An electromagnetic multipole expansion beyond the long-wavelength approximation

1 MB·

by  R. Alaee

An electromagnetic multipole expansion beyond the long-wavelength approximation
R. Alaee,∗1,2 C. Rockstuhl1,3 and I. Fernandez-Corbaton3
1Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany
2Max Planck Institute for the Science of Light, Erlangen 91058, Germany
3Institute of Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany and
∗Corresponding author:[e-mail]
Themultipoleexpansionisakeytoolinthestudyoflight-matterinteractions. Alltheinformation
about the radiation of and coupling to electromagnetic fields of a given charge-density distribution
iscondensedintofewnumbers: Themultipolemomentsofthesource. Thesenumbersarefrequently
computedwithexpressionsobtainedafterthelong-wavelengthapproximation. Here,wederiveexact
expressions for the multipole moments of dynamic sources that resemble in their simplicity their
approximatecounterparts. Wevalidateournewexpressionsagainstanalyticalresultsforaspherical
7 source, and then use them to calculate the induced moments for some selected sources with a non-
1 trivial shape. The comparison of the results to those obtained with approximate expressions shows
0
a considerable disagreement even for sources of subwavelength size. Our expressions are relevant
2
for any scientific area dealing with the interaction between the electromagnetic field and material

n systems.
a
J PACSnumbers: 78.67.Pt,13.40.Em,78.67.Bf,03.50.De

3

] The multipolar decomposition of a given charge- directional light emission6–9, manipulating and con-
s current distribution is taught in every undergraduate trolling spontaneous emission10–12, light perfect ab-
c
i courseinphysics. Theresultingsetofnumbersarecalled sorption13–15, electromagnetic cloaking16,17, and optical
pt the multipolar moments. They are classified according (pulling, pushing, and lateral) forces18–22. In all these
o to their order, i.e. dipoles, quadrupoles etc... For each cases, an external field induces displacement or conduc-
. order, there are electric and magnetic multipolar mo- tive currents into the samples. These induced currents
s
c ments. Each multipolar moment is uniquely connected are the source of the scattered field. But: How can we
i to a corresponding multipolar field. Their importance calculatethemultipolemomentsoftheseinducedcurrent
s
y stems from the fact that the multipolar moments of a distributions?
h charge-current distribution completely characterize both
Exact expressions exists and can be found in standard
p theradiationofelectromagneticfieldsbythesource,and
textbooks, e.g. Eq. (7.20) in Ref. 23 or Eq. (9.165) in
[
the coupling of external fields onto it. The multipolar Ref. 24 (without the magnetization current therein) and

1 decomposition is important in any scientific area dealing a new formulation have been recently derived in Ref. 25.
v with the interaction between the electromagnetic field
However, up to now they are not frequently used in the
5 and material systems. In particle physics, the multipole
literature. One reason for this may be their complexity,
5
moments of the nuclei provide information on the distri-
7 i.e. theyfeaturedifferentialoperatorslikethecurland/or
bution of charges inside the nucleus. In chemistry, the
0 vector spherical harmonics. Instead, a long-wavelength
dipole and quadrupolar polarizabilities of a molecule de-
0 approximation that considerably simplifies the expres-
. termine most of its properties. In electrical engineering, sions is very often used. Their integrands contain al-
1
themultipoleexpansionisusedtoquantifytheradiation
0 gebraic functions of the coordinate and current density
from antennas. And the list goes on.
7 vectors. Moreover,theapproximateexpressionsresemble
1 InthisLetter,wepresentnewexactexpressionsforthe those for the multipole moments derived in the context
: multipolar decomposition of an electric charge-current of electro-statics and magneto-statics. To set a starting
v
i distribution. Theyprovideastraightforwardpathforup- point, these expressions are documented in Table I. The
X
grading analytical and numerical models currently using so-called toroidal moments are also included in these ex-
r the long-wavelength approximation. After the upgrade, pression as the second term in the electric multipole mo-
a
the models become exact. The expressions that we pro- ments26.
vide are directly applicable to the many areas where the
Let us investigate the range of validity of the expres-
multipoledecompositionofelectricalcurrentdensitydis-
sions in Table I by comparing them with Mie theory. In
tributions is used. For the sake of concreteness, in this
Mietheory,thesolutionforthescatteringofaplanewave
article we apply them to a specific field: Nanophotonics.
by a sphere is obtained without any approximation, i.e.
In nanophotonics, one purpose is to control and ma- it is valid for any wavelength and size of the sphere. For
nipulate light on the nanoscale. Plasmonic or high- example, Mie theory allows to compute the individual
index dielectric nanoparticles are frequently used for contributionsofeachinducedelectricandmagneticmul-
this purpose 1,2. The multipole expansion provides in- tipole moment to the total scattering cross-section. We
sight into several optical phenomena, such as Fano res- will compare those exact individual contributions to the
onances3,4, electromagnetically-induced-transparency5, ones obtained using the formulas in Table I.
2
TABLEI:Multipolemomentsinlong-wavelength approximation;electricdipolemoment(ED,i.e. p ),magneticdipolemoment
α
(MD, i.e. m ), electric quadrupole moment (EQ, i.e. Qe ) and magnetic quadrupole moment (MQ, i.e. Qm ) where α,β =
α αβ αβ
x,y,z.
ED: p ≈− 1 (cid:110)´ d3rJω+ k2 ´ d3r(cid:2)(r·J )r −2r2Jω(cid:3)(cid:111) (T1−1)
α iω α 10 ω α α
´
MD: m ≈ 1 d3r(r×J ) (T1−2)
α 2 ω α
ˆ
(cid:26)
Qe ≈− 1 d3r(cid:2)3(cid:0)r Jω+r Jω(cid:1)−2(r·J )δ (cid:3)
αβ iω β α α β ω αβ
EQ: ˆ (T1−3)
+k2 d3r(cid:2)4r r (r·J )−5r2(r J +r J )+2r2(r·J )δ (cid:3)(cid:27)
14 α β ω α β β α ω αβ
´ (cid:110) (cid:111)
MQ: Qm ≈ d3r r (r×J ) +r (r×J ) (T1−4)
αβ α ω β β ω α

(a) (c)

2 2

(b) (d)

2 2

FIG.1: ContributionofeachmultipolemomenttothescatteringcrosssectioncalculatedwithMietheoryandcalculatedwith
the approximate expressions (Table I): a) For a dielectric sphere as a function of the particle’s size parameter 2a/λ. b) For a
goldspherewithafixedradiusofa=250nm. c)andd)RelativeerrorbetweenthemultipolemomentscalculatedwiththeMie
theoryandcalculatedwiththeapproximateexpressions. Notethatthecontributionofeachmultipolemomenttothescattering
cross section is normalized to λ2/2π. For spherical particle, there is a universal limit for each multipole, i.e. (2j+1)λ2/2π.
For example, for a dipolar particle (i.e. j=1), the maximum cross section is 3λ2/2π22,27.
We consider a high-index dielectric nanosphere and tivity of the dielectric sphere is assumed to be (cid:15) =2.52.
r
a gold nanosphere. Both are illuminated with a lin- Dispersive material properties as documented in the lit-
early x-polarized plane wave that propagates in the z- erature are considered for gold28. We assume air as the
direction. The induced multipole moments in both cases host medium. We used a numerical finite element solver
can be computed using the expressions in Table I. The to obtain the electric field distributions29.
induced electric current density is obtained by using
J (r) = iω(cid:15) ((cid:15) −1)E (r), where E (r) is the electric
ω 0 r ω ω
fielddistribution,(cid:15)0 isthepermittivityoffreespace,and Using the multipole moments, it is easy to obtain the
(cid:15)r is the relative permittivity of the sphere. The permit- total scattering cross section, i.e. the sum of the contri-
3

butions from different multipole moments, as24:
Ctotal = Cp +Cm +CQe +CQm +··· (1)
sca sca sca sca sca
(cid:34) (cid:32) (cid:33)
= k4 (cid:88) |p |2+ |mα|2 +
6πε2|E |2 α c
0 inc α
1210(cid:88)(cid:32)(cid:12)(cid:12)kQeαβ(cid:12)(cid:12)2+(cid:12)(cid:12)(cid:12)(cid:12)kQcmαβ(cid:12)(cid:12)(cid:12)(cid:12)2(cid:33)+···

αβ
2
where, p , m are the electric and magnetic dipole
α α
moments, respectively. Qe , Qm are the electric and
αβ αβ
magneticquadrupolemoments,respectively. |E |isthe
inc
amplitudeoftheincidentelectricfield,kisthewavenum-

ber, and c is the speed of light.
Figure 1 shows the contribution of each multipole mo-
ment to the scattering cross section for a high-index di-
electric as well as a gold nanosphere. The results ob-
tained using the approximate expression are compared
2
with those obtained from Mie theory. It can be seen
that, uponincreasingthea/λratio, thereisalargedevi-
ation between the scattering cross section obtained from
the expressions in Table I and the Mie theory. The rela-
tive error between the two approaches is shown in Fig. 1
(c) and (d). The relative error is more than 100% for
the dielectric sphere at 2a/λ ≈ 0.75 for both electric FIG. 2: Contribution of each multipole moment to the scat-
and magnetic dipole moments. This large deviation oc- teringcrosssectioncalculatedwithMietheoryandcalculated
curs because the expressions in Table I are obtained in withtheexactexpressions(TableII).a)Foradielectricsphere
the long-wavelength approximation24, i.e. they are only with a relative permittivity of (cid:15)r = 2.52 as a function of the
valid for particles small compared to the wavelength of particle’s size parameter 2a/λ b) For a gold sphere with a
fixed radius of a=250 nm.
the incident light (i.e. D (cid:28) λ where D is the biggest
dimension of the particle).
Thus, the long-wavelength expressions in Table I can
for all the multipolar moments of a spatially confined
not be used for large particles (compared to the wave-
electric current density distribution. They are valid for
length). The large deviation observed in Fig. 1 (c) and
any size of the distribution. Crucially, the Fourier space
(d) for different multipole moments will significantly af-
partoftheintegralsdoesnotdependonthecurrentden-
fect the quantitative prediction of multipolar interfer-
sity. The results in Tab. II are obtained after carrying
ence,whichisthemainphysicalmechanismbehindFano
resonances3,4,directionallightemission6–9,andlightper- out the Fourier space integrals for the electric and mag-
fectabsorption13,14. Moreover,anyphysicalquantityob- neticdipolarandquadrupolarorders(seethesupplemen-
tary material). Our results have two main advantages
tained using the multipole moments of TableI, e.g. ab-
with respect to other exact expressions23–25. One is that
sorption/extinctioncrosssection,oropticaltorque/force,
ourformulasaresimpler: Thepreviouslyexistingexpres-
carries a corresponding error. Therefore, the application
sions contain differential operators and/or vector spher-
oftheexactexpressionsforthemultipolemomentsisim-
ical harmonics inside the integrands, while ours contain
portantsinceitprovidesabetterunderstandingofallthe
algebraic functions of the coordinate and current den-
highlighted optical phenomena and enables its quantita-
sity vectors, and spherical Bessel functions. The other
tive prediction.
advantage is that the previous expressions lack the simi-
To improve the situation and indeed to provide error-
larity to their long-wavelength approximations that ours
free expressions, we now derive exact expressions for the
have(compareTabs. IandII).Therefor,ourexpressions
induced electric and magnetic multipole moments that
allowastraightforwardupgradeofanalyticalandnumer-
arevalidforanywavelengthandsize (seeTableII).They
ical models using the approximated long-wavelength ex-
can be used to compute the multipole moments of arbi-
pressions. After the upgrade, the models become exact.
trarily shaped particles. Our exact expressions for mul-
tipole moments are very similar to the well-known ex- Basically, any code that has been previously imple-
pression obtained in long-wavelength approximation(see mented to compute the multipole moments with the ap-
Table I). proximate expression can be made to be accurate with a
Our starting point are the hybrid integrals in Fourier marginal change.
andcoordinatespaceinEq.14ofRef.26(seethesupple- In order to show the correctness of the expressions in
mentary material). Theseintegralsareexactexpressions TableII,wecomputethecontributionsofdifferentmulti-
4
TABLE II: Exact multipole moments; electric dipole moment (ED, i.e. p ), magnetic dipole moment (MD, i.e. m ), electric
α α
quadrupolemoment(EQ,i.e. Qe )andmagneticquadrupolemoment(MQ,i.e. Qm )whereα,β =x,y,z. Thederivationcan
αβ αβ
be found in the supplementary material.
ED: p =− 1 (cid:110)´ d3rJωj (kr)+ k2 ´ d3r(cid:2)3(r·J )r −r2Jω(cid:3)j2(kr)(cid:111) (T2−1)
α iω α 0 2 ω α α (kr)2
´
MD: m = 3 d3r(r×J ) j1(kr) (T2−2)
α 2 ω α kr
ˆ
(cid:26)
Qe =− 3 d3r(cid:2)3(cid:0)r Jω+r Jω(cid:1)−2(r·J )δ (cid:3)j1(kr)
αβ iω β α α β ω αβ kr
EQ: ˆ (cid:27) (T2−3)
+2k2 d3r(cid:2)5r r (r·J )−(r J +r J )r2−r2(r·J )δ (cid:3)j3(kr)
α β ω α β β α ω αβ (kr)3
´ (cid:110) (cid:111)
MQ: Qm =15 d3r r (r×J ) +r (r×J ) j2(kr) (T2−4)
αβ α ω β β ω α (kr)2

Uptonow,wehaveconsideredonlysphericalparticles
(a)
that could also be studied with Mie theory. We now use
the new expressions in Table II to calculate the induced
moments of a canonical particle made of two coupled
nanopatches. Its geometry and the results are shown
in Fig. 3. The coupled nanopatches support a strong
electric and magnetic response. The radius and thick-

ness of the coupled disk is assumed to be a = 250 nm,
t = 80 nm, respectively. The spacer between the two
disks is g =120 nm. It can be seen that there is a signif-
icant deviation between the contributions to the scatter-
ingcrosssectionfromthedifferentmultipolemomentsas
2
predicted by the approximate (Table I) and by the ex-
(b)
act (Table II) expressions. The relative error is shown in

Fig. 3 (b). Some of them reach 25% for a particle size of
about half the wavelength.
FIG. 3: a) Contribution of each multipole moment to the Finally, there are a few important facts about the ex-
scattering cross section calculated with the approximate ex- pressions shown in Table II that are worth highlighting:
pressions (Table I) and calculated with the exact expressions
(Table II) for a coupled nanopatch with given geometrical • Theexactmultipolemomentsarevalidforanypar-
parameters as a function of the wavelength. b) Relative er-
ticle’s size (i.e. a/λ) and arbitrarily shaped parti-
ror between the multipole moments calculated with the ap-
cles. Note, any physical quantities obtained from
proximateexpression(TableI)andcalculatedwiththeexact
the these multipole moments will be exact.
expression (Table II).
• Thereisno needtointroduceathirdfamilyofmul-
tipole (i.e. toroidal multipole moments). Our new
polemomentstothescatteringcrosssectionandcompare expressionsrevealthattoroidalmultipolemoments
them to those obtained with Mie theory. Figure 2 shows are only the higher order terms in the expansion of
the different contributions as a function of the particle’s the electric multipole moments30.
size parameter 2a/λ for both the previously considered
dielectrid and gold spheres. It can be seen that the re- • The well known approximate multipole moments
sults from our exact expressions are in excellent agree- in Table I can be obtained from the expressions in
mentwiththosefromMietheory,irrespectiveofthepar- TableIIbyusingalong-wavelengthapproximation.
ticle’s size parameter. Indeed, they are indistinguishable ThismeansthattheapproximateexpressioninTa-
up to a numerical noise level. ble I can be easily recovered by making a small ar-
5
gumentapproximationtothesphericalBesselfunc- canbedirectlyappliedinthemanyareaswherethemul-
tions (see the supplementary material): tipole decomposition of electrical current density distri-
butions is used.
j (kr) ≈ 1−(kr)2/6,
0
j (kr) ≈ kr/3,
1
Acknowledgements
j (kr) ≈ (kr)2/15.
2
In summary, we have introduced new expressions for TheauthorswarmlythankDr. ZeinabMokhtari,Ren-
multipole moments [Table II] which are valid for arbi- wen Yu and Burak Gürlek for their constructive com-
trarily sized particles of any shape. The well-known ments and suggestions. We acknowledge the German
long-wavelengthexpression(TableI)arerecoveredasthe Science Foundation for support within the project RO
lowest order terms of our new exact expressions (Table 3640/7-1.
II). We have shown the correctness of our expressions by
comparingtheirresultswiththoseofMietheoryandob-
tainingacompleteagreement. Weareconfidentthatour
new exact expressions in Table II have the potential to Appendix A: Exact expressions for the multipole
moments
be used in every electrodynamics textbook and actually
should be taught in undergraduate courses in physics.
Beyond the particular case of multipolar moments in- Let us start with the hybrid integrals in Fourier and
duced by an incident field in a structure, our expressions coordinate space (see our previous work26):
(cid:113) ˆ ˆ
(2π)3
4π aωjm = (cid:88)(−i)¯l dp(cid:98)Z†jm(p(cid:98))Y¯lm(p(cid:98)) d3rJω(r)Y¯l∗m((cid:98)r)j¯l(kr), (A1)
¯lm¯
(cid:113) ˆ ˆ
(2π)3
4π bωjm = (cid:88)(−i)¯l dp(cid:98)X†jm(p(cid:98))Y¯lm(p(cid:98)) d3rJω(r)Y¯l∗m((cid:98)r)j¯l(kr), (A2)
¯lm¯
where aω and bω are exact expressions for the multipole moments (electric and magnetic, respectively) of a
jm jm
spatially localized electric current density distribution J (r). These expressions are valid for any size of the current
ω
distribution. As shown in Ref.26, only terms with ¯l = j contribute to the bω , whereas aω has contributions from
jm jm
both ¯l=j−1 and ¯l=j+1. X (p) and Z (p) are the multipolar functions in momentum space and defined as
jm (cid:98) jm (cid:98)
1
X (p) = LY (p), (A3)
jm (cid:98) (cid:112) jm (cid:98)
j(j+1)
Z (p) = ip×X (p), (A4)
jm (cid:98) (cid:98) jm (cid:98)
whereY (p)isthesphericalharmonicsandthreecomponentsofthevectorLaretheangularmomentumoperators
jm (cid:98)
for scalar function. p is the angular part of the momentum vector p (|p|= ω).
(cid:98) c
Inthefollowingsections,weuseEq.A1andEq.A2toderiveasimplifiedexactexpressionsfortheelectricmultipole
moments in both spherical and Cartesian coordinates. We start by documenting a few auxiliary expressions that will
be frequently used at a later stage and consecutively work out afterwards the details for a specific multipolar order of
either the electric and magnetic multipole moments. Following the tradition of Jackson, we treat expressions up to
the quadrupolar order. If higher orders would be needed, the discussion would be analogous.
Appendix B: Useful expressions in spherical and Cartesian coordinates
A vector a in spherical basis defined as
a = a eˆ +a eˆ +a eˆ , (B1)
1 1 0 0 −1 −1
with
6
xˆ+iyˆ
eˆ = − √ ,
1
2
eˆ = zˆ, (B2)
0
xˆ−iyˆ
eˆ = √ .
−1
2
Therefore, the relation between the Cartesian and spherical coordinates of vector a reads as:
 a  −√1 √i 0a 
1 2 2 x
 a0  =  0 0 1 ay , (B3)
a−1 √1 √i 0 az
2 2
and the cross product of two vector in spherical basis can be defined as:
 
a b −a b
1 0 0 1
a×b = ia1b−1−a−1b1 . (B4)
a b −a b
0 −1 −1 0
Let us now introduce a few useful relations between spherical and Cartesian coordinates which will be used in the
following sections.
The vector r and J in spherical basis can be written as:
ω
J = (cid:2)Jω Jω Jω (cid:3)T , r=(cid:2)r r r (cid:3)T , (B5)
ω 1 0 −1 1 0 −1
according to Eq. B3 for vectors r, J , r×J the relation between the spherical and Cartesian coordinates read as:
ω ω
x  √1 (r−1−r1) 
2
 y = √1 (r−1+r1), (B6)
2i
z r
0
Jω   √1 (cid:0)Jω −Jω(cid:1) 
Jxyω = √122i(cid:0)J−−ω11+J11ω(cid:1), (B7)
Jzω J0ω
(r×J )   (r×Jω)−√1−(r×Jω)1 
ω x 2
 (r×Jω)y  =  (r×Jω)−√1+(r×Jω)1 . (B8)
(r×J ) 2i
ω z (r×J )
ω 0
Using Eq. B4, the cross product of vector r×J reads as:
ω
   
(r×J ) Jωr −Jωr
ω 1 0 1 1 0
r×Jω = (r×Jω)0 = iJ−ω1r1−J1ωr−1 , (B9)
(r×J ) Jω r −Jωr
ω −1 −1 0 0 −1
and the scalar product is
r·J = xJω+yJω+zJω. (B10)
ω x y z
Finally, the spherical harmonics has following relations with vector r (ˆr= r)31:
r
7
   
Y (cid:114) −rˆ
11 1 3 −1
 Y10  =  rˆ0 , (B11)
2 π
Y −rˆ
1−1 1
 Y22   √rˆ−21 
 YY2210  = 12(cid:114)21π5 (cid:114)23 (cid:0)r−ˆ02√+2rrˆˆ0−rˆ1−rˆ11(cid:1), (B12)
Y2−1   − 2rˆ0rˆ1 
Y2−2 rˆ12
 −rˆ3 
√ −1
 Y33   3rˆ0rˆ−21 
 Y32  −(cid:113)3(cid:0)2rˆ2+rˆ rˆ (cid:1)rˆ 
 Y  (cid:114)  5 0 1 −1 −1 
 31  1 35  (cid:113) 
 Y  =  2(cid:0)rˆ2+3rˆ rˆ (cid:1)rˆ . (B13)
 30  2 2π  5 0 0 −1 0 
Y   (cid:113) 
 3−1   − 3(cid:0)2rˆ2+rˆ rˆ (cid:1)rˆ 
Y3−2   5 √0 1 −1 1 
Y  3rˆ rˆ2 
3−3 0 1
−rˆ3
1
Appendix C: Electric dipole moment
In this subsection, we derive the exact electric dipole moment for the spherical and Cartesian coordinates.
1. Spherical coordinates
The electric dipole moment in spherical coordinate can be found by using Eq. A2 (j =1, i.e. l=j±1=0,2) :
 aω   aω ¯l=0  aω ¯l=2
11 11 11
 aω10  =  aω10  + aω10  , (C1)
aω aω aω
1−1 1−1ˆ 1−1 ˆ
= − √1 d3rJωj (kr)− k√2 d3r(cid:2)3(cid:0)r†J (r)(cid:1)r−r2J (r)(cid:3)j2(kr),
π 3 0 2π 3 ω ω (kr)2
the derivation for the above expression can be found in our previous work26. In the next section, we introduce the
exact Cartesian electric dipole moment.
2. Cartesian coordinates
Each components of the electric dipole moment (Eq. C1) in spherical basis can be written as:
ˆ ˆ
aω = − √1 d3rJωj (kr)+ k√2 d3r(cid:2)3Jω r2−3Jωr r +Jω(cid:0)r2+r r (cid:1)(cid:3)j2(kr),
11 π 3 1 0 2π 3 −1 1 0 0 1 1 0 −1 0 (kr)2
ˆ ˆ
aω = − √1 d3rJωj (kr)+ k√2 d3r(cid:2)3Jωr r +3Jω r r −2Jω(cid:0)r2+r r (cid:1)(cid:3)j2(kr),
10 π 3 0 0 2π 3 1 −1 0 −1 0 1 0 0 −1 0 (kr)2
ˆ ˆ
aω = − √1 d3rJω j (kr)+ k√2 d3r(cid:2)Jω (cid:0)r2+r r (cid:1)−3Jωr r +3Jωr2 (cid:3)j2(kr), (C2)
1−1 π 3 −1 0 2π 3 −1 0 −1 0 0 −1 0 1 −1 (kr)2
8
Now,theCartesianelectricdipolemomentcanbefoundbyusingthefollowingtransformationbetweenthespherical
and Cartesian coordinates:
aω −aω
pω = Ce 1−1√ 11,
x 1
2
aω +aω
pω = Ce 1−√1 11,
y 1
2i
pω = Ceaω , (C3)
z 1 10
√
where Ce = 3π, which is obtained by comparing the electric field expressions in spherical24 and Cartesian coor-
1 iω
dinates, i.e. Eq. G1. Finally, we substitute Eqs. C2 in Eqs. C3 and by using Eqs. B6 and Eqs. B7, we get:
(cid:40)ˆ ˆ (cid:41)
pω = − 1 d3rJωj (kr)+ k2 d3r(cid:2)3(r·J )x−r2Jω(cid:3)j2(kr) ,
x iω x 0 2 ω x (kr)2
(cid:40)ˆ ˆ (cid:41)
pω = − 1 d3rJωj (kr)+ k2 d3r(cid:2)3(r·J )y−r2Jω(cid:3)j2(kr) , (C4)
y iω y 0 2 ω y (kr)2
(cid:40)ˆ ˆ (cid:41)
pω = − 1 d3rJωj (kr)+ k2 d3r(cid:2)3(r·J )z−r2Jω(cid:3)j2(kr) ,
z iω z 0 2 ω z (kr)2
which can be written in a short form:
(cid:40)ˆ ˆ (cid:41)
pω =− 1 d3rJωj (kr)+ k2 d3r(cid:2)3(r·J )r −r2Jω(cid:3)j2(kr)
α iω α 0 2 ω α α (kr)2
where α=x,y,z. Note that the above expression for the electric dipole moment is valid for any wavelength. This
expression is documented in Tab. 2 of the main manuscript.
3. Long-wavelength approximation
We now can make a long-wavelength approximation by using the small argument approximation to the spherical
Bessel function, i.e.
(kr)2
j (kr) ≈ 1− ,
0 6
(kr)2
j (kr) ≈ , (C5)
2 15
and obtain the expression for the approximate electric dipole moments that are only valid for sources sufficiently
small in their spatial extent with respect to the wavelength:
(cid:40)ˆ ˆ (cid:41)
pω = − 1 d3rJωj (kr)+ k2 d3r(cid:2)3(r·J )r −r2Jω(cid:3)j2(kr) ,
α iω α 0 2 ω α α (kr)2
ˆ ˆ
≈ − 1 (cid:26) d3rJω+ k2 d3r(cid:2)(r·J )r −2r2Jω(cid:3)(cid:27). (C6)
iω α 10 ω α α
This expression is documented in Tab. 1 of the main manuscript. It is important to note that the first term, i.e.
ˆ
1
− d3rJω, (C7)
iω α
9
can be found in any electrodynamics textbook24. It is often considered as the electric dipole moment. However,
we shall always keep in mind that this expression is only valid for very small particles compared to wavelength. The
second term
ˆ
− 1 k2 d3r(cid:2)(r·J )r −2r2Jω(cid:3), (C8)
iω10 ω α α
is the so-called toroidal dipole moment. They have been incorrectly called the third family of multipole moments.
However,basedonourderivation,itisobviousthatthefirstandsecondtermsbelongtotheelectricdipolemoment(see
Ref.26,30).
Appendix D: Magnetic dipole moment
In this subsection, we derive the exact magnetic dipole moment for the spherical and Cartesian coordinates.
1. Spherical coordinates
The magnetic dipole moment in spherical coordinate can be found by using Eq. A2(j =1, i.e. l=j =1) :
 bω   bω ¯l=1
11 11
 bω10  =  bω10  , (D1)
bω bω
1−1 1−1
√ ˆ
3k j (kr)
= − d3rr×J 1 ,
2π ω kr
In contrast to the electric dipole moment, it has only one term (i.e. ¯l=1). The derivation for the above expression
can be found in our previous work26. In the next subsection, we introduce the exact magnetic dipole moment in
Cartesian coordinates.
2. Cartesian coordinates
Each components of the magnetic dipole moment (Eq. C1) can be written as:
√ ˆ
3k j (kr)
bω = − d3r(r×J ) 1 ,
11 2π ω 1 kr
√ ˆ
3k j (kr)
bω = − d3r(r×J ) 1 ,
10 2π ω 0 kr
√ ˆ
3k j (kr)
bω = − d3r(r×J ) 1 . (D2)
1−1 2π ω −1 kr
TheCartesianmagneticdipolemomentcanbeobtainedbyusingthefollowingtransformationbetweenthespherical
and Cartesian coordinates:
bω −bω
mω = Cm 1−1√ 11,
x 1
2
bω +bω
mω = Cm 1−√1 11,
y 1
2i
mω = Cmbω , (D3)
z 1 10
√
where Cm = − 3π, which is obtained by comparing the electric field expressions in spherical24 and Cartesian
1 k
coordinates, i.e. Eq. G1. Finally, we substitute in Eqs. D2 in Eqs. D3 and by using Eq. B6 and Eq. B7, we get:
10
ˆ
3 j (kr)
mω = d3r(r×J ) 1 ,
x 2 ω x kr
ˆ
3 j (kr)
mω = d3r(r×J ) 1 ,
y 2 ω y kr
ˆ
3 j (kr)
mω = d3r(r×J ) 1 , (D4)
z 2 ω z kr
which can be written in a short form:
ˆ
3 j (kr)
mω = d3r(r×J ) 1 (D5)
α 2 ω α kr
where α=x,y,z. This expression is documented in Tab. 2 of the main manuscript.
3. Long-wavelength approximation
We now can make the long-wavelength approximation by using the small argument approximation to the spherical
Bessel function, i.e.
kr
j (kr) ≈ , (D6)
1 3
and obtain the well-known long-wavelength approximation expression for the magnetic dipole moments:
ˆ
3 j (kr)
mω = d3r(r×J ) 1 ,
α 2 ω α kr
ˆ
1
≈ d3r(r×J ) . (D7)
2 ω α
This expression is documented in Tab. 1 of the main manuscript. It is important to mention that the long-
wavelength expression can be found in any electrodynamics textbook24 and only valid at the small object compared
to wavelength (i.e. kr (cid:28)1), whereas our new expression is valid irrespective of the size of the source.
Appendix E: Magnetic quadrupole moment
In this section, we derive the exact magnetic quadrupole moment for the spherical and Cartesian coordinates by
using Eq. A2. They have not been reported neither in spherical nor in Cartesian coordinates
1. Spherical coordinates
The magnetic quadrupole moment in spherical coordinate can be found by using Eq. A1 (j =2, i.e. l=j =2), i.e.
ˆ ˆ
2
4π (cid:88)
bω = − dpX† (p)Y (p) d3rJ (r)Y∗ (r)j (kr). (E1)
2m (cid:113) (cid:98) 2m (cid:98) 2m (cid:98) ω 2m (cid:98) 2
(2π)3 m=−2
By using the explicit expression for the multipolar functions in momentum space, i.e.
 (cid:113) 
− j(j+1)−m(m−1)Y (p)
X†jm(p(cid:98)) = (cid:112)j(j1+1) (cid:113) m2Yjm(p(cid:98))j(m−1) (cid:98) , (E2)
j(j+1)−m(m+1)Y (p)
2 j(m+1) (cid:98)