Download The macroscopic Maxwell equations

Chap. 1 Macroscopic Maxwell’s equations
LALANNE Philippe (IOGS 2nd année)
Chapter 1
Macroscopic Maxwell’s equations
To describe optical radiation and propagation in waveguides, it is mostly sufficient to adopt the
wave picture. This allows us to use classical field theory based on Maxwell’s equations. This
section summarizes the fundamentals of electromagnetic theory forming the necessary basis for
the course. The presentation is oriented towards the theoretical results that are needed to
establish the fundamentals of integrated optics. Here only the basic properties are discussed
and for more detailed treatments the reader is referred to standard textbooks on
electromagnetism such as the books by Jackson [Jac99], and others. The starting point is
Maxwell’s equations established by James Clerk Maxwell in 1873.
1.1 Macroscopic electrodynamics
In macroscopic electrodynamics the singular character of charges and their associated currents
is avoided by considering charge densities ρ and current densities j. In differential form and in SI
units the macroscopic Maxwell’s equations have the form
∇ × E(r , t ) = − ∂B(r , t ) ∂ t ,
(1.1)
∇ × H(r , t ) = ∂D(r , t ) ∂ t + j (r , t ) ,
(1.2)
∇ ⋅ D(r , t ) = ρ(r , t ) ,
(1.3)
∇ ⋅ B(r , t ) = 0 ,
(1.4)
where E denotes the electric field, D the electric displacement, H the magnetic field, B the
magnetic induction, j the current density, and ρ the charge density. The components of these
vector and scalar fields constitute a set of 16 unknowns. Depending on the considered medium,
the number of unknowns can be reduced considerably.
For example, in linear, isotropic, homogeneous and source-free media the electromagnetic
field is entirely defined by two scalar fields. Maxwell’s equations combine and complete the laws
formerly established by Faraday, Ampère, Gauss, Poisson, and others. Since Maxwell’s
equations are differential equations, they do not account for any fields that are constant in space
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Chap. 1 Macroscopic Maxwell’s equations
LALANNE Philippe (IOGS 2nd année)
and time. Any such field can therefore be added to the fields. It has to be emphasized that the
concept of fields was introduced to explain the transmission of forces from a source to a
receiver. The physical observables are therefore forces, whereas the fields are definitions
introduced to eliminate the troublesome phenomenon of the “action at a distance”. Notice that
the macroscopic Maxwell’s equations deal with fields that are local spatial averages over
microscopic fields associated with discrete charges. Hence, the microscopic nature of matter is
not included in the macroscopic fields. Charge and current densities are considered as
continuous functions of space. In order to describe the fields on atomic scale it is necessary to
use the microscopic Maxwell’s equations which consider all matter made of charged and
uncharged particles.
The conservation of charge is implicitly contained in Maxwell’s equations. Taking the
divergence of Eq. (1.2), noting that ∇ ·∇ × H is identical zero, and substituting Eq. (1.3) for
∇ · D one obtains the continuity equation
∇ ⋅ j(r , t ) + ∂ρ(r , t ) ∂ t = 0 .
(1.5)
The electromagnetic properties of the medium are most commonly discussed in terms of the
macroscopic polarization P and magnetization M according to
D(r , t ) = ε 0 E(r , t ) + P(r , t ) ,
(1.6)
H(r , t ) = µ 0−1 B(r , t ) − M(r , t ) ,
(1.7)
where ε 0 and µ 0 are the permittivity and the permeability of vacuum, respectively. These
equations do not impose any conditions on the medium and are therefore always valid.
1.2 Wave equations
After substituting the fields D and B in Maxwell’s curl equations by the expressions (1.6) and
(1.7) and combining the two resulting equations we obtain the inhomogeneous
wave equations
∇ × ∇ ×E +
∂ ( j + ∂P ∂ t + ∇ × M)
1 ∂ 2E
,
= −µ 0
2
2
∂t
c ∂ t
(1.8)
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Chap. 1 Macroscopic Maxwell’s equations
LALANNE Philippe (IOGS 2nd année)
∇ × ∇ ×H +
1 ∂ 2H
∂P 1 ∂ 2M
.
=
∇
×
j
+
∇
×
−
∂ t c2 ∂2 t
c2 ∂2 t
(1.9)
The constant c was introduced for (ε 0 µ 0 )−1 / 2 and is known as the vacuum speed of light. The
expression in the brackets of Eq. (1.8) can be associated with the total current density
J t = J s + Jc +
∂P
+ ∇ ×M,
∂t
(1.10)
where the current density j has been split into a source current density Js and an induced
conduction current density Jc. The terms
∂P
and ∇ × M are recognized as the polarization
∂t
current density and the magnetization current density, respectively. The wave equations as
stated in Eqs. (1.8) and (1.9) do not impose any conditions on the media considered and hence
are generally valid.
1.3 Constitutive relations
Maxwell’s equations define the fields that are generated by currents and charges in matter.
However, they do not describe how these currents and charges are generated. Thus, to find a
self-consistent solution for the electromagnetic field, Maxwell’s equations must be
supplemented by relations that describe the behavior of matter under the influence of the fields.
These material equations are known as constitutive relations. In a non-dispersive linear and
anisotropic medium they have the form
D = ε 0 εE
(P = ε 0 χ eE) ,
(1.11)
B = µ 0 μH
(M = χ mH) ,
(1.12)
J c = σE ,
(1.13)
with χ e , χ m and σ denoting the electric susceptibility, the magnetic susceptibility and the
conductivity, respectively. For nonlinear media, the right hand sides can be supplemented by
terms of higher power. ε and µ are second-rank tensors. Isotropic media can be considered
using scalar forms. In order to account for general bianisotropic media, additional terms relating
D and E to both B and H have to be introduced. For such complex media, solutions to the wave
equations can be found for very special situations only. The constituent relations given above
account for inhomogeneous media if the material parameters ε, µ and σ are functions of
space. The medium is called temporally dispersive if the material parameters are functions of
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Chap. 1 Macroscopic Maxwell’s equations
LALANNE Philippe (IOGS 2nd année)
frequency, and spatially dispersive if the constitutive relations are convolutions over space. An
electromagnetic field in a linear medium can be written as a superposition of monochromatic
fields of the form
E(r , t ) = E(k , ω) cos(kr − ω t ) ,
(1.14)
where k and ω are the wavevector and the angular frequency, respectively. In its most general
form, the amplitude of the induced displacement D(r , t ) can be written as1
D(k , ω) = ε 0 ε(k , ω)E(k , ω) .
(1.15)
Since E(k , ω) is equivalent to the Fourier transform Ê of an arbitrary time-dependent field
E(r , t ) , we can apply the inverse Fourier transform to Eq. (1.15) and obtain
D(r , t ) = ε 0 ∫∫ εˆ (r − r ′, t − t ′)E(r ′, t ′) dr ′d t ′ .
(1.16)
Here, ε̂ denotes the response function in space and time. The displacement D at time t
depends on the electric field at all times t’ previous to t (temporal dispersion or causality).
Additionally, the displacement at a point r also depends on the values of the electric field at
neighboring points r’ (spatial dispersion). A spatially dispersive medium is therefore also called a
non-local medium. Non-local effects can be observed at interfaces between different media or in
metallic objects with sizes comparable with the mean-free path of electrons. In general, it is very
difficult to account for spatial dispersion in field calculations. In most cases of interest the effect
is very weak and we can safely ignore it. Temporal dispersion, on the other hand, is a widely
encountered phenomenon and it is important to take it accurately into account.
1.4 Spectral representation of time-dependent fields
ˆ (r , ω) of an arbitrary time-dependent field E(r,t) is defined by the Fourier
The spectrum E
transform
ˆ (r , ω) = 1 ∞ E(r , t ) exp(iω t ) d t .
E
2π ∫−∞
(1.17)
In order that E(r,t) is a real valued field we have to require that
ˆ (r ,−ω) = E
ˆ ∗ (r , ω) .
E
(1.18)
Applying the Fourier transform to the time-dependent Maxwell’s equations (1.1)-(1.4) gives
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Chap. 1 Macroscopic Maxwell’s equations
LALANNE Philippe (IOGS 2nd année)
ˆ (r , ω) = iωB
ˆ (r , ω) ,
∇ ×E
(1.19)
ˆ (r , ω) = −iωD
ˆ (r , ω) + J
ˆ (r , ω) ,
∇ ×H
(1.20)
ˆ (r , ω) = ρ
ˆ (r , ω) ,
∇ ⋅D
(1.21)
ˆ (r , ω) = 0 .
∇ ⋅B
(1.22)
ˆ (r , ω) has been determined, the time-dependent field is calculated by the
Once the solution for E
inverse transform as
E(r , t ) =
∞
∫−∞ Eˆ (r , ω) exp(- iω t ) dω .
(1.23)
Thus, the time dependence of a non-harmonic electromagnetic field can be Fourier transformed
and every spectral component can be treated separately as a monochromatic field. The general
time dependence is obtained from the inverse transform.
2.5 Time-harmonic fields
The time dependence in the wave equations can be easily separated to obtain a harmonic
differential equation. A monochromatic field can then be written as 1
E(r , t ) = Re[E(r ) exp(- iω t )] =
[
]
1
E(r ) exp(- iω t ) + E ∗ (r ) exp(iω t ) ,
2
(1.24)
with similar expressions for the other fields. Notice that E(r,t) is real, whereas the spatial part
E(r) is complex. The symbol E will be used for both, the real, time-dependent field and the
complex spatial part of the field. The introduction of a new symbol is avoided in order to keep the
notation simple. It is convenient to represent the fields of a time-harmonic field by their complex
amplitudes. Maxwell’s equations can then be written as
∇ × E(r ) = iωB(r ) ,
(1.25)
∇ × H(r ) = −iωD(r ) + j(r ) ,
(1.26)
∇ ⋅ D(r ) = ρ(r ) ,
(1.27)
1
This can also be written as E(r, t) = Re{E(r)} cos(ωt)+Im{E(r)} sin(ωt) = |E(r)| cos[ωt+ϕ(r)],
where the phase is determined by ϕ(r) = atan[Im{E(r)}/Re{E(r)}].
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Chap. 1 Macroscopic Maxwell’s equations
LALANNE Philippe (IOGS 2nd année)
∇ ⋅ B(r ) = 0 .
(1.28)
which is equivalent to Maxwell’s equations (1.19)–(1.22) for the spectra of arbitrary
ˆ (r , ω) of an
time-dependent fields. Thus, the solution for E(r) is equivalent to the spectrum E
arbitrary time-dependent field. It is obvious that the complex field amplitudes depend on the
ˆ (r , ω) . However, ω is usually not included in the argument. Also
angular frequency ω, i.e. E(r ) = E
the material parameters ε, µ and σ are functions of space and frequency, i.e. ε = ε(r,ω),
µ = µ(r,ω) and σ = σ(r,ω). For simpler notation, we will often drop the argument in the fields and
material parameters. It is the context of the problem that determines which of the fields, E(r,t),
ˆ (r , ω) is being considered.
E(r), or E
2.6 Complex dielectric constant
With the help of the linear constitutive relations we can express Maxwell’s curl equations (1.25)
and (1.26) in terms of E(r) and H(r). The harmonic Maxwell’s equations can then be written as
∇ × E(r ) = iωµ 0 μH ,
(1.29)
∇ × H(r ) = −iωε 0 [ε + iσ ωε 0 ]E + J s .
(1.30)
∇ ⋅ D(r ) = ρ(r ) ,
(1.31)
∇ ⋅ B(r ) = 0 .
(1.32)
It is common practice to replace the expression in the brackets on the left hand side by a
complex dielectric constant, i.e.
[ε + iσ ωε 0 ] → ε .
(1.33)
In this notation one does not distinguish between conduction currents and polarization currents.
Energy dissipation is associated with the imaginary part of the dielectric constant. With the new
definition of ε, and assuming that the charge and source terms, ρ and Js, are equal to zero, the
Maxwell’s equations for the complex fields E(r) and H(r) in linear, anisotropic, but inhomogeneous
media are
∇ × E(r ) = iωµ 0 μH ,
(1.34)
∇ × H(r ) = −iωε 0 ε E ,
(1.35)
where we have omitted the null divergence equations since they are implicitly encompassed by
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Chap. 1 Macroscopic Maxwell’s equations
LALANNE Philippe (IOGS 2nd année)
the curl equations (let us remember that ∇ ⋅ ∇ × (•) ≡ 0 ). The complex dielectric constant will be
used throughout this course.
2.7 Reciprocity
We consider two solutions (labeled by the subscripts 1 and 2) at two frequencies ω1 and ω2 for
the same permittivity and permeability distributions,
∇ × E1 (r ) = iω1µ 0 μ1H1 and ∇ × H1 (r ) = −iω1ε 0 ε 1E1 + J1δ(r − r1 ) ,
(1.36a)
∇ × E 2 (r ) = iω 2 µ 0 μ 2H 2
(1.36b)
and ∇ × H 2 (r ) = −iω 2 ε 0 ε 2E 2 + J 2 δ(r − r2 ) ,
We have introduced two dipole sources (J1 and J2 located at r1 and r2) as Dirac distributions for
the sake of generality. Applying the Green-Ostrogradski formula 2 to the vector E2 × H1 on a
closed surface Σ of volume Ω enclosing the sources and by using the classical formula
∇ ⋅ (a × b ) = b ⋅ (∇ × a ) − a ⋅ (∇ × b ) , one gets
∫∫Σ (E 2 × H1 ) ⋅ dΣ = ∫∫∫Ω i (ω1ε 0E 2 ⋅ ε 1E1 + ω2 µ 0H1 ⋅ μ 2H2 ) dΩ − J1 ⋅ E 2 (r1 ) ,
(1.37)
By subtracting to Eq. (1.37) the related relation obtained by exchanging the indices 1 and 2, we
further obtain
∫∫Σ (E 2 × H1 − E1 × H2 ) ⋅ dΣ = i∫∫∫Ω [ω1 (ε 0E 2 ⋅ ε1E1 − µ 0H2 ⋅ μ1H1 ) − ω2 (ε 0E1 ⋅ ε 2E 2 − µ 0H1 ⋅ μ 2H2 )]dΩ
− [J1 ⋅ E 2 (r1 ) − J 2 ⋅ E1 (r2 )] .
(1.38)
If the materials are reciprocal, i.e. if bi-anisotropy and non-locality can be neglected, for any r we
have
ε = T ε and μ= T μ ,
2
(1.39)
∫∫∫Ω (∇ ⋅ F) dΩ = ∫∫Σ (F ⋅ n) dΣ . The left side is a volume integral over the volume Ω, the right side is the
surface integral over the boundary of Ω. The surface Σ is quite generally the boundary of V oriented by
outward-pointing normals, and n is the outward pointing unit normal field of the boundary Σ (dΣ may be
used as a shorthand for n dΣ.) By the symbol within the two integrals, it is stressed once more that Σ is a
closed surface.
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Chap. 1 Macroscopic Maxwell’s equations
LALANNE Philippe (IOGS 2nd année)
where the upperscript
T
denotes tensor-transposition. Since
E 2 ⋅ ε 2E1 = E1 ⋅ ε 2E 2
and
H1 ⋅ μ 2H 2 = H 2 ⋅ μ 2H1 , Eq. (1.38) becomes
∫∫Σ (E 2 × H1 − E1 × H2 ) ⋅ dΣ = i∫∫∫Ω [ε 0 (ω1E 2 ⋅ ε 1E1 − ω2E 2 ⋅ ε 2E1 ) + µ 0 (ω2H2 ⋅ μ 2H1 − ω1H2 ⋅ μ1H1 )]dΩ
− [J1 ⋅ E 2 (r1 ) − J 2 ⋅ E1 (r2 )] ,
(1.40a)
or
∫∫Σ (E 2 × H1 + E1 × H2 ) ⋅ dΣ = i∫∫∫Ω [ε 0 (ω1E 2 ⋅ ε1E1 + ω2E 2 ⋅ ε 2E1 ) + µ 0 (ω2H2 ⋅ μ 2H1 + ω1H2 ⋅ μ1H1 )]dΩ
− [J1 ⋅ E 2 (r1 ) + J 2 ⋅ E1 (r2 )] ,
(1.40b)
if instead of subtracting to Eq. (1.37) the related relation obtained by exchanging the indices 1
and 2, one simply adds the two relations.
Equations (1.40a) and (1.40b) are known as the generalized Lorentz reciprocity theorem
and can be directly used to derive a number of general properties in electrodynamics (whenever
materials are reciprocal), the most famous property being the Poynting theorem.
2.8 Poynting’s theorem
In electrodynamics, Poynting's theorem is a statement of energy conservation for the
electromagnetic field, in the form of a partial differential equation, due to the British physicist
John Henry Poynting. Poynting's theorem, in its general form in the presence of source, is
analogous to the work-energy theorem in classical mechanics, and mathematically similar to the
continuity equation, because it relates the energy stored in the electromagnetic field to the work
done on a charge distribution (i.e. an electrically charged object), and to the dissipation loss.
We consider the scattering problem shown in Fig. 1.1. An incident radiation (created by a far
field source) at a frequency ω is illuminating a complex scatterer, possibly a metallic one with a
complex relative permittivity ε(r ) and relative permeability μ(r ) . The Maxwell’s equations can
be written
∇ × E(r ) = iωµ 0μ(r )H(r ) ,
(1.41a)
∇ × H(r ) = −iωε 0 ε(r )E(r ) ,
(1.41b)
where (E,H) is the total electromagnetic field, including the incident wave created by the
source and the scattered field. Let us now consider a second set of Maxwell’s equations, just
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Chap. 1 Macroscopic Maxwell’s equations
LALANNE Philippe (IOGS 2nd année)
obtained by complex conjugating the previous equations
(
)
∇ × E ∗ = iωµ 0 − μ ∗ H∗ ,
(1.42a)
( )
∇ × H∗ = −iωε 0 − ε ∗ E ∗ .
(1.42b)
Figure 1-1. Sketch of a basic scattering problem used for deriving the Poynting
theorem. The closed surface Σ is the boundary of an arbitrary volume Ω. The source
responsible for the incident field is assumed to be outside the volume Ω.
We now apply the generalized Lorentz reciprocity theorem, Eq. (1.40b), taking for solution 1,
E1 = E, H1 = H, ω1 = ω, ε1 = ε and µ1 = µ, and for solution 2, E2 = E∗, H2 = H∗, ω2 = ω, ε2 = −ε∗ and
µ2 = −µ∗. Since the source is outside the volume Ω, we get
∫∫Σ (E
∗
)
[ (
)
)]
(
× H + E × H∗ ⋅ dΣ = i∫∫∫ ε 0 ωE ∗ ⋅ εE − ωE ∗ ⋅ ε ∗E + µ 0 − ωH∗ ⋅ μ∗H + ωH∗ ⋅ μH dΩ ,
Ω
or equivalently
∫∫Σ (E
∗
)
[ ( (
))
) )]
( (
× H + E × H∗ ⋅ dΣ = iω∫∫∫ ε 0 E ∗ ⋅ ε − ε ∗ E + µ 0 H∗ ⋅ μ − μ∗ H dΩ .
Ω
(
(1.43)
)
Noting that ε − ε ∗ = 2i Im(ε ) and E ∗ × H + E × H∗ = 2 Re E × H∗ , we finally obtain
(
)
[ (
)
(
)]
ω
1
Re E × H∗ ⋅ dΣ = − ∫∫∫ ε0 E∗ ⋅ Im(ε )E + µ 0 H∗ ⋅ Im(μ)H dΩ .
∫∫
Σ
2
2 Ω
(1.44)
The flux of the Poynting vector (right-hand side), i.e. the power flowing out through the surface Σ,
is equal to the Ohmic power loss due to absorption (left-hand side). Note that with our exp(−iωt)
notation, material absorption implies that Im(ε*) > 0 and Im(µ*) > 0; thus the right-hand term in
Eq. (1.44) is negative, consistently with the fact that the power flow through the oriented surface
Σ is negative since more power goes in than it goes out (because a fraction of the incoming
power is absorbed).
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Chap. 1 Macroscopic Maxwell’s equations
LALANNE Philippe (IOGS 2nd année)
2.9 Quality factor of an electromagnetic resonator
In physics and engineering the quality factor or Q-factor is a dimensionless parameter that
describes how under-damped an oscillator or resonator is, or equivalently, characterizes a
resonator's bandwidth relative to its center frequency. Higher Q indicates a lower rate of energy
loss relative to the stored energy of the oscillator; the oscillations die out more slowly. A
pendulum suspended from a high-quality bearing, oscillating in air, has a high Q, while a
pendulum immersed in oil has a low one. Oscillators with high quality factors have low damping
so that they ring longer.
In the context of resonators, Q is defined in terms of the ratio of the energy stored in the
resonator to the energy supplied by a generator, per cycle, to keep signal amplitude constant, at
a frequency (the resonant frequency), ωr, where the stored energy is constant with time [Jac99]
Q = 2π
Energy stored
.
Energy dissipated per cycle
(1.45)
The factor 2π makes Q expressible in simpler terms, involving only the coefficients of the
second-order differential equation describing most resonant systems, electrical or mechanical.
Electromagnetic resonators play a crucial role in photonics, as they allow us to spectrally
filter a photonic signal, to store the energy, to reduce the power required to produce
nonlinearities… Let us start with a very simple case of a dielectric cavity composed of lossless
and non-dispersive materials, see Fig. 1.2. Actually, because the mirrors are not perfect
reflectors, some energy leaks out of the resonator, and this leakage is responsible for the finite
Q-factor (we assume that there is no Ohmic loss).
Figure 1-2. Sketch of an electromagnetic resonator with its quasi-normal mode. Because
the quasi-normal mode is a solution of Maxwell’s equations for a complex frequency, its field
distribution is a stationary wave inside the cavity and a diverging outgoing wave outside. The
Q-factor of the mode can be easily derived from the field distribution by a simple energy balance
over any closed surface that either surrounds the cavity (surface Σ1), or is located entirely
outside the resonator in the clad (surface Σ2), or encloses part of the mode (surface Σ3).
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Chap. 1 Macroscopic Maxwell’s equations
LALANNE Philippe (IOGS 2nd année)
By definition, resonator modes are electromagnetic field distributions that are solutions of
Maxwell’s equations in the absence of source
~
~
∇ ×E = i ~
ωµH
~
~,
∇ ×H = −i ~
ω ε(r ) E
(1.46)
and that satisfy outgoing wave boundary conditions (the Sommerfeld radiation condition) at
~ ~
|r| = ∞). In Eq. (1.46), E, H are the electric and magnetic fields of a resonator mode (for
( )
instance the fundamental one with the smallest frequency) and the dielectric permittivity
distribution ε(r) is real (lossless materials). Because the cavity is an open system, the mode
energies leak and possess a finite lifetime τm. The eigenfrequency ~
ωm is thus complex with
Im(~
ω) = −1/τ, exp(− i~
ω) = ω r .
ω t ) = exp(− i[Re(~
ω) + iIm(~
ω)]t ) ≡ exp(− t τ ) exp(− iωr t ) with Re(~
Consistently with the literature on non-conservative open systems for which the time-evolution
operator is not Hermitian [Cam96], the solutions of Eq. (1.46) with outgoing wave boundary
conditions will be referred to as quasi-normal modes hereafter, rather than normal mode, to
emphasize that they are modes of a non-conservative systems.
Let us first examine the consequence of Im( ~
ωm ) ≠ 0. In a simple Fabry-Perot picture, light is
bouncing back and forth between the mirrors, and because of energy leakage through the
mirrors, the reflected intensity is smaller than the incident one. Energy is thus lost over one
round trip. The imaginary part of the frequency restores a stationary state; actually, as light
propagates inside the cavity, it is amplified (note that for a complex frequency the wavevector
becomes complex with a negative imaginary part) and the amplification exactly compensates for
the loss incurred at the mirror. The amplified propagation (think in terms of laser modes)
associated with complex frequencies also exists outside the resonator and therefore the
cavity-mode field that leaks out in the clad material is exponentially diverging, as sketched in
Fig. 1.2.
The definition of Eq. (1.45) is based on an energy balance and it is instructive to relate it to
the complex frequency using Poynting’s theorem. For that purpose, we first conjugate Eq. (1.46)
~~~
~ ~ ~*
to show that, since E, H, ω
is a solution of Maxwell’s equations, E* ,−H* , ω
is also a solution
(
)
(
)
with the same permittivity distribution since ε(r) = ε*(r). Second we apply the divergence theorem
~ ~ ~ ~
to the vector E × H* + E* × H on an arbitrary closed surface Σ defining a volume Ω, to obtain
~ ~
∫∫ (E × H
*
Σ
)
(
~ ~
+ E* × H ⋅ dΣ = i ~
ω−~
ω*
)∫∫∫  ε E~
Ω
2
~2
+ µ H  dΩ .

11
(1.47)
Chap. 1 Macroscopic Maxwell’s equations
LALANNE Philippe (IOGS 2nd année)
Introducing the Poynting-vector flux through the closed surface Σ (energy leakage),
(
)
~ ~
1/2 ∫∫ Re E × H* ⋅ dΣ , and the time-averaged electromagnetic energy stored in the volume Ω,
Σ
~2
 ~2
1/4 ∫∫∫  ε E + µ H  dΩ , one immediately gets that the usual definition of Q based on an
Ω

energy balance leads to
Q=−
Re(~
ω)
.
2 Im(~
ω)
(1.48)
Amazingly, any closed surface can be used to derive Eq. (1.48). It may surround the cavity
(surface Σ1 in Fig. 1.2) or may be located outside the physical resonator in the clad (surface Σ2).
References
[Cam96] R.K. Chang and A.J. Campillo, Optical processes in microcavities, Chap. 1, (World
Scientific, London, 1996).
[Jac99] J.D. Jackson, Classical Electrodynamics, 3nd ed. (John Wiley, New York, 1999).
[Mar91] D. Marcuse, Theory of dielectric optical waveguides, 2nd Ed. (Academic, 1991).
[Vas91] C. Vassallo, Optical waveguide concepts (Elsevier, Amsterdam, 1991).
[Yeh88] P. Yeh, Optical waves in layered media (J. Wiley and Sons eds., New York, 1988).
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