Open Access
Add to BibSonomy
Abstract
Electromagnetic waves propagating through plasma can interact nonlinearly through a variety of different mechanisms. The excitation of a plasma beat wave (ions or electrons) can create a refractive index modulation that changes the dispersion of the interacting beams. Alternatively, high-intensity beams can enter the regime where relativistic nonlinearities influence the propagation dynamics. In recent studies [Opt. Express 29, 1162 (2021) [CrossRef] ], it was proposed that two beams propagating along the same axis can exchange their polarization state due to nonlinear interaction. Here we present a numerical analysis of two laser beams intersecting in a nonlinear medium at varying angles. Polarization transfer is observed as predicted by analytical theory for a range of angles. For small angles, it is found that filamentation of the interacting beams becomes important. Analytical estimates of the filamentation threshold are presented, and good agreement is found with the simulation data.
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. INTRODUCTION
In recent years it has become increasingly important to control high-power laser beams with intensities above the damage threshold of conventional optical media. Using plasmas to modify such beams has shown to be a promising route to achieve this goal [1]. In contrast to conventional optics, a plasma exists in an ionized state, which means that it is not susceptible to damage in the traditional sense. Numerous applications of plasma mirrors have been proposed. The plasma mirror [1,2] is a transient optical device that improves the contrast of high-intensity pulses and is routinely used in short pulse experiments. High harmonic generation using the relativistically oscillating mirror concept [1,3] promises the realization of high-intensity attosecond pulses. Short pulse amplification can be achieved through stimulated Raman scattering [4,5] or stimulated Brillouin scattering [6]. The generation of beams with high-order orbital angular momentum beams has been proposed using a variety of methods [7–11]. Plasma gratings may be created at the surface of an overdense plasma [12], and photonic crystals may be generated throughout the volume of an underdense plasma [13]. Even the creation of holographic structures in a plasma has been proposed [14].
Recently, Michel et al. [15] proposed a method to control the polarization of a witness beam in a nonlinear medium. Here, the origin of the nonlinearity is arbitrary. In the context of plasma optics, the nonlinear response of the plasma can be caused by relativistic effects or by local density variations, and thus refractive index variations, due to the ponderomotive force. The modification of the polarization state of the witness beam can be explained by the formation of a beating pattern between the two beams. The method assumes that the intensity of the pump is large compared to that of the witness so that the pump polarization remains unaffected. Experimental evidence of the scheme was presented in Ref. [16]. The effect was further investigated in the context of cross-beam energy transfer in Ref. [17].
Polarization transfer between two beams of equal intensity was investigated by Kur et al. [18]. In their model, two parallel propagating beams were shown to exchange their polarization state periodically. The one-dimensional model was used to construct a description of two obliquely interacting beams. However, the resulting equations could not be solved analytically. For two finite-width beams crossing at an angle, it was found that the downstream polarization of each beam was not uniform throughout the beam profile. In this contribution, we present simulations of two beams crossing at arbitrary angles using a full nonlinear Maxwell solver. The simulations generally confirm the theory but also show its limitations and present additional features. The theory in Ref. [18] was developed for two parallel beams. Here, we show that a minimum angle between the beams is necessary in order to suppress filamentation seeded by the transverse beating of the beams. This paper is organized as follows. Section 2 presents the model used in the simulations. Section 3 presents the results. Conclusions are drawn in Section 4.
2. MODEL
We perform two-dimensional simulations of the nonlinear Maxwell equations using the MPulse code [19]. MPulse is an open-source nonlinear finite-differnce time-domain (FDTD) code built using the Schnek library [20]. The code simulates the macroscopic equations
Here ${\textbf E}$ and ${\textbf B}$ are the electric field and the magnetic flux, and ${\textbf D}$ and ${\textbf H} = {\textbf B}/{\mu _0}$ are the displacement field and the magnetic field. ${{\textbf J}_f}$ is the free current, which in all simulations presented here is not considered: ${{\textbf J}_f} = 0$. The displacement field is given byTo keep the model general, we do not specify the origin of the nonlinearity. For example, for beams at relativistic intensities propagating through a plasma below the critical density, the linear refractive index is
For a single frequency beam, ${n_0}$ contains the effects of the linear plasma response, i.e., the influence of the nonrelativistic part of the free electron current on the wave propagation. For relativistic laser intensities and a fully ionized plasma, the nonlinear term can be quantified by [21]On the other hand, for a beam propagating through a transparent medium such as fused silica, the nonlinearity is due to the Kerr effect and $\chi$ can be related to the nonlinear refractive index, ${n_2}$, by
where ${n_0}$ is the linear refractive index and ${n_2}$ is typically in the order of ${10^{- 20}}\;{{\rm m}^2}/{\rm W}$. Here, ${n_2}$ is related to the total refractive index $n$ through In all cases, we neglect the influence of losses. For plasmas, this can be justified when the electron temperatures are sufficiently hot and the plasma can be considered collisionless. For transparent media, the interaction lengths considered in our investigation are short such that no appreciable losses will occur.In the remainder of the paper, we use normalized quantities. All lengths are normalized to the vacuum wavelength of the beams, and we set $c = 1$. This means that time is normalized by the laser period. Without loss of generality, we also set ${n_0} = 1$ for all the simulations presented here. The two beams are injected on the left at $x = 0$,
Here $i = 1,2$ is the index of the beam. ${{\textbf E}_i}(y,0)$ is a top hat profile approximated by a super-Gaussian of order 32 with a beam width of 120. The maximum field amplitude is ${E_0} = 1$. We assume single frequency beams, which require long pulse durations. This is justified for numerous high-energy lasers used in inertial confinement fusion and high-energy density physics [22–24]. These beams typically deliver nanosecond pulses in the 100 TW to PW power regime. Extending this investigation to shorter pulses would require consideration of the frequency dependence of both linear and nonlinear responses of the medium. This will be the topic of a future investigation. The beams are linearly polarized and make an angle of ${\pm}\vartheta$ with the $x$ axis, such that ${k_{i,y}} = \pm \sin \vartheta$. The $x$ component of the unperturbed wavevector is then ${k_{i,x}} = \cos \vartheta$. The beams propagate from left to right through the simulation box, and in all simulations we have $|\vartheta | \le \pi /4$, i.e., $|{k_{i,y}}| \le {k_{i,x}}$. Around the outside of the simulation domain, a perfectly matched layer [25,26] is added to absorb any outgoing waves. The layer has a thickness of 10 grid cells.3. SIMULATION RESULTS
Simulations for small crossing angles are carried out in a box of size $300 \times 300$ with a resolution of $6000 \times 6000$ grid cells. Each vacuum wavelength is resolved by 20 grid points. The beam traveling upward is $s$-polarized, while the beam traveling downward is polarized at 45° with respect to the simulation plane. The nonlinearity is chosen to be $\chi = 0.01$. For these parameters, the theory predicts an exchange of the polarization states between the two beams after an interaction length of ${L_{{\rm swap}}} = 100\sqrt 2$ [18].
Fig. 1. Out-of-plane electric field ${E_z}$ at the end of the simulation for a crossing angle of $\vartheta ={ 1^ \circ}$ (top left), $\vartheta ={ 3^ \circ}$ (top right), $\vartheta ={ 5^ \circ}$ (bottom left), and $\vartheta ={ 10^ \circ}$ (bottom right).
Figure 1 shows the out-of-plane electric field ${E_z}$ for selected crossing angles $\vartheta$. For $\vartheta ={ 5^ \circ}$ and $\vartheta ={ 10^ \circ}$ one can see a regular interference pattern on the left side of the simulation box where the two beams cross. This beating of the waves is expected with a wavevector in the $y$ direction of
For $\vartheta \ge {5^ \circ}$ we observe deviations from the ideal interference pattern for $x \gt 150$, i.e., to the right of the point where the central beam axes intersect. These disturbances can be attributed to edge effects due to the strong transverse gradients of the super-Gaussian beam profile near the edges. Ultimately they also lead to filamentation of the beams. These claims will be confirmed by additional simulations, which will be discussed below.
Fig. 2. Orbits of the $E_y^ \pm$ and $E_z^ \pm$ field components for the two beams at various positions ${x_n}$ along the central axis $y = 150$ for $\vartheta ={ 5^ \circ}$.
To analyze the polarization states of the two beams, we record the electric field at various fixed positions ${x_n}$ over time. The resulting fields ${\textbf E}({x_n},y,t)$ are then Fourier transformed in $y$ and $t$, resulting in ${\boldsymbol {\hat E}}({x_n},{k_y},\omega)$. In these Fourier-transformed fields, the upward and downward propagating beams can be clearly distinguished. We define ${\boldsymbol {\hat E}} = {{\boldsymbol {\hat E}}^ +} + {{\boldsymbol {\hat E}}^ -}$, where ${{\boldsymbol {\hat E}}^ +}$ is nonzero only in the upper right and the lower left quadrants and ${{\boldsymbol {\hat E}}^ -}$ is nonzero only in the upper left and the lower right quadrants,
Fig. 3. The $y$ component of the projection of the normalized Stokes vector against the position $x$ for different interaction angles.
Fig. 4. Out-of-plane electric field ${E_z}$ at the end of the simulation for a crossing angle of $\vartheta ={ 10^ \circ}$ and beam waist of 240 (left). The $y$ component of the projection of the normalized Stokes vector against the position $x$ for the same simulation (right).
Figure 2 shows the orbits of the $E_y^ \pm$ and $E_z^ \pm$ field components along the along the central axis $y = 150$ for $\vartheta ={ 5^ \circ}$. The two beams are injected at $x = 0$, and both beams are linearly polarized. The upward propagating beam is polarized in the $z$ direction, and the downward propagating beam is polarized at 45°. Note that the image shows some ellipticity of the orbits because the first orbits are sampled a few grid points to the right of the location where the beams are injected into the simulation box. For $0 \lt x \lt 1$ both beams become elliptically polarized until, for $x \approx 1$, they have exchanged their polarization state almost completely. For $x \gt 1$ the two polarization states start to switch again, but, because of the filamentation apparent in Fig. 1, the exchange is not perfect and both beams remain elliptically polarized to some extent for all positions in the simulation box.
To further compare the behavior of the polarization transfer between the two beams for different interaction angles, we calculate the Stokes vector,
For linearly polarized beams, ${P_y} = 0$ and the evolution of ${P_y}$ for both beams is expected to be symmetric, i.e., $P_y^ + = - P_y^ -$ for all positions $x$. Figure 3 shows the plots of ${P_y}$ along the horizontal position for different interaction angles. In the left half of the simulation, one can observe regular oscillations indicating complete polarization transfer between the two beams. For angles $\vartheta ={ 7.5^ \circ}$ and 10° the first complete swap occurs at $x \approx {L_{{\rm swap}}}$, as predicted by the theory. For smaller angles, the two beams transfer their polarization state more rapidly. For $\vartheta ={ 3^ \circ}$ the first swap is completed at $x \approx 0.55{L_{{\rm swap}}}$, and one can observe a full cycle at $x \approx 1.1{L_{{\rm swap}}}$. This can be attributed to the onset of self-focusing. As one can see in to top right panel of Fig. 1, for $\vartheta ={ 3^ \circ}$ the beamlets are subject to some self-focusing but the central beamlets exhibit quite regular periodic focusing and defocusing cycles. On average, the intensity on the central axis is increased due to this effect resulting in an increase in nonlinear effects. For $x \gt {L_{{\rm swap}}}$ the behavior of ${P_y}$ becomes more irregular due to the beams being subject to filamentation.
Above, we argued that the filamentation in the beams for $\vartheta { \gt 5^ \circ}$ is mostly caused by self-focusing effects from the beam edges. To show that these effects are indeed caused by the finite width of the beams, we have performed simulations with $\vartheta ={ 10^ \circ}$ in which we doubled the beam waist to 240. Figure 4 shows the results of this simulation. In the left panel, one can clearly observe that the regions where filamentation occurs have moved away from the center when compared to the bottom right panel of Fig. 1. They appear at approximately the same distance from the edges of the two beams. In the right panel of Fig. 4 the oscillation of the Stokes vector also shows that the simulation results follow the theoretical predictions for a much longer distance for the wider beam when compared to the base case.
Fig. 5. Out-of-plane electric field ${E_z}$ at the end of the simulation for a crossing angle of $\vartheta ={ 20^ \circ}$ (top left), $\vartheta ={ 25^ \circ}$ (top right), $\vartheta ={ 35^ \circ}$ (bottom left), and $\vartheta ={ 45^ \circ}$ (bottom right).
Fig. 6. Orbits of the electric field at the after interaction for a crossing angle of $\vartheta ={ 25^ \circ}$ at transverse positions 0 (top left), 40 (top right), 80 (bottom left), and 120 (bottom right) from the leading edge.
We now perform simulations with larger interaction angles $\vartheta ={ 20^ \circ}$, 25°, 35°, and 45°. For these angles, the width of the simulation box was increased to 400 in the $y$ direction, but, for the sake of comparison, only the central region of width 300 is shown in the figures. These larger angles allowed the beams to be fully separated before and after the interaction. In contrast to the small-angle interaction, one cannot make the approximation of parallel propagation. Each longitudinal ray of one beam interacts with a different transverse slice of the other beam. As was already pointed out in [18], this leads to different amounts of polarization change across the transverse profile of each beam.
Figure 5 shows the out-of-plane electric field at the end of these simulations. One can see the finite size of the interaction region. No filamentation is observed within the central part of the beams, but self-focusing is again responsible for the distortion of the beam edges. In order to analyze the polarization of the outgoing beams, we first select a position along the beam axis where the beams have cleared the interaction region. Given this location, a number of transverse positions across the beam profile are chosen, providing starting points for creating electric field orbits. Each orbit is then produced by scanning the electric fields along three wavelengths in the direction of the beam propagation. We record the out of the simulation plane, ${E_z}$, and the in-plane electric field perpendicular to the beam direction, ${E_ \bot}$.
Figure 6 shows the orbits of the out-of-plane component, ${E_z}$, and the in-plane component ${E_ \bot}$ of the transverse field for the two beams for an interaction angle of $\vartheta ={ 25^ \circ}$. Unlike the previous plots, in which the orbits were evaluated as a function of time, here the orbits are evaluated as a function of position along the beam propagation direction outside the interaction region. The figure shows the orbits for different transverse positions 0, 40, 80, and 120 from the leading edge. The orbits at transverse positions 0 and 120 exhibit a significantly lower amplitude compared to the orbits in the central beam region. These positions are just at the edges of the beam envelope, and a lower intensity is expected here. For position 120 the orbits are not closed. This can be explained by the fact that the beams show effects of self-focusing at the trailing edge as can be seen in Fig. 5. The other main observation here is that the polarization states of the two beams are different for the different transverse positions. Again, this is expected because the various transverse rays of one beam interact with different polarization states of the other beam.
The above picture changes when an orthogonal interaction is considered. Figure 7 shows the orbits of the out-of-plane component, ${E_z}$, and the in-plane component ${E_ \bot}$ of the transverse field for the two beams for an interaction angle of 45°. Note, that the interaction angle measures the angle between each beam and the $x$ axis. This means that an interaction angle of 45° corresponds to a right angle between the two beams. As predicted in Ref. [28], in this case, the beam that is polarized in the out-of-plane direction is transmitted through the interaction region without any modification in its polarization state. The beam that is diagonally polarized experiences a change in its polarization state and is elliptically polarized upon exiting from the interaction region.
Fig. 7. Orbits of the electric field at the after interaction for a crossing angle of $\vartheta ={ 45^ \circ}$ at transverse positions 0 (top left), 40 (top right), 80 (bottom left), and 120 (bottom right) from the leading edge.
4. CONCLUSION
In this investigation, we studied the interaction of two intense laser beams in a nonlinear medium. When the two beams have polarizations that are not orthogonal, they will create a static beating pattern at the difference between the two wavevectors. The resulting interaction allows the exchange of the polarization states between the two beams. Two-dimensional nonlinear FDTD simulations have been carried out for a range of interaction angles. For a range of interaction angles ${\theta _{{\rm cr}}} \lt \theta \ll 1$, the one-dimensional model [18] could be confirmed. Here ${\theta _{{\rm cr}}}$ is given by Eq. (13). The two beams periodically and reversibly exchange their polarization state over a length
For values only slightly above the critical angle, $\theta \gtrsim {\theta _{{\rm cr}}}$, we observe a systematic reduction of the exchange length, which can be attributed to a local enhancement of the intensity due to the onset of self-focusing effects. The transverse intensity variations caused by the beating of the two beams can be interpreted as a collection of parallel propagating beamlets, each subject to nonlinear self-focusing. For $\theta \lt {\theta _{{\rm cr}}}$ the energy in each beamlet is sufficient to fully collapse the beamlets and the interaction of the two beams results in rapid filamentation seeded by the transverse beating pattern.For large interaction angles, the one-dimensional theory breaks down and the finite width of the beams has to be taken into account. As the two beams cross, each beam experiences a different part of the other beam [18] and interacts with a different polarization profile. As the exit of the interaction region, the beams will have a range of different polarization states across their individual transverse profiles. The only exception is the crossing of two beams at right angles. Here, the in-plane polarization component of one beam is longitudinal with respect to the other beam and can, therefore, not interact. This leads to the out-of-plane component to experience an overall phase shift upon exit while the in-plane component is unaffected. An $s$-polarized beam will remain $s$-polarized, but a diagonally linear polarization will be converted to an elliptical polarization.
The results presented here could be used to provide fast control over the polarization states by means of nonlinear media. By using a control beam, the polarization state of a witness beam can be manipulated. If the nonlinearity originates from relativistic effects in a plasma, this scheme opens up opportunities for controlling high-intensity beams using plasma optics at intensity ranges for which conventional optical devices cannot be used. On the other hand, nonlinear polarization transfer could have detrimental effects on energy deposition in inertial confinement fusion schemes, and understanding these mechanisms will be important for their success.
Funding
U.S. Department of Energy (DE-AC52-07NA27344); Lawrence Livermore National Laboratory (18-ERD-046).
Disclosures
The authors declare no conflicts interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
REFERENCES
1. H. C. Kapteyn, A. Szoke, R. W. Falcone, and M. M. Murnane, “Prepulse energy suppression for high-energy ultrashort pulses using self-induced plasma shuttering,” Opt. Lett. 16, 490–492 (1991). [CrossRef]
2. G. Doumy, F. Quéré, O. Gobert, M. Perdrix, P. Martin, P. Audebert, J. C. Gauthier, J. P. Geindre, and T. Wittmann, “Complete characterization of a plasma mirror for the production of high-contrast ultraintense laser pulses,” Phys. Rev. E 69, 026402 (2004). [CrossRef]
3. R. Lichters, J. M. ter Vehn, and A. Pukhov, “Short-pulse laser harmonics from oscillating plasma surfaces driven at relativistic intensity,” Phys. Plasmas 3, 3425–3437 (1996). [CrossRef]
4. G. Shvets, N. Fisch, A. Pukhov, and J. M. ter Vehn, “Superradiant amplification of an ultrashort laser pulse in a plasma by a counterpropagating pump,” Phys. Rev. Lett. 81, 4879–4882 (1998). [CrossRef]
5. V. Malkin, G. Shvets, and N. Fisch, “Fast compression of laser beams to highly overcritical powers,” Phys. Rev. Lett. 82, 4448–4451 (1999). [CrossRef]
6. A. A. Andreev, C. Riconda, V. T. Tikhonchuk, and S. Weber, “Short light pulse amplification and compression by stimulated Brillouin scattering in plasmas in the strong coupling regime,” Phys. Plasmas 13, 053110 (2006). [CrossRef]
7. X. Zhang, B. Shen, Y. Shi, X. Wang, L. Zhang, W. Wang, J. Xu, L. Yi, and Z. Xu, “Generation of intense high-order vortex harmonics,” Phys. Rev. Lett. 114, 173901 (2015). [CrossRef]
8. J. Vieira, R. M. Trines, E. P. Alves, R. A. Fonseca, J. T. Mendonça, R. Bingham, P. Norreys, and L. O. Silva, “Amplification and generation of ultra-intense twisted laser pulses via stimulated Raman scattering,” Nat. Commun. 7, 10371 (2016). [CrossRef]
9. J. Vieira, R. M. Trines, E. P. Alves, R. A. Fonseca, J. T. Mendonça, R. Bingham, P. Norreys, and L. O. Silva, “High orbital angular momentum harmonic generation,” Phys. Rev. Lett. 117, 265001 (2016). [CrossRef]
10. L. Zhang, B. Shen, Z. Bu, X. Zhang, L. Ji, S. Huang, M. Xiriai, Z. Xu, C. Liu, and Z. Xu, “Vortex harmonic generation by circularly polarized Gaussian beam interacting with tilted target,” Phys. Rev. Appl. 16, 014065 (2021). [CrossRef]
11. K. Qu, Q. Jia, and N. J. Fisch, “Plasma q-plate for generation and manipulation of intense optical vortices,” Phys. Rev. E 96, 053207 (2017). [CrossRef]
12. S. Monchocé, S. Kahaly, A. Leblanc, L. Videau, P. Combis, F. Réau, D. Garzella, P. D’Oliveira, P. Martin, and F. Quéré, “Optically controlled solid-density transient plasma gratings,” Phys. Rev. Lett. 112, 145008 (2014). [CrossRef]
13. G. Lehmann and K. H. Spatschek, “Laser-driven plasma photonic crystals for high-power lasers,” Phys. Plasmas 24, 056701 (2017). [CrossRef]
14. A. Leblanc, A. Denoeud, L. Chopineau, G. Mennerat, P. Martin, and F. Quéré, “Plasma holograms for ultrahigh-intensity optics,” Nat. Phys. 13, 440–443 (2017). [CrossRef]
15. P. Michel, L. Divol, D. Turnbull, and J. D. Moody, “Dynamic control of the polarization of intense laser beams via optical wave mixing in plasmas,” Phys. Rev. Lett. 113, 205001 (2014). [CrossRef]
16. D. Turnbull, P. Michel, T. Chapman, E. Tubman, B. B. Pollock, C. Y. Chen, C. Goyon, J. S. Ross, L. Divol, N. Woolsey, and J. D. Moody, “High power dynamic polarization control using plasma photonics,” Phys. Rev. Lett. 116, 205001 (2016). [CrossRef]
17. D. Turnbull, A. Colaïtis, R. K. Follett, J. P. Palastro, D. H. Froula, P. Michel, C. Goyon, T. Chapman, L. Divol, G. E. Kemp, D. Mariscal, S. Patankar, B. B. Pollock, J. S. Ross, J. D. Moody, E. R. Tubman, and N. C. Woolsey, “Crossed-beam energy transfer: Polarization effects and evidence of saturation,” Plasma Phys. Control. Fusion 60, 054017 (2018). [CrossRef]
18. E. Kur, M. Lazarow, J. S. Wurtele, and P. Michel, “Nonlinear polarization transfer and control of two laser beams overlapping in a uniform nonlinear medium,” Opt. Express 29, 1162–1174 (2021). [CrossRef]
19. H. Schmitz and V. Mezentsev, “Full-vectorial modeling of femtosecond pulses for laser inscription of photonic structures,” J. Opt. Soc. Am. B 29, 1208–1217 (2012). [CrossRef]
20. H. Schmitz, “Schnek: A C++ library for the development of parallel simulation codes on regular grids,” Comput. Phys. Commun. 226, 151–164 (2018). [CrossRef]
21. W. B. Mori, “The physics of the nonlinear optics of plasmas at relativistic intensities for short-pulse lasers,” IEEE J. Quantum Electron. 33, 1942–1953 (1997). [CrossRef]
22. P. A. Arnold, G. F. James, D. E. Petersen, D. L. Pendleton, G. B. McHale, F. Barbosa, A. S. Runtal, and P. L. Stratton, “An update on NIF pulsed power,” in 2009 IEEE Pulsed Power Conference (IEEE, 2009), pp. 352–356.
23. D. D. Meyerhofer, J. Bromage, C. Dorrer, J. H. Kelly, B. E. Kruschwitz, S. J. Loucks, R. L. McCrory, S. F. Morse, J. F. Myatt, P. M. Nilson, J. Qiao, T. C. Sangster, C. Stoeckl, L. J. Waxer, and J. D. Zuegel, Performance of and Initial Results from the Omega EP Laser System (Institute of Physics Publishing, 2010).
24. S. Weber, S. Bechet, S. Borneis, et al., “P3: An installation for high-energy density plasma physics and ultra-high intensity laser–matter interaction at ELI-beamlines,” Matter Radiat. Extremes 2, 149–172 (2017). [CrossRef]
25. J. P. Bérenger, “Improved PML for the FDTD solution of wave-structure interaction problems,” IEEE Trans. Antennas Propag. 45, 466–473 (1997). [CrossRef]
26. J. A. Roden and S. D. Gedney, “Convolution PML (CPML): An efficient FDTD implementation of the CFS–PML for arbitrary media,” Microw. Opt. Technol. Lett. 27, 334–339 (2000). [CrossRef]
27. R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).
28. P. Michel, E. Kur, M. Lazarow, T. Chapman, L. Divol, and J. S. Wurtele, “Polarization-dependent theory of two-wave mixing in nonlinear media, and application to dynamical polarization control,” Phys. Rev. X 10, 021039 (2020). [CrossRef]
