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We solve the nonlinear Maxwell equations in an InP-based dielectric metamaterial, considering both two-photon absorption and photo-induced free-carrier absorption. We obtain the intensity-dependent reflection, absorption, and effective permittivity and permeability of the metamaterial. Our results show that nonlinear absorption dampens both the electric and magnetic Mie resonance, although the magnetic resonance is more affected because it occurs at longer wavelengths where the free-carrier absorption cross section is larger. Owing to field concentration in the metamaterial at resonance, the threshold intensity for nonlinear absorption is reduced by a factor of about 30 compared to a homogeneous layer of the same thickness. Our results have implications on the use of dielectric metamaterials for nonlinear applications such as higher harmonic generation, optical limiting, and ultrafast modulation.
Materials with strong nonlinear response are desired for applications such as optical limiting1–3, ultrafast modulation4–6, frequency conversion7,8, and optical isolation9,10. Traditional bulk materials have a weak interaction between electrons and sub-bandgap photons, as measured by nonlinear absorption coefficients11–14, and thus require high intensities or long interaction lengths to achieve an efficient nonlinear response.
One way to enhance the nonlinear response is to incorporate metamaterial elements or plasmonic structures to concentrate the electric fields within a nonlinear material15,16. Metamaterials containing plasmonic nanostructures have been integrated with nonlinear materials to enhance second-harmonic generation17–23 and achieve analog electromagnetically induced transparency24,25. The drawback of plasmonic approaches is that the fields are highly localized at the metal dielectric interface, leading to small interaction volumes26. Also, the finite conductivity of metals at optical frequencies leads to undesirable losses. An alternative approach is to induce nonlinearity in dielectric metamaterials using Mie resonances27. Field enhancement in dielectric metamaterials has been demonstrated27 and applied to achieve third-harmonic generation26,28–32, optical limiting33, and ultrafast optical modulation34–36.
To obtain efficient harmonic generation or optical modulation, the materials involved must be transparent at high intensities. However, at high intensities nonlinear absorption processes such as two photon absorption (TPA) and free-carrier absorption (FCA) become significant1,14. Therefore, in order to optimize nonlinear effects in dielectric metamaterials it is important to understand the role of nonlinear absorption. To date, the research has focused primarily on the experimental aspects26,28–32, with relatively few examples of theoretical studies. A few examples employ the recently-developed generalized source method for nonlinear materials37–39, which calculates the diffraction by periodic structures containing linear and nonlinear media. Although these models provide important insights, they decouple the linear and nonlinear responses, and thus do not represent self-consistent solutions of the nonlinear Maxwell equations. Also, existing models33 do not account for the nontrivial frequency dependence of the nonlinear parameters, such as the TPA coefficient and the FCA cross section14.
In this work, we develop a full-wave model to solve the nonlinear Maxwell equations in a dielectric metamaterial accounting for both TPA and photo-induced FCA. We apply the model to study the optical properties and effective parameters of a representative indium phosphide (InP)-based dielectric metamaterial operating in the near infrared spectral band. We find that nonlinear absorption leads to damping of the electric and magnetic Mie resonances at high intensities. For continuous wave illumination, the onset of nonlinear absorption occurs at intensities of 1MW/cm2, while the Mie resonances are almost completely diminished for intensities approaching 5MW/cm2. We also find several unexpected results. First, due to the wavelength dependence of FCA, the nonlinear absorption at the magnetic resonance is larger than at the electric resonance. Second, assuming FCA is independent of wavelength, the absorption at the two resonances is found to be nearly equal, despite the electric field being heavily localized at the electric resonance and more uniformly distributed at the magnetic resonance. Third, owing to the enhancement of the electric field at resonance, we find that the intensity threshold of nonlinear absorption in the metamaterial is nearly 30 times lower compared to a homogeneous material of the same thickness.
To clearly illustrate the effects of nonlinearity on Mie resonances, we consider a large index-contrast metamaterial consisting of 360nm InP spheres with large refractive index (3.3 at 1 μm) in air medium, arranged in a square lattice with a periodicity of 800nm, as shown in inset of Fig. 1(a). InP is chosen for its transparency in the near-infrared band of interest (0.9–1.3 μm). Note the conclusions drawn in this article remain valid when the air medium is replaced by a polymer and the metamaterial layer is placed on a low index substrate such as silica. The size and periodicity of the InP spheres are optimized to position the electric and magnetic Mie resonances, identified as narrowband peaks in the reflection spectrum, in the band of interest as shown in Fig. 1(a).
The origin of nonlinear absorption in InP can be understood from the band structure, shown in Fig. 1(b). The valence bands (VBs) consist of heavy-hole (HH), light-hole (LH), and spin-orbit (SO) bands. The conduction band (CB) is separated from the HH band by the band gap. In the absence of light, states in the VB are filled with electrons and the CB states are empty. Electrons in the VB can enter the CB by absorbing photons with energy larger than the band gap. Since the band gap of InP (1.45eV) is larger than the photon energies in the band of interest (0.95–1.4eV), at low intensities photons transmit through InP without being absorbed. However, when the incident intensity is large, the probability for valence electrons to absorb two photons (shown as two stacked vertical blue arrows) is high, resulting in reduced transmission. In addition, this TPA is followed by FCA in which the holes left behind in the HH band can be filled by one-photon absorption of electrons in the LH and SO bands, shown by the red arrows in Fig. 1(b). Because the strength of TPA and FCA depend on intensity, they are referred to as nonlinear absorption processes. There can also be Drude-like FCA by conduction electrons (not shown). However, this process requires both a photon and phonon to conserve energy and momentum, whereas the FCA considered in our studies is a one-photon process with a cross section two orders of magnitude larger14. We also note that for the highest intensities considered in this work, the carrier density in InP is less than 1018 cm−3, corresponding to a plasma wavelength greater than 50 μm, which is considerably higher than near-infrared wavelengths. Therefore, we neglect FCA due to conduction electrons and the electron-hole plasma. Figure 1(b) also shows the previously calculated values of the TPA coefficient β and FCA cross section σ of InP14. The value of β is relatively constant with wavelength, which is typical for wide-bandgap materials, while σ increases by an order of magnitude over the band. The FCA increases with increasing wavelength because the corresponding photon energy decreases, and the energy-momentum conservation condition for FCA (red lines in Fig. 1(b)) is satisfied only near the center of the Brillouin zone, where a larger number of holes are present.
where n is the refractive index, c is the speed of light, N and σ are the free-carrier concentration and absorption cross section, ε 0 is the free-space permittivity, and PNL(r, t) is the nonlinear polarization. Taking the Fourier transform of Eq. (1), assuming an exp( − iωt) time dependence, and using the relation for the third-order nonlinear polarization
where is the nonlinear susceptibility, we obtain
where I = ½ncε0|E(r, ω)|2 is the intensity, ℏω is the photon energy and τ is the photo-carrier relaxation time. The first term in Eq. (5) describes free-carrier generation via TPA, and the second term describes free carrier recombination. For continuous-wave illumination, the free-carrier concentration will reach steady state conditions (∂N/∂t = 0) and thus
In pure InP, defect-mediated Shockley–Read–Hall lifetimes exceed 10 ms, and radiative recombination lifetimes are effectively long due to photon recycling (i.e., emission and re-absorption)14. The limiting lifetime is associated with Auger recombination, which dominates at higher carrier densities, or equivalently, higher intensities. The lifetime associated with nonradiative Auger recombination is given by 1/(CN 2), where C is the Auger recombination coefficient (=10−30 cm6/s for InP)14. Using the expression in Eq. (6) for the carrier concentration under steady state conditions, for the highest intensity consider in this work (5MW/cm2), we calculate an equilibrium carrier density of 1018cm−3, which corresponds to a lifetime of 0.6 μs. At lower intensities, the lifetime is longer owing to lower carrier densities. We assume a constant lifetime of 1 μs, near the lower end of the range.
We solve Eq. (7) using the full-wave finite-element frequency domain solver in COMSOL Multiphysics. This was accomplished by assigning the two nonlinear terms in Eq. (7) to the imaginary part of n 2. Before solving Eq. (7) in the metamaterial in Fig. 1, we apply it to a homogeneous nonlinear medium and compare the results with the solution to the well-known rate equation40,41
The transmitted intensity, as a function of thickness for InP at a wavelength of 1 μm for various intensities, calculated by solving Eqs (7) and (8), respectively, are shown by dots and solid lines in Fig. 2. The two calculations are in excellent agreement, thus validating our full-wave nonlinear model. In this validation the index of InP is set equal to 1 to avoid interference effects, which are not included in Eq. (8).
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
We now apply our validated nonlinear full-wave model [(Eq. (7)] to understand the role of nonlinear absorption in the dielectric metamaterial shown in Fig. 1. First, we studied the reflection and absorption for different incident intensities, shown in Fig. 3. For all intensities, the reflection spectrum contains narrowband peaks near 1.2 and 0.95 μm, corresponding to the magnetic and electric dipole Mie resonances, respectively. For a low intensity of 1W/cm2, the nonlinear processes are negligible, resulting in low absorption and nearly 100% reflection at the two resonances. As the intensity increases, the absorption at the resonances increases and the reflection decreases.
We note that for a given intensity, the absorption is larger at the magnetic resonance (1.2 μm) than at the electric resonance (0.95 μm). This is a surprising result considering the electric field distributions at the resonances, shown in the inset of Fig. 4, normalized to the incident electric field for an intensity of 1MW/cm2. At the magnetic dipole resonance, the electric field is ring shaped and relatively uniform, while the electric field at the electric dipole resonance is highly concentrated at the center of the sphere. Note the field concentration outside the sphere arises from the boundary condition on the normal component electric field, which is discontinuous by the ratio of the dielectric constants of the sphere and free space42. Thus, based on the field distributions, one might expect the absorption at the electric resonance to be larger due to the larger field concentration. However, we find more absorption at the magnetic resonance. We attribute this to the FCA cross section being about 5 times larger at the magnetic resonance than at the electric resonance [Fig. 1(b)]. To validate this claim, we recalculated the spectral absorption for 1MW/cm2 intensity, assuming a constant FCA cross section. The results, shown in Fig. 4, indicate that the absorption is approximately equal at the two resonances, confirming that the wavelength-dependent FCA cross section is responsible for the larger absorption at the magnetic resonance.
The fact that the absorption is equal at the two resonances for constant FCA is also counterintuitive, since more absorption is expected at the electric resonance because of the larger field concentration. In fact, we find that |E|2 integrated over the sphere is 1.5 times larger at the electric resonance than at the magnetic resonance. Despite this, the absorption is equal at the two resonances for constant free carrier absorption cross section. In materials with complex permittivity (ε = ε′ + iε′′), absorption is given by the volume integral of ε′′|E|2. Noting that ε′′ = 2nk, where n and k are the components of the complex refractive index n+ik, where k is given by and from Eq. (6), , we find that ε′′ is proportional to wavelength squared. Thus, ε′′ is (1.2 μm/0.97 μm)2=1.5 times larger at the magnetic resonance, leading to equal absorption at the two resonances when σ is independent of wavelength.
Since the electric field is enhanced at both resonances, we expect more absorption per unit length in the metamaterial than in a homogeneous material. To illustrate this, we calculated the reflection and absorption of a 360 nm-thick slab of InP, equal in thickness to the InP sphere metamaterial in Fig. 1. We see from Fig. 5 that 100MW/cm2 of intensity is needed to obtain 40% absorption near 1.2 μm in the homogeneous layer, whereas the metamaterial obtains a similar level of absorption for 3MW/cm2. Thus, the homogeneous layer requires much higher intensities to achieve absorption values comparable to the metamaterial. This factor of 30 higher intensity is consistent with the five-fold field enhancement at the magnetic resonance shown in Fig. 4.
It is also important to understand the impact of nonlinear absorption on the effective permittivity (ε) and permeability (μ) of the metamaterial, as ε and μ are often used to obtain unique properties such as negative refraction43 and perfect reflection44. The calculated real parts of ε and μ, shown in Fig. 6, were obtained using S-parameter inversion, assuming a layer thickness of 1.24 μm. At low intensities, shown as the blue line in Fig. 6(a), we see a strong resonance in μ near 1.2 μm, which arises from the magnetic resonance. The weaker resonance near 0.95 μm is the anti-resonance associated with the strong electric resonance at that wavelength, clearly seen in Fig. 6(b) for the permittivity. We also find a strong anti-resonance in ε at 1.1 μm, which arises from the magnetic resonance at 1.2 μm. These anti-resonances are an artifact of S-parameter retrieval that arises from approximating a Bloch wave by a plane wave45. In the resonance regions, either ε or μ is negative, resulting in single-negative regions and high reflectivity, as shown in Fig. 3(a). As the intensity increases to 1MW/cm2 and 3MW/cm2, we find that the magnetic resonance, near 1.2 μm in Fig. 6(a), begins to dampen while the electric resonance, near 1 μm in Fig. 6(b), is largely unchanged. Only when the intensity exceeds 3MW/cm2 does the electric resonance begin to dampen. For 5MW/cm2, both resonances are completely dampened. As the resonance in the real part of ε and μ broadens, the corresponding imaginary parts of ε and μ (not shown) also broaden, as per the Kramer-Kronig relationship, which results in broader-band absorption with increasing intensity, as shown in Fig. 3(b).
In summary, we developed a self-consistent full-wave model to study the effects of two-photon absorption and photo-induced free-carrier absorption on the effective parameters and optical properties of a dielectric metamaterial operating in the near infrared spectral band. As expected, we find that nonlinear absorption leads to damping of the electric and magnetic Mie resonances at high intensities, with an onset around 1MW/cm2 for continuous wave illumination. The resonances are almost completely dampened for intensities around 5MW/cm2. Surprisingly, we find that the nonlinear absorption at the magnetic resonance is larger than at the electric resonance, despite the electric field being more concentrated at the electric resonance. We find this is because the free-carrier absorption cross section is considerably larger at the longer wavelengths near the magnetic resonance. We also find that the metamaterial provides absorption comparable to a homogeneous layer of the same thickness at approximately 30 times less intensity. The lower threshold intensity and smaller footprint for nonlinear absorption have implications on the use of dielectric metamaterials for nonlinear applications such as optical limiting, higher harmonic generation, and ultrafast modulation.
This work was funded by the Office of Naval Research (ONR) through Contract No. N00014-16-C-1023.
B.S. and S.K. developed the nonlinear full-wave model. B.S. and L.Z. performed the simulations. B.S. wrote the manuscript. All authors reviewed the manuscript.
The authors declare that they have no competing interests.
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