All-linear time reversal by a dynamic artificial crystal

Abstract

The time reversal of pulsed signals or propagating wave packets has long been recognized to have profound scientific and technological significance. Until now, all experimentally verified time-reversal mechanisms have been reliant upon nonlinear phenomena such as four-wave mixing. In this paper, we report the experimental realization of all-linear time reversal. The time-reversal mechanism we propose is based on the dynamic control of an artificial crystal structure, and is demonstrated in a spin-wave system using a dynamic magnonic crystal. The crystal is switched from an homogeneous state to one in which its properties vary with spatial period a, while a propagating wave packet is inside. As a result, a linear coupling between wave components with wave vectors k[approximate]π/a and k'=k-2ππ/a[approximate]-π/a is produced, which leads to spectral inversion, and thus to the formation of a time-reversed wave packet. The reversal mechanism is entirely general and so applicable to artificial crystal systems of any physical nature.

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Received 17 Aug 2010 | Accepted 17 Nov 2010 | Published 14 Dec 2010 DOI: 10.1038/ncomms1142

Andrii V. Chumak1, Vasil S. Tiberkevich2, Alexy D. Karenowska3, Alexander A. Serga1, John F. Gregg3, Andrei N. Slavin2 & Burkard Hillebrands1

The time reversal of pulsed signals or propagating wave packets has long been recognized to have profound scientic and technological signicance. Until now, all experimentally veried time-reversal mechanisms have been reliant upon nonlinear phenomena such as four-wave mixing. In this paper, we report the experimental realization of all-linear time reversal. The time-reversal mechanism we propose is based on the dynamic control of an articial crystal structure, and is demonstrated in a spin-wave system using a dynamic magnonic crystal. The crystal is switched from an homogeneous state to one in which its properties vary with spatial period a, while a propagating wave packet is inside. As a result, a linear coupling between wave components with wave vectors k/a and k = k 2/a/a is produced, which leads to spectral inversion, and thus to the formation of a time-reversed wave packet. The reversal mechanism is entirely general and so applicable to articial crystal systems of any physical nature.

All-linear time reversal by a dynamic articial crystal

1 Fachbereich Physik and Forschungszentrum OPTIMAS, Technische Universitt Kaiserslautern, Kaiserslautern 67663, Germany. 2 Department of Physics, Oakland University, Rochester, Michigan 48309, USA. 3 Department of Physics, Clarendon Laboratory, University of Oxford, Oxford OX1 3PU, UK. Correspondence and requests for materials should be addressed to A.V.C. (email: [email protected]).

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Mathematically, conversion between a time-dependent process f(t) and a time-reversed version of this process f( t) is a matter of trivial substitution; but the physical realization of such a

transformation has profound signicance. Not only are time-reversal mechanisms immensely powerful technological tools but their existence challenges our understanding of such fundamental notions as thermodynamic irreversibility, causality, and the perceived arrow of time1.

Articial crystals are wave transmission systems which belong to a category of man-made media known as metamaterials. A meta-material is a synthetic material that has properties derived from an engineered mesoscopic (rather than atomic scale, or molecular) structuring. The dening feature of an articial crystal is an articial lattice formed by wavelength-scale modulations in one or more of its material properties. Prominent examples include photonic213 and

magnonic1420 crystals; the former are optical systems and the latter operate in conjunction with spin waves (the propagating or standing magnetic oscillationsthe quanta of which are known as magnons which can be observed in ordered magnetic materials such as ferromagnets). The transmission spectra of articial crystals abound with interesting features, most notably bandgaps: frequency intervals over which wave propagation is prohibited. The physical origin of these bandgaps lies in resonant wave-lattice interactions analogous to the atomic-scale Bragg scattering processes encountered in the context of X-ray or electron crystallography of natural crystals21.

Research in the eld of articial crystals began with the development of static systems; crystals having a permanent articial lattice dened at the time of fabrication. Recently, however, a new class of dynamic articial crystals has emerged. A dynamic articial crystal has a lattice with properties which can be modied while a wave packet propagates inside it715. It has been suggested that such systemswhich oer unique opportunities for the manipulation of propagating wavesmay provide linear means to perform spectral transformations, including time reversal2,3,12,22,23, which, until now,

have only been possible through nonlinear mechanisms2225. Several linear methods of time reversal in dynamic photonic crystals based on, for example, Bloch oscillations10 and refractive index-or loss-tuning of coupled cavity waveguides2,3 have been proposed

theoretically, but none has been observed experimentally.

The results we report in this paper constitute both the rst demonstration of the spectral transformation of a waveform of nite spectral width using a dynamic articial crystal, and the rst experimental realization of all-linear time reversal of a propagating wave packet. The time-reversal mechanism we proposewhich is a principle of universal application to waves of any natureis predicated on the manipulation of a dynamic articial crystal structure on a timescale comparable with the temporal characteristics of a wave packet travelling through it. Its underlying physical basis is a linear non-stationary process, making it entirely distinct from known nonlinear mechanisms of time reversal (for example, optical four-wave mixing2224).

We demonstrate our proposed time-reversal mechanism experimentally using spin waves in a dynamic magnonic crystal (DMC)14 (Fig. 1a). The use of spin-wave systems as model environments for the study of general wave and quasi-particle phenomena is well established2531 and, indeed, in this context, magnonic crystals are increasingly recognized as an important route to fundamental understanding of wave dynamics in metamaterials1420. The

process of time reversing a signal is equivalent, in the frequency domain, to an inversion of its spectrum about a certain reference frequency2. Accordingly, for clarity, we reveal our time-reversal mechanism through two separate experiments. The rst demonstrates frequency inversion of quasi-monochromatic signals and the second demonstrates time reversal of complex waveforms.

ResultsExperimental DMC system. A thin-lm Yttrium Iron Garnet (YIG)

spin-wave waveguide forms the basis for our one-dimensional DMC

system. The waveguide is biased by a static magnetic eld B0 parallel to the direction of spin-wave propagation (see Methods section for further details and justication). Adjacent to the surface of the YIG is a planar metallic meander structure14,32 with 20 periods of lattice constant a = 300 m (Fig. 1a). If a current is applied to the meander structure, the Oersted eld that results (amplitude B) spatially modulates the waveguides magnetic bias eld and leads to the formation of a magnonic bandgap in its transmission characteristics. Microstrip antennae are positioned at each end of the system: an input antenna to launch a spin-wave signal AS(t) and to detect the signal AR(t) reected by the DMC, and an output antenna to receive the signal AT(t) transmitted through it.

Figure 1b,c provides a quantitative illustration of the dierence between the characteristics of the system with and without a current owing in the DMC meander structure. In the absence of an applied current, the meander structure has negligible inuence on spin-wave transmission through the waveguide (dotted line, Fig. 1c) and the dispersion relationship f(k) (dotted line, Fig. 1b) is continuous. Under these conditions, the DMC is said to be o. Note that the negative slope of the dispersion curve is a practical detail that has no bearing on the results

AS(t)

B0

AR(t)

AT(t)

a

I0(t)

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6,600

6,500

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6,500

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/a

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60

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Figure 1 | Experimental dynamic magnonic crystal (DMC) system. (a) The DMC comprises a planar current-carrying meander structure with 20 periods of lattice constant a = 300 m (10 shown), positioned closeto the surface of an Yttrium Iron Garnet thin-lm spin-wave waveguide (thickness 5 m, width 2 mm). Two spin-wave antennae are arrangedon the surface of the lm 8 mm apart: one to launch a spin-wave signal, amplitude AS(t) and to detect the signal AR(t) reected by the DMC, anda second to detect the signal AT(t) transmitted through it. (b) Theoretical spin-wave dispersion relationship for the waveguide with no applied current (dotted, light blue) and with a static current of 1 A in the meander structure (solid curve, dark blue). (c) Experimental spin-wave transmission characteristics with no current (dotted, light blue) and with a current of 1 A (solid curve, dark blue) applied to the meander structure. Application of the current leads to the formation of a bandgap of width approximatelyfa = 30 MHz, centred on fa = 6,500 MHz.

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f

we go on to describe (see Methods section for elucidation). The application of a static current of 1 A to the meander structure results in the formation of a magnonic bandgap of width approximately fa = 30 MHz,

centred on the DMCs Bragg resonance frequency fa = 6,500 MHz, which corresponds to the Bragg wave vector ka = /a = 105 rad cm 1. This bandgap manifests itself as a deep rejection band in the experimental spin-wave transmission characteristics (solid line, Fig. 1c) and appears as a discontinuity in the dispersion relationship (Fig. 1b).

Frequency inversion experiments. Frequency inversion was studied in the following manner. With the DMC o, a spin-wave signal packet of carrier frequency fS, duration 200 ns (bandwidth 5 MHz), and amplitude AS(t) was launched into the waveguide. Aer a delay, a rectangular current pulse I0(t) of duration 25 ns was supplied to the meander structure (Fig. 1a). Timing parameters were chosen so that the packet was contained within the DMC at the time of the application of the current pulse and its spectrum lay entirely within the bandgap. The carrier frequency fS of the packet was varied over a range of 15 MHz about the resonance frequency fa = 6,500 MHz while its power was maintained at < 1 mW. The spectrum of the reected wave packet AR(t) was captured using a spectrum analyser.

The spatially modulated magnetic eld created in the DMC by the current pulse I0(t) leads to a strong coupling between (initially independent) spin waves having wave vectors k and k = k 2ka. This coupling is especially efficient for kka, k = k 2ka ka when the frequencies of the coupled waves are close to each other, and lie within the bandgap of the DMC. By analogy with a system of two coupled oscillators33, the DMC-induced coupling leads to a transfer of energy between the initially excited signal wave of wave vector kS

and a reected wave of wave vector

kR = kS 2ka.

Aer a certain time (~25 ns under the conditions of our experiment), a substantial fraction of the energy is transferred from the signal (kS) to the reected (kR) wave. If the current, and therefore the eld modulation, is switched o at this time, removing the coupling, the two waves become once again independent and the reected wave propagates back towards the input antenna.

The reected frequency fR = f(kR) can be determined using a linear Taylor expansion of the dispersion relationship f(k) about kSka. The incident signal frequency is given by fS = f(kS)fa + v(kS ka)/(2), where fa = f(ka) is the centre frequency of the DMC bandgap and v = 2df (ka)/dka is the spin-wave group velocity. Using equation (1) and the inversion symmetry of the dispersion curve, the frequency of the reected wave can be written as fRfa v(kS ka)/(2). Thus, the frequencies of the counter-propagating

incident and reected waves are connected by the simple relation

fR = 2fa fS,

that is, they are inverted with respect to the centre of the bandgap fa.

This process of dynamic reection and linear frequency inversion is illustrated schematically in Figure 2.

Experimental demonstration of frequency inversion in the DMC system is shown in Figure 3. The efficiency of the process is maximal in the case of fS = fa, and decreases symmetrically with positive and negative detuning (Fig. 3a). Figure 3b shows the spectrum of the reected signal as a function of the incident signal carrier frequency fS. The spectral width of the reected packet is of order 5 MHzthe bandwidth of the 200 ns input pulseindicating that no signicant spectral broadening is introduced in the dynamic reection process.

Time-reversal experiments. The frequency inversion described by equation (2) directly implies reversal of the time prole of the signal packet2. This can be seen by writing the time prole of the incident waveform as a Fourier series AS(t)~f exp(i2), where f = f fa

(1)(1)

(2)(2)

f

Inversion

Intensity

k

Intensity

/a

0

k

/a k = 2/a

Figure 2 | Schematic diagram illustrating frequency inversion by the dynamic magnonic crystal. The diagonal lines (green and red) represent the spin-wave dispersion curve. Incident signal waves have positive wave vectors (green section) and those reected by the DMC have negative wave vectors (red section). These two groups of waves are counter propagating. Black dots mark the reference frequency fa lying in the centre of the bandgap and corresponding to the Bragg wave vectors ka = /a.

The green open circle and square illustrate two spectral components of an incident signal waveform. The spatially periodic magnetic modulation of the waveguides magnetic bias eld brought about by the applicationof the current pulse to the DMC meander structure couples these components to corresponding components of a reected waveform (red open circle and square). The difference between the wave vectors of the signal and reected waves is xed by the lattice constant a of the DMC so that the k-spectrum of the reected waveform is uniformly shifted to the left by k = 2/a = 2ka (lower panel). This uniform shift in k-space results in spectral inversion in the frequency domain (right panel). The reference frequency fa (black dots) is not shifted and provides the axis of inversion.

is the frequency detuning of a given spectral component from the bandgap centre frequency fa. In the process of frequency inversion, the sign of the detuning f inverts, and thus the dynamically reected signal has a reversed time prole AR(t)~f exp( i2f t)S( t).

To observe the eect of time reversal experimentally, a train of two input spin-wave pulses with carrier frequency fS = fa = 6,500 MHz, 70 ns duration, and spacing 40 ns was used.

Figure 4a showsfor referencetime proles of input (green) and transmitted (blue) signals detected in the absence of DMC activation. Here, if the rst spin-wave pulse is switched o (middle frame), the rst pulse in the transmitted signal disappears. Similarly, the switching o of the second pulse (lower frame) leads to the disappearance of the second transmitted envelope.

The signal reected by the DMC is shown in Figure 4b (red). When both spin-wave pulses are applied, two corresponding reected pulses are observed (upper frame). Here, however, unlike in the case of the transmitted signal, the switching o of the rst pulse results in the disappearance of the second reected signal (middle frame). The corresponding situation when the second pulse is absentthe disappearance of the rst reected signalis indicated in the lower frame. This behaviour provides direct evidence of time reversal. The relative intensity of the two reected signals in the upper frame of Figure 4b is consistent with the fact that, in the process of dynamic reection, the second reected pulse has a longer propagation time and thusdue to dissipationis attenuated more signicantly than the rst one.

Discussion

In this paper, we have demonstrated that the fast manipulation of an articial crystal structure while a propagating wave packet is inside it leads to a previously unforeseen linear coupling between the wave

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25

Reflected power (a.u.)

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6,500

10

0.015

6,490

5

6,490

6,490 6,500 6,510 6,490 6,500 6,510

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0

Signal frequency (MHz)

Figure 3 | Experimental demonstration of frequency inversion by the dynamic magnonic crystal. (a) When the carrier frequency of the incident spin-wave signal packet is detuned from the resonance value of 6,500 MHz (the bandgap centre frequency), the frequency of the reected signal is inverted about this value (left ordinate axis, red diamonds). The solid red line is the theoretical curve of equation (2). The efciency of the reection process (right ordinate axis, blue circles) is maximum at the resonance condition and decreases symmetrically with detuning. (b) Two-dimensional map of reected signal spectra as a function of the incident signal carrier frequency fS (31 spectra taken at 1 MHz intervals of fS). Dark red is indicative of the highest signal intensities, dark blue the lowest; the intervening colour scale is linear. Frequency inversion is clearly demonstrated by the diagonal stripe of high signal intensity with a negative slope (running from the upper left to the lower right corner of the gure). The weak diagonal stripe with a positive slope (from the lower left to the upper right corner) is due to weak conventional (that is, frequency conserving) reection from inhomogeneities in the spin-wave waveguide.

Applied signal

Transmitted signal

Applied signal

Reflected signal

1

1

0.1

0.1

Signal intensity (a.u.)

Signal intensity (a.u.)

DMC onDMC onDMC on

1

1

0.1

1

1

0.1

0.1

0 100 200 300 400 500

0.1

0 100 200 300 400 500

Time (ns) Time (ns)

Figure 4 | Experimental demonstration of time reversal by the dynamic magnonic crystal. A train of two spin-wave pulses (70 ns wide and spacing 40 ns) with carrier frequency fS = 6,500 MHz was applied to the input antenna. The applied signals shown (green) are the envelopes of those supplied by the microwave generator. (a) The transmitted spin-wave signals with no current applied to the DMC meander structure appear after a delay of ~300 ns (upper frame, blue). When the rst of the two input pulses is switched off, the rst transmitted pulse is absent (middle frame) and the same is true for the second pulse (lower frame). (b) The reected spin-wave signals obtained after a brief interval of magnetic eld modulation within the DMC (shaded area shows the time interval of the current pulse applied to the meander structure). In the case that two spin-wave pulses are applied, two corresponding reected pulses are observed (upper frame, red). When the rst of the two pulses is switched off, the second reected pulse is absent (middle frame), and vice versa (lower frame), conrming time reversal.

packets spectral components. The coupling occurs between spectral components that correspond to counter-propagating waves with wave vectors related by the lattice constant a of the crystal: k/a and k = k 2/a /a and accordingly, gives rise to an inversion of the incident packets spectrum about the centre frequency of the crystals bandgap, and the formation of a time-reversed wave packet.

The work reported here establishes that dynamic phenomena in articial crystals can provide a substitute for the nonlinear mechanisms traditionally used to perform complex spectral transformations such as time reversal. As the overwhelming majority of real materials

exhibit only weak nonlinearity, we suggest that this capability may provide an enabling route towards the development of low-power signal and information processing systems with complex functionalities.

Established time-reversal mechanisms that operate by phase-matched nonlinear mixing typically have a bandwidth which is restricted by that of a resonant enhancement or absorption phenomenon in the signal transmission medium. The linear time-reversal mechanism proposed here operates by virtue of a dynamic geometrical interaction between an incident wave packet and an articial crystal lattice and, accordingly, is subject to no such bandwidth constraint.

Although we stress that the key result presented in this article is an entirely general one, it is nevertheless the case that our experimental results also have signicance as a specic and exciting development in the eld of magnonics. Magnonics26,31, the study of spin-wave propagation phenomena in magnetic lms and nanostructures, is increasingly widely recognized as an important route to technological opportunity in the eld of spintronics30. Spintronics3437, or spin transport electronics, is concerned with the development of functional electronic devices which exploit not just the charge-carrying properties of the electron but also its intrinsic magnetic moment or spin.

One of the foremost aspirations of contemporary spintronics is the development of an integrated spin-information platform combining signal processing, transport, and storage. In this context, magnonics has a great deal of potential; not only are spin waves capable of transporting spin angular momentum but their small group velocities, short wavelengths, and highly tunable dynamic properties make them well suited to the requirements of nanoscale device development2531. The results presented in this article demonstrate that magnonic crystals are capable of providing previously unanticipated spin-information processing functionality.

Methods

DMC system. An Yttrium Iron Garnet (YIG) thin-lm spin-wave waveguide was chosen as the basis for our one-dimensional DMC system. Monocrystalline YIG has the narrowest magnetic resonance linewidth of any known material (typically ~0.04 mT in thin-lm samples), allowing spin-wave propagation over centimetre distances and making it the ideal choice for experimental spin-wave studies. Our waveguide is a strip of YIG lm (of width 2 mm, thickness 5 m, and saturation magnetization approximately 0M0 = 0.175 mT), which was grown by high-temperature liquid-phase epitaxy on a Gallium Gadolinium Garnet substrate (YIG surface normal parallel to the 111 crystallographic axis). The DMC comprises a planar metallic meander structure14,32 with 20 periods of lattice constant a = 300 m, separated from the surface of the waveguide by a 100 m glass spacer layer(Fig. 1a). This spacer layer is used to prevent direct inductive interaction between

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the metallic structure and propagating spin waves in the YIG. The width of the metallic structure and the glass spacer layer is large compared with the width of the waveguide14. A current pulse I0(t) of amplitude ~1 A applied to the meander

structure creates a periodic Oersted eld of amplitude B~1 mT which spatially modulates the waveguides static magnetic bias eld B0 = 180 mT. Two microstrip antennae are positioned 8 mm apart, ~1 mm from each end of the DMC: an input antenna to launch a spin-wave signal AS(t) and to detect the signal AR(t) reected by the DMC, and an output antenna to receive the signal AT(t) transmitted through it.

Several dynamically distinct types of spin waves can propagate in YIG lms. These types are categorized according to the angle between their propagation direction and the magnetic bias eld. The in-plane, on-axis orientation of the static bias eld used in the experiment (Fig. 1a) corresponds to the propagation of backward volume magnetostatic spin waves (BVMSW)26,31. BVMSW feature a dispersion curve f(k) with a negative slope as indicated in Figure 1b, but it is stressed that this practical detail has no bearing on the entirely general result we describe. BVMSW were used in the experimental investigations as they are reciprocal waves and are readily excited and received by inductive microstrip antennae26.

Frequency inversion experiments. Frequency inversion experiments involved launching a spin-wave signal packet of carrier frequency fS, duration 200 ns (bandwidth 5 MHz) and amplitude AS(t) into the waveguide with the DMC o.

The 200 ns packets were produced using an (externally triggered) Agilent 8673B Synthesized Signal Generator (frequency range 226.5 GHz) and applied by meansof a broadband 3 port (input, output, and return) microwave Y-circulator. Aer a delay, a rectangular current pulse I0(t) of duration 25 ns, amplitude ~1 A, and rise/fall time < 5 ns was supplied to the DMC meander structure from an Agilent 8114A High Power Pulse Generator (Fig. 1a) through a matching network. To generate the data of Figure 3, the carrier frequency fS of the input packet was varied over a range of 15 MHz about the bandgap centre frequency fa = 6,500 MHz of the DMC. Timing parameters (all controlled by a single programmable pulse generator) were such that, in all experiments, the packet was contained within the magnonic crystal at the time of the application of the current pulse to the meander structure, and its spectrum lay entirely within the DMC bandgap. Throughout, the input microwave signal power was maintained below 1 mW (two orders of magnitude lower than that known to be required to observe the eects of nonlinear terms in YIGs microwave susceptibility26). The spectrum of the reected wave packet AR(t) was captured by means of the return signal port of the Y-circulator using a spectrum analyser (Agilent E4446A,3 Hz44 GHz). The experiments were conducted with a 500 s repetition rate.

Time-reversal experiments. In our time-reversal experiments, the spin-wave input signal was a train of two pulses with carrier frequency fS = fa = 6,500 MHz, 70 ns duration, and spacing 40 ns. As in the frequency inversion investigation, the 1 mW signal pulses were produced using an externally triggered Agilent 8673B Generator and applied using a broadband 3 port microwave Y-circulator with the DMC o.

A current pulse identical in amplitude and duration to that used in the frequency inversion experiments was used to time reverse the signals. To capture the data of Figure 4, the Y-circulator return port at the spin-wave input antenna (connectedto the spectrum analyser in the frequency inversion experiments) was provided toa digital storage oscilloscope (Agilent InniVision DSO7034A, 350 MHz) throughan amplier and broadband detector diode. A second channel of the oscilloscope was connected to the output antenna through a similar amplication and rectifying circuit. Again, the experiments were conducted with a 500 s repetition rate.

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Acknowledgments

Financial support from the Deutsche Forschungsgemeinscha (Grant no. SE 1771/1), from the ECCS Program of the USA National Science Foundation, and from the U.S. Army TARDEC, RDECOM (Grant no. W56HZW-09-PL564) is gratefully acknowledged. DMCs were fabricated with technical help from the Nano + Bio Center, TU Kaiserslautern. A.D.K. is grateful for the support of Magdalen College, Oxford. A.D.K. and J.F.G. thank Paul Ewart and Andrew Turbereld for helpful discussions.

Author contributions

V.T. proposed the linear time reversal mechanism and developed the theory. A.V.C. and A.A.S. designed the dynamic magnonic crystal. A.V.C., V.T., A.D.K., A.A.S. and B.H. planned the experiments. A.V.C. and A.D.K. took the measurements. All authors analysed the data and participated in the preparation of the manuscript. A.A.S., J.F.G., A.N.S. and B.H. supervised the project.

Additional information

Competing nancial interests: The authors declare no competing nancial interests.

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How to cite this article: Chumak, A. V. et al. All-linear time reversal by a dynamic articial crystal. Nat. Commun. 1:141 doi: 10.1038/ncomms1142 (2010).

License: This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 3.0 Unported License. To view a copy of this license, visit http:// creativecommons.org/licenses/by-nc-sa/3.0/

NATURE COMMUNICATIONS | 1:141 | DOI: 10.1038/ncomms1142 | www.nature.com/naturecommunications

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