Multipole Expansion for Radiation;Vector Spherical Harmonics

Physics 214 2013, Electricity and Magnetism

Michael Dine Department 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 multipole expansion for radiation. The strategy will be to consider three regions: 1 Radiation zone: r λ d. This is the region we have already considered for the dipole radiation. But we will see that there is a deeper connection between the usual multipole 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 time derivatives are of order 1/λ (c = 1), while derivatives with respect to r are of order 1/r, so in this region time derivatives 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 simple approximations for the fields.

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics Our goal is to match the solutions in the intermediate and radiation zones. We will see that in the intermediate zone, because of the static nature of the ﬁeld, there is a multipole expansion identical to that of electrostatics (where moments are evaluated at each instant). This solution will match onto eikr−iωt outgoing spherical waves, all falling as r , but with a sequence of terms suppressed by powers of d/λ. So we have two kinds of expansion going on, a different one in each region.

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics Let’s ﬁrst review some of the special functions we will need for this analysis. The basic equation which will interest is the Helmholtz equation (similar to the Schrodinger equation):

(∇~ 2 + ω2)ψ(~x, ω) = 0. (1)

One can expand the solutions for ﬁxed r, in spherical coordinates: X ψ(~x, ω) = f`,m(r)Y`,m(θ, φ). (2) `,m f`,m obeys:

d 2 2 d `(` + 1) + + k 2 − f (r) = 0. (3) dr 2 r dr r 2 `

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics The f`(r)’s are “spherical Bessel functions," f`(r) = g`(kr). The g`’s are rather nice functions (you will encounter them in quantum mechanics in studying scattering theory). Typically they are presented as four types, the j`’s, n`’s, and h`’s, spherical Bessel, Neumann, and Hankel functions. They are distinguished by their behaviors at the origin and at ∞. The j`’s are regular at the origin, n`’s are singular; both behave like sines at ∞. There are actually two types of h’s,

(1) (2) h` = j` + in`; h` = j` − in`. (4) These behave as e±ix at ∞, corresponding to outgoing and incoming spherical waves. They are both singular at the origin (a suitable linear combination is not).

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics sin x cos(x) eix j (x) = n (x) = − h(1)(x) = (5) 0 x 0 x 0 ix with relatively simple asymptotics:

x` (2` − 1)!! x → 0 : j (x) → n (x) → − . (6) ` (2` + 1)!! ` x`+1

1 `π 1 `π x → ∞ : j (x) → sin(x − ) n (x) → − cos(x − ). (7) ` x 2 ` x 2 ix (1) `+1 e (2) `+1 −ix h → (−i) h → (i) e x (8) ` x ` −iωt (1) −iωt+ikr With e time dependence, h` ∼ e is an outgoing spherical wave.

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics Green’s function for the Helmholtz equation:

(∇~ 2 + k 2)G(~x, ~x0) = −δ(~x − ~x0) (9)

0 eik|~x−~x | G(~x, ~x0) = (10) 4π|~x − ~x0|

X 0 ∗ 0 0 = g`(r, r )Y`,m(θ φ )Y`m(θφ) `,m

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

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics Jackson proves this by looking at the equations for the coefﬁcients of each Y`m. But we can see why this must be correct rather simply. First, for r < r 0, say, for each ` the equation for g` is the same equation we studied above, in both r and r 0. As r → 0, the result should be non-singular, so we 0 0 must take the j`’s. For r , as r → ∞, we would like outgoing (1) spherical wave behavior, which ﬁxes h` . The argument is 0 symmetric in r and r , which explains the r<, r>.

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics Now we can ﬁx the coefﬁcients by requiring that as k → 0, this should go over to

` X 1 1 r< ∗ 0 0 Y`m(θ , φ )Y`m(θ, φ). (11) 2` + 1 r> r> `,m

Examining the expansions of h,j for small argument, yields the coefﬁcient above: (kr )` (2` − 1)!! < − ik `+1 (12) (2` + 1)!! (kr>)

1 1 r ` = < 2` + 1 r> r>

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics So we are ready to attack the problem. We seek a generalization of the static multipole expansion. We will follow a slight modification of Jackson’s strategy. We start by considering the intermediate zone. It turns out to be simpler to work 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 by considering the scalar quantities ~r · E~ and ~r · H~ . The first, in electrostatics, can be completely determined from the charge distribution; this is the same problem we will encounter in the intermediate zone. ~r · H~ is similar. We will focus mostly on the first.

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics Since ∇~ 2~r · E~ = ~r · ∇~ 2E~ + 2∇~ · E~ = ~r · ∇~ 2E~ (13) and similarly for H~ , it follows that: ( ) ~r · H~ (∇~ 2 + ω2) = 0. (14) ~r · E~

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

~ X p ~r · E = k `(` + 1)aE (`, m)g`,m(kr)Y`,m(Ω). (15) `,m and similarly for ~r · H~ .

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics Now for the problems which typically interest us, in which we have a localized source of radiation, of size much smaller than a wavelength, we can rather easily figure out the coefficients aE . Consider, first, the “intermediate zone", λ r d. In this regime, the fields are essentially static. You can see this by comparing derivatives:

∂ 1 1 ∂ 1 ∼ ∼ . (16) ∂r r c ∂t λ So we can neglect time derivatives in this regime. But this means that E~ ≈ −∇~ φ (17)

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics In this static regime,

Z ρ(~x0) φ ≈ − d 3x0 . (18) |~x − ~x0|

(this also follows from the form of the Green’s function of the Helmholtz equation in the limit that we can neglect k). Then ∂φ ~r · E~ = −~r · ∇~ φ = − (19) ∂r

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics But we know the form of φ far from a localized charge distribution: X 4π q`,mY`,m(Ω) φ(~x) = . (20) 2` + 1 r `+1 `,m ∂φ We can construct ∂r , so for small r,

p ` + 1 1 a `(` + 1)g (kr) ≈ 4π q . (21) `,m ` 2` + 1 r `+1 `,m

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics Now we want to match to the general solution. Away from r = 0, we have (1) g`(kr) = h` (kr) (22) due to the requirement of outgoing spherical wave boundary conditions. Now we match, noting that (eqns. 9.85,9.88 in Jackson) (2` − 1)!! h(1) ≈ −i (23) ` x`+1 so ck `+2 ` + 11/2 a (`, m) = q . (24) E i(2` + 1)!! ` `m

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics Knowledge of ~r · E~ determines H~ and E~ , through the usual equations for plane waves in free space:

i i H~ = − ∇~ × E~ E~ = ∇~ × H~ (25) k k

So, e.g., for the“TM" (~r · H~ = 0) case, let’s solve the full set of equations. Take (a guess) ~ ~ H`m = A`m(r)LY`,m (26)

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

x ∂A(r) −i x ∂ A(r) = −i x k = 0. (27) ijk j k ijk j r ∂r

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics Then it is easy to see that:

i i ~r · E~ = ~r · (∇~ × H~ ) = r A ∂ L Y (28) k k i ijk `m j k `m 1 1 ~L2A (r)Y = A `(` + 1)Y . k `m `,m k `m `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 can check that this conﬁguration (with the appropriate H~ , as above) solves the full set of Maxwell’s equations; similarly for the "TE" modes.

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics It is natural to deﬁne “vector spherical harmonics":

~ 1 ~ X`m(θ, φ) = LY`m(θ, φ). (29) p`(` + 1)

From the properties of the spherical harmonics, it is easy to see that: Z ~ ∗ ~ X`m · X`0m0 dΩ = δ`,`0 δm,m0 (30) Z ~ ∗ ~ X`m · (~r × X`0m0 )dΩ = 0 (31)

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

~ X p ~ H = A`m(r) `(` + 1)X`m. (32) `m

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics Following your text, we can derive this more directly. Rather than 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~ 0 which is divergence free, i E~ 0 = E~ + ~J (33) ω0 we have the Maxwell equations (in Jacksons 9.160, set Z0 = 1, and M = 0) ∇~ · H~ = 0 ∇~ · E~ 0 = 0 (34) i ∇~ × E~ 0 − ikH~ 0 = ∇~ × ~J ∇~ × H~ + ikE~ 0 = 0. ω0

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics Now take the curl of the third equation, substitute for ∇~ × H~ from the fourth, to give:

(∇2 + k 2)E~ 0 = −i/k∇~ × (∇~ × ~J). (35)

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

(∇2 + k 2)~r · E~ 0 = −i/k (~L · ∇~ × ~J). (36)

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics So now we can solve for ~r · E~ 0 using the Green’s function for the Helmholtz equation:

0 1 Z eik|~x−~x | ~r · E~ 0(x) = − ~L0 · ∇~ 0 × ~J(~x0)d 3x0. (37) 4πk |~x − ~x0|

Now ~L · ∇~ × ~J = −i(~r × ∇~ ) · (∇~ × ~J) (38) can be simpliﬁed using our identity trick.

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics We need to be careful about the ordering of the derivatives. The expression above is:

−iijk klm(ri ∂j ∂`Jm) (39)

= −i(ri ∂j ∂i Jj − ri ∂j ∂j Ji ) 2 ~ ~ ~ = iri ∇ Ji − i~r · ∇(∇ · J) 2 ~ ~ ~ ~ ~ = i∇ (ri Ji ) − 2i∇ · J − i~r · ∇(∇ · J)

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics This can be rewritten: i ∂ ~L · ∇~ × ~J = i∇2(~r · ~J) − (r 2∇~ · A~ ) (40) r ∂r

Physics 214 2013, Electricity and Magnetism Multipole Expansion for Radiation;Vector Spherical Harmonics So using this, we have (using the expansion of the Green’s function in spherical Bessel functions, eqns. 9.95-9.97 in Jackson: 2 Z k ∗ 1 2 ~ ic 1 ∂ 2 3 aE (`, m) = j`(kr)Y`, ∇ (~r · J) − (r ρ) d x. p`(` + 1) m k 2 k r ∂r (41) Integrating by parts on the ∇2 term twice, and using the wave equation (without sources) gives a factor of k 2; but we can actually drop this term in the case of a localized source, leaving our earlier expression for aE (using the small argument limit of the Bessel functions).

Exercise: Verify the matching for the electric dipole and quadrupole moments, i.e. check that these results reproduce our calculation of the dipole and quadrupole radiation, checking the connection between the large and small distance expressions.

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

Jackson considers two applications of this method. First are cases where low moments vanish. This happens in some atoms and nuclei. Alternatively, it is interesting, for example, for antennae when the wavelength is not too different than the size of the antenna. (Think of FM, 100 MHz). Jackson also (in 9.4) discusses the problem of determining the current in an antenna, and shows that this is actually a boundary value problem, where one must solve simultaneously for the current and the ﬁeld.

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