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VECTOR FIELDS
CHAPTER 7
VECTOR FIELDS
7.1 Rotational Properties Consider a vector field described by V(r) = (x + 2y)T+xj.
(7.1-1)
Now let the coordinates be rotated by 45° in the positive sense about the z axis (Fig. 7.1). The two coordinate systems are related by *' = -7= (* + y\
V = -7= ( - x + Λ
(7.1-2a)
or r' = RT
(7.1-2b)
with - * ( - : :> The unit vectors also transform according to (7.1-2b) and (7.1-2c) so that
ί'=4=(ί+ϊ), V2
r=4=(-i+j).
(7-1-3)
V2
The inverse transformations corresponding to (7.1-2) and (7.1-3) are
x = 4=(*'-/),
f
182
i = -L{r-n
f
(7 Μ)
·
7.1 ROTATIONAL PROPERTIES
183
V(r),V'(r')
z, z FIG. 7.1 Coordinate transformations for a vector field.
Substituting (7.1-4) into (7.1-1) we obtain V'(r') = 2x'V - (x' + y') J'
(7.1-5)
where V'(r') is the description of the vector field (7.1-1) relative to the new coordinate system. As in Section 2.1, we are interested in constructing an operator PR that will transform (7.1-1) into (7.1-5). Such an operator is defined by PR\(r) =
R\(R-'r).
(7.1-6)
To see how this works in the example just cited we have
R=
JR- 1
Ti{-i 1;* U _
1 (\
V2V1
-\\(x\_\_(x-y
MW Ji\* + y.
Vx(r) = x + 2y, VX(R- *r) = ^ = (x - y) + V2(x + y) = 4 = (3x + y), Vy{r) = x, Vy{R-ir) =
j=(x-y),
V2V-1 i)\vj
φ\-ν,+
ν,)
184
7. VECTOR FIELDS
Therefore PRVx(r) = RV^R-1!) = — \_Vx(R-ir) + 7,(/T ^ ]
= ji[72(3X PRVy(r) =
RV^R-'T)
=~
V2
+ y) +
j=2iX-y\
2x,
(7.1-7a)
[ - Κ,ίΛ-Μ + K,(/T*r)]
x/2L V 2L V2V ^ ^ V * ^
-(* + y). (7.1-7b)
Equations (7.1-7a) and (7.1-7b) provide the form for the components of the transformed vector (7.1-5); thus the operator defined in (7.1-6) is capable of generating the desired transformation. When the coordinate system is rotated through an infinitesimal angle da about the z axis, / 1 /?(da,z) = ( - d a \ 0 /l R- (S
da 0\ 1 0, 0 1/ -da 1
(7.1-8)
0\/x\ /χ-γδα\ 0 ( y ] = (xda + A
(7.1-9)
«5α 0\ /ΚΛ / Vx+V,Sa\ 1 0 \\vy = - F ^ a + κ Λ
(7.1-10)
and / 1 R{da.,z)V= - d a
\θ
0 \]\vj
\
Vz I
Therefore PR(öoL,z)Vx(r) = RVx(R-1r) = Vx(R-1r) + SoiVy(R-1r) = T^.(x — y da, x da + v, z) + da Vy(x — y da, x da + y, z). (7.1-11) If one writes to first order, dV
dV
dV
(7.1-12)
7.1 ROTATIONAL PROPERTIES
185
a comparison of (7.1-11) with (7.1-12) yields PR(5a, z)Vx(r) = Vx(r) + - ^ (-yd«)
+ - ^ (χδα) + Vyb«.
= (1 + iöocLz)Vx + VyöoL.
(7.1-13a)
Similarly, PÄ(&x,z)7,(r) = (1 + iöocLz)Vy - δα Vx9
(7.1-13b)
PR(öa,z)Vz(r) = (1 + i 5 a L J 7 z .
(7.1-13c)
The three expressions in (7.1-13) may be combined into 0 PR(öa,z)V(r) = (l + iöaLz)V(r) + Soi[-l
1 0\/Vx(r)\ 0 0 K,(r)) 0 0 0 \VZ(T)
(7.1-14)
and, on defining (7.1-15) and JZ = LZ + SZ,
(7.1-16)
Eq. (7.1-14) takes the form Ρκ(δα, z)V(r) = (1 + ίδα JJV(r).
(7.1-17)
It is instructive to compare (7.1-17) with (2.2-8); the two expressions are very similar in form. The transformation of the scalar function is governed by the operator Lz whereas the transformation of the vector function is governed by the operator Jz which is the sum of Lz and the new operator Sz whose properties remain to be explored. By considering rotations about x and y, matrices analogous to 5Z are generated. They are
/0 0 Sx = (0 0 \0 i
0\ -i], 0/
/ 0 0 r Sy = i 0 0 0 \ - i 0 o;
(7.1-18a)
Similarly, corresponding to (7.1-16) one obtains JX = L X + ύχ,
J y = Ly + Oy
(7.1-18b)
186
7. VECTOR FIELDS
The three matrices Sx, Sy, and Sz are manifestly Hermitian; they also satisfy the commutation rules S x S = fS
(7.1-19)
from which it may be inferred that the components of S qualify as angular momentum operators. The eigenvalues of Sz are readily determined from the secular equation -λ i 0
-i -λ
0 0 =0 0 - 2
(7.1-20)
which yields λ= 1,0, - 1 . For the eigenvalue λ = 1, the eigenvalue equation
provides the normalized eigenvector (λ = 1).
(7.1-21)
In similar fashion it is found that (λ = 0);
e,i
=
(λ= -1).
(7.1-22)
These are just the unit vectors in the spherical basis (6.1-22) and are related to the unit vectors ex, e^,, and e2 along the coordinate axes by (6.1-23). Other relations among the unit vectors are given in Appendix 4. One may also construct the operator S2 defined by ύ
— ύχ
+ Oy + (3Z
(7.1-23)
Combining (7.1-15) and (7.1-18) /2 0 0\ S2 = [0
2 O]
(7.1-24)
\0 0 2) and, from the general properties of angular momentum operators, it is expected that e + 1 , e0, and e_ t will also be eigenvectors of S2. Thus, from
7.2 VECTOR SPHERICAL HARMONICS
187
(7.1-21)-(7.1-23),wehave S \ = 2e, = 1(1 + 1)8,
(q = 1,0, -1).
(7.1-25)
In an alternative notation one may write
ί
|H>=e+1
\SMS} = 1 |10> = e 0
(7.1-26)
111 - l > = 8 - i
in which case
SZ\SMS> = MS\SMS} (Ms = 1,0, -1), S2|SMS> = S(S + 1)|SMS> = 1(1 + 1)|SMS>.
(7.1-27)
In summary, the transformation properties of a vector field are governed by an orbital angular momentum operator L, a spin angular momentum operator S, and that the two operators are coupled according to the relation J = L + S.
(7.1-28)
On the basis of (7.1-27) it is seen that a vector field has unit spin (S = 1). On the other hand, as was shown in Section 2.2, the transformation of a scalar field is describable entirely on the basis of an orbital angular momentum operator L, or in other words, the spin of a scalarfieldis zero. Considerations such as these retain their significance even when the fields are quantized as, for example, in the case of photons arising from the quantization of an electromagnetic field. 7.2 Vector Spherical Harmonics It was shown in the previous section that the transformation of a vector field is characterized by an angular momentum operator J which arises from the coupling of an orbital angular momentum operator L and a spin angular momentum operator S. The simultaneous eigenfunctions of L 2 and Lz are the spherical harmonics Ylm(9, φ) and the simultaneous eigenfunctions of S2 and Sz are |SMS> with S = 1 and Ms = 1,0, — 1. It is then possible to construct simultaneous eigenfunctions of J2 and Jz according to the general rules for the coupling of angular momentum eigenfunctions (1.7-23). Adopting the notation ji = l j2 = S=l j=J
ml = m m2 = Ms= 1,0,-1 m=M
|Λ^1> = Ylm(6>
(7.2-1)
188
7. VECTOR FIELDS
one obtains |JM> = |/1 JM} = X lime,
=<-i>-<-*v27nz(m'; _$**.. a**
Equation (7.2-2) is the definition of a vector spherical harmonic usually designated YJiM; that is, YJ,M = \JM}. Since M = m + q, the double sum in (7.2-2) may be replaced by the single sum
ν,„ = (-ΐ)'-"727ΠΣ( Μ !. 9 \ _4)>Ϊ»-Α
Μ
with the following properties: J = / + 1,/,/ - 1, M = m + 9 = J, J - 1 , . . . , - J, (7.2-4) ^ M = /(/ + DYJUf, S2YJ/M = 1(1 + \)YJlM = 2YJZM, ΛΥ/ΙΜ = AfYJIM. ' % ! * = J(J + l)Y,«f, The vector spherical harmonic YZim with J = I and m = M = I, I — 1,...,— / is a particularly important one in electromagnetic theory. For this case the 3/ symbol will vanish unless the triangle condition Δ(/1/) is satisfied; the possible values of / are then / = 1,2,3,... but not / = 0. Using the formulas in Table 1.5 one may write Y
/(/ + 1) - m(m - 1)
_
"™---l
äöTIj 1(1 + 1) - m(m + I)
+ .1
2ΐ(7ΤΪ)
,m 1+1 +
-
m
^w+T)
^"^*-1·
(7 2 6)
· -
But
L(yiim = ynYlm,
L = —L +1 e_! + L0e0 — L_!e + 1 .
Therefore V'(i + 1) 1
Yi-lfLJ^.^ yfiT+T)t
LY
y/W+ΐ)
(7-2-7)
7.2 VECTOR SPHERICAL HARMONICS
189
It is observed that Yllm is orthogonal to the radius vector r since r.Y„m =r
. L - J ^ = = r . r x p — J = = 0.
(7.2-8)
y/l(l + 1) hy/l(l + 1) A number of vector spherical harmonics are listed in Table 7.1. For example,
/T
Yi i o = / ^ - ΐ sin 0 (— sin
(7.2-9b)
Suppose 0 = 90° so that the radius vector lies in the xy plane. Thefieldis parallel to e0; hence we have a linearly polarized field along the z axis, perpendicular to the radius vector. When, however 0 = 0, the field direction is given by e+1. The interpretation of such a field is based on the property e / x iv = i&o K'
(«>«' = 1,0, - 1 )
(7.2-10)
where e0 is the unit vector in the z direction. With q = +1 the cross product is a vector parallel to the z axis whereas the cross product with q = — 1 is antiparallel to the z axis. For this reason, fields specified by e +1 or e_i are said to have positive or negative helicity, respectively. In the context of wave propagation e + 1 corresponds to a counter clockwise rotation of the polarization vector when the observer is facing the oncoming beam. In optical terminology this is a left circularly polarized wave. Hence positive and negative helicity correspond to left and right circular polarization (see also Section 22.1). Finally, when 0 < 0 < 90°, thefieldin elliptically polarized. One often needs the scalar product of a vector spherical harmonic with itself, i.e., the absolute square. To compute this quantity, it is recalled that, according to (6.1-18),
A.A = £(-iM € >i- € . =Σ Λ % = Ul2.
(«=1,0,-1)
(7.2-11)
TABLE 7.1 Vector Spherical Harmonics0 Yo l o — *1 0 1
=
—
- ^ι -i^+i M) Oe + 1 >
*100
/T/sin0 .
=
/~ ^ι o^o + */'! ^ i i ^ - i
e
-VdvF '*
=
M) OeO»
g o + c „o s
*10-l
=
M)Oe-
\
H
/3 / —
sin 0 . e_ + ii — e ' ^ + cosfl-
'8«V
V2
V2
1 /3 .
Υι.ο=-λ/^ι-.«+ι+^1Ίι6-.
4>/π
[ s i n ö ^ e . ! + ei
— I—isin 0( — sin φ ex + cos φ ey V 8π =
/—isinöi*
λ/8π
=
/T/
sin0
/~3~ / /—
sin 0 η e x - ieA — g-'yez+cosö * Ί
Λ
\
= - - /-e- , >(r 2 + icos0r3) 4Vπ
Yl 2 1 = JTTj ^2 0^+i — / — ^21^0 + J - ^22^-1
1 /T 3 /Γ 3 /T = - / — ( 3 c o s 2 0 - l ) e + 1 + - /-cos0sin0
/-(3sin 2 0e 2 , > - 3cos 2 0 + l)e x
- ^ / - (3 sin2 0 e 2i * + 3 cos 2 0 - 1)6, f - / - cos 0 sin 0 £>'>ez y 8 >/π 4>/π 1
The unit vectors ΐι,ί 2 ,Ϊ3 are shown in Fig. A4.1.
TABLE 7.1 (continued) •y2_e+1
Yl20 = ,
/ - y2oeo + J —
-
^21^-1
3 /T 1/2 3 /l = - /—cos0sin0iT l,p e + 1 - - / - ( 3 c o s 2 0 - l ) e 0 - - /—cos 0 sin 0 £?**£_! 4λ/2π 4>/π 3 /Γ = - /-cos0sinO(excos
Yl 2 - 1 =
=
χ
/ 7 ^2 -2^+1 — / — ^2 -1^0 + χ / Τ Τ ^2 0^-1
! /l s in 2 öe- 2 '>e+ 1+ 1 -^ /I.cosösinöiT^o + - /—-(3cos 2 ö-l)e_! 4>/2π
4>/π
4 V 2π
= - -1 /fr i (3 sin2 ö f T 2 ' » - 3 cos2 0 + 1)6*
8>/π ^2 1 2
=
^Ι
_
e
-^-lJl
1^+1
-1^-1
^ 2 1 - 2 — ^Ι
^22 2 -
(3sin 2 0iT 2i * + 3cos2
ί1
/2
J ^ *2 1^+1 + J z ^2 2^0
Υ22 i = - ^
η οδ + ι + ^
>2 ie 0 + ^ 3
Υζ
2g-i
/-y 2 _ 1 e + 1 +J-y 2 1 e_!
Α
/ϊ
/ϊ"
.
Υ22 - l — J : *2 - 2 ^ + ι — / - y 2 - i e 0 +
^ 3 2 ±3
=
^2
±2e±\
Y32 ±2 — / r y2 ± i ^ ± i +
Y32 ±1 =
.
/ - y2o^-i
rz ^2 0*+i +
Y 3 2 0 — / 7 ^2 -1^+1 +
/ - y2
±2e0
/ T T ^2 ±1^0
/ τ ^2 0^0 +
+
λ/77
*2 ± 2 ^ τ ι
/ 7 ^2 l ^ - i
cos0 sin 0e'i9ez
192
7. VECTOR FIELDS
Therefore using (7.2-6) together with the orthogonality property of the unit vectors e^ as given by (6.1-26) one obtains f(Z + 1) m2
+
m
( m - 1)
ΐ ν / Λ xi2 / ( / + 1 ) - m ( m + 1 ) ,UM +
/(7Ti)i
|TZ
,_ , l 2 2
—2Ϊ(ΪΤΊ)—^-(M
(7.2-12) which is seen to be independent of the azimuthal angle φ. For illustration and for future reference one may cite: Yi i ±i · Yi i ±1 = \(\Yio\2
+ \Yi ±i| 2 ) = J ^ ( c o s 2 Ö + 1), (7.2-13)
Yi i o · Yt i o = ^ (l^i - i | 2 + | n + ιΓ) = ^ sin2 Θ, Y 2 2 ±2 · Y 2 2 ±2 = \\Yl
3
±1|
2
(7.2-14)
+ 1 ^ 2 ± 2 | 2 = T | " ( 1 - COS40),
3
i l y |2 + i | y
16π
(7.2-15)
I2 + 1
V .Y —-lv l2_u_|y I 2 _L l y I2 *2 2 ±1 * 2 2 + 1 — 0 |22 0 " r / : | I 2 ± l | + ^ | 2 2 ± 2 |
= - ^ - (1 - 3 cos 2 Θ + 4 cos 4 0), 16π Y I
2 2 0
.Y I 2 2 0
(7.2-16)
— - ly I 2 -L _ l y I2 ^|I2-l| "^^|22+l|
—
= ^sin20cos20.
(7.2-17)
These angular distributions are shown in Fig. 7.2. Those with I = 1 are known as dipole fields and those with Z = 2 are quadrupole fields. When 0 = 0, the spherical harmonics vanish except for Yl0, which has the value
* - . / ^ . In that case, (7.2-12) simplifies to 2/+1 ^llm * Yfim — *Π ±1 * *ll ±1 — o '
(7.2-19)
7.3
PLANE WAVE EXPANSION
193
FIG. 7.2 Angular distribution in (a) dipole and (b) quadrupole fields. (Jackson, 1962.)
that is, when 0 = 0 Y„m · Y//m has a nonvanishing value only when m = ± 1. Referring to (7.2-3) it is seen that the only spherical harmonics that appear in the definition of YJlM are those of order /. YJlM therefore has odd parity when / is odd and even parity when / is even. Also, as a consequence of the orthogonality property of the spherical harmonics and the unit vectors, the vector spherical harmonics satisfy the orthogonality relation JVJlM · YrVM. sin θάθάφ = 8jr διν δΜΜ,
(7.2-20)
with J + I+l+Mv
YJiif = ( - ! ) '
If
(7.2-21)
7.3 Plane Wave Expansion An important application of vector spherical harmonics is in the expan sion of plane waves. According to (1.2-28), Jk · r _
4π Σ 1=0
Σ ilJ,(kr)Ylm(er,
where jt(kr) is a spherical Bessel function (Appendix 5) and (θ„φΧ (6k,(pk) are the angular coordinates of the position and wave vectors, respectively (Fig. 1.4). If it is assumed that the propagation is along the z direction, 0k = 0 in which case only Yf0 is nonzero and is given by (7.2-18). The sum over m then consists of only one term, namely the one with m = 0. Therefore eik* = X iV4n(2/ + l)j,(kr)Yl0(e)
(7.3-1)
1=0
where Θ = 6r. Now let A = eYeikz
(7.3-2)
194
7. VECTOR FIELDS
Apart from an exponential time factor, (7.3-2) represents a plane wave propagating along the z axis and polarized along the x axis. By writing
A=-4=(A + 1 -A-Os-4=(i + i-i-i)* t o we have A ± 1 = e ±1 e* z ,
(7.3-3)
which, we saw, represents right and left circularly polarized waves. Using (7.3-1), A ± ! = Σ « V W + l);,(fcr)^o(0)e±! ·
(7.3-4)
I
We shall now express Yl0(6)e± t in terms of vector spherical harmonics. From (1.5-54), \ml
JM
m
2
—MJ
with M = m1 + m 2 . In the present case |7i"*i> = Yio,
\J2m2> = |1«> = % = e ± i ,
\jJ2JM>
= YJlM = Y,, ± 1 .
Therefore
1ί0β+1 = Σ ( - 1 ) " ' ν ^ Τ Τ ^ j J [ ) Y „ + I .
(7.3-5)
Since the possible values of J are I - 1, /, Z + 1, / 1 /- 1 l 1 ϊ 0 8 + ι = ( - 1 ) "[j2T Ί ν 2 / - 1 ( 0 1 _ ! )Y,-iu
+
^ ( ό ί -ΐ) Υ "< + ^(θ ί '-I1)5"«} (7.3-6)
From Table 1.5, ' 0
* 1
/ _ 1 λ
-I
I
= v(-ΐ)'
/ _ 1 ' ' V 2(2/ + 1)(2/ - 1)'
ό ! -0- ( - ,r 'Vjprn5·
(7 3 7,
·"
7.3 PLANE WAVE EXPANSION
195
Therefore 1 IEHY Y + Ι1+ΆY i , + 2^ΠΤ)Ύι-ιη~>βΥαι >12(21 + 1)
io8 + i = J ^ ^ Y | - i i i - - ^ Y « i + . / ^ f r k
+
i«i.
(7·3"8)
and in the same fashion
Yl 1 =
^-
yJ2W^)Yl-u~1
+
^Yu~1 + ylw+T)Yt+lt~1'
(7 3 9)
'"
Note that when YJIM = Yl^lll the value of M is 1; therefore J cannot be smaller than 1. But since J — I — 1 we must have / ^ 2. Similarly, when YJIM = Y n i , it is necessary that / > 1, but when YJ(M = Y,+1 n, the smallest value of / can be zero. With (7.3-8) and (7.3-9) substituted into (7.3-4) we obtain the plane wave expansion in terms of vector spherical harmonics: A ± 1 = e ±1 e' fe = £ i'j2n(l + 2)jl(kr)Yl+ll
±1
ilj2n(2l+l)jt(kr)yll±i
+ Σ
+ Σ ίψ2π(Ζ-!)./,(&■)¥,_ 1 I ± 1 .
(7.3-10)
1=2
For physical applications one is usually interested in the first few terms of expansion (7.3-10), particularly under conditions where kr « 1. The leading term is the one containing j0(kr\ that is, A<±1)1=>/ii/o(fcr)Y1o±i·
(7.3-11)
From Table 7.1 and the asymptotic values of the spherical Bessel functions ((A5-17) in Appendix 5) *10±1
=
^00^±1
=
r—
e
±l»
χ/4π
j0(kr)=l
(kr« 1),
so that (7.3-11) becomes A^=e±1.
(7.3-12)
Hence the approximation of A ± 1 by A(+\ is equivalent to setting eikz = 1 in (7.3-3). The next higher-order terms are those containing j^kr). These are A<±2> = i46nj1{kr)Y2 t
±1
+ ί^^ήΥ, 1 ± 1 .
(7.3-13)
196
7. VECTOR FIELDS
Again using Table 7.1 and (A5-17) ^ 2 1 ±1 = V2L^10^±l + Yl ±l^o]> *1 1 ±1 = V2"L+ ^10^11 i
h(kr) = ^kr
^1 ±l^o]?
(/cr«l).
Therefore /4π /~3~ /4π A(±\ = i — krY10e±i = i l—kr /-—cos0e ±1 = ikr cos Ö e ± ! = ikze± t
(7.3-14)
and Α(±υι + A(±2\ = (1 + ikz)e±l
(7.3-15)
which corresponds to the first two terms in the power series expansion of the exponential in (7.3-3). 7.4 Multipole Expansion of the Electromagnetic Field The free-field Maxwell equations are V · B = 0,
V · E = 0,
IdE „ n VxB = -—,
^ 13B VxE= — —
c ot
c ot
(7·4-!)
with 1 ί)\ E=—— c ot
B-VxA,
(7.4-2)
where A is the vector potential. The time dependence of thefieldvectors may be taken to be of the form e~i(0t in which case (7.4-1) and (7.4-2) become V.B-* V x B = - ikE, B = V x A,
V.E-a V x E = ikB, E = ikk,
(7.4-4)
where k = ω/c. From (7.4-3) and (7.4-4) and the assumption that V · A = 0 (Coulomb gauge) one obtains the vector Helmholtz equations: (V2 + k2)B(r) = 0,
(7.4-5a)
2
2
(7.4-5b)
2
2
(7.4-5c)
(V +fc )E(r)= 0, (V +fc )A(r)= 0.
7.4 MULTIPOLE EXPANSION OF THE ELECTROMAGNETIC FIELD
197
As a step toward the solution of (7.4-5) we consider first the scalar Helmholtz equation (V2 + fc2)^(r) = 0.
(7.4-6)
Equation (7.4-6) when written in spherical coordinates is separable into two equations one of which depends on r only and the other on Θ and φ. Solutions to the angular equation are expressible in terms of the spherical harmonics. We therefore write = ΣΜ')γι*(°><Ρ)>
ΦΜφ)
(7·4"7)
Im
and on substituting (7.4-7) into (7.4-6), one obtains the radial equation
^ii
\_dr2
r dr
+
^ ^2 ' r
j
u,(r) = 0
(7.4-8)
where ut(r) = yfrfi(r). Equation (7.4-8) is a form of the Bessel equation (Appendix 5). Therefore the solutions to (7.4-6) may be written •AW = Σ lAftWKkr) + ^hf\kr)-\ Im
Υ1η(θ, φ)
(7.4-9)
where h\l\kr) and h{2\kr) are spherical Hankel functions of the first and second kind, respectively, and the coefficients A\m] and A\m] are determined from the boundary conditions. In place of the Hankel functions one may write ^(r) in (7.4-9) in terms of j^kr) and n^kr) since h\2\kr) = jt(kr) - in^kr).
hfKkr) = jt(kr) + inx(kr\
(7.4-10)
Thus any product of the form fi(kr)Ylm(e9q>\ where ft(kr) stands for any one of the spherical Bessel functions h^ikrXhffXkrXj^kr), or n^kr) will satisfy (7.4-6), i.e., (W2 + k2)fl(kr)Ylm(e^) = 0.
(7.4-11)
2
Now consider the commutator [L, V ]. In spherical coordinates .
d2 dr2
2d r dr
L2 r2'
and since the components of L operate only on angular coordinates and L2 commutes with every component of L, we have [L,V 2 ]=0.
(7.4-12)
198
7. VECTOR FIELDS
Therefore L(V2 +fc2W(/cr)>U0,φ) = (V2 +fc2)ü(/cr)L^m(ö,φ) = 0, (7.4-13) from which we conclude that λ(Ια%Υ1ηι{θ,φ) satisfies the vector Helmholtz equation (7.4-5). Using (7.2-7), a solution to (7.4-5a) may be written as (V2 +fc2)BE(/,m,r)= 0
(7.4-14)
where BE(/, m, r) = MkryLYJie, φ) = JÜ^mW^
(7.4-15)
and the corresponding electric field, according to the Maxwell equations, is EE(/, m, r) = l- V x BE(/, m, r).
(7.4-16)
The transversality relation (7.2-8) implies that r.B E (/,m,r) = 0;
(7.4-17)
therefore an electromagneticfieldcharacterized by (7.4-15)-(7.4-17) is known as a TM (transverse magnetic) field of order /. However, the transversality condition does not apply to EE which may then have non-vanishing radial components. Such a field is therefore also designated as an electric multipole of order /. Since ft(kr)LYlm(e, φ) is a solution to the vector Helmholtz equation it could equally well serve as a solution to (7.4-5b). We therefore have another pair of solutions EM(lm9r) = fl(kr%Ylm(e,cpl
(7.4-18)
BM(/, m, r) = ^ V x EM(/, m, r),
(7.4-19)
r.E M (/,m,r) = 0, (7.4-20) and such afieldis known as a magnetic multipole or a TE (transverse electric) field of order /. In this case BM may have radial components. The general solution to the vector Helmholtz equations for the electro magnetic field may now be expressed as a linear superposition of electric and magnetic multipoles of all orders: B(r) = Σ M'> rn)BE(l, m, r) + aM(/, m^l,
m, r)],
(7.4-21)
lm
E(r) = Σ [%('> W)EE('> rn, r) + aM(/, m)EE(/, m, r)],
(7.4-22)
lm
in which the coefficients depend on the sources and the boundary conditions.
7.4 MULTIPOLE EXPANSION OF THE ELECTROMAGNETIC FIELD
199
The two types of multipoles are distinguished by the fact that the roles of the electric and magnetic fields in the two cases are interchanged. Thus from the fact that the parity of YJlM is (— 1)' we see that the parities of BE(/,m,r) and EM(/, m, r), which we designate by π\_ΒΕ(1, m, r)] and π[£Μ(Ζ, m, r)], respec tively, are both (-1)*. Since EE is proportional to V x BE according to (7.4-3), the parity of EE(Z, m, r) must be opposite to that of BE(Z, m, r), or (— 1)'+* because the curl operation transforms a polar vector into an axial vector and vice versa. The same applies to BM(/,m,r) whose parity is also ( — 1) I+1 . Also in view of the proportionality between A and E [Eq. (7.4-4)], the parity of A is the same as the parity of E. Thus we have π[ΒΕ(/, m, r)] = π[ΕΜ(Ζ, m, r)] = π[ΑΜ(ί, m, r)] = (-1)', 7r[EE(/,m,r)] = 7i[BM(/,m,r)] = 7r[AE(/,m,r)] = ( - 1 ) ' + 1 .
(
''
Since Yllm = 0 for / = 0 (Section 7.2), multipoles of order zero vanish iden tically. This ensures the transversality of the fields to the direction of propaga tion in an electromagnetic wave in free space. It is customary to employ the following conventions and notations: (1) The parity of an electromagnetic field nem is defined as the parity of B, i.e., 7rem = π(Β). (2) The polarization of an electromagnetic field is in the direction of E. (3) The multipoles are named according to the values of 2l; thus dipole (/ = 1 , 2 ' = 2), quadrupole (J = 2,2' = 4), octapole (Z = 3,2* = 8), etc. Addi tional designations are E/ for electric multipole of order I and Ml for magnetic multipole of order /. Thus El, E 2 , . . . are electric dipole, quadrupole,... and Ml, M 2 , . . . are magnetic dipole, quadrupole,.... The angular distribution of a multiple may be calculated from the fact that the energy density of a radiation field is proportional to E · E or B · B. Therefore for magnetic multipoles the angular distribution may be obtained from EM(/, m, r) · EM(Z, m, r) which is proportional to Yllm · Y„m whose general form is given by (7.2-12). For electric multipoles the angular distribution can be obtained from BE(/, m, r) · BE(/, m, r) which again is proportional to Yüm · YUm. Thus the angular distribution is the same for electric and magnetic multipoles of the same order. For dipole and quadrupole radiation the patterns are shown in Fig. 7.2. One should note, however, that since EM(Z, m, r) has the same form as BE(/, m, r) but EE(Z, m, r) is proportional to V x BE(Z, m, r), the polarizations of electric and magnetic multipoles of the same order are perpendicular to one another. Let the vector potential be given by A±i =
e±lei
(7.4-24)
200
7. VECTOR FIELDS
as in (7.3-3). The expansion of A ± x was given by (7.3-10). Our objective now is to examine the multipolarity of the electromagnetic fields associated with the first few terms in the expansion of A+ x. According to (7.3-11), the· leading term (kr « 1) is Α(±υ! = ν4π/ο(^)Υι ο ±ι = 8± i
(7-4-25)
and
Β^νχΑΪΙ. To evaluate the curl of quantities such as those on the right side of (7.4-25) the following special formulas (Edmonds, 1960) are useful: V x Mkr)Yllm = ik
Vx
fi-akrm,.^
2/+1
h+iykiyiii+i,
+ ly^-lf,-i(kr)Yll-irn
(7.4-26a)
Yllm.
(7.4-26b)
= -ikf^kr)^^
Thus n \ = - i / yfc/i(fcr)Yii ±i ·
(7.4-26c)
Since / = 1 the field is a dipole field. Also, since the parity of YJlM is (— 1)', the parity of B(+\ is (— I)1 = — 1. From Table 7.2 the electromagnetic field is further identified as El, that is, electric dipole. We conclude that A(+\ in (7.4-25) is the vector potential for an electric dipole field. Following the TABLE 7.2 Multipole Relations0
Multipole
Field vector
Proportional to
El
BE(/,m,r) EE(/,m,r) AE(/,m,r)
fi(kr)Yllm V x fi(kr)Yllm V x ft(kr)YllM
B M (/,m,rr EM(/,m,r) AM(/,m,r)
V xfAkrWum fi(kr)Yllm fi(kr)Yllm
Ml
a
Parity
Electromagnetic field or photon parity
(-1)'
. (-D ,+I (-D' +1 (-D' +1 (-1)' (-1)'
The parity of an electromagnetic field is defined by the parity of B.
(-1)'
7.4 MULTIPOLE EXPANSION OF THE ELECTROMAGNETIC FIELD
201
argument in Section 7.3 it is seen that the approximation whereby A ± 1 = e±1eikz is replaced by Ad) _ £
that is, eikz = 1 corresponds to the approximation whereby a plane wave is replaced by an electric dipole field. The next term in expansion (7.3-10) is A<±2> = iVfeA(*r)[Y 2 i ± i + Yi i ±i] = ikze±±
(7.4-27)
as shown in (7.3-13) and (7.3-14). The term containing Yx x ± 1 has a parity of — 1. Since the vector potential A is proportional to the electric field E, the parity of E is — 1. With / = 1 such a field is identified (Table 7.2) as a magnetic dipole (Ml) field. To identify the term containing Y 2 i ±i we note from (7.4-26b) that V x j1(kr)Y2
1±1
= -ikj2(kr)Y2
2±2.
(7.4-28)
(
Therefore V x A + \ contains Y22±2 which means that there is a B field with / = 2 or (— I)1 = + 1 . From Table 7.2 such a field is identified as electric quadrupole (E2). Thus we have the result that A(+\ in (7.4-27) represents a combination of a magnetic dipole (Ml) and an electric quadrupole (E2) field. The results we have obtained may be summarized by writing eikz = 1(E1) + ikz{E2 + Ml) + · · · .
(7.4-29)
In further detail the M l part of the vector potential is (A ( ± 2 \)MI
= TiyßÜMkryY,!
±1
= il^kr(Y10e±1
- Yx ± 1 e 0 )
(7.4-30)
in which the approximation
h(kr) = $kr
(kr«l)
has been made. The rectangular components are (A^2))M1 = i \ (zex - xe z ),
(A<2>)M1 = i ± (z% - ytt).
(7.4-31)
The E2 part of the vector potential is (A<±2»)E2 = i^zj1(kr)Y2
! ± 1 = ifckr(Y10e±1
+ Y, ± 1 e 0 ),
(7.4-32)
202
7. VECTOR FIELDS
and the rectangular components are (A{x2))E2 =
k i-(zex+xez\
(A<2,)E2
= i\(2% + yU
(7.4-33)
We also note that A<2) = (A<2>)M1+(A<2))E2 = i/cz^, (7.4-34)
VECTOR FIELDS
7.1 Rotational Properties Consider a vector field described by V(r) = (x + 2y)T+xj.
(7.1-1)
Now let the coordinates be rotated by 45° in the positive sense about the z axis (Fig. 7.1). The two coordinate systems are related by *' = -7= (* + y\
V = -7= ( - x + Λ
(7.1-2a)
or r' = RT
(7.1-2b)
with - * ( - : :> The unit vectors also transform according to (7.1-2b) and (7.1-2c) so that
ί'=4=(ί+ϊ), V2
r=4=(-i+j).
(7-1-3)
V2
The inverse transformations corresponding to (7.1-2) and (7.1-3) are
x = 4=(*'-/),
f
182
i = -L{r-n
f
(7 Μ)
·
7.1 ROTATIONAL PROPERTIES
183
V(r),V'(r')
z, z FIG. 7.1 Coordinate transformations for a vector field.
Substituting (7.1-4) into (7.1-1) we obtain V'(r') = 2x'V - (x' + y') J'
(7.1-5)
where V'(r') is the description of the vector field (7.1-1) relative to the new coordinate system. As in Section 2.1, we are interested in constructing an operator PR that will transform (7.1-1) into (7.1-5). Such an operator is defined by PR\(r) =
R\(R-'r).
(7.1-6)
To see how this works in the example just cited we have
R=
JR- 1
Ti{-i 1;* U _
1 (\
V2V1
-\\(x\_\_(x-y
MW Ji\* + y.
Vx(r) = x + 2y, VX(R- *r) = ^ = (x - y) + V2(x + y) = 4 = (3x + y), Vy{r) = x, Vy{R-ir) =
j=(x-y),
V2V-1 i)\vj
φ\-ν,+
ν,)
184
7. VECTOR FIELDS
Therefore PRVx(r) = RV^R-1!) = — \_Vx(R-ir) + 7,(/T ^ ]
= ji[72(3X PRVy(r) =
RV^R-'T)
=~
V2
+ y) +
j=2iX-y\
2x,
(7.1-7a)
[ - Κ,ίΛ-Μ + K,(/T*r)]
x/2L V 2L V2V ^ ^ V * ^
-(* + y). (7.1-7b)
Equations (7.1-7a) and (7.1-7b) provide the form for the components of the transformed vector (7.1-5); thus the operator defined in (7.1-6) is capable of generating the desired transformation. When the coordinate system is rotated through an infinitesimal angle da about the z axis, / 1 /?(da,z) = ( - d a \ 0 /l R- (S
da 0\ 1 0, 0 1/ -da 1
(7.1-8)
0\/x\ /χ-γδα\ 0 ( y ] = (xda + A
(7.1-9)
«5α 0\ /ΚΛ / Vx+V,Sa\ 1 0 \\vy = - F ^ a + κ Λ
(7.1-10)
and / 1 R{da.,z)V= - d a
\θ
0 \]\vj
\
Vz I
Therefore PR(öoL,z)Vx(r) = RVx(R-1r) = Vx(R-1r) + SoiVy(R-1r) = T^.(x — y da, x da + v, z) + da Vy(x — y da, x da + y, z). (7.1-11) If one writes to first order, dV
dV
dV
(7.1-12)
7.1 ROTATIONAL PROPERTIES
185
a comparison of (7.1-11) with (7.1-12) yields PR(5a, z)Vx(r) = Vx(r) + - ^ (-yd«)
+ - ^ (χδα) + Vyb«.
= (1 + iöocLz)Vx + VyöoL.
(7.1-13a)
Similarly, PÄ(&x,z)7,(r) = (1 + iöocLz)Vy - δα Vx9
(7.1-13b)
PR(öa,z)Vz(r) = (1 + i 5 a L J 7 z .
(7.1-13c)
The three expressions in (7.1-13) may be combined into 0 PR(öa,z)V(r) = (l + iöaLz)V(r) + Soi[-l
1 0\/Vx(r)\ 0 0 K,(r)) 0 0 0 \VZ(T)
(7.1-14)
and, on defining (7.1-15) and JZ = LZ + SZ,
(7.1-16)
Eq. (7.1-14) takes the form Ρκ(δα, z)V(r) = (1 + ίδα JJV(r).
(7.1-17)
It is instructive to compare (7.1-17) with (2.2-8); the two expressions are very similar in form. The transformation of the scalar function is governed by the operator Lz whereas the transformation of the vector function is governed by the operator Jz which is the sum of Lz and the new operator Sz whose properties remain to be explored. By considering rotations about x and y, matrices analogous to 5Z are generated. They are
/0 0 Sx = (0 0 \0 i
0\ -i], 0/
/ 0 0 r Sy = i 0 0 0 \ - i 0 o;
(7.1-18a)
Similarly, corresponding to (7.1-16) one obtains JX = L X + ύχ,
J y = Ly + Oy
(7.1-18b)
186
7. VECTOR FIELDS
The three matrices Sx, Sy, and Sz are manifestly Hermitian; they also satisfy the commutation rules S x S = fS
(7.1-19)
from which it may be inferred that the components of S qualify as angular momentum operators. The eigenvalues of Sz are readily determined from the secular equation -λ i 0
-i -λ
0 0 =0 0 - 2
(7.1-20)
which yields λ= 1,0, - 1 . For the eigenvalue λ = 1, the eigenvalue equation
provides the normalized eigenvector (λ = 1).
(7.1-21)
In similar fashion it is found that (λ = 0);
e,i
=
(λ= -1).
(7.1-22)
These are just the unit vectors in the spherical basis (6.1-22) and are related to the unit vectors ex, e^,, and e2 along the coordinate axes by (6.1-23). Other relations among the unit vectors are given in Appendix 4. One may also construct the operator S2 defined by ύ
— ύχ
+ Oy + (3Z
(7.1-23)
Combining (7.1-15) and (7.1-18) /2 0 0\ S2 = [0
2 O]
(7.1-24)
\0 0 2) and, from the general properties of angular momentum operators, it is expected that e + 1 , e0, and e_ t will also be eigenvectors of S2. Thus, from
7.2 VECTOR SPHERICAL HARMONICS
187
(7.1-21)-(7.1-23),wehave S \ = 2e, = 1(1 + 1)8,
(q = 1,0, -1).
(7.1-25)
In an alternative notation one may write
ί
|H>=e+1
\SMS} = 1 |10> = e 0
(7.1-26)
111 - l > = 8 - i
in which case
SZ\SMS> = MS\SMS} (Ms = 1,0, -1), S2|SMS> = S(S + 1)|SMS> = 1(1 + 1)|SMS>.
(7.1-27)
In summary, the transformation properties of a vector field are governed by an orbital angular momentum operator L, a spin angular momentum operator S, and that the two operators are coupled according to the relation J = L + S.
(7.1-28)
On the basis of (7.1-27) it is seen that a vector field has unit spin (S = 1). On the other hand, as was shown in Section 2.2, the transformation of a scalar field is describable entirely on the basis of an orbital angular momentum operator L, or in other words, the spin of a scalarfieldis zero. Considerations such as these retain their significance even when the fields are quantized as, for example, in the case of photons arising from the quantization of an electromagnetic field. 7.2 Vector Spherical Harmonics It was shown in the previous section that the transformation of a vector field is characterized by an angular momentum operator J which arises from the coupling of an orbital angular momentum operator L and a spin angular momentum operator S. The simultaneous eigenfunctions of L 2 and Lz are the spherical harmonics Ylm(9, φ) and the simultaneous eigenfunctions of S2 and Sz are |SMS> with S = 1 and Ms = 1,0, — 1. It is then possible to construct simultaneous eigenfunctions of J2 and Jz according to the general rules for the coupling of angular momentum eigenfunctions (1.7-23). Adopting the notation ji = l j2 = S=l j=J
ml = m m2 = Ms= 1,0,-1 m=M
|Λ^1> = Ylm(6>
(7.2-1)
188
7. VECTOR FIELDS
one obtains |JM> = |/1 JM} = X lime,
=<-i>-<-*v27nz(m'; _$**.. a**
Equation (7.2-2) is the definition of a vector spherical harmonic usually designated YJiM; that is, YJ,M = \JM}. Since M = m + q, the double sum in (7.2-2) may be replaced by the single sum
ν,„ = (-ΐ)'-"727ΠΣ( Μ !. 9 \ _4)>Ϊ»-Α
Μ
with the following properties: J = / + 1,/,/ - 1, M = m + 9 = J, J - 1 , . . . , - J, (7.2-4) ^ M = /(/ + DYJUf, S2YJ/M = 1(1 + \)YJlM = 2YJZM, ΛΥ/ΙΜ = AfYJIM. ' % ! * = J(J + l)Y,«f, The vector spherical harmonic YZim with J = I and m = M = I, I — 1,...,— / is a particularly important one in electromagnetic theory. For this case the 3/ symbol will vanish unless the triangle condition Δ(/1/) is satisfied; the possible values of / are then / = 1,2,3,... but not / = 0. Using the formulas in Table 1.5 one may write Y
/(/ + 1) - m(m - 1)
_
"™---l
äöTIj 1(1 + 1) - m(m + I)
+ .1
2ΐ(7ΤΪ)
,m 1+1 +
-
m
^w+T)
^"^*-1·
(7 2 6)
· -
But
L(yiim = ynYlm,
L = —L +1 e_! + L0e0 — L_!e + 1 .
Therefore V'(i + 1) 1
Yi-lfLJ^.^ yfiT+T)t
LY
y/W+ΐ)
(7-2-7)
7.2 VECTOR SPHERICAL HARMONICS
189
It is observed that Yllm is orthogonal to the radius vector r since r.Y„m =r
. L - J ^ = = r . r x p — J = = 0.
(7.2-8)
y/l(l + 1) hy/l(l + 1) A number of vector spherical harmonics are listed in Table 7.1. For example,
/T
Yi i o = / ^ - ΐ sin 0 (— sin
(7.2-9b)
Suppose 0 = 90° so that the radius vector lies in the xy plane. Thefieldis parallel to e0; hence we have a linearly polarized field along the z axis, perpendicular to the radius vector. When, however 0 = 0, the field direction is given by e+1. The interpretation of such a field is based on the property e / x iv = i&o K'
(«>«' = 1,0, - 1 )
(7.2-10)
where e0 is the unit vector in the z direction. With q = +1 the cross product is a vector parallel to the z axis whereas the cross product with q = — 1 is antiparallel to the z axis. For this reason, fields specified by e +1 or e_i are said to have positive or negative helicity, respectively. In the context of wave propagation e + 1 corresponds to a counter clockwise rotation of the polarization vector when the observer is facing the oncoming beam. In optical terminology this is a left circularly polarized wave. Hence positive and negative helicity correspond to left and right circular polarization (see also Section 22.1). Finally, when 0 < 0 < 90°, thefieldin elliptically polarized. One often needs the scalar product of a vector spherical harmonic with itself, i.e., the absolute square. To compute this quantity, it is recalled that, according to (6.1-18),
A.A = £(-iM € >i- € . =Σ Λ % = Ul2.
(«=1,0,-1)
(7.2-11)
TABLE 7.1 Vector Spherical Harmonics0 Yo l o — *1 0 1
=
—
- ^ι -i^+i M) Oe + 1 >
*100
/T/sin0 .
=
/~ ^ι o^o + */'! ^ i i ^ - i
e
-VdvF '*
=
M) OeO»
g o + c „o s
*10-l
=
M)Oe-
\
H
/3 / —
sin 0 . e_ + ii — e ' ^ + cosfl-
'8«V
V2
V2
1 /3 .
Υι.ο=-λ/^ι-.«+ι+^1Ίι6-.
4>/π
[ s i n ö ^ e . ! + ei
— I—isin 0( — sin φ ex + cos φ ey V 8π =
/—isinöi*
λ/8π
=
/T/
sin0
/~3~ / /—
sin 0 η e x - ieA — g-'yez+cosö * Ί
Λ
\
= - - /-e- , >(r 2 + icos0r3) 4Vπ
Yl 2 1 = JTTj ^2 0^+i — / — ^21^0 + J - ^22^-1
1 /T 3 /Γ 3 /T = - / — ( 3 c o s 2 0 - l ) e + 1 + - /-cos0sin0
/-(3sin 2 0e 2 , > - 3cos 2 0 + l)e x
- ^ / - (3 sin2 0 e 2i * + 3 cos 2 0 - 1)6, f - / - cos 0 sin 0 £>'>ez y 8 >/π 4>/π 1
The unit vectors ΐι,ί 2 ,Ϊ3 are shown in Fig. A4.1.
TABLE 7.1 (continued) •y2_e+1
Yl20 = ,
/ - y2oeo + J —
-
^21^-1
3 /T 1/2 3 /l = - /—cos0sin0iT l,p e + 1 - - / - ( 3 c o s 2 0 - l ) e 0 - - /—cos 0 sin 0 £?**£_! 4λ/2π 4>/π 3 /Γ = - /-cos0sinO(excos
Yl 2 - 1 =
=
χ
/ 7 ^2 -2^+1 — / — ^2 -1^0 + χ / Τ Τ ^2 0^-1
! /l s in 2 öe- 2 '>e+ 1+ 1 -^ /I.cosösinöiT^o + - /—-(3cos 2 ö-l)e_! 4>/2π
4>/π
4 V 2π
= - -1 /fr i (3 sin2 ö f T 2 ' » - 3 cos2 0 + 1)6*
8>/π ^2 1 2
=
^Ι
_
e
-^-lJl
1^+1
-1^-1
^ 2 1 - 2 — ^Ι
^22 2 -
(3sin 2 0iT 2i * + 3cos2
ί1
/2
J ^ *2 1^+1 + J z ^2 2^0
Υ22 i = - ^
η οδ + ι + ^
>2 ie 0 + ^ 3
Υζ
2g-i
/-y 2 _ 1 e + 1 +J-y 2 1 e_!
Α
/ϊ
/ϊ"
.
Υ22 - l — J : *2 - 2 ^ + ι — / - y 2 - i e 0 +
^ 3 2 ±3
=
^2
±2e±\
Y32 ±2 — / r y2 ± i ^ ± i +
Y32 ±1 =
.
/ - y2o^-i
rz ^2 0*+i +
Y 3 2 0 — / 7 ^2 -1^+1 +
/ - y2
±2e0
/ T T ^2 ±1^0
/ τ ^2 0^0 +
+
λ/77
*2 ± 2 ^ τ ι
/ 7 ^2 l ^ - i
cos0 sin 0e'i9ez
192
7. VECTOR FIELDS
Therefore using (7.2-6) together with the orthogonality property of the unit vectors e^ as given by (6.1-26) one obtains f(Z + 1) m2
+
m
( m - 1)
ΐ ν / Λ xi2 / ( / + 1 ) - m ( m + 1 ) ,UM +
/(7Ti)i
|TZ
,_ , l 2 2
—2Ϊ(ΪΤΊ)—^-(M
(7.2-12) which is seen to be independent of the azimuthal angle φ. For illustration and for future reference one may cite: Yi i ±i · Yi i ±1 = \(\Yio\2
+ \Yi ±i| 2 ) = J ^ ( c o s 2 Ö + 1), (7.2-13)
Yi i o · Yt i o = ^ (l^i - i | 2 + | n + ιΓ) = ^ sin2 Θ, Y 2 2 ±2 · Y 2 2 ±2 = \\Yl
3
±1|
2
(7.2-14)
+ 1 ^ 2 ± 2 | 2 = T | " ( 1 - COS40),
3
i l y |2 + i | y
16π
(7.2-15)
I2 + 1
V .Y —-lv l2_u_|y I 2 _L l y I2 *2 2 ±1 * 2 2 + 1 — 0 |22 0 " r / : | I 2 ± l | + ^ | 2 2 ± 2 |
= - ^ - (1 - 3 cos 2 Θ + 4 cos 4 0), 16π Y I
2 2 0
.Y I 2 2 0
(7.2-16)
— - ly I 2 -L _ l y I2 ^|I2-l| "^^|22+l|
—
= ^sin20cos20.
(7.2-17)
These angular distributions are shown in Fig. 7.2. Those with I = 1 are known as dipole fields and those with Z = 2 are quadrupole fields. When 0 = 0, the spherical harmonics vanish except for Yl0, which has the value
* - . / ^ . In that case, (7.2-12) simplifies to 2/+1 ^llm * Yfim — *Π ±1 * *ll ±1 — o '
(7.2-19)
7.3
PLANE WAVE EXPANSION
193
FIG. 7.2 Angular distribution in (a) dipole and (b) quadrupole fields. (Jackson, 1962.)
that is, when 0 = 0 Y„m · Y//m has a nonvanishing value only when m = ± 1. Referring to (7.2-3) it is seen that the only spherical harmonics that appear in the definition of YJlM are those of order /. YJlM therefore has odd parity when / is odd and even parity when / is even. Also, as a consequence of the orthogonality property of the spherical harmonics and the unit vectors, the vector spherical harmonics satisfy the orthogonality relation JVJlM · YrVM. sin θάθάφ = 8jr διν δΜΜ,
(7.2-20)
with J + I+l+Mv
YJiif = ( - ! ) '
If
(7.2-21)
7.3 Plane Wave Expansion An important application of vector spherical harmonics is in the expan sion of plane waves. According to (1.2-28), Jk · r _
4π Σ 1=0
Σ ilJ,(kr)Ylm(er,
where jt(kr) is a spherical Bessel function (Appendix 5) and (θ„φΧ (6k,(pk) are the angular coordinates of the position and wave vectors, respectively (Fig. 1.4). If it is assumed that the propagation is along the z direction, 0k = 0 in which case only Yf0 is nonzero and is given by (7.2-18). The sum over m then consists of only one term, namely the one with m = 0. Therefore eik* = X iV4n(2/ + l)j,(kr)Yl0(e)
(7.3-1)
1=0
where Θ = 6r. Now let A = eYeikz
(7.3-2)
194
7. VECTOR FIELDS
Apart from an exponential time factor, (7.3-2) represents a plane wave propagating along the z axis and polarized along the x axis. By writing
A=-4=(A + 1 -A-Os-4=(i + i-i-i)* t o we have A ± 1 = e ±1 e* z ,
(7.3-3)
which, we saw, represents right and left circularly polarized waves. Using (7.3-1), A ± ! = Σ « V W + l);,(fcr)^o(0)e±! ·
(7.3-4)
I
We shall now express Yl0(6)e± t in terms of vector spherical harmonics. From (1.5-54), \ml
JM
m
2
—MJ
with M = m1 + m 2 . In the present case |7i"*i> = Yio,
\J2m2> = |1«> = % = e ± i ,
\jJ2JM>
= YJlM = Y,, ± 1 .
Therefore
1ί0β+1 = Σ ( - 1 ) " ' ν ^ Τ Τ ^ j J [ ) Y „ + I .
(7.3-5)
Since the possible values of J are I - 1, /, Z + 1, / 1 /- 1 l 1 ϊ 0 8 + ι = ( - 1 ) "[j2T Ί ν 2 / - 1 ( 0 1 _ ! )Y,-iu
+
^ ( ό ί -ΐ) Υ "< + ^(θ ί '-I1)5"«} (7.3-6)
From Table 1.5, ' 0
* 1
/ _ 1 λ
-I
I
= v(-ΐ)'
/ _ 1 ' ' V 2(2/ + 1)(2/ - 1)'
ό ! -0- ( - ,r 'Vjprn5·
(7 3 7,
·"
7.3 PLANE WAVE EXPANSION
195
Therefore 1 IEHY Y + Ι1+ΆY i , + 2^ΠΤ)Ύι-ιη~>βΥαι >12(21 + 1)
io8 + i = J ^ ^ Y | - i i i - - ^ Y « i + . / ^ f r k
+
i«i.
(7·3"8)
and in the same fashion
Yl 1 =
^-
yJ2W^)Yl-u~1
+
^Yu~1 + ylw+T)Yt+lt~1'
(7 3 9)
'"
Note that when YJIM = Yl^lll the value of M is 1; therefore J cannot be smaller than 1. But since J — I — 1 we must have / ^ 2. Similarly, when YJIM = Y n i , it is necessary that / > 1, but when YJ(M = Y,+1 n, the smallest value of / can be zero. With (7.3-8) and (7.3-9) substituted into (7.3-4) we obtain the plane wave expansion in terms of vector spherical harmonics: A ± 1 = e ±1 e' fe = £ i'j2n(l + 2)jl(kr)Yl+ll
±1
ilj2n(2l+l)jt(kr)yll±i
+ Σ
+ Σ ίψ2π(Ζ-!)./,(&■)¥,_ 1 I ± 1 .
(7.3-10)
1=2
For physical applications one is usually interested in the first few terms of expansion (7.3-10), particularly under conditions where kr « 1. The leading term is the one containing j0(kr\ that is, A<±1)1=>/ii/o(fcr)Y1o±i·
(7.3-11)
From Table 7.1 and the asymptotic values of the spherical Bessel functions ((A5-17) in Appendix 5) *10±1
=
^00^±1
=
r—
e
±l»
χ/4π
j0(kr)=l
(kr« 1),
so that (7.3-11) becomes A^=e±1.
(7.3-12)
Hence the approximation of A ± 1 by A(+\ is equivalent to setting eikz = 1 in (7.3-3). The next higher-order terms are those containing j^kr). These are A<±2> = i46nj1{kr)Y2 t
±1
+ ί^^ήΥ, 1 ± 1 .
(7.3-13)
196
7. VECTOR FIELDS
Again using Table 7.1 and (A5-17) ^ 2 1 ±1 = V2L^10^±l + Yl ±l^o]> *1 1 ±1 = V2"L+ ^10^11 i
h(kr) = ^kr
^1 ±l^o]?
(/cr«l).
Therefore /4π /~3~ /4π A(±\ = i — krY10e±i = i l—kr /-—cos0e ±1 = ikr cos Ö e ± ! = ikze± t
(7.3-14)
and Α(±υι + A(±2\ = (1 + ikz)e±l
(7.3-15)
which corresponds to the first two terms in the power series expansion of the exponential in (7.3-3). 7.4 Multipole Expansion of the Electromagnetic Field The free-field Maxwell equations are V · B = 0,
V · E = 0,
IdE „ n VxB = -—,
^ 13B VxE= — —
c ot
c ot
(7·4-!)
with 1 ί)\ E=—— c ot
B-VxA,
(7.4-2)
where A is the vector potential. The time dependence of thefieldvectors may be taken to be of the form e~i(0t in which case (7.4-1) and (7.4-2) become V.B-* V x B = - ikE, B = V x A,
V.E-a V x E = ikB, E = ikk,
(7.4-4)
where k = ω/c. From (7.4-3) and (7.4-4) and the assumption that V · A = 0 (Coulomb gauge) one obtains the vector Helmholtz equations: (V2 + k2)B(r) = 0,
(7.4-5a)
2
2
(7.4-5b)
2
2
(7.4-5c)
(V +fc )E(r)= 0, (V +fc )A(r)= 0.
7.4 MULTIPOLE EXPANSION OF THE ELECTROMAGNETIC FIELD
197
As a step toward the solution of (7.4-5) we consider first the scalar Helmholtz equation (V2 + fc2)^(r) = 0.
(7.4-6)
Equation (7.4-6) when written in spherical coordinates is separable into two equations one of which depends on r only and the other on Θ and φ. Solutions to the angular equation are expressible in terms of the spherical harmonics. We therefore write = ΣΜ')γι*(°><Ρ)>
ΦΜφ)
(7·4"7)
Im
and on substituting (7.4-7) into (7.4-6), one obtains the radial equation
^ii
\_dr2
r dr
+
^ ^2 ' r
j
u,(r) = 0
(7.4-8)
where ut(r) = yfrfi(r). Equation (7.4-8) is a form of the Bessel equation (Appendix 5). Therefore the solutions to (7.4-6) may be written •AW = Σ lAftWKkr) + ^hf\kr)-\ Im
Υ1η(θ, φ)
(7.4-9)
where h\l\kr) and h{2\kr) are spherical Hankel functions of the first and second kind, respectively, and the coefficients A\m] and A\m] are determined from the boundary conditions. In place of the Hankel functions one may write ^(r) in (7.4-9) in terms of j^kr) and n^kr) since h\2\kr) = jt(kr) - in^kr).
hfKkr) = jt(kr) + inx(kr\
(7.4-10)
Thus any product of the form fi(kr)Ylm(e9q>\ where ft(kr) stands for any one of the spherical Bessel functions h^ikrXhffXkrXj^kr), or n^kr) will satisfy (7.4-6), i.e., (W2 + k2)fl(kr)Ylm(e^) = 0.
(7.4-11)
2
Now consider the commutator [L, V ]. In spherical coordinates .
d2 dr2
2d r dr
L2 r2'
and since the components of L operate only on angular coordinates and L2 commutes with every component of L, we have [L,V 2 ]=0.
(7.4-12)
198
7. VECTOR FIELDS
Therefore L(V2 +fc2W(/cr)>U0,φ) = (V2 +fc2)ü(/cr)L^m(ö,φ) = 0, (7.4-13) from which we conclude that λ(Ια%Υ1ηι{θ,φ) satisfies the vector Helmholtz equation (7.4-5). Using (7.2-7), a solution to (7.4-5a) may be written as (V2 +fc2)BE(/,m,r)= 0
(7.4-14)
where BE(/, m, r) = MkryLYJie, φ) = JÜ^mW^
(7.4-15)
and the corresponding electric field, according to the Maxwell equations, is EE(/, m, r) = l- V x BE(/, m, r).
(7.4-16)
The transversality relation (7.2-8) implies that r.B E (/,m,r) = 0;
(7.4-17)
therefore an electromagneticfieldcharacterized by (7.4-15)-(7.4-17) is known as a TM (transverse magnetic) field of order /. However, the transversality condition does not apply to EE which may then have non-vanishing radial components. Such a field is therefore also designated as an electric multipole of order /. Since ft(kr)LYlm(e, φ) is a solution to the vector Helmholtz equation it could equally well serve as a solution to (7.4-5b). We therefore have another pair of solutions EM(lm9r) = fl(kr%Ylm(e,cpl
(7.4-18)
BM(/, m, r) = ^ V x EM(/, m, r),
(7.4-19)
r.E M (/,m,r) = 0, (7.4-20) and such afieldis known as a magnetic multipole or a TE (transverse electric) field of order /. In this case BM may have radial components. The general solution to the vector Helmholtz equations for the electro magnetic field may now be expressed as a linear superposition of electric and magnetic multipoles of all orders: B(r) = Σ M'> rn)BE(l, m, r) + aM(/, m^l,
m, r)],
(7.4-21)
lm
E(r) = Σ [%('> W)EE('> rn, r) + aM(/, m)EE(/, m, r)],
(7.4-22)
lm
in which the coefficients depend on the sources and the boundary conditions.
7.4 MULTIPOLE EXPANSION OF THE ELECTROMAGNETIC FIELD
199
The two types of multipoles are distinguished by the fact that the roles of the electric and magnetic fields in the two cases are interchanged. Thus from the fact that the parity of YJlM is (— 1)' we see that the parities of BE(/,m,r) and EM(/, m, r), which we designate by π\_ΒΕ(1, m, r)] and π[£Μ(Ζ, m, r)], respec tively, are both (-1)*. Since EE is proportional to V x BE according to (7.4-3), the parity of EE(Z, m, r) must be opposite to that of BE(Z, m, r), or (— 1)'+* because the curl operation transforms a polar vector into an axial vector and vice versa. The same applies to BM(/,m,r) whose parity is also ( — 1) I+1 . Also in view of the proportionality between A and E [Eq. (7.4-4)], the parity of A is the same as the parity of E. Thus we have π[ΒΕ(/, m, r)] = π[ΕΜ(Ζ, m, r)] = π[ΑΜ(ί, m, r)] = (-1)', 7r[EE(/,m,r)] = 7i[BM(/,m,r)] = 7r[AE(/,m,r)] = ( - 1 ) ' + 1 .
(
''
Since Yllm = 0 for / = 0 (Section 7.2), multipoles of order zero vanish iden tically. This ensures the transversality of the fields to the direction of propaga tion in an electromagnetic wave in free space. It is customary to employ the following conventions and notations: (1) The parity of an electromagnetic field nem is defined as the parity of B, i.e., 7rem = π(Β). (2) The polarization of an electromagnetic field is in the direction of E. (3) The multipoles are named according to the values of 2l; thus dipole (/ = 1 , 2 ' = 2), quadrupole (J = 2,2' = 4), octapole (Z = 3,2* = 8), etc. Addi tional designations are E/ for electric multipole of order I and Ml for magnetic multipole of order /. Thus El, E 2 , . . . are electric dipole, quadrupole,... and Ml, M 2 , . . . are magnetic dipole, quadrupole,.... The angular distribution of a multiple may be calculated from the fact that the energy density of a radiation field is proportional to E · E or B · B. Therefore for magnetic multipoles the angular distribution may be obtained from EM(/, m, r) · EM(Z, m, r) which is proportional to Yllm · Y„m whose general form is given by (7.2-12). For electric multipoles the angular distribution can be obtained from BE(/, m, r) · BE(/, m, r) which again is proportional to Yüm · YUm. Thus the angular distribution is the same for electric and magnetic multipoles of the same order. For dipole and quadrupole radiation the patterns are shown in Fig. 7.2. One should note, however, that since EM(Z, m, r) has the same form as BE(/, m, r) but EE(Z, m, r) is proportional to V x BE(Z, m, r), the polarizations of electric and magnetic multipoles of the same order are perpendicular to one another. Let the vector potential be given by A±i =
e±lei
(7.4-24)
200
7. VECTOR FIELDS
as in (7.3-3). The expansion of A ± x was given by (7.3-10). Our objective now is to examine the multipolarity of the electromagnetic fields associated with the first few terms in the expansion of A+ x. According to (7.3-11), the· leading term (kr « 1) is Α(±υ! = ν4π/ο(^)Υι ο ±ι = 8± i
(7-4-25)
and
Β^νχΑΪΙ. To evaluate the curl of quantities such as those on the right side of (7.4-25) the following special formulas (Edmonds, 1960) are useful: V x Mkr)Yllm = ik
Vx
fi-akrm,.^
2/+1
h+iykiyiii+i,
+ ly^-lf,-i(kr)Yll-irn
(7.4-26a)
Yllm.
(7.4-26b)
= -ikf^kr)^^
Thus n \ = - i / yfc/i(fcr)Yii ±i ·
(7.4-26c)
Since / = 1 the field is a dipole field. Also, since the parity of YJlM is (— 1)', the parity of B(+\ is (— I)1 = — 1. From Table 7.2 the electromagnetic field is further identified as El, that is, electric dipole. We conclude that A(+\ in (7.4-25) is the vector potential for an electric dipole field. Following the TABLE 7.2 Multipole Relations0
Multipole
Field vector
Proportional to
El
BE(/,m,r) EE(/,m,r) AE(/,m,r)
fi(kr)Yllm V x fi(kr)Yllm V x ft(kr)YllM
B M (/,m,rr EM(/,m,r) AM(/,m,r)
V xfAkrWum fi(kr)Yllm fi(kr)Yllm
Ml
a
Parity
Electromagnetic field or photon parity
(-1)'
. (-D ,+I (-D' +1 (-D' +1 (-1)' (-1)'
The parity of an electromagnetic field is defined by the parity of B.
(-1)'
7.4 MULTIPOLE EXPANSION OF THE ELECTROMAGNETIC FIELD
201
argument in Section 7.3 it is seen that the approximation whereby A ± 1 = e±1eikz is replaced by Ad) _ £
that is, eikz = 1 corresponds to the approximation whereby a plane wave is replaced by an electric dipole field. The next term in expansion (7.3-10) is A<±2> = iVfeA(*r)[Y 2 i ± i + Yi i ±i] = ikze±±
(7.4-27)
as shown in (7.3-13) and (7.3-14). The term containing Yx x ± 1 has a parity of — 1. Since the vector potential A is proportional to the electric field E, the parity of E is — 1. With / = 1 such a field is identified (Table 7.2) as a magnetic dipole (Ml) field. To identify the term containing Y 2 i ±i we note from (7.4-26b) that V x j1(kr)Y2
1±1
= -ikj2(kr)Y2
2±2.
(7.4-28)
(
Therefore V x A + \ contains Y22±2 which means that there is a B field with / = 2 or (— I)1 = + 1 . From Table 7.2 such a field is identified as electric quadrupole (E2). Thus we have the result that A(+\ in (7.4-27) represents a combination of a magnetic dipole (Ml) and an electric quadrupole (E2) field. The results we have obtained may be summarized by writing eikz = 1(E1) + ikz{E2 + Ml) + · · · .
(7.4-29)
In further detail the M l part of the vector potential is (A ( ± 2 \)MI
= TiyßÜMkryY,!
±1
= il^kr(Y10e±1
- Yx ± 1 e 0 )
(7.4-30)
in which the approximation
h(kr) = $kr
(kr«l)
has been made. The rectangular components are (A^2))M1 = i \ (zex - xe z ),
(A<2>)M1 = i ± (z% - ytt).
(7.4-31)
The E2 part of the vector potential is (A<±2»)E2 = i^zj1(kr)Y2
! ± 1 = ifckr(Y10e±1
+ Y, ± 1 e 0 ),
(7.4-32)
202
7. VECTOR FIELDS
and the rectangular components are (A{x2))E2 =
k i-(zex+xez\
(A<2,)E2
= i\(2% + yU
(7.4-33)
We also note that A<2) = (A<2>)M1+(A<2))E2 = i/cz^, (7.4-34)