We study electromagnetic wave propagation in a system that is periodic both in space and in time, namely, a discrete (``lumped'') transmission line with capacitors (``varactors'') that are modulated in time harmonically. These periodicities result in exotic electromagnetic band structures that are periodic in the angular frequency ω and in the phase advance ka of the wave. Depending on the strength of modulation m and the reduced modulation frequency Ω/ω0 (where ω0 is the resonant frequency of a unit cell of the transmission line), this band structure can display frequency or wave vector band gaps, both, or neither. Moreover, minor changes in or the modulation strength can control the aperture or closure of a gap and even transform a k-gap to an ω-gap. Such phase transitions are intimately associated with exceptional or critical points in the (ω, k, Ω, m) space.
2. Joannopoulos, J. D., S. G. Johnson, J. N. Winn, and R. D. Meady, Photonic Crystals, 2nd Ed., Princeton University Press, 2008.
3. Solymar, L. and E. Shamonina, Waves in Metamaterials, Oxford University Press, 2009.
4. Lai, A., T. Itoh, and C. Caloz, "Composite right/left-handed transmission line metamaterials," IEEE Microwave Magazine, Vol. 5, No. 3, 34-50, September 2004.
5. Kozyrev, A. B., H. Kim, A. Karbassi, and D. W. van der Weide, "Wave propagation in nonlinear left-handed transmission line media," Applied Physics Letters, Vol. 87, No. 12, 121109, September 2005.
6. Syms, R. R. A., E. Shamonina, V. Kalinin, and L. Solymar, "A theory of metamaterials based on periodically loaded transmission lines: Interaction between magnetoinductive and electromagnetic waves," Journal of Applied Physics, Vol. 97, No. 6, (064909)1-6, March 2005.
7. Gil, I., J. Bonache, M. Gil, J. Garcia-Garcia, F. Martin, and R. Marqes, "Accurate circuit analysis of resonant-type left handed transmission lines with inter-resonator coupling," Journal of Applied Physics, Vol. 100, No. 7, 074908, October 2006.
8. Kozyrev, A. B., H. Kim, and D. W. van der Weide, "Parametric amplification in left-handed transmission line media," Applied Physics Letters, Vol. 88, No. 26, 264101, June 2006.
9. Algredo-Badillo, U. and P. Halevi, "Negative refraction and focusing in magnetically coupled L-C loaded transmission lines," Journal of Applied Physics, Vol. 102, No. 8, 086104, October 2007.
10. Ou, Y. and G. M. Rebeiz, "Lumped-element fully tunable bandstop filters for cognitive radio applications," IEEE Transactions on Microwave Theory and Techniques, Vol. 59, No. 10, 2461-2468, October 2011.
11. Chaudhary, G., Y. Jeong, and J. Lim, "Microstrip line negative group delay filters for microwave circuits," IEEE Transactions on Microwave Theory and Techniques, Vol. 62, No. 2, 234-243, February 2014.
12. Chaudhary, G. and Y. Jeong, "Distributed transmission line negative group delay circuit with improved signal attenuation," IEEE Microwave and Wireless Components Letters, Vol. 24, No. 1, 20-22, January 2014.
13. Chaudhary, G. and Y. Jeong, "A design of power divider with negative group delay characteristics," IEEE Microwave and Wireless Components Letters, Vol. 25, No. 6, 394-396, June 2015.
14. Zurita-Sanchez, J. R., P. Halevi, and J. C. Cervantes-Gonzalez, "Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ϵ(t)," Physical Review A, Vol. 79, 053821, May 2009.
15. Martinez-Romero, J. S., O. M. Becerra-Fuentes, and P. Halevi, "Temporal photonic crystals with modulations of both permittivity and permeability," Physical Review A, Vol. 93, 063813, June 2016.
16. Koutserimpas, T. T. and R. Fleury, "Electromagnetic waves in a time periodic medium with step-varying refractive index," IEEE Transactions on Antennas and Propagation, Vol. 66, No. 10, 5300-5307, October 2018.
17. Martinez-Romero, J. S. and P. Halevi, "Standing waves with infinite group velocity in a temporally periodic medium," Physical Review A, Vol. 96, 063831, December 2017.
18. Zurita-Sanchez, J. R. and P. Halevi, "Resonances in the optical response of a slab with time-periodic dielectric function ϵ(t)," Physical Review A, Vol. 81, 053834, May 2010.
19. Martinez-Romero, J. S. and P. Halevi, "Parametric resonances in a temporal photonic crystal slab," Physical Review A, Vol. 98, 053852, November 2018.
20. Miller, J. L., "Exceptional points make for exceptional sensors," Physics Today, Vol. 70, No. 10, 23, October 2017.
21. 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, 187-191, November 2017.
22. Chen, W., S. K. Ozdemir, G. Zhao, J. Wiersig, and L. Yang, "Exceptional points enhance sensing in an optical microcavity," Nature, Vol. 548, 192-196, August 2017.
23. Kazemi, H., M. Y. Nada, F. Capolino, and F. Maddaleno, "Experimental demonstration of exceptional points of degeneracy in linear time periodic systems and exceptional sensitivity," Arxiv.org., Vol. 1908, 08516, September 2020.
24. Miri, M. A. and A. Alu, "Exceptional points in optics and photonics," Science, Vol. 363, No. 6422, January 2019.
25. Morgenthaler, F. R., "Velocity modulation of electromagnetic waves," IRE Transactions on Microwave Theory and Techniques, Vol. 6, No. 2, 167-172, April 1958.
26. Currie, M. R. and R. W. Gould, "Coupled-cavity traveling-wave parametric amplifiers: Part I. Analysis," Proceedings of the IRE, Vol. 48, No. 12, 1960-1973, December 1960.
27. Cassedy, E. S. and A. A. Oliner, "Dispersion relations in time-space periodic media: Part I --- Stable interactions," Proceedings of the IEEE, Vol. 51, No. 10, 1342-1359, October 1963.
28. Holberg, D. and K. Kunz, "Parametric properties of fields in a slab of time-varying permittivity," IEEE Transactions on Antennas and Propagation, Vol. 14, No. 2, 183-194, March 1966.
29. Chamanara, N., S. Taravati, Z.-L. Deck-Leger, and C. Caloz, "Optical isolation based on space-time engineered asymmetric photonic band gaps," Physics Review B, Vol. 96, 155409, October 2017.
30. Taravati, S., N. Chamanara, and C. Caloz, "Nonreciprocal electromagnetic scattering from a periodically space-time modulated slab and application to a quasisonic isolator," Physics Review B, Vol. 96, 165144, October 2017.
31. Chamanara, N., Z.-L. Deck-Leger, C. Caloz, and D. Kalluri, "Unusual electromagnetic modes in space-time-modulated dispersion-engineered media," Physical Review A, Vol. 97, 063829, June 2018.
32. 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," Applied Physics Letters, Vol. 107, 074101, August 2015.
33. Reyes-Ayona, J. R. and P. Halevi, "Electromagnetic wave propagation in an externally modulated low-pass transmission line," IEEE Transactions on Microwave Theory and Techniques, Vol. 64, No. 11, 3449-3459, November 2016.
34. Stearrett, R. and L. Q. English, "Experimental generation of intrinsic localized modes in a discrete electrical transmission line," Journal of Physics D: Applied Physics, Vol. 40, No. 17, 5394-5398, August 2007.
35. Sato, M., S. Yasui, M. Kimura, T. Hikihara, and A. J. Sievers, "Management of localized energy in discrete nonlinear transmission lines," EPL (Europhysics Letters), Vol. 80, No. 3, 30002, October 2007.
36. English, L. Q., F. Palmero, P. Candiani, J. Cuevas, R. Carretero-Gonzalez, P. G. Kevrekidis, and A. J. Sievers, "Generation of localized modes in an electrical lattice using subharmonic driving," Physical Review Letters, Vol. 108, 084101, February 2012.
37. Gomez-Rojas, A. and P. Halevi, "Discrete breathers in an electric lattice with an impurity: Birth, interaction, and death," Physical Review E, Vol. 97, 022225, February 2018.