Noncollinear emission occurs in a free running type I parametric oscillator beyond the theoretical tuning curve for collinear oscillation. The noncollinear emission is described in terms of the emission angle and the corresponding angle of divergence. Experimentally recorded single captures of the transverse profile for the noncollinear regime are presented. A theoretical description of the degenerate tuning is described to explain the noncollinear regime.
© Optical Society of America
Most applications (and hence research) have been devoted to the use of pulsed singly resonant nondegenerate Optical Parametric Oscillators (OPOs)  but there has been a growing trend to characterise the more problematic doubly resonant and degenerate oscillators. In this paper the noncollinear regime obtained beyond the theoretical tuning curve for collinear oscillation is characterised in terms of the emission angle.
The crystal used in this paper was a 12 mm type I cut BBO crystal (θ=35°, Cleveland Crystals Inc, USA). The cavity was formed by two planar mirrors and was 17 mm in length. The pumping laser used was a Q-switched Nd:YAG laser with a pulse length of 14ns. The fundamental wavelength was amplified and frequency converted to produce 532 and 355 nm wavelengths. For the 355 nm wavelength greater than 40 mJ was obtainable at 10 Hz, with a beam diameter of 3 mm. A colour filter was used to eliminate the residual UV pump beam.
Fig. 1 presents the tuning curve from 550nm to degeneracy for the signal wave corresponding to a range from 950 nm to degeneracy for the idler. The data presented in Fig. 1 is characteristic of a type I OPO operating in the collinear regime. All data presented was collected at 3 times the average oscillation threshold with an average of 3% optical conversion for the presented region.
Theory of noncollinear interactions
The phase matching conditions which describe the production of the signal and idler wavelengths, for the non-degenerate case, may be expressed as frequency conservation ,
and photon momentum conservation,
where the subscripts p, s, i represent the pump, signal and idler waves respectively. The noncollinearity, α, is the angle between the noncollinear and the corresponding collinear interactions. The frequencies, wavelengths and refractive indices of the three interacting waves are given by ωj, λj and nj respectively. Tuning is achieved in the same way as in the collinear case by rotation of the non-linear optic crystal.
The noncollinear interaction results in a larger gain for the signal and idler waves when compared to the corresponding collinear case. This can be explained in terms of the Poynting vector walk-off experienced by the pump wave . This limits the effective interaction length of the non-linear crystal as the pump and resonant OPO wavelengths become spatially separated. Ordinary waves do not experience this walk-off, so the effective gain is smaller in type I than type II interactions. Increasing the noncollinearity in the parametric interaction has the effect of reducing the walk-off as one of the generated wavelengths propagates along a direction closer to that of the pump. This reduced walk-off increases the possible interaction length and hence increases the gain for the interaction.
Observations of noncollinear emission
Using the crystal described above noncollinear emission was observed beyond the degenerate phase matching angle of 33.12 degrees. The resulting far field profile and experimental arrangement are illustrated in Fig. 2. The noncollinear angle of emission, α was observed to increase as the phase matching angle was increased beyond degeneracy. This behaviour can be explained by considering the effect of the final term into the phase matching condition in Eq. (2). This term increases the phase matching angle at which a specific signal/idler wavelength pair can oscillate. Fig. 3 shows the experimental near degenerate tuning curve of the OPO. Also shown are the theoretical collinear and noncollinear phase matching curves for a noncollinearity of 10 and 20 mrad respectively. The experimental output wavelength collapses to degeneracy for angles beyond the collinear regime (i.e. θ>33.12°). As the phase matching angle increases so the noncollinearity imposed on the degenerate wavelength also increases. For example the theoretical degenerate point for a noncollinearity of 10 mrad is 33.14° and for 20 mrad it is 33.18°. Thus, increasing the phase matching angle beyond the collinear regime results in the production of an off-axis, noncollinear emission. There is a small systematic error in the experimental tuning curve - the data points are slightly displaced to the right of the theoretical tuning curve due to an inaccuracy in the cut of the polished faces of the crystal.
Inset. Degenerate collinear and noncollinear phase matching geometries. Noncollinear emission results in the off-axis production of signal and idler wavelengths. This noncollinear geometry results in a cone of emission with angle α - viewed as an annular pattern in the far field.
As the emission angle is increased the corresponding angle of divergence, Δα decreases, although at large emission angles the energy density of the transverse structure is significantly reduced (due to the large size of the annulus) and hence is harder to observe accurately. The experimentally measured emission angles are presented in Fig. 4, along with the theoretically calculated emission angle for degenerate noncollinear emission. It is apparent that the noncollinear emission described follows the theoretical trend for degenerate noncollinear emission (Eq. (1) and Eq. (2), ωs=ωi). The noncollinear emission angles presented in figure 4 are less than a factor of 2 smaller than the theoretical walk-off  experienced by the extraordinarily polarised pump beam, of 74 mrad, when passing through the non-linear BBO crystal. The theoretical pump walk-off changes less than 0.5 % over the non-collinear range presented. The divergence of the noncollinear emission is presented in Fig. 5. As can be deduced from Fig. 4 and 5 the larger the emission angle the smaller the divergence.
As described the far field pattern for this noncollinear emission is an annulus or ring, which has previously been reported for spontaneous parametric generation in the picosecond [3–5] and CW regimes. The far field transverse profiles were detected using a CCD camera with two different lens configurations. First a single lens was used in a ‘2-f’ configuration, with the crystal exit face at the object plane and the CCD at the image plane of the lens. Using this simple configuration it was possible to follow the change in the emission angle as the crystal angle was adjusted. Representative single shot captures of the noncollinear emission can be seen in Fig. 6a–6c. The phase matching angle increases for Fig. 6a through to 6c. The noncollinear interaction results in an annulus which is localised around the emission angle, α. From these pictures the decrease in the angle of divergence, Δα, can be seen as the phase matching angle is increased – the increasing size of the annulus is the overriding factor in this behaviour, although other factors (such as phase mismatch) may also play a part. Fig. 6c also demonstrates the principle of momentum conservation when applied to parametric interactions. When a collection of photons are emitted in a particular direction the corresponding photons from the interaction lie on the emission angle of opposite sign - this is particularly apparent at the sides of Fig. 6c.
The second collimating lens system consisted of two identical focal length lenses in a ‘3-f’ arrangement. The second lens was placed at the image plane of the first, and the CCD at the second lens’s focal plane. A representative profile of the transverse structure can be seen in Figure 6d. From this image a ‘roll pattern’ type interference image with three prominent peaks in the intensity can be seen. While this pattern resembles the roll patterns predicted for the degenerate OPO [7,4] and other non-linear systems there are some major differences. Firstly the pattern was not produced by the interference of two beams, but rather from the most intense parts of an annulus. Although when one (or both) of the cavity mirrors are adjusted the noncollinear annulus collapses into two oppositely orientated spots and hence the corresponding interference patterns more closely resemble the theoretical description in the literature. On further mirror adjustment the spots rotate around the annulus described by the emission angle. The production of two oppositely orientated spots or ‘twin beams’ can also be achieved by near threshold pumping. In this case a particular pair of directions dominate which results in two spots with the same emission angle as the highly excited case. Secondly the production of this interference pattern is invariant to cavity length variations, in contrast to the predictions of transverse structures in degenerate parametric oscillators , due to the large numbers of oscillating longitudinal modes.
A short animation (Fig. 7) shows the increase in emission angle for the degenerate noncollinear emission as the crystal angle is varied. The animation was produced by taking single images via a digital video camera. The video starts with the usual collinear emission and progresses through the noncollinear emission region in steps of 1 minute (internal angle).
Degenerate noncollinear emission from a collinear OPO has been shown to obey the theoretically predicted behaviour. Angular tuning of this noncollinear emission causes a smooth increase in the emission angle. However at certain crystal angles a bistability was observed between two competing angular emission directions (third and eighth images in Fig. 7). In this regime the output power was nominally split between the two annuli, although on occasion a pulsing of the two annuli could be observed as the power alternated between the two emission angles. Although the authors are not sure of the exact nature of the bistability it would appear that the pulsing behaviour could be linked to the changes in the temporal and spatial characteristics of the Q-switched pump pulses.
Also, although the exact wavelength and linewidth of the noncollinear emission were not determined due to the fact that at (or near) degeneracy the two wavelengths produced become (nearly) indistinguishable, some preliminary comments can be made. Firstly, the wavelength of the non-collinear emission was not observed to change for different emission angles and degenerate emission can be assumed as the recorded profiles in Fig. 6(a–c) do not display a second ring. Secondly the linewidth although large due to the degeneracy was observed to decrease for increasing noncollinearity. Finally it is unclear whether injection seeding which would allow more accurate measurements of the wavelength and linewidth would destroy the transverse structure of this noncollinear emission.
The noncollinear emission only occurs beyond the phase matching angles for collinear emission. Even though the single pass gain for the noncollinear emission is larger than that for collinear emission the reduction in the cavity finesse for noncollinear emission precludes its production in the collinear regime. Only when collinear emission is no longer possible, and hence there is no longer the competition for the pump energy between these two transverse modes, can the noncollinear interaction proliferate. This argument also provides an explanation as to why only degenerate noncollinear emission oscillates - in contrast to the cases of spontaneous parametric generation in the literature. From Fig. 4 it can be seen that for a given noncollinearity (say 10 mrad) there are a large number of non-degenerate frequencies which can oscillate with a larger noncollinearity. For example when θ=33.14° (10 mrad degenerate point) a noncollinearity of 20 mrad would allow the production of wavelengths at 678 and 745 nm for the signal and idler respectively. By the argument above, in the case of two competing noncollinear emissions the emission with the largest round trip gain will proliferate, which in this case is the emission with the larger finesse within the optical cavity, i.e. with the smallest noncollinearity. The use of curved mirrors in the cavity design would increase the gain for all noncollinear interactions. Thus the fact that only degenerate emission is observed is a result of the optical cavity’s role in the determination of the OPO gain.
Finally, it is worth noting that the data presented above is characteristic of a ‘transverse structure’ in the sense that it is only produced for ‘large’ diameters of the pump beam. For example when the OPO is tuned such that the non-collinear emission is just observable (i.e. at the degenerate edge of the collinear regime) the collinear emission can be ‘forced’ to oscillate by the introduction of a small aperture (ϕ<1 mm) into the pump beam.
Experimental evidence has been presented for the degenerate noncollinear emission from a collinear type I OPO. The results presented are consistent with the theoretical predictions.
The authors gratefully acknowledge the financial support of the Engineering and Physical Sciences Research Council, UK.
References and links
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