## Abstract

An iterative threshold model for a pulsed singly resonant Gaussian-reflectivity-mirror (GRM) confocal unstable optical parametric oscillator (OPO) has been proposed. It is found that OPO threshold is determined by important parameters such as GRM central reflectance, Gaussian reflectivity profile, cavity magnification factor, cavity physical length, crystal length, pump pulsewidth. It is demonstrated that this model can be extended to plane-parallel resonator or uniform-reflectivity-mirror (URM) unstable resonator when some specific values are taken. Experimental results show excellent agreement with values calculated from theoretical model. Both theoretical calculations and experimental data illustrate that GRM is a useful solution to reduce threshold of unstable OPO.

©2005 Optical Society of America

## 1. Introduction

Optical parametric oscillators (OPOs) are frequently used for the generation of eyesafe wavelength radiation that is of particular interest in many lidar systems. Many long-distance measurement applications require further laser radiation with high pulse energy and good beam quality. Confocal unstable resonators have been shown to be useful to improve beam quality of laser radiation compared with the conventional plane-parallel resonators [1]. The imaging properties of the unstable resonator increase the propagation angles of the laser, effectively filtering out the high-spatial-frequency components and resulting in a good-beam-quality output laser.

By the radial profile of output mirror reflectivity to resonated wavelength, unstable resonators can be classified into uniform-reflectivity-mirror (URM) ones and nonuniform-reflectivity-mirror (NURM) ones. Owing to simple structure, URM unstable resonators abstract more interest of researchers. It has been demonstrated theoretically [2, 3] and experimentally [4–7] that a confocal URM unstable resonator is useful to produce output beams with good beam quality and desirable energy in the OPOs. The threshold of OPO devices is of paramount importance in most practical applications. However due to the high geometric loss, the URM unstable resonator usually possesses high threshold that impedes the unstable OPO’s development although its advantage in improving beam quality has indeed been outstanding.

Gaussian reflectivity mirror (GRM) that is most frequently used among NURMs, is at first introduced to reduce the detrimental effects of edge diffraction in unstable resonators, enhance transverse mode discrimination and improve the output beam focusing properties [8]. GRM can change the spatial distribution of the loss to make the higher order modes possess larger losses that will inhibit the development of these modes in the resonated field and result in a near-diffraction-limited beam. An unstable resonator employing an output coupler with a Gaussian reflectivity profile may produce single transverse mode of large volume with excellent beam quality [9–11]. The GRM unstable OPO has been demonstrated to improve the output signal brightness as compared to the plane-parallel resonator [12].

In our study, the GRM unstable OPO is observed to perform lower threshold in contrast with the corresponding URM unstable resonator with equal effective output coupling. Comparatively the GRM unstable OPO realizes better mode matching with Gaussian spatial profile pump and changes the mode distribution, thereby resulting in an effective energy extraction from pump. Generally, oscillation begins first at the center of resonated field. Under the condition of equal effective output coupling, the output mirror of a GRM unstable resonator possesses higher central reflectance than that of the corresponding URM unstable resonator. Therefore the oscillation turns on more easily in GRM unstable OPO than in corresponding URM one, consequently producing a low threshold in GRM unstable OPO. Detailed discussion and experimental demonstration will be presented in Section 3.

Theoretical threshold modeling of OPOs has far lagged behind the experimental advances and numerical simulations. To the best of our knowledge there is no detailed theoretical threshold study for a GRM unstable OPO. The classical threshold model of OPOs described in Ref.13 is limited to the plane-parallel resonators and not appropriate to the URM or GRM unstable resonators. Previously, we reported a theoretical threshold model that has been experimentally demonstrated to exactly describe threshold property of a URM confocal unstable resonator [14]. In this paper, a threshold theoretical model, incorporating the coupled wave equations based on spherical wave assumption given in Ref. [14], is proposed to investigate the threshold of a GRM confocal unstable non-critically phase matched or quasiphase matched OPO as a function of important parameters such as the central signal reflectivity of output coupler *R*_{max}
, 1/e radius of Gaussian reflectivity profile *w*, magnification factor *M*, cavity physical length *L*, crystal length *l*_{c}
, full width half maximum (FWHM) of the input pump pulse intensity *T*, and effective gain aperture *r*_{max}
.

The organization of this paper is as follows. In Section 2 the theoretical threshold model for a GRM confocal unstable OPO is proposed. The theoretical results are presented and compared with the experimental data in Section 3.

## 2. Theoretical threshold model

We consider a GRM confocal positive-branch unstable OPO, as schematically shown in Fig. 1, which consists of two spherical mirrors separated by a distance *L*=(*R*
_{1}+*R*
_{2})/2+*l*_{c}
(1-1/*n*_{s}
) to satisfy confocal condition, where *R*_{1}
and *R*_{2}
are curvature radii of input mirror M_{1} and output mirror M_{2} respectively (*R*_{2}
is with a minus sign for the convex mirror case), and *n*_{s}
is the refractive index of the nonlinear crystal to signal. The signal is assumed to be the singly resonated field. The reflectance profile *R*(*r*) of output coupler at the signal wavelength has a Gaussian dependence on the radial variable *r*,

where *R*
_{max} is the signal reflectance at the output mirror center and *w* is the radius at which the reflectance is reduced to 1/e of central value.

The collimated pump is coupled in through M_{1}, and the signal is partially coupled out through M_{2}. Because of 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 separations 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 forward wave in the resonator is considered as plane wave and backward wave is regarded as spherical wave from the confocal point *F*.

This model is appropriate for non-critically phase matched or quasi-phase matched interactions for which there is no walk-off. All fields are assumed to be radially symmetric around z-axis. In the slowly varying amplitude approximation and no consideration of diffraction, the coupled wave equations derived from Maxwell’s equations based on the spherical wave [14], assuming no pump depletion that is valid near threshold, are given by

$$\frac{\partial {E}_{i}(r,z)}{\partial z}+{\alpha}_{i}{E}_{i}(r,z)=i{N}_{i}{E}_{p}(r,z){E}_{s}^{*}(r,z){e}^{i\Delta {k}_{r}r}{e}^{i\Delta {k}_{z}z}$$

where *α*’*s* are the field absorption coefficients and the interaction coefficient is given by

where *d*_{eff}
is the effective nonlinear coefficient of the crystal and *n*_{j}
is the refractive index of the nonlinear crystal at angular frequency *ω*_{j}
. The wavevector mismatch Δ$\overrightarrow{k}$
consists of transverse component Δ*k*
_{r}
and longitudinal component Δ*k*_{z}
. *θ*_{j}
=2*φ*/*n*_{j}
is the refractive angle inside the crystal for the incidence 2*φ* at angular frequency *ω*_{j}
. 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. Due to ignorable small difference, *θ*’*s* of pump, signal and idler waves for the same incidence can be considered equal, written as *θ*, as shown in Fig. 1.

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.

For simplicity, we assume perfect phase matching and *α*_{s}
=*α i*≡

*α*. With Eq. (2), taking zero idler field at the entrance to the crystal, the signal field at the end of a crystal is given by [14]

where *E*_{s}
(*r*,0) is the signal field at the entrance to crystal and

is a time-dependent parametric gain coefficient. ${M}_{s}^{\prime}$
(*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 ${M}_{s}^{\prime}$
(*z*)=*r*/*r*
_{0}. Eq. (4) is still valid for the specific case of the plane wave when *M*
^{′}(*l*_{c}
)=1 and *θ*=0 are taken.

According to the reflectance of output coupler to pump, the GRM unstable OPOs fall into two categories: single-pass pumped one corresponding to the case that output coupler is highly transmitting at the pump wavelength, and double-pass pumped one corresponding to the case that output coupler is highly reflective at the pump wavelength. The iterative threshold models appropriate for single-pass and double-pass pumped cases will be presented in following two subsections, respectively.

## 2.1 Single-pass pumped case

When the output mirror is highly transmitting at the pump wavelength, there is no pump radiation on the backward transit. Thus there is only forward gain and no backward gain for the signal field.

The signal is supposed to grow from a level of background Gaussian noise. For the *m*th transit, the initial signal field is described by ${E}_{\mathit{start}}\left(r\right)={E}_{\mathit{start}0}{e}^{-{(r\u2044\sqrt{2}{r}_{s})}^{2}}$. So the initial signal power is given by

As discussed above, the forward waves traveling in the resonator are the plane waves propagating along z-direction. Let ${M}_{s}^{\prime}$
(*l*_{c}
)=1 and *θ*=0 in Eq. (4), the signal field at the end of the nonlinear crystal on the forward transit is

where the forward parametric gain coefficient *β*_{f}
is defined by

are the forward interaction coefficients.

Here we call *r*
_{max} effective gain aperture defined as the maximum radius within which resonated field can obtain parametric gain. Generally *r*
_{max} is taken as the maximum radial size of the nonlinear crystal. In our model, the signal field portion outside effective gain aperture is considered as zero because that firstly this portion of signal field can not obtain parametric gain, then will gradually deplete as traveling in the cavity due to cavity loss until to zero, so has few contribution to signal power. Secondly, oscillation usually begins at the centre of signal field. The signal field outside effective gain aperture has negligibly weak effect on threshold. Therefore this portion isn’t within consideration in our threshold model.

After reflecting by the output mirror with Gaussian reflectivity profile, the signal field before the output mirror is given by

Here the additional phase factor introduced by refection of spherical mirror has been neglected because there is no consideration of diffraction in our model, and additional phase factor has no any effect on signal power, thereby has no effect on threshold.

Due to highly transmission of the output mirror at the pump wavelength, no signal gain is obtained on the backward transit in the crystal. Hence after one round trip inside the resonator, the signal field becomes

$$={E}_{\mathit{start}0}\frac{\sqrt{{R}_{max}}{e}^{-2\alpha {l}_{c}}}{M}\mathrm{cosh}\left({\beta}_{f}{l}_{c}{e}^{-{\left(\frac{r}{\sqrt{2}{\mathit{Mr}}_{p}}\right)}^{2}}\right)\xb7{e}^{-{\left(\frac{r}{\sqrt{2}{\mathit{Mr}}_{s}}\right)}^{2}}\xb7{e}^{-{\left(\frac{r}{\sqrt{2}\mathit{Mw}}\right)}^{2}},r:0\sim {\mathit{Mr}}_{max}$$

where *M* is the magnification factor of the confocal unstable resonator.

After one round trip in the unstable resonator, the signal spot size *r*_{s}
is narrowed by the Gauss-profile parametric gain and Gauss-profile reflectivity of output mirror, and broadened by the geometrical magnification of unstable resonator and diffraction. There is no strong parametric gain near threshold, so here we simply assume the proper balance of the above four influence factors is achieved. Using Eq. (11), the signal power after the *m*th round trip is given by

$$=\frac{{R}_{max}}{{M}^{2}}{e}^{-4\alpha {l}_{c}}\left(\frac{1}{2}{n}_{s}c{\epsilon}_{0}\right)\xb72\pi \xb7{\mid {E}_{\mathit{start}0}\mid}^{2}\xb7{\int}_{0}^{{\mathit{Mr}}_{max}}{e}^{-{\left(\frac{r}{{\mathit{Mr}}_{s}}\right)}^{2}}{e}^{-{\left(\frac{r}{\mathit{Mw}}\right)}^{2}}{\mathrm{cosh}}^{2}\left({\beta}_{f}{l}_{c}{e}^{-{\left(\frac{r}{\sqrt{2}{\mathit{Mr}}_{p}}\right)}^{2}}\right)r\mathit{dr}$$

Due to high reflectivity of the input mirror to signal, the initial signal power before the input mirror of the mth transit is equal to the signal power after the (*m*-*1*)th transit, that is ${P}_{{m}_{0}}={P}_{m-1}$. Using Eq. (6), making the exponential approximation that cosh^{2}
*φ*≈*e*
^{2φ}/4+1/2 for Eq. (12), we get

where

Eq. (13) can be easily iterated numerically to compute threshold peak intensity for single-pass pumped OPO, incrementing pump peak intensity until a defined threshold is reached. Following the definition used by Byer and Brosnan [13], threshold of pulsed OPO in this study is defined as a signal energy of 100µJ, giving a threshold power to noise power ratio of 0 ln(*P*_{m}
/*P*
_{0})=33.

## 2.2 Double-pass pumped case

A reduction in pump threshold energy and some enhancement in OPO energy efficiency may be obtained by back reflecting the pump radiation to double pass the OPO, thus creating signal gain on both the forward and backward transits of the crystal [13]. The forward transit in the double-pass pumped case is entirely same as that in the single-pass pumped case. In this subsection we will detailedly discuss backward transit in the double-pass pumped case.

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

where ${E}_{\mathit{\text{sr}}}^{b}$
(r) is still described by Eq. (10) and *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 *γ*
_{0} be the ratio of backward to forward pump field amplitude inside the crystal, then the backward parametric gain coefficient is given by

where

are the backward interaction coefficients.

Due to the divergent property of the spherical mirror, not all-profile signal field can obtain backward parametric gain. Hence the signal field at the end of the crystal on the backward transit is piecewise described by

where ${M}_{2}=1+\frac{2}{{n}_{s}\left(\mid {R}_{2}\mid +2{L}_{2}\right)}{l}_{c}$ describes the magnification factor to signal wave traveling from the entrance to the crystal to the end of the crystal.

Due to the magnification of unstable resonator, phase mismatch will be introduced on the backward transit in the cavity. However, 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. Moreover, phase mismatch increases as the incident position *r*
_{0}. For a Gauss-profile gain, the contribution from the edge portion of signal beam to output power is far less than that from the central portion. Therefore, the assumption of perfect phase matching is still reasonable on the backward transit for the noncritically phase-matched operation.

Using Eq. (15), (18), the property of spherical wave determines the signal field propagating to the input mirror on the backward transit is

$$=\{\begin{array}{cc}\frac{{E}_{\mathit{start}0}\sqrt{{R}_{max}}{e}^{-2\alpha {l}_{c}}}{M}{e}^{-{\left(\frac{r}{\sqrt{2}M}\right)}^{2}\left(\frac{1}{{w}^{2}}+\frac{1}{{r}_{s}^{2}}\right)}\mathrm{cosh}\left({\beta}_{f}{l}_{c}{e}^{-{\left(\frac{r}{\sqrt{2}{\mathit{Mr}}_{p}}\right)}^{2}}\right)\mathrm{cosh}\left({\beta}_{b}{l}_{c}{e}^{-{\left(\frac{r}{\sqrt{2}{\mathit{Mr}}_{p}}\right)}^{2}}\right)& r:0\sim {M}_{3}{r}_{max}\\ \frac{{E}_{\mathit{start}0}\sqrt{{R}_{max}}{e}^{-2\alpha {l}_{c}}}{M}{e}^{-{\left(\frac{r}{\sqrt{2}M}\right)}^{2}\left(\frac{1}{{w}^{2}}+\frac{1}{{r}_{s}^{2}}\right)}\mathrm{cosh}\left({\beta}_{f}{l}_{c}{e}^{-{\left(\frac{r}{\sqrt{2}{\mathit{Mr}}_{p}}\right)}^{2}}\right)& r:{M}_{3}{r}_{max}\sim {\mathit{Mr}}_{max}\end{array}$$

where *M*
_{3}=1+2*L*
_{1}/*M*
_{1}
*M*
_{2}|*R*
_{2}| describes the magnification factor to the signal wave traveling from the end of the crystal to the input mirror. It is easily proved that the cavity magnification factor *M* is equal to the product of *M*
_{1}, *M*
_{2} and *M*
_{3}, that is *M*=*M*
_{1}
*M*
_{2}
*M*
_{3}.

For the *m*th transit, the signal power after one round trip in the cavity for the case of double-pass pumped OPO by integrating *E*_{sround}
(*r*) over the spatial radial profile, is given by

$$=[{\int}_{0}^{{M}_{3}{r}_{max}}{e}^{-\frac{{r}^{2}}{{M}^{2}}\left(\frac{1}{{w}^{2}}+\frac{1}{{r}_{s}^{2}}\right)}{\mathrm{cosh}}^{2}\left({\beta}_{f}{l}_{c}{e}^{-{\left(\frac{r}{\sqrt{2}{\mathit{Mr}}_{p}}\right)}^{2}}\right){\mathrm{cosh}}^{2}\left({\beta}_{b}{l}_{c}{e}^{-{\left(\frac{r}{\sqrt{2}{\mathit{Mr}}_{p}}\right)}^{2}}\right)r\mathit{dr}$$

$$+{\int}_{{M}_{3}{r}_{max}}^{{\mathit{Mr}}_{max}}{e}^{-\frac{{r}^{2}}{{M}^{2}}\left(\frac{1}{{w}^{2}}+\frac{1}{{r}_{s}^{2}}\right)}{\mathrm{cosh}}^{2}\left({\beta}_{f}{l}_{c}{e}^{-{\left(\frac{r}{\sqrt{2}{\mathit{Mr}}_{p}}\right)}^{2}}\right)r\mathit{dr}]\xb7\left(\frac{1}{2}{n}_{s}c{\epsilon}_{0}\right)\xb72\pi \frac{{\mid {E}_{\mathit{start}0}\mid}^{2}{R}_{max}{e}^{-4\alpha {l}_{c}}}{{M}^{2}}$$

If we let *γ*=*γ*
_{0}/*M*
_{1}, the backward parametric gain coefficient can be written as

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

According to Eq. (6) and using the exponential approximation that cosh^{2}
*φ*≈*e*
^{2φ}/4+1/2, we get the iterative relation analogous to Eq. (13) for the double-pass pumped case that

$$\frac{{e}^{2{\beta}_{f}{l}_{c}}}{8}\frac{{r}_{3}^{2}}{{r}_{s}^{2}}\left(1+{e}^{-{\left(\frac{{M}_{3}{r}_{max}}{{\mathit{Mr}}_{3}}\right)}^{2}}-2{e}^{-{\left(\frac{{r}_{max}}{{r}_{3}}\right)}^{2}}\right)+\frac{1}{4}\frac{{r}_{4}^{2}}{{r}_{s}^{2}}\left(1+{e}^{-{\left(\frac{{M}_{3}{r}_{max}}{{\mathit{Mr}}_{4}}\right)}^{2}}-2{e}^{-{\left(\frac{{r}_{max}}{{r}_{4}}\right)}^{2}}\right)\left]\right\}$$

where

The threshold peak intensity for double-pass pumped GRM unstable OPO is easily obtained through iterating Eq. (22) numerically. Note that Eq. (13) and Eq. (22) are both suitable to the version of the GRM plane-parallel OPO when *M*=1, *R*
_{1}→∞ and *R*
_{2}→-∞ are taken. When *w*→∞, the reflectivity profile of output mirror trends to uniformity, then Eq. (13), (14), (22), (23) are simplified to the threshold models for confocal URM unstable OPOs just as described in Ref.14. Hence the URM unstable resonator can be regarded to be a specific version of GRM unstable resonator to some extent.

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

In our study, both input mirror and output mirror are assumed to be ideally fully transmitting at the idler wavelength. 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, we will investigate threshold of a double-pass pumped confocal GRM unstable OPO dependent of the central signal reflectivity of output coupler, 1/e radius of the Gaussian reflectivity profile, magnification factor, cavity physical length, crystal length and pump pulsewidth. The nonlinear crystal here considered is KTP, where the type-II noncritical phase matching is satisfied at 1.57µm signal and 3.3µm idler wavelengths when a 1.064µm pump wavelength is used. The maximum radial size of nonlinear crystal is taken as 2.5mm. 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]. For fundamental transverse mode operation, the effective reflectivity of GRM is given by [8]

Threshold results of the corresponding URM unstable OPOs with uniform output signal reflectance which is equal to the effective reflectivity of GRM ones will be plotted for comparison.

Figure 2 illustrates threshold energy against central signal reflectance of output coupler with *L*=54mm, *w*=1.45 mm, *l*_{c}
=20 mm, a 2.8-mm-diameter (1/e intensity width), 13.5-ns-duration (FWHM) pump beam under various cavity magnification factors. The GRM plane-parallel resonator is shown as *M*=1.00. Threshold decreases as the central signal reflectance increases and increases as cavity magnification factor increases. From threshold aspect, large central signal reflectance is preferable. Furthermore, the far-field brightness is maximized when *R*
_{max} is close to 1. Whereas when *R*
_{max} is greater than 1/*M*
^{2}, a central dip appears in the near-field distribution [8]. Therefore a proper central signal reflectance at the given magnification factor is needed.

From energy aspect, smaller magnification factor is preferable. However, it is shown theoretically [2, 3] and experimentally [4] that unstable resonator with large cavity magnification factor generally produces better beam quality. Hence choosing cavity magnification factor should consider both energy and beam quality. It is reported that unstable resonator with cavity magnification factor of *M*=1.20 generally exhibits good energy and beam quality [3, 7].

The threshold results of URM unstable OPOs with uniform output signal reflectance values equal to effective output coupling of corresponding GRM unstable OPOs for magnification factor of 1.20 are included for comparison. As dashed line shown in Fig. 2, threshold of URM unstable resonator decreases with output mirror reflectivity to signal and is obviously higher than the corresponding GRM unstable resonator (~30% higher in this case). The excellent mode matching and relatively high central signal reflectance of output coupler enable the GRM resonator to reach threshold earlier than the corresponding URM resonator. Therefore employing Gaussian reflectivity output coupler is an effective method to reduce threshold of unstable OPO.

The threshold energy dependence on the 1/e radius of the Gaussian signal reflectivity profile of output coupler under various cavity magnification factors is shown in Fig. 3. Threshold energy increases as the 1/e radius of Gaussian reflectivity increases. Using Eq. (13), (14), (22), (23), this can be explained by small 1/e radius of Gaussian reflectivity resulting in small signal beam, consequently large (*r*_{j}
/*r*_{s}
)^{2}, where *j*=1,2,3,4 for double-pass pumped case or *j*=1,2 for single-pass pumped case. Therefore, when a small 1/e radius of Gaussian reflectivity is used, a large round-trip parametric gain is obtained, so exhibiting a low threshold. However small-size signal beam usually possesses relatively high intensity, so easily damages the crystal or other optical elements inside the resonator. Moreover, requirement of small 1/e radius of Gaussian reflectivity will add realization difficulty of Gaussian reflectivity profile. Hence a proper 1/e radius of Gaussian reflectivity should be chosen with the considerations of threshold property, other cavity parameters and damage intensity of crystal or optical elements and coating technology.

Figure 4 shows the threshold energy against cavity physical length for the confocal GRM unstable OPO with *l*_{c}
=20 mm, 2*r*_{p}
=2.8 mm, *R*_{max}
=0.85, *w*=1.45 mm, *T*=13.5 ns. A threshold comparison between GRM and corresponding URM unstable resonator with equal effective output coupling under the same cavity magnification factor of 1.20 is also plotted in Fig. 4. The same trend of the threshold energy increasing with cavity physical length is observed both in URM and GRM unstable resonator. For a resonator with given pump, cavity magnification factor and nonlinear crystal, the parametric gain obtained on one transit of resonator is constant. Under the same pulsewidth, larger cavity physical length means longer round-trip time, resulting in less transit numbers, then less total parametric gain, consequently higher threshold. As shown in Fig. 4, the threshold energy of the URM unstable resonator is far higher than that of the corresponding GRM unstable resonator with the equal effective output coupling.

The threshold energy dependence on the pump pulsewidth is shown in Fig. 5. Notice that the threshold peak intensity decreases as the pump pulsewidth increases. However the threshold energy is determined by both peak intensity and pulsewidth, just as given in Eq. (24). Finally the consequence of balance between two conflicting factors is that as pump pulsewidth increases, the threshold of a confocal GRM unstable resonator increases.

We plot threshold energy versus the crystal length with *L*=60mm, 2*r*_{p}
=2.8mm, *R*_{max}
=0.85, *w*=1.45mm, *T*=13.5ns in Fig. 6. As crystal length is increased, the threshold energy decreases. However a longer crystal will result in a degraded output beam quality [3]. Firstly higher parametric gain from longer crystal would excite additional high spatial frequency modes. Secondly the back conversion will happen when parametric gain is large enough to result in an entirely depleted pump, thus wavefront distortions will be introduced onto the transverse profiles of all three interacting beams. The back conversion usually begins at the center of the beam where the peak pump intensity is highest, even unstable resonator can not be effective in filtering out these low spatial frequency components. For this reason, the OPO should not be over driven with an excessively long crystal. It is noted that in thus cases our theoretical model is no long valid because the assumption of no pump depletion is not valid.

## 3.2 Experimental results

Threshold properties of three cases were investigated experimentally: a GRM plane-parallel resonator with *L*=53mm, a GRM confocal unstable resonator with *M*=1.092 and *L*=53.5mm, and a URM unstable resonator with *M*=1.06 and *L*=22.0mm. The KTP crystal available for these experiments is 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 2.8-mm-diameter (1/e intensity width), 13.5-ns-width pulses (FWHM) with tunable energy from several millijoules to ~150mJ. 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.

The input mirrors M1 for two GRM cases are coated for 97% transmission at the pump wavelength of 1.064µm, 96% transmission at the idler wavelength of 3.3µm and 99.6% reflectance at the signal wavelength of 1.57µm. The output mirrors M2 for two GRM cases are coated for 96% reflectance at the pump wavelength and 95% transmission at the idler wavelength. At the signal wavelength, the output mirrors for two GRM cases have Gaussian reflectivity profile as $R\left(r\right)=0.85\xb7{e}^{-{(r\u20441.45)}^{2}}$. Following Eq. (25), the effective reflectivity to signal of the GRM unstable resonator with M=1.092 is equal to 0.71. In the URM case, the input mirror is coated slightly differently from those of GRM cases. The input mirror of URM case has 98% transmission at the pump, 95% transmission at the idler and 99.8% reflectance at the signal. The output mirror of URM case has 96% reflectance at the pump, 99% transmission at the idler, and a radially uniform signal reflectivity of 0.82.

As shown in Table 1, maximum relative error of theoretical results to experimental data is 5% or so, which illustrates our theoretical model is in excellent agreement with experiment. It is observed from Table 1 that GRM plane-parallel resonator (shown as *M*=1.00) perform slightly better than the GRM unstable resonator in threshold energy under the close cavity length. This is because unstable resonators possess geometric losses that will increase as cavity magnification factor. However because of the geometric expansion of the unstable resonator to mode fields, the most divergent components of the oscillation radiation are filtered out, resulting in a near diffraction limited output beam.

As discussed above, threshold energy of the confocal unstable OPO with URM or GRM increases as cavity magnification factor or physical length increase, and decreases as signal reflectance of output coupler increases. Hence threshold energy of a URM unstable resonator with *L*=53.5mm, *M*=1.092 and *R*=0.71 must be far higher than that of a URM unstable resonator with *L*=22.0mm, *M*=1.06 and *R*=0.82 (~60% higher according to numerical calculation). However as shown in Table 1, both experimental data and theoretical calculations illustrate threshold of the GRM unstable resonator with a long cavity of *L*=53.5mm, *M*=1.092 and relatively low effective signal reflectivity *R*_{eff}
=0.71 produces slightly lower threshold than that of the URM unstable resonator with a very short cavity of *L*=22.0mm, *M*=1.06 and relatively high output mirror reflectance to signal of *R*=0.82. This approach in threshold energy is the result of better mode matching and higher central signal reflectivity of output mirror in the GRM unstable resonator relative to the URM unstable resonator. Therefore it can be concluded experimentally and theoretically that the GRM unstable resonator performs lower threshold than the corresponding URM unstable resonator with equal effective output coupling.

## 4. Conclusion

The theoretical threshold models of pulsed singly resonant GRM confocal unstable OPOs for the single-pass pumped version and also for the double-pass pumped version have been proposed, including the effects of transverse effective gain aperture. With these iterative models, it is direct and effective to design and optimize the OPO parameters that affect threshold, such as central reflectance to signal, Gaussian reflectivity radius, cavity magnification, physical cavity length, crystal length, and pump pulsewidth and radius. It is found that the threshold energy of GRM unstable resonator decreases with the central signal reflectivity and crystal length, and increases with the 1/e radius of Gaussian reflectivity, cavity physical length, cavity magnification factor and pump FWHM pulsewidth. Accurateness of our threshold model is strengthened by the good agreement with experimental results. Our threshold model is suitable for the plane-parallel resonator and the URM unstable resonator in the specific cases, which has been verified experimentally. It has been theoretically and experimentally demonstrated that GRM is an effective solution to reduce the threshold of unstable OPO.

## References and links

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