Hyperbolic metamaterial based on aperiodic graphene-dielectric and application of hyperbolic metamaterial

Abstract

The invention provides a hyperbolic metamaterial based on aperiodic graphene-dielectric and application of the hyperbolic metamaterial, and belongs to the technical field of optical sensors. The graphene single layer A and the dielectric layer B form a hyperbolic metamaterial heterostructure according to the rule of a cantor sequence; incident light is emitted to the surface layer plane of the hyperbolic metamaterial heterostructure, and for a given incident wavelength, the dispersion space of the hyperbolic metamaterial heterostructure shows hyperbolic dispersion characteristics related to the weight occupied by the dielectric thickness by modulating the Fermi level of graphene; by adjusting the Fermi level, the relaxation time, the number of layers and the thickness of a dielectric medium of the graphene single layer A, the phase of a reflection coefficient changes greatly near a phase change point from hyperbolic dispersion to elliptic dispersion, so that huge GH displacement is realized at a resonance angle, and the angle displacement sensor utilizes the corresponding relation between the GH displacement and an incident angle.

Description

Technical Field

[0001] The present invention belongs to the technical field of optical sensors, and relates to aperiodic graphene-dielectric based hyperbolic metamaterials and their applications, specifically to the application of hyperbolic metamaterials in angular displacement sensors. Background Art

[0002] In recent years, metamaterials have attracted extensive attention in the fields of optics and photonics due to their unique electromagnetic properties. Among them, hyperbolic metamaterials (HMMs), as anisotropic materials, have a sign reversal between one principal component and the other two principal components in the dielectric constant or magnetic permeability tensor, resulting in hyperbolic isofrequency curves. Hyperbolic metamaterials have important application values in super-resolution imaging, negative refraction, optical sensing, etc.

[0003] Graphene, as a two-dimensional structural material composed of single-layer carbon atoms, its optical response is mainly controlled by the surface conductivity, and the surface conductivity of graphene can be flexibly adjusted by the gate voltage. Therefore, graphene provides an ideal material for designing novel hyperbolic metamaterials.

[0004] Goos-Hänchen ( GH) shift is the lateral displacement in space of the reflected light relative to the position predicted by geometric optics when a light beam is reflected at the interface between two media. The GH shift phenomenon has appeared on the surfaces of electro-optic materials, photonic crystals, and metamaterial structures. Covering graphene on the dielectric surface and using the surface plasmon polaritons (SPPs) excited by graphene and weak losses can enhance the GH shift. The GH shift has relatively wide applications in optoelectronic fields such as optical switches, optical biosensors, and integrated optics.

[0005] Aperiodic photonic crystals have strong electric field localization for the electric field and also cause optical fractal effects. The strong electric field localization and optical fractal effects will cause a sharp change in the phase of the light wave reflection coefficient. Since the GH shift is proportional to the change rate of the light wave reflection coefficient phase, for example, it will cause a huge GH shift.

[0006] Different from traditional periodic hyperbolic metamaterials, the present invention introduces the concept of weight in the photonic multilayer material, and uses the alternating structure of graphene and dielectric to obtain an aperiodic hyperbolic metamaterial composed of Cantor sequences. Near the phase transition point from hyperbolic dispersion to elliptical dispersion, for example, it causes a drastic change in the phase of the reflection coefficient, and then analyzes the enhancement and regulation of the GH shift in the aperiodic hyperbolic metamaterial, and designs a high-sensitivity angular displacement sensor. Summary of the Invention

[0007] The object of the present invention is to provide a hyperbolic metamaterial based on aperiodic graphene-dielectric and its application in view of the above problems existing in the prior art. The technical problem to be solved by the present invention is to design an optical multilayer structure that can be used for an angular displacement sensor.

[0008] A hyperbolic metamaterial heterostructure is formed by arranging a single layer of graphene A and a dielectric layer B according to the law of the Cantor sequence; the layer structure rule of the hyperbolic metamaterial heterostructure is: S1 = ABA, S2 = S1(BBB)S1,..., S N = S N-1 (BBB) N-1 S N-1 ,(BBB) N-1 represents 3 N-1 Bs, where S0 = A and N is not less than 2;

[0009] A beam of incident light is incident on the surface plane of the hyperbolic metamaterial heterostructure. For a given incident wavelength, by modulating the Fermi level of graphene, its dispersion space exhibits a hyperbolic dispersion characteristic related to the weight of the dielectric thickness; by adjusting the Fermi level, relaxation time, number of layers of the single layer of graphene A, and the thickness of the dielectric, near the phase transition point from hyperbolic dispersion to elliptical dispersion, a huge change occurs in the phase of the reflection coefficient, thereby achieving a huge GH shift at the resonance angle. The angular displacement sensor utilizes the corresponding relationship between the GH shift and the incident angle.

[0010] Further, the incident light is an electromagnetic wave.

[0011] Further, the incident angle of the incident light on the surface plane of the hyperbolic metamaterial heterostructure is 45° - 55° (degrees).

[0012] Further, the wavelength of the incident light λ = 1923.8 nm (nanometers).

[0013] Further, the relaxation time of the single layer of graphene A is controlled between 9 ps and 99 ps (picoseconds).

[0014] Further, the Fermi level of the single layer of graphene A is between 0.4 eV and 0.44 eV (electron volts).

[0015] Further, the dielectric layer B is BaF2.

[0016] A single-layer graphene A and a dielectric layer B are combined to form a hyperbolic metamaterial heterostructure according to the Cantor sequence. Different from traditional periodic hyperbolic metamaterials, the concept of weight is introduced in this work for photonic multilayer materials. For a given incident wavelength, by modulating the Fermi level of graphene, its dispersion space exhibits hyperbolic dispersion characteristics related to the weight occupied by the dielectric thickness. By adjusting the Fermi level, relaxation time, number of layers of graphene, and the thickness of the dielectric, near the phase transition point from hyperbolic dispersion to elliptical dispersion, a huge change occurs in the phase of the reflection coefficient, thereby achieving a huge GH shift at the resonance angle, up to 4000 times the incident wavelength (λ). When the dielectric thickness is 50 nm, the optimal sensitivity can be obtained as 3×10 4 λ / deg. (degree). Based on this effect, a high-sensitivity angular displacement sensor can be designed. Brief Description of the Drawings

[0017] Figure 1 It is a schematic diagram of the graphene / dielectric aperiodic structure.

[0018] Figure 2 In (a), the real part of ε varies with the wavelength at different Fermi levels; || In (b), the real part of ε varies with the wavelength at different Fermi levels; ⊥ In (c), the reflectivity varies with the wavelength at different Fermi levels; (d) is a close-up of the variation of the real part of ε with the wavelength when E F = 0.4 eV. ⊥ In (a), the reflectivity corresponding to different Fermi levels; (b) is the reflection phase corresponding to different Fermi levels; (c) is the GH shift corresponding to different Fermi levels. Other parameters: T = 300 K, τ = 9 ps, t

[0019] Figure 3 = 8 nm, t d = 0.5 nm, N g = 1. g In (a), the reflection phase corresponding to different dielectric thicknesses; (b) is the GH shift corresponding to different dielectric thicknesses. Other parameters: E

[0020] Figure 4 = 0.4 eV, T = 300 K, τ = 9 ps, t F = 0.5 nm, N g = 1. g In (a), the reflection phase corresponding to different relaxation times; (b) is the GH shift corresponding to different relaxation times. Other parameters: E

[0021] Figure 5 = 0.4 eV, T = 300 K, t F = 0.5 nm, Nd = 8 nm, t g = 0.5 nm, N g = 1.

[0022] Figure 6 Among them, (a) is the reflection phase corresponding to different numbers of graphene layers; (b) is the GH shift corresponding to different numbers of graphene layers. Other parameters: E F = 0.4 eV, T = 300 K, τ = 9 ps, t d = 8 nm, t g = 0.5 nm.

[0023] Figure 7 is the sensor sensitivity under different dielectric thicknesses, where figure (a) is when t d = 8 nm; figure (b) is when t d = 22 nm; figure (c) is when t d = 36 nm; figure (d) is when t d = 50 nm; other parameters: E F = 0.4 eV, T = 300 K, τ = 9 ps, t g = 0.5 nm.

[0024] In the figure, A is a single layer of graphene; B is a dielectric layer. Detailed implementation manners

[0025] The following are specific embodiments of the present invention and in combination with the accompanying drawings, the technical solutions of the present invention are further described, but the present invention is not limited to these embodiments.

[0026] A single layer of graphene A and a dielectric layer B are combined according to the composition of the Cantor sequence to form a hyperbolic metamaterial heterostructure. The iterative rule of the Cantor sequence is: where S0 = A, S1 = ABA, S2 = S1(BBB)S1,..., S N = S N-1 (BBB) N- 1 S N-1 ,..., (BBB) N-1 represents 3 N-1 Bs, where N (N = 2, 3, 4...) is the serial number of the sequence, as Figure 1 shown. Among them, label 1 is the incident light beam, label 2 is the position of the reflected light beam predicted by geometric optics when the light beam is completely reflected at the interface, label 3 is the actual reflected light beam, label 4 is graphene, and label 5 is the dielectric. In the figure, θ is the incident angle of light, label G is the GH shift, ε g and ε d are the dielectric constants of graphene and the dielectric respectively, t g and td The thicknesses of the graphene and the dielectric are respectively selected. The dielectric selected is BaF2, and the number of structural periods N = 2 is set. At this time, the structure is ABA(BBB)ABA.

[0027] The transverse magnetic (TM) wave is used as the incident wave, that is, there is only an electric field component and no magnetic field component in the direction of light wave propagation. The effective dielectric constant of the graphene sheet can be expressed as

[0028]

[0029] where ε0 and ω are the vacuum dielectric constant and the incident light angular frequency respectively, and σ g is the surface conductivity of graphene, expressed as σ g =σ intra +σ inter , which can be obtained through the Kubo formula. Among them, the intraband conductivity σ intra and the interband conductivity σ inter are respectively expressed as follows

[0030]

[0031] where e and are the electron charge and the reduced Planck constant respectively, E F and τ are the Fermi level of graphene and the relaxation time of electrons respectively, K B and T are the Boltzmann constant and the ambient temperature respectively.

[0032] When the incident light is in the near-infrared band, when the incident wavelength λ is much larger than the thickness of the dielectric layer, that is, λ>>t g +t d , the effective medium theory can be used to describe the optical response of this system. The direction perpendicular to the graphene layer is defined as the Z axis. The dielectric constant tensor of this structure has a diagonalized form and can be expressed as [ε]=diag[ε xx ,ε yy ,ε zz , where ε xx =ε yy =ε || , ε zz =ε ⊥ , ε || and ε ⊥ represent the dielectric constants parallel and perpendicular to the graphene layer respectively, and are expressed as follows

[0033]

[0034] where t g1 =4 / 9t g ,t d1 =5 / 9t d, the coefficients 4 / 9 and 5 / 9 are the ratios of the number of dielectric and graphene layers to the total number of layers respectively. Through such weight distribution, the respective thicknesses of the dielectric and graphene are combined proportionally to obtain parameters such as the equivalent thickness of the entire multi-layer structure, enabling the effective medium theory to more realistically reflect the performance of the actual multi-layer structure.

[0035] For TM (transverse magnetic) incident waves, its dispersion surface can be expressed as:

[0036]

[0037] where k x and k z are the wave vectors of the structure in the X and Z directions respectively, k0 is the wave vector of free space. If ε || ·ε ⊥ < 0, then the dispersion curve is hyperbolic. If ε || ·ε ⊥ > 0, the dispersion curve is elliptical at this time.

[0038] For the multi-layer structure, the transfer matrix method is used to analyze the reflection phase and reflectivity of the structure. For a given dielectric layer l, when an electromagnetic wave is incident on the structure, the transfer matrix can be expressed as

[0039]

[0040] Here,

[0041]

[0042] where n d and n g are the refractive indices of the dielectric and graphene respectively, and can be obtained through and respectively. η l (l = d, g) is the effective optical admittance in the dielectric and graphene. λ is the wavelength of the incident light wave. The transfer matrix of the multi-layer material can be obtained by multiplying the transfer matrices of each layer, that is

[0043]

[0044] Then the reflection coefficient of the structure can be calculated as

[0045]

[0046] where the optical admittance η o ’ = η0 = (ε0 / μ0) 1 / 2 , and the reflectivity R = |r| 2 of the structure can be obtained through the reflection coefficient, as well as the reflection phase Φ rAccording to the static phase method, the GH displacement can be obtained as

[0047]

[0048] It can be seen from formula (13) that the GH displacement is proportional to the slope of the reflection phase with respect to the incident angle.

[0049] In the calculation, the environmental temperature T = 300K, the relaxation time τ = 9ps, the thickness and dielectric constant of the dielectric are set to t d = 8nm and ε d = 2.2, and the thickness of monolayer graphene is taken as t g = 0.5nm. A TM-polarized light beam is incident normally from the air. Figure 2 (a) shows the variation of the real part of the dielectric constant component ε F parallel to the graphene layer direction with respect to the wavelength at different Fermi levels (E || = 0.4eV, 0.42eV, 0.44eV). As the incident wavelength gradually increases, the real part of ε || gradually decreases, and the real part of ε || is positive for all wavelengths.

[0050] Figure 2 (b) shows the variation of the real part of the dielectric constant component ε ⊥ perpendicular to the graphene layer direction with respect to the wavelength at different Fermi levels. It can be seen that at different Fermi levels, the variation trend of ε ⊥ is the same, but the incident wavelength corresponding to the sign change of the real part of ε ⊥ is significantly different. As the Fermi level gradually increases, the incident wavelength corresponding to the sign change of the real part of ε ⊥ gradually decreases.

[0051] Figure 2 (c) shows the variation of the reflectivity with respect to the wavelength at different Fermi levels. It can be seen that when the wavelength is near 2000nm, the reflectivity is at the trough. At this time, the energy of the incident light can be more fully coupled into the structure, which may bring a relatively large GH displacement.

[0052] According to the above theoretical analysis, it can be known that the dielectric constant of the graphene asymmetric structure is related to the incident wavelength, and the hyperbolic dispersion characteristics of this structure are also affected by the incident wavelength. Therefore, it is necessary to first determine an appropriate incident wavelength. Figure 2 (d) shows a close-up of the variation of the real part of ε F with respect to the wavelength at E ⊥ = 0.4eV. It can be seen that when the wavelength λ is near 1924nm, ε ⊥The real part of undergoes a drastic change, jumping from a negative value to a positive value. More specifically, when the wavelength λ is greater than 1923.8 nm, the real part of ε ⊥ is positive, and at this time, ε || ·ε ⊥ > 0, and the structure exhibits elliptical dispersion characteristics. When the wavelength λ ≤ 1923.8 nm, the real part of ε ⊥ is negative, and at this time, ε || ·ε ⊥ < 0, and the structure exhibits hyperbolic dispersion characteristics, which provides the possibility of obtaining a large and flexibly tunable GH shift. In subsequent studies, TM polarized light with an incident wavelength λ = 1923.8 nm was selected.

[0053] The conductivity of graphene can be flexibly regulated by changing the Fermi level, and the conductivity of graphene is an important factor affecting the GH shift. Therefore, the GH shift can be flexibly regulated by changing the Fermi level.

[0054] Figure 3 (a) shows the curves of the phase of the reflection coefficient versus the incident angle when the E F values are 0.4 eV, 0.42 eV, and 0.44 eV respectively. It can be seen from the figure that when the incident angle varies between 45° and 55°, the reflectivity under different Fermi levels undergoes a sudden change.

[0055] Figure 3 (b) shows the curves of the phase of the reflection coefficient versus the incident angle when the E F values are 0.4 eV, 0.42 eV, and 0.44 eV respectively. It can be seen that as the incident angle changes, the phase of the reflection coefficient undergoes a jump. According to formula (13), the GH shift of the reflected beam is proportional to the change rate of the phase of the reflection coefficient. Therefore, a large GH shift may occur in the reflected beam.

[0056] Figure 3 (c) shows the variation of the GH shift with the incident angle under different Fermi levels. The black box in the figure is an enlarged view of the GH shift. It can be seen that when E F = 0.4 eV, a maximum positive GH shift of 150 times the wavelength can be obtained. When E F = 0.42 eV, a maximum negative GH shift of approximately 12000λ can be obtained.

[0057] The dielectric thickness is also an important parameter affecting the GH shift. The GH shift under different dielectric thicknesses was studied. In the calculation, the dielectric thickness range was set to 8 nm to 50 nm.

[0058] Figure 4(a) shows the curves of the phase of the reflection coefficient varying with the incident angle at different dielectric thicknesses. It can be seen that as the incident angle increases, there is a phase jump in each curve of the reflection coefficient phase. This may generate a relatively large GH shift.

[0059] Figure 4 (b) shows the variation of the GH shift with the incident angle at different dielectric thicknesses. It can be seen that as the dielectric thickness increases, the GH shift gradually increases. When t d = 50 nm, a forward GH shift of approximately 600λ can be obtained.

[0060] The relaxation time of graphene is also an important parameter affecting the GH shift. The GH shifts at different graphene relaxation times were studied. The range of graphene relaxation time was set to 9 ps - 99 ps in the calculation.

[0061] Figure 5 (a) shows the curves of the phase of the reflection coefficient varying with the incident angle at different relaxation times. It can be seen that as the relaxation time increases, there is a phase jump in each curve of the reflection coefficient phase and the slope of the reflection phase curve is negative, and the slope of the reflection phase gradually increases. This may generate a relatively large positive GH shift.

[0062] Figure 5 (b) shows the variation of the GH shift with the incident angle at different relaxation times. It can be seen that as the relaxation time increases, the GH shift gradually increases. When τ = 99 ps, a forward GH shift of approximately 1500λ can be obtained.

[0063] Next, consider the influence of the number of graphene layers on the GH shift, restricting the change in the number of graphene layers to 1 - 4 layers.

[0064] Figure 6 (a) shows the curves of the reflection phase varying with the incident angle at different numbers of graphene layers (N g = 1, 2, 3, 4). It can be seen that as the incident angle changes, there is also a jump in each phase curve, which may generate a relatively large GH shift.

[0065] Figure 6 (b) shows the GH shifts corresponding to different numbers of graphene layers. It can be seen that the GH shift always shows a positive value. When N g = 1, a forward GH shift of up to approximately 150λ can be obtained (within the black box in the figure). When N g = 4, a forward GH shift of up to approximately 4000 times the wavelength can be obtained. Therefore, the magnitude of the GH shift can be flexibly regulated by changing the number of graphene layers.

[0066] Previous studies have shown that the dielectric thickness has a significant influence on the GH shift. When the dielectric thickness td When \(t = 50\ nm\), a positive GH shift of approximately \(600\lambda\) can be achieved. Based on this, this section focuses on exploring the modulation mechanism of the dielectric thickness on the sensor sensitivity when the graphene-dielectric aperiodic composite structure is applied to an angular displacement sensor.

[0067] In an angular displacement sensor, the sensitivity is obtained by taking the derivative of the GH shift with respect to the incident angle, and this derivative reflects the sensitivity of the GH shift to the change in the incident angle. To visually present the effect of the dielectric thickness on the sensitivity, Figure 7 (a)(b)(c)(d) are plotted, corresponding to the sensitivity curves at dielectric thicknesses \(t\) d = 8 nm, 22 nm, 36 nm, and 50 nm, respectively. From the trend of the curves, as the dielectric thickness increases, the optimal sensitivity of the sensor shows a gradually increasing trend. In particular, when \(t\) d = 50 nm, the optimal sensitivity of the sensor can reach \(3\times10\) 4 \(\lambda / \text{deg}\) (where "deg." represents the angular unit "degree").

[0068] It should be noted that the trend of the sensitivity with the dielectric thickness is exactly the same as that of the GH shift. Therefore, an increase in the GH shift can directly improve the sensitivity of the angular displacement sensor. Thus, optimizing the GH shift is the key path to improving the sensor sensitivity. In addition, the Fermi level and relaxation time of graphene can also be used as tuning parameters. By adjusting these parameters to change the GH shift characteristics, the design and optimization of a high-sensitivity angular displacement sensor can also be achieved.

[0069] In summary, graphene A and dielectric B are combined according to the Cantor sequence to form a hyperbolic metamaterial heterostructure. In the near-infrared band, by adjusting the Fermi level of graphene, the hyperbolic dispersion characteristics of the composite structure can be achieved. Further, by changing the Fermi level, relaxation time, number of layers of graphene, and dielectric thickness, at the phase transition point from elliptical dispersion to hyperbolic dispersion, the GH shift can be effectively regulated and a large GH shift can be obtained. Research shows that when the Fermi level is \(0.42\ eV\), a negative GH shift of \(12000\lambda\) can be obtained. The GH shift increases with the increase in the dielectric thickness. When the dielectric thickness is \(50\ nm\), the positive GH shift can reach \(600\lambda\). The GH shift is also affected by the number of layers of graphene. When the number of layers of graphene is 4, a positive GH shift of approximately \(4000\) times the wavelength can be obtained. When the relaxation time \(\tau\) of graphene is \(99\ ps\), a positive GH shift of approximately \(1500\lambda\) can be obtained. Based on the GH shift effect, it can be used to develop an angular displacement sensor in hyperbolic metamaterials. When \(t\) d = 50 nm, the optimal sensitivity can be obtained as \(3\times10\) 4 \(\lambda / \text{deg}\).

[0070] The specific embodiments described herein are merely illustrative of the spirit of the present invention. Those skilled in the art to which the present invention pertains may make various modifications or supplements to the described specific embodiments or use similar means for substitution, but will not deviate from the spirit of the present invention or exceed the scope defined by the appended claims.

Claims

1. Application of an aperiodic graphene-dielectric based hyperbolic metamaterial, characterized in that, A single-layer graphene A and a dielectric layer B are combined into a hyperbolic metamaterial heterostructure according to the law of the Cantor sequence; the layer structure rule of the hyperbolic metamaterial heterostructure is: S1 = ABA, S2 = S1(BBB)S1,..., S N = S N-1 (BBB) N-1 S N-1 , (BBB) N-1 represents 3 N-1 Bs, where S0 = A and N is not less than 2; An incident light is incident on the surface plane of the hyperbolic metamaterial heterostructure. For a given incident wavelength, by modulating the Fermi level of graphene, its dispersion space exhibits hyperbolic dispersion characteristics related to the weight of the dielectric thickness; by adjusting the Fermi level, relaxation time, number of layers of monolayer A of graphene, and the thickness of the dielectric, near the phase transition point from hyperbolic dispersion to elliptical dispersion, a huge change occurs in the phase of the reflection coefficient, thereby achieving a huge GH shift at the resonance angle. The angle displacement sensor utilizes the corresponding relationship between the GH shift and the incident angle.

2. The application of a hyperbolic metamaterial based on aperiodic graphene-dielectric according to claim 1, characterized in that The incident light is a TM electromagnetic wave.

3. The application of a hyperbolic metamaterial based on aperiodic graphene-dielectric according to claim 1, characterized in that, The incident angle of the incident light on the surface plane of the hyperbolic metamaterial heterostructure is 45° - 55°.

4. The application of a hyperbolic metamaterial based on aperiodic graphene-dielectric according to claim 1, characterized in that, The wavelength λ of the incident light is 1923.8 nm.

5. The application of a hyperbolic metamaterial based on aperiodic graphene-dielectric according to claim 1, characterized in that, The relaxation time of monolayer A of graphene is controlled between 9 ps and 99 ps.

6. The application of a hyperbolic metamaterial based on aperiodic graphene-dielectric according to claim 1, characterized in that, The Fermi level of monolayer A of graphene is between 0.4 eV and 0.44 eV.

7. The application of a hyperbolic metamaterial based on aperiodic graphene-dielectric according to claim 1, characterized in that, The dielectric layer B is BaF2.

8. The application of a hyperbolic metamaterial based on aperiodic graphene-dielectric according to claim 1, characterized in that, The thickness of the dielectric layer B is 8 nm - 50 nm.

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