* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download The macroscopic Maxwell equations
History of quantum field theory wikipedia , lookup
Nordström's theory of gravitation wikipedia , lookup
Quantum vacuum thruster wikipedia , lookup
Electrostatics wikipedia , lookup
Vasiliev equations wikipedia , lookup
Woodward effect wikipedia , lookup
Magnetic monopole wikipedia , lookup
Anti-gravity wikipedia , lookup
History of general relativity wikipedia , lookup
Lagrangian mechanics wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Noether's theorem wikipedia , lookup
Euler equations (fluid dynamics) wikipedia , lookup
Introduction to gauge theory wikipedia , lookup
Photon polarization wikipedia , lookup
Navier–Stokes equations wikipedia , lookup
Field (physics) wikipedia , lookup
Lorentz force wikipedia , lookup
Equations of motion wikipedia , lookup
Partial differential equation wikipedia , lookup
Kaluza–Klein theory wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Electromagnetism wikipedia , lookup
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 1 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) 2 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 3 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 4 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)}]. 5 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 6 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. 7 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 8 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). 9 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). 10 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). 12