ńņš. 36 |

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 1ā— , 2ā— , and 2

2*

3ā— are output ports.

Ā¾

Isolators and circulators

(Y )mn ā” 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

ā” ā¤

y11 y12 y12

Y = ā£y12 y11 y12 ā¦ , (8.88)

y12 y12 y11

where

y11 = |y11 | eiĪø11 , y12 = |y12 | eiĪø12 . (8.89)

The unitarity conditions

|y11 |2 + 2 |y12 |2 = 1 , (8.90)

2 |y11 | cos (Īø11 ā’ Īø12 ) + |y12 | = 0 (8.91)

relate the diļ¬erence between the reļ¬‚ection phase Īø11 and the transmission phase Īø12 to

the reļ¬‚ection and transmission coeļ¬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, 1ā— ), 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 8.4.2ā”see

Exercise 8.6ā”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 2ā— and 3ā— are said

to be unused. Thus the partition noise can again be ascribed to vacuum ļ¬‚uctuations

entering through the unused ports.

8.6 Isolators and circulators

In this section we brieļ¬‚y describe two important and closely related devices: the optical

isolator and the optical circulator, both of which involve the use of a magnetic ļ¬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 reļ¬‚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,

Ā¾ 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 ļ¬eld B0 . The source of this change is the response of the

atomic electrons to the combined eļ¬ect of the propagating optical wave and the static

ļ¬eld. Since every propagating ļ¬eld can be decomposed into a superposition of plane

waves, we will consider a single plane wave. The linearly-polarized electric ļ¬eld E of the

wave is an equal superposition of right- and left-circularly-polarized waves E + and E ā’ ;

consequently, the electron velocity vā”which to lowest order is proportional to Eā”can

be decomposed in the same way. This in turn implies that the velocity components

v+ and vā’ experience diļ¬erent Lorentz forces ev+ Ć— B0 and evā’ Ć— B0 . This eļ¬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 diļ¬erent indices of refraction, n+ and nā’ . For a given polarization

s, the change in phase accumulated during propagation through a distance L in the

dielectric is 2Ļns L/Ī», so the phase diļ¬erence between the two circular polarizations is

āĻ = (2Ļ/Ī») (n+ ā’ nā’ ) L, where Ī» is the wavelength of the light. The superposition of

phase-shifted, right- and left-circularly-polarized waves describes a linearly-polarized

ļ¬eld that is rotated through āĻ relative to the incident ļ¬eld.

The rotation of the direction of polarization of linearly-polarized light propagating

along the direction of a static magnetic ļ¬eld is called the Faraday eļ¬ect (Landau

et al., 1984, Chap. XI, Section 101), and the combination of the dielectric with the

magnetic ļ¬eld is called a Faraday rotator. Experiments show that the rotation angle

āĻ for a single pass through a Faraday rotator of length L is proportional to the

strength of the magnetic ļ¬eld and to the length of the sample: āĻ = V LB0 , where V

is the Verdet constant. Comparing the two expressions for āĻ shows that the Verdet

constant is V = 2Ļ (n+ ā’ nā’ ) / (Ī»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 +45ā—¦ with respect to the

ļ¬rst, at the output face. When the magnetic ļ¬eld strength is adjusted so that āĻ =

45ā—¦ , 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 reļ¬‚ected from a mirror

placed beyond the output polarizer, will undergo a polarization rotation of ā’45ā—¦, since

k has been replaced by ā’k. This is a counterclockwise rotation, as seen when looking

along the reversed propagation direction ā’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 +45ā—¦ 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 ļ¬eld. This property has led

to the name optical diodes for such devices.

Ā¾

Isolators and circulators

Instead of linear polarizers, one could as well use anisotropic, linearly polarizing,

single-mode optical ļ¬bers placed at the two ends of an isotropic glass ļ¬ber. If the

polarization axis of the output ļ¬ber is rotated by +45ā—¦ with respect to that of the

input ļ¬ber and an external magnetic ļ¬eld is applied to the intermediate ļ¬ber, then the

net eļ¬ect of this all-ļ¬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) ā’ (ā’k, s) for the wave, and B0 ā’ ā’B0

for the magnetic ļ¬eld, combine to yield āĻ ā’ āĻ for the rotation angle. Thus the

time-reversed wave is rotated by +45ā—¦ clockwise. This is a counterclockwise rotation

(ā’45ā—¦ ) when viewed from the original propagation direction, so it cancels the +45ā—¦

rotation imposed on the incident ļ¬eld. This guarantees that the polarization of the

time-reversed ļ¬eld exactly matches the setting of the input polarizer, so that the

wave is transmitted. The transformation (k, s) ā’ (ā’k, s) occurs automatically upon

reļ¬‚ection from a mirror, but the transformation B0 ā’ ā’B0 can only be achieved by

reversing the currents generating the magnetic ļ¬eld. This is not done in the operation

of the isolator, so the time-reversed ļ¬nal state of the ļ¬eld does not evolve into the time-

reversed initial state. This situation is described by saying that the external magnetic

ļ¬eld violates time-reversal invariance. Alternatively, the presence of the magnetic ļ¬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 ļ¬‚uctuations entering

through an unused port. We will next study another deviceā”the optical circulator,

shown in Fig. 8.4(a)ā”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 conļ¬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 insulatorā”called a ferrite pillā”with a magnetization (a permanent

internal DC magnetic ļ¬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 ļ¬eld energy ļ¬‚ows in a clockwise sense around the cavity, and counterclock-

wise (ā’)-modes, in which the energy ļ¬‚ows in the opposite sense (Jackson, 1999, Sec.

8.7). The (Ā±)-modes both possess a transverse electric ļ¬eld E Ā± , i.e. a ļ¬eld lying in the

plane perpendicular to the cavity axis and therefore also perpendicular to the static

ļ¬eld B0 . In the Faraday-eļ¬ect optical isolator the electromagnetic ļ¬eld propagates

along the direction of the static magnetic ļ¬eld B0 , which acts on the spin degrees of

freedom of the ļ¬eld by rotating the direction of polarization. By contrast, the ļ¬eld in

Ā¾ Linear optical devices

(a) (b)

Port 3

Walls of

Ī±

Port 3 Path

waveguide

(OUT)

C

Ferrite pill

Port 1

A

(magnetization Port 1

(IN)

out of page)

Ī²

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 ā˜ferrite pillā™ 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) Magniļ¬ed view of central portion of (a). Wave energy can

only ļ¬‚ow around the ferrite pill in an anti-clockwise sense, since the clockwise energy ļ¬‚ow

from port 1 to port 3 is forbidden by the destructive interference at point C between paths

Ī± and Ī² (see text).

the circulator propagates around the cavity in a plane perpendicular to B0 , and the

polarizationā”i.e. the direction of the electric ļ¬eldā”is ļ¬xed by the boundary condi-

tions. Despite these diļ¬erences, the underlying mechanism for the action of the static

magnetic ļ¬eld is the same. An electron velocity v has components vĀ± proportional to

E Ā± , and the corresponding Lorentz forces v+ Ć— B0 and vā’ Ć— B0 are diļ¬erent. This

means that the (+)- and (ā’)-modes experience diļ¬erent indices of refraction, n+ and

nā’ ; consequently, they possess diļ¬erent resonant frequencies Ļn,+ and Ļn,ā’ . In the

absence of the static ļ¬eld B0 , time-reversal invariance requires Ļn,+ = Ļn,ā’ , since the

(+)- and (ā’)-modes are related by a time-reversal transformation. Thus the presence

of the magnetic ļ¬eld in the circulator violates time-reversal invariance, just as it does

for the Faraday-eļ¬ect isolator. There is, however, an important diļ¬erence between the

isolator and the circulator. In the circulator, the static ļ¬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 ļ¬eld. As a simpler alternative, we oļ¬er

a wave interference model that is based on the fact that the cavity radius Rc is large

compared to the optical wavelength. This argumentā”which comes close to violating

Einsteinā™s ruleā”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 ļ¬xed values of the material properties,

the available design parameters are the ļ¬eld strength B0 and the cavity radius Rc .

Our ļ¬rst task is to impedance match the cavity by ensuring that there are no

reļ¬‚ections from port 1, i.e. y11 = 0. A signal entering port 1 will couple to both of the

modes (+) and (ā’), which will each travel around the full circumference, Lc = 2ĻRc ,

Ā¾

Isolators and circulators

of the cavity to arrive back at port 1. In our wave interference model this implies

y11 ā eiĻ+ + eiĻā’ , where ĻĀ± = nĀ± (B0 ) k0 Lc and k0 = 2Ļ/Ī»0 . The condition for no

reļ¬‚ection is then

eiĻ+ + eiĻā’ = 0 or eiāĻ + 1 = 0 , (8.92)

where

āĻ = Ļ+ ā’ Ļā’ = [n+ (B0 ) ā’ nā’ (B0 )] k0 Lc = ān (B0 ) k0 Lc . (8.93)

The impedance matching condition (8.92) is imposed by choosing the ļ¬eld strength

B0 and the circumference Lc to satisfy

ān (B0 ) k0 Lc = Ā±Ļ, Ā±3Ļ, . . . . (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 ļ¬eld, ān (B0 ) is a

linear function of B0 and

ān (B0 )

nĀ± (B0 ) = n0 Ā± , (8.95)

2

where n0 is the index of refraction at zero ļ¬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

ļ¬rst way, the (+)-mode propagates along path Ī±. In the second way, the (ā’)-mode

propagates along the path Ī². Consequently, the matrix element y31 is proportional to

eiĻĪ± + eiĻĪ² , where

Lc Lc ān (B0 ) Lc

ĻĪ± = n+ (B0 ) k0 = n0 k0 + k0 (8.96)

3 3 2 3

and

2Lc 2Lc ān (B0 ) 2Lc

ā’

ĻĪ² = nā’ (B0 ) k0 = n0 k0 k0 . (8.97)

3 3 2 3

= 0 is then imposed by requiring ĻĪ² ā’ ĻĪ± to be an odd multiple of

The condition y31

Ļ, i.e.

Lc ān (B0 )

ā’ k0 Lc = Ā±Ļ, Ā±3Ļ, . . . .

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 ļ¬‚ows along the shortest arc length from one port to the next, this device only

allows net energy ļ¬‚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

ā

ā

001

C = ā1 0 0 ā (8.99)

010

for the circulator is nonreciprocal but still unitary. By using the inputā“output relations

for this matrix, one can showā”as in Exercise 8.7ā”that the noise in the output signal

is the same as the noise in the input signal.

Ā¾Ā¼ Linear optical devices

In one important application of the circulator, a wave entering the IN port 1 is

entirely transmittedā”ideally without any lossā”towards an active reļ¬‚ection device,

e.g. a reļ¬‚ecting ampliļ¬er, that is connected to port 2. The ampliļ¬ed and reļ¬‚ected

wave from the active reļ¬‚ection device is entirely transmittedā”also without any lossā”

to the OUT port 3. In this ideal situation the nonreciprocal action of the magnetic ļ¬eld

in the ferrite pill ensures that none of the ampliļ¬ed wave from the device connected to

port 2 can leak back into port 1. Furthermore, no accidental reļ¬‚ections from detectors

connected to port 3 can leak back into the reļ¬‚ection device. The same nonreciprocal

action prevents vacuum ļ¬‚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 ļ¬‚uctuations to the noise can typically be reduced also by a factor of 1000.

Fiber versions of optical circulators were ļ¬rst demonstrated by Mizumoto et al. (1990),

and ampliļ¬cation by optical parametric ampliļ¬ers connected to such circulatorsā”

where the ampliļ¬er noise was reduced well below the standard quantum limitā”was

demonstrated by Aytur and Kumar (1990).

8.7 Stops

An ancillaryā”but still importantā”linear device is a stop or iris, which is a small,

usually circular, aperture (pinhole) in an absorptive or reļ¬‚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 ļ¬eld. This process is called beam cleanup or spatial ļ¬ltering.

The problem of transmission through a stop is not as simple as it might appear.

The only known exact treatment of diļ¬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 ļ¬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 reļ¬‚ected and diļ¬racted ļ¬elds for a

general incident plane wave, but the calculations required are too cumbersome for our

present needs. The interesting quantum eļ¬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 diļ¬raction eļ¬ects are not important, at least if the distance to the

detector is small compared to the Rayleigh range deļ¬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. Ī±k exp (ikz), with k > 0, will scatter according to

Ā¾Ā½

Stops

Ī±k exp (ikz) ā’ Ī±k exp (ikz) + Ī±ā’k exp (ā’ikz) ,

(8.100)

Ī±k = t Ī±k , Ī±ā’k = r Ī±k ,

where the amplitude transmission coeļ¬cient t is determined by the area of the stop.

This deļ¬nes the scattering matrix elements Sk,k = t and Sā’k,k = r. Performing this

calculation for a plane wave of the same frequency propagating in the opposite direction

(k < 0) yields Sā’k,ā’k = t and Sk,ā’k = r. In the limit of negligible diļ¬raction, the

counter-propagating waves exp (Ā±ikz) can only scatter between themselves, so the

scattering matrix for this problem reduces to

t r

. (8.101)

S=

r t

Consequently, the coeļ¬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 Ī±ā’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 reļ¬‚ected) amplitudes as Ī±k = t Ī±k and Ī±ā’k = r Ī±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

Ī±ā’k = 0 is to interpret it as a condition on the quantum ļ¬eld state. As a concrete

example, consider a source of light, of frequency Ļ = Ļk , 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 ļ¬eld for this situation is

described by a density matrix of the form

ā

|n; k Pnm m; k| ,

Ļin = (8.102)

nk ,mk =0

where |n; k = (n!)ā’1/2 aā |0 is a number state for photons in the mode exp (ikz).

n

k

The density operator Ļin is evaluated in the Heisenberg picture, so the time-independent

ā—

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

ā

Pnn = 1 . (8.103)

n=0

Fig. 8.5 A stop of radius a Ī». The ar-

rows represent a normally incident plane wave

together with the reļ¬‚ected and transmitted

waves. The surfaces P1 and P2 are ports.

Ā¾Ā¾ Linear optical devices

Every one of the number states |n; k is the vacuum for aā’k , therefore the density

matrix satisļ¬es

aā’k Ļin = Ļin aā = 0 . (8.104)

ā’k

This is the quantum analogue of the classical condition Ī±ā’k = 0. Since we are not

allowed to impose aā’k = 0, it is essential to use the general relation (8.27) which

yields

ak = t ak + r aā’k ,

(8.105)

aā’k = t aā’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 reļ¬‚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 ļ¬‚uctuation strength in the transmitted ļ¬eld is an example of the general

relation between dissipation and ļ¬‚uctuation. In this connection, we should mention

that the model of a stop as an aperture in a perfectly conducting, dissipationless screen

simpliļ¬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 reļ¬‚ections, which are often a source of experimental diļ¬culties. The

theory in this case is more complicated, since the absorption of the incident light

leads ļ¬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 ļ¬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 coeļ¬cients guaranteeing that the scattered operators

satisfy the canonical commutation relations.

(2) Model an asymmetric beam splitter by coating a symmetric beam splitter (coeļ¬-

cients r and t) with phase shifting materials on each side. Denote the phase shifts

for one transit of the coatings by ĻU and ĻL and derive the scattering relations.

Use your results to express tU , rL , rU , and tL in terms of ĻU , ĻL , r, and t, and

show that the conditions derived in part (1) are satisļ¬ed.

ńņš. 36 |