## Abstract

A theoretical threshold model based on the spherical wave assumption for a pulsed double-pass pumped singly resonant confocal positive-branch unstable optical parametric oscillator (OPO) has been proposed. It is demonstrated that this model is also applicable to the plane-parallel resonator in the special case. The OPO threshold as a function of important parameters such as the cavity magnification factor, cavity physical length, crystal length, pump pulsewidth, output coupler reflectance and crystal position inside the resonator has been presented. Experimental data show the good agreement with the results obtained from the theoretical model.

©2004 Optical Society of America

## 1. Introduction

Optical parametric oscillators (OPOs) are very attractive solid-state sources for the generation of eye-safe wavelengths. Significant improvements in the performance of the OPOs have taken place in the past few years, largely owing to the development of high-quality nonlinear optical crystal and efficient pump laser sources [1–3]. One of the greatest challenges for OPOs in the long-range laser applications is to obtain good beam quality along with high energy, i.e. high-brightness OPOs are required.

Plano-parallel resonators are the most commonly used cavities for OPOs with simple construction and excellent efficiency. However, they typically produce large beam divergence and consequently low brightness. A confocal positive-branch unstable resonator is a wonderful optical solution to the conflicting demands of high beam quality and high energy in OPO [4]. The unstable resonator effectively filters out laser modes with high-spatial-frequency components by a combination of laser-mode magnification and feedback of only the lowest-order spatial modes. But the high-loss characteristic limits the unstable resonator to high-gain systems. Thanks to the advances in high-nonlinearity crystal growth technology, unstable resonator becomes the ideal cavity for high-energy OPO with large-volume mode and high beam quality. It has been theoretically [5, 6] and experimentally [7–11] demonstrated that the confocal positive-branch unstable resonators generate near-to-diffraction-limited signal beams.

The threshold of nanosecond OPO devices is an important parameter for many practical applications. Compared with plane-parallel resonator, although advantage in improving beam quality has indeed been outstanding, threshold of unstable resonator remains rather high [10]. This is mainly due to geometrical loss caused by mode expansion in the unstable resonator. The theoretical threshold model given by Brosnan and Byer (for short as BB model) is based on the plane-wave assumption and applicable to a singly resonant plane-parallel resonator OPO [12]. Although the threshold analysis in Ref. [13] introduces a loss coefficient, the plane-wave basis is also assumed there. It is demonstrated that there is only one unique pair of conjugate points for a given unstable resonator and wave traveling in any direction inside the resonator is considered as a spherical wave coming from the corresponding conjugate point [4]. Therefore the threshold theoretical model based on plane-wave assumption is no longer valid for the unstable OPO. To date there has been no detailed theoretical threshold study for a confocal unstable OPOs as far as we know.

In this paper, Section 2 details the derivation of our threshold model based on the spherical wave assumption for a confocal positive-branch unstable resonator OPO. Our model gives the threshold pump peak intensity and threshold pump energy as a function of important parameters such as the cavity magnification factor *M*, cavity physical length *L*, full width half maximum (FWHM) of the input pump pulse intensity *T*, crystal length *l*_{c}
, output mirror reflectance to signal *R*, pump beam radius *r*_{p}
and separation between crystal to output mirror *L*
_{2}. The results obtained from our model are presented and compared with those of BB model and experimental values in Section 3.

## 2. The theoretical model

In this section a model is proposed to describe the threshold pump intensity and threshold pump energy of a pulsed double-pass pumped confocal unstable singly resonant oscillator (SRO).

The confocal positive-branch unstable OPO, formed by concave mirror M_{1} and convex mirror M_{2}, is schematically in Fig. 1, where signal field *E*_{s}
is assumed to be resonant. The pump is collimatedly coupled in through M_{1}, and the signal is partially coupled out through M_{2}. *R*_{1}
and *R*_{2}
are curvature radii of input mirror and output mirror, respectively (*R*_{2}
is with a minus sign for the convex mirror case). The cavity physical length is set by *L*=(*R*
_{1}+*R*
_{2})/2+*l*_{c}
(1-1/*n*_{s}
) to satisfy confocal condition, where s n is the refractive index to signal of the nonlinear crystal. Owing to the geometrical magnification of the unstable resonator, the signal beam is expanded during one round trip by the cavity magnification factor *M*=-*R*
_{1}/*R*
_{2}. As shown in Fig. 1, *L*
_{1} and *L*
_{2} describe separation between input mirror center to front surface of nonlinear crystal and between back surface of nonlinear crystal to output mirror center, respectively.

The property of confocal positive-branch unstable resonator determines that one of the conjugate points is positioned at infinity and the other one is confocal point *F*. Therefore forward wave in the resonator is considered as plane wave and backward wave is regarded as spherical wave from the confocal point. A primary advantage of this configuration is that it automatically produces a collimated output beam.

#### 2.1 SRO single pass gain solution based on spherical wave

In our model, the wave field profile is assumed to be circular symmetrical around z-axis. The spherical wave field traveling in the unstable resonator with the wave-vector *k*⃗ which has two components: longitudinal component *k*_{z}
and transverse component *k*_{r}
, is defined by

where

is a Fourier component of the optical electric field of pump, signal or idler. Here *r* and *z* are not independent variables, which have the relation of *r*=*r*
_{0}+*θ*_{j}*z. r*
_{0} is the corresponding transverse coordinate at the entrance to the crystal, i.e., *z*=0. *θ*_{j}
=2*α/n*_{j}
shown in the Fig. 1 is the refractive angle in the crystal for the incidence 2*α* and *n*_{j}
is the refractive index of the nonlinear crystal at angular frequency *ω*_{j}* . M*′_{j}(*z*) describes the magnification factor to beam diameter, and is defined as the ratio of distance between considered point (*r,z*) and confocal point to distance between the corresponding incident point (*r*
_{0},0) and confocal point, or equivalently *M*′_{j}(*z*)=*r/r*
_{0}.

From Maxwell’s equations [14], we can get

Substitute Eq. (1) into Eq. (3) for each frequency component, in the slowly varying amplitude approximation and no consideration of diffraction and walk-off which can be ignored in the case of the noncritically phase-matched operation, the equations describing the nonlinear optical parametric oscillation are given by

where

is the absorption coefficient and the interaction coefficient is

where *d*_{eff}
is the effective nonlinear coefficient of the crystal.

Here the wave-vector mismatch Δ*k*⃗ contains two components: longitudinal component Δ*k*_{z}
=*k*_{pz}
-*k*_{sz}
-*k*_{iz}
and transverse component Δ*k*_{r}
=*k*_{pr}
-*k*_{sr}
-*k*_{ir}
. Due to ignorable small difference, *θ*′*s* of pump, signal and idler waves for the same incident can be considered equal. Therefore the factor ${e}^{i\left(\Delta {k}_{r}r+\Delta {k}_{z}z\right)}$ describing the influence of wave-vector mismatch in Eq. (4) is rewritten as ${e}^{i\left(\Delta {k}_{r}{r}_{0}\right)}{e}^{i\Delta k\prime z}$, where Δ*k*′=(1+*θ*
^{2})Δ*k*_{z}
is defined as equivalent longitudinal wave-vector mismatch.

In the assumption of no pump depletion which is reasonable near threshold, *A*_{p}
(*r*
_{0},*z*) maintains constant along *z*-direction, it can be written as *A*_{p}
(*r*
_{0}). Substitute Eq. (2) into Eq. (4), yields

where *M*′
_{j}
(*z*)/*M*′_{p}(*z*)*M*′_{k}(*z*)≈1, *j,k*=*s,i* is employed.

We assume the absorption coefficients of signal and idler waves are equal, or *α*_{s}
=*α*_{i}
≡*α*. Satisfying the boundary condition of zero idler field at the entrance to the crystal, the solution to Eq. (7) is

where

The input pump intensity is temporal-spatial Gaussian defined by ${I}_{p}(r,t)={I}_{p0}\xb7{e}^{-{(t\u2044{\tau}_{p})}^{2}}\xb7{e}^{-{(r\u2044{r}_{p})}^{2}}$, where *τ*_{p}
is the 1/e intensity halfwidth of the pump pulse. For a pulsed OPO operation where pump duration is much longer than one round-trip time of the cavity, or *τ*_{p}
≫2[*L*+(*n*_{p}
-1)*l*_{c}
]/*c*, the pump intensity can be assumed to be constant during a single cavity transit.

The phase mismatch Δ*k*′ in Eq. (8) can be ignored for the non-critical phase matching case. That is reasonable because 1) for the noncritically phase-matched operation, the acceptance of the crystal is generally rather large. As shown in Fig.1, the refractive angle *θ* in the crystal increases with the incident position *r*
_{0} and its maximum value is generally no more than several tens milliradian and commonly within the acceptance of the crystal. 2) The phase mismatch Δ*k*′ can be written as a function of *θ*
^{2} in the case of the noncritical phase matching. Therefore it is concluded that the phase mismatch increases with the incident transverse position *r*
_{0}. Generally the threshold pump intensity is high enough to make ${g}_{0}\left({r}_{0}\right)=\sqrt{{N}_{s}{N}_{i}{\mid {A}_{p}\left({r}_{0}\right)\mid}^{2}}\gg \Delta k\prime \u20442$ when a small *r*
_{0} is considered. The phase mismatch influences the edge part of a spatially Gaussian gain determined by the Gauss-profile pump more strongly than the central part. However signal field is generally assumed Gaussian. So the contribution to total signal power from the edge part is far less than power from the central part. Therefore the assumption of no phase mismatch is generally reasonable for the noncritical phase-matched case.

Taking Eq. (2) into Eq. (8) and ignore phase mismatch, the signal field at the end of the crystal is

where

is a time dependent gain coefficient.

Note that although Eq. (10) is derived from the nonlinear parametric interaction equations based on spherical wave, it is also applicable to the case of plane wave when *M*′(*z*)=1 and *θ*=0.

#### 2.2 One round-trip in the resonator

As discussed above, the forward waves traveling in a confocal unstable resonator are the plane waves propagating along z-direction. Let ${M}_{s}^{f}$
(*z*)≡1 and *θ*=0 in Eq. (10) and assume that perfect phase matching is satisfied, the signal field at the end of the nonlinear crystal on the forward transit is

where *α*_{f}
is the forward field absorption coefficient. The forward parametric gain coefficient is

where

is the forward interaction coefficients

Here ${E}_{s}^{f}$
(*r*,0) describe the signal field at the entrance to the crystal on the forward transit. The signal wave grows from an initial Gaussian noise. For *m* th cavity transit, suppose the initial signal field before the input mirror is ${E}_{\mathit{start}}\left(r\right)={E}_{\mathit{start}0}{e}^{-{(r\u2044\sqrt{2}{r}_{s})}^{2}}$. For the case of plane wave and no air depletion, we know ${E}_{s}^{f}$
(*r*,0)=*E*_{start}
(*r*).

After the reflection of output mirror, the signal field before the output mirror is

and we let *γ*
_{0} be the ratio of backward to forward pump field, the pump field before the output mirror at the start of the backward transit is

According to characteristic of spherical wave, the wave field at the entrance of the crystal on the backward transit is

where *M*
_{1}=1+2*L*
_{2}/|*R*
_{2}| is defined as the magnification factor to wave traveling from output mirror to the back surface of nonlinear crystal. If we let wave field in the crystal on the backward transit be ${E}_{j}^{b}$
(*r*(*z*),*z*)=${A}_{j}^{b}$
(*r*
_{0},*z*)/*M*
_{2j}(*z*), where

describes the magnification factor to wave traveling from the back surface of nonlinear crystal to the arbitrary z-plane in the crystal.

From Eq. (10), in the assumption of no wave-vector mismatch, or Δ*k*′=0, the signal field at the end of the crystal on the backward transit is

where *α*_{b}
is the backward field absorption coefficient. The backward parametric gain coefficient is

where

is the backward interaction coefficients

The property of spherical wave determines the signal field propagating to the plane before the input mirror is

where *M*
_{3s}=1+2*n*_{s}
_{L}
1/(*M*
_{1}
*n*_{s}
|*R*
_{2}|+2*l*_{c}
) is defined as the magnification factor to wave traveling from the front surface of the nonlinear crystal to input mirror on the backward transit.

According to Eqs. (12), (15), (17) and (19), the signal field after one round trip in the confocal unstable resonator is given by

where *M*=*M*
_{3s}
*M*
_{2s} (*l*_{c}
) *M*
_{1} is the cavity geometrical magnification factor and has been verified to be equal to -*R*
_{1}/*R*
_{2}.

If we let *γ*=*γ*
_{0}/*M*
_{1} and assume *θ*_{s}
=*θ*_{i}
=2*r/Mn*_{s}
|*R*
_{2}|, the backward parametric gain coefficient can be written as

where approximation ${\left(\sqrt{1+{\theta}^{2}}\right)}^{-1}\approx 1-{\theta}^{2}\u20442$ has been used.

After one round trip in the unstable resonator, the signal spot size *r*_{s}
is narrowed by the Gauss-profile parametric gain and broadened by the geometrical magnification of unstable resonator and diffraction. Here we simply assume the balance of three influence factors is achieved. For *m* th transit, the initial signal power is

and the signal power after one round trip using Eq. (23) is

$$=\frac{R}{{M}^{2}}{e}^{-2\left({\alpha}_{f}+{\alpha}_{b}\right){l}_{c}}\left(\frac{2\pi}{2}{n}_{s}c{\epsilon}_{0}\right)\xb7{\mid {E}_{\mathit{start}0}\mid}^{2}\xb7\underset{0}{\overset{\infty}{\int}}{e}^{-{\left(\frac{r}{{\mathit{Mr}}_{s}}\right)}^{2}}{\mathrm{cosh}}^{2}\left({\beta}_{f}{l}_{c}{e}^{-{\left(\frac{r}{\sqrt{2}{\mathit{Mr}}_{p}}\right)}^{2}}\right){\mathrm{cosh}}^{2}\left({\beta}_{b}{l}_{c}{e}^{-{\left(\frac{r}{\sqrt{2}{\mathit{Mr}}_{p}}\right)}^{2}}\right)\mathit{rdr}$$

The signal power using Eq. (24) and the relation cosh^{2}
*φ*≈*e*
^{2φ}/4+1/2 is

where

Equation (27) can be easily iterated numerically to compute threshold, incrementing pump peak intensity until a defined threshold is reached. We adopt the threshold definition in Ref. [12], where threshold has been defined as a signal energy of 100µJ, giving a threshold power to noise power ratio of ln(*P*_{m}*/P*
_{0})=33. Note that Eq. (27) is also suitable to the case of plane-parallel resonator when *M*=1 and *R*
_{2}=-∞ are taken.

The threshold energy is calculated by integration over the temporal and spatial pump intensity profile and given by

In our threshold model, there are no mirrors acting on the idler field. If there were some weak idler feedback, the backward gain would be affected by this but it would be too small to consider after that.

## 3. Results and discussion

#### 3.1 Theoretical calculations

In this section, the threshold model results are compared to BB model and experimental measurements for a 1.064µm pumped KTP OPO. We investigate the threshold is determined as a function of cavity magnification factor, cavity physical length, crystal length, output coupler reflectance to signal, pump pulsewidth and crystal position in the unstable resonator.

The crystal considered is KTP, where the type-II non-critical phase matching is satisfied at 1.57µm signal and 3.3µm idler wavelengths when a 1.064µm pump wavelength was used. We take the effective nonlinear coefficient to be 2.9 pmV^{-1} and the refractive indices to signal, idler and pump wave are 1.7348, 1.7793 and 1.7474, respectively [15].

Figure 2 shows the threshold energy against cavity physical length with *R*=0.82, *l*_{c}
=20 *mm*, a 4mm-diameter, 13.5ns-duration (FWHM) pump beam and different cavity magnification factors. The overall trend of the threshold energy is increasing with cavity physical length and cavity magnification factor. In a confocal unstable resonator, the divergence of the signal beam is reduced by 1/*M* in each round trip. Therefore larger cavity magnification is preferable concerning high beam quality. However that means larger geometrical loss and consequently higher threshold. In the practical application, a proper cavity magnification factor should be chosen to balance the conflicting requirement between high beam quality and low threshold. Our theoretical model is to be expected to be helpful to design of a confocal unstable OPO. The threshold energy of *M*=1 corresponding to the case of plane-parallel resonator is also plotted in Fig. 2 compared with the BB model. Our model result agrees well with the BB model, with the differences between them most likely due to the fact that we use a Gaussian pump, whereas a square pulse shape is used in BB model and difference from approximation and mathematical deal.

The threshold energy versus KTP crystal length is shown in Fig. 3. The threshold decreases with crystal length. The unstable resonator with large cavity magnification factor possesses high threshold, but the threshold differences among the unstable resonators with various cavity magnification factors are decreased with crystal length. When a KTP crystal with length long than 40mm is used, the threshold difference between *M*=1.5 and *M*=1 is within 10%. Therefore there is no need for longer crystal concerning the reduction of threshold. Our threshold result of *M*=1 is very close to the BB model.

Figure 4 shows the unstable threshold energy against output mirror reflectance to signal with *L*=60 mm, *l*_{c}
=20 mm, 2*r*_{p}
=4 mm, *T*=13.5 ns. We also plot threshold energy versus the pump FWHM pulsewidth for *L*=60 mm, *l*_{c}
=20 mm, 2*r*_{p}
=4 mm and *R*=0.82 in Fig. 5. The threshold energy decreases with signal reflectance and increases with pump pulsewidth. The signal reflectance of the output mirror must therefore be relatively large to maintain a reasonable threshold.

In Fig. 2~5, the nonlinear crystal is placed in the middle of the resonator, i.e., *L*
_{1}=*L*
_{2}=(*L*-*l*_{c}
)/2. We present the threshold energy dependent of crystal position in the resonator in Fig. 6. For an unstable resonator with the given cavity physical length, crystal length and cavity magnification, the threshold increases with separation *L*_{2}
between crystal and output mirror. As shown in Fig. 6, a threshold increment of close to 20% occurs when the ratio of *L*_{2}
to (*L*-*l*_{c}
) changes from 0 to 1 for a confocal unstable resonator with *M*=1.50, *L*=60 mm and *l*_{c}
=20 mm. This is because that smaller magnification factor *M*_{1}
due to smaller distance *L*_{2}
leads to larger backward parametric gain, then lower threshold. This threshold change is also obvious for a short-cavity OPO. This conclusion shows that the nonlinear crystal is better to be placed close to output mirror in the unstable resonator application in order to reduce threshold. No threshold variation with *L*_{2}
is found in the plane-parallel resonator and our calculations are consistent with the results of BB model.

#### 3.2 Experimental results

BB model is supported by abundant experimental data [12]. As shown above, accurateness of our model is strengthened by the comparison with the BB model. In this section, the threshold energy versus cavity length of the plane-parallel resonator is presented to confirm the agreement of our model with experimental results. We also investigate the threshold property varying with cavity magnification factor, cavity physical length and crystal position in the unstable resonator experimentally. The KTP crystal available for these experiments was 20mm long and had a 5mm×5mm cross section. Both surfaces of the crystal were antireflection coated at the pump and signal wavelength. The pump laser was Nd:YAG laser, which generated 13.5-ns pulsewidth pulses with tunable energy from several millijoules to ~150 mJ. The input mirrors M_{1} for all resonators were coated for 98% transmission at the pump wavelength of 1.064µm, 95% transmission at the idler wavelength of 3.3µm, and 99.8% reflectance at the signal of 1.57µm. The output mirrors M_{2} for all resonators were highly reflecting at the 1.064µm pump wavelength, 82% reflecting at the 1.57µm signal wavelength, and 99% transmission at the 3.3µm idler wavelength. Energy was measured with EPM2000 two-channel joulemeter/power meter and J50HR energy probe (Molectron, Inc.). For every case the threshold value is gained through taking an average of 20 measurements.

As shown in Fig. 7, the threshold energy increases with cavity physical length of a plane-parallel resonator. Experimental results are closely consistent with our model and BB model.

Three different magnifications factors are investigated in our experiments: *M*=1.33 with *R*_{1}
=380mm and *R*_{2}
=-357mm, *M*=1.19 with *R*_{1}
=303mm and *R*_{2}
=-254mm, and *M*=1.06 with *R*_{1}
=219mm and *R*_{2}
=-163mm. In three cases above, the cavity physical length was set to satisfy the confocal condition, and the nonlinear crystal is placed in the middle of the resonator. The experimental threshold properties of three cases are shown in Table 1 compared with the predicted ones by our model. We select the case with *M*=1.33 and *L*=36.0 mm to observe the threshold energy varying with the crystal position inside the unstable resonator, and the experimental data are shown in Fig. 8 compared with our model results. The data presented in Table 1 and Fig. 8 both show good agreement between experiment values and predicted ones by our theoretical model. The theoretical results are generally lower than the corresponding experimental ones. This is explained by no consideration of the diffraction and phase mismatch in our threshold model.

## 4. Conclusions

We have proposed a theoretical model for a double-pass pumped singly resonant confocal unstable OPO which relies on the spherical wave theory. The model includes the influence of the cavity magnification factor, cavity physical length, pump pulsewidth, output coupling and crystal position in the resonator on the threshold of unstable resonator. In the special case this model can be applied to calculate the threshold of the plane-parallel resonator, which has been verified through the comparison with the established BB model. The threshold of a confocal unstable resonator increases with the cavity magnification factor, cavity physical length and pump pulsewidth, and decreases with signal reflectance of output mirror and crystal length. We also conclude that the nonlinear crystal is placed close to the output mirror benefits to reduce the threshold. Experimental results show the excellent agreement with theoretical calculations.

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