1. Introduction

Artificial, near-zero permittivity (NZP) materials were first predicted more than 50 years ago [1, 2]. Since then much attention has been devoted to their linear properties and applications, such as the improvement of antenna directivity [3, 4] and perfect coupling through arbitrarily shaped spatial regions [5]. In addition, the longitudinal (normal to the surface) component of an incident, TM-polarized electric field becomes singular at the interface and inside the material [6]. These materials may then be exploited for nonlinear optical processes [7] such as harmonic generation [8–10], optical bistability [11, 12], and soliton excitation [13]. All naturally occurring materials have two zero-crossing points for the real part of the permittivity in proximity of any electronic, Lorentzian resonance. The positions of the crossing points vary from the far infrared for fluorides and glasses, to the visible and ultraviolet for metals and semiconductors [14]. Absorption, usually not negligible near these resonances, limits electric field enhancement and frustrates both linear and nonlinear optical properties. For this reason solutions to mitigate losses have been discussed in the context of artificial NZPs: active materials may be introduced inside metal-based composites to tailor the electric or magnetic properties to significantly decrease absorption losses [15–18]. The inclusion of metals in artificial NZP media has additional implications due to the nonlinear response arising from symmetry breaking at the surface, the magnetic Lorentz force, inner-core electrons, convective sources, and electron gas pressure [10, 19]. Metals also display a large third-order nonlinearity that may significantly contribute to the generated signals, especially under circumstances where the electric field is dramatically enhanced [20–26].

In this paper we present a brief overview of the linear properties of slabs with NZP properties, and discuss the impact of losses on linear and nonlinear processes. We then consider a scenario where bulk nonlinearities are removed and evaluate the efficiency of harmonic generation arising only from surface, magnetic, and quadrupolar contributions. Our findings suggest that NZP materials are good candidates for low-intensity nonlinear optics, as we show that bulk nonlinearities are not necessary to achieve relatively efficient harmonic generation.

2. Linear properties of NZP materials: the impact of losses

If a monochromatic, TM-polarized plane wave impinges on an interface between a generic medium and a material with relative permittivity that tends to zero, the longitudinal component of the electric field inside the NZP material $\left(E_{\text{z}}\right)$becomes singular. This follows immediately from the condition $E_z=E_{z,0}{\epsilon }_0/{\epsilon }_{\text{M}}$, where $E_{z,0}$ and ${\epsilon }_0$ are the longitudinal component of the electric field and permittivity, respectively, in the half-space of incidence, and ${\epsilon }_{\text{M}}$is the permittivity of the NZP medium. This condition is achieved when the angle of incidence matches the Brewster or critical angle [6]. In contrast, in slabs of finite thickness $E_{\text{z}}$ may become singular by either (i) reducing the thickness of the slab, or (ii) approaching normal incidence [6]. We first consider the linear properties of a homogenous slab of material of thickness d = 200 nm. The slab is surrounded by air and is illuminated by a TM-polarized plane wave incident at an angle ${\vartheta }_i$ with respect to the z-axis, with electric field and wave-vector on the x-z plane, as shown in Fig. 1(a).The slab has relative permittivity${\epsilon }_{\text{M}}(\omega )$ modeled by a superposition of classical Lorentz oscillators:

 

Fig. 1 (a) A TM-polarized pump impinges on a slab with thickness d and angle of incidence ${\vartheta }_i$; (b) Transmission, (c) Reflection and (d) Absorption when d = 200nm. Fundamental (red, dashed line marked λ_ω), second (green, dashed line marked λ_2ω) and third (blue, dashed line marked λ_3ω) harmonic wavelengths are shown on the absorption map.

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Transmission and reflection frequency selectivity and asymmetry depend on the slope and sign of $\mathrm{Re}\left[{\epsilon }_{\text{M}}(\omega )\right]$. In contrast, both angular and spectral selectivity of the slab’s absorption [Fig. 1(c)] depend on$\mathrm{Im}\left[{\epsilon }_{\text{M}}(\omega )\right]$, which in turn depends on the damping coefficient ${\gamma }_1$. No other spectral and angular features are discernible away from the pump wavelength λ_ω = 1064nm [red, dashed line in Fig. 1(d)]. Absorption is low at the second [green, dashed line marked λ_2ω = 2λ_ω in Fig. 1(d)] and third harmonic wavelengths, [blue, dashed line marked λ_3ω = 3λ_ω in Fig. 1(d)]. It is well-known that electric field enhancement [Fig. 2(a) ] is related to the absorption profile [Fig. 1(d)] and is thus linked to the choice of damping coefficient ${\gamma }_1$. However, in this system absorption values depend weakly on$\mathrm{Im}\left[{\epsilon }_{\text{M}}(\omega )\right]$: for example, we find that decreasing ${\gamma }_1$ by two orders of magnitude does not result in substantial decrease in maximum absorption, which is in fact proportional to the product$\mathrm{Im}\left[{\epsilon }_{\text{M}}(\omega )\right]{\left|E\right|}^2$. At the crossing point near λ_ω, a decreasing $\mathrm{Im}\left[{\epsilon }_{\text{M}}(\omega )\right]$is associated with an increasing electric field enhancement: more specifically the electric field enhancement grows proportionally to $1/\sqrt{\mathrm{Im}\left[{\epsilon }_{\text{M}}(\omega )\right]}$. Even if absorption losses were not negligible at the pump wavelength, nonlinear optical interactions are expected to persist even for relatively large ${\gamma }_1$. Figure 2 shows the electric field enhancement maps for different values of ${\gamma }_1$ as a function of wavelength and angle of incidence.

 

Fig. 2 Electric field enhancement [$\max \left(\left|E_{\text{z}}\right|/\left|E_{\text{z,0}}\right|\right)$] vs. wavelength and angle of incidence for a d = 200nm thick, modeled with ${\omega }_{\text{p1}}^{}=0.906049{\omega }_{\text{r}}$, ${\omega }_{\text{01}}^{}=0.25{\omega }_{\text{r}}$and (a) ${\gamma }_1=0.01{\omega }_{\text{r}}$, (b) ${\gamma }_1=0.001{\omega }_{\text{r}}$and (c) ${\gamma }_1=0.0001{\omega }_{\text{r}}$.

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3. Impact of losses on nonlinear dynamics in the presence of bulk nonlinearities

Second and third order nonlinear effects are accounted for by expressing the leading contributions of the nonlinear polarization densities in the k-direction at the second (SH) and third harmonic (TH) frequencies as $P_{\text{2ω,k}}={\epsilon }_0{\displaystyle \sum _{\text{l,m}=1}^3{\chi }_{\text{klm}}^{(2)}(\text{2ω},\text{ω},\text{ω})}{E}_{\text{ω,l}}{E}_{\text{ω,m}}$ and $P_{\text{3ω,k}}={\epsilon }_0{\displaystyle \sum _{\text{l,m,n}=1}^3{\chi }_{\text{klmn}}^{(3)}(\text{3ω,ω,ω,ω})}{E}_{\text{ω,l}}{E}_{\text{ω,m}}{E}_{\text{ω,n}}$, where k,l,m,n are the Cartesian axes,${\chi }_{\text{klm}}^{(2)}$ and${\chi }_{\text{klmn}}^{(3)}$are the instantaneous second and third order susceptibility tensor components, respectively. We assume ${\chi }_{\text{xxx}}^{(2)}={\chi }_{\text{yyy}}^{(2)}={\chi }_{\text{zzz}}^{(2)}=10\text{pm/V}$, ${\chi }_{\text{xxxx}}^{(3)}={\chi }_{\text{yyyy}}^{(3)}={\chi }_{\text{zzzz}}^{(3)}=$ ${10}^{-20}{\text{m}}^{\text{2}}{\text{/V}}^{\text{2}}$ and a continuous wave pump with irradiance ${\text{I}}_{\text{ω}}=100\text{MW}/{\text{cm}}^{\text{2}}$, and evaluate SHG and THG for slabs d = 200 nm thick for different damping coefficients (Fig. 3 ). The total SH (${\text{I}}_{\text{2ω}}/{\text{I}}_{\text{ω}}^{}$) and TH (${\text{I}}_{\text{3ω}}/{\text{I}}_{\text{ω}}^{}$) conversion efficiencies in the undepleted pump approximation are of the order of 10⁻⁵ and 10⁻⁷ [Figs. 3(a)–3(b)] when ${\gamma }_1=0.01{\omega }_{\text{r}}$. SH and TH efficiencies increase by one and two orders of magnitude, respectively, if the damping coefficient is reduced to ${\gamma }_1=0.001{\omega }_{\text{r}}$[Figs. 3(c) and 3(d)]. The efficiency maps resemble the field enhancement map [Figs. 2(a) and 2(b)] in spectral and angular features. The results in Fig. 3 may also be obtained using pulses 400 fs or longer for ${\gamma }_1=0.01{\omega }_{\text{r}}$, or at least 800 fs when${\gamma }_1=0.001{\omega }_{\text{r}}$to resolve the respective spectral features.

 

Fig. 3 Total (a) SH and (b) TH conversion efficiencies for a slab d = 200 nm thick as a function of incident angle and pump wavelength. The slab is modeled assuming ${\omega }_{\text{p1}}^{}=0.906049{\omega }_{\text{r}}$, ${\omega }_{\text{01}}^{}=0.25{\omega }_{\text{r}}$, and ${\gamma }_1=0.01{\omega }_{\text{r}}$. (c) and (d) same as in (a) and (b) for ${\gamma }_1=0.001{\omega }_{\text{r}}$.

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4. Nonlinear dynamics of NZP materials in the absence of bulk nonlinearities

If ${\chi }_{}^{(2)}$and ${\chi }_{}^{(3)}$ are either sufficiently small or absent, only surface, magnetic [19], and electric quadrupole sources contribute to harmonic generation. One may then write equations of motion for the polarizations at the pump, SH and TH frequencies that take into account all of the above contributions, as described in details in Ref [27]. Choosing the same two sets of oscillator parameters described in Sec. 3 one may then quantify SH and TH efficiencies and assess the role of losses. By assuming ${\text{I}}_{\text{ω}}=100\text{MW}/{\text{cm}}^{\text{2}}$, 400fs pulses, and tuning the pump such that ${\epsilon }_{\text{M}}=\text{1}\text{.703}\times {\text{10}}^{\text{-7}}+i0.0115$ at λ_ω = 1064 nm, we obtain a new total SH conversion efficiency maximum of ~10⁻¹⁰ [Fig. 4(a) , blue, solid line], and a vanishingly small TH efficiency maximum of ~10⁻¹⁹ when ${\gamma }_1=0.01{\omega }_{\text{r}}$ [Fig. 4(b), blue, solid line]. By reducing the damping coefficient to ${\gamma }_1=0.001{\omega }_{\text{r}}$and increasing pulse width to at least 800fs, maximum SH and TH efficiencies are of order 10⁻⁹ [Fig. 4(a), red, dashed line] and 10⁻¹⁷ [Fig. 4(b), red, dashed line], respectively. For comparison, we note that the maximum SH conversion efficiency shown in Fig. 4(a) is still ~100 times larger than the SH response of a silver grating where surface plasmons or resonant cavity modes are excited [19, 21]. In Fig. 5 we plot the new conversion efficiencies that exploit the NZP condition both at the pump and SH or TH wavelengths, accomplished by introducing resonances in the oscillators as indicated in the Fig. 5 caption. Our calculations, carried out using pulses having durations up to ~800fs to resolve the resonances for different damping coefficients, show that while SH conversion efficiency grows by an additional two orders of magnitudes to 10⁻⁷, the impact of this condition is more dramatic when the NZP condition occurs for the TH frequency, producing a more than remarkable improvement of eleven orders of magnitude in THG, notwithstanding the fact that the bulk ${\chi }^{(3)}=0$.

 

Fig. 4 (a) SH efficiency vs. angle of incidence for a slab d = 200 nm thick, modeled with ${\omega }_{\text{p1}}^{}=0.906049{\omega }_{\text{r}}$, ${\omega }_{\text{01}}^{}=0.25{\omega }_{\text{r}}$ and ${\gamma }_{\text{1}}^{}=0.01{\omega }_{\text{r}}$ (blue, solid line) and ${\gamma }_{\text{1}}^{}=0.001{\omega }_{\text{r}}$(red, dashed line). (b) Same as in (a) for TH.

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Fig. 5 (a) SH efficiency vs. angle of incidence for a slab d = 200 nm thick, modeled with ${\omega }_{\text{p1}}^{}={\omega }_{\text{p2}}^{}={\omega }_{\text{p3}}^{}=0.906049{\omega }_{\text{r}}$, ${\omega }_{\text{01}}^{}=0.25{\omega }_{\text{r}}$, ${\omega }_{\text{02}}^{}=1.652{\omega }_{\text{r}}$, ${\omega }_{\text{03}}^{}=2.678{\omega }_{\text{r}}$ ${\gamma }_{\text{1}}^{}={\gamma }_{\text{2}}^{}={\gamma }_{\text{3}}^{}=0.01{\omega }_{\text{r}}$ (blue, solid line) and ${\gamma }_{\text{1}}^{}={\gamma }_{\text{2}}^{}={\gamma }_{\text{3}}^{}=0.001{\omega }_{\text{r}}$(red, dashed line); (b) Same as in (a) for TH.

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5. Conclusions

We have discussed SHG and THG in slabs of material with near-zero permittivity. We showed that losses can dramatically influence the nonlinear response in the presence or absence of bulk nonlinearities. If bulk nonlinearities are sufficiently small or absent nonlinear surface, magnetic, and quadrupolar sources cannot be neglected, especially if crossing points are engineered at the harmonic wavelengths. Our findings suggest that sub-wavelength slabs of low-permittivity materials or plasmas may be effectively used to achieve efficient harmonic generation even at low intensities and in the presence of weak nonlinearities.