volume | PIER Journals

Abstract

We studied electromagnetic wave propagation in a system that is periodic in both space and time, namely a discrete 2D transmission line (TL) with capacitors modulated in tandem externally. Kirchhoff's laws lead to an eigenvalue equation whose solutions yield a band structure (BS) for the circular frequency ω as function of the phase advances k_xa and k_ya in the plane of the TL. The surfaces ω(k_xa, k_ya) display exotic behavior like forbidden ω bands, forbidden k bands, both, or neither. Certain critical combinations of the modulation strength m_c and the modulation frequency Ω mark transitions from ω stopbands to forbidden k bands, corresponding to phase transitions from no propagation to propagation of waves. Such behavior is found invariably at the high symmetry X and M points of the spatial Brillouin zone (BZ) and at the boundary ω = (1/2)Ω of the temporal BZ. At such boundaries the ω(k_xa, k_ya) surfaces in neighboring BZs assume conical forms that just touch, resembling a South American toy ``diábolo''; the point of contact is thus called a ``diabolic point''. Our investigation reveals interesting interplay among geometry, critical points, and phase transitions.

1. Brillouin, L., Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices, Dover Publications, 1953.

2. Kittel, C., Introduction to Solid State Physics, 8th Ed., Wiley, 2005.

3. Joannopoulos, J. D., S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals, Princeton University Press, 2008.

4. Caloz, C. and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications, Wiley, 2006.

5. Pozar, D. M., Microwave Engineering, Wiley, 2012.

6. Momeni, O. and E. Afshari, "Electrical prism: A high quality factor filter for millimeter-wave and terahertz frequencies," IEEE Trans. Microw. Theory Techn., Vol. 57, No. 11, 2790-2799, 2009.
doi:10.1109/TMTT.2009.2032343 Google Scholar

7. Afshari, E., H. S. Bhat, A. Hajimiri, and J. E. Marsden, "Extremely wideband signal shaping using one- and two-dimensional nonuniform nonlinear transmission lines," J. Appl. Phys., Vol. 99, 054901, 2006.
doi:10.1063/1.2174126 Google Scholar

8. Lilis, G. N., J. Park, W. Lee, G. Li, H. S. Bhat, and E. Afshari, "Harmonic generation using nonlinear LC lattices," IEEE Trans. Microw. Theory Techn., Vol. 588, 1713-1723, 2010.
doi:10.1109/TMTT.2010.2049678 Google Scholar

9. Bhat, H. S. and E. Afshari, "Nonlinear constructive interference in electrical lattices," Phys. Rev. E, Vol. 77, 066602, 2008.
doi:10.1103/PhysRevE.77.066602 Google Scholar

10. Tousi, Y. M. and E. Afshari, "2-D electrical interferometer: A novel high-speed quantizer," IEEE Trans. Microw. Theory Techn., Vol. 58, 2549-2561, 2010.
doi:10.1109/TMTT.2010.2063830 Google Scholar

11. Lee, W., M. Adnan, O. Momeni, and E. Afshari, "A nonlinear lattice for high-amplitude picosecond pulse generation in CMOS," IEEE Trans. Microw. Theory Techn., Vol. 60, 370-380, 2012.
doi:10.1109/TMTT.2011.2178255 Google Scholar

12. Iyer, A. K. and G. V. Eleftheriades, "Negative refractive index metamaterials supporting 2-D waves," 2002 IEEE MTT-S International Microwave Symposium Digest, 1067-1070, 2002. Google Scholar

13. Eleftheriades, G. V., A. K. Iyer, P. C. Kremer, and , "Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microw. Theory Techn., Vol. 50, 2702-2712, 2002.
doi:10.1109/TMTT.2002.805197 Google Scholar

14. Milford, G. N. and G. V. Eleftheriades, "2D multiplier with left-handed focusing lens for terahertz signal generation," 2013 IEEE Antennas and Propagation Society International Symposium, 1182-1183, 2013. Google Scholar

15. Algredo-Badillo, U. and P. Halevi, "Negative refraction and focusing in magnetically coupled L-C loaded transmission lines," J. Appl. Phys., Vol. 102, 086104, 2007.
doi:10.1063/1.2794558 Google Scholar

16. Eleftheriades, G. V. and O. F. Siddiqui, "Negative refraction and focusing in hyperbolic transmission-line periodic grids," IEEE Trans. Microw. Theory Techn., Vol. 53, 396-403, 2005.
doi:10.1109/TMTT.2004.839944 Google Scholar

17. Zurita-Sánchez, J. R., P. Halevi, and J. C. Cervantes-González, "Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ϵ(t)," Phys. Rev. A, Vol. 79, 053821, 2009.
doi:10.1103/PhysRevA.79.053821 Google Scholar

18. Zurita-Sánchez, J. R. and P. Halevi, "Resonances in the optical response of a slab with time-periodic dielectric function ϵ(t)," Phys. Rev. A, Vol. 81, 053834, 2010.
doi:10.1103/PhysRevA.81.053834 Google Scholar

19. Martínez-Romero, J. S., O. M. Becerra-Fuentes, and P. Halevi, "Temporal photonic crystals with modulations of both permittivity and permeability," Phys. Rev. A, Vol. 93, 063813, 2016.
doi:10.1103/PhysRevA.93.063813 Google Scholar

20. Martínez-Romero, J. S. and P. Halevi, "Parametric resonances in a temporal photonic crystal slab," Phys. Rev. A, Vol. 98, 053852, 2018.
doi:10.1103/PhysRevA.98.053852 Google Scholar

21. Reyes-Ayona, J. R. and P. Halevi, "Observation of genuine wave vector (k or β) gap in a dynamic transmission line and temporal photonic crystals," Appl. Phys. Lett., Vol. 107, 2015. Google Scholar

22. Reyes-Ayona, J. R. and P. Halevi, "Electromagnetic wave propagation in an externally modulated low-pass transmission line," IEEE Trans. Microw. Theory Techn., Vol. 64, 3449-3459, 2016.
doi:10.1109/TMTT.2016.2604319 Google Scholar

23. Halevi, P., U. Algredo-Badillo, and J. R. Zurita-Sánchez, "Optical response of a slab with time-periodic dielectric function ε(t): Towards a dynamic metamaterial," Active Photonic Materials IV, 61-75, 2011. Google Scholar

24. Mohammad-Ali, M. and A. Alù, "Exceptional points in optics and photonics," Science, Vol. 363, No. 6422, 2019.
doi:10.1126/science.aat3158 Google Scholar

25. Marsden, J. E. and A. J. Tromba, Vector Calculus, Pearson, 2004.

26. Yakovlev, A. B. and G. W. Hanson, "On the nature of critical points in leakage regimes of a conductor-backed coplanar strip line," IEEE Trans. Microw. Theory Techn., Vol. 45, No. 1, 87-94, 1997.
doi:10.1109/22.552036 Google Scholar

27. Miller, J. L., "Exceptional points make for exceptional sensors," Phys. Today, Vol. 70, No. 10, 23-26, 2017.
doi:10.1063/PT.3.3717 Google Scholar

28. Seyranian, A. P., O. N. Kirillov, and A. A. Mailybaev, "Coupling of eigenvalues of complex matrices at diabolic and exceptional points," J. Phys. A, Vol. 38, 1723-1740, 2005.
doi:10.1088/0305-4470/38/8/009 Google Scholar

29. Heiss, W. D., "The physics of exceptional points," J. Phys. A, Vol. 45, 444016, 2012.
doi:10.1088/1751-8113/45/44/444016 Google Scholar

30. Chen, W., S. Kaya Özdemir, G. Zhao, J. Wiersig, and L. Yang, "Exceptional points enhance sensing in an optical microcavity," Nature, Vol. 548, 192-196, 2017.
doi:10.1038/nature23281 Google Scholar

31. Hodaei, H., A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, "Enhanced sensitivity at higher-order exceptional points," Nature, Vol. 548, 1476-4687, 2017. Google Scholar

32. El-Ganainy, R., K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, "Non-Hermitian physics and PT symmetry," Nat. Phys., Vol. 14, 1745-2481, 2018. Google Scholar

33. Kazemi, H., Y. N. Mohamed, M. Tarek, F. Ahmed, and C. Filippo, "Exceptional points of degeneracy induced by linear time-periodic variation," Phys. Rev. Applied, Vol. 11, 14007, 2019.
doi:10.1103/PhysRevApplied.11.014007 Google Scholar

34. Berry, M. V. and M. Wilkinson, "Diabolical points in the spectra of triangles," Proc. R. Soc. A, Vol. 392, 15-43, 1984. Google Scholar

35. Dubbers, D. and H.-J. Stöckmann, Quantum Physics: The Bottom-up Approach, Springer, 2013.
doi:10.1007/978-3-642-31060-7

Dr. Jose Salazar-Arrieta

Dr. Peter Halevi