IKEE JOURNAL OF QUASTUM ELECTROSICS, VOL. Q E - ~ ,N O. 8, ACGUST 1968 491 Theory of Optical Parametric Oscillation Internal to the Laser Cavity

Abstract-Since the fields inside a laser cavity are much higher focusing [5], [ H I , [19], doubler efraction [71[ ,1 71[ ,2 01, than the external fields, an analysis of a parametric oscillator with [21], and optical frequency resonators [22], 1231. A num- the nonlinear crystal internal to the laser is performed. Using self- ber of experiments have been performed involving para- consistency equations as the starting point, the equations of motion of such an oscillator are derived.D epending on various cavity, metric amplification [8], [24]-[27], and in1965, Giordmaine pumping, and nonlinearity parameters, these lead to several types and Miller [as] reported the first observation of tunable of oscillation with distinctlyd ifferento peratingc haracteristics : parametrico scillationS. ince that times, e veral other (1) efficient parametric oscillation similar to that of previous anal- observations of pulsed parametric oscillation have been yses; (2) inefficient parametric oscillation resulting from the fact reported [29]-[31] and recently Smith et al. [32] have thatt he nonlineari nteraction drivest hep hasesr athert hant he amplitudes of the signal, idler, and pump; and (3) a pulsing output observed CW parametric oscillation. from the oscillator with repetitivep ulses of the signal and idler. Fig. 1 is a schematicr e presentation of a possible A stabilitya nalysis of these variousr egionss hows that they are internal parametric oscillator with the pump, signal, and mutually exclusive and can be experimentally chosen by changing idler simultaneously resonated. In this noncollinear con- the laser gain, the oscillator output coupling, or the strength of the figuration, the interaction region is reduced;h owever, nonlinear interaction. It is shown that the internal oscillator effi- ciency rapidly approaches the Manley-Rowe limit, as the available the problems of mult'iple wavelength antireflection coat- pump power becomes several times greater than that required for ings and mirrocr oatingsc haracteristic of a collinear threshold. The efficiency of ane xternal oscillator havinga triply configuration are avoided.T wo-dimensional parametric resonant optical cavity is found to be generally less than that of amplification of this sort has been observed external to the corresponding internal oscillator. the laser by hkhmanove t al. [26]. For analytical simplicity we consider the case of collinear interactions,a lthough I. IXTRODUCTION the extension to the noncollinear case is straightforward. N THIS PAPER we examine the theory of optical Such an oscillator might be achieved experimentallyusing frequency parametric oscillation with the nonlinear a dichroic beams plitter t o divertt he signal and idler crystal placed inside the cavity of t,he pumping laser. while allowing thep umpt ot ravelt hrought he laser Such an oscillatorp, roposedo riginally by Kroll [I], medium. takcs advantage of the high pump fields inside the laser The analysisl eads tot he surprisingr esult thata n cavity, compared to outside, in overcoming threshold and internapl arametric oscillatocr a no perate in several in leading to efficient conversion of pump power to signal previouslyu npredicted regimes. Briefly, there are three and idler power. Several authors have discussed t'he en- regions of operat'ion: 1) an efficient regime with operating hancement of second harmonic generation under similar characteristics similar to those of previous analyses; 2) an conditions [a], [3]. inefficient regime in which the parametric coupling drives With the theory of optical parametric oscillation well the phases rather than the amplitudes of the oscillations known at lower frequencies [4], the extensions to optical and wherein an interesting shift of signal, idler, and pump frequencies have been considered by a number of authors frequencies from their normal positions is observed; and 111, [5]-[12]. Much of the theoretical apparatus necessary 3) ar epetitivelyp ulsing regime characterized bys hort for understandingo pticalf requencyp arametric oscilla- pulses of output power at thesignal and idler, accompanied tion has been developed to handle the specific problem by nearly simultaneous decreases in power at the pump. of harmonicg enerationa lthoughe xtension tot he case These results are due basically to the fact that thes ignal of parametric oscillation is straightforward. This includes and idler are coupled directly tot hes aturating gain calculations of the effects of phase matching [7],[ 13]-[17], mechanism of the pumping laser through the nonlinear interaction. The choice of parameters necessary for opera- tion in a particular regime is discussed in terms of the Manuscript received March 29, 1968; revised May 10, 1968. The work reported in this paper was sponsored by the National stability conditions for the various types of operation. Aeronautics and Space Adminstration under NASA Grant NGR- After considering the internal oscillator, one is led t o 05-020-103. M. K. Oshman's workw as partiallys upportedb y SylvaniaE lectronicS ystemst hrought heI ndependent Research question the fundamental difference between it and an Program. Portions of this paper were presented at the 1966 Con- oscillator consisting of cavities external to thel aser, which ference on Electron Device Research. M. K. Oshmanw as witht heD epartment of ElectricalE ngi- again resonate signal, idler, and pump. Again for high-Q neering, Stanford University. He is now with Sylvania Electronic cavities, thep ump fields could bev ery highE. xperi- Systems, Mountain View, Calif. S. E. Harris is with the Department of Electrical Engineering, mentally, the differences are immediately apparent. The Stanford University, Stanford, Calif. difficulties of placing the crystal inside the laser cavity 492 IEEE. J OURNALO FQ UANTUME LECTRONICS, AUGUST 1968 f ' SIGNAL OUTPUT

Here Qii s the Q of the ith mode resulting from the losses '-PUMP CAVITY LASER MEDIUM due to transmission, scattering,a bsorption,a nd diffrac- IDLERC AVITY tion. I n derivingt hesee quations, Q ia rises through the ' PNONLINEA R CRYSTAL definition of a ficticious volumec onductivit,y ui,w hich Fig. 1. One geometical arrangement of ani n t,erndp arametric account's for these losses. The Qi and ui are related by oscillator. are large; however, based on other work that uses cryst,als inside a laser cavity, these seem to be problems that can We will eventually divide uc into two part>so, ne resulting be overcome. On the other hand, difficultythe of efficiently fromv olume loss effects andt heo ther resu1t)ing from coupling the pump into an external oscillat,or is tremen- transmissiont hrough the mirrors. The Ci and X i are dous. Nevertheless, assuming one could optimally couple relatedt o t h et o tapl olarization P(z, 1) throught he the pump into such ano scillator, the question remains as following equations: to the relative performance of the internal and external oscillator. In particular,a st hee xternal oscillator goes P,(t) = P(z, t)ZL,(.> dx above Ohreshold, thep ump experiences reflections from I11 L thea ctivec avity,t ,hus reducing the efficiency of the = C"(4 cos b , t + dJi(t>l oscillatorS. imilar reflections at microwavef requencies were observed by Ho and Siegman [33]. In Section IV, + Si(t) sin [wit + cbZ(t)l. (6) the efficiency of the externalo scillator is compared to In writingt hese self-consistency equations, I C i (t) and that of thei nternal oscillator, and it is apparentt ,hat &(t) are assumed slowly varying with respect to wi. reflections do render the external oscillator less efficient. In considering parametric interactions inside the laser. the polarization at a particular frequency will have two 11. EQUATIOONF SM OTION FOR THE contributions. One primarily affects the pump and results INTERNAL 1'ARAMETRIC OSCILLATOR from t8he presenceo f the laser medium. The other results I n deriving the equations of motion of the internal from the nonlineari nteractions of the electric fields at parametric oscillator, we use a Oechnique similar to that ot,her frequencies. used by Harris and McDuff [ X ] to analyze the FM laser. The contribution to the polarization due to the laser We use Dhe self-consistency equations developed by Lamb mediumc anb ei ntroduceda s macroscopic quadrature [35], whichd escribe the effect' of ana rbitraryo ptical and in-phase components of susceptibility. Then we have polarization on the optical frequency electric fields of a high-Q multi-mode resonator. We direct, our attention to Ci(t) = E0X{Ei(1) (74 three modes of interest with circular frequencies ol,u p,w g and satisfying the relation w1 + w2 = w3. In the usualt er- minology of parametric interactions then a,, w2, and wy x,@)= E0X/'E;(t). (7b) are the signal,i dler, and pump frequency, respectively. We assume t'hat the signal and idler frequencies are well We shall label all quant'it'ies referring to the signal mode, removedf rom anyt ransitions of the laserm edium, so idler, and pump with the subscripts 1, 2, and 3, respec- contributions to them are essentially negligible. For t,he t.ively. pump xi' accountsf org aina nds aturationd ue to t'he Neglecting t,ransverse variations, the total electric field laserm edium; xi result'si nf requency shiftls and mode in the cavity can be written as pulling and pushing effects.

3 In proceeding tot he calculation of the parametric !$(x, f) = E , ( f ) nos [a,t + dJ*(L)]ui(z), (I) contribution to the polarization, we shall use the specific i = l example of lithiumn iobate( IJiNbOg)a s the nonlinear where Ei is thea mplitude of thei th mode and ui(x) element [36]. This is useful inp erformingt 'hea nalysis; is the spatial variationo f the ithm ode, which has circular however, the technique can easily be generalized to any frequency Qi and is that mode lying closest in frequency parametrics ystem. In particular,w ith LiNbO,, we can to w i . We normalize uis uch that assumep hasem atchingi sa chievedf orp ropagation at 90" to the optic axis [37],a s used in the cxperiments of Giordmaine and Miller [25]. I n this way, we cana void the problems of double refraction, whichw ould needlessly where we have let the cavity extendfr om z = 0 Do z = L. complicate this analysis. The self-consistency equations then are For IiiNb03 orientedw ith its crystallographic z-axis OSHMAN AND HARRISI: N TERNALO PTICALP ARAMETRICO SCILLATION 493 along the x-axis of thec avity,t he polarizationi n the the fact that a polarization wave is 90” out of phase with crystal is related to the electric fields by the electromagnetic wave it radiates. We define the phase mismatch in terms of the wave vectors as P~(xt’) = 2 d l S E 3 , (8) Ak = IC, - k, - k,, and (13) andr etain onlyl ow-frequency termsi n (11)w ith the t )P Z ( x , = d3,E;. (9) result The subscripts y and x refer to fields polarized along the ’I2L ’ sin A ~ L . y- and x-axes of the crystal.W es halla ssume that the 6 = -dl,--. -(;) L AkL signal and idler areo rdinaryr aysp olarizeda long the y-axis and the pump is ane xtraordinaryr ay polarized In practicep hasem atchingc anb ea chieved at 90” t o along the z-axis.W e shouldp oint out that (8) and (9) the optic axis by varying the temperature of the crystal relate the time-dependentn onlinearp olarization to t’he [411, [421. time-dependente lectric fields. Normally, the value of d We now combine (6)) (7)) and (10) with (3) and (4). is defined by relat’ing the Fourier amplitude of the non- Since the dielectric constant in the crystalis substantially lineapr olarization tot h e Fouriear mplitudes of the differentf rom e o , whenu sing (10) in (3) and (4)) we electric fields. Therefore, as discussed byP ershan [38], replace eo with e, where E is the dielectric constant of the the numerical value of d in these equations i s a factor of 2 crystal and is approximately the same at w,, w2’ and w,. larger than commonly quoted values. Then one finds Using (8) and (9) in conjunct’ion with (1) and (6)) we therefore find the following parametric contributions to the polarization( K leinman’s symmetry condition [39] is assumed so that d15 = CE3*):

Cdt) = 6E,E’3 cos (43 - 41 - 4 2 ) (loa) Sl(t) = - 6E2E, sin (d3 - - +J (lob)

c Z ( j ) = 6ElE3 cos ( 4 3 - dl - $ 2 ) (10c)

X z ( t ) = - 6EIEz sin (43 - 61 - 42) ( W

G j t ) = 6ElE, cos ( 4 3 - dl - &) (10e)

X 3 ( t ) = 6ElEz sin (4, - dl - d2). (10f) where Here 6 is defined by L’ 6 = dl, 1 ul(x)udz)u3(dz x) , 1(1 ) where we have assumed the crystal extends from x = 0 to z = L’. All contributions at frequencies well removed = - from the frequencies wl, w,, and w3 are neglected. A, -+x;‘ (16c) w2 3 [.Q13 1 In evaluating 6 we shall assume that the mirrors of the resonator represent a short-circuit surface for the signal 0’ = [@ax; - wzx: - w1x:1/2. (17) and the idler and an open-circuit surface for the pump. In treatingt he gain ands aturation of thel aser, we Then the spatial variation of the modes inside the crystal need to assume some functional form for x;’. There have is of the form been several different saturation functions suggested for 1/2 describing the observed effects of various lasers. In this ul(x) = ($) sin k l x (124 analysis we use a small-signal approximation similar to

1/2 that derived byL amb 1351. Witht hisa pproximation, ua(x) = (f) sin k2x (12b) x;‘ is related to the single-pass power gain by

1/2 w,L -x;’ = -goU - DE’:), (18) % ( X ) = );( cos k3x. (124 C where go is the single-pass unsaturated power gain and p The reasonf or choosing theseb oundaryc onditions is is a parameter accounting for the effects of saturat’ion. apparent uponi nspection of the expression for 6. If all A more realistics aturation functionf or inhomogeneously three modesh ads hort-circuitb oundaries, then 6 would broadened lasers is represented by [43] be zero withno parametric interaction. Thiiss the problem met in second harmonic generation where both the fun- damental and pump are resonated 11401, and results from 494 IEEE JOURNAL OF QUANTUM ELECTRONICS, AUGUST 1968

The saturation function of (IS) produces essentially the 111. OPERBTINGC HARACTERISTICS OF THE same results as (19) under the condition that INTERNAL 1'ARSMETRIC OSCILLATOR Using (20a) to (20d) we are in a position to investigate ( y - 1 << 1, the operatingc haract'eristics of the int'ernalp arametric oscillator. Previous analyses of parametric oscillators have where a3 is the single-pass power loss. There are a few resulted in thewell-known threshold conditions foroscilla- low-gain lasers, such as Nd : YAG, under which this condi- tion and one steady-state region of parametric oscillation. tion might well be satisfied. Further, we argue that for Below threshold, the signal and idler remain zero, whereas internal parametric oscillation, the signal and idler drain above threshold, the pump limits at the threshold level much of the laser power, thus not permitting the pump anda dditionalp ump power isc onverted to signal and to saturate fully. Nevertheless, the reason for using the idler power. saturationf unction of (IS) is its simplicity.W eh ave As opposed tot h is previous o lution, the int'ernal carried out hee ntires ubsequent analysis using (19) parametric oscillat'or displays three distinct steady-state and find the same qualitatfive results. Since the objective regions of operation. One region of operation results in herei s to pointo utt he generalb ehavior of internal the expected behavior, which predicts efficient conversion parametric oscillation, we believe the use of the small- of pump power to signal power. In addition t,hereis anot'her signal saturation function is justified. In any particular CW output of the oscillator wherein the phases rather application, one would have to calculatet 'he following than the amplitudes of t'he signal and idler are driven by result,s, using the appropriates aturationf unctiona nd the nonlinear interaction, thereby resulting inle ss efficient appropriate parameters. operation. Finally, there is a continuous relaxation oscilla- If we change thct imev ariable so t8hatT = (c/2L)t, tiont ype of solution. In this case tjhe signal and idler then (15) and (16) become spike on and off, resulting in a repet'itivcly pulsing output _dE_, _ from the oscillator. - -alE, + W,KI

E , = E2 = 0 (23a)

where (2%)

Equation (23b)p redicts t h e steady-statea mplitude of the laser field. For future calculations it is convenient to (2110) find the output power of the laser. To do so we divide a:<, t,he single-pass power loss of the laser,i nt'o two parts and let (21c) f f 3 = a 3 c + %,1, (24) where aQci s that part of the loss resulting from output coupling and aadis Ohat part of the loss attributed to all and otherd issipative mechanisms. Thent heo utput power I, 6 of the laser P, ips roportional to so /( -- EC P3 = Yff3&, (25)

Here ai is the single-pass power loss for the ith mode. where y is the constant of proportionality. Maximizing These equations are similar in form to the general equa- P, withr espect to aSc,t he maximum output power of tions of motion for parametric interactions described by the laser is Siegman [44]. The main difference arises duet o t h e introduction of a saturating gainm echanism. For that reason we believe the form of the subsequent results is generally applicable to other parametric systems coupled Int he presence of parametrici nteraction, the first directly to the gain mechanism of the pump. steady-state solution is found by taking K # 0 and 43 - OSCPIALRLA MTIEO TPNRT IICC AINL T ERNAH LOA RSHRIMSA: N AND 495

dl - d2 = ~ / 2 .T hent he cosine term of (15d) is zero, The second conditionr equires that if the single-pass the sine terms of (15a) to (15d) are one, and gain of the laser is greater than the sum of the loss to

then for stability, I$; = (go - 4%- SOP G Z] (27bj W I W ~ K2 [ y" - a, - a., - cy,J a,ap ______< ----' (31) SOP WlWpK p' - LYL% J 3 - 2 ' (27c) This second conpdroitv ioens to be the condition that WI%K the steady-state solution of (28a) to (2Sd) not be stable. This is the usual efficient solution for parametric oscilla- Finally, there is a third stability condition given by the rather tion. relation complicated

The final steady-stat,e solut,ion is found by assuming For a better feeling of what this relation entails we assume (p3 - (pl - & = a const'ant # a/2: that the signal loss equals the idler loss, i.e., a1 = a2, in which case (32) becomes

go - a? 3a,ac> < --3. SOP W ] W * K This shows that the free running laser power should be less than three times thatr equired for parametric thresh- (28 c) old. This last stability requirement is connectedw ith the qc,.Ba,a, third type of operation of the oscillatorp reviously de- sirl' (& - @1 - $-) = _____ i28d) go - a, - G2 - a3 WlWLK scribed-the pulsing output. If this condition (32) is not satisfied, then the output of the oscillator consists of a B. Stability Criteria for SZeady-State Solutions continuous train of pulses. The solutionf ort his region Since therea re several steady-state solutionsf or the cannot be solved in closed form; however, we have internal oscillator, we must find the necessary conditions performeda d etailedc omputers olution of this region, that a particular solution be stable. There are a number which is described in Section 111-E. of techniques for approaching this problemi, ncluding Finally, for t'he second steady-state parametric oscilla- Liapunov's stabilityc riteria [45] andt he Hurwitx test t'ion solution (28a) t o (2Sd) to be stable, it is necessary 1461, both of which were used. We quote only the results. that At theo utset we should point out thatwe find the st'ability gn > ay1 + 012 + a3 (33) conditions of the various regions to be mutually exclusive. and That is, two steady-state solutions are not simultaneously go - a1 - ff2 - ff3 ala2 possible. >7 (34) ,4s is expected, the stability of the free running laser SOP W W2K solution is guaranteed so long asp arametric oscillation C . Eficient Internal Parametric Oscillation is not possible. That is, below ac ertain level of pump In describing the characteristics of the various regions power, the signal and idler are zero. This condition (for of operation of thei nternalp arametric oscillator, it is stability of the laserw ith no parametric oscillation) is convenient to define ap arameter K ~ T~ h.isp arameter just the inverse of the threshold condition for parametric represents that value of K necessary to overcome threshold oscillation and is given by in the absence of any output coupling loss to the pump, signal, or idler and is given therefore by the relation

Stability of the first steady-state parametric oscillation (35) solution (27a) to (27c) requires that hree conditions where we have divided into two components such that be satisfied. First there is the threshold condition: ai ai = aic + a i d . (36)

(30) Experimentally, K is variable by adjusting phase-matching 496 IEEE JOURNAL OF QUANTUM ELECTRONICS, AUGUST 1968

9,

Fig. 4. Signal power insider esonatorv ersus ( K / K t h ) ' in region of efficient parametric oscillation.

0 I I I I I I I l l I I I I I I I l l I 2 3 4 5 6 7 8 10 20 30 40 50 70 100 Fig. 3. Pump power inside resonator versxs ( K / K ~ I , ) ~i n region of efficient parametric oscillat,ion. -" 2 ("thl' Fig. 5. Optimally coupled signal output power. conditions, focusing, orc rystaln onlinearity.T herefore, the ratio of K to K~~ is a realistic variable and represcnts thes trength of the nonlineari nteractionw ithr espect decrease, much in the same way that increasing output to the strength required for threshold. coupling losses to a laser can produce a similar maximum Figs. 2 to 4 display some charact,eristicso f the oscillator in output power. for the efficient steady-state operation of (27s) to (27c). In order to determine the efficiency of the oscillator, These figures show the form of the onset of oscillation one must determine the proper output coupling to the and the buildup of power for the signal for typical laser signal for given operatingc onditions. In terms of the operatingp arameters. In each case thed otted lines ratio K"/K;,, the efficiency of the oscillator is independent indicate the onset of the pulsing output of the oscillator of all other paramet'ers. and are cont'inued to show where the steady-state solu- The optimum coupling to the signal is found by maxi- tion would have operated. mizing the output signal power P, withr espect to ale. In Fig. 2, .E; and E: arep lott,ed versusl aserg ain. In a manner analogous to (25), PI is given by ]+'or the parameters chosen in this case, the laser reaches its oscillation threshold at a gain of 3 percent per pass and increases in power untilt hep arametric oscillator In terms of the maximum available pump power P3,mnx, reaches threshold. Att hatp oint,t he laserl imits and the maximization produces, when optimally coupled, the further increases in gain result in signal (and idlerpower.) result In Figs. 3 and 4 the pump ands ignal power are plotted versus K~/KE:~:,;, ,, is the value that the square of pump electric field amplitude has in the absence of parametric oscillation. As the parametrici nteraction increases, the With perfect conversion, the Manley-Rowe relations pump power continually decreases. Simultaneously, the limit the signal power to be wI/w3 times the pump power. signal power first increases, goes through a maximum, Therefore, we see that the internal oscillator satisfies this and begins to decrease. Qualitatively, this can be under- condition since that is thea symptoticv aluet hatt he stood as the result of the fact that the parametric inter- signal power approaches. Fig. 5 is a plot of the relation actiona ppears to be an increasing loss tot he laser. given by (3s). From this graph it is apparent that once Eventually, thet otal available power from the gain the oscillator is above threshold it rapidly becomes very mechanism goes through a maximum and begins to efficient. OSHMANA KDH ARRIS : INTERNALO PTICALP ARAMETRICO SCILLATION 497

I. o 8 > a i +a2+a3 0.9 - - 7 - 0.8 E$ 0.7- QlC,OPl 6 - __ - Qld 5 - E23,,max0.50'6 - 4- .\ 0.4 - \ 3- 0.3 - .\. 2 - ---- 0.2 - I - - O 1 l 1 1 1 l 1 1 1 1 1 1 0.1 0 I 2 3 4 5 6 7 8 9 1011 1 2 1 3 1 4 1 5 -K2 0 ( X , #

Fig. 6. Ratio of optimum signal coupling loss to signal dissipative Fig. 8. Pump power versus (K/Kth)' showingo nset of inefficient loss. region of operation.

g,>a,+ap+a,

0.45

0.40

0.35 030

0.15

0.1 0 0.05 I I I U 2 3 4 -K2 K2 t h

Fig. 9. Signal powTer versus ( K / K t h ) ' showingo nset of inefficient Fig. 7. Frequency shift, of signal versus ( K / K ~ ~ ) * . region of operat,ion.

Fig. 6 is a plot of the optimum ratio of coupling loss The fact that c $ ~ + & - 4, = a constant results from the to dissipative loss versus K ~ / K ~ a? n, d is given by relations that dl = Awl (4 1a) (39) 4, = Ao2 (4W

Combiningt heser esults witht hes tability condition 43 = Aw3, (41 c) given by (32), it is possible to show that should efficient where Awl Aw, - Aw, = 0. That is, is a constant, operat'ion be desiredno, pulsing output will be experienced + representinga s hift in frequency of wi to wi A w ~ . so long as optimum coupling t o the signal is maintained. + These frequency shifts are given by the following equa- D. Ineficient Steady-State Region of OpeYatio,n tions: Thes teady-state region of operation described by (2Sa) to (2Sd) represents a case in which the parametric interaction drives the phaseso f the signal, idler, and pump rathert hant heira mplitudes, Ohereby resulting in less efficient operation. In general, this is probably a region to bea voided in almost any practiccaal se by appropriately C adjustingt h eo peratingp arameters of the oscillator. Am3 = (a1 + 0 12) cot (43 - 41 - 4 2 ) X 5~ , (42~ ) Nevertheless, its characteristics are important in under- where standing the behavior of the internal oscillator. In (20d) of Section I1 only one equation was written g"Pffla, __1- . sin2 @ 8 - +1 - = 2 (43) for all the phases. This equation is the sum of three equa- go - CY1 - f f z - ff3 WlW2K tions of the form The shifts are on the order of a megahertz as shown by Fig. 7 for typical operating parameters. Figs. 8 and 9 show the behavior of the internaol scillator, 498 IEEE JOURNAL O F QCANTUM ELECTRONICS, AUGUST 1968

t (0.1 p s / d l v ) - Fig. 10. Pump power versus time in pulsing region of operation.

1.0,

::[ K 2 .9.P ~~ -> 95% 0.7 wtw., go - c y , - a._-, a:< one t,hen finds a result similar to the maximum-cmissiolr principle proposed byS tatz,D eMars,a ndT ang [4T]. The systems pick t'he mode of operation which maximizes the pump power.

E. Pulsing Output of I?~ie~vnPala rametric Oscillatw t (0.1 p s / d i v ) The purpose of thiss ectioni s to describe the( har- Fig. 11. Signal power versus time in plllsing region of operation. acteristics of the internal oscillator in its pulsing mode of Operation. I t has been impossible t o find the solutions for which is able to operatei nt his inefficient region. The this region of operation in closed formT. herefore, a dotted curvesr epresent thc behavior had the oscillator computera nalysis of (20a) to (20d) was used to solve remained in the ot,her steady-state region. Once the value for t,he behavior in this region. The objectjive has been of K ' is large enough t o reach t,hreshold for this second to outline the qualitative behavior of this spiking regime, type of st,eady-state operation, t'hen the pump and signal and laserp arameters were chosen that we believc, are power no longer change for increasing parametric interac- representative of existing lasers and materials. tion. The fact that the signal power limits at a maximum Before proceeding, it is pointed out that we have not in Fig. 9 is a coincidence of the particularp aramet,ers used a,ny rat'e equation approach [4S] in developing the chosen for plotting the curve. equations of motion. Therefore, for some very slow atomic It is difficult to decide exactly what is the physical mechanisms (c.g., for use with CO, lasers) another equa- reason for this type of operation. However, we can make tion mustbe included to account for population changes of some commentst hat do aid in understanding t,his behavior.t he energy levels. Kevertheless, for most lasers, the results First we notice that t8heexpression for the pump power of this analysis appear to be consistent with the assump- (proport,ional to E:) has very much the same form as t'he tion that no rate equation is necessary. similar expression forp ump power int he case of no Figs. 10 and 11 show typical results for the signal and parametric oscillation. The only difference is that a3 pump fields as a function of time in the pulsing region of is replaced by al + a2 + a3. In addition, a threshold operation. The dotted lines represent the corresponding condition for this region of operation is qo > al + a2 + a3, levels had pulsing not begun. The computer analysis has which is similar to the threshold condition for the laser been carried out through many pulses and these appear with, once more, the laser loss replaced by the sumo f the to be constant in peak height and period. We therefore losses to signal, idler, and pump. For these reasons, one is characterize this region of operationa sa r epetitively led t o speculate that in this region of operation, the laser pulsing regime. appears to have its gain mechanism coupled directly to Fig. 12 shows an expanded version of one pulse, pointing all thr,ee circuits: signal, idler, and pump. Then all three outt he relationb etween the signal andp ump during frequencies oscillate withm any characteristics imilar the course of one pulse. Using this graph we can qual- to a laserr ather thana laser pumped parametric oscillator. itatively explain the physical reason for this typeo f opera- Since there are two solutions possible under the condi- tion. tion that go > a1 + a2 + a3,t he question then arises as Since the oscillator is well above threshold (because of to why the system picks one mode of operation as opposed the stability condition of (32)),t he signal and idler build to theo ther. By investigating the second threshold condi- up very rapidly. (As is well known, the buildup time of tion, i.e., ap arametric oscillator decreases ast he margina bove OSHMAN AND HARRIS : INTERNALO PTICALP ARAMETRICO SCILLATION 499

a = 5% 0.50 a3= I %

0.45 g = IO % 040- @_ = I -3 w3 4 - 0.35 - E ,: mox 0.30 - 0.25- ', \ - \ 0.20 \ .. 0.15- .. 0.10- .. go (PERCENT) .. 0.05 ------______Fig. 13. Peak signal power versus laser gain. '0 h b 6 ' b ,b !lZ lk lb ;I 2b 2; 24 26 28 30 22 2 (&I2

Fig. 16. Peak signal power versus ( K I K t h ) ' .

Fig. 14. Pulse period versus laser gain. '0 I2 4 6 8 I O 12 14 16 18 20 22 262 4 28 30 32 34

Fig. 17. Pulse periodv ersus (K/Kth)'.

10.115 \ c I 0.20 I l l l l l l l l ~ l 0 1 2 3 4 5 6 7 8 9 1 0 g o (PERCENT)

Fig. 15. Pulse width versus laser gain. thresholdi ncreases.) Thep ump is slowed ini ts decay somewhat by the bandwidth of its resonance. Therefore the signal and idler buildup past the point to which they should have approached, thus draining more power from the pump. As ar esult, thep ump experiences sufficient effective loss to go below threshold. The continuing decrease of the pump is eventually felt by the signal and pulses and AT is the full width of the pulsesb etween idler and they then decrease rapidly. With no signal and one-half power points. As the gain of the laser is increased, idlerp resent thep ump can once moreb uildup to its the peak power in the pulses increases and their period free running value and the process repeats. and width decrease. Figs. 13 to 15 display the change in pulse parameters In Figs. 16 to 18 the same characteristics of the pulses as the gain of the laser is changed. The dotted linei n are plotted versus K'/K;,,. As K~ increases, again the peak Fig. 13 is proportional to the signal power in the steady- power of the pulses increases, and their period and width state region of operation. Here T is the period between decrease. 500

IV. EFFICIENCOYF AN EXTERNAL PARAMETRIC OSCILLATOR I n this section we briefly describe the results of a cal- culation of the efficiency of an external parametric oscil- lator having pump, signal, and idler resonated [49]. Once the oscillator is above threshold, reflections of the pump from the active cavity can lead to reduced efficiency from /rea,, the oscillator. Therefore, a cent,ral concern of the calcula- tion is that these reflections be talcen into account. I I I 1 I I l l The details of the calculationa ren oti ncludedh ere, 10 20 50 100 alhhough thet echniquei ss ummarized. Beginning with the normalm odef ornmlat'ion of Slater [SO], thei nput Fig. 19. Efficiency of external parametric oscillator. fields att hep umpa ppeara ss urfacei ntegralso nt he cavity boundaries. Reflections are accounted for by energy losses of the internal fields at thcm irrors. The calculation in bot'h t'he internal and external cases. Any real cryst,al leads to second order partial differential equations for the will present some loss, however. amplitudes of the normal modes at the signal, idler, and The effecto n the internal oscillator will be to lowcr pump frequencies. The procedure is very similar to the the pump power through its dependence on losses. Thc technique used by Gordon and Iiigdcn [51] in analyzing extent of thisr eductiond ependsb asicallyo n the gain the Fabryl'erot electrooptic modulator, and the partial of the laser medium and the relative magnitude of the differentiale quationsf or the modea mplitudes of t,he crystal losses with respect to other dissipat'ive losses. parametric oscillator arc as traightforwarde xtension of Typically, the insertion of a crystal inside a laser cavity their worlc. mayr educet he laserp ower by 25 to 50 percent [521 The result of the calculation of signal power P, yields with a comparable reduction in (K/K~,,)'.

The effect of crystal losses at the pump wavelength is

(--+~-- p-, - w_1_ f-f ? C ~ f f l < Kt11 - 5 readily apparenti nt he case of thee xternal oscillator. [ f~), ]' (44) P, w3 f f 3 f f l c f f l d ) a ! l d OL1d First (K/Kth)' is reduced byt hef actor ( c ~ , ~ / a+, ~a, ,)' and at a given value of K/Kth, the output power is reduced where P, is the incident pump power and a,, is the trans- by the factora 3c/aa+, Q , ~ It. seems reasonable to assume mission loss necessary for coupling pump power into the that in many cases the net effect of losses will therefore resonator.T herefore, CY,c onsists of twop arts, coupling result in comparable percentage reductions of power in loss , x Q c and dissipative loss agd.I n this expression, K / K ~ , , the internal and ext'ernal cases. Based on this reasoning is reduced from its previously defined value by the ratio we cant herefore use Fig. 19 to compare theo utput f f z c / f f 3 . power of the internal and external oscillators in the case Defining of no losses at thep ump and conclude that this is similar to their relative fficiencyi n thec ase with losses. However, (4.5) we caution that this is only a rule of thumb, which could be slightly different for very high-gain or very low-gain the expression for P,/P, can be written as lasers. We can draw some fairly fircmon clusions on t'he relative merits of the internal and external oscillators from these curves. The efficiency of the external oscillator is greatly where reduced due to reflect,ions of the pump from the active cavity. As a result, in cases where the crystal losses in- serted into the laser do not substantially reduce the laser power (for example to the point that the laser itself is Maximizing f(p) yields the coupling tot he signal that onlys lightlya bovet h reshold), the int'ernaol scillator produces maximum output power. Fig. 19 is a plot of this shouldp roduce the higher output power. In addition, function maximized witho ptimum coupling f(&J. On the difficulty of coupling thep umpi ntot hee xternal the same scalewe have plotted thefficiencye of the internal oscillat,or will make thei nternal oscillator more useful parametric oscillator ( w , P ~ / w ~ Palso~ , i~nt e ~rm~s) of in practice. (K/Kth)2 for that oscillator. There could be some objection to displaying the d a h in v. SUMMARY ANDC ONCLUSIONS t'his form. However, with a brief explanation it becomes Having solved the equations of motion for a parametric useful in comparing the relative efficiencies of the two oscillator internal to the laser cavity and thereby coupled oscillators. To the extent that the crystal represents no directly to the saturating gain mechanism of the laser, dissipative loss to the pump (in the form of absorption, we have found t'hat such aons cillator can operate in three scattering, or reflection), the value of K / K ~is~ t he same distinctly different regimes: an efficient regime, ar e - OSHMAN AND HARRIS: INTERNAL OPTICAL PARAMETRIC OSCILLATION 50 1 petitively pulsing regime, and an inefficient regime char- pp. 439-484, March 1966 (transl.: “Parametric amplifiers and generators of light,” SovietP hys.-Usp., vol. 9, pp. 210-222, acterizedb yf requencys hifts of thep ump, signal, and September-October 1966). idler. The necessary conditions for operation in one par- [12] A. Yariv and W. H. Louisell, “Theory of the optical parametric oscillator,” IEEE J . Quantum Electronics, vol. &E-2, pp. 418- ticular mode have been derived in terms of the stability 424, September 1966. criteria for that regime. The regimes arem utually ex- [13] J. A. Giordmaine, “Mixing of light beams in crystals,” Phys. Rev. Letters, vol. 8, pp. 19-20, January 1, 1962. clusive and may be chosen by varying the laser .gain, the [14] P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, oscillator output coupling, or thec rystal nonlinearity. “Effects of dispersion and focusing on the productiono f optical harmonics,” Phys.R ev.L etters, vol. 8, pp. 21-22, January 1, To some extent, each regime may be useful in practice. 1962. The pulsing regime offersp eakp owerss lightlyg reater [15] S. A. Akhmanov, A. I. Kovrigin, R. V. Khokhlov, and 0. N. Chunaev, Zh. Eksper. i Theor.F iz., vol. 45, pp. 1336-1343, than the average power of the CW regimes, although the November 1963 (transl.: “Coherent interaction length of light average power is somewhat lower. Some control of repeti- waves in a nonlinear dielectric,” Soviet Phys.-JETP, vol. 18, pp. 919-924, April 1964). tion rate and length of the pulsesi sa vailable through [16] J. E. Midwinter and J. Warner, “The effects of phase matching changes in the laserg ain and crystal nonlinearity. The method and of uniaxial crystal symmetry of the polar distribu- tion of second-order non-linear optical polarization,” Brit. J . inefficient regime is generally less useful than the other AppZ. PhyS., V O ~ .16, pp. 1135-1142, August 1965. two regimes, although some applications might make use [17] D. A. Kleinman, “Theory of secondh armonicg eneration of light,” Phys. Rev., vol. 128, pp. 1761-1775, November 15, 1962. of the several megahertz frequency shifts of the pump, [lS] J. E. Bjorkholm, “Optical second-harmonic generatlon usmg a signal, and idler. focused gaussian laser beam,” Phys. Rev., vol. 142, pp. 126-136, February 1966. The efficient regime isp robably of greatesti nterest. [19] D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic An important result of the analysis showst hat this regime generation of light by focused laser beams,” Phys. Rev., vol. 145, pp. 338-379, May 6, 1966. is capable of approaching 100 percent efficiency in con- [20] N. Bloembergen and P. S. Pershan, “Light waves at the bound- version of pump power to signal and idler power. By way ary of non-linear media,” Phys. Rev., vol. 128, pp. 606-622, October 15, 1962. of comparison, the efficiency of an externalo scillator, with [21] G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, pump,s ignal, and idlerr esonated,i sl imited toa bout “Second-harmonic generation of light with double refraction,” Phys. Rev., vol. 137, pp. A1305-A1320, February 15, 1965. 25 percent. [22] A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second-harmonlc generatlon and mixing,” I E E E J. Quantum Electmzics, vol. &E-2, pp. 109-124, June 1966. ACKNOWLEDGMEXT [23] S. A. Akhmanov, V. G. Dmitriev, and V. P. Modenov, Radio- tekhn. i Electron., vol. 10, p. 649, 1965 (transl.: “On the theory It is a pleasure to acknowledge many helpful discussions of frequency multiplication in a cavity resonator filled with a nonlinear medium,” Radio Eng. Electron., vol. 10, pp. 552-559, with B. Byer, A. E. Siegman, E. 0. Ammann, and W. B. April 1965). Tiffany. Mrs. Cora Barry expertly prepared the computer [24] C. C. Wang and C. W. pacette, “Measurement of parametric gain accompanying optlcal dlfference frequency generation,” programs. Appl. Phys. Letters, vol. 6, pp. 169-171, April 15, 1965. [25] S. A. Akhmanov, A. I. Kovrigin, A. S. Piskarskas, V. V. Fadeev, and R. V. Khokhlov, Zh. Eksperim. i Theor. Fiz., Pis’ma REFERENCES Redakt., vol. 2, pp. 300-305, 1965 (transl.: “Observation of parametric amplification in the optical range,” JETP Letters, ill N. M. Kroll, “Parametric amplification in spatially extended vol. 2, pp. 191-193, October 1, 1965). media and applications to the design of tuneable oscillators at [26] S. A. Akhmanov, A. G. Ershov, V. V. Fadeev, R. V . Khokhlov, optical fiequencies,” Phys. Rev., vol. 127, pp. 1207-1211, August 0. N. Chunaev, and E. M. Shvom, Zh. Eksperim. i Theor. Fiz., 15, 1962. Pis’ma Redakt., vol. 2, pp. 458-463, 1965 (transl.: “Observation 121 J. K. Wright, “Enhancement of second harmonic power gen- of two-dimensional parametrici nteraction of light .waves,” erated bya dielectric crystal inside a laser cavity,” Proc. ZEEE, JETP Letters, vol. 2, pp. 285-288, November 15, 1965). vol. 51, p. 1663, November 1963. [27] J. E. Midwinter and J. Warner“Up-conversion, of near infrared [3] R. G. Smith, K. Nassau, and M. F. Galvin, “Efficient contin- to visible radiation in lithium-meta-niobate,” J . Appl. Phys., uous optical second-harmonic generation,” Appl. Phys. Letters, vol. 38, pp. 519-523, February 1967. vol. 7, pp. 256-258, November 15, 1965. [28] J. A. Giordmaine and R. C. Miller, “Tunable coherent para- [4] Ag eneral treatment of parametrlc amplifier and oscillator metric oscillation in LiNbOa at optical frequencies,” Phys. Rev. techniques with an extensive bibliography is given by W. H. Letters, vol. 14, pp. 973-976, June 14, 1965. Louisell, Coupled Mode and Parametric Electronics. Sew York: [29] S. A. Akhmanov, A. I. Kovrigin, V. A. Kolosov, A. S. Piskarskas, Wiley, 1960. V. V. Fadeev, and R. V-. Khokhlov; Zh. Eksperim. i Theor. Fiz., !5] R. H. Kingston, “Parametric amplification and oscillation at Pis’maR edakt., vol. 3, pp. 372-378, 1966 (transl.:“ Tunable optical frequencies,” Proc. I R E , vol. 50, p.4 72, April 1962. parametric light generator with KDP crystal,” JETP Letters, [6] X. A. Akhmanov and R. V. Khokhlov, Zh. Eksperim i Theor. vol. 3, pp. 241-245, May I, 1966). Fiz vol. 43, pp. 351-353, July 1962 (transl.: “Concerning one 1301 R. C. Miller and W. A. Sordland. “Tunable LiNbOn outical pos2bility of amplification of light waves,’’ Soviet Phys.-JETP, oscillator with external mirrors,” A>pL Phys. Letters,-vd. 10, vol. 16, pp. 252-254, January 1963). pp. 53-55, January 15, 1967. 171 J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. L. B. Kreuzer,“ Ruby-laser-pumped opticalp arametric oscil- Pershan,“ Interactions between light waves in an onlinear latorw ith electro-optic effect tuning,” Appl. Phys. Letters, dielectric,” Phys. Rev., vol. 127, pp. 1918-1939, September 15, vol. 10, pp. 336-338, June 15, 1967. 1962. R. G. Smith, J. E. Geusic, H. J. Levinstein, J. J. Rubin, S. Singh, 181 G. D. Boyda nd A. Askin, “Theory of parametric oscillator and L. G. 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ondence

Current Noise Spectrao f GaAs Laser Diodes in the Lumi- nescence Mode + 013 Abstract-The current noise of GaAs laser diodes in the Iumines- I cence mode has been measured at frequencies between 0.47 kHz and 1.5 MHz. A l//behavior is found at relatively lower frequencies. In the investigated frequency range, the noise level is considerably higher than 2- e*1-A/.

In a recent paper, Haug [l] calculated the current noise spectrum of a semiconductor laser junction for freqilencies small compared to the reciprocal radiative lifetime but high enough for theflicker noise to be unimportant. Current noise measurements [a] in luminescent junctions are reported for frequencies up to about 10 kHz. This cor- respondence reports some results of our investigations on cllrrent noise of GaAs laser diodes in the luminescence mode at an ambicut temperature of 293°K. The diodes were forward biased, a,nd the measurements were made in the frequency range between 0.47 kTIz and 1.5 MHz. The noise measuring setup consisted of a high-input impedance - Fig. 1. Mean-square noise currents in’ at an effectiveb andwidth of 1 Hi; as a preamplifier followed by the main amplifier, filters, and a true rms func:ion of frequency f for the diodes DQ and D13. voltmeter. In the lower frequency range, active filters [a] have been used in contrast with the high-frequency range where a heterodyne cent diode is many orders of magnitude larger than the shot noise voltmeter has been employed. The diode dc current was supplied bp given by the formula 2e.I.Af. This statement is confirmed by our a filtered battery. investigations. However, the comparison of the measured amount of Several diffused GaAs diodes have been tested. The donor concen- noise with 2e.I.Ajmay lead to confusion. Indeed, the shot noise in trations ranged from 4 10’7e rn-3 to 3 X 10’8 (3111-3. When driven x p-n junctions, where the current is carried by diffusion, is given by with pulses at 77”K, the diodes had a lasing threshold current’ be- 2e. ( I + 21,)’ Af, I , representing the saturation current[ B]. This for- tween 0.85 A and 2.4 A. The junction areas ranged fro0m.3 -X 10-3 emZ mula holds as long as the frequency j , associated with Af, is small to 1X lO-3 cm2. The measured mean-square noise current i: at an ef- compared to the reciprocal lifetime of the minority carriers and the fective bandwidth of 1 H z as af unction of Bequency is shown in Figs. reciprocal of their transit time through thed epletion layer. Now, in 1 and 3 for four def orward biased laser diodes The currentnoise spec- forward biased Ga.4~p-n, jlmctions, the current is mainly composed trum of the diodes D9 and 013 (Fig. 1 ) is approximately 1/’ noise. of the following three components: 1) diffusion current, proportional In the vicinity of about 0.7 MHz, the mean-square noise current to exp (eV/IcT); 2) recombination current in the space-charge region, tends to become constant. This results in a time constant of about proportional to exp(eV/2k7’); and 3) tunneling current, proportional = 2.3 X 10-7 seconds, see [4], [5]. It is tempting to compare the rm to exp (a.V),where a is independent of the temperature. Surface re- measured amount of noise with some noise level associated with the combination may also cause a contribution. For the currendte nsities diode current I. It has been pointed out[ 2] that i: for a GaAs lumines- of interest, the current-voltage characteristics of our diodes showed the tunneling component of the current to bnegligible.e The current 3fanuscript received January 25 , 1968. was proportional to exp (eV/mk7’) with m ranging between 1 and 2