devices are called traveling-wave ampli¬ers.

16.2 Regenerative ampli¬ers

In this section we take advantage of the remarkable versatility of the spontaneous

down-conversion process to describe three regenerative ampli¬ers, two phase insensi-

tive and one phase sensitive.

16.2.1 Phase-insensitive ampli¬ers

A modi¬cation of the degenerate OPA design of Section 15.2 provides two examples

of phase-insensitive ampli¬ers. In the modi¬ed design, shown in Fig. 16.1, the signal

and idler modes are frequency degenerate, but not copropagating. In the absence

of the mirrors M1 and M2, down-conversion of the pump radiation would produce

symmetrical cones of light around the pump direction, but this azimuthal symmetry

is broken by the presence of the cavity axis joining the two mirrors. This arrangement

picks out a single pair of conjugate modes: the idler and the signal.

The boundary conditions at the mirrors de¬ne a set of discrete cavity modes, and

the fundamental cavity mode”which we will call the idler”is chosen to satisfy the

phase-matching condition ω0 = ωP /2. The discrete idler mode is represented by a

single operator a (t). On the other hand, the signal mode is a traveling wave with

propagation direction determined by the phase-matching conditions in the nonlin-

ear crystal. Thus the signal mode is represented by a continuous family of operators

bsig,„¦ (t).

¼¿

Regenerative ampli¬ers

Fig. 16.1 Two examples of phase-insensitive optical ampli¬ers based on down-conversion in

a χ(2) crystal: (a) taking the signal-mode in- and out-¬elds as the input and output of the

ampli¬er de¬nes a phase-preserving ampli¬er; (b) taking the signal-mode in-¬eld as the input

and the out-¬eld through mirror M2 as the output de¬nes a phase-conjugating ampli¬er.

The ¬rst step in dividing the world into sample and reservoirs is to identify the

sample. From the experimental point of view, the sample in this case evidently consists

of the atoms in the nonlinear crystal, combined with the idler mode in the cavity. The

theoretical description is a bit simpler, since”as we have seen in Chapter 13”the

atoms in the crystal are only virtually excited. This means that the e¬ect of the

atoms is completely accounted for by the signal“idler coupling constant; consequently,

the sample can be taken to consist of the idler mode alone. There are then three

environmental reservoirs: the signal reservoir represented by the operators bsig,„¦ (t)

and two noise reservoirs represented by the operators b1,„¦ (t) and b2,„¦ (t) describing

radiation entering and leaving the cavity through the mirrors.

Analyzing this model requires a slight modi¬cation of the method of in- and out-

¬elds described in Section 14.3. The new feature requiring the modi¬cation is the form

of the coupling between the idler (sample) mode and the signal (reservoir) mode. This

term in the interaction Hamiltonian HSE does not have the generic form of eqn (14.88);

instead, it is described by eqn (15.7). In a notation suited to the present discussion:

∞

D (ω)

vP („¦) e’iωP t b† a† ’ vP („¦) eiωP t absig,„¦ , (16.12)

—

sig’idl

HSE =i d„¦ sig,„¦

2π

0

where vP („¦) is the strength of the coupling”induced by the nonlinear crystal”

between the signal mode, the idler mode, and the pump ¬eld. The presence of the

products b† a† and absig,„¦ represents the fact that the signal and idler photons are

sig,„¦

created and annihilated in pairs in down-conversion.

After including this new term in HSE , the procedures explained in Section 14.3 can

be applied to the present problem. The interaction term in eqn (16.12) leads to the

modi¬ed Heisenberg equations

∞ ∞

2

D (ω) D (ω)

d †

a (t) = d„¦ vP („¦) bsig,„¦ (t) + d„¦ vm („¦) bm,„¦ (t) ,

dt 2π 2π

0 0

m=1

(16.13)

D (ω)

d

vP („¦) a† (t) ,

bsig,„¦ (t) = ’i („¦ ’ ω0 ) bsig,„¦ (t) + (16.14)

dt 2π

Linear optical ampli¬ers—

¼

where vm („¦) describes the coupling of the idler to the noise modes, and a (t) =

a (t) exp (iω0 t), etc. The equations for the noise reservoir operators bm,„¦ (t) have the

generic form of eqn (14.89). The retarded and advanced solutions of eqn (16.14) for

the signal mode are respectively

t

bsig,„¦ (t) = bsig,„¦ (t0 ) e’i(„¦’ω0 )(t’t0 ) + vP („¦) dt a† (t ) e’i(„¦’ω0 )(t’t ) (16.15)

t0

and

t1

’i(„¦’ω0 )(t’t1 )

dt a† (t ) e’i(„¦’ω0 )(t’t ) .

’ vP („¦)

bsig,„¦ (t) = bsig,„¦ (t1 ) e (16.16)

t

The corresponding results for the noise reservoir operators, bm,„¦ (t), are given by eqns

(14.94) and (14.105).

After substituting the retarded solutions for bsig,„¦ (t) and bm,„¦ (t) into the equation

of motion (16.13), we impose the Markov approximation by assuming that the idler

mode is coupled to a broad band of excitations in the two mirror reservoirs and in

the signal reservoir. The general discussion in Section 14.3 yields the broadband rule

√

vm („¦) ∼ κm for the noise modes. The signal mode must be treated di¬erently, since

vP („¦) is proportional to the classical pump ¬eld, which has a well-de¬ned phase θP .

√

In this case the broadband rule is vP („¦) ∼ gP exp (iθP ), where gP is positive.

The contributions from the noise reservoirs yield the expected loss term ’κC a (t) /2,

but the contribution from the signal reservoir instead produces a gain term +gP a (t) /2.

This new feature is another consequence of the fact that the down-conversion mech-

anism creates and annihilates the signal and idler photons in pairs. Emission of a

photon into the continuum signal reservoir can never be reversed; therefore, the asso-

ciated idler photon can also never be lost. On the other hand, the inverse process”in

which a signal“idler pair is annihilated to create a pump photon”does not contribute

in the approximation of constant pump strength. Consequently, in the linear approx-

imation the coupling of the signal and idler modes through down-conversion leads

to an increase in the strength of both signal and idler ¬elds at the expense of the

(undepleted) classical pump ¬eld.

After carrying out these calculations, one ¬nds the retarded Langevin equation for

the idler mode:

√ √

√

d 1 †

a (t) = ’ (κC ’ gP ) a (t)+ gP eiθP bsig,in (t)+ κ1 b1,in (t)+ κ2 b2,in (t) , (16.17)

dt 2

where ∞

d„¦

bsig,„¦ (t0 ) e’i(„¦’ω0 )(t’t0 )

bsig,in (t) = (16.18)

2π

’∞

is the signal in-¬eld, and the in-¬elds for the mirrors are given by eqn (14.98). For

gP > κC , eqn (16.17) predicts an exponential growth of the idler ¬eld that would

violate the weak-¬eld assumptions required for the model. Consequently”just as in

the treatment of squeezing in Section 15.2-A”the pump ¬eld must be kept below the

threshold value (gP < κC ).

¼

Regenerative ampli¬ers

We now imitate the empty-cavity analysis of Section 14.3.3 by transforming eqn

(16.17) to the frequency domain and solving for a (ω), with the result

√ √

√ †

eiθP gP bsig,in (’ω) + κ1 b1,in (ω) + κ2 b2,in (ω)

a (ω) = . (16.19)

(κC ’ gP ) ’ iω

1

2

The input“output relation for the signal mode is obtained by equating the right sides

of eqns (16.15) and (16.16) and integrating over „¦ to get

√

gP eiθP a† (t)

bsig,out (t) = bsig,in (t) + (16.20)

in the time domain, or

√

gP eiθP a† (’ω)

bsig,out (ω) = bsig,in (ω) + (16.21)

in the frequency domain. The input“output relations for the mirror reservoirs are given

by the frequency-domain form of eqn (14.109):

√

b1,out (ω) = b1,in (ω) ’ κ1 a (ω) , (16.22)

√

b2,out (ω) = b2,in (ω) ’ κ2 a (ω) . (16.23)

A Phase-transmitting OPA

The ¬rst step in de¬ning an ampli¬er is to decide on the identity of the input and

output ¬elds. In other words: What is to be measured? For the ¬rst example, we choose

the in-¬eld and out-¬eld of the signal mode as the input and output ¬elds, i.e. bin (ω) =

bsig,in (ω) and bout (ω) = bsig,out (ω). The idler ¬eld and the two mirror reservoir in-

¬elds are then internal modes of the ampli¬er. Substituting these identi¬cations and

the solution (16.19) into eqn (16.21) yields the ampli¬er input“output equation

bout (ω) = P (ω) bin (ω) + · (ω) , (16.24)

where the coe¬cient

(κC + gP ) ’ iω

1

2

P (ω) = (16.25)

(κC ’ gP ) ’ iω

1

2

has the symmetry property

P (ω) = P — (’ω) , (16.26)

and the operator

√ †

gP eiθP ξC (’ω)

· (ω) = 1

2 (κC ’ gP ) ’ iω

√

√

√

gP eiθP † †

=1 κ1 b1,in (’ω) + κ2 b2,in (’ω) (16.27)

2 (κC ’ gP ) ’ iω

is called the ampli¬er noise.

Linear optical ampli¬ers—

¼

This result shows that the noise added by the ampli¬er is entirely due to the noise

reservoirs associated with the mirrors. The absence of noise added by the atoms in

the nonlinear crystal is a consequence of the fact that the excitations of the atoms

are purely virtual. In most applications, only vacuum ¬‚uctuations enter through M1

and M2, but the following calculations are valid in the more general situation that

both mirrors are coupled to any phase-insensitive noise reservoirs. In particular, the

vanishing ensemble average of the noise operator · (ω) implies that the input“output

equation for the average ¬eld is

bout (ω) = P (ω) bin (ω) . (16.28)

Subtracting this equation from eqn (16.24) yields the input“output equation

δbout (ω) = P (ω) δbin (ω) + · (ω) (16.29)

for the ¬‚uctuation operators.

The ¬rst step in the proof that this ampli¬er is phase insensitive is to use eqn

(16.28) to show that the e¬ect of a phase transformation applied to the input ¬eld is

bout (ω) = P (ω) bin (ω) = eiθ bout (ω) . (16.30)

In other words, changes in the phase of the input signal are simply passed through the

ampli¬er. Ampli¬ers with this property are said to be phase transmitting. The ¬eld

2

strength bout (ω) is evidently unchanged by a phase transformation; therefore the

ampli¬er satis¬es condition (i) of Section 16.1.1.

Turning next to condition (ii), we note that the operators δbin (ω) and · (ω) are lin-

ear functions of the uncorrelated reservoir operators bsig,„¦ (t0 ) and bm,„¦ (t0 ) (m = 1, 2).

This feature combines with eqn (16.29) to give

Kout (ω, ω ) = P (ω) P — (ω ) Kin (ω, ω ) + Kamp (ω, ω ) , (16.31)

where

1

· (ω) · † (ω ) + · † (ω ) · (ω)

Kamp (ω, ω ) = (16.32)

2

is the ampli¬er“noise correlation function. Since · (ω) is a linear combination of the

mirror noise operators, the assumption that the mirror noise is phase insensitive guar-

antees that

Kamp (ω, ω ) = Namp (ω) 2πδ (ω ’ ω ) , (16.33)

where Namp (ω) is the ampli¬er noise strength. If the correlation function Kin (ω, ω )

satis¬es eqn (16.9), then eqns (16.31) and (16.33) guarantee that Kout (ω, ω ) does

also. The output noise strength is then given by

Nout (ω) = |P (ω)|2 Nin (ω) + Namp (ω) . (16.34)

It is also necessary to verify that the output noise satis¬es eqn (16.11), when the

input noise does. This is an immediate consequence of the phase insensitivity of the

ampli¬er noise and the input“output equation (16.29), which together yield

δbout (ω) δbout (ω ) = P (ω) P (ω ) δbin (ω) δbin (ω ) . (16.35)

Putting all this together shows that the ampli¬er is phase insensitive, since it satis¬es

conditions (i) and (ii) from Section 16.1.1.

¼

Regenerative ampli¬ers

For this ampli¬er, it is reasonable to de¬ne the gain as the ratio of the output ¬eld

strength to the input ¬eld strength:

2

| bout (ω + ω0 ) |2

bout (ω)

G (ω) = = . (16.36)

2 2

| bin (ω + ω0 ) |

bin (ω)

Using eqn (16.28) yields the explicit expression

2 2

(κC + gP ) /4 + (ω ’ ω0 )

G (ω ’ ω0 ) = , (16.37)

2 2

(κC ’ gP ) /4 + (ω ’ ω0 )

which displays the expected peak in the gain at the resonance frequency ω0 . An alter-

native procedure is to de¬ne the gain in terms of the quadrature operators, and then

to show”see Exercise 16.1”that the gain is the same for all quadratures.

B Phase-conjugating OPA

The crucial importance of the choice of input and output ¬elds is illustrated by using

the apparatus shown in Fig. 16.1 to de¬ne a quite di¬erent ampli¬er. In this version the

input ¬eld is still the signal-mode in-¬eld bsig,in (ω), but the output ¬eld is the out-¬eld

b2,out (ω) for the mirror M2. The internal modes are the same as before. The input“

output equation for this ampli¬er”which is derived from eqn (16.23) by using the

solution (16.19) and the identi¬cations bin (ω) = bsig,in (ω) and bout (ω) = b2,out (ω)”

has the form

†

bout (ω) = C (ω) eiθP bin (’ω) + · (ω) . (16.38)

The coe¬cient C (ω) and the ampli¬er noise operator are respectively given by

√

κ2 g P

C (ω) = ’ 1 (16.39)

(κC ’ gP ) ’ iω

2

and

√

(κ1 ’ κ2 ’ gP ) ’ iω

1

κ1 κ2

b2,in (ω) ’

2

· (ω) = b1,in (ω) . (16.40)

2 (κC ’ gP ) ’ iω 2 (κC ’ gP ) ’ iω

1 1

The important di¬erence from eqn (16.24) is that the output ¬eld depends on the

adjoint of the input ¬eld. Note that C (ω) has the same symmetry as P (ω):

C (ω) = C — (’ω) . (16.41)

The ensemble average of eqn (16.38) is

†

bout (ω) = C (ω) bin (’ω) , (16.42)

so the phase transformation bin (ω) ’ bin (ω) = exp (iθ) bin (ω) results in

†

bout (ω) = e’iθ C (ω) bin (’ω) = e’iθ bout (ω) . (16.43)

Instead of being passed through the ampli¬er unchanged, the phasor exp (iθ) is re-

placed by its conjugate. A device with this property is called a phase-conjugating

ampli¬er.

Linear optical ampli¬ers—

¼

This ampli¬er nevertheless satis¬es condition (i) of Section 16.1.1, since

2 2

bout (ω) = bout (ω) . (16.44)

The argument used in Section 16.2.1-A to establish condition (ii) works equally well

here; therefore, the alternative design also de¬nes a phase-insensitive ampli¬er. The

form of the input“output relation in this case suggests that the gain should be de¬ned

as

2

bout (ω) κ2 g P

= |C (ω)|2 = 1

G (ω) = . (16.45)

2

(κC ’ gP )2 + ω 2

†

bin (’ω) 4

16.2.2 Phase-sensitive OPA

In the design shown in Fig. 16.2 the ¬elds entering and leaving the cavity through

the mirror M1 are designated as the input and output ¬elds respectively, i.e. bin (t) =

b1,in (t) and bout (t) = b1,out (t). The degenerate signal and idler modes of the cavity

and the input ¬eld b2,in (t) for the mirror M2 are the internal modes of the ampli¬er.

The input“output relation is obtained from eqn (15.117) by applying this identi¬cation

of the input and output ¬elds:

†

bout (ω) = P (ω) bin (ω) + C (ω) eiθP bin (’ω) + · (ω) . (16.46)

The phase-transmitting and phase-conjugating coe¬cients are respectively

κ1 (κC /2 ’ iω)

P (ω) = 1 ’ (16.47)

2 2

(κC /2 ’ iω) ’ |„¦P |

and

|„¦P | κ1

C (ω) = ’ . (16.48)

2 2

(κC /2 ’ iω) ’ |„¦P |

2, out

1, in

Pump

2, in

1, out

M2

M1

1 2

Fig. 16.2 A phase-sensitive ampli¬er based on the degenerate OPA. The heavy solid arrow

represents the classical pump; the thin solid arrows represent the input and output modes

for the mirror M1; and the dashed arrows represent the input and output for the mirror M2.

¼

Regenerative ampli¬ers

The functions P (ω) and C (ω) satisfy eqns (16.26) and (16.41) respectively. The am-

pli¬er noise operator,

√

κ1 κ2 †

· (ω) = ’ (κC /2 ’ iω) b2,in (ω) + „¦P b2,in (’ω) , (16.49)

2 2

(κC /2 ’ iω) ’ |„¦P |

only depends on the reservoir operators associated with the mirror M2, so the ampli¬er

noise is entirely caused by vacuum ¬‚uctuations passing through the unused port at

M2.

According to eqn (16.46), the output ¬eld strength is

2

†

2 2

2 2

= |P (ω)| + |C (ω)|

bout (ω) bin (ω) bin (’ω)

—

†

+ 2 Re P (ω) C — (ω) bin (ω) bin (’ω) . (16.50)

We ¬rst test condition (i) of Section 16.1.1, by applying the phase transformation

(16.5) to the input ¬eld and evaluating the di¬erence between the transformed and

the original output intensities to get

2