1. Introduction

Intracavity second harmonic generation is one of the most fundamental frequency conversion techniques, and as such has served as the basis for several useful applications. Some example applications include laser ranging, medicine, satellite communication, and laser projection [1–4]. The advantage of the intracavity design over the simpler single pass design, where the crystal is located outside of the laser cavity, is tied to the intensity dependence of nonlinear frequency conversion. The intensity of the fundamental wave is significantly higher when it is confined inside of a resonator structure, and thus efficient SHG can be achieved. Therefore, it was a natural conclusion to place the nonlinear crystal inside the laser cavity for higher conversion efficiencies. Many nonlinear crystals have been used for intracavity SHG applications, such as KH₂PO₄ (KDP), LiB₃O₅ (LBO), $\beta $-BaB₂O₄ (BBO), KTiOPO₄ (KTP), and LiNbO₃ (LN). These crystals have varying advantages and disadvantages, but all can be used to successfully frequency double light in the visible and near-infrared wavelength ranges. As a result of intracavity SHG’s nature as a fundamental nonlinear process, it has been extensively studied. This has resulted in an evolving model of the physics of intracavity SHG over the last ∼50 years. Early models developed by Polloni et al., Smith, Kennedy et al., and Volosov et al. focussed on single mode operation and derived several useful relations, such as the maximum output power of SHG being equal to the maximum output power at the fundamental wavelength and the well known “phase problem” resulting in half the generated SHG power being lost to absorption or destructive interference in the cavity [5–9]. Further developments of the model by Rice et al., Baer, and Oka et al. expanded the model to include multiple modes and polarization of the fundamental and second harmonic waves and showed that multiple longitudinal modes could cause instabilities through both SHG and sum frequency generation [10–12]. These models have been instrumental in furthering the understanding of intracavity SHG for both theoretical and practical purposes. It is well known that two mechanisms can cause the depletion of the fundamental wave in the intracavity SHG process. First, the energy conversion from the fundamental wave to SHG wave in the SHG nonlinear process results depletion of the fundamental wave. Second, as a result of the first process, the circulating fundamental power within the laser cavity is reduced due to the additional loss of the fundamental wave caused by the SHG nonlinear process. However, to date, the depletion of the fundamental wave due to these two mechanisms and its dependence on operation temperature and pump power in intracavity SHG has not been well studied.

To date, a model that comprehensively examines the depletion of the fundamental wave in intracavity SHG and shows good agreement with experimental results has not been reported. In the models mentioned previously, the depletion effect was accounted for by fitting a nonlinear coupling coefficient to a rate equation model of the laser cavity, and this coefficient was used as a constant fitting parameter and was independent of operation temperature [5–12]. As a result of this approach, the previous models could not explain simultaneously both the main peak and side peaks in the measured SHG temperature tuning curve (in terms of SHG power and bandwidth). More specifically, the reported models could not predict all three main features of an intracavity SHG temperature tuning curve: the peak power of the main peak [7,9–12], the width of the main peak (temperature tolerance, measured as full width half maximum of the main peak) [5,7–9,11,12], and the height of the side peaks [5–12].

In this paper, for the first time, we present a model that comprehensively and intuitively models the depletion of the fundamental wave in intracavity SHG due to both the mechanisms mentioned above. It is shown that by considering the depletion of the fundamental wave as a function of temperature and pump power, it is now possible to explain the reduced peak power, wider temperature bandwidth, and taller side peaks observed in SHG temperature tuning curves. The model is validated against experimental data for periodically poled lithium niobate (PPLN).

2. Theory

In this model, we are considering the pump power incident on a laser crystal (e.g. Nd:YVO₄) to generate a fundamental wave within a laser cavity formed by two mirrors. The fundamental wave will be incident on a nonlinear crystal (e.g. PPLN) to generate an SHG wave. Linear and nonlinear losses in the cavity were considered in the model. The linear loss is a result of imperfect coatings at the facets of the laser crystal, nonlinear crystal, and mirrors, while the nonlinear loss is caused by both the depletion of the fundamental wave (due to the energy conversion from the fundamental wave to SHG wave) and reduction of the fundamental wave within the laser cavity due to the cavity loss as a result of the SHG nonlinear process. In Fig. 1, a high level block diagram of the model is presented, demonstrating the different components under consideration. The novel part of this model is that we consider the nonlinear loss as a function of operation conditions (i.e., PPLN temperature and pump power) in simulating the temperature tuning curve, instead of treating the nonlinear loss as a fitting parameter to match the simulations with the measured experimental data as reported in the literature.

 

Fig. 1. The intracavity SHG structure being simulated consists of a pump diode (808 nm), a gain medium (Nd:YVO₄), all sources of loss (absorption, scattering, nonlinear depletion, etc.) and the nonlinear crystal producing the SHG output (PPLN).

Download Full Size | PDF

Thus, the first step is to determine the intracavity power for the fundamental wave, which was done using a plane wave solid state laser rate equation. Since the gain medium is the well-studied Nd:YVO₄, a four-level laser model is suitable, and results in the two following differential equations [13]:

The first equation represents the population of atoms in the upper level of the gain medium, while the second equation represents the photon density in the resonator. The value of each parameter used in the simulations will be presented in Section 3. These equations can be numerically integrated to provide a steady-state photon density, which can be converted to an intracavity power using the following relation:

The novel part of the intracavity model with regards to previous models is introducing the nonlinear loss in the photon lifetime term ${\tau _c}$. The photon lifetime is determined by dividing the round-trip time of the resonator with a loss term $\delta $:

In all cases, the various loss terms are typically represented as decimal percentages (i.e. a 1% round trip loss would be $\alpha $=0.01). The next step is to determine the SHG power using the coupled wave equations as derived by Boyd [14]:

In which the first equation represents the evolution of the fundamental wave and the second equation represents the generated second harmonic wave. In the equations, ${A_1}$ and ${A_2}$ are the field amplitudes of the fundamental and second harmonic waves, respectively. d$\; $ is the nonlinear coefficient of the nonlinear crystal along the propagation direction of the fundamental wave, which for PPLN is typically ${d_{33}}$, $\textrm{z}$ is the propagation distance into the nonlinear crystal, ${\omega _1}$ and ${\omega _2}$ are the angular frequencies of the fundamental and second harmonic waves, respectively, and $\Delta k = 2{k_1} - {k_2} - 2\pi /\Lambda $ is the phase mismatch between the fundamental and second harmonic waves, where ${k_i} = 2\pi {n_i}/{\lambda _i}$ is the wave number and the last term represents the effective momentum imparted by the periodic poling of the nonlinear crystal with period $\Lambda $. The temperature dependence of the phase matching can be modelled by the Sellmeier equation for magnesium doped PPLN [15]:

The coupled wave equations can then be numerically integrated by selecting an operating temperature for the nonlinear crystal, where we have assumed that the temperature is held constant via an attached heat sink and thermoelectric cooler in experiments. The step size for the numerical integration is half the poling period of the nonlinear crystal. This is due to the fact that the sign of the nonlinear coefficient will change when the domain of the nonlinear crystal is inverted. As we are assuming equal positive and negative poling (i.e. the poling follows a square wave pattern with a 50% duty cycle, alternating between positive and negative), this leads to a step size of one half the period as the domain flips each half-period. It is also assumed that the initial SHG power is zero.

Now that all the individual parts of the model are accounted for, they can be combined into the final numerical model. This is done by first determining the circulating fundamental wave power in the resonator with rate equations, and then using this fundamental power to calculate the appropriate SHG power and corresponding nonlinear loss. Then, the intracavity power is updated using the new loss, generating a new SHG power and loss, and the process continues until the SHG power is no longer changing between iterations. The flowchart in Fig. 2 illustrates each step of the model, which is carried out at multiple temperatures to generate temperature tuning curves that can be easily replicated in the experiments using a thermoelectric cooler attached to the nonlinear crystal.

 

Fig. 2. A flowchart illustrating the steps carried out by the Python script to generate the final temperature tuning curves. This process is repeated at each temperature until a sufficient temperature tuning curve is generated.

Download Full Size | PDF

Equations (1), (2), (9), and (10), were implemented in Python and numerically integrated using a fourth order Runge-Kutta solver from the SciPy library. We can infer from the flowchart that the initial nonlinear loss will be quite high, as the estimated SHG power will be highest in the first iteration. The nonlinear loss was determined using Eq. (8), where the total generated SHG power is compared to the initial fundamental power. While the forwards and backwards power circulating in the cavity contribute to the nonlinear loss, it is assumed that only the forward power generates useable SHG power, as Smith showed that the backwards SHG power is typically absorbed in the gain medium [6]. In the following iteration, the addition of the nonlinear loss greatly decreases the circulating cavity power, and thus the next estimate of the SHG power will be significantly lower. This leads to a lower nonlinear loss term than the first iteration. This process will continue, and each iteration the nonlinear loss will come closer to converging on the true “final” value of nonlinear loss, and thus the expected SHG power, at each temperature. In the previous models, it is typically assumed that the circulating fundamental power at each temperature is a constant determined by a fitting parameter called the nonlinear coupling coefficient. The novelty of our approach is that a unique nonlinear loss and thus a unique fundamental power is associated with each temperature, greatly increasing the height of the side peaks, and increasing the temperature bandwidth of the main peak. The predicted temperature tuning curves generated by the model are compared to experimental results in Section 4.

3. Experimental setup

The experimental setup, illustrated in Fig. 3, is a diode-pumped solid-state (DPSS) laser-based resonator with a frequency doubling nonlinear crystal inserted into the cavity. An 808 nm pump diode emitting up to 5.5 W optical power was launched into a 1.1 atomic percent Nd:YVO₄ laser crystal with dimensions of 3 × 3x5 mm. The Nd:YVO₄ input facet had a high reflection coating at 1064 nm and high transmittance coating at 808 nm that serves as the end mirror for the cavity. The output facet of the Nd:YVO₄ had an anti-reflection coating at 1064 nm, followed by a PPLN crystal which was mounted to a heat sink with a thermoelectric cooler that can be controlled to within 0.1 $^\circ \textrm{C}$. There was an output coupler positioned after the output facet of the PPLN that had high reflectivity at 1064 nm and high transmittance at 532 nm. The facets of PPLN were anti-reflection coated at 1064 nm to minimize round-trip losses in the cavity. The mirror-to-mirror cavity length was set to 40 mm. This forms the basic SHG intracavity structure used in the experiments. For the simulation inputs, some values that could not be easily measured via experiment such as the absorption and stimulated emission cross sections were taken from the reported papers [15–20]. The inputs to the simulation are listed in Table 1.

 

Fig. 3. Experimental intracavity SHG setup

Download Full Size | PDF

Table 1. The parameters used in the coupled rate equation model. Parameters were measured experimentally where possible.

View Table

4. Discussion and results

4.1 Model validation

To validate the model, simulations are compared with the observed experimental results. The first thing that must be validated is the plane wave rate equation model, as modelling the laser cavity accurately is required to ensure accurate SHG results. The first experiment that was performed was measuring the leaking 1064 nm power from the laser cavity with a dummy PPLN crystal inserted. The dummy PPLN was a crystal of equivalent length to the PPLN used in the SHG experiments, however the temperature of the PPLN crystal was tuned to ensure that no SHG light was being generated. This allowed us to be confident that the experimental results included the additional insertion loss of the PPLN crystal without including any nonlinear loss from SHG. This was verified by measuring the output power of the laser with and without a 1064 nm optical filter placed in front of the power meter. The leaking pump power from the 808 nm pump diode was measured to be ∼3 mW and was subtracted off all measurements, and the experimental results include the loss from the output coupler and the wavelength filter. This additional loss is accounted for in the simulation as well. In Fig. 4, the x-axis is the 808 nm optical pump power, and the y-axis is the leaking 1064 nm power. The squares represent the experimental data while the solid line represents the predicted leaking power from the simulation. The simulation parameters used are shown in Table 1. As shown in Fig. 4, after overcoming the laser threshold at ∼0.15W pump power, the leaking 1064 power increases linearly with pump power as expected. As shown in Fig. 4, the simulation correctly predicts both the slope efficiency of the laser as well as the laser pump threshold, implying that the input parameters of the cavity simulation are correct. The simulation curve was generated by setting the pump power and output coupler reflectivity according to the experimentally measured values, and then varying the linear loss ${\alpha _l}$ until the slope and threshold predicted by the simulation is correct.

 

Fig. 4. Leaking 1064 nm power vs 808 nm pump diode power.

Download Full Size | PDF

The second verification was to measure the leaking 1064 nm power when the PPLN was temperature tuned to produce maximum SHG power. This metric allows a direct comparison between the estimated total intracavity power of the simulation versus the experimental results, as the output power can be used to derive the total circulating intracavity power. These results were gathered by placing a 1064 nm filter in front of the power meter that has effectively 0% transmission at 532 nm and subtracting off the 808 nm pump power as mentioned previously. In Fig. 5, the x-axis is the 808 nm optical pump power, and the y-axis is the leaking 1064 nm power. The squares represent the experimental data while the solid line represents the predicted leaking power from the simulation. The simulation parameters used are shown in Table 1. The “Correct” PPLN referred to in the title is a PPLN crystal that has been temperature tuned for optimal SHG conversion efficiency. As shown in Fig. 5, with the increase of 808 nm pump power, the leaked 1064 nm fundamental wave power increases linearly. The simulations agree well with the experimental results, implying that there is a significant reduction in the circulating 1064 nm power once SHG begins due to the noticeable drop (∼200 mW) in leaking 1064 nm power.

 

Fig. 5. Leaking 1064 nm power vs. 808 nm pump power when SHG power was maximized.

Download Full Size | PDF

The final validation was the comparison of the temperature tuning curves predicted by the simulation versus the experimental temperature tuning curve for a 1.1 mm long PPLN crystal. In Fig. 6, the x-axis is the temperature of the nonlinear crystal in degrees Celsius and the y-axis is the 532 nm SHG power measured at the detector. The squares represent the experimental data while the solid line represents the predicted SHG power from the simulation. The simulation parameters used are shown in Table 1, and the experimental setup is described in Section 3. As shown in Fig. 6, the changing temperature of the nonlinear crystal significantly affects the phase matching and thus the efficiency of the nonlinear process. We can see excellent agreement between the predicted temperature tuning curve and the experimental results, with the only discrepancy being the non-symmetrical nature of the experimental data versus the simulated results. This is most likely a combination of two problems, which is defects in the PPLN due to manufacturing limitations, as well as the attached TEC being unable to completely eliminate temperature gradients in the crystal, causing thermal lensing which may differ with temperature. It can also be observed that the simulation correctly predicts the temperature bandwidth of the main peak and the peak SHG power. These results are promising, as it implies that the main factor that must be considered to properly model intracavity SHG is the induced depletion of the fundamental wave and the resultant loss of the circulating fundamental power within the cavity, which we have considered in our model. The results in Fig. 6 were collected by introducing a 532 nm filter in front of the power meter to filter out the leaking 1064 nm light as well as accounting for the transmission of the output coupler for 532 nm (T∼99%).

 

Fig. 6. Experimental results for 1.1 mm long PPLN sample compared to the predicted results from the simulation. The asymmetry of the curve is likely due to imperfections in the PPLN and the thermal lensing in the PPLN as a result of temperature gradients in the crystal.

Download Full Size | PDF

The experimental data in Fig. 6 exhibit the three main features of intracavity SHG, with a lower peak power than that predicted from single pass theory, a noticeably higher temperature tolerance, and pronounced side peaks. As shown in Fig. 6, our model successfully predicts the peak value and temperature bandwidth and matches the side peaks relatively well. For comparison’s sake, the temperature bandwidth observed for intracavity SHG is ∼30 degrees, while single pass theory predicts a temperature bandwidth of ∼18 degrees for the same crystal. This illustrates the importance of modelling the intracavity SHG processes well, as the temperature bandwidth increases by ∼67%.

4.2 Depletion of fundamental wave

We can use the results to intuitively explain the physical phenomena that causes these three main effects for intracavity SHG. The higher the efficiency of the nonlinear process, the greater the depletion of the fundamental light and the larger the nonlinear loss is. This results in a wider temperature bandwidth, as the lower SHG efficiency due to phase matching as we move away from the peak temperature is partially offset by a larger input fundamental power due to the lower loss. This also explains the larger side peaks that are observed, as while the phase matching is much poorer at the side peaks, the fundamental power at the side peak is ∼370W versus the ∼265W present at the main peak, again partially offsetting the lower efficiency. The fundamental powers referenced here can be observed in Fig. 9, specifically the dashed line which represents the total intracavity power as a function of temperature.

The importance of the nonlinear loss is further reinforced when comparing the SHG temperature tuning curve with and without considering the nonlinear loss. In Fig. 7, the x-axis is the temperature of the nonlinear crystal in degrees Celsius and the y-axis is the 532 nm SHG power predicted by the single pass theory. The solid line represents the simulation prediction when there is no nonlinear loss and so the single pass input power is equivalent to the circulating cavity power with no nonlinear loss, while the dashed line represents the predicted SHG power when a temperature independent nonlinear loss of 0.45% was added to the single pass model, reducing the input power to the nonlinear crystal significantly. The 0.45% value was calculated using Eq. (8) using only the peak SHG power for ${P_{NL}}$.. As an example, if the total forward and backward generated SHG power is 2.4W and the total intracavity power is 265W, then the loss would be 0.45% according to Eq. (8). It is important to keep in mind that only the forward SHG power (1.2 W from the example) would be observed experimentally, but both the forward and backward SHG power contribute to the nonlinear loss in the cavity. The main observation in Fig. 7 is that if we do not account for nonlinear loss and the resulting drop in fundamental input power, the expected SHG power is significantly higher than what is experimentally observed. We also see that by adding a nonlinear loss term and using the reduced fundamental power results in a prediction that is much closer to the observed experimental values. As a result, it is clear that the nonlinear depletion must be accounted for to obtain accurate predictions of SHG power.

 

Fig. 7. Single pass results when assuming no nonlinear loss versus incorporating the nonlinear loss.

Download Full Size | PDF

It is important to point out that the current model is not perfect, as it tends to slightly overestimate or underestimate the side peaks while correctly predicting the peak value and temperature bandwidth of the main peak. This could be due to a number of things, the first being the plane wave assumption used in the cavity rate equation model. This assumption was used as we were only interested in the total circulating intracavity power, and the plane wave assumption allows us to calculate this in a simple manner. However, the lack of a Gaussian model means that it is not possible to account for the intensity variation in both the transverse plane of the laser and the photon density distribution throughout the cavity. This also means we cannot fully account for the difference in the fundamental beam diameter versus the SHG beam diameter in the PPLN, and so perfect beam overlap was assumed. Another potential issue is that we have assumed that the beam diameter in the PPLN is a constant with temperature. However, Rice et al. showed that even with an appropriate heat sink and temperature controller, significant temperature gradients can arise in the nonlinear crystal, and there is potential for non-uniform thermal effects to change the beam diameter in the PPLN [6]. This could partially account for the slightly incorrect side peaks, although it is still clear that the current model offers an improvement over previous models in this regard. As a result, there is room for improving the model in the future.

There are also two interesting physical phenomena that arise as a result of the new model. The first is that the fundamental wave depletion and nonlinear loss depend on the 808 nm pump power significantly. In Fig. 8, the x-axis is the 808 nm pump optical power, and the y-axis is calculated nonlinear loss term due to fundamental wave depletion via SHG. The solid line represents the calculated nonlinear loss from the simulation. The simulation parameters used are shown in Table 1. The nonlinear loss increases quickly as the pump power increases, and the overall trend is that the rate of change of the nonlinear loss slowly decreases as pump power is added. We can also observe that at high pump powers (7–8 W) the loss saturates as we reach the maximum SHG power attainable for the crystal. This is a result that, to the best of our knowledge, has not been reported previously and has implications on finding an optimal pump power for a given intracavity SHG application.

 

Fig. 8. The nonlinear loss as a function of the pump power

Download Full Size | PDF

 

Fig. 9. Total circulating intracavity power (right axis) and nonlinear loss (left axis) as a function of temperature.

Download Full Size | PDF

The second interesting physical phenomenon is that the nonlinear loss and thus the circulating fundamental power has a strong dependence on the temperature. This is to be expected for an intracavity laser, as a small increase in round-trip loss can drastically decrease the fundamental power. As we change the temperature of the PPLN crystal, we also change the SHG efficiency. This greatly affects the induced nonlinear loss and the depletion of the fundamental wave. In Fig. 9, the x-axis is the temperature of the nonlinear crystal in degrees Celsius, the left y-axis is calculated nonlinear loss term due to fundamental wave depletion via SHG, and the right y-axis is the circulating fundamental power within the cavity. The solid line represents the calculated nonlinear loss from the simulation and corresponds with the left y-axis, while the dashed line represents the circulating fundamental power and corresponds with the right y-axis. The simulation parameters used are shown in Table 1. We can see at the points with zero nonlinear loss, the circulating fundamental power is close to 400W, while with a maximum nonlinear loss of 0.45%, the circulating fundamental power drops to ∼265W. This is a significant drop in circulating fundamental power and greatly reduces the expected SHG power and the leaking 1064 nm power, which is a result that cannot be predicted by previous models that use a fixed nonlinear loss at all temperatures. We can see from the agreement of the model with the experimental results that the depletion of the fundamental wave is by far the most important physical phenomena to consider when attempting to model intracavity SHG, as it explains the three effects explained earlier that can be observed in experimental SHG results.

5. Conclusion

In conclusion, a plane wave DPSS laser model has been combined with the coupled nonlinear wave equations for second harmonic generation to create a simple and intuitive model of the physics of intracavity generation of green light using PPLN. The simulation results for intracavity SHG agree well with the experimental results. It has been shown that the dominant factor differentiating intra-cavity SHG from single pass SHG is the addition of accounting for the nonlinear loss and the corresponding drop in circulating fundamental power. As a result, the simulation correctly predicts the main peak SHG power, the wider temperature bandwidth, and the more prominent side peaks observed in intracavity SHG experiments. It has been found that the nonlinear loss and fundamental wave depletion depend heavily on both the 808 nm pump power and PPLN temperature. It is expected that the presented model can be applied to the other nonlinear processes such as three wave mixing.