Effects of resonator design on the efficiency of singly-resonant optical parametric oscillator with intracavity sum-frequency generation

Abstract

We present a theoretical model to investigate the performance of an intracavity sum-frequency generation (SFG) inside the ring resonator of a singly-resonant optical parametric oscillator (SR-OPO) scheme. The effect of the resonator specifications on the efficiency of such SFG–OPO configuration is modeled and simulated by solving the coupled equations in a normalized from which describe simultaneous SFG and OPO interactions inside two individual phase-matched nonlinear crystals. Numerical simulation is carried out using real practical values associated with experimental results that are recently reported by Devi et al. (Opt Express 21:24829–24836, 2013). We found very good agreement and consistent with the used experimental data, indicating the validity of our simulation. The influence of the beam waist of interacting beams on the conversion efficiency of resultant SFG–OPO radiation is investigated through introducing a magnifying factor that relates optical and physical characteristics of the designed SFG–OPO configuration. It is found that when this factor is taken equal to 3, the maximum of normalized SFG power reaches to ~0.0034 for an OPO loss of ~10 %. This is very consistent with the measured value of ~0.0030, confirming that our represented model is valid. The SFG–OPO arrangement is then optimized by extracting a mathematical formula that relates the optimum magnifying factor to the gain parameters of OPO and SFG interactions.

Appendix 1

In the derivation of coupled Eqs. (3) and (4) following assumptions have been made:

• The interacting fields are simplified by slowly varying amplitude approximation,

• Diffraction of propagating waves is neglected, and

• The variation of beam waist through propagation along the nonlinear crystal is assumed to be negligible.

Subsequently, a simplified form of the coupled equations for establishing the OPO radiation can be written as (Moore and Koch 1993)

$$ \begin{aligned} \frac{{dE_{s} (r,z)}}{dz} = i\frac{{2d_{eff}^{opo} \omega_{s} }}{{cn_{s}^{opo} }}E_{p} (r,z)E_{i}^{*} (r,z)e^{{ - i\Delta k_{OPO} z}} \hfill \\ \frac{{dE_{i} (r,z)}}{dz} = i\frac{{2d_{eff}^{opo} \omega_{i} }}{{cn_{i}^{opo} }}E_{p} (r,z)E_{s}^{*} (r,z)e^{{ - i\Delta k_{OPO} z}} \hfill \\ \frac{{dE_{p} (r,z)}}{dz} = i\frac{{2d_{eff}^{opo} \omega_{p} }}{{cn_{p}^{opo} }}E_{s} (r,z)E_{i} (r,z)e^{{i\Delta k_{OPO} z}} . \hfill \\ \end{aligned} $$

(8)

Following the above assumptions, the same derivation can be carried out for the intracavity SFG as

$$ \begin{aligned} \frac{{dE_{p} (r,z)}}{dz} = i\frac{{2d_{eff}^{SFG} \omega_{p} }}{{cn_{p}^{SFG} }}E_{s}^{*} (r,z)E_{SF} (r,z)e^{{ - i\Delta k_{SFG} z}} \hfill \\ \frac{{dE_{s} (r,z)}}{dz} = i\frac{{2d_{eff}^{SFG} \omega_{s} }}{{cn_{s}^{SFG} }}E_{p}^{*} (r,z)E_{SF} (r,z)e^{{ - i\Delta k_{SFG} z}} \hfill \\ \frac{{dE_{SF} (r,z)}}{dz} = i\frac{{2d_{eff}^{SFG} \omega_{SF} }}{{cn_{SF}^{SFG} }}E_{p} (r,z)E_{s} (r,z)e^{{i\Delta k_{SFG} z}} . \hfill \\ \end{aligned} $$

(9)

In order to consider the effect of resonator on the circulating pump and signal waves, it is further assumed that the spot sizes $w_{p,OPO}$ and $w_{s,OPO}$ of respectively pump and signal and beams inside the OPO crystal are typically characterized by Gaussian profile as

$$ \begin{aligned} E_{p} (r,z) = E_{p} (z)e^{{ - \left( {\frac{{x^{2} + y^{2} }}{{w_{p,OPO}^{2} }}} \right)}} \hfill \\ E_{s} (r,z) = E_{s} (z)e^{{ - \left( {\frac{{x^{2} + y^{2} }}{{w_{s,OPO}^{2} }}} \right)}} . \hfill \\ \end{aligned} $$

(10)

The same description is also applied to the pump and signal spot sizes $w_{p,SFG}$ and $w_{s,SFG}$ inside the SFG crystal which are

$$ \begin{aligned} E_{p} (r,z) = E_{p} (z)e^{{ - \left(\frac{{x^{2} + y^{2} }}{{w_{p,SFG}^{2} }}\right)}} \hfill \\ E_{s} (r,z) = E_{s} (z)e^{{ - \left(\frac{{x^{2} + y^{2} }}{{w_{s,SFG}^{2} }}\right)}} . \hfill \\ \end{aligned} $$

(11)

Therefore, by substituting Eqs. (10) and (11) into Eq. (8) and (9) the coupled Eqs. (3) and (4) can be easily derived. Moreover, the advantage of using the described coupled equations is the study of the profiles of idler and SFG beams when emerging from M₂ and M₄ mirors, respectively, in the single pass scheme shown in Fig. 1.