Tunable Optical Parametric Oscillators

2096 PROCEEDINGS OF THE IEEE,V OL. 57, NO. 12, DECEMBER 1969

Abstract-This paper reviews progress on tunable optical parametric oscillators. Topics considered include: parametric amplifica- wL g 1 - ] =:; tion of Gaussian beams; thresholdt; u ning techniques, spectral NONLINEARC RYSTAL output, and stability:s aturation and powero utput: spontaneous Fig. 1 . Schematic of optical parametric oscillator. The mirrors are parametric emission; nonlinear materials: and far infrared genera- highly reflecting at either thsei gnal of idler, or both. tion.

I. INTRODUCTION are nearly independent of frequency, and, unlike the case ORK on optical parametric oscillators began in for a laser transition, very wide tunability is possible. 1961 when Franken et al. [l ] demonstrated second Though in principle, the xijk allow any three optical fre- harmonic generation of light, and thus the exis- quencies to interact, in order to achieve significant para- tencew of substantial nonlinear optical coefficients.Following metric amplification it is required that at each of the three a number of proposals and theoretical studies [2]-[6] frequencies (i.e., at the signaidler,l and pump) the generated Giordmaine and Miller, in1 965, constructed the first polarization travel at the same velocity as a freely propagat- tunable optical parametric oscillator [7]. Since then, work ing electromagnetic wave. This will be the case if the refrac- has proceeded rapidly and it is now possiblteo tune through tive indiceso f the material are such that the k vectors satisfy most of the visible and near infrared; to obtaingreater than the momentum matching condition lis+ Ei=k, [lo]. For 50 percent conversion efficiency of the light from the laser collinearly propagating waves this may be written pump; and tob tain linewidths of lesst han a wavenumber. o,n, (up- o,)q = o p n p , (2) With careful construction, threshold for a CW oscillator + may be as low as 3 mW. where n,, ni, and np are the refractive indices at the signal, Many of the basic ideas of parametric amplification and idler, and pump. Once the pumping laser is chosen, and oscillation have been extensivelyexplored in the microwave thus opf ixed, then if the refractive indices at the signal, frequency range [8]. If some upper frequency op,termed as idler, or pump frequencies are varied, the signal and idler the pump, is incident on a material possessing a nonlinear frequencies will tune. Considerable control of the refrac- reactance, then an incident signal frequency w, mayb e tive indices, and very wide tuning, is possible by making amplified. In the process a third frequency ai,t ermed as the useo f the angular dependence of the birefringenceo f idler frequency, and such that os+ mi=c op is generated. anisotropic crystals, and also by temperature variation. Irrespective of the phase of the incoming signal frequency, Rapid tuning over a limited range is possible by electro- the phase of the idler may adjust such that the signal and optic variation of the refractive indices. idler are amplified, and the pumpis depleted. A schematic of a typical parametric oscillator is shown in In the optical frequency range the nonlinear reactance is Fig. 1. The oscillator consists of a nonlinear crystal and a obtained via the nonlinear polarizability of noncentrosym- pair of mirrors. As will be discussed later, the mirrors may metric crystals [9]. This nonlinear polarizability is described be reflecting at either the signal or idler frequency, or at by a 27 component tensor xi,* which relates the three com- both frequencies. Ideally, 100 percent conversion of inci- ponents of the generated polarization Pi to the nine possible dent pumppower to tunable signal and idler power is possi- combinations of applied field EjE,. That is, ble. The output of an optical parametric oscillator is very much like that of a laser. It is highlymonochromatic with a = 2 1 xijkE$k> (1) spectrum consisting of one or a number of longitudinal j k modes. It is often a fundamental Gaussian transverse mode where i, j , and k may be x, y, or z. Typically over the trans- and may be highly collimated. To the eye, the output of a parency range of the crystal, the nonlinear coefficients x i j k CW parametric oscillator exhibits the same sparkle effect as does a He-Ne gas laser. One principal difference between a laser and an optical Manuscriptr eceivedA ugust 13, 1969; revisedS eptember1 8,1 969. parametric oscillator is the ability of the former to collect Preparation of this paper was sponsored jointly by the National Aero- nautics and Space Administration under NASA Grant NGR-05-020-103; and store wide-band uncollimated spectral energy. A laser’s and by the Air Force Cambridge Research Laboratories, Office of Aero- wavelength and linewidth are determined by the pertinent space Research, under Contract F19628-67-C-0038. This inoited paper is atomic transition and are not affected by the spectral or one of a series planned on topics of general interest.-The Editor. The author is with the Department of Electrical Engineering, Micro- spacial distribution of the pumping radiation. However, wave Laboratory, Stanford University, Stanford, Calif. 94305. in an optical parametric oscillator, phase coherence between HARRIS:T UNABLE OPTICAL PARAMETRIC OSCILLATORS 2097

TABLE I REPRESENTATIVE PARAMETROISCC ILLATOR EXPERIMENTS

RefereP nocwese rO utpuR t angeT uninCg r NyWsotna lvP ienu lem eanrpg itnhg

0.53 p LiNbO, 0.73p-1.93/1 103 w (doubled CaW0,:Nd”) (temperature tuning)

0.53 p KDP 0.96 p- 1.18 p 105 w (doubled Nd3+- Glass) LiNbO, 0.68 p - 2.36 I/ 5W0 (angle tuning)

0.35 p KDP 0.53 pk IO percent -104 w (tripled Nd3+- Glass) 1.06 p f 10 percent (angle tuning)

LiNbO, 1.05/.~-1.20p 4 105 w 1 . 6 4 /A-2.05 p (angle. temperature, and electrooptic tuning)

0.53 p Ba,NaNb,O,, 0.98 p- 1.16 p 3 mW (CW) (doubled Nd3+ :YAG) (temperature tuning)

0.5145 p LiNbO, 0.68 p-0.71 I/ 3 mW (Cw) (argon) 1.9 p-2.1 p (temperature tuning)

LiNbO, 50 p-238 p -70 W 0.696 p - 0.704 105 w (angle tuning)

1.06 p LiNbO, 1.95 p-2.35 p 170 W peak (Nd”:YAG) (temperature and angle 17 mW average tuning) (repetitively pulsed)

the signal, idler, and pump is very important; and either in the crystal. The equations describing this process [8], spectral or angular spread of the pump may increase its [lo]-[12], in MKS units, are threshold or widen its linewidth. dEs Based on crystals presently being developed, it is likely -- - - j q , o , d E x exp - jAkz that within a few years, narrow-band tunable sources will d z be available over the entire spectral region from 0.2 p to dEi - = - j q i o i d E x e xp - jAkz greater than 100 p. Like fixed frequencylasers, these sources dz (3b) should provide at least lo6 times as much power per bandwidth per steradian asd o traditional spectroscopic sources. = - jqpo,dE,Ei exp jAkz, Such sources are likely to have significant impact on many 5dz types of excited state spectroscopy, optical pumping, semi- where the quantities E,, Ei,a nd E , are the envelopes of the conductor studies, and photochemistry. Table Is ummarizes plane waves; e.g., E,(z, t)= Re [E, exp j ( a , t - k , z ) ] . The the characteristics of a number of parametric oscillator quantities q,, vi, and q, are the plane wavei mpedances experiments which have been performedto date. (377,’refractive index) of the three waves, and d is the effec- 11. PARAMETRAMPLIFICATIONIC tive nonlinear coefficient. In general, d depends on the direc- A . Amplification of Plane Waves tion of propagation and othen polarization of the respective waves, and will be considered further in Section VII. We Consider waves witah pumping frequencyw pa nd a signal allow for a E vector mismatch frequency o,t o be incident on a nonlinear material having a polarizability 9-E ’. Mixing of these waves generates a Ak = k, - k, - ki. (4) traveling polarization wave at the difference frequency ai. We first note thatb y taking the complex conjugate of (3a) By adjusting the birefringence of the crystal, the polariza- and (3b) and multiplying (3a), (3b), and (3c) by EJqsws, tion wave may be made to travel at the same velocity as a E J q i o i , a nd q/qpopre,s pectively, [ 5 ] that freely propagating idler wave, thus resulting in cumulative growth. The idler wave also mixes withthe pump top roduce a traveling polarization wave at the signal frequency, phased such that growth of the signal field also results. The I F \ process continues with the signal and idler fields both grow- ing, and the pumping field decayingas a function of distance 2098 PROCDEECDEINMGBSE ROF THE IEEE, 1969

Growth atthe signal impliesgrowth at the idler. Depending on the relative phases of the three frequencies, power may flow either from the-lower frequencies to the upper frequency as is the case for secondharmonic or sum frequency generation; or alternately from the pump to the lower frequencies as in difference frequency generation and parametric amplification. If we neglect depletion of the pump, then (3a) and (3b) mayb es olveds ubject to the boundary conditions that E,=E,(O) and Ei=Ei(0)a t z=O. For a crystal length L, we obtain \ AkL E,(L) = E,(O) exp ( - j i)[cosh SL+ j -sinh SL A2ks 1 (64 AkL - j 5 e(0)exp (-j I> [sinh sL], S and

Ei(L) = Ei(0)exp - j )[:c~' ash SL+ j ( 2s (6b) - j K2S. c(0) exp ( - j y ) [ s i n h sL] where

K, = vsw,dEp (A+) = VioidE, Fig. 2. Normalized gain versus (ML/2).(From Byer [96].)

r2 = K,K: = osoiqAipl2~,~2 erator with a fundamental frequency op/2. wI f e let and os= (op/2)(1 + 6)a nd oi= ( 4 2 )(1 - 6), i.e., s = (I?' - Ak2/4)"2. 6 = h s - o P - 21P - 2, , (9) We first examine the single pass power gain when only a U P 1, signal frequencyi s incident, i.e., take E,(O)=O. Defining then the parametric gain degeneracy is reduced from G = IE,(L)/E,(0)12- 1, we find from (6a) off that on degeneracy by the factor 1 - a2. The lineshape or dependence of the parametric gain on the optical frequency is determined by the variation of Ak withf r equency. Noting that for Ak2L2/4>>r2L2 and small T L

For a given crystal temperature and orientation, the center G ~ ~pia , z, r 2L2 EAky;yT, of the parametric gain linewidth occurs at that signal and idler frequency whereA k = 0. At line center the gain is thus it is seen that for small gain, the full half-power gain line- sinh2rL,w hich for small gain is approximately r2L2.Th us width is determined by IAkLIZ 2a. For a 4 cm crystal of LiNbOJ aat pump wavelength of 4880 A and a signal wavelength of 6328 & the dispersion is such that the half-power As an example, r2L2fo r a 1 cm crystal of 90" cut LiNbO, linewidth is about 1.4 cm-'o r about0.56A. However, near for A,= Ai= 1 p is approximately 0.1 Pp/A,w here P,JA has degeneracy, linewidths may be much larger. From (7) it is units of MW/cm2. From (8) it is seen that the gain of a non- seen that thegain linewidth is also somewhat dependent on linear material is proportional to ldI2/n3, where n is the r2LZan d thus onthe strength of the pump. Fig. 2shows the refractive index. This figure of merit, together with their quantity G r 2 L 2v ersus (Ak/2). transparency range, is shown for a number of nonlinear materials in Fig. 24. It is usefult o note thawt hen os= oi= wp/2 B. Amplijication of Gaussian Beams (ad2 is termed the degenerate frequency), that the single In determining oscillator thresholds we will beconcerned pass gain of an optical parametric amplifier is equal to the with the gain experienced by a fundamental Gaussian mode conversion efficiency (PsH/PFo)f a second harmonic gen- of the oscillator. In general, if a beam having a Gaussian HARRIS:T UNABLE OPTICAL PARAMETRIC OSCILLATORS cross section is incident on a nonlinear crystal and para- 5 = - j q p o p d g ~ s o Eexi op j Akz, metrically amplified, the output beam will no longer be a dz simple Gaussian. Thisoccurs as a result of Poynting vector walk-off (i.e.t,h e fact that in an anisotropiccrystal, the direc- where the spatial coupling factors gs, gi ,a nd g p are tion of power flow and I; vector need not be the same) ; and also, in the absence of Poynting vector walk-off, as a result of the form of the nonlinear polarization generated in the parametric process. These coupling factors are a measure of the failure of the The power gain experienced by an incident Gaussian driving polarizations to completely overlap the desired mode may be found by evaluating an integral of the form Gaussian modes. If (though this cann everb e the case) JE* . 3 dV , w here E is the electric field of the given mode, F=W ,, e=w ,an d Wp=W p,then g s = g i = g p = 1. Except and @ is its driving polarization. A general analysis of this for these coupling factors, (11) is identical to (5); and the problem, allowing for arbitrarily tight focusing, and also for solutions of the previous section may be employed. Poynting vector walk-off has beeng ivenb yB oyd and The parametricgain coefficient T2 of (8) thus becomes Kleinman [13], and some oftheir results will be summarized in the latter part of this section. We first consider the more restricted but important case of near-field focusing and 90“ phase matching (for 90” phase matching, Poynting vector where, from (10) and (12) the factor gsg i is walk-off is absent). An analysis of this case was first given W,%Wp gsgi = 4w2 by Boyd and Ashkin [14]. wfw’ + wfw; + w’wf1 ’- (14) A near-field analysis keeps track of the transverse de- [ pendence of the signal, idler, and pumpm odes ;but assumes The power of theG aussianp ump beam isg ivenb y that this transverse dependence does not change over the (EiO/2qp)(zWi/2a)n,d thus (13) may berewritten length of the nonlinearcrystal. It thusrequires that the confocal parameter of the focus (of all three beams) be as long or longer than the length of the nonlinear crystal. To make the appropriate modification of the previous where analysis we allow the signal, idler, and pump fields to have Gaussian cross-sectional dependencies of the form M 2 = [ K K W p E,, exp -r2/Wf, Eio exp -r2/W’, and EPPe xp -r2/Wi, wfw: + wfwi + w;w; respectively. The generated driving polanzations at the and Pp is the incident pump power. signal, idler, and pump arethen also Gaussians [14] having Ashkin and Boyd [14] have shown that to maximize M 2 beam waist radii F, Ffa nd Wpg iven by for W,a nd W,f ixed, that thep umping beam size W i should 1 1 -=1 ---+- be Wf w’ w; 1 1 1 1 1 1 W : = w f + j @ , = - + - 5 w: w ,f in which case M 2 will be 1 1 1 ---++. 1 1 - wf w’ M & = - w ; 4 wf + w’ For instance, the idler and pump mix to yield a polarization at the signal frequency of the form In order to maximize the parametric gain at a given pump power, Ws and W, should be chosen as small as possible. r2 r2 r2 However, sincet he present analysis is restricted to the near exp - -2 = exp - -exp - -a WS W’ w; field, the smallest allowed spot sizes are approximately those Note that these polarization radii are always smaller than of the confocal condition, i.e., Wf = L&/2nnsW, ;= LlJ2zni. the radius of either of the Gaussian beams which mix to From (16a), if botht he signal and idler are confocally produce them. We take the appropriate projections of these focused, it is seen that the pump should also be confocally Gaussian polarizations by multiplying them by exp - r2/W,”, focused [14]. exp - r2/W:,a nd exp - r2/Wi,r espectively, and integrating Combining the previous equations, and making useo f the over the transverse cross sections. The result of this is the degeneracy factor of (9), the single pass parametric gain set of equations coefficient G = r 2 L 2f or a confocally focused signal, idler, and pump may be written dEso -= - jqso&gJpoE$ exp - jAkz (1W dz

dEi0 - - - - jq,uidgiEpoE:oe xp - jAkz (1 1b) dz where ooi s the degenerate frequency and Wg = (LA0)/(2nn) 2100 OF PRO CDEEECDEINMGBSE R THE IEEE, 1969

IO + 1 TABLE I1 REPRFSENTATIVE 90" PIIASE MATCHABLEM A W

0.0 P, for 30

focusing) ADP 2573 A 5.0cm 0.76 MW/cm' 39watts 8 cm-' LiNbO, 5300A 4.0cm 0.18 MW/cmz 15watts 1 cm-' Ba,NaNb,O,,5 3WA 0.5cm 1.6 MW/cm2 17watts 8cm-l cdse 2 . 5 ~02. 7.07 cMmW /cmZ 14-15w1 axt1ts1 -'

ing 5 = L/bo should be 2.84. However, the increase in gain I 6 over that obtained for L/bo = 1, used in the previous near- 0 - 2 10-1 I IO 0 2 0 3 field analysis, is only about 20 percent and may not be E worth the increased pump power density at the tighter focus. Fig. 3. Reduction factor 6,(B, 5 ) versus 5 . It should dso be mentioned thata t the tighter focusing, Ak (From Boyd and Keelan [ 131.) should be slightly differentf rom zero. This results sincea, t a tight focus, the mixing of the noncollinear components of is the degenerate confocal spot size. The price for moving the signal and idler fields requires slightly longer E vectors off degeneracyi s the .factor (1 -d2)'. For example, if than does the collinear mixing. This optimized fi vector L,=0.473 p and Ls=0.54 p, then 6=0.75a nd (1-6')' match [13] is included in the functionh(B, 5). =0.€88;a nd the threshold is about five times highert han if In many practical cases B will be sufficiently large so that, 1, = &= 21,. Substituting for W,, (17) becomes as Seen in Fig.3 , the maximum value ofl;,(B, 5 )w ill be rather independent of 5. Boyd and Kleinman [13] haves hown that for large B,

hm(B,5 ) + a/4B2 (B2/4 > 5 > 2/Bz), (21) For a crystal of LiNbO, at a degenerate wavelength of 1 p, where for the range of B and 5 shown in parenthesis, (21) is t h i s yields correct to within 10 percent. Since B2 is proportional to L, rzLzE 0.005 q , ( i - 62)2, (19) the L in the numerator of (18) is cancelled, and for large birefringence the gain is nearly independent of the length where is in and Pp in watts. Note that for optimum t c m , of the nonlinear crystal; and subject to the criteria of (211, to focusing rZL2i ncreases only linearly with crystal length. the degree of ocusingS. incef or 90" phasem atching Thus 1 watt of pumping power in a 4 cm LiNbO, crystal h,(B, 5 ) 1, ~the factor a/4BZ is approximately the gain re- provides 2.0 percent single pass gain at a degenerate wave- duction factor which is experienceda s a result of walk-off. length of 1 p. As an example, to phase match a LiNbO, parametric We now proceed with some the of results of the Boyd and oscillator directly pumped by a 1.06 p Nd" :YAG laser at Kieinman analysis [13]. Allowing for both Poyntingv ector 90" requires a crystal temperature of about 750°C [13]. walk-off and arbitrarily tight focusing they show that the Room temperature phase matching at degeneracy is ac- effective single passgain of interacting Gaussian modes all complished by propagating aat n angle of 43"w ith respect o having the same confocal parameter is given by (18) multi- the optic axis. This yields a walk-off angle p = 0.037 radians plied by a reduction factor hJB, 5). The parameter 5 is the and Br4.7 L"'. For a 1 cm crystal the maximum gain at ratio of the length of the nonlinear crystal to the common a given pump -power is then 28 times smaller than it would confocal parameter bo ;a nd Bi s a double refraction param- have been had 90" phase matching been possible. eter [13] defined as It shouldb e noted that the above discussion is concerned with gain maximization at ag iven pump power as opposed to at a given pump power density. If maximization is with regard to power density, walk-off need not be of conse- where p is the walk-off angle, L is the length of the non- quence. If the beam radii of all modes are greater than linear crystal, Lo is the degenerate wavelength, and no and Wo= d p L , then the gain reduction due tow alk-off will be n3 are the refractive indices at the degenerate wavelength less than 15 percent and may be made essentially negligible and pump, respectively. B is approximately the ratio of the for still bigger beams [14]. walk-off angle p to the far field diffraction angle of the Table I1 shows the approximate pump power densities Gaussian beam. and optimized pump powers necessary to obtain 30 percent The reduction factor h,,,(B, 5 ) as a function of 5 with B as single pass gain with typically available lengths of some a parameter is shown in Fig3. . In the absence of double re- nonlinear crystals. Approximate gain linewidths for these fraction (90" phase matching), B = 0, and for optimum focus- crystal lengths are also shown. HARRIS:T UNABLE OPTICAL PARAMETRIC OSCILLATORS 2101

111. THRESHOLDPUM PING POWER TOconstruct an oscillator it is necessartyo resonate either the signal or the idler, or both. The latter case, where both the signal and idler are resonant, yields the lowest threshold, but posess evere stability problems andm irror require- ments. Since they have the lowest threshold, oscillators of this type have thus farb een the most prevalent. To determine threshold of the oscillator we require that the single pass parametric gain (notet hat there iso nly gain when the signal and idler travel in the direction of the us pump) be sufficient to offset the round-trip cavity loss. We Fig. 4. Pumping with a multimode pump. define a, and ai as the round-trip E field losses at the signal and idler frequencies, e.g., E,(O)=(l -asp&). From (6) for 20 mW. The possibility of CW parametric oscillators with Ak = 0, we require such low thresholds wasf irst pointed out byB oyd and Ashkin [14], and first demonstrated by Smith et al. [15]. 1 Their first oscillator employed a 5 mm crystal of E,,(O) = E,,(O) cosh TL - j 5 E;,(O) sinh TL (22a) 1 - a, S Ba,NaNb,O,, and was pumped by a doubled 1.06 p Nd3+ :YAGlaser. Threshold was observed at 45 mW of 1 K . Eio(0)= Eio(0)cosh TL - j .L E,*,(O) sinh TL, (22b) multimode power. More recently, Smith [16] has con- 1 - ai S structed a CW argon pumped oscillator with a threshold of where E,, and Ei, are the peak amplitudes of the Gaussian about 2 mW. modes as defined in the previous section ; r is defined by The formulas of this section have impliedt hat the pump- (13); and K , and xi are given by q,o,dE,g, and qioidEpgi, ing radiation consists of only a single longitudinal mode. respectively. Taking the complex conjugate of (22b) and For a parametric oscillator with both its signal and idler setting the determinant of the resulting two simultaneous cavities resonant, Hams has shown [17] that if the axial equations equal to zero, we obtain mode interval of the idler frequency is set equal to that of the pumping laser, then all of the modes of the pumping cash I‘L - - laser may act in unison to produce gain at a single signal 1 - a, frequency mode. Though the pump modes are randomly asai . phased, the idler modes develop compensating phases cosh T L = 1 + (23) 2 - a, - ai which maximize the gain of the system [18]. A schematic of this idea is shown in Fig. 4. Using it, Byer et al. [19] For low loss resonators at both the signal and idler fre- demonstrated an argon pumped visible CW oscillator with quencies, (23)i s satisfied by a tuning range of 0.68 p . 7 1 in the visible and corresponding 1.9 p-2.1 p range in the IR. The oscillator used a 1.65 r2L2 z a,ai. (24) signal and idler resonant cm long crystal of LiNbO, and had a threshold of about small loses 500 mW. Alternately, if only the signal is resonant and it is assumed It has also been shown that if the dispersion of the non- thatn o idler radiation is returned to the crystal input linear crystal is sufficiently small that the axial mode interval of both thes ignal and idler may be set equal to that of (ai= 1) than for small a, the pump, then the peak power, as opposed to the average r2L2 = 2a,. (25) power of the pump drives the oscillator [20].The pump signal only resonant could then be phase locked and threshold obtained at a small loses lower average power. The ratio of threshold pump power with the signal only For the case of the signal-only resonant oscillator, the resonant as compared to both signal and idler resonant is idler modes are notconstrained, and thusthough it has not 2/ai. Thus for a 2 percent idler cavity loss, one-hundred been formally proven, the full average power of a multi- times as much power irequireds for the signal-only-resonant mode laser should be useful. Further, if the axial mode in- case. It should be noted that for small losses the round-trip terval of the signal frequency is set equal to that of the E field losses a, and a, are also the single pass power losses pump, then peak power should be the pertinent quantity. at the respective frequencies. It should perhaps be mentioned that formany parametric From (24) and (25) it is seent hat parametric gains which oscillators pumped with Q-switched lasers, the problem is are far too small to be useful for tunable amplScation are not oneo f attaining threshold, but of having sufficient gain sufficient to attain threshold in a parametric oscillator [14]. for a sufficient time for the oscillation to build out of the For single pass signal and idler power losses of 2 percent noise. To deplete the pump requires about 140 dB of total each, (24)y ields T2L2=4x Equation (19) shows that gain. If we assume that the round-trip transit time of the to attain this gain in a 4 cm crystal of LiNbO, at a de- oscillator is 1 ns, and if the length of the Q-switched pulse generate wavelength of l p requires a pump power of only of the pumping laser is 20 ns; then the apparent threshold 2 102 PROCEEDINGS OF THE IEEE, DECEMBER 1969

InL CRYSTAL I I A CRYSTAL 2 '"9[ CRYSTAL 3 J'

56. m-- 1

I 1 I I I I I -500 H)o xx) 300 400 500 O C TEMPE!X4TURE Fig. 5. Temperature tuning of LiNbO, for several pump w v h g t h s . Data were obtained from the Sellmeier equatiom and [213. (From Byer [%I.)

PHOTON ENEW IN ev Fig.8. A n ~ ~ t u n i n g o f A D P a t 3 4 7 2 k a i s t h e a a g t e ~ ~ t h e p u m p ing beam and the -direction of observation of spontaneoudy emitred light. Thus, a=O" gives the tuning curve of a @inear osdhx. The right-hand ordinate gives the approximate an& 6, betaraea the optic 45050000 1 ''kaxis and pum.p direction. (Magde a nd Mahr Mi.) D(TERNltL FIcm FlEcD (Kv/an) 25

4000 -a -'=; . . $ -20 -10 0 IO 20 x ) 40 VI r **C + -25 I Fig. 6. Temperature tuning of ADP and KDP. These curves were ob- W w-1 tained by Dowley [24] using the spontaneous parametric emission 3 technique discussed in Section VI. it d o o 86 (MINUTES) Fig. 9. Electrooptic tuning. Wavelength shift as a function d t h e e xternal dc electric field; and oscillator signal wavelength as a function of 68, the change in angle between 5 and the crystal @c ais. ~ 7 1 . 1 HARRIS:T UNABLE OPTICAL PARAMETRIC OSCILLATORS 2103 of the oscillator willb e a pumping powerw hichy ields P U M P somewhat less than about7 dB gain per pass.

IV. TUNINGS,P ECTRAL OUTPUANDT, STABILITY C - A X I S A . Tuning L tNb O l As noted earlier, the position of the center of the para- IDLER metric gain linewidth is determined by satisfaction of the I? $P. vector matching condition E , + E i =I?,,and the frequency condition os+ oi= 0,. For collinearly propagating forward waves these are combined to yield (2). With the pump frequency fixed, any process which changes the refractive k , indices at the signal, idler, or pump wavelengths wiltl une the \% oscillator. Tuning methods include : temperature, angular PUMP S I G N A L variation of the extraordinary refractive index, electro- Fig. 10.S chematic of noncollinear oscillator. Tuningi sa ccomplished optic variation of the refractive indices, and, perhaps, byv arying thea ngle betweent hep umpa nds ignal cavity. (From pressure tuning via the photoelastic effect. Of these, Falk and Murray [29].) temperature or angular tuning may be used to tune over broad ranges, and pressure or electric fields may be used 1.190- for fine tuning. Temperature tuning curves for LiNbO, for a number of different pump wavelengths are shown in Fig. 5. The curves 1.170 - were obtained numerically from the Sellmeier equations of - Hobden and Warner [21]. They may be shifted by 30" to 1.150 - 100" by changes in crystal composition [22]. Giordmaine - 1\3/ and Miller [23] have experimentally temperature tuned a LiNbO, oscillator over the range 7300 A- 19 300 A. Fig. 6 shows the temperature tuning curves for ADP and KDP pumped with the doubled 5145 A line ofargon. These curves were obtained experimentally by Dowley [24] by means of the parametric spontaneous emission method which will be discussed in Section VI. Note that the full visible spectrum is tuned by a variation of only 50°C. Angular tuning curves for a LiNbO, parametric oscillator pumped by doubled 1.06 p [25], and for anA DP oscillator pumped by doubled ruby, are shown in Figs. 7 and 8 [26]. The first of these was obtained in an oscillator experiment, and the second was obtained via spontaneous e ( D E G R E E S ) parametric emission. The angle C#I in Fig. 7 is the comple- ment of the internal angle between the optic axis and the Fig. 1 1 . Tuning of the noncollinear oscillator at different temperatures. (From Falk and Murray [29].) direction of propagation of the pump, i.e., I$ = O for 90" phase matching. For the ADP oscillator a change of about nately, as was done in the experiment of Fig. 10, the entire 8 O of the angle between the optic axis and pumpbeam tunes cavity containing the nonlinear crystal is rotated. Results of most of the visible spectrum. Though the angular tuning this experiment are shown in Fig. 11 [29]. method is mechanical and potentially fast compared If we let [ denote any variable which may be usedto vary to temperature tuning, its disadvantage is the reduced gain the refractive indices,a nd if we assume the pump frequency which results from Poynting vector walk-off (Section 11-B). fixed, then for collinear phase matching the rate of signal Experimental results of electrooptic tuning are shown in frequency tuning with ( is given by Fig. 9 [27]. The oscillator was LiNb03 pumped by ruby and had a tuning rate of about 6.7 A per kV per cm of applied electric field. The angular tuning rateo f this oscillator (26) is also shown. Electrooptic tuning has also been demonstrated by Krivoshchekov et al. [28]. where b is a dispersive constant [31] given by Another tuning technique is shown schematically in Fig. ak. 10 [29]. In this case, the I; vector matching is not collinear b = > - - . ak, [30] and the oscillator is tuned by varying the angle between aoi am, the incoming pump beam and thes ignal cavity. This type of In Section 11-A, the half-power gain linewidth was shown tuning has theadvantage that thenonlinear crystal need not to be determined approximately by the condition (AkLI = 27r. be rotated inside the optical cavity. Instead either the angle Noting that doi= -Ams, then from (4), Ak= bAo,. Thus of the input pump is varied via a beam deflector or alter- the full half-power gain linewidth in Hz is 2104 THE OF PRO CEEDINGS IEEE, DECEMBER 1969

; i I / -IDLER FREQUENCY wt Fig. 13. Longitudinal modes of signal and idler cavities. Frequencies o1 and o2v ertically in line on thed iagram satisfyo 1+ w, =os.Th e vertical dashed line farthest to the left indicates the frequenccyo mbination for index matching. The adjacent line represents the nearest frequency combination forw hich oscillation is possible. Sincet hef requency / $2 oftkt comparable to the cavity linewidthA mc, oscillation actually 1 , ?* bw is occurs at the vertical dashed line at the right where &=O. (From O ' L A i & & 7 b o L Giordmaine and Miller [ 1 1 I.) TEMPERATURE - (OK) Fig. 12. Dspemion constant b versust emperaturef or LNi bO , at a n u m k of pump waveiengths. Data were obtained from the Sellmeier equations of Hobdea and Warner [21]. (From Byer [%I.)

1 lAfl - * bL 'OUT J . From (26)a nd (28)it is seen that xhaterials with small b in general have large tuning rates, but also correspondingly large linewidths. Also,near degeneracy wherethe linewidth of an oscillator is large, it will, in general,tune much more '02 t rapidly than when far from degeneracy. As an example, / KDP has a b that is about 4t hat of LiNbO,, and asa result tunes much more rapidly (see Fig. 5 and 6). On the other hand, the linewidth of a 1 cm crystal of KDP is about 40 cm", as compared to about 5 cm- ' for LiNbO,. The rate of change of the center of the parametric linewidth with respect to fluctuation of the pump frequency has been examined by Kovrigin andB yer [32]. B. Spectral Properties The spectral character of the output of an oscillator is determined by the width and saturating behavior of the P /PTHRESHOLD gain lineshape, and by the interaction of its signal and idler Fig. 14. Power output and linewidth versus PJP, (threshold) of a CW modes. Even for materials with relatively large b the half- argon pumped LiNbO, oscillator. (From Byer et al. [19].) powerg ainw idth will typically be greater than 1 cm-' (30 GHz); and a number of axial modes at the signal and mode whose sum frequency is equal to the frequency of a idler frequencies will lie within the linewidth. For instance pump mode, can only occur for certain axial modes. As for a 4 cm crystal of L i m o , with a pump at 4880 A and a shown in Fig. 13 the particular modes which happen to signal at 6328 A, b=6.2 x lo-'' s/m yielding a half-power align may be far from the center of the gain linewidth. width of 1.34c m-'. With mirrors placed on the ends of the Furthermore,a s a result of temperature changes and crystal the axial mode spacing would bea bout 1.6 GHz and vibration, the modesw hich happent o alignt y pically about 25 signal and idlerm odesw oulde xperiences ig- change very rapidly. This is particularly true when com- nificant gain. Near degeneracy, typical observed linewidths bined with the fact that the pump frequency is itself usually are greater than 100 cm- '. Fig. 12 shows the dispersive fluctuating [15]. For oscillators thus far constructed this constant b for LiNbO, versus temperature at a number of fluctuation has had a time constant between about 10 p s pump wavelengths. Corresponding signal and idler fre- . and 1 ms, and has often severelyr educed their average quencies as a function of temperature are shown in Fig. 5. power output. Since the modes which align or nearly align Parametric oscillators with both their signal and idler are typically clustered together in groups having a spacing frequencies resonant pose a particularly severep roblem which is large compared to the axial mode spacing, this [7], [ll]. Since, as a result of dispersion, the axial mode effect has been termed a "cluster effect" [7], [l 11 . spacing of the signal and idler frequencies are slightly dif- The right scale of Fig1. 4 shows the total spectral width of ferent, simultaneous resonance of a signal and an idler a 5145 A argon pumped CWoscillator 1191. The data were HARRIS:T UNABLE OPTICAL PARAMETRIC OSCILLATORS 2105

Fig. 16. Schematic of frequency locking technique. The letters s, i, and p denote the signal, idler, and pump. respectively. The gas transition is assumed to absorb at only the signal frequency; and the mirrors are assumedt oh aveh ighr eflectivitya t only thei dlerf requency.T he vertical arrows in the LiNbOB crystals denote the direction of their Fig. 15. Spectra of doubly and singly resonant oscillators.(a), (b), and (c) positive z axes. (From Hams [37].) Spectra of the doubly resonant oscillator for increasing pump power. (d) Spectra of a singly resonant oscillator. (From Bjorkholm [34].) optical pumping of gaseous vibrational or rotational lines will require frequency control of better than 0.03 cm- l. obtained by taking a long-term exposure with a scanning One proposed approach which may make it possible to Fabry-Perot etalon. With the oscillator operated about lock the output of an optical parametric oscillator onto a two times above threshold, the spectral width approaches gaseous atomic absorption line is shown in Fig. 16 [37]. the theoretical 4 wavenumber linewidtho f the 1.65 cm The usual single nonlinear crystal is replaced by two non- LiNbO, crystal. For weaker pump drives, only modes linear crystals which have the direction of their + z axes closer to were above threshold, andt he spectral A k = O reversed. Between the reversed nonlinear crystals is placed width is reduced. Average signal power output is also the cell containing the gas to which it is desiredt o lock the shown. output frequency of the oscillator. If the absorbing transi- The cluster problem may be eliminated by constructing tion is at the signal frequency of the oscillator, then the the oscillator with only its signal or its idler cavity, but not oscillator is made resonant at only the idler frequency. As a both, resonant. Feedback at the nonresonated frequency result of the reversed positive axes, the parametric gain of can be prevented by careful choice of mirrors, by the use of the first crystal is partially cancelled by the second crystal. an appropriate absorbing material in the oscillator, or by That is, the relative phases of the signal, idler, and pumpon means of noncollinear vector matching. A collinear singly E entering the second crystal are such that instead of further resonant oscillator this type was first demonstrated by of gain the signal and idler decay to the values which they Bjorkholm [33], [34], and some of his results are shown in had on entering the first crystal. The pressure of an absorb- Fig, 15. Spectra of a doubly resonant oscillator are shown ing gas is then adjusted until it is nearly opaque at the in (a), (b), and (c) for increasing pump power. Oscillation pertinent atomic transition. The loss and phase shift of this occurred in three clusters which had aspacing of about 12 A. gas prevents the gain cancellation in the second crystal, with Part (d) shows the spectra of his singlyresonant oscillator in a resultant sharply peaked gain function centered at the which, in all cases, clusters were absent. Other oscillators frequency of the atomic transition. having only their signal or idler frequencies resonant have Bjorkholm has shown that it is possible to lock, at leasont been constructed by Falk and Murray and by Belyaev [29] a transient basis, the output of a high-power pulsed optical et The oscillator of Falk and Murray was non- al. [35]. parametric oscillator to an incident low level signal. In a collinear and is shown schematically in Fig. 10. Though no recent experiment hes ucceeded in locking a LiNbO, clusters were observed, an additional unexplained spectral oscillator, pumped by a ruby laser, to astabilized CW YAG component was often found a number of angstroms away laser [38]. A minimum locking power of 1 mW, and a from the primary component. Belyaev et al. also found a locking range of about 15 A were obtained. spectrum consisting of one or two lines spaced by a few angstroms ; with the width of each individual line not ex- ceeding 0.1 cm - '. V. SATURATIOAND POWERO UTPUT At this time an experimental study of the competition If the level of the pumping field is above threshold, the between the longitudinal modes of a singlec avity para- signal and idler fields build up and deplete the pump as it metric oscillator has not been accomplished A theoretical passes throught he crystal. Saturation occurs when the study by Kreuzer [36] has shown that if the oscillator is pump intensity, appropriately averaged over the length of operated less than 4.81 times above threshold, that it should the nonlinear crystal, is reduced to the point where single saturate uniformly and in the steady state have only a single pass gain equals single pass loss. oscillating longitudinal mode. For an oscillator operated We fist consider the case where onlythe signal frequency further above threshold, the steady-state solution may con- is resonant and determine the pumping power necessary sist of twoo r more longitudinal modes. For the latter situa- to attain appreciable conversion efficiency. We assumet hat tion, the pump is periodically depleted and restored inside the round-trip cavity lossa nd thusthe saturated single pass the nonlinear crystal resulting in a saturated lineshape gain at the signal frequencya re sufficiently smallso that the which is broadened and double peaked. signal field may beassumed constant over the length of the nonlinear crystal. It should be noted that this is a small gain C. Locking as opposed to a small power assumption. Equation (3a) and Many experiments require an oscillator output with far (3c) mayb es olved subject to E,=O, and Ep=Ep(0) at greater stability than that presently available. For instance, z= 0. Taking E, as an undetermined constant, we find 2106 PROCEEDINGS OF THE IEEE, DECEMBER 1969

800 I E , = E,(O) exp j -AkL[ cos PL - j - sin BL 3 2 A28k 1 where j? = [~iWpli~pdZI+E Ask122/4 ]”’. From ( 5 ) the generated signal power is related to the generated idler power, and thus

The single pass power gain G=(IE,(L)I’ - ~E,(O)~’)/)E,(O)~’ is now set equal to the round-trip power loss 2as.T hus

With the pump level Ep(0)f ixed, \ES[i s determined by the Fig. 17. Output power of noncollinear single cavity oscillator. The dashed solution of (31). At threshold, the [ES=\ O and therefore curve shows the pump in the absence of the parametric interaction. = O ( A k = O ) , yielding a threshold power in agreement with (From Fatk and Murray [29].) (25). At line center, we see from (29), that the pump will be completely depleted whenf lL= 4 2; f rom (31), that this will also be 100 percent efficient With powerdependent occur at a pumping power equal to (7~/2t)im~ es the threshold [a]. pump reflections absent, the generated signal and idler pumping power [36]. If the value of the pumping field is in- powers are given by creased further, the pump will again begin to grow at the expense of the signal and idler field. As noted at the end of the last section, this spatially varying pump field creates an interesting type of line broadening first pointed out by Kreuzer [36]. where P, is the threshold pumping power. At four times A parametric oscillator with only its signal frequency above threshold, 100 percent conversion efficiency is ob- resonant has the advantage thati f the desired output of the tained. Fig. 18 shows the efficiency and the ratio of trans- oscillator is taken at the idler frequency, then an optimum mitted-to-incident pump powerf or the doubly resonant coupling problem is avoided. That is, at any drive level the oscillator with and without powerdependent reflections. signal cavity should be made as lossless as possible. If the Byer et al. have recentlyconstructed a CW argon pumped pump .is adjusted to (42)’ times its threshold value, then ringcavity parametric oscillator [41], showni nF ig. 19. wi/W, of the incident pump power will be obtained at the The oscillator builds up in the direction in which the pump- output [33]. In a recent experiment, Falk and Murray [29] ing wave is traveling and powerdependent reflections are have obtained about 70 percent peakp ower conversion avoided. Though 60 percent depletion of the incident and 50 percent energy conversion from the incident ruby pumping beam wasobserved, as aresult of insul€icient out- beam. As seen from the power-versus-time plots in Fig. 17, put coupling only a few milliwatts of output power were greater energy conversion was prevented by the build-up obtained. time of the oscillator. The schematic of this oscillator is Ammann et al. have recently obtained about 7 percent shown in Fig. 10. average power conversion in a doubly resonant LiNbO, We next consider the case where both the signal and idler oscillator directly pumped by a repetitivelyQ -switched cavities are resonant. Here the signal and idler waves travel Nd3+ :YAGl aser [42]. through the nonlinear media inb oth the forward andb ack- Another interesting type of optical parametric oscillator ward directions. When traveling in the backward direction, is obtained when the nonlinear crystal is placed inside the they mix to produce a pump wave traveling in the opposite cavity of the pumping laser. The mirrors for the signal and direction from the incident pump wave. As &stpointed out idler cavity may be coincident with those of the pumping by Siegman [39], this results m a powerdependent reflec- laser or may be positioned using various types of beam tion of the pump and a limiting oft he transmitted pump to splitters. Oshman andH ams [43]h ave shown theoretically its threshold value. As a resuit, the maximum efficiency of that this type of oscillator may operate in several types of such an oscillator is 50 percent and occurs at a pumpp ower regimes. These are : an efficient regime withoperating char- equal to fourt imes the thmhold pump power. acteristics similar to those of the previouslyd escribed Bjorkholm has recently shown that if this backward re- oscillators; an inefficient regime in which the parametric flection is avoided, the signal and idler resonant case may coupling in effectdrives the phase rather than the amplitude HARRIS:T UNABLEO PTICALP ARAMETRIC OSCILLATORS 2107

APERTURE

TOC HART RECORDER

ARGON LASER

LASER IicNc u1 ( p )

PIP, - Fig. 18. Efficiency and transmitted pump power of the doubly resonant oscillator, with and without power dependent pump reflection. PjP is the ratio of transmitted-to-incident pump power. (From Bjorkholm (401.) kP ( b ) Fig. 20. (a) Apparatus for viewing spontaneous parametric emission. y M 2 r O V E N (b) k vector matching. (From Byer and Hams [31].)

The transition rate fospontaneousr parametric emission ‘LINbOs CRYSTAL may be calculated by forming an interaction Hamiltonian Fig. 19. Ring cavity parametric oscillator where M,a nd M, are 5 cm based on the nonlinear susceptibility, and then applying dielectric mirrors, and M, is a flat gold mirror. (Byer et nl. [41],) first order perturbation theory. Kleinman [51] and Tang [52] have shown that this approach yields the now-accepted of the oscillation and where a shift of the signal, idler, and result that the parametric emission may be considered to pump frequencies from their normal positions is observed; arise as the result of the mixing of a fictitious zero-point and a repetitively pulsing regime, characterized by short flux, at both the signal and idler frequency, with the in- pulses of output power at the signal and idler. A stability coming pump beam [31]. The effective zero-point flux is analysis of these various regions shows that they are mu- obtained by allowing one half-photon to be present in each tually exclusive and canb e experimentally chosen by chang- blackbody mode of a quantizing volume. The result is a ing the laser gain, the oscillator output coupling, or the generated polarization which attempts to radiate at all fre- strength of the nonlinear interaction. quencies and in all directions. Its ability to radiate effec- VI. SPONTANEOPUASR AMETRIC tively is determined by the degree of velocity synchronism EMISSION with the free wave at the given frequency and in the given When light from a pumping laser is incident on a non- direction. linear crystal, there is spontaneous probability that pump A typical experiment for viewingspontaneous parametric photons will split into signal and idler photons. Without emission is shown in Fig. 20. The pump propagates along the need for optical resonators at either the signal or idler the length of a L i m o 3 crystal and is polarized along its wavelengths,e mission at thesew avelengthsm ayb e ob- optic axis. The signal and idler waves are ordinary waves served. This emission has alternately been termed as spon- and make angles 4 and $, respectively, witht he pumpw ave. taneous parametric emission, parametric fluorescence, For a plane wave pump, for small 4 and +, the incremental parametric noise, parametric luminescence, and parametric srontaneously radiated signal power [31] in a bandwidth ah scattering [MI-[49]. It is analogous to laser fluorescence and angle d4 is given by or more exactly to spontaneous Raman and Brillouin scat- dP, = BL2Ppf( os4,) 4d4 do, (33) tering. It is important since even at pump fields which are far tool ow to attain oscillation it may still be observeda nd where used to obtain temperature, angular, or electrooptic tuning curves of potential oscillator materials. The datafor Figs. 6 and 8 were obtained using this technique. The fact that the spontaneously emitted power varies linearly with pump and f (us4,) is the velocity synchronism reduction factor power and is independent of both thearea and coherence of given by the pumping beam, also makes it a useful tool for the measurement of optical nonlinearities. (34) Spontaneous parametric emissionw as predicted and studied by Louise11 et al. [47] and others, and was first ob- where l A k l is the length of the wave vector mismatch taken served at optical frequencies by Akhmanov et ai. [48], in the direction of the pump. For small angles and small Magde and Mahr [26], and Harris et al. [50]. dispersion Ak may be written 2108 PROCEEDINGS OF THE IEEE, DECEMBER 1969

70 - x=, 4880 A X,= 6328 a Pp=I w 60 - , J - - - m h )

50 - 8.0.49' 8.0. 0- 7-0 40- -.I an -10 0 IO 20 30 0 IO 20 30 ' 30- p" - 4r 4r 'E . 3 t A 3 I f ?

Oo 0.2 0.4 0.6 0.8 0.1 1.2 -10 0 10 20 30 0 IO 20 X) -82 (deg2) Fig. 21. Total spontaneously emitted power versus e', showing theoretical and experimental results. (FromB yer and Hams [31].) , I , I , ,) ' ,e=0 .1i00 ,, e=o.070I l Ak = + bdo, + g+', (35) I I where -10 0 IO 20 30 0 IO 20 30 9 = k&pPki, AX; (cm-') AA;'(crn+) and b is defined in (27). As a result of normal dispersion, b Fig. 22. Spectral distribution of spontaneous power at different accep- is negative, and thus higher frequencies will be obtained tance angles. Part (c) shows experimental points normalized to peak of the theoretical curve.( From Byer and Hams [31].) farther off angle. Combining (33), (34), and (351 the total radiated power over all frequencieins a given acceptance angle 8 is

P, = /3L2Ppj+mjoe sinc2.[ #os + g+')L]+d+ do,, (36) - m which may be integrated [31] to yield P, = (/3LPp/b)d2. (37) The total spontaneously emitted signal power thus varies Fig.2 3. Kleinman'sm atchings urface [51]. linearly with the accepted solid angle ne2,p ump power, and crystal length. Noting (33) it is seen to vary as the fourth case of emission tangent to this surface both the power and power of signal frequency and the first power of the idler bandwidth of the spontaneously emitted signal are greatly frequency. enhanced. This effect can be considered to arise as a result The ratio of the spontaneously emitted power to the of a greater number of idler modes which mayc ontribute to incident pumping power as a function of O2 is shown in the emission in the given direction and is analogous to the Fig. 21 for a 1.1 cm crystal of LiNbO, and a CW 4880 A increased spontaneous emission which occurs when near pump. The triangular points are experimental and the solid degeneracy. line is theoretical. Kleinman also discusses a background spontaneous emis- The spectral distribution of the spontaneously emitted sion which is independent of crystal length and which oc- light as a function of the accepted angle 8 is shown in Fig. curs in directions where phase matching is not possible 22. The theoretical curves were obtained by numerically [51]. This nonphase matched emission is muchsmaller than integrating (36). As 8 is decreased the bandwidth is at first the phase matched emission and has thus far not been ob- reduced, but then approaches the limiting bandwidth (l/bL) served. of a collinear interaction. Giallorenzi and Tangh ave observed and discussed spon- A detailed study of spontaneous parametric emission has taneous parametric emission for the case where the idler been given by Kleinman [51]. He makes use of a matching frequency liesi n the infrared absorbing region of the surface, such as is shown in Fig. 23, which is the locus of crystal [53], and they find the intensity to be approximately signal and idler E vectors such that M=O. For the special the same as it would have been had there been no absorp- HARRIS:T UNABLE OPTICAL PARAMETRIC OSCILLATORS 2109

10,000 In t F BOULE 103 1.65. cm LONG t Sc lqS a K W 3 8

-a - HI 0 LiNbO3

TEMPERATURE (DEGf3EE.S C ) (a) I 1 I a Q ) I 1 1 ' 1 I I I ' l l l l i I I I I I loo+ 10 P I P 0.Ip TRANSPARENCYW AVELENGTH Fig. 24. d2/n3a nd transparency of some phase matchable nonlinear optical materials. tion. With one frequency in the absorbing region, spontaneous emission provides a convenient means to obtain the TEMPERATURE (DEGREES C I dispersion curve in the lossy region. (b) Fig. 25. Second harmonic generation versus temperature for good and Recently four-photon parametric noise, corresponding to bad crystals of LiNb03. A high quality crystal should have a half- satisfaction of frequency and L vector conditions of the form power width of about 0.64" C / L where L is its length in centimeters. wp+ up= w, + mia nd E , + E,= E , + E j , has been observed in water by Weinberg [54] and in calcite by Meadors et al. which phase matching occurs, there is some uncertainty to [55]. The potential advantage of a four-frequency process of the values in this figure. this type is that the pump may be a lower frequency than The KDP-ADP type materials are theonly ones with UV the signal. Using a ruby pump, Meadors er al. tuned from transparency [57]-[59] and, as seen in Figs. 6 and 8, allow 4300 A to 5900 A. For a four-photon process of this type, convenient visible tunability. These crystals might also be unlike the three-photon process, focusing of the pump is of used to double a tunable visible source into the 2700 h- consequence. 3000 A region [57]. Unfortunately, Dowley has found that If the parametric gain is sufficient,spontaneous emission these crystals exhibit some form of UV damage which limits may be directly amplified without using optical resonators. the average power which may be obtained when doubling One technique, demonstrated by Akmanov et al. [56] uses the 5145 A line of argon [58]. KDP-ADP type materials multiple reflections between roof top prisms. With a pump have been found to withstand very high optical power den- density of 70 MW/cm2 of doubled 1.06 ,u light, 100 kW of sities (>400 MW/cm2) before exhibiting surface damage tunable radiation was obtained. The nonlinear crystal used [ 6 0 ] and are particularly useful for high power Q-switched was ADP, andtuning was accomplished by crystal rotation. applications. When operated near their Curie temperature Observed linewidths were about 1-2 A. these materials should also allow wide electrooptic tuning [61 I. VII. NONLINEAROPTICAL M ATERIALS LiNbO, has a very useful visible and IR transparency From theconsiderations of the previous sections, a num- range, and is the most widely used oscillator material at ber of desirable qualities for materials to be used in optical this time[ 62]-[65]. Crystals ofe xcellent optical quality parametric oscillators may be formulated. These are: high are now available in 4 to 5 cm lengths. Great care must be nonlinearity; phase matchability and, in particular, 90" taken to grow this material such that its refractive index phase matchabilityn; arrow linewidth (large b); high does not vary with distance in the crystal [ a ] , [67] k vector transparency and freedom fromd amage; and large variation matching for a 4 cm crystal requires that the variation of of refractive indices with temperature, angle, pressure, or refractive indices be lesst han 10- '. It has been found that electric field. Fig. 24 shows d2/n3 (see Section 11) and the this is best achieved bygrowing from a melt which isa bout transparency range of a number of phase matchable mate- 2 percent lithium deficient. Fig. 25 shows second harmonic rials. Normalization is to d2/n3f or KDP. Since both dand n power versustemperature for a high quality and al ow qual- are dependent on the particular angle and frequencies at ity LiNbO, crystal. For the poor crystal, the refractive 2110 DECEMBER IEPEREO, CEEDIN GS OF THE 1969 index varies withdistance, and different parts of the crystal xijk(-o3, 01, O Z ) = xJ"Ik (o1 3 -O3, phase match at different temperatures. For second har- (40) = Xkji(wZ3 ~ 1-03, ). monic generation from 1.15 p, the half-power width for a high quality crystal should be about 0.64"C/L, where L is the Overall permutation symmetry in effect states that if three length in centimeters of the nonlinear crystal. As a result of frequencies are involved in a lossless nonlinear process, optical refractive index damage, LiNbO, must be main- that irrespective of which is doing the generating, or being tained at a temperature greater than about 170°C when in generated, the nonlinear coefficient governing the process is the presence of intense visible radiation [68]. In a repeti- the same. tively pulsed systemsurface damage probably occurs some- The above x i j k are related to the diJi which are used to where int he vicinity of2 0 MW/cmZ. describe second harmonic generation by BazNaNbSOIS has a similar transparency range as LiNbO, and a d2/n3 whichi s almost tent imes greater Xijk(-20?w, = 2dijk(-20, o, (41) [69]-[71]. However, available crystals havel engths of 4 mm or less, and typically exhibit severe striations. At room Since diJx is symmetric in the subscripts j and k, it is ex- temperature the material does not exhibit refractive index pressed in the usual abbreviated notation where dijk=dil damage, but on the other handits phase transition at about according to 1 = 1, 2, 3,4, 5,6 for j k = 11, 22, 33, 23, 13, 12, 3 W , and the decrease of its nonlinearity above this transi- respectively. For lossless media these coefficients may be tion, limit its temperature tuning range. shown to be real, and their small dispersion over the trans- Two possible crystals for oscillation further in the in- parency range of the crystal is usually neglected. In cases frared are proustite (Ag,AsS,) and CdS[e 7 2]-[74]. where the direction of optical propagation is not along the Though proustite has been usedf or mixing experiments, an principal crystal axes, it is necessary to take the projection oscillator using it has not yet been constructed. As a result of the generated polarizations in the directions of the of its large birefringence, only off angle E vector matching respective optical E fields. The resulting coefficients have will be possible. Also, the material is reported to damage been termed as effective nonlinear coefficients, and their relatively easily, i.e., at about 1 MW/cm2. An absorption value for a number of uniaxial crystal classes have been band with absorption of.about 1 cm- at 10.6 p will prob- tabulated by Boyd and Kleinman [13]. ably prevent working with the COz laser. CdSe has a high nonlinearity [74] and should be phase VIII. TUNABLEFA R-INFRAREDG ENERATION matchable near 90"f or a pump at abou2t .5 p. There is great interest in extending tunable source tech- Tellurium had attracted earlier interest when Pate1 used it niques to the farI R region of the spectrum, where relatively both fordoubling COz [75] and too btain a parametric gain conventional sources are at their poorest. Assuming other of 3 dB at 18 p [76]. d2/n3i s about 1.7 lo6o n the normalized factors constant, the pump power necessary to achieve a scale of Fig. 21. However, the material is hard to handle given gain increases inversely as the square of the lower fre- and has a relatively high loss. quency. Also, in the far IR losses are typically somewhat Some other possible nonlinear crystals for use in oscil- greater than in the visible or near IR, andthus significantly lator applications are discussed in [77]-[84]. larger nonlinearities are required. Such nonlinearities may The conventions for specifying optical nonlinearity have be obtained either by making use of a very high indexmate- caused some confusion, and may be summarized as follows rial, such as tellurium, or by operating with the IR fre- [13]. If the electric field and polarization waves are written quency below the Reststrahl frequencies, and thus gaining in the form the contribution of the lattice to the nonlinear coefficient. For instance, in LiNbO,, the nonlinear coefficient governing E,(?) = Re [Ei(w) exp jwt] the interaction between a microwave frequency and two and optical frequencies is approximately 30 times ,greater than the nonlinear coefficient relating three similarly polarized 9Jt) = Re [9,(w) expjwt], (38) optical frequencies. (This may be deduced by converting the then for three interacting frequencies with w, = w1 +w2, light modulation coefficient rsl into an equivalent d co- the generated polarization is given by efficient.) Two other principal changes may occur when one of the interacting frequencies lies below the lattice absorp- 9ii(w3) = Xijk(-w3? O1, 02)Ej(01)Ek(oZ) j k tion frequencies. First, the lattice contribution to the low- frequency dielectric constant often creates the situation 9ii(wZ) = 12 xi,i(-w2, -01, w3)E7(01)Ek(w,) (39) where the sum of the signal and idler li vectors is greater j k than the pump I; vector. This allows noncollinear phase g i ( W 1 ) = Xijk(-wl, - 0 2 9 w3)Ef(w2)Ek(w3)* matching of three waveso f the same polarization. (By j k contrast, as a result of normal dispersion, at optical fre- The nonlinear susceptibility coefficients xijk satisfy what is quencies (k,(+ lkil < lk,l.) Second, it is usually necessaryt o termed as overall permutation symmetry, which states that include the effect of losso n the low-frequency wave. the subscripts and frequencies may be permuted in any As the IR frequency approaches one of the vibrational order. For instance modes of the lattice an increasing fraction of its energy is HARRIS:T UNABLE OPTICAL PARAMETRIC OSCILLATORS 2111

5.0 4.03 I I 1 I I \ x 104 \ \ n 0 \ -- 300 I -4.0 - \ '5 351L N v) - ,-s o- \ 1 a v) \ I + - z w - 0 3.0 E W 0 z +e 0 - 2.0 a 0 VI m

K W eA - 1.0

0 I I I I I I 1 ) 0 5.000 10.000 15,000 20,000 ICI (crn-11 Fig. 26. Dispersion of the A , symmetry 248 cm- polariton mode of I \. 0 ' 480 400 320 240 LiNbO,. The vertical lines intersecting the dispersion curve denote thev alue of 6 necessary forv ectorm atching.( FromY arborough R w k I n - ~ ) et al. [89].) Fig. 27. Parametric gain,a bsorption mfficient, andr atio of W to Stokes power, as a function of frequency near the 366 cm-' lattice resonance of gallium phosphide. (a) Relative parametric gain. (b) IR absorptionc oefficient. (c)R atio of IR to Stokes power densities mechanical rather than electromagnetic, and in this region (SI&). The solid portion of the curve denotes the region over which it is often termed a polariton mode [MI. Gain results from phase matching is possible. (From Henry and Garrett [%I.) the interaction with both the vibrational and electromagnetic portions of the mode; and toth e extent that the vibrational portion is important, it mayb e considered as a The slow variation of the gain coefficient and its approach tunable Raman gain [86], [87]. to zero at about 250 cm- ' is a result of destructive inter- In two recent experiments, Gelbwachs et al. [88] and ference between the parametric and Raman type portions Yarborough et al. [89] have obtained tunable radiation in of the gain coefficient. This interference has been verified the vicinity ofthe A symmetry 248 cm- polariton mode of experimentally by Faust and Henry [91]. LiNbO,. The dispersion diagram for this mode is shown in Though the resonant behavior sf the infrared absorption Fig. 26. The vertical lines intersecting the dispersion curve does not substantially affect the gain,i tm ay greatly in- denote the value of the angle 6 between the pump and the fluence the production of infrared radiation. Henry and Stokes beams which is necessary for li vector matching. Garrett [90] have shown that the ratio of infrared to Stokes (In analogy with the usual Raman process, the upper fre- powers is approximately quency is termed as the Stokes wave). As 8 is varied, the mi G oscillator is tuned. In the Gelbwachs et al. experiment [MI, &RIfLokes = - -' tuning was accomplished by varying the angle between the 0, aIR laser beam and the axis of a high-Q resonator. In the Yar- where G is the gain and am is the infrared loss coefficient. borough er al. experiment [89]: opposite faces of the crystal As seen in Fig. 27, though in the vicinity of the resonance were polished flat and parallel and an external resonator the gain is relatively unaffected,the IR power drops sharply. was not employed. Infrared tuning from 50 p to 238 p was Far-infrared radiation has also been obtained by differ- obtained with a conversion efficiency to the Stokes fre- ence frequency mixing ofhigher frequencies. Van Tran and quency of greater than 50 percent. Though infrared powers Patel [92] have recently reported the useo f a magnetic were not measured, it was estimated that about 70 watts field to achieve phase matched difference frequency genera- were generated inside the crystal at E. z 50 p , tion in InSb. Phase matching was accomplished by varying As the infrared frequency approaches the lattice reso- the cyclotron frequency and thus the free carrier contribu- nance, both the absorption coefficient and the nonlinear tion to the refractive index. Discrete tuning from 95 p to susceptibility become resonately large, though in such a 105 p was obtained by mixing a number of wavelengths of manner as to leave the gain relatively unchanged. Fig. 27 two synchronously Q-switched COz lasers. Previously, shows theoretical results of Henry and Garrett [90]as the nonphase matched far-infrared generation has beenr e- infrared frequency is swept through the 366 cm- ' lattice ported by Zernike [93]. resonance of gallium phosphide. The absorptionc oefficient, Two other approaches to tunable generation in the far parametric gain, and the ratio of generated infrared to infrared are backward wave oscillation in a material with Stokes radiation areshown. The solid portion of the curves low birefringence such as LiTaO, [94], [95], and Raman indicate the region over which phase matching is possible. down shifting of higher frequency tunable radiation. 2112 PROCEEDINGS OF THE IEEE, DECEMBER 1969

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