232 J. Opt. Soc. Am. B/Vol. 14, No. 2 /February 1997 J. H. Shapiro and A. Shakeel
Optimizing homodyne detection of quadrature-noise squeezing by local-oscillator selection
Jeffrey H. Shapiro and Asif Shakeel*
Department of Electrical Engineering and Computer Science, and Research Laboratory of Electronics,Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307
1. INTRODUCTIONIn optical homodyne detection1 the field to be measured iscombined on the surface of a photodetector with the fieldof a strong local-oscillator (LO) laser of the same centerfrequency. The resulting baseband photocurrent con-tains a frequency-translated replica of that signal-fieldcomponent that is copolarized and coherent in space andtime with the LO. In the semiclassical theory of homo-dyne detection2 the signal and LO fields are classical en-tities, and the baseband photocurrent comprises the clas-sical beat term just described plus additive whiteGaussian LO shot noise. Quantum mechanically,3,4 how-ever, homodyne detection is a phase-sensitive, LO mode-matched measurement of the signal beam’s field operator.So the noise seen in homodyne detection is really signal-field quantum noise. When the total field illuminatingthe photodetector is in a Glauber coherent state or in aclassically random mixture of such states, the quantumtheory reproduces the semiclassical statistics. The mosttelling validation of quantum homodyne theory is, there-fore, the observation of quadrature-noise squeezing, i.e.,homodyne measurements of appropriate (squeezed-state)light sources that yield photocurrent noise levels lowerthan values permitted by semiclassical theory; see, e.g.,Ref. 5.It has long been recognized that optimizing an optical
homodyne measurement requires careful choice of the LOfield; see, e.g., Refs. 6–8 for a sample of the extensive lit-erature on coherent laser radar mixing efficiency. Mostsuch LO-optimization studies address the semiclassicalregime, wherein the task is to match best the LO field tothe polarization and spatiotemporal characteristics of theclassical signal field impinging upon the photodetector.
0740-3224/97/020232-18$10.00
LO selection is also crucial in quadrature-noise squeezingmeasurements. Nevertheless, although specific sugges-tions have been made for improved LO selections for par-ticular three-wave mixing9 and four-wave mixing10
squeezing experiments, a general theory has not been es-tablished for LO optimization in quadrature-noise squeez-ing applications. In the present paper we correct thisomission. Our approach separates the generation ofsqueezed light from its detection. In particular, we per-form LO optimization under the presumption that wehave perfect knowledge of signal field’s normally orderedand phase-sensitive covariance functions. Because thesecovariance functions must be found from an understand-ing of the signal-field generation process, we illustrateour methodology with two four-wave mixing examples:continuous wave (cw) squeezed-state generation in a bulkKerr medium with a Gaussian spatial-response functionand pulsed squeezed-state generation in a single-modeoptical fiber whose Kerr nonlinearity has a noninstanta-neous response function. The former, which representsan idealized version of bulk-medium self-phase modula-tion in the four-wave mixing limit, is presented for its il-lustrative value. The latter, which draws on recenttheory for quantum propagation in lossless dispersionlessfiber,11,12 is of considerable physical importance. First, itreveals the nonoptimality of previously suggested LO se-lections for fiber-based squeezing experiments. More im-portantly, it identifies a new Raman-squeezing regime forfiber four-wave mixing.
2. LOCAL-OSCILLATOR OPTIMIZATIONTHEORYConsider the idealized balanced homodyne detection ar-rangement shown in Fig. 1, in which signal and LO
J. H. Shapiro and A. Shakeel Vol. 14, No. 2 /February 1997 /J. Opt. Soc. Am. B 233
beams of common center frequency vc are combined,through a lossless 50/50 beam splitter, upon the surfacesof a pair of identical unity quantum-efficiency photodetec-tors and the resulting photocurrents are subtracted. Werely on the photon-units formulation of quantumphotodetection,4 and for notational simplicity we employa scalar wave treatment. In particular, we useES(x, t)exp(2ivct) and ELO(x, t)exp(2ivct) to denote thepositive-frequency signal and LO field operators enteringthe beam splitter, where x 5 (x, y) is the two-dimensional (2-D) coordinate vector transverse to theirnominal directions of propagation and t is time. Thesefield operators have units of (photons/m2s)1/2 and delta-function commutators:
@EK~x, t !, EK† ~x8, t8!# 5 d~x 2 x8!d~t 2 t8!
for K 5 S, LO. (1)
For generality, the signal will be allowed to be in an ar-bitrary quantum state. The LO, however, will be as-sumed to be in a coherent state, uELO(x, t)&, of sufficientstrength that the statistics of the differenced photocur-rent, i(t), are identical to those of the quantum measure-ment:
i~t ! [ 2q ReF EAd
dxES~x, t !ELO* ~x, t !G ,2 T/2 < t < T/2, (2)
where q is the electron charge, Ad is the sensitive regionof the photodetector, and @2T/2, T/2# is the observationinterval.4
Equation (2) is the foundation for all that follows, so itis appropriate to indicate that little generality has beenlost in starting from this expression. For example, it is asimple matter to modify Eq. (2) to account for such non-idealities as asymmetry in the beam splitter, detectorswith subunity quantum efficiencies, and classical excessnoise on the LO.4 The infinite electrical bandwidth pre-sumed in Eq. (2) is also easily dealt with by invoking thefinite postdetector bandwidths of our quadrature-noisemeasurement schemes. Neglecting polarization, on theother hand, is a more significant matter. In the paraxial-propagation regime the vector wave version of Eq. (2) is
Fig. 1. Balanced homodyne detection. Quantum signal and LOfields, with operator-valued complex envelopes ES(x, t) andELO(x, t), are combined through a lossless 50/50 beam splitter onthe surfaces of two photodetectors, D1 and D2. The resultingphotocurrents are then subtracted.
i~t ! [ 2q ReF EAd
dxES~x, t ! • ELO* ~x, t !G ,2 T/2 < t < T/2, (3)
where ES(x, t) and ELO(x, t) are 2-D vector quantum andclassical fields, respectively. It follows that the scalarwave theory is a natural consequence of the vector formal-ism when the excited (nonvacuum-state) component ofES(x, t) is copolarized with the classical LO field,ELO(x, t). Perfect polarization matching is possible, inprinciple, when the excited component of ES(x, t) is fullypolarized for all x P Ad and t P @2T/2, T/2#. Whenthe excited component of ES(x, t) is only partially polar-ized, the scalar wave formulation is, in general, insuffi-cient to achieve full LO optimization. Nonetheless, manysqueezed-state experiments will not suffer from polariza-tion randomness, so they can be dealt with by means ofscalar theory. Moreover, the vector wave development ofLO optimization for quadrature-noise squeezing does notprovide much additional physical insight, and it comes atthe expense of appreciable notational complication.Thus, with no further apology, we shall hew to the scalartheory path. In essence, we are presuming that the ex-cited component of ES(x, t) is copolarized with ELO(x, t).
A. General Spatiotemporal FormulationSuppose that the signal field in Fig. 1 and Eq. (2) hasknown normally ordered and phase-sensitive covariancefunctions, namely,
KS~n !~x, t, x8, t8! [ ^DES
† ~x, t !DES~x8, t8!&, (4)
KS~ p !~x, t, x8, t8! [ ^DES~x, t !DES~x8, t8!&, (5)
where
DES~x, t ! [ ES~x, t ! 2 ^ES~x, t !& (6)
is the fluctuation operator associated with the signal field.In terms of these covariance functions it is a simple mat-ter to calculate the covariance function of the differencedphotocurrent:
Ki~t, t8! [ ^D i~t !D i~t8!& (7)
5 q2EAd
dxuELO~x, t !u2d~t 2 t8!
1 2q2 ReF EAd
dxEAd
dx8ELO~x, t !
3 KS~n !~x, t, x8, t8!ELO* ~x8, t8!G
1 2q2 ReF EAd
dxEAd
dx8ELO* ~x, t !
3 KS~p !~x, t, x8, t8!ELO* ~x8, t8!G . (8)
When ES(x, t) is in a coherent state, its normally orderedand phase-sensitive covariances vanish, i.e.,
KS~n !~x, t, x8, t8! 5 K
S~ p !~x, t, x8, t8! 5 0, (9)
234 J. Opt. Soc. Am. B/Vol. 14, No. 2 /February 1997 J. H. Shapiro and A. Shakeel
reducing Eq. (8) to
K iCS~t, t8! [ ^D i~t !D i~t8!&
5 q2 EAd
dxuELO~x, t !u2d~t 2 t8!, (10)
where we have used the superscript CS to emphasize thatthis result assumes a coherent-state signal. Equation(10) provides the coherent-state (LO shot-noise) limit thatmarks the divide between semiclassical and quantumphotodetection: Any signal field for which Ki(t, t8)2 Ki
CS(t, t8) is not a positive semidefinite kernel is non-classical. In this regard it is worth remembering thatKS
(n)(x, t, x8, t8) is positive semidefinite for all signalstates. Hence signal states that do not lead to positivesemidefinite Ki(t, t8) 2 K i
CS(t, t8) must have nonzerophase-sensitive covariance functions. As we shall seemore explicitly below, this phase sensitivity of the non-classical signature in photocurrent covariance is due tothe Heisenberg uncertainty principle.Rather than focusing on photocurrent covariances, we
concentrate on the variance of the associated homodynecharge measurement:
Q [ E2T / 2
T / 2
dt i~t !. (11)
For an arbitrary signal-field state the homodyne chargevariance can be found from the covariance function of thedifferenced photocurrent by means of
sQ2 [ var~Q ! 5 E
2T /2
T /2
dt E2T/2
T /2
dt8Ki~t, t8!. (12)
The coherent-state value for this variance follows fromEq. (10):
sCS2 5 E
2T /2
T /2
dt E2T /2
T /2
dt8K iCS~t, t8! 5 q2NLO , (13)
where
NLO [ E2T/2
T/2
dt EAd
dxuELO~x, t !u2 (14)
is the number of photons in the classical LO field. Notethat all signal-field states with finite average photonnumbers have positive s Q
2 values and all classical-statesignal fields satisfy sQ
2 > sCS2 . We can obtain further in-
sight into this result by introducing Q' , the homodynecharge measurement that is in quadrature with Q:
Q' [ 2q ImF E2T /2
T/2
dt EAd
dxES~x, t !ELO* ~x, t !G ,(15)
whose associated measurement variance we denote bysQ'
2 . The charge measurements Q and Q' are canoni-cally conjugate, and sQ
2 sQ'
2 > sCS4 is their associated
Heisenberg inequality. Clearly, phase-sensitive noise isa prerequisite to reaching the nonclassical, sQ
2 , sCS2 re-
gime.Within the preceding homodyne detection framework
the LO optimization problem is easy to state: Given thesignal field’s normally ordered and phase-sensitive cova-riance functions, find the normalized (unity square inte-gral) LO field:
jLO~x, t ! [ ELO~x, t !/ANLO,
x P Ad , t P @2T/2, T/2#, (16)
that minimizes the normalized charge variance:
sN2 [ sQ
2 /sCS2 (17)
5 1 1 2 E2T /2
T/2
dt E2T/2
T/2
dt8 EAd
dx EAd
dx8
3 KS~n !~x, t, x8, t8!jLO~x, t !jLO* ~x8, t8!
1 2 ReF E2T/2
T/2
dt E2T/2
T/2
dt8 EAd
dx EAd
3 dx8KS~ p !
~x, t, x8, t8!jLO* ~x, t !jLO* ~x8, t8!G .(18)
By introducing the real-valued vector–matrix notation
j~x, t ! [ FRe@jLO~x, t !#Im@jLO~x, t !#G , (19)
K~x, t, x8, t8! [ FKR~n !~x, t, x8, t8! 1 KR
~ p !~x, t, x8, t8!
KI~ p !~x, t, x8, t8! 2 KI
~n !~x, t, x8, t8!
KI~n !~x, t, x8, t8! 1 KI
~ p !~x, t, x8, t8!
KR~n !~x, t, x8, t8! 2 KR
~ p !~x, t, x8, t8!G , (20)
we can rewrite Eq. (18) in the following quadratic form:
sN2 5 1 1 2 E
2T /2
T/2
dt E2T /2
T /2
dt8 EAd
dx EAd
3 dx8j T~x, t !K~x, t, x8, t8!j~x8, t8!, (21)
where KR(k) and KI
(k), for k 5 n, p, are the real and theimaginary parts, respectively, of the normally orderedand the phase-sensitive signal-field covariances and T de-notes transpose.The power of the vector–matrix version of the LO opti-
mization problem is that its formal solution is easily de-veloped, as we now demonstrate. All we need are someresults from linear integral equation theory. The nor-mally ordered covariance is Hermitian, i.e.,
KS~n !~x8, t8, x, t ! 5 KS
~n !* ~x, t, x8, t8!, (22)
and the phase-sensitive covariance is symmetric, viz.,
KS~ p !~x8, t8, x, t ! 5 KS
~ p !~x, t, x8, t8!, (23)
implying that the real-valued matrix kernel,K(x, t, x8, t8), is also symmetric:
K~x8, t8, x, t ! 5 KT~x, t, x8, t8!. (24)
J. H. Shapiro and A. Shakeel Vol. 14, No. 2 /February 1997 /J. Opt. Soc. Am. B 235
As a result, K(x, t, x8, t8) has real-valued vector eigen-functions, $fn(x, t), n 5 1, 2, 3, ...%, with associated real-valued scalar eigenvalues, $ln , n 5 1, 2, 3, ...%, that onecan find by solving the Fredholm integral equation13,14
E2T /2
T /2
dt8 EAd
dx8K~x, t, x8, t8!fn~x8, t8!
5 lnfn~x, t !, x P Ad ,
t P @2T/2, T/2#, n 5 1, 2, 3, ... . (25)
Eigenfunctions associated with distinct eigenvalues areorthogonal; linearly independent eigenfunctions associ-ated with degenerate eigenvalues can be orthogonalizedthrough a Gram–Schmidt procedure. Moreover, everyeigenfunction can be normalized to have unity square-integral. Thus we can assume that
E2T /2
T /2
dt EAd
dxfnT~x, t !fm~x, t ! 5 dnm ,
n, m 5 1, 2, 3, ... . (26)
By including eigenfunctions whose associated eigenvaluesare zero, we can take the $fn(x, t)% to be complete,namely,
(n51
`
fn~x, t !fnT~x8, t8! 5 d~x 2 x8!d~t 2 t8!,
x, x8 P Ad , t, t8 P @2T/2, T/2#. (27)
The preceding eigenfunction properties permit us to ex-pand an arbitrary j(x, t) in terms of the $fn(x, t)%, i.e.,
j~x, t ! 5 (n51
`
jnfn~x, t !, (28)
where
jn [ E2T /2
T /2
dt EAd
dxfnT~x, t !j~x, t !, (29)
(n51
`
j n2 5 E
2T /2
T /2
dt EAd
dxjT~x, t !j~x, t ! 5 1. (30)
Substituting Eq. (28) into Eq. (21), we find that
sN2 5 1 1 2 (
n51
`
j n2ln . (31)
Inasmuch as sN2 . 0 prevails for all signal-beam states
with finite average photon numbers and for all LO fields,Eq. (31) reveals that ln . 21/2 for all n. In conjunctionwith Eq. (30), Eq. (31) then implies that
smin2 [ min
j
~sN2 ! 5 1 1 2 min
n~ln!, (32)
with this optimum s N2 value achieved when j(x, t) is the
fn(x, t) with the minimum eigenvalue. Aside from thenontrivial problem of solving Eq. (25) for this optimumeigenfunction–eigenvalue pair, our LO optimization prob-lem is now solved and its performance quantified. InSections 3 and 4 we consider explicit solutions to Eq. (25)
in two physically based special cases: cw squeezed-stategeneration in a bulk Kerr medium with a Gaussianspatial-response function and pulsed squeezed-state gen-eration in a single-mode optical fiber whose Kerr nonlin-earity has a noninstantaneous response function. Beforeconsidering these examples, we produce specializations ofEq. (25) that provide more-appropriate starting points forthem.
B. Stationary Signal, Spectrum Analysis LimitOur general LO optimization formalism encompassesmultiple-spatial-mode nonstationary signal fields, in thatwe have not constrained the normally ordered and phase-sensitive covariance functions beyond the essential re-quirement that sN
2 . 0 prevail for all LO fields. Manysqueezed-state experiments employ cw sources, for whichthese covariance functions will be stationary in time.This means that KS
(n)(x, t 1 t, x8, t) and KS( p)(x, t
1 t, x8, t) are independent of t for all x, x8, and t. Thecovariance functions then have simple Fourier decompo-sitions:
KS~n !~x, t 1 t, x8, t ! 5 E
2`
` dv
2pSS
~n !~x, x8, v!
3 exp~2ivt!, (33)
KS~p !~x, t 1 t, x8, t ! 5 E
2`
` dv
2pSS
~p !~x,x8, v!
3 exp~2ivt!, (34)
in terms of the signal field’s frequency-dependent, nor-mally ordered and phase-sensitive spatial covariancesSS(n)(x, x8, v) and SS
( p)(x, x8, v).In cw-source experiments it is the conventional ap-
proach to use a cw LO field too—in our notation thismeans that ELO(x, t) 5 ELO(x) is time independent—even though, as will be seen below, better performancecan sometimes be obtained with a modulated LO. For acw LO, Eqs. (8), (33), and (34) can be combined to showthat the differenced photocurrent’s covariance isstationary—Ki(t 1 t, t) is independent of t for allt—and hence is characterized by its spectral densityfunction
Si~v! 5 E2`
`
dtKi~t!exp~ivt! (35)
5 q2PLO 1 2q2PLO EAd
dx EAd
dx8
3 SS~n !~x, x8, v!jLO~x!jLO* ~x8!
1 2q2PLO ReF EAd
dx EAd
dx8
3 SS~p !~x, x8, v!j LO* ~x!j LO* ~x8!G , (36)
where we have introduced the normalized (unity squareintegral) cw LO field
jLO~x! [ ELO~x!/APLO for x P Ad , (37)
236 J. Opt. Soc. Am. B/Vol. 14, No. 2 /February 1997 J. H. Shapiro and A. Shakeel
with
PLO [ EAd
dxuELO~x!u2 (38)
being the LO’s classical photon flux.Equation (36) is significant because squeezing experi-
ments with stationary signal-beam statistics and cw LOfields usually measure quadrature-noise squeezing bymeans of spectrum analysis of the differenced photocur-rent noise; see, e.g., Ref. 5. Specifically, they comparethe spectral density of the differenced photocurrent noise,Si(v), with its coherent-state (LO shot-noise) limit,
SiCS~v! 5 q2PLO . (39)
Equation (39) is the familiar Schottky formula of semi-classical photodetection.1 For a classical-state signalfield, Si(v) > q2PLO prevails at all frequencies v and forall choices of the normalized LO field, jLO(x).Quadrature-noise squeezing—manifest as Si(v), q2PLO for some v and jLO(x)—is then a nonclassicalsignature. Evidently, in the stationary signal spectrum-analysis limit the LO optimization problem is to find thejLO(x) and v that minimize the normalized spectrum ofthe differenced photocurrent, SN(v) [ Si(v)/q
2PLO .We can quickly construct a formal solution to this prob-lem by paralleling our optimization approach for the gen-eral case.First, we introduce some vector–matrix notation by de-
fining
j~x! [ FRe@j LO~x!#Im@j LO~x!#G , (40)
S~x, x8, v! [ FSR~n !~x, x8, v! 1 SR
~ p !~x, x8, v!
SI~ p !~x, x8, v! 2 SI
~n !~x, x8, v!
SI~n !~x, x8, v! 1 SI
~ p !~x, x8, v!
SR~n !~x, x8, v! 2 SR
~ p !~x, x8, v!G , (41)
where SR(k) and SI
(k) for k 5 n, p are the real and theimaginary parts, respectively, of the frequency-dependent, normally ordered and phase-sensitive signal-field spatial covariances. Next, using this notation, wewrite SN(v) in the quadratic form
SN~v! 5 1 1 2 EAd
dx EAd
dx8jT~x!S~x, x8, v!j~x8!.
(42)
For every v we have that S(x, x, v) is a symmetric, real-valued matrix kernel in x and x8. It therefore has a real-valued vector-eigenfunction–scalar-eigenvalue expansionof the form
S~x, x8, v! 5 (n51
`
ln~v!fn~x, v!fnT~x8, v!,
x, x8 P Ad , (43)
where, for each v, the $fn(x, v)% constitute a complete or-thonormal 2-D function set on x P Ad .We find the preceding frequency-dependent eigenfunc-
tions and eigenvalues by solving the Fredholm integralequation
EAd
dx8S~x, x8, v!fn~x8, v! 5 ln~v!fn~x, v!,
x P Ad , n 5 1, 2, 3, ... . (44)
Expanding j(x) in the fn(x, v) basis, we can mimic the ar-gument leading from Eq. (28) to Eq. (32) to show that
minj
@SN~v!# 5 1 1 2 minn
@ln~v!#, (45)
with this optimum SN (v) value achieved when j(x) is thefn(x, v) with the minimum eigenvalue over n for thegiven frequency v. To complete our stationary signalspectrum-analysis LO optimization we minimize over fre-quency to obtain
Smin [ minj,v
SN~v! 5 1 1 2 minn,v
@ln~v!#, (46)
where we achieve Smin by choosing j(x) to be the fn(x, v)that has the minimum eigenvalue over both n and v.The stationary signal spectrum-analysis LO optimiza-
tion that we have just performed can also be developed,with very little effort, from our general formulation.First, we substitute the Fourier decompositions from Eqs.(33) and (34) into our normalized charge variance for-mula, Eq. (18). Then, when we choose a normalized LOfield of the form
jLO~x, t ! 5 ~2/T !1/2 cos~v t !jLO~x!, t P @2T/2, T/2#,(47)
a straightforward calculation demonstrates that
limT→`
sN2 5 SN~v!, (48)
with SN(v) given by Eq. (42). Subsequent optimizationover jLO(x) completes the proof that our general formula-tion reproduces the stationary signal spectrum-analysisspecial case.15
There is an interesting byproduct of this sN2 → SN(v)
reduction, which arises from the fact that a constrainedoptimization cannot produce performance superior to thatobtained without the constraint. Consider a cw-sourceexperiment in which the signal field has stationary nor-mally ordered and phase-sensitive covariance functionsand the normalized charge variance is measured over theinfinite interval, 2` , t , `. The optimum spatiotem-poral normalized LO field, jLO(x, t) obtained by findingthe minimum eigenvalue solution to the stationary cova-riance form of Eq. (25) need not take the (constrained)spectrum-analysis form specified in Eq. (47). Indeed, inSection 5 we develop an explicit cw-source case in whichspectrum analysis is dramatically suboptimal: Ramansqueezing in single-mode fiber.
C. Single-Spatial-Mode LimitIn single-mode fiber-squeezing experiments the signalfield has only one nonvacuum-state spatial mode, as willthe optimum LO field; see, e.g., Refs. 16 and 17. It is
J. H. Shapiro and A. Shakeel Vol. 14, No. 2 /February 1997 /J. Opt. Soc. Am. B 237
then possible to reap substantial notational simplificationby focusing on the time-dependent parts of thenonvacuum-state signal and optimum LO spatial modes.To do so we write
EK~x, t ! 5 EK~t !z~x! 1 EKvac~x, t !, K 5 S, LO,
(49)
where z(x) is the normalized (unity square integral)c-number spatial mode of the nonvacuum-state parts ofthe signal and LO fields; the field operators ES(t) andELO(t) have units of (photons/s)1/2 and delta-functioncommutators:
@EK~t !, EK† ~t8!# 5 d~t 2 t8!, K 5 S, LO, (50)
and the field operators ESvac(x, t) and ELO
vac(x, t) are theunexcited (vacuum-state) terms necessary to ensure thatEq. (49) obeys the spatiotemporal commutator relationsgiven earlier in Eq. (1).With Ad large enough to encompass the entire z(x)
mode, single-spatial-mode homodyne detection yields adifferenced photocurrent operator given by
with the obvious definitions of the real and the imaginaryparts of the normally ordered and phase-sensitive covari-ances of ES(t) and of the LO’s normalized temporal mode.We achieve the minimum s N
2 by setting j(t) equal to theminimum-eigenvalue eigenfunction of the Fredholmequation
E2T/2
T/2
dt8K~t, t8!fn~t8! 5 lnfn~t !, t P @2T/2, T/2#,
n 5 1, 2, 3, ..., (54)
and it satisfies
smin2 [ min
j
~sN2 ! 5 1 1 2 min
n~ln!. (55)
Treatment of the cw-source version of the single-spatial-mode regime (with or without the spectrum-analysis mea-surement assumption) can be handled with the tech-niques used in Subsection 2.B.
3. FOUR-WAVE MIXING IN A BULK KERRMEDIUMTo exercise our LO optimization methodology we need ac-cess to physically relevant examples of signal-field nor-mally ordered and phase-sensitive covariance functions.
The simplest squeezed-state example arises from the spa-tiotemporal Bogoliubov transformation
ES~x, t ! 5 m~x, t !EP~x, t ! 1 n~x, t !EP† ~x, t !, (56)
where
um~x, t !u2 2 un~x, t !u2 5 1, x P Ad ,
t P @2T/2, T/2# (57)
and EP(x,t) is a pump-field operator in its vacuum state.Mathematically, Eqs. (56) and (57) represent a linear,phase-sensitive, commutator-preserving transformationfrom $EP(x, t), EP
† (x, t)% to $ES(x, t), ES† (x, t)%. Physi-
cally, such a transformation describes idealized three-wave or four-wave mixing configurations that have re-sponse functions that are pointwise in space andinstantaneous in time, have no loss, and do not incurpump depletion. The implied normally ordered andphase-sensitive signal-field fluctuations are delta corre-lated in space and time, viz.,
KS~p !~x, t, x8, t8! 5 m~x, t !n~x, t !d~x 2 x8!d~t 2 t8!,
(59)
from which the normalized charge variance can be shownto become
sN2 5 E
2T/2
T/2
dtEAd
dxum~x, t !jLO* ~x, t !
1 n* ~x, t !jLO~x, t !u2. (60)
From Eq. (60) it is apparent that
sN2 > E
2T/2
T/2
dtEAd
dx@ um~x, t !u 2 un~x, t !u#2ujLO~x, t !u2
(61)
> minx, t
@ um~x, t !u 2 un~x, t !u#2. (62)
Equality in relation (61) occurs when, for each x P Adand t P @2T/2, T/2#, the phase of jLO(x, t) is chosen tominimize the integrand on the right-hand side of Eq. (60).Equality is approached in relation (62) when
ujLO~x, t !u2 → d ~x 2 x0!d ~t 2 t0!, (63)
where
@ um~x0 , t0!u 2 un~x0 , t0!u#2 5 minx, t
@ um~x, t !u 2 un~x, t !u#2.
(64)
This result can also be divined from our general LO opti-mization formalism, as we now show.According to Subsection 2.A, the optimum LO should
be the minimum-eigenvalue eigenfunction of Eq. (25).For the covariances given in Eqs. (58) and (59) thisamounts to finding the minimum ln solution to
238 J. Opt. Soc. Am. B/Vol. 14, No. 2 /February 1997 J. H. Shapiro and A. Shakeel
L~x, t !j~x, t ! 5 lnj~x, t !, x P Ad ,
t P @2T/2, T/2#, (65)
where
L~x, t ! [ F un~x, t !u2 1 Re@m~x, t !n~x, t !# Im@m~x, t !n~x, t !#Im@m~x, t !n~x, t !# un~x, t !u2 2 Re@m~x, t !n~x, t !#G . (66)
Because the $ln% must be independent of x and t, Eq. (65)has, in general, no nontrivial (nonzero eigenvalue) solu-tions with square integrable j(x, t). This absence of so-lutions is a legacy of the nonhomogeneous, nonstationary,delta-function covariance functions that we are workingwith.14 If m and n are independent of space and time—making Eq. (56) the simplest multimode version of Yuen’saS 5 maP 1 naP
† single-mode, squeezed-state-producingBogoliubov transformation18—Eq. (65) will have non-trivial solutions. In that case the optimum normalizedcharge variance is min(sN
2 ) 5 (umu 2 unu)2, in agreementwith relation (62). Now let us return to m and n beingspace–time dependent. Physically, Eq. (56) affords eachspace–time point (x, t) its own independent Bogoliubovtransformation. Thus, if we arbitrarily allow the $ln% inEq. (65) to be (x, t) dependent, we find that
minx, t
@sN2 ~x, t !# [ 1 1 2 min
x, t@ln~x, t !# (67)
5 @ um~x0 , t0!u 2 un~x0 , t0!u#2. (68)
Furthermore, if m(x, t) and n(x, t) are continuous at(x0 , t0), there are sequences of unity square integralj(x, t) whose normalized charge variances converge tothis optimum value. Basically, the functions in these se-quences have the phase characteristic needed to achieveequality in relation (61) and squared magnitudes that in-creasingly approximate d (x 2 x0)d (t 2 t0). In essence,this says that the optimum LO field probes the low-noisequadrature component of the strongest [maximumun(x, t)u] of the Bogoliubov transformations that constituteEq. (56).By the preceding limiting argument, LO optimization
for the delta-function covariances of Eqs. (58) and (59), asdirectly carried out in relations (60)–(64), is subsumed byour general framework. In the rest of the paper we shallstudy squeezed-state generators with nonpointwise spa-tial responses, noninstantaneous time responses, or both,
Fig. 2. Schematic of FWM in a bulk Kerr medium with a Gauss-ian spatial-response function.
beginning, in the present section, with four-wave mixing(FWM) in a bulk Kerr medium.
A. Derivation of the Covariances
Consider the propagation of a 1z-going, cw, linearly po-larized beam through a bulk Kerr medium with a Gauss-ian spatial-response function, as sketched in Fig. 2 anddescribed, for the moment, by classical nonlinear optics.Modifying the standard theory of self-focusing19 to incor-porate a spatial-response function for the Kerr effect, wehave that
¹T2E~x, z ! 1 2ik
]E~x, z !
]z
5 22kk E dx8g~x 2 x8!uE~x8, z !u2E~x, z !,
0 < z < L (69)
is the equation of motion for the slowly varying spatialcomplex-field envelope in the L–m-long nonlinear me-dium. Here ¹T
2 is the transverse Laplacian operator, k isthe wave number at the light beam’s frequency vc , k. 0 is the Kerr medium’s plane-wave nonlinear phaseshift per unit length per unit photon flux density,20 and
g~x! [ exp~2x • x/ rG2 !/prG
2 (70)
is the normalized spatial-response function of the Kerrmedium. Our assumption that g(x) has this Gaussianform is made for analytical convenience; standard self-focusing theory employs g(x) 5 d (x), i.e., a pointwise re-sponse. As shown in Fig. 2, we shall pump this Kerr me-dium with a strong pump field E(x, 0) 5 EP(x) atz 5 0, and we shall use the output field ES (x)[ E(x, L) as the signal for our homodyne squeezingmeasurement.We are not interested in the self-focusing behavior that
Eq. (69) predicts. Instead, we make a thin-medium ap-proximation, wherein the transverse Laplacian term canbe neglected, reducing Eq. (69) to a bulk-medium self-phase modulation (SPM) form
]E~x, z !
]z5 ik E dx8g~x 2 x8!uE~x8, z !u2E~x, z !,
0 < z < L. (71)
In conjunction with the input–output field definitionsgiven in the previous paragraph, the solution to Eq. (71)is
ES~x! 5 expF ikL E dx8g~x 2 x8!uEP~x8!u2GEP~x!.
(72)
To pass to the quantum form of Eq. (72) we assume thatEP(x, t) and ES(x, t), the delta-function commutatorquantum pump and signal fields, comprise strong c-
J. H. Shapiro and A. Shakeel Vol. 14, No. 2 /February 1997 /J. Opt. Soc. Am. B 239
number mean fields EP(x) and ES(x) plus quantum fluc-tuation operators DEP(x, t) and DES(x, t). The formerare related by Eq. (72). For the latter we assume opera-tion within the FWM regime, wherein linearization of thequantum equation of motion can be performed around themean field solution,21 and assert that
DES~x, t ! 5 expF ikL E dx8g~x 2 x8!uEP~x8!u2G3 FDEP~x, t ! 1 ikL E dx8EP~x!
3 g~x 2 x8!EP* ~x8!DEP~x8, t !
1 ikL E dx8EP~x!
3 g~x 2 x8!EP~x8!DEP† ~x8, t !G . (73)
Equation (73) is a lossless, linear, phase-sensitive,commutator-preserving transformation from $DEP(x, t),DEP
† (x, t)% to $DES(x, t), DES† (x, t)%, hence it is quantum-
mechanically consistent and should produce aquadrature-noise squeezed DES(x, t) when DEP(x, t) hascoherent-state quantum noise.We need explicit expressions for the signal field’s nor-
mally ordered and phase-sensitive covariances to proceedwith LO optimization for this Kerr-medium example.Without appreciable loss of generality, let us suppose thatthe pump field is in the multimode coherent state speci-fied by the classical mean field
EP~x! 5 AIP exp~2x • x/rP2 !, (74)
where IP . 0 is the pump’s peak photon-flux density.We then have that
KS~n !~x, t, x8, t8! 5 FNL
2 eP~x!eP~x8!d~t 2 t8!E dx9
3 g~x 2 x9!eP2 ~x9!g~x8 2 x9!, (75)
KS~p !~x, t, x8, t8! 5 expF iFNL E dx9g~x 2 x9!eP
2 ~x9!G3 expF iFNLE dx9g~x8 2 x9!eP
2 ~x9!G3 d~t 2 t8!F iFNLeP~x8!
3 g~x8 2 x!eP~x! 2 FNL2 E dx9
3 g~x 2 x9!eP2 ~x9!g~x8 2 x9!G , (76)
where
FNL [ kIPL (77)
is the peak nonlinear phase shift incurred by the meanpump field and
eP~x! [ exp~2x • x/rP2 ! (78)
is the pump field’s spatial profile normalized to unitypeak photon-flux density.
B. Local-Oscillator Optimization and SqueezingPerformanceThere were three main assumptions used in reachingEqs. (75) and (76): the Kerr nonlinearity has a Gaussianspatial response, there is no self-focusing, and the FWMapproximation to SPM is accurate. Our concern here isnot with carefully delineating the region of validity ofthese assumptions; indeed, in studying the LO-optimizedsqueezing performance obtained from Eqs. (75) and (76)we shall employ these covariances at FNL values that vio-late the no-self-focusing condition. But, having used thepreceding assumptions to develop a pair of signal-field co-variances that are consistent with quantum mechanics atall FNL values—they never violate Smin . 0—we have anopportunity to explore the utility of our LO optimizationmethodology in a reasonable (though not rigorously justi-fied) physical setting.Equations (75) and (76) are stationary in time, so we
shall address LO optimization for them in the spectrumanalysis limit established in Subsection 2.B. However,before we proceed to the solution of the relevant Fredholmequation, several points are worth noting. First, thedelta-function time dependence in the stationary covari-ances that we are working with confers two useful prop-erties on the LO optimization problem at hand: no perfor-mance loss is incurred by restricting ourselves to thespectrum analysis limit and the kernel, eigenfunctions,and eigenvalues in Eq. (44) will be frequency indepen-dent, obviating the need for the optimization over v in ourspectrum analysis. Second, the classical nonlinear phaseshift of the mean field in Eq. (76) can be rotated out byoptimizing over
jLO8 ~x! [ jLO~x!expF2iFNL E dx8g~x 2 x8!eP2 ~x8!G ;
(79)
the real-valued 2-D vector form of jLO8 (x) will be denotedj8(x). Finally, we allow Ad to become the entire x plane,which amounts to assuming that the photodetectors inFig. 1 have active areas much larger than the extent ofthe LO field that we employ.After the above preliminaries, finding the optimum
jLO8 (x) is reduced to solving for the minimum-eigenvalueeigenfunction of the Fredholm integral equation
E dx8S8~x, x8!fn~x8! 5 lnfn~x!, (80)
where
240 J. Opt. Soc. Am. B/Vol. 14, No. 2 /February 1997 J. H. Shapiro and A. Shakeel
S8~x, x8! [ eP~x!eP~x8!F 0FNLg~x8 2 x!
FNLg~x8 2 x!
2FNL2 E dx9g~x 2 x9!eP
2 ~x9!g~x8 2 x9!G . (81)
With this optimum LO choice, Smin 5 1 1 2 minn(ln) isthe frequency-independent optimum squeezing that re-sults. Thankfully, by our judicious assumption of Gauss-ian forms for g(x) and eP(x), closed-form solutions to Eq.(80) are available. With these solutions we examine thedependence of LO-optimized squeezing on the two key di-mensionless physical parameters of our bulk Kerr-medium problem: FNL , the pump’s peak nonlinearphase shift, and r [ rP /rG , the ratio of the pump beam’sbeam waist to the Kerr medium’s spatial-response length.Our route to solving Eq. (80) begins with the recogni-
tion that the off-diagonal elements of S8(x, x8) from Eq.(81) equal FNLKG(x, x8)KG (y, y8) and the nonzero diag-onal element of this matrix equals 2FNL
2 K G(2)(x, x8)
3 K G(2)(y, y8), where
KG~2 !~x, x8! [ E dx9KG~x, x9!KG~x8, x9!,
2` , x, x8 , ` (82)
and KG(x, x8) is the symmetric, positive-definite spatialkernel
KG~x, x8! [ exp$2~x/rP!2
2 @~x 2 x8!/rG#2 2 ~x8/rP!2%/prG2 ,
2` , x, x8 , `. (83)
Next, with $fGn(x), n 5 0, 1, 2, ...% to denote the com-plete orthonormal eigenfunctions of KG(x, x8) and$lGn ,n 5 0, 1, 2, ...% to denote their associated eigenval-ues, a straightforward calculation will verify that
j8nm~1!~x! [ Fsin~unm!
cos~unm!GfGn~x !fGm~ y !,
n, m 5 0, 1, 2, ..., (84)
jnm8~2!~x! [ F cos~unm!
2sin~unm!GfGn~x !fGm~ y !,
n, m 5 0, 1, 2, ... (85)
are the complete orthonormal eigenfunction solutions toEq. (80), whose associated eigenvalues are
lnm~6! [ 6 unnmu~ umnmu 6 unnmu!, (86)
where tan(2unm) [ 1/unnmu, mnm [ 1 1 nnm , and nnm[ iFNLlGnlGm .When jnm8
(6)(x) is used as the pump-phase-rotated, nor-malized LO field for optical homodyne detection ofDES(x, t), the resulting frequency-independent normal-ized noise spectral density of the differenced photocurrentis
SN~6! 5 1 1 2lnm
~6! 5 ~ umnmu 6 unnmu!2. (87)
We minimize Eq. (87) by choosing the minus and maxi-mizing unnmu over n and m. Because maxn,m(unnmu)
5 maxn(FNLlGn2 ) and because we can arrange the $lGn ,
n 5 0, 1, 2, ...% to occur in decreasing order, we see thatthe optimum normalized LO field is
with Feff [ FNLlG02 being the effective nonlinear phase
shift obtained with the optimum LO. Making use of Eq.(7.374.8) of Ref. 22, we find that the KG eigenfunctionsare Gauss–Hermite functions given by
fGn~x ! 5exp@2~x/rGP!2#Hn~A2x/rGP!
~prGP2 /2!1/4~2nn! !1/2
, (91)
with eigenvalues given by
lGn 5r~1 1 r2 2 A1 1 2r2!n/2
~1 1 r2 1 A1 1 2r2!~n11 !/2, (92)
where rGP [ rP /(1 1 2r2)1/4 and the arrangement in or-der of decreasing eigenvalue has been performed. Put-ting all the preceding results together, we get the follow-ing explicit form for the optimum LO for our Gaussianspatial-response, bulk Kerr-medium FWM example:
jopt~x! 5 exp@iFNL~rGP /rP!2 exp~2x • xrGP2 /rP
4 !#
3 exp~2iu00!exp~2x • x/rGP2 !/AprGP
2 /2,
(93)
Feff 5 r2FNL /~1 1 r2 1 A1 1 2r2! (94)
for its effective nonlinear phase shift. Note that Eq. (93)has an explicit spatially varying phase that is FNL depen-dent, and an absolute phase u00 that is also FNL depen-dent. Although the latter can easily be accommodated inexperiments, the former is more problematic: It impliesthat a significantly different LO—not just a different LOabsolute phase shift—is required to stay on the Smin ver-sus FNL curve as the pump’s peak photon flux density isincreased.Before we further quantify the preceding results, it is
germane to examine more closely how the KGeigenfunctions–eigenvalues decompose the Gaussianresponse FWM problem at hand. Let us expandDEP(x, t) in the spatial basis $fGn(x)fGm(y), n, m5 0, 1, 2, ...%, and let us expand DES(x, t) in the spatialbasis $fGnm8 (x), n, m 5 0, 1, 2, ...%, where
J. H. Shapiro and A. Shakeel Vol. 14, No. 2 /February 1997 /J. Opt. Soc. Am. B 241
fGnm8 ~x! [ expF iFNL E dx8g~x 2 x8!
3 eP2 ~x8!GfGn~x !fGm~ y !. (95)
Fig. 3. Optimum squeezing performance for the bulk Kerr-medium example as a function of the effective nonlinear phaseshift.
Fig. 4. Ratio of the optimum LO’s effective nonlinear phaseshift to the pump’s peak nonlinear phase shift, Feff /FNL , versusnormalized pump-beam beam waist, r [ rP /rG , for the bulkKerr-medium example.
Fig. 5. Ratio of the optimum LO’s normalized beam waist,rGP /rG , to the pump beam’s normalized beam waist, r[ rP /rG , for the bulk Kerr-medium example.
Using DEPnm(t) and DESnm(t) to denote the respectiveq-number expansion coefficients that result, we havefrom Eq. (73) that
DESnm~t ! 5 mnmDEPnm~t ! 1 nnmDEPnm† ~t !. (96)
Equation (96) is a modal Bogoliubov transformation,analogous to the pointwise form seen in Eq. (56); hence itexplains the expressions found above for our optimumKerr-medium LO and its squeezing performance.In Fig. 3 we have plotted the squeezing in decibels:
smin [ 10 log10~1 1 2Feff2 2 2FeffA1 1 Feff
2 !, (97)
versus the effective nonlinear phase shift, Feff . In Fig. 4we plot Feff versus the pump beam’s normalized beamwaist, r 5 rP /rG . In Fig. 5 we plot the optimum LObeam’s normalized beam waist, rGP /rG , versus r. Col-lectively these figures summarize the optimum squeezingbehavior of the Kerr-medium example that we have ana-lyzed in this section. Figure 3 and Eq. (90) demonstratethat strong squeezing occurs when Feff > 1, with Smin' 1/4Feff
2 5 1/4FNL2 lG0
4 for Feff @ 1. Figure 4 and then 5 0 case from Eq. (92) indicate that Feff < FNL , withequality in the limit r → `. Physically, this limit corre-sponds to a plane-wave pump beam, rP → `, for which itis no surprise that the effective nonlinear phase obtainedwith the optimum LO field approaches the plane-wavenonlinear phase shift, FNL . In the plane-wave pumpbeam limit, Fig. 5 shows that the optimum LO is also aplane wave, because we have rGP /rG → (r /A2)1/25 (rP /A2rG)1/2→ ` in this regime. When r ! 1 pre-vails, the pump field has a spatial-frequency content ofmuch higher bandwidth than that of the Kerr medium’sspatial-response function. As a result, Fig. 4 that showsFeff /FNL→ 0 as r → 0.
4. NONINSTANTANEOUS FOUR-WAVEMIXING IN SINGLE-MODE FIBERWe now turn to a more physically significant applicationof our LO optimization theory: pulsed FWM in a single-mode optical fiber. FWM in optical fiber provided one ofthe earliest successful demonstrations of quadrature-noise squeezing.16 Significant advantages accrue in fibersqueezing if the cw pump is replaced by a periodic streamof short pulses, with the fiber arranged in a Sagnac-loopMach–Zehnder interferometer and the output pumppulse—the interferometer’s bright fringe—used as the LOfield for balanced homodyne detection.23 For atransform-limited Gaussian pump pulse, however, suchan arrangement can produce at most 8.73 dB ofquadrature-noise reduction, owing to its LO choice.Pulse compression10 and weak-signal FWM (along thelines of the spatial LO choice for three-wave mixing em-
Fig. 6. Schematic of FWM in a lossless, dispersionless, single-mode fiber with a noninstantaneous Kerr nonlinearity.
242 J. Opt. Soc. Am. B/Vol. 14, No. 2 /February 1997 J. H. Shapiro and A. Shakeel
ployed in Ref. 9) have been suggested as improved LOsources for fiber FWM squeezing. We shall use ourtheory from Section 2 to seek the optimum LO selection,and we shall compare the performance of our LO choicewith those of the bright-fringe, pulse-compression, andweak-FWM LO’s.
A. Derivation of the CovariancesThe fiber FWM configuration that we consider is shown inFig. 6. A single-spatial-mode quantum pump beam,EP(t), is applied at z 5 0 to an L–m-long lossless, dis-persionless fiber. The single-spatial-mode quantumfield, ES(t), that emerges at z 5 L is used as the signalfield for a balanced homodyne detector. Several recenttheoretical studies have shown that proper quantum de-scription of self-phase modulation in the Fig. 6 configura-tion requires a causal, noninstantaneous Kerr-effect re-sponse function.11,24–26 Of these developments, themodel of Boivin et al.11 is the most comprehensive in thatit provides a continuous-time, commutator-preservingquantum theory that explicitly accounts for the Ramannoise that accompanies a causal, noninstantaneous Kerrnonlinearity. Most theoretical studies of quantum self-phase modulation tacitly assume that the noninstanta-neous nature of the Kerr nonlinearity can be ignoredwithin the FWM regime if the classical pump field has amuch lower bandwidth than the Kerr response. Such isnot the case: Raman noise limits fiber FWM squeezingeven in the cw-source, spectrum analysis limit.12 Thuswe employ the pulsed version of the fiber FWM modelfrom Ref. 12 to obtain the signal-field covariance func-tions needed for LO optimization.We suppose that the pump field is in a coherent state,
uEP(t)&. Under conditions similar to those discussed inRef. 12 we have that the resulting signal field ES(t) has ac-number mean value given by
ES~t ! 5 expF ikL E dt8h~t 2 t8!uEP~t8!u2GEP~t !
(98)
and a q-number fluctuation component DES(t) that obeys
DES~t ! 5 expF ikL E dt8h~t 2 t8!uEP~t8!u2GFDEP~t !
1 ikL E dt8EP~t !h~t 2 t8!EP* ~t8!DEP~t8!
1 ikL E dt8EP~t !h~t 2 t8!EP~t8!DEP† ~t8!
1 iU~t !EP~t !G , (99)
where we have suppressed the group-velocity delay. Inthese expressions k . 0 is the fiber’s cw nonlinear phaseshift per unit length per unit photon flux; h(t) is the nor-malized, real-valued, causal response function of the fi-ber’s Kerr nonlinearity; and U(t) is a Hermitian fluctua-tion operator, representing Raman noise, that commuteswith DEP(t8) and DEP
† (t8) for all t and t8.Equation (99) preserves the delta-function commuta-
tors required of our single-spatial-mode photon-units fieldoperators [see Eq. (50)] when
where Hi(v) is the imaginary part of the Kerr nonlineari-ty’s frequency response:
H~v! [ E dth~t !exp~iv t !. (102)
The Raman noise is assumed to have stationary, zero-mean statistics with a symmetrized correlation functiongiven by
KU~t! [ ^U~t 1 t!U~t ! 1 U~t !U~t 1 t!&/2
5 ~kL/2p!E dvHi~v!coth~\v/2kBTR!cos~vt!,
(103)
where kBTR is the thermal fluctuation energy of a noise-oscillator reservoir at absolute temperature TR .Mathematically, a causal, noninstantaneous response
function has a Fourier transform whose real and imagi-nary parts are a Hilbert transform pair.27 This is whyEqs. (101) and (103) have nontrivial effect when h(t)Þ d(t), i.e., when the Kerr nonlinearity is not instanta-neous. Physically, a positive Hi(uvu) represents a Ramangain, in that it is an imaginary component of the nonlin-ear refractive index that, for a cw pump, couples energyfrom the pump frequency vc to the Raman-shifted fre-quency vc 2 uvu, as shown, e.g., in Ref. 12.As discussed in Ref. 11, Hi(v) . 0 must prevail for v
. 0 to ensure that Eq. (103) constitutes a proper repre-sentation of a symmetrized quantum-correlation function.We also require that Hi(v) be absolutely integrable, to en-sure that ^U2(t)& , `. Following Boivin et al.,11 we posita simple single-resonance model for h(t) and H(v):
h~t ! 5v0 exp~2g t/2!sin~Av0
2 2 g2/4t !
Av02 2 g2/4
, t > 0,
(104)
H~v! 5v02
v02 2 v2 2 ivg
, 2` , v , `, (105)
with resonance frequency v0 and damping constant g.This model does not capture the full, multiresonance na-ture of the Kerr response function of fused-silica fiber(see, e.g., Ref. 28), but its simplicity will enable us to ex-hibit the essential physics of LO optimization without un-due numerical complications.In what follows we shall assume that the pump field’s
coherent-state eigenfunction is a transform-limitedGaussian pulse, EP (t) 5 APPeP(t), with peak classicalphoton flux PP and normalized temporal profile eP (t)5 exp@2(t/tP)
2#. Defining the peak pump nonlinearphase shift by
FNL [ kPPL, (106)
J. H. Shapiro and A. Shakeel Vol. 14, No. 2 /February 1997 /J. Opt. Soc. Am. B 243
we find that the fiber FWM signal field’s normally orderedand phase-sensitive temporal covariance functions takeforms similar to those seen for the spatial covariances inSubsection 3.A, i.e.,
where KU8 (t) [ KU(t)/kL.Because h(t) is a causal, noninstantaneous response
function, Eqs. (107) and (108) differ in two importantways from the bulk Kerr-medium FWM covariances inEqs. (75) and (76). First, of course, there is the presenceof a Raman-noise term in both covariances. Second,there is the fact that KH(t, t8) [ eP (t)h(t 2 t8)eP (t8) isnot a symmetric kernel [cf. Eq. (83), which demonstratesthat eP(x)g(x 2 x8) eP (x8) 5 KG(x, x8)KG(y, y8) is sym-metric]. Both of these effects are direct consequences ofh(t) being causal and noninstantaneous. Together theycombine to make explicit determination of the optimumLO and its performance much harder than it was in ourbulk-medium spatial example. Hence, in Subsection 4.Bwe retreat somewhat from full LO optimization by deriv-ing upper and lower bounds on optimum squeezing. Aswill be seen in Subsection 4.C, these bounds afford con-siderable insight into the LO optimization problem, withonly a modest investment in numerical evaluation.
B. Upper and Lower Bounds on Optimum SqueezingAs in Subsection 3.B it is convenient to rotate out thepump’s classical nonlinear phase at the outset. Thus weshall optimize over
and use j8(t) to denote the real-valued 2-D vector form ofjLO8 (t). We shall also allow the time duration of the LOfield to establish the integration interval for the homo-dyne charge measurement, so finding the optimumjLO8 (t) becomes finding the minimum-eigenvalue eigen-function of the Fredholm integral equation:
where $fHn(t),n 5 1, 2, 3, ...% and $FHn(t),n5 1, 2, 3, ...% are both complete orthonormal functionsets and the $lHn ,n 5 1, 2, 3, ...% are the associated non-negative singular values. We find the $FHn(t)% and the$lHn% by solving the Fredholm integral equation:
E dt8KH~2 !
~t, t8!FHn~t8! 5 lHnFHn~t !; (114)
the $fHn(t)% may then be obtained by
E dtKH~t, t8!FHn~t ! 5 AlHnfHn~t8!. (115)
Without loss of generality, we can assume that smin2 is
realized by using the following pump-phase-rotated LO;
jopt8 ~t ! [ F cos~u!c~t !2sin~u!C~t !G , (116)
where c (t) and C(t) are real-valued, unity square inte-gral functions that, along with u, are implicit functions ofFNL . Expanding c(t) in the $fHn(t)% basis and C(t) inthe $FHn(t)% basis, we have that
smin2 5 1 2 2 sin~2u!FNL(
n51
`
cnCnAlHn
1 2@1 2 cos~2u!#S FNL2 (
n51
`
Cn2lHn 1 FNLlRoptD ,
(117)
with the obvious notation for the expansion coefficientsand
244 J. Opt. Soc. Am. B/Vol. 14, No. 2 /February 1997 J. H. Shapiro and A. Shakeel
lRopt [ E dt E dt8C~t !eP ~t !KU8 ~t 2 t8!eP ~t8!C~t8!.
(118)
We can choose the absolute phase of c (t) to ensure that(n51
` cnCnAlHn is nonnegative, so that 0 < u < p/4must hold. Then, using the Schwarz inequality, we canobtain the cn-independent lower bound:
smin2 > 1 2 2 sin~2u!FNLS (
n51
`
Cn2lHnD 1/2
1 2@1 2 cos~2u!#
3 S FNL2 (
n51
`
Cn2lHn 1 FNLlRoptD .
(119)
Minimizing the right-hand side of relation (119) over uthen yields
smin2 > 1 1 2FNL
2 (n51
`
Cn2lHn 1 2FNLlRopt
2 2FFNL2 (
n51
`
Cn2lHn 1 S FNL
2 (n51
`
Cn2lHn
1 FNLlRoptD 2G 1/2. (120)
To proceed further, we define lRopt5 lRopt /(n51
` Cn2lHn and then use straightforward dif-
ferentiation to prove (for FNL . 0) that the right-handside of relation (120) is strictly increasing with increasinglRopt . By the following steps, we can establish a lowerbound on lRopt without explicitly needing C(t):
lRopt 5E dt E dt8C~t !eP ~t !KU8 ~t 2 t8!eP ~t8!C~t8!
E dt E dt8C~t !eP ~t !KH~2 !~t 2 t8!eP ~t8!C~t8!
(121)
>E dt E dt8C~t !eP ~t !KU8 ~t 2 t8!eP ~t8!C~t8!
E dt E dt8C~t !eP ~t !F E dt9h~t 2 t9!h~t8 2 t9!G eP ~t8!C~t8!
(122)
5
E dvuC8~v!u2Hi~v!coth~\v/2kBTR!
E dvuC8~v!u2uH~v!u2(123)
> minv
@Hi~v!coth~\v/2kBTR!/uH~v!u2# (124)
5 minv
@vg coth~\v/2kBTR!/v02# 5 lRL [ 2kBTRg/\v0
2, (125)
where C8(v) is the Fourier transform of C(t)eP(t), thesubscript RL means Raman lower bound and the mini-mum in Eq. (125) occurs at v 5 0.To complete our lower-bound derivation, we replace
lRopt in relation (120) with its lower bound (LB),lRL (n51
` Cn2lHn . Then we use differentiation to prove
(for FNL . 0) that the resulting smin2 bound is minimized
by maximizing (n51` Cn
2lHn , whence, assuming that the$lHn% are arranged in decreasing order,
smin2 > s LB
2 [ 1 1 2FNL2 lH1 1 2FNLlH1 lRL
2 2AFNL2 lH1 1 ~FNL
2 lH1 1 FNLlH1lRL!2.(126)
To place an upper bound on smin2 , we can evaluate sN
2
for any particular LO field. The following choice is espe-cially apt:
j8~t ! 5 F cos~ u !fH1~t !2sin~ u !FH1~t !
G , (127)
with u chosen to minimize the resulting normalizedcharge variance. With this LO choice we find that
smin2 < sUB
2 [ 1 1 2FNL2 lH1 1 2FNLlRU
2 2AFNL2 lH1 1 ~FNL
2 lH1 1 FNLlRU!2,
(128)
where
lRU [ E dt E dt8FH1~t !eP ~t !
3 KU8 ~t 2 t8!eP ~t8!FH1~t8!, (129)
and the subscript RU means Raman upper bound.At this juncture, several points should be made about
our bounds. First, it is the Raman noise, which arisesfrom our having Hi(v) Þ 0, that forces 0 , lRL< lRU /lH1. Indeed, were we to set these Raman coeffi-
cients to zero, our bounds would coincide, yielding
smin2 5 1 1 2Feff
2 2 2AFeff2 1 Feff
4 , (130)
J. H. Shapiro and A. Shakeel Vol. 14, No. 2 /February 1997 /J. Opt. Soc. Am. B 245
with Feff [ FNLlH1, in perfect analogy with our bulkKerr-medium result from Eq. (90).The second point to note about sLB
2 and sUB2 , where the
subscript UB indicates the upper bound, is that evaluat-ing them is a modest numerical task: for the lower bound,we need only lH1, the maximum singular value ofKH(t, t8); for the upper bound we need also FH1(t), andwe must perform the double integral in Eq. (129).Another important feature of these bounds is their a-
symptotic behavior. The lower bound satisfies s LB2
' lRL /FNL , and the upper bound obeys sUB2
' lRU /FNLlH1, in the FNL @ 1 regime. These Raman-limited asymptotes are inferior to the Smin ' 1/4FNL
2 as-ymptotic behavior that holds in the instantaneous-interaction limit with a cw source and a spectrum-analysis measurement. Raman-limited squeezingbehavior for the cw-source–spectrum-analysis arrange-ment—in which a noninstantaneous Kerr nonlinearityleads to SN(0) } 1/FNL at high FNL values—was pre-dicted by Shapiro and Boivin.12 We shall revisit the cwcase in Section 5.Our final observation concerns what our bounds imply
about the optimum LO field: Whenever s LB2 ' sUB
2
holds, we can say that jopt8 (t) is given by Eq. (127). Fur-thermore, to make the functional form of this fieldexplicit—once lH1, lRU , and FH1(t) have been found—requires only use of Eq. (115), to get fH1(t), and thetrivial evaluation of u from tan(2u) 5 AlH1FNL /(lH1FNL
2
1 lRUFNL).
C. Local-Oscillator Performance ComparisonWe want to compare the normalized charge varianceachieved by use of the optimum LO—as bounded by sLB
2
and sUB2 —with the normalized charge variances sBF
2 ,sPC2 , and sFWM
2 that result, respectively, from use of thebright-fringe LO:
where the C ’s are normalizing constants, r . 1 is thepulse-compression factor, and f is an input phase shift.Remember that h(t) is still the simple single-resonancenormalized response function given earlier in Eq. (104)and that ep(t) is still the normalized Gaussian pumppulse specified above Eq. (106). Moreover, in evaluating
sPC2 and sFWM
2 we shall optimize the corresponding LO’sover the pulse-compression factor r and the input phaseshift f.We have already noted that the single-resonance model
does not reflect the multiresonance nature of the Kerrnonlinearity in single-mode fiber. Our purpose, however,is to study an illustrative physically based example. Itturns out that all the calculations that we must performcan be couched in terms of the following four dimension-less parameters: FNL [ kPPL, V0 [ 2v0 /g, G [ g tP/2,and h [ \/2kBTRtP . We shall be plotting our normal-ized charge variances versus FNL . To accentuate the ef-fect of the noninstantaneous Kerr nonlinearity on LOoptimization—without incurring an extraordinary com-putational burden—we somewhat cavalierly assume thatV0 5 4, G 5 4, and h 5 0.07 in all that follows.29 InFig. 7 we have plotted h(t) and eP(t) versus t/tP ; this fig-ure demonstrates that the Kerr nonlinearity is obviouslynoninstantaneous for the assumed pump pulse.To find lH1, FH1(t), and fH1(t) we employ an iterative
procedure based on the singular-value decomposition, Eq.(113). First, we numerically integrate to find
LH [ (n51
`
lHn 5 E dtKH~2 !~t, t !, (134)
Fig. 7. Normalized-time plots of the normalized Kerr-nonlinearity response function, (tP /LH)
1/2h(t), and the normal-ized pump pulse, eP(t).
Fig. 8. Normalized-time plot of the maximum-singular-value in-put eigenfunction, t P
1/2fH1(t).
246 J. Opt. Soc. Am. B/Vol. 14, No. 2 /February 1997 J. H. Shapiro and A. Shakeel
so that we can work with the normalized KH(t, t8) ker-nel:
kH~t, t8! [ KH~t, t8!/ALH (135)
5 (n51
`
AlHn8 FHn~t !fHn~t8!, (136)
where $lHn8 [ lHn /LH ,n 5 1, 2, 3, ...% behaves as aprobability distribution, i.e., each term is bounded be-tween zero and one and they sum to unity. Starting froma seed function, fH1
Fig. 9. Normalized-time plot of the maximum-singular-valueoutput eigenfunction, t P
1/2FH1(t).
Fig. 10. Normalized-frequency plots of the frequency content ofthe maximum-eigenvalue eigenfunctions, tP
21/2M(v) [ tP21/2
3 ufH1(v)u 5 tP21/2uFH1(v)u; the imaginary part of the Kerr non-
linearity’s frequency response, Hi(v); and the Raman noise’sspectral density, Hi(v)coth(\v/2kBTR).
and then
E dtkH~t, t8!FH1~n !~t ! 5 AlH1
~n11 !fH1~n11 !~t8!,
n 5 1, 3, 5, ... . (138)
In these iterations the integrations are performed nu-merically, the $FH1
(n)(t)% and the $fH1(n)(t8)% are normalized,
numerically, to have unity square integrals, and the pro-cess is continued until it converges. As long as the seedfunction fH1
(0) (t8) is not orthogonal to fH1(t), this proce-dure accomplishes our objective.30
In Figs. 8 and 9 we have plotted our numerically deter-mined fH1(t) and FH1(t) versus t/tP . Interestingly,these functions turn out to be a time-reversal pair, viz.,fH1(t) 5 FH1(2t). In Fig. 10 we have plotted the com-mon magnitude of their Fourier transforms, M(v)[ ufH1(v)u 5 uFH1(v)u, versus vtP . Also included inthis figure are Hi(v) and Hi(v) coth (\v /2kBTR). Fig-ures 7–10 reveal that the pump-phase-rotated LO thatachieves sN
2 5 sUB2 is very different from the Gaussian
pump pulse. They also show that this pump-phase-rotated LO has substantial frequency content in commonwith Hi(v).
Fig. 11. Normalized charge variances (in decibels) sK[ 10 log10(sK
2 ), versus nonlinear phase shift, FNL : bright-fringe (BF) LO, K 5 BF; weak FWM LO, K 5 FWM; pulse-compression (PC) LO, K 5 PC; upper bound on optimum-LO per-formance, K 5 UB; lower bound on optimum-LO performance,K 5 LB.
J. H. Shapiro and A. Shakeel Vol. 14, No. 2 /February 1997 /J. Opt. Soc. Am. B 247
The dominant singular value that we obtained from ournumerics is lH1 5 2.7246, and our FH1(t) leads to lRU5 1.8139. Thus the high-FNL asymptote of our upperbound is sUB
2 ' 0.666/FNL . Because our normalized-parameter choices make the corresponding asymptote ofour lower bound s LB
2 ' 0.446/FNL , we see that for allFNL our bounds are within 1.74 dB of each other. Hencethe pump-phase-rotated LO constructed from Figs. 8 and9 by means of Eq. (127) is essentially optimum. It re-mains to see how its performance stacks up against theother LO contenders from Eqs. (131)–(133).In Fig. 11 we have plotted the normalized charge vari-
ances for the bright-fringe LO, the pulse-compression LOwith r 5 6 (essentially the optimum r), and thef-optimized weak-FWM LO. This figure also containsour upper and lower bounds. It is easily shown,31 ana-lytically, that sBF
2 and sPC2 both approach subunity con-
stants in the limit FNL→ `; Fig. 11 hints at these behav-iors. On the other hand, sFWM
2 eventually increases withincreasing FNL ; Fig. 11 shows a glimmer of this behavior.We conclude that none of the previously suggested(bright-fringe, pulse-compression or FWM) local oscilla-tors is optimum over the full range of FNL values that wehave considered. Nevertheless, pulsed fiber FWM ex-periments have usually been confined to FNL< 2p, so the pulse-compression and FWM LO’s shouldnot be dismissed in real applications. Indeed, the pulse-compression LO slightly outperforms the LO that gener-ates our upper bound for 0.45 < FNL < 5.75. However,the weak-FWM LO, which is also quasi-optimal in this re-gime, is probably the most practical alternative. This isbecause our optimum LO depends in detail on the Kerrresponse function—which is coarsely approximated by thesingle-resonance form that we are using—and generatinga pulse-compressed LO with the proper pump-phase rota-tion is not easily accomplished.32
5. RAMAN SQUEEZING IN SINGLE-MODEFIBEROur final task in this paper will be to revisit the cw-sourcefiber FWM problem addressed in Ref. 12. When thespectrum-analysis measurement is employed, the LO-optimized normalized homodyne spectrum at frequency vturns out to be
SN~v! 5 1 1 2FNL2 uH~v!u2
1 2FNLHi~v!coth~\v/2kBTR!
2 2$FNL2 Hr
2~v! 1 @FNL2 uH~v!u2
1 FNLHi~v!coth~\v/2kBTR!#2%1/2, (139)
where Hr(v) [ Re@H(v)#. At v 5 0, Eq. (139) reduces tothe form shown on the right-hand side of relation (126)with lH1 5 1 (Ref. 12); for high FNL this gives SN(0)' 0.446/FNL . In contrast, for v . 0 we can show thatSN(v) → @Hi(v)/uH(v)u#2 . 0 as FNL→ `.31 This sug-gests that Raman gain (and its associated noise) is hostileto squeezing. Support for this assertion can be had byevaluation of Eq. (139) at v 5 v0. Inasmuch as the Kerrnonlinearity’s frequency response is purely imaginary at
this frequency—H(v0) 5 iHi(v0)—we find that SN(v0)5 1; i.e., there is no squeezing at v 5 v0.It may be appealing to argue that no squeezing can be
had at the Raman-resonance frequency v0 because thereis no real-valued nonlinear refractive index at this fre-quency. Any such argument, unfortunately, is an intrin-sically flawed consequence of the spectrum-analysis mea-surement assumption. To see that this is so, we needonly let ep(t) 5 1 in relations (126) and (128) to realizethat smin
2 } 1/FNL when FNL @ 1 for a cw source mea-sured by its optimum LO. Evidently, the spectrumanalysis measurement is not optimum for v . 0. It isgermane, therefore, to seek the optimum LO for the cwsource.When ep(t) 5 1, the kernel in Eq. (110) is stationary
and is given by
K8~t!
5 F 0
FNLh~t!
FNLh~2t!
2FNL2 E dth~t 1 t!h~t ! 1 2FNLKU8 ~t!G .
(140)
For vT @ 1 the functions
jv8~1!~t ! [ A2
T F sin~uv!cos~vt !cos~uv!cos~vt 2 av!G ,
t P @2T/2, T/2#, (141)
jv8~2!~t ! [ A2
T F cos~uv!cos~vt !2sin~uv!cos~vt 2 av!G ,
t P @2T/2, T/2# (142)
are then eigenfunctions of Eq. (110), where uH(v)uexp(iav)is the polar form of H(v) and
tan~2uv! 5FNLuH~v!u
FNL2 uH~v!u2 1 FNLHi~v!coth~\v/2kBTR!
.
(143)The normalized charge variances produced by use of theseeigenfunctions as LO’s are
s 62 ~v! 5 1 1 2FNL
2 uH~v!u2
1 2FNLHi~v!coth~\v/2kBTR!
6 2$FNL2 uH~v!u2 1 @FNL
2 uH~v!u2
1 FNLHi~v!coth~\v/2kBTR!#2%1/2, (144)
which shows that jv8(2)(t) leads to squeezing at observa-
tion frequency v, with
s 22 ~v! ' Hi~v!coth~\v/2kBTR!/uH~v!u2FNL ,
(145)
when FNL @ 1, in keeping with the behavior expectedfrom our smin
2 bounds. Relations (124) and (125) showthat the minimum slope for this high-FNL asymptote oc-curs at v 5 0. But, because maxv@uH(v)u2# does not occurat v 5 0, we know that minv@s2
2 (v)# does not occur at v5 0 for all values of FNL . For the normalized param-eters that we have assumed, i.e., V0 5 4 and hG5 0.28, it turns out that zero-frequency spectrum analy-
248 J. Opt. Soc. Am. B/Vol. 14, No. 2 /February 1997 J. H. Shapiro and A. Shakeel
sis is essentially optimum for cw-source squeezing bymeans of fiber FWM. This is illustrated in Fig. 12, wherewe have plotted SN(0) 5 s 2
2 (0) and s22 (v0) versus FNL .
In this plot minv@s22 (v)# would be indistinguishable from
SN(0), although for 0 , FNL , 1.8 we find thatminv@s2
2 (v)# occurs at v . 0.33
Even though optimum LO squeezing at the Raman fre-quency v0 does not outperform zero-frequency spectrumanalysis, it is remarkable that the former is within 1.2 dBof the latter over the range shown in Fig. 12, given thatSN(v0) 5 1 for all FNL . It is easy to understand whythe spectrum-analysis measurement is not optimum forv . 0 and why the jv8
(2)(t) local oscillator is. Fouriertransforming the cw pump version of Eq. (99) shows that
D ES~v! [ E dtDES~t !exp~ivt !
5 exp~iFNL!$@1 1 iFNLH~v!#D EP~v!
1 iFNLH~v!D EP† ~2v! 1 iAFNLU8~v!%,
(146)
where U8(v) is the Fourier transform of U8(t)[ U(t)/(kL)1/2. Both the spectrum analysis and opti-mum LO measurements at frequency v sense noise thatis a linear combination of fluctuations embedded inD ES(v) and D ES(2v). The spectrum-analysis measure-ment comprises an equal-amplitude measurement ofthese fluctuations, whereas the optimum LO from Eq.(142) constitutes an unequal-amplitude measurement ofthem. Phase is optimized in both cases, but the couplingasymmetry that occurs in Eq. (146) owing to the Hermit-ian nature of H(v) makes the unequal-amplitude mea-surement superior. Indeed, the contrast between the twomeasurements can be pronounced; cf. SN(v0) 5 1 withs 22 (v0) from Fig. 12.Squeezing by means of fiber FWM at zero frequency re-
lies on a real-valued frequency response, i.e., H(0)5 Hr(0). Squeezing at v0 , however, is due to a purelyimaginary frequency response, H(v0) 5 iHi(v0), which
Fig. 12. Cw squeezing curves versus nonlinear phase shift,FNL : Raman-peak frequency, sR [ 10 log10@s2
we have already noted represents a Raman gain. It istherefore fitting to refer to our s 2
2 (v0) , 1 curve in Fig.12 as a theoretical prediction of Raman squeezing in op-tical fiber.Our final comments concern the practicality of the LO
from Eq. (142). Although knowledge of H(v) is necessaryto calculate jv8
(2)(t) from Eq. (142), all that is reallyneeded to implement this LO for a given v is an appara-tus capable of generating an optical beam with frequencycomponents vc 6 v whose amplitudes and phases can becontrolled. Recent progress in optical frequencydivision34 and optical frequency comb generation35 sug-gests that LO’s of this class—even with v/2p values aslarge as the frequency shift of the peak Raman gain 13.2THz—may be realizable.
ACKNOWLEDGMENTSJ. H. Shapiro acknowledges useful technical discussionswith L. G. Joneckis and P. Kumar. He is grateful to J. K.Bounds for identifying and correcting several significanterrors in the original manuscript.
7. B. J. Rye, ‘‘Antenna parameters for incoherent backscatterheterodyne lidar,’’ Appl. Opt. 18, 1390–1398 (1979).
8. J. H. Shapiro, ‘‘Heterodyne mixing efficiency for detector ar-rays,’’ Appl. Opt. 26, 3600–3606 (1987).
9. O. Aytur and P. Kumar, ‘‘Squeezed-light generation with amode-locked Q-switched laser and detection by using amatched local oscillator,’’ Opt. Lett. 17, 529–531 (1992).
10. J. H. Shapiro and L. G. Joneckis, ‘‘Enhanced fiber squeez-ing via local-oscillator pulse compression,’’ in Proceedings of1994 IEEE Nonlinear Optics (Institute of Electrical andElectronics Engineers, New York, 1994), pp. 347–349.
11. L. Boivin, F. X. Kartner, and H. A. Haus, ‘‘Analytical solu-tion to the quantum field theory of self-phase modulationwith a finite response time,’’ Phys. Rev. Lett. 73, 240–243(1994).
12. J. H. Shapiro and L. Boivin, ‘‘Raman-noise limit on squeez-ing in continuous-wave four-wave mixing,’’ Opt. Lett. 20,925–927 (1995).
13. H. L. Van Trees, Detection, Estimation, and ModulationTheory (Wiley, New York, 1968), Part I, Chap. 3.
14. Strictly speaking, we should require that K(x, t, x8, t8) besquare integrable over Ad 3 @2T/2, T/2# to vouchsafe theeigenfunction–eigenvalue properties that we shall assert.For finite area photodetectors and finite measurement in-tervals, however, the principal pathological cases that can
J. H. Shapiro and A. Shakeel Vol. 14, No. 2 /February 1997 /J. Opt. Soc. Am. B 249
occur in Eq. (25) arise from covariance functions that aredelta correlated in space, time, or both. In Section 3 weshall see how such a case can be handled directly from Eq.(18), with a solution that can be understood from Eq. (25).
15. Note that the cos(vt) term in Eq. (47) can be phase shiftedby an arbitrary amount without invalidating Eq. (48).Physically, this is because the spectrum analysis measure-ment is insensitive to radio-frequency (frequency v) phaseshifts.
16. R. M. Shelby, M. D. Levenson, R. G. DeVoe, S. H. Perlmut-ter, and D. F. Walls, ‘‘Broad-band parametric deamplifica-tion of quantum noise in an optical fiber,’’ Phys. Rev. Lett.57, 691–694 (1986).
17. K. Bergman and H. A. Haus, ‘‘Squeezing in fibers with op-tical pulses,’’ Opt. Lett. 16, 663–665 (1991).
18. H. P. Yuen, ‘‘Two-photon coherent states of the radiationfield,’’ Phys. Rev. A 13, 2226–2243 (1976).
19. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, NewYork, 1984), Chap. 17.
20. In anticipation of the photon-units formulation that weshall employ for the quantum theory to come, we are ex-pressing the classical fields in these units too; see Ref. 4.
21. See L. G. Joneckis and J. H. Shapiro, ‘‘Quantum propaga-tion in a Kerr medium: lossless, dispersionless fiber,’’ J.Opt. Soc. Am. B 10, 1102–1120 (1993), for an analogous re-duction of full SPM to FWM in a single-spatial-mode,multiple-temporal-mode quantum setting. Note, however,that this reference also indicates the necessity of the Kerrnonlinearity’s having a noninstantaneous temporal re-sponse to ensure that quantum SPM—the quantized ver-sion of Eq. (72)—properly includes the classical limit. Inthis regard, Eq. (73) should include a causal, noninstanta-neous temporal response, with its attendant Ramannoise.12 We shall eschew the inclusion of these temporaleffects in our bulk Kerr medium example because our prin-cipal aim, in this section, is to develop quantum-mechanically consistent signal-field covariances with whichto explore the issue of LO optimization. Our fiber FWMtreatment, in Section 4, will specifically address the nonin-stantaneous Kerr nonlinearity.
22. I. S. Gradshteyn and I. M. Rhyzhik, Table of Integrals, Se-ries, and Products (Academic, New York, 1980).
23. M. Shirasaki and H. A. Haus, ‘‘Squeezing of pulses in a non-linear interferometer,’’ J. Opt. Soc. Am. B 7, 30–34 (1990).
24. K. J. Blow, R. Loudon, and S. J. D. Phoenix, ‘‘Exact solutionfor quantum self-phase modulation,’’ J. Opt. Soc. Am. B 8,1750–1756 (1991).
25. K. J. Blow, R. Loudon, and S. J. D. Phoenix, ‘‘Quantumtheory of nonlinear loop mirrors,’’ Phys. Rev. A 45, 8064–8073 (1992).
26. L. G. Joneckis and J. H. Shapiro, ‘‘Quantum propagation ina Kerr medium: lossless, dispersionless fiber,’’ J. Opt. Soc.Am. B 10, 1102–1120 (1993).
27. W. M. Siebert, Circuits, Signals, and Systems (McGraw-Hill, New York, 1986), Chap. 15.
28. L. Boivin, ‘‘Sagnac-loop squeezer at zero dispersion with aresponse time for the Kerr nonlinearity,’’ Phys. Rev. A 52,754–766 (1995).
29. The peak Raman gain in fused silica fiber occurs at v0/2p5 13.2 THz, so our V0 choice implies that g 5 4.153 1013 s21. With this g value our G choice dictates thattP 5 0.19 ps. Computing the peak Raman gain from theseparameter values, along the lines laid out in Ref. 12, thenleads to a value ;10 times larger than the actual 1.23 10213 m/W value found in fused-silica fiber. Likewise,the parameter values that we are using predict a low-frequency Raman limit, kBTRg/\v0
2, that is ;10 timeslarger at TR 5 300 K than the 0.03 value of a typical fiber;cf. Ref. 12.
30. If the maximum singular value is degenerate, this proce-dure will still yield a lH1 value that we can use in our upperand lower bounds. With the Gaussian-pulse seed, conver-gence was complete at n 5 20. The same singular valueand eigenfunctions resulted when we used the second-orderGauss–Hermite seed: fH1
(0) (t8) 5 @1 2 2(t8/tP)2#exp@2(t8/
tP)2] / (9p tP
2 /32)1/4. When we used the first-order Gauss–Hermite function as the seed, i.e., fH1
(0) (t8)(t8/tP)3 exp@2(t8/tP)
2#/(ptP2 /32)1/4, our iteration procedure con-
verged to essentially the same lH1 value but had a some-what different set of eigenfunctions. Unlike the Fouriertransforms of the eigenfunctions obtained by use of thezeroth-order and second-order seed functions, those foundwith the first-order seed were zero at zero frequency. Inas-much as any odd symmetry seed function can be shown,analytically, to converge to eigenfunction estimates with nozero-frequency content, we believe that the results obtainedfrom the zeroth-order and second-order seeds correctly cap-ture the values that we need; viz., they provide accurate es-timates of $lH1 , fH1(t8), FH1(t)%.
31. A. Shakeel, ‘‘Enhanced squeezing in homodyne detectionvia local-oscillator optimization,’’ Master’s thesis (Massa-chusetts Institute of Technology, Cambridge, Mass., 1995).
32. Pulse compression without pump-phase rotation was previ-ously suggested as a practical LO choice, in which case com-pression factor much than the r 5 6 value that sufficeswith pump-phase rotation is apt to be needed.10
33. It should be remembered that Fig. 12 presumes a single-resonance H(v) with V0 [ 2v0 /g 5 4 and hG 5 \g/4kBTR 5 0.28. These parameters do not accurately repre-sent fused-silica fiber. Specifically, in conjunction withv0/2p 5 13.2 THz, the preceding parameters make thepeak Raman gain too large. Reducing the peak Ramangain, in our calculations, to a more realistic value wouldhave the effect of increasing s2
2 (v0) in Fig. 12 relative toSmin(0).
34. D. Lee and N. C. Wong, ‘‘Tunable optical frequency divisionusing a phase-locked optical parametric oscillator,’’ Opt.Lett. 17, 13–15 (1992).
35. L. R. Brothers, D. Lee, and N. C. Wong, ‘‘Terahertz opticalfrequency comb generation and phase locking of an opticalparametric oscillator at 665 GHz,’’ Opt. Lett. 19, 245–247(1994).
