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Photonics of time-varying media

Galiffi, Emanuele, Tirole, Romain, Yin, Shixiong, Li, Huanan, Vezzoli, Stefano, Huidobro, Paloma A., Silveirinha, Mário G., Sapienza, Riccardo, Alù, Andrea, Pendry, J. B.·2022·Advanced Photonics
DOI 10.1117/1.ap.4.1.014002
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Abstract

Time-varying media have recently emerged as a new paradigm for wave manipulation, due to the synergy between the discovery of highly nonlinear materials, such as epsilon-near-zero materials, and the quest for wave applications, such as magnet-free nonreciprocity, multimode light shaping, and ultrafast switching. In this review, we provide a comprehensive discussion of the recent progress achieved with photonic metamaterials whose properties stem from their modulation in time. We review the basic concepts underpinning temporal switching and its relation with spatial scattering and deploy the resulting insight to review photonic time-crystals and their emergent research avenues, such as topological and non-Hermitian physics. We then extend our discussion to account for spatiotemporal modulation and its applications to nonreciprocity, synthetic motion, giant anisotropy, amplification, and many other effects. Finally, we conclude with a review of the most attractive experimental avenues recently demonstrated and provide a few perspectives on emerging trends for future implementations of time-modulation in photonics.

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Figures

Fig 1 Scattering at (a) a spatial discontinuity, generating transmitted and reflected waves propagating in different media with refractive indices n 1 and n 2 , and (b) a temporal discontinuity, generating forward and backward waves propagating in the same medium. (c,d) Illustration of the respective scattering processes on a dispersion diagram: the blue and red lines denote the dispersion cone for the two media. Note how the frequency ω is conserved in a spatial scattering process, and coupling occurs to forward and backward modes in the two different media, whereas in a time-scattering process the momentum k is conserved, and the two scattered waves are both embedded in the new medium.
Fig 1 Scattering at (a) a spatial discontinuity, generating transmitted and reflected waves propagating in different media with refractive indices n 1 and n 2 , and (b) a temporal discontinuity, generating forward and backward waves propagating in the same medium. (c,d) Illustration of the respective scattering processes on a dispersion diagram: the blue and red lines denote the dispersion cone for the two media. Note how the frequency ω is conserved in a spatial scattering process, and coupling occurs to forward and backward modes in the two different media, whereas in a time-scattering process the momentum k is conserved, and the two scattered waves are both embedded in the new medium.CC-BY · From the PDF
Fig 2 (a) Schematic of antireflection temporal coatings proposed in Ref. 38. (b) Numerical results for the field distribution before and after the temporal scattering, showing the incident and transmitted waves, with minimized reflection due to the temporal coating shown in (a). (c) Spectra of the incident, forward and backward waves. (d) Schematic of temporal parity-time (TPT) symmetric structures proposed in Ref. 41. (e) and (f): The time evolution of the normalized energy (solid curves) and its two constitutions (circle symbols for interferences and the dashed curves for another). (e) depicts the case of maximum total power while (f) shows that of minimum total power. Insets are the field distributions in simulations before, during and after the TPT slabs. Figures adapted from Refs. 38, 41.
Fig 2 (a) Schematic of antireflection temporal coatings proposed in Ref. 38. (b) Numerical results for the field distribution before and after the temporal scattering, showing the incident and transmitted waves, with minimized reflection due to the temporal coating shown in (a). (c) Spectra of the incident, forward and backward waves. (d) Schematic of temporal parity-time (TPT) symmetric structures proposed in Ref. 41. (e) and (f): The time evolution of the normalized energy (solid curves) and its two constitutions (circle symbols for interferences and the dashed curves for another). (e) depicts the case of maximum total power while (f) shows that of minimum total power. Insets are the field distributions in simulations before, during and after the TPT slabs. Figures adapted from Refs. 38, 41.CC-BY · From the PDF
Fig 3 (a) A conventional prism decomposing white light into its frequency components in different directions. (b) An inverse prism that maps the light with different momentum into different frequencies. (c) Implementation of the inverse prism proposed in Ref. 28. (d) Temporal aiming proposed in Ref. 42. (e) The conventional Brewster angle. (f) Temporal Brewster angle described in Ref. 43. Figures adapted from Refs. 28,42,43.
Fig 3 (a) A conventional prism decomposing white light into its frequency components in different directions. (b) An inverse prism that maps the light with different momentum into different frequencies. (c) Implementation of the inverse prism proposed in Ref. 28. (d) Temporal aiming proposed in Ref. 42. (e) The conventional Brewster angle. (f) Temporal Brewster angle described in Ref. 43. Figures adapted from Refs. 28,42,43.CC-BY · From the PDF
Fig 4 (a) Temporal reflection of graphene plasmons reported in Ref. 53. (b) Impedance matching using a timeswitched transmission line. 55 (c) Time-switched thin absorber in Ref. 59. (d) Unitary excitation transfer between two coupled circuit resonators. 60 (e) Temporal deflection caused by isotropic-to-anisotropic switching on meta-atoms. 63 (f) Temporal switching of structural dispersion for ultrafast frequency-shifts, proposed in Ref. 64. (g) Corresponding frequency spectra for the switching in panel (f). Figures adapted from Refs. 53, 55, 59, 60, 63, 64.
Fig 4 (a) Temporal reflection of graphene plasmons reported in Ref. 53. (b) Impedance matching using a timeswitched transmission line. 55 (c) Time-switched thin absorber in Ref. 59. (d) Unitary excitation transfer between two coupled circuit resonators. 60 (e) Temporal deflection caused by isotropic-to-anisotropic switching on meta-atoms. 63 (f) Temporal switching of structural dispersion for ultrafast frequency-shifts, proposed in Ref. 64. (g) Corresponding frequency spectra for the switching in panel (f). Figures adapted from Refs. 53, 55, 59, 60, 63, 64.CC-BY · From the PDF
Fig 5 (a) Example of a (finite) layered PTC. (b) Band structure of a PTC, showing k-gaps and Zak numbers of the different bands. (c) Change of Zak phase across the first k-gap. (d) Example of an edge between two PTCs with opposite Zak phase. (e) Numerical demonstration of the effect of a temporal edge between two topologically inequivalent PTCs: the parametric amplification process is interrupted and the wave amplitude depleted for a few periods following the temporal edge, forming a temporally localized state. Figures reproduced from Ref. 69.
Fig 5 (a) Example of a (finite) layered PTC. (b) Band structure of a PTC, showing k-gaps and Zak numbers of the different bands. (c) Change of Zak phase across the first k-gap. (d) Example of an edge between two PTCs with opposite Zak phase. (e) Numerical demonstration of the effect of a temporal edge between two topologically inequivalent PTCs: the parametric amplification process is interrupted and the wave amplitude depleted for a few periods following the temporal edge, forming a temporally localized state. Figures reproduced from Ref. 69.CC-BY · From the PDF
Fig 6 (a) The resonant modes of a structure form a lattice. (b) Hopping through its sites can be enabled by time modulation, which can couple their different frequencies. (c) The resulting Hamiltonian is analogous to that of electrons in a periodic ionic potential. (d) A ring resonator combined with a modulator can reproduce such a model, (e) enabling hopping between its equally spaced resonant modes. 74 (f-h) A synthetic dimension may be constructed in a tight-binding model by introducing long-term coupling between the elements of a chain. In the time-domain, this can be realized by modulating the system at multiple frequencies. 75 (i-h) A spatially 1D chain of ring resonators can be combined with temporal modulation to form a synthetic 2D lattice. 76 Figures adapted from Refs. 74-76.
Fig 6 (a) The resonant modes of a structure form a lattice. (b) Hopping through its sites can be enabled by time modulation, which can couple their different frequencies. (c) The resulting Hamiltonian is analogous to that of electrons in a periodic ionic potential. (d) A ring resonator combined with a modulator can reproduce such a model, (e) enabling hopping between its equally spaced resonant modes. 74 (f-h) A synthetic dimension may be constructed in a tight-binding model by introducing long-term coupling between the elements of a chain. In the time-domain, this can be realized by modulating the system at multiple frequencies. 75 (i-h) A spatially 1D chain of ring resonators can be combined with temporal modulation to form a synthetic 2D lattice. 76 Figures adapted from Refs. 74-76.CC-BY · From the PDF
Image 7
CC-BY · From the PDF
Fig 8 (a) Light scattered from a flat, time-varying metasurface can access a locus of states which form a hyperbola in phase space, as opposed to an ellipse in spatial metasurfaces. (b) Implementation of a time-modulated metasurface at GHz frequencies. (c) A 1 MHz modulation performs efficient frequency conversion through time-modulation. 94 (d) Illustration of surface-wave excitation via Wood anomalies using a spatial (e) and temporal (f) modulation of a surface (g). (h) Example of Wood anomaly observed in transmission, the surface-wave excitation is explicitly shown in (i) from a Finite-Element-Time-Domain simulation. 95 Figures adapted from. 94, 95
Fig 8 (a) Light scattered from a flat, time-varying metasurface can access a locus of states which form a hyperbola in phase space, as opposed to an ellipse in spatial metasurfaces. (b) Implementation of a time-modulated metasurface at GHz frequencies. (c) A 1 MHz modulation performs efficient frequency conversion through time-modulation. 94 (d) Illustration of surface-wave excitation via Wood anomalies using a spatial (e) and temporal (f) modulation of a surface (g). (h) Example of Wood anomaly observed in transmission, the surface-wave excitation is explicitly shown in (i) from a Finite-Element-Time-Domain simulation. 95 Figures adapted from. 94, 95CC-BY · From the PDF
Fig 9 (a) Spacetime diagram of a space-time crystal with alternating refractive indices n i and n j , spatiotemporal lattice vector p and velocity v m . (b) Double-period space-time crystal, characterized by two different lattice vectors p A and p B , and two different velocities v m 1 and v m 2 . 130 (c) The band structure for a spatial crystal with ε ∼ cos( gz -Ω t ) becomes asymmetric as (d) a finite temporal component Ω , and therefore a finite modulation velocity v m = Ω /g , is introduced. This is caused by reciprocal lattice vectors (dashed arrows) acquiring a non-zero frequency component. The angle formed by the reciprocal lattice vectors with the slope of the bands determines a subluminal (c-d) and a superluminal (e-f) regime. 129 (g) Nonreciprocal mode-coupling in a space-time modulated waveguide: the modulation can couple forward modes between each other as their frequencies and momenta are matched by the space-time modulation, while two backward waves are not coupled, so that an impinging backward wave is effectively unchanged. 123 Figures adapted from 123,129,130.
Fig 9 (a) Spacetime diagram of a space-time crystal with alternating refractive indices n i and n j , spatiotemporal lattice vector p and velocity v m . (b) Double-period space-time crystal, characterized by two different lattice vectors p A and p B , and two different velocities v m 1 and v m 2 . 130 (c) The band structure for a spatial crystal with ε ∼ cos( gz -Ω t ) becomes asymmetric as (d) a finite temporal component Ω , and therefore a finite modulation velocity v m = Ω /g , is introduced. This is caused by reciprocal lattice vectors (dashed arrows) acquiring a non-zero frequency component. The angle formed by the reciprocal lattice vectors with the slope of the bands determines a subluminal (c-d) and a superluminal (e-f) regime. 129 (g) Nonreciprocal mode-coupling in a space-time modulated waveguide: the modulation can couple forward modes between each other as their frequencies and momenta are matched by the space-time modulation, while two backward waves are not coupled, so that an impinging backward wave is effectively unchanged. 123 Figures adapted from 123,129,130.CC-BY · From the PDF
Fig 10 (a) Angular momentum biasing: a directional modulation along a ring resonator splits the degeneracy between clockwise and anti-clockwise travelling states, (b) in analogy with the Zeeman splitting of electronic states in a magnetic field. 131 (c) This can be realized by discretizing the elements of the ring, e.g. periodically modulating three strongly coupled resonators, with a phase of 120 deg between them. 132 (d) Example of operation of a non-reciprocal ring resonator coupled to two waveguides: excitation from the bottom (top) of the channel waveguide (on the right of the ring) excites a counterclockwise (clockwise) mode, whose resonant frequency has been offset from the one of the clockwise (counterclockwise) mode, so that the direction of excitation determines the efficiency of the coupling to the resonator (e), and hence of the transmission to the output port of the channel waveguide. 131 (f-g) Theoretical and (h) experimental performance of an RF circulator made with angular-momentum biased resonators: panels (f) and (g) show the reciprocal response in the absence (f) and presence (g) of angular biasing, while panel (h) shows the performance of the experimental implementation of the circulator. 132 Figures reproduced from Refs. 131,132.
Fig 10 (a) Angular momentum biasing: a directional modulation along a ring resonator splits the degeneracy between clockwise and anti-clockwise travelling states, (b) in analogy with the Zeeman splitting of electronic states in a magnetic field. 131 (c) This can be realized by discretizing the elements of the ring, e.g. periodically modulating three strongly coupled resonators, with a phase of 120 deg between them. 132 (d) Example of operation of a non-reciprocal ring resonator coupled to two waveguides: excitation from the bottom (top) of the channel waveguide (on the right of the ring) excites a counterclockwise (clockwise) mode, whose resonant frequency has been offset from the one of the clockwise (counterclockwise) mode, so that the direction of excitation determines the efficiency of the coupling to the resonator (e), and hence of the transmission to the output port of the channel waveguide. 131 (f-g) Theoretical and (h) experimental performance of an RF circulator made with angular-momentum biased resonators: panels (f) and (g) show the reciprocal response in the absence (f) and presence (g) of angular biasing, while panel (h) shows the performance of the experimental implementation of the circulator. 132 Figures reproduced from Refs. 131,132.CC-BY · From the PDF
Fig 11 (a) and (b) A synthetic spacetime modulated crystal (panel a) is formally equivalent to a fictitious moving bianisotropic time-invariant crystal (panel b) when v m < c 0 . The velocity of the equivalent moving crystal is identical to the modulation speed v m . (c) In the long wavelength limit the system response can be homogenized and the crystal behaves as a uniaxial dielectric moving with speed v D . The sign of v D is not necessarily coincident with the sign of v . (d) Dispersion diagram of the effective medium ω vs k x . When only one of the material parameters is spacetime modulated the dispersion is symmetric, and the equivalent drag velocity vanishes (blue lines). When both ε and µ are spacetime modulated, the dispersion is asymmetric (red lines, assuming v D > 0 ). The waves that propagate in the direction of v D have a larger group velocity than the waves that propagate in the opposite direction. 153 (e) Drag velocity v D and (f) effective bianisotropic coefficient ξ ef as a function of the modulation speed v m for a representative spacetime modulated crystal: the velocity of the equivalent moving medium and the magneto-electric coupling coefficient flip sign in the transition from the subluminal to the superluminal regime. Adapted from 155.
Fig 11 (a) and (b) A synthetic spacetime modulated crystal (panel a) is formally equivalent to a fictitious moving bianisotropic time-invariant crystal (panel b) when v m < c 0 . The velocity of the equivalent moving crystal is identical to the modulation speed v m . (c) In the long wavelength limit the system response can be homogenized and the crystal behaves as a uniaxial dielectric moving with speed v D . The sign of v D is not necessarily coincident with the sign of v . (d) Dispersion diagram of the effective medium ω vs k x . When only one of the material parameters is spacetime modulated the dispersion is symmetric, and the equivalent drag velocity vanishes (blue lines). When both ε and µ are spacetime modulated, the dispersion is asymmetric (red lines, assuming v D > 0 ). The waves that propagate in the direction of v D have a larger group velocity than the waves that propagate in the opposite direction. 153 (e) Drag velocity v D and (f) effective bianisotropic coefficient ξ ef as a function of the modulation speed v m for a representative spacetime modulated crystal: the velocity of the equivalent moving medium and the magneto-electric coupling coefficient flip sign in the transition from the subluminal to the superluminal regime. Adapted from 155.CC-BY · From the PDF
Fig 12 (a) Electric field intensity at the output port vs time transmitted through different thicknesses (blue to red) d of luminal crystal. (b) In the gain region the permittivity gradient is positive, and vice versa. The gain is maximum at the points (in time or space) where the velocity of the waves matches the velocity of the grating. (c-d) Power spectra of the transmitted waves for input frequency (c) ω 0 = 8Ω , with Ω the modulation frequency, and (d) for a DC input field ( ω 0 = 0 ), showing the frequency generation responsible for the overall gain. 129, 168 (e-f) Trajectories of a point of constant phase along the spacetime variable X = x -v m t . (e) outside of the localization regime (resulting in oscillations which periodically compress and decompress the pulse, panel g), and (f) within the localization regime, resulting in the amplification and compression described in (a-d). 169 Figures adapted from Refs. 129,168,169.
Fig 12 (a) Electric field intensity at the output port vs time transmitted through different thicknesses (blue to red) d of luminal crystal. (b) In the gain region the permittivity gradient is positive, and vice versa. The gain is maximum at the points (in time or space) where the velocity of the waves matches the velocity of the grating. (c-d) Power spectra of the transmitted waves for input frequency (c) ω 0 = 8Ω , with Ω the modulation frequency, and (d) for a DC input field ( ω 0 = 0 ), showing the frequency generation responsible for the overall gain. 129, 168 (e-f) Trajectories of a point of constant phase along the spacetime variable X = x -v m t . (e) outside of the localization regime (resulting in oscillations which periodically compress and decompress the pulse, panel g), and (f) within the localization regime, resulting in the amplification and compression described in (a-d). 169 Figures adapted from Refs. 129,168,169.CC-BY · From the PDF
Fig 13 Examples of applications of space-time metasurfaces: (a) An isolator based on the unidirectional excitation of evanescent modes; (b) Waves scattering off a STM experience a nonreciprocal phase shift; (c) A STM-based circulator. 99 (d) Illustration of an experimentally realized space-time coding metasurface, consisting of a square array of individual voltage-controlled elements with binary impedance values, enabling encoding and imprinting of arbitrary phase and amplitude modulation onto different scattered harmonics for (e) beam-steering and shaping. 109 (f) Illustration of wave power combining using STMs: incoming waves with frequency ω 0 coming from a discrete range of different angles can be scattered towards the same outgoing direction by imprinting the necessary frequencywavevector shifts via the STM. (g) A proposal of an angular momentum-biased metasurface for generation of orbital angular momentum states: each azimuthal section is temporally modulated with a different relative phase, imparting the necessary angular-momentum bias. Figures adapted from Refs. 99,109,173
Fig 13 Examples of applications of space-time metasurfaces: (a) An isolator based on the unidirectional excitation of evanescent modes; (b) Waves scattering off a STM experience a nonreciprocal phase shift; (c) A STM-based circulator. 99 (d) Illustration of an experimentally realized space-time coding metasurface, consisting of a square array of individual voltage-controlled elements with binary impedance values, enabling encoding and imprinting of arbitrary phase and amplitude modulation onto different scattered harmonics for (e) beam-steering and shaping. 109 (f) Illustration of wave power combining using STMs: incoming waves with frequency ω 0 coming from a discrete range of different angles can be scattered towards the same outgoing direction by imprinting the necessary frequencywavevector shifts via the STM. (g) A proposal of an angular momentum-biased metasurface for generation of orbital angular momentum states: each azimuthal section is temporally modulated with a different relative phase, imparting the necessary angular-momentum bias. Figures adapted from Refs. 99,109,173CC-BY · From the PDF
Fig 14 (a,b) Concept of photocarrier excitation and time-refraction in ITO: 210 (a) A pump pulse transfers energy to the electrons in the conduction band, leading to a temporal variation in the refractive index. (b) As the probe experiences different indices at different delays, its spectral content will be shifted (redshift for negative delay, blueshift for positive). (c-f) Measured time-refraction, phase conjugation and negative refraction signals from a 500 nm thick AZO slab pumped by 105 fs pulses, at an energy of 770 GW/cm 2 in a degenerate pump-probe experiment at 1400 nm. 209 (g) Strong coupling between a plasmonic antenna and an ENZ thin film: 216 electric field | E | 2 distribution obtained from FDTD for an ITO layer thickness of 40 nm and Au antenna length of 400 nm. (h) Resulting field distribution as a function of metasurface depth. (i-k) Probe spectrum at various delays for a central wavelength of 1304 nm for 50 fs pulses in a strongly-coupled plasmonic antenna-ENZ system. 217 The increase in modulating pump intensities (0.5, 1 and 2 GW/cm 2 ) leads to a shift near zero delay. Figures adapted from Refs. 209, 210, 216, 217.
Fig 14 (a,b) Concept of photocarrier excitation and time-refraction in ITO: 210 (a) A pump pulse transfers energy to the electrons in the conduction band, leading to a temporal variation in the refractive index. (b) As the probe experiences different indices at different delays, its spectral content will be shifted (redshift for negative delay, blueshift for positive). (c-f) Measured time-refraction, phase conjugation and negative refraction signals from a 500 nm thick AZO slab pumped by 105 fs pulses, at an energy of 770 GW/cm 2 in a degenerate pump-probe experiment at 1400 nm. 209 (g) Strong coupling between a plasmonic antenna and an ENZ thin film: 216 electric field | E | 2 distribution obtained from FDTD for an ITO layer thickness of 40 nm and Au antenna length of 400 nm. (h) Resulting field distribution as a function of metasurface depth. (i-k) Probe spectrum at various delays for a central wavelength of 1304 nm for 50 fs pulses in a strongly-coupled plasmonic antenna-ENZ system. 217 The increase in modulating pump intensities (0.5, 1 and 2 GW/cm 2 ) leads to a shift near zero delay. Figures adapted from Refs. 209, 210, 216, 217.CC-BY · From the PDF
Fig 15 (a) Schematic of an Si metasurface engineered to support collective high-Q Fano resonances. 220 (b) Schematic of the self-induced blue shift of harmonics in the resulting Si nanoantenna cavity: photons in the antenna will undergo a blueshift due to the rapidly changing permittivity of the medium before being upconverted via third-harmonic generation. (c) Measured spectrum evolution for a 80 fs pump pulse length in a GaAs highQ metasurface. 221 The fringes are measured on only one side of the spectrum due to the breaking of time-reversal asymmetry caused by the modulation. (d) Schematic of a nonreciprocal Si metasurface: 222 the travelling wave modulation breaks the space-time symmetry of the reflection phase change. Figures adapted from Refs. 220-222
Fig 15 (a) Schematic of an Si metasurface engineered to support collective high-Q Fano resonances. 220 (b) Schematic of the self-induced blue shift of harmonics in the resulting Si nanoantenna cavity: photons in the antenna will undergo a blueshift due to the rapidly changing permittivity of the medium before being upconverted via third-harmonic generation. (c) Measured spectrum evolution for a 80 fs pump pulse length in a GaAs highQ metasurface. 221 The fringes are measured on only one side of the spectrum due to the breaking of time-reversal asymmetry caused by the modulation. (d) Schematic of a nonreciprocal Si metasurface: 222 the travelling wave modulation breaks the space-time symmetry of the reflection phase change. Figures adapted from Refs. 220-222CC-BY · From the PDF

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Metadata

Authors
Galiffi, E., Tirole, R., Yin, S., Li, H., Vezzoli, S., Huidobro, P. A., Silveirinha, M. G., Sapienza, R., Alù, A., & Pendry, J. B.
Venue
Advanced Photonics
Publisher
SPIE-Intl Soc Optical Eng
Published
2022-02-14
Volume
4
Issue
01
Pages
1-32
ISSN
2577-5421
DOI
10.1117/1.ap.4.1.014002
Cited by
591
References
4
Licence
creativecommons.org/licenses/by/4.0/ (vor)
Funders
Air Force Office of Scientific Research (MURI)

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

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