where »qn = 2πc/ωqn . The mode frequency is

q 2 + (nπ/d)2 ,

ωqn = c (B.84)

and the expansion of a general real ¬eld is

∞ Cn

E (r) = i aqns E qns (z) eiq·r ’ CC , (B.85)

q n=0 s=1

where C0 = 1 and Cn = 2 for n 1.

B.5 Macroscopic Maxwell equations

The macroscopic Maxwell equations are given by (Jackson, 1999, Sec. 6.1)

∇ · D (r, t) = ρ (r, t) , (B.86)

‚D (r, t)

∇ — H (r, t) = J (r, t) + , (B.87)

‚t

‚B (r, t)

∇ — E (r, t) = ’ , (B.88)

‚t

∇ · B (r, t) = 0 , (B.89)

D (r, t) = 0 E (r, t) + P (r, t) , (B.90)

1

H (r, t) = B (r, t) ’ M (r, t) . (B.91)

µ0

In these equations ρ and J respectively represent the charge density and current

density of the free charges, P is the polarization density (density of the electric

dipole moment), M is the magnetization (density of the magnetic dipole moment),

D is the displacement ¬eld, and H is the magnetic ¬eld.

½

Macroscopic Maxwell equations

After Fourier transforming in r and t, Maxwell™s equations reduce to the algebraic

relations

k · D (k, ω) = ’iρ (k, ω) , (B.92)

k — H (k, ω) = ’iJ (k, ω) ’ ωD (k, ω) , (B.93)

k — E (k, ω) = ωB (k, ω) , (B.94)

k · B (k, ω) = 0 , (B.95)

0E (k, ω) + P (k, ω) ,

D (k, ω) = (B.96)

1

H (k, ω) = B (k, ω) ’ M (k, ω) . (B.97)

µ0

The microscopic Poynting™s theorem (B.9) is replaced by

‚D ‚B

E· +H· +∇· S = 0, (B.98)

‚t ‚t

where S = E — H (Jackson, 1999, Sec. 6.7).

For a nondispersive medium, i.e.

Di (r, t) = ij Ej (r, t) , Bi (r, t) = µij Hj (r, t) , (B.99)

where and µij are constant tensors, eqn (B.98) takes the form

ij

‚u (r, t)

+ ∇ · S (r, t) = 0 , (B.100)

‚t

with the energy density

1

{E · D + B · H}

u= (B.101)

2

1

Ei ij Ej + Bi µ’1 Bj .

= (B.102)

ij

2

The most important materials for quantum optics are nonmagnetic dielectrics with

µij (ω) = µ0 δij . In this case eqns (B.86)“(B.91) can be converted into a wave equation

for the transverse part of the electric ¬eld:

1 ‚2 ‚2 ‚

⊥

= µ0 2 P ⊥ + µ0 J⊥ .

∇’2 2 E

2

(B.103)

c ‚t ‚t ‚t

B.5.1 Dispersive linear media

We consider a medium which interacts weakly with external ¬elds. This can happen

either because the ¬elds themselves are weak or because the e¬ective coupling constants

are small. In general, the polarization and magnetization at a space“time point x =

(r, t) can depend on the action of the ¬eld at earlier times and at distant points

¾ Classical electrodynamics

in space. Combining this with the weak interaction assumption leads to the linear

constitutive equations (Jackson, 1999, p. 14)

(1)

Pi (r, t) = dt χij (r ’ r , t ’ t ) Ej (r , t ) ,

d3 r (B.104)

0

(1)

Mi (r, t) = dt ξij (r ’ r , t ’ t ) Hj (r , t ) ,

d3 r (B.105)

(1) (1)

where χij (r ’ r , t ’ t ) and ξij (r ’ r , t ’ t ) are respectively the (linear) electric

and magnetic susceptibility tensors. Thus the relation between the polarization P (r, t)

(magnetization M (r, t)) and the ¬eld E (r, t) (H (r, t)) is nonlocal in both space and

time. The principle of causality prohibits P (r, t) (M (r, t)) from depending on the

¬eld E (r, t ) (H (r, t )) at later times, t > t, so the susceptibilities must satisfy

(1)

χij (r ’ r , t ’ t ) = 0 ,

for t > t . (B.106)

(1)

(r ’ r , t ’ t ) = 0

ξij

This leads to the famous Kramers“Kronig relations (Jackson, 1999, Sec. 7.10).

The four-dimensional convolution theorem, obtained by combining eqns (A.55) and

(A.57), allows eqns (B.104) and (B.105) to be recast in Fourier space as

(1)

Pi (k, ω) = (k, ω) Ej (k, ω) ,

0 χij (B.107)

(1)

Mi (k, ω) = ξij (k, ω) Ej (k, ω) . (B.108)

Combining these relations with the de¬nitions (B.90) and (B.91) produces

Di (k, ω) = (k, ω) Ej (k, ω) (B.109)

ij

and

Bi (k, ω) = µij (k, ω) Hj (k, ω) , (B.110)

where

(1)

(k, ω) ≡ δij + χij (k, ω) (B.111)

ij 0

and

(1)

µij (k, ω) ≡ µ0 δij + ξij (k, ω) (B.112)

are respectively the (electric) permittivity tensor and the (magnetic) permeabil-

ity tensor. The classical ¬elds, the polarization, the magnetization, and the (space“

time) susceptibilities are all real; therefore, the Fourier transforms satisfy

P — (k, ω) = P (’k, ’ω) , E — (k, ω) = E (’k, ’ω) ,

M— (k, ω) = M (’k, ’ω) , B— (k, ω) = B (’k, ’ω) , (B.113)

(1)— (1) (1)— (1)

(k, ω) = χij (’k, ’ω) , ξij (k, ω) = ξij (’k, ’ω) .

χij

(1) (1)

The dependence of χij (k, ω) and ξij (k, ω) on k is called spatial dispersion, and

the dependence on ω is called frequency dispersion. Interactions between atoms at

¿

Macroscopic Maxwell equations

di¬erent points in the medium can cause the polarization at a point r to depend on the

¬eld in a neighborhood of r, de¬ned by a spatial correlation length as . In gases, liquids,

and disordered solids as is of the order of the interatomic spacing, which is generally

very small compared to vacuum optical wavelengths »0 . Thus the polarization at r

can be treated as depending only on the ¬eld at r. Since the medium is assumed to

(1) (1)

be spatially homogeneous, this means that χij (r ’ r , t ’ t ) = χij (t ’ t ) δ (r ’ r ),

(1) (1)

which is equivalent to χij (k, ω) = χij (ω). Similar relations hold for the magnetic

susceptibility. These three types of media are also isotropic (rotationally symmetric),

so the tensor quantities can be replaced by scalars which depend only on ω:

(k, ω) ’ (ω) δij ,

ij

(B.114)

1 + χ(1) (ω) ,

(ω) = 0

µij (k, ω) ’ µ (ω) δij ,

(B.115)

µ (ω) = µ0 1 + ξ (1) (ω) .

Using eqn (B.114) in eqn (B.109) and transforming back to position space produces

the useful relation

D (r, ω) = (ω) E (r, ω) . (B.116)

For crystalline solids, rotational symmetry is replaced by symmetry under the

crystal group, and the tensor character of the susceptibilities cannot be ignored. In

this case as is the lattice spacing, so the ratio as /»0 is still small, but spatial dispersion

cannot always be neglected. The reason is that the relevant parameter is n (ω0 ) as /»0 ,

where n (ω0 ) is the index of refraction at the frequency ω0 = 2πc/»0 . Thus spatial

dispersion can be signi¬cant if the index is large.

In a rare stroke of good fortune, the crystals of interest for quantum optics satisfy

the condition for weak spatial dispersion, n (ω0 ) as /»0 1 (Agranovich and Ginzburg,

1984); therefore, we can still use a permittivity tensor that only depends on frequency:

(k, ω) ’ (ω) . (B.117)

ij ij

For most applications of quantum optics, we can also assume that the permittivity

tensor is symmetrical: ij (ω) = ji (ω). Physically this means that the crystal is both

transparent and non-gyrotropic (not optically active) (Agranovich and Ginzburg, 1984,

Chap. 1). We also assume the existence of the inverse tensor ’1 ij .

There are other situations, e.g. propagation in a plasma exposed to an external

magnetic ¬eld, that require the full tensors ij (k, ω) and µij (k, ω) depending on k

(Pines, 1963, Chaps 3 and 4; Ginzburg, 1970, Sec. 1.2). For nearly all applications of

quantum optics, we can neglect spatial dispersion and assume the forms (B.114) or

(B.117) for the permittivity tensor.

The inertia of the charges and currents in the medium, together with dissipative

e¬ects, imply that the medium cannot respond instantaneously to changes in the ¬eld

at a given point r. Thus the polarization at the position r and time t will in general

depend on the ¬eld at earlier times t < t. Since the response times of gases, liquids,

Classical electrodynamics

and solids exhibit considerable variation, it is not generally possible to ignore frequency

dispersion.

B.5.2 Isotropic linear dielectrics

Here we assume that µij (ω) = µ0 δij , so that H = B/µ0 , and set (ω) = δij (ω)

ij

and ρ = J = 0 in eqns (B.92)“(B.95) to get

k · E (k, ω) = 0 , (B.118)

ω

k — B (k, ω) = ’ r (ω) E (k, ω) , (B.119)

c2

k — E (k, ω) = ωB (k, ω) , (B.120)

k · B (k, ω) = 0 , (B.121)

where r (ω) = (ω) / 0 is the relative permittivity. The ¬nal equation follows from

eqn (B.120), and eliminating B between eqn (B.119) and eqn (B.120) leads to

ω2

k — [k — E (k, ω)] = ’ 2 (ω) E (k, ω) . (B.122)

r

c

The identity a — (b — c) = (a · c) b ’ (a · b) c, together with eqn (B.118), reduces this

to

ω2 2

n (ω) ’ k 2 E (k, ω) = 0 , (B.123)

c2

where n (ω) = r (ω) is the index of refraction. In general r (ω) can be complex,

corresponding to absorption or gain at particular frequencies (Jackson, 1999, Chap. 7),

but for frequencies in the transparent part of the spectrum r (ω) is real and positive.

The relation E = ’‚A/‚t implies E (k, ω) = iωA (k, ω), so the vector potential

satis¬es the same equation

ω2 2

n (ω) ’ k 2 A (k, ω) = 0 . (B.124)

2

c

For a transparent medium the general transverse solution of eqn (B.124) is

A— (’k) e— (’k) δ (ω + ω (k)) ,

A (k, ω) = As (k) es (k) δ (ω ’ ω (k)) + s s

s s

(B.125)

where ω (k) is a positive, real solution of the dispersion relation

ωn (ω) = ck . (B.126)

Thus the fundamental plane-wave solution in position“time is

ei(k·r’ω(k)t) es (k) , (B.127)

and the positive-frequency part has the general form

Macroscopic Maxwell equations

d3 k

A As (k) es (k) ei(k·r’ω(k)t)

(+)

(r, t) = (B.128)

3

(2π) s

for the vector potential, and

d3 k

E As (k) es (k) ei(k·r’ω(k)t)

(+)

(r, t) = i 3 ω (k) (B.129)

(2π) s

for the electric ¬eld.

B.5.3 Anisotropic linear dielectrics

We again assume that µij (ω) = µ0 δij and set ρ = J = 0 in eqns (B.92)“(B.95), but

we drop the assumption ij (ω) = δij (ω). In this case E and D are not necessarily

parallel, so we combine eqn (B.93) with eqn (B.94) to get

k 2 ∆⊥ Ej = µ0 ω 2 Di . (B.130)

ij

In the following we will use a matrix notation in which a second-rank tensor Xij is

←

’

represented as a 3 — 3 matrix X and a vector V = V1 ux + V2 uy + V3 uz is represented

by column or row matrices according to the convention

⎛⎞

V

’

’ ⎝ 1⎠ ’ T ’

V = V2 , V = (V1 , V2 , V3 ) . (B.131)

V3

The polarization properties of the solution are best described in terms of D, since

’

’

eqn (B.92) guarantees that it is orthogonal to k. Thus we solve eqn (B.109) for E and

substitute the result into the left side of eqn (B.130) to ¬nd

←’

’’ ← 1 ’ ’1 ’

’ ’ ← 1 ’ ’1 ← ’

’ ’’

k 2 ∆ ⊥ E = k 2 ∆ ⊥ [← r ] D = k 2 ∆ ⊥ [← r ] ∆ ⊥ D , (B.132)

0 0

where ( r )ij (ω) = (ω) / 0 is the relative permittivity tensor. The last form

ij

’

’

depends on the fact that D is transverse. Putting this together with the right side of

eqn (B.130) yields

←’

’’ ω2 ’’

S D = 2 2D, (B.133)

kc

where the transverse impermeability tensor,1

←

’ ←’ ’1 ←

’

S k, ω = ∆ ⊥ k [← r (ω)] ∆ ⊥ k ,

’ (B.134)

depends on the frequency ω and the unit vector k = k/k along the propagation vector.

←’ ’

’

The real, symmetric matrix S annihilates k :

←’

’’

S k = 0, (B.135)

←

’

so S has one eigenvalue zero, corresponding to the eigenvector k. From eqn (B.133),

’

’

it is clear that the transverse vector D is one of the remaining two eigenvectors that

are orthogonal to k.

1 This is a slight modi¬cation of the approach found in Yariv and Yeh (1984, Chap. 4).

Classical electrodynamics

If ω lies in the transparent region for the crystal, the tensor ← r (ω) is positive

’

←’

de¬nite, so that the nonzero eigenvalues of S are positive. We write the positive

eigenvalues as 1/n2 , so that the corresponding eigenvectors satisfy

s

←’

’’ 1’

S µ s = 2 ’ s (s = 1, 2) .

µ (B.136)

ns

’

’ ’

’

If D is parallel to an eigenvector, i.e. D = Ys ’ s , one ¬nds the dispersion relation

’

µ

c2 k 2 = ω 2 n2 (ω) ; (B.137)

s

in other words, ns is the index of refraction associated with the eigenpolarization

’ (k). Since the matrix ← depends on the direction of propagation k, the indices

’

’ S

µs

ns (ω) generally also depend on k. In order to simplify the notation, this dependence is

not indicated explicitly, e.g. as ns ω, k , but is implicitly indicated by the dependence

of the refractive index on the polarization index. An incident wave with propagation

vector k exhibits birefringence, i.e. it produces two refracted waves corresponding

to the two phase velocities c/n1 and c/n2 . Since ’ s (k) is real, the eigenpolarizations

’

µ

are linear, and they can be normalized so that

’ T (k) ’ (k) = µ (k) · µ (k) = δ .

’ ’

µs µs (B.138)

s s ss

Radiation is described by the transverse part of the electric ¬eld, and for the special

’

’

solution D = Ys ’ s the transverse electric ¬eld in (k, ω)-space is

’µ

’

’ 1← ’

’’ Ys ’ ’,

E s (k, ω) = S Ys µ s = µs (B.139)

2

0 ns

0

where ωs (k) is a solution of eqn (B.137) and ns (k) ≡ ns (ωs (k)). The general space“

time solution,

2

’ (+)

’ d3 k

1 Ys (k) ’

’ (k) ei(k·r’ωs (k)t) ,

E (r, t) = µs (B.140)

3 2 (k)