’

δ bout (ω) = bout (ω) bout (ω)

—

†

e2iθ ’ 1 P (ω) C — (ω) bin (ω)

= 2 Re bin (’ω) .

(16.51)

Satisfying condition (i) would require the right side of this equation to vanish as an

identity in θ. The generic assumption (16.2) combined with the explicit forms of the

functions P (ω) and C (ω) makes this impossible; therefore, the ampli¬er is phase

sensitive.

This feature is a consequence of the fact that P (ω) and C (ω) are both nonzero, so

†

that the right side of eqn (16.46) depends jointly on bin (ω) and bin (’ω). A straight-

forward calculation shows that condition (ii) of Section 16.1.1 is also violated, even

for the simple case that the reservoir for the mirror M2 is the vacuum. Choosing an

appropriate de¬nition of the gain for a phase-sensitive ampli¬er is a bit trickier than

for the phase-insensitive cases, so this step will be postponed to the general treatment

in Section 16.4.

The alert reader will have noticed that the ampli¬ed signal is propagating back-

wards toward the source of the input signal. Devices of this kind are sometimes called

re¬‚ection ampli¬ers. This is not a useful feature for communications applications;

therefore, it is necessary to reverse the direction of the ampli¬er output so that it

propagates in the same direction as the input signal. Mirrors will not do for this task,

since they would interfere with the input. One solution is to redirect the ampli¬er

output by using an optical circulator, as described in Section 8.6. This device will

redirect the output signal, but it will not interfere with the input signal or add further

noise.

Linear optical ampli¬ers—

½¼

16.3 Traveling-wave ampli¬ers

The regenerative ampli¬ers discussed above enhance the nonlinear interaction for a

relatively weak cw pump beam by means of the resonant cavity formed by the mirrors

M1 and M2. This approach has the disadvantage of restricting the useful bandwidth to

that of the cavity. An alternative method is to remove the mirrors M1 and M2 to get

the con¬guration shown in Fig. 16.3, but this experimental simpli¬cation inevitably

comes at the expense of some theoretical complication.

The mirrors in the regenerative ampli¬ers perform two closely related functions.

The ¬rst is to guarantee that the ¬eld inside the cavity is a superposition of a discrete

set of cavity modes. In practice, the design parameters are chosen so that only one

cavity mode is excited. The position dependence of the ¬eld is then entirely given by

the corresponding mode function; in e¬ect, the cavity is a zero-dimensional system.

The second function”which follows from the ¬rst”is to justify the sample“reservoir

model that treats the discrete modes inside the cavity and the continuum of reservoir

modes outside the cavity as kinematically-independent degrees of freedom.

Removing the mirrors eliminates both of these conceptual simpli¬cations. Since

there are no discrete cavity modes, each of the continuum of external modes propagates

through the ampli¬er and interacts with the gain medium. Thus all ¬eld modes are

reservoir modes, and the sample consists of the atoms in the gain medium.

The interaction of the ¬eld with the gain medium could be treated by generalizing

the scattering description of passive, linear devices developed in Section 8.2, but this

approach would be quite complicated in the present application. The fact that the

sample occupies a ¬xed interval, say 0 z LS , along the propagation (z) axis

violates translation invariance and therefore conservation of momentum. Consequently,

the scattering matrix for the ampli¬er connects each incident plane wave, exp (ikz),

to a continuum of scattered waves exp (ik z).

We will avoid this complication by employing a position“space approach that

closely resembles the classical theory of parametric ampli¬cation (Yariv, 1989, Chap.

17). This technique can also be regarded as the Heisenberg-picture version of a method

developed to treat squeezing in a traveling-wave con¬guration (Deutsch and Garrison,

1991b).

16.3.1 Laser ampli¬er

As a concrete example, we consider a sample composed of a collection of three-level

atoms”with the level structure displayed in Fig. 16.4”which is made into a gain

Fig. 16.3 A black box schematic of a travel-

ing-wave ampli¬er. The shaded box indicates

the gain medium and the ¬elds at the two ports

are the input and output values of the signal.

The vacuum ¬‚uctuations entering port 2 are

not indicated, since they do not couple to the

signal.

½½

Traveling-wave ampli¬ers

3

ω32

Fig. 16.4 A three-level atom with a popula-

2

42 tion inversion between levels 1 and 2, main-

tained by an incoherent pump (dark double ar-

Signal

ω21 ω31 row) with rate RP . The solid arrow, the dashed

arrow, and the wavy arrows respectively rep-

resent the ampli¬ed signal transition, a nonra-

1 diative decay, and spontaneous emission.

medium by maintaining a population inversion between levels 1 and 2 through the

use of the incoherent pumping mechanism described in Section 14.5. By virtue of

the cylindrical shape of the gain medium, the end-¬re modes”i.e. ¬eld modes with

frequencies ω ω21 and propagation vectors, k, lying in a narrow cone around the

axis of the cylinder”will be preferentially ampli¬ed.

This new feature requires a modi¬cation of the reservoir assignment used for the

pumping calculation. The noise reservoir previously associated with the spontaneous

emission 2 ’ 1 is replaced by two reservoirs: (1) a noise reservoir associated with

spontaneous emission into modes with propagation vectors outside the end-¬re cone;

and (2) a signal reservoir associated with the end-¬re modes.

In the undepleted pump approximation, the back action of the atoms on the pump

¬eld can be ignored. This certainly cannot be done for the interaction with the signal

reservoir; after all, the action of the gain medium on the signal is the whole purpose of

the device. Thus the coupling of the entire collection of atoms to the signal reservoir

must be included by using the interaction Hamiltonian

(n)

HS1 = ’ S21 (t) d21 · E(+) (rn , t) + HC , (16.52)

n

where the sum runs over the atoms in the sample and the coordinate, rn , of the nth

atom is treated classically.

The description of the signal reservoir given above amounts to the assumption that

the Heisenberg-picture density operator for the input signal is a paraxial state with

respect to the z-axis; consequently, the contribution of the end-¬re modes to the ¬eld

operator can be represented in terms of the slowly-varying envelope operators φs (r, t)

appearing in eqn (7.33). We will assume that the ampli¬er has been designed so that

only one polarization will be ampli¬ed; consequently, only one operator φ (r, t) will be

needed.

Turning next to the input signal, we recall that a paraxial state is characterized

by transverse and longitudinal length scales Λ = 1/ (θk0 ) and Λ = 1/ θ2 k0 re-

spectively, where θ is the opening angle of the paraxial ray bundle. The scale lengths

Λ and Λ correspond respectively to the spot size and Rayleigh range of a classical

Gaussian beam. We choose θ so that Λ > 2RS and Λ LS , where RS and LS are

respectively the radius and length of the cylinder. This allows a further simpli¬cation

in which di¬raction is ignored and the envelope operator is approximated by

Linear optical ampli¬ers—

½¾

1

φ (r, t) = √ φ (z, t) , (16.53)

σ

2

where σ = πRS . In this 1D approximation, the ¬eld expansion (7.33) and the commu-

tation relation (7.35) are respectively replaced by

ω0 (vg0 /c)

E(+) (r, t) = i e0 φ (z, t) ei(k0 z’ω0 t) (16.54)

2 0 n0 σ

and

φ (z, t) , φ† (z , t) = δ (z ’ z ) . (16.55)

The discretely distributed atoms and the continuous ¬eld are placed on a more

even footing by introducing the spatially coarse-grained operator density

1

Sqp (t) χ (z ’ zn ) .

(n)

Sqp (z, t) = (16.56)

∆z n

The averaging interval ∆z is chosen to satisfy the following two conditions. (1) A slab

with volume σ∆z contains many atoms. (2) The envelope operator φ (z, t) is essentially

constant over an interval of length ∆z. The function

χ (z ’ zn ) = θ (∆z/2 ’ z + zn ) θ (z ’ zn + ∆z/2) (16.57)

serves to con¬ne the n-sum to the atoms in a slab of thickness ∆z centered at z. The

atomic envelope operators are de¬ned by

Sqp (z, t) = S qp (z, t) eiωqp t ei[ψq (z,t)’ψp (z,t)] , (16.58)

where the phases satisfy

ψ2 (z, t) ’ ψ1 (z, t) = ∆0 t ’ k0 z . (16.59)

Using this notation, together with eqn (16.54), allows us to rewrite eqn (16.52) as

LS

HS1 = ’i dz f S 21 (z, t) φ (z, t) ’ HC , (16.60)

0

where

(vg0 /c) ω0 d21 · e0

f≡ (16.61)

2 0σ

is the coupling constant.

The total electromagnetic part of the Hamiltonian for this 1D model is, therefore,

∞

dzφ† (z, t) vg0 ∇z φ (z, t) + HS1 .

Hem = (16.62)

i

’∞

This leads to the corresponding Heisenberg equation

‚ ‚

φ (z, t) = f— S 12 (z, t) for 0

+ vg0 z LS , (16.63)

‚t ‚z

‚ ‚

+ vg0 φ (z, t) = 0 for z < 0 or z > LS (16.64)

‚t ‚z

for the ¬eld.

½¿

Traveling-wave ampli¬ers

The atomic operators are coupled to the reservoirs describing the incoherent pump

and spontaneous emission into o¬-axis modes; therefore, we insert eqn (16.60) into the

coarse-grained version of eqn (14.177) to ¬nd

d

S 12 (z, t) = [i∆0 ’ “12 ] S 12 (z, t) ’ f S 11 (z, t) ’ S 22 (z, t) φ (z, t) + ξ12 (z, t) .

dt

(16.65)

The coarse-grained noise operator

1 (n)

ξ12 (t) χ (z ’ zn )

ξ12 (z, t) = (16.66)

∆z n

has the correlation function

†

ξ12 (z, t) ξ12 (z , t ) = nat σC12,12 δ (t ’ t ) δ (z ’ z ) , (16.67)

where δ (z ’ z ) is a coarse-grained delta function, nat is the density of atoms, and

C12,12 is an element of the noise correlation matrix discussed in Section 14.6.2.

In the strong-pump limit, the dephasing rate “12 = (w21 + RP ) /2 is large com-

pared to the other terms in eqn (16.65); therefore, applying the adiabatic elimination

rule (11.187) provides the approximate solution

S 22 (z, t) ’ S 11 (z, t) ξ12 (z, t)

S 12 (z, t) = f φ (z, t) + . (16.68)

“12 ’ i∆0 “12 ’ i∆0

We have to warn the reader that this procedure is something of a swindle, since

ξ12 (z, t) is not a slowly-varying function of t. Fortunately, the result can be justi¬ed”

see Exercise 16.3”by interpreting δ (t ’ t ) in eqn (16.67) as an even coarser-grained

delta function, that only acts on test functions that vary slowly on the dephasing time

scale T12 = 1/“12 .

In the linear approximation for eqn (16.68), the operator S 22 (z, t) ’ S 11 (z, t) can

be simpli¬ed in two ways. The ¬rst is to neglect the small quantum ¬‚uctuations,

i.e. to replace the operator by its average S 22 (z, t) ’ S 11 (z, t) . The next step is to

solve the averaged form of the operator Bloch equations (14.174)“(14.177), with the

approximation that HS1 = 0. The result is

S 22 (z, t) ’ S 11 (z, t) ≈ S 22 (z, t) ’ S 11 (z, t) = nat σD , (16.69)

where D is the steady-state inversion for a single atom. With these approximations,

the propagation equation (16.63) becomes

f—

2

|f| nat σD

‚ ‚

+ vg0 φ (z, t) = φ (z, t) + ξ12 (z, t) . (16.70)

“12 ’ i∆0 “12 ’ i∆0

‚t ‚z

This equation is readily solved by transforming to the wave coordinates:

„ = t ’ z/vg0 (the retarded time for the signal wave) ,

(16.71)

z = z,

Linear optical ampli¬ers—

½

to get

f—

‚

φ (z, „ ) = gφ (z, „ ) + ξ12 (z, „ ) , (16.72)

vg0 (“12 ’ i∆0 )

‚z

where

|f|2 nat σD k0 |d21 · e0 |2 nat D

g= = (16.73)

vg0 [“12 ’ i∆0 ] 2 0 [“12 ’ i∆0 ]

is the (complex) small-signal gain. The retarded time „ can be treated as a parameter

in eqn (16.72), so the solution is

z

f—

gz

dz1 eg(z’z1 ) ξ12 (z1 , „ ) ,

φ (z, „ ) = φ (0, „ ) e + (16.74)

vg0 (“12 ’ i∆0 ) 0

which has the form

f— z ’ z1

z

φ (z, t) = φ (0, t ’ z/vg0 ) e dz1 eg(z’z1 ) ξ12 z1 , t ’

gz

+

vg0 (“12 ’ i∆0 ) vg0

0

(16.75)

in the laboratory coordinates (z, t).

Setting z = LS and letting t ’ t + LS /vg0 gives the ¬eld value at the output face:

f— LS

z1

gLS

dz1 eg(z’z1 ) ξ12 z1 , t +

φ (LS , t + LS /vg0 ) = φ (0, t) e + .

vg0 (“12 ’ i∆0 ) vg0

0

(16.76)

In order to recover the standard form for input“output relations we introduce the

representations

1 dω

bin (ω) e’iω(t’z/vg0 ) for z < 0

φ (z, t) = √ (16.77)

vg0 2π

and

1 dω

bout (ω) e’iω(t’z/vg0 ) for z > LS ,

φ (z, t) = √ (16.78)

vg0 2π

√

for the solutions of eqn (16.64) outside the crystal. The factor 1/ vg0 is inserted to

guarantee that the commutation relation (16.55) for φ (z, t) and the standard input“

output commutation relations

bγ (ω) , b† (ω ) = 2πδ (ω ’ ω ) (γ = in, out) (16.79)

γ

are both satis¬ed. Substituting eqns (16.77) and (16.78) into eqn (16.76) and carrying

out a Fourier transform produces the input“output relation

bout (ω) = egLS bin (ω) + · (ω) , (16.80)

where

f— LS

dz1 eg(z’z1 ) e’iωz1 /vg0 ξ12 (z1 , ω)

· (ω) = √ (16.81)

vg0 (“12 ’ i∆0 ) 0

½

Traveling-wave ampli¬ers

is the ampli¬er noise operator. By using the frequency-domain form of eqn (16.67),

one can show that the noise correlation function is

1

· (ω) · † (ω ) + · † (ω) · (ω )

Kamp (ω, ω ) =

2

= Namp 2πδ (ω ’ ω ) , (16.82)

where the noise strength is

2

nat k0 |d21 · e0 | e2gLS ’ 1 1

Namp = (C12,12 + C21,21 ) . (16.83)

2 0 (“2 + ∆2 ) 2g 2

12 0

Comparing eqn (16.80) to eqn (16.46) shows that P (ω) = egLS and C (ω) = 0. Con-

sequently, this ampli¬er is phase insensitive and phase transmitting.

16.3.2 Traveling-wave OPA

For this example, we return to the down-conversion technique by removing the mir-

rors from the phase-sensitive design shown in Fig. 16.2. Even without the mirrors,

appropriately cutting the ends of the crystal will guarantee that the pump beam and

the degenerate signal and idler beams copropagate along the length of the crystal. We

assume a Gaussian pump beam with spot size w0 and Rayleigh range ZR focussed on

a nonlinear crystal with radius RS and length LS .

If w0 > 2RS and ZR LS , the e¬ects of di¬raction are negligible; consequently,

the problem is e¬ectively one-dimensional. In this limit, the classical pump ¬eld can

be expressed as

E P (r, t) = eP |EP 0 | eiθP fP (t ’ z/vgP ) ei(kP z’ωP t) , (16.84)

where we have assumed that the medium outside the crystal is linearly index matched.

The temporal shape of the pump pulse is described by the function fP („ ), with max-

imum value fP (0) = 1 and pulse duration „P . In the long-pulse limit, „P ’ ∞, the

problem is further simpli¬ed by setting fP (t ’ z/vgP ) = 1.

In the 1D limit the signal“idler mode is described by a paraxial state, so the

¬eld can again be represented by eqn (16.54), with ω0 = ωP /2. The polarization of

the signal“idler mode is ¬xed, relative to that of the pump, by the phase-matching

conditions in the nonlinear crystal. Applying the 1D approximation and the long-pulse

limit to the expressions (7.39) and (13.30) yields the e¬ective ¬eld Hamiltonian

∞ LS

†

dz e’iθP φ2 (z, t) + HC . (16.85)

dzφ (z, t) vg0 ∇z φ (z, t) + g (3)

Hem =

i 2

’∞ 0

The special form of the interaction Hamiltonian”which represents the pair-produc-

tion aspect of down-conversion”produces a propagation equation

‚ ‚

φ (z, t) = ’ig (3) eiθP φ† (z, t)

+ vg0 (16.86)

‚t ‚z

Linear optical ampli¬ers—

½

that couples the ¬eld φ (z, t) to its adjoint φ† (z, t). This means that the propagation

equation and its adjoint must be solved together. In the wave coordinates de¬ned by

eqn (16.71) the equations to be solved are

‚

φ (z, „ ) = ’igeiθP φ† (z, „ ) , (16.87)

‚z

‚†

φ (z, „ ) = ige’iθP φ (z, „ ) , (16.88)

‚z

where g = g (3) /vg0 is the weak signal gain. Since the retarded time „ only appears as

a parameter, these equations can be solved by standard techniques to ¬nd

φ (z, t) = φ (0, t ’ z/vg0 ) cosh (gz) ’ ieiθP φ† (0, t ’ z/vg0 ) sinh (gz) , (16.89)

where we have reverted to the original (z, t)-variables. Thus the solution at z, t is

expressed in terms of the ¬eld operators evaluated at the input face, z = 0, and the

retarded time „ = t ’ z/vg0 .

The time-domain, input“output relation for the traveling-wave ampli¬er is obtained

by evaluating this solution at the output face, z = LS , and letting t ’ t + LS /vg0 :

φ (LS , t + LS /vg0 ) = φ (0, t) cosh (gLS ) ’ ieiθP φ† (0, t) sinh (gLS ) . (16.90)

Fourier transforming this equation yields

e’iLS /vg0 φ (LS , ω) = φ (0, ω) cosh (gLS ) ’ ieiθP φ† (0, ’ω) sinh (gLS ) , (16.91)

which can be brought into the standard form for input“output relations by using the

representations (16.77) and (16.78) to ¬nd

bout (ω) = P bin (ω) ’ ieiθP Cb† (0, ’ω) , (16.92)

in

with

P = cosh (gLS ) and C = sinh (gLS ) . (16.93)