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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.

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