Time domain radiation and absorption by subwavelength sources
aa r X i v : . [ phy s i c s . c l a ss - ph ] D ec Time domain radiation and absorption by subwavelength sources
E. Bossy and R. Carminati ∗ Institut Langevin, ESPCI ParisTech, CNRS, 10 rue Vauquelin, 75231 Paris Cedex 05, France
Radiation by elementary sources is a basic problem in wave physics. We show that the time-domain energy flux radiated from electromagnetic and acoustic subwalength sources exhibits re-markable features. In particular, a subtle trade-off between source emission and absorption underliesthe mechanism of radiation. This behavior should be observed for any kind of classical waves, thushaving broad potential implications. We discuss the implication for subwavelength focusing by timereversal with active sources.
PACS numbers: 42.25.-p, 43.20.+g, 41.20.Jb, 03.50.-z
Any textbook on wave physics or field theory containsa chapter on radiation by elementary sources in homo-geneous media [1]. Frequency-domain analyses of theradiated fields and the associated energy fluxes are themost widespread. In these approaches, the far-field en-ergy flux is usually defined as the contribution that sur-vives time averaging, corresponding to power that contin-uously leaks away from the source. Conversely, radiatednear fields generate oscillating terms in the energy flux,that are discarded in the time-averaging process. Time-domain expressions of radiated fields are also common inthe context of electromagnetic radiation [2–5], includingthe optical regime [6], and in acoustics [7–9]. Neverthe-less, time-domain expression of the energy flux have beengiven much less consideration. In the case of electromag-netic waves, the time-domain energy flux radiated froman electric dipole at rest may be found in some textbooks(see Ref. [4] for instance). Its expression is also at thecore of interesting studies of the time decay of classicaloscillating dipoles [10, 11], but that do not describe thefull contribution of the near-field terms that is discussedin the present study. In most textbooks, the discussion islimited to harmonic oscillations and time averages, sincethe focus is usually on far-field radiation [3, 6]. In acous-tics, although time-domain expressions of the fields radi-ated by monopole or dipole sources are widespread [7–9],we are not aware of any discussion of the time-domain en-ergy flux, and in particular of its near-field and far-fieldcomponents.In this Letter, we revisit the basic problem of radiationby elementary subwavelength sources, from the point ofview of emission and absorption of energy in the time do-main. Considering time-domain expressions of the energyflux for the acoustic monopole and the electromagneticdipole, and analyzing carefully the energy balance, weshow that there is a subtle trade-off between emission ofenergy and subsequent reabsorption by the source, thedifference between emission and reabsorption giving theamount of energy that is irreversibly radiated to the farfield. This result reveals some important features of thedynamic interchange of energy between a subwavelength ∗ Electronic address: [email protected] source and a wavefield, that have not been discussed sofar, to the best of our knowledge. It also suggests anovel point of view on near-field radiation. Since theconclusions hold for both acoustic and electromagneticwaves (with striking similarities), they underline a be-havior that should be found with any kind of classicalwaves, thus having broad implications. We illustrate animplication in the context of subwavelength focusing us-ing time reversal with active sources [12, 13].The propagation of electromagnetic waves generatedby a spatially localized source in an otherwise homo-geneous medium is described by the following equa-tion [2, 3]1 c ∂ E ∂t ( r , t ) + ∇ × ∇ × E ( r , t ) = S em ( r , t ) (1)where E ( r , t ) is the electric field at point r and time t ,and c is the speed of light in the medium. The sourceterm S em ( r , t ) is often written in the form S em ( r , t ) = − µ ( ∂/∂t ) j ( r , t ), where j ( r , t ) is the electric current den-sity and µ the vacuum magnetic permeability. Theelectromagnetic energy current is given by the Poynt-ing vector Π ( r , t ) = E ( r , t ) × H ( r , t ), where E ( r , t ) isthe retarded solution of Eq. (1) and H ( r , t ) the associ-ated magnetic field. The energy flux φ em ( R, t ) across asphere with radius R centered at the origin is φ em ( R, t ) = R sphere Π ( r , t ) · u d r , where u = r / | r | .For acoustic waves in the linear regime, the acousticpressure field p ( r , t ) generated by a spatially localizedsource in a homogeneous medium obeys [7, 8]:1 c s ∂ p∂t ( r , t ) − ∇ p ( r , t ) = S ac ( r , t ) (2)where c s is the acoustic velocity in the medium and S ac ( r , t ) the source term. The acoustic energy current is q ( r , t ) = p ( r , t ) v ( r , t ), p ( r , t ) being the retarded acousticpressure field solution of Eq. (2) and v ( r , t ) the associ-ated acoustic velocity field. The energy flux follows from φ ac ( R, t ) = R sphere q ( r , t ) · u d r .In this Letter we study the radiation produced bysources of size much smaller than the characteristiclength scale of the wavefield, that will be denoted by“subwavelength sources”. In the case of electromagneticwaves, we use a point electric dipole model, with dipolemoment p ( t ) = f ( t ) p , f ( t ) being the dimensionlesstime-domain amplitude and p a time-independent vec-tor accounting for the source polarization. This modeldescribes, e.g., a dipole moment p ( t ) = q e L ( t ) corre-sponding to an oscillating charge q e with oscillation am-plitude L ( t ) much smaller than all other relevant charac-teristic lengths [3]. For a dipole centered at r = 0, theelectromagnetic source term reads: S em ( r , t ) = − µ d p ( t )d t δ ( r ) (3)where δ ( r ) is the three-dimensional Dirac delta function.In the case of acoustic waves, we use a point mass sourcemodel describing a radially oscillating sphere with radius a ( t ) = a + ξ ( t ), in the limit of of vanishingly small ra-dius [8]. For a source centered at r = 0, the acousticsource term reads: S ac ( r , t ) = ρ s d ξ ( t )d t δ ( r ) (4)where ρ is the mass density of the unperturbed homo-geneous medium and s = 4 πa . For the sake of formalsimilarity with the electromagnetic case, we will write ξ ( t ) = f ( t ) ξ with ξ a time-independent length drivingthe acoustic source strength.The time-domain solutions of Eqs. (1) and (2) with thesource terms given by Eqs. (3) and (4) can be found intextbooks on electromagnetic and acoustic waves propa-gation [2–4, 7, 8]. From the field expressions, the energyflux across a sphere with radius R can be deduced af-ter tedious but straightforward algebra. In the case ofelectromagnetic waves, one obtains: φ em ( R, t ) = µ p π c (cid:26) (cid:16) cR (cid:17) (cid:20) d f d t (cid:21) + 12 (cid:16) cR (cid:17) (cid:20) d f d t (cid:21) + (cid:16) cR (cid:17) " dd t (cid:18) d f d t (cid:19) + (cid:20) d f d t (cid:21) ) . (5)For acoustic waves, the explicit calculation of the energyflux leads to: φ ac ( R, t ) = ρ s ξ π c s ( (cid:16) c s R (cid:17) " dd t (cid:18) d f d t (cid:19) + (cid:20) d f d t (cid:21) ) . (6)In Eqs. (5) and (6) all terms within square brackets [ ... ]denote retarded values, and have to be evaluated at time t − R/c (electromagnetic waves) or t − R/c s (acousticwaves). Although their derivation is a rather simple ex-ercise, we will see that these expressions bring to lightfundamental aspects of the mechanism of radiation bysubwavelength sources that have not been discussed sofar.¿From a qualitative point of view, the structure ofEqs. (5) and (6) deserves several comments. The far-field limit, obtained for R → ∞ , leads in both cases toan energy flux proportional to the square of the second derivative of the source amplitude, in agreement with awell-established result in classical wave theory [1]. For amonochromatic source oscillating at a frequency ω , with f ( t ) = sin( ωt ), this far-field term is the only one thatsurvives a time-averaging of Eqs. (5) and (6). The far-field behavior is extensively discussed in textbooks, bothfor monochromatic and pulse sources. Nevertheless thetime-domain electromagnetic and acoustic energy fluxescontain additional near-field terms whose amplitude de-pend on the distance R to the source. The first near-fieldterm scales as R − and is identical in Eqs. (5) and (6),except for a factor of two, while additional terms scal-ing as R − and R − appear only in the expression forthe electromagnetic case. These near-field contributionsexhibit remarkable properties that induce specific behav-iors of the time-domain energy flux. A first result is thatthe time-dependent amplitudes of the near-field terms inEqs. (5) and (6) read as first-order derivatives of func-tions that are positive (squares) and that recover theirinitial values after a finite time interval (the pulse dura-tion, or the period for monochromatic excitation). As aresult, these amplitudes necessarily change sign duringtheir time evolution, meaning that the near-field termslead alternatively to outgoing or incoming contributionsto the energy flux. Conversely, the far-field term onlycontributes to an outgoing energy flux. While this seemsto be a commonly accepted result in the harmonic regime(for electromagnetic waves, it is known that the Poyntingvector in the near field changes sign during one cycle ofoscillation), the above result precisely demonstrates thatthe change of sign in the near-field energy flux also ex-ists for a pulsed source with finite duration (i.e., with anamplitude starting from zero and vanishing after a finitetime).In order to study the behavior of the time-domain en-ergy flux on a quantitative basis, we need to specify thesource amplitude function f ( t ). In the present work, weconsider pulses with two requirements. First, f ( t ) has tobe of strictly finite duration (denoted as T in the follow-ing), in order to define exactly a pulse onset ( t = 0)and a pulse end ( t = T ). Second, f ( t ) and its timederivatives have to vanish continuously to zero at t = 0and t = T , in order to avoid temporal singularities inthe energy flux, as made clear from Eqs. (5) and (6).A broadband pulse matching these two requirement isfor instance f ( t ) = exp[2 T / ( t ( t − T ))] for t ∈ ]0 , T [ and f ( t ) = 0 otherwise. The temporal shape of the sourceamplitude f ( t ) and the shape of the associated far fieldamplitude are shown in Fig. 1. For such a broadbandpulse, the period is on the order of the duration. Moreprecisely, for the function f ( t ) given above, the period isclose to half the duration, and the corresponding wave-length is λ = cT / c and c s are referred to as c .The knowledge of f ( t ) and its derivatives allows us toplot the time evolution of φ em ( R, t ) and φ ac ( R, t ) for dif- source term f ( t ) time t 0 Ttime t far field amplitude d f d t ( t ) FIG. 1: Time evolution of the source amplitude f ( t ) (left)and of its second derivative d f ( t ) / d t (right). The latterrepresents the time-dependence of the far-field amplitude forboth electromagnetic and acoustic waves. ferent observation distances, covering the near-field, theintermediate and the far-field regimes. We show in Fig. 2the time evolution of the energy flux in the electromag-netic (top) and acoustic (bottom) situations, and for fourdifferent distances. In the far field ( R ≫ λ ), the energyflux is always positive and describes the radiated energyflowing irreversibly from the source. In the near field( R ≪ λ ), a completely different behavior is observed.The energy flux oscillates, and takes negative values onsome time intervals. This means that part of the energythat has flowed outside the sphere of radius R at a giventime flows back into the sphere at subsequent times.At this stage, conservation of energy states that a neg-ative energy flux corresponds to an increase of energystored inside the sphere with radius R , or to reabsorp-tion into the source (or both). In order to quantitativelysettle this point, we introduce U x ( R, t ) defined as the en-ergy stored outside the sphere with radius R at time t inthe electromagnetic or acoustic field (the subscript “ x ”stands for em or ac ). It reads: U x ( R, t ) = Z t φ x ( R, t ′ ) d t ′ . (7)The time evolution of U em ( R, t ) is shown in Fig. 3 forthe same distance regimes as in Fig. 2. Although notshown for the sake of brevity, the same behavior is ob-served for acoustic waves. As expected from the changesin sign of the energy flux, we see that U em ( R, t ) is not amonotonic function of time except in the far field. Thisnon-monotonic behavior of the time evolution of the en-ergy stored in the field can be characterized by splitting U x ( R, t ) into U x ( R, t ) = U ∞ x + ∆ U x ( R, t ). The first term U ∞ x = R ∞ φ x ( R, t ) d t corresponds to the overall time-averaged energy eventually radiated irreversibly throughthe sphere of radius R to the far field, and is independentof R . The second term describes the time variations ofthe energy stored in the field beyond the distance R , andeither increases or decreases U x ( R, t ) with respect to theasymptotic value U ∞ x . This dynamic behavior is fullydescribed by the curves in Fig. 3. One clearly sees thatat some time range, for R ≪ λ , the energy stored outside the sphere with radius R exceeds the final energy thatremains in the field after the source has been turned off( t > T ). This proves that part of the energy of the fieldhas been reabsorbed by the source, which constitute themain result of this work. This result, derived here usinga pulse of finite duration and finite energy, remains validfor monochromatic and quasi-monochromatic waves. Itshows without ambiguity that a negative energy flux ob-served in the near field corresponds to reabsorption bythe source.This conclusion puts forward new features of the nearfield. Although it is known that on average, near-fieldterms correspond to non-radiative energy [7, 14, 15], ourwork shows that this non-radiative energy is dynami-cally exchanged between the field and the source, atthe time scale of the main oscillation. This subtle dy-namic process is hidden in the first-place when compu-tations are restricted to time-averaged values. We alsostress that a time-domain analysis reveals behaviors thatcannot be seen in the frequency domain. For example,in near-field optics or acoustics, it is often stated thatsome information is lost in the far field due to the lossof non-radiative components that remain spatially local-ized close to the sources (in the near field zone). With anon-stationary source, one could question what happensafter the source has been turned off. Is the field finallyradiated into the far field, and if so, where is the loss ofinformation? Our work provides an unexpected answer:in the near field, some energy is constantly dynamicallyexchanged between the field and the source, and even-tually most of it is absorbed by the source while onlya small part is radiated into the far field. The discus-sion has been limited in this study to a subwavelengthsource emitting in a homogeneous medium, so that onlynear fields produced by the source itself have been con-sidered. A more general analysis including near fieldsproduced by scattering from subwavelength objects (sec-ondary sources) should also reveal interesting dynamicbehaviors. In particular, understanding, in the time do-main, the concept of non-radiative components that ap-pear in the frequency-domain angular spectrum decom-position of scattered fields [16] would be another stepforward. This is left for future work.It is also interesting to have a look at the distancedependence in the near field of the maximum value of theenergy stored in the field ∆ U maxx ( R ) = max { ∆ U x ( R, t ) } .Conserving only the dominant terms as R → U maxem ( R ) ∼ R − and∆ U maxac ( R ) ∼ R − . Therefore for a quasi point sourcemodel, the energy transiently stored in the field becomesarbitrarily large at short distance. In practice, the energymust be limited somehow by the limitations on the sourcemodel itself. Another peculiar behavior, observable onlywith broadband pulses of strictly finite duration, is thatthe energy flux exhibits a slight sign inversion even attimes t > T , i.e., after the source has become inactive(see the insets in Fig. 2 and 3). This sign inversion doestherefore not correspond to reabsorption in the source −1−0.500.51 R ≫ λ −1−0.500.51 R = 2 λ −202 x 10 −5 −2−1012 R = λ /5 R = λ /50 −10 −3 FIG. 2: Time evolution of the electromagnetic and acoustic energy flux φ em ( R, t ) (top row) and φ ac ( R, t ) (bottom row) for fourdifferent distance regimes. Far-field regime R ≫ λ , limit of the source free regime R = 2 λ (2 λ = cT in this particular case),near-field regimes R < λ and R ≪ λ . For R = 2 λ , the insets show the sign inversion of the energy flux. R/c T+R/c00.51 time t R ≫ λ R = 2 λ R = λ /20∆ U maxem ( R ) R = λ /50 time t/T ∆ U maxem ( R ) FIG. 3: Time evolution of the electromagnetic energy U em ( R, t ) stored outside the sphere of radius R at time t , for the samedistance regimes as in Fig. 2. The inset shows the dip due to the sign inversion of the energy flux. in this case, but to a small part of the energy flowingback and forth through the sphere of radius R . This“anomaly” becomes insignificant (although non strictlyzero) in the far field since it is due to the contribution ofterms in the energy flux that decay as R − or faster.To our knowledge, the near-field contributions in thetime-domain energy flux have been first discussed in op-tics by Mandel [10], in the context of the decay rate ofa classical electric dipole in vacuum. The discussion wasconstrained by the fact that for a freely decaying atomicdipole, “the total field energy could not exceed the max-imum amount of energy of the dipole that ultimatelyemerges by radiation” [10]. A major difference with thepresent work is that Mandel’s approach considered thetime variation of the envelope of the emitted wavefield,but terms varying in time at the scale of the optical pe-riod were discarded. In a more recent study, Schantzconsidered the time variations of the energy flux emittedby a decaying electric dipole keeping all time-dependentterms, with an initial condition corresponding to an elec-trostatic dipole [11]. He concluded that the eventually radiated energy had to correspond to electrostatic energyinitially stored in the far field. Although it is out of thescope of this Letter to further discuss this unexpectedand interesting result, we point out that the situationstudied by Schantz is very different from that consideredhere. Indeed, we considered as a fundamental assump-tion the case of a medium initially free of energy, with asource amplitude starting exactly from zero, and vanish-ing rigorously after a finite time (as opposed to an initialnon-zero static field).The results presented in this Letter were derived inthe case of electromagnetic and acoustic radiation, butthey certainly underline general behaviors that shouldbe found for any classical subwavelength sources. There-fore, the peculiar dynamics of the energy exchange be-tween a subwavelength source and the radiated field haspotentially broad implications. Here, we discuss one im-portant consequence in the context of subwavelength fo-cusing by time reversal. Experimental realizations oftime-reversed wavefields have been demonstrated both inacoustics and electromagnetism, by use of close 2D or 3Dcavities [13, 17, 18]. When the field emitted by a point-like source is time-reversed in the source-free medium,refocusing is limited by diffraction [17]. However, whenboth the wavefield and the source are time reversed, per-fect refocusing can be obtained [12, 16]. Accordingly, ex-periments in acoustics have demonstrated subwavelengthrefocusing with an active time-reversed source [13], thefocal spot size being limited only by the finite size of thesource itself. Intuitively, the role of the time-reversedsource is seen as that of a sink, i.e., of an absorber ofthe incoming time-reversed wave. Our work shows thatthe role of the time-reversed source is more subtle, andthat it necessarily involves both absorption and emissionof energy. Indeed, the time-domain evolution of the fieldenergy in a perfect time-reversal experiment (with re-versed field and source) is directly given by the curves inFig. 3 read backwards. Therefore, the energy in the fieldis transiently larger than the energy carried by the time-reversed wavefield, so that in some time range, the sinkactually behaves as a source. The time-reversed sourceis both an absorber and an emitter. The term ”sink”therefore only makes sense when one considers the over-all energy balance, obtained after time integration. Ourwork has two important consequences for practical ex-periments. First, the focusing performances cannot bediscussed without considering the energy point of view,in particular because for a sink of vanishingly small size,the transient energy that has to be stored in the field be- comes arbitrarily large. Second, perfect subwavelengthrefocusing (i.e., without energy scattered away from thefocal spot) cannot be achieved by use of a passive sub-wavelength absorber, as efficient as it may be, since thedynamic exchange of energy is a necessary condition fora localized absorption of the full energy of the wavefield.In summary, from the study of time-domain expres-sion of the energy flux radiated by pulsed electromag-netic and acoustic elementary sources, we have shownthat the non-radiative energy predominant in the near-field is dynamically exchanged between the source andthe field. We have discussed implications for subwave-length focusing and imaging. Since the results hold forboth electromagnetic and acoustic waves, we believe thatthey underly a universal process of radiation by any kindof subwavelength sources, although demonstrated hereonly for the acoustic monopole and the electromagneticdipole. In the case of electromagnetic waves emitted bya single classical dipole emitter, a giant transient storageof electromagnetic energy is necessary in order to radiatea (much smaller part) in the far field. It would be in-teresting to clarify the way quantum theory handles thispoint in the computation of spontaneous emission by asingle atom.We acknowledge A.C. Boccara, J.J. S´aenz and A. Sen-tenac for helpful discussions. This work was supportedby the Agence Nationale de la Recherche (grant JCJC07-195015). [1] L.D. Landau and E.M. Lifshitz, The Classical Theory ofFields (Pergamon Press, Oxford, 1975).[2] J.A. Stratton
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