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d3 r¦† (r) r — k0 + r — k0 ∇z + z k0 — ∇
L0 = ¦j (r) , (7.84)
i i

and the longitudinal component is

d3 r¦† (r) r ∇ ’r ∇
L0z = ¦j (r) . (7.85)
1 2 2 1
i i

The transverse part L0 is dominated by the term proportional to k0 . After
expressing the integral in terms of the scaled variable r and scaled ¬eld ¦, one ¬nds
that L0 = O (1/θ). The similar terms ω0 N0 and k0 N0 in the momentum and energy
are O 1/θ2 , so they are even larger. This apparently singular behavior is physically
harmless; it simply represents the fact that all photons in the wave packet have energies
close to ω0 and momenta close to k0 .
For the angular momentum the situation is di¬erent. The angular momenta of in-
dividual photons in plane-wave modes k0 +q must exhibit large ¬‚uctuations due to the
tight constraints on the polar angle ‘k given by eqn (7.4). These ¬‚uctuations are not
conjugate to the longitudinal component J0z , since rotations around the z-axis leave
‘k unchanged. On the other hand, the transverse components L0 generate rotations
around the transverse axes which do change the value of ‘k . Thus we should expect
large ¬‚uctuations in the transverse components of the angular momentum, which are
described by the large transverse term L0 . Thus only the longitudinal component L0z
is meaningful for a paraxial state. By combining eqns (7.85) and (7.79), we see that
the lowest-order paraxial angular momentum operator is purely longitudinal,

J0 = k0 [L0z + S0 ] . (7.86)

Approximate photon localizability—
Mandel™s local number operator, de¬ned by eqn (3.204), displays peculiar nonlocal
properties. Despite this apparent ¬‚aw, Mandel was able to demonstrate that N (V )
»3 , where »0
behaves approximately like a local number operator in the limit V 0
is the characteristic wavelength for a monochromatic ¬eld state. The important role
played by this limit suggests using the paraxial expansion to investigate the alternative
de¬nitions of the local number operator in a systematic way. To this end we ¬rst
introduce a scaled version of the Mandel detection operator by
M (r) = √ M (r) eik0 z . (7.87)

By combining the de¬nition (3.203) with the expansion (7.64), the identity (7.81), and
the scaled gradient
Approximate photon localizability— ¾¿¿

∇ 1 1 ‚
∇ + u3
k0 k0 k0 ‚z
= θ∇ + θ2 u3 ∇z , (7.88)

one ¬nds
(0) (1) (2)
+ θ2 M + O θ3 ,
M=M + θM (7.89)
(0) (1) (1)
where M = ¦, M =¦ , and

(2) (2) 2
’ ∇ + 2i∇z ¦ .
M =¦ (7.90)
The corresponding expansion for N (V ) is

N (V ) = N (0) (V ) + θ2 N (2) (V ) + O θ4 , (7.91)

(0)† (0)
(r) · ¦
N (0) (V ) = d3 r¦ (r) ,
(1)† (1) (0)† (2)
·M ·M
(2) 3
N (V ) = dr M +M + HC .

A simple calculation using the local commutation relations (7.34) for the zeroth-
order envelope ¬eld yields

N (0) (V ) , N (0) (V ) = 0 (7.93)

for nonoverlapping volumes, and

N (0) (V ) , ¦† (r) = χV (r) ¦† (r) , (7.94)

where the characteristic function χV (r) is de¬ned by

1 for r ∈ V ,
χV (r) = (7.95)
0 for r ∈ V .

Thus N (0) (V ) acts like a genuine local number operator. The nonlocal features dis-
cussed in Section 3.6.2 will only appear in the higher-order terms. It is, however,
important to remember that the delta function in the zeroth-order commutation rela-
tion (7.34) is really coarse-grained with respect to the carrier wavelength »0 . For this
»3 .
reason the localization volume V must satisfy V 0
The paraxial expansion of the alternative operator G (V ), introduced in eqn (3.210),
shows (Deutsch and Garrison, 1991a) that the two de¬nitions agree in lowest order,
G(0) (V ) = N (0) (V ), but disagree in second order, G(2) (V ) = N (2) (V ). This disagree-
ment between equally plausible de¬nitions for the local photon number operator is a
consequence of the fact that a photon with wavelength »0 cannot be localized to a
¾¿ Paraxial quantum optics

volume of order »3 . Since most experiments are well described by the paraxial approx-
imation, it is usually permissible to think of the photons as localized, provided that
the diameter of the localization region is larger than a wavelength.
The negative frequency part Ai (r) is a sum over creation operators, so it is
tempting to interpret Ai (r) as creating a photon at the point r. In view of the
impossibility of localizing photons, this temptation must be sternly resisted. On the
other hand, the cavity operator a† can be interpreted as creating a photon described by
the cavity mode E κ (r), since the mode function extends over the entire cavity. In the
same way, the plane-wave operator a† can be interpreted as creating a photon in the
(box-normalized) plane-wave state with wavenumber k and polarization eks . Finally
the wave packet operator a† [w] can be interpreted as creating a photon described by
the classical wave packet w, but it would be wrong to think of the photon as strictly
localized in the region where w (r) is large. With this caution in mind, one can regard
the pulse-envelope w (r) as an e¬ective photon wave function, provided that the pulse
duration contains many optical periods and the transverse pro¬le is large compared
to a wavelength.
There are other aspects of the averaged operators that also require some caution.
The operator N [w] = a† [w] a [w] satis¬es

N [w] , a† [w] = a† [w] , [N [w] , a [w]] = ’a [w] , (7.96)

so it serves as a number operator for w-photons, but these number operators are not
mutually commutative, since

[N [w] , N [u]] = (w,u) a† [u] a [w] ’ a† [w] a [u] . (7.97)

Thus distinct w photons and u photons cannot be independently counted unless the
classical wave packets w and u are orthogonal. This lack of commutativity can be
important in situations that require the use of non-orthogonal modes (Deutsch et al.,

7.9 Exercises
7.1 Frequency spread for a paraxial beam
(1) Show that the fractional change in the index of refraction across a paraxial beam
ω0 dn
∆n ∆k n0 dω 0
= ,
k0 1 + ω0 dω 0
n0 n0

where n0 = n (ω0 ) = (ω0 ) / and (dn/dω)0 is evaluated at the carrier fre-
k0 + |q |2 + qz with eqns (7.5) and (7.7) to get
2 2
(2) Combine the relation k =

∆k 1 ∆q 12
θ + ··· .
+ O θ4 =
k0 2 k0 2

(3) Combine this with ∆ω = vg0 ∆k to ¬nd
∆ω n0 vg0 1 n0 12 1
k0 θ2 = θ < θ2 .
= dn
ω0 ck0 2 2 2
n0 + ω 0 dω 0

7.2 Distinct paraxial Hilbert spaces are e¬ectively orthogonal
Consider the paraxial subspaces H (k1 , θ1 ) and H (k2 , θ2 ) discussed in Section 7.2.2.
(1) For a typical basis vector |{qs}κ in H (k1 , θ1 ) show that as (k) |{qs}κ ≈ 0 when-
ever |k ’ k1 | θ1 |k1 |.
(2) Use this result to argue that each basis vector in H (k2 , θ2 ) is approximately or-
thogonal to every basis vector in H (k1 , θ1 ).

7.3 Distinct paraxial ¬elds are independent
Combine the de¬nition (7.46) with the de¬nition (7.14) for distinct beams to show that
eqn (7.47) is satis¬ed in the same sense that distinct paraxial spaces are orthogonal.

An analogy to many-body physics—
Consider a special paraxial state such that the z-dependence of the ¬eld φs (r) can
be neglected and only one polarization is excited, so that φs (r) ’ φ (r ) . De¬ne an
e¬ective photon mass M0 such that the paraxial Hamiltonian HP for this problem
is formally identical to a second quantized description of a two-dimensional, nonrela-
tivistic, many-particle system of bosons with mass M0 (Huang, 1963, Appendix A.3;
Feynman, 1972). This feature leads to interesting analogies between quantum optics
and many-body physics (Chiao et al., 1991; Deutsch et al., 1992; Wright et al., 1994).

Paraxial expansion—
(1) Expand Xs (q, θ) through O θ2 .
(r) = ik0 ∇ · ¦(0) .
(2) Show that ¦
(r) = 1 ∇ ∇ · ¦(0) + ∇2 + 2i∇z ¦(0) .
(3) Show that ¦ 2 4

Distinct paraxial wave packet spaces are e¬ectively orthogonal—
Consider two paraxial wave packets, w ∈ P (k1 , θ1 ) and v ∈ P (k2 , θ2 ), where k1 and
k2 satisfy eqn (7.14).
(1) Apply the de¬nitions of q (Section 7.5) and w — (q) (Section 7.6) to show that

d3 q
w — (q) v s q + ∆k ,
(w, v) = s
V1 (2π) s

where ∆k = k1 ’ k2 and the arguments of w— and v s are scaled with θ1 and θ2
|q|, and combine this with the rapid fall o¬
(2) Calculate ∆k, explain why ∆k
condition in eqn (7.68) to conclude that θ2 (w, v) ’ 0 as θ2 ’ 0 for any value of
¾¿ Paraxial quantum optics

(3) Show that θ2 a [w] , a† [v] ’ 0 as θ2 ’ 0.
Linear optical devices

The manipulation of light beams by passive linear devices, such as lenses, mirrors,
stops, and beam splitters, is the backbone of experimental optics. In typical arrange-
ments the individual devices are separated by regions called propagation segments
in which the light propagates through air or vacuum. The index of refraction is usually
piece-wise constant, i.e. it is uniform in each device and in each propagation segment.
In most arrangements each device or propagation segment has an axis of symmetry
(the optic axis), and the angle between the rays composing the beam and the local
optic axis is usually small. The light beams are then said to be piece-wise paraxial.
Under these circumstances, it is useful to treat the interaction of a light beam with a
single device as a scattering problem in which the incident and scattered ¬elds both
propagate in vacuum. The optical properties of the device determine a linear relation
between the complex amplitudes of the incident and scattered classical waves. After
a brief review of this classical approach, we will present a phenomenological descrip-
tion of quantized electromagnetic ¬elds interacting with linear optical devices. This
approach will show that, at the quantum level, linear optical e¬ects can be viewed”in
a qualitative sense”as the propagation of photons guided by classical scattered waves.
The scattered waves are a rough analogue of wave functions for particles, so the asso-
ciated classical rays may be loosely considered as photon trajectories. These classical
analogies are useful for visualizing the interaction of photons with linear optical de-
vices but”as is always the case with applications of quantum theory”they must be
used with care. A more precise wave-function-like description of quantum propagation
through optical systems is given in Section 6.6.2.

8.1 Classical scattering
The general setting for this discussion is a situation in which one or more paraxial
beams interact with an optical device to produce several scattered paraxial beams.
Both the incident and the scattered beams are assumed to be mutually distinct, in
the sense de¬ned by eqn (7.14). Under these circumstances, the paraxial beams will
be called scattering channels; the incident classical ¬elds are input channels and
the scattered beams are output channels. Since this process is linear in the ¬elds, the
initial and ¬nal beams can be resolved into plane waves. The conventional classical de-
scription of propagation through optical elements pieces together plane-wave solutions
of Maxwell™s equations by applying the appropriate boundary conditions at the inter-
faces between media with di¬erent indices of refraction, as shown in Fig. 8.1(a). This
procedure yields a linear relation between the Fourier coe¬cients of the incident and
¾¿ Linear optical devices

k6 ’k6
k4 ’k4
Fig. 8.1 (a) A plane wave ±kI exp (ikI · r) ’k64
incident on a dielectric slab. The re-
k1 ’k1
¬‚ected and transmitted waves are respec-
tively ±kR exp (ikR · r) and ±kT exp (ikT · r).
(b) The time reversed version of (a). The ex-
= >
tra wave at ’kT R is discussed in the text.

scattered waves that is similar to the description of scattering in terms of stationary
states in quantum theory (Bransden and Joachain, 1989, Chap. 4). From the viewpoint
of scattering theory, the classical piecing procedure is simply a way to construct the
scattering matrix relating the incident and scattered ¬elds. Before considering the
general case, we analyze two simple examples: a propagation segment and a thin slab
of dielectric.
For the propagation segment, an incident plane wave ± exp (ik · r)”the input
channel”simply acquires the phase kL, where L is the length of the segment along
the propagation direction, i.e. the relation between the incident amplitude ± and the
scattered amplitude ± ”representing the output channel”is

± = eikL ± = eiωL/c ± . (8.1)

In some applications the propagation segment through vacuum is replaced by a length
L of dielectric. If the end faces of the dielectric sample are antire¬‚ection coated, then
the scattering relation is

± = eik(ω)L ± = ein(ω)ωL/c ± , (8.2)

where n (ω) is the index of refraction for the dielectric. Since the transmitted wave
can be expressed as
± eik(z’ωt) = ±ei[kz’ω(t’∆t)] , (8.3)
where ∆t = n (ω) L/c, the dielectric medium is called a retarder plate, or sometimes
a phase shifter.
We next turn to the example of a plane wave incident on a thin dielectric slab”
which is not antire¬‚ection coated”as shown in Fig. 8.1(a). Ordinary ray tracing,
using Snell™s law and the law of re¬‚ection at each interface between the dielectric
and vacuum, determines the directions of the propagation vectors kR and kT (where
R and T stand for the re¬‚ected and transmitted waves respectively) relative to the
propagation vector kI of the incoming wave. Since the transmitted wave crosses the
dielectric“vacuum interface twice, we ¬nd the familiar result kT = kI , i.e. the incident
and transmitted waves are described by the same spatial mode.
The plane of incidence is de¬ned by the vectors kI and n, where n is the unit vector
normal to the slab. Every incident electromagnetic plane wave can be resolved into two
Classical scattering

polarization components: the TE- (or S-) polarization, with electric vector perpendic-
ular to the plane of incidence, and the TM- (or P-) polarization, with electric vector
in the plane of incidence. For optically isotropic dielectrics, these two polarizations
are preserved by re¬‚ection and refraction. Since scattering is a linear process, we lose
nothing by assuming that the incident wave is either TE- or TM-polarized. This allows
us to simplify the vector problem to a scalar problem by suppressing the polarization
vectors. The three waves outside the slab are then ±kI exp (ikI · r), ±kR exp (ikR · r),
and ±kT exp (ikT · r). The solution of Maxwell™s equations inside the slab is a linear
combination of the transmitted wave at the ¬rst interface and the re¬‚ected wave from
the second interface. Applying the boundary conditions at each interface (Jackson,
1999, Sec. 7.3) yields a set of equations relating the coe¬cients, and eliminating the
coe¬cients for the interior solution leads to

±kR = r ±kI , ±kT = t ±kI , (8.4)

where the complex parameters r and t are respectively the amplitude re¬‚ection and
transmission coe¬cients for the slab. This is the simplest example of the general piecing
procedure discussed above.
Important constraints on the coe¬cients r and t follow from the time-reversal
invariance of Maxwell™s equations. What this means is that the time-reversed ¬nal
¬eld will evolve into the time-reversed initial ¬eld. This situation is shown in Fig.
8.1(b), where the incident waves have propagation vectors ’kR and ’kT and the
scattered waves have ’kI and ’kT R . The amplitudes for this case are written as ±T , q
where T stands for time reversal. The usual calculation gives the scattered waves as

±T I = r ±’kR + t ±’kT ,
±T T R = t ±’kR + r ±’kT .

In Appendix B.3.3 it is shown that the linear polarization basis can be chosen so that
the time-reversed amplitudes are related to the original amplitudes by eqn (B.80). In
the present case, this yields ±T I = ±— I , ±’kR = ±kR , ±’kT = ±kT , and ±T T R =
— —
’k k ’k
±— T R . Substituting these relations into eqn (8.5) and taking the complex conjugate
gives a second set of relations between the amplitudes ±kI , ±kR , and ±kT :

±kI = r— ±kR + t— ±kT ,
±kT R = t— ±kR + r— ±kT .

There is an apparent discrepancy here, since the original problem had no wave with
propagation vector kT R . Time-reversal invariance for the original problem therefore
requires ±kT R = 0. Using eqn (8.4) to eliminate ±kR and ±kT from eqn (8.6) and
imposing ±kT R = 0 leads to the constraints

|r|2 + |t|2 = 1 ,
r t— + r— t = 0 .

The ¬rst relation represents conservation of energy, while the second implies that
the transmitted part of ’kR and the re¬‚ected part of ’kT interfere destructively as
¾¼ Linear optical devices

required by time-reversal invariance. These relations were originally derived by Stokes
(Born and Wolf, 1980, Sec. 1.6).
Setting r = |r| exp (iθr ) and t = |t| exp (iθt ) in the second line of eqn (8.7) shows us
that time-reversal invariance imposes the relation
θr ’ θt = ±π/2 ; (8.8)
in other words, the phase of the re¬‚ected wave is shifted by ±90—¦ relative to the
transmitted wave. This phase di¬erence is a measurable quantity; therefore, the ± sign
on the right side of eqn (8.8) is not a matter of convention. In fact, this sign determines
whether the re¬‚ected wave is retarded or advanced relative to the transmitted wave.
In the extreme limit of a perfect mirror, i.e. |t| ’ 0, we can impose the convention
θt = 0, so that
θr = ±π/2 , |r| = 1 . (8.9)
For given values of the relevant parameters”the angle of incidence, the index of re-
fraction of the dielectric, and the thickness of the slab”the coe¬cients r and t can
be exactly calculated (Born and Wolf, 1980, Sec. 1.6.4, eqns (57) and (58)), and the
phases θr and θt are uniquely determined.
Let us now consider a more general situation in which waves with kI and kT R are
both incident. This would be the time-reverse of Fig. 8.1(b), but in this case ±kT R = 0.
The standard calculation then relates ±kT and ±kR to ±kI and ±kT R by

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