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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

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