Abstract

Conventionally, time-dependent systems add energy to electromagnetic waves by parametric amplification. Here we identify a distinct, alternative mechanism—compression of lines of force.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Mechanical forces in electromagnetic systems are transmitted through “lines of force”, the electric and magnetic fields postulated by Michael Faraday and deployed by Maxwell in his equations. For example, a superconductor repels a magnet (Fig. 1) because it expels magnetic lines of force, compressing them and hence increasing their energy density. In this example, the lines are conserved and it is their compression rather than addition of new lines that is the key to understanding these quasi-static forces. Although this compression is a familiar concept at very low frequencies, it has so far played no role in explaining work done when amplifying electromagnetic waves. Here the familiar mechanisms are provided by gain media in the case of lasers or by parametric amplification. In these systems, no conservation of lines of force exists, and work is done on radiation by uniformly adding more lines. Hence little light is shed on the process by speaking in terms of lines of force. However, with new experimental possibilities emerging for systems whose parameters change on the same time scale as the frequency of the radiation both at optical [1,2] and terahertz frequencies [35] a different mechanism for amplification comes into play. Well known to computational studies [6,7], it has been little studied from an analytic point of view. Its modus operandi is the focus of this memorandum and constitutes the alternative mechanism for gain alluded to in our abstract by compressing lines of force.

 figure: Fig. 1.

Fig. 1. Superconductor repels a magnet by excluding its lines of force. As the superconductor approaches the magnet, the repulsive force stores energy by compressing the lines of force, which are conserved.

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We illustrate this new mechanism using a simple model: a Bragg grating synthetically moving with a uniform velocity, ${c_g} = \Omega /g $,

$$\begin{split}& \varepsilon ({x - {c_g}t} ) = {\varepsilon _1}[{1 + 2{\alpha _\varepsilon}\cos ({gx - \Omega t} )} ]\\&\mu ({x - {c_g}t} ) = {\mu _1}[{1 + 2{\alpha _\mu}\cos ({gx - \Omega t} )} ]\end{split},$$
where $g$ and $\Omega$ are spatial and temporal modulation frequencies, and ${\alpha _\varepsilon},{\alpha _\mu}$ the strength of electric and magnetic modulations, respectively. ${c_1} = {{c_0}} /{\sqrt {{\varepsilon _1}{\mu _1}}}$ is the velocity of light in the background medium. The modulation frequency, $\Omega$, may in principle be many orders of magnitude smaller that of the incident radiation. The model has been extensively deployed in other studies of time-dependent systems, and a recent review is to be had in [8]. We assume that $\varepsilon ,\mu$ are both real. In this model, the medium itself does not move but is locally modulated so as to provide the time-dependent profile described above and hence there is no restriction on the grating velocity.

In Fig. 2, we show the three regimes supported by this model: Fig. 2A shows the dispersion, $\omega (k)$, of radiation in a uniform dielectric; Fig. 2B shows ${c_g} \ll {c_1}$, where the system is parity-time symmetric and Bloch waves are defined. Bandgaps in $\omega$ appear where bands intersect and within which the Bloch waves either grow or decay in space. In Fig. 2C, ${c_g} \gg {c_1}$ and Bloch waves are also defined, but the bandgaps are different. They are gaps in $k$ within which waves either grow or decay in time. In other words, in these gaps we find parametric amplification. For this mechanism to operate, it is essential to have bandgaps in $k$ where backscattering ensures conservation of momentum when a forward-traveling photon is generated and the reaction deposited in the grating. In the limit $g \to 0$, the model reduces to standard parametric amplification, where the entire system is modulated at twice the input frequency.

 figure: Fig. 2.

Fig. 2. Sketch of dispersion relation of light traveling through (A) a uniform dielectric; (B) a grating modulated as shown in Eq. (1), where ${c_g} \lt {c_0}/\sqrt {{\varepsilon _1}{\mu _1}}$; (C) a grating where ${c_g} \gt {c_0}/\sqrt {{\varepsilon _1}{\mu _1}}$; (D) the case of ${c_g} = {c_0}/\sqrt {{\varepsilon _1}{\mu _1}}$ where all the forward-traveling states are degenerate and therefore strongly coupled. Arrows show displacement of the bands by a space-time reciprocal lattice vector $({g,\Omega})$ with slope ${c_g}$.

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Figure 2D illustrates the third regime, the luminal one where ${c_g} \approx {c_1}$. Here Bloch waves are not defined, and computational studies on waves incident on a finite slab of grating show each period of the grating capturing a group of waves, which it compresses into pulses. The pulses grow ever more intense and narrow while locked to a specific point in the grating period until finally ejected from the far side of the slab as a supercontinuum. Their height, $h$, grows exponentially with length and with the strength of modulation and the pulse width scales as $1/\sqrt h$ so that there is overall amplification as well as local amplification. Their spacing is the period of the grating because they are trapped therein. Trapping occurs for a range of grating velocities close to the speed of light in the background medium [9]. This is illustrated in a schematic fashion in Fig. 3. This is not parametric amplification. In fact, we can turn off all parametric amplification by eliminating backscattering. This we do by ensuring that the grating is everywhere impedance-matched, ${\alpha _\varepsilon} = {\alpha _\mu}$. This closes all bandgaps and hence cuts off all parametric processes, but in the ${c_g} \approx {c_1}$ regime, amplification and compression remain.

 figure: Fig. 3.

Fig. 3. Schematic figure of the effect of a luminal grating, traveling to the right, on a plane wave incident from the left. The grating has little effect on waves incident from the right. Note the compression of lines of force shown schematically as gray lines.

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We stay with the impedance-matched model to ensure that no parametric processes intrude into our calculations, not because the new mechanism is specific to impedance matching. With no backscattering operating forward and backward, waves are independent, each obeying a first-order differential equation. The forward waves are of interest to us because they more nearly keep pace with the grating and they obey

$${+}\frac{{\partial\! {D_y}}}{{\partial t}}\; = - \frac{1}{Z}\frac{{\partial\! {E_y}}}{{\partial x}},$$
where ${E_y}$ is the electric field and ${D_y}$ the displacement field. We assume a wave traveling forward in the $x$ direction. This equation describes the capture and amplification of pulses of radiation. It can be used to prove that the number of lines of force captured in each pulse is a constant,
$$\frac{\partial}{{\partial t}}{\int _{{\rm{period}}}}|{D_y}|{\rm{d}}x = 0,$$
where the integral is evaluated over one grating period. This conservation law forbids adding lines of force, and the only route to increasing energy is the same as in the example of the superconductor shown in Fig. 1. To amplify, the system must compress the lines.

In this memorandum, we have demonstrated that conventional descriptions of amplification fail to explain gain experienced in a luminal medium. A new mechanism, compression of lines of force, explains the processes at work with a highly intuitive picture. The model correctly predicts the relationship between amplification and pulse width. Further details can be found in [10].

Funding

Fundação para a Ciência e a Tecnologia (CEECIND/03866/2017, UIDB/EEA/50008/2020); Engineering and Physical Sciences Research Council (EP/L015579/1, EP/T51780X/1); Gordon and Betty Moore Foundation.

Acknowledgment

We thank the following for support: P. A. H. acknowledges funding from Fundação para a Ciência e a Tecnología and Instituto de Telecomunicaçõoes. E. G. is supported through a studentship in the Center for Doctoral Training on Theory and Simulation of Materials at Imperial College London funded by the EPSRC.

Disclosures

The authors declare no conflicts of interest.

REFERENCES

1. V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020). [CrossRef]  

2. H. Lira, Z. Yu, S. Fan, and M. Lipson, Phys. Rev. Lett. 109, 033901 (2012). [CrossRef]  

3. A. C. Tasolamprou, A. D. Koulouklidis, C. Daskalaki, C. P. Mavidis, G. Kenanakis, G. Deligeorgis, Z. Viskadourakis, P. Kuzhir, S. Tzortzakis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, ACS Photon. 6, 720 (2019). [CrossRef]  

4. C. T. Phare, Y.-H. D. Lee, J. Cardenas, and M. Lipson, Nat. Photonics 9, 511 (2015). [CrossRef]  

5. W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, Nano Lett. 14, 955 (2014). [CrossRef]  

6. N. Chamanara and C. Caloz, “Laser pulse compansion using co-propagating space-time modulation,” arXiv:1810.04129v1 [physics.optics] (2018).

7. E. Galiffi, P. A. Huidobro, and J. B. Pendry, Phys. Rev. Lett. 123, 206101 (2019). [CrossRef]  

8. C. Caloz and Z. Deck-Léger, IEEE Trans. Antennas Propag. 68, 1569 (2020). [CrossRef]  

9. E. S. Cassedy, Proc. IEEE 55, 1154 (1967). [CrossRef]  

10. J. B. Pendry, P. A. Huidobro, and E. Galiffi, “A new mechanism for gain in time dependent media,” arXiv:2009.12077 (2020).

References

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  1. V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
    [Crossref]
  2. H. Lira, Z. Yu, S. Fan, and M. Lipson, Phys. Rev. Lett. 109, 033901 (2012).
    [Crossref]
  3. A. C. Tasolamprou, A. D. Koulouklidis, C. Daskalaki, C. P. Mavidis, G. Kenanakis, G. Deligeorgis, Z. Viskadourakis, P. Kuzhir, S. Tzortzakis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, ACS Photon. 6, 720 (2019).
    [Crossref]
  4. C. T. Phare, Y.-H. D. Lee, J. Cardenas, and M. Lipson, Nat. Photonics 9, 511 (2015).
    [Crossref]
  5. W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, Nano Lett. 14, 955 (2014).
    [Crossref]
  6. N. Chamanara and C. Caloz, “Laser pulse compansion using co-propagating space-time modulation,” arXiv:1810.04129v1 [physics.optics] (2018).
  7. E. Galiffi, P. A. Huidobro, and J. B. Pendry, Phys. Rev. Lett. 123, 206101 (2019).
    [Crossref]
  8. C. Caloz and Z. Deck-Léger, IEEE Trans. Antennas Propag. 68, 1569 (2020).
    [Crossref]
  9. E. S. Cassedy, Proc. IEEE 55, 1154 (1967).
    [Crossref]
  10. J. B. Pendry, P. A. Huidobro, and E. Galiffi, “A new mechanism for gain in time dependent media,” arXiv:2009.12077 (2020).

2020 (2)

V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
[Crossref]

C. Caloz and Z. Deck-Léger, IEEE Trans. Antennas Propag. 68, 1569 (2020).
[Crossref]

2019 (2)

E. Galiffi, P. A. Huidobro, and J. B. Pendry, Phys. Rev. Lett. 123, 206101 (2019).
[Crossref]

A. C. Tasolamprou, A. D. Koulouklidis, C. Daskalaki, C. P. Mavidis, G. Kenanakis, G. Deligeorgis, Z. Viskadourakis, P. Kuzhir, S. Tzortzakis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, ACS Photon. 6, 720 (2019).
[Crossref]

2015 (1)

C. T. Phare, Y.-H. D. Lee, J. Cardenas, and M. Lipson, Nat. Photonics 9, 511 (2015).
[Crossref]

2014 (1)

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, Nano Lett. 14, 955 (2014).
[Crossref]

2012 (1)

H. Lira, Z. Yu, S. Fan, and M. Lipson, Phys. Rev. Lett. 109, 033901 (2012).
[Crossref]

1967 (1)

E. S. Cassedy, Proc. IEEE 55, 1154 (1967).
[Crossref]

Bao, J.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, Nano Lett. 14, 955 (2014).
[Crossref]

Boltasseva, A.

V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
[Crossref]

Bruno, V.

V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
[Crossref]

Caloz, C.

C. Caloz and Z. Deck-Léger, IEEE Trans. Antennas Propag. 68, 1569 (2020).
[Crossref]

N. Chamanara and C. Caloz, “Laser pulse compansion using co-propagating space-time modulation,” arXiv:1810.04129v1 [physics.optics] (2018).

Cardenas, J.

C. T. Phare, Y.-H. D. Lee, J. Cardenas, and M. Lipson, Nat. Photonics 9, 511 (2015).
[Crossref]

Cassedy, E. S.

E. S. Cassedy, Proc. IEEE 55, 1154 (1967).
[Crossref]

Chamanara, N.

N. Chamanara and C. Caloz, “Laser pulse compansion using co-propagating space-time modulation,” arXiv:1810.04129v1 [physics.optics] (2018).

Chen, B.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, Nano Lett. 14, 955 (2014).
[Crossref]

Clerici, M.

V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
[Crossref]

Cumming, D. R. S.

V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
[Crossref]

Daskalaki, C.

A. C. Tasolamprou, A. D. Koulouklidis, C. Daskalaki, C. P. Mavidis, G. Kenanakis, G. Deligeorgis, Z. Viskadourakis, P. Kuzhir, S. Tzortzakis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, ACS Photon. 6, 720 (2019).
[Crossref]

Deck-Léger, Z.

C. Caloz and Z. Deck-Léger, IEEE Trans. Antennas Propag. 68, 1569 (2020).
[Crossref]

Deligeorgis, G.

A. C. Tasolamprou, A. D. Koulouklidis, C. Daskalaki, C. P. Mavidis, G. Kenanakis, G. Deligeorgis, Z. Viskadourakis, P. Kuzhir, S. Tzortzakis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, ACS Photon. 6, 720 (2019).
[Crossref]

DeVault, C.

V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
[Crossref]

Economou, E. N.

A. C. Tasolamprou, A. D. Koulouklidis, C. Daskalaki, C. P. Mavidis, G. Kenanakis, G. Deligeorgis, Z. Viskadourakis, P. Kuzhir, S. Tzortzakis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, ACS Photon. 6, 720 (2019).
[Crossref]

Faccio, D.

V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
[Crossref]

Fan, S.

H. Lira, Z. Yu, S. Fan, and M. Lipson, Phys. Rev. Lett. 109, 033901 (2012).
[Crossref]

Fang, W.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, Nano Lett. 14, 955 (2014).
[Crossref]

Ferrera, M.

V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
[Crossref]

Galiffi, E.

E. Galiffi, P. A. Huidobro, and J. B. Pendry, Phys. Rev. Lett. 123, 206101 (2019).
[Crossref]

J. B. Pendry, P. A. Huidobro, and E. Galiffi, “A new mechanism for gain in time dependent media,” arXiv:2009.12077 (2020).

Hu, Z.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, Nano Lett. 14, 955 (2014).
[Crossref]

Huidobro, P. A.

E. Galiffi, P. A. Huidobro, and J. B. Pendry, Phys. Rev. Lett. 123, 206101 (2019).
[Crossref]

J. B. Pendry, P. A. Huidobro, and E. Galiffi, “A new mechanism for gain in time dependent media,” arXiv:2009.12077 (2020).

Huq, T.

V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
[Crossref]

Jacassi, A.

V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
[Crossref]

Kafesaki, M.

A. C. Tasolamprou, A. D. Koulouklidis, C. Daskalaki, C. P. Mavidis, G. Kenanakis, G. Deligeorgis, Z. Viskadourakis, P. Kuzhir, S. Tzortzakis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, ACS Photon. 6, 720 (2019).
[Crossref]

Kenanakis, G.

A. C. Tasolamprou, A. D. Koulouklidis, C. Daskalaki, C. P. Mavidis, G. Kenanakis, G. Deligeorgis, Z. Viskadourakis, P. Kuzhir, S. Tzortzakis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, ACS Photon. 6, 720 (2019).
[Crossref]

Koulouklidis, A. D.

A. C. Tasolamprou, A. D. Koulouklidis, C. Daskalaki, C. P. Mavidis, G. Kenanakis, G. Deligeorgis, Z. Viskadourakis, P. Kuzhir, S. Tzortzakis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, ACS Photon. 6, 720 (2019).
[Crossref]

Kudyshev, Z.

V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
[Crossref]

Kuzhir, P.

A. C. Tasolamprou, A. D. Koulouklidis, C. Daskalaki, C. P. Mavidis, G. Kenanakis, G. Deligeorgis, Z. Viskadourakis, P. Kuzhir, S. Tzortzakis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, ACS Photon. 6, 720 (2019).
[Crossref]

Lee, Y.-H. D.

C. T. Phare, Y.-H. D. Lee, J. Cardenas, and M. Lipson, Nat. Photonics 9, 511 (2015).
[Crossref]

Li, W.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, Nano Lett. 14, 955 (2014).
[Crossref]

Li, X.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, Nano Lett. 14, 955 (2014).
[Crossref]

Lipson, M.

C. T. Phare, Y.-H. D. Lee, J. Cardenas, and M. Lipson, Nat. Photonics 9, 511 (2015).
[Crossref]

H. Lira, Z. Yu, S. Fan, and M. Lipson, Phys. Rev. Lett. 109, 033901 (2012).
[Crossref]

Lira, H.

H. Lira, Z. Yu, S. Fan, and M. Lipson, Phys. Rev. Lett. 109, 033901 (2012).
[Crossref]

Liu, W.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, Nano Lett. 14, 955 (2014).
[Crossref]

Maier, S. A.

V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
[Crossref]

Mavidis, C. P.

A. C. Tasolamprou, A. D. Koulouklidis, C. Daskalaki, C. P. Mavidis, G. Kenanakis, G. Deligeorgis, Z. Viskadourakis, P. Kuzhir, S. Tzortzakis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, ACS Photon. 6, 720 (2019).
[Crossref]

Meng, C.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, Nano Lett. 14, 955 (2014).
[Crossref]

Mignuzzi, S.

V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
[Crossref]

Pendry, J. B.

E. Galiffi, P. A. Huidobro, and J. B. Pendry, Phys. Rev. Lett. 123, 206101 (2019).
[Crossref]

J. B. Pendry, P. A. Huidobro, and E. Galiffi, “A new mechanism for gain in time dependent media,” arXiv:2009.12077 (2020).

Phare, C. T.

C. T. Phare, Y.-H. D. Lee, J. Cardenas, and M. Lipson, Nat. Photonics 9, 511 (2015).
[Crossref]

Saha, S.

V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
[Crossref]

Sapienza, R.

V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
[Crossref]

Shah, Y. D.

V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
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Shalaev, V. M.

V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
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W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, Nano Lett. 14, 955 (2014).
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A. C. Tasolamprou, A. D. Koulouklidis, C. Daskalaki, C. P. Mavidis, G. Kenanakis, G. Deligeorgis, Z. Viskadourakis, P. Kuzhir, S. Tzortzakis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, ACS Photon. 6, 720 (2019).
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Tasolamprou, A. C.

A. C. Tasolamprou, A. D. Koulouklidis, C. Daskalaki, C. P. Mavidis, G. Kenanakis, G. Deligeorgis, Z. Viskadourakis, P. Kuzhir, S. Tzortzakis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, ACS Photon. 6, 720 (2019).
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W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, Nano Lett. 14, 955 (2014).
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Tzortzakis, S.

A. C. Tasolamprou, A. D. Koulouklidis, C. Daskalaki, C. P. Mavidis, G. Kenanakis, G. Deligeorgis, Z. Viskadourakis, P. Kuzhir, S. Tzortzakis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, ACS Photon. 6, 720 (2019).
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V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
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A. C. Tasolamprou, A. D. Koulouklidis, C. Daskalaki, C. P. Mavidis, G. Kenanakis, G. Deligeorgis, Z. Viskadourakis, P. Kuzhir, S. Tzortzakis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, ACS Photon. 6, 720 (2019).
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W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, Nano Lett. 14, 955 (2014).
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W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, Nano Lett. 14, 955 (2014).
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H. Lira, Z. Yu, S. Fan, and M. Lipson, Phys. Rev. Lett. 109, 033901 (2012).
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A. C. Tasolamprou, A. D. Koulouklidis, C. Daskalaki, C. P. Mavidis, G. Kenanakis, G. Deligeorgis, Z. Viskadourakis, P. Kuzhir, S. Tzortzakis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, ACS Photon. 6, 720 (2019).
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V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha, Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Faccio, R. Sapienza, and V. M. Shalaev, Phys. Rev. Lett. 124, 043902 (2020).
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Figures (3)

Fig. 1.
Fig. 1. Superconductor repels a magnet by excluding its lines of force. As the superconductor approaches the magnet, the repulsive force stores energy by compressing the lines of force, which are conserved.
Fig. 2.
Fig. 2. Sketch of dispersion relation of light traveling through (A) a uniform dielectric; (B) a grating modulated as shown in Eq. (1), where ${c_g} \lt {c_0}/\sqrt {{\varepsilon _1}{\mu _1}}$ ; (C) a grating where ${c_g} \gt {c_0}/\sqrt {{\varepsilon _1}{\mu _1}}$ ; (D) the case of ${c_g} = {c_0}/\sqrt {{\varepsilon _1}{\mu _1}}$ where all the forward-traveling states are degenerate and therefore strongly coupled. Arrows show displacement of the bands by a space-time reciprocal lattice vector $({g,\Omega})$ with slope ${c_g}$ .
Fig. 3.
Fig. 3. Schematic figure of the effect of a luminal grating, traveling to the right, on a plane wave incident from the left. The grating has little effect on waves incident from the right. Note the compression of lines of force shown schematically as gray lines.

Equations (3)

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ε ( x c g t ) = ε 1 [ 1 + 2 α ε cos ( g x Ω t ) ] μ ( x c g t ) = μ 1 [ 1 + 2 α μ cos ( g x Ω t ) ] ,
+ D y t = 1 Z E y x ,
t p e r i o d | D y | d x = 0 ,

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