Maxwell Garnett approximation (advanced topics): tutorial · Maxwell Garnett approximation (advanced topics): tutorial ... This is the second part of the tutorial on Maxwell ... is

Maxwell Garnett approximation(advanced topics): tutorialVADIM A. MARKEL1,2

1Aix-Marseille Université, CNRS, Centrale Marseille, Institut Fresnel UMR 7249, 13013 Marseille, France ([email protected])2On leave from Department of Radiology, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA ([email protected])

Received 5 August 2016; revised 16 September 2016; accepted 18 September 2016; posted 6 October 2016 (Doc. ID 273172);published 28 October 2016

In the second part of this tutorial, we consider several advanced topics related to the Maxwell Garnettapproximation. © 2016 Optical Society of America

OCIS codes: (000.1600) Classical and quantum physics; (160.0160) Materials.

http://dx.doi.org/10.1364/JOSAA.33.002237

1. INTRODUCTION

This is the second part of the tutorial on Maxwell Garnettapproximation. The first, introductory part was published in[1], and we will refer to the equations of [1] in the form (FP#),where # is the equation number.

In the second part, several more advanced topics will be dis-cussed. The overall goal is to justify and generalize the MaxwellGarnett approximation and to place it more solidly in the con-text of modern classical electrodynamics of continuous media.In particular, we will take into consideration the effects of finitefrequencies and electromagnetic interaction of inclusions. Wewill also develop a rigorous homogenization theory for periodiccomposites. We will see that the Maxwell Garnett mixingformula is obtained from this result by neglecting a term thatis responsible for the interaction of inclusions.

We will start with laying out in Section 2 a general theoreticalframework of the macroscopic Maxwell’s equations in continu-ous media. This old topic has attracted renewed attentionrecently [2,3]. Traditionally, the derivation of the macroscopicMaxwell’s equations from the microscopic ones was doneprimarily for pedagogical purposes and the averaging proceduretypically involved in such derivations was not used construc-tively. However, the development of modern computationalapproaches to the microscopic theory and, especially, the emer-gence of the density-functional theory (DFT)-based ab initiomethods for computing the microscopic electromagnetic quan-tities have motivated re-examination of some basic assumptions.

One important idea that requires a fresh look is the classicalinterpretation of the polarization and magnetization fields asthe volume densities of the electric and magnetic dipole mo-ments. Ab initio calculations of the microscopic-induced chargedensity in some solids such as crystalline silicon have shownthat this interpretation can be incorrect [4]. In Section 3, wediscuss the range of applicability of the classical interpretation

and identify two main reasons for its breakdown: charge leakagethrough the cell boundaries (in crystalline materials) and accu-mulation of a nonnegligible electromagnetic phase between theneighboring cells. The two causes for the classical interpretationbreakdown become important in different physical situations.Charge leakage can be relevant for natural materials in whichthe elementary cells are microscopic but not for macroscopiccomposites. On the other hand, the phase shift plays almostno role in natural materials but can come to the fore in themacroscopic homogenization theories.

In Section 4, we consider propagation of waves on three-dimensional cubic lattices of polarizable particles. It is shownthat the Clausius–Mossotti relation follows from the law ofdispersion for such waves in the long wavelength approximation.The Maxwell Garnett mixing formula is then obtained from theClausius–Mossotti relation if we assume that the particles aresmall dielectric spheres and use the appropriate expression fortheir polarizabilities. We also discuss briefly what happens be-yond the long wavelength approximation and, in particular, givea few leads to the extended homogenization theories that includedynamic corrections and effects of nonlocality.

A drawback of the point-particle model is that it does notallow one to account for the shape and physical size of the in-clusions in a mathematically consistent manner. In Section 5,we overcome this limitation by considering a general three-dimensional photonic crystal. A rigorous homogenization theoryis obtained in the long wavelength limit. The resulting mixingformula is very similar to Maxwell Garnett’s but contains anadditional term, which is responsible for the electromagneticinteraction of inclusions. This term also allows one to computetensorial effective permittivities of anisotropic composites.

In Section 6, we show several numerical illustrations of therigorous homogenization formula derived in Section 5. Herewe consider various two-component composites consisting of

Tutorial Vol. 33, No. 11 / November 2016 / Journal of the Optical Society of America A 2237

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http://dx.doi.org/10.1364/JOSAA.33.002237

the same materials (a conductor and a dielectric) as were used inthe first part of this tutorial to illustrate the Wiener andBergman–Milton bounds. The results are compared to theMaxwell Garnett’s and Bruggeman’s predictions, and it isshown that the Maxwell Garnett approximation is quite robustfor isotropic composites, albeit we do not consider strong res-onance effects that can render simple mixing formulas grosslyinaccurate. However, in the case of anisotropic composites,Maxwell Garnett’s predictions are much less robust, even inthe absence of strong resonances.

Consistently with the first part of this tutorial, we will usethe Gaussian system of units.

2. MACROSCOPIC MAXWELL’S EQUATIONS INCONTINUOUS MEDIA

The starting point for our exposition is the set of microscopicMaxwell’s equations for the electric and magnetic fields Eand B, respectively, and the electric current J,

∇ × B � 1

c∂E∂t

� 4π

cJ; ∇ × E � −

1

c∂B∂t

; (1)

where c is the speed of light in vacuum. Equation (1) is sup-plemented by the continuity equation

∂ρ∂t

� ∇ · J � 0; (2)

where ρ is the density of electric charge. The two equations∇ · E � 4πρ and ∇ · B � 0 are not independent but followfrom Eqs. (1) and (2).

The fields E and B are fundamental in the electromagnetictheory because they appear in the expression for the Lorentzforce F acting on a point charged particle, viz.,

F � qE� qcv × B: (3)

In classical physics, electric and magnetic fields are measurableonly through the action of the Lorentz force [5].

Some part of the electric current J can be viewed as a knownand externally controlled source. One example is a radiatingantenna, which is fed in a prescribed way. We denote this termby Jext and refer to it as to the external current. The electromag-netic fields created by Jext will exert the force given by Eq. (3)and set the charges of any material object in motion. This willcreate an induced current Jind in that object. The total currenteverywhere in space is then given by

J�r; t� � Jext�r; t� � Jind�r; t�: (4)

In any sensible application of the theory, the external and theinduced currents do not overlap spatially. Consequently,Jext�r; t� does not overlap with the medium. One can considera more general case mathematically, but such models do notdescribe the physical reality.

The induced current will create additional electromagneticfields, which will, in turn, act back on the moving charges. Evenif we can neglect the back-action of the induced fields on theexternal source, this interaction will result in a big unsolvableset of nonlinear equations. To complicate things further, it isknown that any set of charged particles cannot hold itselftogether by the forces of the form given by Eq. (3) if we apply

the Newtonian laws of motion. Therefore, stability of mattercannot be understood from classical first principles.

Naturally, one wishes to develop an electromagnetic theoryof continuous media that avoids these complications. This goalis achieved by the macroscopic Maxwell’s equation. In a typicaltextbook exposition, these equations are obtained by averagingthe microscopic Eq. (1) over a physically small volume. The der-ivation is based on the observation that h∇ × Ei � ∇ × hEiand similarly for other fields, where

hf �r�i � 1

D

ZDf �r� s�d3s; (5)

where D is a small region of space around the origin and D is itsvolume.We then replace all quantities in Eq. (1) by their averages.The nontrivial task in this case is to compute hJindi, which de-scribes the response of the medium to the electromagnetic fields.

Unfortunately, there exist several problems with the averagingapproach. First, for the method to be mathematically sound,we need to make sure that the integrals of the form given byEq. (5) exist for all quantities of interest and are independentof the shape of D. However, the field of a truly point chargeis not integrable and therefore it cannot be averaged. This prob-lem is not rectified by introducing any number of point chargesor by excluding small spherical regions around each charge fromintegration (this does not suppress the dependence on D).The only solution to the divergence problem is to introducea continuous density of charge. In particular, the quantum-mechanical DFT and the classical hydrodynamic model operatewith continuous densities, which are already in some senseaverages. However, these smooth functions are not obtainedby actual averaging of some (generally, unknown) microscopicdensities; they are simply postulated to exist and are computedaccording to some approximations.

Second, the microscopic quantities of interest are very diffi-cult if not impossible to compute in continuous media. Ofcourse, there exist some simple models such as the Lorentz orthe Drude model. However, these models describe adequatelyonly some special cases and, moreover, they do not employaveraging of the field of point charges. In the first part of thistutorial, we have averaged the field of elementary dipoles(in Section 2.B). This is a somewhat better-defined procedurealbeit not without its own mathematical ambiguities, whichwe have encountered while considering anisotropy in Section 4.

We can conclude that the operation of microscopic fieldaveraging is a purely formal procedure; it is not constructivein the sense that it does not yield any useful predictions. Theaveraging is simply postulated for pedagogical reasons and notused for anything. Instead, a phenomenological approach toconstructing the macroscopic theory is implicitly used. Indeed,all we need to “derive” the macroscopic Maxwell’s equations arethe following three postulates:

(i) The medium is a true continuum, possibly, with sharpboundaries or interfaces where the properties can jump abruptly.(ii) The external source excites a continuous density of

induced current Jind (possibly, with singular contributions atthe surfaces of discontinuity), which is a functional of thefundamental electromagnetic fields.(iii) The induced current is zero in vacuum.

2238 Vol. 33, No. 11 / November 2016 / Journal of the Optical Society of America A Tutorial

In the case of local and linear media, we can describe thefunctional dependence of Postulate (ii) by the equation

Jind �∂P∂t

� c∇ ×M; (6)

where

P�r; t� �Z

∞

0

f e�r; τ�E�r; t − τ�dτ; (7a)

M�r; t� �Z

∞

0

f m�r; τ�B�r; t − τ�dτ: (7b)

Here f e�r; τ� and f m�r; τ� are the electric and magnetic influ-ence functions (in general, symmetric tensors), respectively. Wecan say that Eq. (7) is causal and local: for example, J�r; t� de-pends only on the electric field at the same point r (locality) andat the moments of time t 0 ≤ t (causality). Note that accordingto Postulate (iii), f e�r; τ� � f m�r; τ� � P�r; t� � M�r; t� �0 if r is in vacuum.

Now we need to make a few important comments.

(i) Equations (6) and (7) are not the most general form of alinear functional relating the induced current to the fundamen-tal fields. In bi-isotropic and bi-anisotropic media, Eq. (7) isreplaced by a more general relation in which P is coupledto B and M is coupled to E. However, Eqs. (6) and (7) areof the most general form if we restrict consideration to mediathat are completely characterizable by two symmetric tensors ϵand μ (the dielectric permittivity and magnetic permeability,defined below). Physically, our assumptions mean that themedium is reciprocal and nonchiral. There exists a subtle point:the gyrotropic (magneto-optical) effect is described by a non-symmetric tensor ϵ; it may seem that this phenomenon can beformally described by Eqs. (6) and (7). In fact, the off-diagonalelements of ϵ in gyrotropic media are proportional to theamplitude of a stationary magnetic field and Eqs. (6) and (7)do not allow for such a dependence. Thus, our description doesnot include gyrotropy and nonreciprocity caused by a station-ary magnetic field.(ii) Just like the microscopic Maxwell’s equations (1), the

expressions (6) and (7) are invariant under coordinate inver-sion. Recall that the true vectors such as E, J, and P changesign, while the pseudo-vectors B and M are invariant undercoordinate inversion. However, invariance under time inver-sion is lost in the macroscopic theory. Time inversion canbe understood as an instantaneous change of direction ofthe classical velocities of all particles (we do not discuss hereantimatter and other quantum concepts). The fields J and Band the time derivatives such as ∂E∕∂t change sign under timeinversion, while E and ρ are invariant. Invariance under timeinversion is lost in the macroscopic equations because P�t� inEq. (7a) depends on E�t 0� for all moments t 0 such that t 0 < t .If this dependence was local, e.g., P�t� � χeE�t� (andsimilarly for M), the macroscopic equations would havebeen time-reversible. The temporal nonlocality in Eq. (7), alsoknown as temporal or frequency dispersion, is a phenomeno-logical way of accounting for thermodynamically irreversibleprocesses such as absorption.(iii) The familiar equation formalizing Ohm’s law, Jind � σE,where σ is the conductivity, is a special case of Eqs. (6) and (7).Analysis is most straightforward in the frequency domain,

but we can also use the equation ∂P∕∂t � σE to writeP�t� � σ

Rt−∞ E�t 0�dt 0; comparing this to (7a), we find that

f e�τ� � σ, that is, f e�τ� is independent of τ for conductors.We can also write f e�τ� � σΘ�τ�, where Θ�x� is the stepfunction and expand integration in Eq. (7a) to the whole realaxis. Note that Ohm’s law is a prime example of time-reversibility breaking since the fields J and E transform differ-ently under time inversion.(iv) We have used the same letters E, B, etc., to denote themicroscopic and macroscopic electromagnetic fields. We thusbreak off with the tradition to denote the microscopic fieldsby small letters such as e or b. The latter approach usuallyimplies that E � hei, etc. We do not follow the tradition sincewe associate a given notation with a physical quantity it rep-resents rather than with a particular approximation. Thus, Eis the electric field that appears in the expression for theLorentz force Eq. (3), and the macroscopic theory simply offersus a mathematically tractable approximation for computingthis quantity. Moreover, the small-letter notations are not reallyused for anything: as soon as the averaging operation is formallyintroduced, all calculations are performed with the large-letterquantities, including the calculations of physical measurablesthat are quadratic in the fields. These calculations bear no traceof the averaging procedure and, therefore, the existence ornonexistence of the more fundamental small-letter fields is ofno observable consequence. However, we will use below thenotations ϱ and j for the microscopic densities of charge andcurrent. These quantities are introduced in various theoreticalmodels of continuous media.(v) The term c∇ ×M can be viewed as a spatially nonlocal

correction to the purely local term ∂P∕∂t. We can envisageterms with higher-order derivatives or a more general nonlocalfunctional dependence. In this case, the linear relation betweenJind and E becomes a very general form Jind�r; t� �Rd3r 0

R∞jr−r 0 j∕c dτf �r; r 0; τ�E�r 0; t − τ�, where f can have

singularities that describe derivatives of arbitrary order. Thisdescription requires detailed knowledge of the kernelf �r; r 0; τ�. It is often implied that f is a function of the shiftr − r 0, but this is true only sufficiently far from the boundariesand interfaces. If f is known only far from the interfaces (as isusually the case), introduction of the so-called additional boun-dary conditions and additional medium parameters is required.In natural materials, the effect of nonlocality, although inprinciple unavoidable, is very weak, and f quickly approacheszero for the shifts jr − r 0j that are significantly larger than theatomic scale. On the other hand, nonlocality can result insome physical effects (such as optical activity), which are notpresent in purely local materials. A detailed exposition of thenonlocal optics of crystals is given in [6]. In macroscopic com-posites, nonlocality can play a much more profound role; seeSection 4.E below.(vi) Substitution of Eq. (7) into Eq. (2) results in the follow-ing expression for the induced charge density:

ρind�r; t� � −∇ · P�r; t�: (8)

We could have added to the right-hand side of Eq. (8) anytime-independent function ρ0�r�. It is usually assumed thatthe medium is locally and globally neutral in the absence ofexternal excitation so that ρ0 � 0. We have used a similarassumption in Point (iii) above to determine the constant ofintegration. However, in some cases, this assumption is not


valid. Examples include charged objects and segnetoelectrics. Inwhat follows, we will assume that ρ0 � 0.(vii) The quantities P and M define uniquely Jind throughEq. (6) but the inverse is not true. Indeed, the pair �P 0;M 0�,where P 0 � P� ∇ × F and M 0 � M − ∂F∕∂t , yield the sameJind as the pair �P;M�. Here F is an arbitrary pseudo-vectorfield. This fact has historically led to the proposition that,in order to build the correct macroscopic theory, one mustresort to some additional conditions to define P and Muniquely [7]. However, in the phenomenological approach de-scribed here, one does not really need to determine P and Mfrom a given Jind. Instead, the physically relevant problem isto determine the functions f e�τ� and f m�τ� for a givenmaterial. This can be done uniquely by performing severalmeasurements of the transmitted and reflected field in somesimple geometries such as a slab. A particular example of anexperimental technique for determining the dielectric functionof a homogeneous material is called ellipsometry [8]. Oncef e�τ� and f m�τ� (in practice, their Fourier transforms withrespect to τ) have been found, any boundary value problemcan be solved uniquely for any illumination and any shape ofthe sample. Correspondingly, the fields P and M are also de-fined uniquely in each particular case. Therefore, we do notneed to apply the additional condition that P and M are thedensities of electric and magnetic dipole moments, respec-tively. In fact, we will see that this condition does not holdin general.

We now substitute Eq. (6) into Eq. (4) and the result intothe right-hand side of Eq. (1) and obtain the macroscopicequations

∇ ×H � 1

c∂D∂t

� 4π

cJext; ∇ × E � −

1

c∂B∂t

; (9)

where we have defined two auxiliary fields

D � E� 4πP; H � B − 4πM: (10)

The second equations in the sets (1) and (9) are the same.To complete the construction of the theory, we transform

the macroscopic Maxwell’s equations to the frequency domain.Let us assume that the source Jext�r; t� is strictly monochro-matic. Then we can write

Jext�r; t��Re�Jext�r�e−iωt �; E�r; t��Re�E�r�e−iωt �; (11)

and similarly for all other fields. We then obtain the followingresult from Eqs. (7) and (10):

D�r� � ϵ�r�E�r�; B�r� � μ�r�H�r�; (12)

where

ϵ�r� � 1� 4π

Z∞

0

χe�r; τ� exp�iωτ�dτ; (13a)

μ−1�r� � 1� 4π

Z∞

0

χm�r; τ� exp�iωτ�dτ: (13b)

Substituting Eqs. (11) and (12) into Eq. (9), we arrive at

∇ ×H�r� � −ikϵ�r�E�r� � 4π

cJext�r�; (14a)

∇ × E�r� � ikμ�r�H�r�; (14b)

k � ω∕c � i0: (14c)

Here k is the free-space wave number. We have added to it aninfinitesimal imaginary part to satisfy the boundary conditionsat infinity (this will also prove useful for computing variousintegrals). We have explicitly indicated the dependence ofall functions of position on r. However, the frequency ω issuppressed in the lists of formal arguments of various functionsbecause we focus on just one working frequency. Considering adiscrete or continuous superposition of frequencies will natu-rally lead us to the theory of partially coherent fields, which isvery important in optics but is outside of the scope of thistutorial.

Equation (14) is the closed set of the macroscopicMaxwell’s equations. If the functions ϵ�r� and μ�r� areknown and sufficiently “nice” mathematically, we can solveEq. (14) uniquely. On the other hand, if we have a sampleof material that is known to be homogeneous internally withsome fixed ϵ and μ, we can perform a set of experimentalmeasurements with quasi-monochromatic radiation that willdetermine uniquely these quantities at the working frequency.For example, to determine the properties of a homogeneousisotropic slab, one can perform transmission and reflectionmeasurements at several incidence angles, various states ofpolarization of the incident beam, and, perhaps, variouswidths of the slab. It should be kept in mind that the experi-ments of this kind are usually carried out with partiallycoherent sources of radiation and one should use the appro-priate mathematical description of the polarization states ofpartially coherent radiation.

Finally, it is often more convenient to consider a givenexternal (incident) field rather than a given external current.This is explained in Section 4.B.

3. PHYSICAL MEANING OF POLARIZATION ANDMAGNETIZATION

It is almost universally accepted that the polarization and mag-netization fields P and M are the differential volume densitiesof the electric and magnetic dipole moments in continuous me-dia. However, in a typical textbook exposition of the subject,the mathematical arguments in favor of this belief are eithermissing or flawed. Sometimes it is implied that two functionsare pointwise equal if their integrals over a given region areequal [see Eq. (16) below and its discussion]. Sometimes itis remarked that P andM cannot be determined uniquely fromthe induced current Jind and we must, therefore, apply someadditional conditions to define these quantities unambiguously[this argument is discussed in Point (vii) of Section 2 above]. Inyet other cases, it is argued that P and M must have some“physical meaning,” and then pictures are drawn of little polar-ized atoms or current loops. The problem with this approach isthat it is relevant only to some very special cases.

The classical interpretation of P and M, as we will call it,has gained wide acceptance for several reasons. First, it can beviewed as pedagogical since it gives a visual qualitative explan-ation of the polarization and magnetization phenomena.Second, the approach seemed to work fine when it couldbe applied in practice, that is, for molecular gases or simplecomposites. In other cases, which include pretty much all con-densed matter and composites of complex geometry, there was


not enough information to say anything. The microscopiccharge density in the condensed phase could not be computedfrom simple models.

The situation changed in 1990s with the development of thedensity-functional theory and related computational methods.When the microscopic charge density was computed in somesolids, it became obvious that the classical interpretation doesnot reproduce the experimental results with an acceptable ac-curacy. This motivated the development of the modern theoryof polarization by King-Smith, Resta, and Vanderbilt [9–12].The main idea of this new theory is that the polarization P itself(however defined) is not physically important; it is the changeof P that is measurable and can be related to the macroscopicproperties of matter. The point of view taken in this tutorial isvery similar: the fundamental quantity of interest is the electriccurrent—the time derivative of P (assuming M � 0).However, since our discussion is purely classical, we do nottouch here the geometric quantum phase, which is centralto the modern theory of polarization. We note that recentintroductory-level reviews of this theory are available inRefs. [4,13].

Still, it is hard not to notice that the classical interpretation issurprisingly robust, at least for the electric polarization, andwe naturally wish to understand its limits and conditions ofapplicability. Considering the importance of this subject, wewill discuss it in detail in this section, primarily, in relationto the electric polarization (magnetization is also discussedbriefly). In particular, we will see that the classical interpretationworks fine for the Maxwell Garnett theory as long as the char-acteristic scale of inhomogeneities is much smaller than thewavelength. However, one should be conscious of the condi-tions of applicability of the classical interpretation, especiallywhen high-frequency phenomena are considered.

A. Polarization and the Density of the Electric DipoleMoment

Consider a simple connected region D of volume D and let theclosed and regular surface S be its boundary. The region D islocated arbitrarily with respect to a given sample of continuousmedium, which occupies the spatial regionV. We wish to com-pute d�t�—the macroscopic dipole moment of the matter con-tained in D. To this end, we can start from Eq. (8) and write

d�t� ≡ZDrρind�r; t�d3r � −

ZDr�∇ · P�r; t��d3r

�ZDP�r; t�d3r �

ISr�P�r; t� · dS�: (15)

We have used integration by parts in the second line ofEq. (15), where dS � nd 2r is the vector element of surface,and n is an outward unit normal to S.

If the surface integral in Eq. (15) was zero, we could con-clude that the classical interpretation is correct. Indeed, we cantake D to be sufficiently small so that

RD P�r; t�d 3r ≈ P�r0�D,

where r0 is a point inside D. Then P�r0� ≈ d∕D is, indeed, thedipole moment per unit volume. However, there is no reason tobelieve that the surface integral in Eq. (15) is zero. An obviouscounterexample is a statically polarized piece of metal. Since themacroscopic charge can accumulate in conductors only on the

surface (this result holds at arbitrary frequencies if the conduc-tor is internally homogeneous), the dipole moment of anyinterior volume element is zero while the polarization P isnot. In fact, the macroscopic dipole moment of any internalregion of a homogeneous material is zero.

Thus, Eq. (15) written for the general case does not tell usmuch. To make a more useful statement, we can takeD ⊃ V sothat the surface S is in vacuum and completely encloses theobject. The surface integral in this case vanishes, and we obtainthe following for the total dipole moment of the object:

dtot�t� �ZVP�r; t�d3r: (16)

Therefore, the total dipole moment of any electrically neutralobject is equal to the integral of P over its volume. This result isquite general but, of course, it does not mean that P is the dif-ferential density of dipole moment. Indeed, even if the integralsof two functions evaluated over the same region are equal, thefunctions can still be different.

However, we can use the result given by Eq. (16) construc-tively if we have a microscopic model or some “more fundamen-tal” expression for the density of charge. As discussed above, thisexpression can come from one of the theoretical models such asthe Lorentz model, the Drude model, the hydrodynamic model;it can result from an ab initioDFT calculation and so on. Let ussay that somehow we have computed a fine-scale density ofcharge ϱ. We expect the following equality to hold:Z

VP�r; t�d3r �

ZVrϱ�r; t�d3r � dtot: (17)

In other words, the total dipole moment of the object should bethe same whether we compute it by using the macroscopicpolarization function P or the microscopic charge density ϱ.We hope to use this condition to establish a relation betweenP and ϱ. Here is how we can proceed.

(i) Let us first apply the macroscopic theory to a geometry inwhich the polarization function inside the object is constantand equal to P. Examples include a sphere, an ellipsoid, ora relatively thin slab at a zero or low frequency. Thendtot � V P.(ii) Let us break the object into N elementary cells Cn so

that V � ∪nCn. Let us assume that the function ϱ�r; t� isperiodic. Then the microscopic dipole moment of each cellis dC � R

C rϱ�r�d3r.(iii) From the condition given by Eq. (17), we conclude thatV P � NdC and, consequently, P � dC∕C , where C � V ∕Nis the volume of an elementary cell.

We thus have made an identification of P as the density ofmicroscopic dipole moment in a very special case. In particular,we have assumed that the geometry and frequency are such thatthe macroscopic polarization is spatially uniform inside thesample, the sample is dividable into completely identical cells,and the electromagnetic response of all cells is identical, includ-ing the cells at the boundary and in the interior of the sample.

All the conditions described above are important. For exam-ple, consider the physical situation sketched in Fig. 1. Here thecharge distribution is perfectly periodic. However, if we forgetabout the left and right ends of the chain, there is no unique


definition of the unit cell. We can define the unit cell so thatits dipole moment takes any value in the interval �−qh; qh�.Therefore, if we want to make a meaningful choice,Condition (ii) must be used. Application of this condition de-pends on the left and right ends of the chain, which are notshown in the figure. If the medium is electrically neutral, thereare only two possible terminations: either the left end is neg-ative and the right end is positive or vice versa. In each case,we can break the medium completely into identical elementarycells. In one case, the macroscopic polarization of the mediumPz (along the chain) is�q∕h2 and, in the other case, it is −q∕h2(we assume that each cell is cubic); the intermediate values ofPz are not possible.

The above example suggests that what happens at themedium boundaries is really important. It is not possible touse a microscopic model of a medium without implicit orexplicit consideration of the boundaries. Moreover, the familiarcorrespondence between P and the dipole moment of a cell caneasily break down if the cells adjacent to the boundaries are insome way not equivalent to the cells in the interior.

Here is a simple one-dimensional example illustrating theimportance of boundaries. Consider a microscopic charge dis-tribution ϱ�z� of a sample contained between the planes z � 0and z � L � Nh, which is given by the expression

ϱ�z�� ϱ0

�cos

�2π

z�Δh

�

� h2π

sin

�2π

Δh

��δ�z�−δ�z −L��

�; 0≤ z ≤ L: (18)

This function in Eq. (18) is obtained by taking the unperturbedcharge density ϱ0 cos�2πz∕h� in the interval 0 ≤ z ≤ L and“shifting” it to the left by Δ. We assume that there is a hardwall at z � 0 and the charge cannot leak through it. Thus,all positive charge that would otherwise go through the surfaceis accumulated in the plane z � 0 and forms a surface chargedensity. An analogous process occurs at the other boundary z �L except that there the charge of opposite sign is accumulated.We can imagine that the shift Δ is proportional to the macro-scopic field Ez inside the medium, say,Δ � �χ∕ϱ0�Ez , where χis a proportionality constant. Then the response of the medium

is linear as long as Δ ≪ h. The function in Eq. (18) is illus-trated in Fig. 2 for different values of Δ.

Let us now break the medium into the elementary cellsnh ≤ z < �n� 1�h, n � 0; 1;…; N − 1. It is easy to see thateach elementary cell in the interior of the medium (that is,for 1 ≤ n ≤ N − 2) acquires a dipole moment per unit surfacedC � ϱ0�h2∕2π� sin�2πΔ∕h�. According to the conventionalwisdom, the medium should be assigned the macroscopicpolarization

Pz �ϱ0h2π

sin

�2π

Δh

��!Δ∕h→0

ϱ0Δ: (19)

Comparing this with Pz � ��ϵ − 1�∕4π�Ez and Δ � �χ∕ϱ0�Ez , we would conclude that the medium has the permittivityϵ � 1� 4πχ.

However, the conventional wisdom is wrong in this case.The problem is that the boundary cells are not equivalent tothe cells in the interior. Of course, the boundary cells arenot electrically neutral and, therefore, we cannot compute un-ambiguously their dipole moments. However, we can computethe total dipole moment of the whole structure, and it turnsout to be equal to zero. It so happens that the dipole momentof the first and last cells (considered together) cancels exactlythe dipole moment of all internal cells. If we assign the mediumthe dielectric permittivity ϵ � 1� 4πχ, we would get an ob-viously incorrect result. In fact, and somewhat paradoxically,the correct value of ϵ is 1; the medium is not macroscopicallydifferent from vacuum.

It is interesting to note that the total dipole moment of thesample remains zero if we replace the delta functions in Eq. (18)by Lorentzians of arbitrary width and unit integral weight.

Another way to understand the above result is to considerthe microscopic current jz�z�. Let us assume that Δ is afunction of time with the derivative _Δ. Then we can usethe continuity equation ∂ϱ∕∂t � ∂jz∕∂z � 0 to compute thecurrent. The result is jz � −ϱ0 _Δ cos�2π�z � Δ�∕h�. We haveRL0 jzdz � 0 and, moreover, the average of jz computed overan arbitrarily defined cell is also zero, viz.,

Ra�ha jzdz � 0

for 0 ≤ a ≤ L − h.

+q -q +q -q q-q+q-

d hq

+q -q +q -q q-q+q-

d hq

h

+q -q +q -q q-q+q-

d

(a)

(b)

(c)

Fig. 1. One-dimensional illustration of the ambiguity of definingthe dipole moment of an element of volume. The idea of the figureis borrowed from Ref. [13].

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Fig. 2. Illustration of the charge density given by Eq. (18) forL � 10h. Here Lorentzians of the width γ � 0.01h are plotted insteadof true delta functions.


We have just obtained an illustration of one important fact:unlike the dipole moment, the cell average of the microscopiccurrent is independent of the choice of elementary cell.However, this is only natural: polarization and magnetizationare introduced as auxiliary quantities while the current is fun-damental. Analogously, various multipole moments that areused in the classical interpretation are auxiliary quantitieswhose introduction can be avoided altogether.

We now need to clarify that the example of Eq. (18) andFig. 2 illustrates just one particular reason why the classicalinterpretation can be wrong: the possibility of charge leakagethrough the cell boundaries. This possibility is the mainobjection to the classical interpretation that is raised in themodern theory of polarization. We can, however, envisage aphysical situation in which every elementary cell contains alocalized distribution of charge that cannot leak through thecell boundaries. This situation was referred to as theClausius–Mossotti picture in Ref. [4]. It is well understoodnow that the Clausius–Mossotti picture is not a realistic modelfor polarization of solids. Still, the model may seem to be not sobad for the problem of homogenization of macroscopiccomposite materials. For example, we can consider isolatedmetallic inclusions in a dielectric host or even in vacuum. Inthe latter case, the charge can definitely not leak through thecell boundaries.

However, there is another reason why the classical interpre-tation can fail. According to Condition (ii) above, we should beable to break a macroscopic sample into identical elementarycells. However, if the size of a unit cell is not very small com-pared to the free-space wavelength, such partitioning of themedium is not possible. The fields in two adjacent cells willnever be identical in this case. Therefore, the assumption thatdtot � NdC will not hold. One can also say that an elementarycell is not a physically small volume in this case. For theproblem of homogenization of macroscopic composites, thisobjection is much more important than the charge leakage.

We can now make the following conclusions:

(i) There are two main objections to the classical interpre-tation: charge leakage through the cell boundaries and accumu-lation of non-negligible phase between two neighboringcells. These objections come to the fore in different physicalsituations. The first objection can be unimportant for macro-scopic composite media, while the second objection is almostnever important for the theory of molecular polarization ofnatural materials.(ii) In some cases, neither of the two objections apply and we

can safely use the classical interpretation to compute ϵ. In par-ticular, the Maxwell Garnett theory relies on this approach.However, one should keep in mind that the classical interpre-tation has conditions of applicability.(iii) To be used constructively, the classical interpretation isnot required to be valid for all possible samples. In fact, thisis hardly ever possible. For example, even at a very low fre-quency, we can envisage a sample that is so large that theelectromagnetic fields in its elementary cells are not equivalentand the equality dtot � NdC does not hold. What we actuallyneed is the existence of at least some sample for which all theconditions of applicability of the classical interpretation aremet. We can compute ϵ for this particular sample by using

the classical interpretation and then rely on the fact that ϵ isindependent of the overall shape and size of a sample.

B. Magnetization and the Density of the MagneticDipole Moment

Although we do not discuss magnetic effects in any detail inthis tutorial, we will now look briefly at the vector of magneti-zation M. The main point that we wish to make is that thereis no direct analogy between P and M and the argumentsused above for P cannot be applied to M without additionalrestrictions.

The classical interpretation of magnetization is even moreproblematic because an integral relation of the form of Eq. (16)does not generally hold for M. Indeed, the macroscopic-induced magnetic moment m of the region D is by definition

m � 1

2c

ZDr × Jindd3r; (20)

where Jind is given by Eq. (6). We can write m � m1 �m2,where

m1 �1

2c

ZDr×

∂P∂t

d3r; m2 �1

2

ZD�r×∇×M�d3r: (21)

The term m1 contains the electric polarization vector and isusually disregarded at low frequencies. However, at highfrequencies, or even in the case when the medium supportsa steady current, this term is not zero. To evaluate the secondterm, we can write identically r × ∇ ×M � ∇�r ·M� −M−�r · ∇�M. Therefore,

m2 �1

2

�IS�r ·M�dS −

ZDMd3r −

ZD�r · ∇�Md3r

�: (22)

The last integral can be evaluated by parts asZD�r · ∇�Md3r �

ISM�r · dS� − 3

ZDMd3r: (23)

Collecting everything together, we obtain after some additionalrearrangement

m2 �ZDMd3r −

1

2

ISr ×M × dS: (24)

Again, we see that there is a surface term, which disappears onlyif S is drawn outside of the body.

Let us now take D ⊃ V so that the surface integral vanishes.Then the total magnetic dipole moment of the object is

mtot �ZVMd3r � 1

2c

ZVr ×

∂P∂t

d3r: (25)

This equation is more complicated than the analogous resultfor dtot in Eq. (16). We can use the arguments of previoussubsection to establish a mathematical connection betweenthe magnetic moment of an elementary cell mC and the mac-roscopic magnetization M only if the second term in theright-hand side of Eq. (25) is negligibly small, which canbe the case at sufficiently low frequencies. Moreover, thetransformation of mtot under a coordinate shift r → r� ais given by


mtot → mtot �1

2ca ×

∂∂t

ZVPd3r � mtot �

1

2ca ×

∂dtot∂t

:

(26)

It can be seen that mtot is defined unambiguously only if theterm proportional to ∂dtot∕∂t is negligibly small.

The above discussion concerns magnetization caused by theelectric currents, which are often referred to as orbital, althoughthey can also include the conductivity current. Of course, theorbital currents obey quantum laws at the microscopic level.However, macroscopically, they can be viewed as classical.Further details of the modern theory of orbital magnetizationcan be found in [14,15]. In contrast, ferromagnetism is amacroscopic quantum effect, which cannot be fully understoodwithin the theoretical framework of classical physics.Nevertheless, the classical interpretation of M works fine forferromagnetism. This is so, first, because the last term inEq. (26) is negligible at the low frequencies for which the ferro-magnetic effect is non-negligible and, second, because theelementary amperian current loops (the quantum-mechanicalexpectations of the electric current associated with an electronspin) fit well into the Clausius–Mossotti picture outlinedabove. In other words, the conditions of applicability of theclassical interpretation that were discussed above for the caseof electric polarization also hold for the elementary currentloops in ferromagnetics.

4. WAVES ON LATTICES OF POLARIZED POINTPARTICLES

In the first part of this tutorial, we have derived the MaxwellGarnett approximation for a collection of point noninteractingdipoles at zero frequency. We now wish to construct a moregeneral theory. Specifically, we will consider arbitrary frequen-cies and allow the dipoles to interact. We will, however, restrictour attention to a very simple geometry: point-like, sphericallysymmetric particles arranged on a cubic lattice. We will showthat the Maxwell Garnett mixing formula follows from thedispersion relation for the electromagnetic waves propagatingon such a lattice in the long wavelength limit. We will notuse the classical interpretation of polarization as the dipole mo-ment per unit volume for arriving at this conclusion. However,the classical interpretation is valid in the long wavelength limitand can be used to derive the same result.

The model of electromagnetically interacting point-like po-larizable particles arranged on a three-dimensional lattice pos-sesses an intuitive physical appeal. Historically, many authorshave considered this model [16–21]. Most if not all of thesereferences assume that the particles are embedded in vacuum.We will make an unfortunate compromise and work under thesame assumption here, although it could have been more logicalto start with the model of small inclusions in a host of permit-tivity ϵh ≠ 1. The main reason for our choice is simplicity.Introduction of a host medium is not conceptually difficultbut will shift attention away from the key points, which wewish to avoid. However, we hope that anyone who has followedthrough the derivations of this section will be able to repeatthem for a more general host. Alternatively, the end resultof such a calculation can be obtained from the less general result

of this section by the substitution ϵeff → ϵeff∕ϵh, ϵi → ϵi∕ϵh,where ϵeff is the effective permittivity of the structure and ϵi isthe permittivity of inclusions. Toward the end of this section,we will apply this transformation to derive the Maxwell Garnettapproximation for a general two-component mixture.

A. Polarizability of a Small Particle Beyond theStatic Limit

Consider a small particle illuminated by an external monochro-matic field of the complex amplitude E�r� and frequency ω.If the particle size a is much smaller than the wavelength ofthe incident radiation, it can be characterized by the dipolepolarizability tensor α, which is defined as the linear coefficientin the expression d � αE�r0�. Here d is the complex ampli-tude of the dipole moment, and r0 is a point inside the particle.If the particle has cubic symmetry, we can write α � αI , whereα is a scalar and I is the identity tensor. For simplicity, we willconsider the scalar case in this section. However, the foregoingdiscussion also applies to all principal components of the tensorα, assuming it can be diagonalized by an orthogonal transfor-mation.

The polarizability is not arbitrary but satisfies certain con-straints. A well-known result is that the extinction cross sectionof the particle (if excited by a plane wave) is σe � 4πk Im α.Since σe > 0, we conclude that Im α > 0, but this is notthe most interesting constraint. Another inequality can be ob-tained by noting that the scattering cross section under thesame excitation conditions is σs � �8π∕3�k4jαj2. We knowthat σe ≥ σs, where the equality holds for an ideally nonabsorb-ing particle. From this, we find that

Im�1∕α� ≤ −2k3∕3: (27)

The above result may seem strange if we recall the quasi-staticformula for the polarizability of a sphere (FP3). This formulayields a real-valued result if ϵ is real at the working frequency.Even if we allow ϵ to have a small imaginary part to satisfy thecondition Im α > 0, inequality (27) might still not hold.

The apparent contradiction can be resolved by noting that(FP3) is not quite accurate at finite frequencies: it does not ac-count for the leading-order radiative correction (to the inversepolarizability), which is equal to −2ik3∕3. This correction canbe relatively very small. However, it is needed to satisfy the op-tical theorem. When the electromagnetic interaction of severalsmall particles is considered, not accounting for the radiativecorrection can yield strange results that contradict energy con-servation [22,23].

It turns out that corrections to Re�1∕α� also exist and areeven of lower order in k. We can refer to these corrections asto nonradiative since they are not associated with scattering.Unfortunately, these corrections are not uniquely defined.Different authors give different expressions for the nonradiativecorrections [24]. The reason for this ambiguity is an irremov-able singularity of the Green’s tensor for Maxwell’s equations;we will encounter this singularity momentarily. As a result, thenonradiative corrections depend not only on the particle shape(which could be expected) but also on the form of the incidentfield and on the reference point r0.

On the other hand, the first nonvanishing radiative correc-tion is universal. This is so because the imaginary part of the


Green’s tensor does not experience the singularity mentionedabove. One can argue that it is the correction to Im�1∕α� thatis physically important and should be retained, even in the limitka → 0. In accord with this argument, we will decompose theinverse polarizability as

1∕α � 1∕α − 2ik3∕3; (28)

where α is the polarizability with all the corrections up to theorder of O�k2�. We do not really know the exact form of thesecorrections without formulating the problem more precisely.All we can say for sure is that α → αLL when ka → 0, whereαLL is the quasi-static “Lorentz–Lorenz” polarizability of theparticle. For example, αLL is given by (FP3) for a sphere orby (FP32) for an ellipsoid. The difference αLL − α is of the orderof O�k2� if the particle has a center of symmetry and of theorder of O�k� otherwise. Note that α can be complex or real(in nonabsorbing particles).

One last question that we need to address is the following. Ifwe account for the effects that are of the order of O�k2�, is itcorrect to neglect all higher-order multipole moments? Strictlyspeaking, the answer is “no.” The magnetic dipole and electricquadrupole moments of small particles are quantities of theorder of O�k2�, the same as the difference 1∕α − 1∕αLL. Wetherefore will work in the regime when the nonradiativecorrections and the higher-order multipole moments can beneglected.

As for the radiative correction, we will see that, in the par-ticular problem we are solving, it exactly cancels. The cancel-lation takes place because a plane wave can propagate on aninfinite, perfectly ordered lattice without scattering. Formally,the same result could be obtained if we simply forgot about theradiative correction and neglected the terms that are of the or-der of O�k2� and higher in all formulas. It may seem thereforethat our discussion of the radiative correction was redundant,but this is not truly so. In problems that actually involve scat-tering, e.g., for finite or disordered sets of dipoles, accountingfor the radiative correction can be crucial; its neglect can resultin the appearance of nonphysical resonances or divergencesand, ultimately, in violations of energy conservation. On theother hand, neglect of the nonradiative corrections does notlead to anything dramatic or unphysical. One can also say thatRe�1∕α� and Im�1∕α� are responsible for different physicaleffects and should be considered separately.

B. Green’s Tensor Beyond the Static Limit

We next need an expression for the electric field radiated by apoint dipole. This result is given by the free-space Green’s tensorfor Maxwell’s equations. We have already encountered theGreen’s tensor in the static limit in the first part of this tutorial[see (FP8)]. Now we need to generalize this result to finitefrequencies.

We start from the frequency-domain macroscopic Maxwell’sequations (14). We consider a nonmagnetic medium with μ �1 and let ϵ�r� � 1� 4πχ�r�, where the susceptibility χ�r� iszero in vacuum. Upon substitution of H from the second intothe first equation of (14), and by using the definitions inEqs. (10) and (12) to write χ�r�E�r� � P�r�, we obtain

��∇ × ∇×� − k2�E�r� � 4πk2P�r� � 4πikc

Jext�r�: (29)

From linearity, we can write the solution to the above equationin the following form:

E�r� � Eext�r� �Z

G�r; r 0�P�r 0�d3r 0; (30)

where the field Eext and the Greens tensor G satisfy

��∇ × ∇×� − k2�Eext�r� �4πikc

Jext�r�; (31a)

��∇ × ∇×� − k2�G�r; r 0� � 4πk2Iδ�r − r 0�; (31b)

and I is the identity tensor.We will refer to the solution of Eq. (31a), Eext, as to the

external or incident field. Here Eext is the field that would haveexisted everywhere in space in the absence of the sample. Notethat

Eext�r� � −1

iω

ZG�r; r 0�Jext�r 0�d3r 0: (32)

Electromagnetic problems are typically formulated in termsof a known external field rather than external current.Equation (30) is one possible mathematical formulation of thisapproach.

In contrast, the second term in Eq. (30) is the field that isgenerated or scattered by the sample; we denote this term byEscatt so that

Escatt�r� �Z

G�r; r 0�P�r 0�d3r 0: (33)

The physical significance of the Green’s tensor can be under-stood as follows. Consider scattering of an external field by asmall particle occupying the regionV whose characteristic size ais much smaller than the free-space wavelength so that ka ≪ 1.Let the point r0 ∈ V be the “center” of V. Then, forjr − r0j ≫ a, we can write

Escatt�r� ≈ G�r; r0�ZVP�r�d3r � G�r; r0�d; (34)

where we have used Eq. (16) to obtain the second equality.Therefore, the Green’s tensor G�r; r0� gives the radiation fieldat point r of an elementary dipole at point r0.

We will now use Eq. (31b) to compute G. Accounting forthe translational invariance of free space, we can write

G�r; r 0� � F�r − r 0� �Z

K �p�eip·�r−r 0� d3p�2π�3 : (35)

Substitution of this ansatz and the similar Fourier representa-tion of the delta function δ�r − r 0� into Eq. (31b) results in theequation

��p × p×� � k2�K �p� � −4πk2I : (36)

This is a 3 × 3 matrix equation for K �p�. We can use theidentity �p × p×� � p ⊗ p − p2I to facilitate its solution. Theresult is


K �p� � 4πk2I − p ⊗ p

p2 − k2: (37)

Here ⊗ denotes tensor product of two vectors.The real-space Green’s tensor can be obtained by computing

the Fourier integral in Eq. (35). To this end, we can replace thetensor p ⊗ p by ∇r ⊗ ∇r 0. Then we can perform the angularintegration by using the elementary integration rules and theingratiation over p by residues. We can use the symmetry ofthe integrand to extend integration to the whole real axis; also,we should keep in mind that k has an infinitesimal imaginarypart according to Eq. (14c). The result of this calculation is

G�r; r 0� � �k2I − ∇r ⊗ ∇r 0 �exp�ikjr − r 0j�

jr − r 0j ; (38)

or

F �r� � �k2I � ∇ ⊗ ∇� exp�ikr�r

: (39)

Note that the signs in Eqs. (38) and (39) are correct. Similar tothe static case, Eq. (38) is singular due to the differentiation ofthe nonanalytical expression jrj. We can write

F�r� � −4π

3δ�r� � F R�r�; (40)

where F R is “regular” in the sense thatZjrj<a

F R�r�d3r�!a→00: (41)

We have written the word “regular” in quotes because the prop-erty expressed by Eq. (41) depends on the integration regionand F R is not truly regular; it still contains a singularity. As aresult, some physically important integrals involving F R areconditionally convergent. Indeed, by performing the differen-tiation carefully everywhere except for r � 0, we obtain thefollowing result:

F R�r� ��

k2

r� ik

r2−1

r3

�I �

�−k2

r−3ikr2

� 3

r3

�r ⊗ rr2

�eikr :

(42)

We thus see that F R�r� diverges as r−3 when r → 0. However,the real and imaginary parts of F R behave differently. Theexpansion of F �r� in powers of r near r � 0 is of the form

Re F R�r� ��−1

r3� k2

2r

�I �

�3

r3� k2

2r

�r ⊗ rr2

� O�r�;

(43a)

Im F R�r� ��2 k3

3−2 k5r2

15

�I � k5r2

15

r ⊗ rr2

� O�r4�:

(43b)

It can be seen that Im F R�r� has no singularity. This is why thefirst nonvanishing radiative correction is universal; in fact, it isgiven by the first term in the right-hand side of Eq. (43b).

Let us now evaluate the integral of F R�r� over a small ball ofradius a. We can use the identity

R4π�r ⊗ r�dΩ � 4π

3 r2I,

where dΩ is the element of solid angle, to compute the angularintegrals. We then obtain

Zjrj<a

F R�r�d3r�8π

3I ��1− ika�eika −1�

�!ka→0

4π

3I��ka�2� i

2�ka�33

�…�: (44)

This result is in agreement with Eq. (41).On the other hand, if we evaluate the integral in Eq. (44)

over a more general nonspherical region, the term O�r−3�would not integrate to zero. We will observe a physical mani-festation of this fact when we consider waves on lattices of pointpolarizable particles. We have also seen a similar difficulty whenwe considered anisotropy in the first part of this tutorial (inSection 4). All we can say now is that the “regularizing” decom-position in Eq. (40) is appropriate when spherical particles andregions are considered.

We can also define the “regular” part of the Green’s tensor inthe spatial Fourier domain. That is, we can write

K �p� � −�4π∕3�I � K R�p�; (45)

where F R and K R are spatial Fourier transforms of each other.We can easily find that

K R�p� �4π

3

�2 k2 � p2�I − 3p ⊗ pp2 − k2

: (46)

This result will be used in the calculations below.

C. Coupled-Dipole Equations on an Infinite Lattice

Consider an assembly of identical, point-like particles of scalarpolarizability α located at the positions rn. Let the whole systembe illuminated by the external field Eext�r�. Then the dipolemoments of the particles dn are coupled to the external fieldand to each other by the coupled-dipole equation

dn � α

�Eext�rn� �

Xm≠n

GR�rn; rm�dm�: (47)

Note that the term with m � n is left out of the summation inthe right-hand side of Eq. (47). This reflects our idea that theelectric field acting on the nth particle is a superposition of theexternal field Eext�rn� and the fields scattered by all particlesexcept for the nth particle itself.

Unfortunately, our exposition of the coupled-dipole equa-tion is by necessity brief. We note only that it is importantto use the correct definitions for all quantities appearing inEq. (47). In particular, we should use the polarizability α asit was defined in Section 4.A above and not the “bare polar-izability” [25] or the quasi-static polarizability αLL. Also, wehave used the “regular” part of the Green’s tensor, GR , inEq. (47) rather than G. In the real-space representation, thisis unimportant as long as the m � n term is not includedin the summation. However, upon spatial Fourier transforma-tion, the difference will show up. Additional details about thecoupled-dipole equations can be found in Refs. [19,24,26].

Our goal is to find an eigensolution to Eq. (47) on an in-finite lattice and to analyze the corresponding dispersionrelation. To this end, we assume that rn are the vertices ofan infinite cubic lattice of pitch h, set the external field Eext

to zero and seek the solution to Eq. (47) in the form


dn � d exp�iq · rn�. Upon substitution of this ansatz intoEq. (47), we obtain the equation

1

αd � S�q�d; (48)

where

S�q� �Xrn≠0

F R�rn�e−iq·rn (49)

is the lattice sum, sometimes also referred to as the dipole sum.Lattice sums are ubiquitous in physics [27] but not always

convergent. Correspondingly, a lot of effort has been devoted tocomputing and regularizing diverging lattice sums. Two ap-proaches to this task are common: allowing the point particlesto have higher-order multipole moments or allowing theparticles to have a finite size and shape. Ultimately, the twoapproaches are equivalent since they describe the same physicalreality. We will resort to the second approach—that is, we willassume that the particles are small spheres.

We now proceed with the calculation. First, we use theFourier representation in Eq. (35) (applied to GR) to rewriteEq. (49) as

S�q� �Z

d3p�2π�3 K R�p�

Xrn≠0

ei�p−q�·rn : (50)

To evaluate this expression, we need to complete the summa-tion. This can be accomplished by adding and subtracting unityto the series, which leads us to the expression

S�q� � 1

h3Xn

K R�q� gn� −Z

d3p�2π�3 K R�p�; (51)

where we have used the Poisson summation formula

Xneip·rn �

�2π

h

�3X

nδ�p−gn�; gn�

2π

h�nx;ny;nz�: (52)

Here gn are the reciprocal lattice vectors and we can viewn � �nx; ny; nz� as a composite index.

As expected, the expression in Eq. (51) is divergent: the lastterm in the right-hand side is just F R�0�, but F R�0� is not welldefined. Thus, we will use the regularizing substitutionZ

d 3p�2π�3 K R�p� � F R�0� →

3

4πa3

Zjrj<a

F R�r�d3r

≈�k2

a� i

2a3

3

�I ; (53)

where we have used Eq. (44). As mentioned above, this regu-larization is appropriate in the case when the particles are smallspheres of radius a.

We now put everything together. We substitute Eq. (53)into Eq. (51) and Eq. (51) into Eq. (48), use the decompositionin Eq. (28) for 1∕α, and obtain�

1

α� k2

a

�d � 1

h3Xn

K R�q� gn�d: (54)

As expected, the radiative correction has canceled.

D. Long Wavelength Approximation

The dispersion relation given by Eq. (54) is still too compli-cated for analysis. To proceed, we will need to make the longwavelength approximation. Namely, we will assume that kh ≪1 and jqpjh ≪ 1, where p � x; y; z. We should keep in mindthat, in our model, a ≪ h so that ka ≪ 1 also holds. Then,with the precision of the approximation used, we can write1∕α� k2∕a ≈ 1∕αLL, where αLL is the quasi-static polarizabil-ity of a particle.

We thus have simplified the left-hand side of Eq. (54). Inthe right-hand side, we will write the term with gn � 0 and therest of the sum separately. We will thus obtain

1

αLLd � 1

h3K R�q�d� 1

h3Xgn≠0

K R�q� gn�d: (55)

We will now show that the last term in the right-hand side ofEq. (55) can be neglected in the long wavelength approxima-tion. Indeed, for gn ≠ 0, we can write

K R�q� gn��!kh;jqpjh→0

4π

3Qn; (56)

where

Qn � I − 3gn ⊗ gn

g2n; gn ≠ 0: (57)

Now, if the particles are arranged on a cubic lattice, the recip-rocal lattice is also cubic. We can define a discrete cubic boxwith excluded center, B�L�, as the set of all reciprocal latticevectors with the indexes n � �nx; ny; nz� such that −L ≤ nx ,ny, nz ≤ L except for n � �0; 0; 0�. ThenX

gn≠0Qn � lim

L→∞

Xgn∈B�L�

Qn; (58)

but each individual sumP

gn∈B�L�Qn is zero. This can beproved by noting that the sum over the surface of each boxis zero and then representing a box of size L as a set of con-centric surfaces with the sizes L 0 � 1; 2;…; L. From this, weconclude that the limit in Eq. (58) is also zero and the secondterm in the right-hand side of Eq. (55) can be neglected in thelong wavelength approximation.

We now use the above fact and the expression in Eq. (46) forKR�q� to obtain the following equation:

1

αLLd � 4π

3 h3�2 k2 � q2�I − 3q ⊗ q

q2 − k2d: (59)

This is the dispersion relation for electromagnetic waves propa-gating on a cubic lattice of small, spherical, polarizable particlesin the long wavelength approximation.

The only thing left for us to do is to consider the polariza-tion states. The effective permittivity of a cubic lattice of smallspherical particles cannot be anisotropic (from symmetry, theprincipal axes of the tensor ϵeff must coincide with the crystallo-graphic axes, and the directions along all three crystallographicaxes are equivalent). Therefore, we must assume that themedium is characterizable by a scalar effective permittivityϵeff and q2 � k2ϵeff (we have assumed here that μeff � 1, moreon this later). Under the same conditions, the medium can


support transverse waves for which q · d � 0. Then it followsfrom Eq. (59) that the dispersion equation is

vαLL

� 4π

3

ϵeff � 2

ϵeff − 1; (60)

where we have introduced the specific volume per one particlev � h3. Note that longitudinal waves can also propagate on thelattice but only if the special condition h3∕α � −8π∕3 holds(the structure is similar in this case to a nondissipative plasma).Usually this condition does not hold and we will ignore thelongitudinal waves.

Equation (60) is the inverted Clausius–Mossotti relation[compare to (FP16)]. We therefore have derived theClausius–Mossotti relation as the long wavelength limit ofthe dispersion relation for electromagnetic waves on a cubiclattice of small spherical particles.

The Maxwell Garnett approximation can be obtained if weuse a specific expression for αLL. Since we have assumed that theparticles are small spheres, we will use (FP3). Substitutingeverything into Eq. (60) and inverting the equation, we obtain

ϵeff − 1

ϵeff � 2� f

ϵ − 1

ϵ� 2; (61)

where f � 4πa3∕3v is the volume fraction of inclusions. Thisis the Maxwell Garnett mixing formula for inclusions ofthe permittivity ϵ in vacuum. We can generalize this resultto inclusions of the permittivity ϵi in a host medium of thepermittivity ϵh by using the substitution ϵeff → ϵeff∕ϵh andϵ → ϵi∕ϵh. We thus obtain

ϵeff − ϵhϵeff � 2ϵh

� fϵi − ϵhϵi � 2ϵh

: (62)

This is the Maxwell Garnett mixing formula written in theform of (FP25).

E. Beyond the Long Wavelength Approximation

We now discuss briefly what can happen beyond the long wave-length approximation. The material of this subsection is not aself-contained exposition but a collection of a few basic ideasand leads for further reading and exploration.

Recall that we have neglected the nonradiative corrections tothe inverse polarizability and replaced the term 1∕α� k2∕a by1∕αLL. We cannot make a more precise statement about theinverse polarizability without abandoning the model ofpoint-like dipoles. However, we can still use this model andintroduce physically meaningful corrections to the long wave-length approximation if a ≪ h. In this case, the inequalityka ≪ 1 can hold quite strongly, but the inequalities kh ≪ 1and jqpjh ≪ 1 cannot be so strong. We then expand the termK R�q� gn� with gn ≠ 0 [in Eq. (55)] in powers of k and q.The long wavelength result (56) will give the zeroth order inthis expansion. Some of the expansion terms will sum to zerodue to symmetry but others will remain. If we make all expres-sions dimensionless by multiplying the numerator and denom-inator in the expression for K R�q� gn� by h, the correctionswill be of the order of O��kh�n� and O��qph�n�, where n iseven. This is true for all systems with a center of symmetry.

The terms proportional to �kh�2, �kh�4, etc., are the dynamiccorrections to the dispersion equation. Accounting for these

terms does not change the algebraic structure of the equation;it remains quadratic in q and its solutions remain similar tothose in local, isotropic, homogeneous media. Generally, thesesolutions can be written as q2 � ϵeff �k�k2. However, if we ac-count for the dynamic corrections, ϵeff �k� will no longer satisfythe scaling laws discussed in Section 7 of [1]. In particular, re-scaling the composite as r → βr will change ϵeff �k�; the law ofunaltered ratios will also not hold.

Keeping the terms proportional to �qph�2, �qph�4, orq2xq

2y q

2z h

6 (anisotropic terms of this form do not contradictthe cubic symmetry), etc., will fundamentally change the alge-braic form of the dispersion equation: it will become muchmore complicated than the quadratic equation characteristicof local and isotropic media. The dispersion equation can bewritten in this case as q2 � ϵeff �k; q�k2. The explicit andsometimes very complicated dependence of ϵeff on q can beconsidered as an effect of spatial nonlocality of the constitutiverelations.

Many important physical effects of nonlocality can beglimpsed from the dispersion equation. This includes additionalwaves, birefringence, and optical activity (in systems without acenter of symmetry) [6]. Unfortunately, the dispersion equationdoes not give us the complete information about the medium.Indeed, solving boundary value problems in finite samples re-quires the knowledge of the influence function f e�r; r 0; τ�[the nonlocal generalization of the function f e�r; τ� that appearsin Eq. (7a)]. However, f e�r; r 0; τ� is not the spatial Fouriertransform of ϵeff �k;q� nor is it a function of the shift r − r 0;the latter simplification is obtained only sufficiently far fromthe medium boundaries. Generally, spatial nonlocality and boun-dary value problems do not come together easily.

Things are further complicated by a widespread belief thatweak nonlocality of the dielectric permittivity is physically in-distinguishable from nontrivial magnetic response. For the pre-cise statement of this equivalence principle and relevantreferences, see [28]. In fact, the equivalence holds with respectto the dispersion relation but not the medium impedance. Adetailed analysis of the equivalence principle is given inRef. [29]. Here we only mention an obvious counterexample:an analytical solution can be easily obtained for a plane waveincident on a slab or a sphere with purely local ϵ and μ.However, the same problem cannot be solved so easily if theobject has a nonlocal permittivity and μ � 1. If we define somefunction f e�r; r 0; τ� and solve somehow the resultant integro-differential equations, the electromagnetic fields scattered bythe object will be not the same as in the first case.

The above discussion underscores the fundamental impor-tance of what happens at the medium boundary. In Section 3.A, we have already seen that disregard of the boundaries canyield an incorrect result, even in very simple physical situations.In the more complicated scenarios considered by the extendedhomogenization theories, boundaries are crucial.

To illustrate the importance of boundaries one more time,consider an elementary but frequently overlooked example. Ifwe know only the dispersion relation of an isotropic and localmedium, there is no good way to define the parameters ϵ and μseparately. Indeed, the law of dispersion in this medium isq2 � n2k2, where n2 � ϵμ. Assume that we have computed


n2. All we can gain from this knowledge is the product of ϵ andμ, not these two quantities separately. This fact is frequentlyoverlooked when extended homogenization theories are devel-oped to predict both ϵ and μ but infinite unbounded media areconsidered. The main difficulty of the extended homogeniza-tion theories is accounting for the lack of translational invari-ance that occurs near the medium boundaries.

Now we have to backtrack a little and recall that, in the pre-vious subsection, we have assumed without a justification thatμeff � 1. This was, however, not a random guess. In the longwavelength limit, one can consider the medium with a planarboundary and compute the reflection and transmission coeffi-cients [30]. Alternatively, one can compute the total magneticmoment of the medium in some simple geometry and apply theresults of Section 3.B. Both approaches will indicate that thecorrect choice is μeff � 1. Also, all experimental evidence tellsus that there is no artificial magnetism in composites whoseheterogeneity size is much smaller than the wavelength.

Beyond the long wavelength approximation, one enters agray area in which the scaling laws discussed in Section 7 of[1] do not work, where the magnetic effects can be consideredas important and the introduction of a nontrivial effective mag-netic permeability is feasible, even in intrinsically nonmagneticcomposites. The big question is whether the medium can bereasonably characterized by local tensors ϵ and μ or, perhaps,by a more general but still local constitutive tensor. The ex-tended (nonasymptotic) electromagnetic homogenization the-ories are subjects of active ongoing research [31–38]. The abovereferences and many references therein will give an interestedreader useful leads for further exploration of the subject.

5. WAVES IN PHOTONIC CRYSTALS IN THELOW FREQUENCY LIMIT

We have seen in the previous section that the model of an in-finite lattice of point polarizable particles is not well-behavedmathematically unless we assume implicitly or explicitlysomething about the particle shape and size. All methods ofregularizing the divergent lattice sums boil down to makingthese assumptions, often in a disguised form. It makes sensetherefore to abandon the point-particle model and considerthe actual physical inclusions arranged periodically. We willsee in this section that this approach is ultimately rewardingsince it is free from any mathematical difficulties and allowsone to account self-consistently for the effects of inclusions in-teraction, anisotropy, etc. After all, Maxwell’s equations are notself-contradictory and all mathematical difficulties grow frommaking ill-justified or unphysical approximations.

The mathematical developments shown in this section arenot fundamentally different from those used for the case ofpoint polarizable particles. However, some more advancedalgebraic manipulations will be used.

A. Bloch Waves in a Photonic Crystal

Consider a three-dimensional periodic medium shown sche-matically in Fig. 3. Inclusions of arbitrary shape are arrangedperiodically on an infinite cubic lattice of pitch h. We denotethe elementary cells of the lattice by Cn and the center of Cn by

rn � h�nx; ny; nz�. As above, n � �nx; ny; nz� is a compositeindex.

Each elementary cell contains an inclusion. Let the spatialregion occupied by the nth inclusion be Vn ∈ Cn. All regionsVn are of exactly the same shape and the same volume V anddiffer only by translations. In the figure, we have shown Vn asconnected, but this assumption is not important for us. Also,Vn can touch the boundary of Cn and the center of the cell, rn,can lie inside or outside of Vn.

Further, we assume that the material of inclusions has thepermittivity ϵ ≠ 1 and the material of the host is vacuum. Lateron, we will generalize the consideration to the case of a two-component mixture by the usual substitution ϵ → ϵi∕ϵhand ϵeff → ϵeff∕ϵh.

We start from the integral in Eq. (30). Since we are lookingfor an eigensolution in an infinite medium, we set the externalfield Eext to zero. We also use the regularizing decompositionEq. (40) for G�r; r 0� � F �r − r 0� and obtain the equation

E�r� � −4π

3P�r� �

ZF R�r − r 0�P�r 0�d3r 0: (63)

This equation is valid everywhere in space. However, the func-tion P�r� is nonzero only inside the inclusions. Since all inclu-sions are equivalent, it is possible to reformulate the integral inEq. (63) so that it involves only one inclusion. Indeed, let usrestrict our attention to r ∈ Vn. We then multiply Eq. (63) by�ϵ − 1�∕4π and obtain, after some additional rearrangement,

P�r�� 3χ

4π

Xm

Zr 0∈Vm

F R�r−r 0�P�r 0�d3r 0; r∈Vn; (64)

where

χ � ϵ − 1

ϵ� 2(65)

is the spectral parameter of the theory. We will seek a solutionto Eq. (64) in the form of a Bloch wave P�r� � exp�iq · r�P�r�,where q is the Bloch wave number (to be determined) and P�r�is a lattice-periodic function, which satisfies P�r� rn� � P�r�for any lattice vector rn.

Fig. 3. A schematic illustration of a two-component photonic crys-tal. Identical inclusions are periodically arranged in a host medium.For simplicity, we assume that the host medium is vacuum. Thepattern is three-dimensional.


If r ∈ Vn, we can make a change of variables r � rn � R,where R ∈ V0 ∈ C0, and C0 is the elementary cell centeredaround the point r0 � 0. Then the Bloch wave ansatz takesthe form P�r� � exp�iq · �rn � R��P�R�. Upon substitutionof this expression into Eq. (64), we obtain

P�R�� 3χ

4π

ZR 0∈V0

W �R;R 0�P�R 0�d3R 0; R∈V0; (66)

where

W �R;R 0��XmFR�rn −rm�R−R 0�e−iq·�rn−rm�R−R 0�: (67)

Note that the right-hand side of Eq. (67) is independent of nsince all cells of an infinite lattice are equivalent.

Next, we substitute the Fourier representation in Eq. (35) ofF R into the right-hand side of Eq. (67) and use the Poissonsummation formula in Eq. (52). This yields

W �R;R 0� � 1

h3Xn

K R�q� gn�eign·�R−R 0�: (68)

This expression still contains a summation over the reciprocallattice vectors gn � �2π∕h��nx; ny; nz�. The advantage of usingthe expression in Eq. (68) is the availability of a simple analyti-cal expression for K R in Eq. (46).

We now wish to reformulate the integral in Eq. (66) as analgebraic equation, which is amenable to numerical solution.To this end, we recall that P�R� is a lattice-periodic functionand therefore it can be expanded as

P�R� �Xn

Pneign·R: (69)

Here Pn are the amplitudes of the Bloch harmonics; for a planewave in a truly homogeneous medium, we have Pn ∝ δn0. Bysubstituting ansatz (69) into Eq. (66), we obtain an algebraicequation for the Bloch amplitudes:

Pn � f3χ

4πK R�q� gn�

XmM�gn − gm�Pm; (70)

where f � V ∕h3 is the volume fraction of inclusions, andM �g� is the shape function; it is defined by the Fourierintegral

M �g� � 1

V

ZR∈V0

e−ig·Rd3R: (71)

This integral can be computed analytically for many regularshapes.

Thus, we have written a set of algebraic equations [Eq. (70)]for the Bloch amplitudes Pn. This equation should be viewed asan eigenproblem but not of a traditional kind. Assume that wehave fixed the frequency ω and the shape of the inclusions.Then the parameter χ is also fixed (χ is the only variable inthe equation that depends on ϵ). The volume fraction f is alsofixed, and the function M�g� is uniquely defined by the shapeof the inclusions. Note thatM �g� is invariant under coordinaterescaling r → βr. Then the set of all values of q for whichEq. (70) has a nontrivial solution forms the isofrequency surface.The word “surface” should not be understood literally becauseq can be complex. Generally, the vectors q that allow for

nontrivial solutions of Eq. (70) fill a very complicatedsix-dimensional manifold Ω. For each q ∈ Ω, there can beseveral distinct solutions to Eq. (70)—we can refer to thesesolutions as to the polarization states.

B. Long Wavelength Approximation

Equation (70) is too complicated for analysis and we will nowsimplify it by using the long wavelength approximation. As wasdone in Section 4.D, we will consider the equations for P0

(with g0 � 0 in the left-hand side) and Pn (with gn ≠ 0) sep-arately. We will use the limit given by Eq. (56) in the latter caseand write

P0 � f3χ

4πK R�q�

�P0 �

Xn≠0

M�−gn�Pn

�; (72a)

Pn � f χQn

�M�gn�P0 �

Xm≠0

M �gn − gm�Pm

�; n ≠ 0:

(72b)

Here the index n � 0 is equivalent to the triplet �nx; ny; nz� ��0; 0; 0� and similarly for m.

In deriving Eq. (72), we have assumed that kh; jqpjh ≪ 1 or,equivalently, we have taken the limit h → 0 under the condi-tion that the proportions of all inclusions relative to the cell sizeare fixed. Equation (72a) does not involve any approximations.The only place where the reciprocal lattice vectors (and the var-iable h) appear in Eq. (72a) is M�gn�. However, the functionsM�g� are invariant with respect to rescaling and, therefore,do not change when we take the limit h → 0. In derivingEq. (72b), we have encountered the expression K R�q� gn�with gn ≠ 0, which depend on h. The limit of this term is givenby Eq. (56).

The crucial observation that we need is this: (72b) is a set oflinear equations with respect to the amplitudes Pn with n ≠ 0in which P0 can be viewed as a free term. To be sure, the set isinfinite and any numerical solution will require truncation. Theimportant point is that we can view (72b) as an equation with anonzero free term, not an eigenproblem. Then, from linearity,it follows that Pn � T nM �gn�P0. The set of tensors T n can bedetermined by solving the system of equations in (72b) forthree different right-hand sides: P0 � ex , P0 � ey, andP0 � ez , where ep (p � x; y; z) are unit vectors along the axesof a Cartesian frame. Consequently, we can writeX

n≠0M �−gn�Pn � ΣP0; (73)

where

Σ �Xn≠0

M�−gn�T nM�gn�: (74)

By substituting Eq. (73) into Eq. (72a), we find that thedispersion equation takes the form�

I − f χ�2 k2 � q2�I − 3q ⊗ q

q2 − k2�I � Σ�

�P0 � 0; (75)

where we have used the explicit expression (46) for K R�q�.


The derivations of this section involved a couple of math-ematically nontrivial steps, in particular, the splitting of the setof equations (70) into two subsets (72) and subsequent alge-braic manipulations. It can be mentioned that the final resultfollows directly from the fundamental theorem of determinantsof block matrices. Here we have followed a less formal ap-proach.

We conclude the subsection with a few remarks.

(i) The tensor Σ depends on the shape and orientation ofinclusions and on the spectral parameter χ and therefore on ϵ.(ii) The tensor Σ can be computed by solving a well-defined

system of algebraic equations on a computer. Theoretically, theset is infinite, but we can truncate it by restricting the reciprocallattice vectors to the cubic box with excluded center B�L�,which was defined in Section 4.D. Convergence can be inves-tigated by repeatedly increasing the box size L, e.g., by thefactor of 2.(iii) In general, the tensor Σ is complex symmetric and notnecessarily diagonalizable by an orthogonal transformation.If it is not, then there is no Cartesian frame in which Σ is diago-nal. We have not considered this (perhaps unusual) case andwill not discuss it any further.(iv) Note the limit limf →1 Σ � 0. However, limf →0 Σ � Σ0

where Σ0 is zero for spherical inclusions but not for othershapes.

C. Mixing Formula

The dispersion equation (75) has nontrivial solutions if the de-terminant of the 3 × 3matrix in the square brackets is zero. Theparticular values of q for which this is the case are the Blochwave vectors. There can be several distinct polarization statescorresponding to a given Bloch wave vector. The considerationis rather involved mathematically for a general anisotropicmaterial. We will derive therefore the homogenization resultfor an isotropic composite and then write without proof a moregeneral formula applicable to the case when Σ is tensorial butdiagonal in some Cartesian frame.

The composite is isotropic for any inclusions that obey thesame cubic symmetry as the lattice. For example, such inclu-sions can be cubes or spheres centered around the points rn.However, much more complicated shapes can also obey thecubic symmetry. The major simplification that one obtainsin this case is Σ � ΣI , where Σ is a scalar. Then there aretwo linearly independent transverse polarization states with q ·P0 � 0 for each q. A longitudinal polarization state (such thatP0‖q) exists only if ϵ � 1 − 3∕�2f �1� Σ� � 1�. Usually, thisequality does not hold and we can safely ignore longitudinalwaves. Under these conditions, Eq. (75) is simplified to

1 − f χ2 k2 � q2

q2 − k2�1� Σ� � 0: (76)

We now introduce the effective permittivity through the rela-tion q2 � ϵeff k2, use the explicit definition of χ in Eq. (65) andtransform Eq. (76) to

ϵeff − 1

ϵeff � 2� f

ϵ − 1

ϵ� 2�1� Σ�: (77)

We finally generalize the above result for the case of a two-component mixture. To this end, we apply the usual

transformation ϵeff → ϵeff∕ϵh, ϵ → ϵi∕ϵh and obtain the famil-iar-looking formula

ϵeff − ϵhϵeff � 2ϵh


�1� Σ�: (78)

It should be kept in mind that the spectral parameter χ inEq. (72b) (which is used to compute Σ) should also be changedto χ � �ϵi − ϵh�∕�ϵi � 2ϵh�.

If not for the factor Σ, Eq. (78) would be the MaxwellGarnett mixing formula written in the form of (F25). A non-zero factor Σ accounts for the deviation of the rigorous homog-enization result from the Maxwell Garnett prediction.

Just like the Maxwell Garnett mixing formula, Eq. (78) isnot symmetric and, in fact, we do not expect the rigoroushomogenization result to be symmetric. The definition ofthe mixing formula symmetry and a more detailed discussionare given in Section 3 of [1]. Equation (78) captures correctlythe effects of shape of the inclusions and therefore it is notsymmetric.

Another interesting feature of Eq. (78) is this. Consider cu-bic inclusions. Then we can change the size of the inclusionfrom 0 to h and it will correspond to changing f from 0 to1. At f � 0 we have ϵeff � ϵh, and at f � 1 we haveϵeff � ϵi. If we plot ϵeff �f � given by Eq. (78) parametricallyin the complex ϵ-plane, we will obtain a smooth curve thatconnects the points ϵh and ϵi and lies inside the Wienerbounds. In fact, there are two different curves that connectϵh and ϵi in this manner, depending on whether we assumethat f � 0 corresponds to a homogeneous medium withthe permittivity ϵh [as is assumed in Eq. (78)] or that it cor-responds to a homogeneous medium with the permittivity ϵi.The two curves will not intersect except at the ends, which is amanifestation of the lack of symmetry of the mixing formula.The Maxwell Garnett approximation has exactly the sameproperty.

Therefore, Eq. (78) is a corrected or a generalized version ofthe isotropic Maxwell Garnett mixing formula, which accountsproperly for the electromagnetic interaction of inclusions (in aperiodic system) but has the same general mathematical proper-ties as the classical mixing formula.

We now state without proof the result for the case when Σ isdiagonal in some Cartesian frame XYZ but its principal valuesare different. The result is rigorously derived in Ref. [30] and isa straightforward generalization of Eq. (78), viz.,

�ϵeff �p − ϵh�ϵeff �p � 2ϵh


�1� Σp�; p � x; y; z: (79)

Here Σp is one of the principal values of the tensor Σ. Solvingfor �ϵeff �p, we obtain the following mixing formula:

�ϵeff �p � ϵh1� 2f ϵi−ϵh

ϵi�2ϵh�1� Σp�

1 − f ϵi−ϵhϵi�2ϵh

�1� Σp�

� ϵhϵh � 1�2f �1�Σp�

3 �ϵi − ϵh�ϵh � 1−f �1�Σp�

3 �ϵi − ϵh�: (80)

This is the rigorous homogenization result for anisotropic com-posites that have three well-defined optical axes.


The second expression in Eq. (80) can be compared to theclassical anisotropic Maxwell Garnett mixing formula (FP34).It can be seen that the two formulas cannot be made equivalentby some special choice of the depolarization factors νp. Indeed,the condition of equivalence of the two expressions is

νp �1

3

�1 −

Σp

1� Σp

1

χ

�; where χ � ϵi − ϵh

ϵi � 2ϵh: (81)

However, in the Maxwell Garnett theory, νp is a purely geomet-rical parameter, which is independent of ϵi and ϵh.Equation (81) can hold only if Σp � βpχ∕�1 − βpχ�, whereβp is a material-independent constant. If the latter equalityholds, then it follows from Eq. (81) that νp � �1 − βp�∕3.

In other words, for the Maxwell Garnett approximation tobe precise, it is required that Σ has only one resonance for agiven polarization with the resonance value of χ being1∕βp � 1∕�1 − 3νp�. However, this property does not generallyhold except if �ϵeff �p lies on one of the Wiener bounds, that is,for composites of some very special geometry. Of course, onthe Wiener bounds, the anisotropic Maxwell Garnett mixingformula is precise. In a more general case, the functionaldependence of the form Σp�χ� �

Pnβnpχ∕�1 − βnpχ� is valid,

and the sum is not reduced to just one term. This follows fromthe algebraic form of Eq. (72b), which determines Σ�χ�.Therefore, in reality, the function Σ�χ� has multiple resonan-ces. The physical significance of the resonance values of χ,χnp � 1∕βnp is the following. For χ in the vicinity of one ofits resonance values, electromagnetic interaction of inclusionsbecomes very strong. In a sense, the Maxwell Garnett approxi-mation assumes that only one such resonance exists. If this isnot so, the approximation can break down dramatically.

We can conclude that the Maxwell Garnett’s treatment ofanisotropy is based on a single geometrical parameter νp while,in fact, the rigorous description involves many such factors(βnp, using the above notations) or the function Σp that de-pends on both geometry and composition. The ambiguityof the anisotropic formula discussed in Section 4 of [1] is math-ematically related to this fact. In the next section, we will showan example of anisotropy that is not captured accurately by theMaxwell Garnett theory.

6. NUMERICAL EXAMPLES

Let us truncate the infinite set of equations in Eq. (72b) byrestricting the reciprocal lattice vectors to the box with excludedcenter B�L�. In this case, the number of equations that must besolved to compute Σ is N � 3��2L� 1�3 − 1�. Note that theequations must be solved with three different right-hand sidesto gain access to all elements of Σ.

It can be seen that the number of equations scales as O�L3�.This makes the direct solution of 3D problems difficult and wemust resort to iterative methods. One such fast-converging iter-ative algorithm is based on a continued-fraction expansion[30]. The method is spectral, which means that the computa-tionally intensive part of this algorithm is independent of thespectral parameter χ and, therefore, of ϵi and ϵh. This can beuseful if we are interested in the spectral dependence of ϵeff �ω�for a fixed geometry of the composite and dispersive ϵh�ω� andϵi�ω�. However, in the examples shown below, we fix ϵh and ϵi

and compute ϵeff �f � by changing the size of the inclusions rel-ative to h up to the geometrical limit. In this case, spectralmethods are not computationally advantageous. Nevertheless,the iterative method mentioned above is efficient enough tohandle 3D problems. Below, numerical results will be shownfor L � 32. Convergence was verified by doubling the box sizeL in some of the cases and repeating the computationswith L � 64.

Simulations were performed for spherical and cubic inclu-sions and for two types of parallelepipeds shown in Fig. 4.Numerical results are shown in Fig. 5. We have used thesame permittivities of the composite constituents as in the firstpart of this tutorial (in Section 6): ϵ1 � 1.5� 1.0i andϵ2 � −4.0� 2.5i. We now briefly describe the results.

In Fig. 5(a), we show the effective permittivity of a com-posite consisting of spherical inclusions of the radius a thatvaries from zero to the point of touching, a � h∕2. Thecomposite is isotropic. The data points computed accordingto Eq. (80) are shown by circles. We consider two cases: metalinclusions in the dielectric host and vice versa. Rigorous homog-enization results are compared to the Maxwell Garnett andBruggeman’s mixing formulas. Of course, in the analytical mix-ing formulas f is not limited by the geometry and we haveshown the respective curves for 0 ≤ f ≤ 1. As was mentionedin [1], these curves always connect the two points ϵ1 and ϵ2 inthe complex ϵ-plane. However, the value of f for the rigorouslycomputed numerical data are limited by the condition of non-intersection of the inclusions. Due to this reason, the numericaldata terminate inside the Wiener bounds. The last data pointcorresponds to touching spheres.

It can be seen that the Maxwell Garnett formula is accurateat low-to-moderate volume fractions, while the Bruggeman’sformula does not provide a meaningful correction. Note thatthe comparison of the numerical results and analytical approx-imations in Fig. 5 is not quite definitive: even if two pointsoverlap in the complex ϵ-plane, they can correspond todifferent values of f . Still, it is true that the MaxwellGarnett mixing formula provides a good approximation forspherical inclusions. Admittedly, we did not consider the caseof strong electromagnetic resonances. The materials consid-ered have large imaginary parts, which suppress the resonantphenomena.

In Fig. 5(b), we show similar data for cubes. Now thevolume fraction f can vary from 0 to 1 in the analytical

Fig. 4. Illustration of the anisotropic inclusions and polarization la-bels used in the simulations shown in Fig. 5(c) below.


expressions as well as for the actual inclusions. Again, theMaxwell Garnett mixing formula yields surprisingly accurateresults, while the Bruggeman’s formula can be seen as providinga meaningful correction at low volume fractions and thenbreaking down completely.

The most interesting cases are shown in Fig. 5(c). Here theinclusions are parallelepipedal, as shown in Fig. 4. The effectivemedium is anisotropic and uniaxial. We show all principal val-ues of the effective permittivity tensor. In the case of InclusionA, the geometrical limit of touching inclusions corresponds to aone-dimensional layered medium of alternating metal anddielectric layers. We know that the effective permittivity ofthis medium lies on one of the Wiener bounds. This is indeedthe case. Just as expected, the effective permittivity starts fromϵ1 or ϵ2 (depending on the choice of the host) at f � 0and then terminates at the linear or circular Wiener bound forthe s- or p-polarization, respectively, at f � 0.5. Moreover, atf � 0.5, the layered medium is invariant under the permuta-tion ϵ1↔ϵ2. Therefore, the curves AsDH and AsMH startfrom ϵ1 and ϵ2 at f � 0 and then meet at the linear Wienerbound at f � 0.5. Analogously, the curves ApDH and ApMHmeet at the circular Wiener bound at f � 0.5. This behaviorcannot be captured by the Maxwell Garnett or Bruggeman’smixing formulas, which always produce curves that connectϵ1 and ϵ2 without touching the Wiener bounds. The analyticalpredictions are not plotted in Fig. 5(c) because we knowa priori that these predictions cannot be possibly accurate.

As for Inclusion B, its geometrical limit is a set of infiniterods of a square cross section and the volume fractionf max � 0.25. For the s-polarization, the electric field in thismedium is smooth (see [1], Section 5). We therefore expectthe corresponding values of the effective permittivity to lie

on the linear Wiener bound. However, the composite is notinvariant under the permutation ϵ1↔ ϵ2. Therefore, thecurves BsDH and BsMH indeed terminate at the linear Wienerbound but at different points. For the p-polarization, the dis-placement field D is not smooth. Therefore, the curves BpDHand BpMH stay well inside the Wiener bounds and do nottouch the circular bound at f � f max.

7. CONCLUDING REMARKS

As was promised in the first part of this tutorial, we have de-rived a rigorous homogenization formula by considering a gen-eral photonic crystal in the long wavelength limit. The result isnot entirely analytical. The homogenization formula contains atensor Σ, which is a function of the contrast and geometryand must be computed numerically. Setting Σ � 0 results inthe isotropic Maxwell Garnett mixing formula. The anisotropicformula is obtained if the principal values of Σ are of the specialform Σp�χ� � βpχ∕�1 − βpχ�, where βp are constants. Hereχ � �ϵi − ϵh�∕�ϵi � 2ϵh� is the spectral parameter of thetheory. We can identify the depolarization factors appearingin the classical formula as νp � �1 − βp�∕3. If βp � 0, νp �1∕3 and the classical isotropic mixing formula is obtained.

However, there is a class of composites that are isotropic butwhose rigorous effective permittivity is not equal to theMaxwell Garnett’s prediction. For these composites, a moregeneral functional dependence of the form Σp�χ� �

Pnβnpχ∕

�1 − βnpχ� is valid, where the sum is not reduced to justone term. The composite is isotropic if Σp is independentof p. This is a more general condition than the assumptionthat βnp � 0, which is, essentially, the sufficient and necessarycondition for the isotropic Maxwell Garnett mixing formula

–4 –2 2 4

2

4

6

1

2

Re( )

Im( )

MGDH

MGMH

BG

LWB

CWB

SDH

SMH

–4 –2 2 4

2

4

6

1

2

Re( )

Im( )

MGDH

MGMH

BG

LWB

CWB

CDH

CMH

–4 –2 2 4

2

4

6

1

2

Re( )

Im( )

LWB

CWB

AsDH

ApDH

BsDH

BpDH

AsMH

ApMH

BsMH

BpMH

(c)(b)(a)

Fig. 5. Numerical and analytical results for the effective permittivity of different composites. The inclusions are (a) spherical, (b) cubic, and(c) parallelepipedal. The shapes of the parallelepipeds and the corresponding polarization labels (directions of the fundamental Bloch harmonicP0), as shown in Fig. 4. The volume fraction of the inclusions is varied from zero to the maximum value f max allowed by the geometry. In particular,(a) f max � π∕6, (b) f max � 1, and (c) f max � 0.5, Inclusion A, and f max � 0.25, Inclusion B. Results are shown for both cases when ϵh � ϵ1,ϵi � ϵ2 (dielectric host) and ϵh � ϵ2, ϵi � ϵ1 (metal host). The labeling of various curves and data sets is of the form X or XY or X vY . Here X takesthe values X � MG (isotropic Maxwell Garnett mixing formula), X � BG (isotropic Bruggeman’s mixing formula), X � S [Eq. (80) for sphericalinclusions], X � C [Eq. (80) for cubic inclusions], and X � A; B [Eq. (80) for parallelepipedal inclusions A or B shown in Fig. 4]. The polarizationlabel takes two values v � s or v � p and is applicable only to X � A; B (see Fig. 4). Finally, the label Y takes two values: Y � DH for the dielectrichost and Y � MH for the metal host. LWB and CWB are the linear and circular Wiener bounds, respectitvely. Note that some data points shown ascircles were computed for the values of f slightly different from the exact limiting values (for example, f � 0.01 instead of f � 0 ) to avoidcomplete overlap and obscuration of two or more data points.


to be precise. In general, the Maxwell-Garnett-typefamily of approximations assume that there is only oneresonance of the function Σp�χ�. In actual composites, severalresonances can be present simultaneously for any givenpolarization.

The problem we have considered in Section 5 is known inthe homogenization literature as the cell problem. There existmany mathematical formulations of the cell problem andmany different approaches to its solutions. The specific ap-proach of Section 5 utilizes the basis of plane waves. It alsoreduces the numerical part to computing the tensor Σ, whichwas discussed above. Other approaches may use local bases orcompute the effective permittivity directly, without casting itfirst in a special form resembling the Maxwell Garnett mixingformula.

When we considered waves on lattices of polarized particlesor in photonic crystals, we have used integral equations as themathematical point of departure in order to arrive at the desiredresult in the most direct and economical way. However, there isnothing special in this approach. One can as well start fromthe differential equations and arrive at exactly the same result.Still, the fact that the Maxwell Garnett mixing formula seems tobe “embedded” in the integral equations of classical electrody-namics is quite remarkable and we have intentionally used theintegral equations to emphasize this point.

The numerical examples of Section 6 show that the isotropicMaxwell Garnett mixing formula is quite robust and canwork reasonably well, even beyond the limit of very low volumefractions. Of course, this statement only concerns compositeswithout strong resonance phenomena (which we have notdiscussed in the tutorial) and should not be understood toogenerally.

Finally, many of the ideas discussed in this tutorial (espe-cially in the first part) and many interesting and relevantdiscussions, in particular, of the concept of the local field,can be found in the book [39]. Unfortunately, as of this writing,this book has not been translated into English.

Funding. Agence Nationale de la Recherche (ANR) (ANR-11-IDEX-0001-02).

Acknowledgment. This work has been carried outthanks to the support of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir”French Government program and managed by the FrenchNational Research Agency (ANR).

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