An Integrodifferential Equation for Electromagnetic Fields in Linear Dispersive Media
V.A. Coelho, F.S.S. Rosa, Reinaldo de Melo e Souza, C. Farina, M.V. Cougo-Pinto
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An integro-differential equation for electromagnetic fieldsin linear dispersive media
V.A. Coelho · F.S.S. Rosa · Reinaldode Melo e Souza · C. Farina · M.V.Cougo-Pinto
Received: date / Accepted: date
Abstract
We extend the usual derivation of the wave equation from Maxwell’sequations in vacuum to the case of electromagnetic fields in dispersive homo-geneous isotropic linear media. Usually, dispersive properties of materials arestudied in Fourier space. However, it can be rewarding to consider these prop-erties as causal functions of time. Due to temporal non locality, this proceduregives rise to an integro-differential equation for the electromagnetic fields, thatwe also call a wave equation. We have not found this equation in the literatureand we show in this paper why it can be useful.
Keywords
Electromagnetism · Dispersive Wave Equation · Integro-Differential Equation
The behavior of electromagnetic fields in material media is not always intu-itive and depends substantially on the particular properties of each medium.This is due to the fact that every real medium presents dispersion, both intime and in space, and the amount of dispersion can drastically change howa medium responds to electromagnetic fields. Throughout history, dispersionhas fascinated many scientists and it was carefully studied by giants like New-ton, Cauchy, Helmholtz, Lorentz, Sommerfeld and Brillouin ([1,2,3]). In fact,Lorentz’s atomistic approach and Sommerfeld’s mathematical treatments forwave propagation in dispersive media still constitute the basis for modern ap-proaches ([4]). In spite of its long history, many important aspects of dispersive
V.A. Coelho, F.S.S. Rosa, C. Farina and M.V. Cougo-PintoInstituto de F´ısica, UFRJ, CP 68528, Rio de Janeiro, RJ 21941-972E-mail: [email protected] de Melo e SouzaInstituto de F´ısica, UFF, Niter´oi, RJ 24210-346E-mail: [email protected]ff.br V.A. Coelho et al. media are still at the forefront of research. Amongst many contributions, wecan mention the analysis of, the angular momentum of optical fields and theelectromagnetic helicity in these media ([5,6]), the investigation of the verydefinition of light momentum (in the context of the hundred year old Abraham-Minkowski controversy, see [7] and references therein), and the new resultson polariton excitations in dispersive media ([8,9]). Since Sommerfeld, mosttreatments of electromagnetic wave propagation are given in Fourier space, incontrast to what is usually done in (non-dispersive) free space. The advantagesof formulating the problem in Fourier space are twofold: (i) For most materialsand not very intense fields, the constitutive equations are linear and (ii) thedispersion relation relates frequency and wavector for each Fourier componentof the field. However, there are also some shortcomings. Often, one needs theelectromagnetic field in real space and in order to obtain it from its Fouriercomponents it is necessary to perform an integration in frequency domain.The integrand involves the index of refraction n ( ω ) = p ǫ ( ω ) µ ( ω ), which is re-sponsible for branch-points and branch-cuts in the integrand, making a directevaluation of the integral a difficult task([3]). There are several techniques toavoid this problem, as for instance the modal expansion which has received alot of attention recently ([10,11,12]).Another possibility is to work directly in the real space. Furthermore, it isconceptually interesting to develop a formalism which involves only the realvalued electromagnetic fields E ( r , t ) and H ( r , t ) instead of the complex auxil-iary Fourier modes. In this letter we develop one such method, and show thatinstead of the usual wave equation we obtain in free space we are led to anintegro-differential equation. That dispersion leads to integro-differential equa-tions is well-known for general ondulatory phenomena ([13,14,15,16]) and ithas also been demonstrated in some particular instances for electromagnetism([17,18,19]).In this work, we generalize these results and show that an integro-differen-tial equation follows directly from Maxwell’s equations. We emphasize thatsuch a result consists of a unique wave equation that governs the dynamics ofall possible electromagnetic fields in linear dispersive media (in the frequencydomain there is a different equation for each Fourier component), potentiallyleading to a deeper understanding of the dispersive properties of materialmedia. At the end we discuss some particular examples. Time dispersion is directly related to temporal non-locality, i.e., the fact that amedium does not respond instantaneously to a given electromagnetic field. Inthis sense, for instance, the polarization P ( r , t ) depends not only on the electricfield at instant t but also on earlier instants. Analogously, spatial dispersionmeans that the polarization P ( r , t ) is spatially non-local, i.e., depends notonly on the electric field at position r but also at neighboring points. Theconstitutive equation relating the vector D ( r , t ) = ǫ E ( r , t ) + P ( r , t ) (we are itle Suppressed Due to Excessive Length 3 using SI units) and the electric field E ( r , t ) for an isotropic linear dispersivemedium is given by D ( r , t ) = Z dt ′ Z d r ′ g ( r , r ′ ; t, t ′ ) E ( r ′ , t ′ ) , (1)where the electric response function g is a real function, which is non-localin both space and time due to dispersion. By assuming spatial and temporalhomogeneity, we constrain the dependence of the function g to the differences r − r ′ and t − t ′ , and therefore D ( r , t ) = Z dt ′ Z d r ′ g ( r − r ′ ; t − t ′ ) E ( r ′ , t ′ ) , (2)For the sake of simplicity, we additionally assume that the material mediaconsidered in this work do not exhibit spatial dispersion (in fact, for manypurposes spatial dispersion can be neglected in dielectric materials). In thiscase, the response function can be written as g ( r − r ′ ; t − t ′ ) = δ ( r − r ′ ) ǫ ( t − t ′ ),known as the spatial local limit, so that the previous equation reduce to ([20,21]) D ( r , t ) = Z ∞−∞ dt ′ ǫ ( t − t ′ ) E ( r , t ′ ) , (3)where ǫ is a real function of t − t ′ . The overlined notation ǫ is to avoid theuse of ǫ commonly reserved in the literature to denote electric permittivityas a function of frequency. Furthermore, we impose a causality requirementfor ǫ , meaning that it must vanish for τ := t − t ′ < ǫ ( τ ) → τ → ∞ (the remote past canbe neglected) ([20,21]). Finally, it should be said that an analogous equationrelates the magnetic induction B ( r , t ) = µ [ H ( r , t ) + M ( r , t )], where M ( r , t )is the magnetization, to the magnetic field H ( r , t ), namely, B ( r , t ) = Z ∞−∞ dt ′ µ ( t − t ′ ) H ( r , t ′ ) , (4)where the magnetic response function µ is also a real function that satisfiesconditions analogous to those satisfied by the electric response function ǫ .The past interval of time in which the electromagnetic fields have influenceon a specific medium is of the order of its relaxation time [21].Notice that in vacuum we do not have dispersion and thus ǫ ( t − t ′ ) = ǫ δ ( t − t ′ ) and µ ( t − t ′ ) = µ δ ( t − t ′ ), so that the constitutive relations (3) and(4) reduce to the vacuum constitutive relations.In order to derive the wave equation for electromagnetic fields in a temporaldispersive medium, we start by writing Maxwell’s equations for the electro-magnetic fields E , B , D and H in the absence of free charges and currents insuch a medium, namely, ∇ · B ( r , t ) = 0 , ∇ × E ( r , t ) = − ∂ B ( r , t ) ∂t (5) ∇ · D ( r , t ) = 0 , ∇ × H ( r , t ) = ∂ D ( r , t ) ∂t . (6) V.A. Coelho et al.
These equations must be complemented by the two constitutive relations writ-ten in Eq(s) (3) and (4). To obtain the desired wave equation in dispersivemedia we follow essentially the same procedure used to obtain the wave equa-tion in vacuum. A first thing to notice is that since ∇ · D ( r , t ) = 0 then, fromEq.(3), we obtain R t −∞ dt ′ ǫ ( t − t ′ ) ∇ · E ( r , t ′ ) = 0. Since this must be validfor any t we see that ∇ · E ( r , t ) = 0. In a completely analogous way it can beshown that ∇ · B ( r , t ) = implies ∇ · H ( r , t ) = . Taking the curl of bothsides of the second equation in (5) and using the constitutive equation (4), weobtain ∇ E ( r , t ) = ∂∂t ∇ × Z ∞−∞ dt ′ µ ( t − t ′ ) H ( r , t ′ )= Z ∞−∞ dt ′ ∂µ ( t − t ′ ) ∂t ∂ D ( r , t ′ ) ∂t ′ = Z ∞−∞ dt ′ Z ∞−∞ dt ′′ ∂µ ( t − t ′ ) ∂t ∂ǫ ( t ′ − t ′′ ) ∂t ′ E ( r , t ′′ ) , (7)where in the last step we used the constitutive equation (3). Switching theorder of integration in this equation and using the identity ∂µ ( t − t ′ ) /∂t = − ∂µ ( t − t ′ ) /∂t ′ , we get ∇ E ( r , t ) = − Z ∞−∞ dt ′′ (cid:20)Z ∞−∞ dt ′ ∂µ ( t − t ′ ) ∂t ′ ∂ǫ ( t ′ − t ′′ ) ∂t ′ (cid:21) E ( r , t ′′ )= Z ∞−∞ dt ′′ Z ∞−∞ dt ′ µ ( t − t ′ ) ∂ ǫ ( t ′ − t ′′ ) ∂t ′′ E ( r , t ′′ ) . (8)where in the last step we made an integration by parts in the variable t ′ andused the fact that ∂ǫ ( t ′ − t ′′ ) /∂t ′ = − ∂ǫ ( t ′ − t ′′ ) /∂t ′′ . Integrating by parts twicemore, we finally obtain ∇ E ( r , t ) − Z ∞−∞ dt ′′ (cid:20)Z ∞−∞ dt ′ µ ( t − t ′ ) ǫ ( t ′ − t ′′ ) (cid:21) ∂ E ( r , t ′′ ) ∂t ′′ = . (9)By an entirely analogous method we obtain for the magnetic field ∇ H ( r , t ) − Z ∞−∞ dt ′′ (cid:20)Z ∞−∞ dt ′ ǫ ( t − t ′ ) µ ( t ′ − t ′′ ) (cid:21) ∂ H ( r , t ′′ ) ∂t ′′ = . (10) Equations (9) and (10) are the main results of this work. After consulting a vastlist of graduate and undergraduate textbooks([4],[22]-[36]), we were unable tofind in the literature the above integrodifferential equation. They generalizethe wave equation for the electromagnetic fields in vacuum to the case where Note that we used causality in order to rewrite the limits of integration Note that there are no surface terms, precisely due to our assumptions regarding ǫ ( t − t ′ ).itle Suppressed Due to Excessive Length 5 the fields propagate in an isotropic homogenous linear dispersive medium (withno spatial dispersion). The price to be paid is that now instead of a differentialequation we have to deal with an integrodifferential equation. A few commentsabout these equations are in order here: (i) In first place, as a selfconsistencycheck, the above integrodifferential equations reduce to the usual differentialwave equations in vacuum in the local limit where ǫ ( t − t ′ ) → ǫ δ ( t − t ′ )and µ ( t − t ′ ) → µ δ ( t − t ′ ), as expected. (ii) Note that wave equations (9)and (10) are linear in the electromagnetic fields so that linear superpositionsof their solutions are also solutions of these equations. At this point, it isimportant to emphasize that the Fourier components of the fields, E ( r , ω ) = R ∞−∞ E ( r , t ) e iωt dt , satisfy the well-known Helmholtz equation ∇ E ( r ) + ω µ ( ω ) ǫ ( ω ) E ( r ) = , (11)as can be straightforwardly verified. In the above expression ǫ ( ω ) and µ ( ω )are the Fourier transforms of ǫ ( t − t ′ ) and µ ( t − t ′ ) respectively. An analogousresult holds for the Fourier components of the magnetic field. In this way, theinfinite set of differential (Helmholtz) equations, one for each frequency, areencoded in a single integrodifferential wave equation. Actually, the integrod-ifferential equations (9) and (10) are satisfied by any electromagnetic fieldswhich are solutions of Maxwell’s equations (5)-(6) in isotropic homogeneouslinear dispersive media without free charges and currents.One important situation is when dispersion effects are weak, i.e., the re-sponse functions are very narrow in time. For the sake of simplicity let usassume a non-magnetic material, that is µ ( t − t ′ ) = µ δ ( t − t ′ ) and supposethe timescale for a significative variation of the electric field is much greaterthan the characteristic timescales of the the material. In these cases, we canperform a Taylor expansion in the electric field in Eq.(9), namely, ∂ E ( r , t ′ ) ∂t ′ = ∂ E ( r , t ) ∂t + ∂ E ( r , t ) ∂t ( t ′ − t ) + O (cid:16) ( t ′ − t ) (cid:17) , (12)that once inserted into Eq.(9) yields ∇ E ( r , t ) − (cid:20) µ Z ∞−∞ dt ′ ǫ ( t − t ′ ) (cid:21) ∂ E ( r , t ) ∂t ++ (cid:20) µ Z ∞−∞ dt ′ ǫ ( t − t ′ )( t ′ − t ) (cid:21) ∂ E ( r , t ) ∂t = , (13)where we used that µ ( t − t ′ ) = µ δ ( t − t ′ ). This is a partial differential equationwith a third-order time derivative. If we include more and more dispersivecorrections, higher and higher order time derivatives in the electromagneticfields will show up. The above expression can be written in a more appealingform ∇ E ( r , t ) − µ ǫ (0) ∂ E ( r , t ) ∂t + iµ ǫ ′ ( ω ) | ω =0 ∂ E ( r , t ) ∂t = , (14) V.A. Coelho et al. where ǫ (0) corresponds to the zero frequency component of the Fourier trans-form of ǫ ( t − t ′ ) and ǫ ′ ( ω ) = dǫ ( ω ) /dω . This procedure can be easily generalizedto obtain higher order terms and it can be shown that all the coefficients inthe above equation are real. In this work we have studied the dynamics of electromagnetic fields in a disper-sive homogeneous isotropic linear medium directly in the real space (insteadof in the Fourier frequency space). Since dispersion entails a time memory inthe medium response functions, the electromagnetic fields satisfy an integro-differential equation which subsume the infinite set of Helmholtz differentialequations for each Fourier component. We illustrated our equation in two ex-amples, the monochromatic limit and the weak dispersive regime. In the latter,we showed that the effect of dispersion consists in introducing extra terms tothe standard wave equation with higher-order time derivatives of the fields.We believe that this equation may provide new approaches in dealing withelectromagnetic dispersion phenomena.
Acknowledgements
We are indebted to Prof(s) P. A. Maia Neto, C. E. Magalh˜aes deAguiar and F.A. Pinheiro for helpful discussions. The authors also thank Prof. RichardPrice for a constructive criticism on this work. The authors thank the Brazilian agenciesNational Council for Scientific and Technological Development (CNPq) and Carlos ChagasFilho Foundation for Research Support of Rio de Janeiro (FAPERJ) for support.
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