Actively controlled super-resolution using graphene-based structure

Bo Han Cheng; Kai Jiun Chang; Yung-Chiang Lan; Din Ping Tsai

doi:10.1364/OE.22.028635

1. Introduction

The diffraction limitation [1] is one of the most important issues and remains to be solved for the demand of high-density optoelectronic components, especially for the photolithography industry. Because the lowest diffraction orders of the fine feature of the objects to be resolved (the size is smaller than half of the illuminating wavelength λ) possess evanescent wave form, they cannot be captured by using conventional glass-based optical lens, and hence the corresponding fine feature is lost. For overcoming this problem, substitutions of anisotropic material with the ability of transferring those evanescent signals have already been explored, such as superlens [2–4], multi-layered based hyperlens [5–8] and hybrid-superlens hyperlens [9–11]. Objects with the size of λ/3~λ/6 can be successfully resolved by the mentioned plasmonic-based lens. However, for fulfilling the requirements of super-resolution, the constituent condition of the used material is quite rigorous and the working wavelength band region is severely limited.

Recently, graphene has been widely investigated for its several unique properties [12, 13] and easily large-area fabrication [14]. Especially, graphene can support electromagnetic surface wave under mid-infrared light source illumination, which is similar to that the surface plasmon polaritons (SPPs) exist on the surface of the noble metal in visible region [15, 16]. Comparing to the metal case, the energy loss of the tightly confined surface wave on graphene is quite low. The propagation length is about a few dozens of wavelengths of the excited mode [16]. And, the effective refractive index of the excited surface wave on graphene is able to reach around 100, which allows us to resolve nano-scale objects using mid-infrared source (it will be shown in this work). Most importantly, the propagation length and the effective refractive index can be dynamically tuned by applying temperature [17], magnetic field [18] and voltage [19] to graphene. It results from the tunability of conductivity via changing the chemical potential. The corresponding optical constant (relative permittivity) could be manipulated. In this paper, we focus on the manipulation of optical permittivity under control of chemical potentials of graphene.

Besides the tunability of optical properties, the capability of realization would be one of the most essential factors to judge the practicability of newly proposed components. Comparing to the totally three dimensional fabrications, recently a class of optical devices named metasurface [20] with a reduced dimensionality and low-cost manufacturing characteristics would have a high potential to create the next-generation ultra-compact optoelectronic components because of its easy fabrication. Monolayer graphene with one atom thickness will pave the way due to such indispensable features. Up to now, there are many graphene-based planar optoelectronic components have been investigated, such as transformation optics device [19], directional coupler [21], cloaking metasurface [22], hyperbolic metamaterial [23], active plasmonic device [18, 24], and subwavelength imaging [25–27].

This paper proposes and investigates an actively controlled, super-resolution imaging device based on graphene supported surface waves. This imaging device consists of a monolayer graphene sandwiched between two dielectric materials along with two alternate chemical potentials in graphene (which can be obtained by alternately applying two biased voltages to graphene). The resolution of λ/50 at mid-infrared region in the proposed structures is demonstrated by the finite element method (FEM) electromagnetic simulations. The super-resolution function can be achieved at various operation wavelengths using different applied voltages to change the chemical potentials of graphene. The corresponding dispersion relations of the constituent elements are also examined to unravel the mechanism of super-solution. This investigation provides a conceptual idea to build up an ultra-flat optoelectronic device with the ability to resolve nano-scale objects using mid-infrared source.

The rest of this paper is organized as follows. Section 2 describes the simulation structure and method. Section 3 then presents the mechanism of super-resolution using graphene supported surface waves. Next, Section 4 demonstrates the super-resolution in the proposed device. Conclusions are finally drawn in Section 5.

2. Investigated structure and simulation method

Figure 1 plots the investigated structure. It consists of a monolayer graphene sandwiched between two dielectric materials along with two alternate chemical potentials in graphene. Engheta et al. reported that the graphene layer with spatially inhomogeneous chemical potentials can be obtained by applying spatially inhomogeneous biased voltages to the graphene [19]. Therefore in our design, the two alternate chemical potentials in graphene can be achieved by alternately applying two biased voltages to graphene (Fig. 1). Each constituent of the alternately arranged structure can be viewed as a dielectric/graphene/dielectric (DGD) component with relative permittivity $ε_{1} / ε_{g} / ε_{2}$ where $ε_{1}$ ( $ε_{2}$ ) and $ε_{g}$ are the relative permittivity of the upper (lower) dielectric and monolayer graphene, respectively. The numerical formula for conductivity (from Kubo formula) and relative permittivity of the simulated monolayer graphene can be described as below, respectively [28–31].

σ_{g} = \frac{i e^{2}}{4 π ℏ} \ln [\frac{2 μ_{c} - (ω + i \frac{2 Γ}{ℏ}) ℏ}{2 μ_{c} + (ω + i \frac{2 Γ}{ℏ}) ℏ}] + \frac{i e^{2} k_{B} T}{π ℏ^{2} (ω + i \frac{2 Γ}{ℏ})} [\frac{μ_{c}}{k_{B} T} + \ln (e^{\frac{- μ_{c}}{k_{B} T}} + 1)]

ε_{g n} = (ε_{0} + i \frac{σ_{g}}{ω Δ}) / ε_{0},

where

e,

ℏ,

and

k_{B}

are the electron charge, Plank’s constant and Boltzman’s constant, respectively. In Eq. (1),

T

is the temperature in Kelvin, which is fixed to 300 K in this investigation.

μ_{c}

is the chemical potential, which can be electrically controlled via applying biased voltage on graphene, and hence leads to a voltage-tuned value.

Γ = ℏ / 2 τ

= 0.1 meV, which is related to the electron-phonon coupling loss [32, 33]. Notably, even considering various values

Γ,

the super-resolution function of this proposed device is still observed. In Eq. (2),

Δ

denotes the graphene thickness. Since the effective refractive index obtained from

Δ

= 1 nm (shown in Fig. 2(a)) is almost the same as that from theoretical calculation in literature, for saving simulation computer resource, the graphene thickness is set as 1 nm in this simulation work.

Fig. 1 Concept of investigated structure.

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Fig. 2 (a) Left: dispersion relations ( $ω - k$ diagrams) of monolayer graphene sandwiched between two dielectric materials for different chemical potentials in graphene, where solid (dashed) lines are the dispersion curves with positive (negative) slope. For red (blue, green, and black) line, chemical potential is 0.15 eV, (0.2 eV, 0.3 eV, and 0.4 eV), which comes from theoretical calculation. Solid circle symbols come from FEM simulation. Right: schema of monolayer graphene sandwiched between two dielectric materials. $ε_{1}$ ( $ε_{2}$ ) is the relative permittivity of upper (lower) dielectric material. For simplification, we set $ε_{1} = ε_{2} =$ 6.3 in all investigated work. (b) and (c): Simulated Ex contours of excited surface waves at various operating frequencies for symmetric and ant-symmetric, respectively, modes. The chemical potential is 0.3 eV. (d) Side-view of the investigated system (Fig. 1). (e) Equivalent structure of (d). $n_{e 1}$ ( $n_{e 1}$ ) denotes effective refractive index obtained from formula $n_{e} = c \cdot Re [k] / ω,$ and the inset shows effective relative permittivity tensor of (d). $ε_{/ /}$ and $ε_{⊥}$ are effective relative permittivities in parallel (y and z) and perpendicular (x), respectively, directions.

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Here, all the simulated results are acquired from the commercial electromagnetic software COMSOL Multiphysics based on the finite element numerical method. In our investigation, all the monolayer graphene and its surrounding dielectric materials are modeled in COMSOL. In order to excite the surface wave on the graphene, the source plane with electric field in the xz-plane is launched, which would induce the non-radiative surface wave at the graphene-dielectric interface. The perfectly matched layers (PMLs) are used in all propagating directions. It is worth mentioning that exerting different voltages on different regions of graphene to vary graphene’s optical property has been proposed and investigated in literature [19, 34].

3. Mechanism of super-resolution

To provide more physical understanding about the working principle of the proposed device, we will present the design strategy first and show the simulation results later. First, for super-resolution, building a material with an anisotropic relative permittivity tensor is needed [5]. The proposed graphene-based design can meet the requirement. Figure 2(a) shows the $ω - k$ diagram of the surface waves in the DGD structures. Using the formula $n_{e f f} = c \cdot Re [k] / ω,$ the effective refractive index of the surface waves can be obtained. The sign of slope ( $ω / k$ ) decides the sign of the effective refractive index. Figures 2(b) and 2(c) present the simulated contours of Ex component of excited surface wave. According to Figs. 2(b) and 2(c), the excited surface waves can be divided into two branches: symmetric and anti-symmetric. For symmetric (anti-symmetric) modes, the wavelength of excited surface wave will decrease (increase) as the operation frequency increases. It means that the symmetric (anti-symmetric) mode has a positive (negative) effective refractive index due to the positive (negative) slope of the $ω - k$ diagram. The dispersion relations of anti-symmetric and symmetric modes of excited surface waves can be respectively described as [15, 35]

T a n h [\sqrt{k^{2} - ε_{g} \times {(ω / c)}^{2}} Δ] = - \frac{ε_{g}}{ε_{d}} \sqrt{\frac{k^{2} - ε_{d} \times {(ω / c)}^{2}}{k^{2} - ε_{g} \times {(ω / c)}^{2}}};

C o t h [\sqrt{k^{2} - ε_{g} \times {(ω / c)}^{2}} Δ] = - \frac{ε_{g}}{ε_{d}} \sqrt{\frac{k^{2} - ε_{d} \times {(ω / c)}^{2}}{k^{2} - ε_{g} \times {(ω / c)}^{2}}},

where

ε_{d} = ε_{1} = ε_{2} = 6.3

, see caption of Fig. 2(a)).

Further, a monolayer graphene exerted by a biased voltage can be viewed as a tunable material. Hence, the DGD structure with two alternate chemical potentials ( $μ_{c 1}$ and $μ_{c 2}$ ) in graphene (as shown in Fig. 2(d)) is equivalent to an anisotropic material which is shown in Fig. 2(e). The corresponding effective optical constant can be determined by the dispersion relation derived from the transfer matrix method as [36]

\cos (k_{⊥} Λ) = \cos (k_{1 ⊥} d_{1}) \cos (k_{2 ⊥} d_{2}) - \frac{1}{2} (\frac{n_{e 1}^{2} k_{2 ⊥}}{n_{e 2}^{2} k_{1 ⊥}} + \frac{n_{e 2}^{2} k_{1 ⊥}}{n_{e 1}^{2} k_{2 ⊥}}) \sin (k_{1 ⊥} d_{1}) \sin (k_{2 ⊥} d_{2}),

where

k_{1 ⊥} = \sqrt{n_{e 1}^{2} {(ω / c)}^{2} - k_{/ /}^{2}}

and

k_{2 ⊥} = \sqrt{n_{e 2}^{2} {(ω / c)}^{2} - k_{/ /}^{2}}

denote the x-directional (perpendicular) wavevectors of the effective material with refraction indices

n_{e 1}

and

n_{e 2},

respectively. Here

k_{/ /}

is the y and z-directional (parallel) wavevector;

Λ = d_{1} + d_{2}

is the period of the alternatively biased system; and d1 and d2 are the unit sizes of DGD elements with the chemical potentials of

μ_{c 1}

and

μ_{c 2},

respectively.

Eq. (5) can be further simplified by: (1) using Taylor expansion (up to second order), (2) assuming the long wavelength approximation $λ > > Λ$ ( $λ$ is incident wavelength in free space), and (3) considering $n = \sqrt{ε} .$ Then the dispersion relation of the alternative DGD structures will become as below

\frac{k_{/ /}^{2}}{ε_{⊥}} + \frac{k_{⊥}^{2}}{ε_{/ /}} = k_{0}^{2},

where

ε_{⊥} = {(\frac{d_{1}}{d_{1} + d_{2}} \frac{1}{ε_{e 1}} + \frac{d_{2}}{d_{1} + d_{2}} \frac{1}{ε_{e 2}})}^{- 1}

and

ε_{/ /} = \frac{ε_{e 1} d_{1} + ε_{e 2} d_{2}}{d_{1} + d_{2}}

are the effective relative permittivities of the equivalent anisotropic material in the perpendicular and parallel, respectively, directions;

k_{0}

is the free space wavevector. We restate that the optical property of this effective anisotropic material can be tuned.

For instance, if the chemical potentials $μ_{c 1}$ and $μ_{c 2}$ ( $μ_{c 1} \neq μ_{c 2}$ ) result in two symmetric modes alternately distributing on monolayer graphene, the DGD structure then can be viewed as a birefringence material with two alternate positive refractive indices along x direction. Inversely, if the incident frequency equals to the critical values (i.e. dotted orange lines in Fig. 2(a)), the DGD structure will become a metamaterial with an anisotropic refractive index tensor. And the signs of the components of the refractive index tensor in parallel and perpendicular directions are opposite to each other.

For the proposed device achieving super-resolution, all scattering signals that come from the object need to be transferred and cannot interference with one another until they arrive at the detection position. Basing on our previous work [5], an anisotropic medium with ultra-flat hyperbolic isofrequency curve is required, which can be fulfilled for the multilayered metamaterials that is composed of alternately arranged two dielectric materials with opposite signs in their dielectric constants (i.e., $ε_{/ /} \cdot ε_{⊥} < 0$ ). For the proposed device operating at some specific frequency, this requirement can be met by suitably changing the two alternate chemical potentials of monolayer graphene. As the incident frequency changes, the chemical potentials of graphene and the accompanied applied voltages also need to be altered.

4. Simulation results

With the above physical understanding, in this section, the super-resolution function of proposed device is demonstrated. Figure 3 plots the simulated structure, which consists of a monolayer graphene sandwiched between two dielectric materials.

Fig. 3 Simulated structure. (a) Schematic 3D structure. (b) and (c) xz-plane and yz-plane side views, respectively. Total spatial space (x $\times$ y $\times$ z) in simulation is 60 nm $\times$ 140 nm $\times$ 7 nm. The volume fractions of graphene with chemical potentials $μ_{c 1}$ and $μ_{c 2}$ are 0.5 and 0.5, respectively), for periodicity $Λ$ = 10 nm. Mask material is Chromium with thickness of 2 nm. Two elliptical holes on Cr mask represent the object to be resolved, which are filled with air. Length of major (minor) axis is 8 nm (2 nm). The center-to-center distance of the two holes is 80 nm. The blue arrow denotes the preferred propagation of excited surface wave, which can be fulfilled for proposed devices with flat hyperbolic isofrequency curves.

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The monolayer graphene has the two alternate chemical potentials of $μ_{c 1}$ and $μ_{c 2} .$ (The applied voltages V1 and V2 in Fig. 3(a) are chosen to provide the required $μ_{c 1}$ and $μ_{c 2} .$ ) The schematic 3D structure is displayed in Fig. 3(a). The xz-plane and yz-plane side view are shown in Fig. 3(b) and (c), respectively. Two elliptical holes (with the major and minor axes of 8 nm and 2 nm, respectively) are carved on the chromium (Cr) metal mask and play the role of object to be resolved. Comparing with the wavelength of SPPs supported on noble metal (typical value of several hundred nanometers [15]), the wavelength of excited surface wave on the monolayer graphene can be greatly shrunk from micrometer to nanometer scale [19]. Therefore, it is much easier to resolve the fine structures using such small wavelength. The end of the mask directly contacts the monolayer graphene. A linearly polarized light (with polarization in yz plane) is incident on the Cr mask. As the incident light illuminated the mask, the diffracted lights from the tiny holes will excite the surface waves on the monolayer graphene. To successfully resolve each tiny hole, the excited surface wave from one hole will propagate along the x direction but cannot interfere with that from the other hole. Finally, those signals will transfer to the end of this structure and then be distinguished. In the following paragraphs, two operating frequencies 71.195 THz and 55.2 THz are considered to demonstrate the super-resolution function of the proposed devices.

Fig. 4(a) and 4(b) present the simulated time-averaged power flow contours for the structure of Fig. 3 with the chemical potentials ( $μ_{c 1}, μ_{c 2}$ ) equal to ( $μ_{c 1}$ = 0.2 eV, $μ_{c 2}$ = 0.3 eV) and ( $μ_{c 1}$ = 0.3 eV, $μ_{c 2}$ = 0.3 eV), respectively, and the incident frequency of 71.195 THz. Similarly, Figs. 4(c) and 4(d) plot the same contours for ( $μ_{c 1}$ = 0.15 eV, $μ_{c 2}$ = 0.2 eV) and ( $μ_{c 1}$ = 0.2 eV, $μ_{c 2}$ = 0.3 eV), respectively, with the incident frequency of 55.2 THz. Figures 4(e), 4(f), 4(g) and 4(h) present the corresponding power flow intensities versus y position at the end of the device (x = 47 nm) for Figs. 4(a), 4(b), 4(c) and 4(d), respectively. Figures 4(a), 4(c), 4(e), 4(g) display that the two tiny holes can be resolved at the end (x = 47 nm) of the device. Conversely, Figs. 4(b), 4(d), 4(f) and 4(h) show that the two holes cannot be distinguished at the end of the device. Figures 4(i), 4(j), 4(k) and 4(l) plot the isofrequency dispersion curves for Figs. 4(a), 4(b), 4(c) and 4(d), respectively. They exhibit that, when the two holes can (cannot) be resolved, the isofrequency dispersion curves have the flat hyperbolic (elliptical) form. With the elliptical isofrequency dispersion curves, the excited surface waves of the higher-order diffraction signals that emit from the holes cannot transfer on the monolayer graphene. And the excited surface waves of the lower-order diffraction signals from the two holes will immediately interfere with each other after passing through the holes. Figures 4(a) – 4(l) imply that, only the excited surface waves on the monolayer graphene are not sufficient to resolve two tiny holes with a subwavelength distance between them. The hyperbolic isofrequency dispersion relation is required for the proposed super-resolution devices. (Notably, the chemical potentials of graphen in Figs. 4 are designed according to the super-resolution mechanism discussed in Sect. 3.)

Fig. 4 (a), (b), (c), and (d): Simulated time-averaged power flow contours. They are extracted at xy plane and 1.5 nm above the upper surface of the graphene. (e), (f), (g), and (h): Power flow intensities versus y position at x = 47 nm. (i), (j), (k), and (l): Isofrequency dispersion relations. For (a), (e) and (i), $μ_{c 1}$ = 0.2 eV, $μ_{c 2}$ = 0.3 eV, and incident frequency f = 71.195 THz. For (b), (f) and (j), $μ_{c 1}$ = 0.3 eV, $μ_{c 2}$ = 0.3 eV, and f = 71.195 THz. For (c), (g) and (k), $μ_{c 1}$ = 0.15 eV, $μ_{c 2}$ = 0.2 eV, and f = 55.2 THz. For (d), (h) and (l), $μ_{c 1}$ = 0.2 eV, $μ_{c 2}$ = 0.3 eV, and f = 55.2 THz. Orange lines in (e) – (h) denote the center positions of two tiny elliptical holes.

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Figs. 4(a), 4(c), 4(i) and 4(k) further reveal that, when the two tiny holes are resolved, the isofrequency dispersion curves of the proposed devices are hyperbolic and very flat. Hence, the excited surface waves that come from each hole will propagate in the x direction without interfering with each other. Under this condition, the product of parallel and perpendicular relative permittivities ( $ε_{/ /} \cdot ε_{⊥}$ ) is negative and the value of $\tan^{- 1} (- ε_{/ /} / ε_{⊥}) \approx 0$ [5]. Here $θ =$ $\tan^{- 1} (- ε_{/ /} / ε_{⊥})$ denotes the angle between the propagation direction of delivered surface waves and x axis (it is shown in Fig. 3(a)). (For elliptical isofrequency dispersion curves, $ε_{/ /} \cdot ε_{⊥}$ becomes positive.) The calculated values of $(ε_{/ /}, ε_{⊥})$ for Figs. 4(a), 4(b), 4(c) and 4(d) are (−1.39152, 1.03 $\times$ 10¹⁰), (120040, 120040), (−0.0079373, 5.43855 $\times$ 10¹¹), and (5.601 $\times$ 10⁶, 3.85 $\times$ 10⁶), respectively. Furthermore, the calculated values of $\approx 0^{\circ},$ $45^{\circ},$ $\approx 0^{\circ},$ and ${55.5}^{\circ},$ respectively, which agree with the simulation results very well.

Finally, we will remark that the proposed device can reach λ/50 resolution but has only one-dimensional super-resolution ability since it uses the surface waves that propagate on the monolayer graphene. However, our investigations also reveal that, when the z positions of the objects (i.e. the tiny holes on Cr mask) leave the graphene layer (see Fig. 3(a)), the objects can be still resolved at the end of the device since the surface waves can be excited by the evanescent waves that emit from the objects. This feature enables the proposed device to transform the two-dimensional objects into the one-dimensional image. The proposed devices have potential applications in multi-functional material, real-time subwavelength imaging, and high-density optoelectronic components for using the abnormal diffraction feature.

5. Conclusion

A super-resolution device (with λ/50 resolution ability at mid-infrared region) that consists of a monolayer graphene sandwiched between two dielectric materials along with two alternate chemical potentials in graphene (which can be obtained by alternately applying two biased voltages to graphene) is proposed and analyzed. The proposed device utilizes the surface waves that propagate on the graphene monolayer. The surface waves are excited by the diffraction waves that emit from the objects. When the subwavelength resolution is achieved, the graphene-based device can be viewed as an effective optical medium with alternate arrangement of positive and negative refractive indices for the excited surface waves. And the isofrequency dispersion curves of the effective optical medium have the hyperbolic form. The super-resolution of the proposed device at different desired frequencies can be reached by merely changing the chemical potentials of graphene. The proposed device has only one-dimensional super-resolution ability. However, it can transform the two-dimensional objects into one-dimensional image since the surface waves on graphene can be excited by the objects that leave the graphene monolayer in the z direction. The proposed devices have potential applications in multi-functional material, real-time subwavelength imaging, and high-density optoelectronic components for using the abnormal diffraction feature.

Acknowledgment

The authors acknowledge financial support from Ministry of Science and Technology, Taiwan (Grant Nos. 101-2112-M-006-002-MY3, 103-2745-M-002-004-ASP and 102-2911-I-002-505) and Academia Sinica (Grant No. AS-103-TP-A06). They are also grateful to National Center for Theoretical Sciences, Taipei Office, Molecular Imaging Center of National Taiwan University, National Center for High-Performance Computing, Taiwan, and Research Center for Applied Sciences, Academia Sinica, Taiwan for their support.

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Actively controlled super-resolution using graphene-based structure

Abstract

1. Introduction

2. Investigated structure and simulation method

3. Mechanism of super-resolution

4. Simulation results

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (4)

Equations (6)

Optics Express