Wave propagation in pure-time modulated step media with applications to temporal-aiming

Mourad Sini; Haibing Wang; Qingyun Yao

doi:10.3934/cac.2024004

Article Contents

2024, Volume 2, Issue 1: 48-70. Doi: 10.3934/cac.2024004

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Wave propagation in pure-time modulated step media with applications to temporal-aiming

1.
RICAM, Austrian Academy of Sciences, A-4040, Linz, Austria
2.
School of Mathematics, Southeast University, Nanjing 210096, China
3.
Nanjing Center for Applied Mathematics, Nanjing 211135, China
4.
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

^*Corresponding author: Mourad Sini
^*Corresponding author: Mourad Sini

Received: January 2024

Revised: March 2024

Early access: March 13, 2024

Published online: March 13, 2024

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Abstract

We analyzed the propagation of an incident electromagnetic wave in a purely-time modulated medium. Precisely, we assumed that the permeability is unchanged while the permittivity has a multiple-step profile in time and is uniformly constant in space. For this, we used the time-dispersive Drude's model with time-dependent plasma frequency and highly concentrated values near the centers of the steps' intervals. Under certain regimes, linking the number of steps to the contrasts of the permittivity, we can generate effective permittivity having positive and high values on a finite interval of time which behaves as a 'wall' that kills the forward waves and keeps only the backward waves (i.e. enabling full reflection). But, this happens if these high values are away from a discrete set. If these high values are close to the elements of this discrete set, then the effective medium behaves as a 'well' that absorbs all the forward waves and hence there are no backward waves (i.e. enabling full transmission). Such results are reminiscent to the 'wall' and 'well' effects known in the quantum mechanics theory.

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Mathematics Subject Classification: 35L05, 35C20.

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Figure 1. The model $ \omega_p^2(t) $ with $ C = 1 $, $ h = 0.5 $ $ \delta = 0.05 $, $ l = 0.5 $, and $ d = \delta^l $

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Figure 2. Numerical results for the effective field $ \tilde{E}_{eff} $ when $ 1-h-l = 0 $: for $ \lambda $ close to even $ n\pi $ (left), for $ \lambda $ is away from $ n\pi $ and moderate (middle), for $ \lambda $ is away from $ n\pi $ and large (right)

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Figure 3. Numerical results for the effective field $ \tilde{E}_{eff} $ when $ 1-h-l<0 $: for $ \lambda $ close to odd $ n\pi $ (left), for $ \lambda $ close to even $ n\pi $ (middle), for $ \lambda $ away from $ n\pi $ (right)

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Figure 4. Numerical results for the effective field $ \tilde{D}_{eff} $ and the asymptotic field: when $ 1-h-l>0 $ (first line), $ 1-h-l = 0 $ and small $ C $ (second line), $ 1-h-l = 0 $ and moderate $ C $ (third line), $ 1-h-l<0 $ (fourth line)

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Table 1. The parameter settings for three effective fields

$T$	$l$	$h$	$\delta$	$\kappa$	$C$	$\lambda$	$\lambda T$	$\tan(\lambda T)$
10	0.9	0.1	1e-03	1	8.8696	3.1416	10$\pi$	-7.0046e-06
10	0.9	0.1	1e-03	1	10	3.3166	10.5571$\pi$	-5.5103
10	0.9	0.1	1e-03	1	1000	31.6386	100.7087$\pi$	-1.2998
10	0.9	0.717	1e-03	1	1	8.4828	27.0016$\pi$	0.0049
10	0.9	0.369	1e-07	1	1	8.7968	28.0011$\pi$	0.0033
10	0.9	0.538	1e-07	1	1	34.1139	108.6517$\pi$	-1.9368

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Table 2. The parameter settings in Example 4.1

$T$	$h$	$l$	$\delta$	$\kappa$	$C$	$\omega_p^2$	$\lambda$
10	0.1	0.1	1e-03	1	1	1.9953	1.0020
10	0.1	0.9	1e-03	1	0.1	0.1995	1.0488
10	0.1	0.9	1e-03	1	1.4674	2.9278	1.5708
10	0.342	0.9	1e-03	1	1	10.6170	2.5142

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