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This shows that the paraxial Hamiltonian generates the time translation of the en-
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)


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

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

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