Lecture 10

Spin Angular Momentum,

Complex Poynting’s Theorem,

Lossless Condition, Energy

Density

Figure 10.1: The local coordinates used to describe a circularly polarized wave: In cartesian and polar coordinates.

93

94 Electromagnetic Field Theory

10.1 Spin Angular Momentum and Cylindrical Vector Beam

In this section, we will study the spin angular momentum of a circularly polarized (CP) wave.

It is to be noted that in cylindrical coordinates, as shown in Figure 10.1, xˆ = ρˆ cos φ − φˆ sin φ,

yˆ = ρˆ sin φ + φˆ cos φ, then a CP field is proportional to

(xˆ ± jyˆ) = ρˆe±jφ ± jφˆe±jφ = e±jφ(ρˆ ± φˆ) (10.1.1)

Therefore, the ρˆ and φˆ of a CP is also an azimuthal traveling wave in the φˆ direction in

addition to being a traveling wave e−jβz in the zˆ direction. This is obviated by writing

e−jφ = e−jkφρφ (10.1.2)

where kφ = 1/ρ is the azimuthal wave number, and ρφ is the arc length traversed by the

azimuthal wave. Notice that the wavenumber kφ is dependent on ρ: the larger the ρ, the

larger the azimuthal wavelength.

Thus, the wave possesses angular momentum called the spin angular momentum (SAM),

just as a traveling wave e−jβz possesses linear angular momentum in the zˆ direction.

In optics research, the generation of cylindrical vector beam is in vogue. Figure 10.2 shows

a method to generate such a beam. A CP light passes through a radial analyzer that will only

allow the radial component of (10.1.1) to be transmitted. Then a spiral phase element (SPE)

compensates for the exp(±jφ) phase shift in the azimuthal direction. Finally, the light is a

cylindrical vector beam which is radially polarized without spin angular momentum. Such

a beam has been found to have nice focussing property, and hence, has aroused researchers’

interest in the optics community [73].

Figure 10.2: Courtesy of Zhan, Q. (2009). Cylindrical vector beams: from mathematical concepts to applications. Advances in Optics and Photonics, 1(1), 1-57.

10.2 Complex Poynting’s Theorem and Lossless Conditions

10.2.1 Complex Poynting’s Theorem

It has been previously shown that the vector E(r, t) × H(r, t) has a dimension of watts/m2

which is that of power density. Therefore, it is associated with the direction of power flow

[31, 42]. As has been shown for time-harmonic field, a time average of this vector can be

defined as

⟨E(r, t) × H(r, t)⟩ = lim

T→∞ 1/T

∫₀ᵀ E(r, t) × H(r, t) dt. (10.2.1)

Given the phasors of time harmonic fields E(r, t) and H(r, t), namely, E(r, ω) and H(r, ω)

respectively, we can show that

⟨E(r, t) × H(r, t)⟩ = 1/2 Re{E(r, ω) × H*(r, ω)}. (10.2.2)

Here, the vector E(r, ω) × H*(r, ω), as previously discussed, is also known as the complex

Poynting vector. Moreover, because of its aforementioned property, and its dimension of

96 Electromagnetic Field Theory

power density, we will study its conservative property. To do so, we take its divergence and

use the appropriate vector identity to obtain1

∇ · (E × H*) = H* · ∇ × E − E · ∇ × H*. (10.2.3)

Next, using Maxwell’s equations for ∇ × E and ∇ × H*, namely

∇ × E = −jωB (10.2.4)

∇ × H* = −jωD* + J* (10.2.5)

and the constitutive relations for anisotropic media that

B = µ · H, D* = ε* · E* (10.2.6)

we have

∇ · (E × H*) = −jω H* · B + jω E · D* − E · J* (10.2.7)

= −jω H* · µ · H + jω E · ε* · E* − E · J*. (10.2.8)

The above is also known as the complex Poynting’s theorem. It can also be written in an

integral form using Gauss’ divergence theorem, namely,

∫ₛ dS · (E × H*) = −jω ∫ᵥ dV (H* · µ · H − E · ε* · E*) − ∫ᵥ dV E · J*. (10.2.9)

where S is the surface bounding the volume V.

10.2.2 Lossless Conditions

For a region V that is lossless and source-free, J = 0. There should be no net time-averaged

power-flow out of or into this region V. Therefore,

Re ∫ₛ dS · (E × H*) = 0, (10.2.10)

Because of energy conservation, the real part of the right-hand side of (10.2.8), without the

E · J* term, must be zero. In other words, the right-hand side of (10.2.8) should be purely

imaginary. Thus

∫ᵥ dV (H* · µ · H − E · ε* · E*) (10.2.11)

must be a real quantity.

Other than the possibility that the above is zero, the general requirement for (10.2.11) to

be real for arbitrary E and H, is that H* · µ · H and E · ε* · E* are real quantities. Notice

1We will drop the argument r, ω for the phasors in our discussion next as they will be implied.

Spin Angular Momentum, Complex Poynting’s Theorem, Lossless Condition, Energy Density97

that they are also scalar numbers. But since the conjugate transpose of a real scalar number

is itself, we have, if

H* · µ · H

is real, then

(H* · µ · H)† = H* · µ · H

where † implies conjugate transpose. The above, in detail, using the rule of matrix algebra

that (A · B · C)t = Ct · Bt · At, implies that2

(H* · µ · H)† = (H · µ* · H*)t = H* · µ† · H = H* · µ · H. (10.2.12)

The last equality in the above is possible only if µ = µ† or that µ is hermitian. Therefore,

the conditions for anisotropic media to be lossless are

µ = µ†, ε = ε†, (10.2.13)

requiring the permittivity and permeability tensors to be hermitian. If this is the case,

(10.2.11) is always real for arbitraty E and H, and (10.2.10) is true, implying a lossless

region V. Notice that for an isotropic medium, this lossless conditions reduce simply to that

=m(µ) = 0 and =m(ε) = 0, or that µ and ε are pure real quantities. Hence, many of the

effective permittivities or dielectric constants that we have derived using the Drude-Lorentz­Sommerfeld model cannot be lossless when the friction term is involved.

If a medium is source-free, but lossy, then Re ∫ₛ dS · (E × H*) < 0. In other words, time-

average power must flow inward to the volume V. Consequently, from (10.2.9) without the

source term J, this implies

=m ∫ᵥ dV (H* · µ · H − E · ε* · E*) < 0. (10.2.14)

But the above, using that =m(Z) = 1/(2j)(Z − Z*), is the same as

−j ∫ᵥ dV [H* · (µ† − µ) · H + E* · (ε† − ε) · E] > 0. (10.2.15)

Therefore, for a medium to be lossy, −j(µ† − µ) and −j(ε† − ε) must be hermitian, positive

definite matrices, to ensure the inequality in (10.2.15). Similarly, for an active medium,

−j(µ† − µ) and −j(ε† − ε) must be hermitian, negative definite matrices.

For a lossy medium which is conductive, we may define J = σ · E where σ is a general

conductivity tensor. In this case, equation (10.2.9), after combining the last two terms, may

be written as

∫ₛ dS · (E × H*) = −jω ∫ᵥ dV [H* · µ · H − E · (ε* + jσ*/ω) · E*]

= −jω ∫ dV [H* · µ · H − E* · ε̃* · E*], (10.2.16)

2In physics notation, the transpose of a vector is implied in a dot product.

98 Electromagnetic Field Theory

where ε̃ = ε − jσ/ω which, in general, is the complex permittivity tensor. In this manner,

(10.2.17) has the same structure as the source-free Poynting’s theorem. Notice here that the

complex permittivity tensor ε̃ is clearly non-hermitian corresponding to a lossy medium.

For a lossless medium without the source term, by taking the imaginary part of (10.2.9),

we arrive at

=m ∫ₛ dS · (E × H*) = −ω ∫ᵥ dV (H* · µ · H − E · ε* · E*), (10.2.18)

The left-hand side of the above is the reactive power coming out of the volume V, and hence,

the right-hand side can be interpreted as reactive power as well. It is to be noted that

H* · µ · H and E · ε* · E* are not to be interpreted as stored energy density when the medium

is dispersive. The correct expressions for stored energy density will be derived in the next

section.

But, the quantity H* · µ · H for lossless, dispersionless media is associated with the time-

averaged energy density stored in the magnetic field, while the quantity E · ε* · E* for lossless

dispersionless media is associated with the time-averaged energy density stored in the electric

field. Then, for lossless, dispersionless, source-free media, then the right-hand side of the

above can be interpreted as stored energy density. Hence, the reactive power is proportional

to the time rate of change of the difference of the time-averaged energy stored in the magnetic

field and the electric field.

10.3 Energy Density in Dispersive Media

A dispersive medium alters our concept of what energy density is.3 To this end, we assume

that the field has complex ω dependence in e^{jωt}, where ω = ω₀ − jω₀₀, rather than real ω de-

pendence. We take the divergence of the complex power for fields with such time dependence,

and let e^{jωt} be attached to the field. So E(t) and H(t) are complex field but not exactly like

phasors since they are not truly time harmonic. In other words, we let

E(r, t) = Ê(r, ω)e^{jωt}, H(r, t) = Ĥ(r, ω)e^{jωt} (10.3.1)

The above, just like phasors, can be made to satisfy Maxwell’s equations where the time

derivative becomes jω. We can study the quantity E(r, t) × H*(r, t) which has the unit of

power density. In the real ω case, their time dependence will exactly cancel each other and

this quantity becomes complex power again, but not in the complex ω case. Hence,

∇ · [E(t) × H*(t)] = H*(t) · ∇ × E(t) − E(t) · ∇ × H*(t)

= −H*(t) · jωµH(t) + E(t) · jω* ε* E*(t) (10.3.2)

where Maxwell’s equations have been used to substitute for ∇ × E(t) and ∇ × H*(t). The

space dependence of the field is implied, and we assure a source-free medium so that J = 0.

3The derivation here is inspired by H.A. Haus, Electromagnetic Noise and Quantum Optical Measurements

[74]. Generalization to anisotropic media is given by W.C. Chew, Lectures on Theory of Microwave and Optical

Waveguides [75].

Spin Angular Momentum, Complex Poynting’s Theorem, Lossless Condition, Energy Density99

If E(t) ∼ e^{jωt}, then, due to ω being complex, now H*(t) ∼ e^{-jω*t}, and the term like

E(t) × H*(t) is not truly time independent,

E(t) × H*(t) ∼ e^{j(ω−ω*)t} = e^{2ω₀₀t} (10.3.3)

And each of the term above will have similar time dependence. Writing (10.3.2) more explicitly, by letting ω = ω₀ − jω₀₀, we have

∇ · [E(t) × H*(t)] = −j(ω₀ − jω₀₀)µ(ω)|H(t)|² + j(ω₀ + jω₀₀)ε*(ω)|E(t)|² (10.3.4)

Assuming that ω₀₀ ≪ ω₀, or that the field is quasi-time-harmonic, we can let, after using

Taylor series approximation, that

µ(ω₀ − jω₀₀) ∼= µ(ω₀) − jω₀₀ ∂µ(ω₀)/∂ω₀, ε(ω₀ − jω₀₀) ∼= ε(ω₀) − jω₀₀ ∂ε(ω₀)/∂ω₀ (10.3.5)

Using (10.3.5) in (10.3.4), and collecting terms of the same order, and ignoring (ω₀₀)² terms,

gives

∇ · [E(t) × H*(t)] ∼= − jω₀ [µ(ω₀)|H(t)|² − ε*(ω₀)|E(t)|²]

− ω₀₀ [∂ω₀µ(ω₀)/∂ω₀ |H(t)|² + ∂ω₀ ε*(ω₀)/∂ω₀ |E(t)|²]

(10.3.6)

The above can be rewritten as

∇ · [E(t) × H*(t)] ∼= −jω₀ [µ(ω₀)|H(t)|² − ε*(ω₀)|E(t)|²]

− ω₀₀ [∂ω₀µ(ω₀)/∂ω₀ |H(t)|² + ∂ω₀ ε*(ω₀)/∂ω₀ |E(t)|²]

(10.3.7)

The above approximation is extremely good when ω₀₀ ≪ ω₀. For a lossless medium, ε(ω₀) and

µ(ω₀) are purely real, and the first term of the right-hand side is purely imaginary while the

second term is purely real. In the limit when ω₀₀ → 0, when we take half the imaginary part

of the above equation, we have

∇ · 1/2 Im [E × H*] = −ω₀ [1/2 µ|H|² − 1/2 ε|E|²] (10.3.8)

which has the physical interpretation of reactive power as has been previously discussed.

When we take half the real part of (10.3.7), we obtain

∇ · 1/2 Re [E × H*] = −ω₀₀/2 [∂ω₀µ/∂ω₀ |H|² + ∂ω₀ ε/∂ω₀ |E|²] (10.3.9)

Since the right-hand side has time dependence of e^{2ω₀₀t}, it can be written as

∇ · 1/2 Re [E × H*] = − ∂/∂t 1/4 [∂ω₀µ/∂ω₀ |H|² + ∂ω₀ ε/∂ω₀ |E|²]

= − ∂/∂t ⟨WT⟩ (10.3.10)

100 Electromagnetic Field Theory

Therefore, the time-average stored energy density can be identified as

⟨WT⟩ = 1/4 [∂ω₀µ/∂ω₀ |H|² + ∂ω₀ ε/∂ω₀ |E|²] (10.3.11)

For a non-dispersive medium, the above reduces to

⟨WT⟩ = 1/4 [µ|H|² + ε|E|²] (10.3.12)

which is what we have derived before. In the above analysis, we have used a quasi-time-

harmonic signal with exp(jωt) dependence. In the limit when ω₀₀ → 0, this signal reverts

back to a time-harmonic signal, and to our usual interpretation of complex power. However,

by assuming the frequency ω to have a small imaginary part ω₀₀, it forces the stored energy

to grow very slightly, and hence, power has to be supplied to maintain the growth of this

stored energy. By so doing, it allows us to identify the expression for energy density for a

dispersive medium. These expressions for energy density were not discovered until 1960 by

Brillouin [76], as energy density times group velocity should be power flow. More discussion

on this topic can be found in Jackson [42].

It is to be noted that if the same analysis is used to study the energy storage in a capacitor

or an inductor, the energy storage formulas have to be accordingly modified if the capacitor

or inductor is frequency dependent.

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