# 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

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**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