Finite Element Method for Sensing Applications

  • Khaled S. R. Atia
  • Souvik Ghosh
  • Ahmed M. Heikal
  • Mohamed Farhat O. Hameed
  • B. M. A. Rahman
  • S. S. A. Obayya


In this chapter, the fundamentals of the nodal finite element method (FEM) are presented, including the first-order element and second-order element. The nodal FEM is introduced for the scalar concept of the propagation constant of 2D waveguide cross section. Then, it is extended to include the time domain analysis under perfectly matched layer absorbing boundary conditions. A simple sensor based on optical grating is thereafter simulated using the time domain FEM. Also, the full vectorial analysis is discussed through the application of the penalty function method on the nodal FEM and the vector finite element method (VFEM). For the penalty function method, a global weighting factor is used to incorporate the effect of the divergence-free equation. In the VFEM, nodes are used to represent the orthogonal component of the field while the edges are used to represent the tangential component for accurate application of the boundary conditions. Finally, surface plasmon resonance photonic crystal fiber biosensor is introduced as an example of the full vectorial analysis using the VFEM.


Finite element method Finite element time domain Penalty function method Full-vector finite element method Edge finite element method 


  1. 1.
    M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, 1992)CrossRefGoogle Scholar
  2. 2.
    Zienkiewitz, The Finite Element Method (New York, McGraw-Hill, 1973)Google Scholar
  3. 3.
    M.V.K. Chari, P.P. Silvester, Finite Elements in Electrical and Magnetic Field Problems (Chechester, Wiley, 1980)Google Scholar
  4. 4.
    E. Yamashita, Analysis Methods for Electromagnetic Wave Problems (Boston, Artech House, 1990)Google Scholar
  5. 5.
    D.B. Davidson, Computational Electromagnetics for RF and Microwave Applications (Cambridge, Cambridge University Press, 2005)Google Scholar
  6. 6.
    A. Taflov, Computational Electrodynamics: The Finite Difference Time Domain Method (Artech, 1995)Google Scholar
  7. 7.
    D. Pinto, S.S.A. Obayya, Improved complex-envelope alternating-direction-implicit finite-difference-time-domain method for photonic-bandgap cavities. J. Lightwave Technol. 25(1), 440–447 (2007)CrossRefGoogle Scholar
  8. 8.
    B. Rahman, J. Davis, Finite-element solution of integrated optical waveguides. J. Lightwave Technol. 2(5), 682–688 (1984)CrossRefGoogle Scholar
  9. 9.
    B.M. Azizur Rahman, Finite-element analysis of optical and microwave waveguide problems. IEEE Trans. Microwave Theor. Techniq. 32(1), 20–28 (1984)Google Scholar
  10. 10.
    K. Kawano, T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equations and the Schrodinger’s Equation (New York, wiley, 2001)Google Scholar
  11. 11.
    M. Koshiba, H. Saitoh, M. Eguchi, K. Hirayama, Simple scaler finite element approach to optical waveguides. IEE Proc. J. 139, 166–171 (1992)Google Scholar
  12. 12.
    S.S.A. Obayya, Computational Photonics (Wiley, 2011)Google Scholar
  13. 13.
    S.S.A. Obayya, Efficient finite-element-based time-domain beam propagation analysis of Optical integrated circuits. IEEE J. Quant. Electron. 40(5), 591–595 (2004)CrossRefGoogle Scholar
  14. 14.
    T. Itoh, R. Mittra, Spectral domain approach for calculation the dispersion characteristics of microstrip lines. IEEE Trans. Microwave Theor. Tech. MTT21 496–499 (1973)Google Scholar
  15. 15.
    A. Abdrabou, A.M. Heikal, S.S.A. Obayya, Efficient rational Chebyshev pseudo-spectral method with domain decomposition for optical waveguides modal analysis. Opt. Express 24(10), 10495–10511 (2016)CrossRefGoogle Scholar
  16. 16.
    D.M. Pozar, Microwave Engineering (Wiley, Hoboken, NJ, 2012)Google Scholar
  17. 17.
    J. Berenger, A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200 (1994)MathSciNetCrossRefGoogle Scholar
  18. 18.
    S.D. Gedney, An anisotropic perfectly matched layer absorbing media for the truncation of FDTD latices. Antennas Prop. IEEE Trans. 44, 1630–1639 (1996)Google Scholar
  19. 19.
    W.C. Chew, W.H. Weedon, A 3D perfectly matched medium from modifie Maxwell’s equations with stretched coordinates. Microwave Opt. Technol. Lett. 7, 590–604 (1994)CrossRefGoogle Scholar
  20. 20.
    W.C. Chew, J.M. Jin, E. Michielssen, complex coordinate stretching as a generalized absorbing boundary condition. Microwave Opt. Technol. Lett. 15(6), 363–369 (1997)CrossRefGoogle Scholar
  21. 21.
    M. Koshiba, Y. Tsuji, M. Hikari, Time-domain beam propagation method and its application to photonic crystal circuits. J. Lightwave Technol. 18(1), 102–110 (2000)CrossRefGoogle Scholar
  22. 22.
    V.F. Rodríguez-Esquerre, M. Koshiba, Finite element analysis of photonic crystal cavities: time and frequency domain. J. Lightwave Technol. 23(3), 1514–1521 (2005)CrossRefGoogle Scholar
  23. 23.
    T. Fujisawa, M. Koshiba, time-domain beam propagation method for nonlinear optical propagation analysis. J. Lightwave Tech. 22(2), 684–691 (2004)CrossRefGoogle Scholar
  24. 24.
    V.F. Rodríguez-Esquerre, M. Koshiba, E.H.-Figueroa, Frequency-dependent envelope finite element time domain analysis of dispersion materials. Microwave Opt. Tech. Lett. 44(1), 13–16 (2004)CrossRefGoogle Scholar
  25. 25.
    A. Niiyama, M. Koshiba, Y. Tsuji, An efficient scalar finite element formulation for nonlinear optical channel waveguides. J. Lightwave Technol. 13(9), 1919–1925 (1995)CrossRefGoogle Scholar
  26. 26.
    G.R. Liu, A Generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods. Int. J. Comput. Methods (2008)Google Scholar
  27. 27.
    K.S.R. Atia, S.S.A. Obayya, Novel gradient smoothing method-based time domain beam propagation analysis of optical integrated circuits. Signal Process. Photon. Commun. JM3A–23 (2015)Google Scholar
  28. 28.
    G.R. Liu, Meshfree Methods: Moving Beyond the Finite Element Method (CRC Press, 2009)Google Scholar
  29. 29.
    J.R. LeVeque, Finite Volume Methods for Hyperbolic Problems (Cambridge, Cambridge, 2002)Google Scholar
  30. 30.
    K.S.R. Atia, A.M. Heikal, S.S.A. Obayya, Efficient smoothed finite element time domain beam propagation method for photonic devices. Opt. Exp. 23(17), 22199–22213 (2015)CrossRefGoogle Scholar
  31. 31.
    K.S.R. Atia, A.M. Heikal, S.S.A. Obayya, Time-domain beam propagation method based on gradient smoothing technique for dispersive materials, in Progress in Electromagnetics Research symposium (PIERS) (2015)Google Scholar
  32. 32.
    P.L. Liu, Q. Zhao, F.S. Choa, Slow-wave finite-difference beam propagation method. IEEE Photon. Technol. Lett. 7(8), 890–892 (1995)CrossRefGoogle Scholar
  33. 33.
    G.H. Jin, J. Harari, J.P. Vilcot, D. Decoster, An improved time domain beam propagation method for integrated optics components. IEEE Photon. Technol. Lett. 9(3), 117–122 (1997)CrossRefGoogle Scholar
  34. 34.
    J. Lee, B. Fornberg, A split step approach for the 3-D Maxwell’s equations. J. Comput. Appl. Math. 158(2), 485–505 (2003)MathSciNetCrossRefGoogle Scholar
  35. 35.
    M. Movahhedi, A. Abdipour, Alternating direction implicit formulation for the finite element time domain method. IEEE Trans. Microwave Theor. Technol. 55(6), 1322–1331 (2007)CrossRefGoogle Scholar
  36. 36.
    J.F. Lee, WETD-A finite element time-domain approach for solving Maxwell’s equations. IEEE Microwave Guided Wave Lett. 4(1), 11–13 (1994)CrossRefGoogle Scholar
  37. 37.
    V.F. Rodríguez-Esquerre, H.E. Hernández-Figueroa, Novel time-domain step-by-step scheme for integrated optical applications. IEEE Photon. Technol. Lett. 13(4), 311–313 (2001)CrossRefGoogle Scholar
  38. 38.
    H.A. Van der Vorst, Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. 13(2), 631–644 (1992)MathSciNetCrossRefGoogle Scholar
  39. 39.
    A.D. Berk, Variational principles for electromagnetic resonators and waveguides. IRE Trans. Antennas Propagat. 4(2) (1956)CrossRefGoogle Scholar
  40. 40.
    K.T.V. Grattan, B.T. Meggitt, Optical Fiber Sensor Technology: Fundamental (US, Springer, 2000)Google Scholar
  41. 41.
    T. Dar, J. Homola, B.M.A. Rahman, M. Rajarajan, Label-free slot-waveguide biosensor for the detection of DNA hybridization. Appl. Opt. 51(34) (2012)CrossRefGoogle Scholar
  42. 42.
    C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets et al., All-optical high-speed signal processing with silicon–organic hybrid slot waveguides. Nat. Photonics. 3(4) (2009)CrossRefGoogle Scholar
  43. 43.
    Barrios CA, Banuls MJ, Gonzalez-Pedro V, Gylfason KB, Sanchez, Griol A, et al. Label-free optical biosensing with slot-waveguides. Opt. Lett. 33(7) 2008CrossRefGoogle Scholar
  44. 44.
    M. Koshiba, K. Hayata, M. Suzuki, Vectorial finite-element formulation without spurious solutions for dielectric waveguide problems. Electron. Lett. 20, 409–410 (1984)CrossRefGoogle Scholar
  45. 45.
    Sh Birman, M. The, Maxwell operator for a resonator with inward edges. Vestnik Leningradskogo Universiteta. Matematika. 19, 1–8 (1986)zbMATHGoogle Scholar
  46. 46.
    S.M. Birman, Z.M. Solomyak, Maxwell operator in regions with nonsmooth boundaries. Siberian Malh. J. 28, 12–24 (1987)CrossRefGoogle Scholar
  47. 47.
    F. Kikuchi, Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism. Compur. Methods Appl. Mech. Eng. 64, 509–521 (1987)MathSciNetCrossRefGoogle Scholar
  48. 48.
    M.F.O. Hameed, Y.K.A. Alrayk, S.S.A. Obayya, Self-calibration highly sensitive photonic crystal fiber biosensor. IEEE Photon. 8(3) (2016)CrossRefGoogle Scholar
  49. 49.
    M.F.O. Hameed, M. El-Azab, A.M. Heikal, S.M. El-Hefnawy, S.S.A. Obayya, Highly sensitive plasmonic photonic crystal temperature sensor filled with liquid crystal. IEEE Photon. Technol. Lett. 28(1) (2015)CrossRefGoogle Scholar
  50. 50.
    S.I. Azzam, R.E.A. Shehata, M.F.O. Hameed, A.M. Heikal, S.S.A. Obayya, Multichannel photonic crystal fiber surfrace plasmon resonance based sensor. J. Opt. Quant. Electron. 48(142) (2016)Google Scholar
  51. 51.
    F.F.K. Hussain, A.M. Heikal, M.F.O. Hameed, J. El-Azab, W.S. Abdelaziz, S.S.A. Obayya, Dispersion characteristics of asymmetric channel plasmon polariton waveguide. IEEE J. Quant. Electron. 50(6) (2014)CrossRefGoogle Scholar
  52. 52.
    M.F.O. Hameed, S.S.A. Obayya, H.A. El-Mikati, Passive polarization converters based on photonic crystal fiber with L-shaped core region. IEEE J. Lightwave Technol. 50(6) (2012)Google Scholar
  53. 53.
    M.F.O. Hameed, A.M. Heikal, S.S.A. Obayya, Novel passive polarization rotator based on spiral photonic crystal fiber. IEEE Photon. Technol. Lett. 25(16) (2013)CrossRefGoogle Scholar
  54. 54.
    M.F.O. Hameed, S.S.A. Obayya, R.J. Wiltshire, Beam propagation analysis of polarization rotation in soft glass nematic liquid crystal photonic crystal fibers. IEEE Photon. Technol. Lett. 22(3) (2010)CrossRefGoogle Scholar
  55. 55.
    M.F.O. Hameed, A.M. Heikal, S.S.A. Obayya, Passive polarization converters based on photonic crystal fibers. IEEE Photon. Technol. Lett. 22(3) (2010)Google Scholar
  56. 56.
    S.I. Azzam, M.F.O. Hameed, N.F.F. Areed, S.S.A. Obayya, H. El-Mikati et al., Proposal of ultracompact CMOS compatible TE-/TM-pass polarizer based on SOI platform. IEEE Photon. Technol. Lett. 33(13) (2015)Google Scholar
  57. 57.
    A.M. Heikal, F.F.K. Hussain, M.F.O. Hameed, S.S.A. Obayya, Efficient polarization filter design based on plasmonic photonic crystal fiber. IEEE J. Lightwave Technol. 33(13) (2015)CrossRefGoogle Scholar
  58. 58.
    S.S.A. Obayya, M.F.O. Hameed, N.F.F. Areed, Computational Liquid Crystal Photonics: Fundamentals (Wiley, Modelling and Applications, 2016)CrossRefGoogle Scholar
  59. 59.
    M.F.O. Hameed, S.S.A. Obayya, K. Al-Begain, A.M. Nasr, M.L. Abo el Maaty, Coupling characteristics of a soft glass nematic liquid crystal photonic crystal fiber coupler. IET Optoelectron. 3(6) (2009)Google Scholar
  60. 60.
    M.F.O. Hameed, A.M. Heikal, B.M. Younis, M.M. Abdelrazzak, S.S.A. Obayya, Ultra-high tunable liquid crystal plasmonic photonic crystal fiber polarization filter. Opt. Exp. 23(6), 7007–7020 (2015)CrossRefGoogle Scholar
  61. 61.
    B.M. Younis, A.M. Heikal, M.F.O. Hameed, S.S.A. Obayya, Enhancement of plasmonic liquid photonic crystal fiber. Plasmonics p. 1–7 (2016)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Khaled S. R. Atia
    • 1
    • 2
  • Souvik Ghosh
    • 3
  • Ahmed M. Heikal
    • 1
    • 4
  • Mohamed Farhat O. Hameed
    • 5
    • 6
  • B. M. A. Rahman
    • 3
  • S. S. A. Obayya
    • 1
    • 4
  1. 1.Centre for Photonics and Smart MaterialsZewail City of Science and TechnologyGizaEgypt
  2. 2.Advanced Research ComplexUniversity of OttawaOttawaCanada
  3. 3.School of Mathematics, Computer Science and EngineeringCity University of LondonLondonEngland
  4. 4.Faculty of Engineering, Department of Electronics and Communication EngineeringMansoura UniversityMansouraEgypt
  5. 5.Center for Photonics and Smart Materials and Nanotechnology Engineering ProgramZewail City of Science and TechnologyGizaEgypt
  6. 6.Mathematics and Engineering Physics Department, Faculty of EngineeringMansoura UniversityMansouraEgypt

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