1. INTRODUCTIONOptical parametric amplifiers (OPA’s) have been studiedboth theoretically and experimentally ever since the ad-vent of the field of nonlinear optics.1 Of particular inter-est from a practical point of view is their ability tostrongly amplify weak tunable signals with a fixed-frequency pump-laser beam.2–5 The aim of most of theresearch has been to optimize the OPA parameters to ob-tain the highest possible parametric gain per unit ofpump power. The OPA’s are generally used in thefrequency-nondegenerate mode, and the idler beam gen-erated during the amplification process is discarded.When operated near frequency degeneracy, i.e., when
the signal-beam frequency is one half the pump-beam fre-quency, the gain response of an OPA can become phasesensitive. Amplification of the signal beam then dependson the optical phase difference between the pump beamand the signal beam. This phase-sensitive gain behavioris intimately related to the ability of the OPA’s to gener-ate squeezed states of light.6 Furthermore, OPA’s areideal amplifiers.7 In the phase-sensitive configuration,OPA’s noise figure can approach 0 dB, whereas in thephase-insensitive configuration, they behave like ideal la-ser amplifiers with a 3-dB noise figure. This noise-figurebehavior has been verified in recent experiments that em-ployed a frequency-degenerate but polarization-nondegenerate type II phase-matched OPA, which couldbe operated in both the phase-insensitive as well as thephase-sensitive configurations.8 In the former case theOPA was shown to behave like a classical amplifier with anoise figure that approached 3 dB at high gains; and inthe latter case, it behaved approximately as a noiselessamplifier with a noise figure well below 3 dB.8 Moreover,the noiseless amplifier, when used as a preamplifier, was
shown to improve the quantum efficiency of photo-detection.9
On the other hand, experiments that have demon-strated the generation of quadrature-squeezed light bymeans of the traveling-wave OPA indicate that the ob-servable squeezing is largely limited by the effect of dif-fraction in the parametric process6,10 and by the mis-match between the generated squeezed mode and thelocal-oscillator (LO) mode that is employed in homodynedetection of squeezing.11 In one recent experiment, byusing an LO whose spatio-temporal profile is matched tothe generated squeezed mode, we were able to measure ahigh degree of quadrature squeezing (5.8 dB).12 Duringthe process of maximizing the observable squeezing wecarefully studied the gain behavior of a frequency-degenerate but polarization-nondegenerate type II phase-matched OPA, paying particular attention to the phase-sensitive gain response, and we compared the data withthe theory. In this paper we present the details of ourexperimental observations. We could easily obtain gainsof .100 (20 dB) in the amplified quadrature, which is lim-ited only by the available pump power. However, we findthat the phenomenon of gain-induced diffraction (GID)13
and phase fluctuations conspire to degrade deamplifica-tion in the quadrature that is orthogonal to the amplifiedquadrature. Our experimental observations are in goodagreement with the theory of an OPA in which theGaussian-beam nature of the various fields, along withthe diffraction of the signal beam and the fluctuations ofthe pump phase relative to the signal phase, are takeninto account.The paper is organized as follows: We start with the
experimental details in Section 2 and present the rel-evant OPA theory with Gaussian beams in Section 3.
1997 Optical Society of America
Choi et al. Vol. 14, No. 7 /July 1997 /J. Opt. Soc. Am. B 1565
The amplification response of the OPA, in both the phase-sensitive and the phase-insensitive configurations, is dis-cussed in Section 4, whereas the deamplification responseis covered in Section 5. Finally, the experimental resultsobtained with a decreased confocal distance are presentedin Section 6, followed by our conclusions in Section 7.
2. EXPERIMENTAL DETAILSFigure 1 shows a schematic of our experimental setup,which is similar to the one employed in our recent squeez-ing measurements.12 The traveling-wave OPA consistsof a type II phase-matched (u 5 90°, f 5 21.3°)14
KTiOPO4 (KTP) crystal that is pumped by the frequency-doubled 532-nm output from a Q-switched and mode-locked Nd-doped yttrium-aluminum-garnet (Nd:YAG) la-ser (Quantronix, Model 416). As Section 4 explains,three different crystal lengths are used, each with lateraldimensions of 3 mm33 mm. The Nd:YAG laser is Qswitched at a repetition rate of 1.1 kHz. The resultingQ-switched envelopes of the pump and the signal pulsesare ;145 ns and ;200 ns in duration, respectively. Themode-locked pulses underneath these Q-switched pulseenvelopes are estimated to be ;85 and ;120 ps long forthe pump and the signal beams, respectively.A portion of the fundamental Nd:YAG laser beam at
1064 nm is separated by a prism and used as an input tothe OPA. The intensity and the polarization of this beamare adjusted by a set of half-wave plates (HWP1 andHWP2) and polarizers (Ps and Pp). The intensity of thepump beam is adjusted with a half-wave plate (HWP) anda polarizer (PPs), and it is interferometrically alignedwith the input signal beam through a dichroic beam split-ter (DBS). Because of the critical nature of the type IIphase matching in the KTP crystal, the pump and the sig-nal beams walk away from the idler beam as they propa-gate through the crystal. As precompensation for thiswalkoff, the signal and the idler input beams are sepa-rated and recombined by polarizing beam splitters (PBS1and PBS2) before they enter the OPA. The pump and
Fig. 1. Schematic of the experimental setup to measure the gainresponse of the traveling-wave OPA. In the PSA configurationthe pump phase is controlled by applying a voltage to the PZT.
the signal–idler beams are focused into the KTP crystalby two identical 20-cm focal-length lenses. The locationof the resulting phase fronts and the beam waists relativeto the crystal are shown in Fig. 2. The e21 intensity ra-dius at the signal-beam waist is 44.3 mm. The pathlengths of the various beams from the Nd:YAG laser tothe OPA are made equal to each other within a few mil-limeters to ensure that the various mode-locked pulsesenter the KTP crystal at the same time. The pump andthe signal beams in the crystal are s polarized, whereasthe idler beam is polarized orthogonal (p polarization) tothe signal beam as a result of the type II phase matching.To operate the OPA as a phase-sensitive amplifier
(PSA), we adjust wave plate HWP2 to make the orthogo-nally polarized signal and idler input pulses to the OPAequal in amplitude and in phase with each other. Alter-natively, to operate the OPA as a phase-insensitive am-plifier (PIA), we block the idler path between beam split-ters PBS1 and PBS2 so that the idler input to the OPA iszero. We adjust the phase of the pump pulses relative tothe input signal and idler pulses by moving a mirror thatis mounted on a piezoelectric transducer (PZT). A feed-back servo is implemented to drive this PZT. The pumpphase can be set and locked to amplify or to deamplify theinput signal when the OPA is operated in the PSA mode.After the pump beam is dispersed away with the use of
a prism at the output of the OPA, the signal–idler beamsare detected with an InGaAs PIN photodiode (D1). Thephotocurrent from D1 is passed through a low-pass filter(80-MHz cutoff) and then measured by a boxcar averager(Stanford, Model SR250). Gain measurements are madeby recording the output of D1 with the pump beam on andthen off. The pump power is relatively measured by a SiPIN photodiode (D2) detecting a small portion of thepump beam that is transmitted through the DBS. Thephotocurrent from D2 is also passed through a low-passfilter (5-MHz cutoff) and is then sent to the second chan-nel of the boxcar averager. The 12-ns boxcar gate is ad-justed to measure the peak of the Q-switched envelopesfor both the signal–idler and the pump beams. For eachsetting of the pump power, data are recorded for a largenumber (2000–10 000) of Q-switched pulses. The meanvalue of each data set is used to compute the measuredamplification–deamplification response, whereas thestandard deviation leads to the resulting error bar.
Fig. 2. Schematic of the pump-beam (solid curves) and thesignal-beam (dashed curves) phase fronts plotted as a function ofz inside the nonlinear medium. The bounding hyperbolas markthe contours of the field radii of the two beams at the 6e21 level.
1566 J. Opt. Soc. Am. B/Vol. 14, No. 7 /July 1997 Choi et al.
3. OPA THEORY WITH GAUSSIAN BEAMSAlthough the behavior of an OPA with Gaussian beamshas been studied before,2 no analysis exists in the litera-ture for its phase-sensitive gain response. Therefore inthis section we first develop the relevant theory of anOPA for cw Gaussian beams. To compare with our ex-perimental data, we later generalize to include the pulsednature of the pump and signal fields (see Subsection 3.D).In the slowly varying envelope and undepleted pump ap-proximations, a frequency-degenerate but polarization-nondegenerate type II phase-matched OPA, e.g., the KTPamplifier of Section 2, can be described by the followingcoupled equations15:
]E(
]z1
12ik
¹'2E( 5 KEpEl* exp~ifp!, (1a)
]El]z
112ik
¹'2El 5 KEpE(* exp~ifp!.
(1b)
Here, E( is the electric field associated with thes-polarized signal beam and El is for the p-polarized idlerbeam; Ep is the amplitude of the pump field and fp
is its phase; ¹'2 5 ]2/]x2 1 ]2/]y2; K 5 xeff
(2) (vsvivp /nsninp)
1/2/c with the effective second-order susceptibilityxeff(2) assumed real; and k 5 2pns /l with ns as the refrac-
tive index at the signal wavelength l (we ignore a smalldifference between ns and ni , the latter being the refrac-tive index at l along the idler polarization direction). Inthe plane-wave theory, i.e., where one considers the vari-ous beams to be plane waves, the second terms on the left-hand sides of Eqs. (1a) and (1b) are neglected.15 In ourexperiment, however, the various beams incident on theparametric amplifier have Gaussian spatial profiles;therefore, the theory must take these terms into account.A polarization-nondegenerate OPA can be considered
equivalent to two independent orthogonally polarized de-generate OPA’s16 whose polarization directions make645° angles relative to the signal-beam polarization. Infact, by transforming the signal and the idler electricfields into two 645°-polarized orthogonal field compo-nents,
E↗ [ ~E( 1 El!/A2, (2a)
E↖ [ ~E( 2 El!/A2, (2b)
one can decouple Eqs. (1a) and (1b) into two independentequations:
]E↗]z
112ik
¹'2E↗ 5 K exp~ifp!EpE↗* , (3a)
]E↖]z
112ik
¹'2E↖ 5 K exp@i~fp 1 p!#EpE↖* ,
(3b)
which describe the two independent degenerate OPA’s.We analyze the solution for E↗ ; the one for E↖ will be
similar. We solve Eq. (3a) for our experimental geometrywherein z varies from 2l/2 to l/2, l being the length of theKTP crystal. The pump and the signal (idler) beams arealigned in such a way that their waists are colocated atz 5 0, and their confocal distances are equal, i.e., the
pump-beam waist is smaller than the signal–idler beamwaist by A2. Therefore we can write
Ep~r, z ! 5Ep0
1 1 2iz/z0expS 2r2/2a0
2
1 1 2iz/z0D , (4)
E↗~↖ !~r, z ! 5E↗0~↖0 !~r, z !
1 1 2iz/z0expS 2r2/4a0
2
1 1 2iz/z0D ,
(5)
where a0 is the intensity radius of the pump-beam waistat z 5 0, and z0 5 8pa0
2ns /l is the confocal distance.In the undepleted pump approximation, Ep0 can be set asa real constant. This approximation is valid in our ex-periments since generally uEp0u2 . 102uE↗0(↖0)3 (r, 2l/2)u2. The solution of Eq. (3a) with Eqs. (4) and(5) was derived in Ref. 17 and is given here to second or-der in l/z0 :
E↗0~r, l/2! 512 Xexp~z! 1 exp~2z! 1 @exp~z!
2 exp~2z!#exp~iu! 2 ilz0
$ f1~z, F!
1 f1~2z, 2F! 1 @f1~2z, 2F!
2 f1~z, F!#exp~iu!% 1 S lz0D2
$ f2~z, F!
3 @1 1 exp~iu!# 1 f2~2z, 2F!
3 @1 2 exp~iu!#%CE↗0~2l/2!, (6)
where
z 5 KEp0 l exp~2r2/2a02!, (7a)
F 5 KEp0 l, (7b)
f1~z, F! 5 exp~z!S 1z lnF
z2
12
212z D
1 exp~2z!F 12z1
32
1 S 2z 2 2 21z D ln F
z G ,(7c)
f2~z, F! 5 exp~2z!H 34 21
2z1
3
4z21 ln
F
zF z 2 2
14
z1
6
z21 ln
F
zS 2 1
2
z1
1
z2D G J
1 exp~z!H 231
6z 2
13
41
2
z2
3
4z2
1 lnF
zF8z2 1 23z 2 2 1
8
z2
6
z2
2 lnF
zS 85 z3 1 10z2 1
28
3z 1
1
z2D G J ,
(7d)
and u [ fp 2 2f with f as the phase of E↗0(2l/2) andfp that of the pump field at z 5 2l/2. The solution for
Choi et al. Vol. 14, No. 7 /July 1997 /J. Opt. Soc. Am. B 1567
the orthogonally polarized OPA, Eq. (3b), is found by re-placing E↗0 by E↖0 and fp by fp 1 p in Eq. (6) above.
A. Phase-Sensitive OPAIn the phase-sensitive configuration of the OPA, wechoose the input field polarization to be
E(0~r, 2l/2! 5 El0~r, 2l/2! 5E in
A2. (8)
Therefore we have E↗0(r, 2l/2) 5 E in andE↖0(r, 2l/2) 5 0. Furthermore, if the pump phase isadjusted to set u 5 0, then the input field is amplified.In this case, from Eq. (6) we obtain
uE↗0~r, l/2!u2 > H exp~2z! 1 S lz0D2
@ f1~2z, 2F!2
1 2 exp~z!f2~z, F!#J uE↗0~2l/2!u2,
(9)
to second order in l/z0 .To model a more realistic experimental situation, we
must take into account the effect of fluctuations of thepump phase relative to the signal phase. These fluctua-tions, which are due to acoustic and thermal perturba-tions in the surrounding environment of the experiment,lead to imperfect phase locking. Therefore we set u5 0 1 du, where du is a classical random variable withmean ^du& 5 0 and variance ^du2&. Treating du as asmall parameter similar to l/z0 , we can write exp(idu)> 1 1 idu 2 du 2/2. Then, similar to Eq. (9), from Eq.(6) we obtain
uE↗0~r, l/2!u2 > H exp~2z! 1 S lz0D2
@ f1~2z, 2F!2
1 2 exp~z!f2~z, F!#
1^du2&4
@exp~22z! 2 exp~2z!#J3 uE↗0~2l/2!u2. (10)
We see from Eq. (10) that the correction terms, theterms owing to GID and phase fluctuations, are second or-der in l/z0 and ^du2&1/2, respectively. Hence they makeinsignificant contributions [relative to the exp(2z) term]when the input field is amplified. Therefore we neglectthe correction terms in Eq. (10) and obtain the PSA powergain:
GPSA 5E uE↗~r, l/2!u2dr
E uE↗~r, 2l/2!u2dr
>exp~2F! 2 1
2F. (11)
If, on the other hand, the pump phase is adjusted todeamplify the input field, i.e., if we choose u 5 p, thenEq. (6) yields at the output of the OPA
uE↗0~r, l/2!u2 > H exp~22z! 1 S lz0D2
@ f1~z, F!2
1 2 exp~2z!f2~2z, 2F!#
1^du2&4
@exp~2z! 2 exp~22z!#J3 uE↗0~2l/2!u2. (12)
As we will see in Section 5, in this case we need to keepthe correction terms owing to GID and phase fluctuationsbecause they contribute significantly to the deamplifiedfield. From Eq. (12), the phase-sensitive deamplification(PSD) factor is
GPSD 5
E uE↗~r, l/2!u2dr
E uE↗~r, 2l/2!u2dr
>1 2 exp~22F!
2F1
1
FS l
z0D 2E
0
F
@ f1~z, F!2
1 2 exp~2z!f2~2z, 2F!#dz 1^du2&
2Fsinh2 F.
(13)
B. Phase-Insensitive OPAWhen the OPA is operated as a PIA, the input fields arechosen such that E(0(r, 2l/2) 5 E in and El0(r, 2l/2)5 0 [or E(0(r, 2l/2) 5 0 and El0(r, 2l/2) 5 E in]. Inthis case one can show from Eq. (6) and its counterpart forE↖0 with use of Eqs. (2a) and (2b) that the output signaland idler fields are
E(0~r, l/2! > E inH cosh~z! 2i2 S lz0D @ f1~z, F!
1 f1~2z, 2F!# 112 S lz0D
2
3 @ f2~z, F! 1 f2~2z, 2F!#J , (14a)
El0~r, l/2! > E in exp~iu!H sinh~z! 2i2 S lz0D
3 @ f1~2z, 2F!
2 f1~z, F!# 112 S lz0D
2
3 @ f2~z, F! 2 f2~2z, 2F!#J . (14b)
Keeping terms to second order in l/z0 , we obtain
1568 J. Opt. Soc. Am. B/Vol. 14, No. 7 /July 1997 Choi et al.
uE(0~r, l/2!u2 > uE inu2Xcosh2~z! 114 S lz0D
2
$@ f1~z, F!
1 f1~2z, 2F!#2 1 4 cosh~z!
3 @ f2~z, F! 1 f2~2z, 2F!#%C, (15a)
uEl0~r, l/2!u2 > uE inu2Xsinh2~z! 114 S lz0D
2
$@ f1~z, F!
2 f1~2z, 2F!#2 1 4 sinh~z!
3 @ f2~z, F! 2 f2~2z, 2F!#%C, (15b)
which, as expected, do not depend on u. The contributionof the second-order correction terms is, once again, negli-gible because both the signal and the idler fields are am-plified. We therefore obtain the PIA gain:
GPIA 5
E uE(~r, l/2!u2dr
E uE(~r, 2l/2!u2dr
>1
21
1
4
exp~2F! 2 exp~22F!
2F. (16)
C. Plane-Wave ApproximationWhen the pump and the signal beam spatial profiles areplane waves, one can ignore the ¹'
2 terms in Eqs. (1a)and (1b). In this case, Eqs. (1a) and (1b) can be triviallysolved to yield
E(~l/2! 5 E(~2l/2!cosh~F!
1 El* ~2l/2!exp~ifp!sinh~F!, (17a)
El~l/2! 5 E(* ~2l/2!exp~ifp!sinh~F!
1 El~2l/2!cosh~F!. (17b)
One can derive the following expressions for the PSAgain, the PSD factor, and the PIA gain:
GPSA 5 exp~2F!, (18)
GPSD 5 exp~22F!, (19)
GPIA 512
114
@exp~2F! 1 exp~22F!#.
(20)
When the fluctuations of the pump phase relative to thesignal phase are considered, Eq. (19) is modified to
GPSD 5 exp~22F! 1^du2&4
@exp~2F! 2 exp~22F!#.
(21)
Note that these plane-wave results cannot be obtained asa limiting case of the previous Gaussian-beam results be-cause the integrals in Eqs. (11), (13), and (16) go to infin-ity for plane waves.
D. Temporal VariationsIn our experiment the pump and the signal beams aregenerated in the form of 100-MHz mode-locked pulsetrains beneath the Q-switched pulse envelopes. Thedetector D1 is not fast enough to resolve the individualpulses in the mode-locked train. Therefore we use an80-MHz bandwidth low-pass filter (LPF) to integrateover the ;120-ps-long mode-locked pulses. The;200 ns-long Q-switched pulse envelopes, however, passthrough the LPF largely undistorted. Therefore a mea-surement at the peak of the Q-switched pulse envelope isproportional to the energy of a single mode-locked pulsenear the center of the pulse train. To compare the abovetheory with the experimental data, we thus need to inte-grate the various gain expressions over time as well.18
We assume that the temporal profiles of the input sig-nal and the pump pulses are Gaussian. This is a validassumption for the pulses generated by a mode-lockedNd:YAG laser. Because the pump pulses are generatedby low-efficiency (.10%) second-harmonic generation(SHG) of the fundamental Nd:YAG laser beam (fromwhich the signal pulses are derived), the pump pulsewidth is shorter than the signal pulse width by a factor ofA2. Therefore we can generalize the pump and the sig-nal fields of Eqs. (4) and (5) as
Ep~r, z ! 5Ep0
1 1 2iz/z0expS 2r2/2a0
2
1 1 2iz/z02 t2D , (22)
E↗~↖ !~r, z ! 5E↗0~↖0 !~r, z !
1 1 2iz/z0expS 2r2/4a0
2
1 1 2iz/z02
t2
2 D ,(23)
where t is a scaled time. We then perform the integra-tions over space as well as time to obtain the averagepower gains that can be directly compared with the ex-perimental data.Equation (11) for GPSA is modified to
GPSA >1
p1/2aI1/2E0
`
$exp@2aI1/2 exp~2t2!# 2 1%dt,
(24)
where we set F 5 aI1/2 with I [ Ep02 as the peak pump
power. Similarly, Eq. (13) for GPSD changes to
GPSD >1
p1/2aI1/2 XE0`$1 2 exp@22aI1/2 exp~2t2!#%dt
1 2S l
z0D 2E
0
`E0
F
@ f1~z, F; t !2
1 2 exp@2z exp~2t2!# f2~2z, 2F; t !#
3 exp~2t2!dzdt
1^du2&
4E0
`
$exp@2aI1/2 exp~2t2!#
1 exp@22aI1/2 exp~2t2!# 2 2%dtC, (25)
where f1(z, F; t) and f2(2z, 2F; t) are obtained fromf1(z, F) and f2(2z, 2F), respectively, by the replace-
Choi et al. Vol. 14, No. 7 /July 1997 /J. Opt. Soc. Am. B 1569
ment of z with z exp(2t2). When the OPA is operated inthe PIA configuration, the expression for GPIA is modifiedto
GPIA >1
21
1
4p1/2aI1/2E0
`
$exp@2aI1/2 exp~2t2!#
2 exp@22aI1/2 exp~2t2!#%dt. (26)
Similar to the signal and idler pulses, the power of thepump pulses is measured by a photodetector (D2) whoseresponse time is slow compared with the mode-lockedpulse width but faster than the width of the Q-switchedpulse envelope. The photocurrent generated owing tothe pump pulses is subsequently passed through a 5-MHzLPF. The peak photocurrent measured after the LPF istherefore proportional to the pump-pulse energy near thecenter of the Q-switched pulse envelope, which, in turn, isproportional to the peak mode-locked pump-pulse powerfor a fixed pulse shape. Because the various measuredpulse energies are not absolutely calibrated (they are onlyrelatively accurate to within 5%), we use the proportion-ality constant a in Eqs. (24)–(26) as an adjustable param-eter.To compare the Gaussian-wave theory with the plane-
wave case, we also integrate over time the gain expres-sions in Eqs. (18)–(21). The results are given by
GPSA 52
p1/2 E0
`
exp@2t2 1 2aI1/2 exp~2t2!#dt, (27)
GPSD 52
p1/2 H E0
`
exp@2t2 2 2aI1/2 exp~2t2!#dt
1^du2&
4E0
`
exp~2t2!$exp@2aI1/2 exp~2t2!#
2 exp@22aI1/2 exp~2t2!#%dtJ , (28)
GPIA 51
21
1
2p1/2 E0
`
$exp@2t2 1 2aI1/2 exp~2t2!#
1 exp@2t2 2 2aI1/2 exp~2t2!#%dt, (29)
which then represent the case of a spatially plane-wavepump but with a Gaussian temporal profile.If the PSA gain is plotted as a function of the PIA gain,
then Eqs. (18) and (20) yield
GPSA 5 2GPIA 2 1 1 @~2GPIA 2 1 !2 2 1#1/2. (30)
As discussed in the next section, Eq. (30) turns out to bevalid even when the spatial and temporal profile of thevarious beams is Gaussian.
4. AMPLIFICATION RESPONSE OF THEOPAIn Fig. 3(a) we show the Q-switched envelopes, as cap-tured by an oscilloscope, at both the input and the outputof the OPA when operated in the PSA mode. The pump
power was set to obtain a PSA gain of .6 and the pumpphase was adjusted to either amplify or deamplify the in-put pulse. The 10-ns-period modulation in the pulse en-velope results from the mode-locked pulse train becausethe cutoff of the 80-MHz LPF is not sharp enough to com-pletely integrate over the mode-locked pulses. For GPSA. 6 the plane-wave theory [Eq. (19)] predicts that theoutput should be deamplified by a factor of .6 when thepump phase is properly set. However, note that thedeamplification is less than a factor of 2. In Fig. 3(b) weshow the phase-sensitive nature of the PSA gain. TheOPA peak gain is plotted as a function of the pump phase.We obtained the trace by setting the boxcar gate at thepeak of the Q-switched pulse envelope and scanning thepump phase by applying a voltage ramp to the PZT. Thediscontinuities in the trace are due to the nonlinear re-sponse of the PZT. The dotted–dashed curve is obtainedwhen the input signal polarization is adjusted for zeroidler input. As shown, in this case, the OPA gain be-comes insensitive to the pump phase, i.e., the OPA oper-ates as a PIA. Slight remnant variations are due to theresidual birefringence in the KTP crystal.To understand the gain response of the OPA, we mea-
sured its dependence on the pump power in both thephase-sensitive and the phase-insensitive configurations.In Fig. 4(a) we show the measured variation of GPSA with
Fig. 3. (a) Q-switched envelopes at the input and the output ofthe OPA. The pump-beam phase is adjusted to either amplify ordeamplify the input signal beam. The pump power was set toobtain a peak PSA gain of ;6. (b) Peak gain of the OPA plottedas a function of the pump-beam phase for GPSA .4 in both thePSA and the PIA configurations.
1570 J. Opt. Soc. Am. B/Vol. 14, No. 7 /July 1997 Choi et al.
pump power for three different crystals, A, B, and C,whose lengths were measured to be lA 5 1.576 0.04 mm, lB 5 3.25 6 0.04 mm, and lC 5 5.216 0.04 mm, respectively. For each data point, thepump phase was locked to maximally amplify the input.As shown, PSA gains of over 20 (13 dB) could be observedby increasing the pump power. The error bars are alsoshown, which in many cases are smaller than the data-symbol sizes. To compare the experimental data withthe theory presented in Section 3, plots of Eq. (24) areshown by the solid curves in Fig. 4(a). As there is no ab-solute calibration on the pump power, the parameter awas varied independently for each experimental curve(one for each crystal) to obtain the minimum least-squares sum of the difference between the theoretical andthe experimental values of the gain at each data point.We thus find the theoretical fits to be such that a equals0.067, 0.136, and 0.211 for crystals A, B, and C, respec-tively. In Fig. 4(b) we show the measured dependence ofGPIA on the pump power. Once again, the solid curvesare theoretical fits to Eq. (26), which were obtained by thesame least-squares-sum method as above, with a equal to0.059, 0.133, and 0.210 for the three crystals, respec-tively. Note that both the PSA and the PIA data fit Eqs.
Fig. 4. Dependence of the peak OPA gain on pump power forKTP crystals of length 1.57 mm (triangles), 3.25 mm (diamonds),and 5.21 mm (squares). The OPA is configured as a PSA in (a)and as a PIA in (b) with z0 5 41.1 mm. In both (a) and (b) arelative pump-power value of 100 corresponds to the mode-lockedpump-pulse peak power of ;46 kW corresponding to an intensityof ;1.5 GW/cm2 at the beam waist in the crystal. The solid andthe dashed curves are theoretical fits to Eqs. (24) and (27), re-spectively, in (a) and Eqs. (26) and (29), respectively, in (b).
(24) and (26), respectively, extremely well with approxi-mately the same value of the adjustable parameter a foreach crystal length. The dashed curves in Figs. 4(a) and4(b) are fits to the plane-wave theory, Eqs. (27) and (29),respectively. By applying the above least-squares-summethod, we found the a values for the three crystals to be0.039, 0.092, and 0.141, respectively, in the PSA case, and0.035, 0.089, and 0.141, respectively, in the PIA case.Clearly, the Gaussian-wave theory, which takes thespatio-temporal profile of the pump pulse into account,gives a better fit to the experimental data compared withthe plane-wave theory, which considers the temporal pro-file of the pump but ignores its spatial profile. Moreover,from Eq. (7b) and the definition of the parameter a, weknow that the a values for the three crystals are propor-tional to their respective lengths (see Appendix A). Thea values obtained when fitting the Gaussian-wave theoryhave ratios RBA [ aB /aA 5 2.03, RCB [ aC /aB 5 1.55(PSA case), and RBA 5 2.25, RCB 5 1.58 (PIA case), re-spectively, between the crystals A, B, and C. By directmeasurement of the crystal lengths, the ratios were foundto be 2.07 and 1.60, respectively, with an estimated errorof 5%. Therefore we see that the agreement between theratios of the a values and the crystal lengths is quitegood.In Fig. 5 we show GPSA plotted as a function of GPIA for
the three crystals. When plotted this way, the space–time dependence of the pump beam, which is common inboth configurations of the OPA, cancels out, and the datafor each crystal should follow the same curve. Thedotted–dashed line in Fig. 5 is a fit to Eq. (30), a result ofthe plane-wave theory, without any adjustable param-eter. It is also possible to apply the Gaussian-wavetheory by numerically evaluating GPSA and GPIA fromEqs. (11) and (16), respectively, and plotting GPIA versusGPSA (dotted line in Fig. 5). Furthermore, one can alsoplot GPIA versus GPSA , in either the Gaussian-wave or theplane-wave theories, with the temporal profile of thepulses taken into account by numerically evaluating Eqs.(24) and (26) (solid line in Fig. 5) or Eqs. (27) and (29)(dashed line in Fig. 5), respectively. For the range ofmeasured GPSA and GPIA , the resulting lines are almost
Fig. 5. PSA gain of the OPA plotted as a function of the PIAgain. The solid line is a theoretical fit to the parametric plot ofEqs. (24) and (26), the dashed line to that of Eqs. (27) and (29),and the dotted line to that of Eqs. (11) and (16). The dotted–dashed line is a plot of Eq. (30) without any adjustable param-eter. The various theoretical plots are virtually indistinguish-able from one another.
Choi et al. Vol. 14, No. 7 /July 1997 /J. Opt. Soc. Am. B 1571
indistinguishable from each other. As shown, the agree-ment between the theory and the experimental data isfairly good; the discrepancies are due to the remnant bi-refringence in the KTP crystals, which causes the ampli-fication to depend slightly on the pump phase relative tothe signal phase even in the PIA configuration. We thusconclude that the simple relationship between the PIAgain and the PSA gain, Eq. (30) derived for the case ofplane-wave fields, holds even when the spatio-temporaldependence of the various fields is Gaussian.
5. DEAMPLIFICATION RESPONSE OF THEOPAIn Fig. 3(a) we saw that in the PSA configuration of theOPA, when the pump phase was set to deamplify the in-put signal beam, the observed deamplification was lessthan that expected from the simple plane-wave theory,Eq. (19). To understand this behavior of the OPA in thePSA configuration, we investigated the deamplificationresponse as a function of the pump power. Figure 6shows the measured deamplification factor, GPSD , plottedas a function of the pump power for crystals A, B, and C,respectively. For each data point the pump phase waslocked to maximally deamplify the input signal field. Asshown, for all three crystals, GPSD quickly attains a mini-mum value of .0.6 as the pump power is increased. Afurther increase of the pump power causes GPSD to in-crease instead of decreasing, eventually becoming greaterthan one; i.e., deamplification turns into amplification.At high pump powers, if the pump-phase setting is variedfrom the locked position, then the signal-beam power atthe OPA output increases, indicating that, at these pumppowers, deamplification of the input signal is not possible.The lack of deamplification at high pump powers,
where GPSA is large, is caused by both the GID effect andthe fluctuations of the pump phase relative to the signalphase. Let us first consider the GID effect. As shown inFig. 7(a), when the input signal beam is amplified, its spa-tial extent (for example, as measured by its full-width athalf maximum) decreases owing to the Gaussian trans-verse dependence of the pump intensity. This is becausethe portion of the signal beam that is closer to the propa-gation axis is amplified more than the portion that is far-ther away since the parametric gain is proportional to thepump intensity. The reduced spatial width modifies thediffraction behavior of the signal beam as it propagatesthrough the nonlinear medium. The resulting distortionof the phase fronts, as qualitatively sketched in Fig. 7(b),causes different transverse portions of the signal beam toacquire different phase shifts relative to the pump-beamphase. Thus, after some initial deamplification, alltransverse portions of the signal beam are not simulta-neously deamplified; this leads to an increase in the over-all power deamplification factor GPSD . At sufficientlyhigh pump powers, the transverse variation in the phasefronts is large enough to cause net amplification, i.e.,GPSD . 1, even when the pump phase is set to maximallydeamplify the input. This GID-induced lack of deampli-fication of the input-signal beam at high pump powers is
quantitatively taken into account by the second-order cor-rection terms in Eqs. (13) and (25), which are proportionalto (l/z0)
2.As noted in Section 3, fluctuations of the pump phase
relative to the signal phase, which are caused by acousticand thermal perturbations in the surroundings of the ex-periment, also contribute to degradation of deamplifica-tion. The effect of phase fluctuations is negligible at lowpump powers, i.e., when the PSA gain is low. However,
Fig. 6. Dependence of the peak deamplification factor, GPSD , onpump power for KTP crystals of length (a) 1.57 mm, (b) 3.25 mm,and (c) 5.21 mm. The OPA is configured as a PSA with z05 41.1 mm. The pump-power calibration is the same as that inFig. 4. In each case the solid curves are theoretical fits to Eq.(25) for the measured values of l/z0 and ^du2&1/2, whereas thedashed curves are fits to Eq. (28) for the measured values of^du2&1/2. See text for description of the dotted–dashed and dot-ted curves.
1572 J. Opt. Soc. Am. B/Vol. 14, No. 7 /July 1997 Choi et al.
as the pump power is increased, a slight change in thepump-signal relative phase from the optimum valuecauses a significant change in the deamplification factorGPSD . Random phase fluctuations lead to a net increasein GPSD , and the effect is quantitatively accounted for bythe correction terms in Eqs. (13) and (25) that are propor-tional to ^du2&.To compare the Gaussian-wave theory with the experi-
mental data, we need to evaluate the parameters l/z0 and^du2&1/2. The confocal distance z0 was measured to be41.1 6 3.6 mm by mapping the Gaussian spatial profileof the signal beam at two different points along the opti-cal path. Therefore l/z0 5 0.127 for the longest (l5 5.21 mm) of the three crystals. Hence the use of Eq.(25) is valid, as it was derived under the assumption thatl/z0 ! 1.One can evaluate the phase fluctuations by analyzing
the deamplification response of the OPA in the low pump-power region wherein the GID effect is negligible. By ne-glecting the GID term in Eq. (25), we obtain
Fig. 7. (a) Representative one-dimensional spatial profiles forthe signal beam. The dashed curve is the input profile, and thesolid curves are profiles of the amplified and the deamplifiedquadratures, as marked, for a PSA gain of 8. The fullwidth athalf-maximum, shown by the horizontal dotted line, for the am-plified quadrature is much smaller (by almost a factor of 2) thanthat for the input signal. (b) A qualitative sketch of the gain-induced diffraction phenomenon that is responsible for the de-crease in the peak deamplification factor.
GPSD 51
p1/2F S E0
`
$1 2 exp@22F exp~2t2!#%dt
1^du2&
4E0
`
$exp@2F exp~2t2!#
1 exp@22F exp~2t2!# 2 2%dt D , (31)
where, as before, F 5 aI1/2. Application of classicalnoise analysis to Eq. (31) near the mean-phase value u5 p gives the following result:
dIout 5 GPSDdI in 1 I inFdu2
4A 1
dEp0
Ep0~B 2 C !G ,
(32)
where I in and Iout are the energies of the input and theoutput mode-locked signal pulses, respectively, with dI inand dIout as the corresponding fluctuations; Ep0 is thepeak amplitude of the mode-locked pump pulse with fluc-tuation dEp0 ; and
A 5 E0
`
$exp@2F exp~2t2!#
1 exp@22F exp~2t2!# 2 2%dt, (33a)
B 52
p1/2 E0
`
exp@2t2 2 2F exp~2t2!#dt,
(33b)
C 51
p1/2FE0
`
$1 2 exp@22F exp~2t2!#%dt.
(33c)
Substituting Eq. (31), without the ^du2& term, into Eq.(32), we obtain
^du2&1/2 > ^du4&1/4 5 2SGPSD
A D 1/2H F ^dIout2&
Iout2 2
^dI in2&
I in2
21
4 S B
GPSD2 1 D 2 ^dIp
2&
Ip2 G2J 1/8, (34)
where Ip is the energy of the mode-locked pump pulsewith the corresponding fluctuation dIp . The approxima-tion ^du2&1/2 > ^du4&1/4 is valid if the variance of du2 ismuch smaller than its mean. In the experiment thephase-locking servo mechanism ensures a good approxi-mation to this condition, and the reasonably small stan-dard deviations of the PSD data points lend further sup-port. ^dI in
2&/I in2 and ^dIp
2&/Ip2 were evaluated from
histograms of the signal and the pump pulse energies, re-spectively, measured by the detectors D1 and D2 in con-junction with the corresponding LPF’s (cf Fig. 1). Simi-larly, with the pump phase locked for maximumdeamplification, histograms of the output signal-pulse en-ergies were generated for various values of the pumppower in the low-gain region of the OPA. From these his-tograms, ^dIout
2&/Iout2 was obtained for each value of
GPSD , and the resulting ^du2&1/2 was calculated with useof Eq. (34). The values of ^du2&1/2 obtained in this waywere then averaged over the various data points for each
Choi et al. Vol. 14, No. 7 /July 1997 /J. Opt. Soc. Am. B 1573
crystal length, and the resulting standard deviation wasevaluated. For the data presented in Fig. 6, we obtained^du2&1/2 5 0.362 6 0.156, 0.539 6 0.130, and 0.4786 0.071 for crystals A, B, and C, respectively.The solid curves in Fig. 6 are plots of Eq. (25) for the
measured values of l/z0 and ^du2&1/2 for each crystal.Once again, the fits to the experimental data were foundby the least-squares-sum method described in Section 4.This resulted in a equal to 0.058, 0.122, and 0.200 for therespective crystals. Note that these values of the a pa-rameter for the PSD data are fairly close to the values ob-tained for the PSA and the PIA data in Section 4. Theratios of the a values are RBA 5 2.10 and RCB 5 1.64, re-spectively, which are once again very close to the ratios,2.07 and 1.60, of the crystal lengths. The dotted–dashedcurves in Fig. 6 were obtained by substituting the respec-tive a values into Eq. (28), which is a result of the plane-wave theory with phase fluctuations and time integration.Clearly, the dotted–dashed curves do not fit the experi-mental data. Applying the least-squares-sum method di-rectly to Eq. (28) to fit the data led to a different set of avalues (a equals 0.024, 0.086, and 0.144, respectively) andbetter fits as shown by the dashed curves. However, thesolid curves derived from the Gaussian-wave theory arenoticeably in better agreement with the data. Thereforethe GID effect and phase fluctuations must be consideredwhen analyzing the phase-sensitive deamplification be-havior of an OPA.If the experimental setup is upgraded to reduce the
phase fluctuations, then the deamplification response ofthe OPA can be improved. To find the maximum deam-plification possible in the absence of phase fluctuations(limited only by the GID effect), we show by the dottedcurves in Fig. 6 plots of Eq. (25) with ^du2&1/2 arbitrarilyset equal to zero. We see from Fig. 6(c) that GPSD
min
5 0.36 for the 5.21-mm crystal. For the 3.25-mm crys-tal, GPSD
min 5 0.32 can be achieved by further increasing thepump power [relative value of 565 in Fig. 6(b)]. In thecase of the 1.57-mm crystal, GPSD
min 5 0.27, but the pumppower necessary [relative value of 3746 in Fig. 6(a)] islarger than the damage threshold of the KTP crystal.
6. OPA RESPONSE WHEN l/z0 ! 1 IS NOTVALIDThe theory in Section 3 was developed under the condi-tion that l/z0 ! 1. However, in the experimental setup itis easy to violate this condition. For example, by using10-cm focal-length lenses in place of the 20-cm focal-length lenses (cf Fig. 1), we obtain z0 . 9.19 mm. Withsuch an exchange, much larger OPA gains are obtainedbecause the pump beam is focused more tightly, leadingto a much higher pump intensity in the KTP crystal. Tosee what the maximum OPA gains were that could be ob-served without damaging the KTP crystals, and to com-pare the data with the theory under conditions where thelatter was expected to break down, we also measured thePSA, the PIA, and the PSD response of the OPA with z0.9.19 mm.Figures 8–10 show the experimental data along with
the optimum theoretical fits (solid curves) that were ob-
tained by use of the least-squares-sum method. Asshown in Fig. 8, GPSA . 100 (20 dB) is obtainable withcrystal C of length 5.21 mm. At first glance, it appearsthat the solid curves in Figs. 8(a) and 8(b), which are plotsof Eqs. (24) and (26), respectively, fit the experimentaldata extremely well; however, a more careful inspectionshows otherwise. For the PSA–PIA data in Fig. 8 thevalues of the fitting parameter a are 0.045, 0.079, and0.117 for crystals A, B, and C, respectively. Because wedid not measure phase fluctuations for the PSD data inFig. 9 (data with the 10-cm focal-length lenses were takenbefore our systematic development of the theory pre-sented in Section 3), we varied both a and ^du2&1/2 in Eq.(25) to obtain the optimum fit. The resulting a valueswere found to be 0.046, 0.063, and 0.131 for crystals A, B,and C, respectively. Therefore the a values for the twolonger crystals are quite inconsistent in the PSA–PIAcase (Fig. 8) with those in the PSD case (Fig. 9). Further-more, in Figs. 8 and 9, although good fits to Eqs. (24)–(26)are found for all crystal lengths, the agreement betweenthe ratios of the a values, RBA and RCB (1.76 and 1.48 forthe PSA–PIA case, 1.37 and 2.08 for the PSD case), andthe ratios of the crystal lengths (2.07 and 1.60) is worsecompared with the data shown in Figs. 4–6. This, we be-lieve, is because the Gaussian-wave theory breaks downin the limit where the assumption l/z0 ! 1 is violated.
Fig. 8. Dependence of the peak OPA gain on pump power forKTP crystals of length 1.57 mm, 3.25 mm, and 5.21 mm. TheOPA is configured as a PSA in (a) and as a PIA in (b) with z0.9.19 mm. The solid and the dashed curves are theoretical fitsto Eqs. (24) and Eq. (27), respectively, in (a) and Eqs. (26) andEq. (29), respectively, in (b).
1574 J. Opt. Soc. Am. B/Vol. 14, No. 7 /July 1997 Choi et al.
For the measurements in Figs. 8–10, l/z0. 0.354 and0.567 for crystals B and C, respectively.Disregarding the inconsistency of the a parameters,
the PSA–PIA data in Fig. 8 fit the Gaussian-wave theory,Eqs. (24) and (26), better than the plane-wave theory,Eqs. (27) and (29), shown by the dashed curves, as wasthe case for the data in Section 4 wherein l/z0 ! 1. Theonly exception are the data for l 5 1.57 mm (crystal A),which is to be expected since the two theories coincidewhen the OPA gain is small. Nevertheless, as shown bythe solid lines in Fig. 10, the quasi-linear relationship be-tween GPIA and GPSA , Eq. (30), is still satisfied in allcases.
Fig. 9. Dependence of the peak deamplification factor on pumppower for KTP crystals of length (a) 1.57 mm, (b) 3.25 mm, and(c) 5.21 mm. The OPA is configured as a PSA with z0.9.19 mm. The solid curves are theoretical fits taking into ac-count the spatio-temporal profile of the pulses in the Gaussian-wave theory.
7. CONCLUSIONSWe have experimentally investigated the gain response ofa traveling-wave OPA that consists of a type II phase-matched KTP crystal pumped by a frequency-doubledQ-switched mode-locked Nd:YAG laser. In the phase-sensitive configuration, gains of .100 (20 dB) can beeasily obtained in the amplified quadrature. Because ofgain-induced diffraction and phase fluctuations, however,maximum deamplification in the orthogonal quadratureis limited to ,0.5 (23 dB). To our knowledge, this is thefirst time that the deamplification response of an OPAconfigured as a PSA has been systematically measured.Our experimental results are in excellent agreement withthe theory of an OPA in which the Gaussian-beam natureof the various fields along with diffraction of the amplifiedsignal beam and fluctuations of the pump phase relativeto the signal phase are taken into account. In the ab-sence of phase fluctuations the deamplification responseis limited by the GID effect; a minimum deamplificationfactor of 0.32 is achievable with a .3-mm crystal.
APPENDIX A: COMPARISON WITHSECOND-HARMONIC-GENERATION DATAWe measured for all three crystals the second-harmonicpower that was generated by a strong input-signal beam
Fig. 10. (a) PSA gain of the OPA plotted as a function of the PIAgain for z0 .9.19 mm. (b) An expanded plot of the low-gaindata in (a). In both cases, the solid curve is a plot of Eq. (30)without any adjustable parameter.
Choi et al. Vol. 14, No. 7 /July 1997 /J. Opt. Soc. Am. B 1575
in the PSA configuration of the OPA with the pump beamblocked. It is known19 that, for focused Gaussian beams,the second-harmonic power is proportional to the squareof the length of the nonlinear medium if l/z0 ! 1. As-suming xeff
(2) to be the same for the three KTP crystals,their length ratios deduced from our SHG measurementsare 2.10 6 0.34 and 1.48 6 0.08, respectively, for the firsttwo and the second two in our set of crystals. Thus wesee that the ratios obtained from the SHG measurementsare in reasonable agreement with the directly measuredcrystal-length ratios of 2.07 6 0.13 and 1.60 6 0.05.The slight discrepancy between the two kinds of measure-ments for the ratio of the two larger crystals is probablybecause, in this case, l is insufficiently small comparedwith z0 . Therefore we are justified in our analysis to as-sume that the various crystals have the same values ofxeff(2) .
ACKNOWLEDGMENTSThis work was supported in part by the U.S. Office of Na-val Research and the National Science Foundation. Theauthors acknowledge useful discussions with S. Youn andD. O. Caplan.
†Present address, Department of Information and Com-munication, SoongSil University, 1-1 SangDo-DongDongJak-Ku, Seoul 156-743, Korea.
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