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Absorption at the idler wavelength in an optical parametric oscillator (OPO) is often considered detrimental. We show through simulations that pulsed OPOs with significant idler absorption can perform better than OPOs with low idler absorption both in terms of conversion efficiency and beam quality. The main reason for this is reduced back conversion. We also show how the beam quality depends on the beam width and pump pulse length, and present scaling relations to use the example simulations for other pulsed nanosecond OPOs.
©2011 Optical Society of America
1. Introduction
Wavelength conversion of laser beams in optical parametric oscillators (OPOs) is an effective method to generate tunable radiation as well as radiation at wavelengths outside the range of available laser materials. The parametric process in an OPO generates two beams whose frequencies add up to the input (or ‘pump’) frequency. The generated beams are often denoted signal and idler, for the high and low frequency beams, respectively, and this notation will also be used in this work.
In principle, the signal frequency can be tuned close to the pump frequency, but because all nonlinear crystals absorb below some frequency in the infrared, the resulting low idler frequency will eventually be absorbed. This does not only reduce the idler energy itself, but also the gain for the signal, so idler absorption is an important consideration even in applications where only the signal beam is being used. It is therefore of interest to know how idler absorption affects the performance the OPO and how much idler absorption can be tolerated. In this paper we address these questions by numerical simulation of pulsed OPOs.
A fundamental problem with OPOs is that the parametric amplification process is mediated by the same nonlinear coupling as sum frequency generation (SFG), and the actual direction of energy transfer between the beams depends on the boundary conditions. If the pump beam becomes fully depleted, the process switches to SFG and regenerates light at the pump wavelength at the expense of signal and idler energy. This is known as back conversion, and it obviously reduces the output energy of the OPO. If the intensity varies across the beam, back conversion can take place in some areas and not in others, and this also reduces the beam quality. Since back conversion requires both signal and idler to be present, it can be reduced by removing the idler by filters or dichroic mirrors. This technique has been found to improve OPO performance [1–3]. In this work, we investigate if idler absorption can play the same role. This is not obvious, as idler absorption will also reduce the gain for the signal.
Previously, Lowenthal studied continuous wave OPOs through plane wave simulations and showed that some idler absorption may improve the signal output power [4], while Lyons et al. studied pulsed singly resonant OPOs through numerical simulations and found that back conversion has significant effect on the beam quality of the output of the OPO and that idler absorption affects this [5]. In this work, we present an extensive analysis of how idler absorption affects OPO performance in terms of conversion efficiency and beam quality for a generic pulsed OPO. We start with a brief theoretical analysis to deduce the equations, reduce the number of independent parameters and define the parameter space to be simulated. After a description of the simulation model, we present simulation results, discuss thermal effects of absorption, and conclude by scaling the simulations to real examples.
2. Theory
We start by deducing an expression for the small signal gain for plane waves in a χ (2)-based optical parametric amplifier (OPA) with absorption losses included. We number the beams 1-3 by increasing frequency so that the idler is beam 1, signal is beam 2 and pump is beam 3. The steady-state equations for the signal and idler in the small signal case are
where the superscript * means complex conjugate, the amplitudes are scaled so that equals the intensity Ij, z is the coordinate corresponding to the propagation direction, ωj is the angular frequency, αj is the absorption coefficient for the intensity, hence the factor 1/2, and nj is the refractive index of beam j. The pump amplitude a 3 is taken to be constant, and is the phase mismatch, where is the wave number of beam j in the medium, and c is the speed of light in vacuum. The coupling coefficient γ is defined aswhere deff is the effective nonlinear coefficient for the interaction and ε 0 is the permittivity of vacuum. The pump amplitude, a 3, can be taken to be real without loss of generality, and Eq. (1) can easily be solved to find an expression for the small signal gain coefficient g, :where we have defined . Let us assume perfect phase matching to investigate absorption in isolation, and consider two special cases of absorption.- • Case 1: Equal signal and idler absorption, . Then , i.e. the small signal gain coefficient is reduced by the absorption coefficient.
- • Case 2: only idler absorption, . Then (3) simplifies to
- In the high gain limit, idler absorption reduces the gain half as much as in case 1, where signal and idler absorption are equal. Interestingly, in the low gain limit, the gain is positive, but small, even when the idler absorption is large. There is thus potential for high conversion efficiency in OPOs and OPAs with high idler absorption provided that the nonlinear crystal is long enough.
In an OPO it is also necessary to include the resonator, which is characterized by the reflectance and curvature of the mirrors and the length of air gaps on each side of the crystal. For the sake of tractability, we restrict our analysis to a singly resonant OPO with single pass pump and plane mirrors, and we take the air gaps to be negligibly short. An OPO can then be modeled simply by applying Eq. (8) for the forward pass of each round trip, reflect a part of the signal beam, and propagate it passively back through the crystal. The reflectance of the output coupler at the signal wave, Roc, affects the threshold level, slope efficiency and intracavity fluence of the OPO. Typical values of Roc in pulsed OPOs are in the range 0.3 to 0.8. Changing Roc while keeping other parameters fixed can change the OPO performance dramatically, but it turns out that when other parameters are optimized, the effect of idler absorption on the achievable performance is not very sensitive to Roc. Therefore, we have used Roc = 0.5 in most of this work.
We take the pump beam to have a Gaussian pulse shape with duration tp (FWHM) and a Gaussian transversal profile with beam radius W (e−2M). We take W to be sufficiently large for the Rayleigh length of the beams to be long compared to the crystal, so we do not need to vary the longitudinal position of the waist. For the pulse length, it is convenient to introduce the quantities
where tr is the round trip time for the signal beam and Nrt is the pump pulse length measured in number of resonator round trips.It is usually desirable to operate an OPO with the highest pump intensity that the crystal can withstand without damage, so that the beam width can be minimized to suppress transverse multi-mode operation. The damage threshold depends on pulse length, so when comparing results for different pulse lengths it is important to include the variation in peak pump intensity. It has been found that the damage threshold scales approximately as tp 1/2 for nanosecond pulses [7, section 11.4], so we take the peak pump intensity (in space and time) to be
where I 0 is the peak intensity for the reference pulse length t 0 which has been taken to be 10 ns in this work.To summarize, the parameters we vary systematically are Ld (by varying W and keeping k 3 fixed), σ (by varying deff), α 1, and tp. Given some OPO that satisfies our assumptions and has parameters Ld', σ', α 1', and tp', we find the equivalent OPO in our examples by the procedure
Note that wavelengths and refractive indices are included in σ and Ld, so they do not enter directly in the scaling calculations. An OPO with air gaps can be represented approximately by increasing the crystal length Lc' to get the same Fresnel number and adjusting α 1' and σ ' so that the round trip gain and idler absorption are maintained. The contribution of the air gaps to the round trip time can of course be included exactly.3. The simulation model
The OPO is simulated using an in-house numerical model called Sisyfos (Simulation System for Optical Science) that includes all relevant effects: Nonlinear coupling, diffraction, and idler absorption. The beam propagation method is described in [8], and the signal grows from realistically modelled quantum noise with an average intensity of 1/2 photon per mode [9]. Dispersion and bandwidth effects can be handled by generalizing Eqs. (5) or (8) to sets of equations for each frequency component [9]. The nonlinear polarization term is computed in the time domain, and thus couples the different frequency components. Thermal effects can be included by use of temperature-dependent Sellmeier equations or more simply by use of thermo-optic coefficients. A steady-state temperature distribution can be computed by a separate finite difference or finite element program. A transient temperature contribution (neglecting heat conduction during the pulse) can be calculated by Sisyfos itself from the absorbed energy and the heat capacity of the material. Dispersion and thermal effects will be discussed in Sections 4.3 and 5, respectively. Previous simulation results have been compared to experiments and found to agree well [1, 10, 11].
We use M2 [12] as our beam quality measure because it is mathematically convenient and does not depend on arbitrary parameters. In the simulations it can be computed directly from the second moments of the fluence distributions in the near and far fields, where the far field is obtained by a Fourier transform of the complex amplitude in the near field. M2 does have the weakness of being sensitive to noise far from the center of the beam, but in our examples there is usually a rather sharp transition from low to high M2 values, so in this context small errors in M2 are not critical.
Throughout this work, we have simulated a monolithic OPO consisting of a single crystal with plane and parallel end faces and with non-critical phase matching. This was done to isolate the effect of idler absorption from other effects that may be present in OPOs, such as spatial walk-off. The simulations in this section are performed for single-shot operation, where steady-state thermal effects are not important. Operation with an absorbing crystal at high average power will eventually be limited by thermal effects, and in Section 5 we give estimates for the maximum pulse rate for different idler absorptions.
A sketch of the simulated OPO is shown in Fig. 1 . The peak pump fluence was fixed at 1 J/cm2 for 10 ns FWHM pump pulse length in the simulations, and scaled with (tp/10 ns)0.5 for other pulse lengths. The pump was single passed through the crystal, and the OPO was singly resonant on the signal wavelength with 50% output coupling. The crystal length was fixed to 10 mm while the effective nonlinearity, idler absorption, beam diameter and pump pulse length was varied as explained above.
Pulsed OPOs can have large pulse to pulse fluctuations in output energy and beam quality [9], so ideally, we should have run many simulations to obtain statistics for each point in the parameter space. We have done this for a few points, and for narrow beams and sufficiently long pulse lengths we found the spread to be small enough that the effect of idler absorption can be adequately represented by a single simulation at each point. However, to reduce random variations between neighboring parameter points, all these single simulations were initiated by identical noise. For some of the simulations with shorter pulse lengths, we have run multiple simulations at each point and show the average values. Since our main interest is the qualitative effect of idler absorption, we have not computed detailed statistics. Further, to save computing time, the simulations were performed with single frequency pump, signal and idler. A few simulations were performed allowing the signal and idler to have a bandwidth, and these showed no significant difference from the rest of the simulations (see black curves in Fig. 10 ).
Fig. 10 a) Comparison of performance of OPOs with small (black curves) and large (red curves) pump group velocity mismatch for no (solid curves) and large (dashed curves) idler absorption with W = 0.5 mm and tp = 10 ns. b) Signal spectral for the same cases for η ≈ 0.35 for zero idler absorption and η ≈ 0.5 for α = 300 m−1.
The simulations considered the conversion of 1.064 µm pump to 1.6 µm signal and 3.176 µm idler. Table 1 lists the parameters that were used in the simulations, and the range of variation for the parameters varied. We emphasize that, because of the scaling relations described in Section 2, the simulation results represent a much greater part of the parameter space than just the ranges show in the table.The performance of the OPO is characterized in terms of signal energy and beam quality. Figure 2 shows the set of pulse lengths and beam radii we simulate. For each point in this plane, the nonlinear gain (deff) and the idler absorption were varied in the ranges shown in Table 1.
Table 1. List of parameters used in the simulations
4. Simulation results
4.1 Signal performance
To compare simulations with different beam widths and pulse lengths, and hence pulse energies, we present the results in terms of signal conversion efficiency (η = Esignal/Epump) and signal beam quality. Figure 3 summarizes the simulations for W = 0.5 mm and tp = 10 ns. Figure 3a shows the signal conversion efficiency and signal beam quality as functions of deff for different values of the idler absorption. We see that near threshold, idler absorption reduces the output energy. Well above threshold, on the other hand, idler absorption actually improves both conversion efficiency and beam quality. The main reason for this is that absorption of the idler reduces back conversion, so that the OPO can operate with much higher conversion before the beam quality deteriorates.
Fig. 3 a) Signal conversion efficiency (solid curves) and beam quality (dashed curves) as functions of deff for different idler absorption levels (units m−1) for W = 0.5 mm and tp = 10 ns. b) Signal beam quality and conversion efficiency as a parametric plot with deff as parameter for different idler absorption for the same OPO as in a). The gray curve connects the points with deff = 16 pm/V. The deff -values for the other points are listed in the text.
It is worth mentioning that this OPO has a Fresnel number of about 25, so one might expect multi-transverse mode operation and correspondingly poor beam quality even without back conversion. The high beam quality can be explained by the cumulative spatial filtering by multiple passes through the OPO: In order to grow large, a higher order mode must not only overlap the pump beam in a single pass, but during a substantial fraction of the pump pulse. From this point of view, L c in the expression for F should be replaced by 2LcN, where the factor 2 accounts for passing the resonator in both directions, and N is the number of round trips before the OPO reaches high conversion. In pulsed OPOs, N is typically in the range 1/5 to 1/2 times Nrt. If for simplicity we take N ≈Nrt/2, we obtain F/Nrt as an approximate indicator on which beam quality to expect, and we find below that this ratio does in fact predict beam quality quite well.
To increase the readability of the figures, Fig. 3b shows signal conversion efficiency and beam quality as a parametric plot with deff as parameter. The area of high performance is in the lower right-hand corner, and the graphs show clearly how idler absorption improves the achievable performance. The rest of results will be presented in this way. The following values of deff have been simulated (units pm/V): 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, and in some cases 25, 30, 35, and 40. To make it possible to identify a particular value of deff in a figure, most figures show a line of constant deff and its corresponding value. The other values in the list can be found by counting up or down from this line.
Figure 4 shows the performance of an OPO for different pump pulse lengths, from 2 ns FWHM (Nrt = 17) to 50 ns FWHM (Nrt = 425) for 0.5 mm beam radius. Two major observations can be made from these results: 1) The tendency seen in Fig. 3b with higher possible performance with higher idler absorption, is the same for all pulse lengths. 2) In agreement with the argument above, the beam quality tends to improve for longer pulse lengths. Random pulse to pulse fluctuations start to become important for the W = 0.5 mm beam for 2 ns pulse length, and the results shown for this case are averages of 5 simulations.
Fig. 4 Simulation results for 0.5 mm beam radius for varying pump pulse length, as indicated in the graphs. Similar results for 10 ns pulse length were shown in Fig. 3b.
The effect of a varying pump pulse length has been isolated in Fig. 5 where the performance at the same idler absorption is compared. We notice that the beam quality improves significantly when the pulse length increases from 2 ns to 5 ns (Nrt = 17 to 43), and that the improvement in moving to even longer pulse length is modest. The ratio F/Nrt decrease from 1.6 to 0.6 with the increase in pulse length. We also notice that the point of optimal performance (i.e. high η and BQ, lower right hand corner of the figures) does not vary much for pulse lengths above 5 ns.
Fig. 5 Comparison of OPO performance for fixed idler absorption at different pump pulse lengths for 0.5 mm pump beam radius.
Figure 6 shows the simulation results when varying the pump beam radius for fixed 10 ns pump pulse length. For the smallest beam radius (50 µm) the Rayleigh length is ~12 mm inside the 10 mm long OPO crystal. One might suspect that the position of the beam waist would affect OPO performance in this case, but we have investigated this and found no significant effect.
Fig. 6 Simulation results for 10 ns pulse length varying the beam radius as indicated in the graphs. The corresponding graph for W = 0.5 mm is shown in Fig. 3b.
We notice that the effect of idler absorption is the same for all these beam widths. We also note that the beam quality is slightly reduced for wider beams, but remains near diffraction limited even for W = 1 mm. The Fresnel number of the resonator in this case is high (>100) clearly indicating that gain guiding in multiple round trips is the most important means for mode selection in this case. In this case, the ratio F/Nrt = 1.2. Figure 7 shows how the OPO performance depends on W for two fixed absorptions. The maximum conversion efficiency seems to be approximately independent of the beam size.
Fig. 7 Comparison of OPO performance for fixed absorption levels for varying pump beam width for 10 ns pulse width.
This behavior was also investigated for other points in the parameter space, see Fig. 2. In particular it was of interest to see if the beam quality at short pulse lengths could be improved by a smaller beam. The results are summarized in Fig. 8 . We see that for this configuration, reducing the beam radius from 0.5 mm to 0.25 mm leads to an improvement in beam quality. For smaller beam diameters there does not seem to be a similar effect. This can be understood from the argument above as F/Nrt is 1.6 and 0.4 for W = 0.5 mm and 0.25 mm, respectively, and moving to even smaller beams should therefore not be expected to have an effect on the beam quality.
Fig. 8 Comparison of OPO performance in the case of 2 ns pump pulse length and varying beam width. Solid curves are for zero idler absorption and dashed curves are for α=300 m−1.
4.2. Effect of output coupling
The simulations so far have been performed with Roc = 0.5 on the signal wavelength. Typical signal reflectivity in pulsed OPOs range from Roc = 0.3 to 0.8. Figure 9 shows the effect of idler absorption in OPOs with these two values of Roc, W = 0.5 mm and tp = 10 ns, and it can be compared with Fig. 3b for Roc = 0.5. It is clear that the pattern with increased performance with increased idler absorption is maintained in this range of output coupling.
Fig. 9 Simulation results for 0.5 mm beam radius and 10 ns pulse length with a) 30% and b) 80% signal reflectivity on the output mirror in the OPO. The curves can be compared to Fig. 3b for 50% output coupling.
4.3. Effect of group velocity mismatch for the pump
It has been shown that differences in group velocity have important effects on back conversion [13, 14]. If the group velocity of the pump is within the interval spanned by the group velocities of the signal and idler, back conversion occurs more easily than if the group velocity of the pump is outside this interval. We refer to these two cases as small and large pump group velocity mismatch (GVM), respectively. The main reason for the difference is that a large pump GVM leads to an instability that causes strong modulation of the signal and idler beams, and the resulting wide spectra partially suppress back conversion.
In the simulations so far, pump GVM has been neglected. In this section we compare the results with small and large pump GVM for zero and strong idler absorption. The group indices are ng, 1 =1.7 for the idler, ng, 2 =1.76 for the signal and ng, 3 =1.73 or 1.9 for the pump in the case of small or large pump walk-off, respectively. Apart from ng, 3, the systems are identical and have 0.5 mm beam radius and 10 ns pulse widths, as shown in Fig. 3. The results are shown in Fig. 10a. Idler absorption improves the possible performance for both cases of pump GVM. In contrast, large pump GVM does not always improve the performance: The curves for small pump GVM have sharp transitions to poor beam quality, but just below these transitions they have better beam quality then the corresponding curves for large pump GVM. Figure 10b shows the corresponding signal spectra at or close to the points where the beam quality starts to deteriorate in Fig. 10a. As expected, the signal spectrum is much wider for the case with large pump walk-off.
5. Thermal effects of idler absorption
An important side effect of idler absorption is heating of the nonlinear material. This leads to a transverse temperature gradient which causes thermal lensing and possibly a varying phase mismatch across the beam, and both these effects can reduce the performance of the OPO. The thermal lensing depends on the thermo-optic coefficient dn/dT, whereas the phase mismatch depends on the difference between dn/dT for the interacting beams. For simplicity, we restrict our attention to the former effect and set dn/dT equal for all the beams. A justification for this is that the phase mismatch in a singly resonant OPO acts independently in each pass, whereas the thermal focusing of the signal beam accumulates over all the passes. We find that a single-pass wavefront distortion of a fraction of a wavelength is enough to have severe effects on the beam quality. Since the difference in dn/dT is typically smaller than dn/dT itself, the corresponding thermal phase mismatch .
Figure 11 shows the heat load (as a fraction of the pump energy) as function of signal conversion efficiency for different idler absorption, W = 0.5 mm, and tp = 10 ns. The corresponding curves for other pulse lengths and beam widths are similar. For strong idler absorption the curves approach the linear relation (i.e. all generated idler is absorbed). For 40% signal conversion efficiency, the corresponding heat load is 15 - 18% for α = 200 – 400 m−1.
For nanosecond pulses the thermal conduction during the pulse can be ignored, and the temperature change can be separated in a transient part that depends on the heat capacity and a steady-state part that depends on the thermal conductivity. We consider first the transient part, which is independent of pulse rate, and then we estimate how the steady-state heating sets an upper limit to the pulse repetition rate.
The effect of the transient heating depends on the parameter K = (dn/dT)/Cv, where Cv is the heat capacity per unit volume. Typical values are dn/dT ≈ 10−5 K−1 and Cv ≈ 2×106 J/(m3K). Transient heating becomes important for wide enough beams and long enough pump pulses. Figure 12 compares the performance of an OPO with α = 300 m−1, W = 1 mm and tp = 20 ns with K = 0 and 5×10−12 J/m3. The transient heating reduces the beam quality by focusing the trailing part of the pulse. This is clearly seen in Fig. 12b-c, which shows the near-field profile through the beam centre versus time. In the simulations, a similar effect of transient lensing was not seen for narrower beams or shorter pump pulses.
Fig. 12 Comparison of OPOs simulated with and without transient thermal lensing. W = 1 mm, α = 300 m−1 and tp = 20 ns. K = 5×10−12 J/m3 in the case of transient lensing. a) Comparison of performance. b-c) Comparison of temporal evolution of the near-field intensity profile through the beam center for deff = 12 pm/V.
For the steady state temperature, we start with the expression for the focal length of the thermal lens in a uniformly heated cylinder is [7, Chapter 7]
where κ is the thermal conductivity of the material and Q is the absorbed power per unit volume. In our case the heating is not uniform, but this formula can nevertheless be used as an estimate of the strength of the thermal perturbation. For the purpose of an estimate, we take Q to be the absorbed power density in the center of the beam. We expect that thermal effects become important when fT is similar to Ld, the characteristic length for diffraction (we do not use the term Rayleigh length because the beams are not necessarily Gaussian). When we equate fT and Ld and insert Q = q·Fr, where q is the absorbed energy density per pulse and Fr is the repetition rate, we obtain an estimate for the maximum repetition rateWe note that q depends on idler absorption, conversion efficiency and pump fluence, and is independent of W.Equation (13) is admittedly approximate, so we regard it as a scaling relation. The important features are the dependence on the material parameters κ and dn/dT, and the scaling with W −2. In order to use this scaling relation we need some reference points, which we find by simulation. The crystal is now treated as 10 longitudinal slices and the transversal profile of the dissipated energy is stored for each slice. The absorbed energy distribution is multiplied by the pulse repetition frequency (PRF) to give the heat load, Q(r), and the resulting thermal profile is found by finite difference calculations and fed into a new OPO simulation. This process is iterated until output energy and beam quality converge. The crystal is taken to have a square cross-section (which is significantly larger than the pump beam size), to be cooled through all four side surfaces and to have heat conductivity κ = 2 W/m×K. We set dn/dT = 10−5 K−1 for all beams. Thermal expansion is neglected.
Figure 13a shows the signal conversion efficiency and beam quality as functions of pulse repetition rate for W = 0.5 mm, tp = 10 ns, and α = 100 or 300 m−1. The operating points (chosen by deff) correspond to efficiency η = 0.36 or η = 0.4, respectively, in the limit of low pulse rate. The absorbed heat per pulse is about 0.09 or 0.17 of the pump energy, respectively. Figure 13b shows the corresponding results for W = 1 mm and tp = 20 ns. Transient thermal lens was included in Fig. 13b by using K = 5×10−12 J/m3, but was negligible in Fig. 13a. In both cases, increasing the pulse rate has a stronger effect on the beam quality than on the conversion efficiency.
Fig. 13 Calculated performance from OPOs with idler absorption as function of pulse repetition frequency when thermal lensing is accounted for. a) W = 0.5 mm tp = 10 ns, η(prf = 0) ~0.4. b) W = 1 mm tp = 20 ns, η(prf = 0) ~0.4. For illustration, we have taken the values from single shot simulations and plotted them as 0.1 Hz in b).
Because of the different pulse length, q in the second example is about 1.4 times greater than in the first. With the ratio of beam radii of 2, this gives a total scaling factor for Fmax (Eq. (13)) of about 6, which is consistent with the graphs.
6. Scaling to real examples
In our simulations we vary deff (i.e. σ), α 1, W and t p, and keep Lc, and the wavelengths and refractive indices fixed. In this section we will use the scaling relations in Eq. (9)-(10) and the procedure listed in Section 2 to illustrate how the results in this work can be used for two OPOs with parameters from real nonlinear crystals.
First, consider a KTP based OPO that is type 2 noncritically phase matched (NCPM) for the conversion of 1.064 µm to 1.571 µm and 3.298 µm. Absorption at the idler wavelength in this case is 52 m−1 [15], and the effective nonlinearity for the interaction is 2.9 pm/V [16]. We take the OPO to be pumped by a 1 mm radius Gaussian beam with 30 ns FWHM pulse length and 100 MW/cm2 peak intensity (i.e. 50 mJ pump energy), and the KTP crystal to be 20 mm long. Table 2 lists the parameters for this OPO (‘OPO 1’) scaled to the simulations in this work.
Table 2. Original and scaled parameters for two realistic example OPOs. The columns labeled OPO 1 and OPO 2 show the physical parameters, and the columns labeled “scaled” show the parameters for the equivalent OPOs in our simulations. The shaded cells show the values we use to look up the expected efficiency and beam quality in our graphs
The scaled OPO data do not correspond exactly to any of the simulations presented in this work, but the simulations with the closest parameters can nevertheless give an estimate of its expected performance. The scaled absorption is ≈100 m−1, so we can use the 100 m−1 curves. From Fig. 5 we notice that the difference between 10 ns and 20 ns pump pulse length is marginal. Further, studying Fig. 3b for W = 0.5 mm and Fig. 6c for W = 1.0 mm, both indicate that a deff value in the 7-9 pm/V range seems to be optimal for this configuration. The scaled deff value is in this range, so high performance can be expected. If the pump intensity is reduced, this corresponds to reducing deff in the scaled parameters, so the operating point in Fig. 3b/6c would then move towards lower conversion efficiency.As a second example, consider a PPLN OPO with quasi phase matching for conversion of 1.52 µm to 2.13 µm + 5.30 µm (i.e. ~26 µm grating period). The idler absorption is in this case ~120 m−1 [17] and the effective nonlinearity is 11 pm/V ([18] using Miller’s rule [19]). Further, we take the OPO to be pumped by a Gaussian beam with 0.3 mm radius, 20 ns pulse length and 80 MW/cm2 peak intensity, and the PPLN crystal to be 15 mm long. The scaled OPO parameters are listed as OPO 2 in Table 2.
If we study the 200 m−1 curve in Fig. 6b, we see that the optimal value of the scaled deff is 10-12 pm/V. The scaled value for the example is only 8.3 pm/V, so a conversion efficiency around 35% can be expected from this OPO. If the OPO is tuned to a shorter idler wavelength, idler absorption is reduced while the other parameters in Table 2 are not significantly changed. For example, for 4.5 µm idler wavelength, the idler absorption is 10 m−1. In this case the curve for zero idler absorption in Fig. 6b is applicable, and we notice that the beam quality has deteriorated because the pump intensity is higher than optimal.
7. Discussion
Poor beam quality can always be described in terms of multiple transverse modes, but it may be caused by different mechanisms that require different methods for suppression.
The first and most fundamental reason for multi-mode operation is simply that the resonator and the gain medium do not give sufficient spatial filtering to suppress the higher order modes. Because the mode amplitudes grow from random noise, this situation is also characterized by random pulse-to-pulse fluctuations of the beam shape. A simple resonator can give enough filtering if the ratio of the Fresnel number to the number of resonator roundtrips during signal build up is small, but when this ratio is large it gives very little suppression of higher order modes. This kind of multi-mode operation can occur even at low power. It can be suppressed to some extent by unstable resonators [20], image-rotating resonators [21] or by exploiting walk-off in orthogonal directions [11], but with the simple resonator in our examples it can be seen for the greatest beam diameter or the shortest pulse length.
Even if a resonator has fair suppression of higher order modes, it can give poor beam quality if some mechanism couples power from the fundamental mode into the higher order modes [22]. Back conversion is one such mechanism, and under bad conditions it can couple a large fraction of the signal power into higher order modes in a single pass through the crystal. This is where idler absorption can help, and our results show clear improvement for a wide range of parameters. However, idler absorption has little effect on the beam quality until the conversion efficiency exceeds about 30%.
In order to avoid the complications of random fluctuations and transient thermal lens, which also becomes important for wider beams, we have only shown examples with beam radius W ≤ 1 mm. However, our simulations indicate that idler absorption can improve performance even in this regime.
8. Conclusions
In conclusion, we have shown that OPOs with significant absorption of the idler beam, may perform better than without idler absorption both in terms of signal conversion efficiency and signal beam quality. The reason is that idler absorption can suppress back conversion.
We have investigated this effect in a simple OPO with plane mirrors and found consistent results over wide ranges of pulse length and beam widths. The details would be different with other resonators, but because back conversion is a problem regardless of the resonator, we believe that the effect of idler absorption is also general. Although the simulations are necessarily limited to cover a small part of the parameter space, the results can be scaled to other parameters. Our graphs can therefore be used to obtain quick performance estimates for real OPOs. Idler absorption is, of course, not beneficial for the idler beam from the OPO. For optimum performance on the idler wavelength, idler absorption in the OPO should be minimized.
Idler absorption results in a heat load and, consequently, a thermal lens in the nonlinear material. The detrimental effect of the heating increases with beam width, and we have presented an estimate of how the maximum pulse rate for high performance scales with beam width and other parameters.
Our results also illustrate how the beam quality depends on the pump pulse length, and that an OPO with a large Fresnel number can give high beam quality provided the number of resonator round-trips during the build-up phase is comparable to or larger than the Fresnel number.
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