Narrow-pass-band amplification of THz radiation by dielectric-metal nanostructures with optically active graphene-based inclusions

Denis Iakushev; Servando Lopez-Aguayo

doi:10.1364/OME.7.003093

1. Introduction

Using structures composed of alternating dielectric and metal layers, it is possible to observe optical phenomena that are markedly different from optical phenomena emerging due to the light interaction with ordinary materials. For this reason, dielectric-metal structures can be used in such applications as high-resolution imaging, hyperlens, and nanolithography, see Ref. [1]. Implementation of these applications, however, remains a challenge, particularly in the THz and infrared frequency range, because of the strong electromagnetic absorption in metals.

Of course, the decrease in the electromagnetic absorption can be achieved by clearing of metal from extraneous defects, such as impurities and dislocations, which are the origin of the collisional electromagnetic absorption. However, improving the quality of metal has a natural limit imposed by the fundamental collisionless Landau damping. Because of this, very pure metals also strongly absorb the electromagnetic energy [2], especially within THz and infrared frequency ranges. A more efficient way allowing one to overcome the limitations imposed by the electromagnetic absorption in metals is to incorporate gain media into layered materials with metal constituents. This possibility was studied in a number of works, for instance in Refs. [3–6].

In this study we consider periodic dielectric-metal superlattices in the context of their remarkable ability to transmit incident electromagnetic radiation within narrow frequency ranges. It is possible due to the strong contrast between the impedances of dielectric and metal, that leads to the formation of the narrow pass bands in the photonic spectrum of dielectric-metal superlattices. The electromagnetic absorption prevents the effective use of such systems, so that even a structure containing only a few fairly thin metal layers transmits a small part of the incident electromagnetic flux, whereas the number of metal layers constituting the system has to be sufficient in order to provide the fairly narrow pass bands. Even though for some applications such a low transmission can be enough, nevertheless, it is of interest to understand if one can obtain a periodic dielectric-metal structure that can transmit a considerable part of incoming electromagnetic radiation.

We analyze a possibility of the use of optically active two-dimensional inclusions on the basis of graphene, in order to compensate the electromagnetic losses in metal layers of superlattices. Such insertions possess sufficient freedom to be situated anywhere inside a dielectric layer of a dielectric-metal optical structure. Due to this, photonic properties of the superlattice become dependent on the position of such inclusions, and this additional parameter can be used in order to control the location of the photonic pass bands. Note that the possibility of the use of optically active graphene was discussed in Refs. [7,8].

We show that optical activation of the two-dimensional inclusions can drastically improve photonic transport via periodic dielectric-metal superlattices. Particularly, one can achieve the perfect transparency: the value of the transmission coefficient of superlattice can reach unity. Moreover, optically activated dielectric-metal superlattices can considerably amplify incoming electromagnetic radiation, so that the value of the transmission coefficient can greatly exceed unity. The degree of the optical activation necessary to achieve such effects depends strongly on the position of the optically activated insertion inside the unit cell of the layered structure. For these reasons, the optically activated dielectric-metal superlattices with the two-dimensional active inclusions can be used as narrow-pass-band filters as well as narrow-pass-band amplifiers of the electromagnetic radiation, with possibility to control both photonic spectrum and photonic transport by changing the position of the two-dimensional inclusions in the superlattice.

2. Problem formulation

We study the transmission of the electromagnetic wave through a one-dimensional dielectric-metal nanostructure, i.e. superlattice, consisting of the identical unit cells composed of a- and b-layers. The a-layer, being filled with a dielectric, contains a graphene-based two-dimensional inclusion, and the b-layer is metallic. The a- and b-layers have the constant thicknesses, d_a and d_b, respectively. Thus, the size d of any unit (a, b)-cell is also constant,

d = d_{a} + d_{b} .

The two-dimensional insertion divides the dielectric a-slab into the two dielectric sublayers. The left and right sublayers have thicknesses l₁ and l₂, respectively. It is evident that l₁ + l₂ = d_a.

An electromagnetic wave exciting the superlattice propagates perpendicular to the layers, in the direction of the x-axis, with the electric E(x, t) and magnetic H(x, t) components

E (x, t) = {0, E (x, 0)} exp (- i ω t), H (x, t) = {0, 0, H (x)} exp (- i ω t),

as indicated in Fig. 1. Here ω is the wave frequency.

Fig. 1 Schematic of the dielectric-metal superlattice with inclusions of graphene.

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The optical characteristics of the dielectric a-layer are permittivity ε_a, permeability μ_a, corresponding refractive index n_a, impedance Z_a, wave number k_a, and wave phase shift φ_a,

n_{a} = \sqrt{ε_{a} μ_{a}}, Z_{a} = μ_{a} / n_{a}, k_{a} = n_{a} k, φ_{a} = k_{a} d_{a}, k = ω / c .

The phase wave shift

φ_{a} = φ_{a}^{(1)} + φ_{a}^{(2)}

is the sum of the phase shifts

φ_{a}^{(1)} = k_{a} l_{1}

and

φ_{a}^{(2)} = k_{a} l_{2}

of the left and right dielectric sublayers, respectively. The permittivity ε_a and permeability μ_a are assumed to be of real positive values. The two-dimensional inclusion in the dielectric a-layer is specified by effective complex surface conductivity σ.

The incident electromagnetic wave induces the electric current in the conducting two-dimensional insertions, so that inside each dielectric a-layer the longitudinal current density j_a(x) emerges,

j_{a} (x) = σ E_{a} (x) δ (x - x_{n}^{(i)}),

where E_a(x) is the electric field inside the dielectric a-layer,

x_{n}^{(i)}

is the coordinate of the insertion inside the dielectric layer of the nth unit cell, and

δ (x - x_{n}^{(i)})

is the Dirac delta function. It should be noted that field-current relation (4) is commonly used in order to describe electrodynamics of systems with the inclusions of graphene. Particularly, such a form of the current density in graphene can be found in Refs. [9,10].

According to the Maxwell equations, the equation for electric field E_a(x) takes the following form,

E_{a}^{″} (x) + k_{a}^{2} E_{a} (x) + \frac{4 π i μ_{a} ω}{c^{2}} j_{a} (x) = 0,

and the magnetic field H_a(x) inside the a-layer is expressed via the electric field E_a(x),

H_{a} (x) = \frac{1}{i k μ_{a}} E_{a}^{'} (x) .

Here the prime means the derivative with respect to x.

As known [11], the imaginary part of permittivity is positive for passive and negative for active materials. The permittivity ε_a of the dielectric in the a-layer is assumed to be positive, so that the dielectric sublayers neither amplify nor absorb the electromagnetic radiation, and all effects associated with the electromagnetic amplification are described by the effective complex surface conductivity σ of the optically activated inclusions of graphene.

In order to understand the meaning of the real part of σ, let us interpret Eq. (5a) with current (4) as a wave equation for the electric field in an inhomogeneous medium with x-dependent permittivity ε,

ε = ε_{a} + \frac{4 π i σ}{ω} δ (x - x_{n}^{(i)}) .

Since

Im ε = (4 π Re σ / ω) δ (x - x_{n}^{(i)})

, one sees that the positive real part of effective surface conductivity σ describes the electromagnetic dissipation, whereas the negative one is responsible for the gain. It should be noted that the graphene absorbs the electromagnetic energy, and the real part of the surface conductivity of graphene is positive. However, if such a material is optically activated, one can introduce some effective surface conductivity that counts for both the attenuation and gain. When the latter predominates over the former, the graphene-based inclusion becomes optically active and the real part of its effective surface conductivity is negative.

The distribution of the electromagnetic field inside the dielectric sublayers has fairly simple form, being the superposition of two waves propagating into the opposite directions,

E_{a_{n}}^{(1)} (x) = A_{n +}^{(1)} exp [i k_{a} (x - x_{a_{n}})] + A_{n -}^{(1)} exp [- i k_{a} (x - x_{a_{n}})],

\begin{array}{l} H_{a_{n}}^{(1)} (x) = Z_{a}^{- 1} {A_{n +}^{(1)} exp [i k_{a} (x - x_{a_{n}})] - A_{n -}^{(1)} exp [- i k_{a} (x - x_{a_{n}})]}, \\ inside the left sublayer, where x_{a_{n}} ⩽ x < x_{n}^{(i)}, \end{array}

and

E_{a_{n}}^{(2)} (x) = A_{n +}^{(2)} exp [i k_{a} (x - x_{n}^{(i)})] + A_{n -}^{(2)} exp [- i k_{a} (x - x_{n}^{(i)})],

\begin{array}{l} H_{a_{n}}^{(2)} (x) = Z_{a}^{- 1} {A_{n +}^{(2)} exp [i k_{a} (x - x_{n}^{(i)})] - A_{n -}^{(2)} exp [- i k_{a} (x - x_{n}^{(i)})]}, \\ inside the right sublayer, where x_{n}^{(i)} < x ⩽ x_{b_{n}} . \end{array}

Here

A_{n \pm}^{(1)}

and

A_{n \pm}^{(2)}

are the complex amplitudes of the waves traveling into the positive (+) and negative (−) directions of the x axis. The coordinates x_{a_n} and x_{b_n} refer to the left-hand edges of successive a_n- and b_n-layers, respectively.

The problem of determining the distribution of the electromagnetic field within the metal layer is rather complicated, but in spite of this, such a problem admits an analytical solution [14]. It is important that for our subsequent analysis, we do not need to know how the electromagnetic field is distributed inside the metal layer, since the external response of the metal layer to an electromagnetic excitation is completely defined by surface impedances ζ₀ and ζ_d, correspondingly, of the left-hand and right-hand boundaries of the metal slab.

3. Dispersion equation and transmission coefficient

One sees from Eqs. (4) and (5) that the electric field within the dielectric a-layer is continuous,

E_{a_{n}}^{(1)} (x_{n}^{(i)}) = E_{a_{n}}^{(2)} (x_{n}^{(i)}),

whereas the magnetic field H_a(x) undergoes the jump induced by the current (4) in the two-dimensional inclusion,

H_{a_{n}}^{(2)} (x_{n}^{(i)}) - H_{a_{n}}^{(1)} (x_{n}^{(i)}) = - \frac{4 π σ}{c} E_{a_{n}}^{(1)} (x_{n}^{(i)}) .

Note that boundary condition (10) is usually exploited in studies devoted to photonics of structures with inclusions of graphene, see for instance Refs. [9,12].

Using boundary conditions (9) and (10), along with Eqs. (7) and (8), one finds that the amplitudes of the electric field $A_{n \pm}^{(1)}$ and $A_{n \pm}^{(2)}$ , within the neighboring left and right dielectric sublayers, respectively, are related by

(\begin{matrix} A_{n +}^{(2)} \\ A_{n -}^{(2)} \end{matrix}) = {\hat{Q}}^{(i)} (\begin{matrix} A_{n +}^{(1)} \\ A_{n -}^{(1)} \end{matrix}),

and the matrix Q̂⁽ⁱ⁾ has such elements,

{(Q^{(i)})}_{11} = (1 - \frac{D Z_{a}}{2}) exp (i φ_{a}^{(1)}),

{(Q^{(i)})}_{12} = - \frac{D Z_{a}}{2} exp (- i φ_{a}^{(1)}),

{(Q^{(i)})}_{21} = \frac{D Z_{a}}{2} exp (i φ_{a}^{(1)}),

{(Q^{(i)})}_{22} = (1 + \frac{D Z_{a}}{2}) exp (- i φ_{a}^{(1)}),

where we have introduced quantity D,

D = 4 π σ / c .

Applying the continuous boundary conditions to the electromagnetic field at the right surface x = x_{b_n} of the dielectric a-layer, and taking into account that electric field E_{b_n} (x_{b_n}) at the left surface of the neighboring metal b-layer is expressed via the magnetic fields H_{b_n} (x_{b_n}) and H_{b_n} (x_{a_n+1}) on, respectively, the left and right metal layer surfaces as

E_{b_{n}} (x_{b_{n}}) = H_{b_{n}} (x_{b_{n}}) ζ_{0} - H_{b_{n}} (x_{a_{n + 1}}) ζ_{d},

one obtains the matrix equation that expresses the magnetic fields on the left and right metal layer boundaries via the amplitudes of the electric field within the left adjacent dielectric sublayer,

(\begin{matrix} H_{b_{n}} (x_{a_{n + 1}}) \\ H_{b_{n}} (x_{b_{n}}) \end{matrix}) = {\hat{Q}}^{(a)} (\begin{matrix} A_{n +}^{(2)} \\ A_{n -}^{(2)} \end{matrix}),

where the elements of the matrix Q̂^(a) are given by

Q_{11}^{(a)} = \frac{ζ_{0} - Z_{a}}{Z_{a} ζ_{d}} exp (i φ_{a}^{(2)}),

Q_{12}^{(a)} = - \frac{ζ_{0} + Z_{a}}{Z_{a} ζ_{d}} exp (- i φ_{a}^{(2)}),

Q_{21}^{(a)} = \frac{exp (i φ_{a}^{(2)})}{Z_{a}},

Q_{22}^{(a)} = - \frac{exp (- i φ_{a}^{(2)})}{Z_{a}} .

Analogously, applying the continuous boundary conditions to the electromagnetic field at the right surface x = x_{a_n+1} of the metal b-layer, with the use of the formula

E_{b_{n}} (x_{a_{n + 1}}) = H_{b_{n}} (x_{b_{n}}) ζ_{d} - H_{b_{n}} (x_{a_{n + 1}}) ζ_{0},

one obtains the matrix equation

(\begin{matrix} A_{(n + 1) +}^{(1)} \\ A_{(n + 1) -}^{(1)} \end{matrix}) = {\hat{Q}}^{(b)} (\begin{matrix} H_{b_{n}} (x_{a_{n + 1}}) \\ H_{b_{n}} (x_{b_{n}}) \end{matrix}),

that gives the amplitudes

A_{(n + 1) \pm}^{(1)}

of the electric field within the nearest right, to the metal slab, dielectric sublayer, in terms of the magnetic fields on the left and right metal layer boundaries. Here the matrix Q̂^(b) has the following elements,

Q_{11}^{(b)} = (Z_{a} - ζ_{0}) / 2,

Q_{12}^{(b)} = ζ_{d} / 2,

Q_{21}^{(b)} = - (Z_{a} + ζ_{0}) / 2,

Q_{22}^{(b)} = ζ_{d} / 2 .

Combining Eqs. (11), (15), and (18), we obtain the matrix equation which describes the wave transmission through the whole unit (a, b)-cell,

(\begin{matrix} A_{(n + 1) +}^{(1)} \\ A_{(n + 1) -}^{(1)} \end{matrix}) = \hat{Q} (\begin{matrix} A_{n +}^{(1)} \\ A_{n -}^{(1)} \end{matrix}),

where the unit-cell transfer matrix Q̂ is expressed via matrices Q̂⁽ⁱ⁾, Q̂^(a), and Q̂^(b),

\hat{Q} = {\hat{Q}}^{(b)} {\hat{Q}}^{(a)} {\hat{Q}}^{(i)} .

The unit-cell transfer matrix defines all the photonic characteristics of the system under consideration. Particularly, the dispersion relation for the Bloch wave number κ of the superlattice is defined by the trace of the unit-cell transfer matrix Q̂,

cos (κ d) = (Q_{11} + Q_{22}) / 2 .

Calculating the trace of the unit-cell transfer matrix (21), one finds the dispersion relation of the periodic dielectric-metal superlattice with the periodically distributed two-dimensional insertions of graphene,

\begin{array}{l} cos (κ d) = \frac{ζ_{0}}{ζ_{d}} cos φ_{a} - i \frac{Z_{a}^{2} + ζ_{0}^{2} - ζ_{d}^{2}}{2 Z_{a} ζ_{d}} sin φ_{a} \\ + \frac{D}{4 ζ_{d}} [(Z_{a}^{2} + ζ_{0}^{2} - ζ_{d}^{2}) cos φ_{a} - 2 i Z_{a} ζ_{0} sin φ_{a} - (Z_{a}^{2} - ζ_{0}^{2} + ζ_{d}^{2}) cos (α φ_{a})] . \end{array}

Here we have introduced a quantity

α = (l_{1} - l_{2}) / d_{a}

that shows how far the insertion is from the middle of the dielectric a-layer. From its definition it is clear that the absolute value of α can vary from zero to unity, 0 ⩽ |α| ⩽ 1. When the layer of graphene is situated in the middle of the dielectric a-layers, l₁ = l₂ = d_a/2, this quantity is zero, α = 0. When the insertion is located at the dielectric-metal interface, l₁ = 0 and l₂ = d_a, or l₁ = d_a and l₂ = 0, the absolute value of α is unity, |α| = 1. The fractional values of α correspond to some intermediate positions of the insertions inside the dielectric a-layers.

In the limit of nonconducting two-dimensional insertions, when the effective surface conductivity goes to zero, σ → 0, the dispersion equation (23) transforms to the dispersion equation of the periodic dielectric-metal superlattice without the inclusions of graphene, see Ref. [13],

cos (κ d) = \frac{ζ_{0}}{ζ_{d}} cos φ_{a} - i \frac{Z_{a}^{2} + ζ_{0}^{2} - ζ_{d}^{2}}{2 Z_{a} ζ_{d}} sin φ_{a} .

The transfer matrix of the whole system, which consists of N unit (a, b)-cells, is the product of the transfer matrices of all the unit-cells constituting the stack-structure, Q̂^N. The transmission coefficient T_N of the stack of N unit (a, b)-cells, in the terms of the total transfer matrix Q̂^N, is

T_{N} = {| {(Q^{N})}_{22} |}^{- 2} .

Below we analyze Eqs. (23) and (26), which give the complete description of photonic properties of the periodic dielectric-metal superlattices with the periodically distributed two-dimensional inclusions.

4. Photonic band structure

In order to analyse the transmission spectrum of the system under consideration, we have to understand where the photonic pass bands are situated. So first we have to carry out a comprehensive analysis of the dispersion equation (23), which includes the new terms depending on the effective surface conductivity σ entering Eq. (23) via quantity D = 4πσ/c. To do this, we do not need to know the value of impedance Z_a of the dielectric a-layer as well as the values of the surface impedances ζ₀ and ζ_d of the metal b-layer. It is sufficient to know that the surface impedances ζ₀ and ζ_d of the metal b-layer, taking the values of the same order, are small in comparison with the impedance Z_a of the dielectric a-layer,

| ζ_{0} |, | ζ_{d} | ≪ Z_{a},

whose value is of order of unity. For this reason, as seen from Eq. (25), the photonic pass bands of the dielectric-metal superlattice without the inclusions occupy narrow frequency ranges in the vicinity of the Fabry-Perot resonances emerging in the dielectric a-layers when φ_a = j_aπ, where j_a = 1, 2, 3, . . . is the resonance index.

For subsequent analytical analysis of Eq. (23), it is appropriate to start from the case of an insertion situated in the middle of the dielectric a-layer. For such a configuration α = 0 and the dispersion relation (23) reduces to

\begin{array}{l} cos (κ d) = \frac{D (ζ_{0}^{2} - ζ_{d}^{2} - Z_{a}^{2})}{4 ζ_{d}} + [\frac{D (ζ_{0}^{2} - ζ_{d}^{2} + Z_{a}^{2})}{4 ζ_{d}} + \frac{ζ_{0}}{ζ_{d}}] cos φ_{a} \\ + \frac{Z_{a}^{2} (D ζ_{0} + 1) + ζ_{0}^{2} - ζ_{d}^{2}}{2 i Z_{a} ζ_{d}} sin φ_{a} . \end{array}

One sees that this dispersion equation contains three new, with respect to Eq. (25), terms depending on the surface conductivity σ. It is seen, that due to the strong contrast condition (27), the absolute value of the new term that does not contain the phase wave shift φ_a, can be much greater than unity when |D| = 4π|σ|/c ≳ 1. The same refers to the D-dependent term in the square brackets, which, under the condition |D| = 4π|σ|/c ≳ 1, is much greater than the D-independent term ζ₀/ζ_d. Note that these two D-dependent terms are approximately equal each other in the absolute value, and they have the different signs.

To be specific, let us consider an optically passive graphene sheet as the two-dimensional insertion inside the dielectric a-layer. The surface conductivity of the graphene σ = σ^(intra) + σ^(inter), is a sum of the intraband,

σ^{(intra)} = \frac{2 i e^{2} k_{B} T}{π ℏ^{2} ω} ln [2 cosh (\frac{μ_{c}}{2 k_{B} T})],

and interband,

σ^{(inter)} = \frac{e^{2}}{4 ℏ} {\frac{1}{2} + \frac{1}{π} arctan \frac{ℏ ω - 2 μ_{c}}{2 k_{B} T} - \frac{i}{2 π} ln \frac{{(ℏ ω + 2 μ_{c})}^{2}}{{(ℏ ω - 2 μ_{c})}^{2} + {(2 k_{B} T)}^{2}}},

transition contributions [9]. Here, e is the elementary charge, T is temperature, and μ_c is the chemical potential. The frequency dependencies of the real and imaginary parts of quantity D for a graphene sheet with temperature T = 300 K and chemical potential μ_c = 1.2577 eV are shown in Fig. 2. As seen, the requirement |D| = 4π|σ|/c ≳ 1 is satisfied within the THz frequency range, but the D-dependent terms become relevant even when the value of |D| is quite small. Thus, due to the smallness of the surface impedances ζ₀ and ζ_d, the D-dependent term in the square brackets of Eq. (28) can be of order of the term ζ₀/ζ_d when the value of |D| is much smaller than unity. Therefore, in the particular case of the optically passive graphene inclusions, the dispersion relation of the dielectric-metal superlattice changes significantly: the new terms are relevant because they can exceed the D-independent terms.

Fig. 2 The real and imaginary parts of quantity D = 4πσ/c as a function of the wave frequency, for a graphene sheet with temperature T = 300 K and chemical potential μ_c = 1.2577 eV.

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Generally speaking, in the presence of the inclusions, the positions of the photonic pass bands change. In order to determine the shift of the bands, let us assume that it is small with respect to the distance between the neighboring a-resonances. Due to this, the surface impedances ζ₀, ζ_d, as well as the surface conductivity σ can be replaced with their values at the corresponding Fabry-Perot resonance φ_a = j_aπ. In addition, one can neglect the real part of the surface conductivity σ with respect to its imaginary part, see Fig. 2. To simplify Eq. (28) even more, we can take into account that the values of the real parts of the surface impedances ζ₀ and ζ_d are small with respect to the values of their imaginary parts. One can see this in Fig. 3 showing the real and imaginary parts of the surface impedances of a thin aluminum layer. Then, as it follows from Eq. (28), in the vicinity of every a-resonance, when the resonance detuning (φ_a − j_aπ) is small, |φ_a − j_aπ| ≪ 1, the photonic bands are described by the approximate expression

φ_{a} - j_{a} π = \frac{1}{2} [1 - cos (j_{a} π)] Z_{a} Im D + \frac{2}{Z_{a}} [Im ζ_{0} - Im ζ_{d} cos (j_{a} π) cos (Re κ d)] .

From this approximate equation it is seen that the insertion-induced shift Δω_{j_a} of the j_ath pass band is given by

Δ ω_{j_{a}} = 2 π [1 - cos (j_{a} π)] \frac{Z_{a}}{n_{a}} \frac{Im σ}{d_{a}} .

Expression (32) shows that when the insertion is in the middle of the dielectric a-layer, the displacement occurs for the odd bands only, j_a = 1, 3, . . ., whereas for the even ones, j_a = 2, 4, . . ., this effect is absent and the photonic bands do not shift. We can also see that the shift is greater for the greater values of the imaginary part of the surface conductivity and smaller values of the thickness of the dielectric a-layer.

Fig. 3 (Color Online) The real and imaginary parts of the surface impedances ζ₀ and ζ_d for an aluminum layer of the thickness d_b = 25 nm.

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Although the expression (32) is approximate, the inference that the insertions do not impact the pass bands in the vicinity of the even a-resonances is general, since in the vicinity of the even a-resonances cos φ_a = cos(j_aπ) = +1 and, due to this, the great D-dependent terms in Eq. (28) cancel each other out, so that dispersion equation (28) practically does not differ from dispersion equation (25) for the superlattice without inclusions.

In the case where the inclusions are situated at the interfaces separating the dielectric and metal layers, |α| = 1, and dispersion equation (23) becomes

cos (κ d) = [\frac{ζ_{0}}{ζ_{d}} + \frac{D (ζ_{0}^{2} - ζ_{d}^{2})}{2 ζ_{d}}] cos φ_{a} - i (\frac{Z_{a}^{2} + ζ_{0}^{2} - ζ_{d}^{2}}{2 Z_{a} ζ_{d}} + \frac{D Z_{a} ζ_{0}}{2 ζ_{d}}) sin φ_{a} .

As can be seen, because of the strong contrast condition (27), the terms containing the quantity D = 4πσ/c cannot compete with the corresponding D-independent terms even when the absolute value of D is of order of unity. Thus, the D-dependent term in the square brackets is small with respect to the ratio ζ₀/ζ_d. The same refers to the terms in the parentheses. This means that the positions of the photonic bands do not differ considerably from those of the bands in the case of the superlattice without the insertions. Therefore, in the presence of the insertions placed at the dielectric-metal interfaces, the photonic bands are situated in the vicinity of the Fabry-Perot resonances arising in the dielectric a-layers, as in the case of the superlattice without the insertions.

In the intermediate case, when the inclusion is located between the middle of the a-layer and its boundaries, quantity |α| takes the fractional values belonging to the interval from zero to unity, 0 <|α| < 1. From the general dispersion relation (23) it follows that in the vicinity of the j_ath Fabry-Perot resonance the photonic spectrum, in the linear with respect to the resonance detuning approximation, has the following form,

φ_{a} - j_{a} π = \frac{4}{Z_{a}} \frac{Im ζ_{0} - cos (j_{a} π) Im ζ_{d} cos (Re κ d)}{2 - cos (j_{a} π) Z_{a} Im D α sin (j_{a} π α)} + \frac{1 - cos (j_{a} π) cos (j_{a} π α)}{2 {(Z_{a} Im D)}^{- 1} - cos (j_{a} π) α sin (j_{a} π α)} .

Under the derivation of this equation we have taken into account that |Dζ₀| ≪ 1 for |D|≲ 1. The second ratio in Eq (34) describes the inclusion-induced frequency shift Δω_{j_a} of the j_ath photonic band,

Δ ω_{j_{a}} = \frac{1 - cos (j_{a} π) cos (j_{a} π α)}{2 {(Z_{a} Im D)}^{- 1} - cos (j_{a} π) α sin (j_{a} π α)} \frac{c}{n_{a} d_{a}} .

One sees from Eq. (35) that the presence of the inclusions does not induce the displacement of the j_ath pass band provided quantity α satisfies the equation cos(j_aπα) = cos(j_aπ). Since −1 ⩽ α ⩽ 1, one obtains j_a + 1 values of parameter α for each j_ath a-resonance: for even a-resonances α_m = 2m/j_a, where m is the integer from −j_a/2 to j_a/2, and for odd a-resonances α_m = (2m + 1)/j_a, where m is the integer from −(j_a + 1)/2 to (j_a − 1)/2.

5. Transmission spectrum

In order to calculate the transmission coefficient (26), we have to know the surface impedances ζ₀ and ζ_d of the metal b-layer. They are given by the general expressions counting for both the collision and collisionless damping, see Ref. [14],

ζ_{0} = - \frac{i k}{d_{b}} \sum_{s = - \infty}^{\infty} \frac{1}{k_{s}^{2} - k^{2} ε (k_{s})}, ζ_{d} = - \frac{i k}{d_{b}} \sum_{s = - \infty}^{\infty} \frac{cos (k_{s} d_{b})}{k_{s}^{2} - k^{2} ε (k_{s})} .

The summation in Eqs. (36) is due to the quantization of the electromagnetic field inside the metal b-layer, and k_s = πs/d_b is the wave number of the sth quantized electromagnetic mode. The quantisation of the electromagnetic field is the manifestation of the strong spatial dispersion, which is inherent in metals, especially within the THz and infrared frequency range. Under the strong spatial dispersion, each sth electromagnetic mode has its own dielectric permittivity ε(k_s), which depends on the thickness d_b of the metal b-layer,

ε (k_{s}) = 1 - \frac{ω_{p}^{2}}{ω (ω + i ν)} 𝒦 (k_{s} l_{ω}),

where ν is the electron relaxation frequency, ω_p is the electron plasma frequency, and 𝒦 (k_sl_ω) is the nonlocality factor,

𝒦 (k_{s} l_{ω}) = \frac{3}{2} {[\frac{1}{k_{s} l_{ω}} + \frac{1}{{(k_{s} l_{ω})}^{3}}] arctan (k_{s} l_{ω}) - \frac{1}{{(k_{s} l_{ω})}^{2}}} .

The quantity l_ω = V_F/(ν − iω) entering the nonlocality factor has the meaning of the complex mean-free-path of the electrons due both to their collisions with the scattering centers and to the phase change of the electromagnetic field; V_F is the electron velocity on the Fermi surface. Although there is a simplified version of Eqs. (36), which is based on the Drude-Lorentz model, it is important to note that one has to use the exact, spatially dispersive expressions (36), since the disregard for the spatial dispersion can lead to qualitatively incorrect results, see Refs. [2,13,15].

The real and imaginary parts of the surface impedances (36) for an aluminum layer of the thickness d_b = 25 nm are shown in Fig. 3. The plasma frequency and the Fermi velocity for aluminum are ω_p = 3.82 × 10¹⁵ Hz and V_F = 2.03 × 10⁸ cm/s; the relaxation frequency ν = 10⁻⁴ × ω_p. Impact of the optically activated inclusions of graphene on the transmission coefficient of a superlattice consisting on such aluminum layers is analysed below.

Figure 4 depicts transmission coefficient T₇ of a vacuum-aluminum superlattice consisting on N = 7 unit cells, within the second lower band, when the insertions are placed in the middle of the dielectric a-layer, α = 0. The thicknesses of the a- and b- layers are d_a = 288.176 × δ and d_b = 2 × δ = 25 nm, where δ = c/ω_p is the minimal skin depth of the electromagnetic field in bulk metal. The red solid curves in panels (a), (b), (c), and (d) correspond to the different values of the real part of the effective surface conductivity σ, which is responsible for attenuation, when Reσ > 0, or gain, when Reσ < 0. The imaginary part of the effective surface conductivity 4πImσ/c ≈ 0.1 is the same for all the panels. Such a value is calculated according to Eqs. (29) and (30) for the surface conductivity of the optically passive graphene. Note that this value virtually does not change within the narrow photonic pass band. The red solid curve in panel (a) shows the transmission coefficient when the inclusions are optically passive graphene sheets. The real part of quantity D for such an inclusions is plotted in Fig. 2. Within the second pass band it is ReD ≈ 0.00017. The blue circles show the transmission coefficient of the superlattice without the inclusions. As seen, the transmission spectrum does not change in the presence of the optically passive graphene sheets situated in the middle of the dielectric layer. Also it is seen that the superlattice transmits only a small part of the incident electromagnetic flux: the maximal value of the transmission coefficient within the band is approximately $T_{7}^{(\max)} = 6 \times 10^{- 3}$ . The transmission peaks emerge due to the Fabry-Perot resonances associated with the total length Nd of the superlattice and the total number of such resonances inside the pass band is N − 1 = 6. In panels (b), (c), and (d) the transmission coefficient of the superlattice with the optically activated insertions is shown. Panel (b), corresponding to ReD = −6.622, shows that the superlattice can exhibit the perfect transparency: the maximal value of the transmission coefficient within the fourth peak is unity, $T_{7}^{(\max)} = 1$ . Panel (c) shows that the increase in the absolute value of the effective surface conductivity to ReD =−7 leads to the considerable amplification: the maximal value of the transmission coefficient within the fourth peak becomes $T_{7}^{(\max)} = 9.6$ . It is seen in panel (d) that when the optical activation is increased to the level ReD = −8, the maximal value of the transmission coefficient considerably decreases: within the third peak it becomes approximately $T_{7}^{(\max)} = 3.1$ .

Fig. 4 (Color Online) Transmission coefficient T₇ within the second pass band for vacuum-aluminum superlattice of N = 7 unit cells with α = 0.

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Figure 5 displays the transmission coefficient of the same vacuum-aluminum superlattice, within the second lower band, however in the case where the insertions are placed at the left boundaries of the metal b-layers, α = 1. The red solid curve in panel (a) depicts the transmission coefficient when the inclusion is an optically passive graphene layer. The blue circles indicate the transmission coefficient of the superlattice in the absence of the inclusions. As in the previous case where α = 0, the optically passive graphene sheets do not change the transmission spectrum of the structure, so that the maximal value of the transmission coefficient is fairly small, $T_{7}^{(\max)} = 6 \times 10^{- 3}$ . Panels (b), (c), and (d) depict the transmission coefficient of the superlattice when the insertions are optically activated. Panel (b) shows that when ReD = −0.3706, the maximal value of the transmission coefficient within the second peak reaches unity, $T_{7}^{(\max)} = 1$ . Panel (c) indicates that when ReD = −0.62, the significant amplification is observed: the maximal value of the transmission coefficient within the fifth peak becomes $T_{7}^{(\max)} = 32$ . In panel (d) one sees that a subsequent increase in the absolute value of the real part of the effective surface conductivity, ReD = −0.63, gives rise to the considerable decrease in the transmission coefficient: within the fifth peak, the maximum value of the transmission coefficient becomes approximately $T_{7}^{(\max)} = 4.6$ .

Fig. 5 (Color Online) Transmission coefficient T₇ within the second pass band for vacuum-aluminum superlattice of N = 7 unit cells with α = 1.

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Figure 6 shows what happens to the transmission in the intermediate case $α = \frac{1}{2}$ . Panel (a) displays the transmission coefficient within the frequency region shown in Fig. 5, for the different values of the effective surface conductivity corresponding to Fig. 5. As seen, the transmission is forbidden when $α = \frac{1}{2}$ . This completely confirms our inherence that photonic pass band changes its position if the parameter α is not equal to some special value determined in Section 4. The photonic band structure is altered, and the second pass band occupies a new frequency interval. Panel (b) shows the transmission coefficient within this interval, which turns out to be shifted towards higher frequencies, as compared with the frequency region shown in Fig. 5.

Fig. 6 (Color Online) Transmission spectra in the intermediate case $α = \frac{1}{2}$ . Panel (a): Transmission coefficient within the frequency range shown in Fig. 5. Panel (b): Transmission coefficient within the second pass band for optical activation 4πReσ/c = −0.63.

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6. Conclusion

We have studied the effect of the two-dimensional optically active inclusions on the propagation of the electromagnetic waves through the dielectric-metal periodic superlattices characterized by narrow Fabry-Perot resonance bands. We have deduced the unit-cell transfer matrix and dispersion relation of the system under consideration. Basing on the analytical analysis of the dispersion relation, we have studied the transmission spectrum of the optically activated vacuum-aluminum superlattice, using numerical calculations.

By analysing the dispersion relation, we have obtained the photonic spectrum of the optically activated superlattices and determined the inclusion-induced shift of the photonic pass bands. We have found that at certain positions of the inclusion in the dielectric layer, the positions of the photonic pass bands remain unchanged. In particular, the positions of the pass bands do not change if the inclusions are situated at the left or right boundaries of the dielectric layer. If the inclusions are located in the middle of the dielectric layer, the positions of the even pass bands do not change. This is completely confirmed by the transmission spectrum, which shows that the position of the pass band located in the vicinity of the second a-resonance does not change in the presence of the insertions situated in the middle or at the boundary of the dielectric a-layer, although the effective surface conductivity of the inclusions varies within wide limits.

The maximal value of the transmission coefficient within the band is a strongly non-monotonic function of the real part of the effective surface conductivity, which is responsible for the amplification. Due to this, generally speaking, an increase in the optical activation of the inclusions does not guarantee an increase in the transmission coefficient. The levels of optical activation necessary to obtain the perfect transparency and the considerable amplification strongly depend on the position of the inclusion. In the case of the inclusions situated at the boundaries of the metal b-layers, both the perfect transparency and the considerable amplification can be reached at much smaller levels of optical activation of the inclusions.

Funding

CONACYT (243284).

References and links

1. L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Progress in Quantum Electronics 40, 1–40 (2015). [CrossRef]

2. A. Paredes-Juárez, D. A Iakushev, B. Flores-Desirena, N. M. Makarov, and F. Pérez-Rodríguez, “Landau damping of electromagnetic transport via dielectric-metal superlattices,” Opt. Lett. 40(15), 3588–3591 (2015). [CrossRef] [PubMed]

3. Shumin Xiao, Vladimir P. Drachev, Alexander V. Kildishev, Xingjie Ni, Uday K. Chettiar, Hsiao-Kuan Yuan, and Vladimir M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010). [CrossRef] [PubMed]

4. X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011). [CrossRef]

5. R. S. Savelev, I. V. Shadrivov, P. A. Belov, N. N. Rosanov, S. V. Fedorov, A. A. Sukhorukov, and Y. S. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87, 115139 (2013). [CrossRef]

6. C. Argyropoulos, N. M. Estakhri, F. Monticone, and A. Alù, “Negative refraction, gain and nonlinear effects in hyperbolic metamaterials,” Opt. Express 21(12), 15037–15047 (2013). [CrossRef] [PubMed]

7. Kian Ping Loh, Qiaoliang Bao, Goki Eda, and Manish Chhowalla, “Graphene oxide as a chemically tunable platform for optical applications,” Nature Chemistry 2, 1015–1024 (2010). [CrossRef] [PubMed]

8. K. S. Novoselov, V. I. Fal’ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, “A roadmap for graphene,” Nature 490, 192–200 (2012). [CrossRef] [PubMed]

9. L. A. Falkovsky, “Optical properties of doped graphene layers,” JETP 106(3), 575–580 (2008). [CrossRef]

10. Mohamed A. K. Othman, Caner Guclu, and Filippo Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express 21(6), 7614–7632 (2013). [CrossRef] [PubMed]

11. P. Markoš and C. M. Soukoulis, Wave Propagation. From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University Press, 2008).

12. Ruey-Lin Chern and Dezhuan Han, “Nonlocal optical properties in periodic lattice of graphene layers,” Opt. Express 22(4), 4817–4829 (2014). [CrossRef] [PubMed]

13. A. Paredes-Juárez, D. A Iakushev, B. Flores-Desirena, N. M. Makarov, and F. Pérez-Rodríguez, “Nonlocal effect on optic spectrum of a periodic dielectric-metal stack,” Opt. Express 22(7), 7581–7586 (2014). [CrossRef] [PubMed]

14. A. Paredes-Juárez, F. Díaz-Monge, N. M. Makarov, and F. Pérez-Rodríguez, “Nonlocal effects in the electrodynamics of metallic slabs,” JETP Letters 90(9), 623–627 (2009). [CrossRef]

15. F. Diáz-Monge, A. Paredes-Juárez, D. A Iakushev, N. M. Makarov, and F. Pérez-Rodríguez, “THz photonic bands of periodic stacks composed of resonant dielectric and nonlocal metal,” Opt. Mater. Express 5(2), 361–372 (2015). [CrossRef]

Narrow-pass-band amplification of THz radiation by dielectric-metal nanostructures with optically active graphene-based inclusions

Abstract

1. Introduction

2. Problem formulation

3. Dispersion equation and transmission coefficient

4. Photonic band structure

5. Transmission spectrum

6. Conclusion

Funding

References and links

Cited By

Figures (6)

Equations (50)

Optical Materials Express