled cells coated by indium-tin-oxide.

Efficiency enhancement in polymer light-emitting diodes via embedded indium-tin-oxide nanorods.

Enhanced third-harmonic generation in Si-compatible epsilon-near-zero indium tin oxide nanolayers.

Three dimensional indium-tin-oxide nanorod array for charge collection in dye-sensitized solar cells.

Ultrafast modulation of the plasma frequency of vertically aligned indium tin oxide rods.

Intensification of the electrochemiluminescence of luminol on hollow TiO₂ nanoshell-modified indium tin oxide electrodes.

Electrical stimulation of cultured neurons using a simply patterned indium-tin-oxide (ITO) glass electrode.

A functional assay for gap junctional examination; electroporation of adherent cells on indium-tin oxide.

gold nanocages onto indium tin oxide glass as a reagentless plasmonic mercury sensor.

Conductivity control of as-grown branched indium tin oxide nanowire networks.