Time-varying Medium Research Articles

Time division interferometer setups

Abstract A time division interferometer (TDI) in the optical regime is presented, where a wavetrain is temporally split into two distinct parts that are superimposed at a subsequent stage. At any one time, the photons have no alternatives, only one possible path is possible. There is no path bifurcation in the setup, i.e. no ‘Y’ where the photons can take left or right. The photons choice, if any, is to arrive/collapse before or after the time t 1 when the time modulated medium changes its state. In time-varying media, as energy can be exchanged between the wave and medium, the wave frequency is modified. Thus, the diverted photons become doubly tagged, in their wave vector direction as well as their frequency. A streak camera detector array resolves both, the fringe pattern as well as the frequency beats, in a space versus time interferogram. This apparatus has 0.66 ns temporal resolution and 70 μ m spatial resolution. Several TDI configurations are described, two of them are implemented: a spatio-temporal interferometer where the ‘which path’ criterion is readily observed and an all temporal TDI where only temporal interference is present. Novel possibilities of TDI schemes are discussed.

Numerical analysis for temporal and spectral responses of electromagnetic waves in spatially homogeneous time varying medium

This paper presents simple numerical solutions for electromagnetic plane waves in spatially homogenous time varying medium. The solution is based on converting the resulting second order differential equation into two combined ordinary differential equations which are solved numerically by using the built-in ode113 function in Matlab. By using this method, the time domain responses of the electric and magnetic fields at fixed point in space are obtained. The proposed method is applied on two cases: linearly time varying medium and sinusoidally time varying medium. The corresponding frequency domain response is obtained by using inverse Fourier transformation of the obtained time domain response. The proposed method is compared with FDTD solution. It is found that the proposed method has the same accuracy of FDTD with much less computational time.

Time-varying materials for analog optical computing

Subject of study. In this study, a new class of artificial electromagnetic media, known as time-varying materials, is explored. These materials are characterized by a rapid modulation of their optical parameters on ultrashort timescales that are comparable to or shorter than the wave period. Aim of study. The aim of this study was to develop a new platform for implementing analog optical computers. Specifically, we investigated the dependence of the amplitudes and frequencies of waves, formed through interaction with a time-varying medium, on the permittivity switching time and magnitude of spectral dispersion. Method. The amplitudes of reflected and refracted electromagnetic waves were calculated using the time-domain finite-element method. A Lorentz model with a time-dependent plasma frequency was employed to analyze the time-varying medium with spectral dispersion. The optical response of the medium, which exhibited noninstantaneous switching, was studied by considering a sigmoidal switching profile. Main results. Our findings show that a dispersive time-varying medium can generate several spectral components that are shifted from the frequency of the incident light. By controlling the speed and depth of permittivity switching, the amplitudes and frequencies of the time-reflected/refracted waves can be adjusted. To observe the reflected wave, the duration of switching should be comparable to or less than the period of the incident wave. Practical significance. The results of this study are valuable for developing next-generation optical devices. The ability to flexibly control the amplitude and frequency of light waves through the temporal modulation of homogeneous media offers new opportunities for the realization of analog optical computers.

Time-varying media, dispersion, and the principle of causality [Invited

The article reviews general properties of time-varying dispersive materials. We first briefly discuss different dispersion models that have been proposed in the literature, the role of the causality principle, and the generalization of the Kramers-Kronig relations for the susceptibility of linear time-dependent media. Furthermore, we discuss the Kramers-Kronig-like relations for nonlinear optical processes, with a focus on those processes that may be used to realize strong and fast temporal modulations at optical frequencies to enable, for instance, photonic time-crystal phenomena.

Time-varying media, relativity, and the arrow of time

Research on time-varying media has recently enjoyed renewed interest, especially in photonics. Despite the large amount of research done in this field in the past few years, the attention has been focused on electromagnetic waves solely, while a comprehensive framework describing how wave phenomena in general are influenced by time-varying media has not been fully developed yet. To this aim, we study the implications of time-varying wave mechanics, and show how the standard wave equation is modified if the speed of a wave is not constant in time. In particular, waves that experience longitudinal acceleration are shown to have clear relativistic properties when a constant reference speed exists. Moreover, the accelerating wave equation admits only solutions propagating forward in time, which are continuous across material interfaces. We then consider the special case of electromagnetic waves, finding that the Abraham–Minkowski controversy is caused by relativistic effects, and the momentum of light is in fact conserved between different media. Furthermore, we show that the accelerating waves conserve energy when the wave is moving along a geodesic and demonstrate two example solutions. We conclude with some remarks on the role of the accelerating wave equation in the context of the arrow of time.

Incandescent temporal metamaterials

Regarded as a promising alternative to spatially shaping matter, time-varying media can be seized to control and manipulate wave phenomena, including thermal radiation. Here, based upon the framework of macroscopic quantum electrodynamics, we elaborate a comprehensive quantum theoretical formulation that lies the basis for investigating thermal emission effects in time-modulated media. Our theory unveils unique physical features brought about by time-varying media: nontrivial correlations between fluctuating electromagnetic currents at different frequencies and positions, thermal radiation overcoming the black-body spectrum, and quantum vacuum amplification effects at finite temperature. We illustrate how these features lead to striking phenomena and innovative thermal emitters, specifically, showing that the time-modulation releases strong field fluctuations confined within epsilon-near-zero (ENZ) bodies, and that, in turn, it enables a narrowband (partially coherent) emission spanning the whole range of wavevectors, from near to far-field regimes.

Inverse design of optical pulse shapes for time-varying photonics.

Recent advancements in materials and metamaterials with strong, time-varying, nonlinear optical responses have spurred a surge of interest in time-varying photonics. This opens the door to novel optical phenomena including reciprocity breaking, frequency translation, and amplification that can be further optimized by improving the light-matter interaction. Although there has been recent interest in applying topology-based inverse design to this problem, we propose a novel approach in this article. We introduce a method for the inverse design of optical pulse shapes to enhance their interaction with time-varying media. We validate our objective-first approach by maximizing the transmittance of optical pulses of equal intensity through time-varying media. Through this approach, we achieve large, broadband enhancements in pulse energy transmission, including gain, without altering the incident pulse energy. As a final test, we maximize pulse transmission through thin films of indium tin oxide, a time-varying medium when strongly pumped in its ENZ band. Our work presents a new degree of freedom for the exploration, application, and design of time-varying systems and we hope it inspires further research in this direction.

Towards nonreciprocal pulse dynamics in a time-varying medium

We demonstrate the nonreciprocal propagation of optical pulses in a linear dispersive medium with a time-varying refractive index. The change of refractive index with time breaks the time-translation symmetry of the spatially homogeneous medium compelling it to be inherently nonreciprocal. A time-reversal asymmetric field ratio of the counter-propagating optical pulses along with a frequency shift leading to spectral nonreciprocity in such dynamic media have been observed. The findings have emerged a new avenue to realize nonreciprocal light flow in dynamically tuned integration-feasible photonic systems.

Numerical Modelling of Dynamic Electromagnetic Problems Based on the Time-Domain Finite Integration Technique

Developing numerical methods to solve dynamic electromagnetic problems has broad application prospects. In computational electromagnetics, traditional numerical methods are commonly used to deal with static electromagnetic problems. However, they can hardly be applied in the modeling of time-varying materials and moving objects. So far, the studies on numerical methods that can efficiently solve dynamic electromagnetic problems are still very limited. In this paper, a numerical method called the time-domain finite integration technique (TDFIT) is extended to tackle this problem via the introduction of time-varying iterative coefficients. In order to validate the effectiveness of the proposed algorithm, three numerical examples are demonstrated, including two microstrip structures with a time-varying medium and a rapidly rotating structure. The numerical results reveal that the time-varying medium can induce a nonlinear spectrum shift, and the radar cross section (RCS) of a rapidly rotating structure is highly dependent on the rotating speed. The proposed algorithm opens a new avenue for the exploration of many intriguing phenomena in fundamental physics, including frequency conversion and magnetless nonreciprocity. Meanwhile, it can also lead to a wide range of promising practical applications, such as active electron devices, space-time metamaterials, and hypersonic vehicles.

A technique for measurement of ultrasonic waves propagating in time-varying media

Laser Doppler Vibrometry (LDV) is often used in nondestructive testing to make high-fidelity, non-contact structural dynamic measurements. However, synchronized measurement on slow, structural dynamics (SD) and fast, ultrasonic (US) time scales can yield additional useful information about the overall condition of elastic structures. This work presents a technique, Dynamic Asymmetric Transmission Measurement (DATM), that enables time-aligned measurements of large-scale structural dynamics via LDV and small-scale ultrasonic signals to provide information at a particular global dynamic state of an elastic structure. This approach is applicable to structural health monitoring and generalizable to the measurement of guided ultrasonic waves propagating in structural components subjected to time-varying stress states. We demonstrate the utility of DATM by detecting and characterizing time-varying stress conditions caused by local nonlinearity due to surface-breaking cracks resulting in time-varying asymmetric transmission of US signals in beam and plate structures. The test methodology to synchronize LDV and US data is presented together with example measurements. [Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.]

Static-to-dynamic field conversion with time-varying media

We theoretically demonstrate that a uniform static electric field distribution can be partially converted to radiation fields when a portion of the medium undergoes a temporal change of its permittivity. An in-depth theoretical investigation of this phenomenon is developed for a dielectric block with a steplike temporal change located inside a waveguide charged with a DC voltage source. Closed analytical expressions are derived for the radiated electric and magnetic fields. The exchange of energy between the electrostatic and electromagnetic fields is discussed. The reconciliation between the seemingly contradictory temporal and spatial boundary conditions for the electric and magnetic fields at the interface of the time-varying dielectric block is analyzed and elucidated. Our findings may provide an alternative solution for generating electromagnetic radiation based on time-varying media.

Spontaneous emergence of a spin state for an emitter in a time-varying medium

Time-varying media can dramatically modify the emission of embedded sources by producing time reversed waves refocusing on the source. Here, we show that such a back action can create an angular momentum by setting the source in a spontaneous spin state. We experimentally implement this coupling using self-propelled bouncing droplets sources coupled to the surface waves they emit on a parametrically excited bath. The spin state dynamics result from a self-organized interplay between the source motion and the time reversed waves. The discrete stability analysis agrees with the experimental observations. In addition, we show that these spin states provide a unique opportunity for an experimental access to parameters enabling comparison and calibration of the various existing models.

Analytical transient analysis of temporal boundary value problems using the d'Alembert formula.

Temporal boundary value problems (TBVPs) provide the foundation for analyzing electromagnetic wave propagation in time-varying media. In this paper, we point out that TBVPs fall into the category of unbounded initial value problems, which have traveling wave solutions. By dividing the entire time frame into several subdomains and applying the d'Alembert formula, the transient expressions for waves propagating through temporal boundaries can be evaluated analytically. Moreover, unlike their spatial analogs, TBVPs are subject to causality. Therefore, the resulting analytical transient solutions resulting from the d'Alembert formula are unique to temporal systems.

Universal Statistics of Waves in a Random Time-Varying Medium.

We study the propagation of waves in a medium in which the wave velocity fluctuates randomly in time. We prove that at long times, the statistical distribution of the wave energy is log-normal, with the average energy growing exponentially. For weak disorder, another regime preexists at shorter times, in which the energy follows a negative exponential distribution, with an average value growing linearly with time. The theory is in perfect agreement with numerical simulations, and applies to different kinds of waves. The existence of such universal statistics bridges the fields of wave propagation in time-disordered and space-disordered media.

Exploiting space-time duality in the synthesis of impedance transformers via temporal metamaterials

Abstract Multisection quarter-wave impedance transformers are widely applied in microwave engineering and optics to attain impedance-matching networks and antireflection coatings. These structures are mostly designed in the spatial domain (time harmonic) by using geometries of different materials. Here, we exploit such concepts in the time domain by using time-varying metamaterials. We derive a formal analogy between the spectral responses of these structures and their temporal analogs, i.e., time-varying stepped refractive-index profiles. We show that such space-time duality grants access to the vast arsenal of synthesis approaches available in microwave engineering and optics. This allows, for instance, the synthesis of temporal impedance transformers for broadband impedance matching with maximally flat or equi-ripple responses, which extend and generalize the recently proposed quarter-wave design as an antireflection temporal coating. Our results, validated via full-wave numerical simulations, provide new insights and deeper understanding of the wave dynamics in time-varying media, and may find important applications in space-time metastructures for broadband frequency conversion and analog signal processing.

Characteristics of Nonstatic Quantum Light Waves: The Principle for Wave Expansion and Collapse

Nonstatic quantum light waves arise in time-varying media in general. However, from a recent report, it turned out that nonstatic waves can also appear in a static environment where the electromagnetic parameters of the medium do not vary in time. Such waves in Fock states exhibit a belly and a node in turn periodically in the graphic of their evolution. This is due to the wave expansion and collapse in quadrature space, which manifest a unique nonstaticity of the wave. The principle for wave expansion and collapse is elucidated from rigorous analyses for the basic nonstatic waves which are dissipative and amplifying ones. The outcome of wave nonstaticity can be interpreted in terms of the coefficient of the quadratic exponent in the exponential function appearing in the wave eigenfunction; if the imaginary part of the coefficient is positive, the wave expands, whereas the wave collapses when it is negative. Using this principle, we further analyze novel nonstatic properties of light waves which exhibit complicated time behaviors, i.e., for the case that the waves not only undergo the periodical change of nodes and bellies but their envelopes exhibit gradual dissipation/expansion as well.

Scalable non-invasive imaging through dynamic scattering media at low photon flux

When the intensity of light reduces to single-photon level, the shot noise becomes dominant. Random scatter, especially the time-varying media will highly increase the photon-limited imaging challenge that the traditional imaging means are inadequate to cope with. In this paper, we develop a new scalable “one to all” imaging approach which focuses on dynamic scattering imaging under photon-limited condition. When the dynamic media optical density is five, we apply this principle to inverse scattering problem and retrieve high-quality images in a real-time way using as little as ∼0.4 valid detected photons per pixel on average successfully. The effects of lighting intensities and perturbations are analyzed to highlight special significance of the work. The ultimate results demonstrate that one trained strategy is robust to a wide range of statistical variations in dynamic media, which outperforms common “one to one” method and improves the practical utility efficaciously. This non-invasive imaging work promises a wide prospect in photon-limited scattering applications such as in vivo bioimaging and so on.

Electromagnetic Fields in a Time-Varying Medium: Exceptional Points and Operator Symmetries

In this article, we study the interactions of electromagnetic waves with a non-dispersive dynamic medium that is temporally dependent. Electromagnetic fields under material time-modulation conserve their momentum but not their energy. We assume a time-variation of permittivity, permeability, and conductivity, and derive the appropriate time-domain solutions based on the causality state at a past observation time. We formulate a time-transitioning state matrix and connect the unusual energy transitions of electromagnetic fields in time-varying media with the exceptional point theory. This state-matrix approach allows us to analyze further the electromagnetic waves in terms of parity and time-reversal symmetries and signify parity-time symmetric wave-states without the presence of a spatially symmetric distribution of gain and loss, or any inhomogeneities and material periodicity. This article provides a useful arsenal to study electromagnetic wave phenomena under time-varying media and points out novel physical insights connecting the resulting energy transitions and electromagnetic modes with exceptional point physics and operator symmetries.

Non-reciprocity using quadrature-phase time-varying slab resonators

In this paper, it is shown that non-reciprocity can be observed in time-varying media without employing spatiotemporal modulated permittivities. We show that by using only two one-dimensional Fabry–Perot slabs with time-periodic permittivities having quadrature-phase difference, it is possible to achieve considerable non-reciprocity in transmission at the incidence frequency. To analyze such a scenario, generalized transfer matrices are introduced to find the wave amplitudes of all harmonics in all space. The results are verified by in-house finite-difference time-domain simulations. Moreover, in order to have a simple model of such time-varying slab resonators, a time-perturbed coupled-mode theory is developed for multiple resonances, and it is shown that the results obtained by this method and the analytical method are in excellent agreement.

Numerical Treatment for Electromagnetic Wave in Time-Variant Medium Using Generalized PITD Method

In this letter, the propagation of an electromagnetic (EM) wave in a time-variant medium is treated with a novel methodology stemming from the precise integration time domain (PITD) method. This methodology is a generalized version of the conventional PITD method (thus GPITD, for short). In the GPITD method, the electric displacement and the magnetic induction serve as the iteratively calculated components. In order to overcome the difficulty in implementing the precise integration (PI) routine caused by the time variation of the original coefficient matrix, an effective piecewise constant coefficient matrix is constructed. Both the numerical dispersion relation and the numerical stability condition of the GPITD method are described. Successful application of the GPITD method to solve the EM propagation problem associated with a layer whose permittivity suffers: 1) an abrupt change to a new value and 2) a sinusoidal modulation, confirms its effectiveness.