A 2 + 1 dimensional nonlinear pulse propagation model is presented, illustrating the weighting of different effects for the parametric amplification of ultra-broadband spectra in different regimes of energy scaling. Typical features in the distribution of intensity and phase of state-of-the-art OPA-systems can be understood by cascaded spatial and temporal effects.
© 2013 OSA
Ultra-short laser pulses with several hundred MWs of peak power are becoming increasingly popular for a variety of applications, e.g. high harmonic generation for attosecond spectroscopy , taking advantage of the enormous intensities and the high temporal resolution. Optical parametric amplifiers (OPA’s) in combination with ultra-broadband seed sources are ideally suited to generate these ultra-short pulses especially with high output powers at high repetition rates .
Traditional laser-based amplifier systems are restricted by the amplification bandwidth of the active laser medium. For example, Ti:Sapphire with the broadest gain bandwidth known for a laser crystal supports an amplification bandwidth of 230 nm (FWHM) around the central wavelength at 850 nm . Beside the limited choice of the spectral amplification window the gain narrowing effect during amplification limits the pulse duration of high-energy pulses to some tens of fs. Furthermore, the rather small single pass gain, determined by the maximum reachable population inversion in the laser crystal, requires a complex multipass arrangement. Finally, the thermal load from the multi-level quantum defect limits the power scalability of these laser amplifiers especially for high repetition rates.
In contrast, the amplification principle of an OPA is based on the parametric second-order nonlinearity driven by a strong pump pulse transferring the energy to the signal and idler pulses without energy storage in the nonlinear crystal . Due to energy conservation there is basically no thermal load in the crystal, allowing for a relaxed thermal management and opening an enormous potential in terms of energy and average power scaling. Furthermore, the small signal gain on the seeded signal pulse can easily exceed a factor of 104 which allows for a very compact and simple setup . Contrary to the limited amplification bandwidth of a laser crystal, the amplification bandwidth of an OPA is only limited by the transparency window of the chosen crystal and the phase-matching conditions. For a variety of non-linear crystals, e.g. Beta-barium borate (BBO), the transparency range spans from the UV up to the infrared spectral region. The problematic of a rather small phase-matching bandwidth as it is the case in collinear geometry can be overcome if the signal beam is propagating under a certain noncollinear angle to the pump beam. When this angle matches the so called magic angle, which varies with the specific crystal and the chosen pump wavelength, the phase-matched spectral bandwidth becomes broadest up to the few optical cycle level . Furthermore, due to the flexibility in choosing the amplification window by using different pump wavelengths or different phase matching conditions, the OPA concept allows for wide tunability. This allows for arranging a chain of parametric amplifiers to amplify different spectral regions of an ultra-broadband seed pulse in different stages (see e.g. a quite recently published 1.5 octave spanning amplification system based on the superposition of two non-collinear OPA stages pumped in the green and in the blue ). A deeper look into the details of this OPA reveals remarkable complex spatio-temporal pulse shaping dynamics. Especially phase and intensity distribution in the spectral overlap region between the two stages rises fundamental questions about their precise origin that cannot be explained by the simple non-collinear phase matching picture neglecting spatial propagation effects . For example, the birefringence of the crystal causes a difference between the direction of the wave vector and the Poynting vector of the extraordinary wave, the walk-off; hence the spatial overlap between pump, signal, and idler beam depends on the orientation of the non-collinear geometry. In one orientation, the non-collinear angle between pump and signal pulse is compensated by the walk-off; this geometry is called Poynting vector walk-off compensation geometry (PVWC ,). The opposite orientation with the uncompensated non-collinear propagating signal pulse is the tangential phase-matching geometry (TPM). Additionally, parasitic effects have to be taken into account. In case of BBO pumped at a wavelength of 515 nm the broadband phase matched signal pulses in the PVWC geometry are simultaneously phase-matched to be frequency doubled by a second harmonic generation process (SHG) to wavelengths around 450 nm. The same affects also the idler wave around 1.1 µm . Finally, for rather long crystals and a high pump peak power, the spatially and temporally distributed cascaded back- and forward conversion between signal, idler, and pump pulse gains importance. E.g. absorption of the idler pulse can lead to a suppression of the back conversion especially for long crystals . All these complex nonlinear interactions shape the spectral phase of the signal pulse significantly; we refer to it as the parametric phase [10, 11].
In recent years some models were presented to clarify the complex interplay between the nonlinear effects to some extent [8, 12]. However, no sufficient explanation of some prominent performance characteristics in recent work (e.g. Harth et al. ) is offered by these treatments yet. This is mainly due to the absence of spatial effects in most of these models. The work done by Arisholm et al.  includes spatial effects, but parasitic effects are neglected.
In this paper we present a new 2 + 1 dimensional model including all possible second-order interaction processes as well as phase-matching, diffraction, and walk-off for all involved pulses. As described in more details in the following section, due to the generalization of pump, signal and idler pulses to only one ordinary and one extraordinary field, even unexpected mixing products as well as the mentioned temporal, spatial, cascaded, and parasitic effects are automatically taken into account. This is the main distinctive feature of this novel simulation tool. Supplementary, the neglect of the third – rather unimportant – spatial dimension allows for moderate computing times on standard desktop computers without substantial loss of accuracy. In fact, the simulation enables extensive parameter scans, allowing for precise prediction of beam profiles in one plane, the compressibility of the amplified pulses, the peak intensities, as well as a good estimation of the expected pulse energy.
In the following sections, we introduce our model and compare the results to experimental observations from different OPA’s. In section 4 we refer to the results of the second stage of the two-color OPA. In section 5 we compare the first stage of the two-color OPA and our new experimental high power OPA to the calculations. Finally, we discuss the simulation results of three OPA-systems working in different energy regimes and phase-matching geometries. A closer look to the details of these comparisons helps for a better understanding of the interplay between the contributing linear and nonlinear effects, and allows for the formulation of guidelines for future OPA development.
2. Numerical model
The numerical treatment of second-order nonlinear interactions, especially for parametric difference frequency generation, is typically based on split-step Fourier methods solving the three coupled nonlinear differential equations [8, 12]. Each equation describes the change for pump, signal and idler wave separately. Hence, each phase-matched nonlinear process beyond the considered difference mixing is needed to be implemented individually by adding additional differential equations for each mixing product . Especially for new phase-matching concepts and new nonlinear crystals, the relevance of a specific phase-matched mixing-process is difficult to foresee.
In contrast, the presented (2 + 1)D simulation handles the second-order nonlinear interaction of ultra-short light pulses with only two coupled differential equations for the two orthogonal linear polarisations. We restrict our model to one spatial transverse dimension forming the phase matching anglewith the optical axis of the crystal . Both, the ordinaryand the extra-ordinary simulation field array are chosen to be large enough to include all pulses and their mixing products in time and in space. The Fourier transform of the fields reveals the temporal spectrum as a function of the optical frequenciesand the spatial spectrum as a function of the spatial frequencies. The spatial frequencies can be easily translated to propagation anglesrelative to the main angle  via
The nonlinear interaction between the two fieldsand inside each field itself, is modelled in time/space domain by the differential equations15]. Each term is responsible for one type of nonlinear second-order interaction. Usually, the treatment of OPA simulations is limited to the Type I (oo-e) forward- and back conversion processes. Of course, the final relevance of the additional terms is determined by phase matching. Since the nonlinear interaction parameterdepends on the optical frequency and the direction dependent refractive index especially for the extraordinary field, the factor is applied in Fourier domain.
In summary, the simulation includes all possible second-order interaction processes as well as phase-matching, diffraction and walk-off for all involved pulses. Higher order effects such as self-phase modulation, self-steepening and self-focusing could be easily added. They have been tested but play a minor role in the cases considered here; we omit them for clarity. Due to the restriction of the spatial dimensions to one essential plane (2048x1024) the computing time for 500 steps in 5 mm BBO on a low cost GPU (NVIDIA GeForce GTX 680) was about 10 seconds.
3. Interpretation of the simulation data
Signal, idler, pump, and the parasitic mixing products can be distinguished and separated either by their polarisation, the optical spectrum, or by the direction of propagation. An illustrative way of interpreting the data can be obtained by plotting the fields in Fourier domain against the optical frequency axis and the relative propagation angle calculated by Eq. (1). The example in Fig. 1(a) shows the phase-matched waves from typical non-collinear amplification in 5 mm BBO. The map contains both, the spectral intensities of the extraordinary field (green labels), with strong pump pulse and the second harmonic (SH)-mixing products, and the ordinary field (red labels). The ordinary field includes the broadband signal spectrum, seeded at the magic angle = 2.4° and the angularly dispersed idler at negative angles aroundas it is generated in walk-off compensation geometry .
The corresponding phase-matching conditions for the chosen pump direction ( = 24.35°) and wavelength (515 nm) are indicated by the black double curves in Fig. 1(a), calculated from the simple non-collinear phase-matching picture. The green dotted curve represents the phase-matching for the second harmonic generation in general; the parasitic signal and idler SHG at their respective emission angle are clearly observed. However, the simulation also reveals non-obvious phase-matched mixing products, e.g. the difference frequency generation (DFG) signal at 550 nm / 5° which is pumped by the parasitic signal-SH and seeded by the idler wave at negative angles. The right hand side of Fig. 1 shows intensity plots of the individual electrical fields in time/space regime. In Fig. 1(b) the cascaded back- and forwardconversion and the parasitic mixing products can be seen clearly by the modulation of the depleted part of the pump pulse and the fringed hunches outside the overlap-region. Figure 1(c) and (d) show the effect of the asymmetrically distributed conversion efficiency caused by the temporal overlap (at −500 fs) of the chirped signal pulse and the self-compressed idler pulse which is generated with the opposite chirp of the signal. Note, the second-order dispersion of BBO becomes negative at wavelengths beyond 1.5 µm, as a result the long idler wavelengths at large anglesare stretched in time (see Fig. 1(d)). The inset in Fig. 1(c) reveals the signal power spectrum and phase.
Due to the differentiated and clear access to the full spectral/temporal and spatial/angular information in amplitude and phase at each point of the nonlinear propagation through the crystal, the simulation allows for a thorough observation of the complex spatial and temporal pulse shaping dynamics. With this tool, various effects such as e.g. the unexpected broadness of an amplified spectrum as well as distinct spectral shapes as described in the following sections, well known from various broadband and high-power OPA’s, can be explained for the first time.
4. Experiment and simulation: two-color pumped OPA-system
The two-color OPA-system from Harth et al.  is pumped at 515 nm and 343 nm in two subsequent OPA stages with an intermediary white light generation. As shown in Fig. 2(a) by the red shaded area, the measured output spectrum covers a range of 1.5 octaves from 430 nm to 1200 nm supporting a sub-3-fs pulse. An important part of the spectrum ranging from 600 nm to 1 µm, which includes the overlap region between the two stages, has already been compressed by double-chirped mirrors (DCM’s) to 4.6 fs. The resulting spectral phase measured with a SPIDER apparatus is shown by the red curves in Fig. 2. The successful measurement raises hope to compress the whole spectrum to the single-cycle in the near future. However, there have been legitimate questions about the origin of the surprisingly broad gain which cannot be explained by the very simple phase-matching arguments  (shown in Fig. 2(a) by the blue (pumped) and the green (pumped) shaded areas). It cannot explain the large bandwidth and not at all the prominent intensity peak between 600 nm and 700 nm. Furthermore, also the modulation in the spectral phase (red curve) close to the intensity peak cannot be clarified by the analytically calculated parametric phases (blue, green curves) [10, 11].
In contrast to the simple picture, which excluded spatial and temporal effects, the results of our (2 + 1)D simulation (see Fig. 2(b), black curves) are in excellent agreement with the measured data (red shaded). The calculation refers to the blue-pumped second OPA stage in 5 mm BBO, seeded with the actually measured spectrum as it is shown by the black dashed line in Fig. 2(a). The parameters are: phase-matching-angle of 37.2°, non-collinear angle of 4.5° in the walk-off compensation geometry, pump pulse energy of 7 µJ with a pulse duration of 900 fs, and a pump spot size of 90 µm (1/e2 radius). The spectral phase of the seed is assumed to be given only by the dispersion in the 3 mm BaF2 plate used for white-light generation. Neglecting the phase effects of the filamentation process during white-light generation driven by an already ultra-broadband pulse coming from the first amplification stage (see Fig. 4(a)) works surprisingly well in comparison with the experimental data.
The simulated signal pulses after the second stage reveal pulse energies of 1.1 µJ and pulse durations of 4.3 fs similar to the measurement after compression by 23 bounces on DCM’s and the fine tuning with CaF2-wedges. Also the spectral phase of the compressed simulated pulses (Fig. 2(b), black dashed) shows qualitatively the same characteristic features as the measurement (red dashed). The residual phase deviation is attributed to the simplification in the seed phase. Nevertheless, the shape of the compressed pulses of the SPIDER measurement and the simulation are in good agreement as shown in the inset in Fig. 2(b).
The details of the measured spectra are well understood by reconsidering the temporal and spatial evolution of the signal pulse. As shown in Fig. 3(a) the generated idler-pulses are strongly angularly dispersed in contrast to the signal spectral components propagating just straight in one direction 4.5° relative to the pump. Due to moderate pump pulse energies the pump-spot size is small (90 µm) and the crystal rather long (5 mm). The large angles of the idler spectral components reduce the spatial overlap region with the pump pulse during propagation. Consequently, the back conversion from signal/idler to the pump - especially at long idler wavelengths - is suppressed, leading to the surprisingly broad amplification bandwidth. Unfortunately, the overall signal gain is limited by the same reason.
Another key process for understanding the measurement is the self-compression of the idler which is generated with an opposite chirp of the positively dispersed seed pulses. Especially for spectral components near degeneracy the spatial overlap between pump, signal, and the compressed idler is significantly longer. However, the temporal stretching of the signal on one hand and the compression of the idler on the other hand reduces the overlap in time. Due to the equal phase velocity of signal and idler near degeneracy, the high intensities of the compressed idler are located at a temporal/spectral position of the dispersed signal pulse between 600 nm and 700 nm. This leads to the prominent intensity peak in the measured signal-spectrum (Fig. 2). The peak results in strong pump depletion and cascaded back- and forward conversion, which is responsible for the modulation of the spectral phase around 700 nm. The right hand side of Fig. 3 shows the temporal/spatial profiles of all involved pulses. Figure 3(a) and Fig. 3(c) reveal that the complete spectrum of the signal pulse propagates in the same direction at an angle at 4.5° which almost coincides with the Poynting vector of the pump pulse, leading to a near Gaussian beam profile. In contrast, in the TPM geometry with the walk-off compensated idler pulse, the counter-propagation of signal and pump leads to a separation of the spectral components of the signal in space as it is shown in the inset of Fig. 3(c) for comparison.
5. Limits of phase-matching geometries
As discussed in the previous section, spatial effects play an important role for broad-band parametric amplification, especially for small pump spot sizes and long crystal lengths as required for moderate pump pulse energies / high repetition rates. In fact, recently published results by Demmler et al. , considering short crystal lengths and rather soft focusing, have shown good agreement between experiment and a (1 + 1)D simulation. Hence, the importance of spatial effects in ultra-broadband parametric amplification is strongly linked to the available pump energy via the optimum focusing and the right choice of the phase-matching concept (PVWC or TPM). To formulate some guidelines we compare three different OPA systems in different phase matching geometries.
The first scenario is the first stage of the two-color OPA-system from section 4, which is pumped at 515 nm by tightly focused 1 ps pulses with energies of 10 µJ. The pulse energy of the seed pulses is 0.5 nJ and the phase-matching angles are chosen in correspondence to the experimental conditions with a non-collinear angleof 2.65° and the phase-matching angleof 24.65° in the PVWC-geometry. These values can be extracted precisely by comparing the phase-matching of the parasitic SHG to the gap in the measured signal spectrum. This OPA stage with a small pump spot size of 50 µm and a 5 mm long BBO crystal, needed for sufficient conversion efficiency, forms a good example for a regime with strong spatial and cascaded effects. As shown in Fig. 4(a) these effects are observed as strong modulations in the power spectrum and in the space/time distribution. The figure shows the measured (red shaded) and the calculated signal power spectrum (black solid) integrated across all signal propagation angles. The angular distribution of the signal wavelength components is given in the top of each figure. The group delay from the model is extracted along the incident propagation direction of the seed pulse (red line). It is shown as a black dotted curve; the comparison to the measurement from a SPIDER apparatus (red dotted) reveals a quite good agreement. To compare the different examples and give a more precise description of the compressibility of the simulated pulses with regards to experimental conditions, a chirped mirror compressor has been modeled. By using 5 bounces on each mirror, the simulated signal spectrum integrated within an angular range of 1 mrad (along the red line) has been compressed to a maximum peak power of 100 GW/µJ with a pulse duration of 6 fs. Although the spectral power is angularly distributed especially for wavelengths above 1 µm caused by the hard focus and the generation of new frequency components in time and space from cascaded processes close to degeneracy, the xt-plot on the left hand side of Fig. 4(a) reveals a good beam profile with a radius of 47 µm. If the size of the signal beam in the unconsidered third dimension is assumed to be similar, with the uncompressed peak intensity of 160 GW/cm2 and the temporal pulse shape, a pulse energy of 1.3 µJ can be calculated, which is in good agreement with the measured value of 1 µJ.
However, in TPM geometry (Fig. 4(b)) the beam profile of the signal is strongly disturbed, and the pulses cannot be efficiently compressed by chirped mirrors because of the phase front distortion across the beam profile and the diverging parametric GD at short wavelengths. For these wavelengths (and the corresponding long idler wavelengths) the parametric process is of superior efficiency because the mutual generation process (e.g. idler generates signal and signal generates idler) leads to fields continuously crossing and slowly shifting backwards through a large undepleted part of the pump pulse. The spatial separation of the generated fields prevents back conversion in spite of residual phase mismatch. This phase mismatch becomes apparent as the prominent GD slope at 700 nm. This complex spatio-temporal dynamics during the amplification is visualized in a motion sequence (see Media 1(a,d)).
Concluding the first scenario, the PVWC geometry results in good conversion efficiencies of more than 10% for pump energies around 10 µJ. Despite the strong modulations in the amplified spectrum and phase and the broad angular distribution for higher wavelengths (see Fig. 4(a)), the pulses show a good compressibility down to 6 fs and a good beam profile in the near field. The TPM-geometry results in bad compressibility and focusability due to the disturbed near field of the signal pulse.
The second scenario describes a novel high power OPA system from our lab, which is pumped by chirped pulses (40000 fs2) with 27 µJ of energy and 500 fs duration at the wavelength of 515 nm. The conditions of the simulation ( = 24.35°, = 2.4°) are chosen to be similar to the experiment presented by the red shaded power spectrum and the parametric GD (red dashed) in Fig. 4(c). The larger peak power allows for a more relaxed focusing of 175 µm compared to the first scenario, which leads to a significant reduction of the back- and forward conversion and a less critical pump depletion. The result is a significantly smoother power spectrum and symmetric beam profile with a clean angular distribution. The calculated pulse energy of 5.4 µJ is in good agreement with the measured pulse energies around 5 µJ from the experiment (it has to be taken into account the simulation assumes a perfect Gaussian distribution of the pump pulse which might explain the slight difference). By using 5 and 4 bounces on the mirror compressor, the simulated spectrum within an angular range of 1 mrad can be compressed to 6.5 fs with a peak power of 110 GW/µJ. However, because of the same reasons explained in the first scenario, in the TPM geometry the beam profile is still strongly disturbed and the parametric GD for short signal-wavelengths is still strongly modulated.
The third scenario, which is a very good example for a regime with less spatial influences, is inspired by a recently published high-power OPA system . Here, we simulate the two phase-matching geometries with the same parameters as used for scenario two, but with 120 µJ of pulse energy of the almost unfocused pump beam with a diameter slightly smaller than half a millimeter in just 2 mm of BBO. As it is shown in Fig. 4(e) and (f) the spectra of both geometries do not differ strongly. Only the dip in spectrum and phase around 900 nm caused by the parasitic SHG of the signal pulses at PVWC makes a small difference. In the xt-plots, the tangential geometry provides a reasonable beam profile, and the moderate pump depletion leads to a compressible parametric GD. Using again the compressor model, with just three bounces the simulated spectrum can be compressed down to 6.5 fs and 120 GW/µJ for the PVWC and 6.4 fs and 130 GW/µJ in the TPM-geometry. In contrast to scenario one and two, in this case the tangential geometry is preferable because of the higher output energy of 12 µJ compared to the lower 9 µJ of pulse energy in the PVWC-geometry due to the parasitic SHG.
In conclusion, a novel powerful tool was presented for the systematic investigation of second-order parametric processes. Due to the calculation in 2 + 1 dimensions and the generalization of the coupled propagation equations, all essential effects can be considered at moderate computing times. In contrast to theoretical models published before, this enables extensive parameter scans and precise statements about the expected beam profile and the compressibility as well as a good approximation of the conversion efficiency under consideration of all second order nonlinear interactions with respect to the spatial and temporal evolution of the two fields, respectively.
The simulation helps for better understanding of the complicated pulse shaping dynamics including unexpected cascaded effects and other mixing products, especially for novel high power and ultra-broadband multi-stage OPA-systems. In particular, prominent features of our 1.5 octave spanning two-color pumped OPA-system could be attributed to a temporal self-compression of the idler pulse in connection with the spatial co-propagation effects in this particular phase-matching geometry. Also other prominent features, e.g. the strong intensity peak and parametric phase for short wavelengths in the TPM or the spatial separation of seed an amplified signal in the PVWC, which have been observed in high repetition rate OPA’s, could be explained for the first time by this model. A variety of different OPA-systems have been studied, revealing the optimum phase-matching geometry as function of the available pump power connected to the optimum spot sizes. This knowledge helps for designing future OPA’s avoiding or even taking advantage of the described effects.
The authors thank VENTEON Femtosecond Laser Technologies GmbH for the support. This work was funded by Deutsche Forschungsgemeinschaft within the Cluster of Excellence QUEST, Centre for Quantum Engineering and Space-Time Research.
1. K.-H. Hong, S.-W. Huang, J. Moses, X. Fu, G. Cirmi, C.-J. Lai, S. Bhardwaj, and F. Kärtner, “CEP-stable, few-cycle, kHz OPCPAs for attosecond science: Energy scaling and coherent sub-cycle pulse synthesis, in Multiphoton Processes and Attosecond Physics,” in Springer Proceedings in Physics (Springer Berlin Heidelberg, 2012), Vol. 125, pp. 33–40.
2. J. Rothhardt, S. Demmler, S. Hädrich, J. Limpert, and A. Tünnermann, “Octave-spanning OPCPA system delivering CEP-stable few-cycle pulses and 22 W of average power at 1 MHz repetition rate,” Opt. Express 20(10), 10870–10878 (2012). [CrossRef] [PubMed]
3. S. Backus, C. G. Durfee, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69(3), 1207–1223 (1998). [CrossRef]
4. G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74(1), 1–18 (2003). [CrossRef]
5. S. Witte and K. S. E. Eikema, “Ultrafast optical parametric chirped-pulse amplification,” IEEE Sel. Top. Quantum Electron 18(1), 296–307 (2012). [CrossRef]
6. A. Harth, M. Schultze, T. Lang, T. Binhammer, S. Rausch, and U. Morgner, “Two-color pumped OPCPA system emitting spectra spanning 1.5 octaves from VIS to NIR,” Opt. Express 20(3), 3076–3081 (2012). [CrossRef] [PubMed]
7. A. L. Oien, I. T. McKinnie, P. Jain, N. A. Russell, D. M. Warrington, and L. A. W. Gloster, “Efficient, low-threshold collinear and noncollinear β-barium borate optical parametric oscillators,” Opt. Lett. 22(12), 859–861 (1997). [CrossRef] [PubMed]
8. J. Bromage, J. Rothhardt, S. Hädrich, C. Dorrer, C. Jocher, S. Demmler, J. Limpert, A. Tünnermann, and J. D. Zuegel, “Analysis and suppression of parasitic processes in noncollinear optical parametric amplifiers,” Opt. Express 19(18), 16797–16808 (2011). [CrossRef] [PubMed]
10. I. N. Ross, P. Matousek, G. H. C. New, and K. Osvay, “Analysis and optimization of optical parametric chirped pulse amplification,” J. Opt. Soc. Am. B 19(12), 2945–2956 (2002). [CrossRef]
11. D. Herrmann, R. Tautz, F. Tavella, F. Krausz, and L. Veisz, “Investigation of two-beam-pumped noncollinear optical parametric chirped-pulse amplification for the generation of few-cycle light pulses,” Opt. Express 18(5), 4170–4183 (2010). [CrossRef] [PubMed]
12. M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81(5), 053841 (2010). [CrossRef]
13. G. Arisholm, “General numerical methods for simulating second-order nonlinear interactions in birefringent media,” J. Opt. Soc. Am. B 14(10), 2543–2549 (1997). [CrossRef]
14. J. J. A. Fleck Jr and M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. 73(7), 920–926 (1983). [CrossRef]
15. R. Sutherland, Handbook of Nonlinear Optics, Optical Engineering (Taylor & Francis, 2003).
16. S. Demmler, J. Rothhardt, S. Hädrich, J. Bromage, J. Limpert, and A. Tünnermann, “Control of nonlinear spectral phase induced by ultra-broadband optical parametric amplification,” Opt. Lett. 37(19), 3933–3935 (2012). [CrossRef] [PubMed]