## Abstract

We demonstrate numerically that one-dimensional metamaterials with a gain medium can exhibit a near-unity-refractive-index for TE-modes, which resembles the effect of electro-magnetically induced transparency. Our results are further supported by rigorous coupled wave analysis simulations. Additionally, we analyze the behavior of the modal gain and derive an approximate analytical solution of the dispersion equation that is superior to earlier solutions. Finally, a generalization to the case of ultralow refractive index has been considered.

© 2017 Optical Society of America

## 1. Introduction

Electromagnetically induced transparency (EIT) is a well-known quantum interference effect first discovered in atomic physics. This effect occurs when an atomic medium, which was originally opaque for a weak “probe” field, becomes transparent within a narrow frequency band in the presence of a much stronger “coupling” electromagnetic field. EIT is closely related to such phenomena as slow light and stopped light, and shows promise in applications, such as laser cooling, sensorics, and data storage [1]. Different approaches were proposed to implement EIT in solid-state media; these approaches significantly extend the range of potential applications, which could include the development of various optoelectronic devices [2–8].

Traditionally, EIT is considered as a phenomenon relied on maintaining quantum mechanical coherence between different states of the medium which can suppress different absorption paths [3]. However, optical transparency can also be achieved by other means, namely, using loss compensation by gain which, in turn, results from external optical pumping. This effect can be treated as an analogue of EIT [9–11].

In this paper, using numerical simulation we demonstrate that the analoque of EIT in one-dimensional metal/dielectric or metal/semiconductor metamaterials (MMs) with gain can result in near-unity-refractive-index (NURI) metamaterials. Specifically, we show that one of the diagonal components of the tensor of the nonlocal effective refractive index, or effective permittivity, can be purely real; in particular, it can be unity. Furthermore, various approximate analytical expressions for this component of the tensor have been analyzed. One of them is shown to be superior to earlier considered approximations.

The present study could be of importance from both an applied and fundamental point of view. NURI MMs possess antireflection properties [12]. The dielectric response, which is sensitive to changes in the external irradiation (pump power), could assist in the design of various nanocircuits and elements controlled by light, such as light switches, dimmers, modulators, processors, etc. Alternately, under some conditions NURI MMs can simultaneously exhibit both positive and negative refraction [13], and a slab made of such MM can refract either positively or negatively, depending on the slab thickness [14]. Although here we consider transparent NURI MMs only, it is worth highlighting that for binary 1d superlattices, a small deviation from the transparency threshold, resulted in complex-conjugated layer permittivities, allows one to realize the so-called PT-symmetry MMs, for which a second-order phase transition can occur (see [15] and references therein).

## 2. Formalism

Let us consider a two-phase MM (superlattice) composed of alternating nonmagnetic layers of the thickness *d*_{1} and *d*_{2} with the local permittivities *∊*_{1} and *∊*_{2}, respectively (see Fig. 1). One of the phases (phase 1) is considered to be isotropic and dissipative, while another one (phase 2) - isotropic and energetically active [16]. The dielectric response of phase 1 is characterized by a complex-valued scalar permittivity with a positive imaginary part, and the dielectric response of phase 2 is characterized by a complex-valued scalar permittivity with negative imaginary part.

Optical properties of such MM can be described in terms of the nonlocal effective permittivity tensor, which, generally speaking, is nondiagonal [17]. The effective permittivity of this MM should satisfy the dispersion equation for its eigenmodes. We consider a specific case, wherein the wave propagates parallel to the layers (in-plane propagation) and hence the Bloch vector *k*_{⊥} = 0 with the electric field which is also parallel to the layers (TE polarization). One (diagonal) component *∊*_{‖} of the nonlocal effective permittivity tensor is enough to describe the optical properties.

As shown by Rytov [18] in the case of in-plane propagation, the dispersion equation for the eigenmodes of this superlattice can be split into two equations. One of them characterizes the behavior of the ’even’ mode, for which the local electric field is distributed symmetrically with respect to the middle of the layers. The other equation describes the behavior of the ’odd’ mode, for which the local electric field is asymmetric with respect to the middle of the layers. Only the even mode is of interest for our study, as it allows one to obtain MMs with NURI [19]. The profile of the electric field (*E _{y}* component) for the even mode is schematicalyl shown in Fig. 1. The dispersion equation for this mode is of the form [18]

*j*= 1, 2,

*k*

_{0}= 2

*π*/

*λ*

_{0}, and

*λ*

_{0}is the wavelength of light in vacuum.

For metal/dielectric and metal/semiconductor superlattices, Eq. (1) is a complex-valued transcendental equation, which has no analytical solutions. However, it is easy to find its solutions numerically. To do this, we take, as in [21], *λ*_{0} = 1.5*μm* (the optical telecommunication range) and *∊*_{1} = −122.19 + *i*3.115 (this corresponds to silver). As to the energetically active material, gain can be achieved in various structures, such as, e.g., dye molecules, rare-earth doped glasses, semiconducting polymers, quantum dots and quantum wells. Generally speaking, different physical mechanisms can be responsible for gain, and different models can be used to describe the dielectric response of these materials. For simplicity, in our study we limit ourselves to the steady-state regime, in which the time-independent permittivity *∊*_{2} can vary, depending on the pump power. At the emission frequency, the change of this permittivity, Δ*∊*_{2}, can be considered as purely imaginary, and Δ*∊*_{2} ∝ *i*{1 − [1 + (|*E*|/*e _{s}*)

^{2}]

^{−1}}, where

*E*is the applied electric field and

*e*is the saturation constant [20]. At the same time, the real part can take several values, depending on the composition, but it is supposed to be independent of pumping. In particular, for InGaAsP multiple quantum wells, Re

_{s}*∊*

_{2}≈ 12 [21].

## 3. Results and evaluation

#### 3.1. Numerical solutions to the dispersion equation

To begin, we take *∊*_{‖} = 1 and solve Eq. (1) for three different values of Re*∊*_{2}. This allows us to determine the dependencies of Im*∊*_{2} and *f* on *k*_{0}*d*, for which the metamaterial becomes completely transparent and non-reflecting. Figure 2 shows the results obtained.

An increase in Re*∊*_{2} lowers the fitted value of Im*∊*_{2}, at which transparency is reached ($\text{Im}{\u220a}_{2}^{(0)}$), while the rise of the period *d* allows one to increase it. However, at large *k*_{0}*d*, this value of Im*∊*_{2} becomes only slightly dependent on Re*∊*_{2}. At the same time, the fitted value of *f* lowers with *d*, as well as with Re*∊*_{2}. At Re*∊*_{2} = 12, the value of *∊*_{‖} = 1 can be achieved at Im*∊*_{2} = −0.2358, *f* = 0.905 for *d* = 100 nm and Im*∊*_{2} = −0.114, *f* = 0.827 for *d* = 200 nm. This yields for the gain coefficient $g=-{k}_{0}\text{Im}{\u220a}_{2}/\text{Re}\sqrt{{\u220a}_{2}}$ an estimation of about 2850 cm^{−1} in the former case and 1400 cm^{−1} in the latter case, that is feasible with today’s technology [22].

#### 3.2. The case of a finite slab

The above results were obtained for an infinite medium. To study a finite slab we used rigorous coupled waves analysis and calculated the thickness dependence of the transmittance *T*, reflectance *R*, and absorbance *A* for normal light incidence on the slab at the above fitted values of Im*∊*_{2} and *f* for *d* = 100 nm. The results are shown in Fig. 3.

As can be seen, in the considered range of *h/λ*_{0} values (0; 0.67), the reflectance oscillates close to zero, the transmittance oscillates close to unity, and the absorbance varies within 0.5%. Thus, all three quantities are nearly thickness-independent.

#### 3.3. Modal gain

It is of interest to consider the modal gain for the structure under study. The modal gain coefficient can be introduced as *G*_{2} = *g*Γ_{2}, where Γ_{2} is the modal confinement factor [22]. The latter quantity gives the fraction of the total power in the gain region relative to the total power in the mode itsels (see Appendix). The modal confinement factor (not shown here) monotonically rises with *d*, taking values of the order of 0.91 – 0.97. This means that the power concentrates mainly within gain (semiconductor) layers. At the same time, the modal gain monotonically drops with *d* (see Fig. 4).

It should be noted that the modal gain coefficient can depend differently on Re*∊*_{2}. At small values of the lattice period, the modal gain rises with Re*∊*_{2}, while at large *d* it drops with Re*∊*_{2}.

#### 3.4. Approximate solutions taking into account nonlocal corrections

Although Eq. (1), as well as the more general Rytov dispersion equation [18] are easily to solve numerically, there are various approximate solutions which are gaining recognition [23]. Taking in Eq. (1) tan *z _{j}* ≈

*z*,

_{j}*j*= 1, 2, one has the well-known local approximation (local effective medium theory, or EMT) for the longitudinal component of the effective permittivity tensor

*f*

_{1}=

*d*

_{1}/

*d*and

*f*

_{2}=

*d*

_{2}/

*d*= 1 −

*f*

_{1}. At the same time, the nonlocal corrections to the local EMT for both TE and TM modes are known. In particular, expanding the original dispersion equation of Rytov into Taylor series up to the second order in |

*k*

_{1}

*d*

_{1}| ≪ 1 and |

*k*

_{2}

*d*

_{2}| ≪ 1, Elser

*et al.*[24] obtained an equation where $\delta ={\left({k}_{0}d\right)}^{2}{f}_{1}^{2}{f}_{2}^{2}{\left({\u220a}_{1}-{\u220a}_{2}\right)}^{2}/12{\u220a}_{\Vert}^{(0)}$.

A somewhat different result was derived by Chern [25]. For our case, it can be written as

*δ*| ≪ 1, they, as will be shown below, have different areas of applicability. The applicability of Eqs. (3) and (4) depends not only on the condition |

*k*

_{0}

*d*| ≪ 1, but also on the dielectric contrast of

*∊*

_{1}−

*∊*

_{2}, which enters into the coefficients

*δ*and

*B*

_{2}. Even if

*k*

_{0}

*d*≪ 1, Eq. (3) and Eq. (4) can remain crude. At the same time, it would be useful to find a reasonable approximation for

*∊*

_{‖}at moderate values of the parameter

*k*

_{0}

*d*. In this connection, of interest is to take into account higher-order corrections to the local approximation (2). To do this, we take that tan ${z}_{j}\approx {z}_{j}+{z}_{j}^{3}/3$,

*j*= 1, 2, that allows one to expand Eq. (1) into a series of

To obtain one more approximate solution of the dispersion equation for the even TE mode, we note that the tangent squared can be represented as a Padé approximant: $\text{tan}{z}_{j}^{2}\approx {z}_{j}^{2}/\left(1-2{z}_{j}^{2}/3\right)$. After substituting this into Eq. (1), one has the quadratic equation of the form

with ${c}_{2}=({f}_{1}{\u220a}_{1}-{f}_{2}{\u220a}_{2}){\u220a}_{\Vert}^{(0)}-\eta {\u220a}_{1}{\u220a}_{2}$, ${c}_{1}=2\left({f}_{2}^{2}{\u220a}_{2}-{f}_{1}^{2}{\u220a}_{1}\right)+\eta ({\u220a}_{1}+{\u220a}_{2})$,*c*

_{0}=

*f*

_{1}−

*f*

_{2}−

*η*, and $\eta ={\left({k}_{0}d\right)}^{2}{f}_{1}^{2}{f}_{2}^{2}({\u220a}_{1}-{\u220a}_{2})/6$. Only one solution of this equation has a physical meaning. It yields the sought approximation It should be noted that in a similar way, an approximate solution for the odd TE mode can be also derived. This issue, however, does not fall within the scope of this paper.

Let us now compare the accuracy of the above approximations. To do this, in Fig. 5 we show the discrepancies |*∊*_{‖} − 1|, calculated with the use of Eqs. (2), (3), (4), (5), and (7).

As would be expected, the local approximation (2) is rather crude, and it can be used only at *k*_{0}*d* ≪ 1, that is in the quasi-static limit. The approximation of Elser *et al.* (3) is even worse in the considered range of the values of the parameter of *k*_{0}*d.* This happens because ${\u220a}_{\Vert}^{(0)}$ here can take small values, close to zero, and then |*δ*| becomes large. It means that great care must be taken in using this equation. The approximation of Chern (4), which can be written in the form of ${\u220a}_{\Vert}\approx {\u220a}_{\Vert}^{(0)}(1+\delta )$, is moderately good. The inclusion of higher-order corrections in Eq. (5) allows further accuracy as compared to Eq. (4). Finally, our approximation, based on a Padé approximant, Eq. (7), is the most accurate for all values of *k*_{0}*d* under consideration.

## 4. Closing remarks

As shown earlier [26], within the framework of the local approximation, broadband NURI MMs can be designed using metallodielectric composites with a graded geometry. It would be of interest to consider a similar problem with allowance for nonlocal effects. However, solving this problem requires a more sophisticated technique. An example of broadband plasmon induced transparency, based on the use of U-shaped resonators, has recently been demonstrated in the microwave range [4].

It is important to note that our consideration is not limited to the case of *∊*_{‖} = 1. Due to the scaling of the permittivity, in principle, the case of any real *∊*_{‖} = *r* can be considered after substituting *r∊*_{1} → *∊*_{1} and *r∊*_{2} → *∊*_{2}. If *∊*_{1} and Re*∊*_{2} are fixed, as before, the dependencies of *∊*_{‖}(*f*) and Im*∊*_{2}(*f*) can be found as the solutions to Eq. (1). For example, in Fig. 6 we show such dependencies for the case of the ultralow refractive index (0 ≤ *∊*_{‖} ≤ 1) for two values of the period *d*. In particular, the epsilon-near-zero regime (*∊*_{‖} = 0) can be realized at *f* = 0.895 and Im*∊*_{2} = −0.253 for *d* = 100 nm, and *f* = 0.802 and Im*∊*_{2} = −0.117 for *d* = 200 nm, that is feasible with current technology.

We have shown that the near-unity-refractive-index can be realized for the symmetric TE-mode in one-dimensional superlattices with the use of the solid-state analogue of electromagnetically-induced transparency. Our results, obtained by solving the dispersion equation, are in solid agreement with rigorous coupled wave analysis simulations. Additionally, the behavior of the modal gain was analyzed and an approximate analytical solution to the dispersion equation for the symmetric TE-mode was derived, which is superior to earlier solutions. Finally, we predict that the ultralow-refractive-index MMs may also be designed with realistic materials, using the same 1d geometry.

## Appendix: modal confinement factor

The modal confinement factor for our structure can be defined as

*E*=

*E*. The electric field

_{y}*E*in our case can be found in a closed form (see, e.g., [18]):

_{y}## Funding

Y.C.C. and V.U.N acknowledge support from the Ministry of Science and Technology, Taiwan, Grants 103-2221-E-001-011-MY3 and 104-2923-M-001-001-MY3, 105-2112-M-001-010, 106-2923-M-002-MY3, respectively. A.P. acknowledges partial support from NATO SPS (grant NUKR.SFPP 984617) and AF SFFP Program.

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