Introduction

With the rapid progress of modern technology, natural materials are more and more hard to satisfy the requirements of scientific development, and consequently, new kinds of artificially synthetic materials take advantage of the opportunity to exhibit their exotic features and applications [1,2,3,4,5,6]. At present, many researchers have been turning their attention to metamaterials, which are a kind of artificially composite electromagnetic structures consisting of subwavelength unit cells [7,8,9,10,11,12,13,14,15,16,17]. Artificially engineered metamaterials have shown many unusual electromagnetic properties, for instance, negative refraction [18], ultrahigh-resolution subwavelength imaging [11], self-induced torque [12], spatial filtering, no diffraction transmission [13], and so on. These novel properties are attributed to the fact that they may possess rich dispersion characteristics due to with indefinite permittivity or/and permeability, i.e., not all their tensor components are of the same sign. Owing to the great progress in fabrication technologies of artificial materials, electromagnetic metamaterials are developed from microwave frequency to terahertz frequency and even to light frequency [1, 5, 19].

It is well known that the earliest proposed metamaterial is an array of thin metallic wires [20,21,22,23,24,25,26,27,28]. This kind of anisotropic metamaterials (AMMs) have shown that the dielectric constant can be negative in the direction of the metallic wires, as well as many advantages. On the one hand, because the dielectric response in such metamaterials does not require any resonance, the negative refraction for all incident angles has low loss in a broad spectral range, which is a few orders of magnitude lower than that of single-layer metamaterials [18]. Furthermore, this kind of metamaterials can support propagating waves with large wave vectors, while electromagnetic waves are evanescent in usual dielectrics, enabling manipulation of visible light at subwavelength scale. On the other hand, nonlinear metamaterials are also easy to implement, as long as the wire array is embedded into a nonlinear medium with different nonlinear response types [29]. This provides the possibility for us to dynamically manipulate electromagnetic transmission characteristics in nonlinear metamaterials. In many studies of metamaterial systems, their electromagnetic behaviors are based on a local effective medium model, under this condition that the excitations involve a vanishingly small momentum exchange, i.e., a vanishing wave vector [30]. The well-known local response model for the dielectric function of the wire medium is an excellent approximation at high frequencies [31].

At present, controlling electromagnetic characteristics of metamaterials are getting more and more attentions [5, 18]. The applications of metamaterials in all-optical switching and routing have attracted many interests [32,33,34]. For instance, Shoaei et al. [32] proposed a periodic stack of metal/Al2O3/ graphene /Al2O3/metal nano-layers metamaterial, where light can be switched between a positive/negative or a negative/no-transmission regime via tuning its chemical potential. But there are still some basic issues to be clarified. For instance, how to control dynamically electromagnetic characteristics of metamaterials? We should also expect dynamic regulation may be achieved by nonlinear effects in metamaterials. In the present paper, we consider the case of nonlinear metamaterials, confirming that it is entirely feasible to dynamically control interconversion between the total reflection state (OFF state) and the total transmission state (ON state), as well as control light propagation route through an external field in such metamaterials. In addition, we demonstrate that these dynamically tunable characteristics have wide applications in all-optical switches, all-optical router, and other nanophotonic devices.

The paper is organized as follows. In section “Basic Theory of Electromagnetic Wave Propagation in AMMs”, an anisotropic nonlinear metamaterial is constructed by the periodic array of thin metallic nanowires, and the reflection coefficient and transmission coefficient at the nonlinear metamaterial interface are deduced. In section “Results and Discussion”, the numerical results show that there exist anomalous transmission and reflection in such metamaterial. In particular, these properties can be controlled by an applied electric field and have many potential applications in the field of optoelectronics. Some brief conclusions are presented in section “Conclusions”.

Basic Theory of Electromagnetic Wave Propagation in AMMs

In the following, we consider a nonlinear electromagnetic metamaterial, which can be constructed by an array of ideally conducting nanowires embedded into a nonmagnetic host medium with a Kerr-type nonlinearity such as εh = 𝜖D0(1 + δε) with δε = ϰe∗⋅e, where e is the microscopic electric field and ϰ = 3χ(3)/2𝜖D0 is a constant that determines the strength of the nonlinear effects. The geometry of the parallel nanowire array is illustrated in Fig. 1a. It is assumed that the metallic nanowires are cylindrical, and the radius of nanowires and the period of the array are r and a, respectively. Both of them are much smaller than the working wavelength of incident light so that the electromagnetic properties of the component metamaterial can be analyzed by an effective medium theory. As shown schematically in Fig. 1, xoz and x′oz′ denote the transmission and principal coordinate systems, respectively. The principal axes of the metamaterial crystal lie along the Cartesian axes x′, y′, and z′. xz is the plane of incidence, and z is normal to the slab surface. Without losing generality, we assume that the parallel metallic nanowires are oriented along the z′ direction.

In our model, there are two fields, one is the incident field (signal field), and the other is the applied electrostatic field (modulation field). We assume that the incident field (signal field) is a weak field, which only causes the linear response of the medium and does not cause any nonlinear polarization of the medium. However, the applied electrostatic field (modulation field) has a strong intensity. It will cause the third-order nonlinear polarization of the medium with Kerr nonlinearity. In other words, it will cause the nonlinear change of the dielectric constant (or refractive index) of the medium. This refractive index change will modulate the propagation characteristics of the incident field (signal field) with a weak intensity. It is well known that the relative dielectric permittivity tensor \(\widehat {{{\varepsilon }}} \) of the anisotropic metamaterial has the following diagonalizable form [35, 36]:

$$ \boldsymbol{\widehat{{{\varepsilon}}} }=\left( \begin{array}{ccc} {{\varepsilon}}_{x} & 0 & 0 \\ 0 & {{\varepsilon}}_{y} & 0 \\ 0 & 0 & {{\varepsilon}}_{z} \end{array} \right) $$
(1)

in the principal coordinate system. If an electrostatic field E0 is applied in the y direction, the elements of the dielectric tensor \( \widehat {{{\varepsilon }}} \) of the electromagnetic metamaterial, caused by the modulation field E0, can be expressed as the following forms [7]:

$$ {{\varepsilon}}_{x}={\epsilon}_{D}({E_{0}^{2}}{})={\epsilon}_{D0}+\zeta {E_{0}^{2}}{}/{E_{c}^{2}} $$
(2)
$$ {{\varepsilon}}_{y}={\epsilon}_{D0}+\zeta {E_{0}^{2}}{}/E_{cy}^{2} $$
(3)
$$ {{\varepsilon}}_{z}={\epsilon}_{D}({E_{0}^{2}})-\frac{{\omega_{p}^{2}}}{\omega (\omega -i\gamma_{{\epsilon}} )} $$
(4)

according to the theory of nonlinear optics and Drude model, where εz denotes the effective nonlinear dielectric permittivity of the electromagnetic metamaterial along the direction of the nanowires (namely, the z′ direction), and ωp ≈ (c/a)[2π/ln(a/r)]1/2 is the effective plasma frequency, γ𝜖 = c2/2σS ln(a/r) (in the case of low losses, γ𝜖 ≪ ω), c is the speed of light, σ is the conductivity of the metal wires, and S is the effective area of the wire cross section, respectively [7]. 𝜖D0 is the linear dielectric constant of the dielectric substrate, Ec and Ecy are two characteristic parameters that determine the strength of the nonlinear effects, which are related to the third-order nonlinear susceptibility χ(3) of the host medium, namely, \(1/{E_{c}^{2}}~=~3\left \vert \chi _{xxyy}^{(3)}\right \vert /2~=~3\left \vert \chi _{zzyy}^{(3)}\right \vert /2\), and \(1/E_{cy}^{2}~=~3\left \vert \chi _{yyyy}^{(3)}\right \vert /2\). In addition, ζ = ± 1 stand for a focusing and defocusing nonlinearity, respectively. For the sake of simplicity, we will not consider losses and spatial dispersion effect of the system in the following.

Based on Eqs. 2 and 4, Fig. 2 shows the dependence of εx and εz on the normalized intensity of the applied electric field \({E_{0}^{2}}/{E_{c}^{2}}\), in which (a) and (b) correspond to a focusing and defocusing nonlinearity, respectively. It is not hard to see that there exists different relationship between εz (or εx) and the external electric field E0 for different nonlinear response types. It is worth noting that it is accessible to control the value and sign of the effective dielectric permittivity εz via changing the normalized intensity of external electric field, \({E_{0}^{2}}/{E_{c}^{2}}{}\). In Fig. 2, we can see that with \({E_{0}^{2}}/{E_{c}^{2}}{}\) increasing, both εx and εz increase linearly for a focusing nonlinearity (ζ = 1 ), while they decrease linearly for a defocusing nonlinearity (ζ = − 1). It is important to note that the sign of εz can be changed for both focusing and defocusing nonlinearity with the external electric field intensity increasing, providing a feasible way for us to dynamically control the transmission characteristics of electromagnetic wave in the nonlinear metamaterials. The underlying physics is due to the excitation of surface plasmon polariton (SPP) along the direction of the metallic wires in the metamaterials, which makes it is possible that the effective dielectric permittivity εz of the metamaterials is negative. But it is worth emphasizing that εx should always be positive due to the lack of SPP excitation in the x′ direction.

In what follows, we consider that a transverse magnetic (TM) wave propagates from air into the composite metamaterial structure with an incident angle 𝜃 and angular frequency ω, as shown in Fig. 1b. The time-harmonic magnetic field polarizing along the y axis has the form

$$ \mathbf{H}_{i}=H_{0}\mathbf{e}_{y}\exp (ik_{x}x+ik_{z}z-i\omega t) $$
(5)

where H0 is the amplitude of the incident magnetic field. kx and kz are the x- and z-axis components of the incident wave vector in air dielectric, respectively. By transforming the principal coordinate system (x′oz′) into transmission coordinate system (xoz ) via Euler transformation [37], then the relative dielectric permittivity tensor \(\widehat {{{\varepsilon }}} \) in the transmission axis coordinate system can be written as

$$\begin{array}{@{}rcl@{}} \boldsymbol{\widehat{{{\varepsilon}}} } &=&\left( \begin{array}{ccc} {{\varepsilon}}_{xx} & 0 & {{\varepsilon}}_{xz} \\ 0 & {{\varepsilon}}_{yy} & 0 \\ {{\varepsilon}}_{zx} & 0 & {{\varepsilon}}_{zz} \end{array} \right) \ \\ &=&\left( \begin{array}{ccc} {{\varepsilon}}_{x}\cos^{2}\varphi +{{\varepsilon}}_{z}{\sin}^{2}\varphi & 0 &({{\varepsilon}}_{x}-{{\varepsilon}}_{z}){\sin} \varphi \cos \varphi \\ 0 & {{\varepsilon}}_{y} & 0 \\ ({{\varepsilon}}_{x}-{{\varepsilon}}_{z}){\sin} \varphi \cos \varphi & 0 &{{\varepsilon}}_{z}\cos^{2}\varphi +{{\varepsilon}}_{x}{\sin}^{2}\varphi \end{array} \right) \\ \end{array} $$
(6)

Based on Eq. 6 and Maxwell’s equations, it is not hard to get the dispersion equation in the composite anisotropic metamaterial as follows:

$$ \alpha k_{tz}^{2}+\beta k_{tx}^{2}+{\eta} k_{tz}k_{tx}=\frac{\omega^{2}}{c^{2}} $$
(7)

where ktx and ktz are the x and z components of the refraction wave vector, respectively, and

$$ \alpha =\frac{1}{\mu_{y}{{\varepsilon}}_{x}{{\varepsilon}}_{z}}({{\varepsilon}} _{x}{\sin}^{2}\varphi +{{\varepsilon}}_{z}\cos^{2}\varphi ) $$
(8)
$$ \beta =\frac{1}{\mu_{y}{{\varepsilon}}_{x}{{\varepsilon}}_{z}}({{\varepsilon}} _{x}\cos^{2}\varphi +{{\varepsilon}}_{z}{\sin}^{2}\varphi ) $$
(9)
$$ {\eta} =\frac{1}{\mu_{y}{{\varepsilon}}_{x}{{\varepsilon}}_{z}}({{\varepsilon}} _{x}{\sin} 2\varphi -{{\varepsilon}}_{z}{\sin} 2\varphi ) $$
(10)

where φ is the angle between the transmission axis and principal axis, and μy is the relative permeability of the metamaterial in the y direction [35]. From the dispersion Eq. 7, we can derive that the z component of the refraction wave vector has the following form

$$ k_{tz}=\frac{-{\eta} k_{tx}+\delta \sqrt{({\eta}^{2}-4\alpha \beta )k_{tx}^{2}+ 4\alpha \omega^{2}/c^{2}}}{2\alpha} $$
(11)

where ktx = kx due to the phase matching condition at the interface; δ = ± 1 to ensure compliance with the law of energy conservation.

From Eq. 11, we can know that if the incident electromagnetic wave can transmit inside the right-side composite metamaterial structure, as shown in Fig. 1b, it is necessary to satisfy the inequation [35]

$$ {{\varepsilon}}_{r}\mu_{r}({\eta}^{2}-4\alpha \beta ){\sin} {\theta} + 4\alpha >0 $$
(12)

where εr and μr are the relative permittivity and permeability of air, respectively. By Fresnel equations, we can deduce the reflection coefficient R and transmission coefficient T at the interface between air and the composite metamaterial structure as follows:

$$ R=-1+\frac{4k_{z}}{2k_{z}+{{\varepsilon}}_{r}\mu_{y}k_{tx}\gamma + 2{{\varepsilon}} _{r}\mu_{y}\alpha k_{tz}} $$
(13)
$$ T=\frac{4k_{z}}{2k_{z}+{{\varepsilon}}_{r}\mu_{y}k_{tx}\gamma + 2{{\varepsilon}} _{r}\mu_{y}\alpha k_{tz}} $$
(14)

respectively. Based on the definition of the time averaged Poynting vector \( \langle \mathbf {S\rangle } ~=~\frac {1}{2}\)Re[E ×H∗], the averaged Poynting vector of incident wave can be deduced as follows:

$$\begin{array}{@{}rcl@{}} \langle \mathbf{S_{i}\rangle} &=&\frac{1}{2}\text{Re}[\mathbf{E}_{i}\times \mathbf{H}_{i}^{\ast} ] \\ &=&\text{Re}\left[\frac{{H_{0}^{2}}k_{x}}{2{{\varepsilon}}_{0}{{\varepsilon}}_{r}\omega} \mathbf{e}_{x}+\frac{{H_{0}^{2}}k_{z}}{2{{\varepsilon}}_{0}{{\varepsilon}}_{r}\omega} \mathbf{e}_{z}\right] \end{array} $$
(15)

and the time averaged Poynting vector of refracted wave is

$$\begin{array}{@{}rcl@{}} \langle \mathbf{S_{t}\rangle}\! &\mathbf{=}&\frac{1}{2}\text{Re}[\mathbf{E} _{t}\times \mathbf{H}_{t}^{\ast} ] \\ &=&\frac{1}{2}\mu_{y}{H_{0}^{2}}|T|^{2} \\ &&\times\text{Re}\left\{ \left[\left( \frac{k_{tx}\beta} {{{\varepsilon}}_{0}\omega} \!+\!\frac{ k_{tz}{\eta}} {2\omega {{\varepsilon}}_{0}}\right)\mathbf{e}_{x}+\left( \frac{k_{tz}\alpha} {\omega {{\varepsilon}}_{0}}\!+\!\frac{k_{tx}{\eta}} {2\omega {{\varepsilon}}_{0}}\right) \mathbf{e}_{z}\right]\right.\\ &&\left.\times\exp [i(k_{tz}-k_{tz}^{\ast} )]\vphantom{\left( \frac{k_{tx}\beta} {{{\varepsilon}}_{0}\omega} +\frac{ k_{tz}{\eta}} {2\omega {{\varepsilon}}_{0}}\right)}\right\} \end{array} $$
(16)

From Eq. 16, we can derive that

$$ \mathbf{e}_{z}\cdot \langle \mathbf{S_{t}}\rangle \!=\frac{1}{2}\text{Re}\!\left[ \frac{k_{tx}{{\varepsilon}}_{xz}+k_{tz}{{\varepsilon}}_{zz}}{{{\varepsilon}} _{0}\omega {{\varepsilon}}_{x}{{\varepsilon}}_{z}}{H_{0}^{2}}|T|^{2}\exp [i(k_{tz}-k_{tz}^{\ast} )z\right] $$
(17)
Fig. 1
figure 1

a Schematic of a nonlinear metamaterial constructed by the periodic array of conducting nanowires embedded into a nonlinear host dielectric. a is the periodic distance between the metallic nanowires; r is the radius of metallic nanowires. b Schematic of TM wave in the metamaterial interface. Ki, Kt, and St represent incident wave vector, refraction wave vector, and energy flow vector, respectively; 𝜃i is the incident angle; φ is the angle between the transmission coordinate Z axis and principal axis Z′, called the orientation angle of the nanowire array; 𝜃s and 𝜃t denote the energy-flow and wave-vector refraction angles, respectively. E0 is an external electric field oriented along the y direction

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Then in Eq. 11, when \({\Delta } =({\eta }^{2}-4\alpha \beta )k_{tx}^{2}+ 4\alpha \omega ^{2}/c^{2}<0\)

$$ k_{tz}=\frac{-{\eta} k_{tx}+\delta \sqrt{|{\Delta} |}i}{2\alpha} $$
(18)

where δ = ± 1, and i is the imaginary unit, therefore ktz is a complex number. What’s more, from Eqs. 17 and 18, we can obtain

$$\mathbf{e}_{z}\cdot \langle \mathbf{S_{t}}\rangle =\text{Re}\left[\frac{\pm {{\varepsilon}}_{zz}\sqrt{|{\Delta} |}i}{4\alpha {{\varepsilon}}_{0}\omega {{\varepsilon}}_{x}{{\varepsilon}}_{z}}{H_{0}^{2}}|T|^{2}\exp i(k_{tz}-k_{tz}^{\ast })z\right]= 0 $$

This means, for the transmission wave, there is no energy flow along the z-axis. In other words, when the condition \({\Delta }~=~({\eta }^{2}~-~4\alpha \beta )k_{tx}^{2}~+~4\alpha \omega ^{2}/c^{2}~<~0\) is satisfied, all of incident waves are reflected totally.

What’s more, we obtain the transmissivity and energy-flow angle from Eq. 16 as follows:

$$ t=\frac{\mu_{y}{{\varepsilon}}_{r}|T|^{2}(k_{tx}{\eta} + 2k_{tz}\alpha )}{2k_{z}} $$
(19)
$$ {\theta}_{s}\!=tan^{-1}\frac{(4\alpha \beta \!-\!{\eta}^{2})\sqrt{{{\varepsilon}} _{r}\mu_{r}}{\sin} {\theta}_{i}\!+\!{\eta} \sqrt{({\eta}^{2}\!-\!4\alpha \beta ){{\varepsilon}}_{r}\mu_{r}{\sin}^{2}{\theta}_{i}\!+\!4\alpha} }{2\alpha \sqrt{({\eta} ^{2}\!-\!4\alpha \beta ){{\varepsilon}}_{r}\mu_{r}{\sin}^{2}{\theta}_{i}\!+\!4\alpha} } $$
(20)

respectively [35]. In addition, based on the definition of wave-vector refraction angle, 𝜃t = tan− 1kx/ktz, we can get its specific expression from Eq. 11 as

$$ {\theta}_{t}=\tan^{-1}\!\frac{\sqrt{{{\varepsilon}}_{r}\mu_{r}}2\alpha {\sin} {\theta}_{i}}{-{\eta} \sqrt{{{\varepsilon}}_{r}\mu_{r}}{\sin} {\theta}_{i}\!+\!\sqrt{ ({\eta}^{2}\!-\!4\alpha \beta ){{\varepsilon}}_{r}\mu_{r}{\sin}^{2}{\theta} _{i}\!+\!4\alpha} } $$
(21)

Results and Discussion

In what follows, some parameters used in our numerical calculations are adopted as εr = μr = 1, μy = 1, and incident light frequency ω = ωp/2. At first, we need to define the positive and negative value of angle. Generally, it is based on this rule that beginning from the normal line of interface and pointing to incident rays or refracted lines, if it is clockwise, then the angle is negative; otherwise, the angle is positive. And thus, we can determine whether it is a positive or negative refraction for both energy flow and wave vector. When the incident angle and refraction angle is the same sign, the incident ray and the refracted one are located on both sides of the normal line, then it is regarded as positive refraction. If the incident angle and refraction angle has different sign, the incident ray and the refracted one are located at the same side of the normal line, then it is negative refraction [35].

Comparing Eqs. 20 and 21, we can see that the direction of the energy flow and wave vector should be inconsistent with each other in anisotropic metamaterials generally. To see the difference between them, we plot the energy-flow refraction angle 𝜃s, the wave-vector refraction angle 𝜃t, and their included angle |𝜃s − 𝜃t| as a function of incident angle 𝜃i for different orientation angle φ of the nanowire array in the absence of nonlinear response for two cases: (1) 𝜖D0 = 2.4; (2) 𝜖D0 = 6 in Fig. 3. It is worth emphasizing that when 𝜖D0 = 6.0, εx > 0 and εz > 0; when 𝜖D0 = 2.4, εx > 0 and εz < 0. The two cases correspond to the dielectric properties in the limit of \( {E_{0}^{2}}/{E_{c}^{2}}~=~0\) in Fig. 2b, a, respectively.

Fig. 2
figure 2

Effective relative dielectric permittivities εx and εz versus the normalized intensity of external electric field \({E_{0}^{2}}/{E_{c}^{2}}{}\) for two cases: aζ = 1 (for self-focusing nonlinearity), and εD0 = 2.4, bζ = − 1 (for self-defocusing nonlinearity), and εD0 = 6

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Linear Response with ε x > 0 and ε z > 0

In Fig. 3a–c with 𝜖D0 = 6, corresponding to the case of εx > 0 and εz > 0, we can see that the refraction wave-vector angle 𝜃t has always the same sign as the incident angle 𝜃i for any orientation angle φ, as shown in Fig. 3a; therefore, the wave vector is always positive refraction at the metamaterial interface, which is consistent with that in usual anisotropic materials. However, whether the energy flow is positive or negative refraction depends on the orientation angle φ. From Fig. 3b, one can see that when φ = 0∘, the energy-flow angle 𝜃s has always the same sign as 𝜃i for any incident angle; therefore, the energy flow is always positive refraction when φ = 0∘. But when φ ≠ 0∘, 𝜃s has the same sign as 𝜃i except for the cases of \(0~<~{\theta }_{I}~<~{\theta }_{c}(\varphi )~=~{\sin }^{-1}\sqrt {\frac {{\eta }^{2}}{{{\varepsilon }}_{r}\mu _{r}\beta (4\alpha \beta ~-~{\eta }^{2})}}\), as shown in the green shaded area in Fig. 3b. Here, it is worthy noting that the critical incident angle 𝜃c depends on the orientation angle φ. This indicates that if we choose a suitable orientation angle φ, it is also possible that the energy flow is negative refraction in the case of εx > 0 and εz > 0, and the region of negative refraction is tunable by the orientation angle φ.

Fig. 3
figure 3

The energy-flow refraction angle 𝜃s, the wave-vector refraction angle 𝜃t, and their included angle \( \left \vert \protect {\theta }_{s}-\protect {\theta }_{t}\right \vert \) versus the incident angle 𝜃i with different orientation angles φ of the nanowire array in the limit of \( {E_{0}^{2}}/{E_{c}^{2}}~=~0\) for two cases: (1) 𝜖D0 = 6 (the left column (a, b, c)); (2) 𝜖D0 = 2.4 (the right column (d, e, f))

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Linear Response with ε x > 0 and ε z < 0

In Fig. 3d–f with 𝜖D0 = 2.4, corresponding to the case of εx > 0 and εz < 0, it is not hard to see some important properties as follows: (1) when φ = 0∘, 𝜃t has the same sign as 𝜃i for any incident angle; therefore, the wave vector is always positive refraction. But 𝜃s has opposite sign with 𝜃i, therefore, the energy flow is always negative refraction, which is evidently different from the case of 𝜖D0 = 6 (namely εx > 0 and εz > 0). In the particular case of 𝜃i = 0∘, i.e., the normal incidence, both 𝜃s and 𝜃t are 0, meaning the refraction wave vector is coincident with the Poynting vector, and light travels in straight lines at the metamaterial interface. Therefore, the energy flow belongs to all-angle negative refraction for any incident angle except for 𝜃i = 0∘. The property is absent in usual anisotropic materials. (2) When φ = − 30∘, the wave vector is always positive refraction for any incident angle, but the refractive properties of the energy flow depend on the sign of incidence angle. Specifically, when − 90∘ < 𝜃i < 0∘, 𝜃s has the different sign with 𝜃i; therefore, the energy flow is negative refraction; when 0∘ < 𝜃i < 90∘, the energy-flow angle is positive; consequently, it is positive refraction. (3) For a fixed orientation angle φ = − 45∘, we can get when \(-\pi /2<{\theta }_{i}<-{\sin }^{-1} \sqrt {\frac {-4\alpha } {{\eta }^{2}~-~4\alpha \beta } }\), 𝜃s < 0 and the energy-flow refraction would be positive, but 𝜃t > 0 and the wave vector refraction is negative; if \({\sin }^{-1}\sqrt {\frac {-4\alpha } {{\eta } ^{2}~-~4\alpha \beta } }<{\theta }_{i}<\pi /2\), then 𝜃s > 0 and 𝜃t > 0; therefore, both the wave vector and energy flow are positive refraction; if \(-{\sin }^{-1}\sqrt {\frac {-4\alpha } {{\eta }^{2}~-~4\alpha \beta } } <{\theta }_{i}<{\sin }^{-1}\sqrt {\frac {-4\alpha } {{\eta }^{2}~-~4\alpha \beta } }\), there exist not any wave vector and energy-flow refraction angles, meaning no energy can be transmitted and total reflection phenomena occur. This property is also absent in usual anisotropic materials. Here, \({\theta } _{c1}(\varphi )=\pm {\sin }^{-1}\sqrt {\frac {-4\alpha } {{\eta }^{2}~-~4\alpha \beta } }\) are two critical angles, and when φ = − 45∘,𝜃c1 = ± 39.22∘.

Figure 3c, f shows the included angle between the refraction wave vector kt and the average Poynting vector 〈St〉. We can see that the included angle in anisotropic metamaterials may be much larger than that in usual anisotropic materials. For instance, in the case of 𝜖D0 = 2.4 in Fig. 3f, when φ = − 45∘, the included angle between kt and 〈St〉 is in the region from 90∘ to 270∘ for − 90∘ < 𝜃i < − 39.22∘, implying a backward wave will occur according to the definition of backward wave. What’s more, when φ = 0∘, the included angle between the refraction wave vector and the Poynting vector is symmetrical distribution for any anisotropic materials. Especially, the direction of the refraction wave vector is coincident with that of the Poynting vector when 𝜃i = 0∘ and φ = 0∘. After further studying, we find that when the incident angle 𝜃i and the metallic tilted angle φ satisfy the relationship as follows:

$$ {\theta}_{i}(\varphi )=\pm {\sin}^{-1}\sqrt{\frac{2\alpha \left[\sqrt{1+ \frac{({\eta}^{2}-4\alpha \beta ){\eta}^{2}}{4\alpha^{2}((\alpha -\beta )^{2}+{\eta}^{2})}}-1\right]}{({\eta}^{2}-4\alpha \beta ){{\varepsilon}}_{r}\mu_{r}}} $$
(22)

𝜃s = 𝜃t, meaning their directions of the refraction wave vector and the Poynting vector are coincident. In the case of εx > 0 and εz > 0 with φ < 0 in Fig. 3c, it is not hard to find that when 𝜃i < 0, 𝜃s = 𝜃t. In the case of εx > 0 and εz < 0 with φ < 0 in Fig. 3f; however, it is obvious that when 𝜃i > 0, 𝜃s = 𝜃t.

Exotic Properties in Nonlinear Response

In Figs. 4 and 5, we plot the transmissivity t and the energy-flow angle 𝜃s as a function of the normalized intensity of external electric field \({E_{0}^{2}}/{E_{c}^{2}}{}\) for different incident angles 𝜃i, and their isofrequency dispersion curves in the metamaterials of Kerr-type nonlinearity with φ = − 45∘. Figures 4 and 5 correspond to the self-focusing nonlinearity (ζ = 1) and self-defocusing nonlinearity (ζ = − 1), respectively. From Figs. 4 and 5, we can see that the propagation properties of waves/light, including their propagation direction and transmissivity, can be controlled by the external electric field. In Fig. 4a, b, we find that for a fixed incident angle 𝜃i = 0∘ or 30∘ (in general, when 0 ≤ 𝜃i < 𝜃c1(φ)), the TM-polarized light transmission in the composite structure is cut-off when E0 = 0 or \( {E_{0}^{2}}/{E_{c}^{2}}\) less than a certain value (about 1.6). However, if the external field intensity is increased to \(2<{E_{0}^{2}}/{E_{c}^{2}}<6\), the TM-polarized light break-overs and has very high transmissivity, and the energy flow is negative refraction (𝜃s < 0), as shown in Fig. 4e, f. However, for a fixed incident angle 𝜃 = 45∘ or 60∘ (in general, when 𝜃c1(φ) < 𝜃i < 90∘), as shown in Fig. 4c, d, some different features occur compared with Fig. 4a, b. To be specific, when E0 = 0 or \( {E_{0}^{2}}/{E_{c}^{2}}{}\) is greater than the critical value (about 1.6), there are high transmittance and what’s more different is that the energy-flow angle has different sign for E0 = 0 and \({E_{0}^{2}}/{E_{c}^{2}}>1.6\), as shown in Fig. 4g, h, which means that the energy-flow angle (i.e., the direction of the energy flow) is tunable by controlling the intensity of the external electric field. In other words, we can realize the dynamic transformation of energy flow from positive refraction to cut-off and then to negative refraction by regulating the intensity of the applied electric field.

It is well known that the energy-flow direction is coincident with the direction of the group velocity for plane waves, Vg = ▽Kω(K). They are not necessarily parallel to the wave vector in anisotropic materials. ▽Kω(K) must lie normal to the isofrequency contour, ω(K) = const. The general relationship between the directions of energy flow and refraction wave vector for waves propagating in the metamaterial can also be demonstrated by the isofrequency dispersion contour, as illustrated in Fig. 4i, j, and they show a good agreement with the other figures in Fig. 4. From Fig. 4i, we can see that in the limit of \({E_{0}^{2}}/{E_{c}^{2}}~=~0\), when the incident angle is less than a certain critical angle, there exists no refraction wave vector; consequently, there is no energy-flow transmission in the metamaterial, due to the phase mismatching between the refraction wave vector and incident wave vector at the boundary interface. When the incident angle is larger than the critical angle 𝜃c1(φ), however, there exists refraction wave vector; consequently, there is energy-flow transmission into the metamaterial, and both the refraction wave vector and energy flow are positive refraction, as illustrated in Fig. 4i, which is coincident with the energy flow shown in Fig. 4g, h. From Fig. 4j, we can see that when \({E_{0}^{2}}/{E_{c}^{2}}= 4\), there exists refraction wave vector, as well as energy-flow transmission into the metamaterial; what’s more, the refraction wave vector is positive refraction, while the energy flow is negative refraction.

Similarly, there exist some analogous properties in the case of self-defocusing nonlinearity (ζ = − 1), shown in Fig. 5. When the intensity of the applied electric field is smaller than a certain value (\( 0\leq {E_{0}^{2}}/{E_{c}^{2}}<2\)), that the energy-flow angle is negative and belongs to negative refraction; when it is increased to \( 2<{E_{0}^{2}}/{E_{c}^{2}}<4-{\sin }^{2}({\theta }_{i})\), the energy flow is switched off and there is no refraction wave, as shown in the yellow area. Then increasing further the intensity of the applied electric field, the energy flow is switched on again, and its refraction angle becomes positive, belonging to positive refraction. Therefore, it is also possible that the dynamic transformation of energy flow from negative refraction to cut-off and then to positive refraction by regulating the applied electric field in the metamaterial of self-defocusing nonlinearity. These properties can be demonstrated through the isofrequency dispersion curves, as illustrated in Fig. 5c–e. From Fig. 5c, we can see that when \( {E_{0}^{2}}/{E_{c}^{2}}~=~0\), there exists always refraction wave vector; consequently, there is energy-flow transmission into the metamaterial, and the energy flow are negative refraction, as shown by the red arrow. But when \({E_{0}^{2}}/{E_{c}^{2}}~=~3\), as shown in Fig. 5d, there exists not any refraction wave; consequently, there is no energy-flow transmission into the metamaterial. When \({E_{0}^{2}}/{E_{c}^{2}}~=~4\), there exist refraction wave vector and energy-flow transmission into the metamaterial again, and the energy flow are positive refraction. These properties make it possible for nonlinear metamaterials to be widely used in all-optical switching, all-optical routing, and other nanophotonic devices. For the above analysis about Figs. 4 and 5, we can know that there exists all-angle total reflection region (the yellow area), meaning that there is no energy transmission into the metamaterial for any incident angles, which is dependent on the orientation angle φ and the applied electric field E0. To illustrate the dependencies between them, the all-angle total reflection region (the area enclosed by the red lines) is shown in Fig. 6a, b for the self-focusing and self-defocusing cases, respectively. For instance, when φ = − 45∘, if all-angle total reflection phenomena occur, the required external electric field is \( 0.6<{E_{0}^{2}}/{E_{c}^{2}}<1.33\) (yellow area) for self-focusing nonlinearity, and \(2<{E_{0}^{2}}/{E_{c}^{2}}~<~3\) (yellow area) for self-defocusing nonlinearity, which are consistent with those in Figs. 4 and 5, respectively. The property of all-angle total reflection has wide potential applications in optoelectronic fields such as wave cut-off devices.

Fig. 4
figure 4

The transmissivity t, the energy-flow angle 𝜃s, and the isofrequency dispersion curves versus the normalized intensity of external electric field \({E_{0}^{2}}/{E_{c}^{2}}{}\) in the case of self-focusing nonlinearity, with φ = − 45∘, ζ = 1, 𝜖D0 = 2.4, kc = ω/c. a–d and e–h correspond to the transmissivity and energy flow angle for four different incident angles 𝜃i = 0∘,30∘,45∘,60∘, respectively. i and f are the isofrequency dispersion curves for self-focusing case in the condition of \({E_{0}^{2}}/{E_{c}^{2}}= 0\) and \({E_{0}^{2}}/{E_{c}^{2}}= 4\), respectively, where dark cyan and blue lines denote the dispersion curves in air and the nonlinear metamaterial, respectively. Ki, Kr, Kt and St represent incident wave vector, reflection wave vector, refraction wave vector and energy-flow vector, respectively. Red dashed lines denote the continuity of tangential wave vector

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Fig. 5
figure 5

The transmissivity t, the energy-flow angle 𝜃s, and the isofrequency dispersion curves versus the normalized intensity of external electric field \({E_{0}^{2}}/{E_{c}^{2}}{}\) in the case of self-defocusing nonlinearity, with φ = − 45∘, ζ = − 1, 𝜖D0 = 6, and kc = ω/c. a and b correspond the transmissivity and energy-flow angle for four different incident angles 𝜃i = 0∘,30∘,45∘,60∘ respectively. c, d, and e are the dispersion curves for self-defocusing case in the condition of \({E_{0}^{2}}/{E_{c}^{2}}~=~0\), \( {E_{0}^{2}}/{E_{c}^{2}}~=~3\) and \({E_{0}^{2}}/{E_{c}^{2}}~=~4\), respectively, where dark cyan and blue lines, Ki, Kr, Kt, and St represent the same meaning as Fig. 4

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Fig. 6
figure 6

The all-angle total reflection region enclosed by the red lines for different intensity of external electric field \({E_{0}^{2}}/{E_{c}^{2}}\) and the tilted angle φ for two cases. aζ = 1 (for self-focusing nonlinearity), 𝜖D0 = 2.4. bζ = − 1 (for self-defocusing nonlinearity), 𝜖D0 = 6. The yellow area corresponds to the total reflection region when φ = − 45∘

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In addition, from Figs. 4 and 5, we can see that the transmissivity can equal to 1 in the limit of lossless. When the total transmission phenomena occur, at this point, the incident angle is called the Brewster angle. In Eq. 19, if t = 1, the Brewster angle can be obtained as follows:

$$ {\varepsilon}_{b}={\sin}^{-1}\frac{2\sqrt{\mu_{r}-{{\varepsilon}}_{r}\mu _{y}^{2}\alpha} }{\sqrt{({\eta}^{2}-4\alpha \beta )\mu_{r}{{\varepsilon}} _{r}^{2}{\mu_{y}^{2}}+ 4\mu_{r}}} $$
(23)

Figure 7 shows the changes of Brewster angle with the intensity of the external electric field for self-defocusing and self-focusing cases with φ = 0∘,− 30∘,− 45∘, and − 60∘, respectively. From Fig. 7 and Eq. 23, it is not hard to see that the mutual dependency relation between applied electric field intensity and incident angle when total transmission phenomena occurs for different orientation angles φ. This is helpful for us to analyze and design all-optical switches and routers. It is worth noting that the materials used in the metamaterial are not specified, but we can find some materials of the strong focusing nonlinearity such as n − InSb, and those of the strong defocusing properties such as BaxSr1−xTiO3 [7].

Fig. 7
figure 7

Brewster angle versus the intensity of external electric field \( {E_{0}^{2}}/{E_{c}^{2}}\) for φ = 0∘,− 30∘,− 45∘,− 60∘ in two cases. aζ = − 1 (for self-defocusing nonlinearity), 𝜖D0 = 6. bζ = 1 (for self-focusing nonlinearity), 𝜖D0 = 2.4

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All-Optical Switching and Routing Optimal Designs

From the above features, we can see that this kind of nonlinear metamaterials would have widely potential application in the field of nanophotonic functional materials and devices. In order to optimize their characteristics, facilitating device design and application, Table 1 shows light states, the transmissivity t, the energy-flow refraction angle 𝜃s, the needed intensity of external electric field \( {E_{0}^{2}}/{E_{c}^{2}}\), and φ in the case of normal incidence to the interface between air and the metamaterial for self-defocusing and self-focusing nonlinearity, respectively. From Table 1, we can see that, on the one hand, under the conditions of normal incidence and φ = − 90∘, when \({E_{0}^{2}}/{E_{c}^{2}}~=~1\) for self-defocusing nonlinearity or \({E_{0}^{2}}/{E_{c}^{2}}~=~2.6\) for self-focusing nonlinearity, light propagates perpendicular to the interface and without deflection (𝜃s = 0, called normal refraction states), and it is full transmission (t = 1). However, when \({E_{0}^{2}}/{E_{c}^{2}}\in \lbrack 2,6]\) for self-defocusing nonlinearity or \({E_{0}^{2}}/{E_{c}^{2}}\in \lbrack 0,1.60]\) for self-focusing nonlinearity, light does not transmit, and is reflected totally (t = 0), called cut-off states. These characteristics can be applied to all-optical switching design. On the other hand, under the conditions of normal incidence, by designing suitable orientation angles φ, the interconversion among positive refraction, negative refraction, and cut-off states is feasible by adjusting the intensity of the applied electric field. The simulation with Gaussian wave [38, 39] has been made to validate the correctness of the above theoretical calculations as shown in Fig. 8. For example, in Fig. 8a, b, when φ = − 45∘ and light propagates perpendicular to the interface for self-defocusing nonlinear metamaterials , we can adjust the refraction directions of light from negative (𝜃s = − 38.0∘) to positive (𝜃s = 52.0∘) states by changing the normalized intensity of the applied electric field from 1.44 to 5.56. These characteristics can be applied to all-optical routing design. What’s more, in Fig. 8c, d, when φ = − 90∘ and light propagates perpendicular to the interface for self-defocusing nonlinear metamaterials, we can change the light states from cut-off to normal refraction states (𝜃s = 0∘) by adjusting the normalized intensity of the applied electric field from 3 to 1. It can be applied to all-optical switching design. In addition, there are similar characteristics for self-focusing nonlinear metamaterials.

Table 1 Light states, the transmissivity t, the energy-flow refraction angle \(\protect {\theta }_{s}\), the needed intensity of external electric field \(E_{0}^{2}/E_{c}^{2}\) and φ in the case of normal incidence to the interface between air and the metamaterial for self-defocusing and self-focusing nonlinearity, respectively
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Fig. 8
figure 8

The numerical simulations of a Gaussian wave propogation at an Air-Nonlinear metamaterials (Self-defocusing) interface. Arrows indicate the directions of motion. Red lines indicate the interface between air and self-defocusing nonlinear metamaterials. a Negative refraction state for \( \protect \varphi =-~45^{\circ }, {E_{0}^{2}}/{E_{c}^{2}}= 1.44\). b Positive refraction state for \(\protect \varphi =-~45^{\circ }, {E_{0}^{2}}/{E_{c}^{2}}= 5.56\). c Cut-off state for \(\protect \varphi =-~90^{\circ }, {E_{0}^{2}}/{E_{c}^{2}}= 3\). d Normal refraction state for \(\protect \varphi =-~90, {E_{0}^{2}}/{E_{c}^{2}}~=~1\)

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Conclusions

We have investigated exotic propagation characteristics of TM-polarized light in the nonlinear metamaterials composed of conducting nanowire array embedded into a nonmagnetic host medium with a Kerr-type nonlinearity. We have demonstrated that controlling the propagation direction and propagation states (ON or OFF) of light is possible by adjusting the intensity of applied electric field. In addition, all-angle total reflection phenomena exist in such nonlinear metamaterials by choosing a suitable tilted angle of nanowires and the intensity of external electric field. These properties make this kind of nonlinear metamaterials have widely potential applications in all-optical switches, all-optical routers, wave cut-off devices, and other nanophotonic devices.