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‚ 2 E (r, t) ‚E (r, t)
ω0 E (r, t)
2
ω0 (7.2)
‚t2 ‚t
and
‚ 2 E (r, t) ‚E (r, t)
k0 E (r, t) ;
2
k0 (7.3)
‚z 2 ‚z
in other words, E (r, t) has negligible variation in time over an optical period and
negligible variation in space over an optical wavelength. As we have already seen in
the discussion of monochromatic ¬elds, these conditions cannot be applied to the ¬eld
operator E(+) (r, t); instead, they must be interpreted as constraints on the allowed
states of the ¬eld.

7.2 Paraxial states
7.2.1 The paraxial ray bundle
A paraxial beam associated with the carrier wavevector k0 , i.e. a bundle of wavevectors
k clustered around k0 , is conveniently described in terms of relative wavevectors q =
k ’ k0 , with |q| k0 . For each k = k0 + q the angle ‘k between k and k0 is given by
|k0 — k| |k0 — q| |q | q
sin ‘k = = = 1+O , (7.4)
k0 |k0 + q|
k0 k k0 k0

where q = q ’ qz k0 and qz = q · k0 . This shows that ‘k |q | /k0 , and further
suggests de¬ning the small parameter for the paraxial beam as the maximum opening
angle,
∆q
θ= 1, (7.5)
k0
where 0 < |q | < ∆q is the range of the transverse components of q. Variations in
the transverse coordinate r occur over a characteristic distance Λ de¬ned by the
Fourier transform uncertainty relation Λ ∆q ∼ 1; consequently, a useful length scale
for transverse variations is Λ = 1/∆q = 1/ (θk0 ).
A natural way to de¬ne the characteristic length Λ for longitudinal variations
is to interpret the transverse length scale Λ as the radius of an e¬ective circular
¾¾¼ Paraxial quantum optics

aperture. The conventional longitudinal scale is then the distance over which a beam
waist, initially equal to Λ , doubles in size. At this point, a strictly correct argument
would bring in classical di¬raction theory; but the same end can be achieved”with
only a little sleight of hand”with geometric optics. By combining the approximation
tan θ ≈ θ with elementary trigonometry, it is easy to show that the geometric image
of the aperture on a screen at a distance Λ has the radius Λ = Λ + θΛ . The trick
is to choose the longitudinal scale length Λ so that Λ = 2Λ , and this requires
Λ 1
= k0 Λ2 = 2 .
Λ= (7.6)
θ θ k0
We will see in Section 7.4 that Λ = k0 Λ2 is twice the Rayleigh range”as usually
de¬ned in classical di¬raction theory”for the aperture Λ . Thus our geometric-optics
trick has achieved the same result as a proper di¬raction theory argument. Since
propagation occurs along the direction characterized by Λ , the natural time scale is
T = Λ / (c/n0 ) = 1/ θ2 ω0 .
The spread, ∆qz , in the longitudinal component of q satis¬es Λ ∆qz ∼ 1, so the
longitudinal and transverse widths are related by
2
∆qz ∆q
= θ2 ,
= (7.7)
k0 k0
and the q-vectors are e¬ectively con¬ned to a disk-shaped region de¬ned by
Q0 = q satisfying |q | θ2 k0 .
θk0 , qz (7.8)
In a dispersive medium with index of refraction n (ω) the frequency ωk is a solution
of the dispersion relation ck = ωk n (ωk ), and wave packets propagate at the group
velocity vg (ωk ) = dωk /dk. The frequency width is therefore ∆ω = vg0 ∆k, where vg0
is the group velocity at the carrier frequency. The straightforward calculation outlined
in Exercise 7.1 yields the estimate
∆ω 1
≈ θ2 1, (7.9)
ω0 2
which is the criterion for a monochromatic ¬eld given by eqn (3.107).

7.2.2 The paraxial Hilbert space
The geometric-optics picture of a bundle of rays forming small angles with the central
propagation vector k0 is realized in the quantum theory by a family of states that only
contain photons with propagation vectors in the paraxial bundle. In order to satisfy
the superposition principle, the family of states must be chosen as the paraxial space,
H (k0 , θ) ‚ HF , spanned by the improper (continuum normalized) number states
M
a† m (qm ) |0 , M = 0, 1, . . . ,
|{qs}M = (7.10)
0s
m=1

where a0s (q) = as (k0 +q), {qs}M ≡ {q1 s1 , . . . , qM sM }, and each relative propagation
vector is constrained by the paraxial conditions (7.8). If the paraxial restriction were
¾¾½
Paraxial states

relaxed, eqn (7.10) would de¬ne a continuum basis set for the full Fock space, so the
paraxial space is a subspace of HF . The states satisfying the paraxiality condition
(7.8) also satisfy the monochromaticity condition (3.107); consequently, H (k0 , θ) is
a subspace of the monochromatic space H (ω0 ). A state |Ψ belonging to H (k0 , θ) is
called a pure paraxial state, and a density operator ρ describing an ensemble of
pure paraxial states is called a mixed paraxial state. A useful way to characterize
a paraxial state ρ in H (k0 , θ) is to note that the power spectrum

a† (k) as (k) = Tr ρa† (k) as (k)
p (k) = (7.11)
s s
s s

is strongly concentrated near k = k0 .
In the Schr¨dinger picture, a general paraxial state |Ψ (0) has an expansion in the
o
basis {|{qs}M }, and the time evolution is given by

|Ψ (t) = e’itH/ |Ψ (0) , (7.12)

where H is the total Hamiltonian, including interactions with atoms, etc. It is clear on
physical grounds that an initial paraxial state will not in general remain paraxial. For
example, a paraxial ¬eld injected into a medium containing strong scattering centers
will experience large-angle scattering and thus become nonparaxial as it propagates
through the medium. In more favorable cases, interaction with matter, e.g. transmis-
sion through lenses with moderate focal lengths, will conserve the paraxial property.
The only situation for which it is possible to make a rigorous general statement
is free propagation. In this case the basis vectors |{qs}M are eigenstates of the total
Hamiltonian, H = Hem , so that

d3 q1 d3 qM
|Ψ (t) = ··· F ({qs}M )
(2π)3 (2π)3
s1 sM
M=0
(7.13)
M
— exp ’i ω (|k0 + qm |) t |{qs}M ,
m=1

where F ({qs}M ) = {qs}M |Ψ (0) . Consequently, the state |Ψ (t) remains in the
paraxial space H (k0 , θ) for all times.
For the sake of simplicity, we have analyzed the case of a single paraxial ray bun-
dle, but in many applications several paraxial beams are simultaneously present. The
reasons range from simple re¬‚ection by a mirror to complex wave mixing phenomena
in nonlinear media. The necessary generalizations can be understood by considering
two paraxial bundles with carrier waves k1 and k2 and opening angles θ1 and θ2 . The
two beams are said to be distinct if the vector ∆k = k1 ’ k2 satis¬es

|∆k| max [θ1 |k1 | , θ2 |k2 |] , (7.14)

i.e. the two bundles of wavevectors do not overlap. The multiparaxial space,
H (k1 , θ1 , k2 , θ2 ), for two distinct paraxial ray bundles is spanned by the basis vec-
tors
¾¾¾ Paraxial quantum optics

M K
a† m a† k (pk ) |0
(qm ) (M, K = 0, 1, . . .) , (7.15)
1s 2s
m=1 k=1

where a† (q) ≡ a† (kβ + q) (β = 1, 2) and the qs and ps are con¬ned to the respective
s
βs
regions Q1 and Q2 de¬ned by applying eqn (7.8) to each beam. The argument sug-
gested in Exercise 7.6 shows that the paraxial spaces H (k1 , θ1 ) and H (k2 , θ2 )”which
are subspaces of H (k1 , θ1 , k2 , θ2 )”may be treated as orthogonal within the paraxial
approximation. This description is readily extended to any number of distinct beams.

7.2.3 Photon number, momentum, and energy
The action of the number operator N on the paraxial space H (k0 , θ) is determined by
its action on the basis states in eqn (7.10); consequently, the commutation relation,
N, a† (q) = a† (q), permits the use of the e¬ective form
0s 0s


d3 q
a† (q) a0s (q) .
N N0 = (7.16)
0s
3
(2π)
Q0 s

Applying the same idea to the momentum operator, given by the continuum version
of eqn (3.153), leads to Pem = k0 N0 + P0 , where

d3 q
a† (q) a0s (q)
P0 = q (7.17)
0s
3
(2π)
Q0 s

is the paraxial momentum operator.
The continuum version of eqn (3.150) for the Hamiltonian in a dispersive medium
can be approximated by

d3 q
a† (q) a0s (q) ,
Hem = ω|k0 +q| (7.18)
0s
3
(2π)
Q0 s

when acting on a paraxial state. The small spread in frequencies across the paraxial
bundle, together with the weak dispersion condition (3.120), allows the dispersion
relation ωk = ck/n (ωk ) to be approximated by

ck
ωk = , (7.19)
(ωk ’ ω0 )
dn
n0 + dω 0

and a straightforward calculation yields

q
’ 1 + ··· .
ω|k0 +q| = ω0 + vg0 k0 k0 + (7.20)
k0

The conditions (7.8) allow the expansion

q2
q qz
+ 2 + O θ2 ,
k0 + =1+ (7.21)
k0 k0 2k0
¾¾¿
The slowly-varying envelope operator

which in turn leads to the expression Hem = ω0 N0 + HP + O θ2 , where
vg0 q 2
d3 q
a† (q) a0s (q)
HP = vg0 qz + (7.22)
0s
3 2k0
(2π)
Q0 s

is the paraxial Hamiltonian for the space H (k0 , θ).
The e¬ective orthogonality of distinct paraxial spaces”which corresponds to the
distinguishability of distinct paraxial beams”implies that the various global operators
are additive. Thus the operators for the total photon number, momentum, and energy
for a set of paraxial beams are
N= Nβ , Pem = ( kβ Nβ + Pβ ) , Hem = ( ωβ Nβ + HP β ) , (7.23)
β β β

where Nβ , Pβ , and HP β are respectively the paraxial number, momentum, and energy
operators for the βth beam.

7.3 The slowly-varying envelope operator
We next use the properties of the paraxial space H (k0 , θ) to justify an approximation
for the ¬eld operator, A(+) (r, t), that replaces eqn (7.1) for the classical ¬eld. In order
to emphasize the relation to the classical theory, we initially work in the Heisenberg
picture. The slowly-varying envelope operator ¦ (r, t) is de¬ned by

(vg0 /c)
¦ (r, t) ei(k0 ·r’ω0 t) .
A(+) (r, t) = (7.24)
2 0 k0 c
Comparing this de¬nition to the general plane-wave expansion (3.149) shows that
d3 q
a0s (q) es (k0 + q) ei(q·r’δq t) ,
¦ (r, t) = 3 fq (7.25)
(2π)
Q0 s

where
vg (|k0 + q|) k0
δq = ω|k0 +q| ’ ω0 and fq = . (7.26)
|k0 + q|
vg0
The corresponding expressions in the Schr¨dinger picture follow from the relation
o
(+) (+)
A (r) = A (r, t = 0).
The envelope operator will only be slowly varying when applied to paraxial states
in H (k0 , θ), so we begin by using eqn (7.10) to evaluate the action of the envelope
operator ¦ (r) = ¦ (r, 0) on a typical basis vector of H (k0 , θ):
M
a† m (qm ) |0
¦ (r) |{qs}M = ¦ (r) 0s
m=1
M
a† m (qm ) |0
= ¦ (r) , 0s
m=1
M M
¦ (r) , a† m (1 ’ δlm ) a† l (ql ) |0 ,
= (qm ) (7.27)
0s 0s
m=1 l=1
¾¾ Paraxial quantum optics

where the last line follows from the identity (C.49). Setting t = 0 in eqn (7.25) produces
the Schr¨dinger-picture representation of the envelope operator,
o

d3 q
a0s (q) es (k0 + q) eiq·r ,
¦ (r) = 3 fq (7.28)
(2π)
Q0 s

and using this in the calculation of the commutator yields

¦ (r) , a† m (qm ) = fqm es (k0 + qm ) eiqm ·r
0s

= es (k0 ) eiqm ·r + O (θ) . (7.29)

Thus when acting on paraxial states the exact representation (7.28) can be replaced
by the approximate form

¦ (r) = φs (r) e0s + O (θ) , (7.30)
s

where e0s = es (k0 ), and

d3 q
(q) eiq·r .
φs (r) = 3 a0s (7.31)
(2π)
Q0

The subscript Q0 on the integral is to remind us that the integration domain is re-
stricted by eqn (7.8). This representation can only be used when the operator acts on
a vector in the paraxial space. It is in this sense that the z-component of the envelope
operator is small, i.e.
Ψ1 |¦z (r)| Ψ2 = O (θ) , (7.32)
for any pair of normalized vectors |Ψ1 and |Ψ2 that both belong to H (k0 , θ). In the
leading paraxial approximation, i.e. neglecting O (θ)-terms, the electric ¬eld operator
is
ω0 (vg0 /c)
e0s φs (r, t) ei(k0 ·r’ω0 t) .
E(+) (r, t) = i (7.33)
2 0 n0 s

The commutation relations for the transverse components of the envelope operator
have the simple form

¦i (r, t) , ¦† (r , t) = δij δ (r ’ r ) (i, j = 1, 2) , (7.34)
j


which shows that the paraxial electromagnetic ¬eld is described by two independent
operators ¦1 (r) and ¦2 (r) satisfying local commutation relations. This re¬‚ects the
fact that the paraxial approximation eliminates the nonlocal features exhibited in the
exact commutation relations (3.16) by e¬ectively averaging the arguments r and r
over volumes large compared to »3 . By the same token, the delta function appearing
0
on the right side of eqn (7.34) is coarse-grained, i.e. it only gives correct results
when applied to functions that vary slowly on the scale of the carrier wavelength. This
feature will be important when we return to the problem of photon localization.
¾¾
The slowly-varying envelope operator

In most applications the operators φs (r, t), corresponding to de¬nite polarization
states, are more useful. They satisfy the commutation relations

φs (r, t) , φ† (r , t) = δss δ (r ’ r ) (s, s = ± or 1, 2) . (7.35)
s

The approximate expansion (7.31) can be inverted to get

d3 rφs (r) e’iq·r = d3 re— (k0 ) · ¦ (r) e’iq·r ,
a0s (q) = (7.36)
s


which is valid for q in the paraxial region Q0 . By using this inversion formula the
operators N0 , P0 , and HP can be expressed in terms of the slowly-varying envelope
operator:
φ† (r) φs (r) ,
d3 r
N0 = (7.37)
s
s

φ† (r) ∇φs (r) ,
d3 r
P0 = (7.38)
s
i
s

vg0 ∇2
φ† (r) vg0 ∇z ’
3
HP = dr φs (r) . (7.39)
s
i 2k0
s
We can gain a better understanding of the paraxial Hamiltonian by substituting
eqns (7.24) and (7.22) into the Heisenberg equation
‚ (+)
A (r, t) = A(+) (r, t) , Hem
i (7.40)
‚t
to get

ω0 ¦ (r, t) + i ¦ (r, t) = ω0 [¦ (r, t) , N0 ] + [¦ (r, t) , HP ] . (7.41)
‚t
Since the envelope operator ¦ (r, t) is a sum of annihilation operators, it satis¬es
[¦ (r, t) , N0 ] = ¦ (r, t). Consequently, the term ω0 [¦ (r, t) , N0 ] is canceled by the
time derivative of the carrier wave. The Heisenberg equation for the envelope ¬eld
¦ (r, t) is therefore

i ¦ (r, t) = [¦ (r, t) , HP ] . (7.42)
‚t

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