Broadband Subwavelength Imaging with a Wire Medium Slab Loaded with Graphene Sheets Ali Forouzmand and Alexander B. Yakovlev Center for Applied Electromagnetic Systems Research (CAESR) Department of Electrical Engineering, University of Mississippi 9th European Conference on Antennas and Propagation (EUCAP) April 12-17, 2015
Outline: Introduction and Motivation Formulation and Theoretical Analysis Analytical and Full-Wave Simulation Results Magnetic Line Source Double-Slit Source Conclusions
Perfect Lens: Pendry-Veselago Lens: Veselago pointed out a possibility of the existence of a negative refractive index material (NIM). J. B. Pendry, Phys. Rev. Lett. 85, 03966 (2000). NIM slab could focus both the evanescent and propagating spectra. V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968). Highly Sensitive to Loss Narrow Spectral Bandwidth Pendry Lens: Only negative permittivity (ε < 0) The amplification can be obtained by resonantly exciting surface plasmons on the surfaces of silver. S. A. Ramakrishna and et. Al., J. Mod. Opt. 50, 1419 (2003).
Resonant Grids or Conjugate Sheets : Two Metal Particles Source Dipole S. Maslovski, S. Tretyakov, and P. Alitalo, J. Appl. Phys. 96, 1293 (2004).
Stacked Graphene Sheets: Parallel Graphene Sheets: P. Li and T. Taubner, ACS nano 6, 10107 (2012). Broadband Imaging: w μ c ε h τ h T 800 nm 1.5 ev 3 0.5 ps 800 nm 300 The enhancement of the evanescent waves for subwavelength imaging is realized by well-coupled surface plasmons on graphene sheets. Resolution = λ 0 7 Broadband Subwavelength Imaging f= 26-30 THz
Stacked Graphene Sheets: P. Li and T. Taubner, ACS nano 6, 10107 (2012). Low Sensitivity to Loss: small loss (τ = 0.5 ps and Im[ε h ] = 0.3), a reasonable loss (τ = 0.3 ps and Im[ε h ] = 0.3), and a large losses (τ = 0.05 ps and Im[ε h ] = 0.3) at the frequency f = 27.2 THz. Stacked Graphene Sheets: Stacked Graphene sheets increase the structure s resolution. In the 4-layered case the peak shifts to about k x =14 k 0 and k x =17 k 0 for the 8-layered case. Therefore, their resolutions are improved to over λ/10.
Wire Medium: M. Silveirinha, et.al., Phys. Rev. B 75 035108 (2007) M. Silveirinha, et.al., Opt. Lett. 33, 1726, (2008) Geometry : Dispersion Relation : Transmission/Reflection : The loss sensitivity of this structure isremarkably small. The resolution is restricted only by the spacing between the wires. The properties of this lens cannot be tuned after fabrication. The thickness of wire medium should be chosen in such a way that it is an integer number of half wavelength. P. A. Belov, et.al., Phys. Rev. B, 77, 193108, (2008)
Wire Medium: P. A. Belov, et.al., Phys. Rev. B, 77, 193108, (2008) Subwavelength Imaging f=30-36thz M. Silveirinha, et.al., Phys. Rev. B 75 035108 (2007) M. Silveirinha, et.al., Opt. Lett. 33, 1726, (2008) Source Plane Image Plane Resolution = λ 0 10
Bilayer Mushroom Structure : Metallic Patch Arrays Wire Medium Slab Bilayer Mushroom Structure The thickness of wire medium should be chosen in such a way that it is an integer number of half wavelength to satisfy the subwavelength imaging condition. The dispersion of WM slab is ultimately flat at the corresponding frequency. The properties of WM slab cannot be tuned after fabrication. WM slab loaded with two metallic patches. How is it possible to control the properties of a WM slab??? A lumped load has been considered at the middle of the wires which help to control the dispersion behavior.
Bilayer Mushroom Structure: C. S. R. Kaipa, et,al., J. Appl. Phys., 109, 044901, 2011 Subwavelength Imaging Bilayer mushroom structure consists of two metallic patches connected by loaded metallic wires. Impedance loadings provides controllable coupling between the charge density waves supported by each grids. Strong enhancement= Higher Resolution = subwavelength Imaging in further distance = dispersion curve maximally flat In order to have a strong enhancement of the evanescent waves, we would like the dispersion curve to become maximally flat at the resonant frequency ( i.e., to approach a vertical line). In these ideal circumstances, the density of guided modes at the resonant frequency is extremely large, and the free charges in the metallic grids interact resonantly with the radiation field.
Bilayer Mushroom Structure: The operating frequency of the structure should be chosen slightly above the frequency corresponding to the stopband for the proper real mode. The resolution of the structure is higher than λ/6 even if the thickness of the bilayer mushroom is a significant fraction of the wavelength. The imaging properties can be controlled by changing the structural parameters of the bilayer mushroom. Narrow spectral bandwidth. Negligibly sensitive to loss. D g ε r f L r 0 L 2d 2 0.2 1 6.67GHz 10 0.05 5nH 0.1λ
WM slab Loaded with Graphene Sheets: Parallel Graphene Sheets WM Slab WM Slab Loaded with Graphene Sheet Tunable Broadband Intensively sensitive to loss Impossible to enlarge the lens Cannot be tuned after fabrication Narrow spectral bandwidth Negligibly sensitive to loss Extreme large thickness The both subwavelength imaging properties.
Theoretical Analysis and Formulation: Three regions are k x 2 2 0 k0 k x k 0 sin i Region I Region II: Region III: z > 0 h < z < 0 z < h TEM jktem jk0 TM TM zz k h k 2 p 2 x k /( k 2 p 2 x k k 2 x 2 0 ) h h In Region I : H y (1) = e γ 0 z Re γ 0z In Region II : H 2 y = A + TM e γ TM z+ h 2 + A TM e γ TM z+ h 2 + B + TEM e γ TEM z+h/2 + B TEM e γ TEM(z+h/2) In Region III : H y 3 = Te γ 0 z+h +, A + TM, B TEM,and B TEM A TM are the unknown amplitudes associated with TM and TEM fields. R is the reflection coefficient and T is the transmission coefficient for WM slab loaded with graphene sheets.
Theoretical Analysis and Formulation: Unknown Coefficients R, T, +, A + TM, B TEM A TM B TEM! we need 6 boundary conditions. 2 Two-sided impedance boundary Conditions 2 Generalized Additional Boundary Conditions 2 Two-sided impedance boundary Conditions
Graphene Conductivity: The graphene sheet impedance Z g is given by Z g = 1 σ σ is the graphene s complex surface conductivity σ intra = j K B e 2 T πħ 2 ω j2γ σ inter = je2 4πħ ln 2 μ c 2 μ c σ=σ inter +σ intra μ c μ c + 2 ln etk B + 1 TK B ω + jτ 1 ħ + ω + jτ 1 ħ G. W. Hanson, J. App. Phys, 103, 064302, (2008) ћ : the reduced Plank constant e : the electron charge K B : the Boltzmann s constant E F : the Fermi energy τ : the relaxation time T : the temperature f d ε = (e ε μ c k B T + 1) 1 Fermi-Dirac distribution μ c : the chemical potential
Conventional and Generalized Additional Boundary Conditions: Two-sided sheet impedance boundary conditions at z= 0 and h. A. B. Yakovlev et.al., IEEE Trans. Microwave Theory Tech., 57, Nov. 2009 E x (1) E x (2) z=0 += E (2) x z= h += E (3) x z=0 = Z g[ z= h = Z g[ H y (2) H y (3) z=0 H (1) y z= h H (2) y z=0 +] z= h +] Generalized Additional Boundary condition (GABC) σ di z jωε 0 dz I z z = 0, h + = 0 and in terms of macroscopic fields: 1 σ jωε 0 z k 0 E z (2) + kz η 0 H y (2)!!! z = 0, h + = 0. z = h z = 0 E h H r k i z Ground plane Graphene Monolayer a x
Dispersion behavior - Even Modes: In Region I : H y 1 = e γ 0z + R even e γ 0z e ik xx In Region II : H y 2 = [B TEM cos k TEM z + A TM cos h( γ TM z)]e ik xx R even, B TEM, and A TM are the unknown coefficients. Two-sided Impedance Boundary Conditions Generalized Additional Boundary Condition Additional Boundary Condition
Dispersion behavior - Even Modes: Brown Curve: Re Im k x k 0 1 k x k 0 0 a μ c ε h τ h r 0 T Mode 215 nm 1.5 ev 1 0.5 ps 800 nm 21.5 nm 300 Even Blue and Black Curves: Re Im k x k 0 k x k 0 Blue and Black Curves: Re Im k x k 0 < 1 k x k 0 >> 1
Dispersion Behavior and Transmission Coefficients: Proper regime to obtain Subwavelength Imaging a μ c ε h τ h r 0 T Mode 215 nm 1.5 ev 1 0.5 ps 2400 nm 21.5 nm 300 odd Re(k x /k 0 ) Small Re(k x /k 0 ) Small Im(k x /k 0 ) Small Im(k x /k 0 ) Large
Subwavelength Imaging with Different Approaches with the Same Structural Parameters: a μ c ε h τ h r 0 T d w b f 215 nm 1.5 ev 1 0.5 ps 2400 nm 21.5 nm 300 150 nm 500 nm 750 nm 19 THz Uniform and nearly unity for large range. The transmission of two graphene sheets has two resonances at k x k 0 = 3. 623 and 3. 816, then it drops drastically 18k 0 Decays rapidly
Odd Mode Subwavelength Imaging with Different Approaches with the Same Structural Parameters: a μ c ε h τ h r 0 215 nm 1.5 ev 1 0.5 ps 2400 nm 21.5 nm 62.5 THz nλ 2, n = 1 T d w b f 300 150 nm 500 nm 750 nm 19 THz At low frequencies: Perturbation of the surface plasmons of two parallel graphene sheets. At High frequencies: Similar to the dispersion of bound modes in the WM slab resulting in a Fabry-Perot stopband.
Dispersion behavior - Odd Modes: The left bound of the stopband for the proper complex mode occurs at lower frequency. a μ c ε h τ r 0 T Mode 215 nm 1.5 ev 1 0.5 ps 21.5 nm 300 Odd The right bound of the stopband for the proper complex mode occurs at higher frequency. The resonance has a remarkable shift to the lower frequencies. It leads to the expansion of the stopband regime wherein the propagation of the proper (bound) complex modes stops. By a careful study of the dispersion relation for odd excitation, it concludes that the significant resonance of the structure is occurred at low-thz frequencies
Dispersion behavior - Odd Modes: The left bound of the stopband for the proper complex mode occurs at lower frequency. a ε h τ h r 0 T Mode 215 nm 1 0.5 ps 2400 nm 21.5 nm 300 odd The significant resonance frequency versus the chemical potential of graphene.
Schematics of SubWavelength Imaging with Magnetic Line Source: The amplification of the decaying evanescent fields from the source. Intensity Magnetic Line Source Subwavelength Imaging Device Image Plane Intensity x y x y
Subwavelength Imaging: λ/6.58 Operating Frequency The optimum result for obtaining the high resolution and low distortion. a μ c ε h τ h r 0 T d f 215 nm 1.5 ev 1 0.5 ps 2400 nm 21.5 nm 300 150 nm 19 THz Half Power Beam Width (HPBW)
Subwavelength Imaging: Operating Frequency a μ c ε h τ h r 0 T d f 215 nm 1.5 ev 1 0.5 ps 2400 nm 21.5 nm 300 150 nm 19 THz The canalization of the near field along the wires.
Subwavelength Imaging: a μ c ε h τ h r 0 T d f 215 nm 1.5 ev 1 0.5 ps 2400 nm 21.5 nm 300 150 nm 19 THz The resolution of the proposed lens is λ/10. It shows the canalization of the near field along the wires. The thickness of the lens is λ/6.58 which is large fraction of the wavelength.
Broadband Subwavelength Imaging: a μ c ε h τ h r 0 T d 215 nm 1.5 ev 1 0.5 ps 2400 nm 21.5 nm 300 150 nm For f=18.5-21 THz, the resolution of proposed structure is higher than λ/5.
Tunable Subwavelength Imaging: Operating Frequency a μ c ε h τ h r 0 T d f 215 nm 0.5 ev 1 0.5 ps 2400 nm 21.5 nm 300 150 nm 11 THz Half Power Beam Width (HPBW) Resolution = λ 15
Schematics of SubWavelength Imaging with Double-Slit Source: E i k x = e ik xx+ik z z k x z k z x E obj = 0 dk x v kx e ik zz v kx = 4 πk x z cos k x x + x ik z k x sin k x x sin k x w co s( k x b) E t k x = Te ikxx ) e +ik z (z+h k x z k z x The transmitted near electric field can be obtained by the integration of the E t k x with v kx over k x.
Subwavelength Imaging with Double-Slit Source: Operating Frequency a μ c ε h τ h r 0 T d w b f 215 nm 1.5 ev 1 0.5 ps 2400 nm 21.5 nm 300 150nm 500 nm 750 nm 18.5 THz 2b 2w 2b= 1500 nm λ/10 The maxima do not locate exactly at the center of slit. The physical principle behind this phenomenon can be described as over amplification of the near field and the imperfection of transmission.
Broadband Subwavelength Imaging with Double-Slit Source: Rayleigh Criterion The Rayleigh criterion states that the total intensity at the mid-point of the sum intensity profile of two images of slit sources is 81% of the maximum intensity. At f=18 THz, A bump is appeared in the middle of two maxima which is arising due to the imperfection of transmission. Broadband Subwavelength Imaging f= 18.5-22 THz
Subwavelength Imaging: Operating Frequency a μ c ε h τ h r 0 T d w b f 215 nm 1.5 ev 1 0.5 ps 2400 nm 21.5 nm 300 150 nm 500 nm 750 nm 19 THz Double-Slit Source All subwavelength information is lost a short distance away from the slits. Double-Slit Source + WM Slab Loaded with Graphene Sheets Two slits are completely resolved.
Subwavelength Imaging with Different Approaches with the Same Structural Parameters: a μ c ε h τ h r 0 T d w b f 215 nm 1.5 ev 1 0.5 ps 2400 nm 21.5 nm 300 150 nm 500 nm 750 nm 19 THz Graphene Sheets WM slab WM slab Loaded with Graphene Sheets
Fabrication Guide: Fabrication of Graphene Monolayer: Chemical Vapor Deposition (CVD) Epitaxy Joining the WM slab and graphene methods: Ohmic Contact Soldering Plasmonic Welding Embedding the wire medium in a dielectric slab and considering a small gap between wires and graphene sheets.
Effect of Gap: a μ c ε h τ h r 0 T d f 215 nm 1.5 ev 1 0.5 ps 2400 nm 21.5 nm 300 150 nm 19 THz
Effect of Gap: a μ c ε h τ h r 0 T d f 215 nm 1.5 ev 1 0.5 ps 2400 nm 21.5 nm 300 150 nm 19 THz 10 nm Gap 15 nm Gap 25 nm Gap
Effect of Gap in Presence of Dielectric Slab: a μ c ε h τ h r 0 d f 215 nm 1.5 ev 4 0.5 ps 2400 nm 21.5 nm 150 nm 9.8 THz The resolution of structure is λ/8 which is 2.5 times better than propagation in the air. a) 1 nm Gap b) 2 nm Gap c) 5 nm Gap
Conclusions: The possibility of obtaining subwavelength imaging by utilizing the WM slab loaded with graphene sheets has been studied analytically and validated by full-wave simulation. Transmission characteristics of the structure have been obtained by applying the conventional boundary conditions and generalized additional boundary conditions at the connection of the wires to graphene sheets. The resolution of the proposed structure has been studied by two well-known approaches (Half power beam width (HPBW) and Rayleigh Criterion). Low-loss sensitivity, frequency tunability by tuning the chemical potential of graphene, broadband subwavelength imaging, and the possibility of increasing the thickness are the special advantages of our proposed structure.
Tunable Dual-Band Subwavelength Imaging with a Wire Medium Slab Loaded with Nanostructured Graphene Metasurfaces
WM slab Loaded with Nanostructured Graphene Metasurfaces: Nanostructured Graphene Metasurfaces WM Slab WM Slab Loaded with Graphene Sheet Tunable Capacitive/Inductive (Dual-Band) Intensively sensitive to loss Impossible to enlarge the lens Cannot be tuned after fabrication Single Operating Frequency Negligibly sensitive to loss Extreme large thickness The both subwavelength imaging properties
Graphene Nanopatches Surface Impedance: The graphene nanopatches impedance Z g is given by: σ is the graphene s complex surface conductivity K B e 2 T σ intra = j πħ 2 ω j2γ μ c μ c + 2 ln etk B + 1 TK B σ inter = je2 4πħ ln 2 μ c 2 μ c σ=σ inter +σ intra ω + jτ 1 ħ + ω + jτ 1 ħ G. W. Hanson, J. App. Phys, 103, 064302, (2008) Z s = R s + jx s = D σ s D g j 2ωε 0 ε h + 1 2 π D ln csc πg 2D Y. R. Padooru and et al., Phys. Rev. B, vol. 87, pp. 115401, 2013. ћ : the reduced Plank constant e : the electron charge K B : the Boltzmann s constant E F : the Fermi energy τ : the relaxation time T : the temperature f d ε = (e ε μ c k B T + 1) 1 Fermi-Dirac distribution μ c : the chemical potential
Graphene Nanopatches Surface Impedance: Z s = R s + jx s = D σ s D g j 2ωε 0 ε h + 1 2 π D ln csc πg 2D D g ε h μ c τ T 215 nm 21.5 nm 1 0.5 and 1.5 ev 0.5 ps 300 K μ c = 1. 5 ev μ c = 0. 5 ev
Dispersion Relation and Transmission Coefficients: (a) Dispersion behavior of the natural modes of the structure. The solid line represents the real part of the normalized propagation constant, Re( k x k 0 ), and the dashed line represents the imaginary part of the normalized propagation constant, Im k x k 0. (b) Magnitude of the transmission coefficient as a function of Re( k x k 0 ) calculated for the structure at different operating frequencies (f = 22, 22. 8, 25, and 25. 9 THz).
Dual-Band Subwavelength Imaging: The square normalized amplitude of the magnetic field H y calculated at the image plane at (a) f = 22.8 THz and (b) f = 25.9 THz. CST results for the magnetic field distribution H y at (c) f = 22.8 THz and (d) f = 25.9 THz.