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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 ,
(8.7)
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
tr
±kT ±kI
= . (8.10)
rt
±kR ±kT R
The meaning of the conditions (8.7) is that the 2 — 2 scattering matrix in this equation
is unitary.
Having mastered the simplest possible optical elements, we proceed without hes-
itation to the general case of linear and nondissipative optical devices. The incident
¬eld is to be expressed as an expansion in box-quantized plane waves,

fks (r) = eks exp (ik · r) / V . (8.11)
For the single-mode input ¬eld E in = fks e’iωk t , the general piecing procedure yields an
output ¬eld which we symbolically denote by (fks )scat . This ¬eld is also expressed as
an expansion in box-quantized plane waves. For a given basis function fks , we denote
the expansion coe¬cients of the scattered solution by Sk s ,ks , so that

(fks )scat = fk s Sk s ,ks . (8.12)
ks

Repeating this procedure for all elements of the basis de¬nes the entire scattering
matrix Sk s ,ks . The assumption that the device is stationary means that the frequency
ωk associated with the mode fks cannot be changed; therefore the scattering matrix
must satisfy
Sk s ,ks = 0 if ωk = ωk . (8.13)
In general, the sub-matrix connecting plane waves with a common frequency ωk = ω
will depend on ω.
¾½
Classical scattering

The incident classical wave packet is represented by the in-¬eld

ωk
±ks fks (r) e’iωk t ,
(+)
E in (r, t) = i (8.14)
20
ks

where the time origin t = 0 is chosen so that the initial wave packet E in (r, 0) has not
reached the optical element. For t (> 0) su¬ciently large, the scattered wave packet
has passed through the optical element, so that it is again freely propagating. The
solution after the scattering is completely over is the out-¬eld

ωk
± fk s (r) e’iωk t ,
(+)
E out (r, t) = i (8.15)
2 0 ks
ks

where the two sets of expansion coe¬cients are related by the scattering matrix:

±k s = Sk s ,ks ±ks . (8.16)
ks

Time-reversal invariance can be exploited here as well. In the time-reversed prob-
lem, the time-reversed output ¬eld scatters into the time-reversed input ¬eld, so

±T = T
S’ks,’k s ±’k s , (8.17)
’ks
ks

where ’ks is the time reversal of ks. Time-reversal invariance requires

S’ks,’k s = Sk s ,ks , (8.18)

where the transposition of the indices re¬‚ects the interchange of incoming and outgoing
modes. The classical rule (see Appendix B.3.3) for time reversal is

±T = ’±— , (8.19)
’ks ks

so using eqn (8.18) in the complex conjugate of eqn (8.17) yields

±ks = Sk s ,ks ±k s . (8.20)
ks

Combining this with eqn (8.16) leads to


±ks = Sk s ,ks Sk s ,k ±k , (8.21)
s s
ks ks

which must hold for all input ¬elds {±ks }. This imposes the constraints

Sk s ,ks Sk s ,k = δkk δss , (8.22)
s
ks

that are generalizations of eqn (8.7). In matrix form this is S † S = SS † = 1; i.e. every
passive linear device is described by a unitary scattering matrix.
¾¾ Linear optical devices

8.2 Quantum scattering
We will take a phenomenological approach in which the classical amplitudes are re-
placed by the Heisenberg-picture operators aks (t). Let t = 0 be the time at which
the Heisenberg and Schr¨dinger pictures coincide, then according to eqn (3.95) the
o
operator
aks (t) = aks (t) eiωk t (8.23)
is independent of time for free propagation. Thus in the scattering problem the time
dependence of ak s (t) comes entirely from the interaction between the ¬eld and the
optical element. The classical amplitudes ±ks represent the solution prior to scattering,
so it is natural to replace them according to the rule
±ks ’ lim aks (t) eiωk t = aks (0) = aks . (8.24)
t’0

Similarly, ±k s represents the solution after scattering, and the corresponding rule,
±k s ’ ak s = lim = lim {ak s (t)} ,
ak s (t) eiωk t
(8.25)
t’+∞ t’∞

implies the asymptotic ansatz
ak s (t) ’ ak s e’iωk t . (8.26)
At late times the ¬eld is propagating in vacuum, so this limit makes sense by virtue
of the fact that ak s (t) is time independent for free propagation.
Thus aks and ak s are respectively the incident and scattered annihilation op-
erators, and they will be linearly related in the weak-¬eld limit. Furthermore, the
correspondence principle tells us that the relation between the operators must repro-
duce eqn (8.16) in the classical limit aks ’ ±ks . Since both relations are linear, this
can only happen if the incident and scattered operators also satisfy

ak s = Sk s ,ks aks , (8.27)
ks

where Sk s ,ks is the classical scattering matrix. The in-¬eld operator Ein and the
out-¬eld operator Eout are given by the quantum analogues of eqns (8.14) and
(8.15):
ωk
aks fks (r) e’iωk t ,
(+)
Ein (r, t) = i (8.28)
20
ks

ωk
ak s fk s (r) e’iωk t .
(+)
Eout (r, t) = i (8.29)
20
ks
The operators {aks } and {ak s } are related by eqn (8.27) and the inverse relation

S† —
aks = ak s = Sk s ,ks ak s . (8.30)
ks,k s
ks ks

The unitarity of the classical scattering matrix guarantees that the scattered operators
{ak s } satisfy the canonical commutation relations (3.65), provided that the incident
operators {aks } do so.
¾¿
Quantum scattering

The use of the Heisenberg picture nicely illustrates the close relation between the
classical and quantum scattering problems, but the Schr¨dinger-picture description
o
of scattering phenomena is often more useful for the description of experiments. The
¬xed Heisenberg-picture state vector |Ψ is the initial state vector in the Schr¨dinger
o
picture, i.e. |Ψ (0) = |Ψ , so the time-dependent Schr¨dinger-picture state vector is
o

|Ψ (t) = U (t) |Ψ , (8.31)

where U (t) is the unitary evolution operator. Combining the formal solution (3.83) of
the Heisenberg operator equations with the ansatz (8.26) yields

aks (t) = U † (t) aks U (t) ’ aks e’iωk t as t ’ ∞ , (8.32)

which provides some asymptotic information about the evolution operator.
The task at hand is to use this information to ¬nd the asymptotic form of |Ψ (t) .
Since the scattering medium is linear, it is su¬cient to consider a one-photon initial
state,
Cks a† |0 .
|Ψ = (8.33)
ks
ks

The equivalence between the two pictures implies

0 |aks | Ψ (t) = 0 |aks (t)| Ψ , (8.34)

where the left and right sides are evaluated in the Schr¨dinger and Heisenberg pictures
o
respectively. Since there is neither emission nor absorption in the passive scattering
medium, |Ψ (t) remains a one-photon state at all times, and

0 |aks | Ψ (t) a† |0 .
|Ψ (t) = (8.35)
ks
ks

The expansion coe¬cients 0 |aks | Ψ (t) are evaluated by combining eqn (8.34) with
the asymptotic rule (8.26) and the scattering law (8.27) to get 0 |aks (t)| Ψ =
e’iωk t Cks , where
Cks = Sks,k s Ck s . (8.36)
ks

The evolved state is therefore

e’iωk t Cks a† |0 .
|Ψ (t) = (8.37)
ks
ks

In other words, the prescription for the asymptotic (t ’ ∞) form of the Schr¨dingero
’iωk t
state vector is simply to replace the initial coe¬cients Cks by e Cks , where Cks is
the transform of the initial coe¬cient vector by the scattering matrix.
In the standard formulation of scattering theory, the initial state is stationary”
i.e. an eigenstate of the free Hamiltonian”in which case all terms in the sum over
ks in eqn (8.33) have the same frequency: ωk = ω0 . The energy conservation rule
(8.13) guarantees that the same statement is true for the evolved state |Ψ (t) , so the
¾ Linear optical devices

time-dependent exponentials can be taken outside the sum in eqn (8.37) as the overall
phase factor exp (’iω0 t). In this situation the overall phase can be neglected, and the
asymptotic evolution law (8.37) can be replaced by the scattering law

Cks a† |0 .
|Ψ ’ |Ψ = (8.38)
ks
ks

An equivalent way to describe the asymptotic evolution follows from the observa-
tion that the evolved state in eqn (8.37) is obtained from the initial state in eqn (8.33)
by the operator transformation

a† ’ e’iωk t a† s Sk s ,ks . (8.39)
ks k
ks

When applying this rule to stationary states, the time-dependent exponential can be
dropped to get the scattering rule

a† ’ aks =

a† s Sk s ,ks . (8.40)
ks k
ks

For scattering problems involving one- or two-photon initial states, it is often more
convenient to use eqn (8.40) directly rather than eqn (8.38). For example, the scattering
rule for |Ψ = a† |0 is
ks
a† |0 ’ aks |0 .

(8.41)
ks
The rule (8.39) also provides a simple derivation of the asymptotic evolution law
for multi-photon initial states. For the general n-photon initial state,
Ck1 s1 ,...,kn sn a† 1 s1 · · · a† n sn |0 ,
|Ψ = ··· (8.42)
k k
k1 s1 kn sn

applying eqn (8.39) to each creation operator yields
n
ωkm t Ck1 s1 ,...,kn sn a† 1 s1 · · · a† n sn |0 ,
|Ψ (t) = ··· exp ’i (8.43)
k k
k1 s1 kn sn m=1

where
··· Sk1 s1 ,p1 v1 · · · Skn sn ,pn vn Cp1 ν1 ,...,pn νn .
Ck1 s1 ,...,kn sn = (8.44)
p1 v1 pn vn

For scattering problems the initial state is stationary, so that
n
ωkm = ω0 , (8.45)
m=1

and the evolution equation (8.43) is replaced by the scattering rule
† †
|Ψ ’ |Ψ = ··· Ck1 s1 ,...,kn sn ak1 s1 · · · akn sn |0 . (8.46)
k1 s1 kn sn

It is important to notice that the scattering matrix in eqn (8.27) has a special
property: it relates annihilation operators to annihilation operators only. The scattered
¾
Paraxial optical elements

annihilation operators do not depend at all on the incident creation operators. This
feature follows from the physical assumption that emission and absorption do not occur
in passive linear devices. The special form of the scattering matrix has an important
consequence for the commutation relations of ¬eld operators evaluated at di¬erent
times. Since all annihilation operators”and therefore all creation operators”commute
with one another, eqns (8.28), (8.29), and (8.27) imply
(±) (±)
Eout,i (r, +∞) , Ein,j (r , ’∞) = 0 (8.47)

for scattering from a passive linear device. In fact, eqn (3.102) guarantees that the
positive- (negative-) frequency parts of the ¬eld at di¬erent ¬nite times commute, as
long as the evolution of the ¬eld operators is caused by interaction with a passive
linear medium. One should keep in mind that commutativity at di¬erent times is
not generally valid, e.g. if emission and absorption or photon“photon scattering are
(+) (’)
possible, and further that commutators like Ei (r, t) , Ej (r , t ) do not vanish
even for free ¬elds or ¬elds evolving in passive linear media. Roughly speaking, this
implies that the creation of a photon at (r , t ) and the annihilation of a photon at
(r, t) are not independent events.
Putting all this together shows that we can use standard classical methods to cal-
culate the scattering matrix for a given device, and then use eqn (8.27) to relate the
annihilation operators for the incident and scattered modes. This apparently simple
prescription must be used with care, as we will see in the applications. The utility of
this approach arises partly from the fact that each scattering channel in the classi-
cal analysis can be associated with a port, i.e. a bounding surface through which a
well-de¬ned beam of light enters or leaves. Input and output ports are respectively
associated with input and output channels. The ports separate the interior of the de-
vice from the outside world, and thus allow a black box approach in which the device
is completely characterized by an input“output transfer function or scattering ma-
trix. The principle of time-reversal invariance imposes constraints on the number of
channels and ports and thus on the structure of the scattering matrix.
The simplest case is a one-channel device, i.e. there is one input channel and one
output channel. In this case the scattering is described by a 1—1 matrix, as in eqn (8.2).
This is more commonly called a two-port device, since there is one input port and one
output port. As an example, for an antire¬‚ection coated thin lens the incident light
occupies a single input channel, e.g. a paraxial Gaussian beam, and the transmitted
light occupies a single output channel. The lens is therefore a one-channel/two-port
device.

8.3 Paraxial optical elements
An optical element that transforms an incident paraxial ray bundle into another parax-
ial bundle will be called a paraxial optical element. The most familiar examples
are (ideal) lenses and mirrors. By contrast to the dielectric slab in Fig. 8.1, an ideal
lens transmits all of the incident light; no light is re¬‚ected or absorbed. Similarly an
ideal mirror re¬‚ects all of the incident light; no light is transmitted or absorbed. In
the non-ideal world inhabited by experimentalists, the conditions de¬ning a paraxial
¾ Linear optical devices

element must be approximated by clever design. The no-re¬‚ection limit for a lens is
approached by applying a suitable antire¬‚ection coating. This consists of one or
more layers of transparent dielectrics with refractive indices and thicknesses adjusted
so that the re¬‚ections from the various interfaces interfere destructively (Born and
Wolf, 1980, Sec. 1.6). An ideal mirror is essentially the opposite of an antire¬‚ection
coating; the parameters of the dielectric layers are chosen so that the transmitted
waves su¬er destructive interference. In both cases the ideal limit can only be approx-
imated for a limited range of wavelengths and angles of incidence. Compound devices
made from paraxial elements are automatically paraxial.
For optical elements de¬ned by curved interfaces the calculation of the scattering
matrix in the plane-wave basis is rather involved. The classical theory of the interaction
of light with lenses and curved mirrors is more naturally described in terms of Gaussian
beams, as discussed in Section 7.4. In the absence of this detailed theory it is still
possible to derive a useful result by using the general properties of the scattering
matrix. We will simplify this discussion by means of an additional approximation. An
incident paraxial wave is a superposition of plane waves with wavevectors k = k0 + q,
where |q| k0 . According to eqns (7.7) and (7.9), the dispersion in qz = q·k0 and ω for
an incident paraxial wave is small, in the sense that ∆ω/ (c∆q ) ∼ ∆qz /∆q = O (θ),
where q = q’ q· k0 k0 is the part of q transverse to k0 and θ is the opening angle of
the beam. This suggests considering an incident classical ¬eld that is monochromatic
and planar, i.e.

ω0 iq ·r
(+)
E in (r, t) = ei(k0 z’ω0 t) .
i ±k +q ,s e0s e (8.48)
2 0V 0
q ,s

In the same spirit the scattering matrix will be approximated by

Sks,k s ≈ δkz k0 δkz k0 Sq , (8.49)
s,q s


with the understanding that the reduced scattering matrix Sq s,q s e¬ectively con-
¬nes q and q to the paraxial domain de¬ned by eqn (7.8). In this limit, the unitarity
condition (8.22) reduces to

Sq s ,q s Sq s ,q s = δq δss . (8.50)
q
q ,s

Turning now to the quantum theory, we see that the scattered annihilation opera-
tors are given by
ak0 +q ,s = Sq s,q s ak0 +q ,s . (8.51)
P ,v

Since the eigenvalues of the operator a† aks represent the number of photons in the
ks
plane-wave mode fks , the operator representing the ¬‚ux of photons across a transverse
plane located to the left (z < 0) of the optical element is proportional to

a† 0 +q
F= ,s ak0 +q ,s , (8.52)
k
q ,s
¾
The beam splitter

and the operator representing the ¬‚ux through a plane to the right (z > 0) of the
optical element is

F= ak0 +q ,s ak0 +q ,s . (8.53)
q ,s

Combining eqn (8.51) with the unitarity condition (8.50) shows that the incident and
scattered ¬‚ux operators for a transparent optical element are identical, i.e. F = F .
This is a strong result, since it implies that all moments of the ¬‚uxes are identical,

Ψ |F n | Ψ = Ψ |F n | Ψ . (8.54)

In other words the overall statistical properties of the light, represented by the set of
all moments of the photon ¬‚ux, are unchanged by passage through a two-port paraxial
element, even though the distribution over transverse wavenumbers may be changed
by focussing.

8.4 The beam splitter
Beam splitters play an important role in many optical experiments as a method of
beam manipulation, and they also exemplify some of the most fundamental issues in
quantum optics. The simplest beam splitter is a uniform dielectric slab”such as the
one studied in Section 8.1”but in practice beam splitters are usually composed of
layered dielectrics, where the index of refraction of each layer is chosen to yield the
desired re¬‚ection and transmission coe¬cients r and t . The results of the single-slab
analysis are applicable to the layered design, provided that the correct values of r
and t are used. If the surrounding medium is the same on both sides of the device,
and the optical properties of the layers are symmetrical around the midplane, then
the amplitude re¬‚ection and transmission coe¬cients are the same for light incident
from either side. This de¬nes a symmetrical beam splitter. In order to simplify the
discussion, we will only deal with this case in the text. However, the unsymmetrical
beam splitter”which allows for more general phase relations between the incident and
scattered waves”is frequently used in practice (Zeilinger, 1981), and an example is
studied in Exercise 8.1.
In the typical experimental situation shown in Fig. 8.2, a classical wave,
±1 exp (ik1 · r), which is incident in channel 1, divides at the beam splitter into a




Fig. 8.2 A symmetrical beam splitter. The
surfaces 1, 2, 1 , and 2 are ports and the mode
amplitudes ±1 , ±2 , ±1 , and ±2 are related by
the scattering matrix.
¾ Linear optical devices

transmitted wave, ±1 exp (ik1 · r), in channel 1 and a re¬‚ected wave, ±2 exp (ik2 · r),
in channel 2 . In the time-reversed version of this event, channel 2 is an input channel
that scatters into the output channels 1 and 2, where channel 2 is associated with port
2 in the ¬gure. The two output channels in the time-reversed picture correspond to
input channels in the original picture; therefore, time-reversal invariance requires that
channel 2 be included as an input channel, in addition to the original channel 1. Thus
the beam splitter is a two-channel device, and the two output channels are related
to the two input channels by a 2 — 2 matrix. The beam splitter can also be described
as a four-port device, since there are two input ports and two output ports. In the
present book we restrict the term ˜beam splitter™ to devices that are described by the
scattering matrix in eqn (8.63), but in the literature this term is often applied to any
two-channel/four-port device described by a 2 — 2 unitary scattering matrix.
In the classical problem, there is no radiation in channel 2, so ±2 = 0, and port 2
is said to be an unused port. The transmitted and re¬‚ected amplitudes are then

±2 = r ±1 , ±1 = t±1 . (8.55)

The materials composing the beam splitter are chosen to have negligible absorption in
the wavelength range of interest, so the re¬‚ection and transmission coe¬cients must
satisfy eqn (8.7). Combining eqn (8.7) and eqn (8.55) yields the conservation of energy,
2 2
|±1 | + |±2 | = |±1 |2 . (8.56)

In many experiments the output ¬elds are measured by square law detectors that are
not phase sensitive. In this case the transmission phase θt can be eliminated by the
rede¬nition ±1 ’ ±1 exp (’iθt ), and the second line of eqn (8.7) means that we can
set r = ±it, where t is real and positive. The important special case of the balanced

(50/50) beam splitter is de¬ned by |r| = |t| = 1/ 2, and this yields the simple rule
±i 1
r= √ , t= √ . (8.57)
2 2
Beam splitters are an example of a general class of linear devices called optical
couplers”or optical taps”that split and redirect an input optical signal. In practice
optical couplers often consist of one or more waveguides, and the objective is achieved
by proper choice of the waveguide geometry. A large variety of optical couplers are in
use (Saleh and Teich, 1991, Sec. 7.3), but their fundamental properties are all very
similar to those of the beam splitter.

8.4.1 Quantum description of a beam splitter
A loose translation of the argument leading from the classical relation (8.16) to the
quantum relation (8.27) might be that classical amplitudes are simply replaced by
annihilation operators, according to the rules (8.24) and (8.26). In the present case,
this procedure would replace the c-number relations (8.55) by the operator relations

a2 = r a1 , a1 = t a1 ; (8.58)

consequently, the commutation relations for the scattered operators would be
¾
The beam splitter


a2 , a2† = |r|2 , a1 , a1† = |t|2 . (8.59)

These results are seriously wrong, since they imply a violation of Heisenberg™s uncer-
tainty principle for the scattered radiation oscillators. The source of this disaster is
the way we have translated the classical statement ˜no radiation enters through the
unused port 2™ to the quantum domain. The condition ±2 = 0 is perfectly sensible
in the classical problem, but in the quantum theory, eqn (8.59) amounts to claiming
that the operator a2 can be set to zero. This is inconsistent with the commutation
relation a2 , a† = 1, so the classical statement ±2 = 0 must instead be interpreted as
2
a condition on the state describing the incident ¬eld, i.e.

a2 |¦in = 0 (8.60)

for a pure state, and
a2 ρin = ρin a† = 0 (8.61)
2

for a mixed state. It is customary to describe this situation by saying that vacuum
¬‚uctuations in the mode k2 enter through the unused port 2. In other words, the correct
quantum calculation resembles a classical problem in which real incident radiation
enters through port 1 and mysterious vacuum ¬‚uctuations1 enter through port 2. In
this language, the statement ˜the operator a2 cannot be set to zero™ is replaced by
˜vacuum ¬‚uctuations cannot be prevented from entering through the unused port 2.™
Since we cannot impose a2 = 0, it is essential to use the general relation (8.27)
which yields
a1 a
=T 1 , (8.62)
a2 a2
where
tr
T= (8.63)
rt
is the scattering matrix for the beam splitter. The unitarity of T guarantees that the
scattered operators obey the canonical commutation relations, which in turn guarantee
the uncertainty principle.
We can see an immediate consequence of eqns (8.62) and (8.63) by evaluating the
number operators N2 = a2† a2 and N1 = a1† a1 . Now

N2 = r— a† + t— a† (r a1 + t a2 )
1 2

= |r| N1 + |t| N2 + r— t a† a2 + r t— a† a1 .
2 2
(8.64)
1 2

The corresponding formula for N1 is obtained by interchanging r and t:

N1 = |t| N1 + |r| N2 + r t— a† a2 + r— ta† a1 ,
2 2
(8.65)
1 2

and adding the two expressions gives
1 The
universal preference for this language may be regarded as sugar coating for the bitter pill of
quantum theory.
¾¼ Linear optical devices


N2 + N1 = N1 + N2 + (r— t + t r— ) a† a2 + a† a1 = N1 + N2 , (8.66)
1 2

where the Stokes relation (8.7) was used again. This is the operator version of the
conservation of energy, which in this case is the same as conservation of the number
of photons.
We now turn to the Schr¨dinger-picture description of scattering from the beam
o
splitter. In accord with the energy-conservation rule (8.13), the operators {a1 , a2 , a1 , a2 }
in eqn (8.62) all correspond to modes with a common frequency ω. We therefore begin
by considering single-frequency problems, i.e. all the incident photons have the same
frequency. For the beam splitter, the general operator scattering rule (8.40) reduces to

a† a† t a † + r a†
’T
1 1 1 2,
= (8.67)
a† a† r a1 + t a †

2 2 2

and to simplify things further we will only discuss two-photon initial states. With these
restrictions, the general input state in eqn (8.42) is replaced by
2 2
Cmn a† a† |0 .
|Ψ = (8.68)
mn
m=1 n=1

Since the creation operators commute with one another, the coe¬cients satisfy the
bosonic symmetry condition Cmn = Cnm .
A simple example”which will prove useful in Section 10.2.1”is a two-photon state
in which one photon enters through port 1 and another enters through port 2, i.e.

|Ψ = a† a† |0 . (8.69)
12

Applying the rule (8.67) to this initial state yields the scattered state

a†2 + a†2 |0 + r2 + t2 a† a† |0 .
|Ψ = r t (8.70)
1 2 12

Some interesting properties of this solution can be found in Exercise 8.2.
The simpli¬ed notation, am = akm sm , employed above is useful because the Heisen-
berg-picture scattering law (8.62) does not couple modes with di¬erent frequencies and
polarizations. The former property is a consequence of the energy conservation rule
(8.13) and the latter follows from the fact that the optically isotropic material of the
beam splitter does not change the polarization of the incident light. There are, however,
interesting experimental situations with initial states involving several frequencies and
more than one polarization state per channel. In these cases the simpli¬ed notation is
less useful, and it is better to identify the mth input channel solely with the direction
of propagation de¬ned by the unit vector km . Photons of either polarization and
any frequency can enter and leave through these channels. A notation suited to this
situation is
ω
ams (ω) = aqs with q = km , (8.71)
c
where m = 1, 2 is the channel index and s labels the two possible polarizations. For the
following discussion we will use a linear polarization basis eh km , ev km for each
¾½
The beam splitter

channel, where h and v respectively stand for horizontal and vertical. The frequency ω
can vary continuously, but for the present we will restrict the frequencies to a discrete
set. With all this understood, the canonical commutation relations are written as
ams (ω) , a† (ω ) = δmn δsr δωω , with m, n = 1, 2 and r, s = h, v , (8.72)
nr

and the operator scattering law (8.67)”which applies to each polarization and fre-
quency separately”becomes
a† (ω) t a† (ω) + r a† (ω)

1s 1s 2s . (8.73)
a† (ω) r a1s (ω) + t a† (ω)

2s 2s

Since the coe¬cients t and r depend on frequency, they should be written as t (ω) and
r (ω), but the simpli¬ed notation used in this equation is more commonly found in the
literature.
We will only consider two-photon initial states of the form
2
Cms,nr (ω, ω ) a† (ω) a† (ω ) |0 ,
|Ψ = (8.74)
ms nr
m,n=1 r,s ω,ω

where the sums over ω and ω run over some discrete set of frequencies, and the bosonic
symmetry condition is
Cnr,ms (ω , ω) = Cms,nr (ω, ω ) . (8.75)
Just as in nonrelativistic quantum mechanics, Bose symmetry applies only to the simul-
taneous exchange of all the degrees of freedom. Relaxing the simplifying assumption
that a single frequency and polarization are associated with all scattering channels
opens up many new possibilities.
In the ¬rst example”which will be useful in Section 10.2.1-B”the incoming
photons have the same polarization, but di¬erent frequencies ω1 and ω2 . In this
case the polarization index can be omitted, and the initial state expressed as |Ψ =
a† (ω1 ) a† (ω2 ) |0 . Applying the scattering law (8.67) to this state yields
1 2

|Ψ = t r a† (ω1 ) a† (ω2 ) + a† (ω1 ) a† (ω2 ) |0
1 1 2 2
(8.76)
t 2 a† (ω1 ) a† r2 a† (ω1 ) a† (ω2 ) |0 .
+ (ω2 ) +
1 2 2 1

This solution has a number of interesting features that are explored in Exercise 8.3.
An example of a single-frequency state with two polarizations present is
1
|Ψ = √ a† a† ’ a† a† |0 , (8.77)
2v 1v 2h
2 1h
where the frequency argument has been dropped. In this case the expansion coe¬cients
in eqn (8.74) reduce to
1
(δm1 δn2 ’ δn1 δm2 ) (δsh δrv ’ δrh δsv ) .
Cms,nr = (8.78)
4
The antisymmetry in the polarization indices r and s is analogous to the antisymmetric
spin wave function for the singlet state of a system composed of two spin-1/2 particles,
¾¾ Linear optical devices

so |Ψ is said to have a singlet-like character.2 The overall bosonic symmetry then
requires antisymmetry in the spatial degrees of freedom represented by (m, n). More
details can be found in Exercise 8.4.

8.4.2 Partition noise
The paraxial, single-channel/two-port devices discussed in Section 8.3 preserve the
statistical properties of the incident ¬eld. Let us now investigate this question for the
beam splitter. Combining the results (8.64) and (8.65) for the number operators of the
scattered modes with the condition (8.61) implies

2 2
N2 = Tr (ρin N2 ) = |r| N1 , N1 = |t| N1 . (8.79)

The intensity for each mode is proportional to the average of the corresponding number
2
operator, so the quantum averages reproduce the classical results, I2 = |r| I1 and
2
I1 = |r| I1 . There are no surprises for the average values, so we go on to consider
the statistical ¬‚uctuations in the incident and transmitted signals. This is done by
comparing the normalized variance,

2
N12 ’ N1
V (N1 )
V (N1 ) = = , (8.80)
2 2
N1 N1

of the transmitted ¬eld to the same quantity, V (N1 ), for the incident ¬eld. The cal-
culation of the transmitted variance involves evaluating N12 , which can be done by
combining eqn (8.65) with eqn (8.61) and using the cyclic invariance property of the
trace to get
N12 = |t|4 N1 + |r|2 |t|2 N1 .
2
(8.81)

Substituting this into the de¬nition of the normalized variance leads to

r 2 1
V (N1 ) = V (N1 ) + . (8.82)
t N1

Thus transmission through the beam splitter”by contrast to transmission through a
two-port device”increases the variance in photon number. In other words, the noise
in the transmitted ¬eld is greater than the noise in the incident ¬eld. Since the added
noise vanishes for r = 0, it evidently depends on the partition of the incident ¬eld into
transmitted and re¬‚ected components. It is therefore called partition noise.
Partition noise can be blamed on the vacuum ¬‚uctuations entering through the
unused port 2. This can be seen by temporarily modifying the commutation relation
for a2 to a2 , a† = ξ2 , where ξ2 is a c-number which will eventually be set to unity.
2
This is equivalent to modifying the canonical commutator to [q2 , p2 ] = i ξ2 , and this

2 The spin-statistics connection (Cohen-Tannoudji et al., 1977b, Sec. XIV-C) tells us that spin-1/2
particles must be fermions not bosons. This shows that analogies must be handled with care.
¾¿
The beam splitter

in turn yields the uncertainty relation ∆q2 ∆p2 ξ2 /2. Using this modi¬cation in the
previous calculation leads to
r 2 1
V (N1 ) = V (N1 ) + ξ2 . (8.83)
t N1
Thus partition noise can be attributed to the vacuum (zero-point) ¬‚uctuations of the
mode entering the unused port 2. Additional evidence that partition noise is entirely a
quantum e¬ect is provided by the fact that it becomes negligible in the classical limit,
N1 ’ ∞. Note that if we consider only the transmitted light, the transparent beam
splitter acts as if it were an absorber, i.e. a dissipative element. The increased noise in
the transmitted ¬eld is then an example of a general relation between dissipation and
¬‚uctuation which will be studied later.
8.4.3 Behavior of quasiclassical ¬elds at a beam splitter
We will now analyze an experiment in which a coherent (quasiclassical) state is incident
on port 1 of the beam splitter and no light is injected into port 2. The Heisenberg
state |¦in describing this situation satis¬es
a1 |¦in = ±1 |¦in ,
(8.84)
a2 |¦in = 0 ,
where ±1 is the amplitude of the coherent state. The scattering relation (8.62) combines
with these conditions to yield
a1 |¦in = (r a2 + t a1 ) |¦in = t ±1 |¦in ,
(8.85)
a2 |¦in = (t a2 + r a1 ) |¦in = r ±1 |¦in .
In other words, the Heisenberg state vector is also a coherent state with respect to a1
and a2 , with the respective amplitudes t ±1 and r ±1 . This means that the fundamental
condition (5.11) for a coherent state is satis¬ed for both output modes; that is,

V a1† , a1 = V a2† , a2 = 0 , (8.86)

where the variance is calculated for the incident state |¦in . This behavior is exactly
parallel to that of a classical ¬eld injected into port 1, so it provides further evidence
of the nearly classical nature of coherent states.
8.4.4 The polarizing beam splitter
The generic beam splitter considered above consists of a slab of optically isotropic
material, but for some purposes it is better to use anisotropic crystals. When light
falls on an anisotropic crystal, the two polarizations de¬ned by the crystal axes are
refracted at di¬erent angles. Devices employing this e¬ect are typically constructed
by cementing together two prisms made of uniaxial crystals. The relative orientation
of the crystal axes are chosen so that the corresponding polarization components of
the incident light are refracted at di¬erent angles. Devices of this kind are called
polarizing beam splitters (PBSs) (Saleh and Teich, 1991, Sec. 6.6). They provide
an excellent source for polarized light, and are also used to ensure that the two special
polarizations are emitted through di¬erent ports of the PBS.
¾ Linear optical devices

8.5 Y-junctions
In applications to communications, it is often necessary to split the signal so as to
send copies down di¬erent paths. The beam splitter discussed above can be used for
this purpose, but another optical coupler, the Y-junction, is often employed instead.
A schematic representation of a symmetric Y-junction is shown in Fig. 8.3, where the
waveguides denoted by the solid lines are typically realized by optical ¬bers in the
optical domain or conducting walls for microwaves.
The solid arrows in this sketch represent an input beam in channel 1 coupled to
output beams in channels 2 and 3. In the time-reversed version, an input beam (the
dashed arrow) in channel 3 couples to output beams in channels 1 and 2. Similarly, an
input beam in channel 2 couples to output beams in channels 1 and 3. Each output
beam in the time-reversed picture corresponds to an input beam in the original picture;
therefore, all three channels must be counted as input channels. The three input chan-
nels are coupled to three output channels, so the Y-junction is a three-channel device.
A strict application of the convention for counting ports introduced above requires us
to call this a six-port device, since there are three input ports (1, 2, 3) and three output
ports (1— , 2— , 3— ). This terminology is logically consistent, but it does not agree with
the standard usage, in which the Y-junction is called a three-port device (Kerns and
Beatty, 1967, Sec. 2.16). The source of this discrepancy is the fact that”by contrast
to the beam splitter”each channel of the Y-junction serves as both input and output
channel. In the sketch, the corresponding ports are shown separated for clarity, but it
is natural to have them occupy the same spatial location. The standard usage exploits
this degeneracy to reduce the port count from six to three.
Applying the argument used for the beam splitter to the Y-junction yields the
input“output relation
⎛⎞ ⎛⎞
a1 a1
⎝ a2 ⎠ = Y ⎝ a2 ⎠ , (8.87)
a3 a3

where Y is a 3 — 3 unitary matrix. When the matrix Y is symmetric”(Y )nm =

3



3*
1
Fig. 8.3 A symmetrical Y-junction. The in-
ward-directed solid arrow denotes a signal in-
jected into channel 1 which is coupled to the
output channels 2 and 3 as indicated by the
1*
outward-directed solid arrows. The dashed ar-
rows represent the time-reversed process. Ports
1, 2, and 3 are input ports and ports 1— , 2— , and 2
2*
3— are output ports.
¾
Isolators and circulators

(Y )mn ” the device is said to be reciprocal. In this case, the output at port n from
a unit signal injected into port m is the same as the output at port m from a unit
signal injected at port n.
For the symmetrical Y-junction considered here, the optical properties of the
medium occupying the junction itself and each of the three arms are assumed to
exhibit three-fold symmetry. In other words, the properties of the Y-junction are un-
changed by any permutation of the channel labels. In particular, this means that the
Y-junction is reciprocal. The three-fold symmetry reduces the number of independent
elements of Y from nine to two. One can, for example, set
⎡ ¤
y11 y12 y12
Y = ⎣y12 y11 y12 ¦ , (8.88)
y12 y12 y11

where
y11 = |y11 | eiθ11 , y12 = |y12 | eiθ12 . (8.89)
The unitarity conditions
|y11 |2 + 2 |y12 |2 = 1 , (8.90)
2 |y11 | cos (θ11 ’ θ12 ) + |y12 | = 0 (8.91)
relate the di¬erence between the re¬‚ection phase θ11 and the transmission phase θ12 to
the re¬‚ection and transmission coe¬cients |y11 |2 and |y12 |2 . The values of the two real
parameters left free, e.g. |y11 | and |y12 |, are determined by the optical properties of the
medium at the junction, the optical properties of the arms, and the locations of the
degenerate ports (1, 1— ), etc. For the symmetrical Y-junction, the unitarity conditions
place strong restrictions on the possible values of |y11 | and |y12 |, as seen in Exercise
8.5.
In common with the beam splitter, the Y-junction exhibits partition noise. For
an experiment in which the initial state has photons only in the input channel 1,
a calculation similar to the one for the beam splitter sketched in Section 8.4.2”see
Exercise 8.6”shows that the noise in the output signal is always greater than the
noise in the input signal. In the classical description of this experiment, there are no
input signals in channels 2 and 3; consequently, the input ports 2— and 3— are said
to be unused. Thus the partition noise can again be ascribed to vacuum ¬‚uctuations
entering through the unused ports.

8.6 Isolators and circulators
In this section we brie¬‚y describe two important and closely related devices: the optical
isolator and the optical circulator, both of which involve the use of a magnetic ¬eld.

8.6.1 Optical isolators
An optical isolator is a device that transmits light in only one direction. This prop-
erty is used to prevent re¬‚ected light from traveling upstream in a chain of optical
devices. In some applications, this feedback can interfere with the operation of the
light source. There are several ways to construct optical isolators (Saleh and Teich,
¾ Linear optical devices

1991, Sec. 6.6C), but we will only discuss a generally useful scheme that employs
Faraday rotation.
The optical properties of a transparent dielectric medium are changed by the pres-
ence of a static magnetic ¬eld B0 . The source of this change is the response of the
atomic electrons to the combined e¬ect of the propagating optical wave and the static
¬eld. Since every propagating ¬eld can be decomposed into a superposition of plane
waves, we will consider a single plane wave. The linearly-polarized electric ¬eld E of the
wave is an equal superposition of right- and left-circularly-polarized waves E + and E ’ ;
consequently, the electron velocity v”which to lowest order is proportional to E”can
be decomposed in the same way. This in turn implies that the velocity components
v+ and v’ experience di¬erent Lorentz forces ev+ — B0 and ev’ — B0 . This e¬ect
is largest when E and B0 are orthogonal, so we will consider that case. The index of
refraction of the medium is determined by the combination of the original wave with
the radiation emitted by the oscillating electrons; therefore, the two circular polar-
izations will have di¬erent indices of refraction, n+ and n’ . For a given polarization
s, the change in phase accumulated during propagation through a distance L in the
dielectric is 2πns L/», so the phase di¬erence between the two circular polarizations is
∆φ = (2π/») (n+ ’ n’ ) L, where » is the wavelength of the light. The superposition of
phase-shifted, right- and left-circularly-polarized waves describes a linearly-polarized
¬eld that is rotated through ∆φ relative to the incident ¬eld.
The rotation of the direction of polarization of linearly-polarized light propagating
along the direction of a static magnetic ¬eld is called the Faraday e¬ect (Landau
et al., 1984, Chap. XI, Section 101), and the combination of the dielectric with the
magnetic ¬eld is called a Faraday rotator. Experiments show that the rotation angle
∆φ for a single pass through a Faraday rotator of length L is proportional to the
strength of the magnetic ¬eld and to the length of the sample: ∆φ = V LB0 , where V
is the Verdet constant. Comparing the two expressions for ∆φ shows that the Verdet
constant is V = 2π (n+ ’ n’ ) / (»B0 ). For a positive Verdet constant the polarization
is rotated in the clockwise sense as seen by an observer looking along the propagation
direction k.
The Faraday rotator is made into an optical isolator by placing a linear polarizer
at the input face and a second linear polarizer, rotated by +45—¦ with respect to the
¬rst, at the output face. When the magnetic ¬eld strength is adjusted so that ∆φ =
45—¦ , the light transmitted through the input polarizer is also transmitted through the
output polarizer. On the other hand, light of the same wavelength and polarization
propagating in the opposite direction, e.g. the original light re¬‚ected from a mirror
placed beyond the output polarizer, will undergo a polarization rotation of ’45—¦, since
k has been replaced by ’k. This is a counterclockwise rotation, as seen when looking
along the reversed propagation direction ’k, so it is a clockwise rotation as seen from
the original propagation direction. Thus the counter-propagating light experiences a
further polarization rotation of +45—¦ with respect to the input polarizer. The light
reaching the input polarizer is therefore orthogonal to the allowed direction, and it
will not be transmitted. This is what makes the device an isolator; it only transmits
light propagating in the direction of the external magnetic ¬eld. This property has led
to the name optical diodes for such devices.
¾
Isolators and circulators

Instead of linear polarizers, one could as well use anisotropic, linearly polarizing,
single-mode optical ¬bers placed at the two ends of an isotropic glass ¬ber. If the
polarization axis of the output ¬ber is rotated by +45—¦ with respect to that of the
input ¬ber and an external magnetic ¬eld is applied to the intermediate ¬ber, then the
net e¬ect of this all-¬ber device is exactly the same, viz. that light will be transmitted
in only one direction.
It is instructive to describe the action of the isolator in the language of time reversal.
The time-reversal transformations (k, s) ’ (’k, s) for the wave, and B0 ’ ’B0
for the magnetic ¬eld, combine to yield ∆φ ’ ∆φ for the rotation angle. Thus the
time-reversed wave is rotated by +45—¦ clockwise. This is a counterclockwise rotation
(’45—¦ ) when viewed from the original propagation direction, so it cancels the +45—¦
rotation imposed on the incident ¬eld. This guarantees that the polarization of the
time-reversed ¬eld exactly matches the setting of the input polarizer, so that the
wave is transmitted. The transformation (k, s) ’ (’k, s) occurs automatically upon
re¬‚ection from a mirror, but the transformation B0 ’ ’B0 can only be achieved by
reversing the currents generating the magnetic ¬eld. This is not done in the operation
of the isolator, so the time-reversed ¬nal state of the ¬eld does not evolve into the time-
reversed initial state. This situation is described by saying that the external magnetic
¬eld violates time-reversal invariance. Alternatively, the presence of the magnetic ¬eld
in the dielectric is said to create a nonreciprocal medium.

8.6.2 Optical circulators
The beam splitter and the Y-junction can both be used to redirect beams of light,
but only at the cost of adding partition noise from the vacuum ¬‚uctuations entering
through an unused port. We will next study another device”the optical circulator,
shown in Fig. 8.4(a)”that can redirect and separate beams of light without adding
noise. This linear optical device employs the same physical principles as the older
microwave waveguide junction circulators discussed in Helszajn (1998, Chap. 1). As
shown in Fig. 8.4(a), the circulator has the physical con¬guration of a symmetric
Y-junction, with the addition of a cylindrical resonant cavity in the center of the
junction. The central part of the cavity in turn contains an optically transparent
ferromagnetic insulator”called a ferrite pill”with a magnetization (a permanent
internal DC magnetic ¬eld B0 ) parallel to the cavity axis and thus normal to the
plane of the Y-junction. In view of the connection to the microwave case, we will use
the conventional terminology in which this is called a three-port device. If the ferrite
pill is unmagnetized, this structure is simply a symmetric Y-junction, but we will see
that the presence of nonzero magnetization changes it into a nonreciprocal device.
The central resonant cavity supports circulating modes: clockwise (+)-modes, in
which the ¬eld energy ¬‚ows in a clockwise sense around the cavity, and counterclock-
wise (’)-modes, in which the energy ¬‚ows in the opposite sense (Jackson, 1999, Sec.
8.7). The (±)-modes both possess a transverse electric ¬eld E ± , i.e. a ¬eld lying in the
plane perpendicular to the cavity axis and therefore also perpendicular to the static
¬eld B0 . In the Faraday-e¬ect optical isolator the electromagnetic ¬eld propagates
along the direction of the static magnetic ¬eld B0 , which acts on the spin degrees of
freedom of the ¬eld by rotating the direction of polarization. By contrast, the ¬eld in
¾ Linear optical devices


(a) (b)
Port 3
Walls of
±
Port 3 Path
waveguide
(OUT)
C
Ferrite pill
Port 1
A
(magnetization Port 1
(IN)
out of page)

β
Path

Port 2
Port 2
(to and from
device)



Fig. 8.4 (a) A Y-junction circulator consists of a three-fold symmetric arrangement of three
ports with a ˜ferrite pill™ at the center. All the incoming wave energy is directed solely in an
anti-clockwise sense from port 1 to port 2, and all the wave energy coming out of port 2 is
directed solely into port 3, etc. (b) Magni¬ed view of central portion of (a). Wave energy can
only ¬‚ow around the ferrite pill in an anti-clockwise sense, since the clockwise energy ¬‚ow
from port 1 to port 3 is forbidden by the destructive interference at point C between paths
± and β (see text).

the circulator propagates around the cavity in a plane perpendicular to B0 , and the
polarization”i.e. the direction of the electric ¬eld”is ¬xed by the boundary condi-
tions. Despite these di¬erences, the underlying mechanism for the action of the static
magnetic ¬eld is the same. An electron velocity v has components v± proportional to
E ± , and the corresponding Lorentz forces v+ — B0 and v’ — B0 are di¬erent. This
means that the (+)- and (’)-modes experience di¬erent indices of refraction, n+ and
n’ ; consequently, they possess di¬erent resonant frequencies ωn,+ and ωn,’ . In the
absence of the static ¬eld B0 , time-reversal invariance requires ωn,+ = ωn,’ , since the
(+)- and (’)-modes are related by a time-reversal transformation. Thus the presence
of the magnetic ¬eld in the circulator violates time-reversal invariance, just as it does
for the Faraday-e¬ect isolator. There is, however, an important di¬erence between the
isolator and the circulator. In the circulator, the static ¬eld acts on the spatial mode
functions, i.e. on the orbital degrees of freedom of the traveling waves, as opposed to
acting on the spin (polarization) degrees of freedom.
The best way to continue this analysis would be to solve for the resonant cavity
modes in the presence of the static magnetic ¬eld. As a simpler alternative, we o¬er
a wave interference model that is based on the fact that the cavity radius Rc is large
compared to the optical wavelength. This argument”which comes close to violating
Einstein™s rule”begins with the observation that the cavity wall is approximately
straight on the wavelength scale, and continues by approximating the circulating mode
as a plane wave propagating along the wall. For ¬xed values of the material properties,
the available design parameters are the ¬eld strength B0 and the cavity radius Rc .
Our ¬rst task is to impedance match the cavity by ensuring that there are no
re¬‚ections from port 1, i.e. y11 = 0. A signal entering port 1 will couple to both of the
modes (+) and (’), which will each travel around the full circumference, Lc = 2πRc ,
¾
Isolators and circulators

of the cavity to arrive back at port 1. In our wave interference model this implies
y11 ∝ eiφ+ + eiφ’ , where φ± = n± (B0 ) k0 Lc and k0 = 2π/»0 . The condition for no
re¬‚ection is then
eiφ+ + eiφ’ = 0 or ei∆φ + 1 = 0 , (8.92)
where
∆φ = φ+ ’ φ’ = [n+ (B0 ) ’ n’ (B0 )] k0 Lc = ∆n (B0 ) k0 Lc . (8.93)
The impedance matching condition (8.92) is imposed by choosing the ¬eld strength
B0 and the circumference Lc to satisfy
∆n (B0 ) k0 Lc = ±π, ±3π, . . . . (8.94)
The three-fold symmetry of the circulator geometry then guarantees that y11 = y22 =
y33 = 0.
The second design step is to guarantee that a signal entering through port 1 will
exit entirely through port 2, i.e. that y31 = 0. For a weak static ¬eld, ∆n (B0 ) is a
linear function of B0 and
∆n (B0 )
n± (B0 ) = n0 ± , (8.95)
2
where n0 is the index of refraction at zero ¬eld strength. A signal entering through
port 1 at the point A will arrive at the point C, leading to port 3, in two ways. In the
¬rst way, the (+)-mode propagates along path ±. In the second way, the (’)-mode
propagates along the path β. Consequently, the matrix element y31 is proportional to
eiφ± + eiφβ , where
Lc Lc ∆n (B0 ) Lc
φ± = n+ (B0 ) k0 = n0 k0 + k0 (8.96)
3 3 2 3
and
2Lc 2Lc ∆n (B0 ) 2Lc

φβ = n’ (B0 ) k0 = n0 k0 k0 . (8.97)
3 3 2 3
= 0 is then imposed by requiring φβ ’ φ± to be an odd multiple of
The condition y31
π, i.e.
Lc ∆n (B0 )
’ k0 Lc = ±π, ±3π, . . . .
n0 k0 (8.98)
3 2
The two conditions (8.94) and (8.98) determine the values of Lc and B0 needed
to ensure that the device functions as a circulator. With the convention that the net
energy ¬‚ows along the shortest arc length from one port to the next, this device only
allows net energy ¬‚ow in the counterclockwise sense. Thus a signal entering port 1
can only exit at port 2, a signal entering port 3 can only exit at port 1, and a signal
entering through port 2 can only exit at port 3. The scattering matrix


001
C = ⎝1 0 0 ⎠ (8.99)
010
for the circulator is nonreciprocal but still unitary. By using the input“output relations
for this matrix, one can show”as in Exercise 8.7”that the noise in the output signal
is the same as the noise in the input signal.
¾¼ Linear optical devices

In one important application of the circulator, a wave entering the IN port 1 is
entirely transmitted”ideally without any loss”towards an active re¬‚ection device,
e.g. a re¬‚ecting ampli¬er, that is connected to port 2. The ampli¬ed and re¬‚ected
wave from the active re¬‚ection device is entirely transmitted”also without any loss”
to the OUT port 3. In this ideal situation the nonreciprocal action of the magnetic ¬eld
in the ferrite pill ensures that none of the ampli¬ed wave from the device connected to
port 2 can leak back into port 1. Furthermore, no accidental re¬‚ections from detectors
connected to port 3 can leak back into the re¬‚ection device. The same nonreciprocal
action prevents vacuum ¬‚uctuations entering the unused port 3 from adding to the
noise in channel 2.
In real devices conditions are never perfectly ideal, but the rejection ratio for wave
energies traveling in the forbidden direction of the circulator is quite high; for typical
optical circulators it is of the order of 30 dB, i.e. a factor of 1000. Moreover, the
transparent ferrite pill introduces very little dissipative loss (typically less than tenths
of a dB) for the allowed direction of the circulator. This means that the contribution
of vacuum ¬‚uctuations to the noise can typically be reduced also by a factor of 1000.
Fiber versions of optical circulators were ¬rst demonstrated by Mizumoto et al. (1990),
and ampli¬cation by optical parametric ampli¬ers connected to such circulators”
where the ampli¬er noise was reduced well below the standard quantum limit”was
demonstrated by Aytur and Kumar (1990).


8.7 Stops
An ancillary”but still important”linear device is a stop or iris, which is a small,
usually circular, aperture (pinhole) in an absorptive or re¬‚ective screen. Since the
stop only transmits a small portion of the incident beam, it can be used to eliminate
aberrations introduced by lenses or mirrors, or to reduce the number of transverse
modes in the incident ¬eld. This process is called beam cleanup or spatial ¬ltering.
The problem of transmission through a stop is not as simple as it might appear.
The only known exact treatment of di¬raction through an aperture is for the case of
a thin, perfectly conducting screen (Jackson, 1999, Sec. 10.7). The screen and stop
combination is clearly a two-port device, but the strong scattering of the incident ¬eld
by the screen means that it is not paraxial. It is possible to derive the entire plane-wave
scattering matrix from the known solution for the re¬‚ected and di¬racted ¬elds for a
general incident plane wave, but the calculations required are too cumbersome for our
present needs. The interesting quantum e¬ects can be demonstrated in a special case
that does not require the general classical solution.
In most practical applications the diameter of the stop is large compared to optical
wavelengths, so di¬raction e¬ects are not important, at least if the distance to the
detector is small compared to the Rayleigh range de¬ned by the stop area. By the
same token, the polarization of the incident wave will not be appreciably changed by
scattering. Thus the transmission through the stop is approximately described by ray
optics, and polarization can be ignored. If the coordinate system is chosen so that the
screen lies in the (x, y)-plane, then a plane wave propagating from z < 0 at normal
incidence, e.g. ±k exp (ikz), with k > 0, will scatter according to
¾½
Stops

±k exp (ikz) ’ ±k exp (ikz) + ±’k exp (’ikz) ,
(8.100)
±k = t ±k , ±’k = r ±k ,
where the amplitude transmission coe¬cient t is determined by the area of the stop.
This de¬nes the scattering matrix elements Sk,k = t and S’k,k = r. Performing this
calculation for a plane wave of the same frequency propagating in the opposite direction
(k < 0) yields S’k,’k = t and Sk,’k = r. In the limit of negligible di¬raction, the
counter-propagating waves exp (±ikz) can only scatter between themselves, so the
scattering matrix for this problem reduces to
t r
. (8.101)
S=
r t

Consequently, the coe¬cients automatically satisfy the conditions (8.7) which guaran-
tee the unitarity of S. This situation is sketched in Fig. 8.5.
In the classical description, the assumption of a plane wave incident from z < 0 is
imposed by setting ±’k = 0, so that P1 and P2 in Fig. 8.5 are respectively the input
and output ports. The explicit expression (8.101) and the general relation (8.16) yield
the scattered (transmitted and re¬‚ected) amplitudes as ±k = t ±k and ±’k = r ±k .
Warned by our experience with the beam splitter, we know that the no-input condition
and the scattering relations of the classical problem cannot be carried over into the
quantum theory as they stand. The appropriate translation of the classical assumption
±’k = 0 is to interpret it as a condition on the quantum ¬eld state. As a concrete
example, consider a source of light, of frequency ω = ωk , placed at the focal point of
a converging lens somewhere in the region z < 0. The light exits from the lens in the
plane-wave mode exp (ikz), and the most general state of the ¬eld for this situation is
described by a density matrix of the form

|n; k Pnm m; k| ,
ρin = (8.102)
nk ,mk =0


where |n; k = (n!)’1/2 a† |0 is a number state for photons in the mode exp (ikz).
n
k
The density operator ρin is evaluated in the Heisenberg picture, so the time-independent

coe¬cients satisfy the hermiticity condition, Pnm = Pmn , and the trace condition,

Pnn = 1 . (8.103)
n=0




Fig. 8.5 A stop of radius a ». The ar-
rows represent a normally incident plane wave
together with the re¬‚ected and transmitted
waves. The surfaces P1 and P2 are ports.
¾¾ Linear optical devices

Every one of the number states |n; k is the vacuum for a’k , therefore the density
matrix satis¬es
a’k ρin = ρin a† = 0 . (8.104)
’k
This is the quantum analogue of the classical condition ±’k = 0. Since we are not
allowed to impose a’k = 0, it is essential to use the general relation (8.27) which
yields
ak = t ak + r a’k ,
(8.105)
a’k = t a’k + r ak .
The unitarity of the matrix S in eqn (8.101) guarantees that the scattered operators
obey the canonical commutation relations.
Since each incident photon is randomly re¬‚ected or transmitted, partition noise is
to be expected for stops as well as for beam splitters. Just as for the beam splitter, the
additional ¬‚uctuation strength in the transmitted ¬eld is an example of the general
relation between dissipation and ¬‚uctuation. In this connection, we should mention
that the model of a stop as an aperture in a perfectly conducting, dissipationless screen
simpli¬es the analysis; but it is not a good description of real stops. In practice, stops
are usually black, i.e. apertures in an absorbing screen. The use of black stops reduces
unwanted stray re¬‚ections, which are often a source of experimental di¬culties. The
theory in this case is more complicated, since the absorption of the incident light
leads ¬rst to excitations in the atoms of the screen. These atomic excitations are
coupled in turn to lattice excitations in the solid material. Thus the transmitted ¬eld
for an absorbing stop will display additional noise, due to the partition between the
transmitted light and the excitations of the internal degrees of freedom of the absorbing
screen.

8.8 Exercises
8.1 Asymmetric beam splitters
For an asymmetric beam splitter, identify the upper (U ) and lower (L) surfaces as
those facing ports 1 and 2 respectively in Fig. 8.2. The general scattering relation is
a1 = tU a1 + rL a2 ,
a2 = rU a1 + tL a2 .
(1) Derive the conditions on the coe¬cients guaranteeing that the scattered operators
satisfy the canonical commutation relations.
(2) Model an asymmetric beam splitter by coating a symmetric beam splitter (coe¬-
cients r and t) with phase shifting materials on each side. Denote the phase shifts
for one transit of the coatings by ψU and ψL and derive the scattering relations.
Use your results to express tU , rL , rU , and tL in terms of ψU , ψL , r, and t, and
show that the conditions derived in part (1) are satis¬ed.
(3) Show that the phase shifts can be adjusted so that the scattering relations are


a1 = 1 ’ Ra1 ’ Ra2 ,
√ √
Ra1 + 1 ’ Ra2 ,
a2 =
¾¿
Exercises

2
where R = |r| is the re¬‚ectivity and = ±1. This form will prove useful in Section
20.5.3.

8.2 Single-frequency, two-photon state incident on a beam splitter
(1) Treat the coe¬cients Cmn in eqn (8.68) as a symmetric matrix and show that

C = SCS T ,

where S is given by eqn (8.63) and S T is its transpose.
√ √
(2) Evaluate eqn (8.70) for a balanced beam splitter (r = i/ 2, t = 1/ 2). If there
are detectors at both output ports, what can you say about the rate of coincidence
counting?
(3) Consider the initial state |Ψ = N0 cos θ a†2 + sin θ a†2 |0 .
1 2

(a) Evaluate the normalization constant N0 , calculate the matrices C and C , and
then calculate the scattered state |Ψ .
(b) For a balanced beam splitter, explain why the values θ = ±π/4 are especially
interesting.

8.3 Two-frequency state incident on a beam splitter
(1) For the initial state |Ψ = a† (ω1 ) a† (ω2 ) |0 , calculate the scattered state for the
1 2
case of a balanced beam splitter, and comment on the di¬erence between this
result and the one found in part (2) of Exercise 8.2.
(2) For the initial state |Ψ no photons of frequency ω2 are found in channel 1, but
they are present in the scattered solution. Where do they come from?
(3) According to the de¬nition in Section 6.5.3, the two states

1
|˜± (0) = √ a† (ω1 ) a† (ω2 ) ± a† (ω2 ) a† (ω1 ) |0
1 2 1 2
2

are dynamically entangled. Evaluate the scattered states for the case of a balanced
beam splitter, and compare the di¬erent experimental outcomes associated with
these examples and with the initial state |Ψ from part (1).

8.4 Two-polarization state falling on a beam splitter
Consider the initial state |Ψ de¬ned by eqn (8.77).
(1) Calculate the scattered state for a balanced beam splitter.
(2) Now calculate the scattered state for the alternative initial state

1
|Ψ = √ a† a† + a† a† |0 .
2v 1v 2h
2 1h

Comment on the di¬erence between the results.
¾ Linear optical devices

8.5 Symmetric Y-junction scattering matrix
Consider the symmetric Y-junction discussed in Section 8.5.
(1) Use the symmetry of the Y-junction to derive eqn (8.88).
(2) Evaluate the upper and lower bounds on |y11 | imposed by the unitarity condition
on Y .

8.6 Added noise at a Y-junction
Consider the case that photons are incident only in channel 1 of the symmetric Y-
junction.
(1) Verify conservation of average photon number, i.e. N1 + N2 + N3 = N1 .
(2) Evaluate the added noise in output channel 2 by expressing the normalized vari-
ance V (N2 ) in terms of the normalized variance V (N1 ) in the input channel 1.
What is the minimum value of the added noise?

8.7 The optical circulator
For a wave entering port 1 of the circulator depicted in Fig. 8.4(b), paths ± and β
lead to destructive interference at the mouth of port 3, under the choice of conditions
given by eqns (8.94) and (8.98).
(1) What conditions lead to constructive interference at the mouth of port 2?
(2) Show that the scattering matrix given by eqn (8.99) is unitary.
(3) Consider an experimental situation in which a perfect, lossless, retrore¬‚ecting
mirror terminates port 2. Show that the variance in photon number in the light
emitted through port 3 is exactly the same as the variance of the input light
entering through port 1.
9
Photon detection

Any experimental measurement sensitive to the discrete nature of photons evidently
requires a device that can detect photons one by one. For this purpose a single photon
must interact with a system of charged particles to induce a microscopic change, which
is subsequently ampli¬ed to the macroscopic level. The irreversible ampli¬cation stage
is needed to raise the quantum event to the classical level, so that it can be recorded.
This naturally suggests dividing the treatment of photon detection into several sec-
tions. In Section 9.1 we consider the process of primary detection of the incoming
photon or photons, and in Section 9.2 we study postdetection signal processing, in-
cluding the quantum methods of ampli¬cation of the primary photon event. Finally in
Section 9.3 we study the important techniques of heterodyne and homodyne detection.

9.1 Primary photon detection
In the ¬rst section below, we describe six physical mechanisms commonly employed
in the primary process of photon detection, and in the second section we present a
theoretical analysis of the simplest detection scheme, in which individual atoms are
excited by absorption of a single photon. The remaining sections are concerned with
the relation of incident photon statistics to the statistics of the ejected photoelectrons,
the ¬nite quantum e¬ciency of detectors, and some general statistical features of the
photon distribution.

9.1.1 Photon detection methods
Photon detection is currently based on one of the following physical mechanisms.
(1) Photoelectric detection. These detectors fall into two main categories:
(i) vacuum tube devices, in which the incident photon ejects an electron, bound
to a photocathode surface, into the vacuum;
(ii) solid-state devices, in which absorption of the incident photon deep within the
body of the semiconductor promotes an electron from the valence band to the
conduction band (Kittel, 1985).
In both cases the resulting output signal is proportional to the intensity of the
incident light, and thus to the time-averaged square of the electric ¬eld strength.
This method is, accordingly, also called square-law detection.
There are several classes of vacuum tube devices”for example, the photomultiplier
tubes and channeltrons described in Section 9.2.1”but most modern photoelectric
detectors are based on semiconductors. The promotion of an electron from the
valence band to the conduction band”which is analogous to photoionization of an
¾ Photon detection

atom”leaves behind a positively charged hole in the valence band. Both members
of the electron“hole pair are free to move through the material.
The energy needed for electron“hole pair production is substantially less than the
typical energy”of the order of electron volts”needed to eject a photoelectron
into the vacuum outside a metal surface; consequently, semiconductor devices can
detect much lower energy photons. Thus the sensitivity of semiconductor detectors
extends into the infrared and far-infrared parts of the electromagnetic spectrum.
Furthermore, the photon absorption length in the semiconductor material is so
small that relatively thin detectors will absorb almost all the incident photons. This
means that quantum e¬ciencies are high (50“90%). Semiconductor detectors are
very fast as well as very sensitive, with response times on the scale of nanoseconds.
These devices, which are very important for quantum optics, are also called single-
photon counters.
Solid-state detectors are further divided into two subcategories: photoconduc-
tive and photovoltaic. In photoconductive devices, the photoelectrons are re-
leased into a homogeneous semiconducting material, and a uniform internal elec-
tric ¬eld is applied across the material to accelerate the released photoelectrons.
Thus the current in the homogeneous material is proportional to the number of
photo-released carriers, and hence to the incident intensity of the light beam falling
on the semiconductor. In photovoltaic devices, photons are absorbed and photo-
electrons are released in a highly inhomogeneous region inside the semiconductor,
where there is a large internal electric ¬eld, viz., the depletion range inside a p“n
or p“i“n junction. The large internal ¬elds then accelerate the photoelectrons to
create a voltage across the junction, which can drive currents in an external cir-
cuit. Devices of this type are commonly known as photodiodes (Saleh and Teich,
1991, Chap. 17).
(2) Rectifying detection. The oscillating electric ¬eld of the electromagnetic wave
is recti¬ed, in a diode with a nonlinear I“V characteristic, to produce a direct-
current signal which is proportional to the intensity of the wave. The recti¬ca-
tion e¬ect arises from a physical asymmetry in the structure of the diode, for
example, at the p“n junction of a semiconductor diode device. Such detectors
include Schottky diodes, consisting of a small metallic contact on the surface
of a semiconductor, and biased superconducting“insulator“superconducting
(SIS) electron tunneling devices. These rectifying detectors are used mainly in the
radio and microwave regions of the electromagnetic spectrum, and are commonly
called square-law or direct detectors.
(3) Photothermal detection. Light is directly converted into heat by absorption,
and the resulting temperature rise of the absorber is measured. These detectors
are also called bolometers. Since thermal response times are relatively long,
these detectors are usually slower than many of the others. Nevertheless, they are
useful for detection of broad-bandwidth radiation, in experiments allowing long
integration times. Thus they are presently being used in the millimeter-wave and
far-infrared parts of the electromagnetic spectrum as detectors for astrophysical
measurements, including measurements of the anisotropy of the cosmic microwave
background (Richards, 1994).
¾
Primary photon detection

(4) Photon beam ampli¬ers. The incoming photon beam is coherently ampli¬ed
by a device such as a maser or a parametric ampli¬er. These devices are primarily
used in the millimeter-wave and microwave region of the electromagnetic spectrum,
and play the same role as the electronic pre-ampli¬ers used at radio frequencies.
Rather than providing postdetection ampli¬cation, they coherently pre-amplify
the incoming electromagnetic wave, by directly providing gain at the carrier fre-
quency. Examples include solid-state masers, which amplify the incoming signal
by stimulated emission of radiation (Gordon et al., 1954), and varactor parametric
ampli¬ers (paramps), where a pumped, nonlinear, reactive element”such as a
nonlinear capacitance of the depletion region in a back-biased p“n junction”can
amplify an incoming signal. The nonlinear reactance is modulated by a strong,
higher-frequency pump wave which beats with the signal wave to produce an idler
wave at the di¬erence frequency between the pump and signal frequencies. The
idler wave reacts back via the pump wave to produce more signal wave, etc. This
causes a mutual reinforcement, and hence ampli¬cation, of both the signal and
idler waves, at the expense of power in the pump wave. The idler wave power is
dumped into a matched termination.
(5) Single-microwave-photon counters. Single microwave photons in a supercon-
ducting microwave cavity are detected by using atomic beam techniques to pass
individual Rydberg atoms through the cavity. The microwave photon can cause
a transition between two high-lying levels (Rydberg levels) of a Rydberg atom,
which is subsequently probed by a state-selective ¬eld ionization process. The re-
sult of this measurement indicates whether a transition has occurred, and therefore
provides information about the state of excitation of the microwave cavity (Hulet
and Kleppner, 1983; Raushcenbeutal et al., 2000; Varcoe et al., 2000).
(6) Quantum nondemolition detectors. The presence of a single photon is de-
tected without destroying it in an absorption process. This detection relies on
the phase shift produced by the passage of a single photon through a nonlinear
medium, such as a Kerr medium. Such detectors have recently been implemented
in the laboratory (Yamamoto et al., 1986).
The last three of these detection schemes, (4) to (6), are especially promising for
quantum optics. However, all the basic mechanisms (1) through (3) can be extended,
by a number of important auxiliary methods, to provide photon detection at the
single-quantum level.

9.1.2 Theory of photoelectric detection
The theory presented here is formulated for the simplest case of excitation of free
atoms by the incident light, and it is solely concerned with the primary microscopic
detection event. In situations for which photon counting is relevant, the ¬elds are
weak; therefore, the response of the atoms can be calculated by ¬rst-order perturbation
theory. As we will see, the ¬rst-order perturbative expression for the counting rate is
the product of two factors. The ¬rst depends only on the state of the atom, and the
second depends only on the state of the ¬eld. This clean separation between properties
of the detector and properties of the ¬eld will hold for any detection scheme that can
¾ Photon detection

be described by ¬rst-order perturbation theory. Thus the use of the independent atom
model does not really restrict the generality of the results. In practice, the sensitivity
function describing the detector response is determined empirically, rather than being
calculated from ¬rst principles.
The primary objective of the theory is therefore to exhibit the information on the
state of the ¬eld that the counting rate provides. As we will see below, this information
is naturally presented in terms of the ¬eld“¬eld correlation functions de¬ned in Section
4.7. In a typical experiment, light from an external source, such as a laser, is injected
into a sample of some interesting medium and extracted through an output port. The
output light is then directed to the detectors by appropriate linear optical elements. An
elementary, but nonetheless important, point is that the correlation function associated
with a detector signal is necessarily evaluated at the detector, which is typically not
located in the interior of the sample being probed. Thus the correlation functions
evaluated in the interior of the sample, while of great theoretical interest, are not
directly related to the experimental results. Information about the interaction of the
light with the sample is e¬ectively stored in the state of the emitted radiation ¬eld,
which is used in the calculation of the correlation functions at the detectors. Thus
for the analysis of photon detection per se we only need to consider the interaction
of the electromagnetic ¬eld with the optical elements and the detectors. The total

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