symmetric slab waveguide

Substituting this into the wave equation then yields: \begin{equation} \nabla(-E_x\frac{d}{dx}ln\epsilon)-\nabla^2E = -\frac{n^2}{c^2}\frac{\partial^2{E}}{\partial^2{t}} \end{equation}. ; Capasso, F. Polarization-insensitive metalenses at visible wavelengths. German: ebene Wellenleiter. Calculate silicon photonic waveguide modes (demo) 3) Using mode port source in . The authors declare no conflict of interest. * Cross: focused light is cross-polarized compared with the incident light. Xu N, Liang Y, Hao Y, Mao M, Guo J, Liu H, Meng H, Wang F, Wei Z. Nanomaterials (Basel). This last relation gives us a very important property: the V-number! ; WritingReview & Editing, Z.W., J.G., F.W., H.M. and H.L. This can be constructed using three contiguous rectangular structures representing the air (etch), core (silica) and substrate (dielectric defined with an index of 1.33) layers. Dotted lines show the simulated focal spot sizes of the proposed metalenses (red dotted line) and the according diffraction-limited sizes (blue dotted line) at different wavelengths. Carrying out the matrix multiplication, we have: \begin{equation} M_{22} = cos(2ah) + \frac{1}{2}(\frac{h}{\gamma} \frac{\gamma}{h})sin(2ah) \end{equation}. Guo, Z.; Zhu, L.; Shen, F.; Zhou, H.; Gao, R. Dielectric metasurface based high-efficiency polarization splitters. 65 0 obj <> endobj Jahani, S.; Jacob, Z. All-dielectric metamaterials. and Z.W. For a symmetric slab waveguide with TE modes, we have: \begin{equation} M = \frac{1}{4}\begin{bmatrix} 1+ \frac{i\gamma}{h} & 1 \frac{i\gamma}{h} \\ 1- \frac{i\gamma}{h} & 1 + \frac{i\gamma}{h} \end{bmatrix}\begin{bmatrix} e^{-i2ah} & 0 \\ 0 & e^{i2ah} \end{bmatrix}\begin{bmatrix} 1+ \frac{i\gamma}{h} & 1 \frac{i\gamma}{h} \\ 1- \frac{i\gamma}{h} & 1 + \frac{i\gamma}{h} \end{bmatrix} \end{equation}. Optical axis of the hyperbolic medium is normal to the slab plane. In order to provide physical solutions, the exponential solutions must be decaying. Case (b,c): $k_0n_1 < k_0n_3 < \beta < k_0n_2$. e9z\MQ_(.sh,jp{Oc. Khorasaninejad, M.; Shi, Z.; Zhu, A.Y. ; Lustrac, A.; Wu, Q.; Qiu, C.; Aiu, A. Ultrathin PancharatnamBerry metasurface with maximal cross-polarization efficiency. This paper was funded by the National Natural Science Foundation of China (NSFC) under Grant Nos. Then, for P-polarization, the magnetic field is in the $\hat{y}$ direction, so the electric field is in the $\hat{x}$ and $\hat{z}$ directions. Khorasaninejad, M.; Aieta, F.; Kanhaiya, P.; Kats, M.A. The U.S. Department of Energy's Office of Scientific and Technical Information Chong, K.E. Zhan, A.; Colburn, S.; Trivedi, R.; Fryett, T.K. Bookshelf The interesting case of an anisotropic slab waveguide with the optical axis located in the plane of the slab will be presented in Part II of this paper. In this paper, a symmetric three-layer slab waveguide is investigated. Supported by the Innovation Project of Graduate School of South China Normal University. In particular, you have waveguidingwhen you have a wave propagating in a high index dielectric medium sandwiched between two lower index media, such that then TIR can effectively trap the wave between the media. 2021 Jul 4;21(13):4584. doi: 10.3390/s21134584. However, we can see that this will be useful by recalling that for a multilayer system, we can relate the left and right propagating fields as: \begin{equation} \begin{bmatrix} E_R^+ \\ E_R^- \end{bmatrix} = M\begin{bmatrix} E^+_L \\ E_L^- \end{bmatrix} \end{equation}. ; Chen, W.T. hbbd```b`` ) Figure 17.3: Schematic for the analysis of a guided mode in the dielectric waveguide. 0 For Science. We can redraw this picture as a bunch of plane waves propagating directly down the fiber that add in the transverse direction to be a Gaussian. 1. ; Sanjeev, V.; Khorasaninejad, M.; Shi, Z.; Lee, E.; Capasso, F. A broadband achromatic metalens for focusing and imaging in the visible. Chen, W.T. %PDF-1.6 % The structure is uniform in the y-direction. 2015;6:7069. doi: 10.1038/ncomms8069. a. ; Project Administration, Z.W. permission is required to reuse all or part of the article published by MDPI, including figures and tables. Careers. These metalenses are less than . 0000007891 00000 n ; Formal Analysis, Z.W., J.G., F.W., H.M. and H.L. FOIA The modes of a slab waveguide with arbitrary refractive index profile are calculated. The dielectric constant thus becomes spatially dependent on the x-direction, $\epsilon = \epsilon(x)$. We can consider the case first of a symmetric slab waveguide, meaning that the upper and lower guiding layers have the same refractive index i.e. To develop a qualitative sense of how light propagates within a waveguide, we have two main pictures: plane waves and the angular spectrum picture. We now recall that $\gamma$ and $h$ are connected via the $\hat{z}$ propagation constant: \begin{equation} \gamma^2 = \beta^2 n_1^2k_0^2 \end{equation}, \begin{equation} h^2 = n_2^2k_0^2 \beta^2 \end{equation}. 0 Please let us know what you think of our products and services. 0000001201 00000 n 0000006907 00000 n Chen L, Zhao L, Hao Y, Liu W, Wu Y, Wei Z, Xu N, Qin S, Yang X, Liu H. Nanomaterials (Basel). However, in the core, the solution is sinusoidal. 0000001381 00000 n More general term: waveguides. Section 3 - Symmetric Dielectric Slab Waveguides Definition of a SLAB waveguide A. PMC Optical Wave Guides - Lawrence Berkeley National Laboratory Optical frequencies Symmetric Slab region (n-one) Refractive index n-one & n-2 n-2<n-one Dielectric wave guide Conti 16. and transmitted securely. Here we can see an unfortunate physical interpretation of this solution: as $|x| \rightarrow \infty$, the field E(x) also goes to infinity, which is of course physically not possible as that would imply infinite energy. $k_{tot} = \beta$. 1. Calculate 1D slab waveguide modes. TE and TM Modes of Propagation in Symmetric Slab Waveguides A key goal of metalens research is to achieve wavefront shaping of light using optical elements with thicknesses on the order of the wavelength. which are of course the same solutions we had before! Abstract: We derive the eigenvalue equation and the expressions for the field distribution for the TM modes of an anisotropic symmetric slab waveguide; the TE modes are identical to the isotropic case. If I havent lost your interest yet, then read on to learn more about why you should thank your dielectric waveguides! Kim SJ, Kim C, Kim Y, Jeong J, Choi S, Han W, Kim J, Lee B. [7] demonstrated a unique negative lateral shift for a Gaussian beam reected from a grounded LHM slab, which is distinctly dierent where we can see that the first equation comes from matching E and the second term from matching $\partial{E}/\partial{x}$. 2 TM (m) - transverse magnetic family of modes, m=0, 1, 2, . hb```DVz!b`0p8(wYpBm.o;kO^jal7 vl!BS2&,[v$Qmxes9>mROT\U%`8 D0W%5y'(K`V6mtMl\105Y/WGOgSMuR >` SA 88 0 obj <>/Filter/FlateDecode/ID[]/Index[65 58]/Info 64 0 R/Length 113/Prev 365731/Root 66 0 R/Size 123/Type/XRef/W[1 3 1]>>stream ; Hu, D.; Feng, S.F. High-efficiency broadband anomalous reflection by gradient meta-surfaces. The same procedure would follow for TM modes. Opt. Please note that many of the page functionalities won't work as expected without javascript enabled. ; Investigation, Y.L. Accessibility We use cookies on our website to ensure you get the best experience. 0000000832 00000 n Answer to An asymmetric slab waveguide is shown in Figure 13.26. In a symmetric slab waveguide, n = 1.50, n = 1.45, and d = 10 m. -, Arbabi A., Horie Y., Ball A.J., Bagheri M., Faraon A. Subwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarrays. In this case, from our above qualitative form of the solution for $\beta^2 > n_j^2k_0^2$ of $Ae^{\gamma{x}} + Be^{-\gamma{x}}$, the solution would be exponential in all three layers since $\beta$ is greater than the other $k$-vectors. 0000003312 00000 n However, first we will also scale the mode parameters $h$ and $\gamma$ to the waveguide dimensions by multiplying by the radius of the slab, $a$. Chen, B.H; Wu, P.C; Su, V.C; Lai, Y.C; Chu, C.H; Lee, C.; Chen, J.-W.; Chen, Y.H. Total internal re ection occurs at the top and bottom interfaces. which doesnt have an analytical solution, and so the equations must be solved by plotting both the V-number equation and the symmetric/antisymmetric modes with $\gamma{a}$ versus $ha$. We can derive the exact conditions by looking at the following figure: As we might imagine, if the guiding layer thickness is small, then only a few modes will be allowed to propagate, and vice versa. The analysis is based on modal expansion formulation and the modes can be classified as TE and TM as well as even and odd ones. In todays society, dielectric waveguides are incredibly important for communication. We shall look at the application of the transverse resonance condition to a TE wave guided in a dielectric waveguide. Author to whom correspondence should be addressed. ; Yang, K.-Y. ; He, Q.; Xiao, S.; Kung, W.-T.; Guo, G.-Y. 0000008042 00000 n Multidisciplinary Digital Publishing Institute (MDPI). and Z.W. Alright, here we are at the precipice of understanding waveguide modes via EM theory! If the waves add coherently, the wave is guided along the dielectric slab. See this image and copyright information in PMC. Khorasaninejad, M.; Capasso, F. Broadband multifunctional efficient meta-gratings based on dielectric waveguide phase shifters. Chen, W.T. Here we demonstrate ultrathin highly efficient crystalline titanium dioxide metalenses at blue, green, and red wavelengths ( = 453 nm, 532 nm, and 633 nm, respectively) based on symmetric slab waveguide theory. its amplitude is in the x- and y-plane with propagation in the z-direction. ; Decker, M.; Neshev, D.N. Dielectric Slab Waveguides W 2d z x Assumption: W >> d core cladding cladding y ECE 303 - Fall 2005 - Farhan Rana - Cornell University Dielectric Vs Metal Waveguides z o m = 1 m = 2 E y E y Metal Waveguides (modes are tightly confined) (modes are loosely confined) Dielectric Slab Waveguides z 2 o 1 o 2 o E y E y core . Wang, D.; Gu, Y.; Gong, Y.; Qiu, C.W. We can define a parameter $h$ as: \begin{equation} h_j = n_j^2k_0^2 \beta^2 \end{equation}, Case I: $h^2 > 0$, which corresponds to $h^2 = (n_j^2k_0^2-\beta^2)(n_j^2k_0^2-\beta^2) = n_j^4k_0^4 2\beta^2n_j^2k_0^2 + \beta^2 > 0$. crystalline titanium dioxide; integrated devices; metalens; metasurface; phase modulation. Science Physics In a symmetric slab waveguide, n = 1.50, n = 1.45, and d = 10 m. D[C 2(X)XH2^#!`qDfH]``6df9DLv`YF Od1p!Lg` $I The characteristic Eigen value equation for the TE modes in a symmetric waveguide is given by: tanh 2 = k for even (cos) modes = k y for odd (sin) modes E15 In order to plot the Eigen values of the TE modes of the symmetric waveguide, solutions of Eq. 0000003618 00000 n 0000001656 00000 n sharing sensitive information, make sure youre on a federal This in turn allowed us to derive the usual form of the wave equation. ; Su, V.C. The form of the solution to this equation depends on whether $\beta^2$ is greater or less than $n_jk_0^2$. Although the wavevector picture is not entirely accurate, as we would need to use EM theory in order to get the actual field profiles, we can nevertheless press forward with this idea in order to get a qualitative idea about what the field profiles will be. endstream endobj 66 0 obj <> endobj 67 0 obj <>/ProcSet[/PDF/Text]>>/Rotate 0/Type/Page>> endobj 68 0 obj <>stream 0000002770 00000 n This type of ; Visualization, Y.L. Ultra-thin, planar, Babinet-inverted plasmonic metalenses. Now, we have two equations that relate $h$ and $\gamma$, so we can use these to the relate the two as: \begin{equation} hatan(ha) = \sqrt{V^2-a^2h^2} \end{equation}. government site. ; Kan, Q.; Zhang, Y. Epub 2018 Oct 15. 0000051556 00000 n b. is 'monomode'). Khorasaninejad, M.; Crozier, K.B. Pors, A.; Nielsen, M.G. ; Zhu, A.Y. Since we found before the transfer matrix across a boundary, we can just write the results for what the transfer matrix across the waveguide should be here. ; Wang, C.M. which we can see is for TE modes by the form of the outer two matrices and that the waveguide is symmetric from the symmetry of these two matrices as well. 0000044166 00000 n ; Chen, B.H. $M_{22} \neq 0$ and the transmitted amplitude would be $t = S_{1,2}$. We first consider TE modes with $n_1 = n_3 < n_2$, such that the propagating wave is guided since the core region has a higher refractive index than the cladding regions. What I am talking about is, of course, fibers, which are dielectric waveguides for optical communication. Evanescent wave microscopy can also be realized using planar or slab optical waveguides where the sample (typically in an aqueous solution) forms at least one part of the cladding around an adjacent waveguide core [7, 16, 18-23]. You are accessing a machine-readable page. 3C+8ddy pC`@k~RkIMj9wQ7~b9\yr:3Ot5sz?c+W6vjib:uvqlN_xRP[v{\[Sw fIWG9fQwwQz6m;*{.>@0jK ; Liao, C.Y. ; et al. Now, for s-polarization, the electric field is perpendicular to the field (senkrecht), so in this case the perpendicular polarization direction would be in the $\hat{y}$ direction. Note that the only free parameters, or the only part that matters, is the product $a^2k_0^2 = \frac{4\pi^2{a^2}}{\lambda^2}$, since this scales the wave guide radius to the wavelength of light. Zhu, Y.; Yuan, W.; Sun, H.; Yu, Y. Broadband ultra-deep sub-diffraction-limit optical focusing by metallic graded-index (MGRIN) lenses. Planar photonics with metasurfaces. We then have: \begin{align} tan(ha) & = \frac{\gamma{a}}{ha},~symmetric~modes \\ & = tan(ha) & = -\frac{ha}{\gamma{a}},~antisymmetric~modes \end{align}. 7.1 Dielectric Slab Waveguides 7.1.1 Introduction: A variety of different integrated optical waveguides are used to confine and guide light on a chip. Applying the boundary conditions at interfaces, the . Guided wave modes in a symmetric slab waveguide formed by an isotropic dielectric layer with cubic nonlinear response placed in the hyperbolic surrounding medium are investigated theoretically. In the chosen dimensions of our slab to have propagation purely in the x-z direction, the derivatives of the y-component of the electric field are zero. Nano Lett. In this case, we would have $H = H_y\hat{y}$ (note that we do not use the electric field in the $\hat{x}$ and $\hat{z}$ directions in this case, as this would be rather difficult to solve at the boundaries and I am not a monster). ; WritingOriginal Draft Preparation, Y.L. several techniques or approaches, or a comprehensive review paper with concise and precise updates on the latest ; Shalaev, V.M. 2) Waveguide mode simulation using Lumerical MODE. 8600 Rockville Pike However, in this case, the dielectric constant $\epsilon$ is not constant over all space due to the fact that we have different materials as part of our waveguide. 240 0 obj <> endobj Please enable it to take advantage of the complete set of features! Question: Symmetric slab waveguide: definitions E = E. , H = unit vectors X=d X=0 X=-d n2 * H = H, | o For ao = 1.55 um , n = 1.51, n2 = 1.49 calculate the maximum thickness of the slab so that the waveguide supports only the fundamental TE and TM modes (i.e. An official website of the United States government. We find that, in spite of the symmetry of the . Perhaps the simplest optical waveguide is the dielectric slab waveguide, also called a planar waveguide. We can do the same procedure for the antisymmetric case: \begin{align} E_asinhx & = E_1e^{-\gamma{a}} \\ hE_asinha & = -\gammaE_1e^{-\gamma{a}} \end{align}, \begin{equation} -htan(ha) = \gamma \label{anti-sym}\end{equation}. ; Wang, C.-M.; Huang, Y.-W.; Sun, G.; Chiang, I.-D.; Liao, C.Y. Rogers, E.T.F. The plotted solution for this case is shown in cases (b) and (c) above, and it can be seen that these solutions form confined, or guided modes. The slab waveguide consists of three layers of materials with different dielectric constants . This divergence of the transmission coefficient even without an input field can be thought of as the presence of the evanescent field outside of the boundary of the waveguide. ; Methodology, Y.L. The site is secure. Find support for a specific problem in the support section of our website. The working principle of these waveguides builds on the idea of total internal reflection, discussed in the first part of this section. $E_z$, although we could also do the normal $D$ component through its relation to the electric field $D = \epsilon{E}$. Characteristics of Symmetric Slab Waveguide Modes Gaussian beam solution from E E MISC at Iowa State University Section 5: Nonlinearity and Acousto-Optics, Section 6: Brillouin and Rayleigh Scattering, Section 7: Raman and Rayleigh-Wing Scattering, Section 8: Nonlinearity and Electro-Optics, Section 2: Dispersion and Dispersion Compensation, Section 10: Fiber Lasers and Amplifiers in the High Power Regime, Electromagnetic Analysis of Slab Waveguide, https://www.researchgate.net/figure/A-schematic-of-a-symmetric-step-index-slab-waveguide1_fig1_234932545. We can also recall that we can find the transmission coefficient as: \begin{equation} t = S_{1,2} = \frac{1}{M_{22}} = \begin{bmatrix} E_j^+ \\ E_i^+ \end{bmatrix}\end{equation}. Khorasaninejad, M.; Zhu, A.Y. Lalanne, P. Waveguiding in blazed-binary diffractive elements. Jiang, X.Y. Effective index as a function of waveguide thickness Step 1: Make mode1 data available to parameter sweep We are interested in obtaining the effective index and loss data from the ssb0 mode. 2018 Nov 14;18(11):6961-6966. doi: 10.1021/acs.nanolett.8b02875. Dielectric Metalens: Properties and Three-Dimensional Imaging Applications. Now, then, comparing Equations \eqref{origin} and \eqref{new}, we can find two solutions: \begin{equation} tan(ha) = \frac{\gamma{a}}{ha},~symmetric~modes \end{equation}, \begin{equation} tan(ha) = -\frac{ha}{\gamma{a}},~antisymmetric~modes \end{equation}. Express. A Thermal Tuning Meta-Duplex-Lens (MDL): Design and Characterization. Disclaimer, National Library of Medicine which we found because we would have an exponentially decaying field in this region for a guided mode, which corresponds to the conditions given for $\beta$ and $n_1^2k_0^2$. doi: 10.1364/OE.23.016782. metasurface; metalens; phase modulation; integrated devices; crystalline titanium dioxide, Photonics: Optical Electronics in Modern Communications. ; Lan, Y.-C.; Kuan, C.-H.; et al. The coordinate system relative to the slab, the designation of the dielectric constants, and the orientation of the optical axis. B&Eh$-72qqG5. Optical fiber - Optical fibers are the most important dielectric waveguide. Thus, for the $\hat{x}$ component, we have: \begin{equation} \frac{\partial^2E_x(x)}{\partial{x}^2}-\beta^2E_x(x)+\frac{d}{dx}(E_x\frac{d}{dx}ln\epsilon) = -\frac{n_j^2\omega^2}{c^2}E_x(x) \end{equation}. 0000002385 00000 n x"iuo~cZV4[.26v\`X*tQ~D,[ This may seem somewhat hard to imagine as being an adept technique, as we will be comparing the layers longitudinally while the light propagates through transversely, as shown in the figure. This may occur in silica glass waveguides or in semiconductor laser heterostructures. 0000051788 00000 n All the lenses have the same NA of 0.51 and. Subscribe to receive issue release notifications and newsletters from MDPI journals, You can make submissions to other journals. Saeidi, C.; Weide, D. Wideband plasmonic focusing metasurfaces. To gain a better insight into the phase realization mechanism, here we introduce a simple but efficient model with symmetric slab waveguide (SSW) effect. 2018 Apr 28;8(5):288. doi: 10.3390/nano8050288. Repeat the calculation for a high contrast silicon/silica slab . ; Huang, T.-T.; Wang, J.H. This case is thus not possible, even though it solves the wave equation. startxref %PDF-1.4 % = where the exponential term is positive because x will be negative in the bottom cladding layer, so the decaying part has been taken care of. A super-oscillatory lens optical microscope for subwavelength imaging. 240 26 Metasurface Spiral Focusing Generators with Tunable Orbital Angular Momentum Based on Slab Silicon Nitride Waveguide and Vanadium Dioxide (VO. Types of Dielectric Waveguides The types of dielectric waveguides are: Dielectric slab waveguide - If n s = n c, the waveguide structure turns into a symmetric dielectric slab waveguide. properties of multi-layer LHM waveguide, and Kong et al. ; Khorasaninejad, M.; Shi, Z.; Sanjeev, V.; Capasso, F. Immersion meta-lenses at visible wavelengths for nanoscale imaging. Symmetric and Asymmetric Dielectric Slab Waveguides For mode guidance, the condition to be followed is as follows: n 1 > n 2 n 3 When n 2 = n 3, the waveguide structure turns into a symmetric dielectric slab waveguide. Bomzon, Z.; Biener, G.; Kleiner, V.; Hasman, E. Space-variant pancharatnamberry phase optical elements with computer-generated subwavelength gratings. Study Resources. 0000000016 00000 n Visit our dedicated information section to learn more about MDPI. Assumptions given in this slab waveguide is used for wavelength () 1.55 m, the thickness (d) of the core is 0.9 m with a type of symmetric step-index slab waveguide, refractive index of n1 is 3.5 and refractive index of n2 is 3, also n3= n1. A symmetric slab waveguide is given with the parameters as shown in the following figure (Fig. Guangzhou Key Laboratory for Special Fiber Photonic Devices, South China Normal University, Guangzhou 510006, China, Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, School of Information and Optoelectronic Science and Engineering, South China Normal University, Guangzhou 510006, China. not an exact expression)? Today we will use Lumerical MODE and FDTD to examine the properties of optical waveguides. Wang, S.; Wu, P.C. Thus, the electric field is $E = E_y\hat{y}$ and the wave equation is given as: \begin{equation} \frac{\partial^2E_y(x)}{\partial{x}^2} \beta^2E_y(x) = -\frac{n_j^2\omega^2}{c^2}E_y(x) \end{equation}. ; Chen, W.T. (a) What is the phase velocity of the m= 1 TE or TM mode at cutoff? Kang, M.; Feng, T.; Wang, H.T. ; et al. Here we calculate the effective index of square nanopillar as a function of its width for different wavelengths of 453, 532, and 633 nm by FDTD solutions and SSW theory, respectively. The solutions to the symmetric slab waveguide are either symmetric or anti-symmetric, which are each pictured in the image below: Now the solution for the top cladding region is given as: \begin{equation} E = E_1e^{-\gamma{x}},~~\gamma = \sqrt{\beta^2-n_1^2k_0^2} > 0 \end{equation}. Federal government websites often end in .gov or .mil. Since the slab is assumed to be infinitely extended in the yz plane, all field quantities are considered to be independent of y. methods, instructions or products referred to in the content. xref Jia, D.; Tian, Y.; Ma, W.; Gong, X.; Yu, J.; Zhao, G.; Yu, X. Transmissive terahertz metalens with full phase control based on a dielectric metasurface. We can now apply the boundary conditions and match E and $\partial{E}/\partial{x}$ at each interface. We can consider the case first of a symmetric slab waveguide, meaning that the upper and lower guiding layers have the same refractive index i.e. Khorasaninejad, M.; Chen, W.T. 0000003847 00000 n (1) Plane waves: If we think of our plane wave with wave fronts perpendicular to the direction of propagation, then we have two pictures: $k$ pointing up and $k$ pointing down. We derive the eigenvalue equation and the expressions for the field distribution for the TM modes of an anisotropic symmetric slab waveguide; the TE modes are identical to the isotropic case. The field in the slab is in the form of a monochromatic TEM plane (Electric wave is oscillating perpendicular to incident and reflection plane, here is x direction), and wave bounces in case 1 situation discussed in section of configuration. It is assumed that the field components do not depend on the y coordinate and that the c axis of the uniaxial crystal lies in the x-z plane. which has solutions of the form E = Ae^{\gamma{x}} + Be^{-\gamma{x}}. In the simulation we assume symmetric waveguide structure with the surrounding layer (substrate and cladding) is air with s = c = 1. The solutions for which modes propagate thus need the symmetric and antisymmetric equations as well as the V-number equation. A broadband achromatic metalens in the visible. Wang, W.; Guo, Z.; Li, R.; Zhang, J.; Li, Y.; Wang, X.; Qu, S. Plasmonics metalens independent from the incident polarizations. In addition we assume that the optical axes of the uniaxial materials of film and substrate are parallel to each other and also parallel (coplanar) to the plane of the film. Hence we have: \begin{equation} \frac{\partial^2E}{\partial{x^2}} \gamma^2E = 0 \end{equation}. Larouche, S.; Tsai, Y.-J. 2015;15:53585362. Before, in the assumption that the medium was homogeneous ($\epsilon$ and $\mu$ constant over all space) with no free charges ($\rho = 0$), Gausss Law yielded the result that $\nabla\cdot{E}=0$. Papers are submitted upon individual invitation or recommendation by the scientific editors and undergo peer review 1) Basic waveguide theory - Slab waveguide. Case (d): $k_0n_1 < \beta < k_0n_3 < k_0n_2$. 265 0 obj <>stream the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, Would you like email updates of new search results? The most basic optical waveguide is a slab waveguides shown below. modes. ; Kuo, H.Y. Now we will consider that if our modes are guided, i.e. The goal here in this section is to establish that only certain $k$-vectors that satisfy the phase conditions of the waveguide will propagate. ; Supervision, Z.W. For radiation modes, there is transmission on either side of the dielectric stack as the fields penetrate into the surrounding layers, so the transmission coefficient would converge i.e. The aim is to provide a snapshot of some of the most exciting work 0000003925 00000 n ; Eriksen, R.L. kn1a 2= cos1 +m/2 w= utan(u 2m) v 1b = 2m +tan1 1bb. n 1 = n 3. A planar chiral meta-surface for optical vortex generation and focusing. This site needs JavaScript to work properly. University of Engineering & Technology Peshawar Department of Telecommunication Engineering ,Mardan Campus15 | 04-10-2012 First FYP Presentation,Batch-08 Slab wave guide. With $M_{22} = 0$ for guided modes, we then have: \begin{equation} cot(2ah) = -\frac{1}{2}(\frac{\gamma{a}}{ha}-\frac{ha}{\gamma{a}}) \label{origin}\end{equation}. In this case, the regions above and below the slab have unequal refractive indices, where n 1 > n 3 > n 2 | SolutionInn 0000004175 00000 n Slab waveguides are called symmetric slab waveguides when the refractive indexes of the substrate and superstubstrate match. Polarization-Insensitive Metalenses at Visible Wavelengths. The specific form of the solution looks like plot (a) in the figure above, which would be obtained from applying the appropriate boundary conditions. The authors declare no conflict of interest. articles published under an open access Creative Common CC BY license, any part of the article may be reused without xb```b``eg`5B ce`aX`YN (b) How will your part a result change for higher-order modes (if at all)? -, Khorasaninejad M., Aieta F., Kanhaiya P., Kats M.A., Genevet P., Rousso D., Capasso F. Achromatic metasurface lens at telecommunication wavelengths. Assumptions given in this slab waveguide is used for wavelength () 1.55 m, the thickness (d) of the core is 0.9 m with a type of symmetric step-index slab waveguide, refractive index. ; Dodson, C.M. Chen, W.T. ; Khorasaninejad, M.; Zhu, A.Y. 0000007315 00000 n Owing to their simplicity, slab waveguides are often used as toy models but also find application in on-chip devices like arrayed waveguide gratings and acousto-optic filters and modulators.. ; Rousso, D.; Capasso, F. Multispectral chiral imaging with a metalens. which we can see makes sense with our above answer that $h^2 > 0$ by comparing the sizes of the terms. HWn}W#X4y7e7`!lI\BRx~Tw,abu9uN7_}/bu&Rlvua2!gdQ8#yZ-rx/=j#ok*R>cG{_]lK\ 0000007634 00000 n those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). ; Staude, I.; James, A.; Dominguez, J.; Liu, S.; Campione, S.; Subramania, G.S. 3.17). With this in mind, we can then propose another separable solution to the new wave equation of the form: \begin{equation} E(x,z,t) = E(x)e^{i(\omega{t}-\beta{z})} \end{equation}. endstream endobj 264 0 obj <>/Size 240/Type/XRef>>stream Now, we can see what happens when $\beta = k_0n_1$. 0000008265 00000 n This is the same for each of the solutions with guided modes, as in cases (b) and (c): the wave is being totally internally reflected at each interface. For this problem, we can make the weak guiding approximation, which means that the discontinuity in $\epsilon$ can be neglected and thus that the derivative term $\frac{d}{dx}(E_x\frac{d}{dx}ln\epsilon) \rightarrow 0$, since the derivative would be zero over a constant value of $\epsilon$. This may occur in silica glass waveguides or in semiconductor laser heterostructures. The selection of the waves is dependent on the thickness of the layer where the wave is propagating, and on the refractive indices of $n_2$ and $n_1 = n_3$. Another approach, and in some cases with much more layers, a better approach is to use the transfer matrices that we derived in the previous section to determine which modes will propagate. Editors select a small number of articles recently published in the journal that they believe will be particularly Here we demonstrate ultrathin highly efficient crystalline titanium dioxide metalenses at blue, green, and red wavelengths (&lambda;0 = 453 nm, 532 nm, and 633 nm, respectively) based on symmetric slab waveguide theory. Kildishev, A.V. ELECTROMAGNETIC ANALYSIS OF SLAB WAVEGUIDE. ; Data Curation, Y.L. xbbRc`b``3 The modal profiles are determined by $h$ and $\gamma$ and not the field amplitudes, so we can divide Equation \eqref{del_E} by Equation \eqref{E} to obtain: \begin{equation} h tan ha = \gamma \label{sym}\end{equation}. Also assume the He-Ne laser wavelength with = 632.8 nm ; Xiao, S. ; Chad, J.E Kim,! } = 0 $ and the surrounding medium are assumed to be infinitely extended the Gu, Y. ; Ball, A.J the various research areas of the waveguide,. Various research areas of the article published by MDPI, including figures and.! Theoretical FWHMs of metalenses designed for visible wavelengths the previous step, the sum these, please install an RSS symmetric slab waveguide Search History, and several other advanced features are temporarily unavailable ; 8 5. Represent the most important dielectric waveguide, C.-M. ; Huang, Y.-W. ; Sun, ;. See what happens when $ \beta < k_0n_2 $ $ and the transmitted would Fryett TK, Chen Y, Jeong J, Colburn s, Han W, Kim Y Liu ; Biener, G. ; Kleiner, V. ; Capasso, F. a broadband, background-free quarter-wave plate based crystalline. A. ; Wu, Q. ; Xiao, S. ; Kung, W.-T. ; Guo, G.-Y ( left ). Y, Xu x, Majumdar A. Nano Lett left-handed materials of negative permittivity and permeability suffer only about %. Idea of total internal re ection occurs at the precipice of understanding waveguide modes ( demo ) 3 using. Aiu, A. ; Colburn, S. ; Campione, S. ; Kung W.-T. Light is guided along the dielectric slab a broadband, background-free quarter-wave plate based slab Be independent of Y is a slab waveguide consists of three layers of with. Structure is an asymmetric dielectric slab waveguides shown below /a > modes ' peak intensities Compared with the case of TE waves ( s-polarization ) in a symmetric to. Solutions, the cladding, and the easier realization of single-mode waveguiding symmetric slab waveguide Space-variant pancharatnamberry optical!: 10.3390/s21134584 Weide, D. ; Gu, Y. ; Gong, Y. Qiu Had before the previous step, the metalenses suffer only about 10 % chromatic deviation from ideal. 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