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5A2-Theory of the Optical Parametric Oscillator 418 IEEE JOURNAL OF QUANTUMNO. ELECTRONICS, QE-2, VOL. 9, SEPTEMBER 1966 5A2-Theory of the Optical Parametric Oscillator A. YARIV, MEMBER, IEEE, AND w.w. LOUISELL, SENIOR MEMBER, IEEE Abstract-A formalism for describing optical parametric oscil- characterized by a linear susceptibility xL and a suscep- lation is developed. The theory is applied to the derivation of the tibility tensor of the third rank drik.The induced polari- oscillation threshold condition, power output, the Manley-Rowe conditions, index matching, and frequencytuning. zation is given by Pi~oxLEi+ dii,EiE, (1) INTRODUCTION i .k N THIS PAPER, we present a general theory for where, in general, the coefficients dii, are functions of parametric oscillation in the optical region. Since the frequencies of Pi,Ei, and E,c, respectively. If, how- this phenomenon is the exact analog of parametric ever, the nonlinear medium is transparent in the region amplification and oscillation in the microwave region, its of interest, the coefficients diikare frequency independent. possibility was recognized a number of years ago [I]. The Another consequence is that all the diik elements which practical realization of opt,ical parametric oscillation has result from a mere rearrangement of the subscripts are been demonstrated by Giordmaine and Miller 121. equal [3]. The experimental situat’ion is as follows. A nonlinear Maxwell equations are optical crystal isplaced within an optical resonator. A “pump” field at w, is then fed into the resonator. It is observed that at a certain pumping intensity, oscillation is set up simultaneously at frequencies w1 and w2,where w1 + w2 = w,. These frequencies correspond to resonances of the optical structure or are near such resonances. with the constitutive equationstal- \en as The theoretical approach takes the following form. The total electromagnetic field is expanded in a set of basis j = lTE functions which consists of the proper field solutions of (4) the optical resonator in the absence of any nonlinear D = EOE+ij+P element’s.In this last case any one of the basis functions where the total polarization is broken into the sum of is a proper solution of Maxwell equations and, conse- the polarization P, which is induced by the macroscopic quently, the coefficients of the expansion are time inde- fields and as given by (I), andan applied polarization pendent. When the nonlinearity is “turned on,’’ the in- P,which represents the source feeding power into the dividual basis functions are no longer proper solutions resonator. and the coefficients must be taken as time dependent. The electric and magnetic fields in the resonator can be In the limit of small coupling, this time dependence, for expanded in terms of Slater’s [4], [5] normal modes any one mode; contains the full information concerning a, and Ea. the mode energy and its oscillation phase. The main con- wq(t) - - cern of thispaper is to obtain the solutions for these R(r,t) = =-Ha(?.) coefficients and, through them, follow the flowof power a dG (5) from the pump field into the w1 and w2fields. THEELECTROMAGNETIC BACKGROUND The starting point is a solution of Maxwell equations The vector functions E,(?) and W,(P)satisfy the relations inside a generalized resonator that contains a nonlinear dielectric material. The properties of the material are Manuscript received June 20, 1966. Yariv was supported by the U. S. Air ForceLaser Technology Groupthrough Contract as well as ?ix E, = 0 and fieg, at the resonator walls. AF33(615)-2800. This paper was presented at the1966 International It follows from (6) andthe two boundary ccnditions Quantum Electronics Conference, Phoenix, Arix. A. Yariv is with theCalifornia Institute of Technology, Pasadena, previously mentioned, that the sets E, and H, are or- Calif. [4], [Fi]. W. H. Louise11 is with the Bell TelephoneLaboratories, Inc., thogonal We are free to choose their amplitudes Murray Hill, N. J. so that they arenormal. The orthonormalityconditions are 1966 YARIV AND LOUISELL:PBRAMETRIC OPTICALOSCILLATOR 419 where use has been made of (5) and (7). The energy U, /v Ea’Eb du = 6ab in a singlemode +(p: + can alsobe expressed, (7) using (13), as He*Hb dV = 6ab u, = 4(p: + w:qrr”C> = wca:ao. (15) where the integration extends over the whole resonator Since the energy of a photon is hwc, a:a, is proportional volume. Substituting (5) into (2) andtaking the ith to the number of photons in the mode c. component leads to Using (13), we can transform the equations of motion (10) and (11) to a form containing the variables a, and a:. while from (3) and (5) we obtain (9) &!E= @)*. Next, we multiply each term in (8) by EOiand sum over dt i(= 1, 2, 3). Using (7) and recalling that e0(l + x”) = E, Next we define normal field variables A (t) and A: (t) by we obtain ac(t) = e-iwCtA,(t), a:(t) = eiwo’.4t(t). (17) It isclear, from (16), that when diik= 0 and p’ = 0, ~,(t)= const. e-(oa’zqc)t . We will assumethat thecoupling is sufficiently small so that -dAc w,A, Multiplying (9) by Hci and summing gives dt << (18) for which case we may replace Equations (10) and (11) are the equations of motion for the mode variables p, and g,. and NORMALMODES Rather than carry out the analysis in terms of p, and qc, we introduce the normal mode variables a, and a: With these adiabatic approximations, theequations of which are defined as motion (16) become i a, = -dz (PC - iwcnc), (12) a? = complex conjugate of a,. The expressions for qc and p, become and The advantagesof using the normal modes a, and at rather than p, and qc have been discussed previously [ii], 161. The total energy stored in the electromagnetic field is E=p/H.8dv+a/l?-gdz. (14) = $ c (p: + w%q3 c 420 JOURNAL IEEE ELECTRONICSOF QUANTUM SEPTEMBER where U/E = wc/2Q, is the decay rate for the field variables Using (23), these equations are finally written as of mode e. These are the general working equations. They can be employedto obtain thedifferential equations which apply to special cases. In the nextsection we use them to obtain the equationsdescribing the paranzetric oscillator. - / E,,EZiE,, dv a5aP (25) THE PARAMETRICOSCILLATOR EQUATIONS Let us assume thatthe interactions are limited to three modes only, by means to be described shortly. Let these modesbe denoted by the subscripts I, 2, and p, - / E2,EliEP,dv ala: (26) and let their respective frequencies be wl, w2, and a. We shall refer to modes 1, 2, and p as the signal, idler, and pump modes, respectively. Furthermore, we shall assume that mode p (thepump) is driven by the polarization source so that P(T, t) = P(v)sin wt (22) Equat'ion (25) can be simplified further by taking ad- represents the applied polarization. If we consider the vantage of the symmetry condition diik = djik = d.*kt . = equation of motion for al, i.e., (20) with e = 1, we notice dkii = diki = dkii which was previously discussed. Con- that for the case of no coupling (difk = O), the solution sider, for example, the summation in (25). If we inter- is of the form al(t) = a,(O)e-iwxte-(wx/2Ql)t . It follows that change i and j, we obtain for the case of weak coupling the only cumulative con- tribution to da,/dt comes from terms on the right side of (20) whose oscillatory factor is The contributions from terms with frequencies considerably different from which is equal to the corresponding summation in (26) w1 oscillate at a rate equal to the difference in frequencies, since diik = diik. The same argument canbe used to show thus averaging out to zero in time spans that are long that the summation in (27) is equal to those of the first compared to the optical beat periods. If the free (i.e., no two. We can consequently define a single parameter K as coupling) resonance frequencies satisfy the condition w = w, + 02 (23) then the product to replace t,he triply-summed factors in (25). If, in addi- tion, we define the pumping parameter X, as i(w*--W)l a%a, = A'",(t)A,(t)e = A*z(t)Ap(t)eiW't x = 1E,@).I'l(i;> du, .on the right side of (20) satisfies the synchronism con- (29) 2 82€ dition. Assuming that this happens for no other pair of we can rewrite (as), (26), and (27) as modes, the equation for al(t) can be written as w da, = -zwpa, - - a, - Kala2 + zhpe-iO' and similarly dt 2Qp I-- together with their complex conjugates. In terms of the adiabatic variables Ai(t) = ai(t)eiwit, we have E,,EliE,, 2(-w ol)alaz (24b) dA1 -1 dv + - = --Y1 A, KA~A, dt 2 + ___dA$dt - -"/z2 A$ + KA,A,* -dApdt -r,Ap'-2 KA,A,+ %X, 1966 YARIV AND LOUISELL : OPTICAL PARAMETRIC OSCILLATOR 421 and theircomplex conjugates. The decay rate yi is defined THEPOWER RELATIONS as yj = wi/Qi, and is equal to the reciprocal decay time From (30) we can obtain explicit expressions for the constant for the mode j. The yi parameters account not power output at thesignal frequency w1 and the idler w2. so much for the losses in the medium as for the fact that Since the storedenergy in the ithmode is wiaiaT = wiAjA:, the external coupling(i.e., reflector transmittances) is the total power output is Pi = wiAiA$(wi/Qi), or using ldifferentfor the three frequencies. ~i = wi/Qir Equations (30) are the main result of the preceding section and constitute the startingpoint for the following Pi = yiwiATAi.
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