Leaky waves in planar dielectric waveguide

Cover Page

Cite item

Full Text

Abstract

A new analytical and numerical solution of the electrodynamic waveguide problem for leaky modes of a planar dielectric symmetric waveguide is proposed. The conditions of leaky modes, corresponding to the Gamow-Siegert model, were used as asymptotic boundary conditions. The resulting initial-boundary problem allows the separation of variables. The emerging problem of the eigen-modes of open three-layer waveguides is formulated as the Sturm-Liouville problem with the corresponding boundary and asymptotic conditions. In the case of guided and radiation modes, the Sturm-Liouville problem is self-adjoint and the corresponding eigenvalues are real quantities for dielectric media. The search for eigenvalues and eigenfunctions corresponding to the leaky modes involves a number of difficulties: the problem for leaky modes is not self-adjoint, so the eigenvalues are complex quantities. The problem of finding eigenvalues and eigenfunctions is associated with finding the complex roots of the nonlinear dispersion equation. To solve this problem, we used the method of minimizing the zero order. An analysis of the calculated distributions of the electric field strength of the first three leaky modes is given, showing the possibilities and advantages of our approach to the study of leaky modes.

About the authors

Dmitriy V. Divakov

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
Email: divakov-dv@rudn.ru

Candidate of Physical and Mathematical Sciences, assistant of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow 117198, Russian Federation

Alexandre A. Egorov

A. M. Prokhorov General Physics Institute

Email: yegorov@kapella.gpi.ru

Doctor of Physical and Mathematical Sciences, Chief Researcher of Department of Oscillations

Russian Academy of Sciences, Moscow 119991, Russian Federation

Konstantin P. Lovetskiy

Peoples’ Friendship University of Russia (RUDN University)

Email: lovetskiy-kp@rudn.ru

Associate Professor, Ph.D., Associate Professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow 117198, Russian Federation

Leonid A. Sevastianov

Peoples’ Friendship University of Russia (RUDN University); Bogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research

Email: sevastianov-la@rudn.ru

professor, Doctor of Physical and Mathematical Sciences, professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University); leading researcher of the Bogoliubov Laboratory of Theoretical Physics

6, Miklukho-Maklaya St., Moscow 117198, Russian Federation; 6, Joliot-Curie St., Dubna, Moscow region 141980, Russian Federation

Andrey S. Drevitskiy

Peoples’ Friendship University of Russia (RUDN University)

Email: drevitskiy-as@rudn.ru

PhD student of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow 117198, Russian Federation

References

  1. D. Marcuse, Light Transmission Optics. New York: Van Nostrand Reinhold, 1972.
  2. D. Marcuse, Theory of Dielectric Optical Waveguides. New York: Academic, 1973.
  3. M. Adams, An Introduction to Optical Waveguides. Chichester: Wiley, 1981.
  4. A. Snyder and J. D. Love, Optical Waveguide Theory. London: Chapman and Hall, 1983.
  5. T. Tamir, Integrated Optics. Berlin: Springer-Verlag, 1982.
  6. L. O. Goldstone and A. A. Oliner, “Leaky-wave antennas i: Rectangular waveguides,” IRE Transactions on Antennas and Propagation, vol. 7, no. 4, pp. 307-319, 1959, doi: 10.1109/TAP.1959.1144702.
  7. L. O. Goldstone and A. A. Oliner, “Leaky-wave antennas ii: Circular waveguides,” IRE Transactions on Antennas and Propagation, vol. 9, no. 3, pp. 280-290, 1961, doi: 10.1109/TAP.1961.1144995.
  8. T. Tamir and A. A. Oliner, “Guided complex waves, part i: Fields at an interface,” Proc. inst. Elec. Eng., vol. 110, no. 2, pp. 310-324, 1963, doi: 10.1049/piee.1963.0044.
  9. T. Tamir and A. A. Oliner, “Guided complex waves, part ii: Relation to radiation patterns,” Proc. inst. Elec. Eng., vol. 110, no. 2, pp. 325-334, 1963, doi: 10.1049/piee.1963.0045.
  10. S.-T. Peng and A. Oliner, “Guidance and leakage properties of a class of open dielectric waveguides: Part i-mathematical formulations,” IEEE Transactions on Microwave Theory and Techniques, vol. 29, no. 9, pp. 843- 855, 1981, doi: 10.1109/TMTT.1981.1130465.
  11. A. Oliner, S.-T. Peng, T.-I. Hsu, and A. Sanchez, “Guidance and leakage properties of a class of open dielectric waveguides: Part ii-new physical effects,” IEEE Transactions on Microwave Theory and Techniques, vol. 29, no. 9, pp. 855-869, 1981, doi: 10.1109/TMTT.1981.1130466.
  12. F. Tamir and F. Y. Kou, “Varieties of leaky waves and their excitation along multilayered structures,” IEEE Journal of Quantum Electronics, vol. 22, no. 4, pp. 544-551, 1986, doi: 10.1109/JQE.1986.1072991.
  13. R. Sammut and A. W. Snyder, “Leaky modes on a dielectric waveguide: Orthogonality and excitation,” Applied optics, vol. 15, no. 4, pp. 1040- 1044, 1976, doi: 10.1364/AO.15.001040.
  14. R. Sammut, C. Pask, and A. W. Snyder, “Excitation and power of the unbound modes within a circular dielectric waveguide,” Proc. inst. Elec. Eng., vol. 122, no. 1, pp. 25-33, 1975, doi: 10.1049/piee.1975.0004.
  15. R. Sammut and A. Snyder, “Contribution of umbound modes to light absorption in visual photoreceptors,” J. Opt. Soc. Am., vol. 64, no. 12, pp. 1171-1174, 1974, doi: 10.1364/JOSA.64.001711.
  16. A. W. Snyder, “Leaky-ray theory of optical waveguides of circular cross section,” Appl. Phys., vol. 4, pp. 273-298, 1974, doi: 10.1007/BF00928381.
  17. A. W. Snyder and D. J. Mitchell, “Leaky rays on circular optical fibers,” J. Opt. Soc. Am., vol. 69, no. 5, pp. 599-607, 1974, doi: 10.1364/JOSA.64.000599.
  18. A. W. Snyder and D. J. Mitchell, “Leaky mode analysis of circular optical waveguides,” Optoelectronics, vol. 6, pp. 287-296, 1974, doi: 10.1007/BF01423378.
  19. W. W. Hansen, “Radiating electromagnetic waveguide,” [U.S. Patent No. 2 402 622], 1940.
  20. S. Barone, “Leaky wave contributions to the field of a line source above a dielectric slab,” Microwave Research Institute, Polytechnic Institute of Brooklyn, Report R-532-546, PIB-462, 1956.
  21. S. Barone and A. Hessel, “Leaky wave contributions to the field of a line source above a dielectric slab-part ii,” Microwave Research Institute, Polytechnic Institute of Brooklyn, Report R-698-58, PIB-626, 1958.
  22. E. S. Cassedy and M. Cohn, “On the existence of leaky waves due to a line source above a grounded dielectric slab,” IRE Trans. Microwave Theory Tech., vol. 9, no. 3, pp. 243-247, 1961, doi: 10.1109/TMTT.1961.1125314.
  23. N. Marcuvitz, “On field representations in terms of leaky modes or eigenmodes,” IRE Trans. Antennas Propag., vol. 4, no. 3, pp. 192-194, 1956, doi: 10.1109/TAP.1956.1144410.
  24. V. V. Shevchenko, “On the behavior of wave numbers beyond the critical value for waves in dielectric waveguides (media with losses),” Radiophys. Quantum Electron., vol. 15, pp. 194-200, 1972, doi: 10.1007/BF02209117.
  25. V. V. Shevchenko, “The expansion of the fields of open waveguides in proper and improper modes,” Radiophys. Quantum Electron., vol. 14, no. 8, p. 972, 1974, doi: 10.1007/BF01029499.
  26. A. A. Egorov, “Theoretical and numerical analysis of propagation and scattering of eigenand non-eigenmodes of an irregular integratedoptical waveguide,” Quantum Electronics, vol. 42, no. 4, pp. 337-344, 2012, doi: 10.1070/QE2012v042n04ABEH014809.
  27. K. Ogusu, M. Miyag, and S. Nishida, “Leaky te modes in an asymmetic three-layered slab waveguide,” J. Opt. Soc. Am., vol. 70, no. 1, pp. 68- 72, 1980, doi: 10.1364/JOSA.70.000048.
  28. S. Yamaguchi, A. Shimojima, and T. Hosono, “Analysis of leaky modes supported by a slab waveguide,” Electronics and Communications in Japan, vol. 73, no. 11, pp. 20-31, 1990, doi: 10.1002/ecjb.4420731103.
  29. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: Reflection pole method and wavevector density method,” J. Lightwave Technol., vol. 17, no. 5, pp. 929-941, 1999, doi: 10.1109/50.762914.
  30. J. Petracek and K. Singh, “Determination of leaky modes in planar multilayer waveguides,” IEEE Photon. Technol. Lett., vol. 14, no. 6, pp. 810-812, 2002, doi: 10.1109/LPT.2002.1003101.
  31. A. G. Rzhanov and S. E. Grigas, “Numerical algorithm for waveguide and leaky modes determination in multilayer optical waveguides,” Technical Physics, vol. 55, no. 11, pp. 1614-1618, 2010, doi: 10.1134/S1063784210110113.
  32. A. B. Manenkov, “Orthogonality conditions for leaky modes,” Radiophysics and Quantum Electronics, vol. 48, pp. 348-360, 2005, doi: 10.1007/s11141-005-0076-8.
  33. A. A. Romanenko and A. B. Sotskii, “The solution of the dispersion relations for planar waveguides in the case of complex roots,” Tech. Phys., vol. 43, no. 4, pp. 427-433, 1998, doi: 10.1134/1.1258999.
  34. E. I. Golant and K. M. Golant, “New method for calculating the spectra and radiation losses of leaky waves in multilayer optical waveguides,” Technical Physics, vol. 51, no. 8, pp. 1060-1068, 2006, doi: 10.1134/S1063784206080160.
  35. A. B. Sotsky, L. Steingart, J. Jackson, and et al., “Prism excitation of leaky modes of thin films,” Tech. Phys., vol. 58, no. 11, pp. 1651-1660, 2013, doi: 10.1134/S106378421311025X.
  36. G. Gamow, “Zur quantentheorie des atomkernes,” Z. Phys., vol. 51, pp. 204-212, 1928, doi: 10.1007/BF01343196.
  37. O. Civitarese and M. Gadella, “Physical and mathematical aspects of gamow states,” Phys. Rep., vol. 396, no. 2, pp. 41-113, 2004, doi: 10.1016/j.physrep.2004.03.001.
  38. A. J. F. Siegert, “On the derivation of the dispersion formula for nuclear reactions,” Phys. Rev., vol. 56, pp. 750-752, 1939, doi: 10.1103/PhysRev.56.750.
  39. O. I. Tolstikhin, V. N. Ostrovsky, and H. Nakamura, “Siegert pseudostates as a universal tool: Resonances, s matrix, green function,” Phys. Rev. Lett., vol. 79, pp. 2026-2029, 1997, doi: 10.1103/PhysRevLett.79.2026.
  40. O. I. Tolstikhin, V. N. Ostrovsky, and H. Nakamura, “Siegert pseudostate formulation of scattering theory: One-channel case,” Phys. Rev. A, vol. 58, pp. 2077-2096, 1998, doi: 10.1103/PhysRevA.58.2077.
  41. D. Stowell and J. Tausch, “Guided and leaky modes of planar waveguides: Computation via high order finite elements and iterative methods,” PIERS Online, vol. 6, no. 7, pp. 669-673, 2010, doi: 10.2529/PIERS091216124247.
  42. D. Divakov, A. Tiutiunnik, and A. Sevastianov, “Symbolic-numeric computation of the eigenvalues and eigenfunctions of the leaky modes in a regular homogeneous open waveguide,” MATEC Web of Conferences, vol. 186, p. 01 009, 2018, doi: 10.1051/matecconf/201818601009.
  43. R. E. Bellman, Introduction to matrix analysis. New York: McGraw-Hill, 1960.
  44. J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice Hall, 1983.
  45. R. Fletcher, Practical Methods of Optimization, 2nd. Wiley, 1987.
  46. J. Nocedal and S. Wright, Numerical Optimization, 2nd. Springer, 2006.
  47. J. Zhu and Y. Y. Lu, “Leaky modes of slab waveguides asymptotic solutions,” Journal of Lightwave Technology, vol. 24, no. 3, pp. 1619- 1623, 2006, doi: 10.1109/JLT.2005.863275.
  48. C. T. Kelley, Iterative Methods for Optimization. SIAM, 1999.
  49. J. Hu and C. R. Menyuk, “Understanding leaky modes: Slab waveguide revisited,” Advances in Optics and Photonics, vol. 1, no. 1, pp. 58-106, 2009, doi: 10.1364/AOP.1.000058.
  50. A. J. Martinez-Ros, J. L. Gómez-Tornero, F. J. Clemente-Fernandez, and J. Monzo-Cabrera, “Microwave near-field focusing properties of width-tapered microstrip leaky-wave antenna,” IEEE Trans. on Antennas and Propagation, vol. 61, no. 6, pp. 2981-2990, 2013, doi: 10.1109/TAP.2013.2252138.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download