In the one-dimensional case, the amplitude of a pulse that propagates in a homogeneous material whose properties are instantaneously changed in time will undergo an exponential increase due to the interference between the reflected and transmitted pulses generated at each sudden switch. Here, we resolve the issue by designing suitable reciprocal PT-symmetric space-time microstructures so that the interference between the scattered waves is such that the overall amplitude of the wave will be constant in time in each constituent material. Remarkably, for the geometries proposed here, a pulse will propagate with constant amplitude regardless of the impedance between the constituent materials, and for some, regardless of the wave speed mismatch. We extend, then, these results to the two-dimensional case, by proposing suitable geometries that avoid the blow up of the wave amplitude at the source point due to the scattering associated with time modulation. Given that the energy associated with the wave will increase exponentially in time, this creates the possibility to exploit the stable propagation of the pulse to accumulate energy for harvesting.
References
1.
A. L.
Cullen
, “A travelling-wave parametric amplifier
,” Nature
181
(4605
), 332
(1958
).2.
R. C.
Honey
and E. M. T.
Jones
, “A wide-band UHF traveling-wave variable reactance amplifier
,” IRE Trans. Microwave Theory Tech.
8
(3
), 351
–361
(1960
).3.
W. H.
Louisell
and C. F.
Quate
, “Parametric amplification of space charge waves
,” Proc. IRE
46
(4
), 707
–716
(1958
).4.
F.
Morgenthaler
, “Velocity modulation of electromagnetic waves
,” IRE Trans. Microwave Theory Tech.
6
(2
), 167
–172
(1958
).5.
P. K.
Tien
, “Parametric amplification and frequency mixing in propagating circuits
,” J. Appl. Phys.
29
(9
), 1347
–1357
(1958
).6.
V.
Pacheco-Peña
and N.
Engheta
, “Antireflection temporal coatings
,” Optica
7
(4
), 323
–331
(2020
).7.
Y.
Xia
, E.
Riva
, M. I. N.
Rosa
, G.
Cazzulani
, A.
Erturk
, F.
Braghin
, and M.
Ruzzene
, “Experimental observation of temporal pumping in electromechanical waveguides
,” Phys. Rev. Lett.
126
, 095501
(2021
).8.
H.
Nassar
, X.
Xu
, A.
Norris
, and G.
Huang
, “Modulated phononic crystals: Non-reciprocal wave propagation and Willis materials
,” J. Mech. Phys. Solids
101
, 10
–29
(2017
).9.
Y.
Sharabi
, A.
Dikopoltsev
, E.
Lustig
, Y.
Lumer
, and M.
Segev
, “Spatiotemporal photonic crystals
,” Optica
9
(6
), 585
–592
(2022
).10.
J.
Zhang
, P. W.
Hess
, A.
Kyprianidis
, P.
Becker
, A.
Lee
, J.
Smith
, G.
Pagano
, I.-D.
Potirniche
, A. C.
Potter
, A.
Vishwanath
, N. Y.
Yao
, and C.
Monroe
, “Observation of a discrete time crystal
,” Nature
543
, 217
–220
(2017
).11.
M. W.
McCall
, A.
Favaro
, P.
Kinsler
, and A.
Boardman
, “A spacetime cloak, or a history editor
,” J. Opt.
13
(2
), 024003
(2010
).12.
E.
Galiffi
, M. G.
Silveirinha
, P. A.
Huidobro
, and J. B.
Pendry
, “Photon localization and Bloch symmetry breaking in luminal gratings
,” Phys. Rev. B
104
, 014302
(2021
).13.
A.
Akbarzadeh
, N.
Chamanara
, and C.
Caloz
, “Inverse prism based on temporal discontinuity and spatial dispersion
,” Opt. Lett.
43
(14
), 3297
–3300
(2018
).14.
C.
Caloz
and Z.-L.
Deck-Léger
, “Spacetime metamaterials—Part I: General concepts
,” IEEE Trans. Antennas Propag.
68
(3
), 1569
–1582
(2020
).15.
C.
Caloz
and Z.-L.
Deck-Léger
, “Spacetime metamaterials—Part II: Theory and applications
,” IEEE Trans. Antennas Propag.
68
(3
), 1583
–1598
(2020
).16.
K. A.
Lurie
, An Introduction to the Mathematical Theory of Dynamic Materials
(Springer Cham
, 2017
).17.
V.
Bacot
, M.
Labousse
, A.
Eddi
, M.
Fink
, and E.
Fort
, “Time reversal and holography with spacetime transformations
,” Nat. Phys.
12
, 972
–977
(2016
).18.
J.
Marconi
, E.
Riva
, M.
Di Ronco
, G.
Cazzulani
, F.
Braghin
, and M.
Ruzzene
, “Experimental observation of nonreciprocal band gaps in a space-time-modulated beam using a shunted piezoelectric array
,” Phys. Rev. Appl.
13
, 031001
(2020
).19.
H.
Moussa
, G.
Xu
, S.
Yin
, E.
Galiffi
, Y.
Radi
, and A.
Alù
, “Observation of temporal reflections and broadband frequency translations at photonic time-interfaces
,” arXiv:2208.07236 [physics.app-ph] (2022
).20.
J. T.
Mendonca
, Theory of Photon Acceleration
(CRC Press
, Boca Raton
, 2000
).21.
K.
Lurie
and S.
Weekes
, “Wave propagation and energy exchange in a spatio-temporal material composite with rectangular microstructure
,” J. Math. Anal. Applicat.
314
(1
), 286
–310
(2006
).22.
K.
Lurie
and V.
Yakovlev
, “Energy accumulation in waves propagating in space- and time-varying transmission lines
,” IEEE Antennas Wireless Propag. Lett.
15
, 1681
–1684
(2016
).23.
D.
Torrent
, W. J.
Parnell
, and A. N.
Norris
, “Loss compensation in time-dependent elastic metamaterials
,” Phys. Rev. B
97
, 014105
(2018
).24.
P. A.
Huidobro
, E.
Galiffi
, S.
Guenneau
, R. V.
Craster
, and J. B.
Pendry
, “Fresnel drag in space-time-modulated metamaterials
,” Proc. Natl. Acad. Sci. U. S. A.
116
(50
), 24943
–24948
(2019
).25.
O.
Mattei
and G. W.
Milton
, “Field patterns without blow up
,” New J. Phys.
19
(9
), 093022
(2017
).26.
O.
Mattei
and G. W.
Milton
, “Field patterns: A new type of wave with infinitely degenerate band structure
,” Europhys. Lett.
120
(5
), 54003
(2017
).27.
G. W.
Milton
and O.
Mattei
, “Field patterns: New mathematical object
,” Proc. R. Soc. A
473
, 20160819
(2017
).28.
A. B.
Movchan
, N. V.
Movchan
, I. S.
Jones
, G. W.
Milton
, and H.-M.
Nguyen
, “Frontal waves and transmissions for temporal laminates and imperfect chiral interfaces
,” Philos. Trans. R. Soc. A
380
, 20210385
(2022
).29.
C. M.
Bender
and S.
Boettcher
, “Real spectra in non-Hermitian Hamiltonians having PT symmetry
,” Phys. Rev. Lett.
80
(24
), 5243
–5246
(1998
).30.
K. J.
Deshmukh
and G. W.
Milton
, “An energy conserving mechanism for temporal metasurfaces
,” Appl. Phys. Lett.
121
(4
), 041702
(2022
).31.
K.
Lurie
, D.
Onofrei
, and S.
Weekes
, “Mathematical analysis of the waves propagation through a rectangular material structure in space-time
,” J. Math. Anal. Applicat.
355
(1
), 180
–194
(2009
).32.
V.
Gulizzi
and R.
Saye
, “Modeling wave propagation in elastic solids via high-order accurate implicit-mesh discontinuous Galerkin methods
,” Comput. Methods Appl. Mech. Eng.
395
, 114971
(2022
).33.
B.
Apffel
, S.
Wildeman
, A.
Eddi
, and E.
Fort
, “Experimental implementation of wave propagation in disordered time-varying media
,” Phys. Rev. Lett.
128
, 094503
(2022
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.