velope ¬eld. By using the explicit form (7.22) of HP and the commutation relations

(7.34), it is simple to see that the Heisenberg equation can be written in the equivalent

forms

1‚ 12

i ∇z + ∇ ¦ (r, t) = 0

¦ (r, t) + (7.43)

vg0 ‚t 2k0

or

1‚ 12

i ∇z + ∇ φs (r, t) = 0 .

φs (r, t) + (7.44)

vg0 ‚t 2k0

Multiplying eqn (7.43) by the normalization factor in eqn (7.24) and passing to the

classical limit (A(+) (r, t) ’ A (r, t) exp [i (k0 · r ’ ω0 t)]) yields the standard paraxial

wave equation of the classical theory.

¾¾ Paraxial quantum optics

The single-beam argument can be applied to each of the distinct beams to give the

Schr¨dinger-picture representation,

o

(vgβ /c)

eβs φβs (r) eikβ ·r ,

A(+) (r) = (7.45)

2 0 kβ c

βs

where eβs = es (kβ ), ωβ = ω (kβ ) = ckβ /nβ , vgβ is the group velocity for the βth

carrier wave,

d3 q iq·r

φβs (r) = 3 aβs (q) e , (7.46)

Qβ (2π)

and

φβs (r) , φ† (r ) ≈ δββ δss δ (r ’ r ) (s, s = ± or 1, 2) . (7.47)

β s

The last result”which is established in Exercise 7.3”means that the envelope ¬elds

for distinct beams represent independent degrees of freedom.

The corresponding expression for the electric ¬eld operator in the paraxial approx-

imation is

ωβ (vgβ /c)

eβs φβs (r) eikβ ·r .

E(+) (r) = i (7.48)

2 0 nβ

βs

The operators for the photon number Nβ , the momentum Pβ , and the paraxial Hamil-

tonian HβP of the individual beams are obtained by applying eqns (7.37)“(7.39) to

each beam.

7.4 Gaussian beams and pulses

It is clear from the relation E = ’‚A/‚t that the electric ¬eld also satis¬es the

paraxial wave equation. For the special case of propagation along the z-axis through

vacuum, we ¬nd

12 1 ‚E

‚E

∇ E +i + = 0. (7.49)

2k0 ‚z c ‚t

For ¬elds with pulse duration much longer than any relevant time scale”or equiva-

lently with spectral width much smaller than any relevant frequency”the time depen-

dence of the slowly-varying envelope function can be neglected; that is, one can set

‚E/‚t = 0 in eqn (7.49). The most useful time-independent solutions of the paraxial

equation are those which exhibit minimal di¬ractive spreading. The fundamental solu-

tion with these properties”which is called a Gaussian beam or a Gaussian mode

(Yariv, 1989, Sec. 6.6)”is

w0 e’iφ(z) ρ2 ρ2

E (r, t) = E 0 (r , z) = E0 e0 exp ’ 2

exp ik0 , (7.50)

w (z) 2R (z) w (z)

where the polarization vector e0 is in the x“y plane and ρ = |r |. The functions of z

on the right side are de¬ned by

The paraxial expansion— ¾¾

2

z ’ zw

w (z) = w0 1+ , (7.51)

ZR

2

ZR

R (z) = z ’ zw + , (7.52)

z ’ zw

z ’ zw

φ (z) = tan’1 , (7.53)

ZR

where the Rayleigh range ZR is

2

πw0

ZR = > 0. (7.54)

»0

The function w (z)”which de¬nes the width of the transverse Gaussian pro¬le”has

the minimum value w0 (the spot size) at z = zw (the beam waist). The solution is

completely characterized by e0 , E0 , w0 , and zw . The function R (z)”which represents

the radius of curvature of the phase front”is negative for z < zw , and positive for

z > zw . The picture is of waves converging from the left and diverging to the right of

the focal point at the waist. The de¬nition (7.51) shows that

√

w (zw + ZR ) = 2w0 , (7.55)

so the Rayleigh range measures the distance required for di¬raction to double the area

of the spot. There are also higher-order Gaussian modes that are not invariant under

rotations around the beam axis (Yariv, 1989, Sec. 6.9).

The assumption ‚E/‚t = 0 means that the Gaussian beam represents an in¬nitely

long pulse, so we should expect that it is not a normalizable solution. This is readily

veri¬ed by showing that the normalization integral over the transverse coordinates has

the z-independent value

2 2

d2 r |E 0 (r , z)| = πw0 |E0 | ,

2

(7.56)

so that the z-integral diverges. A more realistic description is based on the observation

that

E P (r, t) = FP (z ’ ct) E 0 (r , z) (7.57)

is a time-dependent solution of eqn (7.49) for any choice of the function FP (z).

If FP (z) is normalizable, then the Gaussian pulse (or Gaussian wave packet)

E P (r, t) is normalizable at all times. The pulse-envelope function is frequently chosen

to be Gaussian also, i.e.

2

(z ’ z0 )

FP (z) = FP 0 exp ’ , (7.58)

L2

P

where LP is the pulse length and TP = LP /c is the pulse duration.

¾¾ Paraxial quantum optics

The paraxial expansion—

7.5

The approach to the quantum paraxial approximation presented above is su¬cient

for most practical purposes, but it does not provide any obvious way to calculate

corrections. A systematic expansion scheme is desirable for at least two reasons.

(1) It is not wise to depend on an approximation in the absence of any method for

estimating the errors involved.

(2) There are some questions of principle, e.g. the issue of photon localizability, which

require the evaluation of higher-order terms.

We will therefore very brie¬‚y outline a systematic expansion in powers of θ (Deutsch

and Garrison, 1991a) which is an extension of a method developed by Lax et al. (1974)

for the classical theory. In the interests of simplicity, only propagation in the vacuum

will be considered.

In order to construct a consistent expansion in powers of θ, it is ¬rst necessary

to normalize all physical quantities by using the characteristic lengths introduced in

Section 7.2.1. The ¬rst step is to de¬ne a characteristic volume

3

»0

’4

2

V0 = Λ Λ = θ , (7.59)

2π

and a dimensionless wavevector q = q + q z k0 , with q = q Λ and q z = qz Λ . In

terms of the scaled wavevector q, the paraxial constraints (7.8) are

Q0 = {q satisfying |q | 1 , qz 1} . (7.60)

The operators a† (k) have dimensions L3/2 , so the dimensionless operators a† (q) =

s s

’1/2 †

V0 as (k0 + q) satisfy the commutation relation

as (q) , a† (q ) = δss (2π) δ (q’q ) .

3

(7.61)

s

In the space“time domain, the operator ¦ (r, t) has dimensions L’3/2 , so it is

√

natural to de¬ne a dimensionless envelope ¬eld by ¦ r, t = V0 ¦ (r, t), where r =

r + z k0 and r = r /Λ , z = z/Λ . The scaled position-space variables satisfy

q · r = q · r = q · r + q z z. The operator ¦ r, t is related to as (q) by

d3 q

as (q) Xs (q, θ) eiq·r ,

¦ (r) = (7.62)

3

(2π)

Q0 s

where Xs (q, θ) is the c-number function:

∞

k0

θn X(n) (q) .

Xs (q, θ) = es (k0 + q) = (7.63)

|k0 + q| s

n=0

Substituting this expansion into eqn (7.62) and exchanging the sum over n with the

integral over q yields

Paraxial wave packets— ¾¾

∞

(n)

θn ¦

¦ (r) = (r) , (7.64)

n=0

where the nth-order coe¬cient is

d3 q

(n)

as (q) X(n) (q) eiq·r .

¦ (r) = (7.65)

s

3

(2π) s

The zeroth-order relation

d3 q

(0)

as (q) es (k0 ) eiq·r

¦ (r) = (7.66)

3

(2π) s

agrees with the previous paraxial approximation (7.31), and it can be inverted to give

(0)

(r) · e— (k0 ) e’iq·r .

d3 r¦

as (q) = (7.67)

s

Carrying out Exercise 7.5 shows that all higher-order coe¬cients can be expressed in

(0)

terms of ¦0 (r).

We can justify the operator expansion (7.64) by calculating the action of the exact

envelope operator on a typical basis vector in H (k0 , θ), and showing that the expansion

of the resulting vector in θ agrees”order-by-order”with the result of applying the

operator expansion. In the same way it can be shown that the operator expansion

reproduces the exact commutation relations (Deutsch and Garrison, 1991a).

Paraxial wave packets—

7.6

The use of non-normalizable basis states to de¬ne the paraxial space can be avoided

by employing wave packet creation operators. For this purpose, we restrict the polar-

ization amplitudes, ws (k), (introduced in Section 3.5.1) to those that have the form

1/2

ws (k0 + q) = V0 w s (q). Instead of con¬ning the relative wavevectors q to the re-

gion Q0 described by eqn (7.60), we de¬ne a paraxial wave packet (with carrier

wavevector k0 and opening angle θ) by the assumption that w s (q) vanishes rapidly

outside Q0 , i.e. w s (q) belongs to the space

n

P (k0 , θ) = lim |q| |w s (q)| = 0 for all n

w s (q) such that 0. (7.68)

|q|’∞

The inner product for this space of classical wave packets is de¬ned by

d3 q —

(w, v) = ws (k0 + q) vs (k0 + q) . (7.69)

3

(2π) s

Since the two wave packets belong to the same space, this can be written in terms of

scaled variables as

d3 q

w — (q) v s (q) .

(w, v) = (7.70)

s

3

(2π) s

¾¿¼ Paraxial quantum optics

For a paraxial wave packet, we set k = k0 + q in the general de¬nition (3.191) to

get

d3 q d3 q

a† (k0 + q) ws (k0 + q) =

a† [w] = a† (q) w s (q) . (7.71)

s

0s

3 3

(2π) (2π)

s s

The paraxial space de¬ned by eqn (7.10) can equally well be built up from the vacuum

by forming all linear combinations of states of the form

P

a† [wp ] |0 ,

|{w}P = (7.72)

p=1

where {w}P = {w1 , . . . , wP }, P = 0, 1, 2, . . ., and the wp s range over all of P (k0 , θ).

The only di¬erence from the construction of the full Fock space is the restriction of the

wave packets to the paraxial space P (k0 , θ) ‚ “em , where “em is the electromagnetic

phase space of classical wave packets de¬ned by eqn (3.189).

The multiparaxial Hilbert spaces introduced in Section 7.2.2 can also be described

in wave packet terms. The distinct paraxial beams considered there correspond to the

wave packet spaces P (k1 , θ1 ) and P (k2 , θ2 ). Paraxial wave packets, w ∈ P (k1 , θ1 )

and v ∈ P (k2 , θ2 ), are concentrated around k1 and k2 respectively, so it is eminently

plausible that w and v are e¬ectively orthogonal. More precisely, it is shown in Exercise

7.6 that

1

n |(w, v)| = 0 for all n

lim 1, (7.73)

θ2 ’0 (θ2 )

i.e. |(w, v)| vanishes faster than any power of θ2 . The symmetry of the inner product

guarantees that the same conclusion holds for θ1 ; consequently, the wave packet spaces

P (k1 , θ1 ) and P (k2 , θ2 ) can be treated as orthogonal to any ¬nite order in θ1 or θ2 .

The approximate orthogonality of the wave packets w and v combined with the

general rule (3.192) implies

a [w] , a† [v] = 0 (7.74)

whenever w and v belong to distinct paraxial wave packet spaces. From this it is easy

to see that the quantum paraxial spaces H (k1 , θ1 ) and H (k2 , θ2 ) are orthogonal to any

¬nite order in the small parameters θ1 and θ2 . In the paraxial approximation, distinct

paraxial wave packets behave as though they were truly orthogonal modes. This means

that the multiparaxial Hilbert space describing the situation in which several distinct

paraxial beams are present is generated from the vacuum by generalizing eqn (7.72)

to

Pβ

a† [wβp ] |0 ,

{w1 }P1 , {w2 }P2 , . . . , = (7.75)

β p=1

where Pβ = 0, 1, . . ., and the wβp s are chosen from P (kβ , θβ ).

Angular momentum—

7.7

The derivation of the paraxial approximation for the angular momentum J = L + S

is complicated by the fact”discussed in Section 3.4”that the operator L does not

Angular momentum— ¾¿½

have a convenient expression in terms of plane waves. Fortunately, the argument used

to show that the energy and the linear momentum are additive also applies to the

angular momentum; therefore, we can restrict attention to a single paraxial space. Let

us begin by rewriting the expression (3.58) for the helicity operator S as

d3 q k0 + q/k0

a† (q) a+ (q) ’ a† (q) a’ (q) .

S= (7.76)

’

+

3

(2π) k0 + q/k0

P

The ratio q/k0 can be expressed as

Λ qz

q Λq

k0 = θq + θ2 q z k0 ,

= + (7.77)

k0 Λ k0 Λ k0

so expanding in powers of θ gives the simple result

S0 = k0 S0 + O (θ) , (7.78)

where

d3 q

a† (q) a+ (q) ’ a† (q) a’ (q)

S0 = ’

+

3

(2π)

P

d3 r φ† (r) φ+ (r) ’ φ† (r) φ’ (r) .

= (7.79)

’

+

Thus, to lowest order, the helicity has only a longitudinal component; the leading

transverse component is O (θ). This is the natural consequence of the fact that each

photon has a wavevector close to k0 .

To develop the approximation for L we substitute the paraxial representation (7.24)

and the corresponding expression (7.48) for E(+) (r, t) into eqn (3.57) to get

1

(’) (+)

r — ∇ Aj

d3 rEj

L0 = 2i 0

i

1

d3 r¦† (r, t) e’ik0 ·r r — ∇ ¦j (r, t) eik0 ·r

= j

i

1

d3 r¦† (r, t) r — k0 + r — ∇ ¦j (r, t) ,

= (7.80)

j

i

where the last line follows from the identity

e’ik0 ·r ∇eik0 ·r ¦j (r, t) = (∇ + ik0 ) ¦j (r, t) . (7.81)

This remaining gradient term can be written as

r— ∇ = r— k0 ∇z + ∇

i i

= r — k0 ∇z + z k0 — ∇ + r — ∇ , (7.82)

i i i

¾¿¾ Paraxial quantum optics

so that

L0 = L0 + k0 L0z , (7.83)

where the transverse part is given by