digital lens report

8/13/2019 Digital Lens Report

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Digital Lens Simulation

Ekaterina Chernobrovkina

Davood Ansari

November 1, 2013

Supervisor: Professor Hossein Mosallaei

Abstract

This report includes guidelines which can help one design a digital metamaterial (MTM) graded-index (GRIN) lenswith a given profile using only two materials.

1 Objective

In this paper, a standard dielectric GRIN lens with the hyperbolic profile will be represented as a set of core-shell spheresmade of two materials. For simplicity, lets call them material #1 and material #2.

2 GRIN Lens ModelThe GRIN lens model used for the simulation is presented in Figure 1.

Figure 1: GRIN Lens: permittivity function & geometrical parameters

The lens refractive index varies according to the following formula:

nr = n0

1

Ar2

2

(1)

where nr the refractive index at a distance r from the optical axis; n0 is the design index on the optical axis, and A is apositive constant.

Consequently, GRIN lens permittivity can be written as follows:

r=0

1

Ar2

2

2(2)

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3 Digital GRIN Lens

The variation of lens permittivity can be obtained by arranging core-shell spheres with required relative effective permittivitywithin the lens body as presented in Figure 2. The outer radii are equal; the inner radius of each core-shell affects itsrelative effective permittivity. So, by varying the ratio between the core-shell spheres inner and outer radii and materialsarrangement, we can obtain the desired lens profile.

Figure 2: Digital GRIN Lens

4 Materials Arrangement

In the proposed design, only two materials are used to build all requited core-shell spheres that the digital lens consist of.

Two possible material arrangements are demonstrated on Fig 3.

(a) Core-shell #1 (b) Core-shell #2

Figure 3: Materials Arrangements

5 Core-Shell Sphere Effective Permittivity

The relative effective permittivity of each core-shell can be found as follows [1]:

eff1 = r11 + 2

r2r1

3r2r1r2+2r1

1 r2r1

3r2r1r2+2r1

(3)

for the core-shell #1 (see Figure 3a),

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eff2 = r21 + 2

r2r1

3r1r2r1+2r2

1 r2r1

3r1r2r1+2r2

(4)

for the core-shell #2 (see Figure 3b).

Figure 4: Core-Shell Sphere Geometry

In general, this problem requires three-stage homogenization process. The first stage affects core of the radius r2 and shell othe radius r1; second the result of homogenization on the first stage and the background coating of outer radius r3; third the result of homogenization on the second stage and the corners of the cube of the side d (see Figure 4).

The equations 3 and 4 represent the relative effective permittivities of the core-shell #1 and core-shell #2 (see Figure 3)on the first stage of homogenization.

Applying the homogenization concept of the first stage to the second one,

eff1 = r31 + 2

r1r3

3eff1r3eff1+2r3

1 r1r3

3eff1r3eff1+2r3

(5)

and

eff2 = r31 + 2

r1r3

3eff2r3eff2+2r3

1 r1r3

3eff2r3eff2+2r3

(6)

where

eff1 and

eff2 are the relative effective permittivities of the core-shells #1 and #2 consisting of the core of the radiusr2, shell of the radius r1 and the background coating of the outer radius r3.

Finally, the relative effective permittivity of the core-shell sphere of the outer radius r1 sitting in a cube of the side d can befound from the following relation:

Vsphere

eff1+ Vbr3 = Vcube

eff1 (7)

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where Vsphere= 4

3r33 is the volume of the core-shell particle coated with the background layer; Vcube= d

3 is the volume othe cube; Vb= Vcube Vsphere is the background volume including the corners (leftovers).

Analogously,

Vsphere

eff2+ Vbr3 = Vcube

eff2 (8)

Thus, the relative effective permittivities

eff1 and

eff2 can be written as follows:

eff1 =

Vsphere

eff1+ Vbr3

Vcube (9)

and

eff2 =Vsphere

eff2+ Vbr3

Vcube(10)

6 Example

6.1 Designing the Building Blocks

Consider two materials, silica (SiO2) and silver (Ag), which at the frequency of interest have the following relative permit-tivities:

rSiO2 = 2.4247 (11)

Using the Drude model of the silver [2],

rAg = 2p

( j) (12)

where = 5; p= 22175THz; = 24.35THz.

The following parameters were used for the simulation:

a. the operating frequencyf0 = 670THz,

b. the operating wavelength0 = 447.451nm,

c. the lattice size (the cube side) is equal to d = 05

= 89.4903nm,

d. the shell radius is equal to r1 = 0.9d2

= 40.2706nm,

e. the background layer outer radius is equal to r3 = d2

= 44.7451nm

For the chosen parameters of material #1 and material #2, the curves defined using equations 3 and 4 are presented onFigure 5. These figures show the relative effective permittivities of the pure core-shell spheres.

(a) Ag coated with S iO2 (b) SiO2 coated with Ag

Figure 5: Relative Effective Epsilon of the Core-Shell Sphere

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The curves defined using equations 5 and 6 are presented on Figure 6. These figures show the relative effective permittivitieof the core-shells #1 and #2 coated with the background layer of outer radius r3.


Figure 6: Relative Effective Epsilon of the Core-Shell Sphere Coated with the Background Layer

Finally, the curves defined using equations 9 and 10 are presented on Fig 7.


Figure 7: Relative Effective Epsilon of the Core-Shell Sphere Sitting in a Cube

6.2 Designing the Digital GRIN Lens

For the simulation, the parameters of the lens presented on the Figure 2 are the following:

a. the sideD is equal to 4.70

b. the thicknessL is 0.80

c. the focal lengthFis equal to 30

Taking into consideration lens parameters and the chosen size of the single building block, the digital GRIN lens consistsof 4 layers of 23 23 core-shell spheres. The proposed design (core-shells numbering starts from the central element, element#1, to the side of it) is presented on the Figure 8.

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Figure 8: Proposed Digital Lens Design

Figure 9: Desired Digital Lens Profile

Knowing the desired lens profile, presented on the figure 9, and applying the equations 9 and 10, one can easily find theunknown core-shell radii r2 presented in the Table 1.

Element # r2, nm1 22.48922 22.41243 22.20674 21.87055 21.37476 20.68777 19.74768 18.48729 16.6968

10 14.035211 8.2722912 0

(a) Ag coated with S iO2

Element # r2, nm1 561.952 42452.33 182210.4 65055.65 25837.76 47018.47 77.98788 59.62979 50.6233

10 45.067211 41.062812 38.068

(b) S iO2 coated with Ag

Table 1: Required r2 [nm] for the digital GRIN Lens

As the outer core-shell radius r1 is equal to 40.2706nm, the choice between the Table 1a vs. Table 1b is very straightforwardAg coated with S iO2 for the core-shells from #1 through #11 and SiO2 coated with Ag for the core-shell #12.

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6.3 Digital GRIN Lens Simulation Results

For the simulation, the following incident field was considered: E0y, H0x, and k0z; the amplitude of the incident field is 1

The electric and magnetic field intensity distributions are presented on the Figures 10 through 12; the electric field intensityoutside of the lens structure is shown on the Figures 13 and 14.

(a) Re(E), Vm

(b)Re(H), Am

Figure 10: Simulation Results: Electric & Magnetic Field Intensity

(a) Re(E), Vm

(b) Abs(E), Vm

Figure 11: Simulation Results: Electric Field Intensity Distribution

(a) Re(H), Am

(b)Abs(H), Am

Figure 12: Simulation Results: Magnetic Field Intensity Distribution

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(a) Simulation Results: Field Probe Along theZ-axis

(b) Simulation Results: Field Probe Along theY-axis

Figure 13: Simulation Results: Electric Field Intensity Distribution

(a) Abs(Ey), Vm

(b) Arg(Ey), radians

Figure 14: Simulation Results: Electric Field Distribution on the Y0ZPlane

References

[1] H. Panaretos and Douglas H. Werner. Analysis of a plasmonic core-shell particle exhibiting high-impedance and highadmittance characteristics. Proceedings of the 2013 IEEE International Symposium on Antennas & Propagation anUSNC/URSI National Radio Science Meeting, pages 16001601, July 2013.

[2] Andrea Alu M. Silveirinha and Nader Engheta. Infrared and optical invisibility cloak with plasmonic implants based on

scattering cancellation. Physical Review B, 78(075107), August 2008.

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