Multipole Expansion for Radiation;Vector Spherical...

Multipole Expansion for Radiation;VectorSpherical Harmonics

Physics 214 2013, Electricity and Magnetism

Michael DineDepartment of Physics

University of California, Santa Cruz

February 2013

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics

We seek a more systematic treatment of the multipoleexpansion for radiation. The strategy will be to consider threeregions:

1 Radiation zone: r λ d . This is the region we havealready considered for the dipole radiation. But we will seethat there is a deeper connection between the usualmultipole moments and the radiation at large distances(which in all cases falls as 1/r ).

2 Intermediate zone (static zone): λ r d . Note that timederivatives are of order 1/λ (c = 1), while derivatives withrespect to r are of order 1/r , so in this region timederivatives are negligible, and the fields appear static.Here we can do a conventional multipole expansion.

3 Near zone: d r . Here is is more difficult to find simpleapproximations for the fields.


Our goal is to match the solutions in the intermediate andradiation zones. We will see that in the intermediate zone,because of the static nature of the field, there is a multipoleexpansion identical to that of electrostatics (where momentsare evaluated at each instant). This solution will match ontooutgoing spherical waves, all falling as eikr−iωt

r , but with asequence of terms suppressed by powers of d/λ. So we havetwo kinds of expansion going on, a different one in each region.


Let’s first review some of the special functions we will need forthis analysis. The basic equation which will interest is theHelmholtz equation (similar to the Schrodinger equation):

(~∇2 + ω2)ψ(~x , ω) = 0. (1)

One can expand the solutions for fixed r , in sphericalcoordinates:

ψ(~x , ω) =∑`,m

f`,m(r)Y`,m(θ, φ). (2)

f`,m obeys: [d2

dr2 +2r

ddr

+ k2 − `(`+ 1)

r2

]f`(r) = 0. (3)


The f`(r)’s are “spherical Bessel functions," f`(r) = g`(kr). Theg`’s are rather nice functions (you will encounter them inquantum mechanics in studying scattering theory). Typicallythey are presented as four types, the j`’s, n`’s, and h`’s,spherical Bessel, Neumann, and Hankel functions. They aredistinguished by their behaviors at the origin and at∞. The j`’sare regular at the origin, n`’s are singular; both behave likesines at∞. There are actually two types of h’s,

h(1)` = j` + in`; h(2)

` = j` − in`. (4)

These behave as e±ix at∞, corresponding to outgoing andincoming spherical waves. They are both singular at the origin(a suitable linear combination is not).


j0(x) =sin x

xn0(x) = −cos(x)

xh(1)

0 (x) =eix

ix(5)

with relatively simple asymptotics:

x → 0 : j`(x)→ x`

(2`+ 1)!!n`(x)→ −(2`− 1)!!

x`+1 . (6)

x →∞ : j`(x)→ 1x

sin(x − `π

2) n`(x)→ −1

xcos(x − `π

2). (7)

h(1)` → (−i)`+1 eix

xh(2)` → (i)`+1e

−ixx (8)

With e−iωt time dependence, h(1)` ∼ e−iωt+ikr is an outgoing

spherical wave.


Green’s function for the Helmholtz equation:

(~∇2 + k2)G(~x , ~x ′) = −δ(~x − ~x ′) (9)

G(~x , ~x ′) =eik |~x−~x ′ |

4π|~x − ~x ′|(10)

=∑`,m

g`(r , r ′)Y ∗`,m(θ′φ′)Y`m(θφ)

where g`(r , r ′) = ikj`(kr<)h(1)` (kr>).


Jackson proves this by looking at the equations for thecoefficients of each Y`m. But we can see why this must becorrect rather simply. First, for r < r ′, say, for each ` theequation for g` is the same equation we studied above, in bothr and r ′. As r → 0, the result should be non-singular, so wemust take the j`’s. For r ′, as r ′ →∞, we would like outgoingspherical wave behavior, which fixes h(1)

` . The argument issymmetric in r and r ′, which explains the r<, r>.


Now we can fix the coefficients by requiring that as k → 0, thisshould go over to

∑`,m

12`+ 1

1r>

(r<r>

)`Y ∗`m(θ′, φ′)Y`m(θ, φ). (11)

Examining the expansions of h,j for small argument, yields thecoefficient above:

ik(kr<)`

(2`+ 1)!!

(−(2`− 1)!!

(kr>)`+1

)(12)

=1

2`+ 11r>

(r<r>

)`


So we are ready to attack the problem. We seek ageneralization of the static multipole expansion. We will follow aslight modification of Jackson’s strategy. We start byconsidering the intermediate zone. It turns out to be simpler towork with ~E and ~H (I will follow Jackson in using ~H rather than~B).~E and ~H are vector quantities. It is helpful to start byconsidering the scalar quantities ~r · ~E and ~r · ~H. The first, inelectrostatics, can be completely determined from the chargedistribution; this is the same problem we will encounter in theintermediate zone. ~r · ~H is similar. We will focus mostly on thefirst.


Since~∇2~r · ~E = ~r · ~∇2~E + 2~∇ · ~E = ~r · ~∇2~E (13)

and similarly for ~H, it follows that:

(~∇2 + ω2)

~r · ~H~r · ~E

= 0. (14)

These are scalar quantities, so they can readily be expanded inspherical harmonics:

~r · ~E =∑`,m

k√`(`+ 1)aE (`,m)g`,m(kr)Y`,m(Ω). (15)

and similarly for ~r · ~H.


Now for the problems which typically interest us, in which wehave a localized source of radiation, of size much smaller thana wavelength, we can rather easily figure out the coefficientsaE . Consider, first, the “intermediate zone", λ r d . In thisregime, the fields are essentially static. You can see this bycomparing derivatives:

∂

∂r∼ 1

r1c∂

∂t∼ 1λ. (16)

So we can neglect time derivatives in this regime. But thismeans that

~E ≈ −~∇φ (17)


In this static regime,

φ ≈ −∫

d3x ′ρ(~x ′)|~x − ~x ′|

. (18)

(this also follows from the form of the Green’s function of theHelmholtz equation in the limit that we can neglect k ). Then

~r · ~E = −~r · ~∇φ = −∂φ∂r

(19)


But we know the form of φ far from a localized chargedistribution:

φ(~x) =∑`,m

4π2`+ 1

q`,mY`,m(Ω)

r `+1 . (20)

We can construct ∂φ∂r , so for small r ,

a`,m√`(`+ 1)g`(kr) ≈ 4π

`+ 12`+ 1

1r `+1 q`,m. (21)


Now we want to match to the general solution. Away fromr = 0, we have

g`(kr) = h(1)` (kr) (22)

due to the requirement of outgoing spherical wave boundaryconditions. Now we match, noting that (eqns. 9.85,9.88 inJackson)

h(1)` ≈ −i

(2`− 1)!!

x`+1 (23)

so

aE (`,m) =ck `+2

i(2`+ 1)!!

(`+ 1`

)1/2

q`m. (24)


Knowledge of ~r · ~E determines ~H and ~E , through the usualequations for plane waves in free space:

~H = − ik~∇× ~E ~E =

ik~∇× ~H (25)

So, e.g., for the“TM" (~r · ~H = 0) case, let’s solve the full set ofequations. Take (a guess)

~H`m = A`m(r)~LY`,m (26)

Note first that ~LA(r) = 0. Using the language of quantummechanics, this is because A(r) is rotationally invariant, and ~Lis the generator of infinitesimal rotations. But it follows from:

−iεijkxj∂kA(r) = −iεijkxjxk

r∂A(r)

∂r= 0. (27)


Then it is easy to see that:

~r · ~E =ik~r · (~∇× ~H) =

ik

riεijkA`m∂jLkY`m (28)

1k~L2A`m(r)Y`,m =

1k

A`m`(`+ 1)Y`m.

so that the connection of A(r) and the expansion coefficients of~r · ~E can be read off immediately (Jackson’s 9.122). You cancheck that this configuration (with the appropriate ~H, as above)solves the full set of Maxwell’s equations; similarly for the "TE"modes.


It is natural to define “vector spherical harmonics":

~X`m(θ, φ) =1√

`(`+ 1)~LY`m(θ, φ). (29)

From the properties of the spherical harmonics, it is easy to seethat: ∫

~X ∗`m · ~X`′m′dΩ = δ`,`′δm,m′ (30)∫~X ∗`m · (~r × ~X`′m′)dΩ = 0 (31)

(the last follows from an integration by parts and use of thecommutation relations of ~L and ~r ).

~H =∑`m

A`m(r)√`(`+ 1)~X`m. (32)


Following your text, we can derive this more directly. Ratherthan work with ~A and φ, it is convenient to work with ~r · ~E and~r · ~H directly. We focus on the latter. Defining a field, ~E ′ which isdivergence free,

~E ′ = ~E +iωε0

~J (33)

we have the Maxwell equations (in Jacksons 9.160, set Z0 = 1,andM = 0)

~∇ · ~H = 0 ~∇ · ~E ′ = 0 (34)

~∇× ~E ′ − ik ~H ′ =iωε0

~∇× ~J ~∇× ~H + ik ~E ′ = 0.


Now take the curl of the third equation, substitute for ~∇× ~Hfrom the fourth, to give:

(∇2 + k2)~E ′ = −i/k ~∇× (~∇× ~J). (35)

(There is a similar equation for ~H). So

(∇2 + k2)~r · ~E ′ = −i/k (~L · ~∇× ~J). (36)


So now we can solve for ~r · ~E ′ using the Green’s function for theHelmholtz equation:

~r · ~E ′(x) = − 14πk

∫eik |~x−~x ′|

|~x − ~x ′|~L′ · ~∇′ × ~J(~x ′)d3x ′. (37)

Now~L · ~∇× ~J = −i(~r × ~∇) · (~∇× ~J) (38)

can be simplified using our ε identity trick.


We need to be careful about the ordering of the derivatives.The expression above is:

−iεijkεklm(ri∂j∂`Jm) (39)

= −i(ri∂j∂iJj − ri∂j∂jJi)

= iri∇2Ji − i~r · ~∇(~∇ · ~J)

= i∇2(riJi)− 2i ~∇ · ~J − i~r · ~∇(~∇ · ~J)


This can be rewritten:

~L · ~∇× ~J = i∇2(~r · ~J)− ir∂

∂r(r2~∇ · ~A) (40)


So using this, we have (using the expansion of the Green’sfunction in spherical Bessel functions, eqns. 9.95-9.97 inJackson:

aE (`,m) =k2√

`(`+ 1)

∫j`(kr)Y ∗`,m

[1k2∇

2(~r · ~J)− ick

1r∂

∂r(r2ρ)

]d3x .

(41)Integrating by parts on the ∇2 term twice, and using the waveequation (without sources) gives a factor of k2; but we canactually drop this term in the case of a localized source, leavingour earlier expression for aE (using the small argument limit ofthe Bessel functions).

Exercise: Verify the matching for the electric dipole andquadrupole moments, i.e. check that these results reproduceour calculation of the dipole and quadrupole radiation, checkingthe connection between the large and small distanceexpressions.


Applications

Jackson considers two applications of this method. First arecases where low moments vanish. This happens in someatoms and nuclei. Alternatively, it is interesting, for example, forantennae when the wavelength is not too different than the sizeof the antenna. (Think of FM, 100 MHz).Jackson also (in 9.4) discusses the problem of determining thecurrent in an antenna, and shows that this is actually aboundary value problem, where one must solve simultaneouslyfor the current and the field.


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