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Group velocity mismatch (GVM) is a major concern in the design of optical parametric amplifiers (OPAs) and generators (OPGs) for pulses shorter than a few picoseconds. By simplifying the coupled propagation equations and exploiting their scaling properties, the number of free parameters for a collinear OPA is reduced to a level where the parameter space can be studied systematically by simulations. The resulting set of figures show the combinations of material parameters and pulse lengths for which high performance can be achieved, and they can serve as a basis for a design.
©2007 Optical Society of America
1. Introduction
Optical parametric amplifiers (OPAs) and generators (OPGs) are useful frequency converters for short pulses because of their simplicity, tunability and the possibility for high performance. A major complication in the design of OPAs for subpicosecond pulses is the group velocity mismatch (GVM) between the interacting beams [1, 2, 3, 4, 5, 6]. The resulting temporal walk-off can limit the effective interaction length and restrict the range of pump pulse lengths that can be used efficiently for a particular configuration. Sukhorukov and Shchednova [1] obtained analytic solutions for short-pulse OPAs in the limit of negligible pump depletion. The gain was found to depend strongly on the direction of temporal walk-off of signal and idler relative to the pump: If signal and idler walk off in the same direction, the gain saturates after a certain propagation length, and high conversion cannot be achieved for short pulses. On the other hand, if the signal and idler walk off in opposite directions, the gain increases with crystal length, even if the pump pulse is shorter than the temporal walk-off. Pump depletion was included in the treatment by Bukauskas et al. [2], and they studied how a signal pulse could break up and form a sequence og shorter pulses under conditions of high conversion.
It is possible to adjust the group velocities by use of noncollinear beams [7, 8] or by tilting the pulse fronts with respect to the beam direction [9, 10], and by combining both techniques, all three group velocities can be matched [11]. However, these approaches also have drawbacks: First, tilted pulse fronts are equivalent to angular dispersion, so prisms or gratings are required to create or recollimate such beams. Second, even if the signal and pump beam entering an OPA are collimated, the idler will have angular dispersion if the beams are noncollinear. Third, noncollinear beams limit the effective interaction length. For these reasons, collinear OPAs are still of practical importance, especially when the pump power is low and the beam must be focused to a small diameter.
In collinear OPAs, the group velocities are determined by the material, the wavelengths and the type of phasematching. Although the effect of GVM on gain [1, 6], pulse shaping [3, 4], and the formation of pulse structure [2, 5] is well known, and many specific short-pulse OPAs have been studied in detail [6, 12, 13], it is of interest to investigate generally and quantitatively how the OPA performance, in terms of conversion efficiency and pulse quality, depends on the group indices, other material parameters, crystal length and pump intensity. The full parameter space has too many dimensions for exhaustive simulations, so in order to obtain general results it is necessary to make some simplifications and to take advantage of the scaling properties of the equations. This is the topic of Section 2. The main simplification is to consider only plane waves, so the effects of diffraction, transverse walk-off and transversely nonuniform beams are not included. Furthermore, group velocity dispersion (GVD) has been neglected. The results, which are discussed in Section 3, are sufficient to obtain fairly general conclusions on the parameter ranges that can give high performance, and Section 4 explains how to scale them to the parameter values for other OPAs. OPGs are discussed briefly in Section 5, and transverse effects and some full 3D simulations are discussed in Section 6.
2. Theory
Consider a nondegenerate second-order interaction of three plane waves with angular frequencies ωj, wave vectors kj, and slowly varying amplitudes ej, where j = 1, 2 and 3 label the idler, signal and pump beams, respectively. Because of energy conservation, ω 1 + ω 2 = ω 3. The real amplitudes in V/m are given by Ej(z, j) = ej(z,t)exp(-i(ωjt - kjz)) + c.c, and ej is related to the intensity by
where Z 0 is the vacuum impedance. In order to simplify the equations, GVD is neglected. Since typical nonlinear crystals have GVD coefficients of the order 10-25 s2, this approximation should be good for pulses as short as about 100 fs and crystals of about 1 cm. Becker et al. [5] found that the behaviour of the nondegenerate OPA they studied was almost the same with or without GVD. Absorption is also neglected, and with these approximations the equations for the slowly varying amplitudes have the form
where ng,j is the group index of beam j, nj is the refractive index of beam j, Δk = k 3 - k 2 - k 1 is the phase-mismatch, and χeff is the effective nonlinear susceptibility. The different ng allow for GVM. Terms of the form ∂ 2 ej/∂z 2 have been neglected based on the slowly varying envelope approximation, i.e. that the pulse is long compared to the wavelength so that ∂ej/∂z ≪ kjej. This assumption is consistent with the omission of GVD. The equations can be transformed to the frame moving with velocity v g,1 = c/n g,1 by using the coordinate t′ = t -z/v g,1 instead of t. The equations can be further simplified by defining the scaled amplitudes
Finally, I take Δk = 0 for the centre frequencies and obtain
where δng,j = ng, j -ng,1. This form of the equations reduces the set of independent parameters to δn g,2, δn g,3, and the crystal length L, while the wavelengths, the refractive index and χeff enter only in the scaling of the amplitudes. The boundary conditions are defined by the input pulses Uj(z = 0,t′). The physical significance of u 3 is that the steady-state small-signal amplitude gain is cosh(u 3 L). The equations are invariant if the GVM parameters δng,j and the time are scaled by the same factor s, or if uj are scaled by a factor s while time and L are scaled by s -1. For simplicity of notation, and without loss of generality, I take n g,1 < n g,2 in the rest of the paper. If the equations are written in the frame moving with beam 2, they read
and these should of course give identical results to Eqs. (6–8). If t′ is now reversed and n g,3 is replaced by n g,1 + n g,2 - n g,3, the original form of the equations is recovered, except that beams 1 and 2 are swapped. It follows that this replacement of n g,3 is equivalent to reversing the pulses and swapping beams 1 and 2. This symmetry is useful for reducing the set of n g,3 values to simulate.
3. Simulations
The examples in this section are restricted to seeded OPAs in order to avoid noise-induced pulse to pulse fluctuations, which are inherent in OPGs. Simulation of OPGs is discussed in Section 5. The simulation program has been described previously [14, 15]. Table 1 shows the physical parameters used in the simulations. They do not correspond to a specific real crystal, but they all have realistic values. Because of the invariance properties of the equations, the whole set of possible behaviours can be covered by keeping δn g,2 and the peak pump amplitude up fixed while varying δn g,3, L, the shape and duration of the pump pulse, and the intensity, shape and duration of the seed pulse. Scaling δn g,2 by s would be equivalent to scaling δn g,3 and t by s -1, and scaling up by s would be equivalent to scaling L and t by s and the seed amplitude us by
It is impractical to repeat the simulations for a large set of different pulse shapes, so I take the pump pulse to be Gaussian and vary only the duration T. The duration is measured as the full width at exp(-2) of the peak power, i.e. u 3(0,t) = up exp(- (2t/T)2). In the high-gain regime, the signal pulse is shaped by the gain, so provided the seed pulse is long enough to overlap the pump, its exact shape is not critical. Furthermore, the simulations show that it makes very little difference whether beam 1 or beam 2 is seeded. For these reasons, the input idler is taken to be zero and only the signal is seeded, and the seed pulse is taken to have the same shape and duration as the pump pulse. With these assumptions for the input pulses, the parameters that must be varied are δn g,3, L, T, and seed intensity.
Table 1. Physical parameters assumed in the simulations. Exceptions are noted in the text or figure captions. The normalised pump amplitude corresponding to these parameters is up = 3.84·103 m-1. n g,3 takes the values 1.63, 1.645, 1.66, 1.69, 1.72, 1.8 and 2, and results for n g,3 < (n g,1 + n g,2)/2 follow from symmetry.
Fig. 1. Power of idler (red), signal (blue) and pump (black) at different positions in the crystal (shown in each graph). The pulses are plotted in the frame moving with beam 1. The input pump pulse is shown by the dashed curve in the top row, and the vertical scale is the same for all the graphs. T and n g,3 are shown above each column. The movie shows the evolution of the pulses in (a) in smaller steps (1.8 MB). [Media 1]
As explained in Section 2, replacing n g,3 by n g,1 + n g,2 - n g,3 is equivalent to swapping beams 1 and 2 and reversing the pulses. Since the input pulses are symmetric and it makes little difference which input beam is seeded, it is only necessary to perform simulations for n g,3 ≥ (n g,1 + n g,2)/2. Results for a smaller n g,3 can be obtained by swapping and reversing the output pulses from the corresponding simulation with n g,1 + n g,2 - n g,3. Pulse quality and energy are of course unaffected by time reversal.
Although the main features of short-pulse parametric amplification have been reported before [1, 2, 3, 4, 5, 6], some examples of pulse evolution and gain are shown as a background for the more general results to follow. Figure 1 shows how signal and idler pulses evolve through the OPA crystal for three examples with different combinations of pump pulse length T and n g,3. In Fig. 1(a), with n g,3 = (n g,1 + n g,2)/2 = 1.63 and a short (200 fs) pump pulse, the peak of the pump pulse has been depleted at z = 4.8 mm. Backconversion regenerates a pump pulse at z = 5.8mm, but because the signal and idler pulses walk off in opposite directions the backcon-version stops before the signal and idler pulses are severely distorted. At position z = 7.2 mm, the regenerated pump has generated weak side pulses for the signal and idler, so the crystal should be kept shorter than this if clean pulses are desired. In Fig. 1(b), with the same n g,3 and a long (4 ps) pump pulse, the middle part of the pump pulse has been depleted at z = 4 mm and backconversion has just started. Because the temporal walk-off is now small compared to the pulse length, it cannot suppress backconversion, and at z = 5 mm the middle part of the pulse has been completely backconverted. At z = 5.8 mm a second round of conversion and backconversion has taken place, and the pulses have broken up completely. Figure 1(c), with n g,3 = 1.72 and long pump pulse, shows similar behaviour, but the pulses become asymmetric because of the temporal walk-off of the pump. The case with n g,3 > n g,2 and a short pump pulse is not shown because the temporal walk-off leads to low gain [1] and very small conversion.
Fig. 2. Logarithm of signal amplitude gain vs. crystal length for various pulse lengths (shown in legend, in ps) and (a) n g,3 = 1.63, (b) n g,3 = 1.66, (c) n g,3 = 1.69, and (d) n g,3 = 2.0. The plateau at gain ≈ 13 corresponds to saturation of the gain by pump depletion.
This reduction of gain by temporal walk-off is illustrated quantitatively in Fig. 2. Pump depletion limits the maximum amplitude gain to about exp(13), and long pulses, for which GVM is not important, reach pump depletion after about 4 mm. For shorter pulses the small-signal gain coefficient (the slope of the lines before pump depletion sets in) is reduced, and pump depletion is not always reached at all. As reported previously [1,6], there is a qualitative difference between situations with n g,3 inside and outside the interval [n g,1,n g,2]. In Fig. 2(a), with n g,3 = 1.63, all pulses grow rapidly to saturation, and only the shortest pulses experience a somewhat reduced gain coefficient. Pulses longer than about 0.3 ps are in the long-pulse regime, i.e. they have a gain coefficient nearly as high as for continuous beams. The high gain, in spite of temporal walk-off can be understood by considering the signal and idler pulses in the frame moving with the pump pulse: Since they walk in opposite directions, the signal light being left behind by the trailing end of the pump pulse has already generated idler light that walks in the other direction. Conversely, when idler light leaves the leading end of the pump pulse, it has generated signal light, and this maintains positive feedback between signal and idler within the pump pulse.
In Fig. 2(b), with n g,3 = n g,2 = 1.66, the gain coefficients for the short pulses are lower, but even the shortest pulses continue growing through the whole crystal [1] and eventually reach pump depletion, so there is not a minimum pulse length for efficient interaction.
When n g,3 > n g,2, as shown in Fig. 2(c) and (d), the temporal walk-off does not only reduce the gain coefficient for short pulses, but it also clamps their total gain at a level below the limit set by pump depletion. Intuitively, any part of the signal or idler pulse can only grow as long as it overlaps the pump pulse, so the maximum effective gain length for a pulse of length T equals the the distance in which the pump pulse separates from the signal and idler pulses by T:
where K is a constant of order 1 and c is the speed of light. δng,p is taken to correspond to the temporal walk-off between the pump and the generated pulse with the best overlap, i.e. δng,p = min(|n g,3 -n g,1|,|n g,3 - n g,2|). The total gain is limited either by this gain length or by pump depletion, whichever occurs first. Corresponding to Le is a minimum pulse length T min required to obtain an amplitude gain of G = cosh(g) (in the absence of pump depletion). This is determined by upLe(T min) = g, and assuming Eq. (12) for Le,
Fig. 3. Simulation results for n g,3 = 1.63. (a) Conversion efficiency as a function of L and T. The T-axis is nonlinear to show every sample point with the same size. (b) Idler pulse quality. The product has been normalised to the value for a transform-limited Gaussian pulse, so unity corresponds to the minimum time bandwidth product. The colour scale stops at 10. (c) Conversion efficiency as a function of crystal length for a few different pulse lengths (same data as in part (a)). The legend shows the pulse length in ps. The oscillations with z (for the long pulses) are due to repeated conversion and back conversion. (d) Time bandwidth products corresponding to part (c) (same data as in part (b)).
Although this is not a rigorous derivation, the expression for T min is at least consistent with the scaling properties of Eqs. (6–8) in that T min ∝ u -1 p. If δn g,2 and δn g,3 vary in proportion, T min is ∝ δn g,3. If δn g,2 is small compared to δn g,3, it can be neglected as far as gain is concerned, although it does determine the signal gain bandwidth. In this case, T min ∝ δn g,3, and the parameter space for simulations can be simplified by keeping δn g,3 fixed and varying only T, L, and the seed intensity.
In addition to the gain, conversion efficiency and pulse quality are important features of a real OPA. The pulse quality is defined as the product of the standard deviations of the pulse intensity and the spectrum [16]. Figure 3 shows the results for n g,3 = 1.63 = (n g,1 + n g,2)/2. The signal and idler have similar pulse quality because they walk off symmetrically with respect to the pump, so only the idler is shown. The same data are shown as images for intuitive interpretation (a, b) and as graphs for quantitative analysis (c, d). The first feature to note is that for sufficiently long pulses, a specific value of L (4 mm in this case), combines moderately high conversion and good pulse quality. The approximate independence of T means that these pulses are in the long-pulse limit, where the effect of temporal walk-off is small. For comparison, the optimal crystal length for a continuous-wave pump with amplitude up would be 3.8 mm. The limited conversion efficiency can be explained by the time-dependent intensity of the pump pulses - no crystal length is optimal for the whole range of pump intensities that occur within the pulse. Longer crystals lead to back conversion, and although conversion can be higher for some crystal lengths, the pulse quality suffers. Short pulses grow more slowly, but for crystals longer than 4 mm they can reach even higher conversion than the long pulses, and the pulse quality remains high for long crystals. This is a beneficial effect of temporal walk-off, as seen in Fig. 1(a): Backconversion is suppressed because the signal and idler pulses walk off in opposite directions. This mechanism appears to work well for pulses up to about 0.2 ps, which have a signal-idler walk-off length Lw = Tc/(n g,2 - n g,1) = 1 mm. For comparison, the inverse of the peak gain coefficient is 1/up = 0.26 mm.
Fig. 4. Like Fig. 3, but with (a,b) n g,3 = 1.645, (c,d) n g,3 = 1.66, (e,f) n g,3 = 1.72, (g,h) n g,3 = 1.8, and (i,j) n g,3 = 2.0. The colour scale for the pulse quality runs from 1 to 10, as in Fig. 3(b).
Fig. 5. Conversion efficiency as a function of n g,3 for T=200 fs. The vertical lines mark n g,1 =1.6 and n g,2 = 1.66 and the legend shows the position in the crystal, in mm.
Figure 4 shows corresponding data for n g,3 in the range 1.645–2. Note that the range of the T-axis is not the same in all the graphs. In these examples, with n g,3 > (n g,1 + n g,2) /2, the pulse quality is often better for the signal than for the idler. Simulations with n g,3 < (n g,1 + n g,2)/2 give better pulse quality for the idler, whereas seeding beam 1 instead of beam 2 makes little difference, as expected in the high-gain regime. Thus, most of the difference in pulse quality can be ascribed to the group velocites – the pulse with group velocity closer to pump has better quality. As the GVM δn g,3 increases, high conversion is no longer possible for short pulses, and the minimum pulse lengths from the figures are in fair agreement with Eq. (13) if K ≈ 0.5. For long pulses, L = 4 mm remains the optimal crystal length. The sensitivity to δn g,3 for short pulses is striking, and this is illustrated in detail in Fig. 5. For short pulses, n g,3 = (n g,1 + n g,2)/2 is the optimal value, and the efficiency drops sharply when n g,3 approaches the ends of the interval [n g,1,n g,2]. The curves are very nearly symmetric about n g,3 = (n g,1 + n g,2)/2, and the slight deviation from symmetry occurs because the signal is seeded and the idler is not.
In a practical situation, the group indices ng,j and the maximum permissible pump amplitude u p,max are given by the material and the desired wavelengths, T is given by the available pump laser, and the parameters that can be adjusted in the design are L and up ≤ u p,max. From the scaling arguments in the beginning of this section, reducing the pump amplitude is equivalent to moving towards the upper left corner in the performance maps of Figs. 3–4. In the case with n g,3 ∈ [n g,1, n g,2], it can be desirable to scale up down (and maybe increase L to make up for the reduced gain) to move into the short-pulse regime where temporal walk-off can reduce back-conversion. For n g,3 ∉[n g,1, n g,2], scaling down up is not advantageous. If the scaled T exceeds T min, high performance can be restored by increasing L, but if the scaled T becomes too short, the reduced pump amplitude cannot be compensated by crystal length. If the pump amplitude is scaled up from a level that works well, high performance can be restored by reducing L, but the figures indicate that there is not much performance to gain by increasing the pump above such a level. Hence, for n g,3 ∉ [n g,1,n g,2], operating with up = u p,max can give nearly optimal performance in most cases.
The seed intensity has not yet been varied. When the OPA operates in the high-gain regime, the seed pulse is amplified linearly in the first part of the crystal, so the output from a long crystal with a weak seed pulse can be expected to resemble the output from a shorter crystal with a correspondingly stronger seed. This is confirmed by simulations with 1000 times higher seed intensity, as shown in Fig. 6 for n g,3 = 1.63 and n g,3 = 2. As expected, the maps of conversion and pulse quality are qualitatively similar to those in Figs. 3 and 4, but they are shifted to shorter crystal lengths. If n g,3 ∉ [n g,1,n g,2], the gain for short pulses saturates after a short propagation distance, and a minimum seed intensity is required in order to reach a significant conversion efficiency. Another way to see this is that T min in Eq. (13) depends on g, which is of course lower for a stronger seed. This is consistent with Fig. 6(c), where T min is slightly smaller than in Fig. 4(i). Simulations with a seed pulse 3 times longer than the pump pulse yielded results nearly identical to those with equal seed and pump pulses.
Fig. 6. Conversion efficiency and pulse quality with seed intensity 10 W/cm2 (1000 times greater than in Figs. 3–4). (a,b) n g,3 = 1.63 (as in Fig. 3), (c,d) n g,3 = 2.0 (as in Fig. 4(i,j)). The colour scale for the pulse quality runs from 1 to 10, as in Fig. 3(b).
4. Application to real crystals
The main importance of Figs. 3–4 is that they can be scaled to apply to other materials and parameters. In the following, the parameters in the figures are denoted by primed symbols and the physical parameters of some other system being studied by unprimed, and I define the scale factors s 1 = up/u′p and S 2 = δn g,2/δn′ g,2. From Section 3, u′p = 3840m-1 and δn′ g,2 = 0.06. δn g,2 can be found from the material data, and from Eqs. (1–5)
s 1 and s 2 can now be found, and from the scaling properties explained below Eq. (8), the remaining primed and unprimed parameters are related by δn′g,3 = δn g,3/s 2,T′ = Ts 1 /s 2, and L′ = Ls 1. If δn′g,3, T, and L are given, the equivalent operating point can be located in the figure for the relevant δn′g,3. Conversely, this figure can be used to identify suitable values for T′ and L′, and these can be transformed to physical parameters to obtain a starting point for a good design.
Table 2 shows data for some examples corresponding to different regimes. In example 1 ng,3 ∈ [n g,1,n g,2], and an OPA should work well with a wide range of pulse lengths. δn′g,3 = 0.016 corresponds approximately (by symmetry) to n g,3 = 1.645, or Fig. 4(a,b). The parameters shown in the table, T = 0.75ps and Lc = 40 mm correspond to a short pulse for which temporal walk-off suppresses backconversion Example 2 is in the regime with n g,3 much greater than n g,1 and n g,2, so T = 6.5ps is the minimum pulse length for efficient conversion. In example 3, n g,3 is just slightly smaller than n g,2. Figure 4(c,d) can be used for a first estimate of the performance, but because of the strong sensitivity when n g,3 ≈ n g,2 as seen in Fig. 5, more accurate calculations should be carried out for a detailed design. Example 4 has n g,3 near the optimal value (n g,1 +n g,2)/2, and Fig. 3 or 4(a,b) should be applied.
Table 2. Parameters, scaling factors and relevant figure for various examples. The beams have been ordered so that n g,1 < n g,2. In the polarization row, F indicates fast and S indicates slow. of-elem. shows the value used for the relevant element of the nonlinear tensor. T′ and L′c are a pair of suitable parameters selected from the figure, and T and Lc are the corresponding scaled parameters for the example.
The conclusion from this section is that Figs. 3–4 represent a large part of the parameter space for collinear high-gain OPAs, and that they include most configurations that combine high conversion and pulse quality. The qualitative behaviour of an OPA with different material parameters, wavelengths or pulse length can be found from the figures by looking up the results for the scaled parameters. The seed intensity has not been varied as systematically as the other parameters, but as long as the OPA remains in the high-gain regime, a moderate change in seed intensity can be compensated by a change of the crystal length, without a great change to the output. If the seed intensity changes by orders of magnitude, a possible change of T min must be taken into account.
5. Optical parametric generators
An OPG is an OPA without a seed input, so the signal and idler grow from quantum noise. Most of the results from the previous sections also apply to OPGs, but because of the noise input the output pulses from an OPG will in general exhibit fluctuations in energy, spectrum, pulse shape and beam shape. The energy fluctuations are reduced by pump depletion, so they tend to be small when the OPG operates with high conversion. A more serious problem may be the pulse quality for long pulses. Because the signal- and idler-spectra are only limited by the acceptance bandwidth of the amplifier, the time-bandwidth product increases with the pulse length, and only pump pulses similar to or shorter than the inverse gain bandwidth can produce high-quality output pulses. In a seeded OPA, on the other hand, the width of the signal and idler spectra is restricted by the seed pulse, so high pulse quality can be maintained for long pulses.
Fig. 7. Conversion efficiency and pulse quality from OPGs, as functions of crystal length and pump pulse length. (a,b) n g,3 = 1.63 (as in Fig. 3), (c,d) n g,3 = 1.66 (as in Fig. 4(c,d)), and (e,f) n g,3 = 1.80 (as in Fig. 4(g,h)).
Figure 7 shows performance maps from plane-wave simulations of OPGs with the same parameters as in Section 3. The simulations were repeated 10 times for each set of parameters. Figure 7(a) (with n g,3 = 1.63) is similar to 3(a), except that the crystal must be about 1 mm longer to give sufficient gain for the OPG. Fig. 7(b) differs noticeably from 3(b) in that the pulse quality is always poor for long pulses, as explained above. For shorter pulses, there is a range of parameters in Fig. 7(a,b) that combine high conversion and high pulse quality.
With n g,3 = 1.66, in Fig. 7(c,d), there is high performance for a parameter range near L = 5mm and T = 1.4 ps, but the range of pulse lengths that combine moderately high conversion and good pulse quality is narrow. With n g,3 = 1.80 (Fig. 7(e,f)), it is no longer possible to combine these features, as the minimum pulse length for high conversion is already too long for high pulse quality. In the OPG, the pulse quality of the signal and idler differ less than in Fig. 4. This is because the spectra of both beams are now mainly determined by the gain bandwidth.
6. Transverse effects
The plane-wave simulation results are directly relevant to waveguide devices and to OPAs operating with wide, flat-top beams, where diffraction and transversely varying intensity can be ignored. However, these are exceptional cases, and most real OPAs operate with beams for which the transverse intensity variation is important. Spatio-temporal simulations of a specific KTP-based short-pulse OPA have been reported [12, 13], and general continuous-wave OPAs with Gaussian beams have also been treated before [20]. As in the present paper, it was possible to obtain general conclusions by exploiting the scaling properties of the equations and performing extensive simulations in the reduced parameter space. High conversion and beam quality could only be combined for narrow beams, for which diffraction could suppress gain narrowing [20]. When both temporal and transverse dimensions are included, it is no longer possible to perform exhaustive simulations, but much insight can be gained by combining the results from the two simplified cases.
Consider first long pulses: When temporal walk-off can be neglected, each point in time can be treated independently, and the OPA can be modelled as a sequence of steady-state OPAs with different pump powers. It is intuitively clear from the steady-state simulations that the OPA cannot be optimised for the whole range of pump power. There will either be low conversion in the tails of the pulse or backconversion and reduced beam quality at the peak.
Now consider the case of wide beams: When diffraction can be neglected each transverse point can be treated independently, and the OPA behaves like a set of plane-wave OPAs with a range of pump intensities. The simulations in Figs. 3 and 4 show that for most parameters, only a narrow range of crystal lengths combine high conversion and pulse quality. By the scaling arguments in Section 2, this corresponds to a narrow range of acceptable pump intensities. Thus, if the system parameters are selected for high performance at peak pump intensity, the area with low pump intensity will have small conversion, and conversely, if there is high conversion in the low-intensity area, other areas will have backconversion, with reduced pulse- and beam-quality. The exception to this is the case with a short pump pulse and n g,3 ≈ (n g,1 + n g,2)/2, where Figs. 3 and 4(a,b) show that the tolerance for crystal length, and hence pump intensity, is considerable. This may allow high conversion, beam- and pulse-quality even for relatively wide beams.
Figure 8 shows conversion efficiency, idler pulse quality and idler beam quality as functions of crystal length for n g,3 = 1.63 or 1.80 and four different combinations of pulse length and pump beam radius. The pulse- and beam-quality of the signal are not shown because they are similar to those of the idler. The pump- and seed-beams are Gaussian with equal waist radii w 0 = 30 or 200 μm. The corresponding Rayleigh length in the nonlinear crystal is about 4 mm or 20 cm, respectively, so these correspond to cases with strong and weak diffraction. In order to avoid unnecessary complications, I have assumed noncritical phase matching (i.e. no transverse walk-off) and cylindrical symmetry.
Fig. 8. Conversion efficiency (blue), pulse quality (black) and beam quality (red) as functions of crystal length z. In order to show all quantities conveniently in the same graph, pulse and beam quality are plotted as the inverse of the time-bandwidth product and M 2, respectively. n g,3 = 1.63 in the top row (a–d) and 1.80 in the bottom row (e–h). The pump pulse length (in ps) and pump beam waist radius (in μm) are given on each graph.
Fig. 9. (a) Signal intensity vs. radial position and time for n g,3 = 1.63, T = 3ps, w 0 = 200 μm, and position z = 7 mm. The movie shows the evolution of all three pulses along z (520 kB). (b) Normalised signal power at z = 7 mm. (c) Normalised signal fluence at z = 7 mm. [Media 2]
In all cases, the pulse- and beam-quality drop sharply as the conversion grows, and only Fig. 8(a) shows high conversion and good pulse- and beam-quality. This is the regime with narrow beam, favourable group velocities and a relatively short pulse. Long pulses and narrow beam (Fig. 8(b,f)) can give moderately high conversion and beam-and pulse quality. For wide beams, comparison of Fig. 8(c) with (d,g,h) shows that a short pulse and favourable group velocities can improve the performance, as expected.
As an illustration of how the pulse- and beam-quality deteriorate, Fig. 9(a) shows the signal pulse in space and time for n g,3 = 1.63, T = 3ps, w 0 =200μm(as in Fig. 8(d)), and L = 7 mm. The conversion efficiency in this case is 0.36, but backconversion breaks up the pulse and gives rise to intensity oscillations in time and along the radial position. Integration over the pulse or beam masks the oscillations, so the poor quality is not so clearly seen in the totals in Fig. 9(b) and (c).
There are of course applications where the time bandwidth product is not important, but as seen in Fig. 8, poor pulse quality usually coincides with poor beam quality. In the steady-state case, the beam quality for wide beams can be improved by use of a two-stage OPA with signal beam expansion between the stages and low gain in the final stage [13, 20]. This method can also be used in the pulsed case, but a detailed discussion is outside the scope of this paper.
7. Conclusion
By assuming plane waves and Gaussian pump and seed pulses, and by exploiting the scaling properties of the coupled wave equations, the set of independent parameters for collinear high-gain OPAs has been reduced to a size that can be handled by exhaustive simulations. The main results in this paper are Figs. 3 and 4, which show performance maps of conversion efficiency and pulse quality as functions of pump pulse length, crystal length and the temporal walk-off parameter δn g,3 = n g,3 - n g,1. The performance of a wide range of OPAs can be estimated by computing the scaled parameters and reading these maps. As reported before [1, 6], there are two main regimes with qualitatively different behaviour: If n g,3 ∈ [n g,1,n g,2], high conversion and pulse quality can be obtained for a wide range of pulse lengths, and short pulses even benefit from temporal walk-off for suppressing backconversion. In the second regime, with n g,3 ∉ [n g,1,n g,2], only pulses longer than a minimum duration T min ∝ δn g,3/up are converted efficiently, where up is the amplitude gain coefficient.
The results are also relevant to OPGs, but in this case long pump pulses always give poor pulse quality because the signal bandwidth is not constrained by a seed. When transverse variation of the beam intensity was taken into account, the combination of high conversion, pulse-and beam-quality in a single-stage OPA was only observed for narrow beams, n g,3 ∈ [n g,1,n g,2], and a pump pulse short enough to take advantage of temporal walk-off for suppressing back-conversion.
Acknowledgments
I thank Magnus Haakestad, Espen Lippert and Knut Stenersen at FFI, Gregor Anstett at FGAN-FOM, and the reviewers for useful comments and corrections.
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