ńņš. 10 |

imposing Ī±kT R = 0 leads to the constraints

|r|2 + |t|2 = 1 ,

(8.7)

r tā— + rā— t = 0 .

The ļ¬rst relation represents conservation of energy, while the second implies that

the transmitted part of ā’kR and the reļ¬‚ected part of ā’kT interfere destructively as

Ā¾Ā¼ 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 (iĪør ) and t = |t| exp (iĪøt ) in the second line of eqn (8.7) shows us

that time-reversal invariance imposes the relation

Īør ā’ Īøt = Ā±Ļ/2 ; (8.8)

in other words, the phase of the reļ¬‚ected wave is shifted by Ā±90ā—¦ relative to the

transmitted wave. This phase diļ¬erence is a measurable quantity; therefore, the Ā± sign

on the right side of eqn (8.8) is not a matter of convention. In fact, this sign determines

whether the reļ¬‚ected wave is retarded or advanced relative to the transmitted wave.

In the extreme limit of a perfect mirror, i.e. |t| ā’ 0, we can impose the convention

Īøt = 0, so that

Īør = Ā±Ļ/2 , |r| = 1 . (8.9)

For given values of the relevant parametersā”the angle of incidence, the index of re-

fraction of the dielectric, and the thickness of the slabā”the coeļ¬cients r and t can

be exactly calculated (Born and Wolf, 1980, Sec. 1.6.4, eqns (57) and (58)), and the

phases Īør and Īøt 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 Ī±kT R = 0.

The standard calculation then relates Ī±kT and Ī±kR to Ī±kI and Ī±kT R by

tr

Ī±kT Ī±kI

= . (8.10)

rt

Ī±kR Ī±kT R

The meaning of the conditions (8.7) is that the 2 Ć— 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

ļ¬eld is to be expressed as an expansion in box-quantized plane waves,

ā

fks (r) = eks exp (ik Ā· r) / V . (8.11)

For the single-mode input ļ¬eld E in = fks eā’iĻk t , the general piecing procedure yields an

output ļ¬eld which we symbolically denote by (fks )scat . This ļ¬eld is also expressed as

an expansion in box-quantized plane waves. For a given basis function fks , we denote

the expansion coeļ¬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 deļ¬nes the entire scattering

matrix Sk s ,ks . The assumption that the device is stationary means that the frequency

Ļk associated with the mode fks cannot be changed; therefore the scattering matrix

must satisfy

Sk s ,ks = 0 if Ļk = Ļk . (8.13)

In general, the sub-matrix connecting plane waves with a common frequency Ļk = Ļ

will depend on Ļ.

Ā¾Ā½

Classical scattering

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

Ļk

Ī±ks fks (r) eā’iĻk 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) suļ¬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-ļ¬eld

Ļk

Ī± fk s (r) eā’iĻk t ,

(+)

E out (r, t) = i (8.15)

2 0 ks

ks

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

Ī±k s = Sk s ,ks Ī±ks . (8.16)

ks

Time-reversal invariance can be exploited here as well. In the time-reversed prob-

lem, the time-reversed output ļ¬eld scatters into the time-reversed input ļ¬eld, so

Ī±T = T

Sā’ks,ā’k s Ī±ā’k s , (8.17)

ā’ks

ks

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

Sā’ks,ā’k s = Sk s ,ks , (8.18)

where the transposition of the indices reļ¬‚ects the interchange of incoming and outgoing

modes. The classical rule (see Appendix B.3.3) for time reversal is

Ī±T = ā’Ī±ā— , (8.19)

ā’ks ks

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

ā—

Ī±ks = Sk s ,ks Ī±k s . (8.20)

ks

Combining this with eqn (8.16) leads to

ā—

Ī±ks = Sk s ,ks Sk s ,k Ī±k , (8.21)

s s

ks ks

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

ā—

Sk s ,ks Sk s ,k = Ī“kk Ī“ss , (8.22)

s

ks

that are generalizations of eqn (8.7). In matrix form this is S ā S = SS ā = 1; i.e. every

passive linear device is described by a unitary scattering matrix.

Ā¾Ā¾ 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 SchrĀØdinger pictures coincide, then according to eqn (3.95) the

o

operator

aks (t) = aks (t) eiĻk 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 ļ¬eld and the

optical element. The classical amplitudes Ī±ks represent the solution prior to scattering,

so it is natural to replace them according to the rule

Ī±ks ā’ lim aks (t) eiĻk t = aks (0) = aks . (8.24)

tā’0

Similarly, Ī±k s represents the solution after scattering, and the corresponding rule,

Ī±k s ā’ ak s = lim = lim {ak s (t)} ,

ak s (t) eiĻk t

(8.25)

tā’+ā tā’ā

implies the asymptotic ansatz

ak s (t) ā’ ak s eā’iĻk t . (8.26)

At late times the ļ¬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-ļ¬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 ā’ Ī±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-ļ¬eld operator Ein and the

out-ļ¬eld operator Eout are given by the quantum analogues of eqns (8.14) and

(8.15):

Ļk

aks fks (r) eā’iĻk t ,

(+)

Ein (r, t) = i (8.28)

20

ks

Ļk

ak s fk s (r) eā’iĻk 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

Sā ā—

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.

Ā¾Āæ

Quantum scattering

The use of the Heisenberg picture nicely illustrates the close relation between the

classical and quantum scattering problems, but the SchrĀØdinger-picture description

o

of scattering phenomena is often more useful for the description of experiments. The

ļ¬xed Heisenberg-picture state vector |ĪØ is the initial state vector in the SchrĀØdinger

o

picture, i.e. |ĪØ (0) = |ĪØ , so the time-dependent SchrĀØdinger-picture state vector is

o

|ĪØ (t) = U (t) |ĪØ , (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 ā (t) aks U (t) ā’ aks eā’iĻk t as t ā’ ā , (8.32)

which provides some asymptotic information about the evolution operator.

The task at hand is to use this information to ļ¬nd the asymptotic form of |ĪØ (t) .

Since the scattering medium is linear, it is suļ¬cient to consider a one-photon initial

state,

Cks aā |0 .

|ĪØ = (8.33)

ks

ks

The equivalence between the two pictures implies

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

where the left and right sides are evaluated in the SchrĀØdinger and Heisenberg pictures

o

respectively. Since there is neither emission nor absorption in the passive scattering

medium, |ĪØ (t) remains a one-photon state at all times, and

0 |aks | ĪØ (t) aā |0 .

|ĪØ (t) = (8.35)

ks

ks

The expansion coeļ¬cients 0 |aks | ĪØ (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)| ĪØ =

eā’iĻk t Cks , where

Cks = Sks,k s Ck s . (8.36)

ks

The evolved state is therefore

eā’iĻk t Cks aā |0 .

|ĪØ (t) = (8.37)

ks

ks

In other words, the prescription for the asymptotic (t ā’ ā) form of the SchrĀØdingero

ā’iĻk t

state vector is simply to replace the initial coeļ¬cients Cks by e Cks , where Cks is

the transform of the initial coeļ¬cient vector by the scattering matrix.

In the standard formulation of scattering theory, the initial state is stationaryā”

i.e. an eigenstate of the free Hamiltonianā”in which case all terms in the sum over

ks in eqn (8.33) have the same frequency: Ļk = Ļ0 . The energy conservation rule

(8.13) guarantees that the same statement is true for the evolved state |ĪØ (t) , so the

Ā¾ Linear optical devices

time-dependent exponentials can be taken outside the sum in eqn (8.37) as the overall

phase factor exp (ā’iĻ0 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 aā |0 .

|ĪØ ā’ |ĪØ = (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

aā ā’ eā’iĻk t aā 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

aā ā’ aks =

ā

aā 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 |ĪØ = aā |0 is

ks

aā |0 ā’ aks |0 .

ā

(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 aā 1 s1 Ā· Ā· Ā· aā n sn |0 ,

|ĪØ = Ā·Ā·Ā· (8.42)

k k

k1 s1 kn sn

applying eqn (8.39) to each creation operator yields

n

Ļkm t Ck1 s1 ,...,kn sn aā 1 s1 Ā· Ā· Ā· aā n sn |0 ,

|ĪØ (t) = Ā·Ā·Ā· exp ā’i (8.43)

k k

k1 s1 kn sn m=1

where

Ā·Ā·Ā· Sk1 s1 ,p1 v1 Ā· Ā· Ā· Skn sn ,pn vn Cp1 Ī½1 ,...,pn Ī½n .

Ck1 s1 ,...,kn sn = (8.44)

p1 v1 pn vn

For scattering problems the initial state is stationary, so that

n

Ļkm = Ļ0 , (8.45)

m=1

and the evolution equation (8.43) is replaced by the scattering rule

ā ā

|ĪØ ā’ |ĪØ = Ā·Ā·Ā· Ck1 s1 ,...,kn sn ak1 s1 Ā· Ā· Ā· 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

Ā¾

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 ļ¬eld operators evaluated at diļ¬erent

times. Since all annihilation operatorsā”and therefore all creation operatorsā”commute

with one another, eqns (8.28), (8.29), and (8.27) imply

(Ā±) (Ā±)

Eout,i (r, +ā) , Ein,j (r , ā’ā) = 0 (8.47)

for scattering from a passive linear device. In fact, eqn (3.102) guarantees that the

positive- (negative-) frequency parts of the ļ¬eld at diļ¬erent ļ¬nite times commute, as

long as the evolution of the ļ¬eld operators is caused by interaction with a passive

linear medium. One should keep in mind that commutativity at diļ¬erent times is

not generally valid, e.g. if emission and absorption or photonā“photon scattering are

(+) (ā’)

possible, and further that commutators like Ei (r, t) , Ej (r , t ) do not vanish

even for free ļ¬elds or ļ¬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-deļ¬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 inputā“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 1Ć—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 antireļ¬‚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 reļ¬‚ected or absorbed. Similarly an

ideal mirror reļ¬‚ects all of the incident light; no light is transmitted or absorbed. In

the non-ideal world inhabited by experimentalists, the conditions deļ¬ning a paraxial

Ā¾ Linear optical devices

element must be approximated by clever design. The no-reļ¬‚ection limit for a lens is

approached by applying a suitable antireļ¬‚ection coating. This consists of one or

more layers of transparent dielectrics with refractive indices and thicknesses adjusted

so that the reļ¬‚ections from the various interfaces interfere destructively (Born and

Wolf, 1980, Sec. 1.6). An ideal mirror is essentially the opposite of an antireļ¬‚ection

coating; the parameters of the dielectric layers are chosen so that the transmitted

waves suļ¬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 deļ¬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 = qĀ·k0 and Ļ for

an incident paraxial wave is small, in the sense that āĻ/ (cāq ) ā¼ āqz /āq = O (Īø),

where q = qā’ qĀ· k0 k0 is the part of q transverse to k0 and Īø is the opening angle of

the beam. This suggests considering an incident classical ļ¬eld that is monochromatic

and planar, i.e.

Ļ0 iq Ā·r

(+)

E in (r, t) = ei(k0 zā’Ļ0 t) .

i Ī±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 ā Ī“kz k0 Ī“kz k0 Sq , (8.49)

s,q s

with the understanding that the reduced scattering matrix Sq s,q s eļ¬ectively con-

ļ¬nes q and q to the paraxial domain deļ¬ned by eqn (7.8). In this limit, the unitarity

condition (8.22) reduces to

ā—

Sq s ,q s Sq s ,q s = Ī“q Ī“ss . (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 aā aks represent the number of photons in the

ks

plane-wave mode fks , the operator representing the ļ¬‚ux of photons across a transverse

plane located to the left (z < 0) of the optical element is proportional to

aā 0 +q

F= ,s ak0 +q ,s , (8.52)

k

q ,s

Ā¾

The beam splitter

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

optical element is

ā

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 ļ¬‚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 ļ¬‚uxes are identical,

ĪØ |F n | ĪØ = ĪØ |F n | ĪØ . (8.54)

In other words the overall statistical properties of the light, represented by the set of

all moments of the photon ļ¬‚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 slabā”such as the

one studied in Section 8.1ā”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 reļ¬‚ection and transmission coeļ¬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 reļ¬‚ection and transmission coeļ¬cients are the same for light incident

from either side. This deļ¬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 splitterā”which allows for more general phase relations between the incident and

scattered wavesā”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,

Ī±1 exp (ik1 Ā· 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 Ī±1 , Ī±2 , Ī±1 , and Ī±2 are related by

the scattering matrix.

Ā¾ Linear optical devices

transmitted wave, Ī±1 exp (ik1 Ā· r), in channel 1 and a reļ¬‚ected wave, Ī±2 exp (ik2 Ā· 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 ļ¬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 Ć— 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 ā˜beam splitterā™ 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 Ć— 2 unitary scattering matrix.

In the classical problem, there is no radiation in channel 2, so Ī±2 = 0, and port 2

is said to be an unused port. The transmitted and reļ¬‚ected amplitudes are then

Ī±2 = r Ī±1 , Ī±1 = tĪ±1 . (8.55)

The materials composing the beam splitter are chosen to have negligible absorption in

the wavelength range of interest, so the reļ¬‚ection and transmission coeļ¬cients must

satisfy eqn (8.7). Combining eqn (8.7) and eqn (8.55) yields the conservation of energy,

2 2

|Ī±1 | + |Ī±2 | = |Ī±1 |2 . (8.56)

In many experiments the output ļ¬elds are measured by square law detectors that are

not phase sensitive. In this case the transmission phase Īøt can be eliminated by the

redeļ¬nition Ī±1 ā’ Ī±1 exp (ā’iĪøt ), and the second line of eqn (8.7) means that we can

set r = Ā±it, where t is real and positive. The important special case of the balanced

ā

(50/50) beam splitter is deļ¬ned by |r| = |t| = 1/ 2, and this yields the simple rule

Ā±i 1

r= ā , t= ā . (8.57)

2 2

Beam splitters are an example of a general class of linear devices called optical

couplersā”or optical tapsā”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

Ā¾

The beam splitter

a2 , a2ā = |r|2 , a1 , a1ā = |t|2 . (8.59)

These results are seriously wrong, since they imply a violation of Heisenbergā™s uncer-

tainty principle for the scattered radiation oscillators. The source of this disaster is

the way we have translated the classical statement ā˜no radiation enters through the

unused port 2ā™ to the quantum domain. The condition Ī±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 , aā = 1, so the classical statement Ī±2 = 0 must instead be interpreted as

2

a condition on the state describing the incident ļ¬eld, i.e.

a2 |Ī¦in = 0 (8.60)

for a pure state, and

a2 Ļin = Ļin aā = 0 (8.61)

2

for a mixed state. It is customary to describe this situation by saying that vacuum

ļ¬‚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 ļ¬‚uctuations1 enter through port 2. In

this language, the statement ā˜the operator a2 cannot be set to zeroā™ is replaced by

ā˜vacuum ļ¬‚uctuations cannot be prevented from entering through the unused port 2.ā™

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 = a2ā a2 and N1 = a1ā a1 . Now

N2 = rā— aā + tā— aā (r a1 + t a2 )

1 2

= |r| N1 + |t| N2 + rā— t aā a2 + r tā— aā 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 tā— aā a2 + rā— taā 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.

Ā¾Ā¼ Linear optical devices

N2 + N1 = N1 + N2 + (rā— t + t rā— ) aā a2 + aā 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 SchrĀØdinger-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 Ļ. 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

aā aā t a ā + r aā

ā’T

1 1 1 2,

= (8.67)

aā aā r a1 + t a ā

ā

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 aā aā |0 .

|ĪØ = (8.68)

mn

m=1 n=1

Since the creation operators commute with one another, the coeļ¬cients satisfy the

bosonic symmetry condition Cmn = Cnm .

A simple exampleā”which will prove useful in Section 10.2.1ā”is a two-photon state

in which one photon enters through port 1 and another enters through port 2, i.e.

|ĪØ = aā aā |0 . (8.69)

12

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

aā 2 + aā 2 |0 + r2 + t2 aā aā |0 .

|ĪØ = r t (8.70)

1 2 12

Some interesting properties of this solution can be found in Exercise 8.2.

The simpliļ¬ed notation, am = akm sm , employed above is useful because the Heisen-

berg-picture scattering law (8.62) does not couple modes with diļ¬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 simpliļ¬ed notation is

less useful, and it is better to identify the mth input channel solely with the direction

of propagation deļ¬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

Ļ

ams (Ļ) = 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

Ā¾Ā½

The beam splitter

channel, where h and v respectively stand for horizontal and vertical. The frequency Ļ

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 (Ļ) , aā (Ļ ) = Ī“mn Ī“sr Ī“ĻĻ , with m, n = 1, 2 and r, s = h, v , (8.72)

nr

and the operator scattering law (8.67)ā”which applies to each polarization and fre-

quency separatelyā”becomes

aā (Ļ) t aā (Ļ) + r aā (Ļ)

ā’

1s 1s 2s . (8.73)

aā (Ļ) r a1s (Ļ) + t aā (Ļ)

ā

2s 2s

Since the coeļ¬cients t and r depend on frequency, they should be written as t (Ļ) and

r (Ļ), but the simpliļ¬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 (Ļ, Ļ ) aā (Ļ) aā (Ļ ) |0 ,

|ĪØ = (8.74)

ms nr

m,n=1 r,s Ļ,Ļ

where the sums over Ļ and Ļ run over some discrete set of frequencies, and the bosonic

symmetry condition is

Cnr,ms (Ļ , Ļ) = Cms,nr (Ļ, Ļ ) . (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 ļ¬rst exampleā”which will be useful in Section 10.2.1-Bā”the incoming

photons have the same polarization, but diļ¬erent frequencies Ļ1 and Ļ2 . In this

case the polarization index can be omitted, and the initial state expressed as |ĪØ =

aā (Ļ1 ) aā (Ļ2 ) |0 . Applying the scattering law (8.67) to this state yields

1 2

|ĪØ = t r aā (Ļ1 ) aā (Ļ2 ) + aā (Ļ1 ) aā (Ļ2 ) |0

1 1 2 2

(8.76)

t 2 aā (Ļ1 ) aā r2 aā (Ļ1 ) aā (Ļ2 ) |0 .

+ (Ļ2 ) +

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

|ĪØ = ā aā aā ā’ aā aā |0 , (8.77)

2v 1v 2h

2 1h

where the frequency argument has been dropped. In this case the expansion coeļ¬cients

in eqn (8.74) reduce to

1

(Ī“m1 Ī“n2 ā’ Ī“n1 Ī“m2 ) (Ī“sh Ī“rv ā’ Ī“rh Ī“sv ) .

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,

Ā¾Ā¾ Linear optical devices

so |ĪØ 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 ļ¬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 (Ļin 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 ļ¬‚uctuations in the incident and transmitted signals. This is done by

comparing the normalized variance,

2

N12 ā’ N1

V (N1 )

V (N1 ) = = , (8.80)

2 2

N1 N1

of the transmitted ļ¬eld to the same quantity, V (N1 ), for the incident ļ¬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 deļ¬nition of the normalized variance leads to

r 2 1

V (N1 ) = V (N1 ) + . (8.82)

t N1

Thus transmission through the beam splitterā”by contrast to transmission through a

two-port deviceā”increases the variance in photon number. In other words, the noise

in the transmitted ļ¬eld is greater than the noise in the incident ļ¬eld. Since the added

noise vanishes for r = 0, it evidently depends on the partition of the incident ļ¬eld into

transmitted and reļ¬‚ected components. It is therefore called partition noise.

Partition noise can be blamed on the vacuum ļ¬‚uctuations entering through the

unused port 2. This can be seen by temporarily modifying the commutation relation

for a2 to a2 , aā = Ī¾2 , where Ī¾2 is a c-number which will eventually be set to unity.

2

This is equivalent to modifying the canonical commutator to [q2 , p2 ] = i Ī¾2 , 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.

Ā¾Āæ

The beam splitter

in turn yields the uncertainty relation āq2 āp2 Ī¾2 /2. Using this modiļ¬cation in the

previous calculation leads to

r 2 1

V (N1 ) = V (N1 ) + Ī¾2 . (8.83)

t N1

Thus partition noise can be attributed to the vacuum (zero-point) ļ¬‚uctuations of the

mode entering the unused port 2. Additional evidence that partition noise is entirely a

quantum eļ¬ect is provided by the fact that it becomes negligible in the classical limit,

N1 ā’ ā. 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 ļ¬eld is then an example of a general relation between dissipation and

ļ¬‚uctuation which will be studied later.

8.4.3 Behavior of quasiclassical ļ¬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 |Ī¦in describing this situation satisļ¬es

a1 |Ī¦in = Ī±1 |Ī¦in ,

(8.84)

a2 |Ī¦in = 0 ,

where Ī±1 is the amplitude of the coherent state. The scattering relation (8.62) combines

with these conditions to yield

a1 |Ī¦in = (r a2 + t a1 ) |Ī¦in = t Ī±1 |Ī¦in ,

(8.85)

a2 |Ī¦in = (t a2 + r a1 ) |Ī¦in = r Ī±1 |Ī¦in .

In other words, the Heisenberg state vector is also a coherent state with respect to a1

and a2 , with the respective amplitudes t Ī±1 and r Ī±1 . This means that the fundamental

condition (5.11) for a coherent state is satisļ¬ed for both output modes; that is,

V a1ā , a1 = V a2ā , a2 = 0 , (8.86)

where the variance is calculated for the incident state |Ī¦in . This behavior is exactly

parallel to that of a classical ļ¬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 deļ¬ned by the crystal axes are

refracted at diļ¬erent angles. Devices employing this eļ¬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 diļ¬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 diļ¬erent ports of the PBS.

Ā¾ 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 diļ¬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 ļ¬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 (1ā— , 2ā— , 3ā— ). 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 thatā”by contrast

to the beam splitterā”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

inputā“output relation

āā āā

a1 a1

ā a2 ā = Y ā a2 ā , (8.87)

a3 a3

where Y is a 3 Ć— 3 unitary matrix. When the matrix Y is symmetricā”(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 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.

(3) Show that the phase shifts can be adjusted so that the scattering relations are

ā

ā

a1 = 1 ā’ Ra1 ā’ Ra2 ,

ā ā

Ra1 + 1 ā’ Ra2 ,

a2 =

Ā¾Āæ

Exercises

2

where R = |r| is the reļ¬‚ectivity and = Ā±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 coeļ¬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.

ā ā

(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 |ĪØ = N0 cos Īø aā 2 + sin Īø aā 2 |0 .

1 2

(a) Evaluate the normalization constant N0 , calculate the matrices C and C , and

then calculate the scattered state |ĪØ .

(b) For a balanced beam splitter, explain why the values Īø = Ā±Ļ/4 are especially

interesting.

8.3 Two-frequency state incident on a beam splitter

(1) For the initial state |ĪØ = aā (Ļ1 ) aā (Ļ2 ) |0 , calculate the scattered state for the

1 2

case of a balanced beam splitter, and comment on the diļ¬erence between this

result and the one found in part (2) of Exercise 8.2.

(2) For the initial state |ĪØ no photons of frequency Ļ2 are found in channel 1, but

they are present in the scattered solution. Where do they come from?

(3) According to the deļ¬nition in Section 6.5.3, the two states

1

|Ī˜Ā± (0) = ā aā (Ļ1 ) aā (Ļ2 ) Ā± aā (Ļ2 ) aā (Ļ1 ) |0

1 2 1 2

2

are dynamically entangled. Evaluate the scattered states for the case of a balanced

beam splitter, and compare the diļ¬erent experimental outcomes associated with

these examples and with the initial state |ĪØ from part (1).

8.4 Two-polarization state falling on a beam splitter

Consider the initial state |ĪØ deļ¬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

|ĪØ = ā aā aā + aā aā |0 .

2v 1v 2h

2 1h

Comment on the diļ¬erence between the results.

Ā¾ 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 Ī± and Ī²

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, retroreļ¬‚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 ampliļ¬ed to the macroscopic level. The irreversible ampliļ¬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 ampliļ¬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 ļ¬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 ļ¬nite quantum eļ¬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 ļ¬eld strength.

This method is, accordingly, also called square-law detection.

There are several classes of vacuum tube devicesā”for example, the photomultiplier

tubes and channeltrons described in Section 9.2.1ā”but most modern photoelectric

detectors are based on semiconductors. The promotion of an electron from the

valence band to the conduction bandā”which is analogous to photoionization of an

Ā¾ Photon detection

atomā”leaves behind a positively charged hole in the valence band. Both members

of the electronā“hole pair are free to move through the material.

The energy needed for electronā“hole pair production is substantially less than the

typical energyā”of the order of electron voltsā”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 eļ¬ciencies are high (50ā“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 ļ¬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 ļ¬eld, viz., the depletion range inside a pā“n

or pā“iā“n junction. The large internal ļ¬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 ļ¬eld of the electromagnetic wave

is rectiļ¬ed, in a diode with a nonlinear Iā“V characteristic, to produce a direct-

current signal which is proportional to the intensity of the wave. The rectiļ¬ca-

tion eļ¬ect arises from a physical asymmetry in the structure of the diode, for

example, at the pā“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 superconductingā“insulatorā“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).

Ā¾

Primary photon detection

(4) Photon beam ampliļ¬ers. The incoming photon beam is coherently ampliļ¬ed

by a device such as a maser or a parametric ampliļ¬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-ampliļ¬ers used at radio frequencies.

Rather than providing postdetection ampliļ¬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

ampliļ¬ers (paramps), where a pumped, nonlinear, reactive elementā”such as a

nonlinear capacitance of the depletion region in a back-biased pā“n junctionā”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 diļ¬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 ampliļ¬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 ļ¬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 ļ¬elds are

weak; therefore, the response of the atoms can be calculated by ļ¬rst-order perturbation

theory. As we will see, the ļ¬rst-order perturbative expression for the counting rate is

the product of two factors. The ļ¬rst depends only on the state of the atom, and the

second depends only on the state of the ļ¬eld. This clean separation between properties

of the detector and properties of the ļ¬eld will hold for any detection scheme that can

Ā¾ Photon detection

be described by ļ¬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 ļ¬rst principles.

The primary objective of the theory is therefore to exhibit the information on the

state of the ļ¬eld that the counting rate provides. As we will see below, this information

is naturally presented in terms of the ļ¬eldā“ļ¬eld correlation functions deļ¬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 eļ¬ectively stored in the state of the emitted radiation ļ¬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 ļ¬eld with the optical elements and the detectors. The total

ńņš. 10 |