Department of Electrical and Computer Engineering, Center for Photonic Communication and Computing,Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3118
Received February 16, 2005; revised September 6, 2005; accepted September 21, 2005; posted October 18, 2005 (Doc. ID 59890)
. INTRODUCTIONiber optical parametric amplifiers (FOPAs) are currentlyhe subject of much research for use in wavelengthonversion1 and efficient broadband amplification.2 Theyre also candidates for performing all-optical networkunctions.3–5 Advances in pumping techniques have per-
itted improvements of the noise figure (NF) of FOPAsperated phase insensitively,1,6 and the manufacture ofigh-nonlinearity and microstructure fibers has improvedhe gain slope7,8 of FOPAs.
To explain our experimental NF result for a FOPA op-rated as a phase-insensitive amplifier (PIA),9 we haveecently published a quantum theory of ��3� parametricmplifiers that takes into account the noninstantaneousonlinear response of the medium and the requisite addi-ion of noise caused by this noninstantaneous nonlinearesponse.10,11 The FOPA operates as a PIA when an idleream is generated wholly in the fiber or when the phaseifference (twice the phase of the pump less the phases ofhe Stokes and anti-Stokes beams) is approximately 0 or
and the Stokes and anti-Stokes beams are approxi-ately equal in amplitude. Our work with FOPAs oper-
ted as PIAs also provides analytical expressions for theF of ��3� phase-insensitive parametric amplifiers10 andavelength converters.11 This theory shows excellentgreement with experiment.9 In addition, we have re-ently experimentally investigated the NF spectra for PIAnd wavelength converter operation of a FOPA, for whichoth show good agreement to an extended theory that in-ludes distributed loss.12 This inclusion of distributed lossas necessary to model the experiment,12 but it also pro-ides the necessary quantum theory to predict the perfor-ance of a distributed amplifier.Phase-sensitive amplifiers (PSAs)13,14 are also of inter-
st because, unlike PIAs, they can ideally provide ampli-cation without degrading the signal-to-noise ratio (SNR)t the input.15 Operation of a FOPA as a PSA occurs whenhe phase difference at the input is approximately � /2nd the signal and idler are approximately equal in am-litude. Experiments with fully frequency degenerate fi-er PSAs have demonstrated a NF of 2.0 dB at a gain of6 dB,16 a value lower than the standard phase-nsensitive high-gain 3 dB quantum limit. A NF below thetandard PIA limit has also been reached in a low-gainSA.17 However, these fully frequency degenerate PSA ex-eriments were impaired by guided-acoustic-wave Bril-ouin scattering (GAWBS)18 requiring pulsed operation19
r sophisticated techniques for partially suppressingAWBS.17 To avoid the GAWBS noise one may obtainhase-sensitive amplification with an improved experi-ental NF by use of a frequency nondegenerate PSA. In
ddition, the nondegenerate PSA, unlike its degenerateounterpart, can be used with multiple channels of data.
nondegenerate PSA is realized by placing the signal inwo distinct frequency bands symmetrically around theump frequency with a separation of several gigahertz, sohat GAWBS noise scattered from the pump is not in therequency bands of the signal. Such a frequency nonde-enerate regime has been demonstrated experimentally,howing good agreement with theory for the average val-es of the signal and idler.20 We have also experimentallyemonstrated an improved bit-error rate by use of a PSAs opposed to a PIA of comparable gain.21 However,uantum-limited NF measurements have yet to be per-ormed. So an analysis of this case is practically useful.ccordingly, we here describe in suitable detail a quan-
um theory of FOPAs that takes into account the nonzeroesponse time of the ��3� nonlinearity along with the effect
006 Optical Society of America
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Voss et al. Vol. 23, No. 4 /April 2006 /J. Opt. Soc. Am. B 599
f distributed linear loss. We present analytical expres-ions for the quantum-limited NF of cw ��3� PSAs in therequency nondegenerate case. We also report the limitingalue of the NF when degeneracy is approached. In addi-ion, we note that this theory is valid for FOPAs used asistributed PSAs.22
A frequency nondegenerate parametric amplifier canlso operate as a phase-sensitive deamplifier (PSD) ofwo-frequency input signals. This occurs when the phaseifference at the input is approximately −� /2 and thetokes and anti-Stokes inputs are approximately equal inmplitude. When a PSD is operated with no input signal,uch a parametric amplifier is said to produceuadrature-squeezed vacuum (parametric fluorescence ofhe PSD) whose two-frequency homodyne detection exhib-ts photocurrent variance less than that of the vacuum for
suitable choice of homodyne phases.23 Quadraturequeezing has been proposed for applications in quantumommunications,24–26 improved measurementensitivity,27,28 and quantum lithography.29 In the case ofOPAs, previous work by Shapiro and Boivin30 used theispersionless theory of self-phase modulation developedy Boivin et al.31 and Kärtner et al.32 that included theoninstantaneous response of the ��3� nonlinearity to ob-ain a limit on quadrature squeezing in the fully four-egenerate-wave case. In this paper we present resultsor frequency nondegenerate cw quadrature squeezing for
noninstantaneous nonlinearity in the presence of dis-ersion. We show that optimal squeezing occurs forlightly different input conditions compared with thoseor optimal classical deamplification. In addition, we showhat, unlike the dispersionless case, the degree of squeez-ng reaches a constant value in the long-interaction-ength limit when the linear phase mismatch is nonzerond a noninstantaneous nonlinear response is present.ur nondegenerate squeezing theory agrees with the pre-ious degenerate squeezing results of Shapiro andoivin30 when degeneracy is approached.This paper is organized as follows. In Section 2 we dis-
uss the solution of the equations describing evolution ofhe mean values of the pump, Stokes, and anti-Stokeselds. In Section 3 we present a quantum mechanicallyonsistent theory of the FOPAs. For calculation of the NFn phase-sensitive operation, we need to obtain only the
ean and variance of the total photocurrent at the Stokesnd anti-Stokes wavelengths. Thus we chose to calculatenly the output Heisenberg annihilation operators for thetokes and anti-Stokes frequency pair of interest. Wehus obtain the desired results in a simpler way than hade used the Wigner or positive-P formalism developed byrummond and Corney33 for propagation of the quantum
tates of pulses in the fiber. In Sections 4 and 5 we applyhis theory to obtain the NF of phase-sensitive amplifica-ion and to obtain the degree of nondegenerate quadra-ure squeezing, respectively. We reemphasize the main re-ults and conclude in Section 6.
. CLASSICAL PHASE-SENSITIVEMPLIFICATION AND DEAMPLIFICATIONe have discussed the ��3� nonlinear response at length11;
nly a brief summary is presented here.
The nonlinear refractive index of the Kerr interactionan be written as
n2 =3��3�
4�0n02c
, �1�
here n0 is the linear refractive index of the nonlinearaterial, �0 is the permittivity of free space, and c is the
peed of light in free space. For clarity, we state that
��3� � �1111�3� �m2
V2� . �2�
he ��3��t� nonlinear response is composed of a time-omain delta-function-like electronic response ��1 fs�hat is constant in the frequency domain over the band-idths of interest and a time-delayed Raman response
�50 fs� that varies over frequencies of interest and isaused by back action of nonlinear nuclear vibrations onlectronic vibrations. Recent experimental and theoreticalesults demonstrate that the nonlinear response function�3��t� can be treated as if it were real in the timeomain,34,35 yielding a real part that is symmetric in therequency domain with respect to pump detuning and anmaginary part in the frequency domain that is antisym-
etric.Although a nonlinear response is also present in the
olarization orthogonal to that of the pump, thisross-polarized nonlinear interaction is ignored becausee assume that the pump, Stokes, and anti-Stokes fieldsf interest stay copolarized as their polarization statesvolve during propagation through the FOPA. Parametricuorescence and Raman spontaneous emission areresent in small amounts in the polarization perpendicu-ar to the pump, but do not affect the NF of the amplifier.
We can write N2��� in the frequency domain as a sumf electronic and molecular contributions:
N2��� = n2e + n2rF���. �3�
e also define a nonlinear coefficient �� to be
�� =2�N2���
�Aeff� 1
Wm� , �4�
here � is the pump wavelength and Aeff is the fiber ef-ective area. Thus our �0 is equivalent to the nonlinearoupling coefficient � used in Agrawal.36 It is the scalingf Aeff with wavelength that mainly causes �� to be noonger antisymmetric with �−� at detunings greater thaneveral terahertz. In what follows, our analytical treat-ent of the mean fields allows for the more general case
f asymmetry in the Raman gain spectrum. However,ther results including graphs assume an antisymmetricaman spectrum as this has a minor effect on the quan-
um noise at large detunings.We next present solutions to the mean-field equations
overning a parametric amplifier. The optical fields aressumed to propagate in a dispersive, polarization-reserving, single-transverse-mode fiber under the slowlyarying envelope approximation. As the involved wavesre quite similar in frequency, to a good approximation allelds can be treated as if their transverse-mode profilesre identical. Even though the fibers used to construct
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600 J. Opt. Soc. Am. B/Vol. 23, No. 4 /April 2006 Voss et al.
OPAs typically support two polarization modes and theolarization state of the waves is usually elliptical at aiven point z in the FOPA, for typical fibers it is still ap-ropriate to describe the system with a scalar theory ifhe detuning is relatively small.12 This is because the in-ut waves are copolarized at the beginning of the ampli-er and the fields of interest remain essentially copolar-
zed during propagation down the fiber.Consider the field
A�t� = Ap + As exp�i�t� + Aa exp�− i�t� �5�
or the total field propagating through a FOPA having arequency and polarization degenerate pump. Here �A�2as units of watts. We refer to the lower-frequency field ashe Stokes field As; the higher-frequency field is referredo as the anti-Stokes field Aa. The classical equation ofotion for the total field37 with the addition of arbitrary
requency-dependent loss is
�A�t�
�z= i�� d� ��t − ��A*���A����A�t�
−� ���
2A���exp�− i�t�d�, �6�
here ��� is the power attenuation coefficient at detun-ng � from the pump and A��� is the Fourier transform ofhe field. Because the involved waves (Stokes, anti-tokes, and pump) are cw, the usual group-velocity dis-ersion term does not explicitly appear in Eq. (6). How-ver, dispersion is included; its effect is simply to modifyhe wave vector of each cw component. Taking the Fourierransform of Eq. (6) and separating into frequency-shiftedomponents that are capable of phase matching, we ob-ain the following differential equations for the meanelds37:
dAp
dz= i�0�Ap�2Ap −
p
2Ap, �7�
dAa
dz= i��0 + ����Ap�2Aa + i��Ap
2As* exp�− ikz� −
a
2Aa,
�8�
dAs
dz= i��0 + �−���Ap�2Aa + i�−�Ap
2Aa* exp�− ikz� −
a
2As.
�9�
ere k=ka+ks−2kp is the phase mismatch. Expandinghe wave vectors in a Taylor series around the pump fre-uency to second order, one obtains k=�2�2 to secondrder, where �2 is the group-velocity dispersion coeffi-ient. The attenuation coefficients are j for j=p ,a ,s athe pump, anti-Stokes, and Stokes wavelengths, respec-ively. The nonlinear coupling coefficients �0, ��, and �−�
re as defined in Eq. (4). Equations (7)–(9) are valid whenhe pump remains essentially undepleted by the Stokesnd anti-Stokes waves and is much stronger than thetokes and anti-Stokes waves. The solution to Eqs. (8)nd (9) can be expressed as
Aa�z,L� = �a�z,L�Aa�z� + a�z,L�As*�z�, �10�
As�z,L� = �s�z,L�As�z� + s�z,L�Aa*�z�, �11�
here we have explicitly written the solution as a func-ion of both a starting point z for the parametric processnd an end point L for the fiber. We do this because weill be interested not only in the input–output relation-
hips of the electromagnetic fields, i.e., Aa�0,L�, but alsovolution of noise generated at a point z that propagateso the end of the fiber L. In the following subsections, werovide expressions for �j�z ,L� and j�z ,L� for the threeain cases of interest.
. Distributed Loss Solutionn the most general case, when there are no restrictionsn k and distributed linear loss is present, Eqs. (7)–(9)an be shown to have a series solution. We here brieflyutline the derivation of this solution. Solving for theean field of the pump, Eq. (7), we obtain
Ap�z� = exp�i�0Ip�0�zeff −pz
2 �Ap�0�, �12�
here the effective length zeff is defined to be zeff= 1exp�−pz� /p. Further defining the initial pump power
n watts to be Ip�0�= �Ap�0��2, and setting the referencehase to be that of the pump at the input of the fiber, weubstitute the resulting expressions into Eqs. (8) and (9).riting
Aa = Ba expi��0 + ���Ip�0�zeff − az/2, �13�
As = Bs expi��0 + �−��Ip�0�zeff − sz/2 �14�
nd making a change of variable from z to zeff, one obtains
dBa
dzeff= i��Ip�0�exp− f�zeff�Bs
*, �15�
dBs*
dzeff= − i�−�
* Ip�0�expf�zeff�Ba, �16�
here
f�zeff� = i�� + �−�* Ip�0�zeff −
�s − a + 2ik�ln�1 − pzeff�
2p.
�17�
fter some algebra and making use of the substitutions
Fa = Ba expf�zeff�, �18�
Fs* = Bs
* exp− f�zeff�, �19�
e can obtain the nonlinear coupled equations
dFa
dzeff− �� +
�
1 − pzeff�Fa = �1Fs
*, �20�
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Voss et al. Vol. 23, No. 4 /April 2006 /J. Opt. Soc. Am. B 601
dFs*
dzeff+ �� +
�
1 − pzeff�Fs
* = �2Fa, �21�
here the following constants are used for calculatingvolution from point z to L : �= i���+�−�
n the nonlinear term in Eqs. (20) and (21), we find a se-ies solution for Fj and then obtain Aj.
The series solution converges in relatively few termshen pzeff is small, which is the case for practical ampli-ers. The following solutions are then obtained:
nd Leff= �1−exp−p�L−z�� /p. The coefficients an and
n* are then calculated through the following recursion re-ations:
an =�an−1 + �1sn−1
* + � j=0
n−1p
j an−1−j
n, �28�
sn* =
− �sn−1* + �2an−1 − � j=0
n−1p
j sn−1−j*
n. �29�
. Lossless and �kÅ0 Solutionhe solution for lossless fiber and k�0 is well known, asre the � and functions that can be expressed as37
a�z,L� = exp�−ik − �2�0 + �� − �−�
* �Ip�L − z�
2 �� � i�
2gsinhg�L − z� + coshg�L − z�� , �30�
a�z,L� = exp�−ik − �2�0 + �−� − ��
* �Ip�L − z�
2 ��� i�*
2g* sinhg*�L − z� + coshg*�L − z�� , �31�
a�z,L� = exp�−ik − �2�0 + �� − �−�
* �Ip�L − z�
2 ��
i��Ap�z�2
gsinhg�L − z�, �32�
s�z,L� = exp�−ik − �2�0 + �−� − �−�
* �Ip�L − z�
2 ��
i�−�Ap�z�2
g* sinhg*�L − z�. �33�
ere Ip�z�=Ip= �Ap�0��2 is the pump power in watts, �
k+ ���+�−�* �Ip, and g=�−�� /2�2+���−�
* Ip2 is the com-
lex gain coefficient.
. Lossless and �k=0 Solutione also state the results for the lossless and k=0 case,hich is useful in our analysis near degeneracy. We havek=0 when the FOPA is pumped at the zero dispersionavelength or if the system is treated as if dispersionless.hen the three frequencies are nearly degenerate, we can
lso make the approximation that k=0. Then the � andfunctions become
�a�z,L� = exp�i�2�0 + �� − �−�*
2 �Ip�L − z���1 + i���L − z�Ip, �34�
�s�z,L� = exp�i�2�0 + �−� − ��*
2 �Ip�L − z���1 + i�−��L − z�Ip, �35�
a�z,L� = exp�i�2�0 + �� − �−�*
2 �Ip�L − z���i���L − z�Ap�z�2, �36�
s�z,L� = exp�i�2�0 + �−� − ��*
2 �Ip�L − z���i�−��L − z�Ap�z�2. �37�
nder the normal assumption of an antisymmetric Ra-an gain profile ���=�−�
* �, we see the power gain for annput only on the anti-Stokes side, Ganti-Stokes= ��a�0,L��2
1–2 Im����IpL+ ����2Ip2L2, has a four-wave-mixing gain
hat is quadratic as a function of fiber length and that theaman loss at the anti-Stokes wavelength is linear inump power and length �−2 Im����IpL�. Similarly, the Ra-an gain at the Stokes wavelength is linear in power and
ength �2 Im�� �I L�.
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602 J. Opt. Soc. Am. B/Vol. 23, No. 4 /April 2006 Voss et al.
. Optimal Classical Phase-Sensitive Amplification andeamplificatione next find the optimal phase-sensitive amplification
nd phase-sensitive deamplification of a mean field con-
isting of a superposition of Stokes and anti-Stokes fields. e
c=mte
stffiottts
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e define optimal phase-sensitive amplification (deampli-cation) as the greatest (least) output signal power pos-ible for a fixed amount of input signal power. Assumingoherent signal inputs �j having powers ��j�2 and phases
xp�i�j� for j=a ,s, the phase-sensitive gain of the PSA is
y properly choosing the relative power of the Stokes andnti-Stokes inputs and their sum phase �=�a+�s relativeo the input pump phase, one achieves maximum (mini-um) phase-sensitive amplification (deamplification).he optimum sum phases �PSA,opt and �PSD,opt are
�PSA,opt = − arg��s s* + �a a
*�, �39�
�PSD,opt = � − arg��s s* + �a a
*� �40�
or amplification and deamplification, respectively. By set-ing the sum input power ��a�2+ ��s�2=C to be some con-tant C, the extrema of Eq. (38) can be found to occurhen the proportion of input anti-Stokes power to the to-
al input power is
��a�2
��a�2 + ��s�2=
1
2�1 ±��s�2 − ��a�2
�4��a a* + �s s
*�2 + ���s�2 − ��a�2�2� ,
�41�
here the negative root corresponds to optimum phase-ensitive amplification and the positive root to optimumhase-sensitive deamplification. The maximum PSA gain,PSA, is found by insertion of Eqs. (39) and (41) into Eq.
38). The result simplifies to
GPSA =���a�2 + ��s�2 + � a�2 + � s�2�
2
+�4��a a
* + �s s*�2 + ���s�2 − ��a�2�2
2
+���s�2 − ��a�2��� a�2 − � s�2�
�4��a a* + �s s
*�2 + ���s�2 − ��a�2�2. �42�
In Fig. 1 all of the plots are optimal in the sense thathe best total phase and relative power of the input fieldsre chosen. It is clear that the Raman effect is negligibles those curves including the Raman effect (circles andquares) are very similar to those neglecting it (solidurve and dashed curve). Figure 1 also shows that theefinition of the effective length is a mathematical onend not a good guide for estimating the gain profile. Theotted and dashed–dotted curves show the gain spectrumf a lossy fiber �0.41 dB/km� 4.44 km in length. The other
urves are for a lossless fiber of effective length Leff3.63 km. Thus distributed loss has a greater effect thanight be supposed: Fibers experience noticeably less gain
han lossless fibers do when the lossless fiber has a lengthqual to the effective length of a lossy fiber.
Some of the characteristics of the classical phase-ensitive response can be seen in Fig. 2, which is a plot ofhe phase-sensitive gain versus fiber length. The primaryeature of phase-sensitive amplification is that the mean-eld gain is relatively insensitive to the relative strengthf the two input fields (this can be seen by the overlap ofhe thin solid lines with the squares and circles). In addi-ion, typical distributed losses do not significantly affecthe gain of the fiber, as can be seen by comparison of thequares (lossless fiber) with the circles, which represent
ig. 1. Gain spectra versus detuning for optimized PSAs maderom (a) lossless fiber of length L=3.63 km and Im����=0 (solidurve); (b) same as (a), but Im���� calculated for dispersion-hifted fiber (DSF) as explained in the text (circles); (c) 4.44 kmf DSF with Leff=3.63 km for a=s=p=0.41 dB/km and Im����alculated for DSF as explained in the text (dotted curve). Gainpectra versus detuning for optimized PSDs made from (d) loss-ess fiber of length L=3.63 km and Im����=0 (dashed curve); (b)ame as (a), but Im���� calculated for DSF as explained in theext (squares); (c) 4.44 km of DSF with Leff=3.63 km for a=sp=0.41 dB/km and Im���� calculated for DSF as explained in
he text (dashed–dotted curve). Input pump power is 0.33 W, �01551.16 nm, pump wavelength is 1551.5 nm, and the disper-ion slope is 57 ps/ �nm2/km�.
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Voss et al. Vol. 23, No. 4 /April 2006 /J. Opt. Soc. Am. B 603
ber with loss of 0.25 dB/km. On the other hand, thechievable degree of phase-sensitive deamplification isuch more sensitive to the relative proportion of the in-
ut fields, which can be seen by comparison of theashed–dotted and dashed curves (equal power splitting,.e., ��a�2= ��s�2) with the dotted curve and thick solid lineoptimum relative proportion). In addition, distributedosses also set a limit on classical deamplification as cane seen by comparison of the dashed curve (lossy) withhe dashed–dotted curves (lossless) and comparison of theotted curve (lossy) with the thick solid line (lossless).
. INPUT–OUTPUT QUANTUM MODERANSFORMATIONS
n this section we discuss the quantum mechanics of the�3� parametric amplifier with a strong, undepleted,oherent-state pump and derive input–output moderansformations in the Heisenberg picture that can besed to calculate the NF of the phase-sensitive operationf a FOPA and the accompanying quadrature squeezing.ere we also extend our previously described quantum
heory10,11 to include the effects of loss. To make our treat-ent consistent with the customary formalism in quan-
um optics, we rescale our field to a photon-flux field, i.e.,he expectation of the field �A�= A=A /��� so that �A�2 hasnits of photons per second. We also rescale our nonlinearoefficient ��=����, where � is Planck’s constant over�.We begin with the continuous-time quantum equation
f motion for a multimode field in the presence of a non-
ig. 2. Gain versus fiber length for a PSA made from dispersion-hifted fiber for (a) phase-sensitive deamplification with opti-um power splitting in lossless fiber �a=s=p=0� (thick solid
ine), (b) phase-sensitive deamplification with optimum powerplitting in a lossy fiber �a=s=p=0.25 dB/km� (dotted curve),c) phase-sensitive deamplification in a lossless fiber with ��a�2��s�2 (dashed–dotted curve), (d) phase-sensitive deamplification
n a lossy fiber with ��a�2= ��s�2 (dashed curve), (e) phase-sensitivemplification in a lossless fiber with ��a�2= ��s�2 (squares) and op-imum input power splitting (solid line under the squares), andf) phase-sensitive amplification in a lossy fiber with ��a�2= ��s�2circles) and optimum input power splitting (solid line under theircles). Input pump power is 4 W, pump–signal detuning isTHz, and phase matching is achieved at the input k=2 Re����Ip�0�.
nstantaneous nonlinearity. This model was presented byrummond and Corney33 and Carter et al.,38 and was
olved for the case of dispersionless self-phase modulationy Boivin et al.,31 and also derived in detail by Kärtner etl.32 We have
�A�t�
�z= −� ���
2A˜ ���exp�− i�t�d�
+ i�� d���t − ��A†���A����A�t�
+ im�z,t�A�t� + l�z,t�, �43�
here ��t� is the causal response function of the nonlin-arity, i.e., the inverse Fourier transform of �� in Eq. (4).n Eq. (43), m�z , t� is a Hermitian phase noise operator
m�z,t� =�0
�
d��2 Im����
2�id�
† �z�exp�i�t�
− id��z�exp�− i�t�, �44�
hich describes coupling of the field to a collection of lo-alized, independent medium oscillators (optical phononodes). In addition, l�t� is a noise operator
l�t� =�0
�
d���
2�exp�i�t�v��t� �45�
hat describes the coupling of the field to a collection ofocalized, independent oscillators in vacuum state. Thisoupling is required to preserve the continuous-time com-utators
A�t�,A†�t�� = ��t − t��, �46�
A�t�,A�t�� = 0. �47�
ote that the time t is in a reference frame traveling atroup velocity vg, i.e., t= tstationary frame− �z /vg�. Theeighting function �2 Im���� must be positive for ��0 so
hat the molecular vibration oscillators absorb energyrom the mean fields rather than providing energy to theean fields. The operators d��z� and d�
† �z� obey the com-utation relation
d��z�,d��† �z�� = ��� − �����z − z��, �48�
nd each phonon mode is in thermal equilibrium:
�d�† �z�d���z��� = ��� − �����z − z��nth �49�
ith a mean phonon number nth= exp��� /kT�−1−1.ere k is Boltzmann’s constant, and T is the temperature.he operators corresponding to vacuum modes mixing
nto the Stokes and anti-Stokes frequencies, v±�, obey theommutation relations
v±��z�, v±�† �z� = ��±� − ± �����z − z�� �50�
nd have no photons in them, i.e., �v±�† �z�v±���z���=0.
We assume that the total field present at the input ofhe fiber contains only a single-frequency pump, a Stokes
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604 J. Opt. Soc. Am. B/Vol. 23, No. 4 /April 2006 Voss et al.
eld, and a symmetrically placed anti-Stokes field. Theperators corresponding to these modes are
A�t� = Ap + As exp�i�t� + Aa exp�− i�t�. �51�
e do not need to consider other modes in the fiber whenhe pump is strong ��Ap�2� �Aa�2 , �As�2� and remains unde-leted, and the Stokes and anti-Stokes frequencies of in-erest are too weak to serve as pumps to other nonlinearrocesses. When this latter condition occurs, fluctuationsrom other higher-order mixing modes are not coupled in.n addition, the vacuum modes at the input do not grow,ue to parametric fluorescence, to become sufficientlytrong to serve as pumps for nonlinear processes. This isecause the input Stokes and anti-Stokes fields wouldaturate the pump long before this would happen. Thushe symmetric pairing of the Stokes and anti-Stokes fieldsf interest is justified and the pairs do not couple throughhe nonlinear process to other frequencies.
To obtain the coupled-wave equations for the three fre-uencies of interest, we insert Eq. (51) into Eq. (43), takehe Fourier transform of the resulting equation, and sepa-ate into different frequencies. The resulting nonlinearuantum operator equations are
dAp
dz= i�0Ap
†ApAp + i��0 + ���As†AsAp + i��0 + �−��Aa
†AaAp
+ i��−� + ���Ap†AsAa exp�ikz�
− �2 Im����Aad�† �z�exp− i�kp − ka�kz
+ �2 Im����Asd��z�exp− i�kp − ks�kz
+ �2 Im���→0�Apd�→0+�z� + d�→0−† �z� −
p
2Ap
+ �pvp�z�, �52�
dAa
dz= i��0 + ���Ap
†ApAa + i�0Aa†AaAa + i��0 + �2��As
†AsAa
+ i��Ap2As
† exp�− ikz� − �2 Im����Ap
�expi�kp − ka�zd��z� −a
2Aa + �ava�z�, �53�
dAs
dz= i��0 + �−��Ap
†ApAs + i�0As†AsAs + i��0 + �−2��Aa
†AaAs
+ i�−�Ap2Aa
† exp�− ikz� + �2 Im����Ap
�expi�kp − ks�zd�† �z� −
s
2As + �svs�z�. �54�
quations (52)–(54) are nonlinear in the quantum opera-ors (except for the last two terms of each equation) andre difficult to solve. However, one may linearize thesequations around the mean values of the operators andbtain solutions accurate to first order in the fluctuations.sing the definition
Aj = Aj + aj, �55�
here Aj represents a c-number mean value of Aj and aj
epresents the quantum fluctuations of Aj, and where jp ,a ,s, we expand Eqs. (52)–(54) to obtain the followingquations for the mean values (i.e., those terms contain-ng no fluctuation operators):
dAp
dz= i�0Ap
*ApAp + i��0 + ���As*AsAp + i��0 + �−��Aa
*AaAp
+ i��−� + ���Ap*AsAa exp�ikz� −
p
2Ap, �56�
dAa
dz= i��0 + ���Ap
*ApAa + i�0Aa*AaAa + i��0 + �2��As
*AsAa
+ i��Ap2As
* exp�− ikz� −a
2, Aa �57�
dAs
dz= i��0 + �−��Ap
*ApAs + i�0As*AsAs + i��0 + �−2��Aa
*AaAs
+ i�−�Ap2Aa
* exp�− ikz� −s
2As. �58�
he equations for those terms that contain one fluctua-ion operator are
dap
dz= i�0�2�Ap�2ap + Ap
2ap†� + i��0 + �����As�2ap + As
*Apas
+ AsApas†� + i��0 + �−����Aa�2ap + Aa
*Apaa + AaApaa†�
+ i��−� + ����Ap*Asaa + Ap
*Aaas + AsAaap†�exp�ikz�
− �2 Im����Aad�† �z�exp− i�kp − ka�kz
+ �2 Im����Asd��z�exp− i�kp − ks�kz
+ �2 Im���→0�Apd�→0+�z� + d�→0† �z� −
p
2ap
+ �pvp�z�, �59�
daa
dz= i��0 + �����Ap�2aa + Ap
*Aaap + ApAaap†� + i�0�2�Aa�2aa
+ Aa2aa
†� + i��0 + �2����As�2aa + As*Asas + AsAaas
†�
+ i���Ap2as
† + 2ApAs*ap�exp�− ikz�
+ �2 Im����Ap expi�kp − ka�zd��z� −a
2aa
� ˆ
+ ava�z�, �60�
WgoW�twcpne
Twetttite
datEbttt�
gFt
FnpRls
tnuSpasdar
Tsce
W(
Voss et al. Vol. 23, No. 4 /April 2006 /J. Opt. Soc. Am. B 605
das
dz= i��0 + �−����Ap�2as + Ap
*Asap + ApAsap†� + i�0�2�As�2as
+ As2as
†� + i��0 + �−2����Aa�2as + Aa*Asaa + AaAsaa
†�
+ i�−��Ap2aa
† + 2ApAa*ap�exp�− ikz�
− �2 Im����Ap expi�kp − ks�zd�† �z� −
s
2s
+ �svs�z�. �61�
hen the amplifier is operating in the unsaturated re-ime, the pump is depleted to a negligible degree. Thusne may neglect the four-wave-mixing terms in Eq. (56).e next make the strong pump approximation, i.e., �Ap�2
�Aa�2 , �As�2, which is valid at the input of the fiber underypically used operating conditions and remains validhen there is essentially no pump depletion. Under these
onditions, we note that the cross-phase modulation of theump wave due to the Stokes and anti-Stokes fields isegligible. We thus obtain the final form of the mean-fieldquations:
dAp
dz= i�0�Ap�2Ap −
p
2Ap, �62�
dAa
dz= i��0 + ����Ap�2Aa + i��Ap
2As*
�exp�− ikz� −a
2Aa, �63�
dAs
dz= i��0 + �−���Ap�2As + i�−�Ap
2Aa*
�exp�− ikz� −s
2As. �64�
he solution of these mean-field equations [Eqs. (7)–(9)]ere examined in Section 2. Examining the fluctuationquations, we note that the first-order fluctuation equa-ions [Eqs. (59)–(61)] are linear in their quantum opera-ors. The higher-order fluctuation terms are nonlinear inhe quantum fluctuation operators, but their contributions negligible because they cannot contain terms propor-ional to Ap
2. We thus neglect the higher-order fluctuationquations and do not report them here.
Similarly, those fluctuation terms in Eqs. (59)–(61) thato not contain two pump factors can be neglected, as theyre much weaker than the other terms. We remark herehat the Langevin noise terms due to the Raman effect inqs. (60) and (61) will not be neglected. This is justifiedecause comparison of the amplitude of the Langevinerms, �2 Im����Ap, in Eqs. (60) and (61) with the ampli-ude of those four-wave-mixing terms in Eqs. (60) and (61)hat contain only one pump term, for example,
A A*a , shows that the Langevin term is much
−� p a p
reater than the one-pump-field four-wave-mixing terms.or example, consider the symmetrized magnitude of the
wo terms:
�2 Im��−���Ap�2��d�† d� + d�d�
† ��/2
4 �−�2 �Aa�2�Ap�2��dp
†dp + dpdp†��/2
=�Im��−���nth + 1�
2��−��2�na�.
�65�
or an anti-Stokes amplitude of up to 10 mW (a flux ofˆ a=7.8�1016 photons/s) this ratio is greater than 25 forure silica and even better for materials with strongeraman gains (such as dispersion-shifted and highly non-
inear fibers). This means that the Langevin noise termhould be included.
One last simplification can be made to the pump equa-ion, Eq. (59). We simply note that all of the remainingoise terms in Eq. (59) affect only the pump field and thatnder the undepleted, strong-pump approximation, thetokes and anti-Stokes fields do not interact with theump fluctuations to first degree. This is the case if onessumes that the pump is, or nearly is, in a coherenttate. Thus, as we are interested only in the noise intro-uced during the amplification process to the Stokes andnti-Stokes fields, we neglect the pump fluctuations andeplace Eq. (59) with
dap
dz= 0. �66�
his allows one to treat the pump field fully classically. Toummarize the steps taken, the undepleted, strong,oherent-state pump approximation yields the followingquations for the first-order fluctuations:
daa
dz= i��0 + ����Ap�2aa + i��Ap
2as† exp�− ikz�
+ �2 Im����Ap expi�kp − ka�zd��z� −a
2aa
+ �ava�z�, �67�
das
dz= i��0 + �−���Ap�2as + i�−�Ap
2aa† exp�− ikz�
− �2 Im����Ap expi�kp − ks�zd�† �z� −
s
2as
+ �svs�z�. �68�
e note that Eqs. (66)–(68) can be combined with Eqs.62)–(64) by using Eq. (55) to yield
dAp
dz= i�0�Ap�2Ap −
p
2Ap, �69�
Tmt
att
o�i�i
4AIp
Wtbtmlcqc
Cgt
wP
Ip�c
w
h
606 J. Opt. Soc. Am. B/Vol. 23, No. 4 /April 2006 Voss et al.
dAa
dz= i��0 + ����Ap�2Aa
+ i��Ap2As
† exp�− ikz� −a
2Aa
+ �2 Im����Ap expi�kp − ka�zd��z� + �ava�z�,
�70�
dAs
dz= i��0 + �−���Ap�2As
+ i�−�Ap2Aa
† exp�− ikz� −s
2As
− �2 Im����Ap expi�kp − ks�zd�† �z� + �svs�z�.
�71�
his final form of the equations allows one to evolve theean fields of all three frequencies and the fluctuations of
he Stokes and anti-Stokes wavelengths simultaneously.The solution of Eqs. (70) and (71) is
As�L� = �a�0,L�Aa�0� + a�0,L�As†�0�
+ �2 Im�����0
L
dzAp�z�expi�kp − ka�z�a�z,L�
− a�z,L�d��z� +�0
L
dz�a�a�z,L�va�z�
+ �s a�z,L�vs†�z�, �72�
As�L� = �s�0,L�As�0� + s�0,L�Aa†�0�
+ �2 Im�����0
L
dzAp�z�expi�kp − ks�z− �s�z,L�
+ s�z,L�d�† �z� +�
0
L
dz�s�s�z,L�vs�z�
+ �a s�z,L�va†�z�. �73�
In the notation used in this paper, the functions �j�z ,L�nd j�L ,z� denote evolution from a point z in the fiber tohe end of the fiber �L� where the intensity and phase ofhe pump at point z must be used.
In this section we have presented a thorough derivationf the input–output mode transformations that govern a�3� parametric amplifier. In Sections 4 and 5 we use thesenput–output mode transformations to obtain the NF of�3� phase-sensitive parametric amplifiers and the squeez-ng parameter for quadrature squeezing.
. NOISE FIGURE OF PHASE-SENSITIVEMPLIFICATION
n this section we discuss the NF of phase-sensitive am-lification, which is defined as
NF =SNRin
SNRout. �74�
e assume for our treatment here that the difference be-ween the Stokes and anti-Stokes frequencies exceeds theandwidth of the detector. Thus beat frequencies of thesewo waves will not be detected and can be neglected. Theean photon flux at the Stokes and anti-Stokes wave-
ength is �nj�= �Aj†�0�Aj�0��= ��j�2, and is assumed to be in a
oherent state. In what follows, we neglect the small fre-uency difference between �a and �s. Thus the input SNRan be written as
SNRin =��na� + �ns��2
�na2� + �ns
2�=
���a�2 + ��s�2�2
��a�2 + ��s�2= ��a�2 + ��s�2.
�75�
alculating the output SNR in a similar way and plug-ing into Eq. (74), the NF for phase-sensitive amplifica-ion can be expressed as
NF =���a�2 + ��s�2���nPI
2 � + �nPS2 ��
�Pa + Ps�2 , �76�
here the mean output power at each wavelength Pa ands is
Voss et al. Vol. 23, No. 4 /April 2006 /J. Opt. Soc. Am. B 607
�ra�2 = 2 Im�����0
L
dz�Ap�z��2��a�z,L� − a�z,L��2, �85�
�rs�2 = − 2 Im��−���0
L
dz�Ap�z��2��s�z,L� − s�z,L��2,
�86�
�ca�s�1�2 =�0
L
dza�s���a�s��z,L��2, �87�
�ca�s�2�2 =�0
L
dzs�a�� a�s��z,L��2, �88�
rx = 2 Im�����0
L
dzAp2�z�exp�− ikz��a�z,L� − a�z,L�
�− �s�z,L� + s�z,L�, �89�
cx1 = a�0
L
dz�a�z,L� s�z,L�, �90�
cx2* = s�
0
L
dz a�z,L�*�s�z,L�*. �91�
In the above expressions, �ra�s��2 represents the inte-rated amplified noise at the anti-Stokes (Stokes) wave-ength seeded by thermally populated optical phonon
odes that are coupled in by the Raman process. Theerms �c �2 represent integrated amplified noise at the
ig. 3. PSA NF versus gain for various detunings for a highlyonlinear fiber. For thick curves, the fiber attenuation is.75 dB/km at pump, Stokes, and anti-Stokes wavelengths; forhin curves, the fiber is lossless. � /2�=13.8 THz (dashedurves); � /2�=1.38 THz (dashed–dotted curves); � /2�40 GHz (solid curves); � /2�=0 Hz (dotted curves). Except for
he dotted curves, phase matching at the input k=2 Re����Ip�0� is achieved. For dotted curves, k=0. The anti-tokes and Stokes relative phase and power splitting at the in-ut is for optimal classical gain. Initial pump power is 340 mW,�0�=9�10−3 W−1 m−1, the peak imaginary part of �� is 3.510−3 W−1 m−1. Fiber length is 1 km.
a�s�1
nti-Stokes (Stokes) wavelength seeded by vacuum noiseixed in through distributed loss at the anti-Stokes
Stokes) wavelength, while the terms �ca�s�2�2 representmplified noise at the anti-Stokes (Stokes) wavelengtheeded by vacuum noise mixed in through distributed losst the Stokes (anti-Stokes) wavelength.In addition, the phase-sensitive terms �a s and �s
* a*
epresent amplified phase-sensitive noise seeded by theacuum noise at the anti-Stokes and Stokes wavelengths.he quantity rx represents amplified phase-sensitiveoise seeded by the thermal phonon fields due to the Ra-an effect, and cx1 and cx2 represent the amplified phase-
ensitive noise seeded by the vacuum noise due to distrib-ted linear losses. Phase-sensitive noise is present whenhe photocurrent variance with both Stokes and anti-tokes waves impinging on a detector is different fromhe sum of the individual noise variances of the Stokesnd anti-Stokes frequencies.
. Degenerate Limity taking the limiting value of the NF as �→0, we find
he NF performance of a fully degenerate FOPA. We findhis limiting value of the NF by expanding the antisym-etric imaginary part of �� in a Taylor series and ex-
anding the exponential in nth= �exp�� / �kT�−1�−1 be-ore allowing �→0. We also use the fact that in this limit,k also approaches 0 and the optimum power splitting ra-
io approaches 0.5. This NF limit is
NFPSA,�→0 = 1 +
4kT�i��0�
��0�1 −
�NL
�1 + �NL2 �
1 + 2�NL2 + 2�NL�1 + �NL
2, �92�
here �NL= �0�Ap�2L is the nonlinear phase shift and �i��0�s the slope of the imaginary part of �� as �→0. We ob-erve that the PSA NF for �→0 increases to a maximumf slightly more than 0 dB and then decreases again and
ig. 4. PSA gain and NF spectrum versus detuning. Anti-Stokesnd Stokes relative phase and input power splitting is for opti-al classical gain. Initial pump power is 300 mW, �0=210−3 W−1 m−1, the peak imaginary part of �� is 0.7510−3 W−1 m−1. Attenuation is 0.41 dB/km at the pump, Stokes,
nd anti-Stokes wavelengths. Fiber length is 4 km. �01551.15 nm, �p=1555.5 nm, and the dispersion slope is7 ps/ �nm/km2�.
ahdmRvtsat
BIulsecbFftIbsttc
PTd
awnnt
fiTcsau
5SWrpso
b
wtsLt
a
pv(
ldlHilas
Tpcnw
Fbmacf=ptcsp�4
608 J. Opt. Soc. Am. B/Vol. 23, No. 4 /April 2006 Voss et al.
pproaches 0 dB in the high-gain limit. This unusual be-avior of decreasing NF versus nonlinear phase shift isue to the relative scaling of the Raman and four-wave-ixing processes when k=0 and the fact that the meanaman gain of the Stokes and anti-Stokes frequenciesanishes. The total Raman noise scales linearly whereashe two-frequency signal undergoes quadratic gain. Con-equently, the Raman noise being created at the Stokesnd anti-Stokes frequencies does not undergo amplifica-ion after it is spontaneously emitted.
. Resultsn Fig. 3 we plot the NF versus PSA gain for several val-es of detuning for a typical highly nonlinear fiber with a
oss coefficient corresponding to 0.75 dB/km. The plotshow that for detunings achievable by use of electro-opticlements (40 GHz detuning, phase-matched, solidurves), the results are almost exactly the same as woulde achieved in the limit of zero detuning (dotted curves).or these simulations, we use realistic values for ��0� and
or the distributed loss. We have additionally assumedhat the highly nonlinear fiber has the same ratio ofm������ to Re������ as the standard dispersion-shifted fi-er, i.e., the two have the same germanium content. Un-urprisingly, we see by comparing the thick curves (lossy)o thin curves (lossless) that the distributed loss increaseshe NF. We also see that as the Raman gain coefficient de-reases, the NF improves.
Interestingly, unlike the PIA case, Fig. 3 shows that theSA NF is greater than 0 dB as the gain approaches 0 dB.his occurs because the Raman gain and loss processesominate in the early parts of the amplifier (Raman gain
ig. 5. Squeezing versus nonlinear phase shift in a lossless fi-er. Lower curves, without Raman effect. Upper curves, with Ra-an effect. Dashed curves signify that phase matching is
chieved at the input k=−2 Re�����Ap�2. The thick dashedurve is for optimal LO power splitting; thin dashed curves areor equal LO power splitting. Dashed–dotted curves are for k−�2/3�Re�����Ap�2. Thick dashed–dotted curve is for optimal LOower splitting; thin dashed curves are for equal LO power split-ing. In all other curves, k=0. Raman effect neglected, dottedurve; Raman effect included and equal LO power splitting arehown with crosses; Raman effect included and optimal LOower splitting are shown with circles; cw limit of Eq. (99) with
nd loss are linear in the early parts of the amplifierhile the four-wave-mixing gain is quadratic), addingoise to both frequencies, while the mean field undergoeso net gain due to the Raman loss at one frequency andhe Raman gain at the other.
In Fig. 4 we show the gain and NF spectrum for a 4 kmber with fiber parameters as described in the caption.his plot shows that the increasing Raman gain coeffi-ient with detuning causes an increasing NF. These re-ults show that for realistic optical fibers, a FOPA oper-ted as a PSA can achieve a NF below 1 dB for detuningsp to 1 THz.
. NONDEGENERATE QUADRATUREQUEEZINGhen no light is injected into the FOPA, the quantum cor-
elations between the Stokes and anti-Stokes modes im-ly the presence of quadrature squeezing, which is mea-urable by homodyne detection with a two-frequency localscillator (LO).
The difference current of the homodyne detector maye written as
I = ba†qa + bs
†qs + H . c . , �93�
here H.c. stands for the Hermitian conjugate of the firstwo terms; qa and qs are the annihilation operators corre-ponding to the anti-Stokes and Stokes components of theO beams, which are in a coherent state, each with pho-
on flux �LO,j�2 relative intensity
yj =�LO,j�2
�LO,a�2 + �LO,s�2, �94�
nd phase �j for j=a ,s.The squeezing parameter is defined as the ratio of the
hotocurrent variance with the pump to the photocurrentariance with vacuum input to the homodyne detectori.e., pump off).
In this paper we concentrate on squezzing results for aossless FOPA. This is because an increased nonlinearrive will, theoretically at least, overpower linear loss,eading to no hard limit on the achievable squeezing.owever, as the Raman effect at each z scales with pump
ntensity as does the four-wave-mixing process, the loss-ess case illustrates a fundamental limit on the achiev-ble squeezing. Using Eqs. (93) and (94), we obtain, afterome simple algebra, for the lossless Raman-active case
S =�I2�
�I2�vac
= 1 + 2�� a�2 + �ra�2nth�ya
+ �1 + 2� s�2 + �rs�2�1 + nth��ys
+ 2��s a�1 + nth� − �a snth
�exp− i��a + �s� + c.c.��yays. �95�
o produce the best squeezing, one must choose the besthase and relative intensity. Once again, these twohoices are independent. To choose the LO phases, weote that the third term in Eq. (95) has a negative signhen
Su
ctt
Tie
i
S
ispsbsEdi=t�pi
wi
imIcpRsitwrqse
Voss et al. Vol. 23, No. 4 /April 2006 /J. Opt. Soc. Am. B 609
�a + �s = � + arg�s a�1 + nth� − �a anth. �96�
ince the total LO power is conserved �ya+ys=1�, we mayse this fact to eliminate y in Eq. (95), which then be-
s
2.
bwwpcam
6WpnftdofN�cgtinfsbfmdlm
ATfit
@EoMpTC
omes quadratic in ya. Maximizing the magnitude of thehird term in Eq. (95) as a function of ya yields the addi-ional condition for maximal squeezing,
hus the power splitting of the LOs for maximal squeez-ng is slightly different from 50% and also slightly differ-nt from that for maximal classical deamplification.
Use of this optimal two-frequency LO yields the follow-ng optimal squeezing result:
To make a connection with previous work, we show thatn the limit of degenerate operation we reach the same re-ult as obtained by Shapiro and Boivin.30 By placing theump at the zero-dispersion wavelength of the fiber, ourolutions are then identical to those in a dispersionless fi-er (assuming that the higher-order terms in an expan-ion of � are negligible). We thus use the expressions inqs. (34)–(37) for �j and j. To evaluate the limit as theetuning approaches zero, we expand the antisymmetricmaginary part of �� as an odd power series around �0 and take the Taylor-series expansion of the exponen-
ial in nth=1/ �exp�� / �kT�−1�. Then taking the limit as→0, the squeezing approaches the limit derived by Sha-
iro and Boivin30 for the fully degenerate case. This limits
Sopt�� → 0� = 1 + 2�NL��NL +2kT�i��0�
��0�
− 2�NL�1 + ��NL +2kT�i��0�
��0�2�1/2
, �99�
here �NL= �0�Ap�2L is the nonlinear phase shift and �i��0�s the slope of the imaginary part of �� as �→0.
In Fig. 5 the main features of this squeezing theory arellustrated for a lossless fiber. First, it is clear that the Ra-
an effect degrades the achievable amount of squeezing.n addition, it can be seen by comparing the dashedurves (phase matched), dashed–dotted curves (partiallyhase matched) and other curves �k=0� that when theaman effect is included, phase matching leads to worsequeezing instead of improved squeezing predicted by annstantaneous nonlinearity model. Finally, by comparinghe Raman-included dashed and dashed–dotted curves,e see that when k�0, precise unequal balancing of the
elative power splitting of the two LO frequencies is re-uired. When this optimum splitting is achieved, thequeezing can be seen to approach a constant value. How-ver, when k=0, the possible amount of squeezing is not
ounded, and squeezing asymptotically scales as 1/�NL,hich is explained by the quadratic scaling of the four-ave-mixing process and the linear scaling of the Ramanrocess. The hard limit of constant squeezing with an in-reasing nonlinear drive in the phase-matched (k�0nd �=0) case occurs when the excess noise due to the Ra-an effect balances the strength of the squeezing process.
. CONCLUSIONe have presented a quantum theory of parametric am-
lification in a ��3� nonlinear medium that includes theoninstantaneous response of the nonlinearity and the ef-ect of distributed linear loss. We have applied this theoryo nondegenerate phase-sensitive amplification andeamplification and have found the input conditions forptimal amplification and deamplification. We have alsoound the input conditions for operation at the minimumF, which for detunings �1 THz is predicted to be0.4 dB in the high-gain limit for FOPAs made from typi-
al dispersion-shifted fibers. We anticipate that nonde-enerate PSAs will produce record NF performance, ashey allow circumvention of GAWBS noise that is presentn the degenerate case. We have also presented a theory ofondegenerate squeezing and found the optimal cw LOor a lossless FOPA with noninstantaneous nonlinear re-ponse. Our results agree with the limit previously foundy Shapiro and Boivin30 as degeneracy is reached. Awayrom degeneracy and with a nonzero linear phase mis-atch, we have shown that optimal squeezing occurs in a
ispersive fiber when k�0 reaches a constant limit, un-ike the 1/�NL scaling that occurs when the linear phase
ismatch vanishes.
CKNOWLEDGMENTShis work was supported by the U.S. Army Research Of-ce, under Multidisciplinary University Research Initia-ive grant DAAD19-00-1-0177.
Correspondence may be directed to kumarpnorthwestern.edu. P. L. Voss is now with the School oflectrical and Computer Engineering, Georgia Institutef Technology, Atlanta, Georgia, Georgia Tech Lorraine,etz, France. K. G. Köprülü is now affiliated with the De-
artment of Electrical and Electronics Engineering,OBB Economics and Technology University, Sögtözüad. No. 43 06530 Ankara, Turkey.
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610 J. Opt. Soc. Am. B/Vol. 23, No. 4 /April 2006 Voss et al.
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