# Broadband Ultra-Deep Sub-Diffraction-Limit Optical Focusing by Metallic Graded-Index (MGRIN) Lenses

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. MGRIN Lens Design

_{1}= (ε

_{d}k

_{0}

^{2}− k

_{x}

^{2})

^{1/2}and k

_{2}= (ε

_{m}k

_{0}

^{2}− k

_{x}

^{2})

^{1/2}, where w is the waveguide width, t is the metal spacing between two adjacent waveguides, k

_{0}is the free space wave vector, and k

_{x}is the propagation wave vector in the x direction. ε

_{m}and ε

_{d}are the permittivities of the metal and dielectric material in the waveguide, respectively. Our nanofocusing structure is built based on the following propagation constant profile of the metallic waveguides along the y axis:

_{s}is the symmetric solution of k

_{x}in Equation (1) for k

_{y}= 0, and β

_{s,}

_{0}is the corresponding value of the central waveguide. a is the gradient parameter. Since the effective refractive index of a waveguide is n

_{e}= β

_{s}/k

_{0}, Equation (2) can be transformed into the following:

_{a}is the antisymmetric solution of k

_{x}in Equation (1) for k

_{y}= π/(w + t), and β

_{a,}

_{0}is the corresponding solution of the central waveguide. From Equations (2) and (4), the Hamiltonian can be deduced:

_{a,}

_{0}− β

_{s,}

_{0})aβ

_{s,}

_{0}]

^{1/2}(w + t), C

_{1}and C

_{2}are constants related to the position and angle of the incident ray. Assuming that at x = 0, the position is y = y

_{0}and the corresponding slope is y’ = y

_{0}’, the above equation can be transformed into:

_{0}’ = 0, a ray trajectory can be further written as:

## 3. Results and Discussion

_{0}= 1 and ε

_{N}= 1.69. At this wavelength, the permittivity of gold is ε

_{m}= −40.764 + 1.261i [43]. We considered a structure with a total of 51 waveguides that have the same width of 10 nm and are uniformly separated by 30 nm of gold. The required permittivity of the dielectric in the nth waveguide (0 ≤ n ≤ 25) is calculated by using Equations (1) and (2), as shown in Figure 2. The maximum variation in the dielectric constant between the adjacent waveguides is less than 0.06. Thus, it is reasonable to consider the structure to be locally periodic.

_{y}with the amplitude of 1.

_{m}= −183.23 + 7.522i; at 3 μm, ε

_{m}= −415.98 + 22.462i; at 4 μm, ε

_{m}= −747.36 + 51.625i) with the same structure as the one designed for the wavelength λ = 1 μm. These simulation results illustrate the similar focusing behavior. For the shorter wavelengths in the visible range down to 650 nm, focusing can also be realized, as shown in Figure 5. Nevertheless, for shorter wavelengths, the nanofocusing effect cannot be observed in the structure due to the losses near the cutoff frequencies for plasma oscillations [44]. Besides the operating wavelength, the propagation losses in the structure also depend on the spacing between metallic waveguides. Losses increase with the metallic spacing. However, the metallic spacing cannot be too small to provide the capability for subwavelength optical confinement. Therefore, the metallic spacing should be appropriately selected for the nanofocusing scheme.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Abbe, E. Resolution of microscopes. Arch. Mikrosk. Anat.
**1873**, 9, 413–425. [Google Scholar] [CrossRef] - Zhang, X.; Liu, Z. Superlenses to overcome the diffraction limit. Nat. Mater.
**2008**, 7, 435–441. [Google Scholar] [CrossRef] [PubMed] - Gramotnev, D.K.; Bozhevolnyi, S.I. Plasmonics beyond the diffraction limit. Nat. Photon.
**2010**, 4, 83–91. [Google Scholar] [CrossRef] - Barnes, W.L.; Dereux, A.; Ebbesen, T.W. Surface plasmon subwavelength optics. Nature
**2003**, 424, 824–830. [Google Scholar] [CrossRef] [PubMed] - Fang, N. Sub-Diffraction-Limited Optical Imaging with a Silver Superlens. Science
**2005**, 308, 534–537. [Google Scholar] [CrossRef] [PubMed] - Schuller, J.A.; Barnard, E.S.; Cai, W.; Jun, Y.C.; White, J.S.; Brongersma, M.L. Plasmonics for extreme light concentration and manipulation. Nat. Mater.
**2010**, 9, 193–204. [Google Scholar] [CrossRef] [PubMed] - Han, Z.; Bozhevolnyi, S.I. Radiation guiding with surface plasmon polaritons. Rep. Prog. Phys.
**2013**, 76, 16402. [Google Scholar] [CrossRef] [PubMed] - Gramotnev, D.K.; Bozhevolnyi, S.I. Nanofocusing of electromagnetic radiation. Nat. Photon.
**2014**, 8, 13–22. [Google Scholar] [CrossRef] - Novotny, L.; Hulst, N.V. Antennas for light. Nat. Photon.
**2011**, 5, 83–90. [Google Scholar] [CrossRef] - Babadjanyan, A.J.; Margaryan, N.L.; Nerkararyan, K.V. Superfocusing of surface polaritons in the conical structure. J. Appl. Phys.
**2000**, 87, 3785–3788. [Google Scholar] [CrossRef] - Stockman, M.I. Nanofocusing of optical energy in tapered plasmonic waveguides. Phys. Rev. Lett.
**2004**, 93, 137404. [Google Scholar] [CrossRef] [PubMed] - Issa, N.A.; Guckenberger, R. Optical nanofocusing on tapered metallic waveguides. Plasmon.
**2007**, 2, 31–37. [Google Scholar] [CrossRef] - Ropers, C.; Neacsu, C.C.; Elsaesser, T.; Albrecht, M.; Raschke, M.B.; Lienau, C. Grating-Coupling of Surface Plasmons onto Metallic Tips: A Nanoconfined Light Source. Nano Lett.
**2007**, 7, 2784–2788. [Google Scholar] [CrossRef] [PubMed] - Gramotnev, D.K.; Vogel, M.W. Ultimate capabilities of sharp metal tips for plasmon nanofocusing, near-field trapping and sensing. Phys. Lett. A
**2011**, 375, 3464–3468. [Google Scholar] [CrossRef] - Kravtsov, V.; Ulbricht, R.; Atkin, J.M.; Raschke, M.B. Plasmonic nanofocused four-wave mixing for femtosecond near-field imaging. Nat. Nanotechnol.
**2016**, 11, 459–464. [Google Scholar] [CrossRef] [PubMed] - Nerkararyan, K.V. Superfocusing of a surface polariton in a wedge-like structure. Phys. Rev. A
**1997**, 237, 103–105. [Google Scholar] [CrossRef] - Durach, M.; Rusina, A.; Stockman, M.I.; Nelson, K. Toward Full Spatiotemporal Control on the Nanoscale. Nano Lett.
**2007**, 7, 3145–3149. [Google Scholar] [CrossRef] [PubMed] - Vernon, K.C.; Gramotnev, D.K.; Pile, D.F.P. Adiabatic nanofocusing of plasmons by a sharp metal wedge on a dielectric substrate. J. Appl. Phys.
**2007**, 101, 104312. [Google Scholar] [CrossRef][Green Version] - Verhagen, E.; Polman, A.; Kuipers, L.K. Nanofocusing in laterally tapered plasmonic waveguides. Opt. Express
**2008**, 16, 45–57. [Google Scholar] [CrossRef] [PubMed] - Verhagen, E.; Spasenović, M.; Polman, A.; Kuipers, L.K. Nanowire plasmon excitation by adiabatic mode transformation. Phys. Rev. Lett.
**2009**, 102, 203904. [Google Scholar] [CrossRef] [PubMed] - Umakoshi, T.; Saito, Y.; Verma, P. Highly efficient plasmonic tip design for plasmon nanofocusing in near-field optical microscopy. Nanoscale
**2016**, 8, 5564–5634. [Google Scholar] [CrossRef] [PubMed] - Pile, D.F.P.; Gramotnev, D.K. Adiabatic and nonadiabatic nanofocusing of plasmons by tapered gap plasmon waveguides. Appl. Phys. Lett.
**2006**, 89, 41111. [Google Scholar] [CrossRef][Green Version] - Ginzburg, P.; Arbel, D.; Orenstein, M. Gap plasmon polariton structure for very efficient microscale-to-nanoscale interfacing. Opt. Lett.
**2006**, 31, 3288–3290. [Google Scholar] [CrossRef] [PubMed] - Gramotnev, D.K.; Pile, D.F.; Vogel, M.W.; Zhang, X. Local electric field enhancement during nanofocusing of plasmons by a tapered gap. Phys. Rev. B
**2007**, 75, 035431. [Google Scholar] [CrossRef] - Vedantam, S.; Lee, H.; Tang, J.; Conway, J.; Staffaroni, M.; Yablonovitch, E. A Plasmonic Dimple Lens for Nanoscale Focusing of Light. Nano Lett.
**2009**, 9, 3447–3452. [Google Scholar] [CrossRef] [PubMed] - Choo, H.; Kim, M.; Staffaroni, M.; Seok, T.J.; Bokor, J.; Cabrini, S.; Schuck, P.J.; Wu, M.C.; Yablonovitch, E. Nanofocusing in a metal–insulator–metal gap plasmon waveguide with a three-dimensional linear taper. Nat. Photon.
**2012**, 6, 838–844. [Google Scholar] [CrossRef] - Volkov, V.S.; Bozhevolnyi, S.I.; Rodrigo, S.G.; Martín-Moreno, L.; García-Vidal, F.J.; Devaux, E.; Ebbesen, T.W. Nanofocusing with Channel Plasmon Polaritons. Nano Lett.
**2009**, 9, 1278–1282. [Google Scholar] [CrossRef] [PubMed] - Sorger, V.J.; Oulton, R.F.; Ma, R.M.; Zhang, X. Toward integrated plasmonic circuits. MRS Bull.
**2012**, 37, 728–738. [Google Scholar] [CrossRef] - Chung, T.; Lee, S.Y.; Song, E.Y.; Chun, H.; Lee, B. Plasmonic nanostructures for nano-scale bio-sensing. Sensors
**2011**, 11, 10907–10929. [Google Scholar] [CrossRef] [PubMed] - Frey, H.; Witt, S.; Felderer, K.; Guckenberger, R. Highresolution imaging of single fluorescent molecules with the optical nearfield of a metal tip. Phys. Rev. Lett.
**2004**, 93, 200801. [Google Scholar] [CrossRef] [PubMed] - Juan, M.L.; Righini, M.; Quidant, R. Plasmon nano-optical tweezers. Nat. Photon.
**2011**, 5, 349–356. [Google Scholar] [CrossRef] - Economou, E.N. Surface plasmons in thin films. Phys. Rev.
**1969**, 182, 539. [Google Scholar] [CrossRef] - Xu, T.; Du, C.; Wang, C.; Luo, X. Subwavelength imaging by metallic slab lens with nanoslits. Appl. Phys. Lett.
**2007**, 91, 201501. [Google Scholar] [CrossRef] - Verslegers, L.; Catrysse, P.B.; Yu, Z.; Fan, S. Planar metallic nanoscale slit lenses for angle compensation. Appl. Phys. Lett.
**2009**, 95, 071112. [Google Scholar] [CrossRef] - Verslegers, L.; Catrysse, P.B.; Yu, Z.; White, J.S.; Barnard, E.S.; Brongersma, M.L.; Fan, S. Planar lenses based on nanoscale slit arrays in a metallic film. Nano Lett.
**2009**, 9, 235–238. [Google Scholar] [CrossRef] [PubMed] - Chen, Q.; Cumming, D.R. Visible light focusing demonstrated by plasmonic lenses based on nano-slits in an aluminum film. Opt. Express
**2010**, 18, 14788–14793. [Google Scholar] [CrossRef] [PubMed] - Zhu, Y.; Yuan, W.; Yu, Y.; Diao, J. Metallic planar lens formed by coupled width-variable nanoslits for superfocusing. Opt. Express
**2015**, 23, 20124–20131. [Google Scholar] [CrossRef] [PubMed] - Gordon, R. Proposal for superfocusing at visible wavelengths using radiationless interference of a plasmonic array. Phys. Rev. Lett.
**2009**, 102, 207402. [Google Scholar] [CrossRef] [PubMed] - Verslegers, L.; Catrysse, P.B.; Yu, Z.; Fan, S. Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array. Phys. Rev. Lett.
**2009**, 103, 033902. [Google Scholar] [CrossRef] [PubMed] - Yariv, A.; Yeh, O. Photonics: Optical Electronics in Modern Communications; Oxford Univercity Press: New York, NY, USA, 2006. [Google Scholar]
- Reino, C.G.; Pérez, M.V.; Bao, C. Gradient-Index Optics: Fundamentals and Applicatioins; Springer: New York, NY, USA, 2002. [Google Scholar]
- Fan, X.; Wang, G.P.; Lee, J.C.W.; Chan, C.T. All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration. Phys. Rev. Lett.
**2006**, 97, 073901. [Google Scholar] [CrossRef] [PubMed] - Babar, S.; Weaver, J.H. Optical constants of Cu, Ag, and Au revisited. Appl. Opt.
**2015**, 54, 477–481. [Google Scholar] [CrossRef] - Ferrell, R.A. Characteristic energy loss of electrons passing through metal foils. ii. Dispersion relation and short wavelength cutoff for plasma oscillations. Phys. Rev.
**1956**, 107, 450–462. [Google Scholar] [CrossRef]

**Figure 1.**The schematic of a metallic graded-index (MGRIN) lens formed by coupled waveguides of uniform width and gold spacing under the normal incidence of a transverse magnetic plane wave. The structure is symmetric with respect to the central waveguide at y = 0, and ε

_{n}(0 ≤ n ≤ N) (n the integer with the values of 0, 1, 2… N) represents the permittivity of the dielectric in the waveguide n.

**Figure 2.**The required permittivity of the dielectric in the nth waveguide of an MGRIN lens working at λ = 1 μm. The MGRIN lens comprises a total of 51 waveguides and the dielectric constant increases from 1 at the center to 1.69 at the sides.

**Figure 3.**Ultra-deep sub-diffraction-limit focusing of an MGRIN lens. (

**a**) FDTD-simulated electric field intensity pattern. The inset shows the enlarged view for the electric intensity distribution of the focus; (

**b**) The derived |E|

^{2}on the optical axis. The FWHM and focal depth of the focus are 8 nm and 1.24 μm, respectively.

**Figure 4.**Ultra-deep sub-diffraction-limit focusing of an MGRIN lens working at longer wavelengths. (

**a**–

**c**) FDTD-simulated electric field intensity patterns for the wavelengths of 2–4 μm, respectively; (

**d**–

**f**) The corresponding |E|

^{2}on the optical axis. The FWHMs of the three foci are all 8 nm. The focal depths at 2–4 μm are 3.18 μm, 5.24 μm, and 7.72 μm, respectively.

**Figure 5.**Ultra-deep sub-diffraction-limit focusing of an MGRIN lens working at the shorter wavelengths. (

**a**–

**c**) FDTD-simulated electric field intensity patterns for the wavelengths of 0.76 μm, 0.65 μm and 0.58 μm, respectively. The permittivity of gold at these three wavelengths is −20.273 + 0.703i, −12.266 + 0.779i, and −7.571 + 1.141i, respectively. The FWHMs of the two foci at 0.76 μm and 0.65 μm are both 8 nm. The corresponding focal depths are 0.86 μm and 0.88 μm, respectively.

**Figure 6.**The focusing performance of an MGRIN lens working at various wavelengths from 0.65 μm to 4 μm. (

**a**–

**c**) Focal length, focal depth, and the maximum intensity at the focus varying as a function of the incident wavelength.

**Figure 7.**Effective indices of the central waveguide filled with a dielectric of ε = 1 and the side waveguide filled with a dielectric of ε = 1.69 for various wavelengths.

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**MDPI and ACS Style**

Zhu, Y.; Yuan, W.; Sun, H.; Yu, Y.
Broadband Ultra-Deep Sub-Diffraction-Limit Optical Focusing by Metallic Graded-Index (MGRIN) Lenses. *Nanomaterials* **2017**, *7*, 221.
https://doi.org/10.3390/nano7080221

**AMA Style**

Zhu Y, Yuan W, Sun H, Yu Y.
Broadband Ultra-Deep Sub-Diffraction-Limit Optical Focusing by Metallic Graded-Index (MGRIN) Lenses. *Nanomaterials*. 2017; 7(8):221.
https://doi.org/10.3390/nano7080221

**Chicago/Turabian Style**

Zhu, Yechuan, Weizheng Yuan, Hao Sun, and Yiting Yu.
2017. "Broadband Ultra-Deep Sub-Diffraction-Limit Optical Focusing by Metallic Graded-Index (MGRIN) Lenses" *Nanomaterials* 7, no. 8: 221.
https://doi.org/10.3390/nano7080221