±kT ±kI

= . (8.10)

rt

±kR ±kT R

The meaning of the conditions (8.7) is that the 2 — 2 scattering matrix in this equation

is unitary.

Having mastered the simplest possible optical elements, we proceed without hes-

itation to the general case of linear and nondissipative optical devices. The incident

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

√

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

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

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

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

the expansion coe¬cients of the scattered solution by Sk s ,ks , so that

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

ks

Repeating this procedure for all elements of the basis de¬nes the entire scattering

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

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

must satisfy

Sk s ,ks = 0 if ωk = ωk . (8.13)

In general, the sub-matrix connecting plane waves with a common frequency ωk = ω

will depend on ω.

¾½

Classical scattering

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

ωk

±ks fks (r) e’iωk t ,

(+)

E in (r, t) = i (8.14)

20

ks

where the time origin t = 0 is chosen so that the initial wave packet E in (r, 0) has not

reached the optical element. For t (> 0) su¬ciently large, the scattered wave packet

has passed through the optical element, so that it is again freely propagating. The

solution after the scattering is completely over is the out-¬eld

ωk

± fk s (r) e’iωk t ,

(+)

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

2 0 ks

ks

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

±k s = Sk s ,ks ±ks . (8.16)

ks

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

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

±T = T

S’ks,’k s ±’k s , (8.17)

’ks

ks

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

S’ks,’k s = Sk s ,ks , (8.18)

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

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

±T = ’±— , (8.19)

’ks ks

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

—

±ks = Sk s ,ks ±k s . (8.20)

ks

Combining this with eqn (8.16) leads to

—

±ks = Sk s ,ks Sk s ,k ±k , (8.21)

s s

ks ks

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

—

Sk s ,ks Sk s ,k = δkk δss , (8.22)

s

ks

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

passive linear device is described by a unitary scattering matrix.

¾¾ Linear optical devices

8.2 Quantum scattering

We will take a phenomenological approach in which the classical amplitudes are re-

placed by the Heisenberg-picture operators aks (t). Let t = 0 be the time at which

the Heisenberg and Schr¨dinger pictures coincide, then according to eqn (3.95) the

o

operator

aks (t) = aks (t) eiωk t (8.23)

is independent of time for free propagation. Thus in the scattering problem the time

dependence of ak s (t) comes entirely from the interaction between the ¬eld and the

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

so it is natural to replace them according to the rule

±ks ’ lim aks (t) eiωk t = aks (0) = aks . (8.24)

t’0

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

±k s ’ ak s = lim = lim {ak s (t)} ,

ak s (t) eiωk t

(8.25)

t’+∞ t’∞

implies the asymptotic ansatz

ak s (t) ’ ak s e’iωk t . (8.26)

At late times the ¬eld is propagating in vacuum, so this limit makes sense by virtue

of the fact that ak s (t) is time independent for free propagation.

Thus aks and ak s are respectively the incident and scattered annihilation op-

erators, and they will be linearly related in the weak-¬eld limit. Furthermore, the

correspondence principle tells us that the relation between the operators must repro-

duce eqn (8.16) in the classical limit aks ’ ±ks . Since both relations are linear, this

can only happen if the incident and scattered operators also satisfy

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

ks

where Sk s ,ks is the classical scattering matrix. The in-¬eld operator Ein and the

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

(8.15):

ωk

aks fks (r) e’iωk t ,

(+)

Ein (r, t) = i (8.28)

20

ks

ωk

ak s fk s (r) e’iωk t .

(+)

Eout (r, t) = i (8.29)

20

ks

The operators {aks } and {ak s } are related by eqn (8.27) and the inverse relation

S† —

aks = ak s = Sk s ,ks ak s . (8.30)

ks,k s

ks ks

The unitarity of the classical scattering matrix guarantees that the scattered operators

{ak s } satisfy the canonical commutation relations (3.65), provided that the incident

operators {aks } do so.

¾¿

Quantum scattering

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

classical and quantum scattering problems, but the Schr¨dinger-picture description

o

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

¬xed Heisenberg-picture state vector |Ψ is the initial state vector in the Schr¨dinger

o

picture, i.e. |Ψ (0) = |Ψ , so the time-dependent Schr¨dinger-picture state vector is

o

|Ψ (t) = U (t) |Ψ , (8.31)

where U (t) is the unitary evolution operator. Combining the formal solution (3.83) of

the Heisenberg operator equations with the ansatz (8.26) yields

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

which provides some asymptotic information about the evolution operator.

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

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

state,

Cks a† |0 .

|Ψ = (8.33)

ks

ks

The equivalence between the two pictures implies

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

where the left and right sides are evaluated in the Schr¨dinger and Heisenberg pictures

o

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

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

0 |aks | Ψ (t) a† |0 .

|Ψ (t) = (8.35)

ks

ks

The expansion coe¬cients 0 |aks | Ψ (t) are evaluated by combining eqn (8.34) with

the asymptotic rule (8.26) and the scattering law (8.27) to get 0 |aks (t)| Ψ =

e’iωk t Cks , where

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

ks

The evolved state is therefore

e’iωk t Cks a† |0 .

|Ψ (t) = (8.37)

ks

ks

In other words, the prescription for the asymptotic (t ’ ∞) form of the Schr¨dingero

’iωk t

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

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

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

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

ks in eqn (8.33) have the same frequency: ωk = ω0 . The energy conservation rule

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

¾ Linear optical devices

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

phase factor exp (’iω0 t). In this situation the overall phase can be neglected, and the

asymptotic evolution law (8.37) can be replaced by the scattering law

Cks a† |0 .

|Ψ ’ |Ψ = (8.38)

ks

ks

An equivalent way to describe the asymptotic evolution follows from the observa-

tion that the evolved state in eqn (8.37) is obtained from the initial state in eqn (8.33)

by the operator transformation

a† ’ e’iωk t a† s Sk s ,ks . (8.39)

ks k

ks

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

dropped to get the scattering rule

a† ’ aks =

†

a† s Sk s ,ks . (8.40)

ks k

ks

For scattering problems involving one- or two-photon initial states, it is often more

convenient to use eqn (8.40) directly rather than eqn (8.38). For example, the scattering

rule for |Ψ = a† |0 is

ks

a† |0 ’ aks |0 .

†

(8.41)

ks

The rule (8.39) also provides a simple derivation of the asymptotic evolution law

for multi-photon initial states. For the general n-photon initial state,

Ck1 s1 ,...,kn sn a† 1 s1 · · · a† n sn |0 ,

|Ψ = ··· (8.42)

k k

k1 s1 kn sn

applying eqn (8.39) to each creation operator yields

n

ωkm t Ck1 s1 ,...,kn sn a† 1 s1 · · · a† n sn |0 ,

|Ψ (t) = ··· exp ’i (8.43)

k k

k1 s1 kn sn m=1

where

··· Sk1 s1 ,p1 v1 · · · Skn sn ,pn vn Cp1 ν1 ,...,pn νn .

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

p1 v1 pn vn

For scattering problems the initial state is stationary, so that

n

ωkm = ω0 , (8.45)

m=1

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

† †

|Ψ ’ |Ψ = ··· Ck1 s1 ,...,kn sn ak1 s1 · · · akn sn |0 . (8.46)

k1 s1 kn sn

It is important to notice that the scattering matrix in eqn (8.27) has a special

property: it relates annihilation operators to annihilation operators only. The scattered

¾

Paraxial optical elements

annihilation operators do not depend at all on the incident creation operators. This

feature follows from the physical assumption that emission and absorption do not occur

in passive linear devices. The special form of the scattering matrix has an important

consequence for the commutation relations of ¬eld operators evaluated at di¬erent

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

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

(±) (±)

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

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

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

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

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

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

(+) (’)

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

even for free ¬elds or ¬elds evolving in passive linear media. Roughly speaking, this

implies that the creation of a photon at (r , t ) and the annihilation of a photon at

(r, t) are not independent events.

Putting all this together shows that we can use standard classical methods to cal-

culate the scattering matrix for a given device, and then use eqn (8.27) to relate the

annihilation operators for the incident and scattered modes. This apparently simple

prescription must be used with care, as we will see in the applications. The utility of

this approach arises partly from the fact that each scattering channel in the classi-

cal analysis can be associated with a port, i.e. a bounding surface through which a

well-de¬ned beam of light enters or leaves. Input and output ports are respectively

associated with input and output channels. The ports separate the interior of the de-

vice from the outside world, and thus allow a black box approach in which the device

is completely characterized by an input“output transfer function or scattering ma-

trix. The principle of time-reversal invariance imposes constraints on the number of

channels and ports and thus on the structure of the scattering matrix.

The simplest case is a one-channel device, i.e. there is one input channel and one

output channel. In this case the scattering is described by a 1—1 matrix, as in eqn (8.2).

This is more commonly called a two-port device, since there is one input port and one

output port. As an example, for an antire¬‚ection coated thin lens the incident light

occupies a single input channel, e.g. a paraxial Gaussian beam, and the transmitted

light occupies a single output channel. The lens is therefore a one-channel/two-port

device.

8.3 Paraxial optical elements

An optical element that transforms an incident paraxial ray bundle into another parax-

ial bundle will be called a paraxial optical element. The most familiar examples

are (ideal) lenses and mirrors. By contrast to the dielectric slab in Fig. 8.1, an ideal

lens transmits all of the incident light; no light is re¬‚ected or absorbed. Similarly an

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

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

¾ Linear optical devices

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

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

more layers of transparent dielectrics with refractive indices and thicknesses adjusted

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

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

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

waves su¬er destructive interference. In both cases the ideal limit can only be approx-

imated for a limited range of wavelengths and angles of incidence. Compound devices

made from paraxial elements are automatically paraxial.

For optical elements de¬ned by curved interfaces the calculation of the scattering

matrix in the plane-wave basis is rather involved. The classical theory of the interaction

of light with lenses and curved mirrors is more naturally described in terms of Gaussian

beams, as discussed in Section 7.4. In the absence of this detailed theory it is still

possible to derive a useful result by using the general properties of the scattering

matrix. We will simplify this discussion by means of an additional approximation. An

incident paraxial wave is a superposition of plane waves with wavevectors k = k0 + q,

where |q| k0 . According to eqns (7.7) and (7.9), the dispersion in qz = q·k0 and ω for

an incident paraxial wave is small, in the sense that ∆ω/ (c∆q ) ∼ ∆qz /∆q = O (θ),

where q = q’ q· k0 k0 is the part of q transverse to k0 and θ is the opening angle of

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

and planar, i.e.

ω0 iq ·r

(+)

E in (r, t) = ei(k0 z’ω0 t) .

i ±k +q ,s e0s e (8.48)

2 0V 0

q ,s

In the same spirit the scattering matrix will be approximated by

Sks,k s ≈ δkz k0 δkz k0 Sq , (8.49)

s,q s

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

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

condition (8.22) reduces to

—

Sq s ,q s Sq s ,q s = δq δss . (8.50)

q

q ,s

Turning now to the quantum theory, we see that the scattered annihilation opera-

tors are given by

ak0 +q ,s = Sq s,q s ak0 +q ,s . (8.51)

P ,v

Since the eigenvalues of the operator a† aks represent the number of photons in the

ks

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

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

a† 0 +q

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

k

q ,s

¾

The beam splitter

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

optical element is

†

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

q ,s

Combining eqn (8.51) with the unitarity condition (8.50) shows that the incident and

scattered ¬‚ux operators for a transparent optical element are identical, i.e. F = F .

This is a strong result, since it implies that all moments of the ¬‚uxes are identical,

Ψ |F n | Ψ = Ψ |F n | Ψ . (8.54)

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

all moments of the photon ¬‚ux, are unchanged by passage through a two-port paraxial