646 J. Opt. Soc. Am. B/Vol. 8, No. 3/March 1991 Eckardt et al. Optical parametric oscillator frequency tuning and control Robert C. Eckardt, C. D. Nabors,* William J. Kozlovsky,t and Robert L. Byer Edward L. Ginzton Laboratory,Stanford University,Stanford, California 94305-4085 Received May 15, 1990; accepted August 7, 1990 The frequency-tuning and -control properties of monolithic doubly resonant optical parametric oscillators are analyzed for stable single-mode pump radiation. Single-axial-mode operation is observed on the idler and the signal for both pulsed and continuous pumping. Projections are made for tuning-parameter tolerances that are required for maintenance of stable single-frequency oscillation. Continuous frequency tuning is possible through the simultaneous adjustment of two or three parameters; thus the synthesis of specific frequencies within the broad tuning range of the doubly resonant optical parametric oscillator is permitted. 1. INTRODUCTION axial modes. Nevertheless, with improved pump sources and nonlinear-optical materials coupled with multiple- An analysis of the frequency-tuning properties of doubly parameter control, DRO's can in principle be operated resonant optical parametric oscillators (DRO's), based on stably and tuned continuously, thus widening their range both experimental observations and theoretical modeling, of applications. is presented. Specific details in this presentation of fre- DRO's can provide highly coherent output, reproduc- quency control and synthesis apply to monolithic DRO's ing the statistical properties of the pump with little addi- constructed from LiNbO3. Where possible, however, re- tional noise. This was shown theoretically by Graham sults are given with more general applicability. The pur- and Haken9 in a quantum-mechanical analysis of the pose is achievement of a quantitative understanding of DRO, and it was demonstrated in experimental measure- the conditions required for stable single-axial-mode para- ments of the coherence properties of the DRO. The metric oscillation and the resulting frequency stability quantum-mechanical analysis showed that the diffusion of the DRO output. Approaches to frequency synthesis of the sum of the signal and the idler wave phases follows and continuous frequency tuning that are based on the the phase diffusion of the pump wave adiabatically. Al- simultaneous adjustment of two or three tuning variables though the phase difference of the signal and the idler are described. may diffuse in an undamped manner, the statistical prop- The potential of optical parametric oscillators (OPO's) erties of a DRO are basically the same as those of an ideal for the generation of tunable coherent radiation was rec- laser. A result of these properties is the addition of only a ognized more than twenty-five years ago.' The complex small amount of phase noise in the output of the DRO tuning properties of DRO's were also revealed in early above that present in the pump. This has been 4 confirmed demonstrations and analyses.2 Optical parametric os- in coherence measurements of the output of a cw DRO.'0 cillation has been discussed in detail in a number of re- For periods of -1 min, the free-running 7 DROthat was not views, and it is a subject treated in more general terms servo locked oscillated on a single mode pair without a in a number of books that discuss nonlinear optics.' Im- mode hop. That the DRO did not add significant excess provements in the quality of nonlinear-optical materials linewidth over that present on the pump was demon- and in the coherence of pump sources led to a number of strated with the measurement of beating between the advances in the performance of OPO's. Using recent ex- DROoutput and an independent diode-laser-pumped solid- perimental results obtained with stable single-mode pump state laser during the periods between mode hops. The sources and monolithic DRO's constructed from high- beat-note linewidth was 13 kHz, which was the expected quality LiNbO3 nonlinear-optical material, we are able value for the typically 10-kHz linewidths of the pump to apply and to extend the earlier analyses. laser and the independent reference laser. Additional co- Resonance of both the signal and the idler frequencies, herence measurements showed that the signal and the double resonance, offers the advantage of a lower thresh- idler were phase anticorrelated when referenced to the old for parametric oscillation than in single resonance. pump laser. Also, the width of the signal-idler beat note Double resonance also provides additional frequency selec- with the DRO near but not at degeneracy was less than tivity in OPO operation. These desirable properties of 1 kHz. The signal-idler beat note indicates the frequency double resonance, however, come with a considerable in- fluctuations added to the DRO output in addition to those crease in the complexity of tuning and with more restric- present on the pump. tive tolerances on pump stability and cavity stability. The results of the classical stationary analysis pre- Diode-pumped solid-state lasers provide the required sented here are consistent with the earlier analyses and pump-frequency stability, and monolithic cavities provide measurements. The main purpose of this presentation is the required mechanical stability in the OPO. Continu- to explain the complex tuning properties of the DRO in ous tuning is difficult in DRO's, which typically tune order to permit fuller use of its remarkable coherence and with axial mode hops and cluster jumps over hundreds of spectral properties. The theoretical presentation of 0740-3224/91/030646-22$05.00 © 1991 Optical Society of America Eckardt et al. Vol. 8, No. 3/March 1991/J. Opt. Soc. Am. B 647 Section 2 begins in Subsection 2.A with a qualitative Phase matching is the major factor in determining overview of DRO tuning. This overview is used to estab- broad tuning properties of an OPO, although cavity reso- lish the extensive terminology required for the discussion. nances have the major effect on details of frequency tun- In Subsection 2.B the threshold condition for paramet- ing. The conditions COP= ct5+ &wiand Ak = 0 define ric oscillation is reviewed and recast in terms that are phase-matching curves. The most commonly shown OPO more easily adapted to tuning calculations. The theoreti- phase-matching curve is the parabolalike shape for type-I cal basis of frequency selection is discussed in Sub- phase matching in a birefringent crystal, for which the section 2.C. Experimental tuning data are presented in signal and the idler waves have the same polarization and Section 3. The degree to which our theoretical model de- the pump wave has the orthogonal polarization. Fig- scribes the observed tuning justifies some confidence in ure 2(a) shows a near-degeneracy (5 - co-)section of the its use for predictive calculations in Section 4. Results temperature-tuning curve for a LiNbO3 noncritically are summarized and discussed in Section 5. Finally, the phase-matched OPO. Propagation is along a crystal prin- properties of MgO:LiNbO3 that are required for modeling cipal axis in noncritical phase matching, which reduces the experimental data are reviewed in Appendix A. dependence on propagation direction and eliminates bire- fringent walk-off. 2. THEORY The spectral width of the parametric gain is also deter- mined by phase matching. A typical spectral distribution A. DRO Tuning Overview for single-pass gain at a fixed temperature is shown in A nonlinear-optical material pumped by intense optical Fig. 2(b). Doubly resonant oscillation also entails simul- radiation at frequency co, can provide gain at two lower taneous signal and idler resonance. Dispersion causes frequencies, called the signal and the idler and related by different cavity axial-mode frequency spacings for the two the conservation-of-energy condition waves, and the simultaneous resonance condition thus occurs only at intervals in frequency. The regions of cop= o. + i(1) simultaneous resonance, called cluster frequencies, are in- The parametric interaction is phase dependent, and dicated in Fig. 2(c). Early DRO's were observed to oscil- proper phasing is required for energy to flow from the late on a group or cluster of adjacent cavity axial modes. from the pump field to the signal and the idler fields. The wavelength of the cluster would at first shift con- Phase-velocity matching ensures that the relative phases tinuously with tuning and then exhibit a discontinuous of the three waves do not change with propagation through the nonlinear material. Phase matching is described by the wave-vector mismatch, which for the case of collinear (o) go IPU) propagation can be expressed by the scalar relationship (a) -- o-II00 Is(0) - Ak = k - k - ki = (npw - nsos - nicoi)1c, (2) 0)P = Cos Coi, JAkJ < d, , Ak = kP - k - ki where kp, k5, and ki are the respective wave-vector magni- tudes of the pump, the signal, and the idler waves, with corresponding indices of refraction given by n,,, n5, and ni, and c is the velocity of light. Useful parametric gain I pW exists in the range of signal and idler frequencies for m E > I~~~~~~~~~(t) which Aki s 7r/l, where I is the length of the nonlinear (b) Kpg material. The parametric gain is maximum near Ak = 0. mirror mirror Phase matching is often achieved by controlling the bire- 2 fringence of a nonlinear crystal through temperature or angle of propagation. An OPO requires feedback at either (or both) the signal g (t) or the idler frequencies. If there is feedback at only one (C)pI~i) g -Is frequency, the device is called a singly resonant oscillator. --II i M 1 M2 Doubly resonant oscillators have feedback at both the sig- nal and the idler frequencies. Feedback can be provided by placing the nonlinear material in a cavity formed by 1pW(t) I(i) two external mirrors, or, in the case of monolithic OPO's, P highly reflecting coatings can be applied directly to the (d) nonlinear material.