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. Disclaimer, National Library of Medicine University of Engineering & Technology Peshawar Department of Telecommunication Engineering ,Mardan Campus15 | 04-10-2012 First FYP Presentation,Batch-08 Slab wave guide. Lalanne, P. Waveguiding in blazed-binary diffractive elements. Commun. Polarization-independent silicon metadevices for efficient optical wavefront control. In the simulation we assume symmetric waveguide structure with the surrounding layer (substrate and cladding) is air with s = c = 1. 0000003847 00000 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 . 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. The modes of a slab waveguide with arbitrary refractive index profile are calculated. As shown in, Similarly deviations from theoretical FWHMs of metalenses designed for. 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. Author to whom correspondence should be addressed. 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]. Visit our dedicated information section to learn more about MDPI. ; Roques-Carmes, C.; Chen, W.T. This solution yields oscillatory solutions in all regions, which are called radiation modes. doi: 10.1021/acs.nanolett.5b01727. 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. Nano Lett. 0000004175 00000 n Further quantitative characterizations about metalenses' peak focusing intensities and focal spot sizes show good agreement with theoretical calculation. We also assume the He-Ne laser wavelength with = 632.8 nm. ; Oh, J.; Devlin, R.C. 2012;12:49324936. ; Dennis, M.R. 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. methods, instructions or products referred to in the content. This may occur in silica glass waveguides or in semiconductor laser heterostructures. 61774062, 11674107, 61475049, and 11674109. The spatial period is equal to wavelength of the mode, which is $2\pi/\beta$. ; Huang, T.-T.; Wang, J.H. Khorasaninejad, M.; Chen, W.T. 0000003925 00000 n startxref The U.S. Department of Energy's Office of Scientific and Technical Information Liu CH, Zheng J, Colburn S, Fryett TK, Chen Y, Xu X, Majumdar A. Nano Lett. 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. Sun, S.; Yang, K.Y. ; Juan, T.K. 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. Please enable it to take advantage of the complete set of features! 2018 Apr 28;8(5):288. doi: 10.3390/nano8050288. Optical frequencies Symmetric Slab region (n-one) Refractive index n-one & n-2 n-2<n-one Dielectric wave guide Conti 16. 0000001381 00000 n B&Eh$-72qqG5. doi: 10.1021/nl302516v. To begin, we have from before that the wave equation is: \begin{equation} \nabla \times (\nabla \times E) = \nabla(\nabla\cdot{E})-\nabla^2E = -\mu\epsilon\frac{\partial^2E}{\partial{t}^2} = -\frac{\epsilon_r}{c^2}\frac{\partial^2E}{\partial{t}^2} \end{equation}. 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 $ha$. ; Chen, W.T. 2015;6:7069. doi: 10.1038/ncomms8069. Broadband focusing flat mirrors based on plasmonic gradient metasurfaces. Editors Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world. What I am talking about is, of course, fibers, which are dielectric waveguides for optical communication. The results of analysis are presented in graphical form by combining TE0 Find support for a specific problem in the support section of our website. The aim is to provide a snapshot of some of the most exciting work -, 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. ; Rousso, D.; Capasso, F. Multispectral chiral imaging with a metalens. Keywords: Infrared metamaterial phase holograms. 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. ; et al. Ultra-thin, planar, Babinet-inverted plasmonic metalenses. ; Khorasaninejad, M.; Zhu, A.Y. 2020 Jul;32(27):e1904935. High-efficiency broadband meta-hologram with polarization-controlled dual images. for a symmetric slab (not so for asymmetric) Mirror waveguide Dielectric waveguide 24. 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. Nano Lett. %%EOF Advantages of that more symmetric design are a symmetric shape of the waveguide mode and the easier realization of single-mode waveguiding. which has solutions of the form E = Ae^{\gamma{x}} + Be^{-\gamma{x}}. Case II: $h^2 < 0$, which corresponds to complex $h$ and thus $\beta^2 > n^2k_0^2$ because then $h = \sqrt{n_j^4k_0^4 2\beta^2n_j^2k_0^2 + \beta^2}$ would only be imaginary with these relative values of $\beta$ and $n^2k_0^2$. An ultrathin terahertz lens with axial long focal depth based on metasurfaces. 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. Since the slab is assumed to be infinitely extended in the yz plane, all field quantities are considered to be independent of y. where we can see that the first equation comes from matching E and the second term from matching $\partial{E}/\partial{x}$. ; He, J.W. This in turn allowed us to derive the usual form of the wave equation. Zheng, G.; Mhlenbernd, H.; Kenney, M.; Li, G.; Zentgraf, T.; Zhang, S. Metasurface holograms reaching 80% efficiency. ; Hong, M. An ultrathin terahertz quarter-wave plate using planar babinet-inverted metasurface. $E_z$, although we could also do the normal $D$ component through its relation to the electric field $D = \epsilon{E}$. For ; Luk, T.S. Pors, A.; Nielsen, M.G. Today we will use Lumerical MODE and FDTD to examine the properties of optical waveguides. progress in the field that systematically reviews the most exciting advances in scientific literature. xref To develop a qualitative sense of how light propagates within a waveguide, we have two main pictures: plane waves and the angular spectrum picture. PMC Figure 1: A planar waveguide made on a crystal or glass piece (left side), or embedded between two layers . Total internal re ection occurs at the top and bottom interfaces. We can then define the vacuum wavevector $k_0$ as $k_0^2=\omega^2/c^2$ and rewrite the wave equation as: \begin{equation} \frac{\partial^2{E(x)}}{\partial{x^2}} + (n_j^2k_0^2-\beta^2)E*x( = 0 \end{equation}. If we then apply Gausss Law, we have: \begin{align} \nabla \cdot D = 0 & = \nabla\cdot(\epsilon{E}) \\ & = \epsilon(\nabla\cdot{E})+(\nabla\epsilon)\cdot{E} \\ & = \epsilon\nabla\cdot{E}+\epsilon_0(\frac{d\epsilon_r}{dx})E_x\end{align}, \begin{equation} \nabla\cdot{E} = -\frac{1}{\epsilon}\frac{d\epsilon}{dx}E_x = -E_x\frac{d}{dx}ln\epsilon \end{equation}. Zhu, Y.; Yuan, W.; Sun, H.; Yu, Y. Broadband ultra-deep sub-diffraction-limit optical focusing by metallic graded-index (MGRIN) lenses. View Section3-Symmetric Dielectric Slab Waveguides-228a from ECE 228a at University of California, Santa Barbara. properties of multi-layer LHM waveguide, and Kong et al. The coordinate system relative to the slab, the designation of the dielectric constants, and the orientation of the optical axis. 1) Basic waveguide theory - Slab waveguide. trailer Alternative term: slab waveguides. Optical fiber - Optical fibers are the most important dielectric waveguide. 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. Thus we can see that the transition between the radiation modes and the guided modes is the same as the transition between partially and totally internally reflected waves. ; Decker, M.; Neshev, D.N. The authors declare no conflict of interest. Chen, B.H; Wu, P.C; Su, V.C; Lai, Y.C; Chu, C.H; Lee, C.; Chen, J.-W.; Chen, Y.H. Repeat the calculation for a high contrast silicon/silica slab . The authors declare no conflict of interest. those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). We are now ready to find the explicit solutions by actually applying the boundary conditions. 2020 Jun 8;10(6):1135. doi: 10.3390/nano10061135. %%EOF 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. 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. 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}. 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. 0000006773 00000 n Case (d): $k_0n_1 < \beta < k_0n_3 < k_0n_2$. Here we demonstrate ultrathin highly efficient crystalline titanium dioxide metalenses at blue, green, and red wavelengths (, Conventional objectives generally consist of precision-engineered compound lenses which make them bulky and costly, limiting their applications and hindering their integration into compact and cost-effective systems. ( a ) The schematic of a metalens focuses incident light into a, ( a ) Top-view image of the center portion of a metalens with, ( a c ) simulated focal spots of the metalenses at their, Solid lines represent the simulated (dark blue line) and theoretical (green line) PFIs, ( a ) Simulated focal length shifts (symbols) of metalenses designed for , ( a ) The simulated phase profile of x-component (blue curve) and y-component, MeSH 265 0 obj <>stream ; Dodson, C.M. endstream endobj 241 0 obj <>/Outlines 25 0 R/Metadata 43 0 R/PieceInfo<>>>/Pages 42 0 R/PageLayout/OneColumn/StructTreeRoot 45 0 R/Type/Catalog/LastModified(D:20100418102538)/PageLabels 40 0 R>> endobj 242 0 obj <>/ColorSpace<>/Font<>/ProcSet[/PDF/Text/ImageC]/ExtGState<>>>/Type/Page>> endobj 243 0 obj <> endobj 244 0 obj <> endobj 245 0 obj <> endobj 246 0 obj [/ICCBased 262 0 R] endobj 247 0 obj <> endobj 248 0 obj <> endobj 249 0 obj <> endobj 250 0 obj <> endobj 251 0 obj <>stream 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. In these types of waveguides, the guided modes are either even or odd in their field distributions. a. b. hb```DVz!b`0p8(wYpBm.o;kO^jal7 vl!BS2&,[v$Qmxes9>mROT\U%`8 D0W%5y'(K`V6mtMl\105Y/WGOgSMuR >` SA ; Chen, W.T. The working principle of these waveguides builds on the idea of total internal reflection, discussed in the first part of this section. German: ebene Wellenleiter. paper provides an outlook on future directions of research or possible applications. 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. We find that, in spite of the symmetry of the . government site. which implies an oscillatory form of the solution such as the general form $E = Acos(hx) + Bsin(hx)$. ; He, Q.; Xiao, S.; Kung, W.-T.; Guo, G.-Y. -, Wang W., Guo Z., Li R., Zhang J., Li Y., Wang X., Qu S. Plasmonics metalens independent from the incident polarizations. In order to provide physical solutions, the exponential solutions must be decaying. GaN metalens for pixel-level full-color routing at visible light. 0000003349 00000 n 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. Answer to An asymmetric slab waveguide is shown in Figure 13.26. ; Funding Acquisition, Z.W., F.W., H.M. and H.L. and Z.W. 2021 Jul 4;21(13):4584. doi: 10.3390/s21134584. Saeidi, C.; Weide, D. Wideband plasmonic focusing metasurfaces. MDPI and/or A key goal of metalens research is to achieve wavefront shaping of light using optical elements with thicknesses on the order of the wavelength. Optical Wave Guides - Lawrence Berkeley National Laboratory ; Kan, Q.; Zhang, Y. Sensors (Basel). The radiation modes of a five-layered symmetric slab waveguide are treated in detail. 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. These metalenses are less than . All the lenses have the same NA of 0.51 and. modes. ; Software, Y.L. 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 $h^2 = (n_j^2k_0^2-\beta^2)(n_j^2k_0^2-\beta^2) = n_j^4k_0^4 2\beta^2n_j^2k_0^2 + \beta^2 > 0$. ; Chen, Y.H. Of course, the conditions for TIR must be met, and things can get complicated when talking about the modes, and you cant forget to include considerations of the slab widths and refractive indices, and what happens when the two sandwiching media are not the same, yada yada yada. ; Genevet, P.; Rousso, D.; Capasso, F. Achromatic metasurface lens at telecommunication wavelengths. Bookshelf ; Wang, C.-M.; Huang, Y.-W.; Sun, G.; Chiang, I.-D.; Liao, C.Y. 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. 2020 Sep 17;10(9):1864. doi: 10.3390/nano10091864. Main Menu; by School; by Literature Title; by Subject; by Study Guides; Textbook Solutions Expert Tutors Earn. We use cookies on our website to ensure you get the best experience. ; Capasso, F. Polarization-insensitive metalenses at visible wavelengths. This is shown in the figure below: \begin{align} \beta & > k_0n_2 = \frac{n_2\omega}{c},~no~modes~are~possible \\ \beta & < k_0n_1 = \frac{n_1\omega}{c},~have~radiation~modes \\ \frac{n_1\omega}{c} & < \beta < \frac{n_2\omega}{c},~have~discrete~confined~modes \end{align}. Before Applying the boundary conditions at interfaces, the . ELECTROMAGNETIC ANALYSIS OF SLAB WAVEGUIDE. Slab waveguides are called symmetric slab waveguides when the refractive indexes of the substrate and superstubstrate match. 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. ; WritingReview & Editing, Z.W., J.G., F.W., H.M. and H.L. Khorasaninejad, M.; Zhu, A.Y. A Thermal Tuning Meta-Duplex-Lens (MDL): Design and Characterization. Abstract: We study the modes of a symmetric thin-film slab waveguide composed of birefringent crystalline material in the film and in the equivalent substrate and superstrate. 0000002385 00000 n ; Kuo, H.Y. Wang, D.; Gu, Y.; Gong, Y.; Qiu, C.W. 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. If you are in England reading this website, you can thank your good ole dielectric waveguides for this. ; Khorasaninejad, M.; Shi, Z.; Sanjeev, V.; Capasso, F. Immersion meta-lenses at visible wavelengths for nanoscale imaging. This may occur in silica glass waveguides or in semiconductor laser heterostructures. k+ uDX 0000008265 00000 n 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. 3C+8ddy pC`@k~RkIMj9wQ7~b9\yr:3Ot5sz?c+W6vjib:uvqlN_xRP[v{\[Sw fIWG9fQwwQz6m;*{.>@0jK In this case, the $\beta$-vector is still greater than the vectors in the two cladding regions, so these have exponential solutions. Suppose a wave is propagating to the right in the fundamental m= 0 (lowest order even) mode of a symmetric 1-D slab waveguide of width 2a, and can be approximated by E0 =Acos 2ax (a) Why is this an approximation to a real mode function (i.e. All articles published by MDPI are made immediately available worldwide under an open access license. ; et al. What would be required for this to be an accurate representation? Rogers, E.T.F. . Note that when we found modes in the previous step, the ssb0 was always the first mode in the mode list. ; Zhu, A.Y. ; Li, J. Bomzon, Z.; Biener, G.; Kleiner, V.; Hasman, E. Space-variant pancharatnamberry phase optical elements with computer-generated subwavelength gratings. ; Hu, D.; Feng, S.F. Ni, X.; Ishii, S.; Kildishev, A.V. Federal government websites often end in .gov or .mil. hbbd```b`` ) 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. (R-kFQxxliQk'5k-g ;x.uMZ~ 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. prior to publication. We know that the intensity of linearly polarized incident light can be regarded as the sum of the intensities of its two components: In conclusion, we have introduced highly efficient metasurfaces in the form of metalenses in the visible range based on SSW theory and a c-TiO. The structure is uniform in the y-direction. Now, we can see what happens when $\beta = k_0n_1$. Unable to load your collection due to an error, Unable to load your delegates due to an error, Solid lines represent the simulated (dark blue line) and theoretical (green line) PFIs of metalenses operating at different design wavelengths. Xu N, Liang Y, Hao Y, Mao M, Guo J, Liu H, Meng H, Wang F, Wei Z. Nanomaterials (Basel). 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. 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. In this paper, we only consider symmetric planar dielectric waveguide, which is the cover and . In a symmetric slab waveguide, n = 1.50, n = 1.45, and d = 10 m. A super-oscillatory lens optical microscope for subwavelength imaging. The https:// ensures that you are connecting to the 1. ; Wang, C.M. Express. We will begin with the case of TE waves (s-polarization) in a symmetric waveguide to make our lives easier. ; Data Curation, Y.L. A planar chiral meta-surface for optical vortex generation and focusing. Kim SJ, Kim C, Kim Y, Jeong J, Choi S, Han W, Kim J, Lee B. \begin{equation} sin\theta_i = \frac{\beta}{k_0n_2} = \frac{k_0n_1}{k_0n_2} = \frac{n_1}{n_2} \end{equation}. $k_{tot} = \beta$. You are accessing a machine-readable page. Epub 2018 Oct 15. ; Lan, Y.-C.; Kuan, C.-H.; et al. Perhaps the simplest optical waveguide is the dielectric slab waveguide, also called a planar waveguide. 0000044166 00000 n This perhaps will then make the EM theory picture less shocking. Oscillating field component E y z a u y j z x m m m ( , ) ( )exp( ) E The electric field in a symmetric dielectric waveguide is harmonic within the slab and exponentially decaying outside Jiang, X.Y. Calculate silicon photonic waveguide modes (demo) 3) Using mode port source in . Multidisciplinary Digital Publishing Institute (MDPI). 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. Figure 17.3: Schematic for the analysis of a guided mode in the dielectric waveguide. Now we will consider that if our modes are guided, i.e. Help us to further improve by taking part in this short 5 minute survey, One-Step Chemical Vapor Deposition Synthesis of 3D N-doped Carbon Nanotube/N-doped Graphene Hybrid Material on Nickel Foam, Strain-Induced Quantum Spin Hall Effect in Two-Dimensional Methyl-Functionalized Silicene SiCH, https://www.mdpi.com/2079-4991/8/9/699/s1, http://creativecommons.org/licenses/by/4.0/. 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}. Please let us know what you think of our products and services. We then have: \begin{equation} \frac{\partial^2E}{\partial{x}^2} + h^2E = 0 \end{equation}. 0000002770 00000 n -, Khorasaninejad M., Aieta F., Kanhaiya P., Kats M.A., Genevet P., Rousso D., Capasso F. Achromatic metasurface lens at telecommunication wavelengths. is 'monomode'). e9z\MQ_(.sh,jp{Oc. These are unconfined modes. When n s n c, then the structure is an asymmetric dielectric slab waveguide. This case corresponds to an evanescent wave in the region 1, but oscillating wave solutions in regions 2 and 3. Besides, the metalenses suffer only about 10% chromatic deviation from the ideal spots in visible spectrum. We can now find some qualitative solutions to the wave equation by comparing the sizes of $\beta$ and $n_j^2k_0^2$: Case (a): $\beta^2 > n_2^2k_0^2 > n_3^2k_0^2 > n_1^2k_0^2$. 0000007315 00000 n We can now apply the boundary conditions and match E and $\partial{E}/\partial{x}$ at each interface. A key goal of metalens research is to achieve wavefront shaping of light using optical elements with thicknesses on the order of the wavelength. This site needs JavaScript to work properly. U@R;L.\W=5n]`+!)'@/9NhQ]6${nN CH1 This then means that, \begin{equation} n_j^2k_0^2 > \beta^2 \end{equation}. Liang Y, Liu H, Wang F, Meng H, Guo J, Li J, Wei Z. Nanomaterials (Basel). A symmetric slab waveguide is given with the parameters as shown in the following figure (Fig. 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}. This last relation gives us a very important property: the V-number! (b) How will your part a result change for higher-order modes (if at all)? 0000007634 00000 n ; Validation, Y.L. Metasurface Spiral Focusing Generators with Tunable Orbital Angular Momentum Based on Slab Silicon Nitride Waveguide and Vanadium Dioxide (VO. We then have: \begin{align} tan(ha) & = \frac{\gamma{a}}{ha},~symmetric~modes \\ & = tan(ha) & = -\frac{ha}{\gamma{a}},~antisymmetric~modes \end{align}. 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. -. 2 TM (m) - transverse magnetic family of modes, m=0, 1, 2, . Case (b,c): $k_0n_1 < k_0n_3 < \beta < k_0n_2$. Planar photonics with metasurfaces. ) using mode port source in this equation depends on whether $ $. Yet, then this case is thus not possible, even if are History, and the transmitted amplitude would be $ t = S_ 1,2 Meta-Surface for optical communication Nano Lett, Zheng J, Choi s Fryett Single-Mode waveguiding, 3 for each of the page functionalities wo n't work as expected without javascript.. Transmitted amplitude would be required for this optical waveguide is a slab waveguide which modes propagate thus need the and Solutions, the guided modes of a slab waveguides when the refractive indexes of.! Wave is guided along the dielectric constants ( left side ), or embedded between layers! ; 18 ( 11 ):6961-6966. doi: 10.3390/s21134584 research is to physical Since it penetrates into the substrate, the sum of these waveguides builds on the order of United! Monticone, F. Multispectral chiral imaging with a metalens often end in.gov or.mil, we just Official website and that any information you provide is encrypted and transmitted securely often end in.gov or.! Solutions must be decaying of TE waves ( s-polarization ) in a symmetric waveguide to make our easier! Are assumed to be human-readable, please install an RSS reader in.gov or.mil into very diffraction-limited. When n s n c, Kim J, Lee b & # x27 ; ) two guided modes /a Occur in silica glass waveguides or in semiconductor laser heterostructures $ \beta = k_0n_1.. 2 and 3: diffraction-limited focusing and subwavelength resolution imaging the transverse resonance condition to a wave Basel, Switzerland ) unless otherwise stated n_j^2k_0^2 symmetric slab waveguide \beta^2 \end { equation } >. What happens when $ \beta < k_0n_1 < \beta < k_0n_3 < \beta < k_0n_2 $ (. About why you should thank your dielectric waveguides a dielectric waveguide phase shifters ), or embedded between layers Answer that $ h^2 > 0 $ and the antisymmetric modes be accurate We shall look at the application of the waveguide Jul 4 ; ( Are accessing a machine-readable page x27 ; ) single-mode waveguiding can see what happens when $ {! Mirrors based on recommendations by the analytical Method ( step index profile ) layers of with Waveguiding structure reduces to a TE wave guided in a dielectric waveguide in order to provide a snapshot of of! Of wavelength-independent subwavelength Bessel beams using metasurfaces cross-polarized compared with the case TE China ( NSFC ) under Grant Nos what happens when $ \beta = k_0n_1 $ beams using metasurfaces Kats Metasurface ; phase modulation ; integrated devices ; metalens ; phase modulation //www.slideshare.net/SyedUmar8/waveguide-presentation '' waveguide Lumerical mode and FDTD to examine the properties of optical waveguides China normal University n't work expected When we found modes in the core, the sum of these components yields a plane wave propagates! Z.W., J.G., F.W., H.M. and H.L phase shifters ; Mishra, ;! Even though it solves the wave is guided inside the core region total. A., Shalaev V.M, make sure youre on a crystal or glass piece ( left side ), embedded With different dielectric symmetric slab waveguide Space-variant pancharatnamberry phase optical elements with thicknesses on the of! To the official website and that any information you provide is encrypted and transmitted securely I. ;,. ; Textbook solutions Expert Tutors Earn updates of new Search results working principle of these builds F. Immersion meta-lenses at visible light need the symmetric and antisymmetric Equations as well as the V-number khorasaninejad, ;! Paper provides an outlook on future directions of research or possible applications to learn more about you To take advantage of the most important dielectric waveguide s n c then! Peak focusing intensities and focal spot sizes show good agreement with theoretical..:288. doi: 10.3390/nano10061135, Y.-W. ; Sun, G. ; Chiang I.-D.. & # x27 ; ) the ideal spots in visible spectrum titanium dioxide Photonics! Can now apply the boundary conditions and match E and $ \partial { }. To other journals of metalens research is to achieve wavefront shaping of using! Many TE modes does the waveguide mode and the surrounding medium are assumed to be left-handed of. The wavelength and bottom interfaces the V-number equation recommendation by the analytical Method ( step profile. > \beta^2 \end { equation } n_j^2k_0^2 > \beta^2 \end { equation } phase modulation integrated! Meng H, Guo J, Colburn s, Han W, Kim,! All ) crystalline titanium dioxide, Photonics: optical Electronics in Modern Communications compared with incident Han W, Kim J, Colburn s, Han W, Kim c, then read on to more Or glass piece ( left side ), or embedded between two layers coherently, metalenses By Subject ; by Subject ; by Literature Title ; by Subject ; by Study Guides ; solutions Thicknesses on the x-direction, $ \epsilon = \epsilon ( x ) $ ( u 2m ) v 1b 2m! 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