Deamplification response of a traveling-wave phase-sensitiveoptical parametric amplifier
Chonghoon Kim, Ruo-Ding Li, and Prem Kumar
Department of Electrical Engineering and Computer Science, Northwestern University,2145 North Sheridan Road, Evanston, Illinois 60208-3118
Received August 23, 1993
We have investigated the phase-sensitive deamplification response of a traveling-wave degenerate opticalparametric amplifier that consists of a type II phase-matched KTP crystal pumped by the second harmonic ofa Q-switched mode-locked Nd:YAG laser. Experimental results are in good agreement with the theory of anoptical parametric amplifier in which the Gaussian-beam nature of the various fields along with the diffractionof the amplified signal beam is taken into account.maximum deamplification is limited to only =3 dB.
Recent experiments that demonstrated the gener-ation of quadrature-squeezed light by means of atraveling-wave optical parametric amplifier (OPA) in-dicate that the observable squeezing is limited largelyby the effect of diffraction in the parametric process",2and by the spatiotemporal mismatch between thegenerated squeezed mode and the local oscillatormode employed in homodyne detection of squeezing.3By using a local oscillator whose spatiotemporal pro-file is matched to the generated squeezed mode, wehave been able to measure >5.5 dB of quadraturesqueezing, which to our knowledge is the highestlevel ever observed in a traveling-wave experiment.4During the process of maximizing the observablesqueezing we have carefully studied the gain behav-ior of a frequency-degenerate OPA, paying partic-ular attention to its phase-sensitive gain response,and have compared the data with the theory. Inthis Letter we report that the phenomenon of gain-induced diffraction prevents the input signal fromdeamplifying in the quadrature that is orthogonalto the amplified quadrature. Our experimental ob-servations are in good agreement with the theoryof an OPA in which the Gaussian-beam nature ofthe various fields along with the diffraction of thesignal beam is taken into account. We believe thatthe lack of deamplification owing to the gain-induceddiffraction effect is what prevented the observation oflarge squeezing in previous experiments.'
Figure 1 shows a schematic of our experimentalsetup. The traveling-wave OPA consists of a type IIphase-matched KTP crystal that is pumped by thefrequency-doubled 532-nm output from a Q-switchedmode-locked Nd:YAG laser. At a repetition rate of1.1 kHz the Q-switched envelopes of the harmonicand fundamental pulses are =4150 and =270 ns,respectively, in duration. The mode-locked pulsesunderneath these Q-switched pulse envelopes areestimated to be n140 and =200 ps long for theharmonic and fundamental beams, respectively.
A portion of the fundamental beam at 1064 nm isseparated by a prism and used as an input to theOPA. The intensity and polarization of this beam
Because of the phenomenon of gain-induced diffraction,
are adjusted with a set of half-wave plates (HVVP1and HWP2) and polarizers (Pp and P). The inten-sity of the pump beam is adjusted by a half-waveplate (HWP) and a polarizer (PP8) and interferometri-cally aligned with the input signal beam by a dichroicbeam splitter (DBS). Because of the critical natureof the type II phase matching in the KTP crystal,the pump and the signal beams walk away from theidler beam as they propagate through the crystal.To precompensate for this walk-off, the signal andthe idler inputs are separated and recombined bypolarizing beam splitters (PBS1 and PBS2) beforethey enter the OPA. We focus the pump and thesignal/idler beams into the KTP crystal by using twoidentical 10-cm focal-length lenses in such a way thatthe resulting beam waists overlapped nearly at thecenter of the KTP crystal. The path lengths of thevarious beams from the Nd:YAG laser to the OPAare made equal to within a few millimeters to ensurethat the various mode-locked pulses enter the KTP
Fig. 1. Schematic of the experimental setup for measur-ing the gain response of the traveling-wave OPA. ThePSA phase is controlled by application of a voltage to thePZT.
crystal at the same time. The pump and the signalbeams in the crystal are polarized along the e axis(s polarization), whereas the idler beam is polarizedorthogonal (p polarization) to the signal beam as aresult of the type II phase matching.
In order to operate the polarization-nondegeneratebut frequency-degenerate OPA as a degenerate OPA[or as a phase-sensitive amplifier (PSA)], we adjustwave plate HWP2 to make the orthogonally polarizedsignal and idler input pulses to the OPA equal inamplitude and in phase. We adjust the phase ofthe pump pulses relative to the input signal/idlerpulses by moving a mirror that is mounted upon apiezoelectric transducer (PZT). A feedback servo isimplemented to drive the PZT. The pump phase canbe set and locked to amplify or to deamplify the inputsignal.
After the pump beam is dispersed away at the out-put of the OPA by use of a prism, the signal and idlerbeams are separated with a beam-splitting polarizer(PBS3) and detected with InGaAs PIN photodiodes.Photocurrent from the signal beam detector (Dl) ispassed through a low-pass filter (LPF; 5-MHz cutoff)and sent to a boxcar averager. The 20-ns boxcar gateis either set to measure the peak of the Q-switchedpulse envelopes or scanned to measure the wholeQ-switched pulse profile.
In Fig. 2 we show the measured Q-switched pulseenvelopes at both the input and the output of theOPA. The dashed curve is the input pulse. Thepump phase was adjusted either to amplify or deam-plify the input pulse. The pump power was set toget a PSA gain of -4. The shoulder in the pulseshape for larger times is an artifact of the timeresponse of the low-pass filter. For GPSA : 4, theplane-wave theory5 predicts that the output should bedeamplified by a factor of =4 when the pump phaseis properly set. Note, however, that the deamplifi-cation is by only a factor of =2. Similar disparitybetween the amplification and deamplification re-sponse of an OPA was reported by Rarity et al.
6
In order to understand this lack of deamplificationby the OPA, we have investigated the deamplifica-tion response as a function of the pump power. InFig. 3 we show the measured deamplification factorDPSA plotted as a function of the pump power for a5-mm-long crystal. For each data point the pumpphase was locked to deamplify maximally the inputfield. We see that, as the pump power is increased,DPSA quickly attains a value of ==0.5. With furtherincrease of the pump power, DPSA starts to increaseinstead of to decrease, eventually becoming greaterthan 1; i.e., deamplification turns into amplification.At high pump powers, if the pump-phase settingis varied from the locked position, the signal beampower at the output of the OPA increases, as is thecase at low pump powers, indicating that, at highpump powers, deamplification of the input beam isnot possible.
The lack of deamplification at high pump powerswhere GPSA is large is due to the phenomenon of gain-induced diffraction. When the input beam is ampli-fied, its spatial extent-as measured by its full widthat half-maximum, for example-decreases because
of the Gaussian transverse dependence of the pumpintensity.2 The portion of the input beam that iscloser to the propagation axis is amplified more thanthe portion that is farther away because the paramet-ric gain is proportional to the pump intensity. Thereduced spatial width modifies the diffraction behav-ior of the beam as it propagates through the nonlinearmedium. The resulting distortion of the phase frontscauses different transverse portions of the amplifiedbeam to acquire different phase shifts relative to thepump beam phase front. Thus, after some initialdeamplification, all transverse portions of the inputbeam are not simultaneously deamplified, limitingthe overall intensity deamplification factor DPSA. Athigh PSA gains there is enough transverse variationof the phase fronts so that there is net amplification,resulting in DPSA > 1, even when the pump phase isset to deamplify maximally the input.
The gain-induced diffraction effect can be quanti-tatively taken into account as follows. In the slowlyvarying envelope and the undepleted-pump approxi-mations a degenerate OPA can be described by
a + 1 V, 2E = K exp(ickp)EpE*, (1)
where E is the positive-frequency part of the electricfield associated with the signal beam; Ep is the am-plitude of the pump field, with Op as its phase; K is
5
4
3
2arx:1C2
0
-1
/ \s~~~~lnput
' ' ' ''-- A c -' Deamplificttion
0 100 200 300 400 500 600Time (ns)
Fig. 2. Q-switched pulse envelopes at the input andoutput of the OPA. The pump phase is adjusted either toamplify or deamplify the input signal beam. The pumppower was set to get a PSA gain of =4.
1.6
1.4
C 1 .2-1.2
0 1.0
0D 0.6
0.4
0.20
5mma
a
.
40 80 120 1 60Pump Power (relative units)
200
Fig. 3. Pump power dependence of the peak deamplifi-cation for a 5-mm crystal. The solid curve is a theoret-ical fit to Eq. (6) below. The dashed curve is a plot ofexp(-20), which is a result of the plane-wave theory andthus excludes the l/zo correction term.
"" ........................... "I .................
proportional to the x(2) susceptibility of the nonlinearmedium; V 1
2= a2 /aX2 + a2 /ay 2 ; k = 27Tn/A, with n as
the refractive index at the signal wavelength A; andwhere we have assumed for simplicity that the pump(at wavelength A/2) and the signal fields are phasematched in the nonlinear medium and that Ep and Kare real. In the plane-wave theory the second termon the left-hand side of Eq. (1) is ignored.'
Decomposing E into the two quadraturesE. = [E exp(-ikp/2) + E* exp(iobp/2)]/2 and Ey =[E exp(-iop/2) - E* exp(iop/2)]/2i, we obtain thefollowing coupled equations for E.,y):
E - 1 _ 1 Ey(.) = KEpEx,). (2)az - TkVE(xNote that the amplified quadrature x couples to thedeamplified quadrature y as the signal wave travelsthrough the amplifier. It is this coupling of the am-plified quadrature to the deamplified quadrature-apurely diffractive effect-that limits the observabledeamplification of the input signal beam.
The general solution of Eqs. (2) for an arbitraryspatial profile of the pump field Ep(p), p = (x, y) isdifficult to obtain. We follow the iterative methodof Belinskil and Chirkin7 to obtain an approximatesolution. As we show below, our experimentaldata agree with this iterative procedure. We firstneglect the V±2 terms in Eqs. (2), as in the plane-wave theory. The resulting zeroth-order equationscan be directly integrated to yield E.f,,)(p, z)EX),(p, O)exp[±KEp(p)z].
In our experiment the transverse spatial profiles ofthe signal and the pump beams are Gaussian. Thepump beam waist is smaller than the signal beamwaist by V2. The confocal distances of the pumpand the signal beams are thus equal, and the phasefronts of the two beams are parallel to each other.Therefore we can write Ep(p) = Epo exp(-p 2 /2a 0
2 )and E(p) = Eo exp(-p 2 /4a 0
2 ), where ao is the inten-sity radius at the pump beam waist and Epo and E0are the peak amplitudes of the pump and the signalfields, respectively.
Inserting the above pump- and signal-field spatialprofiles into the zeroth-order solution, substitutinginto Eqs. (2), and integrating, we obtain the followingfirst-order solutions at the output of the nonlinearmedium:
E.(y)(p, 1) = EX(y)(p, 0)exp(±+)
+ -Ey(X)(p, O)f (±, ±+), (3)
where = ;(p) mKEp(p)l, 1 =KEpol, I is the lengthof the nonlinear medium, zo = 87a0
2n/A is the con-focal distance of the pump and the signal fields,and f(2, (D) = ln(Q/f)exp(2)/2? + [2 + (2; - 2 -1/2>)ln(D/>)]exp(->). Since the additional term inEqs. (3) depends on lzo, the iterative method is thusvalid for l/zo << 1.
Now we return to the positive-frequency part ofthe electric field from the quadrature-field variables.Substituting Eqs. (3) into the defining equations forthe quadratures at the output of the OPA, we have
To deamplify the input signal field we choose Op -2qi = vr, where 0,, is the phase of E(p, 0). Then,from Eq. (4), the output of the OPA is
E(p, 1) = E(p, O){exp[- (p)] - i(l/zo)f (;)}, (5)
from which we obtain the PSA deamplification factor:
DPSA = f IE(p, 1)l2 dpfD E(p, 0)l2dp
1 - exp(-2D)20
2 l 1rb+ D (6)
In the plane-wave theory, on the other hand, thedeamplification factor is given by5 DPSA = exp(-2FD).
To compare our experimental data with the theory,we have shown by the solid curve in Fig. 3 a plotof Eq. (6) that we obtained by setting l/zo = 0.45,D = aI"2 , with I as the pump intensity, and varying a
for the best possible fit. The 0.45 value for l/zo is ingood agreement with that in our experimental setup.The agreement between Eq. (6) and the experimentaldata is quite good. Without the l/zo correction termof Eq. (6) it is impossible to fit the data with thetheory (dashed curve in Fig. 3) at high pump powers.
In conclusion, we have investigated experimen-tally the deamplification response of a traveling-waveOPA and quantitatively compared the data with thetheory.
This research was supported in part by the U.S.Office of Naval Research and the National ScienceFoundation.
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