Some alternatives for calculating multipole expansions of the electromagnetic radiation field

We discuss the multipolar expansion of the electromagnetic field with an emphasis on the radiated field. We investigate if the employment of Jefimenko's equations brings a new insight into the calculation of the radiation field. We show that the affirmation is valid if one finds an interesting example in which inverting the order between spatial derivatives and integration is not allowed. Further, we consider the generalization of the multipolar expansion of the power radiated by a confined system of charges and currents to a higher arbitrary order.


I. INTRODUCTION
Despite the successful and long history of the electromagnetic field theory, there are several topics open to new theoretical and pedagogical contributions. One of them concerns the formalism of multipole expansion of the field, in general, and of the radiated one, in particular. Another issue is related to the importance of Jefimenko's equations in the study of such problems. Motivated by recent publications on the topic, we discuss some features of this type of problems.
The multipole expansion of the electromagnetic field in Cartesian coordinates is exposed in electrodynamics textbooks, as the well-known Refs. 1 and 2 . Ordinarily, these expansions are calculated only in the first two or three orders, the higher-orders being considered too complicated. As Jackson writes in his textbook, the labor involved in manipulating terms in the expansion of the vector potential becomes increasingly prohibitive as the expansion is extended beyond the electric quadrupole terms (see Ref. 2 , pp 415-416). For this reason and due to the applicability only in the long-wavelength range, another treatment, based on the spherical tensors and the solutions of Helmholtz equation is preferred. This alternative has also a larger domain of applications. Actually, starting from the results obtained employing this calculation technique, the reader can verify what effort is involved when returning to the multipole Cartesian moments which offer a higher physical transparency (see Ref. 3 ). A relatively recent textbook 4 and a paper 5 , the last related to the importance of Jefimenko's equations for expressing the electric and magnetic field when discussing the radiation theory, brought our attention on a very hard formalism employed for the calculation of even the first three or four terms of the expansion series. Though there are some prescriptions in the literature 6 , 7 , 8 for calculating higher-order terms of the multipole series based on a simple algebraic formalism of tensorial analysis, it seems that there is some reticence in using this last technique. For this reason, one of the aims of the present paper is to show how one can hide, as much as possible, the higher-order tensors behind some vectors, reducing the calculation technique to the formalism of an ordinary vectorial algebra or analysis.
Another aim of the paper is to investigate if the use of Jefimenko's equations brings, as sometimes presented in the literature, a new insight in the calculation of the radiation field. We show that unless one finds an example where the spatial derivative and the integral operations can not be inverted, it is not always a necessary complication. As already mentioned, we also use the opportunity to explain some advanced features of the multipolar expansions in the field radiation theory.
We start in section II by shortly presenting the notation convention we use and by giving a general formalism for handling multipolar expansions in Cartesian coordinates. In section III, we derive the radiated electric and magnetic field without using the retarded potentials, while in section IV we present characteristics of the calculation for the radiation field when employing Jefimenko's equations. The advantages and disadvantages of different approaches are analyzed. In section V, we further discuss some features of the calculation for the radiated power, with an emphasis on the 4 − th order approximation in d/λ < 1. Finally, in section VI, we give the guidelines for the general tensorial calculus of the electric and magnetic moments. The last section is reserved for conclusions.

II. GENERAL FORMALISM
We write Maxwell equations with a notation independent of the unit system ("system free" Maxwell equations): where ε 0 , µ 0 , α are proportional factors depending on the system of units and are satisfying the equation c is the vacuum light speed. Maxwell equations written in SI units are obtained from equations (1) for α = 1 and the SI values of ε 0 , µ 0 . For the Gauss system of units, α = c, ε 0 = 1/4π, µ 0 = 4π. With this notation, Jefimenko's equations can be written as 2  (3) and (4) the order of the derivative and the integral is inverted, one obtains the well-known relations between fields and potentials: with the retarded potentials In 5 , the authors derive the multipole expansion of the radiation field from equations (3) and (4) claiming to give an original demonstration specific for Jefimenko's equations, without employing the retarded potentials. We should agree with this claim if at least some calculation of the authors is different from those employing the potential multipole expansions which are generally used in literature. In the following, we search for a difference between the calculation presented in Ref. 5 and the standard one making use of potentials.
The goal of the exposition below is to inform on some results regarding multipolar expansion in Cartesian coordinates, too.
Let us derive the multipolar expansion of the field B given by equation (3), and written explicitly as: where e i are the unit vectors of the Cartesian axes. Writing the integral from equation (7) as we obtain the multipolar expansion of the magnetic field about O as function of r ′ using the Taylor series of the integrand: Equation (7) can now be expressed as where we employed the notation We assume one is allowed to invert orders of operations in equation (9) and to write: Equation (10) represents the curl of the multipolar expansion of the vector potential A.
Thus one can perform firstly the multipolar expansion of this potential. It is the usual procedure.
No matter what procedure is employed, a constant in the calculation is the presence of a vector a(r, t; ζ, n) defined by the Cartesian components: Here, ζ can be either an operator or a number. Generalizing to the dynamic case an algorithm used in 9 for the magnetostatic field, we introduce in equation (11) the consequence of the continuity equation, written for t 0 = t − r/c: We obtain where the Cartesian components of the n − th electric moment of the given charge distribution: are introduced. Let us define, for simplifying the notation, the vector Performing partial integration and taking into account that J vanishes on the surface ∂D, equation (12) can be written and processed as follows: Introducing the n − th order magnetic moment, as in Ref. 9 , by its Cartesian components equation (15) becomes Similarly to equation (14), we introduce the vector writing finally equation (17) as a(r, t 0 ; ζ, n) = −αζ × M(r, t 0 ; ζ, n) + 1 n + 1Ṗ (r, t 0 ; ζ, n + 1) .
With this result, the magnetic field from equation (10) can be expressed with the help of the vectors M and P: or, by a change of the summation index in the second sum, From the last expression one has no problem in identifying the multipolar expansion of the vector potential A: The calculation for the electric field can be performed in a similar manner. One obtains: Comparing equations (5) with (21) and (22), we single out the multipole expansion of the potential Φ:

III. RADIATION FIELD
For calculating the radiation field it is sufficient retaining only terms of order 1/r and 1/r 2 for r → ∞. In most textbooks one retains only the terms of order 1/r, the goal being, usually, only the derivation of the radiated energy or of the linear momentum. Actually, when the goal is the complete definition of the radiation field, one must be able to derive all transferring properties, including the angular momentum loss. These are, in fact, minimal conditions for defining a physical system. In the last case, the terms of order 1/r 2 are also necessary (see Ref. 1 -Problem 2 at the end of Section 72, and also Refs. 3 , 10 ). Although the aim of the present paper is different, we also give the formula for introducing terms of order 1/r 2 required for the evaluation of the angular momentum loss. The terms of the orders 1/r and 1/r 2 are selected making use of formula 10 : Again t 0 = t − r/c and ν i = x i /r. By A {i 1 ...in} we understand the sum over all the permutations of the symbols i q that give distinct terms. The coefficients D n are defined by the recurrence relations: The formula from equation (24) can be easily proven by recurrence.
In fact, formula (24) represents the sum of the terms corresponding to l = 0 and 1 of a general formula In this equation, C (n,l) i 1 ...in (ν) are symmetric coefficients expressed as linear combinations of products of components ν i and Kronecker symbols Considering equations (14) and (18), one can see that P(r, t 0 ; ζ, n) and M(r, t 0 ; ζ, n) are solutions of the homogeneous wave equation for r = 0. Consequently, one can apply the formula (24) for these quantities. Let us consider the multiple derivative as, for example, and similarly for P(r, t 0 ; ∇, n). Using equation (27) and the equivalent relation for P(r, t 0 ; ∇, n) in equation (20), we obtain the first approximation of the multipolar expansion of radiated magnetic field, which is sufficient for calculating the radiated energy and the linear momentum: For the field E rad , we obtain We made use of equation (2). From equations (21) and (23), using equation (24), we get easily the expansions of the radiation field potentials: and Given the above expressions, one can verify that the relations between fields and potentials are: These parts (proportional to 1/r) from the radiated electric and magnetic fields are purely transverse fields, satisfying the properties (see also 10 ): IV. IS REALLY THE RADIATION FIELD CALCULATION FROM JEFI-

MENKO'S EQUATIONS A NEW INSIGHT IN THE RADIATION THEORY ?
In Ref. 1 , calculating the radiation field (in the first approximation), the retarded potentials are approximated by the formulae (66.1) and (66.2) of this reference, written here with the "system free" notation: As suggested in Ref. 1 we can write From this last equation we can extract the term of order 1/r which represents the radiated field (precisely, the first approximation of this field): This is, in fact, equation (28)  Introducing also the approximate expression E starting from equation (4), we obtain Using the continuity equation written in the point r ′ at the retarded time t − R/c, it resultṡ The first term from the right-hand side of the last equation gives no contribution to the integral from equation (38) The vector a was defined in equation (11). After introducing its expression as a function of the vectors associated to the electric and magnetic moment, equation (19), the radiated magnetic field becomes The final step is changing n → n − 1 in the second sum of equation (42). We obtain again equation (28).
We can say that the result (42), which is the multipole expansion of the radiation field, was determined via Jefimenko's equations.
Alternatively, after reaching the expression given by equation (9), we can perform the multipolar expansion of the radiation field in such a manner that we can also say that it is obtained via Jefimenko's equations. We extract from the integrand in equation (9) the terms of the order 1/r (and 1/r 2 for a complete definition of this field): arriving, as expected, to equation (41).
We remind the reader that for obtaining the results of equations (20) and (22) we admitted the commutation of the derivative with respect to the spatial coordinates, with the series expansion and the integral on the domain D. In the present section, for the case of the radiation field, we avoid the inversion of the derivative with the integral operation. Regarding the commutation with the Taylor expansion, we consider that such an operation cannot be avoided as long as we want to emphasize the multipolar moments.
As a short extension in the argumentation of the utility of Jefimenko's equations, we show how one can avoid the commutation between derivative and integral for an arbitrary point in the exterior of the domain D. Indeed, using equation (26) in equation (9), we can write the magnetic field with the help of time derivatives Here, Similar to the derivation of equation (17) from equation (11), we get: Inserting this equation in equation (44), we obtain As anticipated, it represents the multipolar expansion of the magnetic field for any point in the exterior of the domain D and it is obtained without inverting the spatial derivative with the integral operation. For the approximation of the radiated field (in 1/r), one can easily verify that writing equation (45) for l = 0 and taking we get equation (28).
In conclusion, employing Jefimenko's equation in the radiation theory could bring a new insight only if the inversion of the spatial derivative and the integral operation is not allowed.
We admit we were unable to find an interesting example where an inversion is not permitted, at least for generalized distributions. However, it might be possible to find such examples, and, in this case, the indispensable character of Jefimenko's equations would be obvious.
Otherwise, for the regular cases, it appears as an unnecessary complication.

V. SOME FEATURES OF THE RADIATED POWER CALCULATION
The Poynting vector is The total radiated power may be written as the limit of a surface integral on a sphere centered in O, of radius r, for r → ∞. Let us express the energy current in the radiation approximation corresponding to the sphere of radius r at the time t: The quantity N(S, Σ r ; t) dt represents, therefore, the energy which crosses the sphere Σ r  µ(t; ν, n) = r M(r, t; ν, n), π(t; ν, n) = r P(r, t; ν, n).
The expression for the radiation intensity I is Employing the notation f , n = ∂ n f (. . . , t)/∂t n , and specifying only the argument n in µ and π when there is not a case of confusion, we can write One can calculate the averaged quantities from the last equation using formula 6 : In equation (51) all terms containing an odd number of factors ν vanish and we can retain only terms with an even number of these factors. We point out that µ(n) or π(n) contain n − 1 factors ν (see equations (14), (18) and (49)).
We have to calculate expressions as, for example, This is cumbersome even for the first approximations.
Before discussing and applying the above results, we stress an essential issue for the existence of a precise approximation criteria when a finite number of terms in equation (82) is retained. Let us consider the source from D being a system of N point-like electric charges q 1 . . . q N . Therefore, where r (i) (t) represents the position vector of the particle i. In this case, and Let us suppose the particles oscillating with a pulsation ω = 2πc/λ, i.e.
For the amplitudes of the source, we identify the orders of magnitude: Obviously, the same relations can be written for any x i . For the time derivatives of the tensors P (n) and M (n) we can conclude that With the approximation criteria considered above, we start by calculating the terms from equation (51) up to the 4−th order in the parameter ζ. For the electric and magnetic dipolar moments we use the usual notation p and m, respectively. We select from equation (51) all nonvanishing terms for (n, m) = (1, 1): Inserting the result for the even combinations ν i ν j , we obtain the well-known expression for the radiation in the dipolar approximation: the vanishing ones : (62) The first term, written explicitly with omission of dots representing the time derivatives of no significance for the tensorial relations, is because of the symmetry of the electric quadrupole moment. For the second term, and finally, Writing equation (51) for (n, m) = (2, 2), we discard the terms of order 6 in ζ as, for example, ··· µ (2)· ··· µ (2). In this approximation, Terms of order 6 in ζ exist also for (m, n) = (1, 3) or (3,1). Retaining only terms of order 4, we get: Adding up equations (61), (63), (64) and (65), we obtain the 4 − th order approximation of the total radiated power: The output can be partially compared with a well-known result from literature (see Refs. 1 and 2 ), but for this we have to introduce the irreducible electric and magnetic momenta defined as symmetric trace-free ("STF") Cartesian tensors. Let us consider a n − th order tensor T (n) and the corresponding projections S(T (n) ) and T (T (n) ) on the subspaces of symmetric and STF tensors. For the case of the electric moment P (n) , this is a symmetric tensor and one has just to establish their STF projection. Let us consider the simplest case of the quadrupolar electric moment P (2) . Writing the components P ij as there is a unique value of the parameter λ such that Π (2) = T (P (2) ). For λ = P qq /3 , In equation (66) the octupolar electric moment P (3) is present. The STF projection can be calculated searching the first order tensor Λ (1) such that the STF projection Π (3) = T P (3) is given by the components From the condition of vanishing traces of the tensor Π (3) , one easily obtains: Concerning the magnetic quadrupolar moment M (2) , we have a simple procedure for STF projection. Let us write the identity where the first bracket represents the symmetric part of this tensor, and the second one, the antisymmetric one. The symmetric part is, for this case (n = 2), a STF tensor Γ (2) = T (M (2) ). Therefore, where Since Combining the magnetic quadrupolar and the electric octupolar terms from equation (66), with their expressions from equations (67), (68), and (70), we get a consequence of the traceless character of Π (2) , Π (3) and Γ (2) . Here, the vector is introduced. The last expression is obtained from equations (71), (69) and applying the continuity equation together with an operation of partial integration. This is the so-called electric toroidal dipole moment 16 , 17 . We can write the radiation intensity I (4) in terms of STF projections of electromagnetic momenta: The toroidal contribution from equation (74) is obtained in Ref. 19 , too, by a calculation close to the procedure from the present paper.
Let us consider the contribution of the electric quadrupolar moment which, as seen from equation (28), is proportional to the third time derivative of the vector The substitution in equation (75) gives One can see that B(r, t) and E(r, t) given by equations (20) and (22) are also invariant if one performs the substitution (76). It is an exercise for the reader to prove that the substitution (76) in the expansions of the potentials A and Φ given by equations (21) and (23) has as effect a gauge transformation of these potentials 8 Unfortunately, this type of invariance is not valid for magnetic moments and, for n ≥ 3, for electric ones. In these cases, the physical results are not invariant with respect to the transformations P (n) → T (P n) ), M (n) → T (M n) ). , P . These new tensors do not necessarily coincide with the corresponding projections T (M (m) ), T (P (n) ) as it will be seen in the following. In this situation, the calculation of the terms from equation (51) obtained by the corresponding substitutions µ(m) → µ(m), π(n) → π(n) is more easily performed. Indeed, if A (n) , B (m) are STF tensors, then 20 Here, since A (n) and B (m) are symmetric tensors, it is of no importance what is the order of factors in the contraction A (n) .B (m) ( noted as A (n) ||B (m) , too). Obviously, the result of a such contraction is a STF tensor of |n − m| − th order. All the terms in equation (51) can be calculated using equation (78) and ν · µ(t; ν, n) ν · µ(t; ν, m) = n! δ nm (2n + 1)!! M (n) . M (m) .
Using these results, we obtain the final expression of the (total) radiated power: ,n+1 (t). P , P infinite series. This is not so catastrophic since for practically applications only a finite number of terms from equation (82) is necessary, as it will be seen in the following. Obviously, one can apply equation (59) for the STFprojections of the multipole momenta.
For an answer to the question how one can use equation (82) for calculating the radiation intensity, let us return to the problem of calculating this quantity up to the 4 − th order in the parameter ζ. Since, together with P (2) , the momenta M (2) and P (3) give contributions in this order, we have to consider in equation (82) the STF projections Γ (2) and Π (3) , too.
This time the field is not invariant for the substitutions Let us calculate the effect of the first substitution from equation (83) in equation (28) where N = e i N i = N (1) . From equation (28) we can see that the alteration of B by the substitution M (2) → Γ (2) is compensated by the following transformation of the dipolar electric moment: Let us consider the substitution P (3) → Π (3) in equation (28). The term affected is: where Λ = e i Λ i . The alteration of B by the substitution P (3) → Π (3) is compensated by the following transformation of the dipolar electric moment: The total alteration of B by the two substitutions (83) is compensated by the transformation with t defined by equation (73). Now, in equation (82) we have to consider P (1) = p given by equation (86), M = m, P = Π (2) , M = Γ (2) , P = Π (3) and, for n ≥ 4, the primitive momenta which contribute only with terms of orders greater than 4 in ζ. We can write: The quadrupolar magnetic and octupolar electric terms are not written because these are of order 6 in ζ. Since This calculation offers a suitable example for understanding the general scheme of tensor reduction for electric and magnetic moments, as well as for the usage of formula (82) when describing the radiated power. Replacing an electric moment of order n by its STFprojection induces in the tensors of inferior order, compensating terms of order ζ n . The general term in the expansion (82) is of order ζ 2n . If we are interested in the approximation of I up to terms of order ζ 2k , then we have to replace the tensors related to the moments P (2k−1) and M (2k−2) by STF-projections and to take into account all compensating terms for the lower-order tensors.

REDUCING THE ELECTRIC AND MAGNETIC MOMENTUM TENSORS
In this section we list the required formulae for the generalized tensor reducing procedure.
The equations are given without the related proofs since they can be found in the literature.
Let S (n) be a symmetric tensor of rank n. Its STF-projection results from the following formula: The operator Λ is defined on the account of a formula for the STF-projection of a symmetric tensor given in 6 (the book 21 is cited as the origin of this formula): The proof can be found in 22 .
[a] stands for the integer part of a and the notation S (n;p) indicates p pairs of contracted indices. In the following, for simplifying the notation, all arguments of the operator Λ should be considered as symmetric tensor i.e. Λ(T (n) ) = Λ(S(T (n) ) for any tensor T n) . The same applies to the operator T : T (T (n) ) = T (S(T (n) ), S being the symmetrization operator.
In the symmetrization process we have to calculate the symmetric projections of some tensors L (n) of the magnetic moment type: they are symmetric in the first n − 1 indices and the contraction of i n with any index i q , q = 1 . . . n − 1 gives a null result. For the symmetric projection of such a tensor, we introduce the formula: where N (λ) i 1 ...i n−1 is the component with the i λ index suppressed. The operator N defines a correspondence between L (n) and a tensor of rank (n − 1) of the same type from the symmetry point of view. Particularly, where a n × b is the tensor defined by the components a n × b i 1 ...in = a i 1 . . . a i n−1 (a × b) in .
The reader is encouraged to apply these formulae for the cases N = 4, 5, 6 and to find convenient intermediary calculation. We point out that in equation (91) the normalisation of the quantities T (n) is chosen such that the quantities T (n) 1 coincide with the electric toroidal moments given in literature at least for n up to 3.

VII. CONCLUSION AND DISCUSSION
After an introduction on the formalism for multipolar expansions of the electric and magnetic field, we dedicated sections III and IV to the main purpose of this article: the analysis of different methods for calculating the radiated field with an emphasis on the novelty the use of Jefimenko's equations can bring. In the last two sections we presented aspects of the radiated power calculations from the tensorial point of view.
From the analysis of sections III and IV we can draw some conclusions on the utility of Jefimenko's equations when considering the multipolar expansion problem. As one can notice, when deriving equation (10) from (9), no matter if we work with potentials or directly with fields, the inversion of the spatial derivative with the integral on the domain D is mandatory. In Section IV it was proven that the multipole expansion of the fields E and B can be obtained generally and directly from Jefimenko's equations. All expanding operations are performed on the integrand, but, still under the assumption that the integration operation is distributive with respect to the Taylor series. It remains only to argue the necessity of the corresponding additional calculation effort for applying this procedure.
As an additional remark, we emphasize that in the same sections we tried to remind the reader that if one wants to completely describe the radiative systems, one has to include besides terms of order 1/r, the 1/r 2 contributions from the field expansions.
Ref. 5 is part of a paper series trying to emphasize the theoretical and practical importance of Jefimenko's equations. These equations are considered as extraordinarily powerful and illuminating as the authors of Ref. 13 write. We have nothing against the open enthusiasm in these papers. We neither dispute the beauty of the result regarding the calculation of the retarded fields E and B directly from Maxwell's equations, without having to introduce and handle the potentials. Maybe a series of applications based on these equations are more efficacious and physically more transparent. Although, we are circumspect concerning the axiomatic treatment of the electromagnetic theory starting from these equations (opposite to opinions from e.g. Refs. 14 , 15 ), but, this subject will be discussed elsewhere.
In the second part of the paper we gave a detailed calculation for the radiated power up to the 4 − th order in d/λ < 1. The method can be similarly applied to higher-orders.
We related our procedure to existing prescriptions in the literature and we underlined what omissions might show when particular terms of the multipolar expansions are neglected. We hope we were able to convince the reader how powerful and not so complicated a consistent vectorial/tensorial computation is.