Tunable trapping and releasing light in graded graphene-silica metamaterial waveguide

Yu Liu; Shuisheng Jian

doi:10.1364/OE.22.024312

1. Introduction

Trapping light, in which the group velocity of light vanishes, is the extreme case of slowing light. The miraculous characteristic of slowing or even trapping light can be used in optical buffers, optical memory, optical switch and optical guiding systems, such as electromagnetically induced transparency [1], coupled resonant structures [2], and optical nonlinearity [3]. In recent years, the “Trapped rainbow” phenomenon has been widely studied as a novel effect in which the light of different frequencies stands still at correspondingly different positions in a waveguide [4–7].

Metamaterials are artificial materials engineered to have extraordinary properties that may not be found in nature such as left-handed behavior [8], cloaking prosperity [9] and zero effective permittivity [10]. Metamaterial technology may be extended to an entirely new class of mid-infrared applications including surface-enhanced molecular spectroscopy [11], highly sensitive chemical and biological sensing [12]. Recently, several groups proposed the slow light structures based on metamaterials. By utilizing the left and right-handed materials for the cladding and the core of the slab waveguide, the light will be trapped [9]. Although it is challengeable to find a natural material of negative refractive index, metamaterials with the novel property could be artificially achieved in experiments [13, 14]. The speed of microwave in wavegide [9] and infrared frequencies [15] can also be slowed down. However, none of these studies as mentioned above has discussed a tunable device that could control the state of light between blocked or transmitted in mid-infrared frequencies.

Graphene, the two-dimensional carbon sheet, is a tunable material whose optical property was substantially studied and utilized in the past decade [16–20]. With its extreme thinness, the band structure of graphene leads to a pronounced electric field effect, which is the variation of a material’s carrier concentration with electrostatic gating [21, 22]. The conductivity of graphene can be tuned by changing chemical potential via electrostatic biasing [23]. By combining graphene and metamaterials, gate-controllable switching [24], ultracompact optical modulator [10] and perfect absorber [25] were achieved.

In this paper, we first use the effective medium theory to investigate the dependency of metamaterial permittivity on the chemical potential of graphene. Afterward, we theoretically discuss the characteristics of the modes in our graded-metamaterial waveguide. Through applying an appropriate voltage on the graphene layers, waveguide has the ability to trap or release the normalized optical power flow. Moreover, the stopping positions of light at different wavelengths along the waveguide are discussed. Finally, we propose that the graded metamaterial waveguide can be the candidate for bandwidth absorber.

2. Graphene–silica metamaterial structure

The structure of metamaterial is composed of alternative layers of graphene and silica. An air cladding surrounds the metamaterial. The polysilicon as the electrodes are inserted between the graphene layers [10, 26] (as shown in Fig. 1(a)). The volume fraction of the graphene is gradually decreased in the direction of light transmission. The thickness of graphene flake d₁ is 0.34 nm, the thickness of the silica layer d₂ gradually increases from 16.5 nm to 24.5 nm and the thickness of polysilicon is 2 nm. The f_g, f_s and f_p are defined as the volume fractions of graphene, silica and polysilicon, respectively. The graphene layers are extended outside to be brought into contact with the metal electrodes. The width of the waveguide d is 200 nm and the length of waveguide l is 120 nm.

Fig. 1 (a) Illustration of waveguide structure, including graded metamaterial core and air cladding. The volume fraction of the graphene is gradually increased along the z-axis (negative direction of transmission). (b) The variation of horizontal effective permittivity with different chemical potential and different volume fraction of graphene when the λ = 5800 nm. (c) Permittivities of metamaterial under different wavelengths and different chemical potentials when f_g = 0.034. The black and green lines are the real parts and imaginary parts of permittivities, respectively.

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From the Kubo formula [23, 27, 28], the dynamic conductivity of graphene can be expressed as

\begin{array}{l} σ (ω, μ_{c}, Γ, T) = \frac{j e^{2} (ω - j τ^{- 1})}{π ℏ^{2}} \\ [\frac{1}{{(ω - j τ^{- 1})}^{2}} \int_{0}^{\infty} ε (\frac{\partial f_{d} (ε)}{\partial ε} - \frac{\partial f_{d} (- ε)}{\partial ε}) d ε - \int_{0}^{\infty} \frac{f_{d} (- ε) - f_{d} (ε)}{{(ω - j τ^{- 1})}^{2} - 4 {(ε / ℏ)}^{2}} d ε] \end{array}

Where ω is the angular frequency of light, e is the charge of an electron, τ = 5 × 10⁻¹³ is the scattering rate, $f_{d} (ε) = {(e^{(ε - μ_{c}) / k_{B} T} + 1)}^{- 1}$ is the Fermi-Dirac distribution, μ_c is the chemical potential, ћ = h/2𝜋≈6.58 × 10⁻¹⁶ eV is the reduced Planck’s constant, T = 300 K is the environmental temperature, and k_B≈8.61 × 10⁻¹⁶ eV/K is the Boltzmann’s constant.

The permittivity of silica is ε_s = 2.1 and the permittivity of polysilicon ε_p = 11.9. Applying voltages on the electrodes can change the carrier density of graphene, which leads to a variation of the tangential permittivity of graphene as [27] $ε_{g x} = ε_{g y} = 2.5 + i σ / (ω ε_{0} d_{1})$ in the calculation. Moreover, the normal component of the graphene’s permittivity should be ε_gz = 2.5.

According to the effective medium theory, the effective permittivity parallel to the metamaterial surface can be defined as ε_x and ε_y. In order to facilitate the description, we assume that the horizontal effective permittivity ε_m = ε_x = ε_y. The effective permittivity normal to the metamaterial surface is ε_z [20]. The vertical effective permittivity of the metamaterial is a constant, which would not alter with the change of chemical potential.

ε_{m} = f_{g} ε_{g x} + f_{s} ε_{s} + f_{p} ε_{p}

ε_{z} = {(\frac{f_{g}}{ε_{g z}} + \frac{f_{s}}{ε_{s}} + \frac{f_{p}}{ε_{p}})}^{- 1}

Here the f_g , $f_{s} = 1 - f_{g} - f_{p}$ and $f_{p} = 5.88 f_{g}$ is the volume fraction of graphene, silica and polysilicon respectively. As shown in Fig. 1(a), It is worth to note that polysilicon replace part of silica in every period of metamaterial.

f_{g} = \frac{d_{1}}{d_{1} + (d_{2} - d_{3}) + d_{3}} = 9.62 \times 10^{- 7} {(z - 120)}^{2} + 0.02655

Next we consider the variation of the permittivity of metamaterial with different chemical potentials. The working wavelength spans from 5400 nm to 5800 nm. Figure 1(b) shows the variation of horizontal effective permittivity with different chemical potentials and different volume fractions (e.g. f_g = 0.028, f_g = 0.034 and f_g = 0.040) of graphene when the λ = 5800 nm. The black and green lines are the real parts and imaginary parts of permittivity, respectively. From the Fig. 1(b) we can see the effective permittivity of the metamaterial gradually increased when λ = 5800 nm. With increasing of the volume fraction of graphene from 0.028 to 0.040, the real part of effective permittivity ε_m goes down gradually. On the other hand, adjusting gate voltages on the graphene sheets can modify ε_m. When the f_g = 0.034, under μ_c<0.4eV, the real permittivity of metamaterial is positive. While in the case of μ_c>0.4eV the permittivity of metamaterial is negative. The Im (ε_m) range is from 0.043 to 0.06, which represents the electro-absorption and loss characteristics.

Figure 1(c) shows the case that f_g = 0.034. It is worth noticing that the imaginary part of the metamaterial permittivity is much less than the real part of permittivity in the work chemical potential. Furthermore, the permittivity ε_m also varies for different wavelengths since the conductivity of graphene is related to the optical wavelength, which is shown in Fig. 1(c) for μ_c = 0.45 eV, μ_c = 0.5 eV and μ_c = 0.55 eV, respectively. We can see the effective permittivity of the metamaterial goes down quickly except a narrow bandwidth, which is from 1000 nm to 2200 nm. It is worth noting that the graphene-silica metamaterial is a kind of hyperbolic metamaterial [6] (i.e. ε_z>0_, ε_x<0) when the wavelength is from 5400 nm to 5800 nm.

To sum up, Figs. 2(a)-(b) plot the permittivity of graded metamaterials along the z-axis i.e. the transmission direction. The function of ε_m is monotonically increasing from the input port to output port. In Fig. 2(a) it is obviously that the decrease of ε_m is with the increasing of λ when μ_c = 0.45eV. This interesting phenomenon means that the light with shorter wavelength is easier to be trapped, because the more closely the effective permittivity approximates to zero, the easier the light can be trapped. Figure 2(b) shows that the higher μ_c will lead to lower ε_m when λ = 5800nm. Based on this conclusion we can alter μ_c to change the range of μ_c i.e. turn the state of mid-infrared between blocked and transmitted. The detailed analysis about these properties would be given in the next section.

Fig. 2 Effective permittivities of graphene-silica metamaterial along transmission direction. (a) The variation of permittivity along transmission direction with different wavelengths the μ_c = 0.45eV. (b) Permittivities of metamaterial along transmission direction under different chemical potentials when λ = 5800nm.

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3. Modes analysis in metamaterial waveguide

The metamaterial waveguide discussed in this paper is simplified as a two-dimensional planar slab waveguide. Researchers in related fields have determined that such a trapping mechanism is applicable for transverse magnetic (TM) waves in insulator-metal-insulator and metal-insulator-metal [6, 29, 30]. Since the propagating properties of wave in the three-dimensional metamaterial waveguide do not become so much different even if the height of the cross section i.e. y-axis is reduced. We only consider the transverse magnetic (TM) polarization which normally incident on the interface, so that H_z is zero. By incorporating gain material [31, 32] and pumped semiconductor [33] into the core dielectric the loss of metamaterial could be compensated. In the following discussion, we assume the loss has been compensated and consider the lossless case. We elucidate the properties of the metamaterial waveguide without loss of generality.

From the continuity of tangential electric and magnetic ðelds, we can get the distribution of propagating modes in the waveguide. According to their field distribution and shape, they could be generally divided into two categories: the plasmonic symmetric mode and the plasmonic anti-symmetric mode. The photonic and plasmonic modes correspond to the sinusoidal and exponential fields respectively. The transcendental characteristic equations are expressed as [34]:

Plasmonic symmetric mode : \frac{ε_{a} κ_{m}}{ε_{m} κ_{a}} = - \tan h (\frac{κ_{m} d}{2})

Plasmonic anti - symmetric mode : \frac{ε_{a} κ_{m}}{ε_{m} κ_{a}} = - coth (\frac{κ_{m} d}{2})

Here ε_a is the effective permittivity of air cladding; ε_m is the effective permittivity of metamaterial core. Given that the equations need to obey the momentum conservation condition, which yields [30]:

β^{2} - κ_{m}^{2} - ε_{m} k_{0}^{2} = 0

β^{2} - κ_{a}^{2} - ε_{a} k_{0}^{2} = 0

The quadratic equations describe the relations between the transverse wave numbers in the air cladding κ_a, in the metamaterial core κ_m, the longitudinal wave numbers β and k vector is along the z-axis. It is noteworthy that β in the core and dielectric cladding are equal due to the phase matching condition. The effective refractive index of the propagating mode is expressed as $n_{e f f} = R e (β / k_{0})$ . Here k₀ denote the wavenumber in free space. In addition, the relation between n_eff and $\sqrt{ε_{a}}$ determines the type of the propagating mode. The mode is classified as photonic mode if $n_{e f f} < \sqrt{ε_{a}}$ whereas plasmonic mode if $n_{e f f} > \sqrt{ε_{a}}$ [35].

Furthermore, anti-symmetric plasmonic mode is not supported in our metamaterial structure. From Eqs. (6)-(8), considering the anti-symmetric plasmonic mode, κ_m and κ_a are the roots of equation

κ_{m}^{2} - k_{0}^{2} (ε_{a} - ε_{m}) = \frac{ε_{a}^{2}}{ε_{m}^{2}} κ_{m}^{2} {coth}^{2} (\frac{κ_{m} d}{2})

In our metamaterial structure, so there must be found that

\frac{ε_{a}^{2}}{ε_{m}^{2}} κ_{m}^{2} {coth}^{2} (\frac{κ_{m} d}{2}) > κ_{m}^{2} > κ_{m}^{2} - k_{0}^{2} (ε_{a} - ε_{m})

It means there are no real solutions in Eq. (9) i.e. the metamaterial waveguide would not support the anti-symmetric plasmonic mode based on our design. So finally the metamaterial waveguide only supports symmetric plasmonic mode when the absolute value of effective permittivity is less than the dielectric cladding’s permittivity.

Then we discuss the normalized power flows, which determine the heading direction and the group velocity of light. According to the direction of energy flow, the propagating modes can be classified as forward modes or backward modes. The symmetric plasmonic wave will be trapped if the normalized power flow vanishes at degenerate point.

Figure 3 plots the magnetic field profiles of the symmetric plasmonic mode in metamaterial waveguide. The y-component magnetic field of the symmetric plasmonic mode in the metamaterial waveguide is given as follows [28–30]. Here a = d/2.

Fig. 3 Magnetic field profiles of the symmetric plasmonic wave in metamaterial waveguide. (a) Schematic diagram of the symmetric plasmonic mode. (b) Front view of the magnetic field in metamaterial waveguide.

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H_{y} = {\begin{matrix} A e x p [- κ_{a} (x - a) + j (β z - ω t)] (x \geq a) \\ B \cos h (κ_{m} x) e x p [j (β z - ω t)] (- a \leq x \leq a) \\ A e x p [κ_{a} (x - a) + j (β z - ω t)] (x \leq - a) \end{matrix}

Using the boundary condition for equations, we can get the coupling coefficients of above equations as

A = B \cos h (κ_{m} x)

Based on the Maxwell’s curl equation, the x-component electric field of the symmetric plasmonic mode is described as

E_{x} = {\begin{matrix} A (β / ω ε_{0} ε_{a}) e x p [- κ_{a} (x - a) + j (β z - ω t)] (x \geq a) \\ B (β / ω ε_{0} ε_{m}) \cos h (κ_{m} x) e x p [j (β z - ω t)] (- a \leq x \leq a) \\ A (β / ω ε_{0} ε_{a}) e x p [κ_{a} (x + a) + j (β z - ω t)] (x \leq - a) \end{matrix}

The Poynting vector is defined as [29]. Then we can derive the power propagating through the metamaterial core and the dielectric cladding as follows:

P_{m} = \frac{β B^{2} [\sin h (2 κ_{m} a) + 2 κ_{m} a]}{2 ω ε_{0} ε_{m} κ_{m}}

P_{a} = \frac{β B^{2} \cos h^{2} (κ_{m} a)}{ω ε_{0} ε_{a} κ_{a}}

The normalized power flow can be expressed as

P_{n} = \frac{\sum^{​} P_{i}}{\sum^{​} | P_{i} |} = \frac{P_{m} + P_{a}}{| P_{m} | + | P_{a} |}

4. Trapping or transmitting light in metamaterial waveguide

In the previous section the propagation mode has been analyzed. Our metamaterial waveguide only supports the symmetric plasmonic mode. The symmetric plasmonic wave will be trapped if the normalized power flow vanishes at degenerate point. We use graphene layers in order to alter the state of light due to the permittivity-tunable properties of graphene. When the metamaterial is applied as absorber, the range of the absorption spectrum can be controlled by the chemical potential of graphene.

As described in the previous section, the working wavelength of light is from 5400 nm to 5800 nm. From the boundary condition and the momentum conservation condition, the characteristic equations of plasmonic symmetric mode have been described in Eq. (5,7,8). For the symmetric plasmonic mode, each κ_m and κ_a is the root of the equation

As for the fundamental symmetric plasmonic mode, when λ = 5400 nm, the intersection of the two curves indicates the propagating mode in the waveguide. With the decrease of ε_m, two intersections come closer to each other, and at a certain point they degenerate into single intersection. From Fig. 4(a), in the case of −1<ε_m<-0.23, two symmetric plasmonic modes are always supported. Figure 4(a) plots two intersections come into a single point in the case of ε_m = −0.23 that means the mode degeneracy occurs. The variation of ε_m may alter the value of β, but the dual plasmonic modes still exist. Finally, when the ε_m is less than −0.23, there is no propagating mode in the waveguide. The symmetric plasmonic mode is denoted as backward mode, in which the normalized energy flow is opposite to the phase velocity. Contrarily, the mode is denoted as a forward mode. The direction of power flow is marked in Fig. 4.

Fig. 4 The relation of transverse wave numbers in symmetric plasmonic mode. (a) In the case of ε_m = −0.5, two symmetric plasmonic modes are always supported. (b) In the case of ε_m = −0.23, two intersections come into single point which means the mode degeneracy occurs.

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As shown in the above analysis, the metamaterial waveguide could support both forward and backward modes. Figures 5(a)-5(b) plot the curve of normalized optical power flow and effective refractive permittivity at λ = 5400 nm. Based on Eq. (16) we can calculate the normalized power flow, which is shown in Fig. 5(a). Intriguingly, dual modes come closer and degenerate together when the effective permittivity reaches a certain critical value along the transmission direction. In other words, there is an appropriate effective permittivity to make the normalized optical power flow vanishes. When the chemical potential of graphene is 0.45 eV, there must be a position in the metamaterial waveguide approaching ε_m = −0.23 where the light can be trapped. On the other hand, P_n is always positive and the reduced flow must be transmitted when μ_c = 0.55 eV, since that the forward mode is larger than backward mode i.e. the normalized optical power will transmit in forward direction. Figure 5(b) illustrates that the effective refractive index of two propagation modes shrinks towards the degeneracy point. As the ε_m decreases, both the effective refractive index of forward and backward modes gradually reach to n_eff = 1.5, which means the intensities of their power flows are equal.

Fig. 5 The variation of normalized energy flow and effective refractive index along the transmission direction. (a) When λ = 5400 nm, the normalized optical power flow under different permittivity. Apparently the forward and backward modes vanish in a certain critical point. (b) The effective refractive index under different permittivity. Following the decrease of permittivity, the difference of effective refractive index is decreased. When μ_c = 0.55 eV, the characteristics is shown as orange area. In the same way when μ_c = 0.45 eV, the characteristics is shown as green area. (c) The stopping position of light at different wavelengths. The graded metamaterial waveguide trap the bandwidth light from 5400 to 5800 nm on an appropriate position.

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In Fig. 5(c), the change of light’s frequency will alter the degenerate position, where the forward and backward mode vanishes. So lights of different frequencies will stop respectively at different guide positions in the graded waveguide. Because of the variation of relative permittivity, long-wavelength light can propagate further than the short-wavelength light. When the chemical potential of graphene reaches to 0.55 eV, all light with the wavelength from 5400 nm to 5800 nm can pass through the waveguide. The design of loss-compensation structure is an important and interesting part in rainbow trapping. We have started the deeper research about compensation method based on gain material structure and prepare to show it in the next paper.

The light cannot pass through the waveguide both in ideal condition and in loss condition [5, 6]. In ideal condition, the waveguide can lead to a complete standstill of light in the critical point without the loss of metamaterial [4]. The propagation status of light is different in loss case: the light will be trapped in a relatively short time and the energy incident from the input port will be reflected due to the strong intermodal coupling between the forward and backward modes [5–7].

We consider the optical absorption property of the waveguide in the lossy condition. In the beginning, the incident light will go forward and decline rapidly. The forward wave will reflect and go back to the input port after it reaches to the degeneracy point. The imaginary part of the permittivity represents the electro-absorption and loss characteristics. In order to enhance the absorbability of waveguide [36], we replace the silica with polyimide (with a frequency independent refractive index of 1.8 + 0.06i) to add a same level imaginary component in vertical direction as the horizontal direction. To explain the absorption process in graded graphene-polyimide metamaterial waveguide, Fig. 6(a) shows the variation of normalized power along the transmission direction. In the forward process, the optical power dropped exponentially and almost 81% of the power has been absorbed. Then the wave reflected back and the rate of power gradually decline with slope getting smaller. Finally the normalized power reaches 6% i.e. absorptivity is 94%. From the Fig. 6(b), we can see that the graded metamaterial waveguide can absorb the over 90% light from 4000 nm to 4300 nm and 6400 nm to 7100 nm when μ_c = 0.3 eV and μ_c = 0.9 eV respectively. The waveguide cannot lead to a complete standstill of light in the critical point due to the loss of metamaterial. Because of the strong intermodal coupling between the forward and backward modes, the energy incident from the input port will be totally reflected [5–7]. Based on light trapping effect, the graded metamaterial waveguide can be the candidate for multi-wavelength absorber or band-stop filter applications in the lossy condition.

Fig. 6 Broadband absorption in metamaterial waveguide. (a) Variation of normalized power in forward process and reflection process. (b) Absorption region under different chemical potential of graphene. Over 90% light is absorbed.

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5. Conclusion

In summary, we propose a tunable graded graphene-silica metamaterial waveguide that can trap or transmit a bandwidth mid-infrared light from 5400 to 5800 nm. We prove that the structure only supports dual symmetric plasmonic mode when the absolute value of metamaterial permittivity is less than the dielectric cladding’s permittivity. Through analyzing the property of graded metamaterial and the conductivity of graphene, we provide the distribution of effective permittivity along the transmission direction in the waveguide. It has been found that adjusting the voltages on graphene layers can lead to the stop or release of light. Also the variation of effective permittivity with different potential chemical on graphene is shown. In lossless condition, the numerical computation shows that the lights of different frequencies will stop respectively at different guide positions in the graded waveguide and this conclusion proves that the board light can be trapped in waveguide. By replacing the silica with polyimide to enhance the absorbability of waveguide, we get a broadband absorber in mid-infrared, which can absorb 94% of the light in center wavelength. Based on the light trapping effect, the graded graphene-polyimide metamaterial waveguide can absorb over 90% mid-infrared light in a large bandwidth by adjusting the chemical potentials, showing potential as the bandwidth absorber with a tunable range of absorption spectrum.

Acknowledgment

This work is supported in part by the Major State Basic Research Development Program of China (Grant No. 2010CB328206).

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Tunable trapping and releasing light in graded graphene-silica metamaterial waveguide

Abstract

1. Introduction

2. Graphene–silica metamaterial structure

3. Modes analysis in metamaterial waveguide

4. Trapping or transmitting light in metamaterial waveguide

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (6)

Equations (16)

Optics Express