Power and spectral characteristics of continuous-wave

1684 J. Opt. Soc. Am. B/Vol. 10, No. 9/September 1993

Power and spectral characteristics ofcontinuous-wave parametric

oscillators: the doubly to singly resonant transition

S. T. Yang, R. C. Eckardt, and R. L. Byer

E. L. Ginzton Laboratory, Stanford University, Stanford, California 94305

Received March 18, 1993; revised manuscript received April 16, 1993

The output power and spectral characteristics of optical parametric oscillators (OPO's) as they evolve from thedoubly resonant to the singly resonant regime are examined. By assuming a high-Q signal cavity, we deriveapproximate analytical solutions of pump depletion and conversion efficiency for OPO's with arbitrary idler QUsing this set of solutions, we examine in detail amplitude and frequency instabilities of doubly resonant oscil-lators with low signal loss and arbitrary idler loss. We find that OPO's with weak idler feedback, when oper-ated at a few times above threshold, exhibit improved spectral stability. Experimental results are presentedfor LiNbO3 and KTP OPO's operating with a fixed low signal loss and various idler losses, including the limit-ing case of the cw singly resonant oscillator with no idler feedback.

1. INTRODUCTION

Continuous-wave doubly resonant optical parametric os-cillators (DRO's) were first demonstrated more than 25years ago."2 The low-threshold-power advantage of theDRO's however, was offset by the necessity for overlap ofsignal- and idler-cavity resonances that placed severe tol-erance limits on cavity length and pump-frequency fluctu-ations.3 4 It was not until recently, with the availabilityof single-axial-mode pump sources such as diode-pumpedsolid-state lasers and stable cavity constructions, that cwDRO's were operated in a single axial mode with narrowlinewidth.5 ` Continuous tuning of DRO's, however, re-mained problematic because of cluster effects and theneed to tune two variables simultaneously.9

It is well known that, by resonating only the signal or theidler in an optical parametric oscillator (OPO), both out-put power stability and the tuning range are improved.'0

Unfortunately the threshold for a singly resonant OPO(SRO) is typically a few hundred times greater than DROthresholds. The high threshold power has precludeddemonstration of the cw SRO until recently." Given thedifficulties in achieving the high cw SRO threshold, wewere motivated to examine the DRO-to-SRO transition todiscover whether one might obtain the benefits of SRO op-eration at reduced threshold from a DRO that operateswith higher loss for one ofthe waves. This question was pre-viously addressed by Falk,2 who analyzed near-thresholdbehaviors of OPO's with arbitrary signal and idler loss.From this analysis Falk concluded that, near threshold,the OPO stability is not expected to improve until the feed-back from either the signal or the idler wave is eliminated.

In this paper we reexamine this issue by investigatingthe well-above-threshold behavior of DRO's with a high-finesse signal cavity and an arbitrary-finesse idler cavity.In Section 2 we present approximate analytical solutionsof pump depletion and conversion efficiency for OPO'swith a high-Q signal cavity and an arbitrary-Q idler cav-ity. The solutions are then used to examine the amplitudeand frequency stability of OPO's at different pumping lev-

els and with various idler losses. Section 3 complementsthe theoretical discussion by presenting experimental re-sults of LiNbO3 and KTP OPO's operating with low signalloss and arbitrary idler losses ranging from near zero to100%. Finally some conclusions concerning practical real-ization of cw SRO's are presented in Section 4.

2. THEORY

A. Pump Depletion and Conversion EfficiencyIn this section we examine pump depletion and conversionefficiency for OPO's with low signal loss and arbitraryidler feedback. Previous plane-wave analyses of OPO'shave treated low-loss DRO's in which both signal and idleramplitudes are assumed to be spatially invariant.' 3 Herewe are interested in examining cases in which the idlerfeedback approaches zero, the SRO condition. With largeidler-output coupling, the constant idler-field approxima-tion is not valid. However, we continue to assume thatthe signal cavity has high Q, and therefore the signal fieldis spatially invariant. By assuming a constant signalfield, we can reduce the set of three coupled-mode equa-tions to two equations, one for the pump and the other forthe idler. With this reduced set of equations, approxi-mate analytical solutions for pump depletion and conver-sion efficiency can be derived. The derivation is identicalto the approach used by Colucci et al. " We repeat a sim-plified version of their derivation in this section, using no-tations that are consistent with previous studies on DRO'sand SRO's.'3 "'

We start with the familiar coupled-wave equations 6 :

az = iKsEp(Z)Ei*(Z)exp(iAkz),

aEi(z) E E (( kaz = KjEp(z)E8*(z)exp(iAkz),

a )= iKpE,,z)E~exp(-iAkz).

(la)

(lb)

(lc)

In these equations K = od/(n.c), K = wid/(nic), and0740-3224/93/091684-12$06.00 © 1993 Optical Society of America

Yang et al.

Vol. 10, No. 9/September 1993/J. Opt. Soc. Am. B 1685

K = wpd/(npc), where w3, coi, and Cw, are the angular fre-quencies for the signal, idler, and pump, respectively, n3 ,ni, and n, are the refractive indices for the signal, idler,and pump, respectively, d is the effective nonlinear coeffi-cient, and c is the speed of light. The momentum mis-match is given by Ak = k - k - ki. Assuming a smallloss at the signal wavelength, we set Es (z) = E =constant. This permits us to ignore Eq. (la), and we canthen proceed to solve Eqs. (lb) and (1c). Subject to theboundary conditions that Ei (z = 0) = Ejo and E, (z =0) = Epo, where Ejo and Epo are complex numbers, solu-tions for Ei (z) and E, (z) are

Ei(z) = {Ejo cos(Fz) - Ei0[iAk/(2r)]sin(Fz)

+ (iKi/F)EpoEso* sin(Lz)}exp(il&kz/2), (2a)

E(z) = {Epo cos(rz) + Epo[iAk/(2F)]sin(rz)

+ (iK/I)EsoEio sin(Fz)}exp(-iAkz/2), (2b)

where r2= KiKPIEs012 + (Ak/2) 2. Equations (2) express

Ei(z) and E,(z) as functions of initial field amplitudes, E3O,Ejo, and Epo. We assume here that the pump passesthrough the crystal once per round trip. Therefore Ep0 isequal to the incident pump field. To solve for E3o and Ego,we invoke self-consistency equations for the signal andidler fields. For the idler field, self-consistency impliesthat Ei(z = 0) = Ei(z = l)ri exp(iTi), where Ti repre-sents the round-trip phase shift for the idler given by'i = -4nil/Ai - 4.r(L - 1,)/Ai where l is the crystallength and L is the one-way cavity length for a standing-wave cavity. Also, ri is the idler-field reflectivity. In thefollowing sections r denotes field-reflection coefficients,whereas R indicates power-reflection coefficients. Im-posing self-consistency on the idler field, we find that theinitial field Ejo is given by

-.EpoEso* sin(Fl,)ri exp(iAkl,/2 + iTi)

1- [cos(Fl) - i sin(kFlc)]ri exp[i(Ake/2 + I)](3)

The solutions for Ei(z) and Ep(z), obtained with thesumption that E.(z) is constant, do not differ significanfrom those obtained from a full numerical treatmenTo improve the accuracy of the signal-field solution,substitute Ei(z) and Ep(z), derived above, into the signfield derivative and integrate over crystal length 1,. resultant E,(z = 4,) is

E3 (z = I = 1 + NCF*sin(x)

Eso x

x 1 + sin(2x) + ikl sin2(x) (l) 2 [1 sin(2x)1. 2x -2 4X2 2x

+ NC{1 - -( 2c) sin2(X)F2

x {2 sin2 (x) + iAkl,[1 - sin(2x)]}

- NCF sin(x) - (A 2 sin(2x)

where x = F1, and N = IEpoI2/JEpth2 is the pumping IErelative to threshold at parametric gain center,'2 whicl

IEpthI2 = (2/KKi l 2)C, where C = [(1 - r)(1 - ri)]/(r +ri). F* denotes the complex conjugate of F, which is de-fined by

F 1 ri exp[i(Akl,/2 + Pi)][cos(x) - iAk1,/(2x)sin(x)]exp[i(Akl/2 + 'i)]ri

(5)

Finally self-consistency is imposed on the signal field:E3 (z = l,) = Esor exp(iPT), where T. = -4irn1l/A 3 -

47r(L- 1,)/A, and r8 is the signal-field-reflection coeffi-cient. Given the pumping level N mirror-field reflec-tances for signal and idler r, and ri, round-trip idler phaseshift Ti, and phase mismatch Akl,, we can calculate thesignal-field intensity iE.012, which is embedded in x2, fromthe real part of the signal self-consistency equation:

1exp - i = 1 + NCF* sin()rS X

1 sin(2x) + ikl sin2 (x) (Akl)2 [1

NCI_ 1 Akl') 2 sin 2(X) L F

- sin(2x)]

2 sin2(x + ik,{1 - sin(2x){ [ 2x ]).2sin-x) ([ kl \2) [ sin(2x)]

- N CF I l - - J i l - IX L \2X /J 2x J (6)

The corresponding round-trip signal phase shift T., is ob-tained from the imaginary part of Eq. (6).

B. Low-Loss DRO and SRO LimitsFor comparison with previous analyses of low-loss DRO'sand SRO's, we assume that Akl = 0 and that signal andidler frequencies coincide with cavity resonances (i.e.,T, = 2mir, Wi = 2nqr, where m and n are integers). Withthese assumptions, Eq. (6) reduces to

1 21 -ri N( sinc(x)sinc(2x)rS rS + ri 1 - cos(x)r

+ 2 sinc2(x){[1-cos(x)ri2 1) (7)

One finds the equation that governs pump depletion bysubstituting E, given by Eq. (3), into Eq. (2b), whichyields

Ep(z) = Epo[cos(z) - 1-cos(Fl)r sin(Fz) . (8)

The depleted pump intensity after a single pass throughthe crystal is then given by

IE (z = I)|21 - cos(x)ri 12= Lco~) (9)

One can also derive the photon-conversion efficiency forthe signal field by starting from Eq. (6). With some alge-bra, photon-conversion efficiency for the signal field isgiven by

(4) &(p n 8E.o12 _ 2 (ri sinc(x)sinc(2x)ws nPIE'oI2 5r \ 1 - cos(x)ri

+ - sinc2(x)1- [1 - cos(xri]2}) (10)

Yang et al.


`0100

C 8080 R=99.9%0~ R=25%

h 60 6 =4%

C40

20 R 100%> ~~~~~~~~~~S

01 1.5 2 2.5 3 3.5 4 4.5 5number of times above threshold

Fig. 1. Theoretical signal-photon-conversion efficiency plottedas a function of pumping intensity relative to threshold. Cal-culation is done for DRO's with fixed round-trip signal-powerreflectance of 100% and various round-trip idler-power reflec-tances Ri. Note that the 100% conversion-efficiency point movescontinuously from N = 4 to N = r/2) 2 as the OPO's evolvefrom the DRO to the SRO regime.

If we set r = 0 and r = 1 (i.e., the SRO condition),Eq. (7) reduces to

IEsthI2

= sinc2

(X) (11)

which is the SRO threshold condition previously derivedby Kreuzer.'5 If we set r = 0, Eq. (9) reduces to

JE(z =I,2 12 = Cos2(x),

which again agrees with the SRO-pump depletion resultobtained previously.5

Similarly, in the limit of low signal and idler loss,Eqs. (9) and (10) converge to become Bjorkholm's solutionsfor DRO's with high-Q signal and idler cavities.'3 Notethat the above derivation is based on the plane-wave ap-proximation, so it does not lend itself to quantitative com-parisons with the experiments except in the near-fieldlimit. However, we expect that the qualitative behaviorsof OPO's will not change significantly with focusing.Also, we assumed that each field (pump, signal, and idler)oscillates in single longitudinal modes.

Figure 1 shows signal-photon-conversion efficiency cal-culated from Eq. (10) for OPO's with R = 100% and forRi's ranging from 99.9% (DRO) to 0% (SRO). As the idler-power reflectivity Ri decreases toward zero, the OPO be-haves more like a SRO. In particular, the pumping levelat which 100% signal-photon-conversion efficiency isachieved moves continuously from N = 4 to N = (/2) asRi decreases from 99.9% to 0%. This transition, however,is nonlinear with respect to change in idler reflectance.For example, the pumping level at which 100% signal-photon-conversion efficiency occurs does not move beyondthe halfway point between N = 4 and N = (7r/2)2 (i.e.,DRO and SRO limits) until Ri has dropped to below 22%.Pump depletion follows the same behavior.

C. Amplitude StabilityAway from exact overlap of signal- and idler-cavity reso-nances, the OPO output intensity drops. One can exam-ine this intensity variation by taking into account the

round-trip phase shift for the signal and idler fields P3 and'Pi. According to Falk's 2 approach, we define the detun-ing from coincidence of signal- and idler-cavity resonancesby the phase-detuned parameter To:

IP + 1P = 'Po + 2kla,

'o = - 2cop [n,1 + (L - 14)]. (13)

The definitions in Eqs. (13) apply for a standing-wave cav-ity. o for a ring cavity is one half of Po for the standing-wave cavity given by Eqs. (13). For fixed pump frequency,note that To depends only on cavity length L. As the cav-ity length changes by half of the pump wavelength, Tovaries from 0 to 27r.

One can calculate the intracavity signal intensity forany detuning from coincidence of signal- and idler-cavityresonances (i.e., T,, Pi $ integer multiples of 27r), usingEq. (6). For simplicity we consider the case of Akl, = 0 inthis subsection. Figure 2 is a plot of the intracavity sig-nal intensity normalized to unity at its peak versus phasedetuning for OPO's with different signal and idler ref lec-tances. The solid curves represent cases in which R, =99%, whereas the single dashed curve denotes the case inwhich R, = Ri = 89.1%. One performs all the calculationsassuming pumping at three times above the respectivethresholds. Without normalization, intensity values arelarger for OPO's with higher idler losses since the pump-ing power required for threshold to be reached is larger inthose cases. As Fig. 2 shows, the tolerance range for op-eration away from coincidence of signal- and idler-cavityresonances, as represented by the half-peak width of T, +Pi, becomes larger as the idler reflectance is lowered. Inparticular, as Ri decreases to 4o, the OPO remains abovethreshold even as signal- and idler-cavity resonances areshifted apart by as much as ±7r rad, assuming no competi-

4-).7 1.2Ca)

4.) 1.0-C:

- 0.8C. 0.6

- 0.4a)Nala 0.2

E" 0.00 -

I-In 07C 0.5n In

phase detuning Vs + viFig. 2. Calculated normalized signal intensity versus detuningof the signal and idler resonances expressed as the sum of theround-trip phase shift T + i. The calculation is done for DRO'swith a signal-power reflectance of 99%, except for the dashedcurve, for which R = 89.1%. The peak of calculated signal inten-sity is normalized to unity. The pumping levels are at threetimes above their respective thresholds. Perfect phase matchingis assumed for these calculations. Note the widely differentphase-detuning tolerance range for Ri ranging from 99% to 0%.For Ri = 99% the tolerance range is so small that it appears as aspike. In the other extreme, when Ri = 0% the SRO amplituderemains flat regardless of phase detuning.

YO %,. *N=3

/ I

, R =89.1%IX I s

'1~ I: R-99% II

l

w w w w l

Yang et al.


N 1.6o ~~~R =99%

- 12 N=4 Rs=25%

1.0-0)

0.8-

o- .6-

E 0.4-*Nl0.2

0.0-0.5it On 05t

phase detuning Vs+ VJ

Fig. 3. Calculated parametric gain, (rB,)2

= KC8KiiEi 2,y

2, versusdetunings from an exact overlap of signal- and idler-cavity reso-nances. The calculation is done for DRO's with fixed round-tripsignal-power reflectance of 99% and idler-power reflectance of25%. The phase-detuning range over which the OPO remainsabove threshold increases as the pumping level is raised. Perfectphase matching is assumed for these calculations.

tion from adjacent axial-mode pairs. In contrast, the tol-erance range for OPO's with R8 = Ri = 99% is only±0.015 rad for pumping at three times above threshold.In addition, the output amplitude modulation versus de-tuning decreases as Ri is decreased. In the limit whenRi = 0, the output amplitude remains flat regardless ofphase detuning. Of particular interest is the tolerancerange difference between the cases in which R, = Ri=89.1% and in which R, = 99% and Ri = 25%. These twocases have the same threshold, and yet the OPO withlower idler Q has a much wider phase-detuning tolerancerange.

Figure 3 shows that, for fixed signal and idler losses,the phase-detuning tolerance range widens as the pump-ing level is increased. Here the parametric gain (rl0 )2, iscalculated for an OPO with R, = 99% and Ri = 25% andat pumping levels N = 1.1, 1.5, 3, and 4. As pump poweris increased, the phase-detuning tolerance range increasesrapidly.

D. Frequency StabilityThe frequency stability of an OPO with arbitrary signaland idler feedback was examined previously by Falk.'2

On the basis of detailed studies of OPO operation nearthreshold, Falk concluded that the nonresonant idler feed-back must be eliminated to prevent frequency instabilitiessimilar to that for DRO's with two high-Q cavities. Inthis subsection we show that Falk's conclusion is validonly near threshold. When the OPO with slight idlerfeedback is pumped at a few times above threshold, thefrequency instabilities resulting from cavity perturba-tions decrease substantially.

According to Falk's' 2 approach, we assume that thepump frequency is sufficiently stable that OPO-frequencyinstability is due primarily to cavity-length fluctuations.As the cavity length drifts randomly over one pump wave-length, Po changes by 2

IT. A measure of OPO-frequencystability is given by the maximum signal- or idler-frequency excursion from parametric-gain center as onevaries Po by 2ir. For DRO's with high signal and idlerfeedback, the maximum frequency excursion fromparametric-gain center is given by half of the cluster spac-

ing.3,4 9 This implies that, for DRO's in a standing-wave

cavity in which the pump passes through the crystal oncebut in which signal and idler traverse the crystal twice perround trip, the extreme OPO operating points occur atANi = +±r/2. For a SRO, the maximum frequency excur-sion is half of the cavity free-spectral range since the SROoscillation occurs on the cavity mode nearest parametric-gain center.

To calculate OPO operating point at each phase detun-ing To corresponding to a particular cavity-length setting,one solves Eq. (6) for ANkl ranging from zero to 2ir at thefixed value of To. The value of Akl corresponding to themaximum signal intensity is then considered to be the op-erating point. Figure 4 is a plot of the OPO oscillationpoint versus Po calculated at four different pumping levelsfor OPO's with signal-power reflectance of 99% and idler-power reflectance of 1%. The solid curves represent cal-culations in which we assumed that the OPO oscillationfrequency coincides with the maximum signal outputpoint. We calculated the dashed curve, using Falk's pre-scription in which the minimum threshold point is consid-ered to be the operating point. Falk's calculation showsthat, near threshold, as To increases from 0 to i, the OPOoperating point shifts linearly from Akl, = 0 toAl, = 1.32 rad. At To = , a cluster hop occurs at whichthe operating point jumps from Ak = -1.32 rad to1.32 rad. For pumping close to threshold and to as highas 1.5 times above threshold, we see that the maximumsignal output points coincide with the minimum thresholdpoints, and the OPO may operate as far as ±1.32 rad fromthe Akl, = 0 point. However, at pumping levels between2.15 and 3 times above threshold, the maximum operating-point excursion from parametric-gain center decreases.For example, at pumping 2.5 times above threshold themaximum signal output point does not deviate from theparametric-gain center by more than ±0.1 rad, whichmeans that the OPO essentially functions as a SRO.

At pumping 2.64 times above threshold, 100% pump de-pletion occurs at the parametric-gain center Akl = 0, andany further increase in pumping results in conversion ofgenerated OPO waves back into the pump. As a result,

1.5

M 1.0

u-E u 0.5

Ca 0.0

= 0

C '2 -0.5 G o i R=9 4). ~R99%8 -1.0 X RI=1%

N=2.15 ------ Falk :-1 .5 , . . . , . . . . . . . . . . .

On 0.5 n I l.5 n 2 nphase detuning V0

Fig. 4. Calculated OPO Akl operating point for OPO's withRs = 99% and Ri = 1% as a function of cavity-detuning parame-ter Po and at different pumping levels. The dashed curve iscalculated by Falk's prescription, which assigns the minimum-threshold point as the OPO operating point. In this study wecalculated the solid curves assuming that the OPO oscillates atthe highest signal output point.

Yang et al.


2.0

,a 1.6

E CU C*.~ 1.2- CO

E 'E. 08

0 2 0.40E

0.0 fi0 0.04 0.08 0.12 0.16 0.2 0.24

r

Fig. 5. Calculated maximum Akl operating-point excursion ofthe OPO relative to the phase-matching peak as a function ofidler-field reflectance r. The OPO has fixed signal power re-flectance of 99%. The dashed curve is from Falk's calculation.The solid curves are calculated assuming that the OPO operatingpoint corresponds to the maximum signal point. They are calcu-lated for pumping at 1.1, 1.5, 2.0, 2.5, and 3.0 times above thresh-old. Pumping at 2.5 times above threshold, the maximumoperating-point excursions are insignificant for r < 0.1.

the OPO actually tends to oscillate away from theparametric-gain center when there is no phase detuning(i.e., To = 0). This behavior is evident in the case ofpumping at three times above threshold. In this case themaximum signal output point lies 0.4 rad from theparametric-gain center at P = 0. If the pumping inten-sity is increased further, the operating point at zero phasedetuning moves farther from the parametric-gain center.

Calculations similar to the ones used to derive Fig. 4 arerepeated for OPO's with the same signal loss but differentidler reflectance and with different pump input. Foreach specified pumping level and idler-field reflectance,the operating point is solved for To varying between zeroand 2 7r, and the maximum operating-point deviation fromthe parametric-gain center is plotted in Fig. 5, in whichthe dashed curve is from Falk's calculation. The solidcurves are results of this study. Near threshold Falk'scalculation shows that the OPO's phase instability persistseven as idler feedback drops to a very low level. For ex-ample, for an idler-electric-field reflectance of 10%, corre-sponding to residual idler-power reflectance of 1%, themaximum operating-point excursion is 1.36 rad, which is87% of the maximum phase-mismatch excursion expectedfor DRO's with two high-Q cavities.'2 However, when thepumping level is increased to 2.5 times above threshold,the operating point excursion for idler-field reflectivityless than 10% is substantially reduced. In particular,maximum operating-point excursion of r = 10% is lessthan 0.1 rad for pumping at 2.5 times above threshold.

We have shown that an OPO with weak idler feedbackexhibits frequency stability similar to that of a SRO whenit is pumped at a few times above its minimum threshold.It is not necessary, therefore, to eliminate idler feedbackto have frequency-stable OPO operation.

3. EXPERIMENTS

We conducted a series of experiments to complement thetheoretical calculations and to ascertain the feasibility of

cw SRO operation. We operated OPO's with low signalloss and various idler output couplings. Figure 6 showsthe experimental setup. In all the experiments the pumpsource that we used was a multiwatt injection-locked cwNd:YAG laser that routinely produces 12-W output powerat 1064 nm with a linewidth in the range of a few tens ofkilohertz. 7 The single-frequency output of the Nd:YAGlaser is resonantly doubled in an external enhancementcavity containing a 6-mm-long LiB305 crystal. 8 With12 W of pump input as much as 4 W of green power isroutinely produced in a diffraction-limited beam.

We used both MgO:LiNbO3 and KTP crystals. TheLiNbO3 crystal that we used has 4.9% magnesium-ion-concentration doping.'9 The MgO:LiNbO3 crystal, whichis 2 mm 2 mm X 12.5 mm, was cut for type-I noncriti-cal phase matching along the crystallographic x axis. Atwo-layer broadband antireflection coating was depositedonto the crystal surface. The coating has residual-powerreflection of 0.2%o/surface for wavelengths ranging from940 to 1070 nm.

The 3 mm 3 mm x 8.6 mm KTP crystal was cut alongthe crystallographic axes with the long dimension alongthe x axis (i.e., = 90 deg, p = 0 deg). The two facets ofthe crystal are antireflection coated at 1090 and 532 nm.With the 532-nm pump beam incident polarized along thecrystal y axis, the signal wavelength of 1090 nm and theidler wavelength at 1039 nm are polarized along they andz axes (i.e., the ordinary and extraordinary axes), respec-tively, by use of type-II noncritical phase matching.20

A. SRO-to-DRO Threshold StudiesWe measured the OPO threshold progression from thedoubly resonant to the singly resonant regime by increas-ing the idler loss of OPO's with a fixed low signal loss.For these threshold measurements we used a bow-tie ringresonator with two curved mirrors (each with a 10-cm ra-dius of curvature) and two flat mirrors as shown in Fig. 6.One of the flat mirrors was mounted upon a piezoelectrictransducer (piezo; PZT) for fine cavity-length control. Wechose a ring cavity to minimize loss through the crystalper round trip. For operation as a DRO with two high-Q

KTP or V VLI~~~bq ~mode -matchingLINbO3 lens

Fig. 6. DRO experimental setup. The two flat mirrors of thebow-tie ring cavity are spaced 45 cm apart between the two flatmirrors, and the space between the two mirrors (each with a 10-cm radius of curvature) is 11 cm. The incident angle on all themirrors is 3 deg. To introduce loss preferentially for the idler ina LiNbO 3 OPO, we replace one of the flat mirrors with aTi:sapphire mirror. For KTP OPO one or two Brewster platesare introduced into the long leg of the cavity to couple out thep-polarized idler waves preferentially.

Yang et al.


cavities, all four mirrors are highly reflecting (>99.9%) inthe range between 970 and 1160 nm and highly transmit-ting (99%) at 532 nm. The cavity layout is designed sothat, at a signal wavelength near 1 rm, a 31-Am beamwaist radius is produced midway between the two curvedmirrors.

For studies that use the MgO:LiNbO3 OPO, the crystalis placed in an oven and is heated to the phase-matchingtemperature of 107 'C. We confirm the temperature uni-formity of the crystal by measuring the phase-matchingtemperature bandwidth when doubling a Nd-dopedgadolinium gallium garnet laser operating at 1.062 jkm.The observed FWHM temperature-acceptance bandwidthof 0.65 deg is close to the theoretical bandwidth of 0.59 degfor a 12.5-mm-long crystal. At the degenerate wave-length of 1064 nm, the OPO-cavity finesse is measured tobe 785, implying a round-trip power loss of 0.8%. Also atdegeneracy, the signal beam waist has a l/e radius of31 ,um. With a 12.5-mm-long crystal length and refractiveindex of n = 2.232, the signal- and idler-beam confocalparameters are equal to the crystal length (i.e., 4s = (i =1). To maximize the spatial-mode coupling between thepump and the OPO waves, the pump confocal parameter isalso set equal to the crystal length. Assuming a round-trip signal- and idler-power loss of 0.8% and that the Boyd-Kleinman2 focusing parameter h(B = 0, = 1) =0.8, we calculate the threshold at degeneracy to be 7.3 mWjust inside the crystal's entrance facet. This calculationuses the Boyd-Kleinman2 ' threshold formula for DRO'swith low signal and idler losses, and it assumes an effec-tive nonlinear coefficient of 4.7 pm/V (Ref. 22) forMgO:LiNbO3. The actual threshold should be 8.1 mW ifwe take into account the 90% pump transmission throughthe crystal's antireflection-coated input face. We experi-mentally measured a minimum threshold of 20 mW Thisdiffers considerably from the predicted threshold of8.1 mW Since the temperature gradient is small, as con-firmed by the second-harmonic-generation experiment,we attribute the higher-than-predicted threshold to im-perfect mode matching of the pump beam into the OPOcavity.

To operate the LiNbO3 OPO with low signal loss andhigh idler loss, we replaced one of the flat mirrors in thecavity by a Ti:sapphire laser mirror that is highly ref lect-ing between 870 and 1130 nm and has a 93% transmissionpeak centered at 1135 nm. To increase idler output cou-pling, we raised the crystal temperature so that the phase-matched idler wavelength tunes toward 1135 nm. Energyconservation then dictates that the signal wavelengthmoves toward 1001 nm. However, since all the mirrorsremain highly reflecting at 1001 nm, the round-trip signalloss remains small. Thus the OPO operates continuouslyin the regime between high-Q DRO and SRO depending onthe crystal temperature.

The threshold measurement was made with the pumpbeam chopped to produce pulses of 28 ,us duration at a 100-Hz repetition rate. The 28-ps pulse duration was longenough compared with the OPO rise time that steady statewas reached during each pulse. By chopping the pumpbeam we considerably reduced the thermal-focusing ef-fects. During measurement the PZT-mounted flat mir-ror was continuously scanned, and the output was sent toa 1-in spectrometer with the slit width set for a resolution

of 1 A. We recorded the signal and idler wavelengths andthe corresponding threshold at each temperature. TheTi:sapphire mirror transmission was measured at eachidler wavelength. Figure 7 shows the measured thresholdat each wavelength correlated with the matching idlerloss. With 4 W of pump power, the OPO remained abovethreshold until the idler loss reached 83%. The curveshown in Fig. 7 was calculated with the formula

PDROth 2(1 - ri)PhiQth a i(l + ri)

(14)

where PhiQth is the threshold of the DRO with low signal-and idler-field losses, a and ai, and PDROth is the OPOthreshold with the same signal-field loss a, but with arbi-trary idler-field reflectance ri. We find that, if we assumethat PhiQth = 17 mW and ai = 0.4%1o, the calculated thresh-old agrees well with the measured values. By extrapolat-ing to 100% idler loss, we find that cw MgO:LiNbO3 SROhas a threshold of 8.5 W This is within the range of high-power cw argon-ion lasers if the crystal can withstand thehigh average pump power. Unfortunately, when thepump beam was not chopped, the MgO:LiNbO3 crystalwas damaged at 3.1 W of input pump power. With apump-focused spot radius of 22 -rm inside the crystal, thisrepresents a damage intensity of 816 kW/cm2 at 532 nm.The damage occurred at the center of the crystal and ap-peared as a blackened track. The damage mechanism isthermal in origin, since chopping the pump to produce 28-,us pump pulses at 100-Hz repetition rate increases thedamage intensity to 1 MW/cm2. The low cw damagethreshold prevented us from reaching the cw SRO thresh-old when we were using the 4.9%-doped MgO:LiNbO3crystal.

The experiment with LiNbO3 prompted us to look for anonlinear crystal that can withstand the high pump inten-sity necessary for cw SRO threshold. Surveying theavailable nonlinear crystals, we found KTP to be the mostpromising material since it has both a high damagethreshold and a relatively high nonlinear coefficient of3 pm/V20 2223 To determine cw SRO threshold by use ofKTP, we conducted threshold experiments similar to thosefor LiNbO3 . For these threshold experiments we usedthe same bow-tie cavity as before, except that in this caseall four mirrors were highly reflecting in the range be-tween 970 and 1160 nm.

10

C)

3

0

a):5

8

6

4

2

O 20 40 60 80 100idler power loss(%)

Fig. 7. Measured LiNbO3 DRO threshold as a function of round-trip idler-power loss. We observed minimum threshold of 20 mWThe curve is calculated by the threshold-ratio formula [Eq. (14)],assuming a 17-mW threshold for round-trip signal- and idler-power loss of 0.8%. The extrapolated SRO threshold is 8.5 W

Yang et al .


The theoretical threshold, as calculated from the Boyd-Kleinman2 threshold formula, is given by Pth = a~ai X289.7 W, where a and ai are round-trip signal- and idler-power losses. In the threshold calculation, we assumedthat the focusing parameter h(B = 0, = 0.86) = 0.72and that deff = 3 pm/V (Refs. 20, 22, and 23) for KTPcrystal.

Since the KTP OPO does not operate at degeneracywhen it is noncritically phase matched at room tempera-ture, we cannot assume that the signal and idler losses arethe same (i.e., a # a). To determine the ratio of round-trip signal to idler losses, we measured the orthogonallypolarized signal and idler output power ratio that leakedout from the highly reflecting cavity mirrors. Using themeasured power ratio of 2.29, one can obtain the loss ratiofrom the formula2 4

P8 _ ys(Yi + Aii,)

Pi Yi(s + 115) (15)

where ys and yi are the output coupling loss of the signaland idler waves, respectively, and A, and Ai are other spu-rious signal and idler losses resulting from crystal absorp-tions, etc. In our case, since we are operating not too farfrom degeneracy and also since the angle of incidenceupon the mirror in the bow-tie cavity is near normal,Vs Vi. In addition, since the round-trip loss of the OPOcavity with the crystal removed is measured to be lessthan 0.2%, we can assume that y << , and yi << ,i. Onthe basis of these assumptions, ,u - 2.29 A,. This is ex-pected because the narrow-band antireflection coating iscentered at a signal wavelength of 1090 nm. When thepump beam was focused into a 22-/Lm beam waist insidethe crystal, we measured a minimum threshold of 30 mWGiven the ratio of signal- to idler-power loss (2.29), themeasured threshold implies that the round-trip signal-and idler-power losses are 0.7% and 1.6o, respectively.

Since the signal and idler waves are orthogonally polar-ized, they can be discriminated by insertion of polarizingelements, such as Brewster plates, inside the OPO cavity.We introduced one or more 3-mm-thick BK7 Brewsterplates into the cavity to produce additional loss for theidler while adding minimal loss for the signal wave. Therefractive index of BK7 glass is 1.517 at the signal wave-length of 1.09 m, and the Brewster angle is 56.6 deg.Each passage through the Brewster plate introduces a28.6% loss for the S-polarized idler waves. We insertedone and two BK7 Brewster plates into the cavity, and theyintroduced 29% and 49% loss, respectively, for the idlerbeams. The OPO threshold power with one and twoBrewster plates in the OPO cavity was 707 mW and2.79 respectively. The threshold ratio between thelow-loss DRO and OPO with one Brewster plate agreeswell when we assume a fixed round-trip signal-power lossof 0.7%. However, with two Brewster plates inserted, themeasured threshold is considerably higher than expected.We believe this is due to the difficulty in aligning twoBrewster plates for minimum signal loss. Extrapolatingfrom the measured threshold for the DRO withoutBrewster plates and the DRO with one Brewster plate, wefound the threshold for a KTP cw SRO is 7.8 W In thesethreshold measurements for KTP OPO the full cw pumppower was incident upon the crystal. With maximum

pump power of 4 W focused to a beam waist radius of22 Apm, we observed no crystal damage.

The KTP DRO threshold experiments revealed the ex-pected cw SRO threshold, and they also showed that theKTP crystal can withstand high average pump power.The extrapolated threshold of 7.8 W, however, exceedsavailable pump power from the resonantly doubled Nd:YAGlaser. To reduce the pump threshold, we chose a standing-wave SRO cavity such that the pump as well as the idler isnonresonantly reflected back through the KTP crystal.As was pointed out earlier by Bjorkholm et al.25 and asexplained in Appendix A, the SRO threshold can be re-duced by as much as a factor of 4 when both the pump andthe idler are nonresonantly reflected back through thecrystal. The cavity design with pump and idler feedbackis shown in Fig. 8. The SRO cavity is a three-mirrorstanding-wave cavity that consists of two curved mirrors(each with a 5-cm radius of curvature) and one flat mir-ror. All three mirrors are highly reflecting from 970 to1160 nm and highly transmitting at 532 nm. Since it isdifficult to obtain a mirror coating that is highly reflect-ing at both the pump and the idler wavelengths, we used aseparate mirror that is highly reflecting at the pumpwavelength and is situated outside the OPO cavity to re-flect the pump back through the crystal. Singly resonantoperation is achieved by insertion of an intracavityLiNbO3 Brewster prism that spatially separates the signaland idler beams. The prism is designed such that the P-polarized signal beam at 1090 nm is incident upon theprism at Brewster's angle and with ordinary polarizationin the LiNbO3 prism, whereas the S-polarized idler beamat 1039 nm traverses the prism with extraordinary polar-ization. Because of the large refractive-index differencebetween the ordinary and extraordinary axes of LiNbO3 ,the signal and idler beams emerge from the prism sepa-rated by 8 deg. The large separation permits the idlerbeam to be completely rejected from the cavity. This en-sures SRO operation.

The KTP crystal that we used is similar to that used inthe DRO experiment, except that the length was increased

pump M2 Do `-- MI f=1 cmmirror

Fig. 8. Cw KTP SRO experimental setup. Mirrors Ml and M2each have a 5iu radius of curvature. Mirror M3 is flat. Mir-rors M2 and M3 and the pump mirror are mounted on PZT's.The pump and idler waves are reflected back through the crystal.The LiNbO3 Brewster prism rejects the idler wave when oneachieves true SRO operation.

kITD ,.vcta V

Yang et al.


Cu

i; 1.2

al 1.0

° 0.8

'- 0.6

: 0.4.! 0.2

a 0.0a) 1

0

..

0

F.

0

.

8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4pump power(watts)

Fig. 9. Generated SRO-idler power versus pump power. Thethreshold is 1.07 W At 3.2 W of 532-nm pump power, the non-resonant idler output power is 1.1 W for an idler conversion effi-ciency of 33%.

threshold. With the insertion of one Brewster plate thepeak modulation depth is reduced to 43% for pumping at1.35 times above threshold as shown in Fig. 10(b). In thelimit of 100% idler loss, Fig. 10(c) shows the SRO outputwhen the flat mirror, M3 (see Fig. 8), is driven. Since theidler beam is rejected by the LiNbO3 prism and emergesfrom the cavity before it reaches the flat mirror, only theresonant signal wave is reflected by mirror M3. In thiscase we would expect a constant SRO output intensity re-gardless of cavity-length setting. This is partially con-firmed in Fig. 10(c), which shows the SRO output forpumping at 1.3 times above threshold. As the figureshows, amplitude fluctuation of the OPO output as a re-

0.3

_ 0.2

X 0.2

C 0.1

° 0.00

to 1 cm. With a measured pump beam waist radius of21 /.Lm and a calculated signal beam waist radius of 31 Am,both the signal and the pump have confocal parametersthat are equal to the crystal length. The finesse of theOPO cavity that was measured along the ordinary axis ofKTP was 743 at 1064 nm, implying a round-trip power lossof 0.85%. The actual loss at signal wavelength of 1090 nmis expected to be less since the crystal antireflection coat-ing is centered at 1090 nm. Assuming a 0.85% round-trippower loss and confocal focusing of pump and signal, wecalculated a SRO threshold of 7.2 W, using the SRO thresh-old expression given in Ref. 26. By nonresonantly re-flecting both the pump and idler back through the crystal,one can reduce the SRO threshold by 4 times to 1.8 W Weexperimentally achieved a threshold of 1.4 W at optimumrelative phase among the three waves. Figure 9 is a plotof the SRO-idler output power versus input pump power.We extrapolated the plotted idler power from actualmeasured values after the beam emerged from the prism,taking into account the 62% reflection loss that the S-polarized idler wave incurred on the LiNbO3 prisms twosurfaces. This reflection loss can, in principle, be elimi-nated, and the total idler power that is generated can beoutcoupled. With 3.2 W of pump power, the maximumidler power generated was 1.07 W The corresponding sig-nal output from each of the three mirrors is 12 mW Thisrepresents a combined signal-plus-idler output power con-version efficiency of 34.6%.

B. Observed OPO Amplitude and Frequency StabilityIn conjunction with the threshold experiments, we also in-vestigated the effect of increasing idler loss on the ampli-tude and frequency stability of OPO's. By lowering theidler-cavity Q, we expect to reduce the intensity modula-tion of the OPO output as the cavity length changes.This is confirmed in the case of KTP OPO with increasingidler loss. Figure 10 shows the output intensity of theKTP OPO with three different idler losses as one scansthe cavity length by driving the PZT-mounted cavity mir-ror. The ramp voltage driving the PZT is superimposedin each graph. The tuning rate of the PZT is 8.3 nm/VWith 100-200 V applied to the PZT in each case, the cav-ity length is scanned over 1 gm. As is shown in Fig. 10(a),the low-loss KTP OPO output intensity varies by as muchas 75% when the OPO is pumped at 1.38 times above

0.0

1.4

0 1 0 20 30 40 50 60 7(tlme(ms)

,i 1.2

2 1.0

3 0.8

= 0.6

00

o 0.2

0.0

Cu

cE 0.8

a 0.6

InC 0.4a)C

o 0.20

U5

SRO (C)

N=1.3

, I ,~~~~

60 70 80tlme(ms)

400

350

300 'aN

250 C

a

150 <<0

100 0

500

o

150

100

50 .0

-50 a

-100

-150D

400

300

200 3100 C0o Do

to-1 00 -Z

0-200 Z,

-300I-qu

90 1 00

Fig. 10. OPO temporal output recorded versus cavity-lengthchange. The superimposed ramp is the voltage drive for thePZT. With 100 V applied to the PZT, the cavity length changesby 0.9 /Lm. (a) Output intensity from a high-Q KTP DRO. Asthe cavity length is scanned, the output intensity modulationreaches a peak of 75% for pumping at 1.38 times above threshold.(b) Output intensity fluctuation of a KTP OPO with one Brewsterplate inserted to discriminate against the idler wave. Note thereduced intensity modulation of 43%. (c) SRO output intensitythe position of mirror M3 is scanned (see Fig. 8). Fast amplitudefluctuations have disappeared in this case. Slow amplitude vari-ations are due to pump-mirror position fluctuations.

I....

79

Yang et al.

I

o


sult of a nanometer change in cavity length disappears.However, output power fluctuates at a millisecond timescale. These slow amplitude changes are due to instabil-ity in the feedback-pump-mirror position, which in turnresults in fluctuating SRO thresholds as discussed in Ap-pendix A.

When PZT scanning is stopped, the cavity length con-tinues to fluctuate by a few tens of angstroms at a 120-Hzrate because of acoustic pickups. The effect of such a fastcavity-length perturbation is to cause power instabilitieson the OPO output accompanied by random jumps in oscil-lation frequency. This is especially severe for a DRO withhigh-finesse signal and idler cavities because of the smalltolerance range. We observed almost 100% intensitymodulation in the output of the low-loss KTP DRO evenwhen the OPO was pumped at 3.6 times above threshold.Corresponding to the rapid intensity fluctuation, we sawthe signal- and idler-oscillation frequency hop every fewmilliseconds. In contrast, when the Brewster plate is in-serted into the OPO cavity, the free-running output am-plitude becomes more stable. For pumping at 2.36 timesabove threshold, the free-running output intensity modu-lation is reduced to 32%. Also, the signal and idler fre-quency remain at one particular axial mode pair for amuch longer time, as we observed with a Fabry-Perot in-terferometer. Typically, for pumping at 2.36 times abovethreshold, single-axial-mode operation was stable for hun-dreds of milliseconds. Note that the cavity-length fluc-tuation is the same with and without insertion of theBrewster plate. The much stabler output from the OPOwith an intracavity Brewster plate is due entirely to thelowering of the idler-cavity Q

We also found that, as the pumping level is increased forOPO's with one Brewster plate, output intensity and fre-quency fluctuation decrease. This behavior is illustratedin Fig. 11, which shows the output power of the KTP DROwith one Brewster plate and with increasing pump inputs.The fluctuation decreases from 100% to 23.1% as thepumping level is increased from near threshold to 2.5times above threshold. The steplike change in the OPOamplitude, as shown in Fig. 11(c), corresponds to axial-mode hops. On average, single-axial-mode operation canbe maintained for a few hundred milliseconds for pumpingat 2.5 times above threshold.

With insertion of an additional Brewster plate to lowerthe idler-cavity Q axial mode hops become more infre-quent. When the limit of 100% idler loss is reached,single-axial-mode operation is consistently observed, eventhough small-cavity-length fluctuation persists. This isexpected since, as Smith pointed out, the SRO will hop toan adjacent axial mode when the resonant-signal-cavityresonance is shifted to ±1/2 free-spectral range awayfrom the parametric-gain center.4 For a standing-wavecavity this implies an axial-mode-hop tolerance of half ofthe resonant signal wavelength, which is orders of magni-tude greater than the tolerance range for DRO's withhigh-Q signal and idler cavities. A typical idler spectrumas observed with a 300-MHz-free-spectral-range scanningFabry-Perot interferometer is shown in Fig. 12. Themeasured idler linewidth of a few megahertz is Fabry-Perot-instrument resolution limited. The actual idlerlinewdith is potentially as narrow as the pump linewidthof 20 kHz.

In addition to having a larger axial-mode-hop toleranceto cavity-length fluctuation, the SRO is also less suscep-tible to pump-frequency fluctuation. To determine toler-ance range for pump-frequency fluctuation, we note that,as the pump frequency is tuned, the signal frequency in a

0

-

0,a

2

0.25 _

0.20 .

0.15

0.10

0.05

0.00 I0

1.4

*- 1.2

.d 1.0

I 0.8

on 0.6

0.4

0 0.20.0

3.0

a)

C 2.5

A 2.0

= 1.5CD

cD 1.0

0050 0-50

N=1.1

1 k iA IM

(a)

L|LeLshJL~zx I IL I U InlA5 10 15 20

time(ms)

I 0.1 0.2 0.3 0.4 0.5time(sec)

0 0.1 0.2 0.3 0.4 0.5tlme(sec)

25 30 35

0.6 0.7

0.6 0.7

Fig. 11. Free-running output of KTP OPO with one Brewsterplate inserted as observed at three different pumping levels.The cavity length is swept in these traces: (a), (b), and (c) repre-sent pumping at 1.1, 1.8, and 2.5 times above threshold, respec-tively. When the pump power is increased, the intensitymodulation resulting from cavity-length fluctuation is reduced.Note that the time interval for (a) is much shorter than the timeduration displayed in (b) and (c). The steplike amplitudechanges shown in (c) correspond to axial mode hops.

,

CU

a,

C

00.0

Fig. 12. KTP SRO output spectrum as observed by a 300-MHzscanning-confocal Fabry-Perot interferometer. The observedsingle-axial-mode output linewidth of 1 MHz is instrument-resolution limited.

300 MHz

I I l-- Il

911

Yang et al.

L


SRO remains fixed at cavity resonance, whereas the idlerfrequency is free to vary to satisfy the energy-conservationcondition. An axial mode hop will occur when the signalfrequency is ±1/2 free spectral range away from theparametric-gain center. The pump-frequency change re-quired for the parametric-gain peak to be shifted by ±1/2free spectral range from the signal-cavity resonance isgiven by the stability factor as defined by Kovrigin andByer 2 7 :

aki _ ak,

av, ap818v,,I = L Av 5. (16)

For KTP crystals the index derivatives can be obtainedfrom the Sellmeier equations given by Kato20:

-= 3.723 X 10-14,

i = 3.922 X 10-14,avi

akp= 3.997 x 10-14. (17)app

These derivatives are evaluated for noncritical phase-matched propagation along the x axis and with As =1.09 ,um and Ai = 1.039 Am. Substituting the numbersinto Eq. (16), we find that the pump-frequency tuningrange is J~vp = 1.32 Av8. For the KTP SRO, the free spec-tral range is 654 MHz. We therefore do not expect anaxial-mode hop to occur until the pump frequency haschanged by 863 MHz. To confirm the less stringentaxial-mode-hop tolerance to pump-frequency fluctuation,we swept the pump frequency over 40 MHz and observedno axial-mode hop. This is in contrast to the 3-MHzpump-frequency fluctuation tolerance for a DRO of simi-lar length.4 '9

Thus far we have concentrated on the fine detail of OPO-frequency stability by examining tolerances for single-axial-mode operation. On a coarser scale, the frequencystability of the OPO is determined by the extent of fre-quency excursion from the parametric-gain center as thecavity length or pump frequency changes. For a SROwith a single-pass pump, the oscillation frequency shouldnot deviate from the parametric-gain center by more than±1/2 signal-cavity free spectral range. When the pumpand idler waves are nonresonantly reflected in a SRO,however, additional frequency instability arises since theminimum-threshold point depends on the changes in therelative phase difference Aqo among the three waves. Asdiscussed in Appendix A, the minimum-threshold pointshifts between Akl = +2.33 rad as ATo varies betweenzero and 27r. For the KTP SRO this implies that the oper-ating point could vary over 0.85 nm as the position of thepump mirror or cavity mirror M2 is changed. To verifythe calculated operating range, we monitored the SRO out-put, using a 1-m monochromator with a resolution betterthan 0.04 nm. The extreme width of the oscillation fre-quency was recorded as we varied ATr between zero and 2'irby shifting the position of the pump mirror. We observedthat, at 2.6 times above threshold and beyond, the tuning

range asymptotically approached 0.76 nm, which is closeto the calculated tuning width of 0.85 nm.

4. CONCLUSION

We have theoretically and experimentally examined thebehavior of OPO's with low signal loss and arbitrary idlerloss. By assuming a longitudinally invariant signal field,we calculated pump depletion and conversion efficiencyfor OPO's with arbitrary idler feedback. In the limits ofnear-unity idler reflectance and no idler reflectance, thederived expression agrees with previous expressions forDRO's and SRO's. In particular, the 100% signal-photon-conversion efficiency point is seen to shift slowly frompumping at four times above threshold to pumping at(0T/2)2 times above threshold as idler loss is increased.This shift is more pronounced as Ri approaches zero.

Using the approximate analytical solutions, we exam-ined amplitude and frequency stability of OPO's with ahigh-Q signal cavity an an arbitrary-Q idler cavity. Weshowed that, for a given pumping level, the range overwhich an OPO remains above threshold as signal- andidler-cavity resonances are detuned from each other in-creases as idler loss is increased. For fixed idler reflec-tance, the tolerance range also widens as one increases thepumping level. The larger phase-detuning tolerancerange is accompanied by reduced modulation of the OPOoutput amplitude when the signal and idler cavity reso-nances are detuned from exact overlap.

Using the approximate analytical solutions for signalpower, we also examined the frequency instabilities ofOPO's at different pumping levels and with various idlerreflectivities. We found that, for residual idler-power re-flectivity of less than 1% and pumping at 2.5 times abovethreshold, frequency jumping of OPO's in the presence ofcavity-length fluctuation is substantially reduced. Thishas important implications for practical realizations of cwSRO's, since it implies that extraordinary effort need notbe made to eliminate all residual idler reflectivity toachieve the frequency stability of the SRO. For example,one can readily construct a DRO with weak idler feedbackby use of a four-mirror ring cavity in which each mirrorhas 90% power transmission at the idler wavelength. Af-ter the idler power traverses once through the cavity, only0.01% of it is fed back. Figure 5 shows that when thisDRO is pumped at two times above threshold it will havethe same frequency stability as SRO.

Complementing the theoretical discussion, we conductedexperiments in which LiNbO3 and KTP OPO's operated asDRO's with a fixed low signal loss and increased idler loss,including the SRO case with 100% idler loss. Carefulthreshold measurement confirmed the expected DROthreshold progression as the SRO limit is approached, andby observation of OPO temporal and spectral output weconfirmed the improved stability of the cw OPO as idlerloss is increased toward the SRO limit.

APPENDIX A: DERIVATION OF THE SROTHRESHOLD EQUATION FORNONRESONANT PUMP AND IDLERREFLECTION

Here we present the derivation of the SRO threshold withboth the pump and the idler fields nonresonantly re-

Yang et al.


flected. Previously, Bjorkholm et al.2 5 treated both theSRO and the DRO cases when only the pump field wasnonresonantly reflected. They mentioned the possibilityof nonresonantly reflecting both the pump and the idlerfields for the SRO but did not present analytic thresholdexpressions. We show below that the threshold for thecase of nonresonant reflection of pump and idler waves isidentical to that of the DRO case with pump reflectionwhen the reflection coefficients for pump and idler areequal to unity.

The starting point is again the coupled-wave equationsfor the pump and idler fields, assuming that the resonantsignal field is spatially invariant:

E) = iKiEp(z)E 8*(z)exp(iAkz),z

OE()= iKE 8 (z)E(z)exp(-iAkz).Oz

(Ala)

(Alb)

Since we are dealing with SRO's, the initial idler field iszero. Using this boundary condition, we find that the so-lutions for the forward-traveling idler and pump fields are

Ei+(z) = -EpoEso* sin(Fz)exp i(Akz/2 + A9 +),F

Ep+(z) = Epo[cos(rz) + 2A sin(Fz)] exp(-iAkz/2), (A2)

where + denotes forward-going waves and A9p+ representsthe initial phase difference among the three waves.Aq,&+ = pp+ - ps+ - Gpo+, where 50p+, sDS+, and Spi+ are initialphases for the pump, signal, and idler at z = 0, and r2 =KiKpIEsOI2 + (Ak/2)2 . For the backward-traveling waves,the initial conditions are

Ei-(z' = 0) = \/R exp[iAsp-]Ej+(z =I),

Ep-(z' = 0) = R/ip exp[iA9p]Ep+(z = C), (A3)

where z' = l - z, Ri and Rp are mirror-power reflectioncoefficients of the idler and pump, respectively, andAp- = Ak + Ap + Ac and A = p - - i representthe phase-shift differences incurred on reflection fromthe mirrors. With these boundary conditions the solu-tions for the backward-going idler and pump fields are

Ei-(Zz,= 1 i - - sin(z')1 Ir

+ iRi NfRF cos(rld) + i-k sin(rl,)IF 2F j

X sin(fz')exp io}

X E*Epo exp[i(Akz/2 + Akl,/2 + Atp+)],

Ep-(z') = {Ni;; [cos(rlc) + iAk sin(Fl)]

cos(rz') + iAk in(rz?)]

K- V- 1E8012 sin(rl,)sin(Fz')exp -iA9 }

x Epo exp - i(Akz/2 + Akl,/2). (A4)

Energy conservation dictates that

nP[EPo12- (1 - Rp)IEp+(z = c)12 - IEp-(Z' = c)12]

= (1 - Ri)nilEi+(z = lc)12 + nEi(z' = lc)12

+ 2asnsIEso2, (A5)

where 2a8 is the round-trip signal-power loss. Substitut-ing the solutions for forward- and backward-travelingpump and idler fields into the energy-conservation equa-tion above, we obtain

nplEpoI2[sin2(rlc) - Ri sin2(rl,)

+ (Ri + RP)cos2(Fl,)sin2(Flc) + (Ri + RP) 2F

+ 2VRi_7_; cos2(Fl)sin 2(rl)cos(Ap)

- 2( )VR72 sin4(rld)cos(A)

-2 A N/--,cos(Fl,)sin(Fl,)sin 2(rl')sin(Aso)]

F,2

= n2a-* (A6)

Near threshold Eo- 0, which implies that rl- Akl,/2.Substituting these approximations into Eq. (A6), we findthat the threshold equation becomes

2 2a, (Akl,/2)2

|Epoth = KsK 1 2 sin2(Akl,/2)

[1 1 1p + \/izcos(klcLi1 + R, + 2VR~7R cos(Akl, + A 9)]

(A7)

When Ri = R = 1, the threshold-reduction expression[the bracketed quantity in Eq. (A7)] is identical to that ofa DRO with nonresonant pump reflection.25 Assumingthat Rp = Ri = 1, we find that the maximum-threshold-reduction factor is 4 and the minimum-threshold-reduction factor is 2.1. For a A9 change of ±7r, theparametric-gain peak shifts between ±2.33 rad.

REFERENCES AND NOTES

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2. R. L. Byer, M. K. Oshman, J. R Young, and S. E. Harris, "Vis-ible cw parametric oscillator," Appl. Phys. Lett. 13, 109-111(1968).

3. J. A. Giordmaine and R. C. Miller, "Optical parametric oscil-lation in LiNbO 3 ," in Physics of Quantum Electronics, P. L.Kelly, B. Lax, and P. E. Tannenwald, eds. (McGraw-Hill, NewYork, 1965), pp. 31-42.

4. R. G. Smith, 'A study of factors affecting the performance ofa continuously pumped doubly resonant optical parametricoscillator," IEEE J. Quantum Electron. QE-9, 530-541(1973).

5. A. Heidmann, R. J. Herowicz, S. Reynaud, E. Giacobino,C. Fabre, and G. Camy, "Observation of quantum noise re-duction on twin laser beams," Phys. Rev. Lett. 59, 2555-2557(1987).

6. C. D. Nabors, R. C. Eckardt, W J. Kozlovsky, and R. L. Byer,"Efficient single-axial-mode operation of a monolithicMgO:LiNbO3 optical parametric oscillator," Opt. Lett. 14,1134-1136 (1989).

Yang et al. )


7. D. Lee and N. C. Wong, "Tunable optical frequency divisionusing a phase locked optical parametric oscillator," Opt. Lett.17, 13-15 (1992).

8. C. D. Nabors, S. T. Yang, T. Day, and R. L. Byer, "Coherenceproperties of a doubly resonant monolithic optical parametricoscillator," J. Opt. Soc. Am. B 7, 815-820 (1990).

9. R. C. Eckardt, C. D. Nabors, W J. Kozlovsky, and R. L. Byer,"Optical parametric oscillator frequency tuning and control,"J. Opt. Soc. Am. B 8, 646-667 (1991).

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Yang et al.

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Power and spectral characteristics of continuous-wave