Introduction.

II. 2 Parametric gain analysis. 27

II. 3 Pump power thresholds. 40

II. 4 Conversion efficiencies. 70

II. 5 Tuning, spectral outputs, and stability requirements. 75 II. 6 Further properties of the doubly-resonant oscillator. 104

II. 7 Mode-matching. 110

II. 8 Conclusions. 114

References. 116

II. 1 Introduction.

This chapter reviews the specific criteria involved when considering the formation of continuous-wave (cw) optical parametric oscillators (OPOs). These criteria are analysed within different sections, each of which describes an important operating condition that must be addressed. The conclusions from each section should be satisfied within the eventual experimental design of a cw OPO.

Before constructing a cw OPO, there are a num ber of critical assessments that must be performed to establish how the device will operate in practise. These requirements include the following key parameters: incident pum p power to reach threshold, optical resonator geometry including material selection, resonance conditions of the cavity, mode- selection properties of the signal and idler frequencies, and effective coupling of the pump radiation into the cavity modes of the signal and idler fields.

In section II. 2, the coupled wave equations of parametric amplification are derived classically. The steady-state equations are presented for the three fields over the length of the gain medium. A nonlinear coupling parameter

is defined in terms of the effective nonlinear coefficient, the phase- synchronism factor, and the spatial overlap of the Gaussian modes over the length of the gain medium. The analysis is valid for the case of small parametric gain, and is restricted to the near field.

Historically, the most important parameter for a cw OPO has been the pum p power required to reach threshold, and this is studied in detail in section II. 3. When the pum p radiation is derived from a cw laser source, with significantly lower pum p powers available, compared to the more familiar pulsed pum p sources used for OPOs, then the design of the OPO cavity becomes critical. In particular, within any phase-matching geometry, the resonance conditions must be selected carefully to maximize the gain of the system to reach threshold. Different optical cavities are studied to show the effects on threshold when resonating different fields within the OPO cavity. In addition, the nature of the interaction of the pump field with the signal and idler fields must be examined. In this context, ring-cavities and standing-w ave cavities provide significantly different threshold values. When standing-wave cavities are analysed, it is important to consider the relative phasings of the forward and backward travelling waves within the resonator, to reduce the pump power threshold to a minimum.

Once above pum p power threshold within an OPO, the conversion efficiency of the pum p radiation to signal and idler powers can be considerable. Once again, these conversion efficiencies are dependent on the type of resonator employed (single-pass travelling-wave ring cavity, or double-pass standing-wave linear cavity), and the number of fields that are brought to resonance within the OPO cavity. These different configurations are reviewed in section II. 4.

The most important consideration in the operating characteristics of cw OPOs is the amplitude and frequency stability of the generated signal and idler output frequencies. These are analysed in section II. 5. The resonance properties of the OPO cavity (finesses and free spectral ranges) play the dominant role in determining the fine frequency outputs of an OPO. In general, stable OPO operation becomes more difficult to obtain as the number of resonant fields within the cavity is increased. Cavities that are resonant for both the signal and idler frequencies place stringent requirements on the frequency stability of the pump source, and the mechanical stability of the OPO resonator, to maintain stable OPO operation. The minimum stability

conditions for the pump frequency and the OPO cavity length to obtain stable OPO outputs are examined for different resonator configurations. Further, methods of generating smooth frequency tuning are analysed. Comparisons are made between different cavity designs, and for pum p sources that are fixed in frequency or can be tuned smoothly.

Another factor determining the fine frequency properties of the signal and idler frequencies is the linewidth of the usable down-converted outputs. This is shown to be dependent on the frequency stability of the pump source and the noise that is inherent in the generation of the signal and idler frequencies from phase-diffusion in the parametric process. In particular, the production of equal numbers of signal and idler photons in parametric down- conversion makes possible a number of fundamental studies of OPOs in quantum optics. The quadrature amplitudes of the signal and idler beams can be regarded as quantum copies of one another, with almost perfect correlation between the signal and idler photon flux. The significance of these properties is explained in section II. 6, with regard to producing squeezed states of light from cw OPOs, for using cw OPOs within studies of high resolution spectroscopy, and for optical frequency division within optical metrology schemes (frequency chains).

Finally, in section II. 7, there is a brief study of the relevant parameters that must be considered to match the radiation from the pump source to the OPO modes, as defined by the OPO resonator. Useful equations are derived that enable effective m ode-m atching by transform ing the Gaussian parameters of a laser source to provide the required focusing of the pump radiation within the nonlinear gain medium. In addition, formulae are presented to enable calculating the required mirror curvatures within the OPO resonator to provide the desired overlap of the pump, signal, and idler fields over the length of the gain medium.